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1,314,259,992,963 | arxiv | \section{Introduction}\label{sec:intro}
The color-magnitude diagrams (CMDs) of most Galactic and extragalactic globular clusters (GCs) are composed of distinct photometric sequences that are detected among stars at different evolutionary stages and correspond to stellar populations with different content of helium and light elements \citep[][and references therein]{milone2012a, milone2020a}. Specifically, most GCs host a first generation (1G) of stars that share the same chemical composition as field stars with similar metallicities and second stellar generations (2G) that are enhanced in helium nitrogen and sodium and depleted in carbon and oxygen \citep[e.g.][and references therein]{kraft1993a, gratton2012a, bastian2018a, milone2018a, marino2019a}.
The star-to-star variation of light elements and the corresponding correlations and anticorrelations observed in GCs \citep[e.g.\,][]{hesser1977a, kraft1993a, Ivans2001a, marino2008a, carretta2009a, meszaros2015a} are understood as the results of CNO cycling and p-capture processes at high temperatures. These processes include the Mg-Al chain, effective at temperatures higher than $\sim 7 \times 10^{7}$ K, which produces Al at the expenses of Mg and is responsible for the Mg-Al anticorrelations observed in some GCs from spectroscopy \citep[e.g.\,][]{carretta2015a, masseron2019a}.
Indeed, Mg abundances are proxies of stellar nucleosynthesis processes occurring at higher temperature than those responsible for the classic, and much more widespread, N-C/Na-O anticorrelations.
In addition to internal variations of light elements, some GCs exhibit stellar populations with different content of heavy elements, including iron and s-process elements \citep[e.g.][]{marino2009a, marino2015a, dacosta2009a, carretta2010a, yong2014a, johnson2017a}.
Clusters with heavy element variations are called Type~II GCs and comprise $\sim$17\% of the total number of Galactic GCs \citep{milone2017a, milone2020a}. A common feature of Type-II GCs is that stars with similar metallicities host stellar populations with different light-elements abundances \citep[e.g.][]{marino2009a, marino2011a, milone2015a, milone2017a}.
The origin of MPs is one of the most-debated open issues of stellar astrophysics, which could have significantly affected the assembly of the Galactic halo and possibly the reionization of the Universe \citep[e.g.][]{renzini2015a, renzini2017a}. \\
According to some scenarios, GCs have experienced multiple star-formation episodes and 2G stars formed out of the material ejected by more-massive 1G stars \citep[e.g.][]{ventura2001a, decressin2007a, dercole2008a, denissenkov2014a, dantona2016a, calura2019a}.
Alternatively, GC stars are coeval and exotic phenomena that occurred in the unique environment of the proto GCs are responsible for the chemical composition of 2G stars \citep[e.g.][]{bastian2013a, gieles2018a}. We refer to \citet{renzini2015a} for critical discussion on the various scenarios.
Various photometric diagrams have been exploited in the past decade to disentangle multiple populations (MPs) in Globular Clusters (GCs) and allowed to identify and characterize MPs in more than seventy Galactic and extragalactic clusters \citep[e.g.][]{milone2017a}. Most of these diagrams are based on $U$-band photometry, which allows to distinguish 1G from 2G stars mainly through the effect of NH molecules on the ultraviolet stellar flux \citep[e.g.][]{marino2008a, yong2008a, milone2012b, milone2015a}.
Some work is based on CMDs made with a wide colour baseline, which is sensitive to stellar populations with different helium abundances. Indeed, main sequence (MS) and red giant branch (RGB) stars enhanced in helium are hotter, hence bluer than stars with the same luminosity and primordial helium (Y$\sim0.25$) \citep[e.g.][]{dantona2002a, bedin2004a, piotto2007a, milone2018a}. Similarly, the $U-V$ or $U-I$ colours are widely used to distinguish stellar populations with different total metal content \citep[Z, e.g.][]{marino2015a, milone2017a}.
\citet{milone2015a} introduced a new photometric diagram called chromosome map (ChM) to identify and characterize the distinct stellar populations of GCs. The ChM is a pseudo two-colour diagram of MS, RGB or asymptotic-giant branch (AGB) stars, where the photometric sequences are verticalized in both dimensions. It maximizes the separation between stellar populations with different He, C, N, O and Fe.
Near infrared photometry is a powerful tool to identify MPs of M-dwarfs with different oxygen abundances and the F110W and F160W filters of the WFC3/NIR camera on board of {\it HST} are the most widely used filters \citep[e.g.\,][]{milone2012a, milone2019a}. Indeed, the F160W band is heavily affected by absorption from various molecules involving oxygen, including H$_{2}$O, while F110W photometry is almost unaffected by the oxygen abundance. As a consequence, second-generation (2G) stars, which are depleted in O with respect to the 1G, have brighter F160W magnitudes and redder F110W$-$F160W colours than the 1G.
In summary, all photometric diagrams used to detect multiple populations are based on colours and magnitudes that are mostly sensitive to the abundances of helium, nitrogen, oxygen and metallicity of stars in the distinct populations.
While photometry has been successful in identifying stellar populations with different C/N/O/Na in GCs, to date, it was almost blind to stars that are composed of material exposed to higher-temperature H burning, e.g. depleted in Mg with respect to the 1G.
In this paper, we introduce new photometric diagrams to disentangle stars based on their magnesium abundances. This is the first time that multiple populations with different [Mg/Fe] are identified from photometry alone.
Our target, $\omega$\,Centauri is the GC where multiple populations have been first detected \citep{woolley1966a}
and one of the most-studied clusters in the context of MPs \citep[e.g.][]{anderson1997a,lee1999a, pancino2000a, bedin2004a, ferraro2004a, sollima2005a, sollima2007a, bellini2010a}.
$\omega$~Centauri is an extreme Type II GC, which hosts at least sixteen populations that span a wide interval of metallicity,
(from [Fe/H] $\lesssim -2.0$ to [Fe/H]$\sim -0.6$)
\citep[e.g.][]{freeman1975a, suntzeff1996a, norris1995a, norris1996a, johnson2010a, marino2011a}, reach extreme abundances of helium and light elements \citep[e.g.][]{norris2004a, tailo2016a, bellini2017a, milone2017b}, and exhibit star-to-star magnesium variations \citep{gratton1982a, norris1995a, dacosta2013a, meszaros2019a}.
In particular, similarly to what is observed in other Type-II GCs, stellar populations of $\omega$~Centauri with any given metallicity span a wide range in light-elements abundances \\ \citep[e.g.][]{marino2010a, marino2011b, marino2012a, johnson2010a, gratton2011a}.
The paper is organized as follows. Section~\ref{sec:data} describes the dataset and the data reduction to derive the photometric diagrams presented in Section~\ref{sec:dia}. The theoretical ChMs and the photometric diagrams from isochrones are discussed in Section~\ref{sec:teo}, while Section~\ref{sec:mpoor} and Section~\ref{sec:mrich} are focused on the metal-poor and metal-rich stellar populations of $\omega$\,Cen, respectively. Summary and conclusions are provided in Section~\ref{sec:summary}.
\section{Data and data analysis}\label{sec:data}
The main dataset used in this work is composed of images collected through the F275W, F280N, F343N, F373N and F814W filters of the Ultraviolet and Visual Channel of the Wide Field Camera 3 (UVIS/WFC3) on board of {\it Hubble Space Telescope} ({\it HST}). The main properties of these images are summarized in Table~\ref{tab:data}.
Photometry and astrometry have been performed from images corrected from the poor charge transfer efficiency \citep[see][]{anderson2010a}. We used the computer program KS2,
that is the evolution of $kitchen\_sink$, originally written to reduce two-filter images collected with the Wide-Field Channel of the Advanced Camera for Survey of {\it HST} \citep{anderson2008a}.
As discussed by \citet{sabbi2016a} and \citet{bellini2017a}, KS2 adopts two different methods to derive high-precision measurements of stars with different luminosities. To determine magnitudes and positions of faint stars we combine information from all exposures and determine the average stellar positions. Once the positions of faint stars are fixed we fit each exposure pixel with the appropriate effective point spread function (PSF) solving for the flux only.
Bright stars are measured in each individual exposure by fitting the best PSF model and the resulting magnitudes and positions are then averaged.
Instrumental magnitudes are calibrated into the Vega-mag system by using the updated photometric zero points\footnote{\url{http://www.stsci.edu/hst/wfc3/analysis/uvis_zpts/} and \url{http://www.stsci.edu/hst/acs/analysis/zeropoints}} and following the recipe by \citet{bedin2005a}. Stellar positions are corrected for geometrical distortion by using the solutions provided by \citet{bellini2009a} and \citet{bellini2011a}. Finally, photometry was corrected for differential reddening as in \citet{milone2012a}.
To increase the number of filters and better constrain the abundances of helium, carbon, nitrogen, oxygen, and magnesium of the stellar populations of $\omega$\,Cen, we used the photometric and astrometric catalogues from \citet{milone2017a} and \citet{milone2018a}, which include photometry obtained from images collected through 31 filters of UVIS/WFC3.
Details of the dataset and the data reduction are provided by \citet{milone2017a} and \citet{milone2018a}.
In summary, our database comprises photometry in 36 bands: F218W, F225W, F275W, F280N, F300X, F336W, F343N, F373N, F390M, F373N, F390M, F390W, F395N, F410M, F438W, F467M, F469N, F475W, F487N, F502N, F555W, F606W, F621M, F625W, F631M, F645M, F656N, F657N, F658N, F673N, F673N, F680N, F689M, F763M, F680N, F689M, F763M, F775W, F814W, F845M and F953N.
\begin{table*}
\caption{Description of the {\it HST} images used in the paper.}
\begin{tabular}{ c c c l l}
\hline \hline
FILTER & DATE & N$\times$EXPTIME & PROGRAM & PI \\
\hline
F275W & Jul 15 2009 & 35$+$9$\times$350s & 11452 & J.\,Kim\,Quijano\\
F275W & Jan 12 -- Jul 4 2010 & 22$\times$800s & 11911 & E.\,Sabbi \\
F275W & Feb 02 2011 & 9$\times$800s & 12339 & E.\,Sabbi \\
F280N & Feb 26 2015 & 600s & 14031 & V.\,Kozhurina-Platais \\
F280N & Feb 01 2016 & 2$\times$800$+$4$\times$850s & 14393 & V.\,Kozhurina-Platais \\
F343N & Feb 26 2015 & 600s & 14031 & V.\,Kozhurina-Platais \\
F343N & Mar 03 2016 & 5$\times$510$+$4$\times$545s & 14393 & V.\,Kozhurina-Platais \\
F373N & Feb 26 2015 & 450s & 14031 & V.\,Kozhurina-Platais \\
F373N & Mar 25 2016 & 5$\times$500s & 14393 & V.\,Kozhurina-Platais \\
F814W & Jul 15 2009 & 35s & 11452 & J.\,Kim Quijano\\
F814W & Jan 10 -- Jul 04 2010 & 27$\times$40s & 11911 & E.\,Sabbi\\
F814W & Feb 15 -- Mar 24 2011 & 9$\times$40s & 12339 & E.\,Sabbi\\
\hline\hline
\end{tabular}
\label{tab:data}
\end{table*}
\begin{centering}
\begin{figure}
\includegraphics[height=10.cm,trim={0.0cm 5cm 5cm 3.0cm},clip]{spettri.pdf}
\caption{Comparison of two synthetic spectra with the same stellar parameters and metallicity quoted in the inset but different abundances of light elements (top panel). The main spectral features that are responsible for the differences between the fluxes of the two spectra are also indicated. In the middle panel we plot the quantity $-2.5 \log{f1/f2}$ as a function of the wavelength, where $f1$ and $f2$ are the fluxes of the red and the blue spectrum, respectively. Lower panel shows the transmission curves of the F275W, F280N, F343N and F373N WFC3/UVIS filters.}
\label{fig:spettri}
\end{figure}
\end{centering}
\section{Photometric diagrams}\label{sec:dia}
In this section we construct the photometric diagrams that will be exploited to analyse stellar populations with different Mg in $\omega$~Cen, starting from of the spectral features most sensitive to this element.
Based on synthetic spectra of RGB stars with different chemical composition, \citet{milone2018a} show that [Mg/Fe] variations have a negligible effect on magnitudes in optical bands but provide significant flux variations in ultraviolet bands \citep[see also][]{sbordone2011a}.
As an example, in the upper panel of Figure~\ref{fig:spettri} we compare the spectra of two RGB stars that have the same chemical abundances in all the elements, except for C, N, O and Mg, as quoted in the figure.
Specifically, the red spectrum is indicative of a 1G stars, whereas the blue spectrum corresponds to the 2G.
We calculated the ratio between the fluxes ($f1$ and $f2$) of the blue and red spectra and plotted in the middle panel of Figure~\ref{fig:spettri} the quantity $-2.5 \log{f1/f2}$ against $\lambda$. Lower panels show the transmission curves of the F275W, F280N, F343N and F373N filters of WFC3/UVIS.
Clearly, the F280N band, which includes the strong Mg-II lines at $\lambda$ $\sim$ 2,800 \AA\ is the most efficient filter on board of {\it HST} to identify stars with different values of [Mg/Fe].
The Mg-rich star is about 0.25 mag fainter in F280N than the Mg-poor one, the flux difference drops to $\sim$0.02 in F275W. Hence, $m_{\rm F275W}-m_{\rm F280N}$ is an efficient colour to identify stellar populations with different magnesium abundances.
Moreover, Figure~\ref{fig:spettri} reveals that the $m_{\rm F343N}-m_{\rm F373N}$ colour is sensitive to stellar populations with different nitrogen abundances. Indeed, the F343N filter includes the NH molecular bands around $\lambda \sim 3370$\AA, whereas the spectral region covered by the F373N filter is poorly affected by light-element variations.
Since $m_{\rm F275W}-m_{\rm F280N}$ and $m_{\rm F343N}-m_{\rm F373N}$ are short colour baselines, they are both poorly sensitive to temperature and reddening differences among stars with similar luminosities.
The observed $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ CMD of $\omega$\,Cen is shown in the upper-left panel of Figure~\ref{fig:cmds}. The RGB exhibits two main parallel sequences in the F814W magnitude interval between $\sim$16.5 and $\sim$13.8, as highlighted in the Hess-diagram plotted in the upper-right panel. The two RGB components seem to merge together at $m_{\rm F814W} \lesssim 13.8$. Additional RGB stars exhibit redder $m_{\rm F275W}-m_{\rm F280N}$ colours than the bulk of $\omega$\,Cen RGB stars. Their colour distance from the main RGB increases when we move from the base of the RGB towards the RGB tip.
The lower panels of Figure~\ref{fig:cmds} show the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD for $\omega$\,Cen RGB stars (left) and the corresponding Hess diagram for RGB stars with $14.0<m_{\rm F814W}<16.5$.
RGB stars of $\omega$\,Cen span a wide range of $m_{\rm F343N}-m_{\rm F373N}$ and exhibit four main stellar overdensities that indicate stellar populations with different nitrogen content.
For completeness, we mark with red crosses the asymptotic giant branch stars selected from the $m_{\rm F814W}$ vs.\,$m_{\rm F606W}-m_{\rm F814W}$ CMD. Since AGB stars have comparable stellar parameters as RGB stars,
a variation in light elements would produce similar broadening
in the AGB sequences. The fact that the AGB is narrower than the RGB in both CMDs of Figure~\ref{fig:cmds}
suggests
that stellar populations with extreme contents of nitrogen and magnesium avoid the AGB phase, in close analogy with what is observed in other GCs \citep[e.g. NGC\,6752, NGC\,2808 and NGC\,6266][]{campbell2013a, wang2016a, marino2017a, lapenna2015a}.
\begin{centering}
\begin{figure*}
\includegraphics[height=6.cm,trim={0.5cm 6.1cm 0cm 9.2cm},clip]{F814WvsF275WF280N.pdf}
\includegraphics[height=6.cm,trim={0.5cm 6.1cm 0cm 9.2cm},clip]{F814WvsF343NF373N.pdf}
\caption{Comparison between the $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ CMD of of $\omega$\,Cen (top), which is very sensitive to stellar populations with different magnesium content, and the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD (bottom), which highlights stars with different nitrogen abundance. RGB stars brighter than $m_{\rm F814W}=16.5$ are marked with black dots, while red crosses indicate AGB stars. Right panels show the Hess diagrams for stars in the rectangular regions shown in the right panels. }
\label{fig:cmds}
\end{figure*}
\end{centering}
\section{Comparison with simulated multiple populations}\label{sec:teo}
As discussed in the Introduction, $\omega$\,Cen exhibits large star-to-star metallicity variations, and the stellar populations with different [Fe/H] host sub stellar populations with distinct content of helium and light element abundance.
To investigate the physical reasons that are responsible for the multiple RGBs shown in Figure~\ref{fig:cmds}, we first compare in Section~\ref{sub:FeTeo} isochrones from the Dartmouth database \citep{dotter2008a} with different metallicities and helium contents.
Then, in Section~\ref{sub:NTeo} we investigate the effect of light-element abundance variations on the CMDs by using mono-metallic isochrones with different content of He, C, N, O and Mg.
\subsection{Stellar populations with different metallicities}\label{sub:FeTeo}
Figure~\ref{fig:iso} shows the $M_{\rm F814W}$ vs.\,$M_{\rm F275W}-M_{\rm F280N}$ (left panel) and the $M_{\rm F814W}$ vs.\,$M_{\rm F343N}-M_{\rm F373N}$ (right panel) CMDs for isochrones from \citet{dotter2008a} with different iron and helium abundances.
Specifically, we considered isochrones with [Fe/H]=$-1.8$ and [Fe/H]=$-1.5$, which correspond to the two main peaks in the metallicity distribution based on high-resolution spectroscopy \citep{marino2011a}, that are represented with blue and green colours, respectively. Moreover, we used isochrones with [Fe/H]=$-1.0$ and [Fe/H]=$-0.6$, which are representative to the most metal rich stellar populations of $\omega$\,Cen.
The isochrones represented with continuous lines have helium abundance Y=0.25+1.5 Z, where Z is the total metal abundance, while those isochrones plotted with dotted lines have helium Y=0.40.
A significant separation along the RGB is apparent at very high-metallicity and in the upper portion of the RGB with the metal-rich isochrones characterised by redder $M_{\rm F275W}-M_{\rm F280N}$ colours than metal-poor ones.
The magnitude where the metal-rich isochrones cross the metal poor ones depends on the value of [Fe/H]. As an example, the isochrones with [Fe/H]=$-1.5$, [Fe/H]=$-1.0$ and [Fe/H]=$-0.6$ intersect the [Fe/H]=$-1.8$ isochrone at $M_{\rm F814W} \sim 0.0$, $\sim 0.5$ and $\sim$2.7, respectively. An exception is provided by the isochrone with [Fe/H]=$-0.6$, which is redder than the used metal-poor isochrones along the entire RGB.
Helium-rich isochrones have almost the same $M_{\rm F275W}-M_{\rm F280N}$ colour as the metal-poor ones and become redder at brighter luminosities.
The right panel of Figure~\ref{fig:iso} shows that the low RGBs of the analyzed metal-rich isochrones typically have redder $M_{\rm F343N}-M_{\rm F373N}$ colours than those of metal poor ones. The most metal rich isochrone, which is bluer than isochrones with [Fe/H]=$-1.5$ and [Fe/H]=$-1.0$, is a remarkable exception.
The isochrones with different [Fe/H] cross each other and change their relative $M_{\rm F343N}-M_{\rm F373N}$ colours in the upper part of the RGB.
Helium-rich RGB stars of the isochrones with [Fe/H]$ \geq -1.5$ are redder than stars with Y$\sim$0.25 and the same F814W magnitude, while the RGBs of the two isochrones with [Fe/H]=$-1.8$ and different helium abundances are nearly coincident for $M_{\rm F814W} \gtrsim -1.5$.
\begin{centering}
\begin{figure*}
\includegraphics[height=9.cm,trim={.7cm 5cm 1cm 5cm},clip]{iso1.pdf}
\includegraphics[height=9.cm,trim={.7cm 5cm 1cm 5cm},clip]{iso2.pdf}
\caption{Dartmouth isochrones \citep{dotter2008a} in the $M_{\rm F814W}$ vs.\,$M_{\rm F275W}-M_{\rm F280N}$ (left panel) and $M_{\rm F814W}$ vs.\,$M_{\rm F343N}-M_{\rm F373N}$ (right panel) planes for 13-Gyr old stellar populations with [$\alpha$/Fe]=0.4 and with the abundances of iron and helium listed in the inset of left panel. The RGB sequences comprise the isochrone segments brighter than $M_{\rm F814W} \sim 3.0$. }
\label{fig:iso}
\end{figure*}
\end{centering}
\subsection{Stellar populations with different light-element abundances }\label{sub:NTeo}
To investigate how the abundances of C, N, O, Mg affect the position of stars in the CMDs of mono-metallic GCs, we derived the colours and magnitudes of isochrones with different abundances of carbon, nitrogen, oxygen and magnesium by using model atmospheres and synthetic spectra, in close analogy with what done in previous papers from our team \citep[e.g.][]{milone2012b, milone2018a}.
In a nutshell, we extracted fifteen points along the isochrones and extracted their effective temperatures, $T_{\rm eff}$, and gravities, $\log{g}$. For each pair of stellar parameters we calculated a reference spectrum with the chemical composition of 1G stars and a comparison spectrum, with the content of C, N, O and Mg of 2G stars.
Specifically, we assumed for 1G stars Y=0.246, solar-scaled abundances of carbon and nitrogen, [O/Fe]=0.4, and [Mg/Fe]=0.4.
We constructed model atmosphere structures by using the computer program ATLAS12 developed by Robert Kurucz \citep{kurucz1970a, kurucz1993a, sbordone2004a}, which exploits the opacity-sampling technique and assumes local thermodynamic equilibrium. We assumed a microturbolent velocity of 2 km s$^{-1}$ for all models.
Synthetic spectra are computed over the wavelength interval between 1,000 and 12,000 \AA\ with SYNTHE \citep{kurucz1981a, kurucz2005a, castelli2005a, sbordone2007a} and have been integrated over the bandpasses of the ACS/WFC and UVIS/WFC3 filters used in this paper to derive the corresponding magnitudes.
We calculated the magnitude difference, $\delta {m}_{\rm X}$, between the comparison and the reference spectrum. The magnitudes of the 2G isochrones are derived by adding to the 1G isochrones the corresponding values of $\delta {m}_{\rm X}$.
We plot in Figure~\ref{fig:ChM280teo} five isochrones with the same iron abundance, [Fe/H]=$-1.7$, but different content of He, C, N, O and Mg in the $M_{\rm F814W}$ vs.\,$M_{\rm F275W}-M_{\rm F280N}$ and $M_{\rm F814W}$ vs.\,$M_{\rm F343N}-M_{\rm F373N}$ planes. The specific chemical composition of each isochrone is quoted in Table~\ref{tab:chimica} and corresponds to constant [(C$+$N$+$O)/Fe]. In particular, we assumed that the green and the red isochrones have [Mg/Fe]=$0.4$, while both the cyan and the blue ones are depleted in magnesium by 0.5 dex. The yellow isochrones have intermediate magnesium abundance ([Mg/Fe]=$+$0.25).
Clearly, the $M_{\rm F275W}-M_{\rm F280N}$ colour of RGB stars mostly depends on the magnesium abundance, with the Mg-poor isochrones having redder RGBs than Mg-rich isochrones. Nitrogen and helium poorly affect the $M_{\rm F275W}-M_{\rm F280N}$ colour of RGB stars. Indeed, the red and green isochrones, which have the same magnesium abundance but different nitrogen content ([N/Fe]=1.21) are superimposed to each other, similarly to the blue and cyan isochrones, which share the same value of [Mg/Fe] but have different helium mass fractions of Y=0.246 and Y=0.34, respectively.
As illustrated in the middle panel of Figure~\ref{fig:ChM280teo}, the $M_{\rm F343N}-M_{\rm F373N}$ colour of RGB stars is mainly a tracer of the nitrogen abundance, and the RGBs moves towards red colours when the value of [N/Fe] increases. The helium abundance of the stellar populations poorly affects the $M_{\rm F343N}-M_{\rm F373N}$ colour of RGB stars, as demonstrated by the fact that the blue and cyan isochrones, which have different helium abundances and the same [N/Fe], are nearly coincident in the $M_{\rm F814W}$ vs.\,$M_{\rm F275W}-M_{\rm F280N}$ CMD.
Right panel of Figure~\ref{fig:ChM280teo} shows the ChM of the RGB stars with $-1<M_{\rm F814W}<2$.
The vectors superimposed on the ChM represent the expected correlated changes of $\Delta_{\rm F343N,F373N}$ and $\Delta_{\rm F275W,F280N}$ when the abundances of He, C, N, O and Mg are changed one at a time.
Specifically, we assumed helium mass fraction variation $\Delta$\,Y=0.154 and elemental variations of $\Delta$[C/Fe]=$-$0.50, $\Delta$[N/Fe]=1.21, $\Delta$[O/Fe]=$-$0.50 and $\Delta$[Mg/Fe]=$-$0.50.
In monometallic GCs, the $\Delta_{\rm F343N,F373N}$ quantity is nearly entirely affected by nitrogen variations whereas the value of $\Delta_{\rm F275W,F280N}$ is mostly dependent on [Mg/Fe].
Star-to-star differences in He, C and O provide small variations of $\Delta_{\rm F343N,F373N}$ and $\Delta_{\rm F275W,F280N}$, that can be appreciated in the inset of Figure~\ref{fig:ChM280teo}, which is a zoom of the region of the ChM around the origin of the vectors.
We conclude that the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM is a efficient tool to identify stellar populations with different [Mg/Fe] and [N/Fe] in monometallic GCs.
\begin{centering}
\begin{figure*}
\includegraphics[height=9.cm,trim={1.0cm 5cm .5cm 12.0cm},clip]{ChM280teo.pdf}
\caption{Green, red, yellow, cyan and blue colours represent the isochrones I1--I5, which have ages of 13 Gyr, [Fe/H]=$-1.8$, [$\alpha$/Fe]=0.4 and different abundances of He, C, N, O and Mg as listed in Table~\ref{tab:chimica}. Right panel show the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM for RGB stars between the horizontal lines. The arrows indicate the effect of changing Y, C, N, O and Mg one at a time in the ChM. Specifically, we assumed $\Delta$\,Y=0.154, $\Delta$[C/Fe]=$-$0.50, $\Delta$[N/Fe]=1.21, $\Delta$[O/Fe]=$-$0.50, $\Delta$[Mg/Fe]=$-$0.50. }
\label{fig:ChM280teo}
\end{figure*}
\end{centering}
\begin{table}
\caption{Chemical composition of the five isochrones shown in Figure~\ref{fig:ChM280teo}. All isochrones have ages of 13 Gyr, [Fe/H]=$-1.8$, [$\alpha$/Fe]=0.4 and the same overall C$+$N$+$O content. }
\begin{tabular}{ c c c c c c}
\hline \hline
Isochrone & Y & [C/Fe] & [N/Fe] & [O/Fe] & [Mg/Fe] \\
\hline
I1 & 0.246 & 0.00 & 0.00 & 0.40 & 0.40 \\
I2 & 0.246 &$-$0.05 & 0.53 & 0.35 & 0.40 \\
I3 & 0.246 &$-$0.10 & 0.93 & 0.20 & 0.25 \\
I4 & 0.246 &$-$0.50 & 1.21 & $-$0.10 & $-$0.10 \\
I5 & 0.340 &$-$0.50 & 1.21 & $-$0.10 & $-$0.10 \\
\hline\hline
\end{tabular}
\label{tab:chimica}
\end{table}
\section{Metal-poor stellar populations}\label{sec:mpoor}
Recent work shows that GCs can be classified into two main classes of clusters with distinct photometric and spectroscopic properties. Type I GCs exhibit single sequences of 1G and 2G stars in their ChMs and have nearly constant metal content. Type\,II GCs are composed of multiple sequences of 1G and 2G stars in the ChM and split SGBs in CMDs made with photometry in optical bands \citep[][]{milone2015a, milone2017a}.
\citet{marino2019a} provided the chemical tagging of multiple populations over the ChM and concluded that Type II GCs correspond to the class of `anomalous' GCs, which exhibit star-to-star variations in some heavy elements, like Fe and s-process elements \citep[e.g.][]{yong2008b, marino2009a, marino2015a, marino2018a, carretta2010a, johnson2015a}. We exploit these findings to identify a sample of metal-poor stars in $\omega$\,Cen, which is a Type II GC with extreme metallicity variations. Indeed, the fact that stars with different metallicities of Type\,II GCs occupy different regions of the ChM makes it possible to identify stellar populations with different iron abundance based on the ChM alone.
The upper panel of Figure~\ref{fig:CMDtI}
shows the ChM of $\omega$\,Cen where the metal-poor stars ([Fe/H]$\sim -1.8$) identified by \citet{milone2017a} are represented with black dots and coloured gray the remaining stars.
The sample of metal poor stars has been selected on the basis of the fact that it define the reddest RGB sequence in $m_{\rm F336W}$ vs.\,$m_{\rm F336W}-m_{\rm F814W}$ CMD, which is sensitive to stellar populations with different metallicities in Type II GCs \citep[e.g.][]{marino2011b, marino2015a, marino2018a}. The selected metal-poor stars also populate the bluest RGB sequence in CMD made with optical filters (e.g.\,$m_{\rm F438W}$ vs.\,$m_{\rm F438W}-m_{\rm F814W}$) and define a distinct sequence in the ChM. Furthermore, the low metallicity of these stars is confirmed by direct spectroscopic measurements of the iron abundance from \citet{johnson2010a, marino2011a, mucciarelli2019a}.
Lower panels
highlight metal-poor stars in the $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ and the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMDs.
The RGB is clearly split in the left-panel CMD and the RGB sequence with redder $m_{\rm F275W}-m_{\rm F280N}$ colours includes $\sim$30\% of the total number of RGB stars. The colour separation between the two main RGBs is about 0.2 mag at $m_{\rm F814W}=16.0$ and decreases towards bright luminosities. The two RGBs merge together at $m_{\rm F814W}\sim12.5$.
The discreteness of the various RGB sequences is less evident in the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD (bottom-right panel of Figure~\ref{fig:CMDtI}). Metal-poor stars span a colour range of about 0.2 mag in the luminosity interval that ranges from the RGB base to $m_{\rm F814W}\sim13.5$, which is narrower than the $m_{\rm F343N}-m_{\rm F373N}$ spanned by metal rich stars. The colour spread decreases for $m_{\rm F814W}\lesssim13.5$ and is comparable with observational errors towards the RGB tip.
\begin{centering}
\begin{figure*}
\includegraphics[height=13.cm,trim={1.5cm 5cm 1.2cm 2.5cm},clip]{CMDtI.pdf}
\caption{This figure illustrates the procedure to identify metal-poor stars in $\omega$\,Cen. the upper panel reproduces the $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM of $\omega$\,Cen from \citet{milone2017a}. Lower panels show the $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ (left) and the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD (right) of $\omega$\,Cen. Metal-poor stars are marked with black dots.
The arrows plotted in the upper panel show the effect of changing He, C, N, O, Mg and Fe, one at a time, on $\Delta_{\rm {\it C} F275W,F336W,F438W}$ and $\Delta_{\rm F275W,F814W}$.
We adopted an iron variation of $\Delta$[Fe/H]=$0.30$ and the same C, N, O and Mg variations of Figure~\ref{fig:ChM280teo}. }
\label{fig:CMDtI}
\end{figure*}
\end{centering}
\subsection{The $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ chromosome map}
The $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ and $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMDs plotted in the bottom panels of Figure~\ref{fig:CMDtI}, are used to derive the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM of RGB stars that we plot in Figure~\ref{fig:ChM} \citep[see][for details]{milone2015a, milone2018a, zennaro2019a}. We only included stars with $13.8<m_{\rm F814W}<16.0$, which is the magnitude interval where multiple populations are clearly visible.
Figure~\ref{fig:ChM} reveals that about 70\% of metal-poor stars define a vertical sequence with $\Delta_{\rm F275W,F280N} \sim 0.05$, while most of the remaining metal-poor stars are clustered around $\Delta_{\rm F275W,F280N} \sim 0.3$ and $\Delta_{\rm F343N,F373N} \sim 0.18$.
As demonstrated in Section~\ref{sub:NTeo}, the abscissa of this ChM is mostly sensitive to magnesium abundance.
Hence, we conclude that about 30\% of the selected metal-poor stars in $\omega$\,Cen are significantly depleted in [Mg/Fe].
\begin{centering}
\begin{figure}
\includegraphics[height=7.5cm,trim={.5cm 5cm .5cm 2.4cm},clip]{ChM.pdf}
\caption{ $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM of RGB stars with $13.8<m_{\rm F814W}<16.0$. Metal-poor stars are represented with black circles, while the remaining stars are coloured gray. The arrows indicate the effect of changing He, C, N, O, Mg and Fe, one at a time, on $\Delta_{\rm F343N,F373N}$ and $\Delta_{\rm F275W,F280N}$.}
\label{fig:ChM}
\end{figure}
\end{centering}
Figure~\ref{fig:popst1} compares the classical ChM of $\omega$ Cen from \citet{milone2017a} with the $\Delta_{\rm F275W,F280N}$ vs.\,$\Delta_{\rm F343N,F373N}$ ChM introduced in this paper. We used green colours to mark the 1G stars defined by Milone and collaborators, which contains 25.9$\pm$1.3\% of stars. We also identified four main groups of 2G stars with different values of $\Delta_{\rm {\it C} F275W,F336W,F438W}$, that we called 2G$_{\rm A--D}$ and include 20.7$\pm$1.2, 23.2$\pm$1.3, 6.3$\pm$0.7 and 23.8$\pm$1.3\% of metal-poor stars, respectively. Their stars are represented in Figure~\ref{fig:popst1} with blue, cyan, orange and magenta colours, respectively.
We find that 2G stars have, on average, higher values of $\Delta_{\rm F275W,F280N}$ and $\Delta_{\rm F343N,F373N}$ than 1G stars.
2G$_{\rm A}$ and 2G$_{\rm B}$ stars share almost the same range of $\Delta_{\rm F275W,F280N}$ as 1G stars, but have higher $\Delta_{\rm F343N,F373N}$ values than the 1G.
2G$_{\rm D}$ stars exhibit significantly higher values of $\Delta_{\rm F275W,F280N}$ and $\Delta_{\rm F343N,F373N}$ than the remaining RGB stars, while the poorly-populated group of 2G$_{\rm C}$ stars exhibits intermediate values of $\Delta_{\rm F275W,F280N}$ and $\Delta_{\rm F343N,F373N}$.
Since $\Delta_{\rm F275W,F280N}$ quantity is mostly affected by magnesium variation, we conclude that 1G stars share similar [Mg/Fe] as 2G$_{\rm A}$ and 2G$_{\rm B}$ stars and correspond to the populations with [Mg/Fe]$\sim$0.5 identified by \citet{norris1995a}, while the remaining stars are significantly depleted in magnesium with respect to the bulk of $\omega$\,Cen stars, with 2G$_{\rm D}$ stars having the lowest magnesium content.
The fact that in monometallic GCs the $\Delta_{\rm F343N,F373N}$ quantity is mostly sensitive to nitrogen variations, indicates that the nitrogen abundance increases when we move from the 1G, which has the lowest value of [N/Fe], to 2G$_{\rm D}$. In the next section, we will exploit photometry in 36 filters to
constrain the chemical composition of the metal-poor stellar populations in $\omega$\,Cen.
\begin{centering}
\begin{figure*}
\includegraphics[height=10.cm,trim={0.5cm 5cm 0.5cm 7.5cm},clip]{popst1.pdf}
\caption{Zoom in of the $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM around the region populated by metal-poor stars (left panel) and $\Delta_{\rm F275W,F280N}$ vs.\,$\Delta_{\rm F343N,F373N}$ ChM (lower panel) of RGB stars of $\omega$\,Cen.
1G stars are represented with green symbols, while the groups of 2G$_{\rm A--D}$ metal-poor stars are coloured blue, cyan, orange and magenta, respectively. Gray dots represent the remaining RGB stars.
}
\label{fig:popst1}
\end{figure*}
\end{centering}
\subsection{The chemical composition of stellar populations}\label{sub:chimica}
To infer the relative abundances of He, C, N, O, and Mg between 2G$_{\rm A--D}$ and 1G stars, we extended the method by \citet{milone2012b, milone2018a} and \citet{lagioia2018a, lagioia2019a} to the sample of metal-poor stars of $\omega$\,Cen.
Briefly, we derived the fiducial lines of 2G$_{\rm A--D}$ and 1G stars in the $m_{\rm F814W}$ vs.\,$m_{\rm X}-m_{\rm F814W}$ CMD in the luminosity interval between $m_{\rm F814W}=13.8$ and $m_{\rm F814W}=16.0$, where X corresponds to each of the 36 filters used in this paper and listed in Section~\ref{sec:data}.
To do this, we divided the RGB into F814W magnitude intervals of
size $\delta m$=0.25 mag which are defined over a grid of points spaced by magnitude $\delta m/2$ bins of size. For each bin we calculated the median
$m_{\rm F814W}$ magnitude and $m_{\rm X}-m_{\rm F814W}$ color and linearly interpolated these median points.
As an example, in the upper panels of Figure~\ref{fig:fidu} we plot $m_{\rm F814W}$ against $m_{\rm X}-m_{\rm F814W}$ for RGB stars of $\omega$\,Cen, where X=F275W, F280N, F343N and F438W and use green, blue, cyan, orange and magenta colours to represent the fiducials of the 1G, 2G$_{\rm A--D}$.
We defined five reference magnitudes along the RGB, namely $m^{\rm ref}_{\rm F814W}=15.9, 15.4, 14.9, 14.4$ and 13.9 to derive five estimates of the chemical composition of stellar populations. These five magnitude levels are marked with dashed horizontal lines in the upper panels of Figure~\ref{fig:fidu}.
For each value of $m^{\rm ref}_{\rm F814W}$ we measured the colour difference between the fiducial of 2G$_{\rm A--D}$ and the fiducial of 1G stars ($\Delta$($m_{\rm X}-m_{\rm F814W}$)).
\begin{centering}
\begin{figure*}
\includegraphics[height=6.cm,trim={0.5cm 5cm 2.25cm 8.9cm},clip]{fid1.pdf}
\includegraphics[height=6.cm,trim={3.15cm 5cm 0.5cm 8.9cm},clip]{fid2.pdf}
\includegraphics[height=7.5cm,trim={.5cm 5cm 0.5cm 2.9cm},clip]{dcol2.pdf}
\includegraphics[height=7.5cm,trim={3.15cm 5cm 0.5cm 2.9cm},clip]{dcol1.pdf}
\caption{Upper panels show the $m_{\rm F814W}$ vs.\,$m_{\rm X}-m_{\rm F814W}$ CMDs for RGB stars, where X=F275W, F280N, F343N and F438W. The fiducial lines of the 1G and 2G$_{\rm D}$ populations identified in the paper are coloured green and magenta, respectively, while the dashed lines mark the five values of $m_{\rm F814W}^{\rm CUT}$ used to estimate the relative chemical compositions of the various stellar populations. Lower panels show the $m_{\rm X}-m_{\rm F814W}$ colour differences between the fiducials of 2G$_{\rm A--D}$ stars and the fiducial of 1G stars for the 36 X filters calculated at $m_{\rm F814W}^{\rm CUT}=15.9$. Observations are represented with coloured points while black crosses correspond to the best-fit models. The procedure, illustrated in the lower panels for $m_{\rm F814W}^{\rm CUT}=15.9$ has been extended to the other four values of $m_{\rm F814W}^{\rm CUT}$.}
\label{fig:fidu}
\end{figure*}
\end{centering}
\begin{centering}
\begin{figure}
\includegraphics[height=9.cm,trim={0.0cm 6cm 0cm 0cm},clip]{chimica.pdf}
\caption{The abundances of helium, carbon, nitrogen and oxygen of the five studied stellar populations relative to the 1G abundances are plotted against the relative magnesium content. Green, blue, cyan, orange and magenta colours represent 1G and 2G$_{A--D}$ stellar populations, respectively.}
\label{fig:chimica}
\end{figure}
\end{centering}
We exploited the isochrones by \citet{dotter2008a} that provide the best match with the observed $m_{\rm F814W}$ vs.\,$m_{\rm F606W}-m_{\rm F814W}$ CMD to estimate the gravities and the effective temperatures of 1G stars corresponding to the five reference points. To do this, we assumed [Fe/H]=$-$1.8, which is similar to the iron abundance inferred by \citet{marino2019a} for metal-poor stars of $\omega$\,Cen, [$\alpha$/Fe]=0.4. We also adopted age of 13.0 Gyr, distance modulus ($m-M$)$_{\rm 0}=13.58$ and reddening E(B$-$V)=0.15, which are similar to the quantities provided by \citet[][updated as in 2010]{harris1996a} and \citet{dotter2010a}.
To constrain the chemical composition of 2G$_{\rm A--D}$ stars from the colors calculated at $m^{\rm ref}_{\rm F814W}=15.9$, we computed a grid of synthetic spectra with different abundances of He, C, N, O and Mg, that we compare with a synthetic spectrum corresponding to the 1G used as reference. The latter has the values of gravity and effective temperature inferred from the best-fit isochrone for $m^{\rm ref}_{\rm F814W}=15.9$, Y=0.246, [C/Fe]=0.0, [N/Fe]=0.0, [O/Fe]=0.4 and [Mg/Fe]=0.4.
We assumed for synthetic spectra a set of values for [C/Fe] that range from $-$0.5 to 0.1, [N/Fe] between 0.0 and 1.5, [O/Fe] that ranges from $-0.6$ to $0.5$ and [Mg/Fe] from $-0.2$ to $0.5$. We used steps of 0.05 dex for all elements.
The helium mass fractions of the comparison spectra range from Y=0.246 to
0.400 in steps of 0.001 and the effective parameters of helium-rich isochrones are taken from the corresponding isochrones from \citet{dotter2008a}. The spectra are computed by using the computer programs ATLAS 12 and Synthe \citep{sbordone2004a, sbordone2007a, kurucz2005a, castelli2005a}.
We convoluted each spectrum with the transmission curves of the 36 WFC3/UVIS and ACS/WFC filters available for $\omega$\,Centauri and derived the colour difference between 2G$_{\rm A--D}$ and 1G stars corresponding to each reference point. The best estimates of Y, C, N, O, and Mg of 2G$_{\rm A--D}$ stars, for $m^{\rm ref}_{\rm F814W}=15.9$ are given by the elemental abundances of the comparison synthetic spectrum that provides the best fit with observed colour differences.
The best fit is derived by means of $\chi$-square minimization, that is calculated by accounting for the uncertainties on color determinations and on the sensitivity of the colors to the abundance of a given element, as inferred from synthetic spectra \citep[see][]{dodge2008a}. In the following cases, we excluded some filters to derive the abundances of certain elements. Specifically, we constrained the helium content from filters that are redder than F438W as they are poorly affected by the effect of C, N, O and Mg on stellar atmosphere. We used the F336W and F343N filters to infer the content of nitrogen alone, as they are largely affected by the abundance of this element. Similarly, we used F280N to derive the magnesium abundance alone.
An example is provided in the lower panels of Figure~\ref{fig:fidu}, where we compare the colour differences between the fiducials of 2G$_{\rm A--D}$ stars and the fiducial of the 1G calculated at $m_{\rm F814W}^{\rm cut}=15.9$ (coloured points) with the colour differences from the best-fit comparison spectra (black crosses).
This procedure, discussed and illustrated for $m_{\rm F814W}^{\rm cut}=15.9$, is extended to the other four reference magnitudes and provides five determinations of He, C, N, O and Mg of population 2G$_{\rm A--D}$-stars relative to the 1G.
The average abundances are listed in Table~\ref{tab:abb}.
\begin{table*}
\caption{Average abundances of He, C, N, O and Mg of population 2G$_{\rm A--D}$-stars relative to the 1G.}
\begin{tabular}{ c c c c c c}
\hline \hline
Population & $\Delta$Y & $\Delta$[C/Fe] & $\Delta$[N/Fe] & $\Delta$[O/Fe] & $\Delta$[Mg/Fe] \\
\hline
2G$_{\rm A}$ & 0.005$\pm$0.004 & $-$0.20$\pm$0.08 & 0.31$\pm$0.08 & $-$0.15$\pm$0.08 & $-$0.03$\pm$0.04 \\
2G$_{\rm B}$ & 0.016$\pm$0.007 & $-$0.20$\pm$0.09 & 0.62$\pm$0.09 & $-$0.30$\pm$0.07 & $-$0.13$\pm$0.06 \\
2G$_{\rm C}$ & 0.051$\pm$0.010 & $-$0.32$\pm$0.11 & 0.75$\pm$0.10 & $-$0.50$\pm$0.09 & $-$0.25$\pm$0.07 \\
2G$_{\rm D}$ & 0.081$\pm$0.007 & $-$0.42$\pm$0.08 & 1.02$\pm$0.07 & $-$0.60$\pm$0.09 & $-$0.44$\pm$0.06 \\
\hline\hline
\end{tabular}
\label{tab:abb}
\end{table*}
To test the robustness of the results and estimate the uncertainties, we repeated the procedure described above on 1,000 simulated $m_{\rm F814W}$ vs.\,$m_{\rm X}-m_{\rm F814W}$ CMDs with fixed light-element variations and the same number of stars and observational errors as our observations of $\omega$\,Centauri. We find that our procedure correctly recovers the input abundances and provides the uncertainties listed in Table~\ref{tab:abb}.
We find that 2G$_{\rm A}$ and 1G stars have similar magnesium content, while 2G$_{\rm D}$ stars exhibit extreme magnesium depletion by $\Delta$[Mg/Fe]$\sim$0.45 dex, with respect to the 1G. 2G$_{\rm B}$ and 2G$_{\rm C}$ exhibit intermediate magnesium abundances.
As shown in Figure~\ref{fig:chimica}, the abundances of helium and nitrogen anti-correlate with magnesium content, while carbon and oxygen correlate with [Mg/Fe].
2G$_{\rm D}$ stars, which exhibit the largest abundance variations, are enhanced in [N/Fe] by $\sim$1.0 dex and in helium mass fraction by $\sim$0.08, with respect to the 1G. Moreover, 2G$_{\rm D}$ stars are depleted in carbon and oxygen by $\sim$0.4 and $\sim$0.6 dex, when compared with the 1G.
2G$_{\rm A--C}$ stars exhibit intermediate abundances of He, C, N and O.
\begin{centering}
\begin{figure*}
\includegraphics[height=13.cm,trim={1.5cm 5cm 1.2cm 2.5cm},clip]{ano1.pdf}
\caption{ $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM (upper panel), $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$ (left) (lower-left panel) and $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD (lower-right panel) of $\omega$\,Cen. Metal-rich stars with $\Delta_{\rm F275W,F814W}<0.65$ are coloured black. The remaining metal-rich stars of the lower- middle- and upper stream are represented with red starred symbols, purple diamonds and purple crosses, respectively. }
\label{fig:ano1}
\end{figure*}
\end{centering}
\section{Metal-rich stellar populations}\label{sec:mrich}
In this section, we investigate the sample of metal-rich stars identified by \citet{milone2017a} that we highlight in the ChM and CMDs of Figure~\ref{fig:ano1}. Specifically, we used red and purple symbols to represent the metal rich stars with $\Delta_{\rm F275W,F814W}>0.6$, which comprise stars with [Fe/H]$\gtrsim -1.0$ \citep[][]{marino2019a}, and we coloured black the remaining metal-rich stars (hereafter metal-intermediate sample).
The RGBs of metal-rich stars clearly exhibit less-steep slopes than those of metal-poor and metal-intermediate stars in the $m_{\rm F814W}$ vs.\,$m_{\rm F275W}-m_{\rm F280N}$, in qualitative agreement with the isochrones plotted in Figure~\ref{fig:iso}.
Metal intermediate stars with $m_{\rm F814W}>13.8$ span a smaller range of $m_{\rm F275W}-m_{\rm F280N}$ than metal poor stars with similar F814W magnitude. Moreover, there is no clear evidence for a bimodal $m_{\rm F275W}-m_{\rm F280N}$ distribution as observed for metal-poor stars.
Based on chemical abundances of stars in the ChM, \citet{marino2019a} identified the lower-, middle-, and upper-streams that are composed of N-poor, N-intermediate and N-rich stars, respectively.
The sample of metal-intermediate stars with $m_{\rm F814W} \gtrsim 13.5$ shown in Figure~\ref{fig:ano1} span an interval of $\sim$0.1 mag in $m_{\rm F343N}-m_{\rm F373N}$ and is composed of three main RGBs. The RGB with the reddest $m_{\rm F343N}-m_{\rm F373N}$ colour is the most-populated one and is composed of stars of the upper stream of $\omega$ Cen. The bluest and the middle RGBs comprise stars that populated the lower and the middle stream.
Stars in the lower, middle and upper stream with $\Delta_{\rm F275W,F814W}>0.65$ are represented with red starred symbols, purple diamonds and purple crosses, respectively.
Most of these metal-rich stars are distributed into two distinct sequences that define the bluest and the reddest boundaries of the low RGB in the $m_{\rm F814W}$ vs.\,$m_{\rm F343N}-m_{\rm F373N}$ CMD, and are composed of stars of the lower and upper stream, respectively. Middle-stream stars exhibit intermediate colours.
All metal-rich stars exhibit bluer $m_{\rm F343N}-m_{\rm F373N}$ colours than the remaining RGB stars at magnitudes brighter than $m_{\rm F814W} \sim 13.0$, as expected from isochrones with different metallicities (see Figure~\ref{fig:iso}).
\begin{centering}
\begin{figure*}
\includegraphics[height=13.cm,trim={1.0cm 5cm 1.cm 2.5cm},clip]{ano2.pdf}
\caption{\textit{Upper-left panels.} Reproductions of the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM introduced in Figure~\ref{fig:ChM}. We used black colours to represent the metal-intermediate stars defined in Figure~\ref{fig:ano1} while metal-rich stars that belong to the upper, middle and lower stream are represented with purple crosses, purple diamonds and red starred symbols, respectively. The remaining RGB stars are coloured gray. The box superimposed on the ChM plotted in the upper-right panel encloses stars with depleted magnesium.
\textit{Lower panel.} $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM.
The 2G$_{\rm B}$, 2G$_{\rm C}$, and 2G$_{\rm D}$ metal-poor stars within the box shown in the upper-left panel, are coloured cyan, orange and magenta, respectively, while candidate Mg-poor metal-intermediate stars marked with aqua symbols in the upper-left and lower panels. }
\label{fig:ano2}
\end{figure*}
\end{centering}
The fact that metal-intermediate and metal-rich stars of $\omega$\,Cen span a wide metallicity interval makes it challenging to estimate their relative abundances of He, C, N, O and Mg.
Indeed, the method adopted in Section~\ref{sub:chimica} for metal-poor stars is based on the comparison between the observed colors of multiple populations and the colors derived by synthetic spectra with appropriate chemical composition. Metallicity variations, including iron abundance and overall C$+$N$+$O abundances, provide significant changes on stellar structure \citep[][]{sbordone2011a}, thus affecting the atmospheric parameters of synthetic spectra. Nevertheless, metallicity is poorly constrained for the various stellar populations of $\omega$\,Cen \citep[][]{marino2019a}.
For this reason, we will only provide a qualitative discussion on the chemical composition of metal-intermediate and metal-rich stellar populations, which are strongly enhanced in [Fe/H] and [(C$+$N$+$O)/Fe] with respect to metal-poor stars \citep[ e.g.][]{johnson2010a, marino2011a, marino2012a}.
To do this, we reproduce in the upper-left panel of Figure~\ref{fig:ano2} the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM and highlight metal-intermediate and metal-rich stars with black and purple+red symbols, respectively.
Most metal-rich stars define two horizontal sequences with nearly constant $\Delta_{\rm F343N,F373N} \sim 0.0$ (populated by lower-stream stars) and one with $\Delta_{\rm F343N,F373N} \sim 0.3$ that hosts upper-stream stars. Middle-stream metal-rich stars exhibit intermediate $\Delta_{\rm F343N,F373N}$ values.
Metal-intermediate stars define three main stellar blobs with $\Delta_{\rm F275W,F280N} \sim 0.05$ but different values of $\Delta_{\rm F343N,F373N} \sim 0.05, 0.13$ and 0.20 that we indicate as M-I$_{\rm A}$, M-I$_{\rm B}$ and M-I$_{\rm C}$, respectively, and are composed of stars in the lower, middle and upper stream.
A fourth group of metal-intermediate stars (M-I$_{\rm D}$) is distributed around ($\Delta_{\rm F275W,F280N}:\Delta_{\rm F343N,F373N}$) $\sim$ (0.15:0.20).
To highlight metal-poor and metal-intermediate stars that, based on their position in the $\Delta_{\rm F343N,F373N}$ vs.\,$\Delta_{\rm F275W,F280N}$ ChM, are strongly depleted in magnesium with respect to the majority of $\omega$\,Cen stars, we draw the gray box in the upper-right panel of Figure~\ref{fig:ano2}.
This box includes the metal-poor populations 2G$_{\rm C}$ and 2G$_{\rm D}$ that we represented with magenta and orange points, respectively, plus a few population 2G$_{\rm B}$ stars coloured cyan.
The evidence that the M-I$_{\rm D}$ stars in the box (aqua dots) have larger values of $\Delta_{\rm F275W,F280N}$ than the Mg-rich metal-poor stars demonstrate that they have low magnesium content.
The $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM plotted in the lower panel of Figure~\ref{fig:ano2} reveals that metal-intermediate Mg-poor stars belong to the upper stream. Hence, as shown by \citet{marino2019a}, they have enhanced He, N and Na and depleted C and O.
\section{Summary and discussion}\label{sec:summary}
We exploited high-precision photometry in the F275W, F280N, F343N and F373N bands of {\it HST} of $\omega$\,Centauri, to introduce a ChM that is able to identify stellar populations with different Mg and N abundances along the RGB.
Based on synthetic spectra with appropriate light-element abundances, we demonstrated that the abscissa of this new ChM, $\Delta_{\rm F275W, F280N}$, is mostly sensitive to the relative content of magnesium of the stellar populations of monometallic GCs and is poorly affected by star-to-star variations of He, C, N, and O. Its ordinate, $\Delta_{\rm F343N, F373N}$, maximises the separation between stellar populations with different nitrogen and is negligibly affected by variation in the other light elements involved in the multiple-population phenomenon, including He, C, O and Mg.
In GCs with iron variations,
both axes of the $\Delta_{\rm F343N, F373N}$ vs.\,$\Delta_{\rm F275W, F280N}$ are affected by changes in metallicity.
To constrain the chemical composition of stellar populations in $\omega$\,Cen, we first considered the sample of metal-poor stars identified by \citet{milone2017a} on the $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM, which comprises stars with nearly homogeneous [Fe/H] \citep{marino2019a}.
We separated 1G and 2G stars as in \citet{milone2017a} and selected four main groups of 2G$_{\rm A}$--2G$_{\rm D}$ stars that span different intervals of $\Delta_{\rm {\it C} F275W,F336W,F438W}$.
1G stars and 2G$_{\rm A}$--2G$_{\rm D}$ stars include 25.9$\pm$1.3\%, 20.7$\pm$1.2\%, 23.2$\pm$1.3\%, 6.3$\pm$0.7\% and 23.8$\pm$1.3\%, respectively, of the total number of studied metal-poor RGB stars.
As expected, the average $\Delta_{\rm F343N, F373N}$ increases from 1G- to 2G$_{\rm D}$-stars. Indeed, both $\Delta_{\rm F343N, F373N}$ and $\Delta_{\rm {\it C} F275W,F336W,F438W}$ strongly depend on the nitrogen abundance.
The fact that population 2G$_{\rm A}$ exhibits similar values of $\Delta_{\rm F275W,F280N}$ as 1G stars, shows that these populations share similar magnesium abundances. The high values of $\Delta_{\rm F275W, F280N}$ of 2G$_{\rm D}$ stars are indicative of their high depletion in [Mg/Fe], while 2G$_{\rm C}$ and and 2G$_{\rm B}$ stars have intermediate magnesium abundances.
We exploited photometry collected through 36 filters of ACS/WFC and WFC3/UVIS on board {\it HST} and compared the relative colours of the various populations with the colours derived from synthetic spectra with appropriate chemical compositions.
We thus inferred the abundances of He, C, N, O and Mg of 2G$_{\rm A--D}$ stars relative to the 1G. We find that 2G$_{\rm A}$ stars have nearly the same magnesium abundance as the 1G, while 2G$_{\rm B}$, 2G$_{\rm C}$ and 2G$_{\rm D}$ stars are depleted in magnesium by $\sim$0.15, $\sim$0.25 and $\sim$0.45 dex, respectively.
All 2G stars are more helium- and nitrogen-rich than 1G stars, with 2G$_{\rm D}$ stars having the highest enhancement of both elements ($\Delta$Y$\sim$0.08 and $\Delta$[N/Fe]$\sim$1.0). The second generations have lower content of carbon and oxygen than the 1G, with 2G$_{\rm D}$ stars having extreme contents of these elements. These results show that the magnesium content of the populations of $\omega$\,Cen correlates with the abundances of oxygen and carbon and anti-correlates with helium and nitrogen.
These correlations are interpreted as the result of proton-capture nucleosynthesis in the CNO cycle and the MgAl- chain of H-burning.
In particular, the MgAl- chain, which is responsible for magnesium variations is only active at temperatures higher than $\sim 7 \times 10^{7}$ K \citep[e.g.][]{denisenkov1990a, langer1993a, renzini2015a, prantzos2006a, prantzos2017a}.
Clearly, the material making the Mg-poor populations was exposed to higher temperatures than the Mg-rich stars, with p-captures having destroyed Mg making Al.
We find that about 70\% of the selected metal-intermediate stars of $\omega$\,Cen exhibit similar values of $\Delta_{\rm F275W,F280N}$ but are clustered around three different values $\Delta_{\rm F343N,F373N}$. We verified that these three groups of stars populate the lower, middle and upper streams defined by \citet{marino2019a} and are composed of stars with different nitrogen contents.
Accurate estimates of the magnesium abundance of metal-intermediate and metal-rich stars of $\omega$\,Cen are challenged by the internal variation of [Fe/H] in these stars. Nevertheless, the fact that a sample of metal-intermediate stars exhibit larger values of $\Delta_{\rm F275W,F280N}$ than Mg-rich metal-poor stars indicates that they are depleted in [Mg/Fe].
\citet{norris1995a} determined abundances of magnesium and of other 19 elements for 40 RGB stars of $\omega$\,Cen, based on high-resolution spectroscopy. They find that the majority of stars have nearly constant [Mg/Fe]$\sim$0.4, with the exception of five stars where magnesium is clearly underabundant ([Mg/Fe]$\sim -0.1$). Two Mg-depleted stars have [Fe/H]$\sim -1.7$ while the other three have slightly higher iron abundances of [Fe/H]$\sim -1.5$. Hence, the group of Mg-depleted stars populate the metal-poor portion of the [Fe/H] distribution of the stars analyzed by \citet{norris1995a}, which span the interval $-1.8\lesssim$[Fe/H]$\lesssim-0.8$.
\citet{norris1995a} noticed that a common property of the group is that it includes only CN-rich and Al-rich stars and that four out five stars are strongly oxygen depleted.
Although {\it HST} photometry is not available for the stars studied by \citet{norris1995a}, their findings that the metal-poor population of $\omega$\,Cen hosts stars depleted by $\sim$0.5 dex in [Mg/Fe] with respect to the bulk of metal-poor stars and that the metal-intermediate stars also host stars with low magnesium abundances are consistent with the conclusions of our paper.
The evidence that $\omega$ Cen hosts stars with depleted magnesium has been recently confirmed by \citet{meszaros2019a} based on spectra of 898 stars collected by the APOGEE survey.
These results, based on high-resolution spectroscopy, demonstrate that the $\Delta_{\rm {\it C} F275W,F336W,F438W}$ vs.\,$\Delta_{\rm F275W,F814W}$ ChM introduced in this work, is an efficient tool to disentangle stellar populations with different [Mg/Fe] and infer their magnesium abundances.
\section*{acknowledgments}
\small
We are grateful to the anonymous referee for a constructive report that has improved the quality of the manuscript.
This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research innovation programme (Grant Agreement ERC-StG 2016, No 716082 'GALFOR', PI: Milone, http://progetti.dfa.unipd.it/GALFOR), and the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie (Grant Agreement No 797100, beneficiary Marino). APM, MT and ED acknowledge support from MIUR through the FARE project R164RM93XW SEMPLICE (PI: Milone). APM and MT have been supported by MIUR under PRIN program 2017Z2HSMF (PI: Bedin).
C.\,L.\. acknowledges support from the one-hundred-talent project of Sun Yat-set University. C.\,L.\, was supported by the National Natural Science Foundation of China under grants 11803048.
This work is based on observations made with the NASA/ESA {\it Hubble Space Telescope}, obtained from data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
\section*{Data availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
|
1,314,259,992,964 | arxiv | \section{Introduction}
Nuclear rings, looking prominent features due to their intense star
formation, are found mostly in barred galaxies so they are commonly
treated as linked to inner Lindblad resonances where all radial gas
inflows are slowed down and where gas is accumulated \citep{butcr93,shlosrings}.
However there are several cases of spectacular nuclear rings in unbarred
galaxies, such as those in NGC 278, NGC 7217, NGC 7702, NGC 7742.
Most of these galaxies are seen face-on, so the conclusion about
the bar absence is quite safe in them. Suggestions about the nature
of nuclear rings in unbarred galaxies includes: resonance effects
produced by weakly triaxial potential \citep{jungpal,bu95};
resonance effects produced by a past bar which is now dissolved \citep{ath};
viscous gas accretion produced by rotation velocity shear in the global disk
and its accumulation at a stagnation point at the turnover radius of the
rotation curve \citep{wegmc2}; finally, minor merger \citep{n278}.
Among these hypotheses, the last provides more opportunities
to explain various combinations of observational facts. Indeed,
\citet{athcoll} have shown that vertical central impact
of a small satellite whose mass is about 10\%\ of the host mass
should produce a nuclear stellar ring which is morphologically
indistinguishable from a resonance ring. On the other hand, if the
merged satellite orbit was close to the main galaxy disk plane, their
gravitational interaction might produce an oval disk distortion which
could in its turn create a resonance nuclear ring. However, certain
combinations of predictions are provided by each theoretical model,
and by collecting more various observational data, both morphological
and kinematical, for every galaxy in question, we would be able at last
to restrict possible mechanisms of nuclear ring generation in any
particular case.
In this paper we will consider NGC 7742 and NGC 7217; both galaxies have
prominent nuclear star-formation rings with a radius of some $10\arcsec$.
We will attempt to find any general features which may be connected to the
nuclear ring origin. As for the latter galaxy, now we are undertaking our
third approach to its study. Earlier we have found a circumnuclear gas polar
ring and two exponential stellar disks with different scalelengths in it
\citep{we97c,we2000}. Also, NGC 7217 is known to possess two counterrotating
stellar subsystems \citep{mk94,we2000}. Recently the SAURON team has also
found a gas-stars counterrotation in the center of NGC 7742 \citep{sau2},
so this fact promises interesting speculations. Both galaxies are
moderate-luminosity unbarred spirals of Sb-type.
\section{Observations and data used}
New observational data which we intend to analyse in this work
concerns mainly NGC 7742:
panoramic spectral data for NGC 7742 have been obtained with
the scanning Fabry-Perot Interferometer (IFP) of the
6m telescope of the Special Astrophysical Observatory of the Russian Academy
of Sciences (SAO RAS) and with two integral-field spectrographs,
the fiber-lens Multi-Pupil Fiber Spectrograph (MPFS) at the 6m telescope
of the SAO RAS and the international Tigre-mode SAURON at the 4.2m William
Herschel Telescope at La Palma (see Table~1). Our 2D spectroscopic data for
NGC 7217 have been described in detail earlier: the Fabry-Perot
data -- by \citet{we97c} and the MPFS data -- by \citet{we2000}.
\subsection{2D spectroscopy with the MPFS}
The present modification of the MPFS of the 6m telescope works
at the prime focus from the summer of 1998 \citep{mpfsref}; see also
http://www.sao.ru/hq/lsfvo/devices/mpfs/).
NGC 7742 was observed with the MPFS several times during 2001--2003.
MPFS is a fiber-lens system: densely packed square microlenses placed
in the focal plane of the telescope create a set of $16\times 15$
micropupils, or $16\times 16$ in 2003,
and the fibers after them transmit the light from the square
elements of the galaxy image to the slit of the spectrograph together
with 16 (17) additional fibers that transmit the sky background light taken
at a distance of $4^{\prime}$ from the galaxy, so the sky spectra
are obtained together with those of the target. The size of one spatial
element is approximately
$1\arcsec \times 1\arcsec$; a CCD TK $1024 \times 1024$ and during the
latest run of October 2003 -- a CCD EEV 42-40 $2048 \times 2048$ were used.
The spectral resolution was about 4~\AA\ varying by about 20\%\ over
the field of view. The wavelength calibration was done with
a He-Ne-Ar lamp before and after the galaxy exposures;
the internal accuracy of linearization was typically 0.25~\AA\ in the
green and 0.1~\AA\ in the red. Also we checked the accuracy of
the wavelength calibration and the absence of a systematic velocity shift
by measuring strong emission lines of the night sky, [\ion{O}{1}]$\lambda$5577
and [\ion{O}{1}]$\lambda$6300. We obtained the MPFS data in two spectral
ranges, green, 4300--5600~\AA, and red, 5900--7200~\AA.
The green spectra were used to obtain the line-of-sight velocity field for
the stellar component and a map of the stellar velocity dispersion
by their cross-correlation with spectra of some template stars,
usually of G8III--K1III spectral type. The red spectral range which
contains strong emission lines H$\alpha$ and [\ion{N}{2}]$\lambda$6583
is appropriate to derive line-of-sight velocity fields of the ionized gas.
\subsection{2D spectroscopy with SAURON}
The other 2D spectrograph which data we use is a rather new
instrument, SAURON, installed at the 4.2m William Herschel Telescope (WHT)
on La Palma -- for its detailed description see \citet{betal01}
and for some preliminary scientific results see \citet{sau2}.
We have taken the data for both our galaxies, NGC 7217 and NGC 7742
observed in October 1999, from the open ING Archive of the UK Astronomy
Data Centre. If to give a brief description, the field of view of this
instrument is $41\arcsec \times 33\arcsec$
with a spatial element size of $0\farcs 94 \times 0\farcs 94$.
The sky background is taken less than 2 arcminutes from the center of the
galaxy and is exposed simultaneously with the target. The spectral
range is fixed as of 4800-5400~\AA, the spectral resolution is about 4~\AA,
also varying over the field of view.
The comparison spectrum is that of pure neon, and to made the
linearization we fit a polynomial
of the 2nd order with an accuracy of 0.07~\AA.
\begin{table*}
\scriptsiz
\caption[ ] {Integral-field spectroscopy of the galaxies studied}
\begin{flushleft}
\begin{tabular}{lllllcc}
\hline\noalign{\smallskip}
Date & Galaxy & Exposure & Configuration & Field
& Spectral range & Seeing \\
\hline\noalign{\smallskip}
14 Oct 99 & NGC~7217, Pos.1 & 120 min & WHT/SAURON+CCD $2k\times 4k$ &
$33\arcsec\times 41\arcsec$ & 4800-5400~\AA\ & $1\farcs 4$ \\
14 Oct 99 & NGC~7217, Pos.2 & 120 min & WHT/SAURON+CCD $2k\times 4k$ &
$33\arcsec\times 41\arcsec$ & 4800-5400~\AA\ & $1\farcs 4$ \\
13 Oct 99 & NGC~7742 & 120 min & WHT/SAURON+CCD $2k\times 4k$ &
$33\arcsec\times 41\arcsec$ & 4800-5400~\AA\ & $1\farcs 1$ \\
22 Sep 01 & NGC~7742 & 45 min & 6m/MPFS+CCD $1024 \times 1024$ &
$16\arcsec \times 15\arcsec $ & 4200-5600~\AA\ & $2\farcs 1$ \\
2 Oct 03 & NGC~7742 & 20 min & 6m/MPFS+CCD $2048 \times 2048 $ &
$16\arcsec \times 16\arcsec $ & 5800-7200~\AA\ & $2\farcs 0$ \\
\hline
\end{tabular}
\end{flushleft}
\end{table*}
\subsection{2D spectroscopy with the IFP}
In November 2003, NGC 7742 has been observed with the scanning Fabry-Perot
Interferometer (IFP) of the 6m telescope installed at the prime focus within
the focal reducer SCORPIO \citep{scorpref}; see also
http://www.sao.ru/hq/moisav/scorpio/scorpio.html.
The total number of 32 spectral channels were exposed, each during 3 minutes,
providing the spectral resolution of 2.5~\AA. The seeing was
$1\farcs 7 -2\farcs 1$; the spatial binning used was $0\farcs 7$ per pixel, and
the full field of view obtained was $6\arcmin \times 6\arcmin$. The narrow
filter centered on the spectral region around redshifted H$\alpha$ and
[\ion{N}{2}]$\lambda$6583 emission lines was used. The velocity field of
the ionized gas obtained by measuring the H$\alpha$ is more precise and
extended whereas the measurements of [\ion{N}{2}] allow to probe the very
center of the galaxy where the H$\alpha$ emission is strongly contaminated by
the absorption line.
As we discuss below, both velocity fields give consistently the orientation
parameters of the gas disk: the kinematical major axis, or the line-of-nodes,
at $PA=128\degr$ and inclination of $i=9\degr$.
\subsection{Long-slit spectroscopy of NGC 7742}
To supplement our 2D spectroscopy by additional data, we have retrieved
some long-slit data for NGC 7742 from the ING Archive: the galaxy was
observed in November 1997 with the two-armed ISIS spectrograph of the
4.2m William Herschel Telescope. However, the quality of these data
seems to be insufficient: evidently, the spectral focus was not checked
promptly, and the spectral resolution was bad. So we have only measured
baricenters of the most prominent emission lines in the long-slit
cross-section of $PA=160\degr$ to determine line-of-sight velocities
of the ionized gas near the center of the galaxy; either gas velocity
dispersion nor stellar kinematics are not probed. This direction of
the slit, $PA=160\degr$, is not very close to the kinematical major axis
of the gas. So to obtain more conclusive data, in November 2004 we have
observed NGC 7742 at the 6m telescope with the focal reducer SCORPIO in the
long-slit mode with a large spectral range of 5700-7200~\AA\ and a spectral
resolution of about 5~\AA. The seeing was about $1\farcs 5$.
The slit of $1\arcsec$ width was aligned with
the kinematical major axis at $PA=128\degr$. Here we analysed both the
emission lines and the absorption line of \ion{Na}{1}D to probe the
kinematics of the
stellar component. The K-giants HD 4744 and 20893 were observed the same
night at the same mode; their spectra were used for cross-correlation
with the spectra of the galaxy.
\subsection{Imaging data}
The same night, on 5th of November, 2004, we obtained a rather deep
V-image of NGC 7742 during 240 s with the focal reducer SCORPIO in the
imaging mode (pixel scale was $0\farcs 36$ and the seeing was $1\farcs 7$).
Besides, we have retrieved large-scale B- and I-filter images for this galaxy
from the ING Archive (the images are obtained with the one-meter Jacobus
Kapteyn Telescope, the scale is $0\farcs 33$ per pixel and the seeing was
$1\arcsec$) as well as small-scale HST/NICMOS2 images (with the scale of
$0\farcs 075$ and the spatial resolution of $0\farcs 2$) and HST/WFPC2
images (with the scale of $0\farcs 1$ and the spatial resolution of
$0\farcs 2$) from the HST Archive. To check the large-scale structure
of the galaxies in the NIR, we have used the 2MASS images taken from the
NASA/IPAC Extragalactic Database.
The data have been mostly analysed by using the software
produced by Dr. V.V. Vlasyuk of the Special Astrophysical Observatory
\citep{vlas}; only primary reduction of the data obtained with the MPFS
and SCORPIO (images and long slit) was done in IDL with various pieces
of software created by one of us (A.V.M.) and by Prof. V. L. Afanasiev.
The V-image of NGC 7742 obtained with the reducer SCORPIO has been
calibrated into the standard Johnson system by using the single
photoelectric aperture measurement by \citet{keel78}.
The data observed with the IFP were reduced with the
IDL-based software described by \citet{ifpref}. Also the ADHOC
package\footnote{ADHOC software is written by J. Boulesteix
(Observatoire de Marseille).
See \texttt{http://www-obs.cnrs-mrs.fr/ADHOC/adhoc.html}}
was involved to smooth the ``data cubes''. The monochromatic images
and velocity fields of the emission lines H$\alpha$ and
[\ion{N}{2}]$\lambda$6583
were constructed by means of fitting the IFP spectra with Gaussians.
We analysed two kinds of data: one with the original spatial resolution
($2\farcs 2$) and the other smoothed by a gaussian filter with
FWHM of $2\times 2$ elements (the spatial resolution of $2\farcs 7$).
The results are mainly the same, but the last data are better
for the low-brightness regions.
\section{Stellar and gaseous kinematics of NGC 7742}
\citet{sau2} having presented the SAURON 2D velocity fields
both for stars and for the ionized gas in NGC 7742 -- the latter obtained
from their sophisticated measurement of the weak [\ion{O}{3}]$\lambda$5007
emission line -- have claimed an appearance of strict counterrotation
of the stars versus the gas. However they have shown only the very central
parts of the velocity fields within the nuclear ring ($R\approx 10\arcsec$),
so it has remained unclear if we deal with a global counterrotation or
with a compact circumnuclear counterrotating gaseous disk.
Figure~\ref{ifp7742} presents
our large-scale Fabry-Perot observations of NGC 7742 in the H$\alpha$
and [\ion{N}{2}] emission lines. In the upper two rows we give the distributions
of the emission-line intensities (left) and the velocity fields (right),
the bottom right plot presents the results of the velocity field analysis
made by a tilted-ring method \citep{begeman}. One can see that despite
the face-on view of the galactic disk, the line-of-sight velocity field
of the ionized gas demonstrates a quite regular rotation up to the
border of the noticeable H$\alpha$ emission at $R=30\arcsec -40\arcsec$,
with the visible amplitude of line-of-sight velocity variations of
about 40 km/s.
By fixing the kinematic center position that coincides reasonably well
with the photometric center and also the systemic velocity and by assuming
the same orientation angles, line-of-nodes position angle $PA_0$ and
inclination $i$, for the whole gaseous disk, we obtain rather sure
estimates of the disk orientation parameters: $i=9\degr \pm 4\degr$ and
$PA_{0,kin,gas}=128\degr \pm 1\degr$. With these parameters of the disk
orientation, the azimuthally averaged circular rotation velocity can be
estimated as 220-230 km/s within the nuclear ring radius, $R\le 10\arcsec$;
outside it decreases smoothly to $\sim 150$ km/s at $R\approx 40\arcsec$.
At the outer edge of the nuclear disk we note a drop of the rotation velocity
by $\sim 30$ km/s; another drop, more prominent in the H$\alpha$ velocity
field than in the [\ion{N}{2}] velocity field, can be detected at the radii of
$25\arcsec -28\arcsec$. As we shall show below, the latter radius is also
distinguished photometrically. So we may conclude that the sense of the
gas rotation (which is opposite to the stars rotation in the center) remains
unchanged up to the large radii. We can even expand the spatial range, over
which the conclusion is valid, beyond the borders of H$\alpha$ emission.
\citet{knapp} measured an emission line of the neutral hydrogen,
$\lambda$21 cm, in several positions near NGC 7742. Though their spacing and
half-beam resolution was rather rough, of about $2^{\prime}$, they detected
a noticeable rotation `in an east-west direction', the eastern side of the
HI disk being receding. So we conclude that NGC 7742 possesses the large
gaseous disk which rotates regularly so that its eastern side is receding;
the line of nodes of this disk is at $PA=128\degr$, and the inclination can
be determined kinematically as $i\approx 9\degr$.
Figure~\ref{mpfs7742} presents the circumnuclear velocity field
of the ionized gas which
we have obtained with the MPFS by measuring the strongest emission line
of this region, [\ion{N}{2}]$\lambda$6583. These data compliment
the large-scale gas velocity field obtained with the IFP. Over this
velocity field also, we see a regular rotation, with the visible amplitude
of the line-of-sight velocity variations of $\pm 60$ km/s, and its kinematical
major axis at radii larger than
$2\arcsec$ can be determined quite certainly as $PA_0=128\degr$, being
completely consistent with the line of nodes of the global gaseous disk.
However, if we apply the inclination of $9\degr$ found for the global gas disk
to the circumnuclear velocity field we would obtain a formal value of the
circular rotation velocity of 400 km/s at $R\approx 3\arcsec$. This seems
improbable for the galaxy of such moderate luminosity,
inconsistent with the Tully-Fisher relation; moreover, the central stellar
velocity dispersion in NGC 7742 estimated by us both with the MPFS and the
SAURON data is less than 80 km/s, so there are no any signs of huge mass
concentration in the nucleus of this galaxy. We should rather conclude that
the inclination of $9\degr$ is not valid for the very central part of the
gaseous disk, $R<4\arcsec$, and that the disk begins to warp when approaching
the nucleus. The SAURON data
were obtained under better seeing conditions than ours, and in the recently
delivered Ph.D. Thesis of Kambiz Fathi \citep{fathithes} the gas velocity
field of NGC 7742 reveals a turn of its kinematical major axis
by $90\degr$ at $R<2\arcsec$. The comment of Dr. Fathi is that we see radial
gas motions. But if the gaseous disk remains to be nearly face-on around the
nucleus, we would not see any noticeable projection of radial velocities
confined within the disk
onto the line of sight. Indeed, if we accept the inclination of $9\degr$
obtained for the whole gaseous disk for the very center of NGC 7742,
the Fathi's results would imply an amplitude of the possible radial motions
exceeding twice the rotation velocities -- namely, of $\pm \sim 400$ km/s.
We do not see any reason to suspect such
supersonic radial gas flows in the morphologically regular galaxy
with the nuclear activity of a rather weak LINER/transition type.
More probably, we see here an inclined circumnuclear disk similar
to those found in some spiral galaxies, and in
particular in NGC 7217 \citep{we97c,we2000}.
As for the stellar rotation, it does not seem to be so fast as that of the
gas and is mostly confined to the very inner, $R<3\arcsec$, region of the
galaxy. Since the seeing conditions during our MPFS observations of NGC 7742
were not good enough to resolve this small region properly, the measured
amplitude of the stellar line-of-sight velocity variations is dropped due to
spatial smoothing, and the orientation of the kinematical major axis cannot
be determined properly from our data. In Fig.~\ref{sau7742} we show our analysis
of the SAURON velocity field for the stars in the center of NGC 7742. We have
obtained $PA_{0,kin,*}=335\degr$ for the stellar component within $R=6\arcsec$,
in some disagreement with $PA_{0,*}=320\degr$ found by Fathi; however,
the possible error may be as large as $10\degr$. If again we formally fix
the inclination of the rotation plane of the stars at the value obtained
for the outer gaseous disk, $i=9\degr$, the peak rotation velocity achieved
at $R=1\arcsec$ would be $v_{rot}\approx 250$ km/s. Farther from the nucleus
it drops to zero at the radius of the ring, and beyond the ring it rises
marginally. Due to low signal-to-noise ratio we are not sure with our results
at $R>10\arcsec$, and to check if the stars continue to counterrotate the
gas outside the ring radius we appeal to the long-slit data.
Figure~\ref{ls7742} presents long-slit velocity profiles for the stars and
ionized gas:
SCORPIO data along the kinematical major axis $PA=128\degr$ and WHT/ISIS
data at $PA=160\degr$. Because of the low surface brightness of NGC 7742
at $R>10\arcsec$, the measurements of the stellar velocities
(Fig.~\ref{ls7742}{\it a}) are
not very extended and are not very precise; however some qualitative
conclusions can be made. The sense of rotation of the stellar component
observed in the center persists up to $R\approx 25\arcsec$ at least; but at
$R\approx 10\arcsec$ -- at the radius of the ring -- we see strong stellar
velocity variations, such that the line-of-sight velocities of the stars at
this radius coincide exactly with those of the ionized gas. We may suggest
that violent star formation in the ring has already produced a substantial
stellar population, including stars of F-G-K type, so that their rotation
coupled with their parent gas contributes significantly into the integrated
LOSVD of the stars at this radius.
The long-slit gas velocity profiles (Figs.~\ref{ls7742}{\it b} and {\it c})
demonstrate different character with respect to the rather smooth rotation
curve obtained by azimuthal averaging of the 2D IFP gas velocity field
(Fig.~\ref{ifp7742}). They
'oscillate' by 70-80 km/s with a characteristics radial period of
$\sim 10\arcsec$, and the locations of the velocity maxima and minima differ
at $PA=128\degr$ and at $PA=160\degr$. We think that these velocity variations
do not relate to regular rotation; they resemble vertical small-scale
oscillations of a tidally perturbed gaseous disk.
Another peculiarity of the long-slit gas velocity profiles which we have
however expected basing on our 2D MPFS data is the very steep central
velocity gradient and the decoupled fast gas rotation within $R=3\arcsec$;
it is confirmed both by the SCORPIO and ISIS data. Moreover, the [\ion{N}{2}]
emission line measurements at the approaching branch of the velocity profile
give even underestimated values of the rotation velocity -- they deviate
toward the systemic velocity not only with respect to the H$\alpha$
measurements which may be affected by underlying
absorption lines, but also with respect to the [\ion{S}{2}] emission line
measurements. Similar differences between
velocity estimates made with different emission lines had been detected more
than once in the centers of other spiral galaxies\citep{mrk744,ourls2} and
might be explained if the H$\alpha$ and
[\ion{S}{2}] emission lines relate to regularly rotating gas ionized
by OB stars and the [\ion{N}{2}] emission lines are formed mostly in shock
wave sites where the ionized gas decelerates. Enormous visible rotation of
the ionized gas within $R=3\arcsec$ revealed by the long-slit data gives
strong evidence for the highly-inclined orientation of the gas
rotation plane in the very center of NGC 7742 ($i_{gas} >35\degr$),
as opposite to the nearly face-on orientation of the global gaseous disk.
\section{Global structure of NGC 7742}
The morphological type of NGC 7742 is
SA(r)b, and taking into account the face-on orientation of the global disk,
the galaxy looks indeed quite round and axisymmetric, except the very
central part (Fig.~\ref{ifp7742}). However, the rather early morphological
type, Sb, deduced perhaps from the appearance of tightly wound, faint spiral
arms, is not supported by the very low stellar velocity dispersion in the
center, $\le 80$ km/s, implying the absence of a large bulge that is obliged
to be a dynamically hot stellar subsystem by definition.
The V-band image obtained with the SCORPIO appears to be very deep: our
surface brightness measurements reaches the radius almost twice that of 25th
B-magnitude. Figure~\ref{iso7742} presents the results of isophotal analysis
of this image, together with the measurements of the I-band image taken from
the ING Archive which is almost similarly deep. The ellipticity behavior reveals
central rise and a peak near the position of the nuclear ring; the
isophotes in the radius range of $12\arcsec - 50\arcsec$ (please note
that $R_{25}=52\arcsec$) are indeed round. However the most interesting
things are seen at $R>R_{25}$: the ellipticity rises to the mean value of
0.15, and the major axis position angle can be measured quite certainly
at $<PA_0>=112\degr$. We cannot be sure that we see a round outer stellar
disk inclined by $\sim 30\degr$ to the line of sight because the kinematical
parameters of the orientation of the more inner {\it gaseous} disk,
$PA_0=128\degr$ and $i=9\degr$, does not coincide with the photometric
parameters found for the outermost part of the broad-band image;
the hypothesis which may be more plausible is that the outer disk is
intrinsically oval. Unfortunately, we have no detailed kinematical
measurements at such large radii.
We have tried to decompose the whole V-image into separate photometric
subcomponents, such as an outer exponential disk and some more inner
components. Two methods were applied: the software GIDRA \citep{mrk315}
which uses 2D surface brightness modelling under constant orientation
parameters over all the image
and iterative 1D brightness profile fitting starting from the outermost
component with subsequent subtraction of the 2D model components from
the original image; the latter method allows to vary orientation parameters
from one component to another according to isophote analysis results.
The GIDRA analysis of the V-image, under the fixed kinematical parameters
of the orientation, $PA(line-of-nodes)=128\degr$ and $i=9\degr$, with an
approximation of the seeing FWHM by $1\farcs 7$, has given TWO exponential
disks superposed, with the scalelengths of $17\farcs 6$ and $7\farcs 2$
and the central surface brightnesses, $\mu _{0,V}$, of 20.6 and 18.2
mag per square arcsecond. The third photometric component seen only in
the very center may be a de Vaucouleurs' bulge with $r_e=4\farcs 2$.
One-dimensional brightness profile fitting made with $PA_0=112\degr$
and with isophote axis ratio of $b/a=0.85$ for the outer component and
with $PA_0=13\degr$ and $b/a=0.93$ for the inner components has also given
two exponential disks: the outermost one being approximated in the radius
range of $50\arcsec - 93\arcsec$ has $\mu _{0,V}=21.04$ and $r_0=20\arcsec$
and the inner one, seen in the radius range of $15\arcsec - 42\arcsec$
after subtracting the outer disk, has $\mu _{0,V}=18.45$ and $r_0=7\farcs 2$
-- see the Fig.~\ref{disk7742}. The most central component which is left after
subtraction of two exponential disks gives a noticeable contribution only
inside $R\approx 5\arcsec$, and since it is affected by spatial resolution
effects we cannot surely determine shape of its profile: it may be exponential
as well as something else. However both our fitting methods indicate
certainly the presence of two exponential disks with different scalelengths.
We would like to stress that it is the outer disk which is `normal':
its $\mu _{0,V}$, 21 V-mag per square arcsec, is very close to the canonical
Freeman's value \citep{fr70}, and the relation between its central
surface brightness and its scalelength in kpc is typical for Sb-galaxy
\citep{dejong3}. The inner disk is more compact and high-surface-brightness
one than spiral galaxies have usually, though not so compact and
bright as circumnuclear disks of early-type galaxies; on the diagram
`$\mu _0$ vs h' collected by \citet{resh} it settles among the large disks
of lenticular galaxies. However the spiral arms and noticeable star formation
(H$\alpha$ emission) in NGC 7742 are confined just to this inner disk.
Interestingly, NGC 7217 -- another galaxy with rings and without a bar --
has very similar structure. We have decomposed its brightness profile
in our work \citep{we2000} and have found two exponential
disks, the outermost disk being the `normal' one, together with the
compact exponential bulge. We compare the structural characteristics
of the components for both galaxies in Table 2. The scalelengths are
very close: 2--3 kpc for the outer disks and $\sim 1$ kpc for the inner
disks. The visible shapes of the outer and inner disks are different
in both galaxies. However, if to compare the derived photometric
characteristics of the disks with the orientation parameters estimated
from the global gas kinematics, we would conclude that in NGC 7217 the outer
disk is round and the inner disk is inclined or is oval (a destroyed bar?),
whereas in NGC 7742 the configuration is opposite: the inner disk is
round and the outer one is oval that may be due perhaps
to an external tidal perturbation.
Recently \citet{n278} have studied another
unbarred ringed galaxy, NGC 278, having the morphological type close to
that of NGC 7217 and NGC 7742, SAB(rs)b. Their graphic presentation of the
surface brightness profile of NGC 278 allows to suggest the same multi-tiers
structure of the global stellar disk as we have found in NGC 7217 and
NGC 7742. All the present star formation
in NGC 278 is confined to the inner disk, within the radius of 1.1 kpc,
as well as in NGC 7742.
\begin{table*}
\scriptsiz
\caption[ ] {Exponential parameters of the brightness profiles fitting}
\begin{flushleft}
\begin{tabular}{lcccccc}
\hline
Disk & Radius range of fitting, arcsec & Radius range of fitting, kpc
& $PA_0$ & $b/a$ &
$r_0$, arcsec & $r_0$, kpc \\
\hline
\multicolumn{7}{l}{NGC 7217}\\
Outer & 60--110 & 5--9 & $90\degr$ & 0.82 & 35.8
& 2.9 \\
Inner & 20--50 & 1.6--4 & -- & 0.92 &
12.5 & 1.0 \\
Central bulge & 5--20 & 0.4--1.6 & $82\degr$ & 0.88 &
3.9 & 0.3 \\
\hline
\multicolumn{7}{l}{NGC 7742}\\
Outer & 50--93 & 6--11 & $112\degr$ & 0.85 & 20
& 2.34 \\
Inner & 15--42 & 1.8--4.9 & -- & 0.93 & 7.2
& 0.8 \\
Central (bulge?) & 1--5 & 0.1--0.6 & -- & 0.93 & 1.3 & 0.15\\
\hline
\end{tabular}
\end{flushleft}
\end{table*}
Some words about the central component of NGC 7742. Its rather high visible
ellipticity was noted earlier, e.g. by \citet{n7742rep} and by
\citet{3bphot}; the former authors mentioned the turn of the isophote major
axis from $110^{\circ}\pm 10^{\circ}$ to $10^{\circ}\pm 10^{\circ}$ between
$r=1\farcs 5$ and $r=5\farcs 1$. We confirm this result and point out
that the rather high ellipticity of the isophotes at some distinct radii,
namely, at $r\approx 1\arcsec$ and at $r\approx 7\arcsec$ (Fig.~\ref{iso7742}),
makes the estimate of the major axis turn quite sure. May be anyone of these
elongated structures a bar or a compact triaxial bulge? If such triaxiality
exists in the center of NGC 7742, it would cause a Z-shaped disturbance of the
gas velocity field; and as we have seen in the previous Section~3,
the orientation of the kinematical major axis of the gas rotation,
$PA_{kin,gas}=128^{\circ}$, stays firmly between $r=2\arcsec$ and
$r\approx 40\arcsec$. So we don't see any signatures of the triaxial potential
in the center of NGC 7742. Instead we may suggest a strong warp of the
rotation and symmetry planes in the center of the galaxy: immediately inside
$R\approx 5\arcsec$ the gas rotation plane conserves the line of nodes
of the outer gaseous disk but probably increases its inclination
that may be deduced from the visible fast rotation, and closer to the center,
at $R< 1\farcs 5$, the kinematical major axis of the gas `switches'
to the `orthogonal' orientation \citep{fathithes}.
\section{The stellar kinematics and structure of NGC 7217}
As we have noted above, two galaxies with rings and without bars --
NGC 7217 and NGC 7742 -- have very similar global structures. As for
the fine features in their centers, in this work we suggest a strong warp
of the rotation plane both for the stars and for the ionized gas in the
center of NGC 7742. In NGC 7217 which is slightly less face-on we have
found a circumnuclear polar gaseous ring with the radius of
$3\arcsec -4\arcsec$ \citep{we2000}. As for the stellar
kinematics in the center of NGC 7217, from our MPFS observations, with our
spatial resolution $\sim 2\arcsec$ we have not
found any deviations from an axisymmetric rotation around the main
symmetry axis of the galaxy at $R\ge 2\arcsec$.
However, the surface distribution of the stellar velocity dispersion in the
center of NGC 7217 looked very strange, with the off-centered minimum,
and we \citep{we2000} were not able to give a reasonable explanation of it.
Now we have in hand the 2D spectral data for NGC 7217 obtained with the
SAURON; these data provide a larger field of view and a slightly better
spatial resolution than the MPFS ones so now we can expand our previous
analysis of the kinematics of the central part in this galaxy.
By applying a tilted-ring method to the whole stellar velocity field
(Fig.~\ref{sau7217}{\it right}) representing a combination of two different
pointings of the telescope, outside $R=3\arcsec$ we obtain the mean parameters
of the rotation plane orientation,
$PA_{0,kin,*}=268\degr \pm 2\degr$ and $i=30\degr \pm 4\degr$,
very stable along the radius, consistent with the axisymmetric rotation
in the main galactic plane. The velocity field of the ionized gas
(Fig.~\ref{sau7217}{\it left})
obtained by measuring the emission line [\ion{O}{3}]$\lambda$5007, strong
in the center of NGC 7217, confirms the orientation of the kinematical major
axis for the ionized gas found by us earlier \citep{we2000}:
$PA_{0,kin,gas}=329\degr \pm 4\degr$ at $R=1\arcsec - 4\arcsec$.
The angular rotation velocity is rather high,
$\omega \sin i_{gas} \approx 29$ km/s/arcsec; being compared to the
stellar rotation velocity at the same radius,
$\omega \sin i_* \approx 14.5$ km/s/arcsec, and taking into account that
$\sin i_*=0.5$, it implies the presence of the circumnuclear edge-on gaseous
disk. The higher spatial resolution of the SAURON data with respect to
the previous MPFS ones allows to notice a turn of the stellar kinematical
major axis inside $R\le 2\arcsec$, and it is a quite new finding.
Inside this radius the stellar kinematical major axis turns
and reaches $PA_{0,kin,*}=309\degr \pm 12\degr$ at $R=1\arcsec$ --
compare to $PA_{0,kin,gas}=329\degr \pm 4\degr$
for the ionized gas inside $R=3\arcsec$. There is a
clear impression that a {\it stellar} inclined disk exists too but it is
much more compact than the gaseous one. We have collected all the available
velocity fields for the central part of NGC 7217 by adding to the data
analysed in this work the Fabry-Perot ionized-gas velocity field presented
by us earlier \citep{we97c} and the CO velocity field presented
by \citet{nuga7217} recently in the frame of the NUGA project.
We have applied the tilted-ring analysis to all of them and have traced
the kinematical major axis orientation from the very center to
$R\approx 30\arcsec$. In Fig.~\ref{majax7217} we compare these
results to the photometric major axis orientation. Outside the nuclear
ring, at $R>10\arcsec$, both the gas -- warm and cold -- and stars
rotate quite axisymmetrically, with their kinematical major axes
agreeing perfectly with the photometric major axis. It is somewhat
strange because earlier \citep{we2000} we supposed the inner disk seen
at $R>20\arcsec$ to be oval because its photometric major axis, after
subtracting the other structural components, deviated by some $30\degr$
from the line of nodes of the outer disk. The only
tentative signature of possible non-circular motions
of the gas within the inner disk of NGC 7217 may be small radial
velocities, of 5--7 km/s, detected by us in the CO velocity field;
but this presence of radial gas motions is not confirmed by the results
of our analysis of the IFP ionized-gas velocity field. We must note here
that the IFP velocity measurements for the ionized gas of NGC 7217 were
made with the weak emission line [\ion{N}{2}]$\lambda$6583 and are rather
noisy and patchy so perhaps we were not able to detect radial motions
of less than 10 km/s. As for the stars, the inner stellar disk of
NGC 7217 is not very cold \citep{we2000}, so the stars may be more stable
against the weak triaxial perturbation. Inside the inner ring,
at $R\le 8\arcsec$, the kinematical major axis of the ionized gas
starts to turn implying an appearance of the inclined disk. At $R=4\arcsec$
the $PA_{0,kin,gas}\approx 320\degr$ that differs by $80\degr$ from the
direction of the photometric major axis which is here at
$PA_{phot}\approx 240\degr$. The kinematical major axis of stars starts
to turn much closer to the center, at $R<3\arcsec$, and at $R=1\arcsec$
it reaches almost the same orientation as that of the ionized gas. So
in NGC 7217 we have two circumnuclear inclined disks, gaseous and stellar,
fairly coplanar to each other, but the latter is much more compact than
the former.
Figure~\ref{sigma7217}{\it left} presents the map of the stellar velocity
dispersion in the center of NGC 7217 obtained from the SAURON data. Now, with
the larger field of view, we can unambiguously recognize the central structure
seen in this map: though the whole distribution is slightly asymmetric,
in the very center there is a certain $\sigma _*$ minimum, perhaps,
shifted by $1\arcsec -2\arcsec$ to the north. Since the color distribution
derived by us from the HST/WFPC2 data is also asymmetric, the color
maximum being shifted in the opposite direction with respect to the
stellar velocity dispersion minimum, we suggest that this asymmetry
may be caused by the dust in the inclined circumnuclear disk. We conclude
that the stellar velocity dispersion distribution is another signature
of the compact circumnuclear stellar disk in this galaxy which must be a
relatively `cold' dynamical component.
\section{Conclusions and Discussion}
By using a variety of 2D kinematical data as well as deep images of
NGC 7742, we analyse stellar and gaseous kinematics in this unbarred
Sb galaxy possessing the nuclear star-forming ring; we compare it to
NGC 7217, another unbarred spiral galaxy with rings which has been
studied by us earlier. We have found some common features in NGC 7217
and NGC 7742.
\begin{enumerate}
\item{
Both galaxies demonstrate global structure consisting of
two exponential stellar disks with different scalelengths; the outer
disks look quite normal whereas the inner disks are compact, with
$r_0\approx 1$ kpc, and have unusual high surface brightness.
We would like to propose the following qualitative scenario to form such a
`multi-tiers' stellar disk. A few Gyrs ago there may be a sudden global gas
redistribution in the disk, due perhaps to external tidal perturbation or
minor merger. Before that event stars should form in the disk with a large,
normal scalelength, and after that when all the gas had been dropped closer
to the center the star formation should continue in the disk with a smaller
scalelength and higher surface density.}
\item{
Both galaxies, NGC 7742 and NGC 7217, have circumnuclear gaseous
disks with the radius of some 300 pc, highly inclined to the global disk
planes; the outer gas disks are, on the contrary, close to the main galactic
symmetry planes. Both galaxies possess also some counterrotationg subsystems.
NGC 7742 has all its gas in counterrotation with respect to all its stars,
with exception of some newly born stellar population in the ring, while
in NGC 7217 the gas outside $R=300$ pc corotates
the bulk of stars, but there are some 30\%\ of all stars in the
inner disk that counterrotates \citep{mk94}.
Three-dimensional dynamical simulations of the self-consistent evolution
of a stellar-gaseous galactic disk unstable with respect to bar-like
perturbations presented by \citet{sec1} proposed a scenario for
the origin of circumnuclear inclined gas rings. If initially the gas
of the global disk counterrotates the stars, then drifting to the center in a
triaxial potential of a transient bar, this gas must leave the disk plane
and accumulate on orbits strongly inclined to this plane -- only these
inclined orbits remain stable for the initially counterrotating gas near the
inner Lindblad resonances. We may suggest that the gas which is now observed
as the circumnuclear strongly inclined disks in NGC 7217 and NGC 7742 has come
from the outer parts of the galaxies, and when it was there, it counterrotated
the stars.}
\end{enumerate}
As for the problem of the origin of initially counterrotating gas, it may
be solved together with the problem of the nuclear star-forming rings origin.
If we suggest past minor merger
of a dwarf gas-rich galaxy from a retrograde orbit, this event had to
supply some amount of counterrotating gas and at the same time it might
cause an oval distortion of the stellar disk of the host galaxy that in
its turn had to produce rapid radial gas re-distribution and the nuclear
star-forming ring appearance -- all the peculiar features observed in
NGC 7217 and NGC 7742.
NGC 7742 demonstrates strong vertical gas oscillations in its
counterrotating gaseous disk implying rather recent gas accretion, NGC 7217
might possess the counterrotating gas in the past, but now it is fully
reprocessed into counterrotating stars.
\citet{n278} have detected strongly peculiar
kinematics of the neutral and ionized hydrogen beyond the optical
stellar disk in the unbarred galaxy with the rings, NGC 278, though
the galaxy is morphologically regular and quite isolated; they conclude
that the galaxy has recently experienced a minor merger.
In absence of the detailed neutral-hydrogen observations well outside the
optical borders of the galaxy, one would treat NGC 278 as a twin for NGC 7217
and NGC 7742. To our opinion, NGC 278 may represent an early stage of the
evolution having followed a minor merger, with respect to two galaxies
considered in our work, and its nearest future is perhaps NGC 7742.
{\it The presence of numerous minor merger signatures in the three unbarred
galaxies with nuclear star-forming rings makes the hypothesis of tidally
induced oval distortion of the global stellar disks the most attactive
scenario for the ring origin in unbarred galaxies.}
\acknowledgements
We thank Prof. V. L. Afanasiev for supporting the Multi-Pupil Fiber
Spectrograph of the 6m telescope and for taking part in some of the
observations which data are used in this work. We are indebted to Dr.
S. Garcia-Burillo who has provided the CO velocity field of NGC 7217
in digital form. The 6m telescope is
operated under the financial support of Science Ministry of Russia
(registration number 01-43). During our data analysis
we used the Lyon-Meudon Extragalactic Database (LEDA) supplied by the
LEDA team at the CRAL-Observatoire de Lyon (France) and the NASA/IPAC
Extragalactic Database (NED) operated by the Jet Propulsion
Laboratory, California Institute of Technology under contract with
the National Aeronautics and Space Administration. The work
is partially based on the data taken from the ING Archive of the
UK Astronomy Data Centre and on observations made with the NASA/ESA
Hubble Space Telescope, obtained from the data archive at the Space
Telescope Science Institute. STScI is operated by the Association of
Universities for Research in Astronomy, Inc. under the NASA contract
NAS 5-26555. The work on the study of global structure of disk galaxies
is partly supported by the grant of the Russian Foundation for Basic
Researches number 05-02-16454.
A.V. Moiseev thanks also the Russian Science Support Foundation.
|
1,314,259,992,965 | arxiv | \section{Introduction}
In recent years, several types of superconducting qubits have been experimentally
proposed.\cite{nakamura,qbit_mooij,vion,martinis}
These systems consist on mesoscopic Josephson devices and they are promising
candidates to be used for the design of qubits for quantum
computation.\cite{nakamura,qbit_mooij,vion,martinis,revqubits,chiorescu,fqubit_recent}
Indeed, a large effort is devoted
to succeed in the coherent manipulation of their quantum states in a controlable way.
The progress made in this case allows to have nowadays Josephson
circuits with small dissipation and large decoherence
times.\cite{vion,martinis,chiorescu,fqubit_recent}
Very recently, it has been proposed that,
due to these developments,
it could also be possible to use mesoscopic Josephson devices for the
study of the quantum signatures of classically chaotic systems.\cite{montangero05,mingo}
In Ref.~\onlinecite{mingo} the quantum dynamics of the Device for the Josephson Flux Qubit (DJFQ)
has been studied. In particular, it has been discussed
how the fidelity (or Loschmidt echo)\cite{jp} of the DJFQ
could be studied experimentally for energies corresponding to the hard chaos regime
in the classical limit.
Here, we extend the work of Ref.~\onlinecite{mingo} by analyzing
the possibility of studying, in the DJFQ, the mixed chaos regime (i.e.,
the energy range where there
is a coexistence of chaotic and regular orbits in the classical limit).
To this end, standard tools of analysis of ``quantum chaos'', like spectral
statistics \cite{metha,bohigas,berryro,seligman,cederbaum84,cederbaum86,prosen,makino01,robnik05}
and phase space distributions,\cite{wigner,husimi,hus-rev,groh} will be used.
It is by now well established that from the analysis of the spectral properties of quantum systems
in the semiclassical regime it is possible to obtain information about
the underlying dynamics of the classical counterpart. \cite{metha,bohigas,berryro,seligman,cederbaum84,cederbaum86,prosen,makino01,robnik05}
The probability distribution $P(s)$ of the spacings $s$ between successive
energy levels - the nearest neighbor spacing distribution
$P(s)$- unveils information on the associated classical dynamics. For integrable systems the levels are uncorrelated, and $P(s)$ obeys a Poisson distribution.
For completely chaotic classical motion, $P(s)$ follows
the prediction of the Random Matrix Theory (RMT) \cite{metha}
and when time reversal symmetry is preserved
$P(s)$ is closely approximated by the Wigner distribution for the Gaussian Orthogonal Ensemble (GOE),
$P(s) \sim s \exp{(-s^{2})}$.\cite{bohigas}
Generic quantum systems do not conform to the
above special cases, the classical phase space typically presents mixed dynamics,
with coexistence of regular orbits and chaotic motion.\cite{berryro,seligman,cederbaum84,cederbaum86,prosen,makino01,robnik05}
In this generic case Berry and Robnik \cite{berryro} proposed an analytical expression for the corresponding $P(s)$, based on the knowledge of pure classical quantities related to the Liouville measure of the chaotic and regular classical regions. The idea behind their calculations is
that each regular or irregular phase space region gives rise to its own sequence of energy levels. For each region the level density results proportional to the Liouville measure of the classical region and the associated level spacing distribution follows the Poisson or the Wigner form for regular and chaotic regions respectively.
In the semiclassical limit these sequences of energy levels can be supposed independent and the complete distribution $P(s)$ is obtained by their random superposition.
Several works have studied numerically the level statistics in systems with mixed dynamics. \cite{seligman,cederbaum84,cederbaum86,prosen,makino01,robnik05}
Systems with two degrees of freedom
have been analyzed by several groups, mostly quartic oscillators\cite{seligman,cederbaum84,cederbaum86}
and billiards,\cite{prosen,makino01,robnik05}
and in some works the Berry-Robnik proposal has
been tested in detail.\cite{cederbaum86,prosen,makino01}
In contrast to the level statistics, the wave functions of quantum chaotic systems have remained relatively less explored. In particular the analysis of wave functions in phase space representations, such as the Wigner function \cite{wigner} or the Husimi distribution, \cite{husimi,hus-rev,groh} allows a direct comparison between the classical and the quantum dynamics. Of particular interest are the zeros of the Husimi distribution
which seem to be organized along regular lines or fill space regions for regular or chaotic classical dynamics
respectively. \cite{leboeuf}
Besides the importance of visualizing the dynamical properties of quantum systems in phase space,
techniques for
measuring these functions, referred as ``quantum tomography " \cite{nielsen,miquel} are subjects of active research in many experimental systems,
like ion traps, optical lattices, entangled photons,\cite{mitchell,kanem}
and also superconducting qubits.\cite{tomo_super}
Josephson junctions have been used for the study of classical chaos since
the early 1980s.\cite{jchaos,jchaos_exp} A single underdamped junction with
a periodic current drive can become chaotic in a wide range of parameter
values.\cite{jchaos} Several
experiments have indeed studied this problem and measured chaotic properties
in current-voltage curves and in voltage noise in
Josephson junctions.\cite{jchaos_exp}
Moreover, networks with several
junctions have been proposed for the study of spatio-temporal chaos.\cite{jchaos_arrays}
All this cases correspond to classical chaos in dissipative systems with
a time-periodic drive.
Much less studied has been the case of classical hamiltonian chaos in Josephson junctions,\cite{parmenter}
mainly due to the fact that dissipation through a shunt resistance and/or coupling
to the external measuring circuitry is typically important.
For the same reason, {\it i.e.}, the difficulty in fabricating Josephson circuits
with negligible coupling to the environment,
few examples of quantum chaos in Josephson systems are found in the literature.
One of them is the work of Graham {\it et al.},\cite{graham}
who considered dynamical localization and level repulsion in a single Josephson junction
with a time periodic drive.
More recently T. D. Clark, M. J.
Everitt and coworkers\cite{everitt} explored chaos and the quantum behaviour
of SQUID rings coupled to electromagnetic field modes.
The recent development of Josephson devices for quantum computation, which need
large coherence times, lead to significant advances in the fabrication of circuits
with small coupling to the external circuit and negligible dissipation.
This opened the possibility of using this type of mesoscopic devices for the
study of quantum chaos.
For example, Montangero {\it et al.} \cite{montangero05} have proposed recently
a Josephson nanocircuit as a realization of the quantum kicked rotator. The
difficulty in realizing experimentally their system resides in that it
needs to move mechanically one superconducting node in a high-frequency
periodic motion.
A different proposal has been put forward in Ref.\onlinecite{mingo},
where it has been shown that
the Device for the Josephson Flux Qubit (DJFQ),\cite{qbit_mooij,chiorescu,fqubit_recent}
which consists on a three-junction SQUID,
is classically chaotic at high energies.
It could be therefore possible to use this system for the experimental study of quantum signatures
of classical chaos.
One possibility
is the analysis of the fidelity or Loschmidt echo\cite{jp} in the quantum dynamics.\cite{mingo}
An experimental setup for the measurement of the Loschmidt echo in the DJFQ has been proposed
in Ref.\onlinecite{mingo}. In the above mentioned work,
the system is prepared initially with a wave packet\cite{nota2}
localized in coordinate (phase) and momentum (charge) with an energy corresponding to
the regime of hard chaos in the classical limit. The quantum evolution of the wave packet
is evaluated in the unperturbed and the perturbed hamiltonians, and the overlap of the
two evolved wave functions defines the Loschmidt echo or fidelity\cite{jp}, which
can be measured experimentally.\cite{mingo}
Different behavior could be observed if the wave packet is initially localized in
a chaotic or in a regular region of the phase space.
Therefore, an interesting case to analyze is
when the wave packet is prepared initially with an energy within the regime where
there is a mixed phase space in the classical limit. In this case, one would expect
that the behavior of the Loschmidt echo could depend
on the location of the average coordinate and momentum of the initial wave packet.
For example, in
Ref.\onlinecite{liu} it has been found a strong dependence of the fidelity
with the initial state for mixed dynamics in the phase space in the case
of Bose-Einstein condensates.
However, in order to be sensitive to the structure of phase space in the
case of mixed dynamics, it is necessary to have a small effective $\hbar$.
The aim of the present work is to analyze the quantum spectra and wave
functions of the DJFQ in order to obtain for which values of
the effective $\hbar$ the quantum physics of this system can show signatures
of the structure of the phase space in the case of mixed dynamics.
To this end, we will use standard tools of quantum chaos theory
by calculating numerically the level statistics of
the DJFQ for different effective $\hbar$
and the Husimi distribution.
Concerning the spectral analysis,
the quantum signatures of chaos have been discussed through the $P(s)$ distribution
in Ref. \onlinecite{kato} for a SQUID with three junctions in the hard chaos regime.
However, the case with only on-site capacitances was considered there
(the capacitance of the junctions was neglected).
Nevertheless, the device for the Josephson flux qubit fabricated by
the Delft goup\cite{chiorescu}
has small on-site capacitances, about two orders of magnitude
smaller than the intrinsic capacitances of the junctions.\cite{nota3}
This fact turns the model hamiltonian
for the DJFQ to be different from the one studied in Refs.\onlinecite{parmenter, kato}.
One of the goals of this paper is to analyze the spectral properties of the DJFQ considering
realistic values of the different capacitances to analyze
the device for the Josephson flux qubit (DJFQ) in the case of mixed dynamics.
In addition we analyze the structure of the Husimi functions for the DJFQ,
an issue that has been so far unexplored.
The paper is organized as follows.
In Sec.\ref{model} we introduce the quantum model for the device for the Josephson flux qubit.
Before presenting the quantum spectral analysis, we will study in Sec. \ref{qchaos} the dynamics of its classical analog. The presence of chaos will be characterized through the analysis of
a measure of the chaotic volume, that will be defined and obtained as a function of the energy.
We devote the rest of Sec. \ref{qchaos} to the analysis of the spectral properties.
The NNS distribution will be obtained for different energies corresponding
to different classical energy regions and dynamics and
for different values of the effective $\hbar$.
In Sec. \ref{distri} we compute the Husimi distribution for the DJFQ in order
to characterize the localization of the quantum states on typical phase space structures
related to the different classical regimes.
Finally in Sec. \ref{conclu} we summarize our results
and discuss possible experimental characterizations of
the quantum manifestations of chaos in this system.
\section{Model for the Device for the Josephson Flux Qubit}
\label{model}
The DJFQ consists of three
Josephson junctions in a superconducting ring \cite{qbit_mooij}
that encloses a magnetic flux $\Phi= f\Phi_0$,
with $\Phi_0=h/2e$, see Fig.\ref{djfq_fig}.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig1.eps}
\caption{Circuit for the Device for the Josepshon Flux Qubit as described in the text.
Josepshon junctions $1$ and $2$ have Josepshon energy $E_J$ and capacitance $C$, and junction $3$
has Josepshon energy and capacitance $\alpha$ times smaller.
The arrows indicate the sign
convention for defining the gauge invariant phase differences. The circuit encloses
a magnetic flux $\Phi = f \Phi_0$.} \label{djfq_fig}
\end{center}\end{figure}
The junctions have gauge invariant phase differences defined
as $\varphi_1$, $\varphi_2$ and $\varphi_3$, respectivily, with
the sign convention corresponding to the directions
indicated by the arrows in Fig.\ref{djfq_fig}.
Typically the circuit inductance
can be neglected and the phase difference of the
third junction is:
$\varphi_3=-\varphi _1 +\varphi _2-2\pi f$.
Therefore the system can be described
with two dynamical variables: $\varphi_1,\varphi_2$.
The circuits that are used for the Josephson flux
qubit have
two of the junctions with the same coupling
energy, $E_{J,1}=E_{J,2}=E_J$, and capacitance, $C_1=C_2=C$,
while the third junction has smaller
coupling $E_{J,3}=\alpha E_J$ and capacitance $C_3=\alpha C$,
with $0.5<\alpha<1$.
The above considerations lead to the Hamiltonian \cite{qbit_mooij,nota}
\begin{equation}
\label{hamil}
{\cal H}=\frac{1}{2}{\vec {P}}^T
{\rm {\bf M}}^{-1}{\vec {P}}
+E_J V(\vec {\bf \varphi})\label{ham_clas}
\end{equation}
where the two-dimensional coordinate is $\vec{\varphi}=(\varphi_1,\varphi_2)$.
The potential energy is given by the Josephson energy of the circuit and,
in units of $E_J$, is:
\begin{equation}
\label{eq:pot}
V(\vec {\bf \varphi})=
2+\alpha -\cos \varphi_1-\cos \varphi_2
- \alpha \cos (2\pi f+\varphi _1 -\varphi _2 ) \; .
\end{equation}
The kinetic energy term is given by the electrostatic energy of the circuit, where
the two-dimensional momentum is
$$\vec{P} = (P_1,P_2)={\rm{\bf M}}\cdot \frac{d{\vec{\varphi}}}{dt},$$
and
$\bf M$ is an effective mass tensor determined by the capacitances of the circuit,
$$
{\rm {\bf M}}= C {\left(\frac{\Phi_0}{2\pi}\right)^2} {\rm {\bf m}}
$$
with
$$
{\rm {\bf m}}=\left(
{{\begin{array}{cc}
{1+\alpha +\gamma } & {-\alpha } \\
{-\alpha } & {1+\alpha +\gamma } \\
\end{array} }} \right).$$
We included in $\bf M$ the on-site capacitance $C_g=\gamma C$.
(Typically $\gamma \sim 10^{-2}-10^{-3}\ll 1$).
In the presence of gate charges $Q_{g,i}$ induced in the islands, the momentum
is
${\vec {P}} \rightarrow {\vec {P}} + \frac{\Phi_0}{2\pi}\vec{Q_g}$.\cite{qbit_mooij}
The system modelled with Eqs.~(\ref{hamil})-(\ref{eq:pot}) is
analogous to a particle with anysotropic mass ${\rm {\bf M}}$
in a two-dimensional periodic potential $V(\vec {\bf \varphi})$.\cite{geisel}
In typical junctions, the Josephson energy scale, $E_J$, is much larger than
the electrostatic energy of electrons, $E_C= e^2/2C$, and the system
is in a classical regime. On the other hand, mesoscopic junctions (with small area) can have
$E_J\sim E_C$, and quantum fluctuations become important.\cite{likharev}
In this case, the quantum momentum operator is defined as
$${\vec {P}} \rightarrow \hat{\vec{P}}= -i\hbar\nabla_\varphi = -i\hbar(\frac{\partial}{\partial\varphi_1},\frac{\partial}{\partial\varphi_2}).$$
After replacing the above defined operator $\hat{\vec{P}}$ in the
Hamiltonian of Eq.(\ref{hamil}), the eigenvalue Schr\"odinger equation becomes
\begin{equation}
\label{eq:Schro}
\left[ -\frac{\eta^2}{2}\nabla_\varphi^T{\rm{\bf m}}^{-1}\nabla_\varphi
+V(\vec {\bf \varphi})\right] \Psi(\vec {\bf \varphi}) = E \Psi(\vec {\bf \varphi}) \; ,
\end{equation}
where we normalized energy by $E_J$ and momentum by
$\hbar/\sqrt{8E_C/E_J}$. We see in Eq.(\ref{eq:Schro})
that the parameter $\eta=\sqrt{8E_C/E_J}$ plays the role of an effective
$\hbar$. It is well-known that the ratio $E_C/E_J$ controls
the effect of quantum fluctuations in single Josephson junctions\cite{schon,haviland}
and in arrays of several Josephson junctions.\cite{fazio,mooij_qjja}
For $E_J\gg E_C$, ($\eta \ll 1$), the
junctions can be described with a classical dynamics;
while for $E_J \sim E_C$, ($\eta \sim 1$) the effect of quantum
fluctuations becomes important.\cite{schon}
Experiments where the Josephson junctions are fabricated for different values of
$E_C/E_J$ have been performed both for single junctions\cite{haviland}
and for junction arrays.\cite{mooij_qjja}
In the last case quantum phase transitions as a function of $E_C/E_J$
have been studied.\cite{fazio,mooij_qjja}
Therefore, the parameter $\eta=\sqrt{8E_C/E_J}$ is a natural choice
for quantifying the effective $\hbar$ in this system.
For quantum computation implementations \cite{qbit_mooij,chiorescu,fqubit_recent}
the DJFQ is operated at magnetic
fields near the half-flux quantum ($f= f_0+\delta f$, with $f_0=1/2$).
For values of $\alpha \ge 1/2$, the potential Eq.(\ref{eq:pot})
has two well defined minima.
At the optimal operation point $f=1/2$, the two lowest (degenerated) energy states are
symmetric and antisymmetric superpositions of two
states corresponding to macroscopic persistent currents
of opposite sign. The offset value $\delta f$ determines the level splitting
between these two states.
These eigenstates are energetically separated
from the others (for small $\delta f$)
and therefore the DJFQ has been used as a qubit \cite{qbit_mooij,chiorescu,fqubit_recent}
({\it i.e.} a two-level truncation of the Hilbert space is performed).
In addition the barrier for quantum tunneling between the states depends strongly on
value of $\alpha$ and its height goes up as $\alpha$ is increased.
The possibility to manipulate the potential landscape by changing $\alpha$
is a crucial point for experimental implementation of qubits. Typical
experiments in DJFQ have values of $\alpha$ in the range $0.6-0.8$.\cite{chiorescu,fqubit_recent}
As we will discuss here, the higher energy states
of the DJFQ show quantum manifestations of classical chaos.
In what follows we focus our study of the DJFQ considering the
realistic case of: (i) small on-site capacitances,
taking $\gamma=0.02$, (ii) a magnetic field corresponding to the optimal operation
point of the DJFQ, $f=1/2$, and (iii) the values of $\alpha= 0.7$ and $0.8 $
in coincidence with the experimental values employed in
Ref. \onlinecite{chiorescu,fqubit_recent}.
\section{Spectral Statistics}
\label{qchaos}
Before entering into the analysis of the quantum spectra we will
focus on the classical dynamics of the DJFQ.
As we already anticipated in the Introduction, generic systems present
mixed classical dynamics and the DJFQ is not the exception.
Therefore for a given energy $E$ our aim is to estimate the chaotic volume $v_{\rm ch}(E)$,
defined as the probability of having a chaotic orbit (i.e. Lyapunov exponent $\lambda > 0$) at energy $E$.
As we will show below, this parameter will be relevant in the statistical analysis of the quantum spectrum.
The classical dynamical evolution was obtained solving the Hamilton equations derived from Eq.(\ref{hamil}):
\begin{equation}
{\rm {\bf m}}\cdot \frac{d^2{\vec{\varphi}}}{dt^2} = -\nabla_\varphi V(\vec {\bf \varphi}),
\end{equation}
where we have normalized energy by $E_J$ and time by $t_c=\sqrt{\hbar^2C/4e^2E_J}
=\hbar/\eta E_J$ (the Josephson plasma frequency is $\omega_p=t_c^{-1}$).
The numerical integration was performed with a second-order leap-frog algorithm
with time step $\Delta t = 0.02 t_c$.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.9\linewidth]{Fig2.ps}
\caption{ (a) Average maximum Lyapunov
exponent $\overline{\lambda} $ and (b) chaotic volume $v_{ch}$
versus energy $E$ for $\alpha=0.8$
and $f=1/2$. } \label{figchaos}
\end{center}\end{figure}
\noindent
For different values of the parameter $\alpha$ and
magnetic flux $ f $ we compute the maximum Lyapunov exponent
$\lambda$ for each classical orbit at different energies $E$. We estimate
the chaotic volume $v_{\rm ch}(E)$ using $10^3$ initial conditions chosen randomly
with uniform probability within the available phase space for each
given energy. Also the average Lyapunov exponent, $\bar{\lambda}(E)$, of
the chaotic orbits is obtained.
These results are shown in Fig.~\ref{figchaos} for $\alpha=0.8$ and $f=1/2$.
We observe that both $v_{\rm ch}(E)$ and $\bar{\lambda}(E)$
increase smoothly with energy, as it is usual in several similar
systems with two degrees of freedom.\cite{benettin,meyer,cejnar}
Above the minimum energy of the potential, $E_{min}$, we find: (i) {\it
regular orbits} for $E_{min} < E < E_{ch}$ ($v_{\rm ch}=0$), (ii) {\it
soft chaos} ({\it i.e.}, coexistence of regular and chaotic
orbits, $0 < v_{\rm ch} < 1$) for $E_{ch} < E < E_{hc}$ with
the average Lyapunov exponent $\bar{\lambda}>0$ above $E_{ch}$
and (iii) {\it hard chaos} (all orbits are chaotic, $v_{\rm ch}=1$) for
$E>E_{ch}$.
The boundaries of these different dynamic regimes
as a function of $\alpha$, in the range $[0.5,1.0]$, and $f$, in the
range $[0,0.5]$, have been obtained in Ref.~\onlinecite{mingo}.
Here we will focus on the case with $f=1/2$ and we will study some different
cases of $\alpha$.
In order to look for signatures of quantum chaos, we follow a standard
statistical analysis of the energy spectrum.
First we calculate the exact spectrum $\{E_n\}$ by diagonalization of the
quantum Hamiltonian. The eigenvalue equation Eq.(\ref{eq:Schro}) is solved
by discretizing the phases with $\Delta \varphi = 2\pi/1000$, and the resulting
hamiltonian matrices of size $10^6 \times 10^6$ are diagonalized using
standard algorithms for sparse matrices. We have verified that
increasing the discretization by a factor of $2$ does not affect the results of the
spectrum within the needed accuracy for the ranges of energies studied here.
As we mentioned we set $\gamma=0.02$ and $f=1/2$,
and we obtain eigenvalue spectra for different values of the parameters
$\eta$ and $\alpha$ defined in the previous section.
The level spectrum is used to obtain the smoothed counting function
$N_{av}(E)$ which gives the cumulative number of states below an
energy $E$.
In order to analize the structure of the level fluctuations
properties one unfolds the spectrum by applying the well
kwown transformation $x_n=N_{av}(E_n)$.\cite{bohigas} From
the unfolded spectrum one can calculate the nearest-neighbor spacing (NNS)
distribution $P(s)$, where $s_i \equiv x_{i+1}-x_i$ is the NNS.
We have taken into account the symmetries of
the Hamiltonian Eq.(\ref{hamil}).
For $f=1/2$ the Hamiltonian has reflection symmetry against
the axis $\varphi_2=\varphi_1$ and against the axis $\varphi_2=-\varphi_1$.
The eigenstates can be chosen with a given parity with
respect to these two symmetry lines. Therefore, we compute the NNS distribution
employing eigenstates of a given parity.
This kind of decomposition is a standard procedure followed in the
analysis of spectral properties of quantum systems whenever the
Hamiltonian of the system possesses a discrete symmetry.\cite{bohigas}
We consider the even-even parity states and
the NNS distribution is computed for different energy regions
inside the classical interval ($E_{ch},E_{hc}$), corresponding to
soft chaos, and for energies $ E > E_{hc}$ ( and $E< 2\Delta$), corresponding
to hard chaos.
The Berry- Robnik theory seems to be suitable to analyze, in the semiclassical regime, sequence of levels
of quantum systems whose classical analogous presents coexistency of regular and chaotic dynamics
({\it i.e.}, soft chaos regime).
If $\rho_{1}$ and $\rho_{2} $ are the relative measures of the regular and chaotic parts of the classical phase
space then, the Berry-Robnik distribution \cite{berryro} reads:
\begin{eqnarray}
\label{pbr}
P^{BR} (s) = \rho_{1}^{2} \exp{(- \rho_{1} s)} \; \mbox{erfc} \left( \frac{1}{2} \sqrt{\pi} \rho_{2} s \right) + \nonumber \\
\left( 2 \rho_{1} \rho_{2} + \frac{1}{2} \pi \rho_{2}^{3} s \right) \exp \left( - \rho_{1} s -
\frac{1}{4} \pi \rho_{2}^{2} s^{2} \right) \; ,
\end{eqnarray}
where $\rho_{1} + \rho_{2} =1$. It is easy to verify that $P^{BR} (s)$ interpolates between the Poisson and Wigner GOE distributions as $0 \rightarrow \rho_{1} \rightarrow 1$, but does not exhibit level repulsion for $\rho_{1} \neq 0$.
\begin{figure}[th]
\begin{center}
\includegraphics[width=1.\linewidth]{Fig3.eps}
\caption{Cumulative distribution $W(s)$ for $\alpha=0.8$ and $f=0.5$. See the text for details.
The continuous line is the fitted Berry-Robnik distribution. We show for comparison
the Poisson (dotted line) and Wigner (dashed line) cumulative distributions.
Top panels correspond to $E=1.6$ with (a) $\eta=0.01$ and (b) $\eta=0.05$.
Bottom panels correspond to $E=2.0$ with (c) $\eta=0.01$ and (d) $\eta=0.05$.
The fitted Berry-Robnik parameters are (a) $\rho_{br}=0.44$,
(b) $\rho_{br}=0.93$,
(c) $\rho_{br}=0.99$ and (d) $\rho_{br}=0.96$.}
\label{nns_fig}
\end{center}
\end{figure}
In Fig.\ref{nns_fig} we show the cumulative level spacing distribution $W(s)= \int P(s) ds$ obtained numerically
following the prescription described before. We have done this in order to describe in some detail the behavior
for small values of $s$, (in the following we denote the cumulative distributions by the same name that the
corresponding NNS distribution).
In all the cases we have fitted the numerically obtained $W(s)$ employing
Eq.(\ref{pbr}) for the NNS distribution,
and we have extracted the fitted quantum parameter $\rho_{1} \equiv \rho_{br}$.
The particular results presented in Fig.~\ref{nns_fig} correspond to a window of
$\sim 100$ eigenvalues around $E_{ch} < E=1.6 < E_{hc}$,
within the soft chaos regime, Fig.~\ref{nns_fig} (a),(b);
and $E_{hc} < E=2$, within the hard chaos regime, Fig.~\ref{nns_fig} (c),(d).
We take the realistic experimental value for the parameter $\alpha=0.8$ and consider different values
of the quantum parameter $\eta$ : the case with
$\eta=0.01$ is shown in Fig.~\ref{nns_fig} (a),(c); and the case with
$\eta=0.05$ is shown in Fig.~\ref{nns_fig} (b),(d).
We should remark that the classical dynamics is independent of the
parameter $\eta$, which has a pure quantum origin
and plays the role of an effective Planck's constant in
the Schr\"odinger equation, as we mentioned before.
In addition in Fig.\ref{nns_fig} we show for comparison the $W(s)$ corresponding to
the Poisson and Wigner GOE distributions.
We first discuss the case with $\eta=0.05$, that is
already smaller than in the cases studied in Ref.\onlinecite{mingo}, where
$\eta=0.07-0.17$ was considered.
The numerical results for $\eta=0.05$, in the case of hard chaos [$E=2.0$,
shown in Fig.~\ref{nns_fig} (d)], are
in good agreement with the Wigner distribution, and we obtain $\rho_{br}=0.96$.
In a case corresponding to mixed classical dynamics
[$E=1.6$, shown in Fig.~\ref{nns_fig} (b)], we find that
the distribution departs slightly from the pure Wigner form.
However, we have obtained
$\rho_{br}=0.93 \gg v_{ch} \sim 0.4$, meaning that
the level distribution in this case does not seem to be very
sensitive to the mixed phase space expected in the classical limit.
The reason is that for increasing $\eta$
the mean energy level spacing increases (proportional to $\eta^2$ for
large energies), and therefore the width of the energy region evaluated for the
statistics with a given number of levels ($\sim 100$ in this case) also increases
in the same way. Since $v_{ch}(E)$ varies rapidly with $E$ within the soft chaos
region, relating its value with the fitted $\rho_{br}$, which is obtained
evaluating the statistics over a wide energy region, becomes meaningless for large
$\eta$.
Indeed, deep in the quantum regime the Berry-Robnik fitted parameters are not expected
to be related to the
classical measure of the chaotic (regular) part of the phase space.\cite{berryro,cederbaum86,prosen}
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig4.eps}
\caption{Fitted Berry-Robnik parameter $\rho_{br}$ as a function of the
dimensionless energy $E/E_{J}$ for $f=0.5$ and $\eta=0.01$. (a) $\alpha=0.7$ and (b) $\alpha=0.8$.
The horizontal error bars in energy are defined
by the interval of eigenenergies used in the statistics,
and it is a decreasing function of the density of states.
The vertical error bars correspond to error in the parameter $\rho_{br}$ as
obtained from the numerical fits.
The dotted line shows the chaotic fraction of the classical phase space $v_{ch}$ obtained
from the classical dynamics.} \label{fig_br}
\end{center}
\end{figure}
We now discuss a smaller value of the effective $\hbar$,
corresponding to $\eta=0.01$.
In Fig.~ \ref{nns_fig} (a), for $E=1.6$ (mixed classical dynamics), we find now
that the $W(s)$ clearly departs from the pure Wigner form, and that it can be fitted with
the Berry- Robnik distribution with $\rho_{br}=0.44$.
This value is very close to the classical chaotic volume for this case, $v_{ch} \approx 0.4$.
In the case for $E=2$ (hard chaos),
shown in Fig.\ref{nns_fig} (c),
we have obtained $\rho_{br}=0.99$, in agreement with $v_{ch}=1$ and also
in good agreement with the Wigner distribution, as expected.\cite{bohigas,seligman}
In general we find that in a nearly semiclassical regime, $\eta=0.01$, the numerical results
for the Berry-Robnik parameter $\rho_{br}$ show a good agreement
with the classical measure $\rho_{1}$,
that by definition is equivalent to the chaotic volume $v_{ch}$.
This is analyzed in Fig. \ref{fig_br} where we plot the quantum parameter $\rho_{br}$
obtained for different sections of the spectra with $\sim 100$ eigenvalues around a given energy $E$.
We show results for two cases of the parameter $\alpha$.
The chaotic fraction of the classical phase space $v_{ch}$ is also plotted.
The results for $\rho_{br}$ and $v_{ch}$ are very close to each other.
When changing the parameter $\alpha$ the location in energy of the onsets
of the soft chaos and hard chaos regimes shifts. We also see that the
curves of $\rho_{br}$ vs. $E$ shift in the same way, giving further support
to the correspondence between $\rho_{br}$ and $v_{ch}$.\cite{seligman,cederbaum86,prosen,makino01}
These results corroborate the validity of the Berry-Robnik theory in the
semiclassical energy region
that corresponds to small effective Planck's constant, as it is the case for $\eta=0.01$.
Besides the cases reported above,
we have also analyzed a few other values of $\alpha$ in the range
$0.5-0.9$ and $f=0.4,0.5$,
obtaining similar results for the spectral statistics.
In general, we observe that in order to obtain a spectral statistics with a Berry-Robnik
paremeter that agrees with the classical measure of the chaotic volume values
of $\eta < 0.05$ are needed.
\section{Phase space and Husimi Distributions for the DJFQ}\label{distri}
In this section we pursue our study of the signatures of quantum chaos
presenting an analysis of the quantum phase-space distributions in the
case of mixed classical dynamics.
Taking into account the analysis of the previous section we focus
on the case $\eta=0.01$.
Quantum phase space distributions are of increasing interest in
studies of quantum chaos because they allow a direct comparison
between classical and quantum dynamics. The Husimi distribution
associated to a quantum wave function $|\Psi \rangle$ (see
definition below, Eq.(~\ref{husi})) it is based on the
coherent-state representation and is well suited to represent wave
functions in phase space because it is always real and
possitive.\cite{husimi,leboeuf,hus-rev,groh} Due to these
properties it is usually referred as a quasi probability
distribution.
In order to compute the Husimi function for the DJFQ we must take
into account the fact that the classical phase space is four
dimensional. The Husimi distribution function for a state
$|\Psi\rangle$ is
\begin{equation} \label{husi}
\rho^H(\vec{P_0},\vec{\varphi_{0}}) = |\langle \vec{P_0},\vec{\varphi_{0}}|\Psi\rangle|^2\;,
\end{equation}
where
$| \vec{P_0},\vec{\varphi_{0}}\rangle$ corresponds to minimum-uncertainty
$2\pi$-periodical wave packets \cite{carruthers} given by
\begin{eqnarray}\label{eq:packet}
|\vec{P_0},\vec{\varphi_{0}}\rangle &=& C\times
\exp[i \vec{K_0}\cdot \left(\vec{\varphi}-\vec{\varphi_0}\right)] \times
\\
& &
\exp\left[\frac{\cos(\varphi_{0,1}-\varphi_1)+\cos(\varphi_{0,2}-\varphi_2)-2}{2\sigma^2}\right]
\nonumber
\end{eqnarray}
where $\vec{K_0}=(k_1,k_2)$ with $k_1,k_2$ integers
and $\vec{P_0}=\eta \vec{K_0}$.
The width of the wave packet is
$\sigma=\sqrt{\eta/s}$, with $s$ the squeezing parameter,
and we choose the value $s=3.23$, which is the
same value used in Ref.\onlinecite{mingo} for the
initial coherent wave packets.
The potential has two minima for $f=1/2$ which are at $(\varphi^*,-\varphi^*)$
and $(-\varphi^*,\varphi^*)$, with $\cos\varphi^* = 1/2\alpha$. To better
analyze the Husimi function, it is convenient to make the following change of variables:
\begin{eqnarray}\label{eq:rotation}
\varphi_x &=& \frac{\varphi_1-\varphi_2}{\sqrt{2}},\;\;\;\;\;\;\;
P_x = \frac{P_1-P_2}{\sqrt{2}};\nonumber\\
\varphi_y &=& \frac{\varphi_1+\varphi_2}{\sqrt{2}},\;\;\;\;\;\;\;
P_y = \frac{P_1+P_2}{\sqrt{2}}.
\end{eqnarray}
In this way the two minima lie along the direction of $\varphi_x$.
The normalization by $\sqrt{2}$ is chosen such that
new variables satisfy $[\varphi_x,P_x]=i\eta$, $[\varphi_y,P_y]=i\eta$
in the quantum regime.
The classical Poincar\'e surface of section is calculated in the
plane $(\varphi_x, P_x)$, taking $\varphi_y = 0$ and $P_y>0$.
We want to compare the Husimi distribution $\rho^H_\nu(\vec{K},\vec{\varphi})$ corresponding to
the eigenstate $|\Psi_\nu\rangle$ with eigenvalue $E_\nu$ with the classical
Poincar\'e section at an energy $E\approx E_\nu$.
To this end, we construct an analog of the surface of section
by obtaining a two-dimensional section of $\rho^H_\nu(\vec{K},\vec{\varphi})$
(which is a four-dimensional density in phase space) in the following way:\cite{groh}
\begin{equation}\label{eq:sec_husi}
\Phi^H_\nu(P_x,\varphi_x) = \rho_\nu^H(P_x,P_y^E;\varphi_x,0)
\end{equation}
where, given the values $P_x, \varphi_x$ and $\varphi_y=0$,
$P_y^E$ is obtained such that the classical energy is equal to $E$ and
the possitive root, $P_y^E>0$, is chosen.
We obtain numerically the eigenstates $|\Psi_\nu\rangle$ of Eq.~(\ref{eq:Schro}),
after using a discretization of $\Delta\varphi= 2\pi/500$. Then, using Eqs.~(\ref{husi})-(\ref{eq:sec_husi}),
we compute the sections of the Husimi distributions, $\Phi^H_\nu(P_x,\varphi_x)$.
In order to characterize the localization of the quantum states on the classical phase space structures,
we choose a few examples of $\Phi^H_\nu$ for eigenstates that lie in energy regions
corresponding to regular classical dynamics $E<E_{ch}$ and
soft chaos region, $E_{ch}<E<E_{hc}$, respectively.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig5.eps}
\caption{(a) Classical Poincar\'e surface of section
for $E=1.52$. Sections are symmetric with respect to
$\varphi_x \rightarrow -\varphi_x$
and $P_x \rightarrow -P_x$;
only the region of $\varphi_x>0$ and $P_x > 0$ is shown.
Section of Husimi phase space distribution,
$\Phi^H_\nu(P_x,\varphi_x)$ for
(b) $E_\nu=1.5219$,
(c) $E_\nu=1.5208$,
(d) $E_\nu=1.5193$.
} \label{fig_hus1}
\end{center}
\end{figure}
In Fig.~\ref{fig_hus1} (a) we plot for $E=1.52 < E_{ch}$ the classical Poincar\'e section in which the stability
islands associated to the regular dynamics are observed.
We have computed the Husimi phase space distributions $\Phi^H_\nu(P_x,\varphi_x)$
for several eigenstates ($\sim 20$) near the energy $E=1.52$.
We show here three cases corresponding to eigenstates with energies $E_\nu=1.5219$,
$E_\nu=1.5208$ and $E_\nu=1.5193$ (panels (a) , (b) and (c) respectively).
The localization of these states on the stability islands and fixed points is clearly observed.
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig6.eps}
\caption{Classical Poincar\'e surface of section
for $E=1.6$.
Section of Husimi phase space distribution,
$\Phi^H_\nu(P_x,\varphi_x)$ for
(b) $E_\nu=1.601$,
(c) $E_\nu=1.6008$,
(d) $E_\nu=1.5993$.} \label{fig_hus2}
\end{center}
\end{figure}
In Fig.\ref{fig_hus2}(a) and Fig.\ref{fig_hus3}(a) we plot for classical energies $E=1.6$ and $E=1.7$
respectively, the classical Poincar\'e sections together with a selection of
some of the calculated
Husimi phase space distributions $\Phi^H_\nu(P_x,\varphi_x)$ for eigenstates with energies
$E_\nu=1.601,1.6008, 1.5993$ (Fig.\ref{fig_hus2} (b), (c) and (d) respectively) and
$E_\nu=1.6992, 1.7004, 1.6994$ (Fig.\ref{fig_hus3} (b), (c) and (d) respectively).
In these cases the soft chaos behavior is evident by the structure of the Poincar\'e sections,
in which regular islands are sorrounded by chaotic regions.
The localization of the states on classical structures like already distroyed
chains of islands is observed in the figures. In addition, the Husimi distribution
of Fig.\ref{fig_hus3} (d) corresponds to a
state localized on the chaotic region of Fig.\ref{fig_hus3}(a).
The above analysis of the Husimi distributions shows that for $\eta=0.01$,
it is possible to use localized wave packets as initial conditions for the experimental measurement of
the Loschmidt echo,\cite{mingo,nota2}
since they can sense the structure of the phase space
with mixed classical dynamics in this case.\cite{liu}
\begin{figure}[th]
\begin{center}
\includegraphics[width=0.8\linewidth]{Fig7.eps}
\caption{Classical Poincar\'e surface of section
for $E=1.7$.
Section of Husimi phase space distribution,
$\Phi^H_\nu(P_x,\varphi_x)$ for
(b) $E_\nu=1.6992$,
(c) $E_\nu=1.7004$,
(d) $E_\nu=1.6994$.} \label{fig_hus3}
\end{center}
\end{figure}
\section{Conclusions}\label{conclu}
In this paper we have characterized the quantum signatures of chaos in the three-junction SQUID device.
For realistic parameter values the classical dynamics exhibits different regimes that go from mixed dynamics to fully developed chaotic motion. As a consequence the spectral statistics, characterized by the distribution of the nearest neigbour energy spacing (NNS) $P(s)$ in the high
energy region, is expected to unveil signatures of the mentioned behavior.
The analysis has been performed for different energy regions inside the classical intervals
corresponding both to the soft chaos ({\it i.e.}, mixed phase space) and hard chaos regimes,
and we considered the even-even parity states to compute the NNS distribution.
Our numerical results show that, for $\eta < 0.05$ (and for $\eta=0.01$ in particular),
in a nearly semiclassical regime, $P(s)$ is well fitted by
Berry-Robnik like formulae, where the pure classical measures of the chaotic and regular regions
have been used as the only free parameters.
We also found that the individual eigenstates can also be intimately linked to the phase space structures
that characterizes the different classical regimes for $\eta<0.05$.
In order to analyze how quantum states are supported or localized on different classical structures that are present in the different regimes in this case,
we have investigated the Husimi phase space distributions
for different eigenstates with energies $E_\nu$ in the classical interval.
We would like to mention that there are few studies
of Husimi distributions for Hamiltonian systems with two degrees of freedom, \cite{groh}
as it is the case of the DFJQ studied in the present work.
One important advantage of Josephson junction devices is that they can be fabricated
with well-controlled parameters.
The effective $\hbar$, is
$\hbar_{\rm eff}=\eta=\sqrt{\frac{8E_C}{E_J}}$, and since $E_J \propto A$ and $E_C \propto 1/A$,
with $A$ the area of the junctions, we have that $\hbar_{\rm eff}\propto 1/A$.
Thus, the fabrication of different DJFQ with junctions with varying area could allow to study
cases with $\hbar_{\rm eff} $ spanning from the semiclassical to the quantum regime.
This is indeed important since different regimes can be accessed experimentally
depending on the magnitude of $\eta$.
The qubit regime of two-level dynamics of the DJFQ is observed experimentally in
devices with $\eta \approx 0.4$.\cite{chiorescu,fqubit_recent}
In Ref.\onlinecite{mingo} it has been found that signatures of chaos in the Loschmidt
echo can be observed at high energies $E \sim 3 E_J$ in devices
with an effective $\hbar$ of the order of $\eta \approx 0.1$.
Here we have shown that the observation of the quantum effects
in the case with mixed chaotic and regular orbits (for an intermediate energy range)
needs the study of devices in a more semiclassical regime
with $\eta\approx 0.01$. This could motivate experimental measurements
looking for the dependence of the Loschmidt echo\cite{mingo} with initial conditions,
due to the phase space structure of the mixed classical dynamics, if
the experiments are performed in devices with $\eta\approx 0.01$.
Considering the values\cite{nota} of $E_J \sim 250 GHz \sim 2 K$
and the operation temperature of $20$ m$K$ reported by the Delft group \cite{chiorescu},
typical level spacings of $ 0.01 E_J \sim 20$ m$K $ can be experimentally resolved
in the device of Ref.\onlinecite{chiorescu}.
This energy resolution is enough for the case of the Loschmidt echo in devices
with $\eta \sim 0.1$, analyzed originally in Ref.\onlinecite{mingo}.
However, the semiclassical regime explored in this work ($\eta=0.01$)
requires a resolution in the level spacings of the order of $5\times10^{-4} E_J$.
Thus, for experiments in the cryogenic range ($20$ m$K$) devices with larger values
of $E_J$ should be employed.
On one hand, a smaller $\eta \sim 0.01$ already requires junctions
with larger area $A$, and therefore larger $E_J$.
On the other hand, Josephson junctions fabricated with high $T_c$ superconductors\cite{jhtc}
can have a large $E_J$.
Therefore, devices designed with high $T_c$ superconductors
can be good candidates for the experimental challenge of studying
the mixed phase space in the semiclassical regime of the DJFQ.
Another possible type of experiment is to start the system in the ground state and apply a constant pulse
in some external parameter (for instance, the magnetic field).
After the pulse is applied, the probability of remaining
in the ground state could be related to the energy level statistics.\cite{cohen}
Also, an interesting experiment could be to perform studies of the
low frequency noise, as it has been done in mesoscopic chaotic cavities.\cite{buttiker,beenakker-rmp}
For example, one could drive the DJFQ into the hard chaos regime with a voltage source
such that $E_V= \frac{1}{2} C V^2 > E_{hc}$ (and $ V < 2\Delta /e$)
and then measure the noise in the current.
How the current noise is related to the spectral
statistics in this case is a very interesting problem,
which could be the subject of future studies.
\acknowledgments
We acknowledge financial support
from ANPCyT (PICT2003-13829, PICT2003-13511 and PICT2003-11609),
Fundaci\'{o}n Antorchas, CNEA and Conicet.
ENP also acknowledges support from U.N. Cuyo.
|
1,314,259,992,966 | arxiv | \section{Introduction}
Suppose we have a time-series $(X_n)$ of real-valued random variables defined on a probability space $(X,\mu)$ and let $M_n:=\max \{X_1,\ldots, X_n\}$ be the sequence of successive maxima of $(X_i)$. There is a well-developed theory for these maximum values in the setting of $(X_n)$ i.i.d \cite{Embrechts, Galambos}. If we consider a dynamical system $(T,X,\mu)$ such that $T\colon X\rightarrow X$ and an observable $\phi: X\to \ensuremath{\mathbb R}$, we can define a stochastic process by
\[
X_n = \phi\circ T^n (x)
\]
for $x\in X$. In the case of modeling deterministic physical phenomenon, $T$ is usually taken as an ergodic, measure-preserving transformation, $\mu$ a probability measure and $\phi$ is a function with some regularity, for example (locally) H\"{o}lder~\cite{V.et.al}. In extreme value literature, it is typically assumed that $\phi$ is a function of the distance $d(x,p)$ to a distinguished point $p$ for some metric $d$ so that $\phi (x)=f(d(x,p))$ for $x\in X$, and $f$ is a monotone decreasing function $f:(0,\infty)\to\mathbb{R}$. In this instance
$\sup_{x\in X}\phi(x)=\lim_{x\to p}\phi(x)$, and hence the set $\{\phi(x)\geq u\}$ corresponds to a neighborhood about $p$. We shall refer to the set of all points $x\in X$ for which $\phi(x)$ achieves its maximum (with $\sup_{x\in X}\phi(x)=\infty$ allowed) as the extremal set $\mathcal{S}$. For convenience (and almost by convention) the observation $\phi (x)=-\log d(x,p)$ is often used, but scaling relations translate extreme value results for one functional form to another quite easily provided the extremal set of $\phi$ is unchanged. If the observable $\phi (x)=-\log d(x,p)$ is changed to another function of $d(x,p)$, then $\mathcal{S}$ remains equal to $\{p\}$.
However, if the underlying extremal set $\mathcal{S}$ is changed, e.g. going from a point to a curve, then the proofs of extreme value results and the results themselves do not translate and new approaches are required. Indeed, even if the extremal set changes from one point to another, then the extreme value laws may change (e.g. $p$ periodic versus $p$ non-periodic give different distributional extreme value laws)~\cite{Dichotomy,Ferguson_Pollicott,FFT3,Keller}.
Since the value of the function $\phi\circ T^n (x)$ is larger the closer $T^n (x)$ is to the extremal set $\mathcal{S}$, there is a close relation between extreme value statistics for the time series $X_n=\phi\circ T^n(x)$ and return-time statistics to nested sets about $\mathcal{S}$~\cite{CC,Collet,DGS,FFT1,FFT3,Gupta,Hirata}. We focus on extreme value theory in this
paper but it would be possible, though computationally very difficult, to derive return time distributions which are simple Poisson (in the
cases in which the extremal index is $\theta=1$) and compound Poisson (in the cases in which the extremal
index $\theta<1$). The parameters in the compound Poisson distribution would in particular be difficult to compute but this would constitute an interesting investigation.
We expect this work could be carried out using basically the same toolkit from extreme value theory.
These parameters are calculated for functions maximized at periodic orbits in the setting of a hyperbolic toral automorphism~\cite{Dichotomy}
and in~\cite{CNZ} for functions maximized at periodic orbits in Sinai dispersing billiard systems. We discuss the concept of extremal index below, it is a number
$0 \le \theta \le 1$ which roughly quantifies the clustering of exceedances. We will say $\theta=1$ is a trivial
extremal index and $\theta<1$ a nontrivial extremal index. For results along these lines see~\cite{Dichotomy,FFT3,FHN}.
Recent literature has focused on the case where the extremal set $\mathcal{S}$ is a single point $\{p\}$. In this paper we address some scenarios of interest where the observable is maximized on sets other than unique points in phase space, and in turn
describe how the extreme value law depends on the geometry of $\mathcal{S}$. We also describe a dynamical mechanism giving rise to a nontrivial extremal index which is not due to periodicity. The recent preprint~\cite{Haydn_Vaienti} provides a different and axiomatic
approach to determining the limit laws (especially simple and compound Poisson distributions) for entry times into neighborhoods of sets of measure zero in dynamical systems. They present similar results to this paper on coupled map lattices and consider other dynamical and statistical examples, including some systems with polynomial decay of correlations. We address here cases that are not easily
captured by axiomatic approaches. This happens for example, if the extremal set $\mathcal{S}$ fails certain transversality assumptions
relative to the local (or global) stable and unstable manifolds of the system. We discuss these situations further in Sections
\ref{sec.anosov} and \ref{sec.discussion}.
\subsection{Background on extremes for dynamical systems}\label{sec.background}
Suppose $(X_n)$ is a stationary process with probability distribution function $F_X(u):=\mu(X\leq u).$ We define an extreme value law (EVL) in the following way. Given $\tau\in\mathbb{R}$, let $u_n(\tau)$ be a sequence satisfying $n\mu(X_0>u_{n}(\tau))\to\tau$, as $n\to\infty$. We say that $(X_n)$ satisfies an extreme value law if
\begin{equation}\label{eq.ev-law}
\mu(M_n\leq u_n(\tau))\to e^{-\theta\tau}
\end{equation}
for some $\theta\in(0,1]$. Here, $\theta$ is called the extremal index and $\frac{1}{\theta}$ roughly measures the average number of exceedances in a time window
given that one exceedance has occurred. When $(X_n)$ is i.i.d. and has a regularly varying tail it can be shown that this limit exists and $\theta=1$.
In the dependent setting for stationary $(X_n)$ the existence of an EVL has been shown provided dependence conditions $D(u_n)$ (mixing condition) and $D'(u_n)$ (recurrence condition) or similar conditions hold for the system~\cite{LLR,FFT1}. Freitas et al~\cite{FnF}, based on Collet's work, in turn gave a condition $D_2(u_n)$ which has the full force of $D(u_n)$ in that together with $D'(u_n)$ it implies the existence of an EVL and is easier to check in the dynamical setting. We describe
more precisely these three conditions below.
There are, however, no general techniques for proving conditions $D_2(u_n)$ and $D'(u_n)$ and checking the latter is usually hard. $D'(u_n)$ is a short returns condition that is not implied by
an exponential decay of correlations. However $D_2(u_n)$ often follows from a suitable rate of decay of correlations.
Collet~\cite{Collet} used the rate of decay of correlation of H\"{o}lder observations to establish $D(u_n)$ for certain one-dimensional
non-uniformly expanding maps.
Condition $D_2(u_n)$ is easier to establish in the dynamical setting by estimating the rate of decay of correlations of H\"older continuous observables
or those of bounded variation and in practice is easier to verify.
For completeness we now state conditions $D(u_n)$, $D_2(u_n)$ and $D^{'}(u_n)$. If $\{X_n\}$ is a stochastic process define
\[
M_{j,l} := \max\{X_j, X_{j+1}, \dots, X_{j+l}\}.
\]
We will often write $M_{0,n}$ as $M_n$. We write $F_{i_1,\ldots,i_n} (u)$ for the joint distribution $F_{i_1,\ldots,i_n} (u)=\mu (X_{i_1} \le u, X_{i_2} \le u,\ldots,
X_{i_n} \le u)$.
\noindent {\bf Condition $D (u_n)$~\cite{LLR}} We say condition $D(u_n)$ holds for the sequence $X_0,X_1,\ldots, $ if for any integers
$i_1<i_2<\ldots < i_p< j_1<j_2<\ldots < j_{p'}\le n$, for which $j_1-i_p>t$
we have
\[
|F_{i_1,i_2,\ldots, i_p, j_1,j_2,\ldots, j_{p'}} (u_n) -F_{i_1,i_2,\ldots, i_p} (u_n) F_{ j_1,j_2,\ldots, j_{p'}} (u_n)| \le \gamma(n,t)
\]
where $\gamma (n,t)$ is non-increasing in $t$ for each $n$ and $n\gamma(n,t_n)\to 0$ as $n\to \infty$ for some sequence $t_n=o(n)$, $t_n\rightarrow \infty$.
\noindent {\bf Condition $D_2 (u_n)$~\cite{FnF}} We say condition $D_2(u_n)$ holds for the sequence $X_0,X_1,\ldots, $ if for any integers $l$,$t$ and $n$
\[
|\mu ( X_0 >u_n, M_{t,l} \le u_n )-\mu (X_0 >u_n)\mu ( M_{l} \le u_n)| \le \gamma(n,t)
\]
where $\gamma (n,t)$ is non-increasing in $t$ for each $n$ and $n\gamma(n,t_n)\to 0$ as $n\to \infty$ for some sequence $t_n=o(n)$, $t_n\rightarrow \infty$.
\noindent {\bf Condition $D^{'} (u_n)$~\cite{LLR}} We say condition $D^{'}(u_n)$ holds for the sequence $X_0,X_1,\ldots, $ if
\begin{equation}\label{cond:dprime}
\lim_{k\to \infty}\limsup_n n\sum_{j=1}^{[n/k]}\mu(X_0>u_n,X_j>u_n)=0.
\end{equation}
Condition $D^{'} (u_n)$ controls the measure of the set of points of $(X_0>u_n)$ which return to the set
relatively quickly, and is a condition that rules out ``short returns''. It is not a consequence of exponential decay of correlations and usually dynamical and
geometric arguments are needed to verify Condition $D^{'} (u_n)$ in specific cases.
In the dynamical case if the time series of observations $X_n=\phi\circ T^n$ satisfy $D(u_n)$ (or $D_2(u_n)$) and $D'(u_n)$
(or some variation thereof) then an EVL holds. In these results, we have extremal index $\theta=1$ for observables of the form $\phi (x)=f(d(x,p))$, maximized at generic $p\in X$ provided $p$ is non-periodic~\cite{Dichotomy,Ferguson_Pollicott,FFT3,FHN,HNT0,Keller}. For periodic $p$, EVLs have been derived for these systems with index $\theta< 1$~\cite{CNZ,Dichotomy,Ferguson_Pollicott,FFT3,Keller,V.et.al}.
For statistical estimation and fitting schemes such as block maxima or peak over thresholds methods \cite{Embrechts}, it is desirable to get a limit along linear sequences of the form $u_n(y)=y/a_n+b_n$. Here the emphasis is changed and the sequence $u_n(y)$ is now required to be linear in $y$. For example suppose $\phi(x)=-\log x$ is
an observable on the doubling map of the interval $[0,1]$, $Tx=(2x)$ mod $1$, which preserves Lebesgue measure $\mu$. The condition $n\mu (\phi > u_n(y))=y$ implies
$u_n(y)=\log n-\log y$. Furthermore we know that $n\mu (\phi > u_n(y))=y$ implies $\mu (M_n \le u_n(y))\to e^{-y}$. This is a nonlinear scaling.
If we change variables to $Y=-\log y$ we obtain $n\mu (-\log x > Y+\log n)\to e^{-y}=e^{-e^{-Y}}$, a Gumbel law.
In general if we restrict to linear scalings $y\in\mathbb{R}$, we obtain a limit $n\mu (X_0>\frac{y}{a_n}+b_n)\to h(y)$ and hence
\[
\mu(a_n(M_n-b_n)\le y)\to e^{-h(y)} =G(y),\quad(n\to\infty).
\]
For i.i.d processes, if $G$ exists and is non-degenerate, then it takes three distinct forms $G(y)=e^{-h(y)}$ with either:
\begin{itemize}
\item[(i)] $h(y)=e^{-y}$, $y\in\mathbb{R}$ (Gumbel);
\item[(ii)] $h(y)=y^{-\alpha}$, $y>0$ and some $\alpha>0$ (Fr\'echet);
\item[(iii)] $h(y)=(-y)^{\alpha}$, $y<0$ and some $\alpha>0$ (Weibull).
\end{itemize}
These three forms can be combined into a unified \emph{generalized extreme
value} (GEV) distribution (up to scale and location $u\to \frac{u-\alpha}{\sigma}$):
\begin{equation}\label{eq.gevlimit}
G_{\xi}(y)=
\begin{cases}
\exp\{-(1+\xi y)^{-\frac{1}{\xi}}\},\text{ if $\xi\neq 0$};\\
\exp\{-e^{-y}\},\text{ if $\xi= 0$}.
\end{cases}
\end{equation}
The case $\xi=0$ corresponds to the Gumbel distribution, $\xi>0$ corresponds to a Fr\'echet distribution,
while $\xi<0$ corresponds to a Weibull distribution.
Numerical fitting schemes for the GEV distribution are renormalized under place and scale transformations so that the extremal index (EI) is $\theta=1$~\cite[Theorem 5.2]{Coles}. Although it is theoretically possible to recover the EI by considering it as a function of these transformations, estimates in this way would have an undetectable level of error. Techniques to directly compute the EI, referred to as \textit{blocks} and \textit{runs} estimators, have been proposed \cite[Section 3.4]{V.et.al}. Both methods utilize the definition of the EI (outlined above) by numerically estimating the ratio of the number of exceedances in a cluster to the total number of exceedances. Where these differ is in their definitions of a cluster; the runs estimator splits the data into fixed blocks of size $k_n$ so that a cluster is defined by the number of exceedances inside each fixed block while the blocks estimator introduces a run length of $q_n$ so that any two exceedances separated by a time gap of less than $q_n$ belongs to the same cluster. The problem with using these estimators in practical applications is their heavy dependence on the choice of
the sequences $k_n$ and $q_n$, respectively.
Recent literature has provided more robust estimates of the extremal index. In particular the S\"{u}veges estimator \cite{suveges} has become more common in extreme value statistics \cite{sandro_coupled,V.et.al}. For a sequence $(X_j)$ $j = 1,\dots,n$ of random variables, let $q$ denote a fixed quantile and $l$ the location of exceedances $\{l:X_l>q\}$ above $q$. We define $T_i = l_{i+1}-l_i$ for $i = 1,\dots,N-1$ as the length of time between each consecutive recurrence. Let $S_i = T_i-1$ and $N_c = \sum_{i=1}^{N-1} {1}_{S_i \ne 0}$, so that $N_c$ is the number of clusters found by counting the set of recurrences separated by a time gap of at least length 1. Then the S\"{u}veges estimator of the extremal index given by,
\[
\hat{\theta} = \frac{\sum_{i=1}^{N-1} qS_i+N-1+N_c-[(\sum_{i=1}^{N-1} qS_i +N-1+N_c)^2-8N_c\sum_{i=1}^{N-1}qS_i]^{1/2}}{2\sum_{i=1}^{N-1}qS_i},
\]
can be viewed as the maximum likelihood estimator for the expected value of the number of recurrences coming from a point process defined by the compound Poisson distribution. We use this method to estimate the EI of the coupled map and the hyperbolic toral automorphism of Section~\ref{sec.numerics}.
For dynamical systems, the corresponding problem of finding scaling constants $a_n, b_n$ depends on both the regularity of
$\mu$ and that of the observable $\phi(x)=f(d(x,p))$ in the vicinity of the point $p$.
For more general dynamical systems, these scaling relations depend on how the invariant measure scales on sets that shrink
to $p$. This problem has been addressed in the case where $\mu$ admits a smooth or regularly varying density function $h$.
However, for general measures (such as Sinai Ruelle Bowen measures) and general observables, estimating $\mu(X>y/a_n+b_n)$ becomes more delicate, see \cite{FFT2, HVRSB}. However, an extreme law can still be obtained along some non-linear sequence $u_n(y)$, with bounds on the growth of $u_n(y)$, see \cite{GHN}.
Furthermore, for deterministic dynamical systems the extremal index parameter $\theta$
may be nontrivial due to periodicity.
For the doubling map discussed above, if $p$ is a periodic point then $\theta=1-\frac{1}{2^q}$ where
$q$ is the period of the period point (see \cite{FFT3,Keller}).
In this article, we consider cases where $\phi$ is maximized on a more general extremal sets $\mathcal{S}$. For general $\mathcal{S}$ we cannot rely on previous methods adapted to observables of the form $\phi=f(d(x,p))$.
\subsection{Physical and energy-like observables.}\label{sec.energy}
In the study of extreme events in dynamical systems, having in mind applications to weather and climate modeling, the notion of a \emph{physical observable} was introduced and described in \cite{HVRSB, LFWK, SHRBV}. By physical observables we mean those of form $\phi(x)=x\cdot v$ or $\phi(x)=x\cdot Ax$, where $A$
is $d\times d$ matrix, and $v$ a specified vector in $\mathbb{R}^d$. The former observable has planar level sets, while the latter has ellipsoidal level sets. In weather applications,
these observables correspond to measuring (respectively) the momentum and kinetic energy of the system. The level geometries of
$\phi$ introduced additional technicalities in establishing extreme laws relative to the cases where
the level sets are metric balls. These issues are discussed in detail in \cite{HVRSB}, where $\mathcal{S}$ had a complicated geometry but its
intersection with the attractor of the system was still a single point.
In this article,
we mainly consider energy-like observables for which the extremal set $\mathcal{S}$ is achieved on a line segment or submanifold.
We also discuss other extremal sets in Section \ref{sec.discussion}.
\subsection{Organization of the paper.} In Section \ref{sec.statement} we describe our main results on: hyperbolic toral automorphisms,
Sinai dispersing billiard maps, and coupled uniformly expanding maps. We calculate the extreme value distribution,
the extremal index and in some cases describe briefly the Poisson return time process. In particular we describe a method for
obtaining a nontrivial extremal index which is not due to periodic behavior but rather self-intersection of a set of non-periodic points under the
dynamics.
Beyond existing approaches, we have to develop arguments that deal with both
the geometry of $\mathcal{S}$, and the recurrence properties of the dynamical systems under consideration. In our examples the underlying invariant measures have regular
densities with respect to Lebesgue measure. This enables us to obtain analytic results
on the GEV parameters and the extremal index. We also compare our results to numerical schemes, see Section \ref{sec.numerics}.
We conclude with a discussion~\ref{sec.discussion} on how the methods we have developed might be applied to general observables
whose extremal sets have more complicated geometries.
\section{Statement of Results}\label{sec.statement}
\subsection{Hyperbolic toral diffeomorphisms}\label{sec.anosov}
We consider hyperbolic toral automorphisms of the two-dimensional torus $\ensuremath{\mathbb T}^2$ induced by a matrix
\[T= \left( \begin{array}{cc}
a & b \\
c & d
\end{array} \right).\]
with integer entries, $\det(T)=\pm 1$ and no eigenvalues on the unit circle. We will assume that both eigenvalues are positive in what follows to simplify the discussion and proofs. Such maps preserve Haar measure $\mu$ on $\ensuremath{\mathbb T}$.
A well-known example is the Arnold Cat map
\[ \left( \begin{array}{cc}
2 & 1 \\
1 & 1
\end{array} \right).\]
We consider $\ensuremath{\mathbb T}^2$ as the unit square with usual identifications with universal cover $\ensuremath{\mathbb R}^2$. $T$ preserves the Haar measure $\mu$ on $\ensuremath{\mathbb T}^2$ and
has exponential decay of correlations for Lipschitz functions, in the sense that there exists $\Lambda \in (0,1)$ such that
\[
|\int \phi \circ T^n \psi d\mu -\int \phi d\mu \int \psi d\mu |\le C\|\phi\|_{Lip} \|\psi \|_{Lip} \Lambda^n
\]
where $C$ is a constant independent of $\phi$, $\psi$ and $\|.\|_{Lip}$ is the Lipschitz norm~\cite{Bowen}.
For a set $D$, we define
$d_{H}(x,D)=\inf\{d(x,y):y\in D\}$ (for Hausdorff distance) , where $d$ is the distance in ambient (usually Euclidean) metric. $\overline{D}$ denotes
the closure of $D$ and we define $D_{\epsilon}=\{x:d_{H}(x,\overline{D})\leq\epsilon\}$ is an $\epsilon$ neighborhood of $D$.
As outlined in Section~\ref{sec.energy}, the observables we consider take the form $\phi (x)=f(d_H(x,L))$ where $x=(x_1,x_2)\in \ensuremath{\mathbb T}^2$ and $L\subset \ensuremath{\mathbb T}$ is a line segment with direction vector $\hat{L}$ and finite length $l(L)$. The function $f:[0,\infty)\to\mathbb{R}$ is a
smooth monotone decreasing function. We will take $f(u)=-\log (u)$. To fix notations, we also need to later consider $\epsilon$-tubes around $\mathcal{S}$. Thus if $\mathcal{S}$ is a line, or curve,
and $\epsilon$ is small, then $\mathcal{S}_{\epsilon}$ is a thin tube.
The matrix $DT$ has two unit eigenvectors $v^{+}$ and $v^{-}$ corresponding to the respective eigenvalues
$\lambda_{+}=\lambda>1$, and $\lambda_{-}=\lambda^{-1}<1$.
We can write $\hat{L}=\alpha v^{+} + \beta v^{-}$ for some coefficients $\alpha$, $\beta$ and so $DT^n \hat{L}=\alpha \lambda_{+}^n v^{+}+ \beta \lambda_{-}^n v^{-}$.
If we let $v^{(n)}$ denote a unit vector in the direction of $DT^n \hat{L}$ and $\alpha\not =0$, $\beta = 0$ then $ v^{(n)}$ aligns with the direction $v^+$ as $n\rightarrow \infty$
If $L$ is aligned with the unstable direction, we may lift $L$ to $\hat{L}$ on a fundamental domain of the cover $\ensuremath{\mathbb R}^2$ of $\ensuremath{\mathbb T}^2$ and write $\hat{L}=\hat{p}_1+ tv^+$, $t\in [0,l(L)]$, $\hat{p}_1\in \ensuremath{\mathbb R}^2$. Thus $L=\pi (\hat{p}_1+ tv^+)$, $t\in [0,l(L)]$ where $\pi: \ensuremath{\mathbb R}^2\to \ensuremath{\mathbb T}^2$ is the usual projection $\pi : \ensuremath{\mathbb R}^2\to \ensuremath{\mathbb R}^2/\ensuremath{\mathbb Z}^2$. We write the
endpoint of $\hat{L}$ as $\hat{p}_2$, i.e. $\hat{p}_2=\hat{p}_1+l(L)v^+$. We will also identify the vectors $\pi \hat{p}_1$ and $\pi \hat{p}_2$ with the corresponding points
$p_1$ and $p_2$ in $\ensuremath{\mathbb T}^2$. Similarly if $L$ is aligned with the stable direction, we may lift $L$ to $\hat{L}$ on a fundamental domain of the cover $\ensuremath{\mathbb R}^2$ of $\ensuremath{\mathbb T}^2$ and write $L=\pi (\hat{p}_1+ tv^{-})$, $t\in [0,l(L)]$, $\hat{p}_1\in \ensuremath{\mathbb R}^2$.
Again we write the
endpoint of $\hat{L}$ as $\hat{p}_2$, i.e. $\hat{p}_2=\hat{p}_1+l(L)v^-$. We will also identify the vectors $\hat{p}_1$ and $\hat{p}_2$ with the corresponding points
they project to under $\pi$, written $p_1$ and $p_2$.
\begin{theorem}\label{thm.anosov} Let $T: \ensuremath{\mathbb T}^2 \to \ensuremath{\mathbb T}^2$ be a hyperbolic toral automorphism with positive eigenvalues
$\lambda^+=\lambda>1$, $\lambda^{-}=\frac{1}{\lambda}<1$. Let $\mu$ denote Haar measure on $\ensuremath{\mathbb T}^2$. Let $L\subset \ensuremath{\mathbb T}^2$ be
the projection $\ensuremath{\mathbb R}^2\to \ensuremath{\mathbb T}^2$ of a line segment $\hat{L}$ with finite length $l(L)$.
Define $\phi (x)=-\log (d_H(x,L))$, $\phi:\ensuremath{\mathbb T}^2 \to \ensuremath{\mathbb R}$. Define $M_n (x) =\max \{ \phi (x), \phi (Tx), \ldots, \phi (T^{n-1} (x))\}$.
Then \begin{equation}\label{eq.maxlimit1-thm}
\lim_{n\to\infty}\mu(M_n\leq y+\log n+l(L))= \exp\{-\theta e^{-y}\}.
\end{equation}
where the extremal index $\theta$ is determined by these cases. If:
\begin{enumerate}
\item $L$ is not aligned with the stable $v^{-}$ or unstable $v^{+}$ direction then $\theta=1$.
\item $L$ is aligned with the unstable direction $v^{+}$ and $\pi (\hat{p}_1+tv^{+})$, $-\infty < t < \infty$ contains
no periodic points then $\theta=1$.
\item If $L$ is aligned with the stable direction $v^{-}$ and $\pi (\hat{p}_1+tv^{-})$, $-\infty < t < \infty$ contains
no periodic points then $\theta=1$.
\item $L$ is aligned with the stable $v^{-}$ or unstable $v^{+}$ direction and $L$ contains a periodic point of
prime period $q$ then $\theta=1-\lambda^{-q}$.
\item\label{extremal-range1} $L$ is aligned with the unstable direction $v^{+}$, $L$ contains no periodic
points but $\pi (\hat{p}_1+tv^{+})$, $-\infty < t < \infty$ contains a periodic point $\zeta$ of prime period $q$; then
$L \cap T^qL=\emptyset$ implies $\theta=1$; otherwise if $ L \cap T^qL\neq \emptyset$ then
$(1-\lambda^{-q}) \le \theta \le 1$ and all values of $\theta$ in this range can be realized depending on the length and placement of $L$;
\item\label{extremal-range2} $L$ is aligned with the stable direction $v^{-}$, $L$ contains no periodic
points but $\pi (\hat{p}_1+tv^{-})$, $-\infty < t < \infty$ contains a periodic point of prime period $q$;
then $L \cap T^qL=\emptyset$ implies $\theta=1$; otherwise if $ L \cap T^qL\neq \emptyset$ then
$(1-\lambda^{-q}) \le \theta \le 1$ and all values of $\theta$ in this range can be realized depending on the length and placement of $L$.
\end{enumerate}
\end{theorem}
\begin{remark}
For cases \ref{extremal-range1}, and \ref{extremal-range2} we may
realize any value of $\theta$ in the range $[(1-\lambda^{-q}),1]$. This will be demonstrated in the proof, where the value of $\theta$ is given as
a function of the locations of $\pi \hat{p}_1$ and $\pi \hat{p}_2$ relative to the period-$q$ point on a continuation of $L$. This formula is difficult to
state in an elegant way in full generality.
\end{remark}
\begin{remark}
In Theorem \ref{thm.anosov} we have focused on
the particular case $f(u)=-\log u$ which gives rise to a Gumbel distribution. For other
functional forms, such as $f(u)=u^{-\alpha}$, $(\alpha>0)$ we obtain
corresponding limit laws.
\end{remark}
\begin{remark}
Since all periodic points of $T$ have rational coefficients $(\frac{p_1}{q_1}, \frac{p_2}{q_2})$ and $v^+$,$v^-$ have irrational slopes it follows that if $\pi (\hat{p}_1+tv^{+})$, $-\infty <t <\infty$ contains a periodic point it contains at most one, and similarly for $\pi (\hat{p}_1+tv^{-})$, $-\infty <t <\infty$.
\end{remark}
Using exponential decay of correlations
of the map, we show that for small $\epsilon$-tubes $L_{\epsilon}$
around $L$, we have (for all $j$ sufficiently large)
$\mu (T^jL_{\epsilon}\cap
L_{\epsilon}) \leq C\mu (L_{\epsilon})^2.$ This enables
us to easily verify the form of the $D(u_n)$ condition of Leadbetter et al~\cite{LLR} that we use.
The argument in the case that $L$ is aligned with $v^{+}$ turns out to be the most subtle. We need a detailed analysis of how the forward images $T^j L$
wrap around the torus. It is clear that these forward images are dense, but
we need quantitative information on how quickly these images
become uniformly distributed. Such considerations are not necessary in the case
where $\mathcal{S}$ is a single point, e.g. as discussed in \cite{Dichotomy}, and furthermore
this scenario is not easily captured by axiomatic approaches, as discussed in \cite{CC, Haydn_Vaienti}.
The close alignment of $\mathcal{S}$ with the unstable manifold appears non-generic in this hyperbolic toral automorphism example.
However, for general observables one could imagine level set geometries failing transversality conditions generically,
e.g. if $\{\phi>u_n\}$ has a non-trivial boundary, which perhaps coils or accumulates upon itself. These scenarios would have
to be treated on a case by case basis.
\subsection{Sinai dispersing billiards maps}\label{sec.billiards}
We now consider another setting in which it is natural to have a smooth observable maximized on a line segment.
Suppose $\Gamma = \set{\Gamma_i, i = 1:k}$ is a family of pairwise disjoint, simply connected $C^3$ curves with strictly positive curvature
on the two-dimensional torus $\mathbb{T}^2$. The billiard flow $B_t$ is the dynamical system
generated by the motion of a point particle in $Q= \mathbb{T}^2/(\cup_{i=1}^k (\mbox{ convex hull of } \Gamma_i))$ which moves with constant unit velocity inside $Q$
until it hits $\Gamma$, then it undergoes an elastic collision where angle of incidence equals angle of reflection.
If each $\Gamma_i$ is a circle and the system is lifted periodically to $\ensuremath{\mathbb R}^2$ then this system is called a periodic Lorentz gas and was a model in the pioneering work of Lorentz on electron motion in conductors.
It is often easier to consider the billiard map $T: \partial Q \to \partial Q$, derive statistical properties for it and then
deduce corresponding properties for the flow. In this paper we will focus on limit laws for the billiard map. Let $r$ be the natural one-dimensional coordinate of
$\Gamma$ given by arc-length and let $n(r)$ be the outward normal to $\Gamma$ at the point $r$. For each
$r\in \Gamma$ the tangent
space at $r$ consists of unit vectors $v$ such that $(n(r),v)\ge 0$. We identify such a unit vector $v$ with an
angle $\vartheta \in [-\pi/2, \pi/2]$. The phase space $M$ is then parametrized by $M:=\partial Q=\Gamma\times [-\pi/2, \pi/2]$, and $M$ consists of the
points $(r,\vartheta)$. $T:M\to M$ is the Poincar\'e map that gives the position and angle $T(r,\vartheta)=(r_1,\vartheta_1)$ after a point $(r,\vartheta)$
flows under $B_t$ and collides again with $M$, according to the rule angle of incidence equals angle of reflection. The billiard map preserves a measure $d\mu=c_{M} \cos \vartheta dr d\vartheta$ equivalent to $2$-dimensional Lebesgue measure $dm=drd\vartheta$ with
density $\rho (x)=c_M \cos \vartheta$ where $x=(r,\vartheta)$.
For this class of billiards the stable and unstable foliations lie in strict cones $C^u$ and $C^s$ in that
the graphs $\vartheta(r)$ of local unstable manifolds have uniform bounds on the slopes of their tangent vectors which lie in the cone $C^u$, $s_0 \le \frac{d\vartheta}{dr} \le s_1$, and similarly tangents to local stable manifolds
lie in a cone $C^s$, $- t_1 \le \frac{d\vartheta}{dr} \le -t_0$, for some strictly positive constants $s_0,t_0,s_1,t_1$.
We will assume a line segment $L$ with direction vector $\hat{L}$ is uniformly transverse to
$C^s$ and $C^u$.
More precisely, we will consider functions maximized on line segments $L=\{x=(r,\vartheta): x\cdot v=c\}$, $v=(v_1,v_2)$, which are transverse to the stable and unstable cone of directions. For example the line segment $r=r_0$, which
is a position on the table rather than the point $(r_0,\vartheta_0)$ (which is in phase space). We note that~\cite{Pene_Saussol}
studied distributional and almost sure return time limit laws to the position $r=r_0$. In our setting the precise extreme law (Weibull, Fr\'echet or Gumbell)
depends upon the observable we choose but results may be transformed from one observable to another in a standard way. We will take the function
$\phi (r,\vartheta)=1-d_H (x,L)$ which because it is bounded will lead to a Weibull distribution. We assume the finite horizon condition, namely that the time of flight of the billiard flow between collisions is bounded above and also away from zero.
Under the finite horizon condition Young~\cite{Y98} proved that the billiard map has exponential decay of correlations for H\"{o}lder observables.
A good reference for background results for this section are the papers~\cite{BSC1,BSC2,CM07,Y98} and the book~\cite{CM}.
Let $L$ be a line segment transverse to the stable and unstable cones and $\phi (r,\vartheta)=1-d_H(x,L)$. Let
$y>0$. We define a sequence $u_n(y)=y/a_n+b_n$ by the requirement $n \mu \{ \phi > u_n(y)\}=y$. Apart from complication arising from the
invariant measure having a cosine term, $a_n$ scales like $\frac{1}{n}$.
The set $\{ \phi > u_n\}$ is a rectangle $U_n$ with center $L$ roughly of width $\frac{Cy}{ n}$ for some constant $C$. Note that we assume $L$ is not aligned in either the
unstable or the stable direction, so the following result is expected from the hyperbolic toral automorphism case.
\begin{theorem}\label{thm:billiards}
Let $T:M\to M$ be a planar dispersing billiard map with invariant measure $d\mu=c_{M} \cos \vartheta dr d\vartheta$. Suppose $x=(r,\theta)$ and $\phi (x)=1- d_H(x,L)$ where $\hat{L}$ is not in the
unstable cone $C^u$ or the stable cone $C^s$.
Let $M_n (x)=\max\{ \phi (x), \phi \circ T(x), \ldots, \phi \circ T^{n-1} (x)\}$.
Then $\mu (M_n \le u_n(y))\to e^{-y}$ as $n\to \infty$. In particular the extreme index $\theta=1$.
\end{theorem}
\begin{remark}
We now make some remarks on what we conjecture in the case that a $C^2$ curve $L$
is contained in, i.e. a piece of, a local unstable or local stable manifold and $\phi (x)=1- d_H(x,L)$. If $L$ is part of a local unstable manifold and $T^n L$ has no self-intersections with $L$ then the extremal index is one. The proofs we give in the case of the hyperbolic toral automorphism for this scenario break down but the techniques of the recent preprint~\cite{Fan_Yang} probably extend to this case. If $L$ contains a periodic point $\zeta$ of period $q$ then the extremal index would be roughly $\theta\sim 1-\frac{1}{|DT_u (\zeta)|^q}$ where $DT_u(\zeta)$ is the expansion in the unstable direction at $\zeta$ with a correctional factor due to the
conditional measure on the unstable manifold which contains $L$. If $L$ does not contain a periodic point but its continuation in the unstable manifold does contain
a periodic point of period $q$ then as in case (5) of Theorem 2.1, if $T^qL\cap L=\emptyset$ then $\theta=1$, otherwise we expect $\theta$
to lie roughly in the range $1-\frac{1}{|DT_u (\zeta)|^q} \le \theta \le 1$ (with all values of $\theta$ being realizable depending on the length and placement of $L$). If $L$ is part of a local stable manifold and $T^n L$ has no self-intersections with $L$ then the extremal index $\theta =1$. If $L$ contains a periodic point $\zeta$ of period $q$ then the extremal index would be roughly $\theta\sim 1- |DT_s (\zeta)|^q$ where $DT_s(\zeta)$ is the expansion in the stable direction at $\zeta$. If $L$ does not contain a periodic point but its continuation in the unstable manifold does contain
a periodic point of period $q$ then as in case (6) of Theorem 2.1, if $T^qL\cap L=\emptyset$ then $\theta=1$, otherwise we expect $\theta$
to lie roughly in the range $1- |DT_s (\zeta)|^q \le \theta \le 1$ (with all values of $\theta$ being realizable depending on the length and placement of $L$).
\end{remark}
\subsection{Coupled systems of uniformly expanding maps.}\label{sec.coupled}
Now we consider a simple class of coupled mixing expanding maps of the unit interval, similar to those examined in~\cite{sandro_coupled}. In fact we were
motivated by the comprehensive work of~\cite{sandro_coupled} (which uses sophisticated transfer operator techniques) to develop in this paper an alternate probabilistic approach in a coupled maps setting. The recent preprint~\cite{Haydn_Vaienti} presents similar results to ours in the case of returns to the diagonal
$\{ x_1=x_2=\ldots =x_n\}$.
Let $T$ be a $C^2$ uniformly expanding map of $S^1$ and suppose that $T$ has an invariant measure $\mu$ with density $h$ bounded above and below from zero. In ~\cite{sandro_coupled} piecewise
$C^2$ expanding maps were considered but we will limit our discussion to smooth maps.
We use all-to-all coupling and first discuss the case of
two coupled maps for clarity.
Let $0<\gamma<1$ and define
\begin{equation}\label{2maps}
F(x,y)= ((1-\gamma)Tx + \frac{\gamma}{2} (Tx+Ty),(1-\gamma)Ty + \frac{\gamma}{2} (Tx+Ty) )
\end{equation}
so that $F: \ensuremath{\mathbb T}^2\to \ensuremath{\mathbb T}^2$. We assume that $F$ has an an invariant measure $\mu$ on $\ensuremath{\mathbb T}^2$ with density $\tilde{h}$ on $\ensuremath{\mathbb T}^2$ bounded
above and also bounded below away from zero almost surely. We will require also that there exists $\epsilon>0$ and $0<\alpha \le 1$ such that
\[
|\tilde{h}|_{\alpha} :=\esssup_{0<\epsilon <\epsilon_0,x\in\ensuremath{\mathbb T}^2} \frac{1}{\epsilon^{\alpha}} \int {\it osc}(h,B_{\epsilon} (x))dm <\infty
\]
where ${\it osc}(h,A)=\esssup_{x\in A}-\essinf_{x\in A}$ for any measurable set $A$. The semi-norm $|.|_{\alpha}$ and this notion of regularity was described in~\cite{sandro_coupled}
and established in several of their examples. An invariant density for $F$ cannot reasonably be assumed to be continuous or Lipschitz. For example a slight perturbation of the doubling map
of the unit circle $T(x)= (2x)$ (mod $1$) to the map $T(x)= ((2+\epsilon)x)$ (mod $1$) gives rise to a map with invariant density which is of bounded variation but not Lipschitz or even continuous.
$|.|_{\alpha}$ can be completed to a norm $\|.\|_{{\it osc},\alpha}$ by defining $\|.\|_{{\it osc},\alpha}=|.|_{\alpha}+\|.\|_{1}$. The value of $\epsilon_0$ and $\alpha$ does not matter in our subsequent discussion. We note that the bounded variation norm and the quasi-H\"older norm $\|.\|_{{\it osc},\alpha}$ are particularly suited to handle
dynamical systems with discontinuities or singularities.
We also assume a strong form of exponential decay of correlations in the sense that for
all Lipschitz $\Phi$, $L^{\infty}$ $\Psi$ on $\ensuremath{\mathbb T}^2$ there exists $C_1>0$ and $C_2>0$ such that for all $n$
\begin{equation}\label{mixing}
\Theta_n(\Phi,\Psi):=\left|\int \Phi\cdot\Psi \circ F^n d\mu-\int \Phi~d\mu\,\int \Psi~d\mu\right| \le C_1 e^{-C_2 n}
\| \Phi\|_{\mathrm{Lip}} \|\Psi \|_{\mathrm{\infty}},
\end{equation}
where $\|\cdot\|_{\mathrm{Lip}}$ denotes the Lipschitz norm and $\|.\|_{\infty}$ denotes the $L^{\infty}$ norm. We note that this assumption is not made for (and does not hold for) hyperbolic toral automorphisms or Sinai dispersing billiards.
The function $\Theta_n(\Phi,\Psi)$ is called the correlation function.
Let $\phi (x,y)=-\log |x-y|$, a function maximized on the line segment (or circle) $L=\{(x,y): y=x\}$. In this setting $L$ is invariant under $F$ and the orthogonal direction to
$L$ is uniformly repelling. Note that the projection of $(x,y)$ onto $L$ is the point $(\frac{x+y}{2},\frac{x+y}{2})$ and the projection on $L^{\perp}$ is $(x-\frac{x+y}{2},y-\frac{x+y}{2})$. Close to
$L$ we have uniform expansion away from $L$ in the $L^{\perp}$ direction under $F$.
This is because $y-x \mapsto (1-\gamma)[Ty-Tx]$ under $F$ so writing $y-x=\epsilon$ we see $\epsilon \rightarrow (1-\gamma)[T(x+\epsilon)-T x]\sim (1-\gamma) DT(x) \epsilon+O(\epsilon^2)$. There is uniform
repulsion away from the invariant line $L$. This observation simplifies many of the geometric arguments we use
to establish extreme value laws.
In the more general case of $m$-coupled maps we define
\begin{equation*}
F(x_1,x_2,\ldots, x_m):=\left(F_1(x_1,x_2,\ldots, x_m),\ldots, F_m(x_1,x_2,\ldots, x_m)\right),
\end{equation*}
with
\begin{equation}\label{m-map}
F_j(x_1,x_2,\ldots, x_m)
= (1-\gamma)T(x_j) + \frac{\gamma}{m} \sum_{k=1}^{m} T(x_k),
\end{equation}
for $j\in[1,\ldots, m]$. For these maps, we assume:
\begin{itemize}
\item[(A)] there exists a mixing invariant measure $\mu$ with density $\tilde{h}$, $\|\tilde{h}\|_{{\it osc},\alpha}<\infty$, on $\ensuremath{\mathbb T}^m$ bounded
above and below away from zero;
\item[(B)] exponential mixing for Lipschitz functions versus $L^{\infty}$ functions as in Equation~\ref{mixing}.
\end{itemize}
\begin{remark}
Using the spectral analysis of the transfer operator of this system as in~\cite{sandro_coupled} and standard perturbation theory it can be shown that
(A) and (B) hold if $\gamma$ is sufficiently small as the uncoupled system is uniformly expanding.
\end{remark}
We consider a function maximized on $L=\{ (x_1,x_2,\ldots ,x_m): x_1=x_2=\ldots =x_m\}$.
The component of a point or vector $x=(x_1,x_2,\ldots, x_m)$ orthogonal to $L$ is
$x^{\perp}=(x_1-\bar{x}, x_2-\bar{x}, \ldots, x_m -\bar{x})$ where $\bar{x}=\frac{1}{m}\sum_{j=1}^m x_j$. We define $\|(x_1,x_2,\ldots, x_m )\|=\max_j |x_j|$ and define for $x=(x_1,x_2,\ldots, x_m)$
\[
\phi (x)=-\log (\|x^{\perp}\|).
\]
The function $\phi$ is maximized on $L$, and large values of $\phi\circ F^n (x)$ indicate the orbit of $x$ is close to full synchrony of the
coupled systems at time $n$. Writing $p_i=x_i -\bar{x}$ we have $\sum_{i=1}^m p_i=0$. Note if we have a vector $(\Delta p_1,
\Delta p_2,\ldots, \Delta p_m)$ orthogonal to $L$ we have $\sum_{i=1}^m \Delta p_i=0$. Thus in a sufficiently small neighborhood of $L$ we may write (for $j\in[1,\ldots,m]$)
$$F_j(x_1-\bar{x}, x_2-\bar{x},\ldots, x_m -\bar{x})
=(1-\gamma) DT\Delta p_j+\frac{\gamma}{m} \sum_{k=1}^m DT \Delta p_k +O(\max_k \Delta p_k)^2, $$
\[
=(1-\gamma) DT\Delta p_j +O(\max_k \Delta p_k)^2
\]
where we have used
twice-differentiability and the fact that $\sum_{i=1}^m \Delta p_i=0$. Hence again there is uniform expansion in a sufficiently small neighborhood of $L$ in
the direction of the $n-1$ dimensional subspace orthogonal to $L$.
For $y>0$
define $u_n(y)$ by $n\mu (\phi >u_n (y))=y$, and $U_n=\{\phi >u_n (y)\}$.
It can be seen that if $F$ is a map of $\ensuremath{\mathbb T}^m$ then $u_n\sim \frac{1}{m}[\log n -\log y]$, the precise relation depends
upon the density $\tilde{h}$ of the invariant measure. The precise functional form of $\phi$ is not important
as a different choice of $\phi$ would lead to a different scaling.
\begin{theorem}\label{thm:coupled}
Let $F:\ensuremath{\mathbb T}^m\to \ensuremath{\mathbb T}^m$ be a coupled system of expanding maps satisfying (A) and (B). Define $p^{\perp}=(x_1-\bar{x}, x_2-\bar{x}, \ldots, x_m -\bar{x})$ where $\bar{x}=\frac{1}{m}\sum_{j=1}^m x_j$ Suppose $\phi (p)=-\log (\|p^{\perp}\|)$.
Let $M_n (x)=\max\{ \phi (x), \phi \circ F(x), \ldots, \phi \circ F^{n-1} (x)\}$.
Then $\mu (M_n \le u_n(y))\to e^{-\theta y}$ as $n\to \infty$
where
\[
\theta=1-[\int_L \frac{1}{[(1-\gamma)DT(x)]^{m-1}} \tilde{h}(x)dx ].
\]
\end{theorem}
We may also consider blocks of synchronization, as in~\cite[Section 7.2]{sandro_coupled} where we take the observable maximized on a set $L$
consisting of synchrony on subsets of distinct lattice sites, for example of form $L=\{ (x_1,x_2,\ldots, x_m): x_{i_1}=x_{i_2}=\ldots =x_{i_k}, x_{j_1}=x_{j_2}=\ldots x_{j_l} \}$.
The main purpose of this section is to illustrate our geometric approach, so we will give one result of this type.
\begin{theorem}\label{thm:block}
Let $F:\ensuremath{\mathbb T}^m\to \ensuremath{\mathbb T}^m$ be a coupled system of expanding maps satisfying $(A)$ and $(B)$. Let $0<k\le m$ and choose
$k$ distinct lattice sites $x_{i_1}$, $x_{i_2}$,$\ldots$, $x_{i_k}$. Define the subspace
$L=\{ (x_1,x_2,\ldots, x_m): x_{i_1}=x_{i_2}=\ldots x_{i_k}\}$ of dimension $m-k+1$ and $\bar{x}=\frac{1}{k}\sum_{j=1}^k x_{i_j}$.
Suppose $\phi (p)=-\log (\max_{j=1,\ldots ,k} |x_{i_j}-\bar{x}|)$.
Let $M_n (x)=\max\{ \phi (x), \phi \circ F(x), \ldots, \phi \circ F^{n-1} (x)\}$.
Then $\mu (M_n \le u_n(\tau))\to e^{-\theta\tau}$ as $n\to \infty$
where
\[
\theta=1-[\int_L \frac{1}{[(1-\gamma)DT(y)]^{k-1}} \tilde{h}(y)dy ]
\]
where $y$ is the natural co-ordinatization of the $m-k+1$ dimensional subspace $L$.
\end{theorem}
\section{Extreme value scheme of proof}\label{sec.extremeproof}
Our proofs are based on ideas from extreme value theory. We will use two conditions, adapted to the dynamical setting, introduced in the important work~\cite{FFT5}
that are based on $D(u_n)$ and $D_2 (u_n)$ but also allow a computation of the extremal index.
Let $X_n=\phi\circ T^n$ and define
\[
A_n^{(q)} :=\lbrace X_0>u_n, X_1 \le u_n,\ldots, X_q\leq u_n\rbrace
\]
For $s,l \in \mathbb{N}$ and a set $B\subset M$, let
\[
\mathscr{W}_{s,l}(B)=\bigcap_{i=s}^{s+l-1} T^{-i}(B^c).
\]
Next we describe the two conditions introduced in~\cite{FFT5}.\\
\noindent {\textbf{ Condition $\DD_q(u_n)$}}: We say that $\DD_q(u_n)$ holds for the sequence $X_0,X_1,\ldots$ if, for every $\ell,t,n\in \mathbb{N}$
\[
\left|\mu\left(A_n^{(q)}\cap
\mathscr{W}_{t,\ell}\left(A_n^{(q)}\right) \right)-\mu\left(A_n^{(q)}\right)
\mu\left( \mathscr{W}_{0,\ell}\left(A_n^{(q)}\right)\right)\right|\leq \gamma(q,n,t),
\]
where $\gamma(q,n,t)$ is decreasing in $t$ and there exists a sequence $(t_n)_{n\in \mathbb{N}}$ such that $t_n=o(n)$ and
$n\gamma(q,n,t_n)\to0$ when $n\rightarrow\infty$.\\
We consider the sequence $(t_n)_{n\in\ensuremath{\mathbb N}}$ given by condition $\DD_q(u_n)$ and let $(k_n)_{n\in\ensuremath{\mathbb N}}$ be another sequence of integers such that as $n\to\infty$,
\[
k_n\to\infty\quad \mbox{and}\quad k_n t_n = o(n).
\]
\noindent {\textbf{ Condition $\DD'_q(u_n)$}}: We say that $\DD'_q(u_n)$
holds for the sequence $X_0,X_1,\ldots$ if there exists a sequence $(k_n)_{n\in\ensuremath{\mathbb N}}$ as above and such that
\[
\lim_{n\rightarrow\infty}\,n\sum_{j=q+1}^{\lfloor n/k_n\rfloor}\mu\left( A_n^{(q)}\cap T^{-j}\left(A_n^{(q)}\right)
\right)=0.
\]
We note that, taking $U_n:=\{ X_0>u_n\}$ for $A_n^{(q)}$, which corresponds to non-periodic behavior, in condition $\DD'_q(u_n)$ corresponds to condition $D'(u_n)$ from~\cite{LLR}. We will abuse notation and consider
$ U_n:=\{ X_0>u_n\}$ as the case of $A_n^{(q)}$ with $q=0$.
Now let
\[
\theta=\lim_{n\to\infty}\theta_n=\lim_{n\to\infty}\frac{\mu(A^{(q)}_n)}{\mu(U_n)}.
\]
\begin{remark}
In a dynamical setting verifying these two conditions picks up the main underlying periodicity or more generally recurrence properties of the system, for example
returns to a periodic point of prime period $q$, and determines the extremal index. However, as we demonstrate, other recurrent phenomena may give rise to an extremal index not equal to unity.
We show below that the self-intersection of a line segment $L$, $T(L)\cap L\not = 0$ (none of whose points are periodic) may lead to a nontrivial extremal index
for functions maximized on $L$. For a more detailed discussion of extremal indices see~\cite{FFT5}.
\end{remark}
From \cite[Corollary~2.4]{FFT4}, it follows that to establish Theorem 2.1 it suffices to prove conditions $\DD_q(u_n)$ and $\DD'_q(u_n)$ for $q=0$ in the
non-recurrent case $\theta =1$ and for
$q>0$ corresponding to the `period' of the cases where there is some recurrence phenomena ($\theta <1$).
In both cases
\[
\lim_{n\to\infty}\mu(M_n\leq u_n (y))= e^{-\theta y}.
\]
The scheme of the proof of Condition $\DD_q(u_n)$ is itself somewhat standard~\cite{Dichotomy,FHN} and is a consequence of suitable decay of
correlation estimates. We outline it for completeness,
indicating the modifications that need to be made for the different geometries of $A^{(q)}_n$.
The main work will be in establishing Condition $\DD'_q(u_n)$.
\subsection{Proof of Theorem~\ref{thm.anosov}}
In the first instance we check condition $\DD_q(u_n)$. We recall some useful statistical properties of hyperbolic toral automorphisms.
In the case where $\Phi$ and $\Psi$ are Lipschitz continuous functions, it is known for hyperbolic toral automorphisms that there exists $C>0$,
$\tau_0\in(0,1)$ such that
\begin{equation}\label{eq.cor-lip}
\left|\int \Phi (\Psi\circ T^n) d\mu-\int \Phi d\mu \int \Psi d\mu \right| \leq C\tau_{0}^n\|\Phi\|_{\mathrm{Lip}}\|\Psi\|_{\mathrm{Lip}},
\end{equation}
Furthermore if $\Psi$ is constant on local stable leaves corresponding to a Markov partition,
then the Lipshitz norm of $\Psi$ on the right-hand side of equation (\ref{eq.cor-lip}) can be replaced by the
$L^{\infty}$ norm~\cite[Section 4]{Y98}. This fact will be useful when checking $\DD_q(u_n)$, see Proposition \ref{prop.ddq} in Section
\ref{sec.ddq} below.
Consider now a set $D$, whose boundary $\partial D$ is a union
of a finite number of smooth curves, so that $\mu(\partial D)=0$. Let $W^{s}_{1}(x)$ denote the local stable manifold through $x$.
We define,
\begin{equation}\label{eq.Hk}
H_{k}(D):= \left\{x \in D:T^{k}(W^s_1(x))\cap \partial D\ne\emptyset\right\}.
\end{equation}
In Section \ref{sec.ddq} we show roughly that $\mu(H_{k} (D))$ decreases exponentially in $k$.
\subsection{Checking condition $\DD_q (u_n)$}\label{sec.ddq}
This argument is a minor adjustment of similar estimates in ~\cite{Dichotomy,FHN}. We state the following proposition.
\begin{proposition}\label{prop.ddq}
For every $\ell,t,n\in \mathbb{N}$, there exists $\lambda_0\in(0,1)$, and $C>0$ such that
\[
\left|\mu\left(A_n^{(q)}\cap
\mathscr{W}_{t,\ell}\left(A_n^{(q)}\right) \right)-\mu\left(A_n^{(q)}\right)
\mu\left( \mathscr{W}_{0,\ell}\left(A_n^{(q)}\right)\right)\right|\leq C(n^{-2}+n^2\lambda^{t}_0).
\]
\end{proposition}
Condition $\DD_q (u_n)$ immediately follows from this. We can take $t_n=(\log n)^5$ so that
$n\gamma(q,n,t)\to 0$.
The proof of Proposition~\ref{prop.ddq} is as follows. To check condition $\DD_q (u_n)$ we use decay of correlations. The main problem
in estimating the correlation function $\Theta_n(\Phi,\Psi)$ (recall equation \eqref{mixing}) is that the relevant indicator
functions $\Phi=1_{A^{(q)}_n}$ and $\Psi=1_{\mathscr W_{0,\ell}\left(\ensuremath{A^{(q)}}_n\right)}$ of the sets $A^{(q)}_n$ and
$\mathscr W_{0,\ell}\left(\ensuremath{A^{(q)}}_n\right)$ are not Lipschitz continuous.
Standard smoothing methods can be used to approximate $\Phi$, but $\Psi$ cannot be uniformly approximated
by a Lipschitz function: the level set $\Psi=1$ has a geometry that becomes increasingly complex (i.e. with multiple connectivity)
as $\ell$ increases. Fortunately, we can employ a further trick to approximate $\Psi$. This is done using a function that is constant on local stable manifolds.
This allows us to use a decay of correlations estimate using the $L^{\infty}$ norm. As part of this approximation we first estimate
$\mu(H_k(D))$ with $D=A^{(q)}_n$. The geometry of the set $A^{(q)}_n$ will be important in calculating this estimate.
\begin{lemma}\label{lemma:annulus1}
Consider the set $D=A^{(q)}_n$. Then there exists $C>0$ such that, for all $k$,
\begin{equation}\label{annulus}
\mu(H_{k}(D))\le C \lambda^{-k},
\end{equation}
where $\lambda^{-1}<1$ is the (uniform) contraction rate along the stable manifolds for the hyperbolic toral automorphism.
\end{lemma}
\begin{proof} We follow~\cite[Proposition 4.1]{Dichotomy}, and consider also the geometrical properties of $D$.
Since the local stable manifolds contract uniformly there exists
$C_1>0$ such that $\ensuremath{\text{dist}}\,(T^n(x), T^n(y))\le C_1 | \lambda|^{-n}$ for all
$y\in W^s_1(x).$ This implies that $|T^k(W_1^s(x))|\le C_1\lambda^{-k}$. Therefore, for every $x\in H_{k}(D)$,
the leaf $T^k(W^s_1(x))$ lies in an tubular region of width $2 /|\lambda |^k$ around $\partial D$. To measure of the size of
this tube we note that $m(D_{\epsilon})\leq \epsilon C_{D}$, where $C=\epsilon c_q\ell_D$. (Again recall the definition
of the tubular region $D_{\epsilon}$ given in Section \ref{sec.anosov}).
The constant $c_q$ depends on
the number of connected components of $A^{(q)}_n$, (which is bounded), and $\ell_D$ is the maximum length of a connected component
of $\partial D$. This is also bounded, since $\partial D$ is formed of straight lines of bounded length. The lemma follows by taking
$\epsilon=\lambda^{-k}$.
\end{proof}
The next lemma also holds for $\{X_0>u_n\}$ in place of $A_n^{q} $, and the proof is the same as~\cite[Lemma 4.2]{Dichotomy}.
Again we give the main steps, indicating the role of Lemma \ref{lemma:annulus1}.
The constant $\tau_1$ in the next lemma comes from the exponential decay of correlations of Lipschitz observables on hyperbolic toral automorphisms.
\begin{lemma}\label{lemma:dun-prelim}
Suppose
$\Phi:M\to \ensuremath{\mathbb R}$ is a Lipschitz map and $\Psi$ is the indicator function
\[
\Psi:= \ensuremath{{\bf 1}}_{\mathscr W_{0,\ell}\left(\ensuremath{A^{(q)}}_n\right)}.
\]
Then there exists $0<\tau_1<1$ such that for all $j\geq 0$
\begin{equation}
\left|\int\Phi\,(\Psi\circ T^j)\, \text{d}\mu - \int\Phi\text{d}\mu \int\Psi\text{d}\mu \right|\le \ensuremath{\mathcal C}\,\left(\|\Phi\|_\infty \,
\lambda^{-\floor{j/2}}+\|\Phi\|_{\text{Lip}}\,\,\tau_{1}^{\floor{j/2}}\right).
\end{equation}
\end{lemma}
\begin{proof}
Following Lemma \cite[Lemma 4.2]{Dichotomy}, we take a version $\overline{\Psi}$ of $\Psi$ that is constant on local stable manifolds,
for example by taking a distinguished point $x^*$
on each local stable manifold $W^s_1 (x)$ and defining $\overline{\Psi}(y) =\Psi (x^*)$ for all $y\in W^s_1 (x)$.
We let $\Psi_j=\Psi\circ T^j$, and again denote $\overline{\Psi}_j$ as the relevant version of $\Psi_j$ (constant on local stable manifolds).
A simple application of the triangle inequality gives the following bound:
\begin{equation}
\Theta_j(\Phi,\Psi)\leq C\left(\|\Phi\|_{\infty}\mu \{\overline{\Psi}_{j/2}\neq\Psi_{j/2}\}+\|\Phi\|_{\mathrm{Lip}}\tau_{0}^{j/2}\right),
\end{equation}
(recall that $\Theta_j$ is defined in equation \eqref{mixing}).
To estimate $\mu \{\overline{\Psi}_{j/2}\neq\Psi_{j/2}\}$, we consider points $x_1,x_2$ on the same stable manifold, and such that
$x_1\in \mathscr W_{i,\ell}\left(\ensuremath{A^{(q)}}_n\right)$, but $x_2\not\in\mathscr W_{i,\ell}\left(\ensuremath{A^{(q)}}_n\right)$, (for $i\geq j/2$).
This set is contained in $\cup_{k=i}^{i+\ell-1}H_k(A^{q}_n)$. Hence
\begin{equation*}
\mu \{\overline{\Psi}_{j/2}\neq\Psi_{j/2}\}\leq\sum_{k=j/2}^{\infty}H_k(A^{(q)}_n)\leq C\lambda^{-j/2}.
\end{equation*}
The conclusion of Lemma \ref{lemma:dun-prelim} follows.
\end{proof}
To continue with the proof of Proposition \ref{prop.ddq}, and hence verify condition $\DD_q (u_n)$, we approximate the characteristic function of the set $\ensuremath{A^{(q)}}_n$ by a suitable Lipschitz function. The key estimate is to bound the Lipschitz norm of the approximation.
Let $A_n=\ensuremath{A^{(q)}}_n$ and $D_n:=\set{x\in \ensuremath{A^{(q)}}_n:\; d_H\left(x, \overline{A_n^c}\right)\geq n^{-2}},$
where $\bar A_n^c$ denotes the closure of the complement of the set $A_n$.
Define $\Phi_n:\mathcal{X}\to\ensuremath{\mathbb R}$ by
\begin{equation}
\label{eq:Lip-approximation}
\Phi_n(x)=\begin{cases}
0&\text{if $x\notin A_n$}\\
\frac{ d_H(x,A_n^c)}{d_H(x,A_n^c)+
d_H(x,D_n)}&
\text{if $x\in A_n\setminus D_n$}\\
1& \text{if $x\in D_n$}
\end{cases}.
\end{equation}
Note that $\Phi_n$ is Lipschitz continuous with Lipschitz constant given by $n^2$.
Moreover $\|\Phi_n-\ensuremath{{\bf 1}}_{A_n}\|_{L^1(m)}\leq C/n^2$ for some constant $C$.
It follows that
\begin{align}
\Big|\int \ensuremath{{\bf 1}}_{\ensuremath{A^{(q)}}_n}\,\left(\Psi_{\floor{j/2}}\circ T^{j -\floor{j/2}}\right) &\,d\mu - \mu (\ensuremath{A^{(q)}}_n)\int \Psi d\mu \Big|\nonumber\\
&\le \abs{\int \left(\ensuremath{{\bf 1}}_{\ensuremath{A^{(q)}}_n} - \Phi_n\right)\Psi_{\floor{j/2}}\,d\mu }+\ensuremath{\mathcal C} \left(\|\Phi_n \|_\infty \,\,j^2\,\,\lambda^{\floor{j/4}}+\|\Phi_n \|_{\text{Lip}}\,\,\tau_{1}^{\floor{j/2}}\right)\nonumber\\
&\quad+ \abs{\int \left(\ensuremath{{\bf 1}}_{\ensuremath{A^{(q)}}_n} - \Phi_n \right)\,d\mu \int \Psi_{\floor{j/2}}\,d\mu },
\end{align}
for some generic constant $\ensuremath{\mathcal C}$. Thus
\[
\abs{\mu \left(\ensuremath{A^{(q)}}_n\cap \mathscr W_{j,\ell}(\ensuremath{A^{(q)}}_n)\right) - \mu (\ensuremath{A^{(q)}}_n)\,\mu \left(\mathscr W_{0,\ell}(\ensuremath{A^{(q)}}_n)\right)}\le \gamma(n,j)
\] where
\[
\gamma(n,j) = \ensuremath{\mathcal C}\,\left(n^{-2}+ n^{2} \,\lambda_1^{\floor{j/2}}\right),
\] and
\[
\lambda_1 = \max\,\set{\tau_1 , \lambda^{-1}}.\]
Thus if, for instance, we choose $j=t_n=(\log n)^{5}$, then $n\gamma(n, t_n)\to 0$ as $n\to\infty.$ This completes the proof.
\subsection{Checking condition $\DD'_q(u_n)$}\label{sec.DDprime}
We make the following decomposition:
\begin{multline*}
n\sum_{j=q+1}^{\lfloor n/k_n\rfloor} \mu (A^{(q)}_n\cap T^{-j}(A^{(q)}_n))=n\sum_{j=q+1}^{R_n} \mu (A^{(q)}_n\cap T^{-j} (A^{(q)}_n))+n\sum_{j=R_n+1}^{(\log n)^5} \mu (A^{(q)}_n\cap T^{-j}(A^{(q)}_n))\\ +n\sum_{(\log n)^5+1}^{\lfloor n/k_n\rfloor} \mu (A^{(q)}_n\cap T^{-j}(A^{(q)}_n)),
\end{multline*}
where the sequence $R_n\to\infty$ (as $n\to\infty$) will be chosen later. Recall that for $q=0$,
$A^{(q)}_n=U_n$. By exponential decay of correlations and a suitable Lipschitz approximation the last sum tends to 0 as $n\to\infty$,
so it suffices to estimate the two sums where $1\le j\le (\log n)^5$.
\noindent {\bf Case $L$ transverse to stable and unstable directions.}
Fix $y$ and define $u_n(y)$ by the requirement $n \mu \{ x: \phi (x) \ge u_n (y) \}=y$. Henceforth we will drop the dependence on $y$ and write
simply $u_n$ for convenience. We define
$U_n:= \{ x: \phi (x) \ge u_n \}$. Geometrically $U_n$ resembles a parallel strip of width $\frac{2}{n}$.
We will verify the short return condition with $q=0$. Consider the set $T^{-j} U_n\cap U_n=\{ x: T^j (x) \in U_n, x\in U_n \}$. $T^{j} U_n$
is a union of parallelogram-like strips corresponding to each winding around the torus and such strip has width $O(\frac{\lfloor\lambda^{-j}\rfloor}{n})$ and
length $O(1)$, the precise constants depending on the angle between $T^j L$ and $L$ as $T^j L$ aligns to the unstable
direction. There are
approximately $\lfloor\lambda^j\rfloor$ such parallelogram strips. Each strip intersects $U_n$ in an area of measure $O(\lfloor \lambda^{-j} \rfloor n^{-2})$ by transversality. See Figure \ref{fig.anosov1}
Hence
\[
n \sum_{j=1}^{(\log n)^5} \mu ( T^{-j} U_n\cap U_n )=O\left(\frac{(\log n)^5}{n}\right).
\]
Thus the extremal index $\theta=1$.
\begin{figure}[h!]\label{fig.transverse}
\centering
\begin{minipage}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale=4]
\begin{scope}
\draw[thick] (0,0) rectangle (1,1);
\clip (0,0) rectangle (1,1);
\draw[thick,rotate=pi/2] (0,0)--(1,1);
\draw[thick,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,->,shorten >=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,<-,shorten <=3.25cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3.25cm,rotate=pi/2] (1,0)--(0,1);
\end{scope}
\node at (0.75,0.28) [right] {$v^{-}$};
\node at (0.9,0.9) [below] {$v^{+}$};
\draw[thick] (0,0.1)--(0.6,1);
\draw[thick,->,dotted] (0,0.8)--(0.53333,0);
\node at (0,0.8) [left] {$v$};
\node at (0.75,1) [above] {$x\cdot v=c$};
\draw[fill = gray,opacity=0.2] (0,0)--(0,0.2)--(0.53333,1)--(0.6667,1)--(0,0);
\draw[thick,|-|] (0,0)--(0,0.2);
\node at (0,0.1) [left] {$U_n$};
\end{tikzpicture}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\raggedright
\begin{tikzpicture}[scale=4]
\begin{scope}
\draw[thick] (0,0) rectangle (1,1);
\clip (0,0) rectangle (1,1);
\draw[thick,opacity=0.1,rotate=pi/2] (0,0)--(1,1);
\draw[thick,opacity=0.1,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=1cm,opacity=0.1,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,->,shorten >=1.25cm,opacity=0.1,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1cm,opacity=0.1,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1.25cm,opacity=0.1,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=3cm,opacity=0.1,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,<-,shorten <=3.25cm,opacity=0.1,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3cm,opacity=0.1,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3.25cm,opacity=0.1,rotate=pi/2] (1,0)--(0,1);
\draw[thick,opacity=0.3,rotate=pi/2] (0,0.1)--(0.6,1);
\draw[fill = gray,opacity=0.3,rotate=pi/2] (0,0)--(0,0.2)--(0.53333,1)--(0.6667,1)--(0,0);
\draw[opacity=0.3,rotate=pi/2] (0,0)--(0,0.2)--(0.53333,1)--(0.6667,1)--(0,0);
\draw[fill = gray, opacity=0.3,rotate=-pi/2] (0,0.3)--(0.7,1)--(1,1.25)--(0,0.25)--(0,0.3);
\draw[rotate=-pi/2] (0,0.3)--(0.7,1)--(1,1.25)--(0,0.25)--(0,0.3);
\draw[fill = gray, opacity=0.3,rotate=-pi/2] (0,0.35)--(0.65,1)--(0.95,1.25)--(0,0.3)--(0,0.35);
\draw[rotate=-pi/2] (0,0.35)--(0.65,1)--(0.95,1.25)--(0,0.3)--(0,0.35);
\end{scope}
\draw[|-|,thick] (0.53333,1)--(0.6667,1);
\node at (0.6,1) [above] {$O(\frac{1}{n})$};
\draw[|-|,thick] (0,0.3)--(0,0.35);
\node at (0,0.325) [left] {$O(\frac{\lambda^{-j}}{n})$};
\draw [<-,very thick] (0.45,0.75)--(0.65,0.55);
\node at (0.65,0.55) [below] {$T^{j} U_n\cap U_n$};
\end{tikzpicture}
\caption*{(b)}
\end{minipage}
\caption{\label{fig.anosov1} (a) The set $U_n$ and line $L$ for $L$ not aligned with $v^{-}$ or $v^{+}$. (b) Iterations $T^j U_n$ and their intersections with $U_n$.}
\end{figure}
\noindent {\bf Case $L$ aligned with unstable direction.}
We lift $L$ to $\hat{L}$ on a fundamental domain of the cover $\ensuremath{\mathbb R}^2$ of $\ensuremath{\mathbb T}^2$ and write $\hat{L}=\hat{p}_1+ tv^+$, $t\in [0,l(L)]$, $\hat{p}_1\in \ensuremath{\mathbb R}^2$. We write the
endpoint of $\hat{L}$ as $\hat{p}_2$, i.e. $\hat{p}_2=\hat{p}_1+l(L)v^+$. The points $\hat{p}_1$ and $\hat{p}_2$ project to the corresponding points
written $p_1=\pi \hat{p_1}$ and $p_2=\pi \hat{p_2} $.
There are 2 main cases, with some subcases.
Case (a): First assume that the line $\hat{p}_1+t v^+$, $-\infty < t<\infty$ contains no point with rational coordinates. This holds for
a measure one set of $\hat{p}_1$ as the set of points in the plane with rational coordinates is countable. In this case $T^nL$, $n\ge 1$, has no intersections with $L$. To see this suppose $p\in L$ and there exists an $n$ such that
$T^n p=q\in L$. If we take a line segment $\tilde{L}$ in direction $v^+$ of length $2l(L)$ centered at $p$ we see
by expansion that $\tilde{L}\subset T^n \tilde{L}$ (since $d(p,q)\le l(L)$) and hence $T^n$ restricted to $\tilde{L}$ has a fixed point $\tilde{p}$ in $\tilde{L}$. However, this implies the lift $\hat{p}_1+t v^+$, $-\infty <t<\infty$ contains a point with rational coordinates, which is a contradiction.
Since $p_1$ is not periodic by assumption and $\hat{p}_1$ is not in the direction of $v^+$ (otherwise the point $(0,0)$
would be contained in $\hat{p}_1+t v^+$, $-\infty <t < \infty$) the iterates $T^j U_n$ are disjoint for large $n$ for small $j$ i.e.
there exists $R_n \to \infty$ such that $\mu (T^{-j} U_n\cap U_n)=0$ for $j<R_n$. Corollary 2.2 of the recent preprint~\cite{Fan_Yang} implies in this case that the extremal index is one.
We include an alternate proof for completeness.
For large $n$ the set $T^j U_n$ comprises
$\lfloor \lambda^j \rfloor$ parallel rectangles (aligned with the unstable direction) of width $O(\frac{\lfloor \lambda^{-j}\rfloor}{n})$.
Identifying $\mathbb{T}^2$ with the unit square the set $T^j L\cap ([0,1]\times \{0\})$ consists of $m(j)\sim[\lambda^j]$ points $x_i^j$, $j=1,\ldots, m(j)$.
If for small iterates $T^i L$ there is no intersection with $([0,1]\times \{0\})$ we extend $T^i L$ in a straight line so that all $x_i^j$, $j=1,\ldots, m(j)$
are defined. Let $\gamma^{-1}$ denote the slope of $v^+$.
The set $\{ x^j_i\}_{i=1,\ldots, m(j)}$ is generated by the relation $x_1^j+k\gamma $(mod $1$) for $k=1,\ldots, m(j)$.
We now estimate $\mu (T^{-j} U_n\cap U_n)$. The set $T^j U_n$ has approximately $[\lambda^j]$ windings around the torus
and we now estimate the fraction of these that intersect $U_n$.
Note that $\gamma$ is a quadratic irrational. This implies that $\gamma$ has low discrepancy in the sense that there exists a constant $C>0$
such that
\[
\sup_{0\le a <b \le 1} \{ \# \{x^j_i \in (a,b)\}/\lfloor \lambda^j\rfloor - (b - a) \} \le C \frac{\log \lfloor \lambda^j\rfloor }{\lfloor \lambda^j\rfloor },
\]
see \cite{N92}.
Hence for $j> R_n$
\[
n\sum_{j=R_n}^{(\log n)^5} \mu (T^{-j} U_n\cap U_n)=O\left(\frac{1}{n}+\frac{\log [\lambda^{R_n}]}{\lambda^{R_n}}\right)=o(1).
\]
This implies that a standard EVL holds with $\theta=1$. See Figure \ref{fig.anosov2}.
Case (b): Assume that $\hat{p}_1+t v^+$, $-\infty <t<\infty$ contains a point with rational coordinates, note that it will contain at most one as the slope of
$v^+$ is irrational. Such a point projects to a point $p_{per}$ periodic under $T$ with period $q$ say.
Case (b1): Assume now that $L $ itself contains $p_{per}$, a periodic point of period $q$.
There will be only one periodic point in $L$ as the slope of $v^+$ is irrational.
Without loss of generality we take $q=1$ by considering $T^q$.
It is easy to see that $\theta=\lim_{n\to\infty}\frac{\mu(A^q_n)}{\mu(U_n)}=1-\frac{1}{\lambda^q}$.
The same discrepancy argument as in the case of no periodic orbits shows that there exists an $R_n\to \infty$ such that $T^{-j} A_n^{(q)} \cap A_n^{(q)}=\emptyset$ for $j<R_n$ and
\[
\sum_{j=R_n}^{(\log n)^5} \mu (T^{-j} A_n^{(q)} \cap A^{(q)}_n)= o\left(\frac{1}{n}\right)
\]
Hence $\theta=1-\frac{1}{\lambda^q}$. See Figure \ref{fig.anosov3}.
Case (b2): $L $ does not contain a periodic point.
We first consider the simplest case where the origin is the fixed point and $\hat{p}_1$ parallel to $v^+$ so that $\hat{p}_1+t v^+$, $-\infty <t<\infty$ contains the fixed point $(0,0)$
but $L$ does not contain $(0,0)$. The line $\hat{p}_1+ t v^+$, $0\le t<\infty$ has a natural ordering by distance from the origin $(0,0)$. If $\lambda \hat{p}_1 > \hat{p}_2$
then it is easy to see all iterates of $T^n L$ on the torus are disjoint and the arguments given in case (a) apply giving $\theta =1$.
Suppose now $\lambda \hat{p}_1 < \hat{p}_2$. We take $q=1$ and calculate
$$\theta =\mu (A_{n}^{(1)})/\mu (U_n)=|\hat{p}_2-\frac{1}{\lambda}\hat{p}_2|/|\hat{p}_2-\hat{p}_1|=(1-\frac{1}{\lambda})\frac{|\hat{p}_2|}{|\hat{p}_2-\hat{p}_1|},$$ as the
stable manifolds are sent strictly into the region of intersection $U_n\cap TU_n$. (See Figure \ref{fig.anosov4}). The condition $\lambda \hat{p}_1 <\hat{p}_2$ implies
$1< \frac{|\hat{p}_2|}{|\hat{p}_2-\hat{p}_1|} < (1-\frac{1}{\lambda})^{-1}$. By varying $\hat{p}_1$ and $\hat{p}_2$ we may obtain all values in this range. Hence
$(1-\frac{1}{\lambda})\le \theta \le 1$.
In the general case of a periodic point $p_{per}$ of period $q$ contained in $\pi (\hat{p}_1+t v^+)$, $-\infty <t<\infty$ we consider
$T^q$ and the analysis proceeds in the same way by considering the expansion on the line segment $[\hat{p}_1-\hat{p}_{per},\hat{p}_2-\hat{p}_{per}]$. We infer that for general $q\geq 1$,
$$(1-\frac{1}{\lambda^q})\le \theta \le 1$$
with all values of $\theta$ in this range being realizable.
The verification
of condition $\DD'_q(u_n)$ is similar to case (b1).
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope}
\draw[thick] (0,0) rectangle (1,1);
\clip (0,0) rectangle (1,1);
\draw[thick,rotate=pi/2] (0,0)--(1,1);
\draw[thick,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,->,shorten >=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,<-,shorten <=3.25cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3.25cm,rotate=pi/2] (1,0)--(0,1);
\node at (0.75,0.28) [right] {$v^{-}$};
\node at (0.9,0.9) [below] {$v^{+}$};
\draw[thick,rotate=pi/2] (0,0.1)--(0.9,1);
\draw[thick,<-,dotted,rotate=pi/2] (0.9,0)--(0,0.9);
\node at (0.8,0.1) [left] {$v$};
\draw[fill = gray,opacity=0.2,rotate=pi/2] (0,0)--(1,1)--(0.8,1)--(0,0.2)--(0,0);
\end{scope}
\node at (0.9,1) [above] {$x\cdot v=c$};
\draw[thick,|-|] (0,0.2)--(0,0);
\node at (0,0.1) [left] {$U_n$};
\end{tikzpicture}
\caption{\label{fig.anosov2} The set $U_n$ and line $L$ for $L$ aligned with $v^{-}$.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope
\draw[dashed] (0,-0.6)--(0,0.6);
\draw (-0.75,0)--(0.75,0);
\draw (-0.5,-0.25) rectangle (0.5,0.25);
\draw[fill=gray,opacity=0.3] (-0.65,-0.15) rectangle (0.65,0.15);
\draw[pattern = north west lines] (0.4,-0.25) rectangle (0.5,0.25);
\draw[pattern = north west lines] (-0.5,-0.25) rectangle (-0.4,0.25);
\draw[|-|] (-0.5,-0.3) -- (-0.4,-0.3);
\node at (-0.45,-0.3) [below] {$A_n^{(1)}$};
\draw[|-|] (0.4,-0.3) -- (0.5,-0.3);
\node at (0.45,-0.3) [below] {$A_n^{(1)}$};
\draw[fill = black] (0,0) circle (0.35pt);
\node at (0.75,0) [right] {$v = v^{+}$};
\node at (0,0.6) [above] {$v^{-}$};
\draw[decorate,decoration={brace,amplitude=10pt}] (-0.5,0.25)--(0.5,0.25);
\node at (0,0.3) [above] {$U_n$};
\draw[decorate,decoration={brace,amplitude=10pt}] (-0.65,-0.15)--(-0.65,0.15);
\node at (-0.7,0) [left] {$T(U_n)$};
\draw[<->] (-0.5,-0.5)--(0.5,-0.5);
\node at (0,-0.5) [above] {$1$};
\draw[<->] (-0.4,-0.55)--(0.4,-0.55);
\node at (0,-0.55) [below] {$\frac{1}{\lambda}$};
\end{scope}
\end{tikzpicture}
\caption{\label{fig.anosov3} Sketch of argument (b1) for $v$ aligned with the unstable direction and $L$ contains a periodic orbit showing intersections of $A_n^{(1)}$ (shown in patterned lines) and $T(U_n)$ (shown in gray). Estimates of the ratio of $A_n^{(1)}$ to $U_n$ (shown in white) give the value of the extremal index.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope
\draw[dashed] (0,-0.6)--(0,0.6);
\draw (-0.6,0)--(0.75,0);
\draw (-0.4,-0.3) rectangle (0.3,0.3);
\draw[fill = gray, opacity = 0.3] (-0.3,-0.15) rectangle (0.65,0.15);
\draw[pattern = north west lines] (0.2,-0.3) rectangle (0.3,0.3);
\draw[fill = black] (-0.4,0) circle (0.35pt);
\draw[fill = black] (0.2,0) circle (0.35pt);
\draw[fill = black] (0.3,0) circle (0.35pt);
\draw[decorate,decoration={brace,amplitude=10pt}] (-0.4,0.3)--(0.3,0.3);
\draw[decorate,decoration={brace,amplitude=10pt}] (0.65,0.15)--(0.65,-0.15);
\node at (-0.05,0.35) [above] {$U_n$};
\node at (0.7,0) [right] {$T(U_n)$};
\node at (-0.6,0) [left] {$v = v^{+}$};
\node at (0,0.6) [above] {$v^{-}$};
\draw [|-|] (0.2,-0.35)--(0.3,-0.35);
\node at (0.25,-0.35) [below] {$A_n^{(1)}$};
\draw[<-] (-0.4,0)--(-0.45,0.2);
\node at (-0.45,0.2) [above] {$\hat{p}_1$};
\draw[<-] (0.2,0) -- (0,0.15);
\node at (0,0.15) [above] {$T^{-1}(\hat{p}_2)$};
\draw[<-] (0.3,0) -- (0.4,0.15);
\node at (0.4,0.15) [above] {$\hat{p}_2$};
\end{scope}
\end{tikzpicture}
\caption{\label{fig.anosov4} Sketch of argument (b2) for $v$ aligned with the unstable direction and $L$ does \textit{not} contain a periodic orbit showing intersections of $A_n^{(1)}$ (shown in patterned lines) and $T(U_n)$ (shown in gray). Estimates of the ratio of $A_n^{(1)}$ to $U_n$ (shown in white) give the value of the extremal index.}
\end{figure}
\noindent {\bf Case $L$ is aligned with the stable direction.}
Suppose now that $L$ aligns with the stable direction $v^{-}$. See Figure \ref{fig.anosov5}. The analysis is similar to the
case where $L$ is aligned with the unstable direction, and again we consider the lift $\hat{L}=\hat{p}_1+ tv^-$, $t\in [0,l(L)]$,
with $\hat{p}_1\in \ensuremath{\mathbb R}^2$, and $\hat{p}_2$ denoting the other endpoint of $\hat{L}$, i.e. $\hat{p}_2=\hat{p}_1+l(L)v^-$. We will make use of the time-reversibility of the system in Case (a) below.
We have the following cases.
Case (a): First assume that the line $\hat{p}_1+t v^-$, $-\infty < t<\infty$ contains no point with rational coordinates. Let $S=T^{-1}$. Then $L$
is aligned with the unstable direction for $S$.
As in the case where $L$ aligned with the unstable direction for $T$, it follows again that
$S^n (L)$ has no intersections with $L$, for all $n\geq 1$. Hence $T^n(L)$ has no intersections with $L$ for all $n\geq 1$.
Thus all the iterates $T^j U_n$ are disjoint for small
$j$, i.e. there exists $R_n \to \infty$ such that $\mu (T^{-j} U_n\cap U_n)=0$ for $j<R_n$. Note that the definition of $U_n$ is the same for $T$ and $S$
and that $\mu (T^{-j} U_n \cap U_n)= \mu ( U_n \cap T^j U_n)=
\mu (U_n\cap S^{-j} U_n)$
by measure-preservation. The argument of Case (a) when $L$ is aligned with the unstable direction shows that
\[
n \sum_{j=R_n}^{(\log n)^5} \mu (S^{-j} U_n \cap U_n)= o(1),
\]
and hence
\[
n \sum_{j=R_n}^{(\log n)^5} \mu (T^{-j} U_n \cap U_n)= o(1),
\]
Thus $\theta=1$.
Case (b): Assume that $\hat{p}_1+t v^-$, $-\infty <t<\infty$ contains a point $\hat{p}_{per}$ with rational coordinates. There will be only
one such point as the slope of $v^{-}$ is irrational. The point $\hat{p}_{per}$ projects to a point $p_{per}$ periodic under $T$ with period $q$ say. We cannot
use time-reversibility in this case as the set $A_n^q$ depends upon the consideration of $T$ or $T^{-1}$ as the transformation.
Case (b1): Assume now that $L$ contains the periodic point $p_{per}$ of period $q$.
Without loss of generality we (again) take $q=1$ by considering $T^q$.
We have $\theta=\lim_{n\to\infty}\frac{\mu(A^{(q)}_n)}{\mu(U_n)}=1-\frac{1}{\lambda^q}$. Geometrically $A_n^{(q)}$ consists of two
strips within $U_n$. Both of these have length $1/n$, (i.e. the same as $U_n$), but their width relative to $U_n$ is
$\frac{1}{2}\left(1-1/\lambda^q\right)$. See Figure \ref{fig.anosov6}.
The same argument as in the case of no periodic orbits shows that there exists an $R_n\to \infty$ such that $T^{-j} A_n^{(q)} \cap A_n^{(q)}=\emptyset$ for $j<R_n$.
We have uniform expansion of $A_n^q$ in the unstable direction and the discrepancy argument of Case (b1) of the previous section (alignment with the
unstable direction) shows that
\[
\sum_{j=R_n}^{(\log n)^5} \mu (T^{-j} A_n^{(q)} \cap A^{(q)}_n)= o\left(\frac{1}{n}\right).
\]
We therefore have $\theta=1-\frac{1}{\lambda^q}$.
Case (b2): $L$ does not contain a periodic orbit.
Again, we illustrate by considering the simplest case of $\hat{p}_1$ parallel to $v^-$ so that $\hat{p}_1+t v^-$, $-\infty <t<\infty$ contains the fixed point $(0,0)$
but $L$ does not contain $(0,0)$. For the lifted line $\hat{p}_1+ t v^-$, $0\le t<\infty$, we use the natural ordering by distance
from the origin $(0,0)$. If $\hat{p}_1 > \lambda^{-1}\hat{p}_2$,
then all iterates of $T^n L$ on the torus are disjoint and the arguments given in case (a) apply giving $\theta =1$.
Suppose now $\hat{p}_1 < \lambda^{-1}\hat{p}_2$. We take $q=1$ and calculate
$$\theta =\frac{\mu (A_n^{(1)})}{\mu (U_n)}
=\left(1-\frac{1}{\lambda}\cdot \frac{|\hat{p}_2-\lambda^{-1} \hat{p}_1|}{\hat{p}_2-\hat{p}_1}\right).$$
See Figure \ref{fig.anosov7}.
The general case where $\hat{p}_1$ is not parallel $v^-$ proceeds the
same way by considering the expansion of $T$ orthogonal to the segment $[\hat{p}_1-\hat{p}_{per},\hat{p}_2-\hat{p}_{per}]$. We infer that for general $q\geq 1$,
$$(1-\frac{1}{\lambda^q})\le \theta \le 1$$
with all values of $\theta$ in this range being realizable.
The verification of condition $\DD'_q(u_n)$ is similar to case (b1).
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope}
\draw[thick] (0,0) rectangle (1,1);
\clip (0,0) rectangle (1,1);
\draw[thick,rotate=pi/2] (0,0)--(1,1);
\draw[thick,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,->,shorten >=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=1.25cm,rotate=pi/2] (0,0)--(1,1);
\draw[thick,black,<-,shorten <=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,<-,shorten <=3.25cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3cm,rotate=pi/2] (1,0)--(0,1);
\draw[thick,black,->,shorten >=3.25cm,rotate=pi/2] (1,0)--(0,1);
\node at (0.75,0.28) [right] {$v^{-}$};
\node at (0.9,0.9) [below] {$v^{+}$};
\draw[thick,->,dotted,rotate=pi/2] (0,0.1)--(0.9,1);
\draw[thick,rotate=pi/2] (0.7,0)--(0,0.7);
\node at (0.7,0.83) [above] {$v$};
\draw[fill = gray,opacity=0.2,rotate=pi/2] (0,0.6)--(0,0.8)--(0.8,0)--(0.6,0)--(0,0.6);
\end{scope}
\node at (0,0.7) [left] {$x\cdot v=c$};
\draw[thick,|-|] (0.6,0)--(0.8,0);
\node at (0.7,0) [below] {$U_n$};
\end{tikzpicture}
\caption{\label{fig.anosov5} The set $U_n$ and line $L$ for $L$ aligned with $v^{+}$.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope
\draw[dashed] (0,-0.6)--(0,0.6);
\draw (-0.6,0)--(0.75,0);
\draw (-0.5,-0.25) rectangle (0.5,0.25);
\draw[fill = gray, opacity = 0.3] (-0.25,-0.5) rectangle (0.25,0.5);
\draw[pattern=north west lines] (-0.5,-0.25) rectangle (0.5,-0.15);
\draw[pattern=north west lines] (-0.5,0.15) rectangle (0.5,0.25);
\draw[fill = black] (0,0) circle (0.35pt);
\node at (0.75,0) [right] {$v = v^{-}$};
\node at (0,0.6) [above] {$v^{+}$};
\draw[<->] (-0.55,0.25)--(-0.55,-0.25);
\node at (-0.55,-0.25) [below] {$1/n$};
\draw[<->] (-0.6,0.15)--(-0.6,-0.15);
\node at (-0.6,0) [left] {$\frac{1}{\lambda n}$};
\draw[|-|] (0.55,0.25) -- (0.55,0.15);
\node at (0.55,0.2) [right] {$A_n^{(1)}$};
\draw[|-|] (0.55,-0.25) -- (0.55,-0.15);
\node at (0.55,-0.2) [right] {$A_n^{(1)}$};
\draw[decorate,decoration={brace,amplitude=8pt},rotate=180] (-0.25,0.5)--(0.25,0.5);
\node at (0,-0.55) [below] {$T(U_n)$};
\draw[decorate,decoration={brace,amplitude=10pt}] (-0.5,0.25)--(0.5,0.25);
\node at (0,0.3) [above] {$U_n$};
\end{scope}
\end{tikzpicture}
\caption{\label{fig.anosov6} Sketch of argument (b1) for $v$ aligned with the stable direction and $L$ contains a periodic orbit showing intersections of $A_n^{(1)}$ (shown in patterned lines) and $T(U_n)$ (shown in gray). Estimates of the ratio of $A_n^{(1)}$ to $U_n$ (shown in white) give the value of the extremal index.}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=4]
\begin{scope
\draw[dashed] (0,-0.6)--(0,0.6);
\draw (-0.75,0)--(0.75,0);
\draw[fill = gray, opacity = 0.3] (-0.4,-0.3) rectangle (0.4,0.3);
\draw[fill=white] (0,-0.15) rectangle (0.6,0.15);
\draw[pattern = north west lines] (0,-0.15) rectangle (0.6,0.15);
\draw[<->] (0.65,0.15)--(0.65,-0.15);
\node at (0.65,0) [right] {$\frac{1}{\lambda n}$};
\node at (-0.75,0) [left] {$v = v^{-}$};
\node at (0,0.6) [above] {$v^{+}$};
\draw[fill = white] (0.3,-0.1) rectangle (0.6,0.1);
\draw[<-] (0.6,0) -- (0.6,0.45);
\node at (0.6,0.45) [above] {$\hat{p}_2$};
\draw[fill=black] (0,0) circle (0.35pt);
\draw[fill=black] (0.3,0) circle (0.35pt);
\draw[fill=black] (0.6,0) circle (0.35pt);
\draw[<-] (0.3,0) -- (0.4,-0.45);
\node at (0.4,-0.45) [below] {$T^{-1}(\hat{p}_2)=\lambda\hat{p}_2$};
\draw[<-] (0,0)--(-0.2,0.45);
\node at (-0.2,0.45) [above] {$\hat{p}_1$};
\draw[decorate,decoration={brace,amplitude=10pt}] (0,0.15)--(0.6,0.15);
\node at (0.3,0.2) [above] {$U_n$};
\draw[decorate,decoration={brace,amplitude=10pt}] (-0.4,0.3)--(0.4,0.3);
\node at (0,0.35) [above] {$T(U_n)$};
\end{scope}
\end{tikzpicture}
\caption{\label{fig.anosov7} Sketch of argument (b2) for $v$ aligned with the stable direction and $L$ does \textit{not} contain a periodic orbit. Showing intersections of $A_n^{(1)}$ (shown in patterned lines) and $T(U_n)$ (shown in gray). Estimates of the ratio of $A_n^{(1)}$ to $U_n$ (shown in white) give the value of the extremal index.}
\end{figure}
\subsection{Proof of Theorem~\ref{thm:billiards}.}\label{sec:billiards}
We will show that conditions $\DD_q(u_n)$ and $\DD'_q (u_n)$ hold with $q=0$ so that the extremal index $\theta=1$. We shall drop the subscript $q$ in this section. The proof of $\DD(u_n)$ follows the same strategy as in the Anosov case, the differences necessary in the planar dispersing billiard setting are addressed in~\cite[Theorem 2.1]{GHN}. To simplify the exposition we will consider the case $L=\{x: r=r_0\}$. The proof in the general case
of a $C^1$ curve is similar.
\subsection{Checking condition $\DD^{'}(u_n)$}
Before checking $\DD^{'}(u_n)$, we note that we need only to consider the sum up to time $(\log n)^{1+\delta}$, for $\delta>0$ since by the exponential decay of correlations of Lemma 3.3 (with $(X_0>u_n)$ equal to $A_n^q$ in this case), the remaining sum $$n\sum_{j= (\log n)^{1+\delta}}^{\lfloor k_n/n\rfloor} \mu (U_n \cap T^{-j} U_n)\to 0.$$
(Note here, we work with $A^{(0)}_n\equiv U_n$).
The set $\{r=r_0\}$ corresponds to a line (call it $L$) which is transverse to the discontinuity set $S^+$ for $T$ and the discontinuity $S^{-}$ for $T^{-1}$. Let $U_n$ be the rectangle centered at $L$ with length $\pi$ and of width roughly $\frac{\tau}{\pi n}$ corresponding to the set $\{\phi > u_n\}$ so that $\mu(U_n) = \frac{\tau}{n}$.
\par\noindent\textbf{Short Returns.}
Let $S_n=\cup_{j=0}^{n-1}T^{-j} S^+$. The number of smooth connected components of $S_{n}$ is bounded above by $\kappa^n$ for some $\kappa>0$. Let $C=\frac{1}{4\log \kappa}$
and then the number of smooth connected components in $S_{[C\log n]}$ is bounded above by $n^{1/4}$. Let $p_i=(r_0,\vartheta_i) \in L$ be the intersection points
$S_{[C\log n]}\cap L$, ordered from lowest $\vartheta$ value to highest and let $\alpha_i=\vartheta_{i+1}-\vartheta_i$. Let $B_1=\{ \alpha_i: \alpha_i < n^{-1/2} \}$.
We estimate $\sum_{\alpha_i \in B_1} \alpha_i
\le n^{1/4}n^{-1/2}=n^{-1/4}$. For each $\alpha_i$ we define the rectangle $R_i=[r_0-\frac{1}{n}, r_0+\frac{1}{n}]\times \alpha_i$ and note that
$\mu (\alpha_i)=O(\frac{\alpha_i}{n})$. Let $B=\{ R_i : \alpha_i \in B_1\}$, then $\mu (B) \le n^{-1}n^{-1/4}=n^{-5/4}$ and so can be neglected.
Let $G= \{ R_i \in B^c\}$. If $R_i \in G$ then
$\mu (R_i)\ge n^{-3/2}$ and is of length $\ge n^{-1/2}$ in the $\vartheta$ direction and width $1/n$ in the $r$-direction. If $R_i \in G$ then $T^{[C\log n]}R_i $ is a connected
`rectangle' which has expanded in the unstable direction, contracted in the stable direction and may wind around the phase space at most once.
$T^{[C\log n]}R_i $ intersects $U_n$ transversely (since $L$ is transverse to the unstable cone) in a connected component of measure
$O(n^{-1/2} \mu (R_i))$. We estimate $\mu (U_n\cap T^{-j} (U_n)) \le \mu (R_i \in B) + \sum_{R_i \in B^c} \mu (U_n \cap T^{j} (R_i))\le C n^{-5/4} \mu (U_n) $.
and conclude,
\[
\lim_{n\to\infty} n\sum_{j=1}^{C\log n}\mu(U_n\cap T^{-j} (U_n)) = 0.
\]
\begin{figure}
\centering
\begin{minipage}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale=4]
\draw[thick] (0,0) rectangle (1,1);
\node at (0,0) [below] {$r$};
\node at (0,0) [left] {$\vartheta$};
\draw[fill=gray,opacity=0.2] (0.4,0) rectangle (0.6,0.15);
\draw[fill=gray,opacity=0.5] (0.4,0.15) rectangle (0.6,0.2);
\draw[fill=gray,opacity=0.2] (0.4,0.2) rectangle (0.6,0.35);
\draw[fill=gray,opacity=0.2] (0.4,0.35) rectangle (0.6,0.55);
\draw[pattern=north west lines] (0.4,0.35) rectangle (0.6,0.55);
\draw[fill=gray,opacity=0.5] (0.4,0.55) rectangle (0.6,0.60);
\draw[fill=gray,opacity=0.2] (0.4,0.60) rectangle (0.6,0.75);
\draw[fill=gray,opacity=0.2] (0.4,0.75) rectangle (0.6,0.85);
\draw[fill=gray,opacity=0.2] (0.4,0.85) rectangle (0.6, 1);
\draw[thick] (0.5,0)--(0.5,1);
\node at (0.5,0) [below] {$U_n$};
\draw [|-|,thick] (0.4,0)--(0.6,0);
\draw [|-|,thick] (0.4,0.35)--(0.4,0.55);
\node at (0.4, 0.45) [left] {$\alpha_i$};
\end{tikzpicture}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\centering
\begin{tikzpicture}[scale=4]
\draw[thick] (0,0) rectangle (1,1);
\node at (0,0) [below] {$r$};
\node at (0,0) [left] {$\vartheta$};
\draw[fill=gray,opacity=0.2] (0.4,0) rectangle (0.6,0.15);
\draw[fill=gray,opacity=0.5] (0.4,0.15) rectangle (0.6,0.2);
\draw[fill=gray,opacity=0.2] (0.4,0.2) rectangle (0.6,0.35);
\draw[fill=gray,opacity=0.2] (0.4,0.35) rectangle (0.6,0.55);
\draw[pattern=north west lines,opacity=0.2] (0.4,0.35) rectangle (0.6,0.55);
\draw[fill=gray,opacity=0.5] (0.4,0.55) rectangle (0.6,0.60);
\draw[fill=gray,opacity=0.2] (0.4,0.60) rectangle (0.6,0.75);
\draw[fill=gray,opacity=0.2] (0.4,0.75) rectangle (0.6,0.85);
\draw[fill=gray,opacity=0.2] (0.4,0.85) rectangle (0.6, 1);
\draw[thick] (0.5,0)--(0.5,1);
\draw[pattern= north west lines,opacity=0.5] (0.25,0)--(0.75,1)--(0.85,1)--(0.35,0)--(0.25,0);
\node at (0.5,0) [below] {$U_n$};
\draw [|-|,thick] (0.4,0)--(0.6,0);
\draw [|-|,thick,xshift=-0.5,yshift=0.5] (0.25,0)--(0.75,1);
\draw[|-|,thick, xshift=-1, yshift=1] (0.4,0.3)--(0.6,0.7);
\end{tikzpicture}
\caption*{(b)}
\end{minipage}
\caption{(a) Intersection of points with $r=r_0$ that will not hit a extremal in $C\log n$ iterates. (b) Expansion of a single rectangle of side-length $\alpha_i$. Lines indicate portion that intersects $U_n$.}
\end{figure}
\textbf{Intermediate Returns.}
The proof of this section is similar to that for a hyperbolic toral automorphism case but with additional complications due to the presence of discontinuities for $T$, causing the unstable manifolds to fragment into small pieces. A scenario which needs to be ruled out is that a large number of small pieces of
fragmented unstable manifolds may find themselves again in $U_n$. To overcome this we use the following property satisfied by the planar dispersing billiard map:
\par\indent\textbf{One-step expansion.} For $\alpha \in (0,1]$,
\[
\lim_{\delta\to 0}\inf\sup_{W:|W|<\delta}\sum_n \Big(\frac{|W|}{|V_n|}\Big)^{\alpha}\cdot\frac{|T^{-1}V_n|}{|W|}<1,
\]
where the supremum is taken over regular unstable curves $W\subset X$, $|W|$ denotes the length of $W$, and $V_n$, $n\ge 1$, the smooth components of $T(W)$, $\alpha\in (0,1]$. The class of regular curves includes our local unstable manifolds~\cite{CM}.
The expansion by $DT$ is unbounded and this may lead to different expansion rates at different points on $W^u (x)$. To overcome this
effect and obtain uniform estimates on the densities of conditional SRB measure it is common to define homogeneous local
unstable and local stable manifolds. This is the approach adopted in~\cite{BSC1,BSC2,CM07,Y98}. Fix a large $k_0$ and
define for $k>k_0$
\[
I_k=\{(r,\vartheta): \frac{\pi}{2} -k^{-2} <\vartheta<\frac{\pi}{2}-(k+1)^{-2} \},
\]
\[
I_{-k}=\{(r,\vartheta): -\frac{\pi}{2} +(k+1)^{-2} <\vartheta<-\frac{\pi}{2}+k^{-2} \},
\]
and
\[
I_{k_0}=\{(r,\vartheta): -\frac{\pi}{2} +k_0^{-2} <\vartheta<\frac{\pi}{2}- k_0^{-2} \}.
\]
In our setting we call a local unstable (stable) manifold $W^u (x)$, ($W^s (x)$) homogeneous if for all $n\ge 0$
$T^n W^u (x)$
($T^{-n} W^s (x)$) does not intersect any of the line segments in $\cup_{k>k_0} (I_k\cup I_{-k})\cup I_{k_0}$. Homogeneous
$W^u (x)$ have almost constant conditional SRB densities $\frac{d\mu_x}{dm_x}$ in the sense that
there exists $C>0$ such that $\frac{1}{C} \le \frac{d\mu_x (z_1)}{dm_x} /\frac{d\mu_x (z_2)}{dm_x} \le C$ for all $z_1,~z_2 \in W^u (x)$ (see ~\cite[Section 2]{CM} and the remarks following Theorem 3.1).
From this point on all the local unstable (stable) manifolds that we consider will be homogeneous.
We may as well suppose all such curves are contained in $R_i \in G$ as
$\mu (B)< n^{-5/4}$.
We now take care of the times $[C\log n] <j< (\log n)^{1+\delta}$. If $W^u (x) \cap U_n \subset R_i \in B^c$ then $T^{[C\log n]}$ has expanded $W^u (x)$ by a factor
$\Lambda^{C\log n}=n^{C\log \Lambda}=n^{\beta}$ for some $\beta >0$ and the iterates of the components of $W^u(x)\cap U_n$
have not hit a extremal set in the first $[C\log n]$ iterates. Let $\gamma_n(x)=W^u (x)\cap U_n$.
By~\cite[Theorem 5.7]{CM07} $\mu ( W^u (x) < n^{-1-\beta/2}) < n^{-1-\beta/2}$ so we may require all $W^u (x) \in \cup_{R_i \in G} R_i$ to satisfy $|\gamma_n (x) | > n^{-1-\beta/2}$.
Now we consider $\mu(U_n \cap T^{-j}(U_n ))$ for $C\log n\le j\le (\log n)^{1+\delta}$. Note that $T^j(\gamma_n (x) )$ consists
of a connected curve for $j\le C\log n$. Recall by expansion under the map we have $|T^j \gamma_n (x)|\ge n^{\beta}|\gamma_n (x)|>n^{-1+\beta/2}$. If we iterate this component further such that
$T^{i+j}\gamma_n(x)$, $i>0 $ intersects a extremal line then we may decompose $T^{i+j}\gamma_n (x)$ into smooth connected components $V_n$ and their preimages $Y_n\subset T^j \gamma_n (x) $ so that $T^i$ maps $Y_n$ onto $V_n$ diffeomorphically and with uniformly bounded distortion. Applying one-step expansion for $p\in \gamma_n (x)$ gives,
\[
\sum_n \Big (\frac{|\gamma_n (x)|}{|V_n(p)|}\Big)^{\alpha}\Big|\frac{Y_n(p)}{\gamma_n (x)}\Big |<1.
\]
Fix $T^j \gamma_n(x)$ and for every point $p\in T^j \gamma_n(x)$ let $d\mu_{\gamma}(p) = \frac{|Y_n(p)|}{|\gamma(x)|}$ be the density of a probability measure $\mu_{\gamma}(p)$ on $T^j \gamma_n(x)$ and $f(p) = \big(\frac{|\gamma_n (x)|}{|V_n(p)|}\big)^{\alpha}$ a function on this probability space. Now $\{p\in T^j \gamma_n (x) : |V_n(p)|<n^{-1+\varepsilon\beta/2} \} \subset\{p\in T^j \gamma_n (x):f(p)>n^{(1-\varepsilon) \beta/2\alpha}\}$ and by Markov's inequality $\mu_{\gamma}\{p \in T^j \gamma_n (x): |V_n(p)|<n^{-(1+\varepsilon)\beta/2} \}\le n^{-(1-\varepsilon)\beta/2\alpha}$.
We choose $\varepsilon$ sufficiently small so that $\rho_1 := 1-(1-\varepsilon)\beta/2\alpha>0$ (since $\beta<1$) and define $\rho = \min\{\rho_1,\beta/2\alpha\varepsilon\}$. With our choice of $\varepsilon$, if $|V_n|\ge n^{-1+\epsilon\beta/2}$ then,
\[
\frac{|V_n\cap U_n |}{|V_n|}\le C_1 n^{-\rho}.
\]
By bounded distortion of the map $T$, after throwing away the $V_n$ such that $|V_n|\le n^{-1+\varepsilon\beta/2}$ we have
\[
\frac{|T^i \gamma_n (x) \cap U_n )|}{|\gamma_n (x) |}\le C_2 n^{-\rho}.
\]
and by bounded distortion again we have,
\[
\frac{|\gamma_n (x) \cap T^{-i}(U_n)|}{|\gamma_n (x) |}\le C_3 n^{-\rho}.
\]
\par This provides a bound on the length of the intersection of a single unstable manifold $\gamma_n (x) $.
We may now use the fact that $\mu$ decomposes as a product measure on $U_n$ so that if we consider all manifolds of $R_i\in G$ we have,
\[
\mu( U_n \cap T^{-j}(U_n))\le C_4 n^{-1-\rho}.
\]
Putting these results together implies,
\[
\lim_{n\to\infty} n \sum_{j=C\log n}^{(\log n)^{1+\delta}}\mu(U_n\cap T^{-j}(U_n)) = 0.
\]
Condition $\DD^{'}(u_n)$ follows.
\begin{remark}
Using essentially the same analysis it is standard to show that the return time statistics to $L=\{ (r,\vartheta): r=r_0\}$ is standard simple Poisson. To see this we need verify
condition $D^{*}_q (u_n)$ of ~\cite[Section 2]{CNZ}, but the proof of this is a minor modification of $\DD(u_n)$. In contrast suppose $(r_0,\vartheta_0)$ is a periodic
point of period $q$, then we would obtain a compound Poisson process as given in~\cite[Theorem 2]{CNZ}.
\end{remark}
\subsection{Proof of Theorem~\ref{thm:coupled} and Theorem~\ref{thm:block}.}
We give the proof in detail only for the case of two coupled maps, as the proofs in the other cases
are the same with obvious modifications. The uniform expansion away from the invariant subspace plays the same role in each setting. Note that the subspace $L$ of Theorem~\ref{thm:block} is invariant, and we will show that there is
uniform expansion in the directions orthogonal to $L$.
Recall $\phi (x,y)=-\log |x-y|$, a function maximized on the line segment or circle $L=\{(x,y):y=x\}$. For $\tau>0$
define $u_n(\tau)$ by $n\mu (\phi >u_n (\tau))=\tau$, and $U_n=\{\phi >u_n (\tau)\}$. Define $A_n=\{\phi > u_n,
\phi\circ F <u_n\}$ and recall for a set $B$, $\mathscr{W}_{s,l}(B)=\bigcap_{i=s}^{s+l-1} F^{-i}(B)$.
Note that the invariant line $L$ is uniformly repelling in the orthogonal direction $(1,-1)$ since writing $y-x=\epsilon$ we see $\epsilon \rightarrow (1-\gamma)[T(x+\epsilon)-T x]\sim (1-\gamma) DT(x) \epsilon+O(\epsilon^2)$ under the map $F$.
Furthermore $A_n$ is a union of two rectangles and $A_n\cap F^{-2} A_n=\emptyset$ as a result of uniform expansion away from the invariant line $L$.
Condition $\DD(u_n)$ follows easily by an approximation argument using exponential decay of correlations of
Lipschitz versus $L^{\infty}$ functions taking $t_n=(\log n)^5$ say.
Now we prove condition $\DD'_q(u_n)$ (for $q=1$), namely
\[
\lim_{n\rightarrow\infty}\,n\sum_{j=1}^{\lfloor n/k_n\rfloor}\mu\left( A_n\cap F^{-j}\left(A_n\right)
\right)=0.
\]
Note that by uniform repulsion from the invariant line $L$ there exists $C_4$ such that
for $j=1,\ldots, C_4 \log n$, $\mu ( A_n\cap F^{-j} A_n )=0$. This follows
since $F^{-1} A_n \cap A_n=\emptyset$ (by definition) and uniform repulsion from the invariant line
ensures also $F^{-j} A_n \cap A_n=\emptyset$ for a certain number of iterates $j=1,\ldots, C_4 \log n$ until for all $(x,y)$ in $A_n$,
$|F^j (x,y)|=O(1)$ (i.e. until the expansion in the $L^{\perp}$ direction is $O(n)$).
As $DF$ is bounded and uniformly expanding, in all directions $A_n$ has been expanded by the map $F^{[C_4\log n]}$ by at least $n^{\alpha}$ for some $0<\alpha<1$. To see this, note that for any expanding map expansion of $A_n$ by the map $F^{[C_4\log n]}$ given by at least $C_5 |DT|_{min}^{C_4\log n}\sim n^\alpha$.
\begin{figure}
\begin{tikzpicture}[scale=4]
\draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0);
\draw[thick] (0,0)--(1,1);
\draw[fill=gray,opacity=0.2] (0,0.15)--(0,0.25)--(0.75 ,1)--(0.85,1);
\draw[fill=gray,opacity=0.2] (0,0.15)--(0,0.55)--(0.45 ,1)--(0.85,1);
\draw[fill=gray,opacity=0.2] (0.15,0)--(0.25,0)--(1,0.75)--(1,0.85);
\draw[fill=gray,opacity=0.2] (0.15,0)--(0.55,0)--(1,0.45)--(1,0.85);
\draw[opacity=0.2] (0,0.15)--(0,0.25)--(0.75 ,1)--(0.85,1);
\draw[opacity=0.2] (0.15,0)--(0.25,0)--(1,0.75)--(1,0.85);
\draw[dotted] (0,1)--(1,0);
\draw[->,>=stealth] (0.5,0.5)--(0.6,0.6);
\draw[->,>=stealth] (0.5,0.5)--(0.4,0.4);
\draw[->,>=stealth] (0.5,0.5)--(0.6,0.4);
\draw[->,>=stealth] (0.5,0.5)--(0.4,0.6);
\node at (1,0)[below]{$x$};
\node at (0,1)[left]{$y$};
\node at (0.75,1)[below]{\small$x=y$};
\draw[|-|] (1,0.85)--(1,0.75);
\node at (1,0.8) [right]{$O(\frac{1}{n})$};
\end{tikzpicture}
\caption{Expansion of the set $A_n$ (given in gray) under the map $F$. The thick, black line represents the line of maximization. Arrows indicate uniform expansion under $F$ in all directions.}
\end{figure}
Choose $C_3\ge C_4$ large enough that $\mu ( A_n\cap F^{-j}(A_n ))\le \frac{1}{n^{3/2}}$, this is possible by exponential decay of correlations and a Lipschitz approximation to $1_{A_n}$.
Thus for $C_4 \log n \le j \le C_3 \log n$, $\mu ( A_n\cap F^{-j} (A_n) )\le \frac{1}{n^{1+\alpha}}$. For $1\le j \le C_4 \log n$, $\mu (F^{-j} (A_n) \cap A_n)=0$ for $C_4 \log n \le j \le C_3 \log n$, $\mu ( A_n\cap F^{-j} A_n )\le \frac{1}{n^{1+\alpha}}$ and for $j\ge C_4 \log n$, $\mu ( A_n\cap F^{-j} (A_n ))\le \frac{1}{n^{3/2}}$.
This implies $\DD'_q(u_n)$ for $q=1$ (corresponding to the fact that $L$ is fixed).
Finally we compute the extremal index, changing coordinates to $v=\frac{x-y}{\sqrt{2}}$, $u=\frac{x+y}{\sqrt{2}}$ we have
\[
\theta=\lim_{n\to\infty}\theta_n=\lim_{n\to\infty}\frac{\mu(A_n)}{\mu(U_n)}.
\]
However
\[
\lim_{n\to \infty}\frac{\mu(A_n)}{\mu(U_n)}
= \lim_{n\to \infty} [1- \int_{0}^{\frac{1}{n[Tv]}} \int_{0}^{\frac{1}{n}}\tilde{h}(u,v)dudv] / \int_{0}^{\frac{1}{n}} \int_{0}^{\frac{1}{n}}\tilde{h}(u,v)du dv. \]
Suppose $m(U_n)=O(\epsilon^{1/4})$, $\epsilon <\epsilon_0$. Since $|\tilde{h}|_{\alpha}<\infty$, $m(x\in U_n : {\it osc} (\tilde{h}, B_{\epsilon} (x)) >\sqrt{\epsilon})<\sqrt{\epsilon}=O(m(U_n)^2)$. We may assume that $\tilde{h}$ is regularized along the diagonal in the sense that for Lebesgue almost every $u$, $\tilde{h}(u,u)$ is the average of the limits of $\tilde{h}(u,v)$ and $\tilde{h}(u,-v)$ as $v\to 0$.
Thus, as expansion along $v$ at $v=0$ is given by $(1-\gamma)DT(u)$, and $\tilde{h}$ is essentially bounded
\[
\theta=1- \int_L \frac{\tilde{h}(u,u)}{(1-\gamma)|DT(u)|} du.
\]
\begin{remark}
Our techniques allow us to obtain similar results to that of ~\cite{sandro_coupled} in a simpler setting through a pure probabilistic approach and extend these results to blocks of synchronization discussion in ~\cite[Section 7.2]{sandro_coupled}.
\end{remark}
\subsection{Numerical Results for the Extremal Index}\label{sec.numerics}
In this section we provide numerical estimates for the extremal index to support the theoretical results for the coupled uniformly expanding map and the hyperbolic toral automorphism provided in Theorems \ref{thm:coupled} and \ref{thm:block} and Theorem \ref{thm.anosov}, respectively. We begin by verifying that the numerical estimates we obtain from the coupled systems agree with that of \cite{sandro_coupled}. Then, we extend these results to include estimates for the extremal index over blocks of synchronization where each block introduces a new invariant direction and changes the value of the extremal index. We end with a numerical investigation for Arnold's cat map as an example of a hyperbolic toral automorphism where the alignment of the singlarity set taken as a line $L$ in the space and existence of periodic orbits along $L$ determine the value of the extremal index.
\subsection{Coupled systems of uniformly expanding maps}
Numerical barriers in computing trajectories in piecewise uniformly expanding maps are given by the fact that
\begin{itemize}
\item[(i)]The periodic orbits are dense making long trajectories not easily computable.
\item [(ii)]Round off errors may produce unreliable results.
\end{itemize}
To overcome (i) we employ a numerical technique adapted from \cite{V.et.al} to prevent trapping of the orbit near the fixed point by adding a small $\varepsilon=O(10^{-2})$ amount to the trajectory. Arguments for this technique are typically given in the form of a shadowing lemma which states the existence of a true orbit that is $\epsilon$-close to the computed orbit; we will support this argument through a more numerical approach. We first note that \cite{FFLTV} proves the existence of an EVL for randomly perturbed piecewise expanding maps provided this perturbation $\varepsilon>10^{-4}$. Futher, \cite{sandro_coupled} provides evidence that the extremal index is qualitatively robust under small $\varepsilon=10^{-2}$ additive noise. To overcome (ii), in light of our long trajectories ($t=10^6$), we refer to \cite{FMT} where the round off error resulting from double precision computation was shown to be equivalent to the addition of random noise of order $10^{-7}$.
\subsubsection*{Estimating the EI for the coupled map system over the whole extremal set.}
We estimate the extremal index in a similar way to that of~\cite{sandro_coupled} for $\phi(\bar{x}) = -\log(|| p^{\perp}||)$ using the formula provided by S\"{u}veges~\cite{suveges}. The code for this estimate can be found in~\cite{V.et.al}. From Theorem~\ref{thm:coupled} we expect,
\[
\theta = 1-\frac{1}{(1-\gamma)^{m-1}}\frac{1}{|DT|^{m-1}}.
\]
We compute the extremal index for fixed $m=2$ and varying values of $\gamma$, and varying values of both $\gamma$ and $m$. Our results coincide with that of \cite{sandro_coupled}; higher values of $m$ and lower values of $\gamma$ produce an extremal index near 1. Low values of $\gamma$ give higher weights to the non-coupled components of the map resulting in a system which behaves more independently. Lower values of $m$ result in a more dependent system since the coupled term is more affected by changes while larger values of $m$ result in a coupled term which is averaged over a larger number of maps and less affected by individual changes. For results see Figure~\ref{fig:ei}.
\begin{figure}[h!]
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{N2}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{MAT_PT1}
\caption*{(b)}
\end{minipage}
\caption{\label{fig:ei} Extremal index $\theta$ estimation for the $m$-coupled map $F$ with $\phi(x) = -\log (||p^{\perp}||)$ where the set of maximization $L=\{(x_1,x_2,\dots,x_m):x_1=x_2=\dots=x_{m}\}$ for (a) fixed $m$ and varying $\gamma$ (10 different realizations $t = 10^6$) and (b) varying $m$ and $\gamma$. The marked line indicates the theoretical value of $\theta$ given.}
\end{figure}
\subsubsection*{Estimating the EI for the coupled map system over blocks of synchronization.}
\par We provide numerical estimates of the extremal index in a more specific setting of block synchronization where $L=\{(x_1,x_2,\dots,x_m):x_1=x_2=\dots=x_{m}\}$ and $L=\{(x_1,x_2,\dots,x_m):x_1=x_2=\dots=x_{m-1},x_m\}$. From Theorem~\ref{thm:block} we expect,
\[
\theta=1-\frac{1}{(1-\gamma)^{m-1}}\frac{1}{|DT|^{m-1}}.
\]
for $L=\{(x_1,x_2,\dots,x_m):x_1=x_2=\dots=x_{m}\}$ and,
\[
\theta=1-\frac{1}{(1-\gamma)^{m-2}}\frac{1}{|DT|^{m-2}}
\]
for $L=\{(x_1,x_2,\dots,x_m):x_1=x_2=\dots=x_{m-1},x_m\}$. Defining $L$ in this way reduces the spacial dimension in which expansion away from $L$ can occur. This results in a extremal index equivalent to that of an $m-1$ coupled system. We give results in the case when $m=5$ (see Figure~\ref{fig:bs2}) .
\begin{figure}[h!]
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{N5_1}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{N5_2}
\caption*{(b)}
\end{minipage}
\caption{\label{fig:bs2} Extremal index $\theta$ estimation (10 different realizations, $t=10^6$) for the $m$-coupled map $F$ with $\phi(x) = -\log(||p^{\perp}||)$ where (a) $L$ is the line $x_1=x_2=\dots=x_m$ and (b) $L$ is the plane $x_1=x_2=\dots=x_{m-1},x_m$. The marked line indicates the theoretical value of $\theta$ given.}
\end{figure}
We also consider blocks of successive indices in the general setting of block synchronization so that $L$ can be defined as any combination of block sequences. From Theorem~\ref{thm:block} we expect the value of the extremal index to be determined by the spacial dimension of expansion for the system. In the following numerical examples we consider $m=5$ and note that the extremal index for that of $L=\{(x_1,\dots,x_5):x_1=x_2=x_3=x_4,x_5\}$ is equivalent to that of $L = \{(x_1,\dots,x_5):x_1=x_2=x_3,x_4=x_5\}$. This is expected since they share the same number of non-invariant directions of expansion.
\begin{figure}[h!]
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{N5_3}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{N5_4}
\caption*{(b)}
\end{minipage}
\caption{\label{fig:bs} Extremal index $\theta$ estimation (10 different realizations, $t=10^6$) for the $m$-coupled map $F$ with $\phi(x) = -\log(||p^{\perp}||)$ where (a) $L$ is the set of two planes $x_1=x_2$ and $x_4=x_5$ so that $\theta = 1-\frac{1}{(1-\gamma)^{2}|DT|^2}$ (b) $L$ is the set of planes $x_1=x_2=x_3$ and $x_4=x_5$, $\theta = 1-\frac{1}{(1-\gamma)^{3}|DT|^3}$. The marked line indicates the theoretical value of $\theta$ given.}
\end{figure}
\subsection{Hyperbolic toral automorphisms.}
We compute trajectories for increasing time intervals of Arnold's cat map given by,
\[
T\begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix}2 & 1\\ 1 & 1\end{pmatrix}\quad\begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mod 1.
\]
The uniformly hyperbolic structure of this map allows us to calculate long trajectories without the risk of points being trapped in a few time steps. The stability of this map ensures that the qualitative behavior is unaffected by small perturbations. We use this to argue the accuracy of the calculated orbit up to $t=10^4$ under double precision.
From Theorem~\ref{thm.anosov} we expect the value of the extremal index $\theta$ to depend on both the alignment of $v$ in the observable $\phi(x) = -\log d(x,L)$, with $d$ the usual Euclidean metric, and the existence of a periodic orbit along $L$. Figure~\ref{fig:cm1} (a) shows the extremal index estimation given by~\cite{suveges} for 10 different initial values where $v$ aligns with the unstable direction and contains a 2-periodic point. Hence, $\theta = 1-\frac{1}{\lambda^2}$. Figure~\ref{fig:cm1}(b) shows the extremal index estimation for 10 different initial values where $v$ is not aligned with the stable or unstable direction. In this setting we expect $\theta=1$. The variation from the expected value for each realization is at most $O(10^{-2})$.
\begin{figure}[h!]
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{cm_vplus_zoomed}
\caption*{(a)}
\end{minipage}
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{cm_v_zoomed}
\caption*{(b)}
\end{minipage}
\caption{\label{fig:cm1} Extremal index $\theta$ estimation (10 different realizations, $t=10^4$) for Arnold's cat map with $\varphi(x) = -\log d(x,L)$ where (a) $L=v^+$ and (b) $L=0.5v^++0.25v^-$. The marked line indicates the theoretical value of $\theta$ given.}
\end{figure}
\section{Discussion: towards more general observables and non-uniformly hyperbolic systems}\label{sec.discussion}
In this article we have focused on hyperbolic systems and considered observables whose level sets $\mathcal{S}_{\epsilon}$
shrink to a non-trivial extremal set $\mathcal{S}$, such as a line segment. We recall that
$\mathcal{S}_{\epsilon}=\{x\in X:\,d_H(x,\mathcal{S})\leq\epsilon\}$, and $d_H(x, \mathcal{S})$ is the Hausdorff distance
from $x$ to $\mathcal{S}$. Thus if $\mathcal{S}$ is a smooth curve, then for this metric $d_H$ we see that
$\mathcal{S}_{\epsilon}$ is a thin tube of width $\epsilon$ around $\mathcal{S}$.
The observable $\phi:X\to\mathbb{R}$ we have assumed to be given by $\phi(x)=f(d_{H}(x,\mathcal{S}))$, for
some smooth function $f:[0,\infty)\to\mathbb{R}$, maximised at 0, e.g. $f(u)=-\log u$.
As explained in Section \ref{sec.statement}, our methods extend to cases where $\mathcal{S}$ is a smooth
curve, assuming some transversality conditions of $\mathcal{S}$ relative to the global stable/unstable manifolds of the system.
We have also considered seemingly non-generic geometrical cases, e.g. where $\mathcal{S}$
aligns precisely with the global stable/unstable manifolds. For hyperbolic toral automorphisms, we established the limit laws that arise in these
scenarios. More generally, it is natural to consider observables whose extremal set $\mathcal{S}$ is no longer (strictly) transverse
to the global stable/unstable manifolds, i.e. there exist points of tangency between $\mathcal{S}$ and the global manifolds.
For the systems we have considered, the ergodic invariant measures are absolutely continuous with
respect to the ambient (two dimensional Lebesgue) measure. For (non-uniformly) hyperbolic systems $(f,\Lambda,\mu)$ where $\Lambda$ is an
attractor the Sinai-Ruelle-Bowen (SRB) measure $\mu$ may not be equivalent to Lebesgue. These systems include Axiom A systems, or H\'enon-like attractors
whose statistical properties (such as mixing rates) are established in \cite{Y98}. As outlined in
Section \ref{sec.background}, there is an established literature on extreme value theory in the non-uniformly hyperbolic setting for observables whose extremal set $\mathcal{S}$ is a point. Recently some progress has been made on more complicated geometries for $\mathcal{S}$~\cite{Haydn_Vaienti} but in a very axiomatic way.
In the case where $\mathcal{S}$ is a line (or in higher dimensions a planar set), then we expect $\mathcal{S}$
to (generically) intersect a fractal attractor $\Lambda$ in a Cantor-like set. For such a set, there are various difficulties that arise
when trying to find the limit extreme value distribution distribution, in the sense of establishing \eqref{eq.ev-law}, or in particular
the limit law given by \eqref{eq.gevlimit}.
If we suspect that a limit law of the form given in equation \eqref{eq.ev-law} is going to exist, then finding the scaling
sequence $u_n$ is a first problem. For a specified observable $\phi$ (i.e. through specifying $f$), the properties of the sequence $u_n$ depend on the asymptotic properties of $\mu(\mathcal{S}_{\epsilon})$ as $\epsilon\to 0$. To estimate this measure, we cannot use local dimension estimates, and finer arguments are required based on the geometric properties of $\mu$. Furthermore, existence of a GEV limit
of the form \eqref{eq.gevlimit} is not guaranteed, as this requires $\mu(\mathcal{S}_{\epsilon})$ to satisfy conditions of regular variation in $\epsilon$ (as $\epsilon\to 0$), see \cite[Chapter 3]{V.et.al}. Axiomatic approaches, e.g. \cite{CC, Haydn_Vaienti, HRS} suggest that once we've found these
scaling laws then an extreme value law holds in the sense of equation \eqref{eq.ev-law}. However, verification of these axioms
still requires fine analysis. This includes verification of axiomatic conditions involving transversality of $\mathcal{S}$ with $\Lambda$,
and conditions involving how $\mu$ behaves on certain shrinking sets (such as thin annuli) on a case-by-case basis.
\clearpage
|
1,314,259,992,967 | arxiv | \section{Introduction}
Person re-identification (re-ID) is a retrieval task of recognizing the same person across images from non-overlapped cameras, which has attracted increasing attention in computer vision community due to its wide application prospects in video surveillance and forensics field \cite{1}. Nevertheless, person re-ID remains a challenge due to some complicated visual variations in real scenarios such as viewpoint, illumination, person pose and background clutter.
Most existing re-ID methods focus on the designment of feature extraction networks or matching distance metrics on the basis of the assumption that all captured images share similar and sufficiently high resolutions. However, this assumption only exists in an absolutely ideal condition. In real and unconstrained scenarios, affected by some objective factors such as shooting distance and camera pixels, the captured images have variable resolutions. The problem of matching person images with variable resolutions is defined as \emph{Cross-Resolution Person Re-ID}.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig1.png}
\caption{Several shortages and limitations of existing cross-resolution person re-ID methods. (a) The recovered HR image with CRGAN may contain a few false details which may mislead the feature extraction. (b) HR images contain plenty of detail information, such as texture and bags, while current methods have not yet exploited complementary global features of LR images. (c) In some cases, resolution is hard to quantify with image pixel size. Some gallery images classified as HR even have worse visual quality than query images classified as LR.}
\label{fig1}
\end{figure}
Recently, a few researchers have paid attention to this problem and proposed some high-performance methods which can be mainly divided into two categories: 1) traditional methods utilizing metric learning or dictionary learning \cite{8,31} and 2) deep learning methods applying super-resolution (SR) technology to restore LR images to HR images, which are most commonly used in cross-resolution person re-ID \cite{10,11,12}.
However, there are still problems in existing SR based methods, just as illustrated in Figure \ref{fig1}. 1) Existing methods mainly focus on recovering higher resolution images and extract HR feature representations. Although the complementary details generated by SR give person images better visual quality, these details may not be real in person appearances. Therefore, in some cases, the features extracted from these generated HR images are not discriminative enough to match correct persons. 2) Although local details are lost in LR images, LR images still can provide some global information, such as body shape and color, as evidenced in the studies of pyramid representation of images \cite{44}. These LR features can complement HR features which may be false details, but all existing methods neglect this useful information. 3) Most existing methods process gallery and query images with different strategies separately, because they tacitly approve all gallery images as HR and all query images as LR. However, in some practical scenarios, the resolutions of gallery or query images are not clearly divided, making it difficult to quantify the image as HR or LR.
In this paper, we investigate the influence of resolution on feature extraction and find that a neural network focuses on more local details in HR person images but more global features in LR person images. Inspired by this, we propose a novel \emph{\textbf{M}ulti-Resolution \textbf{R}epresentations \textbf{J}oint \textbf{L}earning} (\textbf{MRJL}) for cross-resolution person re-ID, which fully utilizes the detail information in HR and complementary information in LR. Our MRJL is made up of two sub-networks named as Resolution Reconstruction Network (RRN) and Dual Feature Fusion Network (DFFN). The RRN adopts a multi-kernel encoder to encode the input image into a feature map, and then applies two different decoders to restore the feature map to HR image and LR image, respectively. The DFFN utilizes a dual-branch structure based on the PCB method \cite{2} to generate person representations from multi-resolution images. It is worth noting that in the testing phase, our MRJL does not need to know the resolution of input image and treats images with different resolutions equally.
The contributions of our work are summarized as follows: 1) As far as we know, it is the first work to detailly explore the influence of resolution on feature extraction in person re-identification. 2) A novel method named as Multi-Resolution Representations Joint Learning (MRJL) is proposed for cross-resolution person re-ID, which fully utilizes features contained in different resolutions.
\section{Related Work}
\subsection{Person Re-ID}
In the past decade, a variety of high-performance methods have sprung up in the field of person re-ID. Most of these existing methods attempt to extract more discriminative features and overcome the difficulties such as pose changes and background clutter. For instance, some methods \cite{2,3} divide person image into several parts and extract local features which contain more discriminative details. Nevertheless, pose changes will affect the feature alignment. To address this problem, some excellent methods adopt pose-transferable GAN \cite{23} or pose estimation \cite{15} to enhance the robustness of network towards pose variations. To attenuate background clutter, some methods apply semantic parsing \cite{4} to remove backgrounds or apply attention mechanism \cite{22} to train the network to focus on more informative areas. However, all above methods are limited in practical use due to the incapability of adaptation to variable image resolutions in unconstrained scenarios.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{Fig2_new.png}
\caption{The architecture of the proposed MRJL. This framework consists of two jointly trained sub-networks, Resolution Reconstruction Network (RRN) and Dual Feature Fusion Network (DFFN). The former is tasked to reconstruct input images into two versions with different resolutions, while the latter is used to extract feature representations from the generated HR and LR images.}
\label{fig2}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig3_new.png}
\caption{The details of the Resolution Reconstruction Network.}
\label{fig3}
\end{figure}
\subsection{Cross-Resolution Person Re-ID}
To meet the challenge of cross-resolution person re-ID, a series of methods have been proposed and can be divided into two categories: 1) methods based on metric learning or dictionary learning and 2) methods based on SR. In the first category, Jing \emph{et al.} \cite{8} develop a semi-coupled low-rank dictionary learning approach to learn the mapping between HR and LR images. Li \emph{et al.} \cite{31} introduces a learning framework which jointly performs cross-scale image domain alignment and distance metric learning. However, the matching capability of these methods is limited due to the lack of fine-grained details in LR images.
The success of super-resolution (SR) technology promotes the development of cross-resolution person re-ID. The key idea of these methods is to restore LR images back to HR images by resolution reconstruction loss or GAN. Both \cite{10} and \cite{11} design a jointly learning framework which simultaneously optimize a SR model and a re-ID model. Wang \emph{et al.} \cite{12} present a cascaded structure to enhance image resolution step by step with the repeated use of SR-GAN \cite{34}. Li \emph{et al.} propose successively RAIN \cite{30} and CAD-Net \cite{29}. The former adopts GAN to generate resolution-invariant representations, while the latter adds the features extracted from recovered images and achieves better performance. Cheng \emph{et al.} \cite{32} introduce a training regularization method which utilizes the underlying association knowledge between SR and re-ID as an extra learning constraint to enhance the compatibility between two networks. Han \emph{et al.} \cite{43} propose an end-to-end PRI framework to adaptively predict the preferable scale factor, recover details for LR images and perform the identification. However, all above methods only focus on the HR features and neglect the useful information in LR images. In this work, we explore the influence of resolutions on feature extraction and verify that LR information matters for cross-resolution person re-ID. Based on the above idea, we develop a novel method fully utilizing features of different resolutions.
\section{Proposed Method}
\subsection{Framework Overview}
As illustrated in Figure \ref{fig2}, our proposed MRJL contains two sub-networks, RRN and DFFN. In the training phase, we define a set of input HR images with associated labels as ${D_H}{\rm{ = }}\left\{ {{x^H},y} \right\}$, where ${x^H} \in {\mathbb{R}^{H \times W \times 3}}$ represents a HR image and $y \in \mathbb{R}$ represents its identity label. To train the RRN with the capability to reconstruct different resolutions of images, we down-sample each HR image with the down-sampling rate $r \in \left\{ {2,3,4} \right\}$ (i.e., the spatial size of the down-sampled image becomes $\frac{H}{r} \times \frac{W}{r}$) and resize them back to the original size. The set of generated LR images obviously share the same identity labels and are denoted as ${D_L}{\rm{ = }}\left\{ {\left( {x_2^L,x_3^L,x_4^L} \right),y} \right\}$ where $x_i^L \in {\mathbb{R}^{H \times W \times 3}}$ is a LR image and the subscript $i \in \left\{ {2,3,4} \right\}$ represents the down-sampling rate. (The subscript $i$ is omitted in following paper for simplicity unless necessary.) In the testing phase, our framework regards the resolution of inputs as unknown, and processes the gallery (HR) and query (LR) equally.
In order to generate both HR and LR images for an input image with unknown resolution, we design the RRN module which is made up of an encoder and two independent decoders. The encoder is utilized to extract feature map from an input image, and the two decoders reconstruct the feature map into HR version and LR version, respectively. The DFFN module adopts a dual-branch structure to extract the feature representations ${f^H} \in {\mathbb{R}^d}$ and ${f^L} \in {\mathbb{R}^d}$ ($d$ denotes the dimension of feature) from the generated HR and LR images, respectively. Note that the two branches don’t share parameter weights. As for testing, feature representations $f^H$ and $f^L$ of all images in gallery and query sets are computed, and then the concatenation $f = \left[ {{f^H},{f^L}} \right] \in {\mathbb{R}^{2d}}$ will be used for distance measure.
\subsection{Resolution Reconstruction Network (RRN)}
Before feature extraction, the quality of generated images greatly affects the discrimination of representations. The proposed RRN module consists of a multi-kernel encoder, a HR decoder and a LR decoder, as shown in Figure \ref{fig3}.
The multi-kernel encoder ($ME$) has a four-branch structure that consists of three feature perception branches and an attention branch. All the perception branches are made up of 8 convolutional layers, and the attention branch has 3 convolutional layers followed by a batch normalization layer and a softmax activation function. To make the network perceive features of different scales, the kernel sizes of these perception branches are different, which are set to $\left\{ {1,3,5} \right\}$, respectively. Motivated by the previous works in SR \cite{11,35}, several skip connections are introduced to RRN to preserve the original visual cues and help reconstruct HR images.
Besides, attention mechanism \cite{36} has widely applied in neural network to make the network focus on parts of interest. In RRN, the attention branch is used to train the encoder to focus on the interested perceptual scale and then learn three attention weights for corresponding perception branches. The output feature map of the encoder is a weighted sum of all the outputs from the individual branch.
The HR decoder ($HD$) and LR decoder ($LD$) adopt the same network structure but don’t share parameter weights. Both decoders have 2 deconvolution layers and 1 convolution layers. For each input image $x$ (It doesn’t matter whether it is ${x^H}$ or ${x^L}$), our RRN can reconstruct both HR and LR images as:
\begin{equation}
{\tilde x^H} = HD\left( {ME\left( x \right)} \right),{\tilde x^L} = LD\left( {ME\left( x \right)} \right)
\end{equation}
According to the formula above, if the training input is a HR image ${x^H}$, its reconstructed HR version and reconstructed LR version are denoted as ${\tilde x^{H2H}}$ and ${\tilde x^{H2L}}$, respectively. Similarly, if the training input is a LR image ${x^L}$, its two reconstructed versions are denoted as $\tilde x_i^{L2H}$ and $\tilde x_i^{L2L}$, where the subscript $i \in \left\{ {2,3,4} \right\}$ represents the corresponding down-sampling rate.
As illustrated in Figure \ref{fig4}, pixel-wise Mean Square Error (MSE) loss \cite{37} is applied in the training strategy of RRN which simultaneously trains the encoder and two different decoders. Since these LR images have variable resolutions with different down-sampling rates picked from $\left\{ {2,3,4} \right\}$, we should set a LR reference standard for RRN. Here we select the median $x_3^L$ in LR images as the standard. To train the encoder and decoders, the HR MSE loss $L_{mse}^H$ and the LR MSE loss $L_{mse}^L$ are calculated as:
\begin{equation}
L_{mse}^H = \left\| {{{\tilde x}^{H2H}} - {x^H}} \right\|_2^2 + \sum\limits_{i = 2}^4 {\left\| {\tilde x_i^{L2H} - {x^H}} \right\|_2^2}
\end{equation}
\begin{equation}
L_{mse}^L = \left\| {{{\tilde x}^{H2L}} - x_3^L} \right\|_2^2 + \sum\limits_{i = 2}^4 {\left\| {\tilde x_i^{L2L} - x_3^L} \right\|_2^2}
\end{equation}
Then we get the joint MSE loss in RRN as:
\begin{equation}
{L_{mse}} = L_{mse}^{LR} + \lambda L_{mse}^{HR}
\end{equation}
where $\lambda$ is a hyper-parameter to control the importance of HR MSE loss.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig4_new.png}
\caption{The training strategy of RRN, consists of two aspects: 1) training the reconstructed HR images to be visually closer to the original HR images, and 2) training the reconstructed LR images to be visually closer to the standard LR images.}
\label{fig4}
\end{figure}
\subsection{Dual Feature Fusion Network (DFFN)}
A dual-branch structure is used in DFFN to extract both detailed information from HR images and complementary information in LR images simultaneously. Two branches in DFFN share the same network but do not share parameter weights since we wish each branch to focus on different types of features from images with different resolution.
Here we adopt the feature extractor on the basis of PCB method \cite{2}. The 3D tensor generated by the backbone network (e.g., ResNet50) is segmented into 4 horizontal stripes. Followed by an average pooling layer and $1 \times 1$ convolutional layers, the feature representation ${f^H} = \left[ {f_1^H,f_2^H,f_3^H,f_4^H,f_5^H} \right]$ (or ${f^L} = \left[ {f_1^L,f_2^L,f_3^L,f_4^L,f_5^L} \right]$) is obtained which is concatenated by 4 256-dimentional local features and a 512-dimentional global feature.
In DFFN, we adopt both triplet loss and cross entropy loss to enhance the discrimination of feature representations:
\begin{equation}
{L_{reid}} = {L_{ce}} + \gamma {L_{trip}}
\end{equation}
where $\gamma$ is a hyper-parameter to control the importance of triplet loss. The cross entropy loss ${L_{ce}}$ can be computed as:
\begin{equation}
{L_{ce}} = - \sum\limits_{i = 1}^5 {\left( {y\log \left( {FC\left( {f_i^H} \right)} \right) + y\log \left( {FC\left( {f_i^L} \right)} \right)} \right)}
\end{equation}
where $y$ denotes the ground truth and the predicted person label can be generated by FC layers. The triplet loss ${L_{trip}}$ can be calculated as:
\begin{equation}
\begin{array}{l}
{L_{trip}} = \sum\limits_{i = 1}^5 {\sum\limits_{f_{a,i}^H,f_{p,i}^H,f_{n,i}^H} {{{\left[ {d\left( {f_{a,i}^H,f_{p,i}^H} \right) - d\left( {f_{a,i}^H,f_{n,i}^H} \right) + m} \right]}_ + }} } \\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ + }}\sum\limits_{i = 1}^5 {\sum\limits_{f_{a,i}^L,f_{p,i}^L,f_{n,i}^L} {{{\left[ {d\left( {f_{a,i}^L,f_{p,i}^L} \right) - d\left( {f_{a,i}^L,f_{n,i}^L} \right) + m} \right]}_ + }} }
\end{array}
\end{equation}
where $f_{a,i}^H$, $f_{p,i}^H$ and $f_{n,i}^H$ indicate the ${i^{{\rm{th}}}}$ sub-features extracted from anchor, positive and negative HR samples ($f_{a,i}^L$, $f_{p,i}^L$ and $f_{n,i}^L$ indicate the corresponding features from LR samples), $d\left( { \cdot , \cdot } \right)$ indicates the Euclidean distance, and $m$ is a margin hyper-parameter to control the differences between intra and inter distances.
\section{Experiment}
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{l c c c c c c c c c c c}
\hline
\multirow{2}{*}{Method} & \multirow{2}{*}{Publication} & \multicolumn{2}{c}{MLR-SYSU} & \multicolumn{2}{c}{MLR-VIPER} & \multicolumn{2}{c}{MLR-CUHK03} & \multicolumn{2}{c}{MLR-Market-1501} & \multicolumn{2}{c}{CAVIAR} \\
\cline{3-4}
\cline{5-6}
\cline{7-8}
\cline{9-10}
\cline{11-12}
& & Rank-1 & Rank-5 & Rank-1 & Rank-5 & Rank-1 & Rank-5 & Rank-1 & Rank-5 & Rank-1 & Rank-5 \\
\hline
JUDEA \cite{31} & ICCV'15 & 18.3 & 41.9 & 26.0 & 55.1 & 26.2 & 58.0 & - & - & 22.0 & 60.1 \\
SLD$^{2}$L \cite{8} & CVPR'15 & 20.3 & 34.8 & 20.3 & 44.0 & - & - & - & - & 18.4 & 44.8 \\
SDF \cite{9} & IJCAI'16 & 13.3 & 26.7 & 9.3 & 38.1 & 22.2 & 48.0 & - & - & 14.3 & 37.5 \\
SING \cite{10} & AAAI'18 & \textcolor{blue}{\textbf{50.7}} & \textcolor{blue}{\textbf{75.4}} & 33.5 & 57.0 & 67.7 & 90.7 & 74.4 & 87.8 & 33.5 & 72.7 \\
CSR-GAN \cite{12} & IJCAI'18 & - & - & 37.2 & 62.3 & 70.7 & 92.1 & 76.4 & 88.5 & 32.3 & 70.9 \\
FFSR+RIFE \cite{11} & IJCAI'19 & - & - & 41.6 & 64.9 & 73.3 & 92.6 & - & - & 36.4 & 72.0 \\
RAIN \cite{30} & AAAI'19 & - & - & 42.5 & 68.3 & 78.9 & 97.3 & - & - & 42.0 & 77.3 \\
CDA-Net \cite{29} & ICCV'19 & - & - & 43.1 & 68.2 & 82.1 & 97.4 & 83.7 & 92.7 & 42.8 & 76.2 \\
PCB+PRI \cite{43} & ECCV'20 & - & - & - & - & 86.2 & \textcolor{red}{\textbf{97.9}} & 88.1 & 94.2 & \textcolor{blue}{\textbf{44.3}} & \textcolor{red}{\textbf{83.7}} \\
INTACT \cite{32} & CVPR'20 & - & - & \textcolor{blue}{\textbf{46.2}} & \textcolor{blue}{\textbf{73.1}} & \textcolor{blue}{\textbf{86.4}} & \textcolor{blue}{\textbf{97.4}} & \textcolor{blue}{\textbf{88.1}} & \textcolor{blue}{\textbf{95.0}} & 44.0 & 81.8 \\
\hline
MRJL (Ours) & & \textcolor{red}{\textbf{73.0}} & \textcolor{red}{\textbf{87.3}} & \textcolor{red}{\textbf{58.7}} & \textcolor{red}{\textbf{84.1}} & \textcolor{red}{\textbf{90.7}} & 95.7 & \textcolor{red}{\textbf{90.1}} & \textcolor{red}{\textbf{95.6}} & \textcolor{red}{\textbf{61.2}} & \textcolor{blue}{\textbf{82.4}} \\
\hline
\end{tabular}}
\caption{Comparisons of our proposed method to the state-of-the-arts (\%). \textcolor{red}{Red} and \textcolor{blue}{blue} bold numbers indicate the ${1^{{\rm{st}}}}$ and ${2^{{\rm{nd}}}}$ top results.}
\label{tab1}
\end{table*}
\subsection{Datasets}
Five person re-ID datasets are used to evaluate our proposed method, including four synthetic Multiple Low Resolutions (MLR) datasets and one real-world dataset. The generation strategy of MLR datasets refers \cite{10,29,32}. Specifically, we down-sample images from one camera by randomly selecting a down-sampling rate $r \in \left\{ {2,3,4} \right\}$, while the images captured by other camera(s) remain unchanged.
1) \textbf{MLR-SYSU} is constructed from the SYSU \cite{38}. SYSU contains 502 identities captured by 2 cameras, and three images per person are randomly selected for each camera. Half of these identities are for training and half are for testing.
2) \textbf{MLR-VIPeR} is a synthetic version built from the VIPeR \cite{39}. VIPeR contains 632 person image pairs taken by 2 cameras. According to the identity labels, these pairs are divided into 2 non-overlapping halves.
3) \textbf{MLR-CUHK03} is based on the CUHK03 \cite{40}. CUHK03 is composed of five different pairs of camera views, and has 14,097 images of 1,467 identities. We use the 1367/100 training/testing identity split.
4) \textbf{MLR-Market-1501} is built from the Market-1501 \cite{41}. Market-1501 comprises more than 32,000 images of 1,501 identities from 6 cameras, and we utilize 751/750 training/testing identity split.
5) \textbf{CAVIAR} \cite{42} is a challenging real-world person re-ID dataset which contains 1220 images of 72 identities captured by 2 cameras. Among them, 22 persons who appear only in the close camera are discarded. Similar to MLR-VIPeR, we split the remaining images into 2 non-overlapping halves.
\subsection{Implementation Details}
In the training phase, all the input images are resized to $128 \times 256$. A mini-batch has 20 images of 5 persons where 2 HR images (each HR image can generate 3 down-sampled LR versions) and 2 original LR images are selected for each person. Noting that original LR samples are only utilized to train the DFFN module. Hyper-parameters $\lambda$, $\gamma$ and $m$ are set to 100, 1, 0.5, respectively. We select Adam to optimize our model with weight decay $5 \times {10^{ - 4}}$. For parameters in the MSE loss, we set a learning rate of $3 \times {10^{ - 3}}$, and for parameters in the re-ID loss, we set a learning rate of $3 \times {10^{ - 4}}$. Our model is trained for 60 epochs in total, and the learning rates are decreased by 0.1 after 30 epochs.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig6_new.png}
\caption{Examples of feature response maps extracted on different resolution samples. All the cases are classified into three groups.}
\label{fig5}
\end{figure}
\subsection{Comparison with State-of-the-art Approaches}
We compare our method with several recent cross-resolution person re-ID methods, and the comparable results are reported in Table \ref{tab1}, which show that the SR based methods commonly achieve better performance than the traditional methods. One important reason is that these SR based methods aid in the recovery of missing spatial information that contains more discriminative features. In contrast, traditional methods are incapable of recovering the lost information, resulting in poor performance.
From Table \ref{tab1}, we can also observe that our proposed MRJL outperforms the state-of-the-arts by \textbf{22.3\%}, \textbf{12.5\%}, \textbf{4.3\%}, \textbf{2.0\%} and \textbf{17.2\%} in Rank-1 on MLR-SYSU, MLR-VIPeR, MLR-CUHK03, MLR-Market-1501 and CAVIAR, respectively. The performance superiority of our method can be mainly attributed to the joint representations of both HR and LR features. All existing SR based methods only extract features from recovered HR images but ignore the complementary information provided by LR ones.
\subsection{Ablation Study}
\subsubsection{Influence of Resolutions on Feature Extraction}
To investigate the influence of resolution on feature extraction, we conduct the following experiments as shown in Table \ref{tab2}. The variant (1.3) extracts the joint HR and LR feature representations, while the variant (1.1) and variant (1.2) only utilize the single branch in RRN and DFFN. The comparison results confirm two assumptions: 1) LR information also matters for cross-resolution person re-ID. LR features can provide complementary information for HR features and further improve the accuracy of matching. 2) Compared with HR images, networks can extract more discriminative features from LR images in some cases, such as on MLR-SYSU and CAVIAR datasets.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l c c c c c}
\hline
\multirow{2}{*}{Resolution} & \multicolumn{4}{c}{MLR-Datasets (Rank-1)} & CAVIAR \\
\cline{2-5}
& SYSU & VIPeR & CUHK03 & Market-1501 & (Rank-1) \\
\hline
(1.1) HR & 68.0 & 54.0 & \textbf{90.7} & 88.9 & 50.1 \\
(1.2) LR & 70.0 & 48.9 & 88.8 & 88.4 & 53.6 \\
(1.3) HR+LR & \textbf{73.0} & \textbf{58.7} & \textbf{90.7} & \textbf{90.1} & \textbf{61.2} \\
\hline
\end{tabular}}
\caption{Effects of different resolutions(\%).}
\label{tab2}
\end{table}
Figure \ref{fig5} visualizes some feature response maps extracted from different resolution samples which further verify the above viewpoints. We classify the different cases into 3 groups. Most cases are similar to the group (a), which reflects that the network can extract similar features from HR and LR images. Group (b) shows that the network extract more global information from LR images compared with HR ones, and group (c) indicates that HR images make it easier for the network to focus on detail information, such as bags and textures. These experiments can provide a reasonable explanation for the pending phenomenon mentioned in \cite{11} that re-ID model achieves lower accuracy when the recovered images become higher resolution. Although the recovered images obtain better visual quality, they have higher risk to generate false details which may mislead feature extraction. In most cases, LR images can still provide discriminative features for matching.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig5_new.png}
\caption{Visual results of the reconstructed HR and LR images. The group (a) and (b) represent the situations that the input is a LR (query) or HR (gallery) image, respectively.}
\label{fig6}
\end{figure}
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l c c c c c}
\hline
\multirow{2}{*}{Encoder Structure} & \multicolumn{4}{c}{MLR-Datasets (Rank-1)} & CAVIAR \\
\cline{2-5}
& SYSU & VIPeR & CUHK03 & Market-1501 & (Rank-1) \\
\hline
(2.1) Single-Branch & 71.9 & 57.8 & 90.6 & 89.8 & 54.0 \\
(2.2) Multi-Kernel & \textbf{73.0} & \textbf{58.7} & \textbf{90.7} & \textbf{90.1} & \textbf{61.2} \\
\hline
\end{tabular}}
\caption{Effects of different encoder structures in RRN (\%).}
\label{tab3}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{Fig7_new.png}
\caption{The method structures of (un)known resolution situations.}
\label{fig7}
\end{figure}
\subsubsection{Analysis on RRN}
To evaluate the validity of RRN module, we conduct the following ablation experiments. Firstly, we compare the single-branch encoder (variant (2.1)) and our multi-kernel encoder (variant (2.2)), and the comparable results are listed in Table \ref{tab3}. It can be observed that, compared with variant (2.1), variant (2.2) achieves higher accuracy on all datasets. The results prove the effectiveness of multi-kernel structure which can perceive features of different scales.
In addition, we test the different LR standards in the training strategy of RRN. During the training phase of RRN, HR images have the definite standard that the original images without down-sampling. Nevertheless, the LR images have variable down-sampling rates which mean different reference standards for reconstructing LR images, as shown in Figure \ref{fig4}. Table \ref{tab4} reports that it is best to choose LR images with down-sampling rate 3 as the LR reference standard in most datasets except for MLR-CUHK03. The LR standard determines the resolution of reconstructed LR images. The results indicate that the discrimination of LR features will decrease if the reconstructed LR images are too close to HR images or too vague to mine features. Note that MLR-CUHK03 achieves the best performance when the LR standard is $x_4^L$. It means $x_3^L$ cannot provide enough complementary information on MLR-CUHK03, which is also reflected on Table \ref{tab2}, and the better LR feature extraction needs lower resolution of reconstructed LR images. One possible explanation is that the samples in MLR-CUHK03 is relatively clearer, which limit the advantages of LR complementary effects.
To further verify the reconstruction capability of RRN, we visualize the reconstructed images. The two groups in Figure \ref{fig6} reflect that RRN is capable of generating HR and LR images regardless the resolution of the input image.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l c c c c c}
\hline
\multirow{2}{*}{Standard} & \multicolumn{4}{c}{MLR-Datasets (Rank-1)} & CAVIAR \\
\cline{2-5}
& SYSU & VIPeR & CUHK03 & Market-1501 & (Rank-1) \\
\hline
(3.1) $x_2^L$ & 72.8 & 58.1 & 90.4 & 90.0 & 57.2 \\
(3.2) $x_3^L$ & \textbf{73.0} & \textbf{58.7} & 90.7 & \textbf{90.1} & \textbf{61.2} \\
(3.3) $x_4^L$ & 71.7 & 55.9 & \textbf{92.3} & 89.9 & 57.6 \\
\hline
\end{tabular}}
\caption{Effects of different LR reference standards in RRN (\%).}
\label{tab4}
\end{table}
\subsubsection{Analysis on Unknown Resolution Strategy}
We divide the cross-resolution person re-ID into two categories: unknown resolution and known resolution, as illustrated in Figure \ref{fig7}. In the case of known situation, we assume the gallery images are all HR and the query images are all LR, and SR networks only need to pre-process the LR images. In the other case, we treat all the images equally in both gallery and query sets since networks needn’t know image
resolutions. Compared with known resolution case, unknown resolution case has two advantages: 1) Unknown resolution methods don’t need the resolution labels which are hard to quantify with pixel size. 2) The unknown resolution case is closer to the unconstrained scenarios, because resolutions of images will not be divided neatly in some practical applications. Table \ref{tab5} reports that our unknown resolution structure (variant (4.2)) achieves a few improvements, which can be explained that a few LR gallery images are mistaken for HR due to the classification by pixel size in structure (b).
\begin{table}[H]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{l c c c c c}
\hline
\multirow{2}{*}{Method} & \multicolumn{4}{c}{MLR-Datasets (Rank-1)} & CAVIAR \\
\cline{2-5}
& SYSU & VIPeR & CUHK03 & Market-1501 & (Rank-1) \\
\hline
(4.1) Known & 72.6 & 56.5 & 90.6 & 89.6 & 60.8 \\
(4.2) Unknown & \textbf{73.0} & \textbf{58.7} & \textbf{90.7} & \textbf{90.1} & \textbf{61.2} \\
\hline
\end{tabular}}
\caption{Effects of different method structures(\%).}
\label{tab5}
\end{table}
\section{Conclusion}
In this paper, we have investigated into the influence of resolutions on feature extraction, and proposed a Multi-Resolution Representation Joint Learning (MRJL) method to solve the cross-resolution person re-ID problem. By a series of experiments, we explore the effectiveness of LR features which is capable of complementing HR features. According to the inspiration, the MRJL utilizes a Resolution Reconstruction Network (RRN) to generate both HR and LR versions no matter what the input resolution is. Besides, a Dual Feature Fusion Network (DFFN) is designed to extract discriminative multi-resolution representations. Extensive experimental results on five challenging datasets demonstrate the superiority of the MRJL over the relevant state-of-the-art methods.
\section*{Acknowledgements}
This research was conducted in collaboration with Singapore Telecommunications Limited and supported by the Singapore Government through the Industry Alignment Fund - Industry Collaboration Projects Grant (No. NTU 2018-0551).
\bibliographystyle{named}
\section{Introduction}
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\section{Tables}
Tables are considered illustrations containing data. Therefore, they should also appear floated to the top (preferably) or bottom of the page, and with the captions below them.
\begin{table}
\centering
\begin{tabular}{lll}
\hline
Scenario & $\delta$ & Runtime \\
\hline
Paris & 0.1s & 13.65ms \\
Paris & 0.2s & 0.01ms \\
New York & 0.1s & 92.50ms \\
Singapore & 0.1s & 33.33ms \\
Singapore & 0.2s & 23.01ms \\
\hline
\end{tabular}
\caption{Latex default table}
\label{tab:plain}
\end{table}
\begin{table}
\centering
\begin{tabular}{lrr}
\toprule
Scenario & $\delta$ (s) & Runtime (ms) \\
\midrule
Paris & 0.1 & 13.65 \\
& 0.2 & 0.01 \\
New York & 0.1 & 92.50 \\
Singapore & 0.1 & 33.33 \\
& 0.2 & 23.01 \\
\bottomrule
\end{tabular}
\caption{Booktabs table}
\label{tab:booktabs}
\end{table}
If you are using \LaTeX, you should use the {\tt booktabs} package, because it produces better tables than the standard ones. Compare Tables \ref{tab:plain} and~\ref{tab:booktabs}. The latter is clearly more readable for three reasons:
\begin{enumerate}
\item The styling is better thanks to using the {\tt booktabs} rulers instead of the default ones.
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\end{enumerate}
\section{Formulas}
IJCAI's two-column format makes it difficult to typeset long formulas. A usual temptation is to reduce the size of the formula by using the {\tt small} or {\tt tiny} sizes. This doesn't work correctly with the current \LaTeX{} versions, breaking the line spacing of the preceding paragraphs and title, as well as the equation number sizes. The following equation demonstrates the effects (notice that this entire paragraph looks badly formatted):
\begin{tiny}
\begin{equation}
x = \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i
\end{equation}
\end{tiny}%
Reducing formula sizes this way is strictly forbidden. We {\bf strongly} recommend authors to split formulas in multiple lines when they don't fit in a single line. This is the easiest approach to typeset those formulas and provides the most readable output%
\begin{align}
x =& \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \nonumber\\
+ & \prod_{i=1}^n \sum_{j=1}^n j_i
\end{align}%
If a line is just slightly longer than the column width, you may use the {\tt resizebox} environment on that equation. The result looks better and doesn't interfere with the paragraph's line spacing: %
\begin{equation}
\resizebox{.91\linewidth}{!}{$
\displaystyle
x = \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i
$}
\end{equation}%
This last solution may have to be adapted if you use different equation environments, but it can generally be made to work. Please notice that in any case:
\begin{itemize}
\item Equation numbers must be in the same font and size than the main text (10pt).
\item Your formula's main symbols should not be smaller than {\small small} text (9pt).
\end{itemize}
For instance, the formula
\begin{equation}
\resizebox{.91\linewidth}{!}{$
\displaystyle
x = \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j + \prod_{i=1}^n \sum_{j=1}^n j_i + \prod_{i=1}^n \sum_{j=1}^n i_j
$}
\end{equation}
would not be acceptable because the text is too small.
\section{Examples, Definitions, Theorems and Similar}
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In \LaTeX{} We strongly recommend you to define environments for your examples, definitions, propositions, lemmas, corollaries and similar. This can be done in your \LaTeX{} preamble using \texttt{\textbackslash{newtheorem}} -- see the source of this document for examples. Numbering for these items must be global, not per-section (e.g.: Theorem 1 instead of Theorem 6.1).
\begin{example}[How to write an example]
Examples should be written using the example environment defined in this template.
\end{example}
\begin{theorem}
This is an example of an untitled theorem.
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You may also include a title or description using these environments as shown in the following theorem.
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\section{Proofs}
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\section{Algorithms and Listings}
Algorithms and listings are a special kind of figures. Like all illustrations, they should appear floated to the top (preferably) or bottom of the page. However, their caption should appear in the header, left-justified and enclosed between horizontal lines, as shown in Algorithm~\ref{alg:algorithm}. The algorithm body should be terminated with another horizontal line. It is up to the authors to decide whether to show line numbers or not, how to format comments, etc.
In \LaTeX{} algorithms may be typeset using the {\tt algorithm} and {\tt algorithmic} packages, but you can also use one of the many other packages for the task.
\begin{algorithm}[tb]
\caption{Example algorithm}
\label{alg:algorithm}
\textbf{Input}: Your algorithm's input\\
\textbf{Parameter}: Optional list of parameters\\
\textbf{Output}: Your algorithm's output
\begin{algorithmic}[1]
\STATE Let $t=0$.
\WHILE{condition}
\STATE Do some action.
\IF {conditional}
\STATE Perform task A.
\ELSE
\STATE Perform task B.
\ENDIF
\ENDWHILE
\STATE \textbf{return} solution
\end{algorithmic}
\end{algorithm}
\section*{Acknowledgments}
The preparation of these instructions and the \LaTeX{} and Bib\TeX{}
files that implement them was supported by Schlumberger Palo Alto
Research, AT\&T Bell Laboratories, and Morgan Kaufmann Publishers.
Preparation of the Microsoft Word file was supported by IJCAI. An
early version of this document was created by Shirley Jowell and Peter
F. Patel-Schneider. It was subsequently modified by Jennifer
Ballentine and Thomas Dean, Bernhard Nebel, Daniel Pagenstecher,
Kurt Steinkraus, Toby Walsh and Carles Sierra. The current version
has been prepared by Marc Pujol-Gonzalez and Francisco Cruz-Mencia.
|
1,314,259,992,968 | arxiv | \section{Introduction \label{sec:intro}}
The $k$-means algorithm \cite{lloyd82} is a technique for assigning each of a collection of observed data to exactly one of $k$ clusters, each of which has a unique center, in such a way that each observation is assigned to the cluster whose center is closest to that observation in an appropriate sense.
The $k$-means method has traditionally been used with limited scope.
Its usual application has been in Euclidean spaces which restricts its application to finite dimensional problems.
There are relatively few theoretical results using the $k$-means methodology in infinite dimensions of which \cite{biau08,canas12,cuesta07, laloe10, lember03, linder02, tarpey03} are the only papers known to the authors.
In the right framework, post-hoc track estimation in multiple target scenarios with unknown data association can be viewed as a clustering problem and therefore accessible to the $k$-means method.
In such problems one typically has finite-dimensional data, but would wish to estimate infinite dimensional tracks with the added complication of unresolved data association.
It is our aim to propose and characterize a framework for the $k$-means method which can deal with this problem.
A natural question to ask of any clustering technique is whether the estimated clustering stabilizes as more data becomes available.
More precisely, we ask whether certain estimates converge, in an appropriate sense, in the large data limit.
In order to answer this question in our particular context we first establish a related optimization problem and make precise the notion of convergence.
Consistency of estimators for ill-posed inverse problems has been well studied, for example \cite{dashti13, osullivan86}, but without the data association problem.
In contrast to standard statistical consistency results, we do not assume that there exists a structural relationship between the optimization problem and the data-generating process in order to establish convergence to true parameter values in the large data limit; rather, we demonstrate convergence to the solution of a related limiting problem.
This paper shows the convergence of the minimization problem associated with the $k$-means method in a framework that is general enough to include examples where the cluster centers are not necessarily in the same space as the data points.
In particular we are motivated by the application to infinite dimensional problems, e.g. the smoothing-data association problem.
The smoothing-data association problem is the problem of associating data points $\{(t_i,z_i)\}_{i=1}^n \subset [0,1]\times\mathbb{R}^\kappa$ to unknown trajectories $\mu_j:[0,1]\to\mathbb{R}^\kappa$ for $j=1,2,\dots,k$.
By treating the trajectories $\mu_j$ as the cluster centers one may approach this problem using the $k$-means methodology.
The comparison of data points to cluster centers is a pointwise distance: $d((t_i,z_i),\mu_j)=|\mu_j(t_i)-z_i|^2$ (where $|\cdot|$ is the Euclidean norm on $\mathbb{R}^\kappa$).
To ensure the problem is well-posed some regularization is also necessary.
For $k=1$ the problem reduces to smoothing and coincides with the limiting problem studied in \cite{hall05}.
We will discuss the smoothing-data association problem more in Section~\ref{sec:MTT:app}.
Let us now introduce the notation for our variational approach.
The $k$-means method is a strategy for partitioning a data set $\Psi_n = \{\xi_i\}_{i=1}^n \subset X$ into $k$ clusters where each cluster has center $\mu_j$ for $j=1,2,\dots, k$.
First let us consider the special case when $\mu_j\in X$.
The data partition is defined by associating each data point with the cluster center closest to it which is measured by a cost function $d:X\times X \to [0,\infty)$.
Traditionally the $k$-means method considers Euclidean spaces $X=\mathbb{R}^\kappa$, where typically we choose $d(x,y)=|x-y|^2=\sum_{i=1}^\kappa(x_i-y_i)^2$.
We define the energy for a choice of cluster centers given data by
\begin{align*}
f_n: X^k & \to \mathbb{R} &
f_n(\mu|\Psi_n) & = \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i,\mu_j),
\end{align*}
where for any $k$ variables, $a_1, a_2,\dots, a_k$,
\( \bigwedge_{j=1}^k a_j := \min\{a_1,\ldots,a_k\}. \)
The optimal choice of $\mu$ is that which minimizes $f_n(\cdot|\Psi_n)$.
We define
\begin{equation*}
\hat{\theta}_n = \min_{\mu\in X^k} f_n(\mu|\Psi_n) \in \mathbb{R}.
\end{equation*}
An associated ``limiting problem'' can be defined
\begin{equation*}
\theta = \min_{\mu\in X^k} f_\infty(\mu)
\end{equation*}
where we assume, in a sense which will be made precise later, that $\xi_i\stackrel{\mathrm{iid}}{\sim} P$ for some suitable probability distribution, $P$, and define
\[ f_\infty(\mu) = \int \bigwedge_{j=1}^k d(x,\mu_j) P(\text{d} x). \]
In Section \ref{sec:cons} we validate the formulation by first showing that, under regularity conditions and with probability one, the minimum energy converges: $\hat{\theta}_n \to \theta$.
And secondly by showing that (up to a subsequence) the minimizers converge: $\mu^n \to \mu^\infty$ where $\mu^n$ minimizes $f_n$ and $\mu^\infty$ minimizes $f_\infty$ (again with probability one).
In a more sophisticated version of the $k$-means method the requirement that $\mu_j \in X$ can be relaxed.
We instead allow $\mu=(\mu_1,\mu_2,\dots,\mu_k)\in Y^k$ for some other Banach
space, $Y$, and define $d$ appropriately.
This leads to interesting statistical questions.
When $Y$ is infinite dimensional even establishing whether or not a minimizer exists is non-trivial.
When the cluster center is in a different space to the data, bounding the set of minimizers becomes less natural.
For example, consider the smoothing problem in which one wishes to fit a continuous function to a set of data points.
The natural choice of cost function is a pointwise distance of the data to the curve.
The optimal solution is for the cluster center to interpolate the data points: in the limit the cluster center may no longer be well defined.
In particular we cannot hope to have converging sequences of minimizers.
In the smoothing literature this problem is prevented by using a regularization term $r:Y^k\to \mathbb{R}$.
For a cost function $d:X\times Y\to [0,\infty)$ the energies $f_n(\cdot|\Psi_n),f_\infty(\cdot):Y^k\to \mathbb{R}$ are redefined
\begin{align*}
f_n(\mu|\Psi_n) & = \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i,\mu_j) + \lambda_n r(\mu) \\
f_\infty(\mu) & = \int \bigwedge_{j=1}^k d(x,\mu_j) P(\text{d} x) + \lambda r(\mu).
\end{align*}
Adding regularization changes the nature of the problem so we commit time in Section \ref{sec:MTT} to justifying our approach.
Particularly we motivate treating $\lambda_n = \lambda$ as a constant independent of $n$.
We are able to repeat the analysis from Section 4; that is to establish that the minimum and a subsequence of minimizers still converge.
Early results assumed $Y=X$ were Euclidean spaces and showed the convergence of minimizers to the appropriate limit \cite{hartigan78,pollard81}.
The motivation for the early work in this area was to show consistency of the methodology.
In particular this requires there to be an underlying `truth'.
This requires the assumption that there exists a unique minimizer to the limiting energy.
These results do not hold when the limiting energy has more than one minimizer \cite{ben-david07}.
In this paper we discuss only the convergence of the method and as such require no assumption as to the existence or uniqueness of a minimizer to the limiting problem.
Consistency has been strengthened to a central limit theorem in \cite{pollard82} also assuming a unique minimizer to the limiting energy.
Other rates of convergence have been shown in \cite{antos05, bartlett98, chou94, linder94}.
In Hilbert spaces there exist convergence results and rates of convergence for the minimum.
In~\cite{biau08} the authors show that $|f_n(\mu^n)-f_\infty(\mu^\infty)|$ is of order $\frac{1}{\sqrt{n}}$, however, there are no results for the convergence of minimizers.
Results exist for $k\to \infty$, see for example~\cite{canas12} (which are also valid for $Y\neq X$).
Assuming that $Y=X$, the convergence of the minimization problem in a reflexive and separable Banach space has been proved in \cite{linder02} and a similar result in metric spaces in \cite{lember03}.
In \cite{laloe10}, the existence of a weakly converging subsequence was inferred using the results of \cite{linder02}.
In the following section we introduce the notation and preliminary material used in this paper.
We then, in Section~\ref{sec:cons}, consider convergence in the special case when the cluster centers are in the same space as the data points, i.e. $Y=X$.
In this case we don't have an issue with well-posedness as the data has the same dimension as the cluster centers.
For this reason we use energies defined without regularization.
Theorem~\ref{thm:cons} shows that the minimum converges, i.e. $\hat{\theta}_n\to \theta$ as $n\to \infty$, for almost every sequence of observations and furthermore we have a subsequence $\mu^{n_m}$ of minimizers of $f_{n_m}$ which weakly converge to some $\mu^\infty$ which minimizes $f_\infty$.
This result is generalized in Section~\ref{sec:MTT} to an arbitrary $X$ and $Y$.
The analogous result to Theorem~\ref{thm:cons} is Theorem~\ref{thm:kmeanscons}.
We first motivate the problem and in particular our choice of scaling in the regularization in Section \ref{sec:MTT:reg} before proceeding to the results in Section \ref{sec:MTT:theory}.
Verifying the conditions on the cost function $d$ and regularization term $r$ is non-trivial and so we show an application to the smoothing-data association problem in Section \ref{sec:MTT:app}.
To demonstrate the generality of the results in this paper, two applications are considered in Section~\ref{sec:examples}.
The first is the data association and smoothing problem. We show the minimum converging as the data size increases.
We also numerically investigate the use of the $k$-means energy to determine whether two targets have crossed tracks.
The second example uses measured times of arrival and amplitudes of signals
from moving sources that are received across a network of three sensors. The
cluster centers are the source trajectories in $\mathbb{R}^2$.
\section{Preliminaries \label{sec:not}}
In this section we introduce some notation and background theory which will be used in Sections~\ref{sec:cons} and \ref{sec:MTT} to establish our convergence results.
In these sections we show the existence of optimal cluster centers using the direct method.
By imposing conditions, such that our energies are weakly lower semi-continuous, we can deduce the existence of minimizers.
Further conditions ensure the minimizers are uniformly bounded.
The $\Gamma$-convergence framework (e.g. \cite{braides02,dalmaso93}) allows us to establish the convergence of the minimum and also the convergence of minimizers.
We have the following definition of $\Gamma$-convergence with respect to weak convergence.
\begin{mydef}[$\Gamma$-convergence]
\label{def:gamcon}
A sequence $f_n :A\to \mathbb{R}\cup \{\pm\infty\}$ on a Banach space $(A,\|\cdot\|_A)$ is said to \textit{$\Gamma$-converge} on the domain $A$ to $f_\infty :A\to \mathbb{R}\cup \{\pm\infty\}$ with respect to weak convergence on $A$, and we write $f_\infty = \Gamma\text{-}\lim_n f_n$, if for all $x\in A$ we have
\begin{itemize}
\item[(i)] (liminf inequality) for every sequence $(x_n)$ weakly converging to $x$
\[ f_\infty(x) \leq \liminf_n f_n(x_n); \]
\item[(ii)] (recovery sequence) there exists a sequence $(x_n)$ weakly converging to $x$ such that
\[ f_\infty(x) \geq \limsup_n f_n(x_n). \]
\end{itemize}
\end{mydef}
When it exists the $\Gamma$-limit is always weakly lower semi-continuous, and thus admits minimizers.
An important property of $\Gamma$-convergence is that it implies the convergence of minimizers.
In particular, we will make extensive use of the following well-known result.
\begin{theorem}[Convergence of Minimizers]
\label{thm:conmin}
Let $f_n: A\to \mathbb{R}$ be a sequence of functionals on a Banach space $(A,\|\cdot\|_A)$ and assume that there exists $N>0$ and a weakly compact subset $K\subset A$ with
\[ \inf_A f_n = \inf_K f_n \quad \forall n>N. \]
If $f_\infty = \Gamma\text{-}\lim_n f_n$ and $f_\infty$ is not identically $\pm\infty$ then
\[ \min_A f_\infty = \lim_n \inf_A f_n. \]
Furthermore if each $f_n$ is weakly lower semi-continuous then for each $f_n$ there exists a minimizer $x_n\in K$ and any weak limit point of $x_n$ minimizes $f_\infty$.
Since $K$ is weakly compact there exists at least one weak limit point.
\end{theorem}
A proof of the theorem can be found in \cite[Theorem 1.21]{braides02}.
The problems which we address involve random observations.
We assume throughout the existence of a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, rich enough to support a countably infinite
sequence of such observations, $\xi_1^{(\omega)},\ldots$.
All random elements are defined upon this common probability space and all stochastic quantifiers are to be understood as acting with respect to $\mathbb{P}$ unless otherwise stated.
Where appropriate, to emphasize the randomness of the functionals $f_n$, we will write $f^{(\omega)}_n$ to indicate the functional associated with the particular observation sequence $\xi_1^{(\omega)},\ldots,\xi_n^{(\omega)}$ and we allow $P_n^{(\omega)}$ to denote the associated empirical measure.
We define the support of a (probability) measure to be the smallest closed set such that the complement is null.
For clarity we often write integrals using operator notation.
I.e. for a measure $P$, which is usually a probability distribution, we write
\[ Ph = \int h(x) \; P(\text{d}x). \]
For a sequence of probability distributions, $P_n$, we say that $P_n$ converges weakly to $P$ if
\[ P_nh \to Ph \quad \quad \text{for all bounded and continuous } h \]
and we write $P_n\Rightarrow P$.
With a slight abuse of notation we will sometimes write $P(U):=P\mathbb{I}_U$ for a measurable set $U$.
For a Banach space $A$ one can define the dual space $A^*$ to be the space of all bounded and linear maps over $A$ into $\mathbb{R}$ equipped with the norm $\|F\|_{A^*} = \sup_{x\in A} |F(x)|$.
Similarly one can define the second dual $A^{**}$ as the space of all bounded and linear maps over $A^*$ into $\mathbb{R}$.
Reflexive spaces are defined to be spaces $A$ such that $A$ is isometrically isomorphic to $A^{**}$.
These have the useful property that closed and bounded sets are weakly compact.
For example any $L^p$ space (with $1<p<\infty$) is reflexive, as is any Hilbert space (by the Riesz Representation Theorem: if $A$ is a Hilbert space then $A^*$ is isometrically isomorphic to $A$).
A sequence $x_n\in A$ is said to weakly convergence to $x\in A$ if $F(x_n)\to F(x)$ for all $F\in A^*$.
We write $x_n\rightharpoonup x$.
We say a functional $G:A\to \mathbb{R}$ is weakly continuous if $G(x_n)\to G(x)$ whenever $x_n \rightharpoonup x$ and strongly continuous if $G(x_n)\to G(x)$ whenever $\|x_n-x\|_A\to 0$.
Note that weak continuity implies strong continuity.
Similarly a functional $G$ is weakly lower semi-continuous if $\liminf_{n\to \infty} G(x_n)\geq G(x)$ whenever $x_n\rightharpoonup x$.
We define the Sobolev spaces $W^{s,p}(I)$ on $I\subseteq \mathbb{R}$ by
\[ W^{s,p} = W^{s,p}(I) = \left\{ f:I\to \mathbb{R} \text{ s.t. } \partial^i f \in L^p(I) \text{ for } i=0,\dots, s \right\} \]
where we use $\partial$ for the weak derivative, i.e. $g=\partial f$ if for all $\phi\in C_c^\infty(I)$ (the space of smooth functions with compact support)
\[ \int_I f(x) \frac{\mathrm{d}\phi}{\mathrm{d}x}(x) \; \text{d} x = - \int_I g(x) \phi(x) \; \text{d} x. \]
In particular, we will use the special case when $p=2$ and we write $H^s=W^{s,2}$.
This is a Hilbert space with norm:
\[ \|f\|_{H^s}^2 = \sum_{i=0}^s \| \partial^i f\|_{L^2}^2. \]
For two real-valued and positive sequences $a_n$ and $b_n$ we write $a_n \lesssim b_n$ if $\frac{a_n}{b_n}$ is bounded.
For a space $A$ and a set $K\subset A$ we write $K^c$ for the complement of $K$ in $A$, i.e. $K^c=A\setminus K$.
\section{Convergence when \texorpdfstring{$Y=X$}{Y=X} \label{sec:cons}}
We assume we are given data points $\xi_i\in X$ for $i=1,2,\dots$ where $X$ is a reflexive and separable Banach space with norm $\|\cdot\|_X$ and Borel $\sigma$-algebra $\mathcal{X}$.
These data points realize a sequence of $\mathcal{X}$-measurable random elements on $(\Omega, \mathcal{F}, \mathbb{P})$ which will also be denoted, with a slight abuse of notation, $\xi_i$.
We define
\begin{align}
f_n^{(\omega)}:X^k & \to \mathbb{R}, \quad f_n^{(\omega)}(\mu) = P_n^{(\omega)} g_{\mu} = \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i^{(\omega)},\mu_j) \label{eq:fn} \\
f_\infty:X^k & \to \mathbb{R}, \quad f_\infty (\mu) = P g_{\mu} = \int_X \bigwedge_{j=1}^k d(x,\mu_j) P(\text{d} x) \label{eq:finfty}
\end{align}
where
\[ g_{\mu}(x) = \bigwedge_{j=1}^k d(x,\mu_j), \]
$P$ is a probability measure on $(X,\mathcal{X})$, and empirical measure
$P_n^{(\omega)}$ associated with $\xi_1^{(\omega)},\ldots,\xi_n^{(\omega)}$ is defined by
\[ P_n^{(\omega)} h = \frac{1}{n} \sum_{i=1}^n h(\xi_i^{(\omega)}) \]
for any $\mathcal{X}$-measurable function $h: X\to \mathbb{R}$.
We assume $\xi_i$ are iid according to $P$ with $P=\mathbb{P}\circ \xi^{-1}_i$.
We wish to show
\begin{equation} \label{eq:limeq}
\hat{\theta}_n^{(\omega)} \to \theta \quad \text{for almost every } \omega \text{ as } n\to \infty
\end{equation}
where
\begin{align*}
\hat{\theta}_n^{(\omega)} & = \inf_{\mu\in X^k} f_n^{(\omega)}(\mu) \\
\theta & = \inf_{\mu\in X^k} f_\infty(\mu).
\end{align*}
We define $\|\cdot\|_k :X^k\to [0,\infty)$ by
\begin{equation} \label{eq:cons:norm}
\| \mu\|_k := \max_j \|\mu_j\|_X \quad \text{for } \mu=(\mu_1, \mu_2,\dots, \mu_k) \in X^k.
\end{equation}
The reflexivity of $(X,\|\cdot\|_X)$ carries through to $(X^k,\|\cdot\|_k)$.
Our strategy is similar to that of~\cite{pollard81} but we embed the methodology into the $\Gamma$-convergence framework.
We show that \eqref{eq:finfty} is the $\Gamma$-limit in Theorem~\ref{thm:gamcon} and that minimizers are bounded in Proposition~\ref{lem:bdd}.
We may then apply Theorem~\ref{thm:conmin} to infer \eqref{eq:limeq} and the existence of a weakly converging subsequence of minimizers.
The key assumptions on $d$ and $P$ are given in Assumptions~\ref{ass:d}.
The first assumption can be understood as a `closeness' condition for the space $X$ with respect to $d$.
If we let $d(x,y)=1$ for $x\neq y$ and $d(x,x)=0$ then our cost function $d$ does not carry any information on how far apart two points are.
Assume there exists a probability density for $P$ which has unbounded support.
Then $f_n^{(\omega)}(\mu)\geq \frac{n-k}{n}$ (for almost every $\omega$), with equality when we choose $\mu_j\in \{\xi_i^{(\omega)}\}_{i=1}^n$.
I.e. any set of $k$ unique data points will minimize $f_n^{(\omega)}$.
Since our data points are unbounded we may find a sequence $\|\xi_{i_n}^{(\omega)}\|_X\to \infty$.
Now we choose $\mu_1^n=\xi_{i_n}^{(\omega)}$ and clearly our cluster center is unbounded.
We see that this choice of $d$ violates the first assumption.
We also add a moment condition to the upper bound to ensure integrability.
Note that this also implies that $Pd(\cdot,0)\leq \int_X M(\|x\|) \; P(\text{d}x)<\infty$ so
$f_\infty(0)<\infty$ and, in particular, that $f_\infty$ is not identically infinity.
The second assumption is slightly stronger condition on $d$ than a weak lower semi-continuity condition in the first variable and strong continuity in the second variable.
The condition allows the application of Fatou's lemma for weakly converging probabilities, see~\cite{feinberg14}.
The third assumption allows us to view $d(\xi_i,y)$ as a collection of random variables.
The fourth implies that we have at least $k$ open balls with positive probability and therefore we are not overfitting clusters to data.
\begin{assumptions} \label{ass:d}
We have the following assumptions on $d:X\times X\to [0,\infty)$ and $P$.
\begin{enumerate}
\item[1.1.] \label{ass:d:norm} There exist continuous, strictly increasing functions $m,M:[0,\infty)\to [0,\infty)$ such that
\[ m(\|x-y\|_X) \leq d(x,y) \leq M(\|x-y\|_X) \quad \text{for all } x,y\in X \]
with $\lim_{r\to \infty}m(r)= \infty$, $M(0)=0$, there exists $\gamma<\infty$ such that $M(\|x+y\|_X)\leq \gamma M(\|x\|_X) + \gamma M(\|y\|_X)$ and finally $\int_X M(\|x\|_X) \; P(\text{d}x) < \infty$ (and $M$ is measurable).
\item[1.2.] \label{ass:d:cts} For each $x,y\in X$ we have that if $x_m\to x$ and $y_n\rightharpoonup y$ as $n,m\to \infty$ then
\[ \liminf_{n,m\to \infty} d(x_m,y_n) \geq d(x,y) \quad \text{and} \quad \lim_{m\to \infty} d(x_m,y) = d(x,y). \]
\item[1.3.] \label{ass:d:meas} For each $y\in X$ we have that $d(\cdot,y)$ is $\mathcal{X}$-measurable.
\item[1.4.] \label{ass:d:fit} There exist $k$ different centers $\mu^\dagger_j\in X$, $j=1,2,\dots,k$ such that for all $\delta>0$
\[ P(B(\mu_j^\dagger,\delta)) > 0 \quad \quad \quad \forall \; j=1,2,\dots,k \]
where $B(\mu,\delta):=\{x\in X: \|\mu-x\|_X< \delta \}$.
\end{enumerate}
\end{assumptions}
We now show that for a particular common choice of cost function, $d$, Assumptions~\ref{ass:d}.1 to~\ref{ass:d}.3 hold.
\begin{remark} \label{rem:dcond}
For any $p>0$ let $d(x,y)=\|x-y\|_X^p$ then $d$ satisfies Assumptions~\ref{ass:d}.1 to~\ref{ass:d}.3.
\end{remark}
\begin{proof}
Taking $m(r)=M(r)=r^p$ we can bound $m(\|x-y\|_X)\leq d(x,y) \leq M(\|x-y\|_X)$ and $m,M$ clearly satisfy $m(r)\to \infty$, $M(0)=0$, are strictly increasing and continuous.
One can also show that
\[ M(\|x+y\|_X) \leq 2^{p-1} \left( \|x\|^p_X + \|y\|^p_X \right) \]
hence Assumption~\ref{ass:d}.1 is satisfied.
Let $x_m\to x$ and $y_n\rightharpoonup y$.
Then
\begin{align*}
\liminf_{n,m\to\infty} d(x_m,y_n)^{\frac{1}{p}} & = \liminf_{n,m\to \infty} \|x_m-y_m\|_X \\
& \geq \liminf_{n,m\to \infty} \left( \|y_n-x\|_X -\|x_m-x\|_X \right) \\
& = \liminf_{n\to \infty} \|y_n-x\|_X \quad \text{since } x_m\to x \\
& \geq \|y-x\|_X
\end{align*}
where the last inequality follows as a consequence of the Hahn-Banach Theorem and the fact that $y_n-x\rightharpoonup y-x$ which implies $\liminf_{n\to \infty} \|y_n-x\|_X\geq \|y-x\|_X$.
Clearly $d(x_m,y)\to d(x,y)$ and so Assumption~\ref{ass:d}.2 holds.
The third assumption holds by the Borel measurability of metrics on complete separable metric spaces.
\end{proof}
We now state the first result of the paper which formalizes the understanding that $f_\infty$ is the limit of $f_n^{(\omega)}$.
\begin{theorem}
\label{thm:gamcon}
Let $(X,\|\cdot\|_X)$ be a reflexive and separable Banach space with Borel $\sigma$-algebra, $\mathcal{X}$; let $\{\xi_i\}_{i\in\mathbb{N}}$ be a sequence of independent $X$-valued random elements with common law $P$.
Assume $d:X\times X\to [0,\infty)$ and that $P$ satisfies the conditions in Assumptions~\ref{ass:d}.
Define $f^{(\omega)}_n:X^k \to\mathbb{R}$ and $f_\infty:X^k\to\mathbb{R}$ by \eqref{eq:fn} and \eqref{eq:finfty} respectively.
Then
\[ f_\infty = \Gamma\text{-}\lim_n f^{(\omega)}_n \]
for $\mathbb{P}$-almost every $\omega$.
\end{theorem}
\begin{proof}
Define $\Omega^\prime$ as the intersection of three events:
\begin{align*}
\Omega^\prime & = \left\{ \omega\in \Omega : P_n^{(\omega)}\Rightarrow P \right\} \cap \left\{ \omega\in \Omega : P_n^{(\omega)}(B(0,q)^c) \to P(B(0,q)^c) \; \forall q\in \mathbb{N} \right\} \\
& \quad \quad \quad \quad \quad \cap \left\{ \omega\in \Omega : \int_X \mathbb{I}_{B(0,q)^c}(x)M(\|x\|_X) \; P_n^{(\omega)} (\text{d} x) \to \int_X \mathbb{I}_{B(0,q)^c}(x) M(\|x\|_X) \; P(\text{d} x) \; \forall q\in \mathbb{N} \right\}.
\end{align*}
By the almost sure weak convergence of the empirical measure the
first of these events has probability one, the second and third are
characterized by the convergence of a countable collection of empirical averages to their
population average and, by the strong law of large numbers, each has probability
one. Hence $\mathbb{P}(\Omega^\prime)=1$.
Fix $\omega \in \Omega^\prime$: we will show that the lim inf inequality holds and
a recovery sequence exists for this $\omega$ and hence for every $\omega\in\Omega^\prime$.
We start by showing the lim inf inequality, allowing $\{\mu^n\}_{n=1}^\infty\in X^k$ to denote any sequence which converges weakly to $\mu\in X^k$.
We are required to show:
\[ \liminf_{n\to \infty} f_n^{(\omega)}(\mu^n) \geq f_\infty (\mu). \]
By Theorem~1.1 in~\cite{feinberg14} we have
\[ \int_X \liminf_{n\to\infty, x^\prime\to x} g_{\mu^n}(x^\prime) \; P(\text{d} x) \leq \liminf_{n\to \infty} \int_X g_{\mu^n}(x) \; P_n^{(\omega)}(\text{d}x) = \liminf_{n\to\infty} P_n^{(\omega)} g_{\mu^n}. \]
For each $x\in X$, we have by Assumption~\ref{ass:d}.2 that
\[ \liminf_{x^\prime\to x,n\to \infty} d(x^\prime,\mu_j^n) \geq d(x,\mu_j). \]
By taking the minimum over $j$ we have
\[ \liminf_{x^\prime\to x,n\to \infty} g_{\mu^n}(x^\prime) = \bigwedge_{j=1}^k \liminf_{x^\prime\to x,n\to \infty} d(x^\prime,\mu_j^n) \geq \bigwedge_{j=1}^k d(x,\mu_j) = g_{\mu}(x). \]
Hence
\[ \liminf_{n\to \infty} f_n^{(\omega)}(\mu^n) = \liminf_{n\to \infty} P_n^{(\omega)} g_{\mu^n} \geq \int_X g_{\mu}(x) \; P(\text{d} x) = f_\infty(\mu) \]
as required.
We now establish the existence of a recovery sequence for every $\omega \in \Omega^\prime$ and every $\mu \in X^k$.
Let $\mu^n=\mu\in X^k$.
Let $\zeta_q$ be a $C^\infty(X)$ sequence of functions such that $0\leq \zeta_q(x) \leq 1$ for all $x\in X$, $\zeta_q(x) = 1$ for $x\in B(0,q-1)$ and $\zeta_q(x) = 0$ for $x\not\in B(0,q)$.
Then the function $\zeta_q(x)g_\mu(x)$ is continuous in $x$ (and with respect to convergence in $\|\cdot\|_X$) for all $q$.
We also have
\begin{align*}
\zeta_q(x) g_\mu(x) & \leq \zeta_q(x) d(x,\mu_1) \\
& \leq \zeta_q(x) M(\|x-\mu_1\|_X) \\
& \leq \zeta_q(x) M(\|x\|_X + \|\mu_1\|_X) \\
& \leq M(q + \|\mu_1\|_X)
\end{align*}
so $\zeta_q g_\mu$ is a continuous and bounded function, hence by the weak convergence of $P_n^{(\omega)}$ to $P$ we have
\[ P_n^{(\omega)} \zeta_q g_\mu \to P \zeta_q g_\mu \]
as $n\to \infty$ for all $q\in\mathbb{N}$.
For all $q\in\mathbb{N}$ we have
\begin{align*}
\limsup_{n\to \infty} |P_n^{(\omega)} g_\mu - Pg_\mu | & \leq \limsup_{n\to \infty} |P_n^{(\omega)} g_\mu - P_n^{(\omega)}\zeta_q g_\mu | + \limsup_{n\to \infty} |P_n^{(\omega)} \zeta_q g_\mu - P \zeta_q g_\mu | + \limsup_{n\to \infty} |P \zeta_q g_\mu - Pg_\mu | \\
& = \limsup_{n\to \infty} |P_n^{(\omega)} g_\mu - P_n^{(\omega)}\zeta_q g_\mu | + |P \zeta_q g_\mu - Pg_\mu |.
\end{align*}
Therefore,
\[ \limsup_{n\to \infty} |P_n^{(\omega)} g_\mu - Pg_\mu | \leq \liminf_{q\to \infty} \limsup_{n\to \infty} |P_n^{(\omega)} g_\mu - P_n^{(\omega)}\zeta_q g_\mu | \]
by the dominated convergence theorem.
We now show that the right hand side of the above expression is equal to zero.
We have
\begin{align*}
|P_n^{(\omega)} g_\mu - P_n^{(\omega)}\zeta_q g_\mu | & \leq P_n^{(\omega)} \mathbb{I}_{(B(0,q-1))^c} g_\mu \\
& \leq P_n^{(\omega)} \mathbb{I}_{(B(0,q-1))^c} d(\cdot,\mu_1) \\
& \leq P_n^{(\omega)} \mathbb{I}_{(B(0,q-1))^c} M(\|\cdot-\mu_1\|_X) \\
& \leq \gamma \left(P_n^{(\omega)} \mathbb{I}_{(B(0,q-1))^c}M(\|\cdot\|_X) + M(\|\mu_1\|_X) P_n^{(\omega)} \mathbb{I}_{(B(0,q-1))^c} \right) \\
& \to \gamma \left( P \mathbb{I}_{(B(0,q-1))^c}M(\|\cdot\|_X) + M(\|\mu_1\|_X) P \mathbb{I}_{(B(0,q-1))^c} \right) \quad \text{as } n\to \infty \\
& \to 0 \quad \text{as } q \to \infty
\end{align*}
where the last limit follows by the monotone convergence theorem.
We have shown
\[ \lim_{n\to \infty} |P_n^{(\omega)} g_\mu - Pg_\mu | = 0. \]
Hence
\[ f_n^{(\omega)}(\mu) \to f_\infty(\mu) \]
as required.
\end{proof}
Now we have established almost sure $\Gamma$-convergence we establish the boundedness condition in Proposition~\ref{lem:bdd} so we can apply Theorem~\ref{thm:conmin}.
\begin{proposition}
\label{lem:bdd}
Assuming the conditions of Theorem~\ref{thm:gamcon} and define $\|\cdot\|_k$ by~\eqref{eq:cons:norm}, there exists $R>0$ such that
\[ \inf_{\mu\in X^k} f_n^{(\omega)}(\mu) = \inf_{\|\mu\|_k\leq R} f_n^{(\omega)}(\mu) \quad \forall n \text{ sufficiently large} \]
for $\mathbb{P}$-almost every $\omega$.
In particular $R$ is independent of $n$.
\end{proposition}
\begin{proof}
The structure of the proof is similar to \cite[Lemma 2.1]{lember03}.
We argue by contradiction.
In particular we argue that if a cluster center is unbounded then in the limit the minimum is achieved over the remaining $k-1$ cluster centers.
We then use Assumption~\ref{ass:d}.4 to imply that adding an extra cluster center will strictly decrease the minimum, and hence we have a contradiction.
We define $\Omega^{\prime\prime}$ to be
\[ \Omega^{\prime\prime} = \cap_{\delta\in\mathbb{Q}\cap(0,\infty), l=1,2,\dots,k} \left\{\omega\in\Omega^\prime: P_n^{(\omega)} (B(\mu^\dagger_l,\delta))\to P(B(\mu^\dagger_l,\delta)) \right\}. \]
As $\Omega^{\prime\prime}$ is the countable intersection of sets of probability one, we have $\mathbb{P}(\Omega^{\prime\prime})=1$.
Fix $\omega\in\Omega^{\prime\prime}$ and assume that the cluster centers $\mu^n\in X^k$ are almost minimizers, i.e.
\[ f_n^{(\omega)}(\mu^n) \leq \inf_{\mu\in X^k} f_n^{(\omega)}(\mu)+
\varepsilon_n \]
for some sequence $\varepsilon_n>0$ such that
\begin{equation} \label{nullseq}
\lim_{n\to \infty} \varepsilon_n=0.
\end{equation}
Assume that $\lim\limits_{n \to \infty}\|\mu^n\|_{k}= \infty$. There exists $l_n \in \{1,\ldots,k\}$ such that $\lim\limits_{n \to \infty}\|\mu^n_{l_n}\|_X= \infty$.
Fix $x\in X$ then
\[ d(x,\mu^n_{l_n}) \geq m(\|\mu^n_{l_n}-x\|_X) \to \infty. \]
Therefore, for each $x\in X$,
\[ \lim_{n \to \infty} \left(\bigwedge_{j=1}^k d(x,\mu^n_{j}) - \bigwedge_{j\neq l_n} d(x,\mu^n_j)\right)= 0. \]
Let $\delta>0$ then there exists $N$ such that for $n\geq N$
\[ \bigwedge_{j=1}^k d(x,\mu^n_{j}) - \bigwedge_{j\neq l_n} d(x,\mu^n_j) \geq -\delta. \]
Hence
\[ \liminf_{n\to \infty} \int \left( \bigwedge_{j=1}^k d(x,\mu^n_j) - \bigwedge_{j\neq l_n} d(x,\mu^n_{j}) \right) \; P_n^{(\omega)}(\text{d} x) \geq -\delta. \]
Letting $\delta\to 0$ we have
\[ \liminf_{n\to \infty} \int \left( \bigwedge_{j=1}^k d(x,\mu^n_j) - \bigwedge_{j\neq l_n} d(x,\mu^n_{j}) \right) \; P_n^{(\omega)}(\text{d} x) \geq 0 \]
and moreover
\begin{equation} \label{eq:contra}\liminf_{n\to \infty} \left( f_n^{(\omega)}\left(\mu^{n}\right) - f_n^{(\omega)}\left((\mu^n_j)_{j\neq l_n} \right)\right)\geq 0,
\end{equation}
where we interpret $f_n^{(\omega)}$ accordingly.
It suffices to demonstrate that
\begin{equation} \label{eq:difffnl}
\liminf_{n\to \infty}\left(\inf_{\mu \in X^k} f_n^{(\omega)}(\mu) - \inf_{\mu\in X^{k-1}} f_n^{(\omega)}(\mu)\right) <0.
\end{equation}
Indeed, if \eqref{eq:difffnl} holds, then
\begin{align*}
& \liminf_{n\to \infty} \left( f_{n}^{(\omega)}\left(\mu^{n}\right) - f_{n}^{(\omega)}\left((\mu^n_j)_{j\neq l_n} \right)\right) \\
= & \lim_{n\to \infty}\bigl(
\underbrace{ f_{n}^{(\omega)}\left(\mu^{n}\right) - \inf_{\mu \in X^{k}} f_n^{(\omega)}(\mu)}_{\leq \varepsilon_n}\bigr) + \liminf_{n\to \infty}\left(\inf_{\mu \in X^{k}} f_n^{(\omega)}(\mu)- f_{n}^{(\omega)}\left((\mu^n_j)_{j\neq l_n} \right)\right)\\
< & 0 \quad \text{by \eqref{nullseq} and \eqref{eq:difffnl}},
\end{align*}
but this contradicts \eqref{eq:contra}.
We now establish \eqref{eq:difffnl}.
By Assumption~\ref{ass:d}.4 there exists $k$ centers $\mu_j^\dagger\in X$ and $\delta_1>0$ such that $\min_{j\neq l} \|\mu^\dagger_j - \mu_l^\dagger\|_X \geq \delta_1$.
Hence for any $\mu\in X^{k-1}$ there exists $l\in\{1,2,\dots,k\}$ such that we have
\[ \| \mu^\dagger_l - \mu_j \|_X \geq \frac{\delta_1}{2} \quad \quad \text{for } j=1,2,\dots,k-1. \]
Proceeding with this choice of $l$, for $x\in B(\mu^\dagger_l,\delta_2)$ (for any $\delta_2 \in (0,\delta_1/2)$) we have
\[ \|\mu_j - x\|_X \geq \frac{\delta_1}{2} - \delta_2 \]
and therefore $d(\mu_j,x) \geq m(\frac{\delta_1}{2}-\delta_2)$ for all $j=1,2,\dots,k-1$.
Also
\begin{equation} \label{minmax}
D_l(\mu) := \min_{j=1,2,\dots, k-1} d(x,\mu_j) - d(x,\mu_l^\dagger) \geq m(\frac{\delta_1}{2}-\delta_2) - M(\delta_2).
\end{equation}
So for $\delta_2$ sufficiently small there exists $\epsilon>0$ such that
\[ D_l(\mu) \geq \epsilon. \]
Since the right hand side is independent of $\mu\in X^{k-1}$,
\[ \inf_{\mu\in X^{k-1}} \max_l D_l(\mu) \geq \epsilon. \]
Define the characteristic function
\[ \chi_\mu(\xi)=\begin{cases} 1 & \text{ if } \|\xi-\mu_{l(\mu)}^\dagger\|_X < \delta_2\\
0 & \text{ otherwise,}\end{cases} \]
where $l(\mu)$ is the maximizer in (\ref{minmax}).
For each $\omega\in\Omega^{\prime\prime}$ one obtains
\begin{align*}
\inf_{\mu\in X^{k-1}} f_n^{(\omega)}(\mu) & = \inf_{\mu\in X^{k-1}} \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^{k-1} d(\xi_i,\mu_j) \\
& \geq \inf_{\mu\in X^{k-1}} \frac{1}{n} \sum_{i=1}^n \left[ \bigwedge_{j=1}^{k-1} d(\xi_i,\mu_j)\left(1-\chi_\mu(\xi_i)\right) + \left( d(\xi_i,\mu^\dagger_{l(\mu)}) + \epsilon \right) \chi_\mu(\xi_i) \right] \\
& \geq \inf_{\mu\in X^k} f_n^{(\omega)}(\mu) + \epsilon \min_{l=1,2,\dots,k} P_n^{(\omega)}(B(\mu^\dagger_l,\delta_2)).
\end{align*}
Then since $P_n^{(\omega)}(B(\mu^\dagger_l,\delta_2))\to P(B(\mu^\dagger_l,\delta_2))>0$ by Assumption~\ref{ass:d}.4 (for $\delta_2\in\mathbb{Q}\cap (0,\infty)$) we can conclude~\eqref{eq:difffnl} holds.
\end{proof}
\begin{remark}
\label{rem:cons:minexist}
One can easily show that Assumption~\ref{ass:d}.2 implies that $d$ is weakly lower semi-continuous in its second argument which carries through to $f_n^{(\omega)}$.
It follows that on any bounded (or equivalently as $X$ is reflexive: weakly compact) set the infimum of $f_n^{(\omega)}$ is achieved.
Hence the infimum in Proposition~\ref{lem:bdd} is actually a minimum.
\end{remark}
We now easily prove convergence by application of Theorem~\ref{thm:conmin}.
\begin{theorem}
\label{thm:cons}
Assuming the conditions of Theorem~\ref{thm:gamcon} and Proposition~\ref{lem:bdd} the minimization problem associated with the $k$-means method converges.
I.e. for $\mathbb{P}$-almost every $\omega$:
\[ \min_{\mu\in X^k} f_\infty (\mu) = \lim_{n \to \infty} \min _{\mu\in X^k} f_n^{(\omega)} (\mu). \]
Furthermore any sequence of minimizers $\mu^n$ of $f_n^{(\omega)}$ is almost surely weakly precompact and any weak limit point minimizes $f_\infty$.
\end{theorem}
\section{The Case of General \texorpdfstring{$Y$}{Y} \label{sec:MTT}}
In the previous section the data, $\xi_i$, and cluster centers, $\mu_j$, took their
values in a common space, $X$. We now remove this restriction and let $\xi_i:\Omega \rightarrow X$ and $\mu_j\in Y$.
We may want to use this framework to deal with finite dimensional data and
infinite dimensional cluster centers, which can lead to the variational
problem having uninformative minimizers.
In the previous section the cost function $d$ was assumed to scale with the underlying norm.
This is no longer appropriate when $d:X\times Y\to[0,\infty)$.
In particular if we consider the smoothing-data association problem then the natural choice of $d$ is a pointwise distance which will lead to the optimal cluster centers interpolating data points.
Hence, in any $H^s$ norm with $s\geq 1$, the optimal cluster centers ``blow up''.
One possible solution would be to weaken the space to $L^2$ and allow this type of behavior.
This is undesirable from both modeling and mathematical perspectives:
If we first consider the modeling point of view then we do not expect our estimate to perfectly fit the data which is observed in the presence of noise.
It is natural that the cluster centers are smoother than the data alone would suggest.
It is desirable that the optimal clusters should reflect reality.
From the mathematical point of view, restricting ourselves to only very weak spaces gives no hope of obtaining a strongly convergent subsequence.
An alternative approach is, as is common in the smoothing literature, to use a regularization term.
This approach is also standard when dealing with ill-posed inverse problems.
This changes the nature of the problem and so requires some justification.
In particular the scaling of the regularization with the data is of fundamental importance.
In the following section we argue that scaling motivated by a simple Bayesian interpretation of the problem is not strong enough (unsurprisingly, countable collections of finite dimensional observations do not carry enough information to provide consistency when dealing with infinite dimensional parameters).
In the form of a simple example we show that the optimal cluster center is unbounded in the large data limit when the regularization goes to zero sufficiently quickly.
The natural scaling in this example is for the regularization to vary with the number of observations as $n^p$ for $p\in[-\frac{4}{5},0]$.
We consider the case $p=0$ in Section \ref{sec:MTT:theory}.
This type of regularization is understood as penalized likelihood estimation \cite{good71}.
Although it may seem undesirable for the limiting problem to depend upon the regularization it is unavoidable in ill-posed problems such as this one: there is not sufficient information, in even countably infinite collections of observations to recover the unknown cluster centers and exploiting known (or expected) regularity in these solutions provides one way to combine observations with qualitative prior beliefs about the cluster centers in a principled manner.
There are many precedents for this approach, including \cite{hall05} in which the consistency of penalized splines is studied using, what in this paper we call, the $\Gamma$-limit.
In that paper a fixed regularization was used to define the limiting problem in order to derive an estimator.
Naturally, regularization strong enough to alter the limiting problem influences the solution and we cannot hope to obtain consistent estimation in this setting, even in settings in which the cost function can be interpreted as the log likelihood of the data generating process.
In the setting of \cite{hall05}, the regularization is finally scaled to zero whereupon under assumptions the estimator converges to the truth but such a step is not feasible in the more complicated settings considered here.
When more structure is available it may be desirable to further investigate the regularization.
For example with $k=1$ the non-parametric regression model is equivalent to the white noise model \cite{brown96} for which optimal scaling of the regularization is known~\cite{agapiou13, zhao00}.
It is the subject of further work to extend these results to $k>1$.
With our redefined $k$-means type problem we can replicate the results of the previous section, and do so in Theorem~\ref{thm:kmeanscons}.
That is, we prove that the $k$-means method converges where $Y$ is a general separable and reflexive Banach space and in particular need not be equal to $X$.
This section is split into three subsections.
In the first we motivate the regularization term.
The second contains the convergence theory in a general setting.
Establishing that the assumptions of this subsection hold is non-trivial and
so, in the third subsection, we show an application to the smoothing-data
association problem.
\subsection{Regularization \label{sec:MTT:reg}}
In this section we use a toy, $k=1$, smoothing problem to motivate an approach to regularization which is adopted in what follows.
We assume that the cluster centers are periodic with equally spaced observations so we may use a Fourier argument.
In particular we work on the space of 1-periodic functions in $H^2$,
\begin{equation} \label{eq:MTT:reg:Y}
Y = \left\{ \mu:[0,1]\to \mathbb{R} \text{ s.t. } \mu(0)=\mu(1) \text{ and } \mu\in H^2 \right\}.
\end{equation}
For arbitrary sequences $(a_n)$, $(b_n)$ and data $\Psi_n=\{(t_j,z_j)\}_{j=1}^n \subset [0,1] \times \mathbb{R}^d$ we define the functional
\begin{equation} \label{eq:5.1fn}
f_n^{(\omega)}(\mu) = a_n \sum_{j=0}^{n-1} \left|\mu(t_j) - z_j\right|^2 + b_n \|\partial^2 \mu\|^2_{L^2}.
\end{equation}
Data are points in space-time: $[0,1]\times \mathbb{R}$.
The regularization is chosen so that it penalizes the $L^2$ norm of the second derivative.
For simplicity, we employ deterministic measurement times $t_j$ in the following proposition although this lies outside the formal framework which we consider subsequently.
Another simplification we make is to use convergence in expectation rather than almost sure convergence.
This simplifies our arguments.
We stress that this section is the motivation for the problem studied in Section~\ref{sec:MTT:theory}.
We will give conditions on the scaling of $a_n$ and $b_n$ that determine whether $\mathbb{E}\min f_n^{(\omega)}$ and $\mathbb{E}\mu^n$ stay bounded where $\mu^n$ is the minimizer of $f_n^{(\omega)}$.
\begin{proposition} \label{prop:boundedmin}
Let data be given by $\Psi_n=\{(t_j,z_j)\}_{j=1}^n$ with $t_j=\frac{j}{n}$ under the assumption $z_j = \mu^\dagger(t_j) + \epsilon_j$ for $\epsilon_j$ iid noise with finite variance and $\mu^\dagger\in L^2$ and define $Y$ by~\eqref{eq:MTT:reg:Y}.
Then $\inf_{\mu\in Y} f_n^{(\omega)}(\mu)$ defined by \eqref{eq:5.1fn} stays bounded (in expectation) if $a_n=O(\frac{1}{n})$ for any positive sequence $b_n$.
\end{proposition}
\begin{proof}
Assume $n$ is odd.
Both $\mu$ and $z$ are 1-periodic so we can write
\[ \mu(t) = \frac{1}{n}\sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}} \hat{\mu}_l e^{2\pi i lt} \quad \quad \text{and} \quad \quad z_j = \frac{1}{n}\sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}} \hat{z}_l e^{\frac{2\pi i lj}{n}} \]
with
\[ \hat{\mu}_l = \sum_{j=0}^{n-1} \mu(t_j) e^{-\frac{2\pi i lj}{n}} \quad \quad \text{and} \quad \quad \hat{z}_l = \sum_{j=0}^{n-1} z_j e^{-\frac{2\pi i lj}{n}}. \]
We will continue to use the notation that $\hat{\mu}_l$ is the Fourier transform of $\mu$.
We write
\[ \hat{\mu}:=\left(\hat{\mu}_{-\frac{n-1}{2}}, \hat{\mu}_{-\frac{n-1}{2}+1}, \dots, \hat{\mu}_{\frac{n-1}{2}} \right). \]
Similarly for $z$.
Substituting the Fourier expansion of $\mu$ and $z$ into $f_n^{(\omega)}$ implies
\[ f_n^{(\omega)}(\mu) = \frac{a_n}{n} \left( \langle \hat{\mu},\hat{\mu} \rangle - 2 \langle \hat{\mu},\hat{z}\rangle + \langle \hat{z},\hat{z} \rangle + \frac{\gamma_n}{n} \langle l^4\hat{\mu},\hat{\mu}\rangle \right) \]
where $\gamma_n = \frac{16\pi^4 b_n}{a_n}$ and $\langle \hat{x},\hat{z}\rangle = \sum_l \hat{x}_l \overline{\hat{z}}_l$.
The Gateaux derivative $\partial f_n^{(\omega)}(\mu;\nu)$ of $f_n^{(\omega)}$ at $\mu$ in the direction $\nu$ is
\[ \partial f_n^{(\omega)}(\mu;\nu) = \frac{2a_n}{n} \left\langle \hat{\mu} - \hat{z} + \frac{\gamma_n l^4}{n} \hat{\mu},\hat{\nu} \right\rangle. \]
Which implies the minimizer $\mu^n$ of $f_n^{(\omega)}$ is (in terms of its Fourier expansion)
\[ \hat{\mu}^n_l = \left(1+\frac{\gamma_n l^4}{n} \right)^{-1} \hat{z} := \left( \left(1+\frac{\gamma_n l^4}{n} \right)^{-1} \hat{z}_l \right)_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}}. \]
It follows that the minimum is
\[ \mathbb{E}\left(f_n^{(\omega)}(\mu^n)\right) = \frac{a_n}{n} \mathbb{E}\left(\left\langle \left( 1 + \frac{n}{\gamma_n l^4} \right)^{-1} \hat{z},\hat{z} \right\rangle \right) \leq a_n \sum_{j=0}^{n-1} \mathbb{E} z_j^2 \lesssim 2a_n n \left( \|\mu^\dagger\|_{L^2}^2 + \text{Var}(\epsilon) \right). \]
Similar expressions can be obtained for the case of even $n$.
\end{proof}
Clearly the natural choice for $a_n$ is
\[ a_n = \frac{1}{n} \]
which we use from here.
We let $b_n = \lambda n^p$ and therefore $\gamma_n = 16\pi^4 \lambda n^{p+1}$.
From Proposition~\ref{prop:boundedmin} we immediately have $\mathbb{E} \min f_n^{(\omega)}$ is bounded for any choice of $p$.
In our next proposition we show that for $p\in[-\frac{4}{5},0]$ our minimizer is bounded in $H^2$ whilst outside this window the norm either blows up or the second derivative converges to zero.
For simplicity in the calculations we impose the further condition that $\mu^\dagger(t)=0$.
\begin{proposition}
In addition to the assumptions of Proposition \ref{prop:boundedmin} let $a_n = \frac{1}{n}$, $b_n=\lambda n^p$, $\epsilon_j \stackrel{\mathrm{iid}}{\sim} N(0,\sigma^2)$ and assume that $\mu^n$ is the minimizer of $f_n^{(\omega)}$.
\begin{itemize}
\item[1.] For $n$ sufficiently large there exists $M_1>0$ such that for all $p$ and $n$ the $L^2$ norm is bounded:
\[ \mathbb{E} \|\mu^n\|^2_{L^2} \leq M_1. \]
\item[2.] If $p>0$ then
\[ \mathbb{E} \|\partial^2 \mu^n\|^2_{L^2} \to 0 \quad \text{as } n\to \infty. \]
\end{itemize}
If we further assume that $\mu^\dagger(t)=0$, then the following statements are true.
\begin{itemize}
\item[3.] For all $p \in [-\frac{4}{5},0]$ there exists $M_2>0$ such that
\[ \mathbb{E}\|\partial^2 \mu^n\|^2_{L^2} \leq M_2. \]
\item[4.] If $p<-\frac{4}{5}$ then
\[ \mathbb{E}\|\partial^2 \mu^n\|^2_{L^2} \to \infty \quad \text{as } n\to \infty. \]
\end{itemize}
\end{proposition}
\begin{proof}
The first two statements follow from
\begin{align*}
\mathbb{E} \|\mu^n\|_{L^2}^2 & \lesssim 2\left( \|\mu^\dagger\|_{L^2}^2 + \text{Var}(\epsilon) \right) \\
\mathbb{E} \|\partial^2\mu^n\|_{L^2}^2 & \lesssim \frac{8\pi^4 n}{\gamma_n} \left( \|\mu^\dagger\|^2_{L^2} + \text{Var}(\epsilon) \right)
\end{align*}
which are easily shown.
Statement 3 is shown after statement 4.
Following the calculation in the proof of Proposition~\ref{prop:boundedmin}, and assuming that $\mu^\dagger(t)=0$, it is easily shown that
\begin{align} \label{exprep} \mathbb{E}\|\partial^2 \mu^n\|_{L^2}^2 = \frac{16\pi^4\sigma^2}{n} \sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}} \frac{l^4}{(1+16\pi^4\lambda n^p l^4)^2} =: S(n)
\end{align}
since $\mathbb{E}|\hat{z}_l|^2 = \sigma^2n$.
To show $S(n)\to \infty$ we will manipulate the Riemann sum approximation of
\[ \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{x^4}{(1+16\pi^4\lambda x^4)^2} \; \text{d} x = C \]
where $0<C<\infty$.
We have
\begin{align*}
\int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{x^4}{(1+ 16\pi^4\lambda x^4)^2} \; \text{d} x & = n^{1+\frac{p}{4}} \int_{-\frac{1}{2} n^{-1-\frac{p}{4}}}^{\frac{1}{2} n^{-1-\frac{p}{4}}} \frac{n^{4+p}w^4}{(1+16\pi^4\lambda n^{4+p}w^4)^2} \; \text{d} w \quad \text{where } x = n^{1+\frac{p}{4}} w \\
& \approx n^{\frac{5p}{4}} \sum_{l=-\left\lfloor\frac{1}{2} n^{-\frac{p}{4}}\right\rfloor}^{\left\lfloor\frac{1}{2} n^{-\frac{p}{4}}\right\rfloor} \frac{l^4}{(1+16\pi^4 \lambda n^pl^4)^2} =: R(n).
\end{align*}
Therefore assuming $p>-4$ we have
\[ S(n) \geq \frac{16\pi^4 \sigma^2}{n^{1+\frac{5p}{4}}} R(n). \]
So for $1+\frac{5p}{4} <0$ we have $S(n)\to \infty$.
Since $S(n)$ is monotonic in $p$ then $S(n)\to \infty$ for all $p<-\frac{4}{5}$.
This shows that statement 4 is true.
Finally we establish the third statement.
If $p = -\frac{4}{5}$ then
\begin{align*}
S(n) & = 16\pi^4\sigma^2 R(n) + \frac{16\pi^4\sigma^2}{n}\left( \sum_{l=-\frac{n-1}{2}}^{\lfloor\frac{n^{\frac{1}{5}}}{2}\rfloor -1} \frac{l^4}{(1+16\pi^4\lambda n^p l^4)^2} + \sum_{l=\lfloor\frac{n^{\frac{1}{5}}}{2}\rfloor +1}^{\frac{n-1}{2}} \frac{l^4}{(1+16\pi^4\lambda n^p l^4)^2} \right) \\
& \leq 16\pi^4\sigma^2 R(n) + \frac{2\pi^4\sigma^2}{n^{\frac{1}{5}}(1+\pi^4\lambda)^2}.
\end{align*}
The remaining cases $p \in [-\frac{4}{5},0]$ are a consequence of (\ref{exprep}) which implies that $p \mapsto \mathbb{E}(\partial^2 \mu)$ is non-increasing.
\end{proof}
By the Poincar\'e inequality it follows that if $p\geq -\frac{4}{5}$ then the $H^2$ norm of our minimizer stays bounded as $n\to \infty$.
Our final calculation in this section is to show that the regularization for $p\in [-\frac{4}{5},0]$ is not too strong.
We have already shown that $\|\partial^2\mu^n\|_{L^2}$ is bounded (in expectation) in this case but we wish to make sure that we don't have the stronger result that $\|\partial^2\mu^n\|_{L^2} \to 0$.
\begin{proposition}
With the assumptions of Proposition \ref{prop:boundedmin} and $a_n=\frac{1}{n}$, $b_n=\lambda n^p$ with $p\in [-\frac{4}{5},0]$ there exists a choice of $\mu^\dagger$ and a constant $M>0$ such that if $\mu^n$ is the minimizer of $f_n^{(\omega)}$ then
\begin{equation} \label{eq:5.1optbound}
\mathbb{E} \|\partial^2 \mu^n\|^2_{L^2} \geq M.
\end{equation}
\end{proposition}
\begin{proof}
We only need to prove the proposition for $p=0$ (the strongest regularization) and find one $\mu^\dagger$ such that \eqref{eq:5.1optbound} is true.
Let $\mu^\dagger(t) = 2\cos(2\pi t) = e^{2\pi i t} + e^{-2\pi i t}$.
Then the Fourier transform of $\mu^\dagger$ satisfies $\hat{\mu}^\dagger_l=0$ for $l\neq \pm 1$ and $\hat{\mu}^\dagger_l=n$ for $l=\pm 1$.
So,
\begin{align*}
\mathbb{E} \|\partial^2 \mu^n\|_{L^2}^2 & = \frac{16\pi^4}{n^2} \sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}} \frac{l^4}{(1+16\pi^4\lambda l^4)^2} \mathbb{E}|\hat{z}_l|^2 \\
& \gtrsim \frac{16\pi^4}{n^2} \sum_{l=-\frac{n-1}{2}}^{\frac{n-1}{2}} \frac{l^4}{(1+16\pi^4\lambda l^4)^2} |\hat{\mu}^\dagger_l|^2 \\
& = \frac{32\pi^4}{(1+16\pi^4\lambda)^2} > 0.
\end{align*}
\end{proof}
We have shown that the minimizer is bounded for any $p\geq -\frac{4}{5}$ and $\|\partial^2\mu^n\|_{L^2} \to 0$ for $p>0$.
The case $p>0$ is clearly undesirable as we would be restricting ourselves to straight lines.
The natural scaling for this problem is in the range $p\in [-\frac{4}{5},0]$.
In the remainder of this paper we consider the case $p=0$.
This has the advantage that, not only $\mathbb{E}\|\partial^2 \mu^n\|_{L^2}$, but also
$\mathbb{E}f_n^{(\omega)}(\mu^n)$ is $O(1)$ as $n \to \infty$.
In fact we will show that with this choice of regularization we do not need to choose $k$ dependent on the data generating model.
The regularization makes the methodology sufficiently robust to have convergence even for poor choices of $k$.
For example, if there exists a data generating process which is formed of a $k^\dagger$-mixture model then for our method to be robust does not require us to choose $k=k^\dagger$.
Of course with the `wrong' choice of $k$ the results may be physically meaningless and we should take care in how to interpret the results.
The point to stress is that the methodology does not rely on a data generating model.
The disadvantage of this is to potentially increase the bias in the method.
Since the $k$-means is already biased we believe the advantages of our approach outweigh the disadvantages.
In particular we have in mind applications where only a coarse estimate is needed.
For example the $k$-means method may be used to initialize some other algorithm.
Another application could be part of a decision making process: in Section \ref{sec:example1} we show the $k$-means methodology can be used to determine whether two tracks have crossed.
\subsection{Convergence For General \texorpdfstring{$Y$}{Y} \label{sec:MTT:theory}}
Let $(X,\|\cdot\|_X)$, $(Y,\|\cdot\|_Y)$ be reflexive, separable Banach spaces
We will also assume that the data points, $\Psi_n =\{\xi_i\}_{i=1}^n \subset
X$ for $i=1,2,\dots,n$ are iid random elements with common law $P$.
As before $\mu=(\mu_1,\mu_2,\dots,\mu_k)$ but now the cluster centers $\mu_j\in Y$ for each $j$.
The cost function is $d:X\times Y\to [0,\infty)$.
The energy functions associated with the $k$-means algorithm in this setting
are slightly different to those used previously:
\begin{align}
g_{\mu}: X & \to \mathbb{R},\quad
g_{\mu}(x) = \bigwedge_{j=1}^k d(x,\mu_j), \notag \\
f_n^{(\omega)}: Y^k & \to \mathbb{R},\quad
f_n^{(\omega)}(\mu) = P_n^{(\omega)}g_{\mu} + \lambda r(\mu), \label{eq:fnreg} \\
f_\infty: Y^k & \to \mathbb{R}, \quad
f_\infty(\mu) = Pg_{\mu} + \lambda r(\mu). \label{eq:finftyreg}
\end{align}
The aim of this section is to show the convergence result:
\[ \hat{\theta}_n^{(\omega)} = \inf_{\mu\in Y^k} f_n^{(\omega)}(\mu) \to \inf_{\mu\in Y^k} f_\infty(\mu) = \theta \quad \text{and} \quad \text{as } n\to \infty \text{ for } \mathbb{P}\text{-almost every } \omega \]
and that minimizers converge (almost surely).
The key assumptions are given in Assumptions~\ref{ass:dandr}; they imply that $f_n^{(\omega)}$ is weakly lower semi-continuous and coercive.
In particular, Assumption~\ref{ass:dandr}.2 allows us to prove the lim inf inequality as we did for Theorem~\ref{thm:gamcon}.
Assumption~\ref{ass:dandr}.1 is likely to mean that our convergence results are limited to the case of bounded noise.
In fact, when applying the problem to the smoothing-data association problem, it is necessary to bound the noise in order for Assumption~\ref{ass:dandr}.5 to hold.
Assumption~\ref{ass:dandr}.5 implies that $f_n^{(\omega)}$ is (uniformly) coercive and hence allows us to easily bound the set of minimizers.
It is the subject of ongoing research to extend the convergence results to unbounded noise for the smoothing-data association problem.
Assumption~\ref{ass:dandr}.3 is a measurability condition we require in order
to integrate and the weak lower semi-continuity of $r$ is needed for the to
obtain the lim inf inequality in the $\Gamma$-convergence proof.
We note that, since $Pd(\cdot,\mu_1)\leq \sup_{x\in \mathrm{supp}(P)} d(x,\mu_1) <\infty$, we have $f_\infty(\mu)<\infty$ for every $\mu\in Y^k$ (and since $r(\mu)<\infty$ for each $\mu\in Y^k$).
\begin{assumptions}
\label{ass:dandr}
We have the following assumptions on $d:X\times Y\to [0,\infty)$, $r:Y^k\to [0,\infty)$ and $P$.
\begin{itemize}
\item[2.1.] For all $y\in Y$ we have $\sup_{x\in \mathrm{supp}(P)} d(x,y)<\infty$ where $\mathrm{supp}(P)\subseteq X$ is the support of $P$.
\item[2.2.] For each $x\in X$ and $y\in Y$ we have that if $x_m\to x$ and $y_n\rightharpoonup y$ as $n,m\to \infty$ then
\[ \liminf_{n,m\to \infty} d(x_m,y_n) \geq d(x,y) \quad \text{and} \quad \lim_{m\to\infty} d(x_m,y) = d(x,y). \]
\item[2.3.] For every $y \in Y$ we have that $d(\cdot,y)$ is $\mathcal{X}$-measurable.
\item[2.4.] $r$ is weakly lower semi-continuous.
\item[2.5.] $r$ is coercive.
\end{itemize}
\end{assumptions}
We will follow the structure of Section~\ref{sec:cons}.
We start by showing that under the above conditions $f_n^{(\omega)}$ $\Gamma$-converges to $f_\infty$.
We then show that the regularization term guarantees that the minimizers to $f_n^{(\omega)}$ lie in a bounded set.
An application of Theorem~\ref{thm:conmin} gives the desired convergence result.
Since we were able to restrict our analysis to a weakly compact subset of $Y$ we are easily able to deduce the existence of a weakly convergent subsequence.
Similarly to the previous section on the product space $Y^k$ we use the norm $\|\mu\|_k:=\max_j \|\mu_j\|_{Y}$.
\begin{theorem}
\label{thm:gamcondiffspace}
Let $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ be separable and reflexive Banach spaces.
Assume $r:Y^k\to [0,\infty)$, $d:X\times Y\to [0,\infty)$ and the probability measure $P$ on $(X,\mathcal{X})$ satisfy the conditions in Assumptions~\ref{ass:dandr}.
For independent samples $\{\xi_i^{\omega)}\}_{i=1}^n$ from $P$ define $P_n^{(\omega)}$ to be the
empirical measure and $f_n^{(\omega)}:Y^k \to \mathbb{R}$ and $f_\infty:Y^k \to
\mathbb{R}$ by \eqref{eq:fnreg} and \eqref{eq:finftyreg} respectively and where $\lambda>0$.
Then
\[ f_\infty = \Gamma\text{-}\lim_n f_n^{(\omega)} \]
for $\mathbb{P}$-almost every $\omega$.
\end{theorem}
\begin{proof}
Define
\[ \Omega^\prime = \left\{ \omega\in\Omega : P_n^{(\omega)} \Rightarrow P \right\} \cap \left\{ \omega \in \Omega : \xi^{(\omega)}_i \in \text{supp}(P) \; \forall i\in \mathbb{N} \right\}. \]
Then $\mathbb{P}(\Omega^\prime)=1$.
For the remainder of the proof we consider an arbitrary $\omega\in\Omega^\prime$.
We start with the lim inf inequality.
Let $\mu^n\rightharpoonup \mu$ then
\[ \liminf_{n\to \infty} f_n^{(\omega)}(\mu^n) \geq f_\infty(\mu) \]
follows (as in the proof of Theorem~\ref{thm:gamcon}) by applying Theorem~1.1 in~\cite{feinberg14} and the fact that $r$ is weakly lower semi-continuous.
We now establish the existence of a recovery sequence.
Let $\mu\in Y^k$ and let $\mu^n=\mu$.
We want to show
\[ \lim_{n\to \infty} f_n^{(\omega)}(\mu) = \lim_{n\to \infty} P_n^{(\omega)}g_\mu + \lambda r(\mu) = Pg_\mu + \lambda r(\mu) = f_\infty(\mu). \]
Clearly this is equivalent to showing that
\[ \lim_{n\to \infty} P_n^{(\omega)} g_\mu = Pg_\mu. \]
Now $g_\mu$ are continuous by assumption on $d$.
Let $M=\sup_{x\in \text{supp}(P)}d(x,\mu_1)<\infty$ and note that $g_\mu(x) \leq M$ for all $x\in\text{supp}(P)$ and therefore bounded.
Hence $P_n^{(\omega)} g_\mu \to Pg_\mu$.
\end{proof}
\begin{proposition}
\label{lem:minset}
Assuming the conditions of Theorem~\ref{thm:gamcondiffspace}, then for $\mathbb{P}$-almost every $\omega$ there exists $N<\infty$ and $R>0$ such that
\[ \min_{\mu\in Y^k} f_n^{(\omega)}(\mu) = \min_{\|\mu\|_k \leq R} f_n^{(\omega)}(\mu) < \inf_{\|\mu\|_k > R} f_n^{(\omega)}(\mu) \quad \forall n\geq N. \]
In particular $R$ is independent of $n$.
\end{proposition}
\begin{proof}
Let
\[ \Omega^{\prime\prime} = \left\{ \omega\in\Omega^\prime : P_n^{(\omega)} \Rightarrow P \right\} \cap \left\{ \omega\in\Omega^\prime : P^{(\omega)}_n d(\cdot,0)\to Pd(\cdot,0) \right\}. \]
Then, for every $\omega \in \Omega^{\prime\prime}$, $f_n^{(\omega)}(0)\to f_\infty(0)<\infty$ where with a slight abuse of notation we denote the zero element in both $Y$ and $Y^k$ by $0$.
Take $N$ sufficiently large so that
\[ f_n^{(\omega)}(0) \leq f_\infty(0) + 1 \quad \quad \text{for all } n\geq N. \]
Then $\min_{\mu\in Y^k} f_n^{(\omega)}(\mu) \leq f_\infty(0) + 1$ for all $n\geq N$.
By coercivity of $r$ there exists $R$ such that if $\|\mu\|_k>R$ then $\lambda r(\mu) \geq f_\infty(0)+1$.
Therefore any such $\mu$ is not a minimizer and in particular any minimizer must be contained in the set $\left\{ \mu\in Y^k : \|\mu\|_k\leq R \right\}$.
\end{proof}
The convergence results now follows by applying Theorem~\ref{thm:gamcondiffspace} and Proposition~\ref{lem:minset} to Theorem~\ref{thm:conmin}.
\begin{theorem}
\label{thm:kmeanscons}
Assuming the conditions of Theorem~\ref{thm:gamcondiffspace} and Proposition~\ref{lem:minset} the minimization problem associated with the $k$-means method converges in the following sense:
\[ \min_{\mu\in Y^k} f_\infty(\mu) = \lim_{n\to \infty} \min_{\mu\in Y^k} f_n^{(\omega)}(\mu) \]
for $\mathbb{P}$-almost every $\omega$.
Furthermore any sequence of minimizers $\mu^n$ of $f_n^{(\omega)}$ is almost surely weakly precompact and any weak limit point minimizes $f_\infty$.
\end{theorem}
It was not necessary to assume that cluster centers are in a common space.
A trivial generalization would allow each $\mu_j\in Y^{(j)}$ with the cost and regularization terms appropriately defined; in this setting Theorem~\ref{thm:kmeanscons} holds.
\subsection{Application to the Smoothing-Data Association Problem \label{sec:MTT:app}}
In this section we give an application to the smoothing-data association problem and show the assumptions in the previous section are met.
For $k=1$ the smoothing-data association problem is the problem of fitting a curve to a data set (no data association).
For $k>1$ we couple the smoothing problem with a data association problem.
Each data point is associated with an unknown member of a collection of $k$ curves.
Solving the problem involves simultaneously estimating both the data partition (i.e. the association of observations to curves) and the curve which best fits each subset of the data.
By treating the curve of best fit as the cluster center we are able to approach this problem using the $k$-means methodology.
The data points are points in space-time whilst cluster centers are functions from time to space.
We let the Euclidean norm on $\mathbb{R}^\kappa$ be given by $|\cdot|$.
Let $X=\mathbb{R} \times \mathbb{R}^\kappa$ be the data space.
We will subsequently assume that the support of $P$, the common law of our observations, is contained within $\tilde{X} = [0,T]\times X^\prime$ where $X^\prime\subseteq [-\tilde{N},\tilde{N}]^\kappa$.
We define the cluster center space to be $Y=H^2([0,T])$, the Sobolev space of functions from $[0,T]$ to $\mathbb{R}^\kappa$.
Clearly $X$ and $Y$ are separable and reflexive.
The cost function $d:X\times Y\to [0,\infty)$ is defined by
\begin{equation} \label{eq:5.2d}
d(\xi,\mu_j) = |z-\mu_j(t) |^2
\end{equation}
where $\mu_j \in Y$ and $\xi=(t,z)\in X$.
We introduce a regularization term that penalizes the second derivative.
This is a common choice in the smoothing literature, e.g. \cite{randall99}.
The regularization term $r:Y^k\to [0,\infty)$ is given by
\begin{equation} \label{eq:5.2r}
r(\mu) = \sum_{j=1}^k \| \partial^2\mu_j\|_{L^2}^2.
\end{equation}
The $k$-means energy $f_n$ for data points $\{\xi_i=(t_i,z_i)\}_{i=1}^n$ is therefore written
\begin{equation} \label{eq:fnap}
f_n(\mu) = \frac{1}{n}\sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i,\mu_j) + \lambda r(\mu) = \frac{1}{n}\sum_{i=1}^n \bigwedge_{j=1}^k |z_i-\mu_j(t_i) |^2 + \lambda \sum_{j=1}^k \| \partial^2\mu_j\|_{L^2}^2.
\end{equation}
In most cases it is reasonable to assume that any minimizer of $f_\infty$ must be uniformly bounded, i.e. there exists $N$ (which will in general depend on $P$) such that if $\mu^\infty$ minimizes $f_\infty$ then $|\mu^\infty(t)|\leq N$ for all $t\in [0,T]$.
Under this assumption we redefine $Y$ to be
\begin{equation} \label{eq:Y}
Y=\{\mu_j\in H^2([0,T]) : |\mu_j(t)|\leq N \, \forall t\in [0,T]\}.
\end{equation}
Since pointwise evaluation is a bounded linear functional in $H^s$ (for $s\geq 1$) this space is weakly closed.
We now minimize $f_n$ over $Y^k$.
Note that we are not immediately guaranteed that minimizers of $f_n$ over $(H^s)^k$ are contained in $Y^k$.
However when we apply Theorem~\ref{thm:kmeanscons} we can conclude that minimizers $\mu^n$ of $f_n$ over $Y_k$ are weakly compact in $(H^s)^k$ and any limit point is a minimizer of $f_\infty$ in $Y^k$.
And therefore any limit point is a minimizer of $f_\infty$ over $(H^s)^k$.
If no such $N$ exists then our results in Theorem~\ref{thm:kmeanscons} are still valid however the minimum of $f_\infty$ over $(H^s)^k$ is not necessarily equal to the minimum of $f_\infty$ over $Y^k$.
Our results show that the $\Gamma$-limit for $\mathbb{P}$-almost every $\omega$ is
\begin{equation} \label{eq:finftyap}
f_\infty(\mu) = \int_X \bigwedge_{j=1}^k d(x,\mu_j) P(\text{d} x) + \lambda r(\mu) = \int_X \bigwedge_{j=1}^k |z-\mu_j(t) |^2 P(\text{d} x) + \lambda \sum_{j=1}^k \| \partial^2\mu_j\|_{L^2}^2.
\end{equation}
We start with the key result for this section, that is the existence of a weakly converging subsequence of minimizers.
Our result relies upon the regularity of Sobolev functions.
For our result to be meaningful we require that the minimizer should at least be continuous.
In fact every $g\in H^2([0,T])$ is in $C^s([0,T])$ for any $s<\frac{3}{2}$.
The regularity in the space allows us to further deduce the existence of a strongly converging subsequence.
\begin{theorem}
\label{thm:strconv}
Let $X=[0,T] \times \mathbb{R}^\kappa$ and define $Y$ by \eqref{eq:Y}.
Define $d:X \times Y\to [0,\infty)$ by \eqref{eq:5.2d} and $r:Y^k\to[0,\infty)$ by \eqref{eq:5.2r}.
For independent samples $\{\xi_i\}_{i=1}^n$ from $P$ which has compact support
$\tilde{X} \subset X$ define $f_n,f_\infty : Y^k\to \mathbb{R}$ by \eqref{eq:fnap} and \eqref{eq:finftyap} respectively.
Then (1) any sequence of minimizers $\mu^n\in Y^k$ of $f_n$ is $\mathbb{P}$-almost surely weakly-precompact (in $H^2$) with any weak limit point of $\mu^n$ minimizes $f_\infty$ and
(2) if $\mu^{n_m}\rightharpoonup \mu$ is a weakly converging (in $H^2$) subsequence of minimizers then the convergence is uniform (in $C^0$).
\end{theorem}
To prove the first part of Theorem~\ref{thm:strconv} we are required to check the boundedness and continuity assumptions on $d$ (Proposition~\ref{prop:B2check}) and show that $r$ is weakly lower semi-continuous and coercive (Proposition~\ref{prop:C1check}).
This statement is then a straightforward application of Theorem~\ref{thm:kmeanscons}.
Note that we will have shown the result of Theorem~\ref{thm:gamcondiffspace} holds: $f_\infty = \Gamma\text{-}\lim_n f_n^{(\omega)}$.
In what follows we check that properties hold for any $x \in \tilde{X}$, which should be understood as implying that they hold for $P$-almost any $x \in X$; this is sufficient for our purposes as the collection of sequences $\xi_1,\ldots$ for which one or more observations lies in the complement of $\tilde{X}$ is $\mathbb{P}$-null and the support of $P_n$ is $\mathbb{P}$-almost surely contained within $\tilde{X}$.
\begin{proposition}
\label{prop:B2check}
Let $\tilde{X} = [0,T]\times[-\tilde{N},\tilde{N}]^\kappa$ and define $Y$ by \eqref{eq:Y}.
Define $d:\tilde{X}\times Y\to [0,\infty)$ by \eqref{eq:5.2d}.
Then (i) for all $y\in Y$ we have $\sup_{x\in \tilde{X}} d(x,y)<\infty$ and (ii) for any $x\in X$ and $y\in Y$ and any sequences $x_m\to x$ and $y_n\rightharpoonup y$ as $m,n\to \infty$ then we have $\liminf_{n,m\to \infty} d(x_m,y_n) = d(x,y)$.
\end{proposition}
\begin{proof}
We start with (i).
Let $y\in Y$ and $x = (t,z)\in [0,T]\times [-\tilde{N},\tilde{N}]^\kappa$, then
\begin{align*}
d(x,y) & = |z-y(t)|^2 \\
& \leq 2|z|^2 + 2|y(t)|^2 \\
& \leq 2\tilde{N}^2 + 2\sup_{t\in [0,T]} |y(t)|^2.
\end{align*}
Since $y$ is continuous then $\sup_{t\in [0,T]} |y(t)|^2< \infty$ and moreover we can bound $d(x,y)$ independently of $x$ which shows (i).
For (ii) we let $(t_m,z_m)=x_m \to x=(t,z)$ in $\mathbb{R}^{\kappa+1}$ and $y_n\rightharpoonup y$.
Then
\begin{align}
d(x_m,y_n) & = \left| z_m - y_n(t_m) \right|^2 \notag \\
& = |z_m|^2 - 2 z_m \cdot y_n(t_m) + |y_n(t_m)|^2. \label{eq:dCts}
\end{align}
Clearly $|z_m|^2\to |z|^2$ and we now show that $y_n(t_m)\to y(t)$ as $m,n\to \infty$.
We start by showing that the sequence $\|y_n\|_Y$ is bounded.
Each $y_n$ can be associated with $\Lambda_n\in Y^{**}$ by $\Lambda_n(\nu)=\nu(y_n)$ for $\nu\in Y^*$.
As $y_n$ is weakly convergent it is weakly bounded.
So,
\[ \sup_{n\in\mathbb{N}} |\Lambda_n(\nu)| = \sup_{n\in \mathbb{N}} |\nu(y_n)| \leq M_\nu \]
for some $M_\nu<\infty$.
By the uniform boundedness principle \cite{conway90}
\[ \sup_{n\in\mathbb{N}} \|\Lambda_n\|_{Y^{**}} < \infty. \]
And so,
\[ \sup_{n\in\mathbb{N}} \|y_n\|_Y = \sup_{n\in\mathbb{N}} \|\Lambda_n\|_{Y^{**}} < \infty. \]
Hence there exists $M>0$ such that $\| y_n\|_Y \leq M$.
Therefore
\begin{align*}
| y_n(r) -y_n(s) | & = \left| \int_s^r \partial y_n(t) \; \text{d} t \right|
\leq \int_s^r \left| \partial y_n(t) \right| \; \text{d} t = \int_0^T \mathbb{I}_{[s,r]}(t) \left| \partial y_n(t) \right| \; \text{d} t \\
& \leq \| \mathbb{I}_{[s,r]} \|_{L^2} \left\| \partial y_n(t) \right\|_{L^2}
\leq M \sqrt{|r-s|}.
\end{align*}
Since $y_n$ is uniformly bounded and equi-continuous then by the Arzel\`a--Ascoli theorem there exists a uniformly converging subsequence, say $y_{n_m}\to \hat{y}$.
By uniqueness of the weak limit $\hat{y}=y$.
But this implies that
\[ y_n(t) \to y(t) \]
uniformly for $t\in [0,T]$.
Now as
\[ |y_n(t_m) - y(t) | \leq |y_n(t_m) - y(t_m) | + | y(t_m) - y(t)| \]
then $y_n(t_m)\to y(t)$ as $m,n\to \infty$.
Therefore the second and third terms of~\eqref{eq:dCts} satisfies
\begin{align*}
2 z_m \cdot y_m(t_m) & \to 2z\cdot y(t) \\
\left|y_n(t_m)\right|^2 & \to \left|y(t)\right|^2
\end{align*}
as $m,n\to \infty$.
Hence
\[ d(x_m,y_n) \to |z|^2 - 2z\cdot y(t) + \left|y(t)\right|^2 = |z-y(t)|^2 = d(x,y) \]
which completes the proof.
\end{proof}
\begin{proposition}
\label{prop:C1check}
Define $Y$ by \eqref{eq:Y} and $r:Y^k\to [0,\infty)$ by \eqref{eq:5.2r}.
Then $r$ is weakly lower semi-continuous and coercive.
\end{proposition}
\begin{proof}
We start by showing $r$ is weakly lower semi-continuous.
For any weakly converging sequence $\mu^n_1\rightharpoonup \mu_1$ in $H^2$ we have that $\partial^2 \mu^n_1\rightharpoonup \partial^2 \mu_1$ weakly in $L^2$.
Hence it follows that $r$ is weakly lower semi-continuous.
To show $r$ is coercive let $\hat{r}(\mu_1)=\|\partial^2\mu_1\|_{L^2}^2$ for $\mu_1\in Y$.
We will show $\hat{r}$ is coercive.
Let $\mu_1\in Y$ and note that since $\mu_1\in C^1$ the first derivative exists (strongly).
Clearly we have $\|\mu_1\|_{L^2}\leq N\sqrt{T}$ and using a Poincar\'e inequality
\[ \left\|\frac{\text{d} \mu_1}{\text{d} t} - \frac{1}{T} \int_0^T \frac{\text{d} \mu_1}{\text{d} t} \; \text{d} t \right\|_{L^2} \leq C \|\partial^2 \mu_1\|_{L^2} \]
for some $C$ independent of $\mu_1$.
Therefore
\[ \left\|\frac{\text{d} \mu_1}{\text{d} t} \right\|_{L^2} \leq C \|\partial^2 \mu_1\|_{L^2} + \left|\frac{1}{T} \int_0^T \frac{\text{d} \mu_1}{\text{d} t} \; \text{d} t \right| \leq C\|\partial^2 \mu_1\|_{L^2} + \frac{2N}{T}. \]
It follows that if $\|\mu_1\|_{H^2}\to \infty$ then $\|\partial^2 \mu_1\|_{L^2}\to \infty$, hence $\hat{r}$ is coercive.
\end{proof}
Finally, the existence of a strongly convergent subsequence in Theorem~\ref{thm:strconv} follows from the fact that $H^2$ is compactly embedded into $H^1$.
Hence the convergence is strong in $H^1$.
By Morrey's inequality $H^1$ is embedded into a H\"older space ($C^{0,\frac{1}{2}}$) which is a subset of uniformly continuous functions.
This implies the convergence is uniform in $C^0$.
\section{Examples \label{sec:examples}}
In this section we give two exemplar applications of the methodology. In
principle any cost function, $d$, and regularization, $r$, (that satisfy the conditions) could be used.
For illustrative purposes we choose $d$ and $r$ to make the minimization simple to implement.
In particular, in Example 1 our choices allow us to use smoothing splines
\subsection{Example 1: A Smoothing-Data Association Problem \label{sec:example1}}
We use the $k$-means method to solve a smoothing-data association problem.
For each $j = 1,2,\dots,k$ we take functions $x^j:[0,T]\times\mathbb{R}$ for $j=1,2,\dots ,k$ as the ``true'' cluster centers, and for sample times $t_i^j$ for $i=1,2,\dots n_j$, uniformly distributed over $[0,T]$, we let
\begin{equation} \label{eq:examples:example1:model}
z_i^j = x^j(t_i^j) + \epsilon_i^j
\end{equation}
where $\epsilon_i^j$ are iid noise terms.
The observations take the form $\xi_i = (t_i,z_i)$ for $i=1,2,\dots,n=\sum_{j=1}^k n_j$ where we have relabeled the observations to remove the (unobserved) target reference.
We model the observations with density (with respect to the Lebesgue measure)
\[ p((t,z)) = \frac{1}{T} \mathbb{I}_{[0,T]}(t) \sum_{j=1}^k w_j p_\epsilon(z-x^j(t)) \]
on $\mathbb{R}\times\mathbb{R}$ where $p_\epsilon$ denotes the common density of the $\epsilon_i^j$ and $w_j$ denotes the probability that an observation is generated by trajectory $j$.
We let each cluster center be equally weighted: $w_j=\frac{1}{k}$.
The cluster centers were fixed and in particular did not vary between numerical experiments.
When the noise is bounded this is precisely the problem described in Section~\ref{sec:MTT:theory} with $\kappa=1$, hence the problem converges.
We use a truncated Gaussian noise term.
In the theoretical analysis of the algorithm we have considered only the minimization problem associated with the $k$-means algorithm; of course minimizing complex functionals of the form of $f_n$ is itself a challenging problem.
Practically, we adopt the usual $k$-means strategy \cite{lloyd82} of iteratively assigning data to the closest of a collection of $k$ centers and then re-estimating each center by finding the center which minimizes the average regularized cost of the observations currently associated with that center.
As the energy function is bounded below and monotonically decreasing over
iterations, this algorithm converges to a local (but not necessarily global) minimum.
More precisely, in the particular example considered here we employ the
following iterative procedure:
\begin{enumerate}[1.]
\item Initialize $\varphi^0: \{1,2,\dots,n\} \to \{1,2,\dots,k\}$ arbitrarily.
\item For a given data partition $\varphi^r: \{1,2,\dots, n\}\to \{1,2,\dots, k\}$ we independently find the cluster centers $\mu^r=(\mu_1^r,\mu_2^r,\dots,\mu_k^r)$ where each $\mu_j^r\in H^2([0,T])$ by
\[ \mu^r_j = \argmin_{\mu_j} \frac{1}{n} \sum_{i:\varphi^r(i)=j} |z_i-\mu_j(t_i)|^2 + \lambda \|\partial^2 \mu_j\|_{L^2}^2 \quad \text{for } j=1,2,\dots, k. \]
This is done using smoothing splines.
\item Data is repartitioned using the cluster centers $\mu^r$
\[ \varphi^{r+1}(i) = \argmin_{j=1,2,\dots, k} |z_i - \mu_j^r(t_i)|. \]
\item If $\varphi^{r+1} \neq \varphi^{r}$ then return to Step 2.
Else we terminate.
\end{enumerate}
Let $\mu^n=(\mu^n_1,\dots,\mu^n_k)$ be the output of the $k$-means algorithm from $n$ data points.
To evaluate the success of the methodology when dealing with a finite sample of $n$ data points we look at how many iterations are required to reach convergence (defined as an assignment which is unchanged over the course of an algorithmic iteration), the number of data points correctly associated, the metric
\[ \eta(n) = \frac{1}{k} \sqrt{\sum_{j=1}^k \| \mu^n_j-x^j \|_{L^2}^2} \]
and the energy
\[ \hat{\theta}_n = f_n(\mu^n) \]
where
\[ f_n(\mu) = \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^k |z_i-\mu_j(t_i)|^2 + \lambda \sum_{j=1}^k \|\partial^2 \mu_j\|^2_{L^2}. \]
\begin{figure
\caption{Smoothed data association trajectory results for the $k$-means method. \label{fig:ex4step}}
\centering
\setlength\figureheight{2.7cm}
\setlength\figurewidth{6cm}
\input{example1kmeansv2.tikz}
\caption*{
The figure on the left shows the raw data with the data generating model.
That on the right shows the output of the $k$-means algorithm.
The parameters used are: $k=3$, $T=10$, $\epsilon_i^j$ from a $N(0,5)$ truncated at $\pm 100$, $\lambda=1$, $x^1(t)=-15-2t+0.2t^2$, $x^2(t)=5+t$ and $x^3(t)=40$.
}
\end{figure}
Figure~\ref{fig:ex4step} shows the raw data and output of the $k$-means algorithm for one realization of the model.
We run Monte Carlo trials for increasing numbers of data points; in particular we run $10^3$ numerical trials independently for each $n=300,600,\dots,3000$ where we generate the data from~\eqref{eq:examples:example1:model} and cluster using the above algorithm.
Each numerical experiment is independent.
\begin{figure
\caption{Monte Carlo convergence results.}
\label{fig:resex4}
\centering
\setlength\figureheight{4cm}
\setlength\figurewidth{2.2cm}
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example1results.tikz}
\end{subfigure}
\quad \quad
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example1resultsiiv3.tikz}
\end{subfigure}
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example1resultsiii.tikz}
\end{subfigure}
\caption*{
Convergence results for the parameters given in Figure~\ref{fig:ex4step}.
In (a) the thick dotted line corresponds to the median number of iterations taken for the method to converge and the thinner dotted lines are the 25\% and 75\% quantiles.
The thick solid line corresponds to the median percentage of data points correctly identified and the thinner solid line are the 25\% and 75\% quantiles.
(b) shows the median value of $\eta(n)$ (solid), interquartile range (box) and the interval between the 5\% and 95\% percentiles (whiskers).
(c) shows the mean minimum energy $\hat{\theta}_n$ (solid) and the 10\% and 90\% quantiles (dashed).
The energy associated with the data generating model is also shown (long dashes).
In order to increase the chance of finding a global minimum for each Monte Carlo trial ten different initializations were tried and the one that had the smallest energy on termination was recorded.
}
\end{figure}
Results, shown in Figure~\ref{fig:resex4}, illustrate that as measured by $\eta$ the performance of the $k$-means method improves with the size of the available data set, as do the proportion of data points correctly assigned.
The minimum energy stabilizes as the size of the data set increases, although the algorithm does take more iterations for the method to converge.
We also note that the energy of the data generating functions is higher than the minimum energy.
Since the iterative $k$-means algorithm described above does not necessarily
identify global minima, we tested the algorithm on
two targets whose paths intersect as shown in Figure \ref{fig:ex4switch}.
The data association hypotheses corresponding to correct and incorrect associations, after the crossing point, correspond to two local minima.
The observation window $[0,T]$ was expanded to investigate the convergence to the correct data association hypothesis.
To enable this to be described in more detail we introduce the crossing and non-crossing energies:
\begin{align*}
E_{\text{c}} & = \frac{1}{T} f_n(\mu_{\text{c}}) \\
E_{\text{nc}} & = \frac{1}{T} f_n(\mu_{\text{nc}})
\end{align*}
where $\mu_{\text{c}}$ and $\mu_{\text{nc}}$ are the $k$-means centers for the crossing (correct) and non-crossing (incorrect) solutions.
To allow the association performance to be quantified, we therefore define the relative energy
\[ \Delta E = E_{\text{c}} - E_{\text{nc}}. \]
\begin{figure
\caption{Crossing tracks in the $k$-means method. \label{fig:ex4switch}}
\centering
\setlength\figureheight{1.8cm}
\setlength\figurewidth{5.8cm}
\input{example1correctswitch.tikz}
\input{example1incorrectswitch.tikz}
\caption*{
Typical data sets for times up to $T_{\text{max}}$ with cluster centers, fitted up till $T$, exhibiting crossing and non-crossing behavior.
The parameters used are $k=2$, $T_{\text{min}}=9.6\leq T\leq 11=T_{\text{max}}$, $\epsilon_i^j\stackrel{\mathrm{iid}}{\sim} N(0,5)$, $x^1(t)=-20+t^2$ and $x^2(t)=20+4t$.
There are $n=220$ data points uniformly distributed over $[0,11]$ with 110 observations for each track.
The crossing occurs at approximately $t\approx 8.6$ but we wait a further time unit before investigating the decision making procedure.
}
\end{figure}
\begin{figure
\caption{Energy differences in the $k$-means method. \label{fig:ex4switchresults}}
\centering
\setlength\figureheight{3cm}
\setlength\figurewidth{5.8cm}
\input{example1energyswitchresults.tikz}
\caption*{
Mean results are shown for data obtained using the parameters given in Figure~\ref{fig:ex4switch} for data up to time $T$ (between $T_{\text{min}}$ and $T_{\text{max}}$).
The thick solid line shows the mean $\Delta E$ and the thinner lines one standard deviation either side of the mean.
The dashed line shows the percentage of times we correctly identified the tracks as crossing.
}
\end{figure}
To determine how many numerical trials we should run in order to get a good number of simulations that produce crossing and non-crossing outputs we first ran the experiment until we achieved at least 100 tracks that crossed and at least 100 that did not.
I.e. let $N_t^\text{c}$ be the number of trials that output tracks that crossed and $N_t^\text{nc}$ be the number of trials that output tracks that did not cross.
We stop when $\text{min}\{N_t^\text{c},N_t^\text{nc}\}\geq 100$.
Let $N_t = 10\left(N_t^\text{c} + N_t^\text{c}\right)$.
We then re-ran the experiment with $N_t$ trials so we expect that we get 1000 tracks that do not cross and 1000 tracks that do cross at each time $t$.
The results in Figure~\ref{fig:ex4switchresults} show that initially the better solution to the $k$-means minimization problem is the one that incorrectly partitions the tracks after the intersection.
However, as time is run forward the $k$-means favors the partition that correctly associates tracks to targets.
This is reflected in both an increase in $\Delta E$ and the percentage of outputs that correctly identify the switch.
Our results show that for $T>9.7$ the energy difference between the two minima grows linearly with time.
However, when we look which minima the $k$-means algorithm finds our results suggest that after time $T\approx 10.25$ the probability of finding the correct minima stabilizes at approximately 64\%.
There is reasonably large variance in the energy difference.
The mean plus standard deviation is positive for all $T$ greater than 9.8, however it takes until $T=10.8$ for the average energy difference to be positive.
\subsection{Example 2: Passive Electromagnetic Source Tracking}
In the previous example the data is simply a linear projection of the trajectories.
In contrast, here we consider the more general case where the measurement $X$ and model $Y$ spaces are very different; being connected by a complicated mapping that results in a very non-linear cost function $d$.
While the increased complexity of the cost function does lead to a (linear in data size) increase in computational cost, the problem is equally amenable to our approach.
In this example we consider the tracking of targets that periodically emit radio pulses as they travel on a two dimensional surface.
These emissions are detected by an array of (three) sensors that characterize the detected emissions in terms of `time of arrival', `signal amplitude' and the `identity of the sensor making the detection'.
Expressed in this way, the problem has a structure which does not fall directly within the framework which the theoretical results of previous sections cover.
In particular, the observations are not independent (we have exactly one from each target in each measurement interval), they are not identically distributed and they do not admit an empirical measure which is weakly convergent in the large data limit.
This formulation could be refined so that the problem did fall precisely within the framework; but only at the expense of losing physical clarity.
This is not done but as shall be seen below, even in the current formulation, good performance is obtained.
This gives some confidence that $k$-means like strategies in general settings, at least when the qualitatively important features of the problem are close to those
considered theoretically, and gives some heuristic justification for the lack of rigor.
Three sensors receive amplitude and time of arrival from each target with periodicity $\tau$.
Data at each sensor are points in $\mathbb{R}^2$ whilst the cluster centers (trajectories) are time-parameterized curves in a different $\mathbb{R}^2$ space.
In the generating model, for clarity we again index the targets in the observed amplitude and time of arrival.
However, we again assume that this identifier is not observed and this notation is redefined (identities suppressed) when we apply the $k$-means method.
Let $x_j(t)\in \mathbb{R}^2$ be the position of target $j$ for $j=1,2,\dots k$ at time $t\in [0,T]$.
In every time frame of length $\tau$ each target emits a signal which is detected at three sensors.
The time difference from the start of the time frame to when the target emits this signal is called the time offset.
The time offset for each target is a constant which we call $o_j$ for $j=1,2,\dots, k$.
Target $j$ therefore emits a signal at times
\[ \tilde{t}_j(m) = m\tau + o_j \]
for $m\in \mathbb{N}$ such that $\tilde{t}_j(m)\leq T$.
Note that this is not the time of arrival and we do not observe $\tilde{t}_j(m)$.
Sensor $p$ at position $z_p$ detects this signal some time later and measures the time of arrival $t_j^p(m)\in [0,T]$ and amplitude $a^p_j(m)\in \mathbb{R}$ from target $j$.
The time of arrival is
\[ t^p_j(m) = m \tau + o_j + \frac{| x_j(m) - z_p |}{c} + \epsilon^p_j(m) = \tilde{t}_j(m) + \frac{| x_j(m) - z_p |}{c} + \epsilon^p_j(m) \]
where $c$ is the speed of the signal and $\epsilon^p_j(m)$ are iid noise terms with variance $\sigma^2$.
The amplitude is
\[ a^p_j(m) = \log\left( \frac{\alpha}{| x_j(m) - z_p |^2 + \beta} \right) + \delta^p_j(m) \]
where $\alpha$ and $\beta$ are constants and $\delta^p_j(m)$ are iid noise terms with variance $\nu^2$.
We assume the parameters $\alpha$, $\beta$, $c$, $\sigma$, $\tau$, $\nu$ and $z_p$ are known.
To simplify the notation $\Pi_q x:\mathbb{R}^2\to \mathbb{R}$ is the projection of $x$ onto it's $q^\text{th}$ coordinate for $q=1,2$.
I.e. the position of target $j$ at time $t$ can be written $x_j(t)=(\Pi_1 x_j(t), \Pi_2 x_j(t))$.
In practice we do not know to which target each observation corresponds.
We use the $k$-means method to partition a set $\{\xi_i=(t_i,a_i,p_i)\}_{i=1}^{n}$ into the $k$ targets.
Note the relabeling of indices; $\xi_i=(t_i,a_i,p_i)$ is the time of arrival $t_i$, amplitude $a_i$ and sensor $p_i$ of the $i^\text{th}$ detection.
The cluster centers are in a function-parameter product space $\mu_j = (\hat{x}_j(t),\hat{o}_j)\in C^0([0,T];\mathbb{R}^2)\times [0,\tau) \subset C^0([0,T];\mathbb{R}^2) \times \mathbb{R}$ that estimates the $j^\text{th}$ target's trajectory and time offset.
The $k$-means minimization problem is
\[ \mu^n = \argmin_{\mu\in (C^0\times [0,\tau))^k} \frac{1}{n} \sum_{i=1}^n \bigwedge_{j=1}^k d(\xi_i,\mu_j) \]
for a choice of cost function $d$.
If we look for cluster centers as straight trajectories then we can restrict ourselves to functions of the form $x_j(t) = x_j(0) + v_j t$ and consider the cluster centers as finite dimensional objects.
This allows us to redefine our minimization problem as
\[ \mu^n = \argmin_{\mu\in (\mathbb{R}^4\times[0,\tau))^k} \frac{1}{n} \sum_{i=1}^j \bigwedge_{j=1}^k d(\xi_i,\mu_j) \]
so that now $\mu_j = (x_j(0),v_j,o_j)\in \mathbb{R}^2\times\mathbb{R}^2\times[0,\tau)$.
We note that in this finite dimensional formulation it is not necessary to include a regularization term; a feature already anticipated in the definition of the minimization problem.
For $\mu_j=(x_j,v_j,o_j)$ we define the cost function
\[ d((t,a,p),\mu_j) = \left( \left( t,a\right) - \psi(\mu_j,p,m) \right) \left( \begin{array}{cc} \frac{1}{\sigma^2} & 0 \\ 0 & \frac{1}{\nu^2} \end{array} \right) \left( \left( \begin{array}{c} t \\ a \end{array} \right) - \psi(\mu_j,p,m)^\top \right) \]
where $m=\max\{n\in \mathbb{N}: n\tau \leq t\}$,
\[ \psi(\mu_j,p,m) = \left(\frac{|x_j+m \tau v_j - z_p|}{c} + o_j + m\tau, \log\left( \frac{\alpha}{|x_j+m\tau v_j-z_p|^2 + \beta} \right) \right) \]
and superscript $T$ denotes the transpose.
We initialize the partitions by uniformly randomly choosing $\varphi^0:\{1,2,\dots, n\}\to \{1,2,\dots ,k\}$.
At the $r^{\text{th}}$ iteration the $k$-means minimization problem is then partitioned into $k$ independent problems
\[ \mu_j^r = \argmin_{\mu_j} \sum_{i\in (\varphi^{r-1})^{-1}(j)} d((t_i,a_i,p_i),\mu_j^0) \quad \text{for } 1\leq j\leq k. \]
A range of initializations for $\mu_j$ are used to increase the chance of the method converging to a global minimum.
For optimal centers conditioned on partition $\varphi^{r-1}$ we can define the partition $\varphi^r$ to be the optimal partition of $\{(t_i,a_i,p_i)\}_{i=1}^{n}$ conditioned on centers $(\mu_j^r)$ by solving
\begin{align*}
\varphi^r: \{1,2,\dots, n\} & \to \{1,2,\dots, k\} \\
i & \mapsto \argmin_{j=1,2,\dots,k} d((t_i,a_i,p_i),\mu_j^r).
\end{align*}
The method has converged when $\varphi^r=\varphi^{r-1}$ for some $r$.
Typical simulated data and resulting trajectories are shown in Figure~\ref{fig:res}.
\begin{figure}[htdp!]
\caption{Representative data and resulting tracks for the passive tracking example. \label{fig:res}}
\centering
\setlength\figureheight{3cm}
\setlength\figurewidth{5.8cm}
\input{example2kmeans.tikz}
\caption*{
Representative data is shown for the parameters $k=2$, $\tau=1$, $T=1000$, $c=100$, $z_1=(-10,-10)$, $z_2=(10,-10)$, $z_3=(0,10)$, $\epsilon^p_j(m)\stackrel{\mathrm{iid}}{\sim} N(0,0.03^2)$, $\delta^p_j(m) \stackrel{\mathrm{iid}}{\sim} N(0,0.05^2)$, $\alpha=10^8$, $\beta=5$, $x_1(t)=\frac{\sqrt{2}t}{400}(1,1)+(0,5)$, $x_2(t)=(6,7)-\frac{t}{125}(1,0)$, $o_1=0.3$ and $o_2=0.6$, given the sensor configuration shown at the top of the figure.
The $k$-means method was run until it converged, with the trajectory component of the resulting cluster centers plotted with the true trajectories at the top of the figure.
Target one is the dashed line with starred data points, target two is the solid line and square data points.
}
\end{figure}
To illustrate the convergence result achieved above we performed a test on a set of data simulated from the same model as in Figure~\ref{fig:res}.
We sample $n_s$ observations from $\{(t_i,a_i,p_i)\}_{i=1}^{n}$ and compare our results as $n_s \to n$.
Let $\hat{x}^{n_s}(t)=(\hat{x}_1^{n_s}(t),\dots,\hat{x}_k^{n_s}(t))$ be the position output by the $k$-means method described above using $n_s$ data points and $x(t)=(x_1(t),\dots, x_k(t))$ be the true values of each cluster center.
We use the metric
\[ \eta(n_s) = \frac{1}{k} \sqrt{\sum_{j=1}^k\|\hat{x}_j^{n_s}-x_j\|_{L^2}^2} \]
to measure how close the estimated position is to the exact position.
Note we do not use the estimated time offset given by the first model.
The number of iterations required for the method to converge is also recorded.
Results are shown in Figure~\ref{fig:resex2}.
In this example the data has enough separation that we are always able to recover the true data partition.
We also see improvement in our estimated cluster centers and convergence of the minimum energy as we increase the size of the data.
Finding global minima is difficult and although we run the $k$-means method from multiple starting points we sometimes only find local minima.
For $\frac{n_s}{n}=0.3$ we see the effect of finding local minima.
In this case only one Monte Carlo trial produces a bad result, but the error $\eta$ is so great (around 28 times greater than the average) that it can be seen in the mean result shown in Figure~\ref{fig:resex2}(c).
\begin{figure}[htdp]
\caption{Monte Carlo convergence results.}
\label{fig:resex2}
\centering
\setlength\figureheight{4cm}
\setlength\figurewidth{2.2cm}
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example2results.tikz}
\end{subfigure}
\quad \quad
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example2resultsiiv3.tikz}
\end{subfigure}
\begin{subfigure}[FIGTOPCAP]{0.3\textwidth}
\caption{}
\input{example2resultsiii.tikz}
\end{subfigure}
\caption*{
Convergence results for $10^3$ Monte Carlo trials with the parameters given in Figure~\ref{fig:res}; expressed with the notation used in Figure~\ref{fig:resex4}.
In (a) we have also recorded the mean number of iterations to converge (long dashes).
The 25\% and 75\% quantiles for the number of iterations to converge is 2 and 4 for all $n$ respectively.
The 25\% and 75\% quantiles for the percentage of data points correctly identified is 100\% in both cases for all $n$.
This is due to large separation in the data space.
To increase the chance of finding a global minimum for each Monte Carlo trial, out of five different initializations, that which had the smallest energy on terminating was recorded.
}
\end{figure}
\section*{Acknowledgments}
The authors are grateful for two anonymous reviewers' valuable comments which significantly improved the manuscript.
MT is part of MASDOC at the University of Warwick and was supported by an EPSRC Industrial CASE Award PhD Studentship with Selex-ES Ltd.
\nocite{cuesta88,wu83,biau08,canas12}
\bibliographystyle{plain}
|
1,314,259,992,969 | arxiv | \section{Introduction}
As demonstrated by a recent experiment,\citep{kiguchi2007} the atomic conductance of a Au nanocontact can be altered by admitting gaseous CO in proximity of the nanocontact, suggesting that CO chemically interacts with the Au atoms in the thinnest part of the contact (a monatomic chain or a single atom).
Indeed, besides a slight renormalization of the typical conductance peak at $1\,G_0$, the conductance histogram of Au after exposition to CO displays an additional peak between $0.5\,G_0$ and $0.6\,G_0$, which can be attributed to CO adsorption.
This is further supported by the analysis of the conductance traces, which clearly pinpoints the existence of an atomic structure with a CO molecule adsorbed on, or incorporated into the monatomic chain.
This structure is invariably characterized by a conductance value close to that of the new histogram peak and also by a small increase of the conductance upon further pulling of the nanocontact, just prior to contact breaking.\citep{kiguchi2007}
In a previous density functional study, we addressed the adsorption of CO on Au monatomic chains using an infinite chain geometry without tips.\cite{sclauzero2012a}
The bridge adsorption site was found to be energetically favored with respect to the atop site, both at the equilibrium spacing of the Au chain and at larger values of the Au-Au spacing.
We characterized the adsorption process by identifying the bonding/antibonding pairs of $5\sigma$ and \tps\ states which arise from the hybridization between CO molecular levels and Au metal states.
Therefore, the electronic structure of this chain/adsorbate system results from a donation/backdonation mechanism analogous to the Blyholder model for CO on transition metal surfaces,\cite{blyholder1964,hammer1996,fohlisch2000} as previously demonstrated also for CO on Pt monatomic chains.\cite{sclauzero2008a,sclauzero2008b}
A strong connection between some electronic structure features of the CO/Au-chain system and the effects produced by CO adsorption on the ballistic transport across the chain was established in that work.\cite{sclauzero2012a}
In particular, the coupling of the $5\sigma$ antibonding state with the $s$ states of the Au chain was found to generate a dip in the $s$ transmission.
Hence, the position of this state turns out to be crucial in determining the reduction of the (tipless) conductance due to the impurity, as well as the strain dependence of the conductance in presence of CO.
In the atop geometry, the $5\sigma$ antibonding state is located very close to the Fermi level (\ef) and approaches \ef\ as the Au chain gets stretched, hence it strongly reduces the conductance of a moderately strained chain and cuts almost completely the conductance of a highly strained chain (close to the rupture Au-Au spacing).\cite{calzolari2004,strange2008,sclauzero2012a}
In the bridge geometry instead, the conductance reduction is much lower than in the atop geometry because the $5\sigma$ antibonding state is more distant from \ef\ at medium strains and the associated transmission dip actually disappears at high strains.\cite{sclauzero2012a,sclauzero2012c}
At variance with the atop geometry, which shows a decrease of the conductance with strain and is therefore not compatible with the experimental conductance traces, the conductance of the bridge geometry shows the correct dependence on strain, namely, a slight increase with increasing strain.\cite{sclauzero2012a}
The calculated conductances of the tipless bridge geometry are however too large compared to the experimental conductance of the histogram peak appearing upon CO adsorption.\cite{kiguchi2007}
Moreover, the nonuniform distribution of the strain, the finite length of the chain, and the effect of the tips are not described in the infinite chain model, therefore those results need to be corroborated by a more realistic modeling of the nanocontact.
Density functional calculations for various kinds of impurities adsorbed on model Au nanocontacts (for instance, molecular hydrogen\citep{csonka2003,csonka2006} and oxygen\citep{thijssen2006}), for CO on other metals, such as Pt,\cite{strange2006} and also for one specific geometry of CO on a Au nanowire contact\cite{xu2006} are already available in the literature.
However, different adsorption geometries of CO on Au nanocontacts have not been compared and the effect of contact stretching on the ballistic conductance of this system still needs to be analyzed.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.9\columnwidth]{COatAuwire_surf-GGA}
\end{center}
\caption{(Color online) Lateral view of the periodic cells for the Au chain between Au(001) surfaces and CO adsorbed at the (a) bridge and (b) atop sites.
The smaller cell in (a) is used to compute the complex band structure of the leads.
The spacing between the bulk Au layers (\db), the outermost interlayer distance (\ds), and the inter-surface distance (\dsurf) are indicated.}
\label{fig:geom}
\end{figure}
In this work, we study the adsorption energetics of CO on a short Au monatomic chain and we examine the ballistic conductance of the system as a function of the contact stretching using model nanocontact geometries.
By comparing the bridge and atop adsorption geometries of CO, we find that the bridge site is energetically favored at all strains, confirming the result obtained within the infinite chain model.\cite{sclauzero2012a}
In the atop geometry, the Au conductance cut due to the $5\sigma$ antibonding states\cite{sclauzero2012a} is observed also in the present nanocontact model when the contact is sufficiently stretched.
The bridge geometry instead, reproduces to a good accuracy the experimental conductance value ($0.5\div0.6\,G_0$),\cite{kiguchi2007} representing a significant improvement with respect to the infinite chain model.
The slope of the conductance as a function of the contact stretching is instead reproduced only qualitatively by our nanocontact model.
We address this discrepancy showing that a conductance slope closer to the experimental one can be obtained by taking into account, even only approximately, the effect of the elastic response of the bulk leads to the external pulling force, not described by our nanocontact model.
This paper is organized as follows: in \pref{sec:method}, we describe the numerical methods and approximations adopted in the calculations; in \pref{sec:geom} and in \pref{sec:cond} we will report on, respectively, the adsorption energetics and the ballistic conductance of the system as a function of the contact stretching; \pref{sec:disc} is devoted to a discussion of the conductance results in the light of the available experimental data; finally, our conclusions will follow in \pref{sec:concl}.
\section{Methods and computational details}\label{sec:method}
All density functional calculations presented in this work are carried out using the plane-wave pseudopotential code \pwscf\ contained in the \qe\ package.\cite{QE-2009}
The exchange and correlations functionals, both for the local density (LDA) and the generalized gradient approximation (GGA), the basis set cutoffs, the pseudopotentials, and the smearing parameters are the same as those used for the infinite chain model.\cite{sclauzero2012a}.
The monatomic chain in the nanocontact geometries is modeled as a row of metal atoms suspended between fcc bulk leads terminated by two facing (001) surfaces.\cite{smogunov2008b}
In \pref{fig:geom}, we show the simulation cells consisting of a periodically repeated slab geometry with 7 fcc-Au layers perpendicular to the [001] direction, which is chosen as $z$ axis and coincides with the electron transport direction (see later).
The spacing between the inner layers corresponds to the theoretical equilibrium value in the bulk \db, which is $2.029\,\ang$ ($2.082\,\ang$) according to our LDA (GGA) calculations.
The clean unreconstructed Au(001) surface shows a significant inward relaxation of about $1.8\%$ ($1.5\%$) for outermost layer within the LDA (GGA), in agreement with recent DFT calculations,\citep{singh-miller2009} while spacings between inner layers deviate from the bulk value by less than $0.4\%$.
Therefore, we keep into account the relaxation of the outermost layer on both sides of the junction by setting the first interlayer distance $\ds=1.992\,\ang$ ($2.051\,\ang$).
The in-plane distances between gold atoms are those of the bulk.
The short chain is freely suspended in the vacuum region between the slabs and the apex atoms of the chain are attached to the surfaces at a 4-fold coordinated hollow site.
The symmetry group of the straight chain between the two (001) surfaces is \Dqh, the same as in the clean surface case studied with a slab geometry having inversion symmetry.
A single CO molecule is adsorbed at the center of the Au chain, at the bridge site of a 4-atom-long chain (\pref{fig:geom}a) and at the atop site of a 5-atom-long chain (\pref{fig:geom}b).
The symmetry group of the system in presence of CO is \Cdv, as in the case of CO adsorbed at the bridge or atop sites of the infinite chain.
The $xy$ in-plane periodicity of the simulation cells corresponds to a $(2\sqrt{2}\times2\sqrt{2})\mathrm{R}45^{\circ}$ surface structure, which gives a chain-chain spacing of about $8.12\,\ang$ in the $x$ and $y$ directions between two adjacent replicas.
The full Brillouin zone (BZ) of these structures is sampled with a uniform mesh of $6\times 6\times 3$ \bfk-points, which can be reduced by symmetry to 12 points (18 when CO is adsorbed).
We relax the atomic positions of the Au chain and of CO until the forces on those atoms drop below $0.026\;\ev/\ang$.
We also check that the optimized atomic positions of the chain and of CO obtained in this way result in atomic forces below $0.08\,\ev/\ang$ when used in a $(3\sqrt{2}\times3\sqrt{2})\mathrm{R}45^{\circ}$ cell (chain-chain spacing of $12.18\,\ang$ along $x$ and $y$).
For this larger cell we reduce the size of the \bfk-point mesh down to $4\times 4\times 3$.
The ballistic conductance is evaluated with the Landauer-B\"uttiker formula, $G = e^2/h\:T(E_F)$, where $T(E_F)$ is the total transmission at the Fermi energy.
We calculate the electron transmission using the scattering-based approach of \citeauthor{choi1999}\cite{choi1999} extended to ultrasoft pseudopotentials,\cite{smogunov2004b} and implemented in the \pwcond\ code.\cite{QE-2009}
In transmission calculations, we include in the scattering region a portion of the leads together with the impurity region (see \pref{fig:geom}).
The semi-infinite bulk leads are modeled using an additional, smaller cell which coincides with two bulk Au(001) layers (smaller cell in \pref{fig:geom}a).\footnote{The lead regions are used to compute the electronic complex band structures (CBSs) needed to solve the scattering problem.
We verified that energy eigenvalues at real $k_z$ in the CBS obtained from the self-consistent potential in the leftmost part of the scattering region (corresponding to the unit cell of the lead) match within $0.05\,\ev$ the eigenvalues in the CBS of the lead region.}
In the $(2\sqrt{2}\times2\sqrt{2})\mathrm{R}45^{\circ}$ nanocontact geometry (which we will also call ``abrupt'' junction), the \kperp-dependent transmission is sampled with a uniform $7\times 7$ shifted mesh of \kperp-points in the 2D-BZ perpendicular to the transport direction, corresponding to $10$ and $16$ \kperp-points in the irreducible 2D BZs of the clean nanocontact and of the nanocontact with CO, respectively.\footnote{The importance of accurately sampling the ballistic transmission of model nanocontacts with extended leads has been discussed by \citeauthor{thygesen2005}.\cite{thygesen2005}
We checked that our sampling gives well converged transmission values in the neighborhood of the Fermi level \ef\ (errors within $1\%$), but can result in larger errors at some scattering energies further away from \ef.
However, we are here more interested in the nanocontact conductance and hence we need a well converged transmission just close to \ef.}
Smoother junctions have been simulated using periodic cells with a $(3\times 3)$ in-plane periodicity and pyramidal tips connecting the chain apexes to the Au(001) surfaces (see \pref{fig:autipstran}).
This geometry is composed of a seven-layer slab with an interlayer spacing equal to \db, plus four additional Au atoms in the positions of an additional layer at a distance \ds\ from the surface planes, with the chain attached to the 4-fold hollow sites formed by these additional atoms.
A $5\times5$ uniform mesh of \kperp-points has been used to sample the transmission in the 2D-BZ.
\section{Geometry and energetics}\label{sec:geom}
For different values of the inter-surface distance \dsurf\ (cf.\ \pref{fig:geom}), we consider straight monatomic chains without CO and chains with CO adsorbed at the bridge or atop sites.
In this way, by increasing \dsurf\ we mimic the increase of strain produced on the chain by the pulling of the contact ends.
We optimize the atomic positions of the C and O atoms, and of the Au atoms belonging to the chain, while the positions of all Au atoms in the (001) planes are kept fixed (as are the distances between the planes).
Since we are mainly interested in the local interaction between CO and the Au chain, which is primarily affected by the chain strain, we will not consider here the atomic relaxations of the Au planes at the two sides of the junction.
The effects of these relaxations on the strain dependence of the ballistic conductance are studied in \pref{sec:disc} through an approximate model of the elastic response of the atomic planes to the contact stretching.
\begin{table}[tb]
\caption{Optimized distances (in \ang) and chemisorption energies (in \ev) of CO at the bridge site of a 4-atom-long chain (cf.\ \pref{fig:geom}a) obtained within GGA for selected values of \dsurf.
The optimized distances of the clean chain are also reported.
Owing to the symmetry,
$d_{3-4}=d_{1-2}$ and $d_{\mathrm{4s}}=d_{\mathrm{s1}}$.}\label{tab:ausurfgeombridge}
\begin{tabular}{c@{\hspace{9pt}}ccc@{\hspace{9pt}}cccccc}
\hline\hline
& \multicolumn{3}{c@{\hspace{9pt}}}{4-Au chain} & \multicolumn{6}{c}{CO at the bridge site} \\
\dsurf& \dsm& \dab& \dbc& \dsm& \dab& \dbc& \dauc& \dco& \echem\\ \hline
11.56& 1.87& 2.61& 2.61& 1.90& 2.59& 2.86& 2.03& 1.17& $-$1.47\\
12.16& 1.99& 2.72& 2.74& 1.93& 2.61& 3.09& 2.02& 1.17& $-$1.61\\
12.76& 2.09& 2.82& 2.94& 1.99& 2.67& 3.44& 2.06& 1.18& $-$1.94\\
\hline\hline
\end{tabular}
\end{table}
\subsection{Clean nanocontact geometries}
We first consider a straight chain without CO and optimize the $z$ coordinate of the \nau\ Au atoms of the chain, thus removing the constraint of uniform interatomic spacing that was adopted in the infinite chain model.\cite{sclauzero2012a}
We did not explore here zigzag or bent configurations, which are expected to become favored only at low values of \dsurf\ in the clean chain.\citep{hakkinen00,sanchez99,skorodumova2005}
The optimized distances between the Au atoms in the chain (\dab, \dbc, \dots) and the distance between the surface plane and the apex atom of the chain (\dsm) are presented in the left parts of \pref{tab:ausurfgeombridge} and \pref{tab:ausurfgeomontop} for a 4-atom-long and a 5-atom-long chain, respectively.
We report here only GGA data, since the LDA results give the same qualitative picture.\cite{sclauzero2010}
In the first row of each table, \dsurf\ is chosen to give Au-Au distances in the short chain close to the equilibrium value in the infinite chain ($2.61\;\ang$), while in the second and third rows the selected $d_{\rm ss}$ values result in moderately or highly stretched Au-Au bonds, respectively.
At the lowest strain considered here, the atoms in the chain are almost equally spaced, but the Au-Au bond length at the extremities of the chain, \dab, adjusts to a value slightly smaller than in the middle (only \dbc\ in the 4-atom and 5-atom chains, but we verified that this holds also for the inner bonds of longer chains).\cite{sclauzero2010}
As the nanocontact is stretched, the distance \dab\ becomes progressively shorter than \dbc.
By studying longer chains ($\nau=6$ and $\nau=7$, not reported here), we observe Au-Au bond lengths which increase while going from the ends toward the center of the unstrained chains.
Instead, when the average Au-Au bond length is above $3.0\,\ang$ the chains show a tendency to dimerization, with alternating longer and shorter bonds, as already reported in the literature.\cite{okamoto1999}
\subsection{Geometries with adsorbed CO}
We now consider the nanocontact geometries with an adsorbed CO molecule.
The atomic structure is partially optimized as described above and the positions of the Au atoms in the chain are fully relaxed.
We first consider the bridge geometry, where the CO is placed upright at the bridge site between the two central Au atoms ($\mathrm{Au}_{(2)}$ and $\mathrm{Au}_{(3)}$), as shown in \pref{fig:geom}a.
In \pref{tab:ausurfgeombridge} (right side), we report the optimized distances for the three \dsurf\ values considered before.
The carbon-oxygen bond length in the adsorbed molecule (\dco) is equal or slightly larger than in the infinite chain model\cite{sclauzero2012a} (by less than $1\%$), while C-Au bond lengths (\dauc) are $1\%$ to $3\%$ larger.
We also notice that the bond length between the two Au atoms in contact with CO (\dbc) is always longer than the other Au-Au bonds in the chain ($d_{1-2} = d_{3-4}$).
This Au-Au bond softening in correspondence of the adsorption site is observed also for longer chains.\cite{sclauzero2010}
At the lowest strain studied here ($\dsurf=11.56\,\ang$), \dbc\ adjusts to a value similar to or slightly larger than the Au-Au distance corresponding to the bridge energy minimum for the infinite chain geometry\cite{sclauzero2012a} at $\dauau=2.87\,\ang$.
The other Au-Au bond lenghts, instead, are closer to the equilibrium spacing of the infinite chain.
At low and moderate strains, the chain in its relaxed geometry bends towards CO, while at larger strains the chain atoms get progressively more aligned forming an almost linear strand with one overstretched Au-Au bond in correspondence of the adsorption site.
Because of the flexibility of the Au-C-Au bond angle, very large Au-Au distances between the two atoms in contact with CO are possible as \dsurf\ is increased, but the tilting of the molecule axis from the perpendicular position could become favorable above a critical value of \dsurf.\cite{strange2006}
However, the upright bridge position in the infinite chain geometry\cite{sclauzero2012a} is the lowest energy configuration for Au-Au distances up to $4.2\,\ang$, a distance longer than the largest $d_{2-3}$ reported in \pref{tab:ausurfgeombridge}.
Hence, one can expect that the perpendicular position of CO is preferred for all values of \dsurf\ considered here.
\begin{table}[tb]
\caption{Optimized distances (in \ang) and chemisorption energies (in \ev) of CO at the atop site of a 5-atom long chain (\pref{fig:geom}b). Here, $d_{4-5}=d_{1-2}$, $d_{3-4}=d_{2-3}$, and $d_{\mathrm{5s}}=d_{\mathrm{s1}}$.}\label{tab:ausurfgeomontop}
\begin{tabular}{c@{\hspace{9pt}}ccc@{\hspace{9pt}}cccccc}
\hline\hline
& \multicolumn{3}{c@{\hspace{9pt}}}{5-Au chain} & \multicolumn{6}{c}{CO at the atop site} \\
\dsurf& \dsm& \dab& \dbc& \dsm& \dab& \dbc& \dauc& \dco& \echem\\ \hline
14.16& 1.86& 2.61& 2.61& 1.98& 2.72& 2.76& 1.98& 1.14& $-$0.78\\
14.96& 2.00& 2.73& 2.75& 1.99& 2.73& 2.83& 1.99& 1.14& $-$0.55\\
15.76& 2.09& 2.87& 2.92& 2.04& 2.77& 3.08& 1.98& 1.14& $-$0.63\\
\hline\hline
\end{tabular}
\end{table}
In \pref{tab:ausurfgeomontop}, we report the optimized distances for CO adsorbed atop the central atom of a five-atom-long Au chain ($\mathrm{Au}_{(3)}$ in \pref{fig:geom}b).
The carbon-oxygen bond distance \dco\ changes very little with strain and is very close to the value found in the infinite chain model.\cite{sclauzero2012a}
Also the C-$\mathrm{Au}_{(3)}$ bond length is not much influenced by strain and is only about $1\%$ larger than the corresponding distance for CO on the infinite chain.
As can be seen from the two nanocontact geometries reported in \pref{fig:geom}, the adsorption of CO in the atop position gives rise to larger distortions of the Au chain compared to the bridge position, with the atom which binds to C, $\mathrm{Au}_{(3)}$, moving towards the molecule and the two lateral atoms, $\mathrm{Au}_{(2)}$ and $\mathrm{Au}_{(4)}$, slightly displaced downwards to create a zigzag geometry.
By comparing \dab\ with \dbc, we see that the binding between C and $\mathrm{Au}_{(3)}$ weakens the metallic bond between $\mathrm{Au}_{(3)}$ and the two neighbouring atoms, $\mathrm{Au}_{(2)}$ and $\mathrm{Au}_{(4)}$.
Indeed, when CO is adsorbed atop the ratio $\dbc/\dab$ is always larger than in the pristine 5-atom-long chain and grows more rapidly with strain (cf.\ \pref{tab:ausurfgeomontop}).
\subsection{Chemisorption energies}
We now turn to examine the chemisorption energies of CO as a function of the surface-surface distance $\echem(\dsurf)$, which are reported in \pref{tab:ausurfgeombridge} and \pref{tab:ausurfgeomontop} for the bridge and atop geometries, respectively.
The large difference between the chemisorption energies of the bridge and atop adsorption sites indicates a strong preference for the former, as already observed in the infinite chain model.\cite{sclauzero2012a}
In the bridge geometry, the chemisorption energy decreases with strain, following the same trend seen for the infinite chain model,\cite{sclauzero2012a} while in the atop geometry this happens only at large enough values of \dsurf.
This can be seen more clearly in \pref{fig:ausurfechem}, where we compare the chemisorption energies computed with the nanocontact geometry as a function of \dsurf, $\echem(\dsurf)$, with those obtained from the infinite chain geometry as a function of the uniform spacing of the chain \dwire, $\echem(\dwire)$.\cite{sclauzero2012a}
To do this, we express $\echem(\dwire)$ as a function of an equivalent inter-surface distance $\tilde{\dsurf}(\dwire)$, which can be directly compared to \dsurf.
Hence, we define: $\tilde{\dsurf}=2\cdot\langle\dsm\rangle + (\nau-1)\cdot\dwire$, where the Au-Au spacing \dwire\ is multiplied by the number of Au-Au bonds in the \nau-atom-long suspended chain and $\langle\dsm\rangle$ accounts for the distance between the surface plane and the apex atom of the chain.
We obtain $\langle\dsm\rangle$ by averaging the \dsm\ values reported in \pref{tab:ausurfgeombridge} for the bridge geometry and in \pref{tab:ausurfgeomontop} for the atop geometry, which give $\langle\dsm\rangle=1.94\,\ang$ and $\langle\dsm\rangle=2.00\,\ang$, respectively.
For the bridge geometry, there is a qualitative agreement between $\echem(\tilde{\dsurf})$ from the infinite straight chain (open squares in \pref{fig:ausurfechem}) and $\echem(\dsurf)$ from the nanocontact geometry with $\nau=4$ (filled squares).
This difference in the numerical values could be mainly imputed to the nonuniform Au-Au spacing, to the bending, or to the short length of the chain in the nanocontact geometry.
For the atop geometry (open and filled circles), there is a larger deviation at low \dsurf\ because of the appearance of zigzag configurations in the nanocontact.
When the height of the Au atom below CO is optimized also in the infinite chain (open diamonds), the agreement with the chemisorption energy obtained in the nanocontact improves considerably, because that Au atom shows a pronounced displacement toward the CO at low strains.
Consequently, energy contributions to \echem\ due to distortions of the chain further away from the adsorption site should be of smaller importance.
\begin{figure}[tb]
\includegraphics[angle=-90,width=0.9\columnwidth]{Echem_surf2-rev}
\caption{(Color online) Chemisorption energies $\echem(\dsurf)$ for CO at the bridge site of a four-Au-atom chain (filled squares) and at the atop site of a five-Au-atom chain (filled circles).
The values of $\echem(\dwire)$ for the infinite straight chain\cite{sclauzero2012a} are reported here as a function of $\tilde{\dsurf}(\dwire)=2\cdot\langle\dsm\rangle + (\nau-1)\cdot\dwire$, where $\nau=4$ for the bridge geometry (open squares) and $\nau=5$ for the atop geometry (open circles).
$\langle\dsm\rangle$ is chosen as described in the text.
For the atop site of the infinite chain we also report $\echem(\tilde{\dsurf})$ after optimizing the position along $x$ of the Au atom below CO (open diamonds).}
\label{fig:ausurfechem}
\end{figure}
Another point that influences the numerical values of \echem\ is the choice of the clean nanocontact geometry which gives the reference energy entering into the calculation of \echem.
Zigzag and bent configurations have been predicted for short monatomic chains between tips,\citep{hakkinen00} while these have not been considered in the present work.
However, we expect that only the \echem\ values for the smallest \dsurf\ value considered here would be affected if using a nonlinear chain geometry as reference.
Indeed, zigzag or bent geometries are preferred to the linear chain only for sufficiently small values of \dsurf, while the linear chain is favored with respect to both zigzag and bent configurations when the average spacing between atoms in the chain is equal or larger than about $2.65\,\ang$ and the three configurations are nearly degenerate for slightly smaller spacings.\cite{hakkinen00}
Finally, this comparison between the infinite chain and the nanocontact geometry shows that a very simplified model such as the straight infinite chain with uniform spacing gives chemisorption energies of CO in qualitative agreement with those obtained through a more realistic and computationally expensive model, leading to the same site preference prediction.
A more quantitative agreement in \echem\ can be obtained by including a few additional degrees of freedom in the geometry optimizations of the infinite chain, such as the Au atom displacement when CO is atop, or a larger spacing between the two Au atoms in contact with CO in the bridge position, which could be inferred from the residual forces after a partial structural optimization.
\begin{figure*}[tp]
\includegraphics[width=0.8\textwidth]{Cond_surf100-LDA}
\caption{(Color online) Transmission as a function of the electron scattering energy for the clean Au nanocontact (diamonds) and for the Au nanocontact with CO (squares).
On the left (right) panels, we report the transmission of the clean Au chain with $\nau=4$ ($\nau=5$) and of the chain with CO adsorbed at the bridge (atop) site.
For each system, two values of the distance between the Au(001) surfaces are considered, a smaller one, corresponding to a moderately strained chain (top panels), and a larger one imposing a high strain on the chain (bottom panels).
In each plot, the Fermi energy is indicated by a solid vertical line and the central part of the scattering region with the impurity is shown in the insets.
The inter-surface distance \dsurf\ and the optimized distances (in \ang), as well as the chemisorption energies of CO (in \ev) are also indicated.
}
\label{fig:ausurftran}
\end{figure*}
\begin{figure*}[t]
\includegraphics[width=0.8\textwidth]{Cond_tips100-LDA}
\caption{(Color online) Transmission of a short chain with CO adsorbed at the bridge (left) or at the atop site (right): dependence on the smoothness of the surface/chain interface.
The transmission of the $(2\sqrt{2}\times2\sqrt{2})\mathrm{R}45^{\circ}$ cell (abrupt interfaces, see \pref{fig:ausurftran}) and that of the $(3\times3)$ cell (smoother interfaces, see insets) are compared for a selected value of \dsurf\ and are reported as a function of the scattering energy (squares and triangles, respectively).
The central region of the smooth junction is built from the atomic positions optimized for the smaller cell (triangles) or by relaxing again the atoms in the constriction (circles).}
\label{fig:autipstran}
\end{figure*}
\section{Ballistic conductance}\label{sec:cond}
In this section, we address the effects of CO adsorption on the electron transport of the Au nanocontact for several values of the inter-surface distance, \dsurf.
In this way, we aim to describe the strain dependence of the CO-induced changes in the ballistic transmission and to simulate the evolution of the conductance as the contact gets gradually stretched.
\subsection{Clean nanocontact}
We start discussing the ballistic transport of the short chain for the abrupt junction without CO.
The ballistic transmission as a function of the electron scattering energy is presented here for the LDA case only.
However, the GGA ballistic conductances reported later in \pref{sec:disc} (\pref{fig:autranexp}) show that the same conclusions could be reached within GGA.
In Figs.~\ref{fig:ausurftran}(a--b), we show the transmission of a 4Au-atom straight chain as a function of the scattering energy for two selected values of the intersurface distance \dsurf, one resulting in a moderately strained chain [$\dsurf=11.76\,\ang$, \pref{fig:ausurftran}(a)], and the other in a highly strained chain [$\dsurf=12.36\,\ang$, \pref{fig:ausurftran}(b)].\footnote{%
A more complete discussion of the LDA geometries can be found in Ref.~\onlinecite{sclauzero2010}.
For the readers' convenience, we report here the LDA optimized distances of the clean nanocontact geometries used for the transmission calculations.
For the 4-Au chain, with $\dsurf=11.76\,\ang$ we obtained $\dsm=1.91\,\ang$, $\dab=2.64\,\ang$, and $\dbc=2.66\,\ang$, while with $\dsurf=12.36\,\ang$ we obtained $\dsm=2.00\,\ang$, $\dab=2.77\,\ang$, and $\dbc=2.81\,\ang$.
For the 5-Au chain, with $\dsurf=14.56\,\ang$ we obtained $\dsm=1.93\,\ang$, $\dab=2.66\,\ang$, and $\dbc=2.69\,\ang$, while with $\dsurf=15.36\,\ang$ we obtained $\dsm=2.01\,\ang$, $\dab=2.81\,\ang$, and $\dbc=2.86\,\ang$.}
In the low strain configuration [\pref{fig:ausurftran}(a)], the conductance is slightly above $0.9\,G_0$, in fair agreement with the theoretical conductance of a 4-atom-chain between Au tips oriented along the [110] direction.\cite{hakkinen00}
Although the number of scattering channels increases with the cross-section of the supercell, for a long enough chain and a small charge transfer between the leads and the chain, the theoretical maximum of the transmission is given by the number of channels in the infinite tipless chain.
Thus, the conductance value close to $1\,G_0$ can be attributed to a single well transmitted spin-degenerate channel of $s$ character.
Around the Fermi energy (\ef) the transmission curve is rather flat, while below \ef\ it is more structured because of a slightly higher reflection of the $s$ channel and the additional contribution from the poorly transmitted $d$ channels of the chain.
This is in line with the common understanding that $d$-type scattering states are more reflected by the presence of an abrupt change in the atomic geometry, like that at the surface/wire interface, owing to the highly directional character of the $d$ wavefunctions.
When a larger strain is considered [\pref{fig:ausurftran}(b)], the conductance changes very little and decreases of just about $1\,\%$.
Previous calculations\cite{hakkinen00,okamoto1999} have also found that the conductance decreases monotonically when the junction gets stretched and that the conductance has not yet started to drop significantly for average Au-Au spacings around $2.80\,\ang$, being still close to $1\,G_0$,\citep{okamoto1999} or slightly below.\citep{hakkinen00}
Above \ef\ the shape of the transmission function is very similar to that obtained for the low-strain configuration, while below \ef\ the reflection slightly increases with strain.
In Figs.~\ref{fig:ausurftran}(c--d), we show the transmission function for the 5-atom-long chain for $\dsurf=14.56\,\ang$ and for $\dsurf=15.36\,\ang$, corresponding to moderate and high strains, respectively.
The conductance of the 5Au-atom chain is about $0.96\,G_0$ for both \dsurf\ values considered here and therefore is slightly larger than that of the 4Au-atom chain, in agreement with the odd-even effect seen in experiments\citep{smit2003} and in theoretical calculations.\citep{delavega2004,skorodumova2005}
For instance, \citet{delavega2004} have calculated the theoretical conductance of short chains between flat Au(111) surfaces showing that the conductance for an even number of atoms in the chain is lower than that for an odd number of atoms, as also seen in experiments.
Their conductance values, about $0.96\,G_0$ and $0.99\,G_0$ for 4-atom and 5-atom long chains, respectively, are somewhat larger compared to the results of this work, probably because their chains are attached to the more compact Au(111) surface.
Also for the 5Au-atom chain, we find that the conductance dependence on strain is quite modest, since it stays almost constant when the average spacing in the chain grows from about $2.74\,\ang$ to $2.90\,\ang$ as $\dsurf$ is increased.
For the 5-atom-long chain between Au(111) surfaces, de la Vega and coworkers have found that the conductance goes from $0.99\,G_0$ to about $1\,G_0$ when the spacing between atoms in the chain grows from $2.70\,\ang$ to $3.00\,\ang$, while the conductance of the 4-atom-long chain does not change appreciably,\citep{delavega2004} as in the case of the Au(001) surface examined here.
\subsection{Nanocontact with adsorbed CO}
We will now describe the changes in the ballistic transmission induced by CO adsorption at the bridge site or at the atop site of the Au chain in the nanocontact.
The transmissions are shown as a function of the scattering energy in Figs.~\ref{fig:ausurftran}(a--b) for the bridge geometry and in Figs.~\ref{fig:ausurftran}(c--d) for the atop one (squares), together with the previously discussed transmissions of the clean nanocontact (diamonds) for the same values of \dsurf.
When CO is at the bridge site, the conductance of the chain at low strains [\pref{fig:ausurftran}(a)] is reduced to about $0.63\,G_0$, a value close to the fractional conductance peak seen in Au nanocontacts experiments in presence of CO gas.\citep{kiguchi2007}
The transmission curve shows only a slight dependence on the scattering energy and decreases monotonically in the range of energies considered here.
The $5d$ transmission peak seen below \ef\ in the pristine chain is suppressed here, but actually for energies slightly above or below it the transmission increases after CO adsorption.
The dependence of the transmission on the scattering energy around \ef\ is similar to that seen in Ref.~\onlinecite{sclauzero2012a} for the tipless infinite chain geometries at medium/high strains, but the transmission values are lower by about $10\,\%$ in the short chain geometries.
At larger strains [\pref{fig:ausurftran}(b)], the transmission of the bridge configuration is still characterized by a smoothly varying and monotonically decreasing behaviour as a function of energy, but shows slightly higher values with respect to the low strain configuration.
The conductance grows to about $0.72\,G_0$ and is therefore compatible with the conductance increase due to the contact stretching observed in the experimental Au conductance traces at fractional conductance values.\citep{kiguchi2007}
Previous conductance calculations for a 3-atom chain between Au(111) leads and a CO molecule at the bridge have found a larger conductance,\cite{xu2006} about $0.9\,G_0$, for a strain level which is roughly intermediate between the two considered here.
This discrepancy may be ascribed to the shorter chain length\citep{grigoriev2006} or to the rather different functional used (B3LYP).
When CO is adsorbed at the atop site, the conductance reduction with respect to the clean nanocontact is much stronger than for the bridge site.
In the low-strain configuration [\pref{fig:ausurftran}(c)], the transmission curve has a wide depression centered just above \ef\ and the conductance is about $0.08\,G_0$, more than one order of magnitude smaller than in the clean nanocontact.
Below \ef, the $d$ peak is suppressed as in the bridge geometry, while at energies higher than $0.5\,\ev$ the $s$ channel is less reflected and has a transmission which approaches that of the pristine chain.
At higher strains [\pref{fig:ausurftran}(d)], the transmission dip shifts towards \ef\ causing a further lowering of the conductance down to about $0.03\,G_0$.
Therefore, the strain dependence of the conductance in the bridge and in the atop configurations are completely different, as already inferred from the transmission of the infinite chain geometries.\cite{sclauzero2012a}
Moreover, the mechanism suppressing the transmission around the Fermi energy in the atop geometry is the same for both nanocontact and infinite chain geometries.
It can be regarded as a result of Fano-like destructive interference due to resonance scattering on the \sga\ antibonding state brought up by CO adsorption and appearing right at \ef\ in the nanocontact geometry.
A more complete picture of the connection between the electronic structure features and the ballistic transmission of the CO/Au chain system was presented in a previous work using a tipless infinite chain model.\cite{sclauzero2012a}
These conclusions are not modified when considering ``smoother'' junctions with pyramidal tips on both sides of the chain.
We use the previously relaxed ``abrupt'' geometries to build a smoother junction by inserting the CO and the Au chain between the two terminal atomic planes of the tips (see insets of \pref{fig:autipstran}).
These planes are made by four atoms each, arranged to form a square, and are spaced by \dsurf\ along $z$.
In \pref{fig:autipstran}, we compare the transmission curves of the so-obtained ``smooth'' interface (triangles) with those of the corresponding ``abrupt'' interface (squares) for a selected value of \dsurf, both for the bridge and for the atop geometry.
A quite good agreement can generally be observed below \ef, while above \ef\ there are larger discrepancies, especially for the bridge geometry.
Nevertheless, the almost perfect correspondence of the conductance values confirms that the striking difference between the bridge and atop conductances is not influenced by the precise atomic geometry chosen to model the wire/surface interface.
We further optimized part of the structure by letting the 4 basal atoms on each side move along the longitudinal direction ($z$ axis) and completely relaxing the atomic positions of the chain and of the CO.
The structural changes are rather small and the total energy is lowered by only $0.3\,\ev$ or less.
The transmissions obtained from these relaxed smooth interfaces (dots in \pref{fig:autipstran}) do not differ appreciably from the non-relaxed structure in the bridge geometry, while the \sga\ transmission dip moves slightly closer to \ef\ in the atop geometry.
In both cases, these conductance values are very similar to those presented above and do not affect the conclusions drawn from the analysis of the abrupt interfaces.
\section{Discussion and comparison with experimental data}\label{sec:disc}
A reproducible behaviour in the conductance traces of Au nanocontacts in presence of CO was reported by a recent experiment:\citep{kiguchi2007} a sharp reduction of the conductance, from the $1\,G_0$ value of the Au monatomic chain down to about $0.5\,G_0$, followed by a slow increase upon further stretching of the nanocontact and by the final drop into the tunneling regime after a small elongation.
It is reasonable to assume that, at some point of the pulling cycle, a CO molecule sticks to the Au chain giving rise to the sudden reduction of the conductance.
Our theoretical conductances for the bridge and atop geometries, together with their strain dependence, allow us to simulate the conductance trace after CO adsorption at the bridge or at the atop site, and hence to make a direct comparison with the experimental traces.
\begin{figure}[tb]
\includegraphics[width=0.9\columnwidth]{Cond-COatAu_surf-cfr_rev}
\caption{(Color online) Ballistic conductance as a function of the tips displacement: comparison between simulations (left panel) and experiment (right panel).
Both LDA (open symbols) and GGA (filled symbols) conductances are shown as a function of \dsurf\ for the bridge (squares) and atop (circles) geometries.
\dsurf\ is shifted by a constant value $d_{\mathrm{ss},0}$ to keep into account the experimental offset in the displacement measurement.
The experimental conductance trace has been adapted from Ref.~\onlinecite{kiguchi2007}.
The same scales for the conductance and the displacement are used in the two panels.}
\label{fig:autranexp}
\end{figure}
In \pref{fig:autranexp}, the conductances of the bridge and atop geometries are shown as a function of the displacements between the Au surfaces and are juxtaposed with an experimental conductance trace\cite{kiguchi2007} which presents the features described here above.
We notice that the bridge configuration gives conductance values close to experimental ones and the correct dependence on strain, while the atop one gives the opposite behaviour with strain and too low conductance values at high strains.
In the bridge geometry, the GGA and LDA conductances show a very good agreement with each other.
In the atop geometry, the GGA conductance decreases more slowly than the LDA one, but the dependence on the contact stretching is the same.
These conductance results, together with the strong energetic preference for the bridge site (\pref{sec:geom}), are compatible with the presence of the fractional peak at about $0.6\,G_0$ in the Au conductance histograms and with the absence of a low conductance tail, which would be possible only in presence of an energetically favored atop geometry.
However, we notice that the theoretical conductance of the bridge geometry reproduces the slope of the experimental conductance trace only qualitatively (\pref{fig:autranexp}).
We are not aware of other experiments reporting about the conductance slope of the CO/Au nanocontact system and we are not able to assess the degree of the experimental reproducibility, but we find anyway interesting to explore further this point.
The discrepancy in the numerical values of the slope might be related to the elastic response of the bulk leads to the external pulling force, which is present in the real nanocontact but has been neglected in the atomic relaxations of our model geometries.
Indeed, a more realistic modeling of the structural modifications during the contact stretching process would require the relaxation of the atomic position in the atomic planes forming the leads, but this can be done only considering much larger supercells.
It is possible to keep into account approximately the mechanical response of the leads to the external stress by treating them as ideal springs with a finite spring-constant $k_s$.\cite{strange2006}
We assume that the position of the electrodes is controlled at two opposite ends far away from the junction, separated by a distance $L \gg \dsurf$.
The force balance between the ideal springs and the elastic response of the junction region can be expressed by the following equation:\citep{strange2006}
\begin{equation}
\frac{1}{2} k_s (L - \dsurf) = \frac{\de \etot(\dsurf)}{\de \dsurf},
\label{eq:springs}
\end{equation}
where $\etot(\dsurf)$ is the total energy of the relaxed nanocontact geometry with inter-surface distance \dsurf.
By solving for $L$ in \pref{eq:springs}, we can convert the inter-surface distance \dsurf\ into an equivalent tips displacement $L(\dsurf;k_s$), which includes the elongation of the leads and has only the electrode stiffness $k_s$ as external parameter.
Notice that in the unrealistic assumption of infinitely stiff leads ($k_s=\infty$), the external stress is totally released in the nanocontact region that has been atomically relaxed (the short monatomic chain) and we have $L=\dsurf$.
Experimental estimates of $k_s$ in Au electrodes have been reported on the basis of the non-exponential dependence of the tunneling current on the distance: values in the range $0.3\,\ev/\ang^2$ to $3.7\,\ev/\ang^2$ have been found, depending on the junction realization.\cite{rubio-bollinger2004}
\begin{figure}[tb]
\includegraphics[width=0.8\columnwidth]{Cond-COatAu_surf-cfr2_rev}
\caption{(Color online) Top panel: ballistic conductance of the 4Au-atom nanocontact with CO at the bridge site as a function of the equivalent tips displacement $L$ for different values of the electrode stiffness $k_s$ (in $\ev/\ang^2$).
The arrows point the start and end points indicated by the arrows in \pref{fig:autranexp} and the solid line between those points approximates the experimental conductance trace\cite{kiguchi2007} with a linearly increasing conductance.
Bottom panel: total energy of the optimized nanocontact geometry as a function of the inter-surface distance \dsurf.}
\label{fig:autranexp2}
\end{figure}
In \pref{fig:autranexp2} (bottom panel), we report the total energy $\etot(\dsurf)$ of the relaxed LDA bridge geometry for some \dsurf\ values in a range of interest and we fit this energy with a $4^{\rm th}$ degree polynomial.
The analytical derivative of the energy fit is used to obtain $L(\dsurf;k_s$) for a few values of the parameter $k_s$ and then express the \dsurf-dependent conductance as a function of $L$ (see \pref{fig:autranexp2}, upper panel).
The case of infinitely stiff leads ($k_s=\infty$) corresponds to the conductance already shown in \pref{fig:autranexp} and is reported here again for comparison.
When taking a finite $k_s$, a fraction of the term depending on the energy derivative is added to \dsurf\ in $L$, thus the calculated conductance points are shifted to higher displacements and the theoretical conductance slope decreases.
For $k_s$ values close to the experimental upper limit ($4.0\,\ev/\ang^2$) the slope does not change significantly, but for $k_s$ values between $0.5\,\ev/\ang^2$ and $1.0\,\ev/\ang^2$ (a realistic range, according to the experiment\cite{rubio-bollinger2004}) the calculated conductance slope is in much better agreement with that of the experimental conductance trace\cite{kiguchi2007} (see thick solid line in \pref{fig:autranexp2}).
This demonstrates that a more accurate description of the leads is needed to obtain a correct slope of the conductance as a function of the contact stretch, but already good estimates can be obtained through this simple elastic-response model.
\section{Conclusions}\label{sec:concl}
We have studied through density functional calculations the adsorption of CO on Au monatomic chains in a model nanocontact geometry and its effects on the ballistic conductance of the nanocontact as a function of the contact stretching.
By comparing the adsorption energies of CO at the bridge and atop sites, we find that the bridge site is energetically favored at all levels of Au strain, as also found in a simpler model without bulk leads.\cite{sclauzero2012a}
The chemisorption energies of CO on the short chain in the nanocontact geometry are comparable to those found in the infinite straight chains\cite{sclauzero2012a} and a fair agreement in the variation of the energetics with strain is obtained when the displacement of the Au atom below the molecule in the infinite chain geometry is taken into account.
This was not clearly predictable, because the finite length of the chain and the non-uniform Au-Au spacing are not encompassed by the infinite chain model, while in the nanocontact geometries these aspects are included and play a role in the structural modifications of the short chain after CO adsorption.
For instance, when CO is at the bridge site, at low strains the chain bends towards the molecule, while at larger strains the Au-Au bond below CO elongates much more rapidly than the others.
With CO atop, the chain forms a zigzag geometry, more pronounced at lower strains, where the Au atom right below the CO moves towards the molecule.
The electron transmission across the Au nanocontact in presence of CO displays some important features which have already been found in the tipless geometries, most notably the transmission dip close to the Fermi level in the atop geometry which causes a strong suppression of the conductance.
This dip is not present in the bridge geometry at those scattering energies, therefore the conductance reduction with respect to the pristine nanocontact is much smaller.
Also the dependence of the ballistic conductance on strain does not change with the inclusion of the tips, modeled here using either abrupt interfaces between the chain and the Au(001) surfaces or smoother pyramidal junctions: the bridge conductance is found to increase slightly with strain, compatible with the experimental findings, while the atop conductance drops rapidly down to zero as the contact is stretched because of the \sga\ transmission dip moving closer to \ef.
With the inclusion of tips, the bridge geometry shows conductance values close to those of the experimentally observed structure which forms in the Au nanocontact after CO exposition.
The slope of the experimental conductance with respect to the contact stretching can also be reproduced with reasonable accuracy by our calculations if the tips displacement is computed taking into account the elastic response of the bulk leads through a simple model.
\begin{acknowledgments}
The authors are grateful to Erio Tosatti for useful and stimulating discussions.
This work has been supported by PRIN-COFIN 20087NX9Y7, as well as by INFM/CNR ``Iniziativa trasversale calcolo parallelo''.
All calculations have been performed on the SISSA-Linux cluster and at CINECA in Bologna.
\end{acknowledgments}
|
1,314,259,992,970 | arxiv | \section{Introduction}
A speech signal can be considered as a variable-length temporal sequence~\cite{graves2012supervised}, and many features have been used to characterize its pattern. Short-term spectral features are used extensively because of the quasi-stationary property of the speech signal. After short-term processing, the raw waveform is converted into a two-dimensional~(2-D) matrix of size $D \times T$, where $D$ represents the frequential feature dimension related to the number of filter coefficients, and $T$ denotes the temporal frame length related to the utterance duration.
For a text-independent speaker verification~(TISV) system, the main procedure is to extract the fixed-dimensional speaker representation from the variable-length spectral feature sequence. One of the widely used spectral features is the Mel-frequency cepstral coefficient ~(MFCC)~\cite{davis1980comparison, Kinnunen2010An}. Typically, MFCC feature vectors from all the frames are assumed to be independent and identically distributed. They can be projected on the Gaussian components or phonetic units to accumulate statistics over the time axis and form a high-dimensional supervector. Then, a factor analysis-based dimension reduction is performed to generate a fixed-dimensional low rank i-vector representation~\cite{dehak2010front}. Recently, with the progress of deep learning, many approaches directly train a deep neural network~(DNN) to distinguish different speakers~\cite{variani2014deep, 2017arXiv170502304L, Snyder2017Deep, xvector, Nagrani2017, Cai_2018_Odyssey}. Systems comprising of x-vector~\cite{xvector} speaker embedding followed by a probabilistic linear discriminant analysis~(PLDA)~\cite{prince2007probabilistic} have shown state-of-the-art performances on multiple TISV tasks~\cite{xvector}. In the x-vector system, a time-delay neural network~(TDNN)~\cite{21701} followed by statistic pooling over the time axis is used for modeling the long-term temporal dependencies from the MFCC features.
\begin{figure*}
\setlength{\abovecaptionskip}{-0.3cm}
\centering
\begin{multicols}{3}
\setlength{\abovecaptionskip}{0.cm}
\subfigure[\label{fig:plot1}]{\includegraphics[width=0.27\textwidth]{only_wb}\par }
\subfigure[\label{fig:plot2}]{\includegraphics[width=0.32\textwidth]{wb_nb_same_domain} \par }
\subfigure[\label{fig:plot3}]{\includegraphics[width=0.37\textwidth]{wb_nb_diff_domain}\par}
\end{multicols}
\caption{Three mixed-bandwidth data training strategies for different scenarios: (a) Only WB training data are given; (b) WB and NB training data are from the same domain; (c) WB and NB training data are from different domains }
\label{fig:losscurve}
\vspace{-0.5cm}
\end{figure*}
For the i-vector, x-vector, and many other speech modeling methods, the feature matrix $D \times T$ is viewed as a multi-channel 1-D time series. Although the duration $T$ may vary among the utterances, the feature dimension $D$ must be a fixed value. In this paper, we consider the feature matrix as a single-channel 2-D image~\cite{lecun1995convolutional}. From this new perspective, the spectral feature is viewed as a ``picture" of the sound, and a 2-D CNN is implemented in the same way as traditional image recognition paradigms. This kind of process brings a type of flexibility, i.e., the size of the input ``image," including the width (frame length) and the height (feature dimension), can be arbitrary numbers. In other words, a 2-D CNN trained with a 64-dimensional spectrogram could potentially also process a spectrogram with 48 dimensions.
We aim to utilize the flexibility of the 2-D CNN to tackle the mixed-bandwidth~(MB) joint modeling problem. Currently, there are many devices and equipment that capture speech data in different sampling rates, thus solving the sampling rate mismatch problem has become a research topic in the speech community. The traditional way to accomplish this goal is to train a specific model for every target bandwidth since the sampling rates are different (typically 8k Hz vs. 16k Hz). An alternative solution is to uniformly downsample the wideband~(WB) speech data or extend the bandwidth of a narrowband~(NB) data, so that they can be combined ~\cite{yamamoto2019speaker,yingxuewang2015}.
In this paper, we present a unified solution to solve the MB joint modeling problem. The key idea is to view the NB spectrogram as a sub-image of the WB spectrogram. The major contributions of this work are summarized as follows.
\begin{itemize}[noitemsep]
\item We leverage the 2-D CNN to tackle the MB joint modeling problem from a novel image classification perspective. We show that speech data with different bandwidths can be naturally combined without any additional downsamping, upsampling, bandwidth extension, padding operation, or auxiliary input.
\item We further investigate several network training strategies targeting various real-world MB application scenarios, including when (a) only the WB training data are given; (b) WB and NB training data are given from the same domain; and (c) WB and NB training data are given but from different domains.
\end{itemize}
\section{Related works}
In~\cite{4032793}, Seltzer~\textit{et al.} present an expectation-maximization algorithm for training with MB data where the missing spectral components of the NB signal are considered additional hidden variables. Li\textit{ et al.} formulate the MB joint training problem as a missing data paradigm and propose training an MB speech recognition system without bandwidth extension in~\cite{li2012improving}. The authors adopt a fully-connected DNN architecture, and thus require a zero-padding or mean-padding operation to ensure all features have the same dimensions. In~\cite{mantena2019bandwidth}, Gautam~\textit{et al.} build a single model for MB speech recognition. The inputs of their network are fixed 40-dimensional features, and their network requires a bandwidth embedding as the auxiliary input.
Recently, since there are more and more open speech databases with speaker labels collected in different sampling rates, the MB joint modeling problem has gained much attention in the speaker recognition community. In~\cite{nidadavolu2018investigation, 8682992}, Nidadavolu~\textit{et al.} investigated several bandwidth extension approaches for speaker recognition with several different network architectures. Meanwhile, the authors of~\cite{yamamoto2019speaker} consider making use of the Mel filter bank coefficients to share acoustic features between WB and NB speech, and implement a new pipeline that uses a DNN-based bandwidth extension as pre-processing of the DNN for speaker embedding extraction.
\section{Methods}
\subsection{Mel-spectrogram}
We adopt the log Mel-filterbank energies as the standard acoustic features. We refer to this feature as the Mel-spectrogram because we process it in an image processing manner.
Here we present an example revealing how to design the filter banks so that the NB spectrogram can be correctly aligned with the low-frequency region of the WB spectrogram. For a given NB speech sampled at 8k Hz, the Mel-spectrogram represents a bandwidth only from 0--4k Hz, and the remaining 4k--8k Hz information is missing compared with the 16k Hz sampled WB data. According to the general formula for converting from Hertz to Mel scale frequency~\cite{deller2000discrete}, the associated relationship between the number of NB and WB filters is computed as follows:
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\begin{equation}\footnotesize
\label{equ:1}
M_N = \floor{M_W \times \frac{\log (1 + f_N / 700)}{\log ( 1 + f_W / 700)}},
\end{equation}
where $f_N$ refers to the NB spectrogram upper limit, $f_W$ represents the WB spectrogram upper limit, and $M_N$ and $M_W$ denote the number of designed filters for the NB and WB data, respectively. In our implementation, we have $f_W=8000$, $f_N=4000$, and $M_W=64$. Therefore, according to Equation~(\ref{equ:1}), we obtain $M_N=48.3$. The feature dimension must be an integer, so we force the $M_N$ to 48 and set $f_N$ to 3978.69 Hz. In other words, the 3979.69 Hz to 4000 Hz information of the NB data is ignored.
\subsection{2-D CNN architecture}
Regarding the Mel-spectrogram as a visual image, we utilize the 2-D convolutions to learn the local time--frequency coupled patterns. For a given feature matrix of size $D \times T$, the 2-D convolution and pooling strategy brings us flexibility: the size of the input image, including the width and height, can be arbitrary. In speech processing, the 2-D CNN not only handles features with variable-length $T$, but also potentially accepts features with different dimensions $D$.
The representation learned by the 2-D CNN is a 3-D tensor of size $C\times H\times W$, where $C$ refers the number of channels, and $H$ and $W$ denote the height and width of the learned feature maps, respectively. We add a global statistics pooling~(GSP) layer after the 3-D feature maps to accumulate the global statistics over the time--frequency axes. We summarize a specific 2-D feature map $ {\bm{F}} \in \mathbb{R}^{H\times W}$ with a global mean and standard deviation statistics $\mu, \sigma$. Since there are $C$ channels of feature maps, we finally get a $2C$-dimensional vector $ {\bm{V}}={\left[\mu_1, \mu_2, \cdots, \mu_C, \sigma_1, \sigma_2, \cdots, \sigma_C\right]}$ to represent an arbitrary duration speech with different bandwidths.
\subsection{Training strategy}
The Mel-spectrogram trained with a 2-D CNN forms a new framework to potentially solve the mixed-bandwidth data problem. The remaining question is how to train a network that fits both the WB and NB spectrograms. Considering different scenarios, we investigate three kinds of training strategies, as described below.
\subsubsection{\textbf{Only WB data are given}}
\label{sec:only}
In this scenario, the evaluation dataset comprises both the WB and NB speech, but only WB speech is provided for training. Figure~\ref{fig:plot1} illustrates the proposed MB system training procedure. Here, WB spectrograms (64 dimensions) are reused to generate the NB spectrogram by selecting its sub-spectrograms (48 dimensions here). There are two rounds of parameter updates for a mini-batch of training data. The first update is from the full-size WB spectrogram, and the second update is from the sub-image of the WB spectrogram. It is desired that the network fits well on both WB and NB data, and the network parameters are shared for these two groups of images with different sizes. After the network is trained, we can feed it with either the full-size image (WB spectrogram) or the sub-image (NB spectrogram). The whole pipeline does not require any downsampling, upsampling, extension, or padding operation.
\subsubsection{\textbf{MB data from the same domain}}
\label{sec:samedomain}
Different from the situation in section~\ref{sec:only} where the NB training is simulated by selecting the sub-image from the WB spectrogram, here we have mixed training data with WB speech as well as NB speech. Therefore, we can first extract 64-dimensional WB spectrograms from the WB data and 48-dimensional NB spectrograms from both the NB and WB data; then, we can train the network as described in section~\ref{sec:only}. As illustrated in Fig.~\ref{fig:plot2}, the network parameters are shared across the WB and NB spectrograms; however, the output units representing the speaker identities are separate assuming there is no speaker overlapping between the WB and NB data. Although the network accepts features with different dimensions, the feature size within a mini-batch should be consistent. Therefore, we maintain two separated data loaders for the data with different bandwidths, and the mini-batch training data are fetched from these two data loaders alternatively to train the network.
\subsubsection{\textbf{MB data from different domains}}
\label{sec:diffdomain}
According to section~\ref{sec:samedomain}, the network parameters are shared across the training data of different bandwidths. In real-world applications, the MB and WB training data may be collected from different domains. Here we give a simple solution as illustrated in Fig.~\ref{fig:plot3}. Our network consists of shared layers and multiple branches of domain-specific layers. Specifically, the bottom CNN is shared for both the NB and WB spectrograms to learn general feature representations. After the GSP layer, the fully-connected layer is learned independently for each domain. Therefore, the speaker embeddings and output units for different bandwidths are in separate branches. After the network is trained, the speaker embedding is extracted from the associated branch corresponding to the sampling rate.
\section{Experimental Results}
\subsection{Datasets}
\subsubsection{VoxCeleb1}
We first conduct simulated experiments on the VoxCeleb1 dataset~\cite{Chung:2018bp}. The training set includes \num{148642} utterances from \num{1211} celebrities. The test set contains \num{4715} utterances from the other \num{40} celebrities. The equal error rate~(EER) is used to measure the system performance.
At the beginning, both the training and test datasets were sampled at 16k Hz. We obtained the 8k Hz evaluation data by downsampling the 16k Hz data using the Sox toolkit.
\newcommand{\blocka}[2]{\multirow{2}{*}{\(\left[\begin{array}{c}\text{3$\times$3, #1}\\[-.1em] \text{3$\times$3, #1} \end{array}\right]\)$\times$#2}
}
\newcommand{\blockb}[3]{\multirow{2}{*}{\(\left[\begin{array}{c}\text{3$\times$3, #2}\\[-.1em] \text{3$\times$3, #1}\end{array}\right]\)$\times$#3}
}
\renewcommand\arraystretch{1.1}
\setlength{\tabcolsep}{3pt}
\begin{table}[t]
\caption{ The proposed network architecture. N/A: Not applicable}
\centering
\begin{adjustbox}{max width=0.99 \columnwidth}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{Layer} &\textbf{Output size} & \textbf{Structure}&\textbf{ \#Params} \\
\hline
Conv1 & $\!16\!\times\!D\!\times\!T$& $3\!\times\!3$, stride 1& \num{176} \\
\hline
\multirow{2}{*}{Res1} & \multirow{2}{*}{$16\!\times\!D\!\times\!T$} & \blockb{16}{16}{3}\multirow{2}{*}{, stride 1} & \multirow{2}{*}{ 14K } \\
&&&\\
\hline
\multirow{2}{*}{Res2} & \multirow{2}{*}{$32\!\times\!\frac{D}{2}\!\times\!\frac{T}{2}$} & \blockb{32}{32}{4}\multirow{2}{*}{, stride 2} &\multirow{2}{*}{70K} \\
& & &\\\hline
\multirow{2}{*}{Res3} & \multirow{2}{*}{$64\!\times\!\frac{D}{4}\!\times\!\frac{T}{4}$} & \blockb{64}{64}{6}\multirow{2}{*}{, stride 2} & \multirow{2}{*}{427K} \\
& & &\\
\hline
\multirow{2}{*}{Res4} & \multirow{2}{*}{$128\!\times\!\frac{D}{8}\times\!\frac{T}{8}$} & \blockb{128}{128}{3}\multirow{2}{*}{, stride 2} & \multirow{2}{*}{821K} \\
& & &\\\hline
GSP&256& Statistics pooling &0\\
\hline
FC1 (embedding) &$ 128$ &Fully-connected & 32K\\
\hline
FC2 (Output) & speakers &Fully-connected & N/A \\
\hline
\end{tabular}
\end{adjustbox}
\label{tab:resnetconfigt}
\vspace{-0.4cm}
\end{table}
\begin{table} [t]
\caption{ Performance on the VoxCeleb1 test data when only the WB VoxCeleb1 training data are given. $\rightarrow$: sub-spectrogram selection operation. SR: Sampling rate.}
\centerline {
\resizebox{0.98\columnwidth}{!}{
\begin{tabular}{c c c c c c c c c}
\toprule
\textbf{ID}&\textbf{Method}& \textbf{Training SR and \#Filters} & \textbf{Testing SR}&\textbf{Testing \#Filters}&\textbf{EER (\%)}\\
\midrule
1&\multirow{4}{*}{ WB baseline} &\multirow{4}{*}{16k and 64}& 16k & 64 &\textbf{4.35}&\\
2& &&8k&64&20.13&\\
3& &&8k&48&8.82&\\
4& &&16k&64$\rightarrow$48&8.87&\\
\cmidrule(lr){1-6}
5& \multirow{4}{*}{ NB baseline} &\multirow{4}{*}{8k and 48}& 8k & 48 &\textbf{4.92}&\\
6&&&16k&48&18.86\\
7&&&16k&64&8.01&\\
8&&&16k&64$\rightarrow$48&4.95&\\
\cmidrule(lr){1-6}
\multirow{2}{*}{ \textbf{9}}& \multirow{2}{*}{ \textbf{Proposed MB}}& \multirow{2}{*}{16k and 64\&48}& 16k&64&\textbf{4.07}\\
&&&8k&48&\textbf{4.37}\\
\bottomrule
\end{tabular}}}
\label{table:onlywb}
\vspace{-0.3cm}
\end{table}
\subsubsection{SITW and NIST SRE 2016}
The SITW dataset consists of unconstrained audio--visual data of English speakers~\cite{mclaren2016speakers}. Our focus is on its core--core protocol for both the development and evaluation sets. For the NIST SRE 2016, the test data are composed of telephone conversations collected outside North America, spoken in Tagalog and Cantonese~\cite{sadjadi20172016}. The development set contains some unlabeled data, which is useful for the unsupervised domain adaptation.
The pooled VoxCeleb1 and VoxCeleb2 datasets were used as our training set for the evaluation on SITW. Finally, a training set of \num{1236567} utterances from \num{7185} celebrities was obtained. We refer to these data as VoxCeleb1\&2 16k data. We also have NB training data from NIST SRE 2004--2010, Mixer 6, Switchboard 2 Phase 1, 2, and 3, as well as Switchboard Cellular. There is a total of \num{99661} utterances from \num{7222} speakers. We refer to this data as SRE 8k data.
\subsection{Implementation details}
First, 64- and 48-dimensional Mel spectrograms are extracted for the WB and NB data, respectively.
Our network is based on the ResNet~\cite{He2016Deep} structure, and the architecture is described in Table~\ref{tab:resnetconfigt}. Dropout with a rate of 0.5 is added before the softmax layer, and the network is trained with a typical cross-entropy loss. We adopt the common stochastic gradient descent algorithm with momentum \num{0.9} and weight decay 1e-4.
\begin{table*} [t]
\caption{ Performance on the SITW and NIST SRE 2016 datasets. }
\centerline {
\begin{tabular}{c c c c c c c c c}
\toprule
\multirow{3}{*}{\textbf{ID}}& \multirow{3}{*}{\textbf{Training data}}& \multicolumn{5}{c}{\textbf{Testing \textbf{EER (\%)} }}\\
\cmidrule(lr){3-7}
& &\multicolumn{2}{c}{\textbf{SITW }} & \multicolumn{3}{c}{\textbf{NIST SRE 2016 }}\\
\cmidrule(lr){3-4} \cmidrule(lr){5-7}
& &\textbf{Dev}&\textbf{Eval}&\textbf{Pool}&\textbf{Cantonese}&\textbf{Taglog}\\
\midrule
1&VoxCeleb1\&2 16k&\textbf{2.17}& \textbf{2.49}& 16.88 & 10.86 &23.02 \\
2& SRE 8k&14.29&17.52&\textbf{6.19}&\textbf{3.66} &\textbf{8.61} \\
\midrule
3& VoxCeleb1\&2 8k + SRE 8k&3.20&3.52&5.69 & 3.39 &8.10 \\
4& \textbf{VoxCeleb1\&2 16k + SRE 8k}&\textbf{2.91}&\textbf{3.18}&\textbf{5.44}& \textbf{3.10} & \textbf{7.61}\\
\bottomrule
\end{tabular}}
\label{table:mixdiffdomain}
\vspace{-0.3cm}
\end{table*}
\begin{table} [t]
\caption{ Performance on the VoxCeleb1 test data when the VoxCeleb1 training data are split into two subsets.}
\centerline {
\begin{tabular}{c c c c c c c c c c c}
\toprule
\multirow{3}{*}{\textbf{ID}} & \multirow{3}{*}{\textbf{VoxCeleb1 Training data}}& \multicolumn{4}{c}{\textbf{VoxCeleb1 Testing}}\\
\cmidrule(lr){3-6}
&&\multicolumn{2}{c}{\textbf{\# filters}}&\multicolumn{2}{c}{\textbf{EER (\%)}}\\
\cmidrule(lr){3-4} \cmidrule(lr){5-6}
&&\textbf{16k}&\textbf{8k}&\textbf{16k}&\textbf{8k}\\
\midrule
1&Subset1 16k&64&48&\textbf{5.66}&11.08\\
2&Subset2 8k&64$\rightarrow$48&48&6.11&\textbf{6.07}\\
\midrule
3& Subset1 8k + Subset2 8k &64$\rightarrow$48&48&4.95&4.92\\
4&\textbf{Subset1 16k + Subset2 8k}&64&48&\textbf{4.53}&\textbf{4.51}\\
\bottomrule
\end{tabular}}
\label{table:mixsamedomain}
\vspace{-0.3cm}
\end{table}
In total, there are three downsampling operations within the convolutional layers. Therefore, the original Mel-spectrograms are downsampled to compact feature maps with size $\frac{D}{8} \times \frac{T}{8}$ before the GSP layer. In the training phase, we implement a variable-length data loader to generate mini-batch training samples on the fly~\cite{caiwch_taslp}. For each step, a dynamic mini-batch of data with size $B \times D \times T$ is generated, where $B$ is the mini-batch size, $D$ is the feature dimension, and $T$ is a batch-wise variable frame length ranging from \num{300} to \num{800}.
After the network is trained, the 128-dimensional speaker embeddings are extracted for evaluation. Simple cosine similarity is adopted to compute the pairwise score for the VoxCeleb1 and SITW evaluation trials. For the NIST SRE 2016 experiments, we adopt the adaptive PLDA backend as implemented in the Kaldi SRE16 recipe~\cite{SRE16v2}.
\subsection{Results and discussion}
\subsubsection{\textbf{Only WB data are given}}
We first train a WB baseline system using the 16k training data. IDs from 1 to 4 in Table~\ref{table:onlywb} show the results for different setups of the evaluation data. It reveals that if we feed the WB model with NB data, then the 48-dimensional NB spectrogram obtains much better results than the 64-dimensional one~(EER 8.87$\%$ vs 20.13$\%$). After selecting the 48-dimensional sub-image from the 64-dimensional WB spectrogram, the system obtains almost the same result as the 8k NB test data. These results suggest that it is more crucial to keep the resolution of the features consistent rather than the dimension the features. We also train an NB system using the downsampled 8k data. We reach similar conclusions as for the WB system, but the best NB results is an EER of 4.92$\%$, which is slightly worse than that of the WB system (4.35\%). This indicates that the WB features might contain more useful information than the NB ones.
By applying corresponding models for the test data with each sampling rate separately, the systems achieve 4.35\% EER for the WB test data and 4.92\% for the NB test data. Following the approach described in section~\ref{sec:only}, we train the proposed MB system using only the 16k WB data. Our single MB model achieves 4.07\% and 4.37\% EER for the WB and NB evaluation data, respectively.
\subsubsection{\textbf{MB data from the same domain}}
Here, the VoxCeleb1 training data are divided into two subsets: the first part consists of \num{74438} utterances from \num{622} speakers, and the second part includes \num{74204} utterances from \num{589} speakers. All of the speech data in the second part are downsampled to 8k Hz to simulate the NB training data. The top half of Table~\ref{table:mixsamedomain} shows the results of the single WB/NB system trained by only subset1 16k or subset2 8k data with the best choices of testing spectrograms. Compared with the results in Table~\ref{table:onlywb}, the performance here is degraded since the scale of the training data is reduced.
To utilize the training data with different bandwidths, we first train a baseline system by downsampling the data from the first part to 8k Hz and then pooling together the data from the two subsets. We can see that the pooled 8k system (ID 3) achieves much better results than the system trained by any single subset (ID 1 and 2). Following the approach described in section~\ref{sec:samedomain}, we train the proposed MB system (ID 4) by jointly modeling the original subset1 16k and subset2 8k data. Compared with the baseline system (ID 3), we obtain consistent EER reduction for both the WB and NB evaluation data.
\subsubsection{\textbf{MB data from different domains}}
The top half of Table~\ref{table:mixdiffdomain} shows the results of the WB system trained with the VoxCeleb1\&2 16k data and the NB system trained with the SRE 8k data. These two systems don't work well on each other's evaluation sets, possibly due to the additional domain mismatch. We further develop a pooled NB system by downsampling the VoxCeleb1\&2 16k data to 8k Hz and pooling the two training datasets in system ID 3. Compared with the SRE 8k system (ID 2), the pooled NB system performs much better on NIST SRE 2016 since the number of training utterances and speakers are increased. However, on the SITW dataset, the performance is degraded significantly compared with the VoxCeleb1\&2 16k WB baseline system (ID 1), which might be due to the domain mismatch.
Following the approach described in section~\ref{sec:diffdomain}, we train an MB system (ID 4) by jointly modeling the VoxCeleb1\&2 16k and SRE 8k data in a single network. The proposed MB system consistently outperforms this baseline (ID 3) on both SITW and NIST SRE 2016 evaluation sets.
\section{Conclusion and future work}
In this paper, we propose a novel multi-bandwidth joint modeling approach for speaker verification. We show that speech data with different sampling rates can be flexibly integrated in a single speaker embedding model based on a 2-D CNN without any additional downsampling, upsampling, extension, or padding operations. Experimental results show that the proposed MB systems achieve significant improvement on both the NB and WB evaluation data. Our future works include comparing the proposed systems with other state-of-the-art MB solutions, such as using a DNN-based bandwidth extension module as the frontend for all NB data.
\section{Acknowledgement}
This research is funded in part by the National Natural Science Foundation of China (61773413), Key Research and Development Program of Jiangsu Province (BE2019054), Six talent peaks project in Jiangsu Province (JY-074), Science and Technology Program of Guangzhou City (201903010040,202007030011).
\newpage
\bibliographystyle{IEEEbib}
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1,314,259,992,971 | arxiv | \section{Introduction}
Throughout the last decade, real-life problems have been solved by deep learning-based algorithms. We have seen in recent years that deep learning has been extensively used in computer vision. Object Tracking is one of the very important tasks in computer vision. It comes just right after object detection. To accomplish the task of object tracking, firstly object needs to be localized in a frame. Then each object is assigned an individual unique id. Then each same object of consecutive frames will make trajectories. Here, an object can be anything like pedestrians, vehicles, a player in a sport, birds in the sky etc. If we want to track more than one object in a frame, then it is called Multiple Object Tracking or MOT. In MOT, we can track all objects of a single class or all objects of said classes. If we only track a single object, it is called Single Object Tracking or SOT. MOT is more challenging than SOT. Thus researchers proposed numerous deep learning-based architectures for solving MOT-related problems.
To make the last three years of research organized, we would like to do a literature review on MOT. This work includes these papers. There are also some review papers on MOT in previous years \cite{pal2021deep, luo2021multiple, park2021multiple, rakai2021data}. But all of them have limitations. Some of them only include deep learning-based approaches, only focus on data association, only analyze the problem statement, do not categorize the paper well, and applications in real life are also missing.
We have tried to overcome all of these issues in this work. We have tried to go through almost every paper from 2020 to 2022 on MOT. After filtering out them, we have reviewed more than a hundred papers in this work. While going through the papers, the first thing that caught our attention is that there are many challenges in MOT. Then we made an attempt to find out different approaches to face those challenges. To establish the approaches, the papers have used various MOT datasets, and to evaluate their work, they have taken help from various MOT metrics. So we have included a quick review of the datasets. Additionally, we have included a summary of new metrics along with previously existing ones. We have also tried to list down some of the MOT applications among the vast use cases of Multiple Object Tracking. Going through these papers, some scope of work has drawn our attention, which was mentioned later on.
Therefore, to sum up, we have organized our work in the following manner:
\begin{enumerate}
\item Figuring out MOT's main challenges
\item Listing down frequently used various MOT approaches
\item Writing a summary of the MOT benchmark datasets
\item Writing a summary of MOT metrics
\item Exploring various applications
\item Some suggestions regarding future works
\end{enumerate}
\section{MOT Main Challenges}
Multiple Object Tracking has some challenges to tackle. Although occlusion is the main challenge in MOT, there are several other challenges that a tracker has to deal with in terms of an MOT problem.
\subsection{Occlusion}
Occlusion occurs when something we want to see is entirely or partially hidden or occluded by another object in the same frame. Most MOT approaches are implemented based only on cameras without sensor data. That’s why it is a bit challenging for a tracker to track the location of an object when they obscure each other. Furthermore, occlusion becomes more severe in a crowded scene to model the interaction between people \cite{xu2021transcenter}. Over time the use of bounding boxes to locate an object is very popular in the MOT community. But in crowded scenes,\cite{dendorfer2020mot20} occlusions are very difficult to handle since ground-truth bounding boxes often overlap each other. This problem can be solved partially by jointly addressing the object tracking and segmentation tasks \cite{meinhardt2021trackformer}. In literature, we can see appearance information and graph information are used to find global attributes to solve the occlusion \cite{huo2021multi, milan2017online, tian2018detection, ullah2018directed}. However, frequent occlusion has a significant impact on lower accuracy in MOT problems. Thus researchers try to attack this problem without bells and whistles. In Figure \ref{fig1a}, occlusion is illustrated. In Figure \ref{fig1b}, the red-dressed woman is almost covered by the lamp post. This is an example of occlusion.
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.45 \textwidth]{Figures/fig1.png}\label{fig1a}}
\subfloat[]{\includegraphics[width=0.48 \textwidth]{Figures/fig2.png}\label{fig1b}}
\caption{(a) Illustration of the occlusion of two objects (green and blue). In frame 1, two objects are separate from each other. In frame 2, they are partially occluded. In frame 3, they are totally occluded.}
\end{figure}
\subsection{Challenges for Lightweight Architecture}
Though recent solution for most of the problems depends on heavy-weight architectures, they are very resource hungry. Thus in MOT, heavy-weight architecture is very counterintuitive to achieving real-time tracking. Therefore researchers always cherish lightweight architecture. For lightweight architecture in MOT, there are some additional challenges to consider \cite{yan2021lighttrack}. Bin et al. mentioned three challenges for lightweight architecture such as,
\begin{itemize}
\item Object tracking architecture requires both pre-trained weights for good initialization and fine-tuned tracking data. Because NAS algorithms need direction from the target task and, at the same time, solid initialization.
\item NAS algorithms need to focus on both backbone network and feature extraction, so that the final architecture can suit perfectly for the target tracking task.
\item Final architecture needs to compile compact and low-latency building blocks.
\end{itemize}
\subsection{Some Common Challenges}
MOT architecture often suffers from inaccurate object detection. If objects are not detected correctly, then the whole effort of tracking an object will go in vain. Sometimes the speed of object detection becomes one major factor for MOT architectures. For background distortion, object detection sometimes becomes quite difficult. Lighting also plays a vital role in object detection and recognition. Thus all of these factors become more important in object tracking. Due to the motion of the camera or object, motion blurring makes MOT more challenging. Many times MOT architecture finds it hard to decide an object as true incomer or not. One of the challenges is the proper association between the detection and tracklet. Incorrect and imprecise object detection is also a consequence of low accuracy in many cases. There are also some challenges, such as similar appearance confuses models frequently, initialization and termination of tracks is a bit crucial task in MOT, interaction among multiple objects, ID Switching (same object identified as different in consecutive frames through the object did not go out of frame). Due to non-rigid deformations and inter-class similarity in shape and other appearance properties, people and vehicles create some additional challenges in many cases \cite{chong2021overview}. For example, vehicles have different shapes and colors than people’s clothes. Last but not least, smaller-sized objects make a variety of visual elements in scale. Liting et al. try to solve the problem with higher resolution images with higher computational complexity. They also used hierarchical feature map with traditional multiscale prediction techniques \cite{lin2021swintrack}.
\section{MOT Approaches}
\begin{figure}[htbp]
\centering
\includegraphics[width = 0.48 \textwidth]{Figures/approaches.pdf}
\caption{Recent MOT approaches categorization}
\end{figure}
The task of multiple object tracking is done normally in two steps: object detection and target association. Some focus on object detection; some focus on data association. There is a diversity of approaches for these two steps. The approaches can not be differentiable whether it is a detection phase or an association phase. Sometimes we can see the overlapping of the approaches. Most of the papers use various combinations of MOT components. So we can not say that the approaches are independent of each other. Yet we have tried to figure out the frequently used approaches so that they can help in making decisions of which one to follow.
\input{Approaches/a.Transformer}
\input{Approaches/b.Graph_Model}
\input{Approaches/c.Detection}
\input{Approaches/d.Attention_Model}
\input{Approaches/e.Motion_Module}
\input{Approaches/f.Siamese_Network}
\input{Approaches/g.Tracklet_Association}
\section{MOT Benchmarks}
A typical MOT dataset contains video sequences. In those sequences, every object is identified by a unique id until it goes out of the frame. Once a new object comes into the frame, it gets a new unique id. MOT has a good number of benchmarks. Among them, MOT challenge benchmarks have several versions. Since 2015, in almost every year, they publish a new benchmark with more variations. There are also some popular benchmarks such as PETS, KITTI, STEPS, and DanceTrack.
\input{Benchmarks/Public_Benchmarks}
\section{MOT Metrics}
\subsection{MOTP}
Multiple Object Tracking Precision. It is a score given based on how precise the tracker is in finding the position of the object \cite{bernardin2008evaluating} regardless of the tracker’s ability to recognize object configuration and maintain consistent trajectories. As $MOTP$ can only provide localization accuracy, it is often used in conjunction with $MOTA$ (Multiple Object Tracking Accuracy), as $MOTA$ alone can not account for localization errors. Localization is one of the outputs of an MOT task. It lets us know where the object is in a frame. Alone it can not provide a thorough idea of the tacker’s performance in object tracking.
\begin{equation*}
MOTP = \frac{ \sum_{i,t} d^i_t}{\sum_t c_t}
\end{equation*}
$d^i_t$: The distance between the actual object and its respective hypothesis at time $t$, within a single frame for each object $o_i$ from the set a tracker assigns a hypothesis $h_i$.\\
$c_t$: Number of matches between object and hypothesis made at time $t$.
\subsection{MOTA}
Multiple Object Tracking Accuracy. This metric measures how well the tracker detects objects and predicts trajectories without taking precision into account. The metric takes into account three types of error \cite{bernardin2008evaluating}.
\begin{equation*}
MOTA = 1 - \frac{ \sum_{t} (m_t + fp_t + mme_t)}{\sum_{{ }_t} {g_t}}
\end{equation*}
$m_t$: The number of misses at time $t$\\
$fp_t$: The number of false positives\\
$mme_t$: The number of identity switches\\
$g_t$: The number of objects present at time $t$
$MOTA$ has several drawbacks. $MOTA$ overemphasizes the effect of accurate detection. It focuses on matches between predictions to ground truths at the detection level and does not consider association. When we consider $MOTA$ without identity-switching, the metric is more heavily affected by poor precision than it is by re-call. The aforementioned limitations could lead researchers to tune their trackers towards better precision and accuracy at detection level whilst ignoring other important aspects of tracking. $MOTA$ can only take into account the short-term associations. It can only evaluate how well an algorithm can perform first-order association and not how well it associates throughout the whole trajectory. But, it doesn’t take into account association precision/ID transfer at all. In fact, if a tracker is able to correct any association mistake, it punishes it instead of rewarding it. While the highest score in $MOTA$ is 1 the is no fixed minimum value for the score, which can lead to a negative $MOTA$ score.
\subsection{IDF1}
The Identification Metric. It tries to map predicted trajectories with actual trajectories, in contrast to metrics like $MOTA$ which perform bijective mapping at the detection level. It was designed for measuring ‘identification’ which, unlike detection and association, has to do with what trajectories are there \cite{luiten2021hota}.
\begin{equation*}
ID-Recall = \frac{\lvert IDTP \rvert }{\lvert IDTP \rvert + \lvert IDFN \rvert }
\end{equation*}
\begin{equation*}
ID-Precision = \frac{\lvert IDTP \rvert }{\lvert IDTP \rvert + \lvert IDFP \rvert }
\end{equation*}
\begin{equation*}
IDF1 = \frac{\lvert IDTP \rvert }{\lvert IDTP \rvert +0.5\lvert IDFP \rvert + 0.5\lvert IDFN \rvert }
\end{equation*}
$IDTP$: Identity True Positive. The predicted object trajectory and ground truth object trajectory match.\\
$IDFN$: Identify False Negative. Any ground truth detection that went undetected and has an unmatched trajectory.\\
$IDFP$: Identity False Positive. Any predicted detection that is false.
Due to $MOTA$’s heavy reliance on detection accuracy, some prefer $IDF1$ as this metric puts more focus on association. However, $IDF1$ has some flaws as well. In $IDF1$, the best unique bijective mapping does not lead to the best alignment between predicted and actual trajectories. The end result would leave room for better matches. $IDF1$ score can decrease even if there are correct detections. The score could also decrease if there are a lot of un-matched trajectories. This incentives researchers to increase the total number of unique and not focus on making decent detections and associations.
\subsection{Track-mAP}
This metric matches the ground truth trajectory and predicted trajectory. Such a match is made between trajectories when the trajectory similarity score, $S_{tr}$, between the pair is greater than or equal to the threshold $\alpha_{tr}$. Also, the predicted trajectory must have the highest confidence score \cite{luiten2021hota}.
\begin{equation*}
Pr_n= \frac{{\lvert TPTr \rvert}_n}{n}
\end{equation*}
\begin{equation*}
Re_n= \frac{{\lvert TPTr \rvert}_n}{\lvert gtTraj \rvert}
\end{equation*}
$n$: The total number of predicted trajectories. Predicted trajectories are arranged according to their confidence score in descending order.\\
$Pr_n$: Calculates the precision of the tracker.\\
$TPTr$: True Positive Trajectories. Any predicted trajectory that has found a match.\\
${\lvert TPTr \rvert}_n$: Number of true positive trajectories among n predicted trajectories.\\
$Re_n$: Measures Re-call\\
$\lvert gtTraj \rvert$: Ground Truth Object Trajectory using the equation for precision and recall further calculation is done to obtain the final $Track-mAP$ score.
\begin{equation*}
InterpPr_n = \smash{\displaystyle\max_{m \geq n}} (Pr_m)
\end{equation*}
We first interpolate the precision values and obtain $InterpPr$ for each value of $n$. Then we plot a graph of $InterpPr$ against $Re_n$ for each value of $n$. We now have the precision-recall curve. The integral from this curve will give us the $Track-mAP$ score.
There are some demerits to track $mAP$ as well. It is difficult to visualize the tracking result for $Track-mAP$. It has several outputs for a single trajectory. The effect of the trajectories with low confidence scores on the final score is obscured.
There is a way to ‘hack’ the metric. Researchers can get a higher score by creating several predictions that have a low confidence score. This would increase the chances of getting a decent match and thus increases the score. However, it is not an indicator of good tracking. $Track-mAP$ can not indicate if trackers have better detection and association.
\subsection{HOTA}
Higher Order Tracking Accuracy. The source paper \cite{luiten2021hota} describes $HOTA$ as, \textit{“HOTA measures how well the trajectories of matching detections align, and averages this overall matching detection, while also penalizing detections that don’t match.”} $HOTA$ is supposed to be a single score that can cover all the elements of tracking evaluation. It is also supposed to be decomposed into sub-metrics. $HOTA$ compensates for the shortcomings of the other commonly used metrics. While metrics like $MOTA$ ignore association and heavily depend on detection($MOTA$) or vice versa ($IDF1$), novel concepts such as $TPA$s, $FPA$s and $FNA$s are developed so that association can be measured just like how $TP$s, $FN$s, and $FP$s are used to measure detection.
\begin{equation*}
HOTA_\alpha = \sqrt{\frac{\sum_{c\in\{TP\}} \textit{A}(c)}{\lvert TP \rvert + \lvert FN \rvert + \lvert FP \rvert}}
\end{equation*}
\begin{equation*}
A(c) = \frac{\lvert TPA(c) \rvert}{\lvert TPA(c) \rvert + \lvert FNA(c) \rvert + \lvert FPA(c) \rvert}
\end{equation*}
$A(c):$ Measures how similar predicted trajectory and ground-truth trajectory are. \\
$TP:$ True Positive. A ground truth detection and predicted detection are matched together given that $S \geq \alpha$. $S$ is the localization similarity and $\alpha$ is the threshold.\\
$FN:$ False Negative. A ground truth detection that was missed\\
$FP:$ False Positive. A predicted detection with no respective ground truth detection.\\
$TPA:$ True Positive Association. The set of True Positives that have the same ground truth IDs and the same prediction ID as a given $TP c$.
\begin{align*}
TPA&(c) = \{k\}, \\
&k \in \{TP|prID(k) = prID(c) \wedge gtID(c) = gtID(c)\}
\end{align*}
$FNA:$ The set of ground truth detections with the same ground truth ID as a given $TP c$. However, these detections were assigned a prediction ID different from $c$ or none at all.
\begin{align*}
FNA&(c) = \{k\}, \\
k & \left. \in
\begin{aligned}
& \{TP \,|\, prID(k) \ne prID(c) \wedge gtID(k) = gtID(c)\} \\
& \cup \{FN \,|\, gtID(k) = gtID(c)\}
\end{aligned}
\right.
\end{align*}
$FPA:$ The set of predicted detections with the same prediction ID as a given $TP c$. However, these detections were assigned a ground-truth ID different from $c$ or none at all.
\begin{align*}
FPA&(c) = \{k\}, \\
k & \left. \in
\begin{aligned}
& \{TP \,|\,prID(k) = prID(c)\wedge gtID(k) \neq gtID(c)\} \\
& \cup \{FP \,|\, prID(k) = prID(c)\}
\end{aligned}
\right.
\end{align*}
$HOTa_\alpha$ means that this is $HOTA$ calculated for a particular value of $\alpha$. Further calculation needs to be done to get the final HOTA score. We find the value of $HOTA$ for different values of $\alpha$, ranging from 0 to 1 and then calculate their average.
\begin{align*}
HOTA = \int_{0}^{1} HOTA\alpha \:d\alpha \approx \frac{1}{19} \sum_{\alpha \in \left \{
\begin{aligned}
& 0.05, 0.1,\\
& ...0.9,0.95
\end{aligned} \right \} } HOTA\alpha
\end{align*}
We are able to break down $HOTA$ into several sub-metrics. This is useful to us because we can take different elements of the tracking evaluation and use them for comparison. We can get a better idea of the kind of errors our tracker is making.
There are five types of errors commonly found in tracking, false negatives, false positives, fragmentations, mergers and deviations. These can be measured through detection recall, detection precision, association recall, association precision, and localization, respectively.
\subsection{LocA}
Localization Accuracy\cite{luiten2021hota}.
\begin{equation*}
LocA = \int_{0}^{1} \frac{1}{\lvert TP\alpha \rvert} \sum_{c\in \{TP\alpha\}} S(c) d\alpha
\end{equation*}
$S(c)$: The spatial similarity score between the predicted detection and ground truth detection.
This sub-metric deals with the error type deviation or localization errors. Localization errors are caused when the predicted detections and ground truth detections are not aligned. This is similar to but unlike $MOTP$ as it includes several localization thresholds. Commonly used metrics like $MOTA$ and $IDF1$ do not take localization into account despite the importance of object localization in tracking.
\subsection{AssA: Association Accuracy Score }
According to MOT Benchmark: \textit{”The average of the association jaccard index over all matching detections and then averaged over localization threshold” \cite{luiten2021hota}}. Association is part of the result of an MOT task that lets us know if objects in different frames belong to the same or different objects. The objects have the same ID and are part of the same trajectories. Association Accuracy gives us the average alignment between match trajectories. It focuses on association errors. These are caused when a single object in ground truth is given two different predicted detections, or a single predicted detection is given two different ground truth objects.
\begin{equation*}
AssA\alpha = \frac{1}{\lvert TP \rvert} \sum_{c\in\{TP\}} A(c)
\end{equation*}
\subsection{DetA: Detection Accuracy}
According to MOT Benchmark: \textit{“Detection Jaccard Index averaged over localization threshold” \cite{luiten2021hota}}. Detection is another output of an MOT task. It is simply what objects are within the frame. The detection accuracy is the portion of correct detections. Detection errors exist when ground truth detections are missed or when there are false detections.
\begin{equation*}
DetA_\alpha = \frac{\lvert TP \rvert}{\lvert TP \rvert + \lvert FN \rvert+ \lvert FP \rvert}
\end{equation*}
\subsection{DetRe: Detection Recall }
The equation is given for one localization threshold. We need to average over all localization thresholds \cite{luiten2021hota}.
\begin{equation*}
DetRe_\alpha= \frac{\lvert TP \rvert}{\lvert TP \rvert +\lvert FN \rvert}
\end{equation*}
Detection recall errors are false negatives. They happen when the tracker misses an object that exists in the ground truth. Detection accuracy can be broken down into Detection recall and Detection precision.
\subsection{DetPr: Detection Precision }
The equation is given for one localization threshold. We need to average over all localization thresholds \cite{luiten2021hota}.
\begin{equation*}
DetPr_\alpha= \frac{\lvert TP \rvert}{\lvert TP \rvert +\lvert FP \rvert}
\end{equation*}
As mentioned previously, detection precision is part of detection accuracy. Detection precision errors are false positives. They happen when the tracker makes predictions that does not exist in the ground truth.
\subsection{AssRe: Association Recall }
We need to calculate the equation below and then average over all matching detections. Finally, average the result over the localization threshold \cite{luiten2021hota}.
\begin{equation*}
AssRe_\alpha= \frac{1}{\lvert TP \rvert} \sum_{c\in
\{TP\}} \frac{\lvert TPA(c)\rvert}{\lvert TPA(c)\rvert + \lvert FNA(c)\rvert}
\end{equation*}
Association Recall errors happen when the tracker assigns different predicted trajectories to the same ground-truth trajectory. Association Accuracy can be broken down into Association Recall and Association Precision.
\subsection{AssPr: Association Precision}
We need to calculate the equation below and then average over all matching detections. Finally, average the result over the localization threshold \cite{luiten2021hota}.
\begin{equation*}
AssRe_\alpha= \frac{1}{\lvert TP \rvert} \sum_{c\in
\{TP\}} \frac{\lvert TPA(c)\rvert}{\lvert TPA(c)\rvert + \lvert FPA(c)\rvert}
\end{equation*}
Association precision makes up part of association accuracy. Association errors occur when two different ground truth trajectories are given the same prediction Identity.
\subsection{MOTSA: Multi Object Tracking and Segmentation Accuracy}
This is a variation of the $MOTA$ metric, so that the trackers performance of segmentation tasks can also be evaluated.
\begin{equation*}
MOTSA= 1-\frac{\lvert FN \rvert +\lvert FP \rvert+\lvert IDS \rvert }{ \lvert M \rvert}
\end{equation*}
\begin{equation*}
=\frac{\lvert TP \rvert - \lvert FP \rvert-\lvert IDS \rvert }{ \lvert M \rvert}
\end{equation*}
Here $M$ is a set of $N$ non-empty ground truth masks. Each mask is assigned a ground truth track Id. $TP$ is a set of true positives. A true positive occurs when a hypothesized mask is mapped to a ground truth mask. $FP$ is false negatives, the set of hypothesized maps without any ground truth maps and $FN$, false negatives are the ground truth maps without any corresponding hypothesized maps. The $IDS$, ID switches are ground truth masks belonging to the same track but have been assigned different ID’s.
The downsides of $MOTSA$ include, giving more importance to detection over association and being affected greatly by the choice of matching threshold.
\subsection{AMOTA: Average Multiple Object Tracking Precision}
This is calculated by averaging the $MOTA$ value over all recall values.
\begin{equation*}
AMOTA= \frac{1}{L} \sum_{r\in
\{\frac{1}{L},\frac{2}{L}...1\}} 1+ \frac{FP_r+FN_r+IDS_r}{num_gt}
\end{equation*}
The value $num_gt$ is the number of ground truth objects in all the frames. For a specific recall value $r$ the number of false positive, number of false negative and the number of identity switches are denoted as $FP_r$, $FN_r$ and $IDS_r$. The number of recall values is denoted using $L$.
\section{Applications}
There is a myriad of applications for MOT. Much work has gone into tracking various objects, including pedestrians, animals, fish, vehicles, sports players, etc. Actually, the domain of multiple object tracking can not be confined to only a few fields. But to get an idea from an application point of view, we will cover the papers depending on specific applications.
\subsection{Autonomous Driving}
Autonomous driving can be said to be the most common task in Multiple Object Tracking. In recent years, this is a very hot topic in artificial intelligence.
Gao et al. have proposed a dual-attention network for autonomous driving where they have integrated two attention modules \cite{gao2020multiple}. Fu et al. have at first detected vehicles by self-attention mechanism and then used multi-dimensional information for association. They have also handled occlusion by re-tracking the missed vehicles \cite{fu2021real}. Pang et al. have combined vehicle detection with Multiple Measurement Models filter (RFS-M3) which is based on random finite set-based (RFS) introducing 3-D MOT \cite{pang20213d}. Luo et al. have also applied 3-D MOT by proposing SimTrack which detects and associates the vehicle from point clouds captured by LiDAR.
Mackenzie et al. have done two studies: one for self-driving cars and the other for sports \cite{mackenzie2022multiple}. They have looked into the overall performance of Multiple Object Avoidance (MOA), a tool for measuring attention for action in autonomous driving. Zou et al. have proposed a lightweight framework for the full-stack perception of traffic scenes in the 2-D domain captured by roadside cameras \cite{zou2022real}. Cho et al. have identified and tracked the vehicles from traffic surveillance cameras by YOLOv4 and DeepSORT after projecting the images from local to global coordinate systems \cite{cho2022autonomous}.
\subsection{Pedestrian Tracking}
Pedestrian Tracking is one of the most frequent tasks of multiple object tracking systems. As streetcam videos are easy to be captured, much work has been done regarding human or pedestrian tracking. Consequently, pedestrian tracking is considered to be an individual field of research.
Zhang et al. have proposed DROP (Deep Re-identification Occlusion Processing) which can re-identify the occluded pedestrians with the help of appearance features of the pedestrians \cite{zhang2021online}. Sundararaman et al. have proposed HeadHunter to detect pedestrians’ heads followed by a re-identification module for tracking \cite{sundararaman2021tracking}. On the other hand, Stadler et al. have proposed an occlusion handling strategy rather than a feature-based approach followed by a regression-based method \cite{stadler2021improving}.
Chen et al. have introduced a framework applied by Faster R-CNN, KCF trackers and Hungarian algorithm to detect vehicle-mounted far-infrared (FIR) pedestrians \cite{chen2021vehicle}. Ma et al. have made a multiple-stages framework for trajectory processing and Siamese Bi-directional GRU (SiaBiGRU) for post-processing them \cite{ma2021deep}. They have also used a Position Projection Network for cross-camera trajectory matching.
Later on, in \cite{wang2022multi}, Wang et al. have tracked pedestrians simply by using YOLOv5 for detection and DeepSORT for tracking. Patel et al. have proposed a number of algorithms regarding different aspects \cite{patel2022motion}. At first, they have created an algorithm to localize objects, then they proposed a tracking algorithm to identify any suspicious pedestrians from the crowd. There are a couple of algorithms for measuring physical distances as well.
\subsection{Vehicle Surveillance}
Vehicle Surveillance is also a very important task along with autonomous driving. To monitor the activities of vehicles, MOT can be applied.
Shi et al. have introduced a motion based tracking method along with a Gaussian local velocity (GLV) modeling method to identify the normal movement of vehicles and also a discrimination function to detect anomalous driving \cite{shi2021anomalous}. Quang et al. have focused more on Vietnamese vehicles’ speed detection. They have at first detected traffic by YOLOv4 and estimated speed by back-projecting it into 3-D coordinate system with Haversine method \cite{quang2021vietnamese}.
Wang et al. have used graph convolutional neural network to associate the bounding boxes of vehicles into tracklets and proposed an embedding strategy, reconstruct-to-embed with global motion consistency to convert the tracklets into tracks \cite{wang2021track}. Zhang et al. have proposed a convolutional network based on YOLOv5 to solve the low recognition rate accuracy problem in tracking vehicles \cite{zhang2022research}. At last, Diego et al. have published a review paper regarding the traffic environment itself discussing various works of multiple object tracking under traffic domain \cite{jimenez2022multi}.
\subsection{Sports Player Tracking}
In the age of artificial intelligence, rigorous analysis of players in any sport is one of the most important tactics. Thus MOT is used in many ways for sports player tracking.
In \cite{kalafatic2022multiple}, Kalafati{\'c} et al. have tried to solve the occlusion problem of football players tracking by typical \textit{tracking by detection} approach. They have also mentioned some challenges like similar appearance, varying size of projection of players, changing illumination, which MOT researchers should keep in mind to solve besides tracking. However, Naik et al. have addressed identity switching in real-world sports videos \cite{naik2022deepplayer}. They have proposed a novel approach DeepPlayer-Track to track players and referees while retaining the tracking identity. They have used YOLOv4 and SORT to some extent.
In \cite{zheng2022soccer}, Zheng et al. have argued that MOT can replace the use of hardware chips for target tracking. For long term real time multicamera multi target tracking of soccer player, they utilize KCF algorithm which has shown good robustness in terms of accuracy. In \cite{cioppa2022soccernet}, Cioppa et al. have proposed a novel dataset of soccer videos. In which they have annotated multiple players, referees, and ball. They have also given some baseline on that dataset. In \cite{vats2021player}, Vats et al. have introduced ice hockey video analysis. Their system can track players, identify teams, and identify individual players. Their work overcomes the challenges of camera panning, zooming of hockey broadcast video.
\subsection{Wild Life Tracking}
One of the potential use cases of MOT is wildlife tracking. It helps wildlife researchers to avoid costly sensors which are not so reliable in some cases.
In \cite{marcos2021animal}, Marcos et al. have developed a uav based single animal tracking system. They have used YOLOv3 with particle filter for object tracking. Furthermore, in \cite{zhang2022animaltrack} Zhang et al. have addressed the challenge of animal motion and behavior analysis for wildlife tracking. Consequently, they have proposed AnimalTrack which is a largescale benchmark dataset for multi-animal tracking. They have also provided some baseline.
In \cite{guo2022video}, Guo et al. have proposed a method to utilize MOT to detect negative behavior of animals. As analysis of the behavior of animals is very important for breeding, they have shown that using two very popular trackers FairMOT \cite{zhang2021fairmot} and JDE \cite{qin2021joint}, they tracked groups of pigs and laying hens. Which further have helped them to analyze the improvement of health and welfare. However, one of the most interesting job is done by Ju et al. In \cite{ju2021turkey}, they have argued that monitoring the turkey health during reproduction is very important. Thus they have proposed a method to identify the behavior of turkeys utilizing MOT. They have introduced a turkey tracker and head tracker to identify turkey behavior.
MOT is also playing a vital role in tracking underwater entities like fish. In \cite{li2022cmftnet}, Li et al. have proposed CMFTNet, which is implemented by applying Joint Detection and Embedding for extracting and associating features. Deformable convolution is applied furthermore to sharpen the features in complex context and finally, with the help of weight counterpoised loss the fish can be tracked accurately. Also, Filip et al. have analyzed some multiple object tracking works on tracking fish accomplished in the past \cite{dvechtverenko2022tracking}.
\subsection{Others}
We can see the real-life application of MOT in other fields, as well as MOT, is not limited to some particular tasks.
In the field of visual surveillance, Ahmed et al. have presented a collaborative robotic framework that is based on SSD and YOLO for detection and a combination of a number of tracking algorithms \cite{ahmed2020towards}. Urbann et al. have proposed a siamese network-based approach for online tracking under surveillance scenarios \cite{urbann2021online}. Nagrath et al. have analyzed various approaches and datasets of multiple object tracking for surveillance \cite{nagrath2022understanding}.
Robotics is a very trendy topic in today’s world. In \cite{wilson2020avot}, Wilson et al. have introduced audio-visual object tracking (AVOT). Peireira et al. have implemented mobile robots and tracked them by typical SORT and Deep-SORT algorithms integrated with their proposed cost matrices \cite{pereira2022sort}.
We can also see the implementation of MOT in agriculture. To track tomato cultivation, Ge et al. have used a combination of YOLO-based shufflenetv2 as a baseline, CBAM for attention mechanism, BiFPN as multi-scale fusion structure, and DeepSORT for tracking \cite{ge2022tracking}. Tan et al. have also used YOLOv4 as detector of cotton seedlings and an optical flow-based tracking method to track the seedlings \cite{tan2022towards}.
MOT can be also utilized in various real-life applications like security monitoring, monitoring social distancing, radar tracking, activity recognition, smart elderly care, criminal tracking, person re-identification, behavior analysis, and so on.
\section{Future Directions}
As MOT is a trending research topic for many years, numerous efforts have been made on it already. But still, there is a lot of scope in this field. Here we would like to point out some of the potential directions of MOT.
\begin{enumerate}
\item Multiple object tracking under multiple cameras is a bit challenging. The main challenge would be how to fuse the scenes. But if scenes from non-overlapping cameras are fused together and projected in a virtual world, then MOT can be utilized to track a target object in a long area continuously. A similar kind of effort can be seen in \cite{ma2021deep}. A relatively new dataset Multi-camera multiple people tracking is also available \cite{han2021mmptrack}. Xindi et al. have proposed a real-time online tracking system for multi-target multi-camera tracking \cite{zhang2019real}.
\item Class-based tracking system can be integrated with multiple object tracking. An MOT algorithm tries to track almost all moving objects in a frame. This will be better applied in real-life scenarios if class-based tracking can be possible. For example, bird tracking MOT system can be very useful in airports, because to prevent the clash of birds with airplanes on the runway some manual preventive mechanism is currently applied. It can be totally automated using a class-based MOT system. Class-based tracking helps in surveillance in many ways. Because it helps to track a certain type of object efficiently.
\item MOT is widely applied in 2D scenes. Though it is a bit challenging task, analyzing 3D videos utilizing MOT will be a good research topic. 3D tracking can provide more accurate tracking and occlusion handling. As in 3D scene depth information is kept, thus it helps to overcome one of the main challenges on MOT which is occlusion.
\item So far in most of the papers transformer is used as a black box. But transformer can be used more specifically in solving different MOT tasks. Some approaches are totally based on detection and further regression is applied to predict the bounding box of the next frame \cite{bergmann2019tracking}. In that case, DETR \cite{carion2020end} can be used to detect as it has very high efficiency in detecting objects.
\item In any application lightweight architecture is very important for real-life applications. Cause lightweight architecture is resource efficient and in real-life scenarios, we have constrained on resources mostly. In MOT lightweight architecture is also very crucial if we want to deploy a model in IoT embedded devices. Also to track in real-time, lightweight architecture plays a very important role. So without decreasing accuracy, if we can achieve more fps then, it can be implemented in real-life applications, where lightweight architecture is very necessary.
\item To apply in real-life scenarios, online multiple object tracking is the only possible solution. Thus inference time plays a very crucial role. We observe the trend of acquiring more accuracy from researchers in recent times. But if we can achieve an inference time of over thirty frames per second, then we can use MOT as real-time tracking. As real-time tracking is the key to surveillance thus it is one of the major future directions for MOT researchers.
\item A trend of applying quantum computing in computer vision can be seen in recent times. Quantum computing can be used in MOT as well. Zaech et al. have published the first paper of MOT using Adiabatic Quantum Computing (AQC) with the help of Ising model \cite{zaech2022adiabatic}. They expect that AQC can speed up the N-P hard assignment problem during association in future. As quantum computing has a very high potential in the near future, this can be a very promising domain to research on.
\end{enumerate}
\subsection{Transformer}
\label{subsection:transformer}
Recently there have been many works regarding Computer Vision \cite{bi2021transformer} implemented by transformer, so as in MOT. It is a deep learning model which has two parts like the other models: Encoder and Decoder \cite{vaswani2017attention}. The encoder captures self-attention, whereas the decoder captures cross-attention. This attention mechanism helps to memorize context for long term. Based on query key fashion, transformer is used to predict the output. Though it has been used solely as a language model in the past, in recent years, vision researchers have focused on it to take advantage of contextual memoization. In most cases, in MOT researchers try to predict the location of the next frame of an object based on previous information, where we think transformer is the best player to handle. As transformer is specialized to handle sequential information, frame by fame processing can be done perfectly by transformer. A whole summary of transformer based approach in MOT is presented in Table \ref{tabtr}.
Peize et al. have built TransTrack \cite{sun2020transtrack}, using the transformer architecture where they produce two sets of bounding boxes from two types of queries, i.e., object query and track query, and by simple IoU matching, they decide the final set of the boxes, which is the tracking box for every object. It is totally as same as \textit{tracking by detection} paradigm. Moreover, it leverages the prediction for tracking from previous detection knowledge utilizing transformer query key mechanism. Tim et al. have done a similar thing by introducing TrackFormer \cite{meinhardt2021trackformer}, excluding some implementation details.
In \cite{chen2022patchtrack}, the patches of images were at first detected, then they took help from probabilistic concepts to get expected tracks and cropped the frames according to the bounding boxes to get the patches. Then using those patches, the tracks of current frames are predicted.
\begin{figure*}[htbp]
\centering
\includegraphics[width = \textwidth]{Figures/TrackFormer.png}
\caption{Utilizing the encoder-decoder architecture of the transformer, TrackFormer \cite{meinhardt2021trackformer} converts multi-object tracking as a set prediction problem performing joint detection and tracking-by-attention.}
\end{figure*}
Later on, En et al. have proposed a method of combining the attention model with a Transformer encoder to make Guided Transformer Encoder (GTE) \cite{yu2022relationtrack} only go through the significant pixels of each frame in a global context.
In \cite{liu2022segdq}, Dynamic Object Query (DOQ) has been applied to make the detection more flexible. Additionally, query-based tracking has also been applied for semantic segmentation. Yihong et al. propose a multi-scaled pixel-by-pixel dense query system \cite{xu2021transcenter}, which generates dense heatmaps for targets that can give more accurate results.
The paper \cite{blatter2021efficient} and \cite{xing2022siamese} focus more on the computation cost for running the architecture in real-time. In \cite{blatter2021efficient}, the transformer layer has been built upon an exemplar attention module which reduces the dimension of input by giving global information. Thus the layer can work in real-time. In \cite{xing2022siamese}, Daitao et al. have improved the computation time by using a lightweight attention layer applied by transformer model which is inserted in a pyramid network.
In \cite{zhou2022global} Zhou et al. introduce the concept of global tracking. Instead of processing frame by frame, they take a window of 32 frames and track within them. It utilizes the transformer’s cross-attention mechanism more efficiently. According to the authors, if an object is lost within a window and reborn again, then their tracker can successfully track it, which makes their tracker lower ID switching which is one of the main challenges of MOT.
In \cite{zeng2021motr} Zeng et al. extend DETR \cite{carion2020end}, which is an object detection transformer. They introduce Query Interaction Module to filter out the output of the decoder of DETR before adding a detection to the tracklet.
In \cite{zhu2021vitt} Zhu et al. use uses encoder of the transformer to generate a feature map and then they use three tracing heads to predict bounding box classification, regression, and embedding. In most of the cases, we show others use convolutional layer to extract features, some use popular CNN architecture to extract features from a frame, but it adds an extra load to the main architecture. This ViTT utilized relatively lightweight transformers encoders compared to others. Moreover, the tracking heads are simple feed-forward networks. Consequently, they produce a lightweight architecture.
\begin{table*}[htbp]
\caption{Summary of Transformer related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{90pt} >{\centering\arraybackslash}p{64pt}}
\hline
\multirow{2}{*}{\bf Reference} & \multirow{2}{*}{\bf Year} & {\bf Detection / Appearance Feature Extraction} & \multirow{2}{*}{\bf Data Association} & \multirow{2}{*}{\bf Dataset} & \multirow{2}{*}{\bf MOTA (\%)} \\
\hline
\cite{sun2020transtrack} & 2020 & Decoder of DETR & Decoder of Transformer & MOT17, MOT20 & 74.5,64.5 \\
\cite{meinhardt2021trackformer} & 2021 & CNN & Decoder of Transformer & MOT17 & 62.5 \\
\cite{chen2022patchtrack} & 2020 & CNN & Transformer & MOT16, MOT17 & 73.3, 73.6 \\
\cite{yu2022relationtrack} & 2022 & Faster R-CNN & Hungarian Algorithm & MOT16, MOT17, MOT20 & 75.8. 74.7, 70.5 \\
\cite{liu2022segdq} & 2022 & CNN + Encoder of Transformer & Decoder + Feed Forward Network & MOT15, MOT16, MOT17 & 40.3, 65.7, 65.0 \\
\cite{xu2021transcenter} & 2021 & DETR & Deformable Dual Decoder & MOT17, MOT20 & 71.9, 62.3 \\
\cite{blatter2021efficient} & 2021 & Exemplar Attention based encoder & Exemplar Attention based encoder & TrackingNet & 70.55 (Precision) \\
\cite{xing2022siamese} & 2022 & Transformer Pyramid Network & Multihead and pooling attention & UAV123 & 85.83 (Precision) \\
\cite{zhou2022global} & 2022 & CenterNet & Tracking transformer & TAO, MOT17 & 45.8 (HOTA), 75.3 \\
\multirow{3}{*}{\cite{zeng2021motr}} & \multirow{3}{*}{2021} & \multirow{3}{*}{DETR} & Decoder and Query Interaction Module + Temporal aggregation network & MOT17, DanceTrack, BDD100k & 57.2 (HOTA), 54.2 (HOTA), 32.0 (nMOTA) \\
\cite{zhu2021vitt} & 2021 & Encoder & Bounding Box Regression Network & MOT16 & 65.7 \\
\hline
\end{tabular}
\end{center}
\label{tabtr}
\end{table*}
\subsection{Graph Model}
Graph Convolutional Network (GCN) is a special kind of convolutional network where the neural network is applied in a graph fashion \cite{kipf2016semi} instead of a linear form. Also, a recent trend has been seen in using Graph models in solving MOT problems where a set of detected objects from consecutive frames are considered as a node, and the link between two nodes is considered as an edge. Normally data association is done in this domain by applying the Hungarian algorithm \cite{kuhn1955hungarian}. An overview of solving MOT problems using graph models is given in Table \ref{tabgcn}.
Guillem et al. detect and track objects globally by using a message passing network (MPN) combined with a graph to extract deep features throughout the graph \cite{braso2020learning}. Later on, \cite{zaech2022learnable} and \cite{ma2021deep} have done something similar. Gaoang et al. have taken the same approach, \cite{wang2021track} but they removed the appearance information because according to them, appearance features can cause more occlusion. Also, they have followed an advanced embedding strategy to design the tacklets.
But Jiahe et al. use two graphs: Appearance Graph Network and Motion Graph Network, to identify the similarity of appearance and motion from among the frames respectively \cite{li2020graph}.
Peng et al. have also used two graph modules to solve the MOT problem, but one of the modules is for generating the proposal, and the other one is for scoring the proposal \cite{dai2021learning}. In the case of proposal generation, they considered small tracklets or detected objects as nodes, and each node is connected with all the others. But for the next module, they have trained a GCN to rank the proposals according to their scores as can be seen in Figure \ref{fig:gcn}.
\begin{figure*}[htbp]
\centering
\includegraphics[width = \textwidth]{Figures/GCN.png}
\caption{(a) Frames with detected objects. (b) Graph constructed with the detected objects or tracklets as each node and proposal generation. (c) Ranking the proposals with GCN. (d) Trajectory Inference. (e) Final output \cite{dai2021learning}}
\label{fig:gcn}
\end{figure*}
In \cite{he2021learnable}, Jiawei et al. have solved two problems: association problem and assignment problem. To solve the association problem, they focused more on matching features within the same frame across the graph rather than finding relationships between two frames. But for the assignment problem, they have integrated a quadratic programming layer to learn more robust features.
So far, the papers have worked with the single-camera MOT problem. But in the next year, in 2022, Kha et al. have worked on multi-camera MOT problem \cite{quach2021dyglip}. They have established a dynamic graph to accumulate the new feature information instead of a static graph like the other papers.
\begin{table*}[htbp]
\caption{Summary of Graph Model related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{100pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{95pt} >{\centering\arraybackslash}p{70pt}}
\hline
\bf Reference & \bf Year & \bf Detection & \bf Association & \bf Dataset & \bf MOTA (\%) \\
\hline
\cite{braso2020learning} & 2020 & ResNet50 & Message Passing & MOT15, MOT16, MOT17 & 51.5, 58.6, 58.8 \\
\cite{li2020graph} & 2020 & ResNet-34 & Hungarian algorithm & MOT16, MOT17 & 47.7, 50.2 \\
\multirow{2}{*}{\cite{ma2021deep}} & \multirow{2}{*}{2021} & \multirow{2}{*}{SeResNet-50} & \multirow{2}{*}{Human-Interaction Model} & MOT15, MOT16, DukeMTMCT & \multirow{2}{*}{80.4, 50.0, 86.7} \\
\multirow{2}{*}{\cite{wang2021track}} & \multirow{2}{*}{2021} & \multirow{2}{*}{CenterNat, CompACT} & Box and Tracklet Motion Embedding & \multirow{2}{*}{MOT17, KITTI, UA-Detrac} & \multirow{2}{*}{56.0, 87.6, 22.5} \\
\cite{li2020graph} & 2020 & ResNet-34 & Hungarian algorithm & MOT16, MOT17 & 47.7, 50.2 \\
\cite{dai2021learning} & 2021 & ResNet50-IBN & Proposal Generation and Scoring & MOT17, MOT20 & 59.0, 56.3 \\
\cite{he2021learnable} & 2021 & CenterNet & Graph Matching & MOT16, MOT17 & 65.0, 66.2 \\
\cite{zaech2022learnable} & 2022 & CenterPoint, MEGVII & Message Passing & nuScenes & 55.4 \\
\hline
\label{tabgcn}
\end{tabular}
\end{center}
\end{table*}
\subsection{Detection and Target Association}
In such kind of approach, detection is done by any deep learning model. But the main challenge is to associate target, i.e. to keep track of the trajectory of the object of interest \cite{bewley2016simple}. Different papers follow different approaches in this regard.
\begin{table*}[htbp]
\caption{Summary of Detection and Target Association related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{100pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{95pt} >{\centering\arraybackslash}p{70pt}}
\hline
\bf Reference & \textbf{Year} & \textbf{Detection} & \textbf{Association} & \textbf{Dataset} & \textbf{MOTA (\%)} \\
\hline
\cite{keuper2018motion} & 2018 & Faster R-CNN & Correlation Co-Clustering & MOT15, MOT16, MOT17 & 35.6, 47.1, 51.2 \\
\cite{karunasekera2019multiple} & 2019 & DPM, F-RCNN, SDP, RRC & Hungarian Algorithm & MOT17, KITTI & 46.9, 85.04 \\
\multirow{2}{*}{\cite{voigtlaender2019mots}} & \multirow{2}{*}{2019} & \multirow{2}{*}{Mask R-CNN} & \multirow{2}{*}{Distance Measurement} & KITTI MOTS, MOTSChallenge & 65.1, KITTI MOTS (MOTSA) \\
\multirow{2}{*}{\cite{song2021multiple}} & \multirow{2}{*}{2021} & \multirow{2}{*}{CenterNet} & \multirow{2}{*}{Hungarian Algorithm} & MOT15, MOT16, MOT17, MOT20 & \multirow{2}{*}{60.6, 74.9, 73.7, 61.8} \\
\cite{wang2021drt} & 2021 & ResNet50 & LSTM-based Motion Model & MOT16, MOT17 & 76.3, 76.4 \\
\cite{kim2021discriminative} & 2021 & CenterNet & Bilinear LSTM & MOT16, MOT17 & 48.3, 51.5 \\
\multirow{2}{*}{\cite{wang2021multiple}} & \multirow{2}{*}{2021} & \multirow{2}{*}{CenterNet} & Correlation Learning & MOT15, MOT16, MOT17, MOT20 & \multirow{2}{*}{62.3, 76.6, 76.5, 65.2} \\
\multirow{2}{*}{\cite{pang2021quasi}} & \multirow{2}{*}{2021} & \multirow{2}{*}{Faster R-CNN} & Quasi-dense Similarity Matching & MOT16, MOT17, BDD100K, Waymo & \multirow{2}{*}{69.8, 68.7, 64.3, 51.18} \\
\cite{sundararaman2021tracking} & 2021 & HeadHunter & HeadHunter-T & CroHD & 63.6 \\
\multirow{3}{*}{\cite{wu2021track}} & \multirow{3}{*}{2021} & \multirow{3}{*}{CenterNet} & \multirow{3}{*}{CVA (Cost Volume based Association} & MOT16, MOT17, nuScenes, MOTS & 70.1, 69.1, 5.9 (AMOTA), 65.5 (MOTSA) \\
\multirow{2}{*}{\cite{liu2022det}} & \multirow{2}{*}{2022} & \multirow{2}{*}{Mask-RCNN} & \multirow{2}{*}{Hungarian Algorithm} & MOT17, MOT20, NTU-MOTD & \multirow{2}{*}{43.21, 57.70, 92.12} \\
\multirow{2}{*}{\cite{tan2022towards}} & \multirow{2}{*}{2022} & \multirow{2}{*}{YOLOv4} & \multirow{2}{*}{Hungarian Algorithm} & TAMU2015V, UGA2015V, UGA2018V & 79.0\%, 65.5\%, 73.4\% \\
\multirow{2}{*}{\cite{kesa2022multiple}} & \multirow{2}{*}{2022} & \multirow{2}{*}{DLA-34} & \multirow{2}{*}{Hungarian Algorithm} & MOT15, MOT16, MOT17, MOT20 & \multirow{2}{*}{55.8, 73.8, 74.0, 60.2} \\
\multirow{2}{*}{\cite{sun2022online}} & \multirow{2}{*}{2022} & DPM and YOLOv5 with detection modifier (DM) & \multirow{2}{*}{Global and Partial Feature Matching} & \multirow{2}{*}{MOT16} & \multirow{2}{*}{46.5} \\
\multirow{2}{*}{\cite{liang2022non}} & \multirow{2}{*}{2022} & \multirow{2}{*}{YOLO X with later NMS} & Kalman Filtering, Bicubic Interpolation and ReID Model & \multirow{2}{*}{MOT17, MOT20} & \multirow{2}{*}{78.3, 75.7} \\
\cite{he2022joint} & 2022 & T-ReDet module & ReID-NMS Model & MOT16, MOT17, MOT20 & 63.9, 62.5, 57.4 \\
\hline
\end{tabular}
\end{center}
\end{table*}
Margret et al. have picked both the bottom-up approach and the top-down approach \cite{keuper2018motion}. In bottom-up approach, point trajectories are determined. But in top-down approach, bounding boxes are determined. Then by combining these two, a full track of objects can be found.
In \cite{karunasekera2019multiple}, to solve the association problem, Hasith et al. have simply detected the objects and used the famous Hungarian Algorithm to associate the information. In the same year 2019, Paul et al. proposed Track-RCNN \cite{voigtlaender2019mots} which is an extension of R-CNN and obviously a revolutionary task in the field of MOT. Track-RCNN is a 3-D convolutional network that can do both detection and tracking along with segmentation.
But in the year 2020, Yifu et al. have done object detection and re-identification in two separate branches \cite{zhang2021fairmot}. The branches are similar in architecture and they both used center to extract features to detect and re-identify respectively. They claim that they have focused equally on the two tasks, that’s why they have named their approach FairMOT.
In the year 2021, we find two papers to improve data association using LSTM. Bisheng et al. propose Detection Refinement for Tracking (DRT), which has done the detection task by semi-supervised learning which produces heatmap to localize the objects more correctly \cite{wang2021drt}. The architecture has two branches where the secondary branch, it can recover occluded objects. Also, the paper has solved the data association problem by LSTM \cite{hochreiter1997long}. Chanho et al. also used bilinear LSTM in this regard \cite{kim2021discriminative}.
Besides in \cite{wang2021multiple}, Qiang et al. have done data association by proposing CorrTracker, which is a correlational network that is able to propagate information across the associations. They have done the part of object detection by self-supervised learning. But Jiangmiao et al. have detected objects by Faster-RCNN extended with residual networks and have combined it with similarity learning and ultimately have proposed Quasi Dense Tracking model (QDTrack) \cite{pang2021quasi}.
In the same year, Yaoye et al. have introduced D2LA network \cite{song2021multiple} which is based on FairMOT as introduced in \cite{zhang2021fairmot} to keep a balance between the trade-off of accuracy and complexity. To avoid occlusion, they have taken measures namely strip attention module. On the other hand, Norman et al. estimate the geometry of each detected object and make a mapping of each object to its corresponding pose so that they can identify the object after occlusion \cite{muller2021seeing}.
Ramana et al. have proposed their own dataset with their own architecture namely HeadHunter for detection and HeadHunter-T for tracking \cite{sundararaman2021tracking}. There are two stages in HeadHunter. In the first stage, they have used FPN and Resnet-50 to extract features. In the second stage, they have used Faster-RCNN and RPN to generate object proposals.
Jialian et al. have proposed two modules \cite{wu2021track}: cost volume based association (CVA) module and motion-guided feature warper (MFW) module to extract object localization offset information and to transmit the information from frame to frame respectively. They have named the integration of the whole process as TraDeS (TRAck to DEtect and Segment). Changzhi et al. have made ParallelMOT \cite{lv2021parallelmot} which have two different branches for detection and re-identification similar to \cite{zhang2021fairmot}.
In 2022, we can see diversity in the problem statements of MOT. \cite{liu2022det} is an exceptional paper where Cheng-Jen et al. have introduced indoor multiple object tracking. They have proposed depth-enhanced tracker (DET) to improve the \textit{tracking-by-detection} strategy along with an indoor MOT dataset. We can again see a different kind of problem statement in \cite{tan2022towards}, which is to track crop seedlings. In this paper, Chenjiao et al. have used YOLOv4 as an object detector and tracked the bounding boxes got from the detector by optical flow.
Oluwafunmilola et al. have done object tracking along with object forecasting \cite{kesa2022multiple}. They have detected bounding boxes using FairMOT \cite{zhang2021fairmot} and then have stacked a forecasting network and have made Joint Learning Architecture (JLE). Zhihong et al. have extracted new features of each frame to get the information globally and have accumulated partial features for occlusion handling \cite{sun2022online}. They have merged these two kinds of features to detect the pedestrian accurately.
No paper has taken any measure to preserve the significant bounding boxes so that they are not eliminated in the data association stage except \cite{liang2022non}. After detecting, Hong et al. applied Non-Maskable Suppression (NMS) in the tracking phase to reduce the probability of the important bounding boxes being removed \cite{he2022joint}. Jian et al. also have used NMS to reduce redundant bounding boxes from the detector. They have re-detected trajectory location by comparing features and re-identified bounding boxes with the help of IoU. The ultimate outcome is a joint re-detection and re-identification tracker (JDI).
\subsection{Attention Module}
To re-identify the occluded objects, attention is needed. Attention means we only consider the objects of interest by nullifying the background so that their features are remembered for long, even after occlusion. The summary of using attention module in MOT field is given in Table \ref{tabatt}.
\begin{figure*}[htbp]
\centering
\includegraphics[width = \textwidth]{Figures/Attention.png}
\caption{The structure of Attention based head of cross-attention \cite{wan2022dsrrtracker}}
\label{fig:attention}
\end{figure*}
\begin{table*}[htbp]
\caption{Summary of Attention related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{155pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{95pt}}
\hline
\textbf{Reference} & \textbf{Year} & \textbf{Attention Mechanism} & \textbf{Dataset} & \textbf{MOTA (\%)} \\
\hline
\cite{song2021multiple} & 2021 & Strip Pooling & MOT15, MOT16, MOT17, MOT20 & 60.6, 74.9, 73.7, 61.8 \\
\multirow{2}{*}{\cite{guo2021online}} & \multirow{2}{*}{2021} & Temporal Aware Target Attention and Distractor Attention & \multirow{2}{*}{MOT16, MOT17, MOT20} & \multirow{2}{*}{59.1, 59.7, 56.6} \\
\cite{liang2021generic} & 2021 & Spatial Transformation Network (STN) & MOT16, MOT17 & 50.5, 50.0 \\
\multirow{2}{*}{\cite{ke2021prototypical}} & \multirow{2}{*}{2021} & \multirow{2}{*}{Spatio-Temporal Cross-Attention} & BDD100K (Validation), KITTI-MOTS (Validation) & 27.4 (MOTSA), 66.4 (mMOTSA) \\
\multirow{2}{*}{\cite{fu2021real}} & \multirow{2}{*}{2021} & \multirow{2}{*}{Self-Attention in Detection} & Custom Dataset: Sparse Scene, Dense Scene & \multirow{2}{*}{70.9, 56.4} \\
\multirow{2}{*}{\cite{quach2021dyglip}} & \multirow{2}{*}{2021} & \multirow{2}{*}{Graph Structural and Temporal Self-Attention} & PETS09, EPFL, CAMPUS, MCT, CityFlow (Validation) & \multirow{2}{*}{93.5, 66.3, 96.7, 95.7, 90.9} \\
\cite{wan2022dsrrtracker} & 2022 & Self- and Cross-Attention as Tracking Head & MOT17, MOT20 & 75.6, 70.4 \\
\hline
\end{tabular}
\end{center}
\label{tabatt}
\end{table*}
In \cite{song2021multiple}, Yaoye et al. have incorporated a strip attention module to re-identify the pedestrians occluded with the background. This module is actually a pooling layer that includes max and mean pooling which extracts features from the pedestrians more fruitfully so that when they are blocked, the model does not forget them and can re-identify further. Song et al. have wanted to use information from object localization in data association and also the information from data association in object localization. To link up between the two, they have used two attention modules, one for target and one for distraction \cite{guo2021online}. Then they finally applied a memory aggregation to make strong attention.
Tianyi et al. have proposed spatial-attention mechanism \cite{liang2021generic} by implementing Spatial Transformation Network (STN) in an appearance model to force the model to only focus on the foreground. On the other hand, Lei et al. have at first proposed Prototypical Cross-Attention Module (PCAM) to extract relevant features from past frames. Then they have used Prototypical Cross-Attention Network (PCAN) to transmit the contrasting feature of foreground and background throughout the frames \cite{ke2021prototypical}.
Huiyuan et al. have proposed self-attention mechanism to detect vehicles \cite{fu2021real}. The paper \cite{quach2021dyglip} also has a self-attention module applied in the dynamic graph to combine internal and external information of cameras.
JiaXu et al. have used both cross and self-attention in a lightweight fashion \cite{wan2022dsrrtracker}. In Figure \ref{fig:attention}, we can see the cross-attention head of that architecture. The self-attention module is used to extract robust features decreasing background occlusion. Then the data is passed to the cross-attention module for instance association.
\subsection{Motion Model}
Motion is an inevitable property of objects. So this feature can be used in the area of multi-object tracking, be it for detection or association. Motion of an object can be calculated by the difference in position of the object between two frames. And based on this measure, different decisions can be taken as we have seen going through the papers. An overview is given in Table \ref{tabmotion}.
Hasith et al. and Oluwafunmilola et al. have used motion to compute dissimilarity cost in \cite{karunasekera2019multiple} and \cite{kesa2021joint} respectively. Motion is calculated by the difference between actual location and predicted location. To predict the location of an occluded object, Bisheng et al. used motion model based on LSTM \cite{wang2021drt}. Wenyuan et al. incorporated motion model with Deep Affinity Network (DAN) \cite{sun2019deep} to optimize data association by eliminating the locations where it is not possible for an object to situate \cite{qin2021joint}.
Qian et al. also have calculated motion by measuring distance from consecutive satellite frames with Accumulative Multi-Frame Differencing(AMFD) and low-rank matrix completion (LRMC) \cite{yin2021detecting} and have formed a motion model baseline (MMB) to detect and to reduce the amount of false alarms. Hang et al. have used motion features to identify foreground objects in the field of vehicle driving \cite{shi2021anomalous}. They have detected relevant objects by comparing motion features with GLV model. Gaoang et al. have proposed a local-global motion (LGM) tracker that finds out the consistencies of the motion and thus associates the tracklets \cite{wang2021track}. Apart from these, Ramana et al. have used motion model to predict the motion of the object rather than data association which hae three modules: Integrated Motion Localization (IML), Dynamic Reconnection Context (DRC), 3D Integral Image (3DII) \cite{sundararaman2021tracking}.
\begin{table*}[htbp]
\caption{Summary of Motion Model related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{200pt} >{\centering\arraybackslash}p{95pt} >{\centering\arraybackslash}p{95pt}}
\hline
\textbf{Reference} & \textbf{Year} & \textbf{Motion Mechanism} & \textbf{Dataset} & \textbf{MOTA (\%)} \\
\hline
\multirow{2}{*}{\cite{karunasekera2019multiple}} & \multirow{2}{*}{2019} & Dissimilarity Distance between Detected and Predicted Object & \multirow{2}{*}{MOT17, KITTI} & \multirow{2}{*}{46.9, 85.04} \\
\multirow{2}{*}{\cite{kesa2021joint}} & \multirow{2}{*}{2021} & Dissimilarity Distance between Detected and Predicted object & MOT15, MOT16, MOT17, MOT20 & \multirow{2}{*}{55.8, 73.8, 74.0, 60.2} \\
\cite{wang2021drt} & 2021 & LSTM-based Model on Consecutive Frames & MOT16, MOT17 & 76.3, 76.4 \\
\cite{qin2021joint} & 2021 & Kalman Filtering & MOT17 & 44.3 \\
\multirow{2}{*}{\cite{yin2021detecting}} & \multirow{2}{*}{2021} & Accumulative Multi-Frame Differencing and Low-Rank Matrix Completion & \multirow{2}{*}{VISO} & \multirow{2}{*}{73.6} \\
\multirow{2}{*}{\cite{shi2021anomalous}} & \multirow{2}{*}{2021} & Distance of Motion Feature and Mean Vector of Gaussian Local Velocity Model & \multirow{2}{*}{NJDOT} & 100 (Anomaly Detection Accuracy) \\
\cite{wang2021track} & 2021 & Box and Tracklet Motion Embedding & MOT17, KITTI, UA-Detrac &56.0, 87.6, 22.5 \\
\multirow{2}{*}{\cite{sundararaman2021tracking}} & \multirow{2}{*}{2021} & Particle Filtering and Enhanced Correlation Coefficient Maximization & \multirow{2}{*}{CroHD} & \multirow{2}{*}{63.6} \\
\multirow{2}{*}{\cite{han2022mat}} & \multirow{2}{*}{2022} & Combination of Camera Motion and Pedestrian Motion (IML), Dynamic Motion-based Reconnection (DRC) & \multirow{2}{*}{MOT16, MOT17} & \multirow{2}{*}{70.5, 69.5} \\
\cite{zou2022compensation} & 2022 & Motion Compensation with Basic Tracker & MOT16, MOT17, MOT20 & 69.8, 68.8. 66.0 \\
\cite{chen2022patchtrack} & 2022 & Kalman Filtering & MOT16, MOT17 & 73.3, 73.6 \\
\hline
\end{tabular}
\end{center}
\label{tabmotion}
\end{table*}
In the year 2022, Shoudong et al. have used motion model for both motion prediction and association by proposing Motion-Aware Tracker (MAT) \cite{han2022mat}. Zhibo et al. have proposed compensation tracker (CT), which can obtain the lost objects having a motion compensation module \cite{zou2022compensation}. But Xiaotong et al. have used motion model to predict the bounding boxes of objects \cite{chen2022patchtrack} so as done in \cite{shi2021anomalous} but to make image patches as discussed in \ref{subsection:transformer}.
\subsection{Siamese Network}
Similarity information between two frames helps a lot in object tracking. Thus the Siamese network tries to learn the similarities and differentiate the inputs. This network has two parallel sub networks sharing the same weight and parameter space. Finally, the parameters between the twin networks are tied up and then trained on a certain loss function to measure the semantic similarity between them. The summary of applying Siamese network in MOT task is given in Table \ref{tabsiam}.
Daitao et al. have proposed a pyramid network that embeds a lightweight transformer attention layer. Their proposed Siamese Transformer Pyramid Network has augmented the target features with lateral cross attention between pyramid features. Thus it has produced robust target-specific appearance representation \cite{xing2022siamese}. Bing et al. have tried to uplift the region based multi object tracking network by incorporating motion modeling \cite{shuai2021siammot}. They have embedded the Siamese network tracking framework into Faster-RCNN to achieve fast tracking by lightweight tracking and shared network parameters.
\begin{figure*}[htbp]
\centering
\includegraphics[width =\textwidth]{Figures/Siamese.png}
\caption{(a) A typical Siamese Network that has symmetric pyramid architecture, (b) A typical Discriminative network, (c) Siamese Transfer Pyramid Network that is proposed in \cite{xing2022siamese}}
\end{figure*}
Cong et al. have proposed a Cleaving Network using Siamese Bi-directional GRU (SiaBiGRU) in post-processing the trajectories to eliminate corrupted tracklets. Then they have established Re-connection Network to link up those tracklets and make a trajectory \cite{ma2021deep}.
In a typical MOT network, there are prediction and detection modules. The prediction module tries to predict the appearance of an object in the next frame, and the detection module detects the objects. The result of these two modules is used in matching the features and updating the trajectory of objects. Xinwen et al. have proposed Siamese RPN (Region Proposal Network) structure as the predictor. They have also proposed an adaptive threshold determination method for the data association module \cite{gao2022multi}. Thus overall stability of a Siamese network has been improved.
In contrast to transformer models \ref{subsection:transformer}, JiaXu et al. have proposed a lightweight attention-based tracking head under the structure of a Siamese network that enhances the localization of foreground objects within a box \cite{wan2022dsrrtracker}. On the other hand, Philippe et al. have incorporated their efficient transformer layer into a Siamese Tracking network. They have replaced the convolutional layer with the exemplar transformer layer \cite{blatter2021efficient}.
\begin{table*}[htbp]
\caption{Summary of Siamese Network Related Papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{160pt} >{\centering\arraybackslash}p{120pt} >{\centering\arraybackslash}p{100pt}}
\hline
\textbf{Reference} & \textbf{Year} & \textbf{Method} & \textbf{Dataset} & \textbf{MOTA (\%)} \\
\hline
\multirow{2}{*}{\cite{xing2022siamese}} & \multirow{2}{*}{2020} & CNN for Apprearance extraction, LSTM and RNN for Motion modelling & \multirow{2}{*}{Duke-MTMCT, MOT16} & \multirow{2}{*}{73.5, 55.0} \\
\cite{shuai2021siammot} & 2021 & Implicit and Explicit motion modelling & MOT17, TAO-person, HiEve & 65.9, 44.3 (TAP@0.5), 53.2 \\
\cite{gao2022multi} & 2021 & Siamese Network with Region Proposal Network & MOT16, MOT17, MOT20 & 65.8, 67.2, 62.3 \\
\cite{blatter2021efficient} & 2021 & Single instance level attention & TrackingNet & 70.55 (Precision) \\
\multirow{2}{*}{\cite{wan2022dsrrtracker}} & \multirow{2}{*}{2022} & Dynamic search region refine and attention based tracking & \multirow{2}{*}{MOT17, MOT20} & \multirow{2}{*}{67.2, 70.4} \\
\cite{xing2022siamese} & 2022 & Transformer based appearance similarity & UAV123 & 85.83 (Precision) \\
\hline
\label{tabsiam}
\end{tabular}
\end{center}
\end{table*}
\subsection{Tracklet Association}
A group of consecutive frames of objects of interest is called a tracklet. In detecting and tracking objects, tracklets are first identified using different algorithms. Then they are associated together to establish a trajectory. Tracklet association is obviously a challenging task in MOT problems. Some papers specifically focus on this issue. Different papers have taken different approaches. Such an overview is as presented in Table \ref{tabtracklet}.
Jinlong et al. have proposed Tracklet-Plane Matching (TPM), \cite{peng2020tpm} where at first short tracklets are created from the detected objects, and they are aligned in a tracklet-plane where each tracklet is assigned with a hyperplane according to their start and end time. Thus large trajectories are formed. This process also can handle non-neighboring and overlapping tracklets. To mitigate the performance, they have also proposed two schemes.
Duy et al. have at first made tracklet by a 3D geometric algorithm \cite{nguyen2021lmgp}. They have formed trajectories from multiple cameras and due to this, they have optimized the association globally by formulating spatial and temporal information.
In \cite{ma2021deep}, Cong et al. have proposed Position Projection Network (PPN) to transfer the trajectories from local to global context. Daniel et al. re-identifies occluded objects by assigning the new-coming object to the previously found occluded object depending on motion. Then they have implemented already found tracks further for regression, thus have taken the \textit{tracking-by-regression} approach. Furthermore, they have extended their work by extracting temporal direction to make the performance better \cite{stadler2021improving}.
In \cite{yu2022towards}, we can see a different strategy from the formers. En et al have considered each trajectory as a center vector and made a trajectory-center memory bank (TMB) which is updated dynamically and calculates cost. The whole process is named multi-view trajectory contrastive learning (MTCL). Additionally, they have created learnable view sampling (LVS), which notices each detection as key point which helps to view the trajectory in a global context. They have also proposed similarity-guided feature fusion (SGFF) approach to avoid vague features.
Et al have developed tracklet booster (TBooster) \cite{wang2022split} to alleviate the errors which occur during association. TBooster has two components: Splitter and Connector. In the first module, the tracklets are split where the ID switching occurs. Thus the problem of assigning the same ID to multiple objects can be resolved. In the second module, the tracklets of the same object are linked. By doing this, assigning the same ID to multiple tracklets can be avoided. Tracklet embedding can be done by Connector.
\begin{table*}[htbp]
\caption{Summary of Tracklet Association related papers}
\begin{center}
\begin{tabular}
{>{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{20pt} >{\centering\arraybackslash}p{240pt} >{\centering\arraybackslash}p{95pt} >{\centering\arraybackslash}p{65pt}}
\hline
\textbf{Reference} & \textbf{Year} & \textbf{Method} & \textbf{Dataset} & \textbf{MOTA (\%)} \\
\hline
\cite{peng2020tpm} & 2020 & Tracklet-plane matching process to resolve confusing short tracklets & MOT16, MOT17 & 50.9, 52.4 \\
\multirow{2}{*}{\cite{nguyen2022lmgp}} & \multirow{2}{*}{2021} & CenterTrack \cite{zhou2020tracking} and DG-Net \cite{zheng2019joint} as tracking graph and GAEC+KLj \cite{keuper2015efficient} heuristic solver for lifted multicut solver & WILDTRACK, PETS-09, Campus & \multirow{2}{*}{97.1, 74.2, 77.5} \\
\cite{ma2021deep} & 2020 & CNN for Apprearance extraction, LSTM and RNN for Motion modelling & Duke-MTMCT, MOT16 & 73.5, 55.0 \\
\cite{stadler2021improving} & 2021 & Regression based two stage tracking & MOT16, MOT17, MOT20 & 66.8, 65.1, 61.2 \\
\multirow{2}{*}{\cite{wang2022split}} & \multirow{2}{*}{2021} & Tracklet splitter splits potential false ids and connector connects pure tracks to trajectory & \multirow{2}{*}{MOT17, MOT20} & \multirow{2}{*}{61.5, 54.6} \\
\multirow{2}{*}{\cite{yu2022towards}} & \multirow{2}{*}{2022} & Learnable view sampling for similarity-guided feature fusion and Trajectory-center memory bank for re-identification & MOT15, MOT16, MOT17, MOT20 & 62.1, 74.3, 73.5, 63.2 \\
\hline
\end{tabular}
\end{center}
\label{tabtracklet}
\end{table*}
\subsection{MOT Challenge Benchmark}
As of now, the MOT challenge has 17 datasets for object tracking, which include MOT15 \cite{leal2015motchallenge}, MOT16 \cite{milan2016mot16}, MOT20, \cite{dendorfer2020mot20} and others. The MOT15 benchmark contains Venice, KITTI, ADL-Rundle, ETH-Pescross, ETH-Sunnyday, PETs, TUD-Crossing datasets. This benchmark is filmed in an unconstrained environment with both static and moving cameras. MOT16 and MOT17 are basically more updated benchmarks from MOT15 with high accuracy of ground truth and strictly followed protocols. MOT20 is a pedestrian detection challenge. This benchmark has 8 challenging video sequences (4 train, 4 test) in unconstrained environments \cite{dendorfer2020mot20}. In addition to object tracking, MOTS dataset has segmentation tasks too \cite{voigtlaender2019mots}. In general, the tracking dataset has a bounding box with a unique identifier for objects in a frame. But in MOTS, every object has a segmentation mask also. TAO \cite{dave2020tao} dataset has a huge size due to tracking each and every object in a frame. There is a dataset called Head Tracking 21. The task for this benchmark is to track the head of every pedestrian. STEP dataset has segmented and tracked every pixel. There are some other datasets; those are included in table \ref{tab1}. Frequency of the datasets used in the papers those we review is shown in Chart \ref{chart}. From the chart, we can see that MOT17 dataset is used more frequently than other datasets.
\begin{table}[htbp]
\caption{Statistics of publicly available datasets}
\begin{center}
\begin{tabular}{>{\centering\arraybackslash}p{60pt} >{\centering\arraybackslash}p{35pt} >{\centering\arraybackslash}p{30pt} >{\centering\arraybackslash}p{32pt} >{\centering\arraybackslash}p{32pt} }
\hline
\multirow{2}{*}{\bf Dataset} & {\bf No. of Frames} & {\bf Size (Bytes)} & {\bf Published year} & \multirow{2}{*}{\bf Reference} \\
\hline
DanceTrack$^{\mathrm{*}}$ & 105000 & 16.5G & 2022 & \cite{sun2021dancetrack} \\
TAO VOS & - & 2.4G & 2021 & \cite{voigtlaender2021reducing} \\
Head Tracking 21 & 11464 & 4.1G & 2021 & \cite{sundararaman2021tracking} \\
STEP-ICCV21 & 2075 & 380M & 2021 & \cite{weber2021step} \\
MOTSynth-MOT\_CVPR22 & \multirow{2}{*}{1381119} & \multirow{2}{*}{-} & \multirow{2}{*}{2021} & \multirow{2}{*}{\cite{fabbri2021motsynth}} \\
MOTSynth-MOTS\_CVPR22 & \multirow{2}{*}{1378244} & \multirow{2}{*}{-} & \multirow{2}{*}{2021} & \multirow{2}{*}{\cite{fabbri2021motsynth}} \\
MOT20 & 13410 & 5.0G & 2020 & \cite{dendorfer2020mot20} \\
3D-ZeF20 & 14398 & 14.0G & 2020 & \cite{pedersen20203d} \\
TAO & 4447038 & 347G & 2020 & \cite{dave2020tao} \\
CTMC-v1 & 152498 & 768M & 2020 & \cite{anjum2020ctmc} \\
OWTB & 4447038 & 350G & 2020 & \cite{dave2020tao} \\
MOTS & 5906 & 783.5M & 2019 & \cite{voigtlaender2019mots} \\
MOT16 & 11235 & 1.9G & 2016 & \cite{milan2016mot16} \\
MOT17 & 33705 & 5.5G & 2016 & \cite{milan2016mot16} \\
PETS 2016 & - & - & 2016 & \cite{patino2016pets} \\
MOT15 & 11283 & 1.3G & 2015 & \cite{leal2015motchallenge} \\
KITTI Tracking & - & 15G & 2012 & \cite{geiger2012we} \\
TUD Multiview Pedestrains & \multirow{2}{*}{179} & \multirow{2}{*}{387M} & \multirow{2}{*}{2010} & \multirow{2}{*}{\cite{andriluka2010monocular}} \\
PETS 2009 & - & 4.9G & 2009 & \cite{ferryman2009pets2009} \\
TUD Campus, Crossing & \multirow{2}{*}{272}& \multirow{2}{*}{100M}& \multirow{2}{*}{2008}& \multirow{2}{*}{\cite{andriluka2008people}}\\
\hline
\multicolumn{5}{l}{$^{\mathrm{*}}$This dataset has scenes indoors only.}
\end{tabular}
\label{tab1}
\end{center}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width = 0.48\textwidth]{Figures/chart.pdf}
\caption{The number of papers for each dataset}
\label{chart}
\end{figure}
\subsection{Benchmark Results}
In table \ref{tab2} we listed down the top 15 public results of MOT20 \cite{dendorfer2020mot20} dataset.
We added KITTI \cite{geiger2012we} benchmark state of art results in table \ref{tab3} \& \ref{tab4}.
\begin{table*}[htbp]
\centering
\caption{Top 15 public results of MOT20 \cite{dendorfer2020mot20} dataset}
\begin{tabular}{cccccccccccc}
\hline
{\bf Tracker} & {\bf MOTA} & {\bf HOTA} & {\bf Rcll} & {\bf Prcn} & {\bf AssA} & {\bf AssRe} & {\bf AssPr} & {\bf DetRe} & {\bf DetPr} & {\bf LocA} & {\bf Hz} \\
\hline
GMOTv2 & 77.07\% & 61.48\% & 85.73\% & 91.14\% & 59.32\% & 66.48\% & 72.19\% & 71.70\% & 76.23\% & 83.67\% & 0.90 \\
MrMOT & 67.72\% & 53.90\% & 74.62\% & 92.23\% & 52.89\% & 58.76\% & 71.74\% & 60.47\% & 74.74\% & 81.71\% & 16.90 \\
OUTrack\_fm\_p \cite{liu2022online} & 65.43\% & 52.09\% & 73.37\% & 90.85\% & 50.74\% & 53.95\% & 77.81\% & 59.42\% & 73.57\% & 81.44\% & 5.10 \\
GGDA & 64.38\% & 53.39\% & 78.62\% & 85.14\% & 52.41\% & 58.07\% & 70.79\% & 63.97\% & 69.28\% & 81.56\% & 0.40 \\
GNMOT\_IA & 63.11\% & 52.49\% & 77.64\% & 84.95\% & 51.26\% & 56.77\% & 70.55\% & 63.22\% & 69.18\% & 81.50\% & 3.60 \\
TransCtr \cite{xu2021transcenter} & 61.04\% & 43.50\% & 71.42\% & 88.25\% & 36.10\% & 44.53\% & 58.99\% & 59.15\% & 73.10\% & 81.70\% & 1.00 \\
GATracker & 60.70\% & 50.57\% & 71.50\% & 88.44\% & 48.23\% & 51.13\% & 78.84\% & 59.46\% & 73.55\% & 81.85\% & 2.20 \\
MPTC \cite{stadler2021multi} & 60.61\% & 48.48\% & 70.24\% & 88.91\% & 46.53\% & 51.61\% & 66.40\% & 56.80\% & 71.99\% & 81.36\% & 0.70 \\
TMOH \cite{stadler2021improving} & 60.13\% & 48.95\% & 67.94\% & 90.23\% & 48.41\% & 52.86\% & 71.02\% & 54.84\% & 72.91\% & 81.24\% & 0.60 \\
STTMOT & 60.10\% & 51.72\% & 67.94\% & 90.32\% & 53.63\% & 57.68\% & 75.95\% & 55.09\% & 73.24\% & 81.36\% & 4.00 \\
OCSORTpublic \cite{cao2022observation} & 59.90\% & 54.30\% & 60.86\% & 98.61\% & 59.46\% & 65.12\% & 76.62\% & 51.56\% & 83.54\% & 85.05\% & 27.60 \\
xumot & 59.65\% & 54.03\% & 81.31\% & 79.81\% & 55.40\% & 60.93\% & 72.76\% & 65.58\% & 64.37\% & 80.73\% & 11.90 \\
mfi\_tst \cite{yang2022online} & 59.25\% & 47.11\% & 66.61\% & 90.51\% & 46.18\% & 50.03\% & 73.88\% & 53.11\% & 72.16\% & 79.91\% & 0.50 \\
MOTer \cite{xu2021transcenter} & 59.11\% & 43.56\% & 71.53\% & 86.12\% & 36.98\% & 45.21\% & 58.84\% & 58.94\% & 70.96\% & 81.22\% & 1.00 \\
ApLift \cite{hornakova2021making} & 58.89\% & 46.61\% & 62.75\% & 94.82\% & 45.23\% & 48.15\% & 76.78\% & 51.30\% & 77.51\% & 82.15\% & 0.40 \\
\hline
\end{tabular}
\label{tab2}
\end{table*}
\begin{table*}[htbp]
\centering
\caption{Top 15 public result of KITTI Car \cite{geiger2012we} dataset}
\begin{tabular}{c c c c c c c c c c}
\hline
{\bf Method} & {\bf HOTA} & {\bf DetA} & {\bf AssA} & {\bf DetRe} & {\bf DetPr} & {\bf AssRe} & {\bf AssPr} & {\bf LocA} & {\bf MOTA}\\ \hline
PC-TCNN \cite{wu2021tracklet} & 80.90\% & 78.46\% & 84.13\% & 84.22\% & 84.58\% & 87.46\% & 90.47\% & 87.48\% & 91.70\%\\
PermaTrack \cite{tokmakov2021learning} & 78.03\% & 78.29\% & 78.41\% & 81.71\% & 86.54\% & 81.14\% & 89.49\% & 87.10\% & 91.33\%\\
PC3T \cite{wu20213d} & 77.80\% & 74.57\% & 81.59\% & 79.19\% & 84.07\% & 84.77\% & 88.75\% & 86.07\% & 88.81\%\\
OC-SORT \cite{cao2022observation} & 76.54\% & 77.25\% & 76.39\% & 80.64\% & 86.36\% & 80.33\% & 87.17\% & 87.01\% & 90.28\%\\
Mono\_3D\_KF \cite{reich2021monocular} & 75.47\% & 74.10\% & 77.63\% & 78.86\% & 82.98\% & 80.23\% & 88.88\% & 85.48\% & 88.48\%\\
DeepFusion-MOT \cite{wang2022deepfusionmot} & 75.46\% & 71.54\% & 80.05\% & 75.34\% & 85.25\% & 82.63\% & 89.77\% & 86.70\% & 84.63\%\\
EagerMOT \cite{kim2021eagermot} & 74.39\% & 75.27\% & 74.16\% & 78.77\% & 86.42\% & 76.24\% & 91.05\% & 87.17\% & 87.82\%\\
DEFT \cite{chaabane2021deft} & 74.23\% & 75.33\% & 73.79\% & 79.96\% & 83.97\% & 78.30\% & 85.19\% & 86.14\% & 88.38\%\\
Opm-NC2 \cite{jiang2022fast} & 73.19\% & 73.27\% & 73.77\% & 80.98\% & 81.67\% & 77.05\% & 89.84\% & 87.31\% & 84.21\%\\
mono3DT \cite{hu2019joint} & 73.16\% & 72.73\% & 74.18\% & 76.51\% & 85.28\% & 77.18\% & 87.77\% & 86.88\% & 84.28\%\\
LGM \cite{wang2021track} & 73.14\% & 74.61\% & 72.31\% & 80.53\% & 82.16\% & 76.38\% & 84.74\% & 85.85\% & 87.60\%\\
CenterTrack \cite{zhou2020tracking} & 73.02\% & 75.62\% & 71.20\% & 80.10\% & 84.56\% & 73.84\% & 89.00\% & 86.52\% & 88.83\%\\
QD-3DT \cite{hu2021monocular} & 72.77\% & 74.09\% & 72.19\% & 78.13\% & 85.48\% & 74.87\% & 89.21\% & 87.16\% & 85.94\%\\
TrackMPNN \cite{rangesh2021trackmpnn} & 72.30\% & 74.69\% & 70.63\% & 80.02\% & 83.11\% & 73.58\% & 87.14\% & 86.14\% & 87.33\%\\
DiTMOT \cite{wang2021ditnet} & 72.21\% & 71.09\% & 74.04\% & 75.98\% & 83.28\% & 76.57\% & 89.97\% & 86.15\% & 84.53\%\\
\hline
\end{tabular}
\label{tab3}
\end{table*}
\begin{table*}[htbp]
\centering
\caption{Top 15 public result of KITTI Pedestrian \cite{geiger2012we} dataset}
\begin{tabular}{cccccccccc}
\hline
{\bf Method} & {\bf HOTA} & {\bf DetA} & {\bf AssA} & {\bf DetRe} & {\bf DetPr} & {\bf AssRe} & {\bf AssPr} & {\bf LocA} & {\bf MOTA}\\ \hline
OC-SORT \cite{cao2022observation} & 54.69\% & 50.82\% & 59.08\% & 55.68\% & 70.94\% & 64.09\% & 73.36\% & 78.52\% & 65.14\%\\
SRK\_ODESA(hp) \cite{mykheievskyi2020learning} & 50.87\% & 53.43\% & 48.78\% & 57.79\% & 72.90\% & 53.45\% & 71.33\% & 78.81\% & 68.04\%\\
PermaTrack \cite{tokmakov2021learning} & 48.63\% & 52.28\% & 45.61\% & 57.40\% & 71.03\% & 49.63\% & 73.28\% & 78.57\% & 65.98\%\\
Opm-NC2 \cite{jiang2022fast} & 46.55\% & 46.82\% & 46.68\% & 53.01\% & 59.38\% & 50.84\% & 65.82\% & 72.07\% & 56.05\%\\
3D-TLSR \cite{nguyen20203d} & 46.34\% & 42.03\% & 51.32\% & 44.51\% & 71.14\% & 54.45\% & 73.11\% & 76.87\% & 53.58\%\\
TuSimple \cite{choi2015near} & 45.88\% & 44.66\% & 47.62\% & 47.92\% & 69.51\% & 52.04\% & 69.88\% & 76.43\% & 57.61\%\\
CAT \cite{nguyen2019confidence} & 45.65\% & 42.43\% & 49.55\% & 45.89\% & 67.79\% & 53.20\% & 71.97\% & 75.90\% & 51.96\%\\
MPNTrack \cite{braso2020learning} & 45.26\% & 43.74\% & 47.28\% & 53.62\% & 58.30\% & 52.18\% & 68.47\% & 75.93\% & 46.23\%\\
NC2 \cite{jiang2021new} & 44.30\% & 42.31\% & 46.75\% & 52.97\% & 52.43\% & 50.91\% & 65.83\% & 72.08\% & 44.18\%\\
SRK\_ODESA(mp) \cite{mykheievskyi2020learning} & 43.73\% & 53.73\% & 36.05\% & 58.01\% & 73.19\% & 40.05\% & 69.44\% & 78.91\% & 67.31\%\\
Be-Track \cite{dimitrievski2019behavioral} & 43.36\% & 39.99\% & 47.23\% & 43.00\% & 69.03\% & 51.28\% & 69.60\% & 76.78\% & 50.85\%\\
Mono\_3D\_KF \cite{reich2021monocular} & 42.87\% & 40.13\% & 46.31\% & 46.02\% & 59.91\% & 52.86\% & 63.50\% & 74.03\% & 45.44\%\\
MDP \cite{xiang2015learning} & 42.76\% & 39.23\% & 47.13\% & 43.83\% & 63.02\% & 50.91\% & 71.04\% & 75.15\% & 47.02\%\\
Quasi-Dense \cite{pang2021quasi} & 41.12\% & 44.81\% & 38.10\% & 48.55\% & 70.39\% & 41.02\% & 72.47\% & 77.87\% & 55.55\%\\
QD-3DT \cite{hu2021monocular} & 41.08\% & 44.01\% & 38.82\% & 48.96\% & 67.19\% & 42.09\% & 72.44\% & 77.38\% & 51.77\%\\
\hline
\end{tabular}
\label{tab4}
\end{table*}
\section{Conclusion}
In this paper, we have tried to compact a summary and review of recent trends in computer vision in MOT. We have tried to analyze the limitations and significant challenges. At the same time, we have found that besides some major challenges like occlusion handling, id switching, there are also some minor challenges that may sit in the driving position in terms of better precision. We have added them too. Brief theories related to each approach are included in this study. We have tried to focus on each approach equally. We have added some popular benchmark datasets along with their insights. We have included some possibilities for future direction based on recent MOT trends. Our observation of this study is that recently researchers have focused more on transformer-based architecture. This is because of the contextual information memorization of transformer. As transformer is resource hungry to get better accuracy with a lightweight architecture, focusing on a specific module is necessary per our study. Finally, we hope this study will serve as complementary to a researcher in the field to start the journey in the field of Multiple Object Tracking.
|
1,314,259,992,972 | arxiv | \section{Introduction}
The {\em order} of a point $x$ in a topological space is the number of connected components
we obtain after removing $x$. A {\em ramification point} is a point which has order at least~3. An {\em endpoint} is a point of order 1. A {\em continuum} is a compact connected topological space.
A {\em dendrite} is a locally connected continuum that contains no simple closed curve. All dendrites we consider
in this article will be metrizable.
A {\em Wa\.zewski dendrite} $W_\omega$ is a dendrite such that each ramification point of $W_\omega$ is of order~$\omega$ and each arc $I$ contained in $W_\omega$
contains a ramification point.
Moreover, for every $P\subseteq\{3,4,\ldots,\omega\}$, there exists a {\em generalized Wa\.zewski dendrite} $W_P$, that is, a dendrite
such that each ramification point of $W_P$ is of order that belongs to $P$ and for every $p\in P$ and an arc $I$ contained in $W_P$, $I$
contains a ramification point of order $p$. For every $P\subseteq\{3,4,\ldots,\omega\}$, a generalized
Wa\.zewski dendrite
is unique up to homeomorphism, see Charatonik-Dilks \cite[Theorem 6.2]{CD}.
Duchesne-Monod \cite{DM} studied structural properties of homeomorphism groups of generalized
Wa\.zewski dendrites, in particular, they showed that these groups are simple.
The homeomorphism group of a generalized Wa\.zewski dendrite is isomorphic (as a topological group)
to the automorphism group of a certain Fra\"{i}ss\'{e}-HP structure (i.e. the Fra\"{i}ss\'{e} limit of a family of finite
first-order structures, which has the joint embedding and the amalgamation properties, but not necessarily the hereditary property), which we now describe.
Let $P$ be fixed and consider $W_P$. Let $M_P$ be the set of all ramification points of $W_P$. Let $\mathcal{L}_P$ be the
first-order language that consists
of a 4-ary relation symbol $D$ and of unary relation symbols $K_p$ for every $p\in P$.
We let ${M_P}$ to be the structure with universe~$M_P$, $D^{{M_P}}(a,b,c,d)$ iff the path in $W_P$
connecting $a$ and $b$ and the path connecting $c$ and~$d$ do not intersect
(we emphasize that we allow here trivial paths, i.e. we allow $a=b$ or $c=d$),
and let $K_p^{{M_P}}(a)$ iff $a$ is a ramification point of the order equal to $p$.
Instead of coding the tree structure using the $D$ relation, we could use the ternary betweenness relation
$B$, where $B(a,b,c)$ iff $b$ belongs to the path $ac$. Indeed, $B(a,b,c)$ holds iff $D(a,c,b,b)$ does
not hold, and $D(a,b,c,d)$ does not hold iff there exists $e$ such that $B(a,e,b)$ and $B(c,e,d)$.
Later on, we will also work with a $C$ relation, which will be defined using the $D$ relation,
moreover, the $D$ relation is used to describe boron trees, see \cite{J},
therefore we decided to work in this article with the $D$ relation rather than with the $B$
relation.
Propositions 2.4 and 6.1 in \cite{DM} imply:
\begin{proposition}\label{unispo}
The homeomorphism group of the generalized Wa\.zewski dendrite $W_P$, equipped with the uniform metric, is isomorphic (as a topological group)
to the automorphism group of $M_P$, equipped with the pointwise convergence metric.
\end{proposition}
A {\em tree} is an acyclic connected undirected graph.
For a tree $T$ we denote by $V(T)$ the set of vertices and by $E(T)$ the set of edges of $T$.
A {\em degree } of a vertex in a graph is the number of edges that come out of that vertex. An {\em endpoint} is a vertex of degree~1.
A~{\em path} is a tree such that each vertex either is an endpoint or it has degree 2.
Note that for any two vertices in a tree there is exactly one path joining them.
A path joining vertices $a$ and $b$ we will often denote by $ab$.
A {\em rooted tree} is a tree with a distinguished point, which we call the {\em root}.
On a rooted tree $T$ with the root $r$ we consider
the tree order $\leq_{T}$ letting $x\leq_{T}y$ iff $x$ belongs to the path $ry$.
A {\em branch} in a rooted tree is a path $ra$, where $r$ is the root and $a$ is an endpoint.
The {\em meet } of $a,b\in T$ is the greatest lower bound of $a$ and $b$ with respect to $\leq_{T}$.
In a rooted tree we can talk about the {\em height} of each vertex.
The root has the height equal to 0 and
the height of $x\in T$ is taken to be the maximum plus 1 of heights of $\{v\in T\colon v<_T x\}$.
The {\em height } of a rooted tree $T$ is the maximum of the heights of all of its vertices,
we denote it by $ht(T)$.
Note that the height of $x\in T$ is equal to the length of the path $rx$,
where the {\em length} of a path is defined to be the number of edges in the path.
A {\em successor} of a vertex $x$ is any point $y\neq x$ such that $x\leq_T y$. A vertex $y$ is an {\em immediate successor} of a vertex $x$ if it is a successor of $x$
and there is no successor $w\neq y$ of $x$ such that $x\leq_T w\leq_T y $.
Let $\mathcal{F}_P$ be the family of all finite structures in the language $\mathcal{L}_P$ such that the universe is a finite tree and the degree of each vertex is different from 2.
If $A\in\mathcal{F}_P$, we let $D^{A}(a,b,c,d)$ iff the path $ab$ and the path $cd$, do not intersect.
Take $K_p$ such that for every $a\inA$ there is exactly one $p\in P$ such that $K^{A}_p(a)$, and if $K^{A}_p(a)$ then
the degree of $a$ is not greater than $p$.
A first-order structure $M$ is {\em ultrahomogeneous} with respect to a family of finite substructures $\mathcal{F}$ if for any finite substructures
$A,B\subseteq M$, $A,B\in\mathcal{F}$, and an isomorphism $p\colon A\to B$, there is an automorphism of $M$
extending $p$.
Proposition 6.1 in \cite{DM} together with Proposition \ref{unispo} imply:
\begin{proposition}
For every $P\subseteq\{3,4,\ldots,\omega\}$, the structure $M_P$ is ultrahomogeneous with respect to $\mathcal{F}_P$.
\end{proposition}
The proposition above implies that $\mathcal{F}_P$ has the joint embedding property and the amalgamation property. Note that $\mathcal{F}_P$ does not have the hereditary property.
Moreover, as additionally for every finite subset $X\subseteq{M_P}$ there is $A\in\mathcal{F}_P$ such that $X\subseteq A\subseteq {M_P}$, we have that ${M_P}$
is the Fra\"{i}ss\'{e} limit of $\mathcal{F}_P$.
\begin{remark}\label{steps}
Let $i\colon (S, D^S)\to (T, D^T)$ be an embedding of trees $S$ and $T$ in which each vertex has degree $\neq 2$. Then every edge in $S$ is mapped to a path in $T$ and
$T$ is obtained from $S$ in a sequence of the following simple steps:
\noindent 1. Start with a tree $T'$. Pick an edge $[a,b]$ in $T'$. Let $c$ and $d$ be points not in $T'$. Get $S'$ by removing edge $[a,b]$,
and by adding points $c$ and $d$, and edges $[a,c]$, $[c,b]$ and $[c,d]$.
\noindent 2. Start with a tree $T'$. Pick an endpoint $e$ in $T'$. Let $c$ and $d$ be points not in $T'$. Get $S'$ by
adding points $c$ and $d$, and edges $[e,c]$ and $[e,d]$.
\noindent 3. Start with a tree $T'$. Pick an vertex $v$ in $T'$ which is not an endpoint.
Let $c$ be a point not in $T'$. Get
$S'$ by adding the point $c$ and the edge $[v, c]$.
\end{remark}
\begin{remark}\label{edge}
Note that the relation $D$ remembers which pairs of vertices are joined by an edge.
Given a tree $T$ and $a,b\in T$, $a\neq b$. Then there is an edge between $a$ and $b$ iff for every $c\in T$, $c\neq a,b$ we have that $c$ does not belong to the path $ab$ iff
for every $c\in T$, $c\neq a,b$, $D^T(a,b,c,c)$ holds.
\begin{remark}
Let $T$ be a tree and let $E$ be the set of endpoints of $T$. Then $D^T\restriction E$ on the set $E$ is an example of a $D$-relation, as defined in \cite[Section 22]{AN}. Moreover, $D^T$ on the tree $T$ satisfies (D1)-(D3) in the definition of a $D$-relation, but not (D4).
\end{remark}
\end{remark}
\section{The universal minimal flow - preliminaries}
Our goal is to compute universal minimal flows of the homeomorphism groups $H(W_P)$, equivalently, of the automorphism groups ${\rm Aut}(M_P)$.
We will work in the framework provided by Kechris-Pestov-Todorcevic. Let us recall relevant definitions and theorems. The presentation below is essentially
copied from \cite{BK}, Section 3.6. Lemma \ref{preco}, Theorem \ref{iden}, and Corollary \ref{ident} are proved there.
A topological group $G$ is {\em extremely amenable} if every $G$-flow has a fixed point.
A~{\em coloring} of a set $X$ is any function $c\colon X\to \{1,2,\ldots,r\}$, for some $r\geq 2$;
we say that $Y\subseteq X$ is {\em $c$-monochromatic} (or just {\em monochromatic}) if $r\restriction Y$ is constant.
Let $\mathcal{G}$ be a family of finite structures in a language $\mathcal{L}$.
For $A,B\in \mathcal{G}$ write $A\leq B$ if $A$ embeds into $B$.
For $A,B$ in $\mathcal{G}$, let ${B \choose A}$~denotes the set of all embeddings of $A$ into $B$.
We say that $A\in\mathcal{G}$ is a
{\em Ramsey object} if for every $B\in \mathcal{G}$ with $A\leq B$ and every integer $r\geq 2$ there exists $C\in \mathcal{G}$ such that for every coloring $c\colon {C \choose A} \to\{1,2,\ldots,r\}$ there exists $h\in {C \choose B}$ such that
$\{ h\circ f\colon f\in {B \choose A} \}$ is monochromatic. Note that to check that $A$ is a Ramsey object it suffices to check it only for $r=2$.
We say that $\mathcal{G}$ is a {\em Ramsey class} (or that it has {\em Ramsey property}) if every structure in $\mathcal{G}$ is a Ramsey object.
A structure $A\in\mathcal{G}$ is {\em rigid} if it has trivial automorphism group.
Kechris-Pestov-Todorcevic \cite{KPT} worked with
Fra\"{i}ss\'{e} families and their ordered Fra\"{i}ss\'{e} expansions, their work was generalized by Nguyen Van Th\'e \cite{NVT} to
Fra\"{i}ss\'{e} families and to arbitrary relational Fra\"{i}ss\'{e} expansions. The Kechris-Pestov-Todorcevic correspondence remains true for Fra\"{i}ss\'{e}-HP families,
which was checked by several people, and it appears in \cite{Z}, see also \cite{BK}.
\begin{theorem}[Kechris-Pestov-Todorcevic \cite{KPT}, see Theorem 5.1 in \cite{Z}] \label{kpt1}
Let $\mathcal{G}$ be a Fra\"{i}ss\'{e}-HP family, let $\mathbb{G}$ be its Fra\"{i}ss\'{e} limit, and let $G={\rm Aut}(\mathbb{G})$.
Then the following are equivalent:
\begin{enumerate}
\item The group $G$ is extremely amenable.
\item The family $\mathcal{G}$ is a Ramsey class and it consists of rigid structures.
\end{enumerate}
\end{theorem}
Let $\mathcal{G}$ be a Fra\"{i}ss\'{e}-HP family in a language $\mathcal{L}$, let $\mathbb{G}$ be its Fra\"{i}ss\'{e} limit, and let $G={\rm Aut}(\mathbb{G})$.
Let $\mathcal{G}^*$ be a Fra\"{i}ss\'{e}-HP family in a language $\mathcal{L}^*\supseteq \mathcal{L}$, $\mathcal{L}^*\setminus \mathcal{L}$ relational,
such that the map defined on $\mathcal{G}^*$ and given by $A^*\mapsto A^*\restriction \mathcal{L}$ is onto $\mathcal{G}$.
In that case we say that $A^*$ is an {\em expansion} of $A^*\restriction \mathcal{L}$ and that $A^*\restriction \mathcal{L}$
is a {\em reduct} of $A^*$, and that $\mathcal{G}^*$ is an {\em expansion} of $\mathcal{G}$.
Let $\mathbb{G}^*$ be the Fra\"{i}ss\'{e} limit of $\mathcal{G}^*$, and let $G^*={\rm Aut}(\mathbb{G}^*)$.
We say that the expansion $\mathcal{G}^*$ of $\mathcal{G}$ is {\em reasonable} if
for any $A,B\in\mathcal{G}$, an embedding $\alpha\colon A\to B$ and an expansion $A^*\in\mathcal{G}^*$ of $A$,
there is an expansion $B^*\in\mathcal{G}^*$ of $B$ such that $\alpha\colon A^*\to B^*$ is an embedding. It is
{\em precompact} if
for every $A\in\mathcal{G}$ there are only finitely many $A^*\in\mathcal{G}^*$ such that $A^*\restriction \mathcal{L}= A$.
We say that $\mathcal{G}^*$ has the {\em expansion property} relative to $\mathcal{G}$ if for any $A^*\in\mathcal{G}^*$ there is
$B\in\mathcal{G}$ such that for any expansion $B^*\in\mathcal{G}^*$, there is an embedding $\alpha\colon A^*\to B^*$.
The following proposition explains the importance of the notion of reasonability.
\begin{proposition}[\cite{KPT}, \cite{NVT}, see Proposition 5.3 in \cite{Z}]\label{kpt_reas}
The expansion $\mathcal{G}^*$ of $\mathcal{G}$ is reasonable if and only if $\mathbb{G}^*\restriction \mathcal{L}= \mathbb{G}$.
\end{proposition}
We say that $\mathcal{G^*}$ has the {\em relative HP} (the relative hereditary property) with respect to~$\mathcal{G}$
if for every $A, B\in\mathcal{G}$ such that $A$ is a substructure of $B$ and for $B^*\in\mathcal{G}^*$, an expansion of $B$,
we have $B^*\restriction A\in \mathcal{G}^*$. This is equivalent to saying that
for any $A\in\mathcal{G}$ and an embedding $i\colon A\to \mathbb{G}$ there is an expansion
$A^*\in\mathcal{G}^*$ of $A$ such that $i\colon A^*\to \mathbb{G}^*$ is an embedding.
The relative HP property is used to show that when an expansion $\mathcal{G}^*$ of $\mathcal{G}$ is precompact, then ${\rm Aut}(\mathbb{G})/{\rm Aut}(\mathbb{G}^*)$ is precompact in the quotient of the right uniformity, the proof is
contained in Section 3.6 in \cite{BK}.
\begin{lemma}\label{preco}
Suppose that $\mathcal{G}^*$ is a reasonable precompact expansion of $\mathcal{G}$
and that the relative HP holds. Then
the right uniform space ${\rm Aut}(\mathbb{G})/{\rm Aut}(\mathbb{G}^*)$ is precompact.
\end{lemma}
Below $(\mathbb{G}, \vec{R})$ denotes an expansion of $\mathbb{G}$ to a structure in $\mathcal{L}^*$.
Instead of $(\mathbb{G}, \vec{R})$ we will often just write $\vec{R}$.
Define
\begin{equation*}
\begin{split}
X_{\mathbb{G}^*}=&\{ \vec{R}:
{\rm\ for\ every\ } A\in\mathcal{G},
{\rm\ and\ an\ embedding\ }
i\colon A\to \mathbb{G} {\rm\ there \ exists\ } \\
& A^*\in\mathcal{G}^*, {\rm such \ that\ } i\colon A^*\to(\mathbb{G},\vec{R})
{\rm \ is\ an\ embedding }\}.
\end{split}
\end{equation*}
The relative HP implies that
the space $X_{\mathbb{G}^*}$ contains $\mathbb{G}^*$.
We make $X_{\mathbb{G}^*}$ a topological space by declaring sets
\[ V_{i, A^*}=\{\vec{R}\in X_{\mathbb{G}^*} : \text{the map }i\colon A^*\to (\mathbb{G},\vec{R}) \text{ is an embedding} \}, \]
where $i\colon A\to \mathbb{G}$ is an embedding, $A^*\in\mathcal{G}^*$, and $A^*\restriction \mathcal{L}= A$, to be open.
The group ${\rm Aut}(\mathbb{G}^*)$ acts continuously on $X_{\mathbb{G}^*}$ via
\[ g\cdot \vec{R}(\bar{a})=\vec{R}(g^{-1}(\bar{a})).\]
Reasonability and precompactness of the expansion $\mathcal{G}^*$ of $\mathcal{G}$ imply that the space $X_{\mathbb{G}^*}$ is compact and
zero-dimensional.
{\bf{From now on till the end of this section,}} we will assume that the expansion $\mathcal{G}^*$ of $\mathcal{G}$ is reasonable, precompact, and satisfies the relative HP.
\begin{theorem}[\cite{KPT}, \cite{NVT}, see Proposition 5.5 in \cite{Z}]\label{kpt_minim}
The following are equivalent:
\begin{enumerate}
\item The flow $G\curvearrowright X_{\mathbb{G}^*} $ is minimal.
\item The family $\mathcal{G}^*$ has the expansion property relative to $\mathcal{G}$.
\end{enumerate}
\end{theorem}
\begin{theorem}[Kechris-Pestov-Todorcevic \cite{KPT}, Nguyen Van Th\'e \cite{NVT}, see Theorem 5.7 in \cite{Z}]\label{kpt2}
The following are equivalent:
\begin{enumerate}
\item The flow $G\curvearrowright X_{\mathbb{G}^*} $ is the universal minimal flow of $G$.
\item The family $\mathcal{G}^*$ is a rigid Ramsey class and has the expansion property relative to~$\mathcal{G}$.
\end{enumerate}
\end{theorem}
A proof of Theorem \ref{iden} is contained in Section 3.6 in \cite{BK}.
Let $\vec{R}^\mathbb{G}$ be such that $\mathbb{G}^*=(\mathbb{G}, \vec{R}^\mathbb{G})$.
\begin{theorem}\label{iden}
The map $g{\rm Aut}(\mathbb{G}^*)\mapsto g\cdot \vec{R}^\mathbb{G}$ from ${\rm Aut}(\mathbb{G})/{\rm Aut}(\mathbb{G}^*)$ to $X_{\mathbb{G}^*}$ is a uniform isomorphism.
\end{theorem}
We will say that flows $G\curvearrowright X$ and $G\curvearrowright Y$ are {\em isomorphic} if there is a homeomorphism from $X$ onto $Y$ which is a $G$-map.
\begin{corollary}\label{ident}
The flow $G\curvearrowright \reallywidehat{{\rm Aut}(\mathbb{G})/{\rm Aut}(\mathbb{G}^*)}$ is isomorphic to the flow $G\curvearrowright X_{\mathbb{G}^*} $.
\end{corollary}
\section{The universal minimal flow - construction}
In this section, we show:
\begin{theorem}\label{main}
For any $P\subseteq\{3,\ldots,\omega\}$ there is a reasonable Fra\"{i}ss\'{e}-HP expansion $\mathcal{F}^*_P$ of $\mathcal{F}_P$, which has the relative HP, the expansion, and the Ramsey properties. In the case when $P$ is finite, this expansion $\mathcal{F}^*_P$ is also precompact.
\end{theorem}
Then using the Kechris-Pestov-Todorcevic correspondence and Proposition \ref{unispo}, we obtain a description of the universal minimal flow of the homeomorphism group of the generalized Wa\.zewski dendrite $W_P$, for all finite $P$. In particular, we will obtain that this universal minimal flow is metrizable, when $P$ is finite.
This answers a question of Duchesne asked
during his talk
at the Workshop ``Structure and Geometry of Polish groups'' in
Oaxaca in 06/2017.
In the special case, when $P=\{\omega\}$, the universal minimal flow of $H(W_{\{\omega\}})$, independently of our work, was identified by Duchesne in~\cite{D2}.
Given $\mathcal{F}_P$ in the language $\mathcal{L}_P$, we first construct a family $\mathcal{T}^*_P$ of rooted trees with ordered and labeled branches, and then we construct
the required family $\mathcal{F}^*_P$ that is Ramsey and has the expansion property with respect to $\mathcal{F}_P$.
{\bf{The family $\mathbf{\mathcal{T}^*_P}$.}}
Take $\mathcal{L}_{\mathcal{T}^*_P}=\mathcal{L}_P\cup \{ C, \prec, G_1, G_2\ldots\}$, where $C$ is a ternary relation symbol, $\prec, G_1,G_2,\ldots$
are binary relation symbols. If $P$ is finite, it suffices to take $G_1,\ldots, G_{m-1}$, where
$m=\max(P\setminus\{\omega\})$.
Let $A\in\mathcal{F}_P$.
\noindent{\bf{Step 1:}} Choosing the root for $A$.
Let $x$ be an edge of $A$ or a vertex of $A$ such that its degree is strictly less than $p$ satisfying $K_p^A(x)$.
In the case when $x$ is a vertex, denote $r=x$ and consider $A$ with the distinguished point $r$, which we call the root. Denote this rooted tree by $T_{A,r}$.
In the case when $x=[a,b]$ is an edge, remove $x$ from $A$, take a new point $r$ and add edges $[a,r]$ and $[r,b]$. The obtained tree with the distinguished
point $r$, which we call the root, denote by $T_{A,r}$ as before.
For simplicity, write $T=T_{A,r}$. Similarly as before, we let for $a,b,c,d\in T$, $D^T(a,b,c,d)$ iff the paths $ab$ and $cd$ do not intersect.
Let for $a,b,c\in T$, $C^T(a,b,c)$ iff $D^T(a,b,c,r)$.
(The relation $C^T$ ``remembers'' that the root $r$ of $T$ is the smallest with respect to $\leq_T$ element of $T$.)
It is crucial that we are allowed to choose the root both with respect to edges and with respect to vertices.
Otherwise, the relative HP would fail, see Remark \ref{relhp}.
\begin{remark}
Let $T$ be a tree and let $E$ be the set of endpoints of $T$. Then $C^T\restriction E$ is an example of a $C$-relation on the set $E$, as defined in \cite[Section 10]{AN}. Moreover, $C^T$ on the tree $T$ satisfies (C1)-(C3) in the definition of a $C$-relation, but not (C4).
\end{remark}
\noindent{\bf{Step 2:}} Labeling the root $r$.
If $r\in A$ (which is exactly in the case when in Step 1 the $x$ we picked was a vertex) then already there is $p\in P$ such that $K^A_p(r)$, i.e. $K^T_p(r)$.
Otherwise, if $r\notin A$, we pick some $p\in P$ and let $K^T_p(r)$.
\noindent{\bf{Step 3:}} Ordering and labeling branches of $T$.
Here we have to do two things: we will
introduce a binary relation that induces an
order of branches of $T$, and then for every $a\in T$ such that for a finite $p\in P$ we have $K_p^T(a)$, we will
put additional labels on the successors of $a$.
The binary relation $\prec^T$: For every $a\in T$ we fix a strict linear order $\prec^T_a$ of its immediate successors.
Then we let $c\prec^T d$ iff for some $a\in T$
there are $i<j$ such that $a_i\leq_T c$ and $a_j\leq_T d$, where $a_1\prec^T_a\ldots\prec^T_a a_n$ are immediate successors of $a$, for some $n$.
The binary relations $G^T_i$: If $a\in T$ and $p\in P\setminus\{\omega\}$ are such that $K^T_p(a)$, and $a_1\prec^T\ldots\prec^T a_n$ are the immediate successors of $a$,
fix an increasing injection $k\colon \{1,\ldots, n\}\to\{1,\ldots,p-1\}$. We let for $b\in T$,
$G_{k(i)}^T(a,b)$ iff $a_i\leq_T b$.
Clearly $\prec^T$ induces an ordering of branches of $T$. Moreover, if $\omega\notin P$, then $\prec^T$ can be recovered from $G_1^T, G_2^T,\ldots$.
Note that if $n<p-1$ in the definition of an injection~$k$, $G_1^T, G_2^T,\ldots$
carry more information that just $\prec^T$.
The reason why we include $\prec^T$ rather than just work with $G_1^T,G_2^T,\ldots$ is that in the case when $\omega\in P$ and $P$ is finite, we do not want to work with infinitely many $G_i$'s
(otherwise precompactness will fail); in the case $\omega\notin P$, it suffices to work only with $G_1^T, G_2^T,\ldots$
and not introduce $\prec^T$.
Finally, put into $\mathcal{T}^*_P$ any structure obtained from $A$ (in a very non-unique way) in the procedure described in Steps 1-3. Note that every vertex in a $T^*\in \mathcal{T}^*_P$, except possibly the root,
has the degree different from 2.
{\bf{The family $\mathbf{\mathcal{F}^*_P}$.}}
Take $\mathcal{L}^*_P=\mathcal{L}_{\mathcal{T}^*_P}\cup \{ R_p\}_{p\in P}\cup \{H_{ij}\}_{1\leq i<j}$, where each $R_p$ and $H_{ij}$ is a binary relation symbol, and $i,j\in\mathbb{N}$.
Start with $A\in\mathcal{F}_P$ and let $T^*\in\mathcal{T}^*_P$ be any rooted tree obtained from $A$. The universes of $A$ and of $T^*$ either are equal or there is
an extra point, the root $r$ of $T^*$, which is not in $A$. All the relations in $\mathcal{L}_{\mathcal{T}^*_P}$ we simply restrict from $T^*$ to $A$. However,
note that in the case $r\notin A$, we "forgot" this way for which $p\in P$ it holds $K^{T^*}_p(r)$ and for which $1\leq i $ it holds
$G_i^{T^*}(r,a)$, whenever $a\in T^*$, $a\neq r$. In order to remember these two pieces of information after removing the root, we set
for any two incomparable with respect to~$\leq^{T^*}$ elements $a,b\in A$ and $c$ equal to the meet of $a$ and $b$ in the rooted tree~$T^*$:
$R_p^{A}(a,b)$ iff $K_p^{T^*}(c)$ and we set $H_{ij}^A(a,b)$ iff $G_i^{T^*}(c,a)$ and $G_j^{T^*}(c,b)$.
\begin{proposition}\label{embeddi}
Let $f\colon S\to T$ be an injection between finite rooted trees $S$ and $T$ with roots $r_S$ and $r_T$, respectively. Then the following are equivalent:
\begin{enumerate}
\item $f$ preserves the relations $C$ (defined with respect to $r_S$ and $r_T$) and $D$;
\item $f$ preserves the relation $C$;
\item $f$ preserves the meet (i.e. for each $a,b\in S$ and their meet $c$, $f(c)$ is the meet of $f(a)$ and $f(b)$).
\end{enumerate}
\begin{proof}
Clearly (1) implies (2). Assume now (2). First notice that then $f$ preserves $\leq_S$ and $\leq_T$. Then note that
if for some $a,b\in S$ and $c$, the meet of $a$ and $b$, we had that $f(c)$ is strictly lower with respect to $\leq_T$
than the meet of $f(a)$ and $f(b)$, then $\neg C^S(a,b,c)$ and $C^T(f(a),f(b),f(c))$, which is impossible. Therefore
we get (3). Now if we assume (3), then $f$ also preserves $\leq_S$ and $\leq_T$. Essentially from the
definitions of the relations $C$ and $D$ it follows that if (3) holds then $f$ preserves $C$ and $D$, and hence
we get (1).
\end{proof}
\end{proposition}
\begin{proposition}
The category $\mathcal{F}^*_P$ with embeddings and the category $\mathcal{T}^*_P$ with embeddings are equivalent via a covariant functor.
\end{proposition}
\begin{proof}
To $A^*\in \mathcal{F}^*_P$ assign $T^*\in \mathcal{T}^*_P$ by adding the root if it is not already in $A^*$.
From the relation $C^{A^*}$ we can recover where the root is.
Recover the information needed about the root
using the relations $R_p$ and $H_{ij}$. To $T^*\in \mathcal{T}^*_P$ assign $A^*\in \mathcal{F}^*_P$ by removing the root if it was added (which is exactly when the degree
of the root is equal to 2).
To an embedding $f\colon A^*\to B^*$ assign an embedding $g\colon S^*\to T^*$, where $S^*$ corresponds to $A^*$ and $T^*$ corresponds to $B^*$
in the following way. If $S^*$ contains the root $r$ which is not already in $A^*$ and this root was added with respect to an edge $x=[a,b]$, we take $g$ to be the
extension of $f$ in which $r$ is mapped to the meet of $f(a)$ and $f(b)$. Again the relations $R_p$ and $H_{ij}$ remember all the information needed
for such a $g$ to be an embedding.
On the other hand, having an embedding $g\colon S^*\to T^*$, we obtain an embedding $f\colon A^*\to B^*$ by simply removing the root from $S^*$,
in case it was added, and restricting $g$.
\end{proof}
Let $\mathcal{T}_P$
denote the set of reducts of elements in $\mathcal{T}^*_P$ to the language $\mathcal{L}_P\cup\{C\}$. We now prove that the
family $\mathcal{F}^*_P$ has all the properties required in Theorem \ref{main}.
\subsection{$\mathcal{F}^*_P$ is reasonable} Let $A, B\in\mathcal{F}_P$ such that $A$ is a substructure of $B$, be given, and fix an expansion $A^*\in\mathcal{F}^*_P$ of $A$.
We will define $B^*\in\mathcal{F}^*_P$, an expansion of $B$ which when restricted to $A$ is equal to $A^*$.
If the root $r=r_{A^*}$ of $A^*$ is a vertex such that there is no $b\in B\setminus A$ and an edge $[r,b]$ in $B$,
we let $r$ to be the root of $B$. If $r$ is a vertex such that there is $b_0\in B\setminus A$ and an edge $[r,b_0]$ in $B$, we let the vertex of $B$
to be any endpoint $e$ of $B$ such that $b_0$ belongs to the path $er$ in $B$.
The resulted rooted tree denote by $T$ and note that $T\in \mathcal{T}_P$.
If the root of $A^*$ was added with respect to an edge $[x_1,x_2]$ in $A$,
then take any edge $[y_1,y_2]\subseteq [x_1,x_2]$ in $B$, and add the root to $B$ with respect to $[y_1,y_2]$.
Take $T\in\mathcal{T}_P$ equal to $B$ with the root $r_T$ added with respect to $[y_1,y_2]$
and let for every $p\in P$, $K_p^T(r_T)$ iff $K_p^{S^*}(r_{S^*})$, where $S^*\in \mathcal{T}^*_P$ with the root $r_{S^*}$ corresponds to $A^*$.
View $S\in\mathcal{T}_P$ equal to the reduct of $S^*$ as embedded in $T$.
In the case when the root $r_{S^*}$ is not in $A$, this embedding takes $r_{S^*}$ to $r_T$.
We still have to define $\prec^T$ and $G^T_1,G^T_2\ldots$, which extend $\prec^{S^*}$ and
$G^{S^*}_1,G^{S^*}_2\ldots$.
For this, for any $b\in T$ and its immediate successors $b_1,\ldots, b_n$, it is enough to define $\prec^T$ on $\{b_1,\ldots, b_n\}$ and
specify for each $i$ and $k$ whether $G^T_i(b,b_k)$ holds or not.
Let $p\in P$ be such that $K_p^B(b)$.
In case $b\notin S^*$, we define $\prec^T$ and $G^T_1,\ldots, G^T_{p-1}$
in an arbitrary way that Step 3 in the construction allows us. In the case $b\in S^*$, we define $\prec^T$ and $G^T_1,\ldots, G^T_{p-1}$
in any way allowed in Step 3 so that additionally if for some $c\in S^*$, $b_k\leq_T c$ and $G_i^{S^*}(b,c)$ then $G_i^T(b, b_k)$ and if for some $c,d\in S^*$,
$b_k\leq_T c$, $b_l\leq_T d$ and $c\prec^{S^*} d$
then $b_k\prec^{T} b_l$.
This defines $T^*\in\mathcal{T}^*_P$, which corresponds to $B^*\in\mathcal{F}^*_P$ we are looking for.
\subsection{$\mathcal{F}^*_P$ is precompact with respect to $\mathcal{F}_P$} Clear. $P$ has to be finite.
\subsection{$\mathcal{F}^*_P$ has the JEP}
For this we can instead work with the family $\mathcal{T}^*_P$. Take $S^*, T^*\in\mathcal{T}^*_P$. Let $r_{S^*}$ be the root of $S^*$, and let $r_{T^*}$ be the root of $T^*$.
Pick a new element $r$, pick $p\in P$, and if $p<\omega$ pick $1\leq i<j\leq p-1$. Let $R^*\in\mathcal{T}^*_P$ be obtained as follows. We take the union of $S^*$ and $T^*$
together with the point $r$ and vertices $[r,r_{S^*}]$ and $[r,r_{T^*}]$. We declare $r$ to be the root of $R^*$, i.e. we define
$C^{R^*}(a,b,c)$ iff $D^{R^*}(a,b,c,r) $, and let $K_p^{R^*}(r)$. For any $r_{S^*}\leq_{S^*} a$ and $r_{T^*}\leq_{T^*} b$, let $a\prec^{R^*} b$.
If $p<\omega$, then if $r_{S^*}\leq_{S^*} a$, we let $G_i^{R^*}(r,a)$ and
if $r_{T^*}\leq_{T^*} a$, we let $G_j^{R^*}(r,a)$.
We also make sure that the degrees of $r_{S^*}$ and $r_{T^*}$ in $R^*$ are at least 3 by adding
additional edges and extending $\prec^{R^*} $ and $G_i^{R^*}$, if needed.
Then this $R^*$ is as required for $S^*$ and $T^*$.
(Note that in this proof we used that the degree of $r_{S^*}$ in $S^*$ is strictly less than $p_0\in P$ such that $K_{p_0}^{S^*}(r_{S^*})$, and similarly for $T^*$.)
\subsection{$\mathcal{F}^*_P$ has the AP} One can show it directly, but it also follows from the rigidity
of each $A\in \mathcal{F}^*_P$ together with the JEP and the Ramsey properties for
$\mathcal{F}^*_P$. The proof of this fact is essentially due to Ne\v{s}et\v{r}il-R\"odl
(see \cite[p. 294, Lemma 1]{NR}), the framework in which they work is somewhat different from ours.
Their proof is for families of structures which are rigid, hereditary, have the JEP and Ramsey properties, see also
\cite[p. 129]{KPT}.
Nevertheless, for a Fra\"{i}ss\'{e}-HP family $\mathcal{F}$,
whenever $A\in\mathcal{F}$ then every structure isomorphic to $A$ is also in $\mathcal{F}$. Therefore the proof presented by
Kechris-Pestov-Todorcevic \cite{KPT} applies to
Fra\"{i}ss\'{e}-HP families as well.
\subsection{$\mathcal{F}^*_P$ has the relative HP}
Fix $A, B\in\mathcal{F}_P$ and $B^*\in\mathcal{F}^*_P$ extending $B$.
Take $T^*\in\mathcal{T}^*_P$ that corresponds to $B^*$ and view $A$ as embedded in the $\mathcal{L}_P$-reduct of $T^*$. There are either one or two minimal elements in $A\subseteq (T^*, \leq_{T^*})$. Let $r$ be this minimal element, if there is exactly one, and otherwise let $r$
be the meet of the two minimal elements. Take $S=A\cup\{r\}$, a rooted tree with the root $r$.
Let $S^*$ be the substructure of $T^*$ such that the universe is $S$. Then $S^*\in\mathcal{T}^*_P$ and hence the corresponding structure $A^*\in\mathcal{F}^*_P$ satisfies $B^*\restriction A\in \mathcal{F}^*_P$.
\begin{remark}\label{relhp}
It is possible to have $A, B\in\mathcal{F}_P$, $A$ embedded into $B$, the expansion $B^*\in\mathcal{F}^*_P$ of $B$,
such that its root was added with respect a vertex, but the root of $A^*$,
the restriction of $B^*$ to $A$, has the root added with respect to an edge.
Let for example $B^*$ be the rooted tree that consists of 4 vertices: $r, a,b, c$, where $r$ is the root,
and edges $[r,a]$, $[r,b]$, $[r,c]$, and let $A$ be the subtree that consists of 2 vertices $a,b$
and the edge $[a,b]$.
Similarly, it is not hard to give an example of $B^*$ and $A$ such that
$B^*$ has the root added with respect to an edge and $A^*$ has the root added with respect
to a vertex.
\end{remark}
\subsection{$\mathcal{F}^*_P$ has the expansion property with respect to $\mathcal{F}_P$}
Let $A^*\in\mathcal{F}^*_P$ be given. Without loss of generality, let the root of $A^*$ belong to $A^*$ (as we can always embed the $A^*$ we started with
in an element of $\mathcal{F}^*_P$ with such a property). Therefore we can think that $A^*\in\mathcal{T}^*_P$.
Take a rooted tree $T\in\mathcal{T}_P$
which has the property that all its expansions to an element in $\mathcal{T}^*_P$
are isomorphic, the degree of the root is $\geq 2$, and $A^*$ embeds in some/every expansion of $T$.
(For this, note that any tree $V\in\mathcal{T}_P$ with the properties: (1) if $x$ and $y$ have the same height, $K_p^V(x)$ and $K_q^V(y)$ hold, then $p=q$;
(2) if $x$ is not an endpoint, $p<\omega$, and $K^V_p(x)$, then $x$ has exactly $p-1$ immediate successors; (3) there is $M\geq 2$ such that every $x$
which is not an endpoint and $K_{\omega}^V(x)$, has exactly $M$ immediate successors; is such that all its expansions to an element in $\mathcal{T}^*_P$
are isomorphic.)
Finally, let $B$ be obtained as follows. Take $T'$ and $T''$, two disjoint copies of $T$. Denote their roots by $r_{T'}$ and $r_{T''}$, respectively.
The disjoint union of $T'$ and $T''$ together with the edge $[r_{T'}, r_{T''}]$ is a required $B$.
This is because whenever we expand $B$ to a $B^*\in\mathcal{F}^*_P$ then we can embed $T^*$
(the unique expansion of $T$) into $B^*$.
If, say, the vertex or edge with respect to which is added the root of $B^*$ lies in $T'$, then the unique expansion of $T''$ embeds into $B^*$.
\subsection{$\mathcal{F}^*_P$ has the Ramsey property}
We generalize the Ramsey theorems by Deuber \cite{D} and by Soki\'c \cite{S} (Theorem 2.2 and Theorem 6.1). For related Ramsey theorems, where it is additionally assumed that endpoints of a rooted tree are mapped to endpoints, see \cite{J}, \cite{BP}, and \cite{Sol}.
\begin{theorem}\label{ramsey}
For any non-empty $P\subseteq\{3,\ldots,\omega\}$, the family $\mathcal{T}^*_P$, and hence the family $\mathcal{F}^*_P$, is Ramsey.
\end{theorem}
Consider $T\in\mathcal{T}^*_P$ with the root $r_T$ and let $q$ be such that $K^T_q(r_T)$.
Let $V\in\mathcal{T}^*_P$
and let $M$ be the maximum of 2 and
the number of immediate successors of all vertices in $V$ labeled with $\omega$.
We are going to define $V[T]\in T^*_P$. First consider $V'\in\mathcal{T}^*_P$ defined as follows.
For every endpoint $e\in V$ take $p_e$ such that $K^V_{p_e}(e)$ and take new points $x_1^e, \ldots, x_{p'_e}^e$,
where $p'_e=p_e-1$ if $p_e<\omega$ and $p'_e=M$ if $p_e=\omega$, and add edges $[e, x_i^e]$, $i=1,\ldots, p'_e$.
Then let $V'\in \mathcal{T}^*_P$ be the tree we obtain by letting $K^{V'}_q(x^e_i)$ for each endpoint $e$, and each $i$,
and (uniquely) choosing $\prec^{V'}$ and
$G_i^{V'}$.
To obtain $V[T]$, to each endpoint of $V'$ attach the tree $T$ by identifying this endpoint with the root $r_T$.
\begin{example}
Let $V=2^{\leq 1}$ with $K_5^V(\emptyset)$, $K_3^V(0)$, and $K_\omega^V(1)$.
Let $T=2^{\leq 1}$ with $K_7^T(\emptyset)$, $K_{10}^T(0)$, and $K_6^T(1)$.
Then $S=V[T]=2^{\leq 3}$ with $K_5^S(\emptyset)$, $K_3^S(0)$, $K_\omega^S(1)$,
$K_7^S(00)$, $K_7^S(01)$, $K_7^S(10)$, $K_7^S(11)$, $K_{10}^S(000)$,
$K_{10}^S(010)$, $K_{10}^S(100)$, $K_{10}^S(110)$,
$K_{6}^S(001)$, $K_{6}^S(011)$, $K_{6}^S(101)$, and $K_{6}^S(111)$.
\end{example}
For a family $\mathcal{G}$ of first-order structures in some language denote by $\mathcal{G}_m$
the family $\{(A_1,\ldots, A_m)\colon A_j\in\mathcal{G}\}$. We say that $(A_1,\ldots, A_m)$ embeds into
$(B_1,\ldots, B_m)$ if for every $j$, $A_j$ embeds into $B_j$.
In the inductive step of the proof of Theorem \ref{ramsey} we will be using the product Ramsey theorem.
\begin{theorem}[Soki\'c, Theorem 2 in \cite{S2}]
Let $\mathcal{G}$ be a family of first-order structures in some language, which is a Ramsey class.
For any $(A_1,\ldots, A_m), (B_1,\ldots,B_m)\in\mathcal{G}_m$
such that $(A_1,\ldots, A_m)$ embeds into $(B_1,\ldots,B_m)$ there is $C\in\mathcal{G}$ such that
for any coloring of embeddings of $(A_1,\ldots, A_m)$ in $(C,\ldots, C)$ into finitely many colors there is a embedding $h=(h_1,\ldots,h_m)$ of $(B_1,\ldots,B_m)$ in $(C,\ldots, C)$
such that the set of all functions $h\circ f$, where $f=(f_1,\ldots, f_m)$ is an embedding of $(A_1,\ldots, A_m)$ into $(B_1,\ldots,B_m)$, is monochromatic.
\end{theorem}
Moreover, from the proof of Soki\'c's theorem it follows that for every $i$: If every $A\in\mathcal{G}$
with $ht(A)\leq i$ is a Ramsey object in $\mathcal{G}$, then every $(A_1,\ldots, A_m)\in\mathcal{G}_m$
with each $A_j$ satisfying $ht(A_j)\leq i$, is a Ramsey object in $\mathcal{G}_m$.
We show that $\mathcal{T}^*_P$ is a Ramsey class , i.e. we show that for every
$S,T\in \mathcal{T}^*_P$ with $S\leq T$ there exists $U\in \mathcal{T}^*_P$ such that for every coloring $c: {U \choose S} \to\{\mathrm{blue}, \mathrm{red}\}$ there exists $h\in {U \choose T}$ such that
$\{ h\circ f: f\in {T \choose S} \}$ is monochromatic.
\begin{proof}[Proof of Theorem \ref{ramsey}]
We show that every $S\in\mathcal{T}^*_P$ is a Ramsey object by induction on the height of $S$.
First let $S\in\mathcal{T}^*_P$ be a one-element structure. Take $T\in\mathcal{T}_P$ such that $S$ embeds into $T$.
Without loss of generality, for any $x\in T$, which is not an endpoint, and $p\in P$ such that $K^T_p(x)$, if $p<\omega$, then the number of immediate successors of $x$ is equal to $p-1$. Moreover assume that there is $M$ such that the for any $x$, which is not an endpoint, such that $K^T_{\omega}(x)$, the number of immediate
successors of $x$ is exactly $M$.
Suppose that $S=\{a\}$ and let $p_S\in P$ be such that
$K^S_{p_S}(a)$.
Let $T_0=T$ and $T_{k}=T_{k-1}[T]$, $1\leq k\leq h=ht(T)$, and we claim that $U=T_{h}$ is as required.
Denote the set of copies of $T$ attached to $T_{k-1}$ in the construction of $T_k$ by $\mathcal{T}_k$, and let $\mathcal{T}_0=\{T_0\}$.
Color embeddings of $S$ into $U$ into two colors: blue and red.
If there is $k$ and $T'\in\mathcal{T}_k$ such that all embeddings of $S$ into $T'$ are in the same color, we are done.
Otherwise, for each $k$ and $T'\in\mathcal{T}_k$ there is a blue embedding of $S$ into $T'$.
We construct the required embedding $f$ of $T$ into $U$ by induction. First we construct $f(r_T)$, where $r_T$ denotes the root of $T$.
Let $p_T\in P$ be such that $K^T_{p_T}(r_T)$. If $p_T\neq p_S$, let $f(r_T)=r_U$, where $r_U$ is the root of $U$.
If $p_T=p_S$, let $f(r_T)$ be an image of any blue embedding of $S$
into $T'=T_0\in \mathcal{T}_0$.
Now let $x\in T$ be of height $k$ and suppose that we constructed $f(x)$ and that $f(x)\in T'$ for some $T'\in \mathcal{T}_k$
Let $x_1\prec^T\ldots\prec^T x_m$
be the list of immediate successors of $x$, and we construct $f(x_1),\ldots, f(x_m)$, each will be in a copy of $T$ that lies in $\mathcal{T}_{k+1}$.
Suppose that $p\in P$ is such that
$K_{p}^T(x)$ and that $y_1\prec^U\ldots \prec^U y_{p'}$ is the list of immediate successors of $f(x)$, where $p'=p-1$
if $p<\omega$ and $p'=M$ when $p=\omega$.
By the construction of $U$, there are $r_1\prec^U\ldots \prec^U r_{p'}$,
such that $r_l$ is a successor of $y_l$ in $U$ and $r_l$ is the root of some $T^l\in\mathcal{T}_{k+1}$.
For each $l$, let $p_l\in P$ be such that $K^T_{p_l}(x_l)$, and if $p_l\neq p_S$, let $f(x_l)$ be equal to the point in $T_l$ corresponding to $x_l$
in the obvious isomorphism between $T$ and $T^l$. Otherwise, if $p_l= p_S$, we let $f(x_l)$ to be the image of a blue embedding of $S$ into $T^l\subseteq U$.
This gives a ``blue'' embedding of $T$ into $U$ and
finishes the base step of the induction.
For the inductive step,
let $S,T\in\mathcal{T}^*_P$ such that $S\leq T$ be given. We assume that every tree in $\mathcal{T}^*_P$ of the height strictly less than the height of $S$ is a Ramsey object.
Let $V\in\mathcal{T}^*_P$ be such that whenever we color embeddings of $\{r_S\}$ into $V$ into two colors,
then there exists an embedding $g\colon T\to V$ such that
$\{ g\circ f\colon f\in {T \choose S} \}$ is monochromatic.
Without loss of generality, we assume that for any $x\in V$, which is not an endpoint, and $p\in P$ such that $K^V_p(x)$, if $p<\omega$, then the number of immediate successors of $x$ is equal to $p-1$.
Let $a_1\prec^S\ldots \prec^S a_k$ be the list of immediate successors of $r_S$, and let $S_i=S_{a_i}=\{b\in S\colon a_i\leq_S b\}$.
Using the well-founded recursion along $V$, for each $x\in V$ we construct a tree $V^x\in\mathcal{T}^*_P$. The $U=V^{r_V}$
will be as needed for $S$ and $T$ and two colors.
For an endpoint $x\in V$, let $V^x=\{x\}$ with $K_p^{V^x}(x)$ iff $K_p^V(x)$ for every $p\in P$.
Now let $x\in V$ not be an endpoint, let $x_1\prec^V\ldots\prec^V x_n$ be the list of all immediate successors of $x$, and assume that we already defined
$V^{x_1},\ldots, V^{x_n}$. Let $V^x_0$ be obtained from the disjoint union of $\{x\}$ and $V^{x_1},\ldots, V^{x_n}$, adding edges $[x, r_{V^{x_i}}]$.
For $a\in V^{x_i}$ and $t=1,\ldots,n$ we let
$G^{V^x_0}_t(x,a)$ iff $G^V_t(x, x_i)$. Similarly, for $a\in V^{x_i}$ and $b\in V^{x_j}$ we let $a\prec^{V^x_0} b$ iff $x_i\prec^V x_j$ and let $K_p^{V^x}(x)$ iff $K_p^V(x)$.
If $S$ does not embed into $V^x_0$ in a way that $r_S$ is mapped to $r_{V^x_0}$,
the root of $V^x_0$, let $V^x=V^x_0$.
Otherwise, and if $p$ such that $K^V_p(x)$ is finite, apply the product Ramsey theorem to $(S_1,\ldots, S_k)$ and $(V^{x_{b(1)}},\ldots, V^{x_{b(k)}})$,
where $b\colon\{1,\ldots,k\}\to \{1,\ldots,n\}$ is an increasing injection such that for any $t$ and $i$, $G_t^S(r_S,a_i)$ iff $G_t^V(x,x_{b(i)})$,
and let $U^x\in\mathcal{T}^*_P$ be the structure we obtain from the product Ramsey theorem.
For each $j\in\rm{rng}(b)$, take any $U^{j}\in \mathcal{T}^*_P$ such that $U^x$ embeds into it and for any $p\in P$, $K^{U^{j}}_p(r_{U^{j}})$ iff
$K^{V^{x_j}}_p(r_{V^{x_j}})$. Finally, let $V^x$ be equal to $V^x_0$ with each $V^{x_{b(i)}}$ replaced by $U^{b(i)}$.
If $K^V_{\omega}(x)$,
take $l$ such that whenever we color increasing injections of $\{1,\ldots, k\}$ to $\{1,\ldots, l\}$ into two colors then there is an increasing injection $g\colon\{1,\ldots, n\}\to \{1,\ldots, l\}$ such that all maps $g\circ f$, where $f\colon \{1,\ldots, k\}\to \{1,\ldots, n\}$ is an increasing injection,
are in the same color.
Let $U_0^x\in\mathcal{T}^*_P$ be any structure that all $V^{x_1},\ldots, V^{x_n}$ embed into it.
Enumerate increasing injections of $\{1,\ldots, k\}$ to $\{1,\ldots, l\}$ into $e_1,\ldots, e_m$.
Define recursively $U_{i+1}^x$, $i=0,\ldots, m-1$ to be the result of applying the product Ramsey theorem to $(S_1,\ldots, S_k)$ and $(U_i^x,\ldots, U_i^x)$.
Define $V^x$ to be the disjoint union of $\{x\}$ and $l$ many $U^x_m$,
we add edges $[x, r_{U^x_m}]$, and specify $\prec^{U^x_m}$, and let $K_p^{V^x}(x)$ iff $K_p^V(x)$.
Observe that for every $x\in V$ with immediate successors $x_1\prec^{V}\ldots\prec^{V} x_n$, which is not an endpoint, we have
for every $p\in P$, $K_p^V(x)$ iff $K_p^{V^x}(r_{V^x})$. If $p$ is finite and such that $K_p^V(x)$
or if $S$ does not embed into $V^x$ in a way that $r_S$ is mapped to $r_{V^x}$, then $x$ has exactly $n$ many immediate successors
$x'_1\prec^{V^x}\ldots\prec^{V^x} x'_n$ in $V^x$ and they are such that for any $t$ and $i$, $G^{V^x}_t(r_{V^x}, x'_i)$ iff $G^{V}_t(x,x_i)$,
and for any $p$ and $i$, $K_p^{V^x}(x'_i)$ iff $K_p^V(x_i)$. Moreover, for any coloring into two colors of embeddings of $S$ into $V^x$
such that $r_S$ is mapped to $r_{V^x}$, there is an embedding $g\colon V^x_0\to V^x$ taking $r_{V^x_0}$ to $r_{V^x}$ such that
$\{ g\circ f\colon f\in {V^x_0 \choose S} \text{ taking $r_S$ to } r_{V^x_0} \}$ is monochromatic.
If $K_{\omega}^V(x)$, set $V^i=\{a\in V^x_0\colon x_i\leq_{V^x_0} a\}$, where $x_1\prec^{V^x_0}\ldots\prec^{V^x_0} x_n$ are the immediate successors of $x$ in $V^x_0$.
Then for any coloring into two colors of embeddings of $S$ into $V^x$ such that $r_S$ is mapped to $r_{V^x}$,
there are immediate successors $y_1\prec^{V^x}\ldots\prec^{V^x} y_n$ of $r_{V^x}$ and
an embedding $g\colon V^x_0\to V^x$ taking $r_{V^x_0}$ to $r_{V^x}$ and satisfying $y_i\leq_{V^x} g(V^i)$, $i=1,\ldots, n$,
such that
$\{ g\circ f\colon f\in {V^x_0 \choose S} \text{ taking $r_S$ to } r_{V^x_0} \}$ is monochromatic.
Color embeddings of $S$ into $U$ into two colors. Using the observations above, find an embedding $h\colon V\to U$ such that any two
embeddings $g_1,g_2\colon S\to U$ whose image is contained in $h(V)$ and with $g_1(r_S)=g_2(r_S)$,
are in the same color. Finally, by the choice of $V$, for the induced coloring of
embeddings of $\{r_S\}$ into $V$ into two colors,
there exists an embedding $g\colon T\to V$ such that
$\{ g\circ f\colon f\in {T \choose S}\}$ is monochromatic.
Then $h\circ g$ is as required.
\end{proof}
We finish this section relating Theorem \ref{ramsey} to the work of Soki\'c \cite{S}.
A {\em semilattice} is a poset such that every 2 elements have an infimum. If $A$ is a semilattice, we define a binary operation $\circ$ on $A$
by $a\circ b=\inf(a,b)$ and a partial order $\leq_A$ by $a\leq_A b$ iff $a\circ b=a$.
Say that $(A,\circ^A)$ is a {\em treeable semilattice} if the induced poset is a rooted tree, i.e. it has the minimum, called the root,
and for each $a\in A$, the set $\{b\in A\colon b\leq_A a\}$ is linearly ordered by $\leq_A$.
Let $\mathcal{T}$ be the family of all finite treeable semilattices in the language $\{\circ\}$.
Let $A\in\mathcal{T}$ and say that $\preceq^A$ is a {\em convex ordering} on $A$ if for every $a,b,c\in A$ with $a\circ b=c$, $a\neq c$ and $b\neq c$,
we have $a\preceq^A b$ iff $a'\preceq^A b'$, where $a',b'$ are immediate successors of $c$, $a'\leq_A a$ and $b'\leq_A b$. Denote the set of all convex
ordering on $A$ by $co(A)$ and let
\[ \mathcal{CT}=\{(A,\circ^A,\preceq^A)\colon (A,\circ^A)\in \mathcal{T}, \preceq^A\in co(A)\}.\]
\begin{theorem}[Soki\'c, Theorem 2.2 in \cite{S}]
$\mathcal{CT}$ is a Ramsey class.
\end{theorem}
The theorem above is a special case of Theorem \ref{ramsey} and is equivalent to the statement that
$\mathcal{T}^*_{\{\omega\}}$ is a Ramsey class. Indeed,
the categories $\mathcal{CT}$ and $\mathcal{T}^*_{\{\omega\}}$ are equivalent via a covariant functor, which follows from
Proposition \ref{embeddi} and an observation that convex orderings on treeable semilattices
correspond to binary relations allowed in Step 3 of the definition of $\mathcal{T}^*_P$.
Similarly, Theorem 6.1 in \cite{S} is equivalent to the statement that each $T^*_{\{k\}}$ is a Ramsey class, $k\geq 3$, therefore again it is a special case of Theorem \ref{ramsey}.
\section{The generalized Wa\.zewski dendrite $W_P$, for an infinite $P\subseteq\{3,4,\ldots,\omega\}$}
In this section, we show that in the case $P$ is infinite, the universal minimal flow of the homeomorphism group of the generalized Wa\.zewski dendrite $W_P$ is non-metrizable, and
we point out two important consequences this fact (see Sections 4.1 and 4.2).
Let $\mathcal{G}$ be a family of finite structures.
Say that $A\in\mathcal{G}$ has {\em Ramsey degree} $\geq t$ iff there exist $A\leq B$ such that for every
$B\leq C$ there exists a coloring
$c_0\colon {C \choose A} \to\{1,2,\ldots,t\}$ such that for every $g\in {C \choose B}$,
$\{ g\circ f: f\in {B \choose A} \}$ assumes $\geq t$ colors.
The Ramsey degree is infinite if for every $t$ it is $\geq t$.
\begin{theorem}[Zucker \cite{Z}, Theorem 8.7]\label{andy2}
Let $\mathcal{G}$ be a Fra\"{i}ss\'{e}-HP family and let $\mathbb{G}$ be its Fra\"{i}ss\'{e} limit. Then some $A\in\mathcal{G}$ has infinite Ramsey degree iff the universal minimal flow of ${\rm Aut}(\mathbb{G})$ is non-metrizable.
\end{theorem}
Theorems \ref{andy2} and \ref{infinite2} imply that when $P$ is infinite then the universal minimal flow of
$H(W_P)$ is non-metrizable.
\begin{theorem}\label{infinite2}
Suppose that $P$ is infinite. Then there is $A\in\mathcal{F}_P$ which has infinite Ramsey degree.
\end{theorem}
\begin{proof}
Let $p_1<p_2<\ldots$ be the increasing enumeration of $P\setminus\{\omega\}$, let $A=\{a,b\}$ be such that $V(A)=\{a,b\}$, $E(A)=\{[a,b]\}$,
$K^A_{p_1}(a)$ and $K^A_{p_1}(b)$, and let $t\geq 2$ be given. Take $B$ constructed as follows. First let $B_1$ consists of a single point $x_0$
such that $K^{B_1}_{p_1}(x_0)$. Let $B_2$ be the tree that consists of vertices $x_0, y_1,\ldots, y_{p_1}$ and edges $[x_0, y_i]$,
$K^{B_2}_{p_1}(x_0)$, and $K^{B_2}_{p_2}(y_i)$, $i=1,\ldots, p_1$. Having constructed $B_k$, $k\leq t$, such that for every endpoint $e$ in $B_k$
it holds $K^{B_k}_{p_k}(e)$, we obtain $B_{k+1}$ from $B_k$ in the following way. For every endpoint $e$ in $B_k$ pick new points $y^e_1,\ldots, y^e_{p_k-1}$
and add vertices $[e, y^e_i]$. Note that $e$ has degree $p_k$ in $B_{k+1}$.
If $k<t$, we let $K^{B_{k+1}}_{p_{k+1}}(y^e_i)$, and if $k=t$, let $K^{B_{k+1}}_{p_1}(y^e_i)$. Take $B=B_t$.
Now take any $C\in\mathcal{F}_P$ such that $B\leq C$. Pick an endpoint $r$ in $C$ and consider $C$ as a rooted tree with the root $r$.
Let $c_0\colon {C \choose A} \to\{1,2,\ldots,t\}$ be the following coloring. For an embedding $f\colon A\to C$, if $f(a) $ and $f(b)$ do not lie on the same branch in the rooted tree $C$,
let $c_0(f)=i$ iff $K^C_{p_i}(c)$, where $c$ is the meet of $f(a)$ and $f(b)$. Otherwise, if $f(a) $ and $f(b)$ do lie on the same branch, let
$c_0(f)$ be an arbitrary color from $\{1,\ldots, t\}$.
Let $g\colon B\to C$ be an embedding. There are $j_1$ and $j_2$ such that $g(x_0)\leq_C g(y_{j_1}), g(y_{j_2})$, $y_{j_1}, y_{j_2} \in B_2$
(in fact, all $j=1,\ldots,p_1$ except one have this property). Clearly in the rooted tree $B'$, obtained from $B$ by
removing all vertices $z$ such that some $y_j\in B_2$, $j\neq j_1, j_2$, is on the path connecting $z$ and $x_0\in B_1$, we have that
for any $i$ there are endpoints $e_1$ and $e_2$ in $B'$ such that the meet of $e_1$ and $e_2$ is a vertex $c$ such that $K^{B'}_{p_i}(c)$.
That implies that $g(B')$ and hence $g(B)$ assumes all $t$ colors.
\end{proof}
\begin{corollary}\label{abcd}
Suppose that $P$ is infinite. Then the universal minimal flow of $H(W_P)$ is non-metrizable.
\end{corollary}
\subsection{$H(W_P)$ has a non-metrizable universal minimal flow and is Roelcke precompact}
A subgroup $H$ of $S(X)$, the group of all permutations of a countable set $X$ with the pointwise convergence topology, is {\em oligomorpic} when for every $n$, the diagonal action of $H$ on $X^n$ has only finitely many
orbits. Note that we do not assume that $H$ is a closed subgroup of $S(X)$.
A topological group $H$ is {\em Roelcke precompact } if for every open neighbourhood $U$ of $1\in H$
there exists a finite set $F\subseteq H$ such that $H=UFU$. As shown by Tsankov \cite[Theorem 2.4]{T}
a subgroup of $S(X)$ is Roelcke precompact if and only if it is an inverse limit of oligomorphic groups.
As observed by Todor Tsankov (private communication in 2013), ${\rm{Aut}}(M_P)$, for each $P\subseteq\{3,4,\ldots,\omega\}$,
is a Roelcke precompact group. This is because when we take
\[M_n=\{m\in M_P\colon K_p(m) \text{ for some } p\in\{3,\ldots,n,\omega \}\}, \]
\[G_n={\rm{Aut}}(M_n)={\rm{Aut}}(M_n, D^{M_n}, (K_p^{M_n})_{p\in P\cap\{3,\ldots,n,\omega\}}),\]
and \[H_n=\{h\in G_n\colon \text{there exists } f\in {\rm{Aut}}(M_P) \text{ such that } h=f\restriction M_n\}, \]
then $H_n$ is an oligomorphic group and the inverse limit of $H_n$ is equal to ${\rm{Aut}}(M_P)$.
Melleray-Nguyen Van Th\'e-Tsankov \cite{MNT} asked:
\begin{question}[Question 1.5 in \cite{MNT}]\label{que}
Is the universal minimal flow of every Roelcke precompact Polish group metrizable?
\end{question}
Moreover, Bodirsky-Pinsker-Tsankov \cite{BPT} asked if every $\omega$-categorical structure has an
$\omega$-categorical expansion which is Ramsey (which by the work of Zucker \cite{Z} is equivalent
to the question above with ``Roelcke precompact Polish'' replaced by ``oligomorphic'').
Evans-Hubi\v{c}ka-Ne\v{s}et\v{r}il \cite{EHN} answered Question \ref{que} in the negative. They provided an example of an oligomorphic
group with a non-metrizable universal minimal flow. Their example is much more involved than ours,
it is based on a very non-trivial construction due to Hrushovski~\cite{Hr}, see also \cite{En}.
\subsection{$H(W_P)$ has a non-metrizable universal minimal flow with a comeager orbit}
Ben Yaacov-Melleray-Tsankov \cite{BMT}, generalizing a result of Zucker \cite{Z}, showed:
\begin{theorem}[Theorem 1.2 in \cite{BMT}]
Let $G$ be a Polish group whose universal minimal flow $M(G)$ is metrizable. Then $M(G)$ has a comeager orbit.
\end{theorem}
They asked if the converse holds:
\begin{question}[Question 1.3 in \cite{BMT}]\label{que2}
Suppose that $G$ is a Polish group such that $M(G)$ has a comeager
orbit. Is it true that $M(G)$ is metrizable?
\end{question}
After this preprint was posted on arXiv, Zucker \cite{Zt} showed:
\begin{theorem}\label{thesis}
Let $\mathcal{G}$ be a Fra\"{i}ss\'{e}-HP family.
Suppose that there exists a reasonable Fra\"{i}ss\'{e}-HP expansion $\mathcal{G}^*$ of $\mathcal{G}$, which has the relative HP, the expansion, and the Ramsey properties, but the precompactness fails. Then the universal minimal flow of ${\rm Aut}(\mathbb{G})$, where
$\mathbb{G}$ is the Fra\"{i}ss\'{e} limit of $\mathcal{G}$, has a comeager orbit.
\end{theorem}
Theorems \ref{main} and \ref{thesis}, together with Corollary \ref{abcd}, provide the negative answer to Question \ref{que2}.
\section*{Acknowledgements}
I thank Todor Tsankov for our discussions on generalized Wa\.zewski dendrites in 2013.
I also thank Miodrag Soki\'c, Bruno Duchesne, and the referee,
for many valuable remarks that improved the presentation of the paper.
The author was supported by Narodowe Centrum Nauki grant 2016/23/D/ST1/01097.
|
1,314,259,992,973 | arxiv | \section{Background and Motivation}
For radiation safety and material accountability, operators need to know the location and amount of hold up, i.e., nuclear material deposited in various systems of a fuel reprocessing facility. Gamma-ray spectroscopy is typically used to estimate holdup activity accurately and requires the use of collimated and shielded gamma-ray detectors, typically inorganic scintillators.
Procedures including quantitative measurements of holdup benefit from assumptions of specific source spatial distribution, enabling operators to perform and analyze thousands of measurements. These procedures are based on the Generalized Geometry Holdup model (GGH)~\cite{russo2005gamma}. However, the bias introduced from the use of the GGH model can lead to a material loss trend spanning years to decades, which may result in adverse regulatory and financial issues~\cite{resolution}.
The manual measurement of holdup in piping is a costly, time consuming, and hard-working process. For this reason, automated robotic systems have been proposed and tested~\cite{jones2019robot}. However, the existing systems are destructive, requiring the operator's physical intervention to insert the robot inside the pipe manually. These current systems are optimized for a single diameter and are not designed to inspect pipes of various curvatures. Furthermore, other types of equipment, such as filters, require extensive modification to be inspected by such robots.
We propose a soft robot that is minimally tethered~\cite{softrobot2} and is equipped with radiation detectors and ultrasound sensors, capable of crawling on surfaces and adjusting to the curvature of the pipe's geometry without the need of \added{re-positioning the robot. The operator could use a telescopic rod to initially place the system at the desired location and then remotely control its movement or let the device autonomously reach the position and perform the measurements. This robot would allow an accurate localization and non-destructive characterization of the holdup material, making measurements faster, more accurate, autonomous, and less demanding on operators.}
In this paper, we focus on the imaging reconstruction and material quantification algorithm, which will result in a 3D mapping and activity of the deposition profile. We will first discuss and present the bias introduced in the GGH model, then propose a new approach to analyze data that can be acquired by such a system. The proposed approach consists of characterizing the detector array's response and solving a linear inverse problem to derive the activity of the holdup. This approach's accuracy is tested in an experimental setup, then in simulations performed using MCNP6.2~\cite{osti_1419730} in 2D and 3D cases.
\section{Methods}
\subsection{GGH Approach}
Holdup analysis usually involves measuring spectra that have overlapping peaks in energy regions between 200 and 800 keV. Thus, the calibration and measurement efforts must be simple, and the analysis must apply to the unique geometry of each measurement. This objective is achieved by adjusting holdup measurement geometries in the field of view of a cylindrical collimated gamma-ray detector so that each holdup deposit geometry can be generalized as a point, line, or area~\cite{russo2005gamma}. After detector calibration, the procedure of holdup measurement follows three steps:
\begin{enumerate}
\item A coarse survey is performed to determine if the deposited geometry can be generalized as a point, line, or area.
\item An accurate measurement of the deposited material is acquired.
\item An analytical model is applied to interpret the data according to the chosen holdup geometry in step 1.
\end{enumerate}
We first reproduced a holdup measurement using laboratory sources to identify the maximum bias due to a misidentification of the source's geometry. We used two 1~$\mu$Ci \added{$^{137}$Cs} point sources at the same location and separated the two sources to investigate the misidentification of the source activity due to the distance. \added{We followed the standard holdup measurement procedure, where a coarse survey of the equipment of interest is performed prior to holdup assessment.} Using a survey process based on measurements from a Victoreen 450 survey meter, we determined that when the sources were located within up to 10~cm from each other, they were identified as a single source.
Thus, the two sources were placed 0, 2, 4, 6, 8, and 10~cm apart. \added{At the expense of a longer measurement time, the use of a collimated, high-efficiency detector instead of the Victoreen survey meter would likely allow to discriminate the two sources even when closer than 10 cm to each other.} We then set a NaI(Tl) detector next to a pipe containing the two \added{$^{137}$Cs} sources to simulate the current method of an operator holding up a detector towards the source, as shown in Fig.~\ref{fig:GGH measurement}. \added{The pipe to detector center distance was chosen to resemble a realistic method of holdup assay where it is not possible to optimize the pipe detector distance based on the source shape estimate.}
\begin{figure}[ht]
\centering
\includegraphics[width=0.75\linewidth]{pics/SingDet.png}
\caption{Setup of single detector measurement with pipe and detector labeled.
}
\label{fig:GGH measurement}
\end{figure}
We measured the distance between the center of the detector and the assumed point source location and the counts at each distance. Based on the initial survey, the sources would be identified as a point source. Thus, the activity $A_H$ of the \added{two $^{137}$Cs sources} was then calculated for each measurement using a point model as follows:
\begin{equation}
A_H = K_pC_Hr^2_H,
\end{equation}
where $r_H$ is the measurement distance between the center of the detector and the point source, $C_{H}$ is the corrected net count rate of the gamma peak, $A_{H}$ is the activity of the point source, and $K_p$ is the calibration constant.
To obtain the calibration constant $K_p$, we measured a \added{$^{137}$Cs} point source of known activity at multiple positions along a line \added{perpendicular to the detector axis} following the standard procedure~\cite{russo2005gamma}, \added{as seen in Fig.~\ref{fig:calibration}. More precisely, the standard procedure is designed to allow the operator to obtain a detector response scalable from one dimension to two dimensions by moving the source to detector distance in a linear pattern. Further details on the standard approach can be found in reference~\cite{russo2005gamma} at \textbf{VIII.1.}}
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{pics/calibration_fig.png}
\caption{\added{Detector responses for the source with mass $m_0$ are measured at successive points along a line perpendicular to the detector axis at a known distance $r_0$ from the detector face.}}
\label{fig:calibration}
\end{figure}
For each measurement, the calibration constant $\overline{K_p}$ can be calculated as
\begin{equation}
\overline{K_p} = \frac{A_{cal}}{C_{cal}r^2_{cal}},
\end{equation}
where $C_{cal}$ is the corrected net count rate of the gamma peak, $A_{cal}$ is the known activity of the point source, and $r_{cal}$ is the distance between the center of the detector and the source. \added{The average of the calculated calibration constants $\overline{K_p}$ is then determined to be used as $K_p$ in Eq. 1 for the single detector measurement.}
\subsection{Linear Inverse Approach}
Suppose the holdup deposited inside a cross section of a pipe is contained in a arbitrary \added{bounding box} ${\Omega}$. Its distribution can be reconstructed by solving an inverse problem where the detector response and the measured counts are known, the solution yielding the most likely source distribution to produce the measured counts.
We first discretize the region of interest ${\Omega}$ into $N$ 1~cm side square pixels and we would like to know the distribution of the radioactivity expressed as $\mathbf{{\mathbf{x}}}$ in $\Omega$ as a function of the pixel number. We measure the counts $\mathbf{y}$ using a detector array encompassing $M$ detectors and assume the system to be linear:
\begin{equation}
\mathbf{y} = \mathbf{R} \mathbf{x} , \mathbf{y} \in \mathbb{R}^{M\times1},\mathbf{x} \in \mathbb{R}^{N\times1}, \mathbf{R} \in \mathbb{R}^{M\times N},\label{eq:linear_eq}
\end{equation}
where ${\mathbf{R}}$ is the system response matrix. ${\mathbf{R}}_{i,j}$ are the counts of detector $i$ when there is a source of unit activity in pixel $j$, which could be obtained by direct calculation or simulation.
For direct calculation, the response matrix ${\mathbf{R}}$ is calculated based on the gamma-ray attenuation law:
\begin{equation} \label{eq:3}
{\mathbf{R}}_{i,j} = f \omega_{\text{int}} \omega_{\text{geo}} e^{-\mu_a d_{a}} e^{-\mu_w d_{w}}
\end{equation}
$f$ is the branching ratio of the 662~keV \added{$^{137}$Cs}, \textit{i.e.}, 0.85. $\omega_{\text{int}}, \omega_{\text{geo}} $ is the intrinsic efficiency and geometric efficiency of detector $i$, respectively. $\mu_a,\mu_w$ are the attenuation coefficients of the 662~keV gamma rays in air and pipe wall, respectively. $d_a,d_w$ are the distances that the gamma rays travelled in air and pipe wall, respectively. If the pixel center is outside the pipe's cross section, the responses from each detector will be assumed to be $0$.
\added{Regardless of the measured pipe cross-section's geometry, the described procedure of creating the response matrix will not change. It would suffice to enter the thickness of the pipe or any other changing parameter to the model to obtain a new response. If the source-to-detector distances or number of pixels are changed due to a different pipe diameter or altered detector positions, the response matrix will also need to be recalculated. Given the dimensions of the pipe are well mapped out, the response matrix will fit to the geometry of the pipe's cross section within its spatial resolution. The spatial resolution of the response can be adjusted by changing the dimensions of the pixels.}
Given a $M\times N$ design matrix ${\mathbf{R}}$ and a target matrix ${\mathbf{y}}$ with $M$ elements, we can solve Eq.~\ref{eq:linear_eq} for the source distribution $\mathbf{x}$. However, this linear system is usually under-determined because $M << N$. We solve the following optimization problem to get the minimum mean square error estimate of $\mathbf{x}$:
\begin{equation}
\hat{{\mathbf{x}}} = \arg \underset{{\mathbf{x}}}{\min} \{\|{\mathbf{R}} {\mathbf{x}} - {\mathbf{y}}\|^2 + \lambda||x||_1\}, {\mathbf{x}}_i \geq 0, i \in \mathbb{Z} \cap [1,N],\label{eq:mmse}
\end{equation}
where $||x||_1$ stands for the sum of the absolute values of the components of ${\mathbf{x}}$. We use a Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) to perform the minimization~\cite{fista2009}, with $\lambda = 10^{-7}$. The minimization process terminates when the error $\|{\mathbf{R}} {\mathbf{x}}\ - {\mathbf{y}}\|$ converges, i.e., two subsequent errors have a relative change less than $10^{-4}$. \added{FISTA was chosen due to its favorable computational time for an operator in a facility setting.}
\subsection{Experimental Validation of a 2D case in a Simplified Geometry}
Fig.~\ref{fig:exp_sketch} shows the experimental setup that we used to validate the linear-inverse approach. Two
\added{$^{137}$Cs} sources with a total activity of 70.80 $\pm$ 14.16~kBq~\added{1SD} are stacked on one another at the center of an aluminum cylindrical pipe of thickness 0.6~cm.
Three 2''~$\times$~2'' NaI(Tl) detectors are then positioned around the center of the pipe. Lead collimators with an aperture of 8~mm and thickness 1.79~cm are used for each detector, which are able to attenuate up to 90\% of the gamma rays.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{pics/model3d.png}
\caption{3D sketch of the experiment. Center of pipe made transparent to show two \added{$^{137}$Cs} sources.}
\label{fig:exp_sketch}
\end{figure}
We measured the gamma-ray spectra for 60 minutes, then calculated the net counts of the 662-keV peak by subtracting the background and Compton continuum from the raw spectra. The collection of the net counts from the three detectors is the target matrix ${\mathbf{y}}$.
As the outside diameter of the pipe is 11.4 cm, the region of interest in this experiment is a square of area $12\times12$~cm${}^2$ encompassing the cross section of the pipe. The square is divided into 144 square pixels of area $1\times1$~cm${}^2$, as shown in Fig.~\ref{fig:2d_pipe}. We are interested in the source activity ${\mathbf{x}}_i$ in each pixel, $i=1,\cdots, 144$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\linewidth]{pics/2d_pipe_discretization.pdf}
\caption{2D discretization of the pipe cross section.}
\label{fig:2d_pipe}
\end{figure}
\added{The geometric efficiency is determined for each detector based on the source-detector geometry in the experimental validation, given by the fractional solid angle subtended by the detector crystal at the assumed point source: }
\begin{equation}
\omega_{\text{geo}} = \frac{1}{2}\left(1- \frac{d}{\sqrt{d^2 + r^2}} \right),
\end{equation}
\added{where $d$ is the source-detector distance and $r$ is the detector radius.
Since the NaI(Tl) detector is a cylinder, the distance $d$ to the source is not uniform. We use the average distance $\overline{d}$ to the detector instead, given by}
\begin{equation}
\overline{d} = \sqrt{\frac{r^2}{2} + \frac{h^2}{12} + c^2},
\end{equation}
\added{where $h$ is the detector length, $r$ is the detector radius and $c$ is the distance from the source to detector center~\cite{knoll2010radiation}.}
\added{The intrisnic efficiency is then determined for each detector using a single measurement with a \added{$^{137}$Cs} source at a known position from the detector. The intrinsic peak efficiency is given according to the standard procedure in reference~\cite{knoll2010radiation} by}
\begin{equation}
\omega_{\text{int}} = \frac{N}{\omega_{\text{geo}} \times BF \times BR \times T \times A},
\end{equation}
\added{where $N$ is the measured counts in the peak area, $\omega_{\text{geo}}$ is the fractional solid angle between the measured source and detector, $BF$ is the branching fraction, i.e., 100\%, $BR$ is the branching ratio, i.e., 85.10\%, $T$ is the acquisition time of the detector, and $A$ is the activity of the measured source in Bq.}
\added{In our case, the calibration source and the experimental test source are both $^{137}$Cs, with the \added{$^{137}$Cs} source used as one of the two sources in the experimental validation measurement. Thus, the intrinsic efficiency factored into the response matrix matches the actual measurement intrinsic efficiency. This favorable condition may not be possible in a real world case where a weighed efficiency as a function of the energy would be used in the response matrix. }
The system response matrix ${\mathbf{R}}$ is then calculated using Eq.~\ref{eq:3}.
We solve the inverse problem for $\mathbf{{\mathbf{x}}}$ using Eq.~\ref{eq:mmse} of our inverse linear approach with $M = 3$ and $N = 144$, i.e., ${\mathbf{R}} \in \mathbb{R}^{3\times 144}$, ${\mathbf{y}} \in \mathbb{R}^{3 \times 1}$, and ${\mathbf{x}} \in \mathbb{R}^{144 \times 1}$. The calculated activity in kBq of the source inside the pipe is then given by
\begin{equation}
\begin{aligned}
A_{cal} = \frac{\sum_{i=1}^{N} {\mathbf{x}}_i}{t} \times \frac{1 \text{ kBq}}{1000 \text{ Bq}},
\end{aligned}
\end{equation}
where $t$ is the measurement time in seconds.
\subsection{Validation via Simulation}
We simulated a two-dimensional (2D) and three-dimensional configuration to examine the inverse approach in greater detail. The simulation is performed using MCNP6.2~\cite{osti_1419730}. We simulated both ${\mathbf{R}}$ and ${\mathbf{y}}$ by recording the detector response and output to a source placed in the pipe. $\mathbf{{\mathbf{x}}}$ is finally calculated using our linear inverse approach. \added{The statistical counting error is not propagated in the reconstruction.}
\subsubsection{2D simulation}
We simulated the experiment with the same geometry and detector positions in our experimental validation, using a single \added{$^{137}$Cs} source with a source strength of $10^8$ gamma rays per second. We similarly obtained the response matrix containing 144 square pixels by simulating the detector response to a source of unit activity in each pixel. Given the simulated counts ${\mathbf{y}} \in \mathbb{R}^{3 \times 1}$ and responses ${\mathbf{R}} \in \mathbb{R}^{3\times 144}$ from the three detectors, our goal is to determine the activity in each pixel ${\mathbf{x}} \in \mathbb{R}^{144}$ and reconstruct a 2D image of the source distribution by solving for the desired source distribution using the linear inverse approach.
\subsubsection{3D simulation}
In this case, we are interested in the 3D source distribution inside the pipe. We simulated multiple source distributions \added{of uniform density} with an overall source strength of $10^8$ gamma rays per second, with all sources located at the bottom of the pipe:
\begin{description}
\item [Source 1]: A point source located at $z = 0~\text{cm}$, as shown in Fig.~\ref{fig:3d_simu_detector1}.
\item [Source 2]: A thin wire of square cross section $1\text{cm}^2$ spanning $-4~\text{cm} \leq z \leq 4~\text{cm}$, as shown in Fig.~\ref{fig:3d_simu_detector2}.
\item [Source 3]: A deposit spanning $-4~\text{cm} \leq z \leq 4~\text{cm}$ of height 1~cm, as shown in Fig.~\ref{fig:3d_simu_detector3}.
\item [Source 4]: \added{In addition to a standard horizontal pipe, we simulated a deposit in an S-shaped pipe as shown in Fig.~\ref{fig:3d_simu_detector4}.}
\end{description}
\begin{figure}[!htbp]
\captionsetup{font=footnotesize}
\centering
\begin{subfigure}{0.8\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=0.65\linewidth]{pics/3DFig.png}
\caption{Source 1}
\label{fig:3d_simu_detector1}
\end{subfigure}%
\newline
\begin{subfigure}{0.9\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=0.7\linewidth]{pics/source1diagram.png}
\caption{Source 2}
\label{fig:3d_simu_detector2}
\end{subfigure}
\newline
\begin{subfigure}{0.9\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=0.7\linewidth]{pics/source2diagram.png}
\caption{Source 3}
\label{fig:3d_simu_detector3}
\end{subfigure}%
\newline
\begin{subfigure}{.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/PipeSourceModel.png}
\caption{\added{Source 4. Detector rings indicate the starting and ending position of the measurements.}}
\label{fig:3d_simu_detector4}
\end{subfigure}%
\caption{(a,b,c) Cross section of the pipe with source and (b,c\added{,d}) 3D source distribution with source, pipe, and collimated detector ring shown. The detector collimators are not shown. \added{The cross section of the aluminum pipe with source 4 is equivalent to source 3.}}
\label{fig: 3d_simu}
\end{figure}
A detector ring containing 36 lead-collimated NaI(Tl) detectors surrounded the pipe and measured the counts at different cross-sections. For sources 1, 2, and 3, the detector ring began at $z = -5$~cm and was moved up to $z = 5$~cm, with the counts recorded in $1$~cm increments.
\added{In source 4, the detector ring traveled in $1$~cm increments along the horizontal and vertical sections of the pipe, measuring once at each corner with an angle of \ang{45} with respect to the horizontal and vertical. To resemble $1$~cm measurements along the corner, we artificially created eight new measurements at each corner. Four measurements were created between the known corner measurement and the nearest vertical/horizontal measurement, with the estimated measured counts of the detectors as the arithmetic means between the known recorded counts.}
To obtain a spatial resolution of the 3D response matrix, we discretized the entire pipe into voxels of 1~cm $\times$ 1~cm $\times$ 1~cm volume, as shown in Fig.~\ref{fig:3d_pipe} for a standard horizontal pipe.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\linewidth]{pics/3d_pipe_discretization.pdf}
\caption{3D discretization of the pipe.}
\label{fig:3d_pipe}
\end{figure}
\noindent Let the voxel whose center is
\begin{equation}
\begin{aligned}
&(x_i,y_j,z_k) = (i-6.5, j-6.5,k), \\
&1\leq i \leq 12, 1\leq j \leq 12, (i,j,k) \in \mathbb{Z}^3
\end{aligned}
\end{equation}
be labeled by $(i,j,k)$ and $\mathbf{M}_n(i,j,k)$ be the response of the $n$th detector at $(x_n, y_n,0)$ to a source of unit activity at voxel $(i,j,k)$. In the creation of the response matrix, we assume that the detector response only depends on the relative distance between the voxel center and the detector front face. Thus, the response of detector $n$ at $(x_n,y_n,l)$ for the voxel $(i,j,k)$ is equivalent to $\mathbf{M}_n(i,j,|k-l|)$. For each detector, a total of $144 \times 10$ simulations are performed to determine the detector's response at each cross-section of the pipe up to 9~cm away. Therefore, we further assume the response for $|k-l| > 9$ is effectively 0 due to the long distance between the source and detector. We then proceed to solve for the source distribution in the measured cross-sections of the pipe using the linear inverse approach.
\added{This method of discretization was assumed to be true for all pipes simulated, regardless of actual geometry.}
\section{Results}
\subsection{Bias Introduced in the GGH model}
According to the standard procedure, we followed the preliminary measurement which resulted in the incorrect
We calculated the source mass based on the point model for each distance between two \added{$^{137}$Cs} sources and compared it to the actual mass, as shown in Table~\ref{table:mass estimate point model}. The assumption that the source deposition geometry is a point leads to an underestimate of source mass up to 30\%, and this effect can be seen to increase as the source-to-source distance increases. \added{This underestimate may additionally be partially due to the source-to-detector distance of approximately 10 cm was not optimized to perform a measurement in good geometry~\cite{knoll2010radiation}.}
\begin{table}[!htbp]
\caption{Percent Differences between Calculated Activity and Real Activity.}
\label{table:mass estimate point model}
\centering
\resizebox{1\linewidth}{!}{
\begin{tabular}{cccc}
\hline
\hline
Source to Source Distance (cm) & \added{Real Activity} (kBq) & \added{Calculated Activity} (kBq) & Percent Difference \\
\hline
0 & 70.80 & 68.24 & 3.69\%\\
2 & 70.80 & 66.22 & 6.69\% \\
4 & 70.80 & 62.89 & 11.83\% \\
6 & 70.80 & 60.57 & 15.58\% \\
8 & 70.80 & 56.31 & 22.80\% \\
10 & 70.80 & 52.31 & 30.04\% \\
\hline
\hline
\end{tabular}}
\end{table}
\begin{figure}[!htbp]
\centering
\begin{subfigure}{.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=\linewidth]{pics/3DetExperiment_v2.png}
\caption{\added{Experiment}}
\label{fig:2d_result}
\end{subfigure}%
\begin{subfigure}{.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=\linewidth]{pics/Sim_3de.png}
\caption{2D simulation}
\label{fig:2dsim_result}
\end{subfigure}%
\caption{2D source distribution inside the pipe, i.e. the elements of the calculated $\mathbf{{\mathbf{x}}}$ on a 2D regular raster. The red circles show the pipe wall cross section. \added{The reconstructed experimental activity distribution exhibits a percentage difference between the green and yellow pixels of approximately 20\%}}
\label{fig:2d_recon}
\end{figure}
\subsection{Experimental Validation in a Simplified Geometry}
Fig.~\ref{fig:2d_result} shows the reconstructed 2D source distribution using FISTA, with the activity estimated as \added{64.00}~kBq.
Compared with the actual source distribution, the source location was accurately determined. Furthermore, the source activity was well within the activity uncertainty provided by the manufacturer, i.e., 70.80±14.16 kBq.
\subsection{Validation via Simulation}
\subsubsection{2D Reconstruction and Activity Quantification}
Fig.~\ref{fig:2dsim_result} shows the reconstructed 2D source distribution based on the simulated detector counts. The activity calculated using FISTA was $99956.15$~kBq, with a relative error of 0.044\%.
Both the source location and activity were accurately determined.
\added{One may notice that the source distribution in Fig.~\ref{fig:2d_result} is slightly wider than Fig.~\ref{fig:2dsim_result}. This artifact may be due to a combination of effects including a slightly incorrect detector and source position assessment, in addition to the choice of iteration parameters in FISTA. As FISTA undergoes more iterations after the stopped iteration count of the experimental validation, the calculated source distribution from Fig.~\ref{fig:2d_result} will become to be only seen in the bottom center two pixels, similar to Fig.~\ref{fig:2dsim_result}.}
\subsubsection{3D Reconstruction and Source Activity Quantification}
We reconstructed the 3D source distributions using FISTA for all four simulated cases, as shown in Fig.~\ref{fig: 3d_simu_result}. The estimated source activities and systematic errors are summarized in Table~\ref{table:activity estimate 3D}. \added{Other than source 4, both the source locations and activities were determined accurately in the 3D reconstructions.}
\begin{figure}[!htbp]
\captionsetup{font=footnotesize}
\centering
\begin{subfigure}{.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/pointSource.png}
\caption{Source 1.}
\label{fig:3d_resultpoint}
\end{subfigure}%
\begin{subfigure}{.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/source1_3Distribution.png}
\caption{Source 2}
\label{fig:3d_resultsource2}
\end{subfigure}
\newline
\begin{subfigure}{0.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/source2_3Distribution.png}
\caption{{Source 3}}
\label{fig:3d_resultsource3}
\end{subfigure}%
\begin{subfigure}{0.5\linewidth}
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/Sshape_3Distribution.png}
\caption{\added{Source 4}}
\label{fig:3d_resultsource4}
\end{subfigure}%
\caption{Reconstructed 3D source distributions.}
\label{fig: 3d_simu_result}
\end{figure}
\begin{table}[!htbp]
\caption{Comparison between the estimated total activity and actual activity.}
\label{table:activity estimate 3D}
\centering
\resizebox{.7\linewidth}{!}{
\begin{tabular}{cccc}
\hline
\hline
Source & Actual activity (kBq) & Estimated activity (kBq) & Relative error \\
\hline
1 & 100000 & 101124.78 & 1.12\%\\
2 & 100000 & 99625.42 & -0.37\% \\
3 & 100000 & 96081.92 & -3.92\% \\
\added{4 (FISTA $\lambda = 10^{-7}$)} & 100000 & 105317.99 & 5.32\% \\
\added{4 (FISTA $\lambda = 10^{-2}$)} & 100000 & 105891.07 & 5.89\% \\
\added{4 (NAG)} & 100000 & 104144.20 & 4.14\% \\
\hline
\hline
\end{tabular}}
\end{table}
\added{The high relative error and partially incorrect source localization seen in source 4 using FISTA is due to the current values set for the parameters in the minimization. The presence of the $l_{1}$ term attempts to induce sparsity in the solution of Eq.~\ref{eq:mmse}~\cite{fista2009}. Thus, as the minimization undergoes more iterations, the optimal solution will contain lower source activity in certain pixels. Adjusting the regularization parameter $\lambda$ and the total number of iterations before terminating the minimization will improve the calculated source activity and location with FISTA. However, we kept the FISTA reconstruction parameters consistent in this analysis and optimized them based on source 2.}
\added{To improve on the results seen in source 4 using FISTA, we tested different reconstruction parameters and found that using $\lambda = 10^{-2}$ and stopping minimization at 30 iterations would result in a favorable source distribution well resembling the simulated one as seen in Fig.~\ref{fig:3d_resultsource4FISTA}, with the calculated activity at Table~\ref{table:activity estimate 3D}.}
\begin{figure}[!htbp]
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/SshapeFISTA.png}
\caption{\added{Source 4 (FISTA with $\lambda = 10^{-2}$ and 30 iterations)}}
\label{fig:3d_resultsource4FISTA}
\end{figure}%
\added{In addition, we believed it would be beneficial to show that using a different method can possibly improve reconstruction results of specific sources. An alternative gradient descent optimization algorithm, i.e., Nesterov accelerated gradient (NAG) descent, was used. Further details on NAG can be found at reference~\cite{NAG}.}
\added{The use of the NAG algorithm resulted in a smaller relative error in the calculated source activity and a cleaner, more accurate source localization as shown in Table~\ref{table:activity estimate 3D} and Fig.~\ref{fig:3d_resultsource4NAG}, respectively.}
\begin{figure}[!htbp]
\captionsetup{font=footnotesize}
\centering
\includegraphics[width=1\linewidth]{pics/SshapeNAG.png}
\caption{\added{Source 4 (NAG)}}
\label{fig:3d_resultsource4NAG}
\end{figure}%
\newpage
\section{Conclusions}
In this work, we demonstrated that current methods of holdup assay based on the GGH model could lead to significant underestimation of holdup mass due to the oversimplification of source geometry. We have developed an inverse-problem based approach, utilizing an array of detectors that rely on the accurate modeling of a system response matrix to reconstruct an image of the source distribution and estimate the source activity to measure the holdup material. We were able to accurately determine the source location in the experiment, and both the source location and activity in the 2D simulation. Similarly, we extended the notion of the response matrix from 2D to 3D, obtaining an accurate estimation of both the source location and activity in varied 3D simulations.
\added{The maximum activity discrepancy was reported in source 4, which is due to the current values set for the parameters in the minimization. Sources inside other pipe geometries could be simulated to test this occurrence with FISTA further, with a possible solution of setting different parameters in FISTA's minimization depending on the measured geometry of the pipe.} Conversely, the method was always able to localize the source accurately within the response matrix's spatial resolution. In principle, smaller voxels could be used to achieve a more accurate source localization. However, this would entail the solution of a linear system with a higher number of unknowns.
\added{Therefore, an array encompassing more detectors may be needed to improve the spatial resolution. The detectors' size may need to be reduced to keep the current system form factor. Robotic automated motion of the detector array would be particularly advantageous in this case since its proximity to the source and an adaptive speed would allow to accumulate the needed integral counts needed to achieve a fine 3D reconstruction. Minimally tethered soft robots~\cite{softrobot2} equipped with such array seem particularly suitable to accomplish this goal, guaranteeing an optimal solid angle regardless of the size or shape of the item inspected by crawling over the surface of the piece of equipment.}
Our method can be generalized to pipes of various curvatures and holdup of varying geometries, implementable into a minimally-tethered soft robot capable of reducing the manual labor and cost associated with holdup measurement. Using an i7-8565U processor, the process of minimization using FISTA for all tested sources had a maximum run-time of approximately \added{4} seconds, \added{heavily dependant on the number of measurements taken before minimization}. Given the fast convergence, solving the source distribution and activity requires little computational cost and could be implemented on off-the-shelf electronic platforms enabling real-time holdup imaging inside a fuel processing facility.
\section*{Acknowledgements}
This work is funded in-part by the Consortium for Verification Technology under Department of Energy National Nuclear Security Administration award number DE-NA0002534 and by the Nuclear Regulatory Commission Faculty Development Grant number 31310019M0011.
\section*{References}
|
1,314,259,992,974 | arxiv | \section{Introduction}
In this paper, we examine vorticity gradient growth in 3D {\it axisymmetric flows without swirl}. Global well-posedness of the 3D Euler equations for this class of flows is shown in the work of Ukhovsksii and Yudovich \cite{Yudovich} (see \cite{majda2002vorticity, Raymond, Shirota} for related results). This class of flows shares similarities with two-dimensional flows where well-posedness is also well-known \cite{majda2002vorticity, Holder, Marchioro, Wolibner}. In two dimensions, vorticity is conserved along particle trajectories while in 3D axisymmetric flows without swirl, the quantity $\omega^\theta(r,z)/r$ plays an analogous role where $\omega^\theta$ is the angular component of vorticity in cylindrical coordinates and $r$ being the radial variable. This fact is key in showing well-posedness for this class of 3D flows as this provides a priori bounds for vorticity.
\smallskip
Our first main result concerns an upper bound on the growth of the gradient of $\omega^\theta/r$. For 2D flows, the upper bound for the gradient of vorticity is double exponential growth in time \cite{Yudovich1}. We will prove a similar upper bound in the axisymmetric case. However, we also show that this upper bound improves to essentially exponential growth near the axis. We make this precise in the next section. This result will serve in contrast to 2D Euler flows as we rule out any double exponential growth
of the gradient at the axis. The special structure of the axisymmetric Biot-Savart law is used.
\smallskip
For our second result, we explore the sharpness of this upper bound and construct an example of double exponential gradient growth. Such growth will occur on the boundary of the unit ball $B(0,1)=\{(r,z): \, r^2+z^2\le 1\}$ away from the axis. For the 2D Euler equations, there have been a number of recent results concerning the gradient growth of vorticity \cite{KiselevSverak, Denisov1, Denisov2, Zlatos}. The techniques we will use bear most resemblance to those of Kiselev and Sverak \cite{KiselevSverak} who construct an example of double exponential vorticity gradient growth on the boundary of a unit disk. Their initial data is inspired by
the ``singular cross" of Bahouri and Chemin \cite{BahouriChemin} and the authors show
that particle trajectories are approximately hyperbolic near the desired point of gradient growth. We will construct an initial data inspired by this scenario for the ball. In order to prove that trajectories have
such structure, we require a closed-form expression for the Green's function of an elliptic operator
to be specified below. This is one of the primary reasons the ball is chosen for our domain rather than the more natural choice of a cylinder. We anticipate that our construction will work on other domains such as a cylinder.
\smallskip
For both of our results, the proofs rely on adequate expressions and estimates for the Biot Savart law for axisymmetric flows without swirl. In the axisymmetric setting, the Biot Savart law is considerably more complicated than the law for fluid velocity $u$ and vorticity $\omega$ in 2D Euler which is
\begin{align*}
u(x,t) = \nabla^\perp \int_D G_D(x,y)\omega(y,t) dy.
\end{align*}
Here, $G_D$ is Green's function for the Laplacian for the Dirchlet problem on a 2D simply connected domain $D$. In section 2.2, we make a precise statement of the Biot Savart law we use. Similar to 2D Euler, one can express the velocity in terms of the vorticity $\omega^\theta$ integrated against some kernel. Away from the axis, this kernel has similar estimates as $\nabla^\perp G_D$, but near the axis, the kernel will have better decay estimates. This similarity away the axis will lead to the double exponential growth at the boundary away from the axis. Additionally, the better kernel decay for points near the axis will lead to our improved upper bound referenced above.
\smallskip
We believe our work is a small step toward bridging ideas of small scale creation in 2D to 3D. For the 3D axisymmetric Euler equations {\it with} swirl, a potential scenario for singularity formation was proposed Luo and Hou \cite{HouLuo1} based upon their numerical simulations. A singularity is reported on the boundary of a cylinder and flow is observed to have hyperbolic structure. In our setting without swirl, we produce an example with double exponential gradient growth where the flow has hyperbolic-type structure. Proving singularity formation for Euler flows with swirl would require many deep new ideas. Another interesting question is the possible singularity formation of the 3D axisymmetric Euler equations with swirl at the axis of symmetry.
\section{The setup}
Consider the 3D Euler equations for a velocity field $u$ and pressure $p$
\begin{align}
\label{euler1}
u_t+ u\cdot \nabla u +\nabla p &=0 \\
\nabla\cdot u &=0 \\
u(\cdot, 0) &= u_0
\end{align}
on $D\times (0,\infty)$ where $D$ is either the unit ball $B(0,1)$ or a
finite radius cylinder with periodic boundary condition in $z$.
In addition, we have no-flow condition on the solid boundary:
\begin{align*}
u\cdot n =0\quad \mbox{on} \quad \partial D.
\end{align*}
Here, we consider $u$ which is axisymmetric without swirl. Specifically, the velocity field $u$ will have the form
\begin{align*}
u(r,z,t)=u^r(r,z,t)\, {\bf e_r}+ u^z(r,z,t)\,{\bf e_z}
\end{align*}
where ${\bf e_r}=\begin{bmatrix} \cos \theta & \sin \theta & 0 \end{bmatrix}^t$ and ${\bf e_z}=\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}^t$ are unit vectors from cylindrical coordinates, the other unit vector being ${\bf e_\theta}= \begin{bmatrix} -\sin \theta & \cos\theta & 0\end{bmatrix}^t$. Since the vector field $u$ has no ${\bf e_\theta}$ component, the flow $u$ is said to be without swirl. After a change of coordinates, $\eqref{euler1}$
becomes
\begin{equation}
\label{euler}
(\partial_t + u^r \partial_r+ u^z \partial_z)w=0, \quad\quad w(r,z,t)= \frac{\omega^\theta(r,z,t)}{r}
\end{equation}
where $\omega^\theta$ is the angular component of vorticity $\omega :=\mbox{curl}\, u$. Throughout, we will assume $w_0(r,z)=w(r,z,0)\in L^\infty$ and so $\|w(\cdot,t)\|_{L^\infty}\le \|w_0\|_{L^\infty}$. The relations between vorticity, velocity, and stream function are as follows
\begin{align}
\label{vorticity}
\omega^\theta=u_z^r-u_r^z, \quad u^r = -\frac{\psi_z}{r}, \quad u^z = \frac{\psi_r}{r}
\end{align}
which can be combined to get
\begin{align}
\label{law}
L\psi := -\frac{\psi_{rr}}{r^2}+\frac{\psi_r}{r^3}-\frac{\psi_{zz}}{r^2}= \frac{\omega^\theta}{r}=w.
\end{align}
Often times, it will be convenient to use an equivalent form of \eqref{law} which is
\begin{align}
\label{law1}
\mathscr{L}\left(\frac{\psi}{r}\right) := -\left(\partial_r^2 +\frac{1}{r}\partial_r +\partial_z^2 - \frac{1}{r^2}\right) \frac{\psi}{r} =\omega^\theta.
\end{align}
From these relations, in the next section, we will derive the Biot-savart law relating $\omega^\theta$ and $u$. Thus, $\omega^\theta$ completely describes the flow.
\medskip
Due to axial symmetry, there are additional conditions at the axis. To ensure solutions remain smooth, the stream function must satisfy
\begin{align}
\label{pole}
\partial_r^{(2m)} \left.\left( \frac{\psi}{r}\right) \right|_{(0+,z)}=0 \quad\mbox{$m=0,1,2,\ldots$}
\end{align}
which implies the following conditions on $u$
\begin{align}
\partial_r^{(2m+1)} u^z(0+,z)=0, \quad \partial_r^{(2m)} u^r(0+,z)=0 \quad\mbox{$m=0,1,2,\ldots$}.
\end{align}
These conditions follow from the below lemma from Liu and Wang \cite{LiuWang}:
\begin{lemma}
\label{wang}
{\bf (a)}
Let $u$ be a $C^k$ smooth (in Cartesian coordinates) 3D axisymmetric vector field $u(r,\theta, z)=u^r(r,z) {\bf e_r} +u^\theta(r,z) {\bf e_\theta} + u^z(r,z){\bf e_z}$. Then $u^r, u^\theta, u^z\in C^k([0,\infty)\times \ensuremath{\mathbb{R}}\xspace)$ and
\begin{align*}
\partial_r^{2\ell +1} u^z(0^+,z) = 0, \quad 1\le 2\ell + 1\le k, \\
\partial_r^{2m} u^r(0^+,z)=\partial_r^{2m} u^\theta(0^+,z) = 0, \quad 0\le 2m\le k.
\end{align*}
\bigskip
{\bf (b)} Suppose $\phi(r,z)\in C^{k+1}([0,\infty),\ensuremath{\mathbb{R}}\xspace), f\in C^k([0,\infty),\ensuremath{\mathbb{R}}\xspace)$ satisfying $\partial_r^{2m} \phi(0^+,z)=0$ for $0\le 2m\le k+1$ and
$\partial_r^{2\ell} f(0^+,z)=0$ for $0\le 2\ell \le k$. Then the vector field
\begin{align*}
u := -\partial_z \phi\, {\bf e_r}+ \frac{\partial_r(r\phi)}{r}\, {\bf e_z} + f\, {\bf e_\theta}
\end{align*}
for $r>0$ is a $C^k$ smooth (in Cartesian coordinates) 3D axisymmetric vector field with a removable singularity at $r=0$.
\end{lemma}
\noindent
The above lemma has a direct analogoue for vector fields over the domains we consider. Also, we note that the incompressibility condition becomes
\begin{align}
\label{divfree}
(ru^r)_r+ (ru^z)_z=0
\end{align}
in cylindrical coordinates.
Recall the similarity between the system \eqref{euler}, \eqref{vorticity}, and \eqref{law} with the 2D Euler equation in vorticity form:
\begin{align*}
\omega_t+ u\cdot \nabla \omega =0, \quad -\Delta\psi= \omega, \,\,\, u= \nabla^\perp \psi.
\end{align*}
\subsection{Statement of Main Results}
We will consider the system on the domain $D$ subject to no-flow boundary condition on solid boundaries
\begin{align*}
u\cdot n =0\quad\mbox{on}\quad \partial D.
\end{align*}
Again, we will only consider the case when $D$ is the unit ball or a finite radius cylinder with periodic boundary condition in $z$. Much of our work will generalize to other axially symmetric domains. We indicate specifically in
the proofs below where our specific domain choice is used.
\medskip
Define the trajectory map $\Phi_t(r,z)=(\Phi_t^r(r,z,t), \Phi_t^z(r,z,t))$ associated with \eqref{euler} by
\begin{align*}
\frac{d}{dt} \Phi_t(r,z) = u(\Phi_t(r,z),t), \quad \Phi_0(r,z)=(r,z).
\end{align*}
As the existence of classical solutions to \eqref{euler} is known \cite{Yudovich, majda2002vorticity, Raymond, Shirota},
the goal of our work is to address the sharpness of a priori bounds for such solutions.
Our first main result is the following upper bound on the gradient of $w$.
\begin{theorem}
\label{upper}
Let $\displaystyle w_0= \frac{\omega^\theta(r,z,0)}{r}\in C^1(D)$ and $\omega_0^\theta(0,z)=0$ for $(0,z)\in \overline{D}$.
Let $w(r,z,t)$ be the corresponding classical solution of \eqref{euler}.
\medskip
{\bf(a)}
We have the following double exponential in time growth estimate for the gradient of $w(r,z,t)$
\begin{align}
\label{upper1}
1+ \log\left(1+ \frac{\|\nabla w(\cdot,t)\|_{L^\infty}}{\|w_0\|_{L^\infty}}\right) \le \left(1+ \log\left(1+ \frac{\|\nabla w_0\|_{L^\infty}}{\|w_0\|_{L^\infty}}\right)\right)\exp(C\|w_0\|_{L^\infty }t).
\end{align}
for some constant $C$.
\medskip
{\bf(b)} Particle trajectories can only approach the axis of symmetry with at most exponential rate. That is, there
exists a constant $C$ dependent on $D$ and $w_0$ such that for every $x\in D$
\begin{align}
\label{lowb}
|\Phi_t^r(x)| \ge r \exp(-Ct)
\end{align}
\medskip
{\bf (c)} The solution $w(r,z,t)$ satisfies the following estimate
\begin{align*}
\sup_{r>0} \frac{|w(r,z,t)-w(0,z,t)|}{r} \le \|\nabla w_0\|_{L^\infty} \exp(Ct)
\end{align*}
\end{theorem}
For the 2D Euler equations, estimates similar to $\eqref{lowb}$ have been shown when additional symmetry is
imposed on the vorticity or the domain has a corner (\cite{Ito1, Elgindi, Ito2}). Next, we provide an example of double exponential growth at the boundary.
\begin{theorem}
\label{lower}
Consider the 3D axisymmetric Euler equations without swirl on the unit ball $B(0,1)$. There exists initial data $w_0$ such that $\|\nabla w_0\|_{L^\infty(B(0,1))}/\|w_0\|_{L^\infty(B(0,1))}>1$ and the solution $w(r,z,t)$ of \eqref{euler} satisfies the following lower bound
\begin{align*}
\frac{\|\nabla w(r,z,t)\|_{L^\infty}}{\|w_0\|_{L^\infty}} \ge \left( \frac{\|\nabla w_0\|_{L^\infty}}{\|w_0\|_{L^\infty}}\right)^{C\exp (C\|w_0\|_\infty t)}
\end{align*}
\end{theorem}
\medskip
\begin{comment}
On the whole space, we'll see that the Biot-Savart law can be written in the form
\begin{eqnarray*}
u^r(\bar{r},\bar{z}) &=& \int_{-\infty}^\infty \int_0^\infty \frac{(z-\bar{z})\sqrt{r}}{\pi \bar{r}^{3/2}} F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) w(r,z)\, dr\, dz \\ u^z(\bar{r},\bar{z}) &=& \int_{-\infty}^\infty \int_0^\infty \mathscr{L}(\bar{r},\bar{z}, r,z) w(r,z)\, dr\,dz
\end{eqnarray*}
where we define
\begin{align*}
\mathscr{L} &=& \frac{1}{\pi}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)\\
&\quad& +\frac{1}{4\pi}\left[ F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)-2\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)\right]\left(\frac{r}{\bar{r}}\right)^{3/2}
\end{align*}
\begin{align*}
F(s)=\int_0^\pi \frac{\cos \theta}{\sqrt{2(1-\cos\theta)+s}}\, d\theta
\end{align*}
\end{comment}
\subsection{Biot-Savart law}
\smallskip
Here, we will give a somewhat general derivation of the axisymmetric Biot-Savart law. Later on, we require rather explicit expressions for $u$ and its derivatives in terms of $w$. We'll emphasize the analogy with the 2D case rather than the general 3D Biot Savart law on domains which is highly non-trivial \cite{Enciso}. In order to derive the relation between $u$ and $\omega^\theta$, one must solve the following system
\begin{align}
\label{biotsavart}
u_z^r-u_r^z &=\omega^\theta \\
\mbox{div}\, u & = 0 \\
u\cdot n &= 0 \quad\mbox{on} \,\, \partial D
\end{align}
Recall, by the divergence-free condition \eqref{divfree}, there exists a scalar stream function $\psi(r,z)$ such that
\begin{align}
\label{psiperp}
u^r= - \frac{\psi_z}{r}\quad \mbox{and}\quad u^z= \frac{\psi_r}{r}
\end{align}
Now, let us find a vector potential $A$ such that
\begin{align}
\label{vectorid}
u(r,z,t)= u^r \, {\bf e_r} + u^z\, {\bf e_z} = \mbox{curl}\, A, \quad \mbox{div}\, A=0
\end{align}
Such an $A$ must satisfy
\begin{align}
\label{Aeq}
\omega^\theta\, {\bf e_\theta} = \mbox{curl}\, u = \mbox{curl}\, \mbox{curl}\, A = - \Delta A.
\end{align}
\begin{comment}
With the following lemma, we have an expression for $A$ in terms of the stream function.
\begin{lemma}
\label{lawform}
Consider \eqref{Aeq} with boundary condition $A|_{\partial D} =0$. Suppose $\omega^\theta(r,z)\in C^1$
and $\omega^\theta(0,z)=0$. Then the vector field $A$ only has ${\bf e_\theta}$ component. In addition, the ${\bf e_\theta}$ component of $A$ is $\theta$-independent. Moreover, $A$ has the following form
\begin{align}
\label{Apsi}
A= \frac{\psi(r,z)}{r}\, e_\theta
\end{align}
where $\psi$ satisfies
\begin{align}
\psi&=0\quad \mbox{on}\,\, \partial D \\
\mathscr{L} \left(\frac{\psi}{r}\right)&:= -\left(\partial_r^2 +\frac{1}{r}\partial_r +\partial_z^2 - \frac{1}{r^2}\right) \frac{\psi}{r}=\omega^\theta.
\end{align}
\end{lemma}
{\bigskip\noindent\bf Proof.\quad}
By lemma \ref{wang} and our hypothesis on $\omega^\theta$, $\omega^\theta\, {\bf e}_\theta$ is a continuous and bounded vector field and we have solvability of
equation $\eqref{Aeq}$.
Let us write $\eqref{Aeq}$ out in its components where $\Delta$ is now the scalar Laplacian
\begin{align*}
\begin{bmatrix} -\omega^\theta(r,z) \cdot \sin\theta \\ \omega^\theta(r,z)\cdot \cos\theta \\ 0 \end{bmatrix} = \begin{bmatrix} \Delta A_x \\ \Delta A_y \\ \Delta A_z \end{bmatrix}
\end{align*}
Now, we analyze each component. Let us expand $A_x$ into Fourier series
\begin{align}
\label{Ax}
A_x = \sum_{n=-\infty}^\infty \left( a_n^1(r,z)\sin(n\theta)+a_n^2(r,z)\cos(n\theta)\right)
\end{align}
and applying the Laplacian in cylindrical coordinates $\displaystyle \Delta f= \frac{1}{r}\partial_r\left(r \partial_rf\right)+\frac{1}{r^2}\partial_\theta^2f+ \partial_z^2 f$ we get
that
\begin{align*}
\partial_{r}^2 (a_n^1)+ \frac{1}{r} \partial_r (a_n^1)+\partial_{z}^2 (a_n^1)
-\frac{n^2}{r^2} a_n^1=0, \quad\mbox{for}\,\, n\ne 1
\end{align*}
\begin{align}
\label{a1}
\partial_{r}^2 (a_1^1)+ \frac{1}{r} \partial_r (a_1^1)+\partial_{z}^2 (a_1^1)
-\frac{1}{r^2} a_1^1= -\omega^\theta
\end{align}
\begin{align*}
\partial_{r}^2 (a_n^2)+ \frac{1}{r} \partial_r (a_n^2)+\partial_{z}^2 (a_n^2)
-\frac{n^2}{r^2} a_n^2=0, \quad\mbox{for all}\,\, n.
\end{align*}
From the boundary conditions, we can conclude the $a_n^2$'s are all zero and also $a_n^1$ for $n\ne 1$ are zero.
Simlarly, we can write
\begin{align}
\label{Ay}
A_y = \sum_{n=-\infty}^\infty \left( b_n^1(r,z)\sin(n\theta)+b_n^2(r,z)\cos(n\theta)\right)
\end{align}
and get following equations for the coefficients
\begin{align*}
\partial_{r}^2 (b_n^1)+ \frac{1}{r} \partial_r (b_n^1)+\partial_{z}^2 (b_n^1)
-\frac{n^2}{r^2} b_n^1=0, \quad\mbox{for all}\,\, n
\end{align*}
\begin{align}
\label{b1}
\partial_{r}^2 (b_1^2)+ \frac{1}{r} \partial_r (b_1^2)+\partial_{z}^2 (b_1^2)
-\frac{1}{r^2} b_1^2= \omega^\theta
\end{align}
\begin{align*}
\partial_{r}^2 (b_n^2)+ \frac{1}{r} \partial_r (b_n^2)+\partial_{z}^2 (b_n^2)
-\frac{n^2}{r^2} b_n^2=0, \quad\mbox{for}\,\, n\ne 1.
\end{align*}
In addition, since $A_z=0$ on the boundary and $\Delta A_z=0$, $A_z=0$. By uniqueness and equations \eqref{a1}, \eqref{b1}, we have $a_1^1 = -b_1^2$. Furthermore, define the function $\psi(r,z)$ by $\psi = r\cdot b_1^2$. Combining everything together, the vector $A$ has the form
\begin{align*}
A = \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix} = \begin{bmatrix} a_1^1 \sin\theta \\ b_1^2 \cos\theta \\ 0 \end{bmatrix} = \frac{\psi(r,z)}{r} e_\theta.
\end{align*}
By construction $\psi$ will satisfy (19) and (20). $\Box$
\end{comment}
\begin{comment}
By using the well-posedness of Poisson equation in each of the three componenst above along with the expression for the scalar Laplacian in cylindrical coordinates, we see that with the boundary condition $A|_{\partial D}=0$, $A= \psi(r,z)/r \cdot e_\theta = \psi(r,z)/r \cdot [-\sin \theta \,\, \cos\theta \,\, 0]^T$ is the unique solution of the system where $\psi$ satisfies (19)-(21). $\Box$
First, we show $A$ only has no $z$ component. The absence of $r$ component can be done similarly. By well-posedness theory for the Poisson equation, we are justified in writing the $z$ component $A_z$ as
\begin{align}
\label{Az}
A_z = \sum_{n=-\infty}^\infty \left( a_n^1(r,z)\sin(n\theta)+a_n^2(r,z)\cos(n\theta)\right)
\end{align}
for some coefficient functions $a_n^{1,2}$.
Using that in cylindrical coordinates, $\displaystyle \Delta f= \frac{1}{r}\partial_r\left(r \partial_rf\right)+\frac{1}{r^2}\partial_\theta^2f+ \partial_z^2 f$, we can use that $\Delta A_z=0$ and derive the following relations for the $a_n^1$'s
\begin{align*}
\partial_{r}^2 (a_n^1)+ \frac{1}{r} \partial_r (a_n^1)+\partial_{z}^2 (a_n^1)
-\frac{n^2}{r^2} a_n^1=0
\end{align*}
Since the boundary value of the $a_n^1$'s are $0$, we can use uniqueness of solutions and conclude $a_n^1=0$ for all $n$. Similarly, $a_n^2=0$ for all $n$. This implies $A_z=0$.
\smallskip
Now that $A$ only has $\theta$ component, we can see quickly that this component is also independent of $\theta$. Indeed, we can take the radial component of the equation \eqref{Aeq} and use the expression for the vector Laplacian in cylindrical coordinates and get
\begin{align*}
\left(\Delta A_r -\frac{A_r}{r^2}-\frac{2}{r^2}\partial_\theta (A_\theta)\right)\, e_r = 0
\end{align*}
Since $A_r=0$, $A_\theta$ is independent of $\theta$.
\smallskip
The equation $\eqref{Apsi}$ follows from $\eqref{psiperp}$ and $\eqref{vectorid}$.
$\Box$
\end{comment}
\medskip
Now, using the Dirchlet Green's function for the (scalar) Laplacian in 3D, we can write an expression for the Biot Savart law.
First, we write the Green's function for the Laplacian as
\begin{align}
\label{3DGreen}
G_D(x,y)= \frac{1}{4\pi|x-y|}+h(x,y)
\end{align}
where for each $y\in D$ the corrector function $h$ solves
\begin{align*}
\Delta_x h(x,y) &= 0 \\
h(x,y) &=-\frac{1}{4\pi|x-y|}, \,\, x\in \partial D
\end{align*}
Writing $y$ in cylindrical coordinates $y=(r,\theta,z)$ and without loss of generality asumming $x=(\bar{r},0,\bar{z})$, we have
\begin{align*}
|x-y| = \sqrt{r^2-2r\bar{r}\cos \theta +\bar{r}^2+(z-\bar{z})^2}.
\end{align*}
Without loss of generality we can refer to $D$ as our given axially symmetric domain in $\ensuremath{\mathbb{R}}\xspace^3$ or express it in coordinates $(r,z)$, $r\ge 0$, $z\in \ensuremath{\mathbb{R}}\xspace$, depending on context. This abuse of notation can be justified since we are in the axisymmetric setting so quantities depend only on their values on one $\theta$ plane of $D$. In cylindrical $(r,z)$ coordinates, $\partial D$ will only be points that correspond to boundary points in 3D. That is, the points on the axis that are not boundary points in 3D do not ``become" boundary points once in $(r,z)$ coordinates.
Then we write the ${\bf e_\theta}$ component of $A$ as
\begin{align*}
A^\theta(\bar{r},\bar{z})=\int_D \mathscr{A}^\theta(\bar{r},\bar{z},r,z) \omega^\theta(r,z) \, dr\, dz
\end{align*}
where we define
\begin{align}
\label{eq:A}
\mathscr{A}^\theta(\bar{r},\bar{z},r,z) = \frac{1}{4\pi}\int_0^{2\pi} r\cos\theta\left(\frac{1}{\sqrt{r^2-2r\bar{r}\cos\theta+\bar{r}^2+(z-\bar{z})^2}}+h(\bar{r},\bar{z},r,\theta,z)\right) \, d\theta.
\end{align}
The $r\cos\theta$ factor above comes from $\eqref{Aeq}$ where the vector Laplacian is used.
By the following lemma, we have a relation between
$A$ and $\psi$.
\begin{lemma}
\label{lawform}
Consider \eqref{Aeq} with boundary condition $A|_{\partial D} =0$. Suppose $\omega^\theta(r,z)\in C^1$
and $\omega^\theta(0,z)=0$. Then the vector field $A$ only has ${\bf e_\theta}$ component. In addition, the ${\bf e_\theta}$ component of $A$ is $\theta$-independent. Moreover, $A$ has the following form
\begin{align}
\label{Apsi}
A= \frac{\psi(r,z)}{r}\, e_\theta
\end{align}
where $\psi$ satisfies
\begin{align}
\label{psi_bdy}
\psi&=0\quad \mbox{on}\,\, \partial D \\
\label{psi_eq}
\mathscr{L} \left(\frac{\psi}{r}\right)&:= -\left(\partial_r^2 +\frac{1}{r}\partial_r +\partial_z^2 - \frac{1}{r^2}\right) \frac{\psi}{r}=\omega^\theta.
\end{align}
\end{lemma}
{\bigskip\noindent\bf Proof.\quad}
By Lemma \ref{wang} and our hypothesis on $\omega^\theta$, $\omega^\theta\, {\bf e}_\theta$ is a continuous and bounded vector field and we have solvability of
equation $\eqref{Aeq}$.
\medskip
First, consider the case when $D$ is the unit ball $B(0,1)$. We can just directly compute.
Recall the Green's function for the ball in cylindrical coordinates
\begin{align}
\label{Green_ball}
G_B(r,z,\bar{r},\bar{z}) =\frac{1}{4\pi\sqrt{r^2-2r\bar{r}\cos\theta+\bar{r}^2+(z-\bar {z})^2}}- \frac{1}{4\pi\sqrt{r^2+z^2} \sqrt{(r^\ast)^2-2r^\ast\bar{r}\cos\theta+\bar{r}^2+(z-\bar{z})^2}}
\end{align}
where $\displaystyle r^\ast = \frac{r}{r^2+z^2}$ and $\displaystyle z^\ast = \frac{z}{r^2+z^2}$. We can use the Green's function $\eqref{Green_ball}$ to solve \eqref{Aeq}.
When integrated against $\sin \theta$, the contribution from the first term of the Green's function $\eqref{Green_ball}$ to $A$ is zero as
$$
\int_0^{2\pi} \frac{r\sin\theta}{\sqrt{r^2-2r\bar{r}\cos\theta+\bar{r}^2+(z-\bar{z})^2}}\, d\theta = 0.
$$
The other term of the Green's function can be handled similarly so
\begin{align*}
A(\bar{r},\bar{z}) &= \int_D G_B(r,z,\bar{r},\bar{z}) \omega^\theta(r,z) \begin{bmatrix} -\sin \theta \\ \cos\theta \\ 0\end{bmatrix} \, r\, d\theta\, dr\, dz \\
&= \int_D G_B(r,z,\bar{r},\bar{z}) \omega^\theta(r,z) \begin{bmatrix}0 \\ \cos\theta \\ 0\end{bmatrix} \, r\, d\theta\, dr\, dz
\end{align*}
Thus, in particular, the vector field $A$ has only ${\bf e_\theta}$ component. Defining $\psi(r,z)$ by $A= \frac{\psi(r,z)}{r} {\bf e_\theta}$,
$\psi$ satisfies \eqref{psi_bdy} and \eqref{psi_eq} as $A$ satisfies \eqref{Aeq}.
\smallskip
If $D$ is a cylinder, the result follows by using the Green's function expansion in terms of Bessel functions. $\Box$
\medskip
Since $\psi|_{\partial D}=0$, $\nabla \psi = \psi_r\, e_r+ \psi_z \, e_z$ is normal to $\partial D$. This implies $(-\psi_z,\psi_r) \cdot n= (ru^r, ru^z)\cdot n=0$. For boundary points not on the axis, this implies $u\cdot n=0$. In the case of the ball, for the boundary points on the axis, using continuity of $u$ and the boundary, we can conclude $u^z=0$ at these points.
\medskip
In addition to showing existence of a stream function, we can use well-known results for the Poisson equation
and Lemma $\ref{wang}$ to conclude that $\displaystyle \frac{\psi(r,z)}{r}$ satisfies axis conditions and has regularity estimates
in $(r,z)$ coordinates. The next theorem allows us to do this.
\smallskip
\begin{lemma}
\label{globalreg}
Let $f\in C^{k,\alpha}(\overline{D})$ and $g \in C^{k+2,\alpha}(\partial D)$ and suppose $g$, $f$ are axially symmetric. Additionally, in cylindrical coordinates $(r,z)$, suppose $f$ and $g$ satisfy
\begin{align*}
\partial_r^{2m} f(0^+,z) &=0, \quad 0\le 2m\le k \\
\partial_r^{2\ell } g (0^+,z) &=0, \quad 0\le 2\ell \le k+2
\end{align*}
Then there exists a unique $\phi(r,z)$ satisfying
\begin{align*}
\mathscr{L}\phi &= f \\
\phi|_{\partial D} &= g.
\end{align*}
In particular, $\phi$ satisfies the following estimate
\begin{align}
\label{g_est}
\|\phi\|_{C_{r,z}^{k+2,\alpha}(\overline{D})} \le C(\|g\|_{C^{k+2,\alpha}(\partial D)} + \|f\|_{C^{k,\alpha}(\overline{D})})
\end{align}
and $\phi$ satisfies
\begin{align}
\label{phipole}
\partial_r^{2m} \phi(0^+,z) &=0, \quad 0\le 2m\le k+2.
\end{align}
Suppose $D' \subset\subset D$ and $d\le \mbox{dist}(D', \partial D)$. Then we have the following interior estimate
\begin{align}
\label{interior_est}
d \|\nabla \phi\|_{L^\infty(D')} + d^2 \| \nabla^2 \phi\|_{L^\infty(D')} \le C (\|\phi\|_{L^\infty(D)} + \|f\|_{C^{\alpha}(D)})
\end{align}
\end{lemma}
{\bigskip\noindent\bf Proof.\quad} Consider the following system
\begin{align*}
-\Delta \Phi &= f\, {\bf e^\theta} \\
\Phi|_{\partial D} &= g\, {\bf e^\theta}
\end{align*}
By Lemma \ref{wang}, $f\, {\bf e^\theta}$ corresponds to a $C^{k,\alpha}$ vector field in Cartesian coordinates and similarly $g\, {\bf e^\theta}$ corresponds to a $C^{k+2,\alpha}$ vector field. Then by well-known results for the Poisson equation (see \cite{Kellogg} or Theorems 6.6, 6.19, and Corollary 6.3 of \cite{GilbargTrudinger}), there exists a unique vector field $\Phi$ in $C^{k+2,\alpha}(\overline{D})$ that satisfies the above system. Additionally, $\Phi = \phi(r,z)\, {\bf e^\theta}$
for some $\phi$. Applying Lemma $\ref{wang}$, we get that $\phi\in C_{r,z}^{k+2}(\overline{D})$ and satisfies $\eqref{phipole}$. In addition by restricting $\Phi$ to the $\theta=0$ plane, the fact that $\Phi$ is H\"{o}lder continuous
implies that $\phi$ is also H\"{o}lder continuous so $\phi \in C_{r,z}^{k+2,\alpha}(\overline{D})$. By construction, $\mathscr{L} \phi=f$. The derivative estimates, \eqref{g_est} and \eqref{interior_est}, for $\phi$ follow from the analogous classical estimates (Corollary 6.3 and Theorem 6.19 of \cite{GilbargTrudinger}) for solutions of the Poisson equation applied to $\Phi$.
$\Box$
\bigskip
By applying the above lemma to the function $\displaystyle \frac{\psi(r,z)}{r}$, \eqref{phipole} gives us the desired axis conditions for the stream function. With the two lemmas above at our disposal, we can continue our calculation of the stream function $\psi$.
Using \eqref{Apsi}, we can write the stream function $\psi$ as
\begin{align*}
\psi(\bar{r},\bar{z}) = \int_D \bar{r}\mathscr{A}^\theta(\bar{r},\bar{z},r,z)\omega^\theta(r, z)\, dr\, dz
\end{align*}
Let $x=\frac{\bar{r}-r}{r}$ and $y=\frac{\bar{z}-z}{r}$. We can write the integral corresponding to the first term in \eqref{eq:A} as
\begin{align*}
\frac{1}{2\pi\sqrt{1+x}}\int_0^\pi \frac{\cos \theta\, d\theta}{\sqrt{2(1-\cos \theta)+\frac{x^2+y^2}{1+x}}}
\end{align*}
Define
\begin{align}\label{Fs}
F(s)=\int_0^\pi \frac{\cos\theta\, d\theta}{\sqrt{2(1-\cos\theta)+s}}.
\end{align}
The function $F$ cannot be expressed in terms of elementary functions. A formula for $F$ in terms of elliptic integrals can be found in Lamb \cite{Lamb}.
After some computation, $ \frac{1}{\sqrt{1+x}} = \sqrt{\frac{r}{\bar{r}}}$ and $ \frac{x^2+y^2}{1+x} = \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}$.
Then the stream function $\psi$ can be written as
\begin{align*}
\psi(\bar{r},\bar{z}) &=\frac{1}{2\pi}\int_D\sqrt{r\bar{r}}F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)\omega^{\theta}(r,z)dr\, dz\\
&\, + \frac{1}{4\pi}\int_D \int_0^{2\pi} r\bar{r} \cos\theta \cdot h(\bar{r},\bar{z},r,\theta,z)\omega^{\theta}(r,z)\, d\theta dr\, dz.
\end{align*}
Define
\begin{align*}
H(\bar{r},\bar{z},r,z)=\frac{1}{4\pi}\int_0^{2\pi} r\bar{r} \cos\theta \cdot h(\bar{r},\bar{z},r,\theta,z) d\theta.
\end{align*}
We can compute and integrate by parts to see that
\begin{align*}
-H_{zz}+\frac{H_r}{r} -H_{rr} &= \frac{\bar{r}r}{4\pi} \int_0^{2\pi} \left(-h_{rr}-\frac{h_r}{r}-h_{zz}+\frac{h}{r^2}\right)\cos\theta\, d\theta \\
&= \frac{\bar{r}r}{4\pi} \int_0^{2\pi} \left(-h_{rr}-\frac{h_r}{r}-h_{zz}-\frac{h_{\theta\theta}}{r^2}\right)\cos\theta\, d\theta \\
&= -\frac{\bar{r}r}{4\pi} \int_0^{2\pi} (\Delta h)(\bar{r},\bar{z},r,\theta,z)\, \cos\theta\, d\theta=0.
\end{align*}
\medskip
We summarize the above calculations with the following proposition.
\begin{proposition}
\label{GREEN}
The function
\begin{align}
\mathscr{G}(\bar{r},\bar{z},r,z) = \frac{\sqrt{r\bar{r}}}{2\pi}F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)+H(\bar{r},\bar{z},r,z).
\end{align}
is the Green's function for the operator $L$ for the domain $D$. The function $H$ satisfies the following:
\begin{align}
\mathscr{L} \left(\frac{H}{r}\right)&= r\cdot LH=-\frac{H_{zz}}{r} + \frac{H_r}{r^2}-\frac{H_{rr}}{r} = 0 \label{Hcorrector}, \quad r>0\\
H(\bar{r},\bar{z},r,z) &= - \frac{\sqrt{r\bar{r}}}{2\pi}
F\left( \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)
, \quad (r,z)\in \partial D \nonumber
\end{align}
with the axis condition
\begin{align*}
\partial_r^{(2m)}\left.\left( \frac{H(\bar{r},\bar{z},r,z)}{r}\right)\right|_{r=0^+}=0, \quad m=0,1,2,\ldots
\end{align*}
In addition, we have
\begin{align*}
u(\bar{r},\bar{z})= \frac{1}{\bar{r}}\int_D \nabla_{(\bar{r},\bar{z})}^\perp \mathscr{G}(\bar{r},\bar{z},r,z) \omega^\theta(r,z) \, dr\, dz = \frac{1}{\bar{r}}\int_D r\nabla_{(\bar{r},\bar{z})}^\perp \mathscr{G}(\bar{r},\bar{z},r,z) w(r,z) \, dr\, dz
\end{align*}
\end{proposition}
\bigskip
\noindent
\begin{remark}
As seen by the above calculations, the Green's function for $L$ is related to the Green's function $G_D$ for the 3D Laplacian by
\begin{align*}
\mathscr{G}(\bar{r},\bar{z},r,z) = \int_0^{2\pi} r\bar{r} \cdot G_D(\bar{r},\bar{z},r,z)\cos \theta d\theta.
\end{align*}
\end{remark}
\bigskip
After some computation, we have the following expressions for $u^r$ and $u^z$ on $D$:
\begin{align}
\label{ur}
u^r(\bar{r},\bar{z}) &= \int_D \left[\frac{(z-\bar{z})\sqrt{r}}{\pi \bar{r}^{3/2}} F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) +\frac{r}{\bar{r}}\partial_{\bar{z}} H\right] w(r,z)\, dr\, dz\\
u^z(\bar{r},\bar{z}) &= \int_D\left[ \widetilde{J}(\bar{r},\bar{z}, r,z) +\frac{r}{\bar{r}}\partial_{\bar{r}} H\right] w(r,z)\, dr\, dz
\end{align}
where we define
\begin{align}
\label{scriptJ}
\widetilde{J}(\bar{r},\bar{z},r,z) = \left(\frac{r}{\bar{r}}\right)^{3/2} \widetilde{\mathscr{J}}(\bar{r},\bar{z},r,z)
\end{align}
and the function $\widetilde{\mathscr{J}}(\bar{r},\bar{z},r,z)$ is given by
\begin{align*}
\widetilde{\mathscr{J}}(\bar{r},\bar{z},r,z)= \frac{1}{\pi}\frac{(\bar{r}-r)}{r}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) &\,&\\
+\frac{1}{4\pi}\left[ F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)-2\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)\right].
\end{align*}
In addition, we set
\begin{align}
\label{scriptK}
\widetilde{K}(r,\bar{r},z,\bar{z}) = \frac{(z-\bar{z})\sqrt{r}}{\pi \bar{r}^{3/2}} F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right).
\end{align}
\medskip
For later use, we also define the kernels in the integrals for $u^r$ and $u^z$
with the corrector
\begin{align}
\label{K_def}
K(\bar{r},\bar{z},r,z) := -\frac{r}{\bar{r}}\partial_{\bar{z}}\mathscr{G} = \widetilde{K}(r,\bar{r},z,\bar{z}) +\frac{r}{\bar{r}}\partial_{\bar{z}} H
\end{align}
\begin{align}
\label{L_def}
J(\bar{r},\bar{z},r,z) := \frac{r}{\bar{r}} \partial_{\bar{r}}\mathscr{G} = \widetilde{J}(\bar{r},\bar{z},r,z) + \frac{r}{\bar{r}}\partial_{\bar{r}} H.
\end{align}
\subsection{Behavior of $F$}
\medskip
We derive estimates for the function $F$ defined in \eqref{Fs} that will be used frequently later. The details for these estimates can be found in the appendix.
\medskip
As $F$ is tied with the Green's function of $L$ from $\eqref{GREEN}$, one may expect that $F$ behaves
roughly logarithmically. However, $F$ will have better asymptotic properties than $\log$ in certain regimes. This will be
key for our estimates later.
First, one can bound $F$ easily with
\begin{align}
|F(s)| \lesssim \left(\frac{1}{s}\right)^{1/2}
\end{align}
but in fact, we have even better asymptotics at $s=0$ and $s=\infty$
\begin{align}
\label{Fexp}
F(s) &= -\frac{1}{2}\log(s)+\log 8-2+O(s\, \log(s))\quad \mbox{near} \quad s=0 \\
F(s) &= \frac{\pi}{2} \frac{1}{s^{3/2}} + O(s^{-5/2}) \quad \mbox{near} \quad s=\infty
\end{align}
and expansions gotten by formally differentiating the series holds. They are as follows:
\begin{align}
\label{F'exp0}
F'(s) &=-\frac{1}{2}\frac{1}{s}+O(\log s) \quad \mbox{near} \quad s=0\\
\label{F''exp0}
F''(s) &=\frac{1}{2} \frac{1}{s^2}+O(1/s)\quad \mbox{near} \quad s=0
\end{align}
\begin{align}
\label{F'expinfty}
F'(s) &= -\frac{3\pi}{4} \frac{1}{s^{5/2}} + O(s^{-7/2}) \quad \mbox{near} \quad s=\infty \\
F''(s) &= \frac{15\pi}{8} \frac{1}{s^{7/2}} + O(s^{-9/2}) \quad \mbox{near} \quad s=\infty.
\end{align}
Let $\epsilon>0$ be a small constant such that for $0<s<\epsilon$, all the expansions above for $F,F'$, and $F''$ near $0$ are valid. We will refer to this $\epsilon$ in our proofs later.
\medskip
We summarize upper bounds on $F$ in the following lemma which is from \cite{FengSverak}
\begin{lemma}
\label{Fest}
For every non-negative integer $k$, for all $s>0$
\begin{align}
|F(s)| & \lesssim_\tau \min \left( \left(\frac{1}{s}\right)^\tau, \left(\frac{1}{s}\right)^{3/2}\right), \quad 0<\tau\le\frac{1}{2} \label{Fbound} \\
|F^{(k)}(s)| &\lesssim_k \min \left( \left(\frac{1}{s}\right)^k, \left(\frac{1}{s}\right)^{k+3/2}\right), \quad k>0. \label{Fprimebound}
\end{align}
\end{lemma}
We will use the above bounds constantly throughout the rest of our proofs.
\begin{comment}
Define $x=(r,z)$ and $y=(r',z')$.
\medskip
\begin{proposition}
We have the following estimate for the Green's function:
\begin{align}
|\partial_{r}\partial_z\mathscr{G}(x,y)| \le C(D) \cdot \frac{1}{|x-y|^2}
\end{align}
\end{proposition}
{\bigskip\noindent\bf Proof.\quad} We can write the Green's function $\mathscr{G}$ in terms of the Green's function $G_D$ for the Laplacian as
\begin{align*}
\mathscr{G}(r,r',z,z')= \int_0^{2\pi} rr'\, \cos \theta'\cdot G_D(r,r',\theta',z,z')\, d\theta'
\end{align*}
where we have written the Laplacian green's function $G_D$ in cylindrical coordinates (without loss of generality
we assumed one of the points has angular coordinate $0$).
\end{comment}
\section{Gradient upper bound}
Our first goal is to prove a Kato type estimate on $\|\nabla u\|_\infty$ (see \cite{Kato}), which will imply an upper bound of $\|\nabla w\|_\infty$. Our estimate will have parallels with the analogous estimate for $\|\nabla u\|_\infty$ for the 2D Euler equations, but the estimates become more tedious due to the more complex Biot-Savart law.
\subsection{Some Green's function computations and derivative estimates}
\medskip
\noindent
Here, we will collect computations concerning the kernels $\widetilde{K}$ and $\widetilde{J}$ arising in the integrals for $u^r$ and $u^z$ which will be useful in our later estimates.
In order to have estimates on $\nabla u$ we will need to bound derivatives of $\widetilde{K}$ and $\widetilde{J}$ which are
\begin{align}
\partial_{\bar{r}}\widetilde{K} (r,\bar{r},z,\bar{z}) &=-\frac{3}{2\pi} \frac{(z-\bar{z}) \sqrt{r}}{\bar{r}^{5/2}} F'\left(s\right) + \frac{z-\bar{z}}{\pi\bar{r}^{3/2}} \sqrt{r} F''\left(s\right)\left( \frac{-2\bar{r}(r-\bar{r})-((r-\bar{r})^2+(z-\bar{z})^2)}{(\bar{r})^2 r}\right) \nonumber \\
&= -\frac{(z-\bar{z})\sqrt{r}}{\pi\bar{r}^{5/2}}\left[\frac{3}{2} F'(s)+sF''(s)\right]-2 \frac{(z-\bar{z})(r-\bar{r})}{\pi\bar{r}^{5/2}\sqrt{r}}F''(s)
\label{Kr} \\
\partial_{\bar{z}} \widetilde{K}(r,\bar{r},z,\bar{z}) &= -\frac{\sqrt{r}}{\pi\bar{r}^{3/2}} \left[ F'\left(s\right)+2\frac{(z-\bar{z})^2}{\bar{r}r} F''\left(s\right) \right] \label{Kz}
\end{align}
\begin{align}
\partial_{\bar{r}} \widetilde{J}(r,\bar{r},z,\bar{z}) &= \frac{1}{\pi} \left(\frac{\sqrt{r}}{\bar{r}^{3/2}}- \frac{3}{2}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{5/2}}\right)F'(s)+ \frac{1}{\pi} \frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}} F''(s) (\partial_{\bar{r}}s) \nonumber \\
&\,\,\, -\frac{3}{8\pi}\frac{r^{3/2}}{\bar{r}^{5/2}} \left[ F\left(s \right)-2s F'\left(s \right)\right]+ \frac{1}{4\pi}\left[-F'(s) (\partial_{\bar{r}} s)-2s F''(s) (\partial_{\bar{r}} s)\right] \left(\frac{r}{\bar{r}}\right)^{3/2} \nonumber \\
&= \frac{1}{\pi} \left(\frac{\sqrt{r}}{\bar{r}^{3/2}}- \frac{3}{2}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{5/2}}\right)F'(s)-
\frac{1}{\pi} \frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}} F''(s) \left(\frac{2(r-\bar{r})}{\bar{r}r} + \frac{s}{\bar{r}}\right)
\nonumber \\
&\,\,\, -\frac{3}{8\pi}\frac{r^{3/2}}{\bar{r}^{5/2}} \left[ F\left(s \right)-2s F'\left(s \right)\right]+ \frac{1}{4\pi}
[F'(s)+2sF''(s)]\left(\frac{r}{\bar{r}}\right)^{3/2}\left(\frac{2(r-\bar{r})}{\bar{r}r} + \frac{s}{\bar{r}}\right) \\
&= \frac{1}{\pi} \frac{r^{3/2}}{\bar{r}^{5/2}}\left[- \frac{3}{8}F(s)
+ sF'(s)+\frac{1}{2}s^2 F''(s)\right]+
\frac{2(r-\bar{r})}{\pi} \frac{\sqrt{r}}{\bar{r}^{5/2}}[F'(s)+sF''(s)] \label{Jr}\\
&\,\,+\frac{1}{\pi}\frac{\sqrt{r}}{\bar{r}^{3/2}}F'(s)+
\frac{2}{\pi}\frac{(r-\bar{r})^2}{\bar{r}^{5/2}\sqrt{r}}F''(s) \\
\partial_{\bar{z}} \widetilde{J}(r,\bar{r},z,\bar{z}) &= \frac{1}{\pi}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}} F''(s) \left(-\frac{2(z-\bar{z})}{\bar{r}r}\right) + \frac{1}{4\pi} \left[-F'(s)-2s F''(s)\right]\left(\frac{r^{3/2}}{\bar{r}^{3/2}}\right)\left(-\frac{2(z-\bar{z})}{\bar{r}r}\right) \label{Jz}
\end{align}
where we have defined $\displaystyle s =\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}$ and $\displaystyle \partial_{\bar{r}} s= -\frac{2(r-\bar{r})}{\bar{r}r} - \frac{s}{\bar{r}}$.
\medskip
\bigskip
\noindent
For later use, we define
\begin{align*}
x=(\bar{r},\bar{z}), \quad \mbox{and}\,\,\, y=(r,z).
\end{align*}
\begin{remark}
Observe that up to factors of $r$ and $\bar{r}$, the most singular terms for the derivatives of $\widetilde{K}$
and $\widetilde{J}$ above are similar to those gotten by computing the second derivatives of the 2D Laplacian Green's function. This will lead
to the double exponential upper bound in Theorem \ref{upper}
\end{remark}
\begin{comment}
Recall that the Dirichlet Green's function in 2 dimensions satisfies the following derivative estimate
$$
| \nabla^2 G(x-y)| \lesssim \frac{1}{|x-y|^2}.
$$
The following lemma will be used in the same way the above estimate is used for 2D Euler gradient estimates. Due in part to the fact the operator $L$ is not symmetric and the bounds on $F$ used, the bounds for the derivatives of $K$ and $J$ are not
the same for each derivative.
\bigskip
\begin{lemma}
\label{kern_est}
The kernels $\widetilde{K}$ and $\widetilde{J}$ as defined in \eqref{scriptK} and \eqref{scriptJ} satisfy the following derivative bounds
\begin{align*}
|\partial_{\bar{r}} \widetilde{K}(\bar{r},\bar{z},r,z) | &\le C(D) \left(\min \left( \left(\frac{r}{\bar{r}}\right)^{3/2} \frac{1}{|x-y|}, \frac{r^3}{|x-y|^4}\right)+ \min \left( \frac{1}{\bar{r}^{1/2}} \frac{r^{3/2}}{|x-y|^2}, \frac{\bar{r}r^3}{|x-y|^5}\right) \right)\\
\end{align*}
\begin{align*}
|\partial_{\bar{z}} \widetilde{K}(\bar{r},\bar{z},r,z) | \le C(D)\min\left( \frac{r^{3/2}}{\bar{r}^{1/2}|x-y|^2}, \frac{\bar{r}r^3}{|x-y|^5}\right)
\end{align*}
\begin{align*}
|\partial_{\bar{z}} \widetilde{J}(\bar{r},\bar{z},r,z)| \le C(D)\left(\min \left( \frac{1}{\bar{r}^{1/2}} \frac{r^{3/2}}{|x-y|^2}, \frac{\bar{r}r^3}{|x-y|^5}\right)+ \min \left( \left( \frac{r}{\bar{r}}\right)^{3/2} \frac{1}{|x-y|}, \frac{r^3}{|x-y|^4}\right)\right)
\end{align*}
and lastly
\begin{align}
\label{Nest}
|\partial_{\bar{r}} \widetilde{J}(\bar{r},\bar{z},r,z)| &\le C(D)\left(\min\left(\frac{r^3}{|x-y|^4},\left( \frac{r}{\bar{r}}\right)^{3/2}\frac{1}{|x-y|}\right)+ \min\left( \frac{r^{3/2}}{\bar{r}^{1/2}}\frac{1}{|x-y|^2}, \frac{\bar{r}r^3}{|x-y|^5}\right)\right. \\
&\, +\left. \left\lbrace \begin{array}{ll} \min \left(\frac{r^4}{|x-y|^5}, \left(\frac{r}{\bar{r}}\right)^2 \frac{1}{|x-y|} \right), & s>M \\ \min \left(\frac{r^3}{\bar{r}|x-y|^3}, \left(\frac{r}{\bar{r}}\right)^2 \frac{1}{|x-y|} \right), & s\le M \end{array} \right.\right). \nonumber
\end{align}
\end{lemma}
\medskip
\noindent
In proving estimates on $\nabla u$, it will be necessary to
divide the integrals defining $\nabla u$ into integrals
over different regions of the domain. Depending on the region, it will become advantageous to choose one expression in the minimums present in the estimates above over the other.
\medskip
\noindent
{\bf Proof of Lemma \ref{kern_est}} The idea is to leverage the bounds of Lemma \ref{Fest} appropriately. First, we bound $\partial_{\bar{r}}\widetilde{K} (r,\bar{r},z,\bar{z})$, the derivative in $\bar{z}$ of $\widetilde{K}$ and $\widetilde{J}$ is similar. Using the expression $\eqref{Kr}$ and $\eqref{Fprimebound}$
\begin{align*}
|\partial_{\bar{r}}\widetilde{K} (r,\bar{r},z,\bar{z})| \lesssim \frac{|z-\bar{z}|\sqrt{r}}{\bar{r}^{5/2}}\min \left( \frac{1}{s^{3/2}}, \frac{1}{s^{5/2}}\right)+\frac{|z-\bar{z}||r-\bar{r}|}{\bar{r}^{5/2}\sqrt{r}}\min \left( \frac{1}{s^{2}}, \frac{1}{s^{7/2}}\right) .
\end{align*}
From the above estimate, we arrive at the desired bound. The estimates for the $\bar{z}$ derivatives of $\widetilde{K}$ and $\widetilde{J}$ are
\begin{align*}
|\partial_{\bar{z}}\widetilde{K}| \lesssim \frac{\sqrt{r}}{\bar{r}^{3/2}} \min \left( \frac{1}{s^{3/2}}, \frac{1}{s^{5/2}}\right)
\end{align*}
\begin{align*}
|\partial_{\bar{z}}\widetilde{J}| \lesssim \frac{|r-\bar{r}||z-\bar{z}|}{\bar{r}^{5/2}\sqrt{r}} \min \left( \frac{1}{s^{2}}, \frac{1}{s^{7/2}}\right)+ |z-\bar{z}|\frac{\sqrt{r}}{\bar{r}^{5/2}}\min \left( \frac{1}{s}, \frac{1}{s^{5/2}}\right)
\end{align*}
The estimate for $\partial_{\bar{r}} \widetilde{J}$ is more complicated. Directly using the bounds for $F$ and its derivatives,
\begin{align*}
\frac{r^{3/2}}{\bar{r}^{5/2}}\left[- \frac{3}{8}F(s)
+ sF'(s)+\frac{1}{2}s^2 F''(s)\right] \lesssim \frac{r^{3/2}}{\bar{r}^{5/2}} \frac{1}{\sqrt{s}} =
\left(\frac{r}{\bar{r}}\right)^2 \frac{1}{|x-y|}
\end{align*}
However, for $s$ large, we can use the expansion of $F$ at infinity to estimate the expression in the brackets above. Indeed for $s>M$, the terms corresponding to $s^{-3/2}$ cancel and we have
\begin{align*}
\frac{r^{3/2}}{\bar{r}^{5/2}}\left[- \frac{3}{8}F(s)
+ sF'(s)+\frac{1}{2}s^2 F''(s)\right] \lesssim\frac{r^{3/2}}{\bar{r}^{5/2}} \frac{1}{s^{5/2}}= \frac{r^4}{|x-y|^5}, \quad s>N
\end{align*}
For $s\le M$, we can also use the estimate $F(s)\lesssim s^{-3/2}$
and the analogous estimates for $F'$ and $F''$, to achieve another bound. The remaining terms of $\partial_{\bar{r}} \widetilde{J}$ can be estimated similarly as in $\partial_{\bar{r}} \widetilde{K}$. $\Box$
\end{comment}
\medskip
Additionally, we will need bounds for the Green's function $\mathscr{G}$ of $L$, which will then allow for bounds on derivatives of $J$ and $K$.
\begin{proposition}
\label{Hest}
The function $\mathscr{G}$ defined by Proposition \ref{GREEN} satisfies the following estimates for $\bar{r}>0$
\begin{align*}
\left| \nabla_{\bar{r},\bar{z}}^2 \left(\frac{\mathscr{G}(\bar{r},\bar{z},r,z)}{\bar{r}}\right)\right| & \le C(D)\min \left( \frac{r}{|x-y|^3}, \sqrt{\frac{r}{\bar{r}}}\frac{1}{|x-y|^2}\right).
\end{align*}
\end{proposition}
{\bigskip\noindent\bf Proof.\quad}
Recall
\begin{align*}
\frac{\mathscr{G}(\bar{r},\bar{z},r,z)}{\bar{r}} = \int_0^{2\pi} r\cos \theta \cdot G_D(\bar{r},\bar{z},r,\theta,z)\, d\theta
\end{align*}
where $G_D$ is defined through $\eqref{3DGreen}$. Using the classical Green's function bound \cite{Widman, Krasovskii}
\begin{align*}
|\nabla^2 G_D(x,y)| \le \frac{C(D)}{|x-y|^3},
\end{align*}
we can arrive at the bound
\begin{align*}
\left|\nabla_{\bar{r},\bar{z}}^2 \left(\frac{\mathscr{G}}{\bar{r}}\right)\right| & \le C(D) \int_0^{2\pi} \frac{r}{(r^2-2r\bar{r}\cos\theta+\bar{r}^2+(z-\bar{z})^2)^{3/2}}\, d\theta \\
&= \frac{C(D)}{r^{1/2}\bar{r}^{3/2}} \int_0^{2\pi} \frac{1}{\left(2(1-\cos \theta) + \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)^{3/2}}\, d\theta.
\end{align*}
As before, we set $\displaystyle s= \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}$. Then applying a similar argument as in section {\bf A.1} of the Appendix we can arrive at
\begin{align}
\label{s_bound}
\int_0^{2\pi} \frac{d\theta}{\left(2(1-\cos \theta) +s\right)^{3/2}} \lesssim \min\left(\frac{1}{s}, \frac{1}{s^{3/2}}\right).
\end{align}
Indeed, easily we have
\begin{align*}
\int_0^{2\pi} \frac{d\theta}{\left(2(1-\cos \theta) +s\right)^{3/2}} \le \frac{2\pi}{s^{3/2}}.
\end{align*}
For the other possible upper bound can rewrite the integral as
\begin{align*}
\int_0^{2\pi} \frac{d\theta}{\left(2(1-\cos \theta) +s\right)^{3/2}} = \frac{1}{2} \int_0^{\pi/2} \frac{1}{(\sin^2 \varphi+s/4)^{3/2}}\, d\varphi
\end{align*}
and then by \eqref{Fprime1} we get an upper bound of a constant times $1/s$.
Using the bound \eqref{s_bound}, we arrive at the desired estimate for $\nabla^2( \mathscr{G}/\bar{r})$. $\Box$
\begin{comment}
\medskip
Next, we need estimates on derivatives of the corrector
function $H$. It suffices to derive estimates in the claim of the lemma for $\partial_{\bar{z}}^2 \frac{H}{\bar{r}}, \partial_{\bar{r}}\partial_{\bar{z}}\frac{H}{\bar{r}}$, $\partial_{\bar{z}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right)$, and $\partial_{\bar{r}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right)$. Estimates for these quantities will imply the desired bounds for $\partial_{\bar{r}}K$, $\partial_{\bar{z}}K$, $\partial_{\bar{z}}J$, and $\partial_{\bar{r}}J$ respectively.
We claim the following estimates
\begin{align*}
\left|\partial_{\bar{z}}^2 \left(\frac{H(\bar{r},\bar{z},r,z)}{\bar{r}}\right)\right| &\le \\
\left|\partial_{\bar{r}}\partial_{\bar{z}}\frac{H}{\bar{r}}\right| &\le \\
\left|\partial_{\bar{z}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right)\right| & \le \\
\left|\partial_{\bar{r}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right)\right| & \le
\end{align*}
\smallskip
Using \eqref{Hcorrector}, we have
\begin{align*}
\partial_{\bar{r}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right) &= - \partial_{\bar{z}}^2 \left( \frac{H}{\bar{r}}\right) \\
\partial_{\bar{z}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right) &= \partial_{\bar{z}}\partial_{\bar{r}} \left(\frac{H}{r}\right) + \frac{1}{\bar{r}} \partial_{\bar{z}}\left(\frac{H}{\bar{r}}\right)
\end{align*}
so it actually suffices to have suitable estimates on derivatives of $H/\bar{r}$. Let $\{B_{\delta_i}(x_i)\}$ be a finite collection of balls centered at points $x_i\in \partial D$ such that for every $i$, there exists a $C^{2,\alpha}$ diffeomorphism $\psi_i$ mapping $B_{\delta_i}(x_i)\cap D$ onto the half plane $\{z>0\}$ such that $\partial D\cap B_{\delta_i}(x_i)$ is mapped onto a subset of $\{z=0\}$. The maps $\psi_i$ locally straighten the boundary. Let $\delta =\min_i \delta_i$ and define $D_\delta= \{x: \mbox{dist}(x,\partial D)>\delta/2\}$. Observe $\delta$ only depends on the domain $D$.
\medskip
Fix $y=(r,z)\in D$. Let $x=(\bar{r},\bar{z})\in D$. We divide the analysis into three cases
\begin{enumerate}
\item $x,y\in D_\delta$
\item $x,y\in B_{\delta_i}(x_i)$ for some $i$
\item $x$ or $y$ is in $D\setminus D_\delta$ but both do not belong to the same ball $B_{\delta_i}(x_i)$
\end{enumerate}
\medskip
For Case (1), by the maximum principle and Corollary \ref{localest}(a),
\begin{align*}
\left|\nabla \left(\frac{H(x,y)}{\bar{r}}\right)\right| + \left|\nabla^2 \left(\frac{H(x,y)}{\bar{r}}\right)\right| \le C(D)
\end{align*}
\smallskip
For Case (2), consider the following decomposition for the Green's function $\mathscr{G}(x,y)$ for $x,y\in B_{\delta_i}(x_i)$
\begin{align}
\mathscr{G}(x,y)&= \frac{\sqrt{r\bar{r}}}{2\pi}F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)+H(\bar{r},\bar{z},r,z)\nonumber\\
&= \frac{\sqrt{r\bar{r}}}{2\pi}F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right)-\frac{\sqrt{r\bar{r}}}{2\pi}F\left(\frac{(r-\bar{r})^2+(z-\psi_i^{-1}(-\psi_i(\bar{z})))^2}{r\bar{r}}\right)+\widetilde{H}(\bar{r},\bar{z},r,z). \label{Htilde}
\end{align}
The second term of \eqref{Htilde}, up to a constant factor depending on $\psi_i$, possesses the analogous derivative estimates as the first term which we have done above in our estimates for $\widetilde{K}$ and $\widetilde{J}$. For $x\in \partial D\cap B_{\delta_i}(x_i)$, $\widetilde{H}(x,y)=0$. Then by local boundary estimate, Corollary \ref{localest}(b),
\begin{align*}
\sup_{(\bar{r},\bar{z})\in B_{\delta_i}(x_i)}\left( \left| \nabla \left(\frac{\widetilde{H}}{\bar{r}}\right)\right| + \left| \nabla^2 \left(\frac{\widetilde{H}}{\bar{r}}\right)\right|\right) \le C(D).
\end{align*}
\smallskip
For Case (3), we first suppose $y\in D_\delta$. Then by Lemma \ref{schauder}(b), we have uniform bound on derivatives of $H/\bar{r}$ by a constant depending only on $D$. If $y\in B_{\delta_i}(x_i)$ for some $i$, then we can argue similarly as to Case (2).
\smallskip
After combining these estimates for $H/\bar{r}$ and the earlier ones for derivatives of $\widetilde{K}$ and $\widetilde{J}$, the proof is complete.
Observe that the function $\displaystyle \frac{H(\bar{r},\bar{z},r,z)}{\bar{r}}$ satisfies
the following elliptic equation
\begin{align*}
\left(\Delta_{(\bar{r},\bar{z})} -\frac{1}{\bar{r}^2}\right) \frac{H}{\bar{r}} = 0.
\end{align*}
It suffices to bound $\partial_{\bar{z}}^2 \left(\frac{H}{\bar{r}}\right)$, $\partial_{\bar{r}}\partial_{\bar{z}} \left(\frac{H}{\bar{r}}\right)$, and $\partial_{\bar{r}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right)$. However, from equation \eqref{Hcorrector}, $\partial_{\bar{r}}\left( \frac{1}{\bar{r}} \partial_{\bar{r}} H\right) = -\partial_{\bar{z}}^2 \left(\frac{H}{\bar{r}}\right)$ so we just
need estimates on second derivatives of $\frac{H}{\bar{r}}$ . We can obtain such estimates from Schauder theory, which together with the estimates of $\widetilde{K}$ and $\widetilde{J}$ imply the desired bounds. At the axis $\bar{r}=0$ there is no issue as we are in the axisymmetric setting.
$\Box$
\end{comment}
\subsection{Kato estimate}
Next, we prove the key estimate that will allow us to deduce Theorem \ref{upper}. For the 2D Euler equations, this type of estimate was proven by Kato \cite{Kato}
\begin{theorem} (Kato type estimate)
\label{kato}
Let $w\in C^\alpha(D)$, $\alpha>0$. Fix $R>0$ and let $D_R=\{(r,z): (r,z)\in D\,\,\mbox{and}\,\, r <R\}$.
\begin{align}
\label{kato_est}
\|\nabla u\|_{L^\infty(D_R)} \le C_1(\alpha,D)\|w_0\|_\infty\left(1+(R+R^3)\log\left(1+\frac{\|w\|_{C^\alpha}}{\|w_0\|_\infty}\right)\right)
\end{align}
\end{theorem}
\begin{remark}
Observe that as we are closer to the axis, the effect of the logarithm on the right hand side is diminished. It is in this respect that the above estimate is different than the estimate for $\|\nabla u\|_\infty$ for solutions of 2D Euler.
\end{remark}
\begin{comment}
Before proving the theorem, we will prove a useful lemma concerning the kernels for $u_r$ and $u_z$.
\begin{lemma}
\label{kern_est}
The kernels $K$ and $J$ as defined in \eqref{K_def} and \eqref{L_def} satisfy the following derivative bounds
\begin{align*}
|\nabla_{(\bar{r},\bar{z})} K(\bar{r},\bar{z},r,z) | &\le C\min \left( \frac{1}{\bar{r}^{1/2}} \frac{r^{3/2}}{|x-y|^2}, \frac{r^2}{|x-y|^3}\right) \\
\end{align*}
\begin{align*}
|\nabla_{(\bar{r},\bar{z})} J(\bar{r},\bar{z},r,z)| \le C\min \left( \frac{1}{\bar{r}^{1/2}} \frac{r^{3/2}}{|x-y|^2}, \frac{r^2}{|x-y|^3}\right) +C \min\left(\frac{r^2}{\bar{r}} \frac{1}{|x-y|^2} , \left(\frac{r}{\bar{r}}\right)^{3/2} \frac{1}{|x-y|}\right)
\end{align*}
\end{lemma}
{\bigskip\noindent\bf Proof.\quad}
Let us consider $u^r$ written in terms of the Green's function for the Laplacian
\begin{align}
u^r(\bar{r},\bar{z})= \int_0^1 \int_0^1\int_0^{2\pi} r^2\cos \theta\cdot \partial_{\bar{z}} G_D(r,\bar{r},\theta, z,\bar{z}) w(r,z)\,d\theta dr\, dz
\end{align}
where we have written the Laplacian green's function $G_D$ in cylindrical coordinates (without loss of generality
we assumed one of the points has angular coordinate $0$). We see that the kernel $K$ can also be expressed as
\begin{align*}
K(r,z,\bar{r},\bar{z}) = \int_0^{2\pi} r^2\cos \theta\cdot \partial_{\bar{z}} G_D(r,\bar{r},\theta, z,\bar{z})\,d\theta
\end{align*}
Using well-known Green's function bounds for derivatives of $G_D$ we have
\begin{align}
\label{Kbound}
\left|\partial_{\bar{r}}K(r,z,\bar{r},\bar{z}) \right| \le C \int_0^{2\pi}\frac{r^2}{(r^2-2r\bar{r}\cos\theta'+r'^2+(z-\bar{z})^2)^{3/2}}\, d\theta
\end{align}
Now, we can use similar estimates as done for $F'$ (we're essentially off by a $\cos \theta$ factor here), one can arrive at
\begin{align*}
\left|\partial_{\bar{r}}K(r,z,\bar{r},\bar{z}) \right| \lesssim \frac{1}{\bar{r}^{3/2}} \int_0^{2\pi} \frac{r^{1/2}}{(2(1-\cos\theta)+s)^{3/2}}\, d\theta &\lesssim \frac{r^{1/2}}{\bar{r}^{3/2}} \min\left( \frac{1}{s}, \frac{1}{s^{3/2}}\right) \\
& \lesssim \min \left( \frac{1}{\sqrt{\bar{r}}} \frac{r^{3/2}}{|x-y|^2}, \frac{r^2}{|x-y|^3}\right)
\end{align*}
where, as before, we set $\displaystyle s=\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}$. An analogous argument
works to bound $\partial_{\bar{z}} K$. Let us give a sketch
for $\partial_{\bar{z}} J$. We can write $u^z$ as
\begin{align}
u^z(\bar{r},\bar{z})= \int_D \int_0^{2\pi} \frac{r^2}{\bar{r}} \cos\theta \cdot \partial_{\bar{r}} \left(\bar{r}G_D(r,\bar{r},\theta, z,\bar{z})\right)\, w(r,z) d\theta dr\, dz
\end{align}
This allows us to write $\partial_{\bar{z}}J$ as
\begin{align*}
\partial_{\bar{z}}J(r,z,\bar{r},\bar{z})= \int_0^{2\pi}
r^2\cos\theta \left(\partial_{\bar{r}\bar{z}} G_D + \frac{\partial_{\bar{z}}G_D}{\bar{r}}\right)\, d\theta
\end{align*}
We only need to estimate the term involving $\partial_{\bar{z}} G_D$ as the other term was bounded earlier. Using Green's function bounds
\begin{align*}
\left| \int_0^{2\pi}
\frac{r^2}{\bar{r}}\cos\theta \cdot \partial_{\bar{z}} G_D \,d\theta\right| & \le \int_0^{2\pi} \frac{r^2}{\bar{r}(r^2-2r\bar{r}\cos\theta'+r'^2+(z-\bar{z})^2)}\, d\theta \\
& \le \frac{r}{\bar{r}^2}\int_0^{2\pi} \frac{1}{(2(1-\cos\theta)+s)}\, d\theta \lesssim \frac{r}{\bar{r}^2}\min\left( \frac{1}{s}, \frac{1}{\sqrt{s}}\right)
\end{align*}
The estimate for $\partial_{\bar{r}}J$ can be done similarly.
$\Box$
\end{comment}
\subsection{Proof of Theorem \ref{kato}}
Let $x=(\bar{r},\bar{z})$. Define $\epsilon_0= ((\sqrt{5}-1)/4)\epsilon $. Recall $\epsilon$ is the radius of the ball centered at $0$ for which our expansions for $F$ in section 2.3 hold. Observe that
\begin{align*}
B_{\epsilon_0\bar{r}}(x) \subset \{ (r,z): s(r,z) <\epsilon\}
\end{align*}
so we can use the expansion for $F(s)$ close to $s=0$ on this ball later.
Let
\begin{align}
\label{delta_choice}
\delta =\min \left(c, \epsilon_0/2, \left( \frac{\|w_0\|_{L^\infty}}{\|w\|_{C^\alpha}}\right)^{1/\alpha}\right)
\end{align}
where $c$ is chosen small so that the set of $x$ such that $\mbox{dist}(x,\partial D)> 2\delta$ is non-empty.
We will bound
\begin{align*}
\partial_{\bar{r}}u^r(\bar{r},\bar{z})= \int_D\partial_{\bar{r}}K(r,z,\bar{r},\bar{z})w(r,z)\, dr\, dz.
\end{align*}
Proposition \ref{Hest} will allow for better decay estimates for the kernel when $|x-y|$ is large
which will lead to the decay factor in front of the logarithm term in $\eqref{kato_est}$.
By the incompressibility condition and axis condition, bounding $\partial_{\bar{r}} u^r$ will imply the desired
bound on $\partial_{\bar{z}} u^z$. We will sketch the proof for the derivatives $\partial_{\bar{r}} u^z$ and $\partial_{\bar{z}} u^r$ in the appendix.
\bigskip
\noindent
{\bf Case 1: $\mbox{dist}(x,\partial D) > 2\delta$}
\medskip
We divide the integral for $\partial_{\bar{r}}u^r$ into three regions
\begin{align*}
\partial_{\bar{r}}u^r(\bar{r},\bar{z}) &=\left( \int_{B_{\delta \bar{r}}(x)} +\int_{\Omega \cap B_{\delta\bar{r}}^c(x)} +\int_{D\setminus \Omega}\right)\partial_{\bar{r}}K(r,z,\bar{r},\bar{z})w(r,z)\, dr\, dz \\
&= I+II+III
\end{align*}
where $\Omega = \{(r,z)\in D: \frac{1}{2}\bar{r} < r < 2\bar{r}, 0\le z\le 1\}$.
Recall from above that we have defined
\begin{align*}
\partial_{\bar{r}}K(r,z,\bar{r},\bar{z}) = \partial_{\bar{r}}\widetilde{K}(r,z,\bar{r},\bar{z}) + \partial_{\bar{r}}\left( \frac{r}{\bar{r}} \partial_{\bar{z}} H\right).
\end{align*}
For $I$, we can use the expansion \eqref{Kr} for $\partial_r \widetilde{K}$ and \eqref{F''exp0} to get that the most dangerous term is
\begin{align}
\label{danger}
-\frac{1}{\pi\sqrt{\bar{r}}} \int_{B_{\delta\bar{r}}(x)} \frac{r^{3/2}(z-\bar{z})(r-\bar{r})}{|x-y|^4} w(r,z)\, dr\, dz.
\end{align}
Indeed, using $\eqref{Kr}$, $\eqref{F'exp0}$, and $\eqref{Fprimebound}$ we can bound the remainder terms
\begin{align*}
\left|\partial_{\bar{r}} \widetilde{K} + \frac{1}{\pi\sqrt{\bar{r}}}\frac{r^{3/2}(z-\bar{z})(r-\bar{r})}{|x-y|^4}\right| \lesssim \left( \frac{|z-\bar{z}|\sqrt{r}}{\bar{r}^{5/2}}\right)\left(\frac{1}{s}\right) \lesssim \frac{r^{3/2}}{\bar{r}^{3/2}} \frac{1}{|x-y|}.
\end{align*}
Thus, the terms other than $\eqref{danger}$ from the expansion of $\partial_r \widetilde{K}$ can be controlled since
\begin{align*}
\frac{1}{\bar{r}^{3/2}}\int_{B_{\delta\bar{r}}(x)} \frac{r^{3/2}}{|x-y|}w(y,t)\, dy \le C\bar{r}^{2} \|w_0\|_\infty.
\end{align*}
By writing the kernel in the integral $\eqref{danger}$ in polar coordinates $\rho, \phi$ centered at $x=(\bar{r},\bar{z})$, we get
\begin{align}
\label{taylor}
- \frac{(\bar{r}+\rho \cos\phi)^{3/2} \cos \phi\sin\phi}{\rho^2} = - \frac{\bar{r}^{3/2} \cos\phi\sin\phi}{\rho^2} + \bar{r}^{1/2} O(1/\sqrt{\rho}).
\end{align}
When integrated over $B_{\delta\bar{r}}(x)$, the second term in \eqref{taylor} is controlled by $C\bar{r}^{2} \|w_0\|_{L^\infty}$. For the other term,
\begin{align*}
\left| \frac{1}{\sqrt{\bar{r}}} \int_{B_{\delta\bar{r}}(x)} \frac{\bar{r}^{3/2}\cos\phi\sin\phi}{\rho^2} w(\rho,\phi) \rho\, d\rho\, d\phi\right| &= \left|\bar{r} \int_{B_{\delta\bar{r}}(x)} \frac{\cos\phi\sin\phi}{\rho} (w(\rho, \phi)-w(\bar{r},\bar{z}))\, d\rho\, d\phi\right| \\
& \le C\bar{r}\|w\|_{C^\alpha} \int_0^{\delta\bar{r}} \rho^{-1+\alpha}\, d\rho \\ &\le C\bar{r}^{1+\alpha} \delta^\alpha \|w\|_{C^\alpha} \le C(\alpha)\bar{r}^{1+\alpha} \|w_0\|_{L^\infty}.
\end{align*}
Now, we estimate the contribution from the corrector function $H$. For $y=(r,z)\in B_{\delta\bar{r}}(x)$, the function $H$ satisfies
\begin{align*}
\mathscr{L}_{\bar{r},\bar{z}} \left( \frac{H(\bar{r}, \bar{z}, r, z)}{\bar{r}}\right) &= 0 \quad\quad\mbox{for} \quad (\bar{r},\bar{z})\in D \\
\frac{H(\bar{r}, \bar{z}, r, z)}{\bar{r}} &= - \frac{1}{2\pi} \sqrt{\frac{r}{\bar{r}}} F\left( \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right) \quad\quad\mbox{for} \quad (\bar{r},\bar{z}) \in \partial D.
\end{align*}
By using bounds for $F$ we obtain,
\begin{align*}
\sup_{(\widetilde{r},\widetilde{z})\in \partial D} \sqrt{\frac{r}{\widetilde{r}}}\left| F\left( \frac{(r-\widetilde{r})^2+(z-\widetilde{z})^2}{r\widetilde{r}}\right)\right| &\le
\sup_{(\widetilde{r},\widetilde{z})\in \partial D, 0\le \widetilde{r}\le \delta} \left|\sqrt{\frac{r}{\widetilde{r}}} F\left( \frac{(r-\widetilde{r})^2+(z-\widetilde{z})^2}{r\widetilde{r}}\right)\right| \\
&\quad +
\sup_{(\widetilde{r},\widetilde{z})\in \partial D, \widetilde{r}\ge \delta} \left|\sqrt{\frac{r}{\widetilde{r}}} F\left( \frac{(r-\widetilde{r})^2+(z-\widetilde{z})^2}{r\widetilde{r}}\right)\right| \\
& \le C\sup_{(\widetilde{r},\widetilde{z})\in \partial D, 0\le \widetilde{r}\le \delta} \left|\frac{\widetilde{r}r^2}{((r-\widetilde{r})^2+(z-\widetilde{z})^2)^{3/2}}\right| \\
& \quad + C \sup_{(\widetilde{r},\widetilde{z})\in \partial D, \widetilde{r}\ge \delta} \sqrt{\frac{r}{\widetilde{r}}} \left| \log\left( \frac{\widetilde{r}r}{(r-\widetilde{r})^2+(z-\widetilde{z})^2}\right) \right|\\
& \le C+C\log \delta^{-1} \le C\log \delta^{-1}
\end{align*}
where we can make $\delta$ smaller if needed.
By relating this equation to the Poisson equation as in the proof of Lemma 2.5, we can use the maximum principle so that for $\displaystyle y\in B_{\delta\bar{r}}(x)$
\begin{align*}
\left| \frac{H(\bar{r},\bar{z},r,z)}{\bar{r}}\right| \le \sup_{(\widetilde{r},\widetilde{z})\in \partial D} \sqrt{\frac{r}{\widetilde{r}}}\left| F\left( \frac{(r-\widetilde{r})^2+(z-\widetilde{z})^2}{r\widetilde{r}}\right)\right| \le C \log \delta^{-1}.
\end{align*}
Using the interior estimate \eqref{interior_est} of Lemma \ref{globalreg}, we get
\begin{align*}
\left|\nabla_{\bar{r},\bar{z}}^2 \left(\frac{H(\bar{r},\bar{z},r,z)}{\bar{r}}\right)\right|
\le C(D) \frac{1}{\delta^2}\sup_{(\bar{r},\bar{z})\in D}\left|\frac{H(\bar{r},\bar{z},r,z )}{\bar{r}}\right|
\le C(D) \frac{1}{\delta^2}\log \delta^{-1}.
\end{align*}
Inserting this estimate to the integral of the corrector over $B_{\delta\bar{r}}(x)$:
\begin{align*}
\left|\int_{B_{\delta\bar{r}}(x)} r \partial_{\bar{r}}
\partial_{\bar{z}} \left( \frac{H}{\bar{r}}\right) w(r,z)\, dr\, dz\right| \le C\bar{r}^3 \|w_0\|_{L^\infty} \log \delta^{-1}.
\end{align*}
Combining estimates for $I$, we get
\begin{align}
\label{I_integral}
|I| \le C(\alpha) (\bar{r}^{2}+\bar{r}^3\log \delta^{-1}+\bar{r}^{1+\alpha}) \|w_0\|_\infty.
\end{align}
Now, we will bound $II$ and we use
\begin{align*}
\partial_{\bar{r}}K(r,z,\bar{r},\bar{z})=-r \partial_{\bar{r}}\partial_{\bar{z}} \left(\frac{\mathscr{G}}{\bar{r}}\right) .
\end{align*}
\begin{comment}
\medskip
To bound the integral against $\widetilde{K}$ we use Lemma \ref{kern_est},
\begin{align}
\int_{\Omega \cap B_{\delta\bar{r}}^c(x)} |\partial_{\bar{r}}\widetilde{K}(r,z,\bar{r},\bar{z})w(r,z)|\, dr\, dz \nonumber &\le C(D)\|w_0\|_\infty \int_{\Omega \cap B_{\delta\bar{r}}^c(x)} \left(\frac{r^{3/2}}{\bar{r}^{1/2}|x-y|^2}+\left(\frac{r}{\bar{r}}\right)^{3/2}\frac{1}{|x-y|}\right)\, dr\, dz \\
&\le C(D) \|w_0\|_\infty \int_{\Omega \cap B_{\delta\bar{r}}^c(x)}\left(\frac{\bar{r}}{|x-y|^2}+\frac{1}{|x-y|}\right)\, dr\, dz
\nonumber \\ &\le C(D)\|w_0\|_\infty+ C(D) \bar{r} \|w_0\|_\infty (1+ \log \bar{r}^{-1} +\log \delta^{-1})\label{II_integral}
\end{align}
\end{comment}
\medskip
\noindent
Using Proposition \ref{Hest},
\begin{align}
|II|\le \int_{\Omega \cap B_{\delta\bar{r}}^c(x)} \left| \partial_{\bar{r}}K(r,z,\bar{r},\bar{z}) w(r,z)\right|\, dr\, dz &\le C(D)\|w_0\|_\infty \int_{\Omega \cap B_{\delta\bar{r}}^c(x)} \frac{r^{3/2}}{\bar{r}^{1/2}} \frac{1}{|x-y|^2}\, dr\, dz \nonumber \\
&\le C(D)\bar{r} \|w_0\|_\infty \int_{\Omega \cap B_{\delta\bar{r}}^c(x)}\frac{1}{|x-y|^2}\, dr\, dz
\nonumber \\ & \le C(D) \bar{r} \|w_0\|_\infty (1+ \log \bar{r}^{-1} +\log \delta^{-1}) \label{II_integral}
\end{align}
\begin{comment}
Then combining the two estimates above,
\begin{align}
\label{II_integral}
|II| \le C(D) \|w_0\|_\infty +C(D) \bar{r} \|w_0\|_\infty (1+ \log \bar{r}^{-1} +\log \delta^{-1})
\end{align}
\end{comment}
\bigskip
\begin{comment}
For the integral $III$, we use the other kernel estimate from Lemma \ref{kern_est} and get
\begin{align}
\int_{D \setminus \Omega } |\partial_{\bar{r}}\widetilde{K}(r,z,\bar{r},\bar{z})w(r,z)|\, dr\, dz &\le C(D) \|w_0\|_\infty \int_{D\setminus \Omega} \left(\frac{r^3}{|x-y|^4}
+\frac{\bar{r}r^3}{|x-y|^5}\right)\, dr\, dz \nonumber \\
& \le C(D)\|w_0\|_\infty \int_{D\setminus \Omega}\left( \frac{1}{|x-y|}+\bar{r}\frac{1}{|x-y|^2}\right)\, dr\, dz \nonumber \\
& \le C(D) \|w_0\|_\infty +C(D)\|w_0\|_\infty\bar{r}\log\delta^{-1} \nonumber.
\end{align}
Above, we have used that $r^2 \le C(r-\bar{r})^2 $ for some constant $C$ on our domain of integration.
\end{comment}
\noindent
Using the other bound from Proposition \ref{Hest}, we can bound $III$
\begin{align}
|III| &\le \int_{D\setminus\Omega } \left|\partial_{\bar{r}}K(r,z,\bar{r},\bar{z}) w(r,z)\right|\, dr\, dz\le C(D) \|w_0\|_\infty \int_{D\setminus\Omega } \frac{r^2}{|x-y|^3}\, dr\, dz \nonumber \\
&\le C(D) \|w_0\|_\infty \int_{D\setminus\Omega } \frac{1}{|x-y|}\, dr\, dz \le C(D) \|w_0\|_\infty. \label{III_integral}
\end{align}
Above, we have used that $r^2 \le C(r-\bar{r})^2 $ for some constant $C$ on our domain of integration.
After combining the estimates \eqref{I_integral}, \eqref{II_integral}, and \eqref{III_integral}, we get the desired estimates at interior points.
\begin{comment}
Thus, for the integral $III$, we get
\begin{align}
|III| \le C(D) \|w_0\|_\infty +C(D)\|w_0\|_\infty\bar{r}\log\delta^{-1} \label{III_integral}
\end{align}
\end{comment}
\medskip
{\bf Case 2: $\mbox{dist} (x,\partial D) < 2\delta$}
\medskip
We can express the derivatives of $u^r$ and $u^z$ as
\begin{align*}
\partial_{\bar{r}} u^r &= -\partial_{\bar{r}}\partial_{\bar{z}} \left( \frac{\psi}{\bar{r}}\right) \\
\partial_{\bar{z}} u^r &= - \partial_{\bar{z}}^2 \left( \frac{\psi}{\bar{r}}\right) \\
\partial_{\bar{r}} u^z &= \partial_{\bar{r}} \left(\frac{\partial_{\bar{r}} \psi}{\bar{r}}\right) = -\partial_{\bar{z}}^2 \left(\frac{\psi}{\bar{r}}\right)- \omega^\theta
\end{align*}
where in the last equality we use $\mathscr{L}(\psi/\bar{r})=\omega^\theta$.
\medskip
Find a point $x'$ such that $\mbox{dist}(x,\partial D) \ge 2\delta$ and $|x'-x|\le C(D) \delta$. By estimate \eqref{g_est} of Lemma \ref{globalreg} applied to $\frac{\psi}{r}$, we know that
\begin{align*}
|\nabla u^r(x)-\nabla u^r(x')| & \le C(\alpha ,D)\delta^\alpha \|\omega^\theta\|_{C^\alpha} \le C(\alpha ,D)\delta^\alpha \|w\|_{C^\alpha} \\
|\partial_{\bar{r}} u^z(x)-\partial_{\bar{r}} u^z(x')|& \le C(\alpha ,D)\delta^\alpha \|w\|_{C^\alpha} .
\end{align*}
Combining these above estimates with our interior estimates above\eqref{I_integral}, \eqref{II_integral}, and \eqref{III_integral} , we can deduce the main estimate $\eqref{kato_est}$. Recall our choice of $\delta$ in $\eqref{delta_choice}$. Then the two estimates we just derived
above become part of the first term on the right side of $\eqref{kato_est}$. The log factor in $\eqref{kato_est}$ will arise
from our earlier estimates $\eqref{I_integral}$, $\eqref{II_integral}$, and choice of $\delta$. We can deduce the $R$ factors in front of the log term of \eqref{kato_est} since our interior
estimate decays with $R$ and our boundary estimate has no $\log $ term. This completes the proof of Theorem \ref{kato} $\Box$.
\subsection{Proof of Theorem \ref{upper}}
With Theorem \ref{kato} now at our disposal, we can derive our first main result. Using \eqref{kato_est}, the proof of estimate \eqref{upper1} is standard and we refer readers to \cite{KiselevSverak} for the details. To prove part (b) of the theorem, using that $u^r(0^+,z)=0$ for all $z$ and \eqref{kato_est},
\begin{align*}
\frac{d}{dt} \Phi_t^r(x) & = u^r(\Phi_t^r(x),t) \ge -\|\nabla u\|_{L^\infty(D_{2\Phi_t(x)})}\Phi_t^r(x) \\
& \ge -C\left(1+ \Phi_t^r(x) \log \left(1+ \frac{\|\nabla w\|_{\infty}}{\|w_0\|_\infty}\right)\right) \Phi_t^r(x) \\
& \ge -C\left(1+ \Phi_t^r(x) \exp(C\|w_0\|_\infty t)\right) \Phi_t^r(x).
\end{align*}
In the last inequality above, we used the double exponential upper bound from part (a). From the above differential inequality,
it is not hard to deduce the desired estimate on $\Phi_t^r(x)$. Part (c) of the theorem follows from part (b)
and $w\circ \Phi = w_0$. $\Box$
\section{Vorticity Gradient Growth on the Boundary Away from the Axis}
In this section, we will provide an example of double exponential gradient growth of vorticity at the boundary using the ideas of Kiselev and Sverak \cite{KiselevSverak}. Observe that away from the axis $\nabla w \approx \nabla \omega^\theta$. We will take our domain to be the unit sphere. We choose the sphere as we have an explicit expression for the Green's function of $L$ for this domain which is
\begin{align}
\label{sphere}
G(r,z,\bar{r},\bar{z})= \frac{\sqrt{r\bar{r}}}{2\pi} \left( F\left(
\frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}\right) -
F\left( \frac{(r^\ast-\bar{r})^2+(z^\ast-\bar{z})^2}{r^\ast\bar{r}}\right)\right)
\end{align}
where we define $\displaystyle r^\ast = \frac{r}{r^2+z^2}$ and $\displaystyle z^\ast = \frac{z}{r^2+z^2}$. Using the methods of \cite{Xiao}, we expect Theorem \ref{lower} to hold for more general domains such as a cylinder.
\medskip
The desired growth of vorticity is achieved by establishing
a ``hyperbolic flow" scenario at the boundary of the sphere. Due to axial symmetry, it is sufficient to consider everything that follows to be on a semi-circular slice of the sphere
$$
D :=\{ (r,\theta,z): \theta=0, \quad r\ge 0, \quad r^2+z^2\le 1\}
$$
and without loss of generality, we will omit the $\theta$ component in our coordinate expressions. We will consider initial data $w_0$ which is odd with respect to $z$ and positive for $z>0$. Our goal is to show that the boundary point $e_1=(r=1,z=0)$ will act as a hyperbolic fixed point of the flow near the boundary. Because of the symmetry assumptions, we can write $u$ as
\begin{align}
u(x) &= \frac{1}{2\pi\bar{r}}\int_D r\nabla^\perp G(y,x)\omega(y,t)\, dy \nonumber \\
\label{Uball}
&= \frac{1}{2\pi\bar{r}}\int_{D^+} r\nabla^\perp \left( \sqrt{r\bar{r}} \left[F\left(\frac{|x-y|^2}{r\bar{r}}\right) -F\left( \frac{|x-y^\ast|^2}{r^\ast\bar{r}}\right)-F\left(\frac{|\widetilde{x}-y|^2}{r\bar{r}}\right)+F\left(\frac{|\widetilde{x}-y^\ast|^2}{r^\ast\bar{r}}\right)\right]\right)\omega(y)\, dy
\end{align}
where we have defined
$$
x=(\bar{r},\bar{z}),\, y=(r,z),\, \widetilde{x}=(r,-z),\, y^\ast=(r^\ast,z^\ast),\, \mbox{and}\, D^+=D\cap \{z\ge 0\}.
$$
\medskip
After some computation, we have the following identities
\begin{align}
\label{identities}
|y^\ast-e_1|^2= \frac{|y-e_1|^2}{|y|^2}, \quad \frac{z^\ast}{|y^\ast-e_1|^2} = \frac{z}{|y-e_1|^2}, \quad \frac{r^\ast-1}{|y^\ast-e_1|^2}=-1-\frac{r-1}{|y-e_1|^2}.
\end{align}
\medskip
The key observation in achieving double exponential growth is an expansion for the Biot Savart law near the fixed point. For the 2D Euler equations, this is the content of Lemma 3.1 of \cite{KiselevSverak}. We will aim to prove a similar expansion in our axially symmetric setting in the lemma below. Choose a constant $N$ such that $N<\min\{1/2,\frac{\epsilon}{8}\}$ let
\begin{align*}
S_N &=\{ 1-N<r<1, 0<z<N\}\cap D \\
Q(\bar{r},\bar{z}) &= \{ 1-N<r<\bar{r}, \bar{z}<z<N\}\cap D.
\end{align*}
Recall we defined $\epsilon$ to be a small constant so that the expansions of $F(s)$ and its derivatives ($\eqref{Fexp}, \eqref{F'exp0}$, and $\eqref{F''exp0}$) can be used for $0<s<\epsilon$.
Define the angular variable $\phi<\pi/2$ to be the angle between the lines $r=1$ and the line through $e_1$ and positive $z$ axis. Also, for any $0<\gamma<\pi/2$, define $D_1^\gamma$ to be the intersection of $D$ with the sector $\pi/2-\gamma\ge \phi\ge 0$. Denote $D_2^\gamma$ to be the intersection of $D$ with the sector $\pi/2\ge \phi\ge \gamma$ and $D^+$.
\medskip
The following lemma will be key in proving Theorem \ref{lower}.
\begin{lemma}
\label{mainlemma1} There exists a small $\delta>0$ such
that for all $x:=(\bar{r},\bar{z})\in D_1^\gamma$ with $|x-e_1|<\delta$,
\begin{align}
\label{uzlaw}
u^z(x) = - \frac{4}{\pi}\bar{z}\cdot \int_{Q(\bar{r},\bar{z})}
\frac{(1-r)z}{((1-r)^2+z^2)^2}w(r,z)\, dr\, dz+ \bar{z} B_1(\bar{r},\bar{z},t)
\end{align}
where $|B_1(\bar{r},\bar{z},t)| \le C(\gamma) \|w_0\|_{L^\infty}$.
Similarly, for all $x\in D_2^\gamma$ with $|x-e_1|<\delta$,
\begin{align}
\label{urlaw}
u^r(x)= \frac{4}{\pi}(1-\bar{r})\cdot \int_{Q(\bar{r},\bar{z})}
\frac{(1-r)z}{((1-r)^2+z^2)^2}w(r,z)\, dr\, dz+ (1-\bar{r}) B_2(\bar{r},\bar{z},t)
\end{align}
where $|B_2(\bar{r},\bar{z},t)| \le C(\gamma) \|w_0\|_{L^\infty}$.
\end{lemma}
\smallskip
\noindent
{\bf Remark:} In the proof of Theorem \ref{upper}, we needed to carefully keep track of the powers of $r$ and $\bar{r}$. However, since we now are examining dynamics in a neighborhood of $(r,z)=(1,0)$ which is away from the axis, in many cases, powers of $r$ and $\bar{r}$ can safely be controlled by uniform constants.
{\bigskip\noindent\bf Proof.\quad} Let us prove the expansion for $u^z$ as the one for $u^r$ can be done similarly. For $x=(\bar{r},\bar{z})\in D_1^\gamma$ with $|x-e_1|<\delta$, we have $1-\bar{r} \le (\cot \gamma)\bar{z}$. Define
\begin{align*}
\rho = 10(1+\cot\gamma) \bar{z}
\end{align*}
so we are assured that $x\in B_\rho(e_1)$. Pick $\delta<1/2$ small enough such that $B_\delta(e_1) \subset S_N$ and $\rho<N/2$. The part of the integral for $u^z$ over the ball $B_\rho(e_1)$ will be the part of the remainder term. Using the bounds $|F'(s)|\lesssim 1/s$ and $\frac{r^{3/2}}{\sqrt{\bar{r}}} \lesssim 1$, one can get that this integral is bounded by
\begin{align*}
C \|w_0\|_{L^\infty} \int_{D^+\cap B_\rho(e_1)} \frac{r^{3/2}}{\sqrt{\bar{r}}} \frac{1}{|x-y|} \, dy \le C(\gamma) \|w_0\|_{L^\infty}\bar{z}.
\end{align*}
Next, we will estimate the integral on $S_N\setminus B_\rho(e_1)$ and will remark on $D^+\setminus S_N$
later.
On the set $S_N\setminus B_\rho(e_1)$, we can use the Taylor expansion of $F(s)$ at $s=0$ to get
\begin{align}
\label{GTaylor}
\frac{1}{2\pi}G(r,z,\bar{r},\bar{z}) &= -\frac{1}{4\pi}\sqrt{r\bar{r}} \left[ \log\left(\frac{|x-y|^2}{r\bar{r}}\right) -
\log\left( \frac{|x-y^\ast|^2}{r^\ast\bar{r}}\right)-\log\left(\frac{|\widetilde{x}-y|^2}{r\bar{r}}\right)+\log\left(\frac{|\widetilde{x}-y^\ast|^2}{r^\ast\bar{r}}\right)\right]\\ &\, + O\left(\frac{|x-y^\ast|^2}{r^\ast\bar{r}}\log\left(\frac{|x-y^\ast|^2}{r^\ast\bar{r}}\right)\right)+O\left(\frac{|x-y|^2}{r\bar{r}}\log\left(\frac{|x-y|^2}{r\bar{r}}\right)\right) \nonumber \\
&= -\frac{1}{4\pi} \sqrt{r\bar{r}} \left[ \log|x-y|^2-\log|x-y^\ast|^2-\log|\widetilde{x}-y|^2+\log|\widetilde{x}-y^\ast|^2\right] \nonumber \\ &\, + O\left(\frac{|x-y^\ast|^2}{r^\ast\bar{r}}\log\left(\frac{|x-y^\ast|^2}{r^\ast\bar{r}}\right)\right)+O\left(\frac{|x-y|^2}{r\bar{r}}\log\left(\frac{|x-y|^2}{r\bar{r}}\right)\right) \nonumber.
\end{align}
\smallskip
Now, we concern ourselves with the four logarithms above as the other terms can be absorbed into the remainder term of \eqref{uzlaw}, which we will remark on later below. We proceed as in \cite{KiselevSverak}. The four logarithms can be written as $-\frac{1}{4\pi}\sqrt{r\bar{r}}$ multiplied with
\begin{align}
\label{logs}
\log \left(1- \frac{2(y-e_1)\cdot(x-e_1)}{|y-e_1|^2} + \frac{|x-e_1|^2}{|y-e_1|^2}\right) - \log\left(1- \frac{2(y^\ast-e_1)\cdot(x-e_1)}{|y^\ast-e_1|^2} + \frac{|x-e_1|^2}{|y^\ast-e_1|^2}\right) \\
-\log \left(1- \frac{2(y-e_1)\cdot(\widetilde{x}-e_1)}{|y-e_1|^2} + \frac{|x-e_1|^2}{|y-e_1|^2}\right)+\log\left(1- \frac{2(y^\ast-e_1)\cdot(\widetilde{x}-e_1)}{|y^\ast-e_1|^2} + \frac{|x-e_1|^2}{|y^\ast-e_1|^2}\right). \nonumber
\end{align}
We use the following expansion for $\log$ for small $q$
\begin{align*}
\log(1+q)= q-\frac{q^2}{2}+O(q^3).
\end{align*}
On the complement of $B_\rho(e_1)$, $|y-e_1|\ge 10 |x-e_1|$ so we can use this expansion. Then $\eqref{logs}$ becomes
\begin{align*}
-4\frac{z\bar{z}}{|y-e_1|^2}+4\frac{\bar{z} z^\ast}{|y^\ast-e_1|^2}-8\frac{(r-1)(\bar{r}-1)z\bar{z}}{|y-e_1|^4}+
8\frac{(r^\ast-1)(\bar{r}-1)z^\ast\bar{z}}{|y^\ast-e_1|^4}+O\left(\frac{|x-e_1|^3}{|y-e_1|^3}\right).
\end{align*}
Using the identities $\eqref{identities}$ to simplify we get
\begin{align*}
-16\frac{(r-1)(\bar{r}-1)z\bar{z}}{|y-e_1|^4}-8\frac{(\bar{r}-1)z\bar{z}}{|y-e_1|^2} + O\left(\frac{|x-e_1|^3}{|y-e_1|^3}\right).
\end{align*}
Then
\begin{align*}
\frac{1}{4\pi}G(r,z,\bar{r},\bar{z})= \frac{4}{\pi}\sqrt{r\bar{r}}\frac{(r-1)(\bar{r}-1)z\bar{z}}{|y-e_1|^4}+\frac{2}{\pi}\sqrt{r\bar{r}}\frac{(\bar{r}-1)z\bar{z}}{|y-e_1|^2}+\sqrt{r\bar{r}}O\left(\frac{|x-e_1|^3}{|y-e_1|^3}\right).
\end{align*}
Differentiating the above expression with respect to $\bar{r}$ we get
\begin{align}
\label{Gdiff}
\frac{4}{\pi}\sqrt{r\bar{r}} \frac{(r-1)z\bar{z}}{|y-e_1|^4}+ \frac{2}{\pi}\sqrt{r\bar{r}}\frac{z\bar{z}}{|y-e_1|^2} +\frac{2}{\pi}\sqrt{\frac{r}{\bar{r}}} \frac{(r-1)(\bar{r}-1)z\bar{z}}{|y-e_1|^4}+\frac{1}{\pi} \sqrt{\frac{r}{\bar{r}}} \frac{(\bar{r}-1)z\bar{z}}{|y-e_1|^2}+O\left(\frac{|x-e_1|^2}{|y-e_1|^3}\right).
\end{align}
The error term is controlled with
\begin{align*}
|x-e_1|^2 \int_{S_N\setminus B_\rho(e_1)} \frac{1}{|y-e_1|^3}\, dy \lesssim |x-e_1|^2 \int_{\rho}^{1} \frac{1}{t^2}\, dt \lesssim |x-e_1|^2 \rho^{-1} \le C(\gamma)\bar{z}.
\end{align*}
In addition, we can control some of the other terms of \eqref{Gdiff} by
\begin{align*}
\bar{z}\int_{S_N\setminus B_\rho(e_1)} \left(\frac{\sqrt{r\bar{r}}z}{|y-e_1|^2}+\sqrt{\frac{r}{\bar{r}}}\frac{|r-1||\bar{r}-1|z}{|y-e_1|^4}+\sqrt{\frac{r}{\bar{r}}} \frac{|\bar{r}-1|z}{|y-e_1|^2}\right)\, dy\\ \le C\bar{z} \int_{S_N\setminus B_\rho(e_1)}\left(\frac{z}{|y-e_1|^2}+\frac{|r-1||\bar{r}-1|z}{|y-e_1|^4}+\frac{|\bar{r}-1|z}{|y-e_1|^2}\right)\, dy \\
\le C\bar{z} \int_{S_N\setminus B_\rho(e_1)} \left(\frac{1}{|y-e_1|}+1\right)\, dy \le C\bar{z} \int_{\rho}^{1} dt \le C(\gamma) \bar{z}
\end{align*}
Therefore, only the first term of $\eqref{Gdiff}$ will contribute to the main term of \eqref{uzlaw}. Next,
\begin{align*}
\int_{S\setminus B_\rho(e_1)} \frac{(r-1)z}{|y-e_1|^4}\, dy=O(1)+ \int_{Q(\bar{r},\bar{z})} \frac{(r-1)z}{|y-e_1|^4} \, dy
\end{align*}
since we have the bound
\begin{align*}
\left|\int_{Q(\bar{r},\bar{z})\cap B_\rho(e_1)} \frac{(r-1)z}{|y-e_1|^4} \, dy\right| &\le \left|\int_{\bar{z}}^{C\bar{z}}\int_1^{C\bar{z}} \frac{(r-1)z}{|y-e_1|^4} \, dr\, dz \right|\le C \int_{\bar{z}}^{C\bar{z}}\int_0^{(C\bar{z}-1)^2} \frac{z}{(t+z^2)^2}\, dt\, dz \\
&\le C\int_{\bar{z}}^{C\bar{z}} \frac{1}{z}\, dz \le C.
\end{align*}
Now, $S_N\setminus (Q(\bar{r},\bar{z})\cup B_\rho(e_1))$ is union of two regions, one along the $r$ axis and one near the boundary. These contributions can be controlled using the following bounds
\begin{align*}
\left|\int_{\bar{r}}^1 \int_{\bar{z}}^{N} \frac{(r-1)z}{|y-e_1|^4}\, dr\, dz\right| \le \int_{\bar{r}}^1 \frac{(1-r)}{(r-1)^2+\bar{z}^2}\, dr\, dz \le C(N)
\end{align*}
\begin{align*}
\left|\int_{N}^{\bar{r}} \int_{0}^{\bar{z}} \frac{(r-1)z}{|y-e_1|^4}\, dr\, dz\right|\le C(N)
\end{align*}
Thus, the proof of $\eqref{uzlaw}$ will be complete as long as we show that the contribution by integration over $D^+ \setminus S_N$ is controlled up to a constant factor by $\bar{z}$. The expansions of $F$ at $0$ are no longer valid. We can just bound these integrals using Lemma \ref{Fest}. After a computation, we have
\begin{align*}
u^z(\bar{r},\bar{z}) = \int_{D^+} \left(\frac{r}{\bar{r}}\right)^{3/2}(\widetilde{\mathscr{J}}(\bar{r},\bar{z},r,z)-\widetilde{\mathscr{J}}(\bar{r},-\bar{z},r,z)-\widetilde{\mathscr{J}}(\bar{r},\bar{z},r^\ast,z^\ast)+\widetilde{\mathscr{J}}(\bar{r},-\bar{z},r^\ast,z^\ast))w(\bar{r},\bar{z},r,z)\, dr\, dz
\end{align*}
On region $D^+ \setminus S_N$, let us bound $ \left(\frac{r}{\bar{r}}\right)^{3/2}(\widetilde{\mathscr{J}}(\bar{r},\bar{z},r,z)-\widetilde{\mathscr{J}}(\bar{r},-\bar{z},r,z)) = \widetilde{J}(\bar{r},\bar{z},r,z)-\widetilde{J}(\bar{r},-\bar{z},r,z)$. The contribution from the other difference of $\widetilde{\mathscr{J}}$'s will be similar. It suffices to bound
\begin{align*}
\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}}\left(F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) - F'\left(\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}\right)\right).
\end{align*}
This can be achieved by using Lemma \ref{Fest} and the Mean Value theorem
\begin{align*}
\left|F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) - F'\left(\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}\right)\right| &\le 2\bar{z} \sup_{\widetilde{z}\in (-\bar{z},\bar{z})} \left|\frac{\partial}{\partial \widetilde{z}} F'\left(\frac{(r-\bar{r})^2+(z+\widetilde{z})^2}{\bar{r}r}\right)\right| \\
& \le C\bar{z} r\bar{r} \frac{1}{(r-\bar{r})^3}.
\end{align*}
Thus, as the other terms satisfy similar bounds, we get
\begin{align*}
\left|\int_{D^+\setminus S_N}\widetilde{J}(\bar{r},\bar{z},r,z)-\widetilde{J}(\bar{r},-\bar{z},r,z)w\, dr\, dz\right| \le C\|w_0\|_\infty\bar{z} \int_{D^+\setminus S_N} \frac{1}{(r-\bar{r})^3}\, dr\, dz \le C(N)\|w_0\|_\infty\bar{z}
\end{align*}
Using a similar argument as directly above, one can show that the terms gotten by formally differentiating the error terms of $\eqref{GTaylor}$
can be similarly controlled by $C\|w_0\|_\infty \bar{z}$.
\medskip
Thus, combining the estimates done above we have that for $x=(\bar{r},\bar{z})\in D_1^\gamma$ with $|x-e_1|<\delta$,
\begin{align*}
u^z(x) &= - \frac{4}{\pi}\frac{\bar{z}}{\sqrt{\bar{r}}}\int_{Q(\bar{r},\bar{z})}
\frac{r^{3/2} (1-r)z}{((1-r)^2+z^2)^2}w(r,z)\, dr\, dz+ \bar{z} B_1(\bar{r},\bar{z},t) \\
&= - \frac{4}{\pi}\bar{z}\int_{Q(\bar{r},\bar{z})}
\frac{(1-r)z}{((1-r)^2+z^2)^2}w(r,z)\, dr\, dz+ \bar{z} B_1(\bar{r},\bar{z},t)
\end{align*}
where $|B_1(\bar{r},\bar{z},t)| \le C(\gamma) \|w_0\|_{L^\infty}$. In the last equality above, we can remove
some factors of $\bar{r}$ and $r$ as the errors they produce can be controlled by the error term $\bar{z} B_1$.
\smallskip
The proof of the lemma is complete $\Box$.
\bigskip
The proof of {\bf Theorem \ref{lower}} now follows from a similar argument as seen in Kiselev and Sverak \cite{KiselevSverak} using Lemma \ref{mainlemma1} in place of their Lemma 3.1. We sketch the details below.
\bigskip
\subsection{Proof of Theorem \ref{lower}}
We start with smooth initial data $w_0$ which is identically $1$ on $D^+$ except on a
strip of width $\delta$ around $z=0$ where $0<w_0(x)<1$. We assume $w_0$ is odd with respect
to $z$ so $w_0=0$ on $z=0$. Below, we will impose more restrictions on $w_0$. As $u$ is
incompressible \eqref{divfree}, the distribution function of $w(r,z,t)$ (with respect to the measure
$r\, drdz$) remains the same for all time. This implies the measure of the region where $0<w_0<1$ does not exceed $2\delta$.
\medskip
Then for $|x-e_1|<\delta$ and $x\in D^+$, we can bound the integral term appearing in $\eqref{urlaw}$ and $\eqref{uzlaw}$ by
\begin{align*}
\int_{Q(\bar{r},\bar{z})}
\frac{(1-r)z}{((1-r)^2+z^2)^2}w(r,z)\, dr\, dz \ge c_1 \int_{c_2\sqrt{\delta}}^{N/2} \int_{\pi/6}^{\pi/3} \frac{1}{\rho}\, d\phi d\rho \ge c_2 \log \delta^{-1}
\end{align*}
Here $c_1$ and $c_2$ are universal postive constants and we can choose $\delta$ small enough (dependent on $N$) such that the rightmost inequality above holds. The coordinates $(\rho,\phi)$ are polar coordinates centered at $e_1$. Now, if necessary,
we can choose $\delta$ smaller such that $c_2 \log \delta^{-1} > 100 \cdot C(\gamma)$ where $C(\gamma)$ is the constant from \ref{mainlemma1}.
\medskip
Let $0<z_1'<z_1'' <1$. Define
\begin{align*}
\mathscr{O} (z_1',z_1'') = \left\lbrace (r,z)\in D^+:\, z > -r+1, \, z_1' < z < z_1''\right\rbrace
\end{align*}
along with the quantities
\begin{align*}
\underline{u^z}(z,t) &= \min\{ u^z(r,z,t): \, (r,z)\in D^+, z > -r+1\} \\
\overline{u^z}(z,t) &= \max\{ u^z(r,z,t): \, (r,z)\in D^+, z > -r+1\}.
\end{align*}
From this we can define quantities $a(t)$ and $b(t)$ by
\begin{align*}
\dot{a}(t) &= \overline{u^z}(a,t), \quad a(0)=\epsilon^{10} \\
\dot{b}(t) &= \underline{u^z}(b,t), \quad b(0) = \epsilon.
\end{align*}
where $0<\epsilon<\delta$. Define $\mathscr{O}_t = \mathscr{O}(a(t),b(t))$. Now, we choose
$w_0$ satisfing the same assumptions as above but also specify $w_0=1$ on $\mathscr{O}_0$ with smooth cutoff to zero
(for example, $\|\nabla w_0 \|_{L^\infty} \sim \epsilon^{-10}$).
\medskip
Now, with these notations in place, one can proceed exactly as in \cite{KiselevSverak}. Using their arguments, one can show that $\mathscr{O}_t$
will be non-empty for all $t>0$ and $w(r,z,t)=1$ on $\mathscr{O}_t$. From this, using lemma \ref{mainlemma1}, one can show
$ a(t) \le \epsilon^{C \exp(Ct)}$ for some positive constant $C$. A particle trajectory starting at $z=\epsilon^{10}$ on $\partial D$ near $e_1$ will
never exceed $a(t)$. From this fact, one can arrive at the main estimate of Theorem \ref{lower}. $\Box$
\section{Acknowledgements}
The author would like to thank Prof. Alexander Kiselev for many helpful comments on the numerous drafts of this manuscript.
\begin{comment}
\section{Main Lemma}
\medskip
Now, we prove the main result of this section.
\begin{lemma}
\label{mainlemma}
Let $0<\gamma<\pi/2$ and $D_1^\gamma$ be the sector $0\le \theta \le \frac{\pi}{2}-\gamma$ intersected with $D$ where $\theta$ is angular variable. Then there exists $\delta$ such that for $x=(\bar{r},\bar{z})\in D_1^\gamma$, $|x|<\delta$,
\begin{align}
\label{mainterm}
u^r(x,t)=-\frac{3}{2\pi}\bar{r} \int_{2\bar{r}}^1\int_{2\bar{z}}^1 \frac{r^3 z}{(r^2+z^2)^{5/2}} w(r,z)\, dr\, dz +
\bar{r}^2(\log \bar{r} +1)\cdot A(x,t)
\end{align}
where $|A(x,t)| \le C(\gamma) \|w_0\|_{L^\infty}$. Similarly, let $D_2^\gamma$ be the sector $\pi/2-\gamma \le \theta \le \frac{\pi}{2}$ intersected with $D$. Then for $x\in D_2^\gamma$, $|x|<\delta$,
\begin{align}
\label{mainterm_z}
u^z(x,t) = \frac{3}{2\pi} \bar{r}\int_{2\bar{r}}^1\int_{2\bar{z}}^1 \frac{r^4}{(r^2+z^2)^{5/2}} w(r,z)\, dr\, dz +
\frac{1}{2\pi} \bar{z} \int_{2\bar{r}}^1\int_{2\bar{z}}^1 \frac{r^3z^2}{(r^2+z^2)^3} w(r,z)\, dr\, dz+ \bar{z}^2(\log \bar{z}+1) \cdot A(x,t)
\end{align}
\end{lemma}
{\bigskip\noindent\bf Proof.\quad} First, let us analyze $u^r$. We will use the notation $\widetilde{y}=(r,-z)$ when $y=(r,z)$. The overall outline for the proof is as follows. We can decompose the kernel $K$ for $u^r$ arising from the Green's function of $L$ as
\begin{align}
\nonumber
K(x,y) &=K(\bar{r},\bar{z},r,z)= -\frac{r}{\bar{r}} \frac{\partial}{\partial \bar{z}} \mathscr{G}(r,z,\bar{r},\bar{z}) \\
\label{decomp}
&= \frac{(z-\bar{z})\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right) - \frac{(z+\bar{z})\sqrt{r}}{\bar{r}^{3/2}} F' \left(\frac{|x-\widetilde{y}|^2}{r\bar{r}}\right)- \frac{\partial}{\partial \bar{z}} \frac{r}{\bar{r}} B(r,z,\bar{r},\bar{z}).
\end{align}
We'll show the first term on the right side of $\eqref{mainterm}$ comes from the first two terms of $\eqref{decomp}$ and using lemma \eqref{green}, the remaining terms can be bounded by $C(\gamma) \bar{r}^2(\log \bar{r} +1) \|w_0\|_\infty$.
\medskip
Let's estimate the integrals from the first term of $\eqref{decomp}$:
\begin{align}
\label{firstterm}
\int_0^1 \int_0^1 \frac{(z-\bar{z})\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right)\, w(r,z)\, dr\, dz.
\end{align}
The analysis for the reflected kernel will be similar. We will divide the $[0,1]\times [0,1]$ into many
regions and estimate each separately. For $\delta$ small and $x$, $y$ not too
close we can use the Taylor series expansion for $F'$ at $\infty$ in $\eqref{F'expinfty}$. We will do this on
one of our regions. For the others, we will just use upper bounds for $F'$ from above.
\medskip
\noindent
\begin{enumerate}
\item \underline{$2\bar{r}\le r\le 1$ and $2\bar{z} \le z\le 1$}
\smallskip
\noindent
We split into two parts. Applying bounds for $F'$, one part can be bounded by $C\cdot \bar{r}^2 \|w_0\|_{L^\infty}$:
\begin{align*}
\left| \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{\bar{z}\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right)\, w(r,z)\, dr\, dz \right|
&\lesssim \bar{r}\bar{z} \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{r^3}{|x-y|^5}\, |w(r,z)|\, dr\, dz \\
&\le C\cdot \|w_0\|_{L^\infty}\cdot \bar{r}\bar{z} \int_{\bar{z}}^1\int_{\bar{r}}^1 \frac{r^3}{(r^2+z^2)^{5/2}}\, dr\, dz \\
&\le C\cdot \|w_0\|_{L^\infty}\cdot \bar{r}\bar{z} \int_{\bar{z}}^1\int_{\bar{r}}^1 \frac{r}{(r^2+z^2)^2}\, dr\, dz
\le C(\gamma)\cdot \bar{r}^2\cdot \|w_0\|_{L^\infty}
\end{align*}
where we have used that since $x\in D_1^\gamma$, $\bar{z} \lesssim_\gamma \bar{r}$. The other term is
\begin{align*}
\bar{r} \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{r^3z}{|x-y|^5}w(r,z)\, dr\, dz
\end{align*}
For the remaining term, we use the Taylor expansion for $F'$ at $\infty$ directly and get
\begin{align*}
\int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{z\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right)\, w(r,z)\, dr\, dz= &\, -\frac{3\pi}{4} \bar{r} \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{zr^3}{|x-y|^5} w(r,z)\, dr\, dz \\
&+ \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{z\sqrt{r}}{\bar{r}^{3/2}} \cdot O\left( \frac{(r\bar{r})^{7/2}}{|x-y|^7}\right)w(r,z)\, dr\, dz
\end{align*}
In this region $r<2(r-\bar{r})$ and $z<2(z-\bar{z})$, we can bound the remainder term by
\begin{align*}
C \bar{r}^2 \|w_0\|_\infty \int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{z r^4}{|x-y|^7} dr\, dz \le C(\gamma)\bar{r}^2 \log \bar{z} \|w_0\|_\infty
\end{align*}
Using that
\begin{align*}
\frac{1}{|x-y|^5} - \frac{1}{(r^2+z^2)^{5/2}} = \bar{r} \cdot O\left( \frac{|r|+|z|}{(r^2+z^2)^{7/2}}\right)
\end{align*}
we can write
\begin{align*}
\int_{2\bar{z}}^1 \int_{2\bar{r}}^1 \frac{zr^3}{|x-y|^5} w(r,z)\, dr\, dz =
\bar{r} \int_{2\bar{r}}^1\int_{2\bar{z}}^1 \frac{r^3 z}{(r^2+z^2)^{5/2}} w(r,z)\, dr\, dz + \bar{r}\log \bar{z} \cdot O(1)
\end{align*}
which is the first term on the right side of $\eqref{mainterm}$.
\medskip
\noindent
\item \underline{$0\le r\le \bar{r}/2$ and $0\le z\le 1$}
\medskip
On this region, using bounds for $F'$, we have
\begin{align*}
\left| \int_0^1 \int_0^{\bar{r}/2} \frac{(z-\bar{z})\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right)\, w(r,z)\, dr\, dz \right|
&\lesssim \bar{r} \int_0^1\int_0^{\bar{r}/2} \frac{r^3|z-\bar{z}|}{|x-y|^5} |w(r,z)|\, dr\, dz \\
&\lesssim \bar{r}\|w_0\|_{L^\infty} \int_{\bar{r}/2}^{\bar{r}} \frac{r^3}{(r^2+\bar{z}^2)^{3/2}}\, dr \lesssim_\gamma \bar{r}^2 \|w_0\|_{L^\infty}
\end{align*}
\medskip
\noindent
\item \underline{$2\bar{r}\le r\le 1$ and $0\le z\le 2 \bar{z}$}
\medskip
Observe that on this region $r<2(r-\bar{r})$. Then on this region, the integral \eqref{firstterm} is bounded
by a constant times
\begin{align*}
\bar{r}\|w_0\|_{L^\infty}\left( \int_{2\bar{r}}^1\int_0^{2\bar{z}} \frac{|r-\bar{r}|^3|z-\bar{z}|}{((r-\bar{r})^2+(z-\bar{z})^2)^{5/2}}\, dz\, dr\right)
\end{align*}
From an easy computation,
\begin{align*}
|z-\bar{z}| \int_{2\bar{r}}^1 \frac{r^3}{(r^2+(z-\bar{z})^2)^{5/2}} \, dr\le C.
\end{align*}
Using this, we get that the contribution over this region is bounded by $C\cdot \bar{r}^2 \|w_0\|_{L^\infty}$.
\medskip
\noindent
\item \underline{$\bar{r}/2 \le r< 2\bar{r}$ and $0\le z\le \bar{z}/2$}
\noindent
\medskip
Here, we can use alternative bound for $F'$, $|F'(s)|\lesssim s^{-1}$, to bound \eqref{firstterm} with
\begin{align*}
C\|w_0\|_{L^\infty} \int_{\bar{r}/2}^{2\bar{r}} \int_0^{1} \frac{r^{3/2}}{\bar{r}^{1/2}} \frac{|z-\bar{z}|}{(r-\bar{r})^2+(z-\bar{z})^2}\, dz\, dr &\le
C\bar{r}\|w_0\|_{L^\infty} \int_{\bar{r}/2}^{2\bar{r}} \log \left(\frac{(r-\bar{r})^2+ \bar{z}^2/4}{(r-\bar{r})^2+\bar{z}^2}\right)\, dz \\
& \le C(\gamma) \bar{r}^2 \log \bar{r} \|w_0\|_{L^\infty}
\end{align*}
\medskip
\noindent
\item \underline{$\bar{r}/2 \le r\le 2\bar{r}$ and $2\bar{z}\le z\le 1$}
\smallskip
Using similar method as the previous region (4), region (5) should be bounded by $C\cdot \bar{r}^2 \|w_0\|_{L^\infty}$ as well.
\medskip
\noindent
\item \underline{$\bar{r}/2 \le r \le 2\bar{r}$ and $\bar{z}/2 \le z \le 2\bar{z}$}
We do a similar bound as in case (4) and get
\smallskip
\begin{align*}
\left|\int_{\bar{r}/2}^{2\bar{r}} \int_{\bar{z}/2}^{2\bar{z}} \frac{(z-\bar{z})\sqrt{r}}{\bar{r}^{3/2}} F'\left( \frac{|x-y|^2}{r\bar{r}}\right)\, w(r,z)\, dr\, dz\right| &\le C\bar{r} \|w_0\|_{L^\infty} \int_{\bar{r}/2}^{2\bar{r}} \int_{\bar{z}/2}^{2\bar{z}} \frac{|z-\bar{z}|}{(r-\bar{r})^2+(z-\bar{z})^2}\, dr\, dz.\\
&\le C\bar{r} \|w_0\|_{L^\infty} \int_{B_{c(\gamma)\bar{r}}(x)} \frac{1}{((r-\bar{r})^2+(z-\bar{z})^2)^{1/2}} dr\, dz\\
&\le C(\gamma) \bar{r}^2 \|w_0\|_{L^\infty}
\end{align*}
\end{enumerate}
\medskip
For $u^z$, we can use a similar argument and achieve \eqref{mainterm_z}. For completeness, we give a sketch
of the differences with $u^r$. Write $J$ as
\begin{align*}
J(x,y) &= \widetilde{J}(\bar{r},\bar{z},r,z)-\widetilde{J}(\bar{r},\bar{z},r,-z) +\frac{r}{\bar{r}} \frac{\partial}{\partial \bar{r}} B(r,z,\bar{r},
\bar{z}) \\
&= \frac{1}{\pi}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right) - \frac{1}{\pi}\frac{(\bar{r}-r)\sqrt{r}}{\bar{r}^{3/2}}F'\left(\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}\right) \\
&\, +\frac{1}{4\pi}\left[ F\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)-2\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}F'\left(\frac{(r-\bar{r})^2+(z-\bar{z})^2}{\bar{r}r}\right)\right]\left(\frac{r}{\bar{r}}\right)^{3/2} \\
&\, - \frac{1}{4\pi}\left[ F\left(\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}\right)-2\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}F'\left(\frac{(r-\bar{r})^2+(z+\bar{z})^2}{\bar{r}r}\right)\right]\left(\frac{r}{\bar{r}}\right)^{3/2} \\
&\, + \frac{r}{\bar{r}} \frac{\partial}{\partial \bar{r}} B(r,z,\bar{r},\bar{z}) \\
&= I+II +III+ IV+V
\end{align*}
For $\delta$ small, we can use expansions of $F'$ and $F$ at $\infty$ and get the leading terms
from above are
\begin{align*}
I \sim -\frac{3}{4\pi}\bar{r}\frac{r^4}{|x-y|^5}, \quad II \sim -\frac{3}{4\pi}\bar{r}\frac{r^4}{|x-\widetilde{y}|^5}
\end{align*}
\begin{align*}
III\sim \frac{1}{2\pi} \frac{r^3}{|x-y|^3}, \quad IV \sim \frac{1}{2\pi} \frac{r^3}{|x-\widetilde{y}|^3}
\end{align*}
and we also have
\begin{align*}
III-IV \sim \frac{1}{2\pi} \bar{z} \frac{r^3z^2}{|x-y|^6}.
\end{align*}
Then one can complete an argument as with $u^r$ and derive \eqref{mainterm_z}.
This completes the proof of the lemma $\Box$.
\end{comment}
\begin{appendix}
\bigskip
\section{Estimates for $F$}
Here we will give a rough derivation of the Taylor expansions for the integral $F(s)$
$$
F(s)=\int_0^\pi \frac{\cos\theta \, d\theta}{\sqrt{2(1-\cos\theta)+s}}.
$$
These expansions can also be found in \cite{Sveraknotes}. We will derive them below and that, while elementary, are nonstandard.
\subsection{Estimates at $0$}
We can write $F$ as
\begin{align*}
F(s)=\int_0^{\pi/2} \frac{1+2\sigma^2}{\sqrt{\sin^2\varphi+\sigma^2}}\, d\varphi -2\int_0^{\pi/2} \sqrt{\sin^2\varphi +\sigma^2}\, d\varphi, \quad \sigma^2=s/4.
\end{align*}
The leading order term above is
$$
f(\sigma)=\int_0^{\pi/2} \frac{d\varphi}{\sqrt{\sin^2\varphi+ \sigma^2}} = \int_0^{\pi/2} \frac{\cos \varphi\, d\varphi}{\sqrt{\sin^2\varphi+\sigma^2}}+ \int_0^{\pi/2} \frac{(1-\cos \varphi)\, d\varphi}{\sqrt{\sin^2\varphi+\sigma^2}} :=I+II.
$$
For $I$ we can compute directly and use Taylor series to get
\begin{align*}
I &= \int_0^1\frac{d\varphi}{\sqrt{\varphi^2+\sigma^2}} =\log\frac{1}{\sigma} + \log(1+\sqrt{1+\sigma^2}) \\ &= \log \frac{1}{\sigma} +\log 2+ O(\sigma^2)= \frac{1}{2}\log \frac{1}{s} +2\log 2+ O(s).
\end{align*}
Similarly for $II$ we can get that for $\sigma\to 0^+$
\begin{align*}
II &= \int_0^{\pi/2} \frac{1-\cos\varphi}{\sin\varphi}\, d\varphi+ \int_0^{\pi/2} (1-\cos\varphi)\left(\frac{1}{\sqrt{\sin^2\varphi +\sigma^2}}-\frac{1}{\sin\varphi}\right) d\varphi\\
&= \log 2+ \sigma^2 \int_0^{\pi/2} \frac{1-\cos\varphi}{\sin\varphi\sqrt{\sin^2\varphi+\sigma^2}(\sqrt{\sin^2\varphi+\sigma^2}+\sin\varphi)}d\varphi\\
&= \log 2 + O\left( \sigma^2 \log \frac{1}{\sigma}\right) = \log 2+ O\left( s\log \frac{1}{s}\right)
\end{align*}
where we use that
\begin{align*}
\int_0^{\pi/2} \frac{1-\cos\varphi}{\sin\varphi\sqrt{\sin^2\varphi+\sigma^2}(\sqrt{\sin^2\varphi+\sigma^2}+\sin\varphi)}d\varphi &\le \int_0^{\pi/2} \frac{1-\cos\varphi}{\sin^2\varphi} \frac{d\varphi}{\sqrt{\sin^2\varphi+\sigma^2}} \\
&\le \int_0^{\pi/2} \frac{1-\cos\varphi}{\sin^2\varphi} \frac{d\varphi}{\sqrt{\varphi^2/4+\sigma^2}} \\
&= O\left(\log \frac{1}{\sigma}\right).
\end{align*}
Similarly, we can also have
\begin{align*}
2\int_0^{\pi/2} \sqrt{\sin^2\varphi +\sigma^2}\, d\varphi &= 2+2\sigma^2 \int_0^{\pi/2} \frac{1}{\sqrt{\sin^2\varphi+\sigma^2}+\sin\varphi}\, d\varphi\\
&=2+O\left(\sigma^2 \log \frac{1}{\sigma}\right) = 2+ O\left(s \log \frac{1}{s}\right).
\end{align*}
Putting these expressions together, we get the desired expansion for $F$:
$$
F(s) = \frac{1}{2} \log \frac{1}{s} + \log 8 - 2 + O\left(s \log \frac{1}{s}\right).
$$
Now, consider the derivative
$$
F'(s)= -\frac{1}{2} \int_0^\pi \frac{\cos \theta \, d\theta}{(2(1-\cos \theta)+s)^{3/2}}.
$$
With $\sigma$ as set above,
\begin{align}
\label{Fprime}
F'(s) = -\frac{1}{8}\int_0^{\pi/2} \frac{1+2\sigma^2}{(\sin^2\varphi+\sigma^2)^{3/2}}\, d\varphi+ \frac{1}{4}\int_0^{\pi/2} \frac{1}{\sqrt{\sin^2\varphi+\sigma^2}}\, d\varphi.
\end{align}
Doing a similar decomposition as in $F$ above
\begin{align}
\label{Fprime1}
\int_0^{\pi/2} \frac{d\varphi}{(\sin^2\varphi+\sigma^2)^{3/2}}&=\int_0^{\pi/2} \frac{\cos\varphi d\varphi}{(\sin^2\varphi+\sigma^2)^{3/2}}+ \int_0^{\pi/2}\frac{(1-\cos\varphi)d\varphi}{(\sin^2\varphi+\sigma^2)^{3/2}} \\
&= \int_0^{1/\sigma} \frac{dt}{\sigma^2(t^2+1)^{3/2}}+O(\log s) = \frac{4}{s} + O(\log s) \nonumber
\end{align}
where for the second integral on the right hand side above we estimate as we did II above.
The expansion for the second integral in $\eqref{Fprime}$ is done above and putting things together
$$
F'(s)= -\frac{1}{2}\frac{1}{s} + O(\log s), \quad s\to 0^+.
$$
\subsection{Estimates at $\infty$}
Write $F$ as
$$
F(s) =s^{-1/2} \int_0^\pi \cos \theta \left( \frac{2-2\cos\theta}{s}+1\right)^{-1/2}\, d\theta.
$$
Then by Taylor expansion,
\begin{align*}
F(s) &= s^{-1/2} \int_0^\pi \cos \theta\, d\theta - \frac{1}{2}s^{-3/2}\int_0^\pi 2\cos\theta(1-\cos \theta)\, d\theta + O(s^{-5/2}) \\
&= \frac{\pi}{2} s^{-3/2} + O(s^{-5/2}), \quad s\to \infty.
\end{align*}
The expansions for the derivatives can be derived similarly.
\section{Estimates for $\partial_{\bar{r}} u^z$ and $\partial_{\bar{z}} u^r$ in Theorem \ref{kato}}
In this section, we sketch the details of the proof of Theorem \ref{kato} for the derivatives
$\partial_{\bar{r}} u^z$ and $\partial_{\bar{z}} u^r$ under the assumption $\mbox{dist}(x, \partial D) >2\delta$.
\medskip
First, let us do $\partial_{\bar{z}}u^r$.
\begin{align*}
\partial_{\bar{z}}u^r(\bar{r},\bar{z}) &=\left( \int_{B_{\delta \bar{r}}(x)} +\int_{\Omega \cap B_{\delta\bar{r}}^c(x)} +\int_{D\setminus \Omega}\right)\partial_{\bar{z}}K(r,z,\bar{r},\bar{z})w(r,z)\, dr\, dz \\
&= I+II+III.
\end{align*}
It suffices to bound the integral $I$ as the other two will be controlled just as in the estimates for $\partial_{\bar{r}} u^r$ earlier.
On $B_{\delta \bar{r}}(x)$, we use the expansions for $F$ and we get
\begin{align*}
\partial_{\bar{z}} \widetilde{K}(r,\bar{r},z,\bar{z}) &= -\frac{\sqrt{r}}{\pi\bar{r}^{3/2}} \left[ F'\left(s\right)+2\frac{(z-\bar{z})^2}{\bar{r}r} F''\left(s\right) \right] = \frac{\sqrt{r}}{2\pi\bar{r}^{3/2}}\left[ \frac{1}{s}-2\frac{(z-\bar{z})^2}{\bar{r}r} \frac{1}{s^2} +O(\log s)\right] \\
&= \frac{r^{3/2}}{2\pi\bar{r}^{1/2}}\left[ \frac{|x-y|^2-2(z-\bar{z})^2}{|x-y|^4} \right] + \frac{\sqrt{r}}{2\pi\bar{r}^{3/2}} O(\log s)
\end{align*}
where as earlier, we set $\displaystyle s:= \frac{(r-\bar{r})^2+(z-\bar{z})^2}{r\bar{r}}$.
Using a similar argument as in $u_r^r$, we can bound the contribution from the first term
\begin{align*}
\left| \int_{B_{\delta \bar{r}}(x)}\frac{r^{3/2}}{2\pi\bar{r}^{1/2}}\left[ \frac{|x-y|^2-2(z-\bar{z})^2}{|x-y|^4} \right] w(r,z)\, dr\, dz \right| \le C(\alpha)\bar{r}^{1+\alpha} \|w_0\|_{L^\infty}.
\end{align*}
We use the fact that the integration of the term in brackets over the ball is $0$.
Furthermore, we can bound the contribution from the error
\begin{align*}
\left| \int_{B_{\delta \bar{r}}(x)}\frac{\sqrt{r}}{2\pi\bar{r}^{3/2}} (\log s) \cdot w(r,z)\, dr\, dz \right| \lesssim \bar{r}(1+\log (\delta^{-1} \bar{r}^{-1})) \|w_0\|_{L^\infty}.
\end{align*}
\bigskip
Now, let us do $u_{\bar{r}}^z$. Again, it will suffice to estimate the integral over $B_{\delta \bar{r}}(x)$.
\begin{comment}
We will need to use the estimate
$\eqref{Nest}$ and require a slightly different split of integrals. The decomposition can be sketched as follows
\begin{align}
\label{urz}
\partial_{\bar{r}}u^z(\bar{r},\bar{z}) &= \left( \int_{B_{\delta \bar{r}}(x)} +\int_{\Omega \cap B_{\delta\bar{r}}^c(x)}+\int_{\{s\le M\}\cap \Omega^c} +\int_{D\cap \{s>M\}}\right)\partial_{\bar{r}}J(r,z,\bar{r},\bar{z})w(r,z)\, dr\, dz.
\end{align}
\end{comment}
Recall
\begin{align*}
\partial_{\bar{r}} \widetilde{J} =\frac{1}{\pi}\frac{\sqrt{r}}{\bar{r}^{3/2}}\left[F'(s)+
2\frac{(r-\bar{r})^2}{\bar{r}r}F''(s)\right]+\frac{2(r-\bar{r})}{\pi} \frac{\sqrt{r}}{\bar{r}^{5/2}}[F'(s)+sF''(s)]+ \frac{1}{\pi} \frac{r^{3/2}}{\bar{r}^{5/2}}\left[- \frac{3}{8}F(s)
+ sF'(s)+\frac{1}{2}s^2 F''(s)\right].
\end{align*}
The first term can be estimated in the same way as for $u_{\bar{z}}^r$. We can bound the other terms as follows
\begin{align*}
\left| \int_{B_{\delta \bar{r}}(x)} \frac{2(r-\bar{r})}{\pi} \frac{\sqrt{r}}{\bar{r}^{5/2}}[F'(s)+sF''(s)]\, w(r,z)\, dr\, dz \right| & \le \int_{B_{\delta \bar{r}}(x)} \left(\frac{r}{\bar{r}}\right)^{3/2} \frac{1}{|x-y|} \, |w(r,z)|\, dr \, dz \\ &\le C \bar{r} \|w_0\|_{L^\infty}
\end{align*}
\begin{align*}
\left| \int_{B_{\delta \bar{r}}(x)} \frac{r^{3/2}}{\bar{r}^{5/2}}\left[- \frac{3}{8}F(s)
+ sF'(s)+\frac{1}{2}s^2 F''(s)\right]\, w(r,z)\, dr\, dz \right| &\le \int_{B_{\delta \bar{r}}(x)} \frac{r^{3/2}}{\bar{r}^{5/2}} \left| \log(s) \cdot w(r,z)\right|\, dr\, dz \\
& \le C \bar{r} (1+\log(\delta^{-1}\bar{r}^{-1}))\|w_0\|_{L^\infty}.
\end{align*}
\begin{comment}
Now, let us estimate the other integrals appearing in \eqref{urz} using Lemma 3.2. We arrive at the following
\begin{align*}
\left| \int_{\Omega\cap B_{\delta\bar{r}}^c(x)}(\partial_{\bar{r}} \widetilde{J} )w \, dr\, dz\right| \lesssim \int_{\Omega\cap B_{\delta\bar{r}}^c(x)} \left(\frac{1}{|x-y|} + \frac{\bar{r}}{|x-y|}\right) |w| \, dr\, dz
\lesssim \|w_0\|_\infty + \bar{r}\|w_0\|_\infty(1+\log (\delta^{-1}r^{-1}))
\end{align*}
\begin{align*}
\left| \int_{\{s\le M\} \cap \Omega^c}(\partial_{\bar{r}}\widetilde{J}) w \, dr\, dz\right| & \lesssim \int_{\{s\le M\} \cap \Omega^c}\left(\frac{r^3}{|x-y|^4} + \frac{\bar{r}r^3}{|x-y|^5}+ \frac{r^3}{\bar{r}|x-y|^3}\right) |w| \, dr\, dz
\\ & \lesssim \int_{\{s\le M\} \cap \Omega^c} \left( \frac{1}{|x-y|}+\frac{\bar{r}}{|x-y|^2} +M \frac{r^4}{|x-y|^5}\right) |w|\, dr \, dz \\
&\le C(D,M) \|w_0\|_\infty + C(D,M)\|w_0\|_\infty \bar{r}\log \delta^{-1}
\end{align*}
\begin{align*}
\left|\int_{D\cap\{s>M\}}(\partial_{\bar{r}} \widetilde{J} )w \, dr\, dz\right| & \lesssim \int_{\{s\le M\} \cap \Omega^c}\left(\frac{r^3}{|x-y|^4} + \frac{\bar{r}r^3}{|x-y|^5}+ \frac{r^4}{|x-y|^5}\right) |w| \, dr\, dz\\
&\lesssim \|w_0\|_\infty + \|w_0\|_\infty \bar{r}\log \delta^{-1}
\end{align*}
\end{comment}
\end{appendix}
|
1,314,259,992,975 | arxiv | \section{Introduction}
In the context of condensed matter physics, the glass transition
is a rather generic phenomenon through which
the state of a system becomes partially frozen below a threshold temperature
\cite{Berthier,Nagel,book-glasses}.
One of the simplest models exhibiting a glass transition
is the Random Energy Model (REM) introduced by Derrida
\cite{Derrida},
in which the energies of microscopic configurations are independent
and identically distributed random variables drawn from a distribution
$\mathcal{P}(E)$,
chosen to be Gaussian in the original version of the model.
The REM provides a simple illustration of the so-called
'one-step replica symmetry breaking' scenario of the glass transition
\cite{BouchMez97}, which is known to hold in more sophisticated mean-field
models \cite{Wolynes87,Wolynes89}.
Several generalizations of the REM have been proposed, mostly to
incorporate correlations in simple ways
\cite{Gardner,Carpentier,Fyodorov,Rosso}.
In addition, interesting connections of the REM with probabilistic issues
such as the convergence properties of sums \cite{BenArous} and extreme values
\cite{Bogachev,Fyodorov,AngelettiEVS} of random variables, as well as
with signal processing issues such as moment estimation
\cite{Angeletti11}, have been pointed out.
Potential connections with string theories have even been recently
outlined \cite{Saakian09}.
In this note, we explore the question whether uncorrelated random
energy levels are enough to generate a glass transition.
To this aim, we consider a generalized version of the REM
offering some freedom in the energy distribution as well as
in the scaling of the number of configuration with system size.
The paper is organized as follows. The model is introduced in
Sect.~\ref{sect-model}, and the necessary framework to analyze
the glass transition is presented in Sect.~\ref{sect-glass-trans}.
Then in Sect.~\ref{sect-no-glass}, which constitutes the core of this
paper, we derive necessary conditions for the absence of glass transition,
and show that these conditions are also sufficient. In addition,
we determine the behavior of the glass transition temperature close to onset.
Finally, Sect.~\ref{sect-discus} summarizes the results,
and discusses some analogies with related problems.
\section{Model}
\label{sect-model}
\subsection{Definition}
We consider a disordered system having a number $M$ of microscopic
configurations, labeled by index $k=1,\ldots,M$; to each configuration
is associated a quenched random energy $E_k$.
The energies $E_k$ are assumed to be independent and identically distributed.
At variance with the standard REM, we do not directly specify the distribution
$\mathcal{P}(E)$ from which the energies $E_k$ are drawn, but we rather assume
that $E_k$ is given by a sum of $N$ individual contributions,
\begin{equation} \label{eq:Es-sum}
E_k = \sum_{i=1}^{N} \eta_{k,i}, \qquad k=1,\ldots,M,
\end{equation}
where the terms $\eta_{k,i}$ are independent and identically distributed
random variables with a finite variance distribution $p(\eta)$.
A standard assumption is that the number $M$ of configurations scales as
\begin{equation} \label{eq:scal-M}
M \sim e^{\alpha N} \qquad (\alpha >0).
\end{equation}
For the standard REM, $\alpha=\ln 2$.
Note that this model may be considered as a simplified version of
the directed polymer problem in a random media \cite{Spohn},
neglecting correlations between local energies on different paths.
One can interpret \Eqref{eq:Es-sum} as the decomposition of the total
energy over the $N$ degrees of freedom of the system, with the strong
assumption that local contributions associated to different microscopic
configurations $k$ are statistically independent.
Such an approach provides an alternative
interpretation (besides the standard one in terms of $p$-spin model
in the limit $p \to \infty$ \cite{Derrida}) of the Gaussian energy
distribution of variance proportional to $N$ used in the standard
REM \cite{Derrida} (though more general energy distributions have
also been considered \cite{BouchMez97}).
But it also gives many ways to depart from the Gaussian distribution,
in the sense that equilibrium at finite temperature is dominated
by energy values in the lower tail of the energy distribution,
far from the maximum of the distribution around which the Gaussian
approximation holds.
As we shall see below, the key ingredient of our generalized version of REM
is the introduction of both the arbitrary distribution $p(\eta)$ and
the free parameter $\alpha$, which leads to a richer behavior than
in the standard REM.
\subsection{Large deviation function}
To characterize the distribution $\mathcal{P}(E)$ of the energies $E_k$ from the distribution $p(\eta)$,
we use large deviation theory \cite{Ellis,Touchette}.
Let us define the energy density $\epsilon=E/N$, and its distribution
$P(\epsilon) = N \mathcal{P}(N\epsilon)$.
The G\"artner-Ellis theorem \cite{Ellis,Touchette}
implies that, if the cumulant generating function
\begin{equation} \label{def-lambda}
\lambda(q) = \ln \Esp{e^{q \eta}}
\end{equation}
is defined and differentiable on the real axis, then the distribution $P(\epsilon)$ is given by
\begin{equation}
\label{eq@LDf}
P(\epsilon) \approx e^{- N \phi(\epsilon) }
\end{equation}
where $\phi$ is the Legendre transform of $\lambda$,
$\phi(\epsilon) = \max_{q} \{\epsilon q - \lambda(q)\}$.
Note that the properties of the Legendre transform imply that $\phi(\epsilon)$
is a convex function. Similarly, since for all $\epsilon$, $(\epsilon q - \lambda(q))_{|q=0} = 0$, we have $\phi(\epsilon) \ge 0
$. Moreover, the lower bound $0$ is attained at $\epsilon=\lambda'(0)$: $\phi(\lambda'(0))=0$.
In the present statistical physics context, it is however more natural
to use the function
\begin{equation} \label{def-mu}
\mu(\beta) = \ln \Esp{e^{-\beta \eta}} =\lambda(-\beta)
\end{equation}
instead of $\lambda(q)$, and we shall thus use $\mu(\beta)$ in the following,
together with the relation
\begin{equation} \label{eq:legendre-mu}
\phi(\epsilon) = -\min_{\beta} \{\epsilon \beta + \mu(\beta)\}.
\end{equation}
\section{Analysis of the glass transition}
\label{sect-glass-trans}
In the present section, we briefly set up the general framework
allowing the glass transition to be studied. This framework
mainly relies on the existence of a finite size cutoff in the
density of state \cite{Derrida}.
\subsection{Finite size cutoff}
\label{sec@Fsc}
Considering a given sample with a finite number of configurations,
the energies $E_k$ are necessarily
confined to a finite subdomain of the support of $\mathcal{P}(E)$.
In the low temperature regime, the lower bound of this domain is
known to play an important role \cite{Derrida}.
Although the boundaries of this domain are also random variables,
it is nevertheless possible to define a sharp lower boundary on the energy
density in the large $M$ limit.
Defining the cumulative $F(\epsilon) = \int_{-\infty}^{\epsilon} d\epsilon' \, P(\epsilon')$,
we consider the probability
\begin{equation} \label{eq:pe1eM}
\Prob{\epsilon_1,\dots,\epsilon_M > \epsilon} = [1-F(\epsilon)]^M
\end{equation}
that all energy densities $\epsilon_k$, $k=1,\ldots,M$, are larger
than a given value $\epsilon$.
We denote as $\epsilon_0$ the value of $\epsilon$ for which $P(\epsilon)$
is maximum (i.e., $\phi(\epsilon_0)=0$).
Note that $\phi(\epsilon)$, being a convex function, necessarily
decreases for $\epsilon<\epsilon_0$.
Using a standard saddle-point method and neglecting non-exponential
prefactors, we have for $\epsilon<\epsilon_0$,
$F(\epsilon) \approx e^{- N \phi(\epsilon)}$ so that
$[1-F(\epsilon)]^M \approx e^{-M F(\epsilon)}$.
\Eqref{eq:pe1eM} can then be rewritten as
\begin{equation} \label{eq:pe1eM2}
\ln \Prob{\epsilon_1,\dots,\epsilon_M > \epsilon} \approx -M\,F(\epsilon)
= - e^{N(\alpha-\phi(\epsilon))},
\end{equation}
using $\ln M = \alpha N$ [see \Eqref{eq:scal-M}].
Let us define $\epsilon^{\dagger}<\epsilon_0$ such that
\begin{equation} \label{eq@ecut}
\phi( \epsilon^{\dagger} ) = \alpha.
\end{equation}
\Eqref{eq:pe1eM2} shows that $\Prob{\epsilon_1,\dots,\epsilon_M > \epsilon}$
exhibits a sharp crossover at $\epsilon=\epsilon^{\dagger}$.
For $\epsilon < \epsilon^{\dagger}$, $\alpha-\phi(\epsilon)<0$,
and $\Prob{\epsilon_1,\dots,\epsilon_M > \epsilon} \to 1$ when $N \to \infty$,
so that there is with probability one no energy levels below $\epsilon$.
In contrast, for $\epsilon > \epsilon^{\dagger}$, $\alpha-\phi(\epsilon)>0$, so that
$\Prob{\epsilon_1,\dots,\epsilon_M > \epsilon} \to 0$ when $N \to \infty$,
which means that with probability one, there are energy levels lower than
$\epsilon$. In the limit $N \to \infty$, the value $\epsilon^{\dagger}$ thus corresponds
essentially to a border below which there are no more energy levels.
In other words, the value $\epsilon^{\dagger}$ can be considered as the ground state energy
density in almost all samples
\footnote{Note that similarly, an upper bound also exists for the energy levels,
but we do not take it into account as it plays no role in the thermodynamic
properties.}.
\subsection{Free energy and glass transition temperature}
To evaluate the disorder averaged free energy $\Esp{F}=-\beta^{-1}\Esp{\ln Z}$,
where $Z$ is the partition function $Z = \sum_{k=1}^M e^{-\beta E_k}$
and $\beta=1/T$ the inverse temperature,
a usual method (beyond the replica trick \cite{BouchMez97}) is
to determine the typical value
$Z_{\mathrm{typ}}$ (rather than the averaged one) of the partition function,
yielding $\Esp{F} \approx -T \ln Z_{\mathrm{typ}}$.
$Z_{\mathrm{typ}}$ is evaluated taking into account the threshold $\epsilon^{\dagger}$, and
approximating the density of states $n(\epsilon)$
by the disorder averaged one $\Esp{n(\epsilon)}=M\, P(\epsilon)$ in the energy
range $\epsilon > \epsilon^{\dagger}$, where $\Esp{n(\epsilon)}$ is large.
One finds, using the large deviation form \Eqref{eq@LDf},
\begin{equation} \label{eq:ztyp}
Z_{\mathrm{typ}} = \int_{\epsilon^{\dagger}}^{\infty} e^{N g(\epsilon)} d\epsilon,
\qquad g(\epsilon) = \alpha-\phi(\epsilon) -\beta \epsilon.
\end{equation}
For large $N$, the partition function \Eqref{eq:ztyp} can be evaluated
through a saddle-point approximation.
Equating the derivative $g'(\epsilon)$ to zero yields
\begin{equation} \label{eq:phiprime-beta}
\phi'(\epsilon_{\mathrm{m}})=-\beta.
\end{equation}
As $g''(\epsilon) \le 0$, $g(\epsilon)$ decreases for
$\epsilon > \epsilon_{\mathrm{m}}$.
If $\epsilon_{\mathrm{m}} > \epsilon^{\dagger}$, the saddle-point evaluation yields
$Z_{\mathrm{typ}} \approx e^{N(\alpha-\phi(\epsilon_{\mathrm{m}}) -\beta \epsilon_{\mathrm{m}})}$.
In the opposite case $\epsilon_{\mathrm{m}} < \epsilon^{\dagger}$, the global maximum of
$g(\epsilon)$ is no longer relevant as it falls outside the integration
interval. The maximum of $g(\epsilon)$ over the interval $[\epsilon^{\dagger},\infty)$
is then $g(\epsilon^{\dagger})$, leading to
$Z_{\mathrm{typ}} \approx e^{N(\alpha-\phi(\epsilon^{\dagger}) -\beta \epsilon^{\dagger})}$.
The border between these two regimes, $\epsilon_{\mathrm{m}} = \epsilon^{\dagger}$,
defines the glass transition temperature $T_g\equiv \beta_{\mathrm{g}}^{-1}$
through the implicit relation
\begin{equation} \label{eq:epsm-ecut}
\epsilon_{\mathrm{m}}(\beta_{\mathrm{g}}) = \epsilon^{\dagger}.
\end{equation}
As $\phi(\epsilon)$ is a convex function, $\phi'(\epsilon)$ is an increasing
function of $\epsilon$. Hence from \Eqref{eq:phiprime-beta},
$\epsilon_{\mathrm{m}}$ is a decreasing function of $\beta$.
The glassy regime $\epsilon_{\mathrm{m}} < \epsilon^{\dagger}$ thus corresponds to $\beta>\beta_{\mathrm{g}}$,
or equivalently to $T<T_g$.
Altogether, the free energy per degree of freedom $f=\Esp{F}/N$ reads
for $N \to \infty$, using \Eqref{eq@ecut},
\bea
f(\beta) &=& \frac{1}{\beta}\, \phi\big(\epsilon_{\mathrm{m}}(\beta)\big) + \epsilon_{\mathrm{m}}(\beta)
- \frac{\alpha}{\beta}, \qquad \beta < \beta_{\mathrm{g}}, \\
f(\beta) &=& \epsilon^{\dagger}, \qquad \beta > \beta_{\mathrm{g}}.
\eea
Inverting the Legendre transform \Eqref{eq:legendre-mu}, one finds
\begin{equation} \label{eq:mu-beta-phi}
\mu(\beta) = -\beta \epsilon_{\mathrm{m}}(\beta) - \phi\big(\epsilon_{\mathrm{m}}(\beta)\big)
\end{equation}
so that the free energy can be rewritten for $\beta < \beta_{\mathrm{g}}$ as
$f(\beta) = -\frac{1}{\beta}\, \mu(\beta) - \frac{\alpha}{\beta}$.
Introducing the function
\begin{equation} \label{eq:def-zeta}
\zeta(\beta) = \beta \mu'(\beta) - \mu(\beta),
\end{equation}
the entropy $s=-df/dT=\beta^2 df/d\beta$ then reads\Comment{
\begin{equation} \label{eq-entropy}
s(\beta) = \begin{cases}
\alpha - \zeta(\beta)& \beta < \beta_{\mathrm{g}}, \\
0 & \beta > \beta_{\mathrm{g}}.
\end{cases}
\end{equation}}
\begin{equation} \label{eq-entropy}
s(\beta) = \left\{ \begin{array}{rl}
\alpha - \zeta(\beta),& \quad \beta < \beta_{\mathrm{g}}, \\
0, & \quad \beta > \beta_{\mathrm{g}}.
\end{array} \right.
\end{equation}
Note that $\zeta'(\beta)=\beta \mu''(\beta)$ and that $\mu(\beta)$ is a
convex function with $\mu(0)=0$, so that $\zeta(\beta)$ is an increasing function of $\beta$,
starting from $\zeta(0)=0$.
Accordingly, the entropy $s$ is a decreasing function of $\beta$
for $\beta < \beta_{\mathrm{g}}$.
We thus recover in this general framework the standard interpretation
of the glass transition in terms of a vanishing entropy per degree
of freedom, meaning
that in the low temperature phase, the probability distribution
concentrates on a few microscopic configurations \cite{Derrida}.
\Eqref{eq-entropy} provides us with an alternative characterization
of the transition temperature $\beta_{\mathrm{g}}$
\begin{equation} \label{eq-betag-alt}
\zeta(\beta_{\mathrm{g}}) = \alpha.
\end{equation}
From \Eqref{eq-betag-alt}, one sees that the glass transition exists
if the function $\zeta(\beta)$, which is defined for all $\beta>0$, reaches
the value $\alpha$ for some finite inverse temperature $\beta_{\mathrm{g}}$.
In the standard REM, which is recovered by choosing for $p(\eta)$
a centered Gaussian distribution of variance $J^2/2$,
one has $\zeta(\beta)=J^2 \beta^2/4$, so that $\zeta(\beta)$ can reach
any value $\alpha$, implying the existence of the glass transition.
More precisely, as shown in the left panel of \figref{fig:frozen}, we have
\begin{equation} \label{eq:betag-gauss}
\beta_{\mathrm{g}}(\alpha) = \sqrt{\frac{4 \alpha}{J^2} }.
\end{equation}
However, in the present more general setting of an arbitrary $p(\eta)$,
$\zeta(\beta)$ may be bounded and
the glass transition may not exist, as we shall see in the next section.
\section{Conditions for the existence or absence of glass transition}
\label{sect-no-glass}
In this section, we study the asymptotic behavior of $\zeta(\beta)$ for
$\beta \to \infty$, to see if $\zeta(\beta)$ converges to a finite limit,
or diverges.
If $\zeta(\beta)$ diverges when $\beta \to \infty$, it will necessarily
cross (assuming continuity) the value $\alpha$ for some finite $\beta_{\mathrm{g}}$.
In constrast, if $\zeta(\beta)$ converges to a finite limit $\zeta_{\infty}$,
\Eqref{eq-betag-alt} has a solution only if $\alpha<\zeta_{\infty}$. Hence
a glass transition exists only if $\alpha < \alpha_{\mathrm{c}} \equiv \zeta_{\infty}$
as illustrated on the right panel of \figref{fig:frozen}.
The question is then to know for which form of the distribution $p(\eta)$
--see \Eqref{def-mu}-- the function $\zeta(\beta)$ can have a finite limit $\alpha_{\mathrm{c}}$.
\begin{figure}
\centerline{ \includegraphics[width=5cm]{figs/gauss} \includegraphics[width=5cm]{figs/without} }
\caption{\label{fig:frozen} Sketch of the phase diagram of the REM in the $(\alpha,\beta)$ plane.
Left: case when the function $\zeta(\beta)$ diverges when $\beta$ goes to infinity. The line corresponds to the glass transition (inverse) temperature $\beta_{\mathrm{g}}(\alpha)$, illustrated here on
the standard REM with a Gaussian energy distribution --see \Eqref{eq:betag-gauss}.
Right: same diagram in the case when the function $\zeta(\beta)$ has a finite limit $\alpha_{\mathrm{c}}$. The vertical dashed line $\alpha=\alpha_{\mathrm{c}}$ corresponds to the asymptote of $\beta_{\mathrm{g}}(\alpha)$.
For $\alpha>\alpha_{\mathrm{c}}$, no glass transition occurs.}
\end{figure}
\subsection{Necessary conditions for the absence of glass transition}
In this section, we wish to derive some necessary conditions for the absence
of glass transition. We thus start by assuming that $\zeta(\beta)$ has a finite limit
$\alpha_{\mathrm{c}}$ and we explore the implications of this assumption on the distribution $p(\eta)$
of the local energies.
\subsubsection{Asymptotic behavior of $\mu'(\beta)$ and $\mu(\beta)$\\}
In order to derive the behavior of $\mu(\beta)$ from the hypothesis that $\zeta(\beta)$ admits a finite limit, it is useful to express $\mu(\beta)$ as a functional of $\zeta(\beta)$ by solving \Eqref{eq:def-zeta} as a differential equation in $\mu$, for a given function $\zeta$.
Using classical ordinary differential equation method, the following result is obtained
\begin{equation} \label{eqn-mu[zeta]}
\mu(\beta)= - \beta \left[ \int_{\beta}^{+\infty} \frac{\zeta(t)} {t^2} dt - \mu'_{\infty} \right],
\end{equation}
where $\mu'_{\infty}$ is a constant.
One should note that the integral in \Eqref{eqn-mu[zeta]} is always well-defined if $\zeta$ admits a finite limit.
The next step is to use this integral form of $\mu$ to show that
$\mu'_{\infty}$ is rightfully the limit of $\mu'(\beta)$ when $\beta \to +\infty$.
Differentiating \Eqref{eqn-mu[zeta]} yields
\begin{equation} \label{eqn-mu'[zeta]}
\mu'(\beta)= - \left[ \int_{\beta}^{+\infty} \frac{\zeta(t)} {t^2} dt - \mu'_{\infty} \right]
+ \frac{\zeta(\beta)}{\beta}.
\end{equation}
Consequently, in the limit $\beta \rightarrow + \infty$, $\mu'(\beta)$ admits a finite limit
\begin{equation} \label{eqn-mu'[zeta]-lim}
\limInf{\beta} \mu'(\beta)= \mu'_{\infty}.
\end{equation}
Note that from the convexity of $\mu(\beta)$, $\mu'(\beta)$ is an increasing
function, so that
\begin{equation} \label{eq:mu'-muinfty}
\mu'(\beta) - \mu'_{\infty} \le 0,
\end{equation}
a property which will prove useful later on.
We shall show that $\mu(\beta)$ has a linear asymptote, which is actually
not obvious from \Eqref{eqn-mu'[zeta]-lim}
\footnote{For instance, a function of the type $f(x)=ax+b\ln x$ has a finite
derivative equal to $a$ when $x \to +\infty$, but has no linear asymptote.}.
\Eqref{eqn-mu[zeta]} can be rewritten as
\begin{equation} \label{eqn-mu[zeta]:0}
\mu(\beta)- \beta \mu'_{\infty} = - \beta \int_{\beta}^{+\infty} \frac{\zeta(t)} {t^2} dt.
\end{equation}
Since $\zeta(\beta)$ admits a finite upper bound $K$, we have
\begin{equation} \label{eq-linasympt-bound}
\mu(\beta) - \beta \mu'_{\infty} \ge - \beta \int_{\beta}^{+\infty} \frac{K} {t^2} dt = -K.
\end{equation}
From \Eqref{eq:mu'-muinfty}, $\mu(\beta) - \beta \mu'_{\infty}$ is a decreasing function.
Therefore, the existence of the lower bound derived in \Eqref{eq-linasympt-bound} implies that $\mu(\beta) - \beta \mu'_{\infty}$ has a finite limit, namely
\begin{equation} \label{eq-mu(infty)}
\limInf{\beta} \mu(\beta) - \beta \mu'_{\infty} = \mu_{\infty}.
\end{equation}
In other words, $\mu(\beta)$ has a linear asymptote with slope $\mu'_{\infty}$.
Furthermore, one necessarily has $\mu_{\infty} \le 0$ since
$\mu(\beta) - \beta \mu'_{\infty}$ is a decreasing function starting from the
value $0$ at $\beta=0$.
\subsubsection{Support of $p(\eta)$\\}
We shall now focus on the case $\mu'_{\infty}=0$, as the case of an arbitrary
value $\mu'_{\infty}$ can be obtained from the case $\mu'_{\infty}=0$
by a shift of the variable $\eta$.
We shall first specify the support of $p(\eta)$.
One of the properties of the cumulant generating function $\lambda$
introduced in \Eqref{def-lambda} is that the image of $\mathbb{R}$ by $\lambda'$
is the support of the probability density function $p(\eta)$, i.e.
\begin{equation}
\{ \lambda'(\beta),\, \beta \in \mathbb{R} \} = \{ \eta,\, p(\eta) > 0,\, \eta \in \mathbb{R} \}.
\end{equation}
Given that $\mu(\beta)=\lambda(-\beta)$ [see \Eqref{def-mu}],
the lower bound of the support of $p(\eta)$ is directly related to $\mu'_{\infty}$
\begin{equation}
\eta_{\min} \equiv \inf\{\eta,\, p(\eta)> 0\} = - \mu'_{\infty}.
\end{equation}
Notably, $\mu'_{\infty}=0$ implies that only the positive values of $\eta$ have a non-zero probability.
Taking into account the fact that the support of $p(\eta)$ is $[0,\infty)$,
$\mu(\beta)$ can be rewritten as
\begin{equation} \label{eq-mu[p]-restricted}
\mu(\beta) = \ln \int_{0}^{\infty} p(\eta) e^{- \beta \eta} d\eta.
\end{equation}
\subsubsection{Characterization of $p(\eta)$ in the neighborhood of $\eta_{\min}$\\}
\Eqref{eq-mu[p]-restricted} implies that the asymptotic behavior of $\mu(\beta)$ (for $\beta \to \infty$) is directly linked to the behavior of $p(\eta)$
in the neighborhood of $\eta_{\min}=-\mu'_{\infty}$. Moreover, as stated by \Eqref{eq-mu(infty)}, for $\mu'_{\infty}=0$, $\mu(\beta)$ admits a finite limit $\mu_{\infty}$. We shall now explore the consequences on $p(\eta)$ of the existence of this finite limit $\mu_{\infty}$.
If $\mu_{\infty}=0$, the property $\mu(0)=0$ and the monotonicity of $\mu(\beta)$ imply that $\mu(\beta) = 0$.
The only possibility for $p(\eta)$ is the degenerate distribution
$p(\eta) =\delta(\eta)$.
This is obviously not a situation of interest, since all configurations
of the model would have the same energy, equal to zero.
If $\mu_{\infty}<0$, $p(\eta)$ has to satisfy
\begin{equation} \label{eq-LTp-eta}
\limInf{\beta} \int_0^{\infty} p(\eta) e^{- \beta \eta} d\eta = e^{\mu_{\infty}},
\qquad 0 < e^{\mu_{\infty}}<1 .
\end{equation}
The integral in \Eqref{eq-LTp-eta} can be rewritten as,
assuming $\beta$ has been made dimensionless,
\begin{equation} \label{eq-integ-split}
\int_0^{\infty} p(\eta) e^{- \beta \eta} d\eta =
\int_0^{1/\sqrt{\beta}} p(\eta) e^{- \beta \eta} d\eta +
\int_{1/\sqrt{\beta}}^{\infty} p(\eta) e^{- \beta \eta} d\eta.
\end{equation}
We first note that the second integral in the r.h.s.~of \Eqref{eq-integ-split}
can be bounded as
\begin{equation}
\int_{1/\sqrt{\beta}}^{\infty} p(\eta) e^{- \beta \eta} d\eta \le
e^{-\sqrt{\beta}} \int_{1/\sqrt{\beta}}^{\infty} p(\eta) d\eta \le e^{-\sqrt{\beta}},
\end{equation}
and thus goes to zero when $\beta \to \infty$ for any distribution $p(\eta)$.
We now focus on the behavior of the first integral in the r.h.s.~of \Eqref{eq-integ-split}.
If $p(\eta)$ does not contain a {Dirac mass} (i.e., a Dirac delta) at $\eta=0$, this integral goes to zero when $\beta \to \infty$, while in the presence
of a {Dirac mass} at $\eta=0$, the integral takes a finite limit
when $\beta \to \infty$.
As a result, \Eqref{eq-LTp-eta}, which is a consequence of the assumption
that $\zeta(\beta)$ has a finite limit when $\beta \to \infty$, can only
be satisfied if $p(\eta)$ contains a {Dirac mass} at $\eta=0$
(we recall that $\eta=0$ is the lowest accessible value of $\eta$).
In this latter case, a natural form for $p(\eta)$ then consists of the following mixture:
\begin{equation} \label{eq-p-eta-delta}
p(\eta) = D\, \delta(\eta) + (1-D)\, c(\eta)
\end{equation}
where $0< D<1$ and $c(\eta)$ is a probability density function with support $[0,\infty)$
and no Dirac mass at $\eta=0$. For instance, $c(\eta)$ can be
a regular distribution (which may have an integrable divergence at $\eta=0$)
or a sum of Dirac masses, meaning that $\eta$ takes discrete values.
Finally, as mentioned above, the case $\mu'_{\infty}\ne 0$ can be easily obtained
through a shift of the variable $\eta$, so that the same results
hold in full generality. The generalization of the distribution
given in \Eqref{eq-p-eta-delta} reads
\begin{equation}
p(\eta) = D\, \delta(\eta+\mu'_{\infty}) + (1-D)\, c(\eta)
\end{equation}
with the support of $c(\eta)$ limited to $[-\mu'_{\infty} ,\infty)$.
In the following subsection, we check that the distribution
\Eqref{eq-p-eta-delta}, obtained through necessary conditions,
indeed leads to a finite $\alpha_{\mathrm{c}}$, and thus to the possibility
of the absence of the glass transition.
\subsection{Distributions with a discrete mass at the minimal energy}
We now wish to show that if one starts from the distribution $p(\eta)$
given in \Eqref{eq-p-eta-delta}, the resulting function $\zeta(\beta)$
converges to a finite limit $\alpha_{\mathrm{c}}$.
This result is not obvious from the previous subsection, where we used
necessary conditions only, and did not study the behavior of the
term $\beta \mu'(\beta)$ in $\zeta(\beta)$.
Starting from \Eqref{eq-p-eta-delta}, $\mu(\beta)$ can be expressed as
\begin{equation} \label{eq-mubeta-D}
\mu(\beta) = \ln \big( D+(1-D)\, I(\beta) \big),
\end{equation}
where we have defined
\begin{equation} \label{def-Ibeta}
I(\beta) = \int_0^{\infty} c(\eta)\, e^{-\beta\eta} d\eta.
\end{equation}
The derivative $\mu'(\beta)$ is then given by
\begin{equation} \label{eq-muprime-D}
\mu'(\beta) = \frac{(1-D)\, I'(\beta)}{D+(1-D)\, I(\beta)}.
\end{equation}
As shown in \Eqref{eq-integ-split}, an integral of the form of $I(\beta)$
converges to zero when $\beta \to \infty$ for any distribution $c(\eta)$
which has no Dirac delta at $\eta=0$.
We thus have $\mu'(\beta) \sim (1-D)\, I'(\beta)/D$ for $\beta \to \infty$.
The evaluation of $\beta \mu'(\beta)$, which is needed to compute
$\zeta(\beta)$, thus boils down to that of $\beta I'(\beta)$.
The behavior of $\beta I'(\beta)$ can be evaluated by performing
an integration by part, yielding
\begin{equation}
\beta I'(\beta) = - \int_0^{\infty} \frac{d}{d\eta} \big( \eta\, c(\eta)\big)
\, e^{-\beta\eta} d\eta.
\end{equation}
Following the same arguments as for $I(\beta)$, one can then show that
$\beta I'(\beta)$ goes to zero when $\beta \to \infty$,
and so does $\beta\mu'(\beta)$.
Hence from \Eqref{eq:def-zeta}, $\zeta(\beta)$ converges to
the finite limit $\alpha_{\mathrm{c}}=-\ln D>0$ when $\beta \to \infty$.
In conclusion, for a distribution $p(\eta)$ of the form
\Eqref{eq-p-eta-delta}, we have shown that the glass transition disappears for
$\alpha > \alpha_{\mathrm{c}}$, with
\begin{equation}
\alpha_{\mathrm{c}}=-\ln D,
\end{equation}
as sketched in \figref{fig:nega}.
\begin{figure}
\centerline{ \includegraphics[width=6cm]{figs/nega} }
\caption{\label{fig:nega} Phase diagram in the $(\alpha,D)$ plane, for the distribution
$p(\eta)$ defined in \Eqref{eq-p-eta-delta}. The line $\alpha_{\mathrm{c}}=-\ln D$ separates regions
where the glass transition exists ($\alpha<\alpha_{\mathrm{c}}$) from regions where
no glass transition occurs.}
\end{figure}
\subsection{Behavior of the glass transition close to the onset threshold} \label{sect-onset-glass}
We explore some interesting consequences of the above results.
For $\alpha<\alpha_{\mathrm{c}}$, the glass transition exists, but the temperature range
of the glassy phase is expected to shrink when $\alpha \to \alpha_{\mathrm{c}}$.
Let us make the argument quantitative.
The glass transition temperature is determined from the relation
$\zeta(\beta_{\mathrm{g}}) = \alpha$ [see \Eqref{eq-betag-alt}].
For $\alpha$ close to (and smaller than)
$\alpha_{\mathrm{c}}$, $\beta_{\mathrm{g}}$ is thus determined by the asymptotic
behavior of $\zeta(\beta)$ for large $\beta$.
To make concrete calculations, we assume that $c(\eta) \sim c_0 \eta^{\nu-1}$
when $\eta \to 0$ ($\nu >0$). Using the change of variables $u=\beta\eta$
in the integral defining $I(\beta)$, see \Eqref{def-Ibeta},
one finds for $\beta \to \infty$ that
\begin{equation} \label{eq-asympt-Ibeta}
I(\beta) \sim \frac{1}{\beta} \int_0^{\infty} c_0 \left(\frac{u}{\beta}\right)^{\nu-1} e^{-u} du = \frac{\Gamma(\nu) c_0}{\beta^{\nu}}.
\end{equation}
$I'(\beta)$ can be computed in the same way, simply replacing $c(\eta)$
by $-\eta c(\eta)$, which amounts to replacing $\nu$ by $\nu+1$
and $c_0$ by $-c_0$.
One then finds $I'(\beta) \sim -\Gamma(\nu+1) c_0/\beta^{\nu+1}$.
Taking into account Eqs.~(\ref{eq-mubeta-D}), (\ref{eq-muprime-D}) and (\ref{eq-asympt-Ibeta}), we obtain
\begin{equation}
-\ln D - \zeta(\beta) \sim \frac{(1-D)(\nu+1) \Gamma(\nu) c_0}{D\, \beta^{\nu}}.
\end{equation}
Using $\zeta(\beta_{\mathrm{g}}) = \alpha$, we get
\begin{equation} \label{eq-betag}
-\ln D - \alpha \sim \frac{(1-D)(\nu+1) \Gamma(\nu) c_0}{D\, \beta_{\mathrm{g}}^{\nu}}.
\end{equation}
At this stage, two different viewpoints can be adopted, namely either
considering $\alpha$ as the control parameter for a fixed $D$, or
considering $D$ as the control parameter for a fixed $\alpha$.
We first fix $D$, and use $\alpha$ as control parameter.
In this case, the glass transition occurs for $\alpha = \alpha_{\mathrm{c}} \equiv -\ln D$. From \Eqref{eq-betag}, we obtain for the glass transition temperature
$T_{\mathrm{g}} \equiv \beta_{\mathrm{g}}^{-1}$
\begin{equation}
T_g \sim A\, (\alpha_{\mathrm{c}} - \alpha)^{1/\nu}, \qquad \alpha \to \alpha_{\mathrm{c}}^-,
\end{equation}
with $A=[(1-D)(\nu+1) \Gamma(\nu) c_0/D]^{-1/\nu}$.
Alternatively, using $D$ as control parameter for a fixed $\alpha$,
the transition exists only for $D<D_c\equiv e^{-\alpha}$.
\Eqref{eq-betag} then leads to
\begin{equation}
T_g \sim \tilde{A}\, (D_c - D)^{1/\nu}, \qquad D \to D_c^-,
\end{equation}
with $\tilde{A}=[(1-e^{-\alpha})(\nu+1) \Gamma(\nu) c_0]^{-1/\nu}$.
\section{Discussion}
\label{sect-discus}
In summary, we have shown that by tayloring the distribution $p(\eta)$
of local energies (and thus modifying the global energy distribution
$\mathcal{P}(E)$), the glass transition can be avoided in some parameter
regimes, so that
the presence of uncorrelated random energy levels is not a
sufficient condition for the emergence of a glass transition.
Reversing the perspective, one could also interpret the onset
of a glass transition when varying either $\alpha$ or the weight $D$
of the Dirac mass, as a kind of critical phenomenon, the order
parameter of this transition being the glass transition itself.
We have seen in Sect.~\ref{sect-onset-glass} that the
glass transition temperature indeed behaves as a power law
close to threshold. However, the corresponding exponent
is non-universal, as it depends on the behavior of the distribution
$p(\eta)$ close to the lower bound of its support.
These results further suggest an interesting analogy with another type
of critical phenomenon. We have seen that if we build the
random energies $E_k$ of the model by adding up random positive terms
$\eta_{k,i}$, such that each term is either equal to zero with probability
$D$, or drawn from a continuous distribution $c(\eta)$ with probability
$1-D$, the glass transition disappears if the fraction of zero terms
exceeds some threshold $D_c$.
This situation is reminiscent of the dilute Ising model, where
the coupling constants between neighboring sites are randomly set to zero
with a given probability. Above a critical fraction of zero couplings,
the transition disappears \cite{Aizenman,Parisi,Vicari}.
The mechanism at play in the dilute Ising model is however different,
as it is related to the percolation of the bonds with nonzero couplings,
and thus has a geometric interpretation.
Besides, it is interesting to study the properties of the large deviation
function $\phi(\epsilon)$, which is related to the microcanonical
entropy $s_\mathrm{m}(\epsilon)$ through
$s_\mathrm{m}(\epsilon)=\alpha-\phi(\epsilon)$ [see \Eqref{eq:ztyp}].
Using \Eqref{eq:mu-beta-phi} and the properties of the Legendre transform,
one can show that $\phi(\epsilon_{\mathrm{m}}(\beta))=\zeta(\beta)$.
Denoting as $\eta_{\min}$ the lower bound of the support of the distribution
$p(\eta)$, one has $\epsilon_{\mathrm{m}}(\beta) \to \eta_{\min}$ when
$\beta \to \infty$ (note that $\eta_{\min}$ may be equal to $-\infty$).
Hence if $\zeta(\beta) \to \infty$ for $\beta \to \infty$,
$\phi(\epsilon)$ also diverges for $\epsilon \to \eta_{\min}$.
In this case, the equation $\phi(\epsilon^{\dagger})=\alpha$ always has a solution
$\epsilon^{\dagger} > \eta_{\min}$, assuming that $\phi(\epsilon)$ is continuous.
Furthermore, as $\phi(\epsilon)$ is regular at $\epsilon=\epsilon^{\dagger}$,
its derivative is finite, and so does the glass transition temperature
$\beta_{\mathrm{g}}=\phi(\epsilon^{\dagger})$ --see Eqs.~(\ref{eq:phiprime-beta}) and
(\ref{eq:epsm-ecut}).
In constrast, if $\zeta(\beta)$ converges to a finite limit,
$\phi(\epsilon)$ also has a finite limit, equal to $\alpha_{\mathrm{c}}$,
for $\epsilon \to \eta_{\min}$.
In this case, as $\phi'(\epsilon_{\mathrm{m}})=-\beta$ (see \Eqref{eq:phiprime-beta}),
the large deviation function necessarily has an infinite negative slope
for $\epsilon \to \eta_{\min}$.
Hence the absence of glass transition in this case does not have the same
origin as in non-disordered systems. For such systems, there is no cut-off
$\epsilon^{\dagger}$, and energies down to $\eta_{\min}$ can be explored.
In the presence of disorder, the glass transition can be avoided
only by tayloring $\phi(\epsilon)$ so that the equation determining the cut-off
$\epsilon^{\dagger}$ has no solution $\epsilon^{\dagger}>\eta_{\min}$.
Let us finally mention that the glass transition in the REM also has
applications in other fields, like the empirical estimation of moments
in statistical signal processing.
It has been recently emphasized that the so-called ``linearization effect''
in multifractal analysis, occuring when empirically determined moments
$S_M(q)=M^{-1}\sum_{i=1}^M x_i^q$ significantly
depart from the theoretical ones $\langle x^q \rangle$,
can be interpreted as an analog of the glass transition in the REM
\cite{Angeletti11,MuzyBacryKozhemiak2006, MuzyBacryKozhemiak2008, BacryGlotterHoffmannMuzy2010}, the inverse temperature $\beta$ being mapped
onto the moment order $q$.
Similar effects also occur even when considering uncorrelated signals,
as long as the marginal distribution of the signal is sufficiently broad,
but with finite moments (for instance a lognormal distribution)
\cite{Angeletti12}.
Along this line of thought, the present version of the REM can be
seen as the analog of the moment estimator of variables built
as products of a large number of independent random variables
(which is a possible way to build broadly distributed variables
with finite moments).
The present study then shows that in most cases, this moment estimator
will also present a linearization effect, and depart from the theoretical
moments for $q$ above some threshold $q^*$.
However, if the underlying distribution of the variables presents a Dirac
mass at its upper bound, this Dirac mass generates a linear branch in the logarithm of the theoretical moments which can mask the empirical linearization effect.
\hspace*{5mm}
|
1,314,259,992,976 | arxiv | \section{Introduction}
Inferring probability density functions (PDFs) from data is a fundamental problem in machine learning, statistics and signal processing\cite{Murphy2012}\cite{Wainwright2019} with applications in varied fields such as conditional inference, samples generation, image reconstruction and many more. The applicability of widely popular methods like Gaussian mixture models (GMMs) are restricted to the case of smooth multi-modal densities where every mode is well approximated by Gaussians. Similar is the case with non-parametric settings such as kernel density estimation (KDE). Moreover, such techniques exhibit lower and lower convergence rates as the data dimensionality increases. For $N$-dimensional data, the convergence rate, in terms of the integrated mean square error (MISE), for the KDE is known to be $\mathcal{O}(N^{-\frac{4}{N+4}}_s)$\cite{Chen2018}.\\%; thus as dimension $N$ increases the number of samples $N_s$ required for trustworthy density reconstruction also increases drastically.
Recently, joint probability mass functions (PMFs) of discrete or discretized random variables (RVs) have been represented as tensors -- in fact \emph{low rank} tensors, using the fact that the different RVs are neither completely dependent nor completely independent \cite{Kargas2018}. There have been significant developments in the estimation of joint PMFs from lower dimesional marginals (eg, 3D marginals) using the Canonical Polyadic Tensor Decompostion (CPD) using these low rank constraints \cite{Kargas2018}. Along similar lines, the work in \cite{Ibrahim2021} uses non-negative matrix factorization (NMF) techniques to estimate the PMF from just pairwise (i.e. 2D) marginals. On the other hand, ideas from tomography were incorporated into this tensor-based framework in \cite{Vora2021} for PMF reconstruction from just 1D marginals. Extending these ideas to the continuous domain, one can discretize the continuous RVs, use the aforementioned techniques for estimating cumulative interval measures (CIMs), followed by an appropriate interpolation technique to recover the joint PDF. For instance, \cite{Kargas2019} uses sinc interpolation assuming that the underlying PDF is band-limited, in keeping with the popular Shannon-Nyquist theorem. Under similar assumptions, there also exists work in the Fourier domain where CIM reconstruction techniques are applied to obtain the `characteristic tensor' and the continuous PDF is then retrieved using the inverse Fourier Transform \cite{Amiridi2020}.\\
In this work, we present a novel approach which combines ideas from the CPD model for tensors and dictionary representations, to reconstruct the joint PDF from just pairwise (2D) marginals, as opposed to 3D marginals. The key idea is to prepare a dictionary of 1D PDFs belonging to various families, with parameters restricted to lie in a carefully chosen range, for each component of the $N$-dimensional data. Reconstruction of the $N$-dimensional PDF using such a dictionary helps us to circumvent restrictive assumptions such as PDF smoothness or band-limitedness as in previous methods. Furthermore, the convergence rate for estimation of 2D marginals is superior to that for 3D marginals used in \cite{Kargas2018}.
\section{Background}
\subsection{Canonical Polyadic Decomposition(CPD) of Tensors}
Any $N$-dimesional tensor $\boldsymbol{\underline{T}}\in \mathbb{R}^{I_1\times I_2\times...\times I_N}$ admits a decomposition in the form of the sum of $F$ rank-1 tensors. This is known as the CPD, and is given by:
\begin{equation}\label{CPD}
\boldsymbol{\underline{T}}=\sum_{r=1}^F \boldsymbol{\lambda}[r] \boldsymbol{A_1}[:,r]\circ \boldsymbol{A_2}[:,r]\circ....\boldsymbol{A_N}[:,r],
\end{equation}
where $F$ is the smallest number for which such decomposition is possible, where for each $n \in [N]$, where $[N] \triangleq [1,2,...,N]$ the matrix $\boldsymbol{A_n} \in \mathbb{R}^{I_n\times F}$ is called a mode factor matrix, $\circ$ denotes the outer product of vectors and $\lambda[r]$ denotes the $r^{\textrm{th}}$ mixing weight. For tensors that represent high-dimensional PMFs, the above decomposition is applicable with the following additional constraint: (1) $\forall n \in [N], r \in [F], \|\boldsymbol{A_n}[;,r]\|_1=1$ with non-negative entries in the mode factor matrices, and (2) $\|\boldsymbol{\lambda}\|_1 = 1$ where $\boldsymbol{\lambda}$ is a vector of $F$ non-negative mixing weights. Recovering the PMF is equivalent to estimating these mode factors and $\boldsymbol\lambda$\cite{Kargas2018}. \\
The above model can be viewed as a naive Bayes model with the latent variable $H$ that takes on $F$ different values such $P(H=r)=\lambda[r]$ for all $r \in [F]$, and the outer product of the mode factor can be viewed as a conditional probability given $H$ \cite{Kargas2018}. With this, the CPD model for the PDF of RV $\boldsymbol{X} = (X_1, X_2, ..., X_N) \in \mathbb{R}^N$ can be formulated as:
\begin{equation}\label{Bayesian_Model}
\boldsymbol{\underline{T}}=\sum_{r=1}^F P(H=r)\prod_{n=1}^{n=N}P(X_n=i_n|H=r).
\end{equation}
\subsection{Continuous RVs: PDF estimation}
For the case of continuous RVs, consider an $N$-dimensional RV $\boldsymbol{X}=\{X_n\}_{n=1}^N$, whose PDF is given by the following mixture of multivariate distributions:
\begin{equation}\label{Cont_case}
f_{\boldsymbol{X}}(x_1,...x_N)=\sum_{r=1}^F \boldsymbol{\lambda}[r]f_{\boldsymbol{X}|H}(x_1,...,x_N|H=r).
\end{equation}
If the RVs are independent given $H$, then each conditional density can be represented by the product of 1D densities and the above equation becomes:
\begin{equation}\label{Cont_Bayesian_Model}
f_{\boldsymbol{X}}(x_1,...,x_N)=\sum_{r=1}^F \boldsymbol{\lambda}[r]\prod_{n=1}^N f_{X_n|H}(x_n|H=r),
\end{equation}
which can be viewed as the continuous analog of the CPD model for PMFs\cite{Kargas2019}. In this case, the problem of estimating the PDF is equivalent to estimating these `continuous' mode factors and their mixing weights.
\subsection{Joint PDF estimation from 3D marginals}
The work in \cite{Kargas2019} exploits the above CPD of densities which are conditionally independent to estimate the joint PDF from the data. They propose to discretize each component $X_n$ of the $N$-dimensional variable into $I_n$ intervals $\{\Delta_n^i \triangleq (d_n^{i-1}, d_n^i)\}_{1\leq i\leq I_n}$ and form the CIM tensor $\boldsymbol{\underline{Z}}$ given by:
\begin{align}\label{PMF}
\boldsymbol{\underline{Z}}(i_1,...,i_N)&=P(X_1\in \Delta_1^{i_1},..., X_N\in \Delta_N^{i_N})\\
&=\sum_{r=1}^F \boldsymbol{\lambda}[r]\prod_{n=1}^N P(X_n\in \Delta_n^i|H=r),
\end{align}
where $P(X_n\in \Delta_n^i|H=r)$ are the CIMs of the corresponding 1D components of the RV. If the PDF/CDF of each component in the product is band-limited, the CIMs can be estimated using the PMF reconstruction techniques described in \cite{Kargas2018}. The reconstruction involves estimating the 3D marginals, $\boldsymbol{\underline{Z}}_{i,j,k}=P(X_i,X_j,X_k)$, using standard histogramming and then minimizing the following cost function:
\begin{gather}
\min_{\boldsymbol{\{A_n\}}_{n=1}^N, \boldsymbol\lambda}\sum_i\sum_{j>i}\sum_{k>j}\|\boldsymbol{\underline{Z}}_{i,j,k}-[\boldsymbol{\lambda}, \boldsymbol{A_i},\boldsymbol{A_j},\boldsymbol{A_k}]\|_F^2 \nonumber\\
\label{eq:Zijkcostfunction}
\textrm{ s.t. } \ \forall n,r \ \|\boldsymbol{A_n}[:,r]\|_1=\|\boldsymbol{\lambda}\|_1=1, \boldsymbol{A_n} \succeq \boldsymbol{0}, \boldsymbol{\lambda} \succeq \boldsymbol{0},
\end{gather}
where $\succeq$ represents the element-wise inequality and $[\boldsymbol{\lambda}, \boldsymbol{A_i},\boldsymbol{A_j},\boldsymbol{A_k}] \triangleq \sum_{r=1}^F \lambda[r]\boldsymbol{A_i}[:,r]\circ \boldsymbol{A_j}[:,r]\circ\boldsymbol{A_k}[:,r]$. After obtaining the discretized samples of the PDFs, the technique in \cite{Kargas2019} invokes the Shannon-Nyquist sampling theorem to use sinc interpolation to produce the original joint PDF using Eqn.~\ref{CPD}. \\
In \cite{Amiridi2020}, under similar band-limitedness of the density, a different kind of approach is considered using the characteristic function of the density, $\boldsymbol{\Phi_{X}}(\boldsymbol\nu)=E[e^{j\boldsymbol{\nu}^T \boldsymbol{X}}],$
where $j \triangleq \sqrt{-1}$. If the true density is given by $f_{\boldsymbol X}(\boldsymbol x)$, then it can be well approximated by truncating the Fourier series below:
\begin{gather}\label{FT}
\hat{f}_{\boldsymbol{X}}(\boldsymbol {x})=\sum_{k_1=-K_1}^{k_1=K_1}...\sum_{k_N=-K_N}^{k_N=K_N}\boldsymbol{\Phi_{X}}(\boldsymbol{k})e^{-2\pi j\boldsymbol{k}^T \boldsymbol {x}}.
\end{gather}
If we furthur impose the CPD model on it, the expression for $\boldsymbol{\underline{\Phi}_{X}}(\boldsymbol{k})$ becomes becomes similar to Eq. \ref{Bayesian_Model}.
\iffalse
\begin{gather}\label{CPD_Fourier}
\boldsymbol{\underline{\Phi}_{X}}(\boldsymbol{k})=\sum_{r=1}^{F} p_H(r)\prod_{n=1}^N\Phi_{X_n|H=r}(k_n),
\end{gather}
\fi
where $F$ is the rank of the tensor. With enough samples $\{\boldsymbol{x}_m\}$, the expectation can be reliably estimated using sample mean $ \boldsymbol{\hat{\underline{\Phi}}_{X}}(\boldsymbol\nu)=\frac{1}{M}\sum_{m=1}^Me^{j\boldsymbol{\nu}^T \boldsymbol{x_m}}$
Minimizing a cost function similar to Eqn.\ref{eq:Zijkcostfunction} for the characteristic tensor $\boldsymbol{\hat{\underline{\Phi}}_{X}}(\boldsymbol\nu)$ followed by inverse Fourier transform, yields $f_{\boldsymbol X}(\boldsymbol x)$.
\subsection{Joint PMF estimation from 2D marginals}
An interesting approach that employs Non-negative Matrix Factorization (NMF) techniques\cite{NMF} to estimate the mode factors from 2D marginals was introduced in \cite{Ibrahim2021}. They estimate $\boldsymbol{Z}_{j,k}$ via sample histogramming and obtain the mode factors using the relation $\boldsymbol{Z}_{j,k}=\boldsymbol{A_j \Lambda A_k}^T$, where $\boldsymbol{\Lambda}$ is a diagonal matrix with diagonal elements obtained from $\boldsymbol{\lambda}$. If the tensor rank $F \gg \textrm{min}(I_j,I_k)$, then NMF techniques cannot be applied \cite{Fu2018}. Therefore, the authors proposed to split the indices of $N$ variables into two sets and construct a matrix $\boldsymbol{\tilde{Z}}$ by row and column concatenation using the indices in the two sets (see \cite[Eqn. 3]{Ibrahim2021}. Then $\boldsymbol{\tilde{Z}}$ is decomposed as $\boldsymbol{\tilde{Z}}=\boldsymbol{WH}^T$ using the successive projection algorithm (SPA) \cite{Gillis2014}. The mode factors are then extracted using the relations $\boldsymbol{W}=[\boldsymbol{A}_{l_1}, \boldsymbol{A}_{l_2},..., \boldsymbol{A}_{l_M}]^T$ and $\boldsymbol{H}^T=\boldsymbol\Lambda[\boldsymbol{A}_{l_{M+1}}, \boldsymbol{A}_{l_{M+2}},..., \boldsymbol{A}_{l_N}]$, where $\{l_1, l_2,...,l_M\}$ and $\{l_{M+1}, l_{M+2}..., l_N\}$ are the two sets of indices.
\section{Problem Statement and Algorithm}
Let $\boldsymbol{X} \triangleq (X_1, X_2,...,X_N)$ be an $N$-d RV, each component of which can be either continuous or discrete. Our aim is to estimate $f_{\boldsymbol{X}}(\boldsymbol{x})$ which follows the CPD model with rank $F$ for the continuous case, given sample values of the RV. Let us further assume that each column of these ``continuous mode factors" $f_{X_n|H}(x_n|H=r)$ are convex combinations of various densities from a given dictionary. Mathematically, we have $f_{X_n|H}(x_n|H=r)=\boldsymbol{\mathcal{A}_n}[:,r] = \boldsymbol{\mathcal{D}_n}\boldsymbol{B_n}[:,r]$, $1\leq r\leq F$, where $\boldsymbol{\mathcal{D}_n}$ is a dictionary of continuous densities (or discrete PMFs in some cases) and $\boldsymbol{B_n}[:,r]\in \mathbb{R}_+^{L_n}\cup\{\boldsymbol{0}\}$ is the non-negative weight vector which sums to 1.
Here $L_n$ is the number of different densities (number of columns) that are present in the dictionary $\boldsymbol{\mathcal{D}_n}$.
We will later see that keeping the dictionary $\boldsymbol{\mathcal{D}_n}$ separate for each component of the RV gives us a lot of flexibility in dealing with RVs defined on disparate domains. Then, the 2D PDFs $\boldsymbol{\mathcal{Z}_{j,k}}$ can be expressed in the form $\boldsymbol{\mathcal{Z}_{j,k}}=\boldsymbol{\mathcal{D}_j}\boldsymbol{B_j}\boldsymbol{\Lambda}\boldsymbol{B_k}^T\boldsymbol{\mathcal{D}_k}^T$ which is straightforward to derive by marginalizing along $X_j$ and $X_k$, and replacing each density in the product by its dictionary representation. We discretize each component of the RV into $I_n$ intervals $\{\Delta_n^i \triangleq (d_n^{i-1}, d_n^i)\}_{1\leq i\leq I_n}$ and form the 2D PMF matrix $\boldsymbol{\bar{Z}_{j,k}}$ given by:
\begin{align}\label{2D_PMF}
\boldsymbol{\bar{Z}_{j,k}}(i_j, i_k)&=P(X_j\in \Delta_j^{i_j}, X_k\in \Delta_k^{i_k})\\ \nonumber
&=\boldsymbol{\bar{D}_j}[i_j,:]\boldsymbol{B_j}\Lambda\boldsymbol{B_k}^T\bar{\boldsymbol{D}}_k[i_k,:]^T,
\end{align}
where the column-wise discretized form of a ``continuous tensor/matrix" $\boldsymbol{\mathcal{Z}}$ is represented as $\boldsymbol{\bar{Z}}$. We can obtain an estimate $\boldsymbol{\hat{Z}_{j,k}}$ of these 2D PMFs by standard histogramming of the samples. It is emphasised here that we do not assume any knowledge of the densities from which the data are generated. For the purpose of estimation, all the dictionaries $\{\boldsymbol{\mathcal{D}}_n\}_{n=1}^N$ are designed by inspecting the samples. We will elaborate on this aspect in more detail in Sec.\ref{sec:exp}. Thus, our task is to estimate the coefficients $\boldsymbol{\Lambda}, \boldsymbol{B}_1,...,\boldsymbol{B_n}$. To this end, we minimize the following cost function:
\begin{gather}\label{eq:jupad_cost_fn}
J(\{\boldsymbol{B_n}\}_{n=1}^N, \boldsymbol{\Lambda})=\sum_{j, j<k}\|\boldsymbol{\hat{Z}_{j,k}}-\boldsymbol{\bar{D}_j}\boldsymbol{B_j}\boldsymbol{\Lambda}\boldsymbol{B_k}^T\boldsymbol{\bar{D}_k}^T\|_F^2\\ \nonumber
\textrm { s.t. } \forall j,r, \|\boldsymbol{B_j}[:,r]\|_1 = 1, \boldsymbol{B_j} \succeq \boldsymbol{0},
\|\textrm{diag}(\boldsymbol{\Lambda})\|_1 = 1, \boldsymbol{\Lambda} \succeq \boldsymbol{0}.
\end{gather}
However, minimization via (say) a simple gradient descent may not be feasible as the solution will not be identifiable for cases when $F>\textrm{min}(L_j,L_k)$, as argued in \cite{Ibrahim2021}. We propose to do the following: (i) We minimize $\|\hat{\boldsymbol{Z}}_{j,k}-\boldsymbol{\bar{D}_j}\boldsymbol{T}_{j,k}\boldsymbol{\bar{D}_k}^T\|_F^2$ for each pair of $(j,k)$ via mirror descent to obtain $\boldsymbol{T}_{j,k} \triangleq \boldsymbol{B}_j \boldsymbol{\Lambda} \boldsymbol{B}_k^T$. For mirror descent, closed-form updates can be derived even with the simplex constraint on $\boldsymbol{T}_{j,k}$ \cite{Kivinen1997}\cite{Beck2003}. (ii) Next, we construct the matrix $\Tilde{\boldsymbol{T}}$ from $\boldsymbol{T}_{j,k}$ using the concatenation approach described in \cite[Eqn. 3]{Ibrahim2021} and determine the matrices $\boldsymbol{W}$ and $\boldsymbol{H}^T$ as outputs from function SPA(.) implemented in \cite{Ibrahim2021}. The weight matrices, $\boldsymbol{B_n}$ and $\boldsymbol{\Lambda}$ can now be identified as submatrices of $\boldsymbol{W}$ and $\boldsymbol{H}^T$. (iii) We further refine our estimates for these weights by using mirror descent on cost function in Eqn.~\ref{eq:jupad_cost_fn}. We name our algorithm \textbf{JUPAD}: \textbf{J}oint density estimation \textbf{U}sing \textbf{P}airwise marginals \textbf{A}nd \textbf{D}ictionaries.\\
The pseudo-code for the algorithm is presented in Alg.\ref{alg:cap}, where $\eta_T, \eta_B$ and $\eta_L$ are the learning rate hyper-parameters chosen via cross-validation, $\otimes$ represents the Hadamard product of two matrices, and $\textrm{vec}(.)$ reshapes a matrix into vector form.
\begin{algorithm}
\caption{Joint probability density estimation using pairwise marginals}\label{alg:cap}
\begin{algorithmic}[1]
\State \textbf{Procedure: JUPAD}
\State Obtain the estimate $\hat{\boldsymbol{Z}}_{j,k}$ for the 2D marginals $\boldsymbol{Z}_{j,k}$ via histogramming
\For{each pair $(j,k), \ j<k$}
\State Randomly initialize $\boldsymbol{T}_{j,k}$
\While {converged==false}
\State $\boldsymbol{T}_{j,k} \gets \boldsymbol{T}_{j,k}\otimes exp(-\eta_T\frac{\partial(\|\hat{\boldsymbol{Z}}_{j,k}-\bar{\boldsymbol{D}}_j\boldsymbol{T}_{j,k}\bar{\boldsymbol{D}}_k^T\|_F^2)}{\partial \boldsymbol{T}_{j,k}})$
\State $\boldsymbol{T}_{j,k} \gets \frac{\boldsymbol{T}_{j,k}}{\|\textrm{vec}(\boldsymbol{T}_{j,k})\|_1}$
\EndWhile
\EndFor
\State Assemble $\Tilde{\boldsymbol{T}}$ from $\{\boldsymbol{T}_{j,k}\}$ following \cite[Eqn. 3]{Kargas2019}.
\State $\{\boldsymbol{B_n}\}_{n=1}^N$, $\boldsymbol\Lambda \gets SPA(\Tilde{\boldsymbol{T}})$
\While{converged==false}
\For {i=1 to N}
\While{converged==false}
\State $\boldsymbol{B}_n \gets \boldsymbol{B}_n\otimes exp(-\eta_B\frac{\partial J}{\partial \boldsymbol{B}_n})$
\State $L_1$ normalize each column of $\boldsymbol{B}_n$
\EndWhile
\EndFor
\While{converged==false}
\State $\boldsymbol\Lambda \gets \boldsymbol\Lambda\otimes exp(-\eta_L\frac{\partial J}{\partial \boldsymbol\Lambda})$
\State $L_1$ normalize the diagonal of $\boldsymbol\Lambda$
\EndWhile
\EndWhile
\end{algorithmic}
\end{algorithm}
\section{Numerical Results}
\begin{figure*}[t]
\begin{subfigure}[t]{.5\linewidth}
\centering
\includegraphics[width=60mm,height=35mm]{plots/sim2.png}
\caption{Mixture of Laplacians}
\label{fig:exp1}
\end{subfigure}
\begin{subfigure}[t]{.5\linewidth}
\centering
\includegraphics[width=60mm,height=35mm]{plots/sim1.png}
\caption{Mixture of Gaussians}
\label{fig:exp2}
\end{subfigure}
\begin{subfigure}[t]{.5\linewidth}
\centering
\includegraphics[width=60mm,height=35mm]{plots/sim3.png}
\caption{Mixture of Gaussians and Laplacians}
\label{fig:exp3}
\end{subfigure}
\begin{subfigure}[t]{.5\linewidth}
\centering
\includegraphics[width=60mm,height=35mm]{plots/sim4.png}
\caption{RV with continuous and discrete components}
\label{fig:exp4}
\end{subfigure}
\caption{$D(\widehat{f}_{\boldsymbol{X}}, f_{\boldsymbol{X}})$ vs Number of samples ($N_s$) for JUPAD (our approach), CPD-3 \cite{Kargas2019} and GMM.}
\end{figure*}
\label{sec:exp}
\subsection{Synthetic Data}
To test our algorithm, we created density functions as a convex combinations of densities from a chosen dictionary. The aim then was to reconstruct the density function from samples of the underlying random variable. We drew sample data from the synthetic density for various sample sizes ($N_s$) and tested the accuracy of the algorithms by averaging the absolute of log likelihood ratio between the estimated ($\widehat{f}_{\boldsymbol{X}}(.)$) and the known true PDF ($f_{\boldsymbol{X}}(.)$). We generated $M\triangleq 1000$ test samples $\{\boldsymbol{z}_k\}_{k=1}^M$ and used the following measure:
\begin{gather}\label{Likelihood}
D(\widehat{f}_{\boldsymbol{X}}, f_{\boldsymbol{X}})\approx \frac{1}{M}\sum_{k=1}^{M}|\log(\widehat{f}_{\boldsymbol{X}}(\boldsymbol{z_k})/f_{\boldsymbol{X}}(\boldsymbol{z_k}))|.
\end{gather}
Notice that the marginalized distribution for the $n^{\textrm{th}}$ component of the RV will be $f_{X_n}(x_n)=\boldsymbol{\mathcal{D}_n}\boldsymbol{B_n}\boldsymbol{\lambda}$. This implies that by looking at the structure of the 1D empirical histogram (obtained from sample values) for each component, we can guess the families of distributions it might belong to. For example, the histogram of an exponential RV will peak at zero and decrease exponentially, that of a mixture of Laplacians will have distinct peaks at the mean of each component with a heavy tail, etc.
For the parameters of the density families, we consider the range of values for samples of that component, say $[a,b]$. Consider that we wish to include Gaussians and Laplacians in our dictionaries. Then we can divide $[a,b]$ into regular intervals, and use the interval boundaries as the mean values for those distributions, while the variance/shape factors are chosen such that the densities with the mean values considered as described here, are sufficiently separated.
In other words, we want the densities in our dictionary to completely cover the range of the samples for that component. If the $n^{\textrm{th}}$ component of the data is discrete, say label, categorical or integer data with $C_n$ states, we simply assign an identity matrix of size $C_n \times C_n$ to $\boldsymbol{\bar{D}_n}$. In this case the PMF of the marginal is given by: $f_{X_n}(x_n=i)=\sum_{r=1}^F \boldsymbol{\lambda}[r]\boldsymbol{B_n}[i,r]$.
For all the experiments we compared our method with the PDF estimation algorithm from \cite{Kargas2019} and the well-known expectation maximization (EM) algorithm for GMM fitting \cite{Dempster1977}. For EM-GMM, GMM-$n$ represents the performance plot for a GMM with $n$ clusters. In all experiments, we started with 5 clusters and went up to the point where increasing the number of clusters resulted in ill-conditioned co-variance. We refer to the algorithm in \cite{Kargas2019} as ``CPD-3". The technique described in \cite{Amiridi2020} was not included in the comparison results, as for many toy experiments it produced results similar to that of CPD-3 but was computationally very expensive for larger number of samples. We generated densities belonging to various families, as described below:
\noindent \textit{Mixture of Laplacians}: In the first experiment, we chose the dimension of the RV to be $N=5$ and $F=10$. Each column in the mode factor is a mixture of 5 Laplacian densities. Thus, if $\mathcal{L(\mu,\alpha)}$ denotes a Laplacian density with mean $\mu$ and shape factor $\alpha$, then each $f_{X_n|H}(x_n|H=r)=\sum_{i=1}^5w_{i,nr}\mathcal{L}(\mu_{i,nr},\alpha_{i,nr})$ where $\mu_{i,nr}\sim\mathcal{U}(-5,5)$ and $\alpha_{i,nr}\sim\mathcal{U}(1,2)$. Here and for all the following cases as well, the mixing weights $w_{i,nr}$ and $\lambda[r]\sim\mathcal{U}(0,1)$ and then $L_1$ normalized. Fig.~\ref{fig:exp1} shows $D(\widehat{f}_{\boldsymbol{X}}, f_{\boldsymbol{X}})$ vs number of samples $N_s$. Clearly, our method works better, be it in low or high sample regime, as Laplacian densities are neither smooth nor band-limited.
\noindent\textit{Mixture of Gaussians}: In the second experiment, we chose $N=6$ and $F=8$. Each column in the mode factor was chosen to be a mixture of 5 Gaussian densities. Thus, $f_{X_n|H}(x_n|H=r)=\sum_{i=1}^5w_{i,nr}\mathcal{N}(\mu_{i,nr},\sigma_{i,nr}^2)$ where $\mu_{i,nr}\sim\mathcal{U}(-5,5)$ and $\sigma_{i,nf}^2\sim\mathcal{U}(1,2)$. Fig.~\ref{fig:exp2} shows us that in lower sample regime, our algorithm is significantly accurate whereas in high sample regime it is not far from the best method. The reason for this performance anomaly is due to the fact that there is a limit to the accuracy with which the dictionary elements can represent a density due to parameter discretization while the algorithm CPD-3 performs sinc-interpolation which does not have any such restriction.
\noindent\textit{Mixture of Gaussians and Laplacians}: In this experiment, we chose $N=7$ and $F=10$. Each column in the first five mode factors is a mixture of 5 Laplacians while mixture of 5 Gaussian densities were used for next 5 mode factors. Again $\mu_{i,nr}\sim\mathcal{U}(-5,5)$ for both Gaussians and Laplacians, $\alpha_{i,nr}, \sigma_{i,nr}^2\sim\mathcal{U}(1,2)$. Just like the first case, our method performs significantly better than other algorithms across all sample regimes, as seen in Fig.~\ref{fig:exp3}.
\noindent\textit{Mixture of Continuous and Discrete RVs}: For the last experiment, we chose $N=4$ and $F=8$. This time the last component in the RV was discrete with 10 states. As explained earlier, the dictionary $\boldsymbol{\bar{D}_4}$ was chosen to be a $10 \times 10$ identity matrix. Each column of the last mode factor $\boldsymbol{B_4}[:,r]$ was generated from $\mathcal{U}(0,1)$ and then $L_1$ normalized. Densities of the other mode factors were generated in the same way as for experiment 2. We could not compare our method with the EM-GMM algorithm because of its incapability to incorporate the discrete components of the RV in its formulation. The results for this experiment are presented in Fig.~\ref{fig:exp4}.
\subsection{Real Data}
\begin{figure}[t]
\centerline{\includegraphics[width=80mm,height=50mm]{plots/real_data.png}}
\caption{Classification accuracy on various datasets}
\label{fig:real_data}
\end{figure}
In the real-world datasets where the true underlying PDF is unknown, we tested our algorithm for a classification task on various datasets taken from UCI repository\footnote{\url{https://archive.ics.uci.edu/ml/datasets.php}} -- `Banknote Authentication' (5D), `Wifi Localization' (8D), `Raisins' (8D) and `Seeds' (8D) datasets, and the KTH TIPS texture dataset\footnote{\url{https://www.csc.kth.se/cvap/databases/kth-tips/download.html}}. The classification results for different methods are summarized in Fig.~\ref{fig:real_data}. The flexibility of our algorithm to adapt to hybrid distributions and learn the joint probability of both discrete and continuous components, endows us to estimate the joint density of the form $p_{\boldsymbol{X},Y}(\boldsymbol{x},y)$, where $\boldsymbol{X}$ denotes the vector of class feature (continuous) and $Y$ is their label (discrete). As described above, the dictionaries $\mathcal{D}_n$ for the features $\boldsymbol{X}$ are chosen from the continuous distribution families after examining their empirical marginals. For the label $Y$, $\mathcal{D}_n$ is set to identity matrix. The value of $F$ was chosen on the basis of accuracy on a validation set which was distinct from the training and test sets. For all the experiments, the classification task was performed by MAP estimation which assigns the label $\hat{y}=\textrm{argmax}_y p(y|\boldsymbol{x})=\textrm{argmax}_y p(\boldsymbol{x},y)/p(\boldsymbol{x})$. We ran the CPD-3 algorithm as implemented by the authors. Treating the label of each class as the latent variable, the classification in CPD-3 is done using the MLE estimate: $\textrm{argmax}_y p(\boldsymbol{x}|y)$, assuming the prior $p(y)$ to be uniform, unlike our algorithm where we learn the complete joint density $p(\boldsymbol{x},y)$. We also compared with GMM having full covariance (`GMM-Full') and diagonal covariance matrices (`GMM-Diag').
\noindent\textit{UCI Dataset}: Here we would like to bring out the generality of our method to model discontinuities in the PDFs. For eg., if the histogram of some component shows abrupt change at some value, then we can model this discontinuity by keeping few uniform distributions spanning the range of the histogram along with other densities. For most of the data, using a dictionary consisting of Gaussians and Uniform distributions yielded satisfactory results (see Fig.~\ref{fig:real_data}).
\noindent\textit{KTH TIPS}: This is a texture dataset which contains images of size $200\times200$ of various textures. We chose three textures - Orange Peel, Bread and Linen for our classification task. We used two training images from each class and divided them into $5\times5$ patches creating 26D (patch size+label) data. Thereafter, we normalized the pixel values so that all of them lie in the range $[0,1]$. Here, we used a dictionary consisting of only Gaussians. For testing, we created a collage of $5\times5$ patches of these textures and classified each patch again by using MAP estimator. Our model performed remarkably well for such a high-dimensional data and outperformed all the other algorithms by a reasonable margin (see Fig.~\ref{fig:real_data}).
\section{Conclusion and Future Work}
We integrated ideas from low-rank tensors CPD and dictionary representation of signals to present a novel joint density estimation technique. Our method is completely general and can be applied to model mixtures of distributions coming from different families. The numerical results demonstrate the efficacy of our algorithm, especially in the low sample regime where other methods under-perform. Some future work may include a theoretical analysis of the proposed method w.r.t. sample complexity.
Our method can also be extended to dictionary learning, where the dictionary elements are themselves learned from the data without any manual inspection.
\bibliographystyle{IEEEbib}
|
1,314,259,992,977 | arxiv | \section{Introduction}
The robust ground state degeneracy (GSD) that arises in topologically
ordered systems\ \cite{wen89,wen90,wen91} has been an object of intense
study over the past quarter-century. Interest in such states of
matter has been motivated in large part by the desire to access
quasiparticles with non-Abelian statistics, whose nontrivial braiding
could be used as a platform for quantum computation.\ \cite{nayak}
Nevertheless, to date there has been no definitive experimental proof
that such non-Abelian quasiparticles exist, nor has there been any
direct observation of topological GSD.
There have been several theoretical proposals for the experimental
detection of topological degeneracy. One set of proposals for the
(putative) non-Abelian $\nu=5/2$ quantum Hall state focuses on
measuring the contribution of the GSD to the electronic portion of the
entropy at low temperatures. Observable signatures of this
contribution include the thermopower\ \cite{halperin,yang} and the
temperature dependence of the electrochemical potential and orbital
magnetization.\ \cite{cooper} The thermopower has been measured on
several occasions\ \cite{chickering1,chickering2} with no conclusive
signatures. Abelian fractional quantum Hall (FQH)
states\ \cite{wen_zee} are also topologically ordered, but the bulk GSD
in these systems is only accessible on closed surfaces (e.g.,~the
torus). This is unnatural for experiments, which are confined to
finite planar systems, although a recent
proposal\ \cite{barkeshli_oreg_qi} suggests a transport measurement in
a bilayer FQH system that avoids this handicap by effectively altering
the topology of the system.
In this paper, we propose that time-reversal-symmetric fractional
topological liquids (FTLs) may constitute a promising alternative
platform for realizing the topological GSD in experimentally
accessible geometries. FTLs with time-reversal symmetry (TRS) have an
effective description in terms of doubled Chern-Simons (CS), or
so-called BF, theories.\ \cite{freedman_nayak} Examples of
time-reversal-symmetric FTLs with topological order include fractional
quantum spin Hall systems,\ \cite{kane_mele1,kane_mele2,bernevig}
certain spin liquids,\ \cite{thomale}
Kitaev's toric code,\ \cite{tc} and even the $s$-wave BCS
superconductor.\ \cite{wen91,hansson} In the present work we emphasize
FTLs whose edge states in planar geometries can be completely gapped
without breaking TRS, which is possible when certain criteria are
satisfied.\ \cite{levin_stern,neupert} In these cases, the degenerate
ground state manifold is well separated from excited states and the
GSD on punctured planar surfaces is accessible experimentally.
Our program for this paper is as follows. We first derive a formula
for the GSD of a doubled CS theory defined on a plane with
$N^{\,}_{\mathrm{h}}$ holes, in cases where all helical edge modes are
gapped by appropriate backscattering terms. This topological
degeneracy increases exponentially with the number of holes, and is
exact in the limit where all holes are infinitely large and infinitely
far apart. We then consider finite-sized systems, where the
degeneracy is split exponentially by quasiparticle tunneling
processes. In this setting, we argue that the holes themselves
realize an effective spin-like system, whose Hilbert space consists of
what was formerly the degenerate ground state manifold. We then
examine calorimetry as a possible experimental probe of the
degeneracy. We argue that, for suitable materials, the contribution
of the GSD to the low-temperature heat capacity could be observed
experimentally, even in the presence of the expected phononic and
electronic backgrounds. Finally, we also briefly revisit the notion
of topological order in $s$-wave superconductors, which was suggested
by Wen\cite{wen91} and investigated in detail by Hansson et al.~in
Ref.~\onlinecite{hansson}. We argue that, for a thin-film
superconductor with (3+1)-dimensional electromagnetism, there is
indeed a ground state degeneracy, which is related to flux
quantization. However, this degeneracy is lifted in a power-law
fashion, rather than exponentially, and is therefore not topological
in the canonical sense of
Refs.~\onlinecite{wen89}--\onlinecite{wen91}.
\section{The topological degeneracy}
\label{sec: The topological degeneracy}
In this section we derive a formula for the ground state degeneracy of
a TRS-FTL with appropriately gapped edges. We begin with some
preliminary information before moving on to the derivation.
\subsection{Definitions and notation}
A general time-reversal-symmetric doubled Chern-Simons theory in
(2+1)-dimensional space and time has the form\ \cite{neupert}
\begin{subequations}
\label{trscs}
\begin{align}
\mathcal{L}^{\,}_{\mathrm{CS}}&\:=
\frac{1}{4\pi}\,
K^{\,}_{\mathsf{ij}}\,
\epsilon^{\mu\nu\rho}\,
a^{\mathsf{i}}_{\mu}\,
\partial^{\,}_{\nu}\,
a^{\mathsf{j}}_{\rho}
+
\frac{e}{2\pi}\,
Q^{\,}_{\mathsf{i}}\,
\epsilon^{\mu\nu\rho}\,
A^{\,}_{\mu}\,
\partial^{\,}_{\nu}\,
a^{\mathsf{i}}_{\rho},
\label{dcs}
\end{align}
where $\mathsf{i},\mathsf{j}=1,\ldots,2N$, $\mu,\nu,\rho=0,1,2$, and
summation on repeated indices is implied. Here, the $2N\times2N$
matrix $K^{\,}_{\mathsf{ij}}$ is symmetric, invertible, and
integer-valued. The fully antisymmetric Levi-Civita tensor
$\epsilon^{\mu\nu\rho}$ appears with the convention
$\epsilon^{012}=1$. The components $A^{\,}_{\mu}$ of the
electromagnetic gauge potential are restricted to (2+1)-dimensional
space and time, and the vector $\bm{Q}$ has integer entries that
measure the charges of the various CS fields $a^{\mathsf{i}}_{\mu}$ in
units of the electron charge $e$. The theory contains $N$ Kramers
pairs of CS fields, which transform into one another under the
operation of time-reversal. We will therefore be particularly
interested in scenarios where the $2N\times 2N$ matrix $K$ has the
following block form, which is consistent with TRS, as was shown in
Ref.~\onlinecite{neupert},
\begin{align}\label{kdef}
K\:=
\begin{pmatrix}
\kappa
&
\Delta
\\
\Delta^{\mathsf{T}}
&
-\kappa
\end{pmatrix},
\end{align}
where the $N\times N$ matrices $\kappa=\kappa^{\mathsf{T}}$ and
$\Delta =-\Delta^{\mathsf{T}}$. TRS further imposes that the charge
vector possess the block form (see Ref.~\onlinecite{neupert})
\begin{align}
\bm{Q}\:=
\begin{pmatrix}
\varrho
\\
\varrho
\end{pmatrix}.
\label{qdef}
\end{align}
\end{subequations}
The theory \eqref{trscs} can also be re-expressed in terms of an
equivalent BF theory\ \cite{santos} by defining the linear
transformation
$\tilde{a}^{\mathsf{i}}_{\mu}\:=
R^{\,}_{\mathsf{ij}}\,a^{\mathsf{j}}_{\mu}$,
where
\begin{subequations}\label{bfdef}
\begin{align}\label{rdef}
R\:=
\begin{pmatrix}
\mathbbm{1}
&
\mathbbm{1}
\\
\frac{\mathbbm{1}}{2}
&
-
\frac{\mathbbm{1}}{2}
\end{pmatrix},
\end{align}
with $\mathbbm{1}$ the $N\times N$ identity matrix. This linear
transformation induces the $K$-matrix and charge vector
\begin{align}
\tilde K &\:=
(R^{-1})^{\mathsf{T}}\,
K\,
R^{-1}=
\begin{pmatrix}
0
&
\varkappa
\\
\varkappa^{\mathsf{T}}
&
0
\end{pmatrix},
\\
\varkappa&\:=
\kappa
-
\Delta,
\label{eq: def varkappa}
\\
\tilde{\bm{Q}}&\:=
(R^{-1})^{\mathsf{T}}\,
\bm{Q}
=
\begin{pmatrix}
\varrho
\\
0
\end{pmatrix}.
\label{ktrans}
\end{align}
\end{subequations}
Note that the transformation \eqref{rdef} preserves $\det\,K$
[c.f.~Eq.~\eqref{ktrans}].
When defined on a manifold with boundary, the CS theory \eqref{dcs}
has an associated theory of $2N$ chiral bosons $\phi^{\,}_{\mathsf{i}}$
at the edge. In the most generic case, the boundary of the system
consists of a disjoint union of an arbitrary number of edges, each
with a Lagrangian density of the form (in the absence of the gauge
field $A^{\,}_{\mu}$)\ \cite{neupert}
\begin{align}\label{edge}
\mathcal L^{\,}_{\mathrm{E}}&=
\frac{1}{4\pi}\,
\left(
K^{\,}_{\mathsf{ij}}\,
\partial^{\,}_{t}\,\phi^{\,}_{\mathsf{i}}\,
\partial^{\,}_{x}\,\phi^{\,}_{\mathsf{j}}
-
V^{\,}_\mathsf{ij}\,
\partial^{\,}_{x}\,\phi^{\,}_{\mathsf{i}}\,
\partial^{\,}_{x}\,\phi^{\,}_{\mathsf{j}}
\right)
+
\mathcal L^{\,}_{\mathrm{T}},
\end{align}
where $K^{\,}_\mathsf{ij}$ is the same $2N\times2N$ matrix as before
and the positive-definite, real-valued, symmetric
matrix $V^{\,}_\mathsf{ij}$ encodes non-universal information
specific to a particular edge. The Lagrangian density
$\mathcal{L}^{\,}_{\mathrm{T}}$ generically contains all inter-channel
tunneling operators,
\begin{align}
\mathcal{L}^{\,}_{\mathrm{T}}\:=
\sum_{\bm{T}\in\mathbb{L}}
U^{\,}_{\bm{T}}(x)\,
\cos
\Big(
\bm{T}^{\mathsf{T}}\,K\,\bm{\phi}(x)
+
\zeta^{\,}_{\bm{T}}(x)
\Big),
\label{tunneling}
\end{align}
where $\bm{T}$ is a $2N$-dimenisonal integer vector,
$\bm{\phi}^{\mathsf{T}}=(\phi^{\,}_{1}\ \ldots\ \phi^{\,}_{2N})$, and
$\mathbb{L}$ is the set of all tunneling vectors $\bm{T}$ allowed by
TRS and charge conservation (if it holds). The real-valued functions
$U^{\,}_{\bm{T}}(x)$ and $\zeta^{\,}_{\bm{T}}(x)$ encode information
about disorder at the edge and are further constrained to be
consistent with TRS (see Ref.~\onlinecite{neupert}). When TRS is
imposed, a necessary and sufficient condition for gapping out the
bosonic modes in the edge theory \eqref{edge} is the existence of $N$
$2N$-dimensional vectors $\bm{T}^{\,}_{i}\in\mathbb{L}$ satisfying\
\cite{neupert,haldane}
\begin{subequations}
\label{tcriteria}
\begin{align}
&
\bm{T}^{\mathsf{T}}_{i}\,\bm{Q}= 0,
\quad\forall\ i\indent\text{(charge conservation)},
\label{chargeconservation}
\\
&
\bm{T}^{\mathsf{T}}_{i}\,K\,\bm{T}^{\,}_{j}= 0,
\quad\forall\ i,j\indent\text{(Haldane criterion)}.
\label{haldanecrit}
\end{align}
\end{subequations}
Strictly speaking, the criterion \eqref{chargeconservation} need not
hold in a general system, such as (for example) in the case of a
superconductor. In this case, one replaces charge conservation with
charge conservation mod 2 (i.e.,~conservation of fermion parity),
so that $\bm{T}^{\mathsf{T}}_{i}\,\bm{Q}$ is only constrained to be even.
In the next section, we will focus on cases where the criteria \eqref{tcriteria}
are satisfied.
\subsection{Gauge invariance in a system with gapped edges}
\label{sec: gluing}
The need for the edge theory \eqref{edge} arises from the failure of
gauge invariance in Chern-Simons theories on manifolds with boundary.
For non-chiral Chern-Simons theories, like those of the form
\eqref{trscs}, the ability to gap out the edge states necessitates an
alternate route to gauge invariance, as we now show. For simplicity,
we will work on the disk, although analogous results hold for
manifolds with multiple disconnected boundaries.
To proceed, we rewrite the Lagrangian density \eqref{dcs}, in the
absence of the electromagnetic gauge potential $A^{\,}_{\mu}$ (which
we ignore hereafter), in terms of two separate sets of $N$ CS fields
$\alpha^{\mathtt{i}}$ and $\beta^{\mathtt{i}}$,
\begin{equation}
\begin{split}
\mathcal{L}^{\,}_{\mathrm{CS}}=&\,
\frac{\epsilon^{\mu\nu\rho}}{4\pi}\,
\left[
\kappa^{\,}_{\mathtt{ij}}\,
\left(
\alpha^{\mathtt{i}}_{\mu}\,\partial^{\,}_{\nu}\,\alpha^{\mathtt{j}}_{\rho}
-
\beta^{\mathtt{i}}_{\mu}\,\partial^{\,}_{\nu}\,\beta^{\mathtt{j}}_{\rho}
\right)
\right.
\\
&\,
\left.
+
\Delta^{\,}_{\mathtt{ij}}
\left(
\alpha^{\mathtt{i}}_{\mu}\,\partial^{\,}_{\nu}\,\beta^{\mathtt{j}}_{\rho}
-
\beta^{\mathtt{i}}_{\mu}\,\partial^{\,}_{\nu}\,\alpha^{\mathtt{j}}_{\rho}
\right)
\right].
\end{split}
\label{annulus}
\end{equation}
Here, ${\mathtt{i}},\mathtt{j}\in\{1,\ldots,N\}$, and the ``new" CS
fields are defined as $\alpha^{\mathtt{i}}_{\mu}(\bm{x},t)\equiv
a^{\mathtt{i}}_{\mu}(\bm{x},t)$ and
$\beta^{\mathtt{i}}_{\mu}(\bm{x},t)\equiv
a^{{\mathtt{i}}+N}_{\mu}(\bm{x},t)$.
We define the CS action on the disk $D$ to be
\begin{equation}
S^{\,}_{\mathrm{CS}}\:=
\int\limits\mathrm{d}t
\int\limits_{D}\mathrm{d}^{2}x\,
\mathcal{L}^{\,}_{\mathrm{CS}}(\bm{x},t).
\label{scs}
\end{equation}
Its transformation law under any local gauge transformation of the form
\begin{subequations}
\begin{equation}
\alpha^{\mathtt{i}}_{\mu}
\mapsto
\alpha^{\mathtt{i}}_{\mu}
+
\partial^{\,}_{\mu}\,
\chi^{\mathtt{i}}_{\alpha},
\qquad
\beta^{\mathtt{i}}_{\mu}
\mapsto
\beta^{\mathtt{i}}_{\mu}
+
\partial^{\,}_{\mu}\,
\chi^{\mathtt{i}}_{\beta},
\end{equation}
where $\mathtt{i}=1,\ldots,N$ and
$\chi^{\mathtt{i}}_{\alpha}$ and $\chi^{\mathtt{i}}_{\beta}$ are
real-valued scalar fields, is
\begin{equation}
S^{\,}_{\mathrm{CS}}\mapsto
S^{\,}_{\mathrm{CS}}
+
\delta S^{\,}_{\mathrm{CS}}
\end{equation}
with the boundary contribution
\begin{align}
\delta S^{\,}_{\mathrm{CS}}\:=&\,
\int\limits\mathrm{d}t
\oint\limits_{\partial D}\mathrm{d}x^{\,}_{\mu}\,
\frac{\epsilon^{\mu\nu\rho}}{4\pi}\,
\left[
\kappa^{\,}_{\mathtt{ij}}
\left(
\chi^{\mathtt{i}}_{\alpha}\,
\partial^{\,}_{\nu}\,
\alpha^{\mathtt{j}}_{\rho}
-
\chi^{\mathtt{i}}_{\beta}\,
\partial^{\,}_{\nu}\,
\beta^{\mathtt{j}}_{\rho}
\right)
\right.\nonumber
\\
&\,
\left.
+
\Delta^{\,}_{\mathtt{ij}}
\left(
\chi^{\mathtt{i}}_{\alpha}\,\partial^{\,}_{\nu}\,\beta^{\mathtt{j}}_{\rho}
-
\chi^{\mathtt{i}}_{\beta}\,\partial^{\,}_{\nu}\,\alpha^{\mathtt{j}}_{\rho}
\right)
\right].
\label{anomaly}
\end{align}
\end{subequations}
Here, the boundary $\partial D$ of the disk $D$ is the circle $S^{1}$, and
$\mathrm{d}x^{\,}_{\mu}\:=\epsilon^{\,}_{\mu0\sigma}\, \mathrm d\ell^{\sigma}$,
with $\mathrm{d}\ell^{\sigma}$ the line element along the boundary.
There are two ways to impose gauge invariance in the doubled
Chern-Simons theory $S^{\,}_{\mathrm{CS}}$. On the one hand, if the criteria
\eqref{tcriteria} do not hold, we must demand that there exist a
gapless edge theory with an action $S^{\,}_{E}$ that transforms as
$S^{\,}_{\mathrm E}\mapsto S^{\,}_{\mathrm E}-\delta S^{\,}_{\mathrm{CS}}$, so that
the total action $S^{\,}_{\mathrm{CS}}+S^{\,}_{\mathrm E}$ is gauge invariant.
On the other hand, if the criteria \eqref{tcriteria} hold, then the
edge fields $\phi^{\,}_\mathsf{i}$ become pinned to the classical
minima of the cosine potentials in $\mathcal{L}^{\,}_{\mathrm{T}}$ for large
$|U^{\,}_{\bm{T}}(x)|$, and are then no longer dynamical degrees of freedom. In
this case, gauge invariance can be achieved by demanding that the
anomalous term $\delta S^{\,}_{\mathrm{CS}}=0$ identically. The latter option
can be accomplished by imposing the boundary conditions
\begin{subequations}
\label{gluing}
\begin{equation}
\chi^{\mathtt{i}}_{\alpha}\vert^{\,}_{\partial D}=
T^{\,}_\mathtt{ij}\,\chi^{\mathtt{j}}_{\beta}\vert^{\,}_{\partial D},
\qquad
\alpha^{\mathtt{i}}_{\mu}\vert^{\,}_{\partial D}=
T^{\,}_\mathtt{ij}\,
\beta^{\mathtt{j}}_{\mu}\vert^{\,}_{\partial D},
\label{gluing1}
\end{equation}
for all $\mathtt{i}=1,\dots,N$, where the invertible $N\times N$
matrix $T$ satisfies the following algebraic criterion:
\begin{equation}
T^{\mathsf{T}}\,
\kappa\,
T
-
\kappa
+
T^{\mathsf{T}}\,
\Delta
-
\Delta\,T=0.
\label{gluing2}
\end{equation}
\end{subequations}
One can show that, in order for the boundary conditions
\eqref{gluing1} to be well-defined and consistent with TRS, the matrix
$T$ must have rational entries and satisfy $T^{2}=\mathbbm{1}$ (see
the Appendix).
It is natural to wonder whether different choices of the matrix $T$ in
Eqs.~\eqref{gluing} correspond to different ways of gapping out the
edge theory, i.e.,~to different choices of the set of $N$ linearly
independent tunneling vectors $\bm{T}^{\,}_{i}$
($i=1,\dots,N$) that satisfy
Haldane's criterion \eqref{haldanecrit}. In the Appendix, we argue
that this is indeed the case, although the correspondence need not be
one-to-one. In particular, while any well-defined choice of the
matrix $T$ implies a particular choice of the set $\{\bm{T}^{\,}_{i}\}$,
\textit{most} (but not \textit{all})
choices of the set $\{\bm{T}^{\,}_{i}\}$ imply a
particular choice of $T$. In the remainder of this paper, we restrict
our attention to cases where the edge theory is gapped in such a way
that this correspondence holds.
We close this section with the observation that the boundary
conditions \eqref{gluing} can be defined on manifolds with multiple
disconnected boundaries. For example, the boundary $\partial A$ of
the annulus
\begin{equation}
A\:=[0,\pi]\times S^{1}
\end{equation}
consists of the disjoint union of two circles ($\partial A = S^{1}\sqcup S^{1}$).
In this case, one imposes independent boundary conditions of
the form \eqref{gluing1} on each copy of $S^{1}$. If both edges are
gapped in the same way, then the boundary conditions \eqref{gluing1}
involve the same matrix $T$ on both edges. It is natural to assume
that this is the case when both boundaries of the annulus separate the
TRS-FTL from vacuum, since both edges have the same symmetries and can
therefore be expected to flow under RG to the same strong-disorder
fixed point with the $N$ most relevant tunneling processes described by
the same set of tunneling vectors $\{\bm T_i\}^{N}_{i=1}$. We will therefore
make this assumption in the derivation below.
\subsection{Calculation of the degeneracy}
The ground state degeneracy on the \textit{torus} of a multi-component
Abelian Chern-Simons theory of the form \eqref{dcs} is known on
general grounds to be given by
$|\det\,K|$.\ \cite{wen89,wen_zee,wesolowski} We now present an argument
that, for a doubled CS theory whose $K$-matrix is of
the form \eqref{kdef}, the ground-state degeneracy of the theory on
the \textit{annulus} is given by the formula
\begin{equation}
\text{GSD}=
\sqrt{|\det\,K|}=
\abs{\text{Pf}
\begin{pmatrix}
\Delta
&
\kappa
\\
-
\kappa
&
\Delta^{\mathsf{T}}
\end{pmatrix}},
\label{gsd}
\end{equation}
provided that both edges of the annulus are gapped by the same
tunneling terms of the form \eqref{tunneling}, and provided that these
terms are chosen appropriately. Note that $|\det\,K|$ is the square
of an integer,\ \cite{neupert,santos} so the GSD in these cases is
also an integer.
The GSD of non-chiral Chern-Simons theories on manifolds with boundary
depends on the details of how the different edges are gapped (see,
e.g., Ref.~\onlinecite{wang}). In our argument, this dependence will
manifest itself in different choices of the boundary conditions
\eqref{gluing1} for the bulk Chern-Simons fields, which affect the
counting of the degeneracy.
\begin{figure}[t]
\includegraphics[width=.4\textwidth]{gluing}
\caption{(Color online) Gluing argument for the special case
$\Delta=0$. In this case, the CS theory consists of two independent
copies, with equal and opposite $K$-matrices. The tunneling
processes (dotted lines) that gap out each pair of
counterpropagating edge modes couple the two annuli, and the
conditions \eqref{gluing} ensure that the two copies of the theory
can be consistently ``glued" together.
\label{fig:gluing}
}
\end{figure}
Using these boundary conditions, it is possible to show that
Eq.~\eqref{gsd} follows in much the same way as its counterpart on the
torus, so long as both edges of the annulus are gapped by the same
tunneling terms of the form \eqref{tunneling}. Before proceeding with
the full argument, we first provide an intuitive picture of why this
is, for the case where $\Delta=0$ in Eq.~\eqref{kdef}. In this case,
Eq.~\eqref{annulus} describes two decoupled CS liquids, one with
$K$-matrix $\kappa$ and the other with $K$-matrix $-\kappa$. We can
imagine that the two CS liquids live on separate copies of the annulus
$A$, which are coupled by the tunneling processes that gap out the
edges. The conditions in Eq.~\eqref{gluing} ensure that the two
coupled annuli can be ``glued" together into a single surface, on
which lives a composite CS theory with a GSD given by $|\det \kappa|$
(see Fig.~\ref{fig:gluing}). Remarkably, these gluing conditions are
also sufficient to treat cases where $\Delta\neq 0$, as we now show.
\subsubsection{Wilson loops, large gauge transformations, and their algebras}
Suppose that we are given a doubled Chern-Simons theory on the annulus
of the form \eqref{trscs}, and that both edges of the annulus are
fully gapped by identical tunneling terms of the form
\eqref{tunneling}. Let us further impose boundary conditions of the
form \eqref{gluing} at each edge, with the matrix $T$ chosen
appropriately (see the Appendix). We can now use these boundary
conditions, arising as they do from the need to cancel the anomalous
boundary term \eqref{anomaly}, to construct Wilson loop operators,
which can in turn be used to determine the dimension of the ground
state subspace.
To do this, we first perform a change of basis on the CS Lagrangian
\eqref{dcs} by defining the linear combinations
\begin{subequations}
\begin{align}\label{change basis}
\begin{split}
a^{\mathtt{i}}_{+,\mu}&\:=
T^{\,}_{\mathtt{ij}}\,\alpha^{\mathtt{j}}_{\mu}
+
\beta^{\mathtt{i}}_{\mu}\\
a^{\mathtt{i}}_{-,\mu}&\:=
\frac{1}{2}
\left(
\alpha^{\mathtt{i}}_{\mu}
-
T^{\,}_{\mathtt{ij}}\,
\beta^{\mathtt{j}}_{\mu}
\right),
\end{split}
\end{align}
where $\mathtt{i}=1,\dots, N$.
In terms of these fields, the transformed CS Lagrangian reads
\begin{align}
\mathcal{L}^{\,}_{\mathrm{CS}}=&
\frac{\epsilon^{\mu\nu\rho}}{4\pi}\,
\(
\varkappa^{\,}_{\mathtt{ij}}\,
a^{\mathtt{i}}_{+,\mu}\,
\partial^{\,}_{\nu}\,
a^{\mathtt{j}}_{-,\rho}
+
\varkappa^{\mathsf{T}}_{\mathtt{ij}}\,
a^{\mathtt{i}}_{-,\mu}\,
\partial^{\,}_{\nu}\,
a^{\mathtt{j}}_{+,\rho}
\right.\nonumber\\
&\qquad\left.+
\widetilde\varkappa^{\,}_\mathtt{ij}\,
a^{\mathtt{i}}_{-,\mu}\,
\partial^{\,}_{\nu}\,
a^{\mathtt{j}}_{-,\rho}
\),
\end{align}
where we have defined the $N\times N$ matrices
\begin{align}
\varkappa&\:=
\kappa\,
T
-
T^{\mathsf{T}}\,
\Delta T,
\\
\widetilde\varkappa &\:=
\kappa
-
\Delta\,
T
-
T^{\mathsf{T}}\,
\Delta^{\mathsf{T}}
-
T^{\mathsf{T}}\,
\kappa T.
\end{align}
Before we continue, note that the linear transformation defined by
Eq.~\eqref{change basis} has determinant $\pm 1$, so that this change
of basis leaves $|\det K|$ invariant. Consequently, we have that
\begin{align}
|\det K|=
|\det\varkappa|^{2}.
\end{align}
Furthermore, observe that, in the case $T=\mathbbm{1}$, the matrix
$\varkappa$ above coincides with the one defined in
Eq.~\eqref{eq: def varkappa}.
For reasons that will be made clear below, we restrict
our attention to cases where the matrix $T$ can be chosen such that
the matrix $\varkappa$ has integer entries.
\end{subequations}
In this new basis, the gluing conditions \eqref{gluing} become
Dirichlet boundary conditions on the $(-)$ fields,
\begin{equation}\label{dirichlet}
\chi^{\mathtt{i}}_{-}\vert^{\,}_{\partial A}=0,
\qquad
a^{\mathtt{i}}_{-,\mu}\vert^{\,}_{\partial A}=0,
\end{equation}
for $\mathtt{i}=1,\ldots,N$. Rewriting the Lagrangian density in the
gauge $a^{\mathtt{i}}_{\pm,0}=0$ (this can be done using a gauge
transformation obeying the gluing conditions), we obtain
\begin{subequations}
\begin{equation}
\begin{split}
\mathcal{L}^{\,}_{\mathrm{CS}}=&\,
\frac{1}{4\pi}\,
\left[
\varkappa^{\,}_\mathtt{ij}
\left(
a^{\mathtt{i}}_{+,2}\,
\partial^{\,}_{0}\,
a^{\mathtt{j}}_{-,1}
-
a^{\mathtt{i}}_{+,1}\,
\partial^{\,}_{0}\,
a^{\mathtt{j}}_{-,2}
\right)
\right.
\\
&\,
\left.
+
\varkappa^{\mathsf{T}}_\mathtt{ij}\,
\left(
a^{\mathtt{i}}_{-,2}\,
\partial^{\,}_{0}\,
a^{\mathtt{j}}_{+,1}
-
a^{\mathtt{i}}_{-,1}\,
\partial^{\,}_{0}\,
a^{\mathtt{j}}_{+,2}
\right)
\right.
\\
&\,
\left.
+
\widetilde\varkappa_\mathtt{ij}\,
\left(
a^{\mathtt{i}}_{-,2}\,
\partial^{\,}_{0}\,
a^{\mathtt{j}}_{-,1}
-
a^{\mathtt{i}}_{-,1}\,
\partial^{\,}_{0}\, a
^{\mathtt{j}}_{-,2}
\right)
\right]
\end{split}
\end{equation}
supplemented by the $2N$ constraints arising from the equations of motion for
$a^{\mathtt{i}}_{0}$
($\mathtt{i}=1,\ldots,N$),
\begin{equation}
\partial^{\,}_{1}\,
a^{\mathtt{i}}_{+,2}
-
\partial^{\,}_{2}\,
a^{\mathtt{i}}_{+,1}=0,
\qquad
\partial^{\,}_{1}\,
a^{\mathtt{i}}_{-,2}
-
\partial^{\,}_{2}\,
a^{\mathtt{i}}_{-,1}=0.
\label{constraints}
\end{equation}
\end{subequations}
The constraints (\ref{constraints}) are met by the decompositions
\begin{subequations}
\label{decomp}
\begin{align}
&
a^{\mathtt{i}}_{\pm,1}(x^{\,}_{1},x^{\,}_{2},t)=
\partial^{\,}_{1}\,
\chi^{\mathtt{i}}_{\pm}(x^{\,}_{1},x^{\,}_{2},t)
+
\bar{a}^{\mathtt{i}}_{\pm,1}(x^{\,}_{1},t),
\\
&
a^{\mathtt{i}}_{\pm,2}(x^{\,}_{1},x^{\,}_{2},t)=
\partial^{\,}_{2}\,
\chi^{\mathtt{i}}_{\pm}(x^{\,}_{1},x^{\,}_{2},t)
+
\bar{a}^{\mathtt{i}}_{\pm,2}(x^{\,}_{2},t),
\end{align}
\end{subequations}
of the CS fields, provided that
$\chi^{\mathtt{i}}_{\pm}(x^{\,}_{1},x^{\,}_{2},t)$
are everywhere smooth functions of $x^{\,}_{1}$ and $x^{\,}_{2}$,
while $\bar{a}^{\mathtt{i}}_{\pm,1}(x^{\,}_{1},t)$
and
$\bar{a}^{\mathtt{i}}_{\pm,2}(x^{\,}_{2},t)$
are independent of $x^{\,}_{2}$ and $x^{\,}_{1}$,
respectively. Furthermore, the geometry of an annulus is implemented by
the boundary conditions
\begin{subequations}\label{conditions}
\begin{equation}
\chi^{\mathtt{i}}_{\pm}(x^{\,}_{1},x^{\,}_{2}+2\pi,t)=
\chi^{\mathtt{i}}_{\pm}(x^{\,}_{1},x^{\,}_{2},t)
\end{equation}
for the fields parametrizing the pure gauge contributions and
\begin{align}
&
\chi^{\mathtt{i}}_{-}(0,x^{\,}_{2},t)=
\chi^{\mathtt{i}}_{-}(\pi,x^{\,}_{2},t)=0,
\\
&
\bar{a}^{\mathtt{i}}_{-,1}(0,t)=
\bar{a}^{\mathtt{i}}_{-,1}(\pi,t)=0,
\\
&
\bar{a}^{\mathtt{i}}_{-,2}(x^{\,}_{2},t)\vert^{\,}_{x^{\,}_{1}=0}=
\bar{a}^{\mathtt{i}}_{-,2}(x^{\,}_{2},t)\vert^{\,}_{x^{\,}_{1}=\pi}=
\bar{a}^{\mathtt{i}}_{-,2}(x^{\,}_{2},t)=0,
\end{align}
\end{subequations}
for the gluing conditions.
The coordinate system employed in these definitions is depicted in
Fig.~\ref{fig:annulus}.
\begin{figure}[t]
\includegraphics[width=.33\textwidth]{annulus}
\caption{(Color online)
Coordinate system on the annulus
$A=[0,\pi]\times S^{1}$.
The inner boundary is at $x^{\,}_{1}=0$, while the
outer boundary is at $x^{\,}_{1}=\pi$.
The coordinate $x^{\,}_{2}$ is defined on the circle $S^{1}$.
\label{fig:annulus}
}
\end{figure}
The next step is to show that the barred variables decouple from the
remaining (pure gauge) degrees of freedom. This can be done by
inserting the decomposition \eqref{decomp} into the action and using
the boundary conditions \eqref{conditions}. In the course of this
calculation, the terms containing $\widetilde\varkappa$ that involve
barred variables are found to vanish due to the fact that
$\bar{a}^{\mathtt{i}}_{-,2}(x^{\,}_{2},t)=0$ for all $x^{\,}_{2}$ and $t$, and to the
periodicity in $x^{\,}_{2}$ of the functions
$\chi^{\mathtt{i}}_{-}(x^{\,}_{1},x^{\,}_{2},t)$.
We then find an action involving only the matrix $\varkappa$ that
governs the barred variables alone,
\begin{subequations}
\begin{align}
S^{\,}_{\mathrm{top}}=
\frac{1}{2\pi}\,
\int\limits\mathrm{d}t\
\varkappa^{\,}_{\mathtt{ij}}\,
A^{\mathtt{i}}_{2}\,
\dot{A}^{\mathtt{j}}_{1},
\label{stop}
\end{align}
where, for all $\mathtt{i}=1,\ldots,N$,
we have defined the global degrees of freedom
\begin{align}
&
A^{\mathtt{i}}_{1}(t)\:=
\int\limits\limits_{0}^{\pi}\mathrm{d}x^{\,}_{1}\,
\bar{a}^{\mathtt{i}}_{-,1}(x^{\,}_{1},t),
\\
&
A^{\mathtt{i}}_{2}(t)\:=
\int\limits\limits_{0}^{2\pi}\mathrm{d}x^{\,}_{2}\,
\bar{a}^{\mathtt{i}}_{+,2}(x^{\,}_{2},t).
\end{align}
\end{subequations}
In Eq.~\eqref{stop},
we employ the notation
$\dot{A}^{\mathtt{j}}_{1}=
\partial^{\,}_{t}A^{\mathtt{j}}_{1}\equiv
\partial^{\,}_{0}A^{\mathtt{j}}_{1}$.
According to the topological action
(\ref{stop}),
the variable
$\varkappa^{\,}_{\mathsf{ij}}\,A^{\mathtt{i}}_{2}/(2\pi)$
is canonically conjugate to the variable
$A^{\mathtt{j}}_{1}$.
Canonical quantization then gives the equal-time commutation relations
\begin{subequations}
\begin{align}
\[A^{\mathtt{i}}_{1},A^{\mathtt{j}}_{2}\]&=
2\pi\mathrm{i}\, \varkappa^{-1}_{\mathtt{ij}},
\\
\[A^{\mathtt{i}}_{1},A^{\mathtt{j}}_{1}\]&=
\[A^{\mathtt{i}}_{2},A^{\mathtt{j}}_{2}\]=0,
\end{align}
\end{subequations}
for $\mathtt{i},\mathtt{j}=1,\ldots,N$.
We may now define the Wilson loop operators
\begin{subequations}
\begin{equation}
W^{\mathtt{i}}_{1}\:= e^{\mathrm{i}A^{\mathtt{i}}_{1}}, \qquad
W^{\mathtt{i}}_{2}\:= e^{\mathrm{i}A^{\mathtt{i}}_{2}},
\label{wilson}
\end{equation}
whose algebra is found to be
\begin{align}
&
W^{\mathtt{i}}_{1}\,W^{\mathtt{j}}_{2}=
e^{
-2\pi\mathrm{i}\,
\varkappa^{-1}_{\mathtt{ij}}
}\,
W^{\mathtt{j}}_{2}\,
W^{\mathtt{i}}_{1},
\label{eq: algebra between W1 and W2}
\\
&
\[W^{\mathtt{i}}_{1},W^{\mathtt{j}}_{1}\]=
\[W^{\mathtt{i}}_{2},W^{\mathtt{j}}_{2}\]=0.
\label{bfalg1}
\end{align}
\end{subequations}
There is still a set of symmetries that imposes constraints on the
dimension of the Hilbert space associated with $S^{\,}_{\mathrm{top}}$. In
particular, the path integral is invariant under the
``large gauge transformations"
\begin{equation}
A^{\mathtt{i}}_{1,2}\mapsto
A^{\mathtt{i}}_{1,2}
+
2\pi
\end{equation}
for any $\mathtt{i}=1,\ldots,N$.
The large gauge transformations are implemented by the operators
\begin{subequations}
\label{bfalg2}
\begin{equation}
U^{\mathtt{i}}_{1}\:=
e^{
+\mathrm{i}\,
\varkappa^{\,}_{\mathtt{ij}}\,
A^{\mathtt{j}}_{2}
},
\qquad
U^{\mathtt{i}}_{2}\:=
e^{
-\mathrm{i}\,
\varkappa^{\,}_{\mathtt{ij}}\,
A^{\mathtt{j}}_{1}
},
\label{bfalg2 a}
\end{equation}
which satisfy the algebra
\begin{align}
&
U^{\mathtt{i}}_{1}\,
U^{\mathtt{j}}_{2}=
e^{
-2\pi\mathrm{i}\,
\varkappa^{\,}_{\mathtt{ij}}
}\,
U^{\mathtt{j}}_{2}\,
U^{\mathtt{i}}_{1},
\nonumber\\
&
\[
U^{\mathtt{i}}_{1},
U^{\mathtt{j}}_{1}
\]=
\[
U^{\mathtt{i}}_{2},
U^{\mathtt{j}}_{2}
\]=
0,
\label{bfalg2 b}
\end{align}
\end{subequations}
for any $\mathtt{i},\mathtt{j}=1,\ldots,N$.
Because we require that
$\varkappa$ is an integer matrix, this means that
\begin{equation}
\[
U^{\mathtt{i}}_{1},
U^{\mathtt{j}}_{2}
\]=
\[
U^{\mathtt{i}}_{1},
U^{\mathtt{j}}_{1}
\]=
\[
U^{\mathtt{i}}_{2},
U^{\mathtt{j}}_{2}
\]=
0
\end{equation}
for all
$\mathtt{i},\mathtt{j}=1,\ldots,N$. Hence, all
$U^{\mathtt{i}}_{1}$,
$U^{\mathtt{i}}_{2}$
with $\mathtt{i}=1,\ldots,N$
can be diagonalized simultaneously. Since any one of $U^{\mathtt{i}}_{1}$
and $U^{\mathtt{i}}_{2}$ generates a transformation
that leaves the path integral invariant, the
vacua of the theory must be eigenstates of any one of
$U^{\mathtt{i}}_{1}$ and $U^{\mathtt{i}}_{2}$ for ${\mathtt{i}}=1,\ldots,N$.
\subsubsection{Dimension of the ground-state subspace}
In order to determine the GSD of the theory,
it suffices to determine the number of eigenstates of
any one of
$U^{\mathtt{i}}_{1}$ and $U^{\mathtt{i}}_{2}$ for ${\mathtt{i}}=1,\ldots,N$.
To do this, we follow the argument of
Wesolowski et al.,\ \cite{wesolowski} which can be adapted to our case
with only minor modifications.
First, we define the eigenstates of
any one of
$U^{\mathtt{i}}_{1}$ and $U^{\mathtt{i}}_{2}$ for ${\mathtt{i}}=1,\ldots,N$
by
\begin{align}
U^{\mathtt{i}}_{1}\,
\ket\Psi=
e^{\mathrm{i}\gamma^{\mathtt{i}}_{1}}\,
\ket\Psi,
\qquad
U^{\mathtt{i}}_{2}\,
\ket\Psi=
e^{\mathrm{i}\gamma^{\mathtt{i}}_{2}}\,
\ket\Psi.
\end{align}
Since $A^{\mathtt{i}}_{1}$ and $A^{\mathtt{j}}_{2}$ do not commute,
we may choose to represent the state $\ket\Psi$
in the basis for which $A^{\mathtt{i}}_{1}$ is diagonal by
\begin{align}
\psi(\{A^{\mathtt{i}}_{1}\})\:=
\braket{\{A^{\mathtt{i}}_{1}\}|\Psi}.
\label{eq: choice basis 1}
\end{align}
The representation
$\psi(\{A^{\mathtt{i}}_{2}\})$ follows from the representation
$\psi(\{A^{\mathtt{i}}_{1}\})$ by a change of basis to the one
in which $A^{\mathtt{i}}_{2}$ is diagonal.
The large gauge transformations
(\ref{bfalg2 a})
are represented by
\begin{equation}
U^{\mathtt{i}}_{1}\:=
e^{2\pi\,\partial/\partial A^{\mathtt{i}}_{1}},
\qquad
U^{\mathtt{i}}_{2}\:=
e^{-\mathrm{i}\,\varkappa^{\,}_{\mathtt{ij}}\,A^{\mathtt{j}}_{1}},
\end{equation}
in the basis (\ref{eq: choice basis 1}).
The eigenvalue problem then becomes
\begin{subequations}
\begin{align}
U^{\mathtt{i}}_{1}\,
\psi(\{A^{\mathtt{i}}_{1}\})\:=&\,
\psi\left(A^{1}_{1},\ldots,A^{\mathtt{i}}_{1}+2\pi,\ldots,A^{N}_{1}\right)
\nonumber\\
\equiv&\,
e^{\mathrm{i}\gamma^{\mathtt{i}}_{1}}\,\psi(\{A^{\mathtt{i}}_{1}\}),
\label{eval1}
\\
U^{\mathtt{i}}_{2}\,
\psi(\{A^{\mathtt{i}}_{1}\})\:=&\,
e^{-\mathrm{i}\,\varkappa^{\,}_{\mathtt{ij}}\,A^{\mathtt{j}}_{1}\,
\psi(\{A^{\mathtt{i}}_{1}\}) }
\nonumber\\
\equiv&\,
e^{\mathrm{i}\gamma^{\mathtt{i}}_{2}}\,
\psi(\{A^{\mathtt{i}}_{1}\}).
\label{eval2}
\end{align}
\end{subequations}
Equation \eqref{eval1} implies that we can write the following series
for $\psi$,
\begin{align}
\psi(\{A^{\mathtt{i}}_{1}\})\equiv
\psi(\bm{A}^{\,}_{1})=
e^{\mathrm{i}\bm\gamma^{\,}_{1}\cdot\bm{A}^{\,}_{1}/2\pi}\,
\sum_{\bm n}
d(\bm{n})\,
e^{\mathrm{i}\bm{n}\cdot\bm{A}^{\,}_{1}},
\label{eq: Fourier expansion psi A's}
\end{align}
where
$\bm{n}=
(n^{\,}_{1},\ldots,n^{\,}_{N})^{\mathsf{T}}\in\mathbb{Z}^{N}$,
$\bm{A}^{\,}_{1}=
(A^{1}_{1},\ldots, A^{N}_{1})^{\mathsf{T}}\in\mathbb{R}^{N}$,
and
$\bm{\gamma}^{\,}_{1}=
(\gamma^{1}_{1},\ldots,\gamma^{N}_{1})^{\mathsf{T}}\in\mathbb{R}^{N}$.
Second, we seek the constraints on the real-valued coefficients
$d(\bm{n})$ entering the expansion (\ref{eq: Fourier expansion psi A's})
that, as we shall demonstrate,
fix the dimension of the ground-state subspace.
To this end, we extract from the $N\times N$
matrix $\varkappa$ that was defined in Eq.~(\ref{eq: def varkappa})
the family
\begin{subequations}
\begin{equation}
\varkappa\=:
\begin{pmatrix}
\bm{k}^{\,\mathsf{T}}_{1}
\\
\vdots
\\
\bm{k}^{\,\mathsf{T}}_{N}
\end{pmatrix}
\end{equation}
of $N$ vectors from $\mathbb{Z}^{N}$ and from its inverse
$\varkappa^{-1}$ the family
\begin{equation}
\varkappa^{-1}\=:
\begin{pmatrix}
\bm{\ell}^{\,}_{1}
&
\ldots
&
\bm{\ell}^{\,}_{N}
\end{pmatrix}
\end{equation}
of $N$ vectors from $\mathbb{Q}^{N}$. By construction, these vectors
satisfy
\begin{equation}
\bm{k}^{\,}_{\mathtt{i}}\cdot\bm{\ell}^{\,}_{\mathtt{j}}=
\delta^{\,}_{\mathtt{ij}}.
\end{equation}
\end{subequations}
Using these vectors, we observe that inserting the series
\eqref{eq: Fourier expansion psi A's}
into the left-hand side of Eq.~\eqref{eval2} gives
\begin{align}
U^{\mathtt{i}}_{2}\,
\psi(\bm{A}^{\,}_{1})=&\,
e^{\mathrm{i}\bm{\gamma}^{\,}_{1}\cdot\bm{A}^{\,}_{1}/(2\pi)}\,
e^{-\mathrm{i}\bm{k}^{\,}_{\mathtt{i}}\cdot\bm{A}^{\,}_{1}}\,
\sum_{\bm{n}}
d(\bm{n})\,
e^{\mathrm{i}\bm{n}\cdot\bm{A}^{\,}_{1}}
\nonumber\\
=&\,
e^{\mathrm{i}\bm{\gamma}^{\,}_{1}\cdot\bm{A}^{\,}_{1}/(2\pi)}\,
\sum_{\bm{n}} d(\bm{n}+\bm{k}^{\,}_{\mathtt{i}})\,
e^{\mathrm{i}\bm{n}\cdot\bm{A}^{\,}_{1}}
\nonumber\\
=&\,
e^{\mathrm{i}\gamma_{2}^{\mathtt{i}}}\, \psi(\bm{A}^{\,}_{1}),
\end{align}
which implies
\begin{align}
d(\bm{n}+\bm{k}^{\,}_{\mathtt{i}})=
e^{\mathrm{i}\gamma^{\mathtt{i}}_{2}}\,
d(\bm{n})
\label{eq: constraint on d of n}
\end{align}
for all ${\mathtt{i}}=1,\ldots,N$. The constraint
(\ref{eq: constraint on d of n})
is automatically satisfied by demanding that
\begin{subequations}
\label{eq: Ansatz for d of n}
\begin{align}
d(\bm{n})=
e^{\mathrm{i}\bm{\gamma}^{\,}_{2}\cdot(\varkappa^{-1})^{\mathsf{T}}\bm{n}}\,
\tilde{d}(\bm{n})
\label{eq: Ansatz for d of n a}
\end{align}
with
\begin{align}
\tilde{d}(\bm{n})=
\tilde{d}(\bm{n}+\bm{k}^{\,}_{\mathtt{i}}),
\label{eq: Ansatz for d of n b}
\end{align}
since
\begin{align}
\bm{\gamma}^{\,}_{2}
\cdot
(\varkappa^{-1})^{\mathsf{T}}\,
\bm{k}^{\,}_{\mathtt{i}}=
\gamma^{\mathtt{j}}_{2}
(\bm\ell^{\,}_{\mathtt{j}}\cdot\bm{k}^{\,}_{\mathtt{i}})=
\gamma^{\mathtt{i}}_{2}.
\end{align}
\end{subequations}
Hence, insertion of
(\ref{eq: Ansatz for d of n a})
into the expansion
(\ref{eq: Fourier expansion psi A's})
that solves the eigenvalue problem
(\ref{eval1})
gives the expansion
\begin{align}
\psi(\bm{A}^{\,}_{1})=
e^{\mathrm{i}\bm{\gamma}^{\,}_{1}\cdot\bm{A}^{\,}_{1}/(2\pi)}
\sum_{\bm{n}}
e^{\mathrm{i}\bm{\gamma}^{\,}_{2}\cdot(\varkappa^{-1})^{\mathsf{T}}\bm{n}}\,
\tilde{d}(\bm{n})\, e^{\mathrm{i}\bm{n}\cdot\bm{A}^{\,}_{1}}
\end{align}
that solves the eigenvalue problem
(\ref{eval2}).
Third, condition \eqref{eq: Ansatz for d of n b}
implies that the set of vectors $\{\bm{n}\}$ forms a
lattice with basis vectors $\{\bm{k}^{\,}_{\mathtt{i}}\}$. The number
of inequivalent points in the lattice is therefore given by
\begin{align}
r\:=
\abs{
\det
\begin{pmatrix}
\bm{k}^{\,}_{1}&\ldots&\bm{k}^{\,}_{N}
\end{pmatrix}
}=
\abs{\det\,\varkappa^{\mathsf{T}}}=
|\det\,\varkappa|.
\end{align}
This means that we can decompose any $\bm{n}$ as
\begin{align}
\bm{n}=
\bm{v}^{\,}_{m}
+
p^{\,}_{\mathtt{i}}\,
\bm{k}^{\,}_{\mathtt{i}},
\end{align}
where $p^{\,}_{\mathtt{i}}\in\mathbb{Z}$ and we have introduced $r$
linearly independent vectors $\bm{v}^{\,}_{m}$. We can therefore
rewrite
\begin{subequations}
\begin{align}
\psi(\bm{A}^{\,}_{1})=
\sum_{m=1}^{r}
\tilde{d}^{\,}_{m}\,
f^{\,}_{m}(\bm{A}^{\,}_{1}),
\end{align}
where
\begin{equation}
\tilde{d}^{\,}_{m}\:=
\tilde{d}(\bm{v}^{\,}_{m}
+
p^{\,}_{\mathtt{i}}\,\bm{k}^{\,}_{\mathtt{i}})=
\tilde{d}(\bm{v}^{\,}_{m}),
\end{equation}
and
\begin{equation}
\begin{split}
f^{\,}_{m}(\bm{A}^{\,}_{1})\:=&\,
e^{\mathrm{i}\bm{\gamma}^{\,}_{1}\cdot\bm{A}^{\,}_{1}/(2\pi)}
\\
&\,
\times
\!\!\!\!
\sum_{p^{\,}_{1},\ldots,p^{\,}_{N}}
\!\!\!\!
e^{
\mathrm{i}\bm{\gamma}^{\,}_{2}\cdot(\varkappa^{-1})^{\mathsf{T}} \(
\bm{v}^{\,}_{m}+p^{\,}_{\mathtt{i}}\,\bm{k}^{\,}_{\mathtt{i}}, \)
}\,
e^{\mathrm{i}(\bm{v}^{\,}_{m}+p^{\,}_{\mathtt{i}}\,\bm{k}^{\,}_{\mathtt{i}})\cdot\bm{A}^{\,}_{1}}.
\end{split}
\end{equation}
\end{subequations}
Since any $\psi(\bm{A}^{\,}_{1})$ in the ground-state manifold can be
written in this way, we have demonstrated that there are
$r=|\det\,\varkappa|$ linearly independent ground-state wavefunctions
$f^{\,}_{m}(\bm{A}^{\,}_{1})$ in the topological Hilbert space. In
other words, we have shown that
\begin{align}
\begin{split}
\text{GSD}&=
|\det\,\varkappa|=
\sqrt{|\det K|},
\end{split}
\label{proof gsd}
\end{align}
with $K$ defined in Eq.~\eqref{kdef}. This is precisely the result
advertised in Eq.~\eqref{gsd}. Note that because $\varkappa$ is an
integer-valued matrix, it has an integer-valued determinant.
Consequently, $\sqrt{|\det K|}=|\det\varkappa|$ is an integer.
\subsubsection{Generalization to manifolds with multiple holes}
It is instructive to consider generalizing these arguments to the case
of a system with the topology of an $N^{\,}_{\mathrm{h}}$-punctured
disk. In this generalization, the boundary can be viewed as the
disjoint union of $N^{\,}_{\mathrm{h}}+1$ copies of $S^{1}$. Since
each of these edges is gapped, anomaly cancellation enforces
independent gluing conditions for each copy of $S^{1}$. In principle,
a different matrix $T$ could be chosen for each boundary. This could
happen if, for example, different edges are gapped by different sets
of tunneling vectors $\bm{T}$ that enter Eq.~\eqref{tunneling}. If
this is the case, then it may not be possible to find a linear
transformation of the form \eqref{change basis} such that $N$ of the
CS fields obey Dirichlet boundary conditions on all edges, as in
Eq.~\eqref{dirichlet}. The remainder of the argument presented here
for counting the degeneracy then breaks down. Finding an alternative
argument that applies in these cases is an interesting problem for
future work, but is beyond the scope of this paper.
In the case where all boundaries are gapped in the same way, however,
one obtains a set of Wilson loops like those in Eqs.~\eqref{wilson}
for each hole. [See, e.g., Eqs.~\eqref{wilson2} in the next section.]
Since these sets of Wilson loops are completely independent, one
obtains a degeneracy of size $|\det\,K|^{N^{\,}_{\mathrm{h}}/2}$.
\section{Applications}
\begin{figure}[t]
(a)\includegraphics[width=.3\textwidth]{holes}\\
\vspace{1cm}
(b)\includegraphics[width=.45\textwidth]{loops}
\caption{(Color online) A punctured TRS-FTL with gapped edges. (a)
Schematic representation of an ``artificial'' spin-like system.
In the limit $D\gg d,R$, each hole (white square) carries with it a
$q$-fold topological degeneracy that is split exponentially by
tunneling processes that encircle (red lines) or connect (green
lines) the holes. (b) Wilson loops defined in Eqs.~\eqref{wilson2}.
The dashed line represents the product of the two Wilson loops above
it, which connects the two holes.
\label{fig: holes}
}
\end{figure}
With the results of Sec.\ \ref{sec: The topological degeneracy}
in hand, we now explore some
of the consequences of Eq.~\eqref{gsd}. We begin by examining the
fate of the topological degeneracy in finite-sized systems, before
considering the possibility of using calorimetry to detect
experimental signatures of the degeneracy. We close the section by
re-evaluating the proposed\ \cite{hansson} topological field theory for
the $s$-wave BCS superconductor in light of the results of this paper.
\subsection{Finite systems: clock models and beyond}\label{parafermions}
On closed manifolds, the topological degeneracy is exact only in the
limit of infinite system size. This is a result of the fact, pointed
out by Wen and Niu,\ \cite{wen90} that quasiparticle tunneling events
over distances of the order of the system size lift the
topological degeneracy by a splitting
that is exponentially small in the linear size of the system. This
observation was also confirmed numerically for the case of the
(2+1)-dimensional Abelian Higgs model on the torus by Vestergren et
al.\ in Refs.~\onlinecite{hansson_splitting} and
\onlinecite{vestergren_prb}. A similar splitting occurs for manifolds
with boundary, like those studied in this work. For a planar system
with many holes, each of which carries a $q$-fold degeneracy (where
$q\:=\sqrt{|\det\, K|}$) in the limit of infinite system size, there
are two kinds of tunneling events that can lift the degeneracy. These
are (1) tunnelings that encircle a single hole and (2) tunnelings
between boundaries. Below we argue that, in a finite-sized system
with $N^{\,}_{\mathrm{h}}$ holes, the array of $N^{\,}_{\mathrm{h}}$
coupled $q$-state degrees of freedom can be modeled as a spin-like
system [see Fig.~\ref{fig: holes}(a)].
To see how this arises, we first note that for a system with
$N^{\,}_{\mathrm{h}}$
holes it is possible to define a set of Wilson loops for each hole.
Analogously to Eqs.~\eqref{wilson},
for any $\mathtt{i}=1,\ldots,N$ we define
\begin{subequations}
\label{wilson2}
\begin{align}
&
W^{\mathtt{i}}_{1,j}\:=
\exp
\Bigg(
\mathrm{i}
\int\limits_{\mathcal{C}^{\,}_{1,j}}
\mathrm{d}\bm\ell\cdot\bar{\bm{a}}^{\mathtt{i}}_{-}
(\bm{x},t)
\Bigg),
\\
&
W^{\mathtt{i}}_{2,j}\:=
\exp
\Bigg(
\mathrm{i}
\oint\limits_{\mathcal{C}^{\,}_{2,j}}
\mathrm{d}\bm\ell\cdot\bar{\bm{a}}^{\mathtt{i}}_{+}
(\bm{x},t)
\Bigg),
\end{align}
where the open curve $\mathcal{C}^{\,}_{1,j}$ connects the $j$-th hole to
the outer boundary, and the closed curve $\mathcal{C}^{\,}_{2,j}$ encircles
the $j$-th hole [see Fig.~\eqref{fig: holes}(b)]. Each set of operators
obeys an independent copy of the algebra \eqref{bfalg1}. Furthermore,
for any pair of holes $j$ and $k$, the Wilson loop
\begin{equation}
W^{\mathtt{i}}_{1,jk}\:=
W^{\mathtt{i}\,\dag}_{1,j}\,
W^{\mathtt{i}}_{1,k}
\end{equation}
\end{subequations}
connects these holes. More generally, any number of holes can be
connected by compositions of the Wilson loops defined in Eqs.\
(\ref{wilson2}).
In an infinite system,
the topological protection of the degeneracy
(\ref{proof gsd})
arises because the Wilson loops defined in Eqs.~\eqref{wilson}
are nonlocal operators and are
therefore forbidden from entering the Hamiltonian. In a finite
system, however, the Wilson loops are no longer nonlocal degrees of
freedom and can therefore enter the effective theory. In principle,
all powers and combinations of the Wilson loops are allowed to enter
the effective Hamiltonian
\begin{align}\label{heff}
H^{\,}_{\mathrm{eff}}\:=&\,
\sum_{{\mathtt{i}}=1}^{N}
\sum_{j=1}^{N^{\,}_{\mathrm{h}}}
\Bigg(
h^{\mathtt{i}}_{1,j}\,
W^{\mathtt{i}}_{1,j}
+
h^{\mathtt{i}}_{2,j}\,
W^{\mathtt{i}}_{2,j}
+
\sum_{k=1}^{N^{\,}_{\mathrm{h}}}
J^{\mathtt{i}}_{jk}\,
W^{\mathtt{i}}_{1,jk}
\nonumber\\
&\,
+
\ldots
\Bigg),
\end{align}
where the omitted terms include higher powers of the Wilson loops as
well as all necessary Hermitian conjugates. In practice, however, all
couplings in $H^{\,}_{\mathrm{eff}}$ are exponentially small in the
shortest available length scale, which limits the tunneling rates.
For example, $J^{\mathtt{i}}_{jk}\propto e^{-c\,d_{jk}/\xi}$, where
$c$ is a constant of order one, $d_{jk}$ is the distance between holes
$j$ and $k$ [see Fig.~\ref{fig: holes}(a)], and $\xi$ is a length scale
associated with quasiparticle tunneling.\ \cite{footnote2}
It is interesting to note that the Hamiltonian $H^{\,}_{\mathrm{eff}}$
admits a certain amount of external control -- the holes can be arranged in
arbitrary ways, and the magnitudes of the couplings can be tuned by
changing the length scales $R$, $d^{\,}_{jk}$, and $D$
defined in Fig.\ \ref{fig: holes}.
In particular,
many terms in $H^{\,}_{\mathrm{eff}}$ can be tuned to zero by varying these
length scales. We will make use of this freedom below.
To illustrate in what sense the effective Hamiltonian \eqref{heff} can be
thought of as a spin-like system,
we consider a specific class of examples.
In particular, we consider the family of TRS-FTLs defined by
\begin{equation}
K\:=
\begin{pmatrix}q&0\\ 0&-q\end{pmatrix},
\qquad
\bm{Q}\:=
\begin{pmatrix}2\\2 \end{pmatrix},
\label{class}
\end{equation}
where $q$ is an even integer. One verifies using
Eq.~\eqref{tcriteria} that a single tunneling term of the form
\eqref{tunneling} with $\bm{T} = (1,-1)^{\mathsf{T}}$ is sufficient to
gap out the counterpropagating edge modes without breaking TRS as
defined in Ref.~\onlinecite{neupert}. (The gluing conditions
\eqref{gluing} can be implemented by the $1\times 1$ gluing ``matrix"
$T=1$.) In this case, Eq.~\eqref{gsd} predicts a $q$-fold degeneracy
per hole. To obtain the explicit effective Hamiltonian, we define
\begin{subequations}
\begin{align}
\sigma^{\,}_{j}\:=
W^{\,}_{1,j},
\qquad
\tau^{\,}_{j}\:=
W^{\,}_{2,j},
\end{align}
whose only nonvanishing commutation relations arise from the algebra
[recall Eq.~(\ref{eq: algebra between W1 and W2})]
\begin{align}
\sigma^{\,}_{j}\,
\tau^{\,}_{j}=
e^{-2\pi\mathrm{i}/q}\,
\tau^{\,}_{j}\,
\sigma^{\,}_{j}.
\end{align}
\end{subequations}
One can check by writing down explicit representations of $\sigma^{\,}_{j}$
and $\tau^{\,}_{j}$ that they also satisfy
\begin{equation}
\sigma^{q}_{j}=\tau^{q}_{j}=\mathbbm{1}.
\end{equation}
For example, in the case $q=2$ we may use Pauli matrices,
e.g.,
\begin{equation}
\sigma^{\,}_{j}=\sigma^{\,}_{z},
\qquad
\tau^{\,}_{j}=\sigma^{\,}_{x},
\end{equation}
and in the case $q=4$
we may use
\begin{subequations}
\begin{align}
&
\sigma^{\,}_{j}\:=
\text{diag}
\left(
1,
\,e^{-\mathrm{i}\,\pi/2},
\,e^{-\mathrm{i}\,\pi},
\,e^{-\mathrm{i}\,3\pi/2}
\right),
\\
&
\tau^{\,}_{j}\:=
\begin{pmatrix}
0&0&0&1
\\
1&0&0&0
\\
0&1&0&0
\\
0&0&1&0
\end{pmatrix}.
\end{align}
\end{subequations}
For a system with $N^{\,}_{\mathrm{h}}$ holes of size $R$ arranged in
a one-dimensional chain with lattice spacing $d$, the effective
Hamiltonian in the limit $D\gg d,R$ (with $D,d,R$ defined in
Fig.\ \ref{fig: holes}) becomes that of a one-dimensional
$\mathbb{Z}^{\,}_{q}$ quantum clock model (see
Ref.~\onlinecite{fendley} and references therein),
\begin{equation}
H^{\,}_{\mathrm{eff}}\:=
\sum_{i=1}^{N^{\,}_{\mathrm{h}}-1}
J^{\,}_{i}\,
\left(
\sigma^{\dag}_{i}\,
\sigma^{\,}_{i+1}
+
\text{H.c.}
\right)
+
\sum_{i=1}^{N^{\,}_{\mathrm{h}}}
h^{\,}_{i}
\left(
\tau^{\,}_{i}
+
\text{H.c.}
\right),
\label{clock}
\end{equation}
where
$J^{\,}_{i}\propto e^{-c^{\,}_{1}\, d/\xi}$
and
$h^{\,}_{i}\propto e^{-c^{\,}_{2}\, R/\xi}$,
with the real constants $c^{\,}_{1}$ and $c^{\,}_{2}$ of order
unity.
For simplicity, we have constrained the
couplings $J^{\,}_{i}$ and $h^{\,}_{i}$ to be real, although their magnitude and
sign is allowed to vary from hole to hole (hence the subscripts $i$).
Note that in the above Hamiltonian, terms linear in $\sigma^{\,}_{j}$ do not
appear, as the associated couplings are suppressed by factors of order
$e^{-c^{\,}_{3}\,D/\xi}\ll e^{-c^{\,}_{1}\,d/\xi},\ e^{-c^{\,}_{2}\,R/\xi}$.
Similarly, longer-range two-body terms, as
well as higher powers of the $\sigma^{\,}_{j}$ and $\tau^{\,}_{j}$, are also
omitted, as they correspond to higher-order tunneling processes.
The Hamiltonian of the clock
model \eqref{clock} is invariant under the symmetry operation
\begin{subequations}
\begin{equation}
H^{\,}_{\mathrm{eff}}\mapsto
\mathcal{S}\,
H^{\,}_{\mathrm{eff}}\,
\mathcal{S}^{-1}
\end{equation}
generated by
\begin{equation}
\mathcal{S}\:=
\prod_{i=1}^{N^{\,}_{\mathrm{h}}}\tau^{\dag}_{i}.
\end{equation}
\end{subequations}
Indeed, under conjugation by
$\mathcal{S}$,
$\tau^{\dag}_{j}\mapsto\tau^{\dag}_{j}$ and
$\sigma^{\dag}_{j}\mapsto e^{-2\pi\mathrm{i}/q}\sigma^{\dag}_{j}$
for all $j$. This $\mathbbm{Z}^{\,}_{q}$
symmetry can be thought of as a remnant of the
$q^{N^{\,}_{\mathrm{h}}}$-fold topological degeneracy of the TRS-FTL,
which would be present in the limit $d,R,D\to\infty$.
Before closing this section, we point out that quantum clock models
like the one discussed in this section have arisen in various contexts
elsewhere in the recent literature, especially in quantum Hall systems
with defects. \cite{barkeshli_oreg_qi,clarke,vaeziPRX,mongPRX}
\subsection{Probing the topological degeneracy with calorimetry}
In this section, we consider experimental avenues to
detect the topological degeneracy of a punctured TRS-FTL. We focus
our attention on calorimetry as a possible probe. In a sample with
$N^{\,}_{\mathrm{h}}$ holes, the ground state degeneracy provides a contribution
$S^{\,}_{\mathrm{GSD}}=N^{\,}_{\mathrm{h}}\,k^{\,}_{\mathrm{B}}\ln q$,
where $k^{\,}_{\mathrm{B}}$ is the Boltzmann constant and
$q=\sqrt{\det K}$, to
the total entropy $S^{\,}_{\mathrm{tot}}$.
If the areal density of holes is kept fixed, then
for a sample of length $L$, we have
$S^{\,}_{\mathrm{GSD}}\sim L^{2}$ for the topological contribution,
which is extensive. This suggests that, were a suitable material to be
discovered, one might be able to detect the topological degeneracy of
a punctured TRS-FTL by measuring its heat capacity. Such a
measurement is feasible with current technology, as membrane-based
nanocalorimeters enable the determination of heat capacities $C^{\,}_{V}$
in microgram samples (and smaller), to an accuracy of
$\delta C^{\,}_{V}/C^{\,}_{V}\sim10^{-4}$--$10^{-5}$
down to temperatures of order
$100$ mK.%
\ \cite{garden_review,ong,tagliati1,tagliati2}
We first determine the topological contribution to the heat capacity
for some particular examples. To do this, we return to the class of
TRS-FTLs defined in Eq.~\eqref{class}.
The heat capacity in this case is easiest to determine from the clock
model of Eq.~\eqref{clock} in the paramagnetic limit $J^{\,}_{i}\to 0$, which
is achieved for $d\gg R$ [see Fig.~\ref{fig: holes}(a)]. Setting $h^{\,}_{i}=h$
for convenience, we see that the clock model can be rewritten,
after a change of basis, as
\begin{equation}
H^{\,}_{\mathrm{eff}}=
h\sum_{i=1}^{N^{\,}_{\mathrm{h}}}
\left(
\sigma^{\,}_{i}
+
\sigma^{\dag}_{i}
\right)=
2h
\sum_{i=1}^{N^{\,}_{\mathrm{h}}}
\cos\(\frac{2\pi}{q}\,n^{\,}_{i}\),
\end{equation}
where $n^{\,}_{i}=0,\ldots,q-1$.
Consequently the partition function is given by
\begin{align}
Z=
\left(
\sum_{n=0}^{q-1}
e^{-2\beta\,h\,\cos(2\pi\,n/q)}
\right)^{N^{\,}_{\mathrm{h}}},
\end{align}
where $\beta\:=1/(k^{\,}_{\mathrm{B}}\,T)$ and $T$ is the temperature.
The topological heat capacity at constant volume, $C^{\mathrm{top}}_{V}$,
is then determined from
the partition function by standard methods. For example,
\begin{equation}
C^{\mathrm{top}}_{V}=
N^{\,}_{\mathrm{h}}\,
\frac{h^{2}}{k^{\,}_{\mathrm{B}}\,T^{2}}
\times
\begin{cases}
4\,
\text{sech}^{2}\(\frac{2\,h}{k^{\,}_{\mathrm{B}}\,T}\),
&
q=2,
\\
2\,
\text{sech}^{2}\(\frac{h}{k^{\,}_{\mathrm{B}}\,T}\),
&
q=4,
\\
\frac{
9\,
\cosh\left(\frac{h}{k^{\,}_{\mathrm{B}}\,T}\right)
+
\cosh\left(\frac{3\,h}{k^{\,}_{\mathrm{B}}\,T}\right)
+
8
}
{
\left[2\,\cosh\left(\frac{h}{k^{\,}_{\mathrm{B}}\,T}\right)
+
\cosh\left(\frac{2\,h}{k^{\,}_{\mathrm{B}}\,T}\right)\right]^{2}
},
&
q=6,
\end{cases}
\end{equation}
and so on.
To date, there has been no experimental realization of a TRS-FTL or
fractional topological insulator. Since background contributions to
the heat capacity are material-dependent, it is difficult to provide a
precise estimate of the observable effect. However, we can
nevertheless identify some constraints on the possible materials that
would favor such a measurement.
To do this, let us estimate the various background contributions
to the heat capacity of a TRS-FTL. First, we note that,
because any TRS-FTL must have a gap $\Delta$, the electronic contribution
$C^{\mathrm{el}}_{V}$ to the heat capacity is
\begin{subequations}
\begin{align}
C^{\mathrm{el}}_{V}\propto
\frac{\Delta}{T}\,
e^{-\eta\,\Delta/(k^{\,}_{\mathrm{B}}\,T)},
\end{align}
where $\eta$ is a constant of order one. The exponential suppression
of $C^{\mathrm{el}}_{V}$ implies that this contribution is always negligible
at sufficiently small temperatures.
However, one must also consider
the phononic contribution, which follows a Debye power law at low
temperatures. This contribution scales with the sample \textit{volume},
which could be three-dimensional if the TRS-FTL is formed in a
heterostructure, as is the case in quantum Hall systems. This fact,
which was noted in Ref.~\onlinecite{cooper}, poses the greatest
challenge to detecting the topological contribution to the heat
capacity, which scales with the \textit{area} of the two-dimensional
sample. In principle, however, one may assume that the TRS-FTL lives
in a strictly two-dimensional sample, or at least in a thin film. In
this case, we have that the phononic contribution $C^{\mathrm{ph}}_{V}$
to the heat capacity is
\begin{equation}
C^{\mathrm{ph}}_{V}\propto
k^{\,}_{\mathrm{B}}\, (T/T^{\,}_{\mathrm{D}})^{2},
\end{equation}
where $T^{\,}_{\mathrm{D}}$ is the Debye temperature (100 K, say).%
\ \cite{footnote3}
We verified numerically, by simulating a square lattice of masses and
springs, that the presence or absence of holes has little effect on
the phonon spectrum as long as the holes are sufficiently small. We
therefore expect the Debye law to hold both with and without holes, as
long as one takes into account the excluded volume due to the holes.
\begin{figure}[t]
\hspace{-.35
cm}
\includegraphics[width=.5\textwidth]{heat_capacity}
\caption{(Color online)
Total heat capacity for a monolayer TRS-FTL
with $N^{\,}_{\mathrm{a}}=10^{14}$.
The topological contribution is shown
(above background) for $q=2,4,$ and $6$.
The parameters used for the topological contribution were
$\nu=5\times 10^{-6}$ ($\sim 22000^{2}$ holes) and
$h/k^{\,}_{\mathrm{B}}\approx 0.321$ K, which leads to a maximum
excess (for $q=6$) of $\sim 30\%$ over the background (blue curve)
near $T=0.1$ K.
\label{heat_capacity}
}
\end{figure}
The total heat capacity is obtained by adding the three contributions:
\begin{align}
C^{\,}_{V}(T)=
N^{\,}_{\mathrm{a}}\,
\left[
C^{\mathrm{top}}_{V}(T)
+
\nu\,
C^{\mathrm{ph}}_{V}(T)
+
\frac{1}{N^{\,}_{\mathrm{a}}}\,
C^{\mathrm{el}}_{V}(T)
\right],
\end{align}
\end{subequations}
where $N^{\,}_{\mathrm a}$ is the number of atoms in the sample and
$\nu\:=N^{\,}_{\mathrm{h}}/N^{\,}_{\mathrm{a}}$ determines the number of holes.
The above formula
leads to the estimate of the specific heat curve presented in
Fig.~\ref{heat_capacity}. A square array of $22000$ holes on a side
produces an excess of up to $30\%$ (for $q=6$) on top of the
background at $T=0.1$ K, which is well above the experimental error
$\delta C^{\,}_{V}/C^{\,}_{V}\sim 10^{-4}$.
We now comment on possible difficulties with this measurement.
Perhaps the most important of these is the fact that the energy scales
$J^{\,}_{i}$ and $h^{\,}_{i}$ entering Eq.~\eqref{clock} are unknown. It may be
possible to circumvent this issue by exploiting the exponential
sensitivity of the couplings to the length scales $R$ and $d$. For
example, one could prepare samples with $d\gg R$ to eliminate the first
term in Eq.~\eqref{clock}, and compare results for different values of
$R$ to determine whether it is possible to resolve the effect. As
long as $h\gtrsim 0.1\,\Delta$, it should be possible to tune $R$ such
that the effect is visible.
The presence of disorder in the sample is another potential source of
difficulty, as localized states due to disorder can also contribute to the entropy.
However, intuition from noninteracting systems, where these states
provide a logarithmic correction to the entropy,\cite{footnote_disorder}
suggests that this contribution would be subleading as compared to the
power-law contribution $S_{\rm GSD}\sim L^2$ that we predict for a
fixed areal density of holes.
\subsection{Are superconductors topologically ordered?}
In an insightful paper, it was argued by Hansson et al.\
in Ref.~\onlinecite{hansson}
that ordinary $s$-wave BCS superconductors are topologically ordered.
In fact, it was shown that, when the electromagnetic gauge field is
treated dynamically and confined to (2+1) dimensional space and time,
the superconductor admits a description in terms of a BF theory like the
one defined in Eqs.~\eqref{bfdef}, with
\begin{align}
\tilde{K}=
\begin{pmatrix}
0
&
2
\\
2
&
0
\end{pmatrix}.
\label{sc}
\end{align}
Furthermore, it was shown that the edge states that arise when
the above theory is defined in a finite planar geometry are generically
gapped by Cooper pair creation terms.
The proposed theory is consistent with the time-reversal symmetry of the
$s$-wave superconductor and captures the statistical phase of $\pi$
that is acquired by an electron upon encircling a vortex. This
effective theory, which is the same as that of the $\mathbbm{Z}^{\,}_{2}$
lattice gauge theory in its deconfined phase, predicts a four-fold GSD
on the torus, whose exponential splitting in finite systems was
verified numerically in Refs.~\onlinecite{hansson_splitting} and
\onlinecite{vestergren_prb}.
Since the theory defined by Eq.~\eqref{sc} falls squarely within the
class of theories studied in this paper, it is tempting to draw the
conclusion that the $s$-wave superconductor exhibits a two-fold GSD on
the annulus. Below we argue that, while this is indeed the case, the
degeneracy is not exponential but power-law in nature, and therefore
is not what one might call a topological degeneracy in the canonical
sense of Refs.~\onlinecite{wen89}--\onlinecite{wen91}. The reason for this
is that the topological nature of the superconductor results from the dynamics
of the electromagnetic gauge field, which, in a real planar superconductor,
is not confined to the sample itself, but rather extends through all
three spatial dimensions.
Consequently, the true electromagnetic gauge field that is present in the
superconductor can be measured by local external probes.
\begin{figure}[t]
\includegraphics[width=.33\textwidth]{sc_flux}
\caption{(Color online) Trapping a flux quantum inside a
superconducting ring. Confining the flux inside the ring costs no
energy for the electrons inside the superconductor, but there is an
electromagnetic energy cost obtained by integrating the enclosed
magnetic field intensity over the interior of the dashed cylinder,
which we denote $\mathcal V$.
\label{flux}
}
\end{figure}
To see how this coupling to the environment lifts the degeneracy in a
power-law fashion, let us consider the origin of the two-fold
degeneracy. Recall that for an annular superconductor (a thin-film
mesoscopic ring, for example), the phase of the superconducting order
parameter winds by $2\pi$ around the hole if a flux quantum
$\phi^{\,}_{0}=h/2e$ is trapped inside. This indicates that the electronic
spectrum of the superconductor cannot be used to distinguish between
cases where an even ($\phi=0$ mod $\phi^{\,}_{0}$) or odd ($\phi=1$ mod
$\phi^{\,}_{0}$) number of flux quanta penetrate the hole. This is precisely
the origin of the degeneracy. However, because the electromagnetic field
also exists outside the sample, there is an additional
electromagnetic energy cost associated with having a flux quantum
trapped in the hole. If we assume for simplicity that the flux is
distributed uniformly over the hole (radius $R$) and does not
penetrate into the superconductor, then the energy cost is
proportional to
\begin{equation}
\int\limits_{\mathcal{V}}
\mathrm{d}^{3}r\,
|\bm{B}|^{2}=
\frac{\phi^{2}_{0}}{2\pi\,R^{2}}\,
L^{\,}_{z},
\end{equation}
where $\mathcal{V}$ is the interior of the cylinder in
Fig.~\ref{flux}, and $L^{\,}_{z}$ is the height of the cylinder.
Strictly speaking, because the magnetic field lines must close outside
the annulus, one needs to replace $L^{\,}_{z}$ by a length scale bounded from
below by the outer radius of the annulus. This energy cost vanishes
as $1/R$ for $R,L^{\,}_{z}\to\infty$, which means that the ground
state degeneracy is lifted as a power law, rather than exponentially.
The reason underlying this power-law splitting is the fact that the
electromagnetic gauge field is not an emergent gauge field in the same
sense as the Chern-Simons fields that are present in, say, a
fractional topological insulator with gapped edges. To elaborate on
this distinction, we first recall that the topological degeneracy
derived in Ref.~\onlinecite{hansson} arises from a dynamical treatment
of the electromagnetic gauge field in (2+1)-dimensional space and
time. The topological sectors in which this degeneracy is encoded
reside in the Hilbert space of the electromagnetic gauge field, which
is in turn entangled with the Hilbert space of the electronic degrees
of freedom. Since the photonic degrees of freedom in a real annular
superconductor also exist outside the sample, there is nothing to
prevent the environment from fixing a topological sector. For
example, the presence of an external magnetic field in the hole can
privilege one topological sector over the other by fixing the flux
through the hole.
It is crucial to contrast this with the case of a ``true" TRS-FTL,
where the Chern-Simons fields arise naturally from electron-electron
interactions. In this case, the topological sectors reside in the
Hilbert space of the electrons alone, and the CS fields do not exist
outside the sample. Inserting an electromagnetic flux through the
hole of an annular TRS-FTL switches between topological sectors, but
does not betray any information about the identity of the initial or
final sector. For this reason, the degeneracy of different
topological sectors is completely protected from the environment in
the limit of infinite system size.
\section{Summary and conclusion}
In this paper we have derived a formula for the topological ground
state degeneracy of a time-reversal symmetric, multi-component,
Abelian Chern-Simons theory. The formula, which holds when the edge
states of the theory are gapped by appropriate perturbations, says
that the GSD of the system on a planar surface with
$N^{\,}_{\mathrm{h}}$ holes is given by
$|\det\,K|^{N^{\,}_{\mathrm{h}}/2}$, where $K$ is the $K$-matrix. We
then examined the situation where this topological degeneracy is split
exponentially by finite-size effects, and found that the set of
$N^{\,}_{\mathrm{h}}$ holes admits a description in terms of an
effective spin-like system whose couplings can be tuned by varying the
sizes and arrangement of the holes. We also considered calorimetry as
a possible means of detecting the topological degeneracy. The
proposed experiment would measure the contribution of the topological
degeneracy to the heat capacity at low temperatures, which we argued
could be visible on top of the expected electronic and phononic
backgrounds as long as the host material is sufficiently thin.
Finally, in light of these results, we revisited the notion that
ordinary $s$-wave superconductors are topologically ordered. We
argued that, while thin-film superconductors do indeed possess a
ground state degeneracy on punctured planar surfaces, this degeneracy
is lifted in a power-law, rather than an exponential, fashion due to
the (3+1)-dimensional nature of the electromagnetic gauge field.
We close by pointing out several possible extensions of this work.
First, we believe that the correspondence suggested in this paper
between gluing conditions \eqref{gluing} and gapped edges of TRS-FTLs
would benefit from further study. Sharpening this correspondence
could provide a viewpoint on fractionalized phases with gapped edges
that is complementary to the classification of such edges in terms of
Lagrangian subgroups.\
\cite{footnote-lagrangian-subgroup,Barkeshli13a,Barkeshli13b,levinPRX}
Second, we
note that our results concerning the ground state degeneracy may still
apply to TRS-FTLs where the backscattering terms of
Eq.~\eqref{tunneling} \textit{do not} respect time-reversal symmetry.
One could therefore also consider extending the results of this paper
to fractional topological insulators whose protected edge modes are
gapped by perturbations that break TRS, as is done in
Refs.~\onlinecite{lindner} and \onlinecite{motruk}. Third, it would
be interesting to determine what other kinds of ``artificial''
spin-like systems could be realized in TRS-FTLs with more complicated
$K$-matrices than those in the class of Eq.~\eqref{class}. It is
conceivable that remnants of the topological degeneracy may manifest
themselves as exotic properties of these less conventional models.
Finally, we must point out that a fractionalized two-dimensional state
of matter with time-reversal symmetry has not yet been discovered
experimentally, and that the search for such a state must remain a
priority.
\section*{Acknowledgments}
We are grateful to Kurt Clausen, Eduardo Fradkin, Hans Hansson,
Shivaji Sondhi, Chenjie Wang, and Frank Wilczek for enlightening discussions.
Upon completion of this work, we were made aware by Shinsei Ryu
of Ref.\ \onlinecite{wang-wen-archives12}, in which related results
were obtained.
T.I. was supported by the National Science Foundation Graduate
Research Fellowship Program under Grant No.~DGE-1247312. T.N. was
supported by DARPA SPAWARSYSCEN Pacific N66001-11-1-4110, and C.C. was
supported by DOE Grant DEF-06ER46316. We also acknowledge support from
the Condensed Matter Theory Visitors' Program at Boston University.
|
1,314,259,992,978 | arxiv | \section{Introduction}
\label{sec:intro}
The subject of this paper is uniqueness and stability
results for properly defined entropy solutions of
mixed hyperbolic-parabolic quasilinear equations appended with
a nonlocal (fractional) diffusion operator.
These equations take the form
\begin{equation}\label{eq1}
\partial_t u + \mathrm{div} f(u) = \mathrm{div} (a(u) \nabla u)
+ \ensuremath{\mathcal{L}}[u],
\end{equation}
where $u=u(t,x)$ is the unknown, $(t,x) \in
Q_T := (0,T) \times {{\mathbb{R}}}^d$, $d\ge 1$, and $T>0$ is a
fixed final time. The operator $\ensuremath{\mathcal{L}}$ is the generator
of a symmetric positivity preserving
pure jump L\'evy semigroup $e^{t\ensuremath{\mathcal{L}}}$ on $L^1({{\mathbb{R}}}^d)$.
Equation \eqref{eq1} is subject to initial data
\begin{equation}\label{in}
u(0,x)= u_0(x) \in (L^1 \cap L^{\infty})({{\mathbb{R}}}^d).
\end{equation}
In \eqref{eq1},
\begin{equation}\label{fregul}
f= (f_1,\ldots, f_d) \in W^{1,\infty}({{\mathbb{R}}};{{\mathbb{R}}}^d)
\end{equation}
is a given vector-valued flux function, $a =(a_{ij})\ge 0$ is a
given symmetric matrix-valued diffusion function of the form
\begin{equation}\label{dif}
a= \sigma^a (\sigma^a)^{\mathrm{tr}}, \qquad
\sigma^a \in {{\mathbb{R}}}^{d \times K},
\quad 1 \leq K \leq d.
\end{equation}
More precisely, the components of $a$ are
$a_{ij} = \sum_{k=1}^K \sigma_{ik}^a \sigma_{jk}^a$ for $i,j= 1,\ldots,d$.
We assume that the matrix-valued function
$\sigma^a = (\sigma_{ik}^a):{{\mathbb{R}}} \to {{\mathbb{R}}}^{d \times K}$ satisfies
\begin{equation}\label{sigreg}
\sigma^a\in W^{1,\infty}({{\mathbb{R}}};{{\mathbb{R}}}^{d\times K}).
\end{equation}
Observe that we do not assume the matrix $a(\cdot)$ to be
strictly positive definite, so the operator
$\mathrm{div} (a(u) \nabla u)$ may be strongly degenerate, and
hence the phrase ``mixed hyperbolic-parabolic" is justified.
In terms of its singular integral representation, the nonlocal
operator $\ensuremath{\mathcal{L}}$ in \eqref{eq1} takes the form
\begin{equation} \label{nlocsplit}
\ensuremath{\mathcal{L}}[u](t,x) = \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\left[ u(t,x+ z) - u(t,x) - z \cdot \nabla u\, \mathbf{1}_{|z|<1}
\right]\, \pi(dz),
\end{equation}
where the singular L\'evy measure $\pi(dz)$ is a positive, $\sigma$-finite Borel measure
on ${{\mathbb{R}}}^d \setminus \{0\}$ satisfying $\pi(\{0\}) = 0$, $\pi(d(-z))=-\pi(dz)$, and
\begin{equation}\label{intcond1}
\int_{{{\mathbb{R}}}^d \setminus \{0\}} \left( \abs{z}^2 \mathbf{1}_{\abs{z}<1}
+ \abs{z} \mathbf{1}_{\abs{z}\ge 1}\right)\, \pi(dz) < \infty,
\end{equation}
where we note that $z$ can be replaced by a certain
regular jump function $j(z)$ easily throughout the analysis.
A typical example is provided by taking
$$
\pi(z)=\frac{1}{|z|^{d+\alpha}}\,
\mathbf{1}_{|z|<1} \, dz,\qquad \alpha \in (0,2).
$$
This example corresponds to the fractional
Laplacian $\Delta_\alpha := -(-\Delta)^{\frac{\alpha}{2}}$
on ${{\mathbb{R}}}^d$, which can also be defined in terms of the Fourier transform as
$$
\widehat{\Delta_\alpha v}(\omega)= \abs{\omega}^{\alpha}
\widehat{v}(\omega), \qquad \omega\in {{\mathbb{R}}}^d.
$$
This definition is employed in \cite{Droniou:2006os} to prove (\ref{nlocsplit}) in this case.
Nonlocal operators like $\Delta_\alpha$ are examples of a
pseudodifferential operator $\mathcal{P}$ with a symbol $a(\omega)\ge 0$ such that
$\widehat{\mathcal{P}v}(\omega)=a(\omega) \widehat{v}(\omega)$.
The function $e^{-ta(\omega)}$ is positive definite, and thus, by the
L\'evy-Khintchine formula, it can be represented as
$$
a(\omega)=i b\cdot \omega + q(\omega)
+ \int_{{{\mathbb{R}}}^d \setminus \{0\}} \left(1-e^{-i z\cdot\omega}
- i z\cdot \omega\, \mathbf{1}_{\abs{z}<1}(z)\right)\,\pi(dz),
$$
where $b\in {{\mathbb{R}}}^d$ represents the drift term, $q(\omega)=\sum_{i,j=1}^d q_{ij} \omega_i \omega_j$
is a positive definite quadratic function representing the pure diffusion
part ($q(\omega)=\abs{\omega}^2$ gives raise to the usual Laplacian $-\Delta$),
and the L\'evy measure $\pi(dz)$ accounts for the jump (non-local) part.
In our setting of $\ensuremath{\mathcal{L}}$, cf.~\eqref{nlocsplit}, we assume
$b\equiv 0$ and $q\equiv 0$, i.e, we are dealing with a pure jump
operator. The key point is that any pseudodifferential operator $\mathcal{P}$ is the generator
of a L\'evy process which is completely characterized in terms of the triplet $\left(b,q,\pi(dz)\right)$.
For more details about the L\'evy-Khintchine formula and L\'evy processes in general, we
refer to \cite{Bertoin:1996ud,Jacob:2001rf,Jacob:2002xp,Jacob:2005ss,Sato:1999qd}.
Integro-partial differential equations, also known as nonlocal, fractional
or L\'evy partial differential equations, appear frequently in many
different areas of research and find many applications in engineering and finance,
including nonlinear acoustics, statistical mechanics, biology, fluid flow,
pricing of financial instruments, and portfolio optimization.
Many authors have recently contributed to advancing the mathematical theory for quasilinear and
fully nonlinear partial differential equations that are supplemented with a
fractional diffusion operator arising as the generator of a L\'evy semigroup, addressing
questions like existence, uniqueness, regularity, formation of
singularities, and asymptotic behavior of solutions.
For results with reference to fully nonlinear equations, such as the Hamilton-Jacobi-Bellman equation,
and the (in this context relevant) theory of viscosity solutions, we refer to \cite{Alibaud:2007dw,Alvarez:1996lq,Arisawa:2006db,Barles:1997xj,Barles:2008lq,Caffarelli:2007hh,Caffarelli:2007qr,Caffarelli:2008bx,Garroni:2002il,Jakobsen:2005jy,Jakobsen:2006aa,Pham:1998zt,Sayah:1991jk,Sayah:1991xy,Silvestre:2006bq,Silvestre:2007qp,Soner:IntegroDif}, see also
\cite{Benth:2001tv,Benth:2002mk,Cont:2004gk} for some concrete applications to finance.
More recently, a number of authors \cite{Alibaud:2007mi,Alibaud:2007qe,Biler:1998ai,Biler:2001th,Bossy:2002eu,Brandolese:2008sp,Droniou:2006os,Karch:2008dp} have studied questions regarding
existence, uniqueness, regularity, and temporal asymptotics for
quasilinear equations, such as the fractal Burgers equation
\begin{equation}\label{eq:fractional-Burgers}
\partial_t u + \partial_x (u^2/2)=-(-\partial_{xx}^2)^{\frac{\alpha}{2}}u,
\end{equation}
and more generally multi-dimensional fractional conservation laws
\begin{equation}\label{eq:fCL}
\partial_t u + \mathrm{div} f(u) = \Delta_\alpha u,
\end{equation}
where the parameter $\alpha$ is assumed lie in the interval $(0,2)$.
Of course, the excluded case $\alpha=2$ corresponds to the already fully understood viscous
conservation law $\partial_t u + \mathrm{div} f(u)=\Delta u$, solutions of which
are always smooth in $t>0$. Regarding the less studied case $\alpha\in [1,2)$, it was recently proved
in \cite{Droniou:2003mz,Kiselev:2008jt} that solutions of the fractional
Burgers equation \eqref{eq:fractional-Burgers} are also smooth in $t>0$.
In the case $\alpha<1$ for the fractional conservation
law \eqref{eq:fCL} the order of the diffusion part
is lower than the first order hyperbolic part, so we do not expect any regularizing
effect to take place. Indeed, for the fractional Burgers
equation \eqref{eq:fractional-Burgers} with $\alpha<1$ it is proved
in \cite{Alibaud:2007qe,Kiselev:2008jt} that solutions can
develop discontinuities in finite time. Consequently, one should employ
a notion of entropy solutions for fractional conservation
laws \eqref{eq:fCL}, i.e., weak solutions satisfying an
additional entropy condition, to ensure the global-in-time well-posedness.
This is well-known for conservation
laws $\partial_t u + \mathrm{div} f(u)=0$, cf.~Kru\u{z}kov \cite{Kruzkov:1970kx}, and
the well-posedness theory of Kru\u{z}kov was recently extended to
fractional conservation laws in \cite{Alibaud:2007mi}.
In recent years the theory of Kru\u{z}kov \cite{Kruzkov:1970kx} has been extended to
quasilinear mixed hyperbolic-parabolic equations of the form
\begin{equation}\label{eq1-local}
\partial_t u + \mathrm{div} f(u) = \mathrm{div} (a(u) \nabla u),
\end{equation}
where $f$ and $a$ satisfy \eqref{fregul} and \eqref{dif}-\eqref{sigreg}, respectively. Since
the diffusion matrix $a(u)$ is not assumed to be strictly positive definite, \eqref{eq1-local}
is strongly degenerate and will in general posses discontinuous solutions.
In the isotropic case (with $a(\cdot)$ being a scalar function) the
first general uniqueness result is due to Carrillo \cite{Carrillo:1999hq}, who developed
an original extension of Kru\u{z}kov's method of doubling variables to prove
his result, cf.~\cite{Karlsen:2002bh,Karlsen:2003za,Mascia:2002dq,Michel:2003tw}
for some additional applications of his techniques. The anisotropic
case ($a(\cdot)$ being a matrix-valued function) was first treated by Chen
and Perthame \cite{Chen:2003td}, who developed a
kinetic formulation and established the uniqueness result using regularization by convolution.
An alternative proof of the result of Chen and Perthame, adapting
the device of doubling variables, was developed
in \cite{Bendahmane:2004go}, cf.~also \cite{Chen:2006oy,Chen:2005wf,Perthame:2003yq}
some other papers dealing with the anisotropic case.
The main purpose of this paper is to extend the uniqueness and
``continuous dependence on the nonlinearities" results of
\cite{Bendahmane:2004go,Chen:2006oy,Chen:2005wf,Perthame:2003yq} to
fractional degenerate parabolic equations of the form \eqref{eq1}.
We introduce the notion of entropy solutions and state
the main results in Section \ref{sec:entform} .
Sections \ref{prexis} (existence), \ref{pruniq} (uniqueness),
and \ref{prcdep} (continuous dependence on the nonlinearities
and the L\'evy measure) are devoted to the proofs of the main results.
\section{Notion of solution and main results}\label{sec:entform}
For $i =1,\ldots,d$ and $k=1,\ldots,K,$ define
$$
\zeta_{ik}^a(z) := \int_0^z \sigma_{ik}^a(\xi)\, d\xi,
\quad \zeta_{ik}^{a,\psi}(z) =
\int_0^z \psi(\xi) \sigma_{ik}^a(\xi)\, d\xi,
\quad z\in {{\mathbb{R}}},
$$
for any $\psi \in C({{\mathbb{R}}})$. Given any convex $C^2$
entropy function $\eta:{{\mathbb{R}}} \to {{\mathbb{R}}}$, we define
the corresponding entropy fluxes $q=(q_i):{{\mathbb{R}}} \to {{\mathbb{R}}}^d$
and $r=(r_{ij}):{{\mathbb{R}}} \to {{\mathbb{R}}}^{d \times d}$ by
$$
q'(z) = \eta'(z) f(z), \qquad r'(z) = \eta'(z) a(z).
$$
We refer to $(\eta, q, r)$ as an entropy-entropy flux triple.
We now introduce the entropy formulation of \eqref{eq1}-\eqref{in}.
\begin{definition}\label{def:entropy}
An entropy solution of the initial value problem \eqref{eq1}-\eqref{in}
is a measurable function $u : Q_T \to {{\mathbb{R}}}$ satisfying the following conditions:
\begin{Definitions}
\item $u \in L^{\infty}(Q_T)$, $u \in L^{\infty}(0,T;L^1({{\mathbb{R}}}^d))$,
\begin{equation}\label{l2regu}
\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)
\in L^2(Q_T), \qquad k=1,\ldots,K,
\end{equation}
and
\begin{equation}\label{frintcnd}
\iint_{Q_T}\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\left(u(t,x+z) - u(t,x) \right)^2 \,
\pi (dz) \, dx \, dt < +\infty.
\end{equation}
\item For $k= 1,\ldots, K$,
\begin{equation}\label{crule}
\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^{a,\psi}(u)
= \psi(u) \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u),
\quad \text{a.e.~in $Q_T$ and in $L^2(Q_T)$},
\end{equation}
for any $\psi \in C({{\mathbb{R}}})$.
\label{def:chainrule}
\item For any entropy-entropy flux triple $(\eta, q, r)$,
\begin{equation}\label{entineq}
\begin{split}
& \iint_{Q_T} \Bigl ( \eta(u) \partial_t \varphi
+ \sum_{i=1}^d q_i(u) \partial_{x_i} \varphi
+ \sum_{i,j=1}^d r_{ij}(u) \ensuremath{\partial_{x_ix_j}^2} \varphi \Bigr) \, dx\, dt
\\ & \qquad
+ \iint_{Q_T} \eta(u) \ensuremath{\mathcal{L}}[\varphi] \, dx\, dt
+ \int_{{{\mathbb{R}}}^d} \eta(u_0) \varphi(0,x) \, dx
\geq n^u + m^u,
\end{split}
\end{equation}
for all non-negative $\varphi \in C_c^{\infty}([0,T) \times {{\mathbb{R}}}^d)$, where
\begin{align*}
n^u & = \iint_{Q_T} \eta''(u) \sum_{k=1}^K
\Bigl(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u) \Bigr)^2 \varphi(t,x)\, dx \, dt, \\
m^u & = \iint_{Q_T} \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{\eta''}(u;z)
\left(u(t,x+z)-u(t,x)\right)^2 \varphi(t,x) \, \pi(dz)\, dx\, dt,
\end{align*}
and
$$
\overline{\eta''}(u;z)
= \int_0^1 (1-\tau) \eta''((1-\tau)u(t,x)+\tau u(t,x+z)) \, d\tau.
$$
If, in addition,
\begin{equation}\label{eq:add-cond}
\left\{
\begin{split}
& |z|\, \pi(dz) \in L^1({{\mathbb{R}}}^d \setminus \{0\}),\\
\text{or}\\
& |z|^2\, \pi(dz) \in L^1({{\mathbb{R}}}^d \setminus \{0\})
\quad \text{and} \quad u\in L^\infty(0,T;BV({{\mathbb{R}}}^d)),
\end{split}\right.
\end{equation}
then we can drop the fractional parabolic dissipation measure $m^u$ and
replace \eqref{entineq} by the simpler condition
\begin{equation}\label{entineq-simpler}
\begin{split}
& \iint_{Q_T} \Bigl ( \eta(u) \partial_t \varphi
+ \sum_{i=1}^d q_i(u) \partial_{x_i} \varphi
+ \sum_{i,j=1}^d r_{ij}(u) \ensuremath{\partial_{x_ix_j}^2} \varphi \Bigr) \, dx\, dt
\\ & \qquad
+ \iint_{Q_T} \eta(u) \ensuremath{\mathcal{L}}[\varphi] \, dx\, dt
+ \int_{{{\mathbb{R}}}^d} \eta(u_0) \varphi(0,x) \, dx\geq n^u.
\end{split}
\end{equation}
\end{Definitions}
\end{definition}
We remark that the chain rule \eqref{crule}
is automatically fulfilled when $a(\cdot)$ is a scalar or
a diagonal matrix, cf.~Chen and Perthame \cite{Chen:2003td}, and in
this case we can drop \ref{def:chainrule} from the definition.
Starting from the definition of $\ensuremath{\mathcal{L}}$ (cf.~calculations in
the upcoming sections), we can replace the term
$$
\iint_{Q_T} \eta(u) \ensuremath{\mathcal{L}}[\varphi] \, dx\, dt-m^u,
$$
occurring in \eqref{entineq} by
\begin{align*}
& \iint_{Q_T} \int_{|z|< r}\eta(u) [\varphi(t,x+z)-\varphi(t,x)
-\nabla \varphi \cdot z]\, \pi(dz) \, dx\, dt,
\\ & \quad
+ \iint_{Q_T} \int_{|z|\geq r}\eta'(u) [u(t,x+z) - u(t,x)]\, \pi(dz) \, dx\, dt,
\quad \forall r\in (0,1),
\end{align*}
This formulation of the nonlocal term is directly related
to the formulation used in \cite{Alibaud:2007mi}
for fractional conservation laws.
Our first result is the expected $L^1$ contraction
property (and thus the uniqueness) of entropy solutions.
\begin{theorem}\label{th:unique}
Suppose $f$ and $a$ satisfy \eqref{fregul}
and \eqref{dif}-\eqref{sigreg}, respectively, and
that the L\'evy measure $\pi(dz)$
satisfies \eqref{intcond1}. Then there exists an
entropy solution of \eqref{eq1}-\eqref{in}.
Let $u,v$ be two entropy solutions of \eqref{eq1} with
initial data $u|_{t=0} =u_0 \in (L^1\cap L^{\infty})({{\mathbb{R}}}^d)$,
$v|_{t=0} = v_0 \in (L^1\cap L^{\infty})({{\mathbb{R}}}^d)$.
For a.e. $t \in (0,T),$ we have
\begin{equation}\label{eq:contraction}
\int_{{{\mathbb{R}}}^d} \left(u(t,x)-v(t,x) \right)^+ \, dx
\leq \int_{{{\mathbb{R}}}^d} \left(u_0-v_0 \right)^+ \, dx.
\end{equation}
Consequently, if $u_0 \leq v_0$ a.e.~in ${{\mathbb{R}}}^d$ then $u \leq v$ a.e.~in $Q_T$, so
whenever $u_0=v_0$ a.e.~in ${{\mathbb{R}}}^d$, then $u=v$ a.e.~in $Q_T$.
Finally, if we replace \eqref{entineq} by \eqref{entineq-simpler}, then the
$L^1$ contraction principle \eqref{eq:contraction}
continues to hold provided \eqref{eq:add-cond} is effective; it
is sufficient that (say) only $v$ belongs to $L^\infty(0,T;BV({{\mathbb{R}}}^d))$ in the case
$\int |z|^2 \,\pi(dz)<\infty$.
\end{theorem}
This theorem generalizes to the ``non-local diffusion" case
the result of Chen and Perthame \cite{Chen:2003td}. The proof follows that
of Bendahmane and Karlsen \cite{Bendahmane:2004go}.
Regarding the last part of Theorem \ref{th:unique}, assuming $v_0\in BV({{\mathbb{R}}}^d)$ it follows
from \eqref{eq:contraction} that $v\in L^\infty(0,T;BV({{\mathbb{R}}}^d))$, as required.
Our second result, which is a refinement of the previous theorem, reveals
how the entropy solution $u$ depends on the L\'evy measure $\pi(dz)$, and the
nonlinear fluxes $f,a$ (i.e., it is a ``continuous dependence" estimate).
\begin{theorem}\label{th:contdep}
Suppose $f$ and $a$ satisfy \eqref{fregul} and
\eqref{dif}-\eqref{sigreg}, respectively, and
that the L\'evy measure $\pi(dz)$ satisfies \eqref{intcond1}.
Let $u \in L^{\infty}(0,T;BV({{\mathbb{R}}}^d))$ be the entropy
solution of \eqref{eq1} with $BV$ initial
data $u_0 \in (L^1\cap L^{\infty} \cap BV)({{\mathbb{R}}}^d)$
and with a L\'evy measure of the form $\pi(dz) = m(z)\, dz$
for some integrable function $m:{{\mathbb{R}}}^d \setminus \{0\}\to {{\mathbb{R}}}_+$.
Replace the data set
$$
(f,a,\pi,u_0), \quad a=\sigma^a(\sigma)^{\mathrm{tr}},
\quad \pi(dz)=m(z)\,dz
$$
by another data set
$$
(\tilde f,\tilde a,\tilde \pi(dz),v_0), \quad
\tilde a=\sigma^{\tilde a}(\sigma^{\tilde a})^{\mathrm{tr}},
\quad \tilde \pi(dz)=\tilde m(z)\,dz,
$$
where $\tilde f,\sigma^{\tilde a}, \tilde\pi, \tilde m$ satisfy the same regularity conditions
as $f, \sigma^a, \pi, m$ and moreover $v_0\in (L^1\cap L^\infty)({{\mathbb{R}}}^d)$.
Denote the corresponding entropy solution by $v$, and assume that $v\in C([0,T];L^1({{\mathbb{R}}}^d))$.
Suppose $u$ and $v$ take values in a closed interval $I \subset {{\mathbb{R}}}$.
For any $t \in (0,T)$,
\begin{equation}\label{contdest}
\begin{split}
&\norm{u(t,\cdot) - v(t,\cdot)}_{L^1({{\mathbb{R}}}^d)}
\\ &\leq \norm{u_0-v_0}_{L^1({{\mathbb{R}}}^d)}
+ C_1 t \norm{f-\tilde f}_{W^{1,\infty}(I);{{\mathbb{R}}}^d)}
+ C_2 \sqrt{t} \norm{\sigma^a-\sigma^{\tilde a}}_{L^{\infty}(I;{{\mathbb{R}}}^{d \times K})} \\
& \quad
+ C_3 \sqrt{t}\sqrt{\left(\int_{|z|<1}\abs{z}^2 \abs{m(z)-\tilde m(z)} \, dz\right)}
+ C_4 t\int_{|z|\geq 1}|z| \abs{m(z)-\tilde m(z)} \, dz,
\end{split}
\end{equation}
where the constants $C_i$, $i=1,\ldots,4$, depend on
the $L^{\infty}(0,T;BV({{\mathbb{R}}}^d))$ norm of $u$.
\end{theorem}
This theorem generalizes results in \cite{Chen:2005wf,Chen:2006oy}
to the ``fractional case".
\section{Proof of Theorem \ref{th:unique} (existence)}\label{prexis}
Although a detailed version of the existence of entropy
solutions to (\ref{eq1}) is presented in \cite{KU2}, to
motivate the entropy condition and to present a brief sketch, let
us consider the following accompanying problem
containing a uniformly parabolic operator depending
on a small parameter $\rho>0$:
\begin{equation}\label{eq1-visc}
\partial_t u_\rho + \mathrm{div} f(u_\rho) =
\mathrm{div}( a(u_\rho) \nabla u_\rho)+ \ensuremath{\mathcal{L}}[u_\rho(t,\cdot)]
+\rho \Delta u_\rho.
\end{equation}
It is standard to construct a smooth solution $u_\rho$
to \eqref{eq1-visc}, for each fixed $\rho>0$.
Indeed, it can be done using the Galerkin method
and the compactness argument, see Chapter 5 in \cite{Ev} and \cite{Kiselev:2008jt}.
As usual, the game is to pass
to the limit as $\rho \to 0$ and identify the entropy
condition satisfied by the limit function $u$.
We will be brief in establishing the following estimates, since
most of them are similar to the ones in \cite{Chen:2003td} and
we will assume $u_0 \in W^{2,1}\cap H^1 \cap L^{\infty}({{\mathbb{R}}}^d)$,
for general $u_0 \in L^1({{\mathbb{R}}}^d)$ one can follow the
approximation procedure presented in \cite{Chen:2003td}.
The following estimates can be established for sufficiently regular initial data:
$$
\norm{u_\rho}_{L^\infty(Q_T)}\le C;\qquad \abs{u_\rho(t,\cdot)}_{BV({{\mathbb{R}}}^d)}\le C;
$$
$$
\norm{u_\rho(t_2,\cdot)-u_\rho(t_1,\cdot)}_{L^1({{\mathbb{R}}}^d)}\to 0,
\quad \text{as $\abs{t_2-t_1}\to 0$, uniformly in $\rho$.}
$$
Hence there is a limit $u$ such that, passing if necessary to a subsequence as $\rho\to 0$,
\begin{equation}\label{eq:ueps-conv}
u_\rho\to u\quad \text{a.e.~in $Q_T$ and in $L^p(Q_T)$ for any $p \in [1,\infty)$.}
\end{equation}
Next, we derive an energy estimate. To this end, fix a convex $C^2$ function $\eta$ and
define $q,r$ by $q'=\eta'f'$, $r'=\eta' a$. Multiplying \eqref{eq1-visc} by $\eta'$ yields
\begin{equation}\label{eq1-visc-entropy}
\partial_t \eta(u_\rho) + \mathrm{div} q(u_\rho)
=\sum_{i,j=1}^d \partial_{ij}^2 r_{ij}(u_\rho) + \ensuremath{\mathcal{L}}[\eta(u_\rho]
+ \rho \Delta \eta(u_\rho) - \nu_\rho
\end{equation}
where $\nu_\rho=\nu_\rho^1+\nu_\rho^2+\nu_\rho^3$ consists of
three parts:
(i) the entropy dissipation term
$$
\nu_\rho^1:=\rho\Delta \eta(u_\rho)-\rho\eta'(u_\rho)\Delta u_\rho
=\rho \eta''(u_\rho) \abs{\nabla u_\rho}^2;
$$
(ii) the parabolic dissipation term
$$
\nu_\rho^2:= \sum_{i,j=1}^d \partial_{ij}^2
r_{ij}(u_\rho)- \eta'(u_\rho)\mathrm{div} (a(u_\rho) \nabla u_\rho)
=\eta''(u_\rho) \sum_{k=1}^K
\left( \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho)\right)^2;
$$
(iii) the fractional parabolic dissipation term
$$
\nu_\rho^3 = \int_{{{\mathbb{R}}}^d \setminus \{0\}}\overline{\eta''}(u_\rho;z)
\left(u_\rho(t,x+z)-u_\rho(t,x)\right)^2\, \pi(dz),
$$
where $\overline{\eta''}(u_\rho;z)= \int_0^1 (1-\tau)
\eta''((1-\tau)u_\rho(t,x) + \tau u_\rho(t,x+z)) \, d\tau$.
In deriving \eqref{eq1-visc-entropy}, the ``new" computation
is the one showing that the commutator
$$
\ensuremath{\mathcal{L}}[\eta(u_\rho)]-\eta'(u_\rho) \ensuremath{\mathcal{L}}[u_\rho]
$$
equals $\nu_\rho^3$, but this follows easily from
Taylor's formula with integral reminder:
\begin{equation}\label{eq:Taylor}
\begin{split}
\eta(b)-\eta(a) &= \eta'(a) \left(b - a\right)
\\ & \qquad
+\left(\int_0^1 (1-\tau) \eta''((1-\tau)a + \tau b) \, d\tau\right)
\left(b - a\right)^2.
\end{split}
\end{equation}
Specifying $\eta(z)=\frac{z^2}{2}$ in \eqref{eq1-visc-entropy} gives
$$
\int_0^T \int_{{{\mathbb{R}}}^d} \sum_{k=1}^K \left( \sum_{i=1}^d \partial_{x_i}
\zeta_{ik}^a(u_\rho)\right)^2\, dx \, dt \le C
$$
and
\begin{equation}\label{eq:weakconv-zeta}
\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho)
\rightharpoonup \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)
\quad \text{in $L^2(Q_T)$}.
\end{equation}
From this we easily see, as in \cite{Chen:2003td}, that
\eqref{l2regu} and \eqref{crule} in Definition \ref{def:entropy} hold.
Regarding the non-local operator $\ensuremath{\mathcal{L}}$, the same choice for $\eta$
reveals that (\ref{frintcnd}) in Definition \ref{def:entropy} holds. Now set
$$
\Pi(dz):= \left( \abs{z}^2 \mathbf{1}_{|z|<1}
+\abs{z}\, \mathbf{1}_{|z|\ge 1}\right)\, \pi(dz),
$$
and note that $\Pi(dz)$ is a bounded Radon measure.
Introducing the short-hand notation
$$
D_\rho(t,x,z)= \frac{u_\rho(t,x+z)
- u_\rho(t,x)}{\abs{z}\mathbf{1}_{_{|z|<1}}
+ \sqrt{\abs{z}} \mathbf{1}_{_{|z|\ge 1}}}
\qquad
d\mu=\Pi(dz)\otimes dx\otimes dt,
$$
\eqref{frintcnd} translates into $D_\rho$ being uniformly bounded
in $L^2((0,T)\times {{\mathbb{R}}}^d\times ({{\mathbb{R}}}^d \setminus \{0\});d\mu)$. Consequently, we may
assume that there is a limit function $D$ such that
$$
D_\rho \rightharpoonup D \quad \text{in $L^2((0,T)\times {{\mathbb{R}}}^d\times ({{\mathbb{R}}}^d \setminus \{0\});d\mu)$.}
$$
Let us identify $D$. To this end, fix a smooth function
$\varphi$ in $C^\infty_c(Q_T)$ and observe
\begin{align*}
&\iint_{Q_T}\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\varphi(t,x)\frac{u_\rho(t,x+z)-u_\rho(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}}
\, \Pi(dz)\, dx\, dt
\\ & = \iint_{Q_T}\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\frac{\varphi(t,x+z)-\varphi(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}} u_\rho(t,x)
\, \Pi(dz)\, dx\, dt.
\end{align*}
Now, using that $u_\rho \overset{\rho\to 0}{\longrightarrow} u$ a.e.~in $Q_T$, we
conclude that
$$
D_\rho \rightharpoonup \frac{u(t,x+z) - u(t,x)}{\abs{z}\mathbf{1}_{|z|<1}
+ \sqrt{\abs{z}} \mathbf{1}_{|z|\ge 1}}
\quad \text{in $L^2((0,T)\times {{\mathbb{R}}}^d\times
({{\mathbb{R}}}^d \setminus \{0\});d\mu)$.}
$$
We are now in a position to pass to the distributional
limit in \eqref{eq1-visc-entropy} to recover the desired
entropy condition satisfied by the limit
$u=\lim_{\rho \to 0} u_\rho$. Note that to
interpret \eqref{eq1-visc-entropy} in the sense of distributions
we use the formula
\begin{equation}\label{eq:IBP}
\int_{{{\mathbb{R}}}^d} \ensuremath{\mathcal{L}}[\Phi(x)] \phi(x) \, dx = \int_{{{\mathbb{R}}}^d} \Phi(x) \ensuremath{\mathcal{L}}[\phi(x)] \, dx,
\end{equation}
which holds for all sufficiently regular (say, $C^2$) functions $\Phi,\phi:{{\mathbb{R}}}^d\to {{\mathbb{R}}}$.
This relation is easily obtained by a change of variables
$(t,x,z) \mapsto (t,x+z,-z)$ and an integration by parts in $x$.
We claim that the entropy condition satisfied by the limit
$u=\lim_{\rho \to 0} u_\rho$ takes the following form:
for any convex $C^2$ entropy function $\eta$ and corresponding entropy
fluxes $q,r$ defined by $q'=\eta'f', r'=\eta' a$,
\begin{equation}\label{eq1-limit-entropy}
\partial_t \eta(u) + \mathrm{div} q(u)
\le \sum_{i,j} \partial_{x_i x_j} r_{ij}(u) + \ensuremath{\mathcal{L}}[\eta(u)] - n^{u,\eta}-m^{u,\eta}
\end{equation}
in the sense of distributions, where
$$
n^{u,\eta} = \eta''(u) \sum_{k=1}^K
\left(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)\right)^2
$$
is the parabolic dissipation measure with respect to $u$ and
$$
m^{u,\eta} = \int_{{{\mathbb{R}}}^d \setminus \{0\}}\overline{\eta''}(u;z)
\left(u(t,x+z) - u(t,x) \right)^2\, \pi(dz),
$$
is the fractional parabolic dissipation measure with respect to $u$.
In view of \eqref{eq:ueps-conv}, to verify \eqref{eq1-limit-entropy}
we only need to argue that
$$
\liminf_{\rho \to 0} \iint_{Q_T} \nu_\rho \, dx \, dt \ge
\iint_{Q_T} \left( n^{u,\eta} + m^{u,\eta} \right) \, dx \, dt.
$$
First, $\iint_{Q_T} \nu_\rho^1 \, dx \, dt \ge 0$ for each $\rho>0$. Second, thanks
to the weak convergence \eqref{eq:weakconv-zeta} and a standard
weak lower semi-continuity result for quadratic functionals,
\begin{align*}
& \liminf_{\rho\to 0}
\int_0^T \int_{{{\mathbb{R}}}^d} \eta''(u_\rho)\sum_{k=1}^K
\left( \sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u_\rho) \right)^2
\varphi \, dx \, dt \\ & \qquad
\ge \int_0^T \int_{{{\mathbb{R}}}^d} \eta''(u)\sum_{k=1}^K \left( \sum_{i=1}^d
\partial_{x_i} \zeta_{ik}^a(u) \right)^2 \varphi \, dx \, dt,
\end{align*}
for all test functions $\varphi\in C^\infty_c$. Similarly,
\begin{align*}
&\liminf_{\rho\to 0}
\iint_{Q_T} \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{\eta''}(u_\rho;z)
\left(u_\rho(t,x+z)-u_\rho(t,x)\right)^2\varphi\, \pi(dz)\, dx\, dt
\\ & \qquad
\ge \iint_{Q_T} \int_{{{\mathbb{R}}}^d \setminus \{0\}}\overline{\eta''}(u;z)
\left(u(t,x+z)-u(t,x)\right)^2\varphi\, \pi(dz)\, dx\, dt,
\end{align*}
for all test functions $\varphi\in C^\infty_c$.
Combining, we deduce that \eqref{entineq}
in Definition \ref{def:entropy} holds. This completes the proof.
\section{Proof of Theorem \ref{th:unique} (uniqueness)}\label{pruniq}
We shall need $C^2$ approximations $\eta_{\ensuremath{\varepsilon}}^\pm(z)$ of the functions
$$
\eta^\pm(z) :=(z)^\pm=\max\left(\pm(z),0\right),
\qquad z\in {{\mathbb{R}}}.
$$
We build these by picking nondecreasing $C^1$
approximations ${\rm sgn}\, _{\ensuremath{\varepsilon}}^\pm(z)$ of
$$
{\rm sgn}\, ^+(z):=
\begin{cases}
0, & \textrm{if $z \leq 0$},\\
1, & \textrm{if $z>0$},
\end{cases}
\qquad
{\rm sgn}\, ^-(z) :=
\begin{cases}
-1, & \textrm{if $z \leq 0$,}\\
0, & \textrm{if $z>0$,}
\end{cases}
$$
and defining
$$
\eta_{\ensuremath{\varepsilon}}^\pm(z)
:= \int_0^z {\rm sgn}\, _{\ensuremath{\varepsilon}}^\pm(\xi) \, d\xi,
\qquad z\in {{\mathbb{R}}}.
$$
For example, we can take
$$
{\rm sgn}\, _{\ensuremath{\varepsilon}}^+(z)=
\begin{cases}
0, & \textrm{if $z<0$},\\
\sin(\frac{\pi}{2\ensuremath{\varepsilon}}z), & \textrm{if $0\le z \leq \ensuremath{\varepsilon}$},\\
1, & \textrm{if $z>\ensuremath{\varepsilon}$}.
\end{cases}
\quad
{\rm sgn}\, _{\ensuremath{\varepsilon}}^-(z)=
\begin{cases}
-1, & \textrm{if $z<-\ensuremath{\varepsilon}$},\\
\sin(\frac{\pi}{2\ensuremath{\varepsilon}}z), & \textrm{if $-\ensuremath{\varepsilon}\le z \le 0$},\\
0, & \textrm{if $z>0$}.
\end{cases}
$$
The functions $\eta_{\ensuremath{\varepsilon}}^\pm$
are $C^2$ and convex. Moreover,
$$
\eta_{\ensuremath{\varepsilon}}^\pm(z)
\overset{\ensuremath{\varepsilon}\to 0}{\longrightarrow}
\eta^\pm(z),
\qquad z\in {{\mathbb{R}}}.
$$
Observe that $\left(\eta_{\ensuremath{\varepsilon}}^\pm(\cdot-c)\right)_{c \in{{\mathbb{R}}}}$
is a family of entropies.
Given these entropies, we introduce the
corresponding entropy fluxes
\begin{align*}
q_{\ensuremath{\varepsilon}}^\pm(z,c) & = \int_c^z (\eta_\ensuremath{\varepsilon}^\pm)'(\xi-c) f'(\xi) d\xi, \qquad z,c \in {{\mathbb{R}}},\\
r_{\ensuremath{\varepsilon}}^\pm(z,c) & = \int_c^z (\eta_{\ensuremath{\varepsilon}}^\pm)'(\xi-c) a(\xi) \, d\xi, \qquad z,c \in {{\mathbb{R}}}.
\end{align*}
Clearly, as $\ensuremath{\varepsilon} \to 0$,
\begin{align*}
q_{\ensuremath{\varepsilon}}^\pm(z,c) \to q^\pm(z,c) & := {\rm sgn}\, ^\pm(z-c) (f(z)-f(c)), \qquad z,c\in {{\mathbb{R}}},\\
r_{\ensuremath{\varepsilon}}^\pm(z,c) \to r^\pm(z,c) & := {\rm sgn}\, ^\pm(u-c) (A(u) - A(c)), \qquad z,c\in {{\mathbb{R}}},
\end{align*}
where the (matrix-valued) function $A(\cdot)$
is defined by $\displaystyle A(z) = \int_0^u a(\xi) \, d\xi$.
Observe that $\left(\eta_{\ensuremath{\varepsilon}}^\pm(\cdot-c),q_{\ensuremath{\varepsilon}}^\pm(\cdot,c),r_{\ensuremath{\varepsilon}}^\pm(\cdot,c)\right)_{c \in {{\mathbb{R}}}}$
is a family of entropy-entropy flux triples, so choosing
$\eta = \eta_{\ensuremath{\varepsilon}}^\pm$ in \eqref{entineq} yields
\begin{equation}\label{in1}
\begin{split}
& \iint_{Q_T} \Bigl( \eta_{\ensuremath{\varepsilon}}^\pm(u-c) \partial_t \varphi
+ \sum_{i=1}^d q_{\ensuremath{\varepsilon},i}^\pm(u,c) \partial_{x_i} \varphi
+ \sum_{i,j=1}^d r_{\ensuremath{\varepsilon},ij}^\pm(u,c) \ensuremath{\partial_{x_ix_j}^2} \varphi \Bigr)\, dx \, dt
\\ & \quad + \iint_{Q_T} \eta_{\ensuremath{\varepsilon}}^\pm(u-c) \ensuremath{\mathcal{L}}[\varphi]\, dx \, dt
+ \int_{{{\mathbb{R}}}^d} \eta^\pm(u_0-c) \varphi(0,x) \, dx
\\ & \geq \iint_{Q_T} (\eta_{\ensuremath{\varepsilon}}^\pm)''(u-c)
\sum_{k=1}^K \Bigl(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \, dx \, dt \\
&\quad + \iint_{Q_T} \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{(\eta_{\ensuremath{\varepsilon}}^\pm)''}(u-c;z)
\left(u(t,x+z)-u(t,x)\right)^2 \varphi \, \pi(dz) \, dx \, dt.
\end{split}
\end{equation}
Moreover,
\begin{align*}
\overline{(\eta_\ensuremath{\varepsilon}^\pm)''}(u-c;z)
& = \int_0^1 (1-\tau) (\eta_\ensuremath{\varepsilon}^\pm)''\Bigl((1-\tau)u(t,x)+\tau u(t,x+z),c\Bigr) \, d\tau
\\ & = \int_0^1 (1-\tau) ({\rm sgn}\, _\ensuremath{\varepsilon}^\pm)'\Bigl((1-\tau)(u(t,x)-c)+\tau (u(t,x+z)-c)\Bigr)\, d\tau.
\end{align*}
To proceed, the following simple observations will be useful:
\begin{itemize}
\item ${\rm sgn}\, _{\ensuremath{\varepsilon}}^-(z-c) =-{\rm sgn}\, _{\ensuremath{\varepsilon}}^+(c-z)$ and $\eta_{\ensuremath{\varepsilon}}^-(z-c) = \eta_{\ensuremath{\varepsilon}}^+(c-z)$;
\item $q_{\ensuremath{\varepsilon}}^-(z,c) = q_{\ensuremath{\varepsilon}}^+(c,z)$ and $r_\ensuremath{\varepsilon}^-(z,c) = r_\ensuremath{\varepsilon}^+(c,z)$;
\item $(\eta_{\ensuremath{\varepsilon}}^-)''(z-c) = (\eta_{\ensuremath{\varepsilon}}^+)''(c-z)$.
\end{itemize}
Employing these observations, we can rewrite the ``$-$" part of \eqref {in1} as
\begin{equation}\label{in3}
\begin{split}
& \iint_{Q_T} \Bigl( \eta_{\ensuremath{\varepsilon}}^+(c-u) \partial_t \varphi
+ \sum_{i=1}^d q_{\ensuremath{\varepsilon},i}^+(c,u) \partial_{x_i} \varphi
+ \sum_{i,j=1}^d r_{\ensuremath{\varepsilon},ij}^+(c,u) \ensuremath{\partial_{x_ix_j}^2} \varphi \Bigr)\, dx \, dt
\\ & \quad + \iint_{Q_T} \eta_{\ensuremath{\varepsilon}}^+(c-u) \ensuremath{\mathcal{L}}[\varphi]\, dx \, dt
+ \int_{{{\mathbb{R}}}^d} \eta_{\ensuremath{\varepsilon}}^+(c-u_0) \varphi(0,x) \, dx
\\ & \geq \iint_{Q_T} (\eta_{\ensuremath{\varepsilon}}^+)''(c-u)
\sum_{k=1}^K \Bigl(\sum_{i=1}^d \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \, dx \, dt \\
&\quad + \iint_{Q_T} \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(c-u;z)
\left(u(t,x+z)-u(t,x)\right)^2 \varphi \, \pi(dz) \, dx \, dt.
\end{split}
\end{equation}
To establish the $L^1$ contraction property \eqref{eq:contraction} we shall
employ the doubling-of-variables device of Kru\u{z}kov \cite{Kruzkov:1970kx}.
Let $u=u(t,x)$, $v=v(s,y)$ be two entropy solutions as stated
in Theorem \ref{th:unique}. Moreover, let $\varphi=\varphi(t,x,s,y)$ be a
test function in the doubled variables $(t,x,s,y)$.
To simplify the presentation, we introduce the following notation (with $\nabla_{x+y}$ being
short-hand for $\nabla_x+\nabla_y$)
\begin{align*}
\ensuremath{\mathcal{L}}_x[\varphi] & := \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\left[ \varphi(t,x+z,s,y) - \varphi
- z \cdot \nabla_x \varphi
\mathbf{1}_{|z|<1} \right]\, \pi(dz), \\
\ensuremath{\mathcal{L}}_y[\varphi] & = \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\left[ \varphi(t,x,s,y+ z) - \varphi
-z \cdot \nabla_y \varphi
\mathbf{1}_{|z|<1} \right]\, \pi(dz), \\
\ensuremath{\mathcal{L}}_{x+y}[\varphi] & = \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\Bigl[ \varphi(t,x+z,s,y+ z)-\varphi
-z \cdot \nabla_{x+y}\varphi
\mathbf{1}_{|z|<1} \Bigr]\, \pi(dz),
\end{align*}
In the ``$+$" part of \eqref{in1} written the entropy solution $u(t,x)$ we choose $c=v(s,y)$
and integrate the result over $(s,y)$, obtaining
\begin{equation}\label{ineqt04}
\begin{split}
& \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \Bigl( \eta_{\ensuremath{\varepsilon}}^+(u-v) \partial_t \varphi
+ \sum_{i=1}^d q_{\ensuremath{\varepsilon},i}^+(u,v) \partial_{x_i} \varphi
+ \sum_{i,j=1}^d r_{\ensuremath{\varepsilon},ij}^+(u,c) \ensuremath{\partial_{x_ix_j}^2} \varphi \Bigr)\, dx \, dt \, dy\, ds
\\ & \quad + \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v) \ensuremath{\mathcal{L}}_x[\varphi]\, dx\, dt\, dy\, ds
+ \int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u_0-v) \varphi(0,x,s,y) \, dx\, dy\, ds
\\ & \geq \iint_{Q_T} (\eta_{\ensuremath{\varepsilon}}^+)''(u-v)
\sum_{k=1}^K \Bigl(\sum_{i=1}^d
\partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2 \varphi \, dx\, dt\, dy\, ds \\
&\quad+ \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int\!\!\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u(t,\cdot)-v;z)
\left(u(t,x+z)-u(t,x)\right)^2 \varphi \, \pi(dz) \, dx\, dt\, dy\, ds.
\end{split}
\end{equation}
Similarly, in \eqref{in3} written for the entropy solution $v(s,y)$ we choose $c=u(t,x)$ and
integrate over $(t,x)$, thereby obtaining
\begin{equation}\label{ineek05}
\begin{split}
& \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \Bigl( \eta_{\ensuremath{\varepsilon}}^+(u-v) \ensuremath{\partial_s} \varphi
+ \sum_{i=1}^d q_{\ensuremath{\varepsilon},i}^+(u,v) \partial_{y_i} \varphi
+ \sum_{i,j=1}^d r_{\ensuremath{\varepsilon},ij}^+(u,v) \ensuremath{\partial_{y_iy_j}^2} \varphi \Bigr)\, dx\, dt\, dy\, ds
\\ & \; + \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v) \ensuremath{\mathcal{L}}_y[\varphi]\, dx\, dt\, dy\, ds
+ \int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v_0) \varphi(t,x,0,y) \, dx\, dt\, dy
\\ & \geq \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int (\eta_{\ensuremath{\varepsilon}}^+)''(u-v)
\sum_{k=1}^K \Bigl(\sum_{i=1}^d\partial_{y_i} \zeta_{ik}^a(v)\Bigr)^2
\varphi \, dx \, dt\, dy\, ds \\
&\; + \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int\!\!\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u-v(s,\cdot);z)
\left(v(s,y+z)-v(s,y)\right)^2 \varphi \, \pi(dz) \, dx\, dt\, dy\, ds.
\end{split}
\end{equation}
Adding \eqref{ineqt04} and \eqref{ineek05} yields
\begin{equation}\label{in6}
I_{\mathrm{time}}(\ensuremath{\varepsilon})+I_{\mathrm{conv}}(\ensuremath{\varepsilon})+I_{\mathrm{diff}}(\ensuremath{\varepsilon})
+I_{\mathrm{fdiff}}(\ensuremath{\varepsilon})+ I_{\mathrm{init}}(\ensuremath{\varepsilon})
\ge I_{\mathrm{diss}}(\ensuremath{\varepsilon})+I_{\mathrm{fdiss}}(\ensuremath{\varepsilon}),
\end{equation}
where
\begin{align*}
I_{\mathrm{time}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v) (\partial_t +\ensuremath{\partial_s}) \varphi \, dx\, dt\, dy\, ds \\
I_{\mathrm{conv}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \sum_{i=1}^d q_{\ensuremath{\varepsilon},i}^+(u,v)
(\partial_{x_i} + \partial_{y_i}) \varphi \, dx\, dt\, dy\, ds \\
I_{\mathrm{diff}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \sum_{i,j=1}^d r_{\ensuremath{\varepsilon},ij}^+(u,v)
(\ensuremath{\partial_{x_ix_j}^2}+\ensuremath{\partial_{y_iy_j}^2}) \varphi \, dx\, dt\, dy\, ds \\
I_{\mathrm{fdiff}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v)
\Bigl(\ensuremath{\mathcal{L}}_x[\varphi]+\ensuremath{\mathcal{L}}_y[\varphi]\Bigr)\, dx\, dt\, dy\, ds\\
I_{\mathrm{init}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u_0-v) \varphi(0,x,s,y) \, dx\, dy\, ds
\\ & \qquad\quad
+\int\!\!\!\!\int\!\!\!\!\int \eta_{\ensuremath{\varepsilon}}^+(u-v_0) \varphi(t,x,0,y) \, dx\, dt\, dy\\
I_{\mathrm{diss}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int (\eta_{\ensuremath{\varepsilon}}^+)''(u-v)
\\ & \qquad \qquad
\times \sum_{k=1}^K \left[ \Bigl(\sum_{i}^d \partial_{x_i} \zeta_{ik}^a(u)\Bigr)^2
+\Bigl(\sum_{i=1}^d\partial_{y_i} \zeta_{ik}^a(v)\Bigr)^2\right]
\varphi \, dx \, dt\, dy\, ds \\
I_{\mathrm{fdiss}}(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int\!\!\int_{{{\mathbb{R}}}^d \setminus \{0\}}
\Biggl [ \overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u(t,\cdot)-v;z) \left(u(t,x+z)-u(t,x)\right)^2
\\ & \qquad\qquad\qquad\qquad\qquad
+ \overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u,v(s,\cdot);z) \left(v(s,y+z)-v(s,y)\right)^2 \Biggr]
\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\times \varphi \, \pi(dz) \, dx\, dt\, dy\, ds.
\end{align*}
In view of the inequality ``$a^2+b^2\ge 2ab$", we have
$I_{\mathrm{diss}}(\ensuremath{\varepsilon}) \ge \widetilde I_{\mathrm{diss}}(\ensuremath{\varepsilon})$, with
$$
\widetilde I_{\mathrm{diss}}(\ensuremath{\varepsilon})
= 2\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int (\eta_{\ensuremath{\varepsilon}}^+)''(u-v)
\sum_{k=1}^K \sum_{i,j=1}^d
\partial_{x_i} \zeta_{ik}^a(u) \partial_{y_j} \zeta_{jk}^a(v)
\varphi \, dx \, dt\, dy\, ds.
$$
Arguing exactly as in \cite{Bendahmane:2004go}, it follows that
\begin{equation}\label{eq:diff-diss-final}
\begin{split}
& \lim_{\ensuremath{\varepsilon}\to 0} \Bigl( I_{\mathrm{diff}}(\ensuremath{\varepsilon})
-\widetilde I_{\mathrm{diss}}(\ensuremath{\varepsilon})\Bigr)
\\ & \quad
\le \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \sum_{i,j=1}^d r_{ij}^+(u,v)
(\ensuremath{\partial_{x_ix_j}^2}+2\ensuremath{\partial_{x_iy_j}^2}+\ensuremath{\partial_{y_iy_j}^2}) \varphi \, dx\, dt\, dy\, ds.
\end{split}
\end{equation}
Fix a small number $\kappa>0$, and let us split $\ensuremath{\mathcal{L}}$ into two parts
\begin{align*}
\ensuremath{\mathcal{L}}[\phi] & =\int_{\abs{z}\le \kappa}\left[ \phi(t,x+ z) - \phi(t,x)
- z \cdot \nabla \phi \mathbf{1}_{|z|<1} \right]\, \pi(dz)
\\ & \qquad +\int_{\abs{z}> \kappa}
\left[ \phi(t,x+ z) - \phi(t,x) - z \cdot \nabla
\phi \mathbf{1}_{|z|<1} \right]\, \pi(dz)
\\ & =: \ensuremath{\mathcal{L}}_\kappa[\phi]+\ensuremath{\mathcal{L}}^\kappa[\phi],
\qquad \forall \phi\in C^2,
\end{align*}
and similarly
$$
\ensuremath{\mathcal{L}}_x = \ensuremath{\mathcal{L}}_{x,\kappa} + \ensuremath{\mathcal{L}}_x^\kappa, \quad
\ensuremath{\mathcal{L}}_y = \ensuremath{\mathcal{L}}_{y,\kappa} + \ensuremath{\mathcal{L}}_y^\kappa, \quad
\ensuremath{\mathcal{L}}_{x+y} = \ensuremath{\mathcal{L}}_{x+y,\kappa} + \ensuremath{\mathcal{L}}_{x+y}^\kappa.
$$
The corresponding splitting of
$I_{\mathrm{fdiff}}(\ensuremath{\varepsilon})$ is written
$$
I_{\mathrm{fdiff}}(\ensuremath{\varepsilon})=
I_{\mathrm{fdiff},\kappa}(\ensuremath{\varepsilon})
+I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon}).
$$
We also need to introduce the
operator $\widetilde \ensuremath{\mathcal{L}}^\kappa$ defined by writing
\begin{align*}
\ensuremath{\mathcal{L}}^\kappa[\varphi]
& = \widetilde \ensuremath{\mathcal{L}}^\kappa [\varphi]
-\left( \int_{\abs{z}>\kappa} z
\mathbf{1}_{|z|<1}\,
\pi(dz)\right) \cdot \nabla_x \varphi,
\end{align*}
with similar definitions for $\widetilde \ensuremath{\mathcal{L}}^\kappa_x$, $\widetilde \ensuremath{\mathcal{L}}^\kappa_y$, and
$\widetilde \ensuremath{\mathcal{L}}^\kappa_{x+y}$. Observe that \eqref{eq:IBP}
continues to hold for all these operators. The function obtained
by replacing $\ensuremath{\mathcal{L}}^\kappa$ with $\widetilde \ensuremath{\mathcal{L}}^\kappa$ in the
definition of $I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})$ will be named
$\widetilde I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})$.
Clearly, in view of \eqref{intcond1},
\begin{equation}\label{eq:kappa-est}
\abs{I_{\mathrm{fdiff},\kappa}(\ensuremath{\varepsilon})}\le
C \norm{D^2\varphi}_{L^1(Q_T\times Q_T)}
\int_{\abs{z}\le \kappa} \abs{z}^2\, \pi(dz)
\overset{\kappa\to 0}{\longrightarrow} 0,
\end{equation}
for some constant $C$ independent of $\kappa$ and $\ensuremath{\varepsilon}$.
Let us analyze $\widetilde I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})$.
By \eqref{eq:IBP},
\begin{align*}
\widetilde I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})
= \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \Bigl(\widetilde \ensuremath{\mathcal{L}}^\kappa_x\left[\eta_\ensuremath{\varepsilon}^+(u-v)\right]
+ \widetilde \ensuremath{\mathcal{L}}^\kappa_y\left[\eta_\ensuremath{\varepsilon}^+(u-v)\right]\Bigr)
\varphi \, dt \, dx \, dy\, ds.
\end{align*}
Specifying $a=u(t,x)-v(s,y)$ and $b=u(t,x+z)-v(s,y)$ in \eqref{eq:Taylor} yields
\begin{equation}\label{eq:Taylor-1}
\begin{split}
&\eta_{\ensuremath{\varepsilon}}^+(u(t,x+z)-v(s,y))-\eta_{\ensuremath{\varepsilon}}^+(u(t,x)-v(s,y))
\\ & = (\eta_{\ensuremath{\varepsilon}}^+)'(u(t,x)-v(s,y)) \left(u(t,x+z) - u(t,x)\right)
\\ & \qquad \qquad
+\overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u(t,\cdot)- v;z)\left(u(t,x+z) - u(t,x)\right)^2.
\end{split}
\end{equation}
Similarly, taking $a=u(t,x)-v(s,y)$, $b=u(t,x)-v(s,y+z)$ in \eqref{eq:Taylor} yields
\begin{equation}\label{eq:Taylor-2}
\begin{split}
&\eta_{\ensuremath{\varepsilon}}^+(u(t,x)-v(s,y+z))-\eta_{\ensuremath{\varepsilon}}^+(u(t,x)-v(s,y))
\\ & = -(\eta_{\ensuremath{\varepsilon}}^+)'(u(t,x)-v(s,y)) \left(v(s,y+z) - v(s,y)\right)
\\ & \qquad \qquad
+\overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u-v(s,\cdot);z) \left(v(s,y+z) - v(s,y)\right)^2.
\end{split}
\end{equation}
Adding the first term on the right-hand side of \eqref{eq:Taylor-1} to the first
term on the right-hand side of \eqref{eq:Taylor-2} yields
\begin{align*}
&(\eta_{\ensuremath{\varepsilon}}^+)'(u(t,x)-v(s,y)) \left(u(t,x+z) - u(t,x)\right)\\
& \qquad -(\eta_{\ensuremath{\varepsilon}}^+)'(u(t,x)-v(s,y)) \left(v(s,y+z) - v(s,y)\right)
\\ & = (\eta_{\ensuremath{\varepsilon}}^+)'(u(t,x)-v(s,y))\Bigl[ \left(u(t,x+z) - v(s,y+z)\right)
-\left(u(t,x) - v(s,y)\right)\Bigr]
\\ & \le \eta_\ensuremath{\varepsilon}^+(u(t,x+z) - v(s,y+z))
-\eta_\ensuremath{\varepsilon}^+(u(t,x) - v(s,y)),
\end{align*}
where we have used the convexity of $\eta_\ensuremath{\varepsilon}$ to derive the last inequality.
In view of these findings, we can rewrite
$\widetilde I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})$ as follows:
\begin{align*}
\widetilde I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})-I_{\mathrm{fdiss}}^\kappa(\ensuremath{\varepsilon})
& \le \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \widetilde \ensuremath{\mathcal{L}}^\kappa_{x+y}
\left[\eta_\ensuremath{\varepsilon}^+(u(t,\cdot)-v(s,\cdot))\right]\varphi \, dt \, dx \, dy\, ds
\\ & \overset{\text{\eqref{eq:IBP}}}{=}
\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_\ensuremath{\varepsilon}^+(u-v)\widetilde \ensuremath{\mathcal{L}}^\kappa_{x+y}[\varphi] \, dt \, dx \, dy\, ds,
\end{align*}
where
\begin{align*}
I_{\mathrm{fdiss}}^\kappa(\ensuremath{\varepsilon}) & = \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int\!\!\int_{\abs{z}>\kappa}
\Biggl [ \overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u(t,\cdot)-v;z) \left(u(t,x+z)-u(t,x)\right)^2
\\ & \qquad\qquad\qquad\qquad\qquad
+ \overline{(\eta_{\ensuremath{\varepsilon}}^+)''}(u-v(s,\cdot);z) \left(v(s,y+z)-v(s,y)\right)^2 \Biggr]
\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\times \varphi \, \pi(dz) \, dx\, dt\, dy\, ds.
\end{align*}
Consequently,
\begin{align*}
I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})-I_{\mathrm{fdiss}}^\kappa(\ensuremath{\varepsilon})
& \le \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_\ensuremath{\varepsilon}^+(u-v)
\ensuremath{\mathcal{L}}^\kappa_{x+y}[\varphi] \, dt \, dx \, dy\, ds,
\end{align*}
The next step is to first send $\kappa\to 0$ and then $\ensuremath{\varepsilon}\to 0$. Related to this, observe that
$$
\lim_{\kappa\to 0} I_{\mathrm{fdiff}}^\kappa(\ensuremath{\varepsilon})= I_{\mathrm{fdiff}}(\ensuremath{\varepsilon}), \quad
\lim_{\kappa\to 0} I_{\mathrm{fdiss}}^\kappa(\ensuremath{\varepsilon})= I_{\mathrm{fdiss}}(\ensuremath{\varepsilon})
$$
for each fixed $\ensuremath{\varepsilon}>0$, by the dominated convergence theorem. Moreover, we clearly have
$\lim_{\kappa\to 0} \ensuremath{\mathcal{L}}^\kappa_{x+y}[\varphi] = \ensuremath{\mathcal{L}}_{x+y}[\varphi]$.
In view of this and \eqref{eq:kappa-est}, we conclude that
\begin{equation}\label{eq:fdiff-fdiss-final}
I_{\mathrm{fdiff}}(\ensuremath{\varepsilon})-I_{\mathrm{fdiss}}(\ensuremath{\varepsilon})
\le \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \eta_\ensuremath{\varepsilon}^+(u-v) \ensuremath{\mathcal{L}}[\varphi] \, dt \, dx \, dy\, ds.
\end{equation}
By \eqref{eq:diff-diss-final} and \eqref{eq:fdiff-fdiss-final}, It follows
from \eqref{in6} and sending $\ensuremath{\varepsilon}\to 0$ that
\begin{equation}\label{in7}
\begin{split}
& \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \Biggl( (u-v)^+ (\partial_t+\ensuremath{\partial_s})\varphi
+ \sum_{i=1}^d q_{i}^+(u,v) (\partial_{x_i}+\partial_{y_i}) \varphi
\\ & \qquad
+\sum_{i,j=1}^d r_{ij}^+(u,v)
(\ensuremath{\partial_{x_ix_j}^2}+2\ensuremath{\partial_{x_iy_j}^2}+\ensuremath{\partial_{y_iy_j}^2}) \varphi
+ \eta^+(u-v) \ensuremath{\mathcal{L}}_{x+y}[\varphi] \Biggr)\, dx\, dt\, dy\, ds
\\ & + \int\!\!\!\!\int\!\!\!\!\int (u_0-v)^+ \varphi(0,x,s,y) \, dx\, dy\, ds
+ \int\!\!\!\!\int\!\!\!\!\int (u-v_0)^+ \varphi(t,x,0,y) \, dx\, dt\, dy \ge 0.
\end{split}
\end{equation}
Let us specify the test function $\varphi=\varphi(t,x,s,y)$. To this end, fix a nonnegative
test function $\phi=\phi(t,x)\in C_c^{\infty}([0,\infty) \times {{\mathbb{R}}}^d)$, and
pick two sequences $\Set{\theta_{\nu}}_{\nu>0} \subset C_c^{\infty}(0,\nu)$,
$\Set{\delta_{\mu}}_{\mu>0} \subset C_c^{\infty}(B(0,\mu))$ of
approximate delta functions, where $B(0,\mu)$ denotes the open ball
centered at the origin with radius $\mu$. Then take
\begin{equation}\label{eq:testfunc}
\varphi(t,x,s,y) = \theta_{\nu}(s-t)
\delta_{\mu}(y-x) \phi(t,x).
\end{equation}
Simple calculations reveal that
\begin{align*}
(\partial_t + \ensuremath{\partial_s}) \varphi
& =\theta_{\nu}(s-t) \delta_{\mu}(y-x) \partial_t \phi(t,x),
\\ (\partial_{x_i} + \partial_{y_i})\varphi
& = \theta_{\nu}(s-t) \delta_{\mu}(y-x) \partial_{x_i} \phi(t,x),
\\ (\ensuremath{\partial_{x_ix_j}^2} +2 \ensuremath{\partial_{x_iy_j}^2} + \ensuremath{\partial_{y_iy_j}^2}) \varphi
& = \theta_{\nu}(s-t) \delta_{\mu}(y-x) \ensuremath{\partial_{x_ix_j}^2} \phi(t,x)
\end{align*}
and
\begin{align*}
&\varphi(t,x+z,s,y+z) - \varphi(t,x,s,y)
\\ & \qquad
= \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\left( \phi(t,x+z)-\phi(t,x) \right).
\end{align*}
Note that $\theta_{\nu} = 0 $ on $(-\infty,0]$ and so $\varphi(t,x,0,y)\equiv 0.$
By the choice of the test function $\varphi$ and the observations above, we
deduce from \eqref{in7} that
\begin{equation}\label{in8}
\begin{split}
& \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int (u-v)^+ \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\partial_t \phi(t,x) \, dx\, dt\, dy\, ds \\
&\quad +\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int
\sum_{i=1}^d q_i^+(u,v) \theta_{\nu}(s-t)
\delta_{\mu}(y-x)\partial_{x_i} \phi(t,x) \, dxdt \, dyds \\
&\quad + \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \sum_{i,j=1}^d r_{ij}^+(u,v)
\theta_{\nu}(s-t) \delta_{\mu}(y-x)\ensuremath{\partial_{x_ix_j}^2} \phi(t,x) \, dx\, dt\, dy\, ds
\\ & \quad + \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int (u-v)^+
\theta_{\nu}(s-t) \delta_{\mu}(y-x) \ensuremath{\mathcal{L}}[\phi] \, dx\, dt\, dy\, ds
+ I_{u_0,v}(\nu,\mu)\geq 0,
\end{split}
\end{equation}
where
\begin{align*}
I_{u_0,v}(\nu,\mu) & :=
\int\!\!\!\!\int\!\!\!\!\int (u_0-v)^+
\theta_{\nu}(s)\delta_{\mu}(y-x) \phi(0,x) \, dx\, dy\, ds
\\ & = -\int\!\!\!\!\int\!\!\!\!\int (u_0-v)^+
\ensuremath{\partial_s}\left(\tilde{\phi}_{\nu}(s)
\delta_{\mu}(y-x)\phi(0,x)\right) \, dx\, dy\, ds,
\end{align*}
with
$$
\tilde{\phi}_{\nu}(s) := \int_s^T \theta_{\nu}(\tau)\, d\tau
= \int_{\min(s,\nu)}^{\nu} \theta_{\nu}(\tau)\, d\tau
\overset{\nu\to 0}{\longrightarrow} 1.
$$
Specifying $\varphi=\tilde{\phi}_{\nu}(s)\delta_{\mu}(y-x)$ in
the entropy inequality for $v$ and noting
that $\theta_{\nu}(s)$ vanishes for $s>\nu$, we obtain
\begin{equation}\label{in9}
\begin{split}
& \iint (u_0-v)^+\ensuremath{\partial_s} \varphi(s,x,y) \, dy\, ds \\ & \qquad
\le \iint (u_0 -v)^+\theta_{\nu}(s)
\delta_{\mu}(y-x)\phi(0,x) \, dy\, ds+o(\nu)
\\ & \quad \overset{\nu\to 0}{\longrightarrow}
\iint (u_0 -v)^+ \delta_{\mu}(y-x) \phi(0,x)\, dy\, ds,
\end{split}
\end{equation}
where the ``$o(\nu)$" term follows from an integrability argument.
Hence, sending $\nu,\mu \to 0,$ we deduce
\begin{equation}\label{in11}
\begin{split}
& \limsup_{\mu \to 0} \limsup_{\nu \to 0} I_{u_0,v}(\nu,\mu)
\\ & \quad \leq \limsup_{\mu \to 0} \iint
(u_0 -v_0)^+ \delta_{\mu}(y-x) \phi(0,x)\, dx\, dy
\\ & \quad
= \int (u_0-v_0)^+\phi(0,x)\, dx,
\end{split}
\end{equation}
with $u_0=u_0(x)$ and $v_0=v_0(x)$.
Keeping in mind \eqref{in11} when sending $\mu,\nu \to 0$
in \eqref{in8}, we conclude that
\begin{equation}\label{in12}
\begin{split}
& \iint_{Q_T} \Biggl( (u -v)^+ \partial_t \phi
+ \sum_{i=1}^d q_i^+(u,v) \partial_{x_i} \phi
\\ & \qquad\qquad\qquad\quad
+ \sum_{i,j=1}^d r_{ij}^+(u,v) \ensuremath{\partial_{x_ix_j}^2} \phi
+(u-v)^+\ensuremath{\mathcal{L}}[\phi] \Biggr) \, dx\, dt
\\ & \qquad\qquad\qquad\qquad\quad
+ \int_{{{\mathbb{R}}}^d} (u_0-v_0)^+ \phi(0,x) \, dx \geq 0,
\end{split}
\end{equation}
where all the involved functions depend on $(t,x)$.
It now only takes a standard argument to conclude from
\eqref{in12} that Theorem \ref {th:unique} holds. Indeed, one chooses
a sequence of functions $0 \leq \phi \leq 1$ from $C^\infty_c([0,T)\times {{\mathbb{R}}}^d)$
that converges to $\mathbf{1}_{[0,t) \times {{\mathbb{R}}}^d}$ for a Lebesgue point $t$ of
$\int_{{{\mathbb{R}}}^d} (u -v)^+\, dx$ and then use the integrability of $u,v$ to conclude the proof.
Finally, let us prove the last part of Theorem \ref{th:unique}, that is, we shall
establish the $L^1$ contraction property for entropy solutions satisfying
the simpler entropy condition \eqref{entineq-simpler} in which the
fractional parabolic dissipation measure has been dropped.
To this end, let us assume that \eqref{eq:add-cond} holds.
Arguing exactly as before we arrive at
\eqref{in6} without the $I_{\mathrm{fdiss}}(\ensuremath{\varepsilon})$ term:
\begin{equation}\label{intineq}
\begin{split}
& \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \Biggl( (u-v)^+ (\partial_t+\ensuremath{\partial_s})\varphi
+ \sum_{i=1}^d q_{i}^+ (u,v) (\partial_{x_i}+\partial_{y_i}) \varphi
\\ & \qquad\qquad\qquad\qquad
+\sum_{i,j=1}^d r_{ij}^+(u,v)
(\ensuremath{\partial_{x_ix_j}^2}+2\ensuremath{\partial_{x_iy_j}^2}+\ensuremath{\partial_{y_iy_j}^2}) \varphi
\\ & \qquad \qquad \qquad\qquad\qquad
+ (u-v)^+ (\ensuremath{\mathcal{L}}_{x}[\varphi]+\ensuremath{\mathcal{L}}_y[\varphi])
\Biggr)\, dx\, dt\, dy\, ds
\\ & \quad
+ \int\!\!\!\!\int\!\!\!\!\int (u_0-v)^+\varphi|_{t=0}\, dx\, dy\, ds
+ \int\!\!\!\!\int\!\!\!\!\int (u-v_0)^+\varphi|_{s=0}\, dx\, dt\, dy \ge 0.
\end{split}
\end{equation}
The new treatment concerns the fractional term
in \eqref{intineq} only, which we denote by $J(\nu,\mu)$,
i.e., $J(\nu,\mu):=\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int(u-v)^+ (\ensuremath{\mathcal{L}}_{x}[\varphi]+\ensuremath{\mathcal{L}}_y[\varphi]) \, dx\, dt\, dy\, ds$; we
employ the same test function $\varphi$ as before, cf.~\eqref{eq:testfunc}.
By letting $z \mapsto -z$
in the $\ensuremath{\mathcal{L}}_y$ term, keeping in mind
that $\pi(d(-z))=-\pi(dz)$, we obtain
\begin{equation}\label{fracinter}
\begin{split}
J(\nu,\mu)=\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \int_{{{\mathbb{R}}}^d \setminus \{0\}}
& (u(t,x)-v(s,y))^+
\theta_{\nu}(s-t)
\\ & \times
\Bigg(\delta_{\mu}(y-x-z) (\phi(t,x+z)-\phi(t,x))
\\ & \qquad\qquad\quad
-\delta_{\mu}(y-x) \nabla \phi(t,x)
\cdot z\, 1_{|z|<1} \Bigg)\, \pi(dz) \, dx\, dt\, dy\, ds
\end{split}
\end{equation}
Sending the ``temporal parameter" $\nu$ to zero in \eqref{fracinter} yields
\begin{align*}
J(\mu) & :=\lim_{\nu\to 0} J(\nu,\mu)
\\ & =\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \int_{{{\mathbb{R}}}^d \setminus \{0\}}
(u(t,x)-v(s,y))^+
\theta_{\nu}(s-t)
\\ & \qquad\qquad\qquad\quad \times
\Bigg(\delta_{\mu}(y-x-z) (\phi(t,x+z)-\phi(t,x))
\\ & \qquad\qquad\quad\qquad\qquad\qquad\quad
-\delta_{\mu}(y-x) \nabla \phi(t,x)
\cdot z\, 1_{|z|<1} \Bigg)\,\pi(dz) \, dx\, dt\, dy\, ds
\end{align*}
Under the additional requirements listed in \eqref{eq:add-cond}
we are also allowed to send the ``spatial parameter" $\mu$ to zero, resulting in
\begin{equation}\label{Jfin1}
\begin{split}
J & := \lim_{\mu\to 0} J(\mu)
\\ & = \iint \int_{{{\mathbb{R}}}^d \setminus \{0\}}
\Biggl( (u(t,x) -v(t,x+z))^+
\left( \phi(t,x+z) - \phi(t,x) \right) \\
&\qquad\qquad\qquad\qquad\quad
-(u(t,x)-v(t,x))^+ \nabla \phi(t,x)
\cdot z\, 1_{|z|<1} \Biggr) \,\pi(dz) \, dx\, dt.
\end{split}
\end{equation}
We divide the remaining discussion into two cases.\\
\textbf{Case 1: $|z|\pi(dz) \in L^1({{\mathbb{R}}}^d \setminus \{0\})$.}
Adding and subtracting identical terms we obtain
\begin{equation}\label{J}
J = \iint (u(t,x)-v(t,x))^+
\ensuremath{\mathcal{L}}[\phi](t,x) \, dt \, dx + E,
\end{equation}
where
\begin{align*}
|E| &\le \iint \int_{{{\mathbb{R}}}^d \setminus \{0\}}
|v(t,x+z)-v(t,x)|\, |\phi(t,x+z)-\phi(t,x)| \,\pi(dz) \, dt \, dx
\\ & \leq \Bigl(2 T\norm{v}_{L^\infty(0,T;L^1({{\mathbb{R}}}^d))}
\norm{\phi'}_{L^\infty(Q_T)}\Bigr) |z|
\in L^1({{\mathbb{R}}}^d \setminus \{0\};\pi(dz)),
\end{align*}
so that we can employ the dominated convergence theorem
to send $\phi \to 1$, and consequently $|E|\to 0$.\\
\textbf{Case 2: $|z|^2 \pi(dz) \in L^1({{\mathbb{R}}}^d \setminus \{0\})$
and $v \in L^\infty(0,T;BV({{\mathbb{R}}}^d))$.} In this case the error
term $|E|$ in (\ref{J}) can be estimated as follows:
$$
|E|\leq \Bigl( T \norm{v}_{L^\infty(0,T;BV({{\mathbb{R}}}^d)}
\norm{\phi''}_{L^\infty(Q_T)}\Bigr)
|z|^2 \in L^1({{\mathbb{R}}}^d \setminus \{0\};\pi(dz)),
$$
and again we can employ the dominated convergence theorem.
This concludes the proof of Theorem \ref{th:unique}.
\section{Proof of Theorem \ref{th:contdep} (continuous dependence)}\label{prcdep}
We again employ the doubling of variables
device as in the previous section, but with a slightly different
choice of the entropy function. For each $\ensuremath{\varepsilon}>0$, define
$$
{\rm sgn}\, _{\ensuremath{\varepsilon}}(\xi) =
\begin{cases}
-1, & \textrm{if $\xi<-\ensuremath{\varepsilon}$}\\
\sin(\frac{\pi}{2\ensuremath{\varepsilon}}\xi), & \textrm{if $ |\xi| \leq \ensuremath{\varepsilon}$}\\
1, & \textrm{if $\xi > \ensuremath{\varepsilon}$},
\end{cases}
$$
which is a $C^1$ approximation of ${\rm sgn}\, (\cdot)$.
This choice gives rise to a $C^2$ approximation
$\eta_\ensuremath{\varepsilon}(z)=\int_0^z {\rm sgn}\, _{\ensuremath{\varepsilon}}(\xi)\,d\xi$ of the entropy flux $\abs{z}$.
As before, we introduce the corresponding entropy flux functions
$\eta^{\ensuremath{\varepsilon}}(u,c)$, $q_i^{\ensuremath{\varepsilon}}(u,c)$, and $r_{ij}^{\ensuremath{\varepsilon}}(u,c)$.
We now employ the doubling variables technique using the test function
$$
\varphi(t,x,s,y) = \theta_{\nu}(s-t) \delta_{\mu}(y-x) \Theta_{\alpha}(t),
$$
where $\theta_{\nu}$, $\delta_{\mu}$ are symmetric approximate
delta functions with support in $(-\nu,\nu)$ and $B(0,\mu)$, respectively.
Fix a time $\tau$ from $(0,T)$.
For any $\alpha>0$ with $0<\alpha<\min(\tau_0,T-\tau)$, we define
$$
\Theta_{\alpha}(t)
=H_{\alpha}(t) - H_{\alpha}(t-\tau),
\quad
H_{\alpha}(t)=\int_{-\infty}^{t}
\theta_{\alpha}(\sigma)\,d\sigma.
$$
so that $\Theta_{\alpha}'(t)=\theta_{\alpha}(t)-\theta_{\alpha}(t-\tau)$.
Proceeding as in the previous section (cf.~also \cite{Chen:2005wf}) and
sending $\ensuremath{\varepsilon} \to 0$, we find
$$
-\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int
\abs{u-v} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\Theta_\alpha'(t)\, dx\, dt\, dy\, ds
\le I_{\mathrm{conv}} - I_{\mathrm{diff}}+I_{\mathrm{fdiff}},
$$
where
$$
I_{\mathrm{conv}} := \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \left[ G(u,v) - F(u,v) \right]\cdot
\nabla_x \delta_{\mu}(y-x) \theta_{\nu}(s-t) \Theta_\alpha(t)\, dx\, dt\, dy\, ds,
$$
$$
F(u,v) := {\rm sgn}\, (u-v) \left(f(u) - f(v)\right), \quad
G(u,v) := {\rm sgn}\, (u-v) \left(g(u) - g(v)\right),
$$
$$
I_{\mathrm{diff}} :=
\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \sum_{i,j=1}^d \Theta_\alpha(t) \theta_{\nu}(s-t)
\ensuremath{\partial_{x_ix_j}^2}\delta_{\mu}(y-x) \int_v^u {\rm sgn}\, (\xi -v)
\ensuremath{\varepsilon}_{ij}^{a-b}(\xi) \, d\xi \, dx\, dt\, dy\, ds,
$$
$$
\ensuremath{\varepsilon}_{ij}^{a-b}(\xi) := \sum_{k=1}^K \left(\sigma_{ik}^a(\xi) \sigma_{jk}^a(\xi)
- 2 \sigma_{ik}^a(\xi) \sigma_{jk}^b(\xi) + \sigma_{ik}^b(\xi)\sigma_{jk}^b(\xi)\right).
$$
and $I_{\mathrm{fdiff}}=
I_{\mathrm{fdiff}_1}+I_{\mathrm{fdiff}_2}$ with
\begin{align*}
I_{\mathrm{fdiff}_{1}} :=
\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \int_{|z|<1} &\abs{u-v}
\theta_{\nu}(s-t)\Theta_{\alpha}(t)
\\ & \quad \times
\Bigl[\delta_{\mu}(y-x-z)-\delta_{\mu}(y-x)
-\nabla \delta_{\mu}(y-x) \cdot z \Bigr]
\\ & \quad \qquad \times (m(z) - \tilde{m}(z))
\, dz\, dx\, dt\, dy\, ds
\end{align*}
and
\begin{align*}
I_{\mathrm{fdiff}_{2}}:=
\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \int_{|z|\geq 1}
\abs{u-v}&\theta_{\nu}(s-t)\Theta_{\alpha}(t)
\Bigl[\delta_{\mu}(y-x-z)-\delta_{\mu}(y-x) \Bigr]
\\ & \times (m(z)-\tilde{m}(z))
\, dz\, dx\, dt\, dy\, ds,
\end{align*}
By triangle inequality
\begin{align*}
& -\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \abs{u(t,x)-v(s,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\Theta_\alpha'(t)\, dx\, dt\, dy\, ds
\\ & \qquad \geq -\int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \abs{u(t,y)-v(t,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\abs{\Theta_\alpha'(t)}\, dx\, dt\, dy\, ds
\\ & \quad \qquad - \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \abs{v(t,y)-v(s,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\abs{\Theta_\alpha'(t)}\, dx\, dt\, dy\, ds
\\ & \quad \qquad - \int\!\!\!\!\int\!\!\!\!\int\!\!\!\!\int \abs{u(t,x)-u(t,y)} \theta_{\nu}(s-t) \delta_{\mu}(y-x)
\abs{\Theta_\alpha'(t)}\, dx\, dt\, dy\, ds
\\ & =: L + R_t + R_x.
\end{align*}
Keeping in mind that $v\in C(L^1)$ and $u\in L^\infty(BV)$, it
is standard to show that
$$
\lim_{\nu \to 0} R_t = 0, \quad
\limsup_{\alpha\to 0}\abs{R_x} \leq C\mu
$$
and moreover, since also $u(t)\to u_0,v(t)\to v_0$ as $t\to 0$,
$$
\lim_{\alpha\to 0} L
= \norm{u(\tau,\cdot) - v(\tau,\cdot)}_{L^1({{\mathbb{R}}}^d)}
-\norm{u_0-v_0}_{L^1({{\mathbb{R}}}^d)}.
$$
Following \cite{Chen:2005wf}, using $u\in L^\infty(BV)$ we conclude that
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0} \abs{I_{\mathrm{conv}}}
\leq C \tau \norm{f-g}_{\mathrm{Lip}(I)},
$$
and, exploiting also that $\int \abs{\partial_{x_i} \delta_\mu}\le C/\mu$,
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0} \abs{I_{\mathrm{diff}}}
\leq \frac{C}{\mu}
\tau \norm{(\sigma^a - \sigma^b)
(\sigma^a -\sigma^b)^{\mathrm{tr}}}_{L^{\infty}(I;{{\mathbb{R}}}^{d\times d})}.
$$
It remains to estimate $\abs{I_{\mathrm{fdiff}}}$.
First, we nconsider $I_{\mathrm{fdiff}_{1}}$.
Using the Taylor and Fubini theorems we obtain
\begin{align*}
\abs{I_{\mathrm{fdiff}_{1}}}
&=\int\!\!\!\!\int\!\!\!\!\int \int_{|z|<1} \int_0^1 (1-\tau)
\theta_{\mu}(s-t) \Theta_{\alpha}(t) (\tilde{m}(z)-m(z))
\\ & \qquad \times
\left( \int_{{{\mathbb{R}}}^d} \abs{u(t,x)-v(s,y)} D^2
\delta_{\mu}(y-x-\tau z)\, z \cdot z
\, dx \right)\, d\tau \, dz \, dy \, ds \, dt.
\end{align*}
Thanks to $|u(t,\cdot)-v(s,y)| \in BV({{\mathbb{R}}}^d)$, an integration by parts yields
\begin{equation}\label{i22bdd}
\begin{split}
I_{\mathrm{fdiff}_{1}} &=
\int\!\!\!\!\int\!\!\!\!\int \int_{|z|<1} \int_0^1 (1-\tau)
\theta_{\mu}(s-t) \Theta_{\alpha}(t) (\tilde{m}(z)-m(z)) \\
&\times \left( \int_{{{\mathbb{R}}}^d} \nabla \delta_{\mu}(y-x-\tau z)\cdot z\,
D_x\left(\abs{u(t,x)-v(s,y)}\right)\cdot z \, dx \right)
\, d\tau \, dz\, dy\, ds\, dt,
\end{split}
\end{equation}
where the inner integral is taken with respect to the bounded Borel
measure $D\left(|u(t,\cdot)-v(s,y)|\right)\cdot z$.
Since $|D(u(t,\cdot) - v(s,y))| \leq |D(u(t,\cdot))|$, the term inside the
parentheses in \eqref{i22bdd}, is upper bounded by
\begin{align*}
|z|^2 \int_{{{\mathbb{R}}}^d} \int_{{{\mathbb{R}}}^d}
|\nabla \delta_{\mu}(y-x-\tau z)|\,|dD(u(t,\cdot))(x)|\, dy
\le |z|^2\abs{u(t,\cdot)}_{BV({{\mathbb{R}}}^d)} \norm{\nabla \delta_{\mu}}_{L^1({{\mathbb{R}}}^d)},
\end{align*}
where we have used that $|D u(t,\cdot)|$ is finite and the Fubini's theorem to
first integrate with respect to $y$. Hence,
$$
\lim_{\alpha \to 0} \lim_{\nu \to 0}
\abs{I_{\mathrm{fdiff}_{1}}}
\leq \frac{C}{\mu}
\tau \int_{|z|<1} |z|^2
\abs{m(z)-\tilde{m}(z)}\, dz,
$$
where $C>0$ is a finite constant.
Similarly, relying again on the $L^\infty(BV)$ regularity of $u$, it
is not difficult to deduce via an integration by parts the estimate
$$
\lim_{\nu \to 0} \lim_{\alpha \to 0}
\abs{I_{\mathrm{fdiff}_{2}}}
\leq C\tau\int_{|z|\geq 1}
|z|\abs{m(z)-\tilde{m}(z)}\, dz.
$$
Finally, we collect the bounds we have obtained so far and then
optimize over $\mu$ to obtain the desired
continuous dependence estimate \eqref{contdest}.
|
1,314,259,992,979 | arxiv |
\section{INTRODUCTION}
Automatic control of complex, infinite-dimensional systems (\textit{i.e.}, dynamically evolving continua) such as soft robots as well as aircraft and underwater vehicles with coupled fluid-structure interactions remains challenging.
Model reduction provides a principled approach to reduce model complexity while capturing the essential physics required for controller synthesis.
In optimal control, we are interested in optimizing over a set of control inputs to track a desired trajectory or stabilize around an operating point.
In these settings, working with the full-order model (FOM) is computationally intractable.
The need to utilize high-fidelity models to control these challenging systems has resulted in significant research efforts towards application of reduced-order models (ROM) for controller design.
In this work we explore recent developments in Spectral Submanifold (SSM) theory \cite{haller2016nonlinear} for nonlinear model reduction and control.
SSMs are the smoothest invariant manifolds that act as nonlinear continuations of the eigenspaces from linearization of a system at a fixed point.
This nonlinear continuation is tangent to a corresponding spectral subspace of the linearized system.
The additional structure given by this continuation allows us to capture highly-nonlinear behavior outside the vicinity of the linear approximation.
Under certain conditions, these nonlinearities can be approximated arbitrarily well without ever increasing the size of the ROM.
\textbf{Contributions:} Motivated by the established theory on SSMs and their successful application to model reduction \cite{jain2021compute}, we propose the adaptation of SSM theory to automatic control.
In particular, we aim to synthesize optimal, nonlinear, periodic, feedback controllers on the reduced-order SSM to exploit the computational speed-up while retaining the fidelity of the high-order model.
Our contributions are threefold:
\begin{enumerate}[label=(\roman*)]
\item We present, for the first time, this novel model reduction technique in the context of control, enabling us to track (quasi-)periodic trajectories. In this setting we restrict ourselves to a special case of trajectory tracking and synthesize optimal control laws that guarantee the existence and persistence of lower-dimensional SSMs on which the true system trajectory lies.
\item We demonstrate the utility of SSMs in accurately capturing the nonlinear ``slow dynamics" of a system while neglecting its ``fast dynamics". This gives a natural setting in which our control effort is focused on the dynamics that persist.
\item We illustrate our approach on a pedagogical example, highlight its advantages and disadvantages, and motivate its application to high-dimensional models.
\end{enumerate}
\textbf{Related work: }
Most of the applications of model reduction for control exploit projection-based methods.
They involve a data-driven procedure to identify a linear subspace from simulation rollouts of the FOM.
This approach has been successfully leveraged in literature for real-time control of infinite-dimensional systems using Model Predictive Control (MPC).
The work in \cite{LorenzettiMcClellanEtAl2021} considers the effect of proper orthogonal decomposition (POD) on the closed-loop error dynamics and the authors develop a constraint-tightening scheme to ensure satisfaction of safety constraints in an MPC framework.
While similar works \cite{ghiglieri2014optimal, alla2015asymptotic, altmuller2014model} adopt POD-based MPC schemes for the control of certain PDE-classes, others explore different combinations of subspace identification and optimal control schemes.
In \cite{alla2017error}, the authors investigate the sub-optimality of LQR due to the projection error introduced by POD, while \cite{antil2010domain} considers linear quadratic optimal control using balanced truncation.
Direct application of projection-based methods for nonlinear systems is difficult since evaluation of the nonlinear terms results in a more expensive procedure than evaluating the full model directly, as the change of reference frame involves high-dimensional matrix multiplications \cite{farhat2015structure}.
To overcome this limitation, much of the literature involves construction of locally approximating \textit{linear} ROMs for which standard linear control techniques can be applied.
In \cite{huang2020balanced}, the authors propose an iterative LQR scheme combined with balanced truncation to control the 1D Burger's Equation.
The authors in \cite{TonkensLorenzettiEtAl2021} apply POD in a piecewise-affine fashion by reducing linear approximations of the high-fidelity model.
They then evaluate the nonlinearities through interpolation of the linear approximations and apply an MPC framework to control a soft robot.
While these approaches have been demonstrated to work well on various real-world systems, their performance and theoretical guarantees are limited to linear ROMs \cite{LorenzettiMcClellanEtAl2021}.
For highly nonlinear systems operating in less constrained workspaces, linear ROMs can result in low-fidelity surrogates that exhibit poor closed-loop performance and even instability.
This necessitates the need to capture the structure of the nonlinearities in a more direct way and we propose a new direction for addressing nonlinear model reduction for control.
\textbf{Organization:} In \Cref{sec:preliminaries}
we introduce our notation and definitions used in this work.
\Cref{sec:problem-statement} defines the optimal control problem where we introduce the tracking error in the periodic orbit of the FOM.
In the methodology in \Cref{sec:methodology}, we describe how we achieve model reduction using SSM in our setting and how we leverage the reduced representation of the dynamics to optimize the tracking error previously introduced in the full-order state-space.
\Cref{sec:validation} showcases the application of the theory on an illustrative example and provides the insights needed for tackling higher-dimensional examples.
We conclude this work in \Cref{sec:conclusions}, highlighting the most promising future avenues.
\section{PRELIMINARIES}
\label{sec:preliminaries}
This section provides the preliminaries that contextualize our approach. We first describe the system dynamics model in Section \ref{section:model} and then define necessary notions in \Cref{sub:spectral-subspace} to lay the groundwork for SSM theory.
\subsection{Notation}
\label{sub:notation}
The set of integers and reals are denoted by $\Z$ and $\R$, with their non-negative counterparts denoted by $\Z_+$ and $\R_+$.
The complex numbers are denoted by $\mathbb{C}$. $\Torus^\nfreq = \R^K / \left(2\pi\Z^K\right)$ represents the circle on the real line.
$C^k$ represents the space of $k$ continuously-differentiable functions and $C^a$ represents the space of analytic functions.
$L^2(V, W)$ is the space of square integrable functions from a complete vector space $V$ to $W$.
$\mathcal{O}(\cdot)$ represents the standard big-O notation. $\otimes$ is the tensor product, where $z^{\otimes 3} = z \otimes z \otimes z$.
\subsection{System Model}
\label{section:model}
\subsubsection{Full Order Model}
Consider the following continuous-time, control-affine, nonlinear dynamics with equilibrium point at the origin
\begin{align}
\label{eq:FOM}
\begin{cases}
\begin{aligned}
\dot{\vect{x}}(t) &= \mat{A} \vect{x}(t) + \vect{f}_0(\vect{x}(t)) + \sum_{i=1}^m \vect{f}_i(\vect{x}(t))\, u_i(t), \\
\vect{y}(t) &= \mat{H} \vect{x}(t),
\end{aligned}
\end{cases}
\end{align}
where the state $\vect{x}(t) \in \R^N$ is high-dimensional, \textit{i.e. } $N $ is large; $A \in \R^{N\timesN}$ is the stability matrix; $f_0(x)$ are the nonlinearities of the uncontrolled system; $f_i : \R^N \rightarrow \R^N$ for $i=1,...,m$ are nonlinear functions that describe the state-dependence of the control effort via an ${n_\ctrl}$-dimensional control input $u(t) \in \R^{n_\ctrl}$; the observed output of the system is denoted as $\vect{y}(t) \in \R^o$; and $H \in \R^{o \times N}$ is the selection matrix of output variables, where $o \ll N$.
In this work, we assume that the performance and output variables are the same and that they are perfectly observable.
We introduce the following assumption on the form of $A$.
\begin{assum}
\label{assum:Astability}
$A$ is negative definite, \textit{i.e. } $A \prec 0$.
\end{assum}
In other words, we assume that the origin $\bar{\vect{x}} = \vect{0}$ is a locally asymptotically stable equilibrium point.
Many physical systems and phenomena of interest such as soft robots and fluid structure interactions satisfy this assumption (possibly up to a shift in origin).
In addition, we introduce the following assumption on the form of $f_i$.
\begin{assum}
\label{assum:analytic}
The functions $f_0,\dots,f_m \in C^a$.
\footnote{We make this assumption for ease of exposition. In general, the right-hand side of System~\eqref{eq:FOM} can have finite smoothness, infinite smoothness, or be analytic; correspondingly the spectral submanifold defined in Section~\ref{sec:SSMprelims} is as smooth as the right-hand side.}
\end{assum}
We remark that this assumption is not particularly limiting since many physical systems (\textit{e.g. } soft robots) generically satisfy this assumption and we are only interested in controlling smooth behavior.
\subsection{Spectral Subspace}
\label{sub:spectral-subspace}
Consider the uncontrolled part of System~\eqref{eq:FOM}
\begin{align}
\label{eq:FOM_uncontrolled}
\dot{\vect{x}}(t) &= \mat{A} \vect{x}(t) + \vect{f}_0(\vect{x}(t)),
\end{align}
whose linearization around the origin is given by
\begin{align}
\label{eq:FOM_uncontrolled_lin}
\dot{\vect{x}}(t) &= \mat{A} \vect{x}(t).
\end{align}
For any eigenvalue $\lambda_j$ of $A$, there exists an eigenspace $\E_j \subset \R^N$ spanned by the (generalized) eigenvectors of $A$.
These eigenspaces are invariant subspaces of the linearized system~\eqref{eq:FOM_uncontrolled_lin}.
\defn{A spectral subspace $\E_{j_1, ..., j_n}$ of System~\eqref{eq:FOM} is defined as the direct sum of an arbitrary collection of eigenspaces of $A$ \textit{i.e. }}
\begin{align*}
\E \vcentcolon= \E_{j_1, ..., j_n} = \E_{j_1} \oplus \E_{j_2} \oplus ... \oplus \E_{j_n}.
\end{align*}
By linearity of System~\eqref{eq:FOM_uncontrolled_lin}, any spectral subspace is an invariant subspace of $A$.
In projection-based methods, ROMs are constructed by projecting the dynamics onto a nested hierarchy of the slowest $k$ spectral subspaces \textit{i.e. } $\E^k = \E_{1,...,k}$ where $\E^1 \subset \E^2 \subset \E^3 \subset \dots \subset \E^k$ and $k \ll N$.
However, such projections of the governing equations to spectral subspaces can be guaranteed to work only for linear systems and do not capture the effects of the nonlinear terms and control inputs of the FOM.
To find a faithful reduction of System~\eqref{eq:FOM}, it is necessary to reason about how the additional nonlinear terms and time dependent forcing influence the structure of the spectral subspace. To this end, we propose using SSMs and their reduced dynamics for reducing the following nonlinear control problem.
\section{PROBLEM STATEMENT}
\label{sec:problem-statement}
In this section we provide a formal problem definition of the full-order, periodic optimal control problem in Section \ref{ROMPOCP}.
\subsection{Periodic Optimal Control Problem}
\label{ROMPOCP}
In this work we design periodic orbits, minimizing the mean distance to some desired trajectory $z^\star(\freq\,t) \in \R^o$, where $\freq \in \R_+^\nfreq$ is the frequency of the reference trajectory.
Our approach is to formulate the following optimal control problem
\begin{align}
\label{eq:PeriodicOCP}
\min_{u(\cdot)} \quad & \frac{1}{T} \int_{0}^{T} \norm{z^\star(\freq\, t) - \vect{y}(t)}_2 dt \nonumber \\
\text{subj. to}& \quad
\text{System~\eqref{eq:FOM}} \\
& \quad \vect{x}(0) = \vect{x}(T) \nonumber .
\end{align}
In Equation~\eqref{eq:PeriodicOCP}, we minimize over a class of periodic feedback control laws of the form
\begin{align}
u(t) = \kappa(\vect{y}(t), \freq t),
\end{align}
where $\kappa \in L^2(\R^o \times \Torus^\nfreq, \R^{n_\ctrl})$.
For ease of notation, throughout the rest of the paper we denote $\vect{\varphi} = \freq t$.
Informally, we minimize the mean-squared trajectory error between our system's periodic orbit and the desired trajectory, after its fast dynamics have sufficiently decayed.
We emphasize that in this work, we are interested in synthesizing control laws that neglect transients and control for a periodic orbit.
We remark that while we consider the case of periodic control laws in this paper, our approach generalizes to the quasi-periodic setting.
\section{METHODOLOGY}
\label{sec:methodology}
\subsection{Spectral Submanifold Preliminaries}
\label{sec:SSMprelims}
An SSM serves as the unique nonlinear continuation of a nonresonant spectral subspace $\E$ for the nonlinear system~\eqref{eq:FOM_uncontrolled} and is defined as follows~\cite{haller2016nonlinear}.
\defn{An autonomous SSM $\mathcal{W}(\E)$, corresponding to a spectral subspace $\E$ of the operator $A$ is an invariant manifold of the nonlinear system~\eqref{eq:FOM_uncontrolled} such that}
\begin{enumerate}
\item $\mathcal{W}(\E)$ is tangent to $\E$ at the origin and has the same dimension as $\E$,
\item $\mathcal{W}(\E)$ is strictly smoother than any other invariant manifold satisfying condition 1 above.
\end{enumerate}
A slow SSM is associated to a spectral subspace containing the slowest decaying eigenvectors of the linearized system. Slow SSMs are ideal candidates for model reduction as typical nearby full system trajectories are exponentially attracted towards these manifolds and synchronize with the slow dynamics on such SSMs.
We synthesize such a controller by focusing on controlling the reduced dynamics along a slow SSM.
As the full system trajectories quickly and automatically synchronize with the dynamics on the slow SSM, we envision a minimal control effort arising from our synthesized controller on the slow SSM.
Hence, we assume a small control input by rescaling the control terms in system~\ref{eq:FOM} by a small scalar parameter $\varepsilon>0$ as
\begin{align}
\label{eq:FOM_epsilon}
\begin{cases}
\begin{aligned}
\dot{\vect{x}}(t) &= \mat{A} \vect{x}(t) + \vect{f}_0(\vect{x}(t)) + \epsilon g(x(t),\freq t), \\
g(x(t),\freq t) &= \sum_i^m \vect{f}_i(\vect{x}(t))\, \kappa_i(\mat{H} \vect{x}(t), \freq t),
\end{aligned}
\end{cases}
\end{align}
where the control input $\kappa_i(\vect{y}(t), \freq t)$ has periodic time-dependence with frequency $\freq$ for all $i=1,\dots,m$.
In this non-autonomous setting of periodic control, SSMs are envisioned similarly to the autonomous setting and the role of the fixed point is taken over by the periodic orbit $\gamma_{\epsilon}$ created by the small-amplitude control force. A nonautonomous, time-periodic SSM $\mathcal{W}(\E,\nnm)$ is then a fibre bundle that perturbs smoothly from the vector bundle $\gamma_{\epsilon}\times \E$ under the addition of the nonlinear and control terms in System~\eqref{eq:FOM_epsilon}. Hence, $\mathcal{W}(\E,\nnm)$ is $\frac{2\pi}{\freq}$-periodic in time.
\defn{A time-periodic SSM $\mathcal{W}(\E,\nnm)$, corresponding to a spectral subspace $\E$ of the operator $A$ is an invariant manifold of the nonlinear system~\eqref{eq:FOM_epsilon} such that}
\begin{enumerate}
\item $\mathcal{W}(\E,\nnm)$ is a subbundle of the normal bundle $N\gamma_{\epsilon}$ of the periodic orbit $\gamma_{\epsilon}$, satisfying $\dim \mathcal{W}(\E,\nnm) = \dim \E + 1$,
\item $\mathcal{W}(\E)$ perturbs smoothly from the spectral subspace $\E$ of the linearized system under the addition of nonlinear and control terms in System~\ref{eq:FOM_epsilon}.
\item $\mathcal{W}(\E,\nnm)$ has strictly more continuous derivatives along $\gamma_{\epsilon}$ than any other invariant manifold satisfying conditions 1 and 2 above.
\end{enumerate}
For any spectral subspace $\E$, the \emph{absolute spectral quotient}~\cite{haller2016nonlinear} is defined as
\begin{align}
\label{eq:spec_qt}
\Sigma(\E) &= \mathrm{Int} \left[ \frac{\min_{\lambda\in \mathrm{Spect}(A)}\mathrm{Re} \lambda}{\max_{\lambda\in \mathrm{Spect}(A|_\E)}\mathrm{Re} \lambda} \right].
\end{align}
This spectral quotient measures the fastest decay exponent outside the spectral subspace $\E$ relative to the slowest decay exponent within $\E$. It is crucial for determining the smoothness class of invariant manifolds in which the SSM uniquely exists. A high-value of the spectral quotient indicates a high-degree of overlap between invariant manifolds tangent to $\E$ at the origin, which is desirable for model reduction over slow SSMs.
For a small-enough control effort, the following theorem guarantees the existence of a time-periodic SSM, whose reduced dynamics provides us an exact nonlinear reduced-order model for control synthesis.
\thm{\label{thm:existence}Consider a spectral subspace $\E$ with $\dim \E = n$ and its associated eigenvalues (counting multiplicities) listed as $\lambda_1,\dots,\lambda_{n}$. Assume that the low-order nonresonance conditions
\begin{equation}
\label{eq:nonres}
\sum_{j=1}^{n} m_j \mathrm{Re} \lambda_j \ne \mathrm{Re} \lambda_l, \quad \lambda_l\not\in \mathrm{Spect}(A|_{\E}),\quad 2\le \sum_{j=1}^{n} m_j\le \Sigma(\E),
\end{equation}
hold for all eigenvalues $\lambda_l$ of $A$ that lie outside the spectrum of $A|_{\E}$ with $m_j \in \N$ and that Assumptions~\eqref{assum:analytic} and \eqref{assum:Astability} are satisfied.
Then the following holds:
\begin{enumerate}
\item There exists a time-periodic SSM, $\mathcal{W}(\E,\nnm)$ for system~\eqref{eq:FOM_epsilon} that depends smoothly on the parameter $\epsilon$ and is unique in the class of $C^{\Sigma(\E) +1}$ invariant manifolds.
\item $\mathcal{W}(\E,\nnm)$ can be viewed as an embedding of an open set $\mathcal{U}$ into the state space of System~\eqref{eq:FOM_epsilon} via the map
\begin{equation}
\vect{W}_\varepsilon(\vect{p},\vect{\varphi}):\mathcal{U}\subset \mathbb{C}^{n}\times \Torus^\nfreq \to \R^{N},
\end{equation}
with the periodic phase variable $\vect{\varphi}\in \Torus^\nfreq$.
\item There exists a polynomial function with respect to $\vect{p}$, $\vect{R}_\varepsilon(\vect{p},\vect{\varphi}):\mathcal{U}\to \mathbb{C}^{n}$ satisfying the invariance equation
\begin{align}
A\vect{W}_\varepsilon(\vect{p},\vect{\varphi}) + f_0(\vect{W}_\varepsilon(\vect{p},\vect{\varphi})) + \epsilon g(\vect{W}_\varepsilon(\vect{p},\vect{\varphi}),\vect{\varphi}) = \nonumber \\\mathrm{D}_p\vect{W}_\varepsilon(\vect{p},\vect{\varphi})\vect{R}_\varepsilon(\vect{p},\vect{\varphi}) + \mathrm{D}_{\vect{\varphi}}\vect{W}_\varepsilon(\vect{p},\vect{\varphi})\freq,
\label{eq:invariance}
\end{align}
such that the reduced dynamics on the SSM is given by
\begin{equation}
\label{eq:ROM}
\dot{\vect{p}} = \vect{R}_\varepsilon(\vect{p},\vect{\varphi}).
\end{equation}
\end{enumerate}
}
\begin{proof}
This is a restatement of Theorem 4 in~\cite{haller2016nonlinear} in our setting, which is deduced from the abstract results on whiskers of invariant tori in \cite{Haro2006}.
\end{proof}
\subsection{Model reduction using SSM}
Theorem~\ref{thm:existence} allows us to approximate $\vect{W}_\varepsilon(\vect{p},\vect{\varphi}),\vect{R}_\varepsilon(\vect{p},\vect{\varphi})$ in a neighbourhood of the origin as a Taylor expansion in the parametrization coordinates $\vect{p}$ with coefficients that depend periodically on the phase variable $\vect{\varphi}$. These periodic cofficients can be further Fourier-expanded resulting in Taylor-Fourier series for $\vect{W}_\varepsilon(\vect{p},\vect{\varphi}),\vect{R}_\varepsilon(\vect{p},\vect{\varphi})$. This means that the SSM and its reduced dynamics can be approximated arbitrarily well without ever increasing the dimension of $\E$. This is a highly desirable property for control since it enables one to faithfully capture the essential nonlinearities in the dynamics without increasing the dimensionality of the model.
As detailed in \cite{jain2021compute}, the solution of the invariance Equation~\eqref{eq:invariance} can be efficiently accomplished by solving the mappings $\vect{W}_\varepsilon$ and $\vect{R}_\varepsilon$ with the ansatz
\begin{align}
\vect{W}_\varepsilon(\vect{p}, \vect{\varphi}) &= \vect{W}_0(\vect{p}) + \varepsilon \vect{W}_1(\vect{p}, \vect{\varphi}) + \mathcal{O}(\varepsilon^2), \\
\vect{R}_\varepsilon(\vect{p}, \vect{\varphi}) &= \vect{R}_0(\vect{p}) + \varepsilon \vect{R}_1(\vect{p}, \vect{\varphi}) + \mathcal{O}(\varepsilon^2),
\end{align}
where the autonomous terms with $\varepsilon = 0$ are expressed as multivariate Taylor expansions:
\begin{align}
\vect{W}_0(\vect{p}) &= \sum_{j \ge 0} \vect{W}_{0,j} \vect{p}^{\otimes j}, \\
\vect{R}_0(\vect{p}) &= \sum_{j \ge 0} \vect{R}_{0,j} \vect{p}^{\otimes j},
\end{align}
with the unknown coefficients $\vect{W}_{0,j}$, $\vect{R}_{0,j}$ being $(j+1)$-tensors.
The $\mathcal{O}(\varepsilon)$ terms are expanded via a Taylor-Fourier series as
\begin{align}
\label{eq:polynomial-eps-ssm}
\vect{W}_1(\vect{p},\vect{\varphi}) &= \sum_{j \ge 0} \vect{W}_{1,j}(\vect{\varphi}) \vect{p}^{\otimes j}, \quad \vect{W}_{1,j}(\vect{\varphi}) = \sum_{\vect{h} \in \Z^K}\vect{W}_{1,j,h} e^{i \langle \vect{h}, \vect{\varphi} \rangle} \\
\label{eq:polynomial-eps-rdyn}
\vect{R}_1(\vect{p},\vect{\varphi}) &= \sum_{j \ge 0} \vect{R}_{1,j}(\vect{\varphi}) \vect{p}^{\otimes j}, \quad \vect{R}_{1,j}(\vect{\varphi}) = \sum_{\vect{h} \in \Z^K}\vect{R}_{1,j,h} e^{i \langle \vect{h}, \vect{\varphi} \rangle},
\end{align}
with $\vect{W}_{1,j,h}, \vect{R}_{1,j,h}$ denoting unknown Taylor-Fourier coefficients at degree $j$ and harmonic $h \in \N$. As detailed in~\cite{jain2021compute}, these unknown coefficients are determined by solving the invariance equation~\eqref{eq:invariance} in a recursive manner, where each recursion involves the solution of a linear system. These computations have been automated and demonstrated on nonlinear finite-element based applications featuring more than 100,000 degrees of freedom~\cite{jain2021compute}. SSMTool, an open-source implementation of this procedure is available at~\cite{SSMTool}.
\subsection{Exploiting the ROM for offline optimization}
We consider a generic periodic feedback control law, expressible through a truncated Taylor-Fourier series:
\begin{align}
\label{eq:ctrl-ansatz-taylor}
\kappa(\vect{y}, \vect{\varphi}) = \sum_{j = 0}^{\Upsilon} \vect{D}_j(\vect{\varphi}) \vect{y}^{\otimes j},
\end{align}
where $\Upsilon \in \N$ is the finite truncation order of the Taylor series. $\vect{D}_j(\vect{\varphi})$ is a tensor of order $j+1$ and dimension ${n_\ctrl}$, \textit{i.e. } $\vect{D}_0(\vect{\varphi}) \in \R^{n_\ctrl}$.
The coefficients are individually determined by the following (truncated) Fourier series:
\begin{equation}
\label{eq:ctrl-ansatz-fourier}
\vect{D}_j(\vect{\varphi}) = \sum_{\vect{h} \in \mathbb{H} \subset \Z^K} \vect{D}_{j,\vect{h}} e^{i \langle \vect{h} , \vect{\varphi} \rangle},
\end{equation}
where $\Gamma = \card{\mathbb{H}}$ is the finite truncation order of the Fourier series.
This allows us to consider the controller family $\kappa_{\vect{D}_{j,\vect{h}}}$ generated by all possible realizations of the parameters $\vect{D}_{j,\vect{h}} \in \R^{\Gamma\times{n_\ctrl}\timeso^j}, \, j \in \{0,\dots,\Upsilon\}$.
The number of parameters to optimize is therefore $n_p = \sum_{j = 0}^\Upsilon\, \Gamma\, {n_\ctrl}\, o^j$.
As a consequence, the previously derived mappings of the $\mathcal{O}(\varepsilon)$-perturbed SSM and its reduced dynamics in \cref{eq:polynomial-eps-ssm,eq:polynomial-eps-rdyn} are now dependent on the control parameters, \textit{i.e. } $\vect{W}_\varepsilon(\vect{p}, \vect{\varphi}, \vect{D}_{j,\vect{h}})$, $\vect{R}_\varepsilon(\vect{p}, \vect{\varphi}, \vect{D}_{j,\vect{h}})$.
We exploit this reduced order presentation, to find the optimal parameters $\vect{D}_{j,\vect{h}}^\star$ in an offline optimization procedure.
The ROM optimization formulation reads:
\begin{align}
\label{eq:reduced-optimization}
\min_{\vect{y}(\cdot), \> \vect{D}_{j,\vect{h}}} \quad \frac{1}{T} & \int_{0}^{T} \norm{z^\star(\freq\, t) - \vect{y}(t)}dt \\
\text{subj.} \quad \dot{\vect{p}} &= \vect{R}_\varepsilon(\vect{p}, \vect{\varphi}, \vect{D}_{j,\vect{h}}) \nonumber \\
\vect{y} &= \mat{H} \, \vect{W}_\varepsilon(\vect{p}, \vect{\varphi}, \vect{D}_{j,\vect{h}}) \nonumber \\
\vect{y}(0) &= \vect{y}(T) \nonumber .
\end{align}
We remark that $\vect{p} \in \R^n$, $n \ll N$.
Hence, this optimization problem is much more tractable than Problem \ref{eq:PeriodicOCP}, motivating the construction of the reduced model.
\subsection{Summary}
\label{sec:summary}
We summarize our method in Algorithm~\ref{alg:nlmr-periodic-ctrl-ssm}.
As an input we process the system matrices of System~\eqref{eq:FOM} with the asymptotically stable fixed point shifted to the origin.
$f_i(\vect{x})$ are defined by supplying the coefficients of their respective multivariate Taylor expansions.
\begin{algorithm}[h]
\caption{Periodic control with SSM}\label{alg:nlmr-periodic-ctrl-ssm}
\begin{algorithmic}[1]
\Require
\hspace*{-1em}\begin{itemize}
\item System~\eqref{eq:FOM}: $\mat{A} \in \R^{N \times N}, \, \vect{f}_i: \R^N \to \R^N,\, \mat{H} \in \R^{o \times N}$
\item $z^\star(\vect{\varphi})$
\end{itemize}
\setstretch{1.5}
\Ensure $ \mat{A} \vect{x}(0) + \vect{f}_0(\vect{x}(0)) + \sum_i^m \vect{f}_i(0)\, u(0) = \vect{0}$
\State $\lambda_i, \vect{v}_i \gets$ \Call{SpectralDecomposition}{$\mat{A}$}
\State $E^n$ $\gets \bigoplus_{k \in \{j_1 \dots j_n\}} v_k $
\Comment{Pick the $n$ slowest dynamics}
\State Define form of $\kappa_{\vect{D}_{j,\vect{h}}}(\vect{y}, \vect{\varphi})$ \Comment{dependent on $\vect{D}_{j,\vect{h}}$}
\State $\vect{W}_{\varepsilon,\vect{D}_{j,\vect{h}}}, \vect{R}_{\varepsilon,\vect{D}_{j,\vect{h}}} \gets$ \Call{ComputeSSMCoeffs}{$\mat{A}, f_i, E^n, \vect{D}_{j,\vect{h}}$}
\State $\vect{D}_{j,\vect{h}}^\star \gets$ \Call{Optimize} {\cref{eq:reduced-optimization}}
\State Apply feedback law $u(t) = \kappa_{\vect{D}_{j,\vect{h}}^\star}(\vect{y}(t), \vect{\varphi})$ to System \ref{eq:FOM}
\end{algorithmic}
\end{algorithm}
The control law is defined by picking the expansion order $\Upsilon$ and the $\Gamma$ integer combinations of desired frequency components, which determines the number of parameters $n_p$ that we optimize over.
Once optimal parameters for following the trajectory $z^\star(\vect{\varphi})$ are found, we apply our optimal feedback periodic control law to the FOM in System~\ref{eq:FOM}.
Assuming the designed periodic orbits are stable, this control strategy guarantees that our system trajectories will asymptotically converge to the $\varepsilon$-perturbed SSM containing this orbit.
\section{VALIDATION}
\label{sec:validation}
\subsection{Overdamped Pendulum Dynamics}
To illustrate the principles of SSM theory for control, we consider the simple example of an overdamped pendulum providing a two-dimensional spectral subspace associated with two distinct stable eigenvalues.
We denote the state space variables with $x_1 = \theta,\, x_2 = \dot{\theta}$.
The dynamics of the system are then given as
\begin{equation}
\label{eq:pendulum_dyn}
\begin{cases}
\dot{x}_1(t) = x_2(t) , \\
\dot{x}_2(t) = -\frac{b}{m\ell^2}\,x_2(t) - \frac{g}{\ell} \sin x_1(t) + \varepsilon \frac{1}{m\ell^2} u(t).
\end{cases}
\end{equation}
Considering the fixed point to be at the origin (corresponding to the pendulum in the downward position with no motion), we convert the system to the form denoted in Equation~\eqref{eq:FOM} by splitting it into a linear part $A$, the nonlinear part $f_0(x)$, and control-affine part:
\begin{equation}
\dot{x}(t) =
\underbrace{\begin{bmatrix}
0 & 1 \\
-\frac{g}{\ell} & -\frac{b}{m\ell^2}
\end{bmatrix}}_{{A}} x(t) + \underbrace{\begin{bmatrix}
0 \\
\frac{g}{\ell} x_1(t) - \frac{g}{\ell}\sin x_1(t)
\end{bmatrix}}_{{f_0}({x(t)})}+ \varepsilon \begin{bmatrix}
0 \\
\frac{1}{m\ell^2}
\end{bmatrix} u(t) .
\label{eq:pendulum-1st-order}
\end{equation}
Note that $\epsilon$ makes explicit that the magnitude of $u(t)$ should be \textit{moderate}; we provide more insight on this later in this section.
In our experiments, we set $\epsilon = 1$.
\begin{figure}
\parbox[t]{.4\linewidth}{\null
\centering
\includegraphics[height=1in,trim=11 15 10 0,clip]{figures/pendulum.pdf}%
\captionof{figure}{Pendulum illustration\label{fig:pendulumfig}}%
}
\parbox[t]{.6\linewidth}{\null
\centering
\vskip-\abovecaptionskip
\captionof{table}[t]{Pendulum parameters\label{tab:pendulumparams}}%
\vskip\abovecaptionskip
\resizebox{!}{.45in}{%
\begin{tabular}{crl}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Unit} \\
\midrule
m & 1 & \si{kg}\\
$\ell$ & 1 & \si{m}\\
g & 9.81 & \si{m/s^2}\\
b & 35 & \si{Nm s/rad}\\
\bottomrule
\end{tabular}}
}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{figures/pendulum-ssm-cdc.eps}
\caption{Overdamped pendulum phase portrait with SSM. $\E_1$ represents the slow spectral subspace to which we attach our SSM (depicted in green) while $\E_2$ represents the fast spectral subspace commensurate with the dynamics that converge quickly to the SSM.}
\label{fig:pendulum-ssm}
\end{figure}
The stability matrix $A$ has two eigenvalues $0 > \lambda_1 > \lambda_2$ with corresponding eigenvectors $\vect{v}_1$ and $\vect{v}_2$.
We pick the spectral subspace spanned by $\vect{v}_1$ (\textit{i.e. } $\E_1 = \text{lin}(\mathbf{v}_1)$) which corresponds to the slowest converging mode in $A$.
The slow and fast spectral subspaces $\E_1$ and $\E_2$, respectively, and the attached SSM to $\E_1$ are shown in Figure~\ref{fig:pendulum-ssm}.
Lastly, using Equation~\eqref{eq:spec_qt}, we compute the spectral quotient to be $\sigma(\E_1) = \mathrm{Int} \left[ \frac{\lambda_2}{\lambda_1}\right] = 122$. Recalling Theorem~\eqref{thm:existence}, we verify that the non-resonance conditions are met.
We consider periodic state feedback controllers for the pendulum of the following form, defined in terms of coefficients $\vect{u_p} \vcentcolon= \begin{bmatrix} u_{p1} \dots u_{p6}\end{bmatrix}$
\begin{align}
\label{eq:periodicinput}
\kappa_{\vect{u_p}}(\vect{x},\,\vect{\varphi}) \vcentcolon= &u_{p1} + u_{p2} \cos\vect{\varphi} + u_{p3} \sin\vect{\varphi} \\
&+ x_1 \left( u_{p4} + u_{p5} \cos\vect{\varphi} + u_{p6} \sin \vect{\varphi} \right) \nonumber
\end{align}
\subsection{SSM Derivation}
To compute an analytic expression of the SSM, we use graph-style parametrization.
Therefore, we first perform a change of basis into the spectral coordinates, \textit{i.e. }
\begin{align*}
\vect{x} = \mat{T} \vect{\xi} = \begin{bmatrix}\vect{v}_1 & \vect{v}_2\end{bmatrix} \begin{bmatrix}\xi_1 \\ \xi_2\end{bmatrix}.
\end{align*}
We express the SSM as a function over $\E_1$, \textit{i.e. } $\xi_2 = h(\xi_{1}, \vect{\varphi})$.
Hence, the mapping back to the FOM state space, is given by:
\begin{equation}
\vect{x} = \vect{W}(\xi_1) = \mat{T} \begin{bmatrix}
\xi_1 \\
h(\xi_1, \vect{\varphi})
\end{bmatrix}
\end{equation}
Using the transformation $\vect{\dot{\xi}} = \mat{T}^{-1}\, \vect{f}(\mat{T}\vect{\xi})$ and taylor expanding the non-polynomial, nonlinear terms (denoted $\tilde{f}_{nl}$) around the origin, we obtain the following dynamics
\begin{align*}
\dot{\xi} =
\Lambda \xi + \tilde{f}_{nl}(\xi_1, \xi_2)
+ \varepsilon \mat{T}^{-1} \begin{bmatrix}
0 \\
\frac{1}{m\ell^2}
\end{bmatrix} \kappa_{\vect{u_p}}(\vect{\varphi},\vect{W}(\xi_1)).
\end{align*}
Denoting the dynamics for
$\dot{\xi}_1$ as
$g_1(\xi, \vect{\varphi})$ and
$\dot{\xi}_2$ as
$g_2(\xi, \vect{\varphi})$,
we state the invariance equation as
\begin{align}
\label{eq:pendulum-invariance}\left. g_2 \right|_{\xi_2 = h(\xi_1,\vect{\varphi})} =
\left. \mathrm{D}_{\xi_1} h(\xi_{1}, \vect{\varphi})\, g_1\right|_{\xi_2 = h(\xi_1, \vect{\varphi})} + \mathrm{D}_{\vect{\varphi}} h(\xi_{1},\vect{\varphi})\, \freq .
\end{align}
The right hand side is given by the derivative in time of our SSM parametrization $h(\xi_{1}, \vect{\varphi})$, similarly to Equation~(\ref{eq:invariance}).
We solve this invariance equation with the ansatz:
\begin{align*}
h(\xi_{1}) &= c_1\,\xi_1^2 + c_2\,\xi_1^3
+ \epsilon h_1(\xi_1, \vect{\varphi}) , \\
h_1(\xi_1,\vect{\varphi}) &= c_3 + c_4 \cos\vect{\varphi} + c_5 \sin\vect{\varphi} + c_6\, \xi_1 \cos\vect{\varphi} + c_7\, \xi_1 \sin\vect{\varphi},
\end{align*}
By coefficient comparison we determine $\begin{bmatrix}c_1 \dots c_7\end{bmatrix}$ as a function of $u_p$.
In this way we obtain $h_{u_p}(\xi_{1}, \vect{\varphi})$, representing the perturbed SSM due to the parametric forcing by the periodic feedback controller.
Let us define $p \vcentcolon= \xi_1$.
Then, the reduced dynamics of the full system in Equation~\eqref{eq:pendulum_dyn} is represented on the SSM as
\begin{align}
\label{eq:reduced-pendulum}
\dot{p} = g_1(p, h_{u_p}(p, \vect{\varphi}))
\end{align}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/pendulum-periodic-cdc-theta-omega-tracking-error.pdf}
\caption{FOM and ROM tracking performance. The top figure represents tracking performance for the desired $\theta^\star(t)$. The bottom figure represents the untracked state $\dot{\theta}$. Simulation is carried out for five time-periods \textit{i.e. } $t_f = 5T$. The steady-state error between the closed-loop system and the desired trajectory is RMSE($\theta$) = $0.49$ deg and RMSE($\dot{\theta}$) = $0.19$ rad/s.}
\label{fig:theta-tracking-error}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/pendulum-periodic-cdc-theta-tracking-far-x0.pdf}
\caption{Convergence to the periodic orbit from $\theta_0$ = -180 deg. The ROM on the SSM perfectly captures the nonlinearities even when far away from the linearization point, \textit{i.e. } the origin.}
\label{fig:pendulum-far-x0}
\end{figure}
\subsection{Controller Performance}
In this section we hope to showcase the predictive capability of SSMs for control synthesis as well as give insights on the limitations of our approach.
We motivate the extension of this work to higher-dimensional problems which satisfy the conditions set forth in our pedagogical experiment.
As shown in Algorithm~\ref{alg:nlmr-periodic-ctrl-ssm}, we first compute the reduced-order representation in Equation~\eqref{eq:reduced-pendulum}.
Our desire is to track the trajectory $\theta^\star(\freq t) = 30 + 60 \sin(\freq t)$ (in degrees), where $\freq = \pi$ (rad/s).
We simulate the trajectory and dynamics for five time periods, ensuring that the dynamics in Equation~\eqref{eq:pendulum_dyn} achieves its periodic orbit.
We then compute the optimal parameters of our periodic feedback control law $u(t)$ in Equation~\eqref{eq:periodicinput} by solving Problem~\ref{eq:reduced-optimization}.
We use the CMA-ES optimization algorithm \cite{hansen2006cma} implemented in the KORALI framework \cite{martin2021korali}.
After two time-periods, System~\eqref{eq:pendulum_dyn} achieves its periodic orbit, hence we set $t_1 = 2T$.
Figure~\ref{fig:theta-tracking-error} shows the closed-loop tracking performance for both the reduced model trajectory and the full order trajectory with the optimal coefficients from the ROM optimization.
In other words, we compare the full system's evolution $\vect{x}(t)$ with the reduced system's evolution $\vect{W}(\vect{p}(t))$.
As expected, after the small initial transient, the full system trajectory is quickly attracted to the periodic orbit induced by our controller.
There is no noticeable difference between FOM and ROM trajectories, meaning that the FOM trajectory lies as expected on the SSM and its $\varepsilon$-perturbation captures the periodic motion well.
In Figure~\ref{fig:pendulum-far-x0} we perform the same experiment, but we initialize the pendulum at $\theta=-180$ deg.
Despite the large distance to the origin and the significant nonlinearities, the ROM still evolves in the same manner as the full-order model.
Furthermore, the system converges to the periodic reference trajectory, as desired.
Figures~\ref{fig:ssm-robustness} and \ref{fig:manifold-transform} highlight the relationship between the spectral quotient and the robustness of the manifold under forcing.
These figures show that the small-$\varepsilon$ assumption on the applied input discussed in Section~\ref{sec:SSMprelims} is nuanced and depends on the dynamics of the system -- specifically the spectral quotient.
Notice that in Figure~\ref{fig:ssm-robustness}, increasing the spectral quotient allows us to increase the allowed forcing amplitude without significant change in error.
This would correspond to a \textit{preservation} of the periodic orbit in Figure~\ref{fig:manifold-transform} as we increase control effort to even larger amplitudes.
As shown in Figure~\ref{fig:manifold-transform}, for systems with low spectral quotient, the SSM quickly disassembles as we increase the amplitude of the input.
We stress that while in this example the spectral quotient and damping are directly related, it is important to distinguish between the two when considering the previous discussion.
Most structural dynamics applications feature small damping, but high spectral quotients because higher frequency modes exhibit higher damping ratios in comparison to low-frequency modes (see~\cite{Jain2018} for an analytic calculation of spectral quotients in a beam, for instance).
The results presented here show promise for applying SSM-based control strategies to robotic systems with continuum-based models such as soft robots.
In these systems the spectral quotients are expected to be high and are in fact, infinite in the continuum limit of structural finite-element models.
Indeed, in our recent work we show the applicability to higher-dimensional models with a data-driven SSM approach, as we discuss in \Cref{sub:future-works}.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{figures/ssm-robustness.eps}
\caption{Error comparison between FOM and ROM for different spectral quotients and forcing amplitudes. As expected, a higher forcing amplitude leads to destruction of the manifold, since the $\varepsilon$-order perturbation is not small anymore. The allowed scale of $\mathcal{O} (\varepsilon)$ is driven by the spectral quotient, and in particular for our pendulum example, the damping coefficient.}
\label{fig:ssm-robustness}
\end{figure}
\section{CONCLUSIONS AND FUTURE WORKS}
\label{sec:conclusions}
In this paper, we presented the first application of SSM theory for control.
In particular, we investigated the periodic setting and extended existing theoretical guarantees to synthesize optimal control policies for the purpose of periodic trajectory tracking.
All existing ROM-based optimal control algorithms project the dynamics onto a linear subspace, resulting in the need to increase the dimension of the ROM to improve predictive capability for closed-loop control.
In contrast, we reason directly about the nonlinearities during the reduction process using the powerful existence and uniqueness guarantees provided by the SSM.
We validated our approach on an illustrative example and provided insights on the persistence of the SSM under control inputs.
\subsection{Future Works}
\label{sub:future-works}
There are numerous extensions and applications of this work.
Direct applications to robotic platforms are appealing, such as highly-nonlinear soft robots or robotic fish with periodic tail actuation and/or undulation due to periodic muscle contraction \cite{wang2015fishaveraging}.
Similar to the pendulum example, we expect the ROM behavior to be coherent with the FOM dynamics even far away from the static equilibrium, enabling larger controllable workspaces, which are difficult to address with current piecewise-linear reduction techniques that have been investigated so far.
It would be interesting to apply more sophisticated control schemes which exploit the embodied intelligence of continuum robots and their in-resonant dynamics to produce hyper-efficient motions.
Our most recent work on applying data-driven Spectral Submanifold Reduction (SSMR) for nonlinear optimal control of a soft robot \cite{alora2022datadriven} shows the predictive capability of SSMs for real-world, high-dimensional robotic systems.
By learning control-oriented models on low-dimensional SSMs, the proposed SSMR-based MPC approach outperforms both model-based and learning-based state-of-the-art methods in tracking performance and computational efficiency.
Further experimental and data-driven validation remains of great value to emphasize the applicability of the SSM-based approaches.
Open questions on generic time-dependent control inputs causing the SSM to lose its invariance provide future avenues of research.
For instance, characterizing model uncertainties can be useful for constraint-tightening schemes in safety-critical applications.
While data-driven SSMR shows significant promise, reliance on experimental data makes it difficult to apply these approaches in the design process.
Thus, it remains worthwhile extending the model-based approach highlighted in this work for the design and control of high-dimensional, exotic robotic systems.
\begin{figure}
\includegraphics[keepaspectratio, width=8.5cm,clip,trim=0 0 0 0 ]{figures/manifold_transform.eps}
\caption{Periodic orbits of a pendulum ($b = 7$) with relatively low spectral quotient ($\sigma(E) = 2$) versus increasing torque amplitude. This explicitly shows that the SSM is less robust to large forcing when the spectral quotient is low, resulting in a disassembling of the SSM.}
\label{fig:manifold-transform}
\end{figure}
\printbibliography
\end{document}
|
1,314,259,992,980 | arxiv | \section{Conclusion}
In this paper, we focus on synthesizing diverse and natural human motions in the given scene environment guided by target action sequence.
We decompose the diversity of scene-aware human motions into three levels, namely the diversity of action-conditioned human-scene interactions, the diversity of obstacle-free paths, and the diversity of body movements.
To comprehensively leverage the inherent diversity of human motions, we propose a novel hierarchy framework with each component accounting for each level of the diversity.
Thanks to the effective decomposition of diversity and elaborated designed modules, our framework is able to produce various vivid human motions in the scene across all three levels with improved efficiency and generality.
Furthermore, the factorized design of our framework make it can be easily incorporated into other human motion synthesizing frameworks.
\section{Related Works}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{Figures/Framework.pdf}
\caption{\small\textbf{Overview of the framework.} Our framework is composed of three stages. Given the target actions and the scene contexts, our framework first generates human-scene interaction anchors by firstly synthesizing scene-agnostic poses via the pose network and then placing the poses into the scene guided by the scene contexts and Pose Refiner. Then the framework produces diverse planning paths through the adapted A$*$ algorithm that is amended with the novel Neural Mapper. At last, the motion completion network is leveraged to synthesize natural human motions guided by the anchors and following the planned paths.}
\label{fig:framework}
\end{figure*}
\paragraph{Motion Synthesis.}
Early works~\cite{barsoum2018hp, cai2018deep, yang2018pose,yan2019convolutional,harvey2020robust, xu2020hierarchical, pavllo2018quaternet} focus on synthesizing natural body poses and neglect the influences of other factors such as action and environments.
Recent studies begin to explore the relationship between human motions with actions and scene contexts.
Recent works~\cite{chuan2020action2motion,cai2021unified} generates human pose sequences with a CVAE model~\cite{sohn2015learning} based on the given action labels.
ACTOR~\cite{petrovich21actor} builds up a transformer based on CAVE to synthesize human motion sequence directly from the given action label.
Cao~\emph{et al.}~\cite{cao2020long} propose a three-stage motion prediction method that can predict different human motions with different destinations.
Wang~\emph{et al.}~\cite{Wang_2021_CVPR} extend CSGN~\cite{yan2019convolutional} to explore the influence of 2D scene contexts on human motion synthesis.
Wang~\emph{et al.}~\cite{wang2021synthesizing} build up a framework to synthesize human motions in the 3D scene controlled by the given pairs of begin-end points.
SAMP~\cite{hassan_samp_2021} extends~\cite{StarkeZKS19} to use 3D oriented objects to facilitate the synthesis of human motions with specific action labels.
Besides, a planning module is incorporated into their framework to find obstacle-free paths.
The limitations of previous works~\cite{hassan_samp_2021,wang2021synthesizing} mainly lie in their reliance on predefined objects or positions, which constrain their ability to explore the inherent interaction diversity of synthesized scene-aware human motions.
In this work, we aim to overcome the limitations of the previous works and synthesize diverse motions guided by target action sequences in the given scenes.
To achieve this, we first synthesize diverse human-interaction anchors, which interacts with different objects in the scene.
Then we plan diverse paths and complete diverse body movements between these anchors.
\paragraph{Motion Prediction.}
Motion prediction is closely related to our problem.
Different from the motion synthesis, the goal of this task is to predict human dynamics in the future with the given moving orientations or previous motions.
Martinez~\emph{et al.}~\cite{martinez2017human} and ERD~\cite{fragkiadaki2015recurrent} proposed motion prediction framework based on the Seq2Seq model~\cite{sutskever2014sequence}.
Ac-LSTM~\cite{li2017auto} mixes synthesized frames and observed frames to enhance the capability of LSTM~\cite{hochreiter1997long} during the training stage.
The graph convolution network~\cite{KipfW17, yan2018spatial}is widely used in recent motion prediction~\cite{Li_2020_CVPR, Cui_2020_CVPR, mao2019learning}.
These methods model dynamic spatial and temporal relationships between the obvious frames and the future frames.
Different from these works, our goal is to synthesize motions without prior knowledge of the previous motions.
\section{Introduction}~\label{sec:intro}
The capability of synthesizing long human motion sequences is essential for a number of real-world applications, such as virtual reality and robotics.
Beyond early attempts that consider body movement synthesis in isolation~\cite{barsoum2018hp, cai2018deep, yang2018pose, yan2019convolutional,zhang2021we}, recent works~\cite{cao2020long,wang2021synthesizing, Wang_2021_CVPR, hassan_samp_2021} begin to explore the influences of surrounding scenes on human motion synthesis for different actions.
Limited by the 2D representation of scene context~\cite{cao2020long,Wang_2021_CVPR} or the reliance on manually assigned interacting targets~\cite{wang2021synthesizing,hassan_samp_2021},
these approaches mainly focus on modeling the body movements and fail to comprehensively investigate the inherent diversity of scene-aware human motions.
In order to synthesize long-term human motions guided by the scene context and the target action sequence,
we propose to model the inherent motion diversity across different granularities, each contributing to different aspects of human motion.
As shown in Figure~\ref{fig:teaser}, the diversity of scene-aware human motions can be factorized into three levels, given the target action sequence~(\eg~A man lies first. Then he sits in different places. At last, he stands somewhere.).
Firstly, given the surrounding scene context and the target action sequence, there exists a distribution of valid locations to realize the actual human-scene interactions for each of these actions~(\eg~We can sit on any chairs or beds and stand on the ground).
Different locations can be sampled from the distribution and serve as the anchors of the whole synthesized motion sequence.
Based on those anchors, we can then follow various paths to bridge them one by one.
Finally, our body poses also differ from case to case when we move along the paths to connect all anchors.
We demonstrate these three levels of diversity in Figure~\ref{fig:teaser}.
Existing attempts for scene-aware human motion synthesis~\cite{wang2021synthesizing,hassan_samp_2021} only emphasize the last level of diversity~(\eg~walking to the pre-defined object or position in the scene) via manually assigning the interaction locations and motion paths.
Consequently, the importance of the scene semantics is substantially muted, as it mainly affects the distribution of valid interaction anchors and the distribution of valid motion paths.
To faithfully capture the diversity of scene-aware human motions, we propose a novel three-stage motion synthesis framework, each stage of which is responsible for modeling one level of the aforementioned diversity.
For \textbf{diverse human-scene interaction anchors}, we design our pose placing framework for the given action sequence.
Different from~\cite{PSI:2019, PLACE:3DV:2020} which only consider the influence of scene context, we first synthesize scene-agnostic poses according to the target action via a conditional VAE (CVAE)~\cite{sohn2015learning}.
Then we follow the practice of POSA~\cite{Hassan:CVPR:2021} to place these poses into the scene.
To be specific, the 3D scene is uniformly split into a set of non-overlapping grids, each of which is associated with a validity score that measures its compatibility as a candidate for placing the poses.
We make two modifications to the original placing method used by POSA
First, we introduce the position relationship between poses with the same action label to enhance the placing diversity by avoiding them being placed to the nearby positions.
Furthermore, we leverage another CVAE model as the placing refiner to produce diverse offsets for each discrete grid.
Examples of generated anchors are depicted in Figure~\ref{fig:teaser} (a).
To produce \textbf{diverse obstacle-free motion paths} following the sampled anchors, we employ an adapted A$^*$ algorithm over the discrete 3D grids as the path planner.
The standard A$^*$ algorithm used by previous works~\cite{hassan_samp_2021} only generates deterministic paths as they only consider collision between objects and distances to the target locations.
To model the inherent diversity of motion paths, we amend the original algorithm with a trainable stochastic module learned in a data-driven manner.
The new module, named Neural Mapper, can provide dynamic scene-conditioned probabilistic guidance to the A$^*$ algorithm, so that the algorithm can automatically produce diverse yet natural paths given the deterministic scenes and location anchors.
We show several examples of generated diverse paths given the same start and end locations in Figure~\ref{fig:teaser} (b).
Lastly, we propose a novel Transformer-based CVAE, called motion completion network, to synthesize \textbf{diverse body movements} guided by the paths generated in the previous step.
Inspired by~\cite{petrovich21actor}, we leverage Transformer as the basic architecture for synthesizing continuous and smooth motions.
Differently, we focus on diverse motion completion of poses with long-term distance and different actions, rather than synthesize motions for the single action~\cite{petrovich21actor}.
Therefore, this motion completion network first generates diverse moving trajectories, loosely following the paths sampled by the aforementioned A$^*$ algorithm.
The body poses are then produced by taking the scene contexts, action labels, human-scene interaction anchors, and synthesized trajectories as inputs.
To summarize our contributions:
1) We analyze the \textbf{inherent diversity} of the human motion and decompose it into three components, namely the diversity on human-scene interaction anchors, paths, and body poses.
2) We propose a novel three-stage framework to \textbf{faithfully capture the diversities} of scene-aware human motions.
This framework can automatically synthesize human motions following these diversities with the condition action labels.
Qualitative and quantitative results on datasets such as PROX~\cite{PROX:2019}demonstrate that our method significantly surpasses previous approaches in terms of diversity and naturalness.
3) In the proposed framework, we make several technique contributions for this task, including the action conditioned pose placing framework for generating diverse human-scene interaction anchors, Neural Mapper for planning diverse paths, and motion completion network for producing diverse and continuous motions. With our decomposition on motion diversity, these technique contributions can achieve our goal efficiently and effectively.
\section{Methodology}
\subsection{Overview}
We first formally define the task of scene-aware 3D human motion synthesis.
We use triangular mesh $S = (v^s, f^s)$ to represent the scene context, where $v^s$ and $f^s$ stand for vertices and faces.
Our task is to synthesize diverse 3D human motions in the given scene context $S$, driven by a sequence of target action labels $A = (a_1, a_2, ..., a_N)$.
Each label stands for one scene-related human action, such as sitting or laying.
The synthesized 3D human motions are represented as a sequence of SMPL-X models~\cite{pavlakos2019expressive} described by their parameters $\{P_0, ..., P_T\}$,
where $P_i$ is composed of $(t_i, \phi_i, \theta_i)$,
$t_i \in \mathbb{R}^3$ is the global translation, $\phi_i \in \mathbb{R}^6$ is the global orientation represented in 6D continuous rotation~\cite{zhou2019continuity}.
$\theta_i \in \mathbb{R}^{32}$ is the body pose parameters, represented in the form of VPoser~\cite{pavlakos2019expressive}.
We use mean values for remaining SMPL-X parameters, including shape parameters, facial parameters, and hand poses.
The overview of our framework is depicted in Figure~\ref{fig:framework}.
We aim to solve this challenging problem in a hierarchical manner via exploiting the inherent properties of the scene-aware human motions.
Our framework first generates diverse human-scene interaction anchors for the given actions.
In this step, the framework first produces scene-agnostic poses corresponding to the action labels and then places these poses into the scene considering the compatibility between the synthesized poses and the scene.
In the next step, we leverage a path planning module to produce diverse obstacle-free paths under the guidance of the synthesized anchors from the first step.
Finally, a motion completion module is adopted to synthesize diverse body movements that fill in the missing motions between consecutive anchors while roughly following the planned paths from the second step.
In the following, we introduce our modules in detail.
\subsection{Human-Scene Interaction Anchor Synthesis}
\label{subsec:interaction}
We first synthesize human-scene interaction anchors.
Unlike previous works~\cite{PSI:2019,PLACE:3DV:2020} that only condition human motion synthesis on the scene context,
we use action labels describing interaction types as an additional condition.
To be specific, we first synthesize scene-agnostic poses corresponding to the action labels.
Then we follow the practice of POSA~\cite{Hassan:CVPR:2021} with several modifications to diversely place the synthesized poses into the scene.
This design affords us more control over the final synthesized motions.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/placing.pdf}
\small\caption{\textbf{Human-scene interaction anchor generation.} There are two steps to synthesize human-scene interaction anchors. The first step is generating diverse scene-agnostic poses conditioned on the target action, as shown in~(a). The second step is placing synthesized poses in the given scene, as depicted in~(b).
}
\label{fig:placing}
\end{figure}
\paragraph{Scene-Agnostic Pose Synthesis.}
As shown in Figure~\ref{fig:placing}~(a), we follow the standard CVAE framework to synthesize scene-agnostic poses $\theta_i$ with the target action $a_i$.
To be specific, we first sample noises from the prior Gaussian distribution and encode them with a fully-connected layer.
Then, we use the one-hot vector $a_i$ to represent the action condition and encode it with another fully-connected layer.
These two features are added up and then served as additional input besides the noise.
The model outputs the synthesized pose $\theta_i$, which is directly used as the body pose for the anchor $P_i$ for $i$-th anchor.
\paragraph{Scene-Conditioned Anchor Placing.}
In this step, we place the scene-agnostic poses $(\theta_1, \theta_2, ..., \theta_N)$ into the given scene.
There are two aspects to be taken into consideration in this step.
The first one is how to place poses to locations with compatible scene structure and interaction semantics.
The other one is how to efficiently find multiple reasonable locations given a pose.
Therefore, we first select our placing candidates following the practice of POSA~\cite{Hassan:CVPR:2021}.
Specifically, each candidate consists of a translation parameter $\bar t_i$ and an orientation parameter $\bar o_i$ for the anchor $P_i$.
We split the given scene into uniform non-overlapping discrete girds as translation candidates.
For each discrete gird, we then uniformly sample eight different orientations that are parallel with the ground plane to build orientation candidates.
Each translation candidate is paired with one of its associated orientations to form one placing candidate.
For each scene-agnostic pose $\theta_i$, we then rank all the placing candidates by their compatibility scores with the pose, which is proposed by~\cite{Hassan:CVPR:2021} that considers both the affordance and penetration.
An intuitive idea is to select the candidate with the best score.
However, our empirical study shows that candidates with the same action labels tend to be located close to each other since the same action usually shares similar physical and semantic structures, as shown in the first row of Figure~\ref{fig:placing_show}.
To increase the placing diversity, we introduce an additional penalty on the locations that have been occupied by anchors with the same action labels.
As shown in Figure~\ref{fig:placing_show}, this new penalty helps produce more diverse placing candidates for similar poses.
In this way, we can sample an initial placing candidate $(\bar t_i, \bar \phi_i)$ for each pose $\theta_i$.
The initial anchor $\bar P_i = (\bar t_i, \bar \phi_i, \theta_i) $ is then constructed subsequently.
In practice, we further adopt another sub-module called Place Refiner to improve the micro diversity of the placing candidates.
Place Refiner is implemented as a CVAE model that takes the noise of $\theta_i$, the scene context encoded by the PointNet~\cite{qi2017pointnet} and the initial anchor $\bar P_i$ as the input.
It outputs the offset $(\Delta t_i, \Delta \phi_i)$ to the sampled position and orientation $(\bar t_i, \bar \phi_i)$.
The final position and orientation are obtained as $t_i = \bar t_i + \Delta t_i$ and $\phi_i = \bar \phi_i + \Delta \phi_i$.
The framework of Place Refiner is depicted in Figure~\ref{fig:placing} (b).
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/Map.pdf}
\small\caption{\textbf{Map building.} The map for our planning algorithm is built based on the collision detection (a) and Neural Mapper (b). Neural Mapper provides diverse moving probability for each neighbor gird to plan diverse paths.}
\label{fig:map}
\end{figure}
\subsection{Diverse Path Planning}~\label{subsec:planning}
In this step, we discuss how to generate diverse obstacle-free paths from human-scene interaction anchors.
Previous works such as SMAP~\cite{hassan_samp_2021} often use standard $A^*$ searching~\cite{hart1968formal} for this purpose.
The $A^*$ algorithm tends to generate deterministic shortest path for practice. However, humans usually move stochastically in the given scene.
To reflect the diversity of human path planning, we incorporate the standard $A^*$ algorithm with scene-aware random information concerning the diversity of human motion.
To begin with, we first discuss how to apply the standard $A^*$ algorithm into our scenario.
We first divide the whole 3D scene into the same set of non-overlapping discrete grids as in Section~\ref{subsec:interaction}.
We then define and calculate the cost function $f$ for each grid in the $A^*$ algorithm~\cite{hart1968formal} as:
\begin{align} \label{eq:astar}
f(q) = g(q) + h(q);q \in \cN(p),
\end{align}
where $g(q)$ measures the cost for moving from the beginning point to grid $q$,
and $h(q)$ measures the cost between grid $q$ and the target grid, during searching points as the next step for $p$ in the neigbourhood $\cN(p)$.
To ensure obstacle-free paths, we further filter out inaccessible grids that might have collisions with the human body.
The collisions are detected via placing a cylinder model that approximates the volume of a human at each grid.
We show an example in the right of the Figure~\ref{fig:map} (a), where red stands for valid and blue stands for invalid.
After calculating $f$ for each grid and excluding invalid grids, an obstacle-free path connecting two human-scene interaction anchors can thus be obtained using the standard $A^*$ algorithm.
It is worth noting that the path obtained in this manner is deterministic and fixed for the same pair of two human-scene interaction anchors.
An intuitive solution to incorporate diversity in path planning is appending the cost function $f$ defined in Equation~\eqref{eq:astar} with a random noise term.
This strategy sounds feasible but fails to generate reasonable paths, which is demonstrated by the examples shown in the top two rows of Figure~\ref{fig:planning_show}.
To this end, we replace the random noise term with a controllable signal $m$ produced by another CVAE, referred as Neural Mapper.
For each grid $p$, Neural Mapper takes sampled latent code and the local scene context feature obtained via BPS~\cite{PLACE:3DV:2020,GRAB:2020} as the input and outputs the feasibility score for each neighbor grid $q \in \cN(p)$.
Based on the Neural Mapper, the cost function is updated as:
\begin{align} \label{eq:astar}
f(p,q) &= g(q) + h(q) + (1 - m(p, q));q \in \cN(p).
\end{align}
The score of $m$ indicates the feasibility of moving from the current grid to this adjacent one so that we can build the cost as $1-m$ to reflect the moving guidance by our Neural Mapper.
The Neural Mapper is trained in a data-driven manner thus it can help the $A^*$ algorithm to generate diverse and reasonable paths.
We show several examples produced by Neural Mapper in the bottom row of Figure~\ref{fig:planning_show}.
Without complex manually designed conditions and constraints,
the proposed Neural Mapper equips the $A^*$ algorithm with the ability to find diverse obstacle-free paths in a flexible and generalizable way.
In Neural Mapper, we can easily change the characteristics of sampled paths by restricting the latent codes,
without hurting their naturalness and coherency.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/planning_show.jpg}
\small\caption{\textbf{Examples for diverse planning.} We sample different paths from $\textcircled{1}$ to $\textcircled{2}$ with different strategies. \textbf{Random Noise} means sampling different weights for each discrete gird. \textbf{Shared Noise} means all discrete girds share the same noise vector. \textbf{Neural Map} refers to probability generated by our Neural Mapper. The results demonstrate that our method is more effective in generating diverse and natural paths than simply adding randomly noise does.}
\label{fig:planning_show}
\vspace{-0.3cm}
\end{figure}
\subsection{Motion Completion}~\label{sec:completion}
With the obstacle-free path obtained from path planning, we are now ready to complete the missing motions between consecutive human-scene interaction anchors.
As shown in Figure~\ref{fig:motion}, our motion completion network consists of two components, namely Path Refiner and Motion Synthesizer.
Although paths for human-scene interaction anchors are planned in Section~\ref{subsec:planning}, this Path Refiner accounts for the gap between diverse real human motions and the path formed by straight lines between the discrete grids.
Both the Path Refiner and the Motion Synthesizer follow the CVAE framework.
Specially, we apply Transformer~\cite{vaswani2017attention} as the basic architecture for both the encoder and decoder of these two networks to synthesize continuous and smooth motions.
Our motion completion network simultaneously synthesizes $M$ frame paths and body poses as~\cite{wang2021synthesizing, Wang_2021_CVPR}, instead of one-by-one in an auto-regressive manner~\cite{hassan_samp_2021, chuan2020action2motion, harvey2020robust}.
For Path Refiner, we take the scene context encoded by PointNet~\cite{qi2017pointnet} to synthesize the refined path.
The refined path is composed of pairs of the translation and orientation sequence$\{(t_1, \phi_1), ..., (t_M, \phi_M)\}$.
Following~\cite{petrovich21actor}, we introduce the positional encoding formed from sinusoidal functions which take time steps $t \in [1,...,M]$ as input to ensure the continuity and smoothness of the refined path.
Moreover, we leverage one more positional encoding obtained from the planned path by encoding each step of the planned path $(t_i, \phi_i)$ in Section~\ref{subsec:planning} by a fully connected layer, to ensure the refined path is still in the obstacle-free regions
The effectiveness of this additional positional encoding is illustrated in Section~\ref{sec:exps},
where our Path Refiner further improves the diversity of synthesized motion.
The motion sequence with $M$ body poses $\{\theta_1, ...,\theta_M \}$ is completed by our Motion Synthesizer.
Same as the Path Refiner, we take the scene context encoded by the PointNet as the condition to complete these scene-aware motions.
The completed motions should fulfill two requirements, namely matching the paths produced by the Path Refiner and naturally transforming between the given human-scene interaction anchors.
To achieve this, the Motion Synthesizer at first takes the refined path as additional position encoding to guide motion synthesis, similar to the practice of Path Refiner.
For the motion transformation, we need to model the relationship between the given two human-scene interaction anchors and the potential motions that could be completed in our Motion Synthesizer.
Inspired by the practice of action token in~\cite{petrovich21actor}, which helps the transformer decoder to build up the relationship between synthesized motions and the given action,
we encode the action labels and poses of human-scene interaction anchors by additional fully connected layers as learnable tokens and add them to the beginning and ending of the positional encoding respectively.
With these tokens, our Motion Synthesizer can directly build up this relationship between and synthesize reasonable and smooth motions.
Following these two steps, the motion completion network can generate natural motions for the given human-scene interaction anchors following the planned path.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/motion.pdf}
\small\caption{\textbf{Completion Network.} Motion completion network is composed of two modules, namely the Trajectory Refiner and the Motion Synthesizer. Trajectory Refiner reconstructs the motion trajectories under the guidance of planned path. The Motion Synthesizer takes the scene context, planned path, the pose and action label of human-scene interaction anchors as the inputs and generates body movements. The two-modules are trained altogether in an end-to-end manner.}
\label{fig:motion}
\vspace{-0.3cm}
\end{figure}
\section{Experiments}
In this section, we first illustrate our experiment settings and metrics for evaluation. Then we discuss the effectiveness of the proposed framework. At last, we demonstrate the qualitative results in different scenes.
\subsection {Experimental Setting}
\noindent \paragraph{Implementation Details.}
All proposed CVAE models in the paper are optimized via ADAM~\cite{kingma2014adam} with learning rate set to $1\mathrm{e}{-4}$
All models are trained for $40$ epochs with batch size set to be $8$.
For better physical plausibility, we perform additional optimization used in~\cite{Hassan:CVPR:2021} and~\cite{wang2021synthesizing} to refine human-scene interaction anchors described in Section~\ref{subsec:interaction} and the completed motions described in Section~\ref{sec:completion}.
More details for the training scheduler and the optimization are included in the supplementary material.
\noindent\paragraph{Dataset.}
Following~\cite{PLACE:3DV:2020,PSI:2019,wang2021synthesizing}, we train our framework on PROX dataset~\cite{PROX:2019}.
We manually label the motions in PROX with action labels~(\ie~sit, lie, stand, walk, and squat) as the action condition.
We do not conduct experiments on GTA-IM~\cite{cao2020long} and SAMP~\cite{hassan_samp_2021} as they do not provide reconstructed 3D real-world scenes.
For the fair comparison, we follow the split of the train and test set as~\cite{PLACE:3DV:2020,PSI:2019,wang2021synthesizing} and synthesize human motions on the unseen scenes during training.
To demonstrate the generalization ability of the proposed framework, we further evaluate it on Matterport3D~\cite{Matterport3D}, which provides large-scale reconstructed 3D scenes.
Please be noted that our framework does not leverage Matterport3D for training.
\noindent \paragraph{Diversity Metric.}
We measure the diversity on synthesized human motions in three aspects, namely human-scene interaction anchors, planned paths, and completed motions.
To evaluate the diversity of the human-scene interacting anchors, we preform \textbf{K-Means} ($K=20$) clustering on the synthesized human-scene interaction anchors, following~\cite{Hassan:CVPR:2021}.
To be specific, we consider two types of the clusters.
The first one considers all parameters $(\theta, t, \phi)$. The second one only considers translation $t$ and orientation $\phi$.
The diversity is measured as the entropy of the cluster sizes and the average distances between the clusters center and the samples belonging to it.
We evaluate path diversity by the standard deviation (\textbf{STD}) of distances between the paths from Neural Mapper and the ones from the standard $A^*$.
To fairly compare with previous works that manually assign anchors or target objects~\cite{PSI:2019, Hassan:CVPR:2021}, we evaluate the diversity of the synthesized human motions with the fixed human-scene interaction anchors.
To measure the ability of our motion completion network in generating diverse results, we do not introduce the diverse sampling strategies as~\cite{zhang2021we, yuan2020dlow}.
Following~\cite{hassan_samp_2021}, we calculate the Average Pairwise Distance (\textbf{APD}) on the SMPL-X parameters of synthesized motions to measure its diversity.
\noindent \paragraph{Naturalness Metric.}
We evaluate the naturalness of synthesized motions via \textbf{user study} and the physical plausibility.
We ask users to compare our results against other methods and score them from 1 to 5 (the higher the better) as the results.
Besides, we involve the \textbf{non-collision} score and \textbf{contact} score~\cite{PSI:2019, PLACE:3DV:2020,wang2021synthesizing} to measure the physical plausibility of synthesized motions between the 3D scenes.
\noindent \paragraph{Motion Metric.}
To evaluate the quality of the whole synthesized motions, we follow~\cite{hassan_samp_2021} to calculate the Frech\'{e}t Distance~(\textbf{FD}) between synthesized motions and ground-truth motions.
This distance is computed using the parameters $P_i=(\theta_i, t_i, \phi_i)$ of each frame.
\subsection{Experimental Results}~\label{sec:exps}
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on human-scene interaction anchors.} We evaluate the diversity of the human-scene interaction anchors (Anchor, considering $\theta$, $t$, and $\phi$) and the placing (Position, considering only $t$ and $\phi$) with/without optimization post-process. $S$ means the sampling strategy based on pose relationship in Section~\ref{subsec:interaction}, and $R$ means our Placing Refiner.}\label{tab:anchors}
\vspace{-0.2cm}
\resizebox{0.48\textwidth}{!}{
\begin{tabular}{l | c c| c c}
\shline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{Anchor} & \multicolumn{2}{c}{Position}\\
& Entropy $\uparrow$ & Cluster $\uparrow$ & Entropy $\uparrow$ & Cluster $\uparrow$\\
\hline
Baseline~\cite{Hassan:CVPR:2021} & 2.62 / 2.60 & 2.44 / 2.40 & 2.63 / 2.61 & 0.68 / 0.67 \\
Baseline~\cite{Hassan:CVPR:2021} + S & 2.74 / 2.73 & 2.55 / 2.53 & 2.69 / 2.68 & 0.79 / 0.78 \\
Baseline~\cite{Hassan:CVPR:2021}+ S + R & \textbf{2.77} / \textbf{2.73} & \textbf{2.57} / \textbf{2.53} & \textbf{2.72} / \textbf{2.70} & \textbf{0.83} / \textbf{0.80} \\
\shline
\end{tabular}}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/results_placing.jpg}
\vspace{-0.5cm}
\small\caption{\textbf{Placing Results.} The first row shows the results with and without the pose related sampling strategy. The second row shows the results where our Place Refiner and the optimization post-processing in~\cite{Hassan:CVPR:2021} works together.}
\label{fig:placing_show}
\vspace{-0.3cm}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.96\textwidth]{Figures/results_final.jpg}
\small\caption{\textbf{Qualitative Results.} In this figure, we show the output of each stage in our framework. The first row is synthesized on PROX dataset, and the second one is on Matterport3D. Our framework can synthesize motions with diverse interactions to the given scene.}
\label{fig:results}
\end{figure*}
In this section, we show quantitative results on PROX dataset. The quantitative results on Matterport3D dataset are included in our supplementary materials. We also show qualitative results on these two datasets in this section.
\noindent\paragraph{Human-Scene Interaction Anchors.}
We first show the diversity of synthesized human-scene interaction anchors in Table~\ref{tab:anchors}.
For this evaluation, we sample 100 poses for each action and employ the placing strategy in POSA~\cite{Hassan:CVPR:2021} as our baseline.
As shown in the table, the position related sampling process and the Pose Refiner can both improve the diversity of interaction anchors.
We further show the effectiveness of these two process in Figure~\ref{fig:placing_show}.
Examples shown in this figure are all generated from the action label ``sit''.
(a) shows the first placed poses.
Without the position related sampling, (a) and (b), which have the same action label, are placed close to each other.
(c) and (d) are the generated anchors using our position related sampling.
It is revealed that they interact with different objects in the scene.
The second row demonstrates the result pairs ((d) V.S. (e) and (f) V.S. (g)), which are produced by our Place Refiner works and optimization post-process~\cite{Hassan:CVPR:2021}.
Using the diverse translations and orientations as initialization states, the optimization algorithm can produce diverse optimal solutions.
In the supplemental material, we show comparison with previous works~\cite{PSI:2019,PLACE:3DV:2020} that are extended to synthesize specific actions using our action condition.
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on diverse planning module}. We compare against the standard $A^*$ algorithm and methods with sampled random noises. The metrics show the diversity and naturalness of the planned paths. }\label{tab:planning}
\vspace{-0.3cm}
\resizebox{0.48\textwidth}{!}{
\small\begin{tabular}{l | ccccc| c}
\shline
Method & 1/6 $\uparrow$ & 1/3 $\uparrow$ & 1/2 $\uparrow$ & 2/3 $\uparrow$ & 5/6 $\uparrow$ & User Study $\uparrow$ \\
\hline
Standard & 0 & 0 & 0 & 0 & 0 & 4.31(0.48)\\
Random Noise & 0.346 & 0.566 & 0.628 & 0.523 & 0.324 & 2.52(0.53) \\
Shared Noise & 0.297 & 0.483 & 0.603 & 0.485 & 0.281 & 3.52(0.45) \\
\hline
Ours & 0.286 & 0.446 & 0.508 & 0.415 & 0.233 & 4.27(0.52)\\
\shline
\end{tabular}}
\end{table}
\noindent\paragraph{Planning.}
We compare the diversity and naturalness of the planned path.
For the evaluation of diversity, We compute the standard deviation of the distances between sampled paths and the paths produced by standard $A^*$ .
In practice, we calculate distances between the discrete points on the paths, which are set as 1/6, 1/3, 1/2, 2/3, and 5/6 of the sample paths.
We also show the results of the user study to reflect the naturalness of the planned paths. The number of samples is set to be $50$ for each method.
The evaluation results are shown in Table~\ref{tab:planning}.
Standard $A^*$ algorithm, which is used in~\cite{hassan_samp_2021} only produces the deterministic path for practice while the proposed Neural Mapper can generate diverse paths with similar naturalness as the standard A$*$ does.
On the other hand, two methods using random noises cannot produce natural results, although they generate more diverse paths than ours.
Similar results are also demonstrated in Figure~\ref{fig:planning_show}.
Compared with the methods using random noises, Neural Mapper can provide consistent and reasonable guidance for the similar local scene context to avoid unnatural moving.
Moreover, Neural Mapper can cope with other manual constraints such as avoiding passing a certain region. We will discuss it in our supplementary materials.
\noindent\paragraph{Motion Synthesis.}
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on motion completion module.} We mainly evaluate the models in two aspects. FD is used to show the completion ability. APD is used to evaluate motion diversity. We compare our framework with several state-of-the-art methods. Specially, ``w/ OPT'' and ``w/o OPT'' refer to the results obtained with/without optimization post-process~\cite{wang2021synthesizing}. ``Ours*'' means our motion completion network without the Path Refiner.}\label{tab:completion-1}
\vspace{-0.3cm}
\resizebox{0.48\textwidth}{!}{\small\begin{tabular}{l | c c | c c }
\shline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{FD $\downarrow$} & \multicolumn{2}{c}{APD $\uparrow$} \\
& w/o OPT & w/ OPT & w/o OPT & w/ OPT\\
\hline
SA-CSGN~\cite{Wang_2021_CVPR} & 176.20 & 175.28 & 2.13 & 2.15 \\
Wang et.al.~\cite{wang2021synthesizing} & 121.22 & 120.01 & 0.00 & 0.00 \\
SAMP~\cite{hassan_samp_2021} & 115.34 & 114.22 & 2.56 & 2.57 \\
\hline
Ours* & 126.46 & 124.52 & 2.46 & 2.46 \\
Ours & \textbf{112.74} & \textbf{111.65} & \textbf{2.77} & \textbf{2.78} \\
\shline
\end{tabular}}
\vspace{-0.5cm}
\end{table}
In this subsection, we compare with other advances on scene-aware motion synthesis~\cite{wang2021synthesizing, hassan_samp_2021, Wang_2021_CVPR}.
We use the official model of~\cite{wang2021synthesizing} trained on PROX dataset and extend~\cite{hassan_samp_2021} and~\cite{Wang_2021_CVPR} to PROX for fair comparison.
In Table~\ref{tab:completion-1}, we first compare against these methods using \textbf{FD} and \textbf{APD} for the motion quality and diversity.
Firstly, for the comparison of \textbf{FD}, we sample 500 motion sequences which begin with the same action, as well as 500 motions which finish the same action.
Besides, for the comparison of \textbf{APD}, we sample $100$ pairs of human-scene interaction anchors and synthesize $10$ motions for each pair.
It is revealed that our method achieves the best results against other methods.
All the comparison results show that our method can synthesize more diverse and natural motions than other methods do.
In the supplementary material, we first compare more naturalness results between these methods, \eg physical compatibility and user study. Then we further discuss our design choice on the motion completion network, including the effectiveness of the Path Refiner and the positional encoding based on planned paths.
\noindent\paragraph{Qualitative Results.}
We show more qualitative results of the proposed method on the PROX~\cite{PROX:2019} and Matterport3D~\cite{Matterport3D} in Figure~\ref{fig:results}.
We show all three aspects of the synthesized scene-aware motions, including the human-scene interaction anchors, diverse planned paths, and completed motions.
These results demonstrate that our framework can synthesize diverse human motions in the specific scene contexts for the given target action sequence.
More qualitative results are included in the supplemental material.
\section{Acknowledgement}
This study is supported under the General Research Fund (GRF) of Hong Kong (No.,14205719), the RIE2020 Industry Alignment Fund–Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s).
{\small
\bibliographystyle{ieee_fullname}
\section{Supplementary Materials}
\subsection{Implementation Details}
In this section, we first discuss the detailed structures of the CVAE models used in our framework. Then we discuss the training and inference details of these models.
\subsubsection{Network Architecture.}
The encoders and decoders of the action conditioned pose generator in Section 3.2, Place Refiner in Section 3.2, and Neural Mapper in Section 3.3 share the exactly same architectures, which are all two-layer Multilayer Perceptron (MLP).
The encoder takes the $256$-dim features encoded by the fully-connected layers and predicts the mean $\mu \in \mathbb{R}^{32}$ and the standard deviation $\sigma \in \mathbb{R}^{32}$ for a Gaussian Distribution.
We sample the latent code $z$ from this distribution for the decoder during training.
For the motion completion network in Section 3.4, we use the Transformer~\cite{vaswani2017attention} as the basic structure as~\cite{petrovich21actor}.
To be specific, we use two fully connected layers to encode all inputs to $256$-dimension features.
The encoder predicts the mean $\mu \in \mathbb{R}^{32}$ and the variance $\sigma \in \mathbb{R}^{32}$ for a Gaussian Distribution as the CVAE model for action conditioned poses.
Following~\cite{petrovich21actor}, we set $8$ layers of the Transformer network for the encoder and decoder.
\subsubsection{Training and Inference Details.}
\noindent\paragraph{Scene-Agnostic Pose Synthesis.}
Firstly, we show how to train the CVAE model for scene-agnostic pose synthesis in Section 3.2.
As the standard VAE~\cite{kingma2013auto} model, the training objective consists of two parts.
The first one is the reconstruction loss between the reconstructed human poses and the input human pose.
The other objective is Kullback-Leibler~(KL) Divergence between the Gaussian Distribution $Q(z|\mu, \sigma)$, where $\mu$ and $\sigma$ are predicted by the encoder, and the standard Gaussian Distribution $N(0, I^2)$.
\noindent\paragraph{Place Refiner.}
The Place Refiner takes the placed body poses, and the scene contexts encoded by the PointNet~\cite{qi2017pointnet} as inputs and predict the offset $\Delta t_i, \Delta o_i$ for the sampled $\bar t_i, \bar o_i$.
To train this network, we first build up the discrete candidates following the same procedure of scene-conditioned anchor placing in Section 3.2 for each scene in our training set.
For practice, we split each scene into non-overlapping discrete grids uniformly as translation candidates and then uniformly sample eight different orientations paralleling with the ground plane as the orientation candidate.
Each pose in the given scene is neighbor to four-position candidates.
We assign each pose in the training set to one randomly sampled neighbor position candidate and one orientation candidate of this position candidate.
Then we predict the offset from these candidates to the original translation and orientation for this pose.
Similar as~\cite{kingma2013auto}, the training objective is the reconstruction loss and the KL-Divergence.
\noindent\paragraph{Neural Mapper.}
The input of Neural Mapper includes the local context encoded by BPS~\cite{PLACE:3DV:2020} and the human moving direction in this local context, which is obtained from ground-truth moving paths of PROX~\cite{PROX:2019} dataset.
To train this model, we first split the motions in the training set of the PROX dataset into different 60 frame sequences.
The local context is cropped as a $2m \times 2m \times 2m$ cubic cage at the motion center as~\cite{PLACE:3DV:2020}.
To compute BPS features, we uniformly sample a set of $N_b=10^4$ basis points within the unit sphere at the center of the local context and then normalize the local scene context into the same unit sphere.
The final BPS feature is the concatenation of these minimal distances $\vx_s \in \mathbb{R}^{N_b \times 1}$ between the sampled unit sphere and normalized scene context.
In practice, we use two additional fully-connected layers to further encode $\vx_s$ as the local context feature.
Similar to the standard VAE~\cite{kingma2013auto}, the training objective of Neural Mapper consists of a KL-Divergence term and a reconstruction term.
Specially, we norm the moving direction between the beginning and ending points of the motion sequence to $[0, 1]$ as the reconstruction target during training.
The reconstruction term estimates the residuals between this normalized moving direction and the expectation of the estimated direction distribution.
\noindent\paragraph{Path Refiner and Motion Synthesizer.}
We train our Path Refiner and Motion Synthesizer together in an end-to-end manner.
Both two models synthesize $M=60$ frames of paths or motions.
Similarly, the training objective consists of the reconstruction loss on the synthesized paths or motions, as well as the KL-Divergence.
Specially, we do not use the planned path obtained from Section 3.3 in training the Path Refiner.
Instead, we use the directions pointing from the beginning to the ending point of the motion as the planned path for practice.
The reason mainly lies in two aspects.
The first one is that repeat running of path planning module to obtain planned paths is not efficient in training.
The second one is that the shortest path from the planning module is similar to the straight line in short-term motions.
During the inference stage, we find that directly synthesizing motions from two consecutive anchors lead to unstable results.
We believe it is majorly caused by the variance of the lengths of the planned paths.
To resolve this, we first split the planned paths into several pieces with equal length.
Each split point is then assigned with an intermediate status action label.
Using the new action labels, the intermediate anchors can be produced following the same method as placing human-scene interaction anchors in Section 3.2.
Given the new anchors and the split paths, our motion completion network synthesizes human motions for each piece and then connects them together as the integrated motion.
For practice, we insert the motions with random poses conditioned on ``walking'', ``standing'', and `` squatting'' action.
\subsubsection{Optimization.}
We perform optimizations to improve the motion quality with the motion and physics constraints. For example, the human should walk on the floor with smooth motions.
We conduct the optimization in~\cite{Hassan:CVPR:2021} and~\cite{wang2021synthesizing} for human-scene interaction anchors and motion sequences, respectively.
For better human-scene interaction anchors, we use the objective functions defined in~\cite{Hassan:CVPR:2021}, that consist of the affordance loss for contacting the specific body parts to the given scene~(\eg~foot to the floor),
penetration loss for the reasonable physical relationship between body meshes and the reconstructed SDF (sign distance field),
and the regularization to keep the optimized pose close to the initial pose.
We optimize all these human-scene interaction anchors for $10$ iterations with $1\mathrm{e}{-3}$ learning rate, using L-BFGS~\cite{LiuN89} algorithm as~\cite{Hassan:CVPR:2021}.
For the synthesized motion obtained in Section 3.4, we follow their optimization objective functions~\cite{wang2021synthesizing} for foot location, environment, and motion smoothness to improve the motion quality.
We optimize all our motions for $100$ iterations with $1\mathrm{e}{-2}$ learning rate, using ADAM~\cite{kingma2014adam} algorithm as~\cite{wang2021synthesizing}.
\subsection{Experiments}
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on naturalness of synthesized motions on PROX~\cite{PROX:2019}.} We measured this by physical plausibity (non-collision and contact score) as well as user study.Specially, w/ and w/o opt means the results with/without optimization post-process~\cite{wang2021synthesizing}. ``Ours*'' means our motion completion network without the Path Refiner.}\label{tab:completion-2}
\resizebox{0.48\textwidth}{!}{\small\begin{tabular}{l | c c | c c | c }
\shline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{Non-Collision $\uparrow$ } & \multicolumn{2}{c|}{Contact $\uparrow$} & User Study $\uparrow$\\
& w/o opt & w/ opt & w/o opt& w/ opt \\
\hline
SA-CSGN~\cite{Wang_2021_CVPR} & 92.37 & 98.21 & 97.16 & 98.72 & 2.74(0.97)\\
Wang et.al.~\cite{wang2021synthesizing} & 93.88 & 98.72 & 98.31 & 99.35 & 3.42(1.06)\\
SAMP~\cite{hassan_samp_2021} & 94.92 & 99.31 & 98.18 & 99.32 & 3.46(0.96) \\
\hline
Ours* & 94.52 & 99.28 & 98.22 & 99.27 & 3.28(0.94)\\
Ours & \textbf{95.93} & \textbf{99.61} & \textbf{98.33} & \textbf{99.35} & \textbf{3.68(0.84)}\\
\shline
\end{tabular}}
\end{table}
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on human-scene interaction anchors for Matterport3D~\cite{Matterport3D}.} We evaluate the diversity of the human-scene interaction anchors (Anchor, considering $\theta$, $t$, and $\phi$) and the placing (Position, considering only $t$ and $\phi$) with/without optimization post-process. $S$ means the sampling strategy based on pose relationship in Section 3.2, and $R$ means our Placing Refiner.}\label{tab:anchors}
\vspace{-0.2cm}
\resizebox{0.48\textwidth}{!}{
\begin{tabular}{l | c c| c c}
\shline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{Anchor} & \multicolumn{2}{c}{Position}\\
& Entropy $\uparrow$ & Cluster $\uparrow$ & Entropy $\uparrow$ & Cluster $\uparrow$\\
\hline
Baseline~\cite{Hassan:CVPR:2021} & 2.54 / 2.50 & 2.45 / 2.44 & 2.51 / 2.50 & 0.58 / 0.56 \\
Baseline~\cite{Hassan:CVPR:2021} + S & 2.63 / 2.61 & 2.53 / 2.53 & 2.54 / 2.54 & 0.64 / 0.65 \\
Baseline~\cite{Hassan:CVPR:2021} + S + R & \textbf{2.70} / \textbf{2.68} & \textbf{2.59} / \textbf{2.58} & \textbf{2.68} / \textbf{2.66} & \textbf{0.72} / \textbf{0.72} \\
\shline
\end{tabular}}
\end{table}
\begin{table}
\centering
\setlength\tabcolsep{9pt}
\caption{\textbf{Evaluation on synthesized motion for Matterport3D~\cite{Matterport3D}.} Comparison on \textbf{APD}, \textbf{non-collision} score and \textbf{contact} score on Matterport3D dataset. Specially, ``w/ OPT'' and ``w/o OPT'' refer to the results obtained with/without optimization post-process~\cite{wang2021synthesizing}. ``Ours*'' means our motion completion network without the Path Refiner.}\label{tab:completion-2}
\vspace{-0.3cm}
\resizebox{0.48\textwidth}{!}{\small\begin{tabular}{l | cc | cc | cc }
\shline
\multirow{2}{*}{Method} & \multicolumn{2}{c|}{APD $\uparrow$} & \multicolumn{2}{c|}{Non-Collision $\uparrow$ } & \multicolumn{2}{c}{Contact $\uparrow$} \\
& w/o OPT & w/ OPT & w/o OPT & w/ OPT & w/o OPT & w/ OPT \\
\hline
SA-CSGN~\cite{Wang_2021_CVPR} & 2.24 & 2.26 & 91.51 & 99.08 & 99.11 & 99.33 \\
Wang et.al.~\cite{wang2021synthesizing} & 0.00 & 0.00 & 93.78 & 99.42 & 99.32 & 99.35 \\
SAMP~\cite{hassan_samp_2021} & 2.46 & 2.48 & 94.35 & 99.32 & 99.28 & 99.35\\
\hline
Ours* & 2.34 & 2.38 & 94.12 & 99.08 & 99.18 & 99.32 \\
Ours & \textbf{2.57} & \textbf{2.60} & \textbf{95.72} & \textbf{99.42} & \textbf{99.32} & \textbf{99.36}\\
\shline
\end{tabular}}
\end{table}
\noindent\paragraph{Naturalness Results on PROX.}
Then we compare the naturalness of these methods in Table~\ref{tab:completion-2}.
For the physical plausibility, we use the same motion as the comparison in Table.3 of our paper.
For user study, we randomly sample motions with $2$, $4$, and $8$ different target actions.
All the comparison results show that our method can synthesize more natural motions than other methods do.
Especially, our method achieves better results without the optimization post-process, because of the guidance from planned obstacle-free paths.
The comparison between the last two rows shows that the guidance of the proposed Path Refiner is advantageous in synthesizing natural motions.
\noindent\paragraph{Results on Matterport3D.}
We show the quantitative results on Matterport3D dataset~\cite{Matterport3D}.
The sampling strategies are the same as our experiments on PROX dataset.
We first perform \textbf{K-Means} ($K=20$) and evaluate the obtained human-scene interaction anchors on Matterport3D with the entropy of cluster sizes and the average distance between the cluster center and the samples belong to it.
As shown in Table~\ref{tab:anchors}, our method enhance the diversity of the anchors for motion synthesis.
Besides, we evaluate the synthesized motion on Matterport3D via the \textbf{APD}, \textbf{Non-Collision} score and \textbf{Contact} score.
The results are listed in Table~\ref{tab:completion-2}.
It is revealed that our method can synthesize better results than previous methods with better diversity and physical plausibility.
Besides, the Path Refiner still can improve the diversity and naturalness on this dataset.
\subsection{Further Discussion}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/supp/bad_case.pdf}
\small\caption{\textbf{Failure cases of the scene-centric paradigm.}
We use the action label as additional condition to extend previous scene-centric paradigms~\cite{PSI:2019,PLACE:3DV:2020}. The first and third column shows results obtained from \cite{PSI:2019} and \cite{PLACE:3DV:2020}, respectively. The second and forth columns shows the results generated by our framework with the same pose as the first and third columns, respectively.
}
\label{fig:failure_case}
\vspace{-0.3cm}
\end{figure}
In this section, we first discuss the reason for using the human-centric paradigm, for human-scene interaction anchors.
The human-centric paradigm means we place the sampled poses to the positions which match the physical structure of these poses.
Then we show how to use our Neural Mapper to work with other manually set constraints.
At last, we show the influence of the planned path on motion synthesis.
\noindent\paragraph{Human-Scene Interaction Anchor.}
Previous works~\cite{PSI:2019, PLACE:3DV:2020} of synthesizing human-scene interaction anchors aim to explore the influence of scene context to place human pose in the given scene and neglect the action labels.
Intuitively, we can incorporate these action labels as an additional condition and incorporate them into their frameworks to synthesize poses.
However, as shown in Figure~\ref{fig:failure_case}, simply extending the previous works cannot guarantee to synthesize the physically plausible poses with the given actions and scene contexts.
We believe it is due to the reason that these methods do not build up the relationship between the action and the scene context explicitly.
For example, method~\cite{PSI:2019} directly uses the pooled 2D image features as the condition and ignores the relationship between the spatial information and the action.
Another method~\cite{PLACE:3DV:2020} first samples different positions to build up the BPS and then synthesize different poses.
However, the poses for each action have their specific physical structure and match different scene structures.
It is difficult to find suitable places for the poses conditioned on the given action label, as shown in the Figure~\ref{fig:failure_case}.
Instead, the human-centric paradigm proposed by us can effectively leverage the explicit relationship between the synthesized 3D human and the scene structure (\eg physical and semantic structures) and thus makes the whole placing process more controllable.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/supp/rebuttal.pdf}
\small\caption{\textbf{ Comparison with randomly sampled intermediate points.} Our method can plan diverse and natural paths without complex manual constraints.}
\label{fig:random_sampling}
\vspace{-0.3cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/supp/manual_avoid.pdf}
\small\caption{\textbf{Neural Map with manual constraints.} We change the valid grids in the blue box as the manual avoiding regions. These regions are the shortest path from these two points. We find that our Neural Map can work with this constraint to sample different planned paths.}
\label{fig:manual_avoid}
\vspace{-0.3cm}
\end{figure}
\noindent\paragraph{Neural Mapper}
Several failure cases generated from randomly sampling intermediate points are included in Figure~\ref{fig:random_sampling}.
In the first row, when sampled intermediate points and the ending points are obstructed, the original $A^\star$ algorithm can not find paths for these points. We adjust $A^\star$ by allowing to search paths in the obstacle regions , and $A^\star$ only produces impractical paths crossing the table as the first row of Figure~\ref{fig:random_sampling}.
In the second row, random sampled points can also lead to unnatural zigzag paths.
One may argue that these failures can be avoided via complex constraints used in previous methods~\cite{vsvestka1998probabilistic,carpin2006randomized}.
However, the proposed Neural Mapper provides an \textbf{automatic and data-driven} way to embed semantic information into natural and diverse path planning, without complex constraints.
Besides, our Neural Mapper also can work with manually set constraints, such as avoiding passing a certain region.
We show the planned results in Figure~\ref{fig:manual_avoid}.
It is revealed that our method can still produce natural and diverse paths under such constraints.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Figures/supp/planning_ablation.jpg}
\small\caption{\textbf{Effect of the planned path.} (a) is the result without planning module and (b) is based on planning module. (c) is the results for only using translations of planned path as the position encoding for our Path Refiner in Section 3.4 and (d) is result from our method.}
\label{fig:motion_ablation}
\vspace{-0.3cm}
\end{figure}
\noindent\paragraph{Planned Path for Motion Synthesis.}
As shown in Figure~\ref{fig:motion_ablation}, we show the effectiveness of using planned paths in the procedure of motion synthesis.
Firstly, without the additional positional encoding from the planned path, the synthesized motion can not follow the planned path and penetrate to the table, as shown in Figure~\ref{fig:motion_ablation} (a).
Besides, we find that both the translation and orientation for the planned path are also crucial for motion synthesis.
As shown in Figure~\ref{fig:motion_ablation} (c), the Path Refiner synthesizes unnatural orientations for human motions without encoding the orientation of the planned path into the positional encoding as Section 3.4.
Instead, as shown in Figure~\ref{fig:motion_ablation} (b) and (d), our method can synthesize natural human motion with the translation and orientation of planned path as the additional positional encoding for our Path Refiner.
|
1,314,259,992,981 | arxiv |
\section{Introduction}
Given a graph and two vertices $s$ and $t$ in it, the problem of determining whether there is a path from $s$ to $t$ in the graph is known as the graph reachability problem. Graph reachability problem is an important question in complexity theory. Particularly in the domain of space bounded computations, the reachability problem in various classes of graphs characterize the complexity of different complexity classes. The reachability problem in directed and undirected graphs, is complete for the classes non-deterministic log-space (\NL) and deterministic log-space (\L) respectively \cite{LP82, Reingold08}. The latter follows due to a famous result by Reingold who showed that undirected reachability is in {\L} \cite{Reingold08}. Various other restrictions of reachability has been studied in the context of understanding the complexity of other space bounded classes (see \cite{RTV06, CRV07, rulcomplete}). Wigderson gave a fairly comprehensive survey that discusses the complexity of reachability in various computational models \cite{widgerson-survey}.
The time complexity of directed reachability is fairly well understood. Standard graph traversal algorithms such as DFS and BFS solve this problem in linear time. We also have a $O(\log^2 n)$ space algorithm due to Savitch \cite{Sav70}, however it requires $O(n^{\log n})$ time. The question, whether there exists a single algorithm that decides reachability in polynomial time and polylogarithmic space is unresolved. In his survey, Wigderson asked whether it is possible to design a polynomial time algorithm that uses only $O(n^{\epsilon})$ space, for some constant $\epsilon < 1$ \cite{widgerson-survey}. This question is also still open. In 1992, Barnes, Buss, Ruzzo and Schieber made some progress on this problem and gave an algorithm for directed reachability that requires polynomial time and $O(n/2^{\sqrt{\log n}})$ space \cite{BBRS92}.
Planar graphs are a natural topological restriction of general graphs consisting of graphs that can be embedded on the surface of a plane such that no two edges cross. {\em Grid graphs} are a subclass of planar graphs, where the vertices are placed at the lattice points of a two dimensional grid and edges occur between a vertex and its immediate adjacent horizontal or vertical neighbor.
Asano and Doerr provided a polynomial time algorithm to compute the \emph{shortest path} (hence can decide reachability) in grid graphs which uses $O(n^{1/2+\epsilon})$ space, for any small constant $\epsilon>0$ \cite{Asano11}. Imai et al extended this to give a similar bound for reachability in planar graphs \cite{INPVW13}. Their approach was to use a space efficient method to design a separator for the planar graph and use divide and conquer strategy. Note that although it is known that reachability in grid graphs reduces to planar reachability in logspace, however since this class (polynomial time and $O(n^{1/2+\epsilon})$ space) is not closed under logspace reductions, planar reachability does not follow from grid graph reachability. Subsequently the result of Imai et al was extended to the class of {\em high-genus} and {\em $H$-minor-free} graphs \cite{CPTVY14}. Recently Asano et al gave a $\tilde{O}(\sqrt{n})$ space and polynomial time algorithm for reachability in planar graphs, thus improving upon the previous space bound \cite{AKNW14}. More details on known results can be found in a recent survey article \cite{Vin14}.
In another line of work, Kannan et al gave a $O(n^{\epsilon})$ space and polynomial time algorithm for solving reachability problem in \emph{unique path graphs} \cite{KKR08}. Unique path graphs are a generalization of {\em strongly unambiguous} graphs and reachability problem in strongly unambiguous graphs is known to be in {\SC} (polynomial time and polylogarithmic space) \cite{BJLR91, logdcflinsc2}. Reachability in strongly unambiguous graphs can also be decided by a $O(\log^2 n/ \log \log n)$ space algorithm, however this algorithm requires super polynomial time \cite{AL98}. {\SC} also contains the class {\em randomized logspace} or {\RL} \cite{Nisan95}. We refer the readers to a recent survey by Allender \cite{allender-update} to further understand the results on the complexity of reachability problem in {\UL} and on certain special subclasses of directed graphs.
\subsection*{Our Contribution}
We show that reachability in directed layered planar graphs can be decided in polynomial time and $O(n^{\epsilon})$ space for any constant $\epsilon >0$. A layered planar graph is a planar graph where the vertex set is partitioned into layers (say $L_0$ to $L_m$) and every edge occurs between layers $L_i$ and $L_{i+1}$ only. Our result significantly improves upon the previous space bound due to \cite{INPVW13} and \cite{AKNW14} for layered planar graphs.
\begin{theorem}
\label{thm:layeredplanarreach}
For every $\epsilon >0$, there is a polynomial time and $O(n^{\epsilon})$ space algorithm that decides reachability in directed layered planar graphs.
\end{theorem}
Reachability in layered grid graphs is in {\UL} which is a subclass of {\NL} \cite{ABCDR09}. Subsequently this result was extended to the class of all planar graphs \cite{BTV}. Allender et al also gave some hardness results the reachability problem in certain subclasses of layered grid graphs. Specifically they showed that, {\oneLGGR} is hard for $\NC^1$ and {\oneoneLGGR} is hard for $\TC^0$ \cite{ABCDR09}. Both these problems are however known to be contained in {\L} though.
Firstly we argue that its enough to consider layered grid graphs (a subclass of general grid graphs). We divide a given layered grid graph into a courser grid structure along $k$ horizontal and $k$ vertical lines (see Figure \ref{fig:gridgraph}). We then design a modified DFS strategy that makes queries to the smaller graphs defined by these gridlines (we assume a solution in the smaller graphs by recursion) and visits every reachable vertex from a given start vertex. The modified DFS stores the highest visited vertex in each vertical line and the left most visited vertex in each horizontal line. We use this information to avoid visiting a vertex multiple number of times in our algorithm. We choose the number of horizontal and vertical lines to divide the graph appropriately to ensure that the algorithm runs in the required time and space bound.
The rest of the paper is organized as follows. In Section \ref{sec:prelim}, we give some basic definitions and notations that we use in this paper. We also state certain earlier results that we use in this paper. In Section \ref{sec:LGGR}, we give a proof of Theorem \ref{thm:layeredplanarreach}.
\section{Reachability in Layered Planar Graphs}
\label{sec:LGGR}
In this section we prove Theorem \ref{thm:layeredplanarreach}. We show that the reachability problem in layered grid graphs, (denoted as {\LGGR}) is in {\ASC} (Theorem \ref{thm:LGGR}). Then by applying Proposition \ref{prop:reductionLGGR} and Theorem \ref{thm:closure} we have the proof of Theorem \ref{thm:layeredplanarreach}.
\begin{theorem}
\label{thm:LGGR}
{\LGGR} $\in$ {\ASC}.
\end{theorem}
To establish Theorem \ref{thm:LGGR} we define an auxiliary graph in Section \ref{sec:H} and give the required algorithm in Section \ref{sec:algo}.
\subsection{The Auxiliary Graph $H$}
\label{sec:H}
Let $G$ be a $n \times n$ layered grid graph. We denote the vertices in $G$ as $(i,j)$, where $0\le i,j \le n$. Let $k$ be a parameter that determines the number of pieces in which we divide $G$. We will fix the value of $k$ later to optimize the time and space bounds. Assume without loss of generality that $k$ divides $n$. Given $G$ we construct an auxiliary graph $H$ as described below.
Divide $G$ into $k^2$ many blocks of dimension $n/k \times n/k$. More formally, the vertex set of $H$ is
\[
V(H) := \{(i,j) \ |\ \textrm{either $i$ or $j$ is a non-negative multiple of $n/k$}.\}
\]
Note that $V(H) \subseteq V(G)$. We consider $k^2$ many {\em blocks} $G_1,G_2,\cdots,G_{k^2}$, where a vertex $(i,j) \in V(G_l)$ if and only if $ i' \frac{n}{k} \le i \le (i'+1) \frac{n}{k} $ and $ j' \frac{n}{k} \le j \le (j'+1) \frac{n}{k} $, for some integer $i' \ge 0$ and $j' \ge 0$ and the vertices for which any of the four inequalities is strict will be referred as \emph{boundary vertices}. Moreover, we have $l= i' \cdot k + j' +1 $. $E(G_l)$ is the set of edges in $G$ induced by the vertex set $V(G_l)$.
For every $i \in [k+1]$, let $L_h(i)$ and $L_v(i)$ denote the set of vertices, $L_h(i)=\{(i',j') | j' = (i-1) \frac{n}{k}\}$ and $L_v(i)=\{(i',j') | i' = (i-1) \frac{n}{k}\}$. When it is clear from the context, we will also use $L_h(i)$ and $L_v(i)$ to refer to the corresponding gridline in $H$. Observe that $H$ has $k+1$ vertical gridlines and $k+1$ horizontal gridlines.
For every pair of vertices $u,v \in V(G_l)\cap V(H)$, for some $l$, add the edge $(u,v)$ to $E(H)$ if and only if there is a path from $u$ to $v$ in $G_l$, unless $u,v \in L_v(i)$ or $u,v \in L_h(i)$, for some $i$. Also for every pair of vertices $u,v \in V(G_l)$, for some $l$, such that $u=(i_1,j_1)$ and $v=(i_2,j_2)$, where $j_1=j' \frac{n}{k}$ and $j_2=(j'+1)\frac{n}{k} $, for some $j'$ or $i_1=i' \frac{n}{k} $ and $i_2=(i'+1)\frac{n}{k} $, for some $i'$, we add edge between $u$ and $v$ in the set $E(H)$ if and only if there is a path from $u$ to $v$ in $G_l$ and we call such vertices as \emph{corner vertices}.
Before proceeding further, let us introduce a few more notations that will be used later. For $j \in [k]$, let $L_h(i,j)$ denote the set of vertices in $L_h(i)$ in between $L_v(j)$ and $L_v(j+1)$. Similarly we also define $L_v(i,j)$ (see Figure \ref{fig:gridgraph}). For two vertices $x,y \in L_v(i)$, we say $x \prec y$ if $x$ is {\em below} $y$ in $L_v(i)$. For two vertices $x,y \in L_h(i)$, we say $x \prec y$ if $x$ is {\em right of} $y$ in $L_h(i)$.
\begin{center}
\input{figure_grid}
\end{center}
\begin{lemma}
\label{lem:auxiliary}
Let $u$ and $v$ be two boundary vertices in $G$ such that they belong to two different vertical or horizontal gridlines or $u\in G_i$ and $v \in G_j$, for $i \ne j$. There is a path from $u$ to $v$ in $G$ if and only if there is path from $u$ to $v$ in the auxiliary graph $H$.
\end{lemma}
\begin{proof}
As every edge $(a,b)$ in $H$ corresponds to a path from $a$ to $b$ in $G$, so if-part is trivial to see. Now for the only-if-part, consider a path $P$ from $u$ to $v$ in $G$. $P$ can be decomposed as $P_1 P_2 \cdots P_r$, such that $P_i$ is a path from $x_i\in V(G_l)$ to $x_{i+1}\in V(G_l)$, where it re-enters $V(G_l)$ for the next time, for some $l$, and is of following two types:
\begin{enumerate}
\item $x_i$ and $x_{i+1}$ belong to different horizontal or vertical gridlines; or
\item $x_i$ and $x_{i+1}$ are two corner vertices.
\end{enumerate}
Now by the construction $H$, for every $i$, there must be an edge $(x_i,x_{i+1})$ in $H$ for both the above cases and hence there is a path from $u$ to $v$ in $H$ as well.
\end{proof}
Now we consider the case when $u$ and $v$ belong to the same vertical or horizontal gridlines.
\begin{claim}
\label{clm:gridreach}
Let $u$ and $v$ be two vertices contained in either $L_v(i)$ or $L_h(i)$ for some $i$. Then deciding reachability between $u$ and $v$ in $G$ can be done in logspace.
\end{claim}
\begin{proof}
Let us consider that $u,v\in L_v(i)$, for some $i$. As the graph $G$ under consideration is a layered grid graph, if there is a path between $u$ and $v$, then it must pass through all the vertices in $L_v(i)$ that lies in between $u$ and $v$. Hence just by exploring the path starting from $u$ through $L_v(i)$, we can check the reachability and it is easy to see that this can be done in logspace, because the only thing we need to remember is the current vertex in the path. Same argument will also work when $u,v\in L_h(i)$, for some $i$ and this completes the proof.
\end{proof}
Now we argue on the upper bound of the length of any path in the auxiliary graph $H$.
\begin{lemma}
\label{lem:pathlength}
For any two vertices $u,v \in V(H) $, any path between them is of length at most $2k+1$.
\end{lemma}
\begin{proof}
Consider any two vertices $u,v\in V(H)$ and a path $u=x_1 x_2\cdots x_r=v$, from $u$ to $v$, denoted as $P$. Now let us consider a bipartite (undirected) graph $K$, where $V(K)=A \cup B$ such that $A=\{x_i|i \in [r]\}$ and $B=\{L_v(i),L_h(j) | i,j \in [k+1]\}$. We add an edge $(a,b)$ in $E(K)$ if and only if $a \in b$, where $b=L_v(i)$ or $b=L_h(i)$, for some $i$. Now observe that by the construction of $H$, each $x_j$ belongs to different $L_v(i)$ or $L_h(i)$ unless $x_j$ is some corner vertex and in that case $x_j \in L_v(i),L_h(i')$, for some $i$ and $i'$. Moreover, if $x_j \in L_v(i)$ (or $L_v(i)$), but is not a corner vertex, then $x_{j+1}$ cannot be a corner vertex on $L_v(i)$ (or $L_h(i)$). As a consequence for every subset $S_A \subseteq A$, its neighbor set $N(S_A):=\{b \in B | \exists a \in S_A \text{ such that }(a,b) \in E(K) \}\subseteq B$ satisfies the condition that $|N(S_A)| \ge |S_A|$. Now we apply the Hall's Theorem \cite{LP86}, which states that a bipartite (undirected) graph $ G=(A \cup B,E) $ has a \emph{matching} if and only if for every $ S \subseteq A $, $ |N(S)| \ge |S| $. Hence there is a matching between $A$ and $B$ and as $|B|\le 2(k+1)$, so is $|A|$. This shows that the path $P$ is of length at most $2k+1$.
\end{proof}
\subsection{Description of the Algorithm}
\label{sec:algo}
We next give a modified version of DFS that starting at a given vertex, visits the set of vertices reachable from that vertex in the graph $H$. At every vertex, the traversal visits the set of outgoing edges from that vertex in an anticlockwise order.
In our algorithm we maintain two arrays of size $k+1$ each, say $A_v$ and $A_h$, one for vertical and the other for horizontal gridlines respectively. For every $i \in [k+1]$, $A_v(i)$ is the {\em topmost} visited vertex in $L_v(i)$ and analogously $A_h(i)$ is the {\em leftmost} visited vertex in $L_h(i)$. This choice is guided by the choice of traversal of our algorithm. More precisely, we cycle through the outgoing edges of a vertex in an anticlockwise order.
We perform a standard DFS-like procedure, using the tape space to simulate a stack, say $S$. $S$ keeps track of the path taken to the current vertex from the starting vertex. By Lemma \ref{lem:pathlength}, the maximum length of a path in $H$ is at most $2k+1$. Whenever we visit a vertex in a vertical gridline (say $L_v(i)$), we check whether the vertex is lower than the $i$-th entry of $A_v$. If so, we return to the parent vertex and continue with its next child. Otherwise, we update the $i$-th entry of $A_v$ to be the current vertex and proceed forward. Similarly when visit a horizontal gridline (say $L_h(i)$), we check whether the current vertex is to the left of the $i$-th entry of $A_h$. If so, we return to the parent vertex and continue with its next child. Otherwise, we update the $i$-th entry of $A_h$ to be the current vertex and proceed. The reason for doing this is to avoid revisiting the subtree rooted at the node of an already visited vertex. The algorithm is formally defined in Algorithm \ref{algo:LGGR}.
\begin{algorithm}
\Input{The auxiliary graph $ H $, two vertices $u,v \in V(H)$}
\Output{{{\tt YES}} if there is a path from $u$ to $v$; otherwise {{\tt NO}}}
Initialize two arrays $A_v$ and $A_h$ and a stack $S$\;
Initialize three variables $curr$, $prev$ and $next$ to {{\tt NULL}}\;
Push $u$ onto $S$\;
\While{$S$ is not empty}{
$curr \leftarrow$ top element of $S$\;
$next \leftarrow$ neighbor of $curr$ next to $prev$ in counter-clockwise order\;
\While{$next \ne {\tt NULL}$}{ \tcc{cycles through neighbors of curr}
\If{$next=v$}{
\Return {{\tt YES}}\;
}
\If{$next \in L_v(i)$ for some $i$ and $A_v[i] \prec next$}{
$A_v[i] \leftarrow next $\;
{\bf break}\;
}
\If{$next \in L_h(i)$ for some $i$ and $A_h[i] \prec next$}{
$A_h[i] \leftarrow next $\;
{\bf break}\;
}
$prev \leftarrow next$\;
$next \leftarrow$ neighbor of $curr$ next to $prev$ in counter-clockwise order\; \tcc{{{\tt NULL}} if no more neighbors are present}
}
\eIf{$next = {\tt NULL}$}{
remove $curr$ from $S$\;
$prev \leftarrow curr$\;
}
{
add $next$ to $S$\;
$prev \leftarrow {\tt NULL}$\;
}
}
\Return {{\tt NO}}\;
\caption{{\tt AlgoLGGR}: Algorithm for Reachability in the Auxiliary Graph $H$}
\label{algo:LGGR}
\end{algorithm}
\begin{lemma}
\label{lem:crossing}
Let $G_l$ be some block and let $x$ and $y$ be two vertices on the boundary of $G_l$ such that there is a path from $x$ to $y$ in $G$. Let $x'$ and $y'$ be two other boundary vertices in $G_l$ such that (i) there is a path from $x'$ to $y'$ in $G$ and (ii) $x'$ lies on one segment of the boundary of $G_l$ between vertices $x$ and $y$ and $y'$ lies on the other segment of the boundary. Then there is a path in $G$ from $x$ to $y'$ and from $x'$ to $y$. Hence, if $(x,y)$ and $(x',y')$ are present in $E(H)$ then so are $(x,y')$ and $(x',y)$.
\end{lemma}
\begin{proof}
Since $G$ is a layered grid graph hence the paths $x$ to $y$ and $x'$ to $y'$ must lie inside $G_l$. Also because of planarity, the paths must intersect at some vertex in $G_l$. Now using this point of intersection, we can easily show the existence of paths from $x$ to $y'$ and from $x'$ to $y$.
\end{proof}
Lemma \ref{lem:reach} will prove the correctness of Algorithm \ref{algo:LGGR}.
\begin{lemma}
\label{lem:reach}
Let $u$ and $v$ be two vertices in $H$. Then starting at $u$ Algorithm \ref{algo:LGGR} visits $v$ if and only if $v$ is reachable from $u$.
\end{lemma}
\begin{proof}
It is easy to see that every vertex visited by the algorithm is reachable from $u$ since the algorithm proceeds along the edges of $H$.
By induction on the shortest path length to a vertex, we will show that if a vertex is reachable from $u$ then the algorithm visits that vertex. Let $B_d(u)$ be the set of vertices reachable from $u$ that are at a distance $d$ from $u$. Assume that the algorithm visits every vertex in $B_{d-1}(u)$. Let $x$ be a vertex in $B_d(u)$.
Without loss of generality assume that $x$ is in $L_v(i,j)$ for some $i$ and $j$. A similar argument can be given if $x$ belongs to a horizontal gridline. Further, let $x$ lie on the right boundary of a block $G_l$. Let $W_x = \{w \in B_{d-1}(u)| (w,x) \in E(H)\}$. Note that by the definition of $H$, all vertices in $W_x$ lie on the bottom boundary or on the left boundary of $G_l$.
Suppose the algorithm does not visit $x$.
Since $x$ is reachable from $u$ via a path of length $d$, therefore $W_x$ is non empty. Let $w$ be the first vertex added to $W_x$ by the algorithm. Then $w$ is either in $L_h(j)$, or in $L_v(i-1)$. Without loss of generality assume $w$ is in $L_h(j)$. Let $z$ be the value in $A_v(i)$ at this stage of the algorithm (that is when $w$ is the current vertex). Since $x$ is not visited hence $x \prec z$. Also this implies that $z$ was visited by the algorithm at an earlier stage of the algorithm. Let $w'$ be the ancestor of $z$ in the DFS tree such that $w'$ is in $L_h(j)$. There must exist such a vertex because $z$ is above the $j$-th horizontal gridline, that is $L_h(j)$.
Suppose if $w'$ lies to the left of $w$ then by the description of the algorithm, $w$ is visited before $w'$. Hence $x$ is visited before $z$. On the other hand, suppose if $w'$ lies to the right of $w$. Clearly $w'$ cannot lie to the right of vertical gridline $L_v(i)$ since $z$ is reachable from $w'$ and $z $ is in $L_v(i)$. Let $w''$ be the vertex in $L_h(j+1)$ such that $w''$ lies in the tree path between $w'$ and $z$ (See Figure \ref{fig:reach}). Observe that all four vertices lie on the boundary of $G_l$. Now by applying Lemma \ref{lem:crossing} to the four vertices $w$, $x$, $w'$ and $w''$ we conclude that there exists a path from $w'$ to $x$ as well. Since $x \prec z$, $x$ must have been visited before $z$ from the vertex $w'$. In both cases, we see that $z$ cannot be $A_v(i)$ when $w$ is the current vertex. Since $z$ was an arbitrary vertex such that $x \prec z$, the lemma follows.
\end{proof}
\begin{center}
\input{figure_reach}
\end{center}
We next show Lemma \ref{lem:visitonce} which will help us to achieve a polynomial bound on the running time of Algorithm \ref{algo:LGGR}.
\begin{lemma}
\label{lem:visitonce}
Every vertex in the graph $H$ is added to the set $S$ at most once in Algorithm \ref{algo:LGGR}.
\end{lemma}
\begin{proof}
Observe that a vertex $u$ in $L_v(i)$ is added to $S$ only if $A_v(i) \prec u$, and once $u$ is added, $A_v(i)$ is set to $u$. Also during subsequent stages of the algorithm, if $A_v(i)$ is set to $v$, then $u \prec v$. Hence $u \prec A_v(i)$. Therefore, $u$ cannot be added to $S$ again.
We give a similar argument if $u$ is in $L_h(i)$. Suppose if $u$ is in $L_v(i)$ for some $i$ and $L_h(j)$ for some $j$, then we add $u$ only once to $S$. This check is done in Line 16 of Algorithm \ref{algo:LGGR}. However we update both $A_v(i)$ and $A_h(j)$.
\end{proof}
Algorithm \ref{algo:LGGR} does not explicitly compute and store the graph $H$. Whenever it is queried with a pair of vertices to check if it forms an edge, it recursively runs a reachability query in the corresponding block and produces an answer. The base case is when a query is made to a grid of size $k \times k$. For the base case, we run a standard DFS procedure on the $k \times k$ size graph.
In every iteration of the {\em outer while} loop (Lines 4 -- 29) of Algorithm \ref{algo:LGGR}, either an element is added or an element is removed from $S$. Hence by Lemma \ref{lem:visitonce} the loop iterates at most $4nk$ times. The {\em inner while} loop (Lines 7 -- 21), cycles through all the neighbors of a vertex and hence iterates for at most $2n/k$ times. Each iteration of the inner while loop makes a constant number of calls to check the presence of an edge in a $n/k \times n/k$ sized grid. Let ${\cal T}(n)$ and ${\cal S}(n)$ be the time and space required to decide reachability in a layered grid graph of size $n \times n$ respectively. Then,
\[
{\cal T}(n) =
\begin{cases}
8n^2 ({\cal T}(n/k) + O(1)) & \text{if } n > k\\
O(k^2) & \text{otherwise}.
\end{cases}
\]
Hence,
\[
{\cal T}(n) = O\left(n^{3\frac{\log n}{\log k}}\right).
\]
Since we do not store any query made to the smaller grids, therefore the space required to check the presence of an edge in $H$ can be reused. $A_v$ and $A_h$ are arrays of size $k+1$ each. By Lemma \ref{lem:pathlength}, the number of elements in $S$ at any stage of the algorithm is bounded by $2k+1$. Therefore,
\[
{\cal S}(n) =
\begin{cases}
{\cal S}(n/k) + O(k \log n) & \text{if } n > k\\
O(k^2) & \text{otherwise}.
\end{cases}
\]
Hence,
\[
{\cal S}(n) = O\left(\frac{k}{\log k} \log^2 n+k^2\right).
\]
Now given any constant $\epsilon > 0$, if we set $k = n^{\epsilon/2}$, then we get ${\cal T}(n) = O(n^{6/\epsilon})$ and ${\cal S}(n) = O(n^{\epsilon})$. This proves Theorem \ref{thm:LGGR}.
\ignore{
Note that the maximum length of a path in $H$ is at most $2k$. Hence when dealing with layered grid graphs, we can easily get rid of the stack and use tape space instead. However for the sake of generality and with the hope that this idea is useful elsewhere as well, we use a stack to simulate the DFS.
Now we would like to mention that instead of deterministic Turing machines if we consider deterministic auxiliary pushdown machine, it will not add any power to the class {\ASC}. Note that deterministic auxiliary pushdown machine is basically a deterministic Turing machines along with a extra ``stack'' space of infinite length. Similar type of class in case of log-space computation is known as {\tt LOGDCFL} and is an important complexity class in the domain of space bounded computation. However this is not required in proving our main theorem of this paper, we provide the details for the interested reader to better understand the class {\ASC}.
\begin{definition}[Complexity Class {\tt Pushdown-ASC} or {\PASC}]
For a fixed $\epsilon > 0$, {\ePASC} denotes the set of languages decided by the deterministic auxiliary pushdown machine simultaneously in $n^{O(1)}$ time and $n^{\epsilon}$ space. Now the complexity class \emph{Pushdown-{\ASC}} or {\PASC} is defined as {\PASC}$:=\bigcap_{\epsilon > 0}${\ePASC}.
\end{definition}
Cook showed the following theorem which relates the class {\PASC} and {\ASC}.
\begin{theorem}[\cite{logdcflinsc2}]
\label{thm:cook}
If a language $L$ is accepted by a deterministic auxiliary pushdown machine simultaneously in $t(n)$ time and $s(n)\ge \log n$ space, then $L$ is also accepted by a deterministic Turing machine simultaneously within $O((t(n))^6)$ time and $O(s(n)+\log t(n)) \log t(n)$ space.
\end{theorem}
The above theorem directly leads to the following consequence.
\begin{proposition}
\label{prop:PDA}
$\PASC = \ASC$.
\end{proposition}
}
\section*{Acknowledgement}
We thank N. V. Vinodchandran for his helpful suggestions and comments. The first author would like to acknowledge the support of Research I Foundation.
\bibliographystyle{alpha}
\section{Preliminaries}
\label{sec:prelim}
We will use the standard notations of graphs without defining them explicitly and follow the standard model of computation to discuss the complexity measures of the stated algorithms. In particular, we consider the computational model in which an input appears on a read-only tape and the output is produced on a write-only tape and we only consider an internal read-write tape in the measure of space complexity. Throughout this paper, by $\log$ we mean logarithm to the base $2$. We denote the set $\{1,2,\cdots,n\}$ by $[n]$. Given a graph $G$, let $V(G)$ and $E(G)$ denote the set of vertices and the set of edges of $G$ respectively.
\begin{definition}[Layered Planar Graph]
A planar graph $G=(V,E)$ is referred as \emph{layered planar} if it is possible to represent $V$ as a union of disjoint partitions, $V=V_1\cup V_2 \cup \dots \cup V_k$, for some $k>0$, and there is a planar embedding of edges between the vertices of any two consecutive partitions $V_i$ and $V_{i+1}$ and there is no edge between two vertices of non-consecutive partitions.
\end{definition}
Now let us define the notion of layered grid graph and also note that grid graphs are by definition planar.
\begin{definition}[Layered Grid Graph]
A directed graph $G$ is said to be a $n \times n$ \emph{grid graph} if it can be drawn on a square grid of size $n \times n$ and two vertices are neighbors if their $L_1$-distance is one. In a grid graph a edge can have four possible directions, i.e., north, south, east and west, but if we are allowed to have only two directions north and east, then we call it a \emph{layered grid graph}.
\end{definition}
We also use the following result of Allender et al to simplify our proof \cite{ABCDR09}.
\begin{proposition}[\cite{ABCDR09}]
\label{prop:reductionLGGR}
Reachability problem in directed layered planar graphs is log-space reducible to the reachability problem in layered grid graphs.
\end{proposition}
\subsection{Class {\ASC} and its properties}
$\TISP(t(n),s(n))$ denotes the class of languages decided by a deterministic Turing machine that runs in $O(t(n))$ time and $O(s(n))$ space. Then, $\SC = \TISP(n^{O(1)},(\log n)^{O(1)})$. Expanding the class {\SC}, we define the complexity class {\ASC} (short for {\tt near-SC}) in the following definition.
\begin{definition}[Complexity Class {\tt near-SC} or {\ASC}]
For a fixed $\epsilon > 0$, we define ${\eASC}:= \TISP(n^{O(1)},n^{\epsilon})$. The complexity class {\ASC} is defined as
\[{\ASC}:=\bigcap_{\epsilon > 0} {\eASC}.\]
\end{definition}
We next show that {\ASC} is closed under log-space reductions. This is an important property of the class {\ASC} and will be used to prove Theorem \ref{thm:layeredplanarreach}. Although the proof is quite standard, but for the sake of completeness we provide it here.
\begin{theorem}
\label{thm:closure}
If $A \le _l B$ and $B \in {\ASC}$, then $A \in {\ASC}$.
\end{theorem}
\begin{proof}
Let us consider that a log-space computable function $f$ be the reduction from $A$ to $B$. It is clear that for any $x \in A$ such that $|x|=n$, $|f(x)| \le n^c$, for some constant $c>0$. We can think that after applying the reduction, $f(x)$ appears in a separate write-once output tape and then we can solve $f(x)$, which is an instance of the language $B$ and now the input length is at most $n^c$. Now take any $\epsilon>0$ and consider $\epsilon' =\frac{\epsilon}{c}>0$. $B \in {\ASC}$ implies that $B \in {\ASC_{\epsilon'}}$ and as a consequence, $A \in {\eASC}$. This completes the proof.
\end{proof}
|
1,314,259,992,982 | arxiv | \section{Introduction}
With 21-cm cosmology \citep{furlanetto_cosmology_2006}, we can potentially probe the earliest phases of the Universe after the cosmic microwave background (CMB) photons decoupled from the dense plasma so that protons and electrons could recombine to form neutral hydrogen when it was energetically favoured. Neutral hydrogen can absorb or emit the hyperfine HI line with a characteristic wavelength of $\lambda = 21$ cm. Finding the 21-cm line in emission or absorption state is dependent on the spin temperature defined through the relative occupancy rates between the excited and neutral state of hydrogen. When the spin temperature is lower than the background radiation in the universe, we will find the 21-cm signal in ``absorption'' and vice versa for emission. This temperature difference is known as the sky-averaged 21-cm signal and it is predicted to have a characteristic absorption feature \citep{pritchard_evolution_2008} which is evolving during Cosmic Dawn (CD) and the Epoch Of Reionisation (EOR). Physical effects affecting the signal shape include the Wouthuysen-Field effect \citep{wouthuysen_excitation_1952, field_excitation_1958} of Lyman-$\alpha$ photon coupling, X-Ray heating through high-energetic X-ray photons, and the progressive reionisation of the hydrogen due to star-forming regions at EOR. For a detailed review of the field see \cite{furlanetto_cosmology_2006, pritchard_21_2012, morales_reionization_2010, liu_data_2020}.
Observatories trying to detect the 21-cm hydrogen line are designed in one of two ways. First, the interferometric approach such as HERA \citep{deboer_hydrogen_2017}, SKA \citep{dewdney_square_2009}, LOFAR \citep{van_haarlem_lofar_2013}, MWA \citep{tingay_murchison_2013} trying to map the spatially fluctuating power spectrum of the 21-cm hydrogen line. Second, the simpler approach uses a wide-beam single antenna system, thus averaging over the whole sky to measure the integrated emission of bright and faint radio objects. However, they are harder to calibrate to detect millikelvin-level signals. These measurements are known as the sky-averaged experiment \citep{liu_global_2013} with observatories such as EDGES \citep{bowman_toward_2008}, SARAS 2 and 3 \citep{singh_saras_2018-1, singh_detection_2021}, LEDA \citep{price_design_2018}, PRIZM \citep{philip_probing_2018}, REACH \citep{de_lera_acedo_reach_2019} and many more in the low radio frequency regime enabling us to access CD and EOR directly.
Researchers from EDGES \citep{bowman_absorption_2018} were the first to claim a detection of an absorption profile at CD with a flattened Gaussian shape, centred at $f_{0,21} = 78 \pm 1$ MHz with an amplitude of $A_{21} = 500^{+500}_{-200}$ mK. This result contradicts the current astrophysical models of \citep{cohen_charting_2017}, who postulated a Gaussian-like profile with a maximum amplitude of around $A_{21} \approx 240$ mK. Moreover, re-examination of the EDGES data analysis by \cite{hills_concerns_2018} concluded that the best fitting profile reported by EDGES is not a unique solution and contains unphysical values in the foreground model with the possibility of an uncalibrated sinusoidal structure in the data. This, therefore, raised concerns and ignited a scientific debate \citep{bowman_reply_2018}
around the results of EDGES. Possible candidates to resolve this issue could include new physics, where we either have excessive cooling of the IGM beyond the adiabatic limit involving dark-matter particle interaction \citep{barkana_possible_2018, barkana_signs_2018, munoz_insights_2018} or an enhanced radio background \citep{jana_radio_2019, fialkov_signature_2019, mirocha_what_2019} with extreme implications on high-redshift star-formation \citep{mittal_implications_2022}.
However, there is also discussion that does not involve new physics i.e. an astrophysical origin of the signal, but rather instrumental effects such as a ground plane artefact \citep{bradley_ground_2019} mimicking an absorption profile or calibration errors \citep{sims_testing_2020} which are seen as sinusoids in the residuals. Furthermore, different data analysis techniques for the foreground involving changing the number of polynomial functions \citep{hills_concerns_2018} or applying maximally smooth functions (MSF) \citep{bevins_maxsmooth_2021, singh_redshifted_2019} could also detect unmodelled sinusoids in the residuals, which are also similarly seen in another sky-averaged experiment LEDA \citep{price_design_2018}. Finally, data obtained by the SARAS 3 radiometer reject the presence of the best-fitting absorption feature reported by EDGES with 95.3 \% confidence and state that there is an uncalibrated systematic structure in the EDGES analysis with a period of 12.5 MHz \citep{singh_detection_2021}.
We investigate the case of systematic effects i.e. a non astrophysical origin for the Bayesian pipeline of REACH. We introduce unmodelled sinusoidal structures inside forward modelled antenna temperature datasets, and study its influence on the resulting sky-averaged 21-cm signal parameter estimation using a Bayesian inference framework. In Section \ref{sec:BayesianInf}, we introduce the Bayesian inference framework, specifically the parameter estimation and model comparison component.
In Section \ref{sec:ForwardModel}, we describe how we generated our antenna temperature datasets using a physically motivated forward model.
In Section \ref{sec:BayesModel}, we discuss how to statistically model the antenna temperature datasets in a Bayesian way using a likelihood function and assigning a prior distribution on the model parameters. In Section \ref{sec:Results}, we quantify our results by using Bayesian evidence-driven model selection and a Goodness-of-Fit test and in Section \ref{sec:Conclusion} we conclude and summarise our results.
\section{Bayesian Inference and Nested Sampling}
\label{sec:BayesianInf}
To analyse the dataset we use Bayesian inference \citep{sivia_data_2006}, a statistical modelling framework where we can assign a prior distribution $p(\theta|M)$ for a model $M$ with its parameters $\theta$ to recover the posterior distribution $p(\theta|D, M)$ of the parameters after the dataset $D$ has been observed. This is achieved by applying Bayes theorem:
\begin{equation}
p(\theta|D,M) = \frac{p(D|\theta, M) p(\theta|M)}{p(D|M)},
\end{equation}
where $L(\theta) \equiv p(D|\theta, M)$ is the likelihood function of the model parameters and $\mathcal{Z} \equiv p(D|M)$ the Bayesian evidence.
To recover the posterior distribution for parameter estimation and the Bayesian evidence $\mathcal{Z}$ for model comparison, we use nested sampling \citep{skilling_nested_2006}. With nested sampling, one computes the Bayesian evidence by gradually shrinking the prior volume fraction $dX$: \begin{equation}
\mathcal{Z} = \int \mathcal{L}(\theta) p(\theta|M) d\theta = \int_0^1 \mathcal{L}(X) dX,
\end{equation}
through sampling from the prior which have likelihoods higher than an evolving likelihood constraint $\mathcal{L}$:
\begin{equation}
X(\mathcal{L}) = \int_{\mathcal{L}(\theta) > \mathcal{L}} p(\theta|M) d\theta.
\end{equation}
In practice, we use $\texttt{PolyChord}$ \citep{handley_polychord_2015, handley_polychord_2015-1} to find samples subject to the likelihood constraint $\mathcal{L}(\theta) > \mathcal{L}$ which implements slice sampling to draw new proposal samples. As \texttt{PolyChord} is a sampling-based algorithm, one can use the computed Bayesian evidence $\mathcal{Z}$ to get posterior samples $\theta^*$, therefore tackling the parameter estimation and model comparison aspects of Bayesian inference simultaneously.
To compare competing models we need to apply Bayes Theorem on the Bayesian evidence $\mathcal{Z}$: \begin{equation}
\label{eqn:modelprob}
p(M|D) = \frac{p(D|M)p(M)}{p(D)},
\end{equation}
to recover the model probabilities $p(M|D)$. We compare competing models $M_1$ and $M_2$ by forming the logarithmic Bayes factor: \begin{equation}
\label{eqn:BayesFactor}
\log \mathcal{K} = \log p(M_1|D) - \log p(M_2|D),
\end{equation}
where we assume a priori unity for the model probabilities: $p(M_1) = p(M_2)$. A positive Bayes factor $\log \mathcal{K} > 0$ indicates a model preference of $M_1$ over the competing model $M_2$.
\section{Forward modelling the dataset $D$}
\label{sec:ForwardModel}
To generate the dataset $D$ for this analysis, the forward model splits the process into a sum of physically motivated components:
\begin{equation}
\label{eqn:dataset}
D \equiv T_{\mathrm{data}}= T_{\mathrm{fg}} + T_{21}+ T_{\mathrm{sys}}+ T_{\mathrm{noise}},
\end{equation}
with $T_{\mathrm{fg}}$ the simulated foreground, $T_{21}$ the sky-averaged 21-cm signal, $T_{\mathrm{sys}}$ the systematic structure and $T_{\mathrm{noise}}$ the noise component of the antenna temperature. The decomposition can be seen in Figure (\ref{fig:dataset}) and the next sections describe how we simulate each component.
\begin{figure}
\centering
\includegraphics{Figures/dataset_composition.pdf}
\caption{Dataset composition, from top to bottom: the foreground $T_{\mathrm{fg}}$ contribution, the Gaussian sky-averaged 21-cm absorption signal $T_{21}$, the heteroscedastic radiometric noise $T_{\mathrm{noise}}$, a sinusoidal systematic structure $T_{\mathrm{sys}}$.}
\label{fig:dataset}
\end{figure}
\subsection{Sky simulation}
To simulate the foreground data we use the 2008 Global Sky Model (GSM \cite{de_oliveira-costa_model_2008}) at 408 MHz, $T_{408}(\Omega)$, and scale it down to 230 MHz, $T_{230}(\Omega)$, through the spatially varying spectral index:
\begin{equation}
\beta(\Omega) = \frac{\log \left(\frac{T_{230}(\Omega) - T_{\mathrm
{CMB}}}{T_{408}(\Omega) - T_{\mathrm
{CMB}}}\right)}{\log \left(\frac{230}{408} \right)},
\end{equation}
with $T_{\mathrm{CMB}}$ the cosmic microwave background at $T_{\mathrm{CMB}} = 2.725$ K. This spatially variable spectral index is a physically motivated choice to realistically model the spatial power distribution pattern through its spectral index. Hence, assuming a simple constant spectral index $\beta(\Omega) = \beta_0 = \mathrm{const}$ is an unfavourable and unphysical choice in the simulation process. We apply the downscaling of the GSM map to 230 MHz so that we utilise an approximated ``empty'' 50-200 MHz representation of the sky without having to worry about contamination introduced by the sky-averaged 21-cm signal as we will add it later on.
With this spectral index we simulate the foreground component:
\begin{equation}
T_{\mathrm{sim}}(\nu, \Omega) = (T_{230} - T_{\mathrm{CMB}} ) \left(\frac{\nu}{230} \right)^{-\beta(\Omega)} + T_{\mathrm{CMB}},
\end{equation}
with $T_{\mathrm{CMB}} = 2.725$ K the cosmic microwave background.
We convolve this simulated foreground map with the beam pattern $D_{\Omega}$ of a conical log-spiral antenna \citep{dyson_characteristics_1965}:
\begin{equation}
T_{\mathrm{fg}}(\nu) =\frac{1}{4 \pi} \int_{\Omega} D(\Omega,\nu) T_{\mathrm{sim}}(\Omega,\nu) d\Omega.
\end{equation}
We choose the conical log-spiral antenna beam pattern as \cite{anstey_informing_2021} reported that this beam pattern has the best performance for correctly extracting a sky-averaged 21-cm signal out of five antenna designs.
The initialisation of this sky-averaged experimental setup is located at the Karoo Radio Reserve in South Africa,
at -30.71131° latitude and 21.4476236° longitude where REACH is being built. For our analysis, we set the observation time to be a snapshot of the sky at `2019-10-01 00:00:00' UTC when the Milky Way centre is not at the zenith.
\subsection{Sky-averaged 21-cm signal}
We add a sky-averaged 21-cm absorption signal with a Gaussian shape to the antenna temperature:
\begin{equation}
\label{eqn:GaussianSignal}
T_{21}(\nu) = -A_{21} \exp\left(-\frac{1}{2 \sigma_{21}^2}(\nu-f_{0,21})^2\right),
\end{equation}
where $\nu$ is the frequency, $A_{21}$ the amplitude of the sky-averaged 21-cm signal, $f_{0,21}$ the central frequency and $\sigma_{21}$ the standard deviation. For the following analysis we set $A_{21} = 155$ mK, $f_{0,21} = 85$ MHz and $\sigma_{21} = 15$ MHz which is an Gaussian approximation of the standard case of the astrophysical models of \cite{cohen_charting_2017}.
We emphasise here the challenge of the sky-averaged 21-cm signal extraction, the Gaussian absorption signal is in the millikelvin-level, whereas the foreground contribution reaches several thousands of kelvin in our frequency band.
\subsection{Antenna temperature noise}
After the sky has been simulated, we add the noise component $T_{\mathrm{noise}}$. We choose a physically motivated radiometric noise model \citep{kraus_radio_1986} which can be modelled through a Gaussian distribution centred at $T_{\mathrm{fg}}(\nu)$ with heteroscedastic radiometric noise:
\begin{equation}
\label{eqn:radiometricNoise}
\sigma_{\mathrm{radio}}(\nu) = \frac{\eta T_{\mathrm{fg}}(\nu) + (1-\eta)T_0 + T_{\mathrm{rec}}}{\sqrt{\tau \Delta\nu}},
\end{equation}
with $T_{\mathrm{fg}}$ the antenna temperature, $\eta$ the antenna radiation efficiency, $T_0$ the ambient temperature, $T_{\mathrm{rec}}$ the antenna receiver temperature, $\tau$ the integration time and $\Delta\nu$ the channel width. For our analysis, we choose $\eta = 0.9$, $T_0 = 293.15$ K, $T_{\mathrm{rec}} = 500$ K, $\tau = 10^5$ s and $\Delta\nu = 0.1$ MHz. This choice of parameters suppresses the noise contribution by a factor of $\sqrt{\tau \Delta \nu} = 10^5$ which is the necessary noise level at the lower frequency end, where the antenna temperature $T_{\mathrm{fg}}$ can reach several thousands of kelvins.
Furthermore, we ``seed'' the noise so that the noise realisation is identical for each dataset. This engenders consistency in the Bayesian evidence when comparing models and across datasets.
\subsection{Systematic structure}
\label{subsec:sinusoidal}
For the systematic component, we choose the general and damped sinusoid:
\begin{equation}
\label{eqn:sinusoidal}
T_{\mathrm{sys,1}}(\nu) = A_{\mathrm{sys}}\sin \left(2\pi \frac{\nu}{P_{\mathrm{sys}}} + \phi_{\mathrm{sys}} \right),
\end{equation}
\begin{equation}
\label{eqn:dampedsinusoidal}
T_{\mathrm{sys,2}}(\nu) = \left(\frac{\nu}{f_0}\right)^{\alpha} A_{\mathrm{sys}}\sin \left(2\pi \frac{\nu}{P_{\mathrm{sys}}} + \phi_{\mathrm{sys}} \right),
\end{equation}
where $A_{\mathrm{sys}}$ is the amplitude, $\nu$ the frequency, $P_{\mathrm{sys}}$ the period of the sinusoid, $\phi_{\mathrm{sys}}$ the phase. The damped sinusoidal systematic has additional parameters $\alpha$ the damping coefficient and $f_0$ the reference central frequency of the antenna band. For our experimental setup we have the frequency band $50-200$ MHz, therefore $f_0 = 125$ MHz. Such systematic structures are seen to be left unmodelled in the EDGES results of \cite{bowman_absorption_2018} after a reanalysis by \cite{hills_concerns_2018, bevins_maxsmooth_2021, singh_redshifted_2019}.
We choose a parameter cube $A_{\mathrm{sys}} \in [0, 60, 100, 150, 200, 500, 1000]$ mK, $P_{\mathrm{sys}} \in [6, 12.5, 25, 50, 100]$ MHz and $\phi_{\mathrm{sys}} \in [0, \frac{\pi}{2}, \pi, \frac{3}{2}\pi]$. For the damped sinusoid, we set the damping coefficient to $\alpha = -2.5$ which is similarly seen in the LEDA \citep{price_design_2018} experiment by an analysis using MSF of \cite{bevins_maxsmooth_2021}. We inject either one of them into the dataset.
\section{Bayesian modelling of the dataset~$D$}
\label{sec:BayesModel}
\subsection{Likelihood function}
Once we have generated the dataset, we define a likelihood function $\mathcal{L}$ for our model and provide prior ranges of the model parameters $\theta$ which \texttt{PolyChord} needs to compute the Bayesian evidence $\mathcal{Z}$ for our chosen model and generate posterior samples $\theta^*$. As we introduce radiometric noise to generate the dataset, we use a radiometric likelihood function:
\begin{equation}
\label{eqn:radiometricLikelihood}
\log \mathcal{L} = \sum_{\nu} -\frac12 \log\left(2 \pi \sigma^2_{\mathrm{radio}}(\nu)\right) - \frac12 \left(\frac{T_{\mathrm{data}}(\nu) - T_{\mathrm{M}}(\nu)}{\sigma_{\mathrm{radio}}(\nu)}\right)^2,
\end{equation}
which is a Gaussian likelihood with heteroscedastic radiometric noise of eq. (\ref{eqn:radiometricNoise}) for the dataset $T_{\mathrm{data}}$ and $T_{\mathrm{M}}$ the modelling component.
We define four models:
\begin{equation}
\label{eqn:signalmodel}
M_1 \equiv T_{M_{1}} = T_{\mathrm{fg}} + T_{21},
\end{equation} the $\emph{signal model}$ and:
\begin{equation}
\label{eqn:nosignalmodel}
M_2 \equiv T_{M_{2}} = T_{\mathrm{fg}},
\end{equation}
the $\emph{no signal model}$. For both models we leave the systematic structure unmodelled to study its influence on the parameter inference of the sky-averaged 21-cm signal in Sections \ref{sec:BayesFac} and \ref{sec:GoFtest}.
We also introduce the models $M_3$ and $M_4$ where the systematic structure is modelled in addition to the foreground, sky-averaged 21-cm signal and noise component:
\begin{equation}
\label{eqn:signalSinusmodel}
M_3 \equiv T_{M_{3}}= T_{\mathrm{fg}} + T_{21} + T_{\mathrm{sys}},
\end{equation}
that includes the sky-averaged 21-cm signal model and:
\begin{equation}
\label{eqn:signalSinusmodel}
M_4 \equiv T_{M_{4}}= T_{\mathrm{fg}} + T_{\mathrm{sys}},
\end{equation}
that does not contain the sky-averaged 21-cm signal. Hence, these models contain a physically motivated component for the antenna electronics e.g. calibration errors \citep{sims_testing_2020} such as direction-dependent gain of the antenna \citep{price_design_2018}. We study the inference results of these models in Section \ref{sec:IncludeModel}.
To model the foreground we use the Bayesian foreground modelling framework with chromaticity correction of \cite{anstey_general_2021}. This model splits the foreground map into $N_{\mathrm{reg}}$ regions of uniform spectral indices to account for chromatic effects. With this subdivided foreground map which is an approximation of the foreground map used for dataset generation, the antenna temperature can be modelled by convolving it with the conical log-spiral beam pattern $D_{\Omega}$. Furthermore, we assume that there is a perfect ground plane which is similarly being built for REACH that uses a $20$ m $\times$ $20$ m metallic ground plane.
For our analysis, we set the foreground region parameter to $N_{\mathrm{reg}} = 14$ as complex structure modelling needs generally more foreground region parameters. One could expand this analysis by studying how the inference changes by varying the number of foreground regions. However, we decide to omit this analysis as it introduces another parameter dimension for an already computationally expensive inference. Moreover, our parameter cube of the sinusoidal structure is extensive enough for the purpose of our study. Additionally, \cite{anstey_general_2021} reported that the inference should in principle not differ much but rather be considered as a tool to search the optimum number of foreground regions with the highest Bayesian evidence. Nevertheless, we note that for a real dataset one should incorporate this extra dimension into the analysis.
For the sky-averaged 21-cm signal component we include a Gaussian signal model of eq. (\ref{eqn:GaussianSignal}) with $\theta_{21} = (f_{0,21 }, \sigma_{21}, A_{21})$ the parameter vector.
For the radiometric noise component, we include the radiometric noise model of eq. (\ref{eqn:radiometricNoise}) through the radiometric likelihood function and parameterise it through $ \theta_{\mathrm{noise}} = (T_{\mathrm{rec}}, \eta, \sigma_{\mathrm{noise}})$, where $\sigma_{noise} = 1/\sqrt{\tau \Delta\nu}$ is the noise level.
For the systematic component, we include a parameterized (damped) sinusoidal model of eq. (\ref{eqn:sinusoidal}) and (\ref{eqn:dampedsinusoidal}) $\theta_\mathrm{sys} = (\alpha_{\mathrm{sys}}, A_{\mathrm{sys}}, P_{\mathrm{sys}}, \phi_{\mathrm{sys}})$. We omit the damping coefficient $\alpha_{\mathrm{sys}}$ for inference when the dataset does not contain a damped structure.
To summarise, we have the following parameterization for each model: $\theta_{M_1} = (\theta_{\mathrm{fg}}, \theta_{21}, \theta_{\mathrm{noise}})$, $\theta_{M_2} = (\theta_{\mathrm{fg}}, \theta_{\mathrm{noise}})$, $\theta_{M_3} = (\theta_{\mathrm{fg}}, \theta_{21}, \theta_{\mathrm{noise}}, \theta_{\mathrm{sys}})$ and $\theta_{M_4} = (\theta_{\mathrm{fg}}, \theta_{\mathrm{noise}}, \theta_{\mathrm{sys}})$.
\subsection{Prior ranges of parameters}
The prior ranges of the model parameters are listed in Table \ref{tab:Priors} and we note that the parameter grid of the sinusoidal structure of Section \ref{subsec:sinusoidal} exceeds the prior upper ranges of $\sigma_{21}$ and $A_{21}$. We set these upper limits to signal strengths which are similarly seen in \cite{bowman_absorption_2018}. This forces \texttt{PolyChord} to look for these specific sky-averaged 21-cm signal within the datasets.
\begin{table}
\centering
\begin{tabular}{lll}
\toprule
{}
Parameter & Type & Range \\
\midrule
$\beta_{1:N_{\mathrm{reg}}}$ & uniform & 2.458-3.146 \\
$f_{0,21}$ & uniform & 50-200 MHz\\
$\sigma_{21}$ & uniform & 10-30 MHz \\
$A_{21}$ & uniform & 0-0.50 K \\
$T_{\mathrm{rec}}$ & log uniform & 100-1000 K \\
$\eta$ & uniform & 0.8-1 \\
$\sigma_{\mathrm{
noise}}$ & log uniform & $10^{-8}$-0.1 \\
\bottomrule
\end{tabular}
\caption{Prior choices for the foreground spectral indices $\beta_{1:N_{\mathrm{reg}}}$, sky-averaged 21-cm signal shape $f_{0,21}$, $\sigma_{21}$, $A_{21}$ and the radiometric noise model $T_{\mathrm{rec}}$, $\eta$, $\sigma_{\mathrm{
noise}}$}
\label{tab:Priors}
\end{table}
\subsection{\texttt{PolyChord} settings}
To compute the Bayesian evidence $\mathcal{Z}$ and to recover the posterior distributions of the model through \texttt{PolyChord} we set the following hyperparameters which are listed in Table \ref{tab:PolyChordSettings}.
\begin{table}
\centering
\begin{tabular}{ll}
\toprule
{}
Parameter & Settings \\
\midrule
\texttt{$n_{\mathrm{live}}$} & $N_{\mathrm{dim}}*25$ \\
\texttt{$n_{\mathrm{prior}}$} & $N_{\mathrm{dim}}*25$ \\
\texttt{$n_{\mathrm{fail}}$} & $N_{\mathrm{dim}}*25$ \\
\texttt{$n_{\mathrm{repeats}}$} & $N_{\mathrm{dim}}*5$ \\
$\texttt{precision criterion}$ & 0.001 \\
$\texttt{do clustering}$ & True \\
\bottomrule
\end{tabular}
\caption{\texttt{PolyChord} settings with \texttt{$n_{\mathrm{live}}$} the number of live points, \texttt{$n_{\mathrm{prior}}$}the number of initial prior samples before compression, \texttt{$n_{\mathrm{fail}}$} the number of failed spawns before stopping the algorithm and \texttt{$n_{\mathrm{repeats}}$} the number of slice sampling repeats which are all proportional to the model/parameter dimension $N_{\mathrm{dim}}$. The \texttt{precision criterion} is the nested sampling evidence termination criterion and \texttt{do clustering} if clustering of samples should be activated.}
\label{tab:PolyChordSettings}
\end{table}
Furthermore, we run \texttt{PolyChord} twice where in the second run we use the posterior sample mean ${\bar \theta^*}$ and sample standard deviation $\sigma^*$ of the first run to construct narrower prior ranges $\bar\theta^* \pm 5 \sigma^*$ (of same shape as in Table \ref{tab:Priors}) for the ``enhanced'' run. This allows us to narrow the initial priors and focus the parameter search around the posterior mode engendering a more tightly constrained posterior distribution of the parameters after the second run. A more detailed description of this method can be found in \cite{anstey_general_2021}.
\section{Results}
\label{sec:Results}
\begin{figure*}
\centering
\includegraphics{Figures/differentSinusoidsModeled.pdf}
\caption{Different sinusoidal structures (black) in the dataset $D$ (parameters in the right panel) and its influence on the foreground subtracted residuals (red), the sky-averaged 21-cm signal inference (blue) in comparison to the true 155 mK Gaussian signal shape (green). The left panel shows the sky-averaged signal recovery when the systematic structure is left unmodelled. The right panel shows the signal recovery when the systematic structure is modelled. One shade in the colorbar represents the $0.5 \sigma$ region.}
\label{fig:VaryingSinusoids}
\end{figure*}
\begin{figure*}
\centering
\includegraphics{Figures/basecase.pdf}
\caption{Top: Sky-averaged 21-cm signal recovery (blue) and foreground subtracted residuals (red) when there is no sinusoid present (black line). ``By eye'' the fit looks reasonable compared to the true signal shape (green).
Bottom: The radiometric noise residuals. This figure is referred to as the base case.}
\label{fig:signalRecovery_0mK}
\end{figure*}
In Figures (\ref{fig:VaryingSinusoids}) and (\ref{fig:signalRecovery_0mK}) we present five sky-averaged 21-cm signal extractions when sinusoidal structures without damping are present in the data and are left unmodelled. Each dataset has a varying sinusoidal structure i.e. different parameter pair values $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ present. Furthermore, these extractions are representative examples of five distinct cases of sinusoidal structures:
\begin{itemize}
\item low amplitude, low period case, (60 mK, 12.5 MHz),
\item low amplitude, high period case, (60 mK, 100 MHz),
\item high amplitude, low period case, (1000 mK, 12.5 MHz),
\item high amplitude, high period case, (1000 mK, 100 MHz),
\item base case, no systematic.
\end{itemize}
In all four cases when there is a sinusoid present inside the data, the resulting sky-averaged 21-cm signal shape is highly deformed. The amplitude, scale and central frequency of the sky-averaged 21-cm signal are all affected by the sinusoid i.e. the true Gaussian signal shape is not accurately recovered. We notice a tendency that the extracted sky-averaged 21-cm signal shape has an enhanced amplitude and broadened profile which is similarly reported by EDGES \citep{bowman_absorption_2018} given the predictions of the astrophysical models of \citep{cohen_charting_2017} and the discussion whether or not this is caused by an unmodelled systematic structure seen in the residuals \citep{hills_concerns_2018, bowman_reply_2018, bevins_maxsmooth_2021, singh_redshifted_2019}.
To statistically quantify the signal recovery in a Bayesian way, we will compare these extractions with the inference results of the no-signal model $M_2$ using the Bayesian evidence $\log \mathcal{Z}$ for all the combinations of parameters of the sinusoidal parameter cube in Section \ref{sec:BayesFac}.
We also discuss the Goodness-of-Fit for these five cases of the sinusoid in Section \ref{sec:GoFtest}.
Moreover, when including a systematic model the true sky-averaged 21-cm signal parameters are accurately recovered with the model having the highest Bayesian evidence of all models considered. This is discussed in more detail in Section \ref{sec:IncludeModel}.
\subsection{Bayes Factor $\log \mathcal{K}$ contours}
\label{sec:BayesFac}
\begin{figure*}
\centering
\includegraphics{Figures/logKradioL_contours.pdf}
\caption{Logarithmic Bayes factor $\log \mathcal{K}$ contour plots for varying parameterisation of the the systematic structure and radiometric noise present in the dataset $D$. The damped sinusoid has a damping coefficient $\alpha_{\mathrm{sys}} = -2.5$.}
\label{fig:logKradioL}
\end{figure*}
For each dataset, we compute the Bayesian evidence $\mathcal{Z}$ of the sky-averaged 21-cm signal model $M_1$ and the no-signal model $M_2$. We then compare these models by forming the Bayes factor $\log \mathcal{K}$ of eq. (\ref{eqn:BayesFactor}). Hence, a positive Bayes factor quantifies that the signal model $M_1$ is preferred for the dataset $D$.
Figure (\ref{fig:logKradioL}) shows the Bayes Factor contour plots for the parameter space $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ of the general sinusoidal structure relative to the sky-averaged 21-cm signal parameters $A_{21}$ and the Full Width at Half Maximum ($\mathrm{FWHM}_{21}$) for different phases $\phi_{\mathrm{sys}}$.
We find in the high amplitude, low period region of the systematics that the Bayes factor $\log \mathcal{K}$ is preferring the no signal model $M_2$. This is due to the low signal-to-noise ratio between the sky-averaged 21-cm signal and the noise-like background of the high amplitude low period oscillations and the increasing radiometric noise in the lower frequency end. This Bayesian way of model selection is based on the principle of Occam's razor to penalise more complicated models with constrained parameters that are not necessarily needed, see a discussion in \cite{hergt_bayesian_2021}.
However, this trend seems to break once we have oscillations with higher periods, where the Bayes factor reaches values of $\log \mathcal{K} > 4$. This indicates a strong preference for the 21-cm signal model $M_1$. For these cases the Bayes factor is always preferring the signal model, however, it is not clear if the actual sky-averaged 21-cm signal has been successful. This will be further investigated and discussed in Section \ref{sec:GoFtest}.
In the low amplitude region there is a clear preference for the sky-averaged 21-cm signal model $M_1$, irrespective of the period of the sinusoid. However, as in the high amplitude, high period case, it is not clear whether the sky-averaged 21-cm signal extraction has been successful.
A similar plot has been generated for the frequency decreasing damped sinusoidal structure which is also shown in Figure (\ref{fig:logKradioL}). Here we observe the same Bayes Factor trends as the general sinusoidal structure and it is also not clear whether the recovery of the sky-averaged 21-cm signal was successful.
\subsection{Goodness-of-Fit test}
\label{sec:GoFtest}
\begin{figure*}
\centering
\includegraphics{Figures/p_values_radioL_contours.pdf}
\caption{Goodness-of-Fit test $p$-value contour plots with the presence of a systemic structure and radiometric noise in the dataset $D$. The damped sinusoid has a damping coefficient $\alpha_{\mathrm{sys}} = -2.5$.}
\label{fig:pvalueContours}
\end{figure*}
\begin{figure*}
\centering
\includegraphics{Figures/triangle_plot.pdf}
\caption{Marginalised posterior distributions for all signal extractions when an unmodelled systematic structure is inside the dataset. The posterior distribution of the base case is in red. The black dotted lines represent the true parameters.}
\label{fig:marginPost}
\end{figure*}
The Bayes Factor analysis is the Bayesian way of quantifying which competing model is preferred, however, this analysis does not quantify if the extraction of the sky-averaged 21-cm signal has been successful i.e. the true sky-averaged 21-cm signal shape has been recovered.
To quantify the parameter recovery we use a Goodness-of-Fit test by evaluating the marginalised posterior probabilities of the true signal parameters. To estimate the probabilities, we learn the marginalised sky-averaged 21-cm signal posterior distribution through normalizing flows \citep{kobyzev_normalizing_2021, papamakarios_normalizing_2021}.
Normalizing flows are a density estimation method that utilise a series of bijections forming an invertible and differentiable transformation $T$ of a base distribution $p_X(x)$ e.g. standard Gaussian to a target distribution $p_Y(y)$:
\begin{equation}
y = T(x) \ \mathrm{where} \ x\sim p_X(x).
\end{equation}
The resulting target distribution is defined through the change of variables formula of probability distributions:
\begin{equation}
p_Y(y)= p_X(x) |\det J_T(x)|^{-1},
\end{equation}
where $J_T$ is the Jacobian of $T$ characterising the warping of the probability density space due to the transformation.
Because the bijections are invertible, the directed transformation from an arbitrarily complex distribution to e.g. a standard Gaussian is coined as the normalizing flow, where the flow represents the series of bijectors.
More complex methods such as masked autoregressive flows \citep{papamakarios_masked_2018} utilise a neural network as a bijective transformation and are starting to be applied in cosmology \citep{alsing_nested_2021} to learn sample distributions.
In practice, we use \texttt{margarine} \citep{bevins_margarine_2022} to train these autoregressive flows on the marginalised posterior distribution of the sky-averaged 21-cm signal parameters and conduct a Goodness-of-Fit test by evaluating the probabilities $P(\theta)$ of the true signal parameters $\theta_{\mathrm{true}}$ and constructing a $p$-value:
\begin{equation}
p = \frac{ \sum_{i=1}^{n} w_i \times \mathbbm{1}\{P(\theta^*_i) < P(\theta_{\mathrm{true}}) \}}{\sum_{i=1}^{n} w_i},
\end{equation}
where $\mathbbm{1}$ is the indicator function for our posterior samples $\theta^*_i$ with weights $w_i$.
We generated an equivalent contour plot of $p$-values using our parameter grid $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ with phase variations $\phi_{\mathrm{sys}}$ of the sinusoidal structure in Figure (\ref{fig:pvalueContours}) for the general and frequency decreasing damped cases. We also show the marginalised posterior distributions of the recovered sky-averaged 21-cm signal parameters in Figure (\ref{fig:marginPost}). The $p$-value contour plots can similarly be divided into two categories of sinusoidal systematics, the high or low period case.
For the base case where there is no systematic structure present inside the data ($A_{sys} = 0$ mK), we observe a $p$-value of $p \approx 0.07$. This is due to the true parameters for the sky-averaged 21-cm signal parameter pair ($\sigma_{21}, f_{0,21}$) being outside of the tightly constrained light red shaded $2\sigma$ region of the marginalised posterior distribution shown in Figure (\ref{fig:marginPost}). Nevertheless, we will consider this fit and $p$-value as well recovered given the compression rate of the posterior relative to the prior.
For the high period cases of the sinusoidal structure, we find $p\approx 0$, therefore, the true signal parameters have unlikely been generated by the posterior distribution of the samples. Combining these findings with the Bayes factor contour plots, we can infer that the pipeline is able to extract a sky-averaged 21-cm signal but not the correct one, as these large oscillations are mimicking a sky-averaged 21-cm signal.
For the lower period cases, we find that the $p$-values are significantly higher than the base case when there is no systematic structure in the dataset. In this region, the dataset is too noisy that one can fit the true sky-averaged 21-cm signal shape into the noise. This can be seen in the marginalised posterior distributions which show weak signs of compression relative to the prior, therefore allowing a wider parameter range to fit this dataset. Furthermore, the Bayes Factor analysis has shown that in this parameter region the no signal model $M_2$ with fewer parameters is preferred. Hence, this region is unsuitable for successful sky-averaged 21-cm signal extraction if the systematics are left unmodelled.
Only when decreasing the amplitude of the sinusoidal significantly $\frac{A_{\mathrm{sys}}}{A_{21}} < 0.5$ the $p$-value combined with the effects of a tighter posterior compression indicates a better recovered fit. Therefore, in this parameter region the sky-averaged 21-cm signal recovery can be partly successful but at the cost of a deformed sky-averaged 21-cm signal which does not accurately resemble the true sky-averaged 21-cm signal shape.
Moreover, we see similar trends regarding the $p$-value for the frequency damped sinusoidal structures.
\subsection{Including a systematic model}
\label{sec:IncludeModel}
The previous analysis has shown that an unmodelled systematic structure can have a dramatic influence on the resulting parameter estimation of the sky-averaged 21-cm signal shape. Hence, we will now study the inference when we include a systematic model i.e. examine the dataset with the systematic models $M_3$ and $M_4$.
The recovery of the sky-averaged 21-cm signal shown in Figure (\ref{fig:VaryingSinusoids}) is now more accurate relative to the true sky-averaged 21-cm signal shape for all four cases of $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ variations. For the high period cases, the central frequency and shape of the sky-averaged 21-cm signal is now more accurately recovered and for the datasets with lower period sinusoids, we see identical success.
More importantly, the Bayesian evidence (shown in Figure (\ref{fig:logZComparison})) for the true model $T_{\mathrm{fg}} + T_{21} + T_{\mathrm{sys}} + T_{\mathrm{Noise}}$ is the highest out of all four competing models for all systematic datasets (extractions) considered. Moreover, for the base case when there is no systematic structure inside the dataset, the Bayesian evidence is the highest for the true signal model $T_{\mathrm{fg}} + T_{21} + T_{\mathrm{Noise}}$. Here, the systematic model has a similar but lower Bayesian evidence than the true model, however, the posterior estimates of the amplitude of the systematic are $\bar A^*_{\mathrm{sys}} \approx 0$, therefore, being in agreement with the true model which does not require the systematic model.
Hence, the Bayesian evidence gives us the capability of accurately distinguishing whether we need a systematic model when there is a systematic structure inside the data and correctly preferring the true model of the dataset, which is critical in a real experimental scenario as we do not know the true underlying model.
We note that the for the damped sinusoidal case with high periods, that the Bayes Factor is $\log \mathcal{K} \approx 0$ when comparing the models $M_3$ and $M_4$. The corresponding $p$-values of these high period cases are also slightly higher than their low period counterpart. This is due to a larger spread of the posterior samples of the sky-averaged 21-cm signal amplitude $A^*_{21}$. This enhanced uncertainty is caused by the correlation with the phase $\phi_{\mathrm{sys}}$ and period $P_{\mathrm{sys}}$ of the sinusoid. The sinewaves maximum can superpose with the amplitude of the sky-averaged 21-cm signal and due to the long periods affecting the sky-averaged 21-cm signal amplitude more significantly when changing the phase and period marginally. Therefore, the Bayesian evidence can not clearly distinguish whether we need a sky-averaged 21-cm signal model $T_{21}$ due to the larger uncertainty of the sky-averaged 21-cm signal amplitude.
However, this degeneracy is disentangled for sinusoids with smaller periods where the posterior samples display uncorrelated marginal posterior distributions between sky-averaged 21-cm signal and sinusoidal parameters. Hence, for these low period sinusoids, one achieves successful sky-averaged 21-cm signal recovery and separation of the two model components with the true model having the highest Bayesian evidence.
\begin{figure*}
\centering
\includegraphics{Figures/logZComparisonModelsBothSinus.pdf}
\caption{Bayesian evidence $\log \mathcal{Z}$ for four competing models: the signal model (blue), the no signal model (red), the signal model with a sinusoid model (green) and the no signal model with a sinusoid model (yellow). The dataset D includes either general (top) and damped (bottom) sinusoidal structures with varying parameters $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ for the $\phi_{\mathrm{sys}}= 0$ case. The errors are in the order of $\sigma_{\log \mathcal{Z}} \approx 0.3$.}
\label{fig:logZComparison}
\end{figure*}
\section{Conclusion}
\label{sec:Conclusion}
Re-examination of the EDGES data analysis has shown that there is an unmodelled systematic structure present inside the data when using various data analysis techniques such as MSFs or changing the number of polynomial functions. These unmodelled structures are possibly evidence that the reported best-fitting shape of the EDGES collaboration is not of astrophysical origin but due to instrumental effects such as calibration errors. We investigate the influence of these unmodelled systematic structures on the sky-averaged 21-cm signal parameter estimation through the Bayesian inference framework by using a nested sampling-based algorithm \texttt{PolyChord}.
We used a physically motivated forward model to generate datasets $D$ representing the antenna temperature. Each dataset has an identical foreground contribution, radiometric noise and a Gaussian sky-averaged 21-cm absorption signal. After generating these datasets, we added varying sinusoidal structures where we parameterised the amplitude $A_{\mathrm{sys}}$, the period $P_{\mathrm{sys}}$, the phase $\phi_{\mathrm{sys}}$ and the damping coefficient $\alpha_{\mathrm{sys}}$ of the sinusoid.
We define four models, the signal model $M_1$ with parameters $ \theta_{M_1} = ( \theta_{\mathrm{fg}}, \theta_{21}, \theta_{\mathrm{noise}})$ and the no signal model $M_2$ with parameters $\theta_{M_2} = ( \theta_{\mathrm{fg}}, \theta_{\mathrm{noise}})$ where we left the systematic structure unmodelled and the systematic models $M_3$ and $M_4$ with the parameterisation $\theta_{M_3} = (\theta_{\mathrm{fg}}, \theta_{21},\theta_{\mathrm{noise}}, \theta_{\mathrm{sys}}),\theta_{M_4} = (\theta_{\mathrm{fg}},\theta_{\mathrm{noise}}, \theta_{\mathrm{sys}}) $, where we include a systematic model. As we used the radiometric noise model to generate our datasets, we use a radiometric likelihood function, which can be modelled through a Gaussian distribution with heteroscedastic radiometric noise. With this radiometric likelihood function and combined with the prior ranges of our parameters, we can recover the marginalised posterior distributions of the parameters using the nested sampling-based algorithm \texttt{PolyChord}. For each dataset, we use \texttt{PolyChord} to generate posterior samples and compute the Bayesian evidence $\log \mathcal{Z}$ for the models considered.
To compare these models, we constructed the logarithmic Bayes factor $\log \mathcal{K}$ by applying Bayes Theorem on the Bayesian evidence $\mathcal{Z}$ to acquire the model probabilities. For our parameterised sinusoids, we computed Bayes factor $\log \mathcal{K}$ contour plots and found that when the systematic structures are left unmodelled the signal model $M_1$ is generally preferred over the no signal model $M_2$ except in the high amplitude, low period regime of the sinusoid. This is due to the low signal-to-noise ratio between the sky-averaged 21-cm signal and the noisy background contributions by the sinusoid and the radiometric noise.
However, the Bayes factor $\log \mathcal{K}$ contour plots do not contain any information if the actual sky-averaged 21-cm signal recovery has been successful i.e. the true values of the sky-averaged 21-cm signal are within the posterior sample estimates. To quantify this statistically we used the Goodness-of-Fit test by learning the posterior distribution of our sky-averaged 21-cm signal parameters $\theta_{21} = (f_{0,21}, \sigma_{21}, A_{21})$ through normalizing flows and computed the corresponding $p$-value. We created a $p$-value contour plot for our sinusoidal parameter cube $(A_{\mathrm{sys}}, P_{\mathrm{sys}}, \phi_{\mathrm{sys}})$ similar to the Bayes Factor $\log \mathcal{K}$ contours.
This analysis has shown that the sky-averaged 21-cm signal recovery is only slightly successful with the cost of high uncertainties in the sky-averaged 21-cm signal parameters for sinusoidal parameter regions where $\frac{A_{\mathrm{sys}}}{A_{21}} <1$ and $\frac{P_{\mathrm{sys}}}{\mathrm{FWHM}_{21}} <1$. For the other regions, the corresponding $p$-value was either close to zero or one indicating a failure of sky-averaged 21-cm signal recovery due to wrong inference or low signal-to-noise ratio, respectively.
However, this picture changes dramatically once we add a systematic model next to the foreground, the sky-averaged 21-cm signal and the noise, hence using the models $M_3$ and $M_4$ for inference. We show that for all four regions of $(A_{\mathrm{sys}}, P_{\mathrm{sys}})$ combinations, the resulting sky-averaged 21-cm signal extraction has been successful and the Bayesian evidence $\log \mathcal{Z}$ for the true model $M_3$ is the highest out of all four competing models. Therefore, the Bayesian evidence provides the tools to decide whether we need a systematic model when there is a systematic structure inside the dataset, and accurately preferring the true model of the dataset.
Furthermore, the marginalised posterior distributions of the parameters show an uncorrelated behaviour between the signal parameters $(f_{0,21}, \sigma_{21}, A_{21})$ and $(A_{\mathrm{sys}}, P_{\mathrm{sys}}, \phi_{\mathrm{sys}})$ for sinusoids with small periods. This indicates a successful separation of these structures in the modelling process, therefore resulting in more precise posterior inferences. For longer periods, there is a tendency that the period and phase of the sinusoid are correlated with the amplitude of the sky-averaged 21-cm signal, hence, making it a crucial task to precisely constrain these sinusoidal parameters through the prior. This is evidence that by including a physically motivated systematic model there is a possibility to do accurate science when there is a systematic structure present in the data.
Overall, this analysis has shown that if there is a systematic structure present inside the dataset, Bayesian inference strongly prefers a model where we include a systematic feature. Additionally, the parameter inference results in a more accurate recovery of the sky-averaged 21-cm signal parameters and more tightly constrained posterior distributions. Finally, this analysis is used to guide the design and the challenging task of calibration of the sky-averaged experiment REACH.
\section*{Acknowledgements}
This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service, provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council. KHS would like to thank Dominic Anstey for developing and providing the core Bayesian REACH pipeline. EdLA is supported by the Science and Technologies Facilities Council Ernest Rutherford Fellowship. WH is supported by a Gonville \& Caius College Research Fellowship and Royal Society University Research Fellowship.
\bibliographystyle{mnras}
|
1,314,259,992,983 | arxiv | \subsection*{Acknowledgements}
We thank
V.~Mart{\' i}n-Lozano for collaboration in the early stages of this work,
as well as F.~Campello
and
M.~Kado
for helpful discussions.
The work of S.~H.\ is supported in part by the
Spanish Agencia Estatal de Investigaci{\' o}n (AEI) and the EU Fondo Europeo de
Desarrollo Regional (FEDER) through the project FPA2016-78645-P and in part by
the “Spanish Red Consolider MultiDark” FPA2017-90566-REDC, in
part by the MEINCOP Spain under contract FPA2016-78022-P and in part by
the AEI through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597.
We acknowledge support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany's Excellence
Strategy -- EXC 2121 ``Quantum Universe'' – 390833306.
\section{Introduction}
\label{sec:intro}
In the year 2012 the ATLAS and CMS Collaborations have discovered a new
particle with a mass of about
$125 \,\, \mathrm{GeV}$~\cite{Aad:2012tfa,Chatrchyan:2012ufa,Khachatryan:2016vau}.
Within the current experimental and theoretical uncertainties
the properties of this particle, which in the
following is denoted as $h_{125}$, agree with the predictions for the
Higgs boson of the Standard Model~(SM) of particle physics,
but they are also
compatible with the interpretation as a Higgs boson in
a variety of SM extensions corresponding to different underlying
physics.
Valid parameter regions in scenarios of physics beyond the SM (BSM) featuring
extended Higgs-boson sectors are established taking into account
the measurements of Higgs-boson couplings, which are known
experimentally to a precision between $10\%$ and
$30\%$~\cite{CMS:2020xwi,ATLAS:2019nkf}, the existing
limits from searches for additional Higgs bosons, as well as other constraints
(see the discussion below).
Consequently, the question whether the observed scalar boson forms
part of an extended Higgs sector is one of the science drivers for
the upcoming run of the LHC beginning in 2022
(Run~3), and beyond.
The most obvious possibility for realisations of
extended Higgs-boson sectors is that all
additional Higgs bosons have masses that are larger than $125 \,\, \mathrm{GeV}$.
But also cases where at least one of the additional Higgs bosons is lighter
than the one at $125 \,\, \mathrm{GeV}$ are phenomenologically viable.
Thus, searches for BSM Higgs bosons ranging from very low to very
high mass scales are crucial in this context.
Among the models with extended Higgs-boson sectors the most studied ones
are models with two Higgs doublets (2HDM), possibly with an additional
(real or complex) Higgs singlet (N2HDM or 2HDMS), as well as their
supersymmetric (SUSY) counter parts, the MSSM and the NMSSM.
The searches at the LHC for BSM Higgs bosons with masses above $125 \,\, \mathrm{GeV}$ have
shown several excesses in the data recorded between 2015 and 2018
(Run~2), which is a finding that by itself is not
unexpected in view of the large number of searches that have been conducted.
However, it is remarkable that several searches show an excess of events
above the background expectation
around the same mass scale of a hypothetical new Higgs boson $\phi$ of
$m_\phi \approx 400 \,\, \mathrm{GeV}$.
The various excesses each reach
the level of
about $3\,\sigma$ local significance or above,
but the global significance of any excess individually, taking
into account the ``look elsewhere effect'', stays below $3\,\sigma$. The
current experimental
situation regarding the mentioned excesses at $m_\phi \approx 400 \,\, \mathrm{GeV}$
can be briefly summarized as follows
(we use here a notation in terms of the CP-even and -odd
eigenstates, $H$ and $A$, respectively, but
interpretations in terms of CP-mixed states would also be possible).
\begin{itemize}[noitemsep,topsep=0pt]
\item[-] $A \to t\bar t$: CMS reported a local
excess of $3.5\,\sigma$ in
their first year Run~2 data~\cite{Sirunyan:2019wph}. There is no
corresponding Run~2 ATLAS analysis available.
\item[-] $\phi \to \tau^+\tau^-$: ATLAS reported a local excess of
$2.7\,\sigma$ in their full Run~2 data~\cite{Aad:2020zxo}.
The corresponding CMS analysis, using only first year
Run~II data, does not show an excess, but also has
substantially weaker expected sensitivities.
\item[-] $A \to Z h_{125}$: ATLAS reported a local
excess of $3.6\,\sigma$ in their first year Run~2
data at a mass of around $440 \,\, \mathrm{GeV}$ in the b-quark
associated production channel~\cite{Aaboud:2017cxo}.
The corresponding CMS analysis is
consistent with the SM background
expectation~\cite{Sirunyan:2019xls}. An updated full Run~2
ATLAS analysis
for the gluon fusion (ggF) production channel also agrees with
the SM expectation~\cite{ATLAS:2020pgp}.
\end{itemize}
\noindent
This experimental situation
(see also the discussion in \citere{Richard:2020cav})
triggers the question whether all or some of these
excesses could be fitted simultaneously.
In a recent publication~\cite{Arganda:2021yms},
in which this question was addressed
in a model-independent approach by assuming
a CP-odd state at $400\,\, \mathrm{GeV}$
with independently adjustable
couplings to the SM particles
as the origin of both excesses,
it was found that the preferred values
of the couplings
would correspond to rather contrived scenarios
that do not lend themselves to
an immediate interpretation within
UV-complete models.
Within the present paper we take a different viewpoint and analyse the observed
excesses in the context of two popular models, namely the
N2HDM and the NMSSM. The symmetry properties of these models give rise to
correlations between the couplings of the CP-odd Higgs boson to different
SM particles, in contrast to the freely adjustable couplings that were
considered in \citere{Arganda:2021yms}.
In our investigation in this paper we will
perform a $\chi^2$ analysis taking into account the
observed excesses in the
$pp \to A \to t \bar t$ channel at CMS and the
$pp \to \phi \to \tau^+\tau^-$
channel at ATLAS, since in both cases the observed excesses over the
background expectation
are not in direct tension with corresponding results
(in terms of search channel and integrated luminosity)
from the other collaboration. The situation is
more ambiguous for the excess in the
$A \to Z h_{125}$ search reported by ATLAS
(as we will discuss in
more detail in \refse{sec:excesses}).
In view of this situation we will focus on the
parameter space that is preferred as a result of the
$\chi^2$ analysis where the $A \to Z h_{125}$ search is not included.
However,
we investigate
whether these parameter regions would also
be compatible with a possible signal in the
$b \bar b \to A \to Z h_{125}$ channel.
As a second step of our analysis we take into account the possibility that a
Higgs boson of an extended Higgs sector could also be lighter than the
observed state at $125\,\, \mathrm{GeV}$.
Searches for Higgs bosons below $125 \,\, \mathrm{GeV}$ have been performed at
LEP~\cite{Abbiendi:2002qp,Barate:2003sz,Schael:2006cr},
the Tevatron~\cite{Group:2012zca} and the
LHC~\cite{Sirunyan:2018aui,Sirunyan:2018zut,ATLAS:2018xad}.
It is intriguing that also
two of those searches show a $2-3\,\sigma$ local excess around the same
mass of about $96 \,\, \mathrm{GeV}$, that is not in tension with other search limits:
\begin{itemize}[noitemsep,topsep=0pt]
\item[-] $pp \to \phi \to \gamma\ga$: CMS reported a local excess of
about $3\,\sigma$ in their first year Run~2 data~\cite{Sirunyan:2018aui}, with a
similar upward deviation of~$2\,\sigma$ local
in the Run\,1 data at a comparable
mass~\cite{CMS:2015ocq}. The ATLAS results based on the data of the
first two years of Run~2~\cite{ATLAS:2018xad} are not sensitive to
the excess.
\item[-] $e^+e^- \to Z\,\phi \to Z\,b\bar b$: LEP reported a local excess of
around $2\,\sigma$~\cite{Biekotter:2019kde}.
\end{itemize}
\noindent
In previous analyses focussing just on the excesses at about $96\,\, \mathrm{GeV}$
it was shown that type~II and type~IV of the N2HDM can fit both
excesses simultaneously~\cite{Biekotter:2019kde}.
This was also investigated in combination with a viable dark-matter
candidate for the case where a complex singlet instead
of a real singlet is considered~\cite{Biekotter:2021ovi}.
It was furthermore
demonstrated that SUSY models like
the NMSSM~\cite{Cao:2016uwt,Domingo:2018uim,Choi:2019yrv}
or the
$\mu\nu$SSM~\cite{Biekotter:2017xmf,Biekotter:2019gtq}
can account for the excesses at a level
of roughly $1\sigma$.
On the other hand, in the MSSM neither a $400 \,\, \mathrm{GeV}$ Higgs boson
can be accomodated in view of the existing constraints (see the
discussion in \refse{sec:nmssm}), nor the CMS excess at around
$96 \,\, \mathrm{GeV}$ can be realized~\cite{Bechtle:2016kui}. Consequently,
in the present paper we focus on the N2HDM and the NMSSM.
We address the question, after separately analyzing
possible
interpretations of the observed excesses at about $400\,\, \mathrm{GeV}$ as described
above (for a discussion of the $t \bar t$ excess
in a general 2HDM, see \citere{Hou:2019gpn}),
whether the observed patterns at $400 \,\, \mathrm{GeV}$ and $96
\,\, \mathrm{GeV}$ can be described simultaneously within the considered models.
Our paper is organized as follows.
In \refse{sec:excesses} we summarize the experimental results in the various
search channels and define the various $\chi^2$ contributions
employed in our analysis. \refse{sec:n2hdm} is devoted to possible
interpretations of
the observed excesses within the N2HDM.
We first investigate
whether the model can fit the
observed excesses at $400 \,\, \mathrm{GeV}$ in the
$pp \to A \to t \bar t$ and $pp \to \phi \to \tau^+\tau^-$ channels
while being in agreement with
the relevant theoretical and experimental constraints.
We find that
the parameter regions that would be preferred by possible signals in the
two channels do not overlap with each other. On the other hand, each of the
two excesses individually can be
described very well by the
N2HDM with Yukawa structure of type~II.
In a second step we then demonstrate that this model
can simultaneously also describe both excesses at $96 \,\, \mathrm{GeV}$.
In \refse{sec:nmssm} we extend our analysis to the case of the NMSSM, where the
structure of the Higgs sector is more rigid than in the
N2HDM. Nevertheless, we demonstrate that also the NMSSM can fit each of
the excesses at $400 \,\, \mathrm{GeV}$ individually, while complying with the
BSM Higgs-boson searches and the signal-rate measurements
of the Higgs boson at $125\,\, \mathrm{GeV}$.
In this analysis, we show that a particularly
well motivated parameter region to accommodate the
$t \bar t$ excess is given by the alignment-without-decoupling
limit of the NMSSM~\cite{Carena:2015moc}.
In this limit a light singlet-like scalar is naturally
present in the Higgs spectrum. Consequently, as a next step we
include the $96 \,\, \mathrm{GeV}$ excesses also into the NMSSM analysis and show
that only the CMS excess can be accommodated alongside with the
$400 \,\, \mathrm{GeV}$ excesses. In \refse{sec:prosp}
these analyses are supplemented with a
discussion on the prospects for
future experimental tests of the parameter space regions of the
N2HDM and the NMSSM that are favored by the experimental excesses.
We conclude in \refse{sec:conclusion}.
In the Appendix we provide supplementary material on different Yukawa types of the
N2HDM and a discussion of the constraints from the signal at $125\,\, \mathrm{GeV}$ in the
alignment-without-decoupling limit of the NMSSM.
\section{Experimental situation}
\label{sec:excesses}
Current LHC Run 2 data of proton-proton collisions at a
center-of-mass energy of $13\,\unit{TeV}$ show some tantalizing local excesses
in various channels with local significances around the
$3\,\sigma$ confidence level (C.L.) which are summarized in the following.
\subsection{Search for \texorpdfstring{\boldmath{$H,A
\to t\bar{t}$}}{HA2tt}}
A search for additional scalar or pseudoscalar
Higgs bosons decaying to a top quark pair was performed by the CMS
experiment~\cite{Sirunyan:2019wph}. The data set analyzed
corresponds to an integrated luminosity of
$35.9\,\unit{fb}^{-1}$. Final states with one or two charged leptons
were considered. The invariant mass of the
reconstructed top quark pair system and angular variables
sensitive to the spin of the particles decaying into the top quark
pair were used to search for signatures of the \ensuremath{H}\ or \ensuremath{A}\ bosons,
taking the interference with the SM \ensuremath{t\bar{t}}\ background into account.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Chisqtt.pdf}
\caption{\small $\chi^2_{t \bar t}$ as a function
of $c_{A t \bar t}$ assuming $m_A = 400\,\, \mathrm{GeV}$ for the
different width hypotheses used in the experimental
analysis~\cite{Sirunyan:2019wph,url400cms}.
$\chi^2_{t \bar t}$ is defined relative
to the best fit point.
}
\label{Chisqtt}
\end{figure}
Constraints on the strength modifiers of the \ensuremath{H}\ensuremath{t\bar{t}}
(\ensuremath{A}\ensuremath{t\bar{t}}) couplings \ensuremath{c_{\ensuremath{H} t\bar{t}}} (\ensuremath{c_{\ensuremath{A} t\bar{t}}}) were derived
as a function of the mass and width of the heavy Higgs boson \ensuremath{H}
(\ensuremath{A}), with the mass of the bosons ranging from 400 to
$750\,\, \mathrm{GeV}$. Here, the modifiers are defined as the coupling strengths
relative to the ones of a SM Higgs boson.
In this study it is assumed that there is no CP violation in the Higgs
sector. As a consequence the \ensuremath{H}\ and \ensuremath{A}\ Higgs bosons are
CP eigenstates and do not mix with each other, which means that there
are no inteference effects between \ensuremath{H}\ and \ensuremath{A}.
The model-independent search was performed for
each CP state separately, setting the coupling modifier for the
other respective CP state to zero.
As a result, a moderate signal-like
deviation was observed for the hypothesis of a pseudoscalar Higgs
boson with the mass ${m_\ensuremath{A} \approx 400\,\, \mathrm{GeV}}$ and a total relative
width of $\Gamma_\ensuremath{A} / m_\ensuremath{A} \approx 4.5\%$, with a local significance of $3.5 \pm
0.3$ standard deviations. This translates to a global significance of
$1.9 \, \sigma$. The excess is also consistent with slightly different masses
and widths of a possible signal, however, with less
statistical significance.
The deviation is driven by the CP-odd Higgs boson, while the impact from the
CP-even Higgs boson is small, because at a mass of ${\approx 400\,\, \mathrm{GeV}}$
the resonant production cross section of a CP-odd Higgs boson is much
larger than that of a CP-even Higgs boson.
This can also be inferred, for instance, from Fig.~3
of~\citere{Bernreuther:1997gs}.
There is no corresponding search at
$13\,\unit{TeV}$ in this channel yet from the ATLAS Collaboration, but
there is a search for heavy pseudoscalar and scalar Higgs bosons
decaying into a \ensuremath{t\bar{t}}\ pair performed with an integrated luminosity
of $20.3\,\unit{fb}^{-1}$ at a center-of-mass energy of
$8\,\unit{TeV}$~\cite{Aaboud:2017hnm}. No significant deviation from
the SM prediction was observed in the \ensuremath{t\bar{t}}\ invariant mass spectrum
in final states with an electron or muon, large missing transverse
momentum, and at least four jets. However, dileptonic final states and angular
variables were not utilized for this analysis, and
cross section limits were set only for masses ranging
from $500\,\, \mathrm{GeV}$ to $750\,\, \mathrm{GeV}$. Consequently, a combined analysis of
the ATLAS and CMS results is not possible.
The \ensuremath{\chi^2}-distribution associated to the CMS
analysis~\cite{url400cms,Sirunyan:2019wph}, \ensuremath{\chi^2_{\ttbar}}\, is shown in
\reffi{Chisqtt} as a function of the coupling strength \ensuremath{c_{\ensuremath{A} t\bar{t}}}\ for
the case of a pseudoscalar Higgs boson with an assumed mass of
$m_{\ensuremath{A}}=400 \,\, \mathrm{GeV}$ and different values of the width,
$\Gamma_{\ensuremath{A}}$, ranging from 0.1\% to 25\% of its mass.\footnote{It
should be noted that in Fig.~7 of \citere{Sirunyan:2019wph} there is a
typo
in the label of the vertical axis, which should read
$-\ln\left(L(g_{At \bar t}) / L_{\mathrm{SM}}\right)$.}
Width values between $1\%$ and $10\%$ have a \ensuremath{\chi^2}\ value close
or below 1. The \ensuremath{\chi^2}\ is given by twice the negative logarithm of
the likelihood function
\begin{linenomath}
\begin{equation}
\label{Eq:likelihood}
\begin{gathered}
\chi^2_{\ensuremath{t\bar{t}}}
= - 2 \cdot \ln\LP L(\ensuremath{c_{\ensuremath{A} t\bar{t}}}; m_{\ensuremath{A}} , \Gamma_{\ensuremath{A}},
\vec{\nu}) \over L_{\rm max} \RP\,, \\[.3em]
L(\ensuremath{c_{\ensuremath{A} t\bar{t}}}; m_{\ensuremath{A}} , \Gamma_{\ensuremath{A}}, \vec{\nu}) = \left(\prod_i
\frac{\lambda_i^{n_i}(\ensuremath{c_{\ensuremath{A} t\bar{t}}}; m_{\ensuremath{A}},
\Gamma_{\ensuremath{A}}, \vec{\nu})}{n_i!}\,
{\mathrm e}^{-\lambda_i(\ensuremath{c_{\ensuremath{A} t\bar{t}}}; m_{\ensuremath{A}} ,
\Gamma_{\ensuremath{A}}, \vec{\nu})}\right) \, G(\vec{\nu}) ,\\[.3em]
\lambda_i(\ensuremath{c_{\ensuremath{A} t\bar{t}}}; m_{\ensuremath{A}}, \Gamma_{\ensuremath{A}}, \vec{\nu}) =
\ensuremath{c_{\ensuremath{A} t\bar{t}}}^4\, s_{R,i}^\ensuremath{A}(m_\ensuremath{A},
\Gamma_\ensuremath{A}, \vec{\nu}) + \ensuremath{c_{\ensuremath{A} t\bar{t}}}^2\,
s_{I,i}^\ensuremath{A}(m_\ensuremath{A}, \Gamma_\ensuremath{A}, \vec{\nu}) +
b_i(\vec{\nu})
,
\end{gathered}
\end{equation}
\end{linenomath}
\noindent
with $b_i$ denoting the combined background yield in a given bin $i$
of the invariant \ensuremath{t\bar{t}}\ mass and spin analyzing distribution,
$s_{R,i}^\ensuremath{A}$ and $s_{I,i}^\ensuremath{A}$ the signal yields in a given bin for
the resonant and interference part, respectively, $\vec{\nu}$
the vector of nuisance parameters on which the signal and background
yields generally depend, and $n_i$ the observed yield in data.
The external constraints on the nuisance parameters
are taken into account in the likelihood via a product of
corresponding probability density functions, $G(\vec{\nu})$.
The \ensuremath{\chi^2_{\ttbar}}\ distribution is normalized by
$L_{\rm max} = L(\ensuremath{c_{\ensuremath{A} t\bar{t}}} = 0.94; m_{\ensuremath{A}} = 400 \,\, \mathrm{GeV} , \Gamma_{\ensuremath{A}}
= 4.5\% \ m_A ,
\vec{\nu}_{\rm max}) $ such that it vanishes for the most likely choice of
mass, width, coupling strength and nuisance parameters,
$\vec{\nu}_{\rm max}$, providing the best description of the data.
\subsection{Search for \texorpdfstring{\boldmath{$\phi
\to \ensuremath{\tau^+ \tau^-}$}}{phi2ll}}
A search for heavy neutral Higgs bosons decaying into two tau
leptons was performed by the ATLAS Collaboration utilizing data corresponding
to an integrated luminosity of
$139\,\unit{fb}^{-1}$~\cite{Aad:2020zxo}. The search covered the mass
range $0.2$--$2.5\,\unit{TeV}$ of the heavy
resonance, decaying into $\ensuremath{\tau^+ \tau^-}$ with at least one $\tau$ lepton decaying into final states with
hadrons.
The natural width of the scalar boson is assumed to be negligible
compared to the experimental resolution.
Upper limits on the production cross section times branching fraction
for a Higgs boson $\phi$ produced
via gluon-gluon fusion (ggF) and
$b$-quark associated production ($b \bar b\phi$) were derived.
The limits were calculated from a statistical
combination of three different final states, involving one $\tau$ lepton
decaying into a neutrino and hadrons and one $\tau$ lepton decaying into
neutrinos and an electron ($e\tau_h$) or into neutrinos and a muon
($\mu\tau_h$), and involving two $\tau$ leptons decaying into a
neutrino and hadrons each ($\tau_h\tau_h$).
For ggF a local excess was observed in the
data at the $2.2 \,\sigma$ C.L.\
at $m_{\phi} = 400\,\unit{GeV}$,
while for $b \bar b\phi$ production a local
excess at the $2.7 \,\sigma$ C.L.\ at $m_{\phi} = 400\,\unit{GeV}$ was found.
We have performed a
combined statistical analysis of the excesses in both
production modes assuming $m_{\phi} = 400\,\, \mathrm{GeV}$
in terms of a
two-dimensional likelihood function as provided by
ATLAS~\cite{Aad:2020zxo,Aad:2020zxoLink}.
The point with the highest excess is located
at $\sigma(gg \to \phi) \times \text{BR}(\phi \to \ensuremath{\tau^+ \tau^-}) = 20.19 \,\fb$ and
$\sigma(b\bar b \to \phi) \times \text{BR}(\phi \to \ensuremath{\tau^+ \tau^-}) = 38.37 \,\fb$.
In our N2HDM and NMSSM analyses
all points receive a contribution $\chi^2_{\ensuremath{\tau^+ \tau^-}}$
relative to the best-fit point
according to the two-dimensional likelihood function.
In both models it is in principle possible
that more than one Higgs boson can
contribute to the observed excess.
Thus, we define the signal as the incoherent
sum of all Higgs bosons with
masses in the range $(400\pm 40)\,\, \mathrm{GeV}$,
and use the $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ values of
the sum according to the experimentally
measured likelihood function for $m_\phi = 400\,\, \mathrm{GeV}$.
It should be noted, however, that
in the N2HDM only the CP-odd Higgs boson with a mass of
$m_A \approx 400\,\, \mathrm{GeV}$ can give rise to a sizable
contribution to the excess. This is due to the fact that for the case where
a second (CP-even)
Higgs boson $h_2$ has a similar mass (i.e.\
$360\,\, \mathrm{GeV} \leq m_{h_2} \leq 440\,\, \mathrm{GeV}$)
the applied constraints require that $h_2$ has to be an almost pure
gauge singlet in the parameter space region that is relevant for
the $\ensuremath{\tau^+ \tau^-}$ excess, and its contributions to the signal cross
section are therefore suppressed
compared to the ones of the CP-odd state.
The corresponding search of the CMS Collaboration for additional neutral
Higgs bosons in the
$\ensuremath{\tau^+ \tau^-}$ final state was done on a dataset corresponding to an
integrated luminosity of
$35.9\,\unit{fb}^{-1}$~\cite{Sirunyan:2018zut}. Model-independent limits were
set on the product of the production cross section times
the branching fraction for the decay into $\tau$ leptons for both the
ggF and the $b \bar b \phi$ production modes.
The search for heavy
resonances was performed over the mass range
$0.09$--$3.2\,\unit{TeV}$, assuming a narrow
width. Final states involving one $\tau$ lepton decaying into
neutrinos and an electron and one $\tau$ lepton decaying into neutrinos
and a muon ($e\mu$) were considered as well as $e\tau_h$, $\mu\tau_h$ and
$\tau_h\tau_h$ final states. No excess over
the SM background was found in this analysis.
However, due to the lower amount of data included,
the expected and observed cross section
limits of the CMS analysis are weaker
than the ones from ATLAS utilizing the full Run~2 dataset.
Thus, the signal intepretation of the ATLAS excess at around $400\,\, \mathrm{GeV}$
at the best-fit point is compatible with the expected (observed)
upper limits from the CMS analysis at the level of $1.1 (1.9) \sigma$.
\subsection{Search for \texorpdfstring{\boldmath{$A
\to Zh_{125}$}}{A2Zh}}
\label{secAZh}
The ATLAS Collaboration performed a search for new resonances decaying
into a $Z h_{125}$
in $\nu\bar{\nu}b\bar{b}$ and $\ell^+\ell^-b\bar{b}$ final states, where
$\ell^\pm = e^\pm$ or $\mu^\pm$~\cite{Aaboud:2017cxo}. The data used
corresponds to a total integrated luminosity of
$36.1\,\unit{fb}^{-1}$. The search was conducted by examining the
reconstructed invariant mass distribution of $Zh$ candidates for evidence
of a localised excess in the mass range of $220\,\unit{GeV}$ up to
$5\,\unit{TeV}$. The results of the search for the CP-odd
scalar boson $A$ in two-Higgs-doublet models
were interpreted in terms of constraints
on the ggF and $b$-quark associated ($b\bar{b}A$) production
cross-sections times the branching fraction of $A \to Zh_{125}$
and the branching fraction of $h_{125} \to b\bar{b}$.
In the search for $b\bar{b}A$ production, an excess of events was
observed for a resonance mass around $440\,\unit{GeV}$, mainly driven
by dimuon final states with three or more $b$-tags required. The local
significance of this excess with respect to the background-only
hypothesis was estimated to be $3.6\,\sigma$, and the global
significance, accounting for the look-elsewhere effect, was estimated to
be $2.4\,\sigma$. In the search for ggF production, agreement between the
data and the background-only hypothesis was found in a preliminary
update analysing the full Run 2 data set corresponding
to an integrated luminosity of $139\,\unit{fb}^{-1}$~\cite{ATLAS:2020pgp}.
The respective analysis performed by the CMS Collaboration for a heavy
pseudoscalar boson $A$ decaying to $Z h_{125}$ similarly
considered final states where the
Higgs boson decays to a bottom quark and antiquark, and the $Z$ boson
decays either into a pair of electrons, muons, or
neutrinos~\cite{Sirunyan:2019xls}. The
analysis was performed using a data sample corresponding to an
integrated luminosity of $35.9\,\unit{fb}^{-1}$. Exclusion limits were
set in the context of 2-Higgs-doublet models in the $A$ boson mass
($m_A$) range between $225$ and $1000\,\unit{GeV}$ on the ggF and
$b\bar{b}A$ production
cross-sections times the branching fraction of $A \to Zh_{125}$
and the branching
fraction of $h_{125} \to b\bar{b}$. In this search, however, the data was
found to be consistent with the background expectations over the whole
range of the reconstructed resonance mass $m_A$.
In view of this somewhat
ambiguous experimental situation we do not
incorporate the excess in the $A \to Zh_{125}$ channel into our
$\chi^2$ analysis. We rather
investigate whether the parameter regions in the N2HDM and the NMSSM
that are favoured as a result of the other observed excesses discussed in this
section could give rise to a possible signal also in the
$A \to Zh_{125}$ channel.
For a discussion of the ATLAS excess in the $A \to Zh_{125}$
channel in the context of the 2HDM
see \citere{Ferreira:2017bnx}, and in the context of
a general NMSSM see \citere{Coyle:2018ydo}.
\subsection{Searches for Higgs bosons in the low mass region}
\label{sec:excesses96}
As discussed above, in the context of searches for additional Higgs
bosons it is important to take into account also the mass region below the
observed state at $125\,\, \mathrm{GeV}$.
Higgs bosons with relatively low masses can be searched
for at the LHC via diphoton resonant searches.
CMS found a deviation between the expected and
observed exclusion limits at a mass of about $96\,\, \mathrm{GeV}$ in
the Run~1 data set, with a local significance of about
$2\,\sigma$~\cite{CMS:2015ocq}. Remarkably, the updated analysis including
the first Run~2 data found an excess of about $3\,\sigma$ at
a comparable mass~\cite{Sirunyan:2018aui}. Taking into account the
accumulated data at 7, 8 and $13\,\, \mathrm{TeV}$ center-of-mass
energy, CMS reported a signal interpretation of
the excesses, corresponding to a signal strength of
\begin{equation}
\mu_{\rm CMS}^{\rm exp} = 0.6 \pm 0.2 \ .
\end{equation}
Here the signal strength $\mu_{\rm CMS}$ is
defined, as usual, as the resonant signal cross
section assuming a scalar particle around $96\,\, \mathrm{GeV}$,
normalized to the expected signal cross section
assuming a SM Higgs boson at the same mass.
Similar searches in the diphoton final state
have also been performed by
ATLAS using $80\ifb$ of the $13\,\, \mathrm{TeV}$
dataset~\cite{ATLAS:2018xad}. Here, only a very small
excess of events over the SM background has
been found around $96\,\, \mathrm{GeV}$.
However, overall the resulting upper limits
on the cross sections found by ATLAS
are substantially above
the corresponding CMS exclusion limits.
Consequently, the ATLAS result cannot exclude the signal
interpretation of the CMS excess.
The CMS excesses have gained considerable interest
in the literature due to the fact that it
is consistent with another excess found
already at the Large Electron Positron collider
(LEP). At LEP, Higgs bosons in this mass range could be searched
for also in hadronic final states due to the
much lower backgrounds. Here an excess was reported
in the search $e^+ e^- \rightarrow Z H \rightarrow
Z b \bar b$ at a mass of about $ 98\,\, \mathrm{GeV}$,
with a local significance of
$2.3\,\sigma$~\cite{Barate:2003sz}.
The signal strength for the excess was found to
be~\cite{Cao:2016uwt}
\begin{equation}
\mu_{\rm LEP}^{\rm exp} = 0.117 \pm 0.057 \ .
\end{equation}
The mass resolution of the LEP excess is rather
limited due to the hadronic final state, such that
both the LEP and the CMS excesses could have
a common origin. Various different scenarios to
accommodate both excesses
at about $96\,\, \mathrm{GeV}$ have already been discussed in the literature.
\footnote{A recent overview about the various BSM models
that can explain both the CMS and the LEP excesses
can be found in \citere{Biekotter:2020cjs}.}
In particular, possible
interpretations comprise
type~II and type~IV of the N2HDM~\cite{Biekotter:2019kde} and
SUSY models like
the NMSSM~\cite{Cao:2016uwt,Domingo:2018uim,Choi:2019yrv} and the
$\mu\nu$SSM~\cite{Biekotter:2017xmf,Biekotter:2019gtq}, where the SUSY models
can account for the excesses at a level
of roughly $1\sigma$.
In the present paper we will first focus on the
observed excesses at
$\approx 400\,\, \mathrm{GeV}$
and then investigate whether a simultaneous
realization of also the excesses at $96\,\, \mathrm{GeV}$
is possible.
In order to quantify
how well the excesses are fitted
for each parameter
point in our numerical analysis, we define
the $\chi^2$ regarding the excesses at
$96\,\, \mathrm{GeV}$ via
\begin{equation}
\chi^2_{96} =
\frac{(\mu_{\rm CMS} - 0.6)^2}{0.2^2} +
\frac{(\mu_{\rm LEP} - 0.117)^2}{0.057^2} \, .
\end{equation}
Here $\mu_{\rm CMS}$ and $\mu_{\rm LEP}$
are the theoretical predictions. For the SM, for
which we have set ${\mu_{\rm CMS}=\mu_{\rm LEP}=0}$,
we find a penalty of $\chi^2_{{\rm SM},96} = 13.2$
from these contributions.
\section{N2HDM interpretation}
\label{sec:n2hdm}
In this section we will discuss possible
realizations of the described
excesses in the Higgs searches within the
context of the
N2HDM. We start by introducing the model
and its parameters, followed
by a discussion of the relevant theoretical
and experimental constraints.
The numerical analysis will be divided
into two parts. First, we discuss how the
excesses at $400\,\, \mathrm{GeV}$ can be accommodated,
and whether a simultaeneous realization of
both the $t \bar t$ and the $\ensuremath{\tau^+ \tau^-}$
excesses is possible.
In a second step, we additionally take into
account the excesses at $96\,\, \mathrm{GeV}$ in order
to address the question whether these can be
realized in the parameter region in which
one of the excesses at $400\,\, \mathrm{GeV}$ is
accommodated.
\subsection{Model definitions}
\label{sec:defn2hdm}
The N2HDM contains two doublet scalar fields
$\Phi_{1,2}$ and a real scalar
singlet field $\Phi_S$~\cite{Chen:2013jvg}.
In the physical basis,
the Higgs particle sector consists of a
total of three CP-even Higgs bosons $h_i$, a
CP-odd state $A$ and two charged
Higgs bosons $H^\pm$.
In order to avoid tree-level flavor-changing
neutral currents (FCNC) a $Z_2$ symmetry is assumed
as in the usual 2HDM, which is only softly broken by
the terms $
- m_{12}^2 \left(
\Phi^\dagger_1 \Phi_2 + \Phi^\dagger_2 \Phi_1
\right)$
appearing in the Lagrangian.
In addition, the scalar potential respects
a second $Z_2$ symmetry, under which only
the singlet field $\Phi_S$ is charged.
This symmetry is however broken spontaneously
when the singlet field has a non zero vacuum expectation value (vev),
${\langle \Phi_S \rangle = v_S}$, which we
will assume throughout the analysis.
The breaking of the electroweak gauge symmetry
originates from the vevs of the doublet
fields $\langle \Phi_{1,2} \rangle =
v_{1,2} / \sqrt{2}$, where
$\sqrt{v_1^2 + v_2^2} = v \approx 246\,\, \mathrm{GeV}$.
A crucial parameter
for the analysis of the experimental excesses is
the ratio of the doublet vevs given by
\begin{equation}
\tan\beta = \frac{v_2}{v_1} \ .
\end{equation}
The three real neutral components
of $\Phi_{1,2,S}$ mix to form the mass
eigenstates $h_{1,2,3}$ with masses
$m_{h_1} < m_{h_2} < m_{h_3}$.
The mixing in the CP-even scalar sector
can be described by the three mixing angles
$\alpha_{1,2,3}$, defining the
matrix
\label{eqmmatrix}
\begin{equation}
R = \begin{pmatrix}
c_{\alpha_{1}}c_{\alpha_{2}} & s_{\alpha_{1}}c_{\alpha_{2}} & s_{\alpha_{2}}\\
-\left(c_{\alpha_{1}}s_{\alpha_{2}}s_{\alpha_{3}}+s_{\alpha_{1}}c_{\alpha_{3}}\right)
&
c_{\alpha_{1}}c_{\alpha_{3}}-s_{\alpha_{1}}s_{\alpha_{2}}s_{\text{\ensuremath{\alpha_{3}}}}
&
c_{\alpha_{2}}s_{\alpha_{3}}\\
-c_{\alpha_{1}}s_{\alpha_{2}}c_{\alpha_{3}}+s_{\alpha_{1}}s_{\alpha_{3}} &
-\left(c_{\alpha_{1}}s_{\alpha_{3}}+s_{\alpha_{1}}s_{\alpha_{2}}c_{\alpha_{3}}\right)
& c_{\alpha_{2}}c_{\alpha_{3}}
\end{pmatrix} \ .
\end{equation}
At least in principle, the properties of
each of the Higgs bosons $h_i$
could be such that it can be identified with the SM-like
Higgs boson at $125\,\, \mathrm{GeV}$. The masses of the
CP-odd Higgs boson
and the charged Higgs bosons
are denoted by $m_A$ and $m_{H^\pm}$,
respectively.
Assigning consistent charges under the $Z_2$ symmetry to the
fermions yields the typical four types of Yukawa structures that
are familiar from the 2HDM. Depending on the type, the couplings of
the scalar particles to the
fermions will be different. The relevant
parameters defining the strength of the
couplings normalized to the one of a
hypothetical SM Higgs boson of the same mass
can be given in terms of $\tan\beta$
and the mixing angles $\alpha_i$
of the CP-even sector.
As already mentioned above, in a first step we will analyze whether
the N2HDM can realize the excesses at $400\,\, \mathrm{GeV}$.
We will go a step further in the subsequent analysis and investigate
whether and how the excesses at $400\,\, \mathrm{GeV}$ and
at $96\,\, \mathrm{GeV}$ can be realized simultaneously.
Our approach for covering
the relevant
parameter space will be discussed in detail
in \refap{n2hdmstrategy}.
Before, we briefly summarize the relevant
theoretical and experimental constraints
that we take into account in our analysis.
\subsection{Theoretical and experimental constraints}
\label{n2hdmconstraints}
Numerous theoretical constraints on the N2HDM
parameter space have to be taken into account
in order
to exclude unphysical parameter configurations.
To assure the presence of a viable electroweak minimum,
described by the vevs $v_1$, $v_2$ and $v_S$
as explained in \refse{sec:defn2hdm},
we demand that the scalar potential is bounded
from below.
The necessary conditions are given
in terms of the quartic scalar couplings
which define the behavior of the potential
for large field values~\cite{Klimenko:1984qx}.
Furthermore, we verified that the electroweak minimum
is either the global minimum of the scalar
potential, or, in case other deeper minima
exist, that the lifetime of the electroweak minimum
is large compared to the age of the observable
universe. Furthermore, in order to verify that the perturbative
treatment of the model is adequate, we checked
that the perturbative unitarity constraints
are fulfilled. These are also given in terms
of the quartic scalar couplings and set upper
limits on their absolute values~\cite{Muhlleitner:2016mzt}.
All the theoretical constraints mentioned
here have been applied using the public code
\texttt{ScannerS}~\cite{Coimbra:2013qq,
Ferreira:2014dya,Costa:2015llh,
Muhlleitner:2016mzt,Muhlleitner:2020wwk}
in combination with the public code
\texttt{EVADE}~\cite{Hollik:2018wrr,Ferreira:2019iqb}
for the calculation of the
lifetime of metastable electroweak minima.
Besides the mentioned theoretical constraints,
we have also included the most relevant experimental constraints on
the N2HDM parameter space
into our analysis:\\
\textit{(i)} We perform a $\chi^2$ test regarding
the measurements of the signal rates of
$h_{125}$ using the public code
\texttt{HiggsSignals v.2.6.0}~\cite{Bechtle:2013xfa,
Stal:2013hwa,Bechtle:2014ewa,Bechtle:2020uwn}.
In the following, we will denote the result
of the \texttt{HiggsSignals} test
by $\chi^2_{125}$, which is constructed taking
into account $n_{\rm obs} = 107$ observables.
The value of $\chi^2_{125}$ for each N2HDM
parameter point will be compared with the SM
prediction $\chi^2_{\mathrm{SM},125} = 84.42$.\\
\textit{(ii)} We verify for each parameter point
that none of the N2HDM scalars is excluded
by searches for additional Higgs bosons at
the LHC, the Tevatron and at LEP by making use
of the public code
\texttt{HiggsBounds v.5.9.0}~\cite{Bechtle:2008jh,
Bechtle:2011sb,Bechtle:2013gu,Bechtle:2013wla,
Bechtle:2015pma,Bechtle:2020pkv}.
For the considered parameter point
this code selects the most sensitive search
channel for each Higgs boson based on the
expected experimental sensitivity. It then
compares the model predictions for the
signal rates $\mu_{\rm theo.}$
with the observed upper limits
at the $95\%$ C.L., $\mu_{\rm excl.}$,
and rejects a point
if for one of the Higgs bosons
the ratio $r \equiv \mu_{\rm theo.} /
\mu_{\rm excl.}$ is larger than one.\\
\textit{(iii)} We include constraints from
electroweak precision observables (EWPO) in terms
of the so-called oblique parameters
$S$, $T$ and $U $~\cite{Peskin:1990zt,
Peskin:1991sw}.
Using the implementation of the code
\texttt{ScannerS} we exclude points
for which the $\chi^2$ value obtained
by comparing to the fit result of
\citere{Haller:2018nnx} deviates by more than
$2\,\sigma$. The N2HDM model predictions
for $S$, $T$ and $U$ are calculated
at the one-loop level
following \citeres{Grimus:2007if,
Grimus:2008nb}.\\
\textit{(iv)} Mainly due to the presence
of the charged Higgs boson in the N2HDM,
constraints from flavor physics are
important in some regions of parameter
space.
Since the extra field of the N2HDM
compared to the 2HDM is a gauge singlet,
it is sufficient to take over
the bounds for most of the flavor
physcis observables from the
2HDM~\cite{Biekotter:2019kde,Muhlleitner:2020wwk}.
We apply the flavor constraints from
\citere{Haller:2018nnx}
as implemented in \texttt{ScannerS}.
They yield a lower limit $m_{H^\pm} \gtrsim 550\,\, \mathrm{GeV}$
in both type~II and type~IV.
In type~I and type~III no such lower
limit on $m_{H^\pm}$ can be established.
In all types the flavor constraints
give rise to a lower limit on $\tb$,
where the specific value is different
in each type, and the strongest constraint is obtained
in the type~IV (N)2HDM.\footnote{Besides the
constraints from processes involving the charged Higgs
boson, also the constraints from ${B_{d,s} \rightarrow \mu^+ \mu^-}$,
involving the neutral scalar sector, can be relevant. However,
since the gauge singlet field does not
couple to the SM fermions directly,
only subleading corrections are expected
to be present in the N2HDM compared to
the limits from the 2HDM.
We therefore apply
the 2HDM constraints also for this
observable.}
Except the $\chi^2$ result of
\texttt{HiggsSignals} regarding the
signal rates of $h_{125}$, the
experimental constraints described in
this section have been taken into account
in terms of a hard cut, i.e.,
a parameter point is regarded to be excluded if the considered
experimental constraints are not fulfilled for this point.
As will
be described in \refse{numlowII},
we combine $\chi^2_{125}$ with the
$\chi^2$ values for the different
observed excesses. In this way we quantify how well the considered models
describe both the experimental results for $h_{125}$
and the potential signals
at $400\,\, \mathrm{GeV}$, as well as in a second step
also the potential signals
at $96\,\, \mathrm{GeV}$.
\subsection{A Higgs boson at \texorpdfstring{\boldmath{$400\,\, \mathrm{GeV}$}}{400gev}
in type~II}
\label{fullII}
In order to accommodate the $t \bar t$ excess
one needs values of $|c_{A t \bar t}| \gtrsim 0.5$
(see \reffi{Chisqtt}). Since in all four Yukawa types of the
N2HDM one finds $|c_{A t \bar t}| = 1 / \tan\beta$,
it is easy to understand that beyond the fact that
different constraints apply in each type, all
four Yukawa types
are very similar regarding the $t \bar t$ excess.
The situation is more complicated for the
$\ensuremath{\tau^+ \tau^-}$ excess, where it is required that
two conditions are fulfilled: Firstly, in order
to accommodate values for the $b \bar b$ associated
production cross section of $A$ of the same size
or larger than the gluon-fusion production cross section,
the condition $|c_{A b \bar b}| \gg |c_{A t \bar t}|$
has to be fulfilled. Secondly, in order to obtain
sufficiently large values for $\text{BR}(A \to \ensuremath{\tau^+ \tau^-})$,
the condition $|c_{A \ensuremath{\tau^+ \tau^-}}| \gg |c_{A t \bar t}|$
has to be fulfilled. The coupling coefficients
$c_{A b \bar b}$ and $c_{A \ensuremath{\tau^+ \tau^-}}$ are different
in each type, either proportional to $\tan\beta$
or its inverse. One finds that only
the type~II Yukawa structure is able to satisfy
both conditions (see \refap{n2hdmstrategy} for a detailed
explanation), such that it is the only type
that can potentially accommodate the $\ensuremath{\tau^+ \tau^-}$ excess.
In the following, we will focus our
numerical discussion on the N2HDM type~II, while
in the appendix we will also provide a discussion
of the N2HDM type~IV.
\begin{table}
\centering
\footnotesize
\def1.5{1.5}
\begin{tabular}{cccccc}
$m_{h_a}$ & $m_{h_b}$ & $m_{h_c}$ &
$m_A$ & $m_{H^\pm}$ & $\tan\beta$ \\
\hline
$[20,1000]$ & $125.09$ & $[20, 1000]$ &
$400$ & $[550, 1000]$ & $[0.5, 12.5]$ \\
\hline
\hline
$c_{h_b VV}^2$ & $c_{h_b t \bar t}^2$ &
$\mathrm{sign}(R_{b3})$ & $R_{a3}$ &
$m_{12}$ & $v_S$ \\
\hline
$[0.6, 1.0]$ & $[0.6, 1.2]$ & $-1,1$ &
$[-1, 1]$ & $[0, 1000]$ & $[10, 1500]$
\end{tabular}
\caption{\small Values of input parameters for the
scan in type~II for the investigation of the excesses
at~$400\,\, \mathrm{GeV}$. The states $h_{a,b,c}$ are
automatically arranged according to the mass-ordered
notation $h_{1,2,3}$ for each point
by \texttt{ScannerS}.}
\label{n2hdmfulltbparas}
\end{table}
In this section we
start by investigating
in which parameter
regions of the type~II N2HDM the excesses at $400\,\, \mathrm{GeV}$
can be accommodated.\footnote{In contrast to
\citere{Arganda:2021yms}, in which
this question was also addressed,
we take into account
the possibility of explaining the
local excess in the $b \bar b \to A \to Z h_{125}$
search.
\citere{Arganda:2021yms} only considered
the cross section limits of this search
as upper limits.
Here it is
important to note that
we used as a constraint
for the ggF production mode
$gg \to A \to Zh_{125}$
the most recent cross
section limits reported by
ATLAS using the full Run~2
data set~\cite{ATLAS:2020pgp}, while
the older result~\cite{Aaboud:2017cxo}
used in \citere{Arganda:2021yms}
(based only on
$36\ifb$ of data)
gives a substantially
weaker limit.
In addition, our analysis includes the
cross section limits from the process
$pp \to t \bar t \, A \to t \bar t \, t \bar t$
as published by CMS using the full Run~2
data set~\cite{Sirunyan:2019wxt}.
Instead, in \citere{Arganda:2021yms} a reinterpreation
of the ATLAS measurement of the SM cross section
for four-top production was
used~\cite{ATLAS:2020hrf}
by simply adding the contribution of a CP-odd Higgs boson
at $400\,\, \mathrm{GeV}$ to the SM
cross section for four-top production.
Consequently, effects on signal acceptances etc.\
through the exchange of the CP-odd Higgs boson are not accounted for.}
We give in \refta{n2hdmfulltbparas} the ranges
of the free parameters used here.
The choice of the
input parameters is the default one of the
public code \texttt{ScannerS}~\cite{Coimbra:2013qq,
Muhlleitner:2020wwk} that was used
to generate the parameter points
respecting the various theoretical and
experimental constraints discussed
in \refse{n2hdmconstraints}.
Each set of
input parameters corresponds to a unique benchmark
point under the assumption
$c_{h_b t \bar t} \cdot c_{h_b V V} > 0$.
This condition arises from the constraints on the
signal rates of
the state $h_{125}$
(here $h_{125} = h_b$,
as defined in \refta{n2hdmfulltbparas}).
In the following, the CP even scalars will
be denoted as $h_{a,b,c}$ when referring to
the input parameters as shown in \refta{n2hdmfulltbparas},
or alternatively as $h_{1,2,3}$, where the
mass ordering $m_{h_1} < m_{h_2} < m_{h_3}$
is implied.
Motivated by the pattern of the observed excesses described in
\refse{sec:excesses}, our scan in the N2HDM is performed such that the observed
excesses at $400 \,\, \mathrm{GeV}$ are mainly associated with the CP-odd
Higgs boson $A$
(a CP-even Higgs boson in this mass
region could yield only a very small contribution to the signal cross
section as a consequence of the constraints and the
required large singlet component),
and we therefore fix $m_A = 400 \,\, \mathrm{GeV}$ in the scan. The scan
range of $\tan\beta = [0.5, 12.5]$ is chosen in view of our qualitative
discussion above of the observed excesses and of the bounds from searches for
additional Higgs bosons.
Moreover, the mass $m_{h_b}$ is fixed to $m_{h_b} = 125.09\,\, \mathrm{GeV}$\cite{Khachatryan:2016vau},
and the corresponding couplings
are restricted to ranges that allow for the presence of
a Higgs boson $h_b$ that resembles the properties
of the Higgs boson that has been detected at $125\,\, \mathrm{GeV}$.
The lower limit of $m_{H^\pm} \geq 550\,\, \mathrm{GeV}$ is chosen in view of the
constraints from flavor physics observables
(see \refse{n2hdmconstraints}). The remaining
mass parameters were varied up to values of $1\,\, \mathrm{TeV}$,
except for $10 \leq v_S \leq 1.5\,\, \mathrm{TeV}$.
The mixing matrix element $R_{a3}$, see \refeq{eqmmatrix},
is scanned over all theoretically possible values,
and both possibilities for
$\mathrm{sign}(R_{b3})$ are taken into account.
Due to the large number of free parameters,
it is not possible to cover the
entire parameter space region defined
in \refta{n2hdmfulltbparas} sufficiently well
without applying
other theoretical prejudices
on the model parameters.
However, since the
properties of the pseudoscalar $A$ are to a large
extent determined by the parameters $m_A$
and $\tan\beta$, robust conclusions can be drawn
with respect to the realization of the excesses
by randomly sampling the parameter space,
without further theoretical restrictions,
for instance, in terms of priors.
We quantify the results of this scan
in terms
of the total $\chi^2$
arising from the comparison of the N2HDM predictions with the observed
results for the excesses at
$400\,\, \mathrm{GeV}$ in the $t \bar t$ and $\ensuremath{\tau^+ \tau^-}$ channels
as well as with the signal-rate
measurements of the Higgs boson at about
$125\,\, \mathrm{GeV}$, such that
\begin{equation}
\chi^2 = \chi^2_{125} +
\chi^2_{t \bar t} +
\chi^2_{\ensuremath{\tau^+ \tau^-}} \ .
\label{eqchisqttno96}
\end{equation}
The evaluation of the individual $\chi^2$ contributions in
\refeq{eqchisqttno96} has been described in \refse{sec:excesses} and
\refse{n2hdmconstraints}.
In our analysis we consider
all points with $\chi^2 \leq \chi^2_{\rm SM}$
as acceptable, where $\chi^2_{\rm SM}$
is evaluated via \refeq{eqchisqttno96}
assuming no signal contributions to
the excesses.
For $\chi^2_{t \bar t}$ and
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$
we remind the reader that the values are
defined relative to the best-fit
points.
One finds $\chi^2_{\mathrm{SM}, t \bar t} = 13.92$
and $\chi^2_{{\rm SM}, \ensuremath{\tau^+ \tau^-}} = 9.99$
assuming the SM hypothesis.
All remaining theoretical and experimental constraints
are included as hard cuts, either allowing or excluding
a parameter point (see \refse{n2hdmconstraints}).
As a consequence of the condition
$\chi^2 \leq \chi^2_{\rm SM}$, the sample of
parameter points
displayed in our plots below will also contain
points with $\chi^2_{125} \approx \chi^2_{\mathrm{SM},125}$
that potentially do not describe
either of the excesses with a confidence level below
$2 \, \sigma$. However, since our
results allow
a point-by-point
comparison\footnote{It should be noted in this context that
the identification of parameter regions with $\chi^2$ values that are lower than
the one of the SM does not automatically indicate a global statistical
preference for the considered model as compared to the SM. We refrain from such
an analysis, which would in particular require to account for the relevant
degrees of freedom, in the present paper where our main focus is on the
description of the
observed excesses rather than on global fits of different models.}
of the individual $\chi^2$ values,
it should be clearly visible which of the displayed
the scan points yield a good description of one or both
of the excesses, i.e.~featuring
small values of
$\chi^2_{t \bar t}$ and/or
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$, while
mantaining an acceptable $\chi^2_{125}$.
\begin{figure}
\centering
\includegraphics[width=0.33\textwidth]{tbeta_cAtt.pdf}~
\hspace*{-0.6cm}~
\includegraphics[width=0.33\textwidth]{mA_cAtt_Gam_n2hdm.pdf}~
\hspace*{-0.6cm}~
\includegraphics[width=0.33\textwidth]{LLxstbGEN.png}
\caption{\small Left: $c_{A t \bar t}$ in dependence of $\tan\beta$.
The colors of the points indicate the value
of $\Gamma_A / m_A$~in~\%. The dashed horizontal
lines indicate the best-fit values of $c_{A t \bar t}$
for different width hypotheses in the
experimental analysis~\cite{Sirunyan:2019wph}.
Middle:
Values of $c_{A t \bar t}$ for the parameter points
with $2.0\% \leq \Gamma_A / m_A \leq 3.0\%$ in comparison to
the observed (blue)
and expected (black dashed) upper limits
at the $95\%$ C.L.\ as well as the corresponding
$1\sigma$ (green) and $2\sigma$ (yellow)
regions around the expected limit assuming
$\Gamma_{A}/m_{A} = 2.5\%$, as
published in \citere{Sirunyan:2019wph}.
Right: Signal cross sections of $A \to \ensuremath{\tau^+ \tau^-}$
for the $gg$ production mode on the
horizontal axis and the $b \bar b$ production mode
on the vertical axis. The colors of the points
indicate the value of $\tan\beta$. The dark blue
and the yellow lines indicate the $1\sigma$
and $2\sigma$ ellipse of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$~\cite{Aad:2020zxo},
respectively.
In both plots the best-fit point, labelled as
$\mathrm{min}(\chi^2)$,
is indicated with a magenta star.
The point with the smallest value of
$\chi^2_{\ensuremath{\tau^+ \tau^-}} + \chi^2_{125}$, labelled as
$\mathrm{min}( \chi^2_{\ensuremath{\tau^+ \tau^-}} + \chi^2_{125} )$,
is indicated with a cyan star.}
\label{figGEN1}
\end{figure}
As explained in \refap{n2hdmstrategy}, the properties of
the $A$-boson
are driven by the dependence of its couplings on $\tan\beta$.
In the left plot of \reffi{figGEN1}
$c_{A t \bar t}$ ($= 1 / \tan\beta$) is displayed as a function
of $\tan\beta$ for different values of $\Gamma_A/m_A$.
One can see that the best-fit values
for $c_{A t \bar t}$ showing the best agreement with the
$t \bar t$ excess,
which are indicated for different width hypotheses by
the horizontal dashed lines,
are reached only for small values of $\tan\beta \lesssim 2$.
As discussed in \refse{sec:excesses} (see \reffi{Chisqtt})
the experimental excess is most pronounced for $\Gamma_A/m_A \approx 4\%$.
This feature manifests itself in the plot by the fact that the largest
displayed values of $\Gamma_A/m_A$ correspond to the largest
best-fit values of $c_{A t \bar t}$.
Overall, the N2HDM of type~II yields a good description of the
$t \bar t$ excess in the region of small values of $\tan\beta$,
$1.2 \lesssim \tan\beta \lesssim 2.5$.
The lower bound on $\tan\beta$ arises here
from the cross section limits on the process
$p p \rightarrow t \bar t (A) \rightarrow t \bar t (t \bar t)$,
as reported by CMS making use of the full
Run~2 data set~\cite{Sirunyan:2019wxt}.
For values of $\tan\beta \gsim 2.5$ we find values of
$c_{A t \bar t}$ substantially below the experimental
best-fit value even for the lowest value
of $\Gamma_A / m_A = 0.5\%$ taken into account
in the analysis. Thus, we conclude that
the $t \bar t$ excess can be well described in the
N2HDM of type~II with low $\tan\beta$, while for
$\tan\beta \gsim 2.5$
the compatibility with the $t \bar t$ excess is reduced.
To further illustrate the fit result
we show in the middle plot of \reffi{figGEN1}
the subset of points for which
we find $2.0\% \leq \Gamma_A / m_A \leq 3.0\%$
in the plane of $m_A$ and $c_{A t \bar t}$, where
we also indicate the expected and observed
exclusion limits from the experimental analysis
assuming a width of $\Gamma_A / m_A = 2.5\%$
as published in \citere{Sirunyan:2019wph}.
One can see that the points stretch over the
values of $c_{A t \bar t} \approx 0.8$ that provide the best
description of the $t \bar t$ excess.
In the right plot of \reffi{figGEN1} we show
the parameter points in the plane of the
signal rates regarding the $\ensuremath{\tau^+ \tau^-}$
excess~\cite{Aad:2020zxo},
where the production cross sections were
calculated with the public code
\texttt{SusHi}~\cite{Harlander:2012pb,Harlander:2016hcx}.
Here, the colors of the points indicate
the value of $\tan\beta$.
Our scan results in two
distinct parameter regions in the displayed plane.
The origin of these regions will be discussed
below.
We also show the
$1\sigma$ and $2\sigma$ ellipses
of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$ with the dark blue
and the yellow lines.
One can see that, as expected following
the discussion in \refap{n2hdmstrategy}, there are only
points with larger values of $\tan\beta > 6$
inside the $1\sigma$ ellipse.
Comparing with
the region of $\tan\beta$
in which the $t \bar t$ excess can be
well described,
we conclude that the N2HDM cannot provide a good simultaneous fit of both
excesses.
This feature can be seen more clearly
in \reffi{figGEN2}, where the
$\chi^2$ distributions of the points
regarding both excesses are displayed, together with their sum.
The values
of
$\chi^2_{t \bar t}$ are shown in blue
and the ones of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$ in orange
for all
the parameter points of this scan.
In agreement with the observations described
above, low values of $\chi^2_{t \bar t} \lesssim 4$
can only be found for $\tan\beta \lesssim 2.5$.
Low values of $\chi^2_{\ensuremath{\tau^+ \tau^-}} \lesssim 4$
are found for $\tan\beta \gtrsim 5.5$.
Consequently, the sum of both $\chi^2$
values, as indicated by the green points,
has the smallest values in both $\tan\beta$
ranges suitable for the $t \bar t$ or the
$\ensuremath{\tau^+ \tau^-}$ excess
(but does not drop below a value of about 10),
while larger values are found
in between both ranges, at $3.5 \lesssim
\tan\beta \lesssim 5.5$.
Therefore,
it is not possible to simultaneously
accommodate both excesses in the N2HDM.
\begin{figure}
\centering
\includegraphics[width=0.68\textwidth]{tbeta_chisq.pdf}
\caption{\small
$\chi^2_{t \bar t}$ (blue) and
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$ (orange)
and $\chi^2_{t \bar t} +
\chi^2_{\ensuremath{\tau^+ \tau^-}}$ (green)
in dependence of $\tan\beta$.
The horizontal blue, orange
and green dashed lines
indicate the value of
$\chi^2_{{\rm SM}, t \bar t}$,
$\chi^2_{{\rm SM}, \ensuremath{\tau^+ \tau^-}}$
and
$
\chi^2_{{\rm SM}, t \bar t} +
\chi^2_{{\rm SM}, \ensuremath{\tau^+ \tau^-}}$, respectively.}
\label{figGEN2}
\end{figure}
The two distinct regions found in the right
plot of \reffi{figGEN1}
correspond to the two separate distributions of points that are visible
for
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$ in the region of large $\tan\beta$.
The left one corresponds to the right region of
points in \reffi{figGEN1}, which have larger
values of $\sigma(gg \rightarrow A \rightarrow \ensuremath{\tau^+ \tau^-})$
and do not cross the center of the ellipses.
Consequently, $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ does
not exactly
reach zero for this set of points. On the contrary, the left band
of points in \reffi{figGEN1} exactly crosses
the center of the ellipses for values
of $\tan\beta \approx 9.5$. Therefore, the
distribution of
points displaying $\chi^2_{\ensuremath{\tau^+ \tau^-}}$
near this value of $\tan\beta$
has its minimum at zero.
The two different regions have their origin
in the presence (or absence) of the decay
of $A$ to a lighter Higgs boson $h_i$ and
a $Z$ boson. If such a decay is possible,
it leads to a reduction of
$\text{BR}(A\rightarrow \ensuremath{\tau^+ \tau^-})$.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{LLxsmh2GEN.png}~
\includegraphics[width=0.48\textwidth]{LLxswycrGEN.png}
\caption{\small
Signal cross sections of $A \to \ensuremath{\tau^+ \tau^-}$
for the $gg$ production mode on the
horizontal axis and the $b \bar b$ production mode
on the vertical axis. The colors of the points
indicate the value of $m_{h_a}$ (left)
and $c_{h_{125} b \bar b} \cdot c_{h_{125} VV}$ (right).
The dark blue
and the yellow lines indicate the $1\sigma$
and $2\sigma$ ellipse of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$~\cite{Aad:2020zxo},
respectively.
In both plots the best-fit point, labelled as
$\mathrm{min}(\chi^2)$,
is indicated with a magenta star.
The point with the smallest value of
$\chi^2_{\ensuremath{\tau^+ \tau^-}} + \chi^2_{125}$, labelled as
$\mathrm{min}( \chi^2_{\ensuremath{\tau^+ \tau^-}} + \chi^2_{125} )$,
is indicated with a cyan star.}
\label{figGEN3}
\end{figure}
The impact of such an
additional decay mode of the CP-odd Higgs
boson $A$
is illustrated in \reffi{figGEN3}, in
which we show the same as in the right
plot of \reffi{figGEN1}, but with the colors
of the points indicating the value of
$m_{h_a}$ (left) and the value of
the product $c_{h_{125} b \bar b} \cdot
c_{h_{125} VV}$ on the right.
In the left plot of \reffi{figGEN3} one
can see that all points with $m_{h_a}
< m_A - M_Z \lesssim 300\,\, \mathrm{GeV}$,
for which the decay mode
$A \to h_a Z$ is kinematically open,
are in the left band of points.
Hence, the presence of a second light
Higgs boson $h_a$ relatively close
in mass to $125\,\, \mathrm{GeV}$ improves the fit
to the $\ensuremath{\tau^+ \tau^-}$ excess within the
N2HDM since it can give rise to a reduction
of $\text{BR}(A\rightarrow \ensuremath{\tau^+ \tau^-})$.
In addition, there are also points with
$m_{h_a} > 300\,\, \mathrm{GeV}$ in the left band of
points. For these points the smaller
values of $\text{BR}(A\rightarrow \ensuremath{\tau^+ \tau^-})$
compared to the points in the right
band are achieved by sizable values of
$\text{BR}(A\rightarrow h_{125} Z)$, with $h_{125} = h_1$.
This branching ratio vanishes in the
alignment limit of the N2HDM, in which
the tree-level couplings of $h_{125}$ reduce to the SM
predictions,
and accordingly
$c_{h_{125} A Z} = 0$.
However, in the N2HDM the
so-called wrong sign Yukawa coupling
scenario can be realized. In this scenario, the absolute
values of the couplings of $h_{125}$ to fermions
are close to the SM prediction, but
$c_{h_{125} b \bar b}$ has the opposite sign
than $c_{h_{125} V V}$. This allows for the
agreement of the properties of $h_{125}$
with the measured signal rates without
vanishing couplings to the other scalars
and pseudoscalars. One can see in
the right plot of \reffi{figGEN3} that
all the points in the left band of
points that do not have $m_{h_a} < 300\,\, \mathrm{GeV}$
are in the wrong sign Yukawa coupling regime.
For these points the presence of the
decay of $A$ into
$h_{125}$
and a $Z$ boson improves the fit to
the $\ensuremath{\tau^+ \tau^-}$ excess.
However, such a signature
has also been probed in direct searches.
For the $gg$ production mode (see below for a discussion of the observed
excess in the $b \bar b A$
production mode~\cite{Aaboud:2017cxo}) the limits obtained from
the search for $gg \to A \rightarrow Z (h_{125}) \rightarrow
Z (b \bar b)$
recently reported by ATLAS~\cite{ATLAS:2020pgp} exclude
parameter points in the wrong sign Yukawa
coupling regime with values of $\tan\beta \lesssim
8.5$. This leads to the lower cut on the left
branch of points in
\reffi{figGEN1} (right) and
\reffi{figGEN3},
and also to the
lower bound on $\tan\beta$ in
the right branch of points
in \reffi{figGEN2}.
It should also be noted that the parameter
point with the smallest value of $\chi^2_{\ensuremath{\tau^+ \tau^-}}
+ \chi^2_{125}$ (indicated by the cyan star)
in \reffi{figGEN3}
lies on the right branch of points. Since this branch
is further away from the center of the ellipses,
this branch has overall slightly larger values of $\chi^2_{\ensuremath{\tau^+ \tau^-}}$.
The fact that we nevertheless find
the cyan star in this branch indicates that
for these points
the compatibility with the measured properties of $h_{125}$ can be better
(yielding a lower $\chi^2_{125}$)
compared to the left branch of points.
Thus, parameter points
in which the additional decay channels $A \to Z h_{1,2}$
are relevant (left branch) are associated with
mildly larger deviations of the signal rates of $h_{125}$
compared to the SM prediction. However, we checked
that the values of $\chi^2_{125}$ can also be very close
to the SM value $\chi^2_{\rm SM, 125} \approx 84$ in the left
branch.
As a consequence, the accuracy of the
signal rate measurements of $h_{125}$ at the (HL)-LHC~\cite{Cepeda:2019klc}
will not be sufficient to fully discriminate between the points in the two
branches.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{AZhGEN.png}~
\includegraphics[width=0.4\textwidth]{AZhwGEN.png}
\caption{\small
Predicted rate for
$\sigma(b \bar b \to A \to Z h_{125}) \times \text{BR}(h_{125} \to b \bar b)$
in comparison with the expected and observed $95\%$ confidence level
upper limits obtained
by ATLAS~\cite{Aaboud:2017cxo}. The colors of the points indicate
the value of $\tan\beta$ (left) and
the value of the product
$c_{h_{125} b \bar b} \cdot c_{h_{125} VV}$ (right).}
\label{figGEN4}
\end{figure}
As already mentioned above,
in the N2HDM type~II
the presence of one of the decays
$A \rightarrow h_{1,2} Z$ with sizable
branching ratio
can have the effect that
the $\ensuremath{\tau^+ \tau^-}$ excess is described so well that one encounters
a vanishing contribution to $\chi^2_{\ensuremath{\tau^+ \tau^-}}$.
Regarding the points in the
wrong sign Yukawa coupling regime, it is
interesting to note that a local $3.6\,\sigma$ excess
at roughly $400\,\, \mathrm{GeV}$ was found
by the ATLAS Collaboration
in the search $b \bar b \to
A \rightarrow h_{125} Z$~\cite{Aaboud:2017cxo}.
The corresponding search
for the $gg$ production mode has meanwhile been
updated including
the full Run~2 dataset, and no excess was found
at $m_A \approx 400\,\, \mathrm{GeV}$.
As explained above
(see also the discussion in \refse{secAZh})
the obtained limit
gives rise to
the lower bound on $\tan\beta$ for the left branch of points
in \reffi{figGEN1} (right) and
\reffi{figGEN3}.
In order to investigate whether the presence of a CP-odd Higgs boson $A$
with $m_A \approx 400\,\, \mathrm{GeV}$ could on the one hand be
the origin of the excess
in the $b \bar b \to A \to Z h_{125}$ search
\footnote{As discussed in \citere{Ferreira:2017bnx}, the
$A \rightarrow Z h_{125}$ excess can be realized
in the wrong sign Yukawa coupling regime
of the 2HDM.}
while on the other hand
being compatible with the cross
section limits from the $g g \to A \to Z h_{125}$ search,
we show in \reffi{figGEN4}
the predicted rate for
$\sigma(b \bar b \to A \to Z h_{125}) \times
\text{BR}(h_{125} \to b \bar b)$
for each parameter point, in combination
with the expected and observed upper limits
from the ATLAS analysis~\cite{Aaboud:2017cxo}.
The colors of the points indicate the value
of $\tan\beta$ (left) and
the value of the product
$c_{h_{125} b \bar b} \cdot c_{h_{125} VV}$ (right),
i.e., the blue points in the right plot
are in the wrong sign Yukawa coupling regime.
One can see that
the points in the
wrong sign Yukawa coupling regime could indeed contribute to the excess
observed by ATLAS in the $b \bar b \to A$ production mode,
while
at the same time being compatible
with the limits obtained for the $gg$ production mode. On the other
hand, for the points that are not in the wrong sign Yukawa coupling
regime in our scan we do not find a sizable contribution to the
$\sigma(b \bar b \to A \to Z h_{125}) \times \text{BR}(h_{125} \to b \bar b)$
rate, since the decays
$A \rightarrow h_{1,2} Z$, if kinematically open,
are suppressed for the case where the Higgs boson in the final state is
$h_{125}$.
For the points that are not in the wrong sign Yukawa coupling regime our
analysis has revealed that the presence of a
second Higgs boson $h_a$ with a mass below $300\,\, \mathrm{GeV}$ can improve the
description of the $\ensuremath{\tau^+ \tau^-}$ excess. It is tempting in this context to
entertain the possibility that this additional Higgs boson could have a mass
of about $96\,\, \mathrm{GeV}$ and be the origin of the observed excesses in this mass
range, see the discussion in \refse{sec:excesses96}. Within the context of the
N2HDM of type~II,
the simultaneous realization of the $\ensuremath{\tau^+ \tau^-}$ excess and the observed
pattern around $96\,\, \mathrm{GeV}$ leads to a very interesting scenario that is very
predictive and can serve as a
benchmark scenario that can be probed in future experiments. We will
investigate this possibility in \refse{numhighII}.
Before turning to this discussion
we will investigate
whether the excesses at $96\,\, \mathrm{GeV}$ can also
be accommodated in combination with
the $t \bar t$ excess. The latter, as
discussed in detail in
\refap{n2hdmstrategy}, can be realized both
in the type~II and the type~IV N2HDM.
A corresponding parameter scan
for the type~II is described in \refse{numlowII},
and for completeness the according scan in type~IV
can be found in \refap{numlowIV}.
\subsection{Higgs bosons at
\texorpdfstring{\boldmath{$96\,\, \mathrm{GeV}$}}{96gev} and
\texorpdfstring{\boldmath{$400\,\, \mathrm{GeV}$}}{400gev}
for low \texorpdfstring{\boldmath{$\tan\beta$}}{tb}
in type~II}
\label{numlowII}
In this section we present the analysis for the scan
in the low $\tan\beta$ regime of the type~II
N2HDM, which is dedicated
to accommodate the $t \bar t$ excess at $400\,\, \mathrm{GeV}$
in combination with the excesses at $96\,\, \mathrm{GeV}$.
The ranges of the free input parameters are given
in \refta{n2hdmlowtbparas},
where for the scan in the present section the $\tan\beta$ range specified
as $\tan\beta_{\rm low}$ is adopted.
\begin{table}
\centering
\def1.5{1.5}
\footnotesize
\begin{tabular}{ccccccc}
$m_{h_1}$ & $m_{h_2}$ & $m_{h_3}$ &
$m_A$ & $m_{H^\pm}$ & $\tan\beta_{\rm low}$ &
$\tan\beta_{\rm high}$ \\
\hline
$[95,98]$ & $125.09$ & $[550, 1000]$ &
$400$ & $[550, 1000]$ & $[0.5, 4]$ & $[6, 12]$ \\
\hline
\hline
$c_{h_2 VV, \rm low}^2$ & $c_{h_2 VV, \rm high}^2$ & $c_{h_2 t \bar t}^2$ &
$\mathrm{sign}(R_{23})$ & $R_{13}$ &
$m_{12}$ & $v_S$ \\
\hline
$[0.6, 0.9]$ & $[0.6, 1.0]$ & $[0.6, 1.2]$ & $-1,1$ &
$[-1, 1]$ & $[0, 1000]$ & $[10, 1500]$
\end{tabular}
\caption{\small Values of input parameters for the
scans in the low and high $\tan\beta$ regime.}
\label{n2hdmlowtbparas}
\end{table}
We start by briefly commenting on our choice of parameter
ranges.\footnote{The requirement of a Higgs boson
at $96\,\, \mathrm{GeV}$ in combination with the
experimental constraints fixes the mass ordering of the
CP even Higgs bosons, i.e.\
$h_{125} = h_2$. We will therefore use
the notation $h_{1,2,3}$ in the following.}
The lightest Higgs boson $h_1$ plays
the role of the candidate
that is associated with
the excesses
at $96\,\, \mathrm{GeV}$. Thus, the mass $m_{h_1}$
was chosen to be in this range. The SM-like
Higgs boson is $h_2$ with a mass of
$m_{h_2} = 125.09\,\, \mathrm{GeV}$. As before,
the $t \bar t$ excess will be
described in terms of the CP-odd Higgs boson with
a mass of $m_A = 400\,\, \mathrm{GeV}$. The scan range
of the charged Higgs-boson mass
$m_{H^\pm} = [550,1000]\,\, \mathrm{GeV}$ is chosen due
to a lower limit of
$m_{H^\pm} \gtrsim 550\,\, \mathrm{GeV}$ arising from the
flavor constraints. The indirect constraints
from the EWPO lead to the restriction that
the third CP-even Higgs boson should have a similar
mass $m_{h_3} \approx m_{H^\pm}$, since
all remaining Higgs bosons have fixed masses
much below the lower limit on $m_{H^\pm}$.
Thus, $m_{h_3}$ is scanned in the same mass
range as $m_{H^\pm}$.
The range of $\tan\beta =
\tan\beta_{\rm low}$ gives rise to sizable
couplings of the CP-odd Higgs boson $A$ to top quarks,
which is desired for the description of the $t \bar t$ excess
(see the discussion in \refap{n2hdmstrategy}).
The ranges for the squared couplings of the
SM-like Higgs boson $h_2$ were taken over from
\citere{Biekotter:2019kde}, where it
was shown that such values are suitable for
explaining the excesses at $96\,\, \mathrm{GeV}$
in the N2HDM. To be precise,
these ranges allow a substantial
mixing between $h_1$ and $h_2$, such that
$h_1$ can have sizable couplings to
quarks and gauge bosons, while
the properties of $h_2$ are still
in agreement with the signal-rate measurements
of the discovered Higgs boson at around $125\,\, \mathrm{GeV}$.
The remaining parameters
in \refta{n2hdmlowtbparas}, $R_{13}$, $m_{12}$ and $v_S$,
were scanned over
a wide range, and both possible signs of $R_{23}$ were taken into account.
We emphasize that the requirement of simultaneously explaining
the $t \bar t$ excess and the excesses at
$96\,\, \mathrm{GeV}$ practically fixes the entire scalar
spectrum of the model. The only remaining free mass
parameter is the common mass scale of $h_3$ and
$H^\pm$. However, this scale cannot be much larger than
$m_A$ (which is fixed to $400\,\, \mathrm{GeV}$) as a consequence of the applied
constraints (see also the discussion below).
The results of this scan will be quantified in terms
of the total $\chi^2$ given by the
contributions arising from the
fit to the excesses at $96\,\, \mathrm{GeV}$,
the fit to the $t \bar t$ and the $\ensuremath{\tau^+ \tau^-}$
excesses at $400\,\, \mathrm{GeV}$, and
the signal-rates
measurements of the Higgs boson at $125\,\, \mathrm{GeV}$, such that
\begin{equation}
\chi^2 = \chi^2_{96} + \chi^2_{125} +
\chi^2_{t \bar t} +
\chi^2_{\ensuremath{\tau^+ \tau^-}} \ .
\label{eqchisqtt}
\end{equation}
We include here for consistency also the contribution from
the $\ensuremath{\tau^+ \tau^-}$ excess even though no
sizable contribution to the correspoding signal
cross sections are expected in the
low $\tan\beta$ regime.
All points with $\chi^2 \leq \chi^2_{\rm SM}$
are considered to be acceptable, where $\chi^2_{\rm SM}$
is evaluated assuming no signal contributions to
the excesses.
Their contribution to $\chi^2_{\rm SM}$ amounts to
${\chi^2_{\rm SM, 96} + \chi^2_{\rm SM, \ensuremath{\tau^+ \tau^-}}
+ \chi^2_{\rm SM, t \bar t} = 37.2}$.
The best-fit point in our scan is determined
by the lowest $\chi^2$ value
according to \refeq{eqchisqtt}.
\begin{figure}
\centering
\includegraphics[width=0.33\textwidth]{tbeta_cAtt_tt.pdf}~
\hspace*{-0.6cm}~
\includegraphics[width=0.33\textwidth]{tbeta_cAtt_96.pdf}~
\hspace*{-0.6cm}~
\includegraphics[width=0.33\textwidth]{mA_cAtt_Gam_n2hdm96.pdf}
\caption{\small
Left and center: $c_{A t \bar t}$ in dependence of $\tan\beta$.
The colors of the points indicate the values
of $\Gamma_A / m_A$ in \% (left) and
the values of $m_{H^\pm}$ (center).
The dashed horizontal
lines indicate the best-fit values of $c_{A t \bar t}$
for different width hypotheses in the
experimental analysis~\cite{Sirunyan:2019wph}.
Right:
Values of $c_{A t \bar t}$ for the parameter points
with $2.0\% \leq \Gamma_A / m_A \leq 3.0\%$ in comparison to
the observed (blue)
and expected (black dashed) upper limits
at the $95\%$ C.L.\ as well as the corresponding
$1\sigma$ (green) and $2\sigma$ (yellow)
regions around the expected limit assuming
$\Gamma_{A}/m_{A} = 2.5\%$, as
published in \citere{Sirunyan:2019wph}.}
\label{figttIItt}
\end{figure}
In \reffi{figttIItt} we show the values of the coupling
coefficient $c_{A t \bar t}$ in dependence of
$\tan\beta$ for all the parameter points with
$\chi^2 \leq \chi^2_{\rm SM}$. One can see that the
parameter points lie in a narrow range of
$1.1 \lesssim \tan\beta \lesssim 1.6$.
The preference for those low values
of $\tan\beta$
is related to the fact that for values of
$\tan\beta \gtrsim 2$ no sizable contribution to
the $t \bar t$ excess is present.
The feature that no points are displayed in our scan
for higher values of $\tan\beta$ although our scan has been performed up to
$\tan\beta = 4$ is mainly due to the sampling used in our scan in combination
with the fact that many points in this region do not fulfill
the condition $\chi^2 \leq \chi^2_{\rm SM}$.
For the allowed points, the size of the couplings
$c_{A t \bar t}$ and the
values for
$\Gamma_A / m_A$, indicated by the colors of the
points in the left plot of \reffi{figttIItt},
provide a very good fit to the $t \bar t$ excess.
In particular, for values slightly above $\tan\beta = 1$
we find $c_{A t \bar t} \approx 0.8$
and $\Gamma_A / m_A \approx 2.5\%$,
which gives rise to values of
$\chi^2_{t \bar t} < 1$.
In the middle plot of
\reffi{figttIItt} the colors of the points indicate
the value of the charged Higgs boson mass
$m_{H^\pm}$. Here it is important to note that no
values above $m_{H^\pm} \approx 750\,\, \mathrm{GeV}$ were found.
This upper limit arises from theoretical constraints
on the allowed values of the quartic couplings
under the condition that $m_{h_1} = 96\,\, \mathrm{GeV}$,
$m_{h_2} = 125\,\, \mathrm{GeV}$ and $m_A = 400\,\, \mathrm{GeV}$.
Hence, the presence of a charged Higgs
boson, and due to the constraints on the $T$ parameter
also of the third CP-even Higgs boson $h_3$, with
masses $m_{h_3} \approx m_{H^\pm}$
below the TeV scale are firm predictions
of the scenario presented here.
In the right plot of \reffi{figttIItt}
we again show the subset of points that
feature values of $2.0\% \leq \Gamma_A / m_A \leq 3.0\%$
in comparison to the experimental expected
and observed exclusion limits~\cite{Sirunyan:2019wph}.
All of the points
lie close to the experimental best-fit value
of $c_{A t \bar t} \approx 0.8$ for a width
of $\Gamma_A / m_A = 2.5\%$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{csgAttII.pdf}~
\includegraphics[width=0.48\textwidth]{hsII.pdf}
\caption{\small
The $\mu_{\rm CMS}$--$\mu_{\rm LEP}$ plane
for the points of the low $\tan\beta$ scan in the
type~II of the N2HDM.
The black ellipse indicates the $1\sigma$ region of
$\chi^2_{96}$ with its center marked with
a black cross.
The best-fit point is highlighted with
a magenta star.
The colors of the points indicate
$\chi^2_{t \bar t}$ in the left
plot and $\Delta \chi^2_{125}$ in the right plot.}
\label{figttII}
\end{figure}
The width of the CP-odd Higgs boson $A$ normalized to its mass ranges
between roughly $1.5\%$ and $3.5\%$, as can be seen in the left plot
of \reffi{figttIItt}. These values are substantially larger than the
ones that one would obtain within the 2HDM for a CP-odd Higgs boson
at $400\,\, \mathrm{GeV}$.
This is due to the fact that in the considered N2HDM scenario
$A$ has additional decay channels available
into $h_{1,2}$ and a $Z$ boson.
In the experimental analysis~\cite{Sirunyan:2019wph}
it was found that
for
values of $\Gamma_A / m_A \approx 2.5\%$
the expected
$95\%$ C.L.\ exclusion limit is at
roughly $c_{A t \bar t} \approx 0.6$,
while the observed exclusion limit varies
(depending on the precise value
of $\Gamma_A / m_A$)
between $c_{A t \bar t} \approx 0.9$ and
$c_{A t \bar t} \approx 1.1$. Thus, our scan essentially covers
the
range of $c_{A t \bar t}$ that is associated with the observed excess,
as can be seen in \reffi{figttIItt}. This is also reflected in the
very small values of $\chi^2_{t \bar t}$, as
discussed in the following.
We show the parameter points of the
scan in the
$\mu_{\rm CMS}$--$\mu_{\rm LEP}$ plane in
\reffi{figttII}.
The color coding of the points indicates the values
of
$\chi^2_{t \bar t}$ and
$\Delta \chi^2_{125} \equiv
\chi^2_{125} - \chi^2_{\mathrm{SM},125}$ in the
left and right plot, respectively.
One can see
that a large
fraction of points lies inside the $1\sigma$
region of $\chi^2_{96}$ while at the same time
reproducing the excess observed for $c_{A t \bar t}$,
since many points have $\chi^2_{t \bar t} \approx 0$.
Thus, we conlcude that the low $\tan\beta$
region of the type~II N2HDM is well suited for accommodating
the excesses at $96\,\, \mathrm{GeV}$ in combination with
the $t \bar t$ excess at $400\,\, \mathrm{GeV}$.
Moreover, we can observe
in the right plot of \reffi{figttII} that several
points have values of $\Delta \chi^2_{125}$ very close
to zero. Consequently, the fit to the
excesses can be realized without being in tension
with the signal rates measurements of the SM
Higgs boson.
The best fit point with a value
of $\chi^2 = 97.97$
is substantially below the
SM value $\chi^2_{\rm SM} = 121.61$.
We also performed
a scan in the low $\tan\beta$ region of type~IV
with the goal of explaining the $t\bar t$
excess and the excesses at $96\,\, \mathrm{GeV}$.
We found that also this type can account
for the excesses, but with a slightly worse
fit result regarding the CMS excess and
the properties of $h_{125}$. This scan
is presented in \refap{numlowIV} for completeness
and to allow for a
comparison to the type~II results.
\subsection{Higgs bosons at
\texorpdfstring{\boldmath{$96\,\, \mathrm{GeV}$}}{96gev} and
\texorpdfstring{\boldmath{$400\,\, \mathrm{GeV}$}}{400gev}
for large \texorpdfstring{\boldmath{$\tan\beta$}}{tb}
in type~II}
\label{numhighII}
As was discussed
before, the type~II N2HDM
is the only candidate of the four different Yukawa
types that can potentially accommodate the
$\ensuremath{\tau^+ \tau^-}$ excess at $400\,\, \mathrm{GeV}$
in combination with the two excesses at
$96\,\, \mathrm{GeV}$.
The existence of a region of
parameter space fulfilling the
relevant criteria
(see the discussion in \refap{n2hdmstrategy})
depends on the simultaneous enhancement
of $\sigma( b \bar b \rightarrow A)$ and
$\text{BR} (A \rightarrow \ensuremath{\tau^+ \tau^-})$, such that both the signal rates
in the $gg$ production mode and the $b \bar b$
production mode have the adequate size for
providing a good description of the data.
Here we present a scan in the
high $\tan\beta$ regime dedicated to answer the
question whether there is a $\tan\beta$ range
for which the observed excesses can be accommodated simultaneously.
The input parameters are the same as the ones in
the low $\tan\beta$ regime (as shown in \refta{n2hdmlowtbparas}), but
with considerably larger values for $\tan\beta$
in the range $6 \leq \tan\beta \leq 12$,
and the upper limit of $c_{h_2 VV} \leq 0.9$ is
removed.\footnote{For larger values of
$\tan\beta$ it is possible to achieve relatively large values
of $\mu_{\rm CMS}$ even for $c_{h_2 VV} > 0.9$.
It is then also possible to find points within the
$1\sigma$ ellipse of $\chi^2_{96}$ even though
hardly any signal is present for
the LEP excess.}
As before, we refer to the point with
the lowest $\chi^2$ value as the best-fit
point, where the same definition as
in \refeq{eqchisqtt} was used
to obtain the total $\chi^2$ value.
Moreover, in the results of our scan we only display
points with a lower value of the overall
$\chi^2$ than the one of the SM, i.e.,
$\chi^2 \leq \chi^2_{\rm SM}$.
Note that for consistency we again include both the
contributions $\chi^2_{\ensuremath{\tau^+ \tau^-}}$
and $\chi^2_{t \bar t}$ in the fit.
However, for the high $\tan\beta$ values
considered here no sizable contribution
to the $t \bar t$ excess is expected.
The condition $\chi^2 \leq \chi^2_{\mathrm{SM}}$
is practically unchanged
depending on whether
$\chi^2_{t \bar t}$ is taken
into account here or not, as can also
be seen in \reffi{figGEN2}, in which a difference
of $\chi^2_{\mathrm{SM},t \bar t} -
\chi^2_{t \bar t} \lesssim 2$ is visible for
$\tan\beta \geq 6$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{LLxstbII.pdf}~
\includegraphics[width=0.48\textwidth]{LLxs96II.pdf}
\caption{\small
Signal cross sections of $A \to \ensuremath{\tau^+ \tau^-}$
for the $gg$ production mode on the
horizontal axis and the $b \bar b$ production mode
on the vertical axis. The colors of the points
indicate the value of $\tan\beta$ (left)
and $\chi^2_{96}$ (right).
The dark blue
and the yellow lines indicate the $1\sigma$
and $2\sigma$ ellipse of $\chi^2_{\ensuremath{\tau^+ \tau^-}}$,
respectively.
The best-fit point is indicated with a magenta star.}
\label{figll1}
\end{figure}
We show the parameter points of
the type~II scan at high $\tan\beta$
in the plane of the signal cross sections regarding
the $\ensuremath{\tau^+ \tau^-}$ excess in \reffi{figll1}
and regarding the signal rates of
the excesses at $96\,\, \mathrm{GeV}$
in \reffi{figll2}.
In the left plot of \reffi{figll1} one can see
that, as anticipated from the couplings of the $A$ boson to fermions
in type~II, the signal rate for the $b \bar b A$ production mode grows with
increasing values of $\tan\beta$, while
$\sigma(g g \rightarrow A \rightarrow \ensuremath{\tau^+ \tau^-})$ shows less sensitivity
to $\tan\beta$.
For values
of $6 \leq \tan\beta \leq 11$ the points lie
within the $1\sigma$ region regarding $\chi^2_{\ensuremath{\tau^+ \tau^-}}$,
which is indicated by the dark blue ellipse.
Moreover, for $\tan\beta \approx 10$ the points
approximately lie at the
center of the ellipses, indicating that
the observed excesses are well described.
We emphasize that this is a non-trivial
result. Given the fixed mass spectrum
in this scenario,
and the fact that
the presence of $h_{96}$
gives rise to the additional decay channel
$A \to Z h_{96}$, the location of the band on which
the points are found in \reffi{figll1} is fixed approximately
(the band corresponds to the left branch of points
in \reffi{figGEN3}).
Hence, a type~II
N2HDM CP-odd Higgs boson $A$ with a mass of
$m_A = 400\,\, \mathrm{GeV}$, in combination with
a CP-even Higgs boson $h_1$ at around $96\,\, \mathrm{GeV}$,
yields a predicted pattern that precisely matches the one of the
excesses observed
by ATLAS in the Higgs searches in the $\ensuremath{\tau^+ \tau^-}$ final state.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{hsIIll.pdf}~
\includegraphics[width=0.48\textwidth]{llII.pdf}
\caption{\small
The $\mu_{\rm CMS}$--$\mu_{\rm LEP}$ plane
for the points of the high $\tan\beta$ scan in the
type~II of the N2HDM.
The black ellipse indicates the $1\sigma$ region of
$\chi^2_{96}$ with its center marked with
a black cross.
The best-fit point is highlighted with
a magenta star.
The colors of the points indicate
$\Delta \chi^2_{125}$
in the left plot,
and $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ in the right plot.}
\label{figll2}
\end{figure}
In the right plot \reffi{figll1} we show the same
parameter points, however with the color of the
points indicating the value of $\chi^2_{96}$.
One can see that a large fraction of points that lie
within the $1\sigma$ ellipse of $\chi^2_{\ensuremath{\tau^+ \tau^-}}$
also have very small values of $\chi^2_{96}$.
Consequently, the parameter space covered in
our scan contains parameter points in which
the excesses at $96\,\, \mathrm{GeV}$ are accommodated
simultaneously with the $\ensuremath{\tau^+ \tau^-}$ excess
at $400\,\, \mathrm{GeV}$.
One can see a slight correlation between
the cross section
$\sigma(g g \rightarrow A \rightarrow \ensuremath{\tau^+ \tau^-})$
and $\chi^2_{96}$. Larger values of the cross section
correspond to, on average, lower
values of $\chi^2_{96}$.
In \reffi{figll2} we show the
results of our scan
in the plane $\mu_{\rm CMS}$--$\mu_{\rm LEP}$.
In the left plot one can see that
$\Delta \chi^2_{125}$, indicated
by the colors of the points, ranges from values
of $\approx 3.5$ to 23. In the scan in the low
$\tan\beta$ regime of type~II also smaller values
closer to zero could be found (see
right plot of \reffi{figttII}).
Thus, the scenario in the high $\tan\beta$ regime
might be associated with larger deviations of
the properties of the SM Higgs boson at $125\,\, \mathrm{GeV}$
compared to the SM prediction.
However, for the points with the
smallest values of $\Delta \chi^2_{125}$
the deviations of the signal rates of $h_{125}$
compared to the SM prediction
are much below the current
experimental uncertainties, and
also more precise future measurements
at the HL-LHC might not be sufficient
to entirely probe this scenario.
One can also see in the right plot of
\reffi{figll2} that many points with low values
of $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ lie within the
$1\sigma$ ellipse regarding $\chi^2_{96}$.
The best-fit point, indicated by the magenta
star, is
close to the central point of
the ellipse. We conclude that the type~II
N2HDM is perfectly capable of accommodating
the excesses at $96\,\, \mathrm{GeV}$ in combination
with the $\ensuremath{\tau^+ \tau^-}$ excess for values
of $\tan\beta \approx 10$.
Since we saw in \refse{fullII} that for the high
$\tan\beta$ region the CP-odd Higgs boson can also
provide an explanation for the excess in
the $Z h$ final state, we show
the corresponding signal cross
sections for the points of this scan in
\reffi{figll3}. As before, the subset
of points that are in
the wrong-sign Yukawa coupling regime
provide relatively large contributions
to the excess, as can be seen in the right
plot of \reffi{figll3},
while the other parameter points do not
give rise to a signal in the
$A \to Zh$ channel.
Overall, the values for the cross sections
are slightly smaller here than what was
found in \refse{fullII}. The reason is
that here $A$ has an additional decay mode
to a $Z$ boson and the lightest Higgs boson
$h_1$ at around $96\,\, \mathrm{GeV}$.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{AZhll.pdf}~
\includegraphics[width=0.4\textwidth]{AZhwll.pdf}
\caption{\small Predicted rate for
$\sigma(b \bar b \to A \to Z h_{125}) \times \text{BR}(h_{125} \to b \bar b)$
in comparison with the expected and observed $95\%$ confidence level
upper limits obtained
by ATLAS~\cite{Aaboud:2017cxo}. The colors of the points indicate
the value of $\tan\beta$ (left) and
the value of the product
$c_{h_{125} b \bar b} \cdot c_{h_{125} VV}$ (right).}
\label{figll3}
\end{figure}
\section{NMSSM interpretation}
\label{sec:nmssm}
Our analysis in the previous section has shown that the N2HDM of type~II
provides an attractive framework for accommodating either the
$t \bar t$ or the $\ensuremath{\tau^+ \tau^-}$ excess at $400\,\, \mathrm{GeV}$ simultaneously with the
observed excesses at $96\,\, \mathrm{GeV}$.
In SUSY models, due
to the holomorphy of the superpotential,
the presence of two Higgs doublet fields
with opposite hypercharge, usually denoted as
$H_u$ and $H_d$, is required for
the mass generation of both up- and down-type
fermions. Identifying $H_d = \Phi_1$ and
$H_u = \Phi_2^*$, there is a
correspondence
between the Yukawa structure of the (N)2HDM type~II
and SUSY extensions of the SM.
Thus, it is an interesting question whether
the excesses can also be realized in SUSY
models. It should be noted in this context that SUSY yields further restrictions
on the
Higgs sector compared to non-supersymmetric models with
a similar structure of the extended Higgs sector.
The simplest SUSY extension of the SM is the
Minimal Supersymmetric Standard Model
(MSSM)~\cite{Nilles:1983ge,Haber:1984rc}.
The Higgs sector of the MSSM comprises only
the two Higgs doublet fields $H_{u,d}$ introduced
above.
In the MSSM the quartic scalar couplings
of the Higgs sector are at lowest order
functions of the gauge couplings due
to SUSY relations.
The rigid structure of the
scalar potential leads to the fact that
deviations w.r.t.\ the SM
of the couplings of the lightest
Higgs boson $h_1 = h_{125}$
scale with the inverse of the
mass parameter
$M_A$~\cite{Carena:2001bg}
of the CP-odd Higgs boson.
Consequently, it is possible to
suppress such deviations in the decoupling
limit $M_A \gg M_Z$, in which $h_{125}$ is
aligned with the SM vev and effectively
indistinguishable from a SM Higgs boson, whereas
for $M_A \lesssim 1\,\, \mathrm{TeV}$ deviations at the
level of $1$--$10\%$ for the coupling
coefficients $c_{h_1 f \bar f}$ are
present~\cite{Gunion:2002zf}.\footnote{In principle,
there is also a possibility of achieving
a SM-like Higgs boson in the so-called
alignment-without-decoupling limit of
the MSSM~\cite{Carena:2013ooa}.
However, this scenario relies on
an accidental cancellation between tree-level
and loop contributions. Moreover,
taking into account LHC constraints, it cannot
be realized in the $\MA$--$\tb$ regime relevant
for the excesses~\cite{Carena:2014nza,Bahl:2018zmf},
and we therefore
do not consider
it in the following discussion.}
Hence, the signal-rate measurements
of
$h_{125}$ at the LHC,
which at the present level of accuracy show no significant deviations from
the SM prediction,
can be cast into a lower limit on $M_A$.
Taking into account the most recent LHC
results~\cite{Aad:2019mbh,CMS:2020gsy},
the current limit was found to be
$M_A \gtrsim 600\,\, \mathrm{GeV}$, (largely) independently of the value of
$\tan\beta$~\cite{Bahl:2018zmf,Bahl:2020kwe}.
Consequently, the MSSM cannot account for
the presence of a CP-odd Higgs boson state around $400\,\, \mathrm{GeV}$
without
being in strong tension with the signal rates
of $h_{125}$, and a SUSY interpretation of the observed excesses
therefore needs to be based on non-minimal SUSY models.
The Next-to Minimal Supersymmetric Standard Model
(NMSSM)~\cite{Maniatis:2009re,Ellwanger:2009dp}
extends the Higgs sector
of the MSSM by a
complex singlet scalar field.
The presence of this singlet field is motivated
in particular
by the so-called $\mu$-problem of the
MSSM~\cite{Ellis:1988er}, in which a dimensionful parameter
$\mu$ is present in the superpotential.
While this parameter would naturally
be expected to be of the order of
the ultraviolet cutoff of the model, it must be
of the order of the EW scale to allow for
a phenomenologically viable Higgs sector.
In the $Z_3$ symmetric NMSSM this problem is
absent, because the $\mu$ term is forbidden
by the global symmetry. Instead, it is generated
dynamically
when the scalar component of the singlet
superfield acquires a vev,
and the $Z_3$ symmetry is
broken spontaneously
(see \citere{Ellwanger:2009dp} for details).
Besides the fact that the singlet field
is complex in the NMSSM,
leading to the presence
of a second CP-odd Higgs boson in addition to the
CP-even scalar singlet state,
and that the singlet field is charged
under a $Z_3$ and not a $Z_2$ symmetry,
the Higgs sector
and the Yukawa sector
resemble the one of the N2HDM type~II introduced
in \refse{sec:defn2hdm}. Thus,
from the point of view of the Higgs phenomenology,
the NMSSM is a promising model for investigating
the possible realization of the
excesses around $400\,\, \mathrm{GeV}$ in the context
of SUSY.
We briefly
summarize some of the key features of
the NMSSM that are relevant for
our analysis. We will focus in particular on the
Higgs sector of the model. The superpotential
of the $Z_3$ symmetric NMSSM is given by
\begin{equation}
W_{\mathrm{NMSSM}} =
W_{\mathrm{MSSM}, \cancel{\mu}} + \lambda \ \hat{s} \ \hat{H}_u
\cdot \hat{H}_d +
\frac{1}{3} \ \kappa \ \hat{s}^3 \ ,
\label{superpot}
\end{equation}
where $W_{\mathrm{MSSM}, \cancel{\mu}}$ denotes the
superpotential of the MSSM except the aforementioned
$\mu$-term.
The second term is the portal coupling between the gauge singlet
superfield $\hat{s}$ and the Higgs doublet
superfields $\hat{H}_{u,d}$ (superfields are
denoted here by a hat).
The (effective) $\mu$-term arises from this term
when the scalar component
of $\hat{s}$ acquires the vev $\langle s \rangle = v_S$,
$\mu := \lambda v_S$.
The third term proportional to $\kappa$ is the
singlet self-coupling and, again once
$v_S$ is non-zero, gives rise to bilinear mass
terms both for the scalar and the fermionic
component of $\hat{s}$.
The complete Higgs potential is then derived from
the usual $F$- and $D$-terms
in combination with the terms arising from
the soft SUSY-breaking Lagrangian.
The fermionic superpartners of the Higgs fields
(called Higgsinos and singlino),
together with the superpartners of the gauge bosons
(called gauginos) give rise to five massive neutral
fermion states (called neutralinos) and two massive
charged fermions (called charginos).
The gauginos obtain their masses via the soft
SUSY-breaking gaugino mass parameters
$M_1$, $M_2$ and $M_3$, while the higgsinos obtain their masses
from the $\mu$ parameter.
Compared to the MSSM, the only additional
fermion is the fifth neutralino
(singlino).
Moreover, extending the matter sector
of the SM, the NMSSM contains the scalar
partners of the leptons (called sleptons
and sneutrinos)
and quarks (called squarks).
The superpotential as defined in
\refeq{superpot} conserves
$R$~parity~\cite{Farrar:1978xj}.
As a result, the lightest SUSY
particle (LSP) is stable.
In case the LSP forms a sizable part
of the dark matter abundance,
bounds from direct detection
experiments exclude a substantial
part of the NMSSM parameter space.
Since we are primarily interested here
in the phenomenology of the Higgs sector,
we will not include these limits in our analysis.
Accordingly, the NMSSM can be considered to be a low-energy
effective model of a more complete
theory in which $R$~parity violating effects
prohibit the existence of a stable SUSY
particle, but in which the Higgs phenomenology
remains effectively unchanged. This can be realized if,
for instance, the $R$~parity violating operators
are non-renormalizable and suppressed by the
inverse of a high energy scale such as the
Planck scale, or if another symmetry suppresses
the amount of $R$~parity breaking.
An example for the latter is the so-called
$\mu$-from-$\nu$ supersymmetric standard model
($\mu\nu$SSM)~\cite{Bratchikov:2005vp,Munoz:2009an}
(see \citere{Lopez-Fogliani:2020gzo} for a
recent review).
It was shown that besides
the fact that there are
more than one gauge singlet
scalar fields present in the $\mu\nu$SSM,
the phenomenology of the SM-like Higgs boson
can be practically unchanged w.r.t.\ the
NMSSM~\cite{Biekotter:2017xmf,Biekotter:2019gtq,
Kpatcha:2019qsz}.
\subsection{The Higgs sector: alignment without decoupling}
\label{sec:align}
In the NMSSM an alignment limit exists
in which no decoupling of the BSM Higgs bosons and no
large cancellations between lowest-order contributions and
radiative corrections to scalar masses
and their couplings are required~\cite{Carena:2015moc}.
We briefly describe here the necessary
conditions that have to be fulfilled in order
to account for the presence of a Higgs boson
resembling the discovered particle state
at $125\,\, \mathrm{GeV}$ even for $M_A \approx 400\,\, \mathrm{GeV}$.
We will also demonstrate that only for low values
of $\tan\beta$ the alignment conditions
can be satisfied. Thus, we expect
that a description of the $t \bar t$ excess can be realized
in the exact alignment limit of the NMSSM, while
a description of the $\ensuremath{\tau^+ \tau^-}$ excess will require departures from
the alignment limit.
In the following, we will denote the CP-even
Higgs bosons either with $h_{1,2,3}$, which
is the mass-ordered notation, such that
$m_{h_1} < m_{h_2} < m_{h_3}$, or we
will use $h_{125}$ for the discovered
Higgs boson, $h_S$ for the Higgs boson with
dominant singlet component and $H$ for the
``heavy'' CP-even Higgs boson.
The latter notation is useful because in the
alignment limit the singlet field
does not mix with $h_{125}$.
For the two CP-odd Higgs bosons
we either use the mass-ordered notation
$A_{1,2}$ or, in order to make connection to the
notation of the MSSM, we write $A_S$ for
the singlet-like state and $A$ for
the MSSM-like CP-odd Higgs boson. As for the
CP-even scalars, the latter notation does
not imply a mass ordering of the particles.
In the alignment limit one of the doublet
fields is aligned in field space with the
Higgs doublet vev $v$ (with $v^2 = v_1^2 + v_2^2$).
Then the tree-level couplings
of the corresponding Higgs state, $h_{125}$,
acquire
their SM values.
This can be achieved
if the mass matrix of the Higgs bosons
is diagonal in the Higgs basis, i.e.,
after a rotation of the $2\times 2$ submatrix
of the doublet components by the
angle $\beta$. In order to ensure that the non-diagonal entry of the
mass matrix between the two doublet
fields vanishes, as a first condition the following relation has to be
fulfilled~\cite{Carena:2015moc},
\begin{equation}
\lambda^2 =
\frac{m_{h_{125}}^2 - M_Z^2
\cos 2 \beta}{v^2 \sin^2\beta} \ .
\label{aligncond1}
\end{equation}
This condition was derived
taking into account the dominant one-loop
corrections to the diagonal mass term
of $h_{125}$, stemming from the (s)top-sector.
As long as $\tan\beta$ is not much larger than $10$,
the other
one-loop corrections
(entering also the non-diagonal mass terms)
are suppressed by a
factor $\mu/M_S$.
Since we will choose $M_S \gg \mu$
in our numerical discussion, they
can safely be neglected.
The same is true also for corrections beyond the one-loop level
in the considered parameter region, where the prediction for the mass of the
SM-like Higgs boson is largely dominated by the tree-level contribution (see
the discussion below).
Given the values for the SM vev
$v \approx 246\,\, \mathrm{GeV}$, the mass of the
$Z$~boson $M_Z \approx 91\,\, \mathrm{GeV}$
and ${m_{h_{125}} \approx
125\,\, \mathrm{GeV}}$, the first alignment condition
determines $\lambda$ as a function of $\tan\beta$.
A second alignment condition
arises from the fact that
also the non-diagonal
mass matrix entry relating the
state $h_{125}$ and the
gauge singlet field $s$ has to vanish
in the alignment limit.
This translates into the requirement~\cite{Carena:2015moc}
\begin{equation}
\frac{M_A^2 \sin^2 2 \beta}{4 \mu^2} +
\frac{\kappa \sin 2 \beta}{2 \lambda} = 1 \ .
\label{aligncond2}
\end{equation}
Here, $M_A^2$ is the squared
CP-odd
mass parameter
defined by
\begin{equation}
\MA^2 =
\frac{\mu \left(
A_\lambda + \kappa v_S
\right)}{\sin\beta \cos\beta} \ ,
\label{defma}
\end{equation}
where $A_\lambda$ is the soft SUSY-breaking trilinear
coupling corresponding to the $\lambda$ term in the superpotential
of \refeq{superpot}.
At tree level $\MA$ is equal to the
mass of the MSSM-like CP-odd Higgs boson $A$,
so that it is a useful input parameter
for our analysis.
In order to make a distinction between $M_A$ and
the loop-corrected physical masses of the
CP-odd Higgs bosons, we will denote the latter
with the lower-case letter $m$ in the following, i.e.\ as
$m_{A_{1}}$, $m_{A_{2}}$ or $m_{A_{S}}$, $m_A$.
Since $M_A$ will be fixed to
$M_A \approx 400\,\, \mathrm{GeV}$ in order to
account for the $t \bar t$ or the $\tau^+\tau^-$ excess, and taking
into account that the value of $\lambda$
is given by \refeq{aligncond1} for a fixed
value of $\tan\beta$, the second
alignment condition can be regarded
as a
relation for $\kappa$ in terms of $\mu$ and $\tan\beta$.
\begin{figure}
\centering
\includegraphics[height=7cm]{align_tb_lam.pdf}~
\includegraphics[height=7cm]{align_mue_kap.pdf}
\caption{Left: $\lambda$
as a function of $\tan\beta$
obtained from
the first alignment condition given
in \refeq{aligncond1}.
The black dashed lines indicate the values
$\tan\beta = 1.7$ and $\tan\beta = 8.0$ used
in our benchmark scenarios (see text).
Right: $\kappa$ as a function of $\mu$
obtained from
the second alignment condition given
in \refeq{aligncond2}
for $\tan\beta = 1.7$ (blue) and
$\tan\beta = 8.0$ (orange), and assuming
that $\lambda$ is chosen such that it
satisfies the first alignment condition. The gray shaded region
is excluded by LEP searches for charginos~\cite{Abdallah:2003xe}.
The dashed lines indicate the maximum
values of $\kappa$ satisfying the condition
that there are no Landau poles below the
GUT scale given in \refeq{kappagut}.}
\label{figsaligncond}
\end{figure}
In the left plot of \reffi{figsaligncond}
we show the values of $\lambda$
as a function of $\tan\beta$
obtained from
\refeq{aligncond1}. One can see that relatively
large values of $\lambda > 0.6$ are required.
The two black dashed vertical
lines indicate the $\tan\beta$ values that we will choose as
benchmark scenarios to realize the $t \bar t$
or the $\ensuremath{\tau^+ \tau^-}$ excesses in our numerical discussion
in \refse{sectanbetaeinssieben} and
\refse{sectanbetaacht},
respectively.
Using the corresponding values for $\lambda$,
we show in the right plot of
\reffi{figsaligncond} $\kappa$
as a function of $\mu$
for the two values of $\tan\beta$.
We also indicate with a gray shaded region
the values of $\mu \lesssim 104\,\, \mathrm{GeV}$
for which the mass of the lightest chargino is below the limits
that were obtained from chargino searches
at LEP~\cite{Abdallah:2003xe}.
Here we have assumed that $M_2$ is sufficiently larger than $\mu$, such
that the light chargino mass is determined by $\mu$.
One can see that for $\tan\beta = 8$
(orange line) the second alignment condition
can only be satisfied for
rather large
values of the singlet self-coupling
$\kappa$.
The required values
are roughly an order of magnitude larger
than the maximum value $\kappa^{\rm max}_{\rm GUT}$
(indicated by the orange dashed line)
based on the condition that there should be no
Landau poles below the GUT scale,
approximately given by~\cite{Miller:2003ay}
\begin{equation}
\kappa^2 +
\lambda^2 \leq 0.7^{\;2} \ .
\label{kappagut}
\end{equation}
We thus conclude that for values of
$\tan\beta$ required to realize the
$\ensuremath{\tau^+ \tau^-}$ excess one cannot
satisfy the alignment conditions
for $\kappa$ values that are in agreement with \refeq{kappagut}.
On the other hand, for $\tan\beta = 1.7$,
which is the value
used for the benchmark scenario
addressing the $t \bar t$ excess
(see \refse{nmssmconstraints}),
one can see that for $\mu \lesssim 200\,\, \mathrm{GeV}$
and $\kappa \lesssim \kappa_{\rm GUT}^{\rm max}
= 0.23$ (blue dashed line in \reffi{figsaligncond}) a region of
parameter space in the alignment limit can be explored
that is in agreement with the LEP limits and the perturbativity constraint
of \refeq{kappagut}.
For the description of the $\ensuremath{\tau^+ \tau^-}$ excess at $\tan\beta = 8$ in
\refse{sectanbetaacht} we will allow for a departure from the alignment limit
in order to satisfy \refeq{kappagut}.
\smallskip
Focusing on the low $\tan\beta$ regime and thus the $t\bar t$
excess,
one can
easily estimate the structure of
the Higgs boson spectrum. Obviously, one has
to require $M_A \approx 400\,\, \mathrm{GeV}$ such
that the CP-odd Higgs boson $A$ can play the role
of the particle explaining the $t \bar t$
excess. Moreover, the tree-level mass of the charged
Higgs bosons $H^\pm$ is related to $M_A$ via the relation
\begin{equation}
M_{H^\pm}^2 = M_A^2 +
M_W^2 - \frac{1}{2} v^2 \lambda^2 \ .
\label{masshpmnmssm}
\end{equation}
For the considered
values of $\lambda$,
the charged Higgs bosons are a few~GeV lighter than the CP-odd state~$A$.
The CP-even doublet state~$H$ is also
close in mass to the $A$~boson as long
as its singlet component is small.
For the squared mass of
the SM-like Higgs boson
at tree level an upper bound is obtained that is
approximately given by
\begin{equation}
(m_{h_{125}}^{(0)})^2 \lesssim
M_Z^2 \cos^2 2 \beta +
\frac{\lambda}{2}
v^2 \sin^2 2 \beta
\ .
\label{treemass}
\end{equation}
In the alignment limit this inequality
is nearly exhausted, and values
of $m_{h_{125}}^2 \approx 125\,\, \mathrm{GeV}$
or even slighly larger are obtained for small $\tb$ and large $\lambda$.
Thus, no large
radiative corrections to $m_{h_{125}}$
are required in order to obtain
a physical mass of $125\,\, \mathrm{GeV}$.
Finally, the masses of the singlet-like
particle states~$h_S$ and~$A_S$
are controlled by $\kappa$ and the
corresponding soft trilinear parameter
$A_{\kappa}$. Due to the small values
of $\kappa$ in order to satisfy
\refeq{aligncond2} (see also
\reffi{figsaligncond}), these two
Higgs bosons have masses substantially
below $M_A$. The dependence of their
masses on $A_\kappa$ is opposite
and approximately given by
\begin{equation}
m_{h_S}^2 \approx
\frac{\kappa \mu}{\lambda} A_\kappa
+ \frac{\kappa^2 \mu^2}{\lambda^2} \ ,
\quad
m_{A_S}^2 \approx - \frac{3 \kappa \mu}{\lambda}
A_\kappa \ ,
\label{signletsmassesAkap}
\end{equation}
where for simplicity possible mixing contributions have not been spelled
out in \refeq{signletsmassesAkap} (those mixing contributions are included in
our numerical analysis below).
Thus, depending on the value of $A_\kappa$
both $h_S$ and $A_S$ can be lighter
than $125 \,\, \mathrm{GeV} / 2$,
giving rise to both a lower and an upper
limit on
$A_\kappa$ in order to avoid
large values of $\text{BR} ( h_{125} \to h_S h_S, A_S A_S)$.
Moreover, the singlino mass term is given by
\begin{equation}
m_{\tilde{s} \tilde{s}} =
\frac{2 \kappa \mu}{\lambda} \ .
\label{chargmass}
\end{equation}
A mixed singlino-higgsino neutralino has
a mass of $m_{\widetilde{\chi}^0_1}
\approx m_{\tilde{s} \tilde{s}}$
at tree level.
For $m_{\widetilde{\chi}^0_1} < m_{h_{125}} / 2$ large
values of $\text{BR}(h_{125} \to \widetilde{\chi}^0_1 \widetilde{\chi}^0_1)$
can spoil the SM-like properties
of $h_{125}$. Thus, \refeq{chargmass} translates
into a lower limit on the possible values of $\mu$.
Further experimental constraints and
their impact on the
parameter space
will be discussed in \refse{nmssmconstraints}.
\subsection{Experimental constraints}
\label{nmssmconstraints}
The experimental constraints that we take
into account in our NMSSM analysis are
focused on the collider phenomenology.
In particular, as
discussed above,
we do not intend to reproduce the observed
dark matter relic abundance, nor
do we
apply
constraints from dark matter direct
detection experiments.
Furthermore, we do not exclude parameter
points based on
constrains from flavor observables, as
the theoretical predictions of these
observables in SUSY models depend on
various different sectors of the model,
while the focus of our analysis is the Higgs
sector phenomenology.
The impact of constraints of this kind on the parameter space
investigated in our analysis would depend on the parameter settings of other
BSM contributions that are not relevant for Higgs physics.
While we do not impose those constraints
for excluding parameter points, in our numerical
discussion we will point out possible tensions with constraints from
observables beyond the Higgs sector.
Tensions with observables from the flavor sector are expected in
particular for
relatively small values of $m_{H^\pm} \lesssim M_A$
and low
values of $\tan\beta$~\cite{Domingo:2015wyn}, see the discussion above.
It should be noted that we treat the mentioned upper limits on possible BSM
effects from other sectors in the same way as the $4.2\,\sigma$ discrepancy
between the experimental results for the anomalous magnetic
moment of the muon $(g-2)_\mu$~\cite{Bennett:2006fi,Abi:2021gix}
and the SM prediction~\cite{Aoyama:2020ynm}.
While in principle SUSY models are capable of explaining this
discrepancy~\cite{Martin:2001st,Stockinger:2006zn}
(see also \citere{Chakraborti:2021dli} for an analysis of the EW MSSM
sector, \citere{Abdughani:2021pdc} for a specific NMSSM analysis,
\citere{Heinemeyer:2021opc} for an analysis in the above discussed
$\mu\nu$SSM, and \citere{Athron:2021iuf} for a recent review),
since the Higgs-boson sector is only marginally involved
we also do not apply this constraint favoring a non-zero BSM contribution.
As
done for the N2HDM analysis,
we confronted all parameter points with
the current
limits from
searches for additional Higgs bosons
using \texttt{HiggsBounds}.
There are several collider searches
which are especially relevant for the
low-$M_A$ region of the NMSSM. We briefly
summarize the most important ones here:
\textit{(i)} For low values of $\tan\beta \lesssim 2$,
the LHC searches for $H^\pm$ decaying into
a $tb$ pair exclude parameter regions with
$m_A \approx m_{H^\pm} \approx 400\,\, \mathrm{GeV}$ in the (N)2HDM
type~II~\cite{ATLAS:2021upq}.
In the alignment limit of the NMSSM,
in which such values of $\tan\beta$ allow for
the realization of the $t \bar t$ excess,
one finds $m_{H^\pm} < M_A$ because
of the relatively large values of $\lambda$
(see \refeq{masshpmnmssm}).
Thus, compatibility with the experimental
limits on the charged Higgs rates requires a
reduction of ${\text{BR}(H^\pm \rightarrow t b)}$ compared
to the (N)2HDM prediction which can occur if additional decay
channels into SUSY particles are kinematically open. Nevertheless,
even for a considerably suppressed
$\text{BR}(H^\pm \rightarrow t b)$
the LHC charged Higgs boson
searches~\cite{Aaboud:2018cwk,Sirunyan:2020hwv,
ATLAS:2021upq} belong to the most constraining
BSM Higgs boson searches
for the analysis dedicated to the $t \bar t$
excess.
\textit{(ii)}
The results for the searches in the $t \bar t$ and $\ensuremath{\tau^+ \tau^-}$
final states are not only relevant in view of a possible description of the
observed excesses, but the limits obtained from those searches are also
important regarding their compatibility with the other Higgs bosons of the
model, namely $h_S$, $H$ and $A_S$.
For small values of $\tan\beta$,
searches for the additional
Higgs bosons in the $t \bar t$ final state
are relevant~\cite{Sirunyan:2019wph,Sirunyan:2019wxt}.
For larger values
of $\tan\beta$,
the limits for the $\ensuremath{\tau^+ \tau^-}$
final state
have the largest impact~\cite{Sirunyan:2018zut,Aad:2020zxo}.
\textit{(iii)} For singlet like states $h_S$ and
$A_S$ in the vicinity of $ 125\,\, \mathrm{GeV}$ or
below, the searches for the SM Higgs boson
at LEP~\cite{Abbiendi:2002qp,Barate:2003sz,Schael:2006cr}
and the Tevatron~\cite{Group:2012zca} are
important. At the LHC, the
most sensitive search for
those light additional Higgs bosons is the diphoton search
in which CMS found the excess at around
$96\,\, \mathrm{GeV}$~\cite{Sirunyan:2018aui}, see \refse{sec:excesses96}.
Outside of the mass interval of this excess,
the CMS searches
place important limits on the
presence of the singlet-like scalars.
The corresponding ATLAS limits are substantially
weaker in almost all mass regions~\cite{ATLAS:2018xad}.
\textit{(iv)} The presence of
rather light singlet
states $h_S$ and $A_S$, in particular
as predicted in the alignment limit,
opens up the possibility of Higgs
cascade decays ${H(A) \rightarrow A_S(h_S)\,Z}$
and ${H^\pm \rightarrow h_S/A_S\,W^\pm}$.
CMS and ATLAS searched for the former processes
in multilepton and $l \bar l b \bar b$
final states~\cite{Khachatryan:2016are,
Aaboud:2018eoy,Sirunyan:2019wrn}.
In our analysis, only the constraints
from the searches regarding
the signature $H(A) \to A_S (h_S) \, Z$ were
capable of excluding parameter points.
On the hand,
for our analysis the presence of the
decay mode
${H^\pm \rightarrow h_S/A_S\,W^\pm}$
turned out to be
relevant,
because additional $H^\pm$ decay channels
into BSM scalars have the potential to further reduce
$\text{BR}(H^\pm \rightarrow t b)$ and help
to avoid the constraints from charged
Higgs boson searches in the $tb$ final
state discussed above.
The experimental signatures mentioned
above were also investigated
under the assumption
that the Higgs boson in the
final state is $h_{125}$,
giving rise to exclusions of
parts of the here analyzed
parameter space~\cite{Sirunyan:2019xls}.
This happens when
a second BSM Higgs boson $h_S$ or $A_S$
is present in the
mass window $(125 \pm 10)\,\, \mathrm{GeV}$, yielding
another contribution that has to be added
to the one of $h_{125}$, such
that there can be a relevant enhancement
of the experimental signature. This case will be discussed
in more detail below.
Finally, also purely scalar cascade decays
of the form $H(A) \to h_{125}\,h_S(A_S)$
are relevant, where CMS searched for such a
signature assuming that $h_{125}$ decays
into a $\ensuremath{\tau^+ \tau^-}$ pair and the other (BSM)
final state particle decays into a
$b \bar b$ pair~\cite{CMS:2021yci}.
In addition to the presence of additional Higgs
bosons, also SUSY particles can be
in the reach of the LHC or previous colliders. The corresponding
search limits
yield further constraints
on the parameter space. We list below the
most relevant searches and their impact
on our analysis:
\textit{(i)}
In our analysis in the alignment limit, the
tree-level mass of $m_{h_{125}}^{(0)} \approx 125\,\, \mathrm{GeV}$
is determined by the alignment conditions
(see \refse{sec:align}) for low
$\tan\beta$ values,
which places a limit on the size of the radiative corrections to the mass
of the SM-like Higgs boson. As a consequence,
the SUSY breaking scale $M_S$,
and therefore the stop masses $m_{\widetilde{t}}$, cannot
be too large (see \citere{Carena:2015moc} for
more details).
This implies that
the LHC stop searches
can be relevant, as they provide (under several assumptions)
a lower limit on $m_{\widetilde{t}}$.
Taking into account the
uncertainties on the predictions
for the Higgs-boson mass(es) in the
NMSSM~\cite{Staub:2015aea,Drechsel:2016htw,Slavich:2020zjv},
we use an interval
of $m_{h_{125}} = (125 \pm 4)\,\, \mathrm{GeV}$ in our
analysis. In order to not exceed this
mass interval given the value of $\tan\beta=1.7$
used in the analysis regarding the $t \bar t$
excess in \refse{sectanbetaeinssieben},
we find that values of
$M_S \approx m_{\widetilde{t}} \lesssim 1.2\,\, \mathrm{TeV}$
are required, which is of the order of the
current experimental lower limit on the stop
masses in simplified
scenarios~\cite{CMS:2021eha}.
Here it should be noted that
in our analysis, due to the
compressed electroweakino spectrum and the
presence of Higgs bosons with masses much
below $M_S$,
the actual experimental lower limit on the stop
masses could be substantially weaker
than $1 \,\, \mathrm{TeV}$.
For larger values of $\tan\beta$, as required for
the $\ensuremath{\tau^+ \tau^-}$ excess discussed in
\refse{sectanbetaacht}, the tree-level enhancement
of $m_{h_{125}}$ given by the term
$\sim \lambda^2$ (see \refeq{treemass})
is suppressed, such that in this case
larger radiative corrections
are needed to achieve a value of
$m_{h_{125}} \approx 125\,\, \mathrm{GeV}$.
Consequently, for the $\ensuremath{\tau^+ \tau^-}$ excess
larger values of $M_S$ are
required and LHC searches for stops
have no significant impact.
\textit{(ii)} As already mentioned in
\refse{sec:align}, LEP searches for charginos yield
lower limits on the chargino masses
of up to
$m_{\widetilde{\chi}^\pm_1} >
104\,\, \mathrm{GeV}$~\cite{Abdallah:2003xe}.
In order to ensure compatibility with the
LEP bounds,
we demand $m_{\widetilde{\chi}^\pm_1} >
104\,\, \mathrm{GeV}$ in our analysis.
\textit{(iii)} LHC searches for light neutralinos
in the context of the NMSSM
are very challenging when
the chargino and
neutralino spectra are compressed.
In addition, the searches for the
neutralinos suffer from the fact that also
the background estimation depends on the
precise form of the spectrum of the SUSY
particles. Therefore, a model interpretation
of a particular experimental search has to
be done not only for each SUSY model
separately, but also for each parameter point
within a certain model. Besides,
neutralino searches critically
depend on whether $R$~parity is
assumed to be conserved, or not. Taking the
above mentioned considerations into account,
we cannot apply general exclusions on the
parameter space of the NMSSM from neutralino
searches in our analysis. The only exception
are exclusion bounds based on
the presence of a neutralino with a mass
below $m_{h_{125}} / 2$, since this
can give rise to large values of
$\text{BR}(h_{125} \to \widetilde{\chi}^0_1 \widetilde{\chi}^0_1)$
(see \refse{sectanbetaeinssieben}),
which are excluded by global constraints
on the signal rates of the $h_{125}$.
Finally, as we did in the N2HDM analysis,
we perform a $\chi^2$ test regarding the
signal rates of $h_{125}$ using
\texttt{HiggsSignals} (which as discussed above excludes large values of
$\text{BR}(h_{125} \to \widetilde{\chi}^0_1 \widetilde{\chi}^0_1)$).
An important difference w.r.t.\ the N2HDM arises from
the fact that the Higgs-boson mass could
be chosen as an input parameter in the
N2HDM, whereas $m_{h_{125}}$ is predicted
in the NMSSM as a function of other model
parameters. The model predictions
are affected by a theoretical uncertainty in particular
for larger values of $\lambda$ due to
radiative corrections beyond the one-loop
level (see
\citere{Dao:2021khm} for a recent account of
this subject).
This is why, as already mentioned
above, we allow for an uncertainty of
$\Delta m_{h_{125}}^{\rm theo} = 4\,\, \mathrm{GeV}$ in our
analysis.
In order to make sure that the mass uncertainty
does not give rise to unacceptably large modifications of
the predicted branching ratios for the
decays of $h_{125}$
into SM particles (which could occur if the branching rations are
evaluated with mass values that are several GeV away from $125
\,\, \mathrm{GeV}$, but still within the $\pm 4 \,\, \mathrm{GeV}$ interval),
we recalculated the corresponding decay widths
by requiring $m_{h_{125}} = 125\,\, \mathrm{GeV}$ for all points in which
the prediction of \texttt{NMSSMTools} for
$m_{h_{125}}$ lies within the mentioned
uncertainty band. This recalculation is especially
relevant for $\text{BR}(h_{125} \rightarrow WW^*,ZZ^*)$,
which have a very sensitive dependence on $m_{h_{125}}$.
For the recalculation of the branching ratios
we made use of the effective coupling coefficients
provided by \texttt{NMSSMTools}, by which the
partial decay widths for $h_{125}$
as predicted by the SM were rescaled.
The values for the SM decay widths were taken from
\citere{deFlorian:2016spz}.
\subsection{Benchmark scenario with
\texorpdfstring{\boldmath{$\tan\beta = 1.7$}}{tb17}}
\label{sectanbetaeinssieben}
As discussed in \refse{sec:align}, the
alignment-without-decoupling limit
of the NMSSM is a theoretically
well motivated scenario for
realizing the $t \bar t$ excess in the context
of SUSY.
Thus, we will investigate in
this section the possibility of accommodating
the $t \bar t$ excess in this limit taking into
account the various different collider constraints
mentioned in the previous section.
As in the N2HDM analysis, we consider only
the CP-odd Higgs boson $A$ as the origin of the excess,
accounting for the fact that the contribution
of CP-even states is much smaller compared
to the one of a CP-odd state.
Moreover, since we restrict our discussion to the CP-conserving case
the contributions of CP-even and CP-odd states do not interfere with each other.
In addition, we pointed out that in this limit
the presence of a second light CP-even Higgs boson
arises naturally.
Thus, although not being the main focus of
this analysis, for the
subset of points that feature a particle
candidate in the relevant mass range for the
LEP and the CMS excesses
(see \refse{sec:excesses96})
we will check in a second step
whether also a simultaneous realization of
these excesses in combination with
the $t \bar t$ excess
is possible.
\smallskip
As can be seen from the first alignment
condition shown in \refeq{aligncond1}, the
choice of the
parameter $\tan\beta$ plays a key role. Starting from the
value of $\tan\beta$, and given $M_A \approx 400\,\, \mathrm{GeV}$,
the remaining parameter values can either be
derived or scanned over. We
begin by choosing an initial value of
$\tan\beta = 1.7$ for our scan. This value
arises from the following considerations.
As described in \refse{sec:align}, in the
alignment limit the singlet CP-odd
state $A_S$ is substantially lighter than
the doublet state $A$. Accordingly, also
the mixing of these states is small. Then
one finds that ${c_{A t \bar t} \approx 1 / \tan\beta}$
(as in the N2HDM type~II),
such that regarding the
$t \bar t$ excess even smaller values of
$\tan\beta \approx 1$ would be
favored.
However, the presence of the charged Higgs
bosons $H^\pm$, having a mass slightly below
$M_A$ in the alignment limit, yields
a lower bound on the possible values
of $\tan\beta$ based on LHC searches
for charged scalars (see \refse{nmssmconstraints}).
Our numerical analysis has revealed that,
by taking into account the possibility
that $H^\pm$ can partially decay
into SUSY particles and lighter
neutral Higgs bosons plus a $W$ boson,
values of $\tan\beta = 1.7$
provide points that are compatible with the constraints
arising from the LHC searches, while also
allowing for sizable values of $c_{A t \bar t}$.
Given the chosen value for $\tan\beta$, one derives
a value of $\lambda = 0.6617$ from
\refeq{aligncond1}. Taking into account that
loop corrections yield a (physical)
mass $m_A$ of the CP-odd Higgs boson that is
slightly below the (tree-level) input parameter $M_A$,
we chose to generate parameter points with
$410\,\, \mathrm{GeV} \leq M_A \leq 430 \,\, \mathrm{GeV}$, in steps of
$5\,\, \mathrm{GeV}$. Given a value for $M_A$ and the ones
for $\tan\beta$ and $\lambda$, we used the
second alignment condition shown in \refeq{aligncond2}
to obtain values of $\kappa$ as a function of $\mu$.
The parameter points were then generated by
scanning over $\mu$ and $A_\kappa$ in steps of $1\,\, \mathrm{GeV}$.
We covered the parameter ranges for which
points could be found that are in agreement with
the experimental and theoretical constraints.
The range
of $\mu$ has a lower limit (for each value
of $M_A$) arising from the restriction that
none of the neutralinos should become substantially lighter
than $125 / 2\,\, \mathrm{GeV}$, since otherwise the decay
$h_{125} \to \widetilde{\chi}^0_1
\widetilde{\chi}^0_1$ would become
kinematically allowed, spoiling agreement of
the properties of $h_{125}$ with the LHC measurements.
An upper limit on $\mu$ is given by
the aforementioned constraints from the
charged Higgs boson searches. These constraints can only
be evaded if there are sizable branching
ratios for the decays $H^\pm \rightarrow
\widetilde{\chi}^0_{1} \widetilde{\chi}^\pm_{1},
h_{S} W^\pm$ and/or $A_{S} W^\pm$.
However, the branching ratios of these decays
get reduced for increasing values of $\mu$, as
the masses of the final state particles,
in particular the mass of the
Higgsino-like chargino $\widetilde{\chi}_1^\pm$,
increase with $\mu$.
Upper and lower limits on $A_\kappa$,
in dependence of the other parameters,
are given by the fact that either $h_S$
or $A_S$ becomes lighter than $125 /2 \,\, \mathrm{GeV}$
(see \refeq{signletsmassesAkap}).
In this case the large value of $\lambda$ gives rise to
unacceptably large values of
$\text{BR}(h_{125} \to h_S h_S / A_S A_S)$.
In addition, we find that all points with
$A_S \lesssim 100\,\, \mathrm{GeV}$ are excluded due to
constraints from the search for the signature
$A \to A_S\,h_{125}$~\cite{CMS:2021yci}.
\begin{table}[t]
\centering
\def1.5{1.5}
\setlength\tabcolsep{4.5pt}
\footnotesize
\begin{tabular}{ccccccccccc}
$M_S$ & $M_1$ & $M_2$ & $M_3$ & $A_{\widetilde{f}}$ &
$M_A$ & $\tan\beta$ & $\lambda(\tan\beta)$ & $\mu$ &
$\kappa(\mu,\tan\beta,M_A)$ & $A_\kappa$ \\
\hline
1200 & 140 & 180 & 2000 & 0 & $[410,430]$ &
1.7 & 0.6617 & $[182,201]$ & $[0.047,0.191]$ &
$[-498,-116]$
\end{tabular}
\caption{\small
Parameter values for the scan in the
alignment limit of the NMSSM with
$\tan\beta = 1.7$
for investigating a possible realization of
the $t \bar t$ excess (see text).
}
\label{catttableparas}
\end{table}
We summarize the parameter ranges of the
scan in \refta{catttableparas}. We emphasize that
the values of $\lambda$ and $\kappa$
are derived from the alignment conditions
for $\tan\beta = 1.7$, $410\,\, \mathrm{GeV} \leq
M_A \leq 430\,\, \mathrm{GeV}$ and $182 \,\, \mathrm{GeV} \leq \mu
\leq 201\,\, \mathrm{GeV}$.
The range of $\mu$ is given by the
experimental constraints related to
$\text{BR}(h_{125} \to \widetilde{\chi}^0_1
\widetilde{\chi}^0_1)$ (lower end) and
$H^\pm$ searches (upper end), and
the range of $A_\kappa$ is given by
the experimental constraints related
to $\text{BR}(h_{125} \to h_S h_S, A_S, A_S)$
(as explained already before).
In \refta{catttableparas} we also
give the values used for the parameters of
the soft SUSY-breaking sector. All soft scalar
masses are set equal to the SUSY breaking scale
$M_S = 1200\,\, \mathrm{GeV}$. This value was chosen in order
to allow for a SM-like Higgs boson mass of
$125\,\, \mathrm{GeV}$ without potentially being in conflict
with experimental constraints from stop
searches (see \refse{nmssmconstraints}).\footnote{We
set the masses of all squarks equal to $M_S$ in our
analysis for simplicity.
For the squarks of the first and the
second generation these values are potentially
excluded by LHC searches. However, the masses of
these quarks have no impact on the mass of $h_{125}$,
such that they could be increased without any impact on
the discussion here.}
The soft trilinear couplings of the
sfermions $A_{\widetilde{f}}$ are set to zero.
This is relevant for the stops in order to
avoid large radiative corrections to $m_{h_{125}}$,
which would yield too large values for $m_{h_{125}}$.
The gaugino mass parameters are set to $M_1 = 140\,\, \mathrm{GeV}$,
$M_2 = 180\,\, \mathrm{GeV}$ and $M_3 = 2000\,\, \mathrm{GeV}$. The values of
$M_1$ and $M_2$ lead to the presence of gaugino-like
neutralinos and charginos with masses below $\approx 200\,\, \mathrm{GeV}$.
For this choice several decays of the kind $H^\pm \rightarrow
\widetilde{\chi}^0_{i} \widetilde{\chi}^\pm_j$
can become relevant, which play a role in avoiding the
constraints from charged Higgs boson searches.
As discussed above,
we checked that the lighter chargino mass is above the
lower limit from LEP constraints. The value for
the gluino mass $M_3$ is large enough to
be compatible with the limits
from direct searches given the compressed
neutralino-chargino spectrum,
independently of the fact whether
$R$~parity is assumed to be conserved or
not~\cite{ATLAS:2021twp}.
Taking into account the experimental constraints
listed in \refse{nmssmconstraints},
we found an allowed parameter
region that is shown in
\reffi{paraspace17}. The left plot shows the $\mu$--$\kappa$ plane,
where the color coding indicates $\MA$,
whereas the right plot shows the $\kappa$--$A_\kappa$ plane, where the
color coding indicates $m_{h_S}$.
Here, we applied the same condition as in the N2HDM analysis, demanding
that $\chi^2 \leq \chi^2_{\rm SM}$, where the
total $\chi^2$ is defined in \refeq{eqchisqtt}.
It should be noted that the contribution
of $\chi^2_{96}$ is included in $\chi^2$,
even though it is not the main focus of this analysis.
For parameter points that feature a
CP-even scalar in the mass interval
$94\,\, \mathrm{GeV} \leq m_{h_1} \leq 98\,\, \mathrm{GeV}$,
we calculate $\chi^2_{96}$ given
the predicted signal strengths of $h_1$,
whereas for the other points we set
$\chi^2_{96} = \chi^2_{\mathrm{SM},96}$.
Thus, only a
subset of points
has a contribution of $\chi^2_{96}$ below the SM one.
However, even though the formal definition
of $\chi^2$ is identical, one should keep
in mind that in the NMSSM analysis the
physical mass $m_A$ of the CP-odd Higgs boson is
not an input parameter, such that it can
differ slightly from the value $m_A = 400\,\, \mathrm{GeV}$
used in the N2HDM. This leads to the fact
that in the NMSSM $\chi^2_{t \bar t}$ is
a function of the
three predicted quantities $c_{A t \bar t}$, $\Gamma_A$
and $m_A$
(see also \refse{sec:excesses}).
As before,
the result for the SM is given by
$\chi^2_{\mathrm{SM}, t \bar t} = 13.98$.
As a consequence
of the alignment conditions, we found
that the properties
of $h_{125}$ are very well in agreement with the
experimental measurements. Even parameter
points in which the decay of $h_{125}$ into
two neutralinos is kinematically allowed were
found to predict $\chi^2 \leq \chi^2_{\rm SM}$.
A more detailed discussion of the properties of
$h_{125}$ can be
found in \refap{secnmssmh125}.
\begin{figure}
\centering
\includegraphics[height=6.2cm]{muekapma.pdf}~
\includegraphics[height=6.2cm]{kapAkap.pdf}
\caption{NMSSM parameter points with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
in the $\mu$--$\kappa$ plane (left)
and in the $\kappa$--$A_\kappa$
plane (right). The colors of the points
indicate the value of the tree-level
mass parameter $M_A$ (left) and the
mass of the CP-even singlet-like Higgs
boson $m_{h_S}$ (right).
}
\label{paraspace17}
\end{figure}
In the left plot of \reffi{paraspace17}
one can see that the allowed range of $\kappa$ is below
the limit from the GUT condition shown in
\refeq{kappagut} in most cases. This indicates
that the considered scenarios correspond
to model realizations that may be well defined
up to the GUT scale. The parameter range of $\mu$ is
close to the EW scale,
which corresponds to the parameter region of the NMSSM
that is favored by naturalness arguments~\cite{Baer:2012up, King:2012tr},
in which no large
fine tuning (corresponding to a ``little hierarchy problem'')
is required in the EW sector of
the model. Concerning the colors of the points, indicating $M_A$,
one should note that each
point in the plot corresponds to a subset
of parameter points with different values of $A_\kappa$.
One can see that for increasing values of $M_A$
also larger values of $\mu$ and $\kappa$ are required
in order to fulfill the second alignment condition.
In the right plot of \reffi{paraspace17} one
can see that only points with negative
values for $A_\kappa$ are allowed.
For values of $A_\kappa > -100\,\, \mathrm{GeV}$ we find
that the condition $\chi^2 < \chi^2_{\rm SM}$
is only fulfilled for a small number of points,
which then are excluded by LHC searches involving
the light CP-odd Higgs boson $A_S$ with $m_{A_S} \lesssim 148\,\, \mathrm{GeV}$.
In particular, the CMS search for the signature
$A \to A_S\,h_{125}$ excludes all such points
with $m_{A_S} \lesssim 100\,\, \mathrm{GeV}$, and points with
slightly larger values of $m_{A_S}$ are excluded
by searches for $H \to A_S\,Z$ (see the discussion
in \refse{nmssmconstraints}).
On the other hand, $m^2_{h_S}$ receives additional contributions
proportional to $\kappa v_S$ that can compensate the
negative contribution $\sim A_\kappa$, so that
$m_{h_S}^2 > 0$ even for values of $A_\kappa \approx -500\,\, \mathrm{GeV}$
(see \refeq{signletsmassesAkap}).
Nevertheless, one can see that, given a certain
value for $\kappa$, a lower bound on $A_\kappa$ is
found
where $m_{h_S}$ (indicated by the color coding)
decreases below $\approx 125 / 2\,\, \mathrm{GeV}$, giving rise to unacceptably
large values of $\text{BR}(h_{125} \rightarrow h_S h_S)$.
In addition, even though $A_S$ is overall heavier
if $A_\kappa$ has large negative values, for the lower values of
$\kappa$ it can still be relatively light,
with masses just above or even below $125\,\, \mathrm{GeV}$.
Hence, for small values of $\kappa$ the searches
for $A_S$ already mentioned above become
relevant again, and
give rise to the fact that none of the points
with $0.05 \leq \kappa \leq 0.07$ pass the
\texttt{HiggsBounds} test. Also the two
remaining isolated lines of points at the lower end
of the $\kappa$ range, which correspond to the two points at the lowest
$\kappa$ values in the left plot of
\reffi{paraspace17},
barely escape the constraints
from the searches for $H \to A_S Z$.
As discussed above, points with
$m_{h_S} \approx 96\,\, \mathrm{GeV}$ can contribute to the
LEP and CMS excesses. These points are found for larger $\kappa$
and smaller $|A_\kappa|$. Given the
condition $\chi^2 \leq \chi^2_{\rm SM}$ in
our scans, for these points somewhat larger
values of $\Delta \chi^2_{125}$
are allowed as compared to
the points for which no candidate at $96\,\, \mathrm{GeV}$
is present.\footnote{It should be remembered
that due to the uncertainty in
the prediction of
the Higgs-boson masses, we calculate
$\chi^2_{96}$ as described in \refse{sec:excesses96} for
points with $94\,\, \mathrm{GeV} \leq m_{h_S} \leq 98 \,\, \mathrm{GeV}$,
while for the other points we set
$\chi^2_{96} = \chi^2_{\mathrm{SM}, 96}$.}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{mA_cAtt_Gam_nmssm_n2hdm.pdf}~
\includegraphics[width=0.48\textwidth]{mA_chtt_Gam.pdf}
\caption{NMSSM parameter points
with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
in the $m_{A}$--$c_{A t \bar t}$ plane (left) and the
$m_{A}$--$\chi^2_{t \bar t}$
plane (right). The colors of the points
indicate the values of $\Gamma_{A}/m_{A}$ in \%.
Left:
Also shown are the observed (blue)
and expected (black dashed) upper limits
at the $95\%$ C.L.\ as well as the
$1\sigma$ (green) and $2\sigma$ (yellow)
regions around the expected exclusion limit assuming
$\Gamma_{A}/m_{A} = 2.5\%$, as
published in \citere{Sirunyan:2019wph},
and the N2HDM parameter points from the
scan discussed in \refse{numlowII}
with $\chi^2 \leq \chi^2_{\mathrm{SM}}$
and $2.0\% \leq \Gamma_A / m_A \leq 3.0\%$.}
\label{cattnmssm}
\end{figure}
We now turn to the discussion of
the description of
the $t \bar t$ excess, which is the
main motivation for the scan
presented here. On the left-hand
side of \reffi{cattnmssm} we show the
value of $c_{A t \bar t}$ of the parameter
points in dependence of the (physical) mass $m_A$.
The color coding shows $\Gamma_A/m_A$ in~\%.
Since the singlet admixture of $A$ is
very small, $c_{A t \bar t}$ is given
to a very good approximation by the
inverse of $\tan\beta$, such that
$c_{A t \bar t} \approx 1 / 1.7 \approx 0.59$.
Consequently, the points are all located
within a tight branch at roughly
this value of $c_{A t \bar t}$, slightly below
the experimental best-fit value of
$c_{A t \bar t} \approx 0.8$ assuming
$m_A = 400\,\, \mathrm{GeV}$ and
$\Gamma_A / m_A = 2.5\%$.
This results in
a description of
the excess at a confidence level of
about $1.5\,\sigma$ for the smallest values of $m_A$,
as further discussed below.\footnote{We
wish to stress that the fact that the
NMSSM parameter points happen to lie along
the curve for the expected limit is a coincidence
from which no direct information about the
fit result can be derived.}
The mass of the CP-odd Higgs boson $A$ ranges
between $400\,\, \mathrm{GeV}$ and $430\,\, \mathrm{GeV}$, where
the range is defined by the scan
range used for the tree-level
parameter $M_A$.
In order to facilitate the
comparison between the N2HDM and
the NMSSM we also show the subset of
N2HDM parameter points featuring
$2.0\% \leq \Gamma_A / m_A \leq 3.0\%$
from the scan performed in \refse{numlowII}.
The N2HDM points
lie at larger values of $c_{A t \bar t} \approx 0.8$,
reflecting the fact that the N2HDM allows for
an even better description of the excess.
The reason for this is that
one cannot further reduce the value of $\tan\beta$
in the NMSSM without violating constraints from
the charged Higgs boson searches. In the N2HDM
these can be avoided by increasing the mass
$m_{H^\pm} \gg 400\,\, \mathrm{GeV}$. In the NMSSM this is not
possible, because of the
relations among the Higgs-boson
masses, as explained in
\refse{nmssmconstraints}.
Thus, also the maximum value of $|c_{A t \bar t}|$
that is possibly realized remains below
the measured best-fit value ($c_{A t \bar t} \approx 0.8$).
The collider phenomenology
of the $H^\pm$ state in the NMSSM
will be discussed in more detail below.
It is also important to note that the
$t \bar t$ excess is most pronounced
for the smallest values of $m_A$
analyzed in the experimental analysis.
In \reffi{cattnmssm} this is reflected by the fact
that the smaller
$m_A$, the larger is the difference
between the observed (blue) and expected
(black dashed) exclusion limit.
It should be remembered here
that $\chi^2_{t \bar t}$ is defined
as the $\chi^2$
difference compared to the best fit value, where the latter
was found for $m_A = 400\,\, \mathrm{GeV}$,
$\Gamma_A / m_A = 4.5 \% $ and $c_{A t \bar t}
= 0.94$ (see also \reffi{Chisqtt}).
Thus, even though the theoretical
predictions for the values of
$c_{A t \bar t}$ are independent of $m_A$,
the improvement of the $\chi^2_{t \bar t}$
values in comparison to $\chi^2_{\mathrm{SM}, t \bar t}$
is substantially larger for
values of $m_A \approx 400\,\, \mathrm{GeV}$, where the smallest
$\chi^2_{t \bar t}$ values found are
$\chi^2_{t\bar t} \approx 5$, i.e.\ somewhat worse than in the
N2HDM (see above).
This is clearly visible
in the right plot of \reffi{cattnmssm},
in which we show the values of $\chi^2_{t \bar t}$
in dependence of $m_A$:
the smallest values of
$\chi^2_{t \bar t} \approx 5$
(corresponding to a fit at about
$1.4 \, \sigma$ confidence level)
are obtained for the points
with the lowest values of $m_A$.\footnote{We
perform a linear interpolation to
obtain a value of $\chi_{t \bar t}$
as a function of $m_{A}$, $\Gamma_{A} / m_{A}$, and
$c_{A t \bar t}$ for each parameter point,
interpolating between the values
considered in the experimental analysis.}
Thus, even though
the NMSSM offers a substantially
better fit to the data than the SM, the NMSSM analysis
does not reach the same level of fit quality as
the N2HDM analysis, where values of
$\chi^2_{t \bar t} < 1$ could
be achieved (see, for instance,
\reffi{figGEN2}).
In addition, the total width
$\Gamma_A$ of the CP-odd Higgs boson, indicated
by the color coding in units of
$m_A$ in \reffi{cattnmssm},
correlates with $m_A$,
where for each branch displayed in the right plot of \reffi{cattnmssm}
(each branch corresponds to a different value of $M_A$)
smaller values of $m_A$ give rise to smaller values of $\Gamma/m_A$.
This correlation has
its origin in the phase space
factor for the $t \bar t$ decay width,
which grows with increasing values of $m_A$
and overcompensates the factor $m_A$ in the denominator.
Since the experimental analysis was carried out
for different hypotheses on $\Gamma_A / m_A$,
$\chi^2_{t \bar t}$ depends on $m_A$ both directly and via its impact on the
ratio $\Gamma_A / m_A$.
On the other hand, one can
see that the overall variation of
$\Gamma_A / m_A$ of the parameter points is small compared to the
step sizes of the different width hypothesis
used in the experimental
analysis, as shown in \reffi{Chisqtt}.
Thus, given the experimental uncertainties,
it would have been sufficient to only
take into account the expected and observed
exclusion limits (and the resulting
$\chi^2_{t \bar t}$ values) obtained
under the assumption $\Gamma_A / m_A = 2.5\%$
(this is the reason why the
left plot of \reffi{cattnmssm}
is displayed for $\Gamma_A / m_A = 2.5\%$).
\begin{figure}
\centering
\includegraphics[height=6.cm]{mu96.pdf}
\caption{NMSSM parameter points with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
and for which
$94\,\, \mathrm{GeV} \leq m_{h_S} \leq 98$
in the
$\mu_{\rm CMS}$--$\mu_{\rm LEP}$
plane. The colors of the points
indicate the values of
$m_{h_S}$.
}
\label{mucmsnmssm}
\end{figure}
\smallskip
As discussed at the beginning of this section,
the alignment conditions lead to the presence of a light
CP-even Higgs boson $h_S \ (=h_1)$,
which is almost entirely singlet-like.
As a result, $h_S$
can be a candidate for
describing
the excesses at $\approx 96\,\, \mathrm{GeV}$.
We show in
\reffi{mucmsnmssm} the parameter points
that fulfill the additional constraint
$94\,\, \mathrm{GeV} \leq m_{h_S} \leq 98\,\, \mathrm{GeV}$ in
the plane of $\mu_{\rm CMS}$ and
$\mu_{\rm LEP}$, where the color coding indicates $m_{h_S}$.
We calculate the signal strengths via
\begin{equation}
\mu_{\rm LEP} \approx c_{h_S VV}^2 \
\frac{\text{BR} (h_S \to b \bar b)}
{\text{BR}_{\rm SM} (H \to b \bar b)} \ , \quad
\mu_{\rm CMS} \approx c_{h_S t \bar t}^2 \
\frac{\text{BR} (h_S \to \gamma \gamma)}
{\text{BR}_{\rm SM} (H \to \gamma \gamma)} \ ,
\end{equation}
hence making use of the fact
that the cross section ratios
can be approximated via the respective
effective coupling coefficients squared.
One can see that the CMS excess can easily be accommodated.
On the other hand, in our analysis the LEP excess cannot
be described. This was expected
as a consequence of the dominantly singlet-like character
of $h_S$ and the resulting suppression
of the Higgsstrahlung production cross
section.\footnote{Scenarios
in the NMSSM and the $\mu\nu$SSM were found
in which both excesses at $\approx 96\,\, \mathrm{GeV}$
can be realized at the level of $1\sigma$,
relying on the fact that a sizable coupling
of $h_S$ to vector bosons can be present
via a mixing with $h_{125}$~\cite{Cao:2016uwt,
Domingo:2018uim,Choi:2019yrv,Biekotter:2017xmf,Biekotter:2019gtq}.
Such scenarios rely on
decoupling effects in order to agree with
the experimental constraints on $h_{125}$. As a consequence, there
is a strong tension with the low values of $m_A
\approx 400\,\, \mathrm{GeV}$ that we focus on in the present analysis.}
Regarding the CMS excess, the obtained relatively large
values of $0.35 < \mu_{\rm CMS} < 0.6$
have their origin in an enhancement
of the diphoton branching ratio of the state $h_S$,
which reaches values that are an order of magnitude
higher than the SM prediction.
This enhancement compensates the
suppression of the gluon fusion production
cross section that is approximately given
by $c_{h_S gg}^2 \approx 0.07$.
The large values of $\text{BR}(h_S \to \gamma \gamma)$ are caused by two
different contributions.
Firstly, the light chargino, which is
close in mass to $h_S$, provides a (positive) BSM
contribution to $c_{h_S \gamma \gamma}$.
Even more important, however, are the
patterns of the effective couplings of $h_S$ to top and bottom quarks. In
our scan we find for the coupling coefficients
$0.220 < |c_{h_S t \bar t}| < 0.237$ and
$0.074 < |c_{h_S b \bar b}| < 0.105$.
The dominant component for the
decay width of the diphoton decay is the
diagram with top quarks in the loop.
Thus, this partial decay width scales with $c_{h_S t \bar t}^2$.
The dominant component of the total width of $h_S$
is given by $\Gamma(h_S \to b \bar b)$, such that the total width
is approximately proportional to $c_{h_S b \bar b}^2$.
As a result, we observe an enhancement of
$\text{BR}(h_S \to \gamma \gamma)$ by
factors of $4.5 < c_{h_S t \bar t}^2 /
c_{h_S b \bar b}^2 < 7.3$,
which can be further increased by a possible
chargino contribution.
Accordingly, the CMS excess can be described with an almost
singlet-like Higgs boson.
\begin{figure}
\centering
\includegraphics[height=7cm]{mHp_xstb_MN1.png}
\caption{NMSSM parameter points with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
in the
$m_{H^\pm}$--${\sigma(pp \to t b H^\pm) \times
\text{BR}(H^\pm \to tb)}$
plane. The colors of the points
indicate the values of
$m_{\widetilde{\chi}^0_1}$.
Also shown are the current observed (black)
and expected (black dashed) upper limits
at the $95\%$ C.L.\ and the expected
$1\sigma$
(green) and $2\sigma$ (yellow)
exclusion regions obtained by ATLAS~\cite{ATLAS:2021upq}.
The red line indicates
the previous ATLAS result for the observed
upper limit using
$35\,\mathrm{fb}^{-1}$~\cite{Aaboud:2018cwk}.
}
\label{hpmnmssm}
\end{figure}
\smallskip
Besides the candidates for describing the discussed
excesses, there is another
very prominent feature of the scenario
presented here that offers promising
possibilities for future
collider searches. As was already mentioned
in \refse{nmssmconstraints},
in the alignment limit the CP-odd Higgs boson $A$
at $\approx 400\,\, \mathrm{GeV}$ is accompanied by slightly
lighter charged Higgs bosons $H^\pm$.
Due to the small value of $\tan\beta$,
the coupling of $H^\pm$ to top quarks is relatively large.
Experimental upper limits
were obtaind on the product
$\sigma(pp \to t b H^\pm) \times \text{BR}(H^\pm \to tb)$
from LHC searches performed both by ATLAS~\cite{ATLAS:2021upq}
and CMS~\cite{Sirunyan:2019arl}.
The currently most stringent limits
for the mass region relevant for our scenario
was reported by ATLAS in \citere{ATLAS:2021upq},
using the full $13\,\, \mathrm{TeV}$ Run~2 data set with
$139\ifb$. We show in \reffi{hpmnmssm} the
parameter points of our scan with the
corresponding cross section times
branching ratio on the
vertical axis and the charged Higgs boson
mass on the horizontal axis. We also include
the ATLAS observed and expected upper limits
at the $95\%$ C.L. Since these limits are
included in the most recent version of
\texttt{HiggsBounds}, which was used to
check for the constraints from direct
searches of BSM Higgs bosons, our set
of parameter points lies below the
observed limit.
As discussed in
\refse{nmssmconstraints},
those experimental limits could be evaded
if decay modes of $H^\pm$ to SUSY
particles are kinematically open.
In \reffi{hpmnmssm} the color
of the points indicates the values
of the lightest neutralino
mass, $m_{\widetilde{\chi}^0_1}$.
One can observe that the signal rate
of $H^\pm$ decreases with decreasing
$m_{\widetilde{\chi}^0_1}$. This indicates
that in particular the decay modes
$H^\pm \to \widetilde{\chi}^0_1 \widetilde{\chi}^\pm_{1,2}$
play an important role in this context, weakening
the collider bounds from
the searches ultilizing the $tb$
final state.\footnote{For the
allowed parameter points we
found values of $3.8\% < \mathrm{BR}(H^\pm
\to \widetilde{\chi}^\pm_1 \widetilde{\chi}^0_1)
< 14\%$.}
Nevertheless, even with such a reduction of the $tb$ mode
our analysis shows that the signal rate is still rather close to
the current observed upper limit.
As a consequence, there are good prospects
that future charged Higgs boson
searches at the LHC and the HL-LHC, both in the $tb$ final state and via
dedicated searches exploring decays into final states involving BSM particles,
will be able to probe this scenario.
Another possiblity for testing the
scenario presented here is given
by the precise measurements of
low energy observables. As was mentioned
in \refse{nmssmconstraints},
these are particularly
sensitive to the presence of the
relatively light charged Higgs bosons
and electroweakinos.
We used \texttt{NMSSMTools} to obtain
predictions for flavour physics observables,
including their theoretical uncertainties,
and the anomalous magnetic moment of the muon.
We find that most of the
flavour observables are
predicted to be in agreement with
the experimentally measured values,
despite the presence of $H^\pm$
below $400\,\, \mathrm{GeV}$.
However, there are a few observables that show
deviations at the $\approx 2\sigma$
level. As an illustrative example,
we briefly state the largest discrepancies
for the best-fit point of this scan.
Similar values are found
for the other points of the scan.
For the $\text{BR}(b \to s \gamma)$ decay,
the predicted range for the best-fit point
within the estimated theoretical uncertainties
is $3.67 \cdot 10^{-4} \leq
\text{BR}_{\rm theo}( b \to s \gamma) \leq 4.82 \cdot
10^{-4}$~\cite{Buras:2002vd,Chetyrkin:1996vx,
Ciuchini:1997xe,Ciuchini:1998xy,Bobeth:1999ww,
Czakon:2006ss,Gambino:2001au,Czakon:2015exa},
which lies just above the experimental
$2\sigma$ interval $3.02 \cdot 10^{-4} \leq
\text{BR}_{\rm exp}( b \to s \gamma) \leq 3.62 \cdot
10^{-4}$~\cite{Amhis:2014hma}.
A similar discrepancy, however this time
in the opposite direction, is found
for the decay $B_d \to \mu^+\mu^-$,
where the predicted range $1.34 \cdot 10^{-11} \leq
\text{BR}_{\rm theo}(B_d \to \mu\bar\mu) \leq 9.15 \cdot
10^{-11}$~\cite{Buras:2002vd,Misiak:2015xwa,
Bobeth:2013uxa,Bobeth:2001jm} is $\approx 2\sigma$
smaller than the experimental range
$11 \cdot 10^{-11} \leq
\text{BR}_{\rm exp}(B_d \to \mu\bar\mu) \leq 71
\cdot 10^{-11}$~\cite{Amhis:2014hma}.
Here the substantially larger experimental
uncertainties should be kept in mind.
\smallskip
Summarizing
the results of
the scan presented here, we
find that the alignment-without-decoupling
limit of the NMSSM
is
a theoretically well
motivated framework
that is capable of describing, at least approximately, the observed
$t \bar t$ excess in the context
of SUSY. In addition, by choosing the
singlet scalar self-coupling $A_\kappa$
appropriately, a simultaenous
description
of the CMS excess at about $96\,\, \mathrm{GeV}$
is possible (but not of the LEP excess).
We have found that the latter scenario is compatible with the experimental
results for the
properties of the $h_{125}$ state
even for the case where the
decay of $h_{125}$
into BSM particles,
for instance into two neutralinos,
is kinematically open.
\subsection{Benchmark scenario with
\texorpdfstring{\boldmath{$\tan\beta = 8$}}{tb8}}
\label{sectanbetaacht}
In this section we
analyze a parameter region
in the NMSSM aiming towards a possible description
of the $\ensuremath{\tau^+ \tau^-}$ excess.
As already discussed in \refse{sec:align}, this requires
larger values of $\tan\beta$ as compared to the
scan in \refse{sectanbetaeinssieben},
which leads to the fact that
the alignment-without-decoupling
conditions cannot be
met anymore for perturbative values
of $\lambda$ and $\kappa$.
This is why we
consider here the usual
decoupling limit, i.e.\ larger values of $M_A$, in order to ensure
that the $h_{125}$
properties are in agreement with the
signal rate measurements at the LHC.
Hence, we use
a wider range of $M_A$ in our scan, extending up to $M_A = 560\,\, \mathrm{GeV}$.
Physical masses of the neutral BSM
Higgs bosons of about $400\,\, \mathrm{GeV}$ are
then achieved via mixing of the doublet states
with the singlet states.
Moreover, in the decoupling limit the
deviations from the alignment limit scale with the factor
$\Delta c_{h_{125} VV} = M_Z^2 \sin{4\beta}/(2 M_A^2)$~\cite{Carena:2001bg}
at tree level,
such that also the larger value of $\tan\beta = 8$
(compared to $\tan\beta = 1.7$ in
\refse{sectanbetaeinssieben}) already
facilitates SM-like behavior
of $h_{125}$
even for $M_A \approx 400\,\, \mathrm{GeV}$.\footnote{One
finds $\sin{4\beta} = -0.85 (-0.48)$ for $\tan\beta = 1.7 (8.0)$.}
However, for the coupling of $h_{125}$
to down-type fermions the decoupling behavior is delayed
for larger values of $\tan\beta$, so that
sizable deviations of $c_{h_{125} b \bar b}$
and $c_{h_{125} \ensuremath{\tau^+ \tau^-}}$ from the respective SM value
can occur (see e.g.\ \citere{Carena:2001bg}).
\begin{table}
\centering
\def1.5{1.5}
\footnotesize
\begin{tabular}{cccccccccccc}
$\tan\beta$ & $\lambda$ & $\kappa$ &
$\mu$ & $M_A$ & $A_\kappa$ &
$A_t$ & $A_{b,\tau}$ & $m_{\widetilde{f}}^2$ &
$M_1$ & $M_2$ & $M_3$ \\
\hline
$8$ & $0.36$ & $0.58$ &
$[110, 170]$ & $[360, 560]$ & $-200$ &
$6200$ & $3000$ & $2500^2$ &
$1000$ & $2000$ & $2700$
\end{tabular}
\caption{\small
Parameter values for the NMSSM scan
with $\tan\beta = 8$
for investigating a possible realization of
the $\ensuremath{\tau^+ \tau^-}$ excess.
}
\label{tablenmssm8}
\end{table}
We show the set of input parameters for
our scan in \refta{tablenmssm8}.
By comparing to the left plot of \reffi{figsaligncond}
one can see that the value for $\lambda$
is far below the value
required to fulfill
the alignment condition given in
\refeq{aligncond1}. In order to avoid a sizable mixing
of $h_{125}$ and $h_S$ we require that
$h_S$ is much heavier than $125\,\, \mathrm{GeV}$
by using a large value of $\kappa = 0.58 > \lambda$.
In combination with the value $A_\kappa = -200\,\, \mathrm{GeV}$
and the parameter range of $\mu$,
we find masses of $280\,\, \mathrm{GeV} < m_{h_S} < 442\,\, \mathrm{GeV}$,
$286\,\, \mathrm{GeV} < m_{A_S} < 367\,\, \mathrm{GeV}$,
$352\,\, \mathrm{GeV} < m_H < 526\,\, \mathrm{GeV}$
and $364\,\, \mathrm{GeV} < m_A < 522\,\, \mathrm{GeV}$.
The masses of the states with dominant doublet component
($H$ and $A$) are more closely related to the value
of $M_A$, and therefore can be even larger
than $500\,\, \mathrm{GeV}$ in this scan.
In order to calculate $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ we
sum up the cross sections of all neutral scalars
in the mass interval $(400 \pm 40)\,\, \mathrm{GeV}$,
where interference effects can be neglected
due to CP conservation and a sizable
mass splitting of $A_S/A$ and $h_S/H$
in each parameter point (see also the discussion below).
Thus, for the upper end of these ranges both singlet-like
states can
contribute to the $\ensuremath{\tau^+ \tau^-}$
excess, in addition to the doublet states
$H$ and $A$.
Here it should be noted that
the pairs $A$/$A_S$ and $H$/$h_S$ are mixed,
such that the classification into doublet
and singlet states is only approximate.
Therefore, in the following
we will adopt the mass-ordered notation
$h_{2,3}$ and $A_{1,2}$,
with $h_1 = h_{125}$.
Taking into account that
we want to exploit the mixing of
the singlet states with the heavy
doublet states in order to obtain
physical masses of $m_{A,H}\approx 400\,\, \mathrm{GeV}$ even
for considerably larger
values of $M_A$, and also that the
singlet-like states can
contribute to the $\ensuremath{\tau^+ \tau^-}$
excess for $m_{h_S,A_S} \lesssim 400\,\, \mathrm{GeV}$,
it is apparent that in the scenario considered
here the excesses at $96\,\, \mathrm{GeV}$
cannot be addressed. We emphasize
that attempting to fit the
excesses at $96\,\, \mathrm{GeV}$ in combination with
the $\ensuremath{\tau^+ \tau^-}$ excess is in any case
not very promising, since (as discussed above)
the description of the
$\ensuremath{\tau^+ \tau^-}$ excess at about $400\,\, \mathrm{GeV}$ in the NMSSM relies
on decoupling effects in order to obtain
a phenomenologically viable state $h_{125}$.
Demanding a CP-even singlet-like state at
$96\,\, \mathrm{GeV}$ that is mixed with $h_{125}$
in order to provide a description of the observed
excesses would give rise to
unacceptably large
modifications of the couplings
of $h_{125}$ compared to the SM.\footnote{See
\citere{Cao:2016uwt,Domingo:2018uim} for discussions of
fits that exclusively address the excesses at $96\,\, \mathrm{GeV}$
in the NMSSM.}
The parameter
ranges as defined in \refta{tablenmssm8}
lead to the presence of
at least two neutral Higgs bosons with masses of
$\approx 400\,\, \mathrm{GeV}$. As mentioned before, for $M_A > 500\,\, \mathrm{GeV}$
the masses $m_{h_3}$ and $m_{A_2}$ can still
be
close to $400\,\, \mathrm{GeV}$
as long as large mixings with the lighter
states are present.
The parameters related to the squark sector are
chosen to obtain a SM-like Higgs boson mass
of $\approx 125\,\, \mathrm{GeV}$. For a value of $\tan\beta = 8$
the additional contribution to the tree-level
mass of $h_1$ proportional to $\lambda^2$ is
small. Thus, sizable radiative corrections to $m_{h_1}$
are required,
in analogy to the case of the MSSM. Compared to the
scan presented in \refse{sectanbetaeinssieben}, we therefore
increase the stop soft SUSY-breaking mass parameters
to $m_{\widetilde{t}_{L,R}}^2 = (2.5\,\, \mathrm{TeV})^2$.
Furthermore, the large value of $A_t = 6.2\,\, \mathrm{TeV}$ yields
a large stop mixing, which further increases
the radiative corrections to the mass of $h_1$.\footnote{The
large value of
$|A_t / m_{\widetilde{t}}|$
can potentially lead to the presence
of color-breaking minima below the EW minimum,
and therefore to problems with vacuum
stability ~\cite{Casas:1995pd,Hollik:2018wrr}.
While the chosen value is near the border between the (phenomenologically
viable) region featuring a meta-stable vacuum and the region where the vacuum
would become unacceptably short-lived, we stress that a prediction for
$m_{h_{125}}$ in accordance with the experimental result
can also be achieved with values
of $A_t$ below $6\,\, \mathrm{TeV}$ and/or larger stop masses.}
For completeness, we list in \refta{tablenmssm8} also
the values of the
remaining soft trilinear parameters $A_{b, \tau}$ and the
soft gaugino mass parameters $M_{1,2,3}$. However,
besides the value of $M_3 = 2.7\,\, \mathrm{TeV}$ on which
$m_{h_1}$ depends mildly, these parameters are
of no relevance in the following discussion.
We performed a simple grid scan over the
parameters $\mu$ and $M_A$. While the former is
proportional to the singlet vev $v_S$ and thus
closely related to the mass scale of the singlet
states, the latter defines the approximate mass scale of the
heavy doublet states.
Since in this scan no Higgs boson at $\approx 96\,\, \mathrm{GeV}$
is present, we used the definition of the total
$\chi^2$ as defined in \refeq{eqchisqttno96},
i.e.\ without $\chi^2_{96}$, in order
to quantify the quality of the fit for each
parameter point. We emphasize that for the chosen value of
$\tan\beta = 8$ no sizable contribution to the
$t \bar t$ excess is present. However, we
keep the quantity $\chi^2_{t \bar t}$
in $\chi^2$ for completeness,
as was done in \refse{sectanbetaeinssieben}
for the $\chi^2_{\ensuremath{\tau^+ \tau^-}}$
contribution ($\chi^2_{\mathrm{SM},\ensuremath{\tau^+ \tau^-}}
= 9.99$).\footnote{Since we find
$m_{A_1} < 365\,\, \mathrm{GeV}$, we only take
into account the contribution of $A_2$ for
the calculation of $\chi^2_{t \bar t}$.}
In comparison to the N2HDM analysis of
the $\ensuremath{\tau^+ \tau^-}$ excess, in which only
one particle gave rise to the excess,
in our NMSSM analysis
we have always at least two Higgs bosons
with masses close to $\approx 400 \,\, \mathrm{GeV}$.
For the calculation of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$, as mentioned above, we
take into account the signal contributions
of all neutral BSM Higgs bosons within a
mass interval of $(400\pm 40)\,\, \mathrm{GeV}$, corresponding
to a mass resultion of $10\%$.
We checked for all parameter points that
the mass differences of $A_1/A_2$ and
$h_2/h_3$ are much larger than the sums
of their total widths.
We thus can neglect interference effects
(see the discussion in \citeres{Fuchs:2014ola,Fuchs:2016swt,Fuchs:2017wkq})
and simply sum the individual contributions. They are each
obtained as the product of the cross sections
for $b \bar b$
and $gg$ production
multiplied by the branching ratio for the decay
into a $\tau$~lepton pair.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{HSChisq.pdf}~
\includegraphics[width=0.48\textwidth]{cH1bb.pdf}
\caption{NMSSM parameter points of the
scan (see \refta{tablenmssm8}) with $\tan\beta = 8.0$
in the plane $\mu$--$M_A$. All shown points have
$\chi^2 \leq \chi^2_{\rm SM}$ and
pass the \texttt{HiggsBounds} test.
The colors of the points indicate
the values of
$\Delta\chi^2_{125}$ (left) and
$c_{h_1 b \bar b}$ (right).}
\label{nmssm8hs}
\end{figure}
We
start the discussion of the scan
by presenting the parameter
ranges of $\mu$ and $M_A$ that pass
the experimental constraints.
In \reffi{nmssm8hs} we show the parameter points,
where $\mu$ is displayed on the horizontal and $M_A$ on the
vertical axis. In the left plot the colors of
the points indicate the value of $\Delta \chi^2_{125}$.
One can see that all points
have very small values of
$\Delta \chi^2_{125} \lesssim 3.5$,
and some points even yield a better description of the properties
of $h_{125}$ than the SM (corresponding to negative values
of $\Delta \chi^2_{125}$).
Accordingly,
the signal rates of the $h_{125} (= h_1)$ state agree very well
with the experimental data. As discussed before, in
this scan we rely on the decoupling effects
in order to obtain a SM-like Higgs boson
at $\approx 125\,\, \mathrm{GeV}$. This is reflected in the
fact that $\Delta \chi^2_{125}$ decreases
with increasing values of $M_A$.
One can also see that, regarding the values
of $\Delta \chi^2_{125}$,
there is a slight preference for lower values of $\mu$.
Comparing to the
right plot of \reffi{nmssm8hs}, in which the colors
of the points indicate the value of
$c_{h_1 b \bar b}$, one can see
that the overall pattern of deviations in $c_{h_1 b\bar b}$ is
very similar to the one of $\Delta \chi^2_{125}$.
For the lowest values of $M_A$ for a given value of $\mu$
the coupling coefficient
$c_{h_1 b \bar b}$ deviates from
the SM prediction by up to $\approx 6\%$.
This leads to a small increase in the total
width of $h_1$, which in turn reduces the
branching ratios $\text{BR}(h_1 \to \gamma\gamma, WW^*)$, and thus
increases
$\Delta \chi^2_{125}$.
The deviations of $c_{h_1 t \bar t}$
and $c_{h_1 VV}$ (not shown here) remain below $1\%$
and are therefore much smaller than the ones
of $c_{h_1 b \bar b}$.
This is due to the fact
that those couplings show a faster decoupling with increasing
values of $\tan\beta$ and/or $M_A$,
while the decoupling behavior
of $c_{h_1 b \bar b}$ is delayed (see the discussion above).
The parameter points shown
in \reffi{nmssm8hs}
furthermore
passed the \texttt{HiggsBounds}
test.
As expected, for the present scan that is targeted to a description of
the $\ensuremath{\tau^+ \tau^-}$ excess we find that
for all the parameter points the most sensitive
constraint is the one from the
searches for neutral Higgs bosons in
the $\ensuremath{\tau^+ \tau^-}$ final state performed
by ATLAS~\cite{Aad:2020zxo}.
The precise shape of the excluded regions
in \reffi{nmssm8hs} arises from a complicated interplay
between the masses of the Higgs bosons, the
mixing between $h_2$/$h_3$ and $A_1$/$A_2$,
and finally the number of Higgs bosons
that \texttt{HiggsBounds}
combines into a single signal
that is confronted with the search limits.
For instance, the lower right triangular shaped
region is excluded because in this parameter region \texttt{HiggsBounds}
combines the signal contributions of all
four neutral BSM Higgs bosons into a single
signal, owing to the fact that, roughly speaking,
their mass differences
decrease
for decreasing values of the ratio
$M_A / \mu$. On the contrary, for larger
values of this ratio, the contributions
of just the two CP-even Higgs bosons $h_{2,3}$
with masses
$m_{h_{2,3}}$ above the region of $\approx 400\,\, \mathrm{GeV}$
where the excess was observed
yield a signal rate that for many points
is excluded by the limits from the search for a
$\ensuremath{\tau^+ \tau^-}$ resonance.
This is the reason for the fact that the highest values of $M_A$ that are
displayed in \reffi{nmssm8hs} are excluded by \texttt{HiggsBounds}.
\begin{figure}
\centering
\includegraphics[height=0.85\textheight]{xsllsep.pdf}~
\caption{NMSSM parameter points of the
scan with $\tan\beta = 8.0$ (see \refta{tablenmssm8})
with $m_\phi$ on the horizontal axis
and $\sigma(b \bar b \to \phi \to \ensuremath{\tau^+ \tau^-})$
(left) and $\sigma(gg \to \phi \to \ensuremath{\tau^+ \tau^-})$
(right) on the vertical axis,
where $\phi = A_1, A_2,
h_2$ and $h_3$ in the first, second, third
and fourth row, respectively.
The colors of the points indicate whether
the parameter point lies inside (colored)
or outside (gray) of the $1\sigma$
ellipse regarding $\chi^2_{\ensuremath{\tau^+ \tau^-}}$.
The best fit values are highlighted with
a magenta star.
The current expected and observed limit~\cite{Aad:2020zxo}
as well as a projection for the expected HL-LHC exclusion
limits~\cite{Cepeda:2019klc} are also shown.
}
\label{nmssm8xsll}
\end{figure}
In order
to get an idea of the composition of
the Higgs bosons that contribute to
the description of the $\ensuremath{\tau^+ \tau^-}$
excess in this scenario, we show
in \reffi{nmssm8xsll} the individual values of
the signal cross sections for each neutral BSM Higgs boson
(i.e., the cross sections for $h_{125}$ are not displayed).
The colored points fit the $\ensuremath{\tau^+ \tau^-}$
excess within the $1\sigma$ level, i.e.\
$\chi^2_{\ensuremath{\tau^+ \tau^-}} < 2.3$, whereas
the gray points lie outside of the $1\sigma$
ellipse regarding $\chi^2_{\ensuremath{\tau^+ \tau^-}}$.
In each row of \reffi{nmssm8xsll}, we also
show the expected (black dashed line with green and yellow bands)
and observed (black dots) exclusion limits
from the ATLAS search
using the full Run~2 dataset~\cite{Aad:2020zxo}.
In addition, the red points indicate
the expected HL-LHC exclusion limits taking
into account $6\iab$~\cite{Cepeda:2019klc}.
For each parameter point
there are always at least two ($h_2$, $A_2$),
sometimes three particles
($h_{2,3}$, $A_2$)
that contribute to the excess within the considered
mass interval of $(400 \pm 40) \,\, \mathrm{GeV}$.
For some points, even $A_1$ has a mass
of $m_{A_1} > 360\,\, \mathrm{GeV}$. However, its signal
contribution is small compared to the other
Higgs bosons as can be seen in the top
row of \reffi{nmssm8xsll}. For $b \bar b$ production,
as shown on the left-hand side of
\reffi{nmssm8xsll}, the most significant
contribution is given by $h_2$ (third row).
For the $gg$ production mode there
is no parameter point for which a single
Higgs boson individually produces a sufficiently
large signal, but signal rates of
up to about $20\ifb$ are achieved when
summing over the contributions of
two or more Higgs bosons within
the required mass window
(see also the discussion below).
Due to the value of $\tan\beta = 8$
chosen for this scan, we obtain cross sections
for the $b \bar b$ production mode that
are roughly twice as large as for the
$gg$ production mode.
This is well in line with
the signal interpretation of the
$\ensuremath{\tau^+ \tau^-}$ excess,
so that one can expect
a good fit to the data.
The best fit point is highlighted in
\reffi{nmssm8xsll} with a magenta star.
For this point,
the signal contribution describing the observed excess arises from
the two Higgs bosons $A_2$ (second row)
and $h_3$ (fourth row)
with masses of $m_{A_2} \approx m_{h_3}
\approx 400\,\, \mathrm{GeV}$.
The remaining two BSM Higgs bosons
$A_1$ (first row) and $h_2$ (third row)
are roughly $100\,\, \mathrm{GeV}$
lighter and have much smaller cross
sections for this parameter point and therefore do not contribute
to the signal. The best fit point
has $\chi^2 = 97.60$, which is substantially
below the SM value of $\chi^2_{\rm SM} = 108.38$.
The difference of these values arises almost
entirely from the values
$\chi^2_{\ensuremath{\tau^+ \tau^-}} = 0.29$ and
$\chi^2_{\mathrm{SM}, \ensuremath{\tau^+ \tau^-}} = 9.99$,
respectively.\footnote{The remaining
difference comes from
$\chi^2_{t \bar t} - \chi^2_{\mathrm{SM}, t\bar t}
\approx 12.90 - 13.97
\approx -1$.} It is also important to note
that
with the projected sensitivity the HL-LHC will be very well suited
for probing the considered scenario. It will either very significantly
confirm or rule out the $\ensuremath{\tau^+ \tau^-}$ excess. In the latter case, it would exclude
the parameter region corresponding to the signal interpretation in our NMSSM
analysis (see in particular the left plot in the second row of
\reffi{nmssm8xsll}).
On the other hand,
in case the observed excess is indeed due to
one ore more BSM Higgs boson(s) at $\approx 400\,\, \mathrm{GeV}$ the HL-LHC has excellent
prospects for discovering those new states.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{MAggbb.pdf}~
\includegraphics[width=0.48\textwidth]{MUggbb.pdf}
\caption{NMSSM parameter points of the
scan with $\tan\beta = 8.0$ (see \refta{tablenmssm8}):
signal cross sections for the $\ensuremath{\tau^+ \tau^-}$
excess for the $gg$ production mode on the
horizontal axis and the $b \bar b$ production mode
on the vertical axis. The colors of the points
indicate the value of $M_A$ (left) and $\mu$ (right).
The dark blue
and the yellow contours indicate the $1\sigma$
and $2\sigma$ ellipse of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$, respectively~\cite{Aad:2020zxo}.
The best-fit point is indicated with a magenta star.}
\label{nmssm8llchsq}
\end{figure}
In \reffi{nmssm8llchsq} we show the parameter
points with $\chi^2 \leq \chi^2_{\rm SM}$ in
the plane of the cross sections for the
two production modes of the $\ensuremath{\tau^+ \tau^-}$
excess. One should keep in mind that in order to
obtain $\chi^2_{\ensuremath{\tau^+ \tau^-}}$ we took into
account the contributions of all BSM Higgs
bosons within the mass intervall
$(400 \pm 40)\,\, \mathrm{GeV}$, which is of the order of
the step sizes of $50\,\, \mathrm{GeV}$ of the different mass
hypotheses of the experimental analysis~\cite{Aad:2020zxo}.
The colors of the points indicate the values
of the NMSSM parameters that were varied
in this scan, where $M_A$ is displayed in the left and
$\mu$ in the right plot. The best fit point is indicated by a magenta star,
and the contours indicate the $1\sigma$
and $2\sigma$ ellipses of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$~\cite{Aad:2020zxo}.
One can see that the majority of points lies
within the $1\sigma$ ellipse of
$\chi^2_{\ensuremath{\tau^+ \tau^-}}$. Taking into
account the scan ranges, the points
in the $1\sigma$ ellipse cover the whole
range of $\mu$. However, for $M_A$ we find
that only values up to $M_A \lesssim 460\,\, \mathrm{GeV}$
yield a description of the excess at the
$1\sigma$ level. For larger values the
particles $h_{2,3}$ and $A_2$ become
heavier than $440\,\, \mathrm{GeV}$, such that only
a smaller overall signal is achieved.
In the left plot of
\reffi{nmssm8llchsq} one can furthermore
observe that there is a slight tendency towards
smaller values of $M_A$. Similarly,
a tendency towards smaller $\mu$ values
can be observed in the right plot.
These slight preferences are however
statistically
insignificant
in view of
the current experimental uncertainties.
\section{Conclusions}
\label{sec:conclusion}
Various searches at the LHC for BSM Higgs bosons with masses above $125 \,\, \mathrm{GeV}$
showed excesses over the background expectation
in the Run~2 data. As a remarkable feature, several of these excesses
were found around a mass scale $m_\phi \approx 400 \,\, \mathrm{GeV}$
of a hypothetical new Higgs boson $\phi$ (or more than one).
In particular, we focused on possible interpretations of the two most
striking excesses:
CMS reported a local excess of more
than $3\,\sigma$ in the channel
$A \to t\bar t$ in their first year Run~2 data~\cite{Sirunyan:2019wph},
while
ATLAS reported a local excess of about
$3\,\sigma$ in the channel
$\phi \to \ensuremath{\tau^+ \tau^-}$ in their full Run~2 data~\cite{Aad:2020zxo}.
In both cases the analysis of the other experiment for the same period
of data taking is not yet available.
In addition, a local excess of more than
$3\,\sigma$ was found in the ATLAS search for a heavy
resonance decaying into a $Z$ boson and $h_{125}$
assuming $b \bar b$-associated
production and utilizing $35\ifb$ of data~\cite{Aaboud:2017cxo}.
While the various excesses reach the level of
$3\,\sigma$ local significance,
all stay below $3\,\sigma$ global significance.
Also the searches at the LHC for BSM Higgs bosons with masses below
$125 \,\, \mathrm{GeV}$ show an interesting excess at about
$96 \,\, \mathrm{GeV}$ in the channel
$pp \to \phi \to \gamma\ga$: CMS reported a local excess of
about $3\,\sigma$ in their first
year Run~2 data~\cite{Sirunyan:2018aui} and a
similar deviation of~$2\,\sigma$ local significance
in their Run~1 data at a comparable
mass~\cite{CMS:2015ocq}. The ATLAS results based on the data of the
first two years of Run~2~\cite{ATLAS:2018xad} are not sensitive to
the excess observed at CMS.
Furthermore,
LEP reported a local excess of about $2\,\sigma$
in the channel $e^+e^- \to Z\,\phi \to Z\,b\bar b$~\cite{Barate:2003sz}.
In this paper we have analyzed whether the observed excesses can be
described by
models comprising an extended Higgs sector,
where we have concentrated on the Next-to-2HDM (N2HDM) and
the Next-to-MSSM (NMSSM) as minimal scenarios for accommodating the
experimentally observed patterns.
Concerning the excesses at about $96 \,\, \mathrm{GeV}$ from CMS and LEP
taken in isolation,
it is known that several models
can fit both excesses simultaneously at the level
of roughly $1\sigma$ or less, including the N2HDM
and the NMSSM.
In the present paper we have analyzed the question
whether the excesses at about $400 \,\, \mathrm{GeV}$ can be accommmodated in these models
and whether the parameter space that is preferred by those analyses in
addition permits a signal interpretation also of the excesses at about
$96 \,\, \mathrm{GeV}$.
In our analysis we have taken into account
the experimental constraints on the properties of the Higgs boson at
$125 \,\, \mathrm{GeV}$, the limits from BSM Higgs-boson searches at the LHC and previous
colliders, constraints from electroweak precision data and flavor observables,
as well as theoretical constraints such as vacuum (meta)stability
and perturbativity.
We first investigated whether the N2HDM can fit the two excesses at
$400 \,\, \mathrm{GeV}$ such that the relevant parameter region is in agreement with
the theoretical and experimental constraints. While we find that
both excesses cannot be fitted simultaneously, each of them can be
described with a very good $\chi^2$
by the type~II N2HDM, while complying with the
other constraints. The $t\bar t$ excess, independently of the N2HDM
type, can be described for
$1 \lesssim \tb \lesssim 2$,
whereas a description of the $\ensuremath{\tau^+ \tau^-}$ excess in the N2HDM type~II
requires
$6 \lesssim \tb \lesssim 11$,
making them mutually incompatible.
In both cases the description of the excesses mainly occurs as a
consequence of the presence of a
CP-odd Higgs boson with $m_A \approx 400 \,\, \mathrm{GeV}$, while the other Higgs
bosons are found to be not much heavier in those scenarios
owing to theoretical constraints.
Interestingly, even though not directly aimed for
in our analysis, a sizable fraction of the parameter
points that fit the $\ensuremath{\tau^+ \tau^-}$ excess
additionally yield a contribution that would be compatible with a
description of the $A \rightarrow Zh_{125}$ excess that
was found at a similar mass.
In a second step, we have demonstrated that
for each of the parameter regions that are preferred by the excesses at about
$400\,\, \mathrm{GeV}$
the N2HDM
can simultaneously also describe both excesses at
about $96 \,\, \mathrm{GeV}$.
This happens through the presence of a
singlet-like scalar at $96 \,\, \mathrm{GeV}$ giving rise to signal rates that are
compatible with the excesses observed at CMS and at LEP.
More specifically,
in type~II either the $t \bar t$ or the $\ensuremath{\tau^+ \tau^-}$ excess
at about $400\,\, \mathrm{GeV}$ can be
described in their respective $\tb$ range, where in the
latter case also a sizable contribution to the $Zh$
excess can be present.
In type~IV the $t \bar t$
excess is compatible with the $96 \,\, \mathrm{GeV}$ excesses, where the CMS excess
can be described at the
level of roughly $1 \sigma$.
In the NMSSM, which has a type~II Yukawa sector,
as a consequence of the underlying symmetry relations of the model
one has much less freedom regarding the choice of parameters in the Higgs
sector as compared to the N2HDM.
Nevertheless, we have demonstrated that also the NMSSM
can fit each of the excesses at $400 \,\, \mathrm{GeV}$ individually, while
complying with the
BSM Higgs-boson searches and Higgs-boson measurements
as well as the other constraints.
Concretely, we discussed exemplary scenarios
in which we scanned over $\MA$, $\mu$, and in
case of low $\tb$ also over $\kappa$ and $A_\kappa$.
The $t\bar t$ excess is described
at the level of $1.5\, \sigma$
for low $\tb$ in the
alignment-without-decoupling limit,
which is a theoretically and, in view of the constraints from
the Higgs boson measurements, experimentally
well motivated scenario.
The $\ensuremath{\tau^+ \tau^-}$ excess, on the other hand, can be described for larger
$\tb$ values somewhat away from the alignment limit,
while the properties of the Higgs boson at about $125\,\, \mathrm{GeV}$ are
nevertheless compatible with the experimental data within the current
uncertainties.
In the NMSSM parameter region which can give rise to the $t \bar t$
excess naturally a light singlet-like scalar is present. Requiring
its mass to be $\approx 96 \,\, \mathrm{GeV}$, we demonstrated that
the CMS excess at about $96 \,\, \mathrm{GeV}$
can be described simultaneously with the $t \bar t$ excess at about $400 \,\, \mathrm{GeV}$,
whereas in this scenario hardly any signal contribution would have been
generated at LEP.
If the observed excesses indeed turn out to be first indications of one or
more BSM Higgs bosons, besides the channels featuring those excesses
particularly promising channels regarding a possible discovery of a BSM Higgs
boson would be
the search for charged Higgs bosons
with masses around $400 \,\, \mathrm{GeV}$
and searches for the
heavier doublet-like Higgs bosons decaying to a $Z$ boson
and either the SM-like Higgs bosons or the CP even/odd
singlet-like scalar with masses below or slightly
above $125\,\, \mathrm{GeV}$.
The scenarios describing the $\ensuremath{\tau^+ \tau^-}$ excess
will entirely be probed by the $\phi \to \ensuremath{\tau^+ \tau^-}$
searches at the HL-LHC.
Thus, there are very good prospects in the near future for clarifying the
tantalizing hints that have been observed in the Higgs searches via new results
from searches, more precise measurements of the properties of the Higgs boson
at $125\,\, \mathrm{GeV}$ and further information, for instance from flavor physics.
\section{Results for other Yukawa types}
\subsection{The excesses at $400\,\, \mathrm{GeV}$ in type I--IV}
\label{n2hdmstrategy}
In the N2HDM with a
CP-odd Higgs boson $A$ at $400\,\, \mathrm{GeV}$
the following requirements need to be fulfilled
in order to describe the observed excesses at about $400 \,\, \mathrm{GeV}$:
\begin{description}
\item[\boldmath{$\hspace*{0.75cm}\quad t \bar t$}-excess:]
$|c_{A t \bar t}| \gtrsim 0.5$
\item[\boldmath{$\quad \ensuremath{\tau^+ \tau^-}$}-excess:]
$|c_{A \ensuremath{\tau^+ \tau^-}}| \gg |c_{A t \bar t}|$ to
get sizable $\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$
\item[{\color{white} \boldmath{$\quad \ensuremath{\tau^+ \tau^-}$}-excess:}]
$|c_{A b \bar b}| \gg |c_{A t \bar t}|$ to
get $\sigma(b \bar b \rightarrow A) \gtrsim
\sigma(gg \rightarrow A)$ .
\end{description}
Here $c_{A f \bar f}$ are the effective coupling
coefficients of $A$ to the SM fermions,
which are defined
as the coupling of $A$ relative to
the one of a hypothetical SM Higgs boson of
the same mass,
see \refse{sec:excesses}.
In the N2HDM, the coefficients
are given depending on the type either by
$\tan\beta$ or $1 / \tan\beta$, as depicted
in \refta{catttable}.
Taking into account that for all four types
$|c_{A t \bar t}| = 1 / \tan\beta$ holds,
one can conclude that regarding the $t \bar t$ excess
all four types are equivalent.
A sufficiently large coupling
$|c_{A t \bar t}| \approx 1$ is obtained for values of $\tan\beta$ not
much larger than one.
\begin{table}[b]
\centering
\def1.5{1.5}
\footnotesize
\begin{tabular}{l|ccc}
Yukawa type & $|c_{A t \bar t}|$ & $|c_{A \ensuremath{\tau^+ \tau^-}}|$ & $|c_{A b \bar b}|$ \\
\hline
I & $1/\tan\beta$ & $1/\tan\beta$ & $1/\tan\beta$ \\
II & $1/\tan\beta$ & $\tan\beta$ & $\tan\beta$ \\
III & $1/\tan\beta$ & $\tan\beta$ & $1/\tan\beta$ \\
IV & $1/\tan\beta$ & $1/\tan\beta$ & $\tan\beta$
\end{tabular}
\caption{\small
Effective coupling coefficients of the CP-odd Higgs boson $A$
to top quarks, bottom quarks and $\tau$-leptons.}
\label{catttable}
\end{table}
The situation is different regarding the $\ensuremath{\tau^+ \tau^-}$ excess.
Here, also the couplings of $A$ to bottom quarks and, obviously,
$\tau$ leptons are important.
In the following, we will separately discuss for each Yukawa
type whether it could realize the
$\ensuremath{\tau^+ \tau^-}$ excess. We will also distinguish between
{the low $\tan\beta$ region,
$\tan\beta \approx 1$,
in which a simultaneous explanation
of the $t \bar t$ excess would be possible,
and the large $\tan\beta$ region,
$\tan\beta \approx 10$.
In the N2HDM type~I, the $\ensuremath{\tau^+ \tau^-}$ excess
cannot be realized
for any
value of $\tan\beta$. The reason is twofold: Firstly,
accommodating the observed pattern of
the excess requires that the $b \bar b$ associated
production cross section ${\sigma(b \bar b \rightarrow A)}$
is roughly of the same size or larger than
the $gg$ production cross section ${\sigma(g g \rightarrow A)}$.
The latter is mainly mediated via the diagram
with the top quark in the loop.
Due to the fact that the Yukawa coupling $Y_t$ is an order
of magnitude larger than $Y_b$, the condition
${\sigma(b \bar b \rightarrow A) \gtrsim \sigma(gg \rightarrow A)}$
requires that $|c_{A b \bar b}| \gg |c_{A t \bar t}|$,
see above. However, in type~I
$c_{A b \bar b} = c_{A t \bar t}$ holds.
In addition, $\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$
is always tiny in type~I, because
$c_{A t \bar t} = c_{A \ensuremath{\tau^+ \tau^-}}$,
and for $m_A = 400\,\, \mathrm{GeV}$
the decay $A \rightarrow t \bar t$ is kinematically open.
Moreover, it was shown in \citere{Biekotter:2019kde}
that the excesses at $96\,\, \mathrm{GeV}$ cannot be
realized in type~I. Thus, the only excess that can
be accommodated in type~I is the $t \bar t$ excess.
In type~II the situation is different. Here
one has
${|c_{A b \bar b} / c_{A t \bar t}| = \tan^2\beta}$.
Thus, the suppression of $\sigma(b \bar b \rightarrow A)$ compared to
$\sigma(gg \rightarrow A)$ due to $Y_b \ll Y_t$
can be compensated for $\tan\beta \approx 10$, so that a possible excess
can occur also for the $b \bar b$ production mode.
Moreover,
${|c_{A \ensuremath{\tau^+ \tau^-}} / c_{A t \bar t}| = \tan^2\beta}$
holds, and therefore
$\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$
can be sizable due to the enhancement of $c_{A \ensuremath{\tau^+ \tau^-}}$
while $\text{BR}(A \rightarrow t \bar t)$ is suppressed.
The question whether the $\ensuremath{\tau^+ \tau^-}$ excess
can be correctly reproduced depends on whether there is
a range of $\tan\beta$ in which both the enhancement
of $\sigma(b \bar b \rightarrow A)$ and the one of
$\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$ are of a size such
that the rates in both production modes are
compatible with the observed patterns.
This issue will be addressed in our scan together with the
question whether
both the $t \bar t$
and the $\ensuremath{\tau^+ \tau^-}$ excess can be accommodated
simultaneously.
Since
it is possible in type~II to realize
the excesses at about $96\,\, \mathrm{GeV}$~\cite{Biekotter:2019kde},
we will subsequently perform also a
dedicated scan with $m_A = 400\,\, \mathrm{GeV}$ and $m_{h_1}
\approx 96\,\, \mathrm{GeV}$ for this type.
In type~III the $\ensuremath{\tau^+ \tau^-}$ excess cannot be accommodated
for any value of $\tan\beta$, because
${c_{A t \bar t} = c_{A b \bar b}}$,
so that the requirement
$\sigma(b \bar b \rightarrow A) \gtrsim \sigma(gg \rightarrow A)$
cannot be realized (as in type~I).
Also the excesses at $96\,\, \mathrm{GeV}$ cannot be
accommodated in type~III~\cite{Biekotter:2019kde}.
Consequently, only the $t \bar t$ excess can be
explained in type~III.
Also in type~IV the $\ensuremath{\tau^+ \tau^-}$ excess cannot
be realized. The coupling coefficient
$c_{A \ensuremath{\tau^+ \tau^-}}$ scales with the inverse
of $\tan\beta$, such that $\tan\beta < 1$ would
be required to enhance $\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$.
On the contrary, in analogy to the argument
given for type~II, the condition
${\sigma(b \bar b \rightarrow A) \gtrsim \sigma(gg \rightarrow A)}$
requires values of $\tan\beta \approx 10$. Thus,
depending on the value of $\tan\beta$,
either $\sigma(b \bar b \rightarrow A)$
or $\text{BR}(A \rightarrow \ensuremath{\tau^+ \tau^-})$ is too small.
Since type~IV is capable of explaining
the excesses at $96\,\, \mathrm{GeV}$~\cite{Biekotter:2019kde},
there is the possibility to accommodate both
excesses at $96\,\, \mathrm{GeV}$ together with the
$t \bar t$ excess at $400\,\, \mathrm{GeV}$ in type~IV.
As a consequence of the above considerations we restrict our parameter
scans investigating exclusively
the excesses at $400 \,\, \mathrm{GeV}$
in the N2HDM to type~II. For the other three types the
$\ensuremath{\tau^+ \tau^-}$ excess cannot be fitted, and the discussion of the
$t \bar t$ excess would be qualitatively very similar to the case of type~II
(the only difference that could arise is that the
total width of $A$ can be different in type~I, III
and IV compared to type~II, leading to slightly
different preferred $\tan\beta$ values due to
different best-fit values of $c_{A t \bar t}$,
see \refse{sec:excesses}).
The results of our scan for
type~II of the N2HDM
are described in \refse{fullII}.
Concerning the question whether there is the
possibility of fitting both the $t \bar t$ excess and the
$\ensuremath{\tau^+ \tau^-}$ excess at $400\,\, \mathrm{GeV}$ simultaneously,
based on our qualitative discussion above
we can already anticipate that there will be a tension
between the $\tan\beta$ values required to fit either
of them. Indeed, we find that
the excesses point towards different
regions of parameter space with either
$\tan\beta \lesssim 3$ for the $t \bar t$ excess
or $\tan\beta \gtrsim 6$ for the
$\ensuremath{\tau^+ \tau^-}$ excess.
In a second step, we will analyze whether the excesses at about $96\,\, \mathrm{GeV}$
can be realized in combination with either of the
excesses at about $400\,\, \mathrm{GeV}$.
Thus, we perform two different scans
with $\tan\beta_{\rm low} = [0.5,4]$
(regarding the $t \bar t$ excess at $400\,\, \mathrm{GeV}$)
and $\tan\beta_{\rm high} = [6,12]$
(regarding the $\ensuremath{\tau^+ \tau^-}$ excess at $400\,\, \mathrm{GeV}$).
The results are detailed in \refse{numlowII}
and \refse{numhighII}. As discussed above, in type~IV one can
potentially fit the excesses at $96\,\, \mathrm{GeV}$ in
combination with the $t \bar t$ excess for
$\tan\beta = \tan\beta_{\rm low}$.
A corresponding parameter scan is described
in \refap{numlowIV}.
\subsection{Higgs bosons at
\texorpdfstring{\boldmath{$96\,\, \mathrm{GeV}$}}{96gev} and
\texorpdfstring{\boldmath{$400\,\, \mathrm{GeV}$}}{400gev}
for low \texorpdfstring{\boldmath{$\tan\beta$}}{tb}
in type~IV}
\label{numlowIV}
We present here the results
for a
scan in the low $\tan\beta$ regime
in the type~IV (flipped) N2HDM.
The input parameters were chosen to be identical
to the scan in type~II as shown in \refta{n2hdmlowtbparas}.
We show the results of the scan in type~IV
in the $\tan\beta$--$c_{A t \bar t}$ plane in
\reffi{figttIVtt} and in the
$\mu_{\rm LEP}$--$\mu_{\rm CMS}$ plane in
\reffi{figttIV}.
\begin{figure}
\centering
\includegraphics[width=0.44\textwidth]{tbeta_cAtt_ttIV.pdf}~
\includegraphics[width=0.44\textwidth]{tbeta_cAtt_96IV.pdf}
\caption{\small
$c_{A t \bar t}$ in dependence of $\tan\beta$.
The colors of the points indicate the values
of $\Gamma_A / m_A$ in \% (left) and
the values of $m_{H^\pm}$ (right). The dashed horizontal
lines indicate the best-fit values of $c_{A t \bar t}$
for different width hypotheses in the
experimental analysis~\cite{Sirunyan:2019wph}.}
\label{figttIVtt}
\end{figure}
Comparing \reffi{figttIVtt} with
the corresponding type~II results
shown in \reffi{figttIItt}, one can see
that the values of $c_{A t \bar t}$
found in the scan
are considerably smaller
in type~IV.
This is related to the smaller
predicted widths
of $A$ for the parameter points that
fulfill $\chi^2 \leq \chi^2_{\rm SM}$
in type~IV compared to type~II, as indicated
in the left plot of
\reffi{figttIVtt}. The experimental result
shows a less pronounced excess for
$\Gamma_A / m_A \approx 1.5\%\dots 2.0\%$~\cite{Sirunyan:2019wph},
and the minimum of
$\chi^2_{t \bar t}$
is shifted towards smaller couplings
(see also \reffi{Chisqtt}),
such that points with low values of
$\chi^2_{t \bar t}$
require smaller couplings. The smaller values of the
width $\Gamma_A$ in type~IV compared to type~II have their
origin in the modified leptonic couplings.
For values of $\tan\beta > 1$ the decay width for the decay
$A \rightarrow \ensuremath{\tau^+ \tau^-}$ is suppressed
in type~IV, while it is
enhanced in type~II (see also \refta{catttable}).
In the right plot of \reffi{figttIVtt}, in which
the color coding indicates the value of $m_{H^\pm}$,
one can observe that $m_{H^\pm}$ is substantially
smaller than the upper limit of the scan range
$1\,\, \mathrm{TeV}$ for all the parameter points.
The same result was found already in \refse{numlowII}
for type~II. The reason for the preference for
relatively small values of $m_{H^\pm}$ lies again in the theoretical
constraints on the absolute values of the quartic
couplings
(where $m_A$ is kept fixed at $400\,\, \mathrm{GeV}$), which are independent of
the Yukawa type.
Comparing \reffi{figttIV} to the results of type~II
(see \reffi{figttII}), one can see that the points
are substantially shifted towards lower values
of $\mu_{\rm CMS}$. As explained in
\citere{Biekotter:2019kde},
in type~IV $\text{BR}(h_1 \rightarrow \ensuremath{\tau^+ \tau^-})$
is enhanced in the
parameter space in which the CMS excess can
be accommodated, giving rise to smaller
values of $\text{BR}(h_1 \to \gamma\gamma)$.
While it is not possible to reach $\chi^2_{96}$ values close to zero,
corresponding to the center of the displayed ellipse,
several
points lie within the $1\sigma$ ellipse
of $\chi^2_{96}$ while in addition accommodating
the $t \bar t$ excess. This is indicated by the low
values of $\chi^2_{t \bar t}$ shown in the
left plot of \reffi{figttIV}.
The lowest values of $\chi^2_{t \bar t}$
are at the level of $\approx 1.1$. This is slightly larger than
the lowest values found in type~II ($\approx 0.25$), as shown in
\reffi{figttII}. This difference has its origin in
the smaller values of the width of $A$ in type~IV, which
yield a worse fit to the data, as can be seen
in \reffi{Chisqtt}.
Taking the issues mentioned above into account,
we conclude that there is a slight tension
between fitting the CMS excess at around $96\,\, \mathrm{GeV}$
and the $t \bar t$ excess at around $400\,\, \mathrm{GeV}$
in type~IV,
but a realization of all three excess is possible
at the $1\sigma$ level of $\chi^2_{96}$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{csgAttIV.pdf}~
\includegraphics[width=0.48\textwidth]{hsIV.pdf}
\caption{\small The $\mu_{\rm CMS}$--$\mu_{\rm LEP}$ plane
for the points of the low $\tan\beta$ scan in the
type~IV of the N2HDM.
The black ellipse indicates the $1\sigma$ region of
$\chi^2_{96}$ with its center marked with
a black cross.
The best-fit point is highlighted with
a magenta star.
The colors of the points indicate
$\chi^2_{t \bar t}$ in the left
plot and $\Delta \chi^2_{125}$ in the right plot.}
\label{figttIV}
\end{figure}
Comparing the right plot
of \reffi{figttIV} to the one in \reffi{figttII},
one can see that the
lowest values of $\Delta\chi^2_{125}$
found in the scan for type~IV are larger than the
ones for type~II.
This suggests that larger deviations of the
properties of the Higgs boson at $\approx 125\,\, \mathrm{GeV}$
compared to the SM prediction can be expected
in type~IV.
The higher values for $\chi^2_{125}$
in combination with a worse fit to
the CMS excess also manifest themselves in the $\chi^2$ value
of the best fit point $\chi^2 = 102.3$, which
is substantially larger than the corresponding
value $\chi^2 = 97.93$ found in the type~II analysis.
Consequently, since
in both the type~II and the type~IV
analysis the condition $\chi^2 \leq \chi^2_{\rm SM}$
was required to be fulfilled for each point,
the total number of points within and near
the $1\sigma$ ellipse for $\chi^2_{96}$ in
\reffi{figttIV} compared to \reffi{figttII}
is substantially smaller.
In summary, our analysis in this section has revealed
that for the low $\tan\beta$ region of the N2HDM
type~II provides a better description of the observed data than type~IV.
\section{\texorpdfstring{\boldmath{$h_{125}$}}{h125}
in the NMSSM alignment-without-decoupling limit}
\label{secnmssmh125}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{chvvchtths.png}~
\includegraphics[width=0.48\textwidth]{chvvchbbhs.png}
\caption{NMSSM parameter points with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
in the
$c_{h_{125} VV}$--$c_{h_{125} t \bar t}$
plane (left) and the
$c_{h_{125} VV}$--$c_{h_{125} b \bar b}$
plane (right).
The colors of the points
indicate the values of
$\Delta\chi^2_{125}$.
}
\label{effcplshsm}
\end{figure}
In \reffi{effcplshsm} we show the effective
coupling coefficients of $h_{125}$,
with $c_{h_{125} VV}$ on the horizontal
axis and $c_{h_{125} t \bar t}$ and
$c_{h_{125} b \bar b}$ on the vertical
axis in the left and right plot, respectively.
One can see that, as
a result of
solving \refeq{aligncond2},
$c_{h_{125} VV} \approx 1$ holds for
all the points, with deviations up to the
level of $1.7\%$.
In addition, also
the couplings to the SM quarks are quite close to the
SM prediction, as a consequence of solving \refeq{aligncond1}
for $\lambda$. The deviations for both
$c_{h_{125} t \bar t}$ and
$c_{h_{125} b \bar b}$ are
at most of the size of $\approx 4$--$6\%$.
Consequently, most of the points have values
of $\Delta \chi^2_{125}$ rather close to zero, as
indicated by the color coding of the points,
and the properties of the particle state
at $125\,\, \mathrm{GeV}$ are
in good agreement
with the experimental constraints given the current
uncertainties.
We find $\Delta \chi^2_{125} \approx 2$
as lowest values, see the discussion below.
One can furthermore see two small clusters of
points
at $c_{h_{125} VV} \approx 0.994$
for which the values of $\Delta \chi^2_{125}$
are considerably larger compared to
the surrounding points. These clusters arise from additional
decay channels of the state $h_{125}$, as will be discussed below,
and they appear distinctively in
\reffi{effcplshsm} because they correspond to the two
isolated points (lines of points) at the lower end
of $\kappa$ in the left (right) plot of \reffi{paraspace17}.
As mentioned before, a point is considered
to be allowed when the
condition $\chi^2 \leq \chi^2_{\rm SM}$
is fulfilled.
Thus, for points with $\Delta \chi^2_{125}$
considerably above zero, it is expected
that they
provide predictions describing
at least one of the
excesses, such that the penalty of $\chi^2_{125}$
within the total $\chi^2$ is compensated
by either $\chi^2_{t \bar t} <
\chi^2_{\mathrm{SM},t \bar t}$,
$\chi^2_{96} < \chi^2_{\mathrm{SM},96}$, or both.
In order to
further elucidate our results for $\Delta \chi^2_{125}$ and to
determine the origin of the larger
values of $\Delta \chi^2_{125}$ in the two
clusters mentioned
above, we show in \reffi{brs17} a selection of $h_{125}$ branching
ratios. On the left-hand
side, one can see that both $\text{BR}(h_{125} \to WW^*)$ and
$\text{BR}(h_{125} \to \gamma \gamma)$ agree with
the SM prediction (indicated by the
blue dashed lines) at the level of $\approx 5$--$10\%$.
For all the points the value of $\text{BR}(h_{125} \to WW^*)$
is slightly below the SM prediction. This has two reasons:
firstly, the coupling coefficient
$c_{h_{125} VV}$ ranges between
$0.983$
and $0.997$, as can be seen
in the left plot of \reffi{effcplshsm}, such
that a
corresponding suppression
is also expected for the
partial widths of the decays to $WW^*$ and $ZZ^*$.
Secondly, as shown in the right plot of
\reffi{effcplshsm}, the coefficients $c_{h_{125}b \bar b}$
are a few percent above the SM prediction
for all of the points. This leads
to a small enhancement of the partial decay width
to $b$~quarks, and therefore also
to an enhancement of the total width of
$h_{125}$, which in turn yields a further suppression of
$\text{BR}(h_{125} \to WW^*)$. Overall, the small deviations of
the branching ratios for $h_{125} \to VV^*,\gamma\gamma$
compared to the SM predictions give
rise to a penalty of $\Delta \chi^2_{125} \approx 2$--$6$.
The two isolated clusters of points mentioned above
are also well visible in the left plot of \reffi{brs17}.
They have
an even slightly smaller BR for the decay to $WW^*$
than the rest of the points. This can be
understood from the right plot of \reffi{brs17}, where we show the
plane $\text{BR}(h_{125} \to \neu1\neu1)$--$\text{BR}(h_{125} \to \ensuremath{\tau^+ \tau^-})$ with
the color coding indicating again $\Delta\chi^2_{125}$. The blue
dotted lines indicate the SM prediction. Most of the points are found
at $\text{BR}(h_{125} \to \neu1\neu1) = 0$. However,
the two isolated clusters of points yield
a neutralino with a mass
$m_{\widetilde{\chi}^0_{1}} \lesssim 62\,\, \mathrm{GeV}$,
such that the decay
$h_{125} \to \widetilde{\chi}^0_{1}\widetilde{\chi}^0_{1}$
becomes kinematically allowed. The corresponding partial decay width
contributes to the total decay width of
$h_{125}$ and reduces the branching ratios
for decays into SM particles.
While for $h_{125} \to WW^*$ this results in a larger
deviation from the SM prediction, for the
decays $h_{125} \to \gamma\ga, \ensuremath{\tau^+ \tau^-}$
this brings the prediction closer to the SM value. Overall, depending on
the size of $\text{BR}(h_{125} \to \widetilde{\chi}^0_{1}\widetilde{\chi}^0_{1})$,
this yields
values of $\Delta \chi^2_{125} \approx 5$--$8$.
In the right plot of
\reffi{brs17} we have also indicated the current upper
limit on \text{BR}($h_{125} \to {\rm inv.})$ as reported by
ATLAS~\cite{ATLAS:2020kdi}.
For our analysis this bound does not yield a restriction since
the global experimental constraints
on the signal rates of $h_{125}$, as
tested by \texttt{HiggsSignals}, already
exclude all points potentially featuring values of
$\text{BR}(h_{125} \to \neu1\neu1) \gtrsim 0.07$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{bhwwbhyyhs.pdf}~
\includegraphics[width=0.48\textwidth]{bhnnbhllhs.pdf}
\caption{NMSSM parameter points with
$\chi^2 \leq \chi^2_{\mathrm{SM}}$
in the plane
$\text{BR}(h_{125} \to WW^*)$--$\text{BR}(h_{125} \to \gamma \gamma)$
(left) and
$\text{BR}(h_{125} \to \widetilde{\chi}_1^0 \widetilde{\chi}_1^0)$--
$\text{BR}(h_{125} \to \ensuremath{\tau^+ \tau^-})$
(right).
The colors of the points
indicate the values of $\Delta\chi^2_{125}$.
The blue dashed lines indicate the SM predictions.
The gray dashed line in the right plot indicates the upper limit
for $\text{BR}(h_{125} \to \mathrm{inv.})$
at the 95\% C.L.\ reported in \citere{ATLAS:2020kdi}.
}
\label{brs17}
\end{figure}
\section{Prospects and outlook}
\label{sec:prosp}
As a final step of our analyses
of the N2HDM and the NMSSM with respect to
possible interpretations of the observed excesses, we briefly discuss the most
relevant future experimental searches and measurements which can probe those
scenarios. In fact, we would like to point out that there are good prospects
for either strengthening the evidence for a possible signal or ruling out the
discussed scenarios in the near future.
For the low $\tan\beta$ region, the searches for
$A \rightarrow t \bar t$ will obviously be important.
The most promising channels are
gluon fusion production, in which the excess was found,
so far only utilizing the first year Run~2
data corresponding to $35.9\ifb$, as well
as the production of $A$
in association with two top quarks, leading to a
final state of four top quarks. The latter is
already published taking into account the full
Run~2 data by both ATLAS and
CMS~\cite{ATLAS:2020hrf,Sirunyan:2019wxt}, and
it is interesting to note that ATLAS measured
a cross section for four-top production roughly
a factor of two larger than the SM
prediction~\cite{ATLAS:2020hrf}. In addition, in the N2HDM
the parameter region in the wrong
sign Yukawa coupling regime will further be probed
via the process ${gg \rightarrow A \rightarrow Z h_{125}}$,
where the ATLAS analysis using the full Run~2
data set puts important constraints on
the parameter space already.
For the scenarios including
the singlet-like Higgs boson at $\approx 96\,\, \mathrm{GeV}$, also
the
searches with $A$ decaying into a $Z$
boson and another BSM Higgs boson can be important.
Moreover, the searches for the charged
Higgs bosons in the $tb$ final state are very
promising. In the NMSSM,
the current limits
exclude
parameter points with valuess for $c_{A t \bar t}$
of the size of the experimental best-fit values,
while in the N2HDM the charged Higgs bosons
can be somewhat heavier,
so that the parameter region yielding the best description of the
$t \bar t$ excess is not affected by the current limits.
For the case of the NMSSM also dedicated searches exploring decays of the
charged Higgs boson into final states involving BSM particles can be promising
(see also the discussion in \refse{sectanbetaeinssieben}).
For the high $\tan\beta$ region, i.e.\ $\tan\beta \approx 8$,
the parameter region corresponding to the signal interpretation will fully
be probed by the HL-LHC Higgs boson
searches in the $\ensuremath{\tau^+ \tau^-}$ final state.
If the CMS search
including the full Run~2 dataset does not confirm the excess observed by ATLAS,
the cross section values that
are preferred by
the $\ensuremath{\tau^+ \tau^-}$ excess in the corresponding
ATLAS search could already be excluded on the basis of the Run~2 data from both
collaborations.
On the other hand, if the observed excess is indeed a first indication of a
signal of one or more more BSM Higgs boson(s),
the prospects for discovering these new
states in future runs at the LHC and the HL-LHC will be excellent.
Furthermore, the cascade decays $A \rightarrow Z h_{125}$,
where the $A$ boson is produced in the $b \bar b$ production mode,
will also probe this scenario. It is important to note in this context that
in contrast to the ATLAS search in the $gg$ production
mode, the ATLAS search assuming $b \bar b$ production
has not yet been updated to include
the full Run~2 data set.
As already mentioned before,
independently of the value of $\tan\beta$ the presence
of
relatively light charged Higgs bosons is a
common prediction of
the scenarios discussed
in this paper, where in the NMSSM the
charged Higgs bosons are even lighter
than the CP-odd Higgs boson at $400\,\, \mathrm{GeV}$,
while in the N2HDM an upper bound of
$m_{H^\pm} \lesssim 750\,\, \mathrm{GeV}$ applies because of
the theoretical constraints.
In the NMSSM the constraints from flavor physics do not result
in a firm lower limit on $m_{H^\pm}$, since
in general the theoretical predictions for the
flavor observables in SUSY models
depend on various different sectors of
the model, and may be weakened without
changing the Higgs-boson phenomenology
discussed here.
In the N2HDM, on the other hand, we included a lower limit of
$m_{H^\pm} = 550\,\, \mathrm{GeV}$, based on
flavor physics constraints~\cite{Haller:2018nnx}.
More recently, a new theoretical
calculation suggested a lower limit of $m_{H^\pm} >
800\,\, \mathrm{GeV}$~\cite{Misiak:2020vlo}, which
however is still
under debate in particular in view of the results
of~\citere{Bernlochner:2020jlt} pointing to possible underestimates
of theoretical uncertainties
that could have the effect of even weakening the
bound on $m_{H^\pm}$ below the value that we have adopted in our scan.
In any case it is obvious that new results concerning flavor
physics observables can have an
important impact on the investigated parameter space.
Regarding the compatibility of the
predicted signal rates of $h_{125}$ with
the experimentally measured values, our
results indicate that projected HL-LHC
measurements of the properties of
$h_{125}$ alone might not be sufficient
to exclude (or confirm) the scenarios
describing the excesses at $400\,\, \mathrm{GeV}$.
The situation is somewhat more promising
when additionally the presence of a
singlet-like Higgs boson at $96\,\, \mathrm{GeV}$
is considered. In this case, the mixing
of the singlet-like Higgs boson with
$h_{125}$, as required to simultaneously
accommodate the LEP and CMS excesses, gives rise
to modifications of the signal rates
of $h_{125}$ compared to the SM
predictions that could be observable
at the HL-LHC.
The NMSSM is additionally constrained
by experimental
searches targeted specifically to the
SUSY particles. For the scenario in
the alignment-without-decoupling limit,
it was found that the stops cannot be
much heavier than $1\,\, \mathrm{TeV}$ in order to
obtain a theoretical prediction for
the mass of $h_{125}$ in agreement
with experiment.
The HL-LHC has a high sensitivity for probing this
mass region.
Another possibility
in this context are searches for the
light electroweakinos, where in
particular searches focusing on
compressed electroweakino spectra
are promising. More exotic signals
that are present in our analysis arise
from the decays of the charged Higgs bosons
into pairs of a neutralino and a chargino,
which as mentioned above could be probed via dedicated searches.
Finally, a small fraction of the analyzed
parameter points feature a neutralino
with masses smaller than $62.5\,\, \mathrm{GeV}$, such that
the decay of $h_{125}$ into a pair of
neutralinos is kinematically allowed.
Such points will be further probed by
direct searches for
invisible decays of $h_{125}$ and by the
global constraints from the signal-rate
measurements of $h_{125}$ at the
HL-LHC.
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1,314,259,992,984 | arxiv | \section{Introduction and Summary}
{}Every time we can learn something new about cancer, the motivation goes without saying. Cancer is different. Unlike other diseases, it is not caused by ``mechanical" breakdowns, biochemical imbalances, etc. Instead, cancer occurs at the DNA level via somatic alterations in the genome structure. A common type of somatic mutations found in cancer is due to single nucleotide variations (SNVs) or alterations to single bases in the genome, which accumulate through the lifespan of the cancer via imperfect DNA replication during cell division or spontaneous cytosine deamination \cite{Goodman}, \cite{Lindahl}, or due to exposures to chemical insults or ultraviolet radiation \cite{Loeb}, \cite{Ananthaswamy}, etc. These mutational processes leave a footprint in the cancer genome characterized by distinctive alteration patterns or mutational signatures.
{}If we can identify all underlying signatures, this could greatly facilitate progress in understanding the origins of cancer and its development. Therapeutically, if there are common underlying structures across different cancer types, then a therapeutic for one cancer type might be applicable to other cancers, which would be a great news.\footnote{\, Another practical application is prevention by pairing the signatures extracted from cancer samples with those caused by known carcinogens (e.g., tobacco, aflatoxin, UV radiation, etc).} However, it all boils down to the question of usefulness, i.e., is there a small enough number of cancer signatures underlying all (100+) known cancer types, or is this number too large to be meaningful or useful? Indeed, there are only 96 SNVs,\footnote{\, In brief, DNA is a double helix of two strands, and each strand is a string of letters A, C, G, T corresponding to adenine, cytosine, guanine and thymine, respectively. In the double helix, A in one strand always binds with T in the other, and G always binds with C. This is known as base complementarity. Thus, there are six possible base mutations C $>$ A, C $>$ G, C $>$ T, T $>$ A, T $>$ C, T $>$ G, whereas the other six base mutations are equivalent to these by base complementarity. Each of these 6 possible base mutations is flanked by 4 possible bases on each side thereby producing $4 \times 6 \times 4 = 96$ distinct mutation categories.} so we cannot have more than 96 signatures.\footnote{\, Nonlinearities could undermine this argument. However, again, it all boils down to usefulness.} Even if the number of true underlying signatures is, say, of order 50, it is unclear whether they would be useful, especially within practical applications. On the other hand, if there are only a dozen or so underlying signatures, then we could hope for an order of magnitude simplification.
{}To identify mutational signatures, one analyzes SNV patterns in a cohort of DNA sequenced whole cancer genomes. The data is organized into a matrix $G_{is}$, where the rows correspond to the $N=96$ mutation categories, the columns correspond to $d$ samples, and each element is a nonnegative occurrence count of a given mutation category in a given sample. Currently, the commonly accepted method for extracting cancer signatures from $G_{is}$ \cite{Alexandrov.NMF} is via nonnegative matrix factorization (NMF) \cite{Paatero}, \cite{LeeSeung}. Under NMF the matrix $G$ is approximated via $G \approx W~H$, where $W_{iA}$ is an $N\times K$ matrix, $H_{As}$ is a $K\times d$ matrix, and both $W$ and $H$ are nonnegative. The appeal of NMF is its biologic interpretation whereby the $K$ columns of the matrix $W$ are interpreted as the weights with which the $K$ cancer signatures contribute into the $N=96$ mutation categories, and the columns of the matrix $H$ are interpreted as the exposures to the $K$ signatures in each sample. The price to pay for this is that NMF, which is an iterative procedure, is computationally costly and depending on the number of samples $d$ it can take days or even weeks to run it. Furthermore, it does not automatically fix the number of signatures $K$, which must be either guessed or obtained via trial and error, thereby further adding to the computational cost.\footnote{\, Other issues include: i) out-of-sample instability, i.e., the signatures obtained from non-overlapping sets of samples can be dramatically different; ii) in-sample instability, i.e., the signatures can have a strong dependence on the initial iteration choice; and iii) samples with low counts or sparsely populated samples (i.e., those with many zeros -- such samples are ubiquitous, e.g., in exome data) are usually deemed not too useful as they contribute to the in-sample instability.}
{}Some of the aforesaid issues were recently addressed in \cite{BioFM}, to wit: i) by aggregating samples by cancer types, we can greatly improve stability and reduce the number of signatures;\footnote{\, As a result, now we have the so-aggregated matrix $G_{is}$, where $s=1,\dots,d$, and $d=n$ is the number of cancer types, not of samples. This matrix is much less noisy than the sample data.} ii) by identifying and factoring out the somatic mutational noise, or the ``overall" mode (this is the ``de-noising" procedure of \cite{BioFM}), we can further greatly improve stability and, as a bonus, reduce computational cost; and iii) the number of signatures can be fixed borrowing the methods from statistical risk models \cite{StatRM} in quantitative finance, by computing the effective rank (or eRank) \cite{RV} for the correlation matrix $\Psi_{ij}$ calculated across cancer types or samples (see below). All this yields substantial improvements \cite{BioFM}.
{}In this paper we push this program to yet another level. The basic idea here is quite simple (but, as it turns out, nontrivial to implement -- see below). We wish to apply clustering techniques to the problem of extracting cancer signatures. In fact, we argue in Section \ref{sec.2} that NMF is, to a degree, ``clustering in disguise". This is for two main reasons. The prosaic reason is that NMF, being a nondeterministic algorithm, requires averaging over many local optima it produces. However, each run generally produces a weights matrix $W_{iA}$ with columns (i.e., signatures) not aligned with those in other runs. Aligning or matching the signatures across different runs (before averaging over them) is typically achieved via nondeterministic clustering such as k-means. So, not only is clustering utilized at some layer, the result, even after averaging, generally is both noisy\footnote{\, By ``noise" we mean the statistical errors in the weighs obtained by averaging. Typically, such error bars are not reported in the literature on cancer signatures. Usually they are large.} and nondeterministic! I.e., if this computationally costly procedure (which includes averaging) is run again and again on the same data, generally it will yield different looking cancer signatures every time!
{}The second, not-so-prosaic reason is that, while NMF generically does not produce exactly null weights, it does produce low weights, such that they are within error bars. For all practical purposes we might as well set such weights to zero. NMF requires nonnegative weights. However, we could as reasonably require that the weights should be, say, outside error bars (e.g., above one standard deviation -- this would render the algorithm highly recursive and potentially unstable or computationally too costly) or above some minimum threshold (which would still further complicated as-is complicated NMF), or else the non-compliant weights are set to zero. As we increase this minimum threshold, the matrix $W_{iA}$ will start to have more and more zeros. It may not exactly have a binary cluster-like structure, but it may at least have some substructures that are cluster-like. It then begs the question: are there cluster-like (sub)structures present in $W_{iA}$ or, generally, in cancer signatures?
{}To answer this question, we can apply clustering methods directly to the matrix $G_{is}$, or, more, precisely, to its de-noised version $G^\prime_{is}$ (see below) \cite{BioFM}. The na\"ive, brute-force approach where one would simply cluster $G_{is}$ or $G^\prime_{is}$ does not work for a variety of reasons, some being more nontrivial or subtle than others. Thus, e.g., as discussed in \cite{BioFM}, the counts $G_{is}$ have skewed, long-tailed distributions and one should work with log-counts, or, more precisely, their de-noised versions. This applies to clustering as well. Further, following a discussion in \cite{StatIndClass} in the context of quantitative trading, it would be suboptimal to cluster de-noised log-counts. Instead, it pays to cluster their normalized variants (see Section \ref{sec.2} hereof). However, taking care of such subtleties does not alleviate one big problem: nondeterminism!\footnote{\, Deterministic (e.g., agglomerative hierarchical) algorithms have their own issues (see below).} If we run a vanilla nondeterministic algorithm such as k-means on the data however massaged with whatever bells and whistles, we will get random-looking disparate results every time we run k-means with no stability in sight. We need to address nondeterminism!
{}Our solution to the problem is what we term {\em *K-means}. The idea behind *K-means, which essentially achieves determinism {\em statistically}, is simple. Suppose we have an $N\times d$ matrix $X_{is}$, i.e., we have $N$ $d$-vectors ${\bf X}_i$. If we run k-means with the input number of clusters $K$ but initially unspecified centers, every run will generally produce a new local optimum. *K-means reduces and in fact essentially eliminates this indeterminism via two levels. At level 1 it takes clusterings obtained via $M$ independent runs or samplings. Each sampling produces a binary $N\times K$ matrix $\Omega_{iA}$, whose element equals 1 if ${\bf X}_i$ belongs to the cluster labeled by $A$, and 0 otherwise. The aggregation algorithm and the source code therefor are given in \cite{StatIndClass}. This aggregation -- for the same reasons as in NMF (see above) -- involves aligning clusters across the $M$ runs, which is achieved via k-means, and so the result is nondeterministic. However, by aggregating a large number $M$ of samplings, the degree of nondeterminism is greatly reduced. The ``catch" is that sometimes this aggregation yields a clustering with $K^\prime < K$ clusters, but this does not pose an issue. Thus, at level 2, we take a large number $P$ of such aggregations (each based on $M$ samplings). The occurrence counts of aggregated clusterings are not uniform but typically have a (sharply) peaked distribution around a few (or manageable) number of aggregated clusterings. So this way we can pinpoint the ``ultimate" clustering, which is simply the aggregated clustering with the highest occurrence count. This is the gist of *K-means and it works well for genome data.
{}So, we apply *K-mean to the same genome data as in \cite{BioFM} consisting of 1,389 (published) samples across 14 cancer types (see below). Our target number of clusters is 7, which was obtained in \cite{BioFM} using the eRank based algorithm (see above). We aggregated 1,000 samplings into clusterings, and we constructed 150,000 such aggregated clusterings (i.e., we ran 150 million k-means instances). We indeed found the ``ultimate" clustering with 7 clusters. Once the clustering is fixed, it turns out that within-cluster weights can be computed via linear regressions (with some bells and whistles) and the weights are automatically positive. That is, we do not need NMF at all! Once we have clusters and weights, we can study reconstruction accuracy and within-cluster correlations between the underlying data and the fitted data that the cluster model produces.
{}We find that clustering works well for 10 out the 14 cancer types we study. The cancer types for which clustering does not appear to work all that well are Liver Cancer, Lung Cancer, and Renal Cell Carcinoma. Also, above 80\% within-cluster correlations arise for 5 out of 7 clusters. Furthermore, remarkably, one cluster has high within-cluster correlations for 9 cancer types, and another cluster for 6 cancer types. These appear to be the leading clusters. Together they have high within-cluster correlations in 11 out of 14 cancer types. So what does all this mean?
{}Additional insight is provided by looking at the within-cluster correlations between signatures Sig1 through Sig7 extracted in \cite{BioFM} and our clusters. High within-cluster correlations arise for Sig1, Sig2, Sig4 and Sig7, which are precisely the signatures with ``peaks" (or ``spikes" -- ``tall mountain landscapes"), whereas Sig3, Sig5 and Sig6 do not have such ``peaks" (``flat" or ``rolling hills landscapes"); see Figures 14 through 20 of \cite{BioFM}.
The latter 3 signatures simply do not have cluster-like structures. Looking at Figure 21 in \cite{BioFM}, it becomes evident why clustering does not work well for Liver Cancer -- it has a whopping 96\% contribution from Sig5! Similarly, Renal Cell Carcinoma has a 70\% contribution from Sig6. Lung Cancer is dominated by Sig3, hence no cluster-like structure. So, Liver Cancer, Lung Cancer and Renal Cell Carcinoma have little in common with other cancers (and each other)! However, 11 other cancers, to wit, B Cell Lymphoma, Bone Cancer, Brain Lower Grade Glioma, Breast Cancer, Chronic Lymphocytic Leukemia, Esophageal Cancer, Gastric Cancer, Medulloblastoma, Ovarian Cancer, Pancreatic Cancer and Prostate Cancer, have 5 (with 2 leading) cluster structures substantially embedded in them.
{}In Section \ref{sec.2} we i) discuss why applying clustering algorithms to extracting cancer signatures makes sense, ii) argue that NMF, to a degree, is ``clustering in disguise", and iii) give the machinery for building cluster models via *K-means, including various details such as what to cluster, how to fix the number of clusters, etc. In Section \ref{sec.3} we discuss i) cancer genome data we use, ii) our application of *K-means to it, and iii) the interpretation of our empirical results. Section \ref{sec.4} contains some concluding remarks, including a discussion of potential applications of *K-means in quantitative finance, where we outline some concrete problems where *K-means can be useful. Appendix \ref{app.code} contains R source code for *K-means and cluster models.
\newpage
\section{Cluster Models}\label{sec.2}
{}The chief objective of this paper is to introduce a novel approach to identifying cancer signatures using clustering methods. In fact, as we discuss below in detail, our approach is more than just clustering. Indeed, it is evident from the get-go that blindly using nondeterministic clustering algorithms,\footnote{\, Such as k-means \cite{Steinhaus}, \cite{Lloyd1957}, \cite{Forgy}, \cite{MacQueen}, \cite{Hartigan}, \cite{HartWong}, \cite{Lloyd1982}.} which typically produce (unmanageably) large numbers of local optima, would introduce great variability into the resultant cancer signatures.\footnote{\, As we discuss below, in this regard NMF is not dissimilar.} On the other hand, deterministic algorithms such as agglomerative hierarchical clustering\footnote{\, E.g., SLINK \cite{SLINK}, etc. (see, e.g., \cite{HAC}, \cite{StatIndClass}, and references therein).} typically are (substantially) slower and require essentially ``guessing" the initial clustering,\footnote{\, E.g., splitting the data into 2 initial clusters.} which in practical applications\footnote{\, Such as quantitative trading, where out-of-sample performance can be objectively measured. There empirical evidence suggests that such deterministic algorithms underperform so long as nondeterministic ones are used thoughtfully \cite{StatIndClass}.} can often turn out to be suboptimal. So, both to motivate and explain our new approach employing clustering methods, we first -- so to speak -- ``break down" the NMF approach and argue that it is in fact a clustering method in disguise!
\subsection{``Breaking Down" NMF}
{}The current ``lore" -- the commonly accepted method for extracting $K$ cancer signatures from the occurrence counts matrix $G_{is}$ (see above) \cite{Alexandrov.NMF} -- is via nonnegative matrix factorization (NMF) \cite{Paatero}, \cite{LeeSeung}. Under NMF the matrix $G$ is approximated via $G \approx W~H$, where $W_{iA}$ is an $N\times K$ matrix of weights, $H_{As}$ is a $K\times d$ matrix of exposures, and both $W$ and $H$ are nonnegative. However, not only is the number of signatures $K$ not fixed via NMF (and must be either guessed or obtained via trial and error), NMF too is a nondeterministic algorithm and typically produces a large number of local optima. So, in practice one has no choice but to execute a large number $N_S$ of NMF runs -- which we refer to as samplings -- and then somehow extract cancer signatures from these samplings. Absent a guess for what $K$ should be, one executes $N_S$ samplings for a range of values of $K$ (say, $K_{\min} \leq K \leq K_{\max}$, where $K_{min}$ and $K_{max}$ are basically guessed based on some reasonable intuitive considerations), for each $K$ extracts cancer signatures (see below), and then picks $K$ and the corresponding signatures with the best overall fit into the underlying matrix $G$. For a given $K$, different samplings generally produce different weights matrices $W$. So, to extract a single matrix $W$ for each value of $K$ one {\em averages} over the samplings. However, before averaging, one must match the $K$ cancer signatures across different samplings -- indeed, in a given sampling X the columns in the matrix $W_{iA}$ are not necessarily aligned with the columns in the matrix $W_{iA}$ in a different sampling Y. To align the columns in the matrices $W$ across the $N_S$ samplings, once often uses a clustering algorithm such as k-means. However, since k-means is nondeterministic, such alignment of the $W$ columns is not guaranteed to -- and in fact does not -- produce a unique answer. Here one can try to run multiple samplings of k-means for this alignment and aggregate them, albeit such aggregation itself would require another level of alignment (with its own nondeterministic clustering such as k-means).\footnote{\, We should point out that at some level of alignment one may employ a deterministic (e.g., agglomerative hierarchical -- see above) clustering algorithm to terminate the malicious circle, which can be a reasonable approach assuming there is enough stability in the data. However, this too adds a(n often hard to quantify and therefore hidden) systematic error to the resultant signatures.\label{fn.alignment}} And one can do this {\em ad infinitum}. In practice, one must break the chain at some level of alignment, either {\em ad hoc} (essentially by heuristically observing sufficient stability and ``convergence") or via using a deterministic algorithm (see fn. \ref{fn.alignment}). Either way, invariably all this introduces (overtly or covertly) systematic and statistical errors into the resultant cancer signatures and often it is unclear if they are meaningful without invoking some kind empirical biologic ``experience" or ``intuition" (often based on already well-known effects of, e.g., exposure to various well-understood carcinogens such as tobacco, ultraviolet radiation, aflatoxin, etc.). At the end of the day it all boils down to how useful -- or {\em predictive} -- the resultant method of extracting cancer signatures is, including signature stability. With NMF, the answer is not at all evident...
\subsection{Clustering in Disguise?}
{}So, in practice, under the hood, NMF already uses clustering methods. However, it goes deeper than that. While NMF generically does not produce vanishing weights for a given signature, some weights are (much) smaller than others. E.g., often one has several ``peaks" with high concentration of weights, with the rest of the mutation categories having relatively low weights. In fact, many weights can even be within the (statistical plus systematic) error bars.\footnote{\, And such error bars are rarely displayed in the prevalent literature...} Such weights can for all practical purposes be set to zero. In fact, we can take this further and ask whether proliferation of low weights adds any explanatory power. One way to address this is to run NMF with an additional constraint that the weights (obtained via averaging -- see above) should be higher than either i) some multiple of the corresponding error bars\footnote{\, This would require a highly recursive algorithm.} or ii) some preset fixed minimum weight. This certainly sounds reasonable, so why is this not done in practice? A prosaic answer appears to be that this would complicate the already nontrivial NMF algorithm even further, require additional coding and computation resources, etc. However, {\em arguendo}, let us assume that we require, say, that the weights be higher than a preset fixed minimum weight $w_{min}$ or else the weights are set to zero. As we increase $w_{min}$, the so-modified NMF would produce more and more zeros. This does not mean that the resulting matrix $W_{iA}$ would have a {\em binary} cluster structure, i.e., that $W_{iA} = w_i~\delta_{G(i), A}$, where $\delta_{AB}$ is a Kronecker delta and $G:\{1,\dots,N\}\mapsto\{1,\dots,K\}$ is a map from $N=96$ mutation categories to $K$ clusters. Put another way, this does not mean that in the resulting matrix $W_{iA}$ for a given $i$ (i.e., mutation category) we would have a nonzero element for one and only one value of $A$ (i.e., signature). However, as we gradually increase $w_{min}$, generally the matrix $W_{iA}$ is expected to look more and more like having a binary cluster structure, albeit with some ``overlapping" signatures (i.e., such that in a given pair of signatures there are nonzero weights for one or more mutations). We can achieve a binary structure via a number of ways. Thus, a rudimentary algorithm would be to take the matrix $W_{iA}$ (equally successfully before or after achieving some zeros in it via nonzero $w_{min}$) and for a given value of $i$ set all weights $W_{iA}$ to zero except in the signature $A$ for which $W_{iA} = \mbox{max}(W_{iA} | A = 1,\dots,K)$. Note that this might result in some empty signatures (clusters), i.e., signatures with $W_{iA} = 0$ for all values of $i$. This can be dealt with by i) ether simply dropping such signatures altogether and having fewer $K^\prime < K$ signatures (binary clusters) at the end, or ii) augmenting the algorithm to avoid empty clusters, which can be done in a number of ways we will not delve into here. The bottom line is that NMF essentially can be made into a clustering algorithm by reasonably modifying it, including via getting rid of ubiquitous and not-too-informative low weights. However, the downside would be an even more contrived algorithm, so this is not what we are suggesting here. Instead, we are observing that clustering is already intertwined in NMF and the question is whether we can simplify things by employing clustering methods directly.
\subsection{Making Clustering Work}
{}Happily, the answer is yes. Not only can we have much simpler and apparently more stable clustering algorithms, but they are also computationally much less costly than NMF. As mentioned above, the biggest issue with using popular nondeterministic clustering algorithms such as k-means\footnote{\, Which are preferred over deterministic ones for the reasons discussed above.} is that they produce a large number of local optima. For definiteness in the remainder of this paper we will focus on k-means, albeit the methods described herein are general and can be applied to other such algorithms. Fortunately, this very issue has already been addressed in \cite{StatIndClass} in the context of constructing statistical industry classifications (i.e., clustering models for stocks) for quantitative trading, so here we simply borrow therefrom and further expand and adapt that approach to cancer signatures.
\subsubsection{K-means}
{}A popular clustering algorithm is k-means \cite{Steinhaus}, \cite{Lloyd1957}, \cite{Forgy}, \cite{MacQueen}, \cite{Hartigan}, \cite{HartWong}, \cite{Lloyd1982}. The basic idea behind k-means is to partition $N$ observations into $K$ clusters such that each observation belongs to the cluster with the nearest mean. Each of the $N$ observations is actually a $d$-vector, so we have an $N \times d$ matrix $X_{is}$, $i=1,\dots,N$, $s=1,\dots,d$. Let $C_a$ be the $K$ clusters, $C_a = \{i| i\in C_a\}$, $a=1,\dots,K$. Then k-means attempts to minimize\footnote{\, Below we will discuss what $X_{is}$ should be for cancer signatures.}
\begin{equation}\label{k-means}
g = \sum_{a=1}^K \sum_{i \in C_a} \sum_{s=1}^d \left(X_{is} - Y_{as}\right)^2
\end{equation}
where
\begin{equation}\label{centers}
Y_{as} = {1\over n_a} \sum_{i\in C_a} X_{is}
\end{equation}
are the cluster centers (i.e., cross-sectional means),\footnote{\, Throughout this paper ``cross-sectional" refers to ``over the index $i$".} and $n_a = |C_a|$ is the number of elements in the cluster $C_a$. In (\ref{k-means}) the measure of ``closeness" is chosen to be the Euclidean distance between points in ${\bf R}^d$, albeit other measures are possible.
{}One ``drawback" of k-means is that it is not a deterministic algorithm. Generically, there are copious local minima of $g$ in (\ref{k-means}) and the algorithm only guarantees that it will converge to a local minimum, not the global one. Being an iterative algorithm, unless the initial centers are preset, k-means starts with a random set of the centers $Y_{as}$ at the initial iteration and converges to a different local minimum in each run. There is no magic bullet here: in practical applications, typically, trying to ``guess" the initial centers is not any easier than ``guessing" where, e.g., the global minimum is. So, what is one to do? One possibility is to simply live with the fact that every run produces a different answer. In fact, this is acceptable in many applications. However, in the context of extracting cancer signatures this would result in an exercise in futility. We need a way to eliminate or greatly reduce indeterminism.
\subsubsection{Aggregating Clusterings}\label{sub.aggr}
{}The idea is simple. What if we {\em aggregate} different clusterings from multiple runs -- which we refer to as samplings -- into one? The question is how. Suppose we have $M$ runs ($M \gg 1$). Each run produces a clustering with $K$ clusters. Let $\Omega^r_{ia} = \delta_{G^r(i),a}$, $i=1,\dots,N$, $a=1,\dots,K$ (here $G^r:\{1,\dots,N\} \mapsto \{1,\dots,K\}$ is the map between -- in our case -- the mutation categories and the clusters),\footnote{\, Note that here the superscript $r$ in $\Omega^r_{ia}$, $G^r(i)$ and $N^r_a$ (see below) is an index, not a power.} be the binary matrix from each run labeled by $r=1,\dots,M$, which is a convenient way (for our purposes here) of encoding the information about the corresponding clustering; thus, each row of $\Omega^r_{ia}$ contains only one element equal 1 (others are zero), and $N^r_a = \sum_{i=1}^N \Omega^r_{ia}$ (i.e., column sums) is nothing but the number of mutations belonging to the cluster labeled by $a$ (note that $\sum_{a=1}^K N^r_a = N$). Here we are assuming that somehow we know how to properly order (i.e., align) the $K$ clusters from each run. This is a nontrivial assumption, which we will come back to momentarily. However, assuming, for a second, that we know how to do this, we can aggregate the binary matrices $\Omega^r_{ia}$ into a single matrix ${\widetilde \Omega}_{ia} = \sum_{r=1}^M \Omega^r_{ia}$. Now, this matrix does not look like a binary clustering matrix. Instead, it is a matrix of occurrence counts, i.e., it counts how many times a given mutation was assigned to a given cluster in the process of $M$ samplings. What we need to construct is a map $G$ such that one and only one mutation belongs to each of the $K$ clusters. The simplest criterion is to map a given mutation to the cluster in which ${\widetilde\Omega}_{ia}$ is maximal, i.e., where said mutation occurs most frequently. A caveat is that there may be more than one such clusters. A simple criterion to resolve such an ambiguity is to assign said mutation to the cluster with most cumulative occurrences (i.e., we assign said mutation to the cluster with the largest ${\widetilde N}_a = \sum_{i=1}^N {\widetilde\Omega}_{ia}$). Further, in the unlikely event that there is still an ambiguity, we can try to do more complicated things, or we can simply assign such a mutation to the cluster with the lowest value of the index $a$ -- typically, there is so much noise in the system that dwelling on such minutiae simply does not pay off.
{}However, we still need to tie up a loose end, to wit, our assumption that the clusters from different runs were somehow all aligned. In practice each run produces $K$ clusters, but i) they are not the same clusters and there is no foolproof way of mapping them, especially when we have a large number of runs; and ii) even if the clusters were the same or similar, they would not be ordered, i.e., the clusters from one run generally would be in a different order than the clusters from another run.
{}So, we need a way to ``match" clusters from different samplings. Again, there is no magic bullet here either. We can do a lot of complicated and contrived things with not much to show for it at the end. A simple pragmatic solution is to use k-means to align the clusters from different runs. Each run labeled by $r=1,\dots,M$, among other things, produces a set of cluster centers $Y^r_{as}$. We can ``bootstrap" them by row into a $(KM) \times d$ matrix ${\widetilde Y}_{{\widetilde a}s} = Y^r_{as}$, where ${\widetilde a} = a + (r - 1)K$ takes values ${\widetilde a}=1,\dots,(KM)$. We can now cluster ${\widetilde Y}_{{\widetilde a}s}$ into $K$ clusters via k-means. This will map each value of ${\widetilde a}$ to $\{1,\dots,K\}$ thereby mapping the $K$ clusters from each of the $M$ runs to $\{1,\dots,K\}$. So, this way we can align all clusters. The ``catch" is that there is no guarantee that each of the $K$ clusters from each of the $M$ runs will be uniquely mapped to one value in $\{1,\dots,K\}$, i.e., we may have some empty clusters at the end of the day. However, this is fine, we can simply drop such empty clusters and aggregate (via the above procedure) the smaller number of $K^\prime < K$ clusters. I.e., at the end we will end up with a clustering with $K^\prime$ clusters, which might be fewer than the target number of clusters $K$. This is not necessarily a bad thing. The dropped clusters might have been redundant in the first place. Another evident ``catch" is that even the number of resulting clusters $K^\prime$ is not deterministic. If we run this algorithm multiple times, we will get varying values of $K^\prime$. Malicious circle?
\subsubsection{Fixing the ``Ultimate" Clustering}\label{sub.ultimate}
{}Not really! There is one other trick up our sleeves we can use to fix the ``ultimate" clustering thereby rendering our approach essentially deterministic. The idea above is to aggregate a large enough number $M$ of samplings. Each aggregation produces a clustering with some $K^\prime\leq K$ clusters, and this $K^\prime$ varies from aggregation to aggregation. However, what if we take a large number $P$ of aggregations (each based on $M$ samplings)? Typically there will be a relatively large number of different clusterings we get this way. However, assuming some degree of stability in the data, this number is much smaller than the number of {\em a priori} different local minima we would obtain by running the vanilla k-means algorithm. What is even better, the occurrence counts of aggregated clusterings are not uniform but typically have a (sharply) peaked distribution around a few (or manageable) number of aggregated clusterings. In fact, as we will see below, in our empirical genome data we are able to pinpoint the ``ultimate" clustering! So, to recap, what we have done here is this. There are myriad clusterings we can get via vanilla k-means with little to no guidance as to which one to pick.\footnote{\, This is because things are pretty much random and the only ``distribution" at hand is flat.} We have reduced this proliferation by aggregating a large number of such clusterings into our aggregated clusterings. We then further zoom onto a few or even a unique clustering we consider to be the likely ``ultimate" clustering by examining the occurrence counts of such aggregated clusterings, which turns out to have a (sharply) peaked distribution. Since vanilla k-means is a relatively fast-converging algorithm, each aggregation is not computationally taxing and running a large number of aggregations is nowhere as time consuming as running a similar number (or even a fraction thereof) of NMF computations (see below).
\subsection{What to Cluster?}
{}So, now that we know how to make clustering work, we need to decide what to cluster, i.e., what to take as our matrix $X_{is}$ in (\ref{k-means}). The na\"{\i}ve choice $X_{is} = G_{is}$ is suboptimal for multiple reasons (as discussed in \cite{BioFM}).
{}First, the elements of the matrix $G_{is}$ are populated by nonnegative occurrence counts. Nonnegative quantities with large numbers of samples tend to have skewed distributions with long tails at higher values. I.e., such distributions are not normal but (in many cases) roughly log-normal. One simple way to deal with this is to identify $X_{is}$ with a (natural) logarithm of $G_{is}$ (instead of $G_{is}$ itself). A minor hiccup here is that some elements of $G_{is}$ can be 0. We can do a lot of complicated and even convoluted things to deal with this issue. Here, as in \cite{BioFM}, we will follow a pragmatic approach and do something simple instead -- there is so much noise in the data that doing convoluted things simply does not pay off. So, as the first cut, we can take
\begin{equation}\label{log.def}
X_{is} = \ln\left(1 + G_{is}\right)
\end{equation}
This takes care of the $G_{is} = 0$ cases; for $G_{is}\gg 1$ we have $R_{is}\approx\ln(G_{is})$, as desired.
{}Second, the detailed empirical analysis of \cite{BioFM} uncovered what is termed therein the ``overall" mode\footnote{\, In finance the analog of this is the so-called ``market" mode (see, e.g., \cite{CFM} and references therein) corresponding to the overall movement of the broad market, which affects all stocks (to varying degrees) -- cash inflow (outflow) into (from) the market tends to push stock prices higher (lower). This is the market risk factor, and to mitigate it one can, e.g., hold a dollar-neutral portfolio of stocks (i.e., the same dollar holdings for long and short positions).} unequivocally present in the occurrence count data. This ``overall" mode is interpreted as somatic mutational {\em noise} unrelated to (and in fact obscuring) the true underlying cancer signatures and must therefore be factored out somehow. Here is a simple way to understand the ``overall" mode. Let the correlation matrix $\Psi_{ij} = \mbox{Cor}(X_{is}, X_{js})$, where $\mbox{Cor}(\cdot,\cdot)$ is serial correlation.\footnote{\, Throughout this paper ``serial" refers to ``over the index s".} I.e., $\Psi_{ij} = C_{ij}/\sigma_i\sigma_j$, where $\sigma_i^2 = C_{ii}$ are variances, and the serial covariance matrix\footnote{\, The overall normalization of $C_{ij}$, i.e., $d-1$ (unbiased estimate) vs. $d$ (maximum likelihood estimate) in the denominator in the definition of $C_{ij}$ in (\ref{cov}), is immaterial for our purposes here.}
\begin{equation} \label{cov}
C_{ij} = \mbox{Cov}(X_{is}, X_{js}) = {1\over {d-1}}\sum_{s=1}^d Z_{is}~Z_{js}
\end{equation}
where $Z_{is} = X_{is} - {\overline X}_i$ are serially demeaned, while the means ${\overline X}_i = {1\over d}\sum_{s=1}^d X_{is}$. The average pair-wise correlation $\rho = {1\over N(N-1)}\sum_{i,j=1;~i\neq j}^N\Psi_{ij}$ between different mutation categories is nonzero and is in fact high for most cancer types we study. This is the aforementioned somatic mutational noise that must be factored out. If we aggregate samples by cancer types (see below) and compute the correlation matrix $\Psi_{ij}$ for the so-aggregated data (across the $n = 14$ cancer types we study -- see below),\footnote{\, So, in this case $d = n = 14$ in (\ref{cov}).} the average correlation $\rho$ is over whopping 96\%. Another way of thinking about this is that the occurrence counts in different samples (or cancer types, if we aggregate samples by cancer types) are not normalized uniformly across all samples (cancer types). Therefore, running NMF, a clustering or any other signature-extraction algorithm on the vanilla matrix $G_{is}$ (or its ``log" $X_{is}$ defined in (\ref{log.def})) would amount to mixing apples with oranges thereby obscuring the true underlying cancer signatures.
{}Following \cite{BioFM}, factoring out the ``overall" mode (or ``de-noising" the matrix $G_{is}$) therefore most simply amount to cross-sectional (i.e., across the 96 mutation categories) demeaning of the matrix $X_{is}$. I.e., instead of $X_{is}$ we use $X^\prime_{is}$, which is obtained from $X_{is}$ by demeaning its columns:\footnote{\, For the reasons discussed above, we should demean $X_{is}$, not $G_{is}$.}
\begin{equation}\label{Xprime}
X^\prime_{is} = X_{is} - {\overline X}_s = X_{is} - {1\over N} \sum_{j=1}^N X_{js}
\end{equation}
We should note that using $X^\prime_{is}$ instead of $X_{is}$ in (\ref{k-means}) does not affect clustering. Indeed, $g$ in (\ref{k-means}) is invariant under the transformations of the form $X_{is} \rightarrow X_{is} + \Delta_s$, where $\Delta_s$ is an arbitrary $d$-vector, as thereunder we also have $Y_{as}\rightarrow Y_{as} + \Delta_s$, so $X_{is} - Y_{as}$ is unchanged. In fact, this is good: this means that de-noising does not introduce any additional errors into clustering itself. However, the actual {\em weights} in the matrix $W_{iA}$ are affected by de-noising. We discuss the algorithm for fixing $W_{iA}$ below. However, we need one more ingredient before we get to determining the weights, and with this additional ingredient de-noising does affect clustering.
\subsubsection{Normalizing Log-counts}\label{sub.norm.log.counts}
{}As was discussed in \cite{StatIndClass}, clustering $X_{is}$ (or equivalently $X^\prime_{is}$) would be suboptimal.\footnote{\, More precisely, the discussion of \cite{StatIndClass} is in the financial context, to wit, quantitative trading, which has its own nuances (see below). However, some of that discussion is quite general and can be adapted to a wide variety of applications.} The issue is this. Let $\sigma^\prime_i$ be serial standard deviations, i.e., $(\sigma_i^\prime)^2 = \mbox{Cov}(X^\prime_{is}, X^\prime_{is})$, where, as above, $\mbox{Cov}(\cdot,\cdot)$ is serial covariance. Here we assume that samples are aggregated by cancer types, so $s=1,\dots,d$ with $d = n = 14$. Now, $\sigma^\prime_i$ are not cross-sectionally uniform and vary substantially across mutation categories. The density of $\sigma_i^\prime$ is depicted in Figure \ref{FigureVolDensity} and is skewed (tailed). The summary of $\sigma_i^\prime$ reads:\footnote{\, Qu. = Quartile, SD = Standard Deviation, MAD = Mean Absolute Deviation.} Min = 0.2196, 1st Qu. = 0.3409, Median = 0.4596, Mean = 0.4984, 3rd Qu. = 0.6060, Max = 1.0010, SD = 0.1917, MAD = 0.1859, Skewness = 0.8498. If we simply cluster $X^\prime_{is}$, this variability in $\sigma_i^\prime$ will not be accounted for.
{}A simple solution is to cluster normalized demeaned log-counts ${\widetilde X}^\prime_{is} = X^\prime_{is}/\sigma_i^\prime$ instead of $X^\prime_{is}$. This way we factor out the nonuniform (and skewed) standard deviation out of the log-counts. Note that now de-noising does make a difference in clustering. Indeed, if we use ${\widetilde X}_{is} = X_{is}/\sigma_i$ (recall that $\sigma_i^2 = \mbox{Cov}(X_{is}, X_{is})$) instead of ${\widetilde X}^\prime_{is} = X^\prime_{is}/\sigma_i^\prime$ in (\ref{k-means}) and (\ref{centers}), the quantity $g$ (and also clusterings) will be different.
\subsection{Fixing Cluster Number}\label{sub.fix.K}
{}Now that we know what to cluster (to wit, ${\widetilde X}^\prime_{is}$) and how to get to the ``unique" clustering, we need to figure out how to fix the (target) number of clusters $K$, which is one of the inputs in our algorithm above.\footnote{\, A variety of methods for fixing the number of clusters have been discussed in other contexts, e.g., \cite{Rousseeuw}, \cite{Pelleg}, \cite{Steinbach}, \cite{Goutte}, \cite{Sugar}, \cite{Hamerly}, \cite{Lleiti}, \cite{DeAmorim}.} In \cite{BioFM} it was argued that in the context of cancer signatures their number can be fixed by building a statistical factor model \cite{StatRM}, i.e., the number of signatures is simply the number of statistical factors.\footnote{\, In the financial context, these are known as statistical risk models \cite{StatRM}. For a discussion and literature on multifactor risk models, see, e.g., \cite{Grinold}, \cite{HetPlus} and references therein. For prior works on fixing the number of statistical risk factors, see, e.g., \cite{Connor} and \cite{Bai}.} So, by the same token, here we identify the (target) number of clusters in our clustering algorithm with the number of statistical factors fixed via the method of \cite{StatRM}.
\subsubsection{Effective Rank}\label{sub.erank}
{}So, following \cite{StatRM} and \cite{BioFM}, we set\footnote{\, Here $\mbox{Round}(\cdot)$ can be replaced by $\mbox{floor}(\cdot) = \lfloor\cdot\rfloor$.}
\begin{equation}\label{eq.eRank}
K = \mbox{Round}(\mbox{eRank}(\Psi))
\end{equation}
Here $\mbox{eRank}(Z)$ is the effective rank \cite{RV} of a symmetric semi-positive-definite (which suffices for our purposes here) matrix $Z$. It is defined as
\begin{eqnarray}
&&\mbox{eRank}(Z) = \exp(H)\\
&&H = -\sum_{a=1}^L p_a~\ln(p_a)\\
&&p_a = {\lambda^{(a)} \over \sum_{b=1}^L \lambda^{(b)}}
\end{eqnarray}
where $\lambda^{(a)}$ are the $L$ {\em positive} eigenvalues of $Z$, and $H$ has the meaning of the (Shannon a.k.a. spectral) entropy \cite{Campbell60}, \cite{YGH}. Let us emphasize that in (\ref{eq.eRank}) the matrix $\Psi_{ij}$ is computed based on the demeaned log-counts\footnote{\, Note that using normalized demeaned log-counts ${\widetilde X}^\prime_{is}$ gives the same $\Psi_{ij}$.} $X^\prime_{is}$.
{}The meaning of $\mbox{eRank}(\Psi_{ij})$ is that it is a measure of the effective dimensionality of the matrix $\Psi_{ij}$, which is not necessarily the same as the number $L$ of its positive eigenvalues, but often is lower. This is due to the fact that many $d$-vectors $X^\prime_{is}$ can be serially highly correlated (which manifests itself by a large gap in the eigenvalues) thereby further reducing the effective dimensionality of the correlation matrix.
\subsection{How to Compute Weights?}\label{sub.reg}
{}The one remaining thing to accomplish is to figure out how to compute the weights $W_{iA}$. Happily, in the context of clustering we have significant simplifications compared with NMF and computing the weights becomes remarkably simple once we fix the clustering, i.e., the matrix $\Omega_{iA} = \delta_{G(i),A}$ (or, equivalently, the map $G:\{i\}\mapsto\{A\}$, $i=1,\dots,N$, $A=1,\dots,K$, where for the notational convenience we use $K$ to denote the number of clusters in the ``ultimate" clustering -- see above). Just as in NMF, we wish to approximate the matrix $G_{is}$ via a product of the weights matrix $W_{iA}$ and the exposure matrix $H_{As}$, both of which must be nonnegative. More precisely, since we must remove the ``overall" mode, i.e., de-noise the matrix $G_{is}$, following \cite{BioFM}, instead of $G_{is}$ we will approximate the re-exponentiated demeaned log-counts matrix $X^\prime_{is}$:
\begin{equation}
G^\prime_{is} = \exp(X^\prime_{is})
\end{equation}
We can include an overall normalization by taking $G^\prime_{is} = \exp(\mbox{Mean}(X_{is}) + X^\prime_{is})$, or $G^\prime_{is} = \exp(\mbox{Median}(X_{is}) + X^\prime_{is})$, or $G^\prime_{is} = \exp(\mbox{Median}({\overline X}_s) + X^\prime_{is})$ (recall that ${\overline X}_s$ is the vector of column means of $X_{is}$ -- see Eq. (\ref{Xprime})), etc., to make it look more like the original matrix $G_{is}$; however, this does not affect the extracted signatures.\footnote{\, This is because each column of $W$, being weights, is normalized to add up to 1.} Also, technically speaking, after re-exponentiating we should ``subtract" the extra 1 we added in the definition (\ref{log.def}) (assuming we include one of the aforesaid overall normalizations). However, the inherent noise in the data makes this a moot point.
{}So, we wish to approximate $G^\prime_{is}$ via a product $W~H$. However, with clustering we have $W_{iA} = w_i~\delta_{G(i),A}$, i.e., we have a block (cluster) structure where for a given value of $A$ all $W_{iA}$ are zero except for $i\in J(A) = \{j|G(j) = A\}$, i.e., for the mutation categories labeled by $i$ that belong to the cluster labeled by $A$. Therefore, our matrix factorization of $G_{is}$ into a product $W~H$ now simplifies into a set of $K$ {\em independent} factorizations as follows:
\begin{equation}
G^\prime_{is} \approx w_i~H_{As},~~~i\in J(A),~~~A=1\dots,K
\end{equation}
So, there is no need to run NMF anymore! Indeed, if we can somehow fix $H_{As}$ for a given cluster, then within this cluster we can determine the corresponding weights $w_i$ ($i\in J(A)$) via a {\em serial} linear regression:
\begin{equation}\label{G.reg}
G^\prime_{is} = \varepsilon_{is} + w_i~H_{As},~~~i\in J(A),~~~A=1\dots,K
\end{equation}
where $\varepsilon_{is}$ are the regression residuals. I.e., for each $A\in\{1,\dots,K\}$, we regress the $d\times n_A$ matrix\footnote{\, The superscript $T$ denotes matrix transposition.} $[(G^\prime)^T]_{si}$ ($i\in J(A)$, $n_A = |J(A)|$) over the $d$-vector $H_{As}$ ($s=1,\dots,d$), and the regression coefficients are nothing but the $n_A$-vector $w_i$ ($i\in J(A)$), while the residuals are the $d\times n_A$ matrix $[(\varepsilon)^T]_{si}$. Note that this regression is run {\em without} the intercept. Now, this all makes sense as (for each $i \in J(A)$) the regression minimizes the quadratic error term $\sum_{s=1}^d \varepsilon^2_{is}$. Furthermore, if $H_{As}$ are nonnegative, then the weights $w_i$ are {\em automatically nonnegative} as they are given by:
\begin{equation}\label{w.reg}
w_i = {{\sum_{s=1}^d G^\prime_{is}~H_{G(i),s}}\over {{\sum_{s=1}^d H^2_{G(i),s}}}}
\end{equation}
Now, we wish these weights to be normalized:
\begin{equation}\label{w.norm}
\sum_{i\in J(A)} w_i = 1
\end{equation}
This can always be achieved by rescaling $H_{As}$. Alternatively, we can pick $H_{As}$ without worrying about the normalization, compute $w_i$ via (\ref{w.reg}), rescale them so that they satisfy (\ref{w.norm}), and simultaneously accordingly rescale $H_{As}$. Mission accomplished!
\subsubsection{Fixing Exposures}
{}Well, almost... We still need to figure out how to fix the exposures $H_{As}$. The simplest way to do this is to note that we can use the matrix $\Omega_{iA} = \delta_{G(i), A}$ to swap the index $i$ in $G^\prime_{is}$ by the index $A$, i.e., we can take
\begin{equation}\label{H}
H_{As} = \eta_A \sum_{i = 1}^N \Omega_{iA}~G^\prime_{is} = {\widetilde \eta}_A~{1\over n_A}\sum_{i\in J(A)} G^\prime_{is}
\end{equation}
That is, up to the normalization constants ${\widetilde\eta}_A$ (which are fixed via (\ref{w.norm})) we simply take cross-sectional means of $G^\prime_{is}$ in each cluster. (Recall that $n_A = J(A)$.) The so-defined $H_{As}$ are automatically positive as all $G^\prime_{is}$ are positive. Therefore, $w_i$ defined via (\ref{w.reg}) are also all positive. This is a good news -- vanishing $w_i$ would amount to an incomplete weights matrix $W_{iA}$ (i.e., some mutations would belong to no cluster.)
{}So, why does (\ref{H}) make sense? Looking at (\ref{G.reg}), we can observe that, if the residuals $\varepsilon_{is}$ cross-sectionally, within each cluster labeled by $A$, are random, then we expect that $\sum_{i\in J(A)}\varepsilon_{is}\approx 0$. If we had an exact equality here, then we would have (\ref{H}) with $\eta_A = 1$ (i.e., ${\widetilde \eta}_A = n_A$) assuming the normalization (\ref{w.norm}). In practice, the residuals $\varepsilon_{is}$ are not exactly ``random". First, the number $n_A$ of mutation categories in each cluster is not large. Second, as mentioned above, there is variability in serial standard deviations across mutation types. This leads us to consider variations.
\subsubsection{A Variation}
{}Above we argued that it makes sense to cluster normalized demeaned log-counts ${\widetilde X}^\prime_{is} = X^\prime_{is}/\sigma^\prime_i$ due to the cross-sectional variability (and skewness) in the serial standard deviations $\sigma_i^\prime$. We may worry about similar effects in $G^\prime_{is}$ when computing $H_{As}$ and $w_i$ as we did above. This can be mitigated by using normalized quantities ${\widetilde G}^\prime_{is} = G^\prime_{is} / \omega_i$, where $\omega_i^2 = \mbox{Cov}(G^\prime_{is}, G^\prime_{is})$ are serial variances. That is, we can define\footnote{\, I.e., here we assume that $\varepsilon_{is}/\omega_i$ are approximately random in (\ref{G.reg}).}
\begin{eqnarray}\label{H1}
&&H_{As} = {\widetilde \eta_A}~{1\over\nu_A} \sum_{i\in J(A)} {\widetilde G}^\prime_{is} = {\widetilde \eta_A}~{1\over\nu_A}\sum_{i\in J(A)} {1 \over \omega_i}~G^\prime_{is}\\
\label{w.reg.1}
&&w_i = \omega_i~{{\sum_{s=1}^d {\widetilde G}^\prime_{is}~H_{G(i),s}}\over {{\sum_{s=1}^d H^2_{G(i),s}}}} = {{\sum_{s=1}^d G^\prime_{is}~H_{G(i),s}}\over {{\sum_{s=1}^d H^2_{G(i),s}}}}
\end{eqnarray}
where $\nu_A = \sum_{i\in J(A)}1/\omega_i$. So, $1/\omega_i$ are the weights in the averages over the clusters.
\subsubsection{Another Variation}
{}Here one may wonder, considering the skewed roughly log-normal distribution of $G_{is}$ and henceforth $G^\prime_{is}$, would it make sense to relate the exposures to within-cluster cross-sectional averages of demeaned log-counts $X^\prime_{is}$ as opposed to those of $G^\prime_{is}$? This is easily achieved. Thus, we can define (this ensures positivity of $H_{As}$):
\begin{equation}
\ln(H_{As}) = \ln({\widetilde \eta}_A) + {1\over n_A}\sum_{i\in J(A)} X^\prime_{is}
\end{equation}
Exponentiating we get
\begin{equation}\label{H2}
H_{As} = {\widetilde \eta}_A \left[\prod_{i\in J(A)} G^\prime_{is}\right]^{1/n_A}
\end{equation}
I.e., instead of an arithmetic average as in (\ref{H}), here we have a geometric average.
{}As above, here too we can introduce nontrivial weights. Note that the form of (\ref{w.reg.1}) is the same as (\ref{w.reg}), it is only $H_{As}$ that is affected by the weights. So, we can introduce the weights in the geometric means as follows:
\begin{equation}
\ln(H_{As}) = \ln({\widetilde \eta}_A) + {1\over \mu_A}\sum_{i\in J(A)} {\widetilde X}^\prime_{is} = \ln({\widetilde \eta}_A) + {1\over \mu_A}\sum_{i\in J(A)} {1\over \sigma^\prime_i}~X^\prime_{is}
\end{equation}
where $\mu_A = \sum_{i\in J(A)}1/\sigma^\prime_i$. Recall that $(\sigma^\prime_i)^2 = \mbox{Cov}(X^\prime_{is}, X^\prime_{is})$. Thus, we have:
\begin{equation}\label{H3}
H_{As} = {\widetilde \eta}_A \prod_{i\in J(A)} (G^\prime_{is})^{1/\mu_A\sigma_i^\prime}
\end{equation}
So, the weights are the exponents $1/\mu_A\sigma_i^\prime$. Other variations are also possible.
\subsection{Implementation}
{}We are now ready to discuss an actual implementation of the above algorithm, much of the R code for which is already provided in \cite{BioFM} and \cite{StatIndClass}. The R source code is given in Appendix \ref{app.code} hereof.
\section{Empirical Results}\label{sec.3}
\subsection{Data Summary}\label{data.summary}
{}In our empirical analysis below we use the same genome data (from published samples only) as in \cite{BioFM}. This data is summarized in Table \ref{table.genome.summary} (borrowed from \cite{BioFM}), which gives total counts, number of samples and the data sources, which are as follows: A1 = \cite{Alexandrov}, A2 = \cite{Love}, B1 = \cite{Tirode}, C1 = \cite{Zhang}, D1 = \cite{Nik-Zainal}, E1 = \cite{Puente2011}, E2 = \cite{Puente2015}, F1 = \cite{Cheng}, G1 = \cite{Wang}, H1 = \cite{Sung}, H2 = \cite{Fujimoto}, I1 = \cite{Imielinski}, J1 = \cite{Jones}, K1 = \cite{Patch}, L1 = \cite{Waddell}, M1 = \cite{Gundem}, N1 = \cite{Scelo}. Sample IDs with the corresponding publication sources are given in Appendix A of \cite{BioFM}. In our analysis below we aggregate samples by the 14 cancer types. The resulting data is in Tables \ref{table.aggr.data.1} and \ref{table.aggr.data.2}.
\subsubsection{Structure of Data}
{}The underlying data consists of a matrix -- call it $G_{is}$ -- whose elements are occurrence counts of mutation types labeled by $i = 1,\dots, N = 96$ in samples labeled by $s = 1,\dots, d$. More precisely, we can work with one matrix $G_{is}$ which combines data from different cancer types; or, alternatively, we may choose to work with individual matrices $[G(\alpha)]_{is}$, where: $\alpha = 1,\dots,n$ labels $n$ different cancer types; as before, $i=1,\dots,N=96$; and $s = 1,\dots, d(\alpha)$. Here $d(\alpha)$ is the number of samples for the cancer type labeled by $\alpha$. The combined matrix $G_{is}$ is obtained simply by appending (i.e., bootstrapping) the matrices $[G(\alpha)]_{is}$ together column-wise. In the case of the data we use here (see above), this ``big matrix" turns out to have 1389 columns.
{}Generally, individual matrices $[G(\alpha)]_{is}$ and, thereby, the ``big matrix", contain a lot of noise. For some cancer types we can have a relatively small number of samples. We can also have ``sparsely populated" data, i.e., with many zeros for some mutation categories. As mentioned above, different samples are not necessarily uniformly normalized. Etc. The bottom line is that the data is noisy. Furthermore, intuitively it is clear that the larger the matrix we work with, statistically the more ``signatures" (or clusters) we should expect to get with any reasonable algorithm. However, as mentioned above, a large number of signatures would be essentially useless and defy the whole purpose of extracting them in the first place -- we have 96 mutation categories, so it is clear that the number of signatures cannot be more than 96! If we end up with, say, 50+ signatures, what new or useful does this tell us about the underlying cancers? The answer is likely nothing other than that most cancers have not much in common with each other, which would be a disappointing result from the perspective of therapeutic applications. To mitigate the aforementioned issues, at least to a certain extent, following \cite{BioFM}, we can aggregate samples by cancer types. This way we get an $N \times n$ matrix, which we also refer to as $G_{is}$, where the index $s=1,\dots,d$ now takes $d=n$ values corresponding to the cancer types. In the data we use $n=14$, the aggregated matrix $G_{is}$ is much less noisy than the ``big matrix", and we are ready to apply the above machinery to it.
\subsection{Genome Data Results}
{}The $96 \times 14$ matrix $G_{is}$ given in Tables \ref{table.aggr.data.1} and \ref{table.aggr.data.2} is what we pass into the function {\tt\small bio.cl.sigs()} in Appendix \ref{app.code} as the input matrix {\tt\small x}. We use: {\tt\small iter.max = 100} (this is the maximum number of iterations used in the built-in R function {\tt\small kmeans()} -- we note that there was not a single instance in our 150 million runs of {\tt\small kmeans()} where more iterations were required);\footnote{\, The R function {\tt\small kmeans()} produces a warning if it does not converge within {\tt\small iter.max}.} {\tt\small num.try = 1000} (this is the number of individual k-means samplings we aggregate every time); and {\tt\small num.runs = 150000} (which is the number of aggregated clusterings we use to determine the ``ultimate" -- that is, the most frequently occurring -- clustering). So, we ran k-means 150 million times. More precisely, we ran 15 batches with {\tt\small num.runs = 10000} as a sanity check, to make sure that the final result based on 150000 aggregated clusterings was consistent with the results based on smaller batches, i.e., that it was in-sample stable.\footnote{\, We ran these 15 batches consecutively, and each batch produced the same top-10 (by occurrence counts) clusterings as in Table \ref{table.occurrence.cts}; however, the actual occurrence counts are different across the batches with slight variability in the corresponding rankings. The results are pleasantly stable.} Based on Table \ref{table.occurrence.cts}, we identify Clustering-A as the ``ultimate" clustering (cf. Clustering-B/C/D).
{}We give the weights for Clustering-A, Clustering-B, Clustering-C and Clustering-D using unnormalized and normalized regressions with exposures computed based on arithmetic averages (see Subsection \ref{sub.reg}) in Tables \ref{table.weights.A.1}, \ref{table.weights.A.2}, \ref{table.weights.B.1}, \ref{table.weights.B.2}, \ref{table.weights.C.1}, \ref{table.weights.C.2}, \ref{table.weights.D.1}, \ref{table.weights.D.2}, and Figures \ref{Figure1A} through \ref{FigureNorm6D}. We give the weights for Clustering-A using unnormalized and normalized regressions with exposures computed based on geometric averages (see Subsection \ref{sub.reg}) in Tables \ref{table.weights.Z.1}, \ref{table.weights.Z.2}, and Figures \ref{FigureGeom1Z} through \ref{FigureGeomNorm7Z}. The actual mutation categories in each cluster for a given clustering can be read off the aforesaid Tables with the weights (the mutation categories with nonzero weights belong to a given cluster), or from the horizontal axis labels in the aforesaid Figures. It is evident that Clustering-A, Clustering-B, Clustering-C and Clustering-D are essentially variations of each other (Clustering-D has only 6 clusters, while the other 3 have 7 clusters).
\subsection{Reconstruction and Correlations}\label{sub.cor}
{}So, based on genome data, we have constructed clusterings and weights. Do they work? I.e., do they reconstruct the input data well? It is evident from the get-go that the answer to this question may not be binary in the sense that for some cancer types we might have a nice clustering structure, while for others we may not. The aim of the following exercise is to sort this all out. Here come the correlations...
\subsubsection{Within-cluster Correlations}
{}We have our de-noised\footnote{\, De-noising per se does not affect cross-sectional correlations. Adding extra 1 in (\ref{log.def}) (recall that we obtain $G^\prime_{is}$ by cross-sectionally demeaning $X_{is}$ and then re-exponentiating) has a negligible effect. So, in the correlations below we can use the original data matrix $G_{is}$ instead of $G^\prime_{is}$.} matrix $G^\prime_{is}$. We are approximating this matrix via the following factorized matrix:
\begin{equation}\label{fac.st}
G^*_{is} = \sum_{A=1}^K W_{iA}~H_{As} = w_i~H_{G(i),s}
\end{equation}
We can now compute an $n\times K$ matrix $\Theta_{sA}$ of {\em within-cluster} cross-sectional correlations between $G^\prime_{is}$ and $G^*_{is}$ defined via ($\mbox{xCor}(\cdot,\cdot)$ stands for ``cross-sectional correlation" to distinguish it from ``serial correlation" $\mbox{Cor}(\cdot,\cdot)$ we use above)\footnote{\, Due to the factorized structure (\ref{fac.st}), these correlations do not directly depend on $H_{As}$.}
\begin{equation}
\Theta_{sA} = \left.\mbox{xCor}(G^\prime_{is}, G^*_{is})\right|_{i\in J(A)} = \left.\mbox{xCor}(G^\prime_{is}, w_i)\right|_{i\in J(A)}
\end{equation}
We give this matrix for Clustering-A with weights using normalized regressions with exposures computed based on arithmetic means (see Subsection \ref{sub.reg}) in Table \ref{table.fit.theta}. Let us mention that, with exposures based on arithmetic means, weights using normalized regressions work a bit better than using unnormalized regressions. Using exposures based on geometric means changes the weights a bit, which in turn slightly affects the within-cluster correlations, but does not alter the qualitative picture.
\subsubsection{Overall Correlations}
{}Another useful metric, which we use as a sanity check, is this. For each value of $s$ (i.e., for each cancer type), we can run a linear cross-sectional regression (without the intercept) of $G^\prime_{is}$ over the matrix $W_{iA}$. So, we have $n=14$ of these regressions. Each regression produces multiple $R^2$ and adjusted $R^2$, which we give in Table \ref{table.fit.theta}. Furthermore, we can compute the {\em fitted} values ${\widehat G}^*_{is}$ based on these regressions, which are given by
\begin{equation}
{\widehat G}^*_{is} = \sum_{A=1}^K W_{iA}~F_{As} = w_i~F_{G(i),s}
\end{equation}
where (for each value of $s$) $F_{As}$ are the regression coefficients. We can now compute the overall cross-sectional correlations (i.e., the index $i$ runs over all $N=96$ mutation categories)
\begin{equation}
\Xi_s = \mbox{xCor}(G^\prime_{is}, {\widehat G}^*_{is})
\end{equation}
These correlations are also given in Table \ref{table.fit.theta} and measure the overall fit quality.
\subsubsection{Interpretation}
{}Looking at Table \ref{table.fit.theta} a few things become immediately evident. Clustering works well for 10 out the 14 cancer types we study here. The cancer types for which clustering does not appear to work all that well are Breast Cancer (labeled by X4 in Table \ref{table.fit.theta}), Liver Cancer (X8), Lung Cancer (X9), and Renal Cell Carcinoma (X14). More precisely, for Breast Cancer we do have a high within-cluster correlation for Cl-5 (and also Cl-4), but the overall fit is not spectacular due to low within-cluster correlations in other clusters. Also, above 80\% within-cluster correlations\footnote{\, The 80\% cutoff is somewhat arbitrary, but reasonable.} arise for 5 clusters, to wit, Cl-1, Cl-3, Cl-4, Cl-5 and Cl-6, but not for Cl-2 or Cl-7. Furthermore, remarkably, Cl-1 has high within-cluster correlations for 9 cancer types, and Cl-5 for 6 cancer types. These appear to be the leading clusters. Together they have high within-cluster correlations in 11 cancer types. So what does all this mean?
{}Additional insight is provided by looking at the within-cluster correlations between the 7 cancer signature extracted in \cite{BioFM} and the clusters we find here. Let ${\cal W}_{i\alpha}$ be the weights for the 7 cancer signatures from Tables 13 and 14 of \cite{BioFM}. We can compute the following within-cluster correlations $(\alpha = 1,\dots,7$ labels the cancer signatures of \cite{BioFM}, which we refer to as Sig1 through Sig7):
\begin{equation}
\Delta_{\alpha A} = \left.\mbox{xCor}({\cal W}_{i\alpha}, W_{iA})\right|_{i\in J(A)}
\end{equation}
These correlations are given in Table \ref{table.cor.delta}. High within-cluster correlations arise for Cl-1 (with Sig1 and Sig7), Cl-5 (with Sig2) and Cl-6 (with Sig4). And this makes perfect sense. Indeed, looking at Figures 14 through 20 of \cite{BioFM}, Sig1, Sig2, Sig4 and Sig7 are precisely the cancer signatures that have ``peaks" (or ``spikes" -- ``tall mountain landscapes"), whereas Sig3, Sig5 and Sig6 do not have such ``peaks" (``flat" or ``rolling hills landscapes"). No wonder such signatures do not have high within-cluster correlations -- they simply do not have cluster-like structures. Looking at Figure 21 in \cite{BioFM}, it becomes evident why clustering does not work well for Liver Cancer (X8) -- it has a whopping 96\% contribution from Sig5! Similarly, Renal Cell Carcinoma (X14) has a 70\% contribution from Sig6. Lung Cancer (X9) is dominated by Sig3, hence no cluster-like structure. Finally, Breast Cancer (X4) is dominated by Sig2, which has a high within-cluster correlation with Cl-5, which is why Breast Cancer has a high within-cluster correlation with Cl-5 (but poor overall correlation in Table \ref{table.fit.theta}). So, it all makes sense. The question is, what does all this tell us about cancer signatures?
{}Quite a bit! It tells us that cancers such as Liver Cancer, Lung Cancer and Renal Cell Carcinoma have little in common with other cancers (and each other)! At least at the level of mutation categories that dominate the genome structure of such cancers. On the other hand, 9 cancers, to wit, Bone Cancer (X2), Brain Lower Grade Glioma (X3), Chronic Lymphocytic Leukemia (X5), Esophageal Cancer (X6), Gastric Cancer (X7), Medulloblastoma (X10), Ovarian Cancer (X11), Pancreatic Cancer (X12) and Prostate Cancer (X13) apparently all have the Cl-1 cluster structure embedded in them substantially. Similarly, 6 cancers, to wit, B Cell Lymphoma (X1), Breast Cancer (X4), Esophageal Cancer(X6), Ovarian Cancer (X11), Pancreatic Cancer (X12) and Prostate Cancer (X13) apparently all have the Cl-5 cluster structure embedded in them substantially. Furthermore, note the overlap between these two lists, to wit, Esophageal Cancer(X6), Ovarian Cancer (X11), Pancreatic Cancer (X12) and Prostate Cancer (X13). We obtained this result purely statistically, with no biologic input, using our clustering algorithm and other statistical methods such as linear regression to obtain the actual weights. It is too early to know whether this insight will aid any therapeutic applications, but that is the hope -- similarities in the underlying genomic structures of different cancer types raise hope that therapeutics for one cancer type could perhaps be applicable to other cancer types. On the other hand, our findings above relating to Liver Cancer, Lung Cancer and Renal Cell Carcinoma (and possibly also Breast Cancer, albeit the latter does appear to have a not-so-insignificant overlap with Cl-5, which differentiates it from the aforesaid 3 cancer types) suggest that these cancer types apparently stand out.
\section{Concluding Remarks}\label{sec.4}
{}Clustering ideas and techniques have been applied in cancer research in various incarnations and contexts aplenty -- for a partial list of works at least to some extent related to our discussion here, see, e.g, \cite{Chen1}, \cite{Chen2}, \cite{Kashuba}, \cite{Nik-Zainal}, \cite{Roberts2012}, \cite {Alexandrov.NMF}, \cite{Alexandrov}, \cite{Burns1}, \cite{Burns2}, \cite{Lawrence}, \cite{Long} \cite{Roberts2013}, \cite{Taylor}, \cite{Xuan}, \cite{AlexStrat}, \cite{Bacolla}, \cite{Bolli}, \cite{Caval}, \cite{Davis}, \cite{Helleday}, \cite{Nik-Zainal2014}, \cite{Poon}, \cite{Qian}, \cite{Roberts1}, \cite{Roberts2}, \cite{Roberts3}, \cite{Sima}, \cite{Chan}, \cite{Pettersen} and references therein. As mentioned above, even in NMF clustering is used at some (perhaps not-so-evident) layer. What is new in our approach -- and hence new results -- is that: i) following \cite{BioFM}, we apply clustering to aggregated by cancer types and de-noised data; ii) we use a tried-and-tested in quantitative finance bag of tricks from \cite{StatIndClass}, which improves clustering; and iii) last but not least, we apply our *K-means algorithm to cancer genome data. As mentioned above, *K-means, unlike vanilla k-means or its other commonly used variations, is essentially deterministic, and it achieves determinism {\em statistically}, not by ``guessing" initial centers or as in agglomerative hierarchical clustering, which basically ``guesses" the initial (e.g., 2-cluster) clustering. Instead, via aggregating a large number of k-means clusterings and statistical examination of the occurrence counts of such aggregations, *K-means takes a mess of myriad vanilla k-means clusterings and systematically reduces randomness and indeterminism without {\em ad hoc} initial ``guesswork".
{}As mentioned above, consistently with the results of \cite{BioFM} obtained via improved NMF techniques, Liver Cancer, Lung Cancer and Renal Cell Carcinoma do not appear to have clustering (sub)structures. This could be both good and bad news. It is a good news because we learned something interesting about these cancer types -- and in two complementary ways. However, it could also be a bad news from the therapeutic standpoint. Since these cancer types appear to have little in common with others, it is likely that they would require specialized therapeutics. On the flipside, we should note that it would make sense to exclude these 3 cancer types when running clustering analysis. However, it would also make sense to include other cancer types by utilizing the International Cancer Genome Consortium data, which we leave for future studies. (For comparative reasons, here we used the same data as in \cite{BioFM}, which was limited to data samples published as of the date thereof.) This paper is not intended to be an exhaustive empirical study but a proof of concept and an opening of a new avenue for extracting and studying cancer signatures beyond the tools that NMF provides.
{}And we do find that 11 out of the 14 cancer types we study here have clustering structures substantially embedded in them and clustering overall works well for at least 10 out of these 11 cancer types.\footnote{\, Breast Cancer possibly being an exception. As mentioned above, it would make sense to exclude Liver Cancer, Lung Cancer and Renal Cell Carcinoma from the analysis, which may affect how well clustering works for Breast Cancer and possibly also the other 10 cancer types.} Now, looking at Figure 14 of \cite{BioFM}, we see that its ``peaks" are located at ACGT, CCGT, GCGT and TCGT. The same ``peaks" are present in our cluster Cl-1 (see Figures \ref{Figure1A} and \ref{FigureNorm1A}). Hence the high within-cluster correlation between Cl-1 and Sig1. On the other hand, Sig1 of \cite{BioFM} is essentially the same as the mutational signature 1 of \cite{Nik-Zainal}, \cite{Alexandrov}, which is due to spontaneous cytosine deamination. So, this is what our cluster Cl-1 describes. Next, looking at Figure 15 of \cite{BioFM}, we see that its ``peaks" are located at TCAG, TCTG, TCAT and TCTT. The first two of these ``peaks" TCAG and TCTG are present in our Cl-5 (see Figures \ref{Figure5A} and \ref{FigureNorm5A}), the third ``peak" TCAT is present in our Cl-1 (see Figures \ref{Figure1A} and \ref{FigureNorm1A}), while the fourth ``peak" TCTT is present in our Cl-4 (see Figures \ref{Figure4A} and \ref{FigureNorm4A}), which is consistent with the high within-cluster correlations between Sig2 and Cl-4 and Cl-5, albeit its within-cluster correlation with Cl-1 is poor. Note that Sig2 of \cite{BioFM} is essentially the same as the mutational signatures 2+13 of \cite{Nik-Zainal}, \cite{Alexandrov}, which are due to APOBEC mediated cytosine deamination. In fact, it was reported as a single signature in \cite{Alexandrov}, however, subsequently, it was split into 2 distinct signatures, which usually appear in the same samples.\footnote{\, For detailed comments, see http://cancer.sanger.ac.uk/cosmic/signatures.} Our clustering results indicate that grouping TCAG and TCTG into one signature makes sense as they belong to the same cluster Cl-5. However, grouping TCAT and TCTT together does not appear to make much sense. Looking at the Figures for Clustering-A, Clustering-B, Clustering-C and Clustering-D, we see that the TCAT ``peak" invariably appears together with the ACGT, CCGT, GCGT and TCGT ``peaks" as in Cl-1 in Clustering-A, Cl-2 in Clustering-B, Cl-1 in Clustering-C, and Cl-1 in Clustering-D, but never with TCTT. So, our clustering approach tells us something new beyond the NMF ``intuition". This may have an important implication for Breast Cancer, which, as mentioned above, is dominated by Sig2. Thus, based on our results in Table \ref{table.fit.theta}, we see that Breast Cancer has high within-cluster correlations with Cl-4 and Cl-5, but not with Cl-1. This may imply that clustering simply does not work well for Breast Cancer, which would appear to put it in the same ``stand-alone" league as Liver Cancer, Lung Cancer and Renal Cell Carcinoma. In any event, clustering invariably suggests that the TCAT ``peak" belongs in Cl-1 with the 4 ``peaks" ACGT, CCGT, GCGT and TCGT related to spontaneous cytosine deamination, rather than those related to APOBEC mediated cytosine deamination.
{}Now, let us check the remaining two signatures of \cite{BioFM} with ``tall mountain landscapes" (see above), to wit, Sig4 and Sig7. Looking at Figure 17 of \cite{BioFM}, we see that its ``peaks" are at CTTC, TTTC, CTTG and TTTG. The same peaks appear in our Cl-6 (see Figures \ref{Figure6A} and \ref{FigureNorm6A}). Hence the high within-cluster correlation between Cl-6 and Sig4. Note that Sig4 is essentially the same as the mutational signature 17 of \cite{Nik-Zainal}, \cite{Alexandrov}, and its underlying mutational process is unknown. Next, looking at Figure 20 of \cite{BioFM}, we see that its ``peaks" for the C $>$ G mutations are essentially the same as in Cl-1. Hence the high within-cluster correlation between Cl-7 and Sig1. So, there are no surprises with Sig1, Sig4 and Sig7. However, based on our clustering results, as we discuss above, with Sig2 we do find -- what we feel is a pleasant -- surprise, that splitting it into two signatures (see above) might be inadequate and the TCAT ``peak" might really belong with the Sig1 ``peaks" (spontaneous v. APOBEC mediated cytosine deamination). This is exciting as it might be an indication of the limitations of NMF (or clustering...).\footnote{\ Or both... Alternatively -- and that would be truly exciting -- perhaps there is a biologic explanation. In any event, it is too early to tell -- yet another possibility is that this is merely an artifact of the dataset we use. More research and analyses on larger datasets (see above) is needed.}
{}In Introduction we promised that we would discuss some potential applications of *K-means in quantitative finance, and so here it is. Let us mention that *K-means is universal, oblivious to the input data and applicable in a variety of fields. In quantitative finance *K-means {\em a priori} can be applied everywhere clustering methods are used with the added bonus of (statistical) determinism.\footnote{\, Albeit with the understanding that it requires additional computational cost.} One evident example is statistical industry classifications discussed in \cite{StatIndClass}, where one uses clustering methods to classify stocks. In fact, *K-means is an extension of the methods discussed in \cite{StatIndClass}. One thing to keep in mind is that in *K-means one sifts through a large number $P$ of aggregations, which can get computationally costly when clustering 2000+ stocks into 100+ clusters.\footnote{\, This can be mitigated by employing top-down clustering \cite{StatIndClass}.} Another potential application is in the context of combining alphas (trading signals) -- see, e.g., \cite{Billion}. Yet another application is when we have a term structure, such as a portfolio of bonds (e.g., U.S. Treasuries or some other bonds) with varying maturities, or futures (e.g., Eurodollar futures) with varying deliveries. These cases resemble the genome data more in the sense that the number $N$ of instruments is relatively small (typically even fewer than the number of mutation categories). Another example with a relatively small number of instruments would be a portfolio of various futures for different FX (foreign exchange) pairs (even with the uniform delivery), e.g., USD/EUR, USD/HKD, EUR/AUD, etc., i.e., FX statistical arbitrage. One approach to optimizing risk in such portfolios is by employing clustering methods and a stable, essentially deterministic algorithm such as *K-means can be useful. Hopefully *K-means will prove a valuable tool in cancer research, quantitative finance as well as various other fields (e.g., image recognition).
|
1,314,259,992,985 | arxiv | \section{Introduction}
\hspace*{10mm}
Theoretical investigations of nucleon-nucleon (NN) transition
amplitudes
in their off-shell domain have a long history in the study
of few and many-nucleon systems. Often those investigations
were inconclusive due to the lack of NN potentials which
describe the NN observables with equally high accuracy. Current interest
in this issue is driven by the recent development of NN potentials which
below pion production threshold describe the NN data base with a
$\chi^2$ per datum $\sim$ 1 \cite{nijm,cdbonn,argonne}. Transition
amplitudes derived from these
potentials can be considered on-shell equivalent. Their
different theoretical derivation gives rise to different off-shell
extrapolations.
\hspace*{10mm}
At intermediate energies elastic nucleon-nucleus (NA) scattering can be
successfully described by the leading term in the spectator expansion
of multiple scattering theory \cite{Corr,Sicil,med2}. Here an optical
potential is derived, which in its most general form is given by the
expectation value of the NN transition amplitude and the ground state
of the target nucleus. This `full-folding' optical potential involves
the convolution of the fully off-shell NN scattering amplitude with a
realistic single particle nuclear density matrix.
\hspace*{10mm}
Recently, significant advances have been made in accurately handling
these off-shell degrees of freedom in elastic NA scattering
\cite{hugo1,hugo2,FFC,ff1,edff}.
Those studies have demonstrated that an accurate treatment of the
off-shell structure of the NN transition amplitude is needed for a
proper account of the theory. In order to cleanly isolate if NA elastic
scattering observables are sensitive to different off-shell structures
of realistic NN transition amplitudes, it is necessary to start from NN
potentials which describe the NN data base with a high degree of
accuracy. Our present study is based on the potential models for the
NN interaction recently developed by
the Nijmegen group \cite{nijm} and the charge-dependent Bonn (CD-Bonn)
potential \cite{cdbonn}. With NN transition amplitudes derived from these
potentials we calculate full-folding optical potentials and elastic
scattering observables for proton scattering from a variety of nuclei in
the energy regime between 100 and 200 MeV projectile energy. Although
the off-shell structure of the NN t-matrices is an important ingredient
in the calculations, we find that off-shell differences between
the models are not discernible by NA elastic scattering.
\hspace*{10mm}
In order to understand this result and obtain more insight
which regions of the off-shell NN t-matrix are
sampled in a calculation of NA elastic scattering observables, we
use the optimum factorized or off-shell $t\rho$ formulation of the
optical potential. This formulation, quite a good approximation
in the energy regime around 200 MeV and higher, has the
advantage that the fully off-shell NN t-matrix enters together with
an on-shell density.
\hspace*{10mm}
The structure of this article is as follows. First we review in Section~II
the relevant expressions for the full-folding optical potential as used
in our calculations. In Section~III we present elastic scattering
results for proton scattering from a variety on nuclei based on
the Nijmegen and CD-Bonn potentials. In Section~IV we present a detailed
study on which off-shell regions of the NN t-matrix are sampled in
a calculation of the elastic scattering observables.
This study is based on the
factorized $t\rho$ approximation to the full-folding optical potential
and is carried out at 200 MeV projectile energy.
We end with concluding remarks in Section~V.
\section{Theoretical Framework for the Optical Potential}
\hspace*{10mm}
The transition amplitude for elastic scattering of a projectile from
a target nucleus is given as \cite{med2}
\begin{equation}
T_{el} = P U P + P U G_0(E) T_{el}, \label{eq:2.1}
\end{equation}
where $P$ is the projector on the ground state $|\Phi_A\rangle$ of
the target, $P = \frac{|\Phi_A\rangle \langle \Phi_A|}
{\langle \Phi_A| \Phi_A\rangle}$, $G_0(E)=(E-H_0 +i\varepsilon)^{-1}$,
and $U$ represents the optical potential.
For the scattering of a single particle projectile from an A-particle
target nucleus the free Hamiltonian is given by $H_0=h_0+H_A$, where
$h_0$ is the kinetic energy operator for the projectile
and $H_A$
stands for the target Hamiltonian. In the spirit of the spectator
expansion the target Hamiltonian is viewed as $H_A=h_i + \sum_{j\neq i}
v_{ij} +H^i$, where $h_i$ is the kinetic energy operator for the
$i$th target nucleon, $v_{ij}$ the interaction between target
nucleon $i$ and the other target nucleons $j$, and $H^i$ is an
(A-1)-body operator containing all higher order effects.
In a mean field approximation $\sum_{j\neq i} v_{ij} \approx W_i$,
where $W_i$ is assumed to depend only on the $i$th particle
coordinate. In this present work we want to concentrate only on the
impulse approximation, which is a good approximation in the intermediate
energy regime (around 200~MeV projectile energy and higher), where the
influence of $W_i$ can be neglected \cite{med2}. Thus the propagator
$G_0(E)$ in the impulse approximation is given as
\begin{equation}
G_0(E) \approx g_i(E)=[(E-E^i)-h_0 -h_i + i \varepsilon]^{-1}.
\label{eq:2.2}
\end{equation}
Here $H^i$, having no explicit dependence on the $i$th particle,
is replaced by an average
energy $E^i$. In the present calculations we set $E_i=0$.
In the energy regime considered in this work, the effect of a value of
$E_i$ of the order of the separation energy of a nucleon from a nucleus
is negligible \cite{edff,swth}.
\hspace*{10mm}
The driving term of Eq.~(\ref{eq:2.1}) denotes the optical potential,
which in first order is given as
\begin{equation}
\langle {\bf k'}|\langle \Phi_A| PUP|\Phi_A\rangle |{\bf k}\rangle
\equiv {\hat U}({\bf k'},{\bf k}) = \sum_{i=n,p}\langle {\bf k'}|
\langle \Phi_A| {\hat \tau}_{0i}({\cal E})|\Phi_A \rangle |{\bf k}\rangle.
\label{eq:2.3}
\end{equation}
Here ${\bf k'}$ and ${\bf k}$ are the external momenta of the system,
${\hat \tau}_{0i}({\cal E})$ represents the NN transition operator
\begin{equation}
{\hat \tau}_{0i}(E) = v_{0i} + v_{0i} g_i(E) {\hat \tau}_{0i}(E) ,
\label{eq:2.4}
\end{equation}
with $g_i(E)$ given in Eq.~(\ref{eq:2.2}) and $v_{0i}$ representing the
NN interaction. The sum over $i$ in
Eq.~({\ref{eq:2.3}) indicates the two different cases, namely when the
target nucleon is one of Z protons, and when it is one of N neutrons.
The energy ${\cal E}$ is the relative energy of the interacting
two-nucleon system.
Inserting a complete set of momenta for the struck target nucleon before
and
after the collision and evaluating the momentum conserving
$\delta$-functions
gives as final expression for the full-folding optical potential
\cite{ff1,swth}
\begin{eqnarray}
\hat{U}({\bf q}, {\bf K})= \sum_{i=n,p} \int d^3P &&
\;\eta({\bf P},{\bf q}, {\bf K})\;\hat{\tau}_{0i}({\bf q},
\frac{1}{2}(\frac{A+1}{A}{\bf K}-{\bf P}), {\cal E}) \nonumber \\
&&\rho_i({\bf P}-\frac{A-1}{A}\frac{{\bf q}}{2},
{\bf P}+\frac{A-1}{A}\frac{{\bf q}}{2}) \label{eq:2.5}.
\end{eqnarray}
Here the arguments of the NN amplitude $\hat{\tau}_{0i}$ are
${\bf q}={\bf k}'-{\bf k}={\bf k_{NN}}'-{\bf k_{NN}}$ and
$\frac{1}{2}({\bf k_{NN}}'+{\bf k_{NN}}) =
\frac{1}{2}(\frac{A+1}{A}{\bf K}-{\bf P})$, where
\begin{equation}
{\bf k'_{NN}}=\frac{1}{2}({\bf k}' - ({\bf P}-{\frac{\bf q}{2}} -
\frac{\bf K}{A}) \label{eq:2.5a}
\end{equation}
and
\begin{equation}
{\bf k_{NN}}=\frac{1}{2}({\bf k} - ({\bf P}+{\frac{\bf q}{2}} -
\frac{\bf K}{A}) \label{eq:2.5b}
\end{equation}
are the nonrelativistic final and initial
nuclear momentum in the zero momentum frame of the NN system,
and ${\bf K}=\frac{1}{2}({\bf k'}+{\bf k})$.
The factor $\eta({\bf P},{\bf q}, {\bf K})$ is the M\o ller
factor for the frame transformation~\cite{Joachain}, and
$\rho_i$ represents the density matrix of the target for either protons
or neutrons.
Evaluating the propagator $g_i(E)$ of Eq.~(\ref{eq:2.2}) in
the nucleon-nucleus (NA) center of mass frame yields for the
energy argument ${\cal E}$ of the NN amplitude
$\hat{\tau}_{0i}$ of Eq.~(\ref{eq:2.5})
\begin{equation}
{\cal E} = E_{NA} - \frac{(\frac{A-1}{A}{\bf K}+{\bf P})^2}{4m_N}.
\end{equation}
Here $E_{NA}$ is the total energy in the NA center of mass frame
and $m_N$ is the nucleon mass.
\hspace*{10mm}
The expression for the optical potential as given in
Eq.~(\ref{eq:2.5}) shows that the evaluation of the full-folding
integral requires the NN t-matrix fully off-shell as well as at positive
energies from $E_{NA}$ to negative energies \cite{hugo1,hugo2,edff}.
\section{Proton Elastic Scattering Observables}
\hspace*{10mm}
In this paper the study of elastic scattering of protons from spin
zero target nuclei at energies between 100 and 200~MeV incident
projectile energy is strictly first order based on the impulse
approximation. The full-folding optical potentials are calculated
according to Eq.~(\ref{eq:2.5}). The details of the calculations are
given in Refs.~\cite{ff1,edff}. As a model for the density matrix
for the target nucleus we employ a Dirac-Hartree (DH) calculation~\cite{DH}.
The Fourier transform of the vector density,
$\rho({\bf r}',{\bf r})$, serves as our
non-relativistic single particle density~\cite{ff1}. The crucial
ingredient under investigation here is the fully off-shell NN t-matrix.
The calculations presented here employ NN t-matrices based on
two different potentials given by the Nijmegen group
\cite{nijm} and the charge-dependent Bonn potential \cite{cdbonn}.
All three potentials are fitted to describe the Nijmegen data base with
a $\chi^2$ per datum $\sim$~1. An essential difference between the two
Nijmegen models is the presence of a momentum dependent, nonlocal term
in the central piece of the NijmI potential, whereas the NijmII model is
strictly local. Both Nijmegen potentials have a $\chi^2$ per datum =~1.03
with respect to both, the neutron-proton and the proton-proton
data base. The CD-Bonn potential is nonlocal due to the structure of the
relativistic meson-nucleon vertices. An additional nonlocality is
contained due to the socalled minimal relativity factors $\sqrt{m/E}$,
which are necessary to maintain the relativistic unitarity condition.
The CD-Bonn potential also describes the Nijmegen data base with
a $\chi^2$ per datum =~1.03. All three potential models describe the
Nijmegen data base with the same high degree of accuracy, thus the NN
t-matrices can be considered on-shell equivalent. From their different
theoretical derivation it can be expected that they have different
extrapolations off-shell.
\hspace*{10mm}
When calculating $\hat{U}({\bf q}, {\bf K})$ as given in
Eq.~(\ref{eq:2.5}), it is to be understood that all spin summations
are carried out. This reduces the required NN t-matrix elements to a
spin independent component (corresponding to the Wolfenstein amplitude A)
and a spin-orbit component (corresponding to the Wolfenstein amplitude
C). Since we are assuming that we have spin saturated nuclei, the
components of the NN t-matrix depending on the spin of the struck
target nucleon vanish. The Coulomb interaction between the projectile
and the target is included using the exact formulation of
Ref.~\cite{coul}.
\hspace*{10mm}
At first we want to concentrate on proton scattering from different
target nuclei at 200~MeV projectile energy. In Fig.~1 we display the
differential cross section $d\sigma/d\Omega$, the analyzing power $A_y$,
and the spin rotation function $Q$ for elastic proton scattering from
$^{16}$O. The solid line represents a calculation based employing the
CD-Bonn t-matrix as input, the dashed line is based on the one derived
from the NijmI potential and the dash-dotted line the one derived from
the NijmII model. All three calculations are remarkably close to each
other, and all three fail to describe the dips in the analyzing power.
The same statement is true for proton scattering from $^{40}$Ca at
200~MeV, which is displayed in Fig.~2. In Fig.~3 we show the elastic
scattering observables for proton scattering from $^{208}$Pb at
200~MeV. Again, all three NN potential models give nearly identical
results, however the spin observables are described slightly better
for $^{208}$Pb.
\hspace*{10mm}
At lower energies the scattering observables may exhibit a somewhat
larger sensitivity to the energy dependence of the NN t-matrix due to
the closer proximity of the deuteron pole and the virtual $^1$S$_0$
state. In order to study the sensitivity of the NA scattering observables
to different NN t-matrices at lower energies we show in Fig.~4
the observables for proton scattering from $^{40}$Ca at 160~MeV and
in Fig.~5 the ones for proton scattering from $^{16}$O at 135~MeV.
Again, all three potential models lead to nearly identical results.
\hspace*{10mm}
We do not want to carry out further studies at lower energies, since it
is well known that the impulse approximation alone is not adequate to
describe the scattering observables at lower energies
\cite{med2,hugo1,edff}. We prefer to pursue further
investigations to find out why expected off-shell differences in the
potential models are not visible in the elastic NA observables.
\section{Investigation of Off-Shell Differences}
\hspace*{10mm}
In the full-folding optical potential as given in Eq.~(\ref{eq:2.5})
the energy of propagation in the NN t-matrix is coupled to the
integration variable. This makes it difficult to access effects
resulting from the off-shell structure of the NN t-matrices separately.
For this reason, we prefer to carry out the following study using the
optimum factorized form of the optical potential, which has been shown
to be quite a good approximation to the full-folding expression at
projectile energies of 200~MeV and higher \cite{edff}. The optimum
factorized form is characterized by two approximations. First, the
energy $\cal E$ of the NN t-matrix in Eq.~(\ref{eq:2.5}) is fixed at
half the projectile energy (in the laboratory frame)
\begin{equation}
{\cal E} \equiv E_0 = \frac{1}{2} \frac{k^2_{lab}}{2m_N}=
\frac{1}{2}\frac{{(\frac{A+1}{A}k_0)}^2}{2m_N}.\label{eq:3.1}
\end{equation}
Here $k_{lab}$ and $k_0$ are the on-shell momenta in the
laboratory and NA system respectively, and
$m_N$ is the mass of a nucleon. Second, the NN t-matrix and
the M\o ller factor are expanded
in ${\bf P}$ around a fixed value ${\bf P_0}$, determined by the
requirement that the contribution of the first derivative term is
minimized. For elastic scattering the contribution vanishes if
${\bf P_0}$ is chosen to be zero \cite{pttw,ernst}. With these
assumptions, the expression for the optical potential in the optimum
factorized form is given as
\begin{equation}
{\hat U}_{fac}({\bf q},{\bf K})=\sum_{i=p,n}
\;\eta({\bf q},{\bf K})\;\hat{\tau}_{0i}
\left({\bf q},{{A+1}\over{2A}}{\bf K} ,E_0 \right)\;
\rho_{i}\left(q \right). \label{eq:4.2}
\end{equation}
In this form the non-local character of the optical potential is
solely determined by the off-shell NN t-matrix and the
M\o ller factor. If we now consider the
integral equation for elastic NA scattering as given in Eq.~(\ref{eq:2.1}),
we see that only the second term in the right hand side of
Eq.~(\ref{eq:2.1}) contains the integration over the optical potential.
The driving term, ${\hat U}_{fac}({\bf k'_0},{\bf k_0},E)$ contains the
NN t-matrix evaluated at the fixed momenta ${\bf k'_0}$ and
${\bf k_0}$, multiplied
with the density profile $\rho_i(q)$. In this case the momentum
vectors ${\bf q}$ and ${\bf K}$ are
${\bf q}={\bf k'}_0 - {\bf k}_0$ and ${\bf K}=\frac{1}{2}
({\bf k'}_0+{\bf k}_0)$.
\hspace*{10mm}
In order to study off-shell effects, we define the following
quantity
\begin{equation}
B({\bf k'}_0,{\bf k}_0,E)= \lim_{\epsilon \rightarrow 0}
\int_0^{\infty} {d^3{\bf k''}}\;\frac{ {\hat U}_{fac}
({\bf k'}_0,{\bf k''},E) \;
T({\bf k''},{\bf k}_0,E)}{E -E(k'')+ i\epsilon}, \label{eq:4.3}
\end{equation}
where $T({\bf k''},{\bf k}_0,E)$ is the solution of
Eq.~({\ref{eq:2.1}}), obtained using the optical potential in the
factorized form. Here $B({\bf k'}_0,{\bf k}_0,E)$ represents the
integral on the right-hand side of Eq.~({\ref{eq:2.1}}), and thus
the quantity in which the optical potential $U$ enters off-shell when
calculating $T_{el}$.
Since the nuclear density in momentum space is a
function strongly peaked for small momenta, we may conjecture that the
density will dominate the fall-off behavior of
${\hat U}_{fac} ({\bf k'}_0,{\bf k''},E)$ for large values of the integration
variable ${\bf k''}$. To investigate this more closely, we write
Eq.~(\ref{eq:4.3}) as
\begin{equation}
B({\bf k'}_0,{\bf k}_0,E)= \lim_{\epsilon \rightarrow 0}
\int d\Omega'' \int_0^{k_{max}}
dk'' \; k''^2 \frac{ {\hat U}_{fac}({\bf k'}_0,{\bf k''},E) \;
T({\bf k''},{\bf k}_0,E)}{E -E(k'')+i\epsilon}, \label{eq:4.4}
\end{equation}
and study the behavior of $B({\bf k'}_0,{\bf k}_0,E)$ as a function of
$k_{max}$. Since $B({\bf k'}_0,{\bf k}_0,E)$ depends on vector variables,
we actually have $B(k_0,k_0,\theta,E)$, where $\theta$ is the
angle between ${\bf k'_0}$ and ${\bf k_0}$. In Fig.~6 we show
the real part $Re \; B(k_0,k_0,\theta,E)$
for different values of $k_{max}$ for neutron
scattering from $^{16}$O at 200~MeV projectile energy, and in
Fig.~7 for neutron scattering from $^{90}$Zr at the same energy.
We see that in the case of $^{16}$O an integration up to $k_{max} =
k_0+1.0$~fm$^{-1}$ is already sufficient to obtain the full result.
In the case of $^{90}$Zr one only needs to integrate to
$k_{max} = k_0+0.5$~fm$^{-1}$ to have a result identical to the
complete integral. In both cases $k_0 \sim 3$~fm$^{-1}$. When considering
the imaginary part of $B(k_0,k_0,\theta,E)$ we arrive at the same
conclusion. We carried out
similar tests at different energies and arrived essentially at the same
values for $k_{max}$ for the two different nuclei. Assuming that the
nuclear density is responsible for the fast fall-off of the optical
potential as function of ${\bf k'}$, this finding is not surprising.
From Figs.~6 and 7 we also see that for a heavier nucleus the
contribution beyond the on-shell value $k_0$ is much less than for a
light nucleus. Again, this is not too surprising, when one recalls the
functional form of the nuclear density profiles. The density profile
$\rho_p (q)$ for the proton distribution of $^{16}$O has its first
minimum at $q \sim 2$~fm$^{-1}$, whereas the proton distribution of
$^{90}$Zr has its first minimum at $q \sim 1$~fm$^{-1}$.
\hspace*{10mm}
In order to verify that the functional form of the density
is the limiting factor for the range of the integration, we
identify in Eq.~(\ref{eq:4.4}) $T({\bf k''},{\bf k}_0,E)$ as well
as ${\hat U}_{fac}({\bf k_0},
{\bf k''},E)$ with the density $\rho(|{\bf k''}-{\bf k_0}|)=\sum_{i=p.n}
\rho_i(|{\bf k''}-{\bf k_0}|)$ to obtain
\begin{equation}
B'({\bf k'_0},{\bf k_0},E)= \lim_{\epsilon \rightarrow 0}
\int d\Omega'' \int_0^{k_{max}}
dk'' \; k''^2 \frac{ \rho(|{\bf k''}-{\bf k_0}|)
\rho(|{\bf k''}-{\bf k'_0}|)}{E -E(k'')+i\epsilon},
\label{eq:4.5}
\end{equation}
and repeat the above study, namely consider $B'(k_0,k_0,\theta,E)$
as a function of $k_{max}$. In Fig.~8 we plot the real part
$Re \;B'(k_0,k_0,\theta,E)$ using different
integration ranges. The result is similar to the one in Fig.~6 and 7.
Considering Fig.~8 the upper bound for the
integral can be constrained to $k_0 +1.5~$fm$^{-1}$. Since
the integral in Eq.~(\ref{eq:4.5}) is symmetric about the on-shell
value, the lower bound of integration can be constrained to
$k_0 -1.5~$fm$^{-1}$.
\hspace*{10mm}
After having found that only a limited region of the NN t-matrix enters
a calculation of NA elastic scattering observables, we need to project
this region, namely $k_0 \pm 1.5~$fm$^{-1}$, on the NN t-matrices and
see if the t-matrices employed in our calculations differ in this
restricted region. Thus we show first in Fig.~9 the real part
of the off-shell Wolfenstein amplitude A,
$Re \;A(k'_{NN},k_{NN},E_0)$, at
200 MeV obtained from the NijmI potential as function of $k_{NN}$
and $k'_{NN}$. Here the angle between ${\bf k_{NN}}$ and
${\bf k'_{NN}}$ is chosen to be zero. The value of the on-shell momentum is
located at $k_{NN}=k'_{NN}=1.55$~fm$^{-1}$. It should be noted that the
values of $Re \;A$ in the plotted domain range between
0.6 and -3~MeVfm$^3$.
Since the study of the integration bounds in the integral
$B(k_0,k_0,\theta,E)$ was carried out using momenta defined in the NA
system, we use Eqs.~(\ref{eq:2.5a}) and (\ref{eq:2.5b}) to transform
the bounds to momenta given in the NN system. As a reminder,
since we work in the optimum factorized form, the momentum ${\bf P}$ in
Eqs.~(\ref{eq:2.5a}) and (\ref{eq:2.5b}) is zero. Using these
transformations, which are explicitly given as
${\bf k'_{NN}}= \frac{1}{4} [(1/A+3){\bf k'_0}+(1/A-1){\bf k''}]$
and ${\bf k_{NN}}= \frac{1}{4} [(1/A-1){\bf k'_0}+(1/A+3){\bf
k''}]$, we obtain the `skew box' given in Fig.~9 as region of
the NN t-matrix whose values enter the NA scattering equation.
\hspace*{10mm}
Next, we display in Fig.~10 the difference between the real parts
of the Wolfenstein amplitudes $Re \;A(k'_{NN},k_{NN},E_0)$
given
by the NijmI and CD-Bonn potentials, again as function of $k_{NN}$,
$k'_{NN}$ and the angle between the two vectors being zero.
First we notice that within the plotted region the off-shell differences
between the two amplitudes is relatively small. Only for $k_{NN}=
k'_{NN} \sim 5$~fm$^{-1}$ there is a difference larger than
0.2~MeVfm$^3$. Again, the on-shell value is located at
$k_{NN}=k'_{NN}=1.55$~fm$^{-1}$. The region which enters a calculation of
NA scattering observables is again indicated by a `skew box'. Within this
box there are essentially no differences between the amplitudes.
The largest difference is located in the upper right corner of
the `skew box' almost opposite the on-shell point, and is about 6\%
of the total value of $Re \;A$.
\hspace*{10mm}
In Fig.~11 we show the difference between the real parts of the
Wolfenstein amplitudes derived from the NijmI
and NijmII potentials. These two amplitudes show off-shell differences of
1~MeVfm$^3$ and larger for values of $k_{NN}=k'_{NN} \sim 4$~fm$^{-1}$.
However, in the region which is sampled by NA scattering calculations
(`skew box') both amplitudes are nearly identical in the lower left
half around the on-shell value. Larger differences between those two
potentials are located in the uppler right corner furthest away from
the on-shell value. For the other Wolfenstein
amplitude, which enters our calculations of elastic NA scattering
observables, we obtain similar conclusions.
A close inspection of the scattering observables for the light nuclei
$^{16}$O and $^{40}$Ca in Figs.~1 and 2 shows that the dash-dotted curves
representing the calculations with the NijmII model can be distinguished
from the other two curves, especially at larger angles. However,
the differences in the observables are still quite small, indicating
the calculations are dominated by the area around the on-shell value.
Thus we can see, that although the Wolfenstein amplitudes A and C
derived from the different NN
potentials under study exhibit differences for large off-shell momenta,
the off-shell region which is sampled in NA elastic scattering
calculations is restricted to an area close to the on-shell value
and thus does not probe those far off-shell regions where the larger
differences occur.
\section{Summary and Conclusion}
\hspace*{10mm}
In this paper we addressed the question if nucleon-nucleus
elastic scattering observables are sensitive to different off-shell
structures of NN transition amplitudes derived from realistic NN potentials.
Our study is based on the recently developed potential models NijmI and
NijmII by the Nijmegen group \cite{nijm} and the charge dependent Bonn
potential \cite{cdbonn}. All three potentials models describe the
Nijmegen NN data base with a $\chi^2$ per datum =1.03. Thus the
transition matrices derived from these models can be considered on-shell
equivalent. The Wolfenstein amplitudes, which enter our NA calculations,
show considerable differences for large off-shell momenta. However,
these differences are not visible in the NA elastic scattering
observables.
\hspace*{10mm}
We calculated elastic scattering observables for proton scattering from
$^{16}$O, $^{40}$Ca, and $^{208}$Pb in the energy regime between
100 and 200~MeV projectile energy. Here we calculated the full-folding
integral for the first order optical potential using the impulse
approximation within the framework of the spectator expansion of
multiple scattering theory. In addition to the NN t-matrices from the
three above mentioned potential models our optical potentials employ a
Dirac-Hartree model for the nuclear density matrix. Recoil and frame
transformation factors are implemented in the calculation in their
complete form. We find that the elastic scattering observables based on
the three different potential models are almost identical. A very
similar result has been obtained in Ref.~\cite{hugo1}. This work
employs different density matrices and is based on the Paris potential
and inversion potentials which are constructed to be phase-shift
equivalent to the Paris potential as well as to the experimentally
extracted phase shifts.
\hspace*{10mm}
In order to better understand our numerical results, we study the regions
of the NN t-matrices, which are sampled in a calculation of NA elastic
scattering observables within the off-shell $t\rho$ or optimum
factorized approximation to the full-folding optical potential. In this
approximation the off-shell character of the optical potential is solely
determined by the off-shell NN t-matrix. This feature allows us to
determine, which region of off-shell momenta for a fixed energy slice of
the NN t-matrix enter the calculation. Our investigation of the
rescattering term of the Lippmann-Schwinger equation shows, that the
off-shell dependence of the optical potential is limited by the nuclear
density, which in momentum space is a strongly peaked function for small
momenta. It is well known that the heavier the nucleus becomes,
the stronger is that forward peaking. This
property of the nuclear density prevents far off-shell momenta of the NN
t-matrix from entering the optical potential and thus the NA scattering
observables.
The coincidence of the calculations based on the
different realistic NN potentials strongly indicates that only off-shell
momenta close to the on-shell value of the NN t-matrix are relevant
for NA scattering. In this region the different potentials still give
very similar results for the NN t-matrix.
\hspace*{10mm}
Comparing our calculations of elastic scattering observables to
experimental data, we still find some systematic inabilities of the
first order full-folding optical potential to describe certain details
for the NA scattering data in the considered energy regime. However,
we find the
limitations of the first order optical potential cannot be attributed
to uncertainties associated with the off-shell behavior of the realistic
NN t-matrices employed.
\vfill
\acknowledgments
The authors want to thank W. Gl\"ockle for many stimulating, helpful and
critical discussion during this project.
This work was performed in part under the auspices of the U.~S.
Department of Energy including contract No. DE-FG02-93ER40756 with
Ohio University. One of us (D.H.) would like to thank the Deutsche
Forschungsgemeinschaft for their support.
We thank the Ohio Supercomputer Center (OSC) for
the use of their facilities under Grant
No.~PHS206
as well as the National Energy Research Supercomputer Center
(NERSC) for the use of their facilities
under the FY1997 Massively Parallel Processing Access Program.
|
1,314,259,992,986 | arxiv |
\section{Supplementary material for \algothree}
\label{app:rsd}
\subsection{Description of the algorithm}
\label{app:rsd_descript}
This section provides a complete description of \algothree[.]Its pseudocode is given in Algorithm~\ref{alg:algo3}. It relies on auxiliary protocols described by Protocols~\ref{proto:init}, \ref{proto:rsd}, \ref{proto:listen}, \ref{proto:sendbit}, \ref{proto:prefsignal} and \ref{proto:punishsemi}.
\begin{algorithm2e}[h]
\DontPrintSemicolon
\KwIn{$T, \delta$}
$\widehat{M}, j \gets \init(T, K)$;\ state $\gets$ ``exploring'' and $\text{blocknumber} \gets 1$ \;
Let $\pmb{\pi}$ be a $M\! \times\! M$ matrix with only $0$ \tcp*{$\pi_k^j$ is the $k$-th preferred arm by $j$}
\While{$t < T$}{
$\text{blocktime} \gets t \ (\text{mod } 5K+MK+M^2K)+1$ \;
\If(\tcp*[f]{new block}){blocktime $= 1$}{blocknumber $\gets \text{blocknumber}\ (\text{mod } M) + 1$;\ $b_k^j(t) \gets \sqrt{2\log(T)/T_k^j(t)}$ \;
Let $\lambda^j$ be the ordering of the empirical means: $\widehat{\mu}_{\lambda^j_k}^j(t) \geq \widehat{\mu}_{\lambda^j_{k+1}}^j(t)$ for any $k$\;
\lIf(\tcp*[f]{send Top-M arms}){(blocknumber, state) $=(j, \text{``exploring''})$ and $\forall k \in [M], \hat{\mu}_{\lambda^j_k}^j - b_{\lambda^j_k}^j \geq \hat{\mu}_{\lambda^j_{k+1}}^j + b_{\lambda^j_{k+1}}^j$\\}{$\pi^j \gets \lambda^j$;\ state $\gets \prefsignal(\pmb{\pi}, j)$}
}
$(l, \text{comm\_arm}) \gets \rsd(\pmb{\pi}, \text{blocknumber})$ \tcp*{$j$ pulls $l^j$}
\vspace{0.5em
\If{state $=$ ``exploring''}{
Pull $l^j$ and update $\widehat{\mu}_{l^j}^j$ \;
\uIf(\tcp*[f]{received signal}){$l^j = \text{comm\_arm}$ and $\eta_{l^j} = 1$}{ \lIf{blocktime $> 4K$}{state $\gets$ ``punishing''}
\lElse{$(\text{state}, \pi^{\text{blocknumber}}) \gets \listen(\text{blocknumber}, \text{state}, \pmb{\pi}, \text{comm\_arm})$}}
}
\vspace{0.5em
\If{state $=$ ``exploiting'' and $\exists i,k \text{ such that } \pi^i_k = 0$}{
Pull $l^j$ \tcp*{arm attributed by RSD algo}
\uIf(\tcp*[f]{received signal}){$l^j \not\in \lbrace l^i | i \in [M]\setminus\{j\} \rbrace$ and $\eta_{l^j}(t) = 1$}{ \lIf{blocktime $> 4K$}{state $\gets$ ``punishing''}
\lElse{ $(\text{state}, \pi^{\text{blocknumber}}) \gets \listen(\text{blocknumber}, \text{state}, \pmb{\pi}, \text{comm\_arm})$}}
}
\vspace{0.5em}
\If(\tcp*[f]{all players are exploiting}){state $=$ ``exploiting'' and $\forall i,k, \pi^i_k \neq 0$}{
Draw inspect $\sim$ Bernoulli$(\sqrt{\log(T)}/T)$ \;
\uIf(\tcp*[f]{random inspection}){inspect $= 1$}{
Pull $l^i$ with $i$ chosen uniformly at random among the other players \;
\lIf(\tcp*[f]{lying player}){$\eta_{l^i} = 0$}{state $\gets$ ``punishing''}}
\uElse{Pull $l^j$;\
\lI
{observed two collisions in a row}{state $\gets$ ``punishing''}
}}
\lIf{state $=$ ``punishing''}{$\punishsemi(\delta)$}
}
\caption{\label{alg:algo3}\algothree}
\end{algorithm2e}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\pmb{\pi}, \text{blocknumber}$}
taken\_arms $\gets \emptyset$ \;
\For{$s = 0, \ldots, M -1$}{
dict $\gets s + \text{blocknumber} - 1 (\text{mod } M) +1$ \tcp*{current dictator}
$p \gets \min \{p' \in [M] \ | \ \pi^{\text{dict}}_{p'} \not\in \text{taken\_arms}\}$ \tcp*{best available choice}
\lIf{$\pi^{\text{dict}}_{p} \neq 0$}{ $l^{\text{dict}} \gets \pi^{\text{dict}}_{p}$ and add $\pi^{\text{dict}}_{p}$ to taken\_arms}
\lElse(\tcp*[f]{explore}){$l^{\text{dict}} \gets t+\text{dict} \ (\text{mod } K) + 1$}
}
comm\_arm $\gets \min [K] \setminus \text{taken\_arms}$ \;
\Return ($l$, comm\_arm)
\caption{\label{proto:rsd}\rsd}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\text{blocknumber}, \text{state}, \pmb{\pi}, \text{arm\_comm}$}
$\text{ExploitPlayers} = \{ i \in [M] \ | \ \pi^i_1 \neq 0 \}$;\quad
$\lambda \gets \pi^{\text{blocknumber}}$ \;
\lIf{$\lambda_1 \neq 0$}{state $\gets$ ``punishing'' \tcp*[f]{this player already sent}}
\lWhile{blocktime $\leq 2K$}{Pull $t+j (\text{mod } K) +1$}
\lIf(\tcp*[f]{repeat signal}){blocktime $= 2K$}{ $\sendbit(\text{comm\_arm}, \text{ExploitPlayers},j)$ }
\lElse{\lWhile{blocktime $\leq 4K$}{Pull $t+j (\text{mod } K) +1$} }
\vspace{-0.5em}
\For{$K$ rounds}{\lIf(\tcp*[f]{signal punishment}){state $=$ ``punishing''}{Pull $j$}
\uElse{Pull $k = t+j (\text{mod } K) + 1$ ;\quad
\lIf{$\eta_k=1$}{state $\gets$ ``punishing'' }}}
\vspace{0.5em}
\For(\tcp*[f]{receive preferences}){$n=1,\ldots,MK$}
{Pull $k = t+j (\text{mod } K) + 1$ \;
$m \gets \left\lceil n/K \right\rceil$ \tcp*{communicating player sends her $m$-th pref.\ arm}
\uIf{$\eta_{k} = 1$}{
\lIf{$\lambda_m \neq 0$}{state $\gets$ ``punishing'' \tcp*[f]{received two signals}}
\lElse{$\lambda_m \gets k$}
}}
\vspace{0.5em}
\For(\tcp*[f]{repetition block}){$n=1,\ldots,M^2 K$}
{$m \gets \left\lceil \frac{n \ (\text{mod } MK)}{K} \right\rceil$ and $l\gets \lceil \frac{n}{MK} \rceil$ \tcp*{$l$ repeats the $m$-th pref.}
\lIf{$j=l$}{Pull $\lambda_m$
}
\uElse{Pull $k = t+j \ (\text{mod } K) + 1$ \;
\lIf{$\eta_k=1$ and $\lambda_m \neq k$}{state $\gets$ ``punishing'' \tcp*[f]{info differs}}}}
\lIf{$\mathrm{Card}\left\lbrace \lambda_m \neq 0 \ | \ m \in [M]\right\rbrace \neq M$}{state $\gets$ ``punishing'' \tcp*[f]{did not send all}}
\Return (state, $\lambda$)
\caption{\label{proto:listen}\listen}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\text{comm\_arm}, \text{ExploitPlayers},j$}
\lIf{$\text{ExploitPlayers} = \emptyset$}{$\widetilde{j} \gets j$}
\lElse{$\widetilde{j} \gets \min \text{ExploitPlayers}$}
\lFor(\tcp*[f]{send bit to exploiting players}){$K$ rounds}{Pull $t+ \widetilde{j} (\text{mod } K) +1$}
\lFor(\tcp*[f]{send bit to exploring players}){$K$ rounds}{Pull comm\_arm}
\caption{\label{proto:sendbit}\sendbit}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\pmb{\pi}, j, \text{comm\_arm}$}
$\text{ExploitPlayers} = \{ i \in [M] \setminus \{j\} \ | \ \pi^i_1 \neq 0 \}$;\
$\lambda \gets \pi^j$ \tcp*{$\lambda$ is signal to send}
state $\gets$ ``exploiting'' \tcp*{state after the protocol}
$\sendbit(\text{comm\_arm}, \text{ExploitPlayers},j)$ \tcp*{initiate communication block}
\vspace{0.5em}
\lFor(\tcp*[f]{wait for repetition}){$2K$ rounds}{Pull $t+j (\text{mod } K) +1$}
\vspace{0.5em}
\For(\tcp*[f]{receive punish signal}){$K$ rounds}{Pull $t+j (\text{mod } K) +1$; \lIf{$\eta_k=1$}{state $\gets$ ``punishing'' }}
\vspace{0.5em}
\lFor(\tcp*[f]{send $k$-th preferred arm}){$n=1,\ldots, MK$}{pull $\lambda_{\left\lceil \frac{n}{K} \right\rceil}$ }
\vspace{0.5em}
\For(\tcp*[f]{repetition block}){$n=1,\ldots,M^2 K$}
{$m \gets \left\lceil \frac{n \ (\text{mod } MK)}{K} \right\rceil$ and $l\gets \lceil \frac{n}{MK} \rceil$ \tcp*{$l$ repeats the $m$-th pref.}
\lIf{$j=l$}{Pull $\lambda_m$
}
\uElse{Pull $k = t+j \ (\text{mod } K) + 1$ \;
\lIf{$\eta_k=1$ and $\lambda_m \neq k$}{state $\gets$ ``punishing'' \tcp*[f]{info differs}}}}
\Return state
\caption{\label{proto:prefsignal}\prefsignal}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\delta$}
\lIf{$\text{ExploitPlayers}=[M]$}{collide with each player twice}
\Else(\tcp*[f]{signal punishment during rounds $3K+1, \ldots, 5K$ of a block}){
\lFor{$3K$ rounds}{Pull $t+j (\text{mod } K)+1$}
$\sendbit(\text{comm\_arm}, \text{ExploitPlayers},j)$}
$\alpha \gets \left(\frac{1+\delta}{1-\delta}\right)^2 \left(1 - 1/K\right)^{M-1}$ and $\delta' = \frac{1-\alpha}{1+3\alpha}$ \;
Set $\widehat{\mu}_k^j, S_k^j, v_k^j, n_k^j \gets 0$\;
\While(\tcp*[f]{estimate $\mu_k^j$}){$\exists k \in [K], \delta' \widehat{\mu}_k^j < 2 s_k^j(\log(T)/n_k^j)^{1/2} + \frac{14 \log(T)}{3 (n_k^j-1)}$}{Pull $k=t+j \ (\text{mod } K) +1$ \;
\uIf{$\delta' \widehat{\mu}_k^j < 2 s_k^j(\log(T)/n_k^j)^{1/2} + \frac{14 \log(T)}{3 (n_k^j-1)}$}{Update $\widehat{\mu}_k^j \gets \frac{n_k^j}{n_k^j+1}\widehat{\mu}_k^j + X_k(t)$ and $n_k^j \gets n_k^j+1$ \;
Update $S_k^j \gets S_k^j + X_k^2$ and $s_k^j \gets \sqrt{\frac{S_k^j - (\widehat{\mu}_k^j)^2}{n_k^j-1}}$ }
}
$p_k \gets \bigg( 1 - \Big(\alpha\frac{\sum_{l=1}^M \widehat{\mu}^j_{(l)}(t)}{M \widehat{\mu}^j_k(t)}\Big)^{\frac{1}{M-1}}\bigg)_+$;\quad
$\widetilde{p}_k \gets p_k/\sum_{l=1}^K p_l$ \tcp*{renormalize}
\lWhile(\tcp*[f]{punish}){$t \leq T$}{Pull $k$ with probability $p_k$}
\caption{\label{proto:punishsemi}\punishsemi}
\end{protocol}
\paragraph{Initialization phase.} \algothree starts with the exact same initialization as \algotwo[,]which is given by Protocol~\ref{proto:init}, to estimate $M$ and attribute ranks among the players. Afterwards, they start the exploration.
\medskip
In the remaining of the algorithm, as already explained in Section~\ref{sec:algo3}, the time is divided into superblocks, which are divided into $M$ blocks of length $5K+MK+M^2K$. During the $j$-th block of a superblock, the dictators ordering for RSD is $(j, \ldots, M, 1, \ldots, j-1)$. Moreover, only the $j$-th player can send messages during this block if she is still exploring.
\paragraph{Exploration.} The exploiting players sequentially pull all the arms in $[K]$ to avoid collisions with any other exploring player. Yet, they still collide with exploiting players.
\algothree is designed so that all players know at each round the $M$ preferred arms of any exploiting players and their order. The players thus know which arms are occupied by the exploiting players during a block $j$. The communication arm is thus a common arm unoccupied by any exploiting player. When an exploring player encounters a collision on this arm at the beginning of the block, this means that another player signaled the start of a communication block. In that case, the exploring player starts \listen[,]described by Protocol~\ref{proto:listen}, to receive the messages of the communicating player.
On the other hand, when an exploring player $j$ knows her $M$ preferred arms and their order, she waits for the next block $j$ to initiate communication. She then proceeds to \prefsignal[,]given by Protocol~\ref{proto:prefsignal}.
\paragraph{Communication block.}
In a communication block, the communicating player first collides with each exploiting and exploring player to signal them the start of a communication block as described by Protocol~\ref{proto:sendbit}. These collisions need to be done in a particular way given by \sendbit so that all players correctly detect the start of a communication block. These players then repeat this signal to ensure that every player is listening.
The communicating player then sends to all players her $M$ preferred arms in order of preferences.
Afterwards, each player repeats this list to ensure that no malicious player interfered during communication. As soon as some malicious behavior is observed, the start of \punishsemi[,]given by Protocol~\ref{proto:punishsemi}, is signaled to all players.
\paragraph{Exploitation.}
An exploiting player starts any block $j$ by computing the attribution of the RSD algorithm between the exploiting players given their known preferences and the dictatorship ordering $(j, \ldots, j-1)$. She then pulls her attributed arm for the whole block, unless she receives a signal.
A signal is received when she collides with an exploring player, while unintended\footnote{She normally collides with exploring players. Yet as she knows the set of exploring players, she exactly knows when this happens.}. If it is at the beginning of a block, it means that a communication block starts. Otherwise, she just enters the punishment protocol. Note that the punishment protocol starts by signaling the start of \punishsemi to ensure that every cooperative player starts punishing.
\medskip
Another security is required to ensure that the selfish player truthfully reports her preferences. She could otherwise report fake preferences to decrease another player's utility while her best arm remains uncontested and thus available. To avoid this, \algothree uses \textit{random inspections} when all players are exploiting. With probability $\sqrt{\log(T)}/T$ at each round, any player checks that some other player is indeed exploiting the arm she is attributed by the RSD algorithm.
If it is not the case, the inspecting player signals the start of \punishsemi to everyone by colliding twice with everybody, since a single collision could be a random inspection.
Because of this, the selfish player can not pull another arm than the attributed one too often without starting a punishment scheme. Thus, if she did not report her preferences truthfully, this also has a cost for her.
\newpage
\input{rsdproof.tex}
\section{Supplementary material for \algotwo}
\label{app:sicmmab}
In this whole section, $M$ is assumed to be at least $3$.
\subsection{Description of the algorithm}
\label{app:sicmmab_descript}
This section provides a complete description of \algotwo[.]The pseudocode of \algotwo is given in Algorithm~\ref{alg:algo2} and relies on several auxiliary protocols, which are described by Protocols~\ref{proto:init}, \ref{proto:meansignal}, \ref{proto:receive}, \ref{proto:send}, \ref{proto:update}, \ref{proto:signalset} and \ref{proto:punishhomo}.
\begin{algorithm2e}[h]
\DontPrintSemicolon
\KwIn{$T, \delta$}
$M, j \gets \init(T, K)$ and punish $\gets$ False\;
$\opt \gets \emptyset$, $M_p \gets M$, $[K_p] \gets [K]$ and $p \gets 1$\;
\While{not punish and $\mathrm{Card} \opt < M$}{
\For{$m=0, \ldots, \left\lceil \frac{K_p 2^p}{M_p} \right\rceil -1 $}{
ArmstoPull $\gets \opt \cup \left\lbrace i \in [K_p] \ \big| \ i-mM_p \ (\text{mod } K_p) \in [M_p] \right\rbrace$ \;
\For{$M$ rounds}{
$k \gets j+t \ (\text{mod } M) +1$ and pull $i$ the $k$-th element of ArmstoPull \;
\lIf(\tcp*[f]{$T_i^j$ pulls on $i$ by $j$ this phase}){$T_i^j(p) \leq 2^p$}{Update $\widehat{\mu}_{i}^j$}
\lIf(\tcp*[f]{collisionless exploration}){$\eta_i = 1$}{punish $\gets$ True}}
}
$(\text{punish}, \opt[,] [K_p], M_p) \gets \meansignal(\widehat{\mu}^j, j, p, \opt, [K_p], M_p)$ \;
$p \gets p+1$}
\vspace{0.5em}
\lIf{punish}{$\punishhomo(p)$}
\Else(\tcp*[f]{exploitation phase}){$k \gets j+ t \ (\text{mod } M) +1$ and pull $i$, the $k$-th arm of OptArms \;
\lIf{$\eta_i = 1$}{punish $\gets$ True}}
\caption{\label{alg:algo2}\algotwo}
\end{algorithm2e}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$T, K$}
$n_{\text{coll}} \gets 0$ and $j \gets -1$ \;
\lFor(\tcp*[f]{estim. $M$}){$12eK^2 \log(T)$ rounds}{Pull $k \sim \mathcal{U}(K)$ and $n_{\text{coll}} \gets n_{\text{coll}} + \eta_k$}
$\widehat{M} \gets 1 + \mathrm{round}\left(\log\left(1 -\frac{n_{\text{coll}}}{12eK^2 \log(T)}\right)/\log\left( 1- \frac{1}{K}\right)\right)$\;
\For(\tcp*[f]{get rank}){$K \log(T)$ rounds}{
\uIf{$j = -1$}{Pull $k \sim \mathcal{U}(\widehat{M})$;\quad
\lIf{$\eta_k = 0$}{$j \gets k$}}
\lElse{Pull $j$}
}
\Return $(\widehat{M}, j)$
\caption{\label{proto:init}\init}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\widehat{\mu}^j, j, p, \opt[,][K_p], M_p$}
punish $\gets$ False\;
\For(\tcp*[f]{receive punishment signal}){$K$ rounds}{
Pull $k = t+j \ (\text{mod } K) +1$;\quad
\lIf{$\eta_k=1$}{punish $\gets$ True}
}
$\widetilde{\mu}_k^j \gets \begin{cases} 2^{-p} \left(\lfloor 2^p \widehat{\mu}_k^j \rfloor +1\right) \text{ with proba } 2^p \widehat{\mu}_k^j - \lfloor 2^p \widehat{\mu}_k^j \rfloor \\
2^{-p} \lfloor 2^p \widehat{\mu}_k^j \rfloor \text{ otherwise } \end{cases}$ \tcp*{quantization}
\For(\tcp*[f]{$i$ sends $\widetilde{\mu}_k^i$ to $l$}){$(i,l,k) \in [M] \times \lbrace 1, 2 \rbrace \times [K]$ such that $i \neq l$}{
\uIf(\tcp*[f]{sending player}){$j =i$}{$\send(j,l,p,\widetilde{\mu}^j_k)$ and
$q \gets \receive(j,p)$ \tcp*{back and forth}
\lIf(\tcp*[f]{corrupted message}){$q \neq \widetilde{\mu}^j_k$}{punish $\gets$ True}}
\lElseIf{$j=l$}{$\widetilde{\mu}^i_k \gets \receive(j, p)$ and $\send(j,i,p,\widetilde{\mu}^i_k)$ }
\lElse{Pull $j$ \tcp*[f]{waiting for others}}}
\For(\tcp*[f]{leaders check info match}){$(i,l,m,k) \in \{(1,2), (2,1)\} \times [M] \times [K]$}{
\lIf{$j=i$}{$\send(j,l,p,\widetilde{\mu}^m_k)$}
\uElseIf{$j=l$}{$q \gets \receive(j, p)$;\quad
\lIf{$q \neq \widetilde{\mu}^m_k$}{punish $\gets$ True \tcp*[f]{info differ}}}
\lElse{Pull $j$ \tcp*[f]{waiting for leaders}}}
\lIf{$j \in \lbrace 1, 2 \rbrace$}{(Acc, Rej) $\gets \update(\widetilde{\mu}, p, \opt, [K_p], M_p)$
}
\lElse(\tcp*[f]{arms to accept/reject}){Acc, Rej $\gets \emptyset$}
(punish, Acc) $\gets \signalset(\text{Acc}, j, \text{punish})$ \;
(punish, Rej) $\gets \signalset(\text{Rej}, j, \text{punish})$ \;
\Return $(\text{punish}, \opt\cup \text{Acc}, [K_p]\setminus \left( \text{Acc} \cup \text{Rej}\right), M_p- \mathrm{Card} \text{Acc})$
\caption{\label{proto:meansignal}\meansignal}
\end{protocol}
\begin{figure*}[h]
\begin{minipage}{0.45\textwidth}
\centering
\begin{protocol}[H]
\DontPrintSemicolon
\KwIn{$j$, $p$}
$\widetilde{\mu} \gets 0$ \;
\For{$n = 0, \ldots , p$ }{
Pull $j$ \;
\lIf{$\eta_{j} (t) = 1$}{$\widetilde{\mu} \gets \widetilde{\mu} + 2^{-n}$}}
\Return $\widetilde{\mu}$ \tcp*{sent mean}
\caption*{\receive}
\caption{\label{proto:receive}\receive}
\end{protocol}
\end{minipage} \hfill
\begin{minipage}{0.45\textwidth}
\centering
\begin{protocol}[H]
\DontPrintSemicolon
\KwIn{$j$, $l$, $p$, $\widetilde{\mu}$}
$\mathbf{m} \gets$ dyadic writing of $\widetilde{\mu}$ of length $p+1$, i.e.,\ $\widetilde{\mu} = \sum_{n=0}^p m_n 2^{-n}$ \;
\For{$n = 0, \ldots , p$}{
\lIf(\tcp*[f]{send $1$}){$m_n = 1$}{Pull $l$}
\lElse(\tcp*[f]{send $0$}){Pull $j$}}
\caption*{\receive}
\caption{\label{proto:send}\send}
\end{protocol}
\end{minipage}
\end{figure*}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\widetilde{\mu}, p, \opt, [K_p], M_p$}
Define for all $k$, $i^k \gets \argmax_{j \in [M]} \widetilde{\mu}_k^j$ and $i_k \gets \argmin_{j \in [M]} \widetilde{\mu}_k^j$\;
$\widetilde{\mu}_k \gets \sum_{j \in [M]\setminus \{i^k, i_k \}} \widetilde{\mu}_k^j$ and $b \gets 4 \sqrt{\frac{\log(T)}{(M-2)2^{p+1}}}$\;
$\text{Rej} \gets \text{set of arms } k \text{ verifying } \mathrm{Card}\left\{ i \in [K_p] \ | \widetilde{\mu}_i - b \geq \widetilde{\mu}_k + b \right\} \geq M_p$ \;
$\text{Acc} \gets \text{set of arms } k \text{ verifying } \mathrm{Card}\left\{ i \in [K_p] \ | \widetilde{\mu}_k - b \geq \widetilde{\mu}_i + b \right\} \geq K_p - M_p$ \;
\Return (Acc, Rej)
\caption{\label{proto:update}\update}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$S, j, \text{punish}$}
length\_S $\gets \mathrm{Card} S$ \tcp*{length of $S$ for leaders, $0$ for others}
\For(\tcp*[f]{leaders send $\mathrm{Card} S$}){$K$ rounds}{
\lIf{$j \in \lbrace 1, 2 \rbrace$}{Pull length\_S}
\uElse{Pull $k = t+j \ (\text{mod } K) +1$\;
\lIf(\tcp*[f]{receive different info}){$\eta_k = 1$ and length\_S $\neq 0$}{punish $\gets$ True}
\lIf{$\eta_k = 1$ and length\_S $= 0$}{length\_S $\gets k$}}
}
\For(\tcp*[f]{send/receive $S$}){$n = 1, \ldots, \mathrm{length\_S}$}{
\For{$K$ rounds}{
\lIf{$j \in \lbrace 1, 2 \rbrace$}{Pull $n$-th arm of $S$}
\uElse{Pull $k = t+j \ (\text{mod } K) +1$;\quad
\lIf{$\eta_k=1$}{Add $k$ to S}}
}}
\lIf(\tcp*[f]{corrupted info}){$\mathrm{Card} S \neq \mathrm{length\_S}$}{punish $\gets$ True}
\Return (punish, $S$)
\caption{\label{proto:signalset}\signalset}
\end{protocol}
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$p$}
\uIf{communication phase $p$ starts in less than $M$ rounds}{\lFor(\tcp*[f]{signal punish to everyone}){$M+K$ rounds}{Pull $j$}}
\lElse{\lFor{$M$ rounds}{Pull the first arm of ArmstoPull as defined in Algorithm~\ref{alg:algo2}}}
%
$\gamma \gets \left(1 - 1/K\right)^{M-1}$ and $\delta = \frac{1-\gamma}{1+3\gamma}$;\quad
Set $\widehat{\mu}_k^j, S_k^j, s_k^j, n_k^j \gets 0$\;
\While(\tcp*[f]{estimate $\mu_k$}){$\exists k \in [K], \delta \widehat{\mu}_k^j < 2 s_k^j(\log(T)/n_k^j)^{1/2} + \frac{14 \log(T)}{3 (n_k^j-1)}$}{Pull $k=t+j \ (\text{mod } K) +1$ \;
\uIf{$\delta \widehat{\mu}_k^j < 2 s_k^j(\log(T)/n_k^j)^{1/2} + \frac{14 \log(T)}{3 (n_k^j-1)}$}{Update $\widehat{\mu}_k^j \gets \frac{n_k^j}{n_k^j+1}\widehat{\mu}_k^j + X_k(t)$ and $n_k^j \gets n_k^j+1$ \;
Update $S_k^j \gets S_k^j + X_k^2$ and $s_k^j \gets \sqrt{\frac{S_k^j - (\widehat{\mu}_k^j)^2}{n_k^j-1}}$ }
}
$p_k \gets \bigg( 1 - \Big(\gamma\frac{\sum_{l=1}^M \widehat{\mu}^j_{(l)}(t)}{M \widehat{\mu}^j_k(t)}\Big)^{\frac{1}{M-1}}\bigg)_+$;\quad
$\widetilde{p}_k \gets p_k/\sum_{l=1}^K p_l$ \tcp*{renormalize}
\lWhile(\tcp*[f]{punish}){$t \leq T$}{Pull $k$ with probability $p_k$}
\caption{\label{proto:punishhomo}\punishhomo}
\end{protocol}
\paragraph{Initialization phase.} The purpose of the initialization phase is to estimate $M$ and attribute ranks in $[M]$ to all the players. This is done by \init[,]which is given in Protocol~\ref{proto:init}. It simply consists in pulling uniformly at random for a long time to infer $M$ from the probability of collision. Then it proceeds to a Musical Chairs procedure so that each player ends with a different arm in $[M]$, corresponding to her rank.
\paragraph{Exploration phase.} As explained in Section~\ref{sec:algo2}, each arm that still needs to be explored (those in $[K_p]$, with Algorithm~\ref{alg:algo2} notations) is pulled at least $M2^p$ times during the $p$-th exploration phase.
Moreover, as soon as an arm is found optimal, it is pulled for each remaining round of the exploration.
The last point is that each arm is pulled the exact same amount of time by any player, in order to ensure fairness of the algorithm, while still avoiding collisions. This is the interest of the ArmstoPull set in Algorithm~\ref{alg:algo2}.
At each time step, the pulled arms are the optimal ones and $M_p$ arms that still need to be explored. The players proceed to a sliding window over these arms to explore, so that the difference in pulls for two arms in $[K_p]$ is at most $1$ for any player and phase.
\paragraph{Communication phase.} The pseudocode for a whole communication phase is given by \meansignal in Protocol~\ref{proto:meansignal}. Players first quantize their empirical means before sending them in $p$ bits to each leader. The protocol to send a message is given by Protocol~\ref{proto:send}, while Protocol~\ref{proto:receive} describes how to receive the message. The messages are sent using back and forth procedures to detect corrupted messages.
After this, leaders communicate the received statistics to each other, to ensure that no player sent differing ones to them.
They can then determine which arms are optimal/suboptimal using \update given by Protocol~\ref{proto:update}. As explained in Section~\ref{sec:algo2}, it cuts out the extreme estimates and decides based on the $M-2$ remaining ones.
\medskip
Afterwards, the leaders signal to the remaining players the sets of optimal and suboptimal arms as described by Protocol~\ref{proto:signalset}. If the leaders send differing information, it is detected by at least one player.
If the presence of a malicious player is detected at some point of this communication phase, then players signal to each other to trigger the punishment protocol described by Protocol~\ref{proto:punishhomo}.
\paragraph{Exploitation phase.} If no malicious player perturbed the communication, players end up having detected the $M$ optimal arms. As soon as it is the case, they only pull these $M$ arms in a collisionless way until the end.
\newpage
\input{proofsicmmab.tex}
\section{Full sensing setting}
\label{sec:collision}
This section focuses on the full sensing setting, where both $\eta_k(t)$ and $X_k(t)$ are always observed as we proved impossibility results for more complex settings. As mentioned before, recent algorithms leverage the observation of collisions to enable some communication between players by forcing them. Some of these communication protocols can be modified to allow robust communication.
This section is structured as follows. First, insights on two new protocols are given for robust communications. Second, a robust adaptation of SIC-MMAB is given, based on these two protocols. Third, they can also be used to reach a logarithmic RSD-regret in the heterogeneous case.
\subsection{Making communication robust}
To have robust communication, two new complementary protocols are needed. The first one allows to send messages between players and to detect when they have been corrupted by a malicious player.
If this has been the case, the players then use the second protocol to proceed to a collective punishment, which forces every player to suffer a considerable loss for the remaining of the game.
Such punitive strategies are called ``Grim Trigger'' in game theory and are used to deter defection in repeated games \citep{friedman1971, axelrod1981, fudenberg2009}.
\subsubsection{Back and forth messaging}
\label{sec:backforth}
Communication protocols in the collision sensing setting usually rely on the fact that collision indicators can be seen as bits sent from a player to another one as follows. If player $i$ sends a binary message $m_{i \to j}=(1, 0, \ldots, 0, 1)$ to player $j$ during a predefined time window, she proceeds to the sequence of pulls $(j, i, \ldots, i, j)$, meaning she purposely collides with $j$ to send a $1$ bit (reciprocally, not colliding corresponds to a $0$ bit).
A malicious player trying to corrupt a message can only create new collisions, i.e.,\ replace zeros by ones. The key point is that the inverse operation is not possible.
If player $j$ receives the (potentially corrupted) message $\widehat{m}_{i \to j}$, she repeats it to player~$i$. This second message can also be corrupted by the malicious player and player~$i$ receives~$\widetilde{m}_{i \to j}$. However, since the only possible operation is to replace zeros by ones, there is no way to transform back $\widehat{m}_{i \to j}$ to $m_{i \to j}$ if the first message had been corrupted.
The player~$i$ then just has to compare $\widetilde{m}_{i \to j}$ with $m_{i \to j}$ to know whether or not at least one of the two messages has been corrupted. We call this protocol \textit{back and forth} communication.
\medskip
In the following, other malicious communications are possible. Besides sending false information (which is managed differently), a malicious player can send different statistics to the others, while they need to have the exact same statistics. To overcome this issue, players will send to each other statistics sent to them by any player. If two players have received different statistics by the same player, at least one of them automatically realizes it.
\subsubsection{Collective punishment}
\label{sec:punish}
The back and forth protocol detects if a malicious player interfered in a communication and, in that case, a collective punishment is triggered (to deter defection). The malicious player is yet unidentified and can not be specifically targeted. The punishment thus guarantees that the average reward earned by any player is smaller than the average reward of the algorithm, $\bar{\mu}_M\coloneqq \frac{1}{M} \sum_{k=1}^M \mu_{(k)}$.
A naive way to \textit{punish} is to pull all arms uniformly at random. The selfish player then gets the reward $(1-1/K)^{M-1} \mu_{(1)}$ by pulling the best arm, which can be larger than $\bar{\mu}_M$. A good punishment should therefore pull arms more often the better they are.
\medskip
During the punishment, players pull each arm $k$ with probability $1 - \big(\gamma\frac{\sum_{l=1}^M \widehat{\mu}^j_{(l)}(t)}{M \widehat{\mu}^j_k(t)}\big)^{\frac{1}{M-1}}$ at least, where $\gamma=\left(1 - 1/K\right)^{M-1}$. Such a strategy is possible as shown by Lemma~\ref{lemma:punishment} in Appendix~\ref{app:punish}. Assuming the arms are correctly estimated, i.e., the expected reward a selfish player gets by pulling $k$ is approximately
$\mu_k (1-p_k)^{M-1}$, with ${p_k = \max \Big( 1 - \big(\gamma\frac{\bar{\mu}_M}{\mu_k}\big)^{\frac{1}{M-1}}, 0\Big)}$.
If $p_k=0$, then $\mu_k$ is smaller than $\gamma \bar{\mu}_M$ by definition; otherwise, it necessarily holds that~$\mu_k (1-p_k)^{M-1} = \gamma \bar{\mu}_M$. As a consequence, in both cases, the selfish player earns at most $\gamma \bar{\mu}_M$, which involves a relative positive decrease of $1-\gamma$ in reward w.r.t. following the cooperative strategy. More details on this protocol are given by Lemma~\ref{lemma:punishhomo} in Appendix~\ref{app:greedyproofsicmmab}.
\section{Conclusion}
We introduced notions of robustness to selfish players and provided impossibility results in hard settings.
With statistic sensing, \algoone gives a rather simple robust and efficient algorithm, besides being optimal among the class of algorithms using no collision information.
On the other hand when collisions are observed, robust algorithms relying on communication through collisions are possible.
Thanks to this, even selfish-robust algorithms can achieve near centralized regret in the homogeneous case, which is not intuitive at first sight.
In the heterogeneous case, a new adapted notion of regret is introduced and \algothree achieves a good performance with respect to it.
\medskip
\algothree heavily relies on collision observations and future work should focus on designing a comparable algorithm in both performance and robustness without this feature.
The topic of robustness to selfish players in multiplayer bandits still remains largely unexplored and leaves open many directions for future work. In particular, punishment protocols do not seem possible for general heterogeneous settings and the existence of robust algorithms for any heterogeneous setting remains open. Also, stronger notions of equilibrium can be considered such as perfect subgame equilibrium.
\section{On harder problems}
\label{sec:harder}
\vspace{-0.5em}
Following the positive results of the previous section (existence of robust algorithms) in the homogeneous case with statistical sensing, we now provide in this section impossibility results for both no sensing and heterogeneous cases.
By showing its limitations, it also suggests a proper way to consider the heterogeneous problem in the presence of selfish players.
\vspace{-0.5em}
\subsection{Hardness of no sensing setting}
\begin{thm}
\label{thm:nosensing1}
In the no sensing setting, there is no profile of strategy $s$ such that, for all problem parameters $(M, \pmb{\mu})$, $ \mathbb{E}[R_T]= o(T)$ and $s$ is an $\varepsilon(T)$-Nash equilibrium with $ \varepsilon(T) =o(T)$.
\end{thm}
\begin{proof}
Consider a strategy $s$ verifying the first property and a problem instance $(M, \pmb{\mu})$ where the selfish player only pulls the best arm. Let $\pmb{\mu'}$ be the mean vector $\pmb{\mu}$ where $\mu_{(1)}$ is replaced by $0$. Then, because of the considered observation model, the cooperative players can not distinguish the two worlds $(M, \pmb{\mu})$ and $(M-1, \pmb{\mu'})$. Having a sublinear regret in the second world implies $o(T)$ pulls on the arm $1$ for the cooperative players. So in the first world, the selfish player will have a reward in $\mu_{(1)} T - o(T)$, which is thus a linear improvement in comparison with following $s$ if $\mu_{(1)} > \mu_{(2)}$.
\end{proof}
Theorem~\ref{thm:nosensing1} is proved for a selfish players who knows the means $\pmb{\mu}$ beforehand, as the notion of Nash equilibrium prevents against any possible strategy, which includes committing to an arm for the whole game. The knowledge of $\pmb{\mu}$ is actually not needed, as a similar result holds for a selfish player committing to an arm chosen at random when the best arm is $K$ times better than the second one.
The question of existence of robust algorithms remains yet open if we restrict selfish strategies to more \textit{reasonable} algorithms.
\subsection{Heterogeneous model}
We consider the full sensing heterogeneous model, where player $j$ receives the reward~$r^j(t) \coloneqq X_{\pi^j(t)}^j(t) (1-\eta_{\pi^j(t)})$ at round~$t$, with $X_k^j \overset{\text{\tiny i.i.d.}}{\sim} \nu_k^j$ of mean $\mu_k^j$. The arm means here vary among the players. This models that transmission quality depends on individual factors such as the localization.
\subsubsection{A first impossibility result}
\begin{thm}
\label{thm:heterimposs}
If the regret is compared with the optimal assignment, there is no strategy $s$ such that, for all problem parameters $\pmb{\mu}$, $ \mathbb{E}[R_T] =o(T)$ and $s$ is an $\varepsilon(T)$-Nash equilibrium with $\varepsilon(T) = o(T)$.
\end{thm}
\begin{proof}
Assume $s$ satisfies these properties and consider a problem instance $\pmb{\mu}$ such that the selfish player unique best arm $j_1$ has mean $\mu^j_{(1)} = 1/2$ and the difference between the optimal assignment utility and the utility of the best one assigning arm $j_1$ to $j$ is $1/3$.
Such an instance is of course possible. Consider a selfish player $j$ playing exactly the strategy $s_j$ but as if her reward vector $\pmb{\mu^j}$ was actually $\pmb{\mu'^j}$ where $\mu_{(1)}^j$ is replaced by $1$ and all other $\mu_k^j$ by $0$, i.e.,\ she fakes a second world $\pmb{\mu'}$ in which the optimal assignment gives her the arm $j_1$. In this case, the sublinear regret assumption of $s$ implies that player $j$ pulls $j_1$ a time $T-o(T)$, while in the true world, she would have pulled it $o(T)$ times.
She thus earns an improvement at least $(\mu^j_{(1)} - \mu^j_{(2)}) T - o(T)$ w.r.t. playing $s_j$, contradicting the Nash equilibrium assumption.
\end{proof}
\vspace{-0.5em}
\subsubsection{Random assignments}
\label{sec:randomass}
We now take a step back and describe ``relevant'' allocation procedures for the heterogeneous case, when the vector of means $\pmb{\mu^j} $ is already known by player $j$.
An assignment is \textit{symmetric} if, when $\pmb{\mu^j} = \pmb{\mu^i}$, players $i$ and $j$ get the same \textbf{expected} utility, i.e.,\ no player is \textit{a priori} favored\footnote{The concept of fairness introduced above is stronger, as no player should be \textit{a posteriori} favored.}. It is \textit{strategyproof} if being truthful is a dominant strategy for any player and \textit{Pareto optimal} if the social welfare (sum of utilities) can not be improved without hurting any player.
Theorem~\ref{thm:heterimposs} is a consequence of Theorem~\ref{thm:zhou} below.
\begin{thm}[\citealt{zhou1990}]\label{thm:zhou}
For $M\geq 3$, there is no symmetric, Pareto optimal and strategyproof random assignment algorithm.
\end{thm}
\citet{liu2019} circumvent this assignment problem with player-preferences for arms. Instead of assigning a player to a contested arm, the latter decides who gets to pull it, following its preferences.
In the case of random assignment, \citet{abdulkadiroglu1998} proposed the Random Serial Dictatorship (RSD) algorithm, which is symmetric and strategyproof. The algorithm is rather simple: pick uniformly at random an ordering of the $M$ players. Following this order, the first player picks her preferred arm, the second one her preferred remaining arm and so on. \citet{svensson1999} justified the choice of RSD for symmetric strategyproof assignment algorithms. \citet{adamczyk2014} recently studied efficiency ratios of such assignments: if $U_{\max}$ denotes the expected social welfare of the optimal assignment, the expected social welfare of RSD is greater than $U^2_{\max}/eM$ while no strategyproof algorithm can guarantee more than $U^2_{\max}/M$. As a consequence, RSD is optimal up to a (multiplicative) constant and will serve as a benchmark in the remaining.
Indeed, instead of defining the regret in comparison with the optimal assignment as done in the classical heterogeneous multiplayer bandits, we are going to define it with respect to RSD to incorporate strategy-proofness constraints. Formally, the RSD-regret is defined as:
\begin{small}
\begin{equation*}
R^{\text{RSD}}_T \coloneqq T{\mathlarger \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)} } \bigg[ \sum_{k = 1}^{M} \mu_{\pi_\sigma(k)}^{\sigma(k)}\bigg] - \sum_{t=1}^T \sum_{j =1}^M \mu^j_{\pi^j(t)}(t) \cdot (1-\eta_{\pi^j(t)}(t)) ,
\end{equation*}
\end{small}with $\mathfrak{S}_M$ the set of permutations over $[M]$ and $\pi_\sigma(k)$ the arm attributed by RSD to player $\sigma(k)$ when the order of dictators is $(\sigma(1), \ldots, \sigma(M))$. Mathematically, $\pi_\sigma$ is defined by:
\begin{small}
\begin{equation*}
\pi_\sigma(1) = \argmax_{l \in [M]} \mu_l^{\sigma(1)} \quad \text{and} \quad \pi_\sigma(k+1) = \argmax\limits_{\substack{l \in [M] \\ l \not\in \lbrace \pi_\sigma(l') \ | \ l' \leq k\rbrace}} \mu_l^{\sigma(k+1)}.
\end{equation*}
\end{small}
\vspace{-1cm}
\section{Introduction}
In the classical stochastic Multi Armed Bandit problem (MAB), a player repeatedly chooses among $K$ fixed actions (a.k.a.~arms). After pulling arm $k \in [K] \coloneqq \lbrace 1, \ldots, K \rbrace$, she receives a random reward in $[0,1]$ of mean $\mu_k$. Her goal is to maximize her cumulative reward up to some horizon $T \in \mathbb{N}$. The performance of a pulling strategy (or algorithm) is assessed by the growth of \textit{regret}, i.e., the difference between the highest possible expected cumulative reward and the actual cumulative reward.
Since the means $\mu_k$ are unknown beforehan
, the player trades off gathering information on under-sampled arms (exploration) vs.\ using her information (exploitation). Optimal solutions are known in the simplest model \citep{LaiRobbins, Agrawal95, Auer2002}. We refer to \citep{survey1, survey2, survey3} for an extensive study of MAB.
This simple model captures many sequential decisions problems including clinical trials \citep{thompson33, robbins52} and online recommandation systems \citep{Li2010} and has therefore known a large interest in the past decades.
\medskip
Another classical application of MAB is cognitive radios \citep{jouini, anandkumar}. In this context, an arm corresponds to a channel on which a player decides to transmit and the reward is its transmission quality. A key feature of this model, is that it involves several players using channels simultaneously. If several players choose the same arm/channel at the same time, then they \textit{collide}
and receive a null reward. This setting remains somehow simple when a central agent controls simultaneously all players \citep{anantharam, centralized2}, which is far from being realistic.
In reality, the problem is indeed completely decentralized: players are independent, anonymous and cannot communicate to each other. This requires the construction of new algorithms and the development of new techniques dedicated to this \textit{multiplayer bandit} problem. Interestingly, there exist several variants of the base problem, depending on the assumption made on observations/feedback received \citep{avner14, musicalchair, besson2018, lugosi2018, magesh2019}.
More precisely, when players systematically know whether or not they collide, this observation actually enables communication between players and a collective regret scaling as in the centralized case is possible, as observed recently \citep{boursier2018, proutiere2019}. Using this idea, it is even possible to asymptotically reach the optimal assignment \citep{GoT2018, tibrewal2019, boursier2019} in the heterogeneous model where the performance of each arm differs among players \citep{kalathil2014, avner15, avner18}. \citet{liu2019} considered the heterogeneous case, when arms also have preferences over players.
\medskip
For the aforementioned result to hold, a crucial (yet sometimes only implicitly stated) assumption is that all players follow cautiously and meticulously some designed protocols and that none of them tries to free-ride the others by acting greedily, selfishly or maliciously. The concern of designing multiplayer bandit algorithms robust to such players has been raised \citep{attar2012}, but only addressed under the quite restrictive assumption of adversarial players called \textit{jammers}. Those try to perturb as much as possible the cooperative players \citep{wang2015, sawant2018, sawant2019}, even if this is extremely costly to them as well.
Because of this specific objective, they end up using tailored strategies such as only attacking the top channels.
We focus instead on the construction of algorithms with ``good'' regret guarantees even if one (or actually more) selfish player does not follow the common protocol but acts strategically in order to manipulate the other players in the sole purpose of increasing her own payoff -- maybe at the cost of other players. This concept appeared quite early in the cognitive radio literature \citep{attar2012}, yet it is still not understood as robustness to selfish player is intrinsically different (and even non-compatible) with robustness to jammers, as shown in Section~\ref{sec:jamvsgreedy}. In terms of game theory, we aim at constructing ($\varepsilon$-Nash) equilibria in this repeated game with partial observations.
\medskip
The paper is organized as follows.
Section~\ref{sec:model} introduces notions and concepts of selfishness-robust multiplayer bandits and showcases reasons for the design of robust algorithms.
Besides its state of the art regret guarantees when collisions are not directly observed, \algoone[,]presented in Section~\ref{sec:algo1}, is also robust to selfish players.
In the more complex settings where only the reward is observed or the arm means vary among players, Section~\ref{sec:harder} shows that no algorithm can guarantee both a sublinear regret and selfish-robustness.
The latter case is due to a more general result for random assignments. Instead of comparing the cumulated reward with the best collective assignment in the heterogeneous case, it is then necessary to compare it with a \textit{good} and appropriate suboptimal assignment, leading to the new notion of \textit{RSD-regret}.
When collisions are always observed, Section~\ref{sec:collision} proposes selfish-robust communication protocols. Thanks to this, an adaptation of the work of \citet{boursier2018} is possible to provide a robust algorithm with a collective regret almost scaling as in the centralized case.
In the heterogeneous case, this communication -- along with other new deviation control and punishment protocols -- is also used to provide a robust algorithm with a logarithmic RSD-regret.
\medskip
Our contributions are thus diverse: on top of introducing notions of selfish-robustness, we provide robust algorithms with state of the art regret bounds (w.r.t. non-robust algorithms) in several settings. This is especially surprising when collisions are observed, since it leads to a near centralized regret. Moreover, we show that such algorithms can not be designed in harder settings. This leads to the new, adapted notion of RSD-regret in the heterogeneous case with selfish players and we also provide a \textit{good} algorithm in this case.
These results of robustness are even more intricate knowing they hold against any possible selfish strategy, in contrast to the known results for jammer robust algorithms.
\section{Problem statement} \label{sec:model}
In this section, we describe formally the model of multiplayer MAB and introduce concepts and notions of robustness to selfish players (or equilibria concepts).
\vspace{-0.5em}
\subsection{Model}
We denote the transmission qualities of the channels by $(X_k(t))_{1\leq k \leq K} \in [0,1]$, drawn i.i.d.\ according to $\nu_k$ of expectation $\mu_k$.
In the following, arm means are assumed to be different and $\mu_{(i)}$ denotes the \mbox{$i$-th} largest mean, i.e.,\ $\mu_{(1)} > \mu_{(2)} > \ldots > \mu_{(K)}$.
At each round $t \in [T]$, all $M$ players simultaneously pull some arms, choice solely based only on their past own observations with $M \leq K$. We denote by $\pi^j(t)$ the arm played by player $j$, that generates the reward
\vspace{-0.5em}
\begin{gather*}
r^j(t) \coloneqq X_{\pi^j(t)}(t) \cdot (1-\eta_{\pi^j(t)}(t)), \\\text{where } \eta_k(t) \coloneqq \mathds{1}\left({\small \mathrm{Card}\{j \in [M] \ |\ \pi^j(t) = k\}>1}\right) \text{ is the collision indicator}.
\end{gather*}
The performance of an algorithm is measured in terms of regret, i.e., the difference between the maximal expected reward and the algorithm cumulative reward after $T$ steps\footnote{As usual, the fact that the horizon $T$ is known is not crucial \citep{degenne2016}.}:
\vspace{-0.5em}
\begin{small}
\begin{equation*}
R_T \coloneqq T \sum_{k = 1}^{M} \mu_{(k)} - \sum_{t=1}^T\sum_{j =1}^M \mu_{\pi^j(t)}(t) \cdot (1-\eta_{\pi^j(t)}(t)).
\end{equation*}
\end{small}
In multiplayer MAB, three different observation settings are considered.
\vspace{-0.5em}
\begin{description}\itemsep-0.2em
\item[Full sensing:] each player observes both $\eta_{\pi^j(t)}(t)$ and $X_{\pi^j(t)}(t)$ at each round.
\item[Statistic sensing:] each player observes $X_{\pi^j(t)}(t)$ and $r^j(t)$ at each round, e.g.,\ the players first sense the quality of a channel before trying to transmit on it.
\item[No sensing:] each player only observes $r^j(t)$ at each round.
\end{description}
Players are not able to directly communicate to each other, since it involves significant time and energy cost in practice. Some form of communication is still possible between players through observed collisions and has been widely used in recent literature \citep{boursier2018, boursier2019, tibrewal2019, proutiere2019}.
\vspace{-0.5em}
\subsection{Considering selfish players} \label{sec:jamvsgreedy}
As mentioned in the introduction, the literature focused on adversarial malicious players, a.k.a.~\textit{jammers}, while considering selfish players instead of adversarial ones is as (if not more) crucial.
These two concepts of malicious players are fundamentally different. Jamming-robust algorithms must stop pulling the best arm if it is being jammed. Against this algorithm, a selfish player could therefore pose as a jammer, always pull the best arm and be left alone on it most of the time.
On the contrary, an algorithm robust to selfish players has to actually pull this best arm if jammed by some player in order to ``punish'' her so that she does not benefit from deviating from the collective strategy.
\medskip
We first introduce some game theoretic concepts before defining notions of robustness. Each player $j$ follows an individual strategy (or algorithm)~$s_j \in \mathcal{S}$ which determines her action at each round given her past observations.
We denote by $(s_1, \ldots, s_M)=s \in \mathcal{S}^M$ the strategy profile of all players and by $(s',s_{-j})$ the strategy profile given by $s$ except for the $j$-th player whose strategy is replaced by $s'$. Let $\mathrm{Rew}^j_T(s) $ be the cumulative reward of player $j$ when players play the profile $s$.
As usual in game theory, we consider a single selfish player -- even if the algorithms we propose are robust to several selfish players assuming $M$ is known beforehand (its initial estimation can easily be tricked by several players).
\vspace{-0.25em}
\begin{defin}
A strategy profile $s\in \mathcal{S}^M$ is an $\varepsilon$-Nash equilibrium if for any $s' \in \mathcal{S}$ and $j \in [M]$:
\vspace{-0.5em}
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^j_T(s',s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^j_T(s)] + \varepsilon.
\end{equation*}
\end{small}
\end{defin} \vspace{-0.5em}
This simply states that a selfish player wins at most $\varepsilon$ by deviating from $s_j$. We now introduce a more restrictive property of stability that involves two points: if a selfish player still were to deviate, this would only incur a small loss to other players. Moreover, if the selfish player wants to incur some considerable loss to the collective players (e.g.,\ she is adversarial), then she also has to incur a comparable loss to herself. Obviously, an $\varepsilon$-Nash equilibrium is $(0, \varepsilon)$-stable.
\vspace{-0.25em}
\begin{defin}
A strategy profile $s \in \mathcal{S}^M$ is $(\alpha, \varepsilon)$-stable if for any $s' \in \mathcal{S}$, $l \in \mathbb{R}_+$ and $i,j \in [M]$:
\vspace{-0.5em}
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^i_T(s',s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^i_T(s)] - l \implies \mathbb{E}[\mathrm{Rew}^j_T(s',s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^j_T(s)] + \varepsilon - \alpha l .\end{equation*}
\end{small}
\end{defin}
\vspace{-1em}
\subsection{Limits of existing algorithms.}
This section explains why existing algorithms are not robust to selfish players, i.e.,\ are not even $o(T)$-Nash equilibria. Besides justifying the design of new appropriate algorithms, this provides some first insights on the way to achieve robustness.
\paragraph{Communication between players.}
Many recent algorithms rely on communication protocols between players to gather their statistics. Facing an algorithm of this kind, a selfish player would communicate fake statistics to the other players in order to keep the best arm for herself. In case of collision, the colliding player(s) remains unidentified, so a selfish player could modify \textit{incognito} the statistics sent by other players, making them untrustworthy.
A way to make such protocols robust to malicious players is proposed in Section~\ref{sec:collision}. Algorithms relying on communication can then be adapted in the Full Sensing setting.
\paragraph{Necessity of fairness}
An algorithm is fair if all players asymptotically earn the same reward \textit{a posteriori} and not only in expectation. As already noticed \citep{attar2012}, fairness seems to be a significant criterion in the design of selfish-robust algorithms. Indeed, without fairness, a selfish player tries to always be the one with the largest reward \textit{a posteriori}.
For example, against algorithms attributing an arm among the top-$M$ ones to each player \citep{musicalchair, besson2018, boursier2018}, a selfish player could easily rig the attribution to end with the best arm, largely increasing her individual reward.
Other algorithms work on the basis of first come-first served \citep{boursier2018}. Players first explore and when they detect an arm as both optimal and available, they pull it forever. Such an algorithm is unfair and a selfish player could play more aggressively to end her exploration before the others and to commit on an arm, maybe at the risk of committing on a suboptimal one (but with high probability on the best arm). The risk taken by the early commit is small compared to the benefit of being the first committing player. As a consequence, these algorithms are not $o(T)$-Nash equilibria.
\subsection{Proof of Section~\ref{sec:algo2}}
\subsection{Regret analysis}
\label{app:proofsicmmabregret}
This section aims at proving the first point of Theorem~\ref{thm:sicmmab1}, using similar techniques as in \citep{boursier2018}. The regret is first divided into three parts:
\begin{equation}
\label{eq:regdec2}
R_T = R^{\text{init}} + R^{\text{comm}} + R^{\text{explo}},
\end{equation}
\begin{equation*}
\text{where } \left\{ \begin{split} \begin{aligned} & R^{\text{init}} = T_{\text{init}} {\mathlarger\sum_{k = 1}^M} \mu_{(k)} - \mathbb{E}_\mu \Big[{\mathlarger\sum_{t=1}^{T_{\text{init}}}} {\mathlarger \sum_{j = 1}^M} r^j(t) \Big]
\text{ with } T_{\text{init}} = (12eK^2 + K) \log(T), \\
& R^{\text{comm}} = \mathbb{E}_\mu \Big[{\mathlarger\sum_{t \in \text{Comm}}}{\mathlarger \sum_{j=1}^M} (\mu_{(j)} - r^j(t)) \Big] \text{ with Comm the set of communication steps,} \\
& R^{\text{explo}} = \mathbb{E}_\mu \Big[{\mathlarger\sum_{t \in \text{Explo}}}{\mathlarger \sum_{j=1}^M} (\mu_{(j)} - r^j(t)) \Big]
\text{ with Explo} = \{T_{\text{init}} +1, \ldots, T \} \setminus \text{Comm.} \end{aligned} \end{split} \right.
\end{equation*}
A communication step is defined as a round where any player is using the \meansignal protocol.
Lemma~\ref{lemma:initfull} provides guarantees about the initialization phase. When all players correctly estimate $M$ and have different ranks after the protocol \init[,] the initialization phase is said successful.
\begin{lemm}
\label{lemma:initfull}
Independently of the sampling strategy of the selfish player, if all other players follow \init[,]with probability at least $1-\frac{3M}{T}$: $\widehat{M}^j = M$ and all cooperative players end with different ranks in $[M]$.
\end{lemm}
\begin{proof}
Let $q_k(t) = \mathbb{P}[\text{selfish player pulls } k \text{ at time } t]$. Then, for any cooperative player $j$ during the initialization phase:
\begin{align*}
\mathbb{P}[\text{player }j \text{ observes a collision at time }t] & = \sum_{k=1}^K \frac{1}{K} (1-1/K)^{M-2}(1 - q_k(t)) \\
& = (1-1/K)^{M-2}(1-\frac{\sum_{k=1}^K q_k(t)}{K}) \\
& = (1-1/K)^{M-1}
\end{align*}
Define $p = (1-1/K)^{M-1}$ the probability to collide and $\hat{p}^j = \frac{\sum_{t=1}^{12eK^2 \log(T)}\mathds{1}_{\eta_{\pi^j(t)}=1}}{12eK^2 \log(T)}$ its estimation by player $j$. The Chernoff bound given by Lemma~\ref{lemma:chernoff0} gives:
\begin{align*}
\mathbb{P}\left[ \left| \hat{p}^j - p \right| \geq \frac{p}{2K} \right] & \leq 2 e^{-\frac{p \log(T)}{e}} \\
& \leq 2/T
\end{align*}
If $\left| \hat{p}^j - p \right| < \frac{p}{2K}$, using the same reasoning as in the proof of Lemma~\ref{lemma:estim1} leads to $1+\frac{\log(1- \hat{p}^j)}{\log(1-1/K)} \in (M-1/2, M+1/2)$ and then $\widehat{M}^j=M$.
With probability at least $1-2M/T$, all cooperative players correctly estimate $M$.
\medskip
Afterwards, the players sample uniformly in $[M]$ until observing no collision. As at least an arm in $[M]$ is not pulled by any other player, at each time step of this phase, when pulling uniformly at random:
\begin{align*}
\mathbb{P}[\eta_{\pi^j(t)} = 0] \geq 1/M.
\end{align*}
A player gets a rank as soon as she observes no collision. With probability at least $1-(1-1/M)^{n}$, she thus gets a rank after at most $n$ pulls during this phase. Since this phase lasts $K \log(T)$ pulls, she ends the phase with a rank with probability at least $1-1/T$. Using a union bound finally yields that every player ends with a rank and a correct estimation of $M$. Moreover, these ranks are different between all the players, because a player fixes to the arm $j$ as soon as she gets attributed the rank $j$.
\end{proof}
Lemma~\ref{lemma:explosicmmab1} bounds the exploration regret of \algotwo and is proved in Appendix~\ref{app:proofsicmmabexploregret}. Note that a minimax bound can also be proved as done in \citep{boursier2018}.
\begin{lemm}
\label{lemma:explosicmmab1}
If all players follow \algotwo[,]with probability $1- \mathcal{O}\left( \frac{KM \log(T)}{T} \right)$,
\begin{small}
\begin{equation*}
R^{\text{explo}} = \mathcal{O}\left(\sum_{k>M} \frac{\log(T)}{\mu_{(M)} - \mu_{(k)}} \right).
\end{equation*}
\end{small}
\end{lemm}
Lemma~\ref{lemma:commsicmmab1} finally bounds the communication regret.
\begin{lemm}
\label{lemma:commsicmmab1}
If all players follow \algotwo[,]with probability $1- \mathcal{O}\left(\frac{KM \log(T)}{T} + \frac{M}{T} \right)$:
\begin{small}
\begin{equation*}
R^{\text{comm}} = \mathcal{O}\left( M^2K \log^2\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2} \right) \right).
\end{equation*}
\end{small}
\end{lemm}
\begin{proof}
The proof is conditioned on the success of the initialization phase, which happens with probability $1- \mathcal{O}\left(\frac{M}{T} \right)$. Proposition~\ref{prop:stopexplor1} given in Appendix~\ref{app:proofsicmmabexploregret} yields that with probability ${1- \mathcal{O}\left(\frac{KM \log(T)}{T} \right)}$, the number of communication phases is bounded by $N = \mathcal{O}\left(\log\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2}\right) \right)$. The $p$-th communication phase lasts $8 MK (p+1) + 3K + K\mathrm{Card} \text{Acc}(p) + K\mathrm{Card} \text{Rej}(p)$, where Acc and Rej respectively are the accepted and rejected arms at the $p$-th phase. Their exact definitions are given in Protocol~\ref{proto:update}. An arm is either accepted or rejected only once, so that $\sum_{p=1}^N \mathrm{Card} \text{Acc}(p) + \mathrm{Card} \text{Rej}(p)=K$. The total length of Comm is thus bounded by:
\begin{align*}
\mathrm{Card} \text{Comm} & \leq \sum_{p=1}^N 8MK(p+1) + 3K + K\mathrm{Card} \text{Acc}(p) + K\mathrm{Card} \text{Rej}(p) \\
& \leq 8MK \frac{(N+2)(N+1)}{2} +3KN + K^2
\end{align*}
Which leads to $R^{\text{comm}} = \mathcal{O}\left( M^2K \log^2\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2} \right) \right)$ using the given bound for $N$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:sicmmab1}.]
Using Lemmas~\ref{lemma:initfull}, \ref{lemma:explosicmmab1}, \ref{lemma:commsicmmab1} and equation~\eqref{eq:regdec2} it comes that with probability $1- \mathcal{O}\left(\frac{KM \log(T)}{T} \right)$:
\begin{small}
\begin{equation*}
R_T \leq \mathcal{O}\left( \mathlarger{\sum_{k>M}} \frac{\log(T)}{\mu_{(M)} - \mu_{(k)}} + M^2K \log^2\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2} \right) + MK^2 \log(T) \right).
\end{equation*}
\end{small}
The regret incurred by the low probability event is $\mathcal{O}(KM^2 \log(T))$, leading to Theorem~\ref{thm:sicmmab1}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma:explosicmmab1}}
\label{app:proofsicmmabexploregret}
Lemma~\ref{lemma:explosicmmab1} relies on the following concentration inequality.
\begin{lemm}
\label{lemma:concentration2}
Conditioned on the success of the initialization and independently of the means sent by the selfish player, if all other players play cooperatively and send uncorrupted messages, for any $k \in [K]$:
\begin{equation*}
\mathbb{P}[\exists p \leq n, \left| \widetilde{\mu}_k(p) - \mu_k \right| \geq B(p)] \leq \frac{4nM}{T}
\end{equation*}
where $B(p)=4 \sqrt{\frac{\log(T)}{(M-2)2^{p+1}}}$ and $\widetilde{\mu}_k(p)$ is the centralized mean of arm $k$ at the end of phase $p$, once the extremes have been cut out. It exactly corresponds to the $\widetilde{\mu}_k$ of Protocol~\ref{proto:update}.
\end{lemm}
\begin{proof}
At the end of phase $p$, $(2^{p+1} - 1)$ observations are used for any player $j$ and arm $k$. Hoeffding bound then gives: $\mathbb{P}\left[\left| \widehat{\mu}_k^j(p) - \mu_k \right| \geq \sqrt{\frac{\log(T)}{2^{p+1}}}\right] \leq \frac{2}{T}$. The quantization only adds an error of at most $2^{-p}$, yielding for any cooperative player:
\begin{equation}\label{eq:hoeffding1}
\mathbb{P}\left[\left| \widetilde{\mu}_k^j(p) - \mu_k \right| \geq 2\sqrt{\frac{\log(T)}{2^{p+1}}}\right] \leq \frac{2}{T}
\end{equation}
Assume w.l.o.g. that the selfish player has rank $M$. Hoeffding inequality also yields:
\begin{small}
\begin{equation*}
\mathbb{P}\left[\bigg| \frac{1}{M-1}\sum_{j=1}^{M-1} \widehat{\mu}_k^j(p) - \mu_k \bigg| \geq \sqrt{\frac{\log(T)}{(M-1)2^{p+1}}}\right] \leq \frac{2}{T}.
\end{equation*}
\end{small}
Since $\sum_{j=1}^{M-1} 2^p(\widetilde{\mu}_k^j(p) -\widehat{\mu}_k^j(p))$ is the difference between $M-1$ Bernoulli variables and their expectation, Hoeffding inequality yields
$\mathbb{P}\left[\left| \frac{1}{M-1}\sum_{j=1}^{M-1} (\widetilde{\mu}_k^j - \widehat{\mu}_k^j(p)) \right| \geq \sqrt{\frac{\log(T)}{(M-1)2^{p+1}}}\right] \leq \frac{2}{T}$ and:
\begin{equation} \label{eq:hoeffding2}
\mathbb{P}\left[\left| \frac{1}{M-1} \sum_{j=1}^{M-1} \widetilde{\mu}_k^j(p) - \mu_k \right| \geq 2\sqrt{\frac{\log(T)}{(M-1)2^{p+1}}}\right] \leq \frac{4}{T}.
\end{equation}
Using the triangle inequality combining equations~\eqref{eq:hoeffding1} and \eqref{eq:hoeffding2} yields for any $j \in [M-1]$:
\begin{align}
\mathbb{P}\left[\Big| \frac{1}{M-2}\sum_{\substack{j' \in [M-1] \\ j' \neq j}} \widetilde{\mu}_k^j(p) - \mu_k \Big| \geq 4 \sqrt{\frac{\log(T)}{(M-2)2^{p+1}}} \right]
& \leq \mathbb{P}\Bigg[ \frac{M-1}{M-2} \Big| \frac{1}{M-1}\sum_{j' \in [M-1]} \widetilde{\mu}_k^j(p) - \mu_k \Big| \nonumber \\ & \phantom{\leq} + \frac{1}{M-2} \left| \widetilde{\mu}_k^j(p) - \mu_k \right|\geq 4 \sqrt{\frac{\log(T)}{(M-2)2^{p+1}}}\Bigg] \nonumber\\
& \leq \mathbb{P}\left[\Big| \frac{1}{M-1} \sum_{j=1}^{M-1} \widetilde{\mu}_k^j(p) - \mu_k \Big| \geq 2\sqrt{\frac{\log(T)}{(M-1)2^{p+1}}}\right]\nonumber\\ & \phantom{\leq } + \mathbb{P}\left[\left| \widetilde{\mu}_k^j(p) - \mu_k \right| \geq 2\sqrt{\frac{\log(T)}{2^{p+1}}}\right] \nonumber\\
& \leq \frac{6}{T}. \label{eq:hoeffding3}
\end{align}
Moreover by construction, no matter what mean sent the selfish player, $$\min_{j \in [M-1]}\frac{1}{M-2}\sum_{\substack{j' \in [M-1] \\ j' \neq j}} \widetilde{\mu}_k^j(p) \leq \widetilde{\mu}_k(p) \leq \max_{j \in [M-1]}\frac{1}{M-2}\sum_{\substack{j' \in [M-1] \\ j' \neq j}} \widetilde{\mu}_k^j(p).$$
Indeed, assume that the selfish player sends a mean larger than any other player. Then her mean as well as the minimal sent mean are cut out and $\widetilde{\mu}_k(p)$ is then equal to the right term. Conversely if she sends the smallest mean, $\widetilde{\mu}_k(p)$ corresponds to the left term. Since $\widetilde{\mu}_k(p)$ is non-decreasing in $\widetilde{\mu}_k^M(p)$, the inequality also holds in the case where the selfish player sends neither the smallest nor the largest mean.
\medskip
Finally, using a union bound over all $j \in [M-1]$ with equation~\eqref{eq:hoeffding3} yields Lemma~\ref{lemma:concentration2}.
\end{proof}
Using classical MAB techniques then yields Proposition~\ref{prop:stopexplor1}.
\begin{prop} \label{prop:stopexplor1}
Independently of the selfish player behavior, as long as the \punishhomo protocol is not used, with probability $1-\mathcal{O}\left(\frac{KM \log(T)}{T} \right)$, every optimal arm $k$ is accepted after at most $\mathcal{O}\left(\frac{\log(T)}{(\mu_{k} - \mu_{(M+1)})^2} \right)$ pulls and every sub-optimal arm $k$ is rejected after at most $\mathcal{O}\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{k})^2} \right)$pulls during exploration phases.
\end{prop}
\begin{proof}
The fact that the \punishhomo protocol is not started just means that no corrupted message is sent between cooperative players. The proof is conditioned on the success of the initialization phase, which happens with probability $1-\mathcal{O}\left(\frac{M}{T} \right)$. Note that there are at most $\log_2(T)$ exploration phases. Thanks to Lemma~\ref{lemma:concentration2}, with probability $1-\mathcal{O}\left(\frac{KM \log(T)}{T} \right)$, the inequality $ \left| \widetilde{\mu}_k(p) - \mu_k \right| \leq B(p)$ thus holds for any $p$. The remaining of the proof is conditioned on this event. Especially, an optimal arm is never rejected and a suboptimal one never accepted.
\medskip
First consider an optimal arm $k$ and note $\Delta_k = \mu_k - \mu_{(M+1)}$ the optimality gap. Let $p_k$ be the smallest integer $p$ such that $(M-2)2^{p+1} \geq \frac{16^2 \log(T)}{\Delta^2_k}$. In particular, $4 B(p_k) \leq \Delta_k$, which implies that the arm $k$ is accepted at the end of the communication phase $p_k$ or before.
Necessarily, $(M-2)2^{p_k+1} \leq \frac{2\cdot 16^2 \log(T)}{\Delta^2_k}$ and especially, $M 2^{p_k+1} = \mathcal{O}\left( \frac{\log(T)}{\Delta_k^2} \right)$. Note that the number of exploratory pulls on arm $k$ during the $p$ first phases is bounded by $M (2^{p+1}+p)$\footnote{During the exploration phase $p$, any explored arm is pulled between $M2^p$ and $M(2^p+1)$ times.}, leading to Proposition~\ref{prop:stopexplor1}.
The same holds for the sub-optimal arms with $\Delta_k = \mu_{(M)} - \mu_{k}$.
\end{proof}
In the following, we keep the notation $t_k = \frac{c \log(T)}{\left( \mu_k-\mu_{(M)} \right)^2}$, where $c$ is a universal constant, such that with probability $1 -\mathcal{O}\left(\frac{KM}{T}\right)$, any arm $k$ is correctly accepted or rejected after a time at most $t_k$.
All players are now assumed to play \algotwo[,]e.g.,\ there is no selfish player. Since there is no collision during exploration/exploitation (conditionally on the success of the initialization phase), the following decomposition holds \citep{anantharam}:
\begin{small}
\begin{equation}\label{eq:exploregdec1}
R^{\text{explo}} = \sum_{k>M} (\mu_{(M)} - \mu_{(k)}) T_{(k)}^{\text{explo}} + \sum_{k \leq M} (\mu_{(k)} - \mu_{(M)}) (T^{\text{explo}} - T_{(k)}^{\text{explo}}),
\end{equation}
\end{small}where $T^{\text{explo}} = \mathrm{Card}\text{Explo}$ and $T_{(k)}^{\text{explo}}$ is the centralized number of pulls on the $k$-th best arm during exploration or exploitation.
\begin{lemm}
\label{lemma:explosicmmab2}
If all players follow \algotwo[,]with probability $1-\mathcal{O}\left(\frac{KM \log(T)}{T} \right)$, it holds:
\begin{itemize}
\item for $k > M$, $ (\mu_{(M)} - \mu_{(k)}) T_{(k)}^{\text{explo}} = \mathcal{O} \left( \frac{\log(T)}{\mu_{(M)} - \mu_{(k)}} \right)$.
\item $\sum_{k \leq M} (\mu_{(k)} - \mu_{(M)}) (T^{\text{explo}} - T_{(k)}^{\text{explo}}) = \mathcal{O}\left(\sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k} \right)$.
\end{itemize}
\end{lemm}
\begin{proof}
With probability $1-\mathcal{O}\left(\frac{KM \log(T)}{T} \right)$, Proposition~\ref{prop:stopexplor1} yields that any arm $k$ is correctly accepted or rejected at time at most $t_k$. The remaining of the proof is conditioned on this event and the success of the initialization phase.
The first point of Lemma~\ref{lemma:explosicmmab2} is a direct consequence of Proposition~\ref{prop:stopexplor1}. It remains to prove the second point.
\medskip
Let $\hat{p}_k$ be the number of the phase at which the arm $k$ is either accepted or rejected and let $K_p$ be the number of arms that still need to be explored at the beginning of phase $p$ and $M_p$ be the number of optimal arms that still need to be explored.
The following two key Lemmas are crucial to obtain the second point.
\begin{lemm} \label{lemma:sicmmabaux1}
Under the assumptions of Lemma~\ref{lemma:explosicmmab2}:
\begin{small}
\begin{equation*}
\sum_{k \leq M} (\mu_{(k)} - \mu_{(M)}) (T^{\text{explo}} - T_{(k)}^{\text{explo}}) \leq \sum_{j > M} \sum_{k \leq M} \sum_{p=1}^{\min (\hat{p}_{(k)}, \hat{p}_{(j)})} (\mu_{(k)} - \mu_{(M)}) 2^p \frac{M}{M_p} + o(\log(T)).
\end{equation*}
\end{small}
\end{lemm}
\begin{lemm} \label{lemma:sicmmabaux2}
Under the assumptions of Lemma~\ref{lemma:explosicmmab2}, for any $j>M$:
\begin{small}
\begin{equation*}
\sum_{k \leq M} \sum_{p=1}^{\min (\hat{p}_{(k)}, \hat{p}_{(j)})} (\mu_{(k)} - \mu_{(M)}) 2^p \frac{M}{M_p} \leq \mathcal{O}\left( \frac{\log(T)}{\mu_{(M)} - \mu_{(j)}} \right).
\end{equation*}
\end{small}
\end{lemm}
Combining these two Lemmas with Equation~\eqref{eq:exploregdec1} finally yields Lemma~\ref{lemma:explosicmmab1}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:sicmmabaux1}.]
Consider an optimal arm $k$. During the $p$-th exploration phase, either $k$ has already been accepted and is pulled $ M \left\lceil\frac{K_p2^p}{M_p} \right\rceil$ times; or $k$ has not been accepted yet and is pulled at least $2^p M$, i.e.,\ is not pulled at most $M \left( \left\lceil \frac{K_p2^p}{M_p} \right\rceil - 2^p \right) $ times.
This gives:
\begin{align*}
(\mu_{(k)} - \mu_{(M)}) (T^{\text{explo}} - T_{(k)}^{\text{explo}}) & \leq \sum_{p=1}^{\hat{p}_k} (\mu_{(k)} - \mu_{(M)}) M \left(\left\lceil \frac{K_p2^p}{M_p}\right\rceil - 2^p\right), \\
& \leq \sum_{p=1}^{\hat{p}_k} (\mu_{(k)} - \mu_{(M)}) M\left(\frac{K_p 2^p}{M_p}- 2^p +1\right),\\
& \leq \hat{p}_k (\mu_{(k)} - \mu_{(M)}) M + \sum_{p=1}^{\hat{p}_k} (\mu_{(k)} - \mu_{(M)}) (K_p - M_p) \frac{M}{M_p} 2^p.\\
\end{align*}
We assumed that any arm $k$ is correctly accepted or rejected after a time at most $t_k$. This implies that $\hat{p}_k=o(\log(T))$. Moreover, $K_p - M_p$ is the number of suboptimal arms not rejected at phase $p$, i.e.,\ $K_p - M_p = \sum_{j>M}\mathds{1}_{p \leq \hat{p}_{(j)}}$ and this proves Lemma~\ref{lemma:sicmmabaux1}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:sicmmabaux2}.]
For $j > M$, define $A_j = \sum_{k \leq M} \sum_{p=1}^{\min (\hat{p}_{(k)}, \hat{p}_{(j)})} (\mu_{(k)} - \mu_{(M)}) 2^p \frac{M}{M_p}$. We want to show $A_j \leq \mathcal{O}\left( \frac{\log(T)}{\mu_{(M)} - \mu_{(j)}} \right)$ with the considered conditions.
Note $T(p) = M(2^{p+1}-1)$ and $\Delta(p) = \sqrt{\frac{c \log(T)}{T(p)}}$. The inequality $\hat{p}_{(k)} \geq p$ then implies $\mu_{(k)} - \mu_{(M)} < \Delta(p)$, i.e.,\
\begin{align*}
A_j & \leq \sum_{k \leq M} \sum_{p=1}^{\hat{p}_{(j)}} 2^p \Delta(p) \mathds{1}_{p \leq \hat{p}_{(k)}} \frac{M}{M_p} = \sum_{p=1}^{\hat{p}_{(j)}} 2^p \Delta(p) M \\
& \leq \sum_{p=1}^{\hat{p}_{(j)}} \Delta(p) (T(p) - T(p-1))
\end{align*}
The equality comes because $\sum_{k \leq M} \mathds{1}_{p \leq \hat{p}_{(k)}}$ is exactly $M_p$. Then from the definition of $\Delta(p)$:
\begin{align*}
A_j & \leq c \log(T) \sum_{p=1}^{\hat{p}_{(j)}} \Delta(p) \left(\frac{1}{\Delta(p)} + \frac{1}{\Delta(p-1)} \right)\left(\frac{1}{\Delta(p)} - \frac{1}{\Delta(p-1)} \right) \\
& \leq (1+\sqrt{2})c \log(T) \sum_{p=1}^{\hat{p}_{(j)}}\left(\frac{1}{\Delta(p)} - \frac{1}{\Delta(p-1)} \right) \\
& \leq (1+\sqrt{2})c \log(T) /\Delta(\hat{p}_{(j)}) \\
& \leq (1+\sqrt{2}) \sqrt{c \log(T) T(\hat{p}_{(j)})}
\end{align*}
By definition, $T(\hat{p}_{(j)})$ is smaller than the number of exploratory pulls on the $j$-th best arm and is thus bounded by $\frac{c \log(T)}{(\mu_{(M)} - \mu_{(j)})^2}$, leading to Lemma~\ref{lemma:sicmmabaux2}.
\end{proof}
\subsection{Selfish robustness of \algotwo}
\label{app:greedyproofsicmmab}
In this section, the second point of Theorem~\ref{thm:sicmmab1} is proven. First Lemma~\ref{lemma:punishhomo} gives guarantees for the punishment protocol. Its proof is given in Appendix~\ref{app:punishhomo}.
\begin{lemm} \label{lemma:punishhomo}
If the \punishhomo protocol is started at time $T_{\mathrm{punish}}$ by $M-1$ players, then for the remaining player $j$, independently of her sampling strategy:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^j | \mathrm{punish}] \leq \mathbb{E}[\mathrm{Rew}_{T_{\text{punish}} + t_p}^j] + \widetilde{\alpha} \frac{T - T_{\mathrm{punish}} - t_p}{M} \sum_{k=1}^M \mu_{(k)},
\end{equation*}
with $t_p = \mathcal{O}\left(\frac{K}{(1-\widetilde{\alpha})^2 \mu_{(K)}}\log(T) \right)$ and $\widetilde{\alpha}=\frac{1+(1-1/K)^{M-1}}{2}$.
\end{lemm}
\begin{proof}[Proof of the second point of Theorem~\ref{thm:sicmmab1} (Nash equilibrium).]
First fix $T_{\text{punish}}$ the time at which the punishment protocol starts if it happens (and $T$ if it does not). Before this time, the selfish player can not perturb the initialization phase, except by changing the ranks distribution. Moreover, the exploration/exploitation phase is not perturbed as well, as claimed by Proposition~\ref{prop:stopexplor1}. The optimal strategy then earns at most $T_{\text{init}}$ during the initialization and $\mathrm{Card}\text{Comm}$ during the communication.
With probability $1-\mathcal{O}\left(\frac{KM \log(T)}{T} \right)$, the initialization is successful and the concentration bound of Lemma~\ref{lemma:concentration1} holds for any arm and player all the time. The following is conditioned on this event.
\medskip
Note that during the exploration, the cooperative players pull any arm the exact same amount of times. Since the upper bound time $t_k$ to accept or reject an arm does not depend on the strategy of the selfish player, Lemma~\ref{lemma:explosicmmab2} actually holds for the cooperative player, i.e.,\ for any cooperative player $j$:
\begin{equation}\label{eq:individualpulls}
\sum_{k\leq M}\left( \mu_{(k)} - \mu_{(M)} \right) \left( \frac{T^{\text{explo}}}{M} - T_{(k)}^j \right) = \mathcal{O}\left(\frac{1}{M}\sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k} \right),
\end{equation}
where $T_{(k)}^j$ is the number of pulls by player $j$ on the $k$-th best arm during the exploration/exploitation. The same kind of regret decomposition as in Equation~\eqref{eq:exploregdec1} is possible for the regret of the selfish player $j$ and especially:
\begin{equation*}
R^{\text{explo}}_j \geq \sum_{k \leq M} (\mu_{(k)} - \mu_{(M)}) \left(\frac{T^{\text{explo}}}{M} - T_{(k)}^j\right).
\end{equation*}
However, the optimal strategy for the selfish player is to pull the best available arm during the exploration and especially to avoid collisions. This implies the constraint $T_{(k)}^j \leq T^{\text{explo}} - \sum_{j \neq j'} T_{(k)}^{j'}$.
Using this constraint with Equation~\eqref{eq:individualpulls} yields $\frac{T^{\text{explo}}}{M} - T_{(k)}^j \geq - \sum_{j\neq j'}\frac{T^{\text{explo}}}{M} - T_{(k)}^{j'}$ and then
$$
R^{\text{explo}}_j \geq -\mathcal{O}\left( \sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k}\right),
$$
which can be rewritten as
$$
\mathrm{Rew}^{\text{explo}}_j \leq \frac{T^{\text{explo}}}{M} \sum_{k=1}^M \mu_{(k)} + \mathcal{O}\left( \sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k}\right).
$$
Thus, for any strategy $s'$ when adding the low probability event of a failed exploration or initialization,
\begin{align*}
\mathbb{E}[\mathrm{Rew}_{t_p + T_{\text{punish}}}^j(s',s_{-j})]& \leq (T_{\text{init}} + \mathrm{Card}\text{Comm} + t_p + \mathcal{O}(KM \log(T))) \\ & \phantom{\leq} + \frac{\mathbb{E}[T_{\text{punish}}] - T_{\text{init}} - \mathrm{Card}\text{Comm}}{M} \sum_{k \leq M} \mu_{(k)} + \mathcal{O}\left( \sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k}\right).
\end{align*}
Using Lemma~\ref{lemma:punishhomo}, this yields:
\begin{align*}
\mathbb{E}[\mathrm{Rew}_{T}^j(s',s_{-j})] & \leq (T_{\text{init}} + \mathrm{Card}\text{Comm} + t_p + \mathcal{O}(KM \log(T))) \\ & \phantom{\leq} + \frac{\mathbb{E}[T_{\text{punish}}] - T_{\text{init}} - \mathrm{Card}\text{Comm}}{M} \sum_{k \leq M} \mu_{(k)} + \mathcal{O}\left( \sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k}\right)\\ & \phantom{\leq} + \widetilde{\alpha} \frac{T - \mathbb{E}[T_{\text{punish}}]}{M} \sum_{k=1}^M \mu_{(k)}.
\end{align*}
The right term is maximized when $\mathbb{E}[T_{\text{punish}}]$ is maximized, i.e.,\ when it is $T$. We then get:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_{T}^j(s',s_{-j})] \leq \frac{T}{M}\sum_{k \leq M} \mu_{(k)} + \varepsilon,
\end{equation*}
\end{small}
where $\varepsilon= \mathcal{O}\bigg(\sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k} + K^2 \log(T) + MK \log^2\left(\frac{\log(T)}{(\mu_{(M)}-\mu_{(M+1)})^2}\right) + \frac{K \log(T)}{(1-\widetilde{\alpha})^2 \mu_{(K)}} \bigg).$
\end{proof}
\begin{proof}[Proof of the second point of Theorem~\ref{thm:sicmmab1} (stability).]
Define $\mathcal{E}$ the \textit{bad event} that the initialization is not successful or that an arm is poorly estimated at some time. Let $\varepsilon' = T \mathbb{P}[\mathcal{E}] + \mathbb{E}[\mathrm{Card} \text{Comm} | \neg \mathcal{E}] + K \log(T)$. Then $\varepsilon' = \mathcal{O} \left( KM\log(T) + KM \log^2\left( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2} \right) \right)$.
Assume that the player $j$ is playing a deviation strategy $s'$ such that for some other player $i$ and $l>0$:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^i(s',s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^i(s)]-l-\varepsilon'
\end{equation*}
First fix $T_{\text{punish}}$ the time at which the punishment protocol starts. Let us now compare $s'$ with the individual optimal strategy for player $j$, $s^*$. Let $\varepsilon'$ take account of the communication phases, the initialization and the low probability events.
\medskip
The number of pulls by any player during exploration/exploitation is given by Equation~\eqref{eq:individualpulls} unless the punishment protocol is started. Moreover, the selfish player causes at most a collision during exploration/exploitation before initiating the punishment protocol, so the loss of player $i$ before punishment is at most $1+\varepsilon'$.
\medskip
After $T_{\text{punish}}$, Lemma~\ref{lemma:punishhomo} yields that the selfish player suffers a loss at least $(1-\widetilde{\alpha})\frac{T - T_{\text{punish}} - t_p}{M} \sum_{k=1}^M \mu_{(k)}$, while any cooperative player suffers at most $\frac{T - T_{\text{punish}}}{M} \sum_{k=1}^M \mu_{(k)}$.
The selfish player then suffers after $T_{\text{punish}}$ a loss at least $(1-\widetilde{\alpha})((l-1) - t_p)$.
Define $\beta = 1-\widetilde{\alpha}$. We just showed:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^i(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^i(s)]-l-\varepsilon' \implies \mathbb{E}[\mathrm{Rew}_T^j(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j})]-\beta (l-1) + \beta t_p
\end{equation*}
Moreover, thanks to the second part of Theorem~\ref{thm:sicmmab1}, $\mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s)] +\varepsilon$ with $\varepsilon= \mathcal{O}\bigg(\sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k} + K^2 \log(T) + MK \log^2\left(\frac{\log(T)}{(\mu_{(M)}-\mu_{(M+1)})^2}\right) + \frac{K \log(T)}{(1-\widetilde{\alpha})^2 \mu_{(K)}} \bigg) $.
Then by defining $l_1 = l + \varepsilon'$, $\varepsilon_1 = \varepsilon + \beta t_p + \beta \varepsilon' +1 = \mathcal{O}(\varepsilon)$, we get:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^i(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^i(s)]-l_1 \implies \mathbb{E}[\mathrm{Rew}_T^j(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s)] + \varepsilon_1 - \beta l_1.
\end{equation*}
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma:punishhomo}.}
\label{app:punishhomo}
The punishment protocol starts by estimating all means $\mu_k$ with a multiplicative precision of~$\delta$. This is possible thanks to Lemma~\ref{lemma:multiplicativeestim}, which corresponds to Theorem~9 in \citep{cesa2019} and Lemma~13 in \citep{berthet2017}.
\begin{lemm}\label{lemma:multiplicativeestim}
Let $X_1, \ldots, X_n$ be $n$-i.i.d. random variables in $[0,1]$ with expectation $\mu$ and define $S_t^2 = \frac{1}{t-1}\sum_{s=1}^t(X_s-\bar{X}_t)^2$. For all $\delta \in (0,1)$, if $n \geq n_0$, where
\begin{equation*}
n_0 = \left\lceil \frac{2}{3 \delta \mu} \log(T) \left(\sqrt{9\frac{1}{\delta^2} + 96 \frac{1}{\delta} +85} +\frac{3}{\delta} +1 \right) \right\rceil +2 = \mathcal{O}\left(\frac{1}{\delta^2 \mu}\log(T) \right)
\end{equation*}
and $\tau$ is the smallest time $t \in \lbrace 2, \ldots, n \rbrace$ such that
\begin{equation*}
\delta \bar{X}_t \geq 2 S_t\left( \log(T)/t \right)^{1/2} + \frac{14 \log(T)}{3(t-1)},
\end{equation*}
then, with probability at least $1-\frac{3}{T}$:
\begin{enumerate}
\item $\tau \leq n_0$,
\item $\left( 1- \delta \right) \bar{X}_\tau < \mu < \left(1+\delta\right) \bar{X}_\tau$.
\end{enumerate}
\end{lemm}
\begin{proof}[Proof of Lemma~\ref{lemma:punishhomo}]
The punishment protocol starts for all cooperative players at $T_{\text{punish}}$. For $\delta = \frac{1-\gamma}{1+3\gamma}$, each player then estimates each arm. Lemma~\ref{lemma:multiplicativeestim} gives that with probability at least $1-3/T$:
\begin{itemize}
\item the estimation ends after a time at most $t_p = \mathcal{O}\left(\frac{K}{\delta^2 \mu_{(K)}}\log(T) \right)$,
\item $(1-\delta) \widehat{\mu}_k^j \leq \mu_k \leq (1+\delta) \widehat{\mu}_k^j$.
\end{itemize}
The following is conditioned on this event. The last inequality can be reversed as $\frac{\mu_k}{1+\delta} \leq \widehat{\mu}_k^j \leq \frac{\mu_k}{1-\delta}$. Then, this implies for any cooperative player $j$
\begin{align*}
1- p_k^j \leq \left(\gamma \frac{(1+\delta)\sum_{m=1}^M \mu_{(m)}}{(1-\delta)M \mu_k} \right)^{\frac{1}{M-1}}.
\end{align*}
The expected reward that gets the selfish player $j$ by pulling $k$ after the time $T_{\text{punish}}+t_p$ is thus smaller than $\gamma \frac{1+\delta}{1-\delta} \frac{\sum_{m=1}^M \mu_{(m)}}{M} $.
Note that $\gamma \frac{1+\delta}{1-\delta} = \frac{1+\gamma}{2} = \widetilde{\alpha}$.
Considering the low probability event given by Lemma~\ref{lemma:multiplicativeestim} adds a constant term that can be counted in $t_p$. This finally yields the result of Lemma~\ref{lemma:punishhomo}.
\end{proof}
\section{Collective punishment proof}
\label{app:punish}
Recall that the punishment protocol consists in pulling each arm $k$ with probability at least $p_k^j= \max \Big( 1 - \Big(\gamma\frac{\sum_{l=1}^M \widehat{\mu}^j_{(l)}}{M \widehat{\mu}^j_k}\Big)^{\frac{1}{M-1}}, 0\Big)$. Lemma~\ref{lemma:punishment} below guarantees that such a sampling strategy is possible.
\begin{lemm}
\label{lemma:punishment}
For $p_k = \max \Big( 1 - \Big(\frac{\gamma \sum_{l=1}^M \widehat{\mu}^j_{(l)}}{M \widehat{\mu}^j_k}\Big)^{\frac{1}{M-1}}, 0\Big)$ with $\gamma=\left(1 - 1/K\right)^{M-1}$: $\sum_{k=1}^K p_k \leq 1$.
\end{lemm}
\begin{proof}
For ease of notation, define $x_k \coloneqq \widehat{\mu}_k^j$, $\bar{x}_M \coloneqq \sfrac{\sum_{l=1}^M x_{(l)}}{M}$ and $S \coloneqq \{k \in [K] \ | \ x_k > \gamma \bar{x}_M\}= \{k \in [K] \ | \ p_k > 0\}$. We then get by concavity of $x \mapsto -x^{-\frac{1}{M-1}}$,
\begin{align}
\sum_{k\in S} p_k & = \mathrm{Card} S \times \left(1- \left(\gamma \bar{x}_M\right)^{\frac{1}{M-1}} \sum_{k \in S} \frac{(x_k)^{-\frac{1}{M-1}}}{\mathrm{Card} S} \right), \\
& \leq \mathrm{Card} S \times \left(1- \left(\frac{\gamma\bar{x}_M}{\bar{x}_{S}}\right)^{\frac{1}{M-1}} \right) \qquad \text{ with } \bar{x}_{S} = \frac{1}{\mathrm{Card} S}\sum_{k \in S} x_k. \label{eq:punish1}
\end{align}
We distinguish two cases.
First, if $\mathrm{Card} S \leq M$, we then get $M \bar{x}_M \geq \mathrm{Card} S \bar{x}_{S}$ because $S$ is a subset of the $M$ best empirical arms. The last inequality then becomes
\begin{equation*}
\sum_{k \in S} p_k \leq \mathrm{Card} S \left(1- \left(\gamma\frac{\mathrm{Card} S}{M}\right)^{\frac{1}{M-1}} \right) .
\end{equation*}
Define $g(x)= \frac{\gamma}{M} - x(1-x)^{M-1}$.
For $x \in (0, 1]$:
\begin{align*}
g(x) \geq 0 & \iff \frac{\gamma}{xM} \geq (1-x)^{M-1},\\
& \iff 1- \left(\frac{\gamma}{xM}\right)^{\frac{1}{M-1}} \leq x, \\
& \iff \frac{1}{x} \left(1- \left(\frac{\gamma}{xM}\right)^{\frac{1}{M-1}}\right) \leq 1.
\end{align*}
Thus, $g(\frac{1}{\mathrm{Card} S}) \geq0$ implies $\sum_{k\in S} p_k \leq 1$. We now show that $g$ is indeed non negative on $[0,1]$.
$x (1-x)^{M-1}$ is maximized at $\sfrac{1}{M}$ and is thus smaller than $\frac{1}{M}(1-1/M)^{M-1}$, and using the fact that $\frac{1}{M}(1-1/M)^{M-1} \leq \frac{\gamma}{M}$ for our choice of $\gamma$, we get the result for the first case.
\medskip
The other case corresponds to $\mathrm{Card} S > M$. In this case, the $M$ best empirical arms are all in~$S$ and thus $\bar{x}_M \geq \bar{x}_{S}$. Equation~\eqref{eq:punish1} becomes:
$$\sum_{k \in S} p_k \leq \mathrm{Card} S \left(1- \gamma^{\frac{1}{M-1}} \right) \leq K (1-(1-1/K)) = 1. $$
\end{proof}
\subsection{Homogeneous case: \algotwo}
\label{sec:algo2}
In the homogeneous case, these two protocols can be incorporated in the SIC-MMAB algorithm of \citet{boursier2018} to provide \algotwo[,]which is robust to selfish behaviors and still ensures a regret comparable to the centralized lower bound.
\citet{boursier2019} recently improved the communication protocol by choosing a leader and communicating all the information only to this leader. A malicious player would do anything to be the leader. \algotwo avoids such a behavior by choosing two leaders who either agree or trigger the punishment. More generally with $n+1$ leaders, this protocol is robust to $n$ selfish players.
The detailed algorithm is given by Algorithm~\ref{alg:algo2} in Appendix~\ref{app:sicmmab_descript}.
\paragraph{Initialization.}
The original initialization phase of SIC-MMAB has a small regret term, but it is not robust. During the initialization, the players here pull uniformly at random to estimate $M$ as in \algoone and then attribute ranks the same way. The players with ranks $1$ and $2$ are then leaders. Since the collision indicator is always observed here, this estimation can be done in an easier and better way.
The observation of $\eta_k$ also enables players to remain synchronized after this phase as its length does not depend on unknown parameters.
\paragraph{Exploration and Communication.} Players alternate between exploration and communication once the initialization is over. During the $p$-th exploration phase, each arm still requiring exploration is pulled $2^p$ times by every player in a collisionless fashion. Players~then communicate to each leader their empirical means in binary after every exploration phase, using the back and forth trick explained in Section~\ref{sec:backforth}. Leaders then check that~their information match. If some undesired behavior is detected, a collective punishment is~triggered.
Otherwise, the leaders determine the sets of optimal/suboptimal arms and send them to everyone.
To prevent the selfish player from sending fake statistics, the leaders gather the empirical means of all players, except the extreme ones (largest and smallest) for every arm. If the selfish player sent outliers, they are thus cut out from the collective estimator, which is thus the average of $M-2$ individual estimates. This estimator can be biased by the selfish player, but a concentration bound given by Lemma~\ref{lemma:concentration2} in Appendix~\ref{app:proofsicmmabexploregret} still holds.
\paragraph{Exploitation.} As soon as an arm is detected as optimal, it is pulled until the end. To ensure fairness of \algotwo[,]players will actually rotate over all the optimal arms so that none of them is favored. This point is thoroughly described in Appendix~\ref{app:sicmmab_descript}.
Theorem~\ref{thm:sicmmab1}, proved in Appendix~\ref{app:sicmmab}, gives theoretical results for \algotwo[.]
\begin{thm}
\label{thm:sicmmab1}
Define $\alpha = \frac{1 - (1-1/K)^{M-1}}{2} $ and assume $M\geq 3$.
\begin{enumerate}\itemsep0em
\item The collective regret of \algotwo is bounded as
\begin{small}
\begin{equation*}
\mathbb{E}[R_T] \leq \mathcal{O}\bigg( \sum_{k>M} \frac{\log(T)}{\mu_{(M)} - \mu_{(k)}} + MK^2 \log(T) + M^2K \log^2\Big( \frac{\log(T)}{(\mu_{(M)} - \mu_{(M+1)})^2} \Big) \bigg).
\end{equation*}
\end{small}
\item Playing \algotwo is an $\varepsilon$-Nash equilibrium and is $(\alpha, \varepsilon)$-stable with
\begin{small}
\begin{equation*}
\varepsilon = \mathcal{O}\bigg(\sum_{k>M}\frac{\log(T)}{\mu_{(M)} - \mu_k} + K^2 \log(T) + MK \log^2\Big(\frac{\log(T)}{(\mu_{(M)}-\mu_{(M+1)})^2}\Big) + \frac{K \log(T)}{\alpha^2 \mu_{(K)}} \bigg).
\end{equation*}
\end{small}
\end{enumerate}
\end{thm}
\subsection{Semi-heterogeneous case: \algothree}
\label{sec:algo3}
The punishment strategies described above can not be extended to the heterogeneous case, as the relevant probability of choosing each arm would depend on the preferences of the malicious player which are unknown (even her identity might not be discovered). Moreover, as already explained in the homogeneous case, pulling each arm uniformly at random is not an appropriate punishment strategy\footnote{Unless in the specific case where $\mu^j_{(1)}(1-1/K)^{M-1} < \frac{1}{M}\sum_{k=1}^M \mu^j_{(k)}$.}.
We therefore consider the $\delta$-heterogeneous setting, which allows punishments for small values of $\delta$ as given by Lemma~\ref{lemma:punishhetero} in Appendix~\ref{app:rsdgreedyproof}.
The heterogeneous model was justified by the fact that transmission quality depends on individual factors such as localization.
The $\delta$-heterogeneous assumption relies on the idea that such individual factors are of a different order of magnitude than global factors (as the availability of a channel). As a consequence, even if arm means differ from player to player, these variations remain relatively small.
\begin{defin}
The setting is $\delta$-heterogeneous if there exists $\{\mu_k ; k \in [K]\}$ such that for all $j$ and $k$, $\mu_k^j \in [(1-\delta)\mu_k, (1+\delta)\mu_k]$.
\end{defin}
In the semi-heterogeneous full sensing setting, \algothree provides a robust, logarithmic RSD-regret algorithm. Its complete description is given by Algorithm~\ref{alg:algo3} in Appendix~\ref{app:rsd_descript}.
\vspace{-0.5em}
\subsubsection{Algorithm description}
\algothree starts with the exact same initialization as \algotwo to estimate $M$ and attribute ranks among the players.
The time is then divided into superblocks which are divided into $M$ blocks. During the $j$-th block of a superblock, the dictators ordering\footnote{The ordering is actually $(\sigma(j), \ldots, \sigma(j-1))$ where $\sigma(j)$ is the player with rank $j$ after the initialization. For sake of clarity, this consideration is omitted here.} is $(j, \ldots, M, 1, \ldots, j-1)$. Moreover, only the $j$-th player can send messages during this block.
\paragraph{Exploration.} The exploring players pull sequentially all the arms. Once player $j$ knows her $M$ best arms and their ordering, she waits for a block $j$ to initiate communication.
\paragraph{Communication.} Once a player starts a communication block, she proceeds in three successive steps as follows:
\vspace{-0.25em}
\begin{enumerate}\itemsep0em
\item she first collides with all players to signal the beginning of a communication block. The other players then enter a listening state, ready to receive messages.
\item She then sends to every player her ordered list of $M$ best arms. Each player then repeats this list to detect the potential intervention of a malicious player.
\item Finally, any player who detected the intervention of a malicious player signals to everyone the beginning of a collective punishment.
\end{enumerate}\vspace{-0.25em}
After a communication block $j$, every one knows the preferences order of player $j$, who is now in her exploitation phase, unless a punishment protocol has been started.
\paragraph{Exploitation.} While exploiting, player $j$ knows the preferences of all other exploiting players. Thanks to this, she can easily compute the arms attributed by the RSD algorithm between the exploiting players, given the dictators ordering of the block.
Moreover, as soon as she collides in the beginning of a block while not intended (by her), this means an exploring player is starting a communication block. The exploiting player then starts listening to the arm preferences of the communicating player.
\subsubsection{Theoretical guarantees}
Here are some insights to understand how \algothree reaches the utility of the RSD algorithm, which are rigorously detailed by Lemma~\ref{lemma:rsdmatching} in Appendix~\ref{app:rsdgreedyproof}.
With no malicious player, the players ranks given by the initialization provide a random permutation $\sigma \in \mathfrak{S}_M$ of the players and always considering the dictators ordering $(1, \ldots, M)$ would lead to the expected reward of the RSD algorithm. However, a malicious player can easily rig the initialization to end with rank $1$. In that case, she largely improves her individual reward w.r.t. following the cooperative strategy.
To avoid such a behavior, the dictators ordering should rotate over all permutations of $\mathfrak{S}_M$, so that the rank of the player has no influence. However, this leads to an undesirable combinatorial $M!$ dependency of the regret.
\algothree instead rotates over the dictators ordering $(j, \ldots,M,1,\ldots, j-1)$ for all $j \in [M]$. If we note $\sigma_0$ the $M$-cycle $(1 \ldots M)$, the considered permutations during a superblock are of the form $\sigma \circ \sigma_0^{-m}$ for $m \in [M]$.
The malicious player $j$ can only influence the distribution of $\sigma^{-1}(j)$: assume w.l.o.g.\ that $\sigma(1)=j$. The permutation $\sigma$ given by the initialization then follows the uniform distribution over $\mathfrak{S}_{M}^{j \to 1} = \lbrace \sigma \in \mathfrak{S}_M \ | \ \sigma(1)=j \rbrace$. But then, for any $m \in [M]$, $\sigma \circ \sigma_0^{-m}$ has a uniform distribution over $\mathfrak{S}_{M}^{j \to 1+m}$. In average over a superblock, the induced permutation still has a uniform distribution over $\mathfrak{S}_{M}$. So the malicious player has no interest in choosing a particular rank during the initialization, making the algorithm robust.
\medskip
Thanks to this remark and robust communication protocols, \algothree possesses theoretical guarantees given by Theorem~\ref{thm:rsd} (whose proof is deterred to Appendix~\ref{app:rsd}).
\vspace{-0.5em}
\begin{thm}
\label{thm:rsd}
Consider the $\delta$-heterogeneous setting and define $r = \frac{1-\left( \frac{1+\delta}{1-\delta}\right)^2 (1-1/K)^{M-1}}{2}$ and ${\Delta = \min\limits_{(j,k) \in [M]^2} \mu_{(k)}^j - \mu_{(k+1)}^j}$.
\begin{enumerate}
\item The RSD-regret of \algothree is bounded as:
$
\mathbb{E}[R_T^{\mathrm{RSD}}] \leq \mathcal{O}\big(MK \Delta^{-2} \log(T) + MK^2 \log(T)\big).
$
\item If $r>0$, playing \algothree is an $\varepsilon$-Nash equilibrium and is $(\alpha, \varepsilon)$-stable with
\begin{itemize}\itemsep0em
\item $\varepsilon = \mathcal{O}\Big(\frac{K \log(T)}{\Delta^2} + K^2 \log(T) + \frac{K\log(T)}{(1-\delta)r^2 \mu_{(K)}}\Big),$
\item $\alpha = \min\Big(r\left( \frac{1+\delta}{1-\delta}\right)^3 \frac{\sqrt{\log(T)}-4M}{\sqrt{\log(T)}+4M},\quad \frac{\Delta}{(1+\delta)\mu_{(1)}},\quad \frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}}\Big).$
\end{itemize}
\end{enumerate}
\end{thm}
\subsection{Regret analysis} \label{app:rsdregretproof}
This section aims at proving the first point of Theorem~\ref{thm:rsd}.
\algothree uses the exact same initialization phase as \algotwo[,]and its guarantees are thus given by Lemma~\ref{lemma:initfull}. Here again, the regret is decomposed into three parts:
\begin{equation}
\label{eq:regdec3}
R_T^{\text{RSD}} = R^{\text{init}} + R^{\text{comm}} + R^{\text{explo}},
\end{equation}
\begin{equation*}
\text{where } \left\{ \begin{split} \begin{aligned} & R^{\text{init}} = T_{\text{init}} {\mathlarger \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)} } \bigg[ \sum_{k = 1}^{M} \mu_{\pi_\sigma(k)}^{\sigma(k)}\bigg] - \mathbb{E}_\mu \Big[{\mathlarger\sum_{t=1}^{T_{\text{init}}}} {\mathlarger \sum_{j = 1}^M} r^j(t) \Big]
\text{ with } T_{\text{init}} = (12eK^2 + K) \log(T), \\
& R^{\text{comm}} = \mathrm{Card} \text{Comm}{\mathlarger \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)} } \bigg[ \sum_{k = 1}^{M} \mu_{\pi_\sigma(k)}^{\sigma(k)}\bigg]- \mathbb{E}_\mu \Big[{\mathlarger\sum_{t \in \text{Comm}}}{\mathlarger \sum_{j=1}^M} r^j(t)) \Big], \\
& R^{\text{explo}} = \mathrm{Card} \text{Explo} {\mathlarger \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)} } \bigg[ \sum_{k = 1}^{M} \mu_{\pi_\sigma(k)}^{\sigma(k)}\bigg] - \mathbb{E}_\mu \Big[{\mathlarger\sum_{t \in \text{Explo}}}{\mathlarger \sum_{j=1}^M} r^j(t)) \Big] \end{aligned} \end{split} \right.
\end{equation*}
with Comm defined as all the rounds of a block where at least a cooperative player uses \listen protocol and $\text{Explo} = \{T_{\text{init}} +1, \ldots, T \} \setminus \text{Comm}$.
In case of a successful initialization, a single player can only initiate a communication block once without starting a punishment protocol. Thus, as long as no punishment protocol is started:
$
\mathrm{Card}\text{Comm} \leq M(5K + M K + M^2 K) = \mathcal{O}(M^3K).
$
Denote by $\Delta^j = \min_{k \in [M]} \mu_{(k)}^j - \mu_{(k+1)}^j$ the level of precision required for player $j$ to know her $M$ preferred arms and their order. Proposition~\ref{prop:stopexplor2} gives the exploration time required for any player $j$:
\begin{prop}\label{prop:stopexplor2}
With probability $1-\mathcal{O}\left( \frac{K}{T} \right)$ and as long as no punishment protocol is started, the player $j$ starts exploiting after at most $\mathcal{O}\left( \frac{K \log(T)}{(\Delta^j)^2} + M^3K \right)$ exploration pulls.
\end{prop}
\begin{proof}
In the following, the initialization is assumed to be successful, which happens with probability $1-\mathcal{O}\left( \frac{M}{T} \right)$.
Moreover, Hoeffding inequality yields: $$\mathbb{P}\left[ \forall t \leq T, \left| \widehat{\mu}_k^j(t) - \mu_k^j(t) \right| \geq \sqrt{\frac{2 \log(T)}{T_k^j(t)}} \right] \leq \frac{2}{T}$$ where $T_k^j(t)$ is the number of exploratory pulls on arm $k$ by player $j$. With probability~${1-\mathcal{O}\left( \frac{K}{T} \right)}$, player $j$ then correctly estimates all arms at each round. The remaining of the proof is conditioned on this event.
During the exploration, player $j$ sequentially pulls the arms in $[K]$. Denote by $n$ the smallest integer such that $\sqrt{\frac{2 \log(T)}{n}} \leq 4 \Delta^j$.
It directly comes that $n = \mathcal{O}\left( \frac{\log(T)}{(\Delta^j)^2} \right)$.
Under the considered events, player $j$ then has determined her $M$ preferred arms and their order after $K n$ exploratory pulls. Moreover, she needs at most $M$ blocks before being able to initiate her communication block and starts exploiting. Thus, she needs at most $\mathcal{O}\left( \frac{K \log(T)}{(\Delta^j)^2} + M^3K \right)$ exploratory pulls, leading to Proposition~\ref{prop:stopexplor2}.
\end{proof}
\begin{proof}[Proof of the first point of Theorem~\ref{thm:rsd}.]
Assume all players play \algothree[.]Simply by bounding the size of the initialization and the communication phases, it comes:
\begin{equation*}
R^{\text{init}} + R^{\text{comm}} \leq \mathcal{O}\left( MK^2 \log(T)\right).
\end{equation*}
Proposition~\ref{prop:stopexplor2} yields that with probability $1-\mathcal{O}\left( \frac{KM}{T} \right)$, all players start exploitation after at most $\mathcal{O}\left(\frac{K \log(T)}{\Delta^2} \right)$ exploratory pulls.
\medskip
For $p=\sqrt{\log(T)}/T$, with probability $\mathcal{O}(p^2M)$ at any round $t$, a player is inspecting another player who is also inspecting or a player receives two consecutive inspections. These are the only ways to start punishing when all players are cooperative.
As a consequence, when all players follow \algothree[,]they initiate the punishment protocol with probability $\mathcal{O}\left(p^2MT\right)$. Finally, the total regret due to this event grows as~$\mathcal{O}\left(M^2 \log(T)\right)$.
\medskip
If the punishment protocol is not initiated, players cycle through the RSD matchings of $\sigma \circ \sigma^{-1}_0, \ldots, \sigma \circ \sigma^{-M}_0$ where $\sigma_0$ is the classical $M$-cycle and $\sigma$ is the players permutation returned by the initialization.
Define $U(\sigma)=\sum_{k = 1}^{M}\mu_{\pi_\sigma(k)}^{\sigma(k)},$ where $\pi_\sigma(k)$ is the arm attributed to the $k$-th dictator, $\sigma(k)$, as defined in Section~\ref{sec:randomass}. $U(\sigma)$ is the social welfare of RSD algorithm when the dictatorships order is given by the permutation $\sigma$. As players all follow \algothree here, $\sigma$ is chosen uniformly at random in $\mathfrak{S}_M$ and any $\sigma \circ \sigma^{-k}_0$ as well. Then $$\mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)}\left[ \frac{1}{M} \sum_{k=1}^M U(\sigma \circ \sigma^{-M}_0) \right] = \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)}\left[U(\sigma) \right].$$
This means that in expectation, the utility given by the exploitation phase is the same as the utility of the RSD algorithm when choosing a permutation uniformly at random. Considering the low probability event of a punishment protocol, an unsuccesful initialization or a bad estimation of an arm finally yields:
\begin{equation*}
R^{\text{explo}} \leq \mathcal{O}\left(\frac{MK \log(T)}{\Delta^2} \right)\ .
\end{equation*}
Equation~\eqref{eq:regdec3} concludes the proof.
\end{proof}
\subsection{Selfish-robustness of \algothree} \label{app:rsdgreedyproof}
In this section, we prove the two last points of Theorem~\ref{thm:rsd}. Three auxiliary Lemmas are first needed. They are proved in Appendix~\ref{app:auxlemmasrsd}.
\begin{enumerate}
\item Lemma~\ref{lemma:meancomparison} compares the utility received by player $j$ from the RSD algorithm with the utility given by sequentially pulling her $M$ best arms in the $\delta$-heterogeneous setting.
\item Lemma~\ref{lemma:punishhetero} gives an equivalent version of Lemma~\ref{lemma:punishhomo}, but for the $\delta$-heterogeneous setting.
\item Lemma~\ref{lemma:rsdmatching} states that the expected utility of the assignment of any player during the exploitation phase does not depend on the strategy of the selfish player. The intuition behind this result is already given in Section~\ref{sec:algo3}.
In the case of several selfish players, they could actually fix the joint distribution of~$(\sigma^{-1}(j), \sigma^{-1}(j'))$. A simple rotation with a $M$-cycle is then not enough to recover a uniform distribution over $\mathfrak{S}_M$ in average. A more complex rotation is then required and the dependence in $M$ would blow up with the number of selfish players.
\end{enumerate}
\begin{lemm} \label{lemma:meancomparison}
In the $\delta$-heterogeneous case for any player $j$ and permutation $\sigma$:
\begin{small}
\begin{equation*}
\frac{1}{M}\sum_{k=1}^M \mu_{(k)}^j \leq \widetilde{U}_j(\sigma) \leq \frac{(1+\delta)^2}{(1-\delta)^2 M}\sum_{k=1}^M \mu_{(k)}^j,
\end{equation*}
\end{small}
where $\widetilde{U}_j(\sigma) \coloneqq\frac{1}{M} \sum_{k=1}^M \mu_{\pi_{\sigma \circ \sigma^{-k}_0}\left(\sigma_0^k \circ \sigma^{-1}(j)\right)}^j$.
\end{lemm}
Following the notation of Section~\ref{sec:randomass}, $\pi_\sigma(\sigma^{-1}(j))$ is the arm attributed to player $j$ by RSD when the dictatorship order is given by $\sigma$. $\widetilde{U}_j(\sigma)$ is then the average utility of the exploitation when $\sigma$ is the permutation given by the initialization.
\begin{lemm} \label{lemma:punishhetero}
Recall that $\gamma=(1-1/K)^{M-1}$. In the $\delta$-heterogeneous setting with $\delta < \frac{1-\sqrt{\gamma}}{1+\sqrt{\gamma}}$, if the punish protocol is started at time $T_{\text{punish}}$ by $M-1$ players, then for the remaining player $j$, independently of her sampling strategy:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^j | \text{punishment}] \leq \mathbb{E}[\mathrm{Rew}_{T_{\text{punish}} + t_p}^j] + \widetilde{\alpha} \frac{T - T_{\text{punish}} - t_p}{M} \sum_{k=1}^M \mu^j_{(k)},
\end{equation*}
\end{small}
with $t_p = \mathcal{O}\left(\frac{K\log(T)}{(1-\delta)(1-\widetilde{\alpha})^2 \mu_{(K)}} \right)$ and $\widetilde{\alpha}=\frac{1+\left(\frac{1+\delta}{1-\delta}\right)^2\gamma}{2}$.
\end{lemm}
\begin{lemm} \label{lemma:rsdmatching}
The initialization phase is successful when all players end with different ranks in $[M]$. For any player $j$, independently of the behavior of the selfish player:
\begin{small}
\begin{equation*}
\mathbb{E}_{\sigma \sim \text{successful initialization}}\left[ \widetilde{U}_j(\sigma) \right] = \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_M\right)}\left[\mu_{\pi_\sigma(\sigma^{-1}(j))}^j \right].
\end{equation*}
\end{small}
where $\widetilde{U}_j(\sigma)$ is defined as in Lemma~\ref{lemma:meancomparison} above.
\end{lemm}
\begin{proof}[Proof of the second point of Theorem~\ref{thm:rsd} (Nash equilibrium).]
First fix $T_{\text{punish}}$ the beginning of the punishment protocol. Note $s$ the profile where all players follow \algothree and $s'$ the individual strategy of the selfish player $j$.
As in the homogeneous case, the player earns at most $T_{\text{init}} + \mathrm{Card}\text{Comm}$ during both initialization and communication. She can indeed choose her rank at the end of the initialization, but this has no impact on the remaining of the algorithm (except for a $M^3K$ term due to the length of the last uncompleted superblock), thanks to Lemma~\ref{lemma:rsdmatching}.
With probability $1-\mathcal{O}\left(\frac{KM + M\log(T)}{T}\right)$, the initialization is successful, the arms are correctly estimated and no punishment protocol is due to unfortunate inspections (as already explained in Section~\ref{app:rsdregretproof}). The following is conditioned on this event.
\medskip
Proposition~\ref{prop:stopexplor2} holds independently of the strategy of the selfish player. Moreover, the exploiting players run the RSD algorithm only between the exploiters. This means that when all cooperative players are exploiting, if the selfish player did not signal her preferences, she would always be the last dictator in the RSD algorithm. Because of this, it is in her interest to report as soon as possible her preferences.
Moreover, reporting truthfully is a dominant strategy for the RSD algorithm, meaning that when all players are exploiting, the expected utility received by the selfish player is at most the utility she would get by reporting truthfully. As a consequence, the selfish player can improve her expected reward by at most the length of a superblock during the exploitation phase. Wrapping up all of this and defining $t_0$ the time at which all other players start exploiting:
\begin{small}
\begin{equation*}
\mathbb{E}\left[\mathrm{Rew}_{T_\text{punish}+t_p}^j(s', s_{-j}) \right] \leq t_0 + (T_{\text{punish}}+t_p-t_0) \mathbb{E}_{\sigma \sim \mathcal{U}(\mathfrak{S}_M)}\left[ \mu^j_{\pi_\sigma(\sigma^{-1}(j))} \right] + \mathcal{O}(M^3K).
\end{equation*}
\end{small}
with $t_0 = \mathcal{O} \left(\frac{K \log(T)}{\Delta^2} + K^2 \log(T)\right)$.
Lemma~\ref{lemma:punishhetero} then yields for $\widetilde{\alpha}=\frac{1+\left(\frac{1+\delta}{1-\delta}\right)^2\alpha}{2}$:
\begin{small}
\begin{equation*}
\mathbb{E}\left[\mathrm{Rew}_{T}^j(s',s_{-j}) \right] \leq t_0 + (T_{\text{punish}}+t_p-t_0) \mathbb{E}_{\sigma \sim \mathcal{U}(\mathfrak{S}_M)}\left[ \mu^j_{\pi_\sigma(\sigma^{-1}(j))} \right] + \widetilde{\alpha}\frac{T- T_{\text{punish}} - t_p}{M} \sum_{k=1}^M \mu_{(k)}^j + \mathcal{O}(M^3K).
\end{equation*}
\end{small}
Thanks to Lemma~\ref{lemma:meancomparison}, $\mathbb{E}_{\sigma \sim \mathcal{U}(\mathfrak{S}_M)}\left[ \mu^j_{\pi_\sigma(\sigma^{-1}(j))} \right] \geq \frac{\sum_{k=1}^M \mu_{(k)}^j}{M}$.
We assume $\delta < \frac{1-(1-1/K)^{\frac{M-1}{2}}}{1+(1-1/K)^{\frac{M-1}{2}}}$ here, so that $\widetilde{\alpha} < 1$. Because of this, the right term is maximized when $T_{\text{punish}}$ is maximized, i.e.,\ equal to $T$. Then:
\begin{equation*}
\mathbb{E}\left[\mathrm{Rew}_{T}^j(s',s_{-j}) \right] \leq T \mathbb{E}_{\sigma \sim \mathcal{U}(\mathfrak{S}_M)}\left[ \mu^j_{\pi_\sigma(\sigma^{-1}(j))} \right] + t_0 + t_p + \mathcal{O}(M^3K).
\end{equation*}
Using the first point of Theorem~\ref{thm:rsd} to compare $T \mathbb{E}_{\sigma \sim \mathcal{U}(\mathfrak{S}_M)}\left[ \mu^j_{\pi_\sigma(\sigma^{-1}(j))} \right]$ with $\mathrm{Rew}_T^j(s)$ and adding the low probability event then yields the first point of Theorem~\ref{thm:rsd}.
\end{proof}
\begin{proof}[Proof of the second point of Theorem~\ref{thm:rsd} (stability).]
For $p_0=\mathcal{O}\left(\frac{KM + M\log(T)}{T}\right)$, with probability at least $1-p_0$, the initialization is successful, the cooperative players start exploiting with correct estimated preferences after a time at most $t_0 = \mathcal{O}\left( K^2 \log(T) + \frac{K \log(T)}{\Delta^2}\right)$ and no punishment protocol is started due to unfortunate inspections.
Define $\varepsilon' = t_0 + Tp_0 + 7M^3K$.
Assume that the player $j$ is playing a deviation strategy $s'$ such that for some $i$ and $l>0$:
\begin{small}
\begin{equation*}
\mathbb{E}\left[ \mathrm{Rew}_T^i(s', s_{-j})\right] \leq \mathbb{E}\left[ \mathrm{Rew}_T^i(s)\right] - l - \varepsilon'
\end{equation*}\end{small}
First, let us fix $\sigma$ the permutation returned by the initialization, $T_{\text{punish}}$ the time at which the punishment protocol starts and divide $l = l_{\text{before punishment}} + l_{\text{after punishment}}$ in two terms: the regret incurred before the punishment protocol and the regret after. Let us now compare $s'$ with $s^*$, the optimal strategy for player $j$. Let $\varepsilon$ take account of the low probability event of a bad initialization/exploration, the last superblock that remains uncompleted, the time before all cooperative players start the exploitation and the event that a punishment accidentally starts. Thus the only way for player $i$ to suffer some additional regret before punishment is to lose it during a completed superblock of the exploitation.
Three cases are possible:
\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]
\item The selfish player truthfully reports her preferences. The average utility of player $i$ during the exploitation is then $\widetilde{U}_i(\sigma)$ as defined in Lemma~\ref{lemma:rsdmatching}.
The only way to incur some additional loss to player $i$ before the punishment is then to collide with her, in which case her loss is at most $(1+\delta)\mu_{(1)}$ while the selfish player's loss is at least $(1-\delta) \mu_{(M)}$.
\medskip
After $T_{\text{punish}}$, Lemma~\ref{lemma:punishhetero} yields that the selfish player suffers a loss at least $(1-\widetilde{\alpha})\frac{T-T_{\text{punish}}-t_p}{M}\sum_{k=1}^M \mu_{(k)}^j$, while any cooperative player $i$ suffers a loss at most $(T-T_\text{punish}) \widetilde{U}_i(\sigma)$. Thanks to Lemma~\ref{lemma:meancomparison} and the $\delta$-heterogeneity assumption, this term is smaller than
$\frac{T-T_{punish}}{M}\left(\frac{1+\delta}{1-\delta} \right)^3\sum_{k=1}^M \mu_{(k)}^j$.
Then, the selfish player after $T_{\text{punish}}$ suffers a loss at least $\frac{(1-\widetilde{\alpha})(1-\delta)^3}{(1+\delta)^3} l_{\text{after punish}} - t_p$.
\medskip
In the first case, we thus have for $\beta = \min(\frac{(1-\widetilde{\alpha})(1-\delta)^3}{(1+\delta)^3}, \frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}})$:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^j(s',s_{-j}) | \sigma] \leq \mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j}) | \sigma] - \beta l + t_p.
\end{equation*}
\item The selfish player never reports her preferences. In this case, it is obvious that the utility returned by the assignments to any other player is better than if the selfish player reports truthfully. Then the only way to incur some additional loss to player $i$ before punishment is to collide with her, still leading to a ratio of loss at most $\frac{\mu_{(M)}^j}{\mu_{(1)}^i}$.
\medskip
From there, it can be concluded as in the first case that for $\beta = \min(\frac{(1-\widetilde{\alpha})(1-\delta)^3}{(1+\delta)^3}, \frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}})$:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^j(s',s_{-j}) | \sigma] \leq \mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j}) | \sigma] - \beta l + t_p.
\end{equation*}
\item The selfish player reported fake preferences. If these fake preferences never change the issue of the \rsd protocol, this does not change from the first case. Otherwise, for any block where the final assignment is changed, the selfish player does not receive the arm she would get if she reported truthfully.
Denote by $n$ the number of such blocks, by $N_{\text{lie}}$ the number of times player $j$ did not pull the arm attributed by \rsd during such a block before $T_{\text{punish}}$ and by $l_{b}$ the loss incurred to player $i$ on the other blocks.
As for the previous cases, the loss incurred by the selfish player during the blocks where the assignment of \rsd is unchanged is at least $\frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}} l_{b}$.
\medskip
Each time the selfish player pulls the attributed arm by \rsd in a block where the assignment is changed, she suffers a loss at least $\Delta$.
The total loss for the selfish player is then (w.r.t. the optimal strategy $s^*$) at least:
\begin{small}
\begin{equation*}
(1-\widetilde{\alpha}) \frac{T-T_{\text{punish}}-t_p}{M} \sum_{k=1}^M \mu_{(k)}^j + \left(\frac{n}{M}\left(T_{\text{punish}}-t_0\right) - N_{\text{lie}} \right) \Delta +\frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}} l_{b}.
\end{equation*}
\end{small}
On the other hand, the loss for a cooperative player is at most:
\begin{small}
\begin{equation*}
\frac{T-T_{\text{punish}}}{M}\left(\frac{1+\delta}{1-\delta}\right)^3\sum_{k=1}^M \mu_{(k)}^j + \frac{n}{M}(T_{\text{punish}}-t_0)(1+\delta)\mu_{(1)} + l_b.
\end{equation*}
\end{small}
Moreover, each time the selfish player does not pull the attributed arm by \rsd[,]she has a probability $\widetilde{p}=1-(1-\frac{p}{M-1})^{M-1} \geq \frac{p}{2}$ for $p=\frac{\sqrt{\log(T)}}{T}$, to receive a random inspection and thus to trigger the punishment protocol. Because of this, $N_{\text{lie}}$ follows a geometric distribution of parameter $\widetilde{p}$ and $\mathbb{E}[N_{\text{lie}}] \leq \frac{2}{p}$.
When taking the expectations over $T_{\text{punish}}$ and $N_{\text{lie}}$, but still fixing $\sigma$ and $n$, we get:
\begin{align*}
l_{\text{selfish}} \geq (1-\widetilde{\alpha}) \frac{T-\mathbb{E}[T_{\text{punish}}]-t_p}{M} \sum_{k=1}^M \mu_{(k)}^j + \left(\frac{n}{M}\left(\mathbb{E}[T_{\text{punish}}]-t_0\right) - 2/p \right) \Delta +\frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}} l_{b},
\\
l \leq \frac{T-\mathbb{E}[T_{\text{punish}}]}{M}\left(\frac{1+\delta}{1-\delta}\right)^3\sum_{k=1}^M \mu_{(k)}^j + \frac{n}{M}(\mathbb{E}[T_{\text{punish}}]-t_0)(1+\delta)\mu_{(1)} + l_b.
\end{align*}
First assume that $\frac{n}{M}(\mathbb{E}[T_{\text{punish}}] -t_0) \geq \frac{4}{p}$. In that case, we get:
\begin{align*}
l_{\text{selfish}} \geq (1-\widetilde{\alpha}) \frac{T-\mathbb{E}[T_{\text{punish}}]-t_p}{M} \sum_{k=1}^M \mu_{(k)}^j + \frac{n}{2M}(\mathbb{E}[T_{\text{punish}}]-t_0) \Delta +\frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}} l_{b},\\
l \leq \frac{T-\mathbb{E}[T_{\text{punish}}]}{M}\left(\frac{1+\delta}{1-\delta}\right)^3\sum_{k=1}^M \mu_{(k)}^j + \frac{n}{M}(\mathbb{E}[T_{\text{punish}}]-t_0)(1+\delta)\mu_{(1)} + l_b.
\end{align*}
In the other case, we have by noting that $(1+\delta)\mu_{(1)} \leq \frac{1+\delta}{1-\delta}\sum_{k=1}^M \mu_{(k)}^j$:
\begin{align*}
l_{\text{selfish}} \geq (1-\widetilde{\alpha}) T\left( 1-\frac{4M}{\sqrt{\log(T)}} -t_p \right)\frac{1}{M} \sum_{k=1}^M \mu_{(k)}^j + \frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}} l_{b},\\
l \leq T\left(1+\frac{4M}{\sqrt{\log(T)}}\right)\frac{1}{M}\left(\frac{1+\delta}{1-\delta}\right)^3\sum_{k=1}^M \mu_{(k)}^j + l_b.
\end{align*}
In any of these two cases, for $\widetilde{\beta} = \min\left((1-\widetilde{\alpha})\left( \frac{1+\delta}{1-\delta}\right)^3 \frac{\sqrt{\log(T)}-4M}{\sqrt{\log(T)}+4M}; \frac{\Delta}{(1+\delta)\mu_{(1)}}; \frac{(1-\delta)\mu_{(M)}}{(1+\delta)\mu_{(1)}}\right)$:
\begin{align*}
l_{\text{selfish}}\geq \widetilde{\beta} l - t_p
\end{align*}
\end{enumerate}
\bigskip
Let us now gather all the cases. When taking the previous results in expectation over $\sigma$, this yields for the previous definition of $\widetilde{\beta}$:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^i(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^i(s)] - l - \varepsilon' \implies \mathbb{E}[\mathrm{Rew}_T^j(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j})] - \widetilde{\beta} l + t_p + t_0.
\end{equation*}
Moreover, thanks to the second part of Theorem~\ref{thm:rsd}, $\mathbb{E}[\mathrm{Rew}_T^j(s^*, s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s)] + \varepsilon$, with $\varepsilon = \mathcal{O}\left(\frac{K \log(T)}{\Delta^2} + K^2\log(T)+ \frac{K\log(T)}{(1-\delta)r^2 \mu_{(K)}}\right)$. Then by defining $l_1 = l + \varepsilon'$, $\varepsilon_1 = \varepsilon + t_p + t_0 + \widetilde{\beta} \varepsilon' = \mathcal{O}(\varepsilon)$, we get:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}_T^i(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^i(s)] - l_1 \implies \mathbb{E}[\mathrm{Rew}_T^j(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}_T^j(s)] - \widetilde{\beta} l_1 + \varepsilon_1.
\end{equation*}
\end{proof}
\subsubsection{Auxiliary lemmas} \label{app:auxlemmasrsd}
\begin{proof}[Proof of Lemma~\ref{lemma:meancomparison}.]
Assume that player $j$ is the $k$-th dictator for an RSD assignment. Since only $k-1$ arms are reserved before she chooses, she earns at least $\mu_{(k)}^j$ after this assignment. This yields the first inequality:
\begin{equation*}
\widetilde{U}_j(\sigma) \geq \frac{\sum_{k=1}^M \mu_{(k)}^j}{M}
\end{equation*}
\medskip
Still assuming that player $j$ is the $k$-th dictator, let us prove that she earns at most $\left(\frac{1+\delta}{1-\delta}\right)^2 \mu_{(k)}^j$. Assume w.l.o.g. that she ends up with the arm $l$ such that $\mu_l^j > \mu_{(k)}^j$. This means that a dictator $j'$ before her preferred an arm $i$ to the arm $l$ with $\mu_l^j > \mu_{(k)}^j \geq \mu_i^j$.
Since $j'$ preferred $i$ to $l$, $\mu_i^{j'} \geq \mu_l^{j'}$. Using the $\delta$-heterogeneity assumption, it comes:
$$\mu_l^j \leq \frac{1+\delta}{1-\delta} \mu_l^{j'}
\leq \frac{1+\delta}{1-\delta} \mu_i^{j'}
\leq \left(\frac{1+\delta}{1-\delta}\right)^2 \mu_i^{j}
\leq \left(\frac{1+\delta}{1-\delta}\right)^2 \mu_{(k)}^{j}
$$
Thus, player $j$ earns at most $ \left(\frac{1+\delta}{1-\delta}\right)^2 \mu_{(k)}^{j}$ after this assignment, which yields the second inequality of Lemma~\ref{lemma:meancomparison}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:punishhetero}.]
The punishment protocol starts for all cooperative players at $T_{\text{punish}}$. Define $\alpha' = \left(\frac{1+\delta}{1-\delta}\right)^2\gamma$ and $\delta' = \frac{1-\alpha'}{1+3\alpha'}$. The condition $r>0$ is equivalent to $\delta' > 0$.
As in the homogeneous case, each player then estimates each arm such that after $t_p = \mathcal{O}\left( \frac{K \log(T)}{(1-\delta) \cdot (\delta')^2 \mu_{(K)}} \right)$\footnote{The $\delta$-heterogeneous assumption is here used to say that $\frac{1}{\mu_{(K)}^j} \leq \frac{1}{(1-\delta)\mu_{(K)}}$.} rounds, $(1-\delta')\widehat{\mu}_k^j \leq \mu_k^j \leq (1+\delta) \widehat{\mu}_k^j$ with probability $1-\mathcal{O}\left(KM/T \right)$, thanks to Lemma~\ref{lemma:multiplicativeestim}. This implies that for any cooperative player $j'$:
\begin{align*}
1-p_k^{j'} & \leq \left( \gamma \frac{(1+\delta')\sum_{m=1}^M \mu_{(m)}^{j'} }{(1-\delta')M\mu_k^{j'} } \right)^{\frac{1}{M-1}} \\
& \leq \left( \gamma \frac{1+\delta'}{1-\delta'} \left(\frac{1+\delta}{1-\delta}\right)^2 \frac{\sum_{m=1}^M \mu_{(m)}^{j} }{M\mu_k^{j} } \right)^{\frac{1}{M-1}}
\end{align*}
The last inequality is due to the fact that in the $\delta$-heterogeneous setting, $\frac{\mu_k^j}{\mu_k^{j'}} \in [\left(\frac{1-\delta}{1+\delta}\right)^2, \left(\frac{1+\delta}{1-\delta}\right)^2]$. Thus, the expected reward that gets the selfish player $j$ by pulling $k$ after the time $T_{\text{punish}} + t_p$ is smaller than $\gamma \frac{1+\delta'}{1-\delta'} \left(\frac{1+\delta}{1-\delta}\right)^2 \frac{\sum_{m=1}^M \mu_{(m)}^{j} }{M }$.
Note that $\gamma \frac{1+\delta'}{1-\delta'} \left(\frac{1+\delta}{1-\delta}\right)^2 = \widetilde{\alpha}$. Considering the low probability event of bad estimations of the arms adds a constant term that can be counted in $t_p$, leading to Lemma~\ref{lemma:punishhetero}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:rsdmatching}.]
Consider the selfish player $j$ and denote $\sigma$ the permutation given by the initialization. The rank of player $j'$ is then $\sigma^{-1}(j')$. All other players $j$ pull uniformly at random until having an attributed rank. Moreover, player $j$ does not know the players with which she collides. This implies that she can not correlate her rank with the rank of a specific player, i.e.,\ $\mathbb{P}_{\sigma}\left[ \sigma(k')=j' | \sigma(k)=j \right]$ does not depend on $j'$ as long as $j' \neq j$.
\medskip
This directly implies that the distribution of $\sigma | \sigma(k)=j$ is uniform over $\mathfrak{S}_{M}^{j \to k}$. Thus, the distribution of $ \sigma \circ \sigma_0^{-l} | \sigma(k)=j$ is uniform over $\mathfrak{S}_{M}^{j \to k+l \ (\text{mod } M)}$ and finally for any $j' \in [M]$:
\begin{align*}
\mathbb{E}_{\sigma \sim \text{successful initialization}}\left[ \frac{1}{M} \sum_{l=1}^M \mu_{\pi_{\sigma \circ \sigma_0^{-l}}\left(\sigma_0^l \circ \sigma^{-1}(j)\right)}^j \ \bigg| \ \sigma(k)=j \right] & = \frac{1}{M} \sum_{l=1}^M \mathbb{E}_{\sigma \sim \mathcal{U}\left(\mathfrak{S}_{M}^{j \to l}\right)}\left[\mu_{\pi_\sigma(\sigma^{-1}(j'))}^{j'} \right], \\
& = \frac{1}{M} \sum_{l=1}^M \frac{1}{(M-1)!}\sum_{\sigma \in \mathfrak{S}_{M}^{j \to l}} \mu_{\pi_\sigma(\sigma^{-1}(j'))}^{j'}, \\
& = \frac{1}{M!}\sum_{\sigma \in \mathfrak{S}_{M}} \mu_{\pi_\sigma(\sigma^{-1}(j'))}^{j'}.
\end{align*}
Taking the expectation of the left term then yields Lemma~\ref{lemma:rsdmatching}.
\end{proof}
\section{Statistic sensing setting}
\label{sec:algo1}
In the statistic sensing setting where $X_k$ and $r_k$ are observed at each round, the \algoone algorithm provides satisfying theoretical guarantees.
\vspace{-0.5em}
\subsection{Description of \algoone}
\vspace{-1em}
\begin{algorithm2e}[h]
\DontPrintSemicolon
\KwIn{$T, \gamma_1\coloneqq \frac{13}{14}, \gamma_2\coloneqq \frac{16}{15}$}
$\beta \gets 39$;\quad $\widehat{M}, t_m \gets \estimateM(\beta, T)$ \;
Pull $k \sim \mathcal{U}(K)$ until round $\frac{\gamma_2}{\gamma_1} t_m$ \tcp*{first waiting room}
$j \gets \getrank(\widehat{M}, t_m, \beta, T)$ and
pull $j$ until round $\left(\frac{\gamma_2}{\gamma_1^2 \beta^2 K^2} + \frac{\gamma_2^2}{\gamma_1^2}\right)\ t_m$ \;
Run $\exploone(\widehat{M}, j)$ until $T$\;
\caption{\label{algo:algo1}\algoone}
\end{algorithm2e}
\vspace{-0.5em}
A global description of \algoone is given by Algorithm~\ref{algo:algo1}. The pseudocodes of \estimateM[,]\getrank and \exploone are respectively given by Protocols~\ref{proto:estimM}, \ref{proto:getrank} and Algorithm~\ref{algo:explo1} in Appendix~\ref{app:algo1} due to space constraints.
\estimateM and \getrank respectively estimate the number of players $M$ and attribute ranks in $[M]$ among the players. They form the initialization phase, while \exploone optimally balances between exploration and exploitation.
\vspace{-0.5em}
\subsubsection{Initialization phase}
\label{sec:estimm}
Let us first introduce the following quantities:\vspace{-0.5em}
\begin{itemize}\itemsep0em
\item $N_k^j(t) =\lbrace t' \leq t \ | \ \pi^j(t')=k \text{ and } X_k(t') >0 \rbrace$ are rounds when player $j$ observed $\eta_k$.
\item $C_k^j(t) = \lbrace t' \in N_k^j(t) \ | \ \eta_k(t') = 1 \rbrace$ are rounds when player $j$ observed a collision.
\item $\hat{p}_k^j(t) = \sfrac{\mathrm{Card} C_k^j(t) }{\mathrm{Card} N_k^j(t)}$ is the empirical probability to collide on the arm $k$ for player~$j$.
\end{itemize}
During the initialization, the players estimate $M$ with large probability as given by Lemma~\ref{lemma:estim1} in Appendix~\ref{app:algo1descr}. Players first pull uniformly at random in $[K]$. As soon as $\mathrm{Card} N_k^j \geq n$ for any $k \in [K]$ and some fixed $n$, player $j$ ends the \estimateM protocol and estimates $\widehat{M}$ as the closest integer to $1 + \log(1-\sfrac{\sum_k \hat{p}_k^j(t_M)}{K})/\log( 1- \frac{1}{K})$.
This estimation procedure is the same as the one of \citet{musicalchair}, except for the following features:
i) Collisions indicators are not always observed, as we consider statistic sensing here. For this reason, the number of observations of $\eta_k$ is random. The stopping criterion $\min_k \mathrm{Card} N_k^j(t) \geq n$ ensures that players don't need to know $\mu_{(K)}$ beforehand, but they also do not end \estimateM simultaneously. This is why a \textit{waiting room} is needed, during which a player continues to pull uniformly at random to ensure that all players are still pulling uniformly at random if some player is still estimating $M$.
ii) The collision probability is not averaged over all arms, but estimated for each arm individually, then averaged. This is necessary for robustness as explained in Appendix~\ref{app:algo1}, despite making the estimation longer.
\paragraph{Attribute ranks.} After this first procedure, players then proceed to a \textit{Musical Chairs} \citep{musicalchair} phase to attribute ranks among them as given by Lemma~\ref{lemma:getrank} in Appendix~\ref{app:algo1descr}. Players sample uniformly at random in $[M]$ and stop on an arm $j$ as soon as they observe a positive reward. The player's rank is then $j$ and only attributed to her. Here again, a \textit{waiting room} is required to ensure that all players are either pulling uniformly at random or only pulling a specific arm (corresponding to their rank) during this procedure. During this second waiting room, a player thus pulls the arm corresponding to her rank.
\vspace{-0.5em}
\subsubsection{Exploration/exploitation}
After the initialization, players know $M$ and have different ranks. They enter the second phase, where they follow \exploone[,]inspired by \citet{proutiere2019}. Player $j$ sequentially pulls arms in $\mathcal{M}^j(t)$, which is the ordered list of her $M$ best empirical arms, unless she has to pull her $M$-th best empirical arm. In that case, she instead chooses at random between actually pulling it or pulling an arm to explore (any arm not in $\mathcal{M}^j(t)$ with an upper confidence bound larger than the $M$-th best empirical mean, if there is any).
Since players proceed in a shifted fashion, they never collide when $\mathcal{M}^j(t)$ are the same for all~$j$. Having different $\mathcal{M}^j(t)$ happens in expectation a constant (in $T$) amount of times, so that the contribution of collisions to the regret is negligible.
\vspace{-0.5em}
\subsection{Theoretical results}
This section provides theoretical guarantees of \algoone[.]Theorem~\ref{thm:algoone} first presents guarantees in terms of regret. Its proof is given in Appendix~\ref{app:regretalgo1}.
\begin{thm}
\label{thm:algoone}
The collective regret of \algoone is bounded as
\begin{small}
\begin{equation*}
\mathbb{E}[R_T] \leq M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(k)}, \mu_{(M)})}\log(T) + \mathcal{O} \left( \frac{MK^3}{\mu_{(K)}} \log(T)\right).
\end{equation*}
\end{small}
\end{thm}
It can also be noted from Lemma~\ref{lemma:ucb1} in Appendix~\ref{app:regretalgo1} that the regret due to \exploone is $M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(k)}, \mu_{(M)})}\log(T) + o(\log(T))$, which is known to be optimal for algorithms using no collision information \citep{besson2019}. \exploone thus gives an optimal algorithm under this constraint, if $M$ is already known and ranks already attributed (as the $\mathcal{O}(\cdot)$ term in the regret is the consequence of their estimation).
On top of good regret guarantees, \algoone is robust to selfish behaviors as highlighted by Theorem \ref{thm:robust1} (whose proof is deterred to Appendix~\ref{app:statisticrobust1}).
\begin{thm}\label{thm:robust1} Playing \algoone is an $\varepsilon$-Nash equilibrium and is ${(\alpha, \varepsilon)\text{-stable}}$
\begin{equation*}
\text{with} \quad {\textstyle \varepsilon = \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(k)}, \mu_{(M)})}\log(T) + \mathcal{O}\left(\frac{\mu_{(1)}}{\mu_{(K)}} K^3 \log(T)\right)} \quad \text{and} \quad {\textstyle
\alpha = \frac{\mu_{(M)}}{\mu_{(1)}}}.
\end{equation*}
\end{thm}
These points are proved for an \textit{omniscient} selfish player (knowing all the parameters beforehand). This is a very strong assumption and a real player would not be able to win as much by deviating from the collective strategy.
Intuitively, a selfish player would need to explore sub-optimal arms as given by the known individual lower bounds. However, a selfish player can actually decide to not explore but deduce the exploration of other players from collisions.
\section{Supplementary material for Section~\ref{sec:algo1}}
\label{app:algo1}
This section provides a complete description of \algoone and the proofs of Theorems~\ref{thm:algoone} and \ref{thm:robust1}.
\subsection{Thorough description of \algoone} \label{app:algo1descr}
In addition to Section~\ref{sec:algo1}, the pseudocodes of \estimateM[,]\getrank and \exploone are given here. The following Protocol~\ref{proto:estimM} describes the estimation of $M$ using the notations introduced in Section~\ref{sec:estimm}.
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\beta, T$}
$t_m \gets 0$ \;
\While{$\min_k \mathrm{Card} N_k^j(t) < \beta^2 K^2\log(T)$}{
Pull $k \sim \mathcal{U}(K)$;\quad
Update $\mathrm{Card} N_k^j(t)$ and $\mathrm{Card} C_k^j(t)$ ;\quad
$t_m \gets t_m+1$}
$\widehat{M} \gets 1 + \mathrm{round}\Big(\frac{\log\left(1-\frac{1}{K}\sum_k \hat{p}_k^j(t_M)\right)}{\log\left( 1- \frac{1}{K}\right)}\Big)$\tcp*{$\mathrm{round}(x)=$ closest integer to $x$}
\textbf{Return} $\widehat{M}, t_m$
\caption{\label{proto:estimM}\estimateM}
\end{protocol}
Since the duration $t_m^j$ of \estimateM for player $j$ is random and differs between players, each player continues sampling uniformly at random until $\frac{\gamma_2}{\gamma_1} t_m^j$, with $\gamma_1=\frac{13}{14}$ and $\gamma_2=\frac{16}{15}$. Thanks to this additional \textit{waiting room}, Lemma~\ref{lemma:estim1} below guarantees that all players are sampling uniformly at random until at least $t_m^j$ for any $j$.
The estimation of $M$ here tightly estimates the probability to collide individually for each arm. This restriction provides an additional $M$ factor in the length of this phase in comparison with \citep{musicalchair}, where the probability to collide is globally estimated. This is however required because of the Statistic Sensing, but if $\eta_k$ was always observed, then the protocol from \citet{musicalchair} would be robust.
Indeed, if we directly estimated the global probability to collide, the selfish player could pull only the best arm.
The number of observations of $\eta_k$ is larger on this arm, and the estimated probability to collide would thus be positively biased because of the selfish player.
\medskip
Afterwards, ranks in~$[M]$ are attributed to players by sampling uniformly at random in~$[M]$ until observing no collision, as described in Protocol~\ref{proto:getrank}. For the same reason, a waiting room is added to guarantee that all players end this protocol with different ranks.
\begin{protocol}[h]
\DontPrintSemicolon
\KwIn{$\widehat{M}, t_m^j, \beta, T$}
$n \gets \beta^2 K^2\log(T)$ and $j \gets -1$\;
\For{$t_m^j \log(T)/(\gamma_1 n)$ rounds}{
\uIf{$j=-1$}{
Pull $k \sim \mathcal{U}(\widehat{M})$;\quad
\lIf(\tcp*[f]{no collision}){$r_k(t) > 0$}{$j\gets k$}
}
\lElse{Pull $j$}}
\textbf{Return} $j$
\caption{\label{proto:getrank}\getrank}
\end{protocol}
The following quantities are used to describe \exploone in Algorithm~\ref{algo:explo1}:
\begin{itemize}
\item $\mathcal{M}^j(t) = \left( l_1^j(t), \ldots, l_M^j(t) \right)$ is the list of the empirical $M$ best arms for player~$j$ at round~$t$. It is updated only each $M$ rounds and ordered according to the index of the arms, i.e.,\ $l_1^j(t) < \ldots < l_M^j(t)$.
\item $\widehat{m}^j(t)$ is the empirical $M$-th best arm for player $j$ at round $t$.
\item $b_k^j(t) = \sup \lbrace q \geq 0 \ | \ T_k^j(t) \textrm{kl}( \widehat{\mu}_k^j(t), q) \leq f(t) \rbrace$ is the kl-UCB index of the arm $k$ for player $j$ at round $t$, where $f(t)=\log(t) + 4 \log(\log(t))$, $T_k^j(t)$ is the number of times player $j$ pulled $k$ and $\widehat{\mu}_k^j$ is the empirical mean.
\end{itemize}
\begin{algorithm2e}[h]
\DontPrintSemicolon
\KwIn{$M$, $j$}
\lIf{$t=0 \ (\mathrm{mod} \ M)$}{Update $\hat{\mu}^j(t), b^j(t), \widehat{m}^j(t)$ and $\mathcal{M}^j(t)= (l_1, \ldots, l_M)$}
$\pi \gets t+j \ (\mathrm{mod} \ M)+1$ \;
\lIf{$l_\pi \neq \widehat{m}^j(t)$}{Pull $l_\pi$ \tcp*[f]{exploit the $M-1$ best empirical arms}}
\Else{$\mathcal{B}^j(t) = \lbrace k \not\in \mathcal{M}^j(t) \ | \ b_k^j(t) \geq \widehat{\mu}^j_{\widehat{m}^j(t)}(t) \rbrace$ \tcp*{arms to explore}
\lIf{$\mathcal{B}^j(t) = \emptyset$}{Pull $l_\pi$}
\lElse{Pull $\begin{cases} l_\pi \text{ with proba } 1/2 \\ k \text{ chosen uniformly at random in } \mathcal{B}^j(t) \text{ otherwise} \quad \tcp*[f]{explore}\end{cases}$ }}
\caption{\label{algo:explo1}\exploone}
\end{algorithm2e}
\subsection{Proofs of Section~\ref{sec:algo1}}
Let us define $\alpha_k \coloneqq \mathbb{P}(X_k(t) > 0) \geq \mu_k$, $\gamma_1 = \frac{13}{14}$ and $\gamma_2 = \frac{16}{15}$.
\subsubsection{Regret analysis}
\label{app:regretalgo1}
This section aims at proving Theorem~\ref{thm:algoone}. This proof is divided in several auxiliary lemmas given below. First, the regret can be decomposed as follows:
\begin{equation}
\label{eq:regdec1}
R_T = R^{\text{init}} + R^{\text{explo}},
\end{equation}
\begin{equation*}
\text{where } \left\{ \begin{split} \begin{aligned} & R^{\text{init}} = T_0 {\mathlarger\sum_{k = 1}^M} \mu_{(k)} - \mathbb{E}_\mu \Big[{\mathlarger\sum_{t=1}^{T_0}} {\mathlarger \sum_{j = 1}^M} r^j(t) \Big]
\text{ with } T_{0} = \left(\frac{\gamma_2}{\gamma_1^2 \beta^2 K^2} + \frac{\gamma_2^2}{\gamma_1^2}\right)\ \max_j t_m^j, \\
& R^{\text{explo}} = (T-T_{0}) {\mathlarger\sum_{k = 1}^M} \mu_{(k)} - \mathbb{E}_\mu \Big[{\mathlarger\sum_{t=T_{0}+1}^{T}} {\mathlarger \sum_{j = 1}^M} r^j(t) \Big]. \end{aligned} \end{split} \right.
\end{equation*}
Lemma~\ref{lemma:estim1} first gives guarantees on the \estimateM protocol. Its proof is given in Appendix~\ref{app:proofestim1}.
\begin{lemm}
\label{lemma:estim1}
If $M-1$ players run \estimateM with $\beta\geq 39$, followed by a waiting room until~$\frac{\gamma_2}{\gamma_1} t_m^j$, then regardless of the strategy of the remaining player, with probability larger than~$1-\frac{6KM}{T}$, for any player:
\begin{equation*}
{\widehat{M}}^j = M \text{ and } \frac{t_m^j \alpha_{(K)}}{K} \in [\gamma_1 n, \gamma_2 n],
\end{equation*}
where $n = \beta^2 K^2 \log(T)$.
\end{lemm}
When $\widehat{M}^j = M$ and $\frac{t_m^j \alpha_{(K)}}{K} \in [\gamma_1 n, \gamma_2 n]$ for any cooperative player $j$, we say that the estimation phase is \textbf{successful}.
\begin{lemm}
\label{lemma:getrank}
Conditioned on the success of the estimation phase, with probability $1-\frac{M}{T}$, all the cooperative players end \getrank with different ranks $j \in [M]$, regardless of the behavior of other players.
\end{lemm}
The proof of Lemma~\ref{lemma:getrank} is given in Appendix~\ref{app:getrankproof}. If the estimation is successful and all players end \getrank with different ranks $j \in [M]$, the initialization is said successful.
Using the same arguments as \citet{proutiere2019}, the collective regret of the \exploone phase can be shown to be $M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(M)}, \mu_{(k)})} \log(T) + o(\log(T))$. This result is given by Lemma~\ref{lemma:ucb1}, whose proof is given in Appendix~\ref{app:ucb1proof}.
\begin{lemm}
\label{lemma:ucb1}
If all players follow \algoone[:]
\begin{small}
\begin{equation*}
\mathbb{E}[R^{\text{explo}}] \leq M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(M)}, \mu_{(k)})} \log(T) + o(\log(T)).
\end{equation*}
\end{small}
\end{lemm}
\begin{proof}[Proof of Theorem~\ref{thm:algoone}.]
Thanks to Lemma~\ref{lemma:ucb1}, the total regret is bounded by $$M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(M)}, \mu_{(k)})} \log(T) + \mathbb{E}[T_0] M + o(\log(T)).$$
Thanks to Lemmas~\ref{lemma:estim1} and \ref{lemma:getrank}, $\mathbb{E}[T_0] = \mathcal{O}\left(\frac{K^3 \log(T)}{\mu_{(K)}}\right)$, yielding Theorem~\ref{thm:algoone}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma:estim1}}
\label{app:proofestim1}
\begin{proof}[]
Let $j$ be a cooperative player and $q_k(t)$ be the probability at round $t$ that the remaining player pulls~$k$. Define $p_k^j(t) = \mathbb{P}[t \in C_k^j(t) \ | \ t \in N_k^j(t)]$. By definition, $p_k^j(t) = 1 - (1-1/K)^{M-2}(1-q_k(t))$ when all cooperative players are pulling uniformly at random. Two auxiliary Lemmas using classical concentration inequalities are used to prove Lemma~\ref{lemma:estim1}. The proofs of Lemmas~\ref{lemma:chernoff1}~and~\ref{lemma:concentration1} are given in Appendix~\ref{app:auxlemmas}.
\begin{lemm}
\label{lemma:chernoff1}
For any $\delta>0$,
\begin{enumerate}
\item $\mathbb{P} \left[\bigg|\frac{\mathrm{Card} C_k^j(T_M)}{\mathrm{Card} N_k^j(T_M)} - \frac{1}{\mathrm{Card} N_k^j(T_M)} \sum_{t \in N_k^j(T_M)} p_k^j(t)\bigg| \geq \delta \ \Big| \ N_k^j(T_M)\right] \leq 2 \exp(-\frac{\mathrm{Card} N_k^j(T_M) \delta^2}{2})$.
\end{enumerate}
For any $\delta \in (0,1)$ and fixed $T_M$,
\begin{enumerate}
\setcounter{enumi}{1}
\item $\mathbb{P} \left[\bigg|\mathrm{Card} N_k^j - \frac{\alpha_k T_M}{K}\bigg| \geq \delta \frac{ \alpha_k T_M}{K} \right] \leq 2\exp(-\frac{T_M \alpha_k \delta^2}{3 K})$.
\item $\mathbb{P} \left[\bigg| \sum_{t=1}^{T_M} (\mathds{1}(t \in N_k^j) - \frac{\alpha_k}{K})p_k^j(t) \bigg| \geq \delta \frac{ \alpha_k T_M}{K} \right] \leq 2 \exp\left( -\frac{T_M \alpha_k \delta^2}{3K}\right)$.
\end{enumerate}
\end{lemm}
\begin{lemm}
\label{lemma:concentration1}
For any $k$, $j$ and $\delta \in (0, \frac{\alpha_k}{K})$, with probability larger than~$1-\frac{6KM}{T}$,
\begin{equation*}
\bigg| \hat{p}_k^j(t_m^j) - \frac{1}{t_m^j} \sum_{t=1}^{t_m^j} p_k^j(t) \bigg| \leq 2\sqrt{\frac{6\log(T)}{n \left(1-2\sqrt{\frac{3}{2\beta^2}(1+\frac{3}{2\beta^2}})\right)}} + 2 \sqrt{\frac{\log(T)}{n}}.
\end{equation*}
And for $\beta \geq 39$:
\begin{equation*}
\frac{t_m^j \alpha_{(k)}}{K} \in \left[\frac{13}{14}n, \ \frac{16}{15}n\right].
\end{equation*}
\end{lemm}
Let $\varepsilon = 2\sqrt{\frac{6\log(T)}{n \left(1-2\sqrt{\frac{3}{2\beta^2}(1+\frac{3}{2\beta^2}})\right)}} + 2 \sqrt{\frac{\log(T)}{n}}$ and $p_k^j = \frac{1}{t_m^j} \sum_{t=1}^{t_m^j} p_k^j(t)$ such that with probability at least $1-\frac{6KM}{T}$, $\big| \hat{p}_k^j - p_k^j \big| \leq \varepsilon$. The remaining of the proof is conditioned on this event. \\
By definition of $n$, $\varepsilon = \frac{1}{K} f(\beta)$ where $f(x)=\frac{2}{x}\sqrt{\frac{6}{1-2\sqrt{\frac{3}{2x^2}(1+\frac{3}{2x^2}})}} + 2/x$. Note that $f(x) \leq \frac{1}{2e}$ for $x\geq 39$ and thus $\varepsilon \leq \frac{1}{2Ke}$ for the considered $\beta$.\\
The last point of Lemma~\ref{lemma:concentration1} yields that $t_m^j \leq \frac{\gamma_2}{\gamma_1} t_m^{j'}$ for any pair $j, j'$. All the cooperative players are thus pulling uniformly at random until at least $t_m^j$, thanks to the additional waiting room. Then,
\begin{equation*}
\frac{1}{K} \sum_k (1-p_k^j(t)) \ = \ (1-1/K)^{M-2} (1 - \frac{1}{K} \sum_k q_k(t))
\ = \ (1-1/K)^{M-1}.
\end{equation*}
When summing over $k$, it follows:
\begin{align*}
\frac{1}{K}\sum_k (1-p_k^j) - \varepsilon & \leq \frac{1}{K}\sum_k (1-\hat{p}_k^j) & \leq \frac{1}{K}\sum_k (1-p_k^j) + \varepsilon \\
(1-1/K)^{M-1} - \varepsilon & \leq \frac{1}{K}\sum_k (1-\hat{p}_k^j) & \leq (1-1/K)^{M-1} + \varepsilon \\
M-1 + \frac{\log(1+ \frac{\varepsilon}{(1-1/K)^{M-1}})}{\log(1-1/K)} & \leq \frac{\log\left(\frac{1}{K}\sum_k (1-\hat{p}_k^j)\right)}{\log(1-1/K)} & \leq M-1 + \frac{\log(1- \frac{\varepsilon}{(1-1/K)^{M-1}})}{\log(1-1/K)} \\
M-1 + \frac{\log(1+ \frac{1}{2K})}{\log(1-1/K)} & \leq \frac{\log\left(\frac{1}{K}\sum_k (1-\hat{p}_k^j)\right)}{\log(1-1/K)} & \leq M-1 + \frac{\log(1- \frac{1}{2K})}{\log(1-1/K)}
\end{align*}
The last line is obtained by observing that $\frac{\varepsilon}{(1-1/K)^{M-1}}$ is smaller than $\frac{1}{2K}$.
Observing that $\max\left(\frac{\log(1-x/2)}{\log(1-x)}, -\frac{\log(1+x/2)}{\log(1-x)}\right) < 1/2$ for any $x>0$, the last line implies:
\begin{equation*}
1+ \frac{\log\Big(\frac{1}{K}\sum_k (1-\hat{p}_k^j)\Big)}{\log(1-1/K)} \in (M-1/2, M+1/2).
\end{equation*}
When rounding this quantity to the closest integer, we thus obtain $M$, which yields the first part of Lemma~\ref{lemma:estim1}. The second part is directly given by Lemma~\ref{lemma:concentration1}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma:getrank}}
\label{app:getrankproof}
The proof of Lemma~\ref{lemma:getrank} relies on two lemmas given below.
\begin{lemm}
\label{lemma:waiting2}
Conditionally on the success of the estimation phase, when a cooperative player~$j$ proceeds to \getrank[,]all other cooperative players are either running \getrank or in a waiting room\footnote{Note that there is a waiting room before \textbf{and} after \getrank[.]}, i.e.,\ they are not proceeding to \exploone yet.
\end{lemm}
\begin{proof}
Recall that $\gamma_1 = 13/14$ and $\gamma_2 = 16/15$. Conditionally on the success of the estimation phase, for any pair $(j, j')$, $\frac{\gamma_2}{\gamma_1} t_m^j \geq t_m^{j'}$. Let $t_r^j = \frac{t_m^j}{\gamma_1 K^2 \beta^2}$ be the duration time of \getrank for player $j$. For the same reason, $\frac{\gamma_2}{\gamma_1} t_r^j \geq t_r^{j'}$. Player $j$ ends \getrank at round $t^j = \frac{\gamma_2}{\gamma_1} t_m^j + t_r^j$ and the second waiting room at round $\frac{\gamma_2}{\gamma_1} t^j$.
As $\frac{\gamma_2}{\gamma_1} t^j \geq t^{j'}$, this yields that when a player ends \getrank[,]all other players are not running \algoone yet. Because $\frac{\gamma_2}{\gamma_1} t_m^j \geq t_m^{j'}$, when a player starts \getrank[,]all other players also have already ended \estimateM[.]This yields Lemma~\ref{lemma:waiting2}.
\end{proof}
\begin{lemm}
\label{lemma:fix1}
Conditionally on the success of the estimation phase, with probability larger than~$1-\frac{1}{T}$, cooperative player $j$ ends \getrank with a rank in $[M]$.
\end{lemm}
\begin{proof}
Conditionally on the success of the estimation phase and thanks to Lemma~\ref{lemma:concentration1}, $t_r^j = \frac{t_m^j}{\gamma_1 K^2 \beta^2} \geq \frac{K \log(T)}{\alpha_{(K)}}$. Moreover, at any round of \getrank[,]the probability of observing $\eta_k(t)=0$ is larger than $\frac{\alpha_{(K)}}{M}$. Indeed, the probability of observing $\eta_k(t)$ is larger than $\alpha_{(K)}$ with Statistic sensing. Independently, the probability of having $\eta_k = 0$ is larger than $1/M$ since there is at least an arm among $[M]$ not pulled by any other player. These two points yield, as $M \leq K$:
\begin{align*}
\mathbb{P}[\text{player does not observe } \eta_k(t)=0 \text{ for } t_r^j \text{ successive rounds}] & \leq \left(1-\frac{\alpha_{(K)}}{M}\right)^{t_r^j} \\
& \leq \exp\left( -\frac{\alpha_{(K)}t_r^j}{M} \right) \\
& \leq \frac{1}{T}
\end{align*}
Thus, with probability larger than $1-\frac{1}{T}$, player $j$ observes $\eta_k(t)=0$ at least once during \getrank[,]i.e.,\ she ends the procedure with a rank in $[M]$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:getrank}.]
Combining Lemmas~\ref{lemma:waiting2}~and~\ref{lemma:fix1} yields that the cooperative player $j$ ends \getrank with a rank in $[M]$ and no other cooperative player ends with the same rank. Indeed, when a player gets the rank $j$, any other cooperative player has either no attributed rank (still running \getrank or the first waiting room), or an attributed rank $j'$. In the latter case, thanks to Lemma~\ref{lemma:waiting2}, this other player is either running \getrank or in the second waiting room, meaning she is still pulling $j'$. Since the first player ends with the rank $j$, this means that she did not encounter a collision when pulling $j$ and especially, $j \neq j'$.
\medskip
Considering a union bound among all cooperative players now yields Lemma~\ref{lemma:getrank}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma:ucb1}}
\label{app:ucb1proof}
Let us denote $T_0^j = \left(\frac{\gamma_2}{\gamma_1^2 \beta^2 K^2} + \frac{\gamma_2^2}{\gamma_1^2}\right)\ t_m^j$ such that player $j$ starts running \exploone at time $T_0^j$. This section aims at proving Lemma~\ref{lemma:ucb1}. In this section, the initialization is assumed to be successful. The regret due to an unsuccessful initialization is constant in $T$ and thus $o(\log(T))$. We prove in this section, in case of a successful initialization, the following:
\begin{equation}
\mathbb{E}[ R^{\text{explo}}] \leq M \sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(M)}, \mu_{(k)})} \log(T) +o(\log(T)).
\end{equation}
This proof follows the same scheme as the regret proof from \citet{proutiere2019}, except that there is no leader here.
Every \textit{bad event} then happens independently for each individual player. This adds a $M$ factor in the regret compared to the follower/leader algorithm\footnote{Which is not selfish-robust.} used by \citet{proutiere2019}.
For conciseness, we only give the main steps and refer to the original Lemmas in \citep{proutiere2019} for their detailed proof.
We first recall useful concentration Lemmas which correspond to Lemmas~1~and~2 in \citep{proutiere2019}. They are respectively simplified versions of Lemma~5 in \citep{combes2015} and Theorem~10 in \citep{garivier2011}.
\begin{lemm}
\label{lemma:proutiere1}
Let $k \in [K]$, $c>0$ and $H$ be a (random) set such that for all $t$, $\lbrace t \in H \rbrace$ is $\mathcal{F}_{t-1}$ measurable. Assume that there exists a sequence $(Z_t)_{t\geq 0}$ of binary random variables, independent of all $\mathcal{F}_t$, such that for $t\in H$, $\pi^j(t)=k$ if $Z_t=1$. Furthermore, if $\mathbb{E}[Z_t] \geq c$ for any $t$, then:
\begin{equation*}
\sum_{t\geq 1} \mathbb{P}[t \in H \ | \ |\widehat{\mu}^j_k(t) - \mu_k| \geq \delta] \leq \frac{4 + 2c/\delta^2}{c^2}.
\end{equation*}
\end{lemm}
\begin{lemm}
\label{lemma:proutiere2}
If player $j$ starts following \exploone at round $T_0^j+1$:
\begin{equation*}
\sum_{t>T_0^j} \mathbb{P}[b_k^j(t) < \mu_k] \leq 15.
\end{equation*}
\end{lemm}
Let $0 < \delta < \delta_0 \coloneqq \min_k \frac{\mu_{(k)} - \mu_{(k+1)}}{2}$. Besides the definitions given in Appendix~\ref{app:algo1descr}, define the following:
\begin{itemize}
\item $\mathcal{M}^*$ the list of the $M$-best arms, ordered according to their indices.
\item $\mathcal{A}^j=\lbrace t > T_0^j \ | \ \mathcal{M}^j(t) \neq \mathcal{M}^* \rbrace$.
\item $\mathcal{D}^j = \lbrace t > T_0^j \ | \ \exists k \in \mathcal{M}^j(t), \ |\widehat{\mu}_k^j(t) - \mu_k| \geq \delta \rbrace$.
\item $\mathcal{E}^j = \lbrace t > T_0^j \ | \ \exists k \in \mathcal{M}^*, \ b_k^j(t) < \mu_k \rbrace$.
\item $\mathcal{G}^j = \lbrace t \in \mathcal{A}^j \setminus \mathcal{D}^j \ | \ \exists k \in \mathcal{M}^* \setminus \mathcal{M}^j(t), \ |\widehat{\mu}_k^j(t) - \mu_k| \geq \delta \rbrace$.
\end{itemize}
\begin{lemm}
\label{lemma:badevent1}
$
\mathbb{E}[\mathrm{Card}(\mathcal{A}^j \cup \mathcal{D}^j)] \leq 8MK^2(6K + \delta^{-2}).
$
\end{lemm}
\begin{proof}
Similarly to \citet{proutiere2019}, we have $(\mathcal{A}^j \cup \mathcal{D}^j) \subset (\mathcal{D}^j \cup \mathcal{E}^j \cup \mathcal{G}^j)$. We can then individually bound $\mathbb{E}[\mathrm{Card}\mathcal{D}^j]$, $\mathbb{E}[\mathrm{Card}\mathcal{E}^j]$ and $\mathbb{E}[\mathrm{Card}\mathcal{G}^j]$, leading to Lemma~\ref{lemma:badevent1}. The detailed proof is omitted here as it exactly corresponds to Lemmas~3 and~4 in \citep{proutiere2019}.
\end{proof}
\begin{lemm}
\label{lemma:ucb2}
Consider a suboptimal arm $k$ and define $\mathcal{H}_k^j = \lbrace t \in \{T_0^j +1, \ldots, T \} \setminus (\mathcal{A}^j \cup \mathcal{D}^j) \ | \ \pi^j(t)=k\rbrace$. It holds
\begin{equation*}
\mathbb{E}\left[\mathrm{Card}\mathcal{H}_k^j\right] \leq \frac{\log T + 4 \log(\log T)}{\mathrm{kl}(\mu_k + \delta, \mu_{(M)}-\delta)} + 4 + 2\delta^{-2}.
\end{equation*}
\end{lemm}
Lemma~\ref{lemma:ucb2} can be proved using the arguments of Lemma~5 in \citep{proutiere2019}.
\medskip
\begin{proof}[Proof of Lemma~\ref{lemma:ucb1}.]
If $t \in \mathcal{A}^j \cup \mathcal{D}^j$, player $j$ collides with at most one player $j'$ such that $t \not\in \mathcal{A}^{j'} \cup \mathcal{D}^{j'}$.
Otherwise, $t \not\in \mathcal{A}^j \cup \mathcal{D}^j$ and player $j$ collides with a player $j'$ only if $t \in \mathcal{A}^{j'} \cup \mathcal{D}^{j'}$. Also, she pulls a suboptimal arm $k$ only on an exploration slot, i.e.,\ instead of pulling the $M$-th best arm. Thus, the regret caused by pulling a suboptimal arm $k$ when $t \not\in \mathcal{A}^j \cup \mathcal{D}^j$ is $(\mu_{(M)} - \mu_k)$ and this actually happens when $t \in \mathcal{H}_k^j$.
This discussion provides the following inequality, which concludes the proof of Lemma~\ref{lemma:ucb1} when using Lemmas~\ref{lemma:badevent1}~and~\ref{lemma:ucb2} and taking $\delta \to 0$.
\begin{small} \begin{equation*}
\mathbb{E} \left[ R^{\text{explo}} \right] \leq \underbrace{2 \sum_{j=1}^M \mathbb{E}\left[\mathrm{Card}(\mathcal{A}^j \cup \mathcal{D}^j)\right]}_{\text{collisions}} + \underbrace{\sum_{j\leq M} \sum_{k>M} (\mu_{(M)} - \mu_{(k)})\mathbb{E}\left[\mathrm{Card}\mathcal{H}_k^j\right]}_{\text{pulls of suboptimal arms}}.
\end{equation*}\end{small}
\end{proof}
\subsubsection{Proof of Theorem~\ref{thm:robust1}}
\label{app:statisticrobust1}
\begin{proof}[\vspace{-1.5em}]
\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]
\item Let us first prove the Nash equilibrium property. Define $\mathcal{E} = [T_0] \cup \Big( \bigcup\limits_{j\in[M]} (\mathcal{A}^j \cup \mathcal{D}^j) \Big)$ with the definitions of $T_0, \mathcal{A}^j$ and $\mathcal{D}^j$ given in Appendix~\ref{app:ucb1proof}.
Thanks to Lemmas~\ref{lemma:estim1}~and~\ref{lemma:getrank}, regardless of the strategy of a selfish player, all other players successfully end the initialization after a time $T_0$ with probability $1 - \mathcal{O}(KM/T)$. The remaining of the proof is conditioned on this event.
\medskip
The selfish player earns at most $\mu_{(1)} T_0$ during the initialization.
Note that \exploone never uses collision information, meaning that the behavior of the strategic player during this phase does not change the behaviors of the cooperative players. Thus, the optimal strategy during this phase for the strategic player is to pull the best available arm. Let $j$ be the rank of the strategic player\footnote{If the strategic player has no attributed rank, it is the only non-attributed rank in $[M]$.}. For $t \not\in \mathcal{E}$, this arm is the $k$-th arm of $\mathcal{M}^*$ with $k = t+j \ (\text{mod } M) +1$. In a whole block of length $M$ in $[T]\setminus \mathcal{E}$, the selfish player then earns at most $\sum_{k=1}^M \mu_{(k)}$.
Over all, when a strategic player deviates from \exploone[,]she earns at most:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^j_T(s', s_{-j})] \leq \mu_{(1)} (\mathrm{Card}\mathcal{E}+M) + \frac{T}{M} \sum_{k=1}^M \mu_{(k)}.
\end{equation*}
\end{small}
Note that we here add a factor $\mu_{(1)}$ in the initialization regret. This is only because the true loss of colliding is not $1$ but $\mu_{(1)}$. Also, the additional $\mu_{(1)}M$ term is due to the fact that the last block of length $M$ of \exploone is not totally completed.
Thanks to Theorem~\ref{thm:algoone}, it also comes:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^j_T(s)] \geq \frac{T}{M} \sum_{k=1}^M \mu_{(k)} -\sum_{k>M} \frac{\mu_{(M)} - \mu_{(k)}}{\mathrm{kl}(\mu_{(k)}, \mu_{(M)})}\log(T) - \mathcal{O} \left( \mu_{(1)} \frac{K^3}{\mu_{(K)}} \log(T) \right).
\end{equation*}
\end{small}
Lemmas~\ref{lemma:getrank} and \ref{lemma:badevent1} yield that $\mathbb{E}[\mathrm{Card}\mathcal{E}] = \mathcal{O}\left(\frac{K^3 \log(T)}{\mu_{(K)}} \right)$, which concludes the proof.
\item We now prove the $(\alpha, \varepsilon)$-stability of \algoone[.]Let $\varepsilon' = \mathbb{E}[\mathcal{E}]+M$. Consider that player $j$ is playing a deviation strategy $s' \in \mathcal{S}$ such that for some other player $i$ and $l>0$:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^i_T(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^i_T(s)] - l - (\varepsilon'+M).
\end{equation*}
\end{small}
We will first compare the reward of player $j$ with her optimal possible reward.
The only way for the selfish player to influence the sampling strategy of another player is in modifying the rank attributed to this other player. The total rewards of cooperative players with ranks $j$ and $j'$ only differ by at most $\varepsilon' + M$ in expectation, without considering the loss due to collisions with the selfish player.
The only other way to cause regret to another player $i$ is then to pull $\pi^i(t)$ at time $t$. This incurs a loss at most $\mu_{(1)}$ for player $i$, while this incurs a loss at least $\mu_{(M)}$ for player $j$, in comparison with her optimal strategy. This means that for incurring the additional loss $l$ to the player $i$, player $j$ must suffer herself from a loss $\frac{\mu_{(M)}}{\mu_{(1)}}$ compared to her optimal strategy $s^*$.
Thus, for $\alpha = \frac{\mu_{(M)}}{\mu_{(1)}}$:
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^i_T(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^i_T(s)] - l - (\varepsilon'+M) \implies
\mathbb{E}[\mathrm{Rew}^j_T(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^j_T(s^*, s_{-j})] - \alpha l
\end{equation*}
The first point of Theorem~\ref{thm:robust1} yields for its given $\varepsilon$:
$
\mathbb{E}[\mathrm{Rew}^j_T(s^*, s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^j_T(s)] + \varepsilon$.
\medskip
Noting $l_1=l + \varepsilon' + M$ and $\varepsilon_1 = \varepsilon + \alpha(\varepsilon' + M) = \mathcal{O}(\varepsilon)$, we have shown:
\begin{small}
\begin{equation*}
\mathbb{E}[\mathrm{Rew}^i_T(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^i_T(s)] - l_1 \implies \mathbb{E}[\mathrm{Rew}^j_T(s', s_{-j})] \leq \mathbb{E}[\mathrm{Rew}^j_T(s)] + \varepsilon_1 - \alpha l_1 .
\end{equation*}
\end{small}
\end{enumerate}
\end{proof}
\subsubsection{Auxiliary lemmas}
\label{app:auxlemmas}
This section provides useful Lemmas for the proof of Lemma~\ref{lemma:estim1}. We first recall a useful version of Chernoff bound.
\begin{lemm}
\label{lemma:chernoff0}
For any independent variables $X_1, \ldots, X_n$ in $[0,1]$ and $\delta \in (0,1)$:
\begin{equation*}
\mathbb{P}\left( \bigg|\sum_{i=1}^n X_i - \mathbb{E}[X_i]\bigg| \geq \delta\sum_{i=1}^n \mathbb{E}[X_i] \right) \leq 2 e^{-\frac{\delta^2 \sum_{i=1}^n \mathbb{E}[X_i]}{3}}.
\end{equation*}
\end{lemm}
\begin{proof}[Proof of Lemma~\ref{lemma:chernoff1}.]
\begin{enumerate}[wide, labelwidth=!, labelindent=0pt]
\item This is an application of Azuma-Hoeffding inequality on the variables ${\mathds{1}(t \in C_k^j(T_M) ) \ | \ t \in N_k^j(T_M)}$.
\item This is a consequence of Lemma~\ref{lemma:chernoff0} on the variables $\mathds{1}(t \in N_k^j)$.
\item This is the same result on the variables $\mathds{1}(t \in N_k^j) p_k^j(t) \ | \mathcal{F}_{t-1}$ where $\mathcal{F}_{t-1}$ is the filtration associated to the past events, using $\sum_{t=1}^{T_M} \mathbb{E}[\mathds{1}(t \in N_k^j) p_k^j(t) | \mathcal{F}_{t-1}] \leq \frac{T_M \alpha_k}{K}$.
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:concentration1}.]
From Lemma~\ref{lemma:chernoff1}, it comes:
\begin{itemize}
\item $\mathbb{P} \left[\exists t \leq T, \Big|\hat{p}_k^j(t) - \frac{1}{\mathrm{Card} N_k^j} \sum_{t' \in N_k^j} p_k^j(t')\Big| \geq 2\sqrt{\frac{\log(T)}{\mathrm{Card} N_k^j}}\right] \leq \frac{2}{T}$,
\item $\mathbb{P} \left[\exists t \leq T, \Big|\frac{K\mathrm{Card} N_k^j}{\alpha_k t} - 1\Big| \geq \sqrt{\frac{6\log(T)K}{\alpha_kt}} \right] \leq \frac{2}{T}, ~\refstepcounter{equation}\hfill(\theequation)\label{eq:chernoff1}$
\item $\mathbb{P} \left[\exists t \leq T, \Big|\frac{K}{\alpha_k t} \sum_{t'\in N_k^j} p_k^j(t') - \frac{1}{t}\sum_{t'\leq t} p_k^j(t')\Big| \geq \sqrt{\frac{6 \log(T) K}{\alpha_k t}} \right] \leq \frac{2}{T}$.
\end{itemize}
Noting that $\sum_{t'\in N_k^j} p_k^j(t') \leq \mathrm{Card} N_k^j$, Equation~\eqref{eq:chernoff1} implies:
\begin{equation*}
\mathbb{P} \left[\exists t \leq T, \bigg|\frac{K}{\alpha_k t}\sum_{t'\in N_k^j} p_k^j(t') - \frac{1}{\mathrm{Card} N_k^j}\sum_{t'\in N_k^j} p_k^j(t')\bigg| \geq \sqrt{\frac{6 \log(T) K}{\alpha_k t}} \right] \leq \frac{2}{T}.
\end{equation*}
Combining these three inequalities and making the union bound over all the players and arms yield that with probability larger than $1-\frac{6KM}{T}$:
\begin{equation}
\label{eq:concentration1} \bigg|\hat{p}_k^j(t_m^j) - \frac{1}{t_m^j}\sum_{t\leq t_m^j} p_k^j(t) \bigg| \leq 2\sqrt{\frac{6\log(T) K}{\alpha_k t_m^j}} + 2 \sqrt{\frac{\log(T)}{\mathrm{Card} N_k^j(t_m^j)}}.
\end{equation}
Moreover, under the same event, Equation~\eqref{eq:chernoff1} also gives that
$$N_k^j(t_m^j) \in \bigg[\frac{\alpha_{k} t_m^j}{K} - \sqrt{\frac{6\alpha_k t_m^j\log(T)}{K}}, \ \frac{\alpha_{k} t_m^j}{K} + \sqrt{\frac{6\alpha_k t_m^j\log(T)}{K}}\bigg].$$
Specifically, this yields $n \leq \frac{\alpha_{k} t_m^j}{K} + \sqrt{\frac{6\alpha_{k} t_m^j\log(T)}{K}}$, or equivalently $\frac{t_m^j \alpha_{k}}{K} \geq n-2\sqrt{\frac{3\log(T)}{2}}\sqrt{n+\frac{3\log(T)}{2}}$. Since $n=\beta^2 K^2 \log(T)$, this becomes $\frac{t_m^j \alpha_{k}}{K} \geq n (1-2\sqrt{\frac{3}{2\beta^2K^2}}\sqrt{1+\frac{3}{2\beta^2K^2}})$ and Equation~\eqref{eq:concentration1} now rewrites into:
\begin{equation*}
\bigg|\hat{p}_k^j(t_m^j) - \frac{1}{t_m^j}\sum_{t\leq t_m^j} p_k^j(t) \bigg| \leq 2\sqrt{\frac{6\log(T)}{n \left(1-2\sqrt{\frac{3}{2\beta^2 K^2}(1+\frac{3}{2\beta^2 K^2}})\right)}} + 2 \sqrt{\frac{\log(T)}{n}}
\end{equation*}
Also, $n \geq \frac{\alpha_{k} t_m^j}{K} - \sqrt{\frac{6\log(T)\alpha_{k} t_m^j}{K}}$ for some $k$, which yields $\frac{t_m^j \alpha_{k}}{K} \leq n(1+\frac{3}{\beta^2 K^2}+2\sqrt{\frac{3}{2\beta^2K^2}}\sqrt{1+\frac{3}{2\beta^2K^2}})$. This relation then also holds for $\frac{t_m^j \alpha_{(K)}}{K}$.
We have therefore proved that:
$$
n \left(1-2\sqrt{\frac{3}{2\beta^2}}\sqrt{1+\frac{3}{2\beta^2}}\right) \leq\frac{t_m^j \alpha_{(k)}}{K} \leq n\left(1+\frac{3}{\beta^2}+2\sqrt{\frac{3}{2\beta^2}}\sqrt{1+\frac{3}{2\beta^2}}\right).
$$
For $\beta \geq 39$, this gives the bound in Lemma~\ref{lemma:concentration1}.
\end{proof}
|
1,314,259,992,987 | arxiv | \section{Introduction} \label{intro}
Particle-particle collisions during small-number fewbody interactions are the cause for several ubiquitous astrophysical phenomena. These include, but are not limited to, blue straggler formation in globular and open star clusters due to stellar collisions \citep[e.g.][]{leonard89,fregeau04,leigh07,hypki16,hypki17}, the production of anomalously blue stars in galactic nuclei \citep[e.g.][]{shara74,davies98,bailey99,yu03,dale09,leigh16b}, the formation of intermediate-mass black holes via runaway stellar collisions that could also serve as the seeds for the formation of supermassive black holes \citep[e.g.][]{portegieszwart04,giersz15,stone17}, the collisional growth of protoplanetary disks \citep[e.g.][]{goldreich04,lithwick07}, the production of runaway stars from young star-forming regions \citep[e.g.][]{blaauw54,perets12,oh15,ryu17a,ryu17b,ryu17c}, the origins of elliptical galaxies \citep[e.g.][]{binney87,balland98,trinchieri03}, etc.
Here, we continue our study of direct collisions between particles during chaotic few-body interactions. We paraphrase the results of our previous works in this series here for completeness. In Paper I \citep{leigh12}, we studied how the collision probability depends on the number of interacting particles. We found a connection between the mean free path approximation and the binomial theorem. We showed that, for identical particles and a given total encounter energy and angular momentum, the collision probability scales roughly as $N^2$, where $N$ is the number of interacting particles. The physical origin of this $N$-dependence comes from the binomial theorem; the number of ways of selecting any pair of particles from a set of $N$ identical particles is ${N \choose 2} = N(N-1)/2$.
In Paper II \citep{leigh15}, we found that, for (near-)identical mass particles, the collision probability is directly proportional to the collisional cross-section for the types of small-number interactions expected to occur in actual star clusters. The dynamics of such gravitationally-bound systems of chaotically-interacting finite-sized particles are analogous to a system of pendulums; the particles oscillate semi-periodically about the system centre of mass. Here, the cross-section for any two particles to collide directly is, to first order for particles with similar masses and large radii, proportional to the square of the sum of their radii. By means of a combinatorics-based approach, it follows that the collision probability can be expressed analytically for any number of particles and any combination of particle radii.
In Paper III \citep{leigh16}, we derived analytic formulae for the time-scales for different collision scenarios to occur, and compared the results to numerical scattering simulations of binary-binary interactions. We showed that the \textit{simulated} relative probabilities for the different collision scenarios are bounded by the corresponding analytic predictions, assuming either purely radial or purely tangential motions for the particles. We further showed that, in the purely radial limit, our analytic time-scales provide good order-of-magnitude estimates for the mean time-scales for direct collisions to occur in our simulations.
In this paper, the fourth in the series, we study the probabilities for different collision scenarios to occur, while simultaneously varying the distribution of particle masses \textit{and} radii. We first describe the framework underlying our model in the Newtonian limit, which is founded on a combinatorics-based backbone and is designed to calculate the time-scales or rates for direct collisions to occur during chaotic gravitational interactions involving finite-sized particles with different (but comparable) masses.
In Section~\ref{method}, we apply the mean free path approximation to derive theoretical collision time-scales and \textit{relative} collision probabilities. We further introduce the Collision Rate Diagram (CRD), which illustratively quantifies how well our derived collision time-scales are able to reproduce the simulated data. We also describe the simulations used in this study to test our model. In Section~\ref{results}, we present and compare the resulting simulated and theoretical collision probabilities and rates, using Collision Rate Diagrams as a guide to constrain to constrain the dominant physics needed to be reproduced via our analytic model. The assumptions and limitations underlying our model are discussed along with their applicability to astrophysical systems in Section~\ref{discussion}. Our key results are summarized in Section~\ref{summary}.
\section{Method} \label{method}
In this section, we re-visit the concept of a Collision Rate Diagram (CRD), first presented in Paper III of this series. In this previous study, we derived different collision time-scales, and from these the various rates and probabilities for different collision scenarios to occur. This will ultimately facilitate our ability to quantify in this paper, via the CRD, the effects of incorporating different assumptions in deriving these time-scales and rates. We go on to present the numerical scattering experiments of binary-binary encounters involving finite-sized particles with different combinations of particle masses and radii. We compare the results of these simulations to our analytic predictions in Section~\ref{results}, for different input assumptions to our model (e.g., setting the collisional cross-section equal to the geometric cross-section, setting it equal to the gravitationally-focused cross-section, with/without the assumption of time-averaged virial equilibrium, etc.), in order to identify the dominant physical processes deciding the relative collision rates and probabilities.
Throughout this paper, we define a direct collision as occurring when the particle radii overlap directly, following the "sticky-star" approximation.
\subsection{Model} \label{model}
In this section, we first re-visit the concept of a Collision Rate Diagram, before going on to present the numerical scattering experiments performed in this paper. Later, these will be used, in conjunction with the CRDs presented in this section, to identify the dominant physical processes deciding the relative collision rates as a function of the the distribution of particle masses and radii, etc.
\subsubsection{Collision Rate Diagram} \label{CRD}
In this section, we re-introduce the concept of a Collision Rate Diagram (CRD), first presented in Paper III of this series. This diagram provides an immediate and visual comparison between the predictions of our analytic derivations for the relative rates of collisions between different particle types (and their underlying assumptions; see Paper III) and the results of numerical scattering experiments. This is because the area corresponding to a particular collision event is directly proportional to the probability of that outcome occurring. Hence, as we will show, it provides a fast and efficient means of comparing theoretical predictions to simulated data. By changing the assumptions underlying a given model, the parameter space indicating the rate dominance for each collision scenario will change. Hence, in this way, the CRD is a potentially efficient tool for isolating the dominant physics deciding the relative rates for different collision scenarios to occur, and can be robustly applied to any astrophysical problem that touches upon the collisional regime of gravitational dynamics.
We begin by describing the original CRD from \citet{leigh16}, and repeat it here for completeness. First, consider interactions involving three different types of particles, labeled A, B, and C. In this case, we can use our derived time-scales to construct a Collision Rate Diagram, using a similar formalism as outlined in Paper III, and earlier in \citet{leigh11} and \citet{leigh13}. A CRD is a diagram that illustrates the parameter space for which the rates of the different types of collisions (e.g., A+A, A+C, B+C, etc.) each dominate over all others. In Paper III of this series, we considered only 3-dimensional CRDs, which have only a single quadrant, facilitated by writing the fraction of single stars as the sum of the fractions of binaries and triples, combined with the critical assumption of mass conservation for the entire system.
The procedure for producing such a CRD is as follows. First, we note that, for three particle types, we can write the total number of particles $N$ involved in an interaction as:
\begin{equation}
\label{eqn:number}
N = N_{\rm A} + N_{\rm B} + N_{\rm C},
\end{equation}
where $N_{\rm A}$, $N_{\rm B}$ and $N_{\rm C}$ denote, respectively, the total number of particles of type A, B and C. Then, the fraction of objects of a given particle type $i$ can be written:
\begin{equation}
\label{eqn:fraction1}
f_{\rm i} = \frac{N_{\rm i}}{N},
\end{equation}
and the sum of their total must of course satisfy the relation:
\begin{equation}
\label{eqn:fraction2}
1 = f_{\rm A} + f_{\rm B} + f_{\rm C}
\end{equation}
Now, to produce a CRD, every pair of collision rates (e.g., $\Gamma_{\rm A+A}$, $\Gamma_{\rm A+B}$, $\Gamma_{\rm A+C}$, etc.) should be equated, and the resulting relation plotted in $f_{\rm B}$-$f_{\rm C}$-space. The region of parameter space in the $f_{\rm B}$-$f_{\rm C}$-plane for which each type of collision occurs at the highest rate can then be identified, and a corresponding boundary can be drawn in the CRD. This produces a diagram that identifies the parameter space in the $f_{\rm B}$-$f_{\rm C}$-plane for which the rates of the different types of collisions (e.g., A+A, A+C, B+C, etc.) each dominate.
Figure~\ref{fig:fig1} shows an example of a 3-D Collision Rate Diagram. To construct this figure, we assume for simplicity that the rate of collisions between particles of type $i$ and $j$ can be written:
\begin{equation}
\label{eqn:gammaij}
\Gamma_{\rm i+j} = f_{\rm i}f_{\rm j}N_{\rm i}n_{\rm j}{\sigma_{\rm i+j}}v_{\rm i+j},
\end{equation}
where we do not yet include gravitational-focusing in our estimate for the collisional cross-section $\sigma_{\rm i+j}$, and instead set it equal to the geometric cross-section for collision. We emphasize that this is the rate for \textit{any} particle of type $i$ to collide with \textit{any} particle of type $j$. This is in contrast to the rate for a \textit{given or specified} object of type $i$ to collide with \textit{any} particle of type $j$, which is smaller than the previous rate by a factor $N_{\rm i}$.
Equation~\ref{eqn:gammaij} can be modified to generate the simplest possible form for this CRD.\footnote{We note that Equation~\ref{eqn:gammaij} is itself not strictly correct, since the relative rates should include a combinatorial correction to avoid over-counting collisions between identical particles. We will return to this later, and properly include this correction in our formulation.} Specifically, we can assume the same particle number density $n = n_{\rm j}$ and relative velocity at infinity $v = v_{\rm i+j}$ for each type of collision (i.e., for all $i$ and $j$). This simplifying assumption neglects the more complicated geometry considered in Paper III, but probes the simplest physical assumptions possible for the problem at hand (i.e., a suitable starting point for the method). Hence, the simplified CRD shown in Figure~\ref{fig:fig1} quantities only the importance of the particle number and the geometric collisional cross-section in determining the relative collision rates.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig1.eps}
\end{center}
\caption[A Collision Rate Diagram for three different types of particles]{The parameter space in the $f_{\rm B}$-$f_{\rm C}$-plane for which different types of collisions dominate. To generate the CRD shown here, we assume three different types of particles with masses $m_{\rm A} =$ 1 M$_{\odot}$, $m_{\rm B} =$ 2 M$_{\odot}$ and $m_{\rm C} =$ 3 M$_{\odot}$, and radii $R_{\rm A} =$ 1 R$_{\odot}$, $R_{\rm B} =$ 2 R$_{\odot}$ and $R_{\rm C} =$ 3 R$_{\odot}$.
\label{fig:fig1}}
\end{figure}
Now, let us add a fourth particle type in to the mix, with label D. Here, we can generate a 4-dimensional CRD, which has four quadrants and ultimately represents a 2-dimensional slice of the over-arching 4-dimensional parameter space. In each of the four quadrants, we set the number of one particle type to be zero. Then, each quadrant is effectively analogous to the CRD shown in Figure~\ref{fig:fig1}. Combining all four quadrants allows for a more thorough comparison between theoretical predictions and the simulations for a larger subset of the total possible parameter space. An example of a 4-D CRD is shown in Figure~\ref{fig:fig2}, adopting the same assumptions as in Figure~\ref{fig:fig1} in order to generate the simplest form of the CRD.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig2.eps}
\end{center}
\caption[A Collision Rate Diagram for four different types of particles]{Each quadrant shows the parameter space in the $f_{\rm i}$-$f_{\rm j}$-plane for which different types of collisions dominate, assuming one of the particle types is not present. To generate the CRD shown here, we assume four different types of particles with masses $m_{\rm A} =$ 1 M$_{\odot}$, $m_{\rm B} =$ 2 M$_{\odot}$, $m_{\rm C} =$ 3 M$_{\odot}$ and $m_{\rm D} =$ 4 M$_{\odot}$, and radii $R_{\rm A} =$ 1 R$_{\odot}$, $R_{\rm B} =$ 2 R$_{\odot}$, $R_{\rm C} =$ 3 R$_{\odot}$ and $R_{\rm D} =$ 4 R$_{\odot}$.
\label{fig:fig2}}
\end{figure}
In the subsequent sections, we will compare the results of numerical scattering simulations to our analytic predictions. This will be done using various forms of the CRD as our guide toward isolating the dominant physics deciding the relative collision rates or probabilities. In this section, we have presented an over-simplified form of the CRD. By introducing additional physics in to our model in the subsequent sections, specifically the gravitationally-focussed cross-section and a combinatorial correction, we will illustrate and quantify the effects of each of these physical components on our over-arching model.
\subsection{Numerical scattering experiments} \label{exp}
As in Paper III of this series, we calculate the outcomes of a series of binary-binary (2+2) encounters using the \texttt{FEWBODY} numerical
scattering code\footnote{The source code can be found at http://fewbody.sourceforge.net.}. As discussed in more detail in \citet{fregeau04}, the code
integrates the usual $N$-body equations in position-space in order to advance the system forward in time. This is done using the
eighth-order Runge-Kutta Prince-Dormand integration method with adaptive time-stepping and ninth-order error estimation.
In this paper, we set $m_{\rm A} =$ 1 M$_{\odot}$, $m_{\rm B} =$ 2 M$_{\odot}$, $m_{\rm C} =$ 3 M$_{\odot}$ and $m_{\rm D} =$ 4 M$_{\odot}$, with $R_{\rm A} =$ 1 R$_{\odot}$, $R_{\rm B} =$ 2 R$_{\odot}$, $R_{\rm C} =$ 3 R$_{\odot}$ and $R_{\rm D} =$ 4 R$_{\odot}$. We then consider different combinations of these four over-arching particle types.
For these simulation sets, all particles are assumed to have finite radii (i.e., spherical) and we adopt the indicated combinations of masses and radii in Table~\ref{table:stats}. In all simulations all binaries have $a_{\rm A} =$ $a_{\rm B} =$ 5 AU initially, and eccentricities $e_{\rm A} = e_{\rm B} =$ 0. We set the impact parameter to zero and the initial relative velocity at infinity $v_{\rm rel}$ to 0.3$v_{\rm crit}$, where $v_{\rm crit}$ is the critical velocity. It is defined as the relative velocity at infinity required for a total encounter energy of zero.\footnote{Note that this choice of relative velocity is typical for dense star clusters.} As found in previous studies \citep[e.g.][]{leigh16}, such low relative velocities at infinity and small impact parameters maximize the probability of long-lived resonant interactions occurring, for which the assumption of ergodicity is upheld. All angles defining the relative configurations of the binary orbital planes and phases are chosen randomly. We perform 4 $\times$ 10$^4$ numerical scattering experiments for every combination of particle masses.
As in previous papers in this series, all simulations are terminated at the instant the first collision occurs. If no collisions occur, we use the same criteria as in \citet{fregeau04} to determine when an encounter is complete. We refer the reader to \citet{fregeau04} and the previous papers in this series for the precise implementation of the stopping criteria used in this paper.
As in previous studies, we adopt a tidal tolerance parameter
$\delta =$ 10$^{-7}$ for all simulations. The reader can refer to previous papers in this series for the full justification underlying this choice for $\delta$ (see also \citealt{geller15} and \citealt{leigh16}).
\section{Results} \label{results}
In this section, we present the results of our numerical scattering experiments and compare them to our theoretical predictions. This is done by plotting the results of our numerical experiments in different manifestations of the Collision Rate Diagram, each time incorporating additional physics in to the model that generates the CRD. These results are summarized below in Table~\ref{table:stats}, and illustrated in Figures~\ref{fig:fig3}, ~\ref{fig:fig4} and~\ref{fig:fig5}.
\subsection{Confronting the analytic rates with simulated data} \label{compare}
In order to compare the results of the simulated data with the predictions of the CRD, we can include points in Figure~\ref{fig:fig3} for those combinations of $f_{\rm A}$, $f_{\rm B}$, $f_{\rm C}$ and $f_{\rm D}$ included in Table~\ref{table:stats} via our simulations. The points are assigned shapes according to the outcome specified in each figure. If the relative collision probabilities in the simulations agree exactly with the analytic predictions shown in the CRD, we plot the points as filled (i.e., the points fall in a region of the CRD corresponding to the dominant outcome found in our simulations). If the points fall in an incorrect region of the CRD, we leave them unfilled.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig3.eps}
\end{center}
\caption[A Collision Rate Diagram for four different types of particles, with the simulated data over-plotted]{The same as Figure~\ref{fig:fig2} but with the simulated data over-plotted.
\label{fig:fig3}}
\end{figure}
Figure~\ref{fig:fig3} clearly shows that the agreement between the simulations and the predictions of the CRD is poor without including additional physics, such as gravitational-focusing and a combinatorial correction. If we include these additional effects in our model, can we improve upon the reported disagreement between theory and simulations? This is directly addressed below.
To address this, we re-construct Figure~\ref{fig:fig3}, but adopting the gravitationally-focused cross-section for collision instead of the geometric cross-section. This is motivated by the derived collision time-scales and rates presented in Paper III of this series. As shown in Figure~\ref{fig:fig4}, the net effect of these alternations to the CRD is to increase the probability of collisions involving heavier particles. The inclusion of this additional physics in our model only slightly improves the agreement between theory and simulations.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig4.eps}
\end{center}
\caption[A Collision Rate Diagram for four different types of particles, with the simulated data over-plotted]{The same as Figure~\ref{fig:fig3}, but adopting the gravitationally-focussed cross-section for collision instead of the geometric cross-section.
\label{fig:fig4}}
\end{figure}
Finally, we include a combinatorial correction in our estimates for the relative collision rates in Equation~\ref{eqn:gammaij}. Specifically, the number of ways of selecting two identical particles from a sample of $N$ identical particles is ${N \choose 2}$. To account for this, we directly correct our assumed number fractions in Equation~\ref{eqn:gammaij} if $i = j$, and leave the rates unchanged otherwise. This corrects for over-estimating the rate of collisions between identical particles in our simplified formulation presented in Section~\ref{CRD}.
As shown in Figure~\ref{fig:fig5}, this additional correction to our base model further improves the agreement between theory and simulations. Now, all data points agree with our theoretical model. It follows that our theoretical model successfully reproduces the simulated data for all of our simulations.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig5.eps}
\end{center}
\caption[A Collision Rate Diagram for four different types of particles, with the simulated data over-plotted]{The same as Figure~\ref{fig:fig4}, but adopting a combinatorial correction to the theoretically-derived collision rates, as discussed in the text.
\label{fig:fig5}}
\end{figure}
\subsection{The validity of the CRD for unexplored regions of parameter space} \label{unexplored}
We consider in total 12 combinations of particle types (A, B, C or D), but many more are also possible. This choice was done for simplicity and to ensure that each simulated data point would appear only once in the CRD, in one of the four quadrants shown in Figure~\ref{fig:fig5}. We do not include combinations of only two particle types, for example, since these simulated data points would be degenerate, and appear in more than one place in the CRD (i.e., two different quadrants).
With that said, we did perform a few additional scattering experiments assuming only two types of particles in a given four-body interaction. Upon comparing the results of these simulations to our model and plotting them in the CRD, we consistently find excellent agreement between the theoretical predictions (i.e., CRD) and the simulations. For example, we performed analogous simulations to those described in Section~\ref{exp} but assuming two particles of type B and two particles of type D. The simulated data predict that B+D collisions should be dominant for $f_{\rm B} = f_{\rm D} = 0.5$. A simple comparison to Figure~\ref{fig:fig5}, and specifically the lower right and left panels, reveals that the simulation outcomes are once again successfully described by the CRD. Upon assuming two particles of type B and two particles of type C, the simulated data predict that B+C collisions should be dominant for $f_{\rm B} = f_{\rm C} = 0.5$. Once again, a simple comparison to Figure~\ref{fig:fig5}, and specifically the upper and lower right panels, confirms that the simulation outcomes agree with the CRD. We also point that, for any set of simulations assuming all identical particles, the resulting data points will always appear at the extremums of the various axes in Figure~\ref{fig:fig5}. Consequently, by definition, these simulated data points will also always agree with the predictions of the CRD. All of these examples further supports the validity the CRD developed and tested in this paper.
\clearpage
\begin{landscape}
\begin{table}
\begin{center}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Particle Combination & \multicolumn{10}{|c|}{Simulated Number} \\
& \multicolumn{10}{|c|}{of Collisions} & \\
($N_{\rm A}$,$N_{\rm B}$,$N_{\rm C}$,$N_{\rm D}$) & $N_{\rm A+A}$ & $N_{\rm B+B}$ & $N_{\rm C+C}$ & $N_{\rm D+D}$ & $N_{\rm A+B}$ & $N_{\rm A+C}$ & $N_{\rm A+D}$ & $N_{\rm B+C}$ & $N_{\rm B+D}$ & $N_{\rm C+D}$ & \\
\hline
(2,0,1,1) & 213 & 0 & 0 & 0 & 0 & 1165 & 2778 & 0 & 0 & 7042 \\
(2,1,0,1) & 408 & 0 & 0 & 0 & 1309 & 0 & 8071 & 0 & 5510 & 0 \\
(2,1,1,0) & 262 & 0 & 0 & 0 & 3305 & 5530 & 0 & 6150 & 0 & 0 \\
(1,2,1,0) & 0 & 2471 & 0 & 0 & 1600 & 1099 & 0 & 6711 & 0 & 0 \\
(1,2,0,1) & 0 & 638 & 0 & 0 & 1115 & 0 & 2954 & 0 & 8144 & 0 \\
(0,2,1,1) & 0 & 1071 & 0 & 0 & 0 & 0 & 0 & 2507 & 7628 & 5668 \\
(0,1,2,1) & 0 & 0 & 2060 & 0 & 0 & 0 & 0 & 3134 & 3481 & 10839 \\
(1,1,2,0) & 0 & 0 & 6004 & 0 & 573 & 2970 & 0 & 7384 & 0 & 0 \\
(1,0,2,1) & 0 & 0 & 2943 & 0 & 0 & 1203 & 1224 & 0 & 0 & 11008 \\
(0,1,1,2) & 0 & 0 & 0 & 5907 & 0 & 0 & 0 & 1625 & 6177 & 6679 \\
(1,0,1,2) & 0 & 0 & 0 & 6535 & 0 & 519 & 1685 & 0 & 0 & 9151 \\
(1,1,0,2) & 0 & 0 & 0 & 7469 & 58 & 0 & 1871 & 0 & 3965 & 0 \\
\end{tabular}
\end{center}
\caption{The simulated numbers of each type of collision for different combinations of particle masses and radii.}
\label{table:stats}
\end{table}
\end{landscape}
\clearpage
\section{Discussion} \label{discussion}
In this paper, we present a method for directly comparing to simulated data the analytic rates for different collision scenarios to occur during chaotic gravitational interactions involving arbitrary numbers of finite-sized particles, with any distributions of particle masses and radii. The method is flexible in the sense that the assumptions underlying the derivations of the analytic rates can be freely modified, and the subsequent impact on the agreement with the simulated data is easily quantified. This facilitates a quick identification of the dominant physics deciding the relative rates of collisions, for the chosen initial conditions and particle properties. To illustrate this, we compare our analytic predictions to the results of numerical scattering experiments of four-body interactions involving finite-sized particles. Overall, the agreement between our analytic predictions and the simulations is excellent.
This paper is meant as a more in-depth introduction to the CRD than given in Paper III of this series, while clearly articulating how to properly use it to construct a robust analytic framework for predicting collision rates, depending on the particle types and important physics (sticky-star approximation for collisions? dissipation? GW emission? etc.) for the problem to which the user wishes to apply their version of the CRD. In other words, the validity of the CRD depends on the astrophysical environment of interest, since this in turn decides the types and properties of the particles (e.g., stars, comets, galaxies, etc.) in addition to the dominant physics affecting the collision rates (e.g., gravitational-focusing, combinatorics, various forms of dissipation such as from tides or GWs, etc.).
\subsection{Caveats} \label{caveats}
A few cautionary notes should be kept in mind when applying the Collision Rate Diagrams presented in this paper. First, we expect our base model to be valid only for mass ratios $\lesssim$ 10 \citep{leigh16}. Above this, we expect the most massive objects to sufficiently dominate the gravitational potential, such that the collision rate begins to enter the loss-cone regime (see \citet{merritt13} for more details) and/or the lowest mass particles are quickly ejected from the system without entering a resonant interaction state. In general, a prolonged resonant state and the assumption of ergodicity being upheld are key requirements for our base model to be applicable \citep{leigh16}. More work will be needed to better understand how to adapt our model to smoothly transition between these two regimes, where these assumptions start to break down. Provided the assumption of ergodicity is upheld, we expect some variation of the collision rate estimates presented in Paper III to accurately capture the physics, in all but the most extreme cases (see below).
Finally, we note that more work is still needed to better understand the dependence of the collision probability on the impact parameter of the encounter, which was consistently set to equal zero throughout this paper. An increase in the impact parameter should mostly act to increase the total angular momentum of the interaction (for a fixed relative velocity at infinity). In future work, we intend to vary this parameter to better understand what adjustments to our base model might be needed to properly accommodate this additional free parameter. The analytic model presented in Paper III of this series for the high-angular momentum regime, which assumes purely tangential motions relative to the system centre of mass, could provide the needed adjustments.
\subsection{Implications for dense stellar environments} \label{implications}
In this section, we describe the implications and applicability of our results to real astrophysical environments, with a focus on dense stellar clusters.
\subsubsection{Old globular clusters} \label{oldGCs}
Old globular clusters are the ideal environments for applying the methods presented in this paper. There are three main reasons for this: 1) direct single-binary and binary-binary encounters (and possibly interactions involving triples as well) occur commonly in the dense cores of old GCs \citep[e.g.][]{leonard89,leigh11,leigh13}; 2) direct collisions between stars occur frequently during 3-body, 4-body, 5-body, etc. interactions in GCs \citep[e.g.][]{leonard89,leigh13,leigh15}; and 3) the range of stellar masses among such old stellar populations are typically $\sim 0.08 - 0.8$ M$_{\odot}$, which limits the mass ratios of the fewbody interactions to q $\gtrsim$ 0.1 and consequently ensures that the assumption of ergodicity (which is needed to apply our model and construct a CRD) should be approximately upheld for most chaotic fewbody interactions in old GCs.
To properly apply the CRD and methods presented in this paper to old GCs, a mass-radius relation for main-sequence stars should be adopted. Typically, for such an old stellar population, the relation $R/R_{\odot} = (m/M_{\odot})^{0.75}$ is used, where $m$ is the main-sequence mass and $R$ is the corresponding stellar radius \citep{hansen04}. This ensures that the particle masses and radii are chosen appropriately when constructing a CRD designed for chaotic fewbody interactions in old GCs. Such a change would likely have the affect of reducing the importance of the particle radius in the CRD, such that the difference in area between the largest and smallest zones (e.g., D+D and A+A) would be reduced.
\subsubsection{Young massive clusters} \label{youngMCs}
Young massive clusters could also be well suited to applying the methods presented in this paper. For example, \citet{portegieszwart04} showed using $N$-body simulations that runaway collisions of massive stars can occur in the dense cores of primordial GCs, possibly forming a supra-massive star or even an intermediate-mass black hole. We emphasize that, since we are only interested in the \textit{relative} collision rates, the model is scale-free for our purposes if we fix the ratio of particle mass to particle radius such that it remains unity (as assumed in Figure~\ref{fig:fig5}, for example). By extension, the CRD shown in Figure~\ref{fig:fig5} is equally well-suited to larger, more massive particles, provided the masses are all directly proportional to the masses assumed in Figure~\ref{fig:fig5}. For example, if we replace the particles with masses of 1, 2, 3 and 4 M$_{\odot}$ with more massive particles with masses of 10, 20, 30 and 40 M$_{\odot}$, then Figure~\ref{fig:fig5} still applies provided the stellar radii are, respectively, 10, 20, 30 and 40 R$_{\odot}$. Thus, our method is just as applicable to treating chaotic interactions involving more massive stars, provided the minimum mass ratio in each interaction satisfies $q \gtrsim 0.1$, as previously discussed.
\section{Summary} \label{summary}
In this paper, the fourth in the series, we push forward in our study of chaotic Newtonian gravity involving small numbers of finite-sized particles. Our focus remains direct collisions between pairs of particles in the "sticky-star" approximation. Our over-arching goal in this series of papers is to develop a method to calculate the probability of any two particles colliding during a chaotic (bound) resonant gravitational interaction involving any number $N$ of particles with any combination of particle masses and radii, as well as to directly compare its predictions to simulated data.
In our previous papers, we showed that (1) the probability of a collision occurring during interactions involving identical particles is approximately proportional to $N^2$, which comes from combinatorics and, specifically, the number of ways of selecting any two particles from a larger set of $N$ identical particles, or ${N \choose 2}$ \citep{leigh12}; (2) for strongly bound gravitational encounters (i.e. with $E \ll$ 0, where $E$ is the total encounter energy, and having small impact parameters) involving small numbers of particles, the collision probability is directly proportional to the collisional cross-section. For identical particle masses and large particle radii, the collisional cross-section is roughly equal to the sum of the cross-sectional areas of the colliding particles \citep{leigh15}; and (3) for different particle masses but identical particle radii, the mean free path approximation can be used in conjunction with (or without) the assumptions of time-averaged virial equilibrium and energy equipartition to derive estimates for the relative collision rates that agree with the simulated data at the order-of-magnitude level \citep{leigh17}.
In this paper, we continue our study by considering interactions involving particles with \textit{both} different masses and radii. As in Paper III, we consider the four-body problem in this paper. This is because, for $N =$ 4, we can run more simulations since we minimize the computational expense due to the small number of particles. This contributes to a significant increase in the statistical significance of the analysis. Using our previous results from Papers I, II and III, we derive Collision Rate Diagrams for our case of interest. This is done first assuming that only the geometric cross-sections for collision affect the relative collision rates, and then again assuming gravitational focusing and including a combinatorial correction. For these cases, we analyze the \textit{relative} collision probabilities as predicted by our analytic formulae and compare them to the results of numerical scattering simulations performed with the \texttt{FEWBODY} code \citep{fregeau04}.
By calculating different manifestations of the CRD, our method illustrates the following key result. While the geometric cross-section captures most of the relevant physics for comparable particle masses, invoking the additional assumptions of a gravitationally-focused cross-section along with a combinatorial correction yield better agreement with the simulations. With these additional assumptions, our analytic estimates reproduce the simulated data for \textit{all} combinations of particle masses and radii considered here.
Our method (or variations thereof) is suitable to direct stellar collisions in dense star clusters, the collisional growth of planetesimals in protoplanetary disks, the growth of super-massive black holes via runaway stellar collisions, the formation of giant elliptical galaxies in the galaxy clusters and groups, and even the growth of large-scale structure in the Universe.
\section*{Acknowledgments}
The authors thank an anonymous reviewer for comments and suggestions that strengthened our manuscript. N.~W.~C.~L. gratefully acknowledges support from the American Museum of Natural History and the Richard Guilder Graduate School, specifically the Kalbfleisch Fellowship Program, as well as support from a National Science Foundation Award No. AST 11-09395.
|
1,314,259,992,988 | arxiv |
\section{Introduction}
\vspace{0.2cm}
One of the most promising technologies to search for the \ensuremath{0\nu\beta\beta}~decay is
an asymmetric high pressure xenon gas (HPGXe) time projection chamber (TPC) with
electroluminescent (EL) amplification. The
NEXT collaboration is building an EL HPGXe TPC capable of holding
100 kg (NEXT-100) of xenon isotopically enriched with \ensuremath{{}^{136}\rm Xe}~. The installation of NEXT-100 at the LSC is planned for 2018. This
technology offers {\bf excellent energy resolution} \cite{Alvarez:2012kua,Lorca:2014sra} ($0.5-0.7\%$ FWHM
at the $Q_{\beta\beta}$), by amplifying the ionization signal with electroluminescent light,
{\bf and tracking capabilities} \cite{Ferrario:2015ina}, as demonstrated by the NEXT
collaboration using two kg-scale prototypes.
The EL amplification is essential to get a linear gain avoiding
avalanche fluctuations and to fully exploit the excellent Fano factor
of xenon in gas to obtain excellent energy resolution. In NEXT, the EL light
is collected by an array of photomultipliers located behind the
cathode (energy measurement) as well as by a dense array of silicon
photomultipliers (topology measurement) located behind the anode. The tracking capability allows for the distinction of signal events (the two electrons
emitted in a \ensuremath{0\nu\beta\beta}~decay), reconstructed as a
continuous track with larger energy depositions (blobs) at both ends, and
background events (mainly due to single electrons, from ${}^{208}$Tl and
${}^{214}$Bi, with kinetic energy comparable to the end-point of the \ensuremath{0\nu\beta\beta}~decay) reconstructed as a
track with only one end-of-track blob \cite{paola}.
NEXT is an international collaboration that includes research groups
from Spain, Portugal, USA, Russia, and Colombia.
The NEXT research program has been organized into four stages: 1)
demonstration of the EL HPGXe TPC technology with $\sim$ 1 kg detectors
(NEXT-DEMO and NEXT-DBDM); 2) characterization of the backgrounds for the \ensuremath{0\nu\beta\beta}~signal and
measurement of the \ensuremath{2\nu\beta\beta}~decay with a 10 kg detector called
NEXT-WHITE (NEW) at the Laboratorio Subterr\'aneo de Canfranc (LSC);
3) search for the \ensuremath{0\nu\beta\beta}~decay with the NEXT-100 detector at the LSC and 4) scale
up and further development to reduce backgrounds and enhance the
topological signature for a 1 tonne-scale EL HPGXe TPC.
\section{Sensitivity of the NEXT-100 detector to \ensuremath{0\nu\beta\beta}~decay}
\vspace{0.2cm}
The most important background source in NEXT comes from radioactive
impurities in the detector components from the uranium and thorium series,
particularly ${}^{208}$Tl and ${}^{214}$Bi, whose photo-peaks lie around the
hypothetical \ensuremath{0\nu\beta\beta}~peak of \ensuremath{{}^{136}\rm Xe}~~(Q$_{\beta\beta}$ = 2.458 MeV).
A thorough campaign of radiopurity measurements have been performed from 2011 to
2016 to estimate the activity of the ${}^{208}$Tl and ${}^{214}$Bi
background sources in the most relevant components of the NEXT-100
detector. On the other hand, an estimate of the \ensuremath{0\nu\beta\beta}~signal and background detection
efficiencies, for the \ensuremath{0\nu\beta\beta}~event selection in the NEXT-100 detector, has
been evaluated using Monte Carlo (MC) simulations.
\begin{wrapfigure}{r}{6.5cm}
\includegraphics[scale=0.35]{img/next100_2.png}
\caption{Sensitivity (at 90 \% C.L) of the NEXT-100 detector to the
\ensuremath{0\nu\beta\beta}~half-life and to the \ensuremath{m_{\beta\beta}}~as a function of the accumulated exposure for an estimated
background rate of $4\times10^{-4}$ counts/keV-kg-yr in the ROI.}
\label{next100}
\end{wrapfigure}
A \ensuremath{0\nu\beta\beta}~candidate event requires that: 1) only one track is
reconstructed fully contained within the fiducial volume of the
detector (defined by excluding a region of 2 cm around the boundaries
of the active volume); 2) the reconstructed track features a blob at both ends; 3) the energy of the event is within the region of interest (ROI)
2.448 $<$ E $<$ 2.477 MeV. This selection gives an efficiency of 28\%
for \ensuremath{0\nu\beta\beta}~signal events, while the natural radioactive
backgrounds, ${}^{208}$Tl and ${}^{214}$Bi, are suppressed by more
than 6 orders of magnitude and the background from \ensuremath{2\nu\beta\beta}~decays is
completely negligible.
Taking into account the contribution of each detector subsystem from
the material-screening measurements and the \ensuremath{0\nu\beta\beta}~event selection, the
estimated overall background rate in NEXT-100 is of
$4\times10^{-4}$ counts/keV-kg-yr or less in the ROI \cite{Martin-Albo:2015rhw}. Assuming an energy
resolution of 0.75\% FWHM at the $Q_{\beta\beta}$ and a \ensuremath{0\nu\beta\beta}~signal efficiency of about 28\%, this
gives an expected sensitivity (at 90\% C.L) to the \ensuremath{0\nu\beta\beta}~decay
half life of $T^{0\nu}_{1/2}>6.0\times10^{25}$ yr for an exposure of
275 kg~yr. Figure \ref{next100} shows the expected sensitivity (at 90\% C.L) of
the NEXT-100 detector to the \ensuremath{0\nu\beta\beta}~half-life and the corresponding
sensitivity to the neutrino Majorana mass, \ensuremath{m_{\beta\beta}}, as a function of
accumulated exposure for the largest and smallest nuclear matrix
element estimates (blue dashed curves).
\section{NEXT-WHITE background expectations and sensitivity to \ensuremath{2\nu\beta\beta}~decay}
\vspace{0.2cm}
The NEW detector is the first phase of the NEXT experiment to operate
underground and is currently being commissioned at the LSC. NEW is a scale 1:2 in size
(1:10 in mass) of NEXT-100 using the same materials and
photosensors \cite{miquel}. A similar background model as for the NEXT-100 detector
has been developed for the NEW detector considering both depleted and
enriched \ensuremath{{}^{136}\rm Xe}~~runs.
The NEW background model is particularly detailed, as it considers
potential contributions from four isotopes (${}^{40}$K, ${}^{60}$Co ,
${}^{214}$Bi, ${}^{208}$Tl) and 17 detector components, for a total of 68
background sources.
A similar event reconstruction and selection as the one developed for
the NEXT-100 \ensuremath{0\nu\beta\beta}~analysis has been performed using MC simulations to search for \ensuremath{2\nu\beta\beta}~in
NEW, the main difference being the looser energy requirement (0.6 $<$
E $<$ 2.8 MeV). Figure \ref{NEW}-left shows the expected energy spectrum
for events passing the \ensuremath{2\nu\beta\beta}~event selection. Taking into account the
material-screening measurements and the \ensuremath{2\nu\beta\beta}~event selection, a
significance of $~$8 sigmas for a 100-day run is estimated for the
\ensuremath{2\nu\beta\beta}~measurement, as shown in Fig. \ref{NEW}-right.
\begin{figure}[htb]
\begin{center}
\includegraphics [scale=0.35]{img/NEW.pdf}
\caption{Left: expected energy spectrum in NEW for the
\ensuremath{2\nu\beta\beta}~selection. Right: contributions to the background rate of NEXT-100 from different detector components. An asterisk (*) indicates that the contribution corresponds to a positive measurement of the activity of the material. }
\label{NEW}
\end{center}
\end{figure}
\section{Prospects}
Further developments to reduce backgrounds and to fully exploit the
potential of the tracking signature (e.g. using deep learning techniques
and different gas mixtures to reduce diffusion) are being studied by the
NEXT collaboration to enhance the sensitivity to the \ensuremath{0\nu\beta\beta}~decay for a one tonne-scale EL HPGXe TPC.
\ack
The NEXT Collaboration acknowledges support from the following agencies and institutions:
the European Research Council (ERC) under the Advanced Grant 339787-NEXT;
the Ministerio de Econom\'ia y Competitividad of Spain and FEDER under grants CONSOLIDER-Ingenio
2010 CSD2008-0037 (CUP), FIS2014-53371-C04 and the Severo Ochoa Program
SEV-2014-0398; GVA under grant PROMETEO/2016/120. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
\bibliographystyle{iopart-num}
\section*{References}
\section{Introduction}
\ack
The NEXT Collaboration acknowledges support from the following agencies and institutions:
the European Research Council (ERC) under the Advanced Grant 339787-NEXT;
the Ministerio de Econom\'ia y Competitividad of Spain and FEDER under grants CONSOLIDER-Ingenio
2010 CSD2008-0037 (CUP), FIS2014-53371-C04 and the Severo Ochoa Program
SEV-2014-0398; GVA under grant PROMETEO/2016/120. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.
\bibliographystyle{iopart-num}
\section*{References}
|
1,314,259,992,989 | arxiv | \section{INTRODUCTION}\label{sect:introduction}
The NASA {\it Kepler} Mission employs a space-based 0.95~m aperture
Schmidt telescope to observe a single 115 square degree field of view
and obtain nearly continuous light curve coverage for over 156,000
stars. The satellite was launched in March 2009 and began science
observations in May 2009 with a primary mission objective of detecting
the transits of small planets orbiting near the habitable zone of
Sun-like stars \citep{boruckietal10}.
Once detrended for instrumental signatures and long-term stellar
variations, the {\it Kepler} light curves are searched for transit
signals that are vetted to eliminate likely false positives
\citep[transit-like signals due to causes other than transiting
planets; see][]{batalhaetal10}. The periodicity and amplitude of
the transits provide initial estimates for orbital periods and sizes
of candidate planets. However, these planet size estimates are
derived from modeling the light curves with a parameter reflecting the
planet-to-star radius ratio and so depend on the uncertainty of the
radius of the host star. Understanding the properties of the host
stars, especially stellar radii, is therefore critical to meeting many
of the mission objectives. In order to identify the most promising
candidates, refine knowledge of the host star properties, and identify
additional false positives, a follow-up observing program was
undertaken to obtain optical spectra of candidate host stars. The
resulting spectra are fitted with models to determine the three
stellar properties T$_{\rm eff}$, log(g), and \rm [Fe/H].\ These parameters are then
used to revise the stellar and candidate planet radii. This program
is one of several providing ground-based follow-up reconnaissance
spectroscopy of candidate exoplanet host stars as part of the {\it
Kepler} Follow-up Program \citep{gautieretal10}.
The target sample is described in \S\ref{sect:sample}, the
observational methods in \S\ref{sect:observations}, and the data
reduction in \S\ref{sect:datareduction}. In
\S\ref{sect:characterization} model fits are used to determine the
stellar properties T$_{\rm eff}$, log(g), and \rm [Fe/H]\ along with an analysis of
their uncertainties. These stellar parameters are used in
\S\ref{sect:radii} to find fits for each star on sets of isochrones
and derive revised stellar and planetary radii. The results are
discussed in \S\ref{sect:discussion} and presented in a table listing
the stellar properties for 220 candidate exoplanet host stars. The
public availability of the data are discussed in
\S\ref{sect:dataavailable} and the findings from these data are
summarized in \S\ref{sect:conclusions}.
\section{TARGET SAMPLE}\label{sect:sample}
The target stars are selected from a list of candidate exoplanet host
stars known as {\it Kepler} Objects of Interest (KOIs) identified by
the mission following a battery of tests that is designed to identify
false positives. These tests include a manual inspection of each
light curve and analysis of any pixel-level flux and centroid
variations during the candidate transits \citep{batalhaetal10}.
Having passed the initial false positive identification tests
unscathed, KOIs can be considered reasonable targets for planet
characterization and confirmation as bona-fide planets using
ground-based follow-up observations. At this point, the KOI list
contains some unidentified false positives with a rate
that depends on the system's properties. Theoretical calculations
have been used to predict the rate of false positives due to eclipsing
binaries, especially cases where flux of a third star is blended with
the eclipsing binary. \citet{mortonjohnson11} predicted an overall
false positive rate of 5\% based on galactic structure models, the
expected binary star population and eclipse depths. Later,
\citet{morton12} pointed out that because the KOI list still contains
some candidates with V-shaped light curves, a higher false positive
rate might be expected. \citet{fressinetal13} carried out a recent
analysis that included simulating eclipsing binaries as background
sources or as members of heirarchical triple systems and systems where
true planets had their light curves blended with the flux of other
stars. Their analysis predicted a higher overall false positive rate
of 9.4\% with a dependence on the presumed planet radius and galactic
latitude. The highest false positive rate of 17.7\% was predicted for
giant planet candidates. Recent observational studies have also
pointed toward a significant false positive rate.
\citet{santerneetal12} conducted a radial velocity survey and
estimated a 35\% false positive rate among short-period giant planet
candidates. \citet{colonetal12} used multi-color light curves to find
two out of a sample of four short-period small planet KOIs were
actually eclipsing binary stars, necessitating a comparably high false
positive rate. Stellar classification spectroscopy can identify false
positives in cases where stellar properties are found to be incorrect,
however other types of observations are typically better suited to
identifying individual false positive candidates.
Our spectra were most often the first follow-up observations taken of
the faint stars of interest. Up to this point, these stars have
normally been characterized based only on modelling of the broadband
photometry contained in the {\it Kepler} Input Catalog, a ground-based
survey of the {\it Kepler} field \citep[KIC;][]{brownetal11}. The
stellar properties determined in the KIC were designed to select
optimal target stars for the mission prior to launch. The ideal
target stars were small (ie. dwarfs) for which transits by a given
size planet produce relatively large signals. The KIC allowed {\it
Kepler} to select mostly small stars, but within the sample, stars
exhibit a range of properties that are not always accurately
determined.
A list of current active KOIs is maintained by the Community Follow-up
Program (CFOP\footnote{https://cfop.ipac.caltech.edu/}) and is
continuously updated as the {\it Kepler} satellite observations are
reduced and vetted for new candidates, or as follow-up observations
help to identify some KOIs as false positives. The properties of the
KOIs in our sample have likewise changed over the course of the
mission. The highest priority targets, and those selected to be
included in the sample, generally fall into one or more of the
following categories: (1) KOIs that are requested for observation as
part of an intensive study of a single star or a small number of host
stars, (2) KOI stars that are candidates to be hosts of small planets
($R_p\lesssim2.5R_{\oplus}$), (3) KOIs in which the candidate planets
orbit in a predicted habitable zone, and (4) KOI stars that harbor
multiple candidate planets.
Because the KOIs are also being pursued by other spectroscopic
follow-up programs and we wish to avoid unnecessary overlap, we have
also selected targets on the basis of apparent brightness. The target
stars span an apparent brightness range ${\rm 8<m_{Kep}<16}$, where
${\rm m_{Kep}}$ is the {\it Kepler} bandpass magnitude
\citep{brownetal11}. Figure~\ref{Fig:kepmags_hist} shows the
magnitude distribution of our target sample along with the current set
of KOI stars. It also includes the magnitude distribution of those
stars with new stellar radius estimates (see \S\ref{sect:radii}). At
the bright end of the magnitude range, ${\rm m_{Kep}<13}$, the
follow-up coverage by other groups is fairly extensive (the {\it
Kepler} Follow-up Program reported approximately 90\% of these stars
as having had spectral follow-up through the 2012 observing season).
To our knowledge, our spectroscopic sample is by far the largest for
candidate host stars of ${\rm m_{Kep}>14}$ ($N=305$). Such faint
stars may prove too difficult or time consuming for other follow-up
methods (e.g., highly precise radial velocity measurements), however
they are quite important to the overall mission goals due to their
large numbers (ie. two thirds of currently-active KOI stars have ${\rm
m_{Kep}>14}$ and two thirds of planet candidates with radii less
than ${\rm 2.5R_\oplus}$ occur around these fainter stars). A full
understanding of {\it Kepler} exoplanet statistics requires large
follow-up studies of the faint stars, or at least those of highest
priority. Finally, note that a few otherwise high priority targets
are excluded from the observations due to visible crowding by other
stars since the modelling described here is not designed for composite
spectra.
Figure~\ref{Fig:PpRp} shows distributions of planet orbital periods
and radii for the same data sets, namely our sample and that of all
KOIs. The entire KOI sample is dominated by planets smaller than
$4R_{\oplus}$. As {\it Kepler} obtains increasingly long time
coverage light curves, the relative fractions of small planets and
those in long-period orbits grows and the lower right hand regions in
the plots of Figure~\ref{Fig:PpRp}, where habitable terrestrial
planets may be located, are becoming increasingly well populated. As
shown in Figure~\ref{Fig:PpRp}, the observed sample has a similar
distribution to the entire KOI sample, but contains relatively fewer
stars harboring large planets and relatively more candidates with
long-period orbits.
\section{OBSERVATIONS}\label{sect:observations}
We observed KOIs in the {\it Kepler} field (115 square degrees
centered at ${\rm \alpha=19^h25^m}$, $\delta=+44.5\arcdeg$) on 48
nights during $2010-2012$ using the National Optical Astronomy
Observatory (NOAO) Mayall 4m telescope on Kitt
Peak and the facility RCSpec long-slit spectrograph with one of its
$2048^2$ pixel CCDs (either T2KA or T2KB). The spectrograph
configuration was the same on each observing run. The slit was
1.0$\arcsec$ wide by 49$\arcsec$ long and oriented with a position
angle of 90$\arcdeg$. The KPC-22b grating in second order was used to
disperse the spectra with 0.72~\AA~pixel$^{-1}$ at a nominal
resolution of $\delta\lambda = 1.7$\AA. The spectra covered a
wavelength range between 3640\AA\ and 5120\AA, but were out of focus
at both ends where the fluxes could not be reliably calibrated. The
effective wavelength range was therefore reduced to a
$3800-4900$\AA\ region. The scale along the slit in each spectrum was
0.69${\rm \arcsec~pixel^{-1}}$.
The observing procedure was basically the same each night. The
telescope autoguider was used during each observation and each pointing
began with an exposure of the instrument's comparison arc lamp
spectrum (HeNeAr or FeAr) for wavelength calibrations. Following
that, normally a single exposure was taken of each target star. The
exposure times ranged between 5 and 20 minutes for most KOIs, although
a few required longer integrations due to faintness or poor observing
conditions. The faintest targets requiring an exposure time exceeding
20 minutes were observed in two exposures to reduce the density of
cosmic ray hits per exposure and aid in their removal during
reduction. The KOIs or other stars (e.g., for flux calibrations) were
all observed at an airmass of less than $\sim1.8$. At the high end of
this airmass range, the atmospheric dispersion for objects in the {\it
Kepler} field remained sufficiently parallel to the slit at the
latitude of Kitt Peak, and permitted efficient operations at a single
instrument rotation. At least one spectrophotometric standard star
selected from \citet{masseyetal88} or \citet{stone77} was observed each
night. Calibration data consisting of bias frames, quartz lamp flat
field exposures, and comparison lamp exposures were taken during the
daytime.
\section{DATA REDUCTION}\label{sect:datareduction}
The data reduction is primarily based on various IRAF\footnote{IRAF is
distributed by the National Optical Astronomy Observatories, which
are operated by the Association of Universities for Research in
Astronomy, Inc., under cooperative agreement with the National
Science Foundation.} packages for performing image reduction and the
{\it onedspec} package for extracting and calibrating the spectra.
The first step is reducing the sets of bias and quartz flat lamp
exposures. The overscan bias level is subtracted from each bias frame
and it is trimmed to a useful data section. These bias frames are
averaged to create a master. The overscan bias level is subtracted
from each flat field frame followed by any (residual) bias pattern in
the master bias. The flat frames are then averaged while rejecting
cosmic ray hits. We normalize this master flat by fitting a smooth
curve to its shape along the dispersion axis (rows) and normalizing
each row of the flat by this curve. Object spectra frames are reduced
by subtracting the overscan bias, trimming them, and subtracting any
residual bias pattern. They are then divided by the normalized flat
field.
The onedspec package task {\it doslit} is the basis of spectral
extraction and calibration using the spectrophotometric standard
stars. To reduce the systematic trends that may result from the
variation in telescope focus with wavelength across the spectrum (a
significant effect with this spectrograph configuration), a relatively
wide aperture is defined to extract each spectrum. This is based on a
cut through the CCD column at ${\rm 4950\AA}$ where the stellar
profile along the slit is broad and representative of the wavelength
region used for much of the spectral modelling. We measure sky flux
in regions extracted from both sides of the stellar spectrum, and
subtract it. The aperture defined by the stellar spectrum is used to
extract a comparison lamp spectrum for each object. A sensitivity
function is found for each night based on the ratio of the standard
star to its standard curve in the IRAF database of KPNO IRS standards
and used to correct the science targets and supply a relative flux
level. The comparison lamp spectra are used to determine wavelength
as a function of columns in order to resample the spectra to a linear
wavelength scale set to closely match the sampling of the 2-D spectra.
\section{STELLAR CHARACTERIZATION}\label{sect:characterization}
\subsection{Overview of the Stellar Characterization Methodology}
We developed specialized software and procedures for this program.
These are first used to find the basic stellar properties T$_{\rm eff}$,
log(g), and \rm [Fe/H], by fitting the observed spectra to theoretical model
spectra (\S\ref{sect:characterization}). Following that, stellar and
planetary radii are estimated based on the best fits of the basic
stellar properties to Yale-Yonsei isochrones in
(\S\ref{sect:radii}).
The model-fitting methods employed here rely on comparisons between
observed spectra and existing synthetic spectra calculated from
stellar atmosphere models and line modelling codes. Model spectra are
available from the literature on a grid of discrete values for T$_{\rm eff}$,
log(g), and \rm [Fe/H]. The process followed here finds the best physical
properties for each observed spectrum by evaluating the rms of the
residuals between the observed spectrum and each model. Following
that, an interpolated value is found for each stellar property by
evaluating the goodness-of-fit over the grid of models. Ideally, a
best-fitting model spectrum could be identified for each star and the
physical properties associated with the model would then be assigned
to the star. In practice, the spectral models fail to accurately
predict all of the features in the observed spectra and fits often
need to be restricted to specific wavelength intervals containing
features that are both well represented by the models and sensitive to
the parameters being sought \citep[e.g., see][]{valentifischer05}.
Furthermore, systematic errors in determining stellar properties can
be introduced by errors in the relative spectral flux calibration and
the discrete values of the model atmosphere sets. Errors in the
stellar parameters can be correlated as well, complicating the
situation.
In order to test the methods employed here and refine them to work on
KOI stars, we observed a large number of ``test stars'' that have
published stellar properties. Experimentation has shown that the fits
can reproduce the relative stellar properties for these stars, but
with systematic offsets from their literature values. The final
fitting methods adopted are ones that best reproduced the literature
values, once corrections for these systematic offsets were made. In
essence, we adopted the test stars and their published properties as a
standard set and worked to find methods that maximized the fitting
precision.
The model fits are confined to a relatively narrow wavelength region
at the long wavelength end of the spectra (see
\S~\ref{subsec:modelfitting}). This region contains the important
H$\beta$ absorption line, which is strong in the hotter stars of our
sample. Its strength and profile is dependent on effective
temperature and surface gravity. Multiple atomic metal lines are also
present, the strongest ones are due to low ionization states of Fe,
Cr, Mn, Ni, Ti and Mg. For cooler stars, a prominent broad molecular
feature appears from MgH (near 4780\AA) and eventually from TiO (near
4760\AA) at the lowest temperatures. These features, and the range of
stellar atmosphere conditions over which they are useful diagnostics,
limit the stars that we can model using these procedures. During the
fitting, the strength of the metal lines drives our estimate of \rm [Fe/H],
while the strength and profile of H$\beta$ is largely responsible for
driving the fits of T$_{\rm eff}$\ and log(g). The strength of MgH is not very
well represented in the synthetic spectra \citep{wecketal03} nor does
this feature appear to be particularly helpful for fitting the cooler
range of our spectra where it appears. The stars that could be fit
most effectively and for which we had representative test stars were
dwarfs within the effective temperature range ${\rm
4750K<T_{eff}<7200K}$ (approximate spectral types
K2V through F0V) as discussed in \S\ref{subsec:teststarfits}.
For this reason, only stellar properties for KOIs within this
temperature range are reported here.
\subsection{Model Spectra}\label{subsec:modelspec}
The fits are based on a set of synthetic model spectra made publicly
available by \citet{coelhoetal05}. These model spectra are calculated
using their extensive line calculation codes along with the model
stellar atmospheres of \citet{castellikurucz03}. They represent
predictions for non-rotating stars with relative metal abundances
set to the solar values of \citet{grevesseandsauval98}.
The model set includes spectra calculated for
stars that lie at discrete points on a 3-D grid defined by the
parameters T$_{\rm eff}$, log(g), and \rm [Fe/H]. The model spectra are calculated
at wavelength steps of 0.02\AA\ and with a range and spacing between
adjacent values of each parameter as follows: T$_{\rm eff}$\ between 3500~K
and 7000~K in steps of 250~K, log(g)\ between 1.0 and 5.0 in steps of
0.5, and \rm [Fe/H]\ between $-2.5$ and $+0.5$ in steps of 0.5 with an
additional set of models at \rm [Fe/H]$=+0.2$.
\subsection{Test Stars}\label{subsec:teststars}
The methods used to fit model spectra to the observations are the
result of experiments fitting models to a set of spectra obtained for
44 test stars while attempting to reproduce the physical parameters
previously published for these stars. These stars were representative
of the majority of the KOIs we planned to target. Physical data taken
from the literature for these stars is given in
Table~\ref{tab:teststars} along with the model fitting results
discussed later. The test stars include a set of 20 exoplanet host
stars characterized by the HATnet project \citep{bakosetal02}. The
HAT stars are dwarfs ranging in T$_{\rm eff}$\ between 4591~K and 6600~K,
log(g)\ between 4.13 and 4.63, and metallicities between $-0.36$ and
$+0.41$. Typical uncertainties in their stellar properties are 80~K
for T$_{\rm eff}$, 0.04 in log(g), and 0.08 for \rm [Fe/H]. The atmospheric
parameters of these stars have been estimated by combining
spectroscopic fits with light curve modelling. The properties
T$_{\rm eff}$\ and \rm [Fe/H]\ were found using the model atmospheres and line
synthesis code provided by the software Spectroscopy Made Easy
\citep[SME;][]{valentipiskunov96} and log(g)\ was found by modelling
the transit light curve parameterized by the ratio of the orbital
semi-major axis to the stellar radius, $a/R_\star$, in the manner of
\citet{sozzettietal07}. A second set of 6 dwarfs from the work of
\citet{valentifischer05} was observed. These stars ranged in
T$_{\rm eff}$\ between 4969~K and 5903~K, log(g)\ between 3.97 and 4.85, and
\rm [Fe/H]\ between $-1.14$ and $+0.22$ based on SME and \citet{kurucz92}
ATLAS9 atmosphere models. Another set of 4 dwarfs are KOIs that had
also been observed by other Kepler follow-up programs using
high-resolution spectroscopy (labelled with KOI or Kepler
designations). These programs utilized SME along with other
constraints. We also observed a number of evolved stars, including a
set of 5 giants in the Kepler field for which properties have been
derived from astroseismological analysis \citep{kallingeretal10} with
updated results as determined in \citet{kallingeretal12}. These stars
ranged in T$_{\rm eff}$\ between 4153~K and 4893~K, log(g)\ between 1.66 and
3.27, and \rm [Fe/H]\ between $-0.29$ and $+0.18$. A set of 8 bright giants
from \citet{luckheiter07} was included. The stellar properties
adopted for this work were those \citet{luckheiter07} derived
spectroscopically using \ion{Fe}{1} and \ion{Fe}{2} lines. Their
spectral line fits were based on MARCS \citep{gustafssonetal03}
atmosphere models and a variant of the MOOG line synthesis code
\citep{sneden73}. These stars had properties determined from
high-resolution spectroscopy and spanned a T$_{\rm eff}$\ range from 4605~K to
7000~K, a log(g)\ range from 2.49 to 3.31, and a \rm [Fe/H]\ range from
$-0.52$ to $+0.31$. In addition to the giants, a single dwarf from
\citet{luckheiter07} is included.
\subsection{Model-Fitting Method}\label{subsec:modelfitting}
To determine stellar properties for both our test stars and KOI stars,
we apply an iterative method of fitting model spectra to our
observations, finding one stellar atmosphere parameter at a time, and
in many cases holding other parameters at fixed values until the
best-fitting set of stellar properties is identified. First we
describe the basic procedures common to every fitting iteration, and
then follow that with the details of each iteration.
To prepare the model spectra, the model data of \citet{coelhoetal05}
are re-binned at a wavelength sampling of 0.3\AA\ for calculation
speed. Then, for each observed spectrum, the models are resampled
onto the wavelength scale of the observed spectra and smoothed using a
Gaussian kernel with a FWHM of 1.5\AA\ to match the observed
resolution. Next, based on experiment, a specific wavelength interval
is chosen for each fitting step. The first procedure during each
iteration is to find the cross-correlation function between the
observed and model spectra, where the mean fluxes of both spectra have
been subtracted. We use the location of the cross-correlation
function peak to shift the model spectra
to match the observations (correcting for any wavelength
calibration errors and, to first order, any Doppler shift). Next,
with the mean flux (F$_\lambda$) of our observed spectrum normalized
(but with no normalization relative to a continuum flux done), we
scale the flux of each model spectrum to minimize the rms residuals of
the fit. This scaling is done with either one or two free parameters:
\begin{equation}\label{eqn:AA}
{\rm F_{scaled} = F_{\lambda}(A + B\lambda)}
\end{equation}
\noindent
Here, ${\rm F_{\lambda}}$ represents the model flux, the parameter A
represents a simple scaling factor, and B an additional term that
corrects slope differences between the observations and model. During
some iterations, B is fixed at zero. The parameter B proved to be
useful in our tests and probably removes systematic errors that might
otherwise adversely affect the fits.
We apply the aforementioned procedures in the following step-by-step
process:
\begin{enumerate}
\item An initial value for \rm [Fe/H]\ is found by fitting over
$\lambda=4600-4890$\AA\ for the full set of models while including B
as a free parameter. The value of \rm [Fe/H]\ for the model having the
minimum rms of fitting residuals is taken as an initial estimate.
\item An initial value of T$_{\rm eff}$\ is found by restricting our fits to
models with \rm [Fe/H]\ equal to that found in step 1. Here, the fit is
done over the wavelength interval $\lambda=4810-4890$\AA\ and B is
fixed at zero. The value of T$_{\rm eff}$\ for the model having the minimum
rms of the fitting residuals is taken as an initial estimate.
\item The spectrum is refit to find \rm [Fe/H]\ in the manner of step 1, but
this time the model set is restricted to include only those models
having T$_{\rm eff}$\ equal to that found in step 2. See
Figure~\ref{Fig:K02931spectra} (top panel) for an example fit to the
spectrum of KOI~2931 where \rm [Fe/H]\ is determined to be $0.0$~dex
during an application of this step.
\item The spectrum is refit to find T$_{\rm eff}$\ in the manner of step 2,
but this time the model set is restricted to include only those
models having \rm [Fe/H]\ equal to that found by step 3. See
Figure~\ref{Fig:K02931spectra} (middle panel) for an example fit to
the spectrum of KOI~2931 where T$_{\rm eff}$\ is determined to be $5000$~K
during an application of this step.
\item The value of log(g)\ is determined while holding fixed the values
of \rm [Fe/H]\ and T$_{\rm eff}$\ at the values found in steps 3 and 4
respectively. This best-fitting model represents the best gridpoint
fit to the observed spectrum. See Figure~\ref{Fig:K02931spectra}
(bottom panel) for an example fit to find ${\rm log(g)=4.5}$ for
KOI~2931 during the application of this step.
\item Finally, an interpolated value for each parameter is found as
described below and illustrated in Figure~\ref{Fig:K02931interp} for
the case of KOI~2931. First, each parameter is fitted in turn,
keeping the values for the two parameters not being fit fixed to
match their values in the model found in step 5. The set of models
fit is thus a function of a single parameter. The rms values of the
fitting residuals for these models are considered as a function of
the parameter value. To find a minimum over a continuous
distribution of the parameter value, a cubic spline is fit through
these data points to locate the minimum. Then a set of $3-4$ points
is selected surrounding this minimum and a quadratic function is fit
through them. The minimum of the quadratic function is taken to be
the interpolated parameter value. In cases where the minimum lies
at the edge of the grid of parameter values, no
interpolation can be done. The spectra in such cases are noted and
their fits are treated with extra caution.
\end{enumerate}
\subsection{Calibrations Using Test Star Fits}\label{subsec:teststarfits}
As mentioned previously, the stellar properties derived from model
fits such as those performed here are subject to systematic errors
that are difficult to resolve. Instead of finding a fitting method
free of such errors (which might not be possible), we have chosen to
calibrate these errors based on fits to a set of test star spectra
with the previously-published spectral properties described in
\S\ref{subsec:teststars}. Once the systematic errors are properly
calibrated, a post-fitting correction is possible. At the same time,
this approach permits an estimate of the uncertainties in the final
stellar properties.
The results of the model fits to the 44 test star spectra are given in
Table~\ref{tab:teststars}. For each star, the previously-published
properties are listed with their uncertainties. Following those are
the values from the fits to our spectra (not yet corrected for the
systematic errors we are attempting to quantify here). The columns
under the heading ``difference in values'' list the value for each
parameter from this work minus the previously-published value. To
make the comparison, we plot the difference between our measured
parameter and those from the other literature as a function of our
parameter values. The results are shown in
Figures~\ref{Fig:feh_trend}, \ref{Fig:teff_trend}, and
\ref{Fig:logg_trend} for \rm [Fe/H], T$_{\rm eff}$, and log(g)\ respectively. In
each figure, the error bars represent the uncertainties quoted for the
previously-published values. Note that in these figures, only some of
the test star data are shown, namely a subset of 24 spectral fits that
satisfy one or both of the following restrictions expressed in terms
of the interpolated parameter values obtained during step 6 of the
model fitting procedure:
\begin{equation}\label{eqn:BB}
{\rm
4761K{\leq}~T_{eff}\leq6998K~\cap~log(g)\geq3.44~\cap~[Fe/H]\geq-0.62}
\end{equation}
\begin{equation}\label{eqn:CC}
{\rm
5446K{\leq}~T_{eff}\leq6998K~\cap~log(g)\geq3.01~\cap~[Fe/H]\geq-0.62}
\end{equation}
\noindent
The values of the parameter limits in equations~\ref{eqn:BB} and
\ref{eqn:CC} are chosen specifically so that after applying the
corrections for systematic errors the same limits are expressed using
convenient parameter values as discussed below. The reason that 20 of
the 44 test star spectra are excluded from the fits is that when all
of the data are plotted it became clear that only the stars falling
within the restricted ranges in equations~\ref{eqn:BB} and
\ref{eqn:CC} behaved in a manner that would make accurate calibration
possible. Furthermore, Equations~\ref{eqn:BB} and \ref{eqn:CC} are
chosen to exclude ranges in stellar properties that some KOIs may
have, but which are not represented among our test stars. The
parameters measured for stars outside of this range were either less
accurate or exhibited large systematic deviations from their
literature values. In any case, it is still possible to distinguish
{\it how} the stars outside of this range differed from those inside
the range (e.g., that they were cooler or had lower log(g) values).
All three of Figures~\ref{Fig:feh_trend}$-$\ref{Fig:logg_trend} show
that there are systematic trends in the differences between the
parameter values fit here and the previously-published values. Note
that there are stars among this set measured using different methods,
but all lie along the same linear trends. To quantify these trends,
an unbiased linear least squares fit to all of the points is found and
shown in the figures. These fits lead to corrective relationships
that can be used to place the measured stellar properties on a scale
defined by the test stars:
\begin{equation}\label{eqn:DD}
{\rm [Fe/H](corr.) = 0.4904{\times}[Fe/H] + 0.0553}
\end{equation}
\begin{equation}\label{eqn:EE}
{\rm T_{eff}(corr.) = 1.0953{\times}T_{eff} - 465K}
\end{equation}
\begin{equation}\label{eqn:FF}
{\rm log(g)(corr.) = 0.3489{\times}log(g) + 2.949}
\end{equation}
\noindent
Here, the corrected value of the parameter is labelled parenthetically
with ``(corr.)'' and is expressed as a function of the uncorrected
parameter obtained during step~6 of the method described if
\S\ref{subsec:modelfitting}. Using
equations~\ref{eqn:DD}$-$\ref{eqn:FF}, the range over which the
corrections are applicable (ie. the range over which the spectral fits
can be calibrated) can now be expressed in terms of the corrected
stellar properties:
\begin{equation}\label{eqn:GG}
{\rm
4750K{\leq}~T_{eff}\leq7200K~\cap~log(g)\geq4.15~\cap~[Fe/H]\geq-0.25}
\end{equation}
\begin{equation}\label{eqn:HH}
{\rm
5500K{\leq}~T_{eff}\leq7200K~\cap~log(g)\geq4.00~\cap~[Fe/H]\geq-0.25}
\end{equation}
The scatter of points around the linear fits in
Figures~\ref{Fig:feh_trend}$-$\ref{Fig:logg_trend} provides an
estimate for the uncertainties in the corrected stellar parameters.
The distribution of the points around the linear fits can be described
in terms of standard deviations, where $\sigma=0.10$~dex for \rm [Fe/H],
$\sigma=59$~K for T$_{\rm eff}$, and $\sigma=0.13$ for log(g). The scatter
reflects a combination of the uncertainties from the previous
measurements and those presented here. To determine the contribution
to the total uncertainty from the latter, one could determine an error
on each parameter that, when added in quadrature to the uncertainties
in the parameters quoted in the literature, would result in the linear
fit having $\chi^{2}_\nu=1$. To do this, the 1-sigma errors on the
new parameter values would need to be ${\rm \sigma([Fe/H])=0.066}$,
${\rm \sigma(T_{eff})=9~K}$, and ${\rm \sigma(log(g))=0.12}$.
Evidently for \rm [Fe/H]\ and T$_{\rm eff}$\ the uncertainties quoted for the
literature values tend to dominate the total uncertainty so that this
method could underestimate the uncertainty of the new parameter fits.
This may be the result of at least some of the uncertainties quoted in
the literature having been overestimated. In contrast, for log(g), the
contribution of the new uncertainties to the total error is larger and
this method is useful to estimate the uncertainty.
However, since there may be unknown effects that could influence the
fits, we have chosen to adopt a more conservative uncertainty on each
measurement. The standard deviations of data around the fitted lines
probably represents an upper limit to uncertainties within this
well-characterized range of stellar properties. With that in mind, we
have adopted a $1\sigma$ uncertainty of 75~K for T$_{\rm eff}$, 0.10~dex for
\rm [Fe/H], and 0.15 for log(g). The stellar properties of our modelled
stars are given in Table~\ref{tab:KOIdata} and referenced by KOI
number and KIC identification number.
\section{REVISED STELLAR AND PLANETARY RADII}\label{sect:radii}
In total, 368 good quality spectra were obtained of 352 stars. From
this master sample, 226 spectra for 220 stars had high enough quality,
were not now known to harbor false positive planets, and had
appropriate stellar atmospheric parameters to allow a new estimate of
stellar radius and hence new estimates of exoplanet candidate radii. A
total of 368 exoplanet candidates orbit these 220 stars. The Kepler
magnitude distribution of these 220 stars is shown in
Figure~\ref{Fig:kepmags_hist}.
To begin, the 226 stellar spectra were separated into three
\rm [Fe/H]\ ranges: ($-0.25:-0.10$), ($-0.10:+0.20$), and
($+0.20:+0.50$). Within each \rm [Fe/H]\ range, measured effective
temperature and surface gravity were used to estimate stellar
luminosity using the so-called Version~2 Yale-Yonsei (YY) isochrones
\citep{demarqueetal04} with solar abundance ratios (i.e. $\alpha = 0$)
and \rm [Fe/H]~$= -0.28$, $+0.04$, and $+0.38$, respectively, as provided in
the on-line
version\footnote{{http://www.astro.yale.edu/demarque/yyiso.html}}.
Stellar luminosity in solar units was estimated for a given
(T$_{\rm eff}$,log(g)) estimate by determining the median stellar luminosity in
the ranges T$_{\rm eff}$\ $\pm75$~K and log(g)\ $\pm0.15$. Given the magnitude
of the (T$_{\rm eff}$,log(g)) uncertainties, it was deemed appropriate to
search the on-line YY~Version~2 isochrone grid without further
interpolation. Stellar radius in solar units was then estimated using
the standard relation
\begin{equation}\label{eqn:II}
R_{\star} = \sqrt{L_{\star}/T_{eff,\star}^4}
\end{equation}
Stellar radii uncertainties can be estimated in two ways. First,
within the search box defined by the (T$_{\rm eff}$,log(g)) uncertainties, the
standard deviation of the mean luminosity $\sigma_L$ can be
computed. The radii uncertainty was estimated as follows:
\begin{equation}\label{eqn:JJ}
\sigma_R = \left (
{\sqrt{(L_{\star}+\sigma_L)/T_{eff,\star}^4} +
\sqrt{(L_{\star}-\sigma_L)/T_{eff,\star}^4}}
\right ) /2
\end{equation}
Second, 6 stars were observed at least twice and sometimes four times
on separate nights during different observing runs separated by
months. Stellar radius uncertainty can be estimated from the
dispersion in stellar radius estimates from these individual
observations. From both methods in combination, a conservative
uncertainty for ${\rm R_\star}$ of $\pm 0.05 R_{\odot}$ is adopted.
The new stellar radii estimates are provided in
Table~\ref{tab:KOIdata}. Only a portion of this long table is
presented here. The entire table is made available in the electronic
version. For stars observed multiple times, individual radii
estimates were averaged into a single value. How do these new
estimates compare to the best previous available radii estimates from
the Kepler Science Analysis System (KSAS)\footnote{The KSAS was a {\it
Kepler} Mission database storing the best available estimates for
stellar and candidate planet properties. Stellar radii were based
on KIC photometry for most stars in the magnitude range of interest
here.}? As Figure~\ref{Fig:YYfits_stellar_radii} illustrates, there
is a formal offset towards larger radii estimates. The radii of 87\%
of these stars are revised upwards (and 13\% downwards), although some
of these revised radii are insignificant given the uncertainties in
the stellar radii. For about 26\% (58) of the stars, the revised
radii are skewed upwards with $R_{\star}^{revised} \geq 1.35 \times
R_{\star}^{KSAS}$. As Figures~\ref{Fig:YYfits_stellar_radii} and
\ref{Fig:YYfits_vs_parameter} illustrate, these stars tend to to be
more evolved than the sample as a whole and relative to their
properties listed by KSAS. In other words, it appears that many KIC
stars are larger than previously assumed. In turn, exoplanet
candidates orbiting those stars must be larger by the same relative
amount.
Revised exoplanet candidate radii estimates can be derived from the
revised stellar radii measurements from the simple geometric
approximation:
\begin{equation}\label{eqn:KK}
R_p^{revised} =
(R_{\star}^{revised} / R_{\star}^{KSAS}) \times R_p^{KSAS}
\end{equation}
\noindent where initial stellar and exoplanet radii come from KSAS.
Stellar radii use solar units ($R_\odot$) while exoplanet candidate
radii use Earth radius units ($R_\oplus$). The on-line
Table~3 provides the revised exoplanet candidate
radii.
Figure~\ref{Fig:planets_vs_parameter} compares revised exoplanet
candidate radii to the characteristics of their host stars. As
previously shown by \citet{buchhaveetal12} and discussed in
\S\ref{sect:discussion}, exoplanet candidates with $R_p^{revised} \geq
4 R_\oplus$ are much more likely to be associated with higher
metallicity stars, while smaller exoplanet candidates are found around
stars spanning the entire metallicity range of our sample.
\section{DISCUSSION}\label{sect:discussion}
The new stellar characteristics derived from this spectroscopic study
have refined the properties of a large sample of KOIs, revealing
statistical trends and identifying a number of individual KOIs as
excellent targets for more detailed follow-up and potential
confirmation as systems harboring small habitable zone planets.
In \S\ref{sect:radii} we found that 26\% of the KOI stars had radii
significantly larger than their values based on the initial
photometric data available to the mission. This effect could be due
to systematic errors in the photometrically-derived stellar properties
like log(g), selection effects in the magnitude-limited KOI sample,
transit detectability dependence on stellar radius, or a combination
of factors. In the case of these data, almost all of the stars with
radii revised upwards by a factor of 1.35 or greater have
T$_{\rm eff}$$>$5200~K, and their positions on the log(g)$-$log(T$_{\rm eff}$) plot of
Figure~\ref{Fig:YYfits_stellar_radii} show that many represent a
population of relatively evolved stars compared to the other stars of
comparable effective temperature.
The lower log(g)\ values measured here may be compared to those of
\citet{verneretal11} who used asteroseismic methods on the {\it
Kepler} light curve data to determine radii for 514 solar-type stars
in the apparent magnitude range $7-12$. For stars with
log(g)$>4.0$~dex and a wide range of effective temperature, the mean
asteroseismic log(g)\ values were 0.23~dex lower than those reported in
the KIC. The corresponding stellar radii were larger as well.
Another sample of stars with asteroseismic log(g)\ values was compared
to KIC log(g)\ by \citet{brunttetal12}. They found asteroseismic
log(g)\ values were lower than those in the KIC by an average of
0.05~dex. They attributed the lower mean difference with respect to
the KIC to the inclusion of stars with log(g)$<4.0$ in the sample, for
which asteroseismic log(g)\ values are in better agreement.
\citet{verneretal11} noted that their asteroseismic sample could be
skewed by a Malmquist bias, which would preferentially select more
evolved and intrinsically brighter stars, as well as by the improved
detectability of the higher amplitude oscillations associated with
stars of lower log(g). In the case of the spectroscopically analyzed
sample presented here, the Malmquist bias would be in effect along
with the counteracting bias favoring detectability of transits across
smaller stars. These two biases were examined by
\citet{gaidosandmann13} who predicted that the Malmquist bias would
have the dominant effect, and the transit sample should be relatively
overabundant in large stars compared to stars at the same temperature
and apparent brightness. In addition to biases in the KOI sample as a
whole, this spectroscopic sample was constructed to include many of
the (relatively rare) smallest planet candidates for follow-up, a
choice that may also select KOI stars with anomalous radii. It is
clear that a full understanding of these biases is necessary to get
better estimates for planet occurrence rates and that large,
spectroscopic samples like the one presented here will play an
important role. A similar spectroscopic study of ``control'' stars,
perhaps {\it Kepler} stars showing no transits, may be of merit as
well.
The revised values for stellar and planet radii have some implications
for the mission goal to determine the frequency of Earth-sized planets
orbiting Sun-like stars in a habitable zone. The radii of some
planets must be revised significantly upwards, perhaps pushing them
outside the size range likely for rocky Earth-like bodies. An
additional effect is that higher luminosities implied by an increase
in stellar radius move the habitable zones for these stars outwards
from the star. As a consequence, the orbital periods of habitable
zone planets must be longer.
Despite the apparent decrease in the number of small planets, these
spectra provide additional evidence to favor certain candidates as
among the most interesting targets for the goals of the {\it Kepler}
Mission. An example candidate host star is KOI2931, the star shown in
Figures~\ref{Fig:K02931spectra} and \ref{Fig:K02931interp}. KOI2931
hosts a single known planet candidate, KOI2931.01, with an orbital
period of 99.248~days. With new stellar properties T$_{\rm eff}$$=4991$~K,
log(g)$=4.49$, \rm [Fe/H]$=-0.03$ and ${\rm R_\star=0.85R_\odot}$, the planet
radius of KOI2931.01 is estimated to be 2.1~$R_\oplus$. The isochrone
fit for this star corresponds to a stellar mass of ${0.78M_\odot}$ and
a planet equilibrium temperature of 326~K is found assuming an albedo
of 0.3 and a circular orbit. KOI2931.01 is one example of a good
candidate for a super-Earth orbiting in the habitable zone.
A correlation between the incidence of relatively large planet
candidates and relatively high host star metallicity (selecting large
planets at ${\rm R_p=4.0R_\oplus}$) was previously seen
spectroscopically in a smaller sample of brighter KOI stars by
\citet{buchhaveetal12}. The KOI stars in their sample were almost all
brighter than ${\rm m_{kep}=14}$, but our and their data sets overlap
in apparent brightness.
There are various ways to examine the significance of the apparent
deficit of large planet candidates around low metallicity host stars
(\rm [Fe/H]$<-0.05$) in this sample. First, note that 5 planet candidates
in this sample (225.01, 998.01, 1067.01, 1226.01 and 1483.01) are all
too large (${\rm >40R_\oplus}$) while the remainder are reasonable
sizes for planets (${\rm <20R_\oplus}$). These 5 objects are
considered likely false positives and excluded from further
consideration. This results in 46 candidate planets orbiting host
stars of \rm [Fe/H]$<-0.05$ and 317 orbiting host stars of \rm [Fe/H]$\geq-0.05$.
A K-S test comparing the planet size distributions of the these two
samples reveals a difference with a confidence level of 98\%. As a
second test, random subsamples of 46 candidate planets are drawn from
the sample of 317 candidates orbiting host stars with \rm [Fe/H]$\geq-0.05$
and compared to the 46 candidate planets orbiting lower metallicity
stars. A set of 1 million random subsamples reveals that the most
probable number of large planet candidates (${\rm R_p>4R_\oplus}$)
orbiting high metallicity host stars is 8 or 9, and that 2 or fewer
large planet candidates occur just 0.4\% of the time (2 is the number
of large planet candidates orbiting the low metallicity host stars).
A fraction of the candidate planet sample may be false positives and
this effect is considered next. Note that 38 known or likely false
positives have already been removed from the sample of 352 stars as
part of creating the candidate planet sample, but others likely
remain. Also, 243 out of 363 planet candidates are members of
multi-planet systems and these have a very high likelihood of being
true planets rather than eclipsing binary stars
\citep{lissaueretal12}. However, multi-planet systems may still be
considered false positives in the sense that their planet radii can be
underestimated due to host star blending \citep{fressinetal13}. A
detailed treatment might be useful to simulate the effects of false
positives in the sample, but a simpler approach is taken here. If a
liberal reduction is made to the sample size in an effort to simulate
the removal of false positives, it will weaken inferences drawn from
these data. \citet{fressinetal13} predict false positive rates in
five planet size ranges: 17.7\% for ${\rm R_p=6-22R_\oplus}$, 15.9\%
for ${\rm R_p=4-6R_\oplus}$, 6.7\% for ${\rm R_p=2-4R_\oplus}$, 8.8\%
for ${\rm R_p=1.25-2R_\oplus}$ and 12.3\% for ${\rm
R_p=0.8-1.25R_\oplus}$. When individual planets are removed from
our sample at these rates, the K-S test signficance of the differences
in planet size distribution on host star metallicity drops to 96\%.
The test of selecting random samples of high metallicity stars to
match the sample size of the low metallicity stars reveals that 2 or
fewer large planet candidates occur around high metallicity stars
1.7\% of the time.
The tests show a dependence between host star metallicity and the
occurance rate of large transiting planets, much like for the sample
of \citet{buchhaveetal12}. It is not surprising to see a similar
pattern in these data, but the fainter stars analyzed here probe a
significantly larger volume of space, showing that these effects
persist across the different stellar populations. The apparent
threshold value of metallicity is chosen at \rm [Fe/H]$=-0.05$ to match the
appearance of the lower left panel in
Figure~\ref{Fig:planets_vs_parameter}, but the discrete and relatively
sparse set of model spectra used to determine \rm [Fe/H]\ may slightly
distort this plot. The lines drawn at ${\rm R_p=4.0R_\oplus}$ were
also chosen by eye, but could have as well been taken at a somewhat
smaller radius (ie. at ${\rm R_p=3-4R_\oplus}$). There are no obvious
trends in the incidence of planet candidates with ${\rm
R_p>4R_\oplus}$ with respect to either log(g)\ or T$_{\rm eff}$. Similarly,
no dependence was found between metallicity and the number of planets
detected around the KOI stars. Given the lack of large planets
detected (in short period orbits) around low metallicity host stars,
the efficiency of planet migration may be dependent on metallicity, or
perhaps large planets simply cannot form around such stars at any
orbital distance.
\section{DATA AVAILABILITY}\label{sect:dataavailable}
The reduced spectra and products from our model fits are made
available on the CFOP website${\rm ^5}$. The CFOP website organizes
data for each KOI and confirmed {\it Kepler} exoplanets, including the
products of many follow-up observations. The data products
contributed from this spectroscopy program include the reduced
spectrum data files, stellar properties, plots of the spectra, fitted
synthetic models plotted alongside the observed spectra (similar to
Figure~\ref{Fig:K02931spectra}) and plots similar to
Figure~\ref{Fig:K02931interp} showing the interpolation between
gridpoint fit values. Additional follow-up spectra and their fits
will be added in the future.
\section{SUMMARY}\label{sect:conclusions}
A spectroscopic analysis of a large sample of stars known as {\it
Kepler} Objects of Interest (KOIs) is presented. In the case of
most of these KOIs, the stellar characterization, and by extension
candidate planet properties, had been based on broadband photometry
available from the pre-launch {\it Kepler} Input Catalog survey.
Spectral follow-up, like that presented here, proves important to
improve the accuracy of the KOI stellar properties, identify
interesting individual planet systems and perform accurate statistical
studies of the KOI list as a whole. The results of model spectra fits
(values for T$_{\rm eff}$, log(g), and \rm [Fe/H]) are given for 268 stars.
Isochrone fits are used to provide revised radii for 220 KOI stars and
their 368 planets. The spectra and results from this survey are made
available to the public through the online CFOP archive.
The spectral and isochrone fits reveal that many of the KOI stars have
larger radii than previously assumed. About 26\% of the stars for
which new radii were determined require corrections to their assumed
radii of a factor of 1.35 or greater, and the isochrone fits for 87\% of
the stars suggest some increase in radius. The stars requiring the
largest upward adjustment in radius represent a relatively evolved
subset of the sample. The increases in stellar radii also require a
reevaluation of the radii derived for the planet candidates hosted by
these stars. The planet radii need to be scaled upwards by
approximately the same ratio as their host stars.
Despite the fact that the revised planet radii are overall larger than
previously assumed, there are candidate planets in this sample that
are now better vetted and continue to be likely small planets in the
habitable zone of Sun-like stars. The example of KOI2931 is presented
as a good candidate for a super-Earth planet orbiting in the habitable
zone of a 4991~K dwarf.
The frequency of large KOI planets in the sample depends on host star
metallicity in a manner similar to that found for a sample of brighter
KOI stars by \citet{buchhaveetal12}. The fainter, larger sample of
$4750-7000$~K dwarf KOIs analyzed in our program shows that these
results extend through a larger volume of space and that the
occurrence of large planets (${\rm R_p>3-4R_\oplus}$) depends on a
threshold metallicity near \rm [Fe/H]$=-0.05$. The large planet
candidates are found almost exclusively around stars with metallicity
higher than this value. In contrast, small planet candidates are found
around stars spanning the full metallicity range examined in this
study.
\acknowledgments
Our work was made possible through the efforts of many others. Among
them are those in the {\it Kepler} Science Office and science team.
At the telescope we always received excellent help from our observing
assistants, and help from additional observers Jay Holberg, Ken
Mighell, and Jason Rowe. Codes used in our modelling were adopted
from work by Greg Doppmann and we received help to compile a list of
properties for our test star sample from Lars Buchhave and Thomas
Kallinger. We also wish to thank the referee for helpful
suggestions that were incorporated into this work. Financial
support for the work was provided by the NASA {\it Kepler} Mission and
Cooperative Agreement AST--0950945 to NOAO.
Facilities: \facility{Mayall}, \facility{Kepler}
\newpage
|
1,314,259,992,990 | arxiv | \section{Introduction}
Technology node scaling in recent decades ushered in gate delay cut-off and rise of interconnection latency \cite{Agarwal2006}. Hence, interconnects have become a major performance bottleneck of high performance \emph{System-on-Chips} (SoC) and \emph{Integrated Circuits} (IC) \cite{Pande2007}. In addition, interconnections have become more susceptible to noises in particular crosstalk \cite{Pande2007}.
On the other hand, the advent of multi-core processors with ever increasing number of cores has highlighted the need for fast and reliable interconnections.
One of the potential solutions to alleviate the interconnection delay problem is the three dimensional integration using \emph{Through-Silicon Vias} (TSV). Vertical integration of IC dies using TSVs offers high density connections between adjacent dies. This technology also allows stacking of dies with nonidentical technologies such as CMOS with high density DRAM which can be used as a solution to mitigate memory wall problem \cite{Loh}. Furthermore, the average and maximum distance between interconnect nodes of the 3D stacked ICs are greatly decreased which leads to significant delay, power, and area improvement. Despite of the TSV advantages, the adjacent, short and bounded TSVs are prone to TSV-to-TSV coupling and crosstalk noise which increases transmission time and power consumption, and more importantly, it threats the signal integrity \cite{Liu2011,Kumar2013}.
As demonstrated in Fig.~\ref{first}, every TSV may be surrounded by neighbour TSVs which cause a big and undesirable coupling noise. This TSV-to-TSV coupling could be very challenging in 3D ICs due to the fact that TSVs are large and thick, thus the coupling between two adjacent TSVs can be huge. Moreover, the effective coupling capacitance between TSVs doubles when the aggressor and the victim signals switch in opposite directions\cite{Liu2011}.
\begin{figure}[!t]
\centering
\includegraphics[width=3.57in]{First}
\centering
\caption{Overview of coupling characteristics in TSVs \cite{Engin2013}; According to ITRS \cite{itritrs}, it is predicted that the height of TSVs will reach to 20-50 $\mu m$ and the via diameter will be 2-8 $\mu m $ till 2018. }
\label{first}
\end{figure}
Plenty of crosstalk minimization methods have been proposed in the literature of 2D design (e.g. \cite{ganguly2008design,Duan2009,Ganguly2009very,Zhang2004,Hirose2000}). However, these methods cannot be directly applied to alleviate TSV-to-TSV crosstalk noise, inasmuch as the TSVs are not placed in the same planar and are greatly affected by more than two aggressors \cite{Zou2014}.
Recent efforts in TSV-to-TSV crosstalk minimization including \cite{Zou2014,Kumar2013,Chang2013} are complex and impose significant area and TSV overhead. SheildUS \cite{Chang2013}, by adding a crossbar, remaps data to TSVs in order to shield more active signals by the signals which predicted to have less transitions in the future. In addition to its complex decision-making circuit, the accuracy of its predictor is under question due to the fact that the signals may not have a regular pattern.
3DLAT \cite{Zou2014} exploits less adjacent codes to limit maximum number of transitions in adjacent TSVs. \emph{Crosstalk Avoidance Codes} (CAC) \cite{Kumar2013} is another coding scheme for TSV-to-TSV crosstalk minimization. These approaches also need a complex and large coder and also suffer from a considerable information redundancy overhead.
In this paper, we propose a TSV-to-TSV crosstalk minimization method, named 3DCAM, which can effectively reduce coupling noise between TSVs with a relatively low area and TSV overhead. In addition, the proposed method uses a small simple coder which reduces run-time performance overhead. In the case of a transition on a target signal, considering the target's neighbours and their coupling effect, 3DCAM decides to whether retain target's value or send its original transition. In the condition that coder decides to retain the value it informs the decoder through a control TSV. The simulation results show that 3DCAM can reduce the transmission delay up to 25.7\% as compared to 3DLAT mechanism. 3DCAM imposes only 30\% TSV overhead which is much less than the 3DLAT TSV overhead (which is 80\% for $\omega=4$).
The rest of this paper is structured as follows. In Section II, related works are reviewed. Section III describes the crosstalk model for TSVs based 3D ICs on which 3DCAM is built. In Section IV we present 3DCAM crosstalk avoidance mechanism. Section V explains the simulations and results and, finally, Section VI concludes the paper.
\begin{figure}[!t]
\centering
\includegraphics[width=2in]{Model}
\centering
\caption{The coupling capacitance crosstalk model for $3\times3$ TSV cluster}
\label{model}
\end{figure}
\section{Related Work} \label{previous}
In the context of \emph{Two Dimensional Network-on-Chips} (2D NoC), there are plenty of works that target power consumption~\cite{performance,toot,smart}, reliability~\cite{performance}, security~\cite{securitry}, or performance~\cite{performance} of the interconnections. Particularly, crosstalk minimization methods can be classified in three categories: physical level, transistor level and, \emph{Register Transfer Level} (RTL) techniques. Wire spacing \cite{Agarwal2006}, active and passive shielding \cite{Zhang2004}, and buffer insertion \cite{Akl2008} are examples of physical level techniques. \cite{Hirose2000} is a transistor level mechanism which reduces the crosstalk noise by skewing the simultaneous opposite transitions. Although this approach reduces the crosstalk, it requires timing adjustment between senders and receivers. Furthermore, this approach suffers from run-time management.
The general idea behind RTL level techniques is to omit some undesirable transition patterns by using coding schemes. There are variety of works that focused on analytical aspect and coding concepts \cite{Duan2009,sridhara2007coding}. Error detection codes and error correction codes \cite{ganguly2008design}, joint crosstalk avoidance mechanism \cite{sridhara2007coding}, and CACs \cite{ganguly2008design} are examples of these coding schemes.
Although the above approaches may cope with crosstalk in 2D ICs, they cannot be directly applied in 3D technologies because the additional dimension makes consequential differences in crosstalk problem analysis. Gathering the long and thick TSVs causes new reliability issues which have been studied recently \cite{Okoro2014,Jung2014}. Several mechanisms have been proposed to make 3D ICs more reliable against crosstalk noise, e.g., \cite{Kumar2013,Eghbal2014,Zou2014,Chang2013,Eghbal2015}.
The TSV-to-TSV capacitance and inductance coupling are two major threats to 3D IC reliability. Previous works have concentrated on these effects from two perspectives. \cite{Kumar2013,Zou2014,Chang2013} proposed capacitance-based mechanisms and \cite{Eghbal2014,Eghbal2015,Motoyoshi2009} proposed inductance-based techniques to reduce crosstalk effects in 3D ICs.
Increasing TSV distances from each other, shielding TSVs, inserting buffers at the victim side, inserting buffers, decreasing driver size at the aggressor side, and increasing load at the wires are the mechanisms examined in \cite{Liu2011} to mitigate TSVs crosstalk noise. According to their experiments, unlike 2D wires, increasing TSV distances is not an effective solution to TSV-to-TSV coupling problem and the other solutions either need high effort at post-design time or have negative impact on timing performance.
RTL mechanisms in 3D IC have been proposed and experimented recently. \cite{Kumar2013} proposed a coding scheme that reduces the maximum crosstalk about 28\% based on their proposed crosstalk model. Two other mentionable mechanisms in 3D IC against crosstalk noise which have been studied recently are ShieldUS \cite{Chang2013} and 3DLAT \cite{Zou2014}. ShieldUS uses data with less transitions as the shield for the more active data. SheildUS tries to minimize average transmission time with run-time mechanism that remaps data to TSVs in order to banish links with specific crosstalk pattern from others. The TSV overhead of this method is not considerable because it uses the same data as shield. But a large crossbar is required for bit shuffling which imposes considerable area overhead. This crossbar will get larger by increasing the number of bits to shuffle. Besides, this method needs the data to be highly regular, as this method tries to predict the activity of the signals.
The authors in \cite{Zou2014} introduce use of less adjacent transition code along with transition signaling to minimize the number of transitions. Furthermore, 3DLAT reduces higher crosstalk class frequency. This scheme has a significant TSV overhead which is not negligible. According to the authors' report, TSV overhead of 3DLAT is about 80\% with $\omega=4$. This mechanism imposes more than 160\% area overhead with $\omega=2$ for the same bitwidth. This method also suffers from significant area overhead which imposed by its complex coder.
\begin{figure}[!t]
\centering
\includegraphics[width=3in]{Editedoverhead_2}
\centering
\caption{Layout of control TSVs for $3\times N$ bus}
\label{control}
\label{layout}
\end{figure}
\section{3D Crosstalk Coupling Noise Model} \label{modelsection}
In 2D integrated circuits, three neighbor wires affect each other and create coupling capacitance. The effective coupling capacitance which imposed on the victim (i.e. middle) wire is modeled by Eq.~(\ref{equation.1}) \cite{Hirose2000}.
\begin{equation} \label{equation.1}
\begin{split}
C_{eff}=C_{G} + C_{C}\mid\frac{\Delta V - \Delta V_{-1}}{V_{dd}}\mid \\
+ C_{C} \mid \frac{\Delta V - \Delta V_{+1}}{V_{dd}}\mid
\end{split}
\end{equation}
Where $\Delta V$ is swing voltage on victim wire, $\Delta V_{-1}$ and $\Delta V_{+1}$ are the voltages that switch on neighbor wires and $V_{dd}$ is the supply voltage. In addition, $C_{c}$ is coupling capacitance that is imposed between the victim wire and its neighbors and, $C_{G}$ is the coupling capacitance between substrate and plate.
Based on the Eq.~(\ref{equation.1}), we can model the transmission delay by the Eq.~(\ref{equation.3}):
\begin{equation} \label{equation.3}
\begin{split}
\tau = (1+\rho \lambda)\pi_{o}
\end{split}
\end{equation}
Where $\rho$ is the coupling coefficient of adjacent wires, $\lambda$ is the coupling capacitance to substrate capacitance ratio ($C_{C} /C_G $) and $\pi_{0}$ is the delay of a wire in the ideal channel, i.e., a channel without any coupling effect such as capacitance and inductance. For instance, when the both aggressors and the victim wire, switch in opposite directions a delay equal to $(1+4\lambda)\pi_{0}$ will be imposed to the channel.
Similar to the 2D IC crosstalk model, we can drive a delay model for TSVs. Akin to previous studies \cite{Kumar2013,Chang2013} and based on TSV's inherent properties, a square model of 9 adjacent TSVs is considered. Fig.~\ref{model} depicts 9 neighbor TSVs from top view. We discuss the crosstalk effect on the victim TSV (specified by red in Fig.~\ref{model}) and we model coupling capacitance noise that is emanated from its neighbor TSVs. Direct neighbors (i.e. north, south, west, and east) are closer to the victim, and thus their coupling capacitances are more destructive than the coupling effects of diagonal neighbors (i.e. northeast, northwest, southeast, and southwest). In order to model the effective capacitance on the victim TSV, we can extend the Eq.~(\ref{equation.1}) to Eq.~(\ref{equation.4}):
\begin{equation} \label{equation.4}
\begin{split}
C_{eff}=C_{G} &+ \sum_{i=-2}^{2} C_{\alpha} { \mid\frac{\Delta V_{I_{0}} - \Delta V_{I_{2i}}}{V_{dd}}\mid} \\
\\
&+ \sum_{i=-1}^{2} C_{\beta} { \mid\frac{\Delta V_{I_{0}} - \Delta V_{I_{2i-1}}}{V_{dd}}\mid}
\end{split}
\end{equation}
Where $C_{\alpha}$ represents coupling capacitance between a direct aggressor and the victim and $C_{\beta}$ is coupling capacitance between diagonal aggressor
and the victim.
Similar to 2D crosstalk delay model, based on Eq.~(\ref{equation.4}) we can extend the Eq.~(\ref{equation.3}) to model 3D TSVs as follows:
\begin{equation} \label{equation.5}
\begin{split}
\tau = (1+\rho_1 \lambda_1+ \rho_2 \lambda_2)\pi_{o}
\end{split}
\end{equation}
Where $\rho_1 $ is the coupling coefficient of the direct aggressors, $\lambda_1$ is the direct coupling capacitance to substrate capacitance ratio ($C_{\alpha} /C_G $), $\rho_2 $ is the coupling coefficient of the diagonal aggressors, and $\lambda_2$ is the diagonal coupling capacitance to substrate capacitance ratio ($C_{\beta} /C_G $). $\rho_1 $ and $\rho_2 $ indicate how the changes in aggressors voltages affect the crosstalk on the victim. For instance, if a direct neighbor switches in opposite direction, $\rho_1 $ would increase by two. In the worst case, all the neighbours switch from $V_{dd}$ to zero and the victim switches from zero to $V_{dd}$. In this case the transmission delay would be $(1+ 8 \lambda_1+ 8 \lambda_2)\pi_{o}$.
\begin{equation} \label{equation.2}
\begin{split}
C_{eff} &= [(C_{\alpha} + C_{\beta}) \times 8 ]+ C_{G} \\
&= [1.5 C_{\beta} + C_{\beta} \times 8] + C_{G} \\
&= 20 \times C_{\beta} + C_{G}
\end{split}
\end{equation}
As reported in \cite{Zou2014,Chang2013} the $ \lambda_1 = 5.54 $ and $ \lambda_2 = 3.92 $. For the sake of simplicity, we assume $ \lambda_1 = 1.5 \lambda_2 $ and consequently $C_\alpha = 1.5 C_\beta$, thus as Eq.~(\ref{equation.2}) shows, we can classify crosstalk patterns in 40 distinct classes which represented in Table.~\ref{table.1}. Indeed, patterns with higher class numbers have higher crosstalk noise and delay.
\begin{table}
\renewcommand{\arraystretch}{2}
\caption{TSV Crosstalk Classes}
\label{table.1}
\centering
\begin{tabular}{| c| c| c|}
\hline
Class & $ C_eff $ & $T_{i-4,...,i+4}$(t)$\to$$T_{i-4,...,i+4}$(t+1)\\
\hline
0 & $C_G$ & 000000000 $\to$ 111111111\\
1 & $C_G+C_\beta$ & 000000000 $\to$ 011111111\\
2 & $C_G+1.5C_\beta$ & 100000000 $\to$ 011111111\\
3 & $C_G+2C_\beta$ & 010000000 $\to$ 101111111\\
\textbf{.} &\textbf{.} & \textbf{.} \\
\textbf{.} &\textbf{.} & \textbf{.} \\
\textbf{.} &\textbf{.} & \textbf{.} \\
39 & $C_G+20C_\beta$ & 000010000 $\to$ 111101111\\
\hline
\end{tabular}
\end{table}
\section{Proposed Mechanism} \label{proposed}
\subsection{Overview}
As mentioned in Section~\ref{modelsection}, we have classified the patterns into 40 different classes based on their crosstalk effects. In this section, we propose 3DCAM mechanism which aims to minimize crosstalk effect on signal integrity and performance. To this end, we need to minimize the occurrences of higher crosstalk classes and maximize the occurrences of lower crosstalk classes. To accomplish which, we have to change the transition patterns on TSVs. As we know about the transmission line, one of the following conditions can occur to a single wire: $0 \rightarrow 1$, $1 \rightarrow 0$, $0 \rightarrow 0$, or $1 \rightarrow 1$.
The motivation behind this work is the fact that retaining the previous value of a victim TSV can significantly affect the transmission's crosstalk class.
Table.~\ref{table.2} presents some examples, in which retaining the victim (middle) TSVs significantly reduces crosstalk class. The first column of this table is the incoming pattern in which the victim TSV could have transition and the second column shows the same pattern except that the victim's transition is eliminated. The up and down arrows show zero to $V_{dd}$ and $V_{dd}$ to zero transitions respectively, and dashes(\lq-\rq) ~ denote no transition on TSVs. The changes in crosstalk classes are represented in third column of this table. For instance, the first row of Table. \ref{table.2} shows the case that falls into crosstalk class of $24C$ because the victim has three direct neighbors with opposite directions ($C_{eff}$ increased by $3 \times 2 \times 1.5 C_\beta$), a direct neighbor without transition ($C_{eff}$ increased by $1 \times 1 \times 1.5 C_\beta$), two diagonal neighbors with same direction transitions (impose no crosstalk) and two diagonal neighbors with no transitions ($C_{eff}$ increased by $2 \times 1 \times 1 C_\beta$). So the $C_{eff}$ will be $C_G + 12.5C_\beta$ and according to Table.\ref{table.1} the pattern falls into 24C class. By similar manner the crosstalk class after eliminating the victim's transition would be 12C.
As depicts in this table, the crosstalk minimization achieved by this simple modification is considerable.
\begin{table}[!h]
\caption{Motivational Example}
\label{table.2}
\includegraphics[width=2.5in]{example}
\centering
\end{table}
\subsection{Control TSVs}
In order to retain victim TSV's value, 3DCAM should by some means inform the receiver side decoder that the victim's transition has been eliminated.
Thus, 3DCAM reserves some TSVs for this purpose. These control TSVs only switch when the value of their corresponding victim TSVs are not valid (i.e. the 3DCAM eliminates their transitions). Fig.~\ref{control} demonstrates the layout of these control TSVs for a $3 \times N$ link. The middle TSVs of every $3\times3$ cluster (including overlapping ones) have a control TSV through which 3DCAM coder informs decoder that the value of the TSV is valid or not. Since the control TSVs may have coupling between themselves, we can repeat the technique and apply 3DCAM on them.
\subsection{Switch Threshold} \label{thresholdsection}
Retaining the victim's value is not always beneficial to crosstalk class. In the cases that the original transitions are good enough, retaining the victim's value may result in a worse crosstalk class. In addition, it has negative impact on control TSVs, considering the fact that retaining a value requires a transition in a control signal. Table.~\ref{table.3} shows an example in which retaining the victim's value leads to a worse crosstalk class and hence a worse transmission delay.
\begin{table}[!h]
\caption{Disruptive Retaining Example}
\label{table.3}
\includegraphics[width=2.5in]{examplenegative}
\centering
\end{table}
To address this issue, we introduce a parameter called \emph{Switch Threshold} (ST). This parameter determines which patterns should be manipulated by 3DCAM. In the other words, 3DCAM retains the value of the victim TSV, only if the transitions of neighbour TSVs make a pattern which has a crosstalk class more than ST.
To find the optimal value for the ST parameter, we swept ST parameter from 0C to 39C and measured average transmission delay of several benchmarks. Fig. \ref{threshold} depicts the results of this experiment. As this figure shows, the average delay is minimum when the ST is set to 20. These results are also consistent with the intuition, seeing that setting ST to 20 bisects the crosstalk classes.
\subsection{CODEC Design}
Fig. \ref{coder} and Fig. \ref{decoder} illustrate the structure of 3DCAM coder and decoder.
In order to send 9 bits of data ($D_{0}$ - $D_{8}$) from die $X$ to die $Y$, the data have to be delivered to the coder which has been placed in die $X$. After that, the coder evaluates the crosstalk class which will be imposed to the victim TSV ($I_0$). Then it checks whether the crosstalk class is greater than the ST parameter, in which case, the coder eliminates the victim's transition and flips the control bit. Next, the data and control bit are sent to the die $Y$ through TSVs. At the die $Y$, decoder receives the data and control signals. Then based on the control signal, the decoder determines the original value of the victim TSV. It is noteworthy that due to straightforward functionality of the 3DCAM coder and decoder, they can be implemented by simple circuits.
\begin{figure}[!t]
\centering
\includegraphics[width=3.2in]{coder}
\centering
\caption{3DCAM coding mechanism}
\label{coder}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.2in]{decoder}
\centering
\caption{3DCAM decoding mechanism}
\label{decoder}
\end{figure}
\subsection{TSV Overhead} \label{TSVoverhead}
Because of the controlling mechanism, we have to reserve an extra TSV for each cluster including 9 TSVs. As a result, 3DCAM suffers from about 30\% TSV overhead. Fig.~\ref{percentage} demonstrates the TSV overhead of proposed method, ShieldUS, and 3DLAT $(\omega=4)$. As Fig.~\ref{percentage} depicts, the TSV overhead of 3DLAT is about 80\% and ShieldUS imposed no TSV overhead to the circuit because it only shuffles and remaps the data. However, ShieldUS needs a considerable $9 \times 9$ crossbar which is used to remap data to TSVs.
\section{Evaluation} \label{evaluation}
In this section, the proposed mechanism is evaluated and compared with two 3D crosstalk reduction schemes, 3DLAT \cite{Zou2014} (with $\omega=4$) and ShieldUS \cite{Chang2013} (with $interval=100$). Based on the crosstalk model that proposed in Section \ref{modelsection}, we measured the amount of crosstalk reduction and average delay on real traces which are taken from SPEC2006 benchmark suite \cite{spec}. We used gem5 simulator \cite{gem5} to capture the transitions of memory data bus of \textit{gcc, mcf, namd, soplex, h264, omnetpp, xalancbmk, perlbench2, bwaves, cactusADM, dealII, lbm}, and \textit{aster} benchmarks.
Without loss of generality, we assume that the TSVs are arranged in $3 \times N$ layout. Also, we suppose that the data bitwidth is 64 and thus we need eight $3 \times 3 $ TSV clusters for the data and three clusters for control lines.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{Threshold}
\centering
\caption{Improvement of 3DCAM for different values of the threshold parameter}
\label{threshold}
\end{figure}
\subsection{Delay Analyisis}
Fig.~\ref{delay} demonstrates the transmission delay for several benchmarks from SPEC2006 suite. The delays are normalized to the case that no crosstalk minimization technique is used. As Fig.~\ref{delay} represents, 3DCAM can reduce the transmission delay of benchmarks by 9\% compared to the base uncoded case and in the best case 3DCAM could reduce transmission delay of \textit{soplex} benchmark by 25.7\% compared to 3DLAT method. Since 3DLAT (with $\omega = 4$) tries to code the input data in the manner that the coded output has no corsstalk class higher than 23C (based on our model), it can have a destructive effect on the transmissions time with lower crosstalk class. As the most transitions of \textit{soplex} benchmark fall into lower crosstalk classes, which is less than the average crosstalk class of 3DLAT coded outputs, 3DLAT even increases the transmission delay of this benchmark.
ShieldUS also could not effectively reduce the delay of experimented benchmarks. Inasmuch as this method can only reduce transmission delay of benchmarks with highly regular data and signals.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{overhead}
\centering
\caption{percentage of extra TSVs needed for 3DCAM, 3DLAT, and ShieldUS mechanisms}
\label{percentage}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=5.5 in]{Delay}
\centering
\caption{Average transmission delay of 12 benchmarks which are normalized to the case that no crosstalk minimization technique is used. }
\label{delay}
\end{figure*}
\subsection{Crosstalk Class Frequency Analysis}
As the occurrence frequency of higher crosstalk classes directly affects the signal integrity and transmission time, the frequency of crosstalk classes is measured before and after applying the 3DCAM mechanism. Fig.~\ref{class2} and Fig.~\ref{class1} show the occurrence frequency of crosstalk classes before and after applying the 3DCAM mechanism, respectively. As these figures show, 3DCAM causes most of crosstalk patterns to fall into the left side of the chart. Namely it pushes the higher crosstalk classes to the lower crosstalk classes.
\subsection{Area Overhead}
As mentioned in Section.~\ref{proposed}, 3DCAM uses a simple coder which leads to less area overhead compared to all previous works.
Table \ref{table.3} demonstrates the coder and decoder area overhead of ShieldUS, 3DLAT, and 3DCAM mechanisms. For $3\times3$ TSV cluster in 3DLAT, the area of coder and decoder is 4264 $\mu m^2$ due to its large comparators. We estimate ShieldUS area overhead with the area of a $9\times9$ crossbar which is 218 $\mu m^2$. Finally, the area of 3DCAM is 116 $\mu m^2$. These mechanisms are implemented and synthesized with 45nm technology using Synopsys Design Compiler.
\begin{figure}[!t]
\centering
\includegraphics[width=3.57in]{classes1}
\centering
\caption{The occurrence frequency of crosstalk classes before applying 3DCAM }
\label{class2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=3.57in]{classes2}
\centering
\caption{The occurrence frequency of crosstalk classes after applying 3DCAM}
\label{class1}
\end{figure}
\begin{table}[!h]
\renewcommand{\arraystretch}{1.5}
\caption{Area overhead of different mechanisms coder}
\label{table.4}
\centering
\begin{tabular}{| c| c|}
\hline
\textbf{Mechanism} & \textbf{Area} ($\mu$$m^2$) \\
\hline
ShieldUS crossbar (only) & 218 \\
\hline
3DLAT & 4264 \\
\hline
3DCAM & 116 \\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
In this paper, a crosstalk avoidance mechanism for TSV-to-TSV coupling capacitance is presented and a different crosstalk model has been discussed. The proposed mechanism decides about sending original data or retaining the TSV value in previous state based on a switch threshold (ST) parameter. 3DCAM reduces the frequency of higher crosstalk patterns which leads to reduce the interconnection delay. As compared to previous work, 3DCAM imposed negligible area overhead. GEM5 simulator is used to extract transitions of the data bus. We used real benchmarks taken from SPEC2006 suite. The simulation results showed that 3DCAM reduces transmission delay up to 25.7\% while it reduces TSV overhead by 62.5\% compared to 3DLAT mechanism.
In our future work, first, we plan to consider a comprehensive crosstalk model based on capacitance and inductance coupling effects. Since the inductive coupling effect will increase in the near future, it has to be considered in conjunction with capacitance coupling effect. Second, presenting a probability model for each crosstalk classes in TSVs is another task for authors. Third, developing an analytical reliability model for TSV-to-TSV coupling effect in 3D ICs is going to be discussed in the future work.
\section*{Acknowledgment}
The authors would like to thank the anonymous reviewers for their comments which were very helpful in improving the quality and presentation of this paper.
\bibliographystyle{abbrv}
|
1,314,259,992,991 | arxiv | \section*{Localised, Finite-time Computation of Network Steady States \label{decentye}}
We will consider a generalised model of consensus dynamics whereby nodes can both store their own `value' or opinion at any point in time, and periodically learn their neighbours values in order to `update' their own opinion.
Mathematically, if ${\bf{x}}_{k} \in \mathbb{R}^n$ is a vector containing the value or state of each node at time-step $k$, all future node values may be described by the iterative process
\begin{equation}
\label{eqnconsdis_main}
{\bf{x}}_{k+1}=W{\bf{x}}_k
\end{equation}
where ${\bf{x}}_{0} \in \mathbb{R}^n$ contains the initial condition for the dynamics, and the weight matrix $W \in \mathbb{R}^{n\times n}$ describes the inter-node relationships giving rise to the dynamical system. All nodes asymptotically (i.e. eventually) converge to the same (consensus) steady state solution under two conditions: $W{\bf 1}={\bf 1}$ (where ${\bf 1}$ corresponds to a vector of ones) and all other eigenvalues of $W$ are within the unit disk.
Depending on the situation, $W$ can take several forms. One of the most common forms is that of the Laplacian consensus dynamics \cite{OlfatiSaber2007}, which describes the process by which a node of a network updates its values based on that of its neighbours. In this case, $W=I-\omega L$, where $L=D-A$ is the Laplacian matrix with $A$ being the adjacency matrix of the network with entries $i,j$ containing the weight of the edge between node $i$ and node $j$, and $D$ being a matrix of zeros everywhere, except on the diagonal where it contains the corresponding node out-degree. For a directed network, if (i) the graph is balanced and strongly connected, and (ii) $0 < \omega < 1/\max(D)$, then all variables in $\bf{x}_k$ asymptotically reach a shared average consensus value. Another key example is that of random walker probability dynamics \cite{Lambiotte09laplaciandynamics}, where $W=AD^{-1}$ and the final state vector represents the long-term transition probabilities of a random walker transitioning through each node. Random walker dynamics on a network forms the basis of a number of algorithms including Google's PageRank \cite{brin1998anatomy}, and community detection algorithms \cite{fortunato2010community, Delvenne2010}.
An example network is shown in the inset of Figure~\ref{fighank}~A, where we simulate Laplacian consensus dynamics as described above, and observe that, despite the small size of the network ($n=8$ nodes), the node dynamics do not converge to a consensus value (for a random initial condition) until step $k\approx 60$. Indeed, in all cases, given the conditions on $W$ above, the rate of asymptotic convergence for such dynamics is upper bounded \cite{OlfatiSaber2007}, yet there is no lower bound to this rate of convergence, i.e., there is no mathematical limit on the number of iterations or steps this could take.
Sundaram and Hadjicostis \cite{Sundaram:2007fk, Sundaram2007} have shown that individual nodes, under certain conditions, can compute their own final state value in a finite number of steps equal to the rank of a node-specific observability matrix (composed of matrix powers of $W$, see SI and \cite{Liu12022013} for a review). However, in most cases, the observability rank is equal to the size of the network, and nodes require knowledge of the full network structure---an assumption not likely to be met for many real-world applications. Yuan \emph{et al.} \cite{Yuan2013} proposed an analogous but localised methodology, employing a `local' Hankel matrix in place of the `global' observability matrix. Hankel matrices are square matrices in which each ascending skew-diagonal from left to right is constant, and are frequently employed in control theory.
Specifically, if $x_i(r)$ for $i=0,1,2,\ldots$ are the state values associated with node $r$, then the Hankel matrix $H_k^{(r)} \in \mathbb{R}^{k \times k}$ of size $k$ for node $r$ is defined as
$$
H_k^{(r)} =
\left[ {\begin{array}{cccc}
x_1(r)-x_0(r) & \hdots & x_k(r)-x_{k-1}(r)\\
x_2(r)-x_1(r) & & x_{k-1}(r)-x_{k-2}(r)\\
\vdots & & \vdots \\
x_k(r)-x_{k-1}(r) & \hdots & x_{2k-1}(r)-x_{2k-2}(r)
\end{array} } \right].
$$
This matrix is increased in size (via addition of rows and columns) until its singular value decomposition \cite{golub2012matrix} includes a single zero eigenvalue and it has dropped rank at $k=\Delta_r+1$, i.e., $\text{rank}\left(H_k^{(r)}\right)=\Delta_r$. The steady state value can be then computed by node $r$ using the formula (see \cite{Yuan2013}):
\begin{equation}
\label{eqnfv}
h^*(r)=\frac{[x_0(r)\hdots x_{\Delta_r}(r)]{\bf v}_{\Delta_r+1}^{(r)}}{{\bf 1}^T{\bf v}_{\Delta_r+1}^{(r)} }.
\end{equation}
where ${\bf v}_{\Delta_r+1}^{(r)} \in \mathbb{R}^{\Delta_r+1}$ is the single nullspace vector of $H_{\Delta_r+1}^{(r)}$ and $x_0(r),\hdots, x_{\Delta_r}(r)$ are successive values of node $r$.
A proof may be found in \cite{Yuan2013}, and is based on a Jordan decomposition \cite{golub2012matrix} of $W$ and a Vandermonde decomposition \cite{golub2012matrix} of $H_{\Delta_r+1}^{(r)}$.
When identical symmetry exists both in the graph and initial condition, the rank of $H_{k}^{(r)}$ does not accurately reflect the number of steps to compute the final value~\cite{hend3, Yuan2013}. Yet, for a random initial condition this is highly unlikely to be an issue.
As mentioned above for the observability method, the number of steps required for each node, given by the corresponding Hankel rank, of most common classes of complex networks is usually trivial (i.e., it equals the graph size $n$) \cite{neavephd}. This implies, firstly, that there is a prohibitive cost for very large networks with millions of nodes in computing the steady state value using the above approach, and, secondly, that all nodes compute the final value in the same number of state values---there is no node-specific predictive advantage. There is also a significant limitation to the accuracy of this approach as the graph size grows due to the fact that the entries of the Hankel matrix tend towards zero as the system converges to the consensus value, i.e., $x_k(r)-x_{k-1}(r) \to 0$ as $k \to \infty$. Determining the matrix rank accurately is therefore numerically unstable for large matrices \cite{oclery}, and particularly so for those composed of small numbers.
We exploit the properties of the singular value decomposition to propose a relaxation of this approach. Specifically, we use the singular vector corresponding to the smallest singular value $\sigma_1$ to approximate the Hankel nullspace vector for increasing Hankel size $k$. More specifically, we compute a sequence of approximations to the steady state solution for node $r$ for steps $k\leq\Delta_r+1$ such that
\begin{equation}
\label{eqndecent_main}
h_{k}^*(r)=\frac{[x_0(r)\hdots x_{k-1}(r)]{\bf v}_k^{(r)}}{{\bf 1}^T{\bf v}_k^{(r)} }
\end{equation}
where ${\bf v}_k^{(r)}$ is the singular vector corresponding to the smallest singular value of the Hankel matrix $H_k^{(r)}$.
As the distance to the singular matrix decreases \cite{dongarra1979linpack}, this sequence of approximates approaches the true solution.
By construction we are guaranteed (in the worst case) to compute the steady state value in a maximum number of steps given by $k=\Delta_r+1$, but, in practice, we typically have $k\ll \Delta_r+1$.
Figure~\ref{fighank} highlights the contrast between the actual number of steps needed for $\epsilon$-convergence of the dynamical iteration (i.e., the dynamic simulation is within an $\epsilon$-distance of the true value, or $|x_k(r)-x^*(r)|\leq \epsilon$), and the significantly fewer number of steps needed for $\epsilon$-convergence of the Hankel approximation to the final value (i.e., $|h_k(r)-x^*(r)|\leq \epsilon$). For practical applications, these state values could be acquired from sensor measurements. For most illustrations below, we compute these initial state values via a short simulation of the dynamical system given by \eqref{eqnconsdis_main}. We then show that the number of steps required to compute the final value of each node, in almost all cases, is significantly less than the number of steps needed for convergence of dynamical system.
We highlight several important features of this example:
\begin{itemize}
\item Convergence of the Hankel approximation is guaranteed and upper-bounded by the Hankel rank of each node, yet in the consensus dynamics example of Figure~\ref{fighank}, the number of steps given by the upper bound was never required---and was significantly less than the number of steps needed for the corresponding dynamical iteration;
\item Each node exhibits a distinct number of steps required to compute the common consensus value---hence some nodes are in a sense more `knowledgeable' than others, and can compute or predict the final value sooner;
\item In the case of consensus models such as the one considered in Figure~\ref{fighank}, a single node, without any information other than a short sequence of its own state values, can determine the final consensus value of the \emph{entire} network.
\end{itemize}
These powerful features, and their potential wide-ranging applications, will be explored in the following section.
\subsection*{Node-specific Predictive Capability in Complex Networks \label{nodeconv}}
\begin{figure}[t!]
\centering
\includegraphics[width=7cm]{Figure2_MB.pdf}
\caption{\footnotesize \emph{Node-specific predictive capability.}
\textbf{(A)} The well-known neuronal network of the \textit{Caenorhabditis Elegans} worm represents the synaptic connections and chemical junctions between sensory (green), motor (grey) and inter- (orange) neurons~\cite{varshney2011structural}.
The layout follows Varshney \emph{et al} \cite{varshney2011structural}: the $y$-coordinate is given by the processing depth, and the $x$-coordinate is the node coordinate of the normalised Fiedler eigenvector.
\textbf{(B)} Each node of the network is coloured according to the number of Hankel steps it needs to approximate its own steady state value to a small tolerance ($\epsilon=10^{-4}$). The values are averaged over dynamics started from $5000$ random initial conditions in the interval $[-1,1]$. We observe that inter-neurons typically compute their steady-state value first. \textbf{(C)} There is a statistically significant difference in the number of Hankel steps needed by the three neuron groups.}
\label{figCelegans}
\end{figure}
The example in the previous section highlights the fact that each node of the network requires a unique number of initial state values to compute its own steady state solution, and that some nodes are better predictors in the sense that they require fewer values. Here we investigate this result by applying our approach to the well-known \emph{C. Elegans} neuronal network which describes the wiring structure of sensory, motor and inter-neurons for the \textit{Caenorhabditis Elegans} worm. \emph{C. Elegans} has been used extensively as a model organism to study neuronal development \cite{watts1998collective} as it is one of the simplest organisms with a nervous system, and is easy to grow in bulk populations.
The \emph{C. Elegans} network is shown in Figure~\ref{figCelegans}~A, with sensory (green), motor (grey) and inter- (orange) neurons located in distinct regions of the network. Key neurons, such as interneurons responsible for mediating signals between input sensory neurons and motor neurons, are labelled explicitly on the network. The edges represent a combination of gap junction (electrical) connections and chemical synapse connections. Following \cite{varshney2011structural} nodes have been positioned with processing depth (i.e., the number of synapses from sensory to motor neurons) on the y-axis, and the node's respective entry in the Fiedler eigenvector on the x-axis. The Fiedler vector is the eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of a graph. The entries of this vector have previously been used to partition the network into two densely connected groups of nodes \cite{newman2006finding} (positive values of the Fiedler vector in one group; negative values in the other), and for spectral embedding, i.e., for finding an optimum one-dimensional representation of a graph \cite{juvan1992optimal}.
We model the communication abilities of the neurons via a simple consensus model, where neurons (nodes) communicate with their neighbours in the network via chemical and electrical signalling \cite{varshney2011structural, macosko2009hub}. We seek to identify nodes which, after receiving a limited number of signals from their neighbours, can predict the final consensus value of all nodes or neurons in the fewest steps. These nodes, we propose, are highly knowledgeable---in the sense that they are influential communicators---about the network dynamics given their position in the network. Figure~\ref{figCelegans}~B shows the network coloured by the number of Hankel steps needed by each node to approximate its own steady state value (the consensus value of the network). These values are averaged over 5,000 random initial conditions in the interval $[-1,1]$, and detected with tolerance $\epsilon=10^{-4}$. We observe that inter-neurons, such as AVER/L, AVKL and RIGL (associated with locomotion in response to stimuli) and sensory neurons ADEL/R located in the head, typically compute their steady-state value first. Overall, inter-neurons exhibit a statistically significant decrease in number of consecutive values needed to approximate the final steady-state value, shown via comparing the three distinct neuron groups in Figure~\ref{figCelegans}~C.
Hence, key inter-neurons, having collected a small number of their own state values based on signals from their neighbours, can compute the common consensus value of the whole network, and broadcast or communicate it to other nodes who have not yet been able to compute this consensus value. In the light of this, it appears that inter-neurons have an increased predictive ability compared to more peripheral nodes, consistent with their functional role as key communicators between sensory and motor neurons in the network.
\begin{figure*}[t!]
\centering
\includegraphics[width=17cm]{ranking5.png}
\caption{\footnotesize \emph{Node ranking from local information.}
\textbf{(A)} The network of the \emph{Karate Club} social network \cite{karate} represents a friendship network within a US karate club dominated by two cliques---one allied to the \emph{President}, and the other to the \emph{Instructor}. The nodes are coloured by their node ranking (darker nodes are ranked higher) confirming the influence of these two key actors.
\textbf{(B)} Starting from a random initial condition (the random initial ranking on the left y-axis), a new node ranking can be obtained for each Hankel step (x-axis). By step 8, all nodes converge to their final ranking (colours of the lines as in A).
\textbf{(C)} The node ranking of the largest connected component of the $2002$ Stanford web network
($n=8929$ nodes, inset)~\cite{leskovec2009community} is computed using the Hankel approximation (with damping factor $\alpha=0.9$, blue circles), and with the full iteration method (grey circles) starting from a random initial condition.
Spearman correlation of both rankings against the ground truth (i.e., the Pagerank achieved asymptotically).
The Hankel approach produces an accurate ranking (i.e., the correlation reaches $1$) in significantly fewer steps than the convergent dynamics.
\textbf{(D)} The number of steps required by the Hankel method compared to the full iteration method when the damping factor $\alpha$ is varied between $0.7$ and $0.925$ (shown is an average over $10$ random initial conditions for each value of $\alpha$ with tolerance $\epsilon=10^{-5}$).
For higher values of $\alpha$ the approximation is closer to the `true' ranking for an undamped system with $\alpha=1$. Hence both the iteration and the Hankel method require more steps to converge.
(Inset) The ratio of Hankel steps to full dynamical iteration steps decreases with increasing $\alpha$, i.e., the Hankel method is increasingly more efficient as we approximate the undamped network ranking.
\label{figdir}
}
\end{figure*}
\subsection*{Global Node Ranking Based on Local Information}
Beyond the consensus framework, the Hankel approximation method may be applied to a wide range of linear models. Here we consider node ranking, which forms the basis of the Google search engine technology \cite{brin1998anatomy}. Analogous to the previous example, using our approach, individual nodes can compute their own ranking while employing significantly fewer iteration steps than traditionally needed---and without the requirement of knowing the full network structure.
The classic PageRank algorithm \cite{googlebook} is based on a model of a random walker moving from node to node along the edges of a network formed by hyperlinked websites. A node's ranking can be seen as the long-run probability of the walker traversing that node relative to other nodes. The PageRank vector (i.e., the node ranking vector) is obtained as the solution of the linear system given in~\eqref{eqnconsdis_main} with $W=\alpha AD^{-1}+\frac{1-\alpha}{n} E$ where $E$ is a matrix of ones. The second term may be seen as a cost or damping term for which choices of $\alpha$ close to 1 yield the most accurate ranking---yet are more computationally expensive in the sense that convergence is slower. In essence, a value of $\alpha$ less than 1 adds constant edges to the network enabling information to diffuse and spread more quickly. For smaller values of $\alpha$, these edges are more heavily weighted and the system reaches an equilibrium state faster.
While details of the state-of-the-art Google ranking algorithms are unavailable, for large networks (the largest being the whole internet), webpage ranking employing this classic algorithm is normally approximated via $50-100$ iterations of the full system, and has been reported to take a few days to complete each month \cite{bryan200625}. Here we show that, for both a well-known social network and a large web network, we can use our Hankel method to compute the ranking value for each node in a localised and efficient manner compared to the traditional approach of iteration of the full dynamics.
\begin{figure*}[t!]
\centering
\includegraphics[width=17.5cm]{fig3a.png}
\caption{\footnotesize
\emph{Community Detection.}
\textbf{(A)} Adjacency matrix of a network with $n=200$ nodes and four equal sized communities (the probability of connection for node pairs within the same community is $P_{i,j}=0.7$, tand across different communities is $P_{i,j}=0.01$).
\textbf{(B)} The Hankel approximation method detects transient clustering of node dynamics, corresponding to the detection of community structure. We visualise the distance matrix $D_k$, where entries correspond to the difference between the Hankel approximations for each node pair at step $k$. The blocks of small values (dark shading) emerging at steps $2$, $3$ and $4$ correspond to the transient clustering of the Hankel approximation within communities. By step 5, the iteration has converged and all nodes have the same value ($D_k$ has only small values).
\textbf{(C)} The corresponding dynamical matrices $S_k$, where entries correspond to the distance between dynamical iteration values for node pairs at step $k$, also detects communities, but at much longer times than the Hankel approach (e.g., community structure only starts to appear at around step $20$).
While the Hankel method computes the final consensus value in just $5$ steps, the full system dynamics takes up to $2500$ steps to converge. \label{figcomm}}
\end{figure*}
Figure~\ref{figdir}~A illustrates the well-known Karate Club network \cite{karate}. This is a social network of friendship links between 34 members of a karate club at a US university in the 1970s. A disagreement between the club's president and main instructor created a split between the members, and ultimately led to the breakup of the club. The nodes are coloured according to the true ranking (dark blue corresponds to high ranking). In Figure~\ref{figdir}~B, for a random initial condition (corresponding to a random initial ranking shown on the left y-axis), we show the node ranking for each step of the Hankel ranking algorithm. We observe that the final ranking is achieved by most nodes by step $k=6$, and all nodes by step $k=8$. The president and instructor are the top-ranked nodes, using their dense friendship links to other club members to cement their position in rival camps.
In order to illustrate the efficiency of our method on a larger scale, we compare the number of steps needed to obtain the node ranking of a large web network, the undirected largest connected component of the Stanford web network which describes hyperlinks between almost $10,000$ web pages in the domain stanford.edu for the year $2002$ with $n=8929$ nodes, and over $25,000$ edges~\cite{leskovec2009community}. Using a damping factor value of $\alpha=0.9$, we compute the node ranking via both the Hankel approximation method and, for comparison, iteration of the full dynamical system. In order to compare the speed of each of these approaches, we compute the Spearman rank correlation \cite{press1982numerical} between the approximate ranking (Hankel or dynamical iteration) and the true ranking (previously computed via a long iteration) at each step. Figure~\ref{figdir}~C shows that our Hankel approximation can obtain the node ranking (i.e., the correlation reaches 1) in significantly fewer steps than the linear dynamic iteration.
For large networks, the quality of this approximation depends on the choice of damping factor $\alpha$ \cite{damping}---the closer $\alpha$ is to $1$, the higher the accuracy of the approximation but also the slower the convergence rate \cite{extrapagerank} ($\alpha=0.85$ is the most commonly used value in the literature~\cite{googlebook}). Figure~\ref{figdir}~D shows that, as we increase $\alpha$ and, thereby, the accuracy of the ranking, both the iteration and the Hankel method require more steps to converge. The inset shows that the ratio of Hankel steps to dynamics steps decreases with increasing $\alpha$, implying that the Hankel method gains in relative efficiency as the damping increases.
These results are important as many applications today contain millions, or sometimes even billions, of nodes. The potential to both locally and efficiently determine metrics such as node rankings, or equivalently a variety of centrality or dynamics-based measures, using our newly proposed method, could render previously intractable problems solvable.
\subsection*{Community Detection from Local Node Transients}
Beyond consensus models and node ranking for networks, linear systems models form the basis for a large class of computational algorithms for the analysis of network structure. Due to the large-scale, complex nature of real-world networks, the detection of communities or groups of nodes, typically tightly connected, can yield powerful insights into network behaviour by revealing the underlying organisation of the network, or providing insight into its function. Furthermore, network size may be reduced via aggregation of nodes in communities, often yielding more a tractable and informative topology \cite{yaliraki2007chemistry, Delvenne2010, oclery}.
Many algorithms have been proposed for the analysis of community structure in graphs \cite{Newman2003,f51} including normalised cut \cite{f52}, modularity \cite{f53,f54} and stability \cite{Lambiotte09laplaciandynamics,Delvenne2010}. Many of these algorithms are based on the idea that a random walker on a network (i.e., a walker that travels from node to node) becomes `trapped in wells', circulating within sub-regions of the network. This can be due, for example, to a region of high connectivity leading the walker to repeatedly traverse the same set of nodes for an extended period of time.
The probabilistic dynamics of a random walker on a graph may be modelled via the construction of a linear system similar to that introduced above in the webpage ranking example \cite{Lambiotte09laplaciandynamics,Delvenne2010}, i.e., $W=AD^{-1}$. Transient clustering of these dynamics provides evidence of community structure in the network as the dynamics of nodes in the same community $\epsilon$-converge temporarily (i.e., they exhibit similar state values) before reaching a (possibly but not necessarily common) steady state. We find that beyond approximating the steady state solution, our Hankel approach can also detect these transient states. This important feature of our method means that, even before computing the final state for each node, Hankel approximations converge for nodes within the same community, thereby allowing communities to be very quickly identified during the Hankel iterations. Hence, as we will see, applying the Hankel method enables us to also detect community structure in a localised manner, and using significantly less successive state values than existing methods.
To illustrate this result, we generate an ensemble of networks with community structure defined by a matrix of probabilities $P$ such that $P_{l,m}$ is the probability of connection between any node in community $l$ and any node in community $m$. Figure~\ref{figcomm}~A shows the entries in the adjacency matrix of such a network, with $n=200$ nodes split into four equal size communities (each with $50$ nodes). In this case, the probabilities of node connection between communities is given by $P_{l,m}=0.7$ for any $l=m$ (i.e., nodes in the same community) and $P_{l,m}=0.01$ for all $l\neq m$.
For each network we compute the iterative dynamics (using $W=AD^{-1}$ and a random initial condition in the interval $[0,1]$ for \eqref{eqnconsdis_main}), and compute a sequence of Hankel values as defined by \eqref{eqnfv}. We seek to detect `distances' between the Hankel values, and the iterative dynamics, for pairs of nodes at each step. We compute distance matrices $D_k$ and $S_k$ at each step $k$ of the iteration, with entries
\begin{align*}
D_{k}(i,j)&=|h^*_k(i)-h^*_k(j)| \\
S_{k}(i,j)&=|x_k(i)-x_k(j)|
\end{align*}
for all pairs of nodes $i$ and $j$, where $h^*_k(i)$ is defined in \eqref{eqndecent_main} and $x_k(i)$ is the ith component of the vector ${\bf x}_k$ in \eqref{eqnconsdis_main}. Groups of nodes exhibiting similar dynamics (as captured by small distance values) signal community structure.
In Figure~\ref{figcomm}~B and C, we visualise the matrices $D_k$ and $S_k$ for increasing values of the iteration number $k$. In both cases, dark shading indicates small distance values---and hence the detection of community structure. We observe that the Hankel approximation transiently detects four communities within just $k=3$ steps, before closely approximating the final steady state value at $k=5$. In contrast, the full dynamics only detect the community structure after around $k=20$ steps, and does not converge to the consensus value until $k>2000$---illustrating the substantial advantage of the Hankel approximation for sparsely connected networks.
\section*{Discussion}
Using local information to obtain global properties from interconnected dynamical systems. our work exhibits a number of distinguishing features. Its localised formulation, whereby nodes need only have access to a limited number of their own successive state values, enables nodes or sensors to independently compute variables of interest, without knowledge of the network structure or of their neighbouring nodes' state values. Its formulation is simple, and is easily adaptable to a large class of network-based dynamical models. It can be scaled to cope with large real-world systems, with significant efficiency compared to commonly used iterative methods as it avoids the need for convergence. Finally, it enables us to identify functional attributes of nodes in networks, i.e., `predictor' or `communicator' nodes that can estimate the full network dynamics ahead of other nodes.
Our algorithm performs well for a range of networks and applications; yet alternative but related approaches could be explored for future work. While we have employed the singular vector corresponding to the smallest singular value as an approximate nullspace vector, it may be possible to use a combination of singular vectors defining a closest subspace in order to obtain `smoother' convergence to the consensus value. Furthermore, computing the SVD or rank update for the Hankel matrix as new rows and columns of data are added (without re-computing the decomposition) in the line of thought explored by \cite{eisenstat1995relative,brand2002incremental} would be beneficial. A closed-form solution for the updated singular vector dependent only on the previous step and the new data does not appear possible using current techniques, but could be feasible under a re-formulation or further relaxation of the problem.
For any such algorithm, it is desirable to have a fully localised convergence criterion that can be computed by each node locally to ascertain when the consensus value is well approximated. In the SI, we show that the difference between successive steps of the algorithm is highly correlated with the `distance' to the true consensus value; hence this criterion can be used as an effective stopping criterion for large graphs. Developing a theoretical convergence criterion would be beneficial, although this is a non-trivial task due to the non-monotonicity of the sequence of steady-state value approximations.
The number of steps needed for an individual agent to approximate its own final value is node-specific, and nodes that require the fewest steps can be seen as key communicators in a network. In the case of consensus dynamics, a subset of influential nodes can `predict' the final value earlier, and potentially communicate that result to other agents in the network. There is much scope to investigate further how these results could be used dynamically by nodes to aid or disrupt consensus, and to develop further applications within the context of network design and autonomous sensing.
\matmethods
\begin{itemize}
\item The adjacency matrix, node type and node position (x and y axis co-ordinates) for the \emph{C. Elegans} neural network \cite{varshney2011structural} in Fig.~\ref{figCelegans} may be requested from Lav Varshney (http://varshney.web.engr.illinois.edu/).
\item The adjacency matrix for the Karate Club \cite{karate} network in Fig.~\ref{figdir} A downloaded from http://www-personal.umich.edu/~mejn/netdata/, and visualised with Gephi (https://gephi.org/).
\item The adjacency matrix for the 2002 Stanford web network \cite{leskovec2009community} in Fig.~\ref{figdir}C downloaded from https://snap.stanford.edu/data/web-Stanford.html, and visualised with Gephi (https://gephi.org/).
\end{itemize}
}
\showmatmethods
\acknow{N.O'C. was funded by a Wellcome Trust Doctoral Studentship. G-B.S. acknowledges the support of EPSRC through the EPSRC Fellowship for Growth EP/M002187/1. M.B. acknowledges support from EPSRC Grants EP/I017267/1 and EP/N014529/1.}
\showacknow
|
1,314,259,992,992 | arxiv | \section{Introduction}
The magnetic field of the Sun and other late-type stars is known to
have, on average, opposite signs of magnetic helicity in the northern
and southern hemispheres \citep{See90,PCM95}.
There is also the possibility of the field being bihelical \citep{BB03}
with a sign change of the magnetic helicity at large length scales.
To detect this in the Sun, one would need to measure {\em spectra} of
magnetic helicity, but this is made complicated by the fact that the
solar surface also displays a systematic north--south variation with
opposite signs in the two hemispheres.
To capture this correctly, a global approach must be adopted that takes
the systematic north--south variation into account.
This is done by utilizing what is known as a two-scale approach
\citep{RS75}.
Here, one scale is that of the large-scale hemispheric modulation, and
the other is the scale of the turbulence, which in itself comprises an
entire range of length scales.
In that approach, one can compute a spectrum covering both north and
south, while taking a systematic north--south variation into account
as if both hemispheres looked just like the northern hemisphere
\citep[][hereafter BPS]{BPS17}.
The problem with the standard two-scale approach is that it is only
a {\em semi-global} one.
Technically, it is still Cartesian in that the solar surface magnetic
field is represented in the Lambert cylindrical equal-area projection.
In a proper global approach, by contrast, one would need to employ a
spherical harmonics decomposition, but this must be done in such a way
that the systematic north--south variation can still be taken into account.
In this paper, a simple heuristic modification to the usual spherical
harmonics spectra is being proposed.
It is based on the idea that in the semi-global two-scale approach, the
helicity spectrum is computed as the product of the magnetic field and
its vector potential at wavenumbers that are offset for the two fields
by a small amount that corresponds to the wavenumber of the large-scale
hemispheric modulation.
Analogously, for spherical harmonics spectra, one should consider the
product of the two terms at spherical harmonic degrees that are shifted
by one.
This idea is then adapted to analyzing also the parity-even and parity-odd
contributions to the linear polarization \citep{Kamion97,SZ97}.
The reason for using such a decomposition is that there are large
uncertainties owing to the $\pi$ ambiguity of the magnetic field in
weak-field regions of the Sun.
This ambiguity reflects the fact that polarization ``vectors'' have
neither head nor tail.
Various disambiguation procedures are available \citep{Sak85,Geo05,
Hoe14,RA14}, but they tend to fail in regions far away from sunspots,
where the magnetic field is weak.
To avoid any bias, the random disambiguation method is often employed
\citep{Liu17}.
This is justified when the Stokes $Q$ and $U$ parameters are dominated
by noise but, if this were indeed the case, it should not be possible
to detect any systematic north--south dependence of the parity-odd $EB$
correlation from weak-field regions.
It is also clear that any magnetic helicity derived from a randomly
disambiguated magnetic field may itself be random and would therefore
be unreliable.
The proper way out of this problem of obtaining a qualitative measure
of the Sun's magnetic helicity from $\pi$-ambiguous magnetic fields is
to work directly with the original linear polarization.
This has already been attempted by determining the rotationally invariant
parity-even and parity-odd contributions, or $E$ and $B$ polarizations,
respectively, from the Stokes $Q$ and $U$ parameters
\citep[][hereafter BBKMRPS]{BBKMR19}.
This decomposition yields a field that is parity even, i.e., statistically
mirror symmetric, and another one that is parity odd, i.e., statistically
mirror antisymmetric \citep{Kamion97,SZ97}.
The relevant diagnostic quantity is usually the cross-correlation
of the spectral representations of $E$ and $B$ \citep{KR05,KMLK14,Bracco19}.
Attempts to analyze solar $E$ and $B$ polarizations have not yet produced
a nonvanishing cross-correlation (BBKMRPS).
However, this could be caused by their method still being provisional
in that only a semi-global approach was used to deal with the fact that
the sign of the cross-correlation is systematically different in the
northern and southern hemispheres.
It was always clear that a proper analysis should involve a decomposition
into spherical harmonics.
More precisely, the linear polarization parameters $Q$ and $U$ must
be decomposed into what is known as spin-2 spherical harmonics, which
have the appropriate transformation properties for linear polarization
\citep{Kamion97,SZ97}; see \cite{Dur08} for a textbook on the subject.
While this method is now routinely applied in cosmology using data from
the {\em Planck} satellite \citep{Akrami18}, it has not yet been adapted to
the case where one expects there to be a global sign change
of magnetic helicity about the equator.
In that case, we employ the spherical harmonics decomposition of
$E$ and $B$, which yields $\tilde{E}_{\ell m}$ and $\tilde{B}_{\ell m}$,
respectively.
We then compute their product at spherical harmonic
degrees that are shifted by one, i.e., we compute
$\tilde{E}_{\ell m}\tilde{B}_{\ell+1\, m}^\ast$.
We also compute $\tilde{E}_{\ell m}\tilde{B}_{\ell-1\, m}^\ast$, which
we shall show to be a better proxy of the expected magnetic helicity
spectrum than the former one.
The work of BBKMRPS suffered from another problem in that the
publicly available polarization data were not cleaned and corrected
to the same extent as those finally used to compute the Sun's magnetic
field \citep{Hughes}.
For example, the quality of the images varied across the solar disk.
Furthermore, proper line fits to solar atmosphere models have not been
performed.
Therefore, there is a possibility of small shifts in frequency that
could affect the resulting $Q$ and $U$ signals.
In particular, the magnetic field can have different strengths at
different geometrical depths, giving rise to more complicated spectral
line profiles that are usually fully accounted for in the inversion
pipelines \citep{Hoe14}, but they were ignored in the more rudimentary
analysis of BBKMRPS.
A legitimate way out of this additional problem is to use the full
solar magnetic field inversion along with its questionable disambiguated
magnetic field and make it ambiguous again!
We can do this by computing a synthetic (or pseudo) linear polarization
from the horizontal magnetic field.
Such work is already in progress (A.\ Prabhu, in preparation), but it
is still local and constrained to finite patches in one hemisphere,
as was done in the works of BPS and \cite{Singh18}.
Here, by contrast, we employ a novel analysis using spin-2 spherical
harmonics to compute a global cross-correlation spectrum.
We begin by testing the global two-scale approach and its ability to
extract a unique spectrum by using data from both hemispheres at the
same time.
In \Sec{Axisymmetric}, we first construct simple axisymmetric fields to
study the effects of a global sign change of the magnetic helicity.
In \Sec{NonAxisymmetric}, we consider nonaxisymmetric magnetic fields
to verify the numerical approach.
In \Sec{Applications}, we use synoptic magnetograms from Carrington
rotations (CRs) 2161 to 2163, for which a semi-global helicity spectrum
was previously determined (BPS).
We discuss the relevance of our results for dynamo theory in
\Sec{Implications} and conclude with the broader implications of the
present work in \Sec{Concl}.
\section{An axisymmetric example}
\label{Axisymmetric}
\subsection{Representation of the magnetic field}
\label{RepresentationAxisymmetric}
It is useful to begin with a simple example that is similar in spirit
to the one-dimensional example used in BPS (see their Figure~1), where
the magnetic helicity density shows a sign change in the middle of
the domain.
For this purpose, we restrict ourselves to an axisymmetric magnetic field,
which can be written in the form
\begin{equation}
\mbox{\boldmath $b$} {}=\mbox{\boldmath $\nabla$} {}\times(a_\phi\hat{\bm{\phi}})+b_\phi\hat{\bm{\phi}},
\label{axisymb}
\end{equation}
where $r$ and $\theta$ are radius and colatitude, $a_\phi(r,\theta)$
is the toroidal component of the magnetic vector potential, and
$b_\phi(r,\theta)$ is the toroidal component of the magnetic field itself.
The proper expansion of $a_\phi$ and $b_\phi$ is in terms of the
associated Legendre polynomials $P_l^1(\cos\theta)$ as
\begin{equation}
a_\phi\hat{\bm{\phi}}=\sum_{\ell=1}^{N_\ell}\tilde{a}_\ell(r)P_\ell^1(\cos\theta),\quad
b_\phi\hat{\bm{\phi}}=\sum_{\ell=1}^{N_\ell}\tilde{b}_\ell P_\ell^1(\cos\theta),
\label{axisymbExpand}
\end{equation}
where $N_\ell$ determines the truncation level.
The two horizontal magnetic field components on the surface of the
sphere at $r=R$, say, are then given by
\begin{equation}
b_\theta(\theta)=-{1\over R}\sum_{\ell=1}^{N_\ell}
\frac{\partial}{\partial r}(r\tilde{a}_\ell)P_\ell^1(\cos\theta),
\label{bt_rder}
\end{equation}
\begin{equation}
b_\phi(\theta)=\sum_{\ell=1}^{N_\ell}
\tilde{b}_\ell P_\ell^1(\cos\theta).
\end{equation}
Even if $\tilde{a}_\ell(r)$ were independent of $r$, the values of
$b_\theta$ would be finite because of the $r$ factor under the derivative.
At the surface, however, it is more likely that
$\tilde{a}_\ell(r)$ decays with $r$ as a power law, for example like
$r^{-(\ell+1)}$, as it would if the exterior magnetic field was a
potential field \citep{KR80}.
In such a case, $b_\phi$ would normally vanish, but this will not be
assumed here, because then the magnetic field would have vanishing
helicity.
Specifically, we are interested in a field with globally antisymmetric
magnetic helicity, so we assume that $b_\phi$ remains finite at $r=R$.
\subsection{Opposite helicities in the two hemispheres}
\label{Opposite}
In BPS, we constructed a magnetic field with globally antisymmetric
helicity by having the two horizontal field components with a relative
wavenumber shift that corresponds to the scale of the latitudinal
variation of the magnetic helicity.
This corresponds to the two components having an $\ell$ value that is
different by one.
In the present case, we choose $b_\ell=b_0$ and $a_\ell=-b_0 R/\ell$,
with some general amplitude factor $b_0$, so
\begin{equation}
b_\theta(\theta)=-b_0 P_\ell^1(\cos\theta),\quad
b_\phi(\theta)=b_0 P_{\ell+1}^1(\cos\theta).
\label{HelField}
\end{equation}
Analogously to BBKMRPS, we compute the complex linear polarization
at $r=R$ as
\begin{equation}
p\equiv Q+{\rm i} U=-\epsilon\,(b_\theta+{\rm i} b_\phi)^2,
\label{Emissivity}
\end{equation}
where $\epsilon$ is the emissivity, which is here assumed to be constant.
The minus sign in front of $\epsilon$ accounts for the fact that
polarization is related to the electric field, which is at right angles
to the magnetic field.
\subsection{Spin-weighted spherical harmonics}
Next, we decompose $p(\theta)$ into spin-weighted spherical harmonics
\citep{Kamion97,SZ97}.
The following expressions readily apply to the nonaxisymmetric case
where the complex polarization also depends on longitude $\phi$,
i.e., $p=p(\theta,\phi)$.
The spin-weighted spherical harmonics are computed as \citep{Goldberg67}
\begin{equation}
_s Y_{\ell m}(\theta,\phi)=_s\!{\cal N}_{\ell m}
\, _s{\cal P}_{\ell m}\big(\!\sin(\theta/2),\cos(\theta/2)\big)
\,e^{{\rm i} m\phi},
\end{equation}
where
\begin{equation}
_s {\cal N}_{\ell m}=(-1)^m
\sqrt{{2\ell+1\over4\pi}\,
{(\ell+m)!\over(\ell+s)!}{(\ell-m)!\over(\ell-s)!}}
\end{equation}
is a normalization factor,
\begin{equation}
_s{\cal P}_{\ell m}(x,y)=x^{2\ell}\,\sum_{r=0}^{\ell-s}
{_{rs}}{\cal M}_{\ell m} \, (y/x)^{2r+s-m}
\label{probl}
\end{equation}
are polynomials of $x$ and $y/x$, and
\begin{equation}
_{rs}{\cal M}_{\ell m}=
\pmatrix{\ell-s\cr r} \pmatrix{\ell+s\cr r+s-m}
(-1)^{\ell-r-s}
\end{equation}
is yet another normalization factor, where the binomials are defined to
be zero when either of the arguments or their difference is nonpositive.
In \Tab{Tab1}, we list a few selected spin-2 spherical harmonics.
\begin{table}[t!] \caption{
The first few spin-2 spherical harmonics.
}\centerline{\begin{tabular}{lcc}
\hline
\hline
$\ell$ & $m$ & $_2 Y_{\ell m}(\theta,\phi)$ \\
\hline
2 & 0 & $(3/4)\sqrt{5/6\pi} \sin^2\theta$ \\
2 & $\pm1$ & $-(1/4)\sqrt{5/\pi} \sin\theta(1\mp\cos\theta)e^{\pm{\rm i}\phi}$ \\
2 & $\pm2$ & $(1/8)\sqrt{5/\pi} (1\mp\cos\theta)^2e^{\pm2{\rm i}\phi}$ \\
3 & 0 & $(1/4)\sqrt{105/2\pi} \sin^2\theta\cos\theta$ \\
4 & 0 & $(15/4)\sqrt{9/10\pi} \sin^2\theta[1-(7/6)\sin^2\theta]$ \\
4 & $\pm3$ & $(1/4)\sqrt{63/2\pi} \sin\theta(1\mp\cos\theta)
[(1\mp\cos\theta)/2-\sin^2\theta]e^{\pm3{\rm i}\phi}$ \\
\label{Tab1}\end{tabular}}\end{table}
The numerical application of \Eq{probl} can become problematic
in the first (second) quadrant for $m>0$ ($m<0$) and $\theta\to0$
($\theta\to\pi$), because the sum has large terms of alternating sign.
This is not the case in the correspondingly other quadrant.
However, for the cases listed in \Tab{Tab1}, we observe that
$_2Y_{\ell m}(\theta,\phi)={_2Y}_{\ell\, -m}(\pi-\theta,\phi)$,
although this relation is not generally true.
\subsection{Spin-2 spherical harmonics decomposition}
\label{Spin2}
We now compute the spin-2 spherical harmonics representation of
$E+{\rm i} B$ in terms of $Q+{\rm i} U$ as \citep{Kamion97,SZ97,Dur08,KK16}
\begin{equation}
\tilde{R}_{\ell m}=\int_{4\pi}
(Q+{\rm i} U)\,_2 Y_{\ell m}^\ast(\theta,\phi)\,
\sin\theta\,{\rm d} {}\theta\,{\rm d} {}\phi,
\label{EBfromQU}
\end{equation}
and define $\tilde{E}_{\ell m}=(\tilde{R}_{\ell m}+\tilde{R}_{\ell,\,-m}^\ast)/2$
as the parity-even part and
$\tilde{B}_{\ell m}=(\tilde{R}_{\ell m}-\tilde{R}_{\ell,\,-m}^\ast)/2{\rm i}$
as the parity-odd part in spectral space, where the asterisk means complex
conjugation, and commas have been used to separate $\ell$ from $-m$.
In the axisymmetric case, we have $m=0$ and drop the index $m$.
Furthermore, $\tilde{E}_{\ell}$ and $\tilde{B}_{\ell}$ are then real.
It should also be noted that our coefficients $\tilde{E}_{\ell m}$ and $\tilde{B}_{\ell m}$ are sometimes defined with the opposite sign
\citep[see, e.g.][]{ZS97}.
Here we follow the sign convention of the textbook by \cite{Dur08}.
The spatial dependencies of $E(\theta,\phi)$ and $B(\theta,\phi)$ are given by
the real and imaginary parts of the inverse transform, $R$, i.e.,
\begin{equation}
E+{\rm i} B\equiv R=\sum_{\ell=2}^{N_\ell}\sum_{m=-\ell}^{\ell}
\tilde{R}_{\ell m} Y_{\ell m}(\theta,\phi).
\end{equation}
It turns out that for a magnetic field given by \Eq{HelField}, finite
values of $\tilde{E}_\ell$ are only obtained for even $\ell$ ($\ell\ge2$),
while finite values of $\tilde{B}_\ell$ are only obtained for odd $\ell$
($\ell\ge3$).
In \Fig{p2_4panels}, we show the $\theta$ dependence of the components
of the two surface components of $\mbox{\boldmath $b$} {}$, as well as the fields $(Q,U)$
and $(E,B)$ for several values of $\ell$.
In \Fig{p2_4panels}, we also show $a_\phi$, which is just
$b_\theta R/\ell$, where the $\ell$ factor comes from the $r$ derivative
in \Eq{bt_rder} and the fact that $\tilde{a}_\ell(r)\propto r^{-(\ell+1)}$.
We choose $\tilde{b}_\ell=-\ell\tilde{a}_\ell/R=b_0$.
In that case, positive contributions to the local magnetic helicity
density, $h(\theta)=2a_\phi b_\phi$ \citep{BDS02}, come from
$\pi/2\leq\theta\leq\pi$, i.e., from the southern hemisphere.
Negative contributions come from the northern hemisphere.
This corresponds to what is seen on the Sun for the small-scale field,
i.e., the field with $k>0.1\,{\rm Mm}^{-1}$.
We emphasize here that the corresponding scale,
$2\pi/0.1\,{\rm Mm}^{-1}\approx60\,{\rm Mm}$, is obviously not small by some
standards, but it is small relative to the large-scale field of the
Sun that manifests itself through the 11 yr cycle and the hemispheric
antisymmetry of the mean toroidal field.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{p2_4panels}
\end{center}\caption[]{
Latitudinal dependence of $a_\phi$ (dashed green), $b_\phi$ (blue), and
$b_\theta$ (red) (left column), $Q$ (blue) and $U$ (red) (middle column),
and $E$ (blue) and $B$ (red) (right column) for the one-dimensional model.
}\label{p2_4panels}\end{figure*}
To distinguish the spherical harmonic degrees of the magnetic field from
those of the $E$ and $B$ polarization, we denote the former with a prime
as $\ell'$.
In order to have negative (positive) contributions to the local magnetic
helicity density in the northern (southern) hemisphere, we now choose
analogously to \Eq{HelField},
\begin{equation}
\tilde{a}_{\ell}=-\delta_{\ell\,\ell'},\quad
\tilde{b}_{\ell}=\delta_{\ell\,\ell'+1},
\label{aellbell}
\end{equation}
for selected values of $\ell'$.
Thus, for $\ell'=1$, for example, we have $\tilde{a}_1=-1$ and
$\tilde{b}_2=1$ as the only two nonvanishing coefficients, so
$b_\theta=-P_1^1(\cos\theta)=\sin\theta$ and
$b_\phi=P_2^1(\cos\theta)=-3\sin\theta\cos\theta$.
\begin{table}[t!] \caption{\vspace{1mm}
Results for $\tilde{E}_\ell\tilde{B}_{\ell+1}$.
The maxima for each $\ell'$ are in bold.
}\centerline{\begin{tabular}{c|rrrrrrc}
\hline
\hline
\backslashbox{$\ell'$}{$\ell$} & $2$~~ & $4$~~ & $6$~~ & $8$~~& $10$~ & $12$~ & $h_{\rm rms}$ \\
\hline
1 &{\bf4.3}& 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.50 \\
2 &{\bf6.5}&$-3.1$& 0.0 & 0.0 & 0.0 & 0.0 & 1.57 \\
3 & 8.3 & 3.9 &$\!\!{\bf-13.6}$& 0.0 & 0.0 & 0.0 & 1.69 \\
4 & 10.1 & 5.3 & 2.1 &$\!\!{\bf-28.0}$& 0.0 & 0.0 & 1.82 \\
5 & 12.0 & 6.5 & 4.0 & 0.0 &$\!\!{\bf-46.4}$& 0.0 & 1.95 \\
6 & 13.9 & 7.5 & 5.0 & 2.9 &${\bf-2.5}$&$\!\!{\bf-65.7}$& 1.95 \\
\hline
\label{Tab2}\end{tabular}}\end{table}
\begin{table}[t!] \caption{\vspace{1mm}
As \Tab{Tab2}, but for $\tilde{E}_\ell\tilde{B}_{\ell-1}$.
}\centerline{\begin{tabular}{c|rrrrrrc}
\hline
\hline
\backslashbox{$\ell'$}{$\ell$} & $4$~~ & $6$~~ & $8$~~& $10$~ & $12$~ & $14$~ & $h_{\rm rms}$ \\
\hline
1 &~~{\bf22.6}& 0.0 & 0.0 & 0.0 & 0.0 & 0.0 &1.50 \\
2 &$ -2.1$&{\bf46.3}& 0.0 & 0.0 & 0.0 & 0.0 &1.50 \\
3 &~~ 3.8 & $-8.4$&{\bf77.9}& 0.0 & 0.0 & 0.0 &1.69 \\
4 &~~ 5.7 & 1.9 & $-16.6$&{\bf117.4}& 0.0 & 0.0 &1.82 \\
5 &~~ 7.2 & 4.1 & 0.0 &$ -26.6$&{\bf165.0}& 0.0 &1.95 \\
6 &~~ 8.6 & 5.4 & 2.8 &$ -2.1$&$ -38.6$&{\bf220.5}&2.08 \\
\hline
\label{Tab3}\end{tabular}}\end{table}
\begin{table}[t!] \caption{\vspace{1mm}
Similar to \Tabs{Tab2}{Tab3}, but now just for $\tilde{B}_{\ell}$.
}\centerline{\begin{tabular}{c|rrrrrrc}
\hline
\hline
\backslashbox{$\ell'$}{$\ell$} & $3$~~ & $5$~~ & $7$~~& $9$~ & $11$~ & $13$~ \\
\hline
1 &~~{\bf 5.9}& 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
2 &$ 5.9$&{\bf 8.9}& 0.0 & 0.0 & 0.0 & 0.0 \\
3 &~~ 7.1 & $ 7.3$&{\bf11.7}& 0.0 & 0.0 & 0.0 \\
4 &~~ 8.6 & 7.9 & $ 8.6$&{\bf 14.6}& 0.0 & 0.0 \\
5 &~~ 10.1 & 9.0 & 8.8 &$ 10.0$&{\bf 17.4}& 0.0 \\
6 &~~ 11.6 & 10.1 & 9.5 &$ 9.8$&$ 11.4$&{\bf 20.3}\\
\hline
\label{Tab4}\end{tabular}}\end{table}
In \Tabs{Tab2}{Tab3}, we list the two-scale polarization spectra
\begin{equation}
K_\ell^+=\tilde{E}_\ell\tilde{B}_{\ell+1}^\ast\quad\mbox{and}\quad
K_\ell^-=\tilde{E}_\ell\tilde{B}_{\ell-1}^\ast,
\label{Kdef1}
\end{equation}
respectively, for different values of $\ell'$.
We note here again that, because $m=0$, $\tilde{E}_\ell$ and
$\tilde{B}_{\ell\pm1}$ are real, so we can drop the asterisk.
In all cases, the integral of $h(\theta)$ over both hemispheres vanishes.
To get a sense of the strength of helicity, we therefore list in
\Tabs{Tab2}{Tab3} the rms value, $h_{\rm rms}$.
We see that $h_{\rm rms}$ increases only mildly with increasing values of $\ell'$.
By contrast, the extrema of $\tilde{E}_\ell\tilde{B}_{\ell+1}$ and
$\tilde{E}_\ell\tilde{B}_{\ell-1}$ increase much faster with $\ell$.
This suggests that the $\ell$-dependence of $K_\ell^-$ does not reflect
the actual $\ell$-dependence of magnetic helicity.
\Tabs{Tab2}{Tab3} also show that the maxima of both $K_\ell^+$ and
$K_\ell^-$ occur for $\ell=2(\ell'+1)$.
An exception is $\tilde{E}_\ell\tilde{B}_{\ell+1}$ for $\ell'=2$,
where the maximum still occurs at $\ell=2$.
It is important to note that the maximum of
$\tilde{E}_\ell\tilde{B}_{\ell-1}$ is much sharper in comparison to the
lower $\ell$ values than that of $\tilde{E}_\ell\tilde{B}_{\ell+1}$.
For this reason, we focus our analysis on the former quantity to
characterize the spectrum of magnetic helicity, because it serves as
the sharpest proxy of the magnetic helicity.
Also, the largest contribution to $\tilde{E}_\ell\tilde{B}_{\ell+1}$ has
the opposite sign for $\ell\geq6$.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pfaraday}
\end{center}\caption[]{
$E(\theta)\,B(\theta)$ for $b_{\rm F}=10$ (red) and $-10$ (blue)
for (a) $l'=1$ and (b) $l'=2$.
Note that for $l'=2$, $EB\neq0$ at the equator ($\theta=90\hbox{$^\circ$}$).
}\label{pfaraday}\end{figure*}
It is in principle also possible to use $\tilde{B}_{\ell,m}$ itself as
a proxy of magnetic helicity.
Its values are listed in \Tab{Tab4} for the same models as above.
We emphasize that $\tilde{B}_\ell$ has contributions only from odd values
of $\ell$.
This is because $B(\theta)$ has a dominant hemispheric $\ell=1$ variation.
By contrast, $\tilde{E}_\ell$ always has contributions for even values
of $\ell$.
This property of $\tilde{E}_\ell$ is also recovered if the field is nonhelical,
which is the case if the magnetic field is purely poloidal or purely toroidal.
On the other hand, if both are present at the same values of $\ell$,
one has helicity without hemispheric modulation.
In that case, $\tilde{B}_\ell$ has contributions only from even values
of $\ell$, while $\tilde{E}_\ell$ vanishes.
\subsection{Analogy with Faraday-rotated fields}
\cite{SF97} calculated the $B$ mode polarization of the cosmic microwave
background radiation in the presence of a uniform magnetic field and found
correlations between the temperature at spherical harmonic degree $\ell$
and the $B$ mode at degrees $\ell+1$ and $\ell-1$; see also \cite{SHM04}.
Such constructs are reminiscent of those in \Eq{Kdef1}.
In their case, the uniform magnetic field led to a superposition of
Faraday-rotated fields with different angles over the depth near the
last scattering surface.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pcombined_f1m1}
\end{center}\caption[]{
$\mbox{\boldmath $b$} {}(\phi,\mu)$ vectors compared with split representations
of $(Q,U)$ and $(E,B)$ for the four combinations $(f_0,g_0)=(1,0)$,
$(0,1)$, and $(1,\pm1)$ with $\ell=4$ and $m=3$.
Individual cross, ring, and swirl-like patterns are highlighted by squares,
along with their positions in the $(Q,U)$ and $(E,B)$ diagrams.
}\label{pcombined_f1m1}\end{figure*}
An analogy with Faraday rotation is indeed justified, because both
magnetic helicity and Faraday rotation lead to similar effects that,
in combination, can either enhance or diminish the resulting polarized
intensity \citep{Soko98,BS14,HF14}.
The presence of magnetic helicity leads either to a correlation or an
anticorrelation between the rotation measure and the total polarized
intensity \citep{VS10}, depending on whether one looks along or against
the direction of the uniform magnetic field.
This explains the analogy with the present case, where we have opposite
signs of magnetic helicity in the two hemispheres.
To demonstrate the effect of Faraday rotation in the present context,
we now include the radial magnetic field.
In fact, the poloidal field associated with the latitudinal component
$b_\theta=-b_0 P_{\ell'}^1(\cos\theta)$ of our earlier examples implies
\begin{equation}
b_r=(\ell'+1)\,b_0 P_{\ell'}(\cos\theta),
\end{equation}
where ${\rm d} {}[\sin\theta P_{\ell'}^1(\cos\theta)]/{\rm d} {}\cos\theta=
-\ell'(\ell'+1)P_{\ell'}(\cos\theta)$ has been used, and the
$\ell'+1$ factor follows from \Eq{aellbell} and the fact that
$-\ell\tilde{a}_\ell/R=b_0$.
We consider models with $\ell'=1$ and $2$.
Faraday rotation rotates the phase angle of the complex polarization,
so \Eq{Emissivity} has to be replaced by
\begin{equation}
p=-\epsilon\,(b_\theta+{\rm i} b_\phi)^2\,e^{2{\rm i} b_r/b_{\rm F}},
\end{equation}
where $b_{\rm F}=(k_{\rm F}n_{\rm e}\lambda^2 d)^{-1}$,
with $k_{\rm F}=2.6\times10^{-17}\,{\rm G}^{-1}$ being a constant
\citep[e.g.][]{ACD94}, $n_{\rm e}$ the mean electron density,
$\lambda$ the wavelength, and $d$ the geometrical depth.
For example, for $n_{\rm e}=10^{14}\,{\rm cm}^{-3}$,
$\lambda=600\,{\rm nm}$, and $d=100\,{\rm km}$, we have $b_{\rm F}\approx10\,{\rm kG}$.
Since the actual surface magnetic field is much weaker, Faraday rotation
will only be a small effect as far as the average field is concerned.
However, given that the effect is highly nonlinear, it is usually not
negligible in active regions and sunspots.
To assess the effects of Faraday rotation on the resulting $EB$
correlation, it is instructive to look at the latitudinal dependence
of the product $E(\theta)\,B(\theta)$ for two representative cases:
one where $\ell'$ is odd and one where it is even.
The result is shown in \Fig{pfaraday} for $\ell'=1$ and $2$, using
$b_0/b_{\rm F}=\pm0.1$ and comparing with the case without Faraday rotation.
For clarity, we only show the range $45\hbox{$^\circ$}\leq\theta\leq135\hbox{$^\circ$}$.
For the Sun, as alluded to above, the actual values of $b_0/b_{\rm F}$
will be much smaller and the Faraday rotation effect hardly noticeable
for the average field.
We see that for $\ell'=1$, Faraday rotation causes an enhancement
(reduction) of the helicity-induced $EB$ correlation if $b_{\rm F}$
is negative (positive); see \Figp{pfaraday}{a}.
This agrees qualitatively with the result of \cite{SF97}, because a
uniform magnetic field corresponds to an odd value $\ell'$.
For $\ell'=2$, on the other hand, we have a mixed hemispheric dependence
of $EB$ with finite values at the equator.
In the case of the Sun, of course, the large-scale magnetic field has
odd symmetry around the equator.
This also applies to the field within sunspots.
The difference between leading and following sunspots would weaken the
net effect, but not its systematic north--south dependence.
We can therefore conclude that Faraday rotation does not compromise the
ability to detect magnetic helicity from $EB$, provided the Faraday
effect remains subdominant compared with the helicity effect, i.e.,
$\lambda$ is small enough.
\section{Nonaxisymmetric examples}
\label{NonAxisymmetric}
\subsection{Two-dimensional patterns of $E$ and $B$}
We now consider two-dimensional examples in the $(\phi,\mu)$ plane,
where $\mu=\cos\theta$.
Analogous to earlier work, we consider the magnetic field
$(b_\phi,b_\mu)\equiv(b_\phi,-b_\theta)$ to be given by $\mbox{\boldmath $b$} {}=\mbox{\boldmath $F$} {}+\mbox{\boldmath $G$} {}$,
where
\begin{equation}
F_i=\nabla_i f\quad\mbox{and}\quad
G_i=\epsilon_{ij}\nabla_j g,
\label{FGformulation}
\end{equation}
using
\begin{equation}
f=-f_0 Y_{\ell m} \quad\mbox{and}\quad
g=g_0 Y_{\ell m}.
\end{equation}
The complex linear polarization is then computed as
$p=-(b_\theta+{\rm i} b_\phi)^2=(b_\phi-{\rm i} b_\theta)^2
=(b_\phi+{\rm i} b_\mu)^2$.
Following BBKMRPS, we consider four combinations, namely
$(f_0,g_0)=(1,0)$, $(0,1)$, and $(1,\pm1)$.
In \Fig{pcombined_f1m1}, we show the result for $\ell=4$ and $m=3$.
All quantities are plotted as a function of $\phi$ and $\mu=\cos\theta$.
This corresponds to the Lambert azimuthal equal-area projection.
We recover familiar structures corresponding to a star-like and
ring-like features for negative and positive $E$ polarizations
and swirly inward clockwise and counter-clockwise patterns
for negative and positive $B$ polarizations.
These structures agree with those in Figure~2 of BBKMRPS.
We recall, however, that we follow here the sign convention
of \cite{Dur08}, in which our \Eq{EBfromQU} becomes
$\tilde{R}(k_x,k_y)=-(\hat{k}_x-{\rm i}\hat{k}_y)^2\tilde{p}(k_x,k_y)$
in the Cartesian limit,
where $k_x$ and $k_y$ are the components of the two-dimensional
wavevector and hats indicate unit vectors.
Equation~(3) of BBKMRPS followed the sign convention of \cite{ZS97},
but their Figure~2 showed polarization vectors, which are at right
angles to the magnetic field vectors, giving therefore the same
orientation as the magnetic vectors in the Durrer convention shown in
our \Fig{pcombined_f1m1}.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pcombined_Dl1}
\end{center}\caption[]{
$\mbox{\boldmath $b$} {}(\phi,\mu)$ vectors compared with split representations of $(Q,U)$
and $(E,B)$, and a representation of the product $EB$, for the model of
\Sec{HemisphericHelicityModulation} with $\ell=4$ and $m=3$.
Note the opposite sense of swirl of eddies in the northern and southern
hemispheres, as highlighted by the squares.
}\label{pcombined_Dl1}\end{figure*}
\subsection{Formulation in terms of superpotentials}
Nonaxisymmetric magnetic fields can no longer be expressed in a form
analogous to \Eq{axisymb}, but we must instead employ the superpotentials
$S$ and $T$ in the form
\begin{equation}
\mbox{\boldmath $b$} {}=\mbox{\boldmath $\nabla$} {}\times\mbox{\boldmath $\nabla$} {}\times(\mbox{\boldmath $r$} {} S)+\mbox{\boldmath $\nabla$} {}\times(\mbox{\boldmath $r$} {} T).
\end{equation}
The first part corresponds to the poloidal field and the second to the
toroidal field.
The two superpotentials are expanded in terms of spherical harmonics, so
\begin{equation}
(S,T)(\theta,\phi)=\sum_{\ell=1}^{N_\ell}\sum_{m=-\ell}^{\ell}
(S_{\ell m},T_{\ell m})\,Y_{\ell m}(\theta,\phi),
\end{equation}
with the inverse transformation given by
\begin{equation}
(\tilde{S}_{\ell m},\tilde{T}_{\ell m})=\int_{4\pi}
(S,T)(\theta,\phi)\,Y_{\ell m}^\ast(\theta,\phi)\,
\sin\theta\,{\rm d} {}\theta\,{\rm d} {}\phi.
\end{equation}
As in \Sec{RepresentationAxisymmetric}, we assume that the radial
dependence of $\tilde{S}_{\ell m}(r)$ is proportional to $r^{-(\ell+1)}$.
This implies that
\begin{equation}
\frac{\partial}{\partial r}(r\tilde{S}_{\ell m})=
-\ell \tilde{S}_{\ell m} \quad\mbox{(for $r=R$)}.
\end{equation}
For chosen values of $\ell$ and $m$, we can then write
\begin{equation}
b_\theta(\theta,\phi)=\mbox{\rm Re}\left(-\ell\tilde{S}_{\ell m}\nabla_\theta Y_{\ell m}
+\tilde{T}_{\ell m}\nabla_\phi Y_{\ell m}\right),
\label{bthetaRep}
\end{equation}
\begin{equation}
b_\phi(\theta,\phi)=\mbox{\rm Re}\left(-\ell \tilde{S}_{\ell m}\nabla_\phi Y_{\ell m}
-\tilde{T}_{\ell m}\nabla_\theta Y_{\ell m}\right).
\label{bphiRep}
\end{equation}
Note in this connection that for axisymmetric models, $b_\theta$ and
$b_\phi$ are related to $Y_{\ell m}(\theta,\phi)$ via $\theta$ derivatives.
This shows that the reason for having expanded $a_\phi(\theta)$ and
$b_\phi(\theta)$ in \Eq{axisymbExpand} in terms of $P_\ell^1(\cos\theta)$
is that the $\theta$ derivative of the Legendre polynomials gives
${\rm d} {} P_\ell(\cos\theta)/{\rm d} {}\theta=P_\ell^1(\cos\theta)$.
Analogously to the axisymmetric case, we choose
$\tilde{T}_{\ell m}=-\ell\tilde{S}_{\ell m}/R=b_0R$.
The formulation given by \Eqs{bthetaRep}{bphiRep} agrees with that given
by \Eq{FGformulation}, provided we replace
\begin{equation}
f\to -\ell \tilde{S}_{\ell m} Y_{\ell m}(\theta,\phi),\quad
g\to \tilde{T}_{\ell m} Y_{\ell m}(\theta,\phi).
\end{equation}
This formulation suggests that the nonaxisymmetric generalization
of \Eq{HelField} is given by
\begin{equation}
f\to -\ell' \tilde{S}_{\ell' m'} Y_{\ell' m'} R,\quad
g\to \tilde{T}_{\ell'+1\, m'} Y_{\ell'+1\, m'},
\end{equation}
and that
\begin{equation}
H_{\ell'}^\pm=\sum_{m'=-\ell'}^{\ell'} 2\ell'(\ell'+1)
\tilde{S}_{\ell' m'} \tilde{T}_{\ell'\pm1\, m'}^\ast
\label{Hpmdef}
\end{equation}
can be used as a global two-scale measure of the magnetic helicity spectrum.
In the following, we use $H_{\ell'm'}^+$ to specify the amplitude of a
single mode; $H_{\ell'm'}^-$, by contrast, vanishes in our single-mode
examples by construction.
We also use $H_\ell^+$ for solar magnetograms.
\subsection{Hemispheric helicity modulation}
\label{HemisphericHelicityModulation}
In the examples considered above, either $E$ or $B$ was zero; see the
gray sub-panels in the split representation of \Fig{pcombined_f1m1}.
We now consider examples where both are nonvanishing.
Specifically, we reconstruct examples where
\begin{equation}
K_{\ell}^\pm\equiv\sum_{m=-\ell}^\ell
\tilde{E}_{\ell m}\tilde{B}_{\ell\pm1\, m}^\ast
\label{KellmpmDef}
\end{equation}
is nonvanishing.
As noted in the previous section, we do this by using fields where
\begin{equation}
-\ell' \tilde{S}_{\ell' m'}/R=\tilde{T}_{\ell'+1\, m'}=-b_0
\end{equation}
is a constant for fixed $\ell'$ and $m'$.
This is equivalent to our choice
$-\ell'\tilde{a}_{\ell'}/R=\tilde{b}_{\ell'+1}=b_0$ in \Sec{Spin2}.
Furthermore, the models of \Fig{p2_4panels} correspond to
$(f_0,g_0)=(1,-1)\times4\pi/\sqrt{(2\ell+1)(2\ell+3)}$.
The result is shown in \Fig{pcombined_Dl1}, again for $\ell'=4$ and $m'=3$.
We see that $E$ is always symmetric about the equator and $B$ is
antisymmetric about the equator.
The product $EB$ is therefore antisymmetric about the equator,
which reflects the opposite signs of magnetic helicity in the
two hemispheres.
The last panel of \Fig{pcombined_Dl1} shows that, although the product
$EB$ is mostly positive in the north and negative in the south, there
are also extended regions of opposite sign.
Quantitatively, we find that $2\bra{EB}/\bra{E^2+B^2}=\pm0.25$, where
the upper (lower) sign applies to the northern (southern) hemisphere.
\begin{table}[t!] \caption{\vspace{1mm}
Values of $\tilde{E}_{\ell m}$, $\tilde{B}_{\ell-1\,m}$, and
$\tilde{E}_{\ell m}\tilde{B}_{\ell-1\,m}$ for $\ell'=4$ and $m'=3$.
}\centerline{\begin{tabular}{c|ccccc}
$\ell$ & $2$ & $4$ & $6$ & $8$ & $10$ \\
\hline
$\tilde{E}_{\ell\;0}$ & $1.00$ & $-0.37$ & $-2.61$ & $2.43$ &$\!\!\!\!\!-0.63$ \\
$\tilde{B}_{\ell-1\;0}$ & $0$ & $-2.37$ & $-3.14$ & $ 3.25$ &$\!\!\!\!\!-0.82$ \\
$\tilde{E}_{\ell 0}\tilde{B}_{\ell-1\,0}^\ast$
& $0$ &$\,\;\;0.89$&$\,\;\;{\bf8.19}$& $ {\bf7.88}$ & $ 0.51$ \\
\hline
$\tilde{E}_{\ell\,6}$ & $ 0 $ & $ 0 $ &$\!\!0.33-0.52{\rm i}$&$-0.19-0.15{\rm i}$ & $1.94$ \\
$\tilde{B}_{\ell-1\;6}$ & $0$ & $ 0 $ & $ 0 $ & $1.67$ & $ 3.18$ \\
$\tilde{E}_{\ell 6}\tilde{B}_{\ell-1\,6}^\ast$
& $0$ & $ 0 $ & $ 0 $ & $-0.31-0.24{\rm i}$ & $ {\bf6.19}$ \\
\hline
$K_\ell^-$ & $0$& $0.89$ & $ 8.19$ & $7.26$ &$\!\!\!{\bf12.89}$ \\
$\sum|\tilde{E}_{\ell m}|^2$ &$1.00$& $0.14$ & $ 7.57$ & $6.02$ &$7.92$ \\
$\!\!\sum|\tilde{B}_{\ell-1\,m}|^2\!\!$&$ 0$& $5.62$ & $ 9.86$ &$\!\!\!\!\!\!16.1$&$\!\!\!\!20.90$ \\
$c_{\rm A}(\ell)$ & $0$& $0.31$ & $ 0.94$ & $0.66$ & $ 0.89$
\label{TabX}\end{tabular}}\end{table}
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pKm}
\end{center}\caption[]{
(a) $c_{\rm S}(\ell)$, (b) $c_{\rm A}^-(\ell)$, and
(c) $c_{\rm A}^+(\ell)$ for the full data set covering CRs~2161--2163
(broad solid lines), compared with the corresponding individual results
for CRs~2161 (red), 2162 (blue), and 2163 (green).
}\label{pKm}\end{figure*}
In \Tab{TabX}, we list all nonvanishing coefficients
$\tilde{E}_{\ell m}$ and $\tilde{B}_{\ell m}$ for
our example with $\ell'=4$ and $m'=3$.
For $m\neq0$, the only nonvanishing contributions come from $m=\pm6$.
Note also that $\tilde{E}_{\ell m}$ is now complex,
while all other coefficients are still real.
The dominant contributions to the parity-odd correlation
come from the product $\tilde{E}_{\ell m}\tilde{B}_{\ell-1\,m}$
with $\ell=2(\ell'-1)=6$ and $\ell=2\ell'=8$ for $m=0$, and
$\ell=2(\ell'+1)=10$ for $m=2m'=6$.
\section{Solar synoptic vector magnetograms}
\label{Applications}
\subsection{Spectra of global two-scale helicity proxies}
\label{GlobalProxies}
We now apply the global two-scale approach to the same solar synoptic
vector magnetograms that were studied by BPS using the semi-global
approach.
As alluded to in the introduction, we use ``$\pi$-ambiguated''
magnetic fields expressed in terms of pseudo-polarization data.
Thus, we only utilize the two horizontal components, $b_\theta$
and $b_\phi$, to compute the complex linear polarization
$p(\theta,\phi)=-(b_\theta+{\rm i} b_\phi)^2$.
The emissivity prefactor in \Eq{Emissivity} has been set to unity
because, in the following, we only work with normalized spectra.
We then compute $\tilde{E}_{\ell m}$ and $\tilde{B}_{\ell m}$
and study the spectra $K_\ell^\pm$; see \Eq{KellmpmDef}.
We normalize them analogously to those in BBKMRPS and write them as
\begin{equation}
c_{\rm A}^\pm(\ell)=\frac
{\sum_{m=-\ell}^{\ell} 2\tilde{E}_{\ell m}\tilde{B}_{\ell\pm1\, m}^\ast}
{\sum_{m=-\ell}^{\ell}\left(|\tilde{E}_{\ell m}|^2+|\tilde{B}_{\ell-1\, m}|^2\right)}.
\end{equation}
Because we sum over positive and negative $m$, the values of
$c_{\rm A}^\pm(\ell)$ are aways real.
They vary between $-1$ and $+1$.
We recall that, based on the comparison of \Tabs{Tab2}{Tab3} in
\Sec{Spin2}, we expect $c_{\rm A}^-(\ell)$ to be a better proxy of
magnetic helicity than $c_{\rm A}^+(\ell)$.
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pBm}
\end{center}\caption[]{
(a) $\tilde{E}_\ell/\tilde{E}_\ell^{\rm rms}$ (for even and odd $\ell$),
(b) $\tilde{B}_\ell^{\rm odd}/\tilde{B}_\ell^{\rm rms}$ (only for odd values
of $\ell$), and (c) $\tilde{B}_\ell^{\rm even}/\tilde{B}_\ell^{\rm rms}$
(for even values of $\ell$), for the full data set covering CRs~2161--2163
(broad solid lines), compared with the corresponding individual results
for CRs~2161 (red), 2162 (blue), and 2163 (green).
}\label{pBm}\end{figure*}
Following BBKMRPS, we also compute the normalized difference of the spectra
of $EE$ and $BB$ polarizations as
\begin{equation}
c_{\rm S}(\ell)=\frac
{\sum_{m=-\ell}^\ell\left(|\tilde{E}_{\ell m}|^2-|\tilde{B}_{\ell m}|^2\right)}
{\sum_{m=-\ell}^\ell\left(|\tilde{E}_{\ell m}|^2+|\tilde{B}_{\ell m}|^2\right)}.
\end{equation}
This quantity varies between $-1$ and $+1$.
It vanishes when the $EE$ and $BB$ polarizations have the same amplitude,
and it is $1/3$ if the amplitude of the $EE$ polarization is twice
that of the $BB$ polarization, as was found in recent dust foreground
measurements of the interstellar medium \citep{Adam16,Akrami18}.
To facilitate comparison with earlier work, we define
\begin{equation}
L^2=\ell(\ell+1),
\label{L2def}
\end{equation}
and plot $c_{\rm S}$ and $c_{\rm A}^\pm$ also versus
the approximate wavenumber $k=L/R$.
As in BPS, we use the combined synoptic vector magnetograms
of three CRs, 2161, 2162, and 2163.
They are based on the full-disk vector magnetograms obtained
from the Helioseismic and Magnetic Imager on board the {\em
Solar Dynamics Observatory} and have been processed by Yang
Liu\footnote{\url{http://hmi.stanford.edu/hminuggets/?p=1689}}
(Stanford).
In \Fig{pKm}, we show $c_{\rm S}(\ell)$ and $c_{\rm A}^\pm(\ell)$ both
for the full data set of all three CRs and also separately for CRs~2161,
2162, and 2163.
For the full data set, the total azimuthal angle is $6\pi$, and the
integration in \Eq{EBfromQU} is carried out over $12\pi$ instead of $4\pi$.
Similar to our earlier semi-global analysis, $c_{\rm S}$ shows large
variations, but is mostly positive for $\ell\leq10$, corresponding to
the wavenumber $k=L/R\leq0.014\,{\rm Mm}^{-1}$.
Furthermore, $c_{\rm A}^-$ shows negative values for similar $\ell$,
while $c_{\rm A}^+$ has the opposite sign, which is in agreement
with our expectations based on the comparison of \Tabs{Tab2}{Tab3}.
For larger $\ell$, both $c_{\rm A}^+$ and $c_{\rm A}^-$ are again very
noisy, although $c_{\rm A}^-$ may be mostly positive, while $c_{\rm A}^+$
may be mostly negative.
To have an estimate of the uncertainty of our results, we also plot the
spectra separately for each of the three CRs.
These results are broadly consistent with those of the full data set.
The tendency of obtaining positive values of $c_{\rm S}$ at $\ell<10$
is also seen individually for all three CRs.
By contrast, the tendency of obtaining negative values of $c_{\rm A}^-$ for
$\ell<10$ is seen for CRs~2161 and 2162, but not for CR~2163 at $\ell=4$.
However, for $\ell=6$, all three data sets give the same (negative) sign
of $c_{\rm A}^-$.
As noted before, $\tilde{B}_\ell$ can itself be used as a helicity proxy,
so we now determine it for the same three CRs.
For completeness, we also analyze $\tilde{E}_\ell$ in a similar fashion.
Owing to nonaxisymmetry, we have contributions from different values
of $m$.
It is then useful to define
\begin{equation}
\tilde{B}_\ell=\sum_{m=-\ell}^{\ell}\tilde{B}_{\ell\,m},\quad
\tilde{B}_\ell^{(2)}=\sum_{m=-\ell}^{\ell}|\tilde{B}_{\ell\,m}|^2.
\end{equation}
In the following, we plot $\tilde{B}_\ell$ and the ratio
$\tilde{B}_\ell/\tilde{B}_\ell^{\rm rms}$, where
$\tilde{B}_\ell^{\rm rms}=[\tilde{B}_\ell^{(2)}]^{1/2}$ is the
rms value.
We define $\tilde{E}_\ell$ and the ratio
$\tilde{E}_\ell/\tilde{E}_\ell^{\rm rms}$ analogously.
For $\tilde{B}_\ell$, we only expect to see a hemispheric modulation
for odd values of $\ell$.
Therefore, to distinguish the contributions from odd and even
values of $\ell$, we denote them as $\tilde{B}_\ell^{\rm odd}$
and $\tilde{B}_\ell^{\rm even}$.
The results are shown in \Fig{pBm} as a function of $\ell$.
We see that both $\tilde{E}_\ell$ and $\tilde{B}_\ell^{\rm odd}$ are
negative for small values of $\ell$, while $\tilde{B}_\ell^{\rm even}$
is positive.
The fact that $\tilde{E}_\ell$ is mostly negative for $\ell<5$ suggests
that, on large length scales, the magnetic field structures are mostly
star-like, but in the range $5<\ell<10$, they are mostly ring-like.
However, no direct visual evidence of this has been reported as yet.
For the $B$ polarization, on the other hand, the negative values
for odd $\ell$, i.e., for $\tilde{B}_\ell^{\rm odd}$, may
reflect a positive magnetic helicity on large length scales; see
\Sec{HemisphericHelicityModulation}.
This agrees with the negative sign found for $c_{\rm A}^-$.
Moreover, as seen in \Tab{Tab2}, $K_\ell^+$ tends to have the opposite
sign.
This agrees with what is found for $c_{\rm A}^+$ in \Figp{pKm}{c}.
\subsection{Spectra of the global two-scale magnetic helicity}
\label{GlobalHelicity}
Finally, we consider $H_\ell^\pm$.
We normalize it by the solar radius $R$, which is set to unity in our
work, so we plot here the ratio $H_\ell^\pm/R$, which has units of $\,{\rm G}^2$;
see \Eq{Hpmdef} for the definition.
To obtain $\tilde{S}_{\ell m}$, we use the observed radial magnetic field
component, $b_r$, compute the spherical harmonics decomposition to find
$\tilde{b}_{r,\ell m}$ and thus $\tilde{S}_{\ell m}=\tilde{b}_{r,\ell m}/L^2$;
see \Eq{L2def}.
Analogously, we compute $\tilde{T}_{\ell m}$ from the radial component
of the current density, $j_r$.
For the vector magnetograms, the components of the magnetic field are
given in uniform intervals of $\mu=\cos\theta$.
We therefore write the radial component of $\mbox{\boldmath $j$} {}=\mbox{\boldmath $\nabla$} {}\times\mbox{\boldmath $b$} {}$ as
\begin{equation}
j_r=\cot\theta\,b_\phi-\sin\theta\,{\partial b_\phi\over\partial\mu}
-{1\over\sin\theta}\,{\partial b_\theta\over\partial\phi}.
\end{equation}
We then compute the spherical harmonics decomposition
to find $\tilde{j}_{r,\ell m}$ and thus compute
$\tilde{T}_{\ell m}=\tilde{j}_{r,\ell m}/L^2$.
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pHp}
\end{center}\caption[]{
$H_\ell^\pm/R$ versus $\ell$ for the full data set covering CRs~2161--2163
(broad solid lines), compared with the corresponding individual results
for CRs~2161 (red), 2162 (blue), and 2163 (green).
The solid (dashed) lines give the results for odd (even) values of $\ell$.
}\label{pHp}\end{figure}
In \Fig{pHp} we plot $H_\ell^\pm/R$ versus $\ell$.
We see that $H_\ell^+$ and $H_\ell^-$ are negative for most values
of $\ell$.
Thus, there is no clear evidence for a positive magnetic helicity at large
length scales.
This is surprising in view of the previous findings based on the $B$
polarization that did suggest positive magnetic helicity on large
length scales.
Of course, previous work has long shown negative magnetic helicity in the
northern hemisphere and positive in the southern \citep{See90,PCM95},
including the work of BPS.
It may therefore indeed be true that there is no sign change in
$H_\ell^\pm$ at the photosphere, and that the sign change in the helicity
proxies may reflect physical properties of the field at some layer above
the photosphere.
However, it could also be an effect of the phase within the solar cycle,
as suggested by \cite{Singh18}, which could then also explain the evidence
for a bihelical field found by \cite{PP14}.
Systematic cycle related magnetic helicity variations are indeed well
documented \citep{KKMRSZ03,ZSPGXSK10,PPLK19}.
It also is surprising that $H_\ell^\pm$ is, for all CRs, consistently
much larger at $\ell=1$ than for any other value of $\ell$.
In BPS, by contrast, we found a rapid decline of power for $0.01\,{\rm Mm}^{-1}$;
see Figure~8 therein, but that work was based on a semi-global approach
which is unable to recover the low $k$ values correctly.
Conversely, it is possible that the global approach overemphasizes the
polar fields.
This may be a concern mainly for the $E$ and $B$ polarization.
Indeed, looking at \Fig{p2_4panels}, we see that the clearest hemispheric
dependence in $B$ is seen at the poles, while at lower latitudes, $E$
and $B$ have no definite correlation.
This may well be a general problem with the $EB$ approach that
should be clarified studying the signs of $E$ and $B$ locally.
It would be important to assess the statistical robustness of these
results by inspecting the magnetic helicity signatures for many more CRs.
\section{Implications for dynamo theory}
\label{Implications}
The $\alpha$ effect in dynamo theory is the main candidate for explaining
the production of large-scale magnetic fields in the Sun.
One of its signatures is the production of magnetic helicity of opposite
signs.
Such a magnetic field is called bihelical.
Figures~\ref{pKm} and \ref{pBm} present some support for this
assertion, in addition to the earlier results of \cite{PP14} and
\cite{Singh18}.
Our inspection of $H_\ell^+$ and $H_\ell^-$ does not support this,
however.
Whether this is indeed related to potential problems regarding the
$\pi$ ambiguity is, however, unclear, and one would like to see more
evidence before continuing to speculate further on this.
There is, however, the possibly that it might be related to the
anticipated sign reversal of magnetic helicity some small distance above
the photosphere.
We elaborate on this possibility next.
To put the various findings into a broader perspective, it is important to
realize that in the solar wind, far away from the solar dynamo, evidence
for a bihelical magnetic field has also been presented \citep{BSBG11}.
However, the sign of magnetic helicity is at all wavenumbers opposite
to what it is at the solar surface.
This was then found to be a generic phenomenon of any system consisting
of a dynamo region adjacent to a nondynamo region; see the work of
\cite{WBM11,WBM12} of a turbulent dynamo simulation with a simple
stellar corona, and the earlier work of \cite{BCC09} in the context of
galactic halos.
We do not know exactly where the sign would swap.
It has been suggested that it could occur in the lower corona,
where the plasma beta crosses unity \citep{BSB18}.
This could be detectable by measuring in situ polarized emission from
within the corona \citep{BAJ17}.
On the other hand, if it happened in the chromosphere, in layers
accessible to a direct face-on measurement of the $EB$ cross-correlation,
this sign change might be detectable using the method discussed in the
present paper.
A major difficulty in detecting an overall sign change of handedness
through the $EB$ cross-correlation lies in the fact that the $E$
polarization is strongly associated with the magnetic field topology.
This particular property could be characterized, for example, by its
correlation with temperature $T$ (related to the intensity or Stokes $I$).
This is a parity-even correlation, which can have either sign, and it
may be this quantity, in addition to $EB$, that also shows a systematic
variation with height.
Not much is known about this, except that in the dust polarization of
the Galactic foreground, the $ET$ correlation is known to be positive
\citep{Akrami18}.
We also know that the $E$ polarization is highly skewed and its skewness
depends systematically on the physics governing the magnetic field.
Ambipolar diffusion, for example, is known to affect the skewness of
$E$ in a systematic way \citep[see Figure~13 of][]{Bra19AD}.
This is also reflected in the fact that the $EE$ correlation can be
different from the $BB$ correlation, i.e., $c_{\rm S}\neq0$, as has been
found in the present work; see \Figp{pKm}{a}.
Addressing these new questions raised above is of direct relevance to
assessing the possibility of a radial sign reversal of the magnetic
helicity, as predicted by dynamo theory and as has been found from
magnetic helicity measurements in the solar wind.
\section{Conclusions}
\label{Concl}
This work has addressed two critical issues in the calculation of a
proxy of solar magnetic helicity spectra: the $\pi$ ambiguity and the
systematic north--south sign change of magnetic helicity.
The problem of the $\pi$ ambiguity has been addressed previously
(BBKMRPS) by calculating the $EB$ cross--correlation from local
Cartesian patches.
This quantity was shown to be a proxy of magnetic helicity under
inhomogeneous conditions, in particular for rotating stratified
convection.
The problem of the systematic north--south variation has also been
addressed previously, but only in a semi-global fashion; see BPS.
Here, we have generalized this approach to a fully global one by first
calculating the parity-even and parity-odd $E$ and $B$ polarizations
globally using spin-2 spherical harmonics, and then correlating them at
spherical harmonic degrees that are shifted by one relative to the other.
This approach is analogous to what was done in the semi-global Cartesian
approach of BPS.
However, unlike their formalism, the present one is heuristic and
has not been derived rigorously from a correlation function that depends
on mean and relative coordinates; see \cite{RS75}.
It is not entirely obvious that this is even possible but, if it is,
the result may well look similar to what has been proposed here.
Through the examples constructed here, we have demonstrated that the correlation
$\tilde{E}_{\ell m}\tilde{B}_{\ell-1\, m}^\ast$ can act as a proxy
of the magnetic helicity, which itself is characterized globally by the product
$\tilde{S}_{\ell m}\tilde{T}_{\ell+1\, m}^\ast$.
In the quest for finding clear evidence of an opposite sign of magnetic
helicity at large length scales, one has to tackle the problem of the
$\pi$ ambiguity in the weak-field regions that occupy the majority of
the solar surface.
A standard approach to $\pi$ disambiguation in those regions is the
random disambiguation, which is problematic and may have been responsible
for the relatively low spectral power at wavenumbers around and below
$0.03\,{\rm Mm}^{-1}$ \citep{Singh18} and also for what looked like a random
sign in the resulting magnetic helicity at those wavenumbers.
In fact, the present results now suggest that there is maximum power
at the very smallest wavenumbers around and below $0.01\,{\rm Mm}^{-1}$.
Our results show that, in the northern hemisphere, where the small-scale
magnetic helicity is negative, $\tilde{E}_{\ell m}\tilde{B}_{\ell-1\, m}^\ast$
is positive.
Likewise, the large-scale field is expected to have positive magnetic
helicity in the northern hemisphere and
$\tilde{E}_{\ell m}\tilde{B}_{\ell-1\,m}^\ast$ is now found to be negative.
Thus, our proxy has the opposite sign to the magnetic helicity.
This agrees with what was found based on the numerical
simulations of BBKMRPS.
This result is not, however, based on the actual helicity $H_\ell^\pm$,
but rather on the helicity proxy.
As mentioned in \Sec{GlobalHelicity}, there could be a general difficulty
with the $EB$ approach in that its highest sensitivity is at the poles.
At lower latitudes, the method suffers a significant amount of
cancellation, as can be anticipated from \Fig{p2_4panels} for $\ell=4$.
Regarding the absence of a clear $EB$ signal in the analysis of solar
$Q$ and $U$ polarization in the work of BBKMRPS, it should be noted
that their results are much more noisy, although in hindsight not so
dissimilar from the present ones.
Tentatively, they found values at small and large length scales that
agree with those here: positive $c_{\rm S}(k)$ at $k=0.01\,{\rm Mm}^{-1}$
along with $c_{\rm A}(k)$ at similar values of $k$.
However, the main reason for their noisy result lies probably in the
fact that their linear polarization data were too contaminated by other
factors, as was already discussed in BBKMRPS.
The present approach of computing the $EB$ signal from the magnetic
field rather than the observed polarization combines the best aspects
of two worlds.
It uses the elaborate inversion technique of spectropolarimetry to obtain
the magnetic field, but is insensitive to the problems associated with
the $\pi$ ambiguity.
What is perhaps unsatisfactory, however, is the fact that the
line-of-sight magnetic field ($b_\|$) or the circular polarization are
not used in the present approach.
No corresponding idea has yet been proposed that would combine these
two pieces of information.
Simply correlating $b_\|$ with $E$ or $B$ may not yield anything useful
because in simple patterns such as those of \Fig{pcombined_f1m1}, the
wavelength of $b_\|$ is always twice that of $E$ or $B$, so it would
lead to a cancellation.
This is because $E$ and $B$ are related to the square of the magnetic
field.
Therefore, the spatial wavelengths of the $E$ and $B$ patterns would
agree with that of $b_\|^2$, but then the potentially useful information
implied by the sign of $b_\|$ is lost.
So, it is not obvious what to do with $b_\|$ in this context.
In this connection, it is useful to remind ourselves that, away from disk
center, $b_\|$ does begin to contribute more strongly to the determination
of $b_\theta$ and $b_\phi$.
One should therefore calculate the complex polarization not from
$b_\theta$ and $b_\phi$, but from the two components of the field
vector $\mbox{\boldmath $b$} {}_\perp$ that is perpendicular to the line of sight.
This would obviously be another next important step to take.
Likewise, it would be highly valuable to inspect the spatial properties
of $E$ and $B$ in much more detail.
This would allow us to study the connection between the sign of $E$
and the topology or structures, and of course between the sign of $B$
and the hemispheric position.
One of the other potential applications of the $EB$ transformation
lies in its potential benefit when regularizing the observed linear
polarization signal.
One could imagine that, instead of applying a random disambiguation for
weak field strengths, one could adopt some type of image reconstruction
in $EB$ space instead of working in $QU$ or $\mbox{\boldmath $b$} {}$ space.
This has not yet been explored and would be a useful target for future
research.
Finally, one may wonder whether the global two-scale helicity proxy
introduced here can be used beyond solar physics.
The answer is probably yes, if one thinks about the technique
of Zeeman Doppler imaging of stellar magnetic fields \cite[see,
e.g.,][]{Donati97,Carroll12,Rosen15}.
Likewise, the magnetic field of our own Galaxy may also be subject to such
an analysis \citep{JF12}.
We therefore expect that these points provide exciting opportunities
for future work.
\acknowledgments
This work was carried out in large parts at the Aspen Center for Physics,
which is supported by National Science Foundation grant PHY-1607611.
I thank Evan Scannapieco for organizing the Aspen program on
the Turbulent Life of Cosmic Baryons.
I also thank Gherardo Valori and Etienne Pariat for organizing the Magnetic
Helicity in Astrophysical Plasmas Team at the International Space Science
Institute in Bern, where some early elements of this work were conceived,
and I also thank Maarit K\"apyl\"a, Alexei Pevtsov, Ilpo Virtanen,
and Nobumitsu Yokoi for providing a splendid atmosphere at the
Nordita program on Solar Helicities in Theory and Observations.
I am grateful to Patrik Sanila for help with the spin-weighted
spherical harmonics, and to Ameya Prabhu for showing me some of his
preliminary results with $\pi$-ambiguated magnetic fields using local
patches around specific active regions.
I also thank Marc Kamionkowski and Kandaswamy Subramanian for useful
discussions on the subject, and an anonymous referee for suggesting
improvements to the paper.
This research was supported in part by the Astronomy and Astrophysics
Grants Program of the National Science Foundation (grant 1615100),
and the University of Colorado through
its support of the George Ellery Hale visiting faculty appointment.
I acknowledge the allocation of computing resources provided by the
Swedish National Allocations Committee at the Center for Parallel
Computers at the Royal Institute of Technology in Stockholm.
|
1,314,259,992,993 | arxiv | \section{Summary}
\IEEEPARstart{T}{he} error floor in modern graph-based error control
codes such as low-density parity-check codes is caused by inherent
structural weaknesses in the code's interconnect network. The
iterative message passing algorithm cannot overcome these weaknesses
and gets trapped in error patterns which are easily identifiable as
erroneous (in LDPC codes), and are thus not valid codewords, but
difficult to overcome or correct \cite{perez08, perez09}. These weaknesses were termed {\em
trapping sets} by Richardson in \cite{Rich04}, a summary definition
for the patterns on which the message passing algorithm fails for
Gaussian channels. These trapping sets are dependent on the code,
the channel used, and to a lesser degree also on the details of the
decoding algorithm. Prior work in identifying the weaknesses of LDPC
codes on erasure channels led to the definition of {\em stopping
sets} in \cite{ChanProiTelaRichUrba02}. Stopping sets, being the
weaknesses of LDPC codes on erasure channels, also play a role on
Gaussian channels, but are not typically the dominant error
mechanisms. In \cite{Zhanetal06} the authors define {\em absorption
sets}, which are the subgraphs of the code graph on which the
Gallager bit-flipping decoding algorithms fail for binary symmetric
channels. The authors observed that these absorption sets also show
up as the dominant trapping sets in certain structured LDPC codes.
In \cite{Zhanetal08} they devise post-processing methods to reduce
the effects of these absorption sets and lower the error floor of
the codes in question.
In this paper we present a linear algebraic approach to the dynamic
behavior of absorption sets. We show that these sets follow a
geometric growth phase during early iterations where messages inside
the absorption set grow towards a largest eigenvector which characterizes
the absorption set. The seemingly erratic behavior
of the messages at early iterations is due to the decreasing
influence of lesser eigenvectors. We define the {\em gain} of an
absorption set and show how it affects the influence of the
extrinsic messages that flow into the absorption set at each
iteration from the remainder of the code network. The importance of
set extrinsic information was already informally observed in
\cite{YanRyaLi04}, who reported a lowering of the error floor with
increased extrinsic connectivity. We use our analysis to produce
accurate error formulas for the error floor BER/FER and support
these results with importance sampling simulations targeting the
absorption sets.
As illustration we carefully identify and classify absorption sets
of the regular $(2048,1723)$ LDPC code recently designed in
\cite{DjuXuAbdLin03}, which is used in the IEEE 802.3an
standard.
Topological features of dominant absorption sets are identified and a search algorithm
is presented which finds the leading dominant sets.
\section{Background}
\IEEEpubidadjcol
Stopping sets completely determine the performance of graph-based
decoding of LDPC codes on erasure channels, i.e., on channels where
the transmitted binary symbols are either received correctly, or are
erased. A complete statistical treatment of stopping sets was given
in \cite{ChanProiTelaRichUrba02}. Aptly named, a stopping set is a
subset of uncorrected variable nodes where the decoder stops, i.e.,
makes no further correction progress. It is simply
defined as:
\begin{defn}
A stopping set $\cal S$ is a set of variable nodes, all of whose
neighboring check nodes are connected to the set $\cal S$ at least twice.
\end{defn}
\figurename \ref{fig:stopping} shows an example of a stopping set.
It is quite straightforward to see that if erasure decoding is
performed following Gallager's decoding algorithm \cite{SchPer04} the
variable values in the stopping set cannot be reconstructed. Valid
codewords are trivially stopping sets, but the set of stopping sets
is larger than the set of valid codewords.
\begin{figure}[!t]
\centering
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\Circle\ar@{.}[drrrrr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[drrr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&&\Circle\ar@{.}[dr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dl]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dl]\ar@{.}[drrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlll]\ar@{.}[drrrrrrr]\ar@{.}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllll]\ar@{.}[drrrrrrr]\ar@{.}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllllll]\ar@{.}[drrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrr]&& \CIRCLE\ar@{-}[dlllllll]\ar@{-}[dl]\ar@{-}[drrrrrrrrrrrrrrr]&& \CIRCLE\ar@{-}[dlllllllll]\ar@{-}[dl]\ar@{-}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllllllllll]\ar@{.}[dr]\ar@{.}[drrrrrrr]&& \Circle\ar@{.}[dlllllllllllll]\ar@{.}[dr]\ar@{.}[drrr]&& \CIRCLE\ar@{-}[dlllllllllllll]\ar@{-}[dlllllllll]\ar@{-}[drrrrr]&& \Circle\ar@{.}[dlllllllllllllll]\ar@{.}[dlllllll]\ar@{.}[drrrrr]&& \CIRCLE\ar@{-}[dlllllllllllllllll]\ar@{-}[dlllllll]\ar@{-}[drrrrr]&& \Circle\ar@{.}[dlllllllllllllllllll]\ar@{.}[dlllllll]\ar@{.}[dlll]&& \CIRCLE\ar@{-}[dlllllllllllllllllll]\ar@{-}[dlllllllllllllll]\ar@{-}[dl]&& \Circle\ar@{.}[dlllllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dlllllllll]&& \CIRCLE\ar@{-}[dlllllllllllllllllllllll]\ar@{-}[dlllllllllllllll]\ar@{-}[dlllllll]&& \Circle\ar@{.}[dlllllllllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dlllll]\\
&\hspace{3.5mm}& &\hspace{3.5mm}& &\boxplus& & \boxplus & & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \boxplus& & \colorbox{gray}{$\boxplus$}& & \boxplus& & \boxplus& & \boxplus& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& &\hspace{3.5mm}& &\hspace{3.5mm}& \\
}$$} \put(40,40){\footnotesize Black: Stopping Set} \put(47,-2){\footnotesize Neighbors}
\end{picture}
\caption{Example of a stopping set.} \label{fig:stopping}
\end{figure}
An absorption set is an extension of the notion of a stopping set to
the binary-symmetric channels \cite{Zhanetal06,Zhanetal08}, and is
defined as:
\begin{defn}\label{def1}
An absorption set $\cal A$ is a set of variable nodes, such that the
majority of each variable node's neighbors are connected to the set $\cal A$
an even number of times.
\end{defn}
\figurename \ref{fig:absorption} shows an example absorption set. It
can be verified that Gallager-type bit flipping decoding will not be
able to correct an absorption set, since a majority of messages
impinging on each variable node will retain the erroneous sign for
each iteration. Consequently, the algorithm locks up.
\begin{figure}[!t]
\centering
\setlength{\unitlength}{0.8mm}
\begin{picture}(130,44)
\setlength{\fboxsep}{0mm} \setlength{\fboxrule}{0mm}
\put(0,37){$$\xymatrix@M=0pt@W=0pt@R=70pt@C=-1.7pt {
\Circle\ar@{.}[drrrrr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[drrr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&&\Circle\ar@{.}[dr]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dl]\ar@{.}[drrrrrrrrrrrrrrr]\ar@{.}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \CIRCLE\ar@{-}[dl]\ar@{-}[drrrrrrr]\ar@{-}[drrrrrrrrrrrrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlll]\ar@{.}[drrrrrrr]\ar@{.}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllll]\ar@{.}[drrrrrrr]\ar@{.}[drrrrrrrrrrrrrrr]&& \CIRCLE\ar@{-}[dlllllll]\ar@{-}[drrrrrrrrr]\ar@{-}[drrrrrrrrrrrrrrr]&& \CIRCLE\ar@{-}[dlllllll]\ar@{-}[dl]\ar@{-}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllllllll]\ar@{.}[dl]\ar@{.}[drrrrrrrrrrrrrrr]&& \Circle\ar@{.}[dlllllllllll]\ar@{.}[dr]\ar@{.}[drrrrrrr]&& \CIRCLE\ar@{-}[dlllllllllllll]\ar@{-}[dr]\ar@{-}[drrr]&& \Circle\ar@{.}[dlllllllllllll]\ar@{.}[dlllllllll]\ar@{.}[drrrrr]&& \Circle\ar@{.}[dlllllllllllllll]\ar@{.}[dlllllll]\ar@{.}[drrrrr]&& \Circle\ar@{.}[dlllllllllllllllll]\ar@{.}[dlllllll]\ar@{.}[drrrrr]&& \Circle\ar@{.}[dlllllllllllllllllll]\ar@{.}[dlllllll]\ar@{.}[dlll]&& \Circle\ar@{.}[dlllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dl]&& \Circle\ar@{.}[dlllllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dlllllllll]&& \Circle\ar@{.}[dlllllllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dlllllll]&& \Circle\ar@{.}[dlllllllllllllllllllllllll]\ar@{.}[dlllllllllllllll]\ar@{.}[dlllll]\\
&\hspace{3.5mm}& &\hspace{3.5mm}& &\boxplus& &\colorbox{gray}{$\boxplus$} & & \colorbox{gray}{$\boxplus$}& & \boxplus& & \boxplus& & \colorbox{gray}{$\boxplus$}& & \boxplus& & \boxplus& & \boxplus& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \boxplus& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& & \colorbox{gray}{$\boxplus$}& &\hspace{3.5mm}& &\hspace{3.5mm}& \\
}$$} \put(40,40){\footnotesize Black: Absorption Set} \put(47,-2){\footnotesize Neighbors}
\end{picture}
\caption{Example of an absorption set.} \label{fig:absorption}
\end{figure}
\section{LDPC Codes on the Gaussian Channel}
The Gaussian channel is different from the binary symmetric and
binary erasure channels and causes a more complicated error behavior
on LDPCs. Richardson \cite{Rich04} first seriously explored the
error floor of LDPCs on Gaussian channels and defined {\em trapping
sets} as the failure mechanism. Noting that typically very few
trapping sets dominate the error floor region he proposed a
semi-analytical method which amounts to a variant of importance
sampling to numerically predict the error floor from the knowledge
of a code's trapping sets.
While finding trapping sets remained a largely open problem,
\cite{Zhanetal06} observed that in certain structured LDPCs the
dominant trapping sets are absorption sets, i.e., the failure
mechanism of the code on binary symmetric channels. In
\cite{Zhanetal08}, algorithmic modifications were proposed to
``eliminate'' the error floor caused by these absorption sets.
Due to its popularity and extensive exposure we will
concentrate on the $(2048,1723)$ regular LDPC code
\cite{DjuXuAbdLin03} used in the IEEE 802.3an standard. This code
has been extensively analyzed. It has a low error floor that appears
at $E_b/N_0\approx 5$dB at a BER of $10^{-12}$, that is too low to
be efficiently explored using conventional simulations\footnote{Even
an FPGA-based simulation running at 100Mb/s requires about a week
for a single data point.}.
\figurename \ref{fig:trapping} shows the structure of the {\em
dominant} absorption set of this code (see also \cite[\figurename
2]{Zhanetal08}). There are $14,272$ such sets in the $(2048,1723)$
code of \cite{DjuXuAbdLin03}. They dominate the error floor since
they are the minimal absorption sets in this code (for definition of
minimal and dominant, see Definition \ref{defn4}).
\begin{figure}[!t]
\centering
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&&&&& \CIRCLE\ar@{-}[ddlllll]\ar@{-}[ddlll]\ar@{-}[ddl]\ar@{-}[ddr]\ar@{-}[ddrrr]&&&& \CIRCLE\ar@{-}[ddr]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlllllllllllll]\ar@{-}[ddlll]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlllllllllllllll]\ar@{-}[ddlllll]\ar@{-}[ddrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlll]\ar@{-}[ddlllllll]\ar@{-}[ddlllllllllllllllll]\ar@{-}[ddrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrr]&&&& \CIRCLE\ar@{-}[ddrrrrrrrrrrrrr]\ar@{-}[ddrrr]\ar@{-}[ddlllll]\ar@{-}[ddlllllllllllllllllll]\ar@{-}[ddlllllllll]&&&& \CIRCLE\ar@{-}[ddl]\ar@{-}[ddlll]\ar@{-}[ddlllll]\ar@{-}[ddlllllll]\ar@{-}[ddlllllllllllllllllllll]&&&& \CIRCLE\ar@{-}[ddlll]\ar@{-}[ddl]\ar@{-}[ddr]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]&&&&& \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\\
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$}&\hspace{9pt}&
\colorbox{gray}{$\boxplus$} }$$} \put(31.1,26.5){\footnotesize Black: Absorption
Set} \put(34.8,-2){\footnotesize Satisfied Checks} \put(33,39){\footnotesize Unsatisfied
Checks}
\end{picture}
\caption{A dominant absorption set of the IEEE 802.3an code (not all
check node connections shown).} \label{fig:trapping}
\end{figure}
\subsection{Finding Dominant Absorption Sets}
The $(2048,1723)$ rate $0.8413$ regular LDPC code
\cite{DjuXuAbdLin03} considered here has a structured parity-check
matrix:
$$\mathbf{H}=\begin{bmatrix}
\mathbf{\sigma}_{11} &\mathbf{\sigma}_{12} &\mathbf{\sigma}_{13} &\cdots &\mathbf{\sigma}_{1,32} \\
\mathbf{\sigma}_{21} &\mathbf{\sigma}_{22} &\mathbf{\sigma}_{23} &\cdots &\mathbf{\sigma}_{2,32} \\
\mathbf{\sigma}_{31} &\mathbf{\sigma}_{32} &\mathbf{\sigma}_{33} &\cdots &\mathbf{\sigma}_{3,32} \\
\vdots &\vdots &\vdots &\ddots &\vdots \\
\mathbf{\sigma}_{61} &\mathbf{\sigma}_{62} &\mathbf{\sigma}_{63} &\cdots &\mathbf{\sigma}_{6,32} \\
\end{bmatrix}_{384\times 2048}$$ where each $\mathbf{\sigma}_{ij}$ is a $64\times
64$ permutation matrix.
\begin{defn}
Let $(a,b)$ denote an absorption set, where $a$ is the size of the
set (number of variable nodes) and $b$ is the extrinsic message
degree (EMD), i.e., the cardinality of the set of the neighboring
check nodes that are connected to the set an odd number of times (the unsatisfied checks).
\end{defn}
For example, \figurename \ref{fig:absorption} and \figurename
\ref{fig:trapping} show $(4,4)$ and $(8,8)$ absorption sets,
respectively.
Table \ref{table1} shows the first few absorption sets of the code in
\cite{DjuXuAbdLin03}.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{Absorption sets of IEEE 802.3an LDPC code.} \label{table1}
\centering
\begin{tabular}{|c|c|c|c|c|}\hline \bfseries $a$ & \bfseries $b$ & \bfseries Existence
& \bfseries Multiplicity & \bfseries Gain\\\hline\hline $<5$ & & No
&&\\\hline $5$ & $10$ & No && \\\hline \multirow{4}{*}{$6$} & $6$ &
\multirow{4}{*}{No} &&\\\cline{2-2}
& $8$ & &&\\\cline{2-2}
& $10$ & &&\\\cline{2-2}
& $12$ & &&\\ \hline
\multirow{8}{*}{$7$} & $0$ & \multirow{6}{*}{No}&&\\\cline{2-2}
& $2$ &&& \\\cline{2-2}
& $4$ &&& \\\cline{2-2}
& $6$ &&& \\\cline{2-2}
& $8$ &&& \\\cline{2-2}
& $10$ &&& \\\cline{2-5}
& $12$ & Yes & $65,472$\footnotemark & $3.29$\\ \cline{2-5}
& $14$ & Yes & ? & $3$ \\ \hline
\multirow{9}{*}{$8$} & $0$ & \multirow{4}{*}{No}&&\\\cline{2-2}
& $2$ &&& \\\cline{2-2}
& $4$ &&& \\\cline{2-2}
& $6$ &&& \\\cline{2-5}
& $8$ & Yes & $14,272$ & $4$\\\cline{2-5}
& $10$ & No &&\\\cline{2-5}
& $12$ & Yes & $44,416$ &$3.5$\\\cline{2-5}
& $14$ & \multirow{2}{*}{Yes} & \multirow{2}{*}{?} &$3.25$\\\cline{2-2}\cline{5-5}
& $16$ & &&$3$\\ \hline
\multirow{7}{*}{$9$} & $0$ & \multirow{2}{*}{No} &&\\ \cline{2-2}
& $2$ & &&\\ \cline{2-5}
& $4\leq b\leq10$ & ? &&\\ \cline{2-5}
& $12$ & \multirow{4}{*}{Yes} & \multirow{4}{*}{?} &$3.67$\\ \cline{2-2}\cline{5-5}
& $14$ & & &$3.44$\\ \cline{2-2}\cline{5-5}
& $16$ & & &$3.22$\\ \cline{2-2}\cline{5-5}
& $18$ & & &$3$\\ \hline
\multirow{7}{*}{$10$} & $\leq8$ & ? &&\\ \cline{2-5}
& $10$ & Yes & $>192$ & $4$\\ \cline{2-5}
& $12$ & \multirow{5}{*}{Yes} & \multirow{5}{*}{?} &$3.8$\\\cline{2-2}\cline{5-5}
& $14$ & & &$3.6$\\\cline{2-2}\cline{5-5}
& $16$ & &&$3.4$\\\cline{2-2}\cline{5-5}
& $18$ & &&$3.2$\\\cline{2-2}\cline{5-5}
& $20$ & &&$3$\\ \hline
\end{tabular}
\end{table}
\begin{defn}\label{defn4}
(i) Let the ratio $b/a$ denote the average EMD for an $(a,b)$ absorption set.
(ii) An $(a,b)$ absorption set is called minimal if no $(a',b')$ absorption set exists with $a'<a$ and $b'/a' \leq b/a$, i.e., less variable nodes and smaller average EMD.
(iii) A minimal $(a,b)$ absorption set is called dominant if no $(a,b')$ absorption set exists with $b'<b$, i.e., smaller EMD.
\end{defn}
The smaller the absorption set, the more severe the effect on the
error floor. Thus our target is to find the dominant absorption sets
in terms of $a$, $b$ and $b/a$. Since the variable node degree
$d_v=6$, $a\geq5$ by the definition of absorption sets and the code
is $4$-cycle free. In addition, $b\in[0,2a]$ and must be even. So
let us start with $a=5$ to develop the numbers in Table
\ref{table1}. The coefficient $5-b/a$ is the gain of the absorption
set, which determines how fast the extrinsic information enters the
set --- see later.
\subsubsection{$a=5$}
Clearly $b$ can only equal $10$ and there is only one possible
connecting topology shown in \figurename \ref{fig1:sub}.
\begin{figure}[!t]
\centering \subfigure[Check nodes shown.]
{
\label{fig1:sub:a}
$\xymatrix@M=0pt@W=0pt@R=20pt@C=-1pt
{
\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus&\hspace{9pt}&\color{blue}\boxplus\\
&\CIRCLE\ar@{--}[ul]\ar@{--}[ur]\ar@{-}[ddl]\ar@{-}[ddr]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]&&&&\CIRCLE\ar@{--}[ul]\ar@{--}[ur]\ar@{-}[ddlllll]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrr]&&&&\CIRCLE\ar@{--}[ul]\ar@{--}[ur]\ar@{-}[ddlllllll]\ar@{-}[ddl]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrr]&&&&\CIRCLE\ar@{--}[ul]\ar@{--}[ur]\ar@{-}[ddlllllllll]\ar@{-}[ddlll]\ar@{-}[ddr]\ar@{-}[ddrrrrr]&&&&\CIRCLE\ar@{--}[ul]\ar@{--}[ur]\ar@{-}[ddlllllllllll]\ar@{-}[ddlllll]\ar@{-}[ddl]\ar@{-}[ddr]&\\
&&&&&&&&&&&&&&&&&&\\
\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus&&\boxplus
}$}\hfil
\subfigure[Check nodes hidden.]
{
\label{fig1:sub:b}
$\xymatrix@M=0pt@W=0pt@R=30pt@C=15pt
{
&&\CIRCLE\ar@{-}[dll]\ar@{-}[drr]\ar@{-}[ddl]\ar@{-}[ddr]&&\\\CIRCLE\ar@{-}[rrrr]\ar@{-}[dr]\ar@{-}[drrr]&&&&\CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\\&\CIRCLE\ar@{-}[rr]&&\CIRCLE&
}$} \caption{The only possible topology of $(5,10)$ absorption
sets.}
\label{fig1:sub}
\end{figure}
\footnotetext{Only sets not contained in $(8,8)$ absorption sets are
counted --- see later.}
\begin{lem}\label{lema=5}
There are no size-$5$ absorption sets.
\end{lem}
\begin{IEEEproof} See Appendix \ref{proofa=5}.
\end{IEEEproof}
\subsubsection{$a=6$}
Let us introduce additional notations needed to prove the existence
of absorption sets.
\begin{defn}\label{constraints}
(i) For any variable node $v$ in an absorption set, let $\mathrm{Deg}(v)$
denote the number of neighboring check nodes of $v$ that are
connected to the set an even number of times. $\mathrm{Deg}(v)$ is the degree
of vertex $v$ in the topology graph with check nodes hidden. (ii)
Let an unordered array $[\mathrm{Deg}(v_i) : i=1,2,\ldots,a]$ denote a class
of $(a,b)$ absorption sets, where $\mathrm{Deg}(v_i)\in \{4,5,6\}$ and
$\sum\limits_{i=1}^a \mathrm{Deg}(v_i)=6a-b$.
\end{defn}
It is difficult to find absorption sets, even by making use of
Algorithm \ref{alg:mine}-type methods (see Appendix \ref{proofa=5}),
due to their extremely low appearance. Hence we need Definition
\ref{constraints} to classify the absorption sets first. For each
pair $(a,b)$, there may be several classes of absorption sets, and
each class may exhibit several topologies. What we are trying to do
is to reduce one unknown absorption set to a smaller absorption set
whose non-existence is known by eliminating nodes from the original
set. We can then argue that there is only a limited number of
topologies that have to be searched algorithmically.
\begin{thm}\label{thma=6}
There are no size-$6$ absorption sets.
\end{thm}
\begin{IEEEproof} See Appendix \ref{proofa=6}.
\end{IEEEproof}
\subsubsection{$a=7$}
\begin{thm}\label{thma=7}
There are no $(7,b)$ absorption sets with $b<12$. $(7,12)$
and $(7,14)$ absorption sets do exist.
\end{thm}
\begin{IEEEproof} See Appendix \ref{proofa=7}.
\end{IEEEproof}
\subsubsection{$a=8$}
First we show
\begin{lem}\label{lema=8}
For $b<8$, there exist no $(8,b)$ absorption sets.\footnote{As a corollary, since there is no $(8,0)$ absorption set, the minimum distance bound of this LDPC code \cite{DjuXuAbdLin03} is strengthened
to $d_{\mathrm{min}}\geq 10$. Therefore, there are no $(9,0)$ absorption sets since a $(9,0)$ absorption set is a length-$9$ codeword.} For $b=8$, there
exists no $(8,8)$ absorption set that contains a degree-$6$ variable node.
\end{lem}
\begin{IEEEproof} See Appendix \ref{prooflema=8}.
\end{IEEEproof}
Then the only possible class of $(8,8)$ absorption set would have connectivity $[5,5,5,5,5,5,5,5]$.
We claim that
\begin{claim}\label{claima=8}
Graphically, there exist five possible topologies
for $[5,5,5,5,5,5,5,5]$ absorption sets, shown in \figurename \ref{figa=8b488:sub:b}, \ref{figa=8b488:sub:g} and \ref{fig12:sub}.
\end{claim}
\begin{IEEEproof} See Appendix \ref{proofclaima=8}.
\end{IEEEproof}
\begin{thm}
The number of $(8,8)$ absorption sets is $14,272$ and they all have the topology of \figurename \ref{fig12:sub:b}.
\end{thm}
\begin{IEEEproof} By searching all topologies in Claim \ref{claima=8} on the $\mathbf{H}$ matrix of \cite{DjuXuAbdLin03}.
\end{IEEEproof}
Since these are the dominant absorption sets, let us sketch their
connections in \figurename \ref{fig13:sub} one more time\footnote{It
took approximately ninety minutes on an AMD Opteron Processor
($64$bits/$2.4$GHz) to search the topology \figurename
\ref{fig13:sub}.}.
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig13:sub:a}
$\xymatrix@M=0pt@W=0pt@R=20pt@C=-2pt
{
&&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&& \color{blue}\boxplus\ar@{--}[d]&&&&& \\
&&&&& \CIRCLE\ar@{-}[ddlllll]\ar@{-}[ddlll]\ar@{-}[ddl]\ar@{-}[ddr]\ar@{-}[ddrrr]&&&& \CIRCLE\ar@{-}[ddr]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlllllllllllll]\ar@{-}[ddlll]\ar@{-}[ddrrrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlllllllllllllll]\ar@{-}[ddlllll]\ar@{-}[ddrrr]\ar@{-}[ddrrrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrrrr]&&&& \CIRCLE\ar@{-}[ddlll]\ar@{-}[ddlllllll]\ar@{-}[ddlllllllllllllllll]\ar@{-}[ddrrrrrrrrrrrrrrr]\ar@{-}[ddrrrrr]&&&& \CIRCLE\ar@{-}[ddrrrrrrrrrrrrr]\ar@{-}[ddrrr]\ar@{-}[ddlllll]\ar@{-}[ddlllllllllllllllllll]\ar@{-}[ddlllllllll]&&&& \CIRCLE\ar@{-}[ddl]\ar@{-}[ddlll]\ar@{-}[ddlllll]\ar@{-}[ddlllllll]\ar@{-}[ddlllllllllllllllllllll]&&&& \CIRCLE\ar@{-}[ddlll]\ar@{-}[ddl]\ar@{-}[ddr]\ar@{-}[ddrrr]\ar@{-}[ddrrrrr]&&&&& \\
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\\
\boxplus& \hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}&
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus }$ }\\
\subfigure[]
{
\label{fig13:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddll]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddr]\ar@{-}[ddrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\ar@{-}[ddll]\\
\CIRCLE\ar@{-}[dr]\ar@{-}[drr]&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dll]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ } \caption{$(8,8)$ absorptions sets.}
\label{fig13:sub}
\end{figure}
The average multiplicity of each variable node appeared in such sets is $14272\times 8/2048=55.75$. Because of the block structure of the $\mathbf{H}$ matrix, certain groups of variable nodes do share the same multiplicity, as listed in Table \ref{table2}. The ratio $b/a=1$. The next possible absorption set with these parameters would be the $(10,10)$ set.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{The multiplicity of each variable node in $(8,8)$ absorption sets.}
\label{table2}
\centering
\begin{tabular}{|c|c||c|c|}\hline
\bfseries Variable Nodes & \bfseries Multiplicities & \bfseries Variable Nodes &
\bfseries Multiplicities\\\hline\hline
$0$---$63$&$63$&$1024$---$1087$&$54$\\\hline
$64$---$127$&$55$&$1088$---$1151$&$66$\\\hline
$128$---$191$&$74$&$1152$---$1215$&$49$\\\hline
$192$---$255$&$60$&$1216$---$1279$&$40$\\\hline
$256$---$319$&$67$&$1280$---$1343$&$75$\\\hline
$320$---$383$&$68$&$1344$---$1407$&$41$\\\hline
$384$---$447$&$54$&$1408$---$1471$&$56$\\\hline
$448$---$511$&$60$&$1472$---$1535$&$59$\\\hline
$512$---$575$&$57$&$1536$---$1599$&$47$\\\hline
$576$---$639$&$66$&$1600$---$1663$&$36$\\\hline
$640$---$703$&$60$&$1664$---$1727$&$59$\\\hline
$704$---$767$&$49$&$1728$---$1791$&$59$\\\hline
$768$---$831$&$39$&$1792$---$1855$&$37$\\\hline
$832$---$895$&$47$&$1856$---$1919$&$52$\\\hline
$896$---$959$&$61$&$1920$---$1983$&$69$\\\hline
$960$---$1023$&$62$&$1984$---$2047$&$43$\\\hline
\end{tabular}
\end{table}
\subsection{Less Dominant Absorption Sets}
There exist larger and less dominant absorption sets. See Appendix \ref{less} for details.
\section{Dynamic Analysis of Absorption Sets}
We now present a linearized analysis to gain insight into the behavior of dominant absorption sets starting with the leading $(8,8)$ absorption set.
First we note that the variable nodes perform simple
addition. Furthermore, the check nodes basically choose the minimum
of the incoming signals. If we make the reasonable assumption that the absorption set converges slower than the remaining nodes in the code, and due to the
fact that each (satisfied) check node is connected exactly to two absorption set variables, the minimum
absolute-value signal into the participating check nodes will come from one of the absorption set variables.
If this is true, the check nodes simple exchange the signals on the connections to the absorption set variable nodes. We will refine this approximation below.
Additionally, each absorption set variable node is singly connected to a ``lone'' floating extrinsic parity
check node, all of whose other connections go to other, set-external variable nodes.
The messages through these eight extrinsic check nodes are the extrinsic messages into the absorption
set, and are of crucial importance. Algorithmically, they play exactly the same role as the intrinsic
channel values which are fed into the variable nodes by virtue of the summation function executed at
the variable nodes.
\figurename \ref{fig:dynamics} shows an example of the dynamic behavior of the absorption set variables
close to its decision threshold boundary. The seemingly erratic behavior resolves after a number
of iterations when all variables follow highly correlated trajectories. This observation is the basis
for the following analysis.
\begin{figure}[!t]
\centering
\setlength{\unitlength}{1mm}
\begin{picture}(130,53)
\put(0,0){\includegraphics[scale=0.414]{Figures/Trapping3C}}
\end{picture}
\caption{Dynamics of an absorption set close to the decision boundary. The different curves are the variable node LLR values for the eight absorption set nodes.}
\label{fig:dynamics}
\end{figure}
\newcounter{MYtempeqncnt}
\begin{figure*}[!b]
\normalsize \setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{6} \vspace*{4pt} \hrulefill
\begin{equation}\label{eq:absorptionset}
P_{\rm AS} = Q \left(\frac{\displaystyle 2 m_\lambda
+ 2 \sum\limits_{j=1}^I \left(\frac{m_{\lambda^{(ex)}}^{(j)}+ m_\lambda}{\mu_{\rm max}^j}
\prod\limits_{l=1}^j \frac{1}{g_l}\right)}
{\sqrt{ \displaystyle \left( 1 + \sum_{j=1}^I \prod_{l=1}^j \frac{1}{g_l \mu_{\rm max} } \right)^2
m_\lambda + \sum_{j=1}^I m_{\lambda^{(ex)}}^{(j)}
\left( \prod_{l=1}^j \frac{1}{g_l \mu_{\rm max}} \right)^2 } } \right)
\end{equation}
\setcounter{equation}{\value{MYtempeqncnt}}
\end{figure*}
Denote the {\em outgoing solid edge values} from the variable nodes (\figurename \ref{fig:trapping}) by $x_i$, i.e., $x_1,\cdots, x_5$ leave variable node $v=0$, $x_6,\cdots, x_{10}$ variable
node $v=1$, etc. Collect the $x_i$ in the length-$40$ column vector $\mathbf{x}$, which is the vector of
outgoing variable edge values in the absorption set. Likewise, and analogously, let $\mathbf y$ be
the {\em incoming edge values} to the variable nodes, such that $y_j$ corresponds to the reverse-direction message. Now, at iteration $i=0$
\begin{displaymath}
\mathbf{x}_0 = \boldsymbol{\lambda}
\end{displaymath}
where the initial input is the vector of channel intrinsics $ \boldsymbol{\lambda} =
\left[\lambda_1, \cdots, \lambda_1, \lambda_2, \cdots, \lambda_2, \cdots, \lambda_8, \cdots, \lambda_8\right]^\mathrm{T}$
duplicated onto the outgoing messages. It undergoes the
following operation at the check node:
\begin{displaymath}
\mathbf{y}_0 = \mathbf{C x}
\end{displaymath}
where $ \mathbf{C}$ is a permutation matrix that exchanges the absorption set signals as discussed
above. At iteration $i=1$ we obtain
\begin{displaymath}
\mathbf{x}_1 = \mathbf{V C}\boldsymbol{\lambda} + \boldsymbol{\lambda}
+ \boldsymbol{\lambda}_1^{(ex)}
\end{displaymath}
where $\mathbf{V}$ is the variable node function matrix, i.e., each output is the sum
of the other four inputs from the check nodes plus the intrinsic input. The extrinsic inputs
from the remainder of the code graph are contained in $\boldsymbol{\lambda}_1^{(ex)}$.
Following the linear model, at iteration $I=j$ extrinsic signals are injected into the
absorption set as
$\boldsymbol{\lambda}^{(ex)}_j= \left[ \lambda^{(ex)}_{j1}, \cdots, \lambda^{(ex)}_{j1},
\lambda^{(ex)}_{j2}, \cdots, \lambda^{(ex)}_{j8}\right]^\mathrm{T}$ via the extrinsic check nodes.
By induction we obtain at iteration $i=I$
\begin{displaymath}
\mathbf{x}_I = \sum_{i=0}^I (\mathbf{V C})^i \boldsymbol{\lambda}
+\sum_{j=1}^I \sum_{i=j}^I (\mathbf{V C})^i \boldsymbol{\lambda}_j^{(ex)}
\end{displaymath}
Applying the spectral theorem we obtain
\begin{displaymath}
(\mathbf{V C})^i \mathbf{\lambda} \rightarrow \mu_{\rm max}^i
\left( \mathbf{\lambda}^\mathrm{T} \mathbf{v}_{\rm max} \right) \mathbf{v}_{\rm max}
\end{displaymath}
where $\mathbf{v}_{\rm max}$ is the unit-length eigenvector of the maximal eigenvalue
$\mu_{\rm max}$ of the matrix $\mathbf{V C}$.
The following lemma holds:
\begin{lem}
The largest eigenvalue of $\mathbf{V C}$ for the $(8,8)$ set is $\mu_{\rm max} = d_v-2=4$, and its associated eigenvector
is $\mathbf{v}_{\rm max} = [1,\cdots, 1]^\mathrm{T}$.
\end{lem}
\begin{IEEEproof} First write $\mathbf{V C} = 4 \mathbf{M}$. By inspection $\mathbf{M}$ is a probability
matrix, i.e., the sum of all rows equals unity. As a special case of the Perron-Frobenius theorem it
is known that the largest eigenvalue of a probability matrix is $1$, therefore the largest eigenvalue
of $\mathbf{V C}$ equals $4$.
By inspection $\mathbf{V C} [1,\cdots,1]^\mathrm{T} = 4 [1,\cdots,1]^\mathrm{T}$.
\end{IEEEproof}
The absorption set in question falls in error if
\begin{equation}\label{eq1}
\beta=\boldsymbol{\lambda}^\mathrm{T} \mathbf{v}_{\rm max}
+ \sum_{j=1}^I \frac{ \left( {\boldsymbol{\lambda}_j^{(ex)}} + \boldsymbol{\lambda} \right)^\mathrm{T}
\mathbf{v}_{\rm max} }
{\mu_{\rm max}^j} \leq 0
\end{equation}
or, in the case of the $(8,8)$ absorption set
\begin{equation}
\beta=\sum_{i=1}^8 \left( \lambda_i + \sum_{j=1}^I \frac{ \lambda^{(ex)}_{ji} + \lambda_i }
{\mu_{\rm max}^j} \right) \leq 0
\label{eq:TSconvergence}
\end{equation}
The eigenvalue $\mu_{\rm max}=d_v-2$ is the {\em gain} of the absorption
set and it is determined by the variable node degree.
Exact knowledge of $\lambda^{(ex)}_{ji}$ is not available to the analysis, since these values depend
on the received signals. However, assuming that the code structure extrinsic to the apsorption set
operates ``regularly'', we may substitute average values for the $\lambda^{(ex)}_{ji}$. Note that
$\lambda_i$ is Gaussian distributed from the channel, and that we may assume that $\lambda^{(ex)}_{ji}$ is also Gaussian distributed as is customary in density evolution analysis \cite{ChuRicUrb01,SchPer04}. Furthermore, like $\lambda_i$, we assume
that $\lambda^{(ex)}_{ji}$ has a {\em consistent} Gaussian distribution with $m=2 \sigma^2$, where
$m$ is the mean. We therefore only need the mean of $\lambda^{(ex)}_{ji}$, which we can
calculate from a Gaussian density evolution calculation\footnote{For details and definitions, see \cite[Chapter 11]{SchPer04}.}, i.e.,
\begin{displaymath}
m_{\lambda^{(ex)}}^{(i)} = \phi^{-1} \left(1 - \left[1-
\phi\left( m_\lambda + (d_v-1) m_{\lambda^{(ex)}}^{(i-1)} \right) \right]^{d_c-1} \right)
\end{displaymath}
where $m_\lambda = 2 E_b/\sigma^2$ is the mean of $\lambda_i$, $m_{\lambda^{(ex)}}^{(i)}$ is the mean of the extrinsic signal $\lambda^{(ex)}_{ji}$, and $\phi$ is the check node mean transfer function \cite{SchPer04}.
With the Gaussian assumptions, the probability of (\ref{eq:TSconvergence}) happening
can be calculated as
\setlength{\arraycolsep}{0.0em}
\begin{eqnarray}
P_{\rm AS} &{ }={ }&
{\rm Pr} \left( \beta \leq 0 \right) \nonumber\\[1mm]
& = & Q \left( \frac{\displaystyle 2 m_\lambda
+ 2 \sum\limits_{j=1}^I \frac{m_{\lambda^{(ex)}}^{(j)}+ m_\lambda}{\mu_{\rm max}^j} }
{\sqrt{ \displaystyle \left( 1 + \sum_{j=1}^I \frac{1}{\mu_{\rm max}^j } \right)^2
m_\lambda + \sum_{j=1}^I \frac{m_{\lambda^{(ex)}}^{(j)}}{\mu_{\rm max}^{2j}} } } \right)
\label{eq:TSerror}
\end{eqnarray}
\setlength{\arraycolsep}{5pt}
Two refinements can be added to this analysis. The exchange of
extrinsics through the matrix $\mathbf{C}$ is an approximation in
two ways: (i) As long as the remaining $d_c-2=30$ inputs to the
check node are relatively small, the entries of $\mathbf{C}$ are
strictly less than unity, and, (ii) in case one of the extrinsic
incoming check node messages has the wrong polarity, the returned
signal to the absorption set switches polarity. Case (i) is
approached as follows. Using a Taylor series approximation we show
that the quintessential check node operation
\begin{equation*}
\tanh^{-1} \left( \tanh(x) \prod_{i=1}^{d_c-2} \tanh(x_i) \right) = \prod_{i=1}^{d_c-2} \tanh(x_i)\, x + O\left[x^3\right]
\end{equation*}
where $\prod\limits_{i=1}^{d_c-2} \tanh(x_i)$ can be interpreted as a ``check node gain''.
If we use for $x_i$ the mean $m_{\mu^{(ex)}}^{(i)}$ of the signals $\mu^{(ex)}$ from the variable
to the check nodes, an average gain can be computed as
\setlength{\arraycolsep}{0.0em}
\begin{eqnarray}
g_i &{ }={ }& E \left[
\prod_{i=1}^{d_c-2} \tanh\left(\frac{m_{\mu^{(ex)}}^{(i)}}{2}\right)\right] \\[1mm]
& = & E \left[\tanh\left(\frac{m_{\mu^{(ex)}}^{(i)}}{2}\right)\right]^{d_c-2} \\ [3mm]
& = & \left( 1 - \phi\left(m_{\mu^{(ex)}}^{(i)}\right) \right)^{d_c-2}
\end{eqnarray}
\setlength{\arraycolsep}{5pt}
\hspace{-3.4pt}where the last equality results from the definition of the density evolution function
$\phi(\cdot)$.
With this result the probability in (\ref{eq:TSerror}) is modified to
(\ref{eq:absorptionset}). In the case of
general sets we need to work with (\ref{eq1}) instead, and compute $\mu_{\rm max}$ and $\mathbf{v}_{\rm max}$
numerically using the set topology.
\addtocounter{equation}{1}
\begin{figure*}[!b]
\normalsize
\setcounter{MYtempeqncnt}{\value{equation}}
\setcounter{equation}{10}
\vspace*{4pt}
\hrulefill
\begin{equation}\label{eq:f(a,b)}
f(a,b) =
\frac{\displaystyle a m_\lambda \left(1+ \sum_{j=1}^I \prod_{l=1}^j\frac{1}{g_l \left(5-\frac{b}{a}\right)^j } \right)
+ b \sum\limits_{j=1}^I \left(\frac{m_{\lambda^{(ex)}}^{(j)}}{\left(5-\frac{b}{a}\right)^j}\prod\limits_{l=1}^j \frac{1}{g_l}\right)}
{\displaystyle \sqrt{ \displaystyle 2a m_\lambda \left( 1 + \sum_{j=1}^I \prod_{l=1}^j \frac{1}{g_l \left(5-\frac{b}{a}\right) } \right)^2
+ 2b\sum_{j=1}^I m_{\lambda^{(ex)}}^{(j)}
\left( \prod_{l=1}^j \frac{1}{g_l \left(5-\frac{b}{a}\right)} \right)^2 } }
\end{equation}
\setcounter{equation}{\value{MYtempeqncnt}}
\end{figure*}
Case (ii) can be handled by the linear analysis as well in the following way. If an external variable
to the absorption set has an incorrect sign, this reverses the polarity of the signal returned to
the absorption set from that particular check node. During the first iteration, these extrinsic signals
are basically the received channel LLRs from the connected variable nodes. The probability that
these are in error is given by the raw bit error rate
\begin{equation}
P_e = Q\left( \sqrt{ 2 \frac{E_s}{N_0} } \right)
\end{equation}
There are $d_c-2=30$ external inputs impinging on each check node of the absorption set, therefore
the probability that a returned signal experiences a polarity reversal is given by
\begin{equation}
P_p = \sum_{k=1}^{14}{30\choose 2k+1} P_e^{2k+1}
\left( 1 - P_e \right)^{29-2k}
\end{equation}
The model in (\ref{eq:absorptionset}) can now be expanded by injecting a correction value
into the absorption set node whenever an external value is in error. We assume that if a polarity
reversal occurs, the minimum value of the check node is likely close to zero, therefore the
injected correction value needs
to cancel the absent feedback signal and is set to
$- \lambda_{{\rm ex},i}/\mu_{\rm max}$.
If $k$ check nodes are in error, $2k$ correction values are injected, one for each message
going back to the absorption set. The injected correction values will alter the mean value of
the decision variable to
\begin{equation*}
{\rm mean} \rightarrow 8 \left( m_\lambda \left(1- \frac{k}{4 \mu_{\rm max}} \right)
+ \sum_{j=1}^I \left(\frac{m_{\lambda^{(ex)}}^{(j)}}{\mu_{\rm max}^j} \prod_{l=1}^j \frac{1}{g_l}\right) \right)
\end{equation*}
and the variance is adjusted accordingly, where care needs to be taken how the correction values
accumulate. We have used an upper bound on the variance.
Note that these modifications only include check node polarity reversal at the first iteration,
but an extension to subsequent iterations is straight-forward if messy. Furthermore, as seen in
\figurename \ref{fig:trappingsetsim2} (dashed curves), the addition of this mechanism has only a minor
effect on the results.
The probability $P_{\rm AS}$ needs to be multiplied with the multiplicity factor of $14,272$ in order
to obtain a union bound. In order to compute a BER estimate, we further
multiply this number by $8/1723$, since there are eight errors that occur in a frame of $1,723$ bit errors
due to this absorption set.
\figurename \ref{fig:f[a,b]} shows $P_{\rm AS}$ for the first most dominant absorption sets. Also shown
are general tendencies of $P_{\rm AS}$ as a function of $a$ and $b$:
\begin{equation}
P_{\rm AS}=Q\left(f(a,b)\right)
\end{equation}
where $f(a,b)$ is defined by (\ref{eq:f(a,b)}). It can be shown that
$\mu_{\rm max}\approx5-b/a$ is a close approximation (exact for
symmetric sets with $a=b$) to the gain of the set, and this was used
in \figurename \ref{fig:f[a,b]} to plot the curves.
It can be seen that the $(8,8)$ absorption set is the most dominant, which is consistent with numerical observations.
Multiplicities also affect a set's impact --- see \figurename \ref{fig:trappingsetsim2}. Additionally, some sets, like the majority
of $(7,12)$ sets, are ``contained'' in larger sets, that is, such $(7,12)$ absorption sets are not stable under bit flipping and will evolve
into $(8,8)$ sets, of which they are subgraphs.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.41]{Figures/fabplot}
\caption{Error probability of dominant absorption sets at
$E_b/N_0=5$dB and approximation functions based on $a$ and $b$.
(Curves are drawn only for possible or
existing parameter combinations.)}
\label{fig:f[a,b]}
\end{figure}
\section{Numerical Verification}
\figurename \ref{fig:trappingsetsim2} shows the analytical error
floor calculation using (\ref{eq:absorptionset}) and the
multiplicity of $14,272$. Lesser absorption sets have an impact more
than an order of magnitude lower. And they are not considered. The
figure also shows hardware simulations using an FPGA platform, as
well as importance sampled simulations using the same absorption
sets as bias targets. Regular mean-shift importance sampling was
utilized and each of the absorption sets containing a specific
variable node was biased separately. As evidenced by the figure, our
linearized analysis provides an accurate picture of the error floor
behavior of this code and illustrates the dominance of the $(8,8)$
absorption sets.
\begin{figure}[!t]
\begin{center}
\setlength{\unitlength}{0.9mm}
\begin{picture}(170,68)
\put(1,0){\includegraphics[scale=0.537]{Figures/ISsim.pdf}}
\put(45,-2){\scriptsize ${E_b}/{N_0}$[dB]}
\put(0,30){\rotatebox{90}{\scriptsize BER}}
\end{picture}
\end{center}
\caption{IS simulations, FPGA hardware simulations, and analytical error floor analysis for
the $(2048,1723)$ regular $(6,32)$ LDPC code.}
\label{fig:trappingsetsim2}
\end{figure}
\section{Conclusion}
We have presented an analytical analysis of the dynamic behavior of
the dominant absorption sets in LDPC message-passing decoders. These
absorption sets cause the infamous error floor at high
signal-to-noise ratios, and we have identified the dominant such
sets for the example regular LDPC code used in the IEEE 802.3an
standard via topological arguments and searches. Using importance
sampling with the dominant sets accurately predicts the error floor
of this code.
\appendices
\section{Proof of Lemma \ref{lema=5}}\label{proofa=5}
Matrix $\mathbf{H}$ is searched observing the constraints imposed by the absorption set topology. In
addition, some of the properties listed in \cite{Zhanetal07, Zhanetal2008} for
array-based LDPC codes apply, as well. The following algorithm
is used:
\begin{algorithm}[H]
\SetVline
\KwIn{Parity-check matrix $\mathbf{H}_{384\times 2048}$ and
\figurename \ref{fig1:sub}.} \KwOut{Absorption sets.}
\ForEach{ variable node $v\in\{0,1,2,\ldots,2047\}$ }{ Pick $4$ out
of $6$ neighboring check nodes of $v$ denoted $c_0,c_1,c_2$ and $c_3$,
respectively\; \ForEach{ one out of $31$ neighboring variable nodes
other than $v$ of $c_0$, denoted as $v_1$ }{ \ForEach{one out of
$31$ neighboring variable nodes other than $v$ of $c_1$, denoted as
$v_2$}{ \eIf{ $v_2$ and $v_1$ are not connected }{ re-pick $v_2$\;
}{ \ForEach{ one out of $31$ neighboring variable nodes other than
$v$ of $c_2$, denoted as $v_3$ }{\eIf{ $v_3$ and $v_1$ or $v_3$ and
$v_2$ are not connected }{ re-pick $v_3$\; }{ \ForEach{ one out of
$31$ neighboring variable nodes other than $v$ of $c_3$, denoted as
$v_4$ }{\eIf{ $v_4$ and $v_1$ or $v_4$ and $v_2$ or $v_4$ and $v_3$
are not connected }{ re-pick $v_4$\; }{ follow Definition \ref{def1}
to determine if this candidate set is an absorption set\; } } } } }
} } } \caption{Algorithm for finding $(5,10)$ absorption sets.}
\label{alg:mine}
\end{algorithm}
\section{Proof of Theorem \ref{thma=6}}\label{proofa=6}
For $a=6$, there are four possible values for $b$ and there is only
one possible topology corresponding to each of them, shown in
\figurename \ref{fig2:sub}.
\begin{figure}[!t]
\centering \subfigure[$(6,6)$]
{
\label{fig2:sub:a}
$\xymatrix@M=0pt@W=0pt@R=25pt@C=20pt
{
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[dd]\ar@{-}[ddr]\ar@{-}[drr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddl]\ar@{-}[dd]\ar@{-}[dr]&\\
\CIRCLE \ar@{-}[dr]\ar@{-}[drr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dll]\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\qquad
\subfigure[$(6,8)$]
{
\label{fig2:sub:b}
$\xymatrix@M=0pt@W=0pt@R=25pt@C=20pt
{
& \CIRCLE\ar@{-}[dl]\ar@{-}[dd]\ar@{-}[ddr]\ar@{-}[drr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddl]\ar@{-}[dd]\ar@{-}[dr]&\\
\CIRCLE \ar@{-}[dr]\ar@{-}[drr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dll]\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\\
\subfigure[$(6,10)$]
{
\label{fig2:sub:c}
$\xymatrix@M=0pt@W=0pt@R=25pt@C=20pt
{
& \CIRCLE\ar@{-}[dl]\ar@{-}[dd]\ar@{-}[ddr]\ar@{-}[drr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddl]\ar@{-}[dd]\ar@{-}[dr]&\\
\CIRCLE \ar@{-}[dr]\ar@{-}[drr]&&& \CIRCLE\ar@{-}[dll]\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\qquad
\subfigure[$(6,12)$]
{
\label{fig2:sub:d}
$\xymatrix@M=0pt@W=0pt@R=25pt@C=20pt
{
& \CIRCLE\ar@{-}[dl]\ar@{-}[dd]\ar@{-}[ddr]\ar@{-}[drr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddl]\ar@{-}[dd]\ar@{-}[dr]&\\
\CIRCLE \ar@{-}[dr]\ar@{-}[drr]&&& \CIRCLE\ar@{-}[dll]\ar@{-}[dl]\\
& \CIRCLE& \CIRCLE&\\
}$ } \caption{Possible topologies of size-$6$ absorption sets.}
\label{fig2:sub}
\end{figure}
By removing any node in \figurename \ref{fig2:sub:a} or either
degree-$4$ node in \figurename \ref{fig2:sub:b}, we obtain the
$[4,4,4,4,4]$ set, one shown in \figurename
\ref{fig1:sub:b}. By Lemma \ref{lema=5}, \figurename
\ref{fig2:sub:a}--\ref{fig2:sub:b} do not exist. After eliminating topologies
\figurename \ref{fig2:sub:c}--\ref{fig2:sub:d} algorithmically,
Theorem \ref{thma=6} follows.
\section{Proof of Theorem \ref{thma=7}}\label{proofa=7}
Now $a$ is large enough for the neighboring check nodes to be connected to the absorption set four times. First, we suppose that all satisfied check nodes are connected to the set twice.
\subsection{$b=0$}
If $b=0$, then the absorption set is a codeword. However, since
$d_{\rm min}\geq 8$ \cite{DjuXuAbdLin03}, $b\not= 0$.
\subsection{$0<b<12$}
We apply the constraints in Definition \ref{constraints} and the
pigeonhole principle to prove this.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $b=2$, there are two classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,6,6,6,5,5]$: removing either degree-$5$ node leaves a $(6,6)$ absorption set.
\item $[6,6,6,6,6,6,4]$: removing the degree-$4$ node generates a $(6,4)$ absorption set.
\end{enumerate}
\item $b=4$, there are three classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,6,5,5,5,5]$: removing any degree-$5$ node generates a
$(6,8)$ absorption set.
\item $[6,6,6,6,5,5,4]$: removing the degree-$4$ node generates a
$(6,6)$ absorption set.
\item $[6,6,6,6,6,4,4]$: this is infeasible since the group of five degree-$6$ nodes requires ten edges emanating from the group of degree-$4$ nodes.
\end{enumerate}
\item $b=6$, there are four classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,5,5,5,5,5,5]$: removing any degree-$5$ node generates a
$(6,10)$ absorption set.
\item $[6,6,5,5,5,5,4]$: removing the degree-$4$ node generates a
$(6,8)$ absorption set.
\item $[6,6,6,5,5,4,4]$: each of the degree-$5$ nodes and each of the degree-$6$ nodes needs at least one and two edges emanating from the two degree-$4$ nodes, respectively. That makes eight. So there is no connection between the degree-$4$ nodes. Thus removing either of them generates a $(6,8)$ absorption set.
\item $[6,6,6,6,4,4,4]$: each of the degree-$6$ nodes needs three edges emanating from the two degree-$4$ nodes. That makes twelve. So there is no connection among the three degree-$4$ nodes. Thus removing any of them generates a $(6,8)$ absorption set.
\end{enumerate}
\item $b=8$, there are four classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[5,5,5,5,5,5,4]$: removing the degree-$4$ node generates a
$(6,10)$ absorption set.
\item $[6,5,5,5,5,4,4]$: let us study the intrinsic
connections between the two degree-$4$ nodes:
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig2degree4:sub:a}
$\xymatrix@M=0pt@W=0pt@R=30pt@C=10pt{
&&\CIRCLE\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dll]\ar@{-}[dr]&&&\CIRCLE\ar@{-}[d]\ar@{-}[dr]\ar@{-}[drr]\ar@{-}[dl]&&\\
&&&&&&&\\}$ }\qquad
\subfigure[]
{
\label{fig2degree4:sub:b}
$\xymatrix@M=0pt@W=0pt@R=30pt@C=10pt{
&\CIRCLE\ar@{-}[rrr]\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dr]&&&\CIRCLE\ar@{-}[d]\ar@{-}[dr]\ar@{-}[dl]&\\
&&&&&\\}$ }\caption{Possible intrinsic connections between two
degree-$4$ nodes.}
\label{fig2degree4:sub}
\end{figure}
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Not connected as shown in \figurename \ref{fig2degree4:sub:a}: removing either degree-$4$ node will reduce it to a $(6,10)$
absorption set.
\item Connected as shown in \figurename \ref{fig2degree4:sub:b}: we consider the
connections between the two degree-$4$ nodes and the other five
nodes in the set. The other five nodes need at least six edges emanating from the two degree-$4$
nodes, so there is no degree-$5$ node connected to both degree-$4$ nodes. Thus removing both of them generates a $(5,10)$ absorption set.
\end{enumerate}
\item $[6,6,5,5,4,4,4]$: the two degree-$5$ nodes and the two degree-$6$ nodes need at least ten edges emanating from the three degree-$4$ nodes. Hence there should be no more than one connection among the group of degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal3:sub:a}: removing any degree-$4$ node will reduce it to a
$(6,10)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal3:sub:b}: removing the topmost degree-$4$ node
will reduce it to a $(6,10)$ absorption set.
\end{enumerate}
\item $[6,6,6,4,4,4,4]$: the three degree-$6$ nodes need twelve edges emanating from the four degree-$4$ nodes. Hence there should be no more than two connections among the group of degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal4:sub:a}: removing any degree-$4$ node will reduce it to a
$(6,10)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal4:sub:b}: removing either topmost degree-$4$ node will
reduce it to a $(6,10)$ absorption set.
\item Two connections: there are two cases:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Removing the bottom-right degree-$4$ node in \figurename \ref{figinternal4:sub:c} will
reduce it to a $(6,10)$ absorption set.
\item Removing either the top or the bottom couple of
degree-$4$ nodes in \figurename \ref{figinternal4:sub:d} will reduce it
to a $(5,10)$ absorption set.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\item $b=10$, there are three classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[5,5,5,5,4,4,4]$: the four degree-$5$ nodes need at least eight edges emanating from the three degree-$4$ nodes. Hence there should be no more than two connections among the group of degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal3:sub:a}: removing any degree-$4$ node will reduce it to a
$(6,12)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal3:sub:b}: removing the topmost degree-$4$ node
will reduce it to a $(6,12)$ absorption set.
\item\label{check1} Two connections as shown in \figurename \ref{figinternal3:sub:c}. No node can be removed to get another absorption set. However, it is straightforward to see that there is only one possible topology to satisfy this:
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=10pt{
& \CIRCLE\ar@{-}[rr]\ar@{-}[drrr]\ar@{-}[dr]\ar@{-}[dl]&& \CIRCLE\ar@{-}[rr]\ar@{-}[drrr]\ar@{-}[dlll]&& \CIRCLE\ar@{-}[dlll]\ar@{-}[dl]\ar@{-}[dr]&\\
\CIRCLE\ar@{-}[rr]\ar@/_0.6pc/@{-}[rrrr]\ar@/_1.2pc/@{-}[rrrrrr]&&
\CIRCLE\ar@{-}[rr]\ar@/_0.6pc/@{-}[rrrr]&& \CIRCLE\ar@{-}[rr]&&
\CIRCLE\\}$$
\bigskip
Therefore we have to go check the $\mathbf{H}$ matrix algorithmically.
\end{enumerate}
\item $[6,5,5,4,4,4,4]$: the two degree-$5$ nodes and the degree-$6$ node need at least ten edges emanating from the four degree-$4$ nodes. Hence there should be no more than three connections among the group of degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal4:sub:a}: removing any degree-$4$ node will reduce it to a
$(6,12)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal4:sub:b}: removing either topmost degree-$4$ node will
reduce it to a $(6,12)$ absorption set.
\item Two connections: there are two cases:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Removing the bottom-right degree-$4$ node in \figurename \ref{figinternal4:sub:c} will
reduce it to a $(6,12)$ absorption set.
\item\label{check2} It is straightforward that there is only one possible topology to satisfy \figurename \ref{figinternal4:sub:d}:
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=10pt{
\CIRCLE\ar@{-}[rr]\ar@{-}[dr]\ar@{-}[drrr]\ar@{-}[drrrrr]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dr]\ar@{-}[drrr]&& \CIRCLE\ar@{-}[rr]\ar@{-}[dr]\ar@{-}[dl]\ar@{-}[dlll]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dlllll]\\
& \CIRCLE&& \CIRCLE\ar@{-}[rr]\ar@{-}[ll]&& \CIRCLE&\\}$$ We have to
turn to $\mathbf{H}$ to show its non-existence.
\end{enumerate}
\item Three connections: there are three cases:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Removing the bottom-right degree-$4$ node in \figurename \ref{figinternal4:sub:e} will
reduce it to a $(6,12)$ absorption set.
\item\label{check3} It is straightforward that there is only one possible topology to satisfy \figurename \ref{figinternal4:sub:f}:
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=10pt{
\CIRCLE\ar@{-}[rr]\ar@{-}[dr]\ar@{-}[drrr]\ar@{-}[drrrrr]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dr]\ar@{-}[rr]&& \CIRCLE\ar@{-}[rr]\ar@{-}[dr]\ar@{-}[dl]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dlllll]\\
& \CIRCLE\ar@/_0.8pc/@{-}[rrrr]&& \CIRCLE\ar@{-}[rr]\ar@{-}[ll]&&
\CIRCLE&\\}$$
\bigskip We have to turn to $\mathbf{H}$ to show its non-existence.
\item\label{check4} It is straightforward that there is only one possible topology to satisfy \figurename \ref{figinternal4:sub:g}:
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=10pt{
\CIRCLE\ar@{-}[rr]\ar@{-}[dr]\ar@{-}[drrr]\ar@{-}[drrrrr]&& \CIRCLE\ar@{-}[dr]\ar@{-}[rr]\ar@/^0.8pc/@{-}[rrrr]&& \CIRCLE\ar@{-}[dlll]\ar@{-}[dl]\ar@{-}[dr]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dlllll]\\
& \CIRCLE\ar@/_0.8pc/@{-}[rrrr]&& \CIRCLE\ar@{-}[rr]\ar@{-}[ll]&&
\CIRCLE&\\}$$
\medskip We have to turn to $\mathbf{H}$ to show its non-existence.
\end{enumerate}
\end{enumerate}
\item\label{check5} $[6,6,4,4,4,4,4]$: it is straightforward to see that there is only one possible topology in this class:
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=30pt{
\CIRCLE\ar@{-}[r]\ar@{-}[dr]\ar@{-}[drrr]\ar@/^0.8pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[drr]& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[dr]& \CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dll]& \CIRCLE \ar@{-}[dl]\ar@{-}[dlll]\\
& \CIRCLE\ar@{-}[rr]&& \CIRCLE&\\}$$ We have to turn to $\mathbf{H}$
to show its non-existence.
\end{enumerate}
\end{enumerate}
\begin{figure}[!t]
\centering \subfigure[]
{
\label{figinternal3:sub:a}
$\xymatrix@M=0pt@W=0pt@R=20pt@C=7pt{
&\CIRCLE&\\
\CIRCLE&&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal3:sub:b}
$\xymatrix@M=0pt@W=0pt@R=20pt@C=7pt{
&\CIRCLE&\\
\CIRCLE\ar@{-}[rr]&&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal3:sub:c}
$\xymatrix@M=0pt@W=0pt@R=20pt@C=7pt{
&\CIRCLE\ar@{-}[dl]\ar@{-}[dr]&\\
\CIRCLE&&\CIRCLE\\}$ } \caption{Possible intrinsic connections
among three nodes.}
\label{figinternal3:sub}
\end{figure}
\begin{figure}[!t]
\centering \subfigure[]
{
\label{figinternal4:sub:a}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE&\CIRCLE\\
\CIRCLE&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal4:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE&\CIRCLE\\
\CIRCLE\ar@{-}[r]&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal4:sub:c}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE\ar@{-}[r]\ar@{-}[d]&\CIRCLE\\
\CIRCLE&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal4:sub:d}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE\ar@{-}[r]&\CIRCLE\\
\CIRCLE\ar@{-}[r]&\CIRCLE\\}$ }\\
\subfigure[]
{
\label{figinternal4:sub:e}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE\ar@{-}[r]\ar@{-}[d]&\CIRCLE\ar@{-}[dl]\\
\CIRCLE&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal4:sub:f}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE\ar@{-}[r]\ar@{-}[d]&\CIRCLE\ar@{-}[d]\\
\CIRCLE&\CIRCLE\\}$ }\qquad
\subfigure[]
{
\label{figinternal4:sub:g}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt{
\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]&\CIRCLE\\
\CIRCLE&\CIRCLE\\}$ } \caption{Possible intrinsic connections among
four nodes.}
\label{figinternal4:sub}
\end{figure}
\subsection{$b=12$}
The extrinsic degree is large now. We will see in the $a=8$ section
that there are smaller $b$'s and $(7,12)$ absorption sets do exist as a reduction from them.
\subsection{$b=14$}
Table \ref{table714} shows the existence of $(7,14)$ sets.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(7,14)$ absorption set.} \label{table714}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes}
\\\hline\hline 0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline 740&21&120&138&236&310&325\\\hline
1801&21&87&184&202&300&374\\\hline
1765&14&125&169&199&266&376\\\hline 43&10&74&138&202&266&330\\\hline
862&10&125&174&242&285&325\\\hline
\end{tabular}\end{table}
\bigskip
If we allow the satisfied check nodes to be connected to the set more more than twice, it is clear that only one check node could be connected to the set four times, as shown in \figurename \ref{figb4}, which is a $(7,14)$ absorption set. Now, there can be no other intrinsic connections among the four variable nodes at the top --- this would create a $4$-cycle. Thus depending on the intrinsic connections among the three variable nodes at the bottom, we could obtain $(7,12)$, $(7,10)$ or $(7,8)$ absorption sets, respectively. Both $(7,14)$ and $(7,12)$ absorption sets exist and the $(7,8)$ can reduce to $(5,10)$ by removing any two degree-$4$ nodes and does not exist therefore. Only $(7,10)$ sets, \figurename \ref{figb41}, need to be searched.
\begin{figure}[!t]
\centering \subfigure[$(7,14)$]
{
\label{figb4}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=0.18pt
{
&&&&&&& \boxplus\ar@{-}[dll]\ar@{-}[dllllll]\ar@{-}[drr]\ar@{-}[drrrrrr]&&&&&&&\\
& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&\\
\boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus\\
&&& \CIRCLE\ar@{-}[ulll]\ar@{-}[ur]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrr]&&&& \CIRCLE\ar@{-}[ull]\ar@{-}[ullllll]\ar@{-}[urr]\ar@{-}[urrrrrr]&&&& \CIRCLE\ar@{-}[ulllllllll]\ar@{-}[ulllll]\ar@{-}[ul]\ar@{-}[urrr]&&&\\
}$ }
\subfigure[$(7,10)$]
{
\label{figb41}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=0.18pt
{
&&&&&&& \boxplus\ar@{-}[dll]\ar@{-}[dllllll]\ar@{-}[drr]\ar@{-}[drrrrrr]&&&&&&&\\
& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&\\
\boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus&& \boxplus& \boxplus& \boxplus\\
&&& \CIRCLE\ar@{-}[rr]\ar@{-}[ulll]\ar@{-}[ur]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrr]&& \boxplus&& \CIRCLE\ar@{-}[ll]\ar@{-}[rr]\ar@{-}[ull]\ar@{-}[ullllll]\ar@{-}[urr]\ar@{-}[urrrrrr]&& \boxplus&& \CIRCLE\ar@{-}[ll]\ar@{-}[ulllllllll]\ar@{-}[ulllll]\ar@{-}[ul]\ar@{-}[urrr]&&&\\
}$}
\caption{One check node connecting to size-$7$ sets four times.}
\label{figbbbb4:sub}
\end{figure}
After searching the topologies in \ref{check1}, \ref{check2},
\ref{check3}, \ref{check4}, \ref{check5} and \figurename \ref{figb41} with $\mathbf{H}$ algorithmically, Lemma
\ref{thma=7} follows.
\section{Proof of Lemma \ref{lema=8}}\label{prooflema=8}
Again, we first suppose that all satisfied check nodes are connected to the
set twice. We apply the constraints in Definition \ref{constraints} and the pigeonhole principle
to prove this.
\subsection{$b=0$}
In other words, a class of $[6,6,6,6,6,6,6,6]$ absorption
sets. We obtain the perfectly symmetric \figurename \ref{fig7} again as \figurename \ref{fig1:sub:b}.
\begin{figure}[!t]
\centering
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]\ar@{-}[ddd]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddll]\ar@{-}[dr]\ar@{-}[ddr]\ar@{-}[ddd]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddr]\ar@{-}[ddrr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\ar@{-}[ddll]\\
\CIRCLE\ar@{-}[dr]\ar@{-}[drr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dll]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ \caption{Only possible topology of $(8,0)$ absorption
sets.}\label{fig7}
\end{figure}
Removing any node will reduce it to a $(7,6)$ absorption set.
\subsection{$b=2$}
There are two classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,6,6,6,6,5,5]$: removing either degree-$5$ node generates a $(7,6)$ absorption set.
\item $[6,6,6,6,6,6,6,4]$: removing the degree-$4$ node generates a $(7,4)$ absorption set.
\end{enumerate}
\subsection{$b=4$}
There are three classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,6,6,5,5,5,5]$: removing any degree-$5$ node generates a $(7,8)$ absorption set.
\item $[6,6,6,6,6,5,5,4]$: removing the degree-$4$ node generates a $(7,6)$ absorption set.
\item $[6,6,6,6,6,6,4,4]$: removing both degree-$4$ nodes generates a $(6,10)$ or a $(6,8)$ absorption set.
\end{enumerate}
\subsection{$b=6$}
There are four classes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,5,5,5,5,5,5]$: removing any degree-$5$ node generates a $(7,10)$ absorption set.
\item $[6,6,6,5,5,5,5,4]$: removing the degree-$4$ node generates a $(7,8)$ absorption set.
\item $[6,6,6,6,5,5,4,4]$: let us study the intrinsic
connections between the two degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Not connected as shown in \figurename \ref{fig2degree4:sub:a}: Removing either degree-$4$ node will reduce it to a $(7,8)$
absorption set.
\item Connected as shown in \figurename \ref{fig2degree4:sub:b}: We consider the
connections between the two degree-$4$ nodes and the other six
nodes in the set. There are six edges emanating from the two degree-$4$
nodes and at least four of the six edges must go to the four
degree-$6$ nodes, respectively. So at most two edges emanating from
the two degree-$4$ nodes can be connected to the two degree-$5$
nodes.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item If either of the two degree-$5$ nodes is connected to the two degree-$4$ nodes at most once:
removing both degree-$4$ nodes generates a $(6,8)$ absorption set.
\item If one degree-$5$ node is connected to both degree-$4$ nodes: removing the other degree-$5$
node generates a $(7,10)$ absorption set.
\end{enumerate}
\end{enumerate}
\item $[6,6,6,6,6,4,4,4]$: let us study the intrinsic
connections among the three degree-$4$ nodes. There are five degree-$6$ nodes, which require ten edges from the
three degree-$4$ nodes. Thus there should be no more than one connection among them.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal3:sub:a}: removing any degree-$4$ node will reduce it to a
$(7,8)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal3:sub:b}: removing the topmost degree-$4$ node
will reduce it to a $(7,8)$ absorption set.
\end{enumerate}
\end{enumerate}
\subsection{$b=8$}
There are four classes by assuming there is a degree-$6$ node:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,5,5,5,5,5,5,4]$: removing the degree-$4$ node
will reduce it to a $(7,10)$ absorption set.
\item $[6,6,5,5,5,5,4,4]$: let us study the intrinsic
connections between the two degree-$4$ nodes:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Not connected as shown in \figurename \ref{fig2degree4:sub:a}: Removing either degree-$4$ node will reduce it to a $(7,10)$
absorption set.
\item Connected as shown in \figurename \ref{fig2degree4:sub:b}: We consider the
connections between the two degree-$4$ nodes and the other six
nodes in the set. There are six edges emanating from the two degree-$4$
nodes and at least two and at most four of the six edges must go to the two
degree-$6$ nodes, respectively. So at most four and at least two edges emanating from
the two degree-$4$ nodes can be connected to the four degree-$5$
nodes.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Two edges between the group of degree-$4$ nodes and the group of degree-$6$ nodes:
$$\xymatrix@M=0pt@W=0pt@R=30pt@C=19pt{
\text{degree-$4$ nodes:}&\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]& \CIRCLE&&\CIRCLE\ar@{-}[r]\ar@{-}[d]& \CIRCLE\ar@{-}[d]\\
\text{degree-$6$ nodes:}&\CIRCLE& \CIRCLE&&\CIRCLE& \CIRCLE\\}$$ Note that under the conditions in the above two cases, the two degree-$6$ nodes have to be connected with each other and either of them has to be connected to all the degree-$5$ nodes. In addition, there are four edges coming from the degree-$4$ nodes to the four degree-$5$ nodes. Thus,
\begin{enumerate}\setlength{\itemsep}{0pt}
\item if there is no degree-$5$ node sharing the two degree-$4$ nodes: removing the two degree-$4$ nodes will reduce it to a $(6,10)$ absorption sets.
\item if there is one and at most one degree-$5$ node sharing the two degree-$4$ nodes: the topologies will be fixed, respectively, as
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=25pt{
\color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]\ar@{-}[drrrrr]& \color{blue}\CIRCLE\ar@{-}[drrr]\ar@{-}[drr]\ar@{-}[drrrr] &&&&\\
\CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]\ar@/_4pc/@{-}[rrrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]& \CIRCLE\ar@{-}[r]& \CIRCLE& \color{blue}\CIRCLE\\
&&&&&\\}$$
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=25pt{
\color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[drrr]\ar@{-}[drrrrr]& \color{blue}\CIRCLE\ar@{-}[drrr]\ar@{-}[d]\ar@{-}[drrrr] &&&&\\
\CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]\ar@/_4pc/@{-}[rrrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]& \CIRCLE \ar@{-}[r]& \CIRCLE& \color{blue}\CIRCLE\\
&&&&&\\}$$
Removing the two degree-$4$ nodes and that degree-$5$ node will reduce them to $(5,10)$ absorption sets.
\end{enumerate}
\item Three edges between the group of degree-$4$ nodes and the group of degree-$6$ nodes:
$$\xymatrix@M=0pt@W=0pt@R=30pt@C=19pt{
\text{degree-$4$ nodes:}&\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]& \CIRCLE\ar@{-}[dl]\\
\text{degree-$6$ nodes:}&\CIRCLE& \CIRCLE\\}$$ Note that under the conditions in the above case, the two degree-$6$ nodes have to be connected with each other and the bottom-right degree-$6$ node has to be connected to all the degree-$5$ nodes. In addition, there are three edges emanating from the degree-$4$ nodes to the four degree-$5$ nodes. Thus,
\begin{enumerate}\setlength{\itemsep}{0pt}
\item if there is no degree-$5$ node sharing the two degree-$4$ nodes: removing the two degree-$4$ nodes will reduce it to a $(6,10)$ absorption sets.
\item if there is one and at most one degree-$5$ node sharing the two degree-$4$ nodes: the topology will be fixed as
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=25pt{
\color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]\ar@{-}[drrrrr]& \color{blue}\CIRCLE\ar@{-}[drrr]\ar@{-}[dl]\ar@{-}[drrrr] &&&&\\
\CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]& \CIRCLE& \color{blue}\CIRCLE\\
&&&&&\\}$$
Removing the two degree-$4$ nodes and that degree-$5$ node will reduce it to a $(5,10)$ absorption sets.
\end{enumerate}
\item Four edges between the group of degree-$4$ nodes and the group of degree-$6$ nodes:
$$\xymatrix@M=0pt@W=0pt@R=30pt@C=19pt{
\text{degree-$4$ nodes:}&\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]& \CIRCLE\ar@{-}[d]\ar@{-}[dl]\\
\text{degree-$6$ nodes:}&\CIRCLE& \CIRCLE\\}$$
\begin{enumerate}\setlength{\itemsep}{0pt}
\item if there is no degree-$5$ node sharing the two degree-$4$ nodes: removing the two degree-$4$ nodes will reduce it to a $(6,10)$ absorption sets.
\item if there is one and at most one degree-$5$ node sharing the two degree-$4$ nodes: the topology will be fixed as
$$\xymatrix@M=0pt@W=0pt@R=40pt@C=25pt{
\color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@{-}[dr]\ar@{-}[drrrrr]& \color{blue}\CIRCLE\ar@{-}[d]\ar@{-}[dl]\ar@{-}[drrrr] &&&&\\
\CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]\ar@/_3pc/@{-}[rrrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]\ar@/_2pc/@{-}[rrr]& \CIRCLE\ar@{-}[r]\ar@/_1pc/@{-}[rr]& \CIRCLE\ar@{-}[r]& \color{blue}\CIRCLE\\
&&&&&\\}$$
Removing the two degree-$4$ nodes and that degree-$5$ node will reduce it to a $(5,10)$ absorption sets.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\item $[6,6,6,5,5,4,4,4]$: There should be no more than two intrinsic connections among the
three degree-$4$ nodes. Otherwise, the group of degree-$4$ nodes will not match the group of the other five nodes remained
in the set.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection as shown in \figurename \ref{figinternal3:sub:a}: removing any degree-$4$ node will reduce it to a
$(7,10)$ absorption set.
\item One connection as shown in \figurename \ref{figinternal3:sub:b}: removing the topmost degree-$4$ node
will reduce it to a $(7,10)$ absorption set.
\item Two connections: Now, there are eight edges coming out the
group of degree-$4$ nodes as shown in \figurename \ref{figinternal3:sub:c}. However, each of the two degree-$5$ nodes
and each of the three degree-$6$ nodes needs one and two connections emanating
from the group of degree-$4$ nodes, respectively. That makes eight.
Thus, removing the three degree-$4$ nodes will reduce it to a $(5,10)$
absorption set.
\end{enumerate}
\item $[6,6,6,6,4,4,4,4]$: The group of degree-$6$ nodes need at
least twelve edges from the group of degree-$4$ nodes. So there should be no more than two connections among the four degree-$4$
nodes.
\begin{enumerate}\setlength{\itemsep}{0pt}
\item No connection: removing any degree-$4$ node in \figurename \ref{figinternal4:sub:a} will reduce it to a
$(7,10)$ absorption set.
\item One connection: removing either topmost degree-$4$ node in \figurename \ref{figinternal4:sub:b} will
reduce it to a $(7,10)$ absorption set.
\item Two connections: there are two cases:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item Removing the bottom-right degree-$4$ node in \figurename \ref{figinternal4:sub:c} will
reduce it to a $(7,10)$ absorption set.
\item Removing either the top or the bottom couple of
degree-$4$ nodes in \figurename \ref{figinternal4:sub:d} will reduce it to a $(6,10)$ absorption set.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\bigskip
If a neighboring check node connecting to the set four times is considered, then the smallest such size-$8$ absorption set would be $(8,4)$, \figurename \ref{figa=8b484}. Removing any two degree-$5$ nodes will reduce it to a $(6,10)$ absorption set. To obtain such $(8,6)$ sets, let us remove one edge from \figurename \ref{figa=8b484}.
\begin{figure}[!t]
\centering
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@/_2pc/@{-}[rrrrrrrrrrrrrrrrrrrrrrrr]|-{\boxplus}\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$
\caption{One check node connecting to $(8,4)$ sets four times.}
\label{figa=8b484}
\end{figure}
\begin{figure}[!t]
\subfigure[]
{
\label{figa=8b486:sub:a}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$
\subfigure[]
{
\label{figa=8b486:sub:b}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@/_2pc/@{-}[rrrrrrrrrrrrrrrrrrrrrrrr]|-{\boxplus}\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$ }
\caption{One check node connecting to $(8,6)$ sets four times.}
\label{figa=8b486:sub}
\end{figure}
By removing either the bottom-left or the bottom-right degree-$5$ node from \figurename \ref{figa=8b486:sub:a}, or the degree-$4$ node from \figurename \ref{figa=8b486:sub:b}, respectively, we obtain a $(7,10)$ absorption set.
Then to obtain such $(8,8)$ absorption sets, first we remove one edge from \figurename \ref{figa=8b486:sub}, which gives us \figurename \ref{figa=8b488:sub}. In addition, the possible topology with two neighboring check nodes connecting to the set four times is shown in \figurename \ref{figa=8b488:sub:g}. Removing the degree-$4$ node, the bottom-left degree-$4$ node or the bottom-right degree-$4$ node in \figurename \ref{figa=8b488:sub:a}, \ref{figa=8b488:sub:c} or \ref{figa=8b488:sub:e}, respectively, reduces it to a $(7,10)$ absorption set, while removing the two degree-$4$ nodes in \figurename \ref{figa=8b488:sub:f} reduces it to a $(6,12)$ absorption set.
Note that \figurename \ref{figa=8b488:sub:b} and \ref{figa=8b488:sub:g} are in the class $[5,5,5,5,5,5,5,5]$, so after searching $\mathbf{H}$ with \figurename \ref{figa=8b488:sub:d}, Lemma \ref{lema=8} results.
\begin{figure}[!t]
\subfigure[]
{
\label{figa=8b488:sub:a}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&&&&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$
\subfigure[]
{
\label{figa=8b488:sub:b}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&&&&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$ }\\
\subfigure[]
{
\label{figa=8b488:sub:c}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$
\subfigure[]
{
\label{figa=8b488:sub:d}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$ }\\
\subfigure[]
{
\label{figa=8b488:sub:e}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \hspace{9pt}\\
&&& \CIRCLE\ar@/_2pc/@{-}[rrrrrrrrrrrrrrrrrrrrrrrr]|-{\boxplus}\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]&&&\\
}$
\subfigure[]
{
\label{figa=8b488:sub:f}
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \hspace{9pt}&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@/_2pc/@{-}[rrrrrrrrrrrrrrrrrrrrrrrr]|-{\boxplus}\ar@{-}[rrrr]\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[rrrr]\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&& \boxplus&&&& \CIRCLE\ar@{-}[llll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
}$ }\caption{One check node connecting to $(8,8)$ sets four times.}
\label{figa=8b488:sub}
\end{figure}
\begin{figure}[!t]
\centering
$\xymatrix@M=0pt@W=0pt@R=40pt@C=-4.7pt
{
&&&&&&&&&&&&&&&\boxplus\ar@{-}[dllll]\ar@{-}[dllllllllllll]\ar@{-}[drrrr]\ar@{-}[drrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&&&&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dlll]\ar@{-}[dr]\ar@{-}[drrr]&&&\\
\boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus&\hspace{9pt}& \boxplus\\
&&& \CIRCLE\ar@{-}[ulll]\ar@{-}[urrrrr]\ar@{-}[urrrrrrrrrrrrr]\ar@{-}[urrrrrrrrrrrrrrrrrrrrr]&&&&&&&& \CIRCLE\ar@{-}[ul]\ar@{-}[ulllllllll]\ar@{-}[urrrrrrr]\ar@{-}[urrrrrrrrrrrrrrr]&&&&&&&& \CIRCLE\ar@{-}[ulllllllllllllll]\ar@{-}[ulllllll]\ar@{-}[ur]\ar@{-}[urrrrrrrrr]&&&&&&&& \CIRCLE\ar@{-}[ulllllllllllllllllllll]\ar@{-}[ulllllllllllll]\ar@{-}[ulllll]\ar@{-}[urrr]&&&\\
&&&&&&&&&&&&&&& \boxplus\ar@{-}[ullll]\ar@{-}[ullllllllllll]\ar@{-}[urrrr]\ar@{-}[urrrrrrrrrrrr]&&&&&&&&&&&&&&&\\
}$\caption{Two check nodes connecting to $(8,8)$ set four times.}
\label{figa=8b488:sub:g}
\end{figure}
\section{Proof of Claim \ref{claima=8}}\label{proofclaima=8}
If a check node connecting to the set an even number, but more than twice is allowed, we obtain \figurename \ref{figa=8b488:sub:b} and \ref{figa=8b488:sub:g}. Let us find the other three by restricting that a satisfied check node can only be connected to the set twice.
We start with one node: $$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \CIRCLE& \CIRCLE&\\
}$$
As Step 2, if the bottom two nodes are
\subsection{not connected}
We must have \figurename \ref{fig88a:sub:a}.
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig88a:sub:a}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uuur]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[ull]\ar@{-}[uull]\ar@{-}[uuu]\ar@{-}[uur]\ar@{-}[ur]&\\
}$ }\quad
\subfigure[]
{
\label{fig88a:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]&\color{blue} \CIRCLE\ar@{-}[dll]\ar@{-}[dr]&\\
\color{blue}\CIRCLE\ar@{-}[d] &&& \color{blue}\CIRCLE\ar@{-}[d]\\
\color{blue}\CIRCLE\ar@{-}[rrr]&&& \color{blue}\CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uuur]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[ull]\ar@{-}[uull]\ar@{-}[uuu]\ar@{-}[uur]\ar@{-}[ur]&\\
}$ } \caption{First possible topology of $(8,8)$ absorption sets.}
\label{fig88a:sub}
\end{figure}
After connecting the remaining five nodes we obtain \figurename \ref{fig88a:sub:b}.
Reorganize \figurename \ref{fig88a:sub:b} into a symmetric form as \figurename \ref{fig12:sub:a}.
\subsection{connected}
There is only one choice for either one of them as shown in \figurename \ref{fig88b:sub:a}.
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig88b:sub:a}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \CIRCLE&\\
}$
\subfigure[]
{
\label{fig88b:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[uull]\ar@{-}[ull]&\\
}$
\subfigure[]
{
\label{fig88b:sub:c}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[uuu]\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[ull]&\\
}$ } \caption{Finding possible topology of $(8,8)$ absorption sets -- Step 3.}
\label{fig88b:sub}
\end{figure}
As Step 3, now there are two choices for the bottom-right node as shown in \figurename \ref{fig88b:sub:b}--\ref{fig88b:sub:c}:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item \figurename \ref{fig88b:sub:b}: as Step 4, the top-right node only has one choice as shown in \figurename \ref{fig88c:sub:a}. Then the other nodes have no choice. We obtain another possible topology as shown in \figurename \ref{fig12:sub:b}.
\item \figurename \ref{fig88b:sub:c}: as Step 4, this top-right node has two choices:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item \figurename \ref{fig88c:sub:b}: By connecting the remaining nodes, we obtain \figurename \ref{fig12:sub:c}.
\item \figurename \ref{fig88c:sub:c}: After connecting the remaining node, this gives us \figurename \ref{fig12:sub:a} again.
\end{enumerate}
\end{enumerate}
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig88c:sub:a}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \color{blue}\CIRCLE\ar@{-}[dr]\ar@{-}[ddr]\ar@{-}[dll]\ar@{-}[ddll]&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[uull]\ar@{-}[ull]&\\
}$
\subfigure[]
{
\label{fig88c:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \color{blue}\CIRCLE\ar@{-}[dll]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[uuu]\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[ull]&\\
}$
\subfigure[]
{
\label{fig88c:sub:c}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \color{blue}\CIRCLE\ar@{-}[ddll]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE &&& \CIRCLE\\
\CIRCLE&&& \CIRCLE\\
& \color{blue}\CIRCLE\ar@{-}[r]\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \color{blue}\CIRCLE\ar@{-}[uuu]\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[ull]&\\
}$ } \caption{Finding possible topology of $(8,8)$ absorption sets -- Step 4.}
\label{fig88c:sub}
\end{figure}
So eventually we obtain the other three
possible $(8,8)$ topologies as shown in \figurename \ref{fig12:sub}.
\begin{figure}[!t]
\centering \subfigure[]
{
\label{fig12:sub:a}
$\xymatrix@M=0pt@W=0pt@R=27pt@C=10pt
{
\CIRCLE\ar@{-}[rr]\ar@{-}[drrrr]\ar@{-}[drr]\ar@{-}[ddr]\ar@{-}[d]&& \CIRCLE\ar@{-}[rr]\ar@{-}[drr]\ar@{-}[dll]\ar@{-}[d]&& \CIRCLE\ar@{-}[dllll]\ar@{-}[dll]\ar@{-}[ddl]\ar@{-}[d]\\
\CIRCLE\ar@{-}[drrr]\ar@{-}[dr]&& \CIRCLE\ar@{-}[dl]\ar@{-}[dr]&& \CIRCLE\ar@{-}[dlll]\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[rr]&& \CIRCLE&\\
}$
\subfigure[]
{
\label{fig12:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddll]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddr]\ar@{-}[ddrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\ar@{-}[ddll]\\
\CIRCLE\ar@{-}[dr]\ar@{-}[drr]&&& \CIRCLE\ar@{-}[dl]\ar@{-}[dll]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$
\subfigure[]
{
\label{fig12:sub:c}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt
{
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddd]& \CIRCLE\ar@{-}[dll]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE \ar@{-}[ddrr]\ar@{-}[d]&&& \CIRCLE\ar@{-}[d]\\
\CIRCLE \ar@{-}[rrr]&&& \CIRCLE\\
& \CIRCLE\ar@{-}[ul]\ar@{-}[uul]\ar@{-}[uurr]\ar@{-}[urr]& \CIRCLE\ar@{-}[uuu]\ar@{-}[ur]\ar@{-}[uur]\ar@{-}[ull]&\\
}$ } \caption{Possible topologies of $(8,8)$ absorption
sets.}
\label{fig12:sub}
\end{figure}
\section{Less Dominant Absorption Sets}\label{less}
We list some examples in this section to show the existence of less dominant absorption sets.
There are two classes of $(8,12)$ absorption sets:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[6,6,4,4,4,4,4,4]$: $11,008$ such sets;
\item $[5,5,5,5,4,4,4,4]$: $33,408$ such sets.
\end{enumerate}
Some topologies are shown in \figurename \ref{top812:sub}.
\begin{figure}[!t]
\centering \subfigure[]
{
\label{top812:sub:a}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt {
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddrr]\ar@{-}[ddd]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddll]\ar@{-}[dr]\ar@{-}[ddr]\ar@{-}[ddd]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddll]\\
\CIRCLE\ar@{-}[dr]&&& \CIRCLE\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\qquad
\subfigure[]
{
\label{top812:sub:sub:b}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt {
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddl]\ar@{-}[drr]\ar@{-}[ddd]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddd]\ar@{-}[dr]\ar@{-}[ddr]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\\
\CIRCLE\ar@{-}[dr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\\
\subfigure[]
{
\label{top812:sub:sub:c}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt {
& \CIRCLE\ar@{-}[r]\ar@{-}[dl]\ar@{-}[ddrr]\ar@{-}[drr]\ar@{-}[ddd]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddd]\ar@{-}[dr]\ar@{-}[ddll]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\\
\CIRCLE\ar@{-}[dr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dl]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ }\qquad
\subfigure[]
{
\label{top812:sub:sub:d}
$\xymatrix@M=0pt@W=0pt@R=15.78pt@C=15.78pt {
& \CIRCLE\ar@{-}[r]\ar@{-}[ddl]\ar@{-}[ddrr]\ar@{-}[drr]\ar@{-}[ddd]& \CIRCLE \ar@{-}[dll]\ar@{-}[ddd]\ar@{-}[ddll]&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddrr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddl]\ar@{-}[ddll]\\
\CIRCLE\ar@{-}[drr]\ar@{-}[rrr]&&& \CIRCLE\ar@{-}[dll]\\
& \CIRCLE\ar@{-}[r]& \CIRCLE&\\
}$ } \caption{Some topologies of $(8,12)$ absorption sets.}
\label{top812:sub}
\end{figure}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(8,14)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
1084&2&120&174&243&301&377\\\hline
116&46&76&184&243&256&372\\\hline
561&39&112&134&248&303&334\\\hline
870&2&76&135&192&264&334\\\hline
1091&46&79&135&237&303&329\\\hline
1970&0&69&134&199&264&329\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(8,16)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
1045&23&110&184&199&306&336\\\hline
1440&23&76&174&248&263&370\\\hline
39&15&79&143&207&271&335\\\hline
1048&19&76&143&253&262&354\\\hline
1444&19&110&140&207&317&326\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(9,12)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
1563&29&120&150&217&264&336\\\hline
1628&40&75&171&248&264&373\\\hline
176&43&78&174&251&267&376\\\hline
314&8&125&150&251&300&368\\\hline
560&40&78&177&199&313&368\\\hline
1258&29&79&137&213&313&370\\\hline
1988&30&125&171&213&267&336\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(9,16)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
1460&2&79&184&238&307&365\\\hline
580&7&102&148&253&307&376\\\hline
890&15&87&181&229&261&349\\\hline
1194&38&87&148&228&319&360\\\hline
1775&2&126&176&228&261&338\\\hline
1881&33&84&176&229&300&360\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(9,18)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
629&15&86&161&248&268&381\\\hline
1063&1&109&178&236&312&349\\\hline
504&20&85&174&197&258&338\\\hline
802&49&86&154&197&300&363\\\hline
1299&6&124&178&247&305&381\\\hline
1789&20&109&150&247&265&363\\\hline
\end{tabular}
\end{table}
There exist two classes of $(10,10)$ absorption sets:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item $[5,5,5,5,5,5,5,5,5,5]$: $192$ such sets;
\item $[6,6,6,5,5,5,5,4,4,4]$: unknown.
\end{enumerate}
\begin{figure}[!t]
\centering $\xymatrix@M=0pt@W=0pt@R=25pt@C=70pt {
\CIRCLE\ar@{-}[r]\ar@{-}[d]\ar@/_1pc/@{-}[dd]\ar@/_2pc/@{-}[ddd]\ar@/_3pc/@{-}[dddd]&\CIRCLE \ar@{-}[ld]\ar@{-}[lddd]\ar@{-}[d]\ar@/^3pc/@{-}[dddd]\\
\CIRCLE\ar@{-}[r]\ar@{-}[rd]\ar@/_1pc/@{-}[dd] &\CIRCLE \ar@{-}[d]\\
\CIRCLE\ar@{-}[r]\ar@{-}[rd]\ar@{-}[rdd]\ar@/_1pc/@{-}[dd]&\CIRCLE \\
\CIRCLE\ar@{-}[r]\ar@{-}[d]&\CIRCLE\ar@{-}[u]\ar@/_1pc/@{-}[uu] \\
\CIRCLE\ar@{-}[r]\ar@{-}[ruuu]&\CIRCLE\ar@{-}[u]\ar@/_1pc/@{-}[uu] \\
}$ \caption{Topology of $[5,5,5,5,5,5,5,5,5,5]$ absorptions
sets.}\label{fig14}
\end{figure}
The average multiplicity of each variable node appeared in class $[5,5,5,5,5,5,5,5,5,5]$
is ${192\times 10}/{2048}=0.9375$. Once again, certain groups of
variable nodes share the same multiplicity, as listed in Table
\ref{table3}. As we can see, some groups are not involved at all.
Therefore, the average multiplicity of each involved variable node
in such sets is ${192\times 10}/{1408}\approx1.3636$. Like the $(8,8)$ ones,
for this class $\mu_{\rm max}=4$ and $\mathbf{v}_{\rm max}$ is an all-$1$ vector, since $\mathbf{V C}$ is a
probability matrix as well.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{The
multiplicity of each variable node in class $[5,5,5,5,5,5,5,5,5,5]$ of $(10,10)$ absorption sets.}
\label{table3}
\centering
\begin{tabular}{|c|c||c|c|}\hline
\bfseries Variable Nodes & \bfseries Multiplicities & \bfseries Variable Nodes &
\bfseries Multiplicities\\\hline\hline
$0$---$63$&$1$&$1024$---$1087$&$1$\\\hline
$64$---$127$&$1$&$1088$---$1151$&$0$\\\hline
$128$---$191$&$1$&$1152$---$1215$&$0$\\\hline
$192$---$255$&$1$&$1216$---$1279$&$2$\\\hline
$256$---$319$&$0$&$1280$---$1343$&$2$\\\hline
$320$---$383$&$2$&$1344$---$1407$&$0$\\\hline
$384$---$447$&$0$&$1408$---$1471$&$1$\\\hline
$448$---$511$&$2$&$1472$---$1535$&$1$\\\hline
$512$---$575$&$1$&$1536$---$1599$&$1$\\\hline
$576$---$639$&$1$&$1600$---$1663$&$0$\\\hline
$640$---$703$&$1$&$1664$---$1727$&$2$\\\hline
$704$---$767$&$2$&$1728$---$1791$&$2$\\\hline
$768$---$831$&$0$&$1792$---$1855$&$1$\\\hline
$832$---$895$&$2$&$1856$---$1919$&$1$\\\hline
$896$---$959$&$1$&$1920$---$1983$&$0$\\\hline
$960$---$1023$&$0$&$1984$---$2047$&$0$\\\hline
\end{tabular}
\end{table}
Table \ref{table1010b} shows the existence of another class of $(10,10)$ absorption sets: $[6,6,6,5,5,5,5,4,4,4]$.
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{Another class of $(10,10)$ absorption sets.}
\label{table1010b} \centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
591&56&65&159&207&302&327\\ \hline
1405&32&120&159&225&314&337\\ \hline
1904&0&119&184&249&314&379\\ \hline
210&42&65&160&248&286&335\\ \hline
732&30&118&157&223&312&335\\ \hline
1676&36&77&183&223&286&376\\ \hline
616&30&119&160&194&275&373\\ \hline
834&42&85&157&249&302&373\\ \hline
892&36&118&142&225&275&327\\ \hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(10,12)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
1460&2&79&184&238&307&365\\\hline
1320&46&67&140&248&307&320\\ \hline
1543&51&72&145&253&312&320\\ \hline
931&45&88&160&252&305&376\\\hline
9&46&110&174&238&302&366\\\hline
1104&33&67&145&252&282&349\\ \hline
1316&51&88&156&199&302&365\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(10,14)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
1460&2&79&184&238&307&365\\\hline
931&45&88&160&252&305&376\\\hline
9&46&110&174&238&302&366\\\hline
1121&15&110&156&198&315&321\\\hline
1316&51&88&156&199&302&365\\\hline
1432&33&121&160&226&315&338\\\hline
1549&45&121&142&198&300&366\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(10,16)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
1045&23&110&184&199&306&336\\\hline
176&43&78&174&251&267&376\\\hline
39&15&79&143&207&271&335\\\hline
305&23&78&132&201&259&335\\\hline
1048&19&76&143&253&262&354\\\hline
1189&43&76&132&234&300&326\\\hline
1444&19&110&140&207&317&326\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(10,18)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
170&49&124&184&196&283&366\\\hline
931&45&88&160&252&305&376\\\hline
253&63&124&174&226&259&336\\\hline
1121&15&110&156&198&315&321\\\hline
1432&33&121&160&226&315&338\\\hline
1549&45&121&142&198&300&366\\\hline
2016&9&113&156&196&259&349\\\hline
\end{tabular}
\end{table}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.1}
\caption{An example of $(10,20)$ absorption sets.}
\centering
\begin{tabular}{|c||c|c|c|c|c|c|}\hline
\bfseries Variable Nodes & \multicolumn{6}{c|}{\bfseries Six Neighboring Check Nodes} \\\hline\hline
0&56&120&184&248&312&376\\\hline
109&56&79&174&199&300&349\\\hline
90&15&120&140&253&305&338\\\hline
358&52&68&184&255&315&327\\\hline
1056&10&115&135&248&300&333\\\hline
574&15&80&169&255&316&333\\\hline
712&52&125&147&198&316&347\\\hline
862&10&125&174&242&285&325\\\hline
1207&25&68&140&232&285&356\\\hline
1837&42&79&147&253&293&356\\\hline
\end{tabular}
\end{table}
Note that $(7,12)$ and $(9,14)$ absorption sets (though not all of
them) can be obtained by removing one node from $(8,8)$ and
$(10,10)$ absorption sets, respectively. For example, there are
$179,648$ $(7,12)$ absorption sets. They all have the topology
\figurename \ref{top712}, which can be obtained from \figurename
\ref{fig13:sub} by removing one node. However, the $(8,8)$ sets
generate $14272\times8=114176$ $(7,12)$ absorption sets (no
duplicates). Hence there are $179648-114176=65472$ $(7,12)$ sets
that are not contained in the $(8,8)$ ones.
\begin{figure}[!t]
\centering $\xymatrix@M=0pt@W=0pt@R=15.78pt@C=3.3pt {
&& \CIRCLE\ar@{-}[rr]\ar@{-}[dll]\ar@{-}[ddll]\ar@{-}[drrrr]\ar@{-}[ddrrrr]&& \CIRCLE \ar@{-}[dllll]\ar@{-}[ddllll]\ar@{-}[drr]\ar@{-}[ddrr]&&\\
\CIRCLE \ar@{-}[d]\ar@{-}[ddrrr]&&&&&& \CIRCLE\ar@{-}[d]\ar@{-}[ddlll]\\
\CIRCLE\ar@{-}[drrr]&\hspace{10pt}&&&&\hspace{10pt}& \CIRCLE\ar@{-}[dlll]\\
&&& \CIRCLE&&&\\
}$ \caption{Topology of $(7,12)$ absorptions sets.}\label{top712}
\end{figure}
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,314,259,992,994 | arxiv |
\section*{Acknowledgments}\label{s:acks}
The work of the first author is partially supported by JSPS KAKENHI under Grant Number 18K11335.
Part of this research was done at Stanford University. This material is based upon the work supported by the Department of Energy under Award Number DE-NA0002373-1.
Some of the computing for this project was performed on the Sherlock cluster 2.0. We would like to thank Stanford University and the Stanford Research Computing Center for providing computational resources and support that contributed to these research results.
\section{Conclusion}\label{s:concl}
To perform IFMM (\autoref{s:ifmm}) efficiently, we proposed to parallelize its factorization and solve phases with respect to nodes in every level of the hierarchy. Unlike in FMM, a loop over nodes cannot be parallelized straightforwardly due to the data dependencies among nodes. To overcome it, we divide nodes in a level into groups (colors) so that any two nodes in each group are separated by $\separation L$ or more, where $\separation$ is a predefined parameter (termed to separation parameter) and $L$ denotes the edge length of the nodes (\autoref{s:grouping}). The number of groups is bounded from above by $\separation^3$. Then, if $\separation$ is 6 or more, we can prove that all the nodes (or computing threads assigned to them) in a group never result in any race condition and, thus, can be handled concurrently (\autoref{s:race}). The grouping (coloring) with $\separation=6$ is our basic strategy (\autoref{s:full}). However, the resulting number of nodes per group is not so large especially for upper levels. The number $\separation$ should be large for a high parallel efficiency. We thus considered to use a smaller value of $\separation$ and chose the minimum value of 3 (\autoref{s:part}). This can increase the number of nodes per group at most $8$ ($=6^3/3^3$) times. In this case, we can still parallelize the loop over nodes by serializing a small fraction of the factorization. We implemented a parallel IFMM code with OpenMP, which can run on a shared memory machine.
In the later part of this paper, we applied the parallel IFMM to a fast iterative BEM, which can be used for 3D acoustic scattering problems, as the preconditioner (\autoref{s:preconditioner}). The numerical results in \autoref{s:num} confirmed that the preconditioner based on the proposed ``n''ode-based parallelization (termed to \textbf{nIFMM}) was indeed faster than the ``m''atrix-based parallelization (termed to \textbf{mIFMM}). Also, nIFMM outperformed the extended block-diagonal preconditioner (termed to \textbf{BD}) for relatively large wavenumbers. Here, the numbers of unknowns are about one million and the size of scatterer is up to ten times wavelength.
In these performance comparisons, we optimized the bottom level (depth) of the hierarchy (i.e., $\ell$ for both nIFMM and mIFMM and $\ell_{\rm BD}$ for BD), which can be chosen differently from the FMM's hierarchy (i.e., $\ell_{\rm FMM}$). We discussed the tendency of the timing result due to $\ell$, but a way to predetermine an optimal choice has not been established yet. Regarding nIFMM, we also investigated the separation parameter $\separation$ and the top level $\tau$ as the tunable parameters. We found that $\tau$ was not so influential to the timing result, while we observed that $\separation=3$ was indeed better than $\separation=6$.
\iffalse
Regarding the numerical accuracy, choosing an optimal value of the IFMM's precision parameter $\varepsilon$ still remains an open question regardless of the parallelization. From the present experiments together with the previous work~\cite{takahashi2017}, we can intuitively say that some difficult test cases (such as complicated B.C., complex boundary shape/topology, large degree of freedom or $N$, large wavenumber) can lead to erroneous solution. \todo{ERIC: You mean eps needs to be decreased more than usual? This sentence is a bit misleading.} We need to investigate this issue from a theoretical viewpoint, but this is not the principal topic of the present work on parallelization.
\fi
Finally, we remark the future directions of this study:
\begin{itemize}
\item First, in order to make full use of threads, we will realize the nested parallelization, that is, parallelize each matrix operation in the parallelized loop over nodes. This is not allowed by the current version of Eigen~\cite{eigenweb}, which is used as the front-end of the MKL. This can be regarded as a hybrid of the node- and matrix-based parallelizations, which are alternative in a level in this study. Such a fine (and complicated) control of threads would be possible by utilizing both Intel MKL~\cite{Intel_MKL} and OpenMP~\cite{OpenMP} (without Eigen) but technically challenging.
\iffalse
\item Formulate and implement the IFMM based on an adaptive tree instead of uniform one. This would minimize the initialization time of IFMM because IFMM can completely reuse the precomputed FMM's operators, although the depth (bottom level) remains as a tunable parameter for IFMM. To this end, we need to modify the algorithm of IFMM and alter the code considerably.
\fi
\item Second, we will implement a parallel IFMM code for memory-distributed systems, using a message passing interface (MPI). This parallelization could resolve the issue of huge memory consumption, which we experienced on shared-memory systems. For this regard, the parallelization proposed for a sparse direct solver (LoRaSp~\cite{pouransari2017}) by Chen et al~\cite{chen2018} would be helpful. However, the modification to dense systems is not trivial because the algorithm of IFMM is significantly different from that of LoRaSp. Once this parallelization using MPI is done, it could be combined with the proposed (or the aforementioned nested) parallelization using OpenMP to construct a so-called MPI-OpenMP hybrid parallelization, which is a good fit for modern cluster computing systems.
\end{itemize}
\section{Introduction}
\file{elsarticle.cls} is a thoroughly re-written document class
for formatting \LaTeX{} submissions to Elsevier journals.
The class uses the environments and commands defined in \LaTeX{} kernel
without any change in the signature so that clashes with other
contributed \LaTeX{} packages such as \file{hyperref.sty},
\file{preview-latex.sty}, etc., will be minimal.
\file{elsarticle.cls} is primarily built upon the default
\file{article.cls}. This class depends on the following packages
for its proper functioning:
\begin{enumerate}
\item \file{pifont.sty} for openstar in the title footnotes;
\item \file{natbib.sty} for citation processing;
\item \file{geometry.sty} for margin settings;
\item \file{fleqn.clo} for left aligned equations;
\item \file{graphicx.sty} for graphics inclusion;
\item \file{txfonts.sty} optional font package, if the document is to
be formatted with Times and compatible math fonts;
\item \file{hyperref.sty} optional packages if hyperlinking is
required in the document.
\end{enumerate}
All the above packages are part of any standard \LaTeX{} installation.
Therefore, the users need not be bothered about downloading any
extra packages. Furthermore, users are free to make use of \textsc{ams}
math packages such as \file{amsmath.sty}, \file{amsthm.sty},
\file{amssymb.sty}, \file{amsfonts.sty}, etc., if they want to. All
these packages work in tandem with \file{elsarticle.cls} without
any problems.
\section{Major Differences}
Following are the major differences between \file{elsarticle.cls}
and its predecessor package, \file{elsart.cls}:
\begin{enumerate}[\textbullet]
\item \file{elsarticle.cls} is built upon \file{article.cls}
while \file{elsart.cls} is not. \file{elsart.cls} redefines
many of the commands in the \LaTeX{} classes/kernel, which can
possibly cause surprising clashes with other contributed
\LaTeX{} packages;
\item provides preprint document formatting by default, and
optionally formats the document as per the final
style of models $1+$, $3+$ and $5+$ of Elsevier journals;
\item some easier ways for formatting \verb+list+ and
\verb+theorem+ environments are provided while people can still
use \file{amsthm.sty} package;
\item \file{natbib.sty} is the main citation processing package
which can comprehensively handle all kinds of citations and
works perfectly with \file{hyperref.sty} in combination with
\file{hypernat.sty};
\item long title pages are processed correctly in preprint and
final formats.
\end{enumerate}
\section{Installation}
The package is available at author resources page at Elsevier
(\url{http://www.elsevier.com/locate/latex}).
It can also be found in any of the nodes of the Comprehensive
\TeX{} Archive Network (\textsc{ctan}), one of the primary nodes
being
\url{http://www.ctan.org/tex-archive/macros/latex/contrib/elsevier/}.
Please download the \file{elsarticle.dtx} which is a composite
class with documentation and \file{elsarticle.ins} which is the
\LaTeX{} installer file. When we compile the
\file{elsarticle.ins} with \LaTeX{} it provides the class file,
\file{elsarticle.cls} by
stripping off all the documentation from the \verb+*.dtx+ file.
The class may be moved or copied to a place, usually,
\verb+$TEXMF/tex/latex/elsevier/+,
or a folder which will be read
by \LaTeX{} during document compilation. The \TeX{} file
database needs updation after moving/copying class file. Usually,
we use commands like \verb+mktexlsr+ or \verb+texhash+ depending
upon the distribution and operating system.
\section{Usage}\label{sec:usage}
The class should be loaded with the command:
\begin{vquote}
\documentclass[<options>]{elsarticle}
\end{vquote}
\noindent where the \verb+options+ can be the following:
\begin{description}
\item [{\tt\color{verbcolor} preprint}] default option which format the
document for submission to Elsevier journals.
\item [{\tt\color{verbcolor} review}] similar to the \verb+preprint+ option, but
increases the baselineskip to facilitate easier review process.
\item [{\tt\color{verbcolor} 1p}] formats the article to the look and feel of the final
format of model 1+ journals. This is always single column style.
\item [{\tt\color{verbcolor} 3p}] formats the article to the look and feel of the final
format of model 3+ journals. If the journal is a two column
model, use \verb+twocolumn+ option in combination.
\item [{\tt\color{verbcolor} 5p}] formats for model 5+ journals. This is always
of two column style.
\item [{\tt\color{verbcolor} authoryear}] author-year citation style of
\file{natbib.sty}. If you want to add extra options of
\file{natbib.sty}, you may use the options as comma delimited
strings as arguments to \verb+\biboptions+ command. An example
would be:
\end{description}
\begin{vquote}
\biboptions{longnamesfirst,angle,semicolon}
\end{vquote}
\begin{description}
\item [{\tt\color{verbcolor} number}] numbered citation style. Extra options
can be loaded with\linebreak \verb+\biboptions+ command.
\item [{\tt\color{verbcolor} sort\&compress}] sorts and compresses the
numbered citations. For example, citation [1,2,3] will become [1--3].
\item [{\tt\color{verbcolor} longtitle}] if front matter is unusually long, use
this option to split the title page across pages with the correct
placement of title and author footnotes in the first page.
\item [{\tt\color{verbcolor} times}] loads \file{txfonts.sty}, if available in
the system to use Times and compatible math fonts.
\item[] All options of \file{article.cls} can be used with this
document class.
\item[] The default options loaded are \verb+a4paper+, \verb+10pt+,
\verb+oneside+, \verb+onecolumn+ and \verb+preprint+.
\end{description}
\section{Frontmatter}
There are two types of frontmatter coding:
\begin{enumerate}[(1)]
\item each author is
connected to an affiliation with a footnote marker; hence all
authors are grouped together and affiliations follow;
\item authors of same affiliations are grouped together and the
relevant affiliation follows this group. An example coding of the first
type is provided below.
\end{enumerate}
\begin{vquote}
\title{This is a specimen title\tnoteref{t1,t2}}
\tnotetext[t1]{This document is a collaborative effort.}
\tnotetext[t2]{The second title footnote which is a longer
longer than the first one and with an intention to fill
in up more than one line while formatting.}
\end{vquote}
\begin{vquote}
\author[rvt]{C.V.~Radhakrishnan\corref{cor1}\fnref{fn1}}
\ead{cvr@river-valley.com}
\author[rvt,focal]{K.~Bazargan\fnref{fn2}}
\ead{kaveh@river-valley.com}
\author[els]{S.~Pepping\corref{cor2}\fnref{fn1,fn3}}
\ead[url]{http://www.elsevier.com}
\end{vquote}
\begin{vquote}
\cortext[cor1]{Corresponding author}
\cortext[cor2]{Principal corresponding author}
\fntext[fn1]{This is the specimen author footnote.}
\fntext[fn2]{Another author footnote, but a little more
longer.}
\fntext[fn3]{Yet another author footnote. Indeed, you can have
any number of author footnotes.}
\address[rvt]{River Valley Technologies, SJP Building,
Cotton Hills, Trivandrum, Kerala, India 695014}
\address[focal]{River Valley Technologies, 9, Browns Court,
Kennford, Exeter, United Kingdom}
\address[els]{Central Application Management,
Elsevier, Radarweg 29, 1043 NX\\
Amsterdam, Netherlands}
\end{vquote}
The output of the above TeX source is given in Clips~\ref{clip1} and
\ref{clip2}. The header portion or title area is given in Clip~\ref{clip1} and
the footer area is given in Clip~\ref{clip2}.
\vspace*{6pt}
\deforange{blue!70}
\src{Header of the title page.}
\includeclip{1}{132 571 481 690}{els1.pdf}
\deforange{orange}
\deforange{blue!70}
\src{Footer of the title page.}
\includeclip{1}{122 129 481 237}{els1.pdf}
\deforange{orange}
\pagebreak
Most of the commands such as \verb+\title+, \verb+\author+,
\verb+\address+ are self explanatory. Various components are
linked to each other by a label--reference mechanism; for
instance, title footnote is linked to the title with a footnote
mark generated by referring to the \verb+\label+ string of
the \verb=\tnotetext=. We have used similar commands
such as \verb=\tnoteref= (to link title note to title);
\verb=\corref= (to link corresponding author text to
corresponding author); \verb=\fnref= (to link footnote text to
the relevant author names). \TeX{} needs two compilations to
resolve the footnote marks in the preamble part.
Given below are the syntax of various note marks and note texts.
\begin{vquote}
\tnoteref{<label(s)>}
\corref{<label(s)>}
\fnref{<label(s)>}
\tnotetext[<label>]{<title note text>}
\cortext[<label>]{<corresponding author note text>}
\fntext[<label>]{<author footnote text>}
\end{vquote}
\noindent where \verb=<label(s)>= can be either one or more comma
delimited label strings. The optional arguments to the
\verb=\author= command holds the ref label(s) of the address(es)
to which the author is affiliated while each \verb=\address=
command can have an optional argument of a label. In the same
manner, \verb=\tnotetext=, \verb=\fntext=, \verb=\cortext= will
have optional arguments as their respective labels and note text
as their mandatory argument.
The following example code provides the markup of the second type
of author-affiliation.
\begin{vquote}
\author{C.V.~Radhakrishnan\corref{cor1}\fnref{fn1}}
\ead{cvr@river-valley.com}
\address{River Valley Technologies, SJP Building,
Cotton Hills, Trivandrum, Kerala, India 695014}
\end{vquote}
\begin{vquote}
\author{K.~Bazargan\fnref{fn2}}
\ead{kaveh@river-valley.com}
\address{River Valley Technologies, 9, Browns Court, Kennford,
Exeter, UK.}
\end{vquote}
\begin{vquote}
\author{S.~Pepping\fnref{fn1,fn3}}
\ead[url]{http://www.elsevier.com}
\address{Central Application Management,
Elsevier, Radarweg 43, 1043 NX Amsterdam, Netherlands}
\end{vquote}
\begin{vquote}
\cortext[cor1]{Corresponding author}
\fntext[fn1]{This is the first author footnote.}
\fntext[fn2]{Another author footnote, this is a very long
footnote and it should be a really long footnote. But this
footnote is not yet sufficiently long enough to make two lines
of footnote text.}
\fntext[fn3]{Yet another author footnote.}
\end{vquote}
The output of the above TeX source is given in Clip~\ref{clip3}.
\vspace*{12pt}
\deforange{blue!70}
\src{Header of the title page..}
\includeclip{1}{132 491 481 690}{els2.pdf}
\deforange{orange}
The frontmatter part has further environments such as abstracts and
keywords. These can be marked up in the following
manner:
\begin{vquote}
\begin{abstract}
In this work we demonstrate the formation of a new type of
polariton on the interface between a ....
\end{abstract}
\end{vquote}
\begin{vquote}
\begin{keyword}
quadruple exiton \sep polariton \sep WGM
\PACS 71.35.-y \sep 71.35.Lk \sep 71.36.+c
\end{keyword}
\end{vquote}
\noindent Each keyword shall be separated by a \verb+\sep+ command.
\textsc{pacs} and \textsc{msc} classifications shall be provided in
the keyword environment with the commands \verb+\PACS+ and
\verb+\MSC+ respectively. \verb+\MSC+ accepts an optional
argument to accommodate future revisions.
eg., \verb=\MSC[2008]=. The default is 2000.\looseness=-1
\section{Floats}
{Figures} may be included using the command, \verb+\includegraphics+ in
combination with or without its several options to further control
graphic. \verb+\includegraphics+ is provided by \file{graphic[s,x].sty}
which is part of any standard \LaTeX{} distribution.
\file{graphicx.sty} is loaded by default. \LaTeX{} accepts figures in
the postscript format while pdf\LaTeX{} accepts \file{*.pdf},
\file{*.mps} (metapost), \file{*.jpg} and \file{*.png} formats.
pdf\LaTeX{} does not accept graphic files in the postscript format.
The \verb+table+ environment is handy for marking up tabular
material. If users want to use \file{multirow.sty},
\file{array.sty}, etc., to fine control/enhance the tables, they
are welcome to load any package of their choice and
\file{elsarticle.cls} will work in combination with all loaded
packages.
\section[Theorem and ...]{Theorem and theorem like environments}
\file{elsarticle.cls} provides a few shortcuts to format theorems and
theorem-like environments with ease. In all commands the options that
are used with the \verb+\newtheorem+ command will work exactly in the same
manner. \file{elsarticle.cls} provides three commands to format theorem or
theorem-like environments:
\begin{vquote}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newdefinition{rmk}{Remark}
\newproof{pf}{Proof}
\newproof{pot}{Proof of Theorem \ref{thm2}}
\end{vquote}
The \verb+\newtheorem+ command formats a
theorem in \LaTeX's default style with italicized font, bold font
for theorem heading and theorem number at the right hand side of the
theorem heading. It also optionally accepts an argument which
will be printed as an extra heading in parentheses.
\begin{vquote}
\begin{thm}
For system (8), consensus can be achieved with $\|T_{\omega z}$
...
\begin{eqnarray}\label{10}
....
\end{eqnarray}
\end{thm}
\end{vquote}
Clip~\ref{clip4} will show you how some text enclosed between the
above code looks like:
\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newtheorem}}
\includeclip{2}{1 1 453 120}{jfigs.pdf}
\deforange{orange}
The \verb+\newdefinition+ command is the same in
all respects as its\linebreak \verb+\newtheorem+ counterpart except that
the font shape is roman instead of italic. Both
\verb+\newdefinition+ and \verb+\newtheorem+ commands
automatically define counters for the environments defined.
\vspace*{12pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newdefinition}}
\includeclip{1}{1 1 453 105}{jfigs.pdf}
\deforange{orange}
The \verb+\newproof+ command defines proof environments with
upright font shape. No counters are defined.
\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newproof}}
\includeclip{3}{1 1 453 65}{jfigs.pdf}
\deforange{orange}
Users can also make use of \verb+amsthm.sty+ which will override
all the default definitions described above.
\section[Enumerated ...]{Enumerated and Itemized Lists}
\file{elsarticle.cls} provides an extended list processing macros
which makes the usage a bit more user friendly than the default
\LaTeX{} list macros. With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list counter
type and its attributes.
\begin{vquote}
\begin{enumerate}[1.]
\item The enumerate environment starts with an optional
argument `1.', so that the item counter will be suffixed
by a period.
\item You can use `a)' for alphabetical counter and '(i)' for
roman counter.
\begin{enumerate}[a)]
\item Another level of list with alphabetical counter.
\item One more item before we start another.
\begin{enumerate}[(i)]
\item This item has roman numeral counter.
\item Another one before we close the third level.
\end{enumerate}
\item Third item in second level.
\end{enumerate}
\item All list items conclude with this step.
\end{enumerate}
\end{vquote}
\vspace*{12pt}
\deforange{blue!70}
\src{List -- Enumerate}
\includeclip{4}{1 1 453 185}{jfigs.pdf}
\deforange{orange}
Further, the enhanced list environment allows one to prefix a
string like `step' to all the item numbers. Take a look at the
example below:
\begin{vquote}
\begin{enumerate}[Step 1.]
\item This is the first step of the example list.
\item Obviously this is the second step.
\item The final step to wind up this example.
\end{enumerate}
\end{vquote}
\deforange{blue!70}
\src{List -- enhanced}
\includeclip{5}{1 1 313 83}{jfigs.pdf}
\deforange{orange}
\vspace*{-18pt}
\section{Cross-references}
In electronic publications, articles may be internally
hyperlinked. Hyperlinks are generated from proper
cross-references in the article. For example, the words
\textcolor{black!80}{Fig.~1} will never be more than simple text,
whereas the proper cross-reference \verb+\ref{tiger}+ may be
turned into a hyperlink to the figure itself:
\textcolor{blue}{Fig.~1}. In the same way,
the words \textcolor{blue}{Ref.~[1]} will fail to turn into a
hyperlink; the proper cross-reference is \verb+\cite{Knuth96}+.
Cross-referencing is possible in \LaTeX{} for sections,
subsections, formulae, figures, tables, and literature
references.
\section[Mathematical ...]{Mathematical symbols and formulae}
Many physical/mathematical sciences authors require more
mathematical symbols than the few that are provided in standard
\LaTeX. A useful package for additional symbols is the
\file{amssymb} package, developed by the American Mathematical
Society. This package includes such oft-used symbols as
$\lesssim$ (\verb+\lesssim+), $\gtrsim$ (\verb+\gtrsim+) or
$\hbar$ (\verb+\hbar+). Note that your \TeX{}
system should have the \file{msam} and \file{msbm} fonts installed. If
you need only a few symbols, such as $\Box$ (\verb+\Box+), you might try the
package \file{latexsym}.
Another point which would require authors' attention is the
breaking up of long equations. When you use
\file{elsarticle.cls} for formatting your submissions in the
\verb+preprint+ mode, the document is formatted in single column
style with a text width of 384pt or 5.3in. When this document is
formatted for final print and if the journal happens to be a double column
journal, the text width will be reduced to 224pt at for 3+
double column and 5+ journals respectively. All the nifty
fine-tuning in equation breaking done by the author goes to waste in
such cases. Therefore, authors are requested to check this
problem by typesetting their submissions in final format as well
just to see if their equations are broken at appropriate places,
by changing appropriate options in the document class loading
command, which is explained in section~\ref{sec:usage},
\nameref{sec:usage}. This allows authors to fix any equation breaking
problem before submission for publication.
\file{elsarticle.cls} supports formatting the author submission
in different types of final format. This is further discussed in
section \ref{sec:final}, \nameref{sec:final}.
\section{Bibliography}
Three bibliographic style files (\verb+*.bst+) are provided ---
\file{elsarticle-num.bst}, \file{elsarticle-num-names.bst} and
\file{elsarticle-harv.bst} --- the first one for the numbered scheme, the
second for the numbered with new options of \file{natbib.sty} and the
last one for the author year scheme.
In \LaTeX{} literature, references are listed in the
\verb+thebibliography+ environment. Each reference is a
\verb+\bibitem+ and each \verb+\bibitem+ is identified by a label,
by which it can be cited in the text:
\verb+\bibitem[Elson et al.(1996)]{ESG96}+ is cited as
\verb+\citet{ESG96}+.
\noindent In connection with cross-referencing and
possible future hyperlinking it is not a good idea to collect
more that one literature item in one \verb+\bibitem+. The
so-called Harvard or author-year style of referencing is enabled
by the \LaTeX{} package \file{natbib}. With this package the
literature can be cited as follows:
\begin{enumerate}[\textbullet]
\item Parenthetical: \verb+\citep{WB96}+ produces (Wettig \& Brown, 1996).
\item Textual: \verb+\citet{ESG96}+ produces Elson et al. (1996).
\item An affix and part of a reference:
\verb+\citep[e.g.][Ch. 2]{Gea97}+ produces (e.g. Governato et
al., 1997, Ch. 2).
\end{enumerate}
In the numbered scheme of citation, \verb+\cite{<label>}+ is used,
since \verb+\citep+ or \verb+\citet+ has no relevance in the numbered
scheme. \file{natbib} package is loaded by \file{elsarticle} with
\verb+numbers+ as default option. You can change this to author-year
or harvard scheme by adding option \verb+authoryear+ in the class
loading command. If you want to use more options of the \file{natbib}
package, you can do so with the \verb+\biboptions+ command, which is
described in the section \ref{sec:usage}, \nameref{sec:usage}. For
details of various options of the \file{natbib} package, please take a
look at the \file{natbib} documentation, which is part of any standard
\LaTeX{} installation.
\subsection*{Displayed equations and double column journals}
Many Elsevier journals print their text in two columns. Since
the preprint layout uses a larger line width than such columns,
the formulae are too wide for the line width in print. Here is an
example of an equation (see equation 6) which is perfect in a
single column preprint format:
\bigskip
\setlength\Sep{6pt}
\src{See equation (6)}
\deforange{blue!70}
\includeclip{4}{134 391 483 584}{els1.pdf}
\deforange{orange}
\noindent When this document is typeset for publication in a
model 3+ journal with double columns, the equation will overlap
the second column text matter if the equation is not broken at
the appropriate location.
\vspace*{6pt}
\deforange{blue!70}
\src{See equation (6) overprints into second column}
\includeclip{3}{61 531 532 734}{els-3pd.pdf}
\deforange{orange}
\pagebreak
\noindent The typesetter will try to break the equation which
need not necessarily be to the liking of the author or as it
happens, typesetter's break point may be semantically incorrect.
Therefore, authors may check their submissions for the incidence
of such long equations and break the equations at the correct
places so that the final typeset copy will be as they wish.
\section{Final print}\label{sec:final}
The authors can format their submission to the page size and margins
of their preferred journal. \file{elsarticle} provides four
class options for the same. But it does not mean that using these
options you can emulate the exact page layout of the final print copy.
\lmrgn=3em
\begin{description}
\item [\texttt{1p}:] $1+$ journals with a text area of
384pt $\times$ 562pt or 13.5cm $\times$ 19.75cm or 5.3in $\times$
7.78in, single column style only.
\item [\texttt{3p}:] $3+$ journals with a text area of 468pt
$\times$ 622pt or 16.45cm $\times$ 21.9cm or 6.5in $\times$
8.6in, single column style.
\item [\texttt{twocolumn}:] should be used along with 3p option if the
journal is $3+$ with the same text area as above, but double column
style.
\item [\texttt{5p}:] $5+$ with text area of 522pt $\times$
682pt or 18.35cm $\times$ 24cm or 7.22in $\times$ 9.45in,
double column style only.
\end{description}
Following pages have the clippings of different parts of
the title page of different journal models typeset in final
format.
Model $1+$ and $3+$ will have the same look and
feel in the typeset copy when presented in this document. That is
also the case with the double column $3+$ and $5+$ journal article
pages. The only difference will be wider text width of
higher models. Therefore we will look at the
different portions of a typical single column journal page and
that of a double column article in the final format.
\vspace*{2pc}
\begin{center}
\hypertarget{bsc}{}
\hyperlink{sc}{
{\bf [Specimen single column article -- Click here]}
}
\vspace*{2pc}
\hypertarget{bsc}{}
\hyperlink{dc}{
{\bf [Specimen double column article -- Click here]}
}
\end{center}
\newpage
\vspace*{-2pc}
\src{}\hypertarget{sc}{}
\deforange{blue!70}
\hyperlink{bsc}{\includeclip{1}{121 81 497 670}{els1.pdf}}
\deforange{orange}
\newpage
\src{}\hypertarget{dc}{}
\deforange{blue!70}
\hyperlink{bsc}{\includeclip{1}{55 93 535 738}{els-3pd.pdf}}
\deforange{orange}
\end{document}
\section{}
\label{}
\section{}
\label{}
\section{Intermediate steps to create the groups in Figure \ref{fig:group_example}(a)}\label{s:group_cc}
\autoref{fig:cc} illustrates the intermediate steps to divide 38 nodes at level 3 into groups so that any two nodes in every group satisfy the condition in \autoref{eq:sep} where $\separation=3$, as mentioned in \autoref{fig:group_example}(a). In each sub-figure, any two nodes with the thick border (in red) satisfy the condition in \autoref{eq:sep} where $\separation=3$. A node in gray has been already included in a previous group. Finally, we have eight groups, i.e., $G_0$ to $G_7$.
\begin{figure}[h]
\begin{tabular}{ccc}
\includegraphics[width=.29\textwidth]{figure/group/fig-cc0}&
\includegraphics[width=.29\textwidth]{figure/group/fig-cc1}&
\includegraphics[width=.29\textwidth]{figure/group/fig-cc2}\\
$G_0$ & $G_1$ & $G_2$\\
\includegraphics[width=.29\textwidth]{figure/group/fig-cc3}&
\includegraphics[width=.29\textwidth]{figure/group/fig-cc4}&
\includegraphics[width=.29\textwidth]{figure/group/fig-cc5}\\
$G_3$ & $G_4$ & $G_5$\\
\includegraphics[width=.29\textwidth]{figure/group/fig-cc6}&
\includegraphics[width=.29\textwidth]{figure/group/fig-cc7}&\\
$G_6$ & $G_7$ &\\
\end{tabular}
\caption{Example of grouping, where 38 nodes are divided to 8 groups, i.e., $G_0$ to $G_7$.}
\label{fig:cc}
\end{figure}
\fi
\iffalse
\section{Complexity of the grouping algorithm}\label{s:group_complexity}
\autoref{fig:group} shows the time to group a given $n$ nodes in a level. The present data was measured in the sphere model (see \autoref{tab:stat_sphereii10}), where the separation parameter $\separation$ was chosen as 3.
\begin{figure}[h]
\centering
\def./181031/sphereii{}
\begin{tabular}{cc}
\includegraphics[width=.5\textwidth]{./181016/sphereii/fig-seek}&
\includegraphics[width=.5\textwidth]{./181031/office/fig-seek-office05}\\
Sphere model & Office model
\end{tabular}
\caption{Time for grouping in seconds.}
\label{fig:group}
\end{figure}
\fi
\section{Proofs of no race-condition under the sufficient conditions in Eqs.~(\ref{eq:sep3}), (\ref{eq:sep3b}), and (\ref{eq:sep6})} \label{s:group_proof}
\subsection{Preliminaries}\label{s:group_pre}
First, we make the following remarks on the notation and the memory location (address) of operators:
\begin{remark
An operator (sub-matrix) $A_{pq}$ stands for an (original or updated) FMM operator between a target node $p$ and source node $q$.
\end{remark}
\begin{remark
The memory location of $A_{pq}$ is denoted by $\&A_{pq}$.
\end{remark}
\begin{remark
The memory location of $A_{pq}$ is different from that of $A_{p'q'}$ (i.e., $\& A_{pq}\ne\& A_{p'q'}$) if $p\ne p'$ or $q\ne q'$. If this is the case, we can rewrite the entities of $A_{pq}$ and $A_{p'q'}$ independently and simultaneously.
\label{remark:memory}
\end{remark}
Second, since the distance in the LHS of \autoref{eq:sep} is based on the maximum norm for 3D vectors, we have the following statement:
\begin{remark}{}
The triangle inequality $\mathop{\mathrm{dist}}(a,b)\le\mathop{\mathrm{dist}}(a,c)+\mathop{\mathrm{dist}}(c,b)$ holds for any nodes $a$, $b$, and $c$ in the same level.
\label{remark:tri}
\end{remark}
Now we will prove the inequalities in Eqs.~(\ref{eq:sep3}), (\ref{eq:sep3b}), and (\ref{eq:sep6}) are sufficient to avoid the data race problems in the cases of (b), (c), and (d), respectively. To this end, we will consider two nodes $i$ and $i'$ in the same level. Then, when we assign threads $T_i$ and $T_{i'}$ to $i$ and $i'$, respectively, we may show that the memory location of the operator where $T_i$ is working does not coincide with that where $T_{i'}$ is working.
\subsection{Proof for $\separation\ge 3$ in Eq.~(\ref{eq:sep3})}\label{s:sep3}
No race condition can occur between $T_i$ and $T_{i'}$ whenever they handle different kinds of operator. Therefore, we may consider a certain kind of neighbor-to-neighbor operator, say $A$. Now we let $T_i$ (respectively, $T_{i'}$) handle $A_{pq}$ and $A_{qp}$ (respectively, $A_{p'q'}$ and $A_{q'p'}$), where $p,q\in\mathcal{N}_i$ (respectively, $p',q'\in\mathcal{N}_{i'}$). Then, since $\mathop{\mathrm{dist}}(i,i')\ge 3L$ follows from \autoref{eq:sep3}, we have $3L\le\mathop{\mathrm{dist}}(i,i')\le\mathop{\mathrm{dist}}(i,p)+\mathop{\mathrm{dist}}(p,p')+\mathop{\mathrm{dist}}(p',i')$ from Remark~\ref{remark:tri}. Here, $\mathop{\mathrm{dist}}(p,i)\le L$ and $\mathop{\mathrm{dist}}(i',p')\le L$ hold from the definition of the neighbor list. Therefore, we have $\mathop{\mathrm{dist}}(p,p')\ge L$. This means that $p\ne p'$. In the same way, we can show $p\ne q'$, $q\ne p'$, and $q\ne q'$. From Remark~\ref{remark:memory}, we can conclude both $\&A_{pq}$ and $\&A_{qp}$ coincide with neither $\&A_{p'q'}$ nor $\&A_{q'p'}$. Hence, $T_i$ and $T_{i'}$ never fall in a data race condition. $\qedsymbol$
\subsection{Proof for $\separation\ge 3$ in Eq.~(\ref{eq:sep3b})}\label{s:sep3b}
In what follows, we consider the child-to-parent operator (i.e., $M2M$)
but the proof is the same for the parent-to-child operator (i.e., $L2L$).
Since $\mathop{\mathrm{dist}}(i,i')\ge 3L$ follows from \autoref{eq:sep3b}, we can write $3L\le\mathop{\mathrm{dist}}(i,i')\le\mathop{\mathrm{dist}}(i,q)+\mathop{\mathrm{dist}}(q,q')+\mathop{\mathrm{dist}}(q',i')$ from Remark~\ref{remark:tri}. Then, from $\mathop{\mathrm{dist}}(q,i)\le L$ and $\mathop{\mathrm{dist}}(i',q')\le L$, we can obtain $3L\le\mathop{\mathrm{dist}}(q,q')+2L~\Rightarrow~\mathop{\mathrm{dist}}(q,q')\ge L~\Rightarrow~q\ne q'$. This means $\&M2M_{\tilde{q}q}\ne\&M2M_{\tilde{q'}q'}$
from Remark~\ref{remark:memory} (even if $\tilde{q}=\tilde{q}'$ holds). $\qedsymbol$
\subsection{Proof for $\separation\ge 6$ in Eq.~(\ref{eq:sep6})}\label{s:sep6}
We may show that (I)~$\&M2L_{pl}\ne\&M2L_{p'l'}$, (II)~$\&M2L_{pl}\ne\&M2L_{l'p'}$, (III)~$\&M2L_{lp}\ne\&M2L_{p'l'}$, and (IV)~$\&M2L_{lp}\ne\&M2L_{l'p'}$ for any $p\in\mathcal{N}_i$, $p'\in\mathcal{N}_{i'}$, $l\in\mathcal{I}_p$, and $l'\in\mathcal{I}_{p'}$\footnote{(I)--(IV) are directly related to the underlined $M2L$ operators regarding $p$ at Line \ref{line:fact_NN1} in \autoref{algo:fact}, but the proof for $q$ at Lines~\ref{line:fact_YN} and \ref{line:fact_NN2} is the same.}. From $\separation\ge 6$ in \autoref{eq:sep6} and Remark~\ref{remark:tri}, we have have $6L\le \mathop{\mathrm{dist}}(i,i')\le \mathop{\mathrm{dist}}(i,p)+\mathop{\mathrm{dist}}(p,p')+\mathop{\mathrm{dist}}(p',i')$. Since $\mathop{\mathrm{dist}}(i,p)\le L$ and $\mathop{\mathrm{dist}}(p',i')\le L$ hold, we can obtain $\mathop{\mathrm{dist}}(p,p')\ge 4L~\Rightarrow~p\ne p'$. Then, (I) and (VI) hold from Remark~\ref{remark:memory}. Further, by combining $\mathop{\mathrm{dist}}(p,l)\le 3L$, which holds from the definition of $\mathcal{I}_p$, with the aforementioned $\mathop{\mathrm{dist}}(p,p')\ge 4L$, we can obtain $4L\le\mathop{\mathrm{dist}}(p,p')\le\mathop{\mathrm{dist}}(p,l)+\mathop{\mathrm{dist}}(l,p')\le 3L+\mathop{\mathrm{dist}}(l,p')~\Rightarrow~L\le\mathop{\mathrm{dist}}(l,p')~\Rightarrow~l\ne p'$. Then, (II) and (III) hold. $\qedsymbol$
\section{IFMM}\label{s:ifmm}
In this section, we describe the sequential IFMM algorithm focusing on the data dependency there and introduce some FMM terminologies used in this paper. For more details, the readers may see the references~\cite{2014_Ambikasaran,coulier2017}.
IFMM is a fast direct solver with $O(N\log^2\frac{1}{\varepsilon})$ complexity for solving a linear system $Ax=b$, where the matrix-vector product with $A \in \bbbr^{N \times N}$ can be evaluated with the FMM. This type of linear system arises from various science and engineering applications. As an example, consider an electrostatic field generated by $N$ charges, where the $i$-th charge is located at $\fat{p}_i\in\bbbr^3$ and the electric potential is given as $b_i$ at $\fat{p}_i$. We are interested in solving for the charge density $x_i$. This problem can be formulated as $Ax=b$, where $x:=[x_1,\ldots,x_N]^T$, $b:=[b_1,\ldots,b_N]^T$, and the $(i,j)$-th element of $A$ is given by the Coulomb interaction, i.e., $\frac{x_i}{4\pi\epsilon\abs{\fat{p}_i-\fat{p}_j}}$ if $i\ne j$ and $0$ otherwise, where $\epsilon$ denotes a permittivity. To solve such a linear system, IFMM takes the following four steps.
\subsection{Hierarchical domain decomposition}\label{s:hdd}
We carry out the standard hierarchical partitioning of the problem domain as in the FMM~\cite{1987_Greengard,Darve2000195,2002_Nishimura} and associate the partitioning with an \textit{octree} (\textit{quadtree}) in a three (two) dimensional space. This partitioning can be easily parallelized with the geometric information of the problem domain. For ease of presentation, we assume the tree is \textit{uniform} (a.k.a., \textit{complete}), so there are $n_{\kappa} = 2^{\kappa \, d}$ nodes at level $\kappa = 0, 1, \ldots, \ell$, where $d$ is the space dimension. Based on the partitioning, we also define the \textit{neighbors} and the \textit{interaction list} of a tree node $i$ as in the FMM (a partition refers to a (cubic) cluster of points in the $d$-dimensional space):
\begin{itemize}
\item[]
\textbf{Neighbors}: tree nodes corresponding to adjacent partitions of node $i$ (excluding $i$);
\item[]
\textbf{Interaction list}: non-adjacent partitions whose parent is a neighbor of node $i$'s parent.
\end{itemize}
\begin{remark}{}
Our definition of neighbors obeys the standard definition in graph theory but differs from one in the FMM, where the neighbors of a node includes the node itself.
\end{remark}
\subsection{Initialization of tree data structure}\label{s:init_tree_data}
Given that the FMM can be used to compute matrix-vector product with $A$ in \autoref{eqn:axb}, we initialize our tree data structure with the six translation operators in the FMM: P2P, P2M, L2P, M2M, M2L, and L2P. In particular, every leaf node stores all six FMM operators associated its partition, and every internal tree node stores M2M, M2L, and L2P associated with its partition (the other three operators are initially empty and will be computed in the algorithm; see lines \autoref{line:fact_transfer1} -- \autoref{line:fact_transfer3} in \autoref{algo:fact}). In the context of BEM, the expressions of the FMM translation operators can be found in~\cite{takahashi2017}. Note this initialization step is embarrassingly parallel with respect to all tree nodes, and this was introduced in our previous work~\cite{takahashi2017}.
\begin{remark}{}
From an algebraic perspective, our tree data structure corresponds to the graph of the extended sparse linear system~\cite{coulier2017} of \autoref{eqn:axb} (see Figure 10 in \cite{coulier2017}). Specifically, the extended sparse linear system is constructed through introducing auxiliary variables into \autoref{eqn:axb}, which are nothing but the ``multipole moments'' and the ``local coefficients'' in the FMM. These auxiliary variables satisfy relations as defined in the FMM, and solving the extended system is equivalent to solving \autoref{eqn:axb}. As an example, assume a three-level hierarchy in the domain decomposition, the extended sparse linear system is the following:
\begin{eqnarray}
\begin{bmatrix}
P2P^{(3)} & L2L^{(3)} & & &\\
{M2M^{(3)}}^{\mathrm{T}} & & -I & &\\
& -I & M2L^{(3)} & L2L^{(2)} &\\
& & {M2M^{(2)}}^{\mathrm{T}} & & -I\\
& & & -I & M2L^{(2)}
\end{bmatrix}
\begin{Bmatrix}
x^{(3)}\\
z^{(3)}\\
y^{(3)}\\
z^{(2)}\\
y^{(2)}
\end{Bmatrix}
=
\begin{Bmatrix}
b\\
0\\
0\\
0\\
0
\end{Bmatrix},
\label{eq:ext}
\end{eqnarray}
where $x^{(3)}$ are exactly the unknowns in \autoref{eqn:axb}, $y^{(\kappa)}$ and $z^{(\kappa)}$ are the multipole moments and the local coefficients associated with all the tree nodes at level $\kappa$, $I$ denotes the identity matrix, and the other matrix blocks correspond to FMM translation operators at different levels.
\end{remark}
\subsection{Factorization}
After initializing our tree data structure, we have effectively transformed the original $N \times N$ dense problem to a (block) sparse linear system with $O(N)$ nonzero entries. To compute an approximate factorization of the sparse matrix in linear complexity, the IFMM employs both Gaussian elimination and low-rank approximation of fill-in blocks.
The factorization phase of the IFMM is summarized in \autoref{algo:fact}, which is a level-by-level traversal of the tree from the leaf level towards the root, as the upward pass in the FMM. For tree nodes at the same level, an elimination-and-compression strategy is conducted iteratively as follows.
At a node $i$, (block) Gaussian elimination is carried out on the associated variables (line \autoref{line:fact_eliminate}), leading to fill-in blocks among its neighboring nodes (a.k.a., \textit{clique}). Then, a compression step is carried out on all pairs of nodes that are neighbors of $i$ (line \autoref{line:fact_pairs}). For such a pair of nodes $(p,q)$, the IFMM may compute a low-rank approximation of the associated fill-in block and update existing FMM operators, depending on whether $p$ or $q$ has been visited/eliminated and whether they are in each other's interaction list (lines \autoref{line:fact_if_zero}, \autoref{line:fact_if_one}, \autoref{line:fact_if_two} and \autoref{line:fact_if_three}). In particular, if a node $q$ is not eliminated and $p$ is in $q$'s interaction list, node $q$ needs to update the stored M2L operators corresponding to its interaction list (lines \autoref{line:fact_YN} and \autoref{line:fact_NN2}). The same principle applies to node $p$ (line \autoref{line:fact_NN1}). Finally, after level $\kappa$ is traversed, the IFMM aggregates the M2M, L2L, and M2L operators of nodes that have the same parent to form the P2M, L2P, and P2P operators of their parent at level $\kappa - 1$ for $\kappa > 2$ (lines \autoref{line:fact_transfer1} and \autoref{line:fact_transfer3}). When $\kappa = 2$, a dense linear system is formed with all M2L operators of the level and factorized with the standard LU factorization; this step is intentionally omitted in \autoref{algo:fact} to save space.
As the above shows, the elimination-and-compression process on a tree node involves only its neighbors and their interaction lists. So two sufficiently distant nodes at the same level can be processed in parallel, which forms the basis of our parallel algorithm in \autoref{s:parallel}.
\begin{remark}{}
Algebraically speaking, we have computed an implicit factorization of the extended sparse linear system (with $O(N)$ nonzeros), which consists of (block) triangular factors and (block) diagonal factors.
\end{remark}
\subsection{Solve}
With the (implicit) factorization, the solve phase follows the standard (sparse) forward and backward substitutions. As \autoref{algo:solve} shows, the in-place solve algorithm consists of three parts: initialization, upward traversal, and downward traversal of our tree data structure. For the initialization, every leaf node extracts corresponding elements in a given right-hand size, and every internal tree node gets a zero vector. It is obvious that this initialization can be easily parallelized among all nodes in the tree.
Both forward and backward substitutions employ the level-by-level traversal as in the factorization, and local updates on the solution vector of node $i$ involves only the neighbors of $i$ (lines \autoref{line:n1} and \autoref{line:n2}). Assume two nodes $i$ and $j$ can be processed in parallel during the factorization, then it is easy to see that they can also be processed in parallel during the two substitutions.
\begin{remark}{}
As in the factorization, we omit the solve with respect to the (small) linear system at level 1, which is done with standard (dense) forward and backward substitutions.
\end{remark}
\begin{remark}{}
\autoref{algo:solve} is a simplification of the algorithm presented in~\cite{2014_Ambikasaran,coulier2017}, where the original solve algorithm involves not only $x^{(\kappa)}$ but also multipole moments $y^{(\kappa)}$ and local coefficients $z^{(\kappa)}$. The data dependency in the solve algorithm is exactly the same.
\end{remark}
\input{algo_factorization.tex}
\input{algo_solve.tex}
\section{Introduction}\label{s:intro}
We are interested in solving a large-scale dense linear system
\begin{eqnarray}
Ax = b
\label{eqn:axb}
\end{eqnarray}
with $N$ unknown variables. Such systems arise from many science and engineering areas, such as discretized integral equations, boundary element methods (BEMs), machine learning, etc. To solve \autoref{eqn:axb}, a naive Gaussian elimination requires $O(N^3)$ computation and $O(N^2)$ memory, which is prohibitive when $N$ is large. While an iterative solver such as CG and GMRES requires $O(N^2)$ computation and memory per iteration, the number of iterations can be large in presence of ill-conditioning and indefinite problems and the $O(N^2)$ costs limit their application to truly large-scale problems. To overcome these disadvantages, the fast multipole method (FMM)~\cite{1985_Rokhlin,1987_Greengard,Darve2000195,2002_Nishimura} can be used to accelerate the computation of matrix-vector products at every iteration of an iterative method, reducing the computation and memory costs to typically $O(N)$.
In order to further reduce the number of iterations of an iterative method, the inverse fast multipole method (IFMM)~\cite{2014_Ambikasaran,coulier2017} has been developed among various fast direct solvers~\cite{2005_Martinsson,2015_Corona,2015_Bremer,2016_Ho} and $\mathcal{H}$-matrix methods~\cite{bebendorf2008hierarchical, 2008_Banjai}. These methods perform Gaussian elimination on a compressed representation of $A$ and computes an approximated factorization of $A$ with nearly linear computational cost. The IFMM is inspired and closely related to the FMM since they use the same hierarchical decomposition of the problem domain (a.k.a. the FMM tree), which corresponds to a partitioning of the unknowns in \autoref{eqn:axb}, and use the same far-field (multipole-to-local) and near-field (particle-to-particle) translation operators, which corresponds to the full-rank and the low-rank matrix blocks in $A$, respectively. This allows the IFMM to solve \autoref{eqn:axb} with the same asymptotic computation and memory complexities as the FMM, i.e., $O(N\log^2\frac{1}{\varepsilon})$, where $\varepsilon$ is a prescribed accuracy. The efficiency of the IFMM has been studied in several previous works for solving \autoref{eqn:axb}: Quaife et al~\cite{quaife2017} and Coulier et al~\cite{coulier2017} applied the IFMM as preconditioners for the GMRES~\cite{saad1986gmres} to solve \autoref{eqn:axb} from the immersed boundary method regarding Stokes flow problems in two and three dimensions (3D); Takahashi et al~\cite{takahashi2017} applied the IFMM together with the low-frequency FMM to accelerate the BEM for the 3D Helmholtz equation; Coulier et al~\cite{ij-cmame-coul-16a} applied the IFMM to reduce the cost of a mesh deformation method which is based on the radial basis function interpolation.
To further reduce the runtime of the IFMM for large problem sizes ($\sim1$ million variables), this paper introduces the parallel IFMM algorithm targeting at shared-memory machines. Although the IFMM uses the same hierarchical tree decomposition as the FMM, the data dependency of the IFMM is much more complicated: the work associated with nodes at the same level in the hierarchy is \textit{not} embarrassingly parallel as it is in the upward pass and the downward pass of the FMM~\cite{2011_Darve,yokota2011biomolecular}. Rather than the FMM, the data dependency in the IFMM is similar to that in the incomplete Cholesky factorization~\cite{saad1996iterative} but has a special structure associated with the FMM hierarchical decomposition. Rather than solving \autoref{eqn:axb} directly, the IFMM solves the equivalent \textit{extended sparse linear system}~\cite{coulier2017}, which is associated with the FMM tree structure. As defined in the original FMM, every node in the FMM tree has a list of neighbors and an interaction list of nodes in the far field. In the IFMM, one step of (block) Gaussian elimination on a node in the FMM tree leads to a dense Schur complement among its neighbors (a.k.a., clique). The Schur complement is then compressed based on the low-rank structure in the FMM, which effectively updates the far-field translation operators of the neighbors and affects the nodes in the neighbors' interaction lists.
Our parallel IFMM algorithm is based on a greedy coloring algorithm (\autoref{algo:group}), which assigns the same color to nodes that are apart from each other by a distance of $\separation$. We proved that the workload for nodes in the same group (or nodes with the same color) is embarrassingly parallel when $\separation \ge 6$, or technically speaking, the neighbors' interaction lists of a node $n_1$ do not intersect with the neighbors of another node $n_2$ if $n_1$ and $n_2$ have the same color. So we derived the basic parallel IFMM algorithm (\autoref{s:full}) associated with $\separation = 6$. For the targeting problem sizes ($\sim1$ million), we found that the number of colors is large (the average group size is small), which limits the parallelism and leads to a significant synchronization cost.
Therefore, we derived the improved parallel IFMM algorithm (\autoref{s:part}) associated with $\separation = 3$ in our coloring algorithm. With $\separation = 3$, there exists a race condition for processing two nodes with the same color: their neighbors may need to update the same far-field translation operator. So these updates are serialized and form a critical section in our improved parallel algorithm. Fortunately, these far-field updates account for only a small fraction of the entire computation (see Fig. 26 in~\cite{coulier2017}), and the overhead of serialization is paid back by the tremendous improvement of parallel efficiency from large group sizes and a small number of groups/colors.
In our previous study~\cite{takahashi2017}, a naive parallel IFMM algorithm was developed by simply replacing all matrix computations such as matrix-matrix multiplication, LU decomposition, and the SVD with their parallel counterparts in the sequential IFMM algorithm. This approach can be realized easily by linking the sequential IFMM code with a multi-threaded linear algebra library such as the Intel MKL library~\cite{wang2014intel} and the OpenBLAS library~\cite{OpenBLAS}. However, this over-fine-grained parallelization would inevitably introduce overwhelming overhead when the number of threads is large. Another closely related approach is the distributed-memory parallel solver introduced by Chen et al~\cite{chen2018}, which parallelizes the LoRaSp algorithm~\cite{pouransari2017}, an analog of the IFMM for solving \autoref{eqn:axb} when $A$ is sparse.
To summarize, this paper presents parallel IFMM algorithms for solving large-scale dense linear systems, and especially, our work makes the following three main contributions:
\begin{enumerate}
\item
analyzing the data dependency and identifying the parallelism in the sequential IFMM algorithm.
\item
deriving two parallel IFMM algorithms (\autoref{s:full} and \autoref{s:part}) based on our coloring scheme (\autoref{algo:group}).
\item
reporting benchmarks and comparisons between our parallel IFMM algorithms and two existing methods for solving real-world problems involving more than a million variables.
\end{enumerate}
The rest of the paper is organized as follows. \autoref{s:ifmm} reviews the basics of the IFMM algorithm. \autoref{s:parallel} describes our parallel strategy and an implementation using OpenMP~\cite{OpenMP}. \autoref{s:preconditioner} incorporates the parallel IFMM to solve dense linear systems from the fast-multipole accelerated BEM (FMBEM) for 3D Helmholtz equation. \autoref{s:num} gives two numerical examples to assess the efficiency of our parallel algorithms. \autoref{s:concl} concludes this study.
\section{Numerical experiments}\label{s:num}
In comparison with mIFMM and BD, we will assess the proposed parallel IFMM-based preconditioner, i.e., nIFMM, through solving two acoustic scattering problems.
\subsection{Details of GMRES, FMM, IFMM, and computers}\label{s:num_setting}
We used the restart GMRES, i.e., GMRES($m$)~\cite{2003_Saad_book}, but no restart took place in the following tests since we used a sufficiently large restart parameter $m$, i.e., $m=1000$ for the IFMM-based preconditioners and $m=3000$ for BD. The iteration was terminated when the relative residual $\norm{Ax-b}_2/\norm{b}_2$ became smaller than the predefined number ``$tol$''. We set $10^{-5}$ to $tol$. We note that our GMRES program is not parallelized except for the preconditioning (\autoref{algo:precon}) and matrix-vector routines, which is performed by the parallelized LFFMM (see below). The parallelization can lead to the reduction of the total computation time if the number of iterations is large.
Regarding the (LF)FMM, its performance can be controlled by two parameters. The first one is the maximum number of boundary elements per leaf node, denoted by $\mu$. Once $\mu$ is given, the depth of FMM's octree is determined as $\ell_{\rm FMM}$ ($\ge 2$). The second parameter is the precision factor $\delta$ to determine the number $p$ of the multipole and local coefficients through the empirical formula~\cite{1993_Coifman}
\begin{eqnarray}
p:=\sqrt{3}Lk+\delta\log\left(\sqrt{3}Lk+\pi\right),
\label{eq:p}
\end{eqnarray}
where $k$ is a prescribed wavenumber. We set $7$ to $\delta$ in all the computations. We note that the row and/or column size of an initial (uncompressed) FMM operator is typically $p^2$ (not $p$). Also, we note that our LFFMM code is parallelized with respect to the loop over nodes in each level.
The major parameter of the IFMM is the relative error $\varepsilon$ that determines the accuracy of the low-rank compression, for which we used the randomized SVD~\cite{2011_Halko}. In general, a smaller $\varepsilon$ makes the elapsed times of factorization and solve phases longer but can reduce the number of iterations~\cite{takahashi2017}. We mainly used $\varepsilon=10^{-3}$ and considered $10^{-1}$ and $10^{-2}$ for comparison.
In the case of the first example in \autoref{s:sphere}, we used 16 cores in a desktop PC (which consists of two Intel Xeon Gold 6134 CPUs) and \unit{512}{GB} memory in total. As a compiler, gcc-7.3.1 was used with the optimizing flags ``-O3 -march=native -mtune=native''\footnote{In addition, we built the IFMM program with the Eigen's macro ``EIGEN\_USE\_MKL\_ALL'' to parallelize each matrix operation by Intel MKL~\cite{Intel_MKL} outside parallel regions. This actually improved the performance of the related codes.}. In the second example in \autoref{s:office}, we needed a much memory because of larger $N$ and $k$ than the first case. Therefore, we used a computing node with \unit{3}{TB} memory and 56 cores (in four Intel Xeon E5-4650v4 CPUs). The compiler was gcc-7.3.0 with the aforementioned flags\footnote{Since both computers are based on the NUMA architecture, we considered to bind the OpenMP threads to specific computing cores. We simply bound the $i$-th thread to the $i$-th core. This improved the timing result in practice.}.
\input{sphere.tex}
\input{office.tex}
\section{Details for the office model}\label{s:office_profile}
\section{Survey of the best settings for the Office model (Section~\ref{s:office}}\label{s:office_profile}
\subsection{BD}\label{s:office_BD}
We first determined the best value of $\ell_{\rm BD}$ for BD. From the result in Table~\ref{tab:office05_BD} and Figure~\ref{fig:office05_BD_total}, \textbf{BDbot4} was nearly the best for $k\le 32$ and \textbf{BDbot3} for $k=64$. For the sake of fair comparison with nIFMM as well as mIFMM, we choose the best result (that is, shortest computation time) of BD for every wavenumber when we draw Figure~\ref{fig:office05_speedup} of the main article.
We note that BDbot2 required too much memory (for precomputing the block diagonals) to run. In addition, we could not perform BDbot5, 6, and 7 in the case of $k=64$ because of the limitation of computation time, i.e., 24 hours.
\begin{table}[h]
\centering
\caption{Profile of \textbf{BD} for the \textbf{office} model. The speedup (``\tsp'') as well as the discrepancy (``\disc'') is relative to BDbot3, which is the best BD for $k=64$.}
\label{tab:office05_BD}
\def./181031/sphereii{./181031/office}
\def\disc{\disc}
\scriptsize
\myinputII{./181031/sphereii/office05-fmm3logBDX0-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/office05-fmm3logBDX1-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logBDX2-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logBDX3-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logBDX4-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logBDX5-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[h]
\centering
\def./181031/sphereii{./181031/office}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=100, ymax=100000,
legend pos=south east,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={BDb7, BDb6, BDb5, BDb4, BDb3},
]
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logBDX0-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logBDX1-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logBDX2-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logBDX3-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logBDX4-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc0_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{BD} for the \textbf{office} model.}
\label{fig:office05_BD_total}
\end{figure}
\subsection{mIFMM}\label{s:office_mIFMM}
As seen in Figure~\ref{fig:office05_mIFMM_total}, the best mIFMM was \textbf{mIFMMbot5} (i.e., $\ell=5$) when we varied $\ell$ from 7 to 4. Although the number of iterations (``\niter'') is relatively small for every wavenumber, the performance of $\ell=4$ was clearly worse than that of $\ell=5$; see Table~\ref{tab:office05_mIFMM}.
Note that $k=64$ of $\ell=7$ did not run due to the out of memory.
\begin{table}[h]
\centering
\caption{Profile of \textbf{mIFMM} for the \textbf{office} model. The speedup (``\tsp'') as well as the discrepancy (``\disc'') is relative to BDbot3, which is the best BD for $k=64$.}
\label{tab:office05_mIFMM}
\def./181031/sphereii{./181031/office}
\def\disc{\disc}
\scriptsize
\myinputII{./181031/sphereii/office05-fmm3logXsub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/office05-fmm3logXsub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[h]
\centering
\def./181031/sphereii{./181031/office}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=1000, ymax=100000,
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={mIFMMb7, mIFMMb6, mIFMMb5, mIFMMb4}
]
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{mIFMM} for the \textbf{office} model.}
\label{fig:office05_mIFMM_total}
\end{figure}
\subsection{nIFMM}\label{s:office_nIFMM}
We tested nIFMM with varying the top level $\tau$ from 2 to 3 and the bottom level $\ell$ from 7 to 4. Basically, $\tau=3$ gave better results than $\tau=2$. Therefore, we show only the result of $\tau=3$ in Table~\ref{tab:office05_nIFMM} From Figure~\ref{fig:office05_nIFMM_total}, we determined \textbf{nIFMMsep3top3bot5} as the best.
The tendency of both initialization (``\init'') and factorization times (``elim} %12 \newcommand{\elim}{elim'') as well as the number of iterations (``\niter'') are similar to that of mIFMM in Table
~\ref{tab:office05_mIFMM}. However, the preconditioning (``prec} %10 \newcommand{\ms}{ms'') and factorization were accelerated by the node-based parallelization.
Note that $k=64$ of $\ell=7$ did not run due to the out of memory.
\begin{table}[h]
\centering
\caption{Profile of \textbf{nIFMM} for the \textbf{office} model. The speedup (``\tsp'') as well as the discrepancy (``\disc'') is relative to BDbot3, which is the best BD for $k=64$.}
\label{tab:office05_nIFMM}
\def./181031/sphereii{./181031/office}
\def\disc{\disc}
\scriptsize
\iffalse
$\separation=3$, $\tau=2$\\
\myinputII{./181031/sphereii/office05-fmm3logXsep3top2sub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top2sub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top2sub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top2sub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{2}
$\separation=3$, $\tau=3$\\
\fi
\myinputII{./181031/sphereii/office05-fmm3logXsep3top3sub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top3sub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top3sub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/office05-fmm3logXsep3top3sub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[h]
\centering
\def./181031/sphereii{./181031/office}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=100, ymax=10000,
legend pos=south east,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={nIFMMs3t3b7, nIFMMs3t3b6, nIFMMs3t3b5, nIFMMs3t3b4}
]
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsep3top3sub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsep3top3sub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsep3top3sub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/office05-fmm3logXsep3top3sub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc0-fillin1e-3-basis1e-3-LR3_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{nIFMM} for the \textbf{office} model.}
\label{fig:office05_nIFMM_total}
\end{figure}
\subsection{Realistic example --- office model}\label{s:office}
\subsubsection{Problem statement}
We consider a complicated problem in comparison with the previous one. Namely, we solve a mixed boundary value problem (BVP) where the scatterer is an office building consisting of four floors (\autoref{fig:model} left and center). Regarding the boundary condition, the Dirichlet boundary condition of $u=1$ is given as the sound source on a wall in the third floor (\autoref{fig:model} right), while the Neumann boundary condition of $q=0$ is given everywhere else. The model is discretized with 1218200 elements.
\begin{figure}[h]
\includegraphics[width=.25\textwidth]{figure/office05_snapshot_lowresolution}
\quad
\includegraphics[width=.25\textwidth]{figure/office05_transparent}
\quad
\includegraphics[width=.4\textwidth]{figure/office05_3rdfloor}
\caption{Office model. (Left) Outlook. The dimensions $\text{width}\times\text{depth}\times\text{height}$ are $0.8\times 0.4 \times 1.025$. (Center) Transparent view. (Right) Two sound sources (colored in red) on the wall in the third floor and the boundary element mesh.}
\label{fig:model}
\end{figure}
We chose the maximum number of elements per node ($\mu$) as $40$, following the previous sphere model. Then, the depth of the FMM hierarchy (i.e., $\ell_{\rm FMM}$) became $7$ as shown in \autoref{tab:stat_office05}.
\input{tab_stat_office.tex}
\subsubsection{Comparison under the best settings}\label{s:office_comp3}
We focus on our prime interest, that is, the performance of nIFMM over BD and mIFMM. Prior to this, we surveyed the best setting of each preconditioner; the details are described in Section~2 of the supplementary material. Basically, since the present model is more complicated, the number of iterations became larger than the previous model. Hence, a longer initialization time can compensate the total computation time. We actually considered $\ell_{\rm BD}\in\{2,7\}$, $\separation=3$, $\tau\in\{2,3\}$, and $\ell\in\{4,\ldots,7\}$ for the survey.
As a result, BDbot4 was nearly the best for $k\le 32$ and BDbot3 for $k=64$. Meanwhile, mIFMMbot5 and nIFMMsep3top3bot5 were nearly the best for higher wavenumbers.
\autoref{fig:office05_speedup} shows the speedup of nIFMMsep3top3bot5 relative to mIFMMbot5 and BD in terms of the total computation time. Here, the time of BD was chosen as the best (shortest) time of BDbot3--7 for every wavenumber; this is because it is difficult to determine the best BD for all the wavenumbers. Clearly, the nIFMM increased its performance with $k$ and achieved about {\nprounddigits{1}\numprint{11.42854921289711542379}} times speedup at the largest wavenumber. Meanwhile, the speedup over the mIFMM was about 4 times.
\begin{figure}[h]
\centering
\iftrue
\begin{tikzpicture}
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Speedup of nIFMMs3t3b5\\relative to BD and mIFMMb5},
xmode=log, ymode=log,
xmin=1, xmax=100,
ymin=0.1, ymax=100.0,
legend pos=north east,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={BD, mIFMMb5},
]
\addplot table [x=k, y=tsp_nIFMM] {./181031/office/fig-tsp-nIFMM-office05-delta7-epsilon1e-3.table};
\addplot table [x=k, y expr=\thisrow{tsp_nIFMM}/\thisrow{tsp_mIFMM}] {./181031/office/fig-tsp-nIFMM-office05-delta7-epsilon1e-3.table};
\end{axis}
\end{tikzpicture}
\else
\includegraphics[width=.5\textwidth]{./181031/office/fig-tsp-nIFMM-office05-delta7-epsilon1e-3}
\fi
\caption{Speedup off the best nIFMM (i.e., nIFMMsep3top3bot5) relative to the best mIFMM (i.e., mIFMMbot5) and BD in terms of the total computation time for the \textbf{office} model.}
\label{fig:office05_speedup}
\end{figure}
\subsubsection{Basic parallel algorithm ($\separation=6$) vs improved parallel algorithm ($\separation=3$)}\label{s:office_sep6}
We simply changed the value of $\separation$ from $3$ to $6$ in the best nIFMM (i.e., nIFMMsep3top3bot5). \autoref{fig:office05_sep6} shows that $\separation=3$ is indeed faster than $\separation=6$ for any wavenumbers. Similarly to the previous example in \autoref{s:sphereii_sep6}, we can confirm that the serialization of a small fraction of the factorization algorithm pays the total efficiency.
\begin{figure}[h]
\centering
\iftrue
\begin{tikzpicture}
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel={Speedup of $\separation=3$ relative to $\separation=6$},
xmode=log,
xmin=1, xmax=100,
ymin=1.0, ymax=3.0, ytick={1.0, 1.5,..., 3.0},
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={factorization phase, solve phase, total}
]
\addplot table [x=k, y=sp_ifmm_elim] {./181031/office/fig-nIFMM_relto_nIFMM6-office05-delta7-epsilon1e-3.table};
\addplot table [x=k, y=sp_msolve_per_iter] {./181031/office/fig-nIFMM_relto_nIFMM6-office05-delta7-epsilon1e-3.table};
\addplot table [x=k, y=sp_main] {./181031/office/fig-nIFMM_relto_nIFMM6-office05-delta7-epsilon1e-3.table};
\end{axis}
\end{tikzpicture}
\else
\begin{tabular}{c}
\includegraphics[width=.5\textwidth]{./181031/office/fig-nIFMM_relto_nIFMM6-office05-delta7-epsilon1e-3}
\end{tabular}
\fi
\caption{Speedup of nIFMM\textbf{sep3}top3bot5 relative to nIFMM\textbf{sep6}top4bot5 in the \textbf{office} model. The meanings of the legends are the same as those in \autoref{fig:sphereii10_sep6}.}
\label{fig:office05_sep6}
\end{figure}
\subsubsection{Accuracy}\label{s:office_accuracy}
Finally, we point out that the IFMM-based preconditioners with using larger values of $\varepsilon$, that is, $10^{-1}$ and $10^{-2}$, produced inaccurate solutions in some cases, where the distribution of the sound pressure $u$ on the boundary were erroneous at specific portions.
The previous work~\cite{takahashi2017} suggested that $\varepsilon=10^{-2}$ was a safer choice by considering the fact that the erroneous solutions were induced by $\varepsilon=10^{-1}$ in a similar mixed BVP of $N=147$k and $k\le 32$. However, this is not true for the present problem of $N=1218$k and $k\le 64$. The present experiments conclude $\varepsilon=10^{-3}$ is safer. In addition, $\varepsilon=10^{-3}$ was faster than $\varepsilon=10^{-1}$ and $10^{-2}$ for most of cases of $k$ because $\varepsilon=10^{-3}$ resulted in more rapid convergence.
Choosing an optimal value of the IFMM's precision parameter $\varepsilon$ still remains an open question regardless of the parallelization.
\section{Parallelization of IFMM}\label{s:parallel}
In this section, we describe the parallelization for each of the four steps in the IFMM. The first step --- hierarchical domain decomposition --- can be done using existing parallel partitioning packages, such as Zoltan~\cite{ZoltanIsorropiaOverview2012}. The second step --- initialization of tree data structure --- is embarrassingly parallel with respect to all tree nodes, and this has been introduced in our previous work~\cite{takahashi2017}, so we skip the details here. The last two steps --- factorization and solve --- are the main challenges and the focus of this section.
As hinted earlier, our parallel strategy is based on coloring: assigning colors to tree nodes at the same level such that nodes of the same color can be processed in parallel during the factorization and the solve. To be more specific, our parallel algorithm replaces the loop over all tree nodes at a level (line \autoref{line:fact_loop_i} in \autoref{algo:fact}, and lines \ref{line:elimination_loop_i} and \ref{line:substitution_loop_i} in \autoref{algo:solve}) with nested loops, where the outer loop goes over all colors \textit{sequentially}, and the inner loop goes over tree nodes with the same color \textit{in parallel}.
To that end, we introduce the separation parameter $\separation$, which characterizes the minimum (normalized) distance between all pairs of nodes having the same color, and also serves as the constraint in our coloring algorithm. Let the partitions be cubes of size $L$ at a level in the FMM-tree, and the normalized (infinity norm) distance between two tree nodes $u$ and $v$ is defined as
\begin{eqnarray*}
\mathop{\mathrm{dist}}(u,v):=\frac{\max(\abs{x_u-x_v},\abs{y_u-y_v},\abs{z_u-z_v})}{L},
\end{eqnarray*}
where $(x_u,y_u,z_u)$ and $(x_v,y_v,z_v)$ are the coordinates of the two partition (cube) centers associated with $u$ and $v$, respectively. Note that by construction, $\mathop{\mathrm{dist}}(u,v)$ is a non-negative integer, and
\begin{eqnarray} \label{eq:dist}
\mathop{\mathrm{dist}}(u,v) =
\begin{cases}
0 & \quad \text{$u=v$,} \\
1 & \quad \text{$u \in \mathcal{N}_v (\Leftrightarrow v \in \mathcal{N}_u)$},\\
2 \text{ or } 3 & \quad \text{$u\in \mathcal{I}_v (\Leftrightarrow v\in \mathcal{I}_u)$.}
\end{cases}
\end{eqnarray}
The separation parameter, or coloring constraint is thus defined as
\begin{eqnarray}
\separation = \min \, \{ \, \mathop{\mathrm{dist}}(u,v) \, | \, \text{color}(u) = \text{color}(v) \, \},
\label{eq:sep}
\end{eqnarray}
where the function ``color()'' returns the color (an integer) of a node.
In the following, we analyze the data dependency in \autoref{algo:fact} and \ref{algo:solve} and show what value $\separation$ needs to take in the coloring algorithm.
\subsection{Data dependency}\label{s:race}
Suppose we have two threads $T_i$ and $T_{i'}$, and want to process two nodes $i$ and $i'$ independently. Apparently, $i \not= i'$ means $\separation \ge 1$. The data dependency in the IFMM falls into three cases as follows.
\begin{enumerate}[label=(\alph*)]
\item \textbf{Local operations}
In \autoref{algo:fact} and \ref{algo:solve}, $T_i$ (resp. $T_{i'}$) updates local data of node $i$ (resp. $i'$) and reads data from neighbors of $i$ (resp. $i'$). For example, $T_i$ updates the ``eliminated'' flag (line \autoref{line:fact_eliminate}) in \autoref{algo:fact} and the ``visited'' flag (line \autoref{line:visit}) in \autoref{algo:solve}, and reads these flags of node $i$'s neighbors (\autoref{algo:fact}: lines \autoref{line:fact_if_zero}, \autoref{line:fact_if_one}, \autoref{line:fact_if_two} and \autoref{line:fact_if_three}, and \autoref{algo:solve}: lines \autoref{line:solve_if_one} and \autoref{line:solve_if_two}). If node $i'$ coincides with $i$'s neighbor $p$, the flags of $i'$ can be updated by $T_{i'}$ while $T_i$ is working on node $i$ according to the original flags of $p (=i')$. This is thus a race condition. To avoid it, $i$ and $i'$ should not be in each other's neighbor list, i.e., $i\notin\mathcal{N}_{i'}$ ($\Leftrightarrow i'\notin\mathcal{N}_i$). According to \autoref{eq:dist}, this is guaranteed if
\begin{eqnarray*}
\separation \ge 2.
\label{eq:sep2}
\end{eqnarray*}
\item \textbf{Neighbor operations}
In \autoref{algo:fact} and \ref{algo:solve}, $T_i$ (resp. $T_{i'}$) updates the local data of $i$'s (resp. $i'$'s) neighbors, i.e., nodes $p$ and $q$ (resp. $p'$ and $q'$). This includes two scenarios as the following. First, $T_i$ updates operators/data associated with a single neighbor of $i$, e.g., $M2M_{\tilde{q}q}$ and $L2L_{q\tilde{q}}$ (lines \ref{line:fact_YN_parent} and \ref{line:fact_NN2_parent}) in \autoref{algo:fact} and $x_p^{(\kappa)}$ (line \autoref{line:elimination_P2L} and \autoref{line:elimination_P2P}) in \autoref{algo:solve}, where $\tilde{p}$ denotes the parent of $p$. Second, $T_i$ updates operators associated with two neighbors of $i$ including $M2L_{pq}$, $M2L_{qp}$, $\PtoL_{pq}$, $\MtoP_{qp}$, $P2P_{pq}$, and $P2P_{qp}$ in \autoref{algo:fact}.
Overall, the neighbor lists of $i$ and $i'$ should not overlap, i.e., $\mathcal{N}_i\cap\mathcal{N}_{i'}=\emptyset$, which is guaranteed if
\begin{eqnarray*}
\separation \ge 3.
\label{eq:sep3}
\end{eqnarray*}
\begin{remark}{}
Proof of $\mathcal{N}_i\cap\mathcal{N}_{i'}=\emptyset$ when $\separation \ge 3$: since $3 \le \separation \le \mathop{\mathrm{dist}}(i, i') \le \mathop{\mathrm{dist}}(i, q) + \mathop{\mathrm{dist}}(q, q') + \mathop{\mathrm{dist}}(q', i')$, where $q \in \mathcal{N}_i$ and $q' \in \mathcal{N}_{i'}$, we have $\mathop{\mathrm{dist}}(q, q') \ge 1$, i.e., $\mathcal{N}_i\cap\mathcal{N}_{i'}=\emptyset$.
\end{remark}
\begin{remark}{}
\autoref{fig:sep3b} and \ref{fig:sep3} show two examples corresponding to the above two scenarios, and illustrate that $\mathcal{N}_i\cap\mathcal{N}_{i'}\not=\emptyset$ when $\separation=2$, while $\mathcal{N}_i\cap\mathcal{N}_{i'}=\emptyset$ when $\separation=3$.
\end{remark}
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[height=.15\textheight]{figure/fig-dd-c-ng}
&\includegraphics[height=.15\textheight]{figure/fig-dd-c-ok}
\end{tabular}
\caption{$T_i$ (resp. $T_{i'}$) updates operators (arrows) associated with $q$ and $\tilde{q}$ (resp. $q'$ and $\tilde{q}'$), where $\tilde{q}$ (resp. $\tilde{q}'$) is the parent, which corresponds to the center of the coarser box. (Left) When $\separation=2$, $q = q'$ and $\tilde{q} = \tilde{q}'$, so there is a race condition when $T_i$ and $T_{i'}$ update the corresponding operators (red arrows) in parallel. (Right) When $\separation=3$, $\tilde{q} = \tilde{q}'$, but $q \not= q'$, so the corresponding operators (green arrows) can be updated in parallel.}
\label{fig:sep3b}
\end{figure}
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[height=.15\textheight]{figure/fig-dd-b-ng}
&\includegraphics[height=.15\textheight]{figure/fig-dd-b-ok}
\end{tabular}
\caption{$T_i$ (resp. $T_{i'}$) updates operators (arrows) associated with $p$ and $q$ (resp. $p'$ and $q'$). (Left) When $\separation=2$, $p = p'$ and $q = q'$, so there is a race condition when $T_i$ and $T_{i'}$ update the corresponding operators (red arrows) in parallel. (Right) When $\separation=3$, $p \not = p'$ and $q \not= q'$, so the corresponding operators (green arrows) can be updated in parallel.}
\label{fig:sep3}
\end{figure}
\item \textbf{Neighbors' interaction list operations}
This operation exists only in \autoref{algo:fact}, not in \autoref{algo:solve}. $T_i$ (resp. $T_{i'}$) updates $i$'s (resp. $i'$'s) neighbors' interaction list (underlined lines in \autoref{algo:fact}). Suppose $q \in \mathcal{N}_i$ and $q' \in \mathcal{N}_{i'}$. The condition that $q \not \in \mathcal{I}_{q'} (\Leftrightarrow q' \not \in \mathcal{I}_{q} )$ needs to be satisfied such that there is no race condition when $T_i$ updates $\mathcal{I}_q$ and $T_{i'}$ updates $\mathcal{I}_{q'}$ in parallel. In other words, the parents of $q$ and $q'$ cannot be neighbors. This is guaranteed if
\begin{eqnarray*}
\separation\ge 6.
\label{eq:sep6}
\end{eqnarray*}
\begin{remark}{}
Proof of $q \not \in \mathcal{I}_{q'}$ when $\separation \ge 6$: since $6 \le \separation \le \mathop{\mathrm{dist}}(i, i') \le \mathop{\mathrm{dist}}(i, q) + \mathop{\mathrm{dist}}(q, q') + \mathop{\mathrm{dist}}(q', i')$, where $q \in \mathcal{N}_i$ and $q' \in \mathcal{N}_{i'}$, we have $\mathop{\mathrm{dist}}(q, q') \ge 4$, i.e., $q \not \in \mathcal{I}_{q'}$ according to \autoref{eq:dist}.
\end{remark}
\begin{remark}{}
\autoref{fig:sep6} illustrates examples that $q \in \mathcal{I}_{q'}$ when $\separation=5$, while $q \not \in \mathcal{I}_{q'}$ when $\separation=6$.
\end{remark}
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
\includegraphics[height=.175\textheight]{figure/fig-dd-d-ng}
&\includegraphics[height=.175\textheight]{figure/fig-dd-d-ok}
\end{tabular}
\caption{$T_i$ (resp. $T_{i'}$) updates $M2L_{ql}$ (resp. $M2L_{q'l'}$) where $q \in \mathcal{N}_i, l \in \mathcal{I}_q$ (resp. $q' \in \mathcal{N}_{i'}, l' \in \mathcal{I}_{q'}$). (Left) When $\separation=5$, $q = l'$, and $q' = l$, so there is a race condition when $T_i$ and $T_{i'}$ update $M2L_{ql}$ and $M2L_{q'l'}$ (red arrows), respectively, in parallel. (Right) When $\separation=6$, $q \not = l'$, and $q' \not= l$, so $M2L_{ql}$ and $M2L_{q'l'}$ (green arrows) can be updated in parallel.}
\label{fig:sep6}
\end{figure}
\end{enumerate}
\subsection{Basic parallel algorithm ($\separation=6$)}\label{s:full}
As the above analysis shows, if nodes $i$ and $i'$ have the same color and $\separation=6$, two threads $T_i$ and $T_{i'}$ have no interference with each other in processing $i$ and $i'$, respectively. Therefore, we can replace the loop over nodes at the same level (line \autoref{line:fact_loop_i} in \autoref{algo:fact}, and lines \ref{line:elimination_loop_i} and \ref{line:substitution_loop_i} in \autoref{algo:solve}) with a double-loop over colors (in serial) and nodes that have the same color (in parallel).
Specifically, we color the tree nodes at every level subject to the constraint that $\separation=6$ in \autoref{eq:sep} (\autoref{algo:group} in \autoref{s:grouping}). Suppose at level $\kappa$, there are $g_\kappa$ colors, and the tree nodes fall into $G_0, G_1, \ldots, G_{g_\kappa-1}$ groups according to their colors. Our basic parallel algorithm can be obtained by replacing
\begin{center}
\hspace*{20pt} \textbf{for} Node $i=0$ \textbf{to} $n_\kappa-1$
\end{center}
at line \autoref{line:fact_loop_i} in \autoref{algo:fact} and line \ref{line:elimination_loop_i} in \autoref{algo:solve} with
\begin{center}
\hspace*{20pt} \textbf{for} Color $I=0$ \textbf{to} $g_\kappa - 1$ \\
\hspace*{20pt} \quad \quad \quad \, \#pragma omp parallel for\\
\hspace*{20pt} \quad \textbf{for all} Node $i \in G_I$
\end{center}
and replacing
\begin{center}
\hspace*{20pt} \textbf{for} Node $i=n_\kappa-1$ \textbf{to} $0$
\end{center}
at line \ref{line:substitution_loop_i} in \autoref{algo:solve} with
\begin{center}
\hspace*{20pt} \textbf{for} Color $I=g_\kappa - 1$ \textbf{to} $0$ \\
\hspace*{20pt} \quad \quad \quad \, \#pragma omp parallel for\\
\hspace*{20pt} \quad \textbf{for all} Node $i \in G_I$
\end{center}
using the ``parallel for'' pragma in OpenMP.
Note that we have to use the same coloring ($\separation$) in \autoref{algo:fact} and \ref{algo:solve} because the ordering in the solve needs to be consistent with that in the factorization; in particular for solving dense linear systems, the ordering of the solve is exactly the reverse of the factorization. Therefore, we cannot apply $\separation=3$ for \autoref{algo:solve} despite the fact thatthe solve is irrelevant to the data dependency of (c).
\subsection{Improved parallel algorithm ($\separation=3$)}\label{s:part}
To improve the efficiency of the basic parallel algorithm, we consider using a smaller $\separation$ in the coloring algorithm at the cost of serializing a part of the algorithm. However, $\separation \le 2$ leads to race conditions in updating neighbors' operators and neighbors' interaction list operators, meaning the factorization and solve are serialized entirely. So, let us consider $\separation=3$, in which case $T_i$ and $T_{i'}$ can work independently in \autoref{algo:solve}, and they interfere with each other only for neighbors' interaction list operations in \autoref{algo:fact}. Therefore, our improved parallel algorithm is based on the coloring with constraint $\separation=3$, and \textit{serializes} the neighbors' interaction list operations (underlined in \autoref{algo:fact}).
Specifically, we still color the tree nodes at every level (\autoref{algo:group} in \autoref{s:grouping}), but with the constraint that $\separation=3$ in \autoref{eq:sep} as apposed to the $\separation=6$ constraint used in the basic parallel algorithm. With the coloring results, we follow the previous parallel algorithm and replace the serial loop over nodes at every level in \autoref{algo:fact} and \ref{algo:solve} with a double-loop, where the outer loop is over all colors in serial and the inner loop is over nodes of the same color in parallel. In addition, we serialize updating neighbors' interaction list operators (underlined in \autoref{algo:fact}) to avoid race condition. This serialization can be implemented with OpenMP locks~\cite{OpenMP} or with OpenMP's critical constructor ``\#pragma omp critical'', which allows only one thread in the critical region at a time. Our implementation adopts the latter for the underlined statements in \autoref{algo:fact}.
Compared with the coloring results ($\separation=6$) used in the basic parallel algorithm, the improved parallel algorithm would have less number of colors and larger group sizes on average for every color, which implies less synchronization cost and more concurrency. The extra serialization cost in the improved parallel algorithm is expected to be small because empirically updating neighbors' interaction list is a small fraction of the total runtime.
\iffalse
In addition, we note that Eigen~\cite{eigenweb} cannot parallelize individual matrix operations in any parallel region, unless the underlying group consists of a single node\footnote{This is determined in the header file \texttt{Eigen/src/Core/products/Parallelizer.h}.}.
\fi
\iffalse
Removing this restriction would provide an opportunity to make full use of threads, but such a fine (and complicated) control of threads is reserved as a future plan.
\fi
\iffalse
\input{algo_factorization_parallel.tex}
\input{algo_solve_parallel.tex}
\fi
\subsection{Coloring algorithm}\label{s:grouping}
Given the tree nodes at the level $\kappa$ in the FMM-tree, we want to assign colors to them so that \autoref{eq:sep} holds for a prescribed $\separation$. The number of colors used is denoted by $g_\kappa$, and the nodes can be classified into groups $G_0, G_1, \ldots, G_{g_\kappa-1}$ based on their colors.
When the FMM-tree is complete, the nodes at the same level lie on a regular grid. It is easy to see that the optimal coloring has $\separation^3$ colors, and nodes of the same color lie on a sub-grid of stride $\separation$, as shown in \autoref{fig:group_example} (a). In general, some of the nodes may be empty and do not need to be colored. In the context of the BEM, only tree nodes corresponding to boundary elements need to be colored. Such an example is shown in \autoref{fig:group_example} (b).
The idea of our greedy algorithm is as follows. We find the first node that is not empty nor colored, and we color it. Then, we try to find other nodes that can be assigned the same color. To find the next node, we first make a stride of $\separation$ and search for a non-empty and un-colored node. Once we find the second node, we make a stride of $\separation$ again and continue the search. After we have exhausted all the remaining nodes, we start over for a new color. This algorithm is shown in \autoref{algo:group}.
\begin{figure}[!htbp]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=.65\textwidth]{figure/group/fig-uniform}
\caption{All nodes are colored}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=.66\textwidth]{figure/group/fig-nonuniform}
\caption{Boundary nodes are colored}
\end{subfigure}
\caption{Coloring results of \autoref{algo:group} in 2D for tree nodes at level $\kappa=3$ and $\separation=3$. (a) The optimal coloring is achieved with $\separation^2=9$ colors. Every color corresponds to a sub-grid of stride 3 in both directions. (b) Only nodes corresponding to the domain boundary are colored. $g_\kappa = 8$ colors are used.}
\label{fig:group_example}
\end{figure}
We note that the computational cost of \autoref{algo:group} is $O(n_\kappa)$. Since up to $\separation^3$ groups are created at line \ref{line:new}, the outer triple-loop over $k_0$, $j_0$, and $i_0$ is executed up to $\separation^3$ times. Since each length of the loops over $k$, $j$, and $i$ is up to $2^\kappa/\separation$, the computational complexity is effectively $\separation^3\times(2^\kappa/\separation)^3=n_\kappa$. In fact, we observed the time to group $n_\kappa$ nodes is proportional to $n_\kappa$ in the two examples in~\autoref{s:sphere} and \autoref{s:office}, where $n_\kappa \lesssim 116$k. In addition, the elapsed time was less than 0.1 seconds in the case of the maximum $n_\kappa$ for each example.
We also note that \autoref{algo:group} does not guarantee to minimize the number of colors or maximize the (average) number of nodes per group. This is due to the fixed stride of $\separation$ (lines \ref{line:g_stride_i}, \ref{line:g_stride_j}, and \ref{line:g_stride_k}). For example, if we look back all the nodes already colored, we would be able to find more nodes in a group but lead to more computational cost.
\input{algo_group.tex}
\section{Application of IFMM to FMBEM}\label{s:preconditioner}
We apply the proposed parallel IFMM to an iterative BEM as the preconditioner. In this section, we first overview the algorithm of the BEM. Then, we define two types of the IFMM-based preconditioners, i.e., mIFMM and nIFMM. Last, we explain the standard preconditioner, that is, the leaf-based block-diagonal preconditioner and its enhancement to be compared with the IFMM-based preconditioners.
\subsection{FMBEM}\label{s:bem}
Following the previous study~\cite{takahashi2017}, we deal with the BEM for 3D Helmholtz equation. Specifically, we solve the acoustic scattering problems with the combined (or Burton-Muller type) boundary integral equations (CBIE)~\cite{1971_Burton_Miller}. We use the preconditioned GMRES~\cite{2003_Saad_book} to iteratively solve the discretized CBIE or the resulting linear system
\begin{eqnarray}
Ax=b,
\label{eq:axb_bem}
\end{eqnarray}
where $A\in\bbbc^{N\times N}$ and $x,b\in\bbbc^N$. Here, $N$ stands for the number of piece-wise constant elements to discretize the surface of a given model (scatterer). To accelerate the matrix-vector product for $A$, we use the low-frequency FMM (LFFMM)~\cite{1995_Epton,2002_Nishimura,2009_Liu_book,2011_Liu,2016_Takahashi,darv04b}. The overall algorithm of our FMBEM is described in \autoref{algo:fmbem}.
\input{algo_fmbem.tex}
The preconditioning is performed through \autoref{algo:precon}. Specifically, the algorithm computes $Mr$, where $M$ denotes a preconditioner such as $M\approx A^{-1}$ and $r$ is a known vector at a given iteration step; therefore, computing $Mr$ means to solve $Av=r$ approximately for $v$. As $M$, we will use nIFMM, mIFMM, and BD introduced below.
\input{algo_precon.tex}
\subsection{IFMM-based preconditioners: nIFMM and mIFMM}\label{s:precon_ifmm}
In the preconditioning routine, we refer to the preconditioner $M$ as \textbf{IFMM-based preconditioner} if $v$ is obtained as a part of $\bar{v}$ that is approximately solved from the IFMM-extended sparse system $\bar{A}\bar{v}=\bar{r}$, where $\bar{r}$ is the extended vector of a given $r$. To use this preconditioner, we first construct $\bar{A}$ and then factorize it according \autoref{algo:fact} or its parallel version before starting GMRES (see lines~\ref{line:fmbem_init} and \ref{line:fmbem_fact} in \autoref{algo:fmbem}). Subsequently, \autoref{algo:solve} or its parallel version is performed when the preconditioning routine is called from GMRES (see \autoref{algo:precon}).
We will denote an IFMM-based preconditioner as \textbf{nIFMM} if IFMM is parallelized with respect to ``n''odes according to \autoref{s:full} (where $\separation=6$ is employed) or \autoref{s:part} (where $\separation=3$). In addition, we will denote another IFMM-based preconditioner as \textbf{mIFMM} if IFMM is parallelized in the grain of individual ``m''atrix operations by means of a multi-threaded linear algebra package; we actually use Intel MKL~\cite{Intel_MKL} through Eigen~\cite{eigenweb}.
We construct the IFMM with the LFFMM; in general, we may handle the IFMM with a different FMM (e.g., the black-box FMM~\cite{fong2007}) from the FMM for the matrix-vector product, i.e., LFFMM in this study. Although it would be simple to manipulate both LFFMMs with the same hierarchy (of boundary elements), the hierarchy of the LFFMM for the matrix-vector products employs the adaptive octree differently from the uniform octree of the IFMM (recall \autoref{s:hdd}). Nevertheless, to share the information as much as possible, we decided that both root nodes have the same position and size. Then, the FMM operators can be shared by the nodes of both FMMs especially in upper levels (see line~\ref{line:reuse} in \autoref{algo:fmbem}). This (irregular) combination of hierarchies was actually used in the previous work~\cite{takahashi2017}.
Yet, the depth of the IFMM's hierarchy (i.e., $\ell$) is not necessarily the same as that of the FMM's hierarchy (i.e., $\ell_{\rm FMM}$), although $\ell\equiv\ell_{\rm FMM}$ was supposed in \cite{takahashi2017}. We will determine the value of $\ell_{\rm FMM}$ by a conventional manner, as described in \autoref{s:num_setting}. Meanwhile, we regard $\ell$ as a tunable parameter, which can affect the performance of nIFMM and mIFMM.
Basically, if we let $\ell$ be less than $\ell_{\rm FMM}$, we no longer perform the factorization at the omitted levels $\ell+1$ to $\ell_{\rm FMM}$. In addition, we can handle a smaller number of nodes at level $\ell$ than $\ell_{\rm FMM}$. The downside is that the matrix manipulation (e.g., low-rank compression) can require more computational cost because the size of individual matrices (FMM operators) increases as $\ell$ decreases; this is because the size is determined by the formula in \autoref{eq:p} in \autoref{s:num_setting}. Further, the initialization cost can become large (respectively, small) if the IFMM cannot (respectively, can) fully reuse the FMM operators, especially the P2P operators, precomputed at leaf nodes of the FMM's hierarchy (see line~\ref{line:reuse} in \autoref{algo:fmbem}, again).
Another parameter of nIFMM is the top level $\tau$ ($\in[2,\ell]$). Namely, the node-based parallelism is applied to level $\tau$ to $\ell$, while the matrix-based parallelism is applied to level 2 to $\tau-1$. By switching the parallelism from the node-based to the matrix-based, we can expect a higher performance at coarse (upper) levels. This is because individual matrix is larger at coarse levels than fine levels and, thus, can be processed more efficiently with multiple threads.
For conciseness, we denote the nIFMM using the parameters $\separation$ (``sep''aration), $\tau$ (``top'' level), and $\ell$ (``bot''tom level) by \textbf{nIFMMsep$\separation$top$\tau$bot$\ell$} or \textbf{nIFMMs$\separation$t$\tau$b$\ell$} for short. Regarding mIFMM, we use the term \textbf{mIFMMbot$\ell$} or \textbf{mIFMMb$\ell$} to specify the bottom level $\ell$. \autoref{tab:names} summarizes the naming scheme of the IFMM-based preconditioners as well as the (extended) block-diagonal preconditioner, which will be explained in the next section.
\subsection{Block-diagonal preconditioner: BD}\label{s:precon_bd}
We will compare mIFMM and nIFMM with the leaf-based block-diagonal preconditioner (referred to as \textbf{BD}), which is frequently used for FMBEM~\cite{2009_Liu_book}. The corresponding preconditioning matrix $M$ is computed as a block-diagonal matrix consisting of the inverse of the self P2P operators (i.e., $P2P_{ii}$) in terms of leaf nodes (see \autoref{algo:precon}), where the FMM that BD relies on is nothing but the LFFMM for the matrix-vector product. Each self P2P operator is LU-decomposed at the initialization stage (see \autoref{algo:fmbem}). It should be noted that BD is fully parallelizable with respect to leaf nodes in each level in both initialization and application (preconditioning) stages.
We can improve the performance of BD by selecting its bottom level (depth), denoted by $\ell_{\rm BD}$, where we actually compute the diagonal blocks of $M$. If we let $\ell_{\rm BD}<\ell_{\rm FMM}$, the corresponding $M$ can have a wider bandwidth and become a better approximation of $A^{-1}$. Then, it can reduce the number of iterations until convergence. On the other hand, the cost and memory for factorizing the individual diagonal blocks becomes more expensive at the initial stage. We denote the enhanced BD using the bottom level of $\ell_{\rm BD}$ by \textbf{BDbot$\ell_{\rm BD}$} or \textbf{BDb$\ell_{\rm BD}$}.
\begin{table}[h]
\centering
\caption{List of the preconditioners. The value of $\ell_{\rm FMM}$ is determined as in \autoref{s:num_setting}.}
\label{tab:names}
\begin{tabular}{|c|c|p{100pt}|}
\hline
Name & Name specifying parameters & \\
\hline
nIFMM & \begin{tabular}{c} nIFMMsep$\separation$top$\tau$bot$\ell$\quad or\quad nIFMMs$\separation$t$\tau$b$\ell$ \\ (where $\separation=3$ or $6$ and $2\le\tau\le\ell\le\ell_{\rm FMM}$) \end{tabular} & Applies the matrix-based parallelism to level 2 to $\tau-1$ and the node-based parallelism to level $\tau$ to $\ell$.\\
\hline
mIFMM & \begin{tabular}{c} mIFMMbot$\ell$\quad or\quad mIFMMb$\ell$ \\ (where $2\le\ell\le\ell_{\rm FMM}$) \end{tabular} & Applies the matrix-based parallelism to all the levels, i.e. 2 to $\ell$.\\
\hline
BD & \begin{tabular}{c} BDbot$\ell_{\rm BD}$\quad or\quad BDb$\ell_{\rm BD}$ \\ (where $2\le\ell_{\rm BD}\le\ell_{\rm FMM}$) \end{tabular} & Handles nodes at level 2 to $\ell_{\rm BD}$.\\
\hline
%
\end{tabular}
\end{table}
\section{Details for sphere model}\label{s:sphereii_details}
\section{Survey of the best settings for the Sphere model (Section~\ref{s:sphere})}\label{s:sphereii_details}
\subsection{BD}\label{s:sphereii_BD}
To make a fair comparison with the IFMM-based preconditioners, we investigated the best value of the bottom level $\ell_{\rm BD}$. The timing results of the FMBEM based on the BD using $\ell_{\rm BD}$ of $3$ to $7$ are compared in Table~\ref{tab:sphereii10_BD}. As $\ell_{\rm BD}$ decreased, the number of iterations (in the column ``\niter'') basically decreased but both initialization (``\init'') and preconditioning times (``prec} %10 \newcommand{\ms}{ms'') increased. As a result, $\ell_{\rm BD}=5$ was the best in terms of the total computation time, as seen in Figure~\ref{fig:sphereii10_BD_total}. We thus determined \textbf{BDbot5} as the best BD.
\begin{table}[h]
\centering
\caption{Profile of \textbf{BD} for the \textbf{sphere} model. The speedup (``\tsp'') is relative to BDbot5. See Section~\ref{s:sphereii_BD}.}
\label{tab:sphereii10_BD}
\def./181031/sphereii{./181031/sphereii}
\scriptsize
\myinputII{./181031/sphereii/sphereii10-fmm3logBDX0-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/sphereii10-fmm3logBDX1-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logBDX2-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logBDX3-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logBDX4-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[h]
\centering
\def./181031/sphereii{./181031/sphereii}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=100, ymax=10000,
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={BDb7, BDb6, BDb5, BDb4, BDb3},
]
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logBDX0-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logBDX1-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logBDX2-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logBDX3-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logBDX4-ngauss4-jpre3-tol1e-5-maxepc040-delta7-inc95_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{BD} for the \textbf{sphere} model.}
\label{fig:sphereii10_BD_total}
\end{figure}
\iffalse
\input{tab_legend}
\fi
\subsection{mIFMM}\label{s:sphereii_mIFMM}
We measured the performance of mIFMM with varying its depth $\ell$ from $4$ to $7$ ($=\ell_{\rm FMM}$). Table~\ref{tab:sphereii10_mIFMM} shows the result. In comparison with BD in Table~\ref{tab:sphereii10_BD}, the number of iterations (in the column ``\niter'') decreased significantly. We can observe that, although the highest speedup relative to the best BD (``\tsp'') was only $1.1$ at $k=32$, $\ell=6$ is the fastest (Figure~\ref{fig:sphereii10_mIFMM_total}). We thus determined \textbf{mIFMMbot5} as the best.
We note that mIFMM could not run $k=32$ with $\ell=7$ because the out of memory occurred during the initialization for level $7$.
\begin{table}[h]
\centering
\caption{Profile of \textbf{mIFMM} for the \textbf{sphere} model. The speedup (``\tsp'') is relative to the best BD (i.e., BDbot5).}
\label{tab:sphereii10_mIFMM}
\def./181031/sphereii{./181031/sphereii}
\scriptsize
%
\myinputII{./181031/sphereii/sphereii10-fmm3logXsub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[]
\centering
\def./181031/sphereii{./181031/sphereii}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=100, ymax=100000,
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={mIFMMb7, mIFMMb6, mIFMMb5, mIFMMb4}
]
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{mIFMM} for the \textbf{sphere} model.}
\label{fig:sphereii10_mIFMM_total}
\end{figure}
\subsection{nIFMM}\label{s:sphereii_nIFMM}
We explored the best setting of nIFMM, with varying $\tau$ from $2$ to $4$ and $\ell$ from $4$ to $7$ and fixing $\separation$ to $3$. Table~\ref{tab:sphereii10_nIFMM} clearly shows that nIFMM reduces its total computation time (``total} %5 \newcommand{\main}{main'') by reducing the number of iterations (``\niter'') at the cost of the preconditioning (``prec} %10 \newcommand{\ms}{ms''), initialization (``\init''), and factorization (``elim} %12 \newcommand{\elim}{elim''). As a result, \textbf{nIFMMsep3top3bot6} was determined as the best from Figure~\ref{fig:sphereii10_nIFMM_total}, where only the result of $\tau=3$ is shown because the remaining $\tau=2$ and $4$ were slightly worse than $\tau=3$ in terms of the total computation time.
Since the initialization routine is common to all the IFMM-based preconditioners, similaly to mIFMM, nIFMM could not perform $k=32$ with $\ell=7$ because the out of memory in the initialization of level $7$.
\begin{table}[h]
\centering
\caption{Profile of \textbf{nIFMM} for the \textbf{sphere} model. The speedup (``\tsp'') is relative to the best BD (i.e., BDbot5).}
\label{tab:sphereii10_nIFMM}
\def./181031/sphereii{./181031/sphereii}
\scriptsize
\myinputII{./181031/sphereii/sphereii10-fmm3logXsep3top3sub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{1}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsep3top3sub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsep3top3sub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{0}
\myinputII{./181031/sphereii/sphereii10-fmm3logXsep3top3sub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-3-basis1e-3-LR3.tex}{}{}{}{}{}{}{}{2}
\end{table}
\begin{figure}[h]
\centering
\def./181031/sphereii{./181031/sphereii}
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Total computation time [sec]},
xmode=log,
ymode=log,
xmin=1, xmax=100,
ymin=100, ymax=100000,
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={nIFMMs3t3b7, nIFMMs3t3b6, nIFMMs3t3b5, nIFMMs3t3b4}
]
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsep3top3sub0-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsep3top3sub1-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsep3top3sub2-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\addplot table [x=k, y=total] {./181031/sphereii/sphereii10-fmm3logXsep3top3sub3-ngauss4-jpre10-tol1e-5-maxepc040-delta7-inc95-fillin1e-1-basis1e-1-LR3_total.pgf};
\end{axis}
\end{tikzpicture}
\caption{Total computation time of \textbf{nIFMM} for the \textbf{sphere} model.}
\label{fig:sphereii10_nIFMM_total}
\end{figure}
\subsection{Test problem --- sphere model}\label{s:sphere}
\subsubsection{Problem statement}
We considered an external Neumann problem for an acoustically hard sphere (with the radius of 0.5 and the center at the origin) irradiated by a point source $f(\fat{x})$ at $\fat{s}=(0.0,0.0,0.8)^{\mathrm{T}}$, i.e., $f(\fat{x}):=\exp(\mathrm{i} k\abs{\fat{x}-\fat{s}})/(k\abs{\fat{x}-\fat{s}})$. We discretized the surface of the sphere with 1003520 triangular piece-wise constant boundary elements. The exact solution is available from the reference~\cite[Section 10.3.1]{Bowman}. In what follows, we considered $k=4$, $8$, $16$, and $32$.; the diameter of the sphere is about five times larger than the wavelength when $k=32$.
\subsubsection{Survey of the best BD, mIFMM, and nIFMM}\label{s:sphereii_best}
First of all, we searched the best value of the parameter $\mu$ (recall \autoref{s:num_setting}) so that the total computation time of the FMBEM based on the standard BD was minimized. As a result, $\mu=40$ was nearly the best. Then, the corresponding depth $\ell_{\rm FMM}$ was $7$.
Consequently, using $\mu=40$, we surveyed the best (fastest) BD, mIFMM, and nIFMM by varying the parameters $\ell_{\rm BD}$, $\ell$, and $\tau$, where the separation parameter $\separation$ was fixed to $3$ (\autoref{s:part}). As a result, we determined \textbf{BDbot5}, \textbf{mIFMMbot6}, and \textbf{nIFMMsep3top3bot6} as the best ones in this example. The details are described in Section~1 of the supplementary material.
\autoref{tab:stat_sphereii10} shows the statistics of the hierarchies used in the following computations. We can observe that the number of nodes of the FMM and IFMM (in the second and fourth columns, respectively) is $\sim 4^\kappa$ at level $\kappa$, which is because the distribution of boundary elements is planar in 3D. Regarding nIFMM, the statistics of the grouping with $\separation=3$ is shown in the fifth to eighth columns. We can see that the group size at level $\kappa$ is $\sim 4^\kappa$ in average. In addition, $p\sim\kappa^{-1}$ is observed in the ninth and tenth columns.
\input{tab_stat_sphere.tex}
\subsubsection{Comparison under the best setting}\label{s:sphereii_comp3}
\autoref{fig:tsp-nIFMM} shows the speedup (in terms of the total computation time) of the best nIFMM with the best BD and mIFMM. Although the best nIFMM was slower than or comparable to the best BD for low wavenumbers, it achieved 2.2 times speedup at the largest $k$. It should be noted that the performance of BD can be improved by parallelizing the present GMRES program. Even if this is done successfully, the best nIFMM would remain faster than the best BD at $k=32$; this is indicated in Tables~I and III of the supplementary material.
In addition, the best nIFMM was about two times faster than the best mIFMM for any $k$. The drop at $k=32$ can be explained by the fact that the performance of mIFMM can increase with $k$, which will be examined in \autoref{s:sphereii_nIFMM_vs_mIFMM}.
\begin{figure}[h]
\centering
\iftrue
\begin{tikzpicture
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel style={align=center},
ylabel={Speedup of nIFMMs3t3b6\\relative to BDb5 and mIFMMb6},
xmode=log,
xmin=1, xmax=100,
ymin=0.0, ymax=3.0, ytick={0.0,0.5,...,3.0},
legend pos=south east,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={BDb5,mIFMMb6},
]
\addplot table [x=k, y expr=\thisrow{nIFMM}/\thisrow{BD}] {./181031/sphereii/fig-tsp-nIFMM-sphereii10-delta7-epsilon1e-3-relto-BD.table};
\addplot table [x=k, y expr=\thisrow{nIFMM}/\thisrow{mIFMM}] {./181031/sphereii/fig-tsp-nIFMM-sphereii10-delta7-epsilon1e-3-relto-mIFMM.table};
\end{axis}
\end{tikzpicture}
\else
\includegraphics[width=.5\textwidth]{./181031/sphereii/fig-tsp-nIFMM-sphereii10-delta7-epsilon1e-3}
\fi
\caption{Speedup of the best nIFMM (i.e., nIFMMsep3top3bot6) relative to the best BD (i.e., BDbot5) and mIFMM (i.e., mIFMMbot6) in terms of the total computation time for the sphere model.
\label{fig:tsp-nIFMM}
\end{figure}
\subsubsection{Comparison of node- and matrix-based parallelizations}\label{s:sphereii_nIFMM_vs_mIFMM}
We compared the two parallel methods, using the best nIFMMsep3top3bot6 and mIFMMbot6 in \autoref{fig:sphereii10_speedup}. The factorization phase was actually boosted by the proposed node-based parallelization. The speedup became smaller with $k$. This is likely due to the fact that each operator becomes larger with $k$ and, thus, individual matrix operation can be performed more efficiently with multiple threads.
Regarding the solve phase, the node-based parallelization achieved about 12 times speedup. This significant speedup is due to the fact that the number of matrix operations involved in the solve phase is absolutely small and therefore the matrix-based parallelization has little chance to work efficiently. In addition, the parallelized solve algorithm is free from the serialization unlike the factorization phase. However, similarly to the factorization phase, the efficiency of the node-based parallelization tends to decrease with $k$.
\begin{figure}[h]
\centering
\iftrue
\begin{tikzpicture}
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel={Speedup},
xmode=log,
xmin=1, xmax=100,
ymin=0.0, ymax=16.0, ytick={0.0,2.0,...,16.0},
legend pos=south west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={factorization phase, solve phase}
]
\addplot table [x=k, y=sp_ifmm_elim] {./181031/sphereii//fig-nIFMM_relto_mIFMM-sphereii10-delta7-epsilon1e-3.table};
\addplot table [x=k, y=sp_msolve_per_iter] {./181031/sphereii//fig-nIFMM_relto_mIFMM-sphereii10-delta7-epsilon1e-3.table};
\end{axis}
\end{tikzpicture}
\else
\includegraphics[width=.5\textwidth]{./181031/sphereii//fig-nIFMM_relto_mIFMM-sphereii10-delta7-epsilon1e-3}
\fi
\caption{Speedup of the node-based parallelization relative to the matrix-based parallelization in terms of the factorization and solve phases. Here, since the number of iterations (denoted by ``\niter'' in Tables II and III in the supplementary material) is preconditioner specific, the time of the solve phase was calculated by dividing the preconditioning time (which is spent for executing the preconditioning in \autoref{algo:precon}) by the number of applications (i.e., $\textrm{\niter}+2$).}
\label{fig:sphereii10_speedup}
\end{figure}
\subsubsection{Influence of nIFMM's parameters}\label{s:sphereii_influence}
\autoref{tab:sphereii10_elim} gives a closer look at the elapsed times of the initialization and factorization phases level by level for the best nIFMM at $k=16$. We analyzed the influence of the parameters as follows:
\begin{enumerate}
\item Influence of $\ell$ on initialization:
To construct the extended matrix, the IFMM's initializer computes the FMM operators for every node and level (at line \ref{line:reuse} in \autoref{algo:fmbem}). If $\ell$ is chosen as $\ell_{\rm FMM}$ ($=7$), the initializer can largely reuse the operators that FMM precomputed (at line \ref{line:precomp}), although reading/writing an operator from/to memory requires $O(p^4)$ cost; this is why the initializer spent about 100 seconds even when $\ell=7$.
If we let $\ell<\ell_{\rm FMM}$, we no longer initialize the levels from $\ell+1$ to $\ell_{\rm FMM}$. On the other hand, we have to newly handle the (uncompressed) FMM operators at the leaf level $\ell$. The additional cost is estimated as $\sim 4^{-\ell}+4^\ell\ell^{-4}$ from the observations\footnote{The cost is proportional to the number of nodes and the square of the individual matrix size at the leaf level $\ell$. The former is $\sim 4^\ell$, while the latter is $\sim(4^{-\ell})^2$ for $P2P^{(\ell)}_{ij}$ and $\sim(p^2)^2\sim\ell^{-4}$ for others such as $M2L^{(\ell)}_{ij}$, where $p\sim\kappa^{-1}$ holds at level $\kappa$.} in \autoref{s:sphereii_best} and increases exponentially as $\ell$ decreases. The trade-off of the positive and negative effects can explain the trend of the initialization time in the table.
\iffalse
We should mention a specific case that the choice of $\ell=\ell_{\rm FMM}$ most likely results in a poorer performance than that of $\ell=\ell_{\rm FMM}-1$. The case is when the number of nodes at the leaf level $\ell_{\rm FMM}$ is considerably smaller than that of level $\ell_{\rm FMM}-1$ (\autoref{fig:bitree-fmm} left). In this case, if we let $\ell=\ell_{\rm FMM}$, IFMM needs to generate vast number of new leaf nodes at level $\ell$ (\autoref{fig:bitree-fmm} center). This can increase the cost of the IFMM's initialization. The memory requirement for the new nodes can be also an issue. To save the initialization cost, the choice of $\ell=\ell_{\rm FMM}-1$ (or less) would be better (\autoref{fig:bitree-fmm} right). In general, we may consider the first candidate of $\ell$ as the level where the population is the largest among all the levels and, thus, IFMM can efficiently reuse the operators that FMM precomputed.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{figure/fig-bitree-fmm}
\caption{Schematic illustration of the case that the choice of $\ell=\ell_{\rm FMM}$ is worse than that of $\ell=\ell_{\rm FMM}-1$. (Left) Example of the FMM's adaptive tree (binary tree for the illustration purpose), where $\ell_{\rm FMM}$ is 4 and leaf nodes are represented by circles colored in black. Leaf nodes can exist at any levels between 2 and $\ell_{\rm FMM}$ ($=4$), but the majority are at level $\ell_{\rm FMM}-1$ ($=3$), as illustrated. (Center) A corresponding IFMM's uniform tree if $\ell$ is set to $\ell_{\rm FMM}$. In this case, leaf nodes colored in red at level $\ell_{\rm FMM}$ must be newly created. (Right) An alternative IFMM's uniform tree if $\ell$ is set to $\ell_{\rm FMM}-1$. In this case, the number of leaf nodes to be newly generated (i.e., 4) is less than the former case of $\ell=\ell_{\rm FMM}$ (i.e., 14).}
\label{fig:bitree-fmm}
\end{figure}
\fi
\item Influence of $\ell$ on factorization:
Similarly to the above observation, if we let $\ell<\ell_{\rm FMM}$, we can omit the factorization from level $\ell+1$ to $\ell_{\rm FMM}$. On the other hand, the time can increase as $\ell$ decreases, because the cost at the leaf level $\ell$ can be estimated as $\sim 4^\ell \times (p^2)^3=4^\ell\ell^{-6}$ from some presumptions\footnote{The first factor $4^\ell$ denotes the number of nodes at leaf level $\ell$ and the second factor $(p^2)^3$ denotes the cost of a matrix-matrix product, which is the typical manipulation in the factorization, at the leaf level, where uncompressed FMM operators of size $\sim p^2\times p^2$ are manipulated}. This cost function has the minimum value at $\ell=\frac{6}{\log 4}\approx 4.3$. This indicates that there exists an optimal value of $\ell$ that minimizes the cost at leaf level.
In fact, \autoref{tab:sphereii10_elim} shows that, when we decreased $\ell$ from $7$ ($=\ell_{\rm FMM}$) to 6, the time for the leaf level almost remained, but the total time was decreased by the time for level 7, i.e., about 47 seconds. This is the positive effect of decreasing $\ell$. On the other hand, when we decreased $\ell$ from 6 to 5 or 4, both time for the leaf level and total time increased significantly. This corresponds to the negative effect and means that the minimizer was actually around 6 rather than 4.3.
\item Influence of $\tau$ on factorization:
In \autoref{tab:sphereii10_elim}, $\tau=3$ has no clear difference from $\tau=2$ at level 2, while $\tau=4$ certainly spoiled the performance at level 3, to which the matrix-based parallelization is applied. This is because the size of each matrix (operator) at level 3 is too small to cultivate an enough parallel efficiency with 16 computing cores.
\end{enumerate}
\input{tab_breakdown.tex}
\subsubsection{Basic parallel algorithm ($\separation=6$) vs improved parallel algorithm ($\separation=3$)}\label{s:sphereii_sep6}
We compared the improved parallel algorithm based on $\separation=3$ with the basic parallel algorithm based on $\separation=6$. The latter is free from any serialized operations, but the group size is about eight ($=6^3/3^3$) times smaller than $\separation=3$ in average.
When we varied $\tau$ from $2$ to $4$ and $\ell$ from $4$ to $7$ in the case of $\separation=6$, the best performance was obtained by nIFMMsep6top4bot6, although the value of $\tau$ was almost insensitive to the timing result.
\autoref{fig:sphereii10_sep6} compares nIFMMsep6top4bot6 with nIFMMsep3top3bot6 (i.e., the best nIFMM using $\separation=3$). Although the difference in total computation time is not significant, this figure shows that $\separation=3$ was indeed better than $\separation=6$ for any $k$.
\begin{figure}[h]
\centering
\iftrue
\begin{tikzpicture}
\begin{axis}[
xlabel={Wavenumber $k$},
ylabel={Speedup of $\separation=3$ relative to $\separation=6$},
xmode=log,
xmin=1, xmax=100,
ymin=1.0, ymax=2.5, ytick={1.0, 1.5,..., 2.5},
legend pos=north west,
legend cell align={left},
ymajorgrids=true,
grid style=dashed,
legend entries={factorization phase, solve phase, total}
]
\addplot table [x=k, y=sp_ifmm_elim] {./181031/sphereii/fig-nIFMM_relto_nIFMM6-sphereii10-delta7-epsilon1e-3.table};
\addplot table [x=k, y=sp_msolve_per_iter] {./181031/sphereii//fig-nIFMM_relto_nIFMM6-sphereii10-delta7-epsilon1e-3.table};
\addplot table [x=k, y=sp_main] {./181031/sphereii//fig-nIFMM_relto_nIFMM6-sphereii10-delta7-epsilon1e-3.table};
\end{axis}
\end{tikzpicture}
\else
\begin{tabular}{c}
\includegraphics[width=.5\textwidth]{./181031/sphereii/fig-nIFMM_relto_nIFMM6-sphereii10-delta7-epsilon1e-3}
\end{tabular}
\fi
\caption{Speedup of the best nIFMM based on $\separation=\textbf{3}$ (i.e., nIFMM\textbf{sep3}top3bot6) relative to the best one based on $\separation=\textbf{6}$ (i.e., nIFMM\textbf{sep6}top4bot6) in the sphere model. Here, the time of the solve phase was calculated by dividing the preconditioning time by $\textrm{\niter}+2$; see the caption of \autoref{fig:sphereii10_speedup}.}
\label{fig:sphereii10_sep6}
\end{figure}
\subsubsection{Influence of $\varepsilon$ and accuracy}\label{s:sphereii_accuracy}
We tested relatively low-precision $\varepsilon$, i.e., $\varepsilon=10^{-1}$ and $10^{-2}$, for mIFMMbot6 and nIFMMsep3top3bot6, which are the best in the case of $\varepsilon=10^{-3}$. A larger $\varepsilon$ led to a shorter initialization and factorization time but resulted in a larger number of iterations. As a result, $\varepsilon=10^{-2}$ maximized the performances of both mIFMMbot6 and nIFMMsep3top3bot6 in the sphere model.
In all the above analyses, the accuracy of the solutions obtained by all the preconditioners were almost the same for every $k$. In particular, $\varepsilon$ did not affect the accuracy; this is shown in Tables~I, II, and III in the supplementary material.
\subsubsection{Memory consumption}
Regarding the largest $k=32$, nIFMMsep3top3bot6 required 381, 393, and \unit{410}{GB} for $\varepsilon=10^{-1}$, $10^{-2}$, and $10^{-3}$, respectively. About 90\% of the required memory was spent for constructing the extended matrix $\bar{A}$ in the initialization phase. Since this phase is common, mIFMMbot6 required almost the same memory as the nIFMM. The huge memory consumption is problematic, but will be less of an issue with distributed memory.
On the other hand, BDbot5 required \unit{25}{GB} for $k=32$, where the precomputation of M2L, P2P, and the preconditioning matrix $M$ spent 9, 6, and \unit{5}{GB}, respectively.
|
1,314,259,992,995 | arxiv | \section{INTRODUCTION}
\label{introduction}
The integrated Sachs-Wolfe (ISW) effect \citep{1967ApJ...147...73S} probes the time variation of gravitational potential, through the induced CMB temperature fluctuation
\begin{equation}
\label{dT2}
\Delta T(\widehat{n})=\frac{2}{c^3}T_0\int\dot{\Phi}(r,\widehat{n})\,a\,dr \ .
\end{equation}
Here $\widehat{n}$ is the line of sight, $T_0$ the mean temperature of CMB, $c$ the speed of light, $\dot{\Phi}$ the time derivative of the gravitational potential along, $a$ the cosmic scale factor and $r$ the comoving radial distance. The gravitational potential $\Phi$ at large/linear scale is time independent, if gravity is GR, the universe is flat and the total matter density $\Omega_m=1$. Observations of primary CMB \citep{2016A&A...594A..13P} show that our universe is flat. Then within the framework of general relativity, any detection of the ISW effect would serve as a smoking gun of dark energy. It can then be used to constrain the dark energy equation of state, and even clustering of dark energy around horizon scale (e.g. \cite{2003MNRAS.346..987W,2004PhRvD..69h3503B,2004PhRvD..70l3002H,10.1111/j.1745-3933.2006.00218.x,2008ApJ...675...29M,2010PhRvD..81j3513D}). Alternatively, it can be used to test GR at cosmological scales \citep{2003AnPhy.303..203H,2006PhRvD..73l3504Z,2006PhRvD..74f3520G,2007MNRAS.381.1347C,2008PhRvD..78h7303F,2012MNRAS.426.2581G}, constrain primordial non-Gaussianities \citep{2012JCAP...06..042N}, or probe GR backreaction \citep{2017MNRAS.469L...1R}.
The major factor limiting the cosmological application of ISW is its weak signal, overwhelmed by the primary CMB. It can be separated from primary CMB, by cross-correlating with the large scale structure (LSS) \citep{1996PhRvL..76..575C,2000ApJ...538...57S}. However, to further suppress the cosmic variance, CMB surveys of nearly full sky coverage and wide and deep galaxy surveys are both required. From the release of the first year WMAP data, there have been various works to measure the ISW effect \citep{2003ApJ...597L..89F,2004Natur.427...45B,2004PhRvD..70h3536A,2004PhRvD..69h3524A,2004MNRAS.350L..37F,2004ApJ...608...10N,2005NewAR..49...75B,2005PhRvD..72d3525P,2005PhRvD..71l3521C,2006MNRAS.365..891V,2006astro.ph..2398M,2007MNRAS.381.1347C,2007MNRAS.377.1085R,2008MNRAS.386.2161R,2010A&A...520A.101H,2010MNRAS.404..532M,2012MNRAS.427.3044S,2012MNRAS.426.2581G,2014A&A...571A..19P,Shajib_2016,2016A&A...594A..21P}. The analyzed CMB experiments include both WMAP and Planck. The analyzed galaxy surveys include SDSS, NVSS, 2MASS, WISE, etc. The LSS tracers include SDSS main galaxies and luminous red galaxies in optical bands, radio galaxies, AGNs, and even weak gravitational lensing reconstructed from CMB \citep{2016A&A...594A..21P}. Although some claimed detection significances ($\sim 2$-$4\sigma$) may be questionable\citep{2012MNRAS.426.2581G,2014MNRAS.438.1724H}, these measurements are in general consistent with the $\Lambda$CDM prediction.
Besides directly using galaxies as LSS tracers, entities derived with galaxy surveys such as superclusters and voids are also explored to measure the ISW effect \citep{2008ApJ...683L..99G,2011ApJ...732...27P,2014A&A...571A..19P}. These measurements have different S/N and different systematics, and therefore are highly complementary to ISW measurements with galaxies. For example, ISW measured from voids may have less contamiantion from radio emission associated with galaxies. However, there are tensions betwen existing measurements and between measurements and theoretical prediction. For example, \cite{2008ApJ...683L..99G} stacked the most significant 50 clusters/voids (superclusters/supervoids) identified with SDSS photo-z data, and found a $4\sigma$ detction of the ISW effect. However, after investigated by other papers, this detection has been found difficult to explain \citep{2010MNRAS.401..547H,2010ApJ...724...12I,2014A&A...572C...2I,2015MNRAS.446.1321H,2015MNRAS.452.1295K}.
Tensions persists in later measurements with different void catalogues. In most cases excess signals are reported compared to the $\Lambda CDM$ prediction \citep{2012JCAP...06..042N}. For example, \cite{2014ApJ...786..110C} found that their detection with SDSS voids is at odds with simulations of a $\Lambda CDM$ universe at $\sim 2\sigma$. In term of the ISW amplitude $A_{ISW}$ ($A_{ISW}=1$ in the standard $\Lambda$CDM cosmology), \citet{2018MNRAS.475.1777K} found with BOSS supervoids $A_{ISW} \approx 9$. \citet{2019MNRAS.484.5267K} found with DES super-voids $A_{ISW} \approx 4.1$.
These tensions are unlikely caused by new physics beyond $\Lambda$CDM, since the ISW measurements with galaxies in the same surveys/cosmic volumes are usually consistent with the $\Lambda$CDM prediction. Furthermore, it is found that the ISW measurements with voids rely heavily on the void catalogues (e.g. \citet{2008ApJ...683L..99G, 2015MNRAS.452.1295K}) and therefore the associated systematics, if uncorrected. In addition, null detections were also reported in observations and favored in simulations \citep{2015MNRAS.446.1321H,2014A&A...572C...2I}.
Among all possibilities leading to the above controversies, void identification in observations likely plays a major role. Voids are defined as low number density regions in the galaxy distribution field. But in low density regions, the galaxy number distribution suffers from relatively larger shot noise. This makes the identification of voids and their centers/radii difficult. The low number density further amplifies the impact of non-uniform galaxy (radial and angular) selection function in spectroscopic redshift surveys, and making its correction more challenging than the case of galaxy clustering. Situation becomes even worse for the photometric data, for which the smearing effect of redshift errors of galaxies in the line-of-sight will ``merge" voids. These issues also complicate the correspondence between voids in observations and in simulations/theory, and make the theoretical interpretation difficult.
Taking these issues into account, in this paper we revisit the measurement of ISW effect by considering a new LSS tracer -- ``low-density-position'' (LDP) \citep{2019ApJ...874....7D}. LDPs are the collection of sky positions, after removing positions within a given radius of any observed galaxy. Statistically speaking, they correspond to low density regions. The density threshold depends on the radius to perform the cut and the mean number density of observed galaxies. So by varying the cut radius, LDPs can also serve as intermediate case between galaxies and voids. So they will provide independent check on the above tensions found between voids and the concordance $\Lambda$CDM, and between observations of voids and galaxies. Furthermore, comparing to voids, LDPs are more straightforward to identify in both observations and in simulations, making the data interpretation more reliable. In \cite{2019ApJ...874....7D}, we have used LDPs to achieve significant detection of weak lensing using CFHTLenS data, and to differentiate dark energy models. In this work, we treat LDPs as density tracers and focus on the LDP-temperature correlation measurement. Our aim is to provide independent ISW measurement, which is then used to test the concordance $\Lambda$CDM cosmology, and cross-check with existing ISW measurements with voids.
The paper is organized as follows: In \S\ref{sec:obs-data}, we introduce the data sets and procedures to measure the LDP-temperature correlations ($\omega_{T\ell}$). In \S\ref{sec:simu-data}, we calculate the theoretical prediction from mocks generated by a $\Lambda CDM$ N-body simulation. \S$\ref{sec:result}$ shows our main results of ISW measurement with the LDP method. In \S\ref{sec:conclusion}, we give conclusion and discussions about related issues.
Some further technical details are presented in the appendix, along with the measured CMB-galaxy correlation (appendix\ref{appendix:ISW-galaxy}).
\section{Operating with the Observational Data}
\label{sec:obs-data}
We choose the DR8 galaxy catalogue of the DESI imaging surveys to construct the LDP field, and then cross with Planck SMICA map to measure the ISW effect in the low-density regions of the universe.
\subsection{Galaxy Catalogue and LDP generation}
\label{sec:obs}
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-galactic.pdf}}
\caption{Source distribution in the DR8 catalogue of the BASS + MzLS + DECaLS + DES imaging surveys. The depth of color represents the number of galaxies per $arcmin^2$.}
\label{fig:DR8}
\end{figure}
The DR8 galaxy catalogue\footnote{\url{http://batc.bao.ac.cn/~zouhu/doku.php?id=projects:desi_photoz:;}} is a combination of four surveys: BASS\citep{2019ApJS..245....4Z}, MzLS\citep{2016AAS...22831702S}, DECaLS\citep{2005astro.ph.10346T,2016AAS...22831701B} and DES\citep{2018ApJS..239...18A}. They are independent optical imaging surveys with close photometric systems. The first three together with the infrared WISE survey \citep{2010AJ....140.1868W} aim at providing galaxy and quasar targets for the follow-up Dark Energy Spectroscopic Instrument survey (DESI; \cite{2016arXiv161100036D}). In the DR8 data release, BASS+MzLS locates in the north Galactic cap, DES locates in the south Galactic cap, and DECaLS locates in both north and south Galactic caps along the equator, resulting in a joint sky coverage $\sim 20000 \;deg^2$. The galaxy catalogue provides the photometrically estimated redshifts (hereafter photo-z), apparent magnitudes in g,r,z bands and stellar masses of galaxies. It is the largest galaxy data set currently available. Its large sky coverage is useful to reduce the statistical errors in our measurements, a key to improve the ISW measurement.
The galaxy catalogue includes those sources which have detections in g, r and z bands, and $r<23$. Furthermore, stars have been excluded through star-galaxy classification. As we can see from Fig.\ref{fig:DR8}, the surface density of galaxies is reasonably uniform across the sky. The residual non-uniformity (inhomogeneous selection function) is a severe issue for measuring the galaxy auto-correlation. But it is less an issue for cross-correlation measurements presented here, since the selection function is largely uncorrelated with the LSS and its impact on ISW-galaxy cross correlation is taken into account in the random catalogue and in the simulation mocks.
Coordinates of galaxies have been converted from equatorial coordinate system into galactic coordinate system, in which all the operations are done in the rest of the paper.
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/mask-ldp.png}}
\caption{The full-sky $f_{ldp}$ (the fraction of low-density points in each cell/pixel) distribution. LDPs are generated with cut radius $R_s=3'$, for the galaxy sample with magnitude cut $Mag_c-5lgh=-21$ and photo-z cut $0.2<z<0.4$. }
\label{fig:ldp}
\end{figure}
\iffalse
\begin{table*}
\footnotesize
\centering
\caption{The galaxy samples we use in the DR8 catalogue.}
\label{table:DR8}
\begin{tabular}{ccccccccccccccc}
\hline
& \multicolumn{4}{c}{$n_{gal}\,(10^6)$ }&&\multicolumn{4}{c}{$\overline{f}_{ldp},\,R_s=3$ arcmin}&&\multicolumn{4}{c}{$\delta_{l,max}$}\\
\hline
{\diagbox[innerleftsep=-0.5cm,innerrightsep=0pt,innerwidth=3.6cm]{$z_m$}{$Mag_c-5lgh$}} & \multicolumn{1}{c}{-20.5} &-21 &-21.5 & -22 && -20.5 &-21 &-21.5 & -22 && -20.5 &-21 &-21.5 & -22\\
\hline
0.1 & 1.7 & 0.81 & 0.28 & - && 0.55 & 0.75 & 0.9 & - && 0.81 &0.34 &0.11 &- \\
0.3 & - & 4.53 & 1.31 & 0.26 && - & 0.22 &0.62 & 0.9 && - &3.55 &0.62 &0.11\\
0.5 & - & - & 1.66 & 0.24 && - & - & 0.55 & 0.91 && - &- &0.83 &0.1\\
\hline
\multicolumn{14}{l}{`{$n_{gal}$}' is the number of galaxies in each galaxy sample.}&\\
\multicolumn{8}{l}{`{$\overline{f}_{ldp}$}' is the average value of $f_{ldp}$.}&\\
\multicolumn{8}{l}{$\delta_{l,max}$ is the maximum $\delta_l$.}&\\
\end{tabular}
\end{table*}
\fi
\begin{table*}[!htp]
\footnotesize
\centering
\caption{The galaxy samples we use in the DR8 catalogue.}
\label{table:DR8}
\begin{tabular}{cccccccccccccccc}
\hline
& \multicolumn{4}{c}{$n_{gal}\,(10^6)$ }&&$R_s$ [arcmin] &\multicolumn{4}{c}{$\overline{f}_{ldp}$}&&\multicolumn{4}{c}{$\delta_{l,max}$}\\
\hline
{\diagbox[innerleftsep=-0.5cm,innerrightsep=0pt,innerwidth=3.6cm]{$z_m$}{$Mag_c-5lgh$}} & \multicolumn{1}{c}{-20.5} &-21 &-21.5 & -22 &&& -20.5 &-21 &-21.5 & -22 && -20.5 &-21 &-21.5 & -22\\
\hline
\multirow{2}{*}{0.1} &\multirow{2}{*}{1.73} &\multirow{2}{*}{0.81} &\multirow{2}{*}{0.28} &\multirow{2}{*}{-} &&3& 0.55 & 0.75 & 0.9 &- && 0.81 &0.34 &0.11 &- \\
&&&&&&5& 0.22 &0.47 &0.75 &- &&3.44 &1.13 &0.33 &-\\
\hline
\multirow{2}{*}{0.3} &\multirow{2}{*}{-} &\multirow{2}{*}{4.53} &\multirow{2}{*}{1.31} & \multirow{2}{*}{0.26} &&3& - & 0.22 &0.62 & 0.9 && - &3.56 &0.62 &0.11\\
&&&&&&5& - &0.02 &0.28 &0.75 &&- &42.7 &2.51 &0.32 \\
\hline
\multirow{2}{*}{0.5} &\multirow{2}{*}{-} &\multirow{2}{*}{-} &\multirow{2}{*}{1.66} &\multirow{2}{*}{0.24} &&3& - & - & 0.55 & 0.91 && - &- &0.83 &0.1\\
&&&&&&5&- &- &0.21 &0.77 &&- &- &3.86 &0.3\\
\hline
\multicolumn{14}{l}{`{$n_{gal}$}' is the number of galaxies in each galaxy sample.}&\\
\multicolumn{8}{l}{`{$\overline{f}_{ldp}$}' is the average value of $f_{ldp}$.}&\\
\multicolumn{8}{l}{$\delta_{l,max}$ is the maximum $\delta_l$.}&\\
\end{tabular}
\end{table*}
\subsubsection{LDP identification}
\label{sec:LDP-def}
We identify LDPs and define the associated LSS field through the following procedures:\\
\noindent \emph{\small $\bullet$ Generating Survey Masks}.
Sets of uniformly distributed random catalogues are provided in the DR8 website\footnote{\url{http://legacysurvey.org/dr8/files/\#random-catalogs}}. Each random point contains the exposure times for g, r, z bands based upon the sky coordinate drawn independently from the observed distribution. We choose random points whose exposure times in all three bands are greater than zero and MASKBITS\footnote{MASKBITS greater than zero means that the source at this position overlaps with bad or saturated pixels, like bright star mask, globular cluster and so on.} equals to zero to produce the survey masks which populate the same sky coverage and geometry with the galaxy catalogue. \\
\noindent \emph{\small $\bullet$ Generating LDPs}. LDPs depend on the galaxy sample, so we need to first select the galaxy sample for LDP generation. (1) First, we calculate the r-band absolute magnitudes of galaxies with their apparent magnitudes and photo-z.\footnote{The absolute magnitudes used have not been K-corrected. Since galaxies within the same photo-z bin have similar K-correction and the LDP generation is only sensitive to relative brightness between these galaxies, this lack of K-correction is not an issue for our purpose. } (2) Then similar to the approach in \cite{2019ApJ...874....7D}, we select galaxies brighter than a certain absolute magnitude, and within a photo-z band $[z_m-0.1,z_m+0.1]$ to form the galaxy sample for LDP generation. $\Delta z=0.2$ is chosen as the photo-z error dispersion $\lesssim 0.1$ on average. (3) Within this galaxy sample, we circle around each galaxy with an angular radius $R_s$, and remove all positions within this radius from the sky. The remaining regions are defined as LDPs candidates. We also remove the LDP candidates lying within the masks. There are no galaxies within radius $R_s$ to any given LDPs, otherwise these points will be excluded by the LDP definition procedure. The underlying density in this region, with $\bar{N}$ galaxy on the average over the survey volume but $N=0$ galaxy for the selected LDPs, has a PDF $P(\delta|N=0)\propto \exp(-\bar{N}(1+\delta))P(\delta)$. $P(\delta)$ peaks at $\delta_{peak}<0$. $P(\delta|N=0)$ then peaks at $\delta<\delta_{peak}<0$. Therefore indeed LDPs occupy under-dense regions. (4) We put LDPs on uniform grids generated with HEALPix\citep{2005ApJ...622..759G}. Although the finer the grids, the more accurate the LDP distribution can be obtained, we adopt $N_{side}=4096$ HEALPix resolution due to the limitation of the number of random points.
\subsubsection{The LDP over-density maps}
Different LDPs correspond to different under-dense regions. For a given LDP, the distance to the nearest galaxy $R_{min}$ satisfies $R_{min}\geq R_s$. Statistically speaking, the larger the $R_{min}$, the more negative the underlying matter overdensity $\delta_m$. Therefore to improve the S/N of ISW-LDP measurement, we need to put larger weight for LDPs with larger $R_{min}$. Theoretically speaking there exists an optimal mapping between $R_{min}$ and $\delta_m$, and we should find and use that relation to figure out the optimal weighting. We leave this issue for futher investigation. Here we take a suboptimal, nevertheless workable, weighting scheme.
We smooth the LDPs with coarse grids, and define the LDP over-density field in each cell as:
\begin{equation}
\label{eqn:LDPoverdensity}
\delta_{l}=\frac{f_{ldp}-\overline{f}_{ldp}}{\overline{f}_{ldp}},
\end{equation}
Here we divide the whole sky into lower resolution cells with $N_{side}=512$ (comparing to $N_{side}=4096$ previously). The corresponding cell size is 6.87'. $f_{ldp}$ is the area proportion of LDPs occupying the given cell. It equals the number of LDPs in each cell devided by $n_{grid}$. $n_{grid}$ is the number of fine grids within the coarse cell, which equals 64 here. The maximum $\delta_l$ occurs for those cells completely occupied by LDPs, corresponding to regions with the most negative $\delta_m$. In order to reduce the impact of survey mask and edge effect on our calculation, we make a selection on the cells being used. We require that the random points which do not satisfy the criteria in \S\ref{sec:LDP-def} should take up less than $\eta=30\%$ of the area of cells. Otherwise, the cells will be disregarded. Smaller $\eta$ results into less cells used for the measurement and therefore larger statistical errors. Larger $\eta$ results into larger misidentification of low-density regions and therefore weaker ISW signal. The adopted $\eta=30\%$ is a balance between the two. As tested in simulation, with this ratio the magnitude of our measured LDP-ISW signal being depressed is less than 4\% compared to the one without mask effect. In observation, we also find that this way of cell-selection helps to enhance the amplitude of $\omega^o_{Tl}$.
In this procedure, the fewer galaxies are selected, the more LDPs would be produced, and vice versa. Taking this into consideration, we control the size of galaxy samples so that their distribution is neither too crowded nor too sparse, otherwise the $\delta_{LDP}$ generation will lack of accuracy in statistics. For $z_m=0.1$, we consider setting $Mag_c-5lgh$ respectively to be -20.5, -21 and -21.5. For $z_m=0.3$, $Mag_c-5lgh$ is set to -21, -21.5 and -22. For $z_m=0.5$, $Mag_c-5lgh$ is set to -21.5 and -22. Galaxies with redshift less than 0.01 are not used due to their too high number density. In Table \ref{table:DR8} we introduce these galaxy samples and LDPs generated with them. Fig.\ref{fig:ldp} shows one example of the $f_{ldp}$ distribution, for which LDPs are generated with parameters $z_m=0.3$, $Mag_c-5lgh=-21$ and $R_s=3'$.
\subsection{CMB Data}
We use the Planck SMICA map for the ISW-LDP measurement. The SMICA map has removed secondary CMB anisotropies such as the thermal Sunyaev Zel'dovich effect, which is also correlated with LSS. It also removes galactic foregrounds, which may be correlated with the galaxy mask/selection and therefore may bias the cross correlation measurement. We downgrade the map from $N_{side}=2048$ to $N_{side}=512$ resolution and adopt WMAP 9-year CMB mask \citep{2013ApJS..208...19H,2019MNRAS.484.5267K} to mask out pixels contaminated by Galactic dust or known points
sources. We find that this resolution downgrading only influences the cross-correlation signals at angular separations less than 10 armcin, as we would expect from the large scale origin of the ISW effect.
\iffalse
All calculations in this work are done under galactic coordinate system. We also repeat the computation under equatorial coordinate system by rotating pixels of CMB map into equatorial coordinate system, and get the same results.
\fi
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/ISW.pdf}}
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/PK_ISW.pdf}}
\caption{The ISW induced $\Delta T$ generated using our simulation.The upper panel shows the full-sky temperature fluctuation map, for which the ray-tracing is done between the radial distance [500, 1500] Mpc/h. While in the lower panel we show its corresponding power spectrum. The blue line shows the prediction from \citet{2010MNRAS.407..201C} based on the linear theory, and the red curve is our result. They are well consistent.}
\label{fig:pk-isw}
\end{figure}
\section{Predicting the ISW effects with a N-body simulation}
\label{sec:simu-data}
Although implementing the LDP analysis is straightforward at the data side, it is highly non-trivial at the theory side due to complicated relation between LDPs and the underlying density/potential field. This is further complicated by various selection effect in observations. Therefore we will use a N-body simulation to generate ISW maps. The same simulation is used to generate mock galaxy catalogues and LDPs, under given observational conditions. They are then used to predict the ISW-LDP correlation signal.
\subsection{N-body Simulation}
The ISW signal is mostly contributed from the large scale mode, so here we use the $1200$Mpc$/h$ N-body simulation from the CosmicGrowth simulation series \citep{2019SCPMA..6219511J}. It contains $3072^3$ simulation particles, and adopts the flat $\Lambda CDM$ cosmology, with $\Omega_c=0.223$, $\Omega_b=0.0445$, $\Omega_\Lambda=0.732$, $\sigma_8=0.83$, h=0.71, and $n_s=0.968$. Halos are identified with FoF group finder, and subhalos are identified with HBT\citep{2012MNRAS.427.1651H}.
\subsection{Construction of Full Sky ISW Map}
\label{sec:full-sky-isw}
As shown in Eq.\ref{dT2}, the ISW induced $\Delta T$ is an integration of $\dot{\Phi}$ along the line-of-sight. Making use of the Poisson equation\footnote{\begin{equation}
\Phi(\vec{k},t)=-\frac{3}{2}\left(\frac{H_0}{k}\right)^2\Omega_m\frac{\delta(\vec{k},t)}{a},
\end{equation}}, we obtain $\dot{\Phi}$ in Fourier space:
\begin{equation}
\label{Phidot}
\dot{\Phi}(\vec{k},t)=\frac{3}{2}\left(\frac{H_0}{k}\right)^2\Omega_m\left[\frac{\dot{a}}{a^2}\delta(\vec{k},t)-\frac{\dot{\delta}(\vec{k},t)}{a}\right].
\end{equation}
Here $\rho(t)$ is the matter density, $\overline{\rho}(t)$ the mean density, $\delta$ the over-density ($\delta\equiv(\rho-\overline{\rho})/\overline{\rho}$), $H_0$ the current Hubble parameter and $\Omega_m$ the present value of matter density parameter. In the linear regime, $\dot{\delta}(\vec{k},t)=\dot{D}(t)\delta(\vec{k},z=0)$, where $D(t)$ is the linear growth factor. For our purpose, it is sufficient to neglect the nonlinear evolution (Rees-Sciama effect). Therefore $\dot{\Phi}(\vec{k},t) \propto k^{-2}(1-\beta(t))H\delta(\vec{k},t)/a$, where $\beta(t)\equiv dlnD(t)/dlna$.
In simulation, we construct the $\dot{\Phi}$ field in the following way. Firstly, we assign dark matter particles into 3D grids under Cartesian coordinate, and construct the density field $\delta(\vec{x})$. During this process, we use a grid of $512^3$ cells for our simulation box. Then we perform the Fast Fourier Transform on the density field to compute its Fourier form $\delta(\vec{k})$. It is then used to yield the $\dot{\Phi}(\vec{k},t)$ field in Fourier space. At last, we perform the inverse Fourier transform to obtain $\dot{\Phi}(x)$ in real space. Above procedures are repeated for eight output snapshots at redshift 0, 0.058, 0.151, 0.253, 0.364, 0.485, 0.616 and 0.76. To avoid the discontinuities of $\Phi$ at boundaries, we assume periodic boundary conditions when using our simulation to construct $\dot{\Phi}(x)$ on a cube whose size length is larger than $1200$ $Mpc/h$.
Next, we choose the center of spliced cube as the location of observer, and generate angular evenly distributed rays from it using HEALPix with $N_{side} = 512$ resolution. For each ray, we accumulate the temperature fluctuations along its line of sight using Eq.(\ref{dT2}) by taking fixed discrete steps. Note that, values of $\dot{\Phi}$ for the same grid generated from different snapshots are different. So for each step, according to its position on the ray we find out the snapshot at that lookback time, and assign it the value of $\dot{\Phi}$ of its nearest grid. In this way, we are able to construct full sky maps of the ISW effect using the density field.
In this paper we construct the ISW induced $\Delta T({\widehat{n}})$ with a maximum redshift 0.7. However, there will be two problems arising from the periodic boundary conditions assumed in the above when constructing the $\Delta T({\widehat{n}})$ map. The first problem is that light rays will pass through the same structure every certain distance, leading to larger fluctuations along these directions. This distance is the shortest for light rays along the main axis of the simulation box, equaling to 1200 Mpc/h. The second problem is that the same structure is seen in multi-directions. As pointed out in \cite{2010MNRAS.407..201C} that the first issue can be solved by generating maps with the radial depth less than the simulation boxsize. The second issue does not matter as long as the angular scales of our analysis are less than the angular size subtended by the simulation box.
In the top panel of Fig.\ref{fig:pk-isw} we show the predicted $\Delta T$ map generated by integrating along the line-of-sight of the observer for the range $500<r_c<1500$ Mpc/h, and in the bottom panel we show its power spectrum ( red solid line). The blue solid line is the linear theoretical prediction from \cite{2010MNRAS.407..201C}, which has assumed a very similar cosmology to us. These two lines are found in good consistency.
\subsection{Generating LDPs with Mock Galaxies}
\begin{figure
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/autogg-DR8-3msate-30-21z0204_normal_LscattergalacticMzerrTest2.pdf}}
\caption{The auto correlation for galaxies brighter than -21 at $z_m=0.3$. The green data points show the measurement in observation, with jackknife error bars. The red solid line is the result for mock galaxies when both $\sigma_{Mag}$ and $\sigma_z$ are introduced. The red dashed line is the result for mock galaxies when only $\sigma_z$ is introduced. The red dotted line is the result when neither $\sigma_{Mag}$ nor $\sigma_z$ is considered.}
\label{fig:auto}
\end{figure}
Similar to the operation in \cite{2019ApJ...874....7D}, we draw correspondence between galaxies and halos/subhalos by matching the galaxy-subhalo abundance (SHAM). With absolute magnitudes of galaxies computed in \S\ref{sec:obs}, we measure the luminosity functions for three redshift bins: [0.01,0.2], [0.2,0.4] and [0.4,0.6]. Then for each used snapshot, we compare the number of halos/subhalos with mass at the accretion time greater than $M$ to the number of galaxies with luminosity less than L at that redshift.
Subhalos from different snapshots are used to fulfill the corresponding radial distance slices. And we adopt the mask of DR8 in simulation in order to ensure the same angular selection of galaxies as in observation.
To better mimic the real situation, we also add scatters to the redshifts and luminosities of mock galaxies. Firstly, with the probability distribution function of redshift dispersion P($\sigma_{z}$, z) measured in observation\footnote{Considering our galaxy sample, for $z_m=$ 0.1, 0.3 and 0.5, we only use galaxies whose absolute magnitudes are respectively lower than -20, -20.5 and -21 to get P($\sigma_{z}$, z).}, we randomly generate $\sigma_z$ for each galaxy. Then for the purposes of this paper we take the assumption that $p(z_{photo}|z)$ follows a Gaussian function with a zero mean and a scatter $\sigma_z$. We randomly move the positions of galaxies in redshift space and update the absolute magnitudes of mock galaxies according to $z-z_{photo}$. Thirdly, we introduce a constant scatter $\sigma_{Mag}=0.375$ dex to the absolute magnitude to mimic the galaxy-halo/subhalo relation\citep{2008ApJ...676..248Y}. After introducing these uncertainties, we redo the SHAM to ensure the same luminosity function of mock galaxies as in observation. In this way, we generate the mock photo-z catalogue for $z<0.6$ in simulation. We don't consider for the redshift bin [0.6,0.8] as the generation of its volume-limited mock galaxy sample may suffer from the incompleteness of galaxies at higher redshifts when $\sigma_z$ is introduced.
To validate our galaxy mocks, we compare the angular distribution of our mock galaxies to observation in Fig.\ref{fig:auto} for $z_m=0.3$. For this purpose, the two-point correlation-function at angular separation $\theta$ is calculated in the simple way of:
\begin{equation}
\omega_{gg}(\theta)\equiv\langle\delta_g(\widehat{n}_1)\delta_g(\widehat{n}_2)\rangle.
\end{equation}
where $\widehat{n}_i\cdot\widehat{n}_j=cos\theta$ and $\delta_g$ is the number over-density of galaxies. $\delta_g$ is obtained as:
\begin{equation}
\delta_g(\widehat{n})=\frac{n_g(\widehat{n})-\overline{n}_g}{\overline{n}_g},
\end{equation}
where $n_g$ is the number density of galaxies in the HEALPix cell with $N_{side}=512$ resolution.
The absolute magnitudes of these galaxies are less than -21, and their redshifts are within the slice [0.2, 0.4]. It shows that the $\omega_{gg}$ of mock galaxies is consistent with it in observation. Adding magnitude uncertainty will slightly suppress the correlation function. Adding redshift uncertainty influences the correlation function more significantly. We also find that the brighter the galaxy, the larger the impact. In \S\ref{sec:conclusion} we will discuss its influence on our LDP-ISW signal measurement.
Notice that the auto-correlation function estimator above is by no means optimal and by no means bias-free, comparing to the standard Landy-Szalay estimator. It is only used for the purpose of comparing the mock and data. Since we use the same estimator for both the mock and the data, the consistency in $\omega_{gg}$ shows that our galaxy mocks well represent the observed galaxy distribution.
Since this paper dose not focus on the auto-correlation analysis, this comparison is sufficient for current purpose.
Then we repeat the operations described in \S\ref{sec:obs} to generate LDPs in simulation. Fig.\ref{fig:gg-bias} shows the cross correlations of LDP overdensity $\delta_l$ with the matter density $\delta_m$. The tight correlation confirms our expectation that $\delta_l$ is indeed a good tracer of LSS. Furthermore, the cross correlation-function has the opposite sign to the matter auto-correlation function or the matter-galaxy cross-correlation function. This confirms that $\delta_l$ is a good tracer of low-density regions of the universe.
\section{ISW-LDP cross correlation measurements}
\label{sec:result}
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/gg-bias.pdf}}
\caption{The cross-correlation between the LDP-matter (red line), galaxy-matter (green line) and matter-matter correlation functions, in our mock. The similarity in shapes of the three correaltions and the significance of the $\omega_{ml}$ signal show that the LDP field is indeed tighly correlated with the matter distribution. The negative sign in $\omega_{ml}$ shows that the LDP field indeed probes under-dense regions.}
\label{fig:gg-bias}
\end{figure}
With the data analysis tools and simulation tools presented in previous sections, we now proceed to the LDP-ISW cross correlation measurements and their theoretical interpretation.
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs3-02-04-08cmberr-30-215LscattergalacticMzerr_norandTest.pdf}}
\caption{ The LDP-CMB cross correlation measured for $z_m=0.3$. LDPs are generated with galaxies whose absolute magnitudes are less than -21.5 and $R_s=\,3^{'}$. The green square points show $\omega^o_{T\ell}$ from observation and the red solid line is the prediction from simulation. The green and red dashed lines show results of the null test ($\omega_{Tr}$) respectively for observation and simulation. They are calculated by correlating randomly disturbed $\delta_l$ with $\Delta T$. Error bars shown for $\omega^o_{Tl}$ are estimated with the CMB rotating strategy. The green (red) shadow area shows one $\sigma$ range of $\omega^o_{Tr}$ ($\omega^s_{Tr}$).}
\label{fig:cross}
\end{figure}
\begin{figure*}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs5-001-02-04cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs5-02-04-08cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\subfigure{
\includegraphics[width=0.68\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs5-04-06-08cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\caption{The LDP-CMB cross correlation measured for three redshifts: $z_m=$ 0.1, 0.3 and 0.5. $R_s$ is set to 5 arcmin. For redshift $z_m=0.1$, the left, middle, and right columns are for $Mag_c-5lgh=$ -21.5, -21, -20.5 respectively. For $z_m=0.3$, the three columns are for $Mag_c-5lgh=$ -22, -21.5 and -21. And for $z_m=0.5$, the two columns are for $Mag_c-5lgh=$ -22 and -21.5. }
\label{fig:cross-all5}
\end{figure*}
\subsection{The cross-correlation measurements}
\label{sec:result-cross}
We adopt a simple estimator for the angular correlation-function $\omega_{T\ell}(\theta)$ between the CMB temperatue map and the LDP over-density $\delta_l$ map,
\begin{equation}
\label{eq-tl}
\omega_{T\ell}(\theta)=\langle\delta_T(\widehat{n}_1)\delta_\ell(\widehat{n}_2)\rangle,
\end{equation}
and we evaluate it with TreeCorr package\citep{2015ascl.soft08007J}. $\delta_T(\widehat{n})=T(\widehat{n}) - T_0$ is the temperature fluctuation, and $\delta_\ell$ the LDP overdensity. With the data in both observation and simulation, we have $4$ cross-correlation measurements. $\omega^o_{T\ell}$ is calculated with DR8 galaxy catalogue. $\omega^o_{Tr}$ is obtained by randomly shuffling $\delta_l$ in different cells and correlated with $\Delta$T. And this operation is repeated for 100 times to obtain the average value and variance. Here the superscript ``o'' denotes observation. $\omega^o_{Tr}$ is useful for the null-test and, if non-zero, should be subtracted from $\omega^o_{T\ell}$ to correct the mean impact of various selection effects such as the survey geometry, mask and mean CMB fluctuations. Namely, the finally estimated ISW-LDP cross correlation is
\begin{eqnarray}
\label{eqn:TTr}
\hat{\omega}_{T\ell}(\theta)=\omega^o_{T\ell}(\theta)-\omega^o_{Tr}(\theta)\ .
\end{eqnarray}
Correspondingly, $\omega^s_{Tl}$ and $\omega^s_{Tr}$ are calculated with mock catalogue in simulation.
\begin{figure*}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs3-001-02-04cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs3-02-04-08cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\subfigure{
\includegraphics[width=0.68\linewidth, clip]{fig/DR8-3msategalactic-isw-cmp-rs3-04-06-08cmberr_fitPJgalactic_normal_Mzerrnorand-norotTestpaper.pdf}}
\caption{Similar to Fig.\ref{fig:cross-all5}, but with $R_s$ shrinked to 3 arcmin.}
\label{fig:cross-all3}
\end{figure*}
Fig.\ref{fig:cross} shows one of such measurements, in which the LDPs are generated with parameters $Mag_c-5lgh=-21.5$, $R_s=$3 arcmin and $z_m=0.3$. The first finding is that $\omega^o_{Tr}\simeq 0$. The scatter is $\sim 2\times 10^{-3}\mu K$, about 10 percent of $\omega^o_{Tl}$. Therefore it does not matter wether $\omega^o_{Tr}$ is subtracted or not. $\omega^s_{Tr}$ is also consistent with zero ($1\times 10^{-3}\mu K$), with much smaller $\sigma$ as no CMB components/foregrounds are considered in our simulation. The second finding is that, the measured cross correlation is in good agreement with the theoretical prediction $\omega^s_{Tl}$.\footnote{When given the prediction of $\omega^s_{T\ell}$ from simulation, there is one issue that needs further attention. Considering that photo-z errors are added to our mock galaxies, there will be galaxies from other redshifts entering the redshift bin $[z_m-0.1, z_m+0.1]$ used for generating LDPs. So the ISW induced temperature fluctuations from the neighboring redshifts will associate with $\delta_l(z_m)$. In this case, it is better to generate $\delta_T(\widehat{n})$ by integrating temperature fluctuations within a broader redshift range. However, to avoid the enlarged $\Delta T(\widehat{n})$ problem caused by the repeated structures discussed in \S\ref{sec:full-sky-isw}, we choose to generate the full-sky map using the $\dot{\phi}$ field within $[z_m-0.2, z_m+0.2]$. The redshift depth 0.4 is considered both from the fact that $\overline{\sigma}_z\sim0.1$ and the boxsize of simulation.}
We repeat the above calculations for three redshift bins: $z_m=$0.1, 0.3, 0.5, and for different choices of critical magnitude. Fig.\ref{fig:cross-all5} \& \ref{fig:cross-all3} show the results of $R_s=5^{'}$ and $3^{'}$ respectively. In general we find good agreement between observation and theory/simulation.
\begin{figure}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/CMBncov-02-04-215-rs5.pdf}}
\caption{The covariance matrix for $\omega^o_{Tl}$. It is calculated with the $\it Rot.$ strategy for parameters $z_m=0.3$, $Mag_c-5lgh=-21.5$ and $R_s=5'$. Further discussions and more results are presented in the appendix. }
\label{fig:cv}
\end{figure}
\subsection{The covariance estimation and the detection significance}
\label{sec:cov}
Error bars shown in Fig.\ref{fig:cross} are directly estimated from the CMB map by rotating it around different axis with angle $\Delta\Phi$. This is based on the consideration that the ISW induced temperature fluctuation is much smaller than the fluctuation from the original CMB. The rotation strategy has been proposed in \cite{10.1111/j.1365-2966.2009.16054.x}, in which the WMAP maps have been rotated around the galactic pole to identify galactic contamination. However, any choice of axis is feasible, since one does not expect correlations for a large rotation angle $\Delta\Phi$. So scatters from independent rotations should reflect the intrinsic variance in the measurements, including the instrumental effect. To perform independent measurements, a minimum rotation $\Delta\Phi=30^\circ$ is required, as suggested in \cite{2012MNRAS.426.2581G}. Therefore, each rotation axis leaves 11 independent samples of cross-correlation measurement. We choose 18 positions on CMB map as the rotation axis \footnote{ We make the following sets of ($\theta$, $\phi$) as rotation axis: $\theta= [90\degree, 120\degree, 150\degree, 180\degree, 210\degree, 240\degree, 270\degree, 300\degree, 330\degree]$ , $\phi=$[30\degree, 60\degree]. }, with angular distances between them no less than $30^\circ$. In this way, we get 198 independent measurements of $\omega_{T\ell}$. The covariance matrix is estimated by
\begin{eqnarray}
\label{eq-cov}
C_{ij}&=&\frac{1}{N}\sum_{n=1}^{N}[(\omega_{T\ell,n}(\theta_i)-\overline{\omega}_{T\ell}(\theta_i))\nonumber \\
&&\times (\omega_{T\ell,n}(\theta_j)-\overline{\omega}_{T\ell}(\theta_j))].
\end{eqnarray}
We show the normalized covariance matrix $C_{ij}/\sqrt{C_{ii}C_{jj}}$ for $\omega^o_{T\ell}$ in the lower panel of Fig.\ref{fig:cv}. It indicates strong correlations between different angular scales, which is a common feature for correlation-function measurement. $\sigma_{\omega}=C^{1/2}_{ii}$ is the error bar shown in Fig.\ref{fig:cross}, for each angular bin $\theta_i$.
\begin{table*}
\footnotesize
\centering
\caption{S/N of $\omega^o_{T\ell(g)}$.}
\label{table:sn}
\begin{tabular}{ccccccccccccc}
\hline
&&&& \multicolumn{4}{c}{S/N}&Max($S/N_t$) \\
\hline
COV &$R_s$[arcmin]& {\diagbox[innerleftsep=-0.5cm,innerrightsep=0pt,innerwidth=3.6cm]{$z_m$}{$Mag_c-5lgh$}} && -20.5 &-21 &-21.5 &-22 &\\
\hline
\multirow{3}{*}{{\it Rot.}} &\multirow{3}{*}{3} &0.1 && 1 & 1.1 & 0.9 & - &\multirow{3}{*}{3.2} \\
&&0.3 && - & 1.9 & 2.2 & 1.5 \\
&&0.5 && - & - & 2.1 & 2.1 \\
\hline
\multirow{3}{*}{{\it Rot.}} &\multirow{3}{*}{5} &0.1 && 1 & 1.2 & 0.9 & - &\multirow{3}{*}{3.2} \\
&&0.3 &&- & 1.2 & 2.1 & 1.4 \\
&&0.5 && - & - &1.8 &2.1 \\
\hline
\multirow{3}{*}{{\it Jack.}} &\multirow{3}{*}{3} &0.1 &&1.5 &1.4 & 1.2 &- &\multirow{3}{*}{3.7}\\
&&0.3 &&- & 2.1 & 2.5 & 2 \\
&&0.5 &&- &- & 2.3 &2.3 \\
\hline
\multirow{3}{*}{{\it Jack.}} &\multirow{3}{*}{5} &0.1 &&1.4 &1.5 &1.2 &- &\multirow{3}{*}{3.6}\\
&&0.3 &&- &1.3 &2.3 & 1.8 \\
&&0.5 &&- &- &1.8 & 2.3 \\
\hline
\multirow{3}{*}{{\it Rot.}}&\multirow{3}{*}{galaxy} &0.1 && 0.9 & 1 & 0.9 &-&\multirow{3}{*}{3.4}\\
&&0.3 && - & 2.4 & 2.3 & 1.5 \\
&&0.5 && - & - & 2.2 & 2.1 \\
\hline
\multicolumn{9}{l}{`{\it Rot.}' represents for using the CMB-rotation technique to estimate the covariance matrix.}&\\
\multicolumn{8}{l}{`{\it Jack.}' represents for using the Jackknife technique.}&\\
\multicolumn{8}{l}{The last three rows are for CMB-galaxy correlation measurement.}&\\
\end{tabular}
\end{table*}
\begin{table*}
\footnotesize
\centering
\caption{The bestfit $A_{ISW}$, the associated $1$-$\sigma$ uncertainty, and $\chi^2_{min}$, combining all three redshifts.}
\label{table:aisw}
\begin{tabular}{ccccccccccc}
\hline
&&& \multicolumn{4}{c}{$\langle A_{ISW}\rangle$}&\multicolumn{4}{c}{$\rm{\chi^2_{min}}$}\\
\hline
COV &$R_s$[arcmin]& &$Max(Mag_c-5lgh)$ &-21.5 &$Min(Mag_c-5lgh)$ &&&$Max(Mag_c-5lgh)$ &-21.5 &$M in(Mag_c-5lgh)$\\
\hline
\multirow{1}{*}{{\it Rot.}} &3. & & 1.14$\pm$0.38 & 1.18$\pm$0.39 &1.07$\pm$0.42 &&& 0.36 & 0.89 & 1.1 \\
\multirow{1}{*}{{\it Rot.}} &5 & & 0.9$\pm$0.4 & 1.07$\pm$0.4 &1.06$\pm$0.42 &&& 0.29 & 0.79 & 1 \\
\multirow{1}{*}{{\it Rot.}} &galaxy & &1.25$\pm$0.38 & 1.21$\pm$0.38 &1.07$\pm$0.42 &&& 0.69 & 1 & 1.2 \\
\hline
\multicolumn{10}{l}{For $z_m=0.1,0.3,0.5$, $Max(Mag_c-5lgh)$ equals to -20.5, -21,-21.5 respectively.}&\\
\multicolumn{10}{l}{For $z_m=0.1,0.3,0.5$, $Min(Mag_c-5lgh)$ equals to -21.5, -22,-22 respectively.}&\\
\end{tabular}
\end{table*}
For each galaxy/LDP sample, the total S/N of the observational signal can be calculated as
\begin{equation}
\label{eq:sn}
\frac{S}{N}=\left[\sum_{i,j}\,\omega^o_{T\ell}(\theta_i)C^{-1}_{ij}\omega^o_{T\ell}(\theta_j)\right]^{1/2}\ .
\end{equation}
The results are shown in Table \ref{table:sn}.
LDP sampes of different photo-z bins are uncorrelated, so the total S/N of the ISW measurement of all three photo-z bins ($\alpha=1,2,3$) is
\begin{equation}
(S/N)_t = \sqrt{\sum_{\alpha=1}^3(S/N)_\alpha^2}.
\end{equation}
Depending on the choice of LDP samples, the total S/N of three photo-z bins varies.
For example, when $R_s=5^{'}$, $(S/N)_t=2.9$ for a universal magnitude cut $Mag_c=-21.5$. For other combinations of magnitude cut, $2.3\leq (S/N)_t\leq 3.2$. For $R_s=3^{'}$, $2.7\leq (S/N)_t\leq 3.2$. Therefore we have achieved a measurement of the ISW effect induced by low-density regions of the universe, at a significane of $3.2\sigma$.
This $3.2\sigma$ detection is comparable to the $3.4\sigma$ detection directly using galaxies (Table \ref{table:sn}). This is an interesting point to address and to further investigate. First, this LDP-ISW cross-correlation is not equivalent to directly using galaxies, since the LDP overdensity $\delta_l$-galaxy overdensity $\delta_g$ relation is nonlinear and non-local. For this reason, it is not subject to the $\sim 7\sigma$ upper bound of S/N in galaxy-ISW cross correlation measurement, for ideal CMB/galaxy surveys \citep{2004PhRvD..70h3536A}. Second, given that the weighting that we adopt to convert LDPs to $\delta_l$ is not optimal, the S/N should further improve if we find and apply the optimal weighting. This is an issue for future investigation.
We caution that the estimated S/N depends on the estimation of covariance matrix and its inversion . We will discuss another estimation of covariance matrix using the Jackknife resampling method, and the inversion of noisy covariance matrix with the SVD method. The results shown in the main text use the covariance matrix estimated by rotating CMB (hereafter $\it Rot.$), and the inversion by keeping the first two eigenmodes of the covariance matrix. Table \ref{table:sn} shows the S/N of other choices.
\begin{figure*}
\centering
\subfigure{
\includegraphics[width=1\linewidth, clip]{fig/3lcut-3z-3msategalacticMzerr_norandTest.pdf}}
\caption{Testing the origin of the detected cross-correlation, by a tophat cut in multipole $\ell$ space ($\ell_{cut}=50$, $100$, and no cut). The cut $\ell_{cut}=100$ has essentially no impact on $\omega_{T\ell}$, showing that the cross-correlation signal mainly arises from $\ell<100$. In contrast, the cut $\ell_{cut}=50$ causes visible difference in $\omega_{T\ell}$, showing that contribution from $50<\ell<100$ is non-negligible. These dependences are consistent with the ISW origin.}
\label{fig:lcut}
\end{figure*}
\subsection{Comparision with theoretical prediction}
Clearly, Fig. \ref{fig:cross-all5} \& \ref{fig:cross-all3} show that the measurements are in good agreement with the prediction of the concordance $\Lambda CDM$ cosmology. To further quantify the agreement, we choose the model of fitting as
\begin{equation}
\omega^o_{T\ell}=A_{ISW}\omega^s_{T\ell}\ .
\end{equation}
Here $A_{ISW}$ is the amplitude to fit and the standard $\Lambda$CDM cosmology has $A_{ISW}=1$. We minimize
\begin{eqnarray}
\chi^2&=&\delta \omega({\theta_i})C^{-1}_{ij}\delta \omega({\theta_j}), \nonumber\\
\delta_\omega(\theta)&\equiv& \omega^o_{T\ell}-\omega^s_{T\ell}\ .
\end{eqnarray}
to obtain the bestfit $A_{ISW}$. The bestfit $A_{ISW}$ and its $1\sigma$ uncertainty is shown in Fig.\ref{fig:cross-all5} \& \ref{fig:cross-all3}. We find that constrained $A_{ISW}$ of all LDP samples are consistent with unity, within 1$\sigma$ statistical uncertainty. Therefore the concordance $\Lambda$CDM indeed describes the data excellently.
We further combine all three redshifts to constrain $A_{ISW}$. The constraint depends on which sample is used for each redshift. If using the largest galaxy sample at each redshift, we obtain
\begin{eqnarray}
A_{ISW}(R_s=3')&=&1.14\pm0.38, \nonumber\\
A_{ISW}(R_s=5')&=&0.9\pm0.4.
\end{eqnarray}
In Table \ref{table:aisw} we show $A_{ISW}$ estimated from various combinations, all consistent with unity. It also shows the minimum chi-square $\chi^2_{min}$ corresponding to are all smaller than $1.2$, demonstrating excellent agreement with $\Lambda$CDM.
In contrast to previous constraints of $A_{ISW}\gg 1$\citep{2010MNRAS.401..547H,2010ApJ...724...12I,2012JCAP...06..042N,2014A&A...572C...2I,2014ApJ...786..110C,2015MNRAS.446.1321H,2015MNRAS.452.1295K,2017MNRAS.465.4166K,2018MNRAS.475.1777K,2019MNRAS.484.5267K}, we find no tension with the concordance $\Lambda$CDM. Furthermore, this agreement holds for different thresholds of $R_s$ (and therefore under-density).
\subsection{Further consistency checks}
The detected signal arises from the low-density regions of the universe, but free of many selection effects of void identification. Nevertheless, given the known difficulties in ISW measurements, we carry out three more tests in order to further validate the measurements.
\begin{itemize}
\item First is to check the origin of the cross-correlation. The ISW effect is expected to arise from large scale and is insensitive to small scale CMB modes. Fig. \ref{fig:lcut} shows the measured LDP-CMB cross correlation with a tophat cut $\ell_{cut}=50$, $100$ in multipole $\ell$ space. We find that the results with $\ell_{cut}=100$ are almost identical to the ones without a cut. Therefore the measured cross-correlation indeed comes from the large angular scale, as expected. Furthermore, the cut $\ell_{cut}=50$ results in minor but visible loss of the cross correlation signal, in particular for the lowest redshift bin $0.01<z<0.2$. This is again expected if the signal arises from the ISW effect. The $k$ corresponding to a given $\ell$ is roughly $k\sim \ell/r(z)$. For the same $\ell$, lower redshift means larger scale (smaller $k$) and therefore larger loss of signal.
\item We also check the measured galaxy density-ISW cross correlation, using the same galaxy samples and the same analysis pipeline. The overall S/N reaches $3.4$ (Table \ref{table:sn}). And the results are also in good agreement with the $\Lambda$CDM prediction, $A_{ISW}=1.21\pm0.38$ for $Mag_c-5lgh=-21.5$. Since the galaxy overdensity field and the LDP overdensity field are defined very differently, both agreements with $\Lambda$CDM further validate our measurements.
\item Besides the Planck SMICA map, we have other CMB maps to analyze. We have tried the Planck 100 GHz map and the V-band WMAP map. After subtracting the foregrounds, the results are consistent.
\end{itemize}
Therefore we believe the robustness of our LDP-ISW detection. The excellent consistency with $\Lambda$CDM then implies hidden systematics in some of the void-ISW cross correlation measurements. Therefore this LDP method is highly complementary to existing methods to cross-check and improve the ISW measurement.
\section{Summary \& Discussion}
\label{sec:conclusion}
We have designed a novel method of ISW measurement, by cross-correlating LDPs (low-density-positions, \citet{2019ApJ...874....7D}) and CMB. We then apply it to the DESI imaging survey DR8 galaxy catalogue of BASS + MZLS + DeCALS + DES, and Planck SMICA map. We achieve a $3.2\sigma$ detection of the ISW effect (Table 2), one of the most significant among existing measurements. Furthermore, the detected signal is fully consistent with the concordance $\Lambda$CDM prediction ($A_{ISW}=1$) for all the galaxy samples that we investigated and the adopted LDP definitions (Table 3), with the bestfit $A_{ISW}$ consistent with $A_{ISW}=1$ and $\chi^2\in (0.3,1.1)$. For example, for $Mag_c-5lgh=-21.5$ and $R_s=3^{'}$, we find $A_{ISW}=1.18\pm0.39$, with $\chi^2_{\min}=0.89$. For $R_s=5^{'}$, $A_{ISW}=1.07\pm0.4$ and $\chi^2_{\min}=0.79$.
The achieved S/N ($3.2\sigma$) is already competitive to that with galaxy-ISW cross correlation ($3.4\sigma$ that we have measured), and there exists room for further improvement. Together with the excellent agreement with the concordance cosmology, we have demonstrated the applicability of the LDP method to measure the ISW effect, for the first time. Our measurement provides an independent check to existing tensions between void ISW and $\Lambda$CDM, and between void ISW and galaxy ISW. Since our LDP ISW measurement has no tension with $\Lambda$CDM prediction and galaxy ISW measurement, we suggest hidden systematics in void ISW measurements.
The measurement can be used to constrain dark energy, in particular given a flat geometry. There are potentially other applications. For example, galaxy overdensity and LDP overdensity probe regions of the universe with statistically different matter density/gravitational potential. So the combination of LDP ISW and galaxy ISW may probe beyond $\Lambda$CDM physics, such as clustered dark energy and screening phenomena in modified gravity models.
Although the measurement S/N is already high among existing ISW detections, there are still possible improvements, related to the LDP definition and the LDP overdensity definition.
\begin{itemize}
\item LDP definition. LDPs depend on the galaxy sample, which is in turn determined by the redshift range and radius threshold, magnitude cut, and other galaxy properties. We have only tried a few configurations and the obtained S/N is unlikely optimal.
\item LDP overdensity $\delta_l$ definition. The underlying matter density $\delta_m$ at LDPs is statistically negative. But the exact value varies with LDPs. For example, $\delta_m$ at LDPs surrounded by LDPs should be on average more negative than LDPs surrounded by non-LDPs. Intuitively speaking, $\delta_m$ should decrease monotonically with increasing $d_{min}$, the distance of a given LDP to the nearest galaxy. The overdensity defintion (Eq. \ref{eqn:LDPoverdensity}) reflects this expectation. It is indeed tightly (negatively) correlated with the underlying matter density field, as verified with our simulation (Fig. \ref{fig:gg-bias}). However, although it has enabled a $3.2\sigma$ detection of the ISW effect, it is still an open question on whether it is the optimal choice. For example, for the adopted definition of $\delta_l$, its value equals to $Max(\delta_l)$ in void regions of size $\gg 6.87^{'}$. So it downweights the contribution from large voids. The optimal $\delta_l$ definition must take it into account. From the viewpoint of ISW measurement, the optimal definition of $\delta_l$ should result in a cross-correlation coefficient with the gravitational potential field as close to unity as possible.
\item The way to populate galaxies. In this work we use the SHAM method to populate galaxies in simulation by allowing a scatter $\sigma_{Mag}$ between galaxy luminosity and halo/subhalo mass and a scatter $\sigma_z$ between the true galaxy redshift and photometrical redshift. Adding $\sigma_{Mag}$ will decrease the nominal absolute magnitude on average, as more fainter galaxies are mistaken for bright ones statistically. While adding $\sigma_z$ will increase the nominal absolute magnitude on average, as more brighter galaxies at higher redshifts are mistaken as fainter galaxies at lower redshifts. So these two uncertainties would lead to the so-called Eddington bias\citep{1913MNRAS..73..359E}. For example, when $z_m=0.3$ and $Mag_c-5lgh=-21$, the number of galaxies increases by $5\%$, while the galaxy sample becomes 1.2 times larger for $Mag_c-5lgh=-22$. Our solution is to redo the SHAM after introducing these uncertainties. Otherwise, the distribution of galaxies with magnitude is changed. Although this is still a rough model to populate galaxies, the comparison of correlation-function in Fig.\ref{fig:auto} show the rationality of our operation.
\item The estimator of cross correlation function. We adopt a simple cross correlation function estimator. More optimal estimator requires input of galaxy selection function. DESI imaging survey galaxy catalogue still contains various imaging systematics, not fully captured by the random catalogue that we use \citep{2020MNRAS.tmp.1781K}. The resulting non-uniform selection function biases the galaxy auto correlation measurement. The problem for the cross correlation that we perform in this paper is much less severe, since the galaxy selection function is uncorrelated with CMB and ISW. Furthermore, to a good approximation, it is uncorrelated to residual foreground in the Planck CMB map. Nevertheless, non-uniform galaxy selection function amplifies statistical error in the cross correlation measurement. Future work needs to suppress such error (and diagnose potential systematics) by improving the cross correlation estimator with the aid of random catalogue.
\end{itemize}
Besides these measurement issues, accurate determination of the covariance matrix and robust estimation of the S/N and $A_{ISW}$ is also important. In the appendix, we have presented our treatments on the covariance matrix and its inverse. We plan to further investigate these issues, with the aid of numerical simulations and mock catalogues.
There are other possibilities to further explore. In principle the cross-correlation in real space is identical to that in Fourier (spherical harmonic) space. But in reality, due to the scale cut, mask and noise/foreground distribution, the two can differ. In this paper we only work on the real space, and leave the analysis in Fourier space elsewhere. For the theory/simulation side, we have used SHAM to populate galaxies into N-body simulation. This exercise turns out to be successful. Nevertheless, there may still room of improvement for higher S/N and better theoretical prediction.
\section*{Acknowledgements}
We thank Yipeng Jing for providing us the N-body simulation. We also thank Jian Yao and Ji Yao for useful discussions. This work is supported by the National Key Basic Research and Development Program of China (No.15ZR1446700, 2018YFA0404504,19ZR1466800), the NSFC grants (11621303,11653003,11673016,11833005,11890692,11773048), the 111 project (No. B20019).
\vspace{6pt}
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|
1,314,259,992,996 | arxiv | \section{Introduction}
\figureone
\figuretwo
\figurethree
\figurefour
The interaction between optical beams and dielectric blocks has always been the subject of great interest, leading, in the past, to formulate the well-known laws of geometric optics\cite{born,sharma,saleh}. In the last century, new phenomena like Goos-H\"anchen shift \cite{goos, artmann, crit1, crit2,crit3} and angular deviations \cite{ra} showed that the optical path predicted by geometric optics only represents an approximation to the real one. Theoretical studies have been undertaken in order to understand in which situations lateral displacements and angular deviations can be amplified and then observed in the laboratory. The omnipresence
of these phenomena\cite{metallic, waveguides, beammodes, sismic1, sismic2} also stimulated their application in technology \cite{micros, highsens, vapor}.
In 1947, Goos and H\"anchen \cite{goos} were the first researchers to experimentally observe the lateral displacement of optical beams transmitted, after many internal reflections, by a dielectric block. The experimental result, today known as Goos-Hänchen shift, was, one year later, explained by Artman\cite{artmann}. Artmann’s observation was that multiple plane waves, contributing to the final electromagnetic field, have rapidly varying phases that cancel each other out. Total internal reflection is indeed characterized by a complex Fresnel coefficient. The stationary condition gives the main term of the phase which is responsible for the additional phase generating the lateral shift in the optical path \cite{s2015}. The divergence in the Artmann formula was later removed \cite{crit1,crit2}. Recently, for incidence in the critical region, an analytical formula, based on the modified Bessel functions, was proposed in \cite{crit3} and, some years later, experimentally confirmed \cite{confirmation}.
In 1973, Ra, Bertoni and Felsen \cite{ra} introduced the phenomenon of angular deviation. This phenomenon appears both for transmission (in this case, we have deviations from the refraction angle predicted by the Snell law) and partial reflection (in this case, we find deviations from the reflected angle predicted by the reflection law). This phenomenon is due, essentially, to the \textit{symmetry breaking} of the Gaussian distribution caused by the Fresnel coefficients modulating the Gaussian distribution in the integral form of the transmitted and reflected beams.
Angular deviations and Goos-H\"anchen sfhits have been investigated in great detail in different fields, not only in optics \cite{metallic,waveguides,beammodes,micros,vapor} but also in seismic data analysis\cite{sismic1, sismic2}. In the critical region, lateral displacements and angular deviations generate oscillatory phenomena, theoretically predicted in \cite{oscill1} and, recently, experimentally confirmed in \cite{oscill2, oscill3}.
In this article, we analyse the combined effect of the angular deviations (caused by the transmission through the incoming and outgoing triangular prism interfaces) and the Goos-H\"anchen shift (caused by the total internal reflection). The study is done outside the critical region. This choice is justified because, outside the critical region, we have the possibility to find an analytic expression for the transmitted beam by using the Taylor expansion of the Fresnel coefficients and, consequently, determine the beam parameters, the incidence angles, and the axial distance for which angular deviations compensate Goos-H\"anchen lateral displacements. The integral form of the beam transmitted through a dielectric prism, see Fig.\,\ref{fig1}(a), is characterised by three Fresnel coefficients: the ones corresponding to the transmission at the left (air/dielectric) and right (dielectric/air) interfaces and the one corresponding to the total internal reflection at the lower (dielectric/air) interface. The upper transmitted beam is, thus, the perfect candidate to study the combined effect of angular deviations and Goos-H\"anchen shifts. In the next section, we fix our notation, introduce the Fresnel coefficients, and calculate the phase of the optical beams. The integral form of the (upper) transmitted beam cannot be analytically solved, so we use the Taylor expansion of the Fresnel coefficients and of the optical phase to obtain a closed form for the transmitted beam. By using this analytic approximation, we obtain a \textit{cubic equation} which allows to determine the peak position of the transmitted beam. {\color{Red}{In a previous paper\cite{alessia}, based on this cubic equation, we studied the phenomenon of pure angular deviations, this implies incidence angle below the critical one. In this paper we analyse incidence greater than the critical one. This allows to investigating both angular deviations and Goss-H\"anchen displacements (only present in the case of total internal reflection). In this incidence region, it is thus possible to study when these optical effect offset each other.}}
Discussions, conclusions, and proposals for experimental implementations appear in the final sections.
\section{The incident beam}
Let us introduce the integral form of the incident beam
\begin{equation}
\label{incB}
E^{^{\mathrm{[inc]}}}(\mathbf{r})\, =\, E_{\mbox{\tiny $0$}}\int\hspace*{-.1cm} \mathrm{d} k_{_x} \,\mathrm{d} k_{_y} \,\,G(k_{_x},k_{_y}) \, \,
e^{{\,i \,\boldsymbol{k}\,\cdot\, \mathbf{r}}}\,\,\,,
\end{equation}
where
\begin{equation*}
G\left(k_{_x},k_{_y}\right) \,=\, \frac{\mathrm{w}_{\0}^{\mbox{\tiny $2$}}}{4\,\pi}\, \exp \left[\,-
\,\left(\,k_{_x}^{^2} +k_{_y}^{^{2}}\,\right)\,\frac{\mathrm{w}_{\0}^{\mbox{\tiny $2$}}}{4}\,\right]
\end{equation*}
is the Gaussian wave number distribution, and
\[\boldsymbol{k}\,\cdot\, \mathbf{r} \,=\, k_{_x} x + k_{_y} y + k_{_z} z\]
is the optical phase with $|\boldsymbol{k}| = 2\,\pi/\lambda$. By using the paraxial approximation,
\[ k_z\,\approx\, |\boldsymbol{k}| - (\,k_{_x}^{^{2}}+k_{_y}^{^2}\,)\,/\,2\,|\boldsymbol{k}|\,\,,\]
the integral in Eq.\,(\ref{incB}) can analytically be solved leading to the following closed expression for
the incident Gaussian beam
\begin{equation}
E^{^{\mathrm{[inc]}}}(\mathbf{r})\,=\, \frac{E_{\mbox{\tiny $0$}}\,e^{{\,i\,|\boldsymbol{k}|\,z}}}{1 + i\,z/z_{_{\mathrm{R}}}}\,\exp \left[\,-\,\,\frac{x^{^2}+y^{^{2}}}{\mathrm{w}_{\0}^{\2}\, (1 + i\,z / z_{_{\mathrm{R}}})}\,\right]\,\,\,,
\end{equation}
where $z_{_{\mathrm{R}}} = \pi\mathrm{w}_{\0}^{^2}/\lambda$ is the Rayleigh axial range {\color{Black}{and $\mathrm{w}_{\0}$ the beam waist}}. The beam intensity is then given by
\begin{equation}
I^{^{\mathrm{[inc]}}}(\mathbf{r}) \,=\, I_{\mbox{\tiny $0$}} \,\frac{\mathrm{w}_{\0}^{^2}}{\mathrm{w}^{^2}(z)} \, \exp \left[\,- \,2 \,\,\frac{ x^{^2} + y^{^2}}{\mathrm{w}^{^2}(z)}\,\right]\,\,\,,
\end{equation}
where $I_{\mbox{\tiny $0$}}=E_{\mbox{\tiny $0$}}\,^{^2}$ and
$\mathrm{w}(z) = \mathrm{w}_{\0} \, \sqrt{1 + (z/z_{_{\mathrm{R}}})^{^2}}$.
\section{The optical phase}
In the integral form of optical beams, an important role is played by the optical phase responsible for the optical path of the beam. In order to calculate the optical phase of the (upper) transmitted beam, it is useful to introduce the coordinate system corresponding to the incident and transmitted beams and the ones corresponding to the left, right, and lower interfaces, see Fig.\ref{fig1}(b),
\begin{equation*}
\label{rotcord}
\begin{pmatrix}
\widetilde{x} \\
\widetilde{z} \\
\end{pmatrix}
=
M\left(\,-\theta\,\right)
\begin{pmatrix}
x \\
z \\
\end{pmatrix}
=
M\left(\,\frac{\pi}{4}\,\right)
\begin{pmatrix}
x_* \\
z_* \\
\end{pmatrix}
=
M\left(\,\theta\,\right)
\begin{pmatrix}
z_{_\mathrm{tra}}\, \\
x_{_\mathrm{tra}} \\
\end{pmatrix}\,\,,
\end{equation*}
where
{\color{Black}{$M(\theta)=\,\{\,\{\,\cos\theta\,,\,-\,\sin\theta\,\}\,,\,\{\,\sin\theta\,,\,\cos\theta\,\}\,\}$}}
represents the anti-clockwise rotation matrix. The optical phase corresponding to the beam propagating from the source to the first interface is given by
\begin{center}
\begin{tabular}{l l}
\circledO{S}{White} $\,\,\rightarrow\,\,$ \circledM{1}{C1} \,\, : & $k_{_x}\, x\,+\,k_{_z}\, z\,=\,k_{\tilde{x}}\,\widetilde{x}\,+\,k_{\tilde{z}}\,\widetilde{z}$ \,\,,
\end{tabular}
\end{center}
where
\[\begin{pmatrix}
k_{\tilde{x}} \\
k_{\tilde{z}} \\
\end{pmatrix}
\, = \,
M\,\left(\,-\theta\,\right)
\begin{pmatrix}
k_{_x} \\
k_{_z} \\
\end{pmatrix}\,\,\,.
\]
After transmission through the left (air/dielectric) interface, the beam moves, into the dielectric, towards the lower (dielectric/air) interface with the following optical phase
\begin{center}
\begin{tabular}{l l}
\circledM{1}{C1} $\,\,\rightarrow\,\,$ \circledM{2}{C2} \,\,: & $q_{\tilde{x}}\,\widetilde{x}\,+\,q_{\tilde{z}}\,\widetilde{z}\,=\,q_{x_{*}}\,x_*\,+\,q_{z_{*}}\,z_*$\,\,,
\end{tabular}
\end{center}
where
\begin{equation*}
\begin{pmatrix}
q_{\tilde{x}} \\
q_{\tilde{z}} \\
\end{pmatrix}
\,=\,
M\,\left(\,\frac{\pi}{4}\,\right)
\begin{pmatrix}
q_{x_{*}} \\
q_{z_{*}} \\
\end{pmatrix}\,\,,
\end{equation*}
with
\[q_{\tilde{x}}\,=\,k_{\tilde{x}}\,\,\,\,\,\mathrm{and}\,\,\,\,\,q_{\tilde{z}}\,=\,\sqrt{\displaystyle n^{^2}\,|\boldsymbol{k}|^{^2}-q_{\tilde{x}}^{^2}-k_{_y}^{^2}}\,\,.\]
The beam is then reflected back and moves between the lower and right interface with an optical phase given by
\begin{center}
\begin{tabular}{l l}
\circledM{2}{C2} $\,\,\rightarrow\,\,$ \circledM{3}{C3} \,\, : & $q_{x_{*}}\,x_*\,-\,q_{z_{*}}\,z_*\,=\,q_{\tilde{x}}\,\widetilde{z}\,+\,q_{\tilde{z}}\,\widetilde{x}$\,\,.
\end{tabular}
\end{center}
Finally, in the integral form of the (upper) transmitted beam appears, as expected, the following optical phase
\begin{center}
\begin{tabular}{l l}
\circledM{3}{C3} $\,\,\rightarrow\,\,$ \circledO{C}{White}\,\, : & $k_{\tilde{x}}\,\widetilde{z}\,+\,k_{\tilde{z}}\,\widetilde{x}\,=\,k_{_x}\,x_{_\mathrm{tra}}\,+\,k_{_z}\,z_{_\mathrm{tra}}$\,\,.
\end{tabular}
\end{center}
\section{The upper transmitted beam}
Once obtained the optical phase of the upper transmitted beam, we can write its integral form:
\begin{align}
E_{_{\mathrm{pol}}}^{^{\mathrm{[tra]}}}(\mathbf{r}_{_{\mathrm{tra}}})\, = \,E_{\mbox{\tiny $0$}}\int\hspace*{-.1cm} \mathrm{d} k_{_x} \,\mathrm{d} k_{_y} \,\,G_{_{\mathrm{pol}}}^{^{\mathrm{[tra]}}}(k_{_x},k_{_y}) \, \,
e^{{\,i \,\boldsymbol{k}\,\cdot\, \mathbf{r}_{_{\mathrm{tra}}}}}\,\,,
\label{Etra0}
\end{align}
where $\,\,\mathbf{r}_{_{\mathrm{tra}}}=(\,x_{_\mathrm{tra}},y,z_{_\mathrm{tra}}\,)$ and
\[G_{_{\mathrm{pol}}}^{^{\mathrm{[tra]}}}(k_{_x},k_{_y}) \,=\,T_{_{\mathrm{pol}}}(k_{_x},k_{_y})\,G(k_{_x},k_{_y})\,\,,\]
with
\begin{eqnarray*}
T_{_{\mathrm{pol}}}(k_{_x},k_{_y})\, =\, \frac{4\,k_{\tilde{z}}q_{\tilde{z}}}{(a_{_{\mathrm{pol}}}k_{\tilde{z}}+q_{\tilde{z}}/a_{_{\mathrm{pol}}})^{^2}}\,
\frac{q_{z_{*}}/a_{_{\mathrm{pol}}}-a_{_{\mathrm{pol}}}k_{\tilde{z}} }{q_{z_{*}}/a_{_{\mathrm{pol}}}+a_{_{\mathrm{pol}}}k_{z_{*}} }\,\,\times \nonumber\\
\exp\{\,i\,[\, q_{z_{*}} d\,\sqrt{2}\,+\,(\,q_{\tilde{z}}\,-\,k_{\tilde{z}}\,)\,(\,l\,-\,d\,)\,]\,\}\,\,,
\end{eqnarray*}
($a_{_{\mathrm{tm}}} = n$ and $a_{_{\mathrm{te}}} = 1$). The additional phase appearing in the Fresnel coefficients is due to the fact that the discontinuities at the air/dielectric and dielectric/air interfaces are located at different points. This phase is responsible for the optical path predicted by geometric optics.
In order to integrate Eq.(\ref{Etra0}), we use the first order Taylor expansion of the transmission coefficient, i.e.
\begin{eqnarray}
T_{_{\mathrm{pol}}}(k_{_x},k_{_x})&=& T_{_{\mathrm{pol}}}(0,0)\, \left[\,1\,+\,\beta_{_{\mathrm{pol}}} \,\frac{k_{_x}}{|\boldsymbol{k}|}\,\right]\,\times \nonumber \\
& & \exp[\,-\,i\,k_{_x}\,x_{_{\mathrm{Snell}}}\,]\,\,,
\label{taylor}
\end{eqnarray}
where
\begin{eqnarray*}
T_{_{\mathrm{pol}}}(0,0)&=&\frac{4\,n\cos\theta\,\cos\psi}{(a_{_{\mathrm{pol}}}\cos\theta+n\cos\psi/a_{_{\mathrm{pol}}})^{^2}}\,\times\\
& &
\frac{n\cos\varphi/a_{_{\mathrm{pol}}}-a_{_{\mathrm{pol}}}\cos\phi}{n\cos\varphi/a_{_{\mathrm{pol}}}+a_{_{\mathrm{pol}}}\cos\phi}\,\times\\
&&\hspace*{-2cm}\exp\{\,i\,[\, n \,\cos\varphi \,d\,\sqrt{2}\,+\,(\,n\,\cos\psi\,-\,\cos\theta\,)
(\,l\,-\,d\,)\,|\boldsymbol{k}|\,]\,\}
\end{eqnarray*}
and
\[x_{_{\mathrm{Snell}}}=(\,\tan\psi\cos\theta\,-\,\sin\theta\,)\,l\,+\,(\,\cos\theta\,+\,\sin\theta\,)\,d\,\,.\]
The $\beta_{_{\mathrm{pol}}}$ factor in (\ref{taylor}) can be expressed in terms of 3 addends, respectively,
corresponding to the transmission through the left (air/dielectric) interface, $\circledM{1}{C1}$, to the reflection by the lower (dielectric/air) interface, $\circledM{2}{C2}$, and, finally, to the transmission through the right (dielectric/air) interface, $\circledM{3}{C3}$,
\[\beta_{_{\mathrm{pol}}}=\beta_{_{\mathrm{pol}}}^{^{[1]}}+\beta_{_{\mathrm{pol}}}^{^{[2]}}+\beta_{_{\mathrm{pol}}}^{^{[3]}}\,\,,\]
with
\begin{eqnarray*}
\beta_{_{\mathrm{te}}}^{^{[1]}}&=&\tan\psi\,-\,\tan\theta\,\,,\\
\beta_{_{\mathrm{te}}}^{^{[2]}}&=& 2\,\tan\phi\,\,\varphi'\,\,,\\
\beta_{_{\mathrm{te}}}^{^{[3]}}&=&(\,\tan\theta\,-\,\tan\psi\,)\,\,\psi'\,\,,\\
\beta_{_{\mathrm{tm}}}^{^{[1]}}&=&(\,\tan\psi\,-\,\tan\theta/\,n^{\2}\,)\,/\,(\,\sin^{\2}\psi\,-\,\cos^{\2}\theta\,)\,\,,\\
\beta_{_{\mathrm{tm}}}^{^{[2]}}&=&2\,\tan\phi\,\,\varphi'\,/\,(\,\sin^{\2}\phi\,-\,\cos^{\2}\varphi\,)\,\,,\\
\beta_{_{\mathrm{tm}}}^{^{[3]}}&=&(\,\tan\theta\,-\,n^{\2}\tan\psi\,)\,\,\psi'\,/\,(\,\sin^{\2}\theta\,-\,\cos^{\2}\psi\,)\,\,,
\end{eqnarray*}
where the different angles which appear in the previous formulas are related to the incidence angle $\theta$ by the Snell law, i.e. $\sin\theta=n\,\sin\psi$ and $n\,\sin\varphi=\sin\phi$, the angle $\varphi$ to
$\psi$ by the geometry of the prism, i.e. $\varphi=\psi+\pi/4$. Finally, we have $\varphi'=\psi'=\cos\theta/n\,\cos\psi$.
By using the Taylor expansion (\ref{taylor}), we can analytically solve the integral of Eq.\,(\ref{Etra0}).
The $k_x$ term in the exponential will be responsible for the shift in the $x_{_\mathrm{tra}}$ coordinate, i.e.
\[\,\widetilde{x}_{_{\,\mathrm{tra}}}\,=\,x_{_\mathrm{tra}}\,-\,x_{_{\mathrm{Snell}}}\,\,,\]
centring the Gaussian beam in the optical path predicted by the Snell and reflection laws.
{\color{Black}{The constant term in (\ref{taylor}), i.e. $T_{_{\mathrm{pol}}} ( 0 , 0 )$, leads to the same integration done
for the incident, consequently we obtain the following contribution
\[
T_{_{\mathrm{pol}}} ( 0 , 0 )\,\,E^{^{\mathrm{[inc]}}} \left(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}} \right)\,\,.
\]
The linear term, i.e. $T_{_{\mathrm{pol}}} ( 0 , 0 )\,\beta_{_{\mathrm{pol}}} \,k_{_x}\,/\,|\boldsymbol{k}|$, is responsible for the breaking of the Gaussian symmetry
for incidence below the critical one and for the Goos-H\"anchen shift in the case of total internal reflection. Observing that $k_x$ in the integrand of (\ref{Etra0}) can be replaced by $-\,i\,\partial/\partial \widetilde{x}_{_{\,\mathrm{tra}}}$, we obtain the following contribution
\[
-\,i\,T_{_{\mathrm{pol}}} ( 0 , 0 )\,\,\frac{\beta_{_{\mathrm{pol}}}}{|\boldsymbol{k}|}\,\,\frac{\partial\,E^{^{\mathrm{[inc]}}} \left(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}} \right)}{\partial \widetilde{x}_{_{\,\mathrm{tra}}}}\,\,.
\]
The analytical expression, for the upper transmitted beam, is then given by
\begin{eqnarray*}
E^{^{\mathrm{[tra]}}}_{_{\mathrm{pol}}}(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}}\,) & = & \left[\,1 +\,2\,i\,\,\frac{ \beta_{_{\mathrm{pol}}} \widetilde{x}_{_{\,\mathrm{tra}}}}{|\boldsymbol{k}|\,\mathrm{w}_{\0}^{^{2}}\,(\,1 + i\, z_{_\mathrm{tra}} / z_{_{\mathrm{R}}}\,)}\, \right]\,\times \nonumber \\
& & T_{_{\mathrm{pol}}} ( 0 , 0 )\,\, E^{^{\mathrm{[inc]}}} \left(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}} \right)\,\,.
\end{eqnarray*}
}}
Finally, after algebraic manipulations, we find
\begin{eqnarray}
\label{Etra}
E^{^{\mathrm{[tra]}}}_{_{\mathrm{pol}}}(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}}\,)
&= & \left(\,1 +\,i\,\, \frac{ \beta_{_{\mathrm{pol}}} \widetilde{x}_{_{\,\mathrm{tra}}} +z_{_\mathrm{tra}}}{z_{_{\mathrm{R}}}}\, \right)\,\times \nonumber \\
& & \frac{T_{_{\mathrm{pol}}} ( 0 , 0 ) }{1 + i\, z_{_\mathrm{tra}} / z_{_{\mathrm{R}}}}\,\, E^{^{\mathrm{[inc]}}} \left(\,\mathbf{\widetilde{r}_{_{\mathrm{tra}}}} \right)\,\,.
\end{eqnarray}
In order to check the validity of our analytical approximation, let us briefly analyse what happens near the critical incidence region. The critical angle is found when $n\,\sin\varphi_c=1$, this implies a critical incidence at
\begin{equation}
\theta_c\,=\,\arcsin\left[\,\left(\,1\,-\,\sqrt{n^{^2}-1}\,\right)\,/\, \sqrt{2}\,\,\right]\,\,.
\end{equation}
In Fig.\,2, we plot the the (upper) transmitted beam shift of the maxima with respect to the path predicted by geometric optics. This is done by numerically integrating Eq.\,(\ref{Etra0}). The plots of the maxima, as a function of
$\delta=\left(\theta-\theta_c\right)|\boldsymbol{k}|\mathrm{w}_{\0}$, refer to a Gaussian laser with $\mathrm{w}_{\0}\,=\,100 \,\mathrm{\mu m}$, $\lambda\,=\,532 \,\mathrm{nm}$ and $n\,=\,1.5195$ (BK7 prism). We can distinguish three regions.
Region I, before the critical region, shows an axial dependence of the shift and this is caused by the modulation of the Gaussian wave number function generated by the real Fresnel coefficients related to the transmission through the first and third interface and the {\color{Black}{partial}} internal reflection. These phenomena represent angular deviations to the Snell and reflection law of geometric optics,
{\color{Black}{For a detailed discussion of pure angular deviations and its amplifications near the Brewster incidence, we refer the reader to the article cited in \cite{alessia}}}.
Region II determines the critical region, in such a region
the infinity in $\beta^{^{[2]}}_{_{\mathrm{pol}}}$ coefficients required a more complicated technique of integration to obtain the analytical expression for the upper transmitted beam \cite{crit3} and new oscillatory phenomena appear \cite{oscill1,oscill3}.
In region III, for incidence greater than the critical one but near enough to amplify the GH shift with respect to angular deviations, this axial depends breaks down. {\color{Black}{Region III will be the region of interest for our discussion because in this region, far enough of the critical region, angular deviations and GH shifts can offset each other. The analysis in this region complements the one presented in ref.\,\cite{alessia}}}. In region III, we have
\[\tan\phi = n\,\sin\varphi\,/\,i\,\sqrt{n^{\mbox{\tiny $2$}}\sin^{\mbox{\tiny $2$}}\varphi-1}\,\,,\]
and consequently the intensity of the upper transmitted beam can be written in the following form
\begin{align}
\label{transmittedintensity}
I^{^{\mathrm{[tra]}}}_{_{\mathrm{pol}}}\,(\mathbf{\widetilde{r}_{_{\mathrm{tra}}}})\,=\,\frac{\mathrm{w}_{\0}^{^2}}{\mathrm{w}^{^2}\,(z_{_\mathrm{tra}})}\,\,T^{^2}_{_{\mathrm{pol}}}\,(0\,,0)\,I^{^{\mathrm{[inc]}}} (\mathbf{\widetilde{r}_{_{\mathrm{tra}}}})\,\times\nonumber
\\
\left[\left(1+\frac{\gamma^{^{[2]}}_{_{\mathrm{pol}}}\,\widetilde{x}_{_{\,\mathrm{tra}}}}{z_{_{\mathrm{R}}}}\right)^{^2}+\left(\frac{z_{_\mathrm{tra}}+\beta^{^{[1+3]}}_{_{\mathrm{pol}}}\,\widetilde{x}_{_{\,\mathrm{tra}}}}{z_{_{\mathrm{R}}}}\right)^{^2}\,\right]\,\,,
\end{align}
where
\begin{eqnarray*}
\gamma_{_{\mathrm{te}}}^{^{[2]}}&=& 2\,\,\frac{n\,\sin\varphi}{\sqrt{n^{\mbox{\tiny $2$}}\sin^{\mbox{\tiny $2$}}\varphi-1}}\,\,\cos\theta\,/\,n\,\cos\psi\,\,,\\
\gamma_{_{\mathrm{tm}}}^{^{[2]}}&=&2\,\,\frac{n\,\sin\varphi}{\sqrt{n^{\mbox{\tiny $2$}}\sin^{\mbox{\tiny $2$}}\varphi-1}}\,\,\frac{\cos\theta\,/\,n\,\cos\psi}{n^{^2}\sin^{^2}\varphi\,-\,\cos^{^2}{\varphi}}\,\,.
\end{eqnarray*}
and
\begin{equation*}
\beta^{^{[1+3]}}_{_{\mathrm{pol}}}=\,\beta^{^{[1]}}_{_{\mathrm{pol}}}+\beta^{^{[3]}}_{_{\mathrm{pol}}}\,\,.
\end{equation*}
\section{GH shifts and angular deviations}
The analytical expression found for the intensity of the upper transmitted beam, see Eq.\, \eqref{transmittedintensity}, allows to calculate its maximum and consequently to
obtain the lateral displacement with respect to the path predicted by geometric optics due to the GH shifts and angular deviations. The intensity $\widetilde{x}_{_{\,\mathrm{tra}}}$ derivative leads to the following cubic equation
\begin{equation}
\label{cubic}
{\left(\frac{\widetilde{x}_{_{\,\mathrm{tra}}}}{\mathrm{w}_{\0}}\right)}^{^3}\,+\,a_{_{\mathrm{pol}}}\,{\left(\frac{\widetilde{x}_{_{\,\mathrm{tra}}}}{\mathrm{w}_{\0}}\right)}^{^2}\,+\,b_{_{\mathrm{pol}}}\,\frac{\widetilde{x}_{_{\,\mathrm{tra}}}}{\mathrm{w}_{\0}}\,=\,c_{_{\mathrm{pol}}}\,\,,
\end{equation}
where
\begin{eqnarray*}
a_{_{\mathrm{pol}}}&=&2\,\, \dfrac{\gamma^{^{[2]}}_{_{\mathrm{pol}}}\,z_{_{\mathrm{R}}}\,+\,\beta^{^{[1+3]}}_{_{\mathrm{pol}}}\,z_{_\mathrm{tra}}}{\left({\gamma^{^{[2]}}_{_{\mathrm{pol}}}}^{^2}+{\beta^{^{[1+3]}}_{_{\mathrm{pol}}}}^{^2}\right)\mathrm{w}_{\0}}\,\,,
\\
\\
b_{_{\mathrm{pol}}}&=&\frac{\mathrm{w}^{^2} (z)}{\mathrm{w}_{\0}^{^2}}\,\left[ \dfrac{z_{_{\mathrm{R}}}^{^2}}{\left({\gamma^{^{[2]}}_{_{\mathrm{pol}}}}^{^2}+{\beta^{^{[1+3]}}_{_{\mathrm{pol}}}}^{^2}\right)\mathrm{w}_{\0}^{^2}}-\frac{1}{2}\right]\,,
\\
\\
c_{_{\mathrm{pol}}}&=&\frac{\mathrm{w}^{^2} (z)}{\mathrm{w}_{\0}^{^2}}\,\dfrac{\gamma^{^{[2]}}_{_{\mathrm{pol}}}\,z_{_{\mathrm{R}}}\,+\,\beta^{^{[1+3]}}_{_{\mathrm{pol}}}\,z_{_\mathrm{tra}}}{2\,
\left({\gamma^{^{[2]}}_{_{\mathrm{pol}}}}^{^2}+{\beta^{^{[1+3]}}_{_{\mathrm{pol}}}}^{^2}\right)\mathrm{w}_{\0}}\,\,.
\end{eqnarray*}
This equation allows to calculate and compare the lateral displacements in region III. When the GH shifts dominate no axial dependence can be seen. When the angular deviations become comparable with GH shifts an axial dependence is seen in the lateral displacements.
Eq.\,(\ref{cubic}) can be reduced to a linear equation by observing that $\widetilde{x}_{_{\,\mathrm{tra}}}\ll \mathrm{w}_{\0}$ and that
{\color{Black}{$b_{_{\mathrm{pol}}}\gg a_{_{\mathrm{pol}}}$}} for axial distance $z_{_\mathrm{tra}}\ll z_{_{\mathrm{R}}}^{^{2}}\,/\,\mathrm{w}_{\0}$. The lateral displacement of the maximum is then given by
\begin{eqnarray}
\label{linear}
\widetilde{x}_{_{\,\mathrm{tra}}}^{^{\mathrm{[max]}}} & = & c_{_{\mathrm{pol}}}\,\mathrm{w}_{\0}\,/\,b_{_{\mathrm{pol}}} \nonumber\\
& \approx & \dfrac{\gamma^{^{[2]}}_{_{\mathrm{pol}}}\,+\,\beta^{^{[1+3]}}_{_{\mathrm{pol}}}\,z_{_\mathrm{tra}}\,/\,z_{_{\mathrm{R}}}}{|\boldsymbol{k}|}\,\,,
\end{eqnarray}
where the axial independent term, proportional to $\lambda$, represents the pure GH shift \cite{goos,artmann} and the axial dependent the angular deviations due to the Fresnel transmission modulation of the Gaussian wave number distribution.
Near the critical region,
\[\theta \,=\, \theta_c\,+\,\delta\,/\,|\boldsymbol{k}| \mathrm{w}_{\0}\:\:\:\:\:\:\:\:\:[\,\delta \geqslant 4\,],\]
we have
\[n^{\mbox{\tiny $2$}}\sin^{\mbox{\tiny $2$}}\varphi\,-\,1\,\approx\, 2\, n\, \cos\varphi_c\,\varphi_c'\,\,\delta\,/\,|\boldsymbol{k}| \mathrm{w}_{\0}\,\,.\]
In the example analysed in this paper, i.e. $\lambda=532\,\mathrm{nm}$ and $\mathrm{w}_{\0}\,=\,100\,\mu\mathrm{m}$,
$\delta\geqslant 4$ implies and incidence angle greater than $\theta_c\,+\,0.2^{^{\mathrm{o}}}$.
Observing that
\[\beta^{^{[1+3]}}_{_{\mathrm{pol}}}\,\ll\,\gamma^{^{[2]}}_{_{\mathrm{pol}}}\,\propto\,\sqrt{|\boldsymbol{k}|\,\mathrm{w}_{\0}}\,\,, \]
and using the approximated expression for the $\gamma$ factors, we obtain
\begin{equation}
\label{amp}
\widetilde{x}_{_{\,\mathrm{tra}}}^{^{\mathrm{[max]}}}\,=\,\frac{\sigma_{_{\mathrm{pol}}}}{n}\,\,\sqrt{\frac{2\,\cos\theta_c}{\delta\,\cos\varphi_c\,\cos\psi_c}\,\,\frac{\mathrm{w}_{\0}}{|\boldsymbol{k}|}}\,\,,
\end{equation}
with $\{\,\sigma_{_{\mathrm{te}}}\,,\,\sigma_{_{\mathrm{tm}}}\,\}\,=\,\{\,1\,,\,n^{\mbox{\tiny $2$}}\,\}$.
Clearly the axial dependence has been removed and this agrees with the numerical calculation shown in Fig.\,2, see region III at the right of the black zone. In this region, Eq.\,(\ref{amp}) also contains the well known $\sqrt{|\boldsymbol{k}|\,\mathrm{w}_{\0}}$ amplification for the GH shift, for details see refs.\cite{crit3,s2013}. The $\sigma$ factor is, finally, responsible for a further amplification of $n^{\mbox{\tiny $2$}}$ for the transverse magnetic wave, see the scale in Fig.\,2 (a) and (b).
For transverse magnetic waves, the pure Goos-H\"anchen shift is found for incidence at the Brewster angle, i.e.
\[\beta^{^{[1+3]}}_{_{\mathrm{tm}}}=0\,\,\,\Rightarrow\,\,\,\theta=\theta_{_{\mathrm{B}}}=\arctan n\,\,,\]
see Fig.\,3(a). For a given axial distance, from Eq.\,(\ref{linear}), we can obtain the incidence angle for which the GH shift is compensated by the angular deviation. For example, for a camera positioned at an axial distance of
\[4,\,8,\,12,\,16,\,20\,\,\,\mathrm{cm}\,\,,\]
for the optical beam considered in this paper, we find incidence angles of
\[69.86^{o},\,66.65^{o},\,64.89^{o},\,63.72^{o},\,62.87^{o}\]
for transverse magnetic waves, see Fig.\,3(a), and of
\[68.36^{o},\,62.37^{o},\,58.37^{o},\,55.32^{o},\,52.82^{o}\]
for transverse electric waves, see Fig.\,3(b). Eq.\,(\ref{linear}) can be also used to find, for a given incidence angle, the axial distance for which
GH lateral displacements and angular deviations offset each other,
\begin{equation}
\label{ztra}
z_{_\mathrm{tra}}\,\,=\,\,-\, \frac{\gamma^{^{[2]}}_{_{\mathrm{pol}}}}{\beta^{^{[1+3]}}_{_{\mathrm{pol}}}}\,\,z_{_{\mathrm{R}}}\,\,.
\end{equation}
For example, for incidence angles of
\[45^{o},\,50^{o},\,55^{o},\,60^{o},\,65^{o},\,70^{o}\,\,,\]
the compensation happens, for transverse electric waves, at the axial distances
\[38.10,\,25.45,\,16.47,\,10.22,\,5.99,\,3.23\,\,\,\mathrm{cm}\,\,,\]
see Fig.\,4(b). For transverse magnetic waves, the compensation happens for incidence angles greater than the Brewster angle, $\theta_{_{\mathrm{B}}}=56.65^{o}$. For incidence angle of
\[60^{o},\,65^{o},\,70^{o}\,\,,\]
angular deviations compensate the GH shifts at the axial distances
\[50.68,\,11.68,\,3.88\,\,\,\mathrm{cm}\,\,,\]
see Fig.\,4(a).
\section{Discussions}
Lateral displacements of optical beams with respect to the path predicted by geometric optics stimulated, in the last decades, both theoretical and experimental investigations. Two types of displacements characterize the transmission through dielectric blocks. The first, known as GH shift, is due to the phase of the total internal reflection coefficient
and it is independent of the axial position of the detector. The second one is due to the modulation of
the transmission coefficients on the wave number distribution of the incident beam and it is dependent on the axial position of the detector.
In region III, far enough to the critical region II, GH shifts are proportional to the wavelength of the optical beam.
{\color{Black}{When the axial distance approaches the Rayleigh axial range also the angular deviations become proportional to the wavelength and this open the doors to the possibility to cancel the lateral displacements induced by the total reflection coefficient. This phenomenon is also known as composite GH effect \cite{WM1,referee1}. In region I, where the partial internal reflection implies the only presence of angular deviations\cite{referee2} an amplification effect happens near the internal Brewster angle, for details see ref.\, \cite{alessia}. Region II represents the region around the critical angle and an amplification by a factor $\sqrt{|\boldsymbol{k}|\,\mathrm{w}_{\0}}$, see Eq.\,(\ref{amp}), is found in proximity of the critical incidence \cite{crit3,s2013}. Such a region is also characterized by oscillatory phenomena \cite{oscill1,oscill2,oscill3} and the analytical analytical formula, obtained in this paper for the intensity of upper transmitted beam, i.e Eq.\,(\ref{transmittedintensity}), fails to reproduce the numerical data. It is important to observe here that region II represents a very small region of the incidence spectrum covering a range of $8/|\boldsymbol{k}|\,\mathrm{w}_{\0}$ around the critical angle. This means, for a beam waist of $100$ $\mu$m and a wavelength of $532$ nm,
a range of $0.4^{\mathrm{o}}$ around the critical angle. Consequently, the analytical formula presented in this paper is in excellent agreement with the numerical data for all the incidence angles greater than $\theta_c\,+\,4\,/\,|\boldsymbol{k}|\,\mathrm{w}_{\0}$ or in the case of the beam parameters used in our simulations for incidence angles greater than $\theta_c\,+\,0.4^{\mathrm{o}}$.
}}
\section{Conclusions and outlooks}
In this paper, by using the Taylor expansion of the Fresnel coefficients of the transmission through the first and third interfaces and of the total reflection by the second interface, we have given an analytical expression for the upper transmitted beam intensity, see Eq.\,(\ref{transmittedintensity}). From this analytical approximation it is immediate to obtain the cubic equation to calculate the intensity maximum. The cubic equation (\ref{cubic}) can then be further reduced to a linear equation (\ref{linear}) from which we can obtain the incidence angles and axial distances for which GH shifts and angular deviations offset each other. For transverse magnetic waves this compensation effect is only possible for incidence greater than the Brewster incidence, $\theta_{_{\mathrm{B}}}=\arctan[n]$.
The analytical expression of the upper transmitted beam given in this paper, see Eq.\,(\ref{Etra}), is also useful in view of experimental implementations done by using the weak measurements technique \cite{WM1, referee3}. This technique is based on the interference between transverse electric and magnetic waves \cite{WM2,WM3}. Consequently the analytical expression for the upper transmitted beam is important to find the main maximum of the combined optical beam which is a function of the different lateral displacements and angular deviations of transverse electric and magnetic waves. For the incidence angles and axial distances for which these optical effects offset each other, the weak measurement breaks down because the double peak phenomenon is no longer present. In a forthcoming paper, we shall revise the weak measurements for transmission through dielectric blocks in view of the analytical expression given in this article.
\subsection*{Acknowledgements}
One of the authors (S.D.L.) thanks the CNPq (grant 2021/307664) and Fapesp (grants
2019/06382-9 and 2021/08848-5) for the financial support and the University of Salento (Lecce)
for the hospitality. The authors are also grateful to A. Alessandrelli, L. Solidoro and A. Stefano for their scientific comments and suggestions during the preparation of this article and to Profs. G. C\'o, L. Girlanda, M. Martino, and M. Mazzeo for their help in consolidating the research BRIT19 project, an international collaboration between the State University of Campinas (Brazil) and the Salento University of Lecce (Italy).
|
1,314,259,992,997 | arxiv | \section*{Acknowledgements}
We would like to acknowledge Thomas Feldker for inspiring this work. We gratefully thank Henning F{\"u}rst, Micha\l\ Tomza and Jook Walraven for fruitful discussions. This work was supported by the Netherlands Organization for Scientific Research (Vidi Grant 680-47-538 and Start-up grant 740.018.008 (R.G.), and Vrije Programma 680.92.18.05) (G.C.G., R.G., J.P.). R.S.L. acknowledges funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 895473.
\clearpage
|
1,314,259,992,998 | arxiv | \section{Introduction}
Transition metal cobaltites are a family of compounds in which the Hund's rule and the crystal field splitting compete fiercely~\cite{Khomskii14}. The process of maximizing the total electronic spin, which is favorable for lowering the exchange energy, gets heavily penalized because of loading electrons onto the $e_g$ orbitals. The outcome of this competition may be neither a high spin state--when the Hund's rule dominates, nor a low spin state--if the crystal field splitting is large. Instead, an intermediate spin state can emerge, with its exemplary manifestation in a cubic perovskite--SrCoO$_3$ ~\cite{Potze95,Zhuang98,Kunes12,Hoffmann15}. Recently, single crystals and epitaxial thin films of SrCoO$_3$ become available~\cite{Long11,HNLee13a,HNLee13b}. In contrast to polycrystalline samples studied earlier~\cite{Balamurugan06}, the epitaxial growth of thin films not only stabilizes the perovskite phase but also allows for substrate engineering~\cite{JHLee11}. They are of great importance for room-temperature multiferroic devices, given the Curie temperature of SrCoO$_3$ being at around 300~K and the Neel temperature of SrCoO$_{2.5}$ exceeding 500~K. The epitaxial thin films also possess a more efficient topotactic transformation from SrCoO$_{2.5}$ to perovskite SrCoO$_3$. Conventionally, such a conversion is achieved either by electrolyte induced long time oxidation~\cite{Bezdicka93} or through annealing at high temperatures and high oxygen pressures~\cite{Toquin06}. In thin films, however, this conversion occurs at much less demanding conditions, i.e. shorter time periods, lower temperatures, and reduced oxygen pressures~\cite{HNLee13a,HNLee13b}. Lately, this transformation has been demonstrated by an electric-field controlled process at room temperature~\cite{Ichikawa12,Tambunan14,NPLu17}.
The structural transitions and magnetic ordering in strontium cobaltites have been studied extensively by employing, for example, the X-ray spectroscopy or magnetic susceptibility measurement. The transport properties of SrCoO$_3$ thin films, however, remain largely unexplored. Such an investigation may shed light on the strongly correlated nature~\cite{Balamurugan06} and unusual magnetic anisotropy of this compound~\cite{Long11}. For example, a possible spin glass state was identified in La$_{1-x}$Sr$_x$CoO$_3$ upon investigating its unusual anomalous Hall resistivity~\cite{Onose06}. SrFeO$_3$, a close cousin of SrCoO$_3$, displays multiple helimagnetic phases at low temperatures. These exotic phases manifest themselves in the magneto-resistivity as kinks and hysteretic jumps~\cite{Ishiwata11,Chakraverty13}. It is therefore of interest to investigate the transport properties of SrCoO$_3$ thin films, given its unique spin state.
Here in this paper, we carry out a systematic magneto-transport study on SrCoO$_3$ thin films down to low temperatures and reveal the existence of persistent spin fluctuation. Through ionic liquid gating~(ILG), we obtain the metallic SrCoO$_3$ with record-low resistivity values from the insulating SrCoO$_{2.5}$. Surprisingly, the anomalous Hall resistivity ($\rho_{AH}$) of SrCoO$_3$ grows linearly as a function of $(\rho_{xx}-\rho_0)$, where $\rho_{xx}$ is the longitudinal resistivity and $\rho_0$ the residual resistivity. We propose theoretically that this behavior is a consequence of a novel type of skew scattering that stems from spin fluctuation with impurity-induced local inversion-symmetry breaking. The scenario of spin fluctuation is supported by the experimentally observed negative magneto-resistance~(MR) in SrCoO$_3$. The MR exhibits a parabolic shape at low magnetic fields and a linear behavior at high fields. Intriguingly, it gets enhanced with a decreasing temperature, well below the Curie transition temperature. After ruling out mechanisms including the surface scattering, anisotropic effect, domain-wall effect and weak localization, we show that the high field negative MR can be reproduced theoretically by considering spin fluctuation. Our work demonstrates that SrCoO$_3$ not only is of importance for applications but also hosts quantum properties that could enrich our understanding on the anomalous Hall effect (AHE).
\section{Sample preparation}
Thin films of SrCoO$_{2.5}$ were grown on (LaAlO$_3$)$_{0.3}$-(SrAl$_{0.5}$Ta$_{0.5}$O$_3$)$_{0.7}$ (001) substrate by a home-designed pulsed laser deposition system. The growth temperature is 750~$^\circ$C with the oxygen pressure of 100~mTorr. The laser energy (KrF, $\lambda$=248~nm) was set at 1.2~J/cm$^2$ with a frequency of 2~Hz. After the growth, samples were cooled down to room temperature with a rate of 5~$^\circ$C/minute. The sample quality was confirmed by X-ray diffraction as well as atomic force microscopy.
Our device under investigation is schematically shown in Figs.~1~(a) and 1~(b). Gold pads were evaporated on the samples as contacts. We carved out the Hall bar structure mechanically. Samples were then immersed together with a Pt counter-electrode into the ionic liquid (DEME-TFSI)~\cite{NPLu17,Ueno14,Yuan10,Jeong13}. The electrochemical reaction and subsequent magneto-transport investigations were carried out in a physical property measurement system (Quantum Design PPMS-9T) with standard lock-in techniques (typically $I_{AC}=1$~$\mu$A, 13~Hz). Pure oxygen gas was filled into the sample chamber to ensure proper oxidization and was later pumped out at around 150~K to avoid the hazardous icing.
As demonstrated in our previous study~\cite{NPLu17}, we can tune from the SrCoO$_{2.5}$ to SrCoO$_3$ through the ILG induced oxygen ion injection. The pristine SrCoO$_{2.5}$~\cite{Munoz08} contains oxygen vacancy chains that run along the [1-10] direction [hexagonal hollow sites in Fig.~1~(b)]. By applying a negative voltage (about -2.5~V) to the gate, oxygen ions can be driven into the sponge-like SrCoO$_{2.5}$ and fill the vacancies to form high-quality SrCoO$_3$~\cite{NPLu17}. The reaction rate is controlled by gating temperature and duration. We achieve fully metallic samples with record-low resistivity values [Fig. ~1~(c)], compared with the previously reported values of the single crystalline bulk~\cite{Long11} and thin films~\cite{HNLee13a}. It indicates high crystalline quality and very low oxygen deficiency: $x\approx3$ in SrCoO$_x$~\cite{HNLee13a,Balamurugan06}.
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig1s}
\caption{(a)~Schematic drawing of the ILG device with the sample in a Hall bar geometry. The Pt coil is the counter-electrode. (b)~Sketch of the strontium cobaltite thin film in contact with the ionic liquid (DEME-TFSI). (c)~Resistivity as a function of temperature for three gated samples with different thicknesses. Dotted curves are parabolic fittings. The dash (dash-dot) curves represent resistivity of bulk single crystal (thin film) SrCoO$_{3-\delta}$ compounds reported previously~\cite{Long11,HNLee13a}.}
\end{figure}
\section{Anomalous Hall effect}
\subsection{Experiment}
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig2s}
\caption{Hall resistivity of three samples with different thicknesses at a set of temperatures [(a): $T=$10, 15, 20, 25, 50, 75, 100, 125, 150, 175 K; (b): 10 to 150 K in a step of 10 K; (c): 10 to 190 K in a step of 10 K]. Dashed lines in (a) illustrate the Hall slopes. Each curve is obtained by carefully removing the contribution from the longitudinal resistivity: $\rho_{yx}(\mu_0 H)=\left[\rho_{\rightarrow}(\mu_0 H)-\rho_{\leftarrow}(-\mu_0 H)\right]/2$, where $\rho_{\rightarrow}(\mu_0 H)$ and $\rho_{\leftarrow}(-\mu_0 H)$ are two Hall traces obtained by sweeping from negative to positive fields and from positive to negative fields, respectively. Linear fits to $\rho_{yx}(\mu_0 H)$ at high fields ($|\mu_0 H|>5$~T) are extrapolated to zero field and the average between the absolute values of the two intercepts is defined as $\rho_{AH}$ [as indicated by the arrows in panel (a)].}
\end{figure}
We carry out detailed investigations in the fully metallic samples. Figure~2 show the Hall resistivity data of three samples with different thicknesses across a large temperature range. All results show step-like behaviors with decreasing anomalous Hall signal at lower temperatures. Figure~3 summarizes $\rho_{AH}$ as a function of $\rho_{xx}(\mu_0 H=0)$ , showing a linear dependence for each sample. To address the relation between $\rho_{AH}$ and $\rho_{xx}$, we use a phenomenological expression $\rho_{AH}=c_0+c_1\rho_{xx}+c_2\rho_{xx}^2$to fit the data (solid curves in Fig.~3). The quadratic terms $c_2$ obtained from the fitting are 0.2 (20-nm), 4 (28-nm), -18 (45-nm) $\Omega^{-1}$~cm$^{-1}$, respectively. These values are two to three orders of magnitude smaller than those in other ferromagnetic thin films such as Fe, Co, etc.~\cite{Tian09,Hou15}, although the obtained quantities of $c_0$ and $c_1$ are comparable. The quadratic term is therefore negligible. We further obtain that $-c_0$ and $c_1\rho_0$ are almost equal (inset to Fig.~3). Essentially, the relation reads: $\rho_{AH}\propto(\rho_{xx}-\rho_0)$.
Conventionally, the AHE depends on the longitudinal resistivity following: $\rho_{AH}=b_0\rho_{xx}+b_1\rho_{xx}^2$, where $b_0$ and $b_1$ are material-dependent parameters. The first term arises from skew scattering; the second term is from side-jump scattering and the nontrivial Berry phase~\cite{Nagaosa10}. It has been demonstrated both theoretically~\cite{Crepieux01} and experimentally~\cite{Tian09} that the conventional skew scattering does not show temperature dependence. Therefore, the formula should read: $\rho_{AH}=b_0\rho_0+b_1\rho_{xx}^2$, where only the second term $b\rho_{xx}^2$ varies with temperature. Clearly, this well-established relation cannot account for the linear dependence on $\rho_{xx}$ in our experiment.
We note that a similar behavior was reported in some other materials such as Yb$_{14}$MnSb$_{11}$ and Pt matrix embedded with Co nanoclusters~\cite{Sales08,Gerber04}. In Yb$_{14}$MnSb$_{11}$, the linear scaling appears only after subtracting a dominant quadratic term. Skew scattering with localized magnetic ions, which is different from the conventional scattering with non-magnetic impurities, was employed to explain the data~\cite{Sales08}. Such a Kondo mechanism may be important in the Co embedded Pt as well~\cite{Gerber04}. However, the Kondo physics is clearly not applicable here, since SrCoO$_3$ is an itinerant ferromagnet.
Recently, it was proposed that the fluctuating, but locally correlated, spins contribute to the AHE~\cite{Ishizuka18}. The mechanism is unlikely to be responsible in SrCoO$_3$ either. In the proposed mechanism, the AHE is proportional to the scalar spin chirality, not to the magnetization. Moreover, the theory considers Dzyaloshinskii-Moriya (DM) interaction as the cause of scalar spin chirality, which is expected to be absent in SrCoO$_3$, since inversion centers exist at the center of the Co-Co bonds.
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig3s}
\caption{Anomalous Hall resistivity as a function of the longitudinal resistivity at zero field. Lines are fits to the data points of each sample. Inset: fitted parameters $-c_0$ (circles) and $c_1\rho_0$ (squares) as a function of the film thickness.}
\end{figure}
\subsection{Theory\label{sec:3B}}
Theoretically, the extrinsic AHE stems from asymmetric scattering processes. The AHE at finite temperature, proportional to the magnetization, is possibly related to the vector spin chirality $\vec{S}_j\times\vec{S}_k$. When a charged non-magnetic impurity is placed into the ferromagnet, the induced electric field couples to the electric dipole of the surrounding spins. It locally breaks the inversion symmetry and causes a chiral spin fluctuation around the impurity (Fig.~4). From the microscopic theory point of view, this is a consequence of the fact that the intermediate spin state of Co ions in SrCoO$_3$~\cite{Potze95,Zhuang98,Kunes12,Hoffmann15} allows the orbital degrees of freedom to play an important role, which may render exotic electromagnetic properties~\cite{Katsura05}. The perturbative interaction to the spins around the impurity is:
\begin{equation}
H_{\mathrm{imp}}\propto V_i\hat{z}\cdot\vec{S}_j\times\vec{S}_k,
\end{equation}
where $V_i$ is the impurity potential, $\vec{S}_j$ and $\vec{S}_k$ are two spins surrounding the impurity; $\hat{z}$ is the unit vector that defines the direction of the uniform magnetization. This interaction is similar to the DM interaction in noncentrosymmetric magnets except that the DM vector depends on the bond [see the Hamiltonian in Eq.~(2)]. Therefore, the impurity-induced interaction may contribute to the anomalous Hall effect by causing spin canting. To demonstrate the chiral fluctuation due to such an interaction, we consider a four-spin model that corresponds to the spins surrounding the non-magnetic impurity:
\begin{eqnarray}
H_S&=&-J\sum_{i=1}^4 \vec{S}_{\tau(i)}\cdot\vec{S}_{\tau(i+1)}-h\sum_{i=1}^4 S_{\tau(i)}^z\nonumber\\
&&-D\sum_{i=1}^4\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z,
\end{eqnarray}
where $J$ is the Heisenberg interaction between the spins, $h$ is the magnetic field, $D$ is the impurity-induced interaction, and $\tau$: $\mathbf{Z}$$\to$ $\mathbf{Z}$ is an integer map that maps $\{1,2,3,4\}$ to the spin index of the four spins surrounding the non-magnetic impurity and $\tau(i+4)\equiv\tau(i)$; the spins are numbered by $\tau$ in the anti-clock order around the impurity. This map is introduced to avoid confusion with the later argument on MR, where we consider all the spins. Here, we ignored the contribution from other spins further away from the impurity as their canting is expected to be much smaller. The qualitative feature of our results is irrespective of the cluster shape of those spins considered. Using the classical spin-wave approximation, we find
\begin{equation}
\langle\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z\rangle=\frac{TD}{(J+h/2)^2-D^2},
\end{equation}
where $\langle\left(\cdot\cdot\cdot\right)_z\rangle$ is the thermal average of the $z$ component of the vector spin chirality. This equation indicates that the impurity-induced interactions give finite vector spin chirality only at finite temperature when the interaction is sufficiently small. Unlike the scalar spin chirality, the vector spin chirality itself does not break the time-reversal symmetry. Therefore, it is expected that the anomalous Hall conductivity is proportional to the magnetization, which is an indicator of time-reversal symmetry breaking.
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig4s}
\caption{Theoretical model of a chiral spin structure around an impurity. Red arrows indicate the tilted spins of Co due to the presence of a central defect. Such an effect is most pronounced for the nearest neighbors. }
\end{figure}
Notably, the skew scattering often appears from the third order in the perturbation (or in the second order in Born approximation). A first Born approximation considering the scattering by two magnetic moments is insufficient. Indeed, a former study considering the vector spin chirality reported that the anomalous Hall effect related to the vector spin chirality vanishes in the bulk~\cite{Taguchi09}. Therefore, the leading order must stem from the process that involves two spins and a nonmagnetic impurity. Considering the two-spin process in Ref.~\cite{Ishizuka18} and its interference with the first-order scattering term by the non-magnetic impurity, we find the scattering amplitude from the electrons with momentum $\vec{k}$ and spin $\sigma$ to that of $\vec{k^\prime}$and $\sigma$ reads
\begin{eqnarray}
W_{k\sigma,k^\prime\sigma}^-&=&-\sigma n_i \frac{16J_K^2V_i ma^2}{(2\pi)^7}k\langle\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z\rangle\nonumber\\
&&\cdot\left(\vec{k}\times\vec{k^\prime}\right)_z,
\end{eqnarray}
where $\sigma=\pm 1$ is the spin index of itinerant electrons, $n_i$ is the density of non-magnetic impurities, $V_i$ is the strength of the impurities, $J_K$ is the exchange coupling between the electrons and the localized moments, and $m$ is the effective mass of electrons.
In our experiment, the resistivity $\rho_{xx}$ consists of two components $\rho_{xx}=\rho_0+\rho_m$, where $\rho_0$ and $\rho_m$ are impurity and the magnetic contributions, respectively. In the Boltzmann theory, the anomalous Hall conductivity induced by the asymmetric scattering $W_{kk^\prime}^-=w(\vec{k}\times\vec{k^\prime})_z$ is: $\sigma_{xy}\propto\tau^2w\propto n_i \rho_m/\rho_{xx}^2$, where $n_i$ is the number of impurities. Here, we used the fact that $w\propto n_i \langle\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z\rangle$, and $\langle\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z\rangle\propto\langle(S^x)^2\rangle$ are proportional to $\rho_m$. Therefore, the Hall resistivity reads $\rho_{xy}\propto\rho_{xx}^2\sigma_{xy}\propto n_i\rho_m\sim n_i[\rho(T)-\rho(T=0)]$, qualitatively consistent with the experiment.
We further estimate the Hall angle $\theta_H\equiv\sigma_{xy}/\sigma_{xx}$ due to the vector spin chirality. We focus on the low temperature region, where the linear spin-wave approximation is accurate. We first estimate the magnitude of the impurity-induced interaction. We assume that: (1) the electric charge of the impurity is of the order of the elementary charge; (2) the scalar potential induced by the impurity has the form of the Coulomb potential; (3) the distance between the impurity and the spins are on the order of the lattice constant $a=4\times10^{-10}$~m. Taking the relative dielectric permittivity $\epsilon/\epsilon_0=10$, the model yields an electric field of $|\vec{E}|\sim 10^9$~V/m. On the other hand, the typical magnitude of the electric polarization induced by spin canting was recently studied in details for the transition-metal oxides~\cite{Jia06}; the calculation showed that the electric polarization of the form $\vec{P}=B\vec{e}_{ij}\times(\vec{S}_i\times\vec{S}_j)$ is about $B\sim 10^2$~nC/cm$^2$ for the nearest-neighbor spins. Hence, the polarization per bond reads: $\vec{p}=\vec{P}a^3\sim 10^{-31}$~C~m. By employing these results, we find the impurity-induced term to be
\begin{equation}
H_{\mathrm{imp}}=-\vec{p}\cdot\vec{E}\sim 10^{-22}(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)})J.
\end{equation}
Based on the classical spin-wave theory, we find $\langle\left(\vec{S}_{\tau(i)}\times\vec{S}_{\tau(i+1)}\right)_z\rangle\sim\frac{TD}{J^2}\sim10^{-2}$, assuming $J\sim$100~K and $T=10$~K. The magnitude of the impurity potential $V_0$ is then estimated via the first Born approximation.
From experiment, we obtain $\sigma_{xx}\sim10^6$~S/m. Using the first Born approximation, we find $\frac{1}{\tau_{\mathrm{imp}}}=\frac{n_i V_i^2}{(2\pi)^2\hbar}\rho(\varepsilon_F)$, where $\rho(\varepsilon_F)$ is the density of states (DOS) at the Fermi energy $\varepsilon_F$. From $\sigma_{xx}\sim q^2n\tau/m$, we find $\tau\sim10^{-14}$~s at $T=10$~K (Here, we ignored the contribution from the magnetic scattering, since at a sufficiently low temperature the impurity scattering dominates over magnetic scatterings.). Using electron density $n\sim 10^{29}$~m$^{-3}$, and the DOS $\rho(\varepsilon_F)\sim\frac{n}{W}\sim10^{48}$~J$^{-1}$m$^{-3}$, we find $n_iV_i^2\sim 10^{-68}$~J~m$^3$. By assuming 0.1\% density of impurity, i.e. $n_i\sim10^{24}$--10$^{25}$~m$^3$, we find $V_i\sim10^{-47}$--$10^{-46}$~J~m$^3$.
We estimate the Hall angle using the above values. In the Boltzmann theory, the Hall angle reads $\theta_H=\tau\rho(\varepsilon_F)W_{k\sigma k^\prime\sigma}^-$ where $W_{k\sigma k^\prime\sigma}^-\sim10^{-36}$~J~m$^3$/s is obtained from the second Born result assuming $k_F\sim10^{10}$~m$^{-1}$. From these results, we find $\theta_H\sim10^{-3}$--$10^{-2}$ at $T=10$~K, consistent with the experiment.
\section{Magneto-resistance}
\subsection{Experiment}
The signature of spin fluctuation can be clearly seen in the magneto-transport data. Figure~5 displays the MR of SrCoO$_3$ samples with different thicknesses at selected temperatures. These metallic samples all possess a parabolic MR (dashed curve) at low fields and a linear MR at high fields (dotted lines). The parabola show little thickness ($d$) dependence. The size effect~\cite{Suzuki98} for a negative MR can be readily excluded because otherwise the MR should depend quadratically on $d$.
Apart from the size effect, negative MR often arises due to the anisotropic magnetization of the material~\cite{Ziese02,Marrows05}. It may account for the parabolic behavior at low fields, since it becomes less distinguishable in a tilted field (see Appendix~\ref{sec:MR}). However, the contribution from the anisotropic MR (AMR) in our thin films is less than 0.5\%, which cannot account for the overall non-saturating MR seen in Fig.~5.
The domain-wall effect also produces large negative MR when sweeping from zero field. We exclude this effect since our sample shows no hysteresis and weak AMR [36], distinctly different from the expected domain-wall driven MR (see Appendix~\ref{sec:MR}). We further exclude the weak localization effect because: (1) the temperature dependent resistivity curve shows no sign of localization [Fig.~1~(c)]; (2) fitting of the magneto-conductivity with the formula for weak localization yields unphysical values (see Appendix~\ref{sec:MR}).
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig5s}
\caption{MR of three samples with different thicknesses at selected temperatures. The dashed curves (dotted lines) are quadratic (linear) fits to the data at T=2 K at low (high) fields.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig6s}
\caption{(a) MR at 8~T as a function of temperature. MR(8T) is the mean of the expected values at $\pm$8~T, if taking linear fits to the MR in the range of $|\mu_0 H|>7$~T. Error bar represents the standard deviation at $\pm$8~T for those linear fits. For most of the data points, the error bar is smaller than the size of the markers. (b) high field slopes of $\rho_{xx}$ for the three samples as a function of temperature.}
\end{figure}
After excluding the above-mentioned mechanisms, we attribute the observed MR to persistent spin fluctuation. First of all, the absolute value of MR becomes larger as the temperature decreases [Fig.~6~(a)]. This behavior is in sharp contrast to the conventional behavior seen in itinerant ferromagnets. There, $|\mathrm{MR}|$ is enhanced at around the Curie temperature due to spin-dependent scattering and gets suppressed at low temperatures as spins align in one direction. The unusually large $|\mathrm{MR}|$ at low temperatures in SrCoO$_3$ therefore indicates that spin-dependent scattering remains prominent. Secondly, the slope of magneto-resistivity ($d\rho_{xx}/d(\mu_0 H)$) at high fields remains finite as $T$ approaches zero [Fig.~6~(b)]. In this high field regime, the magnetization is saturated and the spin wave is expected to be significantly suppressed~\cite{Raquet02}. Previous experiments on Fe, Co and Ni thin films have demonstrated that $d\rho_{xx}/d(\mu_0 H)$ approaches zero super-linearly with decreasing temperature~\cite{Raquet02}. In contrast, our samples exhibit an almost linear decrease of $d\rho_{xx}/d(\mu_0 H)$ with a clear positive intercept as $T\rightarrow 0$.
\subsection{Theory}
To provide further insight into the effect of spin fluctuation on the resistivity, we calculate the magnetic contribution to the relaxation time using first Born approximation considering the exchange coupling $H_K=J_K\sum_i\vec{S}_i\cdot\vec{\sigma}(\vec{r}_i)$, where $\vec{\sigma}(\vec{r}_i)$ is the vector of spin operators for electron spins at $\vec{r}_i$. For the spin Hamiltonian, we consider a 3d Heisenberg model
\begin{equation}
H_S=-J\sum_{\langle i,j\rangle}\vec{S}_i\cdot\vec{S}_j-h\sum_i S_i^z.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=86mm]{Fig7s}
\caption{Theoretically calculated magnetoresistance by considering the spin fluctuation. }
\end{figure}
In this section, we ignore the effective DM interaction induced by non-magnetic impurities, as they only give a higher order correction to the resistivity. We also note that, here, we consider \textbf{all} spins in the system while Sec.~\ref{sec:3B} only considers the four spins around a non-magnetic impurity. In the first Born approximation, the relaxation time $\tau_\mathrm{mag}$ reads:
\begin{equation}
\frac{1}{\tau_\mathrm{mag}}=\frac{J_K^2\rho_\sigma(\varepsilon_{\vec{k}\sigma})}{(2\pi)^5a^3}\left[\langle(S_0^x)^2\rangle+\langle(S_0^y)^2\rangle\right],
\end{equation}
where $\varepsilon_{\vec{k}\sigma}$ is the eigen-energy for electrons with momentum $\vec{k}$ and spin $\sigma$, $\rho_\sigma(\varepsilon)$ is the density of states for electrons with spin $\sigma$ at energy $\varepsilon$, and $\langle\cdot\cdot\cdot\rangle$ represents the thermal average. The field dependence of $\tau_\mathrm{mag}$ comes from the field dependence of $\langle(S_0^x)^2\rangle$ and $\langle(S_0^y)^2\rangle$; here, we set the spin index $i=0$ assuming the translational symmetry of the ferromagnetic order. As this scattering is diagonal in the spin space, we treat the contribution from electrons with different spins independently. In deriving the above formula, we assumed that the magnetic moments are aligned along the $z$-axis, and took into account of the leading order in the fluctuation assuming the fluctuation is small. This situation applies to the high-field region where the magnetic moments are aligned almost along the field direction. In the classical spin-wave approximation, the fluctuation of spins reads
\begin{equation}
\langle(S_0^x)^2\rangle+\langle(S_0^y)^2\rangle=\frac{T}{6J}\int_{-\pi}^\pi \frac{dk^3}{(2\pi)^3}\frac{1}{1+\eta-\Sigma_a\cos k_a},
\end{equation}
where $\eta=h/(6J)$ is the renormalized magnetic field. The sum in the integral is over the three axes $a=$x, y, z. At zero magnetic field and low temperatures (but still higher than the magnetic-field/anisotropy induced gap), the resistivity caused by spin fluctuations increases linearly with respect to $T$. Assuming $J\sim10^2$~K and $J_K\sim10^3$~K, we find $\tau_\mathrm{mag}\sim10^{-14}$~s at $T=100$~K, roughly consistent with the order of resistivity in the experiment.
Under the magnetic field, $\rho_m(h)$ is expected to be suppressed as the field pins the magnetic moments along the field direction. Within the Born approximation, the resistivity of the system follows Matthiessen's rule $\rho_{xx}=\rho_0+\rho_m$, where $\rho_i=\frac{m}{e^2 n \tau_\mathrm{imp}}$ is the contribution from the impurity scattering and $\rho_m=\frac{m}{e^2 n \tau_\mathrm{mag}(h)}$ is the magnetic contribution; $\tau_\mathrm{imp}$ is the relaxation time for the impurity scattering.
Figure~7 plots the field dependence of the magnetoresistance $\rho_m(h)\equiv\rho_m(h)-\rho_m(0)$ renormalized by $\rho_m(h=0)$. The resistivity sharply decreases at the zero field limit, implying that the MR responds sensitively to the spin fluctuation, even when the impurity scattering is larger than the magnetic scattering. Therefore, the MR observed in the experiment is possibly related to the persistent spin fluctuation down to a very low temperature.
\section{Conclusion}
The SrCoO$_3$ thin films realized by ILG exhibit magneto-transport behaviors including: (1) the scaling relation: $\rho_{AH}\propto(\rho_{xx}-\rho_0)$, which is distinctly different from the well-established form of $\rho_{AH}=b_0\rho_0+b_1\rho_{xx}^2$; (2) the negatively enhanced MR at low temperatures, indicating persistent spin fluctuations. We theoretically propose that impurities can induce chiral spin fluctuations in this material. By considering the local spin fluctuation around the impurity, we derive the anomalous Hall effect that is consistent with the experimentally observed relation. We further calculate the negative MR by taking into account the spin fluctuation of all spins, reproducing the non-saturating MR as seen in experiment.
\begin{figure}[!t]
\centering
\includegraphics[width=86mm]{SFig1s}
\caption{MR and AMR of a 36-nm thick SrCoO$_3$ sample. (a) MR at a set of temperatures (10, 25, 50, 75, 100, 125, 150~K). (b) MR at 75~K in a tilted magnetic field. The magnetic field direction is perpendicular to the current (inset). Inset panel summarizes the angular dependence of MR. }
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=86mm]{SFig2s}
\caption{(a) MR of the 20-nm SrCoO$_3$ thin film. The gray curve is obtained by sweeping from 0 to -8 T after the sample is zero-field cooled to 2~K. The blue and red curves are retrieved by sweeping from -8 T to 8 T and back. (b) Magneto-conductivity at 50~K together with the fitted curve (dotted) }
\end{figure}
\acknowledgments
This study was financially supported by the National Basic Research Program of China (grants 2017YFA0304600, 2015CB921700 and 2016YFA0301004); the National Natural Science Foundation of China (grant 51561145005, 11604176); the Initiative Research Projects of Tsinghua University (grant 20141081116); and the Beijing Advanced Innovation Center for Future Chip (ICFC). H.I. and N.N. are supported by CREST JST (No. JPMJCR16F1) and JSPS KAKENHI (No. JP26103006, JP16H06717 and JP18H04222).
Ding Zhang and Hiroaki Ishizuka contributed equally.
|
1,314,259,992,999 | arxiv | \section{introduction}
In the last decades, considerable efforts have been devoted to the study of the complex behavior
of quarks and gluons under the extreme conditions which are reached in heavy-ion collisions.
In principle, the dynamical and thermal properties of a quark-gluon plasma should descend from
the relatively simple Lagrangian of the SU(3) gauge theory which describes QCD. However, things are not so easy because
the standard perturbative approach breaks down in the strong-coupling IR limit and is also plagued by
further resummation problems at any finite temperature. As a matter of fact, we still miss a full
theoretical treatment of the problem.
Even the pure gauge theory, without quarks, is not fully understood, despite its
relevance for describing the quark-gluon plasma. Many important advances have been made by
the numerical simulation of the pure Yang-Mills (YM) Lagrangian on a lattice, providing insights
into the gluon dynamics and the phase diagram.
Among them, the confirmation of a dynamically generated
gluon mass~\cite{duarte,cucchieri08,cucchieri08b,bogolubsky,dudal,binosi12,oliveira12,burgio15},
as predicted by Cornwall in 1982~\cite{cornwall82},
and the occurrence of a phase transition,
with the gluons that become confining below a critical temperature~\cite{lucini,silva,aouane}.
It would be a desirable progress if the dynamical and transport parameters,
like masses, widths, dispersion relations, transport coefficients, etc., which are currently
regarded as phenomenological parameters~\cite{werner2016,castorina2012,alba2012,greco2011},
could be directly evaluated from first principles. That program might be accomplished in part if the elementary correlators
and their analytic properties were known in the Minkowski space.
Unfortunately, all lattice calculations and most numerical works provide information in
the Euclidean space and the analytic continuation is a difficult ill-defined problem for the numerical data~\cite{dudal14}.
In the last years, a very predictive analytical method has been developed~\cite{ptqcd,ptqcd2,analyt,xigauge}
by a mere change of the expansion point of ordinary
perturbation theory (PT) for the exact gauge-fixed Becchi-Rouet-Stora-Tyutin (BRST) invariant YM Lagrangian,
yielding a screened massive expansion which is safe in the IR while
recovering the correct results of ordinary PT in the UV. At one-loop and zero temperature,
the screened expansion provides analytical results which are in
excellent agreement with the lattice and can be easily continued to Minkowski space~\cite{xigauge,scaling,ghost,beta,beta2}.
Thus the method provides a way to extract dynamical details like masses and damping rates from first principles.
In this paper, the formalism is extended to a finite temperature $T\not=0$, with the aim to provide a complementary tool for the study
of the gluon plasma from first principles. As briefly discussed by one of us~\cite{damp},
the screened expansion can be extended to finite temperature, providing a quasi-particle picture
for the gluon which is damped, with a very short finite lifetime, and canceled from the asymptotic states.
Here, we give a full account of the details of the calculation and report a comprehensive set of results for the
gluon sector, including propagators, analytic properties, poles, masses, widths and dispersion relations.
We discuss different optimization strategies and, by a comparison with the available lattice data,
we explore how robust the screened expansion is when it is extended to finite temperature.
While the existence of a screening mass mitigates the effects of the hard thermal loops, several problems arise
at a finite temperature, ranging from the temperature dependence of the optimal mass scale, to the analytic continuation
of the numerical integrals. Actually, even if a formal extension to finite temperature is straightforward and based on
standard thermal Feynman graphs, the ambition to extract analytical results requires a quite tedious and lengthy analytical
calculation of the integrals and, even so, a final one-dimensional numerical integration cannot be avoided.
Nonetheless, the resulting numerical integrals are shown to define analytic functions which can be evaluated in the complex plane.
Then, the poles of the gluon propagator and the resulting dispersion relations can be easily extracted numerically.
Overall, despite the expected difficulties, the one-loop screened expansion seems to be reliable at low temperature, with correct
predictions which become less quantitative at high temperature, especially for the longitudinal sector, when compared with the lattice data.
At $T=0$, the one-loop approximation is quite sensitive to the renormalization scheme and to the subtraction point, but it can be shown
to be basically {\it tangent} to the exact result, which is approached for a special choice of the ratio between the gluon mass parameter $m$ and
the renormalization scale $\mu$.
Here, $m$ is just a mass parameter which defines the shift of the expansion point~\cite{ptqcd,ptqcd2,beta,beta2},
not to be confused with
the physical mass of the gluon. It seems that, for that special ratio $\mu/m$, the higher order terms become negligible, yielding very accurate
analytical expressions for the propagators. While that special ratio is scheme-dependent, it can be determined from first principles
by monitoring some identities which must be fulfilled by the exact propagators,
like the Nielsen identities, which express the gauge-invariance of the poles~\cite{xigauge}.
We must mention that, once the ratio is optimized in the complex {\it Minkowski} space, where the poles are defined, the propagators are found in excellent agreement
with the lattice data in the {\it Euclidean} space. Thus, the optimized analytical expression is not just a good interpolation formula,
but a very good approximation for the whole analytic function which is defined in the complex plane. Moreover, at the optimal ratio $\mu/m$ there
is only one energy scale left in the calculation, say the mass parameter $m$, so that its actual value becomes irrelevant, since it can be
used as energy units and is eventually determined by a comparison with the phenomenology. For instance, sharing the same units of the
lattice data, a value $m=0.656$ GeV was established in previous works~\cite{xigauge,beta}.
At a finite temperature $T\not=0$, there is a third energy scale and the optimal parameters $m$, $\mu$ become two independent functions
of temperature, $m(T)$, $\mu(T)$, since their optimal ratio is expected to depend on $T$. In principle, one could proceed as for $T=0$ and
fix the optimal ratio by monitoring the gauge-invariance of the poles. However, that would at least require a knowledge of the thermal propagators in a generic covariant gauge,
while the present formalism has been developed only in the Landau gauge. Moreover, no lattice data are available for a comparison in a generic gauge
and finite $T$. This is not a theoretical limitation by itself, but leads to a weakening of the control of the accuracy.
That of the gauge-invariance of the poles actually is an additional problem one encounters when extending the theory to finite $T$~\cite{kajantie,heinz,hansson,carrington}. Even though the poles of the propagator are constrained to be non-perturbatively gauge-independent by e.g. the Nielsen identities~\cite{kobes}, in the thermal formalism different powers of the coupling constant coexist at the same loop order when hard-thermal-loop effects are taken into account, so that consistent resummation schemes are needed in order to obtain truly gauge-invariant results for the poles' position. To first order in the coupling, this can be shown to only affect the imaginary part of the dispersion relations, i.e. the gluon's damping rate. In this work no attempt has been made to implement such resummation schemes or to keep under control the accuracy of the approximation with respect to the issue of gauge-invariance. Whereas at low, non-zero temperatures the screening provided by the gluon's mass may somewhat suppress the effects of the required resummed terms, at higher temperatures the latter are expected to become non-negligible, causing our predictions for the gluon damping rate to become less and less reliable as the temperature is increased.
In the Landau gauge, we explored two complementary strategies and checked that the qualitative description which emerges is robust enough
and does not depend on the optimization choice. The first, simpler, strategy consists in using the same $m$ and $\mu$ parameters that work at $T=0$.
That choice was already made in Ref.~~~\cite{damp} (albeit with different values for the parameters) and makes sense at low temperature where we expect that $m(T)\approx m(0)$ and $\mu(T)\approx \mu(0)$.
With this choice, we find the correct qualitative behavior without any adjustment of parameters.
In particular, the longitudinal propagator shows a non-monotonic behavior with a crossover at $T/m(0)\approx 0.15$.
However, the agreement with the lattice data is not quantitative, and the predicted transition temperature is too small ($T\approx 100$ MeV),
thus indicating that we are already outside the safe low-temperature range. Nonetheless, the disagreement can be absorbed in part by a
temperature-dependent optimization of the expansion.
Thus, as a second strategy, we relax the constraints of $m$ and $\mu$ being equal to their $T=0$ values and regard $m(T)$ and $\mu(T)$ as independent
unknown functions. Reversing the argument that led to their optimization at $T=0$, we tune the unknown functions in the Euclidean space by
looking for the best agreement with the lattice data. Then, {\it assuming} that the higher-order terms are smaller when the agreement is
better, the optimized propagators are continued to Minkowski space where the pole location gives information on the dispersion relations of the quasi-gluons
at finite temperature. We anticipate that, from a strictly quantitative point of view, the agreement with the lattice is not comparable
with the excellent result which was reached at $T=0$. Moreover, while the transverse propagator is generally well described, the longitudinal
projection becomes very poor deep in the IR for moderately high temperatures. Since most of the deviation occurs below 500-700 MeV, we expect that
the predictions for the pole position at high momenta might not be affected too much. We stress that there are no data available
in the Minkowski space for a comparison, thus evidencing the power of the method for exploring the analytic properties of the propagators.
Irrespective of the optimization criterion, we confirm the finding of Ref.~~\cite{damp} and the quasi-gluon scenario which was described by
Stingl~\cite{stingl}, with a gluon which has a very short finite lifetime and can only exist as a short-lived
intermediate state at the origin of a gluon-jet event.
This paper is organized as follows. In Sec. II we review the set-up and main features of the screened massive expansion and its extension to finite temperatures. In Sec. III we present our results for the Landau gauge gluon propagator at $T\neq 0$ and vanishing Matsubara frequency, $\omega=0$. In Sec. IV we derive the dispersion relations for the quasi-gluons at finite temperatures. In Sec. V we discuss our results and present our conclusions. In the Appendix we explicitly compute the gluon polarization and ghost self-energy at finite temperatures using the screened massive expansion.
\section{The screened expansion and its extension to finite temperature}
In a linear covariant $\xi$-gauge, the gauge-fixed BRST invariant Lagrangian of pure Yang-Mills SU(N) theory
is
\begin{equation}
{\cal L}={\cal L}_{YM}+{\cal L}_{fix}+{\cal L}_{FP},
\label{Ltot}
\end{equation}
where
\begin{align}
{\cal L}_{YM}&=-\frac{1}{2} \mathrm Tr\left( \hat F_{\mu\nu}\hat F^{\mu\nu}\right),\nonumber\\
{\cal L}_{fix}&=-\frac{1}{\xi} \mathrm Tr\left[(\partial_\mu \hat A^\mu)(\partial_\nu \hat A^\nu)\right],
\label{LYMfix}
\end{align}
and ${\cal L}_{FP}$ is the ghost term
arising from the Faddeev-Popov (FP) determinant.
The tensor operator is defined as
\begin{equation}
\hat F_{\mu\nu}=\partial_\mu \hat A_\nu-\partial_\nu \hat A_\mu
-i g \left[\hat A_\mu, \hat A_\nu\right],
\label{F}
\end{equation}
where the gauge field operators satisfy the $SU(N)$ algebra
\begin{align}
\hat A^\mu&=\sum_{a} \hat X_a A_a^\mu,\nonumber\\
\left[ \hat X_a, \hat X_b\right]&= i f_{abc} \hat X_c,\quad
f_{abc} f_{dbc}= N\delta_{ad}.
\end{align}
In the standard PT formalism, the total action is split as $S_{tot}=S_0+S_I$, where the quadratic part can
be written as
\begin{align}
S_0&=\frac{1}{2}\int A_{a\mu}(x)\delta_{ab} {\Delta_0^{-1}}^{\mu\nu}(x,y) A_{b\nu}(y) {\rm d}^4 x\,{\rm d}^4 y \nonumber \\
&+\int c^\star_a(x) \delta_{ab}{{\cal G}_0^{-1}}(x,y) c_b (y) {\rm d}^4 x\, {\rm d}^4 y,
\label{S0}
\end{align}
while the interaction contains three vertices
\begin{equation}
S_I=\int{\rm d}^4x \left[ {\cal L}_{gh} + {\cal L}_3 + {\cal L}_4\right],
\label{SI}
\end{equation}
\begin{align}
{\cal L}_{3g}&=-g f_{abc} (\partial_\mu A_{a\nu}) A_b^\mu A_c^\nu,\nonumber\\
{\cal L}_{4g}&=-\frac{1}{4}g^2 f_{abc} f_{ade} A_{b\mu} A_{c\nu} A_d^\mu A_e^\nu,\nonumber\\
{\cal L}_{ccg}&=-g f_{abc} (\partial_\mu c^\star_a)c_b A_c^\mu.
\label{Lint}
\end{align}
In Eq.~(\ref{S0}), the standard free-particle propagators for
gluons and ghosts, $\Delta_0$ and ${\cal G}_0$ respectively, are defined by their Fourier transforms
\begin{align}
{\Delta_0}^{\mu\nu} (p)&=\Delta_0(p)\left[t^{\mu\nu}(p)
+\xi \ell^{\mu\nu}(p) \right],\nonumber\\
\Delta_0(p)&=\frac{1}{-p^2}, \qquad {{\cal G}_0} (p)=\frac{1}{p^2},
\label{D0}
\end{align}
where the transverse and longitudinal projectors are used
\begin{equation}
t_{\mu\nu} (p)=g_{\mu\nu} - \frac{p_\mu p_\nu}{p^2},\quad
\ell_{\mu\nu} (p)=\frac{p_\mu p_\nu}{p^2}.
\label{tl}
\end{equation}
Later, we will take the limit $\xi\to 0$ and use the Landau gauge which is a Renormalization Group (RG) fixed point and
is the most studied gauge on the lattice.
In the above equations, the fields and the coupling must be regarded as renormalized objects and the inclusion of
the usual set of counterterms is understood in the total Lagrangian.
\begin{figure}[b] \label{fig:graphs}
\centering
\includegraphics[width=0.25\textwidth,angle=-90]{fig1.eps}
\caption{Two-point graphs with no more than three vertices and no more than one loop.
The cross is the transverse mass counterterm of Eq.~(\ref{dG2}) and is regarded as
a two-point vertex.
In the Appendix, a detailed description of the calculation at finite $T$ is given for all
the polarization graphs in the figure.}
\end{figure}
The massive screened version of PT was developed in Refs.~\cite{ptqcd,ptqcd2,analyt}. At $T=0$ and in a generic
covariant gauge, the method is very accurate and predictive if the expansion is optimized
by the constraints of BRST symmetry~\cite{xigauge,beta,beta2}.
The expansion arises by a mere change of the expansion point of ordinary PT.
Following Refs.~\cite{ptqcd2,xigauge}, the new {\it massive} expansion is recovered by just adding
a transverse mass term to the quadratic part of the action and subtracting it again from the interaction,
leaving the total action unchanged. In more detail, we add and subtract the action term
\begin{equation}
\delta S= \frac{1}{2}\int A_{a\mu}(x)\>\delta_{ab}\> \delta\Gamma^{\mu\nu}(x,y)\>
A_{b\nu}(y) {\rm d}^4\, x{\rm d}^4y,
\label{dS1}
\end{equation}
where the vertex function $\delta\Gamma$ is a shift of the inverse propagator,
\begin{equation}
\delta \Gamma^{\mu\nu}(x,y)=
\left[{\Delta_m^{-1}}^{\mu\nu}(x,y)- {\Delta_0^{-1}}^{\mu\nu}(x,y)\right],
\label{dG}
\end{equation}
and ${\Delta_m}^{\mu\nu}$ is a new {\it massive} free-particle propagator,
\begin{align}
{\Delta_m^{-1}}^{\mu\nu} (p)&=
(-p^2+m^2)\,t^{\mu\nu}(p)
+\frac{-p^2}{\xi}\ell^{\mu\nu}(p).
\label{Deltam}
\end{align}
Adding that term is equivalent to substituting the new massive propagator ${\Delta_m}^{\mu\nu}$ for the
old massless one ${\Delta_0}^{\mu\nu}$ in the quadratic part. Thus, the new expansion point is a massive
free-particle propagator for the gluon, which is much closer to the exact propagator in the IR.
The mass-shift
parameter $m$ is irrelevant in the UV, but acts as a natural cutoff which screens the theory in the IR.
Of course, in order to leave the total action unaffected by the change, the same term is subtracted from the interaction,
providing a new interaction vertex $-\delta\Gamma$, a two-point vertex which can be regarded as a new counterterm.
Dropping all color indices in the diagonal matrices and
inserting Eq.~(\ref{D0}) and (\ref{Deltam}) in Eq.~(\ref{dG}), the vertex is just the transverse mass shift
of the quadratic part,
\begin{equation}
-\delta \Gamma^{\mu\nu} (p)=-m^2 t^{\mu\nu}(p),
\label{dG2}
\end{equation}
and must be added to the standard set of vertices arising from Eq.~(\ref{Lint}). The new vertex
is now part of the interaction, even if it does not depend on the coupling. Thus,
the expansion has the nature of a $\delta$-expansion, since different powers of the coupling
coexist at each order in powers of the total interaction.
The proper gluon polarization and ghost self energy can be evaluated, order by order, by the modified PT.
In all Feynman graphs,
any internal gluon line is a massive free-particle propagator ${\Delta_m}^{\mu\nu}$ and the
new insertions of the (transverse) two-point vertex $\delta \Gamma^{\mu\nu}$ are denoted by a cross, as
shown in Fig.~~1. For further details we refer to Refs.~\cite{ptqcd,ptqcd2,xigauge}.
Since the total gauge-fixed FP Lagrangian is not modified and because of BRST invariance,
the longitudinal polarization is known exactly and is zero.
At $T=0$, the exact polarization and the dressed gluon propagator are defined by a single function,
\begin{equation}
\Pi^{\mu\nu}(p)=\Pi(p)\, t^{\mu\nu}(p),
\label{pol}
\end{equation}
so that, in the Landau gauge, the exact gluon propagator is transverse,
\begin{equation}
\Delta_{\mu\nu}(p)=\Delta (p)\,t_{\mu\nu}(p),
\end{equation}
and defined by the scalar function $\Delta(p)$. This feature is lost at any finite temperature $T>0$,
since Lorentz-invariance is broken, and two scalar functions are required instead. In that perspective, it is convenient to
maintain the Lorentz structure explicit and to switch to the Euclidean
formalism. Then, denoting with $p^2$ the Euclidean squared momentum, the exact (dressed) gluon and ghost propagators can be written as
\begin{align}
{{\Delta}^{-1}}_{\mu\nu} (p)&=(p^2+m^2)t_{\mu\nu}(p)+\frac{p^2}{\xi}\ell_{\mu\nu}(p)-\Pi_{\mu\nu}(p),\nonumber\\
{\cal G}^{-1}(p)&=-p^2-\Sigma (p),
\label{dressprop}
\end{align}
where $t_{\mu\nu}$ and $\ell_{\mu\nu}$ are the Euclidean projectors of Eq.~(\ref{tlE}).
The proper gluon polarization $\Pi_{\mu\nu}$ and the ghost self-energy $\Sigma$ are the
sum of all one-particle-irreducible (1PI) graphs in the screened expansion, including all counterterms.
In Fig.~~1, the two-point 1PI graphs are shown up to one-loop and third order in the delta-expansion.
In the exact self energies, we can single out the tree-level terms and write
\begin{align}
\Pi_{\mu\nu}(p)&=m^2t_{\mu\nu}(p)-p^2t_{\mu\nu}(p)\delta Z_A+\Pi^{loop}_{\mu\nu}(p),\nonumber\\
\Sigma (p)&=p^2\delta Z_c+\Sigma^{loop}(p),
\label{selfs}
\end{align}
where the first term $m^2t_{\mu\nu}(p)$ is the tree graph (1a) in Fig.~~1
and arises from the insertion of the new two-point
vertex $-\delta \Gamma_{\mu\nu}$ of Eq.~(\ref{dG2}). We observe that this first tree term cancels the mass shift
of the gluon propagator in Eq.~(\ref{dressprop}). Indeed, the physical mass of the gluon arises from
the loops and is not merely given by the mass-shift parameter $m^2$.
The other tree-level terms, $-p^2\,t_{\mu\nu}\,\delta Z_A$, $p^2\,\delta Z_c$, are not shown in Fig.~~1 and are the usual
field-strength renormalization counterterms. Their UV diverging parts are not affected by the mass parameter
and are the same of standard PT~\cite{ptqcd,ptqcd2}.
The proper functions, $\Pi^{loop}_{\mu\nu}$, $\Sigma^{loop}$, are given by the sum of all 1PI graphs containing loops.
The finite parts of $\delta Z_A$, $\delta Z_c$ are arbitrary and depend on the scheme and
on the renormalization scale $\mu$~~\cite{beta,beta2}.
The diverging parts of $\delta Z_A$, $\delta Z_c$ cancel the UV divergences of the functions
$\Pi^{loop}_{\mu\nu}/p^2$ and
$\Sigma_{loop}/p^2$ which become finite dimensionless functions of the variable $p_\mu/m$. They are
defined up to a constant which depends on the dimensionless renormalization scale parameter $t=\mu^2/m^2$.
Thus, at $T=0$, there are two energy scales in the calculation, $m$ and $\mu$.
For instance, in a momentum subtraction scheme (MOM) and in the Landau gauge,
the one-loop dressed propagators can be written as
\begin{align}
{\Delta}(p)^{-1}&=p^2-Ng^2\left[\Pi^{(1)}(p) -\Pi^{(1)}(\mu)\right], \nonumber\\
{\cal G}(p)^{-1}&=-p^2-Ng^2\left[\Sigma^{(1)}(p)-\Sigma^{(1)}(\mu)\right],
\label{dressprop1}
\end{align}
having made explicit the dependence on $N$ and $g^2$ as factors in the one-loop functions
$\Pi^{(1)}$, $\Sigma^{(1)}$, according to the notation of Appendix A, where all details of the calculation are reported.
In Eq.~(\ref{dressprop1}), an explicit choice has been made for the finite parts of the renormalization constants
$\delta Z_A$, $\delta Z_c$. Of course, that choice depends on the scheme and on the renormalization scale $\mu$.
A more general way to get rid of all the scheme-dependent parameters, including the renormalized coupling $g^2$,
was discussed in previous papers on the screened expansion~\cite{ptqcd,ptqcd2,xigauge,beta}, where two dimensionless
one-loop functions were defined (see Appendix B.1 for their explicit expressions),
\begin{align}
\pi_1(p^2/m^2)&=-\left(\frac{16\pi^2}{3}\right)\frac{\Pi^{(1)}(p)}{p^2}, \nonumber\\
\sigma_1(p^2/m^2)&=\left(\frac{16\pi^2}{3}\right)\frac{\Sigma^{(1)}(p)}{p^2},
\label{pisigma}
\end{align}
so that the one-loop propagators in Eq.~(\ref{dressprop1}) can be recast as functions of the dimensionless
variable $s=p^2/m^2$,
\begin{align}
p^2\,{\Delta}(p)&=\frac{z_\pi}{\pi_1(s)+\pi_0}, \nonumber\\
p^2\,{\cal G}(p)&=-\frac{z_\sigma}{\sigma_1(s)+\sigma_0},
\label{dressprop2}
\end{align}
where $z_\pi$ and $z_\sigma$ are irrelevant normalization constants while all the scheme-dependent parameters are
embedded in the two constants $\pi_0$ and $\sigma_0$. With some abuse of language, we will refer to them as
{\it renormalization} constants. Eq.~(\ref{dressprop2}) is quite general since it does not require any specific
renormalization scheme to be defined. Of course, our ignorance about those constants reflects a well known weakness
of the one-loop approximation which depends on the details of the renormalization scheme and on the actual value
of the renormalization scale $\mu$. In this sense, we still have two scales, $m$ and $\mu$, and the arbitrary choice
of their ratio $t=\mu^2/m^2$ somehow determines the actual value of the renormalization constants $\pi_0$ and $\sigma_0$.
A nice feature of the one-loop result is its apparent {\it tangency} to the exact result which is approached for special
values of the renormalization constants. Those values are equivalent to a choice of the best renormalization scale
$\mu$, where the approximation is more effective. It is just an example of the
optimized perturbation theory by variation of the renormalization scheme~\cite{stevensonRS1,stevensonRS2}.
There might be a special scale $\mu$ where the expansion converges more quickly and the higher order terms are minimal.
Thus, from first principles, we could determine the optimal constants by monitoring some identities which must
be satisfied by the exact propagators. For instance, in Ref.~~\cite{xigauge},
the Nielsen identities~\cite{nielsen1,nielsen2} were used, which are
a direct consequence of BRST symmetry. From the identities, one can prove the gauge-parameter-independence of the poles
and residues of the exact gluon propagator~\cite{xigauge}. Then, we might expect that the renormalization constants
are optimal when the poles have a minimal sensitivity to the gauge parameter.
It is remarkable that the optimized one-loop propagators turn out to be in excellent agreement with the lattice data in
the IR. Notably, while the comparison with the data requires an analytic continuation to the Euclidean space,
the poles are found in the complex plane. Thus, the one-loop propagators in Eq.~(\ref{dressprop2}) are not just one of the
many interpolation formula for the data, but they provide a very accurate analytic function in the whole complex plane.
The existence of complex poles is one of the most important predictions of the screened expansion.
While a thermal mass and a finite damping rate are expected by PT at high temperature, the existence of finite
intrinsic values at $T=0$ can be regarded as a proof of confinement
as first discussed by Stingl~\cite{stingl}. The quasi-gluon has a finite lifetime and can only exist as a short-lived
intermediate state. However, at finite temperature, the quasi-gluons play an important role for determining
the thermal properties
of the hot plasma. Thus, a finite temperature extension of the screened expansion is required for a full study of the
dispersion relations which emerge from the pole location.
At a finite temperature $T>0$, Eqs.~(\ref{dressprop}),(\ref{selfs}) are still valid, but the one-loop graphs
in Fig.~~1 acquire a finite thermal part which must be added to the vacuum (diverging) contribution at $T=0$.
The thermal parts are finite and no further renormalization is required. We only have to add the thermal parts
to the self-energies in Eq.~(\ref{selfs}).
We write the Euclidean four-vector as
$p^\mu=({\bf p}, \omega)$
where $\omega=p_4=-ip_0$, while the
Lorentz four-vector was $(p_0, {\bf p})$.
In the finite-temperature formalism, $\omega=\omega_n=2\pi n T$ and the
Euclidean integral is replaced by a sum over $n$ and by a three-dimensional
integration,
\begin{equation}
\int \frac{{\rm d}^4p}{(2\pi)^4} \to T\sum_n\int\frac{{\rm d}^3{\bf p}} {(2\pi)^3}.
\end{equation}
Since Lorentz invariance is obviously broken, we introduce a transverse projector $P^T_{\mu\nu}$,
orthogonal to the fourth Euclidean direction, and its longitudinal complement $P^L_{\mu\nu}$, as defined in Eq.~(\ref{TL}),
so that the gluon polarization and propagator in Eqs.~(\ref{dressprop}),(\ref{selfs})
can be written in the Landau gauge, $\xi=0$, as
\begin{align}
\Pi_{\mu\nu}(p,T)&= \Pi_L(p,T)\>P^L_{\mu\nu}(p)+\Pi_T(p,T)\>P^T_{\mu\nu}(p),\nonumber\\
\Delta_{\mu\nu}(p,T)&=\Delta_L(p,T)\>P^L_{\mu\nu}(p)+\Delta_T(p,T)\>P^T_{\mu\nu}(p),
\end{align}
where the projected one-loop dressed functions are
\begin{align}
\Delta_T(p,T)^{-1}&=p^2+p^2\delta Z_A-Ng^2\Pi^{(1)}_T(p,T),\nonumber\\
\Delta_L(p,T)^{-1}&=p^2+p^2\delta Z_A-Ng^2\Pi^{(1)}_L(p,T).
\end{align}
and $\Pi^{(1)}_{L,T}$ are the one-loop projected polarizations, evaluated by projection of the one-loop graphs in
Fig~1, omitting the tree graphs. As discussed in Appendix B, each graph contributing to $\Pi^{(1)}_{L,T}$
can be split as
\begin{equation}
\Pi^{(1)}_{L,T}(p,T)=\left[\Pi^{(1)}_{L,T}\right]_{Th}+\left[\Pi^{(1)}_{L,T}\right]_{V},
\end{equation}
where the vacuum part $\left[\Pi^{(1)}_{L,T}\right]_{V}=\Pi^{(1)}_{L,T}(p,0)$ is the same graph evaluated at $T=0$
and does not depend on $T$, while the thermal part, $\left[\Pi^{(1)}_{L,T}\right]_{Th}$, vanishes at $T=0$. Thus,
we can generalize Eqs.~(\ref{pisigma}),(\ref{dressprop2})
and define dimensionless functions
\begin{align}
\left[\pi_{L,T}(p,T)\right]_V&=-\left(\frac{16\pi^2}{3}\right)
\frac{\left[\Pi^{(1)}_{L,T}(p,T)\right]_V}{p^2} =\pi_1(s), \nonumber\\
\left[\pi_{L,T}(p,T)\right]_{Th}&=-\left(\frac{16\pi^2}{3}\right)
\frac{\left[\Pi^{(1)}_{L,T}(p,T)\right]_{Th}}{p^2},
\end{align}
so that the projections of the one-loop propagator can be recast as
\begin{equation}
p^2\,{\Delta}_{L,T}(p,T)=\frac{z_\pi}{\pi_1(s)+\pi_0+\left[\pi_{L,T}(p,T)\right]_{Th}}.
\label{dresspropLT}
\end{equation}
In this form Eq.~(\ref{dresspropLT}) is quite general since it does not require any specific
renormalization scheme to be defined. All the scheme-dependent parameters are
embedded in the {\it renormalization} constant $\pi_0$.
It is not obvious that the same scale $\mu$ and constant $\pi_0$ which were optimal at $T=0$ are
still optimal at finite $T$. Indeed, they might depend on $T$ and even take a different value
for the different projections. Moreover, the mass parameter $m$, which was the only energy scale
left after optimization at $T=0$, might take a value $m(T)$ which depends on $T$. Thus we have
three energy scales: the optimal $\mu(T)$, the mass parameter $m(T)$ and $T$ itself. In other words,
according to Eq.~(\ref{dresspropLT}), at any $T$ and in units of $m(0)$ we have two free parameters,
the ratio $m(T)/m(0)$ and the optimal renormalization constant $\pi_0(T)$.
Having the role of variational parameters, to be optimized, their best
values might be different for the two projections.
While at $T=0$ the optimal constant $\pi_0$ was determined from first principles~\cite{xigauge}, by requiring
a minimal sensitivity of the poles to any change of the gauge parameter, here we have the less ambitious aim
of exploring {\it if} a set of optimal parameters does exist such that the screened expansion is able to describe the
lattice data with reasonable accuracy. Thus, we work in the Landau gauge and, for each value of $T>0$,
we fix the parameters by a fit of the available lattice data in the Euclidean space.
At low temperature, as we said, we also explored the alternative of maintaining
the parameters fixed at their optimal value for $T=0$, in order to give a general description at finite $T$ from first
principles, without any input from the lattice and from the known phenomenology. Of course, this approach can only be
reliable if $T$ is very low and the thermal effects are small. However, even extrapolating at higher temperatures, the
qualitative predictions turn out to be in agreement with the data. Thus, the screened expansion is able to capture the main
features of gluon thermodynamics at finite temperature. This is a very important aspect, since our final aim will be to extract
some dynamical properties of the quasi-gluons, like the dispersion relations, which cannot be measured on the lattice.
Moreover, even qualitative properties, like the existence of complex poles, are of central interest for understanding
the behavior of the gluon plasma at high temperature and its phase transition.
In order to fulfill that program, once optimized by one of the two alternatives discussed above, the gluon propagator
must be continued to the complex plane. This is a straightforward step if the one-loop graphs are expressed as analytic
functions of the Euclidean momentum. A very detailed but tedious analytical evaluation of the integrals
is reported in the Appendix. Most of the integrals were encountered in a study of the Curci-Ferrari model~\cite{serreau}.
We basically use the same method for decomposing the integrals. However, in the screened expansion there are also
some different graphs, namely the crossed graphs in Fig.~~1, with one insertion of the mass counterterm.
Their explicit expressions are obtained by a derivative in the Appendix.
Unfortunately, at finite $T$, not all the multidimensional integrals can be evaluated analytically
and an external one-dimensional numerical integration cannot be avoided for almost all the one-loop graphs.
Thus, as shown in the Appendix, all the graphs can be written as analytic functions which are defined by
integral representations. The remaining integration can be carried out numerically for any complex value of the
external momentum, provided that no singularity is encountered along the integration path.
Actually, in general, the analytic continuation of integral functions is not trivial.
As discussed in Ref.~~\cite{blaizot2005}, we must check that the external integration on the real axis does not cross any
singular point of the logarithmic functions. Otherwise, a modified path must be chosen before the analytic continuation
can be undertaken. As shown in Ref.~~\cite{damp}, by inspection of the explicit expressions, the existence of singular
points on the integration path can be ruled out in the present case. For instance,
denoting with $\Omega=p_0$ and $p^\mu=(\Omega, {\bf p})$ the external momentum in Minkowski space, the analytic
continuation of the thermal integral $I^{\alpha\beta}(y,-i\Omega)$ is defined by
the integral representation of Eq.~(\ref{Iabth}), where $y$ is the external three vector modulus,
$y=\vert {\bf p}\vert$.
We can continue the external energy $\Omega$ to the complex plane if there are no singular points on the positive
real axis of the integration variable. However,
some branch cuts might be present, originating at the
singular branch point of the logarithmic function in Eq.~(\ref{logalpha}) which reads
\begin{equation}
L_\beta (z_\alpha;y,q)=\log\left[\frac{z_\alpha^2+\epsilon^2_{y+q,\beta}} {z_\alpha^2+\epsilon^2_{y-q,\beta}} \right],
\end{equation}
where the complex
variable $z_\alpha$ is defined as
$z_\alpha=i\Omega\pm i\sqrt{q^2+\alpha^2}$ and $\epsilon^2_{y\pm q,\beta}=(y\pm q)^2+\beta^2$.
Here $\alpha$ and $\beta$ are masses equal to $0$ or $m$ and $q$ is the integration variable.
Assuming the existence of a
branch point at $q=q_0$ on the real axis, the latter must satisfy
\begin{equation}
\pm 2q_0y=\alpha^2-\beta^2-y^2+\Omega^2\pm2\Omega\sqrt{q_0^2+\alpha^2},
\label{zero}
\end{equation}
where the $\pm$ signs are independent of each other. Taking a complex energy $\Omega=\mathop{\rm Re}\Omega+i\Im\Omega$ with $\Im\Omega>0$,
the imaginary part of Eq.~(\ref{zero})
gives
\begin{equation} \mathop{\rm Re}\Omega=\mp \sqrt{q_0^2+\alpha^2},
\end{equation}
and substituting back in the real part we obtain
\begin{equation}
\epsilon^2_{y \pm q_0,\beta}+(\Im\Omega)^2=0,
\end{equation}
which is never satisfied unless $\Im\Omega=\beta=0$. Thus, if $\Omega$ is not real, the branch point $q_0$ cannot be real and
the integral over $q$, on the real axis, defines an analytic function of $\Omega$.
The same argument holds for the other thermal integrals in Appendix B.
Thus, we can safely continue the numerical integrals from
the Euclidean space ($\mathop{\rm Re}\Omega=0$, $\Im\Omega>0$) to the whole upper half-plane.
Moreover, in the large wavelength limit $y\to 0$, there are no branch points at all because the logarithmic
function can be written as $L_\beta (z_\alpha;y,q)\approx \log\left[1+{\cal O} (y)\right]$
and the argument of the log does not vanish if $y$ is small enough.
Having ruled out the existence of singularities along the integration path,
the poles of the gluon propagator and the dispersion relations can be easily extracted numerically in the complex
plane by the integral representation of the thermal integrals which are derived in Appendix B.
\section{The gluon propagator at finite T}
The longitudinal and transverse projections of the polarization graphs entering in Eq.~(\ref{dresspropLT})
are decomposed as the sum of more basic Euclidean integrals in Appendix A, for all the one-loop graphs of Fig.~~1.
The explicit thermal parts of those integrals are presented in Appendix B by integral representations. For any given value
of the external three-momentum $y=\sqrt{{\bf p}^2}$ and Euclidean frequency $\omega=p_4=2\pi n T$,
the one-dimensional integrals are evaluated numerically by a simple integration on the real axis and the result is
inserted in Eq.~(\ref{dresspropLT}).
We will first explore the projected propagators for $\pi_0$ and $m$ fixed at their zero-temperature values
which were determined from first principles in Ref.~~\cite{xigauge}.
Then, we will show how their values can be optimized by a comparison with the available lattice data.
\subsection{Expansion optimized at $\boldsymbol{T=0}$}
In the low-temperature limit, we assume that the optimal renormalization constant $\pi_0(T)$ and mass parameter $m(T)$
can be replaced by their zero-temperature values $\pi_0=-0.876$ and $m(0)=m_0=656$~MeV, as determined in Ref.~~\cite{xigauge}
by requiring a minimal sensitivity of the pole structure to the gauge parameter. Strictly speaking, in the Landau gauge,
that condition fixes $\pi_0$, while $m_0$ is the only energy scale left and is fixed in order to match the energy units
of the lattice data.
\begin{figure}[b] \label{fig:longprop}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig2a.eps}
\
\includegraphics[width=0.32\textwidth,angle=-90]{fig2b.eps}
\caption{Longitudinal propagator $\Delta_L$ in units of $m_0=m(0)$ at $\omega=0$ for the low temperature range $T/m_0<0.15$ (top) and the high temperature range $T/m_0>0.15$ (bottom).
The renormalization constant and the mass parameter are fixed at their optimal $T=0$ values, $\pi_0(T)=\pi_{0}(0)=-0.876$
and $m(T)=m_0=656$~MeV. All the curves are multiplicatively renormalized at $\mu_{0}/m_{0}=6.098$ ($\mu_{0}=4$ GeV in physical units).}
\end{figure}
\begin{figure}[t] \label{fig:transprop}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig3a.eps}\quad\includegraphics[width=0.32\textwidth,angle=-90]{fig3b.eps}
\caption{Transverse propagator $\Delta_T$, with the same notation and parameters of Fig.~~2.}
\end{figure}
Let us first explore the behavior of the gluon propagators as a function of $T$ in the limit $\omega\to 0$,
where $p^2={\bf p}^2$, which is the most studied case on the lattice~\cite{silva,aouane}. The longitudinal and transverse
propagators are shown in units of $m_0$ in Fig.~~2 and Fig.~~3, respectively. The former were multiplicatively renormalized by requiring that
\begin{equation}
\Delta_{L,T}(p,T)\Big|_{\omega=0,|{\bf p}|=\mu_{0}}=\frac{1}{\mu^{2}_{0}}
\end{equation}
with $\mu_{0}/m_{0}=6.098$ (corresponding to $\mu_{0}=4$ GeV for $m_{0}=656$ MeV). We observe that, because of the chosen optimization, in the limit $T\to 0$ the longitudinal and transverse propagators coincide and reproduce the lattice
data extremely well~\cite{ptqcd,ptqcd2,xigauge,scaling,beta,beta2}, so that the low-temperature limit can be
regarded as exact. For reference, in Tab. I we report the physical equivalent of the adimensional temperatures $T/m_{0}$ used for the plots.
\begin{table}[h]
\def1.3{1.3}
\begin{tabular}{|c|c|c|c|c|}
\hline
$T/m_{0}$&0.05&0.08&0.12&0.15\\
\hline
$T\ \text{(MeV)}$&32.80&52.48&78.72&98.40\\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$T/m_{0}$&0.18&0.21&0.25&0.30&0.36&0.44\\
\hline
$T\ \text{(MeV)}$&118.08&137.76&164.00&196.80&236.16&288.64\\
\hline
\end{tabular}
\caption{Dimensionful values of the adimensional temperatures $T/m_{0}$ plotted in Figs.~2 and 3, given $m_{0}=656$ MeV.}
\end{table}
We observe a crossover, in Fig.~~2, with the longitudinal propagator which increases in the IR for increasing $T$ below
$T_c\approx 0.15\cdot m_0$, but sharply decreases above $T_c$. This non-monotonic behavior is a well known feature
which has been reported by several lattice calculations~\cite{silva,aouane}.
The transverse propagator in Fig.~~3, on the other hand, has a monotonic behavior, decreasing for increasing $T$, again in qualitative
agreement with the known predictions of the lattice. Actually, we cannot expect a quantitative agreement at $T\approx T_c$
or larger values, because we are extrapolating the optimization condition which was valid at $T=0$. Thus, the correct
qualitative behavior of the propagators at high temperature is an encouraging result. A crude estimate of $T_c$
is found by using the zero-temperature value $m_0=656$~MeV for restoring the energy units, yielding
at the crossover $T_c\approx 100$~MeV. This value is quite smaller than the known transition temperature
$T_c\approx 270$~MeV which is measured on the lattice~\cite{lucini,silva,aouane}. The difference might well be the consequence
of a sub-optimal choice of the renormalization constant, but it could also arise from a change of the mass parameter
with temperature or from the more general failure of PT at high temperature. Thus, it becomes relevant to explore
whether a more quantitative agreement might be obtained by a tuning of the free parameters.
\subsection{Optimization by a fit of data at finite $\boldsymbol{T}$}
As the temperature increases, our previous assumption, $m(T)=m(0)$, $\pi_{0}(T)=\pi_{0}(0)$, becomes less valid. In what follows, we turn to fixing the optimal value of the parameters at $T\neq 0$ by a fit of the lattice data of Ref.~~~\cite{silva}. Since at non-zero temperatures the projections $\Delta_{L}(p,T)$ and $\Delta_{T}(p,T)$ have different behaviors with respect to a change in $T$, we may expect that the optimal values of the parameters will differ depending on which of the two components of the lattice propagator is used for the fit. This is indeed what we found. Of course, since in the subtracted Lagrangian of the present formalism the gluon mass parameter $m^{2}(T)$ is multiplied by the full four-dimensional transverse projector $t_{\mu\nu}(p)$, choosing different mass parameters/scales for the two components of the propagators is not allowed from first principles. This issue will be addressed at the end of this section.\\
\begin{figure}[t] \label{fig:longpropfit}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig4a.eps}\quad\includegraphics[width=0.32\textwidth,angle=-90]{fig4b.eps}
\caption{Longitudinal propagator $\Delta_L$ at $\omega=0$ below (top) and above (bottom) the critical temperature $T_{c}\approx 270$ MeV. The curves are obtained using the parameters given in Tab.~II. The lattice data were taken from Ref.~~~\cite{silva}.}
\end{figure}
\begin{figure}[t] \label{fig:transpropfit}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig5a.eps}\quad\includegraphics[width=0.32\textwidth,angle=-90]{fig5b.eps}
\caption{Transverse propagator $\Delta_T$ at $\omega=0$ below (top) and above (bottom) the critical temperature $T_{c}\approx 270$ MeV. The curves are obtained using the parameters given in Tab.~II. The lattice data were taken from Ref.~~~\cite{silva}.}
\end{figure}
\begin{figure}[t] \label{fig:longpropfit458}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig6.eps}
\caption{Longitudinal propagator $\Delta_L$ for $\omega=0$, $T= 458$ MeV and different values of the gluon mass parameter. The lattice data were taken from Ref.~~~\cite{silva}.}
\end{figure}
In Figs.~4 and 5 we show, respectively, the longitudinal and transverse components of the gluon propagator at $\omega=0$ (multiplicatively renormalized at $\mu_{0}=4$~GeV), as functions of the three-dimensional momentum $|{\bf p}|=\sqrt{{\bf p}^{2}}$, with $m(T)$ and $\pi_{0}(T)$ as reported in Tab.~II. Such values where obtained by a separate fit of the two components to the lattice data of Ref.~~~\cite{silva}; the mass parameters should be understood to have an uncertainty of about $\pm 50$ MeV.
\begin{table}[h]
\def1.3{1.3}
\begin{tabular}{|c|c|c|}
\hline
$T$ (MeV)&$m(T)$ (MeV) (long., trans.)&$\pi_{0}(T)$ (long., trans.)\\
\hline
121&550, 656&$-0.89$, $-0.84$\\
194&425, 550&$-1.10$, $-0.70$\\
260&425, 450&$-1.42$, $-0.42$\\
290&275, 450&$-0.97$, $-0.48$\\
366&150, 450&$-0.60$, $-0.20$\\
458&$\ \,$//, 450&$\ \ \ \ \,$//, $\, +0.21$\\
\hline
\end{tabular}
\caption{Parameters for the curves in Figs.~4 and 5, obtained by a separate fit of the lattice data for the longitudinal and transverse gluon propagator of Ref.~~~\cite{silva}.}
\end{table}
As we can see, once the parameters are tuned to fit the data, the screened expansion is able to reproduce the lattice propagators quite accurately down to momenta of approximately $0.5$ GeV. Moreover, the longitudinal propagator still shows the characteristic non-monotonic behavior with respect to a change in the temperature, increasing at fixed momentum below $T=T_{c}\approx 270$ MeV and decreasing above $T=T_{c}$.
Below $|{\bf p}|\approx0.5$ GeV, the transverse propagator is still in good agreement with the data, while the longitudinal one shows significant deviations, especially at high temperatures. From a numerical point of view, why this is so is exemplified in Fig.~~6, where we display the longitudinal propagator for $T=458$ MeV and different values of the mass parameter\footnote{For each value of the mass parameter, the renormalization constant $\pi_{0}(T)$ was optimized so as to obtain the best fit with the data at large momenta.}. When tuning the mass parameter $m(T)$, there is a tension between the low- and intermediate-momentum behavior of the propagator: at lower values of $m$, the propagator is enhanced (resp. suppressed) below (resp. above) $|{\bf p}|\approx1$ GeV, so that achieving a good match at low momenta results in a loss of accuracy at intermediate momenta. This behavior is actually shared by both the components of the propagator and at every $T\neq 0$, albeit being less significant for the transverse component and at low temperatures. In particular, already at $T=458$~MeV the optimal longitudinal values of the mass parameter and of the renormalization constant strongly depend on the choice of a lower cutoff momentum for the fit to the lattice data; for this reason, we do not report them.
As anticipated earlier, the optimal mass parameters (and renormalization constants) needed to reproduce the lattice data differ for the two components of the propagator. In Fig.~~7 we plot the parameters of Tab.~II as functions of the temperature. With the exception of the point $T=260$ MeV, which is very close to the critical temperature $T_{c}\approx 270$ MeV, the optimal mass parameter $m(T)$ is a non-increasing function of the temperature for both the projections. When fitted from the transverse propagator, $m(T)$ shows plateaux both at small and at large temperatures, decreasing from $m(T)= m(0)=656$~MeV to $m(T)\approx 450$ MeV. As for the longitudinal propagator, except for $T=260$ MeV, $m(T)$ is approximately linear, with a behavior which is well-described by the equation
\begin{equation}
m(T)\approx 656\ \text{MeV}-1.307\ T\qquad\text{(long.)}.
\end{equation}
At $T=260$ MeV $\approx T_{c}$, the optimal value of $m(T)$ is nearly equal for both the projections, namely $m(T)=425-450$ MeV. As for the renormalization constant, except for the point at $T=290$ MeV $\approx T_{c}$, the optimal $\pi_{0}(T)$ increases with the temperature when fitted from the transverse propagator. When optimized by the longitudinal propagator, on the other hand, it shows a non-monotonic behavior, decreasing below $T_{c}$ and increasing again above $T_{c}$.
\begin{figure}[t] \label{fig:mf0}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig7a.eps}\quad\includegraphics[width=0.32\textwidth,angle=-90]{fig7b.eps}
\caption{Mass parameters (top) and renormalization constants (bottom) of Tab.~II, as extracted from the lattice data of Ref.~~~\cite{silva}.}
\end{figure}
\
The large differences in the optimal values of $m(T)$ and $\pi_{0}(T)$ obtained for the two projections make it clear that, in the present formalism, it is not possible to quantitatively recover both the longitudinal and the transverse component of the gluon propagator by a unique choice of parameters. Thus at $T\neq 0$ the screened expansion appears to be suboptimal as a ``variational'' ansatz. At least in part, this could be expected on the basis of what is known about the high-temperature, low-momentum behavior of the Yang-Mills propagators: at large temperatures and low momenta, the gluons' thermal mass is best described by a momentum- and direction-dependent Hard Thermal Loop (HTL) term in the Lagrangian, given by~\cite{braaten90b}
\begin{equation}
\Delta \mathcal{L}_{\text{HTL}}=-\frac{1}{2}\,m_{el}^{2}(T)\ \text{Tr}\left\{F_{\mu\nu}\int\frac{d\Omega}{4\pi}\frac{\hat{y}^{\nu}\hat{y}^{\lambda}}{(\hat{y}\cdot D)^{2}}\ F_{\lambda}^{\ \mu}\right\},
\end{equation}
where $m_{el}^{2}(T)=g^{2}NT^{2}/3$, $\hat{y}$ is a light-like four-vector and the integration is over the directions of $\hat{y}$. To first order in the coupling, $\Delta\mathcal{L}_{\text{HTL}}$ generates two different thermal masses for the three-dimensional projections $\Delta_{L}(p,T)$ and $\Delta_{T}(p,T)$ of the gluon propagator. By not taking into account this difference, the screened expansion lends itself to a breakdown at large temperatures, which can be partially avoided if the mass parameter and renormalization constant are tuned to separately fit the two projections.
The simplest way of solving this issue in the context of the screened expansion, i.e. without resorting to a HTL resummation, would be to change the expansion point of perturbation theory in such a way that the two three-dimensional projections of the \textit{zero-order} gluon propagator, $\Delta_{m}^{T}$ and $\Delta_{m}^{L}$, have different masses \textit{ab initio}. This can be achieved by redefining the kernel $\delta\Gamma_{\mu\nu}(p;T)=m^{2}(T)\,t_{\mu\nu}(p)$ of the shift of the action $\delta S$ as
\begin{equation}
\delta\Gamma_{\mu\nu}(p;T)\to m_{T}^{2}(T)\,P^{T}_{\mu\nu}(p)+m_{L}^{2}(T)\,P^{L}_{\mu\nu}(p),
\end{equation}
where $m_{T}(T)$ and $m_{L}(T)$ are independent mass-parameter functions for the two projections. With such a prescription, in a general covariant gauge the zero-order Euclidean gluon propagator $\Delta_{m}^{\mu\nu}(p;T)$ would read
\begin{align}\label{newshift}
\nonumber\Delta_{m}(p;T)_{\mu\nu}&\to \Delta_{m}^{T}(p;T)\ P^{T}_{\mu\nu}(p)+\Delta_{m}^{L}(p;T)\ P^{L}_{\mu\nu}(p)+\\
&\quad+\frac{\xi}{p^{2}}\ \ell_{\mu\nu}(p),
\end{align}
where
\begin{equation}
\Delta_{m}^{T,L}(p;T)=\frac{1}{p^{2}+m_{T,L}^{2}(T)}
\end{equation}
are the sought-after zero-order propagators. Setting-up the perturbation theory with independent mass functions for the two projections would give us the freedom to optimize the former separately from first principles, according to the behavior of the respective dressed propagators. Implementing the shift in Eq.~~\eqref{newshift}, however, is a non-trivial task: having different longitudinal and transverse masses running in the loops breaks the Lorentz-invariance even of the simplest vacuum integrals and, more generally, requires a complete recalculation of the gluon polarization.
\section{Dispersion relations at finite T}
Being in possession of analytical expressions (modulo a one-dimensional integration at finite $T$) for the Euclidean gluon propagator allows us to analytically continue the latter to the whole complex plane so as to study its singularities. As is well known, the location of the poles of the propagator gives us information on the dispersion relations of the gluonic quasi-particles: the energy $\varepsilon_{T,L}({\bf p},T)$ and damping rate $\gamma_{T,L}({\bf p},T)$ of the quasi-particles, as functions of the three-dimensional momentum ${\bf p}$ and of the temperature $T$, are obtained by solving the equation
\begin{equation}
\Delta_{T,L}^{-1}(-i\omega_{T,L}({\bf p},T),{\bf p},T)=0,
\end{equation}
where $\omega=\varepsilon-i\gamma$ (modulo a factor of $i$) extends the real and discrete Matsubara frequencies $\omega_{n}=2\pi n T$ to the complex plane and the subscripts $T,L$ refer to the components of the propagator. At non-zero temperatures and momenta, the poles of the two components are expected to be found at different locations, yielding two separate branches of the dispersion relations.
The limit $T\to 0$ of the dispersion relations was already studied in the framework of the screened massive expansion in Refs.~~\cite{analyt,scaling,xigauge}. In~\cite{xigauge} we found that the zero-temperature gluon propagator (whose longitudinal and transverse three-dimensional components are constrained to be equal by Lorentz simmetry) has two complex-conjugate poles at $-p^{2}=m_{\text{pole}}^{2},\,(m_{\text{pole}}^{2})^{*}$, where, setting $m_{0}=656$ MeV by
sharing the same units of the lattice,
\begin{equation}
m_{R}^{2}=0.197\ \text{GeV}^{2}\ ,\qquad m_{I}^{2}=0.436\ \text{GeV}^{2},
\end{equation}
with $m_{\text{pole}}^{2}=m_{R}^{2}+i\,m_{I}^{2}$. In terms of $\varepsilon_{\text{vac}}({\bf p})=\lim_{T\to 0}\varepsilon_{T,L}({\bf p},T)$ and $\gamma_{\text{vac}}({\bf p})=\lim_{T\to 0}\gamma_{T,L}({\bf p},T)$ -- and singling out one of the poles --, this translates into the dispersion relations
\begin{widetext}
\begin{align}\label{vacuumdisp}
\nonumber\varepsilon_{\text{vac}}({\bf p})&=\left[\frac{1}{2}\ \sqrt{({\bf p}^{2}+m_{R}^{2})^{2}+(m_{I}^{2})^{2}}+\frac{1}{2}\,({\bf p}^{2}+m_{R}^{2})\right]^{1/2}\ ,\\
\gamma_{\text{vac}}({\bf p})&=\left[\frac{1}{2}\ \sqrt{({\bf p}^{2}+m_{R}^{2})^{2}+(m_{I}^{2})^{2}}-\frac{1}{2}\,({\bf p}^{2}+m_{R}^{2})\right]^{1/2}\ .
\end{align}
\end{widetext}
Clearly, $m_{R}^{2}=(\varepsilon_{\text{vac}}^{2}-\gamma_{\text{vac}}^{2})|_{{\bf p}=0}$ and $m_{I}^{2}=2\,\varepsilon_{\text{vac}}\,\gamma_{\text{vac}}|_{{\bf p}=0}$, where
\begin{equation}
\varepsilon_{\text{vac}}({\bf 0})=581\ \text{MeV}\ ,\qquad\gamma_{\text{vac}}({\bf 0})=375\ \text{MeV}\ .
\end{equation}
At the other end of the spectrum, as $|{\bf p}|\to \infty$, the gluon's vacuum dispersion relations reduce to those of a massless particle, $\varepsilon_{\text{vac}}({\bf p})\to |{\bf p}|$, $\gamma_{\text{vac}}({\bf p})\to 0$.
Under the assumption that the optimal masses $m(T)$ and renormalization constants $\pi_{0}(T)$ reported in the previous section only depend on the temperature, and not on the Matsubara frequency $\omega_{n}$, the finite-$T$ dispersion relations of the gluon quasi-particles can be easily extracted from the screened expansion's gluon propagator, making use of said parameters (cf. Tab.~II). We remark that, since at low momenta the longitudinal projection was not found to be in good agreement with the lattice data for any value of the parameters, the longitudinal dispersion relations are expected to be reliable only at sufficiently high momenta (say above $|{\bf p}|\approx0.5-0.7$ GeV).
In Figs.~8 and 9 we plot the energy $\varepsilon_{T,L}({\bf p},T)$ and the damping rate $\gamma_{T,L}({\bf p},T)$ of the transverse and longitudinal gluons at fixed $T$, as functions of the momentum $|{\bf p}|$. As we can see, below the critical temperature $T_{c}\approx 270$~MeV both the transverse energy and the transverse damping rate (Fig.~~8) are suppressed with respect to their zero-temperature (vacuum) limit, with the effect being more pronounced for $\varepsilon_{T}$ than for $\gamma_{T}$. Above $T_{c}$ this behavior is reversed; the transverse energy starts to approach again its vacuum limit, while the damping rate grows larger than it. The longitudinal branch (Fig.~~9) shows a more significant suppression in both the energy and the damping rate below $T_{c}$, with $\gamma_{L}$ becoming quite small at high momenta around the critical temperature. At higher temperatures both $\varepsilon_{L}$ and $\gamma_{L}$ start to approach back their vacuum limit.\footnote{Here we are disregarding the low-momentum behavior of the longitudinal dispersion relations due to their lack of reliability, as previously discussed.}\\
\begin{figure*} \label{figdispersion1}
\centering
\includegraphics[width=0.30\textwidth,angle=-90]{fig8a.eps}\qquad\ \includegraphics[width=0.30\textwidth,angle=-90]{fig8b.eps}\\
\includegraphics[width=0.30\textwidth,angle=-90]{fig8c.eps}\qquad\ \includegraphics[width=0.30\textwidth,angle=-90]{fig8d.eps}\\
\includegraphics[width=0.30\textwidth,angle=-90]{fig8e.eps}\qquad\ \includegraphics[width=0.30\textwidth,angle=-90]{fig8f.eps}
\caption{Transverse dispersion relations for the gluon quasi-particles. The broken lines are the vacuum dispersion relations, common to both projections and given by Eq.~~\eqref{vacuumdisp}. The gluon mass parameters $m(T)$ and renormalization constants $\pi_{0}(T)$ used for the plots are reported in Tab.~II.}
\end{figure*}
\begin{figure*} \label{figdispersion2}
\centering
\includegraphics[width=0.30\textwidth,angle=-90]{fig9a.eps}\qquad\ \includegraphics[width=0.30\textwidth,angle=-90]{fig9b.eps}\\
\includegraphics[width=0.30\textwidth,angle=-90]{fig9c.eps}\qquad\ \includegraphics[width=0.30\textwidth,angle=-90]{fig9d.eps}\\
\includegraphics[width=0.30\textwidth,angle=-90]{fig9e.eps}
\caption{Longitudinal dispersion relations for the gluon quasi-particles. The broken lines are the vacuum dispersion relations, common to both projections and given by Eq.~~\eqref{vacuumdisp}. The gluon mass parameters $m(T)$ and renormalization constants $\pi_{0}(T)$ used for the plots are reported in Tab.~II. Except for vanishingly small temperatures, these dispersion relations are not expected to be reliable below $|{\bf p}|\approx 500-700$~MeV.}
\end{figure*}
In the limit ${\bf p}\to 0$ and for any non-zero $\omega$, the longitudinal and the transverse projection of the gluon propagator are known to collapse to a single temperature-dependent function; as a consequence, the corresponding branches of the dispersion relations share the same zero-momentum limit. The ${\bf p}=0$ poles of the gluon propagator are located at $-i(\varepsilon_{0}(T)-i\,\gamma_{0}(T))$, where
\begin{equation}
\varepsilon_{0}(T)=\lim_{|{\bf p}|\to 0}\ \varepsilon_{T,L}({\bf p},T)\ ,\quad \gamma_{0}(T)=\lim_{|{\bf p}|\to 0}\ \gamma_{T,L}({\bf p},T)
\end{equation}
are, respectively, the mass and the (zero-momentum) damping rate of the gluon quasi-particles. With regards to such a constraint, the optimized framework of Sec.~IIIB is inconsistent: using different mass parameters for the longitudinal and the transverse projections of the propagator causes the two branches of the dispersion relations to have unequal ${\bf p}\to 0$ limits. All the same, as previously discussed, the low-momentum limit of the longitudinal gluon propagator was found to be quantitatively unreliable at temperatures which are not vanishingly small. It follows that the ${\bf p}\to 0$ limit of the longitudinal dispersion relations cannot be trusted regardless of the inconsistency. Since only the screened expansion's transverse propagator, with the parameters in Tab.~II, was found to reproduce the lattice data at low momenta, in what follows we will make use of the transverse dispersion relations to study the behavior of $\varepsilon_{0}(T)$ and $\gamma_{0}(T)$. From first principles, it is understood that a good description of the long-wavelength longitudinal gluon excitations must yield the same results.
In Fig.~~10 we display the mass and the zero-momentum damping rate of the gluon quasi-particles as functions of the temperature. Across the critical temperature, both of them show a characteristic behavior, decreasing below $T_{c}$ and increasing again in a linear fashion above $T_{c}$. The mass decreases from $\varepsilon_{0}(0)=\varepsilon_{\text{vac}}({\bf 0})=581$~MeV to $\varepsilon_{0}(T_{c})\approx 450$~MeV, whereas the zero-momentum damping rate slightly decreases from $\gamma_{0}(0)=\gamma_{\text{vac}}({\bf 0})=375$~MeV to about $350$~MeV around $T_{c}$. The increase in the damping rate actually seems to start somewhat below the critical temperature (see the data point $T=260$~MeV in Fig.~~10); we could not determine whether this is a physically meaningful behavior or an artifact due to uncertainties in the parameters of Tab.~II.
The behavior of the gluon mass in Fig.~~10 confirms the picture of a confined gluon -- whose mass is dynamically generated through the strong interactions themselves like in the $T\to 0$ limit -- which becomes deconfined above the critical temperature $T_{c}\approx 270$~MeV. In the deconfined phase, the mass of the gluon is thermal in nature and increases linearly with the temperature. The same qualitative behavior was observed in~\cite{damp}, where the gluon mass and zero-momentum damping rate were studied in the screened expansion at finite $T$ using the same scheme of Sec.~IIIA, i.e.
taking temperature-independent values for both the gluon mass parameter $m$ and the renormalization constant $\pi_{0}$.
\begin{figure}[t] \label{zeromomentum}
\centering
\includegraphics[width=0.32\textwidth,angle=-90]{fig10.eps}
\caption{Mass $\varepsilon_{0}(T)$ and zero-momentum damping rate $\gamma_{0}(T)$ of the gluon quasi-particles, as functions of the temperature. The parameters used for the plot are reported in Tab.~II under the transverse denomination. See text for further details.}
\end{figure}
\section{Discussion}
The comparison with the available lattice data showed that the screened expansion gives a correct
qualitative description of the gluon propagator at finite $T$. The agreement
improves if the renormalization constants are tuned at each value of the temperature. At high temperatures and deep in the IR,
the failure to reproduce the longitudinal projection might arise from the combined effect of several issues like the need of some HTL resummation, a poor optimization and the inadequacy of the single-mass splitting of the action at a finite temperature.
Indeed, the lattice data seem to suggest that a two-mass scheme should be introduced from the beginning for
extending the screened expansion at a finite temperature.
Nonetheless, the qualitative behavior of the propagators seems to be correct and quite robust, irrespective of the optimization scheme.
The pole trajectories can be determined in the complex plane, yielding valuable predictions which cannot be
extracted from the lattice data in the Euclidean space. We have reported
in some detail the dispersion relations of the quasi-gluon for several temperatures across the deconfinement
transition.
An important feature which emerges from our study is a crossover at the deconfinement transition. The energy of the quasi-particle
is suppressed by temperature in the confined phase. On the other hand, above the critical temperature, the
behavior is reversed and the energy increases as a function of temperature. The same effect can be observed for
the physical mass, defined as the long wavelength limit $\varepsilon_{0}(T)$ of the pole's real part, as shown
in Fig.~~10. In the confined phase, the mass decreases like an order parameter being suppressed by
the temperature. This behavior is consistent with that of a dynamical mass which is related to a condensate, the latter being expected
to vanish at the transition temperature. However, at finite temperature the quasi-gluon is also expected to
acquire a thermal mass which increases linearly, like any other quasi-particle. The two effects might coexist across
the transition, yielding a crossover rather than a sharp transition. In the low-temperature limit the dynamical
nature of the mass dominates, while above the deconfinement transition the mass becomes a pure thermal mass.
Thus, we argue that in the low-temperature phase the mass suppression might be a signature of the dynamical
nature of the gluon mass. On the other hand, as discussed in Ref.~~\cite{damp}, the existence of an intrinsic
damping rate, which saturates at a finite value at $T=0$, is a confirmation of the quasi-gluon scenario laid out by Stingl~\cite{stingl}. The massive gluon also has a very short finite lifetime and is canceled
from the asymptotic states~\cite{damp}, suggesting that the gluon quasi-particles of the interacting vacuum can only travel the short
distance of about a Fermi and can only exist as intermediate states at the origin of a gluon-jet event.
\acknowledgments
The authors are in debt to Orlando Oliveira for sharing the lattice data of Ref.~~\cite{silva}.
This research was supported in part by "Piano per
la Ricerca di Ateneo 2017/2020 - Linea di intervento 2" of the University of Catania.
|
1,314,259,993,000 | arxiv | \section{Introduction}
General relativity (GR) is an essential component in the realistic modeling of core collapse supernovae because of the very
strong gravitational fields in the vicinity of the collapsed core of a star. Hydrodynamics and neutrino transport are closely
connected in this problem, and as we will show, GR can have a profound effect on each of these, especially in the critical phase
of shock reheating. The detection of neutrinos from supernova 1987A (Bionta \emph{et al.} 1987, Hirata \emph{et al.} 1987) and
the hope of detecting neutrino signatures from future supernovae, with next-generation detectors, is additional motivation for an
accurate general relativistic treatment of the neutrino transport in numerical simulations.
\begin{figure}[t]
\label{radii}
\includegraphics[angle=-90, scale=0.65]{figure1l.eps}
\caption{Shock and gain radii vs. post-bounce time for model S15s7b. Both cases are calculated with Newtonian radiation
transport.}
\end{figure}
\begin{figure}[t]
\label{rmsr}
\includegraphics[angle=-90, scale=0.65]{figure2l.eps}
\caption{RMS energy vs. radius for the $\nu_{\mathrm{e}}$'s in model S15s7b. This figure contrasts stationary state GR and
Newtonian transport at $t_{\mathrm{pb}}=114$ ms.}
\end{figure}
\begin{figure}[t]
\label{lum}
\includegraphics[angle=-90, scale=0.65]{figure3l.eps}
\caption{Luminosity vs. radius for the $\nu_{\mathrm{e}}$'s in model S15s7b. This figure contrasts stationary state GR and
Newtonian transport at
$t_{\mathrm{pb}}=114$ ms.}
\end{figure}
\begin{figure}[t]
\label{rmst}
\includegraphics[angle=-90, scale=0.65]{figure4l.eps}
\caption{RMS energy vs. post-bounce time for $\nu_{\mathrm{e}}$'s and $\bar{\nu}_{\mathrm{e}}$'s in model S15s7b.}
\end{figure}
We have developed a code for general relativistic multigroup flux-limited diffusion (MGFLD) that computes the neutrino transport
in a static background metric. This is the first step in the development of a fully GR MGFLD code. The GR metric
used is $ds^2 = a^2 c^2 dt^2 - b^2 dr^2 - r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$. This metric is allowed to evolve in the
hydrodynamics calculations, but is ``frozen'' in the transport calculations, which are then performed treating the metric as
static. Two precollapse models, a 15$\mathrm{M}_{\odot}$ model (S15s7b) and a 25$\mathrm{M}_{\odot}$ model (S25s7b) (Woosley
\& Weaver 1995; Weaver \& Woosley 1997), were evolved through core collapse, bounce, and to approximately 800 ms after bounce in
three sets of simulations. The first was with Newtonian hydrodynamics and Newtonian radiation transport, the second was with GR
hydrodynamics and Newtonian transport, and the third was with both GR hydrodynamics and transport. In addition to these
three sets of simulations, stationary-state neutrino distributions were computed for various post-bounce time slices in these
models in order to isolate the effects associated with each of the metric components.
\section{The Role of General Relativity}
The effects of GR are seen quite clearly in the hydrodynamic evolution of the initial models. GR hydrodynamics produces a much
more compact post-bounce structure than Newtonian hydrodynamics. After $t_{\mathrm{pb}} = 0.4$ s in both models ($t_{\mathrm{pb}}$ being the
post-bounce time), the radius of the shock and the gain radii are reduced by a factor of 2 in the GR calculations, as shown in
Figure 1.
Also strongly affected by GR is the flow velocity between the shock and the proto-neutron star. Because matter falls through a
greater potential well to reach the shock in the GR calculation, GR preshock and therefore postshock velocities are larger than
their Newtonian counterparts, again by a factor of approximately 2.
The main effect of GR on the neutrino rms energies is the redshift of the neutrinos after they decouple from the matter, which
is governed by the metric parameter $a$. The rms energies are reduced by a factor of $a$ evaluated at the $\nu$-sphere. A
smaller effect is a slight outward shift of the $\nu$-sphere resulting from the (non-unity) value of the metric parameter
$b^{-1}$, which causes the neutrinos to decouple outside the $\nu$-sphere at a lower temperature. These effects are shown in
Figure 2. Also shown are the independent effects of $a$ and $b$ on the neutrino transport.
GR reduces the neutrino luminosity by three effects: redshift, governed by the metric parameter $a$; the difference in the local
clock rates at the emission surface and the observer radius, also governed by the metric parameter $a$; and the reduction of the
neutrino flux, governed by the metric parameter $b^{-1}$. All three of these effects are of roughly equal magnitude
and reduce the luminosities by a total factor of $a^2 b^{-1}$ evaluated at the $\nu$-sphere, as shown in Figure 3. This figure
also shows the independent effects of $a$ and $b$ on the stationary-state neutrino transport.
The shock heating rate is proportional to the product of the luminosity and the square of the $\nu_{\mathrm{e}}$ rms energy.
This means that any percentage change in these quantities add together. Looking at Figures 2 and 3, we see a decrease of 8\% in
the $\nu_{\mathrm{e}}$ luminosity and a 3\% reduction in the $\nu_{\mathrm{e}}$ rms energy. These combine to give a 14\%
reduction in the heating rate. Similar results are obtained for the $\bar{\nu}_{\mathrm{e}}$'s.
These are significant differences [\emph{e.g.}, see Burrows \& Goshy (1993), Janka \& M\"uller (1996), Mezzacappa \emph{et al.}
(1998), and Messer \emph{et al.} (1998)] and serve to illustrate the point that modeling core collapse supernovae without GR
hydrodynamics and transport leads to results that cannot be interpreted as realistic. For more information on this subject, the
reader is referred to Bruenn, De~Nisco, \& Mezzacappa (1998).
\section{Nucleosynthesis}
r-process nucleosynthesis is believed to occur in a neutrino-driven wind emanating from the proto-neutron star after the
successful launch of the shock. The r-process yields are a function of the $\nu_{\mathrm{e}}$ and $\bar{\nu}_{\mathrm{e}}$
luminosities and rms energies. The luminosities affect the entropy, mass loss rate, and expansion time scale associated with the
wind, and the rms energies determine the neutronization of the wind. As an example of the impact of GR on the r-process,
consider its effect on the rms energies. Because the $\bar{\nu}_{\mathrm{e}}$-sphere lies below the
$\nu_{\mathrm{e}}$-sphere, the $\bar{\nu}_{\mathrm{e}}$'s suffer a greater emergent redshift. As can be seen in Figure 4, general
relativistic transport and hydrodynamics affects the ratio of the $\nu_{\mathrm{e}}$ and $\bar{\nu}_{\mathrm{e}}$ rms energies.
This differential redshifting affects, in turn, the ratio of the number of $\nu_{\mathrm{e}}$'s to
$\bar{\nu}_{\mathrm{e}}$'s, at a given neutrino energy, and therefore, the neutronization of the wind. This suggests that
general relativistic hydrodynamics and transport will be required to obtain accurate r-process yields.
|
1,314,259,993,001 | arxiv | \section{Introduction} \label{Sec0}
A large number of modern astrophysical observations suggest the existence of
scalar fields in our Universe as possible candidates for dark matter.
Pulsons are localized configurations of the fields having oscillating energy
density. Numerical simulations of Seidel and Suen \cit
{{Seidel-Suen1},*{Seidel-Suen2}} have revealed the existence of long-lived
self-gravitating pulsons, so-called oscillating soliton stars or
oscillatons, in the Einstein-Klein-Gordon (EKG) syste
\begin{eqnarray}
R_{\mu \nu }-\frac{1}{2}Rg_{\mu \nu } &=&\varkappa \left[ \phi _{,\mu }\phi
_{,\nu }-\left( \frac{1}{2}\phi _{,\alpha }\phi ^{,\alpha }-U(\phi )\right)
g_{\mu \nu }\right] , \nonumber \\
\phi _{;\alpha }^{;\alpha }+U^{\prime }(\phi ) &=&0 \label{eq1}
\end{eqnarray
with the potential $U(\phi )=(m^{2}/2)\phi ^{2}$ corresponding to a free
massive scalar field. The authors have established that soliton stars can be
formed from rather general initial field distributions due to specific
relaxation process, the gravitational cooling.
Pulsons were first observed numerically by Bogolubsky and Makhankov \cit
{{Bog-Makh1},*{Bog-Makh2}} in the Klein-Gordon (KG) model with $\phi ^{4}$
and $sG$ potentials. In these cases, in the absence of gravity, the formation of
the pulsons occurs solely due to self-coupling effects. In the present-day
literature such configurations are often called oscillons, but below we shall use
their original name, pulsons \cite{Bog1}.
Subsequent investigations have shown that pulsons exist in various models and
spatial dimensions, and that they evolve from the diversity of initial conditions \cit
{Marques, Bog2, Olsen, Geicke, Maslov1, Gleiser1, Copeland, Maslov2, Piette,
Hormuz, Maslov3, Dymnik, Choptuik, Urena, Gleiser2, Kasuya, Koutv1, Koutv2,
Gleiser3} (see \cite{Fodor1}\ for a review). It turns out that pulsons can
arise from both uniform and non-uniform field distributions. Thus pulsons
can emerge in scalar condensates due to the parametric instability of the
spatially uniform background oscillating near a vacuum value \cite{Maslov3,
Gleiser2, Kasuya, Koutv2}. In this case the energy of the background
oscillations is transferred to an incipient pulson via the resonance
mechanism. Quite a different scenario is realized when pulsons are formed
from localized field distributions that appear, e.g., in shrinking
cylindrical domain walls \cite{Geicke}, in collapsing spherical bubbles \cit
{Gleiser1, Copeland}, or at bubble collisions \cite{Dymnik}. In such a case
an initial field lump sheds excessive energy by radiation of scalar waves
(gravitational cooling of the soliton stars) and settles into a quasi-stable
state, the pulson, whose lifetime depends strongly on the initial
conditions. This suggests the existence of such initial conditions that
evolve into very long-lived quasi-periodic, or even infinitely long-lived
periodic pulsons. The latter would imply\ the existence of exact localized
time-periodic solutions. For the $\phi ^{4}$, $\phi ^{3}-\phi ^{4}$, and $sG$
models, certain of these initial configurations have been found numerically
in \cite{Copeland, Piette, Hormuz, Choptuik}. Recently, in Ref. \cite{Fodor2}
small amplitude pulson solutions of the EKG system have been obtained for
the potentials expansible in a power series. This brings up the following question:
How does gravity affect the dynamics of the finite amplitude pulsons? For example,
could gravity turn non-periodic pulson solutions into periodic ones?
Consideration of finite amplitude pulsons takes on great significance in the
case where a scalar field potential is not expansible in a power series in
the small amplitude limit.
In this paper we search for pulsons in the{\Large \ }EKG system (\ref{eq1})
with the potentia
\begin{equation}
U(\phi )=\frac{m^{2}}{2}\phi ^{2}\left( 1-\ln \frac{\phi ^{2}}{\sigma ^{2}
\right) , \label{eq2}
\end{equation
where $\phi$ is a real scalar field, $m$ is a bare mass (in units $\hbar =c=1$), and $\sigma $
is a characteristic amplitude of the field which is assumed to be finite, but not too
large, so that $\varkappa \sigma ^{2}\ll 1$, where $\varkappa$ is the gravitational constant.
The nonlinear KG equation with
the logarithmic potential (\ref{eq2}) was first considered in quantum
field theory by Rosen \cite{Rosen} and later by Bialynicki-Birula and
Mycielski \cite{BBM}. In general, for the nonlinear KG equation the
potential (\ref{eq2}) is the only one which permits real solutions of the
form $\phi =a(t)u(\mathbf{r})$ to exist \cite{Maslov1}. Such singular
potentials currently appear in inflationary cosmology \cite{Barrow} and in
some supersymmetric extensions of the standard model (flat direction
potentials in the gravity mediated supersymmetric breaking scenario) \cit
{Enqvist}. The logarithmic term in parentheses arises due to quantum
corrections to the bare inflaton mass.
The paper is organized as follows. In Sec.~\ref{Sec2}, using the smallness of the
gravitational constant, we obtain the approximate solution of the EKG system (\ref{eq1})
which describes time-periodic pulsons of a finite amplitude in the
Schwarzschild metric $ds^{2}=Bdt^{2}-Adr^{2}-r^{2}(d\vartheta ^{2}+\sin
^{2}\vartheta \;d\varphi ^{2})$. In Sec.~\ref{Sec3} we use the obtained solution to find
the initial conditions for direct numerical integration of the system.
We show that these initial conditions do evolve into a very long-lived
periodic pulson. Stability of the self-gravitating pulsons and their possible
astrophysical meaning are briefly discussed in Sec.~\ref{Sec4}.
\section{Solution} \label{Sec2}
After the scaling $t\rightarrow t/m,\;r\rightarrow r/m,\;\phi /\sigma
\rightarrow \phi ,$ $\varkappa \sigma ^{2}/2\rightarrow \varkappa ,$ the
system (\ref{eq1}) takes the for
\begin{equation}
\frac{A_{r}}{A}+\frac{A-1}{r}=\varkappa r\left( \frac{A}{B}\phi
_{t}^{2}+\phi _{r}^{2}+A\phi ^{2}(1-\ln \phi ^{2})\right) , \label{eq3}
\end{equation
\begin{equation}
\frac{B_{r}}{B}-\frac{A-1}{r}=\varkappa r\left( \frac{A}{B}\phi
_{t}^{2}+\phi _{r}^{2}-A\phi ^{2}(1-\ln \phi ^{2})\right) , \label{eq4}
\end{equation
\begin{equation}
\frac{A}{B}\phi _{tt}-\phi _{rr}-\frac{2}{r}\phi _{r}+\left( \frac{A}{2B
\right) _{t}\phi _{t}+\frac{B}{2A}\left( \frac{A}{B}\right) _{r}\phi
_{r}=A\phi \ln \phi ^{2}, \label{eq5}
\end{equation
where $\varkappa \ll 1$ is the rescaled gravitational constant. Looking for
localized solutions, we impose the boundary conditions $\phi (t,\infty
)=0,\;A(t,\infty )=1,\;B(t,\infty )=1,\;\phi _{r}(t,0)=0,\;A(t,0)=1.$
If we set $\varkappa =0$, from (\ref{eq3})-(\ref{eq5}) we immediately obtai
{\Large \ }$A=B=1$ and arrive at the nonlinear{\Large \ }Klein-Gordon
equation
\begin{equation}
\phi _{tt}-\phi _{rr}-(2/r)\phi _{r}-\phi \ln \phi ^{2}=0. \label{eq5a}
\end{equation}
This equation has a whole family of exact pulson solutions \cite{Marques,
Bog2, Maslov1}. The simplest of them is given b
\begin{equation}
\phi (t,r)=a(t)e^{(3-r^{2})/2}, \label{eq6}
\end{equation
where $a(t)$ satisfies the equation of a nonlinear oscillator
\begin{equation}
a_{tt}=-dV(a)/da,\quad V(a)=(a^{2}/2)(1-\ln a^{2}). \label{eq7}
\end{equation
As is clear from the shape of the potential $V(a)$ depicted in Fig.~\ref{Fig-1}, oscillations are
possible in the range $-1<a(t)<1$, so we shall consider below that the
pulson's amplitude may be finite, $\left\vert \phi \right\vert \lesssim O(1)$.
\begin{figure}[htb]
\includegraphics[width=0.35\textwidth]{Fig1.eps}
\caption{The shape of the potential $V(a)$. Initial conditions
for the nonlinear oscillator (\ref{eq7}) are $a(0)=a_{max},\ a_{t}(0)=0$. }
\label{Fig-1}
\end{figure}
For small $\varkappa \ll 1$\ we construct the Krylov-Bogoliubov-type
asymptotic expansion (see, e.g., \cite{Nayfeh}) near the non-gravitating
pulson,
\begin{eqnarray}
\phi (t,r) &=&\left[ a(\theta )+\varkappa Q(\theta ,r)+O(\varkappa ^{2}
\right] e^{(3-r^{2})/2}, \label{eq8} \\
\theta _{t} &=&1+\varkappa \Omega +O(\varkappa ^{2}), \label{eq9}
\end{eqnarray
where $a(\theta )$\ satisfies Eqs. (\ref{eq7}), with the phase $\theta $\
instead of $t$, and the initial conditions $a(0)=a_{\max }<1$, $a_{\theta
}(0)=0$. The function $Q(\theta ,r)$ and the constant $\Omega $ to be found
describe the deviation of the pulson's shape from the Gaussian one and the
frequency shift $\delta \omega /\omega =\varkappa \Omega $ due to
gravitational effects.
Setting in Eqs. (\ref{eq3}), (\ref{eq4}
\begin{equation}
A(t,r)=(1-r_{g}/r)^{-1},\quad B(t,r)=(1-r_{g}/r)e^{-s} \label{eq9a}
\end{equation
and using (\ref{eq8}), (\ref{eq9}), we fin
\begin{eqnarray}
r_{g}(t,r) &=&\varkappa \int_{0}^{r}\left( \frac{1}{B}\phi _{t}^{2}+\frac{1}
A}\phi _{r}^{2}+\phi ^{2}(1-\ln \phi ^{2})\right) r^{2}dr \nonumber \\
&=&\varkappa \left[ V_{\max }\left( (\sqrt{\pi }/2)e^{r^{2}}\mathrm{erf
\,r-r\right) -a^{2}r^{3}\right] e^{3-r^{2}} \nonumber \\
&&+O(\varkappa ^{2}), \label{eq10} \\
s(t,r) &=&2\varkappa \int_{r}^{\infty }\left( \frac{A}{B}\phi _{t}^{2}+\phi
_{r}^{2}\right) r\,dr \nonumber \\
&=&\varkappa \left( 2V_{\max }+a^{2}\ln a^{2}+a^{2}r^{2}\right) e^{3-r^{2}}
\nonumber \\
&&+O(\varkappa ^{2}), \label{eq11}
\end{eqnarray
where $a=a(\theta (t))$, $V_{\max }=V(a_{\max })$. Substituting (\ref{eq8})
into (\ref{eq5}) leads to the equation for $Q(\theta ,r)$
\begin{equation}
Q_{\theta \theta }-Q_{rr}+(2/r)(r^{2}-1)Q_{r}-(2+\ln a^{2})Q=S(a,r),
\label{eq11a}
\end{equation
wher
\begin{eqnarray}
S(a,r) &=&a\{V_{\max }[\sqrt{\pi }(2-r^{2}-\ln a^{2})(2r)^{-1}e^{r^{2}
\mathrm{erf}\,r \nonumber \\
&&+\,3r^{2}-4-3\ln a^{2}] -5a^{2}r^{2}+2a^{2}r^{4} \nonumber \\
&&+\,a^{2}-2a^{2}\ln ^{2}a^{2}\}e^{3-r^{2}}-2\Omega
a\ln a^{2}. \label{eq11b}
\end{eqnarray
Its solution is given b
\begin{equation}
Q(\theta ,r)=\frac{1}{r}\sum_{n=0}^{\infty }c_{n}X_{n}(\theta )H_{2n+1}(r),
\label{eq12}
\end{equation
where $c_{n}=\pi ^{-1/4}\left[ 2^{2n+1}\left( 2n+1\right) !\right] ^{-1/2}$,
and $H_{2n+1}(r)$ are Hermite polynomials. The functions $X_{n}(\theta )$ must
satisfy the non-homogeneous singular Hill's equatio
\begin{equation}
X_{n_{\,\scriptstyle{\theta \theta }}}+\left( E-2-\ln a^{2}\right)
X_{n}=f_{n}(a), \label{eq13}
\end{equation
where $E=E_{n}=4n$
\begin{equation}
f_{n}(a)=2c_{n}\int_{0}^{\infty}S(a,r)H_{2n+1}(r)e^{-r^{2}}r\,dr. \label{eq13a}
\end{equation
The calculation give
\begin{eqnarray}
f_{0}(a) &=&D_{0}(a)-\sqrt{2}\pi ^{1/4}\Omega a\ln a^{2}, \label{eq14} \\
f_{n}(a) &=&D_{n}(a)\quad (n=1,2,...), \label{eq15} \\
D_{n}(a) &=&\frac{(-1)^{n}(2n)!c_{n}}{2^{n+4}(2n-1)n!}\sqrt{\frac{\pi }{2}
e^{3} \nonumber \\
&&\times \{a^{3}(4n^{2}-1)(4n^{2}-4n-7-16\ln ^{2}a^{2}) \nonumber \\
&&-2V_{\max }a[24n^{3}+20n^{2}-46n-1 \nonumber \\
&&+4(2n-1)(6n+5)\ln a^{2}]\}. \label{eq16}
\end{eqnarray
Note that $f_{n}(a)$ is a $T$-periodic function of $\theta $, while $\ln
a^{2}$ on the left hand side of Eq. (\ref{eq13}) is a $T/2$-periodic one, where $T$
is a period of $a(\theta )$.
Solutions of the homogeneous singular Hill's equation were investigated in
Ref. \cite{Koutv2}. In accordance with the Floquet theory (see, e.g., \cite{Whitt})
Eq. (\ref{eq13}) with $f_{n}=0$ has two linearly
independent solutions of the form $\varphi (\theta )e^{\mu \theta }$ and
\varphi (-\theta )e^{-\mu \theta }$, where $\mu $ is a characteristic
exponent, and $\varphi (\theta )$ is a $T$-periodic ($T/2$-periodic or $T/2
-antiperiodic) function. Obviously, we can set $\varphi (0)=1$. Let $X^{\pm
}\left( \theta \right) $ be two solutions of the homogenious Eq. (\ref{eq13}) (with $f_{n}=0$)
satisfying the conditions $X^{+}\left( 0\right) =1$, $X_{_{\,\scriptstyle
\theta }}}^{+}\left( 0\right) =0$, $X^{-}\left( 0\right) =0$, $X_{_{\
\scriptstyle{\theta }}}^{-}\left( 0\right) =1$. They can be written a
\begin{eqnarray}
X^{+}\left( \theta \right) &=&\frac{1}{2}\left[ \varphi (\theta )e^{\mu
\theta }+\varphi (-\theta )e^{-\mu \theta }\right] , \label{eq16a} \\
X^{-}\left( \theta \right) &=&\frac{1}{2\left( \mu +\varphi _{_{\
\scriptstyle{\theta }}}(0)\right) }\left[ \varphi (\theta )e^{\mu \theta
}-\varphi (-\theta )e^{-\mu \theta }\right] . \label{eq16b}
\end{eqnarray
If $\left| X^{+}\left( T/2\right) \right| >1$, we have the resonance case:
\mu >0$ and is determined by the equation $\cosh (\mu T/2)=\left|
X^{+}\left( T/2\right) \right| $, $\varphi (\theta )$ is a real $T/2
-periodic or $T/2$-antiperiodic function, and hence oscillations of $X^{\pm
}\left( \theta \right) $ grow exponentially with $\theta $. If $\left|
X^{+}\left( T/2\right) \right| <1$, we have the non-resonance case: $\mu
=i\nu $, and $\varphi (\theta )$ is a complex $T/2$-periodic function such that
\varphi ^{\ast }(\theta )=\varphi (-\theta )$. Hence the solutions $X^{\pm
}\left( \theta \right) $ are bounded. They can be periodic (with some
period), or non-periodic depending on $\nu $, which is determined by $\cos (\nu
T/2)=X^{+}\left( T/2\right) $. These cases are realized in different domains
of the $(E,a_{\max }^{2})$ plane that make up a stability-instability chart.
The domains with $\mu >0$ are known as resonance zones. The special case
\left| X^{+}\left( T/2\right) \right| =1$ is realized on their boundaries
where $\mu =0$. Then one of the solutions, either $X^{+}\left( \theta \right) $ or
$X^{-}\left( \theta \right) $,
is a $T$-periodic ($T/2$-periodic or $T/2$-antiperiodic) function, and
another one is proportional to the product of this function times $\theta $
plus some $T$-periodic function ($T/2$-periodic or $T/2$-antiperiodic,
respectively).
The surface $\mu (E,a_{\max }^{2})$ over the resonance zones has been
constructed in Ref. \cite{Koutv2}. For discrete $E=4n$ the above functions
acquire the subscript $n$, so we shall write $\varphi _{n}(\theta )$,
X_{n}^{\pm }\left( \theta \right) $, $\mu _{n}(a_{\max }^{2})$.
Each cross-section of the surface $\mu (E,a_{\max }^{2})$ with the plane $E=4n,\ n=1,2,...$,
gives the characteristic exponent $\mu_{n}$ as a function of $a_{\max }^{2}$.
This function is represented by a series of peaks $\mu_{n}>0$ separated
by intervals of stability.
By superposing the curves $\mu_{n}(a_{\max }^{2})$
for all considered modes $n=1,2,...,N$, one gets the pattern shown in Fig.~\ref{Fig0}.
The mode $n=0$ corresponds to the above special case $\mu=0$ and thus does not contribute
to the pattern.
\begin{figure}[htb]
\includegraphics[width=0.4\textwidth]{Fig2.eps}
\caption{A collection of the resonance peaks obtained by superposition of the functions $\mu_n(a_{max}^ 2)$.}
\label{Fig0}
\end{figure}
The obtained composite plot gives an idea of the existence of unstable and (quasi)stable modes
in different regions of the $a_{\max }^{2}$ axis and demonstrates the tendency
to progressively fill the interval $0<a_{\max }^{2}<1$ by the resonant peaks
as the successively higher energy levels $E_{n}=4n$ are accounted for.
In terms of $X_{n}^{\pm }\left( \theta \right) $ the general solution of Eq.
(\ref{eq13}) is written a
\begin{eqnarray}
X_{n}\left( \theta \right) &=&\left( X_{n}\left( 0\right) -\int_{0}^{\theta
}X_{n}^{-}f_{n}\,d\theta \right) X_{n}^{+}\left( \theta \right) \nonumber
\\
&&\hspace{-10pt}+\left( X_{n_{\,\scriptstyle{\theta }}}\left( 0\right)
+\int_{0}^{\theta }X_{n}^{+}f_{n}\,d\theta \right) X_{n}^{-}\left( \theta
\right) . \label{eq17}
\end{eqnarray
The solutions $X_{0}^{\pm }\left( \theta \right) $\ have the for
\begin{eqnarray}
X_{0}^{+}\left( \theta \right) &=&\xi q^{-1/3}(\xi ^{2})-\xi _{\theta
}\int_{0}^{\theta }K(\xi ^{2})\,d\theta , \label{eq18} \\
X_{0}^{-}\left( \theta \right) &=&-(\omega _{0}^{2}-1)^{-1}\xi _{\theta },
\label{eq19}
\end{eqnarray
where$\;$the notations $\xi (\theta )=a/a_{\max }$, $\omega
_{0}^{2}=1-\ln a_{\max }^{2}$ are introduced
\begin{equation}
\xi _{\theta }^{2}=(\omega _{0}^{2}-1)(1-\xi ^{2})q^{-2/3}(\xi ^{2})
\label{eq19a}
\end{equation
is the first integral of Eq. (\ref{eq7}) in terms of $\xi (\theta )
, and the functions $q(\xi ^{2})$\ and $K(\xi ^{2})$\ ar
\begin{eqnarray}
q(\xi ^{2}) &=&\left( \frac{\omega _{0}^{2}-1}{\omega _{0}^{2}+(1-\xi
^{2})^{-1}\xi ^{2}\ln \xi ^{2}}\right) ^{3/2}, \label{eq20} \\
K(\xi ^{2}) &=&\frac{1-q(\xi ^{2})}{1-\xi ^{2}}q^{-1/3}(\xi ^{2}).
\label{eq21}
\end{eqnarray
Note that $0<\left( 1-\omega _{0}^{-2}\right) ^{3/2}\leqslant q(\xi
^{2})\leqslant 1$, $dq/d\xi ^{2}>0\;(\xi ^{2}\leqslant 1)$, and $q(1)=1$.
Since $K(\xi ^{2})$ is a sign-definite periodic
function of $\theta $, its average $\overline{K
=T^{-1}\int_{0}^{T}K\,d\theta \neq 0$, so the solution (\re
{eq18}) can be represented in the form $X_{0}^{+}\left( \theta \right) =
\overline{K}\xi _{\theta }\theta +\psi (\theta )$, where $\psi
(\theta )$ is a $T/2$-antiperiodic function with
\overline{\psi (\theta )}=0$ [here and elsewhere the bar means
the average over the period $T$ of $a(\theta )$]. Thus
oscillations of $X_{0}^{+}\left( \theta \right) $ grow linearly
with $\theta $ for any $a_{\max }$. This is in agreement
with the fact that $X_{0}^{+}\left( T/2\right) =-1$ and the line
E=0$ is the boundary of a resonance zone on the $(E,a_{\max }^{2})$
plane \cite{Koutv2}. The equality $X_{0}^{+}\left( T/2\right) =-1$
immediately follows from Eq. (\ref{eq18}) if one takes into account that
\xi (T/2)=-1$, $\xi _{\theta }(T/2)=0$ (see Fig.~\ref{Fig-1}).
The requirement of boundedness of the general solution $X_{0}\left(
\theta \right)\ $(\ref{eq17}) determines $\Omega $ and,
hence, the frequency shift in accordance with Eq. (\ref{eq9}). Indeed,
substituting $X_{0}^{+}\left( \theta \right) $, $X_{0}^{-}\left(
\theta \right)$, and $f_{0}(a)$ into Eq. (\ref{eq17}) and
integrating by parts, we find that the linearly growing terms cancel out i
\begin{equation}
\Omega =\frac{\sqrt{2}}{\pi ^{1/4}a_{\max }}\left( \frac{\overline
X_{0}^{+}\left( \theta \right) D_{0}(a)}}{\ln a_{\max }^{2}}-X_{0}\left(
0\right) \overline{K}\right). \label{eq22}
\end{equation
Under the condition (\ref{eq22
) the solution $X_{0}\left( \theta \right) $ is a bounded $T$-periodic
function.
To obtain the corresponding conditions for $n\geqslant 1$, we substitute (\re
{eq16a}), (\ref{eq16b}) into (\ref{eq17}) and require that $X_{n}\left(
\theta \right) =X_{n}\left( \theta +T\right) $. In this equality the
integrals between the limits $0$ and $\theta $ cancel
out. The remaining terms make up a linear combination of the independent
solutions $\varphi _{n}(\theta )e^{\mu _{n}\theta }$ and $\varphi
_{n}(-\theta )e^{-\mu _{n}\theta }$. Equating to zero coefficients
of these solutions and using the identitie
\begin{eqnarray}
e^{-\mu _{n}T/2}\overline{\varphi _{n}(\theta )e^{\mu _{n}\theta }f_{n}(a)}
&=&e^{\mu _{n}T/2}\overline{\varphi _{n}(-\theta )e^{-\mu _{n}\theta
}f_{n}(a)} \nonumber \\
&=&\frac{\overline{X_{n}^{+}\left( \theta \right) f_{n}(a)}}{\cosh \left(
\mu _{n}T/2\right) }, \label{eq22a}
\end{eqnarray
\begin{equation}
\mu _{n}+\varphi _{n_{\,\scriptstyle{\theta }}}(0)=\frac{X_{n_{\,\scriptstyl
{\theta }}}^{+}\left( T\right) }{\sinh \left( \mu _{n}T\right) },
\label{eq22b}
\end{equation
we arrive at the condition
\begin{eqnarray}
X_{n}\left( 0\right) &=&-\frac{T}{X_{n_{\,\scriptstyle{\theta }}}^{+}\left(
T\right) }\overline{X_{n}^{+}\left( \theta \right) f_{n}(a)}, \label{eq23}
\\
X_{n_{\,\scriptstyle{\theta }}}\left( 0\right) &=&0. \label{eq23a}
\end{eqnarray
Note that $X_{n_{\,\scriptstyle{\theta }}}^{+}\left( T\right) \neq 0$
because $\mu _{n}+\varphi _{n_{\,\scriptstyle{\theta }}}(0)$ in (\ref{eq22b})
is proportional to the Wronskian $W\left( \varphi _{n}(\theta )e^{\mu
_{n}\theta },\ \varphi _{n}(-\theta )e^{-\mu _{n}\theta }\right) $. Equation (\re
{eq22b}) can be easily derived if one expresses $\varphi _{n}(\theta )$ from
\ref{eq16a}) in terms of $X_{n}^{+}\left( \theta \right) $ and
X_{n}^{+}\left( \theta +T\right) $ and takes into account that
X_{n}^{+}\left( T\right) =\cosh \left( \mu _{n}T\right) $.
Interestingly, Eq. (\ref{eq23}) is still valid on the boundaries of
resonance zones, Eq. (\ref{eq23a}) being no longer necessary. In particular,
this is true for $n=0$. Indeed, differentiation of (\ref{eq18}) gives
X_{0_{\,\scriptstyle{\theta }}}^{+}\left( T\right) =-T\overline{K}\ln
a_{\max }^{2}$. To calculate $\overline{X_{0}^{+}\left( \theta \right)
f_{0}(a)}$ we substitute $a\ln a^{2}=a_{\theta \theta
}=a_{\max }\ln a_{\max }^{2}X_{0_{\,\scriptstyle{\theta }}}^{-}\left( \theta
\right) $ in (\ref{eq14}) and, integrating by parts, take into account that $W\left(
X_{0}^{+}\left( \theta \right),\ X_{0}^{-}\left( \theta \right) \right) =1$.
As a result, we arrive at the condition (\ref{eq22}) again.
Thus, under the conditions (\ref{eq22}), (\ref{eq23}), (\ref{eq23a}) the
solution (\ref{eq12}) is $T$-periodic with respect to $\theta $. This means
the solution (\ref{eq8}) is also periodic [with the period
(1+\varkappa \Omega )^{-1}T$ with respect to $t$]. Note it involves the free
parameters $a_{\max }$, $X_{0}\left( 0\right) $, and $X_{0\,_{\scriptstyle
\theta }}}(0)$.
To be certain that the obtained solution is correct, we examine the mass
conservation law. The mass of a self-gravitating field lump is defined as
M=4\pi \int_{0}^{\infty }T_{0}^{0}r^{2}dr$, where $T_{0}^{0}$ is the energy
density of the scalar field involved in the EKG system (\ref{eq1}). In terms
of the rescaled variables it can be written as $M=(2\pi \sigma
^{2}/m)\lim_{r\rightarrow \infty }(r_{g}(t,r)/\varkappa )$, where $r_{g}(t,r)
$ is defined in (\ref{eq10}), $\varkappa $ being the rescaled gravitational
constant. This limit must be time independent. To check this, we substitute
the solution (\ref{eq8}) into (\ref{eq10}) and calculate the limit of
r_{g}/\varkappa $ in the first order in $\varkappa $ using the orthogonality of
the Hermite polynomials. The result is given b
\begin{eqnarray}
M&=&\frac{\left( e\sqrt{\pi }\right) ^{3}\sigma ^{2}V_{\max }}{m}\Biggl\{ 1-\varkappa \frac{\sqrt{2}a_{\max }}{\pi ^{1/4}V_{\max }}\biggl[ X_{0}\left(
0\right) \ln a_{\max }^{2} \nonumber \\
&-&\frac{e^{3}\pi ^{1/4}}{128}a_{\max }^{3}\left(1+14\ln a_{\max }^{2}\right) \biggr]+O(\varkappa ^2)\ \Biggr\} , \label{eq23b}
\end{eqnarray
which is evidently constant.
Since $\varkappa \ll 1$, the gravitational field created by this mass is weak,
as is clearly seen from (\ref{eq9a})-(\ref{eq11}). In the limit $\varkappa
\rightarrow 0$\ the gravity vanishes. However, the rescaled scalar field
persists, satisfying Eq. (\ref{eq5a}), and its amplitude may have any value
in the range $0<a_{\max }<1$. As $a_{\max }$ changes from unity to zero, the
pulson's frequency changes from zero to infinity, correspondingly. In
particular, in the small amplitude limit the pulson's frequency is $\omega
_{0}=\left( 1-\ln a_{\max }^{2}\right) ^{1/2}$.
Thus, we have obtained a three-parametric family of the spatially localized
time-periodic solutions (\ref{eq8})-(\ref{eq11}) of the system (\ref{eq3})-
\ref{eq5}), wherein only the smallness of the rescaled gravitational constant
\varkappa \;$has been. Note that the smallness of the pulson's
amplitude, $a_{\max }\ll 1,$\ is not assumed in the above consideration. To
our knowledge, this is the first example of the pulson solutions of the EKG
system, that have an arbitrary frequency. We have named them
gravipulsons.\bigskip
\section{Numerical simulation} \label{Sec3}
Our solution, however, is an approximate one. It was obtained in the first
order in the gravitational constant. Hence its deviation from an exact solution
increases in time, as happens with any asymptotic solution in the theory of
nonlinear oscillations \cite{Nayfeh}. But we can go back in time and take
the initial state of the obtained solution as initial conditions for direct
numerical integration of the starting EKG system. As a result, we have a
three-parametric family of the initial conditions
\begin{eqnarray}
\phi (0,r) &=&a_{\max }e^{(3-r^{2})/2}+\varkappa G(r;a_{\max },X_{0}\left(
0\right) ), \label{eq24a} \\
\phi _{t}(0,r) &=&2\varkappa c_{0}X_{0\,_{\scriptstyle{\theta }}}(0)e^{(3-r^{2})/2}. \label{eq24}
\end{eqnarray
The function $G(r;a_{\max },X_{0}\left( 0\right) )=Q(0,r)e^{(3-r^{2})/2}$
describes admissible deformations of the initial pulson's profile which
evolve into periodic solutions. In calculating $G$ we assume that $a_{\max }$
belongs to one of the intervals of quasi-stability \cite{Koutv1, Koutv2}
where $X_{n}^{\pm }\left( \theta \right) $, with $1\leqslant n\leqslant N$,
are bounded for sufficiently large $N$. This can be easily inspected by
numerical integration of the Hill's equation, taking into account that
the boundedness of $X_{n}^{\pm }\left( \theta \right) $ is equivalent to the
condition $\left| X_{n}^{+}\left( T/2\right) \right| <1$. We restrict
ourselves to the summation from $0$ to $N$ in (\ref{eq12}). In deciding on
N $, it is necessary to take into account that the related error in $Q$ must
not exceed $O(\varkappa )$. Below we take $\varkappa =0.005$, $N=9$, and set
$X_{0}\left( 0\right) =X_{0\,_{\scriptstyle{\theta }}}(0)=0$ for simplicity.
Figure \ref{Fig1} shows the examples of the admissible deformations calculated
for three different values of $a_{\max }$.
\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{Fig3.eps}
\caption{Admissible deformations of the initial pulson's profile calculated
by formula (\ref{eq12}) with $X_{0}(0)=0$ and $X_{n}(0)$ (\ref{eq23}).}
\label{Fig1}
\end{figure}
In Figs. \ref{Fig2} and \ref{Fig3} we compare our solution (\ref{eq8})-
\ref{eq11}) (solid lines) with the results of direct numerical integration
of the EKG system (indicated by dots). We started with one of the
admissible deformations of the pulson's profile that we have found (see Fig. \re
{Fig1}).
Oscillations of the scalar field and metric at the center of the pulson are
shown in Fig.~{\ \ref{Fig2}}. Figure \ref{Fig3} shows the pulson's and metric's
profiles taken in some intermediate moment of time.
\begin{figure*}[thb]
\includegraphics[width=0.9\textwidth]{Fig4.eps}
\caption{Oscillations of the scalar field (top panel) and metric coefficient
(bottom panel) at the center of the pulson for $\varkappa = 0.005,\\
a_{max}=0.64$.}
\label{Fig2}
\bigskip
\end{figure*}
\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{Fig5.eps}
\caption{Profiles of the scalar field and metric coefficients.}
\label{Fig3}
\end{figure}
We have performed the Fourier analysis of the
scalar field oscillations obtained by numerical integration of the EKG
system. The resulting spectrum shown in Fig. \ref{Fig4}(a) demonstrates
periodicity with high accuracy.
Then we violated the condition (\ref{eq23}) by tripling $X_{1}\left(0\right) $
that was calculated before, and integrated the EKG system again. As expected,
the resulting field oscillations were found to be non-periodic.
The corresponding spectrum is presented in Fig. \ref{Fig4}(b).
Nonperiodicity manifests itself as additional peaks in the spectrum which
are absent in Fig. \ref{Fig4}(a).
\begin{figure}[tbh]
\includegraphics[width=0.425\textwidth]{Fig6.eps}
\caption{Fourier spectrum of $\protect\phi (t,0)$ for admissible~(a) and
inadmissible~(b) initial conditions.}
\label{Fig4}
\end{figure}
\begin{figure}[tbh]
\includegraphics[width=0.45\textwidth]{Fig7.eps}
\caption{Fourier spectrum of $\protect\phi (t,0)$ obtained from the solution
of the nonlinear KG equation (\ref{eq5a}) with the initial conditions (\ref{eq24a}),
(\ref{eq24}). All parameters are the same as in Figs. (\ref{Fig2}), (\ref{Fig3}). }
\label{Fig5}
\end{figure}
To clarify the meaning of gravity, we used the obtained initial conditions (\ref{eq24a}), (\ref{eq24})
with $\varkappa $ as a formal parameter for the numerical
integration of the nonlinear KG equation (\ref{eq5a}). The solution was
found to be non-periodic, as is clear from its spectrum which is shown in Fig. \re
{Fig5}. We thus conclude that it is because of gravity that the periodic
pulsons of the considered non-Gaussian shapes exist.
\section{Concluding remarks} \label{Sec4}
Thus we have demonstrated the existence of long-lived time-periodic pulsons
in the EKG system. These pulsons differ from the non-gravitating ones in
their shapes and frequencies and exist only due to gravitational effects.
The question arises as to whether these gravipulsons are stable. While the
stability analysis is out of the scope of the present work, it is worth noting that
the stability of the solution (\ref{eq8}), and hence (\ref{eq9a}), is determined by the stability of the
solutions of the non-homogeneous Hill's equation $X_{n}(\theta )$ involved
in (\ref{eq12}). In turn, as it follows from (\ref{eq17}), the stability of the
general solution $X_{n}(\theta )$ is determined by the behavior of the
functions $X_{n}^{+}(\theta )$ and $X_{n}^{-}(\theta )$.
It is clear that all solutions $X_{n}(\theta )$ satisfying the initial
conditions (\ref{eq23}), (\ref{eq23a}) are unstable in the resonance case
\mu _{n}>0$. Indeed, any perturbation of the initial values $X_{n}(0)$,
X_{n\,_{\scriptstyle{\theta }}}(0)$, determined by (\ref{eq23}), (\ref{eq23a
), leads to the appearance of terms $\sim \exp (\mu _{n}\theta )$ on the
right-hand side of (\ref{eq17}), thus making the corresponding function
X_{n}(\theta )$, and hence the solution (\ref{eq8}), exponentially growing in
time.
On the other hand, in the non-resonance case $\mu _{n}=i\nu _{n}$, the functions
X_{n}^{+}(\theta )$, $X_{n}^{-}(\theta )$ as well as $X_{n}(\theta )$ in
\ref{eq17}) are bounded, and a small perturbation of the initial conditions
\ref{eq23}), (\ref{eq23a}) results in the appearance of only small oscillating
terms in $X_{n}(\theta )$. So we can expect that the solution (\ref{eq8}) is
stable if all modes $X_{n}(\theta )$ are non-resonant. The question is, does
any value of $a_{\max }$ exist such that all modes $E_{n}=4n\ (n>0)$
are stable?
A collection of the peaks $\mu _{n}>0$ with $n=1,2,...,10$, shown in Fig.
\ref{Fig0}, demonstrates the existence of numerous stability intervals
separating the instability ones on the $a_{\max }^{2}$ axis. All modes
$n=1,2,...,10$ are stable in the gaps between the peaks, and thus $\mu_{n}=i\nu _{n}$.
However, if we take into consideration additional modes
with $10<n\leqslant N$, supplementary peaks must be added to this plot. Some of the
new peaks will be overlapped by the existing ones, but the rest will fall
within the stability intervals and erode them. Nevertheless, narrow
stability gaps remain visible on the abscise axis even in the case of large
N$.
While we have no proof that some gaps of stability survive as $N$ goes to
infinity, one should take into account that the amplitude of the peaks in Fig.
\ref{Fig0} decreases with increasing $n$, and in any case, narrow intervals
on the $a_{\max }^{2}$ axis can be found where only high-$n$ modes are
unstable. We refer to them as intervals of quasi-stability. Indeed, while
the solution (\ref{eq8}) with $a_{\max }$ falling in one of
these intervals is unstable, this instability evolves very slowly, and
the gravipulson still remains a long-lived object.
Moreover, as it was demonstrated in our simulation \cite{Koutv1}, in the
case of the non-gravitating pulson, the nonlinear stage of instability
saturates very quickly, resulting in a slightly modified pulson which remains
a compact oscillating object. We expect the same instability behavior
in the case of gravipulsons also, at least at small $\varkappa$, even if this
instability is caused by the action of some other perturbative objects around them.
A few words about possible astrophysical applications of the obtained solution are in order.
There are a number of papers where scalar solitons are considered
as models of galactic halos in hopes of explaining the observational
flatness of the rotation curves (see., e.g., \cite{Mielke} and references therein).
It is easy to see that then the
energy density of a scalar field must not decay faster than $r^{-2}$.
Evidently, our solution does not satisfy this criterion. However, if a
galactic halo is not a single soliton-like object, but is an ensemble
of dark matter lumps, of so-called ''mini-MACHOs'' \cite{Hernandez
, the gravipulsons may be reasonable candidates for these compact
constituents. In this case the gravipulson masses (\ref{eq23b}) need to be
limited by the condition $M\lesssim 10^{-7}M_{\odot }$ following from
microlensing data \cite{Alcock}. This constrains the amplitude of the
gravipulsons and the parameters of the potential (\ref{eq2}).
\begin{acknowledgments}
We are grateful for discussions with participants of the IV International Conference
"Frontiers of Nonlinear Physics" (FNP-2010).
\end{acknowledgments}
\nocite{*}
|
1,314,259,993,002 | arxiv | \section{Introduction}
In recent years, the rapid improvement of high-contrast imaging instrumentation and techniques have led to the
discovery of a number of wide sub-stellar companions to nearby young stars, down to planetary mass
\citep[e.g.][]{2005A&A...438L..29C, 2006ApJ...649..894L,2008Sci...322.1348M, 2010Natur.468.1080M,2009A&A...493L..21L, 2014ApJ...780L..30C}. Several of these discoveries, such as AB Pic~b \citep{2005A&A...438L..29C}, HN~Peg~B \citep{2007ApJ...654..570L}, 1RXS J1609~b \citep{2010ApJ...719..497L, 2008ApJ...689L.153L}, HIP 78530~b \citep{lafreniere2011} and the recently discovered HD~106906~b \citep{2014ApJ...780L...4B} and ROXS~42B~b \citep{2014ApJ...780L..30C},
have mass ratios with respect to their parent stars of only $\sim 1\%$ and
seriously challenge the current planet formation paradigm.
In particular, their large separations are hard to explain and suggest they might be extreme outcomes of their underlying formation mechanism, regardless of whether it is based on core accretion or disk instability.
Our previous survey of 91 stars in the USco region \citep{2014ApJ...785...47L} implies a frequency of wide companion for such regions of 4-5\%, in agreement with other studies \citep{2011ApJ...726..113I}.
This suggests a frequency of wide companions in star forming regions comparable to the values for young moving groups or the field, reported for example by \cite{2007ApJ...670.1367L, 2009ApJS..181...62M, 2010A&A...509A..52C}.
Most recently we also confirmed three new companions with masses of $\sim 40-100~M_{Jup}$ and separations of
$\sim 40-230~AU$ in the Scorpius-Centaurus (Sco-Cen) region \citep{2012ApJ...758L...2J}.
These companions represent an interesting intermediate between stellar companions and the $\sim 10-20~M_{Jup}$ ones described above in the Upper Scorpius (USco) region.
The existence of such a seemingly continuous population might imply that binary formation extends all the way down to planetary
masses for wide separations, or at least that mass alone is not a clear-cut diagnostic for distinguishing
between formation mechanisms.
In order to further address these issues, we conducted a survey of 74 stars in the Taurus star forming region with ALTAIR/NIRI \citep{2000SPIE.4007..115H,2003PASP..115.1388H}.
The results of the full survey will be presented in a dedicated paper (Daemgen et al. 2014, ApJ Submitted).
Here we present the discovery of a $18-50~M_{Jup}$ companion at a projected separation of $\sim 400$~AU from the F8 star HD~284149.
A dedicated analysis of the host properties is also presented in Sec.~\ref{sec:host}, addressing the question of its questionable Taurus membership.
\section{Observations and data reduction}
\label{sec:obsred}
HD\,284149 was observed during six epochs between October 2011 and March 2014 on Gemini North with the adaptive-optics assisted NIRI instrument (Hodapp et al. 2003) in J, H, and Ks-band. The f/32 camera provided a sampling of 21.9~mas/pixel and a field of view (FoV) of 22\arcsec$\times$22\arcsec. Total integration times varied between 9\,sec (J) and $\sim$7\,sec (Ks) and were taken as a series of coadds in a 5-point dither pattern to increase dynamic range and allow sky subtraction.
The details of the observing times for each epoch, together with the mean airmass and seeing at each observing date are reported in Tab.~\ref{tab:comp_char}.
After subtraction of a striping pattern frequently observed in NIRI images, all images were flat fielded, bad pixel corrected, and sky subtracted. The field distortion was corrected as described in \cite{2014ApJ...785...47L} who determine a residual astrometric uncertainty of 15\,mas, 25\,mas, and 50\,mas at radii 4\arcsec, ~8\arcsec, ~and ~12\arcsec\ ~from the center, respectively.
The left panel of Fig.~\ref{fig:cpm} shows one of the fully reduced images of HD\,284149 and its companion obtained with NIRI in 2012B. The achieved full width at half maximum of the point spread function is 0\farcs08, and the companion, at a separation of $3.7\arcsec$ is detected at $\gtrsim$14$\sigma$.
As part of our survey for faint companions in Taurus (Daemgen et al. 2014, subm.), we also obtained deep exposures of HD\,284149 in H and J band, which confirm the presence of the companion with high S/N $>$ 200. These observations, however, saturate the central star and render the relative astrometry and photometry less precise than in the lower-S/N images analyzed here, and are not further used.
\begin{figure*}[]
\centering
\includegraphics[width=110mm]{f1.eps}\label{fig:cpm}
\caption{ {\bf Left:} Candidate companion detected in $K_s$ with GEMINI/NIRI in one of the images obtained in 2012B. {\bf Right:} Evaluation of common proper motion for the detected companion to HD 284149. The continuous line shows the motion of a background star and the filled symbols its positions of at the various epochs. The open symbols represent the corresponding measurements of the position of HD 284149 B at the same epochs. If the star and companion are co-moving, the open symbols with error bars should be close to the filled circle in the origin, which represents our most recent epoch used as reference. If the companion candidate does not move with respect to the background, then the open symbol is consistent with the location of the identical filled symbol.}
\label{fig:cpm}
\end{figure*}
\section{Host star properties
\label{sec:host}
HD 284149 was included among the members of Taurus-Auriga association by \cite{1996A&A...312..439W}, but it is not considered in the compilations by \cite{2008hsf1.book..405K} and \cite{2014ApJ...784..126E}. Therefore, a re-assessment of the stellar properties is needed. HD 284149 was classified as F8 by \cite{2012ApJ...745..119N} and G1 by \cite{2000A&A...359..181W}.
A young age of the star is supported by the large lithium EW \citep{2000A&A...359..181W}, the large X-ray luminosity as revealed by ROSAT, the photometric variability and fast rotation \citep[a period of 1.079 days is reported by ][]{2007IBVS.5752....1G}. The short-term RV monitoring by \cite{2012ApJ...745..119N} was able to exclude the possibility of a tidally-locked binary. However with a 3 Km/s difference between mean RV from \cite{2000A&A...359..181W} and \cite{2012ApJ...745..119N} being about 3 km/s, a binarity with periods of months or years cannot be excluded\footnote{In order to reduce the impact of possible binarity on the space velocity, we decided to use the average velocity.
Efforts are underway to better understand the multiplicity status of HD 284149 and will be presented in further publications. There are no indication from non-detection on line profile alteration in Nguyen et al. 2012 and lack of direct detection in our own images that the companion contributes significantly to the integrated flux. The expected impact on the age indicators is then minor.}
From the G1-F8 spectral classification, an effective temperature of 5970-6100 K is derived following \cite{2013ApJS..208....9P}. Photometric colors are broadly consistent with such temperatures, with a detailed comparison hampered by obervational scatter (e.g. peak-to valley differences larger than 0.2 mag in V band), possibly linked to the photometric variability of the star. Adopting the V magnitude from ASAS ($9.653\pm0.060$), the $V-K_s$ color is 1.55 mag. Comparison with the pre-main sequence (pre-MS) intrinsic colors of young stars by \cite{2013ApJS..208....9P} suggests a reddening E(B-V) of about 0.05-0.08 mag for a G1 and F8 star, respectively.
Slightly smaller amounts of reddening are indicated by the B-V and V-I colors. Such amount of reddening is not unexpected at the distance of the star. The trigonometric parallax from \cite{2007A&A...474..653V} is $9.24\pm1.58$ mas. A comparison with members of young moving groups (MGs) (see left panel of Fig.~\ref{fig:age}) indicates that the lithium equivalent width of HD 284149 \citep[208 m\AA,][]{2000A&A...359..181W} is comparable with that of members of $\beta$ Pic, Tuc-Hor, Columba and Carina moving groups of similar temperatures, and clearly above that of Pleiades open cluster and AB Dor moving group.
A similar result is obtained for a comparison of the X-ray luminosity ($\log L_{X}/L_{bol}=-3.3$ for HD 284149).
The position of HD 284149 on an HR diagram (see right panel of Fig.~\ref{fig:age}) is close to the 25 Myr isochrone using the theoretical models by \cite{2012MNRAS.427..127B}, with ages between 15 to 100 Myr also compatible with the data.
This isochronal age is on average older than that obtained by other authors \citep[14-16 Myr, ][]{2000A&A...359..181W,palla2002} because of the revision in the trigonometric parallax, with respect to the value reported in the Hipparcos catalogue \citep{1997ESASP1200.....P}.
The resulting kinematic parameters, adopting the \cite{2007A&A...474..653V} parallax and proper motion and the mean of the RVs obtained by \cite{2000A&A...359..181W} and \cite{2012ApJ...745..119N}, are $U=-12.3$~km/s, $W=-6.4$~km/s and $W=-8.8$~km/s. This is similar to that of the Octans association discussed in \cite{2008hsf2.book..757T}, whose proposed members are however all very far from HD 284149 on the sky.
Recently \cite{2013ApJ...778....5Z} identified an additional group of young stars with similar kinematics to the Octans association but with a smaller distance from the Sun and a different sky distribution, labeled as Octans-Near. While the link between Octans and Octans-Near groups and the existence of the latter as a true moving group deserves further investigation, we note that one of the Octans-Near proposed members, HIP 19496, is separated on the sky by about 5 deg from HD 284189, it has a comparable distance (98 vs 108 pc), and the space velocities of the two stars differ by just 2.7 km/s. \cite{2013ApJ...778....5Z} estimate an age of 30 Myr for HIP 19496, similar to our determination for HD 284189.
In summary, HD 284149 is significantly older than the bulk of the Taurus-Auriga association. Membership is still possible in case of earlier start of star formation in the outer regions of the association \citep[][]{2002ApJ...581.1194P}. Independently of the Tau-Aur membership, the Li EW well above the Pleiades locus coupled with the position on HR diagram above ZAMS and other age diagnostics indicate an age of about 25~Myr, with minimum and maximum values of about 15 and 50~Myr respectively.
A summary of the stellar parameters is given in Table~\ref{tab:properties}.
\begin{figure*}[]
\centering
\includegraphics[width=110mm]{f2.eps}
\caption{{\bf Left:} Lithium equivalent width vs B-V of HD284149 compared to that of members of nearby groups and clusters of young age. The black star represents our target, orange circles represent members of beta Pic moving gropus and TW Hya association (age 8-16 Myr) red circles members of Tucana, Columba and Carina MG (age 30 Myr);
light blue circles: members AB Dor MG (age 100 Myr); green circles: Pleiades OC (age 125 Myr); purple circles: Hyades OC (age 625 Myr). The filled squares represent the median values of EW Li for the corresponding color bin.
Li EW data are from \cite{2006A&A...460..695T} for young assocition, \cite{1993AJ....106.1059S} for the Pleiades, and \cite{1993ApJ...415..150T} for the Hyades.
{\bf Right:} Position of HD284149 (filled dots) on HR diagram for the temperatures corresponding to F8 and G1 spectral classification. Overplotted are the 5, 12, 20, 30, and 70 Myr isochrones from \cite{2012MNRAS.427..127B}.}
\label{fig:age}
\end{figure*}
\begin{table}[ht!]
\small
\begin{tabular}{l|l|l|l}
\hline
\hline
Parameter & Host Star & Companion & Ref. \\
\hline
\hline
V (mag) & 9.653$\pm$0.060 & & 2 \\
B-V & 0.58 & & 3 \\
V-I & 0.675$\pm$0.088 & & 4 \\
J (mag) & 8.479$\pm$0.043 & 15.516$\pm$0.043 & 5,1 \\
H (mag) & 8.208$\pm$0.021 & 14.715$\pm$0.047 & 5,1 \\
K (mag) & 8.100$\pm$0.029 & 14.332$\pm$0.040 & 5,1 \\
parallax (mas) & 9.24$\pm$1.58 & & 6 \\
E(B-V) & 0.05$\pm$0.05 & & 1 \\
RV (Km/s) & 14.0$\pm$2.0 & & 7,8 \\
Ew Li m($\AA$) & 208 & & 7 \\
Prot (d) & 1.079 & & 9 \\
log Lx/Lbol & -3.3$\pm 0.1$ & & 1 \\
vsini (Km/s) & 27.0$\pm$1.9 & & 8 \\
Age (Myr) & $25^{+25}_{-10}$ & & 1 \\
Sp. Type & G1-F8 & M8-L1 & 7,8,1 \\
$T_{Eff}$~(K) & 5970-6100 & $2537^{+95}_{-182}$ & 1 \\
Mass & $1.14\pm0.05~M_{\odot}$ & $32^{+18}_{-14}~M_{Jup}$ & 1 \\
\hline
\hline
\end{tabular}
\caption{ Summary of properties of both HD 284149 and its companion.\\
{\bf References:} [1] This paper; [2] ASAS \citep{2002AcA....52..397P}; [3] SIMBAD; [4] TASS \citep{2000PASP..112..397R}; [5] 2MASS \cite{2003yCat.2246....0C}; [6] \cite{2007A&A...474..653V}; [7] \cite{2000A&A...359..181W}; [8] \cite{2012ApJ...745..119N}; [9] \cite{2007IBVS.5752....1G} }
\label{tab:properties}
\end{table}
\section{Companion properties}
The relative position of HD~284149 and its companion were determined with PSF photometry using \emph{daophot} in \emph{IRAF}. The bright star HD~284149 was used as PSF reference to obtain relative photometry and astrometry of the companion. Statistical uncertainties were inferred from the rms noise between the individual dither exposures for each epoch and filter. Systematic flux uncertainties are estimated from the residuals after PSF subtraction to be $\lesssim$5\%, and systematic astrometric uncertainties are dominated by the uncertainty of the distortion correction at the position of HD~284149b of $\lesssim$15\,mas. The resulting astrometry and photometry are listed in Tab.~\ref{tab:comp_char}, while a summary of the derived properties is given in Tab.~\ref{tab:properties}.
The right panel of Fig.~\ref{fig:cpm} shows the relative change of separation and position angle of the companion between our previous observations (filled right-facing triangle) with respect to the most recent one (filled circle).
We conclude that the point source we imaged is consistent with a co-moving companion at $\sim 400$~AU from HD284149 with $>99\%$ confidence according to a $\chi^2$ test.
As discussed in Sec.~\ref{sec:host}, the age of this system is controversial, as it appears to be older than other members of the Taurus association.
With an adopted age of $25^{+25}_{-10}~Myr$, the Ks brightness of the companion suggests a mass of $32^{+18}_{-14}~M_{Jup}$, according to the DUSTY models by \cite{2000ApJ...542..464C}.
Using the DUSTY models by \cite{2000ApJ...542..464C} we derived an effective temperature of $2337^{+95}_{-182}$~K. Together with the color measurements (J-H~$\sim 0.8$, H-K~$\sim 0.4$) this suggests a spectral type between M8 and L1 \citep[see e.g.][and www.pas.rochester.edu/$\sim$emamajek/ for the extended table]{2013ApJS..208....9P}, but further measurements are required to better constrain this.
\begin{figure*}[]
\centering
\includegraphics[width=110mm]{f3.eps}
\caption{ {\bf Left:} Absolute magnitude in $K_S$ band vs age of a companion of 20, 30 and 50~$M_{Jup}$ (blue, cyan and purple curve, respectively) according to the COND (solid lines) models by \cite{2003A&A...402..701B} and DUSTY (dashed lines) models by \cite{2000ApJ...542..464C}. The position of HD 284149 B is marked by a filled star. {\bf Right:} Mass ratio vs separation of HD 284149 (filled star), compared to those of other known low-mass companions discovered so far. The companions discovered using the radial velocity (RV) technique are marked with blue crosses, the ones transiting their parent stars with green triangles. Finally, directly imaged companions are represented by red circles if the stellar age is less than 500~Myr (young companions), and with red circles otherwise.}
\label{fig:age_mag}
\end{figure*}
\begin{table*}[]
\tiny
\centering
\begin{tabular}{l|ll|lll|lll|ll|lll}
\hline \hline
UT Date & \multicolumn{8}{|c|}{Observing conditions and setup} & \multicolumn{2}{|c|}{Astrometry} & \multicolumn{3}{|c}{Photometry} \\ \cline{2-14}
& Seeing & Air Mass & \multicolumn{3}{|l|}{Total Exposure Time} & \multicolumn{3}{|l|}{$N_{Coadds}$} & Sep. & PA & $\Delta~J$ & $\Delta~H $ & $\Delta~K_S$ \\
& (arsec)& & J (sec) & H (sec) & K$_S$ (sec) & J & H & K$_S$ & (arcsec) & (deg) & (mag) & (mag) & (mag) \\
\hline
\hline
2011-10-19 & 1.086 & 0.557 & -- & -- & 10.50 & -- & -- & 30 & 3.6826 $\pm$ 0.0038 & 255.065 $\pm$ 0.028 & -- & -- & 6.208 $\pm$ 0.031 \\
2012-08-27 & 1.018 & 0.417 & -- & -- & 8.00 & -- & -- & 40 & 3.6847 $\pm$ 0.0015 & 255.143 $\pm$ 0.025 & -- & -- & 6.171 $\pm$ 0.041 \\
2013-08-14 & 1.241 & 1.061 & -- & -- & 7.00 & -- & -- & 20 & 3.6818 $\pm$ 0.0017 & 254.965 $\pm$ 0.065 & -- & -- & 6.319 $\pm$ 0.045 \\
2013-08-23 & 1.053 & 0.378 & 9.10 & -- & 7.00 & 26 & -- & 25 & 3.6828 $\pm$ 0.0029 & 254.925 $\pm$ 0.032 & 6.99 $\pm$ 0.080 & -- & 6.218 $\pm$ 0.085 \\
2013-11-27 & 1.036 & 0.658 & 8.96 & 8.32 & 7.00 & 28 & 26 & 35 & 3.6847 $\pm$ 0.0049 & 254.922 $\pm$ 0.070 & 6.990 $\pm$ 0.047 & 6.647 $\pm$ 0.061 & 6.262 $\pm$ 0.058 \\
2014-03-11 & 1.267 & 0.716 & 9.00 & 8.32 & 7.00 & 10 & 26 & 35 & 3.6813 $\pm$ 0.0038 & 254.850 $\pm$ 0.157 & 7.06 $\pm$ 0.280 & 6.738 $\pm$ 0.059 & 6.299 $\pm$ 0.074 \\
\hline
\hline
\end{tabular}
\caption{Detailes of the observations setup and conditions, and relative astrometry and photometry of HD 284149 and its companion}
\label{tab:comp_char}
\end{table*}
\section{Discussion and Conclusions}
We presented here the detection of a substellar (~32~$M_{Jup}$ assuming an age of 25~Myr, see Fig.~\ref{fig:age_mag}~a and Tab.~\ref{tab:properties}) companion orbiting the young star HD 284149 at a separation of $\sim 400$~AU.
Fig.~\ref{fig:age_mag} shows a comparison of HD 284149 mass-ratio and separation with the values of the planetary companions found by radial velocities (RV) and transit methods, as well as other directly imaged planetary and brown dwarf companions.
The group of objects with separation $<100~AU$ and mass ratio $<0.01$ seems to be well separated from the one including companions with larger separation and mass ratio, suggesting that different formation mechanisms could be at play.
The small mass ratio of the first group might suggest planet-like formation, but objects with similar mass-ratios at larger separations are difficult to explain.
HD 284149b shows very similar properties to objects like ROXs 42Bb or AB Pic b which, as suggested by \cite{2014ApJ...780L..30C}, places it between the bona-fide planets and the lowest mass brown dwarfs imaged so far.
Together with these and other companions of similar mass and separation, such as HN~Peg~B and HD~106906, HD~284149b represents a challenge for our understanding of the formation of low-mass companions at very wide separations.
The high mass ratio of these systems might suggest a planet-like origin, but at the same time, their estimated mass is well above the deuterium burning limit, suggesting a stellar-like formation.
The existence of such companions suggests that mass ratio alone is not sufficient to distinguish between planet-like and star-like formation, at least for wide companions \citep[see also][]{2012ApJ...745....4J}.
Finally, the findings of dedicated RV campains around young stars seem to suggest a paucity of close-in planetary companions around these targets. Only few close in companions have been detected around young early-G and F-type stars, such as HD 70573 \citep{2007ApJ...660L.145S} and HD 113337 \citep{2014A&A...561A..65B}.
Their small number seem to imply a lower frequency of such companions if compared to the more massive, more distant ones as HD 284149B.
If confirmed, this could suggest that multiple planet formation mechanisms are at play around these objects.
The object brightness and separation makes HD~284149~b a very well suited target for detailed characterization of both the host star and the companion. Efforts toward this direction are already under way and will be presented in further publications.
\subsubsection*{Acknowledgments}
{ This work was supported by grants from the NSERC of Canada and the University of Toronto McLean Award to R.J. SD is supported by a McLean Postdoctoral Fellowship. M.B. is founded through the “Progetti Premiali” funding scheme of the Italian Ministry of Education, University, and Research, and through the PRIN-INAF 2010 “Planetary systems at young ages and the interactions with their active host stars”. Based on observations obtained at the Gemini Observatory, operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France}
|
1,314,259,993,003 | arxiv | \section{Natural models}
The following concept is due to Grothendieck \cite{AGV}.
\begin{definition}\label{def:rep}
Let \ensuremath{\mathbb{C}}\ be a small category. A natural transformation $$f : Y \to X$$ of presheaves on \ensuremath{\mathbb{C}}\ is called \emph{representable} if all of its fibers are representable objects, in the following sense: for every $C\in\ensuremath{\mathbb{C}}$ and $x\in X(C)$, there is a $D\in\ensuremath{\mathbb{C}}$, a $p : D\to C$, and a $y\in Y(D)$ such that the following square is a pullback,
\begin{equation}\label{diag:rep}
\xymatrix{
\ensuremath{\mathsf{y}}{D} \ar[d]_{\ensuremath{\mathsf{y}}{p}} \ar[r]^-{y} \pbcorner & Y\ar[d]^{f}\\
\ensuremath{\mathsf{y}}{C} \ar[r]_{x} & X .}
\end{equation}
As here, we shall freely use the Yoneda lemma to identify elements $x\in X(C)$ with natural maps $x:\ensuremath{\mathsf{y}}{C} \to X$.
\end{definition}
Our first observation is that a representable natural transformation is the same thing as a \emph{category with families} in the sense of Dybjer \cite{CwF}.
Indeed, let us write the objects of \ensuremath{\mathbb{C}}\ as $\Gamma, \Delta, \ldots$ and the arrows as $\sigma : \Delta \to \Gamma, \dots$, thinking of \ensuremath{\mathbb{C}}\ as a ``category of contexts". Let $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ be a representable map of presheaves, and write its elements as:
\begin{align*}
A\in \ensuremath{\mathcal{U}}(\Gamma)\ &\Leftrightarrow\ \Gtypes{A}\\
a\in \ensuremath{\widetilde{\mathcal{U}}}(\Gamma)\ &\Leftrightarrow\ \Gterms{a:A},
\end{align*}
where $A=p\circ a$, as indicated in:
\[
\xymatrix{
& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ensuremath{\mathsf{y}}\Gamma \ar[ru]^{a} \ar[r]_{A} & \ensuremath{\mathcal{U}} .}
\]
Thus we regard \ensuremath{\mathcal{U}}\ as the \emph{presheaf of types}, with $\ensuremath{\mathcal{U}}(\ensuremath{\Gamma})$ the set of all types in context $\ensuremath{\Gamma}$, and \ensuremath{\widetilde{\mathcal{U}}}\ as the \emph{presheaf of terms}, with $\ensuremath{\widetilde{\mathcal{U}}}(\ensuremath{\Gamma})$ the set of all terms in context $\ensuremath{\Gamma}$, while the component $p_{\ensuremath{\Gamma}} : \ensuremath{\widetilde{\mathcal{U}}}(\ensuremath{\Gamma}) \to \ensuremath{\mathcal{U}}(\ensuremath{\Gamma})$ is the typing of the terms in context $\ensuremath{\Gamma}$ (cf.~\cite{Hofmann2} for an early statement of this point of view).
Observe that naturality of $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ means that for any ``substitution" $\sigma:\Delta\to\ensuremath{\Gamma}$, we have an action on types and terms:
\begin{align*}
\Gtypes{A}\ &\Rightarrow\ \types{\Delta}{A\sigma}\\
\Gterms{a:A}\ &\Rightarrow\ \terms{\Delta}{a\sigma : A\sigma}
\end{align*}
While, by functoriality, given any further $\tau: \Delta'\to\Delta$, we have
\[
(A\sigma)\tau = A(\sigma\circ\tau) \qquad (a\sigma)\tau = a(\sigma\circ\tau),
\]
as well as
\[
A1 = A \qquad a1 = a
\]
for the identity substitution $1 : \ensuremath{\Gamma}\to\ensuremath{\Gamma}$.
Finally, the representability of $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ is exactly the operation of \emph{context extension}: given any $\Gtypes{A}$, by Yoneda we have the corresponding map $A : \ensuremath{\Gamma} \to \ensuremath{\mathcal{U}}$, and we let $p_A: \ext{\ensuremath{\Gamma}}{A} \to \ensuremath{\Gamma}$ be the resulting fiber of $p$ as in \eqref{diag:rep}. We therefore have a pullback square:
\begin{equation}\label{diag:conext}
\xymatrix{
\ext{\ensuremath{\Gamma}}{A} \ar[d]_{p_A} \ar[r]^{q_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ensuremath{\Gamma} \ar[r]_{A} & \ensuremath{\mathcal{U}} ,}
\end{equation}
where the map $q_A : \ext{\ensuremath{\Gamma}}{A}\to\ensuremath{\widetilde{\mathcal{U}}}$ determines a term
$$\terms{\ext{\ensuremath{\Gamma}}{A}}{q_A:Ap_A}.$$
In \eqref{diag:conext} and henceforth, we omit the $\ensuremath{\mathsf{y}}$ for the Yoneda embedding, letting the Greek letters serve to distinguish representable presheaves.
\medskip
The fact that \eqref{diag:conext} is a pullback means that given any $\sigma: \Delta\to\ensuremath{\Gamma}$ and $\terms{\Delta}{a:A\sigma}$, there is a map $$(\sigma, a):\Delta\to\ext{\ensuremath{\Gamma}}{A},$$ and this operation satisfies the equations
\begin{align*}
p_A\circ(\sigma,a) &= \sigma \\
q_A(\sigma,a) &= a,
\end{align*}
as indicated in the following diagram.
\begin{equation*}
\xymatrix{
\Delta \ar@/{}_{1pc}/[ddr]_{\sigma} \ar@{.>}[dr]|-{(\sigma,a)} \ar@/{}^{1pc}/[drr]^{a} \\
& \ext{\ensuremath{\Gamma}}{A} \ar[d]^{p_A} \ar[r]_{q_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
& \ensuremath{\Gamma} \ar[r]_{A} & \ensuremath{\mathcal{U}} }
\end{equation*}
Moreover, by the uniqueness of $(\sigma,a)$, for any $\tau: \Delta'\to\Delta$, we also have:
\begin{align*}
(\sigma,a)\circ\tau &= (\sigma\circ\tau,a\tau)\\
(p_A,q_A) &= 1.
\end{align*}
Comparing the foregoing with the definition of a category with families in \cite{CwF}, we have shown:
\begin{proposition}
Let $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ be a natural transformation of presheaves on a small category \ensuremath{\mathbb{C}}\ with a terminal object. Then $p$ is representable in the sense of Definition \ref{def:rep} just in case $(\ensuremath{\mathbb{C}}, p)$ is a category with families.
\end{proposition}
The notion of a category with families is a variable-free way of presenting dependent type theory, including contexts and substitutions, types and terms in context, and context extension. Accordingly, we may think of a representable map of presheaves on a category \ensuremath{\mathbb{C}}\ as a ``type theory over~\ensuremath{\mathbb{C}}" --- with \ensuremath{\mathbb{C}}\ serving as the category of contexts and substitutions (the requirement that \ensuremath{\mathbb{C}}\ should have a terminal object, representing the ``empty context", is purely conventional). As we shall see below, such a map of presheaves is essentially determined by a class of maps in \ensuremath{\mathbb{C}}\ that is closed under all pullbacks; these maps will be the types in context.
\begin{definition}
By a \emph{natural model of type theory} on a small category \ensuremath{\mathbb{C}}\ we mean a representable map of presheaves,
\[
p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}.
\]
\end{definition}
\begin{corollary}
Natural models of type theory are evidently closed under composition, coproducts, and pullbacks along arbitrary maps $\ensuremath{\mathcal{U}}'\to\ensuremath{\mathcal{U}}$.
\end{corollary}
\begin{remark}
As has recently been emphasized (e.g.\ in \cite{BCH}), the notion of a ``category with families" is essentially algebraic, consisting of the four sorts: \emph{contexts, substitutions, types, terms}; operations defined on the sorts, including in particular \emph{context extension}; and equations between terms built from the operations. The same is, of course, true of the equivalent concept of a \emph{natural model of type theory}, in the form of a representable natural transformation $f : A \to B$ over a category $\ensuremath{\mathbb{C}}$, as we now briefly indicate.
Regarded as a many-sorted algebraic theory, a natural model consists of four basic sorts $$C_0,\ C_1,\ A,\ B$$ along with the following operations and equations:
\begin{description}
\item[category:] the usual domain, codomain, identity and composition arrows for the index category:
\[
\xymatrix{
C_1 \times_{C_0} C_1 \ar[r]^-{\circ} & C_1 {\ar@<-2ex>[rr]_{\ensuremath{\mathsf{dom}}} \ar@<2ex>[rr]^{\ensuremath{\mathsf{cod}}}} && \ar[ll]|-{\,\ensuremath{\mathsf{id}}\,} C_0
}
\]
together with the familiar equations for a category.
\item[presheaf:] the indexing and action operations for the presheaves:
\[
\xymatrix{
C_1 \times_{C_0} A \ar[r]^-{\alpha} & A \ar[d]^{p_A} \\
& C_0
}
\qquad
\xymatrix{
C_1 \times_{C_0} B \ar[r]^-{\beta} & B \ar[d]^{p_B} \\
& C_0
}
\]
together with the equations making $\alpha$ a (contravariant) action of $C_1$ on $A$:
\begin{align*}
p_A(\alpha(u, a)) &= \ensuremath{\mathsf{dom}}(u),\\
\alpha(u\circ v, a) &= \alpha(v,\alpha(u,a)),\\
\alpha(1_{p_{A}(a)},a) &= a,
\end{align*}
and similarly for $\beta$.
\item[natural transformation:] an operation
\[
f : A \to B
\]
satisfying the naturality equations: $$p_B \circ f = p_A,\qquad f\circ\alpha = \beta\circ(C_1\times_{C_0}f).$$
\item[representable:] note that a natural transformation $f : A \to B$ is representable just if the associated functor on the categories of elements,
\[
\textstyle{\int_\ensuremath{\mathbb{C}}{f} : \int_\ensuremath{\mathbb{C}}{A} \to \int_\ensuremath{\mathbb{C}}{B}}
\]
has a right adjoint $f^* : \int_\ensuremath{\mathbb{C}}{B} \to \int_\ensuremath{\mathbb{C}}{A}$ (cf.~\cite{ABSS}, \S\,8), which is an algebraic condition.
In more detail, we requiring the following additional structure:
\begin{itemize}
\item an operation
\[
(f^*)_0 : B \to A
\]
taking the objects of $\int_\ensuremath{\mathbb{C}}{B}$ to those of $\int_\ensuremath{\mathbb{C}}{A}$ (not necessarily preserving the indexing over $C_0$),
\item an operation on the arrows in the categories of elements:
\[
(f^*)_1 : C_1 \times_{C_0} B \to C_1 \times_{C_0} A
\]
respecting domains and codomains,
\begin{align*}
\ensuremath{\mathsf{dom}}(\pi_1( (f^*)_1 (u, b) )) &= (f^*)_0(\ensuremath{\mathsf{dom}}(u)),\\
p_A(\pi_2( (f^*)_1 (u, b) )) &= (f^*)_0(\ensuremath{\mathsf{cod}}(u)),
\end{align*}
and satisfying the functoriality equations,
\begin{align*}
(f^*)_1((u,b)\circ(u', b')) &= (f^*)_1(u,b)\circ (f^*)_1(u', b'),\\
(f^*)_1(1,b)&= (1,b).
\end{align*}
\item two further operations
\begin{align*}
\eta &: A \to C_1 \times_{C_0} A \\
\epsilon &: B \to C_1 \times_{C_0} B
\end{align*}
satisfying the standard equations for natural transformations of the form $\eta: 1 \to f^*\circ f$ and $\epsilon: f\circ f^*\to 1$.
\item the familiar triangle identities for an adjunction.
\end{itemize}
The details are left to the reader.
\end{description}
\end{remark}
\section{Modelling the type constructors}\label{sec:modelling}
When does a natural model of type theory also model the various type constructors, such as (dependent) product $\Pi$, sum $\Sigma$, and identity types $\mathsf{Id}$? As the notation $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ suggests, the notion of a natural model is similar to Voevodsky's notion of a \emph{universe} \cite{KLV}, and we shall modify the approach taken there in order to determine conditions ensuring that the usual type-forming operations are modeled in our setting. (A related idea was used already in \cite{StreicherUp}.)
We require the following preliminary observations regarding polynomial functors, for more on which see \cite{GK}.
Given a map $f : B \to A$ in a locally cartesian closed category $\ensuremath{\mathcal{E}}$, there is an associated \emph{polynomial endofunctor} $P_f : \ensuremath{\mathcal{E}} \to \ensuremath{\mathcal{E}}$,
defined for every object $X\in\ensuremath{\mathcal{E}}$ by
\begin{equation}\label{eq:polydef}
P_f(X)\ =\ \sum_{a:A}X^{B_a}
\end{equation}
where, as usual, we write $B_a = f^{-1}(a)$ for the fiber of $f$ at $a:A$, using the internal language of $\ensuremath{\mathcal{E}}$ as explained in \cite{GK}. Formally, this functor is defined from the LCCC structure on $\ensuremath{\mathcal{E}}$ as a composite:
\[
P_f(X)\ =\ \sum_{A}\prod_{f}B^{*}(X)
\]
where:
\begin{align*}
B^* &: \ensuremath{\mathcal{E}} \to \ensuremath{\mathcal{E}}/B && \text{is pullback along $B\to 1$},\\
\prod_{f} &: \ensuremath{\mathcal{E}}/B \to \ensuremath{\mathcal{E}}/A && \text{is right adjoint to pullback along $f : B\to A$},\\
\sum_{A} &: \ensuremath{\mathcal{E}}/A \to \ensuremath{\mathcal{E}} && \text{is composition along $A\to 1$}.
\end{align*}
\begin{lemma}\label{lem:polyclass}
There is a natural bijection between maps $g : Y \to \sum_{a:A}X^{B_a}$ and pairs of maps $\big(g_1:Y \to A,\ g_2 : Y\times_A B \to X\big)$ as indicated in the following diagram.
\begin{equation}\label{diag:polyclass}
\xymatrix{
X & Y\times_A B \ar[l]_-{g_2} \ar[d]\ar[r]\pbcorner & B \ar[d]^{f}\\
&Y\ar[r]_{g_1} & A}
\end{equation}
\end{lemma}
\begin{proof}
Given
\[
\xymatrix{
g: Y \ar[r] & \sum_{a:A}X^{B_a} = \sum_{A}\prod_{f}B^{*}(X)
}
\]
compose with the projection $\pi_1 : \sum_{A}\prod_{f}B^{*}(X)\to A$ to get $g_1 = \pi_1\circ g : Y\to A$, making $g$ a map over $A$,
\[
\xymatrix{
Y \ar[r]^-{g} \ar[rd]_{g_1} & \sum_{A}\prod_{f}B^{*}(X) \ar[d]^{\pi_1} \\
& A.
}
\]
As an object over $A$, the map $\pi_1$ is
\[
\prod_{f}B^*(X) = \prod_{f}f^*A^*(X) = (A^*(X))^f.
\]
We can therefore take the exponential transpose of $g$ to get another map over $A$ of the form:
\[
\xymatrix{
Y\times_A B \ar[r]^{\tilde{g}} \ar[rd]_{g_1\times f} & A^*(X) \ar[d] \\
& A.
}
\]
Composing $\tilde{g}$ with the second projection $A^*(X) = A\times X \to X$ gives $g_2 :Y\times_A B \to X$ as indicated in
\begin{equation}\label{diag:compwithproject}
\xymatrix{
Y\times_A B \ar[r]^{\tilde{g}} \ar[rd]_{g_1\times f} \ar@/{}^{20pt}/[rr]^{g_2} & A \times X \ar[d] \ar[r] & X \\
& A .
}
\end{equation}
This assignment of $(g_1, g_2)$ to $g$ is clearly reversible and natural in $Y$.
\end{proof}
Since the isomorphism of lemma \ref{lem:polyclass} is natural in $Y$, it is convenient to consider the generic case, where $Y = \sum_{a:A}X^{B_a}$ and $g$ is the identity. In that case, we have a diagram of the form,
\[
\xymatrix{
X & \\
G \ar[u]_{u_2} \ar[d] \ar[r] \pbcorner & B \ar[d]^{f}\\
\sum_{a:A}X^{B_a} \ar[r]_-{u_1} & A
}
\]
where $u_1$ is the canonical projection $\pi_1 : \sum_{a:A}X^{B_a} \to A$, and the ``generic pulled-back object" $G$ can be described over $A$ as
$X^{B_a} \times_A B_a$.
The map $u_2 : G \to X$ is then evaluation over $A$, composed with the second projection as in \eqref{diag:compwithproject}.
Now given any $g : Y \to \sum_{a:A}X^{B_a}$, the associated maps $g_1 : Y \to A$ and $g_2 : Y\times_A B \to X$ are given by pullback and composition, as indicated in the following diagram.
\begin{equation}\label{diag:polyclassnat}
\xymatrix{
& X & \\
Y\times_A B \ar[r] \ar[ur]^{g_2} \ar[d] \pbcorner & G \ar[u]_{u_2} \ar[d] \ar[r] \pbcorner & B \ar[d]^{f}\\
Y \ar[r]^-{g} \ar@/{}_{20pt}/[rr]_{g_1} & \sum_{a:A}X^{B_a} \ar[r]^-{u_1} & A
}
\end{equation}
Now consider the case of the polynomial functor $P = P_p$ of a natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$, with the form
\begin{equation}\label{eq:polyuniv}
P(X)\ =\ \sum_{\term{A}{\ensuremath{\mathcal{U}}}}X^{A},
\end{equation}
where we write simply $A = \ensuremath{\widetilde{\mathcal{U}}}_A$ for the fiber of $p$ over $A:\ensuremath{\mathcal{U}}$. Thus is justified by considering the pullback
\begin{equation}\label{diag:conext2}
\xymatrix{
\ext{1}{A} \ar[d] \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
1 \ar[r]_{A} & \ensuremath{\mathcal{U}} ,}
\end{equation}
as the case of \eqref{diag:conext} where $\ensuremath{\Gamma} =1$ is terminal, and therefore $\ext{1}{A} = A$ is just an object of \ensuremath{\mathbb{C}}, i.e.\ a ``closed type".
Applying Lemma \ref{lem:polyclass} to \eqref{eq:polyuniv} in the case $f = p$ and $X=\ensuremath{\mathcal{U}}$ and $Y = \ensuremath{\Gamma}$ representable, we obtain a natural, bijective correspondence:
\begin{equation}\label{class:cont}
\begin{prooftree}
\[
(A,B): \ensuremath{\Gamma} \to \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A}
\Justifies
\thickness=1em
\xymatrix{
\ensuremath{\mathcal{U}} & \ext{\ensuremath{\Gamma}}{A} \ar[l]_{B} \ar[d]\ar[r]\pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
&\ensuremath{\Gamma} \ar[r]_{A} & \ensuremath{\mathcal{U}}
}
\]
\Justifies
\thickness=1em
\Gtypes{A},\quad \types{\ext{\ensuremath{\Gamma}}{A}}{B}
\end{prooftree}
\end{equation}
Thus just as $\ensuremath{\mathcal{U}}$ \emph{classifies types in context} $\Gtypes{A}$, we can say that $P(\ensuremath{\mathcal{U}})\ =\ \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A}$ \emph{classifies types in an extended context} $\types{\ext{\ensuremath{\Gamma}}{A}}{B}$. For the record:
\begin{proposition}\label{prop:polyclasstypes}
The presheaf $P(\ensuremath{\mathcal{U}})\ =\ \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A}$ classifies types in context $\types{A}{B}$, in the sense that there is a natural isomorphism between maps $\ensuremath{\Gamma} \to \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A}$ and pairs $\types{\ensuremath{\Gamma}}{A}$ and $\types{\ext{\ensuremath{\Gamma}}{A}}{B}$, as displayed in the following diagram.
\begin{equation}\label{diag:polyclasstypes}
\xymatrix{
& \ensuremath{\mathcal{U}} & \\
\ext{\ensuremath{\Gamma}}{A} \ar[r] \ar[ur]^{B} \ar[d] \pbcorner & \cdot \ar[u] \ar[d] \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ensuremath{\Gamma} \ar[r] \ar@/{}_{20pt}/[rr]_{A} & \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A} \ar[r] & \ensuremath{\mathcal{U}}
}
\end{equation}
\end{proposition}
\proof{
This is just diagram \eqref{diag:polyclassnat} specialized to the present case.
}
\subsection{Products}
\begin{proposition}\label{prop:prod} Let $P(X) =\sum_{\term{A}{\ensuremath{\mathcal{U}}}}X^{A} $ be the polynomial functor associated to a natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$.
Then $p$ also models the product rules of type theory just in case there are maps
\begin{align}
\lambda :&\ P(\ensuremath{\widetilde{\mathcal{U}}})\to \ensuremath{\widetilde{\mathcal{U}}} \label{prop:prod1}\\
\Pi :&\ P(\ensuremath{\mathcal{U}}) \to \ensuremath{\mathcal{U}} \label{prop:prod2}
\end{align}
making the following diagram a pullback.
\begin{equation}\label{diag:prod}
\xymatrix{
P(\ensuremath{\widetilde{\mathcal{U}}}) \ar[d]_{P(p)} \ar[r]^-{\lambda} & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
P(\ensuremath{\mathcal{U}}) \ar[r]_-{\Pi} & \ensuremath{\mathcal{U}} }
\end{equation}
\end{proposition}
\begin{proof}
Replacing $P$ by its definition, we obtain a diagram of the form:
\begin{equation}\label{diag:prod2}
\xymatrix{
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A} \ar[d] \ar[rr]^-{\lambda} && \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \ar[rr]_-{\Pi} && \ensuremath{\mathcal{U}} }
\end{equation}
Using proposition \ref{prop:polyclasstypes}, which states that $\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}$ classifies pairs $\Gtypes{A} $ and $\types{\ext{\ensuremath{\Gamma}}{A}}{B}$, the operation $\Pi: \sum_{\term{A}{\ensuremath{\mathcal{U}}}}\ensuremath{\mathcal{U}}^{A} \to \ensuremath{\mathcal{U}}$ is seen to be the type-theoretic \emph{formation rule},
\[\tag{\text{$\Pi$-form}}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ext{\ensuremath{\Gamma}}{A}}{B}
\justifies
\Gtypes{\prod_{A}{B}}.
\end{prooftree}
\]
Now just as $P(\ensuremath{\mathcal{U}})\ =\ \sum_{A:\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}$ classifies pairs of the form $$\Gtypes{A}, \quad \types{\ext{\ensuremath{\Gamma}}{A}}{B},$$ so $P(\ensuremath{\widetilde{\mathcal{U}}})\ =\ \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A}$ classifies pairs of the form $$\Gtypes{A}, \quad \terms{\ext{\ensuremath{\Gamma}}{A}}{b:B}.$$ This follows from lemma \ref{lem:polyclass} just as did proposition \ref{prop:polyclasstypes}, but replacing the presheaf of types $\ensuremath{\mathcal{U}}$ by the presheaf of terms $\ensuremath{\widetilde{\mathcal{U}}}$.
Thus the operation $\lambda: \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A} \to \ensuremath{\widetilde{\mathcal{U}}}$ models the type-theoretic \emph{introduction rule},
\[\tag{\text{$\Pi$-intro}}
\begin{prooftree}
\Gtypes{A} \quad \terms{\ext{\ensuremath{\Gamma}}{A}}{b:B}
\justifies
\Gterms{\lambda_A b:\prod_{A}B}.
\end{prooftree}
\]
Consider the \emph{elimination rule}:
\[\tag{\text{$\Pi$-elim}}
\begin{prooftree}
\Gterms{f:\prod_{A}B} \qquad \Gterms{a:A}
\justifies
\Gterms{f(a):B[a]}
\end{prooftree}
\]
and the associated \emph{computation rule}:
\[\tag{\text{$\Pi$-comp}}
\begin{prooftree}
\terms{\ext{\ensuremath{\Gamma}}{A}}{b:B} \qquad \Gterms{a:A}
\justifies
\Gterms{(\lambda_A b)(a) = b[a] : B[a]}
\end{prooftree}
\]
The notation $B[a]$ and $b[a]$ is interpreted as follows: given $\Gterms{a:A}$, we have
\[
\xymatrix{
& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\Gamma \ar[ru]^{a} \ar[r]_{A} & \ensuremath{\mathcal{U}}}
\]
and so, by taking a pullback,
\begin{equation*}
\xymatrix{
\ensuremath{\Gamma} \ar@/{}_{1pc}/[ddr]_{1} \ar@{.>}[dr]|-{(1,a)} \ar@/{}^{1pc}/[drr]^{a} \\
& \ext{\ensuremath{\Gamma}}{A} \ar[d] \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
& \ensuremath{\Gamma} \ar[r]_{A} & \ensuremath{\mathcal{U}} ,}
\end{equation*}
we get a substitution $(1,a) : \ensuremath{\Gamma}\to \ext{\ensuremath{\Gamma}}{A}$ into the context extension $\ext{\ensuremath{\Gamma}}{A}$.
Now $\types{\ext{\ensuremath{\Gamma}}{A}}{B}$ and $\terms{\ext{\ensuremath{\Gamma}}{A}}{b:B}$ are of the form
\[
\xymatrix{
& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ext{\ensuremath{\Gamma}}{A} \ar[ru]^{b} \ar[r]_{B} & \ensuremath{\mathcal{U}},}
\]
so we can set
\begin{align*}
B[a] &= B\circ (1, a) \\
b[a] &= b\circ (1, a),
\end{align*}
as indicated in
\[
\xymatrix{
&& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ensuremath{\Gamma} \ar@/{}^{20pt}/[rru]^{b[a]} \ar[r]^-{(1, a)} \ar@/{}_{20pt}/[rr]_{B[a]} & \ext{\ensuremath{\Gamma}}{A} \ar[ru]^{b} \ar[r]^{B} & \ensuremath{\mathcal{U}},}
\]
to get the terms
\begin{align*}
\Gtypes{B[a]} \\
\Gterms{b[a] : B[a]}.
\end{align*}
Now suppose that \eqref{diag:prod2} is a pullback. We require a term $\Gterms{f(a):B[a]}$, assuming we have the premises $\Gterms{f:\prod_{A}B}$ and $\Gterms{a:A}$. The first premise means there are maps $(A, B)$ and $f$ as indicated in
\begin{equation}\label{diag:pielim1}
\xymatrix{
\ensuremath{\Gamma} \ar@/{}_{1pc}/[ddr]_{(A,B)} \ar@{.>}[dr]|-{(A,\tilde{f})} \ar@/{}^{1pc}/[drr]^{f} \\
& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A} \ar[d]^{P(p)} \ar[r]_{\lambda} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
&\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \ar[r]_{\Pi} & \ensuremath{\mathcal{U}} .}
\end{equation}
Since the square is a pullback, there is a unique map as indicated, and by the classifying property of $\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A}$, it has the form $(A, \tilde{f})$ for a unique $\terms{\ext{\ensuremath{\Gamma}}{A}}{\tilde{f}:B}$,
\[
\xymatrix{
& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ext{\ensuremath{\Gamma}}{A} \ar[ru]^{\tilde{f}} \ar[r]_{B} & \ensuremath{\mathcal{U}}.}
\]
Now set:
\[
f(a) = \tilde{f}[a] = \tilde{f}\circ (1, a),
\]
so that indeed $\terms{\ext{\ensuremath{\Gamma}}{A}}{f(a) = \tilde{f}[a]:B[a]}$, as required.
For the computation rule, suppose $\terms{\ext{\ensuremath{\Gamma}}{A}}{b:B}$. Then $\Gterms{\lambda_A b: \prod_{A}B}$ and diagram \eqref{diag:pielim1} becomes
\begin{equation}\label{diag:pielim2}
\xymatrix{
\ensuremath{\Gamma} \ar@/{}_{1pc}/[ddr]_{(A,B)} \ar@{.>}[dr]|-{(A,\widetilde{\lambda_A b})} \ar@/{}^{1pc}/[drr]^{\lambda_A b} \\
& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A} \ar[d]^{P(p)} \ar[r]_-{\lambda} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
&\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \ar[r]_-{\Pi} & \ensuremath{\mathcal{U}} .}
\end{equation}
for a unique $\terms{\ext{\ensuremath{\Gamma}}{A}}{\widetilde{\lambda_A b}:B}$,
\[
\xymatrix{
& \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\ext{\ensuremath{\Gamma}}{A} \ar[ru]^{\widetilde{\lambda_A b}} \ar[r]_{B} & \ensuremath{\mathcal{U}}.}
\]
But clearly $\terms{\ext{\ensuremath{\Gamma}}{A}}{b:B}$ also satisfies this condition, so we have
\[
\terms{\ext{\ensuremath{\Gamma}}{A}}{\widetilde{\lambda_A b} = b:B}.
\]
But then
\[
\Gterms{(\lambda_A b)(a) = (\widetilde{\lambda_A b})[a] = b[a] : B[a]}.
\]
as required.
The converse is equally straightforward and is omitted
\end{proof}
\begin{remark}\label{remark:substitution}
The type and term constructors $\Pi$, $\lambda$, and $-_1(-_2)$ occurring in the formation, introduction, and elimination rules are required to respect substitutions $\sigma : \Delta\to \ensuremath{\Gamma}$. Specifically, consider e.g.\ the $\Pi$-formation rule:
\begin{equation}\label{eq:piform}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ext{\ensuremath{\Gamma}}{A}}{B}
\justifies
\Gtypes{\prod_{A}{B}}.
\end{prooftree}
\end{equation}
Applying $\sigma : \Delta\to \ensuremath{\Gamma}$ to the premises gives a new instance of the rule
\[
\begin{prooftree}
\types{\Delta}{A\sigma} \qquad \types{\ext{\Delta}{A\sigma}}{B\sigma}
\justifies
\types{\Delta}{\prod_{A\sigma}{B\sigma}}.
\end{prooftree}
\]
On the other hand, one can instead apply $\sigma : \Delta\to \ensuremath{\Gamma}$ to the conclusion of \eqref{eq:piform}
to obtain
\[
\types{\Delta}{(\prod_{A}{B})\sigma}.
\]
It is part of the definition of ``modelling the product rules in a category with families" that these two things should be the same,
\[
\types{\Delta}{(\prod_{A}{B})\sigma = \prod_{A\sigma}{B\sigma} }
\]
as elements of $\ensuremath{\mathcal{U}}(\Delta)$. But indeed, we have
\begin{align*}
(\prod_{A}{B})\sigma &= (\Pi \circ (A,B))\circ\sigma \\
&= \Pi \circ ((A,B)\circ\sigma) \\
&= \Pi \circ (A\sigma, B\sigma) \\
&= \prod_{A\sigma}{B\sigma}
\end{align*}
as indicated in the following diagram
\[
\xymatrix{
\Delta \ar@/{}_{15pt}/[rd]_-{(A\sigma,B\sigma)} \ar[r]^{\sigma} & \ensuremath{\Gamma} \ar[d]_-{(A,B)} \ar@/{}^{10pt}/[rd]^-{\prod_{A\sigma}{B\sigma}}& \\
& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \ar[r]_-{\Pi} & \ensuremath{\mathcal{U}}}
\]
where the equation
\[
(A,B)\circ\sigma \ = \ (A\sigma, B\sigma)
\]
follows easily from proposition \ref{prop:polyclasstypes}. The other two required equations,
\begin{align*}
(\lambda_{A}{b})\sigma &= \lambda_{A\sigma}({b\sigma}) \\
(f(a))\sigma &= (f\sigma)(a\sigma)
\end{align*}
follow similarly.
\end{remark}
\begin{remark}
Our specification differs from that in \cite{KLV} mainly in requiring the square \eqref{diag:prod} to be a pullback, rather than applying the $\Pi$ functor in presheaves to the generic case.
\end{remark}
\subsection{Sums}
For the sum constructor $\Sigma$ we shall replace the family
\[
\xymatrix{
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\widetilde{\mathcal{U}}}^{A} \ar[r] & \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}
}
\]
in diagram \eqref{diag:prod2}
by a different one, corresponding to the different premises of the $\Sigma$-\emph{introduction rule},
\[\tag{\text{$\Sigma$-intro}}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ext{\ensuremath{\Gamma}}{A}}{B} \qquad \Gterms{a:A} \qquad \Gterms{b:B[a]}
\justifies
\Gterms{\pair{a, b}:\sum_{A}B}.
\end{prooftree}
\]
The base object $\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}$ remains the same, corresponding to the fact that the $\Sigma$-\emph{formation rule} has the same form as the one for $\Pi$, namely,
\[\tag{\text{$\Sigma$-form}}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ext{\ensuremath{\Gamma}}{A}}{B}
\justifies
\Gtypes{\sum_{A}{B}}.
\end{prooftree}
\]
But the object over $\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}$ must now classify data of the form
\begin{equation}\label{eq:sigmadata}
\big( \Gtypes{A},\ \types{\ext{\ensuremath{\Gamma}}{A}}{B},\ \Gterms{a:A},\ \Gterms{b:B[a]} \big).
\end{equation}
This is accomplished with the following object (again constructed using the internal language in presheaves):
\[
\sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a)
\]
We have a projection $\pi$ associated to the first two sums,
\[
\xymatrix{
\sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a) \ar[r]^{\cong} \ar[rd]_{\pi} & \sum_{(A, B)\ensuremath{\,:\,}(\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A})} \sum_{(a : A)}B(a) \ar[d] \\
& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}\,,
}
\]
and factorizations of maps of the form $(A,B):\ensuremath{\Gamma} \to \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}$ through $\pi$,
\[
\xymatrix{
& \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a) \ar[d]^{\pi}\\
\ensuremath{\Gamma} \ar[r]_{(A,B)} \ar@{.>}[ru] & \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} }
\]
are then in natural, bijective correspondence with data of the form \eqref{eq:sigmadata}, as can be proved similarly to proposition \ref{prop:polyclasstypes}.
\begin{proposition}\label{prop:sum}
A natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$ satisfies the type theoretic rules for (strong) dependent sums just in case there are maps
\begin{align}\label{eq:sumop1}
\ensuremath{\mathsf{pair}} :& \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a) \to \ensuremath{\widetilde{\mathcal{U}}}\\
\Sigma :& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \to \ensuremath{\mathcal{U}}\label{eq:sumop2}
\end{align}
making the following diagram a pullback.
\begin{equation}\label{diag:sigma}
\xymatrix{
\sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a) \ar[d]_{\pi} \ar[rr]^-{\ensuremath{\mathsf{pair}}} && \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A} \ar[rr]_-{\Sigma} && \ensuremath{\mathcal{U}} }
\end{equation}
\end{proposition}
\begin{proof}
The operations \eqref{eq:sumop1} and \eqref{eq:sumop2} clearly give the $\Sigma$-introduction and formation rules, respectively.
We shall prove the \emph{strong} elimination rule, which has the two parts
\[\tag{\text{$\Sigma$-elim}}
\begin{prooftree}
\Gterms{c:\sum_{A}B}
\justifies
\Gterms{\pi_1(c) : A}
\end{prooftree}
\qquad\qquad
\begin{prooftree}
\Gterms{c:\sum_{A}B}
\justifies
\Gterms{\pi_2(c) : B[\pi_1(c)]}
\end{prooftree}
\]
with associated $\Sigma$-\emph{computation rules}:
\begin{align*}\tag{\text{$\Sigma$-comp}}
\pi_1(\pair{a,b}) &= a \\
\pi_2(\pair{a,b}) &= b \\
\pair{\pi_1(c), \pi_2(c)} &= c
\end{align*}
To show this, assume that \eqref{diag:sigma} is a pullback, let $\ensuremath{\Gamma}$ be any object, and suppose that we have $(A, B): \ensuremath{\Gamma}\to (\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A})$ and $c : \ensuremath{\Gamma}\to\ensuremath{\widetilde{\mathcal{U}}}$ such that $\Sigma\circ a = p\circ c : \ensuremath{\Gamma}\to\ensuremath{\mathcal{U}}$, which means exactly that $\terms{\ensuremath{\Gamma}}{c : \sum_{A}B}$. There is then a unique map
\[
\tilde{c} : \ensuremath{\Gamma} \to \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a)
\]
with $\pi\circ\tilde{c} = (A, B)$ and $\ensuremath{\mathsf{pair}}\circ\tilde{c} = c$. Since $\tilde{c}$ is known to be uniquely of the form \eqref{eq:sigmadata}, we can write it as $$\tilde{c} = \big(A, B, \pi_1(c), \pi_2(c)\big)$$ with $\pi_1(c):A$ and $\pi_2(c): B[\pi_1(c)]$. We then write $$\ensuremath{\mathsf{pair}}\big(A, B, \pi_1(c), \pi_2(c)\big) = \pair{\pi_1(c), \pi_2(c)}$$ accordingly. Thus indeed $c = \pair{\pi_1(c), \pi_2(c)}$, as required. To prove the other two $\Sigma$-computation equations, it suffices by the uniqueness of elements classified to show that, for any $a:A$ and $b:B[a]$, we have
$$(A, B, a, b) = \big(A, B, \pi_1(\pair{a,b}), \pi_2(\pair{a,b})\big).$$
But this is now clear, since
\begin{align*}
\ensuremath{\mathsf{pair}}(A, B, a, b) &= \pair{a,b}\\
&= \pair{\pi_1(\pair{a,b}), \pi_2(\pair{a,b})}\\
&= \ensuremath{\mathsf{pair}}(A, B, \pi_1(\pair{a,b}), \pi_2(\pair{a,b})).
\end{align*}
Again, the converse is just as direct.
\end{proof}
As in the case of products, the operations $\Sigma$, $\pi_1, \pi_2$, and $\ensuremath{\mathsf{pair}}$ can easily be shown to respect substitution $\sigma: \Delta\to\ensuremath{\Gamma}$.
\begin{remark}
The map
\begin{equation}
\xymatrix{
\sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{U}})}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{U}}^{A})}\sum_{(a : A)}B(a) \ar[d]_{\pi} \\
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}\ensuremath{\mathcal{U}}^{A}
}
\end{equation}
from \eqref{diag:sigma} can also be understood in terms of polynomial functors. As in \eqref{eq:polydef}, let $$P_p(X) = \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{U}}}X^{A}$$ be the polynomial functor $P_p : \widehat{\ensuremath{\mathbb{C}}}\to\widehat{\ensuremath{\mathbb{C}}}$ determined by the map $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$. The map $\pi$ above also determines a polynomial functor $P_\pi : \widehat{\ensuremath{\mathbb{C}}}\to\widehat{\ensuremath{\mathbb{C}}}$, again via \eqref{eq:polydef}. These are related by
\[
P_\pi = P_p \circ P_p\ .
\]
Thus in particular the composite ${P_p}^2 = P_p\circ P_p : \widehat{\ensuremath{\mathbb{C}}}\to\widehat{\ensuremath{\mathbb{C}}}$ is also polynomial, and $\pi$ is the map representing it. Moreover, recall from \cite{GK} that pullback diagrams of maps
\begin{equation*}\label{diag:pbpoly}
\xymatrix{
B \ar[d]_{f} \ar[r] \pbcorner & D \ar[d]^{g}\\
A \ar[r] & C }
\end{equation*}
in $\widehat{\ensuremath{\mathbb{C}}}$ correspond to morphisms of the polynomial functors on $\widehat{\ensuremath{\mathbb{C}}}$ that they determine, $P_f \Rightarrow P_g$ (cartesian natural transformations).
Thus the pullback condition \eqref{diag:sigma} says that there is a map of polynomial functors $P_p \circ P_p \Rightarrow P_p$\,. It is easy to see that there is also a map $1_\ensuremath{\mathbb{C}} \Rightarrow P_p$, determined by the terminal object of \ensuremath{\mathbb{C}}, with its unique term,
\begin{equation}
\xymatrix{
1 \ar[d] \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
1 \ar[r] & \ensuremath{\mathcal{U}} .}
\end{equation}
The further investigation of this structure is left to future work (cf.~\cite{PTJ:BD}, section 2, for a related development).
\end{remark}
\begin{remark}\label{remark:catoftypes}
We mention only in passing the full, internal category $\mathbb{U}$ in $\widehat{\ensuremath{\mathbb{C}}}$ determined by $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$, which may be called the \emph{category of types}. This presheaf of categories has $\mathbb{U}_0 = \ensuremath{\mathcal{U}}$ as its object of objects, and as object of arrows $\mathbb{U}_1$ the exponential
\[
\xymatrix{
B^A \ar[d] \\
\ensuremath{\mathcal{U}} \times \ensuremath{\mathcal{U}}}
\]
in $\widehat{\ensuremath{\mathbb{C}}}/(\ensuremath{\mathcal{U}} \times \ensuremath{\mathcal{U}})$, where we have written $A = p_1^*(\ensuremath{\widetilde{\mathcal{U}}})$ and $B = p_2^*(\ensuremath{\widetilde{\mathcal{U}}})$ for the results of pulling $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$ back along the two projections $p_1, p_2 : \ensuremath{\mathcal{U}} \times \ensuremath{\mathcal{U}} \to \ensuremath{\mathcal{U}}$. This internal category can be seen to be cartesian closed in virtue of the rules just given for $1$, $\Sigma$, and $\Pi$. This essentially amounts to saying that the category of all types $\Gtypes{A}$ in a given context $\ensuremath{\Gamma}$ is always cartesian closed, and substitution $\Delta \to \ensuremath{\Gamma}$ preserves the cartesian closed structure.
\end{remark}
\subsection{Extensional identity}
The formation and introduction rues for identity types are as follows.
\[\tag{$\mathsf{Id}$-form}
\begin{prooftree}
\Gtypes{A} \quad
\Gterms{a : A} \quad
\Gterms{b : A}
\justifies
\Gtypes{\id{A}(a,b)}
\end{prooftree}
\]
\smallskip
\[\tag{$\mathsf{Id}$-intro}
\begin{prooftree}
\Gterms{a : A}
\justifies
\Gterms{\mathsf{i}(a) : \id{A}(a,a)}
\end{prooftree}
\]
To interpret these, we use the ``diagonal" map $\delta:\ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}}\ensuremath{\widetilde{\mathcal{U}}}$ of a natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$, formed by first taking the pullback of $p$ against itself, and then factoring the identity morphism $1:\ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\widetilde{\mathcal{U}}}$ as indicated in the following diagram.
\[
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar@/{}_{1pc}/[ddr]_{1} \ar@{.>}[dr]|-{\delta} \ar@/{}^{1pc}/[drr]^{1} &&\\
& \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r] \ar[d] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
& \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_p & \ensuremath{\mathcal{U}}
}
\]
\begin{proposition}\label{prop:extid}
A natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$ satisfies the type theoretic rules for \emph{extensional} identity types just in case there are maps
\begin{align}
\mathsf{i} &: \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\widetilde{\mathcal{U}}} \label{prop:exteq1}\\
\mathsf{Id} &: \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}} \label{prop:exteq2}
\end{align}
making the following diagram a pullback.
\begin{equation}\label{diag:extid}
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar[r]^\mathsf{i} \ar[d]_-\delta & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
\ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_-\mathsf{Id} & \ensuremath{\mathcal{U}}
}
\end{equation}
\end{proposition}
\begin{proof}
Since maps $(A, a,b) : \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}}$ correspond naturally to pairs of terms $\Gterms{a,b:A}$ of the same type $\Gtypes{A}$,
we can set
\[
\id{A}(a,b) = \mathsf{Id}\circ (A,a,b).
\]
Then $\mathsf{Id} : \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$ validates the $\mathsf{Id}$-formation rule.
Moreover, given any element $a : \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}$, the commutativity of \eqref{diag:extid} means exactly that
$\Gterms{ \mathsf{i}\circ a : \id{A}(a,a)}$. So setting
\[
\mathsf{i}(a) = \mathsf{i} \circ a
\]
also gives the introduction rule. The interpretation of the formation and introduction rules is then displayed by the following diagram.
\begin{equation}\label{diag:intformintroextid}
\xymatrix{
&& \ensuremath{\widetilde{\mathcal{U}}} \ar[r]^\mathsf{i} \ar[d]^-\delta & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
\ensuremath{\Gamma} \ar@/{}^{1pc}/[rru]^a \ar[rr]^{(A,a,b)} \ar[rrd]_A && \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[d] \ar[r]_-\mathsf{Id} & \ensuremath{\mathcal{U}} \\
&& \ensuremath{\mathcal{U}}
}
\end{equation}
Suppose \eqref{diag:extid} is a pullback. Then for any $(A,a,b) : \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}}$ and $c: \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}$ such that $\mathsf{Id}(A,a,b) = p\circ c$, meaning that
\[
\Gterms{c : \id{A}(a,b)},
\]
there is a unique
$u : \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}$
with $\delta\circ u = (A,a,b)$ and $\mathsf{i}\circ u = c$. But this means that
\[
\Gterms{a = b : A} \qquad\text{and}\qquad \Gterms{c = \mathsf{i}(a) : \id{A}(a,a)}.
\]
Thus we have the standard rules for \emph{extensional} Identity types:
\[\tag{ext-$\mathsf{Id}$-elim}
\begin{prooftree}
\Gterms{c : \id{A}(a,b)}
\justifies
\Gterms{a = b : A}
\end{prooftree}
\qquad\qquad
\begin{prooftree}
\Gterms{c : \id{A}(a,b)}
\justifies
\Gterms{c = \mathsf{i}(a) : \id{A}(a,a)}
\end{prooftree}
\]
The converse is, again, equally direct.
\end{proof}
Summing up:
\begin{theorem}\label{thm:extid}
A natural model of \emph{extensional} Martin-L\"of type theory with product, sum, and identity types is given by a small category \ensuremath{\mathbb{C}}\ equipped with a representable map of presheaves, $$p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}},$$ together with maps,
\[
\Pi,\ \lambda,\ \Sigma,\ \ensuremath{\mathsf{pair}},\ \mathsf{Id},\ \mathsf{i},
\]
as in propositions \ref{prop:prod}, \ref{prop:sum}, and \ref{prop:extid}, such that the squares \eqref{diag:prod}, \eqref{diag:sigma}, and \eqref{diag:extid} are pullbacks.
\end{theorem}
\subsection{Intensional identity}
Models of extensional type theory can be obtained easily from locally cartesian closed categories by various methods, including \cite{Hofmann}. We are mainly interested here in models of the \emph{intensional} theory. The formation and introduction rules remain the same as in the extensional case, but the elimination rule takes a somewhat more complex form inspired by inductive definitions.
As in \eqref{prop:exteq1} and\eqref{prop:exteq2}, we assume maps
\begin{align*}
\mathsf{i} &: \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\widetilde{\mathcal{U}}} \\%\label{prop:inteq1}
\mathsf{Id} &: \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}
\end{align*}
but now we merely require the resulting square to commute:
\begin{equation}\label{diag:intid}
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar[r]^\mathsf{i} \ar[d]_-\delta & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
\ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_-{\mathsf{Id}} & \ensuremath{\mathcal{U}}
}
\end{equation}
Once again we can set
\[
\id{A}(a,b) = \mathsf{Id}\circ (A,a,b)
\]
to validate the $\mathsf{Id}$-formation rule:
\[\tag{$\mathsf{Id}$-form}
\begin{prooftree}
\Gtypes{A} \quad
\Gterms{a : A} \quad
\Gterms{b : A}
\justifies
\Gtypes{\id{A}(a,b)}
\end{prooftree}
\]
Also as before, given any element $a : \ensuremath{\Gamma} \to \ensuremath{\widetilde{\mathcal{U}}}$, we have
$\Gterms{\mathsf{i}\circ a : \id{A}(a,a)}$. So setting
\[
\mathsf{i}(a) = \mathsf{i}\circ a
\]
again gives the introduction rule:
\[\tag{$\mathsf{Id}$-intro}
\begin{prooftree}
\Gterms{a : A}
\justifies
\Gterms{\mathsf{i}(a) : \id{A}(a,a)}
\end{prooftree}
\]
Now take the pullback of $p$ along $\mathsf{Id}$. We obtain an object $I$ over $\ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}}$, together with a factorization $\rho = (\delta, \mathsf{i})$ of the diagonal $\delta$:
%
\begin{equation*}
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar@/{}_{1pc}/[ddr]_{\delta} \ar@{.>}[dr]|-{\rho} \ar@/{}^{1pc}/[drr]^{\mathsf{i}} &&\\
& I \ar[r] \ar[d] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
& \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_-{\mathsf{Id}} & \ensuremath{\mathcal{U}}
}
\end{equation*}
This structure serves as a ``generic identity type".
Indeed, consider the following diagram, in which the parallel vertical arrows are the evident projections, and the indicated squares are constructed as pullbacks.
\begin{equation}\label{diag:big}
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar[d]_{p} & \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \ar[l]_c \ar[d]^{\rho_A} \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]_{\rho} \ar[dr]^{\mathsf{i}} & \\
\ensuremath{\mathcal{U}} & \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A \ar[l]^-C \ar@{.>}[lu]_{\mathsf{j}(c)} \ar[d] \ar[r] \pbcorner & I \ar[d] \ar[r] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p} \\
& {\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A} \ar@<-1ex>[d] \ar@<.5ex>[d] \ar[r]_{q^2_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_-{\mathsf{Id}} \ar@<-1ex>[d] \ar@<.5ex>[d] & \ensuremath{\mathcal{U}} \\
& \ext{\ensuremath{\Gamma}}{A} \ar[d]_{p_A} \ar[r]_{q_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p} & \\
& \ensuremath{\Gamma} \ar[r]_A & \ensuremath{\mathcal{U}} &
}
\end{equation}
The interpretation of $\types{\ensuremath{\Gamma} \ensuremath{\!\centerdot\!} A \ensuremath{\!\centerdot\!} A}{\mathsf{Id}_A}$ is the center horizontal composite
$$\mathsf{Id}\circ q^2_A\ :\ \ttexdot{\ensuremath{\Gamma}}{A}{A }\to \ensuremath{\mathcal{U}},$$
and so the context extension $\qtexdot{\ensuremath{\Gamma}}{A}{A}{\mathsf{Id}_A}$ is the indicated pullback of $I$ along $q^2_A$.
Observe that $$\ttexdot{\ensuremath{\Gamma}}{A}{A} = (\btexdot{\ensuremath{\Gamma}}{A})\times_\ensuremath{\Gamma} (\btexdot{\ensuremath{\Gamma}}{A}),$$ and that the map
\begin{equation}\label{eq:substrho}
\rho_A : \btexdot{\ensuremath{\Gamma}}{A}\to \qtexdot{\ensuremath{\Gamma}}{A}{A}{\mathsf{Id}_A}
\end{equation}
factors the diagonal $\delta_A : \btexdot{\ensuremath{\Gamma}}{A}\to \ttexdot{\ensuremath{\Gamma}}{A}{A}$, because it is the pullback of ${\rho : \ensuremath{\widetilde{\mathcal{U}}}\to I}$, which factors the diagonal $\ensuremath{\widetilde{\mathcal{U}}}\to \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}}\ensuremath{\widetilde{\mathcal{U}}}$. The map $\rho_A$ interprets the substitution $a \mapsto (a,a,\mathsf{i}(a))$ associated to the introduction term $\mathsf{i}$.
We can now state the \emph{$\mathsf{Id}$-elimination rule} as follows:
\smallskip
\[
\tag{$\mathsf{Id}$-elim}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} \id{A}}{C}
\qquad \terms{\ext{\ensuremath{\Gamma}}{A}}{c : C\rho_A}
\justifies
\terms{\ensuremath{\Gamma}\!\centerdot\! A\!\centerdot\! A\!\centerdot\! \id{A}}{\mathsf{j}(c):C}
\end{prooftree}
\]
where the indicated substitution $C\rho_A$ is taken along the map $\rho_A$ just defined \eqref{eq:substrho}.
The associated \emph{computation rule} then has the form:
\[\tag{$\mathsf{Id}$-comp}
\begin{prooftree}
\Gtypes{A} \qquad \types{\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} \id{A}}{C}
\qquad \terms{\ext{\ensuremath{\Gamma}}{A}}{c : C\rho_A}
\justifies
\terms{\ensuremath{\Gamma}\!\centerdot\! A}{\mathsf{j}(c)\rho_A = c : C\rho_A}
\end{prooftree}
\]
The elimination and computation rules are interpreted in the upper left square of the diagram \eqref{diag:big}, where the dotted arrow $\mathsf{j}(c)$ indicates a choice of diagonal filler interpreting the corresponding term. Since the rules are supposed to hold for all types $C$ and terms $c$, they are evidently equivalent to the following condition.
\begin{quote}
\emph{For any $\ensuremath{\Gamma}:\type$ and $\Gtypes{A}$, the substitution $\rho_A$ has the left-lifting property with respect to $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$}
\end{quote}
\begin{remark}
Let us consider the requirement that the rules must respect substitution, in the sense of remark \ref{remark:substitution}, for the present case. The formation and introduction rules clearly satisfy this condition, since they are modeled by composition with particular maps. Indeed, consider the diagram \eqref{diag:intformintroextid}, which gives the interpretation of formation and introduction, and take any substitution $\sigma:\Delta\to\ensuremath{\Gamma}$.
\begin{equation}\label{diag:intformintroextidsubst}
\xymatrix{
&&& \ensuremath{\widetilde{\mathcal{U}}} \ar[r]^\mathsf{i} \ar[d]^-\delta & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
\Delta \ar@/{}^{1pc}/[rrru]^{a\sigma} \ar[r]_{\sigma} \ar@/{}_{1pc}/[rrrd]_-{(A\sigma,a\sigma,b\sigma)} & \ensuremath{\Gamma} \ar[rru]^a \ar[rr]^{(A,a,b)} \ar[rrd]^A && \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[d] \ar[r]_-\mathsf{Id} & \ensuremath{\mathcal{U}} \\
&&& \ensuremath{\mathcal{U}}
}
\end{equation}
As indicated in the diagram above, we then have the required conditions:
\begin{align*}
\id{A}(a,b)\sigma &= (\mathsf{Id}\circ(A,a,b))\sigma = \mathsf{Id}\circ((A,a,b)\sigma) \\
&= \mathsf{Id}\circ(A\sigma,a\sigma,b\sigma) = \id{A\sigma}(a\sigma,b\sigma)\\
\mathsf{i}(a)\sigma &= (\mathsf{i}\circ a)\sigma = \mathsf{i}\circ (a\circ \sigma) = \mathsf{i}\circ (a\sigma) = \mathsf{i}(a\sigma).
\end{align*}
The corresponding condition for the elimination rule has the form:
\begin{equation*}
\mathsf{j}(c)\sigma = \mathsf{j}(c\sigma).
\end{equation*}
More precisely, for any substitution $\sigma :\Delta\to \ensuremath{\Gamma}$, we require that
\begin{equation}\label{eq:BCforJ}
\mathsf{j}(c)\sigma_{\mathsf{Id}_{A}} = \mathsf{j}(c\sigma_{A}),
\end{equation}
as indicated in the following diagram.
\begin{equation*
\xymatrix{
\Delta\ensuremath{\!\centerdot\!} A\sigma \ar[d]_{\rho_A\sigma} \ar[rr]^{\sigma_A} \pbcorner && \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \ar[d]^{\rho_A} \ar[r]^c & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p} \\
\Delta\ensuremath{\!\centerdot\!} A\sigma\ensuremath{\!\centerdot\!} A\sigma\ensuremath{\!\centerdot\!}\mathsf{Id}_{A\sigma} \ar[rr]_-{\sigma_{\mathsf{Id}_{A}}} \ar@{.>}[urrr]^{\mathsf{j}(c\sigma_A)} \ar[d] \ar[rr] \pbcorner && \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A \ar@{.>}[ur]|-{\mathsf{j}(c)} \ar[r]_-C \ar[d] \pbcorner & \ensuremath{\mathcal{U}} \\
\Delta \ar[rr]_\sigma && \ensuremath{\Gamma} &
}
\end{equation*}
In the cases of $\Pi$ and $\Sigma$, the analogous condition followed from the uniqueness of a certain map into a pullback. But in this case, there is no such uniqueness, and one must instead require the existence of a family of maps $\mathsf{j}(c)$, in all situations of the form
\begin{equation}\label{diag:Idfill}
\xymatrix{
\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \ar[d]_{\rho_A} \ar[r]^c & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p} \\
\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A \ar@{.>}[ur]|-{\mathsf{j}(c)} \ar[r]_-C & \ensuremath{\mathcal{U}},
}
\end{equation}
and selected in such a way as to be compatible with all maps of the form $\sigma :\Delta\to \ensuremath{\Gamma}$, in the sense of \eqref{eq:BCforJ}.
We shall take a different approach in what follows: as in the cases of $\Pi$ and $\Sigma$, we shall specify a single map $\mathsf{j}$ in a suitable universal case, which then gives rise to the individual maps $\mathsf{j}(c)$ in a uniform way, which is then automatically natural in the sense of \eqref{eq:BCforJ}.
\end{remark}
We require a preliminary definition.
Let $f : A\to B$ and $g : C\to D$ be maps in a cartesian closed category $\mathcal{C}$, and consider the following square, which always commutes.
\begin{equation}\label{diag:intLLP}
\xymatrix{
C^B \ar[d]_{g^B} \ar[r]^{C^f} & C^A \ar[d]^{g^A} \\
D^B \ar[r]_{D^f} & D^A.
}
\end{equation}
Taking the pullback of $D^f$ and $g^A$, we obtain a canonical comparison map $c = (g^B, C^f)$ as indicated in the following.
\begin{equation}\label{diag:intLLP2}
\xymatrix{
C^B \ar[dd]_{g^B} \ar[rr]^{C^f} \ar@{.>}[dr]|-{\,c\,} && C^A \ar[dd]^{g^A} \\
& D^B\times_{D^A} C^A \ar[ld] \ar[ru] \pbcorner & \\
D^B \ar[rr]_{D^f} && D^A.
}
\end{equation}
Recall that $f$ has the \emph{left lifting property} with respect to~$g$, written $$f \pitchfork g$$ if every commutative square from $f$ to $g$ has at least one diagonal filler,
\[
\xymatrix{
A \ar[d]_{f} \ar[r] & C \ar[d]^{g} \\
B \ar[r] \ar@{.>}[ur] & D.
}
\]
\begin{definition}\label{def:LLP}
A \emph{left-lifting structure $s$ for $f$ with respect to $g$}, written
$$f\ \pitchfork_s\ g$$
is a section of the comparison map $c = (g^B, C^f)$ in \eqref{diag:intLLP},
\begin{equation}
\xymatrix{
C^B \ar[rr]_-{c} && D^B\times_{D^A} C^A \ar@/_{2pc}/@{.>}[ll]^{s}\,.
}
\end{equation}
\end{definition}
\begin{lemma}\label{lem:extintLLP}
Let $f : A\to B$ and $g : C\to D$ be maps in a locally cartesian closed category $\mathcal{C}$.
The following conditions are equivalent.
\begin{enumerate}
\item\label{lem:intLLP} $f$ has a left-lifting structure $s$ with respect to $g$,
$$f\ \pitchfork_s\ g.$$
\item\label{lem:expLLP} For each object $X$ and maps $\alpha, \beta$ as indicated in the diagram below (making the outer, streched square commute),
\begin{equation}\label{diag:expLLP}
\xymatrix{
X \ar@/{}_{1pc}/[rdd]_{\alpha} \ar@{.>}[rd]|-{\gamma(\alpha,\beta)} \ar@/{}^{1pc}/[rrd]^{\beta} & & \\
& C^B \ar[d]^{g^B} \ar[r]_{C^f} & C^A \ar[d]^{g^A} \\
& D^B \ar[r]_{D^f} & D^A.
}
\end{equation}
there is an associated map $\gamma(\alpha,\beta)$ as shown (making the evident triangles commute), and the assignment is natural in $X$ in the sense that for any $u : Y\to X$,
\[
\gamma(\alpha,\beta)\circ u = \gamma(\alpha\circ u,\beta\circ u).
\]
\item\label{lem:extLLP} For all objects $X$, $$(X\times f) \pitchfork g$$ \emph{naturally in $X$}, in the sense that there exists a family $c(a,b)$ of diagonal fillers
\[
\xymatrix{
X\times A \ar[d]_{X\times f} \ar[rr]^{a} && C \ar[d]^{g} \\
X\times B \ar[rr]_{b} \ar@{.>}[urr]|-{c(a,b)} && D
}
\]
that are natural in $X$, meaning that for every $u : Y \to X$,
\[
c(a,b)\circ(u\times B) = c(a\circ(u\times A),b\circ(u\times B))
\]
as in the following diagram, where we have written $c = c(a,b)$ and $c' = c(a\circ(u\times A),b\circ(u\times B))$.
\begin{equation*
\xymatrix{
Y\times A \ar[rr]^{u\times A} \ar[d]_{Y\times f} && X\times A \ar[d]^{^{X\times f} } \ar[r]^-a & C \ar[d]^{g} \\
Y\times B \ar[rr]_{u\times B} \ar@{.>}[urrr]^{c'} && X\times B \ar@{.>}[ur]_{c} \ar[r]_-b & D
}
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{proof} (Sketch)
To show that \ref{lem:intLLP} implies \ref{lem:expLLP}, suppose $f\ \pitchfork_s\ g$ and take any $\alpha$ and $\beta$ making the following commute.
\begin{equation*
\xymatrix{
X \ar@/{}_{1pc}/[rdd]_{\alpha} \ar@/{}^{1pc}/[rrd]^{\beta} & & \\
& C^B \ar[d]^{g^B} \ar[r]_{C^f} & C^A \ar[d]^{g^A} \\
& D^B \ar[r]_{D^f} & D^A.
}
\end{equation*}
Then as $\gamma(\alpha,\beta) : X\to C^B$ we can take the composite map
\[\xymatrix{
X \ar[r]^-{(\alpha,\beta)} & D^B\times_{D^A}C^A \ar[r]^-{s} & C^B.
}
\]
Conversely, in diagram \eqref{diag:expLLP} let $X = D^B\times_{D^A}C^A$ and let $\alpha$ and $\beta$ be the projections from the pullback. The resulting map $\gamma : D^B\times_{D^A}C^A \to C^B$ is then the required section $s$.
Now assume condition \ref{lem:expLLP}. To prove \ref{lem:extLLP}, take any $X$ and suppose we have $a : X\times A\to C$ and $b: X\times B\to D$ making the following commute.
\[
\xymatrix{
X\times A \ar[d]_{X\times f} \ar[rr]^{a} && C \ar[d]^{g} \\
X\times B \ar[rr]_{b} && D.
}
\]
transposing, we obtain a commutative diagram
\begin{equation*
\xymatrix{
X \ar@/{}_{1pc}/[rdd]_{\tilde{a}} \ar@/{}^{1pc}/[rrd]^{\tilde{b}} & & \\
& C^B \ar[d]^{g^B} \ar[r]_{C^f} & C^A \ar[d]^{g^A} \\
& D^B \ar[r]_{D^f} & D^A.
}
\end{equation*}
Thus there is a map $\gamma(\tilde{a},\tilde{b}) : X \to C^B$ making the evident triangles commute. Transposing again provides the required map
\[
c(a,b) = \widetilde{\gamma(\tilde{a},\tilde{b})} : X\times B \to C.
\]
The converse is just as direct.
\end{proof}
\begin{proposition}\label{prop:intid}
A natural model $p : \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}}$ satisfies the type theoretic rules for \emph{intensional} identity types just in case there are maps
\begin{align}
\mathsf{i} &: \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\widetilde{\mathcal{U}}} \label{prop:inteq1}\\
\mathsf{Id} &: \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \to \ensuremath{\mathcal{U}} \label{prop:inteq2}
\end{align}
with $p\circ \mathsf{i} = \mathsf{Id}\circ \delta$, and such that the canonical map $(\delta,\mathsf{i})$
\begin{equation*
\xymatrix{
\ensuremath{\widetilde{\mathcal{U}}} \ar@/{}_{1pc}/[ddr]_{\delta} \ar@{.>}[dr]|-{(\delta,\mathsf{i})} \ar@/{}^{1pc}/[drr]^{\mathsf{i}} &&\\
& I \ar[r] \ar[d] \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^p \\
& \ensuremath{\widetilde{\mathcal{U}}}\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[r]_-{\mathsf{Id}} & \ensuremath{\mathcal{U}}
}
\end{equation*}
has a left-lifting structure $j$ with respect to $p$, when both are regarded as maps over $\ensuremath{\mathcal{U}}$,
$$(\delta,\mathsf{i})\ \pitchfork_j\ \ensuremath{\mathcal{U}}^*(p).$$
\end{proposition}
\begin{proof}
Let us write $\rho = (\delta,\mathsf{i})$. By lemma \ref{lem:extintLLP}, a left-lifting structure $j$ for $\rho$ with respect to $p$, both regarded as maps over $\ensuremath{\mathcal{U}}$, is equivalent to a natural (in $X$) choice of diagonal fillers $j(a,b)$ for all squares over $\ensuremath{\mathcal{U}}$ of the form
\[
\xymatrix{
X\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}} \ar[d]_{X\times_\ensuremath{\mathcal{U}}{\rho}} \ar[rr]^{a} && \ensuremath{\mathcal{U}}^*\ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{\ensuremath{\mathcal{U}}^*p} \\
X\times_\ensuremath{\mathcal{U}} I \ar[rr]_{b} \ar@{.>}[urr]|-{\ j(a,b)\ } && \ensuremath{\mathcal{U}}^*\ensuremath{\mathcal{U}},
}
\]
where $\ensuremath{\mathcal{U}}^* : \ensuremath{\widehat{\mathbb{C}}} \to \ensuremath{\widehat{\mathbb{C}}}/\ensuremath{\mathcal{U}}$ is the base change. Letting $X = (A : \Gamma \to \ensuremath{\mathcal{U}})$ as an object over $\ensuremath{\mathcal{U}}$, and consulting \eqref{diag:big}, we see that:
\begin{align*}
\ensuremath{\mathsf{dom}}(X\times_\ensuremath{\mathcal{U}} \ensuremath{\widetilde{\mathcal{U}}})\ &=\ \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \\
\ensuremath{\mathsf{dom}}(X\times_\ensuremath{\mathcal{U}} I) \ &=\ \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A \\
\ensuremath{\mathsf{dom}}(X\times_\ensuremath{\mathcal{U}}{\rho})\ &=\ \rho_A : \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \to \ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A
\end{align*}
Thus, transposing the above diagram to forget the base $\ensuremath{\mathcal{U}}$, we arrive at the equivalent filling problem
\[
\xymatrix{
\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A \ar[d]_{\rho_A} \ar[rr] && \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p} \\
\ensuremath{\Gamma}\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!} A\ensuremath{\!\centerdot\!}\mathsf{Id}_A \ar[rr] \ar@{.>}[urr]|-{\ \mathsf{j}\ }&& \ensuremath{\mathcal{U}}.
}
\]
Comparing this to the diagram \eqref{diag:Idfill}, we see that the assumed left-lifting structure $j$ indeed provides a choice of fillers $\mathsf{j}$ that is natural in $\ensuremath{\Gamma}$, as required to correctly interpret the elimination rule.
\end{proof}
\section{Supporting a natural model}
The representability of a natural transformation $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ imposes conditions on the maps in $\ensuremath{\mathbb{C}}$ that are represented (cf.~corollary \ref{cor:repnattransstableclass}), and the requirement that $p$ should model the type-forming operations $\Sigma, \Pi, \mathsf{Id}$ imposes further conditions on those maps. Our goal is to determine conditions on a category $\ensuremath{\mathbb{C}}$ that are sufficient to ensure that it carries a natural model of type theory.
Let $p : \ensuremath{\widetilde{\mathcal{U}}}\to\ensuremath{\mathcal{U}}$ be a representable natural transformation over $\ensuremath{\mathbb{C}}$. For each object $C\in\ensuremath{\mathbb{C}}$ and each element $A\in\ensuremath{\mathcal{U}}(C)$, pick an object $\btexdot{C}{A}$, a map $p_{A}:\btexdot{C}{A} \to C$ and an element $q_A\in \ensuremath{\widetilde{\mathcal{U}}}(\btexdot{C}{A})$, all fitting into a pullback square of the form:
\begin{equation}\label{diag:prep}
\xymatrix{
\ext{C}{A} \ar[d]_{p_A} \ar[r]^{q_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
C \ar[r]_{A} & \ensuremath{\mathcal{U}}}
\end{equation}
Such a map $p_{A}:\btexdot{C}{A} \to C$ arising as a canonical pullback of $p$ will be called a \emph{display map}, and the corresponding pullback square, a \emph{display square for~$p_A$}.
\begin{remark}\label{remark:displaypb}
Observe that a display map has a pullback along any map, even though $\ensuremath{\mathbb{C}}$ is not assumed to have all pullbacks.
Indeed, for any display map $p_{A}:\btexdot{C}{A} \to C$ and any map $s : D\to C$, there is a uniquely determined pullback square with $p_{As} : \ext{D}{As} \to D$ a display map, as shown on the left in the diagram below:
\[
\xymatrix{
\ext{D}{As} \ar@/{}^{2pc}/[rr]^{q_{As}} \ar[d]_{p_{As}} \pbcorner \ar@{.>}[r] & \ext{C}{A} \ar[d] \ar[d]_{p_A} \ar[r]^{q_A} \pbcorner & \ensuremath{\widetilde{\mathcal{U}}} \ar[d]^{p}\\
D \ar@/{}_{2pc}/[rr]_{As} \ar[r]_{s} & C \ar[r]_{A} & \ensuremath{\mathcal{U}}}
\]
because the outer rectangle and the righthand square are canonical pullbacks.
\end{remark}
Conversely, let $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ be a class of maps in $\ensuremath{\mathbb{C}}$ that is closed under all pullbacks along arbitrary maps in $\ensuremath{\mathbb{C}}$, in the sense that given any pullback square
\begin{equation
\xymatrix{
A\ar[d]_{g} \ar[r] \pbcorner & C\ar[d]^{f}\\
B \ar[r] & D}
\end{equation}
if $f\in\ensuremath{\mathcal{D}}$ then $g\in\ensuremath{\mathcal{D}}$. Observe that $\ensuremath{\mathcal{D}}$ is closed under isomorphisms in the arrow category. Call such class $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ a \emph{stable class of maps} in $\ensuremath{\mathbb{C}}$. We define two presheaves $\ensuremath{\mathcal{D}}_0, \ensuremath{\mathcal{D}}_1$ and a natural transformation $\pi : \ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$ between them as follows:
\begin{align*}
\ensuremath{\mathcal{D}}_1(C)\ &=\ \{ (a, d)\in \ensuremath{\mathbb{C}}_1\times \ensuremath{\mathcal{D}}\ |\ \ensuremath{\mathsf{cod}}(a) = \ensuremath{\mathsf{dom}}(d) \} \\
\ensuremath{\mathcal{D}}_0(C)\ &=\ \{ (b, d) \in \ensuremath{\mathbb{C}}_1\times \ensuremath{\mathcal{D}}\ |\ \ensuremath{\mathsf{cod}}(b) = \ensuremath{\mathsf{cod}}(d) \}\\
\pi_C &: \ensuremath{\mathcal{D}}_1(C) \to \ensuremath{\mathcal{D}}_0(C) \\
& \pi_C(a,d)\ =\ (d\circ a, d).
\end{align*}
Schematically, we have the following situation:
\begin{align*}
\ensuremath{\mathcal{D}}_1(C)\ &=\ \left\{ \vcenter{ \xymatrix{& \cdot \ar[d]^{d\,\in\,\ensuremath{\mathcal{D}}} \\
C\ar[ru]^{a} & \cdot} }
\right \} \\
&\quad \xymatrix{ \strut \ar[d]_{\pi_C}\\
\strut } \\
\ensuremath{\mathcal{D}}_0(C)\ &=\ \left\{ \vcenter{ \xymatrix{& \cdot \ar[d]^{d\,\in\,\ensuremath{\mathcal{D}}} \\
C \ar[r]_{b} & \cdot} }
\right\}
\end{align*}
The action of the presheaves is by precomposition in the first factor, thus for $s : C' \to C$, we let:
\begin{align*}
\ensuremath{\mathcal{D}}_1(s)(a, d)\ &=\ (a\circ s, d), \\
\ensuremath{\mathcal{D}}_0(s)(b, d)\ &=\ (b\circ s, d) .
\end{align*}
This is plainly (strictly) functorial. The component $\pi_C$ is simply composition with the arrow $d$ in the second factor,
which is obviously natural.
\begin{remark}
The specification of $\pi: \ensuremath{\mathcal{D}}_1\to \ensuremath{\mathcal{D}}_0$ from $\ensuremath{\mathcal{D}}$ is closely related to a coherence theorem for certain kinds of indexed categories (respectively fibrations): it takes the pseudofunctor $\ensuremath{\mathcal{D}} : \ensuremath{\mathbb{C}}\ensuremath{^\mathrm{op}}\to\ensuremath{\mathsf{Cat}}$ given by pullback of a stable class of maps and returns an equivalent presheaf of categories, i.e.\ a ``strictification" of the pseudofunctor (or ``splitting" of the associated fibration). Several such strictifications have been studied previously: this one is left adjoint to the inclusion of functors into pseudofunctors, and there is also a right adjoint, and others (all three are attributed to Giraud in cf.~\cite{Streicher}, which also gives the relation to work of Benabou). The use of this construction to obtain a model of intensional type theory is the main result of \cite{LW}; see remark \ref{rem:credit} below.
\end{remark}
\begin{proposition}\label{prop:Dtorepnattrans}
Let $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ be a stable class of maps in $\ensuremath{\mathbb{C}}$. Then the natural transformation $\pi : \ensuremath{\mathcal{D}}_1\to \ensuremath{\mathcal{D}}_0$ just defined is representable.
\end{proposition}
\begin{proof}
Let $C\in\ensuremath{\mathbb{C}}$ and $A\in\ensuremath{\mathcal{D}}_0(C)$. We require an object $\btexdot{C}{A}$, a map $p_{A}:\btexdot{C}{A} \to C$ and an element $q_A\in \ensuremath{\mathcal{D}}_1(\btexdot{C}{A})$ fitting into a pullback square of the form:
\begin{equation}\label{diag:prep2}
\xymatrix{
\ext{C}{A} \ar[d]_{p_A} \ar[r]^{q_A} \pbcorner & \ensuremath{\mathcal{D}}_1\ar[d]^{\pi}\\
C \ar[r]_{A} & \ensuremath{\mathcal{D}}_0}
\end{equation}
Now $A\in\ensuremath{\mathcal{D}}_0(C)$ is a cospan of the form, say,
\[
\xymatrix{& A_1 \ar[d]^{d_A} \\
C \ar[r]_{|A|} & A_0}
\]
with $d_A \in \ensuremath{\mathcal{D}}$. So we can take a pullback to define $p_{A}:\btexdot{C}{A} \to C$ and $q'_A $ as indicated in:
\begin{equation}\label{diag:prep3}
\xymatrix{
\ext{C}{A} \ar[d]_{p_A} \ar[r]^{q'_A} \pbcorner & A_1\ar[d]^{d_A}\\
C \ar[r]_{|A|} & A_0}
\end{equation}
Let $q_A \in \ensuremath{\mathcal{D}}_1(\btexdot{C}{A})$ be defined by $q_A = (q'_A, d_A)$. To see that the square \eqref{diag:prep2} commutes, observe that
\[
\pi_{\btexdot{C\,}{\,A}}(q'_A, d_A) = (d_A\circ q'_A, d_A) = (|A|\circ p_A, d_A) = \ensuremath{\mathcal{D}}_0(p_{A})(|A|, d_A) = \ensuremath{\mathcal{D}}_0(p_{A})(A).
\]
The proof that \eqref{diag:prep2} is a pullback is a routine unwinding of the definitions.
\end{proof}
\begin{corollary}\label{cor:repnattransstableclass}
A representable natural transformation determines a stable class of maps $\ensuremath{\mathcal{D}}\subseteq \ensuremath{\mathbb{C}}_1$, namely all those maps isomorphic to display maps, and every stable class of maps $\ensuremath{\mathcal{D}}\subseteq \ensuremath{\mathbb{C}}_1$ is determined in this way by a representable natural transformation $\pi : \ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$.
\end{corollary}
\begin{proof}
Every display map is clearly in \ensuremath{\mathcal{D}}, and a display map has a pullback along any map by Remark \ref{remark:displaypb}. It follows directly that $\ensuremath{\mathcal{D}}$ is stable. Conversely, every map $d : D\to C$ in \ensuremath{\mathcal{D}}\ occurs as a display map for the associated representable natural transformation $\pi : \ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$ of Proposition \ref{prop:Dtorepnattrans}, in the form:
\[
\xymatrix{
\ext{C}{D} \ar[d]_{p_D} \pbcorner \ar[r]^{\ensuremath{\mathsf{id}}} & D\ar[d]^{d}\\
C \ar[r]_{\ensuremath{\mathsf{id}}} & C}
\]
\end{proof}
Note that different representable natural transformations on a category $\ensuremath{\mathbb{C}}$ may give rise to the same stable class of maps $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$. We shall not pursue this line of inquiry further, since it is not required for what follows.
Our task now is to determine conditions on a stable class of maps \ensuremath{\mathcal{D}}\ that will ensure that the associated representable natural transformation $\pi : \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_0$ models the various type-theoretic rules in the sense determined in section \ref{sec:modelling}.
\subsection{Sums and Products}
Recall from proposition \ref{prop:sum} the condition on $\pi : \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_0$ required to model the rules for sum types $\Sigma$:
there should be maps
\begin{align*
\ensuremath{\mathsf{pair}} :& \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0)}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0^{\cors{A}})}\sum_{(a : \cors{A})}B(a) \to \ensuremath{\mathcal{D}}_1\\
\Sigma :& \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}} \to \ensuremath{\mathcal{D}}_0\label{eq:sumop2}
\end{align*}
making the following diagram a pullback,
\begin{equation}\label{diag:sigmapb}
\xymatrix{
\sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0)}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0^{\cors{A}})}\sum_{(a : \cors{A})}B(a) \ar[d]_{\pi_1} \ar[rr]^-{\ensuremath{\mathsf{pair}}} && \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi}\\
\sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}} \ar[rr]_-{\Sigma} && \ensuremath{\mathcal{D}}_0 }
\end{equation}
where $\cors{A}$ denotes the fiber of $\pi$ over $A$, i.e.\ the object given by pullback:
\begin{equation*
\xymatrix{
\cors{A} \ar[d] \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi}\\
X \ar[r]_A & \ensuremath{\mathcal{D}}_0 }
\end{equation*}
Take any $X\in\ensuremath{\mathbb{C}}$ and $(A,B) : X \to \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}}$, and we seek an assignment of a map $\Sigma(A,B) : X \to \ensuremath{\mathcal{D}}_0$, in a way that is natural in $X$.
Using Lemma \ref{lem:polyclass}, the map $(A,B) : X \to \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}}$ uniquely determines maps $A : X \to \ensuremath{\mathcal{D}}_0$ and $B : \cors{A} = X\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_0$, as already suggested by the notation. These in turn correspond uniquely (by Yoneda) to cospans:
\begin{align*}
A\ &= (a\in\ensuremath{\mathbb{C}}, p\in\ensuremath{\mathcal{D}}) \\
B\ &= (b\in\ensuremath{\mathbb{C}}, q\in\ensuremath{\mathcal{D}})
\end{align*}
as indicated in:
\begin{equation}\label{diag:wrongsigma}
\xymatrix{
& B_1 \ar[dd]^{q} &\\
&& \\
& B_0 &\\
X\times_{A_0} A_1 \ar[ru]_-{b} \ar[d]_{p_X} \ar[rr] && A_1 \ar[d]^{p}\\
X \ar[rr]_{a} && A_0}
\end{equation}
Here we have used the following easily proved fact, which we record for later reuse:
\begin{lemma}\label{lem:reindex}
When $A : X\to\ensuremath{\mathcal{D}}_0$ corresponds to the cospan $(a,p)$, then $A$ factors through $a : X \to A_0$ via the map $(\ensuremath{\mathsf{id}}_{A_0}, p) : A_0\to\ensuremath{\mathcal{D}}_0$, and the following is then a pullback:
\[
\xymatrix{
A_1 \ar[d]_{p} \ar[rr] \pbcorner && \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi}\\
A_0 \ar[rr]_{(\ensuremath{\mathsf{id}},\,p)} && \ensuremath{\mathcal{D}}_0 }
\]
Thus
\[
\cors{A}\ =\ X\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1\ =\ X\times_{A_0} A_1\,.
\]
\end{lemma}
Returning to diagram \eqref{diag:wrongsigma}, it might now be expected that the sum $\Sigma(A,B)$ would be built by first pulling $q$ back along $b$ to give $q_X$, and then composing with $p_X$:
\begin{equation}\label{diag:wrongsigma2}
\xymatrix{
& B_1 \ar[dd]^{q} &\\
\cdot \ar@/_8ex /[ddd]_{\Sigma(A,B)?} \ar[ru] \ar[dd]^{q_X} && \\
& B_0 &\\
X\times_{A_0} A_1 \ar[ru]_-{b} \ar[d]^{p_X} \ar[rr] && A_1 \ar[d]^{p}\\
X \ar[rr]_{a} && A_0}
\end{equation}
This is ``morally" what we want to do, since the resulting composite is indeed the display map $\btexdot{X}{\Sigma(A,B)}\to X$, and so the requirement that \emph{$\ensuremath{\mathcal{D}}$ is closed under composition} suggests itself. There is a problem with this construction, however: $\Sigma(A,B)$ must be a cospan $(c\in\ensuremath{\mathbb{C}},d\in\ensuremath{\mathcal{D}})$ of the form: $\xymatrix{& \cdot \ar[d]^{d} \\
X \ar[r]_c & \cdot}$
But the only candidate in sight for $c$ is the identity on $X$, and that assignment would not be natural in $X$!
Instead, we shall use a construction similar to that applied in section \ref{sec:modelling} to devise a ``generic case" in which to perform the operation of pullback-plus-composition, so that all other cases result simply from mapping into the generic one. This construction, however, requires that \ensuremath{\mathcal{D}}\ not only be closed under composition, but also that certain \emph{right} adjoints to pullback exist. To state the required condition precisely, for any object $C\in\ensuremath{\mathbb{C}}$, let us write $\ensuremath{\mathcal{D}}(C)$ for the full subcategory $\ensuremath{\mathcal{D}}(C)\ensuremath{\hookrightarrow}\ensuremath{\mathbb{C}}/C$ on the \ensuremath{\mathcal{D}}-maps into $C$ as objects.
\begin{definition}\label{def:closed}
A stable class of maps $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ is \emph{closed} if the following conditions hold:
\begin{enumerate}
\item \ensuremath{\mathbb{C}}\ has a terminal object $1$, and every map $C\to 1$ is in \ensuremath{\mathcal{D}}.
\item \ensuremath{\mathcal{D}}\ is closed under composition.
\item For any $d : D\to C$ in \ensuremath{\mathcal{D}}, the pullback functor $d^* : \ensuremath{\mathcal{D}}(C) \to \ensuremath{\mathcal{D}}(D)$ has a right adjoint $d_* : \ensuremath{\mathcal{D}}(D) \to \ensuremath{\mathcal{D}}(C)$, and the inclusion functor $\ensuremath{\mathcal{D}}(C)\ensuremath{\hookrightarrow}\ensuremath{\mathbb{C}}/C$ preserves expoentials.
\end{enumerate}
\end{definition}
\begin{proposition}\label{prop:sumprod}
If $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ is a closed, stable class of maps, then the associated representable natural transformation $\pi:\ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$ models the rules for sums $\Sigma$ and products $\Pi$.
\end{proposition}
\begin{proof}
Taking up the argument from diagram \eqref{diag:wrongsigma}, consider the following construction:
\begin{equation}\label{genericcase}
\xymatrix{
& B_1 \ar[dd] |\hole ^>>>>>>>{q} & &\\
\cdot \ar[ru] \ar[dd]_{q_X} \ar[rr] && \ar[lu] G \ar[dd]^{q'} & \\
& B_0 & &\\
X\times_{A_0} A_1\ar[ru]^{b} \ar[d]_{p_X} \ar[rr]_{\overline{b}\times_{A_0}A_1} && \ar[lu]_{\mathrm{ev}} B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
X \ar[rr]^{\overline{b}} \ar@/_5ex /[rrr]_{a} && B_0^{A_1} \ar[r] & A_0}
\end{equation}
We first factor the map $a: X\to A_0$ through the transpose $\overline{b} : X \to B_{0}^{A_1}$ of $b : X\times_{A_0} A_1\to B_0$ over $A_0$ (regarding $B_0$ as a constant object over $A_0$ by base change along $A_0 \to 1$). Here we know that $B_{0}^{A_1}$ exists in $\ensuremath{\mathcal{D}}(A_0)$ because both $p$ and $B_0\to 1$ are in \ensuremath{\mathcal{D}}, and we know that $B_{0}^{A_1}$ is also an exponential in $\ensuremath{\mathbb{C}}/A_0$ by the definition of ``closed".
Pulling $p$ back along $a$ in two stages gives the two lower pullback squares. Next, still working over $A_0$, the map $b$ now factors as $\mathrm{ev}\circ(\overline{b}\times_{A_0}A_1)$ by the exponential adjunction.
The pullback $q_X$ of $q$ along $b$ can therefore also be constructed in two stages, giving first the map $q' : G \to B_{0}^{A_1} \times_{A_0} A_1$ as the pullback of $q$ along the evaluation $\mathrm{ev}$.
The generic case of the ``pullback and compose" construction \eqref{diag:wrongsigma2} that we seek now has the form:
\[
\xymatrix{
B_1 \ar[d] ^{q} & &\\
B_0 & \ar[lu] \ar@/_8ex/ [dd] G \ar[d]^{q'} & \\
& \ar[lu] |<<<<<<\hole B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
& B_0^{A_1} \ar[r] & A_0}
\]
The composite $p'\circ q' : G \to B_0^{A_1}$ is the $\ensuremath{\mathcal{D}}$ component of the desired cospan $(\overline{b},\, p'\circ q')$ defining $\Sigma(A,B) : X\to \ensuremath{\mathcal{D}}_0$. Observe that the pullback of $p'\circ q'$ along $\overline{b}$ is indeed $p_X\circ q_X$, and that the same is true for any given $y : Y\to B_0^{A_1}$, because any such map is uniquely of the form $y=\overline{c} : Y\to B_0^{A_1}$ for $c = \mathrm{ev}\circ(y\times_{A_0}A_1) : Y\times_{A_0} A_1 \to B_0$.
This defines the natural transformation $\Sigma : \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}} \to \ensuremath{\mathcal{D}}_0$.
Explicitly, given $(A, B): X \to \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}}$, where $A = (a, p)$ and $B = (b, q)$, we define $\Sigma(A,B) : X\to \ensuremath{\mathcal{D}}_0$ by $\Sigma(A,B) = (\overline{b},\, p'\circ q')$. This assignment is natural in $X$, for given any $s: Y\to X$, we have
$$\Sigma(A,B)s = (\overline{b}\circ s,\, p'\circ q') = (\overline{b\circ s'},\, p'\circ q') = \Sigma(As,Bs),$$
because the \ensuremath{\mathcal{D}}-component is fixed, and exponential transposition is natural.
\[
\xymatrix{
&& B_1 \ar[d] ^{q} & &\\
&&B_0 & \ar[lu] G_1 \ar[d]^{q'} & \\
Y\times_{A_0} A_1\ar[rru]^{bs'} \ar[d]_{p_Y} \ar[r]_{s'} & X\times_{A_0} A_1\ar[ru]_{b} \ar[d]_{p_X} \ar[rr]_{\overline{b}\times_{A_0}A_1} && \ar[lu]_{\mathrm{ev}} B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
Y \ar[r]^{s} \ar@/_5ex /[rrr]_{\overline{bs'}} & X \ar[rr]^{\overline{b}} && B^{A_1} \ar[r] & A_0}
\]
To define the pairing map,
\[
\ensuremath{\mathsf{pair}} : \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0)}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0^{\cors{A}})}\sum_{(a : \cors{A})}B(a) \to \ensuremath{\mathcal{D}}_1,
\]
take an element $(A,B,c): X\to \sum_{(A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0)}\sum_{(B\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0^{\cors{A}})}\sum_{(a : \cors{A})}B(a)$, and we require an element $\ensuremath{\mathsf{pair}} (A,B,c) : X\to \ensuremath{\mathcal{D}}_1$ via an assignment that is natural in $X$. The map $(A,B,c)$ determines data of the form:
\begin{align*}
A\ &= (a\in\ensuremath{\mathbb{C}}, p\in\ensuremath{\mathcal{D}}) \\
B\ &= (b\in\ensuremath{\mathbb{C}}, q\in\ensuremath{\mathcal{D}})\\
c &= (a',b')
\end{align*}
where:
\begin{equation*
\xymatrix{
& B_1 \ar[d]^{q} &\\
& B_0 &\\
X\times_{A_0} A_1 \ar@/^2ex /[ruu]^{b'} \ar[ru]_-{b} \ar[d]_{p_X} \ar[r] & A_1 \ar[d]^{p}\\
X \ar[r]_{a} \ar[ru]^{a'} & A_0}
\end{equation*}
But this is just a section of the composite $p_X\circ q_X$,
\[
\xymatrix{
& B_1 \ar[dd] |\hole ^>>>>>>>{q} & &\\
\cdot \ar[ru] \ar[dd]_{q_X} \ar[rr] && \ar[lu] G \ar[dd]^{q'} & \\
& B_0 & &\\
X\times_{A_0} A_1\ar[ru]^{b} \ar[d]_{p_X} \ar[rr]_{\overline{b}\times_{A_0}A_1} && \ar[lu]_{\mathrm{ev}} B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
X \ar@/^8ex /[uuu]^{(a',b')} \ar[rr]^{\overline{b}} \ar@/_5ex /[rrr]_{a} && B_0^{A_1} \ar[r] & A_0}
\]
or, equivalently, a section of the composite $p'\circ q'$ over $\overline{b}$. But this in turn is exactly an element $c'$ of the generic $\Sigma$-type,
\[
\xymatrix{
& G \ar[d]^{q'} & \\
& B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
X \ar@/^2ex /[ruu]^{c'} \ar[r]_{\overline{b}} & B_0^{A_1} \ar[r] & A_0}
\]
So we can set $$\ensuremath{\mathsf{pair}}_X(A,B,c) = (c',p'\circ q')\in \ensuremath{\mathcal{D}}_1(X).$$
Again, this is plainly natural in $X$, because the action in the first component is precomposition and second component is fixed.
It is immediate that this assignment makes \eqref{diag:sigmapb} a pullback: for fixed $A = (a, p)$ and $B = (b, q)$, the correspondence $c=(a',b') \mapsto c'$ is clearly reversible.
For the products $\Pi$, we start from the object constructed in \eqref{genericcase}:
\begin{equation*
\xymatrix{
& B_1 \ar[d]_{q} & &\\
& B_0 & \ar[lu] G \ar[d]^{q'} & \\
&& \ar[lu]_{\mathrm{ev}} B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
&& B_0^{A_1} \ar[r] & A_0}
\end{equation*}
But now rather than composing $p'\circ q'$, we use the \emph{right} adjoint $p'_*$ to pullback along $p'$ to build the map $p'_*q' : G' \to B_0^{A_1} $:
\begin{equation}\label{genericcase2}
\xymatrix{
& B_1 \ar[d]_{q} & &\\
& B_0 & \ar[lu] G \ar[d]^{q'} & \\
&G' \ar[rd]_{p'_*q'} & \ar[lu]_{\mathrm{ev}} B_{0}^{A_1} \times_{A_0} A_1 \ar[d]^{p'} \ar[r] & A_1 \ar[d]^{p}\\
&& B_0^{A_1} \ar[r] & A_0}
\end{equation}
Note that $p'_*q'$ (exists and) is in \ensuremath{\mathcal{D}}\ by our assumption that \ensuremath{\mathcal{D}}\ is closed.
Now, as in the previous case, given $(A,B) : X \to \sum_{A\ensuremath{\,:\,}\ensuremath{\mathcal{D}}_0}\ensuremath{\mathcal{D}}_0^{\cors{A}}$, we have $A = (a\in\ensuremath{\mathbb{C}}, p\in\ensuremath{\mathcal{D}})$ and $B = (b\in\ensuremath{\mathbb{C}}, q\in\ensuremath{\mathcal{D}})$, from which we can construct $p'$, $q'$, and $\overline{b}: X\to B_0^{A_1}$. Then set:
\[
\Pi(A,B)\ =\ (\overline{b},\, p'_*q') : X \to \ensuremath{\mathcal{D}}_0
\]
The assignment is again obviously natural in $X$.
The construction of $\lambda$ and verification that the resulting square is a pullback are entirely analogous to the case of $\ensuremath{\mathsf{pair}}$, and are omitted.
Finally, observe that for any $y : Y\to B_0^{A_1}$, the Beck-Chevalley conditions for the left and right adjoints to pullback $y^*$ give:
\begin{align*}
y^*(p'\circ q') &= (p_Y)\circ(q_Y)\\
y^*(p'_*q') &= (p_Y)_*(q_Y)
\end{align*}
This ensures that the context extension operation behaves correctly.
\end{proof}
\subsection{Identity types}
As was the case for sums and products, in order to model intensional identity types $\mathsf{Id}$, we require an additional condition on the stable class of maps $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$. It may be surprising that we also still need the class to be \emph{closed} in the sense of definition \ref{def:closed}; this is used to again construct certain ``generic" cases.
Let $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ be a class of maps in a category \ensuremath{\mathbb{C}}.
We shall say that a map $a : A \to B$ in \ensuremath{\mathbb{C}}\ is \emph{anodyne} if it has the left lifting property with respect to all maps in $\ensuremath{\mathcal{D}}$. The class $\ensuremath{\mathcal{D}}\subseteq\ensuremath{\mathbb{C}}_1$ will be called \emph{factorizing} if every map $f : A\to B$ in \ensuremath{\mathbb{C}}\ factors as $f = d\circ a$ with $a$ anodyne and $d\in\ensuremath{\mathcal{D}}$,
\[
\xymatrix{
& B' \ar[d]^{d}\\
A \ar[ru]^{a} \ar[r]_{f} & B.}
\]
The following is standard.
\begin{lemma}\label{lemma:anodynestable}
If $\ensuremath{\mathcal{D}} \subseteq \ensuremath{\mathbb{C}}_1$ is a closed, stable, factorizing class of maps, then the anodyne maps are preserved by pullback along all maps in \ensuremath{\mathcal{D}}.
\end{lemma}
\begin{proposition}\label{prop:id}
If $\ensuremath{\mathcal{D}} \subseteq \ensuremath{\mathbb{C}}_1$ is a closed, stable, factorizing class of maps, then the associated representable natural transformation $\pi:\ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$ models the rules for intensional identity types $\mathsf{Id}$.
\end{proposition}
\begin{proof}
Recall from proposition \ref{prop:intid} that we require maps
\begin{align}
\mathsf{i} &: \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_1\\
\mathsf{Id} &: \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_0
\end{align}
commuting with $\pi$ and its diagonal $\delta$,
\begin{equation*
\xymatrix{
\ensuremath{\mathcal{D}}_1 \ar@/{}_{1pc}/[ddr]_{\delta} \ar@{.>}[dr]|-{(\delta,\,\mathsf{i})} \ar@/{}^{1pc}/[drr]^{\mathsf{i}} &&\\
& I \ar[r] \ar[d] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi} \\
& \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \ar[r]_-{\mathsf{Id}} & \ensuremath{\mathcal{D}}_0
}
\end{equation*}
and a left-lifting structure $j$ for the map $(\delta,\mathsf{i})$ with respect to $\pi$,
$$(\delta,\mathsf{i})\ \pitchfork_j\ \pi$$
where both are regarded as maps over $\ensuremath{\mathcal{D}}_0$.
Again, we shall write $\rho = (\delta,\mathsf{i}) : \ensuremath{\mathcal{D}}_1 \to I$.
We begin by constructing the map
\[
\mathsf{Id} : \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \to \ensuremath{\mathcal{D}}_0.
\]
For each $A\in\ensuremath{\mathbb{C}}$, pick a factorization of the diagonal,
\[
\xymatrix{
& I_A \ar[d]^{d_A}\\
A \ar[ru]^{r_A} \ar[r]_-{\delta_A} & A\times A}
\]
with $r_A$ anodyne and $d_A\in\ensuremath{\mathcal{D}}$, and do the same for every map $A: A_1 \to A_0$ in \ensuremath{\mathcal{D}},
\[
\xymatrix{
& I_A \ar[d]^{d_A}\\
A_1 \ar[ru]^{r_A} \ar[r]_-{\delta_A} & A_1\times_{A_0} A_1}.
\]
(Of course, the second step subsumes the first.)
For $X\in\ensuremath{\mathbb{C}}$, a map $\alpha: X\to \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1$ consists of a map $A : X \to {\ensuremath{\mathcal{D}}_0}$ together with two maps $a_1,a_2 : X\to \ensuremath{\mathcal{D}}_1$ over ${\ensuremath{\mathcal{D}}_0}$. Now $A$ is a cospan $A = (a\in \ensuremath{\mathbb{C}},p\in\ensuremath{\mathcal{D}})$, and there is a pullback diagram,
\[
\xymatrix{
&& A_1 \ar[d]^{p} \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi}\\
X \ar@<1ex>[urr]^{a_1} \ar[urr]_{a_2} \ar[rr]_{a} \ar@/_4ex /[rrr]_{A} && A_0 \ar[r] & \ensuremath{\mathcal{D}}_0}
\]
with the corresponding elements $a_1, a_2$ fitting in as shown. These in turn determine an element $(a_1,a_2)$ of $A_1\times_{A_0} A_1$, which we could also have constructed directly, as indicated in the following:
\[
\xymatrix{
X \ar@/^6ex /[rrr]^{\alpha} \ar[rr]_-{(a_1,\, a_2)} \ar@/_2ex/[rrdd]_{a} && \ar@<-1ex>[d] \ar@<.5ex>[d] A_1\times_{A_0} A_1 \ar[r] & \ar@<-1ex>[d] \ar@<.5ex>[d] \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1\\
&& A_1 \ar[d]^{p} \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi} \\
&& A_0 \ar[r] & \ensuremath{\mathcal{D}}_0 }
\]
We require an element $\mathsf{Id}(\alpha) : X\to \ensuremath{\mathcal{D}}_0$, by an assignment that is natural in $X$. For this, we take the following cospan:
\[
\xymatrix{
&& I_A \ar[d]^{d_A}\\
X \ar[rr]_-{(a_1,\, a_2)} && A_1\times_{A_0} A_1}
\]
To define $\mathsf{i} : \ensuremath{\mathcal{D}}_1\to \ensuremath{\mathcal{D}}_1$, an element of $\ensuremath{\mathcal{D}}_1(X)$ has the form $(a,p)$ with $a: X \to A_1$ and $p: A_1\to A_0$ with $p\in\ensuremath{\mathcal{D}}$. Compose with $r_A : A_1 \to I_A$ to get
\[
\mathsf{i}(a,p)\ =\ (r_A\circ a, d_A),
\]
which is again an element of $\ensuremath{\mathcal{D}}_1(X)$:
\begin{equation}\label{diag:rho}
\xymatrix{
&& A_1 \ar[d]^{r_A} \\
X \ar[rru]^{a} \ar[rr]_{r_A\circ\,a} \ar@/_2ex/[rrd]_{(a,a)} && I_A \ar[d]^{d_A}\\
&& A_1\times_{A_0} A_1}
\end{equation}
This specification plainly makes $\pi\circ\mathsf{i}(a,p)=\mathsf{Id}\circ\delta(a,p)$, as required.
Next, the presheaf $I:\ensuremath{\mathbb{C}}\ensuremath{^\mathrm{op}}\to \ensuremath{\mathsf{Set}}$ has as elements of $I(X)$ pairs $$\alpha: X\to \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1,\quad \beta : X\to \ensuremath{\mathcal{D}}_1$$ fitting together as follows:
\[
\xymatrix{
&& I_A \ar[d]^{d_A}\\
X \ar@/^2ex /[urr]^{b} \ar[rr]_-{(a_1,a_2)} \ar@/_2ex /[rrdd]_{a} && \ar@<-1ex>[d] \ar@<.5ex>[d] A_1\times_{A_0} A_1\\
&& A_1 \ar[d]^{p}\\
&& A_0 }
\]
where $\alpha = (a,p)$ and $\beta=(b,d_A)$.
The maps $\pi_1: I\to \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1$ and $\pi_2:I\to \ensuremath{\mathcal{D}}_1$ are of course the projections.
Finally, the map $\rho = (\delta,\mathsf{i}) : \ensuremath{\mathcal{D}}_1 \to I$ takes $(a,p): X\to\ensuremath{\mathcal{D}}_1$ with $a: X \to A_1$ and $p: A_1\to A_0$ to the pair:
\[
\rho(a,p) = (\delta(A,p), \mathsf{i}(a,p)),
\]
which is indeed in $I(X)$ by diagram \eqref{diag:rho}.
Now by lemma \ref{lem:extintLLP}, a left-lifting structure $j$ for $\rho$ with respect to $\pi$ over $\ensuremath{\mathcal{D}}_0$ is equivalent to a natural (in $X$) choice of diagonal fillers $j(\alpha,\beta)$ for all squares over $\ensuremath{\mathcal{D}}_0$ of the form
\begin{equation}\label{jnat}
\xymatrix{
X\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \ar[d]_{X\times_{\ensuremath{\mathcal{D}}_0}{\rho}} \ar[rr]^{\alpha} && {\ensuremath{\mathcal{D}}_0}^*\ensuremath{\mathcal{D}}_1 \ar[d]^{{\ensuremath{\mathcal{D}}_0}^*\pi} \\
X\times_{\ensuremath{\mathcal{D}}_0} I \ar[rr]_{\beta} \ar@{.>}[urr]|-{\ j(\alpha,\beta)\ } && {\ensuremath{\mathcal{D}}_0}^*{\ensuremath{\mathcal{D}}_0}
}
\end{equation}
where ${\ensuremath{\mathcal{D}}_0}^* : \ensuremath{\widehat{\mathbb{C}}} \to \ensuremath{\widehat{\mathbb{C}}}/{\ensuremath{\mathcal{D}}_0}$ is the base change. Let the object $X$ over ${\ensuremath{\mathcal{D}}_0}$ be $A : X \to {\ensuremath{\mathcal{D}}_0}$. Using lemma \ref{lem:reindex}, there is a corresponding cospan $(a,p)$ and a double pullback diagram:
\[
\xymatrix{
X\times_{A_0}A_1 \ar[d]_{p_A} \ar[r] & A_1 \ar[d]^{p} \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi}\\
X \ar[r]_{a} \ar@/_4ex /[rr]_{A} & A_0 \ar[r] & \ensuremath{\mathcal{D}}_0}
\]
Thus $X\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 = X\times_{A_0}A_1$ in diagram \eqref{jnat}. Proceding similarly for the other expressions there, we have:
\begin{align*}
\ensuremath{\mathsf{dom}}(X\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1)\ &=\ X\times_{A_0}A_1 \\
\ensuremath{\mathsf{dom}}(X\times_{\ensuremath{\mathcal{D}}_0} I) \ &=\ X\times_{A_0} I_A \\
\ensuremath{\mathsf{dom}}(X\times_{\ensuremath{\mathcal{D}}_0}{\rho})\ &=\ X\times_{A_0}r_A
\end{align*}
as displayed in the following diagram.
\begin{equation}\label{diag:big2}
\xymatrix{
{X}\times_{A_0} A_1 \ar[d]_{{X\times_{A_0}r_A}} \ar[r] & A_1 \ar[d]_{r_A} \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]_{\rho} \ar[dr]^{\mathsf{i}} & \\
{X}\times_{A_0} I_A \ar[d] \ar[r] & I_A \ar[d] \ar[r] \pbcorner & I \ar[d] \ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi} \\
{{X}\times_{A_0} A_1\times_{A_0} A_1} \ar@<-1ex>[d] \ar@<.5ex>[d] \ar[r] & A_1\times_{A_0}A_1 \ar@<-1ex>[d] \ar@<.5ex>[d] \ar[r] & \ensuremath{\mathcal{D}}_1\times_{\ensuremath{\mathcal{D}}_0} \ensuremath{\mathcal{D}}_1 \ar[r]_-{\mathsf{Id}} \ar@<-1ex>[d] \ar@<.5ex>[d] & {\ensuremath{\mathcal{D}}_0} \\
{X}\times_{A_0}{A_1} \ar[d]_{p_A} \ar[r] & A_1\ar[d]_{p}\ar[r] \pbcorner & \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi} & \\
{X} \ar[r]_a & A_0 \ar[r] & {\ensuremath{\mathcal{D}}_0} &
}
\end{equation}
Transposing diagram \eqref{jnat} to forget the base ${\ensuremath{\mathcal{D}}_0}$, we arrive at the equivalent filling problem
\[
\xymatrix{
X\times_{A_0}A_1 \ar[d]_{{X\times_{A_0}r_A}} \ar[rr]^{\alpha} && \ensuremath{\mathcal{D}}_1 \ar[d]^{\pi} \\
X\times_{A_0} I_A \ar[rr]_{\beta} \ar@{.>}[urr] && \ensuremath{\mathcal{D}}_0
}
\]
to be solved naturally in $X$. Now $\beta$ is a cospan of the form:
\[
\xymatrix{
&& B_1 \ar[d]^{q} \\
X\times_{A_0} I_A \ar[rr]_-{b} && B_0
}
\]
with $q\in\ensuremath{\mathcal{D}}$. And $\alpha = (c, q)$ completes the square,
\begin{equation}\label{diag:pause}
\xymatrix{
X\times_{A_0}A_1 \ar[d]_{{X\times_{A_0}r_A}} \ar[rr]^-{c} && B_1 \ar[d]^{q} \\
X\times_{A_0} I_A \ar[rr]_-{b} && B_0.
}
\end{equation}
By lemma \ref{lemma:anodynestable}, ${X\times_{A_0}r_A}$ is anodyne, and $q$ is in \ensuremath{\mathcal{D}}\ by assumption, so there is a diagonal filler $j(c,b)$ for this case, but we need to make a systematic choice that will be natural in $X$. In order to do this, we will again construct a generic case from which all others arise by mapping in. For that, we require the following.
\begin{lemma}
Given maps $p:A_1\to A_0$ and $q:B_1\to B_0$ in a locally cartesian closed category, there is an object $G$ with maps $e_1:G\times A_1 \to B_1$ and $e_0 : G\times A_0\to B_0$ such that $e_0 \circ (G\times p) = q\circ e_1$:
\[
\xymatrix{
G\times A_1 \ar[d]_{G\times p} \ar[r]^-{e_1} & B_1 \ar[d]^{q} \\
G\times A_0 \ar[r]_-{e_0} & B_0
}
\]
and such that, given any object $X$ with maps $f_1:X\times A_1 \to B_1$ and $f_0 : X\times A_0\to B_0$ such that $f_0 \circ (X\times p) = q\circ f_1$, there is a (unique) map $$f : X\to G$$ such that $f_i = e_i\circ(f\times A_i)$ for $i=0,1$:
\[
\xymatrix{
X\times A_1 \ar@/^4ex /[rrr]^{f_1} \ar[d]_{X\times p} \ar[rr]_{f\times A_1} && G\times A_1 \ar[d]_{G\times p} \ar[r]_-{e_1} & B_1 \ar[d]^{q} \\
X\times A_0 \ar@/_4ex /[rrr]_{f_0} \ar[rr]^{f\times A_0} && G\times A_0 \ar[r]^-{e_0} & B_0
}
\]
In other words, $(G,e_0,e_1)$ is a universal object for the presheaf (in $X$) of commutative diagrams of the form
\[
\xymatrix{
X\times A_1 \ar[d]_{X\times p} \ar[r]^-{f_1} & B_1 \ar[d]^{q} \\
X\times A_0 \ar[r]_-{f_0} & B_0.
}
\]
\end{lemma}
\begin{proof}
Using in-line notation $[X,Y] = Y^X$, take $$G = [{A_0}, B_0]\times_{[{A_1},B_0]} [{A_1}, B_1]$$ where the pullback is formed with respect to $p$ and $q$, as in lemma \ref{diag:intLLP}.
\begin{equation}\label{diag:required}
\xymatrix{
[{A_0}, B_0]\times_{[{A_1},B_0]} [{A_1}, B_1] \ar[d] \ar[r] & [{A_1}, B_1] \ar[d]^{[{A_1}, q]} \\
[{A_0}, B_0] \ar[r]_{[p, B_0]} & [{A_1},B_0]
}
\end{equation}
The maps $e_i$ for $i=1,2$ are defined by $e_i\ =\ \ensuremath{\mathrm{ev}}_i \circ (p_i\times A_i)$:
\[
\xymatrix{
\big( [{A_0}, B_0]\times_{[{A_1},B_0]} [{A_1}, B_1]\big) \times A_i \ar[d]_{p_i\times A_i} \ar@/^4ex /[rrd]^-{e_i}&& \\
[{A_i}, B_i] \times A_i \ar[rr]_{\ensuremath{\mathrm{ev}}_i} && B_i
}
\]
We have $e_0 \circ (G\times p) = (\ensuremath{\mathrm{ev}}_0 \circ (p_0\times A_0)) \circ (G\times p)
= q\circ (\ensuremath{\mathrm{ev}}_1 \circ (p_1\times A_1))$
Verification of the construction is left to the reader.
\end{proof}
Returning to the proof of the proposition, we first restore the products on the left in diagram \eqref{diag:pause} by restoring the indexing over $A_0$ and moving $q : B_1\to B_0$ to $\ensuremath{\mathbb{C}}/A_0$ by base change along $A_0\to 1$ (but without explicitly writing $A_0^*(B_1)$, etc.). We now want to apply the lemma to the case of the category $\ensuremath{\mathbb{C}}/A_0$, with $q: B_1\to B_0$ as named in the lemma and $r_A : A_1 \to I_A$ in place of $p: A_1 \to A_0$. Although $\ensuremath{\mathbb{C}}/A_0$ is not locally cartesian closed, the objects $B_0$ and $B_1$ and the maps $p:A_1 \to A_0$ and $I_A\to A_0$ are all in \ensuremath{\mathcal{D}}, and so the required exponentials exists in $\ensuremath{\mathcal{D}}(A_0)$, and thus in $\ensuremath{\mathbb{C}}/A_0$. Moreover, the required pullback \eqref{diag:required} exists because $q$ is in \ensuremath{\mathcal{D}}.
Applying the lemma to the filling problem in diagram \eqref{diag:pause}, we can therefore interpolate the universal case $(G, e_0, e_1)$ to obtain the following (where we have written $\times$ for $\times_{A_0}$):
\[
\xymatrix{
X\times A_1 \ar@/^4ex /[rrr]^{c} \ar[d]_{X\times r_A} \ar[rr]_{f\times A_1} &
& G\times A_1 \ar[d]_{G\times r_A} \ar[r]_-{e_1} & B_1 \ar[d]^{q} \\
X\times I_A \ar@/_4ex /[rrr]_{b} \ar[rr]^{f\times I_A} &
& G\times I_A \ar[r]^-{e_0} & B_0
}
\]
where $f : X \to G$ classifies $(X,b,c)$.
Now $G\times r_A$ is anodyne, since $r_A$ is, so we can find a diagonal filler
$j(e_1,e_0): G\times A_0 \to B_1$ for this generic case.
\[
\xymatrix{
X\times A_1 \ar@/^4ex /[rrrr]^{c} \ar[d]_{X\times r_A} \ar[rr]_{f\times A_1} &
& G\times A_1 \ar[d]_{G\times r_A} \ar[rr]_-{e_1} && B_1 \ar[d]^{q} \\
X\times I_A \ar@/_4ex /[rrrr]_{b} \ar[rr]^{f\times I_A} &
& G\times I_A \ar[rr]^-{e_0} \ar@{.>}[rru]|-{j(e_1,e_0)} && B_0
}
\]
Then for any lifting problem of the form $(X, b, c)$ in \eqref{diag:pause}, we can take as a filler $j(c,b) = j(e_1,e_0)\circ (f\times I_A)$ to have a choice that is natural in $X$. This provides the required left-lifting structure for $(\delta, \mathsf{i})$ with respect to $\pi$.
\end{proof}
\subsection{The main result}
Combining propositions \ref{prop:sumprod} and \ref{prop:id}, we have now reached our goal:
\begin{theorem}\label{thm:natmod}
Let $\ensuremath{\mathcal{D}}$ be any closed, stable, factorizing class of maps in a category \ensuremath{\mathbb{C}}. There is a representable natural transformation $\pi:\ensuremath{\mathcal{D}}_1\to\ensuremath{\mathcal{D}}_0$ over $\ensuremath{\mathbb{C}}$ that models dependent type theory with sums $\Sigma$, products $\Pi$, and intensional identity types $\mathsf{Id}$.
\end{theorem}
\begin{corollary}\label{cor:CwF}
Let $\ensuremath{\mathcal{D}}$ be any closed, stable, factorizing class of maps in a category \ensuremath{\mathbb{C}}. There is a category-with-families model of dependent type theory, with sums $\Sigma$, products $\Pi$, and intensional identity types $\mathsf{Id}$, with the contexts and substitutions being the objects and morphisms of \ensuremath{\mathbb{C}}, and as types and terms in context $X$, a category equivalent to the \ensuremath{\mathcal{D}}-maps into $X$ and their sections.
\end{corollary}
\begin{remark}\label{rem:credit}
A result essentially the same as our corollary \ref{cor:CwF} was announced in 2012 by Lumsdaine and Warren, and has finally appeared in \cite{LW}. Reasoning very similar to that used here is also used in that work, which should be regarded as prior. The main contribution of the present work is the concept of a natural model of type theory as an alternative presentation of the notion of a category with families, and the adaptation of the methods of \cite{LW} to that context.
\end{remark}
\noindent Examples of categories satisfying the conditions of theorem \ref{thm:natmod} include:
\begin{enumerate}
\item the category of Kan simplicial sets, with the (right) weak factorization system of the associated Quillen model structure.
\item any locally cartesian closed model category that is right proper, and in which the cofibrations are the monos; e.g.\ any right proper, Cisinski model category.
\item any weak factorization system on a (pre)sheaf topos in which the left maps are preserved by pullback along the right maps (the ``Frobenius condition" of van den Berg and Garner \cite{GvdB}).
\item non-LCC examples of categories with a weak factorization system for which the right maps are exponentiable, such as (strict, $n$-)groupoids and categories (with iso-fibrations).
\item any $\pi$h-tribe, in the sense of Joyal's categorical axiomatics for homotopy type theory \cite{J}.
\item the syntactic category of contexts $\mathcal{C}(\mathbb{T})$ of a system of type theory $\mathbb{T}$ with $\Sigma, \Pi$ and $\mathsf{Id}$ types (see \cite{GG}).
\end{enumerate}
\begin{remark}
Regarding terminology:
Let $\ensuremath{\mathcal{D}}$ be any closed, stable, factorizing class of maps in a category \ensuremath{\mathbb{C}}. We may call the maps in \ensuremath{\mathcal{D}}\ \emph{typical} (since they are the \emph{types}), and say that \ensuremath{\mathcal{D}}\ is a \emph{typical structure} on \ensuremath{\mathbb{C}}, and that \ensuremath{\mathbb{C}}\ (together with \ensuremath{\mathcal{D}}) is a \emph{typical category}. Our main theorem says that any typical category supports a natural model of basic homotopy type theory.
Assuming a class of maps \ensuremath{\mathcal{D}}\ that is stable and closed, it is enough to require anodyne-\ensuremath{\mathcal{D}}\ factorizations just for the diagonal maps $A\to A\times A$, in order to obtain them for all maps. The notion of a typical category is then closely akin to \emph{first-order logic}: a category of contexts and substitutions, equipped with a system of ``predicates" closed under $\Sigma$, $\Pi$, and $\mathsf{Id}$. A notion of category suitable to model full homotopy type theory, with a (univalent) universe and higher inductive types, will then be a typical category with some additional structure.
\end{remark}
\subsubsection*{Acknowledgements}
The results developed here are an amalgamation of original ideas and ones derived from \cite{KLV} and \cite{LW}. The author has benefitted from conversations with Thierry Coquand, Richard Garner, Andr\'e Joyal, Peter Lumsdaine, Andy Pitts, Michael Shulman, Thomas Streicher, Michael Warren, and Vladimir Voevodsky. The author thanks the Institute for Advanced Study, where this research was mainly conducted and first presented, and the Institut Henri Poincar\'e, where it was concluded. Support was provided by the Air Force Office of Scientific Research and by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the AFOSR or the NSF.
|
1,314,259,993,004 | arxiv |
\section{Conclusions and Future Work}\label{sec:conclusion}
Our work follows Denning's proposal of lattices as the mathematical basis for analysis about secure information flows. We segue to order-preserving morphisms between lattices as a natural framework for a scalable and modular analysis for secure inter-domain flows.
From the basic secure flow requirements that preserved the autonomy of the individual organisations, we identified the simple and elegant theory of Lagois connections as an appropriate formulation.
Lagois connections provide us a way to connect the security lattices of two (secure) systems in a manner that does not expose their entire internal structure and allows us to reason only in terms of the interfaced security classes.
We have also illustrated that the theory of Lagois connections provides a versatile framework for supporting the discovery, decomposition, update and maintenance of secure MoUs for exchanging information between administrative domains.
Compositionality of Lagois connections provides the necessary modularity when chaining connections across several domains, while the canonical decomposition results provide methodological rules within which we can re-establish secure connections when the security lattices are updated.
Moreover, as illustrated here,
we have shown this framework is also applicable in more intricate information flow control formulations such as decentralised IFC and models with declassification \cite{myers-phd-tr-award}.
Ongoing work indicates that the framework works smoothly in formulations with data-dependent security classes \cite{Lourenco2015-ug} as well.
Note that the secure Lagois connection between two domains, especially in the decentralised model, introduces new flows from principals in one domain to those in another, and conversely.
It can be argued that the \textit{sound} and \textit{complete} relabelling rule in the Decentralised Label Model
\[
P \vdash L_1 \sqsubseteq L_2 \mbox{~iff~}
(\forall P' \supseteq P) \textbf{X}(L_1, P') \supseteq \textbf{X}(L_2, P')
\]
(where $\textbf{X}(L, P')$ denotes the set of flows permitted by label $L$ given principals hierarchy $P'$), in a sense already accounts for these new flows.
From the viewpoint of one domain, the permitted flows to and within the other domain can be viewed as an extension $P'$ of its principals hierarchy $P$ that now incorporates principals from the other domain.
The Lagois conditions, however, guarantee that when data flow to another domain and back, \textit{no new flows} are created within each individual domain.
Thus, our connections-based framework provides a modular approach to the static analysis of permitted flows, by partitioning the analysis to flows within each domain and inter-domain flows.
Indeed, the analysis is confined to the \textit{syntactic framework} of the principals hierarchies, and the associated policies and labels.
The proofs of correctness with respect to the semantics do not have to be reworked to consider the slew of new flows.
We believe that it is important to have a framework in which secure flows should be treated in a modular and autonomous manner for the following reason.
The notion of a principal delegating to others the capacity to act on its behalf
(\textit{e.g.}, in the DIFC model of Myers \cite{myers-phd-tr-award}) does not scale well to large, networked systems since a principal may repose different levels of trust in principals on various hosts in the network.
For this reason, we believe that frameworks such as Fabric \cite{liu2009fabric, liu2017fabric} may provide more power than mandated by a principle of least privilege.
In general, since a principal rarely vests unqualified trust in another in all contexts and situations, one should confine the influence of the principals possessing delegated authority to only specific domains.
A mathematical framework that can deal with localising trust and delegation of authority in different domains and controlling the manner in which information flow can be secured deserves a deeper study.
We believe that mathematical theories such as Lagois connections provide the necessary structure for articulating these concepts.
We conclude by noting that it is surprising that Lagois connections have not seen greater use in computer science and particularly in static analysis.
Most applications of Galois connections in fact employ \textit{Galois insertions}, which also happen to be special cases of Lagois connections \cite{MELTON1994lagoisconnections}.
While the duality between closure and interior operators in Galois connections provides them an elegance, the quite different symmetries exhibited by Lagois connections also seem to be natural and useful in many settings.
\section*{References}
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\section{Finding Lagois Connections}\label{sec:revisit-lagois}
Lagois connections go well beyond providing a simple and elegant framework for secure connections between independent security lattices.
They exhibit several properties that support the creation of secure connections, negotiating secure MoUs, and maintaining secure connections when organisational changes occur in the two connected lattices.
Let $\rid{\alpha}[\ld{L}]$ and $\ld{\gamma}[\rid{M}]$ refer to the images of the order-preserving functions $\rid{\alpha}$ and $\ld{\gamma}$, respectively.
Further, let us define the upward closures $\uparrow \rid{m} = \{ \rid{z} \in \rid{M} | \rid{m}\ \rd{\sqsubseteq} \rid{z} \}$ and $\uparrow \ld{l} = \{ \ld{z} \in \ld{L} | \ld{l}\ \ld{\sqsubseteq} \ld{z} \}$.
The two monotone functions
$\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and
$\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ in a Lagois connection
$(\ld{L},\rid{\alpha},\ld{\gamma}, \rid{M})$
uniquely determine each other.
\begin{proposition}[Proposition 3.9 in \cite{MELTON1994lagoisconnections}]\label{prop:fnguniquelydetermin}
If $(\ld{L},\rid{\alpha},\ld{\gamma}, \rid{M})$ is a Lagois connection, then the functions $\rid{\alpha}$ and $\ld{\gamma}$ uniquely determine each other, in fact:
\begin{align}
\ld{\gamma}(\rid{m}) = \bigsqcup \rid{\alpha}^{-1}[\bigsqcap \{ \rid{m^*} \in \rid{\alpha}[\ld{L}] \ | \rid{m} \rd{\sqsubseteq} \rid{m^*}\}] \\
= \bigsqcup \rid{\alpha}^{-1}[\bigsqcap ( \uparrow\rid{m} \cap \rid{\alpha}[\ld{L}])]
\end{align}
and
\begin{align}
\rid{\alpha}(\ld{l}) = \bigsqcup \ld{\gamma}^{-1}[\bigsqcap \{ \ld{l^*}\in \ld{\gamma}[\rid{M}] \ | \ld{l} \ld{\sqsubseteq} \ld{l^*}\}] \\
= \bigsqcup \ld{\gamma}^{-1}[\bigsqcap ( \uparrow\ld{l} \cap \ld{\gamma}[\rid{M}])]
\end{align}
\end{proposition}
\subsection{Negotiating an MoU when given one order-preserving map}\label{sec:lagois-adjoint}
Suppose we are given two security lattices $\ld{L}$ and $\rid{M}$ and an order-preserving function $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$, we can \textit{find a Lagois adjoint} $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$, i.e., an order-preserving function such that $(\ld{L},\rid{\alpha},\ld{\gamma}, \rid{M})$ is a Lagois connection, and thus secure information flow between $\ld{L}$ and $\rid{M}$ is ensured.
\begin{proposition}[Proposition 3.10 in \cite{MELTON1994lagoisconnections}]\label{prop:existlagoisadjoint}
Let $\ld{L}$ and $\rid{M}$ be posets. Then an order-preserving function $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ has a Lagois adjoint $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ iff:
\begin{enumerate}
\item $\rid{\alpha}^{-1}(\rid{m})$ has a largest member, for all $\rid{m} \in \rid{\alpha}[\ld{L}]$.
\item $\uparrow \rid{m} \cap \rid{\alpha}[\ld{L}]$ has a smallest member, for all $\rid{m} \in \rid{M}$.
\item The restriction of $\rid{\alpha}$ from $\{ \bigsqcup \rid{\alpha}^{-1}(\rid{m}) | \rid{m} \in \rid{\alpha}[\ld{L}]\}$ to its image $\rid{\alpha}[\ld{L}]$ is an order isomorphism.
\end{enumerate}
\end{proposition}
Consider the example of the order-preserving function $\rid{\alpha}$ in Figure \ref{fig:WnD}.
First we check whether the given function $\rid{\alpha}$ has a Lagois adjoint or not, using Proposition \ref{prop:existlagoisadjoint}.
The conditions on $\rid{\alpha}$ are obviously satisfied in our example.
We then use the third condition of Proposition \ref{prop:existlagoisadjoint} to identify the order-isomorphic substructures of the participating security lattices.
The security classes which form such an order-isomorphic structure for our example are shown in blue/red in Figure \ref{fig:findingLCfromalpha}.
We allow the information to flow back to domain $\ld{L}$ by mapping first only the \textit{budpoints} in $\rid{M}$ \textit{to the budpoints} in $\ld{L}$, while respecting the conditions for Lagois connections, i.e., $\mathbf{LC3}$ and $\mathbf{LC4}$.
These mappings are shown as dotted brown arrows in Figure \ref{fig:findingLCfromalpha}.
Then we invoke Proposition \ref{prop:fnguniquelydetermin}
to complete the mappings of function $\ld{\gamma}$.
The unmapped security classes in domain $\rid{M}$, say $\rid{m_1}$, are connected to the greatest of those elements of $\ld{L}$ which are mapped by $\rid{\alpha}$ to the least budpoint in $\rid{M}$ greater than $\rid{m_1}$, as shown by solid brown arrows in Figure \ref{fig:findingLCfromalpha2}.
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) {{\color{red}\scriptsize$\top 2$}};
\node(T1) [xshift=-3cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {{\color{blue}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node[align=center] (AS2) [below of=Sec2] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2) [below of=AS2] {{\color{red}\scriptsize$Dean(Colleges)$}};
\node(E2) [below of=Dir2] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2) [below of=E2] {{\color{red}\scriptsize$\bot 2$}};
\draw [OliveGreen, thick] (F1) to (E2);
\draw [OliveGreen, ->, thick] (S1) to (E2);
\draw [OliveGreen, thick] (B1) to (B2);
\draw [OliveGreen, thick] (T1) to (T2);
\draw [OliveGreen, thick] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen, thick] (D1) to (Dir2);
\draw [OliveGreen, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [brown, dotted, thick, <-] (F1)[bend left = 10] to (E2);
\draw [brown, dotted, thick,<-] (B1)[bend left = 15] to (B2);
\draw [brown, dotted, thick, <-] (T1) [bend right = 15] to (T2);
\draw [brown, dotted, thick, <-] (Dir1) [bend left = 20] to (Dir2);
\draw (-3,-2.5)[blue] ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.2cm and 2.7cm);
\draw [dotted] (0,-1.4) ellipse (0.8cm and 1.2cm);
\draw [dotted] (-3,-1.7) ellipse (1.35cm and 0.3cm);
\draw [dotted] (-3,-3.2) ellipse (0.5cm and 0.55cm);
\end{tikzpicture}
\caption{\small Connecting budpoints while finding a viable Lagois \textit{adjoint} for a given order-preserving function $\rid{\alpha}$ (from Figure \ref{fig:WnD}). } \label{fig:findingLCfromalpha}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) {{\color{red}\scriptsize$\top 2$}};
\node(T1) [xshift=-3cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {{\color{blue}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node[align=center] (AS2) [below of=Sec2] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2) [below of=AS2] {{\color{red}\scriptsize$Dean(Colleges)$}};
\node(E2) [below of=Dir2] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2) [below of=E2] {{\color{red}\scriptsize$\bot 2$}};
\draw [densely dashed, black, thick, <->] (F1) to (E2);
\draw [OliveGreen, ->, thick] (S1) to (E2);
\draw [densely dashed, black, thick,<->] (B1) to (B2);
\draw [densely dashed, black, thick, <->] (T1) to (T2);
\draw [densely dashed, black, thick, <->] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen, thick] (D1) to (Dir2);
\draw [OliveGreen, thick] (DS1) to (Dir2);
\draw [brown, thick] (AS2) to (T1);
\draw [brown, thick] (Sec2) to (T1);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (-3,-2.5)[blue] ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.2cm and 2.7cm);
\draw [dotted] (0,-1.4) ellipse (0.8cm and 1.2cm);
\draw [dotted] (-3,-1.7) ellipse (1.35cm and 0.3cm);
\draw [dotted] (-3,-3.2) ellipse (0.5cm and 0.55cm);
\end{tikzpicture}
\caption{\small Defining a viable \textit{Lagois adjoint} for a given $\rid{\alpha}$ (in Figure \ref{fig:WnD})
\label{fig:findingLCfromalpha2}}
\end{minipage}
\end{figure}
\subsection{Negotiating an MoU \textit{ab initio}} \label{sec:lagois-abinitio}
In the absence of any constraints on $\rid{\alpha}$ and $\ld{\gamma}$, it is always possible to define a Lagois connection between two finite security class lattices, e.g., by mapping all elements to the topmost security class of the other lattice.
But such a Lagois connection is of little use to the participating organisations as the shared information, being in the topmost security class, is inaccessible to most principals in the respective organisations. An operative result for creating viable\footnote{We use term ``viable" informally to mean that the natural constraints of the application are taken into account, and that the security level of information is escalated only to the extent required, thus not making its access overly restricted.} secure MoUs is Theorem \ref{lemma:existenceLC}.
\begin{theorem}[Theorem 3.20 in \cite{MELTON1994lagoisconnections}] \label{lemma:existenceLC} Let $(\ld{L},\ld{\sqsubseteq})$ and $(\rid{M},\rd{\sqsubseteq})$ be posets. There is a Lagois connection between $(\ld{L},\ld{\sqsubseteq})$ and $(\rid{M},\rd{\sqsubseteq})$ if and only if the following four conditions hold:
\begin{enumerate}
\item There exist order-isomorphic subsets $\ld{L^*} \subseteq \ld{L}$ and $\rid{{M^*}} \subseteq \rid{M}$.
\item There exists equivalence relations $\ld{\sim_L}$ on $\ld{L}$ and $\rd{\sim_M}$ on $\rid{M}$
such that $\ld{L^*}$ is a system of representatives for $\ld{\sim_L}$ and $\rid{{M^*}}$ is a system of representatives for $\rd{\sim_M}$ respectively\footnote{The members of $\ld{L^*}$ and $\rid{{M^*}}$ are called budpoints and the equivalence classes are called blossoms.}.
\item if $\ld{l_1} \in \ld{L}$ and $\ \ld{l^*} \in \ld{L^*}$ with $\ld{l_1} \ld{\sim_L} \ld{l^*}$, then $\ld{l_1} \ld{\sqsubseteq} \ld{l^*}$; and if $\rid{m_1} \in \rid{M}$ and $\rid{m^*} \in \rid{M^*}$ with $\rid{m_1} \rd{\sim _M}\ \rid{m^*}$, then $\rid{m_1} \rd{\sqsubseteq} \rid{m^*}$.
\item If $\ld{l_1} \ld{\sqsubseteq} \ld{l_2}$ in $\ld{L}$ and $\ld{l^*_1}, \ld{l^*_2} \in \ld{L^*}$ with $\ld{l_1} \ld{\sim_L} \ld{l^*_1}$ and $\ld{l_2} \ld{\sim_L} \ld{l^*_2}$, then $\ld{l^*_1} \ld{\sqsubseteq} \ld{l^*_2}$; and if $\rid{m_1} \rd{\sqsubseteq} \rid{m_2}$ and $\rid{m^*_1}, \rid{m^*_2} \in \rid{M^*}$ with $\rid{m_1} \rd{\sim_M} \rid{m^*_1}$ and $\rid{m_2} \rd{\sim_M} \rid{m^*_2}$, then $\rid{m^*_1} \rd{\sqsubseteq} \rid{m^*_2}$.
\end{enumerate}
\end{theorem}
\begin{corollary}[Corollary 3.21 in \cite{MELTON1994lagoisconnections}] \label{cor:existenceLC2}
Let $\ld{L}$ and $\rid{M}$ be posets, and $\ld{c}: \ld{L} \rightarrow \ld{L}$ and $\rid{i}: \rid{M} \rightarrow \rid{M}$ be closure operators such that $\ld{c}[\ld{L}]$ and $\rid{i}[\rid{M}]$ are isomorphic (with their inherited orders). If $\rid{h}: \ld{c}[\ld{L}] \rightarrow \rid{i}[\rid{M}]$ is such an isomorphism, then $(\ld{L}, \rid{h}\ld{c}, \ld{h^{-1}}\rid{i}, \rid{M})$ is a Lagois connection.
\end{corollary}
Theorem \ref{lemma:existenceLC} suggests the following method, which we illustrate using an example in
Figures \ref{fig:LCfrmScratchA}-\ref{fig:LCfrmScratchWidClosures},
where an agreement is negotiated between \textit{Dorm-Life} and \textit{College}.
\begin{enumerate}
\item Find the maximal order-isomorphic substructures $\ld{L^*} \subseteq \ld{L}$ and $\rid{M^*} \subseteq \rid{M}$ (which include the transfer classes
[\ld{\textit{Student}}-\rid{\textit{Student}}, \ld{\textit{HouseMaster}}-\rid{\textit{Dean(S)}}], the bottom-most and the topmost class) in the given security class lattices $(\ld{L},\ld{\sqsubseteq})$ and $(\rid{M},\rd{\sqsubseteq})$.
These classes in the order-isomorphic structures of the two domains are indicated in bold in Figure \ref{fig:LCfrmScratchA}.
\item Identify an equivalence relation $\ld{\sim_L}$ such that $\ld{L^*}$ is a system of representatives for $\ld{\sim_L}$. Similarly, identify an equivalence relation $\rd{\sim_M}$ such that $\rid{M^*}$ is a system of representatives for $\rd{\sim _M}$. The equivalence relations should be such that the two conditions (3) and (4) of Theorem \ref{lemma:existenceLC} hold, which essentially ensure that one is allowed to reason about information flows from all the security classes in the given security class lattices.
The members of $\ld{L^*}$ (resp. $\rid{M^*}$), called budpoints,
play a significant role in delineating the connection between the transfer classes in the two lattices.
We show the equivalence classes for our example in Figure \ref{fig:LCfrmScratchB}, where the budpoints are the security classes coloured blue and red in the respective domains.
\item Now interconnect the budpoints of both domains to each other while respecting the inherited order, forming an isomorphic structure, as done in Figure \ref{fig:LCfrmScratchC} for our example.
\item
Now keeping in mind Corollary \ref{cor:existenceLC2},
we define closure operators $\ld{c}: \ld{L} \rightarrow \ld{L}$ and $\rd{i}: \rid{M} \rightarrow \rid{M}$, using the equivalence relations $\ld{\sim_L}$ and $\rd{\sim _M}$, such that $\ld{c}[\ld{L}]$ and $\rd{i}[\rid{M}]$ are isomorphic (with their inherited orders). Figure \ref{fig:LCfrmScratchWidClosures} shows the closure operators for our example.
\item Thereafter, define an increasing Lagois connection using Corollary \ref{cor:existenceLC2} as $(\ld{L}, \rid{h}\ld{c}, \ld{h^{-1}}\rd{i}, \rid{M})$. For our example, the Lagois connection is defined as shown in Figure \ref{fig:LCfrmScratchD}, with green arrows completing the mapping from \textit{Dorm-Life} and brown arrows completing the mapping from \textit{College}.
\end{enumerate}
\begin{figure}[!ht]
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T1) { \scriptsize$\mathbf{\top 1}$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {\scriptsize$\mathbf{Dean\ (S)}$};
\node(S1) [below right of = F1] {\scriptsize$\mathbf{Student}$};
\node(B1) [below of=S1] {\scriptsize$\mathbf{\bot 1}$};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw (0,-2.5)[red] ellipse (1.4cm and 2.7cm);
\node(T1') [xshift=-3cm] { \scriptsize$\mathbf{\top 0}$};
\node(Dir1') [below of = T1'] {\scriptsize$\mathbf{HouseMaster}$};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {\scriptsize$\mathbf{Student}$};
\node(B1') [below of=S1'] {\scriptsize$\mathbf{\bot 0}$};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw (-3,-2.5)[blue] ellipse (1.4cm and 3.1cm);
\end{tikzpicture}
\caption{\small Security lattices for two autonomous organisations that want to negotiate a \textit{secure} MoU \textit{ab initio}.
\label{fig:LCfrmScratchA}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T1) {{\color{red} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {{\color{red}\scriptsize$Dean\ (S)$}};
\node(S1) [below right of = F1] {{\color{red}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{red}\scriptsize$\bot 1$}};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw (0,-2.5)[red] ellipse (1.4cm and 2.7cm);
\node(T1') [xshift=-3cm] {{\color{blue} \scriptsize$\top 0$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$HouseMaster$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 0$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw (-3,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\draw [dotted,thick] (-3,-0) ellipse (0.3cm and 0.3cm);
\draw [dotted,thick] (-3,-4.5) ellipse (0.3cm and 0.3cm);
\draw [rotate around={-15:(-0.4,-1.4)},dotted,thick] (-0.4,-1.4) ellipse (0.5cm and 1.7cm);
\draw [dotted,thick] (-0,-4.5) ellipse (0.3cm and 0.3cm);
\draw [dotted,thick] (-3,-3.4) ellipse (0.6cm and 0.3cm);
\draw [dotted,thick] (-0,-3.4) ellipse (0.6cm and 0.3cm);
\draw [dotted,thick] (-3,-1.9) ellipse (1.5cm and 1.1cm);
\draw [dotted,thick] (0.7,-1.7) ellipse (0.6cm and 0.3cm);
\end{tikzpicture}
\caption{\small Identifying equivalence relations in given security lattices for discovering a Lagois connection \textit{ab initio}.
\label{fig:LCfrmScratchB}}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T1) {{\color{red} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {{\color{red}\scriptsize$Dean\ (S)$}};
\node(S1) [below right of = F1] {{\color{red}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{red}\scriptsize$\bot 1$}};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw (0,-2.5)[red] ellipse (1.4cm and 2.7cm);
\node(T1') [xshift=-3cm] {{\color{blue} \scriptsize$\top 0$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$HouseMaster$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 0$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [densely dashed, black, thick, <->] (T1') to (T1);
\draw [densely dashed, black, thick, <->] (B1') to (B1);
\draw [densely dashed, black, thick, <->] (S1') to (S1);
\draw [densely dashed, black, thick, <->] (Dir1') to [bend left = 5] (DS1);
\draw (-3,-2.5)[blue] ellipse (1.4cm and 3.1cm);
\draw [dotted,thick] (-3,-0) ellipse (0.3cm and 0.3cm);
\draw [dotted,thick] (-3,-4.5) ellipse (0.3cm and 0.3cm);
\draw [rotate around={-15:(-0.4,-1.4)},dotted,thick] (-0.4,-1.4) ellipse (0.5cm and 1.7cm);
\draw [dotted,thick] (-0,-4.5) ellipse (0.3cm and 0.3cm);
\draw [dotted,thick] (-3,-3.4) ellipse (0.6cm and 0.3cm);
\draw [dotted,thick] (-0,-3.4) ellipse (0.6cm and 0.3cm);
\draw [dotted,thick] (-3,-1.9) ellipse (1.5cm and 1.1cm);
\draw [dotted,thick] (0.7,-1.7) ellipse (0.6cm and 0.3cm);
\end{tikzpicture}
\caption{\small Connecting budpoints of equivalent security classes.
\label{fig:LCfrmScratchC}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T1) {{\color{red} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {{\color{red}\scriptsize$Dean\ (S)$}};
\node(S1) [below right of = F1] {{\color{red}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{red}\scriptsize$\bot 1$}};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw (0,-2.5)[red] ellipse (1.4cm and 2.7cm);
\node(T1') [xshift=-3cm] {{\color{blue} \scriptsize$\top 0$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$HouseMaster$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 0$}};
\draw [brown,thick, ->] (F1) to [bend right = 10] (T1');
\draw [brown,thick, ->] (D1) to [bend right = 10](T1');
\draw [brown,thick, ->] (Dir1) to (T1');
\draw [OliveGreen,thick, ->] (D1') to [bend right = 20](DS1);
\draw [OliveGreen,thick, ->] (F1') to [bend right = 15] (DS1);
\draw [OliveGreen,thick, ->] (DS1') to [bend right = 15] (DS1);
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [densely dashed, black, thick, <->] (T1') to (T1);
\draw [densely dashed, black, thick, <->] (B1') to (B1);
\draw [densely dashed, black, thick, <->] (S1') to (S1);
\draw [densely dashed, black, thick, <->] (Dir1') to [bend left = 5] (DS1);
\draw (-3,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\end{tikzpicture}
\caption{\small A secure MoU negotiated \textit{ab initio}.
\label{fig:LCfrmScratchD}}
\end{minipage}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T1) [xshift=3cm] {{\color{red} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {{\color{red}\scriptsize$Dean\ (S)$}};
\node(S1) [below right of = F1] {{\color{red}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{red}\scriptsize$\bot 1$}};
\node(T12) [yshift=-1cm] {{\color{red} \scriptsize$\top 1$}};
\node(DS12) [below of = T12] {{\color{red}\scriptsize$Dean\ (S)$}};
\node(S12) [below of = DS12] {{\color{red}\scriptsize$Student$}};
\node(B12) [below of=S12] {{\color{red}\scriptsize$\bot 1$}};
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw (3,-2.5)[red] ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.4cm and 2.7cm);
\node(T1') [xshift=-6cm] {{\color{blue} \scriptsize$\top 0$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$HouseMaster$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 0$}};
\node(T12') [xshift=-3cm,yshift=-1cm] {{\color{blue} \scriptsize$\top 0$}};
\node(Dir12') [below of = T12'] {{\color{blue}\scriptsize$HouseMaster$}};
\node(S12') [below of = Dir12'] {{\color{blue}\scriptsize$Student$}};
\node(B12') [below of=S12'] {{\color{blue}\scriptsize$\bot 0$}};
\draw [purple,thick, ->] (T1) to (T12);
\draw [purple,thick, ->] (B1) to (B12);
\draw [purple,thick, ->] (S1) to (S12);
\draw [purple,thick, ->] (DS1) to [bend left = 20] (DS12);
\draw [purple,thick, ->] (T1') to (T12');
\draw [purple,thick, ->] (B1') to (B12');
\draw [purple,thick, ->] (S1') to (S12');
\draw [purple,thick, ->] (F1) to [bend right = 10] (T12);
\draw [purple,thick, ->] (D1) to [bend right = 10](T12);
\draw [purple,thick, ->] (Dir1) to (T12);
\draw [purple,thick, ->] (T1) to (T12);
\draw [purple,thick, ->] (D1') to [bend right = 20](Dir12');
\draw [purple,thick, ->] (F1') to [bend right = 15] (Dir12');
\draw [purple,thick, ->] (DS1') to [bend right = 15] (Dir12');
\draw [purple,thick, ->] (Dir1') to (Dir12');
\draw [Fuchsia, ->] (B12') to (S12');
\draw [Fuchsia, ->] (S12') to (Dir12');
\draw [Fuchsia, ->] (Dir12') to (T12');
\draw [Fuchsia, ->] (B12) to (S12);
\draw [Fuchsia, ->] (S12) to (DS12);
\draw [Fuchsia, ->] (DS12) to (T12);
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [densely dashed, black, thick, <->] (T12') to (T12);
\draw [densely dashed, black, thick, <->] (B12') to (B12);
\draw [densely dashed, black, thick, <->] (S12') to (S12);
\draw [densely dashed, black, thick, <->] (Dir12') to (DS12);
\draw (-6,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\draw (-3,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\end{tikzpicture}
\caption{\small Using isomorphic images of closure operators to define a Lagois connection. Purple edges define the closure operators for each organisational domain.
\label{fig:LCfrmScratchWidClosures}}
\end{figure}
\subsection{MoUs involving several administrative domains}\label{sec:lagois-composition}
Suppose there is a sequence of administrative domains such that each adjacent pair of domains has negotiated a secure Lagois connection to ensure bidirectional SIF between them.
Theorem \ref{lemma:composinglagoisconnections} and Corollary \ref{lemma:composition2} allow the composition of these Lagois connections using simple functional composition.
\begin{theorem} [Theorem 3.22 in \cite{MELTON1994lagoisconnections}] If $(\ld{L_1}, \rid{\alpha_1}, \ld{\gamma_1}, \rid{M_1})$ and $(\rid{M_1}, \tid{\alpha_2}, \rid{\gamma_2}, \tid{Q})$ are increasing Lagois connections, then the flow defined by the increasing Lagois connection $(\ld{L_1}, \tid{\alpha_2} \circ \rid{\alpha_1} , \ld{\gamma_1} \circ \rid{\gamma_2}, \tid{Q})$ is secure iff
\begin{align}
\rid{\gamma_2} \circ \tid{\alpha_2} \circ \rid{\alpha_1}[\ld{L_1}] \subseteq \rid{\alpha_1}[\ld{L_1}]\ and
\label{cond:composelagois1}
\end{align}
\begin{align}
\rid{\alpha_1} \circ \ld{\gamma_1} \circ \rid{\gamma_2}[\tid{Q}] \subseteq \rid{\gamma_2}[\tid{Q}]
\label{cond:composelagois2}
\end{align}
\label{lemma:composinglagoisconnections}
\end{theorem}
\begin{corollary}[Corollary 3.23 in \cite{MELTON1994lagoisconnections}] \label{lemma:composition2}
If $(\ld{L_1}, \rid{\alpha_1}, \ld{\gamma_1}, \rid{M_1})$ and $(\rid{M_1}, \tid{\alpha_2}, \rid{\gamma_2}, \tid{Q})$ are Lagois connections and if either $\rid{\gamma_2}[\tid{Q}] \subseteq \rid{\alpha_1}(\ld{L_1})$ or $\rid{\gamma_2}[\tid{Q}] \supseteq \rid{\alpha_1}(\ld{L_1})$, then $(\ld{L_1}, \tid{\alpha_2} \circ \rid{\alpha_1}, \ld{\gamma_1} \circ \rid{\gamma_2}, \tid{Q})$ is a Lagois connection.
\end{corollary}
We illustrate how an MoU negotiated between \textit{Dorm-Life} (which has security classes \textit{HouseMaster}, \textit{DiningManager}, \textit{Caretaker}, \textit{Assistant}, and \textit{Student}) and \textit{College} can be composed with the MoU which has been negotiated between the \textit{College} and \textit{University} in Figure \ref{fig:Bi-L-WnD-compose}, to come up with an MoU which allows secure bidirectional information flow between all three domains.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) [xshift=3cm] {{\color{red}\scriptsize$\top 2$}};
\node(T1) {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {{\color{blue}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node[align=center] (AS2) [below of=Sec2] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2) [below of=AS2] {{\color{red}\scriptsize$Dean(Colg)$}};
\node(E2) [below of=Dir2] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(S2) [below of = E2] {{\color{red}\scriptsize$Student$}};
\node(B2) [below of=S2] {{\color{red}\scriptsize$\bot 2$}};
\draw [densely dashed, black, thick, <->] (F1) to (E2);
\draw [densely dashed, black, thick,<->] (B1) to (B2);
\draw [densely dashed, black, thick, <->] (T1) to (T2);
\draw [densely dashed, black, thick, <->] (S2) to (S1);
\draw [densely dashed, black, thick, <->] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen, thick] (D1) to [bend right = 5] (Dir2);
\draw [OliveGreen, thick] (DS1) to (Dir2);
\draw [brown, thick] (AS2) to (T1);
\draw [brown, thick] (Sec2) to (T1);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (S2);
\draw [Fuchsia, ->] (S2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (0,-2.5) [blue] ellipse (1.4cm and 2.7cm);
\draw (3,-3)[red] ellipse (1.2cm and 3.3cm);
\draw (-3.5,-2.5) [Emerald] ellipse (1.5cm and 2.7cm);
\node(T1') [xshift=-3.5cm] {{\color{Emerald} \scriptsize$\top 0$}};
\node(Dir1') [below of = T1'] {{\color{Emerald}\scriptsize$HouseMaster$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Caretaker$};
\node[align=center](F1') [below of = D1'] {\scriptsize $Assistant$};
\node[align=center](DS1') [below right of = Dir1'] {\scriptsize $Dining-$ \\ \scriptsize $Manager$};
\node(S1') [below right of = F1'] {{\color{Emerald}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{Emerald}\scriptsize$\bot 0$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (Dir1') to (DS1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [densely dashed, black, thick, <->] (T1') to (T1);
\draw [densely dashed, black, thick, <->] (B1') to (B1);
\draw [densely dashed, black, thick, <->] (S1') to (S1);
\draw [densely dashed, black, thick, <->] (Dir1') to [bend left = 5] (DS1);
\draw [Mahogany,thick, ->] (Dir1) to [bend right = 10] (T1');
\draw [Mahogany,thick, ->] (F1) to [bend right = 10] (Dir1');
\draw [Mahogany,thick, ->] (D1) to [bend right = 10](Dir1');
\draw [Emerald,thick, ->] (D1') to [bend left = 10] (DS1);
\draw [Emerald,thick, ->] (F1') to [bend right = 3] (DS1);
\draw [Emerald,thick, ->] (DS1') to [bend right = 10] (DS1);
\end{tikzpicture}
\caption{\small Composing Lagois connections.
Here an MoU negotiated between \textit{Dorm-Life} and \textit{College} is composed with another MoU which has been negotiated between \textit{College} and \textit{University}.
\label{fig:Bi-L-WnD-compose}}
\end{figure}
\section{Maintaining MoUs When Security Lattices Change}\label{sec:maintaining-mou}
\subsection{Analysing a Lagois Connection}\label{sec:lagois-decomposition}
Before we discuss how to update an existing MoU, we present Theorem \ref{lemma:decompose}, which provides us a decomposed, analytical view of the anatomy of an increasing Lagois connection.
For example, the Lagois connection given in Figure \ref{fig:findingLCfromalpha2}
can be alternatively viewed as shown in Figure \ref{fig:Bi-L-WnD-decomposedView}.
Let us assume that $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is a Lagois connection. Then, let $\ld{L^*} = \ld{\gamma}[\rid{M}]$ and $\rid{M^*} = \rid{\alpha}[\ld{L}]$. Also, let $\ld{r_1} = (\ld{\gamma} \circ \rid{\alpha})|_{\ld{L}}^{\ld{L^*}}$, $\rid{r_2} = (\rid{\alpha} \circ \ld{\gamma})|^{\rid{M^*}}_{\rid{M}}$, $\rid{i_1} = \rid{\alpha}|^{\rid{M^*}}_{\ld{L^*}}$, $\ld{i_2} = \ld{\gamma}|^{\ld{L^*}}_{\rid{M^*}}$ and let $\ld{e_1}$ be the embedding (inclusion) from $\ld{L^*}$ to $\ld{L}$ and $\rid{e_2}$ be the embedding (inclusion) from $\rid{M^*}$ to $\rid{M}$. Then, $(\ld{L^*}, \rid{i_1}, \ld{i_2}, \rid{M^*})$ is a Lagois isomorphism. This isomorphic substructure is very helpful in reducing the computational effort involved in maintaining the negotiated MoUs based on Lagois connections, as detailed in the sequel.
The key insight is that as long as the inter-domain mappings, i.e., $i_1$ and $i_2$ are not changed, and new security lattices can be connected to old lattices with an increasing Lagois insertion\footnote{A Lagois connection is an insertion if one of the two mappings is injective.}, we will not need to re-negotiate the inter-domain mappings. For example, Figure \ref{fig:Bi-L-WnD-addSC1} shows two such Lagois insertions between old and new security lattices.
\begin{theorem} [Theorem 3.24 in \cite{MELTON1994lagoisconnections}]\label{lemma:decompose}
Every increasing Lagois connection $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is a composite\footnote{For easy left-to-right readability, we have used a composition operator $\diamond$ and reversed the order of the components from how they appear in Theorem 3.24 in \cite{MELTON1994lagoisconnections}. }
$(\ld{L}, \ld{r_1}, \ld{e_1}, \ld{L^*})
\diamond
(\ld{L^*}, \rid{i_1}, \ld{i_2}, \rid{M^*})
\diamond
(\rid{M^*}, \rid{e_2}, \rid{r_2}, \rid{M})$ where
\begin{enumerate}
\item $(\ld{L}, \ld{r_1}, \ld{e_1}, \ld{L^*})$ is an increasing Lagois insertion,
\item $\rid{i_1}$ and $\ld{i_2}$ are isomorphisms that are inverses to each other, and
\item $(\rid{M^*}, \rid{e_2}, \rid{r_2}, \rid{M})$ is an increasing Lagois insertion.
\end{enumerate}
\end{theorem}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) [xshift=3cm, yshift=-1cm] {{\color{red}\scriptsize$\top 2$}};
\node(T1) [yshift=-1cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(F1) [below of = Dir1] {{\color{blue}\scriptsize$Faculty$}};
\node(B1) [below of=F1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Dir2) [below of=T2] {{\color{red}\scriptsize$Dean(Colg)$}};
\node(E2) [below of=Dir2] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2) [below of=E2] {{\color{red}\scriptsize$\bot 2$}};
\draw [densely dashed, black, thick, <->] (F1) to (E2);
\draw [densely dashed, black, thick,<->] (B1) to (B2);
\draw [densely dashed, black, thick, <->] (T1) to (T2);
\draw [densely dashed, black, thick, <->] (Dir1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (F1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (F1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (T2);
\draw (0,-2.5)[blue] ellipse (1.4cm and 2cm);
\draw (3,-2.5)[red] ellipse (1.2cm and 2cm);
\node(T2') [xshift=6cm] {{\color{red}\scriptsize$\top 2$}};
\node(T1') [xshift=-3.5cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Dean\ (F)$};
\node(F1') [below of = D1'] {{\color{blue}\scriptsize$Faculty$}};
\node(DS1') [below right of = Dir1'] {\scriptsize$Dean\ (S)$};
\node(S1') [below right of = F1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2') [below of = T2'] {\scriptsize$Chancellor$};
\node[align=center] (AS2') [below of=Sec2'] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2') [below of=AS2'] {{\color{red}\scriptsize$Dean(Colg)$}};
\node(E2') [below of=Dir2'] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2') [below of=E2'] {{\color{red}\scriptsize$\bot 2$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (S1') to (F1');
\draw [Fuchsia, ->] (S1') to (DS1');
\draw [Fuchsia, ->] (DS1') to (Dir1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [Fuchsia, ->] (B2') to (E2');
\draw [Fuchsia, ->] (E2') to (Dir2');
\draw [Fuchsia, ->] (Dir2') to (AS2');
\draw [Fuchsia, ->] (AS2') to (Sec2');
\draw [Fuchsia, ->] (Sec2') to (T2');
\draw [densely dashed, black, thick, <->] (T1') to (T1);
\draw [densely dashed, black, thick, <->] (B1') to (B1);
\draw [densely dashed, black, thick, <->] (Dir1') to (Dir1);
\draw [densely dashed, black, thick, <->] (F1') to [bend right = 5] (F1);
\draw [OliveGreen, thick, ->] (S1') to [bend right = 15] (F1);
\draw [OliveGreen, thick, ->] (DS1') to (Dir1);
\draw [OliveGreen, thick, ->] (D1') to [bend right = 10] (Dir1);
\draw [densely dashed, black, thick, <->] (T2') to (T2);
\draw [densely dashed, black, thick, <->] (B2') to (B2);
\draw [brown, thick, ->] (Sec2') to (T2);
\draw [brown, thick, ->] (AS2') to (T2);
\draw [densely dashed, black, thick, <->] (Dir2') to (Dir2);
\draw [densely dashed, black, thick, <->] (E2') to (E2);
\draw (-3.5,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\draw (6,-2.5)[red] ellipse (1.3cm and 2.7cm);
\draw [dotted] (6,-1.4) ellipse (0.8cm and 1.2cm);
\draw [dotted] (-3.5,-1.7) ellipse (1.35cm and 0.3cm);
\draw [rotate around={45:(-3.8,-3.4)},dotted] (-3.8,-3.4) ellipse (0.4cm and 0.6cm);
\end{tikzpicture}
\caption{\small A decomposed view of a Lagois connection.
Dashed black arrows define permissible flows between budpoints. \label{fig:Bi-L-WnD-decomposedView}}
\end{figure}
\subsection{Ch-ch-ch-ch-changes}\label{sec:changes}
There are four ways in which security lattices can evolve over time:
\begin{enumerate}
\item Adding security classes to the existing security lattices;
\item Removing security classes from the existing security lattices;
\item Adding new edges to the existing security lattices;
\item Removing edges from existing security lattices.
\end{enumerate}
We examine how to re-establish a secure Lagois connection when changes in the lattice structures occur.
The key observation is that we only need to monitor if the original isomorphic substructure mediating the old Lagois connection between the participating security lattices changes or not.
If the participating isomorphic substructure remains unchanged, and one can find an increasing Lagois insertion between the changed security lattice and the original isomorphic substructure then the MoU need not be re-negotiated.
The necessary updates of the monotone function can be done independently of the other organisation.
\begin{enumerate}
\item \textit{No change in the isomorphic sub-structure}:
\begin{enumerate}
\item The number of equivalence classes remains the same.
This is a simple case.
Use Theorem \ref{lemma:decompose} to first find a Lagois insertion between the new security lattice and the isomorphic substructure, and then update the functions $r_i$ mentioned in Theorem \ref{lemma:decompose}, which can be done independently of the other parts of the diagram.
We illustrate this with an example where both security lattices are updated by adding Teaching Assistants (\textit{TAs}) and student \textit{Mentors} in College and department heads (\textit{HOD}) in \textit{University}, as shown in Figure \ref{fig:Bi-L-WnD-addSC1}.
Using Theorem \ref{lemma:decompose}, we get the updated Lagois connection shown in Figure \ref{fig:Bi-L-WnD-addSC2} --- adding 2 new green arrows on the left, and a new brown arrow on the right.
\end{enumerate}
\item \textit{Change in the isomorphic sub-structure}:
\begin{enumerate}
\item \textit{Increase in number of equivalence classes}: Check the four conditions of Lagois connections (\textbf{LC1}, \textbf{LC2}, \textbf{LC3} and \textbf{LC4}) for the affected equivalence classes in the security lattices.
This is similar to re-negotiating a Lagois connection for those parts of security lattices, as discussed earlier. Use Theorem \ref{lemma:existenceLC} for finding a new Lagois connection.
\item \textit{Decrease in number of equivalence classes}: Use Corollary \ref{lemma:refine} to check if the new mappings (of which the initial mappings are a refinement) form a Lagois connection (discussed in more detail below).
\end{enumerate}
\end{enumerate}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) [xshift=3cm] {{\color{red}\scriptsize$\top 2$}};
\node(T1) {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1) [below of = T1] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [below right of = Dir1] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {{\color{blue}\scriptsize$Student$}};
\node(B1) [below of=S1] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node[align=center] (AS2) [below of=Sec2] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2) [below of=AS2] {{\color{red}\scriptsize$Dean(Colg)$}};
\node(E2) [below of=Dir2] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2) [below of=E2] {{\color{red}\scriptsize$\bot 2$}};
\draw [densely dashed, black, thick, <->] (F1) to (E2);
\draw [OliveGreen, ->, thick] (S1) to (E2);
\draw [densely dashed, black, thick,<->] (B1) to (B2);
\draw [densely dashed, black, thick, <->] (T1) to (T2);
\draw [densely dashed, black, thick, <->] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen, thick] (D1) to (Dir2);
\draw [OliveGreen, thick] (DS1) to (Dir2);
\draw [brown, thick] (AS2) to (T1);
\draw [brown, thick] (Sec2) to (T1);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (0,-2.5) [blue] ellipse (1.4cm and 2.7cm);
\draw (3,-2.5)[red] ellipse (1.2cm and 2.7cm);
\node(T2') [xshift=6cm] {{\color{red}\scriptsize$\top 2$}};
\node(T1') [xshift=-3.5cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Dean\ (F)$};
\node(F1') [below of = D1'] {\scriptsize$Faculty$};
\node(DS1') [below right of = Dir1'] {\scriptsize$Dean\ (S)$};
\node(M1') [below of = DS1'] {{\color{OliveGreen}\scriptsize$Mentors$}};
\node(TA1') [below of = F1'] {{\color{OliveGreen}\scriptsize$TAs$}};
\node(S1') [below right of = TA1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2') [below of = T2'] {\scriptsize$Chancellor$};
\node[align=center] (AS2') [below of=Sec2'] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2') [xshift=5.5cm,yshift=-3.3cm] {{\color{red}\scriptsize$Dean(Colg)$}};
\node(HoD2') [xshift=6.8cm,yshift=-3.3cm] {{\color{Mahogany}\scriptsize$HOD$}};
\node(E2') [below right of=Dir2'] {{\color{red}\scriptsize$Univ.Fac.$}};
\node(B2') [below of=E2'] {{\color{red}\scriptsize$\bot 2$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (TA1') to (F1');
\draw [Fuchsia, ->] (S1') to (TA1');
\draw [Fuchsia, ->] (S1') to (M1');
\draw [Fuchsia, ->] (M1') to (DS1');
\draw [Fuchsia, ->] (DS1') to (Dir1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [Fuchsia, ->] (B2') to (E2');
\draw [Fuchsia, ->] (E2') to (Dir2');
\draw [Fuchsia, ->] (Dir2') to (AS2');
\draw [Fuchsia, ->] (E2') to (HoD2');
\draw [Fuchsia, ->] (HoD2') to (AS2');
\draw [Fuchsia, ->] (AS2') to (Sec2');
\draw [Fuchsia, ->] (Sec2') to (T2');
\draw [densely dashed, black, thick, <->] (T1') to (T1);
\draw [densely dashed, black, thick, <->] (B1') to (B1);
\draw [densely dashed, black, thick, <->] (Dir1') to (Dir1);
\draw [densely dashed, black, thick, <->] (D1') to [bend left = 12] (D1);
\draw [densely dashed, black, thick, <->] (F1') to [bend right = 20] (F1);
\draw [OliveGreen, thick] (TA1') to [bend right = 10] (F1);
\draw [OliveGreen, thick] (M1') to [bend right = 10](DS1);
\draw [densely dashed, black, thick, <->] (S1') to (S1);
\draw [densely dashed, black, thick, <->] (DS1') to [bend right = 12] (DS1);
\draw [densely dashed, black, thick, <->] (T2') to (T2);
\draw [densely dashed, black, thick, <->] (B2') to (B2);
\draw [densely dashed, black, thick, <->] (Sec2') to (Sec2);
\draw [densely dashed, black, thick, <->] (AS2') to (AS2);
\draw [densely dashed, black, thick, <->] (Dir2') to (Dir2);
\draw [densely dashed, black, thick, <->] (E2') to (E2);
\draw [brown, thick] (HoD2') to (AS2);
\draw (-3.5,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\draw (6,-2.5)[red] ellipse (1.3cm and 2.7cm);
\end{tikzpicture}
\caption{\small Organisations can add security classes to their lattice structures autonomously as long as they are able to connect the new lattice structures with the old lattice structures (participating in the MoU) via a Lagois insertion.
Dashed black arrows define permissible flows between budpoints. \label{fig:Bi-L-WnD-addSC1}}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\node(T2') {{\color{blue}\scriptsize$\top 2$}};
\node(T1') [xshift=-3.5cm] {{\color{blue} \scriptsize$\top 1$}};
\node(Dir1') [below of = T1'] {{\color{blue}\scriptsize$CollegePrincipal$}};
\node(D1') [below left of = Dir1'] {\scriptsize$Dean\ (F)$};
\node(F1') [below of = D1'] {\scriptsize$Faculty$};
\node(DS1') [below right of = Dir1'] {\scriptsize$Dean\ (S)$};
\node(M1') [below of = DS1'] {{\color{OliveGreen}\scriptsize$Mentors$}};
\node(TA1') [below of = F1'] {{\color{OliveGreen}\scriptsize$TAs$}};
\node(S1') [below right of = TA1'] {{\color{blue}\scriptsize$Student$}};
\node(B1') [below of=S1'] {{\color{blue}\scriptsize$\bot 1$}};
\node(Sec2') [below of = T2'] {\scriptsize$Chancellor$};
\node[align=center] (AS2') [below of=Sec2'] {\scriptsize $Vice$ \\ \scriptsize $Chancellor$};
\node(Dir2') [xshift=-0.5cm,yshift=-3.3cm] {{\color{blue}\scriptsize$Dean(Colg)$}};
\node(HoD2') [xshift=0.8cm,yshift=-3.3cm] {{\color{Mahogany}\scriptsize$HOD$}};
\node(E2') [below right of=Dir2'] {{\color{blue}\scriptsize$Univ.Fac.$}};
\node(B2') [below of=E2'] {{\color{blue}\scriptsize$\bot 2$}};
\draw [Fuchsia, ->] (Dir1') to (T1');
\draw [Fuchsia, ->] (D1') to (Dir1');
\draw [Fuchsia, ->] (F1') to (D1');
\draw [Fuchsia, ->] (TA1') to (F1');
\draw [Fuchsia, ->] (S1') to (TA1');
\draw [Fuchsia, ->] (S1') to (M1');
\draw [Fuchsia, ->] (M1') to (DS1');
\draw [Fuchsia, ->] (DS1') to (Dir1');
\draw [Fuchsia, ->] (B1') to (S1');
\draw [Fuchsia, ->] (B2') to (E2');
\draw [Fuchsia, ->] (E2') to (Dir2');
\draw [Fuchsia, ->] (Dir2') to (AS2');
\draw [Fuchsia, ->] (E2') to (HoD2');
\draw [Fuchsia, ->] (HoD2') to (AS2');
\draw [Fuchsia, ->] (AS2') to (Sec2');
\draw [Fuchsia, ->] (Sec2') to (T2');
\draw [densely dashed, black, thick, <->] (F1') to [bend right = 10] (E2');
\draw [OliveGreen, ->, thick] (TA1') to [bend right = 5] (E2');
\draw [OliveGreen, ->,thick] (M1') [bend right = 10] to (Dir2');
\draw [brown, ->,thick] (HoD2') [bend left = 3] to (T1');
\draw [OliveGreen, ->, thick] (S1') to [bend right = 5] (E2');
\draw [densely dashed, black, thick,<->] (B1') to (B2');
\draw [densely dashed, black, thick, <->] (T1') to (T2');
\draw [densely dashed, black, thick, <->] (Dir1') [bend left = 20] to (Dir2');
\draw [OliveGreen, thick] (D1') to (Dir2');
\draw [OliveGreen, thick] (DS1') to (Dir2');
\draw [brown, thick] (AS2') to (T1');
\draw [brown, thick] (Sec2') to (T1');
\draw (-3.5,-2.5) [blue] ellipse (1.4cm and 3.1cm);
\draw (0,-2.5)[red] ellipse (1.3cm and 2.7cm);
\draw [rotate around={25:(0,-1.4)}, dotted] (0,-2) ellipse (0.8cm and 1.8cm);
\draw[rotate around={70:(-3.4,-2)}, dotted] (-3.4,-2) ellipse (0.7cm and 1.4cm);
\draw[rotate around={40:(-4.2,-3.4)}, dotted] (-4.2,-4) ellipse (0.55cm and 1.15cm);
\end{tikzpicture}
\caption{\small A \emph{new} viable increasing Lagois connection created by the composition of old security lattices with new security lattices.
Dashed black arrows define permissible flows between budpoints. \label{fig:Bi-L-WnD-addSC2}}
\end{figure}
\subsubsection{Coarsening of inter-domain mappings.}
When the isomorphic structure is changed, the MoU has to be renegotiated. When deleting a security class, we can always ``up-classify'' information pertaining to that class, to cause a stricter flow of information to a higher security class in the other lattice.
When coarsening a lattice, we would like to re-establish a Lagois connection that respects the previous Lagois connection as much as possible (maintaining legacy) while making the necessary up-classifications where necessary.
Let $(\ld{L},\rid{\alpha}, \ld{\gamma}, \rid{M})$ be an existing secure Lagois connection.
Suppose we need to replace $\rid{\alpha}$ by another order-preserving function $\rid{\alpha'}: \ld{L} \rightarrow \rid{M}$.
We look for a suitable
map $\ld{\gamma'}: \rid{M} \rightarrow \ld{L}$ such that $(\ld{L},\rid{\alpha'}, \ld{\gamma'}, \rid{M})$ is a Lagois connection using the following result:
\begin{proposition}[Proposition 2.4 in \cite{melton1991connections}] \label{def:connection}
If $\rid{\alpha_1}: \ld{L} \rightarrow \rid{M}$ is an order-preserving map with semi-inverse $\ld{\gamma_1}: \rid{M} \rightarrow \ld{L}$, i.e.,
$\rid{\alpha_1} \circ \ld{\gamma_1} \circ \rid{\alpha_1} = \rid{\alpha_1}$, then $(\ld{L},\rid{\alpha_1}, \ld{\gamma_1} \circ \rid{\alpha_1} \circ \ld{\gamma_1}, \rid{M})$ is a (Lagois) connection.
\end{proposition}
As mentioned above, when the isomorphic structure is disturbed due to coarsening a lattice, we would like to retain as much of original functions as possible.
We can re-establish a Lagois connection making minimal changes to the old functions $\rid{\alpha}$ and $\ld{\gamma}$ Lagois by considering an (order-preserving)
$\rid{\alpha'}$ which satisfies the assumptions:
\begin{enumerate}
\item \label{maintain-cond2} $Ker(\rid{\alpha}) \subseteq Ker(\rid{\alpha'})$\footnote{$Ker(\rid{\alpha})$ is the equivalence relation on $\ld{L}$ given by $(\ld{a},\ld{b}) \in Ker(\rid{\alpha})$ iff $\rid{\alpha}(\ld{a}) = \rid{\alpha}(\ld{b})$. For $\ld{b} \in \ld{L}$, we denote the equivalence class containing $\ld{b}$ by $[\ld{b}]_{\rid{\alpha}}$, as the equivalence relation on $\ld{L}$ is defined by $\rid{\alpha}$.}
\item \label{maintain-cond3} for each $\ld{l} \in \ld{L}, \rid{\alpha'}(\ld{l}) = \rid{\alpha}(\ld{l^*})$ when $\ld{l^*}$ is the largest element in $\{\ld{l_1} \in \ld{L}| \rid{\alpha'}(\ld{l_1}) = \rid{\alpha'}(\ld{l})\}.$
\end{enumerate}
Typically we would consider as a candidate an $\rid{\alpha'}$ that induces as fine a coarsening of the kernel of $\rid{\alpha}$ as possible while ensuring the second condition (\textit{i.e.}, mapping elements of equivalence classes induced by $\rid{\alpha'}$ (which should be closed with respect to joins) according to how $\rid{\alpha}$ mapped the largest element in that class).
\begin{corollary}[Corollary 3.16 in \cite{MELTON1994lagoisconnections}] \label{lemma:refine
Let $(\ld{L},\rid{\alpha}, \ld{\gamma}, \rid{M})$ be a Lagois connection, and let $\rid{\alpha'}\colon \ld{L} \rightarrow \rid{M}$ be an order-preserving map such that
\begin{enumerate}
\item $\equiv_{\rid{\alpha}}$ is a refinement of $\equiv_{\rid{\alpha'}}$
(i.e., $Ker(\rid{\alpha}) \subseteq Ker(\rid{\alpha'})$)
and
\item for each $\ld{l} \in \ld{L}$, the equivalence class $[\ld{l}]_{\rid{\alpha'}}$ has a largest element -- call it $\ld{l^*}$ -- with $\rid{\alpha'}(\ld{l}) = \rid{\alpha}(\ld{l^*})$.
\end{enumerate}
Then $(\ld{L}, \rid{\alpha'}, \ld{\gamma} \circ \rid{\alpha'} \circ \ld{\gamma}, \rid{M})$ is a Lagois connection.
\end{corollary}
\section{Introduction}
Denning's seminal work \cite{Denning76} established \textit{complete lattices} as the mathematical basis for a variety of analyses regarding \textit{secure information flow} (SIF), \textit{i.e.}, showing only authorised flows of information are possible.
An information flow model (IFM)
$\langle N, P, SC, \sqcup, \sqsubseteq \rangle$ consists of storage objects $N$, which are assigned \textit{security classes} drawn from a (finite) complete lattice $SC$.
The partial ordering $\sqsubseteq$ represents \textit{permitted flows} between classes.
Reflexivity and transitivity capture intuitive aspects of information flow; antisymmetry helps avoid redundancies in the framework.
The join operation $\sqcup$ succinctly captures the combination of information belonging to different security classes in arithmetic, logical and computational operations.
$P$ is a set of processes, which are assigned security classes as ``clearances''.
The ensuing decades have seen a plethora of static and dynamic analysis techniques using that framework for programming languages \cite{sabelfeld2003language,myers1999jflow,Pottier2003-FlowCaml,liu2017fabric,roy2009laminar,Lourenco2015-ug}, operating systems \cite{Krohn2007-aa,zeldovich2006-osdi,cheng2012aeolus,efstathopoulos2005asbestos,roy2009laminar}, databases \cite{schultz2013ifdb}, and hardware architectures \cite{ferraiuolo2018hyperflow, zhang2015secVerilog-asplos}, etc.
However, the question on how information can flow securely between independent organisations each with possible quite different policies has not been adequately addressed.
An answer needs to indicate how security classes from one lattice are mapped to those in another.
In doing so, we wish to abjure \textit{ad hoc} approaches to reclassifying information.
We revisit and expand on our previous work \cite{BhardwajP2019}, where we proposed a simple and versatile mathematical framework involving \textit{monotone} increasing functions between lattices, which guaranteed secure and modular inter-organisational flows of information.
Our work is based on the observation that large information systems are not monolithic, and are often constructed by autonomous organisations, each with its own security lattice, policies and mechanisms, negotiating agreements (MoUs) that promise respecting the security policies of others.
The MoUs typically mention only a small set of security classes, called \textit{transfer classes}, between which \textit{all} information exchange is managed.
\textit{Modularity} and \textit{autonomy} are important: each organisation would wish to retain control over its own security policies and the ability to redefine them.
We identified the elegant theory of Lagois Connections \cite{MELTON1994lagoisconnections} of order-preserving functions between security lattices as the appropriate framework, showing that they guarantee SIF, without the need for re-verifying the security of the application procedures in either of the domains, and confining the analysis to only the transfer classes involved in potential exchange of data.
This paper substantiates this by showing (i) how language-based techniques such as security type systems given by Volpano \textit{et al}. \cite{DBLP:journals/jcs/VolpanoIS96} can be adapted to a setting with different systems having distinct secure flow policies communicating data objects between each other; (ii) how results from Lagois theory provide a robust methodological framework for negotiating and maintaining secure agreements on information flow between autonomous organisations, even when either or both organisations change their security policies.
(ii) how decentralised secure flow systems such as that proposed by Myers \cite{myers-phd-tr-award} can be smoothly and conservatively extended to cross-organisational delegation and decentralisation.
Indeed, one way to view this work is that Lagois connections support a conservative extension of SIF analysis techniques developed on complete lattices to embrace structures involving lattices and morphisms between them.
In \S\ref{sec:Bi-DirFlow}, we identify
intuitive requirements for secure bidirectional flow, present the definition of Lagois connections \cite{MELTON1994lagoisconnections}, and show that Lagois connections between the security lattices satisfy security and other requirements.
We include a brief account of how lattices and Lagois connections can be very succinctly represented to support efficient algorithms.
We present in \S\ref{sec:model} a minimal operational language consisting of a small set of \textit{atomic primitives} for effecting the transfer of data between domains.
The framework is simple and can be adapted for establishing secure connections between distributed systems at any level of abstraction (language, system, database, ...).
We assume each domain uses \textit{atomic transactional operations} for object manipulation and intra-domain computation.
The primitives of our model include reliable and atomic communication between two systems, transferring object data in designated \textit{output} variables of one domain to designated \textit{input} variables of a specified security class in the other domain.
To avoid interference between inter-domain communication and the computations within the domains, we assume that the sets of designated input and output variables are all mutually exclusive of one another, and also with the program/system variables used in the computations within each domain.
Thus by design we avoid the usual suspects that cause interference and insecure transfer of data.
The language should be seen as notation for execution sequences of atomic actions performed by concurrent communicating systems.
So it does not include conditional or iterative constructs, assuming these are absorbed within atomic intradomain transactions.
Thus, we do not have to concern ourselves with issues of implicit flows that arise due to branching structures (\textit{e.g.}, conditionals and loops in programming language level security, pipeline mispredictions at the architectural level, etc.)
The operational description of the language consists of the primitives
together with their execution rules (\S\ref{sec:operations}).
The correctness of our framework is demonstrated by expressing soundness (with respect to the operational semantics) of a type system (\S\ref{sec:typing}), stated in terms of the security lattices and their connecting functions.
In particular, Theorem \ref{thm:soundness} shows the standard semantic property of \textit{non-interference} \cite{DBLP:conf/sp/GoguenM82a} in \textit{both domains} holds of all operational behaviours.
We adapt and \textit{extend} the approach taken by Volpano \textit{et al.} \cite{DBLP:journals/jcs/VolpanoIS96} to encompass systems coupled using the Lagois connection conditions, and (assuming atomicity of the data transfer operations) show that \textit{security is conserved}.
Since our language is a minimal imperative model with atomic operations, and security types are exactly the security classes, our proof pares down the techniques of Volpano \textit{et al.} to the bare essentials.
We revisit in \S\ref{sec:revisit-lagois} several results of Melton \textit{et al.} \cite{MELTON1994lagoisconnections} that help us develop a methodical approach to finding and defining suitable MoUs for secure bidirectional flow, pictorially illustrating the systematic development on an example.
In \S\ref{sec:lagois-adjoint}, we use the fact that the morphisms of a Lagois connection uniquely determine each other to complete a MoU when given one side of a proposed mapping, by finding its \textit{Lagois adjoint}.
\S\ref{sec:lagois-abinitio} discusses a methodical approach to negotiating a viable secure MoU \textit{ab initio}.
In \S\ref{sec:lagois-composition}, we use a compositionality result on Lagois connections to chain secure flows through a sequence of organisations.
\S\ref{sec:maintaining-mou} tackles the issue of renegotiating and re-establishing a secure MoU when either party changes its security lattice.
This is made possible by an decomposition result on Lagois connections discussed in \S\ref{sec:lagois-decomposition}.
\S\ref{sec:changes} details the various techniques for rebuilding a secure MoU when changes are made to the security lattice, using appropriate results from
\cite{MELTON1994lagoisconnections}
We then show that our Lagois connection framework readily accommodates decentralised flow control mechanisms within and across organisations, conservatively extending the model of Myers \cite{Myers1997-ss,myers-phd-tr-award}.
After a brief summary of the DLM framework in \S\ref{sec:DLM-summary}, we show in Theorem \ref{Thm:LC-PH-LC-IFL} (\S\ref{sec:lagois-DLM-IFL}) that a Lagois connection between the principals hierarchies of two domains induces a Lagois connection between their corresponding lattices of labels.
A simple corollary, stated for static principals hierarchies that are connected by a Lagois connection, is that the declassification rule remains safe even with bidirectional information exchange between the domains.
In \S\ref{sec:related}, we briefly review some related work.
We conclude in \S\ref{sec:conclusion} with a discussion on our approach and directions for future work.
\textbf{Note}: This paper substantially expands on our earlier work \cite{BhardwajP2019}.
Contents of that paper included here appear in the preliminaries in \S\ref{sec:Bi-DirFlow}, the technical results of \S\ref{sec:model} and the discussion on related work in \S\ref{sec:related}.
\section{Lagois Connections and All That}\label{sec:Bi-DirFlow}
\paragraph{Motivating Example} \ \
Consider a university $\rid{U}$ in which students study in semi-autonomously administered colleges (\textit{e.g.}, $\ld{C}$) affiliated to the university.
A college has \textit{students}, \textit{faculty} members, \textit{deans}, all of whom work under a \textit{College Principal}.
The university has its own \textit{university professors}, a \textit{dean of colleges}, and a \textit{vice-chancellor} working under a \textit{chancellor}.
Students can take classes with both college faculty members and university professors.
Assume that each institution has established its secure information flow mechanisms and policies, and information flows from $\ld{C}$ to $\rid{U}$, as shown by the blue arrows in Figure \ref{fig:WnD}.
\textit{Monotonicity} (or \textit{order-preservation}) of a function mapping security classes in $\ld{C}$ to $\rid{U}$ suffices for $\rid{U}$ to respect $\ld{C}$'s security policies.
However, as we showed earlier \cite{BhardwajP2019}, when flow is \textit{bidirectional}, composing order-preserving functions between $\ld{C}$ and $\rid{U}$ is insufficient.
Indeed, even a \textit{Galois connection} between the two domains does not ensure security (see Figure \ref{fig:Bi-WnD-yNotGC}).
While \textit{Galois insertions} ensure security, they require one of the functions to be \textit{surjective}, whereas in many situations, an organisation may not wish to expose its entire security class lattice.
Further, we do not wish information flowing between two domains to be reclassified in an overly restrictive manner that makes it inaccessible (we call these ``precision'' and ``convergence'' requirements).
\begin{figure}[t
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=0.9cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [blue, thick] (S1) to (E2);
\draw [OliveGreen, densely dashed, thick] (F1) to (E2);
\draw [OliveGreen, densely dashed, thick] (B1) to (B2);
\draw [OliveGreen, densely dashed, thick] (T1) to (T2);
\draw [blue, thick] (Dir1) [bend left = 10] to (Dir2);
\draw [OliveGreen, densely dashed, thick] (D1) [bend right = 15] to (Dir2);
\draw [OliveGreen, densely dashed, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw (-3,-2.2)[blue] ellipse (1.4cm and 2.4cm);
\draw (0,-2.2)[red] ellipse (1.2cm and 2.5cm);
\end{tikzpicture}
\caption{\small Unidirectional flow: If the solid blue arrows denote identified flows connecting important classes, then the dashed green arrows are constrained by monotonicity to lie between them. \label{fig:WnD}}
\end{minipage}
\quad \quad
\begin{minipage}[t]{.45\textwidth}
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\title{W and D}
\node(T2) {\scriptsize$\top 2$};
\node(T1) [xshift=-3cm] {\scriptsize$\top 1$};
\node(Dir1) [below of = T1] {\scriptsize$CollegePrincipal$};
\node(D1) [below left of = Dir1] {\scriptsize$Dean\ (F)$};
\node(F1) [below of = D1] {\scriptsize$Faculty$};
\node(DS1) [xshift=-2.3cm,yshift=-1.7cm] {\scriptsize$Dean\ (S)$};
\node(S1) [below right of = F1] {\scriptsize$Student$};
\node(B1) [below of=S1] {\scriptsize$\bot 1$};
\node(Sec2) [below of = T2] {\scriptsize$Chancellor$};
\node(AS2) [below of=Sec2] {\scriptsize$Vice\ Chancellor$};
\node(Dir2) [below of=AS2] {\scriptsize$Dean(Colleges)$};
\node(E2) [below of=Dir2] {\scriptsize$Univ.Fac.$};
\node(B2) [below of=E2] {\scriptsize$\bot 2$};
\draw [OliveGreen, thick] (S1) [bend left = 5] to (Dir2);
\draw [red, densely dashdotted, thick] (Dir2) [bend left = 5] to (S1);
\draw [red, densely dashdotted, thick] (F1) to (Dir2);
\draw [OliveGreen, thick] (B1) to (B2);
\draw [OliveGreen, thick, ->] (T1) [bend right=10] to (T2);
\draw [brown, thick, ->] (T2) [bend right=10] to (T1);
\draw [OliveGreen, thick] (Dir1) [bend left= 10] to (AS2);
\draw [red, densely dashdotted, thick] (D1) to (Dir2);
\draw [red, densely dashdotted, thick] (DS1) to (Dir2);
\draw [Fuchsia, ->] (Dir1) to (T1);
\draw [Fuchsia, ->] (D1) to (Dir1);
\draw [Fuchsia, ->] (F1) to (D1);
\draw [Fuchsia, ->] (S1) to (F1);
\draw [Fuchsia, ->] (S1) to (DS1);
\draw [Fuchsia, ->] (DS1) to (Dir1);
\draw [Fuchsia, ->] (B1) to (S1);
\draw [Fuchsia, ->] (B2) to (E2);
\draw [Fuchsia, ->] (E2) to (Dir2);
\draw [Fuchsia, ->] (Dir2) to (AS2);
\draw [Fuchsia, ->] (AS2) to (Sec2);
\draw [Fuchsia, ->] (Sec2) to (T2);
\draw [brown, thick] (B2) [bend left=20] to (B1);
\draw [brown, thick] (Sec2) to (T1);
\draw [brown, thick] (AS2) to (Dir1);
\draw [brown, thick] (E2) [bend left=10] to (S1);
\draw (-3,-2.5)[blue] ellipse (1.4cm and 2.7cm);
\draw (0,-2.5)[red] ellipse (1.2cm and 2.7cm);
\end{tikzpicture}
\caption{\small
The arrows between the domains define a Galois Connection.
However, the red dash-dotted arrows highlight flow security violations when information can flow in both directions.
\label{fig:Bi-WnD-yNotGC}}
\end{minipage}
\label{fig:first-combined}
\end{figure}
\paragraph{Requirements} \ \ Accordingly, we identified the following requirements \cite{BhardwajP2019} for viable secure information flow between two lattices $(\ld{L}, \ld{\sqsubseteq})$ and
$(\rid{M}, \rd{\sqsubseteq})$ with order-preserving functions
$\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and
$\ld{\gamma}: \rid{M} \rightarrow \ld{L}$.
\begin{itemize}
\item \textit{Security:}
\[
\textbf{SC1}~~ \lambda \ld{l}.\ld{l} ~\ld{\sqsubseteq}~
\ld{\gamma} \circ \rid{\alpha}
~~~~~~\hfill~~~~~~
\textbf{SC2} ~~ \lambda \rd{m}.\rd{m} ~ \rd{\sqsubseteq}~
\rid{\alpha} \circ \ld{\gamma}
\]
\item \textit{Precision:}
Let $\rid{\alpha}[\ld{L}]$ and $\ld{\gamma}[\rid{M}]$ denote the images of
set $\ld{L}$ under mapping $\rid{\alpha}$
and set $\rid{M}$ under mapping $\ld{\gamma}$.
\[
\begin{array}{c}
\textbf{PC1}~~\rid{\alpha}(\ld{l_1}) = \rid{\bigsqcup} ~ \{\rid{m_1} ~|~ \ld{\gamma}(\rid{m_1}) = \ld{l_1} \}, \; \; \forall \ld{l_1} \in \ld{\gamma}[\rid{M}]\\
\textbf{PC2} ~~\ld{\gamma}(\rid{m_1}) = \ld{\bigsqcup} ~ \{\ld{l_1} ~|~ \rid{\alpha}(\ld{l_1}) = \rid{m_1}\}, \; \; \forall \rid{m_1} \in \rid{\alpha}[\ld{L}]
\end{array}
\]
\item \textit{Convergence:} (\textbf{CC1} and \textbf{CC2})
\textit{Fixed points} for the compositions $\ld{\gamma} \circ \rid{\alpha}$ and $\rid{\alpha} \circ \ld{\gamma}$
are reached as low in the orderings $\ld{\sqsubseteq}$ and $\rd{\sqsubseteq}$ as possible.
\end{itemize}
\paragraph{Lagois Connections} \ \
We identified the elegant formulation of \textit{Lagois Connections} \cite{MELTON1994lagoisconnections} as an appropriate structure that satisfies these requirements:
\begin{definition}[Lagois Connection \cite{MELTON1994lagoisconnections}]
If $L = (\ld{L},\ld{\sqsubseteq})$ and $M = (\rid{M},\rd{\sqsubseteq})$ are two partially ordered sets, and $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ are order-preserving functions, then we call the quadruple $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ an {\em increasing} Lagois connection, if it satisfies the following properties:
\[
\begin{array}{llcll}
\textbf{LC1}~~ & \lambda \ld{l}.\ld{l} ~\ld{\sqsubseteq}~
\ld{\gamma} \circ \rid{\alpha}
& ~~~~~~~~~~~ &
\textbf{LC2}~~ & \lambda \rd{m}.\rd{m} ~\rd{\sqsubseteq}~
\rid{\alpha} \circ \ld{\gamma} \\
\textbf{LC3}~~ & \rid{\alpha} \circ \ld{\gamma} \circ \rid{\alpha} = \rid{\alpha}
& ~~~~~~~~~~~ &
\textbf{LC4}~~ & \ld{\gamma} \circ \rid{\alpha} \circ \ld{\gamma} = \ld{\gamma}
\end{array}
\]
\end{definition}
\textbf{LC3} ensures that $\ld{\gamma}(\rid{\alpha}(\ld{c_1}))$ is the least upper bound of all security classes in $\ld{C}$ that are mapped to the same security class, say $\rid{u_1} = \rid{\alpha}(\ld{c_1})$ in $\rid{U}$.
Observe that Lagois connections are transposable: if
$(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is a Lagois connection, then so is
$(\rid{M}, \ld{\gamma}, \rid{\alpha}, \ld{L})$.
Lagois connections are fundamentally different from Galois connections in that they relate two linked closure operators in two posets, as opposed to a linking a closure and an interior operator.
We showed that Lagois connections satisfy the desired requirements:
\begin{otheorem}[Theorem in \cite{BhardwajP2019}]\label{thm:secureconnection}
Let $L = (\ld{L},\ld{\sqsubseteq}, \ld{\sqcup}, \ld{\sqcap})$ and $M = (\rid{M},\rd{\sqsubseteq}, \rd{\sqcup}, \rd{\sqcap})$ be two complete security class lattices, and let $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and $\ld{\gamma}: \rid{M} \rightarrow \ld{L}$ be order-preserving functions.
Then the flow of information permitted by $\rid{\alpha}$, $\ld{\gamma}$ satisfies conditions \textbf{SC1}, \textbf{SC2}, \textbf{PC1}, \textbf{PC2}, \textbf{CC1} and \textbf{CC2}
if $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ is an increasing Lagois connection.
\end{otheorem}
In the following discussion, let $(\ld{L}, \rid{\alpha}, \ld{\gamma}, \rid{M})$ be a Lagois connection.
Let $\rid{\alpha}[\ld{L}]$ and $\ld{\gamma}[\rid{M}]$ refer to the images of the order-preserving functions $\rid{\alpha}$ and $\ld{\gamma}$, respectively.
The images $\ld{\gamma}[\rid{M}]$ and $\rid{\alpha}[\ld{L}]$ are in fact isomorphic lattices.
For all $\rid{m} \in \rid{\alpha}[\ld{L}]$ and $\ld{l} \in \ld{\gamma}[\rid{M}]$, $\ld{\gamma}(\rid{m})$ and $\rid{\alpha}(\ld{l})$ exist.
\begin{oproposition}[Proposition 3.7 in \cite{MELTON1994lagoisconnections}]\label{prop:largest-pre}
Let $\rid{m} \in \rid{\alpha}[\ld{L}]$ and $\ld{l} \in \ld{\gamma}[\rid{M}]$.
Then $\rid{\alpha}^{-1}(\rid{m})$ has a largest member, which is $\ld{\gamma}(\rid{m})$, and $\ld{\gamma}^{-1}(\ld{l})$ has a largest member, which is $\rid{\alpha}(\ld{l})$.
\end{oproposition}
We call these dominating members of the pre-images of $\rid{\alpha}$ and $\ld{\gamma}$ \textit{budpoints}.
Indeed:
\begin{align}
\ld{\gamma}(\rid{\alpha}(\ld{l})) ~=~
\ld{\sqcap} \{ \ld{l^*} \in \ld{\gamma}[\rid{M}] ~|~
\ld{l} ~\ld{\sqsubseteq}~ \ld{l^*} \}, \label{TIGHT1} \\
\rid{\alpha}( \ld{\gamma}(\rid{m})) ~=~
\rid{\sqcap} \{ \rid{m^*} \in \rid{\alpha}[\ld{L}] ~|~
\rid{m} ~\rd{\sqsubseteq}~ \rid{m^*} \}.
\label{TIGHT2}
\end{align}
Let $\rid{\thicksim_M}$ and $\ld{\thicksim_L}$ be the equivalence relations induced by the functions $\ld{\gamma}$ and $\rid{\alpha}$.
$\ld{L^*} = \ld{\gamma}[\rid{\alpha}[\ld{L}]] = \ld{\gamma}[\rid{M}]$ and
$\rid{M^*} = \rid{\alpha}[\ld{\gamma}[\rid{M}]] = \rid{\alpha}[\ld{L}]$ define a system of representatives for $\ld{\thicksim_L}$ and $\rid{\thicksim_M}$.
Element $\rid{m^*} = \rid{\alpha}(\ld{\gamma}(\rd{m}))$ in $\rid{M^*}$, which is a \textit{budpoint}, acts as the representative of the equivalence class $[\rd{m}]$ in the following sense:
\begin{align}
\textit{if}~ \rid{m}\in \rid{M} ~\textit{and}~ \rid{m^*} \in \rid{M^*} ~\textit{with}~ \rid{m} ~\rid{\thicksim_M}~ \rid{m^*}~\textit{then}~ \rid{m} ~\rd{\sqsubseteq}~ \rid{m^*}
\end{align}
Symmetrically, $\ld{L^*} = \ld{\gamma}[\rid{\alpha}[\ld{L}]] = \ld{\gamma}[\rid{M}]$ defines a system of representatives for $\ld{\thicksim_L}$.
These budpoints play a significant role in delineating the connection between the transfer classes in the two lattices.
Further, Proposition \ref{prop:meets} shows that these budpoints are closed under meets.
This property enables us to confine our analysis to just these security classes when reasoning about bidirectional flows.
\begin{proposition}[Proposition 3.11 in \cite{MELTON1994lagoisconnections}]\label{prop:meets}
\label{prop:meet-existence}
If $\ld{A} \subseteq \ld{\gamma}[\rid{M}]$, then
\begin{enumerate}
\item the meet of $\ld{A}$ in $\ld{\gamma}[\rid{M}]$ exists if and only if the meet of $\ld{A}$ in $\ld{L}$ exists, and whenever either exists, they are equal.
\item the join $\ld{\hat{a}}$ of $\ld{A}$ in $\ld{\gamma}[\rid{M}]$ exists if the join $\ld{\check{a}}$ of $\ld{A}$ in $\ld{L}$ exists, and in this case $\ld{\hat{a}} = \ld{\gamma}(\rid{\alpha}(\ld{\check{a}}))$.
\end{enumerate}
\end{proposition}
\subsection{Algorithmic Issues} \label{sec:complxley}
We propose using a recently proposed succinct representation for lattices \cite{munro2020space}, in which order-testing comparisons can be answered in $\mathcal{O}(1)$ time, and the meet or join of two elements in $\mathcal{O}(n^{3/4})$, where $n$ is the number of elements in the lattice.
The data structure occupies $\mathcal{O}(n^{3/2}\ log{}\ n)$
bits of space, with pre-processing time $\mathcal{O}(n^{2})$.
The functions $\rid{\alpha}, \ld{\gamma}$ and their inverses are represented using hash-based dictionaries, and equivalence classes induced by $\rid{\alpha}, \ld{\gamma}$ are represented using the union-find data structure.
Thus, the total space for a Lagois connection representation is $\mathcal{O}(n^{3/2})$, where $n$ is $max(n_1,n_2)$, with $n_1$ and $n_2$ being the number of elements in $\ld{L}$ and $\rid{M}$ respectively.
Order-comparisons within a domain are checked using the succinct data structure.
Cross-domain flows, e.g., between $\ld{x} \in \ld{L}$ and $\rid{y} \in \rid{M}$, are permitted if and only if $(\exists \rid{z}\in \rid{M}. \rid{\alpha}(\ld{x}) = \rid{z}\ and\ \rid{z} \rd{\sqsubseteq} \rid{y})$.
Since looking up $\rid{\alpha}(\ld{x})$ or $\ld{\gamma}(\rid{y})$ in a hashing-based dictionary can be done in $\mathcal{O}(1)$ time, all
order-comparisons can be performed in $\mathcal{O}(1)$.
Testing \textbf{LC1} and \textbf{LC3} takes $\mathcal{O}(n_1)$
time, and \textbf{LC2} and \textbf{LC3} $\mathcal{O}(n_2)$
time.
Thus checking whether $(\ld{L},\rid{\alpha},\ld{\gamma}, \rid{M})$ is a Lagois connection takes $\mathcal{O}(n)$ time, where $n = max(n_1,n_2)$.
If the transfer classes have been identified and these are far fewer than the number of lattice points, we can further optimise the pre-processing and the data structure (since the two sets of transfer classes are order-isomorphic to each other as we shall in Theorem \ref{lemma:decompose} of \S\ref{sec:lagois-decomposition}).
\section{Securely Connecting Decentralised Label Models}\label{sec:connecting-DLM}
In this section, we show how Lagois Connections can be used to ensure secure information flow between two organisations, both of which have employed the decentralized label model (DLM) of Myers \cite{Myers1997-ss} for SIF.
This illustrates how the Lagois framework can extend autonomy in IFC within an organisation to secure cross-organisational decentralised control.
We show that if a Lagois Connection $LC$ has been established between the Principals hierarchies\footnote{We prefer the term ``principals hierarchy" to ``principal hierarchy'' for grammatical reasons.} of both organisations, we can establish a Lagois connection $\widehat{LC}$ between the security lattices formed by the labels derived from the respective Principals hierarchies.
The second result we show is that the declassification rule \cite{Myers1997-ss} does not introduce any insecure flows, even when exchanging information between domains.
\subsection{The Decentralised Label Model}\label{sec:DLM-summary}
\paragraph{Principals Hierarchy} \ \
DLM based systems \cite{myers1999jflow, liu2009fabric, liu2017fabric} use abstract \textit{principals} to represent entities that can trust or be trusted,\textit{ e.g.}, users, roles, groups, organizations, privileges, etc.
Principals express trust via \textit{acts-for} relations \cite{Myers1997-ss}.
If a principal $p$ acts-for a principal $q$, then $q$ trusts $p$ completely and $p$ may perform any action (read
declassify) that $q$ may perform (written $p \succeq q$). The acts-for relation is a \textit{pre-order} and the \textit{principals hierarchy} refers to the set of principals under the acts-for ordering.
The operators $\curlywedge$
and $\curlyvee$
can be used to form conjunction and disjunction of principals in more elaborate principals hierarchies.
The conjunctive principal $p \curlywedge q$ represents the joint authority of $p$ and $q$, and acts for both: $p \curlywedge q \succeq p$ and $p \curlywedge q \succeq q$.
The disjunctive principal $p \curlyvee q$ represents the disjoint authority of $p$ and $q$, and is acted for by both: $p \succeq p \curlyvee q$ and $q \succeq p \curlyvee q$.
For convenience we include the most
restrictive and least restrictive principals, denoted as $\top$ and $\bot$ respectively.
\paragraph{Label Model} \ \
The security policies in DLM are expressed using \textit{labels}: each label is the \textit{conjunction} of a \emph{set of policies} each of which expresses privacy\footnote{As observed by Denning, the analysis for the \textit{integrity} of \textit{written} values is dual to the analysis for privacy of values read.} requirements in terms of principals \cite{myers-phd-tr-award}.
In the DLM framework, a privacy policy has two parts: an \textit{owner}, and a \textit{set of readers}, and is written in the form ``owner: readers''.
The owner of a policy is a principal whose data has contributed to constructing the value that is labeled by this policy.
The readers of a policy are a set of principals who are permitted by the owner to read the value so labelled.
For example, in the
label $L = \{o_1 : r_3, r_4;\ o_2 : r_4, r_5\}$, there are two policies (semicolons are separators) --
one owned by owner $o_1$, which permits the set of readers $\{r_3, r_4\}$, and
the second owned by owner $o_2$ that permits the set of readers $\{r_4, r_5\}$.
A principal wishing to access an object is required to satisfy \textit{all} confidentiality policy components in the object's label to be able to learn that object's value.
Thus each policy can be viewed as a constraint placed by the owner on what flows are permitted between principals, and a principal who is not the owner of any policy labelling a value places no constraints on allowed flows for that value.
If a policy $K$ is part of the label $L$ (i.e., $K \in L$), then $\textbf{o}(K): \mathit{policy} \rightarrow \mathit{principal}$ denotes the owner of that policy, and $\textbf{r}(K): \mathit{policy} \rightarrow (\mathit{principal}~\textit{set})$ denotes the set of readers specified by that policy.
The functions $\textbf{o}$ and $\textbf{r}$ completely characterize a label.
These policies then can be \textit{modified} safely by the individual owners -- a form of safe \textit{decentralised} declassification.
An owner may add readers to the reader set of its policy in a label, or remove the entire policy, effectively allowing all readers.
Arbitrary declassification is not possible because flow policies of other principals remain in force
\paragraph{Derived Information Flow Lattice} \ \
A pre-ordering relation on labels is derived from the acts-for relation $\succeq$ \cite{myers-phd-tr-award}.
We write $P \vdash L_1 \sqsubseteq L_2$, when $L_1$ is less or equal to $L_2$, given a principals hierarchy $P$.
The relation
$P \vdash L_1 \sqsubseteq L_2$ is defined formally in \cite{myers-phd-tr-award}, as shown in Figure \ref{fig:completerelabelingrule}.
\begin{figure}[!ht]
\centering
\begin{equation}
\boxed{
\begin{array}{rcl}
P \vdash L_1 \sqsubseteq L_2 & \Longleftrightarrow & \forall(I \in L_1) \exists(J \in L_2) P \vdash I \sqsubseteq J\\
P \vdash I \sqsubseteq J
& \Longleftrightarrow & P \vdash \textbf{o}(J) \succeq \textbf{o}(I) ~\wedge \\
&& ~~\forall(r_j \in \textbf{r}(J)) [ P \vdash r_j \succeq \textbf{o}(I) \vee \exists(r_i \in \textbf{r}(I)) P \vdash r_j \succeq r_i ]\\
& \Longleftrightarrow & P \vdash \textbf{o}(J) \succeq \textbf{o}(I)~ \wedge \\
&& ~~\forall(r_j \in \textbf{r$^+$}(J))~ \exists(r_i \in \textbf{r$^+$}(I)) ~P \vdash r_j \succeq r_i\\
\end{array}
}
\end{equation}
\caption{Definition of complete relabeling rule ($\sqsubseteq$)}
\label{fig:completerelabelingrule}
\end{figure}
The definition says that given the principals hierarchy $P$, it is safe to relabel information tagged with label $L_1$ to $L2$, if each policy
$I \in L_1$ is subsumed by a policy $J \in L_2$. Policy $J$ subsumes $I$ when the owner in $J$ ``acts-for'' the owner of $I$, and for every reader $r_j$ (effectively) permitted by policy $J$, there is a reader $r_i$ (effectively) permitted by $I$ such that $r_j$ can act for $r_i$.
Note that each policy component of a label (e.g., $I,J$) can also be considered a label, so writing $P \vdash I \sqsubseteq J$ is only mild abuse of notation.
The relation $\sqsubseteq$ is a pre-order, i.e, reflexive and transitive, but not necessarily anti-symmetric.
However, since the acts-for pre-order $\succeq$ supports join and meet operations, and because the semantics of labels is given in terms of \textit{sets} of permitted flows, we can construct an \textit{information flow lattice} (IFL) by
defining an equivalence relation $L_1 \equiv_{\sqsubseteq} L_2 ~\Longleftrightarrow~ (L_1 \sqsubseteq L_2 \text{ and } L_2 \sqsubseteq L_1)$ and taking equivalence classes to be lattice elements.
The join and meet on this information flow lattice are called \textit{label join} and \textit{label meet} (written as $\sqcup$ and $\sqcap$ respectively).
Since labels are sets of policies that must be together satisfied, we get the so-called ``\textit{Join rule}'', namely $L_1 \sqcup L_2 ~=~ L_1 \cup L_2$.
The least label and greatest label are $\{\}$ and $\{\top\colon\}$ respectively.
Myers presents a rule for safe \textit{relabelling by declassification} \cite{myers-phd-tr-award}:
Let $A$ be a set of principals in the current authority.
Let $L_A = \bigcup_{p \in A} \{p \colon \}$.
Then $L_1$ can be safely declassified to $L_2$ if
$L_1 \sqsubseteq L_2 \sqcup L_A$.
\subsection{Lagois Connections on Principals Hierarchies
and Derived IFLs}\label{sec:lagois-DLM-IFL}
Assume that two organizations with their own principals hierarchies
$\ld{P_L}$ and $\rid{P_R}$ negotiate an increasing Lagois Connection
\(
LC = (\ld{P_L}, \rid{\alpha}, \ld{\gamma}, \rid{P_R})
\)
between their principals hierarchies, $\rid{\alpha}: \ld{P_L} \rightarrow \rid{P_R}$ and $\ld{\gamma}: \rid{P_R} \rightarrow \ld{P_L}$.
This Lagois connection may be considered as a coupling between the input and outputs channels of the respective individual domains, together with certain strong information flow guarantees.
We show that there exists an increasing Lagois connection, $\widehat{LC}$, between the information flow lattices of the two organisations which is derived from the Lagois Connection between their principals hierarchies (see Figure \ref{fig:LCcommutesDLM}).
\begin{definition}[LC on IFLs]\label{Def:LC_IFL}
\ \ Define
$
\widehat{LC} = ( \ld{\mathit{IFL}_L}, \rid{\hat{\alpha}}, \ld{\hat{\gamma}}, \rid{\mathit{IFL}_R} )
$,
where $\ld{\mathit{IFL}_L}$ and $\rid{\mathit{IFL}_R}$ are the corresponding \textit{information flow lattices}, and
$\rid{\hat{\alpha}}: \ld{\mathit{IFL}_L} \rightarrow \rid{\mathit{IFL}_R}$
and
$\ld{\hat{\gamma}}: \rid{\mathit{IFL}_R} \rightarrow
\ld{\mathit{IFL}_L}$ are specified as:
\begin{flalign*}
&\rid{\hat{\alpha}}(\ld{L_l}) = \bigcup_{\ld{I_l} \in \ld{L_l}}\
\rid{\hat{\alpha}}(\ld{I_l}) \\
&\rid{\hat{\alpha}}(\ld{I_l}) =\ \{ \langle \rid{\alpha}(\textbf{o}(\ld{I_l})): \{\rid{\alpha}(\ld{r_l})\ |\ \ld{r_l} \in \textbf{r}(\ld{I_l})\} \rangle \} \\
&\ld{\hat{\gamma}}(\rid{L_r}) = \bigcup_{\rid{I_r} \in \rid{L_r}}\
\ld{\hat{\gamma}}(\rid{I_r}) \\
&\ld{\hat{\gamma}}(\rid{I_r}) =\ \{ \langle \ld{\gamma}(\textbf{o}(\rid{I_r})): \{\ld{\gamma}(\rid{r_r})\ |\ \rid{r_r} \in \textbf{r}(\rid{I_r})\} \rangle \}
\end{flalign*}
\end{definition}
Observe that $\rid{\hat{\alpha}}$ and $\ld{\hat{\gamma}}$ distribute homomorphically over joins:
\begin{align}
\rid{\hat{\alpha}}(\ld{L_1} \ld{\sqcup} \ld{L_2}) ~=~
\rid{\hat{\alpha}}(\ld{L_1} \cup \ld{L_2}) ~=~
\rid{\hat{\alpha}}(\ld{L_1}) ~\rd{\sqcup}~
\rid{\hat{\alpha}}(\ld{L_2})
\label{LagoisLabelDistr1}
\end{align}
\begin{align}
\ld{\hat{\gamma}}(\rd{L_1} \rd{\sqcup} \rd{L_2}) ~=~
\ld{\hat{\gamma}}(\rd{L_1} \cup \rd{L_2}) ~=~
\ld{\hat{\gamma}}(\rd{L_1}) ~\ld{\sqcup}~
\ld{\hat{\gamma}}(\rd{L_2})
\label{LagoisLabelDistr2}
\end{align}
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[framed,->,node distance=1cm,on grid]
\node(T2) {$\rid{P_R}$};
\node(T1) [xshift=-3cm] {$\ld{P_L}$};
\node(B1) [xshift=-3cm, yshift=-2.5cm] {$\ld{IFL_L}$};
\node(B2) [yshift=-2.5cm] {$\rid{IFL_R}$};
\draw[every loop]
(T1) edge[bend left, auto=left] node {$\rid{\alpha}$} (T2)
(T2) edge[bend left, auto=left] node {$\ld{\gamma}$} (T1) ;
\draw[every loop]
(B1) edge[bend left, auto=left] node {$\rid{\hat{\alpha}}$} (B2)
(B2) edge[bend left, auto=left] node {$\ld{\hat{\gamma}}$} (B1) ;
\draw [OliveGreen, thick, =>] (T1) to (B1);
\draw [OliveGreen, thick, =>] (T2) to (B2);
\end{tikzpicture}
\caption{A Lagois connection between two principals hierarchies induces a Lagois connection between the corresponding Information Flow Lattices}
\label{fig:LCcommutesDLM}
\end{figure}
\begin{theorem}\label{Thm:LC-PH-LC-IFL}
Let $LC = (\ld{P_L}, \rid{\alpha}, \ld{\gamma}, \rid{P_R})$ be an increasing Lagois connection and $\ld{\mathit{IFL}_L}$ be derived from $\ld{P_L}$ and $\rid{\mathit{IFL}_R}$ be derived from $\rid{P_R}$, as described earlier\footnote{Note $\rid{\alpha}: \ld{P_L} \rightarrow \rid{P_R}$ and $\ld{\gamma}: \rid{P_R} \rightarrow \ld{P_L}$ are total functions by
Theorem \ref{prop:fnguniquelydetermin} and Corollary \ref{cor:existenceLC2}.}.
Then $\widehat{LC} = (\ld{\mathit{IFL}_L}, \rid{\hat{\alpha}}, \ld{\hat{\gamma}}, \rid{\mathit{IFL}_R})$ is also an increasing Lagois connection.
\end{theorem}
\begin{proof}
We prove the following properties for $\rid{\hat{\alpha}}$ and $\ld{\hat{\gamma}}$ using Definition \ref{Def:LC_IFL}:
\begin{enumerate}
\item Monotonicity
\begin{enumerate}
\item \label{IFL:prop1}If $\ld{P_L} \vdash \ld{L_{l1}} \ld{\sqsubseteq} \ld{L_{l2}}$ then $\rid{P_R} \vdash \rid{\hat{\alpha}}(\ld{L_{l1}})\ \rd{\sqsubseteq}\ \rid{\hat{\alpha}}(\ld{L_{l2}})$
\item \label{IFL:prop2}If $\rid{P_R} \vdash \rid{L_{r1}} \rd{\sqsubseteq} \rid{L_{r2}}$ then $\ld{P_L} \vdash \ld{\hat{\gamma}}(\rid{L_{r1}})\ \ld{\sqsubseteq}\ \ld{\hat{\gamma}}(\rid{L_{r2}})$
\end{enumerate}
\item Increasing
\begin{enumerate}
\item \label{IFL:prop3} $\ld{P_L} \vdash \ld{L_{l1} \sqsubseteq \hat{\gamma}}(\rid{\hat{\alpha}}(\ld{L_{l1}})),~~~ \forall \ld{L_{l1}} \in \ld{\mathit{IFL}_L}$
\item \label{IFL:prop4} $\rid{P_R} \vdash \rid{L_{r1}} \rd{\sqsubseteq}\ \rid{\hat{\alpha}}(\ld{\hat{\gamma}}(\rid{L_{r1}})), \forall \rid{L_{r1}} \in \rid{\mathit{IFL}_R}$
\end{enumerate}
\item Identity/ Equality/ Fixed Points
\begin{enumerate}
\item \label{IFL:prop5} $\rid{P_R} \vdash \rid{\hat{\alpha}}(\ld{L_{l1}}) \equiv_{\rd{\sqsubseteq}} \rid{\hat{\alpha}}(\ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{L_{l1}}))), ~~~\forall \ld{L_{l1}} \in \ld{\mathit{IFL}_L}$
\item \label{IFL:prop6} $\ld{P_L} \vdash \ld{\hat{\gamma}}(\rid{L_{r1}}) \equiv_{\ld{\sqsubseteq}} \ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{\hat{\gamma}}(\rid{L_{r1}}))), \forall \rid{L_{r1}} \in \rid{\mathit{IFL}_R}$
\end{enumerate}
\end{enumerate}
We show proofs for properties \ref{IFL:prop1}, \ref{IFL:prop3} and \ref{IFL:prop5}, as others are similar. We use notation $\textbf{r}^+(I) = \textbf{r}(I) \cup\ \textbf{o}(I)$ \cite{myers-phd-tr-award}.
\begin{enumerate}
\item[(\ref{IFL:prop1})]
Given $LC =\ (\ld{P_L}, \rid{\alpha}, \ld{\gamma}, \rid{P_R})$ is an increasing Lagois connection, we know that $\rid{\alpha}$ is an order-preserving function.
So, if $\ld{p_i} \preceq \ld{p_j}$ then $\rid{\alpha}(\ld{p_i})\ \rd{\preceq}\ \rid{\alpha}(\ld{p_j})$, for all principals $\ld{p_i,p_j} \in \ld{P_L}$.
\begin{alignat}{3}
& \ld{P_L} \vdash \ld{L_{l1} \sqsubseteq L_{l2}} \Longleftrightarrow
\forall(\ld{I} \in \ld{L_{l1}}) \exists(\ld{J} \in \ld{L_{l2}}) \ld{P_L} \vdash \ld{I \sqsubseteq J} &&~(\text{Def of} \sqsubseteq) \label{given1a} \\
& \ld{P_L} \vdash \ld{I \sqsubseteq J} \equiv \ld{P_L} \vdash \textbf{o}(\ld{I}) \ld{\preceq} \textbf{o}(\ld{J}) ~\wedge~ \\ &\hspace{1.5cm}\forall(\ld{r_j}\in \textbf{r}^+(\ld{J}))\ \exists(\ld{r_i} \in \textbf{r}^+(\ld{I}))\ \ld{P_L} \vdash \ld{r_i \preceq r_j} &&~(\text{Def of} \sqsubseteq) \label{1agivenrd}
\end{alignat}
As $\rid{\alpha}$ is order-preserving, we get \label{1alphamonotone}
\begin{alignat}{3}
& \rid{P_R} \vdash \rid{\alpha}(\textbf{o}(\ld{I})) \rd{\preceq} \rid{\alpha}(\textbf{o}(\ld{J})) ~\wedge \\
&\hspace{1cm}\forall(\ld{r_j}\in \textbf{r}^+(\ld{J}))\ \exists(\ld{r_i} \in \textbf{r}^+(\ld{I}))\ \rid{P_R} \vdash \rid{\alpha}(\ld{r_i}) \rd{\preceq} \rid{\alpha}(\ld{r_j}) &&~~(\ref{1agivenrd},\ref{1alphamonotone}) \label{1aa}\\
&\rid{P_R} \vdash \textbf{o}(\rid{\hat{\alpha}}(\ld{I})) \rd{\preceq} \textbf{o}(\rid{\hat{\alpha}}(\ld{J})) \wedge \\
&\hspace{1cm}\forall(\rd{r_j}\in \textbf{r}^+(\rid{\hat{\alpha}}(\ld{J})))\ \exists(\rd{r_i} \in \textbf{r}^+(\rid{\hat{\alpha}}(\ld{I})))~ \rid{P_R} \vdash \rd{r_i} \rd{\preceq} \rd{r_j} &&~(\ref{1aa},\text{Def}~ \ref{Def:LC_IFL}) \label{1alphanew} \\
& \implies \rid{P_R} \vdash \rid{\hat{\alpha}}(\ld{I}) \rd{\sqsubseteq} \rid{\hat{\alpha}}(\ld{J}) \label{1alph} &&~(\ref{1alphanew},\text{Def}~ \ref{Def:LC_IFL})\\
& \forall(\ld{I}\in \ld{L_{l1}})\ \exists(\ld{J} \in \ld{L_{l2}})\ \rid{P_R} \vdash \rid{\hat{\alpha}}(\ld{I})\ \rd{\sqsubseteq} \rid{\hat{\alpha}}(\ld{J}) && ~(\ref{given1a},\ref{1alph}) \label{1anewpolicies} && \\
&\rid{P_R} \vdash \rid{\hat{\alpha}}(\ld{L_{l1}})\ \rd{\sqsubseteq} \rid{\hat{\alpha}}(\ld{L_{l2}}) && ~(\ref{1anewpolicies},\text{Def}~ \ref{Def:LC_IFL})
\end{alignat}
Hence we have proved Monotonicity.
\item[(\ref{IFL:prop3})]
Assuming $\ld{\gamma} \circ \rid{\alpha}$ is increasing we have for any $\ld{p} \in \ld{P_L}$:
%
\begin{alignat}{3}
& \ld{P_L} \vdash \ld{p} ~\ld{\preceq}~ \ld{\gamma}(\rid{\alpha}(\ld{p})), ~~
(\forall \ld{p} \in \ld{L}) && && \\
& \forall(\ld{I}\in \ld{L_{l1}})\ \ld{P_L} \vdash \textbf{o}(\ld{I}) \ld{\preceq} \ld{\gamma}(\rid{\alpha}(\textbf{o}(\ld{I})) ~\wedge &&
\\
& \hspace{1cm}\forall(\ld{r_i}\in \textbf{r}^+(\ld{I})\ \ld{P_L} \vdash \ld{r_i} ~\ld{\preceq}~ \ld{\gamma}(\rid{\alpha}(\ld{r_i})) &&
\\
& \forall(\ld{I}\in \ld{L_{l1}})\ \ld{P_L} \vdash
\textbf{o}(\ld{I}) ~\ld{\preceq}~
\textbf{o}(\ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{I})))~ \wedge && ~(\text{Def}~\ref{Def:LC_IFL})\\
& \hspace{1cm}\forall(\ld{r_j}\in \textbf{r}^+(\hat{\gamma}(\rid{\hat{\alpha}}(\ld{I})))\ \exists(\ld{r_i} \in \textbf{r}^+(\ld{I})) \ \ld{P_L} \vdash \ld{r_i} \ld{\preceq} \ld{r_j} && ~(\text{Def}~ \ref{Def:LC_IFL})\\
& \forall(\ld{I}\in \ld{L_{l1}})\ \ld{P_L} \vdash \ld{I} ~\ld{\sqsubseteq}~ \ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{I})) && ~(\text{Def of }\sqsubseteq)\\
& \forall(\ld{I}\in \ld{L_{l1}}) \exists(\ld{J} \in \ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{L_{l1}}))\ \ld{P_L} \vdash \ld{I} ~\ld{\sqsubseteq}~ \ld{J} && ~(\text{Def of} \sqsubseteq)\\
& \ld{P_L} \vdash \ld{L_{l1}} ~\ld{\sqsubseteq}~ \ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{L_{l1}})) && ~(\text{Def of} \sqsubseteq)
\end{alignat}
Hence we have proved $\hat{\ld{\gamma}} \circ \rid{\hat{\alpha}}$ is increasing.
\item[(\ref{IFL:prop5})] We have by Definition \ref{Def:LC_IFL}, $\forall \ld{I_l} \in \ld{L_l}$
\begin{align*}
&\rid{\hat{\alpha}}(\ld{I_l}) =\ \langle \rid{\alpha}(\textbf{o}(\ld{I_l})): \{\rid{\alpha}(\ld{r})\ |\ \ld{r} \in \textbf{r}(\ld{I_l})\} \rangle, \\ \\
&\rid{\hat{\alpha}}(\ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{I_l}))) = \langle \rid{\alpha}(\ld{\gamma}(\rid{\alpha}(\textbf{o}(\ld{I_l})))): \{\rid{\alpha}(\ld{\gamma}(\rid{\alpha}(\ld{r}))) ~|~ \ld{r} \in \textbf{r}(\ld{I_l})\} \rangle
\end{align*}
As $LC = (\ld{P_L}, \rid{\alpha}, \ld{\gamma}, \rid{P_R})$ is an increasing Lagois connection, we know that $\rid{\alpha} \circ \ld{\gamma} \circ \rid{\alpha} = \rid{\alpha}$.
Therefore,
$
\rid{\hat{\alpha}}(\ld{L_{l1}}) \equiv_{\rd{\sqsubseteq}} \rid{\hat{\alpha}}(\ld{\hat{\gamma}}(\rid{\hat{\alpha}}(\ld{L_{l1}}))),\ \forall \ld{L_{l1}} \in \ld{\mathit{IFL}_L}
$. \\
Hence proved.
\end{enumerate}
\end{proof}
Assume the principals hierarchies in two domains are static and Laogis-connected.
Let $\ld{A}$ and $\rd{A}$ represent two Lagois-connected sets of principals under which authority two communicating processes are operating in their respective domains.
Then the safe declassification rule continues to apply even with bidirectional communication between the two domains.
\begin{corollary}\label{corollary:safe-declassification}
Let $LC = (\ld{P_L}, \rid{\alpha}, \ld{\gamma}, \rid{P_R})$ be an increasing Lagois connection and
let $\widehat{LC} = (\ld{\mathit{IFL}_L}, \rid{\hat{\alpha}}, \ld{\hat{\gamma}}, \rid{\mathit{IFL}_R})$ be the derived Lagois connection between the corresponding IFLs.
Let $\ld{A}$ and $\rd{A}$ be sets of principals such that $\rid{\hat{\alpha}}[\ld{A}] = \rd{A}$ and
$\ld{\hat{\gamma}}[\rd{A}] = \ld{A}$.
Then (a) $\ld{L_1}$ can be safely declassified to
$\ld{\hat{\gamma}}(\rd{L_2})$ iff
$\rid{\hat{\alpha}}(\ld{L_1})$ can be safely declassified to $\rd{L_2}$, and (b)
$\rd{L_1}$ can be safely declassified to
$\rid{\hat{\alpha}}(\ld{L_2})$ iff
$\ld{\hat{\gamma}}(\rd{L_1})$ can be safely declassified to $\ld{L_2}$.
\end{corollary}
\begin{proof}
Straightforward from monotonicity, expansiveness, homomorphic distribution over joins, and the assumed Lagois connection between $\ld{A}$ and $\rd{A}$.
\end{proof}
\section{An Operational Model}\label{sec:model}
We present an operational model consisting of a \textit{language} and its operational semantics, and then a \textit{security type system}, which we show to be sound in that well-typed programs exhibit non-interference \cite{DBLP:conf/sp/GoguenM82a}.
The objective of this section is to show that under reasonable assumptions of atomicity and isolation, the Lagois connection framework allows SIF analyses performed within systems to be lifted systematically to concurrent systems exchanging information.
\subsection{Computational Model.}\label{sec:operations}
Assume two organisations $\ld{L}$ and $\rid{M}$ which have their own IFMs want to share data with each other.
The two domains comprise storage objects $\ld{Z}$ and $\rd{Z}$ respectively, which are manipulated using their own sets of \textit{atomic} transactional operations, ranged over by $\ld{t}$ and $\rd{t}$ respectively.
We further assume that these transactions within each domain are internally secure with respect to their flow models, and have no insecure or interfering interactions with the environment.
Thus, we are agnostic to the level of abstraction of the systems we aim to connect securely, and since our approach treats the application domains as ``black boxes'', it is readily adaptable to any level of discourse (language, system, OS, database) found in the security literature.
We extend these operations with a minimal set of operations to transfer data between the two domains.
To avoid any concurrency effects, interference or race conditions arising from inter-domain transfer, we augment the storage objects of both domains with a fresh set of \textit{export} and \textit{import} variables into/from which the data of the domain objects can be copied \textit{atomically}.
We designate these sets $\ld{X}, \rd{X}$ as the respective \textit{export} variables, and $\ld{Y}, \rd{Y}$ as the respective \textit{import} variables, with the corresponding variable instances written as $\ld{x_i}$, $\rd{x_i}$ and $\ld{y_i}$, $\rd{y_i}$.
These export and import variables form mutually disjoint sets, and are distinct from any extant domain objects manipulated by the applications within a domain.
These variables are used exclusively for transfer, and are manipulated atomically.
We let $\ld{w_i}$ range over all variables in $\ld{N} ~=~\ld{Z} \cup \ld{X} \cup \ld{Y}$ (respectively $\rd{w_i}$ over $\rd{N} ~=~ \rd{Z} \cup \rd{X} \cup \rd{Y}$).
Domain objects are copied \textit{to export} variables and \textit{from import} variables by special operations $\ld{rd}(\ld{z}, \ld{y})$ and $\ld{wr}(\ld{x}, \ld{z})$ (and $\rd{rd}(\rd{z}, \rd{y})$ and $\rd{wr}(\rd{x}, \rd{z})$ in the other domain).
We assume \textit{atomic transfer} operations (\textit{trusted by both domains}) $T_{RL}, T_{LR}$ that copy data from the export variables of one domain to the import variables of the other domain as the only mechanism for inter-domain flow of data.
Let ``phrase'' $p$ denote a command in either domain or a transfer operation, and let $s$ be any (empty or non-empty) sequence of phrases.
\textbf{Note}: This language should be understood as a notation for describing a distributed system's execution sequences involving computation, communication, input and output, rather than as a programming language.
Thus, we only need to consider sequences of atomic actions.
Hence the absence of constructs such as conditionals, iteration or repetition.
Further, the importance of atomicity of the computational steps and communication-related operations should be evident.
We will later see that the ``types'' are exactly the security classes of the IFMs, and so there are no constructions such as cartesian product, records and function types.
\[
\begin{array}{c}
\text{(command)} ~~~
\ld{c} ::= \ld{t} ~|~ \ld{rd}(\ld{z}, \ld{y}) ~|~ \ld{wr}(\ld{x}, \ld{z}) ~~~~~~
\rd{c} ::=
\rd{t} ~|~ \rd{rd}(\rd{z}, \rd{y}) ~|~ \rd{wr}(\rd{x}, \rd{z}) \\
\text{(phrase)} ~~~ p ::= T_{RL}(\rd{x},\ld{y}) ~|~ T_{LR}(\ld{x},\rd{y}) ~|~
\ld{c} ~|~ \rd{c} ~~~\hfill~~~ \text{(seq)}~~~ s ::= \epsilon ~|~ s_1 ; p \\
\end{array}
\]
\begin{figure}
\[
\begin{array}{c}
\inferrule* [Left = \ld{T}]{\ld{\mu} \vdash \ld{t} \Rightarrow \ld{\nu}
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{t} \Rightarrow \langle \ld{\nu}, \rd{\mu} \rangle}
~~~~\hfill~~~~
\inferrule* [Left = \rd{T}]{\rd{\mu} \vdash \rd{t} \Rightarrow \rd{\nu}
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{t} \Rightarrow \langle \ld{\mu}, \rd{\nu} \rangle} \\[1ex]
\inferrule*[Left = \ld{Wr}]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{wr}(\ld{x},\ld{z}) \Rightarrow \langle \ld{\mu}[\ld{x} := \ld{\mu}(\ld{z})], \rd{\mu} \rangle}
\\[1ex]
\inferrule*[Left = \rd{Wr}]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{wr}(\rd{x},\rd{z}) \Rightarrow \langle \ld{\mu}, \rd{\mu}[\rd{x} := \rd{\mu}(\rd{z})] \rangle}
\\[1ex]
\inferrule*[Left = \ld{Rd}]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \ld{rd}(\ld{z},\ld{y}) \Rightarrow \langle \ld{\mu}[\ld{z} := \ld{\mu}(\ld{y})], \rd{\mu} \rangle} \\[1ex]
\inferrule*[Left = \rd{Rd}]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash \rd{rd}(\rd{z},\rd{y}) \Rightarrow \langle \ld{\mu}, \rd{\mu}[\rd{z} := \rd{\mu}(\rd{y})] \rangle} \\[1ex]
\inferrule*[Left = Trl]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash T_{RL}(\ld{y},\rd{x}) \Rightarrow
\langle \ld{\mu}[\ld{y} := \rd{\mu}(\rd{x})],\rd{\mu} \rangle} \\[1ex]
\inferrule*[Left = Tlr]{
}{\langle \ld{\mu},\rd{\mu} \rangle \vdash T_{LR}(\rd{y},\ld{x}) \Rightarrow
\langle \ld{\mu},\rd{\mu}[\rd{y} := \ld{\mu}(\ld{x})] \rangle} \\[1ex]
\inferrule* [Left = Seq0]{ }{
\langle \ld{\mu},\rd{\mu} \rangle \vdash \epsilon \Rightarrow^* \langle \ld{\mu},\rd{\mu} \rangle} \\[1ex]
\inferrule* [Left = SeqS]{\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\mu_1},\rd{\mu_1} \rangle, ~~~
\langle \ld{\mu_1},\rd{\mu_1} \rangle \vdash p \Rightarrow \langle \ld{\mu_2},\rd{\mu_2} \rangle}{\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 ; p \Rightarrow^* \langle \ld{\mu_2},\rd{\mu_2} \rangle}
\end{array}
\]
\caption{Execution Rules}
\label{fig:evalrules11}
\end{figure}
A \textit{store} (typically $\ld{\mu}, \ld{\nu}, \rd{\mu}, \rd{\nu}$) is a finite-domain function from variables to a set of values (not further specified).
We write, \textit{e.g.}, $\ld{\mu}(\ld{w})$ for the contents of the store $\ld{\mu}$ at variable $\ld{w}$, and $\ld{\mu}[\ld{w} := \rd{\mu}(\rd{w})]$ for the store
that is the same as $\ld{\mu}$ everywhere except at variable $\ld{w}$,
where it now takes value $\rd{\mu}(\rd{w})$.
The rules specifying execution of commands are given in Fig. \ref{fig:evalrules11}.
Assuming the specification of intradomain transactions (\textit{\ld{t}, \rd{t}}) of the form
$\ld{\mu} \vdash \ld{t} \implies \ld{\nu}$ and $\rd{\mu} \vdash \rd{t} \implies \rd{\nu}$, our rules allow us to specify judgments of the form
$\langle \ld{\mu}, \rd{\mu} \rangle \vdash p \implies \langle \ld{\nu}, \rd{\nu} \rangle$ for phrases, and the reflexive-transitive closure for sequences of phrases.
Note that phrase execution occurs \textit{atomically}, and the intra-domain transactions, as well as copying to and from the export/import variables affect the store in only one domain, whereas the \textit{atomic transfer} is only between export variables of one domain and the import variables of the other.
\subsection{Typing Rules}\label{sec:typing}
Let the two domains have the respective different IFMs:
\[ FM_L = \langle \ld{N}, \ld{P}, \ld{SC}, \ld{\sqcup}, \ld{\sqsubseteq} \rangle ~~~\hfill~~~ FM_M = \langle \rd{N}, \rd{P}, \rd{SC}, \rid{\sqcup}, \rd{\sqsubseteq} \rangle, \]
such that the flow policies in both are defined over different sets of security classes $\ld{SC}$ and $\rd{SC}$.\footnote{Without loss of generality, we assume that $\ld{SC} \cap \rd{SC} = \emptyset$, since we can suitably rename security classes.}
The (security) types of the core language are as follows.
Metavariables $\ld{l}$ and $\rd{m}$ range over the sets of security classes, $\ld{SC}$ and $\rd{SC}$ respectively, which are partially ordered by $\ld{\sqsubseteq}$ and $\rd{\sqsubseteq}$.
Note that, with the language being minimal, there are no other type constructions.
A type assignment $\ld{\lambda}$ is a finite-domain function from variables $\ld{N}$ to $\ld{SC}$ (respectively, $\rd{\lambda}$ from $\rd{N}$ to $\rd{SC}$).
The important restriction we place on $\ld{\lambda}$
and $\rd{\lambda}$ is that they map export and import variables $\ld{X}, \ld{Y}, \rd{X}, \rd{Y}$ only to points in the security lattices $\ld{SC}$ and $\rd{SC}$ respectively which are in the domains of $\ld{\gamma}$ and $\rid{\alpha}$, \textit{i.e.}, these points participate in the Lagois connection.
Intuitively, a variable $w$ mapped to security class $\ld{l}$ can store information of security class $\ld{l}$ or lower.
The type system works with respect to a given type assignment. Given a security level, \textit{e.g.}, $\ld{l}$, the typing rules track for each command \textit{within that domain} whether all written-to variables in that domain are of security classes ``above'' $\ld{l}$, and additionally for transactions within a domain, they ensure ``simple security'', \textit{i.e.}, that all variables which may have been read belong to security classes ``below'' $\ld{l}$.
We assume for the transactions within a domain, \textit{e.g.}, $\ld{L}$, we already have a security type system that will give us judgments of the form $\ld{\lambda} \vdash \ld{t}: \ld{l}$.
The novelty of our approach is to extend this framework to work over two connected domains, \textit{i.e.}, given implicit security levels of the contexts in the respective domains.
Cross-domain transfers will require pairing such judgments, and thus our type system will have judgments of the form
\[ \langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle
\ld{l}, \rd{m} \rangle \]
We introduce a set of typing rules for the core language, given in Fig. \ref{fig:syntax-type-rules11}.
In many of the rules, the type for one of the domains is not constrained by the rule, and so any suitable type may be chosen as determined by the context, \textit{e.g.}, $\rd{m}$ in the rules \ld{\sc Tt},
\ld{\sc Trd}, \ld{\sc Twr} and $\ld{TT_{RL}}$,
and both $\ld{l}$ and $\rd{m}$ in {\sc Com0}.
\begin{figure}[!ht]
\centering
\[
\begin{array}{c}
\inferrule*[Left = \ld{Tt}]{~}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{t}: \langle \ld{l}, \rd{m} \rangle} ~\text{if for all $\ld{z}$ assigned in $\ld{t}$, }
\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z}) \\
~~~~~~~~~~~~~~~~~~~~\hfill
\text{ \& for all $\ld{z_1}$ read in $\ld{t}$, }
\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l}
\\
\inferrule*[Left = \rd{Tt}]{~}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{t}: \langle \ld{l}, \rd{m} \rangle} ~\text{if for all $\rd{z}$ assigned in $\rd{t}$, }
\rd{m} ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{z}) \\
~~~~~~~~~~~~~~~~~~~~\hfill
\text{ \& for all } \rd{z_1} \text{ read in $\rd{t}$, }
\rd{\lambda}(\rd{z_1}) ~\rd{\sqsubseteq}~ \rd{m}
\\[1ex]
\inferrule*[Left = \ld{Trd}]{\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{rd}(\ld{z},\ld{y}):
\langle \ld{\lambda}(\ld{z}), \rd{m} \rangle
}
\\
\inferrule*[Left = \rd{Trd}]{\rd{\lambda}(\rd{y}) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{z})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{rd}(\rd{z},\rd{y}):
\langle \ld{l}, \rd{\lambda}(\rd{z}) \rangle
}
\\[1ex]
\inferrule*[Left = \ld{Twr}]{\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{wr}(\ld{x},\ld{z}):
\langle \ld{\lambda}(\ld{x}), \rd{m} \rangle}
\\
\inferrule*[Left = \rd{Twr}]{\rd{\lambda}(\rd{z}) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{x})}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \rd{wr}(\rd{x},\rd{z}):
\langle \ld{l}, \rd{\lambda}(\rd{x}) \rangle} \\[1ex]
\inferrule*[Left = \ld{$TT_{RL}$}]{
\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})
}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{RL}(\ld{y},\rd{x}):
\langle \ld{\lambda}(\ld{y}), \rd{\lambda}(\rd{x}) \rangle}
\\
\inferrule*[Left = \rd{$TT_{LR}$}]{
\rid{\alpha}(\ld{\lambda}(\ld{x})) ~\rd{\sqsubseteq}~ \rd{\lambda}(\rd{y})
}{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{LR}(\rd{y},\ld{x}):
\langle \ld{\lambda}(\ld{x}), \rd{\lambda}(\rd{y}) \rangle }
\\[1ex]
\inferrule*[Left = Com0]{ }{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \epsilon: \langle \ld{l}, \rd{m} \rangle}
\\
\inferrule*[Left = ComS]{
\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_1}, \rd{m_1} \rangle
~~~~~\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s: \langle \ld{l}, \rd{m} \rangle
}{\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s; p:
\langle \ld{l_1} \ld{\sqcap} \ld{l} , \rd{m_1} \rid{\sqcap} \rd{m} \rangle}
\end{array}
\]
\caption{Typing Rules}
\label{fig:syntax-type-rules11}
\end{figure}
For transactions \textit{e.g.}, $\ld{t}$ entirely within domain $\ld{L}$, the typing rule \ld{\sc Tt} constrains the type in the left domain to be at a level $\ld{l}$ that dominates all variables read in
$\ld{t}$, and which is dominated by all variables written to in $\ld{t}$, but places no constraints on the type $\rd{m}$ in the other domain $\rid{M}$.
In the rule \ld{\sc Trd}, since a value in import variable $\ld{y}$ is copied to the variable $\ld{z}$, we have $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$, and the type in the domain $\ld{L}$ is $\ld{\lambda}(\ld{z})$ with no constraint on the type $\rd{m}$ in the other domain.
Conversely, in the rule \ld{\sc Twr}, since a value in variable $\ld{z}$ is copied to the export variable $\ld{x}$, we have $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$, and the type in the domain $\ld{L}$ is $\ld{\lambda}(\ld{x})$ with no constraint on the type $\rd{m}$ in the other domain.
In the rule $\ld{TT_{RL}}$, since the contents of
a variable $\rd{x}$ in domain $\rid{M}$ are copied into a variable $\ld{y}$ in domain $\ld{L}$, we require $\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$, and constrain the type in domain $\ld{L}$ to $\ld{\lambda}(\ld{y})$.
The constraint in the other domain is unimportant (but for the sake of convenience, we peg it at $\rd{\lambda}(\rd{x})$).
Finally, for the types of sequences of phrases, we take the meets of the collected types in each domain respectively, so that we can guarantee that no variable of type lower than these meets has been written into during the sequence.
Note that Proposition \ref{prop:meet-existence} ensures that these types have the desired properties for participating in the Lagois connection.
\subsection{Soundness} \label{sec:soundness}
We now establish soundness of our scheme by showing a non-interference theorem with respect to the operational semantics and the type system built on the security lattices.
This theorem may be viewed as a conservative adaptation (to a minimal secure data transfer framework in a Lagois-connected pair of domains) of the main result of Volpano \textit{et al.} \cite{DBLP:journals/jcs/VolpanoIS96}.
We assume that underlying base transactional languages in each of the domains have the following simple property (stated for $\ld{L}$, but an analogous property is assumed for $\rid{M}$).
Within each transaction $\ld{t}$, for each assignment of an expression $\ld{e}$ to any variable $\ld{z}$, the following holds:
If $\ld{\mu}$, $\ld{\nu}$ are two stores such that
for all $\ld{w} \in vars(\ld{e})$, we have $\ld{\mu}(\ld{w}) = \ld{\nu}(\ld{w})$, then after executing the assignment, we will get $\ld{\mu}(\ld{z}) = \ld{\nu}(\ld{z})$.
That is, if two stores are equal for all variables appearing in the expression $\ld{e}$, then the value assigned to the variable $\ld{z}$ will be the same.
This assumption plays the r\^{o}le of ``Simple Security'' of expressions in \cite{DBLP:journals/jcs/VolpanoIS96} in the proof of the main theorem.
The type system plays the r\^{o}le of ``Confinement''.
We start with two obvious lemmas about the operational semantics, namely preservation of domains, and a ``frame'' lemma:
\begin{lemma}[Domain preservation]\label{lemma:equaldomainoneval}
If $\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_1}, \rd{\mu_1} \rangle$, then $dom(\ld{\mu}) = dom(\ld{\mu_1})$, and $dom(\rd{\mu}) = dom(\rd{\mu_1})$.
\end{lemma}
\begin{proof}
By induction on the length of the derivation of $\langle \mu, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \mu_1, \rd{\mu_1} \rangle$.
\end{proof}
\begin{lemma}[Frame]\label{lemma:notassigned}
If $\langle \mu, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \mu_1, \rd{\mu_1} \rangle, w \in dom(\mu) \cup dom(\rd{\mu})$, and $w$ is \textit{not} assigned to in $s$, then $\ld{\mu}(w) = \ld{\mu_1}(w)$ and
$\rd{\mu}(w) = \rd{\mu_1}(w)$.
\end{lemma}
\begin{proof}
By induction on the length of the derivation of $\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_1}, \rd{\mu_1} \rangle$.
\end{proof}
The main result assumes an ``adversary'' that operates at a security level $\ld{l}$ in domain $\ld{L}$ and at security level $\rd{m}$ in domain $\rid{M}$.
Note however, that these two levels are interconnected by the monotone functions $\rid{\alpha}: \ld{L} \rightarrow \rid{M}$ and
$\ld{\gamma}: \rid{M} \rightarrow \ld{L}$, since these levels are connected by the ability of information at one level in one domain to flow to the other level in the other domain.
The following theorem says that if (a) a sequence of phrases is well-typed, and (b,c) we start its execution in two store configurations that are (e) indistinguishable with respect to all objects having security class below $\ld{l}$ and $\rd{m}$ in the respective domains, then the corresponding resulting stores after execution continue to remain indistinguishable on all variables with security classes below these adversarial levels.
\begin{theorem}[Type Soundness]\label{thm:soundness}
Suppose $\ld{l}, \rd{m}$ are the ``adversarial'' type levels in the respective domains, which satisfy the condition $\ld{l} = \ld{\gamma}(\rd{m})$ and
$\rd{m} = \rid{\alpha}(\ld{l})$.
Let
\begin{enumerate}[label=(\alph*)]
\item \label{assume:1}$\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s:
\langle \ld{l_0}, \rd{m_0}\rangle$; ~~~
($s$ has security type $\langle \ld{l_0}, \rd{m_0}\rangle$)
\item \label{assume:2}$\langle \ld{\mu}, \rd{\mu} \rangle \vdash s \Rightarrow^* \langle \ld{\mu_f}, \rd{\mu_f} \rangle$;
~~(execution of $s$ starting from $\langle \ld{\mu}, \rd{\mu} \rangle $)
\item \label{assume:3}$\langle \ld{\nu}, \rd{\nu} \rangle \vdash s \Rightarrow^* \langle \ld{\nu_f}, \rd{\nu_f} \rangle$;
~~(execution of $s$ starting from $\langle \ld{\nu}, \rd{\nu} \rangle $)
\item \label{assume:4}$dom(\ld{\mu}) = dom(\ld{\nu}) = dom(\ld{\lambda})$ and
$dom(\rd{\mu}) = dom(\rd{\nu}) = dom(\rd{\lambda})$;
\item \label{assume:5} $\ld{\mu}(\ld{w}) = \ld{\nu}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$, and
$\rd{\mu}(\rd{w}) = \rd{\nu}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
\end{enumerate}
Then
$\ld{\mu_f}(\ld{w}) = \ld{\nu_f}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$, and
$\rd{\mu_f}(\rd{w}) = \rd{\nu_f}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
\end{theorem}
\begin{proof}
By induction on the length of sequence $s$.
The base case is vacuously true.
We now consider a sequence $s_1; p$.
$\langle \ld{\mu},\rd{\mu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\mu_1},\rd{\mu_1} \rangle$ and
$\langle \ld{\mu_1},\rd{\mu_1} \rangle \vdash p \Rightarrow \langle \ld{\mu_f},\rd{\mu_f} \rangle$
and
$\langle \ld{\nu},\rd{\nu} \rangle \vdash s_1 \Rightarrow^* \langle \ld{\nu_1},\rd{\nu_1} \rangle$ and
$\langle \ld{\nu_1},\rd{\nu_1} \rangle \vdash p \Rightarrow \langle \ld{\nu_f},\rd{\nu_f} \rangle$
By induction hypothesis applied to $s_1$, we have
$\ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w})$ for all
$\ld{w}$ such that $\ld{\lambda(\ld{w})} ~\ld{\sqsubseteq}~\ld{l}$,
and
$\rd{\mu_1}(\rd{w}) = \rd{\nu_1}(\rd{w})$ for all
$\rd{w}$ such that $\rd{\lambda}(\rd{w}) ~\rd{\sqsubseteq}~ \rd{m}$.
Let $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash s_1: \langle \ld{l_s}, \rd{m_s} \rangle$, and $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_p}, \rd{m_p} \rangle$.
We examine four cases for $p$ (the remaining cases are symmetrical). \\
\textbf{Case} $p$ is $\ld{t}$: \ \ \
Consider any $\ld{w}$ such that
$\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~\ld{l}$.
If $\ld{w} \in \ld{X} \cup \ld{Y}$ (\textit{i.e.}, it doesn't appear in $\ld{t}$), or if $\ld{w} \in \ld{Z}$ but is not assigned to in $\ld{t}$, then
by Lemma \ref{lemma:notassigned} and the induction hypothesis, $\ld{\mu_f}(\ld{w}) =
\ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w}) =
\ld{\nu_f}(\ld{w})$. \\
Now suppose $\ld{z}$ is assigned to in $\ld{t}$.
From the condition $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash p: \langle \ld{l_p}, \rd{m_p} \rangle$, we know that
for all $\ld{z_1}$ assigned in $\ld{t}$, $\ld{l_p} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z_1})$ and
for all $\ld{z_1}$ read in $\ld{t}$, $\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l_p}$.
Now if $\ld{l} \ld{~\sqsubseteq}~ \ld{l_p}$, then since in $\ld{t}$ no variables $\ld{z_2}$ such that
$\ld{\lambda}(\ld{z_2}) ~\ld{\sqsubseteq}~\ld{l}$ are assigned to.
Therefore by Lemma \ref{lemma:notassigned}, $\ld{\mu_f}(\ld{w}) = \ld{\mu_1}(\ld{w}) = \ld{\nu_1}(\ld{w}) = \ld{\nu_f}(\ld{w})$, for all $\ld{w}$ such that $\ld{\lambda}(\ld{w}) ~\ld{\sqsubseteq}~ \ld{l}$. \\
If $\ld{l_p} \ld{~\sqsubseteq}~ \ld{l}$, then for all $\ld{z_1}$ read in $\ld{t}$, $\ld{\lambda}(\ld{z_1}) ~\ld{\sqsubseteq}~ \ld{l_p}$.
Therefore, by assumption on transaction $\ld{t}$, if any variable $\ld{z}$ is assigned an expression $\ld{e}$,
since $\ld{\mu_1}$, $\ld{\nu_1}$ are two stores such that
for all $\ld{z_1} \in \ld{Z_e} = vars(\ld{e})$, $\ld{\mu_1}(\ld{z_1}) = \ld{\nu_1}(\ld{z_1})$, the value of $\ld{e}$ will be the same.
By this simple security argument, after the transaction $\ld{t}$, we have $\ld{\mu_f}(\ld{z}) = \ld{\nu_f}(\ld{z})$.
Since the transaction happened entirely and atomically in domain $\ld{L}$, we do not have to worry ourselves with changes in the other domain
$\rid{M}$, and do not need to concern ourselves with
the adversarial level $\rd{m}$.\\[1ex]
\textbf{Case} $p$ is $\ld{rd}(\ld{z},\ld{y})$: \ \ \
Thus $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{rd}(\ld{z},\ld{y}):
\langle \ld{\lambda}(\ld{z}), \rd{m} \rangle$,
which means $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{z})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{l}$, then since by I.H., $\ld{\mu_1}(\ld{y}) = \ld{\nu_1}(\ld{y})$, we have $\ld{\mu_f}(\ld{z}) =
\ld{\mu_1}[\ld{z} := \ld{\mu_1}(\ld{y})](\ld{z}) =
\ld{\nu_1}[\ld{z} := \ld{\nu_1}(\ld{y})](\ld{z}) =
\ld{\nu_f}(\ld{z})$. \\[1ex]
\textbf{Case} $p$ is $\ld{wr}(\ld{x},\ld{z})$: \ \ \
Thus $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash \ld{wr}(\ld{x},\ld{z}):
\langle \ld{\lambda}(\ld{x}), \rd{m} \rangle$,
which means $\ld{\lambda}(\ld{z}) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{x})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{x}) ~\ld{\sqsubseteq}~ \ld{l}$, then since by I.H., $\ld{\mu_1}(\ld{z}) = \ld{\nu_1}(\ld{z})$, we have $\ld{\mu_f}(\ld{x}) =
\ld{\mu_1}[\ld{x} := \ld{\mu_1}(\ld{z})](\ld{x}) =
\ld{\nu_1}[\ld{x} := \ld{\nu_1}(\ld{z})](\ld{x}) =
\ld{\nu_f}(\ld{x})$. \\[1ex]
\textbf{Case} $p$ is $T_{RL}(\ld{y},\rd{x})$: \ \ \
So $\langle \ld{\lambda}, \rd{\lambda} \rangle \vdash T_{RL}(\ld{y},\rd{x}):
\langle \ld{\lambda}(\ld{y}), \rd{\lambda}(\rd{x}) \rangle$, and
$\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$.
If $\ld{l} ~\ld{\sqsubseteq}~ \ld{\lambda}(\ld{y})$,
there is nothing to prove (Lemma \ref{lemma:notassigned}, again).
If $\ld{\lambda}(\ld{y}) ~\ld{\sqsubseteq}~\ld{l}$,
then by transitivity, $\ld{\gamma}(\rd{\lambda}(\rd{x})) ~\ld{\sqsubseteq}~ \ld{l}$.
By monotonicity of $\rid{\alpha}$:
$\rid{\alpha}(\ld{\gamma}(\rd{\lambda}(\rd{x}))) ~\rd{\sqsubseteq}~ \rid{\alpha}(\ld{l}) ~=~ \rd{m}$
(By our assumption on $\ld{l}$ and $\rd{m}$).
But by \textbf{LC2}, $\rd{\lambda}(\rd{x}) ~\rd{\sqsubseteq}~ \rid{\alpha}(\ld{\gamma}(\rd{\lambda}(\rd{x})))$.
So by transitivity, $\rd{\lambda}(\rd{x}) ~\rd{\sqsubseteq}~ \rd{m}$.
Now, by I.H., since $\rd{\mu_1}(\rd{x}) = \rd{\nu_1}(\rd{x})$, we have
$\ld{\mu_f}(\ld{y}) =
\ld{\mu_1}[\ld{y} := \rd{\mu_1}(\rd{x})](\ld{y}) =
\ld{\nu_1}[\ld{y} := \rd{\nu_1}(\rd{x})](\ld{y}) =
\ld{\nu_f}(\ld{y})$.
\end{proof}
\section{Related Work}\label{sec:related}
The only cited use of the notion of Lagois connections \cite{MELTON1994lagoisconnections} in computer science of which we are aware is the work of Huth \cite{huth1993equivalence} in establishing the correctness of programming language implementations.
To our knowledge, our work is the first to propose their use in secure information flow control.
Abstract Interpretation and type systems \cite{cousot1997types-as-ai} have been used in secure flow analyses, \textit{e.g.}, \cite{cortesi2015datacentricsemantics, cortesi2018} and \cite{zanotti2002sectypingsbyai}, where security types are defined using Galois connections employing, for instance, a standard collecting semantics.
Their use of two domains, concrete and abstract, with a Galois connection between them, for performing static analyses \textit{within a single domain} should not be confused with our idea of secure connections between independently-defined security lattices of two organisations.
At the systems level, there has been quite some work on SIF in a distributed setting.
An exemplar is DStar \cite{zeldovich2008-nsdi}, which uses sets of opaque identifiers to define security classes.
The DStar framework takes a \textit{particular} Decentralized Information Flow Control (DIFC) model \cite{Krohn2007-aa,zeldovich2006-osdi} for operating systems and extends it to a distributed network.
Subset inclusion is the (only) partial order considered in DStar's security lattice.
Thus it is not clear if DStar can work on general IFC mechanisms such as FlowCaml \cite{Pottier2003-FlowCaml}, which can employ any partial ordering.
Nor can the DStar model express the labels of JiF \cite{myers1999jflow} or Fabric \cite{liu2017fabric} completely.
DStar allows bidirectional communication between processes $R$ and $S$ only if $L_R \sqsubseteq_{O_R} L_S$ and $L_S \sqsubseteq_{O_S} L_R$, \textit{i.e.}, when there is an order-isomorphism between the labels.
We have argued that such a requirement is far too restrictive for most practical arrangements for data sharing between organisations.
Fabric \cite{liu2009fabric,liu2017fabric} adds \textit{trust relationships} directly derived from a principals hierarchy to support systems with mutually distrustful nodes and allows dynamic delegation of authority.
It is not immediately clear whether that framework supports modular decomposition and analysis, a topic for future investigation.
Most of the previous DIFC mechanisms \cite{myers1999jflow, zeldovich2006-osdi, Krohn2007-aa, efstathopoulos2005asbestos, roy2009laminar, cheng2012aeolus} including Fabric are susceptible to the vulnerabilities mentioned in the motivating examples of our previous work \cite{BhardwajP2019}.
|
1,314,259,993,005 | arxiv | \section{Introduction}\label{introsec}\eqnoset
In this paper, we study the global existence of following strongly coupled parabolic system of $m$ equations ($m\ge2$) for the unknown vector $u=[u_i]_{i=1}^m$
\beqno{ep1}(u_i)_t=\Delta(u_ip_i(u))+u_ig_i(u),\quad (x,t)\in \Og\times(0,\infty).\end{equation}
Here, $p_i, g_i:{\rm I\kern -1.6pt{\rm R}}^m\to{\rm I\kern -1.6pt{\rm R}}$ are sufficienly smooth functions. Namely, $p_i\in C^2({\rm I\kern -1.6pt{\rm R}}^m)$ and $g_i\in C({\rm I\kern -1.6pt{\rm R}}^m)$. $\Og$ is a bounded domain with smooth boundary in ${\rm I\kern -1.6pt{\rm R}}^N$, $N\ge2$.
The system is equipped with Dirichlet boundary and sufficiently smooth initial conditions
\beqno{ep1bc}\left\{\begin{array}{l} \mbox{$u_i=0$ on $\partial \Og\times(0,\infty)$},\\ u_i(x,0)=u_{i,0}(x) ,\quad x\in \Og. \end{array}\right.\end{equation}
The consideration of \mref{ep1} is motivated by the extensively studied model in population biology introduced by Shigesada {\it et al.} in \cite{SKT}
\beqno{e0}\left\{\begin{array}{lll} u_t &=& \Delta(d_1u+\ag_{11}u^2+\ag_{12}uv)+k_1u+\bg_{11}u^2+\bg_{12}uv,\\v_t &=& \Delta(d_2v+\ag_{21}uv+\ag_{22}v^2)+k_2v+\bg_{21}uv+\bg_{22}v^2.\end{array}\right.\end{equation} Here, $d_i,\ag_{ij},\bg_{ij}$ and $k_i$ are constants with $d_i>0$. Dirichlet or Neumann boundary conditions were usually assumed for \mref{e0}. This model was used to describe the population dynamics of {\em two} species densities $u,v$ which move and interact with each other under the influence of their population pressures.
Of course, \mref{e0} is a special case of \mref{ep1} with $m=2$ and
$$p_i(u,v)=d_i+\ag_{i1}u+\ag_{i2}v,\; g_i(u,v)=k_i+\bg_{i1}u+\bg_{i2}v.$$
We will refer to the functions $p_i$'s (respectively, $g_i$'s) as the diffusion (respectively, raction) rates (see \cite{LM} for further discussions).
Under suitable assumptions on the constant parameters $\ag_{ij}$'s, $\bg_{ij}$'s and that $\Og$ is a planar domain ($N=2$), Yagi proved in \cite{yag} the global existence of (strong) positive solutions, with positive initial data. In this paper, we will extend this investigation to multi-species versions of \mref{e0} for more than two species on bounded domains of arbitrary dimension $N$.
The global existence problem of \mref{ep1}, a fundamental problem in the theory of pdes. We can write \mref{ep1} in its general divergence form
\beqno{ep1div} u_t =\Div(A(u)Du)+f(u).\end{equation}
This a strongly coupled parabolic system with the diffusion matrix $A(u)$, the Jacobian of $[u_ip_i(u)]_1^m$, being a {\em full} matrix. We say that the system is weakly coupled if $A(u)$ is diagonal (i.e., $p_i$ depends only on $u_i$).
The key point in the proof of global existence of strong solutions of \mref{ep1div} is the a priori estimate of their spatial derivatives. In fact, it was established by Amann in \cite{Am2} that \mref{ep1} has a global strong solution $u$ if there is some exponent $p>N$ such that for any $T\in (0,\infty)$
$$\limsup_{t\to T^-}\|Du\|_{L^p(\Og)}<\infty.$$
Thus, we need only prove that $\sup_{t\in(0,T) }\|Du\|_{L^p(\Og)}<\infty$ for all $T\in(0,\infty)$ and some $p>N$. With this a priori estimate, one can alternatively use the homotopy or fixed point approaches in \cite{letrans, dleANS,dlebook}, instead of semigroup theories in \cite{Am2}, to obtain the local/global existence of strong solutions.
The derivation of such estimates for \mref{ep1} is a difficult issue when $A(u)$ is full because the known techniques for scalar equations ($m=1$) are no longer applicable unless the matrix $A(u)$ are of special form, e.g., diagonal or triangular, these techniques can be partly applied together with some ad hoc arguments (see \cite{YW}). In this paper, we will consider \mref{ep1} with {\em full} diffusion matrix $A(u)$ of special forms where some nontrivial modifications of the classic methods can apply and yield new affirmative answers to the problem.
Precisely, we study the case when either the diffusion or reaction rates are identical. Being inspired by the standard (SKT) system \mref{e0} where $p_i$ are a linear function in $u$, we consider a function $\Psi$ on ${\rm I\kern -1.6pt{\rm R}}$, a linear combination $L(u)$ of $u_i$'s, $L(u)=\sum_i a_iu_i$, and assume that for $i=1,\ldots,m$\beqno{pidef} p_i(u)=\llg_0+\Psi(L(u)).\end{equation}
We also assume that the reaction rates $g_i$'s satisfy the control growth $|g_i(u)|\le C+c_0\Psi(|L(u)|)$ for some positive constants $C,c_0$. We will establish the global existence of nonnegative strong solutions to \mref{ep1} with nonnegative initial data.
Clearly, \mref{e0} is the case when $d_1=d_2$, $\ag_{i1}=\ag_{j1}$, $\ag_{i2}=\ag_{j2}$ and $\Psi(s)=s$.
On the other hand, we can relax the assumption that the diffusion rates are identical as in \mref{pidef}. The trade off is that the reaction rates $g_i$'s are identical and satisfying the above control growth.
The paper is organized as follows. In \refsec{scalareqn}, we discuss some regularity positivity results for strong solutions to scalar parabolic equations. Our main results on the system \mref{ep1} will be presented and proved in \refsec{equal}.
\section{Some facts on scalar equations} \label{scalareqn}\eqnoset In this section we consider the following scalar equation \beqno{scalarveqn}v_t = \Delta(P(v))+\Div(vb(v))+vg(v)\end{equation} in $Q=\Og\times(0,T)$ and
and study the smoothness, uniform boundedness and positivity of its {\em strong solution} $v$ under some special conditions on $P,g$ which will serve our purpose in discussing cross diffusion systems later.
To proceed, we first need the following parabolic Sobolev imbedding inequality.
\blemm{parasobolev} Let $r^*=p/N$ if $N>p$ and $r^*$ be any number in $(0,1)$ if $N\le p$. For any sufficiently nonegative smooth functions $g,G$ and any time interval $I$ there is a constant $C$ such that\beqno{paraSobo}\itQ{\Og\times I}{g^{r^*}G^p}\le C\sup_I\left(\iidx{\Og\times\{t\}}{g}\right)^{r^*}\itQ{\Og\times I}{(|DG|^p+G^p)}\end{equation} If $G=0$ on the parabolic boundary $\partial\Og\times I$ then the integral of $G^p$ over $\Og\times I$ on the right hand side can be dropped.
Furthermore, if $r<r^*$ then for any $\eg>0$ we can find a constant $C(\eg)$ such that
\beqno{paraSobo1}\itQ{\Og\times I}{g^{r}G^p}\le C\sup_I\left(\iidx{\Og\times\{t\}}{g}\right)^{r}\itQ{\Og\times I}{(\eg|DG|^p+C(\eg)G^p)}\end{equation}
\end{lemma}
{\bf Proof:~~} For any $r\in(0,1)$ and $t\in I$ we have via H\"older's inequality
\beqno{Sobo1}\iidx{\Og}{g^rG^p}\le \left(\iidx{\Og}{g}\right)^{r}\left(\iidx{\Og}{G^\frac{p}{1-r}}\right)^{1-r}.\end{equation}
If $r=r^*$ then $p/(1-r)=N_*=pN/(N-p)$, the Sobolev conjugate of $p$ if $N>p$ (the case $N\le p$ is obvious), so that the Sobolev inequality gives
$$\left(\iidx{\Og}{G^\frac{p}{1-r}}\right)^{1-r}\le \iidx{\Og}{(|DG|^p+G^p)}.$$
Using the above in \mref{Sobo1} and integrating over $I$, we easily obtain \mref{paraSobo}. On the other hand, if $r<r^*$, then $p/(1-r)<N_*$. A simple contradiction argument and the compactness of the imbedding of $W^{1,p}(\Og)$ into $L^{p/(1-r)}(\Og)$ imply that for any $\eg>0$ there is $C(\eg)$ such that
$$\left(\iidx{\Og}{G^\frac{p}{1-r}}\right)^{1-r}\le \eg\iidx{\Og}{|DG|^p}+C(\eg)\iidx{\Og}{G^p}.$$ We then obtain \mref{paraSobo1}. \begin{flushright} Q.E.D. \end{flushright}
We now have the following a priori boundedness of solution of \mref{scalarveqn}.
\btheo{vboundthm} Consider a (weak or strong) solution $v$ to \mref{scalarveqn} in $Q=\Og\times(0,T)$. Assume that there are a function $\llg(v)$ and a number $\llg_0$ such that $\llg(v)\ge \llg_0>0$ and
\beqno{Qvcond}P_v(v)\ge \llg(v),\end{equation} \beqno{bvcond}|b(v)|\le g_1\llg(v),\end{equation}\beqno{fvcond}|g(v)|\le g_2\llg(v),\end{equation}
where $g_1,g_2$ are functions such that $g_1^2+g_2\in L^q(Q)$ for some $q>N/2+1$.
For $v\in {\rm I\kern -1.6pt{\rm R}}$ and $p\ge1$ consider the function \beqno{Fdef} F(v,p)=\int_0^v\llg^\frac12(s)s^{p-1}\,ds,\end{equation} and assume that \beqno{Fdefcond}|F(v,p)|\sim Cp\llg^\frac12(v)|v|^{p}\quad \mbox{for all $p$ and $v\in{\rm I\kern -1.6pt{\rm R}}$.}\end{equation}
If $\|v\llg(v)\|_{L^1(Q)}$ is finite then $v,Dv$ are bounded and H\"older continuous in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$. Their norms depend on $\|v\llg(v)\|_{L^1(Q)}$.
\end{theorem}
The condition \mref{Fdefcond} is clearly verified if $\llg(v)$ has a polynomial growth in $|v|$.
{\bf Proof:~~} We test the equation with $|v|^{2p-2}v$ and use integration by parts
$$\iidx{\Og}{\Delta(P(v))|v|^{2p-2}v}=-\iidx{\Og}{P_v(v)DvD(|v|^{2p-2}v)},$$
$$\iidx{\Og}{\Div(vb(v))|v|^{2p-2}v}=-\iidx{\Og}{vb(v)D(|v|^{2p-2}v)}.$$
Because $D(|v|^{2p-2}v)=(2p-1)|v|^{2p-2}Dv$ and the assumptions on $Q_v(v)$ and $b(v),g(v)$,
we easily get for {\em all} $p\ge1$
\beqno{vptest}\barrl{\sup_{(0,T)}\frac{1}{2p}\iidx{\Og}{|v|^{2p}}+(2p-1)\itQ{Q}{\llg(v)|v|^{2p-2}|Dv|^2}\le}{3cm} &C\itQ{Q}{g_1|\llg(v)||v|^{2p-1}|Dv|}+C\itQ{Q}{g_2|\llg(v)||v|^{2p}}.\end{array}\end{equation}
Applying Young's inequality $g_1|\llg(v)||v|^{2p-1}|Dv|\le \eg|v|^{2p-2}|Dv|^2+C(\eg)g_1^2|v|^{2p}$ for $\eg$ small,
$$\sup_{(0,T)}\frac{1}{2p}\iidx{\Og}{|v|^{2p}}+(2p-1)\itQ{Q}{\llg(v)|v|^{2p-2}|Dv|^2}\le C\itQ{Q}{(g_1^2+g_2)|\llg(v)||v|^{2p}}.$$
As $\llg(v)|v|^{2p-2}= F_v^2(v,p)$ by the definition \mref{Fdef}, for $g_3=g_1^2+g_2$ the above is
$$\sup_{(0,T)}\frac{1}{2p}\iidx{\Og}{|v|^{2p}}+(2p-1)\itQ{Q}{|D(F(v,p)|^2}\le C\itQ{Q}{g_3|\llg(v)||v|^{2p}}.$$
Thus, for $p\ge1$
$$\sup_{(0,T)}\iidx{\Og}{|v|^{2p}},\; \itQ{Q}{|D(F(v,p))|^2}\le Cp\itQ{Q}{g_3|\llg(v)||v|^{2p}}.$$
Applying the parabolic Sobolev inequality in \reflemm{parasobolev} with $g=|v|^p$ and $G=F(v,p)$, the above estimate yields for $r=2/N$
$$ \left(\itQ{Q}{|v|^{2pr}|F(v,p)|^2}\right)^\frac{1}{1+r} \le Cp^{1+\frac{2}{1+r}}\itQ{Q}{g_3|\llg(v)||v|^{2p}}.$$
As $F(v)\sim Cp^{-1}\llg^\frac12(v)|v|^{2p}$ by \mref{Fdefcond}, we then obtain for $\cg=1+2/N$
$$ \left(\itQ{Q}{|v|^{2p\cg}\llg(v)}\right)^\frac{1}{\cg} \le Cp^{1+\frac{2}{\cg}}\itQ{Q}{g_3\llg(v)|v|^{2p}}.$$
H\"older's inequality yields
$$\itQ{Q}{g_3\llg(v)|v|^{2p}}\le C\left(\itQ{Q}{g_3^q\llg(v)}\right)^\frac{1}{q}\left(\itQ{Q}{|v|^{2pq'}\llg(v)}\right)^\frac{1}{q'}.$$
Let $d\mu=\llg(v)dz$. As we assume that $g_1^2, g_2\in L^q(Q,d\mu)$, $g_3\in L^q(Q,d\mu)$ and the first factor on the right hand side is finite. The above inequality is
\beqno{viterate}\|v\|_{L^{2p\cg}(Q,d\mu)} \le (2Cp)^{(1+\frac{2}{\cg})\frac{1}{2p}}\|v\|_{L^{2pq'}(Q,d\mu)}.\end{equation}
Because $q>N/2+1$, $q'<\cg=1+2/N$. Replacing $p$ by $pq'$ and defining $\cg_0=\cg/q'>1$). It follows that
\beqno{viterate1}\|v\|_{L^{2p\cg_0}(Q,d\mu)} \le (2Cp)^{(\frac{1}{q'}+\frac{2}{\cg_0})\frac{1}{2p}}\|v\|_{L^{2p}(Q,d\mu)}.\end{equation}
Because $\cg_0>1$, we can apply the Moser iteration agument to show that $v$ is bounded. Indeed, by taking $2p=\cg_0^i$ with $i=0,1,\ldots$. to the above estimate implies
$$\|v\|_{ L^{\cg^i}(Q,d\mu)}\le (2C)^{\cg_1}\cg^{\cg_2}\|v\|_{L^1(Q,d\mu)},$$ with $ \cg_1=(\frac{1}{q'}+\frac{1}{\cg_0})\sum_{i=0}^\infty \cg_0^{-i},\cg_2=(\frac{1}{q'}+\frac{1}{\cg_0})\sum_{i=0}^\infty i\cg_0^{-i}$. Letting $i\to\infty$ and using the fact that $\lim{p\to\infty\|v\|_{L^p(Q,d\mu)}=\|v\|_{L^\infty(Q,d\mu)}}$ (we will show that $d\mu$ is finite below) we obtain for some constant $C_0$
that $\|v\|_{L^\infty(Q,d\mu)}\le C_0\|v\|_{L^1(Q,d\mu)}$.
As $\llg(v)$ is bounded below by a positive constant, this implies that $v$ is bounded if $v\in L^1(Q,d\mu)$ is bounded. Furthermore, we now show that $d\mu$ is finite. Because $$ \itQ{|v|\ge1}{\llg(v)}\le \|v\|_{L^1(Q,d\mu)},$$ and $\llg(u)$ is bounded on the set $|v|<1$, we see that $d\mu$ is finite.
Once we show that $v$ is bounded, we obtain the {\em local} Harnack inequality (using both posive and negative power $p$ and cutoff functions) and so that $v$ is H\"older continuous. The argument is now classical and we refer the readers to the classical books \cite{LSU,Lieb} for details. It also follows that $Dv$ is bounded and H\"older continuous in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$. Indeed, we can adapt the freezing coefficient method in \cite{GiaS} to establish this fact. \begin{flushright} Q.E.D. \end{flushright}
\brem{vlarge} The conditions in the theorem and remarks need only hold only for $|v|$ large. This is easily to see if we make use of the cutoff function
\beqno{vcut}\bar{v}_{(k)}=\left\{\begin{array}{ll} v &\mbox{if $|v|\ge k$},\\k &\mbox{if $0<v< k$},\\
-k &\mbox{if $-k<v\le 0 $}\end{array} \right.\end{equation}
with $k$ sufficiently large and observe that $D\bar{v}_k=0$ on the set $|v|<k$. \erem
\brem{vPsieqn} In connection with the systems considered in the next section, we consider the scalar equation
\beqno{veqn1} v_t= \llg_0 \Delta v +\Delta (\Psi(v)v)+ vg(v),\end{equation}
where $\llg_0>0$ and
$\Psi:{\rm I\kern -1.6pt{\rm R}}\to{\rm I\kern -1.6pt{\rm R}}$ be a $C^1$ function and satisfying for $|v|$ large
\beqno{Psicondv}\Psi(v),\;\Psi'(v)v\ge0.\end{equation}
Asume also that for $v\in {\rm I\kern -1.6pt{\rm R}}$ and $p\ge1$ the function \beqno{Fdef1} \hat{F}(v,p)=\int_0^v\Psi^\frac12(s)s^{p-1}\,ds\end{equation} satisfies \beqno{Fdefcond1}|\hat{F}(v,p)|\sim Cp\Psi^\frac12(v)|v|^{2p}\quad \mbox{for all $p$ and $v\in{\rm I\kern -1.6pt{\rm R}}$.}\end{equation}
This condition allows us to apply \reftheo{vboundthm} with $P(v)=\llg_0 v +\Psi(v)v$ and $\llg(v)=\Psi(v)+\Psi'(v)v+\llg_0$. Thanks to \mref{Psicondv}, $\llg(v)$ satisfies \mref{Fdefcond}. Also, \mref{Fdef1} and \mref{Fdefcond1} imply that the function $F$ defined by \mref{Fdef} satisfies \mref{Fdefcond}. We then apply \reftheo{vboundthm} to \mref{veqn1} and obtain that $v,Dv$ are bounded in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$ and their norms are bounded in term of $\|v\|_{L^1(Q)}$ and $\|v\Psi(v)\|_{L^1(Q)})$.
We can also consider the scalar equation
\beqno{veqn2} v_t= \llg_0 \Delta v +\Delta (\Psi(|v|)v)+ vg(v),\end{equation}
and
$\Psi:{\rm I\kern -1.6pt{\rm R}}\to{\rm I\kern -1.6pt{\rm R}}$ be a $C^1$ function and satisfying for $v\ge0$ and large
\beqno{Psicondv1}\Psi(v),\;\Psi'(v)\ge0.\end{equation}
Indeed, we now define $\psi(v)=\Psi(|v|)$. We then have $\psi'(v)v=\Psi'(|v|)\mbox{sign}v v=\Psi'(|v|)|v|\ge0$ because of \mref{Psicondv1} $|v|\ge0$. Thus, $\psi$ satisfies \mref{Psicondv} and the theorem applies.
\erem
In applications we usually prefer that $v$ is nonnegative if the initial is. The following result serves this purpose.
\btheo{vposcoro} Let $a,g$ be $C^1$ functions on ${\rm I\kern -1.6pt{\rm R}}\times Q$ and $b$ be a bounded $C^1$ map from $Q$ into ${\rm I\kern -1.6pt{\rm R}}^N$. Assume that $a(w)\ge \llg_0$ for $w\ge0$ and $\llg_0$ is a positive constant. Also suppose that $a,g$ are bounded by a constant depending on $w$ in $(x,t)\in Q$.
Let $w$ be the strong solution to \beqno{veqn2a} \left\{\begin{array}{ll}w_t= \Div(a(w,x,t)Dw)+\Div(wb) +wg(w,x,t), & \mbox{in $Q$}\\w(x,0)=w_0(x)&\mbox{on $\Og$}.
\end{array} \right.\end{equation}
If $w_0\ge0$ then $w\ge0$ on $Q$. \end{theorem}
{\bf Proof:~~}
Because $w$ is a strong solution, there is a constant $M>0$ susch that $|w|\le M$.
We then truncate $a,g$ to $C^1$ function $\hat{a},\hat{g}$ which are constants for $v$ outside $[-M-1,M+1]$ and consider the equation \beqno{veqn2b} v_t= \Div(\hat{a}(|v|,x,t)Dv) + \Div(vb(x,t))+v\hat{g}(v,x,t),\end{equation}
with initial data $w_0$.
We have $\hat{a}(|v|,x,t)\ge\llg_0$ and is bounded from above and $|v\hat{g}(v,x,t)|\le C|v|$ for some constant $C$. These facts and the classical theory of scalar parabolic equation show that \mref{veqn2} has a strong solution $v$.
Let $v^+,v^-$ be the positive and negative parts of $v$. We test the equation with $v^-$. Using the facts that
$|v|=v^++v^-$, $|v|=v^-$ on the set $v^->0$, $v^+Dv^-=Dv^+Dv^-=0$ on the set $v^->0$, we obtain
$$-\frac{d}{dt}\iidx{\Og}{(v^-)^2}-\iidx{\Og}{\hat{a}|Dv^-|^2}=\iidx{\Og}{[-bv^-Dv^-+(v^-)^2\hat{g}]}.$$
Because $b$ are bounded by a constant $C(M)$, applying Young's inequality $$\iidx{\Og}{|bv^-Dv^-|} \le \eg \iidx{\Og}{|Dv^-|^2} +C(\eg,M)\iidx{\Og}{(v^-)^2}.$$
Because $\hat{g}$ is bounded by a constant $C$ depending on $M$ and $a(v)\ge\llg_0$, we can choose $\eg$ sufficiently small in the above inequality to arrive at
$$\frac{d}{dt}\iidx{\Og}{(v^-)^2}+\iidx{\Og}{|Dv^-|^2}\le C(M)\iidx{\Og}{(v^-)^2}.$$
Thus, we see that the function $$z(t)=\iidx{\Og}{(v^-)^2}$$ satisfies the differential inequality
$z'\le C_1z$ and $z(0)=0$ because the initial data $v_0\ge0$. We then apply comparision theorem to the equation $y'=Cy$ with $y(0)=0$ which has the solution $y(t).=0$ We then have $z(t)=0$ for all $t\in(0,T)$. Hence, $v^-=0$ on $Q$ so that $v\ge0$. It follows that the solution $v$ of \mref{veqn2} also solves \mref{veqn2a}. By the uniqueness of strong solutions, $w=v\ge0$ in $Q$. \begin{flushright} Q.E.D. \end{flushright}
\section{Cross diffusion system with equal diffusion/reaction rates} \label{equal}\eqnoset
In this section, we consider the system \mref{ep1} and assume either that the diffusion rates $p_i$'s or the reaction rates are equal. We will always assume nonngative initial data $u_{i,0}$.
Throughout this section we will consider a nonnegative $C^1$ function $\Psi$ on ${\rm I\kern -1.6pt{\rm R}}$ satisfying \beqno{Psimaincond}\Psi'(s)\ge0 \mbox{ for $s\ge0$}.\end{equation}
\subsection{Equal diffusion rates:}
We first consider the following system of $m$ equations for $u=[u_i]_1^m$
\beqno{uWeqn}\left\{\begin{array}{ll}(u_i)_t=\Delta(\llg_0 u_i + \Psi(L(u))u_i)+u_ig_i(u)&\mbox{in $\Og\times(0,\infty)$},\\u_i(x,0)=u_{i,0}(x)&\mbox{on $\Og$},\end{array}\right.\end{equation} where $\llg_0>0$ and $L(u)$ is a linear combination of $u_i$. That is, $ L(u)=\Sigma_{i=1}^m a_i u_i$ with $a_i>0$.
Assume that there are constants $ C_{ij},c_{ij}\ge0$ such that
\beqno{gicond} |g_i(u)|\le \sum_j (C_{ij}+c_{ij}\Psi(|u_i|)).\end{equation}
We have
\btheo{equaldiffthm} If $c_0=\max c_{ij}$ is sufficiently then \mref{uWeqn} has a unique nonnegative strong solution.\end{theorem}
As we explained in the introduction, we need only establish a priopi the finiteness of $\sup_{(0,T_{max})}\|Du\|_{L^p(\Og)}$, with some $p>N$, for any strong solution $u=[u_i]_1^m$ of \mref{uWeqn} on $\Og\times(0,T_{max})$ for any $T_{max}\in(0,\infty)$. We will do this for $p=\infty$ via several lemmas.
\blemm{uposlemma} $u_i\ge0$ on $\Og\times(0,T_{max})$. \end{lemma}
{\bf Proof:~~} We can use \reftheo{vposcoro} to show first that $u_i\ge0$ on $Q=\Og\times[0,T]$ for any $0<T<T_{max}$ and all $i$. We rewrite the equation of $u_i$ as
\beqno{upsisys}\left\{\begin{array}{ll}(u_i)_t=\Div(a_i(u_i,x,t)Du_i)+\Div(u_ib_i(x,t))+ u_ig_i(u) & \mbox{in $Q$},\\ u_i(x,0)=u_{i,0}(x)&\mbox{on $\Og$},\end{array} \right.\end{equation} where $$a_i(u_i,x,t)=\llg_0 + \Psi(L(u))+\partial_{u_i}\Psi(L(u))u_i,\; b_i(x,t)=\sum_{j\ne i} \partial_{u_j}\Psi(L(u(x,t))).$$
Following the proof of \reftheo{vposcoro}, because $u$ bounded on $Q$, $|L(u)|\le M$ for some constant $M$. We truncate the function $\Psi$ outside the interval $[-M-1,M+1]$ to obtain a bounded $C^1$ function $\psi$ satisfying: $\psi(s),\psi'(s)\ge0$ and $\psi(s)$ is a constant when $|s|\ge M+1$.
Denoting $\hat{v}=[|v_i|]_1^m$ for any vector $v=[v_i]_1^m$. We consider the system
\beqno{uvpsisys}\left\{\begin{array}{ll}(v_i)_t=\Div(\hat{a}_i(v,x,t)Dv_i)+\Div(v_ib_i(x,t))+ v_ig_i(u) & \mbox{in $Q$}, \\v_i(x,0)=u_{i,0}(x)&\mbox{on $\Og$},\end{array} \right.\end{equation} where $$\hat{a}_i(v,x,t)=\llg_0 + \psi(L(\hat{v}))+\partial_{v_i}\psi(L(\hat{v}))v.$$ Because $\psi'(s)\ge0$ for $s\ge0$ and $L(\hat{v})\ge0$, we have $\partial_{u_i}\psi(L(\hat{v}))v_i=\psi'(L(\hat{v}))a_i\mbox{sign}(v_i)v_i=\psi'(L(\hat{v}))a_i|v_i|\ge0$. We also have $\psi(L(\hat{v}))\ge 0$. Thus $\hat{a}_i(v,x,t)\ge\llg_0$ and bounded from above. The system \mref{uvpsisys} is a diagonal system with bounded continuous coefficients and has a unique strong solution $v$ according to the classical theory (e.g., see \cite[Chapter 7]{LSU}).
Applying the argument in the proof of \reftheo{vposcoro} to each equation in \mref{uvpsisys}, the system \mref{uvpsisys} has a nonnegative strong solution $v$, so that $\psi(\hat{v})=\psi(v)$, which also solves \mref{upsisys} by the definition of $\psi$, an extension of $\Psi$. By the uniqueness of strong solutions, $u_i=v_i\ge0$ in $Q$ for all $i$. \begin{flushright} Q.E.D. \end{flushright}
Next, define $W=L(u)$. The following lemma provides bounds of $W,DW$ that are independent of the number $M$, which was used only in establishing that $u_i\ge0$.
\blemm{Wbound} Let $W=L(u)\ge 0$. Assume that
\beqno{Fpsidef} F(v,p):=\int_0^v\Psi^\frac12(s)s^{p-1}\,ds\sim Cp\Psi^\frac12(v)|v|^{2p}\quad \mbox{for all $p$ and $v\ge0$.}\end{equation}
Then $W,DW$ are bounded in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$ by a constant depending only on $\|W\|_{L^1(Q)},\|W\Psi(W)\|_{L^1(Q)}$.
\end{lemma}
{\bf Proof:~~}
Taking a linear combination of the equations, we obtain
\beqno{Weqn1} W_t= \llg_0 \Delta W +\Delta (\Psi(W)W)+ f(u),\end{equation}
where $f(u)=\sum_i a_i u_i g_i(u)$.
Because $u_i\ge0$ and $a_i>0$, $W$ is nonnegative and $|u_i|\le W$. Since $\Psi(s)$ is increasing for $s\ge0$, the assumption on $g_i$'s \mref{gicond} implies
$$|g_i(u)|\le \sum_j (C_{ij}+c_{ij}\Psi(|u_i|))\le \sum_j (C_{ij}+c_{ij}\Psi(W)).$$
Hence, $f$ satisfies for some positive constants $C$ and $c_0=\max c_{ij}$
\beqno{fgcond} |f(u)| \le C|W|(1+c_0\Psi(W)).\end{equation}
We then apply \reftheo{vboundthm} (to be precise, its \refrem{vPsieqn} and the equation \mref{veqn1}) with $v=W$, noting that $v=W\ge0$. The assumption \mref{Fpsidef} on $\Psi$ guarantees that \mref{Fdefcond1} is satisfied. We see that the norms of $W,DW$ are bounded in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$ by constants independent of $M$ but on $\|W\|_{L^1(Q)}$ and $\|W\Psi(W)\|_{L^1(Q)})$. The lemma follows.
\begin{flushright} Q.E.D. \end{flushright}
\brem{WL1bound} If the constant $c_0$ in \mref{fgcond} is sufficiently small then the norms $\|W\|_{L^1(Q)}$ and $\|W\Psi(W)\|_{L^1(Q)})$ are bounded by a constant. Indeed, testing the equation of $W$ by $W$ and using \mref{fgcond}
\beqno{Wptest}\sup_{t\in (0,T)}\iidx{\Og\times\{t\}}{W^{2}}+\itQ{\Og\times(0,t)}{\Psi(W)|DW|^2}\le C\itQ{\Og\times(0,t)}{[1+c_0\Psi(W)]W^{2}}.\end{equation}
Applying the Sobolev inequality to the function $F(W,1)$ (see \mref{Fpsidef}) we find a constant $C(N)$ such that $$\iidx{\Og\times\{t\}}{\Psi(W)W^2}\le C(N)\iidx{\Og\times\{t\}}{\Psi(W)|DW|^2}.$$ Thus, using this, we see that if $c_0$ is sufficiently small then the integral of $Cc_0\Psi(W)W^2$ in the inequality \mref{Wptest} can be absorbed to the left and we get
$$\sup_{t\in (0,T)}\iidx{\Og\times\{t\}}{W^{2}}+\itQ{\Og\times(0,t)}{\Psi(W)|DW|^2}\le C\itQ{\Og\times(0,t)}{W^{2}}.$$
This yields an integral Gr\"onwall inequality for $y(t)=\|W\|_{L^2(\Og\times\{t\})}$ on $(0,T)$ and shows that this norm is bounded by a universal constant on $(0,T)$. This fact and the above inequality show that the left hand side quantities are bounded. We then make use of the parabolic Sobolev inequality to see that $\|W^{2\cg}\Psi(W)\|_{L^1(Q,d\mu)}$ is bounded by a constant. This implies $\|W\Psi(W)\|_{L^1(Q)})$ is bounded because $2\cg>1$.
\erem
{\bf Proof of \reftheo{equaldiffthm}:}
We write the equation of $u_i$ in its divergence form
$$(u_i)_t=\Div(aDu_i)+\Div(u_ib))+u_ig_i(u),$$ where $a=\llg +\Psi(W)$ and $b=D(\Psi(W))$.
Using the facts that $\Psi(W)\ge0$ (because $W\ge0$) and $W$ is bounded, we have $a\ge\llg_0$ and bounded from above. Also, $b=D(\Psi(W))$ are bounded. In addition, $u_ig_i(u)$ is bounded because $0\le u_i\le W/a_i$ which is bounded. We then use
the standard theory of scalar parabolic equation with bounded coefficients to show that $Du_i$ is bounded and H\"older continuous in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$. \begin{flushright} Q.E.D. \end{flushright}
\subsection{Equal reaction rates:}
We now present two examples which relax the assumption of equal diffusion rates $p_i$'s. However, we have to consider equal reaction rates $g_i$'s and restrict ourselves to the case of systems of two equations.
In the sequel, we will always assume that $\Psi$ is a $C^1$ function on ${\rm I\kern -1.6pt{\rm R}}$ such that
\beqno{PsiWpos}\Psi(s), \Psi'(s)\ge 0 \mbox{ and } \Psi(s)\ge s \mbox{ for $s\ge0$}.\end{equation}
We consider first the following system
\beqno{YW}\left\{\begin{array}{lll} u_t&=&\Delta(\llg_0 u +u\Psi(L(u,v)) +\eg_0 a\Delta(uv)+ug(u,v),\\
v_t&=&\Delta(\llg_0 v + v\Psi(L(u,v))-\eg_0 b\Delta(uv)+vg(u,v).\end{array}\right.\end{equation}
Here, $L(u,v)=bu+av$. $\llg_0,\eg_0,a,b$ are positive constants.
Regarding the reaction term, we also assume that there are positive constants $C,c_0$ such that (compare with \mref{gicond})
\beqno{guvcond} |g(u,v)|\le C+c_0\Psi(|L(u,v)|) \mbox{ for all $u,v\in{\rm I\kern -1.6pt{\rm R}}$}.\end{equation}
We consider nonnegative initial data $u_0, v_0$ for $u,v$.
\btheo{uvthm}If $\eg_0, c_0$ are sufficiently small then the system \mref{YW} has a unique global strong solution $(u,v)$ with $u,v\ge0$. \end{theorem}
We need the following proposition which will be useful later.
\bprop{uvprop}
We consider a strong solution $(u,v)$ with nonnegative initial data $u_0,v_0$ to the following system
\beqno{YW0}\left\{\begin{array}{lll} u_t&=&\Delta(\llg_0 u +u\Psi(L(u,v)) +\eg_0 a\Delta(u|v|)+ug(u,v),\\
v_t&=&\Delta(\llg_0 v + v\Psi(L(u,v))-\eg_0 b\Delta(u|v|)+vg(u,v).\end{array}\right.\end{equation}
For any $\eg_0>0$ we have that $u,v$ and $Du,Dv$ are bounded. Also $u\ge0$ in $Q$.
If $\eg_0$ are sufficiently small then $v$ is also nonnegative in $Q$.
\end{propo}
{\bf Proof:~~} The proof will be divided into several steps. First of all, taking a linear combination of the above two equations, we see that $W=L(u,v)$ satisfying
\beqno{Weqn1a} W_t= \llg_0 \Delta W +\Delta (\Psi(W)W)+ Wg(u,v).\end{equation}
{\bf Step 1:} We show that $W,DW$ are bounded and $W\ge0$.
For a given strong solution $(u,v)$ of \mref{YW0} we consider the the equation
\beqno{Weqn1b} w_t= \llg_0 \Delta w +\Delta (\Psi(|w|)w)+ wg(u,v)\end{equation} and the initial data $w_0=au_0+bv_0\ge0$.
We proved in \reftheo{vboundthm} that this equation has a strong solution $w$ and, by \reftheo{vposcoro}, $w\ge0$. By uniqueness of strong solutions, $W=w$ so that $W\ge0$.
Now, from the proof of \reflemm{Wbound} we see that $W,DW$ are bounded in $\Og\times(\tau,T)$ for any $\tau\in(0,T)$ in terms of $\|W\|_{L^1(Q)}$ and $\|W\Psi(W)\|_{L^1(Q)}$ The latter two norms can be bounded by a constant if $c_0$ is sufficiently small (see \refrem{WL1bound}).
We should note that because we already proved that $W\ge0$, hence we do not need here the fact that $u,v\ge0$ (which will be established later) as before in \reflemm{Wbound} but the conditions $\Psi(s),\Psi'(s)\ge0$ for $s\ge0$ in \mref{PsiWpos} and that $|g(u,v)|\le C+c_0\Psi(|W|)$ in \mref{guvcond} (see equation \mref{veqn2} of \refrem{vPsieqn}).
{\bf Step 2:} We prove that $u\ge0$. We write the equation of $u$ in its divergence form
\beqno{udiveqn}u_t=\Div(ADu)+\Div(uB)+ug(u,v)\end{equation}
with $A=\llg_0 +\Psi(W)+\eg_0a|v|$, $B=-\Psi'(W)DW+\eg_0 aD(|v|)$.
Again, we can assume that $u,v$ are locally bounded as in the proof of \reflemm{uposlemma}. Because $W,DW$ are bounded, we apply \reftheo{vposcoro} to prove that $u\ge0$.
{\bf Step 3:} We now prove that $u$ is bounded by using the iteration argument in \reftheo{vboundthm}.
We multiply the above equation \mref{udiveqn} by $u^{2p-1}$, recall that $u\ge0$, and follows the proof of \reftheo{vboundthm} to get
\beqno{uiter}\frac{d}{dt}\iidx{\Og}{u^{2p}}+(2p-1)\iidx{\Og}{Au^{2p-2}|Du|^2}\le \iidx{\Og}{\Div(uB)u^{2p-1}}+\iidx{\Og}{g(u,v)u^{2p}}\end{equation}
From the definition of $B$ we need to study the following two terms on the right of \mref{uiter}
\beqno{adiv}-\iidx{\Og}{\Div(u\Psi'(W)D(W))u^{2p-1}},\;\iidx{\Og}{a\Div(uD(|v|))u^{2p-1}}.\end{equation}
The first one can be treated easily, using the fact that $W$ is bounded (see also below). We consider the second term. We have
$$\iidx{\Og}{a\Div(uD(|v|))u^{2p-1}}
=-(2p-1)\iidx{\Og}{auD(|v|)u^{2p-2}Du}.$$
For each $t>0$ we split $\Og=\Og^+(t)\cup\Og^-(t)$ where $\Og^+(t)=\{x:\,v(x,t)\ge0\}$.
Since $aDv=DW-bDu$ and on $\Og^+(t)$, $D(|v|)=Dv$, we have that the integral over $\Og^+(t)$ of $-auD(|v|)u^{2p-2}Du$ is
$$ \iidx{\Og^+(t)}{u^{2p-1}(-DW+bDu)Du}=-\iidx{\Og^+(t)}{u^{2p-1}DWDu}+\iidx{\Og^+(t)}{bu^{2p-1}|Du|^2}.$$
Because $DW$ is bounded in $\Og\times(\tau,T)$ for any $\tau\in(0,t)$, it follows that for any $\eg>0$ there is $c_1(\eg)$ such that
$$\iidx{\Og}{u^{2p-1}|DWDu|}\le \iidx{\Og}{(\eg u^{2p-2}|Du|^2 +c_1(\eg)u^{2p})}.$$
Choosing $\eg$ small, the integral of $u^{2p-2}|Du|^2$ can be absorbed to the integral of $\llg_0 u^{2p-2}|Du|^2$ in the left of \mref{uiter}. This argument also applies to the first integral in \mref{adiv}.
Meanwhile, on the set $v\ge0$, as $W\ge bu\ge0$ so that $\Psi(W)\ge W\ge bu$ (by the assumption \mref{PsiWpos} on $\Psi$). Thus, the integral over $\Og^+(t)$ of $bu^{2p-1}|Du|^2$ can also be absorbed to the integral over $\Og^+(t)$ of $\Psi(W)u^{2p-2}|Du|^2$ in $Au^{2p-2}|Du|^2$ of the left of \mref{uiter}.
On $\Og^-(t)$, $D(|v|)=-Dv$, we have that the integral over $\Og^-(t)$ of $-auD(|v|)u^{2p-2}Du$ is
$$ \iidx{\Og^-(t)}{u^{2p-1}(DW-bDu)Du}=\iidx{\Og^-(t)}{u^{2p-1}DWDu}-\iidx{\Og^-(t)}{bu^{2p-1}|Du|^2}.$$
The first integral on the right hand side can be handled as before. The second integral is nonnegative and can be dropped.
Putting these togetter, we then obtain for all $p\ge1$
$$\frac{d}{dt}\iidx{\Og}{u^{2p}}+\iidx{\Og}{u^{2p-2}|Du|^2}\le C\iidx{\Og}{u^{2p}}.$$ This allows to obtain a bound for $\|u\|_{L^\infty(Q)}$ in terms of $\|u\|_{L^1(Q)}$ (see \reftheo{vboundthm}). Let $p=1$ in the above inequality to get a Gr\"onwall inequality for $\|u\|_{L^2(\Og)}^2$. We see that $\|u\|_{L^2(\Og)}^2$, so is $\|u\|_{L^1(\Og)}$, is bounded on $(0,T)$.
Once we prove that $u$ and $W,DW$ are bounded we then use a cutoff function and repeat a similar argument to the above one in order to obtain local strong/weak Harnack inequalities. It follows that $u$ is H\"older continuos. This is a standard procedure and the readers are referred to the book \cite{Lieb}. It also follows that $Du$ is bounded.
{\bf Step 4:} We show that $v$ is bounded. This is easy because $v=(W-bu)/a$ and $u,Du$ and $W,DW$ are bounded. We should note that in the above steps we have not imposed any assumptions on $\eg_0$. Thus, the first assertion of the Proposition was proved.
{\bf Step 5:} Finally, we prove that $v\ge0$. First of all, we write the equation of $v$ in its divergence form
$$ v_t=\Div(A_1Dv)+\Div(|v|B_1)+vg,$$ where $A_1=\llg_0 +\Psi(W)-\eg_0bu\mbox{sign}(v)$, $B_1=\Psi'(W)DW+\eg_0 bD(u)$.
Since $u$ is bounded by a constant independent of $\eg_0$ and $\Psi(W)\ge0$, we can choose $\eg_0$ small such that $A_1\ge\llg_0/2$. Also, $B_1$ is bounded because $W,DW$ and $Du$ are. The proof of \reftheo{vposcoro} applies and shows that $v\ge0$. \begin{flushright} Q.E.D. \end{flushright}
{\bf Proof of \reftheo{uvthm}:} From \refprop{uvprop} the system \mref{YW0} has a strong solution $(u,v)$ which also solves \mref{YW}. By uniqueness of strong solutions, we see that strong solution $(u,v)$, and its spatial derivatives, of \mref{YW} are bounded uniformly in terms of the data. Because $\|Du\|_{L^\infty(\Og)},\|Dv\|_{L^\infty(\Og)}$ do not blow up in any time interval $(0,T)$, the solution exists globally. \begin{flushright} Q.E.D. \end{flushright}
We also consider the following system
\beqno{YWz}\left\{\begin{array}{lll} u_t&=&\Delta(\llg_0 u +u\Psi(L(u,v)) +\eg_0 a\Delta(uv)+ug(u,v),\\
v_t&=&\Delta(\llg_0 v + v\Psi(L(u,v))+\eg_0 b\Delta(uv)+vg(u,v).\end{array}\right.\end{equation}
Here, $L(u,v)=bu-av$ and $\eg_0,a,b$ are positive constants.
We then have the following result similar to \reftheo{uvthm} without the assumption on the smallness of $\eg_0$. However, we have to strengthen the condition \mref{PsiWpos} by assuming in addition that
\beqno{PsiWpos1} \Psi(s)\ge0 \quad \forall s\in{\rm I\kern -1.6pt{\rm R}}.\end{equation}
\btheo{uvthmz}If $c_0$ in the assumption \mref{guvcond} is sufficiently small then the system \mref{YW} has a unique global strong solution $(u,v)$ with $u,v\ge0$. \end{theorem}
{\bf Proof:~~} Following the proof of \reftheo{uvthm}, we consider a strong solution $(u,v)$ with the same initial data to the following system
\beqno{YWz0}\left\{\begin{array}{lll} u_t&=&\Delta(\llg_0 u +u\Psi(L(u,v)) +\eg_0 a\Delta(u|v|)+ug(u,v),\\
v_t&=&\Delta(\llg_0 v + v\Psi(L(u,v))+\eg_0 b\Delta(u|v|)+vg(u,v).\end{array}\right.\end{equation}
For any $\eg_0>0$ we will prove that $u,v$ and $Du,Dv$ are bounded. We also show that $u,v\ge0$ in $Q$. We follow the proof of \refprop{uvprop} and provide necessary modifications.
Let $W=bu-av$. Taking a linear combination of the two equations, we can follows Step 1 of the proof of \refprop{uvprop} to show that $W,DW$ are bounded. Note that we cannot prove that $W\ge0$ as before because its initial data $bu_0-av_0$ is not nonnegative.
Similarly, Step 2 also yields that $u\ge0$. We need to change the argument in Step 3 of the proof to prove that $u,Du$ are bounded. We test the equation of $u$ by $u^{2p-1}$. As in Step 3, we need to consider the following term on the right hand side of \mref{uiter}
$$\iidx{\Og}{a\Div(uD(|v|))u^{2p-1}}=-(2p-1)\iidx{\Og}{auD(|v|)u^{2p-2}Du}.$$
We again split $\Og=\Og^+\cup\Og^-$ where $\Og^+=\{v\ge0\}$. Because $av=bu-W$ (instead of $av=W-bu$ as before) we need to interchange $\Og^+,\Og^-$ in the previous argument. Namely, the integral over $\Og^+$ now contributes a nonnegative term to the left and an integral of $u^{2p}$ to the right. Meanwhile, on $\Og^-$ we have $W =bu-av\ge bu\ge0$ so that $\Psi(W) \ge bu$ and the integral over $\Og^-$ of $bu^{2p-1}|Du|^2$ now can be absorbed to the left hand side. The proof then continues to prove that $u,Du$ are bounded.
Using $v=(bu-W)/a$, we see that $v,Dv$ are bounded.
We now show that $v\ge0$, without the assumption that $\eg_0$ is small. We slightly modify Step 5 of \refcoro{uvprop}. We write the equation of $v$ as
$$ v_t=\Div(A_2Dv)+\Div(\eg_0uD(|v|))+\Div(vB_2)
+ vg(u,v).$$
Here, $A_2=\llg_0 +\Psi(W)$, $B_2=\eg_0\mbox{sign}(v)Du+D\Psi(W)$. We follow the proof of \reftheo{vposcoro} and test the equation with $v^-$. We need to consider the integral of $\Div(\eg_0uD(|v|))v^-$ on the right hand side. Using integration by parts and the fact that $D(|v|)=Dv^++Dv^-$,
$$\iidx{\Og}{\Div(\eg_0uD(|v|))v^-}=-\iidx{\Og}{\eg_0uD(|v|)Dv^-}=-\iidx{\Og}{\eg_0u|Dv^-|^2}.$$
Because $u\ge0$, the last term provides a nonnegative term on the left hand side. Meanwhile, we have that $A_2\ge \llg_0$ and $A_2,B_2$ are bounded (as $u,Du,W,DW$ are bounded).
We obtain as in the proof of \reftheo{vposcoro} a Gr\"onwall inequality of $\|v^-\|_{L^2(\Og)}$ and conclude that $v^-=0$ on $Q$. Thus, $v$ is nonnegative. The proof is complete.
\begin{flushright} Q.E.D. \end{flushright}
\bibliographystyle{plain}
|
1,314,259,993,006 | arxiv |
\section{Introduction}
With increasing interest in interactive speech systems such as voice assistants, there is an increased demand for human-like text-to-speech (TTS) systems. While recent technology advancements in speech synthesis have achieved human-like audio quality \cite{wavenet, alignT, attentionA, tacotron2}, the TTS's speaking style does not mimic the naturalness and expressiveness as in human conversations, because conventional speech interfaces respond to input speech queries with default speaking style learned from the TTS training dataset. To make the TTS more interactive, the TTS's response should vary depending on the context and the speaking style of the input speech query. For example, when the user is speaking fast and rushing out the door in the morning, the TTS would match the hurried pace; and when the user is in a quiet place and is speaking softly, the TTS would respond with a soft and quiet voice. By detecting the input speech query's style and generating response accordingly, TTS can provide a more natural and customized user experience. One way to achieve this interaction is to incorporate two key components: a style extraction model that detects the speaking style of the input speech query and generates a style embedding, and a multi-style TTS system that can synthesize styled speech with respect to different style embedding inputs.
The challenge lies in jointly training the style extraction model and the multi-style TTS so that the style embeddings generated by the style extraction model can be genuinely respected by the TTS, even though the two components are trained with different datasets and style labels. In this paper, the TTS dataset is a commissioned dataset recorded with professional voice talents. Only a small part of the TTS dataset has style labels. For the style extraction model, we make use of the external IEMOCAP dataset. These two datasets have different style labels. In order to achieve consistent labels between TTS training data and unseen queries, we incorporated both IEMOCAP dataset and a small portion of the labeled TTS dataset in the style classifier model's training.
We first train a multi-modal style classifier using the IEMOCAP dataset with the model described in \cite{IE2018learning}. Taking the softmax layer of the style classifier as style embedding, the classifier serves as a style embedding extraction model. This model is applied to the unlabelled TTS dataset to generate the style embeddings in a semi-supervised fashion. By using the style embedding as additional auxiliary features for the TTS system, we could train a controllable multi-style TTS system that learns to respect given target styles. During speech synthesis, style embedding is first extracted from the input speech query and then fed into the TTS system to produce response in matching styles. In summary, we developed an interactive multi-style TTS system that could lead to natural, expressive human-machine speech interactions. The multi-style TTS system is evaluated using comprehensive subjective experiments.
\section{Related work}
\subsection{Emotion recognition}
Early approaches on emotion recognition have mostly been inspired by psychology studies \cite{lee2011emotion, mower2010framework}. Recently, deep neural networks (DNNs) have first been used to learn high-level representations for utterance-level emotion recognition \cite{han2014speech}. Trigeorgis \emph{et al.} further applied convolutional neural networks (CNNs) to model context-aware emotion-relevant features, which are then combined with long short-term memory (LSTM) networks aiming towards end-to-end emotion modeling \cite{trigeorgis2016adieu}. Fundamentally, the expression of emotions is usually conveyed through multi-modal behavior channels, including speech, language, body gestures, or facial expressions. Thus, emotion recognition is often formulated as a classification problem of utterances using these multi-modal signals. Reference \cite{Emotionmultimodal} proposed a multi-modal dual recurrent encoder to simultaneously model the dynamics of both text and audio signals within an utterance to predict emotion classes. This architecture has achieved state-of-the-art performance on IEMOCAP\cite{busso2008iemocap} dataset, which is a multi-modal emotion dataset and has been widely used in the affective computing community.
\subsection{Expressive TTS}
One popular topic in the recent research of TTS is expressive TTS. Expressive TTS has been studied for years from the HMM-based synthesis using style modeling with control vector \cite{HMMmodeling,tachibana2004hmm,yamagishi2004speaking} to the state-of-the-art prosody transfer expressive TTS work \cite{latentstylefactor, prosodytransfer, styletoken}, which is aiming at achieving controllable style synthesis in TTS. However, to learn and synthesize specific styles, there are limitations with unsupervised style factorization learning \cite{styletoken}. Since the disentanglement of different styles is heavily influenced by randomness and the choice of hyper-parameters \cite{ICMLbest2018}, the learning of specific target styles is not completely controllable.
Under supervision with explicit prosody labels, the styles could be learned with direct guidance \cite{controlTTSwithlabeldata, Controllableprinciples, rabiee2019adjustingeTTS}. Supervised learning requires a large amount of labeled data, giving difficulties in the development of expressive TTS research and applications. Furthermore, the data labels for styles may not be well overlapped with the needs. An approach to tackle this is proposed in \cite{IE2018learning}. But, the external dataset and the synthesis dataset Blizzard 2017 \cite{king2017blizzard} have differences in background noise, recording environment, speech quality, etc. With the differences between these two datasets, the classifier trained using an external dataset may not be well-adapted to extract representations from the synthesis data. The final emotion synthesis accuracy is 41\% on four emotions \cite{IE2018learning} evaluated by listeners, which may be caused by the domain gap between the TTS dataset and the external dataset.
\section{Datasets}
\subsection{TTS dataset}
\label{sssec:subsubhead}
The TTS dataset was recorded in voice production studio by multiple professional voice talents with $24$kHz sampling rate. It has balanced phonemic and textual information. After labelling to accommodate with the task, $7\%$ of the TTS dataset has utterance level style labels including happy, sad, neutral, angry, rushed, and soft. Details of the data are summarized in Table \ref{tab:data}. These utterances are used, as additional data, to train a multi-speaker style classifier, described in Section \ref{styleModel}. To train the multi-style TTS, we use 40,244 utterances from a single speaker which contains around 3000 style labelled utterances. The style embeddings for unlabelled portion are extracted using the style extraction model, more details in Section \ref{styleModel}.
\subsection{IEMOCAP dataset}
\label{sssec:subsubhead}
To compensate for the limited amount of labeled data in our TTS dataset, we chose IEMOCAP \cite{busso2008iemocap}, which is widely used for emotion recognition, to complement our training data. In this dataset, both video and audio were recorded from ten actors in dyadic sessions under scripted and spontaneous communication scenarios. The dataset contains 12.5 hours of recordings with a sampling rate of 22kHz. Each utterance contains one emotion label, such as neutral, happy, sad, anger, surprise, etc. To be consistent with former research \cite{IE2018learning, Emotionmultimodal} and also be suitable for our own interaction goal, we select the following emotions in our study: neutral, happy, sad, and angry. Similar to the approach in \cite{Emotionmultimodal}, we merge utterances with excited emotion with those of happy emotion.
\begin{table}[t]
\caption{Training data style label statistics}
\label{tab:data}
\vspace{-0.2cm}
\resizebox{\columnwidth}{!}
{
\begin{tabular}{lc | cccccc}
\toprule
Dataset & Split & \multicolumn{1}{c}{Rushed} & \multicolumn{1}{c}{Soft} & \multicolumn{1}{c}{Neutral} & \multicolumn{1}{c}{Happy} & \multicolumn{1}{c}{Angry} & \multicolumn{1}{c}{Sad} \\
\hline
\midrule
TTS Dataset & \begin{tabular}{@{}c@{}@{}c@{}} Train \\ Dev \\ Test \\ All \end{tabular}
& \begin{tabular}{@{}c@{}} 1145 \\105\\124 \\1374 \end{tabular}
& \begin{tabular}{@{}c@{}} 1814 \\161 \\220 \\2195\end{tabular}
& \begin{tabular}{@{}c@{}} 4481 \\ 439 \\506 \\5426 \end{tabular}
& \begin{tabular}{@{}c@{}} 885 \\79 \\93 \\1057 \end{tabular}
& \begin{tabular}{@{}c@{}} 140 \\ 13 \\17 \\170 \end{tabular}
& \begin{tabular}{@{}c@{}} 35 \\ 3 \\2 \\40 \end{tabular}
\\
\hline
\midrule
IEMOCAP & \begin{tabular}{@{}c@{}@{}c@{}} Train \\ Dev \\ Test \\ All \end{tabular}
& \begin{tabular}{@{}c@{}} -- \\-- \\--\\-- \end{tabular}
& \begin{tabular}{@{}c@{}} -- \\--\\--\\-- \end{tabular}
& \begin{tabular}{@{}c@{}} 1390\\ 100 \\218 \\1708 \end{tabular}
& \begin{tabular}{@{}c@{}} 1307 \\90 \\239 \\1636 \end{tabular}
& \begin{tabular}{@{}c@{}} 865 \\ 61 \\177 \\1103 \end{tabular}
& \begin{tabular}{@{}c@{}} 883 \\ 62 \\139 \\1084 \end{tabular} \\% sad
\bottomrule
\end{tabular}
}
\end{table}
\section{Framework and Models}
\label{sec:pagestyle}
\label{sec:format}
\subsection{Semi-supervised style transfer learning}\label{styleModel}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{f2-v8.pdf}
\vspace{-0.8cm}
\caption{Multimodal style extraction model}
\label{fig:Styleframework}
\vspace{-0.3cm}
\end{figure}
For speech style classification, we used the multimodal dual recurrent encoder (MDRE) model adapted from \cite{Emotionmultimodal}. As shown in Figure \ref{fig:Styleframework}, the model is composed of two separate recurrent encoders for audio and text modeling, respectively. The audio model uses 39 dimensions Mel-frequency Cepstral Coefficients (MFCC) features and utterance level prosody feature extracted using openSMILE \cite{openSMILE} as inputs, and the text model uses 300-dimension embeddings to represent each word token. The MFCC, prosody, and text features are the same as described in \cite{Emotionmultimodal}. The audio encoder output is concatenated with the text encoder output, then fed into a fully-connected layer to produce the final classification. We changed the loss function from sigmoid cross-entropy to softmax cross-entropy as it produced significantly better results for our training task. We use the softmax layer output as embedding features, which can be interpreted as a weighted representation of different speaking styles. The softmax feature as embedding is shown in Figure \ref{fig:Styleframework}.
The style classifier is used to generate style embedding from the speech query during inference, as well as to extract style embedding for the TTS training dataset. At first, we trained the style classifier using the IEMOCAP dataset and applied it to generate style features on the TTS dataset. However, the classifier gives inaccurate predictions on the TTS dataset due to domain mismatch between the TTS dataset and the IEMOCAP dataset. Therefore, we labeled a small part of our TTS dataset using one label per utterance as Table \ref{tab:data} and fine-tuned the style classifier using these labels as in Section \ref{ssec:stylec}.
\subsection{Multi-style TTS system}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{f1-v8.pdf}
\caption{Style embedded TTS framework: the style extraction model generates the style embedding based on the user speech query (text + audio), which is used to condition the TTS synthesis.}
\label{fig:framework}
\vspace{-0.7cm}
\end{figure}
Figure \ref{fig:framework} shows the architecture of the expressive TTS system. It consists of a style embedding extraction component that generates the style embedding from speech query and a multi-style TTS, which uses the style embedding to synthesize its response in matching style.
As shown in Figure \ref{fig:framework}, our TTS pipeline is a multi-model framework that consists of a linguistic frontend, a prosody model, an acoustic model, and a conditional neural vocoder. Specifically, the input text is first converted to linguistic features through a text normalization component followed by a joint-sequence grapheme to phoneme model. Then, the linguistic features, along with any conditional features such as style embedding, speaker IDs are used to produce the prosodic features such as duration and F$_{0}$. The prosody model consists of a single layer LSTM model with 256 hidden units with content-based global attention \cite{luong2015effective}, whose context vector contains linguistic features of the entire utterance. It is important to build a separate prosody model in the pipeline because it allows easier control for the speech style during synthesis time. Then, linguistic features combined with prosodic features are used to generate the 13-dim MFCC spectral acoustic features. The acoustic models consist of a two layer uni-directional LSTM with 256 hidden units per layer. At the last stage, a conditional neural vocoder using the WaveRNN \cite{kalchbrenner2018efficient}, takes in the 13-dim MFCC along with the F0 feature to synthesize a 24kHz audio waveform. Our WaveRNN model consists of a single layer gated recurrent unit (GRU) with 1024 hidden units. The speaking style of the synthesized speech is controlled by the conditional style embedding feature, which can be pre-defined or extracted using the style extraction model from the input query, as in Figure \ref{fig:framework}.
\section{Experiments and results}
\label{sec:typestyle}
\begin{table*}[t]
\caption{Style classification on TTS data: AdaBN helps the domain adaption between IEMOCAP dataset and TTS dataset, improving the weighted accuracy of six style classes.}
\label{tab:results}
\vspace{-0.6cm}
\begin{center}
\scalebox{1}{
\begin{tabular}{lccccccccc}
\toprule
\multirow {2}{*}{Dataset} & \multirow{2}{*}{Trick} & \multirow{2}{*}{Neutral} & \multirow{2}{*}{Rushed} & \multirow{2}{*}{Soft} & \multirow{2}{*}{Happy} & \multirow{2}{*}{Angry} & \multirow{2}{*}{Sad} & \multicolumn{2}{c}{{\bf Accuracy}} \\
& & & & & & & & {Weighted} & {Unweighted} \\
\hline
\midrule
Train & \begin{tabular}{@{}c@{}} BN \\ AdaBN \end{tabular}
& \begin{tabular}{@{}c@{}} 0.984 \\ 0.953 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.871 \\ 0.847 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.964 \\ 0.918 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.892 \\ 0.903 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.176 \\ 0.353 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.0 \\ 0.0 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.779 \\ 0.915 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.973 \\ 0.957 \end{tabular}\\
\hline
\midrule
Dev & \begin{tabular}{@{}c@{}} BN \\ AdaBN \end{tabular}
& \begin{tabular}{@{}c@{}} 0.979 \\ 0.927 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.819 \\ 0.8 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.994 \\ 0.963 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.81 \\ 0.873 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.385 \\ 0.538 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.0 \\ 0.333 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.686 \\ 0.766 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.931 \\ 0.904 \end{tabular} \\
\hline
\midrule
Test & \begin{tabular}{@{}c@{}} BN \\ AdaBN \end{tabular}
& \begin{tabular}{@{}c@{}} 0.984 \\ 0.953 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.871 \\ 0.847 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.964 \\ 0.918 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.892 \\ 0.903 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.176 \\ 0.353 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.0 \\ 0.0 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.683 \\ 0.715 \end{tabular}
& \begin{tabular}{@{}c@{}} 0.940 \\ 0.914 \end{tabular}\\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-0.1cm}
\end{table*}
\subsection{Implementation details}
The style classification model is adapted from \cite{Emotionmultimodal} and is shown in Figure \ref{fig:Styleframework}. Specifically, we set the batch normalization layer with 0.9 momentum to help cross-domain adaptation. To compensate for the imbalance among style labels, we weighted the by-class loss function and the per-class accuracy with an inverse of style label prior and capping the neutral label prior to 0.25. Besides, AdaBN \cite{li2018adaptive} is implemented in this model to boost domain adaptation performance between the TTS and multi-style datasets.
The multi-style TTS system is trained using the commissioned TTS dataset with style embedding features as conditional input features. The style embedding labels were generated by passing each utterance through the style classification model, as described in Section \ref{styleModel}. In the synthesis phase, the style embedding features could be automatically extracted from the input query or manually assigned as a combination of different styles.
\subsection{Style classification}
\label{ssec:stylec}
In the style classification task, we first tested the style classifier model performance on the IEMOCAP train/test split. It achieves an overall accuracy of 72.7\%, which is similar to the reported state-of-the-art \cite{Emotionmultimodal}.
To improve the embedding quality on the TTS dataset, the IEMOCAP dataset and the labeled subset of the TTS dataset were combined during training. The results show that the style classifier achieves 91.4\% overall accuracy and 71.5\% weighted accuracy on the TTS labeled dataset.
With a lack of labeled data in anger and sadness in the TTS dataset, the prediction accuracy of these two classes is not high. The style classification accuracy decreased slightly on the IEMOCAP dataset after joint training, likely due to the mismatch between the TTS and IEMOCAP datasets.
We performed normalization on the input features. The normalization is performed corpus-wise to compensate for the domain difference between our TTS dataset and the IEMOCAP dataset. Table \ref{tab:features} shows that normalizing both MFCC and prosody provides the best classification accuracy on the TTS dataset's validation set. So in the final model, we normalized both MFCC features and prosody features. The final classification accuracy for the TTS dataset is in Table \ref{tab:results}.
\begin{table*}[h]
\vspace{-0.2 cm}
\caption{TTS data F0 statistics: the Happy style has higher mean F0 than other styles. And the F0 standard deviations of Angry, Happy and Sad are larger than Neutral, Rushed and Soft styles.}
\vspace{-0.6 cm}
\label{tab:F0}
\begin{center}
\scalebox{1}{
\begin{tabular}{lcccccc}
\toprule
{Style}
& {Angry} & {Happy} & {Sad} & {Neutral} & {Rushed} & {Soft} \\
\hline
\midrule
F0 & \small{195.5$\pm$30.8}
& 214.8$\pm$37.3 & 197.3$\pm$30.8 & 183.7$\pm$10.3 & 181.9$\pm$12.8 & 180.5$\pm$14.7 \\
\bottomrule
\end{tabular}
}
\end{center}
\vspace{-0.3cm}
\end{table*}
\begin{table}[t]
\vspace{-0.3 cm}
\caption{Feature selection: Normalizing MFCC and prosody vector can improve the performance of style classifier.}
\label{tab:features}
\vspace{-0.6cm}
\begin{center}
\scalebox{1}{
\begin{tabular}{lcc}
\toprule
\multirow{2}{*}{Features} & \multicolumn{2}{c}{{\bf Accuracy}} \\
& {Weighted} & {Unweighted} \\
\hline
\midrule
Unnormalized & 0.726
& 0.875 \\
\midrule
Normalized MFCC
& 0.673
& 0.840\\
\midrule
Normalized prosody
& 0.494
& 0.62\\
\midrule
Normalized both
& 0.766
& 0.904 \\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
\subsection{Multi-style TTS with conditional style embedding}
\label{ssec:subhead}
To evaluate our multi-style TTS's performance, we collected subjective evaluation responses from 22 listeners.
As reported in \cite{IE2018learning, banse1996acoustic, scherer1991vocal}, the human perception on the emotions of natural speech is only around 50\%, showing the ambiguity of emotion perception. Hence, instead of evaluating the subjective style accuracy on the multi-style synthesis results, we conducted the ABX test and preference test. Synthesis samples of our system are available at \cite{styleDemo}.
\subsubsection{ABX test}
The ABX test is designed to evaluate whether two styles generated with the same style embedding are perceived to be closer in speaking style when compared to a sample with a different style embedding. Since the style embedding can be used as a probability distribution over the 6 styles, to synthesize audio in a certain style, we construct the style embedding vector to have a value of 0.95 for the selected style and 0.01 for the other five styles. We designed the ABX test as follows. Given two different styles, we randomly choose an example in each style. We denote these two examples as $A$ and $B$. We then randomly choose a different sample $X$ from one of these two styles as reference. We then ask the listener to listen to samples $A$, $B$, and $X$, and then select which of $A$ or $B$ is perceived to be of the same style as the reference $X$.
We created $15$ test sets in total, each of which corresponds to a pair of styles $A$ and $B$, and a reference $X$ of which the linguistic content is emotionally neutral. $22$ listeners participated in the test, which gives a total of $330$ ABX test comparison scores. We achieved an overall accuracy of $82.42\%$ (i.e. total number of matching pairs divided by the total number of ABX tests), indicating that the multi-style TTS is able to generate samples with perceivably distinguishable styles.
\subsubsection{Preference test}
The preference test is designed to compare TTS responses generated by a default TTS without multi-style capability and the multi-style TTS when the style embedding is explicitly provided. Specifically, we ask the listeners to choose between TTS responses synthesized with the same text but different models: baseline TTS model (i.e., TTS without style embedding) or the multi-style TTS model. For the multi-style TTS responses, we provide the utterance with style of either the neutral style or, when appropriate, a hand-crafted style embedding (i.e., other style) based on the style of the text, assigned as a soft probability label whose style weights are determined based on the content of the utterance.
Results in Table \ref{tab:subjective} show that the multi-style TTS is preferred over the baseline TTS 72\% of the time, indicating strong user-preference when an appropriately styled TTS response is provided. It is interesting to note that the neutral style from the multi-style TTS is preferred by the listeners most of the time. This is largely due to the content of the test utterances, which is best spoken with a peaceful and relaxing neutral style. This result is consistent with the findings in \cite{IE2018learning}, which states that listeners prefer appropriate variation over random variation.
\begin{table}[!t]
\vspace{-0.4 cm}
\caption{Subjective Preferences: the proposed TTS model's results are preferred over the baseline TTS model's.}
\label{tab:subjective}
\vspace{-0.6cm}
\begin{center}
\scalebox{1}{
\resizebox{\columnwidth}{!}{
\begin{tabular}{lccc}
\toprule
& \multirow{2}{*}{{\bf Baseline TTS}} & \multicolumn{2}{c}{{\bf Multi-style TTS}} \\
& & {Neutral Style} & {Other Styles} \\
\hline
\midrule
Preference (\%) & 28.0
& 54.2
& 17.8\\
\bottomrule
\end{tabular}
}
}
\end{center}
\vspace{-0.6cm}
\end{table}
\subsubsection{Mimicking real life input query with styled TTS response}
\label{realexp}
We conducted experiments to evaluate the generalization capacity of the close-loop style extraction and multi-style TTS system. We recorded speech queries from multiple speakers who have never been seen in the training of our framework. These speakers read the queries freely in a quiet conference room. We then generated TTS responses for each query by conditioning on its style embedding. Our results show that over 40\% of test pairs are evaluated as good matches by listeners. We noticed that when the speaking style of the input query is strong, the TTS response can match the input style to a certain extent (samples are at \cite{styleDemo}). This can potentially be improved with more coherent style labels between the style extraction model training data and the TTS dataset.
\section{Discussions}
\label{sec:print}
In our proposed system, the soft (probability) style embedding is a weighted representation of different styles such that increasing the weight of a certain style emphasizes that style's effect on the synthesis outputs, shown in \cite{styleDemo}. This demonstrates the multi-style TTS's capability of synthesizing styled-speech with respect to the soft style embedding.
We noticed utterance-mean F0 differs for different styles in the synthesis results, representing the style difference. For example, the inference result of the Happy style has a significantly higher F0 mean than the other styles. This is consistent with the statistics of F0 for different predicted classes in TTS training data, as shown in Table \ref{tab:F0}.
We also noticed that the the Happy style of multi-style TTS has a significantly higher F0 mean than the other styles. This could be due to the reason that the model focused on the most distinguishable feature, such as F0 mean and failed to learn the nuances of the F0 contour. To mitigate this problem, the F0 mean and the F0 contour can be modeled separately.
In addition, the sad and angry styled audio quality was comparatively worse than other styles, which could be due to the lack of anger and sadness samples in the TTS dataset. In the future, the performance of the multi-style TTS system can be further improved with a training dataset that contains more balanced style labels.
\section{Conclusions}
\label{sec:page}
In conclusion, we attempted to develop a style-embedded TTS that is more contextual and interactive. As shown in Section \ref{ssec:subhead}, with perfect style embedding, the system generated preferred TTS responses compared to a single style TTS. With automatically extracted style embeddings from real speech queries, the system demonstrated moderate capability in mimicking the speaking style of the input speech query. The overall quality can be improved with a more balanced multi-style TTS dataset and more coherent style labels between the style extraction model training data and the TTS dataset.
\bibliographystyle{IEEEtran}
|
1,314,259,993,007 | arxiv | \section{Introduction}
In this paper we continue the work that was started in \cite{FKL}.
Our focus is on easy special cases of otherwise difficult to evaluate
polynomials, and their relation to various classes of arithmetic circuits.
It is conjectured that the permanent and hamiltonian polynomials are hard
to evaluate. Indeed, in Valiant's model \cite{Val79,Val82} these families of polynomials
are both ${\rm VNP}$-complete.
In the boolean framework they are complete for the complexity class
${\rm \sharp P}$ \cite{Val79a}. However, for matrices of bounded treewidth the
permanent and hamiltonian polynomials can efficiently be evaluated - the
number of arithmetic operations being polynomial in the size of the
matrix \cite{CMR}.
An earlier result along these lines is related to computing weights
of perfect matchings in a graph:
The sum of weights of all perfect matchings in a weighted (undirected) graph
is another hard to evaluate polynomial, but for planar graphs it can be
evaluated efficiently due to Kasteleyn's theorem \cite{Ka}.
By means of reductions these evaluation methods
can all be seen as general-purpose
evaluation algorithms for certain classes of polynomials.
As an example, if an arithmetic formula represents a polynomial $P$ then
one can construct a matrix $A$ of bounded treewidth such that:
\begin{itemize}
\item[(i)] The entries of $A$ are variables of $P$, or constants from
the underlying field.
\item[(ii)] The permanent of $A$ is equal to $P$.
\end{itemize}
It turns out that the converse holds as well, so with respect
to the computational complexity computing the permanent
of a bounded treewidth matrix is equivalent to evaluating an arithmetic
formula.
In \cite{FKL} the following results (with abuse of notation)
were established:
\begin{itemize}
\item[(i)] permanent/hamiltonian(bounded treewidth matrix) $\equiv$ arithmetic formulas.
\item[(ii)] perfect matchings(planar matrix) $\equiv$ arithmetic skew circuits.
\end{itemize}
One can also by similar techniques show that:
\begin{itemize}
\item[(iii)] perfect matchings(bounded treewidth matrix) $\equiv$ arithmetic formulas.
\end{itemize}
Other notions of graph ``width'' have been defined in the litterature
besides treewidth, e.g. pathwidth, cliquewidth and rankwidth.
Here we would like to study the evaluation methods mentioned above,
but considering matrices $A$ that
have bounded pathwidth or bounded cliquewidth instead of bounded treewidth.
In this paper we establish the following results:
\begin{itemize}
\item[(i)] per/ham/perf. match.(bounded pathwidth matrix) $\equiv$ arithmetic skew
circuits of bounded width $\equiv$ arithmetic weakly skew circuits of bounded width $\equiv$
arithmetic formulas.
\item[(ii)] arithmetic formulas $\subseteq$ per/ham/perfect matchings(bounded
cliquewidth matrix)
$\subseteq$ ${\rm VP}$.
\end{itemize}
\emph{Overview of the paper.} The second section of the paper introduces definitions used throughout the paper
and provides some small technical results related to graph widths.
In particular we
show equivalence between the weighted definitions of cliquewidth, NLC-width and
m-cliquewidth with respect to boundedness.
Sections 3 and 4 are devoted to the
expressiveness of the permanent, hamiltonian, and perfect matchings of the
graphs of bounded pathwidth and bounded weighted cliquewidth respectively.
We prove in Section 3 that permanent, hamiltonian, and perfect matchings limited
to bounded pathwidth graphs express arithmetic formulas. In Section 4, we show
that for all three polynomials the complexity is between arithmetic formulas and
${\rm VP}$ for graphs of bounded weighted cliquewidth.
\section{Definitions and preliminary results}
\subsection{Arithmetic circuits}
\begin{definition}
An {\em arithmetic circuit} is a finite, acyclic, directed graph. Vertices have
indegree 0 or 2, where those with indegree 0 are referred to as {\em inputs}.
A single vertex must have outdegree 0, and is referred to as {\em output}.
Each vertex of indegree 2 must be labeled by either $+$ or $\times$, thus
representing computation. Vertices are commonly referred to as {\em gates}
and edges as {\em arrows}.
\end{definition}
By interpreting the input gates either as constants or variables it is
easy to prove by induction that each arithmetic circuit naturally
represents a polynomial.
In this paper various subclasses of arithmetic circuits will be considered:
For {\em weakly skew} circuits we have the restriction that for every
multiplication gate, at least one of the incoming arrows is from a subcircuit
whose only connection to the rest of the circuit is through this incoming
arrow.
For {\em skew} circuits we have the restriction that for every
multiplication gate, at least one of the incoming arrows is from an
input gate.
For {\em formulas} all gates
(except output) have outdegree 1. Thus, reuse of partial
results is not allowed.
For a detailed description of various subclasses of arithmetic circuits,
along with examples, we refer to \cite{MP}.
\begin{definition}
The {\em size} of a circuit is the total number of {\em gates} in the circuit.
The {\em depth} of a circuit is the length of the longest path from an
input gate to the output gate.
\end{definition}
\subsection{Pathwidth and treewidth}\label{sec:pathwidthandtreewidth}
Since the definition of pathwidth is closely related to the definition
of treewidth (bounded pathwidth is a special case of bounded treewidth)
we also include the definition of treewidth in this paper.
Treewidth for undirected graphs is commonly defined as follows:
\begin{definition}
Let $G = \langle V,E \rangle$ be a graph. A $k$-tree-decomposition of $G$ is:
\begin{itemize}
\item[(i)] A tree $T = \langle V_T, E_T \rangle$.
\item[(ii)] For each $t \in V_T$ a subset $X_t \subseteq V$
of size at most $k + 1$.
\item[(iii)] For each edge $(u,v) \in E$ there is a $t \in V_T$ such that
$\lbrace u,v \rbrace \subseteq X_t$.
\item[(iv)] For each vertex $v \in V$ the set $\lbrace t \in V_T |
v \in X_t \rbrace$ forms a (connected) subtree of $T$.
\end{itemize}
The treewidth of $G$ is then the smallest $k$ such that there exists a
$k$-tree-decomposition for $G$.\\
A $k$-{\em path}-decomposition of $G$ is then a $k$-tree-decomposition where the
``tree'' $T$ is a path (each vertex $t \in V_T$ has at most one child in $T$).
\end{definition}
\begin{example}
Here we show that cycles
have pathwidth at most 2 by constructing a path-de\-com\-po\-si\-ti\-on of $G$
where each $X_t$ has size at most 3. Let $v_1, v_2, \ldots ,v_n$ be the
vertices of a graph $G$ which is a cycle.
The edges of $G$ are $(v_1,v_2),(v_2,v_3),
\ldots,(v_{n-1},v_n),(v_n,v_1)$. The vertex $v_1$ is contained in every
$X_t$ of the path-decomposition. Vertices $v_2$ and $v_3$ are contained
in $X_1$, vertices $v_3$ and $v_4$ are contained in $X_2$, and so on. Finally,
vertices $v_{n-1}$ and $v_n$ are contained in $X_{n-2}$.
This gives a path-decomposition of $G$ of width 2.
\end{example}
The pathwidth (treewidth) of a directed, weighted graph is naturally
defined as the
pathwidth (tree\-width) of the underlying, undirected, unweighted graph.
The pathwidth (treewidth) of an $(n \times n)$
matrix $M = (m_{i,j})$ is defined as
the pathwidth (treewidth) of the directed graph
$G_M = \langle V_M,E_M,w \rangle$ where
$V_M = \lbrace 1, \ldots , n \rbrace$, $(i,j) \in E_M$ iff
$m_{i,j} \neq 0$, and $w(i,j) = m_{i,j}$.
Notice that $G_M$ can have loops. Loops affect neither the pathwidth nor
the treewidth of $G_M$ but are important for the characterization of the
permanent polynomial.
\subsection{Cliquewidth, NLCwidth and m-cliquewidth}
Although there exists many algorithmic results for graphs of bounded
treewidth, there are still classes of ``trivial'' graphs that have
unbounded treewidth. Cliques are an example of such graphs.
Cliquewidth is a different notion of ``width'' for graphs, and it is more
general than treewidth since graphs of bounded treewidth have bounded
cliquewidth, but cliques have bounded cliquewidth and unbounded treewidth.
We recall the definitions of cliquewidth, NLCwidth and m-cliquewidth for
unweighted, undirected graphs.
Then we introduce the new notions of $W$-cliquewidth, $W$-NLCwidth and $W$-m-cliquewidth
which are variants of the preceding ones for {\em weighted, directed} graphs.
These graph widths are all defined using terms over an universal algebra. When
we refer to parse-trees it means the parse-trees of these terms.
\begin{definition}[\cite{CER,CO}]
A graph $G$ has cliquewidth (denoted $cwd(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named clique operations):
\begin{itemize}
\item[(i)] $ver_a$, $a \in {\mathcal S}$
(basic construct: create a single vertex with label $a$).
\item[(ii)] $\rho_{a \rightarrow b} (H)$, $a,b \in {\mathcal S}$
(rename all vertices with label $a$ to have label $b$ instead).
\item[(iii)] $\eta_{a,b} (H)$, $a,b \in {\mathcal S}$, $a\neq b$
(add edges between all couples of vertices where one of them has label $a$
and the other has label $b$).
\item[(iv)] $H \; \oplus \; H'$ (disjoint union of graphs).
\end{itemize}
\end{definition}
\begin{example}
Using the clique algebra, the clique with four vertices $K_4$ is constructed by
the following term using only two source labels; $S = \{a,b\}$:
$$\eta_{a,b} ((\rho_{a \rightarrow b} (\eta_{a,b} ((\rho_{a \rightarrow b} (\eta_{a,b}
(ver_a \; \oplus \; ver_b))) \; \oplus \; ver_a)))\; \oplus \; ver_a).$$
\end{example}
\begin{definition}[\cite{Wa}]
A graph $G$ has NLCwidth (denoted $w_{NLC}(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named NLC operations):
\begin{itemize}
\item[(i)] $ver_a$, $a \in {\mathcal S}$
(basic construct: create a single vertex with label $a$).
\item[(ii)] $\circ_R (H)$ for any mapping $R$ from ${\mathcal S}$ to
${\mathcal S}$ (for every source label $a \in {\mathcal S}$
rename all vertices with label $a$ to have label $R(a)$ instead).
\item[(iii)] $H \;\times_S \; H'$ for any $S\subseteq {\mathcal S}^2$
(disjoint union of graphs to which are added edges between
all couples of vertices $x \in H$ (with label $l_x$),
$y \in H'$ (with label $l_y$) having $(l_x,l_y) \in S$).
\end{itemize}
\end{definition}
One important distinction
between cliquewidth and NLCwidth on one side and m-cliquewidth
(to be defined below) on the other side
is that in the first two each vertex is assigned exactly {\em one} label,
and in the
last one each vertex is assigned a \emph{set} of labels (possibly empty).
\begin{definition}[\cite{CT07}]
A graph $G$ has m-cliquewidth (denoted $mcwd(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named m-clique operations):
\begin{itemize}
\item[(i)] $ver_A$ (basic construct: create a single
vertex with a set of labels $A$, $A\subseteq {\mathcal S}$).
\item[(ii)] $H \;\otimes_{S,h,h'} \; H'$ for any $S\subseteq {\mathcal S}^2$
and any $h,h' : {\mathcal P}({\mathcal S}) \rightarrow
{\mathcal P}({\mathcal S})$
(disjoint union of graphs to which is added edges between
all couples of vertices $x \in H$, $y \in H'$ whose sets of labels $L_x,L_y$
contain a
couple of labels $l_x,l_y$ such that $(l_x,l_y) \in S$.
Then the labels of vertices from $H$ are changed via $h$ and the labels
of vertices from $H'$ are changed via $h'$).
\end{itemize}
\end{definition}
It is stated in \cite{CT07} (a proof sketch of this result is given
in~\cite{CT07}, one of the inequalities is proven in \cite{Jo98}) that
$$mcwd(G)\leq wd_{NLC}(G) \leq cwd(G) \leq 2^{mcwd(G)+1}-1.$$
Hence, cliquewidth, NLC-width and m-cliquewidth
are equivalent with respect to boundedness.
\medskip
We have seen that the definition of pathwidth and treewidth for weighted
graphs straight forward was defined as the width of the underlying,
unweighted graph.
This is a major difference
compared to cliquewidth.
We can see
that if we consider non-edges as edges of weight 0, then every weighted graph
has a clique (which has bounded cliquewidth 2) as its underlying,
unweighted graph.
Our main motivation for studying bounded cliquewidth matrices is to obtain
efficient algorithms for evaluating polynomials like the permanent and
hamiltonian for such matrices.
For this reason, it is not reasonable
to define the cliquewidth of a weighted graph as the cliquewidth of the
underlying, unweighted graph, because then computing the permanent of a
matrix of cliquewidth 2 is as difficult as the general case.
Hence, we put restrictions on how weights are assigned to edges:
Edges added in the same operation between vertices having the same pair
of labels, will all have the same weight.
We now introduce the definitions of $W$-cliquewidth, $W$-NLCwidth and
$W$-m-cliquewidth.
We will consider simple, weighted, directed graphs where the weights are
in some set $W$.
In the three following constructions, an
arc from a vertex $x$ to a vertex $y$
is only added by
relevant operations if there is not already an arc from $x$ to $y$.
The operations that differ from the unweighted case are
indicated by \textbf{bold} font.
\begin{definition}
A graph $G$ has $W$-cliquewidth (denoted $Wcwd(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named $W$-clique operations):
\begin{itemize}
\item[(i)] $ver_a$, $a \in {\mathcal S}$
(basic construct: create a single vertex with label $a$).
\item[(ii)] $\rho_{a \rightarrow b} (H)$, $a,b \in {\mathcal S}$
(rename all vertices with label $a$ to have label $b$ instead).
\item[\textbf{(iii)}]
$\alpha_{a,b}^w (H)$, $a,b \in {\mathcal S}$, $a\neq b$, $w \in W$
(add missing arcs of weight $w$ from all vertices with label $a$
to all vertices with label $b$).
\item[(iv)] $H \; \oplus \; H'$ (disjoint union of graphs).
\end{itemize}
\end{definition}
\begin{definition}
A graph $G$ has $W$-NLCwidth (denoted $Wwd_{NLC}(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named $W$-NLC operations):
\begin{itemize}
\item[(i)] $ver_a$, $a \in {\mathcal S}$
(basic construct: create a single vertex with label $a$).
\item[(ii)] $\circ_R (H)$ for any mapping $R$ from ${\mathcal S}$ to
${\mathcal S}$
(for every source label $a \in {\mathcal S}$ rename all vertices with label
$a$ to have label $R(a)$ instead).
\item[\textbf{(iii)}] $H \;\times_S \; H'$ for any partial function
$S : {\mathcal S}^2 \times \{-1,1\} \rightarrow W $
(disjoint union of graphs to which are added arcs of weight $w$
for each couple of vertices $x\in H$, $y\in H'$ whose labels
$l_x,l_y$ are such that $S(l_x,l_y,s)=w$;
the arc is from $x$ to $y$ if $s=1$ and from $y$ to $x$ if $s=-1$).
\end{itemize}
\end{definition}
\begin{definition}
A graph $G$ has $W$-m-cliquewidth (denoted $Wmcwd(G)$) at most $k$ iff there exists a set of source
labels ${\mathcal S}$ of cardinality $k$ such that $G$ can be constructed using
a finite number of the following operations (named $W$-m-clique operations):
\begin{itemize}
\item[(i)] $ver_A$ (basic construct: create a single
vertex with set of labels $A$, $A\subseteq {\mathcal S}$).
\item[\textbf{(ii)}] $H \;\otimes_{S,h,h'} \; H'$ for any partial function $S : {\mathcal S}^2 \times \{-1,1\} \rightarrow W $
and any
$h,h' : {\mathcal P}({\mathcal S}) \rightarrow {\mathcal P}({\mathcal S})$
(disjoint union of graphs to which is added missing arcs of weight $w$
for each couple of vertices $x\in H$, $y\in H'$
whose sets of labels $L_x,L_y$ contain
$l_x,l_y$ such that $S(l_x,l_y,s)=w$;
the arc is from $x$ to $y$ if $s=1$ and from
$y$ to $x$ if $s=-1$. Then the labels of
vertices from $H$ are changed via $h$ and the labels of vertices
from $H'$ are changed via $h'$).
\end{itemize}
\end{definition}
In the last operation for $W$-m-cliquewidth,
there is a possibility that two (or more) arcs are added from a
vertex $x$ to a vertex $y$
during the same operation
and then the obtained graph
is not simple.
For this reason, we will consider
as well-formed terms only the terms (or parse-trees) where this does not occur.
The three preceding constructions of graphs can be extended to weighted
graphs with loops by adding the
basic constructs $verloop_a^w$ or $verloop_A^w$ which creates a single
vertex with a loop of weight $w$ and label $a$ or set of labels $A$.
If $G$ is a weighted graph (directed or not) with loops
and $Unloop(G)$ denotes the weighted graph (directed or not) obtained from $G$
by removing all loops, then one can easily show the following result.
\begin{itemize}
\item $Wcwd(G) = Wcwd(Unloop(G))$.
\item $Wwd_{NLC}(G) = Wwd_{NLC}(Unloop(G))$.
\item $Wmcwd(G) = Wmcwd(Unloop(G))$.
\end{itemize}
This justifies the fact that we overlook technical details for
loops in the proof of the following theorem. Theorem~\ref{weightedEquiv}
shows that the inequalities between the three widths are
still valid in the weighted case.
It justifies our definitions of cliquewidth for weighted graphs.
For the proof we collect
the ideas in~\cite{CT07,Jo98} and combine them with our definitions
for weighted graphs.
\begin{theorem}\label{weightedEquiv}
For any weighted graph $G$, $$Wmcwd(G)\leq Wwd_{NLC}(G) \leq Wcwd(G) \leq 2^{Wmcwd(G)+1}-1.$$
\end{theorem}
\proof
First inequality:
Let $G$ be a weighted graph of $W$-NLCwidth at most $k$ and $T$ be a parse-tree constructing $G$
with $W$-NLC operations on a set of source labels ${\mathcal S}$ of cardinality $k$.
We can consider without loss of generality that in $T$:
\begin{itemize}
\item[-] there are no two consecutive $\circ_R (H)$
operations, otherwise we can replace $T$ by $T'$ where the two consecutive nodes of $T$
with $\circ_R (H)$ and $\circ_{R'} (H)$ operations on them have been replaced by one node $\circ_{R''} (H)$ ($R''=R'\circ R$).
\item[-] no $ver_a$ operation is followed by a $\circ_R (H)$ operation, otherwise we can replace $T$ by $T'$ where
this two operations are replaced by $ver_b$ where $b=R(a)$.
\item[-] each $H \;\times_S \; H'$ operation is followed by exactly one $\circ_R (H)$ operation, otherwise
we can add an $\circ_{Id} (H)$ operation if there is none ($Id$ is the identity function from ${\mathcal S}$
to ${\mathcal S}$).
\end{itemize}
We can replace the $W$-NLC operation $ver_a$ by the $W$-m-clique operation $ver_{\{a\}}$,
and the consecutives $W$-NLC operation $H \;\times_S \; H'$ and $\circ_R (H)$ by the $W$-m-clique operation
$H \;\otimes_{S,h,h} \; H'$ where $h(\{a\})=\{R(a)\}, \forall a \in {\mathcal S}$.
It is clear that these replacements in $T$ will give a parse-tree constructing $G$
with $W$-m-clique operations on the same set of source labels ${\mathcal S}$ of cardinality $k$.
Hence, we have $Wmcwd(G)\leq Wwd_{NLC}(G)$.
Second inequality:
Let $G$ be a weighted graph of $W$-cliquewidth at most $k$ and $T$ be a parse-tree constructing $G$
with $W$-clique operations on a set of source labels ${\mathcal S}$ of cardinality $k$.
We can consider without loss of generality that in $T$:
\begin{itemize}
\item[-] after a disjoint union operation $H \; \oplus \; H'$
all arcs in $G$ from $x \in H$ to $y\in H'$ (resp. from $y$ to $x$) are added
between the disjoint union operation $H \; \oplus \; H'$ and the
first following operation $O$ of disjoint union or renaming. Otherwise consider the first operation
$\alpha_{a,b}^w (H)$ after $O$ adding an arc between a vertex $x'$ from $H$ and a vertex $y'$ from $H'$. We can add an
operation $\alpha_{a',b'}^w (H)$ before $O$ where $a'$(resp. $b'$) is the label in $H \; \oplus \; H'$
of the tail (resp. head) of the arc added by the operation $\alpha_{a,b}^w (H)$.
\item[-] each operation $\alpha_{a,b}^w (H)$ add at least one arc.
\item[-] all $\alpha_{a,b}^w (H)$ operations are between a disjoint union operation $H \; \oplus \; H'$ and the
first following operation $O$ of disjoint union or renaming.
\end{itemize}
We can replace the $W$-clique operation $ver_a$ by the $W$-NLC operation $ver_a$,
and the $W$-clique operation $\rho_{a \rightarrow b} (H)$ by the $W$-NLC operation $\circ_R (H)$
where $R(a)=b$ and $R(c)=c, \forall c \in {\mathcal S}, c\neq a$.
Finally each group consisting of a
$H \;\oplus \; H'$ $W$-clique operation and the following $\alpha_{a,b}^w (H)$ $W$-clique operations
can be replaced by the $W$-NLC operation
$G \;\times_S \; G'$ where $S(a,b,1)=S(a,b,-1)=w$ if there is an $\alpha_{a,b}^w (H)$ operation
in the group.
It is clear that these replacements in $T$ will give a parse-tree constructing $G$
with $W$-NLC operations on the same set of source labels ${\mathcal S}$ of cardinality $k$.
Hence, we have $Wwd_{NLC}(G) \leq Wcwd(G)$.
Last inequality:
Let $G$ be a weighted graph of $W$-m-cliquewidth at most $k$ and $T$ be a parse-tree constructing $G$
with $W$-m-clique operations on a set of source labels ${\mathcal S}$ of cardinality $k$.
Let ${\mathcal S}'$ be a set of source labels of cardinality $2^{k+1}-1$,
${\mathcal S}' ={\mathcal S}_l\sqcup {\mathcal S}_r \sqcup \{empty\} $ where $|{\mathcal S}_l|=|{\mathcal S}_r|=2^k -1$.
We define three bijections
$l : {\mathcal P}({\mathcal S}) \backslash \emptyset \rightarrow {\mathcal S}_l$,
$r : {\mathcal P}({\mathcal S}) \backslash \emptyset \rightarrow {\mathcal S}_r$,
and $u : {\mathcal S}_l\rightarrow {\mathcal S}_r$ such that $u(l(A))=r(A), \forall A \in {\mathcal P}({\mathcal S})$.
We will denote by $\rho_f$ a sequence of $\rho_{a\rightarrow b}$ $W$-clique operations
realizing a function $f$ from ${\mathcal S}'$ to ${\mathcal S}'$.
We associate to each function $S : {\mathcal S}^2 \times \{-1,1\} \rightarrow W $
a sequence $\alpha_S$ consisting of $\alpha_{l(A),r(B)}^w$ (resp. $\alpha_{r(B),l(A)}^w$) $W$-clique operations for all couples
$(a,b) \in {\mathcal S}^2, (A,B) \in ({\mathcal P}({\mathcal S}) \backslash \emptyset)^2$ such that $S(a,b,1)=w$ (resp. $S(a,b,-1)=w$),
$a \in A$ and $b\in B$.
We can replace the $W$-m-clique operation $ver_A$ by the $W$-clique operation $ver_{l(A)}$ if $A\neq \emptyset$ and
$ver_{empty}$ otherwise.
Each $W$-m-clique operation $H \otimes_{S,h,h'} H'$ will be replaced by the following
$W$-clique operations:
\begin{itemize}
\item[-] apply $\rho_{u}$ to the subtree constructing $H'$.
\item[-] make a $H \;\oplus \; H'$ $W$-clique operation.
\item[-] apply $\alpha_S$.
\item[-] apply $\rho_{l\circ h \circ l^{-1}}$.
\item[-] apply $\rho_{l \circ h' \circ r^{-1}}$.
\end{itemize}
It is clear that these replacements in $T$ will give a parse-tree constructing $G$
with $W$-clique operations on the set of source labels ${\mathcal S}'$ of cardinality $2^{k+1}-1$.
Hence, we have $Wcwd(G) \leq 2^{Wmcwd(G)+1}-1$.
\qed
\subsection{Permanent and hamiltonian polynomials}
In this paper we take a graph theoretic approach to deal with permanent
and hamiltonian polynomials. The reason for this is that a natural way
to define pathwidth, treewidth or cliquewidth of a matrix $M$
is by the width of the graph $G_M$
(see Section \ref{sec:pathwidthandtreewidth}), also see e.g.~\cite{MM}.
\begin{definition}
A {\em cycle cover}
of a directed graph is a subset of the edges, such that
these edges form disjoint, directed cycles (loops are allowed).
Furthermore, each vertex in the
graph must be in one
(and only one)
of these cycles. The {\em weight} of a cycle cover
is the product of weights of all participating edges.
\end{definition}
\begin{definition} \label{permdef}
The {\em permanent} of an $(n \times n)$ matrix $M = (m_{i,j})$ is the
sum of weights of all cycle covers of $G_M$.
\end{definition}
The permanent of $M$ can also be defined by the formula
$${\rm per}(M)=\sum_{\sigma \in S_n} \prod_{i=1}^n m_{i,\sigma(i)}.$$
The equivalence with Definition~\ref{permdef} is clear since any permutation
can be written down as a product of disjoint cycles,
and this decomposition is unique.
The {\em hamiltonian} polynomial
${\rm ham}(M)$ is defined similarly,
except that we only sum over
cycle covers
consisting of a {\em single} cycle (hence the name).
There is a natural way of representing polynomials by permanents.
Indeed, if the entries of $M$ are variables or constants from some field $K$,
then $f={\rm per}(M)$ is a polynomial with coefficients in $K$
(in Valiant's terminology,
$f$ is a projection of the permanent polynomial).
In the next sections we study the power of this representation in the case where
$M$ has bounded pathwidth or bounded cliquewidth.
\subsection{Connections between permanents and
sum of weights of perfect matchings}
Another combinatorial characterization of the permanent is by sum of
weights of perfect matchings in a bipartite graph. We will use this connection
to deduce results for the
permanent from results for the sum of weights of perfect matchings and vice versa.
\begin{definition}
Let $G$ be a directed graph (weighted or not). We define the
{\em inside-outside graph} of $G$,
denoted $IO(G)$, as the bipartite, undirected graph (weighted or
not) obtained as follows:
\begin{itemize}
\item split each vertex $u \in V(G)$ in two vertices $u^+$ and $u^-$;
\item each arc $uv$ (of weight $w$) is replaced by an edge between $u^+$
and $v^-$ (of weight $w$). A loop on $u$ (of weight $w$) is replaced by an
edge between $u^+$ and $u^-$ (of weight $w$).
\end{itemize}
\end{definition}
It is well-known that the permanent of a matrix $M$ can be defined as
the sum of weights of all perfect matchings of $IO(G_M)$.
We can see that the adjacency matrix of $IO(G_M)$ is
$\left(\begin{matrix}
0 & M \\
M^t & 0
\end{matrix} \right)
$.
\begin{lemma}\label{twIO}
If $G$ has treewidth (pathwidth) $k$, then $IO(G)$ has treewidth
(pathwidth) at most $2 \cdot k+1$.
\end{lemma}
\proof
Let $\langle T,(X_t)_{t\in V(T)} \rangle$ be a $k$-tree(path)-decomposition of $G$.
It is clear that $\langle T,(X'_t)_{t\in V(T)} \rangle$, where
$X'_t=\{u^+, u^- | u \in X_t\}$, is a tree(path)-decomposition
of $IO(G)$ of width $2 \cdot k+1$.
\qed
\begin{lemma}\label{cwIO}
If $G$ has $W$-cliquewidth $k$, then $IO(G)$ has $W$-cliquewidth at
most $2 \cdot k$.
\end{lemma}
\proof
Let $T$ be a parse-tree constructing $G$ with $W$-clique operations on a set of source labels ${\mathcal S}$ of cardinality $k$.
We can replace the $W$-clique operation $ver_a$ by the three operations $(ver_{a^+}) \;\oplus \; (ver_{a^-})$,
and the $W$-clique operation $\rho_{a \rightarrow b} (H)$ by the $W$-clique operations $\rho_{a^+ \rightarrow b^+} (H)$ and
$\rho_{a^- \rightarrow b^-} (H)$.
Finally each $\alpha_{a,b}^w (H)$ $W$-clique operation
can be replaced by the $\eta_{a^+,b^-}^w (H)$ $W$-clique operation.
It is clear that these replacements in $T$ will give a parse-tree constructing $IO(G)$
with $W$-clique operations on the set of source labels $\{a^+, a^- | a \in
{\mathcal S}\}$ of size $2 \cdot k$.
\qed
\section{Expressiveness of matrices of bounded pathwidth}
In this section we study the expressive power of permanents, hamiltonians
and perfect matchings of matrices of bounded pathwidth.
We will prove that in each case we capture exactly the families
of polynomials computed by polynomial size skew circuits of bounded width.
A by-product of these proofs will be a proof of the equivalence between
polynomial size skew circuits of bounded width and polynomial size \emph{weakly}
skew circuits of bounded width. This equivalence can not be immediately
deduced from the already known equivalence between polynomial size skew
circuits and polynomial size weakly skew circuits in the unbounded width
case~\cite{To} (the proofs in~\cite{To} use a combinatorial characterization of
the complexity of the determinant as the sum of weights of $s,t$-paths in a
graph of polynomial size with distinguished vertices $s$ and $t$. The additional
difficulties to extend these proofs to circuits and graphs of bounded width
would be equivalent to the ones we deal with).
We will then prove that skew circuits of bounded width
are equivalent to arithmetic formulas.
\begin{definition}
An arithmetic circuit $\varphi$ has {\em bounded width} $k \geq 1$
if there exists a finite
set of totally ordered layers such that:
\begin{itemize}
\item[-] Each gate of $\varphi$ is contained in exactly 1 layer.
\item[-] Each layer contains at most $k$ gates.
\item[-] For every non-input gate of $\varphi$ if that gate is in some layer
$n$, then both inputs to it are in layer $n+1$.
\end{itemize}
\end{definition}
\begin{theorem}\label{bwcircuitToPerm}
The polynomial computed by a weakly skew circuit of bounded width can be expressed as the
permanent of a matrix of bounded pathwidth.
The size of the matrix is polynomial in the size of the circuit.
All entries in the matrix are either 0, 1, constants of the polynomial,
or variables of the polynomial.
\end{theorem}
\proof
Let $\varphi$ be a weakly skew circuit of bounded width $k \geq 1$
and $l>1$ the number of layers in $\varphi$.
The directed graph $G$ we construct will have pathwidth at most $\left\lfloor
\frac{7\cdot k}{2}\right\rfloor -1$
(each bag in the path-de\-com\-po\-si\-ti\-on will contain at most
$\left\lfloor \frac{7\cdot k}{2}\right\rfloor$ vertices) and the number of bags in the
path-decomposition will be $l-1$.
$G$ will have two distinguished vertices $s$ and $t$, and the sum
of weights of all directed paths from $s$ to $t$ equals the value computed
by $\varphi$.
The vertex $s$ will be in all bags of the path-decomposition of $G$.
Since $\varphi$ is a weakly skew circuit we consider a decomposition of it
into disjoint subcircuits defined recursively as follows:
The output gate of $\varphi$ belongs to the {\em main subcircuit}. If a gate in the
main subcircuit is an addition gate,
then both of its input gates are in the
main subcircuit as well. If a gate $g$ in the main subcircuit is a
multiplication gate, then we know that at least one input to $g$ is
the output gate of a subcircuit which is disjoint from $\varphi$ except for its
connection to $g$. This subcircuit forms a {\em disjoint multiplication-input
subcircuit}. The other input to $g$ belongs to the main subcircuit.
If some disjoint multiplication-input subcircuit $\varphi'$
contains at least one multiplication gate, then we make a decomposition of
$\varphi'$ recursively.
Note that such a decomposition of a weakly skew circuit not necessarily is
unique (nor does it need to be), because {\em both} inputs to a
multiplication gate can be disjoint
from the rest of the circuit, and then any one of these two can be chosen as the
one that belongs to the main subcircuit.
Let $\varphi_0, \varphi_1, \dots, \varphi_d$ be the disjoint subcircuits obtained
in the decomposition ($\varphi_0$ is the main subcircuit). The graph $G$ will
have a vertex $v_g$ for every gate $g$ of $\varphi$ and $d+1$ additional
vertices $s=s_0, s_1, \dots, s_d$ ($t$ will correspond to $v_g$ where $g$ is the
output gate of $\varphi$). For every gate $g$ in the subcircuit $\varphi_i$, the
following construction will ensure that the sum of weights of directed paths
from $s_i$ to $v_g$ is equal to the value computed at $g$ in $\varphi$.
For the construction of $G$ we process the {\em decomposition} of $\varphi$
in a bottom-up manner.
Let sub\-cir\-cuit $\varphi_i$ be a leaf in the decomposition of $\varphi$
(so $\varphi_i$ consists solely of addition gates and input gates).
Assume that $\varphi_i$ is located in layers $top_i$ through $bot_i$
($1 \geq top_i \geq bot_i \geq l$) of $\varphi$.
First we add a vertex $s_i$ to $G$ in bag $bot_i - 1$, and
for each input gate with value $w$ in the bottom layer $bot_i$ of $\varphi_i$
we add a vertex to $G$ also in bag $bot_i - 1$ along with an edge of weight $w$
from $s_i$ to that vertex.
Let $n$ range from $bot_i - 1$ to $top_i$:
Add the already created vertex $s_i$ to bag $n-1$ and handle input gates
of $\varphi_i$ in layer $n$ as previously described.
For each addition gate of $\varphi_i$ in layer $n$
we add a new vertex to $G$ (which is added to bags $n$ and $n-1$ of
the path-decomposition of $G$). In bag $n$ we already have two vertices that
represent inputs to this addition gate, so we add edges of weight 1
from both of these to the newly added vertex.
The vertex representing the output gate of the circuit $\varphi_i$ is
denoted by $t_i$. The sum of weighted directed paths from $s_i$ to $t_i$
equals the value computed by the subcircuit $\varphi_i$.
Let $\varphi_i$ be a subcircuit in the decomposition of $\varphi$ that contains
multiplication gates. Addition gates and input gates
in $\varphi_i$ are handled as before.
Let $g$ be a multiplication gate in $\varphi_i$ in layer $n$ and
$\varphi_j$ the disjoint multiplication-input subcircuit
that is one of the inputs to $g$. We know that
vertices $s_j$ and $t_j$ already are in bag $n$, so we add an edge of
weight 1 from the vertex representing the other input to $g$ to
the vertex $s_j$, and an edge of weight 1 from $t_j$ to a newly
created vertex $v_g$ that represents gate $g$,
and then $v_g$ is added to bags $n$ and $n-1$.
For every $b$ ($1 \geq b \geq l-1$) we need to show that only a constant
number of vertices are added to bag $b$ during the entire process.
Every gate in layer $b$ of $\varphi$ is represented by a vertex, and these
vertices may all be added to bag $b$. Every gates in layer $b+1$ are also
represented by a vertex, and all of these are added to bag $b$
(because they are used as input here). So far we have at most $2 \cdot k$ gate
vertices in each bag.
In addition a number of $s_i$ vertices are also added to bag $b$.
For each subcircuit $\varphi_j$ that has a gate in layer $b$ or $b+1$,
we have the corresponding
$s_j$ vertex in bag $b$, so what remains is to show that at most
$\left\lfloor \frac{3 \cdot k}{2} \right\rfloor$ disjoint subcircuits
have a gate in layer $b$ or $b+1$. Each of these subcircuits are in exactly
one of the following 3 sets:
\begin{itemize}
\item[$C_1$:] Subcircuits that have a gate in layer $b$, but NONE
of them are multiplication gates.
\item[$C_2$:] Subcircuits that DO have a multiplication gate in layer $b$.
\item[$C_3$:] Subcircuits that have their root in layer $b+1$.
\end{itemize}
There are at most $\left\lfloor \frac{k}{2} \right\rfloor$ subcircuits
in the set $C_2$.
Otherwise, since two inputs to a multiplication gate are in different
subcircuits and since subcircuits in $C_2$ are disjoint
layer $b+1$ would contain at least $2 \cdot (\left\lfloor
\frac{k}{2} \right\rfloor + 1)$ gates and thus have width more than $k$.
By how subcircuits are constructed, all subcircuits in $C_3$ are considered
as the disjoint multiplication-input subcircuit of distinct
multiplication gates in layer $b$, so there are at least $| C_3 |$
multiplication gates in layer $b$. Since subcircuits in $C_1$ do NOT
have multiplication gates in layer $b$ we have that $|C_1| + |C_3| \leq k$.
Thus, at most $|C_1|+|C_2|+|C_3| \leq \left\lfloor \frac{3 \cdot k}{2}
\right\rfloor$ distinct subcircuits have their $s_i$ vertex added to bag $b$.
Note that in layer $1$ of $\varphi$ we just have the output gate. This gate
is represented by the vertex $t$ of $G$ which is in bag $1$ of the
path-decomposition.
The sum of weights of all directed paths from $s$ to $t$
in $G$ can by induction
be shown to be equal to the value computed by $\varphi$.
The final step in the reduction to the permanent polynomial
is to add an edge of weight
$1$ from $t$ back to $s$ and loops of weight $1$ at all nodes different
from $s$ and $t$.
\qed
\medskip
The proof of Theorem~\ref{bwcircuitToPerm} can be modified to work for
the hamiltonian polynomial as well.
We adapt the idea used to show universality of the hamiltonian polynomial
in \cite{Mal}.
For the permanent polynomial
each bag in the path-decomposition contains at most $\left\lfloor \frac{7\cdot k}{2}\right\rfloor$
vertices; for each of these vertices we now need to introduce one extra
vertex in the same bag. In addition each bag must contain 2 more vertices in
order to establish a connection to adjacent bags in the path-decomposition.
In total each bag now contains at most $7 \cdot k + 2$ vertices.
\begin{theorem}\label{bwcircuitToMatch}
The polynomial computed by a weakly skew circuit of bounded width can be expressed as the
sum of weights of perfect matchings of a symmetric matrix of bounded pathwidth.
The size of the matrix is polynomial in the size of the circuit.
All entries in the matrix are either 0, 1, constants of the polynomial,
or variables of the polynomial.
\end{theorem}
\proof
It is a direct consequence of Theorem~\ref{bwcircuitToPerm} and Lemma~\ref{twIO}.
\qed
\medskip
Now we prove that the permanent, the hamiltonian, and the sum of weights of perfect
matchings of a bounded pathwidth graph can be expressed as a skew circuit of
bounded width.
\begin{theorem}\label{pathwidthHamToCircuit}
The hamiltonian of a matrix of bounded pathwidth can be expressed as a
skew circuit of bounded width.
The size of the circuit is polynomial in the size of the matrix.
\end{theorem}
\proof
Let $M$ be a matrix of bounded pathwidth $k$ and let $G_M$ be the underlying,
directed graph. Each bag in the path-decomposition of $G_M$ contains
at most $k+1$ vertices. We refer to one end of the path-decomposition as
the {\em leaf} of the path-decomposition and the other as the {\em root}
(recall that path-decompositions are special cases of tree-decompositions).
We process the path-decomposition of $G_M$ from the leaf towards the root.
The overall idea is the same as the proof of Theorem 5 in \cite{FKL}
-- namely to consider weighted partial path covers (i.e. partial covers consisting
solely of paths) of
subgraphs of $G_M$ that are induced by the path-decomposition of $G_M$.
During the processing of the path-decomposition of $G_M$ at every level distinct
from the root, new partial path
covers are constructed by taking one previously generated partial path cover
and then add at most ${(k+1)}^2$ new edges, so all the multiplication gates
we have in our circuit are skew. For any bag in the path-decomposition
of $G_M$ we only need to consider a number of partial path covers that depends
solely on $k$, so the circuit we produce has bounded width.
At the root we add sets of edges to partial path covers to form
hamiltonian cycles.
\qed
\begin{theorem}\label{pathwidthMatchToCircuit}
The sum of weights of perfect matchings of a symmetric matrix of bounded pathwidth
can be expressed as a skew circuit of bounded width.
The size of the circuit is polynomial in the size of the matrix.
\end{theorem}
\proof
Let $M$ be a symmetric matrix of bounded pathwidth $k$ and let $G_M$ be
the underlying, undirected graph. Each bag in the path-decomposition of $G_M$ contains
at most $k+1$ vertices.
We process the path-decomposition of $G_M$ from the leaf towards the root.
The proof is very similar to the proof of Theorem \ref{pathwidthHamToCircuit}
-- namely to consider weighted matchings of subgraphs of $G_M$ that are
induced by the matching of $G_M$.
During the processing of the matching of $G_M$ at every level
distinct from the root, new matchings
are constructed by taking one previously generated matching
and then add at most ${(k+1)}^2$ new edges, so all the multiplication gates
we have in our circuit are skew. For any bag in the path-decomposition
of $G_M$ we only need to consider a number of matchings that depends
solely on $k$, so the circuit we produce has bounded width.
At the root we sum only the weights of \emph{perfect} matchings to obtain the
output of the circuit.
\qed
\begin{theorem}\label{pathwidthToCircuit}
The permanent of a matrix of bounded pathwidth can be expressed as a
skew circuit of bounded width.
The size of the circuit is polynomial in the size of the matrix.
\end{theorem}
\proof
It is a direct consequence of Theorem~\ref{pathwidthMatchToCircuit} and Lemma~\ref{twIO}.
\qed
\medskip
\begin{corollary}
A family of polynomials is computable by polynomial size skew circuits of bounded width
if and only if it is computable by polynomial size weakly skew circuits of bounded width.
\end{corollary}
\proof
It is trivial to see that a family of polynomials computed by polynomial size skew circuits of bounded width can
be computed by polynomial size weakly skew circuits of bounded width.
Conversely, if a family of polynomials is computed by polynomial size weakly skew circuits of bounded width then
by Theorem \ref{bwcircuitToPerm} it can be expressed as the permanents of bounded pathwidth graphs
which can be computed by polynomial size skew circuits of bounded width according
to Theorem~\ref{pathwidthToCircuit}.
\qed
\medskip
We need the following Theorem from \cite{BC} to prove the equivalence between polynomial size
skew circuits of bounded width and polynomial size arithmetic formulas.
\begin{theorem}\label{formToLBS}
Any arithmetic formula can be computed by a
linear bijection straight-line program of polynomial size that uses
three registers.
\end{theorem}
Let $R_1,\dots ,R_m$ be a set of $m$ registers,
a linear bijection straight-line (LBS) program is a vector
of $m$ initial values given to the registers plus a
sequence of instructions
of the form
\begin{itemize}
\item[(i)] $R_j \leftarrow R_j + (R_i \times c)$, or
\item[(ii)] $R_j \leftarrow R_j - (R_i \times c)$, or
\item[(iii)] $R_j \leftarrow R_j + (R_i \times x_u)$, or
\item[(iv)] $R_j \leftarrow R_j - (R_i \times x_u)$,
\end{itemize}
where $1\leq i,j\leq m$, $i\neq j$, $1\leq u\leq n$,
$c$ is a constant, and $x_1,\dots, x_n$ are variables ($n$ is the number of variables).
We suppose without loss of generality that the value computed by the LBS program
is the value in the first register after all instructions have been executed.
\begin{theorem}\label{formulaEquiSBW}
A family of polynomials is computable by polynomial size skew circuits of
bounded width if and only if it is computable by a family of polynomial size arithmetic
formulas.
\end{theorem}
\proof
Let $(f_n)$ be a family of polynomials computable by polynomial size skew
circuits of bounded width, then
by Theorem \ref{bwcircuitToPerm} it can be expressed as the permanents of
bounded pathwidth graphs.
Since graphs of bounded pathwith have bounded treewidth, we know by Theorem 5
in~\cite{FKL} that
it can be computed by a family of polynomial size arithmetic formulas.
Conversely, if $(f_n)$ is a family of polynomial size arithmetic formulas,
then by Theorem~\ref{formToLBS}, it is computable by linear bijection
straight-line programs
of polynomial size that use three registers. We will modify these programs
to obtain equivalent skew circuits of width 6. At each step, the set of
indices $\{i,j,k\}$ will be equal to $\{1,2,3\}$.
Suppose the initial values of the three registers are $r_1, r_2, r_3$, then
the first layer of our skew circuit contains three input gates with the three
values $r_1, r_2, r_3$ along with two others inputs which will be defined according
to the next instruction in the straight-line program.
If the next instruction is $R_j \leftarrow R_j + (R_i \times U)$ where $U$ is
a variable or a constant, then we assign the values $0$ and $U$ to the two input
gates not already defined in the current layer $l$ and we create a new layer $l-1$ with
three addition gates corresponding to $R_i,R_j,R_k$ whose inputs are the gate corresponding to $R_i$
(resp. $R_j,R_k$) in layer $l$ and the input with value $0$ in layer $l$. We also put
a multiplication gate whose inputs are the gate corresponding to $R_i$
and the input with value $U$ in layer $l$. And we put again an input gate with value $0$.
Then we create a new layer $l-2$ with three addition gates corresponding to $R_i,R_j,R_k$
whose inputs are the gate corresponding to $R_i$
(resp. $R_j,R_k$) and the input with value $0$ for $i,k$
or the gate computing $(R_i \times U)$ for $j$ in layer $l-1$. We also put two others
inputs which will be defined according
to the next instruction.
If the next instruction is $R_j \leftarrow R_j - (R_i \times U)$, then we need
to create one more layer than in the first case.
We first assign the values $0$ and $U$ to the two input
gates not already defined in the current layer $l$ and we create a new layer $l-1$ with
three addition gates corresponding to $R_i,R_j,R_k$ whose inputs are the gate corresponding to $R_i$
(resp. $R_j,R_k$) in layer $l$ and the input with value $0$ in layer $l$. We also put
a multiplication gate whose inputs are the gate corresponding to $R_i$
and the input with value $U$ in layer $l$. And we put again an input gate with value $0$ and another
one with value $-1$.
Then we create an intermediate new layer $l-2$ with three addition gates corresponding to $R_i,R_j,R_k$
whose inputs are the gate corresponding to $R_i$
(resp. $R_j,R_k$) and the input with value $0$. We also put
a multiplication gate whose inputs are the gate computing $(R_i \times U)$
and the input with value $-1$ in layer $l-1$. And we put again an input gate with value $0$.
Finally we create a new layer $l-3$ with three addition gates corresponding to $R_i,R_j,R_k$
whose inputs are the gate corresponding to $R_i$
(resp. $R_j,R_k$) and the input with value $0$ for $i,k$
or the gate computing $-(R_i \times U)$ for $j$ in layer $l-2$. We also put two others
inputs which will be defined according
to the next instruction.
In both cases, it is clear by induction that the three gates of the current layer
corresponding to $R_i,R_j,R_k$ are computing the values in these registers
if we execute the instructions treated so far. Hence the result.
\qed
\section{Expressiveness of matrices of bounded weighted cliquewidth}
In this section we study the expressive power of permanents, hamiltonians
and perfect matchings of matrices that have bounded weighted clique\-width.
We first prove that every arithmetic formula can be expressed as
the permanent, hamiltonian, or sum of weights of
perfect matchings of a matrix of bounded
$W$-cliquewidth, using the results for the bounded pathwidth matrices and the
following lemma.
\begin{lemma}\label{pwcw}
Let $G$ be a weighted graph (directed or not) with weights in $W$.
If $G$ has pathwidth $k$, then $G$ has $W$-cliquewidth at most $k+2$.
\end{lemma}
\proof
Let $\langle T,(X_t)_{t\in V(T)} \rangle$ be a $k$-path-decomposition of $G$.
We refer to one end of the path-decomposition as
the {\em leaf} of the path-decomposition and the other as the {\em root}.
Let $G_t$ be the subgraph of $G$ induced by the vertices in bags below $X_t$.
We prove by induction on the height of $\langle T,(X_t)_{t\in V(T)} \rangle$
that every graph $G_t$ can be constructed by $W$-clique operations using at most
$k+2$ distinct labels. Moreover, at the end of this construction all vertices in
bag $X_t$ have distinct labels and all other vertices have a \emph{sink} label.
If $|V(T)|=1$ then $G$ has at most $k+1$ vertices. We can create them with $k+1$
distinct labels and add independently each edge between two vertices using
$W$-clique operations.
Suppose $|V(T)|>1$, let $r$ be the root and $t$ be its child. By induction,
$G_t$ can be constructed by $W$-clique operations using at most
$k+2$ distinct labels. For all vertex $v \in X_t \backslash X_r$, we add a renaming
operation which gives \emph{sink} label to $v$ (this renaming operation renames
only $v$ since, by induction, $v$ has distinct label from other vertices).
Since $|X_r| \leq k+1$ and all vertices in $V(G) \backslash X_r$ have \emph{sink}
label, we can create the vertices of $X_r \backslash X_t$ with distinct labels
and add them by disjoint union to the current construction. It is now clear that
all the vertices of $X_r$ have distinct labels thus we can add independently
each edge between two vertices. Hence the conclusion.
\qed
\begin{theorem}\label{formToPermClique}
Every arithmetic formula can be expressed
as the permanent of a matrix of $W$-cliquewidth at most $22$
and size polynomial in $n$,
where $n$ is the size of the formula. All entries in the matrix are
either 0, 1, constants of the formula, or variables of the formula.
\end{theorem}
\proof
Let $\varphi$ be a formula of size $n$. Due to the proof of
Theorem~\ref{formulaEquiSBW}, we know that it can be computed
by a skew circuit of width 6 and size $O(n^{O(1)})$.
Hence it is equal to the permanent of a graph of size $O(n^{O(1)})$, pathwidth
at most $\left\lfloor \frac{7 \cdot 6}{2} \right\rfloor - 1 = 20$ by Theorem~\ref{bwcircuitToPerm},
and $W$-cliquewidth at
most $20+2=22$ by Lemma~\ref{pwcw}.
\qed
\medskip
For the hamiltonian the $W$-cliquewidth
becomes $((7 \cdot 6 +2 ) -1 ) +2 = 45$ instead.
\begin{theorem}\label{formToMatchClique}
Every arithmetic formula can be expressed
as the sum of weights of perfect matchings of a symmetric matrix of
$W$-cliquewidth at most $44$
and size polynomial in $n$,
where $n$ is the size of the formula. All entries in the matrix are
either 0, 1, constants of the formula, or variables of the formula.
\end{theorem}
\proof
It is a direct consequence of Theorem~\ref{formToPermClique} and Lemma~\ref{cwIO}.
\qed
\medskip
Alternatively we can modify the constructions of bounded treewidth
graphs expressing
formulas in~\cite{FKL}. These modifications require more work than the preceding
proofs but we obtain
smaller constants since we obtain graphs of $W$-cliquewidth at most 13/34/26
(instead of 22/45/44) whose permanent/hamiltonian/sum of weights of
perfect matchings are equal to formulas. The proofs of these constants are
given in the Appendix.
\medskip
Due to our restrictions on how weights are assigned in our
definition of bounded $W$-clique\-width it is not true that {\em weighted}
graphs
of bounded treewidth have bounded $W$-cliquewidth. In fact, if one tries to
follow the proofs in \cite{CO,CR} that show that graphs of bounded treewidth
have bounded cliquewidth, then one obtains that a weighted graph $G$ of treewidth $k$
has $W$-cliquewidth at most $3 \cdot (|W_G|+1)^{k-1}$ or $3 \cdot (\Delta + 1)^{k-1}$.
$W_G$ denotes the set of weights on the edges of $G$ and $\Delta$ is the maximum
degree of $G$.
Weighted trees still have bounded weighted cliquewidth (the bound is 3),
but we can show that there exists a family of weighted
graphs with treewidth 2 and unbounded $W$-cliquewidth~\cite{LT}.
\medskip
We now turn to the upper bound on the complexity of the permanent,
hamiltonian, and sum of weights of
perfect matchings of graphs of bounded weighted cliquewidth. We show that in all
three cases the complexity is at most the complexity of {\rm VP}.
The decision version of the hamiltonian cycle problem has been shown to
be polynomial time solvable in \cite{EGW} for matrices of bounded
cliquewidth. Here we extend these ideas in
order to compute the hamiltonian polynomial efficiently (in ${\rm VP}$)
for bounded $W$-m-cliquewidth matrices.
\begin{definition}
A {\em path cover} of a directed graph $G$ is a subset of
the edges of $G$, such that these edges form disjoint, directed,
non-cyclic paths in $G$.
We require that every vertex of $G$ is in (exactly) one path.
For technical reasons
we allow ``paths'' of length 0, by having paths that start and end in
the same vertex.
Such constructions do {\em not} have the same interpretation as a loop.
The {\em weight} of a path cover is the product of
weights of all participating edges (in the special case where there are no
participating edges the weight is defined to be 1).
\end{definition}
\begin{theorem}\label{hamilCliqueToCircuit}
The hamiltonian of an $n \times n$ matrix of bounded $W$-m-cliquewidth
can be expressed as a circuit of size $O(n^{O(1)})$
and thus is in ${\rm VP}$.
\end{theorem}
\proof
Let $M$ be an $n \times n$ matrix of bounded $W$-m-cliquewidth.
By $G$ we denote the underlying, directed, weighted graph for $M$.
The circuit is constructed based on the parse-tree $T$ for $G$.
By $T_t$ we denote the subtree of $T$ rooted at $t$ for some node $t \in T$.
By $G_t$ we denote the subgraph of $G$ constructed from the parse-tree $T_t$.
The overall idea is to produce a circuit that computes the sum of weights
of all hamiltonian cycles of $G$. To obtain this there will be
non-output gates that
compute weights of all path covers of all $G_t$ graphs,
and then we combine these subresults.
Of course, the total number of path covers can grow exponentially
with the size of $G_t$, so we will not ``describe'' path
covers directly by the edges participating in the covers.
Instead we describe a path cover of some $G_t$ graph by the labels
associated with the start- and end-vertices of the paths in the cover.
Such a description do not uniquely describe a path cover, because two different
path covers of the same graph can contain the same number of paths
and all these paths can have the same labels associated.
However, we do not need the weight of each individual path cover. If multiple
path covers of some graph $G_t$ share the same description, then we
just compute the sum of weights of these path covers.
For a leaf in the parse-tree $T$ of $G$ we construct a single gate of
constant weight 1, representing a path cover consisting of a single ``path''
of length 0, starting and ending in a vertex with the given labels.
Per definition this path cover has weight 1.
For an internal node $t \in T$ the grammar rule describes
which edges to add and how to relabel vertices. We obtain new path covers
by considering a path cover from the left child of $t$
and a path cover from the right child of $t$:
For each such pair of path covers consider all subsets of edges
added at node $t$, and for every subset of edges check if the addition
of these edges to the pair of path covers will result in a valid path cover.
If it does, then add a gate that computes the weight of this path cover,
by multiplying the weight of the left path cover, the weight of the
right path cover and the total weight of the newly added edges.
After all pairs of path covers have been processed, check if any of the
resulting path covers have the same description - namely that the number
of paths in some path covers are the same, and that these paths have the
same labels for start- and end-vertices.
If multiple path covers have the same description then add
addition gates to the circuit and produce a single gate which computes
the sum of weights of all these path covers.
For the root node $r$ of $T$ we combine path covers from the children
of $r$ to produce hamiltonian cycles, instead of path covers.
Finally, the output of the circuit is a summation of all gates computing
weights of hamiltonian cycles.
Proof of correctness:
The first step of the proof is by induction over the height of the parse-tree
$T$. We will show that for each non-root node $t$ of $T$
there is for every path cover description of $G_t$ a corresponding gate in the
circuit that computes the sum of weights of
all path covers of $G_t$ with that description.
For the base cases - leaves of $T$ - it is trivially true.
For the inductive step we consider two disjoint graphs that are being connected
with edges at a node $t$ of the parse-tree $T$.
Edges added at node $t$ are {\em only}
added in here, and not at any other nodes in $T$, so every path cover of
$G_t$ can be split into 3 parts: A path cover of $G_{t_l}$, a path cover of
$G_{t_r}$ and a polynomial number of edges added at node $t$.
Consider a path cover description along with all path covers of $G_t$
that have this description.
All of these path covers can be split into 3 such parts,
and by our induction hypothesis the weights of the path covers of $G_{t_l}$
and $G_{t_r}$ are computed in already constructed gates.
In order to complete the proof of correctness we have to handle the root $t$ of
$T$ in a special way. At the root we do not compute weights of
path covers, but instead compute weights of hamiltonian cycles. Every
hamiltonian cycle of $G$ can (similarly to path covers) be split into 3 parts:
A path cover of $G_{t_l}$, a path cover of $G_{t_r}$ and a polynomial number
of edges added at the root of $T$.
By our induction hypothesis all the needed weights are already computed.
The size of the circuit is polynomial since at each step the number of
path cover descriptions is polynomially bounded once the $W$-m-cliquewidth
is bounded.
\qed
\begin{theorem}\label{matchToVP}
The sum of weights of perfect matchings of an $n \times n$ symmetric
matrix of bounded $W$-NLCwidth
can be expressed as a circuit of size $O(n^{O(1)})$
and thus is in ${\rm VP}$.
\end{theorem}
\proof
Let $M$ be an $n \times n$ symmetric matrix of bounded $W$-NLCwidth.
By $G$ we denote the underlying, undirected, weighted graph for $M$.
The circuit is constructed based on the parse-tree $T$ for $G$.
By $T_t$ we denote the subtree of $T$ rooted at $t$ for some node $t \in T$.
By $G_t$ we denote the subgraph of $G$ constructed from the parse-tree
$T_t$. Let $k$ be the $W$-NLCwidth of $G$. We assume without loss of
generality that $T$ is a parse-tree on the set of labels $\{ a_1,\dots ,a_k\}$.
The overall idea is much similar to that of Theorem~\ref{hamilCliqueToCircuit},
namely to produce a circuit that computes the sum of weights
of all perfect matchings of $G$. To obtain this there will be
non-output gates that
compute weights of all matchings of all $G_t$ graphs,
and then we combine these subresults.
Of course, the total number of matchings can grow exponentially
with the size of $G_t$, so we will not ``describe'' matchings
directly by the edges participating in the covers.
Instead we describe a matching of some $G_t$ graph by the labels
associated to the uncovered vertices. More precisely, for each matching of $G_t$ and each
label $a$ we give the number of $a$-vertices which are not covered by the
matching. Such a description do not uniquely describe a matching, because two different
matchings of the same graph can have the same number of uncovered vertices which
have the same labels associated.
However, we do not need the weight of each individual matching. If multiple
matchings of some graph $G_t$ share the same description, then we
just compute the sum of weights of these matchings. It is clear that the number
of description needed is at most $n^k$.
For a leaf $ver_{a_i}$ in the parse-tree $T$ of $G$ we construct a
single terminal gate of
constant weight 1, representing an empty matching. The description associated to
this gate is $((a_1,0),\dots ,(a_i,1),\dots ,(a_k,0))$.
For an internal node $t \in T$ with operation $\circ_R (H)$ we just need to
change the description of terminal gates in the circuit contructed so far.
More precisely, if the description of the gate was
$((a_1,n_1),\dots ,(a_i,n_i),\dots ,(a_k,n_k))$ then it becomes
$$((a_1, \sum_{a_j \in R^{-1}(a_1)} n_j),\dots
,(a_i,\sum_{a_j \in R^{-1}(a_i)} n_j),\dots
,(a_k,\sum_{a_j \in R^{-1}(a_k)} n_j)).$$
For an internal node $t \in T$ with operation $H \;\times_S \; H'$
the grammar rule describes which edges to add.
We first create a multiplication gate using the values of each couple of
terminal gates
of the left child $l$ of $t$ and the right child $r$ of $t$.
It corresponds to the
weights of the disjoint unions of the matchings of $l$ and $r$. There is at most
$n^{2k}$ such gates. To each gate, we associate a left and right
description corresponding to the vertices from $l$ and $r$. Those gates are the
new terminal gates.
We put the following total order
$a_1< a_2 < \dots <a_k$ on the labels and the corresponding
lexicographic order on the couples
$(a_i,a_j)$. We will consider that the edges added via $S$ are added by blocks
corresponding to a couple $(a_i,a_j)$ (All edges in the same block are added at
the same time) and that all blocks of edges are added sequentially in
lexicographic order. Thus we have at most $k^2$ steps of adding edges to
consider. Suppose $S(a_i,a_j)=w_{ij}$.
For the step corresponding to $(a_i,a_j)$ we obtain new matchings by
considering each terminal gate $g_0$. Let $((a_1,n_1),\dots
,(a_i,n_i),\dots ,(a_k,n_k))$ and $((a_1,n'_1),\dots
,(a_j,n'_j),\dots ,(a_k,n'_k))$ be the left and right description of $g_0$.
Let $n_{min} = min\{n_i,n'j\}$. For all matching corresponding to $g_0$
and all $p$ between $0$ and $n_{min}$ we can obtain $\binom{n_i}{p} \cdot
\binom{n'_j}{p}$ matchings by adding $p$ edges of weight $w_{ij}$ between $p$
vertices among $n_i$ of $G_l$ and $p$ vertices among $n'_j$ of $G_r$. Hence,
for all $p\neq 0$ we add a multiplication gate with inputs $g_0$
and the constant $\binom{n_i}{p} \cdot \binom{n'_j}{p} \cdot (w_{ij})^p$.
This new gate $g_p$ has left and right
description $((a_1,n_1),\dots ,(a_i,n_i-p),\dots ,(a_k,n_k))$ and
$((a_1,n'_1),\dots ,(a_j,n'_j-p),\dots ,(a_k,n'_k))$. There are at most
$2\cdot n^{2k+1}$ such new gates since $p < n$.
Finally we make an addition tree computing the addition of the gates $g_p$ which
have the same left and right description. Each such tree needs at most
$O((2k+2)\log(n))$ new gates and there are at most $2 \cdot n^{2k}$ trees.
The outputs of these
trees are the new terminal gates. When all the $k^2$ steps of adding edges are
done we compute the description of each terminal gate as the sum of
its left and right description then we put an addition tree computing the
addition of the terminal gates which have the same global description.
The outputs of these trees are the new terminal gates.
Finally, we obtain the output of the circuit at the root node $r$ of $T$. It is
the output of the terminal gate with description
$((a_1,0),\dots ,(a_i,0),\dots ,(a_k,0))$.
Proof of correctness:
The first step of the proof is by induction over the height of the parse-tree
$T$. We will show that for each node $t$ of $T$
there is for every matching description of $G_t$ a corresponding gate in the
circuit that computes the sum of weights of
all matchings of $G_t$ with that description.
For the base cases - leaves of $T$ - it is trivially true.
For the inductive step we consider two disjoint graphs that are being connected
with edges at a node $t$ of the parse-tree $T$.
Edges added at node $t$ are {\em only}
added in here, and not at any other nodes in $T$, so every matching of
$G_t$ can be split into 3 parts: A matching of $G_{t_l}$, a matching of
$G_{t_r}$ and a polynomial number of edges added at node $t$.
Consider a matching description along with all matchings of $G_t$
that have this description.
All of these matchings can be split into 3 such parts,
and by our induction hypothesis the weights of the path covers of $G_{t_l}$
and $G_{t_r}$ are computed in already constructed gates.
The number of new gates added for each operation $H \;\times_S \; H'$
is at most $O(k^2 \cdot n^{2k+1})$.
Since the number of these operations is at most $n$,
we obtain a circuit of polynomial size.
\qed
\begin{theorem}
The permanent of an $n \times n$ matrix of bounded $W$-m-cliquewidth
can be expressed as a circuit of size $O(n^{O(1)})$
and thus is in ${\rm VP}$.
\end{theorem}
\proof
It is a direct consequence of Theorem~\ref{matchToVP} and Lemma~\ref{cwIO}.
\qed
\section{Acknowledgements}
Much of this work was done while U.~Flarup was visiting the ENS Lyon during
the spring semester of 2007.
This visit was partially made possible by funding from
Ambassade de France in Denmark,
Service de Coop\'eration et d'Action Culturelle,
Ref.:39/2007-CSU 8.2.1.
|
1,314,259,993,008 | arxiv |
\section{Introduction}
\label{sec:intro}
Polar codes are the first provably capacity-achieving channel codes with an explicit construction, low-complexity encoding and decoding algorithms, and easily adaptable coding rate~\cite{arikan2009}. Although the capacity of binary symmetric memoryless channels can be achieved using the low-complexity successive cancellation (SC) decoding algorithm, the sequential nature of SC decoding typically leads to a large decoding latency, which constrains its application in high-throughput and low-latency communication scenarios such as 5G and optical wireless communications~\cite{koonen2020}. A simplified successive cancellation (SSC) decoder was proposed in~\cite{alamdar2011simplified}, where fast decoding methods are described for subcodes of the polar code (called \emph{constituent codes}) that consist either of only information bits or of only frozen bits. Following this idea, other constituent codes with special information bit patterns and their corresponding fast decoders were identified in \cite{sarkis2014fast,hanif2017fast,condo2018generalized,gamage2019low}. The family of these decoding algorithms is often referred to as \emph{fast-SSC} decoding. To increase the number of fast decoding constituent codes, \cite{638mbps,giard2018fast} altered
the polar code construction to further improve latency at the cost of a small error-correcting performance degradation. Methods to optimize the memory footprint of fast-SSC decoders were described in~\cite{Furkan2017}.
The work of \cite{sr2020} proposed a new class of \emph{sequence repetition } (SR) constituent codes, which is a generalization of most existing constituent codes. It was also shown that the decoding of SR constituent codes can be highly parallelized to achieve further latency reduction compared to the state of the art without tangibly affecting the error-correcting performance. However, the work of \cite{sr2020} only focused on the algorithmic aspects of SR constituent codes and no hardware implementation has been reported in the literature.
\subsubsection*{Contribution} In this work, we describe a hardware architecture for a fast-SSC decoder that exploits the SR constituent codes described in \cite{sr2020} and we provide FPGA implementation results. Even though our proposed implementation is not yet highly optimized, it still achieves a $17.9$\% higher decoding throughput than the state of the art.
\section{Background}
\label{sec:pre}
\subsection{Polar Codes}
\label{sec:PC}
A polar code with code length $N=2^n$ and information length $K$ is denoted by $\mathcal{P}\left(N,K\right)$ and has rate $R = K/N$.
The input bit sequence $\boldsymbol{u}$ consists of $K$ information bits whose positions form set $\mathbb{A}$ and $N-K$ frozen bits whose positions form $\mathbb{A}^c$. The values of the frozen bits are usually set to $0$. The encoded bit sequence can be calculated as $\boldsymbol{x}=\boldsymbol{u}\mathbf{G}_N$, where $\mathbf{G}_N=\mathbf{R}_N\mathbf{F}_2^{\otimes n}$ is the generator matrix of the polar code, $\mathbf{R}_N$ is a bit-reversal permutation matrix and $\mathbf{F}_2=\left[\begin{smallmatrix}1&0\\1&1\end{smallmatrix}\right]$.
\subsection{SC and Fast-SSC Decoding}
\label{sec:FSC}
\subsubsection{Algorithm}
\begin{figure}
\centering
\scalebox{0.75}{\includegraphics{./tikz/tikz_compiled0.pdf}}
\caption{SC decoding tree representation of a polar code with $N=8$.}
\label{fig:tree}
\end{figure}
SC decoding of polar codes can be represented as the traversal of a binary tree as in Fig.~\ref{fig:tree}. The $i$-th node at level $j$ ($1\leq i\leq 2^{n-j}$) of the SC decoding tree corresponds to a constituent code with bit index from $2^j\cdot\left(i-1\right)+1$ to $2^j\cdot i$, and is denoted as $\mathcal{N}_j^i$. The left and the right child nodes of $\mathcal{N}_j^i$ are $\mathcal{N}_{j-1}^{2i-1}$ and $\mathcal{N}_{j-1}^{2i}$, respectively. For $\mathcal{N}_j^i$, the symbol $\alpha_j^i\left[k\right]$, $1\leq k\leq2^j$, denotes the $k$-th input logarithmic likelihood ratio (LLR) value, and $\beta_j^i\left[k\right]$, $1\leq k\leq2^j$, denotes the $k$-th output binary hard-valued message. The SC decoding follows a depth-first principle, with priority
to the left branch. When LLR messages pass to the left and right child nodes, $f$ and $g$ functions over the LLR domain are executed, respectively, which are given by
\begin{equation}
\begin{aligned} \label{eq:f_function}
\alpha_{j-1}^{2i-1}\left[k\right]\approx &\text{sign}\left(\alpha_j^{i}\left[{2k-1}\right]\right)\text{sign}\left(\alpha_j^{i}\left[{2k}\right]\right)\\
&\cdot\min\left(\alpha_j^{i}\left[{2k-1}\right],\alpha_j^{i}\left[{2k}\right]\right),
\end{aligned}
\end{equation}
\begin{equation}
\label{eq:g_function}
\alpha_{j-1}^{2i}\left[k\right]=\left(-1\right)^{\beta_{j-1}^{2i-1}\left[k\right]}\alpha_j^{i}\left[{2k-1}\right]+\alpha_j^{i}\left[{2k}\right].
\end{equation}
When the LLR value of the $k$-th bit at level zero $\alpha_0^k,\;1\leq k\leq N$, is calculated, the estimation of $u\left[k\right]$, denoted as ${\hat u}\left[k\right]$, is
\begin{equation}\label{eq:decision}
\centering
\hat{u}\left[k\right]=\hat\beta_0^k=\begin{cases} 0, &\mbox{if } k\in \mathbb{A}^c \text{,} \\
\frac{1-\mathrm{sign}(\alpha_0^k)}{2}, & \mbox{otherwise.} \end{cases}
\end{equation}
The hard messages are propagated back to the parent node as
\begin{equation}
\hat\beta_j^i\left[k\right]=
\begin{cases} \hat\beta_{j-1}^{2i-1}\left[\frac{k+1}2\right]\oplus\hat\beta_{j-1}^{2i}\left[\frac{k+1}2\right], &\mbox{if} \mod(k,2)=1 \text{,} \\
\hat\beta_{j-1}^{2i}\left[\frac{k}2\right], & \mbox{if} \mod(k,2)=0. \end{cases}
\label{eq:hardpropagation}
\end{equation}
The estimation of each bit depends on the estimation of all previous bits in the SC decoding algorithm, which leads to a large latency. It was pointed out in \cite{sarkis2013increasing} that for a node $\mathcal{N}_j^i$, the maximum-likelihood (ML) estimate of the vector $\beta_j^i\left[1:2^j\right]$ can be calculated in parallel by evaluating
\begin{equation} \label{eq:estimate}
\hat\beta_j^i\left[1:2^j\right]= \underset{\beta_j^i\left[1:2^j\right]\in\mathbb{C}_j^i}{\argmax}\sum_{k=1}^{2^j}\left(-1\right)^{\beta_j^i\left[k\right]}\alpha_j^i\left[k\right],
\end{equation}
where $\mathbb{C}_j^i$ is the set of all the codewords associated with node $\mathcal{N}_j^i$. The complexity of evaluating~\eqref{eq:estimate} is generally very high. However, the main idea behind fast-SSC decoding is that for some nodes with special frozen and non-frozen bit patterns the evaluation of~\eqref{eq:estimate} can be simplified significantly. Some prominent examples of such nodes include the Rate-0 node ($2^j$ frozen bits), the Rate-1 node ($2^j$ information bits), the repetition (REP) node (one information bit and $2^j-1$ frozen bits), and the single parity-check (SPC) node ($2^j-1$ information bits and one frozen bit).
The key advantage of using specific parallel decoders for the aforementioned special nodes is that, since the SC decoding tree is not traversed when one of these nodes is encountered, a significant latency reduction can be achieved. For example, if $\mathcal{N}_j^i$ is a Rate-1 node, hard decision decoding can be used to immediately obtain the decoding result as
\begin{equation}
\hat\beta_j^i\left[k\right]=h\left(\alpha_j^i\left[k\right]\right)=
\begin{cases} 0, &\mbox{if } \alpha_j^i\left[k\right]\geq0 \text{,} \\
1, & \mbox{otherwise.} \end{cases}
\label{eq:rate1}
\end{equation}
If $\mathcal{N}_j^i$ is a REP node, all its bits are either equal to one or equal to zero. According to \eqref{eq:estimate}, estimation can be obtained by extracting the sign bit of the sum of its LLR values. If $\mathcal{N}_j^i$ is an SPC node, a hard decision based on~\eqref{eq:rate1} is first performed, which is followed by the calculation of the parity of the output using modulo-2 addition. The hard decision value with the index of the least reliable bit will be flipped if the parity check constraint is not met.
\subsubsection{Fast-SSC Decoder Hardware Architectures}
A typical fast-SSC decoder contains three main modules \cite{sarkis2014fast}: a memory, an arithmetic logical unit (ALU), and a controller. The memory consists of five separate sub-modules. The channel LLR, internal LLR $\alpha$, and estimation $\beta$ sub-modules feed the ALU. The instruction sub-module stores the operations to be executed and is routed into the controller. Finally, the codeword sub-module stores and outputs the final codeword.
The ALU implements the $f$ function given in \eqref{eq:f_function}, the $g$ function given in \eqref{eq:g_function}, the combining operation given in~\eqref{eq:hardpropagation}, as well as the update rules for various special nodes like the rate-$1$ node given in \eqref{eq:rate1}.
Finally, the controller tracks which node in the decoding tree is currently being decoded by using a list of instructions that is pre-compiled based on $\mathbb{A}$ and $\mathbb{A}^c$.
\section{Fast-SSC Decoding with Sequence Repetition Nodes}
\label{sec:sr}
\subsection{Sequence Repetition Node}
\label{sec:sr node}
Let $\mathcal{N}_j^i$ be a node at level $j$ of the binary tree representation of SC decoding as shown in Fig.~\ref{fig:tree}. An SR node is any node at stage $j$ for which all its descendants are either Rate-0 or REP nodes, except the rightmost one at a certain stage $r$, $0\leq r\leq j$, that is a generic node of rate $C$. The general structure of an SR node is depicted in Fig.~\ref{fig:SR}.
The rightmost node $\mathcal{N}_r^{i\times 2^{j-r}}$ at stage $r$ is denoted as the source node of the SR node $\mathcal{N}_j^i$. Let $E = i\times 2^{j-r}$ so the source node can be denoted as $\mathcal{N}_r^E$.
\begin{figure}
\centering
\scalebox{0.8}{\includegraphics{./tikz/Special_Node.pdf}}
\setlength{\abovecaptionskip}{-15pt}
\caption{General structure of a sequence repetition node.}
\label{fig:SR}
\end{figure}
An SR node can be represented by three parameters as $\text{SR}(\boldsymbol{v},\text{SNT},r)$, where $r$ is the level of the SC decoding tree in which $\mathcal{N}_r^E$ is located. SNT is the source node type, and as shown in \cite{sr2020}, $\text{SNT}\in\{\text{Rate-0}, \text{Rate-1}, \text{EG-PC}, \text{Rate-C}\}$. The EG-PC node is a node at level $j$ having all its descendants as Rate-1 nodes except the leftmost one at a certain level $r<j$, that is a Rate-0 or REP node. $\text{Rate-C}$ is a generic node of rate $C$. When $\text{SNT}\in\{\text{Rate-0}, \text{Rate-1}, \text{EG-PC}\}$, the source node is a special node whose bits are all non-frozen except the leftmost $b$ bits, where
\begin{align}
b = \left\{
\begin{matrix}
0, & \text{if\;SNT=Rate-1}, \\
1, & \text{if\;SNT=Rate-0}, \\
2^h \text{ or } 2^h-1, & \text{if\;SNT=EG-PC},
\end{matrix}
\right. \label{eq:a}
\end{align}
and where $h<r-1$ is the level of the leftmost Rate-0/REP node of the EG-PC node.
Note that the source node has a minimum length of 2 as all the possible frozen bit patterns with length 2 fall into the above category. The vector $\boldsymbol{v} = \left(v\left[{j}\right],v\left[{j-1}\right],\ldots,v\left[{r+1}\right]\right)$ has length $\left(j-r\right)$ such that for the left child node of the parent node of $\mathcal{N}_r^E$ at level $k$, $r<k\leq j$, $v\left[{k}\right]$ is calculated as
\begin{equation}
v\left[k\right] = \begin{cases}
0, & \text{if the left child node is a Rate-0 node,}\\
1, & \text{if the left child node is a REP node.}
\end{cases}
\end{equation}
Note that when $r=j$, $\mathcal{N}_j^i$ is a source node and thus $\boldsymbol{v}$ is an empty vector denoted as $\boldsymbol{v} = \emptyset$.
\subsection{Repetition Sequence}
In this subsection, we define \emph{repetition sequences}, which can be used to calculate the output bit estimates of an SR node based on the estimates of its source node. To derive the repetition sequences, $\boldsymbol{v}$ is used to generate all the possible sequences that have to be XORed with the output of the source node to generate the output bit estimates of the SR node. Let $\eta_k$ denote the rightmost bit value of the left child node of the parent node of $\mathcal{N}^E_r$ at level $k+1$. When $v[k+1] = 0$, the left child node is a Rate-0 node so $\eta_k = 0$. When $v[k+1] = 1$, the left child node is a REP node, thus $\eta_k$ can take the value of either $0$ or $1$. The number of repetition sequences is dependent on the number of different values that $\eta_k$ can take. Let $W_{\boldsymbol{v}}$ denote the number of $1$'s in $\boldsymbol{v}$. The number of all possible repetition sequences is thus $2^{W_{\boldsymbol{v}}}$. Let $\mathbb{S} = \{\boldsymbol{s}_1,\ldots,\boldsymbol{s}_{2^{W_{\boldsymbol{v}}}}\}$ denote the set of all possible repetition sequences.
The output bits of SR node $\beta_i^j[1:2^j]$ have the property that their repetition sequence is repeated in blocks of length $2^{j-r}$. Let $\beta_r^E[1:2^r]$ denote the output bits of the source node of an SR node $\mathcal{N}_j^i$. The output bits for each block of length $2^{j-r}$ in $\mathcal{N}_j^i$ with respect to $\beta_r^E[1:2^r]$ can be written as
\begin{equation}
\label{eq:betaS}
\beta_j^i\left[\left(k-1\right)2^{j-r}+1:k2^{j-r}\right]=\beta_r^E\left[k\right]\oplus{\boldsymbol{s}_l},
\end{equation}
where $k\in\left\{1,\dots,2^r\right\}$ and $\boldsymbol{s}_l = \{s_l[1],\ldots,s_l[2^{j-r}]\}$ is the $l$-th repetition sequence in $\mathbb{S}$. To obtain the repetition sequence $\boldsymbol{s}_l$ and with a slight abuse of terminology and notation for convenience, the Kronecker sum operator $\boxplus$ is used, which is equivalent to the Kronecker product operator, except that addition in GF($2$) is used instead of multiplication. For each set of values that $\eta_k$'s can take, $\boldsymbol{s}_l$ can be calculated as
\begin{equation}
\boldsymbol{s}_l=\left(\eta_r,0\right)\boxplus\left(\eta_{r+1},0\right)\boxplus\cdots\boxplus \left(\eta_{j-1},0\right).
\label{eq:S}
\end{equation}
For a given code, the locations of SR nodes in the decoding tree are fixed and can be determined offline. Therefore, the repetition sequences in ${\mathbb{S}}$ of all of the SR nodes can be pre-computed and used in the course of decoding.
\subsection{Decoding of SR Nodes}
To decode SR nodes, the LLR values $\alpha_{r_l}^E[1:2^r]$ of the source node $\mathcal{N}_r^E$ associated with the $l$-th repetition sequence $\boldsymbol{s}_l$ are calculated based on the LLR values $\alpha_{j}^i[1:2^j]$ of the SR node $\mathcal{N}_j^i$ and repetition sequence $\boldsymbol{s}_l$ by the following equation which is proved in~\cite[Proposition 1]{sr2020}
\begin{equation}
\label{eq:alphaS}
\alpha_{r_l}^E\left[k\right]= \sum_{m = 1}^{2^{j-r}} \alpha_{j}^i\left[\left(k-1\right)2^{j-r}+m\right] \left(-1\right)^{s_l[m]}.
\end{equation}
Using \eqref{eq:betaS} and \eqref{eq:alphaS}, \eqref{eq:estimate} can be written as~\cite[(20)]{sr2020}
\begin{equation}
\label{eq:estimate1}
\hat\beta_j^i=\mkern-10mu\underset{\substack{\beta_r^E\left[1:2^r\right]\in\mathbb{C}_r^E\\l\in \{1,\ldots,\left|{\mathbb S}\right|\}}}{\argmax}\sum_{k=1}^{2^r}\left(-1\right)^{\beta_r^E\left[k\right]}\alpha_{r_l}^E\left[k\right].
\end{equation}
Thus, the bit estimates of an SR node $\hat\beta_j^i\left[1:2^j\right]$ can be calculated by finding the bit estimates of its source node $\beta_r^E\left[1:2^r\right]$ and the repetition sequence using \eqref{eq:estimate1}, and then combine them as shown in \eqref{eq:betaS}.
\input{./Algorithm_1}
The decoding algorithm of an SR node $\mathcal{N}_j^i$ is described in Algorithm~\ref{alg:alg1}. The algorithm first calculates $\alpha_{r_l}^E$ to obtain the soft messages that go into the source node for the $l^{th}$ repetition sequence ${\boldsymbol s}_l$, $l\in\left\{1,\dots,\left|{\mathbb S}\right|\right\}$. $\alpha_{r_l}^E$, $\widehat\beta_{r_l}^E$, and $\widehat \beta_{j_l}^i$ are the soft and hard messages associated with ${\boldsymbol s}_l$. Then, the source node is decoded under the rule of the SC decoding. If the source node is a special node, a hard decision is made directly. Parity check and bit flipping will be performed further using Wagner decoding if $\text{SNT}\neq\text{Rate-1}$. Finally, the index of the optimal repetition sequence can be selected according to the comparison in \eqref{eq:comparison} and the decoding result is obtained according to \eqref{eq:betaS}. Based on Algorithm~\ref{alg:alg1}, the SR node-based fast-SSC (SRFSC) decoding algorithm is proposed. It follows the SC decoding algorithm schedule until an SR node is encountered where Algorithm~\ref{alg:alg1} is executed.
\section{Architecture of SRFSC decoder}
\label{sec:Ar}
The top-level architecture of the proposed SRFSC decoder is shown in Fig.~\ref{fig:decoder}. When decoding starts, the instructions for the polar code that is being decoded are fetched by the controller and the channel LLRs are loaded into memory. The controller decodes the instructions to get the node schedule and updates the decoding stage parameters accordingly. The updates in the controller follow the principle of SC decoding until an instruction corresponding to an SR node is reached, where the SR module is activated to process the LLRs. The estimation results from both the SR module and processing module are routed into the partial sum network (PSN) module, from where the estimated codeword is also output when decoding terminates. In the following, the architecture of the various individual modules is discussed in detail.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./tikz/SC.pdf}
\setlength{\abovecaptionskip}{-15pt}
\caption{Top-level architecture of the proposed SRFSC decoder.}
\label{fig:decoder}
\end{figure}
\subsection{Memory, Processing, and PSN Modules}
The architectures of these three modules are identical with those presented in \cite{semi2013} and we thus only describe them on a high level. The memory module stores all soft messages $\alpha$. The update of hard estimates $\beta$ is in the partial sum network (PSN) module. A set of $P$ processing elements (PEs) is instantiated in the processing module to process up to $2P$ LLRs in parallel. A PE implements both the $f$ and the $g$ function using sign-and-magnitude representation and the appropriate output is selected according to the current decoding stage.
\subsection{Controller Module}\label{sec:controller}
The operation in the controller module follows the standard SC decoding schedule until an instruction that indicates an SR node is found. When this occurs, the $2P$ LLRs will be routed to the SR module instead of the processing module to perform the decoding of SR node in Algorithm~\ref{alg:alg1}. The required number of clock cycles to decode the SR node by the SR module is pre-calculated and a counter is initialized to this value. All updates in the controller are suspended until the counter reaches zero. Then, the decoding bit index is added the length of the SR node and the updates resume. Although the Rate-0 and Rate-1 nodes can also be represented as special cases of SR nodes, the controller will bypass the SR module and signal the processing module to execute immediate decoding for these two nodes so that there is no additional latency.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{./tikz/Instruction.pdf}
\setlength{\abovecaptionskip}{-25pt}
\caption{Instruction structure of the proposed SRFSC decoder.}
\label{fig:schedule}
\end{figure}
The structure of the instructions used in the controller is shown in Fig.~\ref{fig:schedule}. The instructions contain all the required information to decode an SR node and they are stored in memory according to the visiting order in the decoding tree. The elements \texttt{SRstage}, \texttt{SourceStage}, \texttt{FroNum}, \texttt{SeqNum} and \texttt{NodeType} in the instruction represent the stage of SR node, the stage of source node, the number of frozen bits in source node, the base 2 logarithm of the number of repetition sequences and the node type, respectively.\footnote{Note that we use \texttt{FroNum} instead of SNT because no SR node with a Rate-C node as its source node is found for the code length $\left(N=1024\right)$ and rates $\left(R=1/2,\;1/4,\;3/4\right)$ that we consider in Section~\ref{sec:performance}}
Moreover, the vector $\boldsymbol{v}$ is replaced with \texttt{SRstage}, \texttt{SeqNum} and \texttt{NodeType} since these three elements can be used directly in the decoder, so that additional calculations (e.g., \eqref{eq:S}) can be avoided. \texttt{NodeType} is in fact a pointer to the memory of repetition sequences. As only nodes with $\texttt{SeqNum}>0$ have non-zero repetition sequences that need to be stored, \texttt{NodeType} refers to these node types and is used as pointer to find their corresponding repetition sequences in the memory.
The different repetition sequences in the SR node are processed in parallel. Since a maximum of $2P$ LLRs are input to the SR module each time, we have the constraint
\begin{equation}
2^{\texttt{SRstage}+\texttt{SeqNum}}\leq2P.
\label{eq:constraint}
\end{equation}
All SR nodes that meet this constraint can be handled, while others are divided into smaller nodes. Therefore, \texttt{SRstage} and \texttt{SourceStage} always have values between $0$ and ${1+\log_2 P}$. \texttt{FroNum} can be calculated according to \eqref{eq:a} and thus have values between 0 and $P/2$. Consider source node with a minimum length of 2. Then, the maximum value of \texttt{SeqNum} is constrained by $2^{1+2\texttt{SeqNum}}\leq2P$.
Thus, \texttt{SeqNum} has values between $0$ and $\frac12\log_2 P$. As for \texttt{NodeType}, it has values between $0$ and $NT\left(N,\;\mathbb{A},\;P\right)$, where $NT$ is a function of $N$, $\mathbb{A}$ and $P$, which depends on the polar code being decoded.
As an example, we consider a set of 5G polar codes \cite{3GPP1} of length $N=1024$ and rates $R=1/2$, $R=1/4$, and $R=3/4$. For a code length of $N=1024$, $P=64$ is shown to be a reasonable choice \cite{Furkan2017}. With these parameters, in Fig.~\ref{fig:schedule}, \texttt{SRstage} and \texttt{SourceStage} take values in $\{0,1,\hdots,7\}$, \texttt{FroNum} takes values in $\{0,1,\hdots,3\}$, and \texttt{SeqNum} takes values in $\{0,1,2\}$. The three considered codes contain a total of six SR nodes with $\texttt{SeqNum}>0$. As such, \texttt{NodeType} takes values in $\{0,1,\hdots,6\}$. Specifically, when $\texttt{NodeType}=0$, the node only has an all-zero repetition sequence and the remaining values represent the six SR nodes with $\texttt{SeqNum}>0$. From the above analysis, the size of each instruction for the considered example is $13$ bits.
\begin{figure*}
\centering
\scalebox{0.8}{\includegraphics{./tikz/SR.pdf}}
\setlength{\abovecaptionskip}{-15pt}
\caption{Example of the SR module architecture for $N=1024$, $R \in \{1/2,1/4,3/4\}$, and $P=64$.}
\label{fig:SRmodule}
\end{figure*}
\subsection{SR Module}
The ranges of some elements in the instructions are variable and depend on the set of supported polar codes. Thus, some of the data widths in the SR module are also variable and it is difficult to give a fully generic explanation of our proposed architecture. For this reason, we consider the previous example of $N=1024$, $R \in \{1/2,1/4,3/4\}$, and $P=64$. The architecture of the SR module for this example is shown in Fig.~\ref{fig:SRmodule}. The submodules with red, blue, and green color correspond to the operations in Step 1, Step 2, and Step 3 in Algorithm~\ref{alg:alg1}, respectively, and are explained in more detail in the sequel.
\textbf{\emph{Step 1:}} This part of the SR decoder is used to calculate the input LLRs into the source node if $\texttt{SRstage}\neq \texttt{SourceStage}$. In the XOR submodule, the first $2^{\texttt{SRstage}}$ LLRs in the $2P$ inputs are repeated $2^{\texttt{SeqNum}}$ times so that the decoding for different repetition sequences can be handled in parallel. The repetition sequences are obtained using \texttt{NodeType}. They will be XORed with the sign bit of the $2^{\texttt{SeqNum}}$ input repetitions according to \eqref{eq:alphaS}. The XOR result of different repetition sequences are concatenated and expanded into a vector of length $2P$ by appending zeros if $2^{\texttt{SRstage}+\texttt{SeqNum}}<2P$.
Then, the LLR vector enters a $\left(1+\log_2 P\right)$-layer adder tree that performs the addition of LLRs in~\eqref{eq:alphaS}. The command signal $\text{Cmd}_1=7-\left(\texttt{SRstage}-\texttt{SourceStage}\right)$ is pre-calculated in the control module and it is used in the adder tree to decide the addition result of which layer will be output by a multiplexer. Those outputs from the adder tree are the input LLRs of the source node for different repetition sequences. In the considered example, there exist $2^{\texttt{SourceStage}+\texttt{SeqNum}}\leq16$ for SR node whose $\texttt{SRstage}\neq \texttt{SourceStage}$. Moreover, all LLRs are quantized using $Q$ bits. Thus, the data width of the adder tree output is $16Q$ bits.
\textbf{\emph{Step 2:}} This part of the SR node is used to perform the parity-check and bit-flipping steps for the source node. The LLRs of the source node first enter a {$\left(1+\log_2 P\right)$-layer} compare-select (CS) tree. Processing units in the CS tree execute the $f$ function to decode SPC node. There are two cases where more than one SPC nodes will be decoded in parallel in our design: 1) when $\texttt{FroNum}=1$ and $\texttt{SeqNum}>0$, there are $2^{\texttt{SeqNum}}$ SPC nodes which correspond to different repetition sequences and are decoded simultaneously, and 2) when $\texttt{FroNum}=2$ and $\texttt{FroNum}=3$, the decoding of source node can be viewed as a parallel decoding of 2 and 4 SPC nodes, respectively \cite{sr2020}. The length of the SPC node decides the layer from which the index of the least reliable input and the $f$ function result are selected. As the length of the SPC node can be calculated as $2^{\texttt{SourceStage}+1-\texttt{FroNum}}$, the output layer selection signal Cmd$_4$ has the following representation
\begin{align}
\text{Cmd}_4=\left\{
\begin{matrix}
7, &\texttt{FroNum}=0,\\
6-\texttt{SourceStage}+\texttt{FroNum}, & \text{otherwise}.
\end{matrix}
\right.
\end{align}
Since the maximum number of parallel SPC nodes in our example is 4, the output indices and LLRs have a data width of $4\times7$ and $4Q$ bits, respectively. Note that the output LLRs goes both to the parity check module and a 2-layer adder tree. This is because all SPC nodes have an even parity constraint except when $\texttt{FroNum}=3$, where SPC nodes can have an even or odd parity constraint which is calculated according to~\cite[(16)]{sr2020} and implemented by a 2-layer adder tree.
The parity constraint type, the output indices, and LLRs are then input into the parity-check submodule to do the parity check and bit flipping on these SPC nodes using~\cite[(13)]{sr2020}. Then, the estimated bits of these SPC nodes are concatenated to form the estimated bits of source node and they are XORed with the repetition sequence to generate the estimated bits of SR node in the SR bits generation submodule according to \eqref{eq:betaS}. Finally, the SR bits corresponding to the repetition sequence with the index value from Step 3 are selected as the output.
\textbf{\emph{Step 3:}} This part of the SR decoder is executed in parallel with Step 2 to evaluate~\eqref{eq:comparison} using a {$\left(\texttt{SourceStage}+\texttt{SeqNum}\right)_{\max}$-layer} adder tree and $\texttt{SeqNum}_{\max}$-layer CS tree, where $\left(\texttt{SourceStage}+\texttt{SeqNum}\right)_{\max}$ is the maximum value of $\left(\texttt{SourceStage}+\texttt{SeqNum}\right)$ for all SR nodes with $\texttt{SeqNum}>0$ and $\texttt{SeqNum}_{\max}$ denotes the maximum value of $\texttt{SeqNum}$.
As only magnitudes of LLRs are used for addition in \eqref{eq:comparison}, all inputs are positive. As a result, the processing unit in the 4-layer adder tree is simpler than that in the 7-layer adder tree in Step 1 because it does not need to compare magnitudes. The output of the adder tree is selected by the output layer selection signal Cmd$_2=4-\texttt{SourceStage}$ and has a bit-width of $4Q$ as there are at most 4 repetition sequences in the considered example. The four sums are then input into the 2-layer CS tree to find the index of the maximum using selection signal Cmd$_3=2-\texttt{SeqNum}$. Finally, the index is obtained from a multiplexer and the value is 0 if $\texttt{SeqNum}=0$ and the output from the CS tree otherwise.
\section{Implementation Results}
\label{sec:performance}
The proposed decoder has been implemented using VHDL and targeting an Altera Stratix IV EP4SGX530KH40C2 FPGA device. Channel LLRs are generated by transmitting random codewords through an additive white Gaussian noise (AWGN) channel after binary phase-shift keying (BPSK) modulation. A quantization scheme $Q\left(6,\;4,\;0\right)$ has
been used, where $Q\left(Q_i,\;Q_c,\;Q_f\right)$ are the quantization bit size
for internal LLRs, channel LLRs, and fraction bit size for
both internal and channel LLRs, respectively. This scheme leads to an error-correcting performance that is very close to that of the floating-point implementation, as shown in Fig.~\ref{fig:SR-performance}.
\begin{figure}
\leftline{
\includegraphics{./tikz/tikz_compiled1.pdf}
\includegraphics{./tikz/tikz_compiled2.pdf}}
\centering
\includegraphics{./tikz/tikz_compiled_crossref0.pdf}
\caption{Floating-point and fixed-point FER and BER performance for SRFSC decoding of 5G polar codes $\mathcal{P}\left(1024,512\right)$ \cite{3GPP1}.}
\label{fig:SR-performance}
\end{figure}
Table~\ref{tab:com2} compares the proposed decoder with other state-of-the-art works. As can be seen, the proposed SRFSC decoder provides a $17.9\%$ and $31.7\%$ throughput improvement compared to the architectures presented in \cite{Furkan2017} and \cite{sarkis2014fast}, respectively. This is mainly due to a $9.9\%$ and $22.4\%$ higher $f_{max}$ with respect to \cite{sarkis2014fast} and \cite{Furkan2017}. The number of CLKs in our work is slightly higher than that in \cite{Furkan2017}. This is because of the insertion of some registers to decrease certain critical paths and because we have not merged $f$ ad $g$ operations as was done in~\cite{Furkan2017}. In addition, a total of 186, 200 CLKs are required at rates $1/4$, $3/4$, respectively. In terms of the used LUTs, this work requires an increase of $23.2\%$ and $187.5\%$ compared to \cite{Furkan2017} and \cite{sarkis2014fast}, respectively. As far as the memory size is concerned, although our decoder uses fewer RAM bits, the required number of registers is about 8 times higher compared to~\cite{Furkan2017,sarkis2014fast}. The big difference in registers can be mostly attributed to the separate storage of channel and internal LLRs in synthesis. Internal LLRs are stored in RAM and channel LLRs are arranged in registers, while in other works both are stored in RAM.
\begin{table}[t]
\centering
\caption{FPGA Implementation Results for $\mathcal{P}\left(1024,512\right)$.}
\renewcommand\arraystretch{1.25}
\begin{tabular}{p{2.0cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}}
\toprule
& \cite{sarkis2014fast} & \cite{Furkan2017} & This Work \\
\midrule
Quantization & $Q\left(6,\;4,\;1\right)$ & $Q\left(6,\;4,\;1\right)$ & $Q\left(6,\;4,\;0\right)$\\
$P$ & 64 & 64 & 64 \\
LUTs & \textbf{6126} & 14300 & 17615 \\
Registers & 1223 & \textbf{1216} & 10505 \\
RAM (bits) & 23592 & 18350 & \textbf{16128} \\
Instruction size & \textbf{5} bits & 6 bits & 13 bits \\
$\#$ of Instruction & 209 & 157 & \textbf{41} \\
$\#$ of CLKs & 266 & \textbf{214} & 222 \\
$f_{\max}$ (MHz) & 99.8 & 89.6 & \textbf{109.6} \\
$T/P$ (Mps) & 384 & 428.6 & \textbf{505.6} \\
\bottomrule
\end{tabular}
\label{tab:com2}
\end{table}
\section{Conclusion}
\label{sec:Conclu}
In this paper, we presented the first FPGA implementation of the SRFSC decoder for polar codes.
To this end, we designed a dedicated architecture for the SR node processor. For a 5G polar code
with length 1024, code rate $1/2$ and $P=64$ processing units, we obtained a $17.9\%$ improvement in throughput over the previous work.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
1,314,259,993,009 | arxiv | \section{Introduction}\label{sec:introduction}
Networks naturally arise in the vast majority of scientific domains from physics to biology in order to model interactions among diverse entities with numerous instances such as collaboration, protein-protein, and brain connectivity networks \cite{newman}. Hence, graph analysis tools have become crucial to extract and analyze the underlying meaningful information from networks. In this direction, Graph Representation Learning (GRL) \cite{GRL-survey-ieeebigdata20} approaches have become a dominant way to carry out various downstream tasks such as node classification, link prediction, and community detection thanks to their superior performance compared to the classical techniques. GRL models mainly aim to map similar nodes in the network to close latent positions in a low dimension space by automatically learning corresponding node features \cite{survey_hamilton_leskovec}.
The initial GRL works aimed to learn representations or features by simulating random walks over networks, taking inspiration from the Natural Language Processing field \cite{deepwalk-perozzi14, node2vec-kdd16, expon_fam_emb, netmf-wsdm18, line}. They mainly extract embeddings by optimizing the co-occurrence probability of node pairs within a certain distance through random walks. In recent years, we have witnessed a tremendous increase in the number of Graph Neural Networks (GNN) \cite{survey_hamilton_leskovec} methods with their usage in supervised tasks. They primarily rely on iterative message-passing operations of node attributes and hidden features around the surroundings of nodes for a given task. The matrix decomposition-based models \cite{netmf-wsdm18, netsmf-www2019} are also a notable class of the GRL methods. They learn node embeddings by decomposing a designed target matrix based on first and higher-order proximities. However, few GRL methods rely on Non-negative Matrix Factorization (NMF), although it is a popular technique for unsupervised signal decomposition and approximation of multivariate non-negative data. NMF techniques have gathered lots of attention since they allow for structure retrieval through the latent factors of the imposed decomposition providing easy interpretable part-based representations\cite{lee99}.
Applications of NMF include network analysis allowing for efficient, unsupervised, and overlapping community detection, as well as GRL \cite{nmf1,nmf2,nmf3,nmf4}. Within the NMF formulation, various works have sought to define mixed-membership frameworks for analysis and community detection purposes. A mixed-membership Stochastic Block Model (SBM) \cite{JMLR:v9:airoldi08a} has been linked to the symmetric-NMF decomposition with uniqueness guarantees in \cite{nmf4}.
Standard least-squares NMF optimization was exchanged to a Poisson likelihood optimization for obtaining the propensity of nodes belonging to different communities in \cite{nmf1}. In addition, a GRL approach for overlapping communities was presented in \cite{nmf2} where NMF was utilized to discover Poisson distributed mixed-memberships. These works, design mixed-memberships vectors for part-based representations \cite{lee99} projected in an NMF constructed space where node similarity, as well as position and metric properties, can be abstract.
The Latent Space Models (\textsc{LSM}s) are also one of the most powerful ways to learn latent representations \cite{nakis2022hierarchical}. These methods employ generalized linear models for constructing latent node embeddings which express important network characteristics. More specifically, the \textsc{LDM} \cite{exp1} employs Euclidean norm for positioning similar nodes closer in the latent space, which comes as a direct consequence of the triangular inequality, naturally representing transitivity and homophily properties. The \textsc{LDM} can be generalized through the Eigenmodel \cite{hoff2007modeling} that can account for stochastic equivalence akin to the SBM \cite{JMLR:v9:airoldi08a} and the mixed membership SBM \cite{JMLR:v9:airoldi08a}.
Furthermore, \textsc{LDM}s have been endowed with a clustering model imposing a Gaussian Mixture Model as prior forming the latent position clustering model \cite{handcock2007model}
In this study, we propose a novel unsupervised representation learning method over graphs, namely, the \modelname \ (\textsc{\modelabbrv}), by bringing together the strengths of LDM and NMF.
Specifically, the \textsc{\modelabbrv} offers a reconciliation between part-based representations of networks and low-dimensional latent spaces satisfying similarity properties such as homophily and transitivity. The choice of these similarity properties is of high significance and one of the key characteristics behind GRL since they allow for easily interpretable discovery of network structure. Additionally, our proposed method permits us to capture the latent community structure of the networks using a simple continuous optimization procedure. Notably, unlike most existing approaches imposing hard community memberships constraints, the assignment of community memberships in our proposed
hybrid model can be controlled and altered through the simplex volume formed by the latent node representations. We extensively evaluate the performance of the proposed method in the ability to perform link prediction, as well as, community discovery over various networks of different types, and we demonstrate that our model outperforms recent methods.
\noindent\textbf{Source code:} \href{https://github.com/Nicknakis/Hybrib-Membership-Latent-Distance-Model}{\textit{Hybrid-Membership Latent Distance
Model}}.
\section{Problem statement and proposed method}\label{sec:method}
Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be an undirected graph where $\mathcal{V}$ shows the vertex set and $\mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$ the edge set. We use $\mathbf{Y}_{N \times N}=\left(y_{i,j}\right)\in \{0,1\}^{N\times N}$ to denote the adjacency matrix of the graph where $y_{i,j}=1$ and $y_{j,i}=1$ if the pair $(i,j) \in \mathcal{E}$ otherwise they are equal to $0$ for all $ 1\leq i< j\leq N := |\mathcal{V}|$. Our main goal is to learn a representation, $\mathbf{w}_i \in \mathbb{R}^{D}$, for each node $i \in \mathcal{V}$ in a lower dimensional space ($D \ll N$) such that similar nodes in the network should have close embeddings. More specifically, we concentrate on mapping the nodes into the unit $D$-simplex set, $\Delta^{D} \subset \mathbb{R}_{+}^{D+1}$.
Therefore, the extracted node embeddings can convey information about latent community memberships. Many GRL approaches also do not provide identifiable or unique solution guarantees, so their interpretation highly depends on the initialization of the hyper-parameters. In this study, we will also address the identifiability problem and seek identifiable solutions which can only be achieved up to a permutation invariance, as reported in Def. \ref{def:identifiabilty}.
\begin{definition}[\textbf{Identifiabilty}]\label{def:identifiabilty}
An embedding matrix $\mathbf{W}$ whose rows indicating the corresponding node representations is called an \textit{identifiable solution up to a permutation} if it satisfies $\widetilde{\mathbf{W}}=\mathbf{W}\mathbf{P}$ for a permutation $\mathbf{P}$ and a solution $\widetilde{\mathbf{W}} \not= \mathbf{W}$.
\end{definition}
We define a Poisson distribution over the adjacency matrix $\mathbf{Y}$ of the network $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ to be conditionally independent given the unobserved latent positions, and write the log-likelihood function as follows:
\begin{equation}
\label{eq:prob_adj}
\log P(\mathbf{Y}|\bm{\Lambda})=\!\!\sum_{\substack{i<j \\ y_{ij}=1}}\!\log(\lambda_{ij})\;-\;\sum_{\substack{i< j }}\Big(\lambda_{ij}+\log(y_{ij}!)\Big) \:.
\end{equation}
where $\bm{\Lambda}=(\lambda_{ij})$ is the Poisson rate matrix which has absorbed the dependency over the model parameters. We here adopted a Poisson regression model similar to the work in \cite{doi:10.1198/016214504000001015}. In this study, we make use of a Poisson likelihood for modelling binary networks, as validated in \cite{nmf2}.
We propose the \modelname \ (\textsc{\modelabbrv}) with a log-rate based on the $\ell^2$-norm as:
\begin{equation}
\label{eq:nmf_rate}
\log \lambda_{ij}=\Big(\gamma_i+\gamma_j-\delta^p\cdot||\mathbf{w}_i -\mathbf{w}_j||_2^p\Big),
\end{equation}
where $\mathbf{w_i} \in [0,1]^{D+1}$ and $\sum_{d=1}^{D+1} w_{id}=1$, $\delta \in \mathbb{R}_+$ and $\gamma_i \in \mathbb{R}$ denotes the node-specific random-effects \cite{doi:10.1198/016214504000001015,KRIVITSKY2009204} describing essentially the tendency of nodes to sending and receiving connections, accounting for degree heterogeneity. In addition, the norm degree $p \in \{1,2\}$ controls the power of the $\ell^2$-norm and combined with the latent embeddings sum-to-one condition constrains the latent space to the $D-$simplex with size equal to $\delta$.
A remarkable property of Eq. \eqref{eq:nmf_rate}, for $p=2$, is that it resembles a positive Eigenmodel with random effects: $ \tilde{\gamma}_i+\tilde{\gamma}_j+(\mathbf{\tilde{w}}_i\bm{\Lambda}\mathbf{\tilde{w}}_j^{\top})$ where $\bm{\Lambda}$ is a diagonal matrix having non-negative elements, i.e. $\tilde{\gamma}_i=\gamma_i-\delta^2\cdot||\mathbf{w}_i||^2_2$, $\tilde{\gamma}_j=\gamma_j-\delta^2\cdot||\mathbf{w}_j||^2_2$ and $\tilde{\mathbf{w}}_i\bm{\Lambda}\tilde{\mathbf{w}}_j^\top=2\delta^2\cdot \mathbf{w}_i\mathbf{w}_j^\top$ thus the squared Euclidean distance reconciles the conventional LDM and non-negativity constrained eigenmodel. The squared Euclidean distance is not fully a metric but it still expresses homophily, leading to an interpretable latent space. Even though the triangle inequality is not exactly satisfied, it preserves the ordering of pairwise Euclidean distances, and it is highly preferred in applications since it is a strictly convex smooth function. By the well-known cosine formula, we have
\begin{align}
||\mathbf{w}_i-\mathbf{w}_j||_2^2 &= ||\mathbf{w}_i-\mathbf{w}_k||_2^2+|| \mathbf{w}_k-\mathbf{w}_j ||_2^2+2||\mathbf{w}_i-\mathbf{w}_k||_2||\mathbf{w}_k-\mathbf{w}_j)||_2\cos(\theta)\nonumber
\end{align}
\noindent for $\theta \in (-\pi/ 2, \pi / 2)$ and the third term also converges to $0$ for similar nodes since we will have close representations. For the other case where $\theta \in [\pi/ 2, 3\pi / 2]$, it holds the triangle inequality: $||\mathbf{w}_i - \mathbf{w}_j ||_2^2 \leq || \mathbf{w}_i - \mathbf{w}_k ||_2^2 + || \mathbf{w}_k - \mathbf{w}_j ||_2^2$.
The embedding vectors, $\{\mathbf{w}_i\}_{i=1}^{N}$ in Eq. \eqref{eq:nmf_rate}, are constrained to non-negative values and to sum to one. Thereby, they reside on a simplex showing the participation of node $i\in\mathcal{V}$ over $D+1$ latent communities. Any \textsc{LDM} can be translated to the non-negative orthant without any loss in performance or in expressive capability. Non-negative embeddings do not affect the distance metric, as it is invariant to translation, as shown by Figure \ref{fig:invariances} (a). In addition, the $D$-dimensional non-negative orthant can be reconstructed by a large enough $D$-simplex. Based on these arguments, it is trivial to show that for large values of the $\delta$ parameter in Eq. \eqref{eq:nmf_rate}, despite the sum-to-one constraint on the embeddings $\mathbf{W}$, we obtain an unconstrained \textsc{LDM}, as distances are unbounded when $\delta\rightarrow+\infty$. In this case, the memberships defined by $\mathbf{W}$ are not uniquely identifiable due to the distance invariance of rotation, as seen in Figure \ref{fig:invariances} (b).
\begin{figure}[]
\centering
\subfloat[Translation Invariances.]{{
\includegraphics[width=0.20\columnwidth]{figures/sim1.png} }}%
\hfill
\subfloat[Rotation Invariances.]{{
\includegraphics[width=0.25\columnwidth]{figures/sim2.png} }}%
\hfill
\subfloat[Decreased simplex volume ensuring identifiability.]{{\includegraphics[width=0.34 \columnwidth]{figures/simplex2.png} }}%
\caption{A $2$-dimensional latent space with the $2$-simplex given as the green and yellow triangles, the blue points denote embedding positions of the $\textsc{LDM}$ and $\delta$ is the simplex size.}\label{fig:invariances}
\end{figure}
However, by shrinking the volume of the simplex (equivalent to decreasing $\delta$), eventually the $D$-dimensional space of \textsc{LDM} will no longer be enclosed inside the $D$-simplex, forcing nodes to start populating the corners of this smaller simplex. We call a node \textit{champion} if its latent representation is a standard binary unit vector.
\begin{definition}[\textbf{Community champion}]\label{def:champion}
A node for a latent community is called \textit{champion} if it belongs to the community (simplex corner) while forming a binary unit vector.
\end{definition}
The champion nodes are of great significance for identifiability because if every corner of the simplex is populated by at least one node (champion), then the solution of the model is identifiable (up to a permutation matrix) (Def. \ref{def:identifiabilty}) as any random rotation does not leave the solution invariant anymore, as shown by Figure \ref{fig:invariances} (c). We observe then, that the scalar, $\delta$, controls the type of memberships of the model and its expressive capabilities. Large enough values lead to the basic \textsc{LDM} but inherits its rotational invariance. Small values of $\delta$ lead to identifiable solutions and ultimately hard cluster assignments. Thereby, for very small values of $\delta$, nodes are solely assigned to the simplex corners. Lastly, we can also find regimes of values for $\delta$ that offer identifiable solutions but also performance similar to \textsc{LDM}, defining a silver lining.
A different take on the identifiability of the model for $p=2$, can also be given under the Non-negative Matrix Factorization (NMF) theory. This is easily shown by a re-parameterization of Eq. \eqref{eq:nmf_rate} by $\tilde{\gamma}_i+\tilde{\gamma}_j+2\delta^2\cdot(\mathbf{w}_i\mathbf{w}_j^{\top})$ as described in Eq. \eqref{eq:nmf_rate}. In this formulation, the product $\mathbf{W}\mathbf{W}^{\top}$ defines a symmetric NMF problem which is an identifiable and unique factorization (up to permutation invariance) when $\mathbf{W}$ is full-rank and at least one node resides solely in each simplex corner, ensuring separability \cite{nmf4,nmf5}. Under this NMF formulation, the product $\mathbf{w}_i\mathbf{w}_j^{\top} \in [0,1]$ achieves its upper bound only if both nodes $i$ and $j$ reside in the same corner of the simplex. The parameter, $\delta$, acts as a simple multiplicative factor in the first term of the objective function of \textsc{\modelabbrv}, given in Eq. \ref{eq:prob_adj}, while in the second term acts as a power of the exponential function. For small values of $\delta$, the model is biased towards hard latent community assignments of nodes since similar nodes achieve high rates only when they belong to the same latent community (simplex corner). On the other hand, nodes heading towards the simplex corners for large values of $\delta$ lead to an exponential change in the second term of the log-likelihood function given in Eq. \ref{eq:prob_adj}. Thus, a possible hard allocation of dissimilar nodes to the same community penalizes the likelihood severely. For this reason, high order of $\delta$ benefits mixed-membership allocations.
\section{Experimental evaluation}\label{sec:experimental_evaluations}
We proceed by evaluating the efficiency and performance of the proposed method.
In our set-up, we make use of networks with unknown community structures, as well as, with ground-truth communities. We employ the former networks to validate the ability of our framework to discover identifiable latent structures and predict missing links. The latter networks are used to verify that the $\textsc{\modelabbrv}$ discovers communities successfully. We consider multiple social and scientific collaboration networks as shown by Table \ref{tab:network_statistics}. In all experiments, we compare against unsupervised methods, and we do not include GNNs since they perform poorly in such settings due to the over-smoothing \cite{gnn_bad}. Finally, we treat all networks as unweighted and undirected.
\begin{table}[!t]
\centering
\caption{Network statistics; $|\mathcal{V}|$: \# Nodes, $ |\mathcal{E}|$: \# Edges, $|\mathcal{K}|$: \# Communities.}
\label{tab:network_statistics}
\resizebox{1\textwidth}{!}{%
\begin{tabular}{rcccccccc}\toprule
& \textsl{AstroPh}\cite{snapnets} & \textsl{GrQc}\cite{snapnets} & \textsl{Facebook}\cite{snapnets} & \textsl{HepTh}\cite{snapnets}& \textsl{Hamilton}\cite{fb_nets} & \textsl{Amherst}\cite{fb_nets} & \textsl{Rochester}\cite{fb_nets} & \textsl{Mich}\cite{fb_nets} \\\midrule
$|\mathcal{V}|$ & 17,903 & 5,242 & 4,039 & 8,638 & 2,118 & 2,021 & 4,145 & 2,933 \\
$|\mathcal{E}|$ & 197,031 & 14,496 & 88,234 & 24,827 & 87,486 & 87,496 & 145,305 & 54,903 \\
$|\mathcal{K}|$& - & - & - & - & 15 & 15 & 19 & 13 \\\bottomrule
\end{tabular}%
}
\end{table}
\textbf{Link prediction:}
For the link prediction experiments, we follow the well-established strategy \cite{deepwalk-perozzi14, node2vec-kdd16} and remove $50\%$ of the network edges while keeping the residual network connected. We consider four networks with unknown community structures and asses performance across different dimensions. In Table \ref{tab:auc_roc}, we compare the results of our method with other prominent GRL and NMF approaches. All the baselines have been tuned and feature vectors for dyads are constructed based on binary operators as in \cite{node2vec-kdd16}. In particular, for the baselines we choose the hyperparameter settings, as well as, the binary operator that returns the maximum AUC-ROC. In contrast, for our methods we adopt an unbiased evaluation, and we choose the first of the considered $\delta$ values which keeps the solution identifiable (at least one champion per community), as $\delta$ decreases. We note though, the existence of identifiable regimes with higher predictive power. The true dimensions for \textsc{\modelabbrv} are $D+1$ but reported as $D$ since they express the true number of model parameters, for a fair comparison with the baselines. For our method, we show the mean performance over five independent runs (error bars were found to be in the scale of $10^{-3}$ and thus not presented). When comparing with the non-NMF models, we observe that our \textsc{\modelabbrv} (either $p=1$ or $p=2$) outperforms the baselines and in most cases significantly, returning favorable results. For the NMF models, we see mostly a big performance gap with the \textsc{\modelabbrv}, showcasing the existence of regimes for $\delta$ where we can successfully achieve identifiable community memberships while also exhibiting the link prediction power of the \textsc{LDM}. (AUC Precision-Recall scores are similar to the AUC-ROC scores and thus not presented)
\begin{table*}[]
\centering
\caption{Area Under Curve (AUC-ROC) scores for varying representation sizes.}
\label{tab:auc_roc}
\resizebox{0.70\textwidth}{!}{%
\begin{tabular}{rcccccccccccc}\toprule
\multicolumn{1}{l}{} & \multicolumn{3}{c}{\textsl{AstroPh}} & \multicolumn{3}{c}{\textsl{GrQc}} & \multicolumn{3}{c}{\textsl{Facebook}}& \multicolumn{3}{c}{\textsl{HepTh}}\\\cmidrule(rl){2-4}\cmidrule(rl){5-7}\cmidrule(rl){8-10}\cmidrule(rl){11-13}
\multicolumn{1}{r}{Dimension ($D$)} & $8$ & $16$ & $32$ & $8$ & $16$ & $32$ & $8$ & $16$ & $32$& $8$ & $16$ & $32$ \\\cmidrule(rl){1-1}\cmidrule(rl){2-2}\cmidrule(rl){3-3}\cmidrule(rl){4-4}\cmidrule(rl){5-5}\cmidrule(rl){6-6}\cmidrule(rl){7-7}\cmidrule(rl){8-8}\cmidrule(rl){9-9}\cmidrule(rl){10-10}\cmidrule(rl){11-11}\cmidrule(rl){12-12}\cmidrule(rl){13-13}
\textsc{DeepWalk}\cite{deepwalk-perozzi14} &.945 &.950 &.952 & .919 &.916 & .929 & .986 & .986 & .984 &.874 &.867 & .873 \\
\textsc{Node2Vec}\cite{node2vec-kdd16} &.950 &\underline{.962} &\underline{.957} & .897 &.913 &.930 & \underline{.988} & \underline{.988} & \underline{.987} &.881 &.882 &.881 \\
\textsc{LINE} \cite{line} &.909 &.938 &.947 & .920 &.925 &.919 & .981 & .987 & .983 &.873 &.886 &.882 \\
\textsc{NetMF} \cite{netmf-wsdm18} &.813 &.823 &.839 & .860 & .866&.877 & .935 & .963 & .971 &.792&.806 &.821\\
\textsc{NetSMF} \cite{netsmf-www2019} & .891 &.901 &.919 & .837&.858 &.886 &.975 &.981 &.985 & .809&.822 &.836 \\
\textsc{LouvainNE}\cite{louvainNE-wsdm20} & .813 &.811 &.819 & .868 &.875 &.873 & .958 &.961 &.963 &.874 &.867 &.873 \\
\textsc{ProNE}\cite{prone-ijai19} & .907 &.929 &.947 & .885 &.911 &.921 & .971 & .982 & .987 &.827 &.846 &.859 \\\midrule
\textsc{NNSED}\cite{NNSED} &.861 &.882 &.891 & .792 &.808 &.828 &.908 &.927 &.935 & .756 & .779 &.796 \\
\textsc{MNMF}\cite{MNMF} & .893 &.925 &.943 & .911 &.928 &.937 &.965 &.978 &.982 & .857 &.880 &.891 \\
\textsc{BigClam}\cite{nmf3} &.500 &.723 &.810 & .752 &.769 &.780 & .744 &.722 &.647 & .776 &.700 &.748 \\
\textsc{SymmNMF}\cite{SymmNMF} &.767 &.779 &.800 & .729 &.772 &.835 & .933 &.942 &.951 & .696 &.727 &.766 \\\midrule
\textsc{HM-LDM($p=1$)} & \underline{.956} &.952 &.952 &\textbf{.944} &\textbf{.948} &\textbf{.951} & .982 & .979 & .974 &\textbf{.916} & \textbf{.921} &\textbf{.924}
\\
\textsc{\modelabbrv ($p=2$)} &\textbf{.972} &\textbf{.973} & \textbf{.963} &\underline{.940} & \underline{.942} & \underline{.946} & \textbf{.992} & \textbf{.993} & \textbf{.993} &\underline{.908} &\underline{.910} &\underline{.911}
\\\bottomrule
\end{tabular}%
}
\end{table*}
\textbf{Performance and simplex sizes:} In Figure \ref{fig:roc_d}, we provide the link prediction performance as a function of $\delta^2$ in terms of the Area Under Curve-Receiver Operating Characteristic (AUC-ROC) scores across various latent dimensions, networks and for both $p=1$ and $p=2$. We here observe that small $\delta$ values provide the minimum scores. This phenomenon is anticipated due to the fact that homophily properties are not sufficiently met (except within clusters) due to the very small simplex volume that these low $\delta$ values define. Rethinking \textsc{\modelabbrv} with $p=2$ as a positive Eigenmodel, we can also notice how the positivity constraint on the $\Lambda$ diagonal matrix does not allow for stochastic equivalence properties which would essentially boost performance even on low simplex volumes. As we increase the values of $\delta$, we naturally reach the performance of an unconstrained \textsc{LDM}. Comparing now, the squared and simple $\ell^2$-norm metric we observe that the former converges to performance saturation more rapidly.
\begin{figure}[!ht]
\centering
\subfloat[\textsl{AstroPh}]{{\includegraphics[width=0.22\columnwidth]{figures/roc_astroph.pdf} }}%
\hfill
\subfloat[\textsl{Facebook}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_facebook.pdf} }}%
\hfill
\subfloat[\textsl{GrQc}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_grqc.pdf} }}%
\hfil
\subfloat[\textsl{HepTh}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_hepth.pdf} }}%
\hfill
\subfloat[\textsl{AstroPh}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_astroph_l2.pdf} }}%
\hfill
\subfloat[\textsl{Facebook}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_facebook_l2.pdf} }}%
\hfill
\subfloat[\textsl{GrQc}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_grqc_l2.pdf} }}%
\hfil
\subfloat[\textsl{HepTh}]{{ \includegraphics[width=0.22\columnwidth]{figures/roc_hepth_l2.pdf} }}%
\caption{AUC-ROC scores as a function of $\delta^2$ across dimensions for \textsc{\modelabbrv}. Top row: $p=2$. Bottom row $p=1$.}\label{fig:roc_d}
\end{figure}
\textbf{Type and quality of latent memberships:} In order to understand how the size of the simplex affects the membership types of \textsc{\modelabbrv}, we provide in Figure \ref{fig:phase_transitions} the total network percentage of community champions as a function of $\delta^2$ across various latent dimensions. As expected, for very small values of $\delta$ almost $100\%$ of nodes are assigned solely to a unique simplex corner, yielding hard cluster assignments. As we increase $\delta$, we observe that more and more nodes are assigned with mixed-memberships; on the other hand, the number of champions goes to zero across all dimensions for large values of $\delta$. Contrasting again, the different powers $p$ of the \textsc{\modelabbrv} formulation, we notice that the decrease in community champions is steeper for $p=2$. This also explains why the squared $\ell^2$ choice leads to faster convergence in the AUC-ROC, as the model converges faster to the classic \textsc{LDM}. Overall, it is evident that the $p=2$ \textsc{\modelabbrv} needs smaller simplex volumes to be identifiable. We continue with assessing unique latent structures of \textsc{\modelabbrv}. For that purpose, in Figure \ref{fig:adj} we provide the reorganized adjacency matrices with respect to the community allocations of \textsc{\modelabbrv} (for mixed-memberships we assign a node based on the maximum membership). We witness how \textsc{\modelabbrv} successfully discovers latent communities, facilitating part-based network representations while choosing appropriate $\delta$ regimes ensure identifiability.
\begin{figure} [!b]
\centering
\subfloat[\textsl{AstroPh}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_astroph.pdf} }}%
\hfill
\subfloat[\textsl{Facebook}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_facebook.pdf} }}%
\hfill
\subfloat[\textsl{GrQc}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_grqc.pdf} }}%
\hfill
\subfloat[\textsl{HepTh}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_hepth.pdf} }}%
\hfill
\subfloat[\textsl{AstroPh}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_astroph_l2.pdf} }}%
\hfill
\subfloat[\textsl{Facebook}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_facebook_l2.pdf} }}%
\hfill
\subfloat[\textsl{GrQc}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_grqc_l2.pdf} }}%
\hfill
\subfloat[\textsl{HepTh}]{{ \includegraphics[width=0.22\columnwidth]{figures/avg_champions_hepth_l2.pdf} }}%
\caption{Total community champions (\%) in terms of $\delta^2$ across dimensions for \textsc{\modelabbrv}. Top row: $p=2$. Bottom row $p=1$.} \label{fig:phase_transitions}
\end{figure}
\begin{figure}
\centering
\hfil
\subfloat[\textsl{GrQc} $(p=2)$]{{ \includegraphics[width=0.20\columnwidth]{figures/grqc.jpg} }}%
\hfil
\subfloat[\textsl{HepTh} $(p=2)$]{{ \includegraphics[width=0.20\columnwidth]{figures/hepth.jpg} }}%
\hfil
\subfloat[\textsl{GrQc} $(p=1)$]{{ \includegraphics[width=0.20\columnwidth]{figures/grqc_l2.jpg} }}%
\hfil
\subfloat[\textsl{HepTh} $(p=1)$]{{ \includegraphics[width=0.20\columnwidth]{figures/hepth_l2.jpg} }}%
\caption{Ordered adjacency matrices based on the memberships of a $D=16$ dimensional \textsc{\modelabbrv} with $\delta$ values ensuring identifiability.}\label{fig:adj}
\end{figure}
\textbf{Experiments using real ground-truth communities:} In order to assess the ability of \textsc{HM-LDM} to discover informative communities, we make use of $4$ networks providing ground-truth community labels.
For the NMF-based methods, including ours, we test the ability of the algorithms to detect valid structures by comparing the inferred memberships with the ground-truth community labels while we set the latent dimensions to be equal to the total number of communities. For the GRL approaches which do not define memberships, we extract latent embeddings and use {\it k-means} (average over 20 runs for robustness) to obtain memberships. We report the Normalized Mutual Information (NMI) score, as well as, the Adjusted Rand Index (ARI), both measures have been validated for community quality assessment in \cite{com_metrics}. Again, all the baselines have been tuned individually for each network in terms of their hyperparameters. In contrast, for our \textsc{HM-LDM} we do not perform any tuning and we just set $\delta=1$ for all networks since this choice provides in general informative and mostly hard cluster assignments. For our method and the classic LDMs we report scores averaged over five independent runs in each of which we run the algorithm five times extracting the model with the lowest training loss to remove the effect of local-minimas. We summarize our findings in Table \ref{tab:nmi_ari}, where we witness mostly favorable or on par performance of \textsc{\modelabbrv} with all of the competitive baselines for the NMI metric. For the ARI metric we observe that our framework outperforms significantly the baselines in all of the considered networks
\begin{table*}[]
\centering
\caption{Normalized Mutual Information (NMI) and Adjusted Rand Index (ARI) scores for networks with ground-truth communities.}
\label{tab:nmi_ari}
\resizebox{0.5\textwidth}{!}{%
\begin{tabular}{rcccccccc}\toprule
\multicolumn{1}{l}{} & \multicolumn{2}{c}{\textsl{Amherst}} & \multicolumn{2}{c}{\textsl{Rochester}} & \multicolumn{2}{c}{\textsl{Mich}}& \multicolumn{2}{c}{\textsl{Hamilton}}\\\cmidrule(rl){2-3}\cmidrule(rl){4-5}\cmidrule(rl){6-7}\cmidrule(rl){8-9}
\multicolumn{1}{r}{Metric} & NMI & ARI & NMI & ARI & NMI & ARI & NMI & ARI \\\cmidrule(rl){1-1}\cmidrule(rl){2-2}\cmidrule(rl){3-3}\cmidrule(rl){4-4}\cmidrule(rl){5-5}\cmidrule(rl){6-6}\cmidrule(rl){7-7}\cmidrule(rl){8-8}\cmidrule(rl){9-9}
\textsc{DeepWalk}\cite{deepwalk-perozzi14} &.498 &.347 &.348 & .205 &.207 &.157 & .447 &.303 \\
\textsc{Node2Vec}\cite{node2vec-kdd16} &.535 &.375 & .364 & .223 & .217 &.161 &.481 &.348 \\
\textsc{LINE} \cite{line} & .549 &.452 & .365 & .217 &\textbf{.249} &.192 &.499 &.411 \\
\textsc{NetMF} \cite{netmf-wsdm18} &.491 &.330 &.377 & .243 & .237 &.136 &.456 & .297 \\
\textsc{NetSMF} \cite{netsmf-www2019} &\underline{.562} &.408 &\underline{.381} & .228 & \underline{.242} &.169 & .494 & .391 \\
\textsc{LouvainNE}\cite{louvainNE-wsdm20} & \underline{.562} &.395 & .347 &.204 &.175 &.114 &.475 & .334 \\
\textsc{ProNE}\cite{prone-ijai19} & .536 & .443 &.356 & .312 &.229 &\underline{.200} &.478 & .396 \\\midrule
\textsc{NNSED}\cite{NNSED} &.295 & .243 &.168 &.116 & .064 &.035 & .335 & .285 \\
\textsc{MNMF}\cite{MNMF} & .542 &.362 & .324 & .171 &.188 &.102 & .466 & .287 \\
\textsc{BigClam}\cite{nmf3} & .091 &.066 & .028 & .022 &.024 &.015 & .053 &.041 \\
\textsc{SymmNMF}\cite{SymmNMF} &\textbf{.596} &.397 & .308 &.175 &.207 & .088 & .437 &.341 \\\midrule
\textsc{HM-LDM($p=1$)} & \underline{.562} &\underline{.502} &\textbf{.400} &\textbf{.392} &.228 &\textbf{.205} & \textbf{.527} & \underline{.485}
\\
\textsc{HM-LDM($p=2$)} &.539 &\textbf{.506} &\underline{.384} & \underline{.373} &.217 &.183 &\underline{.507} & \textbf{.504}
\\\bottomrule
\end{tabular}%
}
\end{table*}
\textbf{Comparison with the \textsc{LDM}:} We further investigate the performance of \textsc{HM-LDM} against the LDM, including random effects for a fair comparison and for both normal and squared $\ell^2$-norm \textsc{LDM-Re} and \textsc{LDM-Re}-$(\ell^2)^2$, respectively. Towards that aim, in Table \ref{tab:auc_roc_comp} and Table \ref{tab:nmi_ari_comp} we provide the performance scores for the link prediction and clustering tasks of each model. We here witness that constraining the latent space in identifiable simplex volumes leads to minor decrease in the predictive power, in terms of the AUC-ROC. For the community detection task, we see favorable NMI scores while the \textsc{\modelabbrv} leads to considerably higher ARI scores. Comparing the classical LDM
with \textsc{\modelabbrv} for $\delta^2=10^3$
provides on par link-prediction performance but the clustering scores drop significantly. This is expected as for large simplex volumes the \textsc{\modelabbrv} approximates almost exactly the LDM with the cost of identifiability
\textbf{Extension to bipartite networks:} Finally, we showcase the extension of our \textsc{\modelabbrv} framework to the analysis of bipartite networks. This is straightforward by introducing a different set of latent variables for the two disjoint set of nodes, as defined by the bipartite structure. In particular,
\textsc{\modelabbrv} for $p=2$,
simply extends the symmetric NMF formulation, obtained for the undirected networks, to the non-symmetric NMF specification. In Figure \ref{fig:bip}, we provide the re-ordered adjacency matrix with respect to the community allocations defined by the learned embeddings of \textsc{\modelabbrv} for a \textsl{Drug-Gene} \cite{snapnets} network ($|\mathcal{V}|=7,341|$, $|\mathcal{E}|=15,138$) where we observe a clear block structure. Importantly, the \textsc{\modelabbrv} offers identifiable joint embedding representations, mixed memberships, and community discovery for bipartite networks, tasks considered to be non-trivial and arduous.
\begin{table*}[]
\centering
\caption{\textsc{\modelabbrv} and \textsc{LDM-Re} comparison for the link prediction task.}
\label{tab:auc_roc_comp}
\resizebox{0.75\textwidth}{!}{%
\begin{tabular}{lcccccccccccc}\toprule
\multicolumn{1}{l}{} & \multicolumn{3}{c}{\textsl{AstroPh}} & \multicolumn{3}{c}{\textsl{GrQc}} & \multicolumn{3}{c}{\textsl{Facebook}}& \multicolumn{3}{c}{\textsl{HepTh}}\\\cmidrule(rl){2-4}\cmidrule(rl){5-7}\cmidrule(rl){8-10}\cmidrule(rl){11-13}
\multicolumn{1}{r}{Dimension ($D$)} & $8$ & $16$ & $32$ & $8$ & $16$ & $32$ & $8$ & $16$ & $32$& $8$ & $16$ & $32$ \\\cmidrule(rl){1-1}\cmidrule(rl){2-2}\cmidrule(rl){3-3}\cmidrule(rl){4-4}\cmidrule(rl){5-5}\cmidrule(rl){6-6}\cmidrule(rl){7-7}\cmidrule(rl){8-8}\cmidrule(rl){9-9}\cmidrule(rl){10-10}\cmidrule(rl){11-11}\cmidrule(rl){12-12}\cmidrule(rl){13-13}
\textsc{LDM-Re} &.973 &.974 &.979 & .949 &.952 &.954 & .993 & .994& .992 &.920 &.923 &.923 \\
\textsc{HM-LDM}($p=1,\delta^2=\text{identifiable}$) & .956 &.952 &.952 &.944 &.948 &.951 & .982 & .979 & .974 &.916 & .921 &.924
\\
\textsc{HM-LDM}($p=1,\delta^2=10^3$) &.967 & .967 & .965 & .956 & .955 & .951 & .985 & .986 & .987 & .932 & .931 & .926
\\\midrule
\textsc{LDM-Re}-$(\ell^2)^2$ &.979 &0.978 &.976 & .944 &.944 & .945 & .990 & .990 & .991 & .913 &.912 &.909\\
\textsc{\modelabbrv}($p=2,\delta^2=\text{identifiable}$) &.972 &.973 & .963 &.940 & .942 & .946 & .992 & .993 & .993 &.908 &.910 &.911
\\
\textsc{\modelabbrv}($p=2,\delta^2=10^3$) & .984& .983 & .980 & .948 & .946 & .946 & .991 & .991 & .992 & .920 & .918 & .913
\\\bottomrule
\end{tabular}%
}
\end{table*}
\begin{table*}[]
\centering
\caption{\textsc{\modelabbrv} and \textsc{LDM-Re} comparison for the clustering task.}
\label{tab:nmi_ari_comp}
\resizebox{0.65\textwidth}{!}{%
\begin{tabular}{lcccccccc}\toprule
\multicolumn{1}{l}{} & \multicolumn{2}{c}{\textsl{Amherst}} & \multicolumn{2}{c}{\textsl{Rochester}} & \multicolumn{2}{c}{\textsl{Mich}}& \multicolumn{2}{c}{\textsl{Hamilton}}\\\cmidrule(rl){2-3}\cmidrule(rl){4-5}\cmidrule(rl){6-7}\cmidrule(rl){8-9}
\multicolumn{1}{r}{Metric} & NMI & ARI & NMI & ARI & NMI & ARI & NMI & ARI \\\cmidrule(rl){1-1}\cmidrule(rl){2-2}\cmidrule(rl){3-3}\cmidrule(rl){4-4}\cmidrule(rl){5-5}\cmidrule(rl){6-6}\cmidrule(rl){7-7}\cmidrule(rl){8-8}\cmidrule(rl){9-9}
\textsc{LDM-Re} & .548 &.366 &.391 &.212 &.230 &.132 & .491 & .320 \\ \textsc{HM-LDM}($p=1,\delta^2=\text{identifiable}$) & .562 &.502 &.400 &.392 &.228 &.205 & .527 & .485\\
\textsc{HM-LDM}($p=1,\delta^2=10^3$) & .439 &.386 & .308 &.303 &.176 & .133 & .405 & .377
\\
\midrule
\textsc{LDM-Re}-$(\ell^2)^2$ &.546 &.370 &.393 &.211 &.231 &.137 &.497 &.327 \\
\textsc{\modelabbrv}($p=2,\delta^2=\text{identifiable}$) &.539 &.506 &.384 & .373 &.217 &.183 &.507 & .504
\\
\textsc{HM-LDM}($p=2,\delta^2=10^3$) &.240 &.133 &.206 & .119 & .116 &.056 & .232 &.209
\\
\bottomrule
\end{tabular}%
}
\end{table*}
\begin{figure}
\centering
\subfloat[$p=1$, $\delta=1$]{{ \includegraphics[width=0.30\columnwidth]{figures/l2_drug_gene_1.jpeg} }}%
\hspace{0.1cm
\subfloat[$p=2$, $\delta=1$]{{ \includegraphics[width=0.30\columnwidth]{figures/drug_gene_1.jpeg} }}%
\caption{\textsl{Drug-Gene} ordered adjacency matrices based on \textsc{\modelabbrv} with $D=8$.}\label{fig:bip}
\end{figure}
\section{Conclusion}\label{sec:conclusion}
In this paper, we have proposed the \textsc{\modelabbrv} that reconciles network embedding and latent community detection. The approach utilizes both the normal and squared Euclidean distance model where the latter integrated the non-negativity constrained latent eigen-model with the latent distance model. We demonstrated that the model could be constrained to the simplex without losing expressive power. The reduced simplex provides unique representations, ultimately resulting in hard clustering of nodes to communities when the simplex is sufficiently shrunk. Notably, the proposed \textsc{\modelabbrv} combines network homophily and transitivity properties with latent community detection enabling explicit control of soft and hard assignment through the volume of the induced simplex. We observed favorable link prediction performance in regimes in which the \textsc{\modelabbrv} provides unique representations while enabling the ordering of the adjacency matrix in terms of prominent latent communities. Finally, we showed the ability of the model to extract correct community structures across multiple networks and showcased how the analysis extends to bipartite networks.
\section{Problem Statement}\label{sec:problem_statement}
In this study, we aim to offer a reconciliation between part-based embedding representations of networks and low-dimensional latent spaces obeying similarity properties such as homophily and transitivity. The choice of these similarity properties are of high importance since they should allow for network projections interpretable and explainable by human perception, a characteristic considered as one of the main intuitions behind GRL. In addition, we aim to lift the hard constraint imposed by prominent GRL methods where models should define either mixed or unique community memberships. For that, we propose a hybrid model where the assignment of community-memberships can be controlled and altered through the volume of the latent space.
\section{Related Work}\label{sec:relate_work}
Under the NMF formulation, various works have tried to define mixed-membership frameworks for the analysis and community detection in complex networks. In \cite{nmf4}, a mixed-membership SBM has been studied under a symmetric-NMF decomposition with uniqueness guarantees, achieving part-based community assignments for networks. In \cite{nmf1}, standard least-squares NMF optimization was exchanged to a Poisson likelihood optimization in order to find the propensity of nodes belonging to different communities. In addition, a GRL approach for overlapping communities was presented in \cite{nmf2} where NMF was used to discover weighted latent memberships under a likelihood optimization scheme of Poisson distributed memberships. All of the previous works, create mixed-memberships vectors for part-based representations projected in an NMF constructed space where node similarity, as well as, position and metric properties can be abstract. |
1,314,259,993,010 | arxiv | \section{Introduction}
Given a graph $G=(V,E)$ with two distinguished vertices $s,t\in V$ and an
integer $L$, an {\em $L$-bounded flow} is a flow between $s$ and $t$ that can
be decomposed into paths of length at most $L$. In the {\em maximum
$L$-bounded flow problem} the task is to find a maximum $L$-bounded flow
between a given pair of vertices in the input graph.
The $L$-bounded flow was first studied, as far as we know, in 1971 by Ad\'amek
and Koubek~\cite{adamek1971remarks}. In connection with telecommunication
networks, $L$-bounded flows in networks with unit edge lengths have been widely
studied and are known as \emph{hop-constrained} flows~\cite{BleyNeto}.
For networks with unit edge lengths (or, more generally, with polynomially
bounded edge lengths, with respect to the number of vertices), the problem can
be solved in polynomial time using linear programming. Linear programming is a
very general tool that does not make use of special properties of the problem
at hand. This often leaves space for superior combinatorial algorithms that do
exploit the structure of the problem. For example, maximum flow, matching,
minimum spanning tree or shortest path problems can all be described as linear
programs but there are many algorithms that outperform general linear
programming approaches. However, as far as we know, no polynomial-time
combinatorial algorithm\footnote{Combinatorial in the sense that it does not
explicitly use linear programming methods or methods from linear algebra or
convex geometry.} for the $L$-bounded flow is known.
\subsection{Related results}
For clarity we review the definitions of a few more terms that are used in this
paper.
A {\em network} is a quintuple $G = (X, R, c, s, t) $, where $G = (X, R)$ is
a directed graph, $X$ denotes the set of vertices, $R$ the set of edges, $c$ is
the edge capacity function $c: R \rightarrow \mathbb{R}^+$, $s$ and $t$ are two
distinguished vertices called the source and the sink. We use $m$ and $n$ to denote the
number of edges and the number of vertices, respectively, in the network $G$, that is,
$m=|R|$ and $n=|X|$.
Given an $L$-bounded flow $f$, we denote by $|f|$ the size of the flow, and for
an edge $e\in R$, we denote by $f(e)$ the total amount of flow $f$ through the
edge $e$.
An {\em $L$-bounded flow problem with edge lengths} is a generalization of the
$L$-bounded flow problem: each edge has also an integer length and the length
of a path is computed not with respect to the number of edges on it but with
respect the sum of lengths of edges on it.
Given a network $G$ and an integer parameter~$L$, an {\em $L$-bounded cut} is a
subset $C$ of edges $R$ in $G$ such that there is no path from $s$ to $t$ of
length at most $L$ in the network ${G = (X, R \setminus C, c, s, t)}$. The
objective is to find an $L$-bounded cut of minimum size. We sometimes
abbreviate the phrase $ L $-bounded cut to {$ L $-cut} and, similarly, we abbreviate the phrase
$ L $-bounded flow to $ L $-flow .
Although the problems of finding an $ L $-flow and an $ L $-cut are easy to define
and they have been studied since the 1970's, still some fundamental open
problems remain unsolved. Here we briefly survey the main known results.
\paragraph{L-bounded flows}
As far as we know, the $L$-bounded flow was first considered in 1971 by Ad\'amek and
Koubek~\cite{adamek1971remarks}. They published a paper introducing the $ L $-bounded flow
s and cuts and describing some interesting properties of them. Among other
results, they show that, in contrast to the ordinary flows and cuts, the
duality between the maximum $ L $-flow and the minimum $L$-cut does not hold.
The maximum $ L $-flow can be computed in polynomial time using linear
programming~\cite{baier2010length,kolman2006improved,baier2010length,mahjoub2010max}.
The only attempt, that we are aware of, to describe a combinatorial algorithm
for the maximum $L$-bounded flow problem was done by Koubek and \v R\'\i ha in
1981~\cite{koubek1981maximum}.
The authors say the algorithm finds a
maximum $ L $-flow in time $ O(m \cdot |I|^{2}\cdot S/\psi(G)) $, where $I$ denotes
the set of paths in the constructed $L$-flow, $S$~is the size of the maximum
$ L $-flow, and $ \psi(G) = \min( |c(e) - c(g)|: c(e) \neq c(g), e, g \in R \cup
\{e'\} ) $, where $ c(e') = 0 $.
Unfortunately, their paper contains
substantional flaws and the algorithm does not work as we show in the first
part of this paper. Thus, it is a challenging problem to find a polynomial time
combinatorial algorithm for the maximum $L$-bounded flow.
Surprisingly, the maximum $L$-bounded flow problem with edge lengths is
NP-hard~\cite{baier2010length} even in outer-planar graphs.
Baier~\cite{baier2003flows} describes a FPTAS for the maximum $L$-bounded flow
with edge lengths that is based on the ellipsoid algorithm. He also shows that
the problem of finding a decomposition of a given $L$-bounded flow into paths
of length at most $L$ is NP-hard, again even if the graph is outer-planar.
A related problem is that of $L$-bounded disjoint paths: the task is to find
the maximum number of vertex or edge disjoint paths, between a given pair
of vertices, each of length at most $L$. The vertex version of the problem is
known to be solvable in polynomial time for $L\leq 4$ and NP-hard for
$L\geq 5$~\cite{itai1982complexity}, and the edge version is solvable
in polynomial time for $L\leq 5$ and NP-hard for $L\geq 6$~\cite{Bley:03}.
\paragraph{L-bounded cuts}
The $ L $-bounded cut problem is NP-hard
\cite{Schieber:1995:CFM:241577}. Baier et al.~\cite{baier2010length} show that
it is NP-hard to approximate it by a factor of
$1.377$ for $L\geq 5$ in the case of the vertex $L$-cut, and for $L\geq
4$ in the case of the edge $L$-cut. Assuming the Unique Games
Conjecture, Lee at al.~\cite{Lee2017ImprovedHF} proved that the minimum $ L $-bounded cut \
problem is NP-hard to approximate within any constant factor. For planar
graphs, the problem is known to be NP-hard~\cite{FHNN:15,ZFMN:17}, too.
The best approximations that we are aware of are by Baier et
al.~\cite{baier2010length}: they describe an algorithm with an
$\mathcal{O}(\min\{L, n/L\}) \subseteq \mathcal{O}(\sqrt{n})$-approximation for
the $L$-bounded vertex cut, and
$\mathcal{O}(\min\{L, n^{2}/L^{2}, \sqrt{m}\}) \subseteq
\mathcal{O}(n^{2/3})$-approximation for the $L$-bounded edge cut.
The approximation factors are closely related with the cut-flow gaps:
there are instances where the minimum edge
$L$-cut (vertex $L$-cut) is $\Theta(n^{2/3})$-times
($\Theta(\sqrt{n})$-times) bigger than the maximum $ L $-flow~\cite{baier2010length}.
For the vertex version of the problem, there is a $\tau$-approximation
algorithm for graphs of treewidth $\tau$~\cite{Kolman2018OnAE}.
The $ L $-bounded cut was also studied from the perspective of parameterized complexity.
It is fixed parameter tractable (FPT) with respect to the treewidth of the
underlying graph~\cite{Dvork2015ParametrizedCO,Kolman2018OnAE}.
Golovach and Thilikos~\cite{Golovach2009PathsOB} consider several
parameterizations and show FPT-algorithms for many variants of the problem
(directed/undirected graphs, edge/vertex cuts).
On planar graphs, it is FPT with respect to the length bound
$L$~\cite{Kolman2018OnAE}.
The $ L $-bounded cut appears in the literature also as the short paths interdiction
problem~\cite{Bazgan2018AMF}, \cite{Kolman2018OnAE}, \cite{Lee2017ImprovedHF}
or as the most vital edges for shortest paths~\cite{Bazgan2018AMF}.
\subsection{Our contributions}
In the first part of the paper, we show that the combinatorial algorithm by
Koubek and {\v R\'\i ha}~\cite{koubek1981maximum} for the maximum $L$-bounded flow is
not correct.
In the second part of the paper we describe an iterative combinatorial
algorithm, based on the exponential length method,
that finds a $(1+\epsilon)$-approximation of the maximum $L$-bounded
flow in time $\mathcal{O}{}(\varepsilon^{-2}m^2L\log L)$
; that is, we describe a fully polynomial approximation scheme (FPTAS)
for the problem.
Moreover, we show that this approach works even for the
NP-hard generalization of the maximum $L$-bounded flow problem in which
each edge has a length.
This approach is more efficient than the FPTAS based on the ellipsoid
method~\cite{baier2003flows}.
Our result is not surprising (e.g., Baier~\cite{baier2003flows}
mentions the possibility, without giving the details, to use the exponential
length method to obtain a FPTAS for the problem); however, considering the
absence of other polynomial time algorithms for the problem that are not based
on the general LP algorithms, despite of the effort to find some, we regard it
as a meaningful contribution. The paper is based on the results in the
bachelor's thesis of Kate\v{r}ina Altmanov\' a~\cite{Altmanova} and in the
master's thesis of Jan Voborn\' \i k~\cite{Vobornik}.
\section{The algorithm of Koubek and~\v{R}\'{i}ha}
\subsection{Increasing an $L$-bounded flow}
Before describing the problem with the algorithm by Koubek and
{\v R\'\i ha}~\cite{koubek1981maximum}, we informally describe the purpose and the
main attributes of {\em an increasing $L$-system}, a key structure used in the
algorithm.
Consider a network $G=(X, R, c, s, t)$ and an arbitrary $ L $-bounded flow $f$ from $s$ to $t$ in $G$,
together with its decomposition into paths of length at most $L$ (say paths
$p_1, p_2, \ldots $ carrying $r_1, r_2, \ldots $ units of flow, resp.)
that is not a~maximum $ L $-bounded flow . Given $G$ and $f$, Koubek
and \v{R}\'{i}ha \cite{koubek1981maximum} build a labeled oriented tree
$T=(V,E,v_{0},LABV,LABE)$ where $V$ is the set of nodes, $E$ is the set of
edges, $ v_{0} $ is the root, $LABV$ is a vertex labelling and $LABE$ is an
edge labeling. The tree is called {\em an increasing $L$ system with respect
to~$f$}.
There are four types of the nodes of the tree $T$; to explain the error in the paper,
it is sufficient to deal with three of them: $1$-son, $3$-son, $4$-son.
With (almost) each node $u$ in $T$, are associated two consecutive paths in $G$:
the first one, denoted by $q(u)$, contains only
edges that are not used by the current $ L $-flow $f$, and the second one, denoted by
$\bar q(u)$, coincides with a subpath of some path from the current $ L $-flow $f$.
\ (Fig.~\ref{fig:concatenation}).
\begin{figure}[h]
\centering
\includegraphics[scale =1]{obrazky/zaklad_bez_34}
\caption{The concatenation of the section $ q(v) $ to $ \overline{q}(v) $.
\label{fig:concatenation} }
\end{figure}
The tree $T$ encodes a combination of these paths with paths in $f$ and this
combination is supposed to yield a larger $L$-flow than the $ L $-flow $f$.
The label of a vertex $v$ in the tree $T$, denoted by LABV in the original
paper, and the label of the edge $e$ connecting $v$ to its immediate ancestor,
if there is one, denoted by LABE, are of the following form:
\hspace{2cm}
\begin{tabular}{|l|l|l|}
\hline
& LABV & LABE\\ \hline
$1$-son & ${(q(v),i(v),a(v),b(v))}$ & none \\ \hline
$3$-son or $4$-son & ${(q(v),i(v),a(v),b(v))}$ & $(h(e),j(e),d(e),o(e))$ \\ \hline
\end{tabular}
\\
where
\begin{itemize}
\setlength\itemsep{0mm}
\item $q(v)$ is a path in $G$ that is edge disjoint with every path in the $L$-flow $f$,
\item $i(v), j(v)$ are indices of paths in the $L$-flow $f$,
\item $a(v), b(v), d(e)$ are positive integers (distances),
\item $o(e)$ is a positive integer, if $v$ is a $3$-son, and $o(v)$ is a pointer to
a $3$-son, if $v$ is a $4$-son,
\item $h(e)$ is a subset of edges in $G$.
\end{itemize}
As for every node $v$ in the tree (except for the root) there is a unique edge
$e$ connecting it to its parent, Koubek and {\v R\'\i ha} often refer to the label of
the edge $e$, and to its attributes, by the name of the vertex $v$, e.g., they
write $h(v)$ instead of $h(e)$; we shall use the same convention.
The tree $T$ is supposed to describe an $L$-flow $f'$ derived from $f$.
In
particular, each path $q(v)$ and $\bar q(v)$ is a subpath of a new path
between $s$ and $t$ of length at most $L$.
Very roughly speaking, the
attributes $a(v)$ and $d(v)$ store information about the distance of the path
segments $q(v)$ and $\bar q(v)$ from $s$ along the paths used in the
new $L$-flow $f'$, the attribute $i(v)$ specifies the index of a path from $f$
s.t. $\bar q(v)$ is a subpath of $p_{i(v)}$, and the attributes $b(v)$ and $o(v)$, resp., specify
the number of edges along which the paths $p_{i(v)}$ and $p_{j(v)}$ are being
followed by some of the new paths.
Consider a node $w$ in the tree $T$ such that at least one edge in $\bar q(w)$,
say an edge $e$, is saturated in the $L$-flow $f$ (i.e., $f(e)=c(e)$).
In this case, the properties of the tree
$T$ enforce that the node $w$ has at least one $3$-son $u$ whose responsibility
is to desaturate the edge $e$ by diverting one of the paths
that use $e$ in $f$ along a new route; the attribute $j(u)$ specifies the index of the path from $f$
that is being diverted by the $3$-son $u$ of $w$ (Fig.~\ref{fig:3-son}),
\begin{figure}[h]
\centering
\includegraphics[scale =1]{obrazky/3syn} \label{obr:3syn}
\caption{Desaturation of a saturated edge $e$ in a $\bar q(w)$ by a $3$-son $u$.
\label{fig:3-son}}
\end{figure}
and $h(w)$ specifies which saturated
edges from $\bar q(v)$ are desaturated by the son $u$ of $w$.
As the definition of the tree $T$ does not pose any requirements on the
disjointness of the $\bar q$-paths corresponding to different nodes of
$T$, it may happen that the paths $\bar q(w)$ and $\bar q(w')$ for two different
nodes $w$ and $w'$ of the tree $T$ overlap in a saturated edge $e$. In this
case, Koubek and {\v R\'\i ha} allow an {\em exception} (our terminology) to the rule
described in previous paragraph: if one of the nodes $w$ and $w'$, say the node
$w$, has a $3$-son $u$ that desaturates $e$, the other node, the node $w'$, need not
have a $3$-son but it may have a $4$-son instead. The purpose of this $4$-son
is just to provide a pointer to the $3$-son $u$ of $w$ that takes care about
the desaturation of the edge $e$.
\subsection{Small mistakes and typos}\label{smallMistakes}
The paper is full of small mistakes and typos which change the meaning.
Here we mention the most striking typo.
On the page 393 in the paper \cite{koubek1981maximum}, there is the rule 3b:
\begin{center}
If $v$ is a $1$-son \textbf{of} a $3$-son then $v$ has a $1$-son
if and only if \\ $(END(q(v)) + b(v)) \mod\ p_{i(v)} \neq t$.
\end{center}
where $v$ is note in the tree $T$, $t$ is a vertex in the graph $G$,
$END(q(v))$ denotes the last vertex of the path $q(v)$, and for a path $p$,
a vertex $w$ on $p$ and an integer $k$, $w+k \mod p$ denotes the vertex on the
path $p$ that is $k$ edges after $w$.
The correct reading of the above rule, with a significantly different meaning, is:
\begin{center}
If $v$ is a $1$-son \textbf{or} a $3$-son, then $v$ has a $1$-son
if and only if \\ $(END(q(v)) + b(v)) \mod\ p_{i(v)} \neq t$.
\end{center}
The difficulty with the original version is that it does not guarantee that the
paths in new $L$-flow $f'$ terminate in the vertex $t$.
\subsection{The main error}\label{mainError}
We start by recalling a few definitions and lemmas from the
original paper~\cite{koubek1981maximum};
for the definition of the increasing system (more than one page long)
we refer to~\cite{koubek1981maximum}.
\begin{defn}[Definition 4.2 in \cite{koubek1981maximum}]\label{d4.2}
Let $ T $ be an increasing $ L $-system with respect to an~$ L $-flow $ f = \{ (p_i,r_i): i \in I\} $ in a network $ G = (X, R, c, s, t) $. Given an edge $ u \in R $, we define:
\begin{itemize}
\item $ T_1(u) $ is the number of vertices $ x $ in the tree $ T $ such that $ u \in \overline{q}(x) $ and if there is a saturated edge $ v \in \overline{q}(x) $ then there is a $ 3 $-son $ y $ of $ x $ with $ v \in h(y) $, $ u \notin p_{j(y)} $.
\item $ T_2(u) $ is the number of vertices $ x $ in the tree $ T $ such that $ u \in q(x) $.
\item $ T_3(u) $ is the number of vertices $ x $ which are $ 3 $-sons or $ 4 $-sons with $ u \in h(x) $.
\end{itemize}
For $ i \in I $ we denote $ m_i = \sup \{T_3(u): u \in p_i \} $, $|T| = \min \{\frac{c(u)}{T_2(u)}: u\in R, f(u) = 0\} \cup \{\frac{c(u) - f(u)}{T_1(u)}: u \in R\} \cup \{\frac{r_i}{m_i}: i \in I\}$, where the expressions that are not defined are omitted.
\end{defn}
\begin{lemma}[Lemma 4.2 in \cite{koubek1981maximum}]\label{l4.2}
If there is an increasing $ L $-system with respect to an $ L $-flow $f$, then there is an $ L $-flow $g$
with $ |g| = |f| + |T| $.
\end{lemma}
\begin{defn}[Definition 4.3 in \cite{koubek1981maximum}]\label{d4.3}
Let $ \overline{R} = R \cup \{u'\} $, where $ u' \notin R $ and $ c(u') = 0 $.
We put $ \psi(G) = \min( |c(u) - c(v)|: c(u) \neq c(v), u, v \in \overline{R} )$.
\end{defn}
\begin{figure}[h]\centering
\includegraphics[scale =1]{obrazky/maly_error_graf}
\caption{A network $G$ with a $2$-bounded flow $f$.}\label{network_5bounded_flow} \label{inputNetwork}
\end{figure}
\begin{figure}[h]\centering
\includegraphics[scale = 1]{obrazky/maly_dekompozice}
\caption{A decomposition of the $2$-bounded flow $f$ into paths
$ p_1, p_2$.
\label{inputFlow} }
\end{figure}
\begin{lemma}[Lemma 4.4 in \cite{koubek1981maximum}]\label{l4.4}
For each increasing $ L $-system $ T $ (with respect to an $ L $-flow ${ f =\{(p_i, r_i): i \in I\}) }$ constructed by the above procedure it holds $ |T| \geq \psi(G)/|I| $.
\end{lemma}
The {\em above procedure} in Lemma \ref{l4.4} refers to a construction of an
increasing $L$-system that is outlined in the original paper.
As Definition \ref{d4.3} implies $\psi(G) > 0$, we also know by Lemma \ref{l4.4}
that for every increasing $L$-system $T$, $|T| > 0$.
Now we are ready to describe the counter example.
Take $k=2$ and consider the following network $G$ with a $2$-bounded flow $f$
of size $2$ (Fig. \ref{inputNetwork} and \ref{inputFlow}); apparently, this is
a maximum $2$-bounded flow.
We are going to show that there exists an increasing system $T$ for $f$.
According to Lemmas~\ref{l4.2} and \ref{l4.4} this implies the existence
of a $2$-bounded flow $g$ of size $|f|+|T| > |f|$.
As the flow $f$ is a maximum $2$-bounded flow in $G$, this is a contradiction.
\begin{figure}[ht]\centering
\includegraphics[scale = 1]{obrazky/maly_strom}
\caption{Increasing $ 2 $-system $ T $.
\label{increasingTree} }
\end{figure}
The increasing system $T$ is depicted in Figure \ref{increasingTree}; for the
sake of simplicity, we list only the most relevant attributes. It is just a
matter of a mechanical effort to check that it meets Definition 4.1 of the
increasing system from the original paper.
In words, the essence of the counter example is the following.
The purpose of the root of the tree, the node $u_0$, is to increase the flow
from $s$ to $t$ along the path $q(u_0)\bar q(u_0)$ which is (accidently)
the path $p_1$. As there is a saturated edge on this path, namely the edge
$sa$, there is a $3$-son of the node $u_0$, the node $u_1$, whose purpose
is to desaturate the edge $sa$ by diverting one of the paths that use the
edge $sa$ along an alternative route; in particular, the node $u_1$ is
diverting the path $p_1$ and it is diverting it from the very beginning,
from $s$, along the path $q(u_1)\bar q(u_1)$ which is (accidently) the
path $p_2$.
As there is a saturated edge on this path, namely the edge
$sb$, there is a $3$-son of the node $u_1$, the node $u_2$, whose purpose
is to desaturate the edge $sb$ by diverting one of the paths that use the
edge $sb$ along an alternative route; in particular, the node $u_2$ is
diverting the path $p_2$ and it is diverting it from the very beginning,
from $s$, along the path $q(u_2)\bar q(u_2)$ which is (accidently)
again the path $p_1$.
As there is a saturated edge on this path, namely the edge
$sa$, and as there is already another node in the tree that is
desaturating $sa$, namely the node $u_1$, the node $u_2$ does not have
a $3$-son but it has a $4$-son $u_3$ instead, which is a pointer to the
$3$-son $u_1$. This way, there is a kind of a deadlock cycle in the
increasing system: $u_1$ is desaturating the edge $sa$ for the node $u_0$ but
it itself needs $u_2$ to desaturate the edge $sb$ in it and $u_2$ in turn needs
$u_3$ to desaturate the edge $sa$, but $u_3$ delegates this task back
to $u_1$.
At this point, we know that Lemma \ref{l4.2} or Lemma \ref{l4.4} is not correct.
By Definition~\ref{d4.2}, one can check that $|T|=1/2$ which implies,
as we started with a maximum flow, that it is Lemma~\ref{l4.2} that does
not hold.
\section{FPTAS for maximum $L$-bounded flow}\label{chap:approximations}
We first describe a fully polynomial approximation scheme for maximum
$L$-bounded flow on networks with unit edge length. The algorithm is based on
the algorithm for the maximum multicommodity flow by Garg and
K\"{o}nemann~\cite{garg2007apx}.
Then we describe a FPTAS for the $L$-bounded
flow problem with general edge lengths. Our approximation schemas for the
maximum $L$-bounded flow on unit edge lengths and the maximum $L$-bounded flow
with edge lengths are almost identical, the only difference is in using an
approximate subroutine for resource constrained shortest path in the general
case which slightly complicates the analysis.
\subsection{FPTAS for Unit Edge Lengths}\label{sec:apxunit}
Let us consider the path based linear programming (LP) formulation of the maximum $L$-bounded flow,
\textbf{P}$_{\text{path}}$, and its dual, \textbf{D}$_{\text{path}}$.
We assume that $G=(V,E,c,s,t)$ is a given network and $L$ is a given
length bound.
Let $\mathcal{P}_L$ denote the set of all $s$-$t$ paths of length at most
$L$ in $G$. There is a primal variable $x(p)$ for each path $p\in \mathcal{P}_L$,
and a dual variable $y(e)$ for each edge $e\in E$.
Note that the dual LP is a relaxation of an
integer LP formulation of the minimum $L$-bounded cut problem.
\begin{minipage}{.5\linewidth}
\centering
\begin{equation*}\label{eq:pathlp}
\begin{alignedat}{2}
&\text{max} \ & \sum_{P\in\mathcal{P}_L}x(P) & \\
&\text{s.t.} \ & \sum_{\substack{P\in\mathcal{P}_L:\\e\in P}}x(P) &\le
c(e)\quad \forall e\in E\\
&& x &\ge 0
\end{alignedat}
\end{equation*}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\centering
\begin{equation*}\label{eq:pathdual}
\begin{alignedat}{2}
&\text{min} \ & \sum_{e\in E}c(e)y(e) & \\
&\text{s.t.} \ & \sum_{e\in P}y(e) &\ge 1\quad \forall P\in
\mathcal{P}_L\\
&& y &\ge 0
\end{alignedat}
\end{equation*}
\end{minipage}
The algorithm simultaneously constructs solutions for the maximum $L$-bounded
flow and the minimum fractional $L$-bounded cut. It iteratively routes flow
over shortest paths with respect to properly chosen dual edge lengths and at
the same time increases these dual lengths; dual edge length of the edge $e$
after $i$ iterations will be denoted by $y_i(e)$. During the runtime of the
algorithm, the constructed flow need not respect the edge capacities; however,
with the right choice of parameters $\varepsilon, \delta$ the resulting flow can be
scaled down to a feasible (i.e., respecting the edge capacities)
flow (Lemma~\ref{lem:feasibleflow}) that is a
$(1+\varepsilon)$-approximation of the maximum $L$-bounded flow
(Theorem~\ref{thm:approxunit}).
For a vector $y$ of dual variables, let $d^L_y(s,t)$ denote the length of the
$y$-shortest $s-t$ path from the set of paths $\mathcal{P}_L$ and let
$\alpha^L(i)=d_{y_i}^L(s,t)$. Note that a shortest $s-t$ path with respect to
edge lengths $y$ that uses at most a given number of edges can be computed in
polynomial time by a modification of the Dijkstra's shortest path algorithm.
\begin{algorithm}
\caption{\textsc{Approx}($\varepsilon, \delta$)}
\label{algo:approx}
\begin{algorithmic}[1]
\STATE $i\leftarrow 0, \ y_0(e) \leftarrow \delta \quad \forall e\in E,\ x_0(P)\leftarrow 0
\quad \forall P\in \mathcal{P}_L$
\WHILE{$\alpha^L(i)<1$}
\STATE $i\leftarrow i+1$
\STATE $x_i \leftarrow x_{i-1}, y_i \leftarrow y_{i-1}$
\STATE $P \leftarrow y_i$-shortest $s$-$t$ path with at most $L$ edges
\STATE $c \leftarrow \min\limits_{e\in P}c(e)$
\STATE $x_i(P)\leftarrow x_i(P)+c$
\STATE $y_i(e)\leftarrow y_{i}(e)(1+\varepsilon c/c(e)) \quad \forall e\in P$
\ENDWHILE
\RETURN $x_i$
\end{algorithmic}
\end{algorithm}
Let $f_i$ denote the size of the flow after $i$ iterations,
$f_i=\sum_{P\in\mathcal{P}_L}x_i(P)$, and let $\tau$ denote the total number of
iterations performed by \textsc{Approx}; then $x_{\tau}$ is the
output of the algorithm and $f_{\tau}$ its size.
\begin{lemma}
\label{lem:feasibleflow}
The flow $x_{\tau}$ scaled down by a factor of $\log_{1+\varepsilon}\frac{1+\varepsilon}{\delta}$
is a feasible $L$-bounded flow.
\end{lemma}
\begin{proof}
By construction, for every $i$, $x_i$ is an $L$-bounded flow. Thus, we only have
to care about the feasibility of the flow
\begin{equation}
\frac{x_{\tau}}{\log_{1+\varepsilon}\frac{1+\varepsilon}{\delta}}\ .
\end{equation}
For every iteration $i$ and every edge $e\in E$, as $\alpha^L(i-1)<1$, we also
have $y_{i-1}(e) < 1$ and so $y_i(e)< 1+\varepsilon$. It follows that
\begin{equation}\label{eqn:lengthtupper}
y_{\tau}(e)<1+\varepsilon \ .
\end{equation}
Consider an arbitrary edge $e\in E$ and suppose that the flow $f_{\tau}(e)$
along $e$ has been routed in iterations $i_1, i_2,\dots,i_r$ and the amount of
flow routed in iteration $i_j$ is $c_j$. Then $f_{\tau}(e)=\sum_{j=1}^r c_j$
and $y_{\tau}(e)=\delta\prod_{j=1}^r (1+\varepsilon c_j/c(e))$. Because each $c_j$ was
chosen such that $c_j\le c(e)$, we have by Bernoulli's inequality that $1+\varepsilon
c_j/c(e)\ge
(1+\varepsilon)^{c_j/c(e)}$ and
\begin{equation}\label{eqn:lengthtlower}
y_{\tau}(e)\ge\delta\prod_{j=1}^r(1+\varepsilon)^{c_j/c(e)} = \delta
(1+\varepsilon)^{f_{\tau}(e)/c(e)}.
\end{equation}
Combining inequalities (\ref{eqn:lengthtupper}) and (\ref{eqn:lengthtlower}) gives
$$\frac{f_{\tau}(e)}{c(e)}\le\log_{1+\varepsilon}\frac{1+\varepsilon}{\delta}$$
which completes the proof.
\end{proof}
\begin{claim} \label{lem:alpha}
For $i=1,\ldots,\tau$,
\begin{align}
\alpha^L(i)& \le\delta L e^{\varepsilon f_i/\beta} \ .
\end{align}
\end{claim}
\begin{proof}
For a vector $y$ of dual variables,
let $D(y)=\sum_e c(e)y(e)$ and let $\beta=\min_y D(y)/d^L_y(s,t)$. Note that
$\beta$ is equal to the optimal value of the dual linear program.
For notational simplicity we abbreviate $D(y_i)$ as $D(i)$.
Let $P_i$ be the path chosen in iteration $i$ and $c_i$ be the value of $c$ in
iteration $i$. For every $i\ge 1$ we have
\begin{align}
D(i) &= \sum_{e\in E} y_i(e)c(e)\nonumber\\
&= \sum_{e\in E} y_{i-1}(e)c(e) + \varepsilon \sum_{e\in P_i} y_{i-1}(e)c_i\nonumber\\
&= D(i-1) + \varepsilon(f_i-f_{i-1})\alpha^L(i-1)\nonumber
\end{align}
which implies that
\begin{equation}\label{eqn:di}
D(i)= D(0)+\varepsilon\sum_{j=1}^i(f_j-f_{j-1})\alpha^L(j-1).
\end{equation}
Now consider the length function $y_i-y_0$. Note that $D(y_i-y_0)=D(i)-D(0)$
and $d^L_{y_i-y_0}(s,t)\ge \alpha^L(i)-\delta L$.
Hence,
\begin{equation}
\label{eqn:beta}
\beta\le\frac{D(y_i-y_0)}{d_{y_i-y_0}^L(s,t)}\le\frac{D(i)-D(0)}{\alpha^L(i)-\delta
L} \ .
\end{equation}
By combining relations~(\ref{eqn:di}) and~(\ref{eqn:beta}) we get
$$\alpha^L(i)\le \delta L
+\frac{\varepsilon}{\beta}\sum_{j=1}^i(f_j-f_{j-1})\alpha^L(j-1) \ .$$
Now we define $z(0)=\alpha^L(0)$ and for $i=1,\ldots,\tau$, $z(i)=\delta L
+\frac{\varepsilon}{\beta}\sum_{j=1}^i(f_j-f_{j-1})z(j-1)$.
Note that for each $i$, $\alpha^L(i) \le z(i)$. Furthermore,
\begin{align*}
z(i) &= \delta L +\frac{\varepsilon}{\beta}\sum_{j=1}^i(f_j-f_{j-1})z(j-1)\\
&= \left(\delta L
+\frac{\varepsilon}{\beta}\sum_{j=1}^{i-1}(f_j-f_{j-1})z(j-1)\right)+\frac{\varepsilon}{\beta}(f_i-f_{i-1})z(i-1)\\
&=z(i-1)(1+\varepsilon(f_i-f_{i-1})/\beta)\\
&\le z(i-1)e^{\varepsilon(f_i-f_{i-1})/\beta}.
\end{align*}
Since $z(0)\le\delta L$, we have $z(i)\le \delta Le^{\varepsilon f_i/\beta}$, and thus also,
for $i=1,\ldots,\tau$,
$\alpha^L(i)\le\delta L e^{\varepsilon f_i/\beta}\ .$
\end{proof}
\begin{theorem}\label{thm:approxunit}
For every $0<\varepsilon<1$ there is an algorithm that computes an
$(1+\varepsilon)$-approximation to the maximum $L$-bounded flow in a network with
unit edge lengths in time $\mathcal{O}{}(\varepsilon^{-2}m^2L\log L)$.
\end{theorem}
\begin{proof}
We start by showing that for every $\varepsilon<\frac13$ there is a constant
$\delta=\delta(\epsilon)$ such that $x_{\tau}$, the output of \textsc{Approx}($\varepsilon,\delta$),
scaled down by $\log_{1+\epsilon}\frac{1+\epsilon}{\delta}$ as in Lemma~\ref{lem:feasibleflow},
is a $(1+3\varepsilon)$-approximation.
Let $\gamma$ denote the approximation ratio of such an algorithm, that is,
let $\gamma$ denote the ratio of the optimal dual solution ($\beta$) to
the appropriately scaled output of \textsc{Approx}($\varepsilon,\delta$),
\begin{equation}
\gamma = \frac{\beta \log_{1+\epsilon}\frac{1+\epsilon}{\delta}}{f_{\tau}} \ ,
\end{equation}
where the constant $\delta$ will be specified later.
By Claim~\ref{lem:alpha} and the stopping condition of the while cycle
we have
\begin{equation}
1\le\alpha^L(\tau)\le\delta L e^{\varepsilon f_{\tau}/\beta}\nonumber
\end{equation}
and hence
$$\frac{\beta}{f_{\tau}}\le\frac{\varepsilon}{\log\frac{1}{\delta L}}.$$
Plugging this bound in the equality for the approximation ratio $\gamma$, we obtain
$$\gamma\le\frac{\varepsilon\log_{1+\varepsilon}\frac{1+\varepsilon}{\delta}}{\log\frac{1}{\delta
L}}=\frac{\varepsilon}{\log(1+\varepsilon)}\frac{\log\frac{1+\varepsilon}{\delta}}{
\log\frac{1}{\delta L}}.$$
Setting $\delta=\frac{1+\varepsilon}{((1+\varepsilon)L)^{1/\varepsilon}}$ yields
$$\frac{\log\frac{1+\varepsilon}{\delta}}{
\log\frac{1}{\delta L}} = \frac{\frac{1}{\varepsilon}\log((1+\varepsilon)L)}{
\left(\frac1\varepsilon-1\right)\log((1+\varepsilon)L)}=\frac1{1-\varepsilon}.$$
Taylor expansion of $\log(1+\varepsilon)$ gives a bound
$\log(1+\varepsilon)\ge\varepsilon-\frac{\varepsilon^2}{2}$ for $\varepsilon<1$ and it follows for
$\varepsilon<\frac13$ that
$$\gamma\le\frac{\varepsilon}{(1-\varepsilon)\log(1+\varepsilon)}\le
\frac{\varepsilon}{(1-\varepsilon)(\varepsilon-\varepsilon^2/2)}\le\frac{1}{1-\frac32\varepsilon}\le 1+3\varepsilon.$$
To complete the proof, we just put $\varepsilon'=\varepsilon/3$ and run
\textsc{Approx}($\varepsilon',\delta(\varepsilon')$).
It remains to prove the time complexity of the algorithm.
In every iteration $i$ of \textsc{Approx}, the length $y_i(e)$ of an edge $e$
with the smallest capacity on the chosen path $P$ is increased by a factor of
$1+\varepsilon'$. Because $P$ was chosen such that $y_i(P)<1$ also $y_i(e)<1$ for
every edge $e \in P$. Lengths of other edges get increased by a factor
of at most $1+\varepsilon'$, therefore $y_{\tau}(e)<1+\varepsilon'$ for every edge $e\in E$.
Every edge has the minimum capacity on the chosen path in at most
$\left\lceil\log_{1+\varepsilon'}\frac{1+\varepsilon'}{\delta} \right\rceil =
\mathcal{O}{}(\frac1\varepsilon\log_{1+\varepsilon}L)$ iterations, so \textsc{Approx} makes at most
$\mathcal{O}{}(\frac m{\varepsilon}\log_{1+\varepsilon}L)=\mathcal{O}{}(\frac{m}{\varepsilon^2}\log L)$ iterations.
Each iteration takes time $\mathcal{O}{}(Lm)$ so the total time taken by \textsc{Approx} is
$\mathcal{O}{}(\varepsilon^{-2}m^2L\log L)$.
\end{proof}
\subsection{FPTAS for General Edge Lengths}\label{sec:apxgeneral}
Now we extend the approximation algorithm to networks with general
edge lengths that are given by a length function $\ell:E\to \mathbb{N}$.
The dynamic programming algorithm for computing shortest paths
that have a restricted length with respect to another length function,
does not work in this case. In fact, the problem of finding shortest path with
respect to a given edge length function while restricting to paths of bounded
length with respect to another length function is \hbox{\rm \sffamily NP}-hard in general~\cite{HZ:80}.
On the other hand, there exists a FPTAS for it~\cite{hassin1992restricted,lorenz2001restricted}.
\iffalse
The restricted shortest path problem appears in many routing problems and has
been studied both theoretically and experimentally. First FPTAS has been given
by Hassin~\cite{hassin1992restricted}. The key idea behind his algorithm is to
get an upper and a lower bounds for the size of the optimum and then scale and
round the input to use dynamic programming similar to the shortest path
algorithm from the previous section. Hassin's result has been improved by
Lorenz and Raz~\cite{lorenz2001restricted}. They found a better way of
computing upper and lower bounds for the optimum and got time complexity of
$\mathcal{O}{}(mn(\log\log n+1/\varepsilon))$.
\fi
We assume that we are given as a black-box an algorithm that for a given
graph $G$, two edge length functions $y$ and $\ell$, two distinguished
vertices $s$ and $t$ from $G$, a length bound $L$ and an error parameter $w>0$,
computes a $(1+w)$-approximation of the $y$-shortest path of $\ell$-length
at most $L$; we denote by $d_{y,\ell}^L(s,t;w)$ the length of such a path
and we also introduce an abbreviation $\bar\alpha^L(i)=d_{y_i,\ell}^L(s,t;w)$.
Note that for every $i$, $\bar\alpha^L(i) \leq (1+w)\alpha^L(i)$.
We can use the FPTAS of Lorenz and Raz~\cite{lorenz2001restricted} for this
task.
The algorithm of Garg and K\"{o}nemann~\cite{garg2007apx} for approximating
maximal multicommodity flow has been improved by Fleischer
~\cite{fleischer2000apx}. The original algorithm computes the shortest path
between every terminal pairs in every iteration. Fleischer divided the
algorithm to phases where she worked with commodities one by one. This way her
algorithm effectively works with approximations of shortest paths while
eliminates the dependency on the number of commodities and still gets a good
approximation ratio.
Using a similar analysis we show that we can work with an
approximation shortest path algorithm to get an FPTAS to otherwise intractable maximum
$L$-bounded flow problem with general edge lengths.
The structure of the $L$-bounded flow algorithm with general edge lengths stays the same as in the unit edge lengths case.
The only difference is that instead of $y$-shortest $L$-bounded paths,
approximations of $y$-shortest $L$-bounded paths are used (steps 2 and 5).
\begin{algorithm}
\caption{\textsc{ApproxGeneral}($\varepsilon, \delta, w$)}
\label{algo:approxgeneral}
\begin{algorithmic}[1]
\STATE $i\leftarrow 0, \ y_0(e) \leftarrow \delta \quad \forall e\in E, \ x_0(P)\leftarrow 0 \quad \forall P\in \mathcal{P}_L$
\WHILE{$\bar\alpha^L(i)<1+w$}
\STATE $i\leftarrow i+1$
\STATE $x_i \leftarrow x_{i-1}, y_i \leftarrow y_{i-1}$
\STATE $P \leftarrow (1+w)$-approximation of the $y_i$-shortest $L$-bounded path
\STATE $c \leftarrow \min\limits_{e\in P}c(e)$
\STATE $x_i(P)\leftarrow x_i(P)+c$
\STATE $y_i(e)\leftarrow y_{i}(e)(1+\varepsilon c/c(e)) \quad \forall e\in P$
\ENDWHILE
\RETURN $x_i$
\end{algorithmic}
\end{algorithm}
The analysis of the algorithm follows the same steps as the analysis of
Algorithm~\ref{algo:approx} but one has to be more careful when dealing with
the lengths.
As in the previous subsection, let $f_i$ denote the size of the
flow after $i$ iterations
and let $\tau$
denote the total number of iterations.
Due to the lack of space, the proofs are given in the Appendix.
\begin{lemma}
\label{lem:feasibleflowgeneral}
The flow $x_{\tau}$ scaled down by a factor of $\log_{1+\varepsilon}\frac{(1+\varepsilon)(1+w)}{\delta}$
is a feasible $L$-bounded flow.
\end{lemma}
\begin{proof}
For every edge $e\in E$ and iteration $i$, as $\bar \alpha^L(i-1)<1+w$, we
also have $y_{i-1}(e) < 1+w$. By description of the algorithms, this implies
$y_i(e)< (1+\varepsilon)(1+w)$, and in particular,
\begin{equation}\label{eqn:lengthtuppergeneral}
y_{\tau}(e)<(1+\varepsilon)(1+w) \ .
\end{equation}
Combining this with $y_{\tau}(e)\ge\delta (1+\varepsilon)^{f_{\tau}(e)/c(e)}$
from inequality~(\ref{eqn:lengthtlower}) in previous subsection,
we derive
$$\frac{f_{\tau}(e)}{c(e)}\le\log_{1+\varepsilon}\frac{(1+\varepsilon)(1+w)}{\delta}$$
which completes the proof.
\end{proof}
\begin{claim} \label{lem:alpha2}
For $i=1,\ldots,\tau$,
\begin{align}
\alpha^L(i)& \le\delta L e^{\varepsilon (1+w)f_i/\beta} \ .
\end{align}
\end{claim}
\begin{proof}
By the same reasoning as in the proof of Claim~\ref{lem:alpha}, we obtain
\begin{equation}\label{eqn:digeneral}
D(i) \le D(0)+\varepsilon\sum_{j=1}^i(f_j-f_{j-1})(1+w)\alpha^L(i-1) \ ,
\end{equation}
where the extra $1+w$ factors stems from the fact that we work, in iteration $i$,
not with a path of length $\alpha(i)$ but with a path of length
$\bar\alpha(i)\leq(1+w)\alpha(i)$.
Combining this with $\beta\le\frac{D(i)-D(0)}{\alpha^L(i)-\delta
L}$ from inequality~(\ref{eqn:beta}), we obtain
$$\alpha^L(i)\le \delta L +\frac{\varepsilon(1+w)}{\beta}\sum_{j=1}^i(f_j-f_{j-1})\alpha^L(j-1) \ .$$
From this point, we proceed again along the same lines as in the proof of
Claim~\ref{lem:alpha} (the only difference is that instead of $\epsilon/\beta$,
we work now with $(1+w)\epsilon/\beta$) and get the desired bound.
\end{proof}
\begin{theorem} \label{thm:gen}
There is an algorithm that computes an $(1+\varepsilon)$-approximation to the
maximum $L$-bounded flow in a graph with general edge lengths in time
$\mathcal{O}{}(\frac{m^2n}{\varepsilon^2}\log L(\log\log n + \frac{1}{\varepsilon}))$.
\end{theorem}
\begin{proof}
We show that for every $\varepsilon\le\frac13$ there are constants
$\delta$ and~$w$ such that $x_{\tau}$, the output of
\textsc{ApproxGeneral}($\varepsilon, \delta, w$), scaled down by
$\log_{1+\varepsilon}\frac{(1+\varepsilon)(1+w)}{\delta}$ as in
Lemma~\ref{lem:feasibleflowgeneral}, is a $(1+5\varepsilon)$-approximation to the
maximum $L$-bounded flow with general capacities; the theorem easily
follows.
Let $\gamma$ denote the approximation ratio of such an algorithm, that is,
let $\gamma$ denote the ratio of the optimal dual solution ($\beta$) to
the appropriately scaled output of \textsc{ApproxGeneral}($\varepsilon,\delta,w$),
\begin{equation}
\gamma = \frac{\beta \log_{1+\epsilon}\frac{(1+\epsilon)(1+w)}{\delta}}{f_{\tau}} \ ,
\end{equation}
where the constants $\delta$ and $w$ will be specified later.
By the stopping condition of the while cycle we have
$1+w \leq \bar\alpha^L(\tau) \leq (1+w)\alpha^L(\tau)$, that is, $1\leq \alpha^L(\tau)$;
combining it with Claim~\ref{lem:alpha2}, we get
$$\frac{\beta}{f_{\tau}}\le\frac{\varepsilon(1+w)}{\log\frac{1}{\delta L }}.$$
Plugging this bound in the equality for the approximation ratio $\gamma$,
we obtain
\begin{equation} \label{eqn:gamma}
\gamma\le
\frac{\varepsilon(1+w)\log_{1+\varepsilon}\frac{(1+\varepsilon)(1+w)}{\delta}}{\log\frac{1}{\delta
L}}=\frac{\varepsilon(1+w)}{\log(1+\varepsilon)}\frac{\log\frac{(1+\varepsilon)(1+w)}{\delta}}{
\log\frac{1}{\delta L}}\ .
\end{equation}
Setting $\delta=\frac{(1+\varepsilon)(1+w)}{((1+\varepsilon)(1+w)L)^{1/\varepsilon}}$
yields
\begin{equation}
\frac{\log\frac{(1+\varepsilon)(1+w)}{\delta}}{
\log\frac{1}{\delta L}} = \frac{\frac{1}{\varepsilon}\log((1+\varepsilon)(1+w)L)}{
\left(\frac1\varepsilon-1\right)\log((1+\varepsilon)(1+w)L)}=\frac1{1-\varepsilon}\ .
\end{equation}
Thus, the bound on the approximation ratio $\gamma$~(\ref{eqn:gamma}) simplifies to
$$\gamma\le\frac{\varepsilon(1+w)}{(1-\varepsilon)\log(1+\varepsilon)}\le
\frac{\varepsilon(1+w)}{(1-\varepsilon)(\varepsilon-\frac{\varepsilon^2}{2})}\le
\frac{1+w}{1-\frac{3}{2}\varepsilon}\ ,$$
where the second inequality follows from the Taylor expansion of $\log(1+\varepsilon)$
and the bound $\log(1+\varepsilon)\ge\varepsilon-\frac{\varepsilon^2}{2}$, for $\varepsilon<1$.
By setting $w=\varepsilon$, for $\varepsilon\le\frac{1}{3}$ we get the promised bound
$$\gamma\le\frac{1+w}{1-\frac{3}{2}\varepsilon}\leq (1+\varepsilon)(1+3\varepsilon)\leq1+5\varepsilon \ .$$
Concerning the running time, we observe that in
every iteration the length of at least one edge gets increased by the ratio
$1+\varepsilon$. For every edge $e\in E$ we have $y_{\tau}(e)\le(1+\varepsilon)(1+w)$. By
the same arguments as in the previous subsection, our choice of the parameters
ensures that the total number of iterations is at most
$\mathcal{O}{}(\frac{m}{\varepsilon}\log_{1+\varepsilon}L)=\mathcal{O}{}(\frac{m}{\varepsilon^2}\log L)$.
The FPTAS approximating the resource bounded shortest path takes time
$\mathcal{O}{}(mn(\log\log n+\frac{1}{\varepsilon}))$. Combining these two bounds completes
the proof.
\end{proof}
We note that the exponential length method can be used for many fractional
packing problems and using the same technique we could get an approximation
algorithm for maximum multicommodity $L$-bounded flow.
\bibliographystyle{abbrv}
|
1,314,259,993,011 | arxiv | \section{Introduction}
Graph neural networks (GNNs) have obtained promising performance in various applications to graph data~\cite{ying2018graph,qiu2018deepinf,chen2019semi}. Recent studies have
shown that GNNs, like other types of deep learning models, are also vulnerable to adversarial attacks~\cite{dai2018adversarial,Z_gner_2018}. However, there still exists a gap between most of the existing attack setups and practice where the capability of an attacker is limited. Instead of directly modifying the original graph (aka., Graph Modification Attack, GMA)~\cite{yang2021derivative}, we focus on a more practical setting to inject extra fake nodes into the original graph (aka., Graph Injection Attack, GIA)~\cite{tdgia}. We perform query-based attack, which indicates that the attacker has no knowledge on the victim model but can access the model with a limited number of queries.
As an example in Figure~\ref{fig: Fake Node Attack}, high-quality users in a social network may be misclassified as low-quality users after being connected with a fraudulent user whose features (maybe the meta data of the account) are carefully crafted utilizing query information from the target model.
\begin{figure}[t]
\centering
\includegraphics[width=8.5cm]{Fake-Node-Attack.png}
\caption{An illustration of query-based graph injection attack with partial information
}
\label{fig: Fake Node Attack}
\end{figure}
Compared to adversarial attacks on image classification~\cite{10.1145/3128572.3140448,ilyas2018blackbox,Dong-2018}, the study of query-based adversarial attacks on graph data is still at an early stage.
Existing attempts include training a surrogate model using query results \cite{wan2021adversarial}, a Reinforcement-Learning-based method~\cite{10.1145/3447548.3467416}, derivative-free optimization \cite{yang2021derivative} and a gradient-based method~\cite{10.1145/3460120.3484796}. However, most of the existing attacks simply adopt the optimization methods from other fields, such as image adversarial attacks, without utilizing the rich structure of graph data, which has considerable potential to achieve a higher performance attack.
Unlike most previous work, we only allow the adversary to access the information of a small part of the nodes since it is usually impossible to observe the whole graph, especially for large networks in practical scenarios. Moreover, we perform a \emph{black-box} attack, which does not allow the adversary to have access to the model structures or parameters. The adversary has only a limited number of queries on the victim model about the predicted scores of certain nodes, which is more practical.
It is also noted that, owing to the non-i.i.d. nature of graph data, connecting victim nodes to fake nodes may have side effects on the accuracy of victim nodes' neighbors, which is not our purpose. To our best knowledge, this is the first attempt to limit the influence to a certain range of victim nodes and protect the other nodes from being misclassified simultaneously.
We propose a unified framework for query-based adversarial attacks on graphs, which subsumes existing methods. In general, the attacker decides on the current perturbation as a conditional distribution on history query results and current graph status.
Under this framework, we propose a novel attack method named \textsl{Cluster Attack}, which considers the graph-dependent priors by better utilizing the unique structure of the graph. In particular,
we try to find an equivalent discrete optimization problem. We first demonstrate that a GIA problem can be formulated as an equivalent graph clustering problem. Because the discrete optimization problem of an adjacent matrix can be solved in the context of graph clustering, we prevent query-inefficient searching in the non-Euclidean space. The resulting cluster serves as a graph-dependent prior for the adjacent matrix, which utilizes the vulnerability of the local structure.
Second, the key challenge in graph clustering is to define the similarity metric between nodes. We propose a metric to measure the similarity of victim nodes, called Adversarial Vulnerability; this is related to how the victim nodes will be affected by the injected fake node, and we cluster the victim nodes accordingly. The Adversarial Vulnerability is only related to the local structure of the graph and thus can be handled with only part of the graph observed.
Our contributions are summarized as follows:
\begin{itemize}
\item We propose a unified framework for query-based adversarial attacks on graphs, which formulates the current perturbation as a conditional distribution on the history of query results and on the current graph status.
\item We propose \textsl{Cluster Attack}, an injection adversarial attack on a graph, which formulates a GIA problem as an equivalent graph clustering problem and thus solves the discrete optimization of an adjacent matrix in the context of clustering.
\item After providing theoretical bounds on our method, we empirically show that our method achieves high performance in terms of success rate of attacks under an extremely strict setting with a limited number of queries and only part of the graph observed
\end{itemize}
\section{Background}
In this section, we present recent works on node classification and adversarial attacks on graphs.
\subsection{Node Classification on a Graph}
Node classification on graphs is an important task, with a wide range of applications such as user classification in financial networks.
It aims to carry out classification by aggregating the information from neighboring nodes~\cite{kipf2016semisupervised,hamilton2017inductive,velikovi2017graph}. Recent work has carried out node classification using graph convolutional networks (GCNs)~\cite{kipf2016semisupervised}, which is one of the most representative GNNs. Specifically, let a graph be $G=(\mathbf{A},\mathbf{X})$, where $\mathbf{A}$ and $\mathbf{X}$ respectively represent the adjacency matrix and the feature matrix. Given a subset of labeled nodes in the graph, GCN aims to predict the labels of the remaining unlabeled nodes in the graph as
\begin{equation}
f(G)=f(A,X)=\mbox{softmax}\left( \hat{\mathbf{A}}\sigma(\hat{\mathbf{A}}\mathbf{X}\mathbf{W}^{(0)})\mathbf{W}^{(1)} \right),\label{GCN}
\end{equation}
where $\hat{\mathbf{A}}$ is the normalized adjacency matrix; $\mathbf{W}^{(0)}$ and $\mathbf{W}^{(1)}$ are parameter matrices; $\sigma$ is the activation function; and $f(G)$ is the prediction corresponding to each node.
\subsection{Graph Adversarial Attacks}
Numerous methods have been developed to perform adversarial attacks on graphs. Early works focused on modifying the original graph (i.e., Graph Modification Attack) \cite{dai2018adversarial,Z_gner_2018}, while some recent works \cite{tdgia,gnia} have focused on a more practical setting to inject extra fake nodes into the original graph (i.e., Graph Injection Attack). For query-based graph adversarial attacks, as shown in Table \ref{unified}, various efforts have been made. Some have focused on attacking the task of node classification \cite{yang2021derivative} while there also have been efforts to attack graph classification \cite{wan2021adversarial,10.1145/3447548.3467416,10.1145/3460120.3484796}, with gradient-based \cite{10.1145/3460120.3484796} or gradient-free methods \cite{yang2021derivative,wan2021adversarial,10.1145/3447548.3467416}. Nevertheless, most of the existing attacks just adopt optimization methods from other fields (especially image adversarial attack), ignoring the unique structure of graph data. In this work, we propose to attack in a graph-specific manner utilizing the inherent structure of a graph.
\section{A Unified Framework for Query-Based Adversarial Attacks on Graphs}
\begin{table*}[tbhp]
\centering
\small
\begin{tabular}{lll}
\toprule
Method & Optimization Step & Target Task \\
\midrule
Random&$\Delta G\sim \mbox{Random}$ & - \\
\midrule
GRABNEL &\multirow{2}{*}{$\Delta G = \underset{\Delta G}{\mbox{argmin}}\,\mathcal{L}_{sur}(G+\Delta G)$}& Graph Classification\\
\cite{wan2021adversarial}&&(GMA+GIA)\\
\midrule
DFO \cite{yang2021derivative}&$\Delta G = F(G, \delta)-G$, $\delta \sim$ DFO & Node Classification (GMA)\\
\midrule
\cite{10.1145/3460120.3484796}&$\nabla p(\Delta A) = \frac{1}{Q}\sum_{1}^{Q}\mbox{sgn}(\frac{p(\Delta A +\mu u_q)-p(\Delta A)}{\mu}u_q)$, $u_q\sim$ Gaussian& Graph Classification (GMA)\\
\midrule
Rewatt \cite{10.1145/3447548.3467416} &$\Delta G\sim p(\cdot|G)$ from Reinforcement Learning Agent & Graph Classification (GMA)\\
\midrule
\multirow{2}{*}{Cluster Attack (Ours)} &$\left\{
\begin{array}{l}
\Delta \mathbf{X}_{fake} = \mathbb{I}(\mathcal{L}(\mathbf{A}^+, \mathbf{X}^+)>\mathcal{L}(\mathbf{A}^+, \begin{bmatrix}\mathbf{X}\\\mathbf{X}_{fake}+\delta \mathbf{X}_{ij}\end{bmatrix}))\cdot\delta \mathbf{X}_{ij}\\
\nabla_{\mathbf{X}_{fake}}\mathbb{E}\,\mathcal{L}(\mathbf{A}^+,\mathbf{X}^+) = \frac{1}{\sigma n}\sum_{i=1}^{n}Z_i\mathcal{L}(\mathbf{A}^+,\begin{bmatrix}\mathbf{X}\\\mathbf{X}_{fake}+\sigma Z_i\end{bmatrix})
\end{array}
\right.$& \multirow{2}{*}{Node Classification (GIA)} \\
&$\Delta A\sim$ cluster prior&\\
\bottomrule
\end{tabular}
\caption{Existing query-based methods on graph adversarial attacks.}
\label{unified}
\end{table*}
We now present a unified framework for query-based adversarial attacks as well as the threat model and loss function.
\subsection{Graph Injection Attack}
Given a small set of victim nodes $\Phi_{\mathbf{A}}\subseteq \Phi$ in the graph, the goal of graph injection attack is to perform mild perturbations on the graph $G=(\mathbf{A},\mathbf{X})$, leading to $G^{+}=(\mathbf{A}^{+},\mathbf{X}^{+})$, such that the predicted labels of the victim nodes in $\Phi_{\mathbf{A}}$ are changed into the target labels. This goal is usually achieved by optimizing the adversarial loss $\mathcal{L}(\cdot)$ under constraints as:
\begin{equation}
\min_{G^{+}}\, \mathcal{L}(G^{+})\,\,s.t.\,\,\mbox{dist}(G, G^{+})\leq \Delta\label{overall0},
\end{equation}
where $\mbox{dist}(G, G^{+})$ denotes the magnitude of perturbation and has to be within the adversarial budget $\Delta$. In this section, it can be specified as $|\Phi_{fake}|\leq \Delta_{fake}$ and $\sum_{u\in \Phi_{fake}} d(u) \leq \Delta_{edge}$ with $d(u)$ being the degree of node $u$.
In graph injection attacks, we have the augmented adjacent matrix $\mathbf{A}^{+}=\begin{bmatrix}\mathbf{A}&\mathbf{B}^{T}\\\mathbf{B}&\mathbf{A}_{fake}\end{bmatrix}$ and the augmented feature matrix $\mathbf{X}^{+}=\begin{bmatrix}\mathbf{X}\\\mathbf{X}_{fake}\end{bmatrix}$. We further use $\Phi^{+}=\Phi\cup \Phi_{fake}$ to denote the node set of $G^{+}$. In particular, $\mathbf{X}_{fake}$ corresponds to the feature of fake nodes; $\mathbf{B}$ denotes the connections between fake nodes and original nodes, and $\mathbf{A}_{fake}$ denotes the mutual connections between fake nodes. The malicious attacker manipulates $\mathbf{A}_{fake},\mathbf{B}$ and $\mathbf{X}_{fake}$, leading to as low classification accuracy on $\Phi_{\mathbf{A}}$ as possible.
In a query-based adversarial attack on graphs, we uniformly formulate the update of a graph at time $t$ as
\begin{equation}
\Delta G_{(t)}\sim p\big(\Delta G_{(t)}|\{f(G_i)|i=1,2,...,q_t\},G_{t})\mbox{,}
\end{equation}
where $\{f(G_i)|i=1,2,...,q_t\}$ denotes the feedback (hard labels or predicted values) of total $q_t$ queries in the history from the target model. A curated list of current query-based graph adversarial attacks is shown in Table \ref{unified}. In general, perturbation $\Delta G_{(t)}$ in time step $t$ is conditioned on the history of query results $\{f(G_i)|i=1,2,...,q_t\}$ and the current graph status $G$. Previous work has focused on using different optimization methods, including reinforcement learning \cite{10.1145/3447548.3467416} and gradient-based optimization \cite{10.1145/3460120.3484796} to decide $\Delta G$ without utilizing the graph structure explicitly
\subsection{Threat Model}
\paragraph{Adversary Capability.} We greatly restrict the attacker's ability so that we can only make connections between victim nodes and fake nodes. No connections can be made between fake nodes because connected malicious fake nodes are easier for detectors to locate. The number of new edges $\Delta_{edge}$ is set as barely the number of victim nodes $\Phi_A$, which means that each victim node is connected by only one new edge.
\paragraph{Protected Nodes.} As mentioned above, owing to the non-i.i.d nature of graph data, attacking victim nodes may have unintended side effects on their neighboring nodes. While attacking victim nodes, we simultaneously aim to keep the labels of untargeted nodes unchanged, to make our perturbation unnoticeable. In our setting, we try to protect $\mathcal{N}_{k}(\Phi_{A})$, which are the neighbors of the victim nodes within $k$-hop.
\paragraph{Partial Information.} It is practical to assume that the attacker has access to only part of the graph when conducting the attack. As mentioned above, we adopt an extremely strict setting so that we can only observe the features and connections of the observed nodes defined as
\begin{equation}
\Phi_o=\Phi_A\cup\mathcal{N}_{k}(\Phi_{A})\cup\Phi_{fake}\label{observed_nodes}.
\end{equation}
\paragraph{Limited Queries.} It is practical in real scenarios that we have a limited number of queries to the victim model rather than full outputs of arbitrarily many chosen inputs. In our setting, we can query at most $K$ times in total for the predicted scores of the observed nodes. The architecture and parameters about the victim model are unknown to the attacker.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{Cluster-Attack.png}
\caption{An illustration of Cluster Attack. We first compute Adversarial Vulnerability for each victim node with a limited number of queries; after that, we cluster the victim nodes and inject fake nodes accordingly; and finally we optimize the fake nodes' features.}
\label{fig: Cluster Attack}
\end{figure*}
\subsection{Loss Function}
We aim to make the classifier misclassify as many nodes as possible in the victim set of $\Phi_{\mathbf{A}}$ . As it is nontrivial to directly optimize the number of misclassified nodes since the objective is discrete, we choose to optimize a surrogate loss function:
\begin{align}
\min_{G^{+}}&\, \mathcal{L}(G^{+})\triangleq \sum_{v\in \Phi_{\mathbf{A}}}{\ell(G^{+},v)}+\lambda\sum_{v\in \mathcal{N}_{k}(\Phi_{\mathbf{A}})}{\ell_{\mathcal{N}}(G^{+},v)},\label{overall_goal}\notag\\
& s.t.\,\,\mbox{dist}(G, G^{+})\leq \Delta,
\end{align}
where $\ell(G^{+},v)$ and $\ell_{\mathcal{N}}(G^{+},v)$ represent the loss functions for each victim node and for a protected node, respectively. A smaller $\ell(G^{+},v)$ means that node $v$ is more likely to be misclassified by the victim model $f$; by contrast, a smaller $\ell_{\mathcal{N}}(G^{+},v)$ means that the predicted label of node $v$ is less likely to be changed during our attack. In particular, we design our loss in the manner of the C\&W loss \cite{carlini2016evaluating}, and define:
\begin{equation}
\begin{aligned}
&\ell(G^{+},v) = \\
&\sigma\left( \max_{y_i\neq y_t} ([f(\mathbf{A}^{+},\mathbf{X}^{+})]_{v,y_i})-[f(\mathbf{A}^{+},\mathbf{X}^{+})]_{v,y_t} \right), \label{l1}
\end{aligned}
\end{equation}where $y_t$ stands for the target label of node $v$ and the attacker succeeds only when node $v$ is misclassified as $y_t$. $\sigma(x)=\max(x,0)$. $[f(G^{+})]_{v,y_i}$ denotes the output logit of node $v$ of class $y_i$. For protected nodes, we define:
\begin{equation}
\begin{aligned}
&\ell_{\mathcal{N}}(G^{+},v) =\\
& \sigma\left( \max_{y_i\neq y_g} [f(\mathbf{A}^{+},\mathbf{X}^{+})]_{v,y_i})-[f(\mathbf{A}^{+},\mathbf{X}^{+})]_{v,y_g} \right)\mbox{,}\label{l2}
\end{aligned}
\end{equation}
where $y_g$ is the ground-truth label of $v$ from the victim model.
\section{Cluster Attack with Graph-Dependent Priors}
\subsection{Cluster Attack}
The combinatorial optimization problem \eqref{overall_goal} is hard to solve owing to the non-Euclidean nature of the adjacent matrix and the complex structure of neural networks. To tackle this, we tried to find an equivalent combinatorial optimization problem and transform our GIA problem into a well-studied one.
Here, we point out that every choice of adjacent matrix has an equivalent representation of a division of victim nodes into clusters. Thus, we get our key insight that this discrete optimization problem can be transformed into an equivalent graph clustering problem,
which is a well-studied discrete optimization problem~\cite{schaeffer2007graph}
\begin{proposition}[GIA/Graph Clustering Equivalence] Given graph $G$ and a set of nodes $\Phi_A \subseteq \Phi$, for a division of the victim nodes $\Phi_{\mathbf{A}}$ into $N_{fake}$ clusters $C=\{C_{1},C_{2},...,C_{N_{fake}}\}$, $\cup_{C_{i}\in C}C_{i}=\Phi_{\mathbf{A}}$, there exists a corresponding $\mathbf{B}$ and vice versa.\label{equivalence}
\end{proposition}
\begin{proof}We provide a one-to-one mapping between cluster $C$ and adjacent matrix $\mathbf{B}$. Specifically, given $C=\{C_{1},C_{2},...,C_{N_{fake}}\}$, $\cup_{C_{i}\in C}C_{i}=\Phi_{\mathbf{A}}$ we get $\mathbf{B}$ from
\begin{equation}
\mathbf{B}_{ij}=\begin{cases}
1, & \mbox{if }v_j\in C_i\\
0, & \mbox{otherwise}\label{B_equation}
\end{cases}.
\end{equation}
We have
\begin{equation}
\Delta_{edge} = \sum_i\sum_j\mathbf{B}_{ij}=\sum_i|C_i|=|\Phi_A|.
\end{equation}
Thus, we resultant $\mathbf{B}$ is valid in our setting.
Given $\mathbf{B}$ where $|\Phi_A| = \Delta_{edge} = \sum_i\sum_j\mathbf{B}_{ij}$, we derive the cluster $C$ from
\begin{equation}
v_j\begin{cases}
\in C_i, & \mbox{if }\mathbf{B}_{ij}=1\\
\notin C_i, & \mbox{if }\mathbf{B}_{ij}=0\label{C_equation}
\end{cases}.
\end{equation}
We have $\sum_i|C_i|=\sum_i\sum_j\mathbf{B}_{ij} = |\Phi_A|$.
In our setting, each victim node is connected to only one fake node, which indicates
\begin{equation}
\sum_i \mathbf{B}_{ij} = 1,\,\,\forall\,v_j\in\Phi_A.
\end{equation}
In this case, each cluster gets disjoint with each other
\begin{equation}
C_i\cap C_j=\emptyset,\,\,\forall\,1\leq i,j\leq N_{fake}.
\end{equation}
Then we get $\cup_{C_{i}\in C}C_{i}=\Phi_{\mathbf{A}}$ and $C$ is a valid division.
\end{proof}
Because $\textbf{A}_{fake}=\textbf{0}$ is fixed in our setting, we formulate our graph injection attack problem as an equivalent graph clustering problem. As a result, the non-trivial discrete optimization problem of the adjacency matrix can be solved in the context of graph clustering. The resulting cluster serves as a graph-dependent prior for adjacent matrix $\mathbf{B}$, which prevents inefficient searching in non-Euclidean discrete space
For graph clustering, the main concern is the metric of the similarity between victim nodes. To investigate how a fake node will affect the performance on a certain node, we propose Adversarial Vulnerability as the similarity metric for graph clustering.
Adversarial Vulnerability of a victim node reflects its ``most vulnerable angle'' towards adversarial features of fake nodes, which is related only to the local structure of the graph and can be handled with only part of the graph observed. We have the insight that victim nodes sharing similar Adversarial Vulnerability are more likely to be affected simultaneously when they are connected to the same fake node
\begin{definition}[Adversarial Vulnerability] For victim node $v\in \Phi$, its \textsl{Adversarial Vulnerability} is defined as
\begin{equation}
\mbox{AV}(v) = \underset{x_{u}}{\mbox{argmin}}\,\,\mathcal{L}(G^{+}),\label{av}
\end{equation}
where $x_u$ denotes the feature of fake node $u$ connected to node $v$. For the fake node itself, the Adversarial Vulnerability is defined as its own feature.
\end{definition}
Here, we adopt Euclid's distance as distance metric between victim nodes' Adversarial Vulnerability which is in Euclidean feature space
\begin{definition}[Adversarial Distance Metric] $\forall v,u\in \Phi^+$, the \textsl{Adversarial Distance Metric} between node $v$ and $u$ is defined as $d(v,u)=||\mbox{AV}(v)-\mbox{AV}(u)||_{2}^{2}$.
\end{definition}
After the Adversarial Vulnerability is computed, the objective of the cluster algorithm is to minimize the following cluster distance as
\begin{equation}
\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i})\label{cluster_loss},
\end{equation}
where the cluster center $c_i$ is the corresponding fake node of cluster $C_i$, and
\begin{equation}
\mbox{AV}(c_{i})=\frac{1}{|C_{i}|}\sum_{v\in C_{i}}\mbox{AV}(v).\label{cluster_center}
\end{equation}
\subsection{Optimization}
To approximate Adversarial Vulnerability, we adopt zeroth-order optimization~\cite{10.1145/3128572.3140448}, which is similar to query-based attacks on image classification, to better utilize limited queries. For graph with discrete features, the optimization of Eq. (\ref{av}) can be
\begin{equation}
\begin{aligned}
\label{discrete}&\Delta \mathbf{X}_{fake} = \\
&\mathbb{I}\left( \mathcal{L}(\mathbf{A}^+, \mathbf{X}^+)>\mathcal{L}\left( \mathbf{A}^+, \begin{bmatrix}\mathbf{X}\\\mathbf{X}_{fake}+\delta \mathbf{X}_{ij}\end{bmatrix} \right) \right)\cdot\delta \mathbf{X}_{ij}\mbox{,}
\end{aligned}
\end{equation}
where $\delta \mathbf{X}_{ij}$ denotes the tentative perturbation in dimension $j$ of a feature of the $i$th fake node and $\mathbb{I}(\cdot)$ is the indicator function. A tentative perturbation is adopted only if it diminishes the adversarial loss. For continuous feature space, we adopt NES~\cite{ilyas2018blackbox} for gradient estimation as
\begin{equation}
\begin{aligned}
\label{continuous}&\nabla_{\mathbf{X}_{fake}}\mathbb{E}\,\mathcal{L}(\mathbf{A}^+,\mathbf{X}^+) = \\ &\frac{1}{\sigma n}\sum_{i=1}^{n}Z_i\mathcal{L}\left( \mathbf{A}^+,\begin{bmatrix}\mathbf{X}\\\mathbf{X}_{fake}+\sigma Z_i\end{bmatrix} \right)\mbox{,}
\end{aligned}
\end{equation}
where $\sigma>0$ is the standard variance, $n$ is the size of the NES population and $Z_i\sim \mathcal{N}(\mathbf{0},\mathbf{I}_{N_{fake}\times D})$ is the perturbation of $\mathbf{X}_{fake}$. After gradient estimation, gradient-based optimization methods can be adopted. Here, we use Projected Gradient Descent (PGD) \cite{madry2017towards} to optimize $\mathbf{X}_{fake}$.
Our method is outlined in Figure \ref{fig: Cluster Attack}. With the resultant Adversarial Vulnerability, we solve the optimization of Eq. (\ref{cluster_loss}) by K-Means clustering. After that, the features of fake nodes, initialized as the cluster center in Eq.~\eqref{cluster_center}, are optimized using Eq. (\ref{discrete}) and Eq. (\ref{continuous}). More details of our algorithm are deferred to the appendix.
\subsection{Theoretical Analysis}
Connecting fake nodes to an original graph brings a victim node (1) one 1-hop neighbor (the fake node connected to it); (2) neighbors at a farther distance connected through this fake node; (3) fake nodes connected to other victim nodes which are at least 2-hop away. It is noted that 1-hop neighbors are often dominant. Here we leave out the influence of farther neighbors caused by the fake node and fake nodes connected to other victim nodes which are at least 2-hop or even farther. The loss function over the $i$th victim node $v_i\in \Phi_A$ in Eq. (\ref{overall_goal}) can thus be seen as a function of fake nodes' features (here we set a trade-off parameter $\lambda=0$ for analysis). We have
\begin{equation}
\mathcal{L}(G)=\sum_{v_i\in \Phi_A} l(G^{+},v_i)= \sum_{i=1}^{|\Phi_A|} l_i(x_i)\label{divide}\mbox{,}
\end{equation}
where $x_i$ denotes fake nodes' features connected to victim node $v_i$. Theoretically, we provide our bounds under certain smooth conditions which hold for numerous neural networks.
\begin{definition}[W-condition] We say that a function $\mathcal{L}(G)=\sum_{i=1}^{|\Phi_A|}l_i(x_i)$ satisfies the \textsl{W-condition}, if and only if $\forall\,1\leq i\leq |\Phi_A|$, $l_i(\cdot)$ satisfies the Lipschitz condition of order 2. In this case, we have
\begin{equation}
m_i||x_i-x_i^{*}||_2^2\leq l_i(x_i)-l_i(x_i^*)\leq M_i||x_i-x_i^{*}||_2^2\mbox{,}\label{wcondition}
\end{equation}
where $0\leq m_i\leq M_i$ are constants and $1\leq i\leq |\Phi_A|$. $x_{i}^{*}$ is the minimum of $l_i(\cdot)$.
\end{definition}
Note that $M$ exists because the loss function satisfies the Lipschitz condition of order 2 under W-condition; and it also includes $m$ because $m=0$ always satisfies Eq. (\ref{wcondition}). Under W-condition, we derive our bounds on the difference in adversarial loss between our results and optimal adversarial examples.
\begin{proposition}\label{pro:bound}
If $\mathcal{L}(\cdot)$ in Eq. (\ref{divide}) satisfies the W-condition, $G^{m}$ is the optimal choice of Eq. (\ref{overall0}) and $G^{'}$ is the optimal given by the Cluster Attack of Eq. (\ref{cluster_loss}). Then, we have
\begin{equation}
\mathcal{L}(G^{'})-\mathcal{L}(G^{m})\leq |M-m|\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i})\mbox{,}\label{bound}
\end{equation}
where $M=\max_{1\leq i\leq |\Phi_A|}M_i$ and $m=\min_{1\leq i\leq |\Phi_A|}m_i$.
\end{proposition}
\begin{proof}
We have
\begin{equation}
\begin{aligned}
&\mathcal{L}(G^{'})-\mathcal{L}(G^{m})\\ =&\sum_{i=1}^{|\Phi_A|}l_i(x_i')-\sum_{i=1}^{|\Phi_A|}l_i(x_i^m)\\
=&\sum_{i=1}^{|\Phi_A|}(l_i(x_i')-l_i(x_i^*))-\sum_{i=1}^{|\Phi_A|}(l_i(x_i^m)-l_i(x_i^*))\\
\leq& \sum_{i=1}^{|\Phi_A|}M_i ||x_i'-x_i^*||_2^2-\sum_{i=1}^{|\Phi_A|}m_i ||x_i^m-x_i^*||_2^2\\
\leq& M\sum_{i=1}^{|\Phi_A|}||x_i'-x_i^*||_2^2-m\sum_{i=1}^{|\Phi_A|}||x_i^m-x_i^*||_2^2\\
=&M\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i})-m\sum_{i=1}^{|\Phi_A|}||x_i^m-x_i^*||_2^2\\
\leq&M\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i})-m\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i})\\
=&|M-m|\min_{C}\sum_{C_{i}\in C}\sum_{v\in C_{i}} d(v,c_{i}),
\end{aligned}
\end{equation}
where $x_i^{m}$ and $x_i'$ are features of the fake node connected to $i$th victim node provided by Eq. (\ref{overall0}) and Cluster Attack of Eq. (\ref{cluster_loss}), respectively.
\end{proof}
Proposition \ref{pro:bound} indicates that the difference in adversarial loss between our results and optimal adversarial examples is bounded by the minimal cluster distance in Eq. \eqref{cluster_loss} and how each $l_i(\cdot)$ is linear to $||x_i-x_i^*||_2^2$
\section{Experiments}
\subsection{Experimental Setup}
\paragraph{Dataset.} We do our experiments on Cora and Citeseer \cite{Sen_Namata_Bilgic_Getoor_Galligher_Eliassi-Rad_2008}, which are two benchmark small citation networks with discrete node features, and on Reddit~\cite{hamilton2017inductive} and ogbn-arxiv~\cite{hu2020open}, which are two large networks with continuous node features. The statistics of the datasets are shown in Table \ref{dataset}
\begin{table}[tb]
\centering
\begin{tabular}{lllll}
\toprule
Name & Nodes & Edges &Features &Classes \\
\midrule
Cora & 2708 & 5429 &1433 &7 \\
Citeseer & 3327 & 4732 &3702 &6 \\
Reddit & 232965 & 11606919 &602 &41 \\
ogbn-arxiv & 169343 & 1157799 &128 &40 \\
\bottomrule
\end{tabular}
\caption{Statistics of the datasets.}
\label{dataset}
\end{table}
\begin{table*}[tbhp]
\centering
\begin{tabular}{l|llll|llll}
\toprule
\multirow{2}{*}{Method} &
\multicolumn{4}{c|}{Cora}&
\multicolumn{4}{c}{Citeseer} \\
& $T=3$ & $T=5$ & $T=7$ & $T=10$ &$T=3$ & $T=5$ & $T=7$ & $T=10$\\
\midrule
Random &0.07&0.08&0.04&0.05&0.04&0.02&0.03&0.03\\
NETTACK &0.61&0.57&0.55&0.53&0.75&0.71&0.66&0.61\\
NETTACK - Sequential &0.68&0.73&0.72&0.70&0.76&0.74&0.72&0.67\\
Fake Node Attack &0.61&0.58&0.54&0.52&0.76&0.68&0.62&0.60\\
G-NIA &-&-&-&-&0.86&0.76&0.70&0.65\\
\midrule
Cluster Attack &\textbf{0.99}&\textbf{0.93}&\textbf{0.84}&\textbf{0.72}&\textbf{1.00}&\textbf{0.89}&\textbf{0.80}&\textbf{0.70}\\
\bottomrule
\end{tabular}
\caption{Success rates of Cluster Attack along with other baselines with discrete feature space. $T$ denotes number of victim nodes.
}
\label{exp-result}
\end{table*}
\paragraph{Parameters.} For each experimental setting, we run the experiment for 100 times and report the average results. In each round, we randomly sample $|\Phi_{A}|$ nodes as victim nodes.
We set $k=1$ in $\mathcal{N}_{k}(\Phi_{A})$, which means we aim to protect the $1$-hop neighbors of victim nodes. Without specification, we compare our method with baselines with a trade-off parameter set as $\lambda=0$ in Eq.~\eqref{overall_goal}.
\paragraph{Comparison Methods.} Since this study is the first to perform query-based injection attack on node classification, most of the previous baselines on graph injection attacks cannot be easily adapted to our problem. We include the following baselines
\textbf{Random Attack}, which decides the fake nodes' features and connections between fake nodes and original nodes randomly.
\textbf{NETTACK}, one of the most effective attacks by first adding several nodes and then adding many edges between the fake nodes and original nodes~\cite{Z_gner_2018}.
\textbf{NETTACK - Sequential}, which is a variant of NETTACK~\cite{Z_gner_2018} by sequentially adding fake nodes
\textbf{Fake Node Attack}, which adds fake nodes in a white-box attack scenario~\cite{wang2018attack}
\textbf{G-NIA}, a white-box graph injection attack~\cite{gnia}. We refer to the reported results on Citeseer.
\textbf{TDGIA}, a black-box GIA method with superior performance to all the baselines in KDD Cup 2020\footnote{https://www.kdd.org/kdd2020/kdd-cup} of graph injection attack. We mainly compare our method with this method. Note that TDGIA is not query-based~\cite{tdgia}.
Among the above baselines, TDGIA is performed in a continuous feature space while NETTACK and Fake Node Attack are performed in discrete feature spaces.
\subsection{Quantitative Evaluation}
Without loss of generality, we uniformly set $N_{fake}=4$ and let the number of victim nodes vary to see the performance under different $N_{fake}:|\Phi_{A}|$.
\subsubsection{\textbf{Performance on Small Datasets with Discrete Features}}
We first evaluate the performance of the Cluster Attack along with other baselines on Cora and Citeseer with discrete features. The number of queries $K$ is set to $K=|\Phi_{A}|\cdot K_{t}+N_{fake}\cdot K_{f}$, where $K_{t}=K_{f}=D$ (feature dimension). The results are shown in Table \ref{exp-result}. Our algorithm outperforms all baselines in terms of success rates. This is because our method prevents inefficiently searching the non-Euclidean space of the adjacent matrix and better utilizes the limited queries in searching the Euclidean feature space. The results also demonstrate that the Adversarial Vulnerability is a good metric for clustering the victim nodes.
\subsubsection{\textbf{Performance on Large Datasets with Continuous Features}}
In this section, we evaluate the performance of the Cluster Attack on Reddit (with $1500{|\Phi_A|}+750N_{fake}$ queries) and obgn-arxiv (with $4000{|\Phi_A|}+2000N_{fake}$ queries), two large networks with continuous feature whose victim nodes have a higher average degree. We compare our method with the state-of-the-art method which has superior performance to other baselines. In these challenging datasets, we perform an untargeted attack, which means attacker successes when the predicted labels of victim nodes are changed. The results are shown in Table \ref{continuous-result}. Our algorithm outperforms the baseline in terms of success rates. This is because, with the cluster prior of the adjacent matrix, our Cluster Attack prevents inefficiently searching in non-Euclidean space and make the best use of the limited queries to search the Euclidean feature space. Another reason is because of the good metric of the Adversarial Vulnerability, which provides an appropriate cluster prior.
\begin{table}[tbhp]
\centering
\begin{tabular}{l|ll|ll}
\toprule
\multirow{2}{*}{Method} &
\multicolumn{2}{c|}{ogbn-arxiv}&
\multicolumn{2}{c}{Reddit} \\
& $T=12$ & $T=16$ &$T=12$ & $T=16$\\
\midrule
TDGIA &0.45&0.38&0.09&0.07\\
\midrule
Cluster Attack &\textbf{0.67}&\textbf{0.59}&\textbf{0.15}&\textbf{0.12}\\%arxiv 4/10 0.74 reddit 4/10 0.15
\bottomrule
\end{tabular}
\caption{Success rates of Cluster Attack along with other baseline with continuous features. $T$ denotes number of victim nodes.}
\label{continuous-result}
\end{table}
\subsection{Ablation Study}
\subsubsection{\textbf{Performance with Different Trade-Off Parameters $\lambda$}}
In this section, we examine the performance of Cluster Attack with different trade-off parameters $\lambda$ between fake nodes and protected nodes in the Cora dataset. We uniformly set $N_{fake}=4$, $|\Phi_{A}|=10$. We choose two competitive baselines and adapt their loss functions to our trade-off format. The results are shown in Figure \ref{lambda}. It can be seen from Figure \ref{lambda} that, when $\lambda$ increases (which means that we pay more attention to the protected nodes), the percentage of protected nodes whose labels remain unchanged during the attack also increases. This is because we try to protect the labels of the protected nodes from being changed in a trade-off formulation in our loss function, Eq. (\ref{overall_goal}). Also, our trade-off formulation can be generalized to other baselines, as shown in Figure \ref{lambda}. This is because we design our loss function in Eq.~(\ref{overall_goal}) in a generalizable manner independent of attack method.
\begin{figure}[bhtp]
\centering
\subfigure[Success rates of attack]{
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=1.5in]{lambda1.pdf}
\end{minipage}%
}%
\subfigure[Percentage of unchanged protected nodes
]{
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[width=1.5in]{lambda2.pdf}
\end{minipage}
}%
\centering
\caption{Cluster Attack in Cora with different $\lambda$.}
\label{lambda}
\end{figure}
\subsubsection{\textbf{Performance with Different Number of Queries}}
In this section, we examine the performance of Cluster Attack with a different number of queries. We set $K_{t}=K_{f}=\alpha \cdot D$ and examine the performance under different $\alpha$ in Cora and Citeseer dataset. We uniformly set $N_{fake}=4$, $|\Phi_{A}|=10$ with $\lambda=0$ and $\lambda=1$. The results are shown in Figure~\ref{query-result}. The success rate of Cluster Attack drops as the number of queries drops. Our algorithm still performs well when the number of queries drops slightly, especially when $\alpha\ge 0.4$. This demonstrates that our Cluster Attack can work in a query-efficient manner. This is because cluster algorithm provides graph-dependent priors for the adjacent matrix and thus prevents inefficient searching. Searching in the Euclidean feature space is more efficient.
\begin{figure}[bhtp]
\centering
\includegraphics[width=1.8in]{query-result.pdf}
\caption{Success rates of Cluster Attack with different number of queries in Cora and Citeseer.}
\label{query-result}
\end{figure}
We provide additional experiments in the appendix. The experiments show that nodes with a lower degree are more likely to get misclassified under attack. Also, when the number of fake nodes increases, the success rate of the attack increases too, which is consistent with our intuitive understanding. We provide an ablation study on the cluster metric of Adversarial Vulnerability. We find that original Cluster Attack performs better than Cluster Attack without Adversarial Vulnerability, i.e., the victim nodes' Adversarial Vulnerabilities are randomly set. This result demonstrates the effectiveness of our Adversarial Vulnerability.
\section{Conclusion}
In this paper, we provide a unified framework for query-based adversarial attacks on graphs. Under the framework, we propose Cluster Attack, a query-based black-box graph injection attack with partial information. We demonstrate that a graph injection attack can be formulated as an equivalent clustering problem.
The difficult discrete optimization problem of the adjacent matrix can thus be solved in the context of clustering. After providing theoretical bounds on our method, we empirically show that our method has strong performance in terms of the success rate of attacking.
\section*{Ethical Statement}
The safety and robustness of AI are attracting more and more attention. In this work, we propose a method of adversarial attack. We hope our work reveals the potential weakness of current graph neural networks to some extent, and more importantly inspires future work to develop more robust graph neural networks.
\section*{Acknowledgments}
This work was supported by the National Key Research and Development Program of China (Nos. 2020AAA0104304, 2017YFA0700904), NSFC Projects (Nos. 62061136001, 61621136008, 62076147, U19B2034, U19A2081, U1811461), the major key project of PCL (No. PCL2021A12),
Tsinghua-Alibaba Joint Research Program, Tsinghua-OPPO Joint Research Center, and the High Performance Computing Center, Tsinghua University.
|
1,314,259,993,012 | arxiv | \section{Introduction}\label{sec:introduction}
Cancer subtype prediction has evidently been of great importance for guiding the treatment of patients suffering from any type of cancer. For several years, researchers and biologists have been studying how subtype identification can be used to plan treatments and diagnosis \cite{10.1007/978-3-030-34139-8_39,10.3389/fgene.2019.00020} . Identifying potential bio-markers in a patients body by studying the gene expression values has been a subject of research for a long time \cite{biomarkers}. With the advancement of machine learning, it is possible to classify cancer subtypes and study the results in an elegant manner. However, since human cells contain thousands of genes, all of which may or may not play a crucial role in the identification of a cancer or its subtypes, the need to filter the genes using various parameters and metrics arises.
The study in \cite{10.1007/978-3-030-34139-8_39} has shown how a feature selection approach can impact the performance of the supervised predictive model when identifying cancer subtypes. In \cite{10.1007/978-3-030-34139-8_39}, the authors used a kernel-based clustering method for gene selection, and selected genes based on their weights. The work in \cite{DBLP:conf/icml/YuL03} proposed a correlation-based filter solution for feature selection in high-dimensional data by introducing a novel concept of predominant correlation. This method also identified redundancy among relevant features. A novel approach to combine feature selection and transductive support vector machine (TSVM) was proposed in \cite{6334430} which resulted in improved accuracy as compared to standard SVM algorithms. In \cite{doi:10.1080/03772063.2021.1878062}, the authors proposed an unsupervised feature selection method for cancer prediction. They used a Singular Value Decomposition (SVD) Entropy method for feature ranking. A novel unsupervised Similarity Kernel Fusion (SKF) was proposed in \cite{10.3389/fgene.2019.00020} which made use of spectral clustering on the integrated kernel to predict cancer subtypes.
There have been studies using deep learning for cancer subtype classification as well. A novel supervised cancer classification framework, Deep Cancer subtype Classification (DeepCC) was proposed in \cite{Gao2019DeepCCAN} which was based on deep learning of functional spectra quantifying activities of biological pathways. A stacking ensemble based deep learning approach was also proposed in \cite{PMID:34341396} which was based on One-dimensional Convolutional Neural Network (1D-CNN) and Least Absolute Shrinkage and Selection Operator (LASSO) as the feature selection methods. The study in \cite{Mostavi2020} presented novel CNN models to predict cancer types based on gene expression profiles and elucidated biological relevance of cancer marker genes. The authors in \cite{ijerph18042197} used deep autoencoder for feature extraction and used an oversampling algorithm to handle data imbalance. Such approaches demonstrate how effective feature selection can be when dealing with high-dimensional data for classification.
However, most of these approaches were applied on a particular type of cancer and omics. In order to understand and compare the effects of common feature selection algorithms for predicting cancer subtypes, we aim to present a comparative study of how the effectiveness of these algorithms vary when tested against different omics of various cancer types. It is important to note that a majority of these aforementioned works used a single feature selection algorithm. Also, in case of data with high dimension and low sample size, deep learning approaches may find it difficult to learn relevant patterns, and in some cases, overfit the training data. Such approaches also take a significant amount of time to produce results.
To overcome these shortcomings, along with improving the performance of a relatively simpler machine learning model, we use three different feature selection techniques sequentially. When these methods are used individually on the complete training data, the estimator takes a significant amount of time going over every single feature to calculate its importance score. One of these methods uses k-fold cross validation to calculate the optimal number of features. This makes the methods computationally expensive and the resultant number of features is still quite high. Using these methods sequentially provides only the relevant features to the next algorithm in the sequence which in turn reduces the computation time and the dimension of the data.
The remainder of this paper is organized as follows. After a brief introduction in \Cref{sec:introduction}, data processing and workflow are presented in \Cref{sec:dataprocessing}. Machine learning modeling of the relevant data is presented in \Cref{sec:modeling}, followed by results and analysis in \Cref{sec:results}. Finally, \Cref{sec:conclusions} presents concluding remarks and discusses some outlines for future work.
\section{Data Processing}\label{sec:dataprocessing}
We use various classification algorithms to compare the effect that the individual feature selection methods have on the omics data. Since omics data sets have a very large number of features, ranging from 1,000 to 50,000, it is necessary to perform various feature selection methods, and dimensionality reduction algorithms to make the data sets more suitable for predictive modeling and analysis. To this aim, we perform normalization of the gene expression values to scale the values of the data set. After preprocessing the values, we use various machine learning models to make cancer subtype predictions. By using advanced dimensionality reduction techniques like Uniform Manifold Approximation and Projection (UMAP), we further analyze the data for additional insights and inferences. A detailed overview of the entire process is depicted in \Cref{fig:workflow}, and discussed next.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{workflow.png}
\caption{Workflow of Cancer Subtype Prediction.}
\label{fig:workflow}
\end{figure}
\subsection{Dataset}
Cancer subtypes are the smaller groups that a particular type of cancer is divided into based on specific characteristics of the cancer cells. It is essential to know the subtype of cancer to plan treatment and determine prognosis. The National Institutes of Health (NIH) created The Cancer Genome Atlas (TCGA) program to obtain a comprehensive understanding of the genomic alterations that underlie all major cancers. TCGA provides access to various omics data like gene expression, DNA methylation, copy number variations, etc. across different cancer types. Such data are highly variable and high-dimensional. Since a particular cancer can have multiple subtypes, their identification is essential for providing patients with the necessary treatment.
In the interest of our analysis, we download ten datasets from the Genomic Data Commons (GDC) Data Portal which includes five cancer types having two omics each. The two omics are miRNA Expression and Gene Expression (RNASeq), and the five cancer types are Head-Neck Squamous Cell Carcinoma (TCGA-HNSC), Kidney Renal Clear Cell Carcinoma (TCGA-KIRC), Kidney Renal Papillary Cell Carcinoma (TCGA-KIRP), Lung Adenocarcinoma (TCGA-LUAD), and Lung Squamous Cell Carcinoma (TCGA-LUSC). We use the harmonized versions of all the datasets for our study.
\subsection{Required Tools}
In our work, we use the R programming language for initial preprocessing and analysis of our data. We use TCGAbiolinks package \cite{tcgabiolinks,tcgaworkflow,gtex} for downloading, preparing, and performing some preprocessing tasks on the data. The initial analysis of data also requires the EDASeq package \cite{edaseq}. We use the SummarizedExperiment package \cite{assays} to use its container which contains one or more assays. The assays are represented by matrix-like objects. The rows typically represent genomic ranges of interest and the columns represent samples. We use the DESeq2 package \cite{deseq2} for further analysis of RNASeq data using DESeqDataSet object and for normalizing the data. For Differential Expression Analysis (DEA), we use the edgeR package \cite{edger}. After getting the normalized data, we use Python programming language for performing feature selection and extraction, and machine learning modeling.
\subsection{Preparing Data}
On a predictive modeling problem, machine learning algorithms are used to map the input variables to a target label. However, we cannot fit these algorithms on raw data. We need to transform the data into such a representation that we can best explore its underlying structure and use a suitable algorithm for getting the satisfactory performance out of the machine learning model with the given resources.
We search the GDC database for parameters like data category, data type, workflow type, and file type to access the data. For RNASeq, we select the data category as Transcriptome Profiling, the data type as Gene Expression Quantification, and the workflow type as HTSeq-Counts. For miRNA, we select the data category as Transcriptome Profiling, and the data type as miRNA Expression Quantification. After getting the queried data, we take all the samples and select only the Primary Solid Tumor (PST) and Solid Tissue Normal (STN) samples. We perform down-sampling and add clinical information, TCGA molecular information, and respective cancer subtype information to the samples.
\subsection{Differential Expression Analysis (DEA)}
DEA involves taking the normalized read count data and performing statistical analysis to discover quantitative changes in expression levels between experimental groups. For example, statistical testing can be used to decide whether, for a given gene, an observed difference in read counts is significant, that is, whether it is greater than what would be expected just due to natural random variation. The goal of DEA is to determine which genes are expressed at different levels between conditions. These genes can offer biological insight into the processes affected by the condition(s) of interest. The count data used for DEA represents the number of sequence reads that originated from a particular gene. The higher the number of counts, the more reads associated with that gene, and the assumption that there was a higher level of expression of that gene in the sample. Before performing DEA on the dataset, we preprocess the data for removing samples (using the Pearson Correlation Coefficient) with low correlation that were possible outliers.
In case of RNASeq count data, major technology-related artifacts and biases affect the expression measurements and therefore normalizing the expression values based on some metric is an important aspect before performing the DEA. The GC-content bias is one such bias which describes the dependence between fragment count and GC-content found in Illumina sequencing data. This bias can dominate the signal of interest for analyses that focus on measuring fragment abundance within a genome, such as RNASeq \cite{10.1093/nar/gks001}. We use \emph{TCGAanalyze\_Normalization} which uses within-lane normalization procedures to adjust for GC-content effect (or other gene-level effects) on read counts by using Loess robust local regression, global-scaling, and full-quantile normalization \cite{edaseq}, and between-lane normalization procedures to adjust for distributional differences between lanes (e.g., sequencing depth) by using global-scaling and full-quantile normalization \cite{Bullard2009EvaluationOS}.
After normalizing the expression values, we use a filtration process to remove all the genes except the ones which have quantile mean values higher than the threshold defined across all samples. Next, we perform DEA on the remaining gene expression values where a negative binomial generalized log-linear model is fit to the read counts for each gene that outputs another set of filtered genes, thereby clearing the cutoff set for the DEA procedure.
\subsection{Data Normalization}
In gene expression data analysis, normalization is used to correct the measurement errors and bias introduced in the acquisition of data. The errors and bias may be introduced due to many factors such as concentration of target RNA sequence, instrumental noise, etc. From the data obtained after performing DEA, we create a \emph{DESeqDataSet} object to store the read counts and the intermediate estimated quantities during statistical analysis. Then, we handle the under-expressed genes by removing the rows which have a sum of less than a certain threshold (here, it is selected as 10). On the resulting data, we calculate Variance Stabilizing Transformation (VST) from the fitted dispersion-mean relation(s) and transform the count data (normalized by division by the size factors or normalization factors), yielding a matrix of values which are approximately homoskedastic (having constant variance along the range of mean values).
\subsection{Feature Selection and Dimension Reduction}
Reducing the number of input variables can reduce the computational cost, improve the performance of the model, and render the underlying algorithm fast and effective. However, the choice of the algorithm depends on the type of data. Problems involving gene expression can be high-dimensional in nature, i.e., they can contain tens of thousands of features. With staggeringly high number of features, the calculations become extremely difficult. For better performance of the machine learning models, we aim to reduce the dimensions of the data.
In our work, we use the python library scikit-learn \cite{scikit-learn} for feature selection, dimensionality reduction, and machine learning modeling. First, we perform tree-based feature selection using random forest classifier. This is used to compute impurity-based feature importance, which in turn are used to discard irrelevant features. We then perform Recursive Feature Elimination with Cross-Validation (RFECV) on the obtained relevant features. We use linear kernel SVC as the estimator to find the optimal number of features based on the accuracy score on the training set. We perform stratified two-fold cross-validation to reduce bias. Finally, we apply Recursive Feature Elimination (RFE) with the optimal number of features using the same estimator as before. RFE selects features by recursively considering smaller and smaller sets of features. We train the estimator on the initial set of features and obtain their importance score and prune the least important features. We select the optimal number of features by recursively repeating this step.
\section{Machine Learning Modeling}\label{sec:modeling}
During the modeling phase, we perform a baseline test on various models and select the models that perform relatively better. We then try to improve the performance by tuning various hyper-parameters and introducing various augmentations to the models. We perform the baseline test on six models, namely Decision Tree Classifier, Random Forest Classifier, KNeighbors Classifier, SVC, Gaussian Naive Bayes, and Logistic Regression.
We use the aforementioned feature selection algorithms to filter out features with low importance in the prediction process of the model. In the tree-based feature selection, we use a Random Forest classifier to select the features with high importance score, which are then used to make the prediction. For RFE, we use SVC as the estimator to recursively select features until the required amount of features are reached. Using all these algorithms separately does improve the result, with the recursive techniques giving comparatively better results. But the recursive techniques are computationally very expensive as well. This can be attributed to the fact that recursive algorithms are generally slower by design and the number of features being very high, it becomes an even more time consuming process. However, the tree-based method is comparatively faster, while stripping the dataset of a decent number of features. Each method filters out the features based on different parameters and metrics. What we require is using all these metrics together for the selection of the best possible features. In addition to that, since the tree based algorithm get rid of a large chunk of features, it makes computing the recursive algorithms much less computationally expensive. Hence, one may infer that instead of using each method separately before predicting the results, all these methods can be used sequentially so that after each pass, a certain metric would be satisfied. Therefore, the end result would be those features that pass the tests and are deemed satisfactory.
The final set of features are then passed through a SVC to get the final predictions. In order to choose the best possible parameters for these models, hyper-parameter tuning is performed, both manually and with the help of an algorithm called GridSearchCV, which is an augmentation to our base estimator (SVC) that accepts a bunch of hyper-parameters and returns the set of parameters that are the most optimum for our base estimator model. We also test an alternate method in which Principal Component Analysis (PCA) is used to reduce the dimensions of the datasets and then SVC is invoked to get the results.
\section{Results and Analyses}\label{sec:results}
To demonstrate the efficacy of our proposed sequential technique, we train our machine learning model on all the datasets and record the results. As stated earlier, we consider ten datasets, five of which are from RNASeq and the other five are from miRNA expression. The dimensions of the datasets after normalization are presented in \Cref{tab:dimensions}.
\begin{table}[!ht]
\centering
\caption{Dimensions of data.}
\label{tab:dimensions}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{miRNA} & \multicolumn{2}{c|}{RNASeq} \\ \hline
& Samples & Features & Samples & Features \\ \hline
HNSC & 278 & 412 & 277 & 4466 \\ \hline
KIRC & 449 & 263 & 443 & 5287 \\ \hline
KIRP & 160 & 273 & 158 & 5308 \\ \hline
LUAD & 236 & 433 & 243 & 5112 \\ \hline
LUSC & 157 & 595 & 179 & 6773 \\ \hline
\end{tabular}
}
\end{table}
We now discuss the results of each test in the coming subsections.
\subsection{Baseline Test}
We perform a baseline test on both omics data using six models, namely Decision Tree Classifier, Random Forest Classifier, Logistic Regression, SVC, K Neighbors Classifier, and Gaussian Naive Bayes Classifier. The results of miRNA expression data and RNASeq data are shown in \Cref{fig:mirna-baseline,fig:rnaseq-baseline}, respectively.
\begin{figure}[h!]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{miRNA_baseline-eps-converted-to}
\caption{miRNA baseline test.}
\label{fig:mirna-baseline}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{RNASeq_baseline-eps-converted-to}
\caption{RNASeq baseline test.}
\label{fig:rnaseq-baseline}
\end{subfigure}
\caption{Baseline test results (LUSC).}
\label{fig:baseline-test-results}
\end{figure}
One may observe the effect of highly varying, high-dimensional data with low sample size from the results in \Cref{fig:mirna-baseline,fig:rnaseq-baseline}. Due to these problems, the accuracy is either quite low or in some cases the test set has a higher accuracy than the train set. Though not shown here, it readily follows from our analyses that the baseline results on other datasets are also similar. The one model that performs better in comparison to the other five models is the SVC. Thus, we pick SVC to further improve its performance.
\subsection{Feature Selection Results}
We use three feature selection algorithms, namely, tree-based feature selection, RFECV, and RFE. We use bagged decision trees using the Random Forest estimator in tree-based feature selection. For RFECV and RFE, we use SVC as the estimator, which we selected after the baseline test. As mentioned before, the results are only slightly better than what we get using the baseline models with RFECV producing somewhat better results but taking longer than the other two methods to execute. All three feature selection algorithms cover certain aspects while selecting or eliminating features, but the number of selected features are still quite high.
\begin{figure}[!ht]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{lusc_rnaseq_fs_accuracy-eps-converted-to}
\caption{miRNA accuracy.}
\label{fig:mirna-accuracy}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{lusc_mirna_fs_features-eps-converted-to}
\caption{miRNA features.}
\label{fig:mirna-features}
\end{subfigure}
\caption{miRNA Accuracy-Features Relation (LUSC).}
\label{fig:mirna-accuracy-feature-relation}
\end{figure}
\begin{figure}[!ht]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{lusc_mirna_fs_accuracy-eps-converted-to}
\caption{RNASeq accuracy.}
\label{fig:rnaseq-accuracy}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{lusc_rnaseq_fs_features-eps-converted-to}
\caption{RNASeq features.}
\label{fig:rnaseq-features}
\end{subfigure}
\caption{RNASeq Accuracy-Features Relation (LUSC).}
\label{fig:rnaseq-accuracy-feature-relation}
\end{figure}
In order to address the shortcomings associated with the aforementioned techniques, we use these techniques sequentially. We take the features selected from the tree-based feature selection method based on their importance scores and use them to eliminate non-relevant features using RFECV. The stratified k-fold cross-validation in RFECV redces algorithm bias towards some of the features to get the results. Next, we use RFE with the remaining features and apply SVC to make predictions. This improves the overall performance and also lowers the computation cost. A comparative analysis of the results is presented in \Cref{fig:mirna-accuracy-feature-relation,fig:rnaseq-accuracy-feature-relation}.
\begin{table}[!ht]
\centering
\caption{Execution time (in seconds).}
\label{tab:execution-time}
\begin{tabular}{|c|c|c|}
\hline
& RNASeq & miRNA \\ \hline
Tree-Based & 0.405 & 0.19 \\ \hline
RFECV & 212 & 2.94 \\ \hline
RFE & 67 & 1.18 \\ \hline
Combined & 11.3 & 1.13 \\ \hline
\end{tabular}%
\end{table}
The respective execution time of all the feature selection techniques in presented in \Cref{tab:execution-time}. Although tree-based feature selection seems to have the lowest execution time, it also gives the least accuracy when compared to the other methods. Out of the three feature selection techniques, RFECV performs the best, but its execution time is also quite high. However, our proposed method reduces the execution time considerably in addition to improving the accuracy.
\subsection{Evaluation Metrics}
Based on the above results and analyses, we proceed with the sequentially applied feature selection model. We present a comparative study of multiple classification metrics in \Cref{tab:rnaseq-metrics,tab:mirna-metrics} to gain additional insights.
\begin{table}[!ht]
\centering
\caption{Evaluation metrics for RNASeq data.}
\label{tab:rnaseq-metrics}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& \multicolumn{6}{c|}{Support Vector Classifier} \\ \hline
& Precision & Recall & Specificity & F1 Score & Accuracy & G-Mean \\ \hline
HNSC & 0.91 & 0.91 & 0.97 & 0.91 & 91.1 & 0.94 \\ \hline
KIRC & 0.76 & 0.75 & 0.91 & 0.75 & 75.0 & 0.83 \\ \hline
KIRP & 0.94 & 0.94 & 0.95 & 0.94 & 93.7 & 0.94 \\ \hline
LUAD & 0.77 & 0.73 & 0.93 & 0.72 & 73.4 & 0.82 \\ \hline
LUSC & 0.98 & 0.97 & 1.00 & 0.97 & 97.2 & 0.98 \\ \hline
\end{tabular}
}
\end{table}
\begin{table}[!ht]
\centering
\caption{Evaluation metrics for miRNA data.}
\label{tab:mirna-metrics}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& \multicolumn{6}{c|}{Support Vector Classifier} \\ \hline
& Precision & Recall & Specificity & F1 Score & Accuracy & G-Mean \\ \hline
HNSC & 0.79 & 0.77 & 0.93 & 0.77 & 76.8 & 0.84 \\ \hline
KIRC & 0.73 & 0.71 & 0.88 & 0.70 & 70.8 & 0.78 \\ \hline
KIRP & 0.76 & 0.75 & 0.75 & 0.74 & 75.7 & 0.73 \\ \hline
LUAD & 0.72 & 0.71 & 0.94 & 0.70 & 70.8 & 0.80 \\ \hline
LUSC & 0.77 & 0.75 & 0.89 & 0.73 & 75.0 & 0.80 \\ \hline
\end{tabular}
}
\end{table}
We use Accuracy, Precision, Recall/Sensitivity, Specificity, F1 score, and G-Mean as the metrics to assess the predictive performance of our model. Accuracy is the total percentage of samples that are correctly predicted. Precision signifies how many samples classified as a particular cancer subtype actually belong to that subtype. Recall (or Sensitivity) signifies how many samples that actually belong to a particular subtype have been predicted as such. Specificity signifies how many samples not classified as a particular cancer subtype actually don't belong to that particular subtype. F1-score is the harmonic mean of precision and recall which balances these metrics. Similarly, G-Mean is the geometric mean of sensitivity and specificity and is used to balance these two metrics. While accuracy can give an overall intuition about how a model is performing, metrics like precision/recall and sensitivity/specificity can better evaluate the usefulness of the model.
As presented in \Cref{tab:rnaseq-metrics,tab:mirna-metrics}, RNASeq datasets have very high values of specificity and quite high values of sensitivity as well. On the other hand, miRNA datasets have relatively low sensitivity but high specificity. High sensitivity means our model identifies maximum number of patients suffering from a particular cancer subtype correctly. High specificity means our model can be used to cross-check if a patient identified as not having a particular cancer subtype is correct. There is usually a trade-off between precision and recall, and sensitivity and specificity. Tuning the model to increase one of these metrics decreases the other, and vice-versa. Thus, F1-score and G-mean are used to evaluate the model properly.
\subsection{Visual Analyses}
We perform techniques like PCA and UMAP \cite{mcinnes2018umap-software} on the data to gain some visual insights regarding the difference between the performance of both miRNA and RNASeq datasets. UMAP clustered the data into different classes comparatively better than PCA. A comparision of PCA and UMAP is presented in \Cref{fig:pca-umap}.
\begin{figure}[!ht]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{miRNA_pca.png}
\caption{miRNA PCA plot.}
\label{fig:mirna-pca}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{miRNA_umap.png}
\caption{miRNA UMAP plot.}
\label{fig:mirna-umap}
\end{subfigure}
\caption{PCA versus UMAP.}
\label{fig:pca-umap}
\end{figure}
\begin{figure}[!ht]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{HNSC_RNASeq_connectivity.png}
\caption{RNASeq connectivity graph.}
\label{fig:hnsc-rnaseq-connectivity}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{HNSC_miRNA_connectivity.png}
\caption{miRNA connectivity graph.}
\label{fig:hnsc-mirna-connectivity}
\end{subfigure}
\caption{HNSC connectivity graph.}
\label{fig:hnsc-connectivity}
\end{figure}
\begin{figure}[!ht]
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{KIRC_RNASeq_connectivity.png}
\caption{RNASeq connectivity graph.}
\label{fig:kirc-rnaseq-connectivity}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.475\columnwidth}
\centering
\includegraphics[width=1\linewidth]{KIRC_miRNA_connectivity.png}
\caption{miRNA connectivity graph.}
\label{fig:kirc-mirna-connectivity}
\end{subfigure}
\caption{KIRC connectivity graph.}
\label{fig:kirc-connectivity}
\end{figure}
We use the result from UMAP to plot weighted graphs which are the intermediate topological representation of the approximate manifold the data has been sampled from. Intuitively, it gives us an idea of how interrelated the data points in the dataset are, both within the clusters and between the clusters. RNASeq showed greater connectivity in the dataset as compared to corresponding miRNA dataset which is depicted in \Cref{fig:hnsc-connectivity,fig:kirc-connectivity}. Visually, this can be inferred by the thickness of the weighted components of the graphs plotted. This connectivity in the manifold shows why the RNASeq datasets outperform the corresponding miRNA datasets.
\section{Conclusions}\label{sec:conclusions}
We studied the transcriptomic data of five different cancer types using the miRNA and RNASeq expression values, and it was found that a combination of multiple feature selection algorithms helps in reducing the computational expense in addition to improving the accuracy of the prediction. Our model was highly specific in case of miRNA data but maintained a good balance between specificity and sensitivity in case of RNASeq data. Therefore, the model can be used for the final confirmation of the disease when used with miRNA data, and for initial diagnois as well as final confirmation of disease when used with RNASeq data. The results in this work provided a strong foundation in cancer subtype classification. One such area is the classification of cancer subtypes using integrated multi-omics data which may be an interesting investigation for future studies.
\bibliographystyle{IEEEtran}
|
1,314,259,993,013 | arxiv | \section{Introduction}
Silvio Ghilardi \citep{Ghil99,Ghil2000modal} studies
unification in propositional logics. More precisely, he
describes all solutions for
$ A(x_1,\ldots,x_n)\leftrightarrow \top$
within a background logic like intuitionistic logic ${\sf IPC}$ or a modal logic containing ${\sf K4}$. By a solution we mean a substitution
$\theta$ such that $\theta(A \leftrightarrow \top)$ holds.
On the other hand, we have a related question for
decidability/characterization
of admissible rules of ${\sf IPC}$.
A rule $A/B$ is admissible to a logic ${\sf L}$ if
${\sf L}\vdash\theta(A)$ implies ${\sf L}\vdash \theta(B)$
for every substitution $\theta$.
Despite classical logic,
in which every admissible rule is also derivable,
the case of modal logic and intuitionistic logic are not trivial.
Probably the first such underivable admissible rule for
${\sf IPC}$ is the following \citep{Harrop60}:
\begin{center}
\AxiomC{$\neg A\to (B\vee C)$}
\UnaryInfC{$(\neg A\to B)\vee(\neg A\to C)$}
\DisplayProof
\end{center}
Using the tools and results in \citep{Ghil99}, Rosalie Iemhoff
proves the completeness of a base for
\textit{all} admissible rules of ${\sf IPC}$
\citep{IemhoffT,Iemhoff-admissibility}, which previously conjectured by
de Jongh and Visser.
Decidability of admissibility for ${\sf IPC}$ was already known
\citep{Rybakov_1987,rybakov_1992,Rybakov_Book}.
There are
similar results in some modal logics extending ${\sf K4}$ both for
unification \citep{Ghil2000modal} and
admissibility \citep{jevrabek2005admissible,iemhoff2009proof}.
There is yet another related notion, \textit{preservativity},
an intuitionistic alternate for the classical notion of interpretability
or conservativity \citep{Iemhoff.Preservativity,Visser02}.
Preservativity is a binary relation $A\pres{{\sf T}}{\Gamma}B$
defined as ``$\Gamma\vdashsub\sft A$ implies $\Gamma\vdashsub\sft B$".
Albert Visser in \citep{Visser02} shows that ${\sf NNIL}$-preservativity and admissibility are tightly related, in which ${\sf NNIL}$, is the class of No
Nested Implications in the Left, introduced in
\citep{Visser-Benthem-NNIL} and more elaborated in \citep{Visser02}.
This class of propositions are proved to be helpful in the realm of
intuitionistic logic. A crucial
result concerning ${\sf NNIL}$ appeared in \citep{Visser02} is to provide
an algorithm that takes $A\in\lcal_{0}$ and returns its best
${\sf NNIL}$ approximation $A^*$ from below, i.e., $ \vdash A^*\ra A$ and
for all ${\sf NNIL}$ formulae $B$ such that $ \vdash B\ra A$, we have
$ \vdash B\ra A^*$.
Later in \cref{sec-pres-2} we also provide an algorithm
which computes $A^\star$, the best $\NNILpar$-approximation of $A$ from below.
The main work of current paper is to extend
\citep{Ghil99,Iemhoff-admissibility} and prove their results relative in
$\NNILpar$ propositions, the class of No Nested Implications in the Left \citep{Visser-Benthem-NNIL} which are made up from set of atomic parameters ${\sf par}$.
First we imitate \citep{Ghil99} and study
projectivity and extendibility relative in $\NNILpar$-propositions
(\cref{Theorem-Ghil-Ext}). This
will lead us to a relativised version of projective approximations
(\cref{Theorem-IPC-nnilp-Finitary}).
Then we take a route similar to \citep{Iemhoff-admissibility} and
provide a base called ${{\sf AR}_{\parr}}\!\,$, for $\NNILpar$-admissibility of ${\sf IPC}$
and prove its completeness (\cref{Characterization-admissibility}).
This last result together with \citep{Sigma.Prov.HA,mojtahedi2021hard},
lead us to the characterization and decidability
of provability logic of Heyting arithmetic $\hbox{\sf HA}{} $,
which is splitted to another manuscript \citep{PLHA}.
Finally we axiomatize two interesting preservativity predicates
$\pres{{\sf IPC}}{\Gamma}\ $: first when $\Gamma$ is considered as the set of
$\NNILpar$-projective propositions (this is same as projectivity relative in $\NNILpar$, as defined in \cref{Gamma-proj-nonmodal}), and second when $\Gamma:=\NNILpar$.
\section{Preliminary definitions and facts}
This section is devoted to preliminaries and conventions.
Among other well-known notions, we
define ${\sf NNIL}$ propositions, admissibility, preservativity and greatest lower bounds.
\subsection{propositional language}\label{lang}
The propositional language $ \lcal_{0} $ includes connectives
$ \vee$, $ \wedge $, $ \to $ and $ \bot $. Negation $ \neg $ is defined
as $ \neg A:= A\to\bot $ and $\top:=\neg\bot$. By default we assume that $ \lcal_{0} $ includes
finite set of atomic variables $ {\sf var} $
and also finite set of atomic parameters
$ {\sf par} $.
The union $ {\sf var}\cup{\sf par} $ is
annotated as $ {\sf atom} $, the set of atomics.
We use ${\bs{p}}$ and ${\bs{q}}$ as a finite set or list of parameters
and ${\bs{x}}$ and ${\bs{y}}$ for a finite set or list of variables.
Finite lists or sets of atomics are annotated by ${\bs{a}}$ and ${\bs{b}}$.
We use $x$, $y$ and $z$ (possibly with subscripts) as meta-variables for variables and also $p$, $q$ and $r$ (possibly with subscripts) for parameters. Also $a$, $b$ and $c$
(again possibly with subscripts) are used for both atomic variables and parameters.
Let $\bs{a}=a_1,\ldots,a_n$ be a list of atomics and
$\bs{B}=B_1,\ldots,B_n$. Then $A[\bs{a}:\bs{B}]$
indicate the simultaneous substitution of
$B_i$ for $a_i$ in $A$.
We also use the notation $ \lcal_{0}(X) $ to indicate the language of all boolean combinations of propositions in $ X $.
We consider $ {\sf IPC} $ as the intuitionistic propositional logic
\citep{TD} and
$\vdash $ indicates derivability in ${\sf IPC}$.
All propositional logics considered in this paper are assumed
to be closed under (1) modus ponens and (2) substitutions
\subsection{Substitutions}
A substitution $\theta$ is a function on propositional language $\lcal_{0}$
which commutes with all connectives, i.e.
\begin{itemize}
\item $\theta(B\circ C)=\theta(B)\circ\theta(C)$ for every $\circ\in\{\vee,\wedge,\to\}$.
\item $\theta(\bot)=\bot$.
\end{itemize}
\textit{By default we assume that all substitutions are identity on the set
${\sf par}$ of parameters.} We say that a substitution is \textit{general},
if we relax this condition on ${\sf par}$ and allow the parameters to be substituted as well.
\subsection{Kripke models for intuitionistic logic}
A Kripke model for intuitionistic logic, is a triple
$ \mathcal{K}=(W,\prec,\mathrel{V}) $ with following properties:
\begin{itemize}
\item $ W\neq\emptyset $.
\item $ (W, \prec) $ is a partial order (transitive and irreflexive). We write $ \preccurlyeq $ for the reflexive closure of $ \prec $.
\item $ \mathrel{V} $ is the valuation on atomics,
i.e.~$ V\subseteq W\times{\sf atom} $.
\item $ w\preccurlyeq u $ and $ w\mathrel{V} a $ implies $ u\mathrel{V} a $ for every $ w,u\in W $ and $ a\in{\sf atom} $.
\end{itemize}
The valuation $ \mathrel{V} $ may be extended to include all propositions as follows:
\begin{itemize}
\item $ \mathcal{K},w\Vdash a $ iff $ w\mathrel{V} a $, for $ a\in{\sf atom} $.
\item $ \mathcal{K},w\Vdash A\wedge B $ iff $ \mathcal{K},w\Vdash A $ and
$ \mathcal{K},w\Vdash B $.
\item $ \mathcal{K},w\Vdash A\vee B $ iff $ \mathcal{K},w\Vdash A $ or
$ \mathcal{K},w\Vdash B $.
\item $ \mathcal{K},w\Vdash A\to B $ iff for every $ u\succcurlyeq w $ if we have
$ \mathcal{K},w\Vdash A $ then
$ \mathcal{K},w\Vdash B $.
\end{itemize}
We also define the following notions for Kripke models:
\begin{itemize}
\item \textit{Finite:} if $ W $ is a finite set.
\item \textit{Rooted:} if there is some node $ w_0\in W $
such that $ w_0\preccurlyeq w $ for every $ w\in W $.
\item \textit{Tree:} if for every $ w\in W $ the set
$ \{u\in W: u\preccurlyeq w\} $ is finite linearly ordered
(by $ \preccurlyeq $) set.
\end{itemize}
By default we assume that \textit{all Kripke models of ${\sf IPC}$ in this paper
are finite rooted and tree.} As we will see in \cref{sec-ARmod},
some other sort of Kripke semantics are used, called ${{\sf AR}_{\parr}}\!\,$-models,
which might not be finite or tree.
Given $A\in\lcal_{0}$, we define $\Mod{A}$ as the class of all
(finite rooted tree) Kripke models of $A$.
\subsection{$ {\sf NNIL} $ propositions}\label{sec-NNIL-def}
The class of {\em No Nested Implications in the Left},
${\sf NNIL}$ formulae, was discovered by Albert Visser and first
published in
\citep{Visser-Benthem-NNIL}, and more explored in \citep{Visser02,NNIL-rev}.
For simplicity of notations, we may write ${\sf N}$ for ${\sf NNIL}$.
The crucial
result of \citep{Visser02} is to provide
an algorithm that
takes $A\in\lcal_{0}$ and returns its best
${\sf NNIL}$ approximation $A^*$ from below, i.e., $ \vdash A^*\ra A$ and
for all ${\sf NNIL}$ formulae $B$ such that $ \vdash B\ra A$, we have
$ \vdash B\ra A^*$.
Later in this paper we define another algorithm $A^\star$
which calculates the best $\NNILpar$-approximation of $A$ from below
(\cref{sec-pres-2}).
The classes ${\sf NNIL}$ and $ {\sf NI} $ of propositions
in $\lcal_{0}$ are defined inductively:
\begin{itemize}
\item $ a\in{\sf NNIL} $ and $a\in{\sf NI} $ for every $ a\in{\sf atom} $.
\item $ B\circ C\in {\sf NNIL} $ if $ B,C\in{\sf NNIL} $.
Also $ B\circ C\in {\sf NI} $ if $ B,C\in{\sf NI} $. ($ \circ\in\{\vee,\wedge\} $)
\item $ B\to C\in{\sf NNIL} $ if $ B\in{\sf NI} $ and $ C\in{\sf NNIL} $.
\end{itemize}
\subsection{Notations on sets of propositions}\label{notation-set}
In rest of the paper we deal with several sets of propositions and following notations make life easier.
Given $A\in\lcal_{0}$, let $ \sub{A} $ be the set of all subformulas of $ A $.
For simplicity of notations, we write
${\sf X_1\ldots X_n}$ for ${\sf X_1}\cap\ldots\cap{\sf X_n}$, when
${\sf X_i}$ are sets of propositions.
For a set $\Gamma$ of propositions define
\begin{itemize}
\item $\Gamma^\vee:=\{\bigvee \Delta:\Delta\subseteq_{\text{fin}} \Gamma
\text{ and } \Delta\neq\emptyset
\}.$ ($X\subseteq_{\text{fin}} Y$ indicates that $X$ is a finite subset of $Y$)
\item $ \Gamma(X) $ indicates the set $ \Gamma\cap\lcal_{0}(X) $.
\item $ {{\downarrow}^{\!\!\sft}\!\Gamma}:= $ the class of all
$ \Gamma $-projective propositions in ${\sf T}$.
We say that a proposition $ A $ is $ \Gamma $-projective in ${\sf T}$,
if there is some substitution $ \theta $ and $ B\in\Gamma $ such that
$ \sft\vdash \theta(A)\leftrightarrow B $ and $ A\vdashsub\sft x\leftrightarrow \theta(x) $ for every $ x\in{\sf var} $ (see \cref{Gamma-proj-nonmodal}).
Whenever ${\sf T}={\sf IPC}$, we may omit the superscript $ {\sf T} $ and simply write $ \darrow \Gamma$.
\end{itemize}
Also define
\begin{itemize}
\item $ {\sf N}:={\sf NNIL}:= $ as defined in \cref{sec-NNIL-def}.
\item ${\sf P}^{\sf T}:={\sf Prime}^{\sf T}:=$ the set of all
${\sf T}$-prime propositions, i.e.~the set of propositions $A$
such that for every $B,C$ with $\sft\vdash A\to (B\vee C)$
either we have $\sft\vdash A\to B$ or $\sft\vdash A\to C$.
Whenever ${\sf T}={\sf IPC}$, we may omit
the ${\sf T}$-superscript from notations.
\end{itemize}
And finally we assume that
$ (.)^\vee $ has the lowest precedence after $ \darrow{(.)} $. This means that $$\darrow{XY}^\vee:=\left(\darrow{\left(XY\right)}\right)^\vee. $$
\subsection{Admissibility and preservativity}\label{pres-admis}
Given a Logic ${\sf T}$, the binary relation $\adsm{{\sf T}}{}$
is defined to hold for those pairs $A$ and $B$ such that the inference
rule $A/B$ is admissible. More precisely $A\adsm{{\sf T}}{}B$ iff
for every substitution $\theta$, $\sft\vdash \theta(A)$ implies
$\sft\vdash \theta(B)$.
The admissibility relationship is trivial when one considers the classical
propositional logic, since every admissible $A/B$ is also derivable.
However this relationship is highly nontrivial when one considers
a modal logic or intuitionistic logic. Probably the first known
non-derivable admissible rule is the following rule \citep{Harrop60}:
\begin{prooftree}
\AxiomC{$\neg A\to (B\vee C)$}
\UnaryInfC{$(\neg A\to B)\vee(\neg A\to C)$}
\end{prooftree}
Harvey Friedman asked in 1975 for decidability of admissibility
in the intuitionistic propositional logic. Then
\citep{Rybakov_1987,rybakov_1992,Rybakov_Book} answers to this question
positively. Although it was shown that no finite base exists for all
admissible rules of the intuitionistic logic ${\sf IPC}$ \citep{Rybakov87},
de Jongh and Visser introduced a recursive base and conjectured it to generate all admissible rules of ${\sf IPC}$. Then Iemhoff proved this
conjecture \citep{Iemhoff-admissibility,IemhoffT}.
Here in this paper, we consider a relativised version of admissibility.
Given a logic ${\sf T}$ and a set $\Gamma$ of propositions define
the $\Gamma$-admissibility relation in ${\sf T}$ as follows
\begin{center}
$A\mathrel{\adsm{{\sf T}}{\Gamma}} B$ iff for every substitution $\theta$
and $C\in\Gamma$:
$ \sft\vdash \theta(C\to A)$
implies $ \sft\vdash \theta(C\to B)$.
\end{center}
Note that there is a hidden role for the language $\lcal_{0}$
in the definition of $\mathrel{\adsm{{\sf T}}{\Gamma}}$, when we consider substitution $\theta$.
However since almost everywhere in the paper we fix the language $\lcal_{0}$,
by default we assume substitutions over this fixed language and we do not explicitly mention $\lcal_{0}$.
There is also another binary relation on propositions, called
preservativity, which is known .
The $\Gamma$-preservativity relation in ${\sf T}$ is defined
as follows:
$$A\mathrel{\pres{{\sf T}}{\Gamma}} B \quad \text{ iff } \quad \forall\,E\in\Gamma(\sft\vdash E\to A \Rightarrow \sft\vdash E\to B).$$
Preservativity could be considered as intuitionistic analogue of classical
interpretability or conservativity.
This notion as a propositional logic, well studied in
\citep{Visser02} and \citep{Iemhoff.Preservativity} provided
Kripke semantics for it. \citep{Zhou-PhD,Iemhoff2005} include some
more elaboration on preservativity and provability, including fixed-point theorem and Beth property.
Following theorem says that $\mathrel{\adsm{{\sf T}}{\Gamma}}$ and $\mathrel{\pres{{\sf T}}{\Gamma}}$ are ascending
on $\Gamma$. All over this paper we may
use this fact without mentioning.
\begin{theorem}\label{pres-admis-asce}
If $\Gamma\subseteq \Gamma'$
then $ {\pres{{\sf T}}{\Gamma'}}\subseteq {\mathrel{\pres{{\sf T}}{\Gamma}}}$ and
${\adsm{{\sf T}}{\Gamma'}}\subseteq {\mathrel{\adsm{{\sf T}}{\Gamma}}}$.
\end{theorem}
\begin{proof}
Left to the reader.
\end{proof}
\begin{theorem}\label{pres-admis-rel}
$A\mathrel{\adsm{{\sf T}}{\Gamma}} B$ implies $A\preslow{{\sf T}}{{{\downarrow}^{\!\!\sft}\!\Gamma}}B$.
\end{theorem}
\begin{proof}
Let $A\mathrel{\adsm{{\sf T}}{\Gamma}} B$ and $E\in{{\downarrow}^{\!\!\sft}\!\Gamma}$ such that $\sft\vdash E\to A$.
Since $E\in{{\downarrow}^{\!\!\sft}\!\Gamma}$ there is some $\theta$ and $E^\dagger\in\Gamma$ such that
$E\vdashsub\sft \theta(a)\leftrightarrow a$ for every $a\in{\sf atom}$ and
$\sft\vdash \theta(E)\leftrightarrow E^\dagger$.
Hence
$\sft\vdash E^\dagger \to \theta(A)$. Then by $A\mathrel{\adsm{{\sf T}}{\Gamma}} B$
we get $\sft\vdash E^\dagger\to \theta(B)$ and thus
$E\vdashsub\sft \theta(E\to B)$. Since $\theta$ is $E$-projective,
we may conclude $E\vdashsub\sft E\to B$ and thus $\sft\vdash E\to B$.
\end{proof}
\begin{question}
What can be said about the other direction of \cref{pres-admis-rel}?
\end{question}
\begin{remark}
By \cref{pres-admis-rel,pres-admis-asce}, $A\mathrel{\adsm{{\sf T}}{\Gamma}} B$ implies
$A\mathrel{\pres{{\sf T}}{\Gamma}} B$, however the converse may not hold. As a counterexample
let $A$ and $B$ are two different variables and $\top\in\Gamma$
and ${\sf T}={\sf IPC}$. Then we have $A\mathrel{\pres{{\sf T}}{\Gamma}} B$ and not $A\mathrel{\adsm{{\sf T}}{\Gamma}} B$.
\end{remark}
Later in this paper we axiomatize $\mathrel{\pres{{\sf T}}{\Gamma}}$ and $\mathrel{\adsm{{\sf T}}{\Gamma}}$
for several pairs $({\sf T},\Gamma)$. Before we continue with this, let us see some basic axioms.
Let ${\sf T}$ be a logic.
The logic $ \BAR{\sf T} $
proves statements $ A\rhd B $ for $A,B\in\lcal_{0}$
and has the following axioms and rules:
\\[4mm]
\textbf{Aximos}
\begin{itemize}[leftmargin=1.5cm]
\item[${\sf Ax}:$] \quad $A\rhd B$,
for every $ \sft\vdash A\to B$.
\end{itemize}
\textbf{Rules}
\begin{center}
\bgroup
\begin{tabular}{c c}
\AxiomC{$A\rhd B$}
\AxiomC{$A\rhd C$}
\RightLabel{Conj}
\BinaryInfC{$A\rhd B\wedge C$}
\DisplayProof \quad \quad \quad
&
\AxiomC{$A\rhd B$}
\AxiomC{$B\rhd C$}
\RightLabel{Cut}
\BinaryInfC{$A\rhd C$}
\DisplayProof \quad \quad \quad
\end{tabular}
\egroup
\end{center}
\vspace{4mm}
The above mentioned axiom and rules are not interesting, because
$\BAR{\sf T}\vdash A\rhd B$ iff $\sft\vdash A\to B$. However we define several interesting additional rules:
\begin{center}
\bgroup
\def1{1}
\begin{tabular}{c c}
\quad \quad \quad \AxiomC{$B\rhd A$}
\AxiomC{$C\rhd A$}
\RightLabel{Disj}
\BinaryInfC{$B\vee C\rhd A$}
\DisplayProof
&
\quad \quad \quad
\AxiomC{$A\rhd B$}
\AxiomC{($ C\in\Delta $)}
\RightLabel{$\mont(\Delta)\xspace$}
\LeftLabel{}
\BinaryInfC{$ C\to A\rhd C\to B$}
\DisplayProof
\end{tabular}
\egroup
\end{center}
\begin{theorem}[\textbf{Soundness}]\label{gen-pres-sound}
If ${\sf T}$ is closed under substitutions, then
$\BAR{\sf T}$ is sound for relative admissibility
interpretations,
i.e.~$\BAR{\sf T}\vdash A\rhd B$ implies $A\mathrel{\adsm{{\sf T}}{\Gamma}} B$ and $A\mathrel{\pres{{\sf T}}{\Gamma}} B$
for every set
$\Gamma$ of propositions and every logic ${\sf T}$. Moreover
\begin{enumerate}
\item if $\Gamma$ is ${\sf T}$-prime i.e.~$\sft\vdash A\to (B\vee C)$
implies either $\sft\vdash A\to B$ or $\sft\vdash A\to C$ for every
$A\in\Gamma$ and arbitrary $B,C$, and
$\Gamma$ is closed under substitutions, then Disj is also sound,
\item if $\Gamma$ is closed under $\Delta$-conjunctions, i.e.~$A\in\Gamma$ and $B\in\Delta$ implies $A\wedge B\in \Gamma$
\uparan{up to ${\sf T}$-provable equivalence relation}, then $\mont(\Delta)\xspace$
is sound.
\end{enumerate}
\end{theorem}
\begin{proof}
Easy induction on the complexity of proof $\BAR{\sf T}\vdash A\rhd B$
and left to the reader.
\end{proof}
\begin{theorem}\label{vee-pres}
${\mathrel{\pres{{\sf T}}{\Gamma}}}={\pres{{\sf T}}{\Gamma^\vee}\ }$ and ${\argt}={\mathrel{\adsm{{\sf T}}{\Gamma^\vee}}\ }$ .
\end{theorem}
\begin{proof}
We only show $A\mathrel{\pres{{\sf T}}{\Gamma}} B$ iff $A\pres{{\sf T}}{\Gamma^\vee} B$ and leave the similar argument for $A\argt B$ iff $A\mathrel{\adsm{{\sf T}}{\Gamma^\vee}}\ B$ to the reader.
The right-to-left direction holds since
$\Gamma\subseteq \Gamma^\vee$. For the other direction assume that
$A\mathrel{\pres{{\sf T}}{\Gamma}} B$ and let $E\in\Gamma^\vee$ such that
$\sft\vdash E\to A$. Then $E=\bigvee_i E_i$ with $E_i\in\Gamma$.
Hence for every $i$ we have $\sft\vdash E_i\to A$.
Then $A\mathrel{\pres{{\sf T}}{\Gamma}} B$ implies $\sft\vdash E_i\to B$. Thus
$\sft\vdash E\to B$, as desired.
\end{proof}
\noindent\textbf{Notation.} Whenever ${\sf T}={\sf IPC}$ we may omit the ${\sf T}$
form notations $\mathrel{\adsm{{\sf T}}{\Gamma}}$ and $\mathrel{\pres{{\sf T}}{\Gamma}}$ and simply write
$\mathrel{ \ar_{_{\!\!\!\text{\fontsize{4}{0} \selectfont\sf $\Gamma$}}}\!}$ and $\mathrel{\pres{}{\Gamma}}$ for them. Also if $\Gamma:=\{\top,\bot\}$ we may omit
$\Gamma$ from notations.
\subsection{Greatest lower bounds}\label{glb}
Given a set $\Gamma\cup\{A\}$
of propositions, and a logic
${\sf T}$, we say that $B$ is a lower bound for
$A$ w.r.t.~$(\Gamma,{\sf T})$, if the following conditions met:
\begin{enumerate}
\item $B\in\Gamma$,
\item $\sft\vdash B\to A$.
\end{enumerate}
Moreover we say that $B$ is the greatest
lower bound (glb) for $A$ w.r.t.~$(\Gamma,{\sf T})$,
if for every lower bound $B'$ for $A$ w.r.t.~$(\Gamma,{\sf T})$
we have $\sft\vdash B'\to B$. Note that up to ${\sf T}$-provable equivalence relation, such glb is unique
and we annotate it as
$\ap{\Gamma}{{\sf T}}{A}$.
\\
We say that $ (\Gamma,{\sf T}) $ is downward compact, if
every $A\in\lcal_{0}$ has glb w.r.t.~$ (\Gamma,{\sf T}) $.
\begin{question}
One may similarly define the notion of least upper bounds
and upward compactness. Does downward compactness
imply upward compactness?
\end{question}
\begin{theorem}\label{Gamma-approx-preserv}
$B$ is the glb for $ A $ w.r.t.~$(\Gamma,{\sf T})$, iff
\begin{itemize}
\item $B\in\Gamma$,
\item $\sft\vdash B\to A$,
\item $A\mathrel{\pres{{\sf T}}{\Gamma}} B$.
\end{itemize}
Hence we have
$A\mathrel{\pres{{\sf T}}{\Gamma}} \ap\Gamma{\sf T} A$.
\end{theorem}
\begin{proof}
Left to the reader.
\end{proof}
\begin{question}
As we saw in \cref{Gamma-approx-preserv}, the glb may be expressed via
preservativity relation $ \mathrel{\pres{{\sf T}}{\Gamma}} $. One may think of its adjoint relation
which best suites for lub's:
$$ A\mathrel{ ^{^*}\!\!\!\prtg} B \quad \text{ iff }\quad \forall\,E\in\Gamma(\sft\vdash A\to E \Rightarrow \sft\vdash B\to E).$$
\citep[Corollary 7.2]{Visser02} axiomatizes $\mathrel{\pres{{\sf T}}{\Gamma}}$ for ${\sf T}={\sf IPC}$ and $\Gamma={\sf NNIL}$.
We ask for an axiomatization for $ \mathrel{ ^{^*}\!\!\!\prtg} $ when we let $ {\sf T}={\sf IPC} $ and
$ \Gamma={\sf NNIL} $.
\end{question}
\begin{corollary}\label{Cor-Gamma-approx-preserv}
If $\ap\Gamma{\sf T} A$ exists, then for every $B\in\lcal_{0}$ we have
$$\sft\vdash \ap\Gamma{\sf T} A \to B\quad \text{ iff }
\quad
A\mathrel{\pres{{\sf T}}{\Gamma}} B.
$$
\end{corollary}
\begin{proof}
First assume that $\sft\vdash \ap\Gamma{\sf T} A$. Also let $E\in\Gamma$
such that $\sft\vdash E\to A$. \Cref{Gamma-approx-preserv} implies
$A\mathrel{\pres{{\sf T}}{\Gamma}}\ap\Gamma{\sf T} A$ and hence $\sft\vdash E\to \ap\Gamma{\sf T} A$.
Then by $\sft\vdash \ap\Gamma{\sf T} A\to B$ we get
$\sft\vdash E\to B$, as desired. \\
For the other direction let
$A\mathrel{\pres{{\sf T}}{\Gamma}} B$. By definition we have $\ap\Gamma{\sf T} A\in\Gamma$
and $\sft\vdash \ap\Gamma{\sf T} A\to A$. Hence by $A\mathrel{\pres{{\sf T}}{\Gamma}} B$
we get $\sft\vdash \ap\Gamma{\sf T} A\to B$, as desired.
\end{proof}
\section{$\NNILpar$-fication: unification to $\NNILpar$}
\label{Sec-Ghil}
Silvio Ghilardi, in \citep{Ghil99} characterizes
projective propositions in the language $\lcal_{0}({\sf var})$
with the aid of Kripke semantics. Then he uses this characterization to prove that the unification type of
${\sf IPC}$ is finitary. Afterwards, Rosalie Iemhoff
\citep{IemhoffT,Iemhoff-admissibility} uses this
result together with a special sort of Kripke models,
called AR-models,
to characterize the admissible rules of ${\sf IPC}$.
In this section we consider a relativised version for those results.
The difference from previous version is that
we are not allowed to substitute parameters (a reserved set of atomics), and also instead of unification, we expect to
simplify the proposition to a
$\NNILpar$ proposition, called $\NNILpar$-fication.
In fact, previous results will be an special case of ours when
${\sf par}=\emptyset$ and hence $\NNILpar=\{\top,\bot\}$.
The methods of our proof follows main roads took in \citep{Ghil99,Iemhoff-admissibility}.
\\
We start with relativised version of projective unification
(\cref{Gamma-proj-nonmodal}) and extension property
(\cref{relative-ext-prop}).
Then (\cref{rel-ext-proj})
we prove a correspondence between relativised
projectivity and extendibility.
Having such Kripke semantical charcterization in hand,
then we prove that every proposition has a finitary projective approximation (\cref{proj-res}).
Actually we prove something more: every proposition
has a finitary projective resolution (\cref{def-proj-res}).
Finally at the end of this section (\cref{NNIL-resol}), we prove that
in the specific case, when $A\in{\sf NNIL}$, this finitary projective
resolution takes an elegant form.
\subsection{Relative Projectivity}\label{Gamma-proj-nonmodal}
Given $A\in\lcal_{0}$, a substitution $\theta$ is
called \textit{$A$-projective} (in ${\sf IPC}$)
if
\begin{equation}
\text{For all atomic $a$ we have } \quad
A\vdash a\leftrightarrow \theta(a).
\end{equation}
When one considers unification for propositional logics,
projectivity is proved to be of great help \citep{Ghilardi97}.
As we will see, our study is not an exception.
\\
If $\Gamma\subseteq\lcalz(\parr)$, a substitution $\theta$
is a \textit{$\Gamma$-fier} (as a generalization for uni-fier) for
$A$, if
$$
\vdash \theta(A)\in\Gamma
\quad
\text{i.e.~$\theta(A)$ is ${\sf IPC}$-equivalent to some
$A'\in\Gamma$.}
$$
In this case we use the notation $ A\xtwoheadrightarrow{}{\theta} \Gamma $. If
$ \Gamma $ is a singleton $ \{A'\} $ we write
$ A\xtwoheadrightarrow{}{\theta} A' $ instead of $ A\xtwoheadrightarrow{}{\theta} \{A'\} $.
$\theta$ is a unifier for $A$ if it is $\{\top\}$-fier for $A$.
We say that a substitution $ \theta $ projects $ A $ to $ \Gamma $
(notation: $ A\xrightarrowtail{}{\theta} \Gamma $)
if $ \theta $ is $ A $-projective and $ \Gamma $-fier.
We say that $ A $ is $ \Gamma $-projective (notation
$ A\xrat{}{} \Gamma $) if there is some $ \theta $ such that
$ A\xrightarrowtail{}{\theta}\Gamma $.
We say that
$A$ is projective, if it is $\{\top \}$-projective.
Also $\dar{\,}{\Gamma}$
indicates the set of all propositions which are
$\Gamma$-projective.
\subsubsection*{Uniqueness of $\Gamma$-projections}
Let $A\xrat{\theta}{} A'$ and $A\xrat{\tau}{}A''$
and $ A',A''\in\Gamma\subseteq\lcalz(\parr) $.
From the $A$-projectivity of $\theta$ and $\tau$,
for every atomic $a $ we have $A\vdash \theta(a)
\leftrightarrow\tau(a)$. Hence $A\vdash \theta(A)\leftrightarrow\tau(A)$
and then $A\vdash A'\leftrightarrow A''$. By applying
$\theta$ to both sides of this derivation,
we have $\theta(A)\vdash \theta(A')\leftrightarrow\theta(A'')$.
Since $\theta$ is identity over parameters and $A',A''\in\lcalz(\parr)$,
we have $A'\vdash A'\leftrightarrow A''$.
Hence $ \vdash A'\to A''$. Similarly we have
$ \vdash A''\to A'$, and hence $ \vdash A'\leftrightarrow A''$.
This argument shows that for every $\Gamma$-projective
$A$, there is a unique (modulo ${\sf IPC}$-provable
equivalence) $A^\dagger\in\Gamma$ such that
for some $A$-projective substitution $\theta$
we have $ \vdash \theta(A)\leftrightarrow A^\dagger$. Such unique
$A^\dagger$ is called the \textit{$\Gamma$-projection}
of $A$.
\begin{lemma}\label{Gamma-projectivity-pres}
Let $A$ be $\Gamma$-projective and $A^\dagger\in \Gamma$
its projection. Then $\vdash A\to A^\dagger$.
\end{lemma}
\begin{proof}
Let $\theta$ be the $A$-projective $\Gamma$-fier for
$A$, i.e.~$A\vdash B\leftrightarrow \theta(B)$ for every $B$,
and $ \vdash \theta(A)\leftrightarrow A^\dagger$.
Hence we have $A\vdash A\leftrightarrow \theta(A)$ and thus $A\vdash A\leftrightarrow A^\dagger$.
This implies $\vdash A\to A^\dagger$, as desired.
\end{proof}
\begin{lemma}\label{proj-pres-admiss}
Let $ \Gamma\subseteq \lcalz(\parr) $.
If $ A\xrightarrowtail{}{\theta} A^\dagger \in\Gamma $
and $ B $ is an arbitrary proposition,
then we have
$$ \vdash A\to B \quad \text{ iff } \quad \vdash \theta(A^\dagger\to B).$$
\end{lemma}
\begin{proof}
The left-to-right direction is obvious. For other direction,
let $\vdash \theta(A^\dagger\to B)$. Hence
$A\vdash \theta(A^\dagger\to B)$ and then $A\vdash A^\dagger\to B$.
\Cref{Gamma-projectivity-pres} implies $\vdash A\to A^\dagger$
and thus $\vdash A\to B$.
\end{proof}
\subsection{Relative Extendibility}\label{relative-ext-prop}
Given a Kripke model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ and $w\in W$,
$\mathcal{K}_w$ indicates the restriction of $\mathcal{K}$ to
the nodes $u\succcurlyeq w$. For a set ${\bs{a}}\subseteq{\sf atom}$,
we say that $\mathcal{K}'=(W',\preccurlyeq',\mathrel{V}')$
is an $ {\bs{a}} $-submodel of $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$,
annotated as $\mathcal{K}'\submodel{\bs{a}} \mathcal{K}$, if
there exists a relation $R\subseteq W'\times W$ such
that
\begin{itemize}
\item $\mathcal{K}',w'\Vdash a$ iff $\mathcal{K},w\Vdash a$,
for every $a\in {\bs{a}}$ and $(w',w)\in R$.
\item $v'\succcurlyeq'w'\mathrel{R} w$ implies
$\exists\, v\in W\ (v'\mathrel{R}v\succcurlyeq w)$, for every $w\in W$
and $w',v'\in W'$.
\item $\forall\, w'\in W'\exists\, w\in W(w'\mathrel{R} w)$.
\end{itemize}
Also we say that $\mathcal{K}'\submodelp{\bs{a}}\mathcal{K}$
if the above relation $R$, is function. In this case the
second condition takes a more readable face:
\begin{itemize}
\item $w'\preccurlyeq' v'$ implies $f(w')\preccurlyeq f(v')$.
\end{itemize}
Moreover define $\mathcal{K}'\submodeli{\bs{a}} \mathcal{K}$ iff we have
\begin{itemize}
\item $W'\subseteq W$,
\item ${\preccurlyeq'}$ is the restriction of $\preccurlyeq$ to $W'$,
\item $w\mathrel{V}' a$ iff $w\mathrel{V} a$ for every $w\in W'$ and $a\in{\bs{a}}$,
\end{itemize}
Also $\mathscr{K}\submodel{\bs{a}}\mathcal{K}$ for
a class $\mathscr{K}$ of Kripke models and a Kripke model
$\mathcal{K}$ indicates that for every $\mathcal{K}'\in\mathscr{K}$
we have $\mathcal{K}'\submodel{\bs{a}}\mathcal{K}$. We have similar notations
for $\mathscr{K}\submodeli{\bs{a}}\mathcal{K}$ and $\mathscr{K}\submodelp{\bs{a}}\mathcal{K}$.
Since we only consider Kripke models with finite rooted tree frames, we have the equivalency of $\submodelp{\bs{a}}$ and
$\submodel{\bs{a}}$:
\begin{lemma}\label{Remark-embed-sub}
$\mathcal{K}'\submodelp{\bs{a}}\mathcal{K}$
is equivalent to $\mathcal{K}'\submodel{\bs{a}} \mathcal{K}$.
\end{lemma}
\begin{proof}
Let $\mathcal{K}'\submodel{\bs{a}} \mathcal{K}$ and $R$ is a relation with above mentioned properties. It is enough to define a function $f\subseteq R$ such that $f$ and $R$
share the same domain $W'$.
For every $\preccurlyeq'$-minimal node $w'\in W'$,
define $f(w')$ an arbitrary node $w$ with $w' \mathrel{R} w$. Note that such $w$ always exists. Since $\mathcal{K}'$ is tree, for every $w'_1\in W'$
which is not minimal, there is a unique predecessor
$w'_0\preccurlyeq' w'_1$. Then by definition for every $w'_1\in W'$ there is some
$w_1\in W$ such that $f(w'_0)\preccurlyeq w_1$. Define $f(w'_1):=w_1$
for some such $w_1$. Then it is not difficult to observe that
this $f$ satisfies the condition
``$w'\preccurlyeq' v'$ implies $f(w')\preccurlyeq f(v')$".
\end{proof}
Although $\submodeli{\bs{a}}$ is not equivalent to
$\submodelp{\bs{a}}$, we have the following partial equivalency:
\begin{lemma}\label{Remark-embed-sub2}
If $\mathcal{K}_0\submodelp {\bs{a}} \mathcal{K}_1\Vdash A$ then
$\mathcal{K}_0\submodeli{\bs{a}} \mathcal{K}_2\Vdash A$ for some $\mathcal{K}_2$.
\end{lemma}
\begin{proof}
The proof is almost identical to the proof of theorem 6.9 in
\citep{Visser-Benthem-NNIL} and we refer the
reader to it for more details.
Let $\mathcal{K}_i=(W_i,\preccurlyeq_i,\mathrel{V}_i)$ for $i\in\{0,1\}$
and $f:W_0\to W_1$ be the embedding of $\mathcal{K}_0$ in $\mathcal{K}_1$.
Moreover we may assume that $f$ is surjective, lest
we add a copy of $\mathcal{K}_1$ to $\mathcal{K}_0$
with a fresh root in beneath of them and then extend
the embedding on the new nodes.
Define $\mathcal{K}_2:=(W_2,\preccurlyeq_2,\mathrel{V}_2)$ as follows:
\begin{itemize}
\item $W_2:=\{ (w_0,f(w_0),w_1): w_0\in W_0 \text{ and } f(w_0)\preccurlyeq_1 w_1\in W_1\}$.
\item $(w_0,f(w_0),w_1)\preccurlyeq_2 (w'_0,f(w'_0),w'_1)$ iff either of the following holds:
\begin{itemize}
\item $w_0\preccurlyeq_0 w'_0$ and $w_1=f(w_0)$,
\item $w_0=w'_0$ and $w_1\preccurlyeq_1 w'_1$.
\end{itemize}
\item $(w_0,f(w_0),w_1)\mathrel{V}_2 a$ iff $w_1\mathrel{V}_1 a$.
\end{itemize}
It is straightforward to show that $\mathcal{K}_2$ is a finite rooted
tree-frame Kripke model with the root $(\rho,f(\rho),f(\rho))$
in which $\rho$ is the root of $\mathcal{K}_0$.
Also
$\mathcal{K}_1$ and $\mathcal{K}_2$ are bisimilar and hence prove the same set of propositions including $A$.
Moreover one may easily show that $g$ as defined in the following,
is a 1-1 embedding of $\mathcal{K}_0$ into $\mathcal{K}_2$:
$g(w_0):=(w_0,f(w_0),f(w_0))$.
\end{proof}
Remember that ${\sf NNIL}({\bs{a}})$ indicate ${\sf NNIL}\cap \lcal_{0}({\bs{a}})$.
The following theorem is a Kripke semantical
characterization of ${\sf NNIL}$ propositions
\citep{Visser-Benthem-NNIL}.
\begin{theorem}\label{Theorem-NNIL-Submodel}
Given ${\bs{a}}\subseteq{\sf atom}$, we have
$A\in{\sf NNIL}({\bs{a}})$ iff the class of Kripke
models of $A$ is closed under $\submodeli{\bs{a}}$.
\end{theorem}
\begin{proof}
See \citep{Visser-Benthem-NNIL} or
\citep{Visser02}.
\end{proof}
\begin{remark}\label{Remark-NNIL-finiteness}
Modulo ${\sf IPC}$-provable equivalence,
${\sf NNIL}$ is finite.
\end{remark}
\begin{proof}
Observe that each proposition can be written
as $\bigvee\bigwedge C$ in which $C$ is an atomic or implication, which we call it a component.
Observe that the number of propositions in $n$ atomics,
$f(n)$, is less than or equal to $2^{2^{g(n)}}$, in which $g(n)$ is the
number of components in $n$ atomics. Then observe that
$g(n+1)\leq (n+1)f(n)+ n+1$, because one may assume that
each component is either of the form $p\to A$ for some
atomic $p$ and some $A$ in $n$ variables, or it is of the
form $p$ for some atomic $p$. Hence the following recursive function is an upperbound for the
number of all formulas in $n$ atomics:
\[f(0):=2\quad \quad , \quad
\quad
f(n+1):=2^{2^{(n+1)(f(n)+1)}}\qedhere
\]
\end{proof}
Define $\ttbrace{\mathcal{K}}_{_\Gamma}:=\{A\in\Gamma: \mathcal{K}\Vdash A\}.$
\begin{theorem}\label{Theorem-NNIL-Submodel2}
Let $\mathcal{K},\mathcal{K}'$ be two Kripke models and ${\bs{a}}\subseteq{\sf atom}$.
Then $\mathcal{K}'\Vdash \ttbrace{\mathcal{K}}_{{\sf N}({\bs{a}})}$ iff
$\mathcal{K}'\submodel{\bs{a}} \mathcal{K}$.
\end{theorem}
\begin{proof}
See \citep[theorem~7.1.2]{Visser-Benthem-NNIL}.
\end{proof}
Given a substitution
$\theta$
and a Kripke model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$, we define
$\theta(\mathcal{K}):=(W,\preccurlyeq,\mathrel{V}')$
as follows.
For every atomic $a$,
define $w\mathrel{V}' a$ iff $\mathcal{K},w\Vdash \theta(a)$.
\begin{lemma}\label{Lem-sub-swap}
Given a general substitution $\theta$,
Kripke model $\mathcal{K}$ and $A\in\lcal_{0}$, we have
$$\mathcal{K},w\Vdash \theta(A) \quad \quad
\text{iff} \quad \quad
\theta(\mathcal{K}),w\Vdash A.
$$
\end{lemma}
\begin{proof}
Use induction on the complexity of $A$.
\end{proof}
\begin{remark}
Let $\theta$ and $\tau$ be two general substitutions
and $(\theta\tau)$ is their composition.
Above lemma, implies that
$(\theta\tau)(\mathcal{K})=\tau(\theta(\mathcal{K}))$. This conflicts
with our standard notation for the composition of
functions. This confliction could be resolved
by choosing another name, e.g.~$\theta^*$ for the operation
on Kripke models corresponding to $\theta$.
However, for the simplicity of notations, we prefer not to do so.
\end{remark}
Given Kripke models $\mathcal{K}$ and $\mathcal{K}'$,
we say that $\mathcal{K}'$ is an
${\bs{a}}$-\textit{variant} of $\mathcal{K}$, if
$\mathcal{K}'$ and $\mathcal{K}$ share the same frame and the same
atomic valuations, except possibly at the root and only
for atomics not in ${\bs{a}}$, for which we may
have different valuations. We also say that $\mathcal{K}'$ is
a variant of $\mathcal{K}$,
if it is $\emptyset$-variant of $\mathcal{K}$.
Also for a Kripke model $\mathcal{K}$ with the root $w_0$ we define
$\mathcal{K}\Vdash^- A$ iff $\mathcal{K},w\Vdash A$ for
every $w\neq w_0$.
\\
We say that a proposition $A$ is ${\bs{a}}$-extendible
if for every Kripke model $\mathcal{K}\Vdash A$ and every
$\mathcal{K}'\submodeli{\bs{a}} \mathcal{K}$ with $\mathcal{K}'\Vdash^- A$
there is an ${\bs{a}}$-variant $\mathcal{K}''$ of $\mathcal{K}'$
such that $\mathcal{K}''\Vdash A$. Also we say that $A$ is
extendible if it is $\emptyset$-extendible.
\\
For later applications in this paper it is helpful to define
${\sf par}$-extendibility also for a class of Kripke models.
Let $\mathscr{K}$ is a class of Kripke models.
Define the Kripke model $\sum(\mathscr{K})$ as the disjoint union of all Kripke models in $\mathscr{K}$ with a fresh root
$w_0$ such that for every atomic $a$ we have
$\sum(\mathscr{K}),w_0\Vdash a$ iff $\mathscr{K}\Vdash a$.
Also for a
Kripke model $\mathcal{K}$ with
$\mathscr{K}\submodeli\mathcal{K}$
define $\sum(\mathscr{K},\mathcal{K})$
as disjoint union
of the Kripke models in $\mathscr{K}$
with a fresh root $w_0$ and following valuation for atomics
$a\in{\bs{a}}$:
$$\sum(\mathscr{K},\mathcal{K}),w_0\Vdash a\quad \text{iff}\quad
\mathcal{K}\Vdash a. $$
We say that $\mathscr{K}$ is
\textit{${\bs{a}}$-extendible}, if
for every finite $\mathscr{K}'\subseteq \mathscr{K}$
with $\mathscr{K}'\submodeli{\bs{a}} \mathcal{K}\in\mathscr{K}$,
there is an ${\bs{a}}$-variant of $\sum(\mathscr{K}',\mathcal{K})$ which
belongs to $\mathscr{K}$. We say that $\mathscr{K}$ is \textit{extendible}
if it is $\emptyset$-extendible.
One may easily observe that $A$ is ${\sf par}$-extendible iff $\Mod A$
is so.
\subsection{${\sf NNIL}({\sf par})$-projectivity and ${\sf par}$-extendibility}
\label{rel-ext-proj}
In this section we will prove \cref{Theorem-Ghil-Ext},
an extension of Ghilardi's
characterization of projective propositions
via the notion of extendibility
(see \cref{Theorem-Ghil}).
\\
For a proposition $A$ and
a set ${\bs{x}}\subseteq {\sf var}$,
define the substitution
$\theta^\xbold_{\!\!A}:\lcal_{0} \longrightarrow\lcal_{0}$ as follows:
$$\theta^\xbold_{\!\!A}(x):=\begin{cases}
A\to x \quad &: x\in{\sf var}\cap{\bs{x}}\\
A\wedge x &: x\in{\sf var}\setminus{\bs{x}}
\end{cases}
$$
Let ${\bs{x}}_{\!1},\ldots,{\bs{x}}_{\!s}$ be a list of all subsets
of ${\sf var}$ such that
${\bs{x}}_{\!i}\subseteq{\bs{x}}_{\!j}$ implies
$i\leq j$. Finally define
$$\ta{} :=\ta{{\bs{x}}_{\!s}}\ta{{\bs{x}}_{\!s-1}}\ldots
\ta{{\bs{x}}_{\!1}} $$
The following theorem, is the main preliminary tool
provided in \citep{Ghil99} to characterize the unification type of ${\sf IPC}$. We refer the reader to
\citep[theorem 5]{Ghil99} for its proof.
We will prove a generalization of this in \cref{Theorem-Ghil-Ext}.
\begin{theorem}\label{Theorem-Ghil}
For $A\in\lcal_{0}$,
the following conditions are equivalent:
\begin{enumerate}
\item $\ta{}$ is a unifier for $A$,
i.e.~$ \vdash\ta{}(A)$,
\item $A$ is projective,
\item $A$ is extendible.
\end{enumerate}
\end{theorem}
Before we continue with a generalization
of above theorem, let us give another definition.
Let $\mathcal{K}$ be a Kripke model and
${\bs{p}}\subseteq{\sf par}$.
Then define $A^\dagger$ as follows: (remember that previously we defined
$ A^\dagger $ for $ \NNILpar $-projective $ A $ as the unique
$ A'\in\NNILpar $ such that $ A\xrat{}{} A' $. As we will see in next theorem, these two definitions are the same up to $ {\sf IPC} $-provable equivalence relation.)
$$
A^\dagger:=
\bigwedge_{{\bs{p}}\subseteq{\sf par}}\left(
\bigwedge{\bs{p}}\to \bigvee_{\mathcal{K}\Vdash {\bs{p}},A}
\bigwedge\tbnpr{\mathcal{K}}
\right)$$
Note that since by \cref{Remark-NNIL-finiteness} the set
$\NNILpar$ is finite and $\tbnp{\mathcal{K}}\subseteq\NNILpar$,
the conjunction $\bigwedge \tbnpr{\mathcal{K}}$ is
a proposition and also the disjunction may considered as a finite disjunction.
\begin{remark}\label{dagger-dist-arrow}
Note that by above definition,
if $ \vdash A\to B $ then $ \vdash A^\dagger\to B^\dagger $.
\end{remark}
\begin{theorem}\label{Theorem-Ghil-Ext}
For $A\in\lcal_{0} $, the following conditions are equivalent:
\begin{enumerate}
\item $A\xtra{\ta{}}{} A^\dagger$,
\item $A\xrat{}{} \NNILpar$,
\item $A$ is ${\sf par}$-extendible.
\end{enumerate}
\end{theorem}
\begin{proof}
$1\to 2$:
From definitions of $A^\dagger$, evidently
$A^\dagger\in{\sf NNIL}$.
Also observe that
$\theta^\xbold_{\!\!A}$ is $A$-projective. Then
since $A$-projective
substitutions are closed under compositions,
$\ta{}$ are $A$-projective.
\\
$2\to 3$: Let $A\xrightarrowtail{}{\theta} A'\in\NNILpar$
and $\mathcal{K}\Vdash A$ and
$\mathcal{K}'\submodeli{\sf par} \mathcal{K}$ and $\mathcal{K}'\Vdash^- A$
seeking some variant $\mathcal{K}''$ of $\mathcal{K}'$ such that
$\mathcal{K}''\Vdash A$.
Let $\mathcal{K}''=\theta(\mathcal{K}')$.
First note that
by $A$-projectivity of $\theta$, $\mathcal{K}''$ is a
${\sf par}$-variant of $\mathcal{K}'$.
Since $\mathcal{K}\Vdash A'$,
$A'\in\NNILpar$ and $\mathcal{K}'\submodeli{\sf par}\mathcal{K}$,
\cref{Theorem-NNIL-Submodel} implies that
$\mathcal{K}'\Vdash A'$. Hence
$\mathcal{K}'\Vdash \theta(A)$, and by
\cref{Lem-sub-swap}
we have $\mathcal{K}''\Vdash A$.
\\
$3\to 1$: Let $A$ is ${\sf par}$-extendible and
show $ \vdash A^\dagger\leftrightarrow \ta{}(A)$.
We use induction on the height of Kripke model $\mathcal{K}$
and show $\mathcal{K}\Vdash A^\dagger\leftrightarrow \ta{}(A)$.
Suppose that $w_0$ is the root of $\mathcal{K}$.
By induction hypothesis,
we have $\mathcal{K}\Vdash^- A^\dagger\leftrightarrow \ta{}(A)$
and hence
by \cref{Lem-sub-swap}, $\ta{}(\mathcal{K})\Vdash^- A^\dagger
\leftrightarrow A$. We show $\ta{}(\mathcal{K})\Vdash A^\dagger\leftrightarrow A$.
If $\ta{}(\mathcal{K})\nVdash^- A^\dagger$, then
$\ta{}(\mathcal{K})\nVdash^- A$ and hence
$\ta{}(\mathcal{K}),w_0\nVdash A^\dagger$
and
$\ta{}(\mathcal{K}),w_0\nVdash A$.
Then
$\ta{}(\mathcal{K}),w_0\Vdash A^\dagger
\leftrightarrow A$ and we are done.
So assume that $\ta{}(\mathcal{K})\Vdash^- A\wedge A^\dagger$.
It is enough to show the following items:
\begin{itemize}[leftmargin=*]
\item $\ta{}(\mathcal{K}),w_0\Vdash A$ implies
$\ta{}(\mathcal{K}),w_0\Vdash A^\dagger$.
By \cref{Gamma-projectivity-pres} we have $\vdash A\to A^\dagger$
and hence we have desired result.
\item $\ta{}(\mathcal{K}),w_0\Vdash A^\dagger$ implies
$\ta{}(\mathcal{K}),w_0\Vdash A$.
Let $\ta{}(\mathcal{K}),w_0\Vdash A^\dagger$.
Also assume that
$\ta{}(\mathcal{K})$ is ${\bs{p}}$-model,
i.e.~$\ta{}(\mathcal{K}),w_0\Vdash{\bs{p}}$ and
$\ta{}(\mathcal{K}),w_0\nVdash \bigvee ({\sf par}\setminus{\bs{p}})$.
Since $\ta{}(\mathcal{K}),w_0\Vdash A^\dagger$,
for some
$\mathcal{K}_1\in\Mod{A}$ with $\mathcal{K}_1\Vdash {\bs{p}}$
we have
$\ta{}(\mathcal{K})\Vdash \tbnpr{\mathcal{K}_1}$.
\Cref{Theorem-NNIL-Submodel2} implies that
$\ta{}(\mathcal{K}) \submodel{\sf par}\mathcal{K}_1$. Then
\cref{Remark-embed-sub} implies
$\ta{}(\mathcal{K}) \submodelp{\sf par}\mathcal{K}_1$
and thus by \cref{Remark-embed-sub2}
there is some $\mathcal{K}_2\Vdash A$ such that
$\ta{}(\mathcal{K}) \submodeli{\sf par}\mathcal{K}_2$.
Since $A$ is ${\sf par}$-extendible, there is a
${\sf par}$-variant $\mathcal{K}'$ of $ \ta{}(\mathcal{K})$
such that $\mathcal{K}'\Vdash A$.
Thus
\cref{Lem-Ghil-1} implies $\ta{}(\mathcal{K})\Vdash A$.
\qedhere
\end{itemize}
\end{proof}
\begin{corollary}\label{Decid-nnilpar-proj}
${\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}$-projectivity is decidable. In other words,
given $A\in\lcal_{0}$, one may algorithmically decide $A\in{\dar{\,}\nnilpar}$.
\end{corollary}
\begin{proof}
Given $A$, by \cref{Theorem-Ghil-Ext}
it is enough to decide ${\sf IPC}\vdash \ta{}(A)\leftrightarrow A^\dagger$, which
is decidable since ${\sf IPC}$ is decidable.
\end{proof}
\begin{lemma}\label{Lem-Ghil-1}
If $\ta{}(\mathcal{K})\Vdash^- A$ and $A$ is valid in a
${\sf par}$-variant of $\ta{}(\mathcal{K})$ then $\ta{}(\mathcal{K})\Vdash A$.
\end{lemma}
\begin{proof}
See proof of the theorem 5 in \citep{Ghil99}.
\end{proof}
\subsection{Projective resolution}\label{proj-res}
The main result in \citep{Ghil99} is that
the unification type
of ${\sf IPC}$ is finitary. It means that for every
$A\in\lcalz(\varr)$, there exists
a finite \textit{complete set of unifiers} for $A$, i.e.~a finite set
$\Theta$ of unifiers for $A$ such that
every unifier of $A$ is less general than
some $\theta\in \Theta$. We say that $\theta$ is less general than $\gamma$ if there is some substitution
$\lambda$ such that for every $x\in{\sf var}$ we have
$$ \vdash \theta(x)\leftrightarrow\lambda(\gamma(x)).$$
The proof of above mentioned fact is based on projective approximations
which later \citep{ghilardi2002resolution} provides a resolution/tableaux method for its computation.
The aim for this subsection is to prove a relativised version
of projective approximations
in \cref{Theorem-IPC-nnilp-Finitary}.
\begin{definition}\label{def-proj-res}
Given $\Gamma,\Pi\subseteq\lcal_{0}$ and $A\in\lcal_{0}$,
we say that $\Pi$ is
$\Gamma$-projective resolution for $A$ if
\begin{itemize}
\item $ \Pi $ is a set of independent propositions, i.e.~for
$ B,C\in\Pi $, $ \vdash B\to C $ implies $ B=C $.
\item Every $B\in \Pi$ is $\Gamma$-projective
\item $ A\mathrel{ \ar_{_{\!\!\!\text{\fontsize{4}{0} \selectfont\sf $\Gamma$}}}\!}\bigvee\Pi $.
\item $ \vdash \bigvee\Pi\to A$.
\end{itemize}
A $\{\top\}$-projective resolution is also called
projective resolution.
\end{definition}
\noindent
Note that $\emptyset $ is a projective resolution for
a proposition which is not unifiable.
The greatest lower bound (glb) for a proposition $A$ is defined in \cref{glb}. Intuitively
a glb for $A$ w.r.t.~$(\Gamma,{\sf T})$ is the best
$\Gamma$-approximation from below inside the logic ${\sf T}$.
\begin{remark}\label{remark-resol-glb}
If $\Pi$ is $\Gamma$-projective resolution of $A$ then
$\bigvee\Pi$ is a glb for $A$ w.r.t.~$({{\downarrow}\Gamma^\vee},{\sf IPC})$.
\end{remark}
\begin{proof}
By \cref{pres-admis-rel} we have $A\pres{{\sf T}}{{\dar{\,}\Gamma}}\bigvee\Pi$ and hence by \cref{vee-pres} we have $A\pres{{\sf T}}{{\dar{\,}\Gamma}^{\!\vee}}\bigvee\Pi$.
Thus
\cref{Gamma-approx-preserv} implies desired result.
\end{proof}
\begin{theorem}\label{Theorem-IPC-Finitary}
Whenever ${\sf par}=\emptyset$,
every $A\in\lcal_{0} $ has projective
resolution.
\end{theorem}
\begin{proof}
See \citep[theorem 5]{Ghil99}.
We will also prove a generalization of this
result in \cref{Theorem-IPC-nnilp-Finitary}.
\end{proof}
\noindent
First some preliminary definitions. We refer the reader
to \citep{Ghil99} for more information on these notions.
Let
$\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ and $\mathcal{K}'=(W',\preccurlyeq',\mathrel{V}')$
are two Kripke models with the roots
$w_0$ and $w'_0$. Also let
$\mathcal{K}(w):=\{a\in{\sf atom} : \mathcal{K},w\Vdash a\} $ and $\lcal_{0}(\mathcal{K})$ be defined as $\lcal_{0}(\bigcup_{w\in W}\mathcal{K}(w))$.
We say that $\mathcal{K}$ is with finite valuations if for every
$w\in W$ we have $\mathcal{K}(w)$ is finite. Also define:
\begin{align*}
\mathcal{K}\sim_0 \mathcal{K}' \quad &\text{iff}\quad
\mathcal{K}(w_0)=\mathcal{K}'(w_0')\\
\mathcal{K}\sim_{n+1} \mathcal{K}' \quad &\text{iff}\quad
\forall\, w\in W\,\exists\, w'\in W' (\mathcal{K}_w\sim_n\mathcal{K}'_{w'})
\text{ and vice versa}\\
\mathcal{K}\leq_0\mathcal{K}' \quad &\text{iff} \quad
\mathcal{K}(w_0)\supseteq
\mathcal{K}'(w'_0)\\
\mathcal{K}\leq_{n+1}\mathcal{K}' \quad
&\text{iff} \quad
\forall\, w\in W\,\exists\, w'\in W' (\mathcal{K}_w\sim_n\mathcal{K}'_{w'})
\end{align*}
Evidently $\sim_n$ is an equivalence relation and
$\leq_n$ is reflexive transitive. One may easily
observe by induction on $n$ that $\mathcal{K}\sim_{n+1}\mathcal{K}'$
implies $\mathcal{K}\sim_n\mathcal{K}'$. Hence
$\mathcal{K}\sim_n\mathcal{K}'$ ($\mathcal{K}\leq_n\mathcal{K}'$) implies
$\mathcal{K}\sim_m\mathcal{K}'$ ($\mathcal{K}\leq_m\mathcal{K}'$) for every $m\leq n$.
\\
Let $c_{\!_\to\!}(A)$ indicate the maximum number of nested implications in $A$:
\begin{itemize}
\item $c_{\!_\to\!}(a)=c_{\!_\to\!}(\top)=c_{\!_\to\!}(\bot)=0$ for atomic $a$.
\item $c_{\!_\to\!}(A\circ B):=\max\{c_{\!_\to\!}(A),c_{\!_\to\!}(B)\}$, for $\circ\in\{\vee,\wedge\}$.
\item $c_{\!_\to\!}(A\to B):=1+\max\{c_{\!_\to\!}(A),c_{\!_\to\!}(B)\}$.
\end{itemize}
Remember that by default we assume the set ${\sf atom}$ to be a finite set.
\begin{remark}\label{Remark-Finiteness-c(A)}
Modulo ${\sf IPC}$-provable equivalence relation, there are finitely many
propositions $A\in \lcal_{0}$ with $c_{\!_\to\!}(A)\leq n$.
\end{remark}
\begin{proof}
By induction on $n$, we define an upper bound $f(n)$
for the number of propositions $A\in\lcal_{0}$ with
$c_{\!_\to\!}(A)\leq n$.
\begin{enumerate}[leftmargin=*]
\item $f(0):$ Observe that any $A$ with $c_{\!_\to\!}(A)=0$ is ${\sf IPC}$-equivalent to a disjunction of conjunctions of atomics.
Hence $f(0)=2^{2^m}$ is an obvious upper bound,
in which $m$ is the number of atomics in ${\sf atom}$.
\item $f(n+1):$ For every implication $B\to C$
with $c_{\!_\to\!}(B\to C)\leq n+1$, we have $c_{\!_\to\!}(B),c_{\!_\to\!}(C)\leq n$,
and hence $f(n)^2$ is an upper bound for the number of
inequivalent such propositions. Then since
{modulo $\IPC$-provable equivalence}{}
every proposition is a disjunction of conjunctions of atomics or implications, the following definition is an upperbound:
\[f(n+1):=2^{2^{[m+f(n)^2]}}.\qedhere\]
\end{enumerate}
\end{proof}
\begin{lemma}\label{Lem-nnilpn-c(n)}
Every $A\in{\sf NNIL}$ has an ${\sf IPC}$-provable equivalent
$A'\in{\sf NNIL}$ with $c_{\!_\to\!}(A')\leq \#{\sf atom}$.
\end{lemma}
\begin{proof}
Observe that every $A\in{\sf NNIL}$ has an ${\sf IPC}$-equivalent $B\in{\sf NNIL}$
such that $B=\bigwedge_i\bigvee_j C_i^j$
and every implication in
$C^j_i$ is of the form $a\to E$, with $a\in{\sf atom}$ and
$E$ does not contain $a$. Then one may easily prove the statement of this lemma by induction on the number of elements in ${\sf atom}$.
\end{proof}
\begin{lemma}\label{Lem-Characteristic-Kripke}
For every Kripke model $\mathcal{K}$,
there exists a proposition $\charn{\kcal}\in\lcal_{0}$
with the following properties:
\begin{itemize}
\item $\mathcal{K}'\Vdash \charn{\kcal}$ iff $\mathcal{K}'\leq_n \mathcal{K}$.
\item $c_{\!_\to\!}(\charn{\kcal})\leq n$.
\end{itemize}
\end{lemma}
\begin{proof}
We only give the definition of $A$ here, and refer the
reader to \citep[proposition 1]{Ghil99} for its proof.
\\
Let
$\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$
and define $\charac{\mathcal{K}}{0}:=\bigwedge\mathcal{K}(w_0)$ and
\[
\charac{\mathcal{K}}{n+1}:=\bigwedge_{\{\mathcal{K}':\forall\, w\in W(\mathcal{K}'\nsim_n\mathcal{K}_w)\}}
\left(
\charac{\mathcal{K}'}{n}\to
\bigvee_{\{\mathcal{K}'': \mathcal{K}'\nleq_n\mathcal{K}''\}}
\charac{\mathcal{K}''}{n}
\right)
.\qedhere
\]
\end{proof}
\begin{corollary}\label{Cor-Kripke-bisim-property}
For every Kripke models $\mathcal{K}$ and $\mathcal{K}'$,
we have $\mathcal{K}'\leq_n\mathcal{K}$ iff
for every $A\in\lcal_{0}$ with $c_{\!_\to\!}(A)\leq n$ we have
$\mathcal{K}\Vdash A$ implies $\mathcal{K}'\Vdash A$.
\end{corollary}
\begin{proof}
For the left to right direction, use induction on $n$.
For the right to left, if $\mathcal{K}$ is with finite
valuations, one may easily use \cref{Lem-Characteristic-Kripke} and have desired result. Otherwise One may restrict $\mathcal{K}$ and
$\mathcal{K}'$ to arbitrary finite atomics and then apply previous argument.
\end{proof}
\begin{corollary}\label{Corollary-n-sim-Kripke}
$\mathcal{K}'\sim_n\mathcal{K}$ iff for every $A$ with $c_{\!_\to\!}(A)\leq n$
we have
\begin{center}
$\mathcal{K}\Vdash A$ iff $\mathcal{K}'\Vdash A$.
\end{center}
\end{corollary}
\begin{proof}
First observe that $\mathcal{K}\sim_n \mathcal{K}'$ is equivalent to
$\mathcal{K}\leq_n\mathcal{K}'\leq_n\mathcal{K}$ and then use
\cref{Cor-Kripke-bisim-property}.
\end{proof}
\begin{lemma}\label{Lem-Mod(A)-Kripke}
A class $\mathscr{K}$ of Kripke models
is of the form $\Mod{A}$
with $c_{\!_\to\!}(A)\leq n$, iff $\mathscr{K}$ is $\leq_n$-downward
closed, i.e.~for every Kripke model $\mathcal{K}'$
with
$\mathcal{K}'\leq_n\mathcal{K}\in\mathscr{K}$ we have $\mathcal{K}'\in\mathscr{K}$.
\end{lemma}
\begin{proof}
For the left-to-right direction, let $\mathcal{K}'\leq_n\mathcal{K}\in\Mod{A}$ for some $A$ with $c_{\!_\to\!}(A)\leq n$. Since
$\mathcal{K}\Vdash A$, \cref{Cor-Kripke-bisim-property} implies
$\mathcal{K}'\Vdash A$ and hence $\mathcal{K}'\in\Mod{A}$.
\\
For the other direction, let $\mathscr{K}$ is $\leq_n$-downward
closed and define
$$A:=\bigvee_{\mathcal{K}\in\mathscr{K}}\charac{\mathcal{K}}{n}.$$
By
\cref{Remark-Finiteness-c(A)}, the disjunction is finite
and hence $A$ is indeed a proposition.
One may easily observe that \cref{Lem-Characteristic-Kripke} implies that
$c_{\!_\to\!}(A)\leq n$ and
$\mathscr{K}=\Mod{A}$.
\end{proof}
\begin{lemma}\label{Lem-extendible}
If a class of Kripke models $\mathscr{K}$ is ${\sf par}$-extendible and $\theta$
is a substitution, then $\theta(\mathscr{K})$
is also ${\sf par}$-extendible.
\end{lemma}
\begin{proof}
Easy and left to the reader.
\end{proof}
\noindent
We say that a class $\mathscr{K}$ of Kripke models
is stable, if for every $\mathcal{K}\in \mathscr{K}$ and every node
$w$ in $\mathcal{K}$ we have $\mathcal{K}_w\in\mathscr{K}$.
\begin{remark}\label{Remark-stable}
A class $\mathscr{K}$ of Kripke models
is ${\sf par}$-extendible iff for
every finite stable
class of models $\mathscr{K}'$ which is ${\sf par}$-submodel of
some $\mathcal{K}\in\mathscr{K}$, a ${\sf par}$-variant of
$\sum(\mathscr{K}',\mathcal{K})$ belongs to $\mathscr{K}$.
\end{remark}
\begin{proof}
Easy and left to the reader.
\end{proof}
\noindent Define
$\langle\scrk\rangle_n:=\{\mathcal{K}: \exists\,\mathcal{K}'\in\mathscr{K}\
(\mathcal{K}\leq_n\mathcal{K}')\text{ and } \mathcal{K}
\text{ is a Kripke model}\}$.
\begin{lemma}\label{Lem-stable-extendible}
If $\mathscr{K}$ is ${\sf par}$-extendible
stable class of Kripke models, then so is
$\langle\scrk\rangle_n$, for every $n> \#{\sf par}$. \uparan{$\#{\sf par}$ indicates the number of elements in ${\sf par}$.}
\end{lemma}
\begin{proof}
We only prove here the ${\sf par}$-extendibility
of $\langle\scrk\rangle_n$ and leave other properties to the reader.\\
Let $\mathscr{F}'=\{\mathcal{K}'_i\}_i$ is a finite set of models
in $\langle\scrk\rangle_n$,
which are ${\sf par}$-submodels of some $\mathcal{K}'\in\langle\scrk\rangle_n$.
By \cref{Remark-stable} we may also assume
that $\mathscr{F}'$ is stable.
We must show that a ${\sf par}$-variant of
$\sum(\mathscr{F}',\mathcal{K}')$ belongs to $\langle\scrk\rangle_n$.
Since $\mathcal{K}'\in\langle\scrk\rangle_n$ and $\mathscr{K}$ is stable,
there is some
$\mathcal{K}\in\mathscr{K}$ such that $\mathcal{K}'\sim_{n-1}\mathcal{K}$.
Similarly, since $\mathcal{K}'_i\in\langle\scrk\rangle_n$, there is some
$\mathcal{K}_i\in\mathscr{K}$
such that $\mathcal{K}'_i\sim_{n-1}\mathcal{K}_{i}$. Let
$\mathscr{F}:=\{\mathcal{K}_i\}_i$.
\\
First we show that $\mathscr{F}$ is a
${\sf par}$-submodel of $\mathcal{K}$.
Since $\mathscr{F}'$ is a ${\sf par}$-submodel of $\mathcal{K}'$,
by \cref{Theorem-NNIL-Submodel2}
we have $\mathcal{K}'_i\Vdash \tbnpr{\mathcal{K}'}$.
From $\mathcal{K}'_i\sim_{n-1}\mathcal{K}_i$, \cref{Lem-nnilpn-c(n),Corollary-n-sim-Kripke} we get
$\mathcal{K}_i\Vdash \tbnpr{\mathcal{K}'}$. Also
since $\mathcal{K}'\sim_{n-1}\mathcal{K}$, by
\cref{Lem-nnilpn-c(n),Corollary-n-sim-Kripke} we have
$ \tbnpr{\mathcal{K}'}=\tbnpr{\mathcal{K}}$. Hence $\mathcal{K}_i\Vdash \tbnpr{\mathcal{K}}$, and by \cref{Theorem-NNIL-Submodel2}
we have $\mathcal{K}_i$ is a ${\sf par}$-submodel of
$\mathcal{K}$. Hence $\mathscr{F}$ is a ${\sf par}$-submodel of
$\mathcal{K}$. \\
We go back to the main proof.
Since $\mathscr{F}$ is
${\sf par}$-submodel of $\mathcal{K}$, by extendibility
of $\mathscr{K}$, there exist a ${\sf par}$-variant $\hat{\mathcal{K}}$
of $\sum(\mathscr{F},\mathcal{K})$ in $\mathscr{K}$.
Let $w_0$ is the root of $\mathcal{K}$
which is also the root of $\hat{\mathcal{K}}$ and
$w'_0$ is the root of $\mathcal{K}'$.
Define the ${\sf par}$-variant $\hat{\mathcal{K}}'$ of
$\sum(\mathscr{F}',\mathcal{K}')$
for atomic $x\not\in {\sf par}$ as follows:
$$\hat{\mathcal{K}}',w'_0\Vdash x \quad \Longleftrightarrow \quad
\hat{\mathcal{K}},w_0\Vdash x.$$
It is enough to show that $\hat{\mathcal{K}}'\in\langle\scrk\rangle_n$.
For this aim it is enough to show
$\hat{\mathcal{K}}'\leq_n \hat{\mathcal{K}}$. From definition of
$\hat{\mathcal{K}}$ and $\hat{\mathcal{K}}'$, it is clear that
it is enough to show that $\hat{\mathcal{K}}'\sim_{n-1}\hat{\mathcal{K}}$.
We use induction on
$k\leq n-1$ and show $\hat{\mathcal{K}}'\sim_{k}\hat{\mathcal{K}}$.
\\
If $k=0$, we must show that for every atomic $a$
we have
$$\hat{\mathcal{K}}',w'_0\Vdash a \quad \Longleftrightarrow \quad
\hat{\mathcal{K}},w_0\Vdash a.$$
For atomic variables $x$, by definition of $\hat{\mathcal{K}}'$, we already have this.
Also since $\hat{\mathcal{K}}$ is a ${\sf par}$-variant of
$\sum(\mathscr{F},\mathcal{K})$,
$\hat{\mathcal{K}}'$ is a ${\sf par}$-variant of
$\sum(\mathscr{F}',\mathcal{K}')$ and $\mathcal{K}\sim_{0}\mathcal{K}'$,
for every $p\in{\sf par}$ we also have
$$\hat{\mathcal{K}}',w'_0\Vdash p \quad \Longleftrightarrow \quad
\hat{\mathcal{K}},w_0\Vdash p.$$
Then let $0<k<n$ and show $\hat{\mathcal{K}}'\sim_k\hat{\mathcal{K}}$.
We have the following items to prove:
\begin{itemize}[leftmargin=*]
\item For every node $w'$ in $\hat{\mathcal{K}}'$, there is some
$w$ in $\hat{\mathcal{K}}$ such that $\hat{\mathcal{K}}'_{w'}
\sim_{k-1}\hat{\mathcal{K}}_w$. If $w'$ is the root of $\hat{\mathcal{K}}'$,
take $w$ also the root of $\hat{\mathcal{K}}$
and we have desired result by
induction hypothesis. If $w'$ is not the root of
$\hat{\mathcal{K}}'$, since $\mathscr{F}'$ is stable, we may let
$w'$ as a root $w'_i$ of some $\mathcal{K}'_i$.
Take $w=w_i$.
Then by definition of $\mathcal{K}_i$, we have
$$\hat{\mathcal{K}}'_{w'}=\mathcal{K}'_i\sim_{n-1}\mathcal{K}_i=\hat{\mathcal{K}}_{w}$$
Since $k-1\leq n-1$, we have the desired result.
\item For every node $w$ in $\hat{\mathcal{K}}$, there is some
$w'$ in $\hat{\mathcal{K}}'$ such that $\hat{\mathcal{K}}'_{w'}
\sim_{k-1}\hat{\mathcal{K}}_w$. Again if $w$ is the root, take $w'$
also the root and we are done by induction hypothesis.
If $w$ is not the root, there is some $i$ such that
$w$ is a node of $\mathcal{K}_i$. Since
$\mathcal{K}_i\sim_{n-1}\mathcal{K}'_i$, there is some $w'$ in
$\mathcal{K}'_i$ such that
$(\mathcal{K}'_i)_{w'}\sim_{n-2}(\mathcal{K}_i)_w$. Since
$k-1\leq n-2$, we have
$$\hat{\mathcal{K}}'_{w'}=(\mathcal{K}'_i)_{w'}\sim_{k-1}(\mathcal{K}_i)_w
=\hat{\mathcal{K}}_w,$$
as desired.\qedhere
\end{itemize}
\end{proof}
\begin{theorem}\label{Theorem-IPC-nnilp-Finitary}
Every $A\in\lcal_{0}$ has
$\NNILpar$-projective resolution $\Pi$.
Moreover for every $B\in\Pi$ we have $c_{\!_\to\!}(B)\leq
\max\{c_{\!_\to\!}(A),1+\#{\sf par}\}$ and $\Pi$ is a computable
function of $A$.
\end{theorem}
\begin{proof}
Given a substitution $\theta$
and $A'\in\NNILpar$ such that
$ \vdash A'\to \theta(A)$,
we will find some $B^{A'}_\theta\in\lcal_{0}$,
with the following properties:
\begin{enumerate}
\item $B^{A'}_\theta$ is $\NNILpar$-projective.
\item $ \vdash A'\leftrightarrow \theta(B^{A'}_\theta)$.
\item $ \vdash B^{A'}_\theta\to A$.
\item $c_{\!_\to\!}(B^{A'}_\theta)\leq n$ for $n:=\max\{c_{\!_\to\!}(A),1+\#{\sf par}\}$ ($\#{\sf par}$ indicates the number of atomics in ${\sf par}$).
\end{enumerate}
Then by items 1-3 (an independent subset of) the following set is a
$\NNILpar$-projective resolution for $A$:
\[
\Pi:=\{
B^{A'}_\theta:
A'\in\NNILpar \text{ and $\theta$ a substitution such that }
\vdash A'\to \theta(A)
\}.
\]
Moreover \cref{Remark-Finiteness-c(A)} and
item (4) implies that $\Pi$ is finite, as desired.
So it remains to find $B^{A'}_\theta$
with mentioned properties.
Define
$$\mathscr{K}:=\theta(\Mod{A'}):=\{\theta(\mathcal{K}): \mathcal{K}\in\Mod{A'}\}$$
Since $\langle\scrk\rangle_n$ has downward $\leq_n$-closure condition,
we may apply \cref{Lem-Mod(A)-Kripke}
and find some proposition, e.g.~$B^{A'}_\theta$,
such that $c_{\!_\to\!}(B^{A'}_\theta)\leq n$ (so item 4 is satisfied)
and $\langle\scrk\rangle_n=\Mod{B^{A'}_\theta}$.
Since $A'\in\NNILpar$, evidently it is
$\NNILpar$-projective. Hence by \cref{Theorem-Ghil-Ext},
$A'$ is ${\sf par}$-extendible.
Hence by \cref{Lem-extendible},
$\mathscr{K}$ is ${\sf par}$-extendible.
Since $\mathscr{K}$ is stable
and $n> \#{\sf par}$, \Cref{Lem-stable-extendible}
implies that $\langle\scrk\rangle_n$ is also ${\sf par}$-extendible.
Hence $B^{A'}_\theta$ is ${\sf par}$-extendible and by
\cref{Theorem-Ghil-Ext}, $B^{A'}_\theta$ is $\NNILpar$-projective.
So item (1) is satisfied.
\\
To show item 3 for $B^{A'}_\theta$,
it is enough to show $\mathcal{K}\Vdash B^{A'}_\theta\to A$ for every finite rooted model $\mathcal{K}$. If $\mathcal{K}\Vdash B^{A'}_\theta$, we have $\mathcal{K}\in\langle\scrk\rangle_n$. Hence $\mathcal{K}\leq_n \mathcal{K}'$ for
some $\mathcal{K}'\in\mathscr{K}$. Then $\mathcal{K}'=\theta(\mathcal{K}'')$
for some finite rooted $\mathcal{K}''$ such that
$\mathcal{K}''\Vdash A'$. Since $\vdash A'\to \theta (A)$,
we have $\mathcal{K}''\Vdash \theta(A)$, and by
\cref{Lem-sub-swap} we get $\theta(\mathcal{K}'')\Vdash A$,
whence $\mathcal{K}'\Vdash A$. Since
$c_{\!_\to\!}(A)\leq n$ and $\mathcal{K}\leq_n\mathcal{K}'$,
\cref{Cor-Kripke-bisim-property} implies that
$\mathcal{K}\Vdash A$, as desired.
\\
It remains to show that item 2 holds.
It is enough to show $\mathcal{K}\Vdash
A'\leftrightarrow \theta(B^{A'}_\theta)$ for arbitrary finite rooted $\mathcal{K}$.
If $\mathcal{K}\Vdash A'$, then $\theta(\mathcal{K})\Vdash A'$ and
hence $\theta(\mathcal{K})\in\mathscr{K}\subseteq\langle\scrk\rangle_n=\Mod{B^{A'}_\theta}$.
Then $\theta(\mathcal{K})\Vdash B^{A'}_\theta$ and hence $\mathcal{K}\Vdash
\theta(B^{A'}_\theta)$. For the other direction, let
$\mathcal{K}\Vdash \theta(B^{A'}_\theta)$. Hence $\theta(\mathcal{K})\Vdash B^{A'}_\theta$ and then $\theta(\mathcal{K})\in\Mod{B^{A'}_\theta}=\langle\scrk\rangle_n$.
So there is some $\mathcal{K}'\in\mathscr{K}$ such that
$\theta(\mathcal{K})\leq_n\mathcal{K}'$. Since $\mathcal{K}'\in\mathscr{K}$,
there is some $\mathcal{K}''$ such that $\mathcal{K}'=\theta(\mathcal{K}'')$ and $\mathcal{K}''\Vdash A'$. Since
$A'=\theta(A')$, we have $\mathcal{K}''\Vdash \theta(A')$
and hence $\mathcal{K}'\Vdash A'$. By
\cref{Lem-nnilpn-c(n)} $c_{\!_\to\!}(A')<n$ and the
\cref{Cor-Kripke-bisim-property} implies $\mathcal{K}\Vdash A'$.
Finally we provide an algorithm which computes
$\Pi$. Given $A$, compute the finite set
$$\Pi':=\{B\in\lcal_{0}: c_{\!_\to\!}(B)\leq\max\{c_{\!_\to\!}(A),1+\#{\sf par}\} \text{ and } \vdash B\to A \text{ and } B\in{\dar{\,}\nnilpar}\}.$$
Note that $\Pi'$ is computable since ${\sf IPC}$ is decidable and
by \cref{Decid-nnilpar-proj} we can decide
$B\in{\dar{\,}\nnilpar}$. Finally one may easily find $\Pi\subseteq\Pi'$
which includes pairwise ${\sf IPC}$-independent propositions, as required
for projective resolutions.
\end{proof}
\subsection{Projective resolution for {\sf NNIL}}\label{NNIL-resol}
In this subsection, we will see that
projective resolution of
a ${\sf NNIL}$-proposition gets a more elegant form.
We will use this form later for characterization of
$\NNILpar$-admissible rules of ${\sf IPC}$, specifically
for the validity of disjunction rule.
By \cref{Theorem-IPC-Finitary} or equivalently
\cref{Theorem-IPC-nnilp-Finitary} with empty ${\sf par}$,
there is a finite projective resolution for every proposition $A$, i.e.~a set $\{A_1,\ldots ,A_n\}$,
with the following properties:
\begin{itemize}
\item Every unifier of $A$, is also a unifier of some
$A_i$, in other words $A \mathrel{\setlength{\unitlength}{1ex \bigvee A_i$.
\item $ \vdash \bigvee A_i\to A$.
\item $A_i$ is projective for every $i\leq n$.
\end{itemize}
We will prove here that if $A\in{\sf NNIL}$, the projective
resolution can be chosen such that
every $A_i$ is ${\sf NNIL}$ and moreover $ \vdash A\leftrightarrow
\bigvee A_i$.
\\
Given $A\in{\sf NNIL}$, we say that $A$ is a ${\sf T}$-component
if $A=\bigwedge \Gamma\wedge\bigwedge\Delta $
with the following properties:
\begin{itemize}
\item Every $B\in \Gamma$ is atomic.
\item Every $B\in\Delta$ is an implication
$C\to D$ for some atomic $C$ such that
${\sf T}\nvdash \bigwedge\Gamma\to C$.
\end{itemize}
\begin{lemma}\label{ipc-component-dec}
Let ${\sf T}$ be a logic extending ${\sf IPC}$.
Every $A\in{\sf NNIL}$ can be decomposed to ${\sf T}$-components,
i.e.~there is a finite set of
${\sf T}$-components $\Gamma_A$ such that
$\sft\vdash A\leftrightarrow \bigvee\Gamma_A$. Moreover,
if $A\in\NNIL(\abold)$ then $\Gamma_A\subseteq\NNIL(\abold)$.
\end{lemma}
\begin{proof}
We use induction on
$\suba A$ (the set of atomic
formulas in $A$) ordered by $\supset$ and find some finite set $\Gamma_A$ of ${\sf T}$-components with
$\suba{\Gamma_A}\subseteq\suba A$ and $\sft\vdash \bigvee\Gamma_A\leftrightarrow A$.
As induction hypothesis assume that for every ${\sf T}$ and
$B\in{\sf NNIL}$ with
$\suba B\subset \suba A$ there is a finite set
$\Gamma_B$ of
${\sf T}$-components such that $\sft\vdash B\leftrightarrow \bigvee\Gamma$
and $\suba{\Gamma_B}\subseteq\suba B$.
For the induction step, assume that $A\in{\sf NNIL}$ is given.
Using derivation in ${\sf IPC}$ one may easily find finite sets
$\Gamma_i$ and $\Delta_i$ for $1\leq i\leq n$ such that
\begin{itemize}
\item ${\sf IPC}\vdash A\leftrightarrow \bigvee_{i=1}^n A_i$,
in which $A_i:=\bigwedge\Gamma_i\wedge\bigwedge\Delta_i$.
\item $\Delta_i$ includes only atomic propositions.
\item $\Gamma_i$ includes implications
with atomic antecedents.
\item $\suba{\Gamma_i\cup\Delta_i}\subseteq \suba A$.
\end{itemize}
It is enough to decompose every $A_i$ to ${\sf T}$-components.
If ${\sf T}\nvdash \bigwedge\Delta_i\to E$ for every antecedent
$E$ of an implication in $\Gamma_i$, then $A_i$ already is a
${\sf T}$-component and we are done. Otherwise, there is some
$E\to F\in\Gamma_i$ such that $\sft\vdash \bigwedge\Delta_i\to E$.
Then let $A_i':=A_i[E:\top]$, i.e.~the replacement of every
occurrences of $E$ in $A_i$ with $\top$. Also let ${\sf T}':={\sf T}+E$.
Hence $\suba{A'_i}\subsetneqq \suba{A}$ and by induction hypothesis
we may decompose $A'_i$ to ${\sf T}'$-components:
$${\sf T},E\vdash A'_i\leftrightarrow \bigvee_j B_j$$
It is not difficult to observe that if $B_j$ is a ${\sf T}'$-component then
$B'_j:=E\wedge B_j$ is a ${\sf T}$-component. Moreover
${\sf T}\vdash E\wedge A'_i\leftrightarrow \bigvee_j B'_j$ and since
${\sf IPC}\vdash (E\wedge A'_i)\leftrightarrow (E\wedge A_i) $ and $\sft\vdash A_i\to E$,
we get $${\sf T}\vdash A_i\leftrightarrow \bigvee_j B'_j$$
Hence we have decomposed $A_i$ to ${\sf T}$-components $B'_j$ with
$\suba{B'_j}\subseteq\suba{A}$, as desired.
\end{proof}
\begin{lemma}\label{Lemma-NNIL-normal-projective}
Every ${\sf IPC}$-component is extendible.
\end{lemma}
\begin{proof}
Let $B=\bigwedge B_i$ is an ${\sf IPC}$-component and
$\mathscr{K}$ be a finite class of finite rooted Kripke models
for $B$. We must show that a variant of
$\sum(\mathscr{K})$ is a model of $B$.
Let $w_0$ be the root of $\sum(\mathscr{K})$ and
define a variant $\mathcal{K}$ of $\sum(\mathscr{K})$ as follows.
$\mathcal{K},w_0\Vdash a$ iff $a=B_i$ for some $i$.
Then it is easy to observe that
$\mathcal{K},w_0\Vdash B$.
\end{proof}
\begin{corollary}\label{Corollary-proj-res-NNIL}
For $A\in{\sf NNIL}$ there is a finite set $\Delta$
of projective
and ${\sf NNIL}$ propositions with
$$ \vdash A\leftrightarrow \bigvee \Delta.$$
\end{corollary}
\begin{proof}
Use \cref{Lemma-NNIL-normal-projective,ipc-component-dec,Theorem-Ghil}.
\end{proof}
\begin{lemma}\label{Remark-extendible-disjun}
Every extendible $A$ is ${\sf IPC}$-prime, i.e.~if
$ \vdash A\to (B\vee C)$, then
either $ \vdash A\to B$ or $ \vdash A\to C$.
\end{lemma}
\begin{proof}
We prove this by contraposition. Let $ \nvdash A\to B$
and $ \nvdash A\to C$. Then there are some Kripke
models $\mathcal{K}_1 $ and $\mathcal{K}_2$ such that
$\mathcal{K}_1\Vdash A$, $\mathcal{K}_1\nVdash B$,
$\mathcal{K}_2\Vdash A$ and $\mathcal{K}_2\nVdash B$. Since
$A$ is extendible, there is some variant
$\mathcal{K}'$ of $\sum(\{\mathcal{K}_1,\mathcal{K}_2\})$
such that $\mathcal{K}\Vdash A$.
Since
$\mathcal{K}_1\nVdash B$ we have $\mathcal{K}\nVdash B$. Similarly
$\mathcal{K}\nVdash C$. Hence $\mathcal{K}\nVdash B\vee C$
and then $\mathcal{K}\nVdash A\to (B\vee C)$.
\end{proof}
\begin{theorem}\label{component-extendible-prime}
Given $A\in\lcal_{0}$, the following are equivalent:
\begin{enumerate}
\item $A$ is an ${\sf IPC}$-component, \uparan{modulo ${\sf IPC}$-provable equivalence relation}
\item $A$ is extendible,
\item $A$ is ${\sf IPC}$-prime.
\end{enumerate}
\end{theorem}
\begin{proof}
$1\Rightarrow 2$: \cref{Lemma-NNIL-normal-projective}.
$2\Rightarrow 3$: \cref{Remark-extendible-disjun}.
$3\Rightarrow 1$: Let $A$ is ${\sf IPC}$-prime.
By \cref{ipc-component-dec} it can be decomposed to
${\sf IPC}$-components $\Gamma_A$.
Thus $\vdash A\leftrightarrow \bigvee\Gamma_A$ and by ${\sf IPC}$-primality of $A$
we have $\vdash A\to B$ for some $B\in\Gamma_A$. Then $\vdash A\leftrightarrow B$
and hence $A$ is ${\sf IPC}$-equivalent to some ${\sf IPC}$-component.
\end{proof}
Remember that $\pNNILpar$ indicates the set of ${\sf IPC}$-prime
and $\NNILpar$-propositions.
\begin{corollary}\label{Lem-nnil-normal-form}
Up to ${\sf IPC}$-provable equivalence relation, we have
${\sf NNIL}=\pNNIL^\vee$ and $\NNILpar=\pNNILpar^\vee$.
\end{corollary}
\begin{proof}
By \cref{ipc-component-dec} every $A\in{\sf NNIL}$ can be decomposed to
${\sf IPC}$-components $ \Gamma_A$ such that
$A\in\NNILpar$ implies
$\Gamma_A\subseteq\NNILpar$. Then \cref{component-extendible-prime}
implies that every $E\in\Gamma_A$ is ${\sf IPC}$-prime. Hence
$\bigvee\Gamma_A\in\pNNIL^\vee$ and moreover
$A\in\NNILpar$ implies $\bigvee\Gamma_A\in\pNNILpar^\vee$.
\end{proof}
\begin{corollary}\label{pnnilpv-nnilp}
${\adsm{{\sf IPC}}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ \ \ }={\adsm{{\sf IPC}}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ }$.
\end{corollary}
\begin{proof}
\Cref{Lem-nnil-normal-form,vee-pres}.
\end{proof}
A consequence of the results in this subsection is that now
we have uniqueness of the projective resolutions:
\begin{theorem}[\bf Projective Resolution]\label{Projec-resol}
Every $A\in\lcal_{0}$ has a $ \pNNILpar $-projective resolution.
Moreover this resolution is computable and
unique up to $ {\sf IPC} $-provable equivalency,
i.e.~for every two $ \pNNILpar $-projective resolutions
$ \Delta=\{B_1,\ldots,B_m\}$ and $\Delta'=\{C_1,\ldots,C_n\} $
for $ A $, we have
$ m=n$ and there is some permutation $\sigma$ such that
for every $ i $, $ \vdash B_i\leftrightarrow C_{\sigma(i)} $.
\end{theorem}
\begin{proof}
Given $A$, by \cref{Theorem-IPC-nnilp-Finitary} there is a
$\NNILpar$-projective resolution $\Delta$ for $A$. Then define
$$\Pi_0:=\{E\wedge E': E\in \Delta\text{ and } E'\in \Gamma_{E^\dagger}\}
$$
in which $E^\dagger\in\NNILpar$ is the $\NNILpar$-projection of $E$
and $\Gamma_{E^\dagger}$ is the decomposition of $E^\dagger$ to
${\sf IPC}$-components, as provided by \cref{ipc-component-dec}. Finally
let $\Pi\subseteq\Pi_0$ be some $\subseteq$-minimal set with
$\vdash \bigvee \Pi_0\leftrightarrow\bigvee\Pi$.
Then by
\cref{pnnilpv-nnilp} and the following fact
one may easily observe that $\Pi$ is a
$\pNNILpar$-projective resolution for $A$: if $ A\xrightarrowtail{}{\theta} A'\in\NNILpar $ and $ E\in\pNNILpar $, then $ (A\wedge E)\xrightarrowtail{}{\theta} (A'\wedge E)$.
\\
For the uniqueness, it is enough to show that for every
$ \pNNILpar $-projective $ E $, if
$ E \mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } \bigvee_i F_i$
then for some $ i $ we have
$ \vdash E\to F_i $. Let $ E\xrightarrowtail{}{\theta} E^\dagger $.
Then by $ E \mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } \bigvee_i F_i$ we have
$ \vdash \theta(E^\dagger\to \bigvee_iF_i) $.
Hence $ \vdash E^\dagger\to\bigvee_i\theta(F_i) $ and
since $E^\dagger $ is ${\sf IPC}$-prime, we have
$ \vdash E^\dagger\to \theta(F_i) $ for some $ i $.
Thus \cref{proj-pres-admiss} implies $ \vdash E \to F_i$, as desired.
\end{proof}
\section{$\NNILpar$-admissible rules of ${\sf IPC}$}
\label{sec-admis}
In \citep{IemhoffT},
the admissibility relation $ \mathrel{\setlength{\unitlength}{1ex $ is characterized by means of preservation relation $\rhd$ and its Kripke semantics, called
${\sf AR}$-models. In this section we will characterize
and prove the decidability of
{$\arn$}\ , the \textit{$\NNILpar$-admissible rules of ${\sf IPC}$}
(see \cref{pres-admis}).
For this end, we imitate the route in
\citep{IemhoffT}, i.e.~we define a system ${{\sf AR}_{\parr}}\!\,$ for the
$\NNILpar$-admissible rules of ${\sf IPC}$ and also introduce
a Kripke semantic for it and prove the soundness and completeness.
Finally using this and the results
in \cref{Sec-Ghil} we prove that ${{\sf AR}_{\parr}}\!\,$ is sound and complete for
both $\NNILpar$-admissibility and ${\dar{\,}\NNILpar}$-preservativity,
i.e.~${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ iff $A\arn B $ iff $A\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ } B$.
\subsection{The system ${{\sf AR}_{\parr}}\!\,$}\label{ARp}
${{\sf AR}_{\parr}}\!\,$ is a system which proves propositions
in the form $A\rhd B$, and $A,B\in\lcal_{0}$.
Before we continue with the axioms and rules of the
system ${{\sf AR}_{\parr}}\!\,$, let us first define a notation.
$$\itp{A}{B}:=\begin{cases}
B \quad &: B\in{\sf par}\cup\{\bot\}\\
A\to B &: \text{otherwise}
\end{cases}
$$
\noindent Then ${{\sf AR}_{\parr}}\!\,$ is defined as $\BAR{\sf IPC}$ (as defined in \cref{pres-admis}) plus the following axiom and rule:
\begin{itemize}[leftmargin=1cm]
\item[${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}:$]\quad $B\to C\rhd \bigvee_{i=1}^{n+m}
\itp{B}{E_i}$, in which
$B=\bigwedge_{i=1}^n (E_i\to F_i)$ and
$C=\bigvee_{i=n+1}^{n+m} E_i$.
\end{itemize}
\begin{center}
\AxiomC{$A\rhd B$}
\RightLabel{$\textup{Mont}({\sf par})$}
\LeftLabel{($p\in{\sf par}$)}
\UnaryInfC{$p\to A\rhd p\to B$}
\DisplayProof
\end{center}
\begin{remark}
The system ${\sf AR}$, as defined in \citep{IemhoffT},
is ${{\sf AR}_{\parr}}\!\,$ with ${\sf par}=\emptyset$. The Visser rule ${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$ in this case
is proved to be of central importance \citep{iemhoff2005intermediate}.
\end{remark}
\begin{remark}
As we will see in \cref{montagna-nnil},
the following extension of the Montagna's rule is
admissible in ${{\sf AR}_{\parr}}\!\,$:
\begin{prooftree}
\AxiomC{$A\rhd B$}
\LeftLabel{\uparan{$E\in\NNILpar$}}
\RightLabel{.}
\UnaryInfC{$E\to A\rhd E\to B$}
\end{prooftree}
\end{remark}
\begin{remark}\label{remark-arp}
${{\sf AR}_{\parr}}\!\,$ is closed under general substitutions $\theta$ with
$\theta(p)\in\{\top,\bot\}\cup{\sf par}$ for every $p\in{\sf par}$, i.e.~${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ implies ${{\sf AR}_{\parr}}\!\,\vdash
\theta(A)\rhd\theta(B)$.
\end{remark}
\begin{proof}
Use induction on the complexity of proof ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$. All cases are easy and left to the reader.
\end{proof}
The following theorem is from \citep{IemhoffT}.
\begin{theorem}
$A \mathrel{\setlength{\unitlength}{1ex B$ iff ${\sf AR}\vdash A\rhd B$.
\end{theorem}
\noindent
\begin{lemma}\label{Lem-ARN-implies-arpn}
${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ implies $A\arn B$.
\end{lemma}
\begin{proof}
We use induction on the complexity of the proof
${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$. All cases are easy except for
the axiom ${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$ and the rules ${\textup{Mont}({\sf par})}$ and Disj.
\begin{itemize}[leftmargin=*]
\item \textit{${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$:} Let $C=\bigwedge_{i=1}^n
(E_i\to F_i)$ and $D=\bigvee_{i=n+1}^{n+m} E_i$.
We show $C\to D\arn \bigvee_{i=1}^{n+m}\itp{C}{E_i}$.
So assume that $\theta$ is a substitution
and $G\in{\NNIL(\pbold)}$ and show that
$ \vdash G\to \theta(C)$ implies
$ \vdash G\to
\theta(\bigvee_{i=1}^{n+m}\itp{C}{E_i})$.
We reason by contraposition.
Let $ \nvdash G\to
\theta(\bigvee_{i=1}^{n+m}\itp{C}{E_i})$.
Hence for every $i\leq n+m$ we have
$ \nvdash \theta(G\to \itp{C}{E_i})$. Then
for every $i$ there is some Kripke model $\mathcal{K}_i$
with the root $w_i$
such that $\mathcal{K}_i\Vdash G$ and
$\mathcal{K}_i\nVdash \theta({E_i})$ and
moreover for every $i$
with $E_i\not\in{\sf par}\cup\{\bot\}$
we have $\mathcal{K}_i\Vdash C$. Since $G$ is projective,
by \cref{Theorem-Ghil} it is extendible.
Let $\mathcal{K}$ be a variant of $\sum(\{\mathcal{K}_i\}_i)$
such that $\mathcal{K}\Vdash G$ and $w_0$ be its root.
Also define $\mathcal{K}'$ as follows:
for every $i$ such that $E_i\in{\sf par}\cup\{\bot\}$,
eliminate $\mathcal{K}_i$ (we mean its nodes) from $\mathcal{K}$.
Then evidently $\mathcal{K}'$ is a submodel of $\mathcal{K}$.
Since $G$ is ${\sf NNIL}$, by \cref{Theorem-NNIL-Submodel},
we have $\mathcal{K}'\Vdash G$.
It is enough to show that $\mathcal{K}'\Vdash \theta(C)$
and
$\mathcal{K}'\nVdash \theta(D)$. Since for every node $w$ in
$\mathcal{K}'$ other than $w_0$, we have $\mathcal{K}',w\Vdash \theta(C)$,
if we show $\mathcal{K}',w_0\nVdash \theta(E_i)$ for every $i$,
we have both $\mathcal{K}'\Vdash \theta(C)$
and $\mathcal{K}'\nVdash \theta(D)$ and we are done.
We have two cases. (1) $E_i\in{\sf par}\cup\{\bot\}$.
In this case we have $\theta(E_i)=E_i$, and since
$\mathcal{K},w_i\nVdash E_i$ we have $\mathcal{K},w_0\nVdash E_i$
and hence $\mathcal{K}',w_0\nVdash E_i$. (2)
$E_i\not\in{\sf par}\cup\{\bot\}$. Since $w_i$ in this case
is a node of $\mathcal{K}'$ and $\mathcal{K}',w_i\nVdash
\theta(E_i)$, we have $\mathcal{K}',w_0\nVdash \theta(E_i)$.
\item $\textup{Mont}({\sf par})$:
Let $A\arn B$ and show
$p\to A\arn p\to B$ for every $p\in{\sf par}$.
Let $\theta$ be a substitution and $E\in{\NNIL(\pbold)}$ such that
$ \vdash E\to \theta(p\to A)$.
Hence $ \vdash (E\wedge p)\to \theta(A)$. Then
by $A\arn B$ we have $ \vdash (E\wedge p)\to
\theta(B)$ and hence $ \vdash E\to \theta(p\to B)$, as desired.
\item \textit{Disj:} Let $B\arn A$ and
$C\arn A$ and show $B\vee C\arn A$.
\Cref{pnnilpv-nnilp} and $B\arn A$ and
$C\arn A$ imply $B\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } A$ and
$C\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } A$.
Let $E\in\pNNILp$ and $\theta$ a substitution
such that $ \vdash E\to \theta(B\vee C)$.
Since $E$ is ${\sf IPC}$-prime,
either we have $ \vdash E\to\theta(B)$
or $ \vdash E\to \theta(C)$. In either of the cases,
by $B\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } A$ and $C\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } A$ we have
$ \vdash E\to \theta(A)$.
So by this argument we may conclude that $(B\vee C)\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } A$ and then
by \Cref{pnnilpv-nnilp} we have $(B\vee C)\arn A$.
\qedhere
\end{itemize}
\end{proof}
\begin{corollary}\label{Cor-ARN-implies-arn}
$A\rhd B$ implies $A\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } B$.
\end{corollary}
\begin{proof}
Use \cref{Lem-ARN-implies-arpn,pnnilpv-nnilp}.
\end{proof}
\begin{corollary}\label{Cor-ARN-implies-IPC-deriv}
For every $A\in\NNILpar$ and $B\in\lcal_{0}$,
if ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ then
$ \vdash A\to B$.
\end{corollary}
\begin{proof}
Let ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$. Then by
\cref{Lem-ARN-implies-arpn}, we have
$A\arn B$. Let $\theta$ be identity substitution. Then we have $ \vdash A\to \theta(A)$. Hence $ \vdash A\to \theta(B)$, which implies
$ \vdash A\to B$, as desired.
\end{proof}
\subsection{${{\sf AR}_{\parr}}\!\,$-models}\label{sec-ARmod}
Before we define ${{\sf AR}_{\parr}}\!\,$-models, the Kripke models for
which ${{\sf AR}_{\parr}}\!\,$ is sound and complete, let us present some
definitions.
Let $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ is a Kripke model, possibly infinite or
not tree. All over the rest of this subsection we assume that
in general a Kripke model might be infinite or not tree.
Given a set $\Gamma$ of propositions,
two nodes $v,w\in W$ are called \textit{$\Gamma$-similar},
notation $v\equiv_\Gamma w$, if for every $A\in\Gamma$
we have $\mathcal{K},v\Vdash A$ iff $\mathcal{K},w\Vdash A$.
Let $W'\subseteq W$ is a set of nodes and $w\in W$.
The notation $w\preccurlyeq W'$, means $w\preccurlyeq w'$ for every
$w'\in W'$.
We say that
$w\in W$ is a \textit{tight predecessor} of $W'$, if
$w\preccurlyeq W'$ and for every $u\succcurlyeq w$,
either $u=w$ or $u\succcurlyeq v$ for some $v\in W'$.
A node $w$ is called a \textit{base}, if for every
finite set $W'\subseteq W$ such that $w\preccurlyeq W'$, there
is some $w'\in W$ such that:
$w\preccurlyeq w'\preccurlyeq W'$ and $w\equiv_\parr w'$ and
$w'$ is a tight predecessor of $W'$. And finally,
a Kripke model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ is an ${{\sf AR}_{\parr}}\!\,$-model if
it is rooted (let $w_0$ be its root) and
there is some set $W_b\subseteq W$ with the following
properties:
\begin{itemize}
\item $w_0\in W_b$,
\item every $w\in W_b$ is a base,
\item for every $w'\in W_b$ and $w\succcurlyeq w'$,
there is some $v\in W_b$ such that $v\equiv_\parr w$ and
$w'\preccurlyeq v\preccurlyeq w$.
\end{itemize}
Such $W_b$ is called a \textit{base-set} for $\mathcal{K}$.
\\
We say that $\mathcal{K}$ is \textit{good}, if for every
finite set of nodes $W'$, and every
$X\subseteq {\sf par}$
such that $\mathcal{K},W'\Vdash X$,
there is some $w'\in W_b$ such that $w'\preccurlyeq W'$ and
$\mathcal{K}(w')\cap{\sf par}= X$.
\begin{remark}\label{Remark-ARN-models}
Let $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ is an ${{\sf AR}_{\parr}}\!\,$-model
with a base-set $W_b$, and $w\in W_b$.
Then $\mathcal{K}_w$ is also an ${{\sf AR}_{\parr}}\!\,$-model with the
base-set $W'_b:=\{v\in W_b: v\succcurlyeq w\}$.
\end{remark}
\begin{theorem}\label{Theorem-ARN-Soundness}
\textup{(\textbf{Soundness})}
${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ implies $\mathcal{K}\Vdash B$,
for every ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}$ with $\mathcal{K}\Vdash A$.
\end{theorem}
\begin{proof}
We use induction on the proof ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$. All
cases are trivial except for the axiom ${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$ and the
rule $\textup{Mont}$. First we treat the Montagna's rule.
As induction hypothesis, let $\mathcal{K}\Vdash A$
implies $\mathcal{K}\Vdash B$, for every ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}$.
Also let $\mathcal{K}\Vdash p\to A$ for some $p\in{\sf par}$
and ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ with the base-set
$W_b$.
We will show $\mathcal{K}\Vdash p\to B$. Let $w\in W$
such that $\mathcal{K},w\Vdash p$.
Since $\mathcal{K}$ is an ${{\sf AR}_{\parr}}\!\,$-model, there is some
$w'\in W_b$ such that $w\equiv_\parr w'$ and $w'\preccurlyeq w$.
Then $\mathcal{K},w'\Vdash p$ and hence $\mathcal{K},w'\Vdash A$.
Observe that $\mathcal{K}_{w'}$ is also an ${{\sf AR}_{\parr}}\!\,$-model and
$\mathcal{K}_{w'}\Vdash A$. Hence by induction hypothesis
$\mathcal{K}_{w'}\Vdash B$, which implies
$\mathcal{K},w\Vdash B$, as desired.
\\
Next we show
$\mathcal{K}\Vdash {\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$ for every ${{\sf AR}_{\parr}}\!\,$-model
$\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ with the root $w_0$.
Let $B=\bigwedge_{i=1}^n (E_i\to F_i)$ and
$C=\bigvee_{i=n+1}^{n+m} E_i$.
Also assume
that $\mathcal{K},w_0\nVdash \bigvee_{i=1}^{n+m}\itp{B}{E_i}$.
We show that
$\mathcal{K},w_0\nVdash B\to C$.
By definition of $\itp{B}{E_i}$, for every
$E_i\in{\sf par}\cup\{\bot\}$, we have
$\mathcal{K},w_0\nVdash E_i$,
and for every $E_i\not\in {\sf par}\cup\{\bot\}$, there is
some $w_i\succcurlyeq w_0$
such that $\mathcal{K},w_i\Vdash B$ and $\mathcal{K},w_i\nVdash E_i$.
Let $W':=\{w_i: E_i\not\in{\sf par}\cup\{\bot\} \}$. There
is some $w'\in W_b$ which is a tight predecessor
of $W'$ and $w'\equiv_\parr w_0$. We show that $\mathcal{K},w'\nVdash
B\to C$ by showing $\mathcal{K},w'\Vdash B$ and
$\mathcal{K},w'\nVdash C$. Let $E_i$ be some disjunct in $C$.
If $E_i\in{\sf par}\cup\{\bot\}$, then since $w'\equiv_\parr w_0$
and $\mathcal{K},w_0\nVdash E_i$, we have
$\mathcal{K},w'\nVdash E_i$. Otherwise, since
$\mathcal{K},w_i\nVdash E_i$ and $w'\preccurlyeq w_i$, we have
$\mathcal{K},w'\nVdash E_i$. This finishes showing
$\mathcal{K},w'\nVdash C$. Then we show $\mathcal{K},w'\Vdash B$.
Let $E_i\to F_i$ is a conjunct in $B$. Consider some
$w\succcurlyeq w'$ such that $\mathcal{K},w\Vdash E_i$.
Since $w'$ is a tight predecessor
of $W'$, either we have $w=w'$ or $w\succcurlyeq w_j$ for
some $w_j\in W'$. If $w\succcurlyeq w_j$, since
$\mathcal{K},w_j\Vdash B$, we have $\mathcal{K},w\Vdash B$ and then
$\mathcal{K},w\Vdash E_i\to F_i$, whence $\mathcal{K},w\Vdash F_i$.
Also if $w=w'$, then by the following argument, we have
$\mathcal{K},w'\nVdash E_i$, a contradiction with our first
assumption $\mathcal{K},w\Vdash E_i$.
Finally, the argument for
$\mathcal{K},w'\nVdash E_i$: if $E_i\in{\sf par}\cup\{\bot\}$,
then since $w'\equiv_\parr w_0$
and $\mathcal{K},w_0\nVdash E_i$, we have
$\mathcal{K},w'\nVdash E_i$. Otherwise, since
$\mathcal{K},w_i\nVdash E_i$ and $w'\preccurlyeq w_i$, we have
$\mathcal{K},w'\nVdash E_i$.
\end{proof}
\noindent
We follow the methods in
\citep{IemhoffT} to prove the completeness theorem.
This proof is almost identical to the one for \citep[proposition 7.2.2]{IemhoffT}.
First some definitions and lemmas.
A set $w$ of propositions is ${\sf IPC}$-saturated if
\begin{itemize}
\item $w\vdash A$ implies $A\in w$,
\item $\bot\not\in w$,
\item $A\vee B\in w$ implies either $A\in w$ or $B\in w$.
\end{itemize}
Also $w$ is called ${{\sf AR}_{\parr}}\!\,$-saturated if
it is ${\sf IPC}$-saturated and
\begin{itemize}
\item If ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ and $A\in w$, then $B\in w$.
\end{itemize}
Let $*(.)$ is a property on sets of propositions.
We say that $*(.)$ is an \textit{extendible} property
if the following conditions hold:
\begin{itemize}
\item If $*(w)$ and
$w\vdash A$, then $*(w\cup\{A\})$.
\item If $*(w\cup\{A\vee B\})$ then
either $*(w\cup\{A\})$ or $*(w\cup\{B\})$ hold.
\end{itemize}
If also the following condition holds, we say that
$*(.)$ is ${{\sf AR}_{\parr}}\!\,$-extendible property.
\begin{itemize}
\item If $*(w)$ and ${{\sf AR}_{\parr}}\!\,\vdash w\rhd A$, then $*(w\cup\{A\})$.
\end{itemize}
In the above expression, ${{\sf AR}_{\parr}}\!\,\vdash w\rhd A$ is
a shorthand for ${{\sf AR}_{\parr}}\!\,\vdash (\bigwedge_i B_i)\rhd A$ for some finite set $\{B_i\}_i\subseteq w$.
\begin{lemma}\label{Lem-Max-Sat}
For every extendible property $*(.)$, if $*(w)$
for some set $w$ of propositions holds,
there is some maximal
${\sf IPC}$-saturated $w'\supseteq w$ such that
$*(w')$. Moreover if $*(.)$ is ${{\sf AR}_{\parr}}\!\,$-extendible,
then $w'$ is also ${{\sf AR}_{\parr}}\!\,$-saturated.
\end{lemma}
\begin{proof}
Let $A_1,A_2,\ldots $ be a list of all propositions
such that each proposition occurs infinitely often.
We define a sequence $w=w_0\subseteq w_1\subseteq
w_2\subseteq \ldots$ and then define $w':=\bigcup_i w_i$.
$$w_{n+1}:=\begin{cases}
w_n\cup\{A_n\} \quad &: *(w_n\cup\{A_n\})
\\
w_n &:\text{otherwise}
\end{cases}$$
It can be easily proved that this $w'$ satisfies all required conditions.
\end{proof}
\begin{corollary}\label{Lem-ARN-saturation}
If ${{\sf AR}_{\parr}}\!\,\nvdash A\rhd B$, then there is some
${{\sf AR}_{\parr}}\!\,$-saturated $w$ such that $A\in w$ and $B\not\in w$.
\end{corollary}
\begin{proof}
Define the property $*(.)$ as follows:
$$*(y): {{\sf AR}_{\parr}}\!\,\nvdash y\rhd B$$
Then it is straightforward to observe that
$*(.)$ is ${{\sf AR}_{\parr}}\!\,$-extendible and $*(\{A\})$ holds. Hence
\cref{Lem-Max-Sat} implies the desired result.
\end{proof}
\begin{theorem}\label{Theorem-ARN-Completeness}
\textup{(\textbf{Completeness})}
${{\sf AR}_{\parr}}\!\,$ is complete for good ${{\sf AR}_{\parr}}\!\,$-models, i.e.~if
for every good ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}$, we have
$\mathcal{K}\Vdash A$ implies $\mathcal{K}\Vdash B$,
then ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$.
\end{theorem}
\begin{proof}
As usual, we reason corapositively. Let
${{\sf AR}_{\parr}}\!\,\nvdash A\rhd B$.
Define the Kripke model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ as follows.
Since ${{\sf AR}_{\parr}}\!\,\nvdash A\rhd B$, by \cref{Lem-ARN-saturation}
there is some ${{\sf AR}_{\parr}}\!\,$-saturated set $w_0$ such that $A\in w_0$ and
$B\not\in w_0$. Then define
$$W:=\{w\supseteq w_0: w\text{ is a
${\sf IPC}$-saturated set of propositions}\}
$$
Also define $u\preccurlyeq v$ iff $u\subseteq v$. Finally define
$w\mathrel{V} a$ iff $a\in w$ for atomic $a$. We will show that this model is a good ${{\sf AR}_{\parr}}\!\,$-model
such that $\mathcal{K}\Vdash A$
and $\mathcal{K}\nVdash B$.
First note that by a standard argument,
one may easily prove by induction on the complexity
of $A$ that $A\in w$ iff $\mathcal{K},w\Vdash A$.
Then since $A\in w_0$ and $B\not\in w_0$, we have
$\mathcal{K}\Vdash A$ and $\mathcal{K}\nVdash B$.
So it remains to show that $\mathcal{K}$ is
a good ${{\sf AR}_{\parr}}\!\,$-model. Let $W_b$ as follows:
$$W_b:=\{w\in W: w \text{ is ${{\sf AR}_{\parr}}\!\,$-saturated}\}$$
We will show that $W_b$ is a base-set, i.e.~has
the following properties:
\begin{itemize}
\item $w_0\in W_b$,
\item every $w\in W_b$ is a base,
\item for every $w'\in W_b$ and $w\succcurlyeq w'$,
there is some $v\in W_b$ such that $v\equiv_\parr w$ and
$w'\preccurlyeq v\preccurlyeq w$.
\end{itemize}
The first property is obvious. For the second property
we will need ${\sf V}^\parr_{\!\text{\fontsize{6}{0}\selectfont\AR}}$ and for the third one we will use
\textup{Mont}'s rule.\\
Let $w\in W_b$ and $w\preccurlyeq\{ w_1,\ldots,w_n\}$. We find
some tight predecessor $u$ such that $u\equiv_\parr w$ and
$w\preccurlyeq u\preccurlyeq \{w_1,\ldots,w_n\}$.
Let $\hat{w}:=\bigcap_i w_i$ and define
$$\Delta:=\{
E\to F: E\to F\in \hat{w}\text{ and }
(E\not\in \hat{w}\vee E\in{\sf par}\setminus w)
\}$$
Define the property $*(.)$ as follows:
$$*(y): y\vdash \bigvee_i A_i \vee\bigvee_i p_i
\text{ and $\forall\, i \ (p_i\in {\sf par})$ implies }\exists\, i\
(A_i\in \hat{w}) \vee \exists\, i\ (p_i\in w).$$
Note that by letting the second disjunction as empty,
from $*(y) $ we have
$y\vdash \bigvee_i A_i$
implies $\exists\, i\ (A_i\in \hat{w})$. Similarly and by considering the first disjunction as empty disjunction,
from $*(y) $ we get
$y\vdash \bigvee_i p_i$
implies $\exists\, i\ (p_i\in w)$.
It is not difficult to observe that $*(.)$ is an
extendible property.
Then we show $*(w\cup\Delta)$.
Let $w\cup\Delta\vdash \bigvee_i C_i\vee
\bigvee_i p_i$ and $p_i\in {\sf par}$.
Then
$w\vdash G \to (\bigvee_i C_i\vee
\bigvee_i p_i)$ in which
$G=\bigwedge_i(E_i\to F_i)$ and $E_i\to F_i\in \Delta$.
Since $w\in W_b$, and
$${{\sf AR}_{\parr}}\!\,\vdash \left(G\to (\bigvee_i C_i\vee
\bigvee_i p_i)\right)\rhd \left(
\bigvee_i \itp{G}{E_i}\vee
\bigvee_i\itp{G}{C_i}\vee\bigvee_i\itp{G}{p_i}\right)
,$$
we have $w\vdash \bigvee_i \itp{G}{E_i}\vee
\bigvee_i\itp{G}{C_i}\vee\bigvee_i\itp{G}{p_i}$.
Since $w$ is ${\sf IPC}$-saturated,
either $w\vdash \itp{G}{E_i}$ or
$w\vdash \itp{G}{C_i}$ or
$w\vdash \itp{G}{p_i}$, for some $i$. If
$w\vdash \itp{G}{E_i}$, then
$w,\Delta\vdash E_i$
and since $w\cup\Delta\subseteq \hat{w}$, we have
$E_i\in \hat{w}$, a
contradiction. So either we have
$w\vdash \itp{G}{C_i}$ or
$w\vdash \itp{G}{D}$, in
which we have $w,\Delta\vdash C_i$ or $w\vdash p_i$.
Hence either $C_i\in \hat{w}$ or $p_i\in w$.
This finishes showing $*(w\cup\Delta)$.
\\
Now let $u\supseteq (w\cup \Delta)$ be a maximal
${\sf IPC}$-saturated set such that $*(u)$, as provided by
\cref{Lem-Max-Sat}.
Then we show that $u$ satisfies all required conditions:
\begin{itemize}
\item $w\preccurlyeq u\preccurlyeq \{w_1,\ldots, w_n\}$. Since
$w\subseteq u$, we have $w\preccurlyeq u$. Also from
$*(u)$, we get $u\subseteq\hat{w}$ and hence for every $i$
we have $u\preccurlyeq w_i$.
\item $w\equiv_\parr u$.
Since $w\subseteq u$, we have
$w\cap{\sf par}\subseteq u\cap{\sf par}$. For the other direction, let $p\in {\sf par}\cap u$. Then
$u\vdash p$ and from $*(u)$ we have $p\in w$.
\item $u$ is a tight predecessor of $\{w_1,\ldots,w_n\}$.
We reason by contraposition.
Let $v\supsetneqq u$ such that for every $i$,
$w_i\not\subseteq v$.
Then for every $i$ there is some
$C_i\in w_i\setminus v$ and hence
$\bigvee C_i\in \hat{w}\setminus v$.
On the other hand,
since $u$ is a maximal saturated set with
$*(u)$ and $v\supsetneqq u$ and $v$ is ${\sf IPC}$-saturated,
we have $\neg *(v)$. Hence
$v\vdash \bigvee_i A_i\vee \bigvee_ip_i$ and
for every $i$ we have $p_i\in{\sf par}$ and $A_i\not\in \hat{w}$ and $p_i\not\in w$. From
$v\vdash \bigvee_i A_i\vee \bigvee_ip_i$,
there is some $E$ such that
either we have $E\in v\setminus \hat{w}$ or
$E\in{\sf par}\setminus w$. In
either of the cases, by definition of $\Delta$
we have $E\to \bigvee C_i\in\Delta$. Hence
$E\to \bigvee C_i\in v$ and
then $\bigvee C_i\in v$, a contradiction.
\end{itemize}
It finishes showing the second property
of base-set $W_b$. Next we show that $W_b$ satisfies
the third condition.
Let $w'\in W_b$ and $w'\preccurlyeq w$.
Define the property $*(.)$ as follows.
$$*(y): \text{ for every $C$, if }
{{\sf AR}_{\parr}}\!\,\vdash y\rhd C \text{, then } C\in w.$$
We show that $*(.)$ is an ${{\sf AR}_{\parr}}\!\,$-extendible property and
$*(w'\cup w_{\sf par})$, in which
$w_{\sf par}:=w\cap{\sf par}$. First let us show why
this finishes the proof. From \cref{Lem-Max-Sat}
we get some ${{\sf AR}_{\parr}}\!\,$-saturated
$v\supseteq (w'\cup w_{\sf par})$ such that $*(v)$.
Hence by definition $v\in W_b$.
Since $v\supseteq w'$, we have $w'\preccurlyeq v$.
Then we show $v\preccurlyeq w$. Let $C\in v$. From $*(v)$
and ${{\sf AR}_{\parr}}\!\,\vdash v\rhd C$, we have $C\in w$, as desired.
So we have $v\preccurlyeq w$. Finally we show $v\equiv_\parr w$.
We must show $v_{\sf par}=w_{\sf par}$, which holds
because $v\supseteq w_{\sf par}$ and $v\subseteq w$.
\\
So it remains to show that $*(.)$ is an ${{\sf AR}_{\parr}}\!\,$-extendible
property and $*(w'\cup w_{\sf par})$. First we show that
$*(.)$ satisfies all required conditions for
${{\sf AR}_{\parr}}\!\,$-extendibility:
\begin{itemize}
\item If $*(y)$ and $y\vdash E$. We must show
$*(y\cup\{E\})$. Let ${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$.
Hence ${{\sf AR}_{\parr}}\!\,\vdash E\wedge \bigwedge_i F_i\rhd C$ for some
finite set $\{F_i\}_i\subseteq y$. Then since
$y\vdash E $, we have ${{\sf AR}_{\parr}}\!\,\vdash y\rhd
E\wedge \bigwedge_i F_i$. Hence ${{\sf AR}_{\parr}}\!\,\vdash y\rhd C$.
Then from $*(y)$ we have $C\in w$, as desired.
\item If neither
$*(y\cup\{E\})$ nor $*(y\cup\{F\})$ hold, then
we show that $*(y\cup\{E\vee F\})$ does not hold.
Let $C,D$ such that ${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$ and
${{\sf AR}_{\parr}}\!\,\vdash y\cup\{F\}\rhd D$ and $C\not\in w$ and
$D\not\in w$. Hence by disjunction rule,
we have ${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\vee F\}\rhd C\vee D$. Since
$w$ is ${\sf IPC}$-saturated, we also have $C\vee D\not\in w$.
Hence $*(y\cup\{E\vee F\})$ does not hold.
\item Let $*(y)$ and ${{\sf AR}_{\parr}}\!\,\vdash y\rhd E$. We
must show that $*(y\cup\{E\})$. Let
${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$. Then from ${{\sf AR}_{\parr}}\!\,\vdash y\rhd E$ we have
${{\sf AR}_{\parr}}\!\,\vdash y\rhd C$. Then from $*(y)$ we have
$C\in w$.
\end{itemize}
Finally we show that $*(w'\cup w_{\sf par})$. Let
${{\sf AR}_{\parr}}\!\,\vdash w'\cup w_{\sf par}\rhd C$.
Hence ${{\sf AR}_{\parr}}\!\,\vdash \bigwedge w_{\sf par}\wedge E\rhd C$, for some $E\in w'$.
Then by
\textup{Mont}'s rule we have ${{\sf AR}_{\parr}}\!\,\vdash
\bigwedge w_{\sf par}\to E\rhd \bigwedge w_{\sf par}\to C$.
Since $E\in w'$, we have $\bigwedge w_{\sf par}\to E\in w'$
and hence by ${{\sf AR}_{\parr}}\!\,$-saturatedness of $w'$ we have
$\bigwedge w_{\sf par}\to C\in w'$. Since $w'\subseteq w$,
we have $\bigwedge w_{\sf par}\to C\in w$ and hence
$C\in w$.
\\
It only remains to show that $\mathcal{K}$ is good. Let
$w_1,\ldots,w_n\in W$ and $\hat{w}:=\bigcap_iw_i$.
Also assume that $X\subseteq \hat{w}\cap{\sf par}$.
We find some
$w\in W_b$ such that
$ w\subseteq \hat{w}$ and $w\cap{\sf par}={X}$.
Define
$$*(y): \text{For every $C_i$ and $p_i\in{\sf par}$, if }
{{\sf AR}_{\parr}}\!\,\vdash y\rhd \bigvee_i C_i\vee\bigvee_i p_i \text{, then }
\exists\, i\ C_i\in \hat{w} \vee \exists\, i\ p_i\in {X}.$$
We show that $*(.)$ is an ${{\sf AR}_{\parr}}\!\,$-extendible
property and $*({X})$. Then by
\cref{Lem-Max-Sat} we have some ${{\sf AR}_{\parr}}\!\,$-saturated $w$
such that ${X}\subseteq w$ and $*(w)$ holds.
From $*(w)$ it is clear that $w\subseteq \hat{w}$. Also
if $p\in {\sf par}\cap w$, then by $*(w)$ we have
$p\in {X}$ and hence $w'\cap{\sf par}={X}$.
Hence $w$ satisfies all required conditions.
It remains only to show that $*(.)$ is
${{\sf AR}_{\parr}}\!\,$-extendible property and $*({X})$.
First the ${{\sf AR}_{\parr}}\!\,$-extendibility of $*(.)$:
\begin{itemize}
\item If $*(y)$ and $y\vdash E$. We must show
$*(y\cup\{E\})$. Let $C=\bigvee_i C_i\bigvee_i p_i$ and
$p_i\in{\sf par}$ and
${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$.
Hence ${{\sf AR}_{\parr}}\!\,\vdash E\wedge \bigwedge_i F_i\rhd C$ for some
finite set $\{F_i\}_i\subseteq y$. Then since
$y\vdash E $, we have ${{\sf AR}_{\parr}}\!\,\vdash y\rhd
E\wedge \bigwedge_i F_i$. Hence ${{\sf AR}_{\parr}}\!\,\vdash y\rhd C$.
Then from $*(y)$ we have $C_i\in \hat{w}$
or $p_i\in {X}$, for some $i$.
\item If neither
$*(y\cup\{E\})$ nor $*(y\cup\{F\})$ hold, then
we show that $*(y\cup\{E\vee F\})$ does not hold.
Let $C=\bigvee_i C_i\vee\bigvee_ip_i$ and
$D=\bigvee_iD_i\vee\bigvee_iq_i$ and $p_i,q_i\in{\sf par}$
such that ${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$ and
${{\sf AR}_{\parr}}\!\,\vdash y\cup\{F\}\rhd D$ and for all $i$ we have
$C_i,D_i\not\in \hat{w}$ and
$p_i,q_i\not\in {X}$.
Hence by disjunction rule,
we have ${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\vee F\}\rhd C\vee D$,
while for all $i $, $C_i,D_i\not\in \hat{w}$ and $p_i,q_i\not\in {X}$.
Hence $*(y\cup\{E\vee F\})$ does not hold.
\item Let $*(y)$ and ${{\sf AR}_{\parr}}\!\,\vdash y\rhd E$. We
must show that $*(y\cup\{E\})$. Let
$C=\bigvee_i C_i\vee\bigvee_ip_i$ and $p_i\in {\sf par}$
and
${{\sf AR}_{\parr}}\!\,\vdash y\cup\{E\}\rhd C$.
Then from ${{\sf AR}_{\parr}}\!\,\vdash y\rhd E$ we have
${{\sf AR}_{\parr}}\!\,\vdash y\rhd C$. Then from $*(y)$ we have
$C_i\in \hat{w}$ or $p_i\in {X}$
for some $i$.
\end{itemize}
It finishes showing that $*(.)$ is an ${{\sf AR}_{\parr}}\!\,$-extendible
property. Then we show $*({X})$. Let
$C=\bigvee_iC_i\vee\bigvee_ip_i$
and $p_i\in{\sf par}$ and
${{\sf AR}_{\parr}}\!\,\vdash {\sf par}\cap w\rhd C$.
Then by \cref{Cor-ARN-implies-IPC-deriv}
we have ${X}\vdash C$.
Since $\bigwedge ({X})$ is extendible,
by \cref{Remark-extendible-disjun}
for some $i$ we have
${X}\vdash C_i$ or
${X}\vdash p_i$.
Since ${X}\subseteq\hat{w}$, for some $i$
either we have $C_i\in\hat{w}$ or $p_i\in{X}$, as desired.
\end{proof}
\subsection{$\NNILpar$-admissibility}
\begin{lemma}\label{Lem-ARN-models-arn}
For every good ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}$ and $n\in\mathbb{N}$,
there is some
${\sf par}$-extendible stable class of finite rooted models
$\mathscr{K}$ such that for every
proposition $A$ with $c(A)\leq n$ we have
$\mathcal{K}\Vdash A$ iff $\mathscr{K}\Vdash A$.
\end{lemma}
\begin{proof}
Given a good ${{\sf AR}_{\parr}}\!\,$-model $\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ with
$W_b\subseteq W$ as its base-set, we define a stable
${\sf par}$-extendible class $\mathscr{K}$ of finite
rooted Kripke models as
follows. $\mathscr{K}$ includes all Kripke models
$\mathcal{K}'=(W',\preccurlyeq',\mathrel{V}')$ with the following properties:
\begin{itemize}
\item $\mathcal{K}'$ is finite rooted with tree frame.
\item $\mathcal{K}'$ is embeddable in $\mathcal{K}$,
i.e.~there is a function $f:W'\longrightarrow W$ such
that $w'\mathrel{V}' a$ iff $f(w')\mathrel{V} a$; and
$w'\preccurlyeq' v'$ implies $f(w')\preccurlyeq f(v')$.
\item For all $A$ with $c(A)\leq n$ and for every
$w'\in W'$ we have $\mathcal{K}',w'\Vdash A$
iff $\mathcal{K},f(w')\Vdash A$.
\end{itemize}
Obviously $\mathscr{K}$ is stable and $\mathcal{K}\Vdash A$ implies
$\mathscr{K}\Vdash A$ for every $A$ with $c(A)\leq n$.
It remains to show:
\begin{enumerate}[leftmargin=*]
\item $\mathscr{K}\Vdash A$ implies $\mathcal{K}\Vdash A$ for every
$A$ with $c(A)\leq n$. It is enough to show that
for a given $n$ and $w_0\in W$, there is a
finite rooted (with the root $w'_0$) tree-frame Kripke
model $\mathcal{K}'=(W',\preccurlyeq',\mathrel{V}')$ which is embeddable in
$\mathcal{K}$ with the embedding $f$ such that
$f(w'_0)=w_0$ and for every
$w'\in W'$ and $A$ with $c(A)\leq n$ we have
$\mathcal{K}',w'\Vdash A$ iff $\mathcal{K},w\Vdash A$.
First we inductively
define sets $W_i$ of sequences of implications
$B\to C$ with $c(B\to C)\leq n$,
for $0\leq i\leq n$ and the function
$f$ from $W_i$ to $W$.
Then let $W':=\bigcup_{i=0}^n W_i$.
Let $W_0:=\{\langle \rangle\}$ and
$f(\langle\rangle):=w_0$. Assume that we already
defined $W_i$ and define $W_{i+1}$ as follows. For
every sequence $\sigma\in W_i$ and
implication $B\to C$ with $c(B\to C)\leq n$ such that
$\mathcal{K},f(\sigma)\nVdash B\vee (B\to C)$, add the new node
$\sigma*\langle B\to C\rangle$ to $W_{i+1}$ and
define $f(\sigma*\langle B\to C\rangle)=u$
for some $u$ such that
$u\succcurlyeq f(\sigma)$ and $\mathcal{K},u\Vdash B$
and $\mathcal{K},u\nVdash C$.
This finishes definition of $W_i$ and $W'$ and the
embedding $f:W'\longrightarrow W$. Finally define
$\sigma\preccurlyeq'\gamma$ iff $\sigma$ is an initial segment of
$\gamma$. Since there are only finitely many
inequivalent propositions $A$ with $c(A)\leq n$,
one may easily observe that $\mathcal{K}'$ is finite. The other
required properties for $\mathcal{K}'$ are easy and left to the
reader.
\item $\mathscr{K}$ is ${\sf par}$-extendible.
Let $\mathscr{K}':=\{\mathcal{K}_{1},\ldots,\mathcal{K}_{n}\}
\subseteq \mathscr{K}$ be finite such that
$\mathscr{K}'$ is a ${\bs{p}}$-submodel of some
$\mathcal{K}_{0}\in \mathscr{K}$ and $w'_i$ be
the root of $\mathcal{K}_i$.
Let $f_i$ be the embedding of $\mathcal{K}_i$ in $\mathcal{K}$
and $w_i:=f_i(w'_i)$.
Since $\mathcal{K}$ is good, there is some $u\in W$ such that
$u\equiv_\parr w_0$ and $u\preccurlyeq w_1,\ldots,w_n$ and $u\in W_b$.
Since $u$ is a base, there is some tight predecessor
$v\in W$ for the set $\{w_1,\ldots,w_n\}$ such
that $u\equiv_\parr v$ and $u\preccurlyeq v\preccurlyeq w_1,\ldots,w_n$.
Define a ${\bs{p}}$-variant $\mathcal{K}''$
of $\mathcal{K}':=\sum(\mathscr{K}',\mathcal{K}_0)$ in this way:
$\mathcal{K}'',w'_0\Vdash a$ iff $\mathcal{K},v\Vdash a$, for every atomic $a$. Then it is not difficult to observe that
$\mathcal{K}''\in\mathscr{K}$.
\end{enumerate}
\end{proof}
\begin{lemma}\label{Lem-arn-p-extendible}
If $A\arn B$ and $\mathscr{K}\subseteq\Mod{A}$ is ${\sf par}$-extendible
and stable, then $\mathscr{K}\Vdash B$.
\end{lemma}
\begin{proof}
Let $A\arn B$ and $\mathscr{K}$ is a stable class of
finite rooted models with tree frames.
Let $\mathscr{K}'$ be the restriction of $\mathscr{K}$ to the atomics
appeared in $A,B,{\sf par}$. Obviously $\mathscr{K}'\subseteq\Mod A$ also
is a ${\sf par}$-extendible
stable class.
Let $n:=\max\{c(A),\#{\sf par}\}$. Then
\cref{Lem-stable-extendible} implies that
$\langle \mathscr{K}'\rangle_n$ is also a
${\sf par}$-extendible stable class of finite rooted models
with tree frames. \Cref{Lem-Mod(A)-Kripke} implies
$\langle \mathscr{K}'\rangle_n=\Mod{C}$ for some
$C$ with $c(C)\leq n$. Moreover, by
\cref{Theorem-Ghil-Ext}, there is a substitution
$\theta$ and $C'\in\NNILpar$ such that
$ \vdash C'\leftrightarrow \theta(C)$ and $C\vdash E\leftrightarrow \theta(E)$ for every proposition $E$.
On the other hand, \cref{Cor-Kripke-bisim-property}
implies $\langle\mathscr{K}'\rangle_n\Vdash A$. Hence $A$ is
valid in $\Mod{C}$, which implies $ \vdash C\to A$.
Hence $ \vdash \theta(C)\to \theta(A)$ and then
$ \vdash C'\to \theta(A)$. From $A\arn B$ infer
$ \vdash C'\to \theta(B)$, or equivalently
$ \vdash \theta(C\to B)$. Hence for every $\mathcal{K}$,
and of course for every
$\mathcal{K}\in \langle\mathscr{K}'\rangle_n$ we have $\mathcal{K}\Vdash
\theta(C\to B)$. Since $\mathcal{K}\Vdash C$ and $\theta$ is
$C$-projective, we have $\mathcal{K}\Vdash C\to B$, and hence
$\mathcal{K}\Vdash B$. So we may deduce
$\mathcal{K}\in \langle\mathscr{K}'\rangle_n\Vdash B$. Since
$\mathscr{K}'\subseteq \mathcal{K}\in \langle\mathscr{K}'\rangle_n$, we
also have $\mathscr{K}'\Vdash B$. Whence $\mathscr{K}\Vdash B$, as
desired.
\end{proof}
\begin{theorem}\label{Characterization-admissibility}
The following statements are equivalent:
\begin{enumerate}
\item ${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$.
\item $A\arn B$.
\item $B$ is valid in every ${\sf par}$-extendible
stable class of
Kripke models of $A$.
\item $B$ is valid in every good ${{\sf AR}_{\parr}}\!\,$-model of $A$.
\end{enumerate}
\end{theorem}
\begin{proof}
$1\to 2$: \cref{Lem-ARN-implies-arpn}.\\
$2\to 3$: \cref{Lem-arn-p-extendible}.\\
$3\to 4$: \cref{Lem-ARN-models-arn}.\\
$4\to 1$: \cref{Theorem-ARN-Completeness}.
\end{proof}
\begin{corollary}
\label{montagna-nnil}
The following rule is admissible in ${{\sf AR}_{\parr}}\!\,$:
\AxiomC{$A\rhd B$}
\LeftLabel{\uparan{$E\in\NNILpar$}}
\UnaryInfC{$E\to A\rhd E\to B$}
\DisplayProof.
\end{corollary}
\begin{proof}
Since $\rhd={\arn}\ \ $ and
\AxiomC{\begin{tabular}{c}
{$A\arn B$}
\end{tabular}
}
\UnaryInfC{$E\to A\arn E\to B$}
\DisplayProof, we have the desired result.
\end{proof}
\subsection{${\dar{\,}\NNILpar}$-preservativity logic}\label{sec-pres-1}
In the following theorem we show that the other direction of
\cref{pres-admis-rel} holds when $ \Gamma=\NNILpar $:
\begin{theorem}\label{arnp-character}
$ {\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ }} = {\arn}$ .
\end{theorem}
\begin{proof}
\Cref{pres-admis-rel} implies that if
$A\arn B$ then $A\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ } B$. For the other direction,
assume that $A\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ } B$ seeking to show $A\arn B$.
By \cref{pnnilpv-nnilp} it is enough to show $A\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } B$.
Let $E\in\pNNILpar $ and substitution $ \theta $ such that
$ \vdash E\to \theta(A) $. Let $ \Pi_A $ be the $ \pNNILpar $-projective resolution for $ A $, as guarantied by
\cref{Projec-resol}. Since $A\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } \bigvee \Pi_A$
we have $\vdash E\to \bigvee \theta(\Pi_A) $ and hence
by primality of $E$,
for some $ F\in\Pi_A $ we have
$\vdash E\to \theta(F)$. On the other hand, since $\Pi_A$
is a projective resolution for $A$ we have
$ \vdash F\to A $. Then
by $ A\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ } B $ we get $ \vdash F\to B $. Hence
$ \vdash \theta(F)\to \theta(B) $, which implies
$ \vdash E\to\theta(B) $, as desired.
\end{proof}
\begin{remark}\label{argt=prtg}
For every $\Gamma$ and a logic ${\sf T}\supseteq{\sf IPC}$ which admits
$\Gamma$-projective resolutions,
i.e.~every $A\in\lcal_{0}$ has a $\Gamma$-projective resolution in ${\sf T}$,
the above proof works and we have ${\mathrel{\adsm{{\sf T}}{\Gamma}}}={\mathrel{\pres{{\sf T}}{\Gamma}}}$. Hence we have
${\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ }}={\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ }}$ .
\end{remark}
\begin{remark}
By \cref{arnp-character,Characterization-admissibility,argt=prtg} we may conclude that:
\begin{center}
${{\sf AR}_{\parr}}\!\,\vdash A\rhd B$ \quad
iff\quad $A\arn B$ \quad
iff\quad $A\mathrel{\adsm{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } B$
\quad iff\quad $A\mathrel{\pres{}{{\dar{\,}\nnilpar}}\ \ \ } B$
\quad iff\quad $A\mathrel{\pres{}{{\dar{\,}\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}}\ \ \ \ } B$.
\end{center}
\end{remark}
\subsection{$\NNILpar$-preservativity logic}\label{sec-pres-2}
In this subsection we axiomatize the
${\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}$-preservativity and
show $ {\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ }}=\ARD\IPC\parr{\sf A} $ in which
$ \ARD\IPC\parr{\sf A} $ is defined as $ {{\sf AR}_{\parr}}\!\, $ plus the following axiom schema
(the substitution axiom):
$${\sf sub}: A\rhd \theta(A) \text{ for every substitution $ \theta $ (which by default is identity on parameters)}$$
The main point of the axiom $ {\sf sub} $ is that we may annihilate
occurrences of atomic variables, and together with other axioms of
$ {{\sf AR}_{\parr}}\!\, $ we may simplify propositions to $ \NNILpar $-propositions.
Before we continue with providing such simplifying algorithm,
let us define $\itap{A}{B}$ and $\itapp{A}{B}$, two variants of
$\ita{A}{B}$:
$$\itap{A}{B}:=\begin{cases}
B\quad &: B\text{ is $\bot$ or parameter}\\
A \to B &: B\in{\sf var}\\
\itap{A}{C}\circ\itap{A}{D} &:
B=C\circ D\text{ and }\circ\in\{\vee,\wedge\}\\
({A{\downarrow}C})\to {B} &: B=C\to D
\end{cases}
$$
$$
\itapp{A}{B}:=
\begin{cases}
A\to\bot\quad &: B\in{\sf var}\\
\itap{A}B &:\text{otherwise}
\end{cases}
$$
In which $A{\downarrow}C$ indicates the replacement of $D$
for every occurrence of an implication $C\to D$ in $A$.
\\
Then define the following variant of Visser rule:
\begin{itemize}
\item[${\sf V}'_{\!\text{\fontsize{6}{0}\selectfont\AR}}:$]\quad $B\to C\rhd \bigvee_{i=1}^{n+m}
\itapp{B}{E_i}$, in which
$B=\bigwedge_{i=1}^n (E_i\to F_i)$ and
$C=\bigvee_{i=n+1}^{n+m} E_i$.
\end{itemize}
Note that ${{\sf AR}_{\parr}}\!\,\vdash (B\to C)\rhd \bigvee_{i=1}^{n+m}
\itap{B}{E_i}$ for $B$ and $C$ as defined in above lines.
\begin{lemma}\label{variant-visser-prop}
$\ARD\IPC\parr{\sf A}\vdash{\sf V}'_{\!\text{\fontsize{6}{0}\selectfont\AR}}$.
\end{lemma}
\begin{proof}
Let $B$ and $C$ as in ${\sf V}'_{\!\text{\fontsize{6}{0}\selectfont\AR}}$. By $\viss\parb$ we have
$(B\to C)\rhd \bigvee_{i=1}^{n+m}
\ita{B}{E_i}$. Also Since
for every $A$ and $B$ we have
$\vdash \itap{A}{B}\to \ita{A}{B}$,
then ${{\sf AR}_{\parr}}\!\,\vdash (B\to C)\rhd \bigvee_{i=1}^{n+m}
\itap{B}{E_i}$. So it is enough to show for every
$1\leq j\leq n+m$:
$$\ARD\IPC\parr{\sf A}\vdash \itap{B}{E_j}\rhd \itapp{B}{E_j} \vee
\bigvee_{j\neq i=1}^{n+m} \itap{B}{E_i}.$$
If $E_j\not\in{\sf var}$ we are trivially done. So assume that
$E_j=x\in{\sf var}$ and hence $\itap{B}{E_j}=B\to x$.
Then by the substitution axiom ${\sf sub}$
we have $\ARD\IPC\parr{\sf A}\vdash (B\to x)\rhd \hat{\theta}(B\to x)$, in which
$\theta$ is the substitution with $\theta(x)=y\vee z$ and
identity elsewhere and $y,z\in{\sf var}$ are fresh variables, i.e.~variables not appeared in $B$ and $C$.
Let $E'_i:=\hat{\theta}(E_i)$ and $F'_i:=\hat{\theta}(F_i)$ and $B':=\hat{\theta}(B)$.
Hence $\ARD\IPC\parr{\sf A}\vdash (B\to x)\rhd (B'\to (y\vee z))$. On the other hand by $\viss\parb$ we have
${{\sf AR}_{\parr}}\!\,\vdash (B'\to (y\vee z))\rhd
\bigvee_{j\neq i=1 }^{n}\itap{B'}{E'_i} \vee (B'\to y)\vee (B'\to z)$.
Let $\alpha$,
$\beta$ and $\gamma$ be substitutions that
$$\alpha(y)=\alpha(z)=x,\quad \beta(y)=\gamma(z)=\bot,\quad
\beta(z)=\gamma(y)=x
$$
and identity elsewhere. Then by ${\sf sub}$ we have
$$\ARD\IPC\parr{\sf A}\vdash (B'\to z)\rhd \hat{\gamma}(B'\to z)
\quad \text{ and hence }\quad
\ARD\IPC\parr{\sf A}\vdash (B'\to z)\rhd (B\to \bot),
$$
$$\ARD\IPC\parr{\sf A}\vdash (B'\to y)\rhd \hat{\beta}(B'\to y)
\quad \text{ and hence }\quad
\ARD\IPC\parr{\sf A}\vdash (B'\to z)\rhd (B\to \bot),
$$
$$\ARD\IPC\parr{\sf A}\vdash \itap{B'}{E'_i} \rhd \hat{\alpha}(\itap{B'}{E'_i} )
\quad \text{ and hence }\quad
\ARD\IPC\parr{\sf A}\vdash \itap{B'}{E'_i} \rhd \itap{B}{E_i}.
$$
Hence $\ARD\IPC\parr{\sf A}\vdash (B'\to (y\vee z))\rhd
\bigvee_{j\neq i=1 }^{n}\itap{B}{E_i} \vee (B\to \bot)$ and thus $\ARD\IPC\parr{\sf A}\vdash \itap{B}{E_j}\rhd
\bigvee_{j\neq i=1 }^{n}\itap{B}{E_i} \vee \itapp{B}{E_j} $.
\end{proof}
\begin{lemma}\label{star-prop}
For every $A\in\lcal_{0}$ one may effectively compute
$A^\star\in\NNILpar$ such that:
\begin{enumerate}
\item $ {\sf IPC}\vdash A^\star\to A $,
\item $ \ARD\IPC\parr{\sf A}\vdash A\rhd A^\star$,
\item $\suba{A^\star}\subseteq \suba{A}$.
\end{enumerate}
\end{lemma}
\begin{proof}
By induction on $\mathfrak{o} A$ we define $A^\star$ with required properties. So let us first define the complexity number
$\mathfrak{o} A\in\mathbb{N}^3$.
\begin{itemize}[leftmargin=*]
\item $I(A):=\{E\to F: E\to F\in\sub{A}\}$.
\item $\ifrak A:=\max\{\#I(B):B\in I(A)\}$. ($\#B$ indicates the number of elements in $B$)
\item $\cfrak A$ is defined as the number of connectives occurring
in $A$.
\item $\vfrak A $ is defined as the number of occurrences of
variables occurring in $A$.
\item $\mathfrak{o} A:=(\ifrak A, \dfrak A,\vfrak A)$. Finally we order triples in $\mathbb{N}^3$ lexicographically.
\end{itemize}
Then by induction on $\mathfrak{o} A$ define $A^\star$ fulfilling the required conditions in the statement of lemma.
\begin{itemize}[leftmargin=*]
\item $A=B\wedge C$: Define $A^\star:=B^\star\wedge C^\star$.
\item $A=B\vee C$: Define $A^\star:=B^\star\vee C^\star$.
\item $A\in{\sf var}$: Define $A^\star:=\bot$. Note that to
show $\ARD\IPC\parr{\sf A}\vdash A\rhd A^\star$, here we need
the substitution axiom ${\sf sub}$.
\item $A\in{\sf par}$: Define $A^\star:=A$.
\item $A=B\to C$: We have several sub-cases:
\begin{itemize}[leftmargin=*]
\item $B$ has outer disjunction, i.e.~a disjunction
$E\vee F$ which is not in
the scope of $\to$. Then there is some
proposition $B_0(x)$ with the following properties:
(1) $x$ is a variable not appearing in $B$,
(2) $x$ occurs only once in $B$,
(3) $x$ has outer occurrence in $B_0$,
i.e.~$x$ is not in the scope of arrows,
(4) $B=B_0[x: E\vee F]$. Then define $B_1:=B_0[x:E]$ and
$B_2:=B_0[x:F]$ and let
$$A^\star:=(B_1\to C)^\star\wedge (B_2\to C)^\star.$$
\item $C$ has outer conjunction, i.e.~a conjunction
$E\wedge F$ which is not in
the scope of $\to$. Then there is some
proposition $C_0(x)$ with the following properties:
(1) $x$ is a variable not appearing in $C$,
(2) $x$ occurs only once in $C$,
(3) $x$ has outer occurrence in $C_0$,
i.e.~$x$ is not in the scope of arrows,
(4) $C=C_0[x: E\wedge F]$. Then define $C_1:=C_0[x:E]$ and
$C_2:=C_0[x:F]$ and let
$$A^\star:=(B\to C_1)^\star\wedge (B\to C_2)^\star.$$
\item $B=\bigwedge^n_{i=1}B_i$ and
$C=\bigvee^{n+m}_{i=n+1}E_i$ in which
every $B_i$ and $E_i$ is either atomic or implication.
Again we have several sub-cases:
\begin{itemize}[leftmargin=*]
\item $B_i=\top$ for some $i$. Remove $B_i$ from the conjunction
$B$ and let $B_0$ be the result. Then define $A^\star:=(B_0\to C)^\star$.
\item $B_i=\bot$ for some $i$. Then define $A^\star:=\top$.
\item $B_i\in{\sf var}$ for some $i$. Let $\theta$ be a substitution
such that $\theta(B_i):=\top$ and $\theta$ is identity elsewhere.
Then define $A^\star:=(\hat{\theta} (A))^\star$. Note that $\vfrak{\hat{\theta}(A)}<\vfrak A$.
\item $B_i\in{\sf par}$ for some $i$. Let $B_0$ results in by removing $B_i$ from the conjunction $B$ and define
$$A^\star:=B_i\to (B_0\to C)^\star.$$
\item $B_i=E_i\to F_i$, for every $1\leq i\leq n$.
Then define
$$
A_1:=\bigwedge_{i=1}^n((B{\downarrow}E_i) \to C)
\quad \text{and} \quad
A^\star:=(A_1\wedge \bigvee_{i=1}^{n+m}\itapp{B}{E_i})^\star.
$$
\end{itemize}
\end{itemize}
\end{itemize}
For last case, we reason for the following facts:
\begin{itemize}[leftmargin=*]
\item $\mathfrak{o}{(B{\downarrow}E_i) \to C}<\mathfrak{o} A$
for every $0\leq i\leq n$. Note that
$\ifrak{(B{\downarrow}E_i) \to C}\leq \ifrak A$ and
$\cfrak{(B{\downarrow}E_i) \to C}<\cfrak A$.
\item $\mathfrak{o}{ \itapp{B}{E_i}}<\mathfrak{o} A$ for every
$1\leq i\leq n+m$. If $E_i\in{\sf var}$, we have this inequality because $\itapp{B}{E_i}=B\to\bot$ and hence
$\vfrak{ \itapp{B}{E_i}}<\vfrak A$. For every other case we may show $\ifrak{\itapp{B}{E_i}}<\ifrak A$. We refer the reader to
\citep[sec.~7]{Visser02}.
\item $\ARD\IPC\parr{\sf A}\vdash A\rhd A^\star$: Use induction hypothesis and
\cref{variant-visser-prop}.
\item ${\sf IPC}\vdash A^\star\to A$: By induction hypothesis,
it is enough to show ${\sf IPC}\vdash (A_1\wedge \itapp{B}{E_i})\to A$
for every $1\leq i\leq n+m$.
So we reason inside ${\sf IPC}$. Assume $A_1$ and $\itapp{B}{E_i}$
and $B$ seeking to derive $C$. If $i>n$, then by definition we have $\itapp{B}{E_i}\to C$ and we are done. Otherwise, by
$A_1$ we have $(B{\downarrow}E_i)\to C$ and hence it is enough to show $B{\downarrow}E_i$. Hence by $B$ it is enough to show
$E_i$, which holds by $B$ and $\itapp{B}{E_i}$.\qedhere
\end{itemize}
\end{proof}
\begin{theorem}\label{Theorem-nnilp-Pres}
For every $A,B$, following items are equivalent:
\begin{enumerate}
\item $ \ARD\IPC\parr{\sf A}\vdash A\rhd B $,
\item $ A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } B $,
\item $\vdash A^\star\to B $.
\end{enumerate}
\end{theorem}
\begin{proof}
We show $1 \Rightarrow 2 \Rightarrow
3\Rightarrow 1$:
\begin{itemize}
\item $1\Rightarrow 2$: By \cref{Lem-con1-ipc}.
\item $2\Rightarrow 3$: Let $ A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } B $.
By \cref{star-prop} we have $ A^\star\in\NNILpar $ and
$ \vdash A^\star\to A $.
Then by $ A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } B $
we get $ \vdash A^\star\to B $.
\item $3\Rightarrow 1$:
From $ \vdash A^\star\to B $ we get
$ \ARD\IPC\parr{\sf A}\vdash A^\star\rhd B $. Also by \cref{star-prop}
we have $\ARD\IPC\parr{\sf A}\vdash A\rhd A^\star$ and then
Cut implies desired result.\qedhere
\end{itemize}
\end{proof}
\begin{lemma}\label{prnp=prpnp}
${\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ }}={\mathrel{\pres{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ }}$ .
\end{lemma}
\begin{proof}
\Cref{Lem-nnil-normal-form,vee-pres}.
\end{proof}
\begin{lemma}\label{Lem-con1-ipc}
$\ARD\IPC\parr{\sf A}\vdash A\rhd B$ implies $ A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } B $.
\end{lemma}
\begin{proof}
We use induction on complexity of the proof
$ \ARD\IPC\parr{\sf A}\vdash A\rhd B $. All steps trivially hold except:
\begin{itemize}[leftmargin=*]
\item ${\sf sub}$: This axiom holds because ${\sf IPC}$
is closed under substitutions and $ \theta(E)=E $ for every $ E\in\NNILpar $.
\item $\viss{\atomb}$: \Cref{VA-validity}.
\item Disj:
Let $A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } C$ and $B\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } C$ seeking to show
$A\vee B\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } C$. By \cref{prnp=prpnp} it is enough to show
$A\vee B\mathrel{\pres{}{\pnnil(\parr)}\def\pNNILpar{\pNNIL(\parr)}\ \ \ } C$.
Let $E\in \pNNILpar$ such that $\vdash E\to (A\vee B)$.
Since $E$ is ${\sf IPC}$-prime, either we have
$\vdash E\to A$ or $\vdash E\to B$.
Then by $A\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } C$ and $B\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } C$,
in either of the cases we have
$\vdash E\to C$.
\qedhere
\end{itemize}
\end{proof}
\begin{lemma}\label{VA-validity}
$B\to C\mathrel{\pres{}{{\nnil(\parr)}}\def\NNILpar{{\NNIL(\parr)}}\ \ \ } \bigvee_{i=1}^{n+m}
\itp{B}{E_i}$, in which
$B=\bigwedge_{i=1}^n (E_i\to F_i)$ and
$C=\bigvee_{i=n+1}^{n+m} E_i$.
\end{lemma}
\begin{proof}
We reason by contraposition. Let $E\in\NNILpar$
be such that $\nvdash E\to (\bigvee_{i=1}^{n+m}
\itp{B}{E_i})$. Hence there is some finite rooted
$\mathcal{K}=(W,\preccurlyeq,\mathrel{V})$ such that
$\mathcal{K},w_0\Vdash E$ and $\mathcal{K},w_0\nVdash \bigvee_{i=1}^{n+m} \itp{B}{E_i}$. Let $I$ be the set of indexes $i$ such that $E_i\in{\sf par}$ or $E_i=\bot$.
Also let $J$ be the complement of $I$. Thus
for every $i\in I$ we have $\mathcal{K},w_0\nVdash E_i$
and for every $j\in J$, there is some $w_j\succcurlyeq w_0$
such that $\mathcal{K},w_j\Vdash B$ and
$\mathcal{K},w_j\nVdash E_j$.
Let $W'$ defined as follows:
$$W':=W\setminus
\{v\in W: \neg\exists j\in J(w_j\preccurlyeq v)\}$$
and define $\mathcal{K}':=(W',\preccurlyeq,\mathrel{V})$.
Then since $E\in{\sf NNIL}$, \cref{Theorem-NNIL-Submodel}
implies $\mathcal{K}',w_0\Vdash E$.
Moreover, it is not difficult to observe that
$\mathcal{K}',w_0\Vdash B$ and $\mathcal{K}',w_0\nVdash C$.
Thus $\mathcal{K}',w_0\nVdash E\to (B\to C))$ and then
$ \nvdash E\to (B\to C) $.
\end{proof}
\section*{Acknowledgement}
Special acknowledgement heartily granted to
Mohammad Ardeshir,
Majid Alizadeh, Philippe Balbiani, Rosalie Iemhoff,
Emil Jeřábek, Dick de Jongh, Deniz Tahmouresi
for stimulating discussions or communications we had on preservativity, unification or admissibility.
|
1,314,259,993,014 | arxiv | \section{Introduction.}}
\par
The Schwarz-Sen electromagnetic dual model \cite{Sen}
is a four-dimensional member of a more general family of
theories which are
manifestly invariant under duality transformations
\cite{Hen1}.
Its two-dimensional version is known as Tseytlin
model \cite{Tsey}, introduced at first in the string theory
context. It is a conformal theory and as such can be
decomposed into two independent (for left and
right movers) Floreanini-Jackiw (FJ) \cite{Flo} chiral
boson models. FJ models have peculiar features which have
been extensively investigated in the last ten years \cite{Gir}.
In particular, despite the fact they
are not-manifestly Lorentz-invariant, they turn out to
be $2$-dimensional Poincar\'e invariant. The
quantum hamiltonian
structure of the FJ model was analyzed in \cite{Gir},
while in \cite{CD} this analysis was extended to the
Tseytlin model, proving in particular the closure of $2D$
Poincar\'e algebra both in the classical and in the
quantum case. In \cite{CD} it was furthermore proposed a
hamiltonian supersymmetric theory which coincides
with a $2$-dimensional reduction of
the supersymmetric extension of the original
Schwarz-Sen model \cite{Sen}. A complete analysis of its symmetries,
as well as a manifest supersymmetric formulation, was
however not carried out in that paper.
The purpose of our present work is to fully investigate
the properties of supersymmetric extensions of both
chiral (FJ) and dual (Tseytlin) models. Our aim is to
provide the algebraic setting underlying
dimensional reductions of supersymmetric $4$-dimensional dual
models. \par
In this paper we construct for both FJ and Tseytlin
models their $N=1$ and $N=2$ supersymmetric
extensions by using a superfield
formalism\footnote{ In reality the ``$N=2$" Tseytlin model as
a quantum mechanical system is globally $N=4$ supersymmetric. We
discuss this point in detail in the following.}. We show
that their symmetries generate $N=1,2$ SuperVirasoro
algebras and are in connection with the $N=1,2$
Coulomb gas formulation (see \cite{scg} and
references therein). The closure of $2D$ ($N=1,2$)
superPoincar\'e algebra is proven in both classical and quantum cases.\par
It is worth mentioning that the construction of
supersymmetric extensions must be carefully performed,
which means their investigation is quite interesting.
As an example we just mention that the equations
of motion for a system involving a chiral boson and a
chiral fermion can be derived by using two
different hamiltonian pictures. Only one of
them leads to a supersymmetric theory,
while the other does not. Further topics of
this kind will be discussed in the text.\par
The scheme of the paper is as follows:
\par
In the next section we review the formulation of
the bosonic FJ and Tseytlin models. Despite the fact that
most of the material presented is nowadays
standard, some results presented are new.\par
In section $3$ we introduce
the $N=1$ superfield formalism and derive the corresponding
super-FJ and superTseytlin models. Since it is not possible
to derive the equations of motion directly from a
manifestly supersymmetric $2$-dimensional action, we supersymmetrize
the space coordinate only, leaving the time an ordinary
bosonic variable. Our formulation differs
from a previously constructed version \cite{PST} in
which one light-cone variable was supersymmetrized,
and is more suitable for analyzing the
Tseytlin model which
deals with both chiralities.
The invariances under $1$-dimensional
superdiffeomorphisms and $2$-dimensional
superPoincar\'e transformations are proven. The Dirac bracket
analysis is performed and the N\"other supercurrents are
derived. They realize an $N=1$ superVirasoro algebra
(for both chiralities in the Tseytlin model) with central
charge $c={\textstyle{3\over 2}}$ in the quantum case.\par
The section $4$ deals with the $N=2$ extensions
of the above models. They are
constructed by making use of $N=2$
chiral and antichiral
superfields\footnote{ The word is here
employed to mean $N=2$ chirality and not
the ordinary space-time chirality discussed above.}
and mimicking the procedure employed in the previous case. The total
field content (in the $N=2$ FJ model)
consists of two ordinary chiral bosons and two ordinary chiral fermions.
Invariances under $2$-dimensional $N=2$ superPoincar\'e
transformations and $1$-dimensional $N=2$
superdiffeomorphisms are proven. The Dirac's brackets for
the conserved N\"other currents generate an $N=2$
superVirasoro algebra with central charge $c=3$ in the quantum case.
\vspace{0.2cm}
\noindent{\section{The bosonic FJ and Tseytlin models.}}
\par
Let us start introducing the bosonic FJ and Tseytlin models.
They are defined in terms of the following lagrangian
densities:
\begin{eqnarray}
{\cal L}_{FJ} &=& \partial_0\phi\partial_1\phi - (\partial_1\phi)^2
\label{fjlagr}
\end{eqnarray}
for the FJ model and
\begin{eqnarray}
{\cal L}_{Ts} &=& \partial_0\phi \partial_1{\tilde\phi}
+\partial_0{\tilde\phi}\partial_1\phi -(\partial_1\phi)^2
-(\partial_1{\tilde\phi})^2
\label{tslagr}
\end{eqnarray}
for the Tseytlin model.\par
The above lagrangians coincide with the ones given in the
literature \cite{{Tsey},{Flo},{CD}} up to an
overall normalization factor.\par
We work in the $2$-dimensional Minkowski spacetime; the time coordinate
$t$ will also be denoted as $x^0$ (${\partial_0}={\textstyle{\partial
\over\partial t}}$)
and the space coordinate $x$ as $x^1$ (${\partial_1}={\textstyle{
\partial\over\partial x}}$). We
will also make use of the light-cone coordinates $z_{\pm}$ defined as
\begin{eqnarray}
&& z_{\pm} = x \pm t,\quad\quad
\partial_{\pm} = {1\over 2}({\partial_1\pm\partial_0}).
\nonumber
\end{eqnarray}
The equations of motion are given by
\begin{eqnarray}
(\partial_0-\partial_1)\partial_1\phi &=& 0
\end{eqnarray}
in the FJ case and
\begin{eqnarray}
&&{\partial_1}^2{\tilde\phi} -\partial_0\partial_1\phi =0
\, ,\quad\quad
{\partial_1}^2\phi -\partial_0\partial_1{\tilde \phi} =0
\end{eqnarray}
in the Tseytlin case.\\
The lagrangian density (\ref{tslagr}) is invariant under
duality transformations, i.e. exchanging $\phi \leftrightarrow
{\tilde\phi}$. The Tseytlin model can be decomposed
into two (chiral and antichiral) FJ models as it is
evident from the positions
\begin{eqnarray}
\phi_\pm&=& {1\over \sqrt{2}} (\phi \pm {\tilde{\phi}})\, .
\end{eqnarray}
The lagrangian ${\cal L}_{Ts}$ can therefore be rewritten as
\begin{eqnarray}
{\cal L}_{Ts} &=& \partial_1\phi_+(\partial_1-\partial_0)\phi_+
+\partial_1\phi_-(\partial_1+\partial_0)\phi_-
\, .\end{eqnarray}
Both the FJ and the Tseytlin model are invariant under
Poincar\'e $2$-dimensional transformations,
with $\phi$, ${\tilde\phi}$ transforming as scalar
fields. A basic difference between the
FJ theory and its dual version consists in
the fact that in the chiral case the
hamiltonian $H\equiv P^0$ coincides with
the space-translation generator $P^1$ while
they are different for the Tseytlin lagrangian.\par
Another set of invariances is provided by a full class
of transformations dependent on a parameter $\lambda$.
Such kind of transformations are very well studied in the
context of Coulomb gas picture \cite{DF}. However, they have not
been considered for the models we are dealing
with. Indeed the action correspondent to the
FJ Lagrangian is invariant under the infinitesimal transformations:
\begin{eqnarray}
\delta_{\lambda} \phi(x,t) &=&
\epsilon(z_+)\partial_1\phi + \lambda
\partial_1\epsilon (z_+)
\label{diff}
\end{eqnarray}
for any value of $\lambda$. The same transformation
applied to the $\phi_+$ field leaves
invariant the Tseytlin action while a similar
transformation applied on $\phi_-$ depends on an
infinitesimal parameter ${\overline\epsilon} (z_-)$.\par
By expanding $\epsilon (z_+)$ in Laurent series
($\epsilon (z_+) = -\sum_{n\in{\bf Z}} \epsilon_n {z_+}^{n+1}$) we can
introduce for any $\lambda$
the operators $l_n(\lambda)$ given by
\begin{eqnarray}
l_n(\lambda) &=& -\left({z_+}^{n+1}\partial_1 + \lambda
(n+1) {z_+}^n\right)\, ,
\end{eqnarray}
therefore
\begin{eqnarray}
\delta_{\lambda}\phi &=& \sum_{n\in{\bf Z}} \epsilon_n
l_n(\lambda )\phi\, .
\end{eqnarray}
For any fixed value of $\lambda$ the algebra generated by the
$l_n(\lambda )$ operators is the $1$-dimensional diffeomorphisms
algebra
(centerless Virasoro algebra):
\begin{eqnarray}
\relax [l_n(\lambda) , l_m(\lambda) ] &=&
(n-m) l_{n+m} (\lambda)\, ,
\end{eqnarray}
therefore the set of
invariances of the FJ model includes the
$1D$-diffeomorphisms, while in the Tseytlin case
we have the direct sum of two copies of $1D$-diffeomorphisms, one
for each
chirality.\par
The analysis of the hamiltonian dynamics, the structure of primary
constraints and the construction of Dirac's brackets has been performed
in \cite{Gir} in the FJ case and in \cite{CD} in the Tseytlin case.\par
These results lead to the hamiltonian\footnote{ In the following formulae
the double dots denote the standard normal ordering;
moreover for our purposes, in order to avoid complications arising
from boundary conditions, we assume the space coordinate $x$ being
compactified on a circle $S^1$ with periodic boundary conditions, or living
on
${\bf R}$ and the fields being fast-decreasing at the infinities.}
\begin{eqnarray}
H_{FJ} &=& \int dx {\theta_{FJ}}^{00} (x)\, ,
\end{eqnarray}
with the current ${\theta_{FJ}}^{00}$ given by (in the quantum case)
\begin{eqnarray}
{\theta_{FJ}}^{00} &=& - : (\partial_1\phi )^2:
\quad .
\end{eqnarray}
The equal-time Dirac's brackets \cite{Dir}
are computed and give
\begin{eqnarray}
\relax [ \phi (x), \phi (y) ]_{D} &=&
{i\over 2}{\partial_y}^{-1}\delta (x-y)
\, ,
\end{eqnarray}
where the standard delta-function appears in the r.h.s.\par
In the Tseytlin case we have respectively
\begin{eqnarray}
H_{Ts} &=& -\int dx (:(\partial_1\phi_+)^2: + :(\partial_1\phi_-)^2:)\, ,
\end{eqnarray}
and the following Dirac's brackets
\begin{eqnarray}
&&\relax [ \phi_{\pm} (x), \phi_{\pm}(y)]_D=
\pm {i\over 2} {\partial_y}^{-1} \delta (x-y)\, ,
\quad\quad
\relax [\phi_+(x),\phi_-(y)]_D
= 0\, .
\end{eqnarray}
The $2$-dimensional Poincar\'e algebra defined by the translation
generators $P^0$, $P^1$ and Lorentz boost $M$, with structure constants
\begin{eqnarray}
&&\relax [M,P^0] = i P^1\, ,\quad\quad
\relax [M,P^1] = i P^0
\label{poi}
\end{eqnarray}
(and vanishing otherwise), is reproduced by the conserved charges computed
with the standard N\"other methods. In the FJ case we
have
\begin{eqnarray}
&&P^1 = P^0 \equiv H_{FJ}\, ,\quad\quad
M = \int dx {\cal M} (x)\, ,
\end{eqnarray}
with ${\cal M} (x)$ given by
\begin{eqnarray}
{\cal M} (x) &=& (x+t){\theta_{FJ}}^{00}(x,t)\, .
\end{eqnarray}
\par
We can compute the commutation
relations of the ${\theta_{FJ}}^{00}(x)$ currents by using
OPE techniques. The result is the following
\begin{eqnarray}
\relax [ {\theta_{FJ}}^{00} (x), {\theta_{FJ}}^{00} (y)] &=&
-{1\over 12}{\partial_y}^3\delta (x-y) +
2 i{\theta_{FJ}}^{00}(y)
\partial_y\delta (x-y) + i \partial_y{\theta_{FJ}}^{00}(y)\cdot
\delta (x-y)\, . \nonumber\\
&&
\label{virfj}
\end{eqnarray}
The above algebra corresponds to the Virasoro algebra and the first term
in the r.h.s. gives the central extension. In the classical case such
term is not present and the algebra coincides
with the algebra of $1$-dimensional diffeomorphisms.
By reexpressing (\ref{virfj}) in standard quantum OPE form we
realize that the value of the central
charge $c$ corresponds to $c=1$.\par
By setting $P^0=P^1=P$ the $2D$-Poincar\'e algebra can be recovered from
the single commutation relation
\begin{eqnarray}
\relax [M,P] &=& i P\, .
\end{eqnarray}
We finish this part devoted to the FJ model by discussing its invariance
properties under $1D$-diffeomorphisms. For any given $\lambda$
the (\ref{diff}) transformations are generated by the N\"other currents
$\theta_\lambda (x,t)$ given by
\begin{eqnarray}
\theta_\lambda (x) &=&
- :(\partial_1\phi)^2: + \lambda (\partial_1)^2 \phi
\, .
\end{eqnarray}
The conserved charges are given by $L_n(\lambda)$,
\begin{eqnarray}
L_n(\lambda) &=& \int dx (x+t)^{(n+1)} \theta_\lambda (x,t)
\, .
\end{eqnarray}
The $L_n(\lambda)$ charges satisfy a closed algebra structure generated
by Dirac's brackets. It coincides with the Virasoro algebra with central
charge $c = 1-6i\lambda^2$:
\begin{eqnarray}
\relax [L_n(\lambda), L_m(\lambda)]&=&
i (n-m) L_{n+m}(\lambda) -{c\over 12} n(n^2-1) \delta_{n+m,0}
\, .
\end{eqnarray}
The extra term $\lambda(\partial_1)^2\phi$ corresponds to
the well-known Feigin-Fuchs term in the Coulomb gas picture \cite{DF}.
Its purpose there consists
in providing a bosonization for conformal theories with any specific value
of the central charge $c$. In the present context it tells us the following
feature of the model under consideration. It keeps an invariance under
$1D$-diffeomorphisms even in the quantum case because it is possible to
find a particular value
of $\lambda$ ($\lambda = \sqrt{{- i\over 6}}$)
in such a way that the central charge is vanishing,
which leads to a non-anomalous quantum
theory.\par
For what concerns the Tseytlin model its N\"other
currents ${\theta_{Ts}}^{00}$, ${\theta_{Ts}}^{01}$
associated to the space-time translations can be decomposed as
\begin{eqnarray}
&&\theta_\pm = {\theta_{Ts}}^{00} \pm {\theta_{Ts}}^{01}=
\theta_\pm = -:(\partial_1\phi_\pm)^2:\quad ,
\end{eqnarray}
while the current ${\cal M}$ associated to the Lorentz boost is
\begin{eqnarray}
{\cal M} &=& x{\theta_{Ts}}^{00} + t {\theta_{Ts}}^{01}
\, .
\end{eqnarray}
The $2$-D Poincar\'e algebra (\ref{poi}) is realized
by the commutators of the conserved charges
\begin{eqnarray}
&& P^0 = \int dx {\theta_{Ts}}^{00},\quad\quad
P^1= \int dx {\theta_{Ts}}^{01}, \quad\quad
M=\int dx {\cal M}
\, .
\end{eqnarray}
\par
The currents $\theta_\pm$ make explicit
the fact that the model under consideration is
conformally invariant since their commutation relations
satisfy the following algebraic relations
\begin{eqnarray}
\relax [\theta_\pm (x), \theta_\pm (y)]
&=& -{1/12} {\partial_y}^3\delta (x-y) + 2i\theta_\pm(y)
\partial_y\delta(x-y) + i\partial_y\theta_\pm (y)\cdot
\delta(x-y)\, ,\nonumber\\
\relax [\theta_+ (x), \theta_- (y)]&=& 0\, ,
\end{eqnarray}
which correspond to two separated copies (one for each chirality) of
the Virasoro algebra, both with central charge $c=1$ in the quantum case.
We wish to mention
that this analysis corrects a statement in \cite{CD}
claiming the absence of the central extension
(the proof there furnished of the Poincar\'e invariance
remains valid because not affected by the presence
of the central term).
\vspace{0.2cm}
\noindent{\section{The $N=1$ supersymmetric FJ and dual models.}}
\par
The theory of a chiral boson $\phi$ and a chiral
fermion $\psi $ consists in the system of equations of motion
\begin{eqnarray}
&&\partial_-\partial_1\phi = 0\, ,\quad\quad \partial_-\psi = 0\, .
\label{eqmo}
\end{eqnarray}
The above system of equations can be recovered from a
single superfield equation where both spacetime
coordinates $x,t$ (or more commonly the lightcone
coordinates $z_\pm$) have been supersymmetrized. However
one can easily realize that such an equation cannot
be derived from a $2D$ manifestly supersymmetric action
principle \cite{IT}. It is neverthless possible to make use
of a superaction principle where only one coordinate has
been supersymmetrized, while the other has
been kept ordinary. Such procedure has been employed in
\cite{PST} to define the super-Siegel model \cite{Sie}
in light-cone coordinates (one made supersymmetric). Here
we adopt the point of view of leaving the time variable
$t$ unchanged while supersymmetrizing the space coordinate
$x$ (now denoted as $X\equiv x, \theta$, with $\theta$ a
Grassmann variable). This approach is quite natural when
dealing with hamiltonian systems which single out the
time coordinate and more suitable for application to
supersymmetric dual models; indeed one can obtain them from a
single dynamics instead of being obliged to introduce two
separate dynamics, one for each chirality.\par
In our conventions $X\equiv x,\theta$ and
the $N=1$ supersymmetric derivative
is
\begin{eqnarray}
D&=& {\partial\over\partial \theta} +i\theta\partial_1\, , \quad\quad
\quad D^2 = i\partial_1\, .
\end{eqnarray}
We introduce the superfield $\Phi (X,t)$:
\begin{eqnarray}
\Phi(X,t) &=& \phi(x) + \theta \psi(x)
\, .
\end{eqnarray}
The $N=1$ supersymmetric action $S$ is given by
\begin{eqnarray}
S&=& -i\int dXdt (\partial_0\Phi-\partial_1\Phi ) D\Phi
\, ,
\label{supera}
\end{eqnarray}
which implies for $\Phi$ the equation of motion, equivalent to (\ref{eqmo}),
\begin{eqnarray}
\partial_- D \Phi &=& 0\, .
\end{eqnarray}
In components the supersymmetric lagrangian ${\cal L}_{SFJ}$ is
\begin{eqnarray}
{\cal L}_{SFJ} &=& \partial_0\phi\partial_1\phi -
(\partial_1\phi)^2 +i \psi
(\partial_0\psi-\partial_1\psi )
\end{eqnarray}
and the global supersymmetry transformation is given by
\begin{eqnarray}
&&\delta\phi = \epsilon \psi\, , \quad\quad
\delta\psi = i \epsilon \partial_1\phi
\, .
\end{eqnarray}
The hamiltonian analysis of the above action can
be straightforwardly done. At first
we compute the supermomentum $\Pi$
\begin{eqnarray}
\label{103}
\Pi +iD\Phi&=&\Omega \approx 0 \, ,
\end{eqnarray}
which represents one primary superconstraint ($\Omega $)
of second class.\par
The total hamiltonian is
\begin{eqnarray}
\label{104}
H_T &=& H_c+\mu \Omega \, ,
\end{eqnarray}
where $\mu$ is an arbitrary multiplier and
\begin{eqnarray}
\label{105}
H_c &=& -i\int dX \partial _1\Phi D\Phi\, .
\end{eqnarray}
The following generalized Poisson algebra
is satisfied by the superfields
\begin{eqnarray}
\label{106}
\{\Phi (X), \Pi (Y)\}&=& \delta (X,Y)\equiv
\delta (x-y) (\theta_x -\theta_y )
\end{eqnarray}
(all the other Poisson brackets being zero).\par
The constraints (\ref{103}) are found to be second class
\begin{eqnarray}
\label{107}
&&\Delta (X,Y) =_{def}\{\Omega (X), \Omega (Y)\}= 2iD_X \delta (X,Y)
\label{delta}
\end{eqnarray}
and they do not generate secondary constraints.\par
In order to find the correct equations of motion we
construct the reduced phase space structure following
Dirac \cite{Dir}. The inverse of (\ref{delta}) is
\begin{eqnarray}
\Delta^{-1} (X,Y) &=& -{1\over 2} D_X{\partial^{-1}}_x
\delta (X,Y)
\, .
\end{eqnarray}
The Dirac brackets can now be computed for
any couple of superfields $A(X)$, $B(X)$:
\begin{eqnarray}
\{ A(X), B(Y)\}_D &=& \{A(X), B(Y)\} - \int dZdW
\{A(X), \Omega (Z)\} \Delta^{-1} (Z,W)\{\Omega(W), B(Y)\}
\, .
\nonumber\\
&&
\end{eqnarray}
In particular we obtain, as fundamental
algebraic relation
\begin{eqnarray}
\label{108}
\{\Phi (X),\Phi (Y)\}_D&=& {1\over 2} D_X\partial ^{-1}_x
\delta (X,Y)\, .
\end{eqnarray}
Finally we get the classical hamiltonian equations of motion
\begin{eqnarray}
\label{109}
&&\partial _0\Phi (X)=\{\Phi (X),H\}_D= \partial _1\Phi (X)
\, .
\end{eqnarray}
\par
At the quantum level the equal-time anti-commutator in
superfield notation is
\begin{eqnarray}
\relax \{ D_X\Phi (X), D_Y\Phi (Y) \}_t &=& -
{1\over 2} D_Y \delta (X,Y)\, ,
\end{eqnarray}
which in components reads
\begin{eqnarray}
&&\relax [\partial_x \phi(x), \partial_y\phi (y)]
= -{i\over 2} \partial_y\delta(x-y)\, ,
\quad\quad
\{ \psi (x), \psi (y) \} = {1\over 2} \delta (x-y)\, .
\end{eqnarray}
\par
The supercurrent
\begin{eqnarray}
\vartheta^{00} (X) &=& i : \partial_1\Phi D\Phi : \quad =
q(x) + \theta l(x)=\nonumber\\
&=& i:(\partial_1\phi)\cdot \psi: +\theta (-:(\partial_1\phi)^2 :
+i :\partial_1\psi\cdot\psi :)
\end{eqnarray}
gives the $c= {3\over 2}$ $N=1$ superVirasoro algebra. \par
The N\"other conserved charges which generates a constrained
($P^0=P^1$) version
of the $2D$ superPoincar\'e algebra are
\begin{eqnarray}
&& P=P^0=P^1= \int dx l(x)\, ,\quad\quad M = \int dx (x+t) l(x)\, ,\quad\quad
Q = \int dx q(x)\, .
\label{superham}
\end{eqnarray}
The non-vanishing (anti)-commutators are
\begin{eqnarray}
&&\relax [ M, P] = i P\, ,\quad\quad [ M, Q] = i{1\over 2} Q\,, \quad\quad
\{ Q,Q \} = {1\over 2} P\, .
\end{eqnarray}
Just like its bosonic counterpart,
the supersymmetric action (\ref{supera})
is classically invariant under a class of
$\lambda$-parametrized $1D$-superdiffeomorphisms transformations:
\begin{eqnarray}
\delta \Phi (X,t) &=& \epsilon (z_+,\theta) \partial_1\Phi
-{i\over 2} D\epsilon (z_+,\theta)\cdot D\Phi - {1\over 2} \lambda
\partial_1\epsilon (z_+,\theta)
\, ,
\label{sdiff}
\end{eqnarray}
where the infinitesimal variation $\epsilon$ is function
of $z_+$, $\theta$ only ($\partial_-\epsilon = 0$).
By performing the same analysis as in the bosonic case
we find the fermionic
supercurrent $\vartheta_\lambda (X)$ which
generates the transformations above (\ref{sdiff}):
\begin{eqnarray}
\vartheta_\lambda (X) &=& i : \partial_1\Phi D\Phi : -i \lambda
\partial_1 D\Phi
\, .
\end{eqnarray}
The (anti)-commutation
relations satisfied by $\vartheta_\lambda$ produce
the $N=1$ superVirasoro algebra
with central extension $ c= {3\over 2 } - 6 i \lambda^2$:
\begin{eqnarray}
\{ \vartheta_\lambda (X), \vartheta_\lambda (Y)\} &=&
-{1\over 8} (i + 4 \lambda^2) D_y(\partial_y )^2 \delta
(X,Y)-{3\over 2} i \vartheta_\lambda (Y) \partial_y \delta (X,Y)-\nonumber\\
&&
-{1\over 2} D\vartheta_\lambda (Y)\cdot D_Y \delta (X,Y) -i \partial_y
\vartheta_\lambda (Y)\cdot \delta (X,Y)
\, .
\end{eqnarray}
The conserved charges are computed as in the bosonic case
and the non-anomalous $1D$-superdiffeomorphisms invariance
is recovered for the value $\lambda = \sqrt{ -i\over 4}$.\par
We point out that, for
the real-valued component fields $\phi$, $\psi$, the equations
of motion (\ref{eqmo}) can also be obtained from the hamiltonian
\begin{eqnarray}
H&=& -\int dx \left( :(\partial_1\phi)^2 : + i :
\partial_1\psi \cdot \psi :\right)
\label{hambis}
\end{eqnarray}
with (anti)-commutation relations
\begin{eqnarray}
&&\relax [\partial_x \phi(x), \partial_y\phi (y)]
= -{i\over 2} \partial_y\delta(x-y)\, , \quad\quad
\{ \psi (x), \psi (y) \} = -{1\over 2} \delta (x-y)\, .
\end{eqnarray}
The above formulas are recovered from the previous ones
after setting $\psi \mapsto i \psi$.\par
The hamiltonian $H$ of eq. (\ref{hambis}) however, unlike
$P^0$ in eq. (\ref{superham}), is not
supersymmetric because the fermionic
hermitian operator $Q= i \sqrt{2} :\partial_1\phi\psi :$ in this case
leads to $\{Q,Q\} = - H$. Notice the presence of the
``wrong" minus sign. When dealing with extended
supersymmetries or supersymmetric dual models one has to be very careful
in picking up the ``correct" supersymmetric hamiltonian.\par
We devote the last part of this section to discuss the
superextension of the Tseytlin model. As in the bosonic
case the supersymmetric dual model can be decomposed
into two independent, respectively chiral and
antichiral, $N=1$ FJ models. The supersymmetric
action which coincides with a dimensional reduction of
the $4$-dimensional super-Schwarz-Sen model is up to a normalizing factor
\cite{CD}
\begin{eqnarray}
\label{100}
S = \int d^2x \left[ \partial_{0} \phi \partial_{1}
{\tilde \phi}+
\partial_{0} {\tilde\phi } \partial_{1} \phi -
(\partial_{1} \phi)^2 - (\partial_{1} {\tilde\phi})^2
\right. \nonumber\\
\left. + i\psi \partial _0\psi
+ i{\tilde \psi}\partial _0{\tilde \psi} -i\psi
\partial _1 {\tilde\psi}
-i{\tilde\psi}
\partial _1 \psi
\right]\,\,\, ,
\end{eqnarray}
and can be rewritten in superfield notations as
\begin{eqnarray}
S &=& -i \int dX dt \left( \partial_0 \Phi_+
-\partial_1\Phi_+ )D\Phi_+ -(\partial_0 \Phi_-
+\partial_1\Phi_- )D\Phi_-\right)
\, ,
\end{eqnarray}
where
\begin{eqnarray}
&&\Phi_+ = \phi_+ + \theta \psi_+\, , \quad\quad
\Phi_- = \phi_- + i\theta \psi_-\,
\end{eqnarray}
are chiral (antichiral) superfields and
\begin{eqnarray}
&& \phi_\pm = {1\over \sqrt{2}} (\phi\pm
{\tilde \phi})\, ,\quad\quad
\psi_\pm = {1\over \sqrt{2}} (\psi\pm {\tilde\psi})
\, .
\end{eqnarray}
Notice that the presence of an extra ``$i$" in
the decomposition of $\Phi_-$ is in order to make
the hamiltonian for the antichiral sector supersymmetric,
as explained above.\par
The duality invariance corresponds to the exchange
$\Phi_\pm \leftrightarrow \pm \Phi_\pm $.\par
The N\"other analysis is recovered from the results
of the $N=1$ FJ model. The anticommutation relations
are
\begin{eqnarray}
\relax [ D\Phi_\pm (X), D\Phi_\pm (Y) ] &=& \mp {1\over 2}
D_Y\delta (X,Y) , \quad\quad [ D\Phi_+ (X), D\Phi_- (Y) ] =0\, .
\end{eqnarray}
Two independent $c={3\over 2} $ superVirasoro algebras
result from the supercurrents $\vartheta_\pm $:
\begin{eqnarray}
&& \vartheta_\pm = \gamma_\pm :\partial_1\Phi_\pm\cdot D\Phi_\pm :\quad =
q_\pm -i \gamma_\pm \theta l_\pm
\, ,
\end{eqnarray}
where $\gamma_+ = i$ and $\gamma_- = -1$.\par
Let us introduce the currents
\begin{eqnarray}
&&\vartheta^{00} = l_++l_-, \quad\quad
\vartheta^{01} = l_+-l_-,\quad\quad q^{01} = q_++q_-,
\quad\quad q^{02} = q_+-q_-\, .
\end{eqnarray}
The superPoincar\'e algebra is realized by the bosonic
conserved charges
\begin{eqnarray}
&& P^0 = \int dx \vartheta^{00},\quad\quad P^1 =\int
dx \vartheta^{01},\quad\quad
M = \int dx (x \vartheta^{00} + t \vartheta^{01})
\, ,
\end{eqnarray}
together with the supercharges
\begin{eqnarray}
Q^1 =\int dx q^{01}, \quad\quad Q^2 =\int dx q^{02}
\, .
\end{eqnarray}
As a quantum mechanical system, the
``$N=1$" Tseytlin model is globally $N=2$ supersymmetric
since $Q^{1,2}$ satisfy
\begin{eqnarray}
\{Q^1,Q^1\}=\{Q^2,Q^2\}= H,\quad\quad \{Q^1,Q^2\} = 0
\, ,
\end{eqnarray}
where $H$ is the hamiltonian.\par
Explicitly $Q^1$, $Q^2$ generate two
supersymmetry transformations \begin{eqnarray}
&&
\delta_1\phi_\pm = {\epsilon_1\over 2} \psi_\pm, \quad\quad \delta_1\psi_\pm
= \pm
{i\over 2}\epsilon_1\partial_1\phi_\pm ; \quad\quad
\delta_2\phi_\pm =\pm {\epsilon_2\over 2} \psi_\pm,
\quad\quad
\delta_2\psi_\pm = {i\over 2}\epsilon_2\partial_1\phi_\pm
\, .
\end{eqnarray}
The model is superconformally invariant and
non-anomalous even in the quantum case as a
trivial consequence of the $1D$ superdiffeomorphisms
invariance of the $N=1$ FJ theory.
\vspace{0.2cm}
\noindent{\section{The $N=2$ extensions.}}
\par
The $N=2$ extensions of the FJ and the (globally $N=4$ invariant)
Tseytlin model can be constructed by
mimicking the previous constructions in a
manifest $N=2$ superfield formulation. Since the analysis proceeds
as before we limit ourselves to write the results.\\
When dealing with $N=2$ superfields we have at first to
establish if real or constrained (anti)-chiral
superfields are employed. It turns out
that chiral-antichiral superfields make the job.\par
The theories will be defined by leaving the time $t$
ordinary while the space coordinate will be $N=2$
supersymmetrized with the introduction of $\theta,
{\overline\theta}$ Grassmann variables. Our $N=2$
conventions (see also \cite{IT}) are as follows. The
fermionic derivatives $D, {\overline D}$ are
\begin{eqnarray}
&& D= {\partial\over\partial\theta}-{i\over 2}
{\overline\theta}\partial_1\, , \quad\quad
{\overline D} = {\partial\over\partial{\overline\theta}}
-{i\over 2} \theta\partial_1\, .
\end{eqnarray}
They satisfy the equations
\begin{eqnarray}
&& D^2={\overline D}^2 = i\partial_1\, , \quad\quad
\{ D, {\overline D}\} = 0\, .
\end{eqnarray}
$N=2$ chiral ($\Phi$) and antichiral
(${\overline\Phi}$) superfields satisfy
the constraints
\begin{eqnarray}
&& D\Phi=0 \, ,\quad\quad {\overline D}{\overline\Phi}=0
\, .
\end{eqnarray}
In components we have
\begin{eqnarray}
&&\Phi = \phi +{\overline\theta} \psi + {i\over 2}
\theta{\overline\theta}\partial_1\phi\, , \quad\quad
{\overline\Phi}= {\overline\phi} + \theta{\overline\psi}
-{i\over 2} \theta{\overline\theta} \partial_1{\overline\phi}\, .
\end{eqnarray}
The $N=2$-invariant action for the FJ model is
given by the sum of two pieces involving
separately $N=2$ chiral and antichiral superfields:
\begin{eqnarray}
S&=& i \int dt dX_L \left(\partial_0\Phi -\partial_1\Phi
) D{\overline\Phi}\right) + i \int dt dX_R \left(
(\partial_0{\overline\Phi}-\partial_1{\overline\Phi})
{\overline D}\Phi\right)
\, ,
\end{eqnarray}
where $dX_L\equiv dxd\theta$, $dX_R\equiv dx d{\overline
\theta}$ denote integration over the chiral (antichiral)
variables.\par
In components the lagrangian ${\cal L}$ is
\begin{eqnarray}
{\cal L} &=& \partial_0\phi \partial_1{\overline\phi}
+\partial_0{\overline\phi}\partial_1\phi - 2\partial_1\phi
\partial_1{\overline\phi} +
i(\partial_0\psi-\partial_1\psi){\overline \psi}
+ i(\partial_0{\overline\psi} -\partial_1{\overline\psi})
\psi
\, .
\end{eqnarray}
The reality condition sets ${\phi}^\dagger = {\overline
\phi}$, ${\psi}^\dagger = {\overline\psi}$.\par
At the level of the equations of motion we obtain two
copies of the supersimmetric FJ equations. Indeed
\begin{eqnarray}
&&\partial_1\partial_-\phi =\partial_1\partial_-{\overline\phi}
=0\, , \quad\quad\partial_-\psi=\partial_-{\overline\psi} =0 \, .
\end{eqnarray}
The (anti)-commutation relations which define the
hamiltonian dynamics are given by
\begin{eqnarray}
&&\relax [ \partial_1{\overline \phi}(x), \partial_1\phi
(y) ] = {i\over 2} \partial_y\delta (x-y)\, , \quad\quad
\{{\overline{\psi}} (x), \psi (y) \} = {1\over 2} \delta
(x-y)\, .
\end{eqnarray}
and vanishing otherwise.\par
In manifest $N=2$ superfield notation they are written as
\begin{eqnarray}
\{ D{\overline\Phi} (X), {\overline D}\Phi(Y) \}
&=& {1\over 2} D_X {\overline D}_Y \delta (X,Y)
\, ,
\end{eqnarray}
here $\delta (X,Y) = \delta (x-y) (\theta_x-\theta_y)
({\overline\theta}_x -{\overline\theta}_y) $ is the
$N=2$ supersymmetric delta-function.\par
The hamiltonian $H$ is given by
\begin{eqnarray}
H &=& \int dt dX : D{\overline \Phi}\cdot{\overline D}\Phi:
\quad .
\end{eqnarray}
The supercurrent $ J(X) = : D{\overline \Phi}\cdot{\overline D}\Phi: =
j + \theta q + {\overline \theta} {\overline q}
+ \theta{\overline\theta} l $,
where
\begin{eqnarray}
&&j(x)= :\psi{\overline\psi}:\, , \quad\quad
q(x) = i:\partial_1\phi\cdot
{\overline\psi} : \, , \quad\quad
{\overline q} (x) = -i: \partial_1{\overline\phi}
\cdot\psi :\, ,\nonumber\\
&&l(x) = :\partial_1\phi\cdot\partial_1{\overline\phi}: +
{i\over 2}:{\partial_1\psi\cdot{\overline\psi}}:+{i\over 2}
:{\partial_1{\overline\psi}}\cdot\psi :
\end{eqnarray}
satisfy the $N=2$ superVirasoro algebra with central
charge $c=3$ in the quantum case (the standard
OPE conventions are recovered by rescaling
$l\mapsto {\tilde l} = - 2 i l,\quad
(q,{\overline q}) \mapsto ({\tilde q}, {\tilde{\overline q}}) =
\sqrt{-8 i} (q, {\overline q}), \quad
j \mapsto {\tilde j} = 2 j$):
\begin{eqnarray}
\relax [ l(x), l(y) ] &=& -{3\over 48}{\partial_y}^3\delta(x-y) + i
l(y)\partial_y\delta(x-y)
+{i\over 2} \partial_y l(y)\cdot \delta (x-y)\, ,\nonumber\\
\relax [l(x), q(y)] &=& {3 i\over 4} q(y) \partial_y\delta
(x-y) +{i\over 2} \partial_y q(y)\cdot \delta (x-y)\, ,\nonumber\\
\relax[l(x), {\overline q}(y)] &=& {3 i\over 4}
{\overline q}(y) \partial_y\delta(x-y)+{i\over 2}
\partial_y {\overline q}\cdot (y)\delta(x-y)\, , \nonumber\\
\relax [ l(x), j(y) ] &=& {i\over 2} j(y) \partial_y\delta
(x-y) +{i\over 2}\partial_yj(y)\cdot\delta (x-y)\, ,\nonumber\\
\{ q(x), {\overline q}(y)\} &=& {i\over 8} {\partial_y}^2
\delta (x-y) +{i\over 2} j(y)\partial_y\delta (x-y) +
({1\over 2} l(y) +{i\over 4} \partial_y j(y))\cdot \delta (x-y)\, ,\nonumber\\
\relax [j(x), q(y) ] &=& {1\over 2} q(y) \delta (x-y)\, ,
\quad\quad [ j(x) ,
{\overline q}(y)] = -{1\over 2} {\overline q}(y)\delta (x-y)\, ,\nonumber\\
\relax[j(x), j(y) ] &=& {1\over 4} \partial_y\delta (x-y)
\end{eqnarray}
and vanishing otherwise.\par
The above currents are the building blocks to
construct the $N=2$ superPoincar\'e
generators just like in the previous cases. \par
In particular the $N=2$ global hermitian charges are
\begin{eqnarray}
Q_1 =_{def} \int dx \left( q(x) +{\overline q} (x)\right)
\, ,
\quad && \quad Q_2 =_{def} i \int dx \left( q(x) -
{\overline q}(x) \right)\, ,\nonumber
\end{eqnarray}
which satisfy $\{Q_1, Q_2\} = 0$, $ \{ Q_1,Q_1\} =\{Q_2,Q_2\} = H$.\par
The (non-anomalous) invariance under $1$-dimensional
$N=2$ diffeomorphisms is
implied in the quantum case by the existence of modified currents
$u_{\lambda,{\overline\lambda}}$ ($u$ denotes either $j$, $q$,
${\overline q}$ or $l$),
which satisfy an $N=2$ superVirasoro where all central
charges are vanishing.
The modified currents cannot be accomodated into an
$N=2$ superfield formalism and we are obliged to write
them in components. We get
\begin{eqnarray}
j_{\lambda ,{\overline\lambda}}(x)&=&:\psi{\overline\psi}:
-i\lambda{\partial_1\phi} -i{\overline\lambda}\partial_1{\overline\phi}\, ,
\nonumber\\
q_{\lambda ,{\overline\lambda}}(x) &=& i:\partial_1\phi\cdot
{\overline \psi} : - i{\overline \lambda} \partial_1
{\overline \psi}\, , \quad\quad
{\overline q}_{\lambda,{\overline\lambda}} (x) = -i:
\partial_1{\overline\phi}
\cdot\psi : - i {\lambda} \partial_1\psi
\, ,\nonumber\\
l_{\lambda,{\overline\lambda}}(x) &=& :\partial_1\phi\cdot\partial_1
{\overline\phi}: +
{i\over 2}:{\partial_1\psi\cdot{\overline\psi}}:+{i\over 2}
:{\partial_1{\overline\psi}}\cdot\psi :
+{\lambda\over 2}{\partial_1}^2 \phi -{{\overline\lambda}\over 2}
{\partial_1}^2 {\overline\phi}
\, .
\end{eqnarray}
If $\lambda$, ${\overline\lambda}$ are chosen in such a way
that $\lambda\cdot {\overline\lambda} = -{i\over 4}$
then all central charges are vanishing; as an example
in particular
\begin{eqnarray}
\relax [j_{\lambda,{\overline\lambda}}
(x), j_{\lambda,{\overline\lambda}} (y) ] &=& -i
(\lambda{\overline\lambda} + {i\over 4})\partial_y\delta (x-y)
\, .
\end{eqnarray}
We notice that the modified current $l_{\lambda,{\overline\lambda}}$ is
no longer
hermitian if the above constraint is taken into
account (however on abstract level the closed algebraic
structure it satisfies is compatible
with a hermitian condition). This feature is not specific
of $N=2$ but is already present in the bosonic and $N=1$ cases.\par
The modified currents $u_{\lambda,{\overline\lambda}}$
are the generators of the $N=2$
$1$-dimensional diffeomorphisms
invariances, the infinitesimal transformation being
given by the commutators with
\begin{eqnarray}
&&\int dx \epsilon_u (z_+) u_{\lambda,{\overline\lambda}} (x)\, .
\end{eqnarray}
The dualized version of the $N=2$ FJ model is now easily
constructed by introducing a second set of superfields
(antichiral in spacetime). The action is given by
\begin{eqnarray}
S&=& 2 i \int dt dX_L \left(\partial_-\Phi_+
\cdot D{{\overline\Phi}_+} -\partial_+\Phi_-\cdot
D{\overline \Phi}_-\right) + 2 i \int dt dX_R \left(
\partial_-{{\overline\Phi}_+}\cdot
{\overline D}\Phi_+ -\partial_+{\overline\Phi}_-\cdot
{\overline D}\Phi_- \right)\, .\nonumber\\
&&
\end{eqnarray}
The correct expansion for $\Phi_\pm$, ${\overline\Phi}_\pm$ in
hermitian component fields
which leads to the supersymmetric hamiltonian (see the remark
in section $3$) is given by:
\begin{eqnarray}
&&
\Phi_\pm = \phi_\pm -i \gamma_\pm {\overline\theta}
\psi_\pm + {i\over 2}\theta{\overline\theta} \partial_1\phi_\pm\, ,
\quad\quad
{\overline\Phi}_{\pm} ={\overline\phi}_\pm -i\gamma_\pm
\theta {\overline\psi}_\pm -{i\over 2}\theta{\overline\theta}
\partial_1{\overline \phi}_\pm
\end{eqnarray}
(here again $\gamma_+ = i$, $\gamma_- = -1$).\par
The lagrangian in components reads
\begin{eqnarray}
{\cal L} &=& 2( \partial_-\phi_+\cdot\partial_1{\overline\phi}_+ +
\partial_-{\overline\phi}_+ \cdot\partial_1\phi_+
-\partial_+\phi_-\cdot \partial_1{\overline\phi}_-
-\partial_+{\overline\phi}_-\cdot\partial_1\phi_-+
\nonumber\\
&& + i \partial_-\psi_+\cdot{\overline\psi}_+ + i
\partial_-{\overline\psi}_+\cdot\psi_+
+i\partial_+\psi_-\cdot{\overline\psi}_- +
i\partial_+{\overline\psi}_-\cdot\psi_-)
\, .
\end{eqnarray}
The non-vanishing (anti)-commutators are
\begin{eqnarray}
&&\relax [ \partial_x{\overline\phi}_\pm (x), \partial_y
{\phi}_{\pm}(y) ] = \pm {i\over 2} \partial_y\delta (x-y)
\, , \quad\quad
\{ {\overline\psi}_\pm (x), \psi_\pm(y) \} = {1\over 2}
\delta (x-y)
\, .
\end{eqnarray}
The conserved currents of the antichiral sector generate a
second $N=2$ superVirasoro algebra with (quantum) central charge $c=3$.\par
Following the same reasoning as in
the previous section we can construct
four global supercharges $Q^i$ ($i=1,...,4$)
which lead to a global $N=4$ supersymmetry ($\{Q^i, Q^i\}= H$ for any $i$ and
$\{Q^i, Q^j \} =0$ for $i\neq j$). The Coulomb gas
realization for the $N=2$ FJ model implies
the full $N=2$ superconformal invariance for the quantum dual model.
\vspace{0.2cm}
\noindent{\section{Conclusions.}}
\par
In this paper $N=1,2$ extensions of chiral and dual
models have been constructed and their symmetry properties
analyzed. In particular their relativistic
character was proven by computing global charges
which close the $2D$-superPoincar\'e algebra.
The invariances under $1D$-superdiffeomorphisms and
respectively superconformal transformations were
furthermore investigated. It was shown that, due to $N=1,2$
Coulomb gas results, modified currents exist
which lead to non-anomalous quantum theories.\par
The present work was mainly motivated to
establish an algebraic framework for dimensional reductions of
higher-dimensional
supersymmetric dual models. The algebraic structures found
however have an interest in their own and further
investigations look promising.
Currently under study, e.g. the $N=4$ supersymmetric
extensions seem related to non-abelian
structures leading to a new $N=4$ realization of the Coulomb gas.\par
Another topic which deserves to be studied, as
suggested in \cite{Sen}, is the coupling of the
above theories with $2D$ supergravity, with special attention to the
presence of anomalies.
\vspace{0.2cm}
|
1,314,259,993,015 | arxiv | \section{Introduction}
At one time, radio sources offered the only readily available
means of locating galaxies at large redshifts ($z > 1$), and were
thus studied as a window on the early history of galaxy evolution.
Radio galaxies obey a tight near--infrared Hubble ($K$--$z$)
relation and are frequently associated with rich cluster environments.
This suggests that there is an evolutionary sequence linking high-redshift
radio galaxies to low-redshift giant ellipticals and cD galaxies.
The discovery that many high-redshift radio galaxies have elongated,
complex UV continuum structures aligned with the radio source
axis (McCarthy et al.\ 1987; Chambers et al.\ 1987) suggested that the
active nucleus might affect the UV morphology and possibly even
the evolutionary history of the host galaxy. It was believed that
some radio galaxies might be true ``protogalaxies'' forming the bulk
of their stars via some process induced by the radio jets.
However, later studies have shown that in many cases the aligned
UV continuum arises largely from scattered AGN emission
(Di Serego Alighieri et al.\ 1989) and/or nebular continuum
emission (Dickson et al.\ 1995).
The spectacular, complex structures seen in optical WFPC2 images of
3CR radio galaxies by (e.g.) Best et al.\ (1997) are, in many
cases, AGN--related ``ephemera'' surrounding a more normal host galaxy.
However, the brightness of this aligned, blue light makes it difficult to
study the properties of the underlying stellar component when observing
at optical wavelengths. Observers have therefore turned to the
near--IR, where the AGN--related emission is fainter and the stellar
component brighter. Rigler et al.\ (1992), Dickinson et al.\ (1994)
and Best et al.\ (1998), among others, have shown that some 3CR radio
galaxies are rounder and more symmetric when observed in the near--IR,
suggesting the presence of relatively ``normal'' host galaxies underlying
the UV--bright, aligned continuum (but see also Eisenhardt \& Chokshi 1990).
But until now these ground--based studies have been limited by angular
resolution. Now, using NICMOS on board HST, we can for the first time
study the near--IR morphologies of high redshift galaxies with resolution
comparable to that of the pioneering WFPC2 studies of these same objects.
\begin{figure}
\plotfiddle{fig1.eps}{4.9in}{0}{75}{75}{-205}{-25}
\caption{WFPC2 and NICMOS images of four galaxies from our NICMOS imaging
sample, along with best--fitting models to the NICMOS host galaixes
and the NICMOS minus model residuals.}
\end{figure}
\section{Observations }
Our sample consists of 11 3CR radio galaxies at $0.8 < z < 1.8$,
imaged with NICMOS Camera 2, which provides diffraction limited
images at 1.6$\mu$m. We used bandpasses (F160W or F165M) which
avoid strong nebular emission lines, which could significantly
contaminate the fluxes and affect the observed morphologies.
Optical WFPC2 imaging is available for the whole sample: in many
cases we have unusually deep and often polarimetric WFPC2 data,
while in others archival data by Best et al.\ 1997 or other sources
were used. In most cases we also have extensive ground--based
supporting data (spectroscopy, polarimetry, etc.) from the
W.M. Keck Observatory and other facilities.
\begin{figure}
\plotone{fig2.eps}
\caption{Rest frame Gunn $r$ size--surface brightness relation for
local cluster ellipticals (small squares; J{\o}rgensen et al. 1995)
and for our NICMOS radio galaxies (lines). The lines connect the
effective radii for q$_{0}$ = 0 and q$_{0}$ = 0.5 cosmologies.
The dotted line shows the relation for constant galaxy luminosity,
as expected for ``standard candle'' galaxies.}
\end{figure}
\section{Discussion}
The NICMOS and WFPC2 images of four galaxies from this sample are shown
in Figure 1. We have fit PSF--convolved models to the NICMOS images
using a hybrid scheme which matches 1D surface brightness profiles and
2D PA~+~ellipticity information. The models and the NICMOS image
residuals after model subtraction are also shown in Figure 1.
In most cases, the NICMOS images show that the rest--frame optical
light from powerful 3CR radio galaxies at $z > 1$ is rounder,
smoother, more symmetric and centrally concentrated than that observed
at rest--frame UV wavelengths. The complex, aligned structures seen
in WFPC2 images are generally much less pronounced in the near--IR,
although in several cases (e.g. 3C 280, 3C 266, 3C 368) the highest
surface brightness regions of the aligned components can still be
detected. In several cases, the near--IR surface brightness peaks
at the position of a local {\it minimum} in the WFPC2 images,
suggesting the effects of dust lanes affecting the near--UV
morphologies. A few galaxies (e.g., 3C~265) appear to have nuclear
point sources in the IR, possibly showing the ``unveiled'' AGN.
Overall, the gross morphologies and surface brightness profiles
of most 3CR hosts are consistent with their being high luminosity
giant elliptical galaxies, already structurally mature. This may be
true as early as $z = 1.8$, although at that redshift 3C~239 appears
to have ``ragged edges'' perhaps suggesting that it is in the process
of accreting material through mergers. However, the most distant
galaxy in our sample, 3C~256 at $z = 1.82$, is radically different
than the others. It is elongated, aligned, diffuse, and underluminous,
and thus may be the exceptional example of a young radio galaxy early
in the stages of its formation (see also Eisenhardt \& Dickinson 1992,
Simpson et al.\ 1999).
In Figure 2, we plot surface brightness vs.\ effective radius
(the ``Kormendy relation'') for 6 galaxies which are well fit
by $R^{1/4}$--law models, converting the NICMOS photometry
(rest--frame $\lambda_0$0.57 to 0.88$\mu$m for our sample)
to rest--frame Gunn~$r$ ($\lambda_0 0.65 \mu$m) for comparison
to nearby cluster ellipticals. The galaxies are physically smaller
than the largest and brightest giant cluster ellipticals at $z = 0$,
and have higher rest--frame surface brightnesses as would be
expected given nominal luminosity evolution. Most fall on the
locus of constant luminosity (see also Best et al.\ 1998),
as might be expected given the small $K$--$z$ scatter.
3C~239 at $z=1.78$ is significantly more luminous for
its size compared to the galaxies at $0.8 < z < 1.3$.
\acknowledgments
Support for this work was provided by NASA grant GO--07454.02--96A.
Thanks to Hy for getting us all into this radio galaxy mess!
|
1,314,259,993,016 | arxiv | \section{Introduction}
\label{introduction}
A very important step toward unveiling the origin of the sources of
UHECR is to identify the range of energies where cosmic rays become
mainly of extragalactic origin. For such extragalactic cosmic rays,
in the hypothesis of a proton dominated spectrum,
the propagation in the intergalactic medium induces three features in
the spectrum observed at the Earth: 1) the GZK feature \cite{GZK},
a suppression of the flux at energies in excess of $\sim 10^{20}$
eV, due to the photopion production interactions of cosmic rays
off the CMB photons; 2) a bump (\cite{HS85} - \cite{Stanev00}),
due to the accumulation of particles below the kinematic threshold
for photopion production. As was shown in \cite{BG88}, while the
bump is present in the calculated spectra of single sources, it almost
disappears
in the diffuse spectrum, because the position of the bump depends
upon the source redshift; 3) A dip (\cite{HS85} - \cite{BGG3}),
generated due to pair production, $p+\gamma_{\rm CMB}
\to p+e^++e^-$, where the target is provided by the CMB photons.
The detection of these features would be a definitive test of the
extragalactic origin of UHECR and of the fact that they are mainly
protons. Since the detection of the GZK feature requires very large
statistics of events and, as stressed above, the bump is almost
absent in the diffuse spectrum, at present the spectral feature that
can be detected more easily is the dip. As we show here (see also
\cite{BGG3} - \cite{BGG}), the dip (see Fig.~\ref{fig:mfactor}) is a
quite robust prediction of the calculation and we claim that in fact
it might have already been observed by the AGASA, Fly's Eye, HiRes
and Yakutsk experiments (see \cite{agasa} - \cite{NW} for the data).
However, the detectability of the dip as a feature of the
propagation of cosmic rays on cosmological scales would also imply
that the transition from galactic to extragalactic cosmic rays
should not take place at the {\it ankle}, as has been postulated
since the end of '70s, when this feature was discovered in the
Haverah Park data (see \cite{ankle} for the recent works).
The traditional explanation of the transition from galactic to
extragalactic cosmic rays invokes the intersection between a steep
($E^{-3.1}$) galactic spectrum and a flat ($E^{-\alpha}$ with
$\alpha=2-2.3$) extragalactic spectrum, at the {\it ankle},
located at an energy $E_a \approx 10^{19}$ eV and identified as a
flattening of the spectrum in the data of AGASA, HiRes and Yakutsk
detectors (see Fig. \ref{fig:dips} and \cite{DeMSt} for a general
discussion of the transition).
It is important to stress that in the dip scenario the predicted
spectrum flattens below and above the dip location (see
Fig~\ref{fig:mfactor}). The high energy flattening, at $E_a \approx
1\times 10^{19}$ eV, reproduces perfectly well the observed {\it
ankle}. The low energy flattening, at $E_{\rm cr} \approx 1\times
10^{18}$ eV, obtained for both cases of rectilinear \cite{BGG} and
diffusive propagation \cite{AB,Lem,AB1}, provides the transition to
a steeper galactic component. Note that this property, the
intersection of steep and flat spectral components, is the same for
both models of transition.
We demonstrate here (see also \cite{AB1}) that $E_{\rm cr}$ is
connected with the energy scale $E_{\rm eq} = 2.3 \times 10^{18}$
eV, where the rates of pair production and adiabatic energy losses
are equal. The visible transition from galactic to extragalactic
cosmic rays occurs at $E_{\rm tr} < E_{\rm cr}$, and this energy
coincides with the position of the {\em second knee} (Akeno -
$6\times 10^{17}$~eV, Fly's Eye - $4\times 10^{17}$~eV, HiRes -
$7\times 10^{17}$~eV and Yakutsk - $8\times 10^{17}$~eV). The
transition at the second knee was also proposed as a consequence of
the study of the propagation of galactic cosmic rays
(\cite{Biermann} - \cite{Dermer}).
The energy region around
$E_{\rm cr} \approx 10^{18}$ is also expected to correspond to a change
in the chemical composition, from a heavy galactic component to a
proton-dominated composition of UHECR. While HiRes \cite{mass-Hires},
HiRes-MIA \cite{Hi-Mia} and Yakutsk \cite{Glushkov00} data support
this prediction and Haverah-Park \cite{HP} data do not contradict it
at $E \gsim (1 - 2)\times 10^{18}$~eV, the Akeno \cite{mass-Akeno}
and Fly's Eye \cite{FE} data favor a mixed composition, dominated
by heavy nuclei (for a review see \cite{NW} and \cite{Watson04}).
In this paper we shall use the number density of UHECR sources estimated from
the small-scale clustering in the angular distribution of the arriving
particles \cite{ssc}. In \cite{ssa1,ssa2} it was found that
the observed small-scale clustering implies, in the case of rectilinear
propagation of particles, a spatial density of UHECR
$n_s \approx (1 - 3)\times 10^{-5}~{\rm Mpc}^{-3}$. Approximately
the same space density was found from a study of the small-scale clustering
for propagation in magnetic fields \cite{Sato}. This density is of the
same order of magnitude as the density of powerful AGN.
It is however worth stressing that the simulated AGASA spectra with
this source density are incompatible with the observed AGASA spectrum
at the $5\sigma$ level \cite{danny1}.
The potential of future detectors such as the Pierre
Auger Observatory to measure the source density from small scale
anisotropies has been discussed in \cite{danny2}.
From the phenomenological point of view,
there are some observational indications of AGN as the sources of
UHECR \cite{TT}, although the subject is still matter of much debate.
The paper is organized as follows: in Section \ref{dip} we discuss
the physics behind the formation of the dip, and compare our
predictions with the data of several experiments. We discuss also
the robustness of the prediction of the dip and some physical
phenomena which modify the shape of the dip. In
Section \ref{transition} we address the more specific issue of the
transition from galactic to extragalactic cosmic rays, stressing the
differences between the {\it dip scenario} and the {\it ankle
scenario}. In the Appendix we discuss the problems of acceleration relevant
for the dip scenario. We conclude in Section \ref{conclusions}.
\section{The dip}
\label{dip}
In this section we describe in detail the physical arguments that
explain the formation of the dip in the spectrum.
\label{sec:dip}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=8.0cm]{mfactor1.eps}
\end{center}
\caption{Modification factor for a power-law generation spectrum
with slope $\gamma_g=2.0$ and 2.7. The horizontal line $\eta=1$ corresponds to
adiabatic energy losses only. The curves $\eta_{ee}$ and $\eta_{tot}$
correspond respectively to the modification factor for adiabatic and pair
production energy losses and the modification factor where all losses
are taken into account.}
\label{fig:mfactor}
\end{figure}
In order to do this, we use the formalism of the {\it modification
factor}, first introduced in \cite{BG88} and defined as the ratio of
the spectrum $J_p(E)$ with all energy losses taken into account, and
the unmodified spectrum $J_p^{\rm unm}$, where only adiabatic energy
losses (red shift) are included: $\eta(E)=J_p(E)/J_p^{\rm unm}(E)$.
The spectrum $J_p(E)$ can be calculated from the conservation of the
number density of particles as
\begin{equation}
n_p(E,t_0)dE= \int_{t_{min}}^{t_0} dt Q_{\rm gen}(E_g,t)dE_g,
\label{conserv}
\end{equation}
where $n_p(E,t_0)$ is the space density of UHE protons at the
present time, $t_0$, $Q_{\rm gen}(E_g,t)$ is the generation rate per
comoving volume at cosmological time t, and $E_g(E,t)$ is the generation
energy at time t for a proton with energy E at $t=t_0$. This energy
is found from the loss equation $dE/dt=- b(E,t)$, where $b(E,t)$ is
the rate of energy losses at epoch t. The spectrum, Eq. (\ref{conserv}),
calculated for a power-law generation spectrum $\propto E^{-\gamma_g}$
and for a homogeneous distribution of sources, is called
{\em universal spectrum}. The important feature of the universal spectrum
is its independence of the mode of propagation: it is the same for
rectilinear propagation and propagation in arbitrary magnetic fields. This
property of the universal spectrum is guaranteed by the propagation theorem
\cite{AB}, according to which the spectra do not depend on the
propagation mode if the distance between sources is less than any
propagation length, e.g. energy attenuation or diffusion length.
For homogeneous distribution of the sources with vanishing distance
between them the propagation theorem is obviously fulfilled.
The generation rate $Q_{\rm gen}(E,t)$ might include the cosmological
evolution of the sources. In the results presented in this section,
we shall not include it in the calculations for two reasons:
(i) The evolution involves at least two free parameters, $m$ and
$z_{\rm max}$, where $m$ is the exponent in the evolution rate
$(1+z)^m$, and $z_{\rm max}$ is the maximum redshift
up to which evolution takes place. This makes the fit to the data
more arbitrary. (ii) Evolution is a very model-dependent phenomenon, and
as such we will discuss it later in Section~ \ref{evolution}, regarding
it as an uncertainty in the predictions.
Since the injection spectrum $E^{-\gamma_g}$ enters both the numerator
and the denominator of $\eta (E)$, one may expect that the
modification factor depends weakly on $\gamma_g$ and numerical
calculations confirm it.
In Fig. \ref{fig:mfactor} we plot the modification factor as a function
of energy for two slopes of the injection spectrum, $\gamma_g=2.0$
and $\gamma_g=2.7$. As expected, the differences are quite small.
\begin{figure}[ht]
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{mfactorAGASA.eps}
\end{minipage}
\hspace{2mm}
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{mfactorHires.eps}
\end{minipage}
\vspace{2mm}
\begin{minipage}{8cm}
\centering
\includegraphics[width=7.6cm,clip]{mfactorYak.eps}
\end{minipage}
\hspace{7mm}
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{mfactorAuger.eps}
\end{minipage}
\caption{\label{fig:dips} Predicted dip in comparison with AGASA, HiRes,
Yakutsk and Auger\protect\cite{Auger} data.}
\end{figure}
In Fig. \ref{fig:dips} we show the comparison of the modification
factor calculated for $\gamma_g=2.7$ with the observational
data of AGASA, HiRes, Yakutsk and Auger. The dip, i.e.
the modification factor $\eta_{ee}(E)$, is well confirmed by the data at
energy below $E \approx 4\times 10^{19}$~eV, above which the photopion
production dominates (see Fig.~\ref{fig:dips}). Fly's Eye data, not
shown here, confirm the dip equally well. Auger spectrum does not
contradict the high energy part of the dip, but needs continuation of
the spectrum to lower energies to test the dip as a whole.
At energy $E\geq 1\times 10^{19}$~eV the dip shows a flattening, which
explains the ankle, seen in the data in Fig.~\ref{fig:dips} at this
energy.
By definition the modification factor must be less than unity.
At energy $E < 1\times 10^{18}$~eV the modification factors of
AGASA-Akeno and HiRes exceed this bound. This signals the appearance
of another component, which is most probably given by galactic cosmic
rays. This is the first indication in favor of a transition from
extragalactic to galactic cosmic rays at $E \sim 1\times 10^{18}$~eV.
The best fit to the data provided by analytical calculations
corresponds to $\gamma_g=2.7$, though $2.6 \leq \gamma_g \leq 2.8$
provide an acceptable description of the data. The detailed
Monte-Carlo simulations of the spectra at $E \geq 3\times 10^{18}$~eV,
accounting for statistical errors in the energy determination of the
events, lead to a best fit injection spectrum with slope
$\gamma_g=2.6$ \cite{DBO}, in rather good agreement with the results
of analytical calculations. In addition to the statistical errors, the
simulations in \cite{DBO} may also account for a systematic error in
the energy determination. For most currently operating experiments
such error is of order 20\% and sometimes in excess of this.
The Monte-Carlo simulations of \cite{DBO} lead to the
conclusion that the alleged discrepancy between the AGASA
and HiRes experiments could be explained in terms of a combination of
statistical, systematic errors in the energy of the events, and
statistically limited number of events at energies above $\sim 10^{20}$
eV. This conclusion was strengthened in \cite{DBO1} where the same
authors showed that the realizations of the simulated propagation of UHECR
that are found to have 11 or more events at energy above $10^{20}$ eV
resemble very closely the AGASA data, although the {\it average}
spectrum has a pronounced GZK feature. In the same paper, the
authors also make an attempt to extract events at random directly from
the AGASA data and calculate the probability of obtaining the HiRes
spectrum by chance. In both cases the alleged discrepancy between the
two experiments is found to be statistically not very significant.
The shape of the dip can be used for the energy calibration of the
detectors. Shifting the energies by a factor $\lambda$ for each detector
in the energy interval of the dip ($(1 - 40)\times 10^{18}$~eV) one
can determine the minimum $\chi^2$ and from it deduce the value of
$\lambda$ that best fits the data. This procedure leads to
$\lambda_{\rm Ag}=0.9$, $\lambda_{\rm Hi}=1.2$ and $\lambda_{\rm
Ya}=0.75$ for AGASA, HiRes and Yakutsk detector, respectively.
It is worth noticing that the required correction factor is less than
unity for ground arrays and exceeds unity for fluorescence experiments.
It is impressive that after this shift (calibration by the dip) the spectra
of these three detectors agree very well (see Fig.~\ref{fig:AgHiYa}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=17.0cm]{AgHiYa.eps}
\end{center}
\caption{The spectra as measured by Akeno-AGASA, HiRes and Yakutsk
arrays (left panel) and after energy calibration by the dip (right panel).
}
\label{fig:AgHiYa}
\end{figure}
The results shown in Fig.~\ref{fig:AgHiYa} have already been presented
by some of us at different conferences starting from 2004 (see
e.g. \cite{Berez05}).
Recently AGASA collaboration \cite{Teshima06} has reduced the energies
by about 10\% as we predicted.
\subsection{Robustness and caveats}
\label{robust}
The prediction that a dip is present in the spectrum of extragalactic
cosmic rays has been obtained using what we called the universal
spectrum. Several assumptions have been used in the calculation (see
below) and we need to assess what could be their role in changing the
basic predictions of the dip scenario. We will prove that within some
reasonable limitations, the universality of the spectrum is not
substantially changed, in particular in the region of energy around
the dip, $1\times 10^{18}~{\rm eV} \leq E \leq 4\times 10^{19}$~eV.
The GZK feature, located at higher energies is expected to exhibit
noticeable deviations from the universal spectrum due to possible local
inhomogeneities in the source distribution (local overdensity or
deficit of the sources \cite{Blanton,BGG}), due to a possible acceleration
related cutoff and discreteness in the source distribution \cite{BGG}.
We start with listing the phenomena which may modify the universal
spectrum:
\begin{itemize}
\item[1)] Discreteness in the source distribution for the case
of propagation in magnetic fields;
\item[2)] Inhomogeneous source distribution;
\item[3)] Cosmological evolution of the sources;
\item[4)] Energetics corresponding to the best fit injection spectrum;
\item[5)] Chemical composition.
\end{itemize}
\subsubsection{Discreteness in the source distribution for propagation
in magnetic fields}
\label{discrete}
Here we discuss how the discreteness in the source distribution affects
the universal spectrum. We distinguish the weak magnetic field case,
when protons propagate with moderate deflections, and the case of
strong magnetic field when propagation is diffusive.
We shall start with the unrealistic case of a rectilinear propagation of
protons in the dip energy region, which is formally valid in the
absence of magnetic field. In Fig.~\ref{fig:discrete} the spectra are
calculated in the case of source distributions with different
distances between them, as indicated in the plot.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm]{discret.eps}
\end{center}
\caption{UHE proton spectra for rectilinear propagation from discrete
sources, located in the vertexes of the cubic grid with spacing
d=60, 40, 20, 10, 5 and 1~Mpc. The calculations are performed
for $z_{\rm max}=4$, $E_{\rm max}=1\times 10^{22}$~eV and $\gamma_g=2.7$.
}
\label{fig:discrete}
\end{figure}
One can see that while the shape of the GZK steepening is noticeably
modified as the distance $d$ between sources changes, the dip is very weakly
affected by it, and the spectrum remains universal (the curve with
d=1~Mpc in Fig.~\ref{fig:discrete}). This result directly follows
from the propagation theorem, because the only propagation length scale
in this problem, the energy attenuation length, exceeds 1000 Mpc in
the region of the dip, being thus considerably larger than the
distance between the sources used in the calculations.
It is clear that with a weak magnetic field the spectra remain universal,
like for rectilinear propagation, if particles are not deflected by large
angles, when propagating from a source to the observer. It is less trivial
that the spectra shown in Fig.~\ref{fig:discrete} should coincide with
the spectra obtained for the tangled trajectories (including
diffusion) for the same $d$ as in the case of rectilinear
propagation. This follows again from the propagation theorem which
states that for diffusive propagation the spectrum remains universal
provided that the attenuation length and the diffusion length are
larger than the mean separation $d$ between sources.
As pointed out in \cite{AB1}, at energy $\sim 1\times 10^{18}$ eV the
maximum distance $R_{\rm max}(E)$ from which particles can reach the
observer suffers a sharp increase (the so-called antiGZK effect
\cite{AB1}). The energy at which this takes place was found to be
independent of the choice of the diffusion coefficient
\cite{AB1}. This is illustrated in Fig. \ref{fig:diffuse} where it is
visible that for different choices of the diffusion coefficient
(Kolmogorov, Bohm and $D(E)\propto E^2$) the curves depart from
the universal spectrum at the same energy $E_{\rm cr} =1\times 10^{18}$~eV.
For rectilinear propagation (Fig.~\ref{fig:discrete}) one may see the
same value of $E_{\rm cr}$.
This universality of $E_{\rm cr}$ value may be explained recalling
the proximity of
$E_{\rm cr}$ to the energy $E_{\rm eq}$, where pair-production
energy losses are as fast as the adiabatic energy losses.
In terms of evolution of the generation energy with the look-back
time, when the proton energy reaches $E_{\rm eq}$, its energy
increases fast, the diffusion length grows even faster and
$R_{\rm max}(E_{\rm cr})$ becomes large in an almost discontinuous way
(see \cite{AB1} for analytical and numerical calculations).
In a semi-quantitative way, the connection between $E_{\rm cr}$
and $E_{\rm eq}$ can be expressed as
$E_{\rm cr}=E_{\rm eq}/(1+z_{\rm eff})^2$, where $z_{\rm eff}$ is an
effective redshift of the sources
contributing to the flux of cosmic rays at energy $\sim E_{\rm cr}$.
A simplified analytical estimate for $\gamma_g=2.6 - 2.8$ gives
$1+z_{\rm eff} \approx 1.5$ and hence $E_{\rm cr} \approx 1\times 10^{18}$ eV.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm]{dip-diff.eps}
\end{center}
\caption{Diffusive energy spectrum for $B_c=1$~nG, $l_c=1$~Mpc,
$d=50$~Mpc and $\gamma_g=2.7$ for the Bohm, Kolmogorov and
$D(E)\propto E^2$ diffusion. The dash-dotted curve shows the
universal spectrum. The data of Akeno-AGASA are also shown. }
\label{fig:diffuse}
\end{figure}
While the position $E_{\rm cr}$ and the high energy behavior of the
'cutoff' (in fact in terms of $J(E)$ this appears as a flattening of
the spectrum) are determined by an increase of energy losses, there
is a more evident spectral steepening and cutoff at lower energies,
caused by the magnetic horizon. In a very rough approximation,
particles cannot reach the observer from distances larger than
$R_{\rm hor}(E) \sim \sqrt{D(E)t_0}$. When $E$ is small enough, so
that $R_{\rm hor}(E)$ is smaller than the distance $d$ to the nearby
sources, the spectrum develops an exponential cutoff. At low
energies, where energy losses of protons are only adiabatic and
diffusion coefficient is $D(E) \propto E^{\alpha}$, the magnetic
horizon can be calculated (see \cite{AB1}) as \begin{equation} R_{\rm
hor}(E)=2\left (\frac{D(E)}{\alpha H_0}\right )^{1/2} \left
(e^{\alpha H_0 t_0}-1 \right )^{1/2}, \label{mhorizon} \end{equation} where
$H_0$ is the Hubble parameter and $t_0$ is the age of the universe.
For $B_c=1$~nG and $l_c=1$~Mpc one finds the energy of the magnetic
horizon cutoff from the condition $R_{\rm hor}(E_{\rm cut})=d$,
where $d$ is the distance between sources, as $E_{\rm cut}=2.3\times
10^{15}$~eV for $d=30$~Mpc and $E_{\rm cut}=4.8\times 10^{16}$~eV
for $d=50$~Mpc, both for the Kolmogorov diffusion. For the case of
Bohm diffusion these energies are larger.
We present in Fig.~\ref{fig:diffuse} the spectra calculated for
diffusive propagation in relatively strong turbulent magnetic field with basic
turbulent length $l_c= 1$~Mpc, with coherent magnetic field on this
scale $B_c= 1$~nG and for separation between sources $d=50$~Mpc. At
energies higher than $E_{\rm cr}$ diffusion proceeds in the regime
where $D(E) \propto E^2$, (see \cite{AB1}), while at lower energies
we assume the appropriate diffusion regimes with $D(E) \propto E^{\alpha}$,
as indicated in
Fig.~\ref{fig:diffuse}. One can see that the dip calculated in the
diffusive approximation differs very little from the one in the
universal spectrum, shown by dash-dot line.
We conclude this section asserting that the presence of magnetic
fields in the universe modify quite weakly, within a wide range of
parameters, the shape of the dip calculated in the form of a
universal spectrum. We provided evidences that a transition from
extragalactic to galactic cosmic rays occurs at energy $E_{\rm
cr}=1\times 10^{18}$~eV, independently of the mode of propagation
(from rectilinear to diffusive). However, for any reasonable
extragalactic magnetic fields the propagation of protons and nuclei
at $E < 1\times 10^{18}$~eV is expected to be diffusive. Even for
random magnetic fields with parameters $l_c= 1$~Mpc and $B_c=
0.1$~nG, taken as average over the universe, the diffusion length at
$E=3\times 10^{17}$~eV, is only $\sim 10$~Mpc, which is considerably
smaller than the size of the region contributing the observed
diffuse flux at this energy. This makes the low-energy diffusive
cutoff, which provides the transition in our model, most reliable.
\subsubsection{Inhomogeneity in the source distribution}
\label{inhomog}
It is often argued that the distribution of matter in the universe is
not homogeneous and that the inhomogeneous distribution of the sources
of UHECR may have an effect on the observed spectrum. This comment
clearly applies to all inhomogeneities on scales smaller than the loss
length of particles with given energy. For instance, at extremely high
energies (around $10^{20}$ eV) the loss length is $20 - 100$~Mpc.
On these scales the universe is indeed inhomogeneous, and this
may result in a shape of the GZK feature which reflects this
inhomogeneity: a local overdensity (deficit) would make the GZK
feature less (more) pronounced.
On the other hand, we know from cosmological observations that the
universe is homogeneous and isotropic on the scale of the cosmological
horizon, and in fact already on scales of the order of $l\gsim 100$ Mpc.
The particles from the dip, i.e. with energies below $4\times 10^{19}$~eV,
have attenuation length of the order of 1000~Mpc, and thus the
dip shape is insensitive to inhomogeneities, provided they occur
on spatial scales smaller than $\sim 1000$~ Mpc. We checked this
conclusion by including in the calculations some spatial
inhomogeneities on scales of 100~Mpc.
\subsubsection{Cosmological evolution of UHECR sources}
The cosmological evolution of the sources, namely the increase in the
source luminosity or comoving density with red-shift $z$, is
observed for many classes of astrophysical objects. The evolution
is reliably observed for the star formation rate in normal galaxies,
but this case is irrelevant for most of the cases of our concern,
because neither stars nor normal galaxies can be the sources of
UHECR due to insufficient cosmic-ray luminosities $L_p$ and maximum
energy at acceleration $E_{\rm max}$. An exception to this
rule might be represented by the case of Gamma Ray Bursts (GRBs).
\label{evolution}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=10.0cm]{evolution.eps}
\end{center}
\caption{Dip calculated in the models with cosmological evolution.
The parameters of evolution used in the calculations for curves 1 and
2 are close to those observed for AGN. The curve 3 is the universal
spectrum with $m=0$.}
\label{fig:evolution}
\end{figure}
Active Galactic Nuclei (AGN), which are most probable candidates for
UHECR sources, exhibit evolution in the radio, optical
and X-ray bands. X-rays are probably the most relevant tracer for
evolution of AGN as the sources of UHECR: X-rays
are produced in accretion disks around massive black holes, and the
X-ray luminosity is connected thus with the accretion rate. UHECR
are probably also connected with the accretion power of massive
black holes through the production of jets and generation
of shock waves in the jets and radio lobes. According to
recent detailed analysis \cite{Ueda,Barger} the
evolution of AGN seen in X-ray radiation can be described in terms of
the factor $(1+z)^m$ up to $z_c \approx 1.2$ and is saturated at larger
$z$. In \cite{Ueda} the pure luminosity evolution and pure density
evolution is allowed with $m=2.7$ and $m=4.2$, respectively and
with $z_c \approx 1.2$ for both cases. In \cite{Barger} the
pure luminosity evolution is considered as preferable with $m=3.2$
and $z_c =1.2$. These authors do not distinguish between
different morphological types of AGN. It is possible that some AGN
undergo weak cosmological evolution, or no evolution at all. For
instance BL Lacs, which are suspected as sources of observed UHECR
\cite{TT}, show {\em negative} evolution ($m<0$) \cite{Morris}, which
should be most probably interpreted as weak or absent evolution.
The effect of the evolution of the sources of UHECR on
the shape of the dip was discussed in \cite{BGG} and in a recent
review \cite{blasirev}.
In the case of UHECR there is no need to distinguish between luminosity
and density evolution, because the diffuse flux is determined by the
comoving energy-density production rate ({\em emissivity})
${\mathcal L}=L_p n_s$ , where $L_p$ is the cosmic ray luminosity and
$n_s$ is the space density of the sources.
In Fig.~\ref{fig:evolution} we present the calculated spectrum for
evolutionary models, inspired by the data cited above. For
comparison we show also the case of absence of evolution $m=0$. As
our calculations show, in most cases the negative evolution ($m
<0$) results in the same shape of the dip as the no-evolution case
$m=0$.
The universal spectra, obtained for sources evolving up to $z_c > 1$,
fit the observational data down to $E \sim 3\times 10^{17}$~eV and
even at lower energies. However, for reasonable magnetic fields in the
intergalactic medium, protons with these energies have small
diffusion lengths and the universal spectrum fails at
$E < 1\times 10^{18}$~eV, exhibiting a diffusion 'cutoff' that starts
at energy $E_{\rm cr}$.
We conclude that at present evolutionary models can fit the shape of
the dip as well as models without evolution ($m=0$).
\subsubsection{Energetics }
\label{energetics}
The universal spectrum, which fits the observed dip, requires an
injection spectrum with a slope $\gamma_g=2.6-2.7$. The normalization
to the observed flux needs the emissivity (energy-density production
rate) at $t=t_0$~~ ${\mathcal L}_0 \propto E_{\rm min}^{-(\gamma_g-2)}$,
where $E_{\rm min}$ is the minimum acceleration energy. For low
$E_{\rm min} \sim 1$~GeV the required emissivity is too high. In order
to prevent this energetic crisis, it was suggested phenomenologically
in Refs.~\cite{BGG} and \cite{BGH} that the generation rate per unit
comoving volume $Q(E_g)$ may have the form
\begin{equation}
Q_{\rm gen}(E_g)=\left\{ \begin{array}{ll}
\propto E_g^{-2} ~&{\rm at}~~ E_g \leq E_c\\
\propto E_g^{-2.7} ~&{\rm at}~~ E_g \geq E_c,
\end{array}
\right.
\label{broken}
\end{equation}
where the spectrum $\propto E^{-2}$ is due to non-relativistic shock
acceleration. Recently, an interesting idea was put forward in
Ref.~\cite{michael}, due to which the broken spectrum Eq. (\ref{broken})
can be realized. The authors of \cite{michael} observed that while
the slope of the acceleration spectrum (e.g. 2.0 for non-relativistic shocks
and 2.2 for relativistic shocks) is universal, the maximum acceleration
energy $E_{\rm max}$ depends on individual characteristics of the sources,
such as magnetic field and/or size. As a result the sources should
be expected to have a distribution of maximum energies at the different
acceleration sites, $n_s(E_{\rm max})$.
This distribution naturally results in a complex spectrum of the form
given in Eq. (\ref{broken}), with $E_c$ being a free parameter.
In section \ref{Emax} we propose another model for the complex
spectrum (Eq. \ref{broken}). It is based on a possible correlation
of $E_{\rm max}$ with source luminosity. The distribution of sources
over luminosities results in the complex spectrum in the form of Eq.
(\ref{broken}) for the generation rate per comoving volume, $Q_{\rm
gen}(E_g)$.
With the complex spectrum Eq.~(\ref{broken}) we obtain for the spectrum
shown in the right panel of Fig.~\ref{fig:AgHiYa}, the emissivity
${\mathcal L}_0=3.7\times 10^{46}$~erg Mpc$^{-3}$ yr$^{-1}$ for
$E_c=1\times 10^{18}$~eV. Using the sources
space density $n_s=2\times 10^{-5}$~Mpc$^{-3}$, as deduced from
small scale anisotropy \cite{ssa1,ssa2}, we arrive at the cosmic ray
luminosity of a source $L_p=5.9\times 10^{43}$,~$2.6\times 10^{44}$
and $1.2\times 10^{45}$~erg/s for $E_c$ equals to $1\times 10^{18}$~eV,
$1\times 10^{17}$~eV and $1\times 10^{16}$~eV, respectively. These
luminosities fit well the energetics potential of AGN.
It is necessary to emphasize that low $E_c$ in Eq.~(\ref{broken})
does not change our
conclusion about the transition from galactic to extragalactic
cosmic rays, provided $E_c$ is below or close to $1\times 10^{18}$~eV.
\subsubsection{Chemical Composition}
\label{chemical}
The presence of nuclei heavier than protons in the primary UHECR flux can
substantially modify the proton dip \cite{BGG3,Allard} and affect the
agreement with observational data shown in Fig.~\ref{fig:dips}.
In Fig~\ref{fig:dip-nucl} the modification factors for helium and iron
nuclei are presented in comparison with the proton modification factor.
One can see that the presence of 15 - 20 \% of nuclei in the primary flux
affects the good agreement with observations.
\begin{figure}[ht]
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{he.eps}
\end{minipage}
\hspace{5mm}
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{iron.eps}
\end{minipage}
\caption{Modification factors for helium and iron
nuclei in comparison with that for protons. Proton modification
factors are given by curves 1 and 2.
Modification factors for nuclei are shown as curve 3 (adiabatic
and pair production energy losses) and curve 4 (with
photodisintegration included).}
\label{fig:dip-nucl}
\end{figure}
The modification factor for a mixed composition can be calculated by
introducing a mixing parameter $\lambda=Q_A^{\rm unm}(E)/Q_p^{\rm
unm}(E)$ as
\begin{equation}
\eta (E)= \frac{\eta_p(E)+\lambda \eta_A(E)}{1+\lambda},
\label{mod-mix}
\end{equation}
where $Q_A^{\rm unm}(E)$ is the injection spectrum of nuclei with mass
number $A$ at the source.
\begin{figure}[ht]
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{pHe01.eps}
\end{minipage}
\hspace{5mm}
\begin{minipage}[h]{8cm}
\centering
\includegraphics[width=7.6cm,clip]{pHe0195.eps}
\end{minipage}
\caption{\label{fig:dip-mix} Modification factors for the mixed composition
of protons and helium nuclei in comparison with the AGASA data. The left
panel corresponds to mixing parameter $\lambda=0.1$, and the right
panel to $\lambda=0.2$.
}
\end{figure}
The mixing parameter $\lambda$ is determined primarily by the ratio of
the number densities $n_A/n_H$ in the gas where acceleration occurs.
The largest ratio $n_A/n_H$ is given by helium which has basically
a cosmological origin. The cosmological mass fraction of helium
$Y_p=0.24$ results in $n_{\rm He}/n_H=0.079$.
In Fig.~\ref{fig:dip-mix} the modification factors for the mixed
composition of protons and helium with $\lambda=0.1$ and
$\lambda=0.2$ are shown. In the former case the agreement is
sufficiently good, in the latter case the agreement is noticeably
worse than for a pure proton composition.
Apart from the density ratio, the mixing parameter depends on the
details of the acceleration mechanism. In principle, if acceleration
takes place in a relatively cold medium, where helium is not ionized
(the ionization potential for helium is very high: 24.4~eV and
54.4~eV for the first and second ionization potentials,
respectively), helium nuclei may be not accelerated at all. This
possibility can occur for the case of induced electric fields in the
plasma. However, for shock acceleration the fraction of accelerated
nuclei is determined by their injection into the shock acceleration
region (see e.g. \cite{berez}), and even for low temperature of the
upstream gas, the rate of injection of nuclei in the downstream
region can be high. For non-relativistic shocks the ratio of
temperatures downstream and upstream can be low if the Mach number
of the shock is low, and injection of nuclei may be suppressed. This
case is considered in more detail in section \ref{sec:injection}.
The more realistic possibility for suppression of nuclei is given by
the case of ultra-relativistic shocks with large Lorentz factor (see
section \ref{sec:injection} for a detailed discussion). The fraction
of heavy nuclei $\lambda$ can be suppressed exponentially in this
case, see Eq.~(\ref{A/p}).
Finally, UHE nuclei can be photo-dissociated
in the photon field of a source \cite{Sigl,Berez05}.
\section{Transition from galactic to extragalactic cosmic rays}
\label{transition}
From the analysis of the dip we obtained the indications that the
transition from galactic to extragalactic cosmic rays takes place
at energy $E_{\rm cr} =1\times 10^{18}$~eV. The first indication
is given by the measured modification factor which at this
energy becomes larger than one (see Fig.~\ref{fig:dips}) in
contradiction with the definition of $\eta$ that forces $\eta (E) \leq
1$. This signals the appearance of a new component at lower energies,
which can be nothing but galactic cosmic rays.
The second indication is given by the low-energy 'diffusion cutoff'
of the extragalactic spectrum (Fig.~\ref{fig:diffuse}), which
inevitably provides the dominance of the galactic component at energy
$E < E_{\rm cr}$.
The energy at which the transition begins, $E_{\rm cr}$, is
completely determined by the equality of the rate of pair production
losses and the rate of adiabatic losses at the energy $E_{\rm eq}=
2.3\times 10^{18}$ eV \cite{BGG}. As stressed in
Section~\ref{discrete} the connection between $E_{\rm cr}$ and
$E_{\rm eq}$ is given by $E_{\rm cr} =E_{\rm eq}/(1+z_{\rm eff})^2$,
and results in $E_{\rm cr} \approx 1\times 10^{18}$~eV. The
transition must be observationally visible at some lower energy. It
coincides thus with the second knee located at energy $E_{\rm 2kn}$,
for which the different experiments give $E_{\rm 2kn} \sim (0.4 -
0.8)\times 10^{18}$ eV. For the other models, where the transition
also occurs at the second knee, see \cite{Biermann}-\cite{Dermer}.
The explanation of the transition given above can be put in a simple
and general way: from the plot of the modification factors
in Fig. \ref{fig:dips}, one can see that the spectrum of
extragalactic cosmic rays reproduces the generation spectrum when
the modification factor tends to unity. The generation spectrum
is always flatter
than the spectrum of galactic cosmic rays $\propto E^{-3.1}$. This
argument is further strengthened by the low-energy 'diffusion cutoff'
in the extragalatic spectrum.
The {\em dip-transition} model works most naturally in the framework
of the rigidity
model for the origin of the first knee. There are two versions of
this model: rigidity-confinement model and rigidity-acceleration model
(e.g. \cite{Biermann}). In the first model, the position of the proton
knee ($E_p \approx 2.5\times 10^{15}$~eV) is determined by magnetic
confinement of protons in the galactic halo. In this case the knee in
the spectrum of nuclei with charge $Z$ is located at $E_Z=ZE_p$.
The highest energy knee, due to iron, must be located at
$E_{\rm Fe}= 6.5\times 10^{16}$~eV. This finding seems to be confirmed
by Fig.~\ref{fig:masses}, where the mean $A$ is that corresponding to
iron nuclei at roughly this energy. At energy $E > E_{\rm Fe}$ the
total galactic flux, consisting mostly of iron, must be steeper. The
acceleration rigidity model predicts the same behaviour of the
spectra, but the knees appear due to different maximum acceleration
energies $E_{\rm max}$ for different nuclei. The data of KASCADE
confirm well the rigidity models when the SYBILL interaction model
is adopted, less well with the QGSJET model \cite{kascade}.
\begin{figure}[ht]
\begin{minipage}[h]{9cm}
\centering
\includegraphics[width=86mm,height=70mm,clip]{masses.eps}
\end{minipage}
\caption{Mean logarithmic mass number of cosmic rays as a function of
energy \cite{horandel}.}
\label{fig:masses}
\end{figure}
In the framework of a rigidity dependent origin of the knees, the
dip model describes the transition as the intersection of a steep
galactic spectrum at $E> E_{\rm Fe}= 6.5\times 10^{16}$~eV with a
flat extragalactic proton spectrum at $E < E_{\rm cr}=1\times
10^{18}$~eV. Numerically this transition is shown in the left panel
of Fig.~\ref{fig:transition} for the Bohm diffusion of extragalactic
protons at energies below $E_{\rm cr}=1\times 10^{18}$~eV.
The beginning of the galactic steepening at $E_{\rm Fe}= 6.5\times
10^{16}$~eV and the extragalactic flattening below $E_{\rm
cr}=1\times 10^{18}$~eV are the results of separate pieces of
Physics and nothing "unnatural" takes place in the proximity of
these energies (they differ by one order of magnitude). The ratio of
the fluxes $J_{\rm gal}(E_{\rm Fe})/J_{\rm extr}(E_{\rm cr}) \sim
7\times 10^{3}$ is large, and we do not see any fine-tuning in this
model of the transition.
The galactic and extragalactic fluxes become equal at $E_{\rm
tr}=5\times 10^{17}$~eV. The transition is very prominent, if the iron
and proton components are resolved, but in the total spectrum the
transition appears as a faint feature known as the {\em second
knee}. This property is the same as for the other knees (proton,
helium, carbon, etc.) observed by KASCADE: while the transition
between the knees is distinct in the spectra with fixed chemical
composition, the resultant total spectrum has a smooth power-law
shape.
For the fraction of
iron and proton fluxes in the energy range $1\times 10^{17} -
1\times 10^{18}$~eV see \cite{BGH,AB1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=14.0cm]{transition.eps}
\end{center}
\caption{{\it Left panel: the second-knee transition }. The
extragalactic proton spectrum is shown for $E^{-2.7}$ generation
spectrum and for propagation in magnetic field with
$B_c= 1$~nG and $l_c=1$~Mpc, with the Bohm diffusion at $E \lsim E_c$.
The distance between sources is $d=50$~Mpc. $E_b=E_{\rm cr}=1\times
10^{18}$~eV is the beginning of the transition, $E_{\rm Fe}$ is
the position of the iron knee and $E_{\rm tr}$ is the energy where
the galactic and
extragalactic fluxes are equal. The dash-dot line shows the
power-law extrapolation of the KASCADE spectrum to higher
energies, which in fact has no physical meaning, because of the
steepening of the galactic spectrum at $E_{\rm Fe}$.
{\it Right Panel: the ankle transition},
for the injection spectrum of extragalactic protons $E^{-2}$.
In both cases the dashed line is obtained as a result of
subtracting the extragalactic spectrum from the observed
all-particle spectrum.}
\label{fig:transition}
\end{figure}
The right panel of Fig.~\ref{fig:transition} shows the traditional
transition from galactic to extragalactic cosmic rays at the ankle
($E_a \approx 1\times 10^{19}$~eV). In this model the extragalactic
component has a very flat generation spectrum $\propto E^{-2}$ which
naturally intersects the steep ($\propto E^{-3.1}$) galactic
component. The most attractive feature of this model is given by the
flatness of the extragalactic generation spectrum, which provides
reasonable luminosities of the sources and a natural interpretation
of the intersection of the galactic and extragalactic cosmic ray
components. Being stimulated by the discovery of the ankle by the
Haverah Park array in the '70s, this model has been considered
recently in Refs.~\cite{ankle,DeMSt}. Both models of the transition,
at the second knee and at the ankle, have some advantages and
problems, as summarized below:
\begin{itemize}
\item The second knee model is inspired by and based on the
numerical confirmation of the existence of the dip as a spectral
feature of extragalactic protons interacting with the CMB (see
Fig.~\ref{fig:dips}). The probability of an accidental agreement,
estimated from the $\chi^2$, the number of free parameters and the
number of energy bins in each of the four experiments, is very
small. The ankle model explains the dip as a possible interplay
between galactic and extragalactic spectra. It looks rather odd that
such feature has exactly the same shape as that of the CMB-induced
dip.
\item The explanation of the transition is more straightforward in
the ankle model: it is the simple intersection of the flat extragalactic
spectrum with the steep galactic spectrum. This model naturally
predicts a rather low luminosity of the sources and allows to
incorporate an arbitrary fraction of heavy nuclei in the total
flux at $E > 1\times 10^{19}$~eV, in case the future experiments
will show that this is needed. The second knee
transition is also based on the intersection of a steep galactic
spectrum with a flat extragalactic spectrum. The flatness of the
extragalactic spectrum (diffusion 'cutoff') appears quite naturally
at energy close to $E_{\rm cr} =1\times 10^{18}$~eV due to diffusion
of protons with $E < E_{\rm cr}$. However, a low luminosity of
the sources can be achieved only by postulating a distribution of
maximum energies at the sources. The energy where the {\it effective}
generation spectrum shows the steepening is a free parameter.
\item
The dip is modified by the presence of heavy nuclei in the primary
radiation and it allows only small admixture of heavy nuclei at the
dip and above it. This may in turn be interpreted as a possible
signature of the model of transition at the dip.
\item
The model of the transition at the ankle requires that the galactic
component of cosmic rays extends to energies in excess of $10^{19}$
eV. This requires a revision of the existing galactic models of
propagation and acceleration, which predict the maximum acceleration
energy to be less than $1\times 10^{18}$~eV for iron nuclei
\cite{Emax}. The model of transition at the dip appears to be in
agreement with observations of the first knee combined with a
rigidity dependent picture (either related to propagation or to
acceleration) of the knees for nuclei more massive than hydrogen.
\item
At energy $E \geq 1\times 10^{17}$~eV , i.e. much higher than the
proton knee, at least 10\% of the observed flux is in the form of
protons. This fraction is most naturally explained in the context of
the second knee model. It also represents a serious challenge for
the ankle model.
\end{itemize}
As stressed above, both models make clear predictions, that can and
should be used to prove or disprove them. The most critical
observation is the measurement of the chemical composition in the
energy region around $E \sim 1\times 10^{18}$~eV. While the dip
model predicts a strong dominance of protons, in the ankle model a
strong dominance of iron nuclei is expected. We think that the
fluorescence measurements provide us with the most promising tool
for discrimination of these two models. At present HiRes elongation
rate data \cite{mass-Hires} show the transition to a
proton-dominated mass composition at $E \approx 1\times 10^{18}$~eV
as the dip model predicts, while HiRes predecessor, Fly's Eye, shows
this transition at higher energies. Referring to this contradiction
we want to emphasize the uncertainties, both experimental and
theoretical, involved in present analysis, and which hopefully will
be overcome in the future.
As we discussed above, the ankle model in its canonical form
predicts a transition from galactic to extragalactic component at
energy $E_a \approx 1\times 10^{19}$~eV, as observational data from
Fig.~\ref{fig:dips} imply, and as most authors of the ankle models
assume. However, there could be an intermediate possibility between
the dip and the ankle models, when transition occurs at $1\times
10^{18} < E < 1\times 10^{19}$~eV. The most elaborated model of this
type is the mixed-composition model \cite{parizot}. The transition
is found to occur at energy $E \sim 3\times 10^{18}$~eV, and thus
it softens the difficulties with the highest energy end of the
galactic cosmic ray spectrum. In this model, the observed dip, as
it is shown in
Fig.~\ref{fig:dips}, is reproduced exactly, provided that the
galactic component of cosmic rays is fitted
{\it a posteriori} by subtracting the calculated extragalactic
component from the observed total spectrum (the model for the
galactic spectrum is not discussed and the reason why the dip in
this model coincides exactly with the dip calculated for
extragalactic protons is left unanswered). Inspired by
observations of galactic cosmic rays, the chemical composition of
extragalactic cosmic rays is assumed to be mixed. The authors show
that elongation rates, especially the ones from the data of Fly's
Eye and Yakutsk, confirm better the mixed model than the pure proton
model of the dip. We can add to this finding that Akeno data
\cite{mass-Akeno} confirm also the mixed composition, while data of
HiRes \cite{mass-Hires}, HiRes-MIA \cite{Hi-Mia} and more reliable
{\em muon data} of Yakutsk \cite{Glushkov00} support the
proton-dominated composition at $E \geq 1\times 10^{18}$~eV. Based
on these many contradictory sets of data we do not feel of sharing
with the authors of \cite{parizot} their trust in the accuracy of
interaction-dependent analysis of the elongation rate at present,
although we hope in its future progress.
\section{Conclusions}
\label{conclusions}
The dip is a feature in the extragalactic cosmic ray spectrum,
that originates from the Bethe-Heitler pair production of protons
on the cosmic microwave background (\cite{BGG3},\cite{BGG}). The
dip appears at energy $1\times 10^{18} - 4\times 10^{19}$~ eV, with
shape practically independent of the discreteness and inhomogeneity in
the source distribution, independent of local overdensity and deficit
of the sources and of the maximum acceleration energy, independent
of the presence or absence of magnetic fields in the intergalactic space,
and independent of fluctuations in $p\gamma$ interactions.
The cosmological evolution of UHECR sources, with the parameters
taken from observations of AGN, do not affect the dip. The only
phenomenon which modifies noticeably the shape of the dip is the presence
of a large fraction of nuclei, heavier than hydrogen, in the generation
spectrum. We consider the small fraction of nuclei, allowed by the dip
shape, as an indication of possible mechanisms of acceleration or
injection, operating in UHECR sources (Section \ref{sec:injection}).
The predicted shape of the dip is in excellent agreement with the
data of Akeno-AGASA, Fly's Eye, HiRes and Yakutsk detectors
(the data of Auger are inconclusive because of the absence of data at
$E< 3\times 10^{18}$~eV, essential for the dip). The energy
calibration of these detectors by the position of the dip results in
excellent agreement between the measured fluxes.
To our knowledge, the precision of the agreement between the predictions
for the dip, which need only two free parameters, and observations
(see Fig.~\ref{fig:dips}) is the best that ever existed in cosmic ray
physics. This, and the results of energy calibration of the detectors
by the position of the dip (see Fig.~\ref{fig:AgHiYa}), makes
improbable that we are observing an accidental agreement between
the predictions and the observations.
All the discussion presented so far in the paper was based on purely
phenomenological and largely model independent grounds. Clearly, in
order to establish a connection with existing theories of
acceleration or propagation of cosmic rays, the dip needs to be
related to models. In passing we notice however that the shape of
the dip agrees well with data when AGN with their observed
cosmological evolution are considered as sources of UHECR (Section
\ref{evolution}). A possible, though model dependent, correlation
between the maximum acceleration energy and the luminosity of the
sources might also solve the problem of the excessive energetics
required by the dip scenario (Section \ref{Emax}). A small fraction
of heavy nuclei can also be accommodated in the model (Section
\ref{chemical}).
In the dip scenario the transition from galactic to extragalactic
cosmic rays takes place at $E_{\rm cr} \approx 1\times 10^{18}$~eV.
The natural character of this transition is guaranteed by the fact
that a flat extragalactic spectrum at $E < E_{\rm cr}$ intersects a
steep galactic spectrum at $E > E_{\rm Fe} \sim 1\times 10^{17}$~eV.
Observationally the transition occurs at the second knee.
An alternative possibility for the transition is given by the {\em
ankle} model (see Section \ref{transition}).
The ankle model has many attractive features. It gives a simple and
natural picture of the transition as the intersection of a steep galactic
spectrum with a very flat extragalactic spectrum, resulting from a
generation spectrum as predicted by the 'standard' acceleration
models ($E^{-2}$ for non-relativistic shock acceleration and
$E^{-2.2}$ for relativistic shocks). These generation spectra
have no problem with energetics. The model can also easily accommodate
a large fraction of heavy nuclei, if observations show that this is
needed.
On the other hand the ankle model has weaknesses: the model requires
that the galactic component extends to energies in excess of $10^{19}$
eV, in apparent contradiction with the KASCADE data. We add also
that it seems to contradict the most current models of cosmic
ray acceleration in galactic sources \cite{Emax}. The model also has
difficulties in explaining the $\sim 10\%$ of protons observed
in Akeno at $E \sim 10^{17}$~eV (in the dip model these protons are
extragalactic).
In conclusion, we want to emphasize the importance of having
detectors working in the energy range $3\times 10^{17} - 1\times
10^{19}$~eV, such as possible low-energy extensions of Auger,
Telescope Array (TA) and low-energy extension of TA (TALE). These
detectors can reliably solve the problem of measuring the mass
composition in the transition region. It is important to observe
that the two basic models of the transition, at the second knee and
at the dip, give drastically different predictions for the chemical
composition in the energy interval $1\times 10^{18} - 1\times
10^{19}$~eV: proton-dominated in the dip model and iron-dominated in
the ankle model. The updated fluorescent method with measurement of
atmosphere transparency for each event can reliably distinguish
these two models, in spite of existing experimental and theoretical
(models of interaction) uncertainties.
One can add another signature common to both the dip and the ankle
scenarios (see \cite{parizot}) namely the appearance of a jump in $x_{max}$ at
energy $E_{tr}$, which equals $\sim 5\times 10^{17}$ eV for the dip
model and $\sim 10^{19}$ eV for the ankle model. This feature appears
because of a sharp transition from a steep galactic (iron dominated)
spectrum to a flat extragalactic (proton dominated) component (see
Fig. 10). Such a jump is practically absent or much smoother in the
mixed composition model. In contrast with the dip and ankle models,
the mixed composition model needs a
higher accuracy of the fluorescent method to distinguish the mixed
extragalactic composition from the proton-dominated or
iron-dominated compositions (there are no reliable predictions for
the mass composition of extragalactic UHECR).
One should keep in mind the great predictive power of the {\em
spectrum measurements} for the determination of the mass composition
of extragalactic UHECR. The modification factors for protons and
different nuclei (see Fig.~\ref{fig:dip-nucl} for comparison) differ
very strongly in the energy region $1\times 10^{18} - 1\times
10^{20}$~eV and even small admixtures of nuclei can be discovered as
a distortion of the proton modification factor. Note that this
possibility exists for all models: dip, ankle and intermediate
transition.
In principle, the {\em anisotropy} discriminates the different
models of transition. While the dip-transition model predicts an
extragalactic anisotropy at $1\times 10^{18} - 1\times 10^{19}$~eV,
the ankle models predicts the galactic anisotropy at this energy
range. However, in the case of an expected iron-dominated
composition of galactic cosmic rays both models predict too small
anisotropy.
We think that measurements of the spectrum and chemical composition
in the energy region $3\times 10^{17} - 1\times 10^{19}$~eV can
resolve the existing problem of establishing where the transition
from galactic to extragalactic cosmic rays takes place and much
experimental effort should be put in aiming at this goal.
\section*{Acknowledgments}
We thank ILIAS-TARI for access to the LNGS research infrastructure and for
the financial support through EU contract RII3-CT-2004-506222. The
work of S.G. is partly supported by Grant No. LSS-5573.2006.2.
|
1,314,259,993,017 | arxiv | \section{Introduction}
Elliptic curves and their geometric and algebraic structure have been a flourishing
field of research in the past. They find prominent applications in cryptography
and played a key role in the proof of Fermat's Last Theorem. A salient feature of the algebraic
structure of an elliptic curve is its rank. Among general elliptic curves,
congruent number curves of high rank are of particular interest (see, \eg,~\cite{HighRank}).
More difficult than finding an individual congruent number curve
of high rank is to find infinite families of such curves. Johnstone and Spearman~\cite{JS}
constructed such a family with rank at least three which is related
to rational points on the biquadratic curve $w^2 = t^4 + 14t^2 + 4$. In the present paper,
we show an elementary construction for an infinite family of
congruent number curves of rank at least two which are related
to the quadratic diophantine equation $m^2=n^2+nl+l^2$, and which
have three integral points with positive $y$-coordinate
on a straight line. Incidentally,
some members of the family exhibit surprisingly high individual rank.
We start by fixing the notions and notations used througout the text.
A positive integer $A$ is called a {\bf congruent number} if
$A$ is the area of a right-angled triangle with three rational
sides. So, $A$ is congruent if and only if there exists a
rational Pythagorean tripel $(a,b,c)$ (\ie, $a,b,c\in\mathds{Q}$,
$a^2+b^2=c^2$, and $ab\neq 0$), such that $\frac{ab}2=A$.
The sequence of integer congruent numbers starts with
$$
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37,\ldots
$$
(see, \eg, the On-Line Encyclopedia of Integer Sequences~\cite{oeisA003273}).
For example, $A=7$ is a congruent number,
witnessed by the rational Pythagorean triple $$\Bigl(\frac{24}{5}\,,
\frac{35}{12}\,,\frac{337}{60}\Bigr).$$
It is well-known that $A$ is a congruent number if
and only if the cubic curve $$C_A:\ y^2=x^3-A^2 x$$
has a rational point $(x_0,y_0)$ with $y_0\neq 0$.
The cubic curve $C_A$ is called a {\bf congruent number elliptic curve}
or just {\bf congruent number curve}.
This correspondence between rational points on congruent number curves and
rational Pythagorean triples can be made explicit as follows:
Let
$$
C(\mathds{Q}):= \{(x,y,A)\in \mathds{Q}\times\mathds{Q}^*\times \mathds{Z}^*:y^2=x^3-A^2x\},
$$
where $\mathds{Q}^*:=\mathds{Q}\setminus\{0\}, \mathds{Z}^*:=\mathds{Z}\setminus\{0\}$, and
$$
P(\mathds{Q}):=\{(a,b,c,A)\in \mathds{Q}^3\times\mathds{Z}^*:a^2+b^2=c^2\ \textsl{and\/}\ ab=2A\}.
$$
Then, it is easy to check that
\begin{equation}\label{psi}
\begin{aligned}
\psi\ :\ \quad P(\mathds{Q})&\ \to\ C(\mathds{Q})\\
(a,b,c,A)&\ \mapsto \ \Bigl(\frac{b(b+c)}{2}\,,\,
\frac{b^2(b+c)}{2}\,,\,A\Bigr)
\end{aligned}
\end{equation}
is bijective and
\begin{equation}\label{psi-1}
\begin{aligned}
\psi^{-1}\ :\qquad C(\mathds{Q})&\ \to\ P(\mathds{Q})\\
(x,y,A)&\ \mapsto\ \Bigl(\frac{2x A}{y}\,,\;
\frac{x^2-A^2}{y}\,,\;\frac{x^2+A^2}{y}\,,\,A\Bigr).
\end{aligned}
\end{equation}
For positive integers $A$, a triple $(a,b,c)$ of
non-zero rational numbers is called a {\bf rational Pythagorean $\boldsymbol{A}$-triple}
if $a^2+b^2=c^2$ and $A=\big{|}\frac{ab}{2}\big{|}$.
Notice that if $(a,b,c)$ is a {rational Py\-tha\-go\-re\-an $A$-triple}, then $A$
is a congruent number and $|a|,|b|,|c|$ are the
lengths of the sides of a right-angled triangle
with area $A$. Notice also that we allow $a,b,c$
to be negative.
If $a,b,c$ are positive integers such that $a^2+b^2=c^2$ and $A=\frac{ab}{2}$
is integral, then the triple $(a,b,c)$ is a called a {\bf Pythagorean $\boldsymbol{A}$-triple}.
For any positive integers $m$ and $n$ with $m>n$, the triple
$$\bigl(\,\underset{\text{\small $a$}}{\underbrace{\,\mathstrut 2mn\,}}\,,\;
\underset{\text{\small $b$}}{\underbrace{\mathstrut m^2-n^2}}\,,\;
\underset{\text{\small $c$}}{\underbrace{\mathstrut m^2+n^2}}\,\bigr)$$
is a {Py\-tha\-go\-re\-an $A$-triple}. In this case, we obtain $A=mn(m^2-n^2)$ and
\begin{equation}\label{eq-psi}
\psi(a,b,c,A)=\bigl(
\underset{\text{\small $x$}}{\underbrace{m^2(m^2-n^2)}}\,,\;
\underset{\text{\small $y$}}{\underbrace{m^2(m^2-n^2)^2}}\,,\;A\bigr)\,.
\end{equation}
In particular, the point $(x,y)$ on $C_A$ which corresponds to the
{Py\-tha\-go\-re\-an $A$-triple} $(a,b,c)$ is an integral point.
Concerning the equation
$$m^2=n^2+nl+l^2\,,$$ we would like to mention the following fact
(see Dickson~\cite[Exercises\,XXII.2, p.\,80]{Dickson} or Cox~\cite[Chapter\,1]{Cox}):
\begin{fct}\label{fct:m2}
Let $p_1<p_2<\ldots <p_j$ be primes, such that
$p_i\equiv 1\mod 6$ for $1\le i\le j$, and let $$m=\prod_{i=1}^j p_i.$$
Then the number of positive, integral solutions $l<n$ of
$$m=n^2+nl+l^2$$ is $2^{j-1}$. By definition of $m$, for each
integral solution of $m=n^2+nl+l^2$, $n$ and $l$ are relatively prime,
denoted $(n,l)=1$.
Moreover, the number of positive, integral solutions $l<n$ of
$$m^2=n^2+nl+l^2$$ is $\frac{3^j-1}2.$
Among the $\frac{3^j-1}2$ integral solutions $l<n$ of
$m^2=n^2+nl+l^2$ we find $2^{j+1}$ solutions with $(n,l)=1$.
In particular, if $j=1$ and $p\equiv 1\mod 6$, then
the solution in positive integers $n<l$ of
$$p^2=n^2+nl+l^2$$ is unique and $(n,l)=1$.
\end{fct}
For a geometric representation of integral solutions of $x^2+xy+y^2=m^2$,
see Halbeisen and Hungerb\"uhler~\cite{HHAnning}.
If $m,n,l$ are positive integers such that $m^2=n^2+nl+l^2$, then, for $k:=n+l$,
each of the following three triples
$$\bigl(\,\underset{\text{\small $a_1$}}{\underbrace{\,\mathstrut 2mn\,}}\,,\;
\underset{\text{\small $b_1$}}{\underbrace{\mathstrut m^2-n^2}}\,,\;
\underset{\text{\small $c_1$}}{\underbrace{\mathstrut m^2+n^2}}\,\bigr)\,,$$
$$\bigl(\,\underset{\text{\small $a_2$}}{\underbrace{\,\mathstrut 2ml\,}}\,,\;
\underset{\text{\small $b_2$}}{\underbrace{\mathstrut m^2-l^2}}\,,\;
\underset{\text{\small $c_2$}}{\underbrace{\mathstrut m^2+l^2}}\,\bigr)\,,$$
$$\bigl(\,\underset{\text{\small $a_3$}}{\underbrace{\,\mathstrut 2mk\,}}\,,\;
\underset{\text{\small $b_3$}}{\underbrace{\mathstrut k^2-m^2}}\,,\;
\underset{\text{\small $c_3$}}{\underbrace{\mathstrut k^2+m^2}}\,\bigr)\,,$$
is a {Py\-tha\-go\-re\-an $A$-triple} for $A=mn(m^2-n^2)=ml(m^2-l^2)=km(k^2-m^2)$ (see Hungerb\"uhler~\cite{Noebi}).
In particular, with $m,n,l$ and~(\ref{eq-psi}) we obtain three distinct integral points on $C_A$.
Let us now turn back to the curve $C_A$.
It is convenient to consider the curve $C_A$ in the
projective plane $\mathds{R} P^2$, where the curve is given by
$$
C_A :\ y^2z = x^3-A^2xz^2.
$$
On the points of $C_A$, one can define a commutative, binary,
associative operation ``$+$'', where $\mathscr{O}$, the neutral
element of the operation, is the projective point $(0,1,0)$
at infinity. More formally, if $P$ and $Q$ are two points on $C_A$,
then let $P\# Q$ be the third intersection point of
the line through $P$ and $Q$ with the curve $C_A$.
If $P=Q$, the line through $P$ and $Q$ is replaced with the tangent in $P$.
Then
$P+Q$ is defined by stipulating
$$P+Q\;:=\;\mathscr{O}\# (P\# Q),$$
where for a point $R$ on $C_A$, $\mathscr{O}\# R$ is the point reflected across the $x$-axis.
The following figure shows the congruent number curve $C_A$ for
$A=5$, together with two points $P$ and $Q$ and their sum $P+Q$.
\begin{center}
\psset{xunit=.6cm,yunit=.4cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=5pt 0,linewidth=1.6pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211)
\psaxes[labelFontSize=\scriptstyle,xAxis=true,yAxis=true,Dx=2.,Dy=2.,ticksize=-2pt 2pt,subticks=1,linewidth=.6pt,]{->}(0,0)(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211)
\psplotImp[linewidth=1.2pt,linecolor=blue,stepFactor=0.1](-9.0,-9.0)(9.0,10.0){1.0*y^2+25.0*x^1-1.0*x^3}
\psplot[linewidth=.6pt]{-7.1329967371431815}{8.234736979744335}{(-43.01566031988557-1.5476943995438228*x)/-7.215284285929719}
\psline[linewidth=.6pt,linestyle=dashed,dash=4pt 4pt](-4.388182328551535,-7.588280903625012)(-4.388182328551535,9.360344080789211)
\begin{small}
\psdots[dotsize=4pt 0](5.824738905621608,7.211160924647724)
\rput[bl](5.3,7.4){$Q$}
\psdots[dotsize=4pt 0](-1.3905453803081107,5.663466525103901)
\rput[bl](-1.3,6){$P$}
\psdots[dotsize=4pt 0](-4.388182328551535,5.020466785549749)
\rput[bl](-6.,5.2){$P\# Q$}
\psdots[dotsize=4pt 0](-4.388182328551535,-5.020466785549749)
\rput[bl](-6.3,-5.4){$P+Q$}
\end{small}
\end{pspicture*}
\end{center}
More explicitly, for two points $P=(x_0,y_0)$ and $Q=(x_1,y_1)$ on
a congruent number curve $C_A$, the point $P+Q=(x_2,y_2)$ is given by
the following formulas:
\begin{itemize}
\item If $x_0\neq x_1$, then
$$x_2=\lambda^2-x_0-x_1,\qquad y_2=\lambda(x_0-x_2)-y_0,$$
where $$\lambda:=\frac{y_1-y_0}{x_1-x_0}.$$
\item If $P=Q$, \ie, $x_0=x_1$ and $y_0=y_1$, then
\begin{equation*}\label{eq:2P}
x_2=\lambda^2-2x_0,\qquad y_2=3x_0\lambda-\lambda^3-y_0,
\end{equation*}
where
\begin{equation*}\label{eq:lambda}
\lambda:=\frac{3x_0^2-A^2}{2y_0}.
\end{equation*}
Below we shall write $2*P$ instead of $P+P$.
\item If $x_0=x_1$ and $y_0=-y_1$, then $P+Q:=\mathscr{O}$. In particular,
$(0,0)+(0,0)=(A,0)+(A,0)=(-A,0)+(-A,0)=\mathscr{O}$.
\item Finally,
we define $\mathscr{O}+P:=P$ and $P+\mathscr{O}:=P$ for any point $P$, in particular,
$\mathscr{O}+\mathscr{O}=\mathscr{O}$.
\end{itemize}
With the operation~``$+$'',
$(C_A,+)$ is an abelian group with neutral element $\mathscr{O}$.
Let $C_A(\mathds{Q})$ be the set of rational points on $C_A$ together
with $\mathscr{O}$. It is easy to see that $\bigl(C_A(\mathds{Q}),+\bigr)$.
is a subgroup of $(C_A,+)$. Moreover, it is well known that
the group $\bigl(C_A(\mathds{Q}),+\bigr)$ is finitely generated.
One can readily check that the three points $(0,0)$ and
$(\pm A,0)$ are the only points on $C_A$ of order~$2$,
and one easily finds other points of finite order on $C_A$.
However, it is well known that if
$A$ is a congruent number and $(x_0,y_0)$ is a rational
point on $C_A$ with $y_0\neq 0$, then the order of $(x_0,y_0)$
is infinite. In particular, if there exists one {rational Py\-tha\-go\-re\-an $A$-triple}, then
there exist infinitely many such triples (for an elementary proof
of this result, which is based on a theorem of Fermat's, see
Halbeisen and Hungerb\"uhler~\cite{Fermat}).
Furthermore, {\sc Mordell's Theorem}
states that the group of rational points on $C_A$ is
finitely generated, and by the
{\sc Fundamental Theorem of Finitely Generated Abelian Groups},
the group of rational points on an elliptic curve is isomorphic
to some group of the form
$$\underset{\text{\scriptsize torsion group}}
{\underbrace{\mathds{Z}/n_1\mathds{Z}\times\ldots\times\mathds{Z}/n_k\mathds{Z}}}\times\mathds{Z}^r,$$
where $n_1,\ldots,n_k$ and $r$ are positive integers. The group
$\mathds{Z}/\mathds{Z}_{n_1}\times\ldots\times\mathds{Z}/\mathds{Z}_{n_k}$, which is generated by rational
points of finite order, is the so-called
\emph{torsion group}, and $r$ is
called the \emph{rank\/} of the curve. Now, since $C_A$ does not
have rational points of finite order besides the points $(0,0)$ and
$(\pm A,0)$, the torsion group of $C_A$ is isomorphic to
$\mathds{Z}/2\mathds{Z}\times\mathds{Z}/2\mathds{Z}$.
Based on integral solutions of $m^2=n^2+nl+l^2$, we will show that there
are infinitely many congruent number curves $C_A$ with rank at least two
(for congruent number curves with rank at least three see
Johnstone and Spearman~\cite{JS}).
\section{Rank at Least Two}
In order to ``compute'' the rank of a curve of the form
$$\Gamma: y^2=x^3+Bx,$$ according to
Silverman and Tate\;\cite[Chapter\,III.6.]{SilvermanTate},
we first have to write down several equations of the form
\begin{eqnarray}
b_1 M^4+b_2e^2&=&N^4\label{eqn:A}\\[1.2ex]
\bar{b}_1\bar{M}^4+\bar{b}_2\bar{e}^2&=&\bar{N}^4, \label{eqn:barA}
\end{eqnarray}
namely one for each factorisation $B=b_1b_2$ and $-4B=\bar{b}_1\bar{b}_2$, respectively,
where $b_1$ and $\bar{b}_1$ are square-free. Then we have to decide,
how many of these equations have integral solutions:
Let $\#\alpha(\Gamma)$ be
the number of equations of the form~(\ref{eqn:A}) for which we find
integral solutions $M,e,N$ with $e\neq 0$, and
let $\#\alpha(\bar\Gamma)$ be
be the corresponding number with respect to equations of the form~(\ref{eqn:barA}).
Then, if $r>0$, $$2^r=\frac{\#\alpha(\Gamma)\cdot \#\alpha(\bar\Gamma)}{4}.$$
Moreover, one can show that if $(x,y)$ is a rational point on $\Gamma$, where $y\neq 0$,
then one can write that point in the form
\begin{equation}\label{eq-x_b1}
x=\frac{b_1 M^2}{e^2},\qquad y=\frac{b_1 M N}{e^3},
\end{equation}
where $M,e,N$ is an integral solution of an equation of the form~(\ref{eqn:A}),
and vice versa. The analogous statement holds for rational points on the curve
$\bar\Gamma: y^2=x^3-4Bx$ with respect to equations of the form~(\ref{eqn:barA}).
Now we are ready to prove
\begin{thm}\label{thm:main}
Let $m,n,l$ be pairwise relatively prime positive integers, where
$m=\prod_{i=1}^j p_i$ is a product of pairwise distinct primes
$p_i\equiv 1\mod 6$ and $m^2=n^2+nl+l^2$. Furthermore, let $k:=n+l$
and let $A:=klmn$. Then the rank of
the curve $$C_A: y^2=x^3-A^2x$$ is at least two.
\end{thm}
\begin{proof} Since we have at least one rational point $(x,y)$ on $C_A$
with $y\neq 0$, we know that the rank $r$ of $C_A$ is positive. So,
to show that the rank of the curve $C_A$ is at least two, it
is enough to show that $\#\alpha(C_A)\ge 9$. For this, we have
to show that there are integral solutions for~(\ref{eqn:A}) for at least~9
distinct square-free integers $b_1$ dividing $-A^2$, or equivalently,
we have to find at least~9 rational points on $C_A$, such that the~9
corresponding integers $b_1$ are pairwise distinct, which we will do now.
Notice that because of~(\ref{eq-x_b1}), to compute $b_1$ from a rational point $P=(x,y)$ on $C_A$
with $x\neq 0$, it is enough to know the $x$-coordinate of $P$ and then
compute $x$~mod~${\mathds{Q}^*}^2$ (\ie, we compute $x$ modulo squares of rationals).
The $x$-coordinates of the three integral points we get by~(\ref{psi}) from the three
{Py\-tha\-go\-re\-an $A$-triples} $(a_1,b_1,c_1)$, $(a_2,b_2,c_2)$, $(a_3,b_3,c_3)$ generated by
$m,n,l,k$, are
$$x_1=m^2(m^2-n^2)=m^2kl\,,\quad x_2=m^2(m^2-l^2)=m^2kn\,,\quad k^2(k^2-m^2)=k^2nl\,,$$
and modulo squares, this gives us three values for $b_1$,
namely $$b_{1,1}=kl\,,\quad b_{1,2}=kn\,,\quad b_{1,3}=nl\,.$$
Now, exchanging in each of the three {Py\-tha\-go\-re\-an $A$-triples} the two catheti $a_i$ and $b_i$ (for $i=1,2,3$),
we obtain again three distinct integral points on $C_A$, whose $x$-coordinates modulo squares
give us $$b_{1,4}=mn\,,\quad b_{1,5}=ml\,,\quad b_{1,6}=mk\,.$$
Finally, if we replace each hypothenuse $c_j$ of these six {Py\-tha\-go\-re\-an $A$-triples} with $-c_j$, we
obtain again six distinct integral points on $C_A$, whose $x$-coordinates modulo squares
give us
$$
\begin{array}{rclrclrcl}
b_{1,7}&\hspace{-1.5ex}=\hspace{-1.5ex}&-kl\,,
& b_{1,8}&\hspace{-1.5ex}=\hspace{-1.5ex}&-kn\,,
& b_{1,9}&\hspace{-1.5ex}=\hspace{-1.5ex}&-nl\,\\[1.6ex]
b_{1,10}&\hspace{-1.5ex}=\hspace{-1.5ex}&-mn\,,
& b_{1,11}&\hspace{-1.5ex}=\hspace{-1.5ex}&-ml\,,&
b_{1,12}&\hspace{-1.5ex}=\hspace{-1.5ex}&-mk\,.
\end{array}
$$
In addition to these~$12$ integral points on $C_A$ (and the
corresponding $b_1$'s), we have the two integral points $(\pm A,0)$ on $C_A$,
which give us $$b_{1,13}=klmn\qquad\text{and}\qquad b_{1,14}=-klmn.$$
Recall that, by assumption, $m$ is square-free and $k,l,n$ are pairwise relatively prime.
Therefore, if for some $i,j$ with $1\le i<j\le 14$, $b_{1,i}\equiv b_{1,j}$ modulo squares,
at least two of the integers $k,l,n$ are squares, say $n=u^2$, and $l=v^2$ or $k=v^2$.
Then $$m^2=u^4+u^2v^2+v^4\quad\text{(in the case when $l=v^2$),}$$
or $$m^2=u^4-u^2v^2+v^4\quad\text{(in the case when $k=v^2$).}$$
If $l=v^2$, this implies that $u^2=1$ and $v=0$, or $u=0$ and $v^2=1$, and
if $k=v^2$, this implies that $u^2=1$ and $v=0$, $u=0$ and $v^2=1$, or $u^2=v^2=1$
(see, for example, Mordell~\cite[p.\,19{\sl f\/}]{MordellBook} or
Euler~\cite[p.\,16]{Euler1780}).
So, at most one of the integers $k,l,n$ is a square, which implies that
$\#\alpha(C_A)\ge 14$ and this completes the proof.
\end{proof}
As an immediate consequence we get the following
\begin{cor} Let $m,n,l$ be as in {\sc Theorem}\;\ref{thm:main} and let
$q$ be a non-zero integer. Then the rank of
the curve $C_{Aq^4}$ is at least two.
\end{cor}
\begin{proof}
Notice that if $m,n,l$ are such that $m^2=n^2+nl+l^2$, then,
for $mq,nq,lq$, we have $(mq)^2=(nq)^2+nq\cdot lq+(lq)^2$,
which implies that for $\tilde A=kq\cdot lq\cdot mq\cdot nq=Aq^4$,
the rank of the curve $C_{\tilde A}$ is at least two.
\end{proof}
\section{Odds and Ends}
As a matter of fact we would like to mention that the
three integral points on $C_A$ which correspond to an integral solution
of $m^2=n^2+nl+l^2$ lie on a straight line.
\begin{fct}
Let $m,n,l$ be positive integers such that $m^2=n^2+nl+l^2$, let
$k=n+l$, and let $A=klmn$. Then the three integral points
\begin{eqnarray*}
\bigl(\,\underset{x_1}{\underbrace{m^2(m^2-n^2)}}\,,\;
\underset{y_1}{\underbrace{m^2(m^2-n^2)^2}}\,\bigr)\,,\\[1.4ex]
\bigl(\,\underset{x_2}{\underbrace{m^2(m^2-l^2)}}\,,\;
\underset{y_2}{\underbrace{m^2(m^2-l^2)^2}}\,\bigr)\,,\\[1.4ex]
\bigl(\,\underset{x_3}{\underbrace{k^2(k^2-m^2)}}\,,\;
\underset{y_3}{\underbrace{k^2(k^2-m^2)^2}}\,\bigr)\,,
\end{eqnarray*}
on the curve $C_A$ lie on a straight line.
\end{fct}
\begin{proof}
For $i=2,3$ let $$\lambda_{1,i}:=\frac{y_i-y_1}{x_i-x_1}\,.$$
It is enough to show that $\lambda_{1,2}=\lambda_{1,3}$, or equivalently,
that $\lambda_{1,3}-\lambda_{1,2}=0$. Now, an easy calculation shows that
$\lambda_{1,2}=k^2$ and that $\lambda_{1,3}-k^2=\frac{0}{k(k-n)^3}=0$.
\end{proof}
As a last remark we would like to mention that with the help
of {\scriptsize \sagelogo} we found that some of the curves
which correspond to an integral solution of $m^2=n^2+nl+l^2$ have
rank~3 or higher. In fact, we found several curves of rank~$3$ or~$4$,
as well as the following curves of rank~$5$:
\begin{comment}
\begin{small}
\begin{center}
\begin{tabular}{rccrr}
$A$\hspace{3ex} & factorisation & $m$ & $l$\hspace{1ex} & $n$\hspace{1ex}{\,}\\
\hline\\
$8\,813\,542\,297\,560$ &
$2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 19 \cdot 29 \cdot 37 \cdot 59 \cdot 61$ &
$7\cdot 13\cdot 37$ & $232$ & $3\,245$\\
$81\,096\,660\,783\,600$ &
$2^4 \cdot 3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 37 \cdot 103$ &
$37\cdot 103$ & $2\,139$ & $2\,261$\\
$225\,722\,120\,463\,840$ &
$2^5 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 19 \cdot 31 \cdot 53 \cdot 101 \cdot 149$ &
$13\cdot 19\cdot 31$ & $505$ & $7\,392$\\
$457\,485\,316\,904\,280$ &
$2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 97 \cdot 179$ &
$7\cdot 31\cdot 37$ & $895$ & $7\,544$\\
$5\,117\,352\,889\,729\,080$ &
$2^3 \cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 19\cdot 31\cdot 47 \cdot 67\cdot 103\cdot 223$ &
$67\cdot 223$ & $1\,551$ & $14\,105$\\
$281\,692\,457\,452\,791\,000$ &
$2^3 \cdot 3^4 \cdot 5^3 \cdot 7 \cdot 11 \cdot 41 \cdot 79 \cdot 103 \cdot 331 \cdot 409$ &
$79\cdot 409$ & $9\,064$ & $26\,811$
\end{tabular}
\end{center}
\end{small}
\end{comment}
\begin{center}
\begin{tabular}{rcrrr}
$A=klmn$\hspace{1ex} & \hspace{3ex}$m=\prod p_{i\mathstrut}$\hspace{3ex} &
\hspace{5ex}$l$\hspace{1ex} & \hspace{9ex}$n$\hspace{1ex} & \hspace{3ex}$k=n+l$\\
\hline\\
$237\,195\,512\,400$ &
$7\cdot 127$ & $464$ & $561$ & $1\,025$\\
$8\,813\,542\,297\,560$ &
$7\cdot 13\cdot 37$ & $232$ & $3\,245$ & $3\,477$\\
$10\,280\,171\,942\,040$ &
$37\cdot 67$ & ${741}$ & $2\,024$ & $2\,765$\\
$81\,096\,660\,783\,600$ &
$37\cdot 103$ & $2\,139$ & $2\,261$ & $4\,400$\\
$225\,722\,120\,463\,840$ &
$13\cdot 19\cdot 31$ & $505$ & $7\,392$ & $7\,897$\\
$457\,485\,316\,904\,280$ &
$7\cdot 31\cdot 37$ & $895$ & $7\,544$ & $8\,439$\\
$5\,117\,352\,889\,729\,080$ &
$67\cdot 223$ & $1\,551$ & $14\,105$ & $15\,656$\\
$281\,692\,457\,452\,791\,000$ &
$79\cdot 409$ & $9\,064$ & $26\,811$ & $35\,875$\\
$24\,666\,188\,870\,481\,576\,600$ &
$13\cdot 31\cdot 223$ & $46\,169$ & $57\,400$ & $103\,569$
\end{tabular}
\end{center}
It is possible that these curves might be candidates
for high rank congruent number elliptic curves (for another approach see
Dujella, Janfada, Salami~\cite{HighRank}).
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
1,314,259,993,018 | arxiv |
\section{Introduction}
\input{introduction}
\section{Pre-existing components}
\input{preexisting}
\section{Results}
\input{results}
\section{Conclusion}
\input{conclusion}
\subsubsection*{Acknowledgments}
\input{acks}
\small
\input{nestrl_manuscript_arxiv.bbl}
\subsection{Extending the ROS - MUSIC toolchain}
\label{sec:zmq-adapter-ros}
We extended the RMT by adding adapters that support communication via ZeroMQ following a publish-subscribe pattern.
ZeroMQ is a messaging library that allows applications to exchange messages at runtime via sockets.
Continuously developed by a large community, it offers bindings for a variety of languages including C++ and Python, and supports most operating systems.
A single communication adapter of the RMT sends (receives) data via a ZeroMQ socket and receives (sends) data via a MUSIC port.
While the adapters can handle arbitrary data, we defined a set of specialized messages in JSON format (see supplementary material) specifically designed to communicate observations, rewards and actions as discrete or continuous real-valued variables of arbitrary dimensions, as used in the OpenAI Gym.
We chose the JSON format due to its simplicity, easy serialization and broad platform support.
In addition to the ZeroMQ adapters dedicated for communication with MUSIC, we developed several further adapters that can perform specific transformations of the data.
As discussed above, environments can be defined in continuous or discrete spaces with arbitrary dimensionality. To generate the required closed-loop functionality, the observations provided by the environment must be consistently transformed to a format that can be fed into neural network simulations. Conversely, the activity of the neural network must be interpreted and transformed into valid actions which can be executed in the environment.
A standard way to address the first issue is to introduce so called {\it place cells}. Each of these cells is tuned to a preferred (multidimensional) observation, i.e., is highly active for a specific input and less active for other inputs \cite{Fremaux2013_e1003024}. The dependence of the activity of a single place cell on observations is described by its tuning curve, often chosen as a multidimensional Gaussian. To perform the transformation of observations to activity of place cells, we implemented a {\it discretize adapter} that allows users to specify the position and width of the tuning curves of an arbitrary number of place cells. While having a certain biological plausibility \cite{Moser08_69}, one disadvantage of this approach is that the number of place cells required to cover the whole observation space evenly scales exponentially in the number of dimensions of the observation. For observations with a small number of dimensions, however, this approach is very suitable.
To perform action selection, we added several adapters that can, respectively, select the most active neuron ({\it argmax adapter}), threshold the activity across neurons to create a binary vector ({\it threshold adapter}) or linearly combine the activity of neurons across many input channels ({\it linear decoder}). Depending on the type of action required (discrete/continuous) by the environment, the user can select a single one or a combination of these. See the documentation of the RMT for detailed specifications of the adapters.
In general, we followed the design principle behind the RMT and developed modular adapters. This makes each individual adapter easy to understand and enables users to quickly extend the toolchain with their own adapters. By combining several adapters, the RMT allows arbitrarily complex transformations of the data and can hence be applied to many use-cases.
\subsection{ZeroMQ wrapper for the OpenAI Gym}
\label{sec:zeromq-gym}
The second part of the toolchain is a Python wrapper around the OpenAI Gym that exposes ZeroMQ sockets for communicating actions, observations and rewards (see \fref{fig:general_interface}). An environment in the OpenAI Gym is updated in steps. In each step, an agent needs to provide an action and receives an observation and reward for its current state.
The wrapper consists of four different threads that coordinate: (i) performing steps in an environment, (ii) receiving actions via a ZeroMQ SUB socket, (iii) publishing observations via a ZeroMQ PUB socket and (iv) publishing rewards via a ZeroMQ PUB socket.
Before spawning the threads, the wrapper starts a user-specified environment and creates the necessary communication buffers. The thread coordinating the environment reads actions from the corresponding buffer, performs single steps in the environment and updates the observation and reward buffers based on the return values of the environment. Upon detecting that a single episode has ended, e.g., by an agent reaching a certain goal position, it resets the environment and allows a user-specified break before starting another episode.
The communication threads continuously send(receive) messages via ZeroMQ and read from(write to) the corresponding buffers. All threads can be run with different update intervals, for example, to slow down movement of the agent by performing steps on a coarse time grid whilst continuously receiving action choices from the neural network simulation running on a fine time grid. The user can specify a variety of parameters via a configuration file in JSON format (see supplementary material). See the documentation for detailed specifications of the wrapper.
\subsection{Applications}
To demonstrate the functionality of the toolchain, we implemented a
neural network model with actor-critic architecture in NEST and
trained it on two different environments simulated in the OpenAI Gym.
In the first task, the agent needs to learn to perform a sequence of
actions in order to reach the top of a hill in a continous
environment. The second task is a classical grid-world in which
an agent needs to learn to navigate to a goal position in a
two-dimensional discrete environment with obstacles. We first
describe the neural network architecture and learning rule and
afterwards discuss the network's performance on the two tasks.
\subsubsection{Neural network implementation}
\label{sec:neural-network}
We consider a temporal-difference learning algorithm \cite{Sutton98} implemented as an actor-critic architecture, originally using populations of spiking neurons \cite{Fremaux2013_e1003024}.
We translated the spike-based implementation to rate neurons, mainly to simplify the implementation by avoiding issues arising from noise introduced by spiking neuron models \cite{Potjans11_e101133,Fremaux2013_e1003024}.
The neuron dynamics we considered here are determined by the following stochastic differential equation:
\begin{align}
\tau \frac{dz_i(t)}{dt}=-z_i(t) + \mu_i + f\left(h_i(t) - \theta_i\right) + \xi_i(t)\, ,
\label{eq:rate_model}
\end{align}
where $\tau$ is some positive time constant, $\mu_i$ a baseline activity
level, $f(\cdot)$ some (arbitrary) activation function, $h_i(t)$ a time
dependent input field, $\theta_i$ an input threshold and $\xi_i(t)$ white
noise with a certain standard deviation $\sigma_\xi$. The input field
$h_i(t)$ is determined by the activity of other neurons according to
$h_i(t) = \sum_j w_{ij}z_j(t)$, with $w_{ij}$ denoting the strength of
the connection (weight) from neuron $j$ to neuron $i$. Here we will
exclusively consider activation functions of the form $f(x)=x$ (linear
case), and $f(x)=\Theta(x)x$ (threshold-linear case, ``relu''). These
models have recently been added to the NEST simulator.
Their dynamics are solved on a fixed time-grid
by a stochastic-exponential-Euler method with a step size determined
by the resolution of the simulation. For more details on the model
implementation see \cite{Hahne16_arxiv}.
The input layer is a population of threshold-linear rate neurons which
receive inputs through MUSIC and encode observations from the environment
(see \fref{fig:NN_implementation}). Via plastic connections these
place cells project to a single neuron representing the value that the
network assigns to the current state (the ``critic''). An additional
neuron calculates the reward-prediction error by combining the reward
received from the environment with input from the critic. Plasticity
of the projections from inputs to the critic is modulated by this
reward-prediction error (see below).
In addition, neurons in the input layer project to a population of
neurons representing the available actions (the ``actor''). To enforce
selection of a specific action, the actor units are arranged in a
winner-take-all (WTA) circuit. This is implemented by recurrent
connections between actor units that correspond to short-range
excitation and long-range inhibition, depending on the similarity of
the action that actor units encode. The activity of actor units is
transformed to a valid action and communicated to the environment via the RMT.
\begin{figure}
\center
\includegraphics[height=0.3\textheight]{\figuredirnestrl/network.pdf}
\caption{{\bf Actor-critic architecture for RL with rate neurons}. Observations are communicated via a MUSIC input port to a population of place cells. These project on the one hand to a critic unit and on the other hand to actor units arranged in a winner-take-all circuit. The critic and an additional MUSIC input port project to a unit representing the reward-prediction error that modulates the plasticity between place cells and critic and actors, respectively. The actor units project to a MUSIC output port encoding the selected action.}
\label{fig:NN_implementation}
\end{figure}
To derive a learning rule for the critic, we followed similar steps as in \cite{Fremaux2013_e1003024} applied to rate models (equation \eqref{eq:rate_model}). The critic activity should approximate a continous-time value function defined by \cite{Doya00}
\begin{align}
V^\pi(\vec{s}(t)):= \int_{t'}^\infty r(\vec{s}^\pi(t'))e^{-\frac{t'-t}{\tau_r}}dt'\, .
\label{eq:cont_value}
\end{align}
Here $\vec{s}(t)$ denotes the state of the agent at time $t$, $r(\vec{s}^\pi(t))$ denotes the reward obtained in state $\vec{s}(t)$, $\tau_r$ a discounting factor for future rewards and $\pi$ the agent's policy.
To achieve this, we define the following objective function which should be minimized by gradient descent on the weights from inputs to the critic:
\begin{align}
E(t) = \frac{1}{2}(V^\pi(t) - z(t))^2\, ,
\label{eq:objective_function}
\end{align}
where $z(t)$ represents the activity of the critic unit.
By performing gradient descent on equation \eqref{eq:objective_function}, using a self-consistency equation for $V^\pi(t)$ from the derivative of equation \eqref{eq:cont_value} and bootstrapping on the current prediction for the value (see supplementary material and \cite{Doya00,Fremaux2013_e1003024}), we obtain the following local Hebbian three-factor learning rule that approximately minimizes the objective function (equation \eqref{eq:objective_function}):
\begin{align}
\Delta w_j = \eta \delta(t) x_j(t) \Theta\left(z(t) - \theta_\text{post} \right)\, ,
\label{eq:learning_rule}
\end{align}
where $\eta$ is a learning rate, $x_j(t)$ represents the activity of the $j$th place cell, $\Theta(\cdot)$ the Heaviside function and $\theta_\text{post}$ a parameter that accounts for noise on the postsynaptic unit (see supplementary material for details). The term $\delta(t) = \dot{v}(t) + r(t) - \frac{1}{\tau_r} v(t)$ corresponds to the activity of the reward-prediction error unit, acting as a neuromodulatory signal for the Hebbian plasticity between the presynaptic ($x_j$) and postsynaptic ($z$) units. To avoid explicit calculation of the derivative, we rewrite $\delta(t)$ as:
\begin{align}
\delta(t) \approx \left(\frac{1}{d} - \frac{1}{\tau_r} \right)v(t) - \frac{1}{d} v(t-d) + r(t)\, .
\end{align}
To approximate the derivative we hence implement two connections from the critic to the reward-prediction error unit: one instantaneous, and one with delay $d > 0$.
To learn an optimal policy, we exploit that the actor units
follow the same dynamics as the critic. Similar to
\cite{Fremaux2013_e1003024}, we hence apply the same learning rule
to the connections between the inputs and the actor units. In order
to assure that at least one actor unit is active, thus preventing a
deadlock, we introduce a minimal weight for each connection between
input and output units and add input noise to the actor units.
\subsubsection{Mountain Car}
\label{sec:mountain-car}
As an example of an environment with continous states, we consider the
\emph{MountainCar} environment. The task is to steer a toy vehicle
that starts at a valley between two hills to the top of the right one
(\fref{fig:mountain_car}{\bf A}, inset). To make the task more challenging,
the car's engine is not strong enough to reach the top in one go, so
the agent needs to learn to gain momentum by swinging back and forth
between the two hills. A single episode in this environment starts
when the agent is placed in the valley and ends when it reaches the
final position on the top of the right hill. The state of the agent is
described by two continous variables: the x-position $x(t)$ and the
x-velocity $\dot{x}(t)$. The agent can choose from three different
discrete actions that affect the velocity of the vehicle (accelerate
left, no acceleration, accelerate right). It receives
punishment from the environment in every step; the goal is to
minimize the total punishment collected over the whole episode. Since
this is difficult to implement in an actor-critic architecture
\cite{Potjans11_e101133}, we provide additional reward when the agent
reaches the final position.
To translate the agent's current state into neuronal activity, we distribute 25 place cells evenly across the two dimensional plane of possible positions and velocities using the \emph{discretize adapter} of the RMT. The actor is implemented by a WTA circuit of three units as described in \fref{sec:neural-network}. The activity of these units is transformed into an action via the \emph{argmax adapter} (\fref{sec:zmq-adapter-ros}).
\begin{figure}[t]
\center
\includegraphics[width=1.0\textwidth]{\figuredirnestrl/mountain_car.pdf}
\caption{{\bf Network performance on an environment with continuous states and discrete actions.}
{\bf A:} Reward obtained by the agent per episode averaged over $10$ simulations with different seeds (solid orange line). Orange band indicates $\pm$ one standard deviation. Light gray line marks average reward per episode for which the environment is considered solved. Inset: screenshot of the environment with agent (stylized vehicle), environment with valley and two hills (black solid line) and goal position (yellow flag). The agent is close to the starting position at the trough.
{\bf B:} Activity traces of place cells (bottom), actor units (second from bottom), critic unit (second from top) and reward-prediction-error unit (top). Shown are neural activities during $6.5\;\mathrm{s}$ early (left) and late (right) in the simulation.
}
\label{fig:mountain_car}
\end{figure}
Initially, the agent explores the environment by selecting random
actions. Due to the WTA circuit dynamics, a single action stays active
over an extended period of time. The constant punishment gradually
decreases the weights from the place cells to the corresponding actor unit,
eventually leading to another actor unit becoming active
(\fref{fig:mountain_car}{\bf B}, left). After a while, the agent
reaches the goal by performing actions that have not been
significantly punished. For this task the stable nature of the WTA is
advantageous, causing the agent to perform the same action repeatedly
allowing efficient exploration of the state space. After the agent has found the
goal once, the number of steps spent on exploring actions in the
following episodes is much smaller. From the sixth episode on, the
performance of the agent is already close to optimal
(\fref{fig:mountain_car}{\bf A}). After learning for about 10
episodes, the agent's performance has converged. The value of the
final state is successfully propagated backwards over different
states, leading to a ramping of activity of the critic unit from the
start of an episode to the end (\fref{fig:mountain_car}{\bf B},
right).
Since the OpenAI Gym offers a variety of environments, we trained the same network model on an additional task with different requirements.
\subsubsection{Frozen Lake}
\label{sec:frozen-lake}
As a second application, we chose the \emph{FrozenLake} environment
consisting of a discrete set of 16 states arranged in a four-by-four
grid (\fref{fig:grid_world}{\bf A}, inset). Each state is either a start
state (S), a goal state (G), a hole (H) or a frozen state (F).
From the start position, the agent has to reach the rewarded state
by navigating over the frozen states without falling into holes which
would reset the agent to the starting position. In each step, the
agent can choose from four different actions: move west, move north,
move east and move south. Usually, the tiles are ``slippery'', i.e.,
there is a chance that a random action is executed irrespective of the
action chosen by the agent. However, to simplify learning for
demonstration purposes, we turned this feature off. Upon reaching the
goal, the agent receives a reward of magnitude one. Since the optimal
path involves six steps from start to goal, the theoretical optimal
reward per step is $\sim0.16$. To encourage exploration, the agent
receives a small punishment in each state, and additionally,
to speed up learning, the agent is punished for falling into holes.
Unlike in the continuous \emph{MountainCar} environment, the tuning curves of
place cells do not overlap in the discrete case, leading to sharp
transitions in the network activity. This leads to severe issues for
associating values and actions with the respective states. To
address this problem we introduced a simple eligibility trace by
evaluating the activity of the pre- and post synaptic units in the
learning rule with a small delay $\delta t$ (see supplementary material). With this addition, the network model is
able to find the optimal solution for this task within roughly 2000
steps (\fref{fig:grid_world}{\bf A}). It also learns to associate
holes with punishment and frozen states with reward if they are on the
path to the goal (\fref{fig:grid_world}{\bf B}). Although there are
two possible paths to the goal, the agent prefers the path with less
corners, since it is easier to navigate using a WTA circuit.
\begin{figure}[t]
\center
\includegraphics[width=1.0\textwidth]{\figuredirnestrl/frozen_lake.pdf}
\caption{{\bf Network performance on a grid-world environment.}
{\bf A}: Average reward collected by the agent over the next 500 steps (orange solid line) averaged over 5 simulations.
Orange band indicates $\pm$ one standard deviation.
Gray line: theoretical optimum.
Inset: screenshot of the environment with start state (S), frozen states (F), holes (H) and goal state (G). The position of the agent is indicated in pink.
{\bf B}: The learned policy and value map of the environment.
Red colors indicate positive, blue colors negative values.
Arrows indicate the preferred direction of movement.}
\label{fig:grid_world}
\end{figure}
\section{Derivation of the learning rule}
\input{derivation}
\section{Network description}
The tables \ref{tab:nordlie} and \ref{tab:params} summarize the network architecture and parameters.
\newlength{\columnfigwidth}
\newlength{\fullfigwidth}
\setlength{\columnfigwidth}{8.6cm}
\setlength{\fullfigwidth}{17.2cm}
\newlength{\columnwidthleft}
\newlength{\columnwidthmiddle}
\newcommand{\modelhdr}[3]{
\multicolumn{#1}{|l|}{
\color{white}
\cellcolor[gray]{0.0}
\textbf{\makebox[0pt][l]{#2}\hspace{0.5\fullfigwidth}\makebox[0pt][c]{#3}}
}
}
\newcommand{\parameterhdr}[3]{
\multicolumn{#1}{|l|}{
\color{black}\cellcolor[gray]{0.8}
\textbf{\makebox[0pt][l]{#2}\hspace{0.5\fullfigwidth}\makebox[0pt][c]{#3}}
}
}
\begin{table*}
\setlength{\columnwidthleft}{0.2\textwidth}
\setlength{\columnwidthmiddle}{0.2\textwidth}
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\modelhdr{2}{A}{Model summary}\\\hline
%
Populations & Seven \\
\hline
Topology & None \\
\hline
Connectivity & Population specific \\
\hline
Neuron model & Linear \& threshold-linear rate \\
\hline
Channel models & None \\
\hline
Synapse model & Instantaneous \& delayed continuous coupling \\
\hline
Plasticity & Three-factor Hebbian \\
\hline
External input & Continuous MUSIC ports \\
\hline
External output & Continuous MUSIC ports \\
\hline
Measurements & Rates of all neurons \\
\hline
%
\end{tabularx} \\
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|p{\columnwidthmiddle}|X|}
\hline\modelhdr{3}{B}{Populations}\\\hline
\bf Name & \bf Elements & \bf Size \\
\hline
%
Observation & MUSIC in port & 1 \\
Reward & MUSIC in port & 1 \\
Action & MUSIC out port & 1 \\
Place cells & Threshold-linear & $16 (25)$ \\
Critic & Threshold-linear & $1$ \\
Actor & Threshold-linear & $4 (3)$ \\
Prediction error & Linear & $1$ \\
\hline
%
\end{tabularx} \\
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|p{\columnwidthmiddle}|X|}
\hline\modelhdr{3}{C}{Connectivity}\\\hline
\bf Source & \bf Target & \bf Pattern \\
\hline
%
Observation & Place cells & One-to-one (by MUSIC channel), instantaneous, static, weight $w_\text{o}$ \\
Reward & Prediction error & One-to-one (by MUSIC channel), instantaneous, static, weight $w_\text{r}$ \\
Actor & Action & One-to-one (by MUSIC channel), instantaneous, static, weight $w_\text{a}$ \\
Place cells & Critic & All-to-all, instantaneous, plastic, initial weight $w_\text{pc}$ \\
Place cells & Actor & All-to-all, instantaneous, plastic, initial weight $w_\text{pa}$ \\
Critic & Prediction error & One-to-one, instantaneous, static, weight $1/d - 1/\tau_r$ \\
Critic & Prediction error & One-to-one, delay $d$, static, weight $-1/d$ \\
Actor & Actor & All-to-all, instantaneous, static, weight $\alpha \exp(-\Delta a / \sigma) + \beta$ \\
\hline
%
\end{tabularx} \\
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\modelhdr{2}{D}{Neuron and synapse model}\\\hline
%
Type & Linear rate neuron \\
Dynamics & $\tau \frac{dz(t)}{dt}=-z(t) + \mu + \left(h(t) - \theta\right) + \xi(t)$ \\
\hline
Type & Threshold-linear rate neuron \\
Dynamics & $\tau \frac{dz(t)}{dt}=-z(t) + \mu + \Theta\left(h(t) - \theta\right)\left(h(t) - \theta\right) + \xi(t)$ \\
\hline
Type & Three-factor Hebbian synapse \\
Plasticity & $\Delta w_{ij} = \eta \delta(t) g x_j(t - \delta t) \Theta(z_i(t - \delta t) - \theta_\text{post})$ \\
\hline
%
\end{tabularx}
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\modelhdr{2}{E}{Input}\\\hline
\bf Type & \bf Description \\
\hline
Observation & Rate $r\in [-1, 1]$ according to tuning of place cell (using {\it discretize} adapter) \\
\hline
Reward & Rate $r\in [-1, 1]$ according to reward provided by the environment \\
\hline
\end{tabularx}
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\modelhdr{2}{F}{Output}\\\hline
\bf Type & \bf Description \\
\hline
Action & Rates $r_i \in [0, \infty)$ according to activities of the actor units \\
\hline
\hline
\end{tabularx}
\caption{
Description of the network model (according to \citep{Nordlie-2009_e1000456}).
\label{tab:nordlie}
}
\end{table*}
\begin{table*}
\setlength{\columnwidthleft}{0.2\textwidth}
\setlength{\columnwidthmiddle}{0.2\textwidth}
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\parameterhdr{2}{B}{Populations: place cells}\\\hline
\bf Name & \bf Values \\
\hline
$\tau$ & 5.0 (1.0) \\
\hline
$g$ & 1.0 \\
\hline
$\mu$ & 0.0 \\
\hline
$\sigma_\xi$ & 0.0 \\
\hline
$\theta$ & -0.5 \\
\hline
%
\hline\parameterhdr{2}{B}{Populations: critic}\\\hline
\bf Name & \bf Values \\
\hline
$\tau$ & 0.1 \\
\hline
$g$ & 1.0 \\
\hline
$\mu$ & -1.0 \\
\hline
$\sigma_\xi$ & 0.0 \\
\hline
$\theta$ & -1.0 \\
\hline
%
\hline\parameterhdr{2}{B}{Populations: reward}\\\hline
\bf Name & \bf Values \\
\hline
$\tau$ & 1.0 \\
\hline
$g$ & 1.0 \\
\hline
$\mu$ & 0.0 \\
\hline
$\sigma_\xi$ & 0.0 \\
\hline
$\theta$ & 0.001 (-0.0999) \\
\hline
%
\hline\parameterhdr{2}{B}{Populations: actor}\\\hline
\bf Name & \bf Values \\
\hline
$\tau$ & 0.1 \\
\hline
$g$ & 1.0 \\
\hline
$\mu$ & 0.0 \\
\hline
$\sigma_\xi$ & 0.2 (0.05) \\
\hline
$\theta$ & 0.0 \\
\hline
%
\hline\parameterhdr{2}{C}{Connectivity}\\\hline
\bf Name & \bf Values \\
\hline
%
$w_\text{o}$ & $0.5$ \\
\hline
$w_\text{r}$ & $0.1$ \\
\hline
$w_\text{a}$ & $1.0$ \\
\hline
$w_\text{pc}$ & $0.0 $ \\
\hline
$w^\text{min}_\text{pc}$ & $-1.0$ \\
\hline
$w^\text{max}_\text{pc}$ & $1.0$ \\
\hline
$\theta_\text{post}^\text{pc} $ & $-1.0$ \\
\hline
$w_\text{pa}$ & $0.9 (0.3)$ \\
\hline
$w^\text{min}_\text{pa}$ & $0.1 (0.05)$ \\
\hline
$w^\text{max}_\text{pa}$ & $1.0$ \\
\hline
$\theta_\text{post}^\text{pa} $ & $0.5 (0.1)$ \\
\hline
$d$ & $1.0$ \\
\hline
$\tau_r$ & $20000.0$ \\
\hline
$\alpha$ & $1.2$ \\
\hline
$\beta$ & $-0.55$ \\
\hline
$\sigma_\xi$ & $0.1$ \\
\hline
$\eta_\text{critic}$ & $0.01 (0.125)$ \\
\hline
$\eta_\text{actor}$ & $0.2 (0.250)$ \\
\hline
$\delta t$ & $19.0 (0.0)$ \\
\hline\parameterhdr{2}{E}{Input: discretize adapter}\\\hline
\bf Name & \bf Values \\
\hline
%
$\sigma_x$ & 0.01 (0.2) \\
$\sigma_y$ & -- (0.2) \\
%
\hline
\end{tabularx} \\
\caption{
Table of the network parameters used for both tasks (according to \citep{Nordlie-2009_e1000456}). Values in brackets are used for the \emph{MountainCar} environment.
\label{tab:params}
}
\end{table*}
\section{JSON Message types}
Listing \ref{lst:message_types} show the standard message types used for communication between the OpenAI Gym and the RMT. All messages are serialized using JSON and communicated via ZeroMQ.
\begin{lstlisting}[caption=Message types used for communication.,label=lst:message_types]
BasicMsg
{
float min
float max
float value
float timestamp
}
ObservationMsg
{
BasicMsg[] observations # one basic msg per dimension
}
RewardMsg
{
BasicMsg[] reward # reward is always one dimensional
}
ActionMsg
{
BasicMsg[] actions # one dimensional for discrete actions
# or one dimension per possible action
}
\end{lstlisting}
\section{Example wrapper configuration file}
Listing \ref{lst:zmq-wrapper-conf} shows an example configuration file for running the mountain car environment.
\begin{lstlisting}[caption=Example configuration file for the wrapper to run the ``MountainCar-v0'' environment.,label=lst:zmq-wrapper-conf]
"All":
{
"seed": 12345,
"time_stamp_tolerance": 0.01,
"prefix": null,
"write_report": true,
"report_file": "./report.json",
"overwrite_files": false,
"flush_report_interval": null
},
"Env":
{
"env": "MountainCar-v0",
"initial_reward": null,
"final_reward": null,
"min_reward": -1.0,
"max_reward": 1.0,
"render": true,
"monitor": false,
"monitor_dir": "./experiment-0/",
"monitor_args":
{
"write_upon_reset": true,
"video_callable": false
}
},
"EnvRunner":
{
"update_interval": 0.01,
"inter_trial_duration": 0.4
},
"CommandReceiver":
{
"socket": 5555,
"time_stamp_tolerance": 0.01
},
"ObservationSender":
{
"socket": 5556,
"update_interval": 0.01
},
"RewardSender":
{
"socket": 5557,
"update_interval": 0.01
}
\end{lstlisting}
\section{Example MUSIC configuration file}
Listing \ref{lst:music-conf} shows an example MUSIC configuration file to run the MountainCar environment. It shows the different processes with parameters which are spawned by MUSIC including RMT adapters and NEST.
\begin{lstlisting}[caption=Example MUSIC configuration file to run the MountainCar environment.,label=lst:music-conf]
stoptime=150.
rtf=1.
[reward]
binary=zmq_in_adapter
args=
np=1
music_timestep=0.001
message_type=GymObservation
zmq_topic=
zmq_addr=tcp://localhost:5557
[sensor]
binary=zmq_in_adapter
args=
np=1
music_timestep=0.001
message_type=GymObservation
zmq_topic=
zmq_addr=tcp://localhost:5556
[discretize]
binary=discretize_adapter
args=
np=1
music_timestep=0.001
grid_positions_filename=grid_pos.json
[nest]
binary=../actor_critic_network/network.py
args=-t 150. -n 25 -m 3 -p network_params.json
np=1
[argmax]
binary=argmax_adapter
args=
np=1
music_timestep=0.001
[command]
binary=zmq_out_adapter
args=
np=1
music_timestep=0.01
message_type=GymCommand
zmq_topic=
zmq_addr=tcp://*:5555
sensor.out->discretize.in[2]
discretize.out->nest.in[25]
reward.out->nest.reward_in[1]
nest.out->argmax.in[3]
argmax.out->command.in[1]
\end{lstlisting}
\section{Environments}
Table \ref{tab:env} shows parameters for the OpenAI Gym environments and the ZeroMQ wrapper.
\begin{table*}
\label{tab:env}
\setlength{\columnwidthleft}{0.4\textwidth}
\setlength{\columnwidthmiddle}{0.2\textwidth}
\begin{tabularx}{\fullfigwidth}{|p{\columnwidthleft}|X|}
\hline\parameterhdr{2}{ }{OpenAI Gym}\\\hline
\bf Name & \bf Values \\
\hline
Version & 0.8.1 \\
\hline\parameterhdr{2}{ }{MountainCar}\\\hline
\bf Name & \bf Values \\
\hline
Version & 0 \\
\hline
Max episode steps & None \\
\hline
Initial reward* & -1.0 \\
\hline
Final reward* & -0.4 \\
\hline
Inter-trial duration* & 0.4 \\
\hline
Update interval (env runner)* & 0.02 \\
\hline\parameterhdr{2}{ }{FrozenLake}\\\hline
\bf Name & \bf Values \\
\hline
Version & 0 \\
\hline
Max episode steps & None \\
\hline
Slippery & False \\
\hline
Final reward null* & -0.1 \\
\hline
Inter-trial duration* & 0.1 \\
\hline
Update interval (env runner)*& 0.1 \\
\hline
\end{tabularx} \\
\caption{Table of the environment parameters. Values marked with * indicate values for the ZeroMQ wrapper.}
\end{table*}
|
1,314,259,993,019 | arxiv | \section{Introduction}
String theory and theories with supersymmetry usually contain many scalar fields which can
play an important role in the early Universe. For example, in string theory they often describe the dynamics
of extra spatial dimensions and other degrees of freedom living in higher dimensions. These
scalar degrees of freedom couple to matter fields propagating in the three large dimensions we
perceive, see for example \cite{Lukas}-\cite{Moduli2}. If these extra dimensions exist, they will have had
an influence on the evolution of the universe at some point. It is usually thought that in particular
the dynamics of the very early universe is affected by the existence of extra dimensions. As such, they
will alter the predictions of inflationary cosmology. If apart from the inflaton field(s) other scalar
fields are present, they will generally alter the evolution of the field(s) driving inflation
and affect the production of cosmological perturbations.
Generally speaking, the existence of multiple fields during the inflationary epoch modifys some of
the single field predictions. For example, apart from the usual adiabatic perturbations produced
during inflation, there could be isocurvature (entropy) perturbations produced, whose existence
is constrained by observations (see, for example \cite{Wands02}-\cite{Byrnes:2006fr} and references
therein).
Related to this, in single field inflation, the curvature perturbation on constant
energy density hypersurfaces, $\zeta$, is constant on super-horizon scales and can be
evaluated at the horizon crossing. However, in the presence of multiple fields during
the inflationary epoch, $\zeta$ does not remain constant and varies since the non-vanishing
isocurvature perturbations act as a source term for the change of $\zeta$~\cite{Bellio-Wands95,Bellio-Wands96}.
In order to distinguish between inflationary models, cosmologists need to extract as much
information as possible from the data. Usually the spectral index and the scalar--tensor ratio
are used to distinguish between inflationary models. With the advent of precision data (in particular
the data obtained by WMAP\cite{Spergel:2006hy}), the running of the spectral index can be added as an additional quantity
in order to distinguish between the models. One can expect considerable improvement with future data
coming from the PLANCK satellite and the mapping of the large scale structures in the universe, as well
as a better understanding of small scale clustering as obtained from the Ly$\alpha$ forest.
Another potential observable which could distinguish between inflationary models is the amount of
non-Gaussianity generated during inflation (see \cite{non-gaussian1} and references therein). Although
it is small for many inflationary models (e.g. \cite{Maldacena02}-\cite{Battefeld:2006sz}), there are examples in which
perturbations show a considerable amount of non-Gaussianity \cite{easther}-\cite{cline}.
In this paper we study the slow-roll regime of generalised two-field inflation,
where the scalar fields are also coupled through kinetic terms~\cite{Bellio-Wands95,Bellio-Wands96,Starobinsky01}.
In general, a multi-field action can be given by
\begin{equation}
S = \int d^{4}x \, \sqrt{-g} \, \left[ M_P^2\frac{R}{2} -\frac12
{\cal G}_{IJ} \, \partial_{\mu} \phi^I \partial^{\mu} \phi^J - W(\phi) \right] .
\end{equation}
For simplicity we consider a system of two scalar fields, whose dynamics are governed by the following action:
\dis{
S=\int d^4 x \sqrt{-g}\left[M_P^2\frac{R}{2}-\frac12g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi
-\frac12e^{2b(\varphi)}g^{\mu\nu}\partial_{\mu}\chi\partial_\nu\chi-W(\varphi,\chi) \right]. \label{action}}
In this expression $M_P=1/\sqrt{8\pi G}$ is the reduced Planck mass. Note that, in this case, ${\cal G}_{IJ}$ is symmetric.
We obtain the general formulae for the running of the spectral indices (adiabatic, non-adiabatic
and correlated), following~\cite{DiMarcoFinelli03} and~\cite{DiMarcoFinelli05}. The running is affected by the presence of
the coupling and therefore future experiments constraining the running of the spectral index will
give vital information about the existence of fields which couple non-trivially to the field(s)
driving inflation. Furthermore, we derive the general formulae for non-Gaussianity and apply them to some specific models.
The paper is organised as follows: In the next Section we summarise the equations governing the background
and perturbation evolution.
In Section~\ref{SecRun} we derive the general expressions for the running of the spectral
indices (for the adiabatic and isocurvature power spectra as well as the correlation spectrum).
In Section~\ref{SecNonG} we derive the expected non--Gaussianity for two cases.
In Sections~\ref{SecTheory1} and
\ref{SecTheory2} we then apply our results to specific examples. Numerical results are presented in Section~\ref{secNumerical}.
Our conclusions can be found in Section~\ref{SecConc}. Useful formulae are collected in the Appendices.
\section{Background and Perturbation Equations}
In this section we set our notation and briefly review some results found in previous work. It will closely follow \cite{DiMarcoFinelli03} and \cite{DiMarcoFinelli05}.
The equations of motion for the two scalar fields in a Friedmann--Robertson--Walker spacetime
follow from the action (\ref{action}) and are given by
\dis{
\ddot{\varphi}+3H\dot{\varphi}+W_\varphi=b_\varphi e^{2b}\dot{\chi}^2,\\
\ddot{\chi}+(3H+2b_\varphi\dot{\varphi})\dot{\chi}+e^{-2b}W_\chi=0,
\label{background}
}
where $b_\varphi=\frac{\partial b(\varphi)}{\partial \varphi}$, $W_\varphi = \frac{\partial W(\varphi,\chi)}{\partial \varphi}$, etc.
Einstein's equations lead to
\dis{
H^2=\frac{1}{3M_P^2}\left[\frac12\dot{\varphi}^2+\frac12e^{2b}\dot{\chi}^2+W\right], \\
\dot{H}=\frac{1}{2M_P^2}\left[\dot{\varphi}^2+e^{2b}\dot{\chi}^2\right]=-\frac{\dot{\sigma}^2}{2M_P^2}.
\label{Friedmann}
}
We define e-folding number, $N$, as
\begin{equation}
N(t_e, t_*) \equiv \int_*^e H dt \label{N}.
\end{equation}
It is useful to separate the perturbations into components of adiabatic and
isocurvature modes.
Following~\cite{DiMarcoFinelli03,Gordon01}, we define the average (adiabatic)
and orthogonal (entropy) fields $\sigma$ and $s$ as
\dis{
d\sigma=&\cos\theta d\varphi +\sin\theta e^b d \chi,\\
ds=&e^b\cos\theta d\chi -\sin\theta d \varphi,
}
with
\dis{
\cos\theta=\frac{\dot{\varphi}}{\sqrt{\dot{\varphi}^2+e^{2b}\dot{\chi}^2}},\qquad
\sin\theta=\frac{e^b\dot{\chi}}{\sqrt{\dot{\varphi}^2+e^{2b}\dot{\chi}^2}}.
}
These fields satisfy the equations of motion
\dis{
\ddot{\sigma}+3H\dot{\sigma}+W_\sigma=0,\\
\dot{\theta}=-\frac{W_s}{\dot{\sigma}}-b_\varphi \dot{\sigma}\sin\theta,
}
where
\dis{
W_\sigma=W_\varphi\cos\theta+e^{-b}W_\chi\sin\theta,\\
W_s=-W_\varphi\sin\theta+e^{-b}W_\chi\cos\theta.
}
We now turn to cosmological perturbations. We work in the longitudinal gauge, in which the perturbed
metric has the form
\dis{
ds^2=-(1+2\Phi)dt^2+a^2(1-2\Phi)d{\bf x}^2,
}
where $a=a(t)$ is the scale factor and $\Phi=\Phi(t,{\bf x})$ is the metric perturbation.
We now calculate the perturbations in the adiabatic and entropy fields.
Instead of working with the field perturbation $\delta \sigma$, it is more convenient to work with the
Sasaki--Mukhanov variable $Q_\sigma=\delta \sigma+\frac{\dot\sigma}{H}\Phi$. The equation of motion for $Q_\sigma$
is given by
\dis{
\ddot{Q}_\sigma &+ 3H\dot{Q}_\sigma +\left[\frac{k^2}{a^2}+W_{\sigma\sigma} +\dot{\theta} \frac{W_s}{\dot{\sigma}}-\frac{1}{M_P^2a^3}\left(\frac{a^3\dot{\sigma}^2}{H} \right)^{\cdot}-b_\varphi\dot{\varphi}\frac{W_\chi e^{-b}}{\dot{\sigma}}\sin\theta \right]Q_\sigma \\
&= -2\left(\frac{W_s}{\dot{\sigma}}\delta s \right)^{\cdot}+2\left(\frac{W_s}{\dot{\sigma}}+\frac{\dot{H}}{H} \right)\frac{W_s}{\dot{\sigma}}\delta s,
}
whereas for $\delta s$ it is given by
\dis{
\ddot{\delta s}+3H\dot{\delta s}+ \left[\frac{k^2}{a^2}+W_{ss}+3\dot{\theta}^2-b_{\varphi\vp}\dot{\sigma}^2 + b_\varphi^2 g(t) + b_\varphi f(t) \right]\delta s = -\frac{k^2}{a^2}\frac{\Phi}{2\pi G}\frac{W_s}{\dot{\sigma}^2},
}
with
\dis{
g(t)&=-\dot{\sigma}^2(1+3\sin^2\theta),\\
f(t)&=W_\varphi(1+\sin^2\theta)-4W_s \sin\theta.
}
We have taken the notation of \cite{DiMarcoFinelli05} and use
\begin{eqnarray}
W_{\sigma\sigma}=W_{\varphi\vp} \cos^2\theta+V_{\varphi\chi} \sin{2\theta}e^{-b}+V_{\chi\chi}\sin^2{\theta} e^{-2b},\\
W_{ss}=W_{\varphi\vp} \sin^2\theta-V_{\varphi\chi} \sin{2\theta}e^{-b}+V_{\chi\chi}\cos^2{\theta} e^{-2b}.
\end{eqnarray}
We define the curvature and isocurvature fluctuations as
\dis{
\zeta=\frac{H}{\dot{\sigma}}Q_\sigma,\qquad S=\frac{H}{\dot{\sigma}}\delta s,
}
then the time derivative of $\zeta$ is related to $S$ by
\dis{
\dot{\zeta}=\frac{H}{\dot{H}}\frac{k^2}{a^2}\Phi-2\frac{W_s}{\dot{\sigma}}S.
}
During inflation, the fields are assumed to follow the slow-roll limit,
\dis{
\dot{\varphi}=\dot{\sigma}\cos\theta\simeq -\frac{W_\varphi}{3H}, \qquad
\dot{\chi}=\dot{\sigma}\sin\theta e^{-b} \simeq -\frac{W_\chi}{3H}e^{-2b},
\label{slow-roll}
}
\dis{
H^2(\varphi,\chi)\simeq \frac{1}{3M_P^2}W(\varphi,\chi).
}
The slow-roll parameters are defined in Appendix \ref{App1} for convenience.
In the slow-roll limit and on large scales, we find the evolutions of
curvature and isocurvature perturbation can be written in terms of slow-roll
parameters,
\dis{
\dot{\zeta}\simeq BHS,\qquad \dot{S}\simeq -\gamma HS,
\label{evol}
}
where
\dis{
B&=-2\eta_{\sigma s}-{\rm sign}(b_\varphi){\rm sign}\left(\frac{W_\chi}{W}\right)\sqrt{\epsilon_b\epsilon_\chi}\sin^2\theta,\\
\gamma&=-\eta_{\sigma\sigma}+2\epsilon+\eta_{s s} +\frac12 {\rm sign}(b_\varphi){\rm sign}\left(\frac{W_\chi}{W}\right)\sqrt{\epsilon_b\epsilon_\chi}\sin\theta\cos\theta\\
&\qquad +\frac12{\rm sign}(b_\varphi) {\rm sign}\left(\frac{W_\varphi}{W}\right)\sqrt{\epsilon_b\epsilon_\vp}(1+\sin^2\theta). \label{Beq}
}
Generally speaking on large scales the evolutions of adiabatic and isocurvature fluctuations
follow the following set of equations
\dis{
\dot{\zeta}=\alpha(t)H(t)S,\\
\dot{S}=\delta(t)H(t)S.\label{evolution-of-fluctuations}
}
We use the formalism of transfer matrix~\cite{Wands02},
\dis{
\left( \begin{array}{c}
\zeta(t)\\ S(t)\end{array}\right)
= \left(\begin{array}{cc} 1&T_{\zeta S} \\ 0& T_{SS}\end{array}\right)
\left( \begin{array}{c}
\zeta(t_*)\\ S(t_*)\end{array} \right),
}
where
\dis{
T_{SS}(t_*,t)=\exp\left(\int^t_{t_*}\delta(t')H(t')dt' \right),\\
T_{\zeta S}(t_*,t)=\int^t_{t_*}\alpha(t')H(t')T_{SS}(t_*,t')dt'.
\label{transferfunction}
}
Then the power spectra are
\dis{
{\cal P}_\zeta=(1+T_{\zeta S}^2)\left.{\cal P}_\zeta\right|_*={\cal P}_\zeta|_*(1+\cot^2\Delta),
\label{Pzeta}
}
\dis{
{\cal P}_S=T^2_{SS}{\cal P}_\zeta|_*,
}
\dis{
{\cal P}_{C}=T_{\zeta S}T_{SS}{\cal P}_\zeta|_*,
}
where the cross-correlation angle $\Delta$ is
\dis{
\cos\Delta=\frac{{\cal P}_{C}}{\sqrt{{\cal P}_{\zeta}{\cal P}_S }}.
}
We note that $T_{\zeta S}=\cot\Delta$ and $\Delta$ has range
$0\le\Delta\le \pi$ giving positive and negative correlation depending
on the sign of $\cos\Delta$.
The spectral indices are defined as
\dis{
n_X-1\equiv\frac{d \ln {\cal P}_X}{d \ln k},\qquad X=\zeta, S, {\cal C}.
}
The spectral indices are best written in terms of slow-roll parameters.
They have been calculated in \cite{DiMarcoFinelli05} and are given by
\begin{eqnarray}
n_\zeta - 1 &=& -6\epsilon + 4\epsilon(\cos\Delta)^2 + 2\eta_{\sigma\sigma}(\sin\Delta)^2
+ 4 \eta_{\sigma s} \sin\Delta \cos\Delta + 2\eta_{ss}(\cos\Delta)^2 \nonumber \\
&+& 2~\rm{sign}(b_\varphi)\rm{sign}\left(\frac{W_\chi}{W}\right)\sqrt{\epsilon_b
\epsilon_\chi}(\sin\theta)^2 \sin\Delta\cos\Delta \nonumber \\
&+& \rm{sign}(b_\varphi)\rm{sign}\left(\frac{W_\varphi}{W}\right)\sqrt{\epsilon_b\epsilon_\varphi}\left(1
+\sin^2\theta\right)\cos^2\Delta \nonumber \\ &-& \rm{sign}(b_\varphi)\rm{sign}\left(\frac{W_\chi}{W}\right)
\sqrt{\epsilon_b \epsilon_\chi}sin\theta\cos\theta\sin^2 \Delta,
\end{eqnarray}
\begin{eqnarray}
n_S - 1 = -2\epsilon + 2 \eta_{ss}
+ \rm{sign}(b_\varphi)\rm{sign}\left(\frac{W_\varphi}{W}\right)\sqrt{\epsilon_b \epsilon_\varphi}(1+\sin^2 \theta),
\end{eqnarray}
and
\begin{eqnarray}
n_C - 1 = n_S - 1 -B\tan\Delta.
\label{n_C}
\end{eqnarray}
In these expressions, all the slow-roll parameters are evaluated at horizon crossing.
\section{Running of the Spectral Indices}
\label{SecRun}
To calculate the running of the spectral indices we need to know the
second-order slow-roll parameters and the derivative of first-order
slow-roll parameters and the transfer functions.
For convenience, we list them extensively in Appendix \ref{App1}.
When $b_\varphi \ne 0$ we have 14 parameters at second-order in general, as listed in Table~\ref{Tab2}.
However, considering the component fields ($\sigma$,$s$), $H$, $T_{SS}$ and $\xi^2_{sss}$
do not appear in the running of spectral indices, which reduces the parameters to 11.
Therefore the running spectral indices are given by: $\theta$, $\Delta$ and 9 slow-roll parameters.
We define the running spectral indices $\alpha_X$ as
\dis{
\alpha_X\equiv \frac{d n_X}{d \ln k},\qquad X=\zeta, S, {\cal C}.
}
To leading order in slow roll, the runnings of the spectral indices
after inflation are:
\dis{
\alpha_\zeta\equiv\frac{d n_\zeta}{d \ln k}=&\left(\frac{d n_\zeta}{d \ln k}\right)_{b=0}
+\frac{1}{16}\Big[5-3\cos2\theta+8(1-\cos2\theta)(\sin2\theta\sin2\Delta+\cos2\theta\cos2\Delta)\\
-2\sin2&\theta(2-3\cos2\theta+\cos^22\theta)\sin4\Delta-\{3+\cos2\theta(3-6\cos2\theta+2\cos^22\theta)\}\cos4\Delta \Big]\epsilon_b\epsilon\\
&-\frac18\Big[1+\cos2\theta+2\cos2\Delta+(1-\cos2\theta)(\sin2\theta\sin2\Delta+\cos2\theta\cos2\Delta) \Big]\xi_b\epsilon\\
&+\Big[2\cos^3\theta+\cos\theta(3-2\cos2\theta)\cos2\Delta+\cos\theta(-2+\cos2\theta)\cos4\Delta\\
&+2\sin^3\theta(2\sin2\Delta-\sin4\Delta) \Big]\textrm{sign}(b_\vp)\textrm{sign}\left(\frac{V_\sigma}{V}\right)\sqrt{\epsilon_b \epsilon}\epsilon\\
&+\Big[\Big\{-\cos\theta+\sin\theta(1-\cos2\theta)(-\sin2\Delta+\frac12\sin4\Delta)+\frac12\cos\theta(-3+2\cos2\theta)\cos2\Delta \\
&+\frac12\cos\theta(2-\cos2\theta)\cos4\Delta \Big\}\eta_{\sigma\sigma}
+\Big\{ \frac52\sin\theta+\frac12\cos\theta(3+\cos2\theta)\sin2\Delta\\
&-\cos\theta(2-\cos2\theta)\sin4\Delta-\frac12\sin\theta(6+\cos2\theta)\cos2\Delta+\sin\theta(1-\cos2\theta)\cos4\Delta \Big\}\eta_{\sigma s}\\
&+\Big\{\frac32\cos\theta+\frac12\sin\theta(3+\cos2\theta)\sin2\Delta-\sin^3\theta\sin4\Delta+\cos^3\theta\cos2\Delta\\
&+\frac12\cos\theta(-2+\cos2\theta)\cos4\Delta \Big\}\eta_{s s}\Big]\textrm{sign}(b_\vp)\textrm{sign}\left(\frac{V_\sigma}{V}\right)\sqrt{\epsilon_b \epsilon},\\
\alpha_S\equiv\frac{d n_S}{d \ln k}=
&\left(\frac{d n_S}{d \ln k}\right)_{b=0} +\sin^2\theta \cos^4\theta \epsilon_b\epsilon -\frac12(1+\sin^2\theta)\cos^2\theta\xi_b \epsilon
+ \left[2\cos\theta \epsilon -(1+ \sin^2 \theta)\cos\theta\eta_{s s}\right.\\
&+\left. \sin\theta(-1+3\sin^2\theta) \eta_{\sigma s}
+2\,\cos^3\theta\eta_{ss} \right]\textrm{sign}(b_\vp)\textrm{sign}\left(\frac{V_\sigma}{V}\right)\sqrt{\epsilon_b \epsilon},\\
\alpha_C\equiv\frac{d n_C}{d \ln k}=
&\frac{d n_S}{d \ln k}+ \tan\Delta\Big[ 4\eta_{\sigma s}(\eta_{\sigma \sigma} - \eta_{ss}) -2\xi_{\sigma\sigma s}^2 + \frac12\sin^2\theta\sin2\theta\cos2\theta\epsilon_b\epsilon
-\frac12\sin^3\theta\cos\theta\xi_b\epsilon\\
& +\Big\{\sin\theta(3\cos^2\theta-1)\eta_{s s}-5\sin^2\theta\cos\theta\eta_{\sigma s} \Big\} \textrm{sign}(b_\vp)\textrm{sign}\left(\frac{V_\sigma}{V}\right)\sqrt{\epsilon_b \epsilon}\Big]\\
&+\tan^2\Delta\Big[-4\eta_{\sigma s }^2-4\eta_{\sigma s }\sqrt{\epsilon_b\epsilon}\sin^3\theta -\sin^6\theta\epsilon_b\epsilon \Big],}
where
\dis{
\left(\frac{d n_\zeta}{d \ln k}\right)_{b=0}=
&2(-7+4\cos2\Delta-\cos4\Delta)\epsilon^2+2(4-3\cos2\Delta+\cos4\Delta)\epsilon\eta_{\sigma\sigma}
+4(2\sin2\Delta-\sin4\Delta)\epsilon\eta_{\sigma s}\\
&+2(2+\cos2\Delta-\cos4\Delta)\epsilon\eta_{s s}
+\sin^22\Delta(\eta_{\sigma\sigma}^2+\eta_{s s}^2)+ 4\cos^22\Delta\eta_{\sigma s}^2-(1-\cos4\Delta)\eta_{\sigma\sigma}\eta_{s s} \\
&+2\sin4\Delta(\eta_{\sigma\sigma}\eta_{\sigma s}-\eta_{\sigma s}\eta_{s s}) -(1-\cos2\Delta)\xi_{\sigma\sigma\sigma}-2\sin2\Delta\xi_{\sigma\sigma s}^2 -(1+\cos2\Delta)\xi_{\sigma s s}^2,\\
\left(\frac{d n_S}{d \ln k}\right)_{b=0} =&
-8 \epsilon^2+4\epsilon(\eta_{\sigma\sigma} +\eta_{ss})+4\eta_{\sigma s}^2 -2\xi_{\sigma ss}^2.
}
All slow-roll parameters are evaluated at horizon crossing.
We note that the running spectral index of isocurvature perturbation, $\alpha_S$, is independent of $\Delta$, which means that the $\alpha_S$ is determined
at horizon crossing and does not change thereafter.
\begin{table}[tb]
\begin{center}
\begin{tabular}{|c||cccccccccccccc|}
\hline
$\varphi$,$\chi$ & \multirow{2}{*}{$H$} & \multirow{2}{*}{$T_{SS}$} & \multirow{2}{*}{$\Delta$} & $\epsilon_\varphi$ & $\epsilon_\chi$ & $\eta_{\varphi\vp}$ & $\eta_{\varphi\chi}$ & $\eta_{\chi\chi}$ &
\multirow{2}{*}{$\epsilon_b$} & \multirow{2}{*}{$\xi_b$} & $\xi^2_{\varphi\vp\varphi}$ & $\xi^2_{\chi\chi\chi}$ & $\xi^2_{\varphi\vp\chi\chi}$ & $\xi^2_{\varphi\chi\chi\chi}$ \\
$\sigma$,$s$ & & & & $\epsilon$ & $\theta$ & $\eta_{\sigma\sigma}$ & $\eta_{\sigma s}$ & $\eta_{s s}$ &
& & $\xi^2_{\sigma\sigma\sigma}$ & $\xi^2_{\sigma\sigma s}$ & $\xi^2_{\sigma s s}$ & $\xi^2_{sss}$ \\
\hline
\end{tabular}
\caption{Full set of model parameters to second order, listed by physical fields ($\varphi$,$\chi$) or component fields ($\sigma$,$s$).
$H$, $T_{SS}$ and $\xi^2_{sss}$ are not required for the spectral running and can be disregarded.}
\label{Tab2}
\end{center}
\end{table}
\section{Non-Gaussianity During Inflation}
\label{SecNonG}
Another potential discriminator between inflationary models is the level of
non--Gaussianity produced during inflation.
Current observations limit the nonlinearity parameter, $|f_{\rm NL}|<100$~\cite{Creminelli:2006rz}. A perfect CMB experiment cannot hope to detect $|f_{\rm NL}|<3~$\cite{Komatsu:2001rj}.
Using the $\delta$N formalism \cite{starob85,ss1,Sasaki:1998ug,lms,lr}, the non-linearity parameter is
given by~\cite{Seery05}
\dis{
-\frac65f_{\rm NL}=\frac{{\cal A}^{IJK}N_{,I} N_{,J} N_{,K}}{( N_{,I} N_{,J}{\cal G}^{IJ})^2\sum_i k_i^3 }
+\frac{{\cal G}^{IM}{\cal G}^{KN} N_{,I} N_{,K} N_{,MN} }{( N_{,I} N_{,J}{\cal G}^{IJ})^2}\label{fNL},
}
which is valid in the slow-roll limit.
The first term, which we call $-\frac65f_{\rm NL}^{(3)}$, can be written as~\cite{VernizziWands},
\dis{
-\frac65f_{\rm NL}^{(3)}=\frac{r}{16}(1+f),
\label{fNL3}
}
where $r\equiv {\cal P}_T/{\cal P}_\zeta$ is tensor-to-scalar ratio and
$f$ is a function of the momentum
triangle with the range of values $0\le f\le\frac56$~\cite{Maldacena02}.
In Appendix \ref{AppC}, we have shown Eqn.(\ref{fNL3}) is valid for non-canonical kinetic terms.
From \cite{DiMarcoFinelli05}, the tensor-to-scalar ratio is calculated to be
$r\lesssim 16\epsilon$
and therefore we can approximate
\begin{equation}
\left| -\frac65f_{\rm NL}^{(3)} \right| \lesssim\epsilon
\end{equation}
which is certainly too small to be observable in CMB experiments.
Also see \cite{Zaballa:2006pv}.
Due to this, we will concentrate on the second term, $f_{\rm NL}^{(4)}$, only in the next sections and consider two cases: separable potentials of product and sum.
\subsection{Product Potential $W(\varphi,\chi)=U(\varphi)V(\chi)$}
We first consider the case of a separable potential by product, for which
we derive the analytic formula for the non-linear parameter $f_{\rm NL}$.
To use Eq.~(\ref{fNL}), we need to know the dependence of the number of
e-foldings on the fields $\varphi$ and $\chi$.
Following the calculations in~\cite{Bellio-Wands96}, we can obtain the
derivatives of $N$ by $\varphi_*$ and $\chi_*$.
First we find the number of e--foldings,
\begin{equation}
N(\varphi_*,\chi_*)= -\frac{1}{M_P^2} \int^e_* \frac{U}{U_\varphi} d\varphi
= -\frac{1}{M_P^2} \int^e_* \frac{V}{V_\chi} e^{2b(\varphi)} d\chi,
\end{equation}
where superscript (or subscript) $^e$ and $^*$ denotes the values evaluated
at the end of inflation and horizon crossing respectively.
From the equations of motion, in the slow-roll regime, it is possible to find a constant of motion along the trajectory, as in~\cite{Bellio-Wands96}:
\begin{equation}
C_1\equiv -\frac{1}{M_P^2} \int e^{-2b(\varphi)} \frac{U d \varphi}{U_\varphi} + \frac{1}{M_P^2} \int
\frac{V d \chi}{V_\chi} .
\label{integral1}
\end{equation}
\noindent Using this constant of motion, in the slow-roll regime we find the
first derivatives of $N(t_e,t_*)$ with respect to the fields,
\dis{
\frac{\partial N }{\partial\varphi_*}&=\frac{{\rm sign}(U/U_\varphi)}{M_P\sqrt{2\epsilon_\vp^*}}\left[1-\frac{\epsilon_\chi^e}{\epsilon^e}
e^{2b_e-2b_*} \right],\\
\frac{\partial N }{\partial\chi_*}&=\frac{{\rm sign}(V/V_\chi)}{M_P\sqrt{2\epsilon_\chi^*}}\left[\frac{\epsilon_\chi^e}{\epsilon^e}
e^{2b_e-b_*} \right].\label{non-gaussian1}
}
\noindent From the first derivatives we can find the second derivatives,
\begin{eqnarray}
M_P^2 \frac{\partial^2 N}{\partial \varphi_*^2} &=&
1
- \frac{\eta^*_{\varphi\vp}}{2 \epsilon^*_\varphi} +
\frac12{\rm sign}\left(b_\varphi\right) {\rm sign}\left(\frac{U_\varphi}{U}\right)\sqrt{\frac{\epsilon^*_b}{\epsilon^*_\varphi}}
\frac{\epsilon^e_\chi}{\epsilon^e} e^{2b_e-2b_*}
- \frac{1}{\epsilon^*_\varphi}e^{4b_e-4b_*} {\cal A}_P
, \nonumber\\
M_P^2 \frac{\partial^2 N}{\partial \chi_*^2} &=&
e^{2b_*} \left[
\left(1
- \frac{\eta^*_{\chi\chi}}{2 \epsilon^*_\chi}\right)
\frac{\epsilon^e_\chi}{\epsilon^e} e^{2b_e-2b_*}
- \frac{1}{\epsilon^*_\chi}e^{4b_e-4b_*} {\cal A}_P
\right]
,\label{non-gaussian2}\\
M_P^2 \frac{\partial^2 N}{\partial \varphi_* \partial \chi_*} &=&
{\rm sign}\left(\frac{U_\varphi}{U}\right){\rm sign}\left(\frac{V_\chi}{V}\right)\frac{1}{\sqrt{\epsilon^*_\varphi \epsilon^*_\chi}} e^{4b_e-3b_*} {\cal A}_P
,\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\eta_{ss} &\equiv&
\frac{(\epsilon_\chi\eta_{\varphi\vp}+\epsilon_\varphi\eta_{\chi\chi})}{\epsilon}, \\
{\cal A}_P&\equiv&-\frac{\epsilon^e_\varphi \epsilon^e_\chi}{(\epsilon^e)^2}
\left[
\eta_{ss}^e -
\frac12{\rm sign}\left(b_\varphi\right) {\rm sign}\left(\frac{U_\varphi}{U}\right) \frac{(\epsilon^e_\chi)^2}{\epsilon^e} \sqrt{\frac{\epsilon^*_b}{\epsilon^*_\varphi}}
- 4 \frac{\epsilon^e_\varphi \epsilon^e_\chi}{\epsilon^e}
\right].
\end{eqnarray}
We have followed the notation of \cite{VernizziWands} as much as possible, for reasons that become apparent in the next section.
From Eqs.~(\ref{non-gaussian1}) and (\ref{non-gaussian2})
we find the second term of Eq.~(\ref{fNL}),
\begin{eqnarray}
-\frac{6}{5} f^{(4)}_{\rm NL}&=&
\frac{2 e^{-2b_e+2b_*}}{\left( \frac{u^2\alpha^2}{\epsilon_\varphi^*}
+ \frac{v^2}{\epsilon_\chi^*} \right)^2}
\left[
\frac{u^3\alpha^3}{\epsilon^*_\varphi}
\left(1
- \frac{\eta^*_{\varphi\vp}}{2 \epsilon^*_\varphi}
\right)
+ \frac{v^3}{\epsilon^*_\chi}
\left(1
- \frac{\eta^*_{\chi\chi}}{2 \epsilon^*_\chi }
\right) \right.
\nonumber
\\
&&
\quad\left.
+ \frac12 {\rm sign}\left(b_\varphi\right) {\rm sign}\left(\frac{U_\varphi}{U}\right)
\frac{vu^2\alpha^2}{(\epsilon_\varphi^*)^2} \sqrt{\epsilon^*_b\epsilon^*_\varphi}
- \left( \frac{u\alpha}{{\epsilon^*_\varphi}}
- \frac{v}{{\epsilon^*_\chi}}
\right)^2
e^{2b_e-2b_*} {\cal A}_P
\right]
,
\label{fNL4}
\end{eqnarray}
with the definitions
\begin{equation}
u\equiv\frac{\epsilon_\varphi^e}{\epsilon^e}, \quad v\equiv\frac{\epsilon_\chi^e}{\epsilon^e},
\quad \alpha\equiv e^{-2b_e+2b_*}\left[1+\frac{\epsilon_\chi^e}{\epsilon_\varphi^e}
\left(1-e^{2b_e-2b_*}\right)\right].
\end{equation}
When $b(\varphi)=0$, then $\alpha=1$ and $\epsilon_b=0$, which means that the symmetry between $\varphi$ and $\chi$ in Eq.(\ref{fNL4}) is restored.
\subsection{Sum Potential $W(\varphi,\chi) = U(\varphi) + V(\chi)$}
The second case we consider consists of separable potential models by sum.
Similar cases with canonical kinetic terms were previously
studied in \cite{VernizziWands}, where it was shown that they do not generate
significant non-Gaussianity. We will now derive the general formula
for $f_{\rm NL}$ in the presence of non-canonical kinetic terms, closely following the
derivation found in the above-mentioned paper.
The total number of e--foldings along a trajectory in slow-roll regime
is given by
\begin{equation}
N(\varphi_*,\chi_*)=- \frac{1}{M_P^2} \int^e_* \frac{U}{U_\varphi} d\varphi -
\frac{1}{M_P^2} \int^e_* e^{2b(\varphi)} \frac{V}{V_\chi} d\chi.
\end{equation}
We note that in principle $\varphi$ in the integration of $\chi$ can be
re-written in terms of
$\varphi_*,\ \chi_*$ and $\chi$ along the trajectory using equations of motion.
For sum potentials, $\varphi$ and $\chi$ have the relation along the
trajectory through
\dis{
\int^\varphi_{\varphi_*}\frac{e^{-2b(\varphi)}}{U_\varphi}d\varphi=\int^\chi_{\chi_*}\frac{1}{V_\chi}d\chi.
}
As before, the integral of motion along the trajectory leads to a constant of motion
\begin{equation}
C_2\equiv -M_P^2 \int e^{-2b(\varphi)} \frac{d \varphi}{U_\varphi} + M_P^2 \int
\frac{d \chi}{V_\chi} .
\label{integral2}
\end{equation}
\noindent This enables to derive the first derivatives of $N$
\begin{eqnarray}
M_P \frac{\partial N}{\partial \varphi_*}&=&
{\rm sign}\left(\frac{U_\varphi}{W}\right) \frac{1}{\sqrt{2\epsilon_\varphi^*}W^*} \left(U^*+Z^e\right) - {\cal G} , \label{dNdp} \nonumber \\
M_P \frac{\partial N}{\partial \chi_*}&=&
{\rm sign}\left(\frac{V_\chi}{W}\right)\frac{e^{b_*}}{\sqrt{2\epsilon_\chi^*}W^*}
\left(V^*- Z^e\right) - {\cal H}, \label{dNdc}
\end{eqnarray}
using the definitions,
\begin{eqnarray}
Z^e &=& \frac{(V^e {\epsilon^e_\varphi} - U^e
{\epsilon^e_\chi})}{\epsilon^e} e^{2b_e-2b_*} , \label{Z2}\\
{\cal G}(\varphi_*,\chi_*) &\equiv&\frac{1}{M_P} \int^e_* 2b_\varphi e^{2b(\varphi)} \frac{V}{V_\chi} \frac{\partial\varphi}{\partial\varphi_*} d\chi \, ,\label{Geqn}\\
{\cal H}(\varphi_*,\chi_*) &\equiv&\frac{1}{M_P} \int^e_* 2b_\varphi e^{2b(\varphi)} \frac{V}{V_\chi} \frac{\partial\varphi}{\partial\chi_*} d\chi \, . \label{Heqn}
\end{eqnarray}
In the same way the second derivatives can be derived:
\begin{eqnarray}
\label{d2Ndp2}
M_P^2 \frac{\partial^2 N}{\partial \varphi_*^2} &=&
1
- \frac{\eta^*_{\varphi\vp}}{2 \epsilon^*_\varphi} \frac{U^*+Z^e}{W^*} +
{\rm sign}\left(\frac{U_\varphi}{W}\right) \frac{M_P}{W^*\sqrt{2\epsilon_\varphi^*}}
\frac{\partial Z^e}{ \partial \varphi^*} - M_P\frac{\partial {\cal G}}{\partial \varphi_*}
, \nonumber \\
\label{d2Ndc2}
M_P^2 \frac{\partial^2 N}{\partial \chi_*^2} &=& e^{2b_*}\left[1 -
\frac{\eta^*_{\chi\chi}}{2 \epsilon^*_\chi} \frac{V^*-Z^e}{W^*} -
{\rm sign}\left(\frac{V_\chi}{W}\right) \frac{M_P e^{-b_*}}{W^*\sqrt{2\epsilon_\chi^*}}
\frac{\partial Z^e}{ \partial \chi^*}\right] - M_P\frac{\partial {\cal H}}{\partial \chi_*},
\nonumber \\
\label{d2Ndcdp}
M_P^2 \frac{\partial^2 N}{\partial \chi_*\partial \varphi_*} &=&
{\rm sign}\left(\frac{U_\varphi}{W}\right)\frac{M_P}{W^*\sqrt{2\epsilon_\varphi^*}}
\frac{\partial Z^e}{ \partial \chi^*} - M_P\frac{\partial {\cal G}}{\partial \chi_*},
\\
\label{d2Ndpdc}
M_P^2 \frac{\partial^2 N}{\partial \varphi_*\partial \chi_*} &=&
e^{b_*}\left[
\frac{1}{2} {\rm sign}(b_\varphi) {\rm sign}\left(\frac{V_\chi}{W}\right)\sqrt{\frac{\epsilon_b^*}{\epsilon_\chi^*}}\frac{V^*-Z^e}{W^*}
- {\rm sign}\left(\frac{V_\chi}{W}\right)\frac{M_P}{W^*\sqrt{2\epsilon_\chi^*}} \frac{\partial Z^e}{ \partial \varphi^*}
\right]
-M_P \frac{\partial {\cal H}}{\partial \varphi_*}. \nonumber
\end{eqnarray}
The order of the second derivatives,
$\frac{\partial^2 N}{\partial \varphi_*\partial \chi_*} = \frac{\partial^2 N}{\partial \chi_*\partial \varphi_*}$,
and the last two equations emphasise that ${\cal G}$ and ${\cal H}$ are not independent
quantities.
In fact, they are related by
\begin{eqnarray}
M_P \frac{\partial {\cal G}}{\partial \chi_*} &=& M_P \frac{\partial {\cal H}}{\partial\varphi_*}
- {\rm sign}(b_\varphi){\rm sign}\left(\frac{V_\chi}{W}\right)\frac{e^{b_*}}{2}\frac{V^*}{W^*}\sqrt{\frac{\epsilon_b^*}{\epsilon_\chi^*}}.
\end{eqnarray}
A similar expression relating $\frac{\partial {\cal G}}{\partial \varphi_*}$ to $\frac{\partial {\cal H}}{\partial \varphi_*}$ and $\frac{\partial {\cal H}}{\partial \chi_*}$ can be obtained, but it is irrelevant for our purposes and is therefore not given.
From the definition of $Z^e$ in Eq.~(\ref{Z2}), we can calculate
\begin{eqnarray}
{\rm sign}\left(\frac{U_\varphi}{W}\right)
\sqrt{\epsilon_\varphi^*}\frac{\partial Z^e}{ \partial \varphi_*}
= \frac{\sqrt2}{M_P} W^*\left[ {\cal A}_S
+ {\cal B}_S
+ {\cal C}_S \right], \\
{\rm sign}\left(\frac{V_\chi}{W}\right)
\sqrt{\epsilon_\chi^*}\frac{\partial Z^e}{ \partial \chi_*} e^{-b_*}
= -\frac{\sqrt2}{M_P} W^*\left[ {\cal A}_S
+ {\cal B}_S \right],
\end{eqnarray}
where we define
\begin{eqnarray}
{\cal A}_S &\equiv& - \frac{W_e^2}{W_*^2} \frac{\epsilon^e_\varphi
\epsilon^e_\chi}{\epsilon^e}
\left(1 - \frac{\eta_{ss}^e}{\epsilon^e} -\frac12{\rm sign}(b_\varphi){\rm sign}\left(\frac{U_\varphi}{W}\right)\frac{\epsilon_\chi^e}{(\epsilon^e)^2} \sqrt{\epsilon_b^e \epsilon_\varphi^e} \right) e^{4b_e-4b_*} \, , \\
{\cal B}_S &\equiv& \frac12 {\rm sign}(b_\varphi){\rm sign}\left(\frac{U_\varphi}{W}\right)\frac{\epsilon_\chi^e}{\epsilon^e} \sqrt{\epsilon_b^*\epsilon_\varphi^*} \frac{W^e}{W_*^2} Z^e e^{2b_e-2b_*} \, , \\
{\cal C}_S &\equiv& -\frac12 {\rm sign}(b_\varphi){\rm sign}\left(\frac{U_\varphi}{W}\right) \frac{Z^e}{W^*} \sqrt{\epsilon_b^*\epsilon_\varphi^*}.
\end{eqnarray}
From the results of the first and second derivatives, it is straightforward to calculate
\begin{eqnarray}
-\frac{6}{5} f^{(4)}_{\rm NL}&=&
\frac{2}{\left( \frac{u^2\alpha_u^2}{\epsilon_\varphi^*}
+ \frac{v^2\alpha_v^2}{\epsilon_\chi^*} \right)^2}
\left[
\frac{u^2\alpha_u^2}{\epsilon^*_\varphi}
\left(1
- \frac{\eta^*_{\varphi\vp}}{2 \epsilon^*_\varphi}
u - M_P\frac{\partial {\cal G}}{\partial \varphi_*}
+\frac{{\cal C}_S}{\epsilon_\varphi^*}
\right)
\right.
\nonumber \\
&&
\left.
- 2M_P {\rm sign}\left(\frac{U_\varphi}{W}\right){\rm sign}\left(\frac{V_\chi}{W}\right) \frac{u\alpha_u}{\sqrt{\epsilon_\varphi^*}} \frac{v\alpha_v e^{-b_*}}{\sqrt{\epsilon_\chi^*}} \frac{\partial {\cal G}}{\partial \chi_*}
\right.
\\
&& \quad \quad \quad \left.
+ \frac{v^2\alpha_v^2}{\epsilon^*_\chi}
\left(1
- \frac{\eta^*_{\chi\chi}}{2 \epsilon^*_\chi }
v -M_P\frac{\partial H}{\partial \chi_*}e^{-2b_*}
\right)
+ \left( \frac{u\alpha_u}{{\epsilon^*_\varphi}}
- \frac{v\alpha_v}{{\epsilon^*_\chi}}
\right)^2
\left({\cal A}_S+{\cal B}_S\right)
\right]
,
\nonumber
\end{eqnarray}
where we have defined
\begin{eqnarray}
u \equiv \frac{U^*+Z^e}{W^*}, \quad \quad
v \equiv \frac{V^*-Z^e}{W^*},
\end{eqnarray}
\begin{eqnarray}
\alpha_u \equiv 1+{\rm sign}\left(\frac{U_\varphi}{W}\right)\frac{\sqrt{2\epsilon_\varphi^*}}{u} {\cal G} , \quad \quad
\alpha_v \equiv 1+{\rm sign}\left(\frac{U_\chi}{W}\right)\frac{\sqrt{2\epsilon_\chi^*}e^{-b_*}}{v} {\cal H}.
\end{eqnarray}
When $b(\varphi)=0$, this calculation of $f^{(4)}_{\rm NL}$ simplifies, since
\begin{eqnarray}
{\cal B}_S={\cal C}_S={\cal G}={\cal H}=0 , \quad \quad \alpha_u=\alpha_v=1,
\end{eqnarray}
and the result is identical to that found in \cite{VernizziWands}.
\section{Examples of Scalar-Tensor Theories with Product Potentials}
\label{SecTheory1}
At first, we consider a separable potential by product with exponents in the $\varphi$ field,
as in the case of a massless dilaton, $\varphi$,
\begin{eqnarray}
W(\varphi,\chi)=U(\varphi)V(\chi)=e^{4c(\varphi)}V(\chi)\label{app1}.
\end{eqnarray}
When considering the physical fields ($\varphi$,$\chi$), only $H$ and $T_{SS}$ can be discarded.
Therefore, the set of parameters consists of $\Delta$ and 11 slow-roll parameters as shown in Table~\ref{Tab2}.
However, with the potential chosen above, the set reduces to only 5 independent slow-roll parameters, which are useful for the running spectral indices:
\beqa{
\epsilon_{\varphi} = 8 M^2_{\rm P} c_{\varphi}^2, && \qquad \epsilon_{\chi} = \frac{ M^2_{\rm P}}{2}\left(\frac{V_{\chi}}{V}\right)^2 e^{-2b}, \label{par_1}
\\
\eta_{\chi \chi} = M^2_{\rm P}\left( \frac{V_{\chi\chi}}{V}\right)
e^{-2b},&&\qquad
\xi_{\chi\chi\chi}^2=M^4_{\rm P}\left( \frac{V_{\chi\chi\chi V_\chi}}{V^2}\right) e^{-4b}, \label{par_4}
\\
\xi_c&=&8 M_{\rm P}^2 c_{\varphi\vp},
}
since the remaining 8 are not independent:
\begin{equation} \epsilon_c=\epsilon_\varphi\, , \end{equation}
\dis{
\eta_{\varphi \varphi} = \frac12 \xi_c + 2 \epsilon_{\varphi} \, , \qquad
\eta_{\varphi \chi} = 2 {\rm sign}(c_{\varphi}){\rm sign}
\left(\frac{V_{\chi}}{V}\right) \sqrt{\epsilon_{\varphi}\epsilon_{\chi}}
=\epsilon\sin2\theta, \label{par_3}
}
\begin{equation} \xi_{\varphi\vp\varphi}^2=4\epsilon_\varphi^2+3\xi_c\epsilon_\varphi+\xi_{cc}\sqrt{\epsilon_\varphi}\, , \quad \quad \xi_{\varphi\vp\chi\chi}^2=4\epsilon_\varphi\epsilon_\chi+\epsilon_\chi\xi_c
\, , \end{equation}
\begin{equation}
\xi_{\varphi\chi\chi\chi}^2=2{\rm sign}(c_{\varphi}){\rm sign}
\left(\frac{V_{\chi}}{V}\right) \sqrt{\epsilon_{\varphi}\epsilon_{\chi}}\eta_{\chi\chi}\, . \end{equation}
\begin{table}
\begin{center}
\begin{tabular}{|c|cc|cc|}
\hline
Model & 1a &1b & 2a& 2b\\ \hline
$b(\varphi)$ & $-\beta\frac{\varphi}{M_P}$ & $-\beta\frac{\varphi}{M_P}$ & $-\beta\frac{\varphi^2}{M_P^2}$ & $-\beta\frac{\varphi^2}{M_P^2}$\\
$V(\chi)$ & $\frac{\lambda}{4}\chi^4$ & $\frac12m_\chi^2\chi^2$ &$\frac{\lambda}{4}\chi^4$ & $\frac12m_\chi^2\chi^2$ \\
$\eta_{\chi\chi}$ & $\frac32\epsilon_\chi$ & $\epsilon_\chi$ &$\frac32\epsilon_\chi$ & $\epsilon_\chi$ \\
$\xi^2_{\chi\chi\chi} $ & $\frac32\epsilon_\chi^2$ & $0$ &$\frac32\epsilon_\chi^2$ & $0$ \\
$\xi_b$ & $0$ & $0$ &$-16\beta$& $-16\beta$\\ \hline
\end{tabular}
\caption{Four Jordan-Brans-Dicke models considered in the text
and their slow-roll parameters for a product potential,
$W=e^{4b(\varphi)}V(\chi)$.
In these specific models, 5 parameters are reduced
further to only 2 and 3 independent parameters for model 1 and 2 respectively.
$\epsilon_\varphi$ and $\epsilon_\chi$ are directly related to $\epsilon$
through $\theta$.
The three remaining parameters ($\eta_{\chi\chi}, \ \xi^2_{\chi\chi\chi}, \ \xi_b$) are shown to be also functions
of $\epsilon$ through $\epsilon_\varphi$ and $\epsilon_\chi$.}
\label{Tab1}
\end{center}
\end{table}
\subsection{Jordan-Brans-Dicke theory}
First we apply our results to the Jordan-Brans-Dicke (JBD) theory with quadratic and quartic
potentials.
In this theory,
\begin{eqnarray}
U(\varphi)=e^{4b(\varphi)}, \quad \quad b(\varphi)=-\beta\frac{\varphi}{M_P},
\label{Model1}
\end{eqnarray}
where we assume $\beta$ is a positive constant. For this potential choice, $\epsilon_\vp=\epsilon_b=\eta_{\varphi\vp}/2=8\beta^2$
and the slow-roll ends when $\epsilon^e=\epsilon_\chi^e+\epsilon_\vp^e=1$.
Two potentials are chosen for $\chi$:
\begin{itemize}
\item {\bf Model 1a} $V(\chi)=\frac{\lambda}{4}\chi^4$
\item {\bf Model 1b} $V(\chi)=\frac12m^2_\chi \chi^2$
\end{itemize}
where $\lambda$ is a dimensionless parameter and $m_\chi$ is the mass of the $\chi$ field.
The slow-roll parameters for both quartic and quadratic potentials for the $\chi$-field are shown in the first two columns of Table~\ref{Tab1}.
As seen in the table, the choice of potentials reduces the 5 independent slow-roll parameters to only two: $\epsilon_\varphi$ and $\epsilon_\chi$.
Equivalently, since $\epsilon_\varphi=\epsilon\cos^2\theta$ and $\epsilon_\chi=\epsilon\sin^2\theta$, we can use the parameters $\epsilon$ and $\theta$.
Hence, all the first and second order slow-roll parameters are proportional
to $\epsilon$ and $\epsilon^2$ respectively
which make the primordial spectral indices proportional
to $\epsilon^2$ in the lowest order.
One final parameter is required, $\tan\Delta$, to describe the evolution after inflation.
Then $\frac{1}{\epsilon^2}\frac{dn_{(\zeta,\ C,\ S)}}{d\ln k}$ are just
a function of $\theta$ and $\Delta$. We note that $\theta$ is calculated
at horizon crossing and depends on $\beta$ and the initial values
of $\varphi$ and $\chi$, whereas $\Delta$ represents the evolution after horizon
crossing and depends on the late time evolution of the Universe.
In Figure~\ref{fig:1-ab} we show the running spectral indices $\alpha_\zeta$ and
$\alpha_C$ for Model 1a.
The result for Model 1b is almost identical and is not shown.
As said before, $\alpha_S$ is independent of $\Delta$, thus we plot
the $\alpha_S$ dependence on $\theta$ separately in Figure~\ref{fig:runningS}.
We plot in the range of $0<\theta <\pi$ since the potential has
the reflection symmetry $\chi \rightarrow -\chi$ which corresponds
to the change of $\theta$ to $-\theta$.
For the correlated spectral running, $\alpha_C$, there is an divergence
at $\Delta\rightarrow\frac{\pi}{2}$. This divergence is not observable.
From Eq.~(\ref{Pzeta}), it can be seen that, for $\Delta=\frac{\pi}{2}$ ($T_{\zeta S}=0$), there is no evolution in ${\cal P}_\zeta$ and the amplitude of the correlation
spectrum is zero.
It is therefore impossible to define a spectral index or running at this point.
As $\Delta\rightarrow\frac{\pi}{2}$, $T_{\zeta S}$ is non-zero, but very small.
In this limit, the correlation amplitude would be also small and therefore unobservable.
As an aside, it can be noted that, for the specific case of Model 1a,
$B$ (defined in Eq.~(\ref{Beq})) is identically zero,
due to a cancelling of the slow-roll parameters.
This means that from Eq.~(\ref{evol}) the curvature perturbation remains
constant after horizon crossing ($\dot{\zeta}=0$)
for this JBD model with quartic potential.
\begin{figure}[!t]
\begin{tabular}{c c}
\includegraphics[width=0.33\textwidth]{./figs/angle_index_case1a.ps}
&
\includegraphics[width=0.66\textwidth]{./figs/case1a.ps}
\end{tabular}
\caption{The spectral index ($(n_\zeta -1)/\epsilon$) and running of $n_\zeta$ and $n_C$ in terms of $\epsilon$ for model 1-a.}
\label{fig:1-ab}
\end{figure}
It is possible to consider the case $V(\chi)= \lambda \chi^n$.
In this JBD theory, and using the fact that at the end of inflation $\epsilon^e=\epsilon_\vp^e+\epsilon_\chi^e=1$, we can analytically solve $N$ in the slow-roll limit,
\dis{
N=\frac{1}{8\beta^2}\ln \left[1+\frac{4\beta^2}{nM_P^2}e^{-2\beta\varphi_*/M_P}\chi_*^2 \right]+\frac{1}{8\beta^2}\ln \left[\frac{1-8\beta^2}{1+(2n-8)\beta^2} \right].
\label{Nsolve}}
We find that $\beta<0.05$ is required to obtain enough e-foldings for $\varphi^*, \chi^* <50M_P$.
From this fact, $8\beta^2 N \lesssim 1$ and the limit of Eq.~(\ref{Nsolve}) gives $\epsilon^*\simeq \frac{n}{4N}< 0.004 n$, where we have taken $N\approx 60$.
In this limit of $\beta$, we can approximate Eq.~(\ref{Nsolve}) further:
\begin{equation}
N\approx\frac{\chi_*^2}{2n}-\frac{n}{4} -\beta \frac{\varphi_*\chi_*^2}{n}.
\label{approxNmod1}
\end{equation}
For Models 1a and 1b, we see from Figure~\ref{fig:1-ab} that
\dis{
-10 (\epsilon^*)^2 \lesssim \alpha_\zeta \lesssim 0.
}
Hence $|\alpha_\zeta|\lesssim 1.6 n^2 \times10^{-4}$
which is quite compatible with observations~\cite{Spergel:2006hy}.
From the slow-roll equations of motion, Eq.~(\ref{slow-roll}), we find the solution of $\varphi$
\dis{
\varphi = \varphi_*+M_P {\rm sign}\left(\frac{U_{\varphi}}{U}\right)\sqrt{2\epsilon_\vp}N,
}
which gives $b_e-b_*=-4\beta^2N$.
Using this we can express Eq.~(\ref{fNL4}) at the end of inflation
(we take $\epsilon^e=\epsilon^e_\chi+\epsilon^e_\varphi=1$) in terms of slow-roll parameters at horizon crossing.
In the limit $8\beta^2N\ll 1$, we find
$\alpha\approx 1+N$, $u\approx\epsilon^e_\varphi$ and $v\approx1-\epsilon^e_\varphi$.
This leads to the simplified form of $f^{(4)}_{NL}$,
\dis{
-\frac65 f^{(4)}_{NL}\simeq 2\epsilon_\chi^*-\eta_{\chi\chi}^*+2\epsilon_\chi^*\epsilon_\vp^*(N+1)-\epsilon_\vp^*.
\label{fNLmodel1}
}
It can be noted that in this small $\beta$ limit, $\epsilon_\vp^*<\epsilon_\chi^*$ and the last
two terms in this equation are negligible compared to the first two terms.
Using Eq.~(\ref{approxNmod1}), $f_{\rm NL}^{(4)}$ can be written in terms of the required e-folding number in the small $\beta$ limit (required for enough inflation)
\begin{equation}
-\frac65 f^{(4)}_{NL} \simeq \frac{1}{2N} + \frac{\beta}{N} \frac{\varphi_*}{M_P}.
\end{equation}
It is clear that, in order to achieve enough inflation, the parameters are required to be small and hence $-\frac65 f^{(4)}_{NL}\approx \frac{1}{2N}$, which is unobservable.
\subsection{Brans-Dicke Type Models with Quadratic Exponent}
In order to observe the effect of higher powers in $b(\varphi)$, we consider a quadratic function, such that $\xi_b\neq0$.
To this end, we consider a Brans-Dicke type model with
\begin{eqnarray}
U(\varphi)=e^{4b(\varphi)}, \quad \quad b(\varphi)=-\beta\varphi^2/M_P^2.
\label{Model2}
\end{eqnarray}
With this choice, $\epsilon_\vp=32\beta^2\varphi^2/M_P^2$ and $\eta_{\varphi\vp}=-8\beta+64\beta^2\varphi^2/M_P^2$.
Once again, two example potentials are chosen for $\chi$:
\begin{itemize}
\item {\bf Model 2a} $V(\chi)=\frac{\lambda}{4}\chi^4$
\item {\bf Model 2b} $V(\chi)=\frac12m^2_\chi \chi^2$
\end{itemize}
where $\lambda$ and $m_\chi$ have the same definitions as before.
\begin{figure}[!t]
\begin{tabular}{c c}
\includegraphics[width=0.33\textwidth]{./figs/angle_index_case2a.ps}
&
\includegraphics[width=0.66\textwidth]{./figs/case2a.ps}
\end{tabular}
\caption{Spectral index ($n_\zeta -1$) and running of $n_\zeta$ and $n_C$ for model 2-a. We used $\beta=0.1\epsilon$ here.}
\label{fig:2ab}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.4\textwidth]{./figs/run_S_line.ps}
\end{center}
\caption{Running of $n_S$ for Models 1 and 2. $\alpha_S$ is independent of $\Delta$.}
\label{fig:runningS}
\end{figure}
The slow-roll parameters for these potentials are shown in the last two columns of Table~\ref{Tab1}. Once again, most of the slow-roll parameters can be written in terms of the single parameter, $\epsilon^2$. The exception is $\xi_b$, which is related to the coefficient of the coupling term, $\beta$. In order to analyse the expected levels of running from these cases, it is necessary to relate $\beta$ to $\epsilon$ and we take the assumption that $\beta=0.1\epsilon$.
For Model 2a, the numeric calculations of the spectra for this special case (in relation to $\epsilon$, $\theta$ and $\Delta$) are shown in Figures~\ref{fig:2ab} and \ref{fig:runningS}. Model 2b is almost identical to Model 2a and the numerical results are not shown.
Models 2a and 2b have a reflection symmetry about $\varphi$
($\theta\rightarrow -\theta$) and/or $\chi$ ($\theta\rightarrow \pi-\theta $).
However since the sign of $\tan\Delta$ is determined by the sign of
$\alpha$ in Eq.~(\ref{evolution-of-fluctuations}) and
$\alpha$ is proportional to $\sin 2\theta$ for the potential in
Eq.~(\ref{app1}), The reflection of $\varphi$ or $\chi$
changes also the sign of $\tan\Delta$ before the end of inflation.
For this reason, we see the symmetry in Figure~\ref{fig:2ab}
around the point ($\theta=\pi/2,\Delta=\pi/2$).
However after inflation, $\Delta$ is no longer dependent on the fields
$\varphi$ and $\chi$, thus this symmetry is not necessarily valid any more.
In this model, the solutions for $\varphi$ and $\epsilon_\varphi^e$ are given by
\dis{
\varphi_e=\varphi_*e^{8\beta N}, \qquad
\epsilon_\varphi^e=32\beta^2\varphi_e^2/M_P^2 = \epsilon_\varphi^* e^{16\beta N}.
\label{model2solphi}
}
Due to the quadratic coupling, the potential is steeper than that of traditional Brans-Dicke models, so that a much smaller
$\beta$ is required to give enough e-folding number.
In this case, if we assume $16\beta N \ll 1$,
then we obtain
\dis{
N\simeq\frac{1}{16\beta}\ln\left[1+\frac{8\beta}{n}e^{-2\beta\varphi_*^2}
\chi_*^2\right]-\frac{1}{16\beta}\ln\left[1+\frac{4\beta n^2}{1-32\beta^2\varphi_*^2} \right].
\label{Nmod2}
}
In order to obtain enough inflation, for $\varphi_*,\chi_*<50M_P$, then we require $\beta<0.0005$. In this very small limit, we can approximate Eq.~(\ref{Nmod2}):
\begin{equation}
N\approx\frac{\chi_*^2}{2n}-\frac{n^2}{4} +\beta
\left(
\frac{n^4}{2}-\frac{\varphi_*^2\chi_*^2}{n} - \frac{2\chi_*^4}{n^2}
\right).
\label{approxNmod2}
\end{equation}
As in Model 1a, it is possible to estimate the non-Gaussianity for small $\beta$.
Again, we take $\epsilon^e=1$, so that $\alpha$, $u$ and $v$ are given as before.
In the limit $16\beta N\ll1$, we find
\dis{
-\frac65 f^{(4)}_{NL}\simeq 2\epsilon_\chi^*-\eta_{\chi\chi}^*+2\epsilon_\chi^*\epsilon_\vp^*(N+1)-\epsilon_\vp^*-\frac12\left(\epsilon_\chi^*\right)^2\epsilon_\vp^*\xi_b(N+1)^3.
\label{fNLmodel2}
}
The final term is directly due to $b_{\varphi\vp}\neq0$. Using the approximation in Eq.~(\ref{approxNmod2}), we find
\begin{equation}
-\frac65 f^{(4)}_{NL} \simeq \frac{1}{2N} + \frac{\beta}{N} \frac{\varphi_*^2}{M_P^2}.
\end{equation}
As in Model 1, in order to achieve enough inflation, the parameters are required to be small and hence $-\frac65 f^{(4)}_{NL}\approx \frac{1}{2N}$.
\section{Examples of Scalar-Tensor Theories with Sum Potentials}
\label{SecTheory2}
The second models we consider are those with sum potentials. If we define the potential
\dis{
W(\varphi,\chi)=U(\varphi)+V(\chi)=\frac12m_\varphi^2\varphi^2+\frac12m^2_\chi\chi^2,
}
then the slow-roll parameters simplify greatly and are given by
\dis{
\epsilon_\varphi=\frac{M_P^2}{2}\left(\frac{m^2_\varphi \varphi}{W}\right)^2, \qquad
\epsilon_\chi=\frac{M_P^2}{2}\left(\frac{m^2_\chi \chi}{W}\right)^2e^{-2b},
}
\dis{
\eta_{\varphi\vp}=M_P^2\frac{m^2_\varphi}{W},\qquad
\eta_{\chi\chi}=M_P^2\frac{m^2_\chi}{W}e^{-2b},\qquad \eta_{\varphi\chi}=0,
}
\dis{
\xi^2_{\varphi\vp\varphi}=\xi^2_{\varphi\vp\chi\chi}=\xi^2_{\varphi\chi\chi\chi}=\xi^2_{\chi\chi\chi}=0.}
It is no longer possible to directly relate the parameters to $\epsilon$ and $\theta$.
However, if we represent the ratio of masses by
\dis{
r=\frac{m_\chi^2}{m_\varphi^2},}
then the slow-roll parameters can be re-written in terms of $\epsilon$, $\beta$ and $r e^{-2b}$ only:
\dis{
\eta_{\varphi\vp}=\epsilon\left(\cos^2(\theta)+\frac{\sin^2(\theta)}{r e^{-2b}}\right), \qquad
\eta_{\chi\chi}=\epsilon\left(\sin^2(\theta)+\cos^2(\theta)r e^{-2b}\right), \qquad
}
\dis{
\epsilon_b=64\frac{\beta^2}{\epsilon}\frac{\cos^2(\theta)}{\left(\cos^2(\theta)+\frac{\sin^2(\theta)}{r e^{-2b}} \right)}.}
\begin{figure}[!t]
\begin{tabular}{c c}
\includegraphics[width=0.33\textwidth]{./figs/angle_index_case3.ps}
&
\includegraphics[width=0.66\textwidth]{./figs/case3.ps}
\end{tabular}
\caption{Spectral index ($n_\zeta -1$) and running of $n_\zeta$ and $n_C$ in terms of $\epsilon$ for model 3. Values of $\beta=0.1\sqrt{\epsilon}$ and $r e^{-2b}=2$ have been assumed.}
\label{fig:34}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.4\textwidth]{./figs/run_S_line_3and4.ps}
\end{center}
\caption{Running of $n_S$ for Models 3 and 4. $\alpha_S$ is independent of $\Delta$.}
\label{fig:runningS34}
\end{figure}
For this sum potential, the remaining choice is in the function of $b(\varphi)$ and we consider the two obvious cases:
\begin{itemize}
\item {\bf Model 3} $b(\varphi)=-\frac{\beta \varphi}{M_P}$
\item {\bf Model 4} $b(\varphi)=-\frac{\beta \varphi^2}{M_P^2}$
\end{itemize}
We can fix $\beta$ and the mass ratio, in order to estimate the spectral index and running. We take $r e^{-2b}=2$ and, for Models 3 and 4 respectively, we assume $\beta=0.1\sqrt{\epsilon}$ and $\beta=0.1\epsilon$.
The results of the spectrum for Model 3 are shown in Figures~\ref{fig:34} and \ref{fig:runningS34}.
The results for Model 4 are very similar to those of Model 3 and are therefore not shown.
\section{Numerical Analysis of Non-Gaussianity}
\label{secNumerical}
In order to verify our analytical results of the previous section, we numerically solved the background equations of motion, Eq.~(\ref{background}) and the Friedmann equation, Eq.~(\ref{Friedmann}), not assuming slow-roll.
Multiple trajectories were considered, with initial values, $\varphi_i$ and $\chi_i$, given by a $\varphi$-$\chi$ grid and $\dot\varphi_i=\dot\chi_i=0$.
The end of inflation is taken to be the time at which the slow-roll parameter $\epsilon=\epsilon_\vp+\epsilon_\chi$ becomes unity and the total number of e-foldings until this point is given by $N_f$.
At horizon crossing, the fields have reached values of $\varphi_*$ and $\chi_*$ and the remaining number of e-folds is denoted by $N$.
The calculation of non-Gaussianity requires the gradient of the number of e-foldings between horizon crossing and the end of inflation, $N$ with respect to each of the fields at crossing, $\varphi_*$ and $\chi_*$.
It is useful to notice, quite trivially, that if for a given trajectory, $\varphi_i=\varphi_*$ and $\chi_i=\chi_*$, then $N_f=N$.
It is therefore possible to calculate $N,_i$, $N,_{ij}$ etc by differentiating the grid of $N_f$ with respect to $\varphi_i$ and $\chi_i$.
Each of the cases considered above (Models 1-4) were modelled numerically.
The analytical slow-roll approximations given in Eq.~(\ref{fNLmodel1}) and (\ref{fNLmodel2}) were seen to agree within $5\%$.
For Models 3 and 4, no analytical approximation was found. For these sum potentials, the numerical level of non-Gaussianity was calculated from the second term in Eq.~(\ref{fNL}) and the results are shown in Figure~\ref{fig:f_nl}.
In previous work, \cite{VernizziWands}, for theories with {\it canonical} kinetic terms and a sum potential, $f_{\rm NL}\approx1/N$.
We find numerically that the $f_{\rm NL}$ is of that order of magnitude, even with non-canonical couplings. Therefore we conclude that $\beta$ has little impact on the level of nonlinearity.
\begin{figure}[!t]
\begin{tabular}{c c}
\includegraphics[width=0.5\textwidth]{./figs/f_NL_case3.ps}
&
\includegraphics[width=0.5\textwidth]{./figs/f_NL_case4.ps}
\end{tabular}
\caption{Level of non-Gaussianity for Models 3 (left) and 4 (right) calculated from the second term of Eq.~(\ref{fNL}). We have assumed $m_\varphi=m_\chi=5\times 10^{-5} M_P$. $\beta=0.1$.}
\label{fig:f_nl}
\end{figure}
\section{Conclusion}
\label{SecConc}
The focus of this paper was slow-roll inflation in theories with two scalar fields,
coupled through a potential as well as their kinetic terms. We have derived the general
formulae for the running of the spectral indices for adiabatic and entropic perturbations
as well as for the cross-correlation spectrum. In addition, using the $\delta N$-formalism and
specialising to the product potential ($W(\varphi,\chi)=U(\varphi)V(\chi)$) and the sum potential
($W(\varphi,\chi)=U(\varphi)+V(\chi)$) we have derived the general expressions for the
nonlinearity parameter $f_{\rm NL}$ during inflation.
We have calculated the spectral indices and runnings for several example models. The results are within present observational ranges.
One of the specific examples we have considered is the Jordan-Brans-Dicke theory with quadratic and
quartic potentials.
In this case, the slow-roll parameter for the field $\varphi$ is
given by the (constant) coupling parameter $\beta$, defined in Eq.~(\ref{Model1}). The slow-roll
condition for the field $\varphi$ implies that $\beta$ itself has to be small. In this case,
an analytical formula for $f_{\rm NL}$ during inflation is given by Eq.~(\ref{fNLmodel1}) and is of order $1/N$, where $N$ is the total number of e-foldings.
Assuming $\beta$ to be positive, the effect of the coupling is to {\it enhance} $f_{\rm NL}$ by an additional term $\frac{\beta\varphi}{NM_P}$.
When the coupling is quadratic in the $\varphi$-field (as in Eq.~(\ref{Model2})), the nonlinearity is {\it enhanced} by a term $\frac{\beta\varphi^2}{NM_P^2}$. Neither of these terms are significant and in the slow-roll limit, $f_{\rm NL}$ is not much enhanced by the coupling.
We also considered models with a sum of two quadratic potentials. Due to the complexity of the analytic formula for $f_{\rm NL}$, it is difficult to find an approximate equation.
We therefore provide numerical results for simple models and showed that $f_{\rm NL}\ll1$ and is unobservable for the slow-roll limit.
To conclude, in order to distinguish between inflationary models, future observations will search for
deviations from standard one-field inflation. In order to do so one has to go beyond the spectral index as
an observable.
Future experiments, such as PLANCK, will tighten the constraints on the spectral running and nonlinearity.
The results of this paper can be used to calculate these parameters in other multi-field models with non-canonical kinetic terms, in order to
constrain the parameter ranges.
\acknowledgments
CvdB, K-YC and LH acknowledge support from PPARC.
|
1,314,259,993,020 | arxiv | \section{Introduction}
We consider the Klein--Gordon--Schr\"odinger system
\begin{equation}
\begin{aligned}
\label{KGS}
&c^{-2}\partial_{tt}z(t,x) - \Delta z(t,x) + c^2z(t,x) = \vert \psi(t,x) \vert^2, \\
&i \partial_t \psi(t,x) + \frac12\Delta\psi(t,x) + \psi(t,x) z(t,x) = 0,
\end{aligned}
\end{equation}
given by a Klein--Gordon equation coupled nonlinearly with a classical Schr\"odinger equation. This setting arises in quantum field theory, representing the dynamics of the interaction between a complex-valued scalar nucleon field $\psi:\mathbb{R}\times\mathbb{R}^d\to\mathbb{C}$ with a neutral real-valued scalar meson field $z:\mathbb{R}\times\mathbb{R}^d\to\mathbb{R}$. For existence and uniqueness of global smooth solutions see \cite{FuTs1}, \cite{FuTs2}, \cite{FuTs3}.
The parameter $c$, proportional to the speed of light, plays a very important role in the behaviour of the solution and gives rise to two different regimes. We distinguish between the so-called \textit{relativistic regime}, where $c=1$, and the \textit{non-relativistic regime} with $c\gg 1$. The former regime is well studied numerically, for instance see \cite{BY}, as its solution is slowly varying. The non-relativistic, on the other hand, brings in a significant additional challenge in terms of its numerical treatment, given the highly oscillatory nature of its solution. It causes classical numerical methods to collapse, as they fail to capture the rapid oscillations, leading to large errors or, in turn, severe time step size restrictions and thus very intense computational efforts. This is even the case for Gautschy-type methods (see \cite{BD}, \cite{HLW}), which were specifically designed to numerically solve highly oscillatory problems. Splitting methods fail to deal with these rapid oscillations in a similar way, see for instance \cite{Fa} for their analysis in the context of Schr\"odinger equations.
In \cite{BZ}, an unconditionally stable method was developed, based on a multi-scale expansion technique, that achieves uniform linear convergence in time, for sufficiently smooth solutions. Quadratic convergence was achieved in this setting, yet only in the case where either $c=\mathcal{O}(1)$ or $c\tau\geq 1$.
In \cite{BaKoS18} an approach was presented that succeeded to capture all regimes in $c$, providing error bounds that were independent of this parameter and without requiring any step size restrictions. This was done by means of the introduction of the so-called \textit{twisted variables}, that were already well known both in physics as "interaction picture", and in the study of partial differential equations at low regularity. The main idea therein was to explicitly filter out the highly oscillatory phases, approximate the slowly varying parts, which does not produce dependency on $c$ in the error constants, and integrate the interacting highly oscillatory phases exactly. In addition, this approach achieves asymptotic consistency, meaning that it preserves the NLS limit on the discrete level.
In comparison to \cite{BaKoS18}, we propose a novel class of exponential-type integrators that equally manages to capture all regimes of $c$, without any time step size restrictions, and is asymptotically consistent. {We introduce a new construction, exploiting the structure of the leading differential operators $\frac12\Delta$ and $c\langle \nabla \rangle_c$, which allows us to establish an explicit relation between a gain of negative powers of the potentially very large parameter $c$ in the error constant versus a loss of derivative. In other words, a gain in accuracy, in the non-relativistic regime, in exchange of a loss in derivative. In addition to this, in the first order scheme, we require one derivative less in the Klein--Gordon part than pre-existing methods up to our knowledge.} We achieve this by employing techniques introduced in \cite{CS} in the context of the Klein--Gordon equation. This recent work couples the ideas of low regularity (in space) approximation presented in \cite{RS} in the context of an abstract class of of evolution equations, with the idea of uniform accuracy achieved in \cite{BaKoS18}.
The underlying strategy is the expansion of suitable filtered functions that allows the embedding of the full spectrum of oscillations into the numerical scheme. Here, the commutator structure of the leading operator $\frac12\Delta$ and
is studied (see Lemma \ref{lemma_comm}), as it plays a crucial role in the achievement of low regularity approximations that do not produce powers of $c$ in our error estimates. {In addition to this, we study the asymptotic behaviour of the leading operator $c\langle \nabla \rangle_c$ in order to resolve the nonlinear frequency interaction caused by the coupled nature of this system.}
In the present setting, however, the fact that the equations are coupled non-linearly supposes an additional challenge. It makes the analysis much more involved, since one has to consider the non-linear interaction of highly oscillatory parts. This rises the need of new, adapted techniques.\\
\\
\noindent{\bf Outline of the paper.}
We begin by expressing \eqref{KGS} as a first order system in time in Section 2. In Section 3 we motivate the new first order scheme and it's second order counterpart will be derived in Section 4. We prove their uniform convergence in Theorems \ref{thm:1} and \ref{thm:2}, respectively. In Section 5 we briefly present the limit system, show that, as $c\to\infty$ formally, we recover the solution to the limit system. Finally, numerical experiments are presented in Section 6, confirming our theoretical results.\\
\\
\noindent{\bf Notation.}
For reasons regarding ease of implementation and clarity of presentation, we impose periodic boundary conditions, i.e. $x\in\mathbb{T}^d$. However, we note that nor the construction nor the analysis of our scheme depends on any Fourier expansion techniques, and thus can be generalised to bounded domains $x\in \Omega\subset \mathbb{R}^d$ equipped with suitable boundary conditions, as well as the full space $x\in\mathbb{R}^d$. For simplicity, we may occasionally make use of $\mathcal{O}$-notation exclusively in the context of constants independent of $c$. For the sake of simplifying future notation, we define the $\varphi_1$-function as
\begin{align}\label{phi1}
\varphi_1(\xi) = \frac{e^{\xi}-1}{\xi},\quad \xi \in \mathbb{C}.
\end{align}
We also note
\begin{align}\label{stability_trick}
\varphi_1(\xi) = 1+\mathcal{O}(\xi).
\end{align}
We refer to \cite{HoOs} for details on this family of functions. In the following we fix $r>\frac{d}{2}$ and we denote by $\| . \|_r$ the standard $H^r=H^r(\mathbb{T}^d)$ Sobolev norm, where, for this choice of $r$, the following well-known bilinear estimate holds
$$
\| fg\|_r \leq C_{r,d} \|f\|_r\|g\|_r,
$$
for some constant $C_{r,d}>0$ independent of $f$ and $g$.
\section{Formulation as a first order system}
We start by defining the following differential operator that will simplify our notation significantly. For a given $c>0$ we define the following operator
\begin{align*}
c\langle \nabla \rangle_c = c\sqrt{c^2-\Delta}.
\end{align*}
One can verify that this differential operator is well-defined, as its corresponding Fourier multiplier has the form $(\langle \nabla \rangle_c)_k=\sqrt{k^2+c^2}$. We may now rewrite \eqref{KGS} as a first order system in time (see \cite{MaNa}). Setting
\begin{align}\label{def_u}
u = z - ic^{-1}\langle \nabla \rangle_c^{-1}\partial_t z,\quad v = z - ic^{-1}\langle \nabla \rangle_c^{-1}\partial_t \overline{z},
\end{align}
a simple calculation shows that $z=\frac12(u+v)$. Furthermore, if we assume that $z(t,x)\in\mathbb{R}$, we have
\begin{align}\label{z_ito_u}
z=\frac12 (u+\overline{u}).
\end{align}
In the following we will restrict our attention to this case purely for the purpose of presenting our main ideas as clearly as possible. Note, however, that this does not impose any significant restriction. Having said this, a short calculation shows that the corresponding first order system in $(u,\psi)$ reads
\begin{align}
\label{KG}
&i \partial_t u + c \langle \nabla \rangle_c u - c \langle \nabla \rangle_c^{-1} \vert \psi \vert ^2 =0, &&u(0)=z(0) -ic^{-1}\langle \nabla \rangle_c^{-1}\partial_tz(0),\\
\label{Schr}
&i \partial_t \psi + \frac12\Delta\psi + \frac12\psi (u+\overline{u}) = 0,&&\psi(0)=\psi_0.
\end{align}
\section{A first order integrator}
In this section we proceed to give a detailed derivation of the numerical scheme for $u^{n+1}\approx u(t_{n+1})$ with $t_{n+1}=t_n+\tau$, followed by a less detailed derivation of the numerical scheme for $\psi^{n+1}\approx\psi(t_{n+1})$ that employs analogous ideas. Duhamel's formula for \eqref{KG} reads
$$
u(t_n+\tau) = e^{i \tau c\langle \nabla \rangle_c}u(t_n) - i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert \psi(t_n+s) \vert^2 ds.
$$
Iterating Duhamel's formula for \eqref{Schr} leads to
\begin{align}
\label{1o_1}
u(t_n+\tau) = e^{i \tau c\langle \nabla \rangle_c}u(t_n) - i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{i\frac12 s\Delta}\psi(t_n) \vert^2 ds + \mathcal{R}_1,
\end{align}
where $\mathcal{R}_1$ fulfills a bound of the form
\begin{equation}\label{R_1}
\begin{aligned}
\|\mathcal{R}_1\|_r &\leq \bigg\| i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{\psi(t_n)} \big) \mathcal{I}_{\psi}(s) \,ds \bigg\|_r \\
&+ \bigg\| i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c}\big( e^{i\frac12s\Delta}{\psi(t_n)} \big) \mathcal{I}_{\psi}(s) \,ds \bigg\|_r \\
&+ \bigg\| i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c}\big| \mathcal{I}_{\psi}(s)\big|^2 ds \bigg\|_r\\
&\leq \tau^2 K\big( \sup_{0\leq\xi\leq\tau} \|\psi(t_n+\xi)\|_r\big),
\end{aligned}
\end{equation}
where
\begin{align}\label{I_psi}
\mathcal{I}_{\psi}(s)=\frac{i}{2} e^{i\frac12 s\Delta}\int_0^{s}e^{-i\frac12\sigma\Delta}\psi(t_n+\sigma)\big( u(t_n+\sigma) + \overline{u(t_n+\sigma)} \big)d\sigma,
\end{align}
thanks to the following remarks
\begin{align}
\label{cnab_liniso_bound}
\|c\langle \nabla \rangle_c^{-1}\|_r \leq 1, \quad \|e^{itc\langle \nabla \rangle_c}\|_r =1,\quad \|e^{it\Delta}\|_r =1 \,\quad\forall t\in\mathbb{R}.
\end{align}
It is left to approximate the present oscillatory integral in a suitable way and, to this end, we introduce the following crucial commutator term, which will appear in our local error estimates.
\begin{definition}
For a function $H(v_1,\dots,v_n)$, $n\geq 1$, and a linear operator $L:H^r\to H^r$ we define the following commutator type term
\begin{align*}
\mathcal{C}[H,L] (v_1,\dots,v_n)= -L(H(v_1,\dots,v_n))+\sum_{i=1}^n D_iH(v_1,\dots,v_n)\cdot Lv_{i},
\end{align*}
where $D_iH$ stands for the partial derivative of H with respect to the variable $v_i$. Furthermore, we set
$$\mathcal{C}^2[H,L] (v_1,\dots,v_n)= \mathcal{C}[ \mathcal{C}[H,L],L] (v_1,\dots,v_n)$$
and
$$
f_{\text{quad}}(v,w)=vw.
$$
\end{definition}
We now establish the necessary bounds for these commutator type terms.
\begin{lemma}
\label{lemma_comm}
We have that
\begin{align*}
&\|\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),\Delta] (v,w)\|_r \leq K_1 \|v\|_{r+1}\|w\|_{r+1},\\
&\|\mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),\Delta] (v,w)\|_r \leq K_2 \|v\|_{r+2}\|w\|_{r+2}
\end{align*}
for some $K_1,K_2>0$.
\end{lemma}
\begin{proof}
We will show the first assertion in detail, the second assertion can be proven iterating this argument. By definition,
\begin{align*}
\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),\Delta] (v,w)= -\Delta (vw) + w\Delta v + v\Delta w.
\end{align*}
The assertion follows by the product rule of the Laplacian
$$
\Delta(vw) = v\Delta w + 2\nabla v \cdot\nabla w + w\Delta v.
$$
\end{proof}
We now aim to capture the oscillatory integral in \eqref{1o_1} in a way that does not trigger dependency on $c$ in the error terms and allows a low regularity approximation. The technique is captured in the following lemma.
\begin{lemma}[First order approximation of the integral in \eqref{1o_1}]
\label{Lemma1ou}
It holds that
\begin{align*}
\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{i \frac12 s\Delta}v \vert^2 ds = \tau \overline{v}\varphi_1(i \tau (\Delta-c^2))v + \mathcal{O}\big(\tau^2(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),\Delta](v,v)+c^{-2\alpha}\Delta^{1+\alpha}v)\big).
\end{align*}
\end{lemma}
\begin{proof}
We introduce the following filtered function defined by
\begin{align}
\label{N}
\mathcal{N}(s,s_1,\Delta,v) :=e^{i \frac12 s_1 \Delta} \big( e^{-i\frac12 s_1\Delta}e^{i s\Delta}v \big)\big( e^{-i\frac12 s_1\Delta}\overline{v} \big).
\end{align}
Then, the integral reads
$$
\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{i \frac12 s\Delta}v \vert^2 ds = \int_0^{\tau} e^{-is c\langle \nabla \rangle_c} e^{-i\frac12 s\Delta} \mathcal{N}(s,s,\Delta,v) ds.
$$
{At this point one needs to first tackle the interactions between the differential operators $c\langle \nabla \rangle_c$ and $\frac12\Delta$. Note that, as it can be shown via fractional Taylor series expansion of the function $x\to c^2\sqrt{c^2+x^2}$, it holds
\begin{align}\label{taylor_cnab}
c\langle \nabla \rangle_c = c^2- \tfrac{1}{2}\Delta+ \mathcal{O}\big(\tfrac{\Delta^{1+\alpha}}{c^{2\alpha}}\big), \quad 0\leq\alpha\leq1,
\end{align}
and thus,
$$
\int_0^{\tau} e^{-is c\langle \nabla \rangle_c} e^{-i\frac12 s\Delta} \mathcal{N}(s,s,\Delta,v) ds = \int_0^{\tau} e^{-is c^2} \mathcal{N}(s,s,\Delta,v) ds + \mathcal{R}_1,
$$
where, using \eqref{taylor_cnab}, we see that we may bound $\mathcal{R}_1$ by
\begin{align}\label{c_trick}
\|\mathcal{R}_1\|_r \leq c^{-2\alpha}K(\|v\|_{r+2(1+\alpha)}),\quad 0\leq \alpha \leq 1,
\end{align}
for some $K'>0$ independent of $c$.}
On the other hand, Taylor series expansion gives the following approximation
$$\mathcal{N}(s,s_1,\Delta,v)=\mathcal{N}(s,0,\Delta,v)+\int_0^{s} \partial_{s_1}\mathcal{N}(s,s_1,\Delta,v) ds_1.$$
Thus, plugging in this expansion and then integrating exactly we obtain
\begin{align*}
\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} e^{i\frac12 s\Delta} \mathcal{N}(s,s,\Delta,v) ds & = \int_0^{\tau} e^{-i s c^2} \mathcal{N}(s,0,\Delta,v) ds+ \mathcal{R}_1\\
& = \int_0^{\tau} e^{-i s c^2} \big( e^{i s\Delta}{v} \big)\overline{v} ds+ \mathcal{R}_1+ \mathcal{R}_2\\
&= \tau \overline{v}\varphi_1(i \tau (\Delta-c^2))v + \mathcal{R}_1+ \mathcal{R}_2,
\end{align*}
where $\mathcal{R}_2$ can be bounded as follows. By \eqref{cnab_liniso_bound} and the remark
\begin{align*}
\partial_{s_1} \mathcal{N}(s,s_1,\Delta,v)_{\vert s_1=s}
= e^{i\frac12 s_1\Delta}\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),-i\tfrac{1}{2}\Delta](e^{-i\frac12 s_1\Delta}e^{is\Delta}v , e^{-i\frac12s_1\Delta}\overline{v}).
\end{align*}
it holds
\begin{align*}
\|\mathcal{R}_2\|_r \leq \bigg\| \int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} e^{i\frac12s\Delta} \bigg(\int_0^s\partial_{s_1} \mathcal{N}(s,s_1,\Delta,v)ds_1\bigg) ds \bigg\|_r\leq \tau^2 K\big(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , \overline{v})\big).
\end{align*}
\end{proof}
The expansion \eqref{1o_1} together with Lemma \ref{Lemma1ou} lead to the following first order uniformly accurate integrator
\begin{align}
\label{KGo1}
u^{n+1} = e^{i \tau c\langle \nabla \rangle_c}u^n - i \tau c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\overline{\psi^n}\varphi_1(i \tau (\Delta-c^2))\psi^n .
\end{align}
Finally, given $u^{n+1}$ by \eqref{KGo1}, one can very easily find $z^{n+1}$, via \eqref{z_ito_u},
$$
z^{n+1}=\frac12\big( u^{n+1} +\overline{u^{n+1}} \big).
$$
Now we proceed as follows for $\psi^{n+1}$. Duhamel's formula for \eqref{Schr} reads
$$
\psi(t_n+\tau) = e^{i\frac12\tau\Delta}\psi(t_n) + i\frac12 e^{i\frac12\tau\Delta}\int_0^{\tau}e^{-i\frac12s\Delta}\psi(t_n+s)\big( u(t_n+s) + \overline{u(t_n+s)} \big)ds.
$$
Iterating Duhamel's formula for \eqref{KG} and \eqref{Schr} respectively leads to
\begin{align}
\label{1o_3}
\psi(t_n+\tau) = e^{i\frac12\tau\Delta}\psi(t_n) + \frac{i}{2} e^{i\frac12\tau\Delta}\int_0^{\tau}e^{-i\frac12s\Delta}\big(e^{i \frac12s\Delta}\psi(t_n)\big)\big( e^{i sc\langle \nabla \rangle_c}u(t_n) + e^{-i sc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds+ \mathcal{R}_2,
\end{align}
where $\mathcal{R}_2$ fulfills the bound
\begin{align*}
\| \mathcal{R}_2 \|_r \leq \|\mathcal{R}_{2,1}\|_r + \|\mathcal{R}_{2,2}\|_r,
\end{align*}
with
\begin{align*}
\mathcal{R}_{2,1}=\frac{i}{2} e^{i\frac12\tau\Delta}\int_0^{\tau}e^{-i\frac12s\Delta}
\mathcal{I}_{\psi}(s)
\big( e^{i sc\langle \nabla \rangle_c}u(t_n) + e^{-i sc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds,
\end{align*}
with $\mathcal{I}_{\psi}$ given by \eqref{I_psi} and
\begin{align*}
\mathcal{R}_{2,2}=\frac{i}{2} e^{i\frac12\tau\Delta}\int_0^{\tau}e^{-i\frac12s\Delta}\big(e^{i \frac12s\Delta}\psi(t_n)\big)\big( \mathcal{I}_u(s)+ \overline{\mathcal{I}_u(s)} \big)ds,
\end{align*}
\begin{align}\label{I_u}
\mathcal{I}_u(s)=i c \langle \nabla \rangle_c^{-1} e^{i s c\langle \nabla \rangle_c}\int_0^{s} e^{-i \sigma c\langle \nabla \rangle_c} \vert \psi(t_n+\sigma) \vert^2 d\sigma.
\end{align}
Thus, we conclude by \eqref{cnab_liniso_bound} that
\begin{align}
\label{R_2}
\|\mathcal{R}_2\|_r \leq \tau^2 K\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_r,\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_r\big).
\end{align}
It is left to approximate the highly oscillatory integral in \eqref{1o_3} and to this end we proceed analogously as in Lemma \ref{Lemma1ou}.
\begin{lemma}[First order approximation of the integral in \eqref{1o_3}]
\label{Lemma1opsi}
It holds that
\begin{align*}
\int_0^{\tau}e^{-i\frac12s\Delta}&\big(e^{i \frac12s\Delta}v\big)\big( e^{i sc\langle \nabla \rangle_c}w + e^{-i sc\langle \nabla \rangle_c}\overline{w} \big)ds \\
&=\tau v\big( \varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))w + \varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{w} \big) + \mathcal{O}(\tau^2 \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta](v, w)).
\end{align*}
\end{lemma}
\begin{proof}
We define the following filtered functions
$$
\mathcal{N}(s,s_1,\Delta,v,w)= e^{-i\frac12s_1\Delta}\big(e^{i\frac12 s_1\Delta}v)(e^{i\frac12 s_1\Delta}e^{-i \frac12s\Delta} (e^{i sc\langle \nabla \rangle_c}w + e^{-i sc\langle \nabla \rangle_c}\overline{w})\big).
$$
Taylor series expansion around the point $s_1=0$
yields the following first order approximation
\begin{align*}
\int_0^{\tau}e^{-i\frac12s\Delta}\big(e^{i \frac12s\Delta}v\big)&\big( e^{i sc\langle \nabla \rangle_c}w + e^{-i sc\langle \nabla \rangle_c}\overline{w} \big)ds\\
&= \int_0^{\tau}\mathcal{N}(s,s,\Delta,v,w)ds =\int_0^{\tau}\mathcal{N}(s,0,\Delta,v,w)ds+ \mathcal{R}
\\&={\tau v\big( \varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))w + \varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{w} \big)} + \mathcal{R}.
\end{align*}
Once again we find a bound for $\mathcal{R}$ thanks to the observation
$$
\partial_{s_1} \mathcal{N}(s,s_1,\Delta,v,w)=e^{-i\frac12s_1\Delta}\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](e^{i\frac12s_1\Delta}v, e^{i\frac12s_1\Delta}(e^{is_1c\langle \nabla \rangle_c}w+e^{-is_1c\langle \nabla \rangle_c}\overline{w} ) ),
$$
thus
$$
\|\mathcal{R}\|_r\leq \tau^2 K\big( \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w)\big).
$$
\end{proof}
Plugging our findings from Lemmata \ref{Lemma1ou} and \ref{Lemma1opsi} into the expansion \eqref{1o_3} motivates the following scheme for $\psi^{n+1}$.
\begin{align}
\label{Schro1}
\psi^{n+1} = e^{i \frac12\tau\Delta}\psi^{n} + \tau \frac{i}{2} e^{i\frac12\tau\Delta}\psi^{n}\big( \varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))u^{n} + \varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{u^{n}} \big).
\end{align}
In the sections that follow we aim to carry out the error analysis of the scheme in $(u^{n},\psi^{n})$ given by \eqref{KGo1} and \eqref{Schro1}. Before we begin we denote by $\varphi^t_{K}$, $\varphi^t_{S}$ the exact flows of \eqref{KG} and \eqref{Schr} respectively and by $\Phi^t_{K}$, $\Phi^t_{S}$ the numerical flows corresponding to \eqref{KGo1} and \eqref{Schro1} respectively, such that in particular it holds
$$
u(t_{n+1})=\varphi^{\tau}_K(u(t_n),\psi(t_n)),\, \psi(t_{n+1})=\varphi^{\tau}_S(u(t_n),\psi(t_n)),\quad u^{n+1}=\Phi^{\tau}_{K}(u^n,\psi^n),\, \psi^{n+1}=\Phi^{\tau}_{S}(u^n,\psi^n).
$$
\subsection{Local error analysis}
\begin{lemma}
\label{localerror1}
Fix $r>\frac{d}{2}$. The local error given by the differences $\varphi^{\tau}_K(u(t_n),\psi(t_n))-\Phi^{\tau}_K(u(t_n),\psi(t_n))$ and $\varphi^{\tau}_S(u(t_n),\psi(t_n))-\Phi^{\tau}_S(u(t_n),\psi(t_n))$ satisfies
$$
\varphi^{\tau}_K(u(t_n),\psi(t_n))-\Phi^{\tau}_K(u(t_n),\psi(t_n)) = \mathcal{O}(\tau^2(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta]+c^{-2\alpha}\Delta^{1+\alpha}))(\psi(t_n) ,\overline{\psi(t_n)}))
$$
and
$$
\varphi^{\tau}_S(u(t_n),\psi(t_n))-\Phi^{\tau}_S(u(t_n),\psi(t_n)) = \mathcal{O}(\tau^2 \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](\psi(t_n) , w)),
$$
for $w\in \{ u(t_n),\overline{u(t_n)} \}$.
\end{lemma}
\begin{proof}
This assertion follows by the bounds we have found for the remainder terms in \eqref{R_1} and \eqref{R_2}, Lemma \ref{Lemma1ou} and Lemma \ref{Lemma1opsi}, together with Lemma \ref{lemma_comm}.
\end{proof}
\subsection{Stability analysis}
\begin{lemma}
\label{stability1}
Fix $r>\frac{d}{2}$. The numerical flows $\Phi^{\tau}_K$ and $\Phi^{\tau}_S$ defined by \eqref{KGo1} and \eqref{Schro1} respectively are stable in $H^r$, namely it holds for any $v_i,w_i\in H^r$, $i\in\{1,2\}$ that
$$
\|\Phi^{\tau}_K(v_1,w_1)-\Phi^{\tau}_K(v_2,w_2)\|_r\leq \|v_1-v_2\|_r + \tau M_1 \|w_1-w_2\|_r,
$$
$$
\|\Phi^{\tau}_S(v_1,w_1)-\Phi^{\tau}_S(v_2,w_2)\|_r\leq \|w_1-w_2\|_r + \tau M_2 (\|v_1-v_2\|_r +\|w_1-w_2\|_r),
$$
where $M_1$ and $M_2$ can be chosen independently of $c$.
\end{lemma}
\begin{proof}
This claim follows by \eqref{cnab_liniso_bound}. In addition, we use the estimate $\vert \varphi_1 (i\xi)\vert\leq 1$ for all $\xi\in\mathbb{R}$.
\end{proof}
\subsection{Global error}
\begin{theorem}\label{thm:1}
Fix $r>\frac{d}{2}$ and assume that the solution $(u,\psi)$ of \eqref{KG}-\eqref{Schr} satisfies $u\in\mathcal{C}([0,T],H^{r+1})$, {$\psi\in\mathcal{C}([0,T],H^{r+2(1+\alpha)})$, $0\leq\alpha\leq1$}. Then there exists a $\tau_0>0$ such that for all $0<\tau\leq\tau_0$ the following estimate holds for $(u^n,\psi^n)$ defined in \eqref{KGo1} and \eqref{Schro1}
\begin{align*}
\|u(t_n)-u^n\|_r + \|\psi(t_n)-\psi^n\|_r &\leq \tau K_1\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_{r+1},\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+1}\big)\\
&{+\tau c^{-2\alpha}K_2\big(\sup_{t_n\leq t\leq t_{n+1}}\|\psi(t)\|_{+2(1+\alpha)}\big)},
\end{align*}
where, in particular, $K_1$ and $K_2$ can be chosen independently of $c$.
\end{theorem}
\begin{proof}
The proof follows by means of a Lady Windermere's fan argument (see, for example \cite{HNW}), after plugging in the results obtained in Lemmata \ref{localerror1} and \ref{stability1}.
\end{proof}
{\noindent{\bf Remark.} We note that, in the fully discrete case of the non-relativistic with highest discrete frequency $\vert K \vert \ll c^{-2\alpha}$, the second term in the global error estimate presented above becomes negligible, as the contribution of the higher Sobolev norm is nearly cancelled by the very small parameter $c^{-2\alpha}$. In this case the error constant is then lead by the $H^{r+1}$ norm of the solution, thus allowing for lower regularity assumptions than in pre-exisiting methods in practice.}
\section{A second order integrator}
We dedicate this section to the derivation of a second order counterpart of the uniformly accurate low regularity integrator we have obtained in the previous section. Iterating Duhamel's formula for \eqref{Schr} yields
\begin{equation}\label{2o_1}
\begin{aligned}
u(t_n+\tau) &= e^{i \tau c\langle \nabla \rangle_c}u(t_n) - i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{is\frac12\Delta}\psi(t_n) \vert^2 ds \\
&-ic\langle \nabla \rangle_c^{-1}e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-is\frac12\Delta}\overline{\psi(t_n)}\big)\mathcal{I}_{\psi}(s)ds\\
&-ic\langle \nabla \rangle_c^{-1}e^{i \tau c\langle \nabla \rangle_c}\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{is\frac12\Delta}\psi(t_n)\big)\overline{\mathcal{I}_{\psi}(s)} ds + \mathcal{R}'_3,
\end{aligned}
\end{equation}
where, if we once again iterate Duhamel's formula, we obtain
\begin{equation}\label{Ipsitilde}
\begin{aligned}
\mathcal{I}_{\psi}(s)&=i\frac12 e^{i\frac12s\Delta }\int_0^se^{-i\frac12\sigma\Delta}\psi(t_n+\sigma)\big(u(t_n+\sigma)+\overline{u(t_n+\sigma)}\big)d\sigma\\
&=i\frac12 e^{i\frac12s\Delta }\int_0^se^{-i\frac12\sigma\Delta}\big(e^{i\frac12\sigma\Delta}\psi(t_n)\big)\big(e^{i\sigma c\langle \nabla \rangle_c}u(t_n)+e^{-i\sigma c\langle \nabla \rangle_c}\overline{u(t_n)}\big)d\sigma +\mathcal{R}''_3\\
&=\Tilde{\mathcal{I}}_{\psi}(s)+\mathcal{R}''_3.
\end{aligned}
\end{equation}
For the remainder terms $\mathcal{R}'_3$ and $\mathcal{R}''_3$, using \eqref{cnab_liniso_bound}, we find that
\begin{equation}
\begin{aligned}
\label{2o_rt1}
\|\mathcal{R}'_3\|_r &\leq \bigg\| ic\langle \nabla \rangle_c^{-1}\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\vert\mathcal{I}_{\psi}(s)\vert^2 ds \bigg\|_r\\
&\leq \tau^3 K\big( \sup_{0\leq\xi\leq\tau}\|u(t_n+\xi)\|_r,\sup_{0\leq\xi\leq\tau}\|\psi(t_n+\xi)\|_r \big),
\end{aligned}
\end{equation}
and, similarly,
\begin{equation}\label{2o_rtt1}
\begin{aligned}
\|\mathcal{R}''_3\|_r &\leq \bigg\| e^{i\frac12s\Delta }\int_0^se^{-i\frac12\sigma\Delta} \mathcal{I}_{\psi}(\sigma) \big(e^{i\sigma c\langle \nabla \rangle_c}u(t_n)+e^{-i\sigma c\langle \nabla \rangle_c}\overline{u(t_n)}\big)d\sigma \bigg\|_r\\
&+\bigg\| e^{i\frac12s\Delta }\int_0^se^{-i\frac12\sigma\Delta}\big(e^{i\frac12\sigma\Delta}\psi(t_n)\big)\big(\mathcal{I}_u(\sigma) + \overline{\mathcal{I}_u(\sigma)}\big)d\sigma \bigg\|_r \\
&+ \bigg\| e^{i\frac12s\Delta }\int_0^se^{-i\frac12\sigma\Delta} \mathcal{I}_{\psi}(\sigma) \big(\mathcal{I}_u(\sigma) + \overline{\mathcal{I}_u(\sigma)}\big)d\sigma \bigg\|_r \\
& \leq s^2 K\big( \sup_{0\leq\xi\leq\tau}\|u(t_n+\xi)\|_r,\sup_{0\leq\xi\leq\tau}\|\psi(t_n+\xi)\|_r \big),
\end{aligned}
\end{equation}
where $\mathcal{I}_u$ is given in \eqref{I_u}. Now, after these considerations, the expansion given in \eqref{2o_1} reads
\begin{equation}\label{2o_2}
\begin{aligned}
u(t_n+\tau) &= e^{i \tau c\langle \nabla \rangle_c}u(t_n) - i c \langle \nabla \rangle_c^{-1} e^{i \tau c\langle \nabla \rangle_c}\mathfrak{I}_u(u(t_n),\psi(t_n)) + \mathcal{R}_3,
\end{aligned}
\end{equation}
where
\begin{align}
\label{I1}
\mathfrak{I}_u(u(t_n),\psi(t_n)) &= \int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{i\frac12s\Delta}\psi(t_n) \vert^2 ds \\
\label{I2}
&+\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{\psi(t_n)}\big)\Tilde{\mathcal{I}}_{\psi}(s)ds\\
\label{I3}
&+\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{i\frac12s\Delta}\psi(t_n)\big)\overline{\Tilde{\mathcal{I}}_{\psi}(s)} ds ,
\end{align}
recall the definition of $\tilde{\mathcal{I}}_{\psi}$ is given in \eqref{Ipsitilde}. Using \eqref{2o_rt1} and \eqref{2o_rtt1}, we see that
\begin{align}
\label{2o_r1}
\|\mathcal{R}_3\|_r\leq \tau^3 K\big( \sup_{0\leq\xi\leq\tau}\|u(t_n+\xi)\|_r,\sup_{0\leq\xi\leq\tau}\|\psi(t_n+\xi)\|_r \big),
\end{align}
where $K>0$ is chosen independently of $c$.
Prior to treating the three highly oscillatory integrals separately, we define the function
\begin{align}\label{psi2}
\Psi_2(\xi)= \frac{e^{\xi}-\varphi_1(\xi)}{\xi},
\end{align}
recall $\varphi_1$ is given in \eqref{phi1}. We again refer to \cite{HoOs} for details on this family of functions.
\begin{lemma}[Second order approximation of the integral \eqref{I1}]
\label{Lemma2ou1}
For $0\leq\alpha\leq1$ it holds that
\begin{align*}
I_1(w,v) &=\int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} \vert e^{i\frac12s\Delta}v \vert^2 ds\\
&= \tau \big[\overline{v}\varphi_1(i \tau (\Delta-c^2)){v} + e^{i\frac12\tau\Delta} \big( e^{-i\frac12\tau\Delta} \Psi_2(i \tau (\Delta-c^2)) v \big)\big(e^{-i\frac12\tau\Delta}\overline{v}\big) - \overline{v}\Psi_2(i \tau (\Delta-c^2)) {v} \big] \\
&- i\tau^2 \varphi_1(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))(c\langle \nabla \rangle_c-c^2+\frac12\Delta) \big(\overline{v}\Psi_2(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))v\big)\\
&+ \mathcal{O}{\big(\tau^3(\mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v})+c^{-4\alpha}\Delta^{2+2\alpha} \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v}) \big)}\\
&=\tilde{I}_1(w,v)+ \mathcal{O}{\big(\tau^3(\mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v})+c^{-4\alpha}\Delta^{2+2\alpha} \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v}) \big)}.
\end{align*}
\end{lemma}
\begin{proof}
For $\mathcal{N}$ defined in \eqref{N} we see that
\begin{equation}\label{2o_i1}
\begin{aligned}
I_1(v)&= \int_0^{\tau} e^{-i s c\langle \nabla \rangle_c} e^{-i\frac12 s \Delta} \mathcal{N}(s,s,\Delta,v) ds\\
&{=\int_0^{\tau} e^{-i s c^2} e^{-i\frac12 s (c\langle \nabla \rangle_c-c^2+\frac12\Delta)} \mathcal{N}(s,s,\Delta,v) ds}\\
&{= \int_0^{\tau} e^{-i s c^2} (1-is(c\langle \nabla \rangle_c-c^2+\frac12\Delta)) \mathcal{N}(s,s,\Delta,v) ds + \mathcal{R}_1}\\
&{= \int_0^{\tau} e^{-i s c^2} \mathcal{N}(s,s,\Delta,v) ds - \int_0^{\tau} e^{-i s c^2} is(c\langle \nabla \rangle_c-c^2+\frac12\Delta)\mathcal{N}(s,s,\Delta,v) ds +\mathcal{R}_1},\\
\end{aligned}
\end{equation}
where, by \eqref{c_trick}, it holds
$$
\|\mathcal{R}_1\|_r \leq \tau^3 K \big(c^{-4\alpha}\Delta^{2+2\alpha}\vert v \vert^2 \big).
$$
We tackle the first integral in \eqref{2o_i1}. A second order expansion of $\mathcal{N}$ reads
$$
\mathcal{N}(s,s,\Delta,v)=\mathcal{N}(s,0,\Delta,v)+s\partial_{s_1}\mathcal{N}(s,s_1,\Delta,v)_{s_1=0} + \int_0^s \int_0^{\sigma} \partial^2_{s_1}\mathcal{N}(s,s_1,\Delta,v) \,ds_1\, d\sigma,
$$
where $\partial^2_{s_1}\mathcal{N}(s,s_1,\Delta,v)$ obeys
\begin{align*}
\partial^2_{s_1}\mathcal{N}(s,s_1,\Delta,v) =e^{i\frac12 s_1\Delta}\mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),-i\tfrac{1}{2}\Delta](e^{-i\frac12 s_1\Delta}e^{is\Delta}v , e^{-i\frac12s_1\Delta}\overline{v}),
\end{align*}
recall the notation
$$
\mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w) = \mathcal{C}[\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta],i\tfrac{1}{2}\Delta](v , w).
$$
In order to guarantee the stability of this scheme, we employ a finite difference approximation of $\partial_{s_1}\mathcal{N}(s,0,\Delta,v)$, namely, for $0\leq \tau\leq s$,
$$
\partial_{s_1}\mathcal{N}(s,0,\Delta,v) = \frac{\mathcal{N}(s,\tau,\Delta,v)-\mathcal{N}(s,0,\Delta,v)}{\tau} + \mathcal{O}(\tau\partial^2_{s_1}\mathcal{N}(s,s_1,\Delta,v)).
$$
With this in mind and using definition \eqref{psi2}, we see that \eqref{2o_i1} reads
\begin{equation}\label{intu11}
\begin{aligned}
\int_0^{\tau} e^{-i s c^2} \mathcal{N}(s,s,\Delta,v) ds &=\int_0^{\tau} e^{-i s c^2} \bigg(\mathcal{N}(s,0,\Delta,v)+\frac{s}{\tau}\big( \mathcal{N}(s,\tau,\Delta,v)-\mathcal{N}(s,0,\Delta,v) \big) \bigg) ds+ \mathcal{R}_1+ \mathcal{R}_2
\\&= \tau \big[\overline{v}\varphi_1(i \tau (\Delta-c^2)){v} + e^{i\frac12\tau\Delta} \big( e^{-i\frac12\tau\Delta} \Psi_2(i \tau (\Delta-c^2)) v \big)\big(e^{-i\frac12\tau\Delta}\overline{v}\big) \\
&- \overline{v}\Psi_2(i \tau (\Delta-c^2)) {v} \big] + \mathcal{R}_1+ \mathcal{R}_2,
\end{aligned}
\end{equation}
where $\mathcal{R}_2$ satisfies
\begin{align*}
\|\mathcal{R}_2\|_r &\leq \tau^3 K\big( \mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v}) \big).
\end{align*}
As for the second integral in \eqref{2o_i1}, we note that it suffices to carry out a Taylor expansion up to first order in $s$. We obtain
\begin{equation}\label{intu12}
\begin{aligned}
&\int_0^{\tau} e^{-i s c^2} is(c\langle \nabla \rangle_c-c^2+\frac12\Delta)\mathcal{N}(s,s,\Delta,v) ds\\ &=\int_0^{\tau} e^{-i s c^2} is(c\langle \nabla \rangle_c-c^2+\frac12\Delta)\mathcal{N}(s,0,\Delta,v) ds + \mathcal{R}_3\\
&=i\tau^2 (c\langle \nabla \rangle_c-c^2+\frac12\Delta) \big(\overline{v}\Psi_2(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))v\big)+ \mathcal{R}_3\\
&=i\tau^2 \varphi_1(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))(c\langle \nabla \rangle_c-c^2+\frac12\Delta) \big(\overline{v}\Psi_2(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))v\big)+ \mathcal{R}_3+ \mathcal{R}_4
\end{aligned}
\end{equation}
where, given \eqref{c_trick} and the above first order Taylor expansion of $N(s,s,\Delta,v)$, $\mathcal{R}_3$ satisfies
$$
\|\mathcal{R}_3 \|_r\leq \tau^3 K\big( c^{-2\alpha}\Delta^{1+\alpha} \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v}) \big).
$$
We note that in the last step of \eqref{intu12} we have introduced the factor $\varphi_1(i\tau(c\langle \nabla \rangle_c-c^2+\frac12\Delta))$ in order to ensure stability and, by \eqref{stability_trick}, \eqref{c_trick} and the estimate obtained for $\mathcal{R}_3$, $\mathcal{R}_4$ satisfies
$$
\|\mathcal{R}_4 \|_r\leq \tau^3 K\big( c^{-4\alpha}\Delta^{2+2\alpha} \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta]( v, \overline{v}) \big).
$$
Using \eqref{intu11} and \eqref{intu12} we achieve the desired second order approximation of \eqref{2o_i1}.
\end{proof}
\begin{lemma}[Second order approximation of the integral \eqref{I2}]
\label{Lemma2ou2}
For $0\leq\alpha\leq1$ it holds that
\begin{align*}
I_2(w,v)&=\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{v}\big)\Tilde{\mathcal{I}}_{\psi}(s)ds \\
&= \frac{\tau}{2c^2}\overline{v} \big[ (\varphi_1(it(\Delta+2c^2))-\varphi_1(it(\Delta+c^2)))vw - (\varphi_1(it\Delta)-\varphi_1(it(\Delta+c^2)))v\overline{w} \big]
\\&+\mathcal{O}{\big(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta](v , w)+\Delta w + c^{-2\alpha}\Delta^{1+\alpha}(vw) )\big)}\\
&=\tilde{I}_2(w,v) +\mathcal{O}{\big(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta](v , w)+\Delta w + c^{-2\alpha}\Delta^{1+\alpha}(vw))\big)}.
\end{align*}
\end{lemma}
\begin{proof}
Applying the result found in \eqref{Schro1} within $\tilde{\mathcal{I}}_{\psi}(s)$, we find that
\begin{align*}
I_2(w,v)&=\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{v}\big)\Tilde{\mathcal{I}}_{\psi}(s)ds\\
&=i \frac12 \int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{v}\big) \big( se^{i\frac12s\Delta}v(\varphi_1(is(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))w + \varphi_1(-is(c\langle \nabla \rangle_c +\tfrac{1}{2}\Delta))\overline{w}) \big) ds\\& +\mathcal{R}_1,
\end{align*}
where, by Lemma \ref{Lemma1opsi} and with \eqref{cnab_liniso_bound}, it holds for $\mathcal{R}_1$ that
\begin{equation*}
\begin{aligned}
\| \mathcal{R}_1\|_r \leq \tau^3 K\big(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v, w)\big).
\end{aligned}
\end{equation*}
Note that, by \eqref{c_trick} it holds formally that
\begin{equation}\label{anastasiya}
\begin{aligned}
s\varphi_1(is(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)) &= \int_0^s e^{i\sigma(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)}d\sigma\\
& =s\varphi_1(isc^2)+ s^2\mathcal{O}(\Delta + c^{-2\alpha}\Delta^{1+\alpha}).
\end{aligned}
\end{equation}
Thus,
\begin{equation*}
\begin{aligned}
I_2(w,v)=i\frac12\int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{-i\frac12s\Delta}\overline{v}\big) \big( se^{i\frac12s\Delta}v (\varphi_1(isc^2)w+ \varphi_1(-isc^2)\overline{w}) \big) ds+\mathcal{R}_1 +\mathcal{R}_2,
\end{aligned}
\end{equation*}
and, by \eqref{anastasiya}, $\mathcal{R}_2$ fulfills the bound
$$
\| \mathcal{R}_2\|_r \leq \tau^3 K(v(\Delta + c^{-2\alpha}\Delta^{1+\alpha})w).
$$
We now let
$$
\mathcal{N}(s,s_1,v,w) = e^{i\frac12s_1\Delta}\big( e^{-i\frac12s_1\Delta}\overline{v}\big) \big( e^{is\Delta}e^{-i\frac12s_1\Delta}v (\varphi_1(isc^2)w+ \varphi_1(-isc^2)\overline{w}) \big),
$$
and the assertion is obtained following the same line of argumentation as in Lemma \ref{Lemma2ou1}.
\end{proof}
As for the third integral, following analogous steps as the ones that lead to the approximation of \eqref{I2}, we obtain the following second order approximation of \eqref{I3} for $0\leq\alpha\leq1$
\begin{equation}
\label{2o_i3_final}
\begin{aligned}
{I_3(w,v)} &= \int_0^{\tau}e^{-i s c\langle \nabla \rangle_c}\big( e^{i\frac12s\Delta}v\big)\bigg( -i\frac12 e^{-i\frac12s\Delta }\int_0^se^{i\frac12\sigma\Delta}\big(e^{-i\frac12\sigma\Delta}\overline{v}\big)\big(e^{i\sigma c\langle \nabla \rangle_c}w+e^{-i\sigma c\langle \nabla \rangle_c}\overline{w}\big)d\sigma \bigg) ds \\
&= \frac{\tau}{2c^2} \big[-\overline{v}w (\varphi_1(i\tau\Delta)-\varphi_1(i\tau(\Delta-c^2)))v+\overline{v}\overline{w}(\varphi_1(i\tau(\Delta-2c^2))-\varphi_1(i\tau(\Delta-c^2)))v\big]\\
&+ \mathcal{O}{\big(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta](v , w)+\Delta w + c^{-2\alpha}\Delta^{1+\alpha}(vw))\big)}\\
&= \tilde{I}_3(w,v) +\mathcal{O}{\big(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\Delta](v , w)+\Delta w + c^{-2\alpha}\Delta^{1+\alpha}(vw))\big)}.
\end{aligned}
\end{equation}
Collecting our results from Lemma \ref{Lemma2ou1}, Lemma \ref{Lemma2ou2} and \eqref{2o_i3_final}, together with the bound proven in Lemma \ref{lemma_comm}, yields the following second order approximation of the oscillatory integral $\mathfrak{I}_u(w,v)$ in \eqref{2o_2}.
\begin{cor}\label{cor:I_u}
For $0\leq\alpha\leq1$ it holds that
\begin{equation*}
\begin{aligned}
\mathfrak{I}_u(w,v) &= \tilde{I}_1(w,v)+\tilde{I}_2(w,v)+\tilde{I}_3(w,v)+\mathcal{O}{ (\tau^3(\Delta w)(\Delta v))}\\
&= \tilde{\mathfrak{I}}_u(w,v)+\mathcal{O}{\big(\tau^3((\Delta w)(\Delta v) + c^{-2\alpha}\Delta^{1+\alpha}(vw) + c^{-4\alpha}\Delta^{2+2\alpha}(vw))\big)},
\end{aligned}
\end{equation*}
\end{cor}
Finally, Corollary \ref{cor:I_u} leads to the following second order approximation based on \eqref{2o_2}:
\begin{equation}\label{KGo2}
\begin{aligned}
u^{n+1}=e^{i \tau c\langle \nabla \rangle_c}u^n-ic\langle \nabla \rangle_c^{-1}e^{i \tau c\langle \nabla \rangle_c}\tilde{\mathfrak{I}}_u(u^n,\psi^n).
\end{aligned}
\end{equation}
We may now consider Duhamel's formula for \eqref{Schr}, where we iterate Duhamel's formula for \eqref{KG} and \eqref{Schr} respectively.
\begin{align*}
\psi(t_n+\tau) &= e^{i\frac12\tau\Delta}\psi(t_n) + i\frac12 e^{i\frac12\tau\Delta}\mathfrak{I}_{\psi}(u(t_n),\psi(t_n)) + \mathcal{R}_4,
\end{align*}
where $\mathcal{R}_4$ fulfills
\begin{align}\label{R_4}
\|\mathcal{R}_4\|_r \leq \bigg\| \int_0^{\tau}e^{-i\frac12s\Delta}\mathcal{I}_{\psi}(s)(\mathcal{I}_u(s) + \overline{\mathcal{I}_u(s)})ds \bigg\|_r
\leq \tau^3 K\big(\sup_{0\leq \xi\leq \tau} \|u(t_n+\xi)\|_r,\sup_{0\leq \xi\leq \tau} \|\psi(t_n+\xi)\|_r\big),
\end{align}
and the integral $\mathfrak{I}_{\psi}$ reads
\begin{align}
\mathfrak{I}_{\psi}(w,v)&= \int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}\psi(t_n) \big)\big( e^{isc\langle \nabla \rangle_c}u(t_n) + e^{-isc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds\label{J1}\\
&+\int_0^{\tau}e^{-i\frac12s\Delta}\mathcal{I}_{\psi}(s)\big( e^{isc\langle \nabla \rangle_c}u(t_n) + e^{-isc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds\label{J2}\\
&+\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}\psi(t_n) \big)(\mathcal{I}_u(s) + \overline{\mathcal{I}_u(s)})ds.\label{J3}
\end{align}
{Recall that $\mathcal{I}_u$ is given by \eqref{I_u} and $\mathcal{I}_{\psi}$ by \eqref{I_psi}.} We may now tackle the three highly oscillatory integrals separately.
\begin{lemma}[Second order approximation of the integral \eqref{J1}]\label{lemma:J1}
It holds
\begin{align*}
J_1(w,v)&=\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}v \big)\big( e^{isc\langle \nabla \rangle_c}w + e^{-isc\langle \nabla \rangle_c}\overline{w} \big)ds\\
&= \tau \big[ v\varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))w + v\varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{w} \\
&+ e^{-i\frac12\tau\Delta}\big(e^{i\frac12\tau\Delta}v\big)\big[e^{i\frac12\tau\Delta}\Psi_2(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))w+ e^{i\frac12\tau\Delta}\Psi_2(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{w}\big]\\
&- v\Psi_2(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)){w} - v\Psi_2(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))\overline{w} \big] + \mathcal{O}{(\tau^3 \mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w))}
\\&=\tilde{J}_{1}(w,v) + \mathcal{O}{(\tau^3 \mathcal{C}^2[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w))}.
\end{align*}
\end{lemma}
\begin{proof}
We define the following filter functions:
\begin{align*}
&\mathcal{N}(s,s_1,v,w)=e^{-i\frac12s_1\Delta}\big( e^{i\frac12s_1\Delta} v \big)\big( e^{i\frac12s_1\Delta}e^{-i\frac12s\Delta}[e^{isc\langle \nabla \rangle_c}w + e^{-isc\langle \nabla \rangle_c}\overline{w}] \big).
\end{align*}
Now, via a second order expansion of these two filter functions and a finite difference approximation of their first derivative with respect to $s_1$, we obtain the assertion.
\end{proof}
As for the integral \eqref{J2}, iterating Duhamel's formula yields
\begin{align*}
J_2(u(t_n),\psi(t_n)) &=\int_0^{\tau}e^{-i\frac12s\Delta}\bigg(
\frac{i}{2} e^{i\frac12s\Delta}\int_0^{s}e^{-i\frac12\sigma \Delta}\big(e^{i\frac12 \sigma \Delta}\psi(t_n)\big)\big( e^{i \sigma c\langle \nabla \rangle_c}u(t_n) \big)d\sigma
\bigg)\big( e^{isc\langle \nabla \rangle_c}u(t_n) \big)ds
\\&+\int_0^{\tau}e^{-i\frac12s\Delta}\bigg(
\frac{i}{2} e^{i\frac12s\Delta}\int_0^{s}e^{-i\frac12\sigma \Delta}\big(e^{i\frac12 \sigma \Delta}\psi(t_n)\big)\big( e^{i \sigma c\langle \nabla \rangle_c}u(t_n) \big)d\sigma
\bigg)\big(e^{-isc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds
\\&+\int_0^{\tau}e^{-i\frac12s\Delta}\bigg(
\frac{i}{2} e^{i\frac12s\Delta}\int_0^{s}e^{-i\frac12\sigma \Delta}\big(e^{i\frac12 \sigma \Delta}\psi(t_n)\big)\big( e^{-i \sigma c\langle \nabla \rangle_c}\overline{u(t_n)} \big)d\sigma
\bigg)\big( e^{isc\langle \nabla \rangle_c}u(t_n) \big)ds
\\&+\int_0^{\tau}e^{-i\frac12s\Delta}\bigg(
\frac{i}{2} e^{i\frac12s\Delta}\int_0^{s}e^{-i\frac12\sigma \Delta}\big(e^{i\frac12 \sigma \Delta}\psi(t_n)\big)\big( e^{-i \sigma c\langle \nabla \rangle_c}\overline{u(t_n)} \big)d\sigma
\bigg)\big( e^{-isc\langle \nabla \rangle_c}\overline{u(t_n)} \big)ds \\
&+ \mathcal{R}_4'
\\&=J_{2,1}(u(t_n),\psi(t_n))+J_{2,2}(u(t_n),\psi(t_n))+J_{2,3}(u(t_n),\psi(t_n))+J_{2,4}(u(t_n),\psi(t_n)) \\
&+ \mathcal{R}_4',
\end{align*}
where $\mathcal{R}_4'$ fulfills
\begin{align}\label{R_4p}
\|\mathcal{R}'_4\|_r \leq \tau^3 K\big(\sup_{0\leq \xi\leq \tau} \|u(t_n+\xi)\|_r,\sup_{0\leq \xi\leq \tau} \|\psi(t_n+\xi)\|_r\big).
\end{align}
We now handle the first term in $J_2$ in detail and the remaining three terms can be handled analogously.
\begin{lemma}[Second order approximation of the integral \eqref{J21}]\label{lemma:J21}
For $0\leq\alpha\leq1$ it holds that
\begin{align*}
J_{2,1}(w,v)&=\frac{\tau}{2c^2} \big[ vw\big( \varphi_1(i\tau(c^2+c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))-\varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)) \big)w
\big] \\
&+ \mathcal{O}{(\tau^3( \Delta(vw)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}
\\&=\tilde{J}_{2,1}(w,v) + \mathcal{O}{(\tau^3( \Delta(vw)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}.
\end{align*}
\end{lemma}
\begin{proof}
Proceeding as in the derivation of \eqref{Schro1} and using \eqref{anastasiya}, we obtain
\begin{align}\label{J21}
J_{2,1}(w,v)&=\frac{i}{2}\int_0^{\tau}s\varphi_1(isc^2)e^{-i\frac12s\Delta}\big(
e^{i\frac12s\Delta} vw \big)\big( e^{isc\langle \nabla \rangle_c}w\big)ds + \mathcal{R},
\end{align}
where $\mathcal{R}$ fulfills
$$
\|\mathcal{R}\|_r \leq \tau^3 K\big( \Delta(vw) +c^{-2\alpha}\Delta^{1+\alpha}(vw)\big).
$$
Finally, the assertion follows with the definition of the filtered function
$$
\mathcal{N}(s,s_1,v,w) = e^{-i\frac12s_1\Delta}\big(
e^{i\frac12s_1\Delta} vw \big)\big( e^{i\frac12s_1\Delta} e^{-i\frac12s\Delta} e^{isc\langle \nabla \rangle_c}w \big),
$$
via a Taylor expansion up to order one.
\end{proof}
Analogously, we obtain that, for $0\leq\alpha\leq1$,
\begin{equation}\label{J22}
\begin{aligned}
J_{2,2}(w,v)&=\frac{\tau}{2c^2} \big[ vw\big( \varphi_1(i\tau(c^2-c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))-\varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta)) \big)\overline{w} \big] \\
&+ \mathcal{O}{(\tau^3( \Delta(vw)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}
\\&=\tilde{J}_{2,2}(w,v) + \mathcal{O}{(\tau^3( \Delta(vw)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))},
\end{aligned}
\end{equation}
\begin{equation}\label{J23}
\begin{aligned}
J_{2,3}(w,v)&=-\frac{\tau}{2c^2} \big[v\overline{w}\big( \varphi_1(i\tau(-c^2+c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta))-\varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)) \big)w \big] \\
&+ \mathcal{O}{(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}
\\&=\tilde{J}_{2,3}(w,v) + \mathcal{O}{(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}
\end{aligned}
\end{equation}
and
\begin{equation}\label{J24}
\begin{aligned}
J_{2,4}(w,v)&=-\frac{\tau}{2c^2} \big[ v\overline{w}\big( \varphi_1(-i\tau(c^2+c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta))-\varphi_1(-i\tau(c\langle \nabla \rangle_c+\frac{1}{2}\Delta)) \big)\overline{w} \big] \\
&+\mathcal{O}{(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}
\\&=\tilde{J}_{2,4}(w,v) + \mathcal{O}{(\tau^3(\mathcal{C}[f_{\text{quad}}(\cdot,\cdot),i\tfrac{1}{2}\Delta](v , w)+c^{-2\alpha}\Delta^{1+\alpha}(vw)))}.
\end{aligned}
\end{equation}
Finally, we may approximate the last integral \eqref{J3} as follows. We iterate Duhamel's formula, obtaining
\begin{align*}
J_3(u(t_n),\psi(t_n))&=\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}\psi(t_n) \big)\bigg( -ic\langle \nabla \rangle_c^{-1}e^{isc\langle \nabla \rangle_c}\int_0^se^{-i\sigma c\langle \nabla \rangle_c}\vert e^{i\frac12\sigma\Delta}\psi(t_n)\vert^2 d\sigma \bigg)ds
\\&+\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}\psi(t_n)\big)\bigg( ic\langle \nabla \rangle_c^{-1}e^{-isc\langle \nabla \rangle_c}\int_0^se^{i\sigma c\langle \nabla \rangle_c}\vert e^{i\frac12\sigma\Delta}\psi(t_n)\vert^2 d\sigma \bigg)ds + \mathcal{R}_4''
\\&= J_{3,1}(u(t_n),\psi(t_n)) + J_{3,2}(u(t_n),\psi(t_n)) +\mathcal{R}_4'',
\end{align*}
where $\mathcal{R}_4''$ fulfills
\begin{align}\label{R_4pp}
\|\mathcal{R}_4''\|_r\leq \tau^3 K\big(\sup_{0\leq \xi\leq \tau} \|u(t_n+\xi)\|_r,\sup_{0\leq \xi\leq \tau} \|\psi(t_n+\xi)\|_r\big).
\end{align}
\begin{lemma}\label{lemma:J31}
For $0\leq\alpha\leq1$ it holds that
\begin{equation}
\begin{aligned}
J_{3,1}(w,v)&=\frac{\tau}{c^2}\big[ v\big( \varphi_1(i\tau(c\langle \nabla \rangle_c-c^2-\tfrac{1}{2}\Delta)) - \varphi_1(i\tau(c\langle \nabla \rangle_c-\tfrac{1}{2}\Delta)) \big)c\langle \nabla \rangle_c^{-1}\vert v\vert^2\big]+ \mathcal{O}{(\tau^3 w\Delta v)}
\\&=\tilde{J}_{3,1}(w,v) + \mathcal{O}{(\tau^3 (\Delta v + c^{-2\alpha}\Delta^{1+\alpha}v ))}.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
As for the first term, proceeding as in the derivation of \eqref{KGo1} and arguing similarly as in \eqref{anastasiya}, we obtain
\begin{align*}
J_{3,1}(w,v)&=\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}v \big)\big( -i s c \langle \nabla \rangle_c^{-1} e^{i s c\langle \nabla \rangle_c} v \varphi_1(-i s (c^2+\Delta)) \overline{v} \big)ds + \mathcal{R}_1 \\
&=\int_0^{\tau}e^{-i\frac12s\Delta}\big( e^{i\frac12s\Delta}v \big)\big( -i s c \langle \nabla \rangle_c^{-1} e^{i s c\langle \nabla \rangle_c} v \varphi_1(-i s c^2) \overline{v} \big)ds + \mathcal{R}_1+ \mathcal{R}_2,
\end{align*}
with
$$
\|\mathcal{R}_1\|_r\leq \tau^3 K\big( \mathcal{C}[f_{\text{quad}}(\cdot,\cdot),\Delta](v,v)+c^{-2\alpha}\Delta^{1+\alpha}v \big),
$$
by Lemma \ref{Lemma1ou} and
$$
\|\mathcal{R}_2\|_r\leq \tau^3 K\big( \Delta v \big),
$$
by \eqref{c_trick}. We define
$$
\mathcal{N}(s,s_1,v)=e^{-i\frac12s_1\Delta}\big(e^{i\frac12s_1\Delta}v\big)\big(e^{i\frac12s_1\Delta}e^{-i\frac12s\Delta}e^{isc\langle \nabla \rangle_c}\overline{v}\big)
$$
and obtain the assertion similarly as in Lemma \eqref{lemma:J21}, via a first order Taylor expansion of $\mathcal{N}(s,s_1,v)$ at $s_1=0$.
\end{proof}
Analogously we obtain the second term
\begin{equation}\label{J32}
\begin{aligned}
J_{3,2}(w,v)&=\frac{\tau}{c^2} \big[ v\big( \varphi_1(i\tau(-c\langle \nabla \rangle_c+c^2-\tfrac{1}{2}\Delta)) - \varphi_1(-i\tau(c\langle \nabla \rangle_c+\tfrac{1}{2}\Delta)) \big)c\langle \nabla \rangle_c^{-1}\vert v\vert^2\big]+ \mathcal{O}{(\tau^3 w\Delta v)}
\\&=\tilde{J}_{3,2}(w,v) + \mathcal{O}{(\tau^3 w\Delta v)}.
\end{aligned}
\end{equation}
Collecting our results, namely Lemma \ref{lemma:J1}, Lemma \ref{lemma:J21}, \eqref{J22}, \eqref{J23}, \eqref{J24}, Lemma \ref{lemma:J31} and \eqref{J32}, leads to the following Corollary, where we use in addition the bound found in Lemma \ref{lemma_comm}.
\begin{cor}\label{cor:I_psi}
For $0\leq\alpha\leq1$ it holds that
\begin{align*}
\mathfrak{I}_{\psi}(w,v)
&= \tilde{J}_1(w,v) + \tilde{J}_{2,1}(w,v) + \tilde{J}_{2,2}(w,v)+\tilde{J}_{2,3}(w,v) +\tilde{J}_{2,4}(w,v)+\tilde{J}_{3,1}(w,v)+\tilde{J}_{3,2}(w,v)\\
&+\mathcal{O}{(\tau^3(\Delta w v +c^{-2\alpha}\Delta^{1+\alpha}vw))}\\
&= \Tilde{\mathfrak{I}}_{\psi}(w,v) + \mathcal{O}{(\tau^3(\Delta w v +c^{-2\alpha}\Delta^{1+\alpha}vw))}.
\end{align*}
\end{cor}
These considerations collected in Corollary \ref{cor:I_psi} lead to the second order uniformly accurate low regularity integrator:
\begin{equation}\label{Schro2}
\begin{aligned}
\psi^{n+1} &= e^{i\frac12\tau\Delta}\psi^n + \frac{i}{2}e^{i\frac12\tau\Delta}\tilde{\mathfrak{I}}_{\psi}(u^n,\psi^n).
\end{aligned}
\end{equation}
In the sections that follow we aim to carry out the error analysis of the scheme in $(u^{n},\psi^{n})$ given by \eqref{KGo2} and \eqref{Schro2}. We recall that we denote by $\varphi^t_{K}$, $\varphi^t_{S}$ the exact flows of \eqref{KG} and \eqref{Schr} respectively and we let $\tilde{\Phi}^t_{K}$, $\tilde{\Phi}^t_{S}$ be the numerical flows corresponding to \eqref{KGo2} and \eqref{Schro2} respectively, such that in particular it holds
$$
u^{n+1}=\tilde{\Phi}^{\tau}_{K}(u^n,\psi^n),\quad \psi^{n+1}=\tilde{\Phi}^{\tau}_{S}(u^n,\psi^n).
$$
\subsection{Local error analysis}
\begin{lemma}
\label{localerror2}
Fix $r>\frac{d}{2}$. The local error given by the differences $\varphi^{\tau}_K(u(t_n),\psi(t_n))-\tilde{\Phi}^{\tau}_K(u(t_n),\psi(t_n))$ and $\varphi^{\tau}_S(u(t_n),\psi(t_n))-\tilde{\Phi}^{\tau}_S(u(t_n),\psi(t_n))$ satisfies
$$
\varphi^{\tau}_K(u(t_n),\psi(t_n))-\tilde{\Phi}^{\tau}_K(u(t_n),\psi(t_n)) = \mathcal{O}\big(\tau^3(\Delta u(t_n){\psi(t_n)}+ c^{-2\alpha}\Delta^{1+\alpha}u(t_n)\psi(t_n) + c^{-4\alpha}\Delta^{2+2\alpha}\partial_x\psi(t_n))\big)
$$
and
$$
\varphi^{\tau}_S(u(t_n),\psi(t_n))-\tilde{\Phi}^{\tau}_S(u(t_n),\psi(t_n)) = \mathcal{O}\big(\tau^3(\Delta u(t_n){\psi(t_n)} + c^{-2\alpha}\Delta^{1+\alpha}u(t_n)\psi(t_n) )\big),
$$
where $0\leq\alpha\leq1$.
\end{lemma}
\begin{proof}
This assertion follows by Corollary \ref{cor:I_u}, \eqref{2o_r1} and Corollary \ref{cor:I_psi}, \eqref{R_4}, \eqref{R_4p} and \eqref{R_4pp} respectively.
\end{proof}
\subsection{Stability analysis}
\begin{lemma}
\label{stability2}
Fix $r>\frac{d}{2}$. The numerical flows $\tilde{\Phi}^{\tau}_K$ and $\tilde{\Phi}^{\tau}_S$ defined by \eqref{KGo2} and \eqref{Schro2} respectively are stable in $H^r$, namely it holds for any $v_i,w_i\in H^r$, $i\in\{1,2\}$ that
$$
\|\tilde{\Phi}^{\tau}_K(v_1,w_1)-\tilde{\Phi}^{\tau}_K(v_2,w_2)\|_r\leq \|v_1-v_2\|_r + \tau M_1 (\|v_1-v_2\|_r +\|w_1-w_2\|_r),
$$
$$
\|\tilde{\Phi}^{\tau}_S(v_1,w_1)-\tilde{\Phi}^{\tau}_S(v_2,w_2)\|_r\leq \|w_1-w_2\|_r + \tau M_2 (\|v_1-v_2\|_r +\|w_1-w_2\|_r),
$$
where $M_1$ and $M_2$ can be chosen independently of $c$.
\end{lemma}
\begin{proof}
This claim follows by \eqref{cnab_liniso_bound}. In addition, we use the fact that it holds $\vert \varphi_1 (i\xi)\vert\leq 1$ for all $\xi\in\mathbb{R}$.
\end{proof}
\subsection{Global error}
\begin{theorem}\label{thm:2}
Fix $r>\frac{d}{2}$ and assume that the solution $(u,\psi)$ of \eqref{KG}-\eqref{Schr} satisfies $u\in\mathcal{C}([0,T],H^{r+2+2\alpha})$ and $\psi\in\mathcal{C}([0,T],H^{r+5+4\alpha})$, $0\leq\alpha\leq1$. Then there exists a $\tau_0>0$ such that for all $0<\tau\leq\tau_0$ the following estimate holds for $(u^n,\psi^n)$ defined in \eqref{KGo2} and \eqref{Schro2}
\begin{align*}
\|u(t_n)-u^n\|_r &\leq \tau^2 K\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_{r+2},\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+2}\big)\\
& + \tau^2 c^{-2\alpha} K\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_{r+2\alpha+2},\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+2\alpha+2}\big)\\
&+\tau^2 c^{-4\alpha} K\big(\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+4\alpha+5}\big),
\end{align*}
and
\begin{align*}
\|\psi(t_n)-\psi^n\|_r &\leq \tau^2 K\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_{r+2},\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+2}\big)\\
&+\tau^2 c^{-2\alpha}K\big(\sup_{t_n\leq t\leq t_{n+1}} \|u(t)\|_{r+2+2\alpha},\sup_{t_n\leq t\leq t_{n+1}} \|\psi(t)\|_{r+2+2\alpha}\big)
\end{align*}
where $0\leq\alpha\leq1$ and, in particular, $K$ can be chosen independently of $c$.
\end{theorem}
\begin{proof}
The proof follows by means of a Lady Windermere's fan argument, after plugging in the results obtained in Lemmata \ref{localerror2} and \ref{stability2} and using the regularity estimates for the commutator terms obtained in Lemma \ref{lemma_comm}.
\end{proof}
\section{Asymptotic consistency}
In this section we show that our novel class of first and second order integrators are asymptotically consistent, meaning that in the limit $c\to\infty$ we recover the solution of the limit system.
The limit system can be for instance derived via Modulated Fourier Expansion techniques, see for example \cite{CHL}, \cite{FaS}, \cite{HL}, \cite{HLW}. We refer to \cite{BaKoS18} for the details of this derivation.
\subsection{Asymptotic convergence of the first order method}
In this section we motivate why the method given by \eqref{KGo1} and \eqref{Schro1} converges towards the solution of the limit system as $c\to\infty$.
We see that formally it holds
\begin{align}\label{ac_1}
e^{i\tau c\langle \nabla \rangle_c} = e^{i\tau (c^2+\frac12\Delta)} +\mathcal{O}(c^{-2}),
\end{align}
as Taylor series expansion of the function $x\mapsto c\sqrt{x+c^2}$ around the point zero shows
\begin{align}\label{h4req}
\big\|c\langle \nabla \rangle_c f - \big(c^2-\frac12 \Delta\big)f\big\|_r\leq K c^{-2}\|f\|_{r+4},
\end{align}
for some $K>0$ independent of $c$. Note that this particular asymptotic bound requires additional regularity for $u$.
It follows by \eqref{cnab_liniso_bound}, the observation
\begin{align}\label{ac_2}
\big\|\tau \varphi_1(\pm i\tau c^2) \big\|_r \leq \frac{2}{c^2},
\end{align}
and by \eqref{anastasiya}, that for $u^n$ given by \eqref{KGo1} it holds
$$
u^{n+1} = e^{-\frac12 i\tau \Delta}u^n + \mathcal{O}(c^{-2}).
$$
As for $\psi^n$ given by \eqref{Schro1}, we see by \eqref{ac_2} that
$$
\psi^{n+1} = e^{i\tau\Delta}\psi^n + \mathcal{O}(c^{-2}).
$$
\subsection{Asymptotic convergence of the second order method}
Analogously to the previous section, using \eqref{cnab_liniso_bound}, \eqref{ac_1} and \eqref{ac_2} together with
\begin{align}\label{ac_3}
\big\|\tau\Psi_2(i\tau c^2)\big\|_r\leq \frac{2}{c^2}
\end{align}
and \eqref{anastasiya}, we are able to see that indeed $\tilde{\mathfrak{I}}_u$ given in Corollary \ref{cor:I_u} fulfills
$$
ic\langle \nabla \rangle_c^{-1}\tilde{\mathfrak{I}}_u(u^n,\psi^n) = \mathcal{O}(c^{-2}),
$$
Note that, by \eqref{h4req}, we also require here $H^4$ for $u$. Similarly, with \eqref{ac_2} and \eqref{ac_3}, we obtain that $\tilde{\mathfrak{I}}_{\psi}$ given in Corollary \ref{cor:I_psi} fulfills
$$
\tilde{\mathfrak{I}}_{\psi}(u^n,\psi^n) = \mathcal{O}(c^{-2}).
$$
This implies that the second order method given by \eqref{KGo2} and \eqref{Schro2} also converges to the corresponding solution of the limit system with order $c^{-2}$ formally.
\section{Numerical Experiments}
{We dedicate this last section to the numerical verification of our results. We mainly concentrate on the convergence of the first order method in order to illustrate the explicit relation in our error estimates between gain in $c^{-2\alpha}$, $0\leq\alpha\leq1$, for large $c$, and consequent loss in derivative. In particular, we observe uniform accuracy and and improvement in convergence for more regular initial data and large $c$, as depicted in Theorem \ref{thm:1}. Then we briefly present the convergence results for the second order method, which verify second order convergence and uniform accuracy, as obtained in Theorem \ref{thm:2}. We leave out the experiments for different regularity assumptions in this case for the sake of brevity. For the spatial discretisation we use a standard Fourier pseudospectral method, choosing $M=200$ as the highest Fourier mode. See \cite{STW} for more information on this technique as well as for some applications.}
In Figure 1 we plot the global error of the first order scheme given by \eqref{KGo1} and \eqref{Schro1} measured in $H^1$ for different values of $c$ and initial data in Sobolev spaces varying in regularity, as well as the convergence of the second order scheme given by \eqref{KGo2} and \eqref{Schro2} measured in $H^1$ for different values of $c$ and smooth initial data.
$$
$$
\begin{figure}[h!]
\includegraphics[width=0.95\textwidth]{order_plot.eps}
\caption{Convergence plot of the first and second order scheme given by \eqref{KGo1}, \eqref{Schro1} and \eqref{KGo2}, \eqref{Schro2}. The blue, orange, yellow, purple and green lines correspond to the values of $c=1$, $c=10$, $c=100$, $c=1000$ and $c=10000$ respectively. The black thick lines are reference lines of slope one and two. The ticker solid lines correspond to the second order scheme with smooth initial data. The thinner dotted, dashed and mixed lines correspond to the first order scheme with $H^2$, $H^3$ and $H^4$ initial data respectively.}
\label{order_plot}
\end{figure}
\newpage
\subsection*{Acknowledgements}
{\small
The author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 850941).
}
|
1,314,259,993,021 | arxiv | \section{Introduction}\label{sec:intro}
The ability of chirped (or graded) arrays to create slow light or sound waves, confine them, and filter them spatially according to frequency is attractive for wave control. This is often connected with rainbow trapping of light \cite{hess2007} or sound \cite{acou_rain,jimenez17a},
and, when combined with the optical properties of surface plasmon polaritons \cite{pendry2004},
has led to the development of surface dispersion engineering devices based on chirped gratings \cite{gan11a} or tapered devices \cite{argy13a}.
A recurring theme is that by using a graded surface grating, different wavelengths are trapped at different locations. Improvements in design and fabrication of nanopatterned metal surfaces, 3D printing of surfaces and optimization of their optical and acoustic properties open new applications to exploit the effect in light storage, energy harvesting, filtering and delayed delivery for novel devices \cite{boriskina13a}.
This comprehensive body of work, primarily in optics, plasmonics and electromagnetism, forms a paradigm for the control of waves more broadly and has been extended to the field of elasticity \cite{celli1,krodel2015,colombi16a}, where it is used to generate mode conversion devices that hybridise surface waves to body waves.
A feature less often exploited is that energy can accumulate in regions where the group velocity tends to zero \cite{romero13a,sensing}. Assuming the local behaviour in the array is dominated by neighbouring elements enables the implied local periodicity to predict band-gaps (or forbidden frequencies), and the band-gap edges
determine a relation between local array properties and frequencies at which group velocities are zero.
This interpretation
empowers design of broadband arrays with spatial filtering by frequency and localised amplification,
and has recently been taken into linear water wave theory \cite{bennetts_graded_2018}.
The approach complements other recent activity,
in which ideas from transformation optics have been applied to water waves at the millimetric amplitude scale to achieve localised amplitude amplification up to a factor of three \cite{li18a},
and constant-spaced arrays involving resonant components, in the sense of metamaterials,
have been used to achieve low-frequency filtering with potential applications to coastal defence \cite{dupont_type_2017}.
\begin{figure}[b!]
\centering
\includegraphics[width=\columnwidth]{Fig01.pdf}
\caption{(a)~Wave flume experimental set-up, involving a chirped array of eight identical cylinders, with spacing increasing in the incident wave direction.
(b)~Side-view photo showing cylinders~3--6,
including probes in cylinder spacings.}
\label{fig:model}
\end{figure}
In a seemingly disparate branch of ocean engineering,
there has been intense activity on methods to harvest the vast quantity of renewable energy carried by ocean waves.
Many wave energy converter (WEC) designs have been proposed \cite{babarit2017},
and a very large class of WECs are tuned to resonate with the dominant frequencies in a sea-state,
but suffer from having small bandwidth compared to the bandwidth of the (random) incident waves \cite{falnes2007}. In any practical application, multiple interacting WECs will be present, and much research has focussed on regular arrangements of these \citep[e.g.][]{tokic19a}.
In this Letter we address the pressing question of whether it is possible to take advantage of generic concepts from rainbow trapping from optics in a (potentially nonlinear and dissipative) water wave system,
to yield amplification of chosen frequencies at different locations,
where (in an absorbing array) the energy could conceivably be captured by suitably tuned WECs.
We do so through careful experimentation on waves of length order metres interacting with
an array of eight identical fixed bottom-mounted cylinders, of radius 0.25\,m
arranged along the centreline of a 1.49\,m wide, 54\,m long, 1.6\,m tall wave flume,
and with chirped spacing, similar to \cite{cebrecos_enhancement_2014}
but with the cylinder spacing increasing in the incident wave direction
(see Fig.\ \ref{fig:model}).
The cylinders span the full height of the flume, which is filled with water to 1.1\,m depth.
A hinged wavemaker
at one end of the flume
generates incident waves,
and those transmitted through the array are absorbed by a passive beach, which reflects $<$ 1\% of the energy reaching it (in the frequency range of interest).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Fig02.pdf}
\caption{(a)~Band diagram (dispersion curves) of symmetric modes for infinite array,
using third cylinder spacing for $x$-periodicity,
where the abscissa covers wavenumber space for normal incidence in the first irreducible Brillouin zone from in-phase ($\Gamma$) to out-of-phase ($X$), and red shading indicates second band-gap. Inset shows eigenmode on second band at $\Gamma$, as indicated by red star.
(b)~Red shading shows evolution of the second band-gap as cylinder spacing increases along the array, where spacing is interpolated between spacing mid-points and extrapolated at ends.
(c,d)~Model predictions of free surface elevation modulus, $\vert\eta\vert$,
normalised by the incident amplitude, $a_\textnormal{inc}$,
for regular incident waves with angular frequency (c)~$\omega=6.60$\,rad\,s$^{-1}$ and (d)~$\omega=6.50$\,rad\,s$^{-1}$.
Dashed curves connecting (b) with (c,d) indicate theoretical predictions of zero group velocity, where $\omega=6.60$\,rad\,s$^{-1}$ (\mydash{\color{magenta}}) and $\omega=6.50$\,rad\,s$^{-1}$ (\mydash{\color{darkorange}}) enter the band-gap.}
\label{fig:band}
\end{figure*}
Analysis of doubly-periodic infinite arrays---with periodicity in the propagation direction ($x$-direction) matching the local cylinder spacing, and the periodicity in the transverse direction matching the flume width (analogous to the walls)---yields valuable information to guide the experimental tests.
The band diagram associated with the third spacing (Fig.\ \ref{fig:band}a) shows well defined band-gaps, with the
first two bands
well separated from the more densely arranged bands above, and informs choices of incident frequencies. We choose to operate towards the upper end of the second band as the wavelength is smaller than on the first band, so the waves scatter more strongly from the cylinders, travel more slowly and are in deep water relative to wavelength. The second band is very flat close to the upper band edge over a large range of wavenumbers, generating a high density of states,
and enabling us to harness ``flat band slow light", thereby reducing dispersion \cite{li08a}.
Simulations using the boundary element method \citep{sauter11a},
with channel walls accounted for using the decomposed tank Green's function \citep{chen1994side},
give the steady-state linear free surface responses around the array shown in Figs.\ \ref{fig:band}(c,d).
The almost flat dispersion curve acts to slow the wave within the array, and the cylinder spacings at which the band-gap causes propagation to cease are predicted (see Fig.\ \ref{fig:band}b).
The linear theory for an infinite array, implicitly assuming adiabatic grading of the array, is remarkably accurate in its predictions even for merely eight cylinders; moreover, the amplitude amplifications in the cylinder spacings before propagation ceases increase by an order of magnitude, i.e., factors 9 and 17 for $\omega=6.50$\,rad\,s$^{-1}$
(wavelength 1.46\,m; Fig.\ \ref{fig:band}d)
and $\omega=6.60$\,rad\,s$^{-1}$
(wavelength 1.41\,m; Fig.\ \ref{fig:band}c), respectively.
Given predictions of such large amplifications at precise localisations,
the wave flume experiments are used to explore the limitations of linear theory, and whether assumptions, such as ignoring viscosity, surface tension and wave breaking,
alter these predictions in practice.
We probe the resonant modes using regular incident waves of
angular frequency $\omega=6.50$\,rad\,s$^{-1}$ and 6.60\,rad\,s$^{-1}$
(for comparison with theoretical responses in
Figs.\ \ref{fig:band}c,d),
and amplitude $a_\textnormal{inc}=0.01$\,m.
An incident wave of frequency $\omega=3.24$\,rad\,s$^{-1}$ and amplitude $a_\textnormal{inc}=0.03$\,m,
which exists on the first band and does not intersect a band edge,
is also tested.
Eulerian point measurements of the free surface elevation
are recorded with wave probes located at the centre of each of the first six spacings
(Fig.\ \ref{fig:band} suggests the maximum response will be at the centre).
Band-pass filtered, normalised, linear time-series responses for
$\omega=6.50$\,rad\,s$^{-1}$ and 6.60\,rad\,s$^{-1}$ (Figs.\ \ref{fig:fig3}d--i)
show large amplifications in spacing~3 and spacing~2, respectively,
and minimal transmission farther along the array,
consistent with theoretical predictions.
Further, similar mode shapes to those predicted by linear theory are indicated by the wetted surface of the cylinders (Fig.\ \ref{fig:fig3}j).
In contrast, the incident wave at frequency $\omega=3.24$\,rad\,s$^{-1}$ propagates through the array almost unchanged (Figs.\ \ref{fig:fig3}a-c).
The time series in Figs.\ \ref{fig:fig3}(a--i) are truncated just prior to the point
when waves reflected by the array then re-reflected by the wavemaker return to contaminate the results.
The resonances have timescales longer than the test window available
\cite[see Supplementary Material,][]{Supp},
so amplitudes are still increasing towards steady-state at the end of the window,
and the maximum possible amplifications are not achieved.
Numerical simulations do not suffer from contamination by spurious reflections,
and suggest substantial increases in amplitude from 50\,s to 100\,s (35\% for $\omega=6.50$\,rad\,s$^{-1}$
and 55\% for $\omega=6.60$\,rad\,s$^{-1}$),
and comparable increases from then till the steady-states are reached for $t>300$\,s
\cite[see][]{Supp},
although it is likely that
viscous dissipation would result in smaller increases in the experiments if the test windows could be enlongated, even if wave breaking could be avoided.
Focussed incident wave packets are used
to investigate the overall spectral structure of the response.
The propagating linear free surface signal created at the wavemaker is
\begin{equation}
\eta_\textnormal{inc}(\tau)=\frac{A\,\sum_n \left\{S(\omega_n)\,\cos(k_n\,x_0-\omega_n\,\tau)\right\}}{
\sum_n S(\omega_n)},
\end{equation}
where $A=0.045$\,m is the nominal amplitude, $x_0=-24$\,m is the mean wavemaker position relative to the linear focal point, $\tau$ is time relative to linear focus time, $k_n$ is the wavenumber,
and $S$ is a Gaussian with
standard deviation $\sigma=0.15$\,rad\,s$^{-1}$ and mean $\omega_\textnormal{p}=6.03$\,rad\,s$^{-1}$, i.e.\ a frequency on the second band. The linear focus position is 2\,m into the array and the sum is over $2^{10}$ equally spaced components.
Figs.\ \ref{fig:fig3}(k--n) show raw surface-elevation time-series responses in spacings~1--4,
with corresponding incident series superimposed for comparison (from tests without cylinders).
The insets show the response amplitude spectra in 0.04\,rad\,s$^{-1}$ frequency bins,
from the (discrete) Fourier transform of the surface elevation time series, $\hat{\eta}=\vert\mathcal{F}(\eta)\vert$.
The spectral peak occurs on the second band,
and is frequency downshifted from
$6.66\pm0.02$\,rad\,s$^{-1}$
to $6.42\pm0.02$\,rad\,s$^{-1}$
over
spacings~1--4,
consistent with the theoretical predictions (cf.~Fig.\ \ref{fig:band}b).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Fig03.pdf}
\caption{(a--i)~Linearised experimental time series for regular incident waves at frequency (a--c)~$\omega=3.24$\,rad\,s$^{-1}$, (d--f)~$\omega=6.50$\,rad\,s$^{-1}$, and (g--i)~$\omega=6.60$\,rad\,s$^{-1}$,
with responses shown in selected spacings,
and where $t=0$ is the incident wave arrival time.
(j)~Photo of resonant mode shape in spacing~3 for $\omega=6.50$\,rad\,s$^{-1}$.
(k--n)~Experimental time series for focussed incident wave packet in spacings~1--4 (\myline{\color{matlabblue}}),
with corresponding incident packets superimposed (\myline{\color[rgb]{0.5,0.5,0.5}}).
Insets show spectral decompositions of responses.
}
\label{fig:fig3}
\end{figure*}
\begin{figure} [!ht
\centering
\includegraphics[width=\columnwidth]{Fig04.pdf}
\caption{(a)~Experimental and (b)~numerical amplitude transfer functions, at wave probe locations along the array.
Grey strips represent spatial locations of cylinders~1--7.}
\label{fig:figure4}
\end{figure}
A transfer function is defined as
\begin{equation}
\mathcal{A} = \frac{\hat{\eta}}{\hat{\eta}_\textnormal{inc}}
\quad\textnormal{where}\quad
\hat{\eta}_\textnormal{inc} = \vert\mathcal{F}(\eta_\textnormal{inc})\vert,
\end{equation}
to quantify amplitude amplifications over the frequency spectrum.
Fig.\ \ref{fig:figure4} shows the amplitude transfer function calculated from the numerical model (for which $\mathcal{A}=\vert\eta\vert\,/\,a_\textnormal{inc}$) and from
focussed wave group experiments (averaged over test
shown in Figs.\ \ref{fig:fig3}k--n and tests with a shifted peak frequency
$\omega_\textnormal{p}=6.55$\,rad\,s$^{-1}$ and shifted amplitude $A=0.029$\,m).
The experimental and numerical transfer functions are in excellent agreement,
notwithstanding the expected larger and sharper amplifications in the
numerical model.
The amplifications on the second band are evident,
along with the spectral downshift along the array.
Cognate amplifications are also shown for the first band \citep[see][]{Supp}.
\begin{figure} [htb!]
\centering
\includegraphics [width=\columnwidth] {Fig05.pdf}
\caption{(a--d)~Experimental (\myline{\color[rgb]{0.85,0.325,0.098}}) and numerical (\myline{\color[rgb]{0,0.447,0.741}}) amplitude transfer functions in spacings~1--4.
(e)~Combined experimental transfer functions for spacings~1--6
(\myline{\color[rgb]{0.85,0.325,0.098}})
with envelope (\protect\tikz \protect\fill [color=gray,opacity=0.5] (0,0) rectangle (8pt,6pt);), and
amplification boundary
(\mydash{\color{black}}).}
\label{fig:figure5}
\end{figure}
The numerical--experimental agreement is more clearly illustrated in Figs.\ \ref{fig:figure5}(a--d) for spacings~1--4.
The transfer functions overlap, except at the most resonant peaks,
where differences can partly be attributed to truncation of the experimental data resulting in under-estimation of the maximum amplifications. In the regular wave tests,
the slow resonant build-ups are truncated (Figs.\ \ref{fig:fig3}e,h),
whereas in the focussed packet tests,
truncations cut off the slow resonant decays.
Dissipative phenomena
also contribute to the disparity, particularly at higher frequencies.
The broadband amplification given by the array is illustrated by Fig.\ \ref{fig:figure5}(e),
which shows the experimental transfer functions for spacings~1--6,
and with the envelope indicated by the grey shading.
The majority of the envelope lies well above the boundary $\mathcal{A}=1$, which divides amplification ($\mathcal{A}>1$) from suppression ($\mathcal{A}<1$),
with the only interval experiencing suppression from 7.45--8.10\,rad\,s$^{-1}$.
The maximum amplifications are $\approx{}5$, and occur on the second band, with corresponding energy amplifications $>20$.
Amplifications $>3$ also occur over multiple frequency intervals,
giving energy amplifications $\approx{}10$.
In this Letter we have presented results from wave-flume experiments
that demonstrate localisation and amplification of water waves in a chirped array of bottom-mounted cylinders,
despite the challenges of working with highly resonant systems in a finite-length flume.
Linear theory was shown to predict the amplification spectrum
accurately in frequency--distance space,
including peak resonance locations and
spectral downshifts along the array.
Moreover, even for the rather small array of eight cylinders used,
the amplification locations,
at which the group velocity slows to zero,
were shown to be consistent with the band structure of infinite periodic arrays using local cylinder spacings.
The outcomes suggest future strategies for ocean wave energy harvesting should consider taking advantage of the broadband response and precise control demonstrated here.
We have not attempted a parametric study to optimise such an array;
this will be a future undertaking.
\section*{Acknowledgements}
AJA was supported by the Wave Energy Research Centre, jointly funded by The University of Western Australia and the Western Australian Government, via the Department of Primary Industries and Regional Development (DPIRD).
HAW acknowledges financial support from Shell Australia.
LGB is supported by an Australian Research Council mid-career fellowship (FT190100404).
RVC thanks the UK EPSRC for their support through Programme Grant EP/L024926/1 and also acknowledges the support of the Leverhulme Trust.
|
1,314,259,993,022 | arxiv | \section{INTRODUCTION}
The analysis of traffic flow and road user behavior become increasingly important for both, infrastructure design and driving applications. Especially, to ensure a high standard of traffic safety with increasing automation, verification and validation have taken a central role in the development of automated driving functions.
Since validation by real tests exclusively is neither economically viable nor feasible in a reasonable time, software-based tests play a decisive role~\cite{importance_of_software_tests}. The data required for these tests cannot be generated completely synthetically. For a broad coverage of scenarios, the use of real recorded data is essential, so that a system can be confronted with diverse situations like highly interactive or unusual road user behavior~\cite{highD}.
For this purpose, multiple trajectory datasets have been published in the last years~\cite{Dataset_revisit_Li}. Thereby, naturalistic trajectories of road users recorded from a bird's-eye view form an important class of datasets.
With this kind of dataset, the interplay of road users at individual intersections or other locally bounded infrastructures can be detected and analyzed within relatively fixed boundary conditions~\cite{pNEUMA}.
The increasing amount of recorded data allows the user to improve testing and verification of systems and applications. But to use this data, it is crucial that the enormous amount can be overlooked.
For this purpose, datasets are typically accompanied by a separate publication. These mostly focus on the methodology used for the trajectory extraction and not on their quality~\cite{inD}. Usually, little more about the given trajectories is provided than rudimentary information. For example, the \textit{highD} paper \cite{highD} solely makes statements about the behavior of cut-in scenarios and velocity distribution. In addition, the respective dataset is usually described qualitatively, but characteristics of trajectories like their velocities, paths or interactions are rarely considered empirically~\cite{pNEUMA}.
There are hardly any quantitative statements about the interaction and none about the anomaly and relevance of tracks or scenarios. Also, not every dataset is well suited for every application as very few interactions take place or nearly all road users behave according to traffic rules. Therefore, experts typically have to review datasets manually and score them subjectively due to their interesting situations like critical, complex or unusual ones. Especially those real road user constellations that might not be considered in simulations, have to be used to validate systems. In order to improve the feasibility and efficiency of this process, automatically derived scores are needed that match the perception of the experts~\cite{driving_factors}.
For that purpose, as the main contribution, an automated analysis framework for trajectory datasets is proposed in this paper.
We define an interaction score, an anomaly score as behavior that highly deviates from the typical behavior in the recording and a relevance score as a combination of interaction and anomaly.
The hierarchical framework evaluates the three scores on four different levels of abstraction of the dataset (\ref{sec:methodology}).
Accumulating these scores over the abstract levels enables the user not only to evaluate individual tracks but also compare complete datasets (see Fig.~\ref{fig:headliner}).
After the presentation of the analysis framework, we compare it with experts and other algorithms for validation. Since there is no such absolute definition of interaction, a survey is set up to test the correlation between score and perception of experts.
Additionally, the framework is compared to the scheme of the \textit{INTERACTION} dataset~\cite{INTERACTION_dataset} that tries to measure interaction using different metrics.
Subsequently, further functions are presented and application fields are shown.
\section{RELATED WORK}
As a basis for the analysis framework developed within this paper, we provide an overview of previous approaches for evaluating traffic scenarios, as well as an overview of their specific use for recent trajectory datasets.
\subsection{Assessment Systems for Traffic Scenarios}
\label{sec:relatedwork_assesment}
Research on traffic scenario analysis began long before the publication of large trajectory datasets. Already in the 1970s, metrics for the description of conflicts between vehicles were presented for accident research:
\begin{itemize}
\item \textbf{Time to collision} (TTC): The metric describes the time to a potential crash of two road users assuming an unchanged driving state~\cite{TTC_Hayward}.
\item \textbf{Time headway} (THW): It measures the time it takes for a following vehicle to reach the current position of the followed vehicle, assuming constant speed~\cite{THW_Vogel}.
\item \textbf{Deceleration rate to avoid a crash} (DRAC): The metric (\ref{eq:DRAC}) describes the minimum average delay ($a_{dec}$) of a road user to avoid an accident at given velocities (v) and distance between the vehicles ($\Delta x$)~\cite{DRAC_explanation}.
\end{itemize}
\begin{align}
DRAC = \frac{(v_2 - v_1)^2}{2 \Delta x}
\label{eq:DRAC}
\end{align}
Despite their development for accident research, they are still used today with regard to automated driving~\cite{usage_of_metrics_automated_driving, risk_measure_ttc}. At the same time, new metrics have been developed that focus on the application for automated vehicles.~\cite{INTERACTION_scheme}:
\begin{itemize}
\item \textbf{Difference of time to conflict point} ($\Delta$TTCP): The $\Delta$TTCP describes the time difference in which two road users pass the same point (a point of conflict).
\item \textbf{Waiting period} (WP): The metric describes the time a road user stands still with the intention to move on.
\end{itemize}
All of these metrics have in common that they are defined for linear scenarios or, in the case of the $\Delta$TTCP, require predefined \textit{Merging Points}~\cite{INTERACTION_scheme}, which makes the availability of map data necessary.
Nevertheless, some metrics are frequently used due to their simplicity and have been validated in experiments~\cite{TTC_THW_experiment}. In the case of the THW, it is currently used in road traffic regulations~\cite{THW_Vogel}. However, most of these metrics can only be applied correctly to special scenarios like following scenarios e.g. on highways.
Due to this kind of limitations, adaptations are needed for using and combining metrics in more complex scenarios.
In addition to metrics, additional approaches were pursued to describe characteristics: for example in the area of behavioral description~\cite{WAN20, Stanford_Drone_Dataset} and trajectory prediction~\cite{trajectory_prediction}. The focus is usually set on the search of patterns and the deviation from the rule in specific infrastructures like e.g. highways. To find these patterns, density-based approaches are used mainly for anomaly detection of trajectories \cite{framework_anomaly_detection, anomaly_detection_score}. Therefore, clustering methods~\cite{trajectory_clustering_CHO} or neuronal networks~\cite{trajectory_neural_network} are set up to group attributes and find unusual data.
Besides these density-based methods, especially trajectory prediction methods combine various influences to determine the best possible trajectory based on complex cost functions including e.g. safety, visibility and comfort. For example~\cite{WAN15} uses an approach based on road conditions, driver behavior and vehicle state. It sums these up by generating a potential field through which the optimal route can be found. This approach allows the combination of different input parameters \cite{vehicle_limit_BEN15, NACHIKET18}. Based on these methods, several approaches were developed in the last few years, but almost none of them is used for actual dataset evaluation (\ref{sec:relatedwork_datasets}).
\subsection{Trajectory Datasets}
\label{sec:relatedwork_datasets}
Trajectory datasets have become increasingly relevant in recent years. This has resulted in a large number of public datasets~\cite{Dataset_revisit_Li}. In general, a distinction can be made between the utilization of offboard and onboard sensors in the dataset creation process. Onboard sensors can only perceive immediate surroundings around the vehicle and are limited by occlusions, so that they perceive traffic situations incompletely. However, because they are fixed to moving vehicles, datasets include a larger variety of infrastructures~\cite{range_onboard_sensors}. In contrast, offboard sensors are typically positioned stationary and thus record the behavioral variety at small sets of selected recording sites. Since they are not tied to vehicles, they typically have a better perspective and can record traffic situations over a long period~\cite{inD}. For example, in the \textit{inD} dataset~\cite{inD} intersections are observed and road users within are tracked. Thus, complex events including several traffic participants over time at fixed locations (e.g. intersections) can be recorded and analyzed in a next step.
With the increase in the number of recorded datasets, the number of relevant trajectory datasets recorded by offboard sensors is growing. In order to give the user an overview, information about the recorded road users is typically provided in a publication. However, these analyses are usually limited to basic quantitative parameters such as metrics or the number and type of traffic participants that are only analyzed independently~\cite{pNEUMA, inD, rounD}. In addition, a series of qualitative descriptions is usually carried out. Beyond that, only a few quantitative descriptions are provided. An example is the INTERACTION dataset~\cite{INTERACTION_dataset}, for which the interaction is analyzed using two predefined metrics~\cite{INTERACTION_scheme}. The authors compare similar datasets with those metrics. However, the employed scheme does not provide detailed information about individual scenarios, because metrics with similar values are just grouped. Those groups are only analyzed on the dataset level. This allows a rough comparison of the general interaction. However, it does not provide a detailed overview, nor a comprehensive interaction analysis of individual road users because only bilateral relations are analyzed but no more complex vehicle constellations.
\section{Methodology}
\label{sec:methodology}
The framework proposed in this paper follows two principles: comprehensiveness and clarity. In order to achieve comprehensiveness, no specific use case is targeted to design the framework as unbiased as possible. So, it is universally applicable to find any kind of interesting situation. For this purpose, three characteristics are defined as essential scores of the framework:
\begin{itemize}
\item \textbf{Interaction}: Interaction describes the interplay of the road users among other traffic participants.
\item \textbf{Anomaly}: Anomaly is driven by rarity~\cite{definition_anomaly}. The anomaly score highlights unusual phenomena or behavior.
\item \textbf{Relevance}: Relevance is defined as the combination of interaction and anomaly.
\end{itemize}
While interaction and anomaly are generally known concepts, relevance is newly defined. It combines both points. Depending on the dataset usage, interactive scenarios may be less relevant if vehicles drive with constant velocity on a highway with other road users. A turning vehicle on an intersection might be unusual but it might not relevant until it interacts with another road user. Therefore, we define the product of both, interaction and anomaly, as relevance.
To ensure that the clarity is not lost despite the amount of information generated by applying these scores for a comprehensive analysis to every point in time, a hierarchical four-level approach is used.
In a first step, a modular set of rules is applied and characteristics (\textit{detections}) are extracted from the track data (chap.~\ref{sec:observed_scenarios}). These are evaluated and weighted to combine them to the three characteristic scores (chap.~\ref{chap:combining_scores}) on four layers (chap.~\ref{sec:hierachical_kombination}):
\begin{itemize}
\item \textbf{Temporal punctual}: On the most basic level, the scores are defined for every road user in each timestep individually.
\item \textbf{Track}: The temporal, punctual scores are merged to accumulate the scores for whole trajectories of road users.
\item \textbf{Spatial region}: Besides the entity-focus, the scores can be derived for spatial regions of the infrastructure. Therefore, the behavior of road users within the region is considered.
\item \textbf{Overall}: On the most abstract level, the scores of all detected road users are summed up to get a comprehensive picture of the whole dataset.
\end{itemize}
Considering the three types of scores on four abstract layers the user is given both: a high degree of clarity and the possibility of variable use.
\section{Traffic Situation Analysis}
\label{sec:observed_scenarios}
As the basis of the hierarchical framework, all road users including pedestrians and bicycles are observed individually at every timestep and are broken down to relevant attributes of its trajectory. Those attributes are called \textit{detections}. They form the foundation of the modular framework. These detections are derived by applying a set of criteria (\textit{detection types}). Fourteen different detection types in three thematic blocks are presented: vehicle relation indicators, individual vehicle state and context-related behavior. These are checked independently for each timestep and detections are assigned to both, the involved traffic participants and the associated infrastructure region. In order to provide a comprehensive picture, a minimum of dataset information needed for analysis is assumed. These requirements are kept general, so that neither country-specific restrictions nor a limitation to a certain type of scenario are considered.
\subsection{Dataset Requirements}
\label{sec:data_requirements}
According to chapter~\ref{sec:relatedwork_datasets}, the type of trajectory datasets can vary in their recording method. However, in order to generate a comprehensive analysis, the following restrictions are made with regard to the recording approach as well as the provided data to ensure that the datasets are comparable. Especially the recording approach offers the advantage of a temporally extended view of selected recording sites. So, complex traffic scenarios can be recorded within a limited spatial region. Those scenarios can go beyond the duration of stay of individual road users. This fact allows linking the infrastructure to road user behaviors.
Furthermore, for the processing by the proposed framework, datasets have to include three key elements:
\begin{itemize}
\item \textbf{Trajectory data}: Information about velocity, acceleration, position and orientation of every road user for each frame
\item \textbf{Meta data}: Information about road user dimensions and type to distinguish vulnerable road users from cars, as well as regulatory permission of region usage
\item \textbf{Semantic map}: Information about the infrastructure including position and definition of characteristics such as lanes, possible driving paths, speed limits and walkways in a reasonable granularity (see OpenStreetMap~\cite{OpenStreetMap} or Lanelets~\cite{Lanelets})
\end{itemize}
The restrictions made are fulfilled by datasets like \textit{inD}~\cite{inD}, \textit{rounD}~\cite{rounD} or \textit{INTERACTION}~\cite{INTERACTION_dataset}). Thus, the request enables a detailed assessment so that datasets matching the requirements can be comprehensively mapped.
\subsection{Vehicle Relation Indicators}
\label{sec:detections_interaction}
An essential point in the evaluation of driving scenarios is the interaction between road users. These can sometimes become quite complex and can be both, temporally and spatially extended. Within our framework, complex scenarios are broken down into a combination of bilateral vehicle interactions to effectively take an unlimited number of traffic participants into account. To comprehensively map the individual interactions, several relation indicators (see Chap.~\ref{sec:relatedwork_assesment}) are used.
However, in order to be able to calculate these metrics automatically within more complex scenarios, the metrics have to be partially adapted because they are mostly designed for straight-line scenarios. Thus, the bilateral metrics are based on at least one extrapolation of a path with constant speed. Usually, the current speed of the object at the time of calculation is kept constant and the time gap is determined by the Euclidean distance to a conflict point with its interaction partner~\cite{calculation_metrics}. Due to the sometimes complex trajectories at intersections, this approach is revised: Instead of using only the Euclidean distance the actual driven path is considered in addition.
Furthermore, the definition of conflict points is adopted for the $\Delta$TTCP. In simple cases like junctions, a conflict point (CP) is originally defined by the road layout. However, if the crossing point is not uniquely given by the road layout, as in the case of a ramp, the conflict point is derived from the trajectories driven. We adopt this concept by considering not only one but all possible conflict points. These are defined by the set of all points that both road users share on their real driven paths including their physical dimensions. Thereby, only the physical dimensions of the road users and not the infrastructure or situation are considered.
After this conflict point definition, the calculation is analog to the $\Delta$TTCP.
This modified Time to conflict point ($\Delta$mTTCP) is also mathematically similar to the $\Delta$TTCP.
Starting from the road users' current position, the Time to conflict point (TTCP) is determined as the time a road user takes to travel to a conflict point by assuming a constant speed and using the actual path driven.
In contrast to the $\Delta$TTCP, the time difference is determined not for one, but all conflict points, so that each vehicle pair has a set of time differences linked to individual conflict points. The smallest of those time differences is defined as the $\Delta$mTTCP (\ref{eq:dmTTCP}) and the concerning conflict point is defined as the critical conflict point (CCP).
\begin{align}
\Delta mTTCP = \min_{p \in CP}{|{mTTCP}^{p}_1 - {mTTCP}^{p}_2|} \label{eq:dmTTCP}
\end{align}
This more comprehensive and accurate definition of conflict points (see Fig.~\ref{figure:dttcp_modification}) requires neither the division of conflict points into static and dynamic nor detailed information about the infrastructure. So, it can be applied not only for vehicles but also for bicycles and pedestrians on sidewalks.
Calculating the TTCP for all possible conflict points causes a higher computational complexity compared to the original approach. However, to limit the complexity, a maximum prediction horizon of 5~seconds is used for trajectory prediction. As following scenarios are already covered by the THW, a minimum angle between the intersection of trajectories is utilized to filter them. For urban scenarios, it is set to 20~degrees because at that angle the mid-line of the rear vehicle would contact the side and not the rear of a car in front of it. Using that threshold merging scenarios or intersecting trajectories can still be detected. For highway scenarios, a much lower angle ($\beta_{min} = 2$ degrees) is used because of the parallelism of the vehicles and smaller yaw rates. This angle is derived from a potential lane change maneuver on a normal street within 100 meters.
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig2.PNG}
\caption{Original (upper)~\cite{INTERACTION_scheme} and modified (lower) calculation of $\Delta$TTCP conflict point (black cross) in order to not define merging points. Instead using a collision angle $\beta$.}
\label{figure:dttcp_modification}
\end{figure}
With these adjustments, a variety of metrics can be applied robustly to any infrastructure. For a comprehensive view, four common metrics are used as detection types. These are chosen to supplement each other both in the type of scenario identification and its criticality assessment without overlapping much (Tab.~\ref{table:scope_metrics}).
These metrics are calculated for all road user pairs and assigned to both respectively. Exceptions to this are THW and DRAC as these are each based primarily on the behavior of one road user. In these cases, the calculations are performed bidirectionally, so that both road users can have different DRACs and THWs.
In addition to these bilateral metrics, the WP is considered, too. It does not map the interaction between two road users, but registers forced stops. Those imply interactions with the traffic in general and therefore cover multiple possible interactions~\cite{INTERACTION_scheme}.
\begin{table}[h]
\caption{Scope of relation indicators}
\label{table:scope_metrics}
\begin{center}
\begin{tabular}{c|c|c|c|c}
\textbf{Metric} & \textbf{Follow-up} & \textbf{Merging} & \textbf{Criticality} & \textbf{Coverage} \\
\hline
THW & ++ & - & o & o \\
TTC & o & + & + & o \\
DRAC & - & - & ++ & o \\
$\Delta$mTTCP & - & ++ & o & + \\
WP & - & + & o & ++ \\
\end{tabular}
\end{center}
\end{table}
\subsection{Individual Vehicle State}
Relation indicators are an important aspect, but can only capture part of a traffic scenario. By adding information about the individual driving states, the framework is extended. The observation is less relevant for interaction but more important for anomaly detection. For the description of the vehicle state, the bicycle model~\cite{bicycle_model} is chosen. So, just a few, but basic and characteristic parameters are observed (Tab.~\ref{table:vehiclestate_limits}).
The most obvious value is the absolute speed. In addition, the different types of acceleration represent essential aspects of the vehicle maneuver. They do not only describe the change in the driving state, but also give a statement about forces and driving behavior. Additionally, the sideslip angle is considered as an extension of the force transmission and characterizes the contact between wheel and street. Finally, the yaw rate is added for a more precise characterization of cornering than just lateral acceleration. So, a set of five parameters is chosen to represent the vehicle state and influence the framework via detections. But since detections are only supposed to record interesting and unusual information, only exceedances of normal driving behavior are taken into account. These limits of normal driving maneuvers are based on results from publications~\cite{vehicle_limit_BEN15, vehicle_limit_DOU11}.
\begin{table}[h]
\caption{Vehicle state detection types with thresholds}
\label{table:vehiclestate_limits}
\begin{center}
\begin{tabular}{c|c|c}
\textbf{Parameter} & \textbf{Threshold} & \textbf{Range [$km/h$]}\\
\hline
Velocity & see chap.~\ref{sec:detections_infrastructure} & -\\[0.5ex]
Lon Acceleration & $4 m/s^2$ & $v \leq 50$ \\[0.5ex]
& $4m/s^2 - 2m/s^2 \frac{v - 50km/h}{100km/h}$ & $50 < v \leq 100$ \\[0.5ex]
& $2m/s^2$ & $ 100 < v$\\[0.5ex]
Lat Acceleration & $2.5m/s^2 + 4.5m/s^2 \frac{v}{40km/h}$ & $ v \leq 40$ \\[0.5ex]
& $7m/s^2$ & $40 < v \leq 50$ \\[0.5ex]
& $7m/s^2 - 4m/s^2 \frac{v-50 km/h}{50 km/h}$ & $50 < v \leq 100$ \\[0.5ex]
&$3m/s^2$ & $v < 100$\\[0.5ex]
Yaw rate & $\frac{50}{180} \pi \deg/\sec $ & $ v \leq 50$ \\[0.5ex]
& $\frac{15}{180} \pi \deg$ / $\sec$ & $50 < v$\\[0.5ex]
Sideslip angle & $10 \deg$ & - \\
\end{tabular}
\end{center}
\end{table}
\subsection{Context-Related Behavior}
\label{sec:detections_infrastructure}
While both previous groups of detection types do not consider traffic rules or guidance, the latter is examined in more detail below. For this purpose, road user behavior is subdivided into two categories: the first with respect to traffic rules and the second to other road users. Therefore, the peculiarity of the infrastructure of the dataset is used. So, it must be divided into logical driving \textit{regions} (see Fig.~\ref{figure:inD_frankenberg_map}) as set out in \ref{sec:data_requirements}. The regions represent possible traffic routes or sections of those. For example, a two-lane street is described at least with two regions. A t-junction is described by at least six regions because of the six possible routes. On an intersection, where several driving routes and walkways cross, an overlapping of regions is also possible.
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig3.png}
\caption{Digital map of the intersection \textit{Frankenberg}~\cite{inD} with colored region types as provided in the dataset (black: street; gray: parking lot; orange: walkway; green: grass verge) and intersecting regions (darker than normal colors)}
\label{figure:inD_frankenberg_map}
\end{figure}
In order not to be bound to national peculiarities when checking traffic rules, they are kept general. For this reason, only three rules are taken into account referring to the dataset requirements:
\begin{itemize}
\item Local speed limit
\item Designated area usage
\item Driving direction
\end{itemize}
These rules are evaluated for all vehicles in their respective regions. If a road user can be assigned to more than one region, it is assumed that the road user behaves correctly, if he behaves correctly in at least one of the assigned regions. This is particularly relevant when considering the direction of travel, for example, if regions overlap in the center of an intersection.
In addition to the rules given by the infrastructure, the behavior of vehicles, pedestrians and other vulnerable road users are examined in relation to the norm. This norm may differ from infrastructure rules due to common rule violations e.g. if a designated lane is blocked. For this purpose, driving behaviors are compared on a local level as a combination of speed ($v$) and orientation ($\psi$). They are collected in each region and grouped by using a DBSCAN clustering approach~\cite{DBSCAN}. Therefore, the distance ($D$) between two data points ($i,j$) is defined using a Mahalanobis norm. \begin{align}
{D}_{i,j} = \sqrt{(\psi_1 - \psi_2)^2 + \Bigl(\frac{v_1 - v_2}{\max\{\max\{v_1, v_2\}, 1.5\}}\Bigl)^2} \label{eq:adaptive_mahalanobis}
\end{align}
The norm is scaled with respect to the velocity, since the usual deviation of the direction of movement decreases with increasing travel speed. So, pedestrians and vehicles on highways can be discussed equally.
Another advantage is, that the approach can be used for sidewalks as well as for freeways.
A detection is only found if a driving state cannot be assigned to a cluster or if the cluster of the state makes up less than 10 percent of the total quantity. Applying this rule, unusual driving maneuvers can be detected with reference to the infrastructure without having a look at usual behavior a priori. So, driving against the direction of travel can be regarded as normal if a significant amount of objects do so. For example, this common misbehavior can be caused by a blockade of the own lane. Since it is might still be an interesting situation, it is detected by infrastructure rules. Nevertheless, it is rated less important because it is only assigned to the infrastructure rules, but not to a specific cluster.
To examine the clustering itself, values for $\epsilon$ ($= 0.7$) and \textit{min\textunderscore samples} (\eqref{eq:min_samples}) are chosen by experiment. Thereby \textit{min\textunderscore samples} is set adaptively with regards to the number of detections ($|M_\text{input}|$) to take care of the wide span of possible input data.
\begin{align}
min\_samples = 2 + 0.01 \cdot |M_\text{input}| \label{eq:min_samples}
\end{align}
On a more abstract level, a similar procedure is used. Not the driving states, but whole trajectories are grouped by the HDBSCAN algorithm~\cite{HDBSCAN}. Therefore, the Fr\'{e}chet norm~\cite{frechet_norm} is used to determine the distance. Just like the previous clustering, the number of trajectories ($|M_\text{Trajectories}|$) can vary. So, the \textit{min\textunderscore cluster\textunderscore size} is set adaptively as well.
\begin{align}
min\_cluster\_size = 1 + 0.005 \cdot |M_\text{Trajectories}| \label{eq:min_cluster_size}
\end{align}
Those points that do not belong to any significant cluster are outliers and declared as detections (see Fig.~\ref{figure:anomaly_in_trajectory_clustering}). For further classifying outliers the possibility that even small deviations can cause a trajectory to be an outlier is taken into account. Therefore, the value of the Fr\'{e}chet distance to the nearest trajectory within a cluster is used as a measure of anomaly.
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig4.PNG}
\caption{ trajectories (different grays) at an intersection with an additional anomaly (red)}
\label{figure:anomaly_in_trajectory_clustering}
\end{figure}
\subsection{Comparability of Detections}
\label{sec:comparability_of_detections}
The fourteen presented detection types (see \ref{sec:detections_interaction}~-~\ref{sec:detections_infrastructure}, Tab.~\ref{table:detection_scoring}) out of the three areas are combined in chapter~\ref{chap:combining_scores} to create interaction, anomaly and relevance scores. For this purpose, each detection is scored according to its characteristics. To create a relative weighting between the detection types, $THW(1s) = 1$ is set as a reference score. Based on this metric, the other components are scaled relative to the THW with regards to their criticality and potential interestingness (Tab.~\ref{table:detection_scoring}). Thereby, binary detections are assigned as fixed values. Correspondingly, real-valued detections are scored as those. To not distort the result, an upper limit is set so that errors in the input data do not dominate. Of course, the scorings tend to be subjective, but they are weighted by simple rules: As the detections refer to the THW scoring, detections of different intensities are evaluated similarly. Nevertheless, situations with high conflict potential are evaluated higher, because they combine both: low and high-intensity detections. In order to give the user the possibility to prefer either the occurrence or the intensity of interaction, the criticality factor $\kappa$ is proposed. This factor is set to 1 by default. An increase leads to a focus on intensive interactions.
Compared to these relation indicators mainly anomaly-based detections are weighted higher, so that anomalies stand out from ordinary interactions. The vehicle state and behavior detection types themselves are weighted equally so that none of them dominates the score.
\begin{table}[h]
\caption{Scoring of detection types}
\label{table:detection_scoring}
\begin{center}
\begin{tabular}{c|c|c}
\textbf{Detection type} & \textbf{Scoring} & \textbf{Max} \\
\hline
THW & $\frac{1}{THW}$ & 2 \\
$\Delta$mTTCP & $\frac{1}{\Delta mTTCP} \frac{4}{{mTTCP}_1 + {mTTCP}_2}$ & 4 \\
TTC & $\frac{2 \kappa}{Value}$ & 2 $\kappa$ \\
DRAC & $\frac{\kappa}{5} \cdot Value $ & 2 $\kappa$ \\
WP & $\sqrt{Value}$ & 7.75 \\
Lon acceleration & $0.1 \cdot |Value - Limit|$ & 10 \\
Lat acceleration & $2 \cdot |Value - Limit| $ & 20 \\
Sideslip angle & $25 \cdot |Value - Limit|$ & 8.725 \\
Yaw rate & $|Value - Limit|$ & 3.141 \\
Area usage & $5$ & - \\
Driving direction & $4$ & - \\
Velocity & $\frac{10}{Limit} (Value - Limit)$ & 10 \\
Driving behavior (\ref{sec:detections_infrastructure}) & $1.2 \cdot Value$ & - \\
Trajectory (\ref{sec:detections_infrastructure}) & $Value $ & 10 \\
\end{tabular}
\end{center}
\end{table}
\section{COMBINING DETECTIONS TO ABSTRACT SCORES}
\label{chap:combining_scores}
As described, the interaction-, anomaly- and relevance scores are based on the introduced detections which can be calculated for each timestep. Within the scope of this paper, each recording was considered individually. A larger set of trajectories like the combination of several recordings within an infrastructure would be also possible to use as an input for the framework. However, due to changing traffic conditions like congestion or time depending behavior, temporally different recordings may only be less meaningful because of the change in boundary conditions.
Based on the determined detections, the scores are calculated for every traffic participant at any given time in a first step (chap.~\ref{sec:interaction_score} - \ref{sec:relevance_score}). A hierarchical combination of these scores is proposed in chap.~\ref{sec:hierachical_kombination}.
\subsection{Interaction Score}
\label{sec:interaction_score}
A core requirement for trajectory datasets is high interactivity between traffic participants. This interplay is represented by the interaction score (see Fig.~\ref{fig:flowchart_interaction_score}). For calculating it, the vehicle relation indicators (Chap.~\ref{sec:detections_interaction}) are used as its base: all detections belonging to the relation indicators of a traffic participant are scored (see Tab.~\ref{table:detection_scoring}) and summed up to a base score.
\begin{align}
{S}_\text{Base} &= \sum_i^\text{Detections}{S}_{i} \label{eq:interaction_base_score}
\end{align}
Since the base score is limited to the interaction between two road users, two further influences are considered: the freedom of movement of the observed vehicle and the opportunity of participating road users to handle a specific situation.
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig5.png}
\caption{Flowchart of the interaction score calculation. Beginning with the road user related detections, three influences are calculated and summed up for interaction score.}
\label{fig:flowchart_interaction_score}
\end{figure}
If a road user interacts with multiple other traffic participants simultaneously, multiple detections are assigned to the road user at the same point in time. Handling such scenarios is more difficult since the road user has to consider the behavior of different traffic participants. Therefore, the higher the number of different interacting partners in the detections of the observed vehicle ($R_\text{Observed}$) is, the higher the interaction is rated.
Just as relevant as the freedom of movement of the observed vehicle are the numbers of interacting vehicles of the partners themselves ($R_i$). For example, if the observed vehicle wants to drive onto a freeway, but the lane is blocked by another vehicle, the interaction could be defused by a lane change of this second vehicle. However, if the second road user itself interacts with other participants who are preventing a lane change, the situation becomes more interactive for the observed vehicle, too. Therefore, this mutual interaction (${S}_\text{Mutual}$) is considered separately (\ref{eq:interaction_mutual}).
After calculating these three aspects, they are combined to the punctual interaction score of a road user (\ref{eq:interaction_overall}).
\begin{align}
{S}_\text{Mutual} &= \sum_{i}^{\substack{\text{Interacting} \\ \text{vehicles}}} \sum_{j}^{\substack{\text{Mutual} \\ \text{detections}}} 0.1 \cdot R_i \cdot S_{j} \label{eq:interaction_mutual} \\
{S}_\text{Interaction, Punctual} &= {S}_\text{Base}\cdot(1 + 0.1 \cdot{R}_\text{Observed}) + {S}_\text{Mutual} \label{eq:interaction_overall}
\end{align}
\subsection{Anomaly Score}
Since the framework follows the goal to describe datasets comprehensively, a sole consideration of interaction is not sufficient. Often, it is the interest to find unusual and interesting situations~\cite{driving_factors}. Therefore, the anomaly score focuses on the rarity of events. In order to follow this principle each detection type is weighted depending on its rarity within the context ($\gamma_i$).
To correctly interpret anomalies even in larger infrastructures and not to bundle different phenomena, an individual context is defined for each region.
Within these regions the number of all detections of a detection type belonging to a region ($M_\text{Type,Region}$) is considered as well as the number of road users passing the same region ($U_\text{Region}$). Those attributes form the weights of the contexts. By using the weights, the detections can be scaled before they are summed up to the anomaly score (see Fig.~\ref{figure:flowchart_anomaly_score}).
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig6.png}
\caption{Flowchart of the anomaly score calculations. Each detection is weighted by a context provided by the belonging regions and all of them are summed up afterwards. If the affiliation is unclear, additionally, the neighbourhood is taken into account.}
\label{figure:flowchart_anomaly_score}
\end{figure}
In accordance with the rarity of anomalies, the amount of detections is considered twice:
Firstly, to normalize the number of detections, so that a higher amount does not directly lead to a higher score because that would lead the anomaly concept ad absurdum.
Secondly, the framework follows the approach, that a rarity is worth more, if the quantity of road users is higher. To limit this effect, the impact of the amount of anomalies is diminished by the root.
\begin{align}
\gamma_i &= \frac{1}{M_{\text{Type,Region}, i}} \frac{U_{\text{Region}, i}}{{M_{\text{Type,Region}, i}}^{\frac{1}{2}}} \\
{S}_\text{Anomaly, Punctual} &= \sum_i^\text{Detections} {S}_{i} \cdot \gamma_{i} \label{eq:anomaly_overall
\end{align}
In contrast to the interaction score, anomaly is not limited to a few detection types, but considers all of them, to ensure a comprehensive view of the dataset.
For example, waiting (waiting period) before a crosswalk is common, but a stationary car on a freeway with fast-moving traffic is unusual.
Since the context is set up for each region, it is possible to differentiate standing on a highway from stopping in front of a crosswalk and tracking those anomalies on both trajectory and dataset level, too.
However, regions may overlap and vehicles may belong to several regions at the same time. In the case of overlapping regions, it is assumed that a road user behaves normally. So, the least unusual context is selected for calculating the detection score. Unluckily, the problem of the affiliation of a road user to a region and the correct context for a detection is more complex since both, road user and region each cover a certain area and can overlap to different degrees. A vehicle is assigned to a region as soon as it overlaps with the region with at least two square meters or occupies more than 50 percent of the region. If this results in a road user being assigned to more than one region, the contexts of those regions are taken into account and the detection is weighted by as many regions as needed to assign 100 percent of the vehicle to regions. Thereby, the regions used for the calculation are chosen depending on their weights to minimize the overall anomaly. To finally calculate the score the detection region weights are proportionally taken into account with regards to the intersecting area between road user and region. This method is used to distinguish vehicles that only touch a sidewalk with their bumper from those that are completely on it.
The same concept applies to vulnerable road users (VRU). However, these are usually only given as points and like to stretch existing rules e.g. when walking on the street directly next to a sidewalk. To take these points into account, an area is defined around the VRU as a circle with a radius of 2.5 meters. It is used for the described calculation as the area vehicles have by default. It seems a lot, but considering, that 2 square meters are enough to fully assign a road user to a region and the least unusual region is chosen, the method reduces the misdetections due to inaccuracy and minor rule violations by VRUs.
\subsection{Relevance Score}
\label{sec:relevance_score}
If interaction and anomaly are considered separately, only limited statements can be made about the relevance of traffic scenarios. For example, abnormal behavior without interacting with a second traffic participant normally has little relevance e.g. for validation. Similarly, an anomaly makes an interactive situation more interesting. To account for this, the relevance score ($S_\text{Relevance}$) is introduced. It closes the gap resulting from the two previous scores. It is not based on the detections directly, but it is defined by the combination of the punctual interaction ($S_\text{I}$) and anomaly score ($S_\text{A}$).
\begin{align}
{S}_\text{Relevance, Punctual} = {S}_\text{I} \cdot {S}_\text{A} + \gamma_\text{I} \cdot {S}_\text{I} + \gamma_\text{A} \cdot {S}_\text{A}
\label{eq:relevance_score}
\end{align}
In order to consider highly interactive scenarios as well as anomalous in the evaluation framework, they are included in the score on a subordinate scale. Thereby, the different scoring of the individual detections in favor of the anomaly-related detection types is compensated through weights ($\gamma_{I} = 5$, $\gamma_{A} = 0.1$). The weights are chosen, so that default maximum values of interaction and anomaly have a similar impact. Depending on the application, other weightings can be quite conceivable.
\subsection{Hierachical Combination of Scores}
\label{sec:hierachical_kombination}
Defined scores were previously determined for timesteps of individual road users. This is not enough for a comprehensive overview, because a few hundred thousand scores can be calculated for a normal recording over 15 minutes. Therefore, these scores are converted into more abstract layers. Starting with the defined scores within one timestep, the scores are determined for trajectories, local regions and finally for whole recordings. Therefore, two different methods are used: One is purely detection-based and the other one only dependent on abstract scores. Interaction and relevance scores have to be considered differently for more abstract scores because they are additionally based on other influences like the number of interacting road users. In order to generate trajectory scores from the punctual scores, only the time series of the punctual score itself is used to derive the trajectory score. Only the positive change of the series is summed up.
\begin{align}
{S}_\text{g, Trajectory} &= \int \max\Biggl(\frac{\text{d}{S}_\text{g, Punctual}}{dt}, 0\Biggl) dt \label{eq:positive_summed_up} \\
g &\in [\text{Interaction}, \text{Relevance}]
\end{align}
This method is preferred to ordinary integration, since an unchanged state is usually not very interesting, regardless of the duration. For example, a continuous distance and velocity between two vehicles are not relevant. However, this scenario would become interesting when the situation changes, e.g. because the vehicle in front brakes and expects a reaction of the following vehicle.
The same approach can be used analogously for regions by summing up the scores of all trajectory parts that are assigned to the region. Additionally, due to the previously determined layers, a dataset score is defined as the sum of the individual trajectories, which characterize the dataset extensively.
\begin{align}
{S}_\text{g, Dataset} = \sum_j^\text{Trajectories} S_{g, \text{Trajectory}, j} \label{eq:summed_up_dataset}
\end{align}
A disadvantage of this method is the elimination of simultaneously occurring opposite effects: If a vehicle increases the distance to a vehicle in front, the THW increase and so the detection score itself decreases. But if the conflict with a third vehicle grows, this second detection score increases. Due to the punctual interaction score definition, both detection scores are summed up and so the changes partially balance each other out in $S_\text{Punctual}$. As a result of this, the trajectory score is underestimated. This issue does not have a big impact, but since the anomaly score is based directly on the detections, another approach can be used. For the creation of the abstract scores ($S_\text{Anomaly}$), the detections within the observed scope are collected and the maximum value of the individual detections are summed up. So all detections can be considered in their maximum intensity. In order to take a temporal extension into account, the detections are evaluated for each region. So it becomes more important if a detection occurs for a longer time.
\begin{align}
{S}_\text{Anomaly, i, j} =\sum_{i}^{\substack{\text{ } \\ \text{Regions}}} \sum_{j}^{\substack{\text{ Road } \\ \text{users}}} \sum_{k}^{\substack{\text{Detection} \\ \text{types}}} \max{S}_{i,j,k} \label{eq:combination_anomaly}
\end{align}
\section{EVALUATION}
In order to check the suitability of the framework, three different approaches are used. In a first step, a study is executed to check the correspondence between human perception and the framework.
Although a study to check human perception of relevance would be best, this is not easily achievable. Reasons for this are that both, that anomaly and as a result of that relevance as well take the context of recordings into account. Therefore, study participants would have to overlook a large part of the record to rate specific situations. However, this long preparation time would lead to fatigue of the participants and thus to inaccurate results.
Therefore, we only test the human perception of interaction as a cornerstone of the framework.
In contrast to the other scores, it offers the possibility of intuitively evaluating individual situations without further knowledge about the whole dataset. So, situations across multiple infrastructures can be checked without exhausting the participants too much.
In a second step, this score is compared with existing approaches. That analysis is distinct from the study since there is no framework analyzing situations similar extensively, but focusing on bilateral interactions.
In a third step, anomaly and relevance are analyzed by applying these scores in different levels of abstraction to three actual datasets in urban scenarios. In this context, existing datasets are analyzed and compared with each other. That approach is used because the scores can not be tested against previous schemes. Anomaly has not yet been considered in such a holistic approach and relevance not at all in the literature. The evaluation is limited to the datasets inD \cite{inD}, rounD \cite{rounD} and INTERACTION \cite{INTERACTION_dataset} because they belong to few actual datasets fulfilling the requirements and represents a broad range of scenarios.
\subsection{Interaction Score Study}
\label{sec:interaction_score_study}
In order to check the correspondence between human perception of interaction and the interaction score, a study is carried out. For this purpose, twenty-four persons with a background in automated driving and automotive engineering estimate the interaction in given situations from the perspective of a selected road user. Therefore, they are provided with twenty aerial images, each for a single traffic situation. The participants score the situations subjectively and with regards to their experience on a scale of 0 - 10. These twenty situations belong each to one of three recording sites from two different datasets (see Fig.~\ref{fig:survey_infrastructure}). As these recording sites include a roundabout, a t-crossing and an intersection with a zebra crossing a wide range of scenarios are considered. For the selection of the situations, the number of participating road users, vehicle constellations and their velocities are varied. To enable the participants to rate the situations adequately, a short video of the traffic recorded in the dataset is shown to get them used to the bird-eye's perspective and the common traffic flow. Thereby, no reference score is given to get the uninfluenced interaction impression of the subjects.
\begin{figure}[thpb]
\centering
\subfloat[Intersection \textit{Frank- enberg}~\cite{inD}]{
\includegraphics[width=0.29\linewidth]{graphics/fig7a.png}
\label{fig:infrastructure_frankenberg}
}
\hspace{0.005\linewidth}
\subfloat[Intersection \textit{Aseag} \cite{inD}]{
\includegraphics[width=0.29\linewidth]{graphics/fig7b.png}
\label{fig:infrastructure_aseag}
}
\hspace{0.005\linewidth}
\subfloat[Roundabout \textit{Neu- weiler}~\cite{rounD}]{
\includegraphics[width=0.29\linewidth]{graphics/fig7c.png}
\label{fig:infrastructure_neuweiler}
}
\caption{Bird-eye perspective of observed infrastructures of the study.}
\label{fig:survey_infrastructure}
\end{figure}
\begin{figure*}[t]
\centering
\subfloat[Overall average]{
\includegraphics[width=0.25\linewidth]{graphics/fig8a.png}
\label{fig:pearson_average}
}
\subfloat[Intersection \textit{Frankenberg}]{
\includegraphics[width=0.237\linewidth]{graphics/fig8b.png}
\label{fig:pearson_frankenberg}
}
\subfloat[Roundabout \textit{Aseag}]{
\includegraphics[width=0.237\linewidth]{graphics/fig8c.png}
\label{fig:pearson_aseag}
}
\subfloat[Roundabout \textit{Neuweiler}]{
\includegraphics[width=0.237\linewidth]{graphics/fig8d.png}
\label{fig:pearson_neuweiler}
}
\caption{Correlation coefficient (Pearson r) of each rater values compared to the calculated interaction scores of each rater (blue bars) and the average value of raters (orange line)}
\label{fig:pearson_overview}
\end{figure*}
Because of the fixed range for the participants' interaction ratings, the interaction score is scaled to this range for the study. Despite a wide span (average 6.7) of the volunteer's ratings within the individual situations, a high correlation between the interaction score and the ratings can be observed for most of the volunteers (see Fig.~\ref{fig:pearson_overview}). Comparing these with a baseline correlation between the number of interacting road users and the human perception of interaction ($r = 0.34$), the interaction score fits the perception significantly better. Thereby, interacting road users are those that are either near to the observed road user or that intersects the road users future path within a reasonable time. Only at the roundabout \textit{Neuweiler} the perception of few participants deviates from the score, but is in average still significantly higher than the correlation to the number of interacting vehicles. However, if the ratings are averaged over the participants, the correlation is noticeably higher. That indicates that the framework reflects the average perception of the subjects better than the opinion of an individual expert. The average is used instead of e.g. the median because the perception of interaction tends to be subjective and the opinion of all participants should be included.
When comparing the absolute values, a similar picture emerges. While the scores normally differ by an average of 2 points between participant and framework, the average score is only 0.8 points higher than the interaction scores. It is noticeable that especially following maneuvers and conflicts with large prediction periods are evaluated differently. Classical scenarios as crosswalk usage or turning conflicts at intersections are evaluated similarly throughout and are better represented by the framework.
However, a difference can be seen not only between different scenarios at the same location but also between infrastructures. In contrast to the classical intersection \textit{Frankenberg}, interaction on straight traffic routes with increased speed is rated higher by participants. In addition, the deviations between raters are smaller at the intersections than at the roundabout (see Fig.~\ref{fig:pearson_neuweiler}). Despite these differences, the interaction score can accurately reflect the participants' opinions. Even at roundabouts, where the opinions of the test subjects are scattered, the framework fits the average opinion of the participants better than the judgment of an individual.
This can also be confirmed by the fact that the evaluation criteria of the test persons fit those of the framework. The participants were asked an open question about the influences of their interaction scoring retrospectively. Most of them mention predicted conflicts (67 percent), the number of surrounding vehicles (67 percent) and compliance with traffic rules (29 percent) as the most important factors. Speed, complexity, type of road user (17 percent each) and stopping of vehicles (4 percent) are less important for their scoring.
\subsection{Comparison with Existing Approaches}
After the correlation between interaction score and human perception has been shown, a comparison of the interaction score is made with a previously proposed framework for the evaluation of trajectory-based datasets. For this purpose, the so-called \textit{INTERACTION} framework is chosen~\cite{INTERACTION_dataset} because it is the most comprehensive one focusing on the interaction in trajectory datasets from bird-eye's view. Besides this metric approach, there is no other method used for dataset evaluation. Compared to our proposed framework, the INTERACTION scheme uses a more simplistic approach since it focuses only on metrics (WP and $\Delta$TTCP) and does not link them to create a comprehensive analysis of road users. For that reason, this scheme can not be compared to the study in chap.~\ref{sec:interaction_score_study} since the study participant were asked to rate the interaction of road users and not bilateral relations. So, the INTERACTION scheme is only compared to our proposed interaction score within a representative sample of usual intersections. The selected recording (\textit{inD 21}) was taken at the intersection \textit{Frankenberg} (see Fig.~\ref{fig:infrastructure_frankenberg}) and contains 550 trajectories derived from a 15 minute long video. Since WP and $\Delta$TTCP are defined punctually, only the punctual interaction score is compared. As a result, it can be observed, that the interaction score of the new framework detects more interactive vehicles and more interaction pairs. This is mostly caused by the detection of following scenarios missed by the two individual metrics. Additionally, the interaction score describes the scenarios much more in detail by observing its environment comprehensively and combining different relation indicators (Tab.~\ref{table:scope_INTERACTION_comparison}).
\begin{table}[h]
\caption{Scope of interaction evaluation in record \textit{inD 21}}
\label{table:scope_INTERACTION_comparison}
\begin{center}
\setlength\tabcolsep{5.2pt}
\begin{tabular}{c|c|c|c}
\textbf{Method} & \textbf{Vehicles} & \textbf{Interacting pairs} & \textbf{Detections} \\
\hline
Interaction score & 407 & 676 & 21528 \\
INTERACTION scheme & 389 & 478 & 2642 \\
\end{tabular}
\end{center}
\end{table}
Furthermore, the characteristics of the detected interactions differ from those of the INTERACTION scheme. Thus, Fig.~\ref{fig:comparison_scope} shows significant differences between the interaction scores and the metrics translated into the scoring system of the framework. These are due to the more comprehensive evaluation of the environment made by the new framework. On the one hand, large deviations between interaction scores and metrics at low scores are mainly caused by the THW detection and by the number of vehicles involved. So, the interaction score does not oversimplify situations by using just individual metrics, but it takes multiple interacting vehicles into account. On the other hand, the differences at high interaction scores, mostly near misses and collisions, are caused by the TTC and DRAC detections. Because of those additionally used metrics, vehicles are not just evaluated more in-depth, but also the complexity of situations and a broader scope of vehicle relations are considered.
\begin{figure}[thpb]
\centering
\subfloat[Comparison Interaction score with $\Delta$TTCP]{
\includegraphics[width=0.45\linewidth]{graphics/fig9a.png}
\label{fig:comparison_dttcp}
}
\hspace{0.03\linewidth}
\subfloat[Comparison Interaction score with WP]{
\includegraphics[width=0.45\linewidth]{graphics/fig9b.png}
\label{fig:comparison_wp}
}
\caption{Comparison of the scope of the INTERACTION scheme (y-axis) and new framework (x-axis) including the scoring function of the metric within the interaction framework (red curve, cf. Tab.~\ref{table:detection_scoring}); interactions only found by the interaction score are marked as vertical gray lines}
\label{fig:comparison_scope}
\end{figure}
The more detailed evaluation can be illustrated by observing the temporal sequence of the interaction score of an individual vehicle (see Fig.~\ref{figure:comparison_dttcp_is_vehicle}). For this comparison, the $\Delta$TTCP detections are derived in each timestep and their scores are summed up for comparability. That has to be done since the $\Delta$TTCP is only determined for bilateral interaction between road users while the interaction score covers all interactions of a specific road user. The diagram shows not only the more detailed course of the interaction score according to the continuously changing traffic constellations but also a time shift of the global maximum compared to the summed $\Delta$TTCP and additional local maxima. In this example, the shift is a result of the different observation spaces. A large number of interactions cannot be covered by the $\Delta$TTCP, whereas these are included in the interaction score through additional detections. This can cause a different interpretation and leads to a more comprehensive picture.
While the scores may partially correlate depending on the infrastructure, the shift and score in the last section of the trajectory are caused by a preceding object that is not recognized by the $\Delta$TTCP in this specific example. However, this detected road user in front of the observed vehicle limits its freedom of movement and has to be considered in a comprehensive interaction scoring. So, the maximum of the interaction score is more accurate and fits the human perception.
\begin{figure}[thpb]
\centering
\includegraphics[width=\linewidth]{graphics/fig10}
\caption{Comparison of $\sum 1 / \Delta$TTCP and punctual interaction score for one vehicle along its trajectory}
\label{figure:comparison_dttcp_is_vehicle}
\end{figure}
As shown, the new framework considers the environment especially in complex scenarios more comprehensively and allows a detailed evaluation of the interaction along the road user trajectories. That mostly includes following scenarios as well as the combination of different relations with multiple traffic participants. These often occur when there is an accumulation of road users and various regulatory elements. Relatively similar coverage and evaluation are reached for simpler traffic scenarios such as the interaction of only two traffic participants at an intersection.
A similar conclusion applies to the rating of entire datasets because it can be seen as a sum of its trajectories. While a summation discretized into fixed time gap intervals is used in~\cite{INTERACTION_dataset} for $\Delta$TTCP, the new framework provides a single value to compare a dataset with others. The hierarchical structure of the new method not only makes a comparison simpler, but also provides detailed information along with several layers with different degrees of abstraction that can be used in a much more detailed way than the compared method using two single metrics.
\subsection{Utilization of the Framework}
Beyond the interaction score, the framework includes two other scores to support the user to find anomalies and relevant situations in a dataset. These functions are not considered in previous dataset evaluations. Therefore, they can not be compared to the INTERACTION- or other schemes since the target to score anomaly and relevance in the scope of vehicles, regions and datasets is unique. However, these scores are essential for a comprehensive assessment within the framework. On the lowest level, the punctually defined anomaly score offers two decisive advantages: overview over unusual situations and comprehensive error detection. Since the anomaly score is based on all fourteen detection types rather than just a few like the interaction score, it gives a more comprehensive view of the traffic situation. This is particularly clear at the punctual level. The highest anomaly scores contain unusual behavior patterns (see Fig.~\ref{fig:buggy_anomaly}), which are highly relevant in the context of safeguarding automated driving functions. The scores are normally significantly impacted by the clustering and infrastructure-related detections because they often occur together. For example, an unusual trajectory often correlates with locally unusual behavior. Nevertheless, the situations found are not always unusual subjectively. If just a single vehicle is driving into a parking bay, it is highly abnormal within the context of the record. If a human anticipates the driver's behavior and includes this knowledge into its evaluation of the situation, the situation might seem usual for him. Anyways, the scoring makes sense from an unbiased point of view when considering the actions within the given record. This gap between objective and subjective evaluation is not necessarily bad since it shows the characteristic of the dataset and creates awareness for rare scenarios. Of course, it is up to the user to subsequently filter out unwanted scenarios such as parking vehicles, considering the given use case.
In addition to behavior detection, errors can be identified as anomalies. Whether it is a bird that has been assigned as a pedestrian (see Fig.~\ref{fig:bird_as_pedestrian}) or a wrongly detected vehicle.
\begin{figure}[thpb]
\centering
\subfloat[Anormal behavior of a car $\mbox{\text{(orange)}}$ passing street, crosswalk and walkway]{
\includegraphics[width=0.45\linewidth]{graphics/fig11a.png}
\label{fig:buggy_anomaly}
}
\hspace{0.03\linewidth}
\subfloat[Flight route of a bird (orange) detected as a trajectory of a pedestrian]{
\includegraphics[width=0.45\linewidth]{graphics/fig11b.PNG}
\label{fig:bird_as_pedestrian}
}
\caption{Examples for highly anomal scored situations.}
\label{fig:anomaly_situations}
\end{figure}
Applying the framework on the \textit{inD} dataset and filtering the data for the most unusual scenarios one out of ten scenarios is caused by false image object detection. Two others include vehicles driving on designated pedestrian crosswalks perpendicular to the normal driving direction (see Fig.~\ref{fig:buggy_anomaly}). One vehicle does a U-turn, two performing parking maneuvers, another vehicle drives in the middle of two lanes and in a next situation one does a left turn resulting in driving in the middle of two lanes, too. Furthermore, two other situations show pedestrians crossing large roads without using designated crosswalks. So, as already shown above, the anomaly score mainly highlights exceptional situations, but also includes scenarios that are not rare generally but in the observed data.
Having a look at the ten most interactive scenarios gives a less diverse picture. Thereby, four situations are cluttered and contain each more than seven interacting road users. Two situations are assigned to wrong detected collision and two more to near misses. The last two are pedestrian-related and include disembarking from a bus and bypassing a construction site. Whereas the groups of pedestrians might be less relevant for automated driving functions, especially complex and near-miss scenarios can help to validate and optimize those functions heavily.
After having a look at anomaly and interaction, the relevance score is elaborated as the product of it.
Although it is composed of both scores, the highest-rated interaction or anomaly situations do not necessarily lead to high relevance scores. Instead, a high relevance score is mainly driven by the interplay of multiple vehicles and complex situations.
Due to the multiplication of predetermined scores, its values are subject to a higher error tolerance than the uncombined ones: Because of the higher sensitivity to accidents, errors in vehicle detection can have a big impact on the relevance score. If vehicle boxes overlap with others or are wrongly assigned to regions, the detection may lead to a virtual collision or the occurrence of other detections.
Therefore, individual situations can require manual review due to errors in datasets. On more abstract levels, the proposed approaches are more resistant to dataset errors or weaknesses of single metrics so that they can provide reliable values for records.
As explained below especially the relevance score can help to find errors in datasets. So, four out of ten of the most relevant situations of the inD dataset are caused by wrong detections and pseudo collisions, three are related to interesting parking maneuvers and the others can be assigned to complex scenarios on intersections, near misses and a vehicle on a sidewalk. Depending on the use case, those scenarios are more or less relevant. Although the wrong detections might seem useless, they are especially useful for publishers of datasets to check for the quality of their data. Besides that, the punctual relevance score, as well as anomaly and interaction, show different and diverse situations helping a user to apply a variety of situations to driving functions and getting an overview about interesting situations in the dataset.
Beyond the detection of individual situations, more abstract statements can be made with all three scores. Information about the general traffic situation can be obtained by looking at the regional distribution of scores. This distribution not only makes abnormal behavior visible in each location, but also allows conclusions to be drawn about infrastructure elements. In Fig.~\ref{fig:anomaly_map}, it is evident that the non-signed crosswalk over the left arm of the intersection has a significantly higher anomaly score compared to the signed crosswalk over the right arm. This finding could lead to the hypothesis that regulatory elements prevent anomalous crossings. With this kind of analysis, it is possible to draw conclusions about individual elements and then select infrastructures for future recordings according to the underlying requirements of e.g. scenario-based testing. This selection can be supported not only by the three scores themselves but additionally also by their composition of belonging detection types.
\begin{figure}[thpb]
\centering
\includegraphics[width=0.9\linewidth]{graphics/fig12.png}
\caption{Heatmap of anomaly region scores of \textit{inD} 21}
\label{fig:anomaly_map}
\end{figure}
Thus, the combination of the three scores also provides an overview of the level of entire trajectory datasets and allows a comparison of them (see Fig.~\ref{fig:dataset_comparison}). Comparing interaction and relevance scores in over 200 recordings, it can be observed, that interaction and relevance are highly linked for individual road users in the given recordings. Because of the high traffic density, unusual behavior causes further interaction with other road users.
While the three scores are similar for different recordings at the same location due to similar road user behavior, they significantly differ between different location sites and different datasets. Across the observed datasets, it can be observed that rounD might suit well for testing complex multi-road user scenarios due to their high interaction. If a user would be more interested in unusual road user behavior, the inD dataset and especially the intersection \textit{Frankenberg} might fit better. This kind of analysis can be used to select appropriate recordings for the individual application and to draw conclusions about the driving behavior in different locations in order to plan future datasets recordings according to one's own needs.
\begin{figure*}[t]
\centering
\subfloat{
\includegraphics[width=0.3\linewidth]{graphics/fig13a.png}
\label{fig:dataset_overview_interaction}
}
\hspace{0.02\linewidth}
\subfloat{
\includegraphics[width=0.3\linewidth]{graphics/fig13b.png}
\label{fig:dataset_overview_anomaly}
}
\hspace{0.02\linewidth}
\subfloat{
\includegraphics[width=0.3\linewidth]{graphics/fig13c.png}
\label{fig:dataset_overview_relevance}
}
\caption{Comparison of overview scores for urban records of \textit{inD}~\cite{inD}, \textit{rounD} (red pentagons)~\cite{rounD} and \textit{INTERACTION} (light blue hexagons)~\cite{INTERACTION_dataset} with detailed view on \textit{inD} infrastructures (\textit{Aseag}: deep blue circles, \textit{Bendplatz}: yellow squares, \textit{Frankenberg}: green diamonds and \textit{Heckstrasse}: gray triangle)}
\label{fig:dataset_comparison}
\end{figure*}
Thus, the framework offers several possibilities for the analysis of trajectory-based trajectory datasets. As the framework utilizes a set of weights for combining the individual scores, use case specific adaptation is possible and necessary for best results. Furthermore, a graphical interface can help both, to get an quick overview as well as analyzing scores quickly and in-depth.
\section{CONCLUSIONS}
In this paper, an automated analysis framework for urban trajectory-based datasets from a bird's eye view was presented. Both, existing and newly derived metrics were combined with approaches of anomaly detection and clustering to take vehicle constellations and driving behavior into account. Based on these fourteen traffic detection types, interaction, anomaly and relevance of the data were determined on different hierarchical levels. Thus, the framework showed a high correlation between the interaction score and human perception of interaction. Mostly, the framework reflects this perception better than asking a single expert. It remains part of future research to find study designs that allow validation also for the anomaly- and relevance score.
The fact of outperforming individual experts and the complete automation of the framework allows different datasets to be compared in-depth. This can be done in more detail and with more clarity than with existing approaches.
The proposed method brings not only a benefit for users of datasets but also for publishers, as they can draw conclusions for future recordings.
Although the evaluation framework has been optimized for urban scenarios in its current form, the modular design of the detections allows adaptations to further use cases so that the framework can be extended.
|
1,314,259,993,023 | arxiv | \section{Introduction}
The symmetric tensor space $\mathrm{Sym}^dV$, with $V=\R^2$ (resp. $V=\C^2$), contains real (resp. complex) binary forms,
which are homogeneous polynomials in two variables. The forms which can be written as $v^d$, with $v\in V$, correspond to polynomials which are the $d$-power of a linear form,
they have rank one.
We denote by $C_d\subset \mathrm{Sym}^dV$ the variety of forms of rank one.
The $k$-secant variety $\sigma_k(C_d)$ is the closure of the set of forms which can be written as $\sum_{i=1}^k\lambda_iv_i^d$ with $\lambda_i\in\R$ (resp. $\lambda_i\in\C$).
We say that a nonzero rank $1$ tensor is a critical rank one tensor for $f\in \mathrm{Sym}^d V$ if it is a critical point of the distance function from $f$ to the variety of rank $1$ tensors.
Critical rank one tensors are important to determine the best rank one approximation of $f$, in the setting of optimization
\cite{FriTam, Lim, Stu}. Critical rank one tensors may be written as $\lambda v^d$ with $\lambda\in\C$ and $v\cdot v=1$, the last scalar product is the Euclidean scalar product.
The corresponding vector $v\in V$ has been called tensor eigenvector, independently by Lim and Qi, \cite{Lim, Qi}. In this paper we concentrate on
critical rank one tensors $\lambda v^d$, which live in $\mathrm{Sym}^dV$ (not in $V$ like the eigenvectors), for a better comparison with critical rank $k$ tensors, see Definition \ref{def:kcritical} .
There are exactly $d$ critical rank one tensor (counting with multiplicities)
for any $f$ different from $c(x^2+y^2)^{d/2}$ (with $d$ even), while
there are infinitely many critical rank one tensors for $f=(x^2+y^2)^{d/2}$ (see Prop. \ref{prop:eigendisc}).
The critical rank one tensors for $f$ are contained in the hyperplane $H_f$
(called the singular space, see \cite{OP}), which is orthogonal to the vector $D(f)=yf_x-xf_y$. We review this statement at the beginning of \S \ref{sec:singularspace}.
The main result of this paper is the following extension of the previous statement to critical rank $k$ tensors, for any $k\ge 1$.
\begin{thm}\label{thm:main} Let $f\in \mathrm{Sym}^d\C^2$ .\hfill
i) All critical rank $k$ tensors for $f$ are contained in the hyperplane $H_f$, for any $k\ge 1$.
ii) Any critical rank $k$ tensor for $f$ may be written as a linear combination of the critical rank $1$ tensors for $f$.
\end{thm}
Theorem \ref{thm:main} follows after Theorem \ref{mainTheorem} and Proposition \ref{prop:main2}. Note that Theorem \ref{thm:main} may applied to the best rank $k$ approximation of $f$, which turns out to be contained in $H_f$
and may then be written as a linear combination of the critical rank $1$ tensors for $f$. This statement may be seen as a weak extension of the Eckart-Young Theorem to tensors. Indeed, in the case of matrices, the best rank $k$ approximation is exactly the sum of the first $k$ critical rank one tensors, by the Eckart-Young Theorem, see \cite{OP}. The polynomial $f$ itself may be written as linear combination of its critical rank $1$ tensors, see Corollary \ref{cor:corf}, this statement may be seen as a {\it spectral decomposition for $f$}. All these statements may be generalized to the larger class of tensors, not necessarily symmetric, in any dimension, see \cite{DOT}.
In \S\ref{sec:lastreal} we report about some numerical experiments regarding the number of real critical rank $2$ tensors in $\mathrm{Sym}^4\R^2$.
\section{Preliminaries}
Let $V=\R^2$ equipped with the Euclidean scalar product.
The associated quadratic form has the coordinate expression $x^2+y^2$,
with respect to the orthonormal basis $x, y$.
The scalar product can be extended to a scalar product on the tensor space $\mathrm{Sym}^dV$ of binary forms,
which is $SO(V)$-invariant.
For powers $l^d$, $m^d$ where $l, m\in V$, we set
$\langle l^d, m^d\rangle : = \langle l, m\rangle^d$
and by linearity this defines the scalar product on the whole $\mathrm{Sym}^dV$ (see Lemma \ref{lema:scalarproduct}).
Denote as usual $\left\|{f}\right\|=\sqrt{\langle f, f\rangle}$.
For binary forms which split in the product of linear forms we have the formula
\begin{equation}\label{eq:decomp}\langle l_1l_2\cdots l_d, m_1m_2\cdots m_d\rangle =
\frac{1}{d!}\sum_{\sigma}\langle l_1,m_{\sigma(1)}\rangle \langle l_2,m_{\sigma(2)}\rangle
\cdots \langle l_d,m_{\sigma(d)}\rangle \end{equation}
The powers $l^d$ are exactly the tensors of rank one in $\mathrm{Sym}^dV$,
they make a cone $C_d$ over the rational normal curve.
The sums $l_1^d+\ldots +l_k^d$ are the tensors of rank $\le k$, and equality holds when the number of summands is minimal. The closure of the set of tensors of rank $\le k$, both in the Euclidean or in the Zariski topology, is a cone $\sigma_kC_d$,
which is the $k$-secant variety of $C_d$.
The Euclidean distance function $d(f,g)=\left\|f-g\right\|$ is our objective function.
The optimization problem we are interested is, given a real $f$, to minimize $d(f,g)$ with the
constraint that $g\in \left(\sigma_kC_d\right)_\R$. This is equivalent to minimize the square function $d^2(f,g)$,
which has the advantage to be algebraic. The number of complex critical points of the square distance function $d^2$
is called the Euclidean distance degree (EDdegree \cite{DHOST}) of $\sigma_kC_d$ and has been computed for small values of
$k, d$ in the rightmost chart in Table 4.1 of \cite{OSS}. We do not know a closed formula for these values,
although \cite[Theorem 3.7]{OSS} computes them in the case of a general quadratic distance function, not
$SO(2)$-invariant.
\section{Critical points of the distance function}
Let us recall the notion of eigenvector for symmetric tensors (see \cite{Lim, Qi},\cite[Theorem 4.4]{OP}).
\begin{defn}\label{def:eigentensor}
Let $f\in \mathrm{Sym}^d V$. We say that a nonzero rank $1$ tensor is a critical rank one tensor for $f$ if it is a critical point of the distance function from $f$ to the variety of rank $1$ tensors. It is convenient to write a critical rank one tensor in the form $\lambda v^d$ with $\left\|{v}\right\|=1$, in this way
$v$ is defined up to $d$-th roots of unity and is called an eigenvector of $f$ with eigenvalue $\lambda$.
\end{defn}
\begin{remark}
Let $d=2$ and let $f$ be a symmetric matrix.
All the critical rank one tensors of $f$ have the form $\lambda v^2$ where $v$ is a classical eigenvector of norm $1$ for the symmetric matrix $f$, with eigenvalue $\lambda$.
\end{remark}
\begin{lem}\label{lem:eigentensor}
Given $f\in\mathrm{Sym}^d V$, the point $\lambda v^d$ of rank $1$, with $\left\|{v}\right\|=1$, is a critical rank one tensor for $f$ if and only if $\langle f,v^{d-1}w\rangle=
\lambda \langle v,w\rangle$ $\forall w\in V$, which can be written (identifying $V$ with $V^\vee$ according to the Euclidean scalar product) as
$$f\cdot v^{d-1}= \lambda v,$$ with $\lambda=\langle f, v^d\rangle $.
\end{lem}
\begin{proof} The property of critical point is equivalent to $f-\lambda v^d$ being orthogonal to
$v^{d-1}w$ $\forall w\in V$, which gives
$\langle f, v^{d-1}w\rangle =\langle \lambda v^d,v^{d-1}w\rangle$
$\forall w\in V$. The right-hand side is $\left\|{v}\right\|^{2d-2} \lambda \langle v,w\rangle=\lambda \langle v,w\rangle$, as we wanted.
Setting $w=v$ we get $\langle f, v^{d}\rangle =\lambda$.
\end{proof}
On the other hand, eigenvectors correspond to critical points of the function
$f(x,y)$ restricted on the circle $S^1=\{(x,y)|x^2+y^2=1\}$ (\cite{Lim, Qi}).
By Lagrange multiplier method, we can compute the eigenvectors of $f$ as the normalized solutions $(x,y)$ of:
\begin{equation}\label{eq1}
\mathrm{rank}
\begin{bmatrix}
f_{x} & f_{y} \\
x & y
\end{bmatrix} \leq 1
\end{equation}
This corresponds with the roots of discriminant polynomial $D(f)=yf_x-xf_y$.
$D$ is a well known differential operator which satisfies the Leibniz rule, i.e. $D(fg)=D(f)g+fD(g)$
$\forall f, g\in \mathrm{Sym}^d V$.
For any $l=ax+by\in V$ denote $l^\perp=D(l)=-bx+ay$. Note that $\langle l,l^\perp\rangle =0$.
We have the following:
\begin{prop}\label{prop:eigendisc} Consider $f(x,y)\in\mathrm{Sym}^dV$:
\begin{itemize}
\item If $v$ is eigenvector of $f$ then $D(v)=v^\perp$ is a linear factor of $D(f)$.
\item Assume that $D(f)$ splits as product of distinct linear factors and $v^\perp|D(f)$, then $\frac{v}{\left\|{v}\right\|}$ is an eigenvector of $f$.
\end{itemize}
\end{prop}
We postpone the proof after Prop. \ref{prop:critical}.
Now let us differentiate some cases in terms of $D(f)$ (see Theorem $2.7$ of \cite{ASS}):
\begin{itemize}
\item if $d$ is odd: $D(f)= 0$ if and only if $f= 0$, in particular $D:\mathrm{Sym}^dV\rightarrow \mathrm{Sym}^dV$ is an isomorphism.
\item if $d$ is even: $D(f)=0$ if and only if $f=c(x^2+y^2)^{d/2}$ for some $c\in\R$. We will show in Lemma \ref{lemma2} which are the eigenvectors in this case. The image of $D:\mathrm{Sym}^dV\rightarrow \mathrm{Sym}^dV$ is the space orthogonal to $f=(x^2+y^2)^{d/2}$.
\end{itemize}
\begin{lem} (\cite{LS}, Section $2$)\label{lema:scalarproduct}
Suppose $f=\sum_{i=0}^{d} \binom{d}{i} a_i x^iy^{d-i}$ and $g=\sum_{i=0}^{d} \binom{d}{i} b_i x^iy^{d-i}$. Then we get:
\begin{equation}\label{eq:scalar}
\langle f,g\rangle:=\sum_{i=0}^{d} \binom{d}{i} a_ib_i
\end{equation}
where $\langle\,,\,\rangle$ is the scalar product defined in the introduction.
\end{lem}
\begin{proof}
By linearity we may assume $f=(\alpha x+\beta y)^d$ and $g=(\alpha'x+\beta' y)^d$. The right-hand side of (\ref{eq:scalar}) gives
$$\langle f,g\rangle=\sum_{i=0}^{d} \binom{d}{i}(\alpha\alpha')^i(\beta \beta')^{d-i}=(\alpha\alpha'+\beta\beta')^d$$
which agrees with $\langle \alpha x+\beta y,\alpha' x+\beta' y\rangle^d$.
\end{proof}
\begin{lem}\label{remark:2}
Let $f=(x^2+y^2)^{d/2}\in \mathrm{Sym}^d V$ with $d$ even, and $v=\alpha x+\beta y\in V$, $v\neq 0$, then $\langle v^d,f\rangle=\left\|{v}\right\|^d$.
\end{lem}
\begin{proof}
By applying (\ref{eq:decomp}) with a grain of salt (e.g. decomposing $x^2+y^2$
into two conjugates linear factors) we get
$$\langle v^d,f\rangle=\langle (x^2+y^2),v^2\rangle^{d/2} =
(\alpha^2+\beta^2)^{d/2}=\left\|{v}\right\|^d.$$
\end{proof}
\begin{lem}\label{lemma2}
If $f=(x^2+y^2)^{d/2}\in \mathrm{Sym}^d V$ then, for every nonzero $v\in V$,
$\langle f, v^{d-1}w\rangle=\left\|{v}\right\|^{d-2}\langle v, w\rangle$.
In particular every vector $v$ of norm $1$ is eigenvector of $f$ with eigenvalue $1$.
\end{lem}
\begin{proof}
As in Lemma \ref{remark:2} we get
$$\langle f, v^{d-1}w\rangle = {\langle (x^2+y^2),v^2\rangle}^{d/2-1}
\langle (x^2+y^2),vw\rangle = \left\|{v}\right\|^{d-2}\langle v, w\rangle.$$
The second part follows by putting $w=v$ and equating with Lemma \ref{remark:2}. We get
$\langle f, v^{d-1}w\rangle=\langle v^d,f\rangle\langle v, w\rangle$ just in the case $|v|=1$.
\end{proof}
\begin{remark}
Lemma \ref{lemma2} extends the fact that every vector of norm $1$ is eigenvector of the identity matrix with eigenvalue $1$.
The geometric interpretation of this lemma
is that the $2$-dimensional cone
of rank $1$ degree $d$ binary forms cuts any sphere centered in $(x^2+y^2)^{d/2}$
in a curve. This curve
\end{remark}
\begin{lem}\label{lem}
The normal space at $l^d\in C_d$ coincides with $\left(l^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$
\end{lem}
\begin{proof}
The tangent space at $l^d$ is spanned by $l^{d-1}V$ and has dimension $2$. The elements in $\left(l^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$ are orthogonal to the tangent space,
moreover the dimension of this space is the expected one $d-1$.
\end{proof}
\begin{defn}\label{def:kcritical}
We say that $g\in\mathrm{Sym}^dV$ is a critical rank $k$ tensor for $f$ if it is a critical point of the distance function $d(f,\_)$ restricted on $\sigma_kC_d$.
\end{defn}
\begin{prop}\label{prop:critical} Let $2k\le d$. A polynomial $g=\sum_{i=1}^k \mu_i l_i^d \in \sigma_kC_d$ is a critical rank $k$ tensor for $f$ if and only if there exist $h\in\mathrm{Sym}^{d-2k}V$ such that
\begin{equation}\label{eq}
f=\sum_{i=1}^k \mu_i l_i^d +h\cdot \prod_{i=1}^k\left(l_i^\perp\right)^2
\end{equation}
\end{prop}
\begin{proof}
By Terracini Lemma, the tangent space of the point $g\in\sigma_kC_d$ is given by the sum of $k$ tangent spaces at $l_i^d=(a_ix+b_iy)^d$. By Lemma \ref{lem} the normal space of each of these tangent spaces are given by $\left(l_i^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$. Hence, the normal space to $g$ is given by intersection of the $k$ normal spaces, which is given by
polynomials $\prod_{i=1}^k\left(l_i^\perp\right)^2 \cdot h$ where $h\in\mathrm{Sym}^{d-2k}V$.
Now suppose that $g$ is a critical rank $k$ tensor for $f$. This means that $f-g$ is in the normal space. Hence, $f-g$ is of the form $\prod_{i=1}^k\left(l_i^\perp\right)^2 \cdot h$ for some $h\in\mathrm{Sym}^{d-2k}V$.
Conversely, if $(\ref{eq})$ holds, we need that $f-g$ belongs to the normal space at $g$ which is also true by the construction of the normal space.
\end{proof}
\begin{proof} [Proof of Prop. \ref{prop:eigendisc}]
If $v$ is eigenvector of $f$ then $\langle f,v^d\rangle v^d$ is critical rank $1$ tensor for $f$ (by Lemma \ref{lem:eigentensor}). By Prop. \ref{prop:critical}
$f=\langle f,v^d\rangle v^d+h \left(v^\perp\right)^2$ where $h\in\mathrm{Sym}^{d-2}V$. Applying the operator $D$ to $f$ we get by Leibniz rule, since $D(v)=v^{\perp}$ and $D(v^{\perp})=-v$:
$$D(f)=\langle f,v^d\rangle dv^{d-1}v^\perp+D(h)\left(v^\perp\right)^2-2vv^\perp h\Longrightarrow v^\perp|D(f)$$
Conversely, since we assume there are $d$ distinct eigenvectors, then we find all the linear factors of $D(f)$.
\end{proof}
This proposition is connected with Theorem $2.5$ of \cite{LS}.
\section{The singular space}\label{sec:singularspace}
In \cite{OP} it was considered the singular space $H_f$ as the hyperplane orthogonal to $D(f)=yf_x-xf_y$. It follows from Prop. \ref{prop:eigendisc}
that the critical rank $1$ tensor for $f$ belong to $H_f$ (since the eigenvectors of $f$ can be computed as the solutions of (\ref{eq1}) that coincides with $D(f)$ for binary forms), see \cite[Def. 5.3]{OP}.
It is worth to give a direct proof that the critical rank $1$ tensors for $f$ belong to $H_f$, the hyperplane orthogonal to $D(f)$,
based on Prop.
\ref{prop:critical}.
Let $\mu l^d$ be a critical rank $1$ tensors for $f$, then by Prop.
\ref{prop:critical} there exist $h\in\mathrm{Sym}^{d-2}V$ such that
$f= \mu l^d +h\left(l^\perp\right)^2$.
We have to prove $\langle D(f), l^d\rangle =0$ which follows immediately from (\ref{eq:decomp})
since $l^{\perp}$ divides $D(f)$ by Prop. \ref{prop:eigendisc}.
\begin{lem}\label{lem:lmperp}
Let $l, m\in V$, Then $\langle l^\perp, m\rangle +\langle m^\perp, l\rangle =0$.
\end{lem}
\begin{proof}
Straightforward.
\end{proof}
Our main result is
\begin{thm}\label{mainTheorem}
The critical points of the form $\sum_{i=1}^{k}\mu_i l_i^d$ of the distance function $d(f,-)$ restricted on $\sigma_kC_d$
belong to $H_f$.
\end{thm}
\begin{proof}
Given a decomposition
$f= \sum_{i=1}^k \mu_i l_i^d +h\cdot \prod_{i=1}^k\left(l_i^\perp\right)^2$, with $h\in\mathrm{Sym}^{d-2k}V$,
we compute
\begin{equation}\label{eq:3sum}
D(f)=d\sum_{i=1}^k \mu_il_i^{\perp}l_i^{d-1}-\sum_{i=1}^k2l_il_i^{\perp}\prod_{j\neq i}^k\left(l_j^\perp\right)^2h+D(h)\prod_{i=1}^k\left(l_i^\perp\right)^2\end{equation}
and we have to prove \begin{equation}\label{eq:dfli}\langle D(f),\sum_{j=1}^k l_j^d\rangle =0.\end{equation}
We compute separately the contribution of the three summands in (\ref{eq:3sum}) to the scalar product with $l_j^d$.
We have for the first summand
$$\langle \left(\sum_{i=1}^kl_i^{\perp}l_i^{d-1}\right), l_j^d\rangle = \sum_{i=1}^k\langle l_i^\perp, l_j\rangle\langle l_i\cdot l_j\rangle^{d-1}$$
Summing over $j$ we get zero by Lemma \ref{lem:lmperp}.
We have for the second summand
$$\langle\left(\sum_{i=1}^kl_i,l_i^{\perp}\prod_{p\neq i}^k\left(l_p^\perp\right)^2h\right),l_j^d \rangle =
\langle\left(l_jl_j^{\perp}\prod_{p\neq j}^k\left(l_p^\perp\right)^2h\right),l_j^d \rangle=0 $$
We have for the third summand
$$\langle\left(D(h)\prod_{i=1}^k\left(l_i^\perp\right)^2\right), l_j^d\rangle = 0$$
Summing up, this proves (\ref{eq:dfli}) and then the thesis.
\end{proof}
\begin{example}
If $f=x^3y+2y^4$ then there are $6$ critical points of the form $l_1^4+l_2^4$ and $x^3y$ which lies on the tangent line at $x^4$. It cannot be written as $l_1^4+l_2^4$ and indeed it has rank $4$.
\end{example}
\section{The scheme of eigenvectors for binary forms}
Suppose $f\in \mathrm{Sym}^d V$ a symmetric tensor and $\mathrm{dim} V=2$. We denote by $Z$ the scheme defined by the polynomial $D(f)$, embedded in $\Pe(\mathrm{Sym}^d V)$ by the $d$-Veronese embedding in $\Pe V$ (see \cite{AEKP} for the case of matrices).
\begin{prop}\label{prop:main2}
$\langle Z \rangle = H_f$.
\end{prop}
\begin{proof}
$(i)$ If $D(f)$ has $d$ distinct roots then it is known that $\langle Z \rangle\subseteq H_f$, since $H_f$ is the hyperplane orthogonal to $D(f)$ (Theorem \ref{mainTheorem} with $k=1$). Hence $\langle Z \rangle\subseteq H_f$.
$(ii)$ Now let us suppose that $D(f)$ has multiple roots but $f\neq (x^2+y^2)^{d/2}$. We show that $\langle Z\rangle\subseteq H_f$ by a limit argument. For every tensor $f$ such that $f\neq 0$ and $f\neq(x^2+y^2)^{d/2}$ there exists a sequence $(f_n)$ such that $f_n\rightarrow f$ and $D(f_n)$ has distinct roots for all $n$. Then, $H_{f_n}\rightarrow H_f$ because the differential operator is continuous. Moreover $H(f_n)$ is a hyperplane for all $n$. On the other hand, by definition we have that $\langle Z_{f_n}\rangle$ is the spanned of the roots of $D(f_n)$. When $f_n$ goes to the limit we get that $\langle Z_{f_n}\rangle\rightarrow \langle Z\rangle$. Hence, $\langle Z\rangle\subseteq H_f$.
$(iii)$ In the case that $f=(x^2+y^2)^{d/2}$ with $d$ even, then by Lemma \ref{lemma2} we know that every unitary vector is an eigenvector and $H_f$ is the ambient space. Hence, $\langle Z\rangle=H_f$.
We prove now that $\mathrm{dim} \langle Z \rangle=\mathrm{dim} H_f$ for $(i)$ and $(ii)$.
Since $\mathcal{I}_{Z,\Pe^1}=\mathcal{O}_{\Pe^1}(-d)$,
$$\mathrm{codim}\langle Z\rangle=h^0(\mathcal{I}_{Z,\Pe^1}(d))=h^0(\mathcal{O}_{\Pe^1}(-d+d))=h^0(\mathcal{O}_{\Pe^1})=1$$
which coincides with the codimension of $H_f$.
\def\niente{
First we suppose that $D(f)$ has $d$ distinct roots.
It is known that $\langle Z \rangle\subseteq H_f$, since $H_f$ is the hyperplane orthogonal to $D(f)$.
We prove that the equality holds by showing that $\mathrm{dim} \langle Z \rangle=\mathrm{dim} H_f$.
If we embedded the $d$ distinct eigenvectors $v_1,\ldots,v_d$ into the rational normal curve of degree $d$, $C_d$, it turns out $d$ independent elements $v_1^d,\ldots,v_d^d$. The embedding of each of the eigenvectors in $C_d$ is of the form:
$$(1:v_i)\mapsto (1:v_i:v_i^2:\ldots:v_i^d)\quad i=1,\ldots,d$$
We know that these points are independent by using \textit{Vandermonde determinant} since
\begin{equation*}
\mathrm{det}
\begin{bmatrix}
1 & v_1 & v_1^2 & \ldots & v_1^d\\
1 & v_2 & v_2^2 & \ldots & v_2^d\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_d & v_d^2 & \ldots & v_d^d
\end{bmatrix}
=\prod_{i<j}(v_j-v_i)\neq 0
\end{equation*}
Hence, the dimension of each of the spaces are equal.
Let us suppose that $D(f)$ has multiple roots but $f\neq (x^2+y^2)^{d/2}$. First we show that $\langle Z\rangle\subseteq H_f$ by a limit argument. For every tensor $f$ such that $f\neq 0$ and $f\neq(x^2+y^2)^{d/2}$ there exists a sequence $(f_n)$ such that $f_n\rightarrow f$ and $D(f_n)$ has distinct roots for all $n$. Then, $H_{f_n}\rightarrow H_f$ because the differential operator is continuous. Moreover $H(f_n)$ is a hyperplane for all $n$. On the other hand, by definition we have that $\langle Z_{f_n}\rangle$ is the spanned of the roots of $D(f_n)$. When $f_n$ goes to the limit we get that $\langle Z_{f_n}\rangle\rightarrow \langle Z\rangle$. Hence, $\langle Z\rangle\subseteq H_f$.
Now we prove the other inclusion. Suppose that we have $r$ distinct eigenvectors $v_1,\ldots,v_r$ with multiplicities $m_1,\ldots,m_r$ and $m_1+\ldots +m_r=d$.
If we embedded the $r$ distinct eigenvectors into the rational normal curve of degree $d$, $C_d$, it turns out $r$ independent elements $v_1^d,\ldots,v_r^d$. The embedding of each of the eigenvectors in $C_d$ is of the form:
$$(1:v_i)\mapsto (1:v_i:v_i^2:\ldots:v_i^{d-1}:v_i^d)\quad i=1,\ldots,r$$
We consider also its derivatives:
$$(1:v_i)\mapsto (0:1:2v_i:\ldots:(d-1)v_i^{d-2}:dv_i^{d-1})$$
$$(1:v_i)\mapsto (0:0:2:\ldots:(d-1)(d-2)v_i^{d-3}:d (d-1)v_i^{d-2})$$
$$\vdots$$
$$(1:v_i)\mapsto (0:0:0:\ldots:\binom{d-1}{m_i-1}v_i^{d-m_i}:\binom{d}{m_i-1}v_i^{d+1-m_i})$$
We know that these points are independent by using the \textit{Confluent Vandermonde determinant} since,
\begin{equation*}
\mathrm{det}
\begin{bmatrix}
1 & v_1 & v_1^2 & v_1^3 & \ldots & v_1^{d-1} & v_1^d\\
0 & 1 & 2v_1 & 3 v_1^2 & \ldots & (d-1)v_1^{d-2} & dv_1^{d-1}\\
0 & 0 & 2 & 6 v_1 & \ldots & (d-1)(d-2)v_1^{d-3} & d (d-1)v_1^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_2 & v_2^2 & v_2^3 & \ldots & v_2^{d-1} & v_2^d\\
0 & 1 & 2v_2 & 3 v_2^2 & \ldots & (d-1)v_2^{d-2} & dv_2^{d-1}\\
0 & 0 & 2 & 6 v_2 & \ldots & (d-1)(d-2)v_2^{d-3} & d (d-1)v_2^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_r & v_r^2 & v_r^3 & \ldots & v_r^{d-1} & v_r^d\\
0 & 1 & 2v_r & 3 v_r^2 & \ldots & (d-1)v_r^{d-2} & dv_r^{d-1}\\
0 & 0 & 2 & 6 v_r & \ldots & (d-1)(d-2)v_r^{d-3} & d (d-1)v_r^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots
\end{bmatrix}
=\prod_{1\leq i<j\leq r}(v_j-v_i)^{m_i m_j}\neq 0
\end{equation*}
Hence, the dimension of each of the spaces are equal.
Finally, if $f=(x^2+y^2)^{d/2}$ with $d$ even, then by Lemma \ref{lemma2} we know that every nonzero vector is an eigenvector and $H_f$ is the ambient space. Hence, $\langle Z\rangle=H_f$.}
\end{proof}
As a consequence we obtain the following corollary, which may be seen
as a {\it Spectral Decomposition} of any binary form $f$.
\begin{cor}\label{cor:corf}
Any binary form $f\in \mathrm{Sym}^d V$ with $\mathrm{dim} V=2$ can be written as a linear combination of the critical rank one tensors for $f$.
\end{cor}
The previous statement holds even in the special case $d$ even
and $f=(x^2+y^2)^{d/2}$, since from \cite[Theorem 9.5]{Rez} there exists $c_d\in{\mathbb R}$ such that the following decomposition holds $\forall\phi\in{\mathbb R}$
$$(x^2+y^2)^{d/2}=c_d\sum_{k=0}^{d/2}\left[\cos(\frac{2k\pi}{d+2}+\phi)x+\sin(\frac{2k\pi}{d+2}+\phi)y\right]^d$$
In this decomposition the summands on the right-hand side correspond to
$(d+2)/2$ consecutive vertices of a regular $(d+2)$-gon.
In the $d=2$ case, the Spectral Theorem asserts any binary quadratic form $f\in\mathrm{Sym}^2\R^2$
can be written as sum of its rank one critical tensors. This statement fails for $d\ge 3$, as it can be checked already on the examples $f=x^d+y^d$ for $d\ge 3$,
where only two among the $d$ rank one critical tensors are used, namely $x^d$ and $y^d$, and the coefficients of the remaining $d-2$ rank one critical tensors
in the Spectral Decomposition of $f$ are zero.
\section{Real critical rank $2$ tensors for binary quartics}
\label{sec:lastreal}
We recall the following result by M. Maccioni.
\begin{thm}\label{thm:maccioni}(Maccioni, \cite[Theorem 1]{Mac})
Let $f$ be a binary form.
$$\# \text{ real roots of f }\leq\# \text{ real critical rank 1 tensors for\ } f$$
The inequality is sharp, moreover it is the only constraint between the number of real roots and the number of real critical rank $1$ tensors, beyond parity mod $2$.
\end{thm}
As a consequence, as it was first proved in \cite{ASS}, hyperbolic binary forms (i.e. with only real roots) have all real critical rank $1$ tensors.
We attempted to extend Theorem \ref{thm:maccioni} to rank $2$ critical tensors.
Our description is not yet complete and we report about some numerical experiments in the space $\mathrm{Sym}^4\R^2$. From these experiments it seems that the constraints
about the number of real rank $2$ critical tensors are weaker than for rank $1$ critical tensors.
For quartic binary forms the computation of the critical rank $2$ tensors
is
easier since the dual variety of the secant variety $\sigma_2(C_4)$ is given by quartics which
are squares, which make a smooth variety.
The number of complex critical rank $2$ tensors for a general binary form of degree d
was guessed in \cite{OSS} to be $3/2d^2-9d/2+1$. For $d=4$
this number is $7$, which can be confirmed by a symbolic computation
on a rational random binary quartic.
In conclusion, for a general binary quartic there are $4$ complex critical rank $1$ tensors
and $7$
complex rank $2$ critical tensors.
The following table reports some computation done for the case of binary quartic forms, by testing several different quartics. The appearance of ``yes'' in the last column means that we have found an example
of a binary quartic with the prescribed number of distinct and simple
real roots, real rank $1$ critical tensors and real critical rank $2$ tensors.
Note that we have not found any quartic with the maximum number of seven real
rank $2$ critical tensors, we wonder if they exist.
\begin{center}
\begin{tabular}{c | c| c |c| c}
&\#\text{real roots}& \#\text{real critical rank 1 tensors} & \#\text{real critical rank 2
tensors} & \\
\hline
& $0$ & $2$ & $3$ & yes\\
& $2$ & $2$ & $3$ & yes\\
& $0$ & $2$ & $5$ & yes\\
& $2$ & $2$ & $5$ & yes\\
& $0$ & $4$ & $3$ & yes\\
& $2$ & $4$ & $3$ & yes\\
& $4$ & $4$ & $3$ & yes\\
& $0$ & $4$ & $5$ & yes\\
& $2$ & $4$ & $5$ & yes\\
& $4$ & $4$ & $5$ & ?\\
& * & * & 7 & ?
\end{tabular}
\end{center}
\section{Acknowledgement} Giorgio Ottaviani is member of INDAM-GNSAGA. This paper has been partially supported by the Strategic Project ``Azioni di Gruppi su variet\'a e tensori'' of the University of Florence.
|
1,314,259,993,024 | arxiv | \section{The characterization of a Kropina metric.}\label{}
\hspace{0.2in}
Let $(M, \alpha)$ be an $n(\ge 2)$-dimensional differential manifold endowed with a Riemannian metric $\alpha$.
A Kropina space $(M, \alpha^2/\beta)$ is a Finsler space whose fundamental function is given by $F=\alpha^2/\beta$,
where $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ and $\beta=b_i(x)y^i$.
Even though Kropina spaces can be studied in more general case ([4]),
in this paper, we suppose that the matrix $(a_{ij})$ is
positive definite.
Let us remark that for a Kropina space $(M, \alpha^2/\beta)$ the Kropina metric $F=\alpha^2/\beta$ can be rewrited
as follows:
\begin{eqnarray*}
\frac{\alpha^2}{F^2}-\frac{\beta}{F}+\frac{b^2}{4}&=&\frac{b^2}{4}.
\end{eqnarray*}
where $b^2=a^{ij}b_ib_j$ and $(a^{ij})=(a_{ij})^{-1}$.
Let $\kappa(x)$ be a function of $(x^i)$ alone. Multiplying the above equation by $e^{\kappa(x)}$, we have
\begin{eqnarray}
e^{\kappa(x)}a_{ij}\frac{y^i}{F}\frac{y^j}{F}
-e^{\kappa(x)}a_{ij}\frac{y^i}{F}b^j
+\frac{1}{4}e^{\kappa(x)}a_{ij}b^ib^j=\frac{e^{\kappa(x)}b^2}{4},
\end{eqnarray}
Defining a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector field $W=W^i(\partial/\partial x^i)$ on $M$ by
\begin{eqnarray}
h_{ij}=e^{\kappa(x)}a_{ij}\hspace{0.1in} and \hspace{0.1in} 2W_i=e^{\kappa(x)}b_i,
\end{eqnarray}
where $W_i=h_{ij}W^j$, the equation (1.1) reduces to
\begin{eqnarray*}
\bigg|\frac{y}{F}-W\bigg|=|W|.
\end{eqnarray*}
In the above equation, the symbol $| \cdot |$ denotes the length of a vector with respect to the Riemannian metric $h$.
We notice that the equation $|W|=1$ holds if and only if the function $\kappa(x)$ satisfies the condition
\begin{eqnarray}
e^{\kappa(x)}b^2=4.
\end{eqnarray}
Suppose that the function $\kappa(x)$ satisfies (1.3), then we have $|W|=1$ and
\begin{eqnarray}
\bigg|\frac{y}{F}-W\bigg|=1.
\end{eqnarray}
Therefore, in each tangent space $T_xM$, the indicatrix of the Kropina metric necessarily goes through the origin.
Conversely, consider a Riemannian space $(M, h)$, where $h=\sqrt{h_{ij}(x)y^iy^j}$,
and a unit vector field $W=W^i(\partial /\partial x^i)$ on it.
Then, we consider the metric $F$ characterized by (1.4).
Solving for $F$ from (1.4), we get
\begin{eqnarray*}
F=\frac{|y|^2}{\{\sqrt{2}h(y,W)\}^2}.
\end{eqnarray*}
Comparing the above equality with a Kropina metric $F=\alpha^2/\beta$, we obtain (1.2) and from the assumption
$|W|=1$ we get (1.3).
Summarizing the above discussion, we obtain\\
{\textbf {Theorem 1}} \hspace{0.5in}
\textit{Let $(M, \alpha)$ be an $n(\ge 2)$-dimensional Riemannian space with a Riemannian metric
$\alpha=\sqrt{a_{ij}(x)y^iy^j}$.
For a Kropina space $(M, F=\alpha^2/\beta)$, where $\beta=b_i(x)y^i$, we define a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector field
$W=W^i(\partial/\partial x^i)$ of constant length 1 by (1.2) and (1.3). Then, the Kropina metric $F$ satisfies the equation (1.4).}
\textit{Conversely, suppose that $h=\sqrt{h_{ij}(x)y^iy^j}$ is a Riemannian metric and $W=W^i(\partial/\partial x^i)$ is
a vector field of constant length 1 on $(M,h)$.
Consider the metric $F$ defined by (1.4).
Then, defining $a_{ij}(x):=e^{-\kappa(x)}h_{ij}(x)$ and $b_i(x):=2e^{-\kappa(x)}W_i$ by (1.2) using some function
$\kappa(x)$ of $(x^i)$ alone, we get
$F=\alpha^2/\beta$ and it follows the function $\kappa(x)$ satisfies (1.3).}
\section{ The coefficients of the geodesic spray in a Kropina space. }\label{}
From the theory of Riemannian spaces, we have the following theorem:\\
{\textbf {Theorem A}}([6])\hspace{0.5in}
Let $(M, g)$ and $(M, g^*=e^{\rho}g)$, where $g=\sqrt{g_{ij}(x)y^iy^j}$ and $g^*=\sqrt{g_{ij}^*(x)y^iy^j}$ respectively,
be two $n$-dimensional Riemannian spaces which are comformal each other.
Furthermore, let ${{\gamma_j}^i}_k$ and ${{\gamma^*_j}^i}_k$ be the coefficients of Levi-Civita connection of $(M, g)$
and $(M, g^*)$, respectively.
Then, we have
\begin{eqnarray*}
g^*_{ij}=e^{2\rho}g_{ij}, \hspace{0.2in} g^{*ij}=e^{-2\rho}g^{ij},\hspace{0.2in}
{{\gamma^*_j}^i}_k={{\gamma_j}^i}_k+\rho_j{\delta^i}_k+\rho_k{\delta^i}_j-\rho^ig_{jk},
\end{eqnarray*}
where $\rho_i=\partial \rho/\partial x^i$ and $\rho^i=g^{ij}\rho_j$.\\
From (1.2), we have $h_{ij}=e^{\kappa}a_{ij}$. Applying Theorem A , we get
\begin{eqnarray}
{{^h \gamma_j}^i}_k={{^\alpha\gamma_j}^i}_k+\frac{1}{2}\kappa_j{\delta^i}_k+\frac{1}{2}\kappa_k{\delta^i}_j-\frac{1}{2}\kappa^ia_{jk},
\end{eqnarray}
where ${{^h \gamma_j}^i}_k$ and ${{^\alpha\gamma_j}^i}_k$ are the coefficients of Levi-Civita connection of $(M, h)$
and $(M, \alpha)$ respectively, $\kappa_i=\partial \kappa/\partial x^i$ and $\kappa^i=a^{ij}\kappa_j$.
Transvecting (2.1) by $y^jy^k$, we get
\begin{eqnarray}
{{^h \gamma_0}^i}_0={{^\alpha\gamma_0}^i}_0+\kappa_0y^i-\frac{1}{2}h_{00}\overline{\kappa}^i,
\end{eqnarray}
where $\overline{\kappa}^i=h^{ij}\kappa_j$.
We denote the covariant derivative in the Riemannian space $(M, \alpha)$ by $(_{;i})$ and put as follows:
\begin{eqnarray*}
s_{ij}:=\frac{b_{i;j}-b_{j;i}}{2},\hspace{0.2in} r_{ij}:=\frac{b_{i;j}+b_{j;i}}{2},\hspace{0.2in}
s_j:=b^is_{ij},\hspace{0.2in} r_j:=b^ir_{ij}.
\end{eqnarray*}
In [1], the authors have shown that the coefficients $G^i$ of the geodesic spray in a Finsler space
$(M, F=\alpha\phi(\beta/\alpha))$ are given by
\begin{eqnarray}
2G^i= {{^\alpha \gamma_0}^i}_0+2\omega \alpha {s^i}_0+2\Theta(r_{00}-2\alpha \omega s_0)
\bigg(\frac{y^i}{\alpha}+\frac{\omega'}{\omega-s\omega'}b^i\bigg),
\end{eqnarray}
where
\begin{eqnarray*}
\omega:=\frac{\phi'}{\phi-s\phi'},\hspace{0.2in}
\Theta:=\frac{\omega-s\omega'}{2\{1+s\omega+(b^2-s^2)\omega'\}}.
\end{eqnarray*}
For a Kropina space, we have $\phi(s)=1/s$, hence we obtain
\begin{eqnarray*}
\phi'=-\frac{1}{s^2}, \hspace{0.2in} \omega=-\frac{1}{2s}, \hspace{0.2in} \omega'=\frac{1}{2s^2}
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{\omega'}{\omega-s\omega'}=-\frac{1}{2s},\hspace{0.1in}
r_{00}-2\alpha \omega s_0=r_{00}+Fs_0,\hspace{0.1in}
1+s\omega+(b^2-s^2)\omega'=\frac{b^2}{2s^2},\hspace{0.1in}
\Theta=-\frac{s}{b^2}.
\end{eqnarray*}
Substituting the above equalities in (2.3) and using (2.2), we get
\begin{eqnarray*}
2G^i&=&{{^h\gamma_0}^i}_0-\kappa_0y^i
+\frac{1}{2}h_{00}\overline{\kappa}^i-F{s^i}_0-\frac{1}{b^2}(r_{00}+Fs_0)(\frac{2}{F}y^i-b^i).
\end{eqnarray*}
From Theorem 1, for a Kropina space $(M, \alpha^2/\beta)$, a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector
field $W=W^i(\partial/\partial x^i)$ are defined by (1.2) and (1.3). So, the vector field $W$ satisfies the condition
$|W|=1$ and we have $F=h_{00}/2W_0$.
Therefore, we get
\begin{eqnarray}
2G^i={{^h\gamma_0}^i}_0+2\Phi^i,
\end{eqnarray}
where
\begin{eqnarray}
2\Phi^i:=-\kappa_0y^i+ \frac{1}{2}h_{00}\overline{\kappa}^i-\frac{h_{00}}{2W_0}{s^i}_0
-\frac{1}{b^2}(r_{00}+\frac{h_{00}s_0}{2W_0})(\frac{4W_0}{h_{00}}y^i-b^i).
\end{eqnarray}
Using (2.1), we have
\begin{eqnarray*}
b_{i;j} =2e^{-\kappa}W_{i||j}+e^{-\kappa}\bigg(\kappa_iW_j-\kappa_jW_i-W_r\overline{\kappa}^rh_{ij}\bigg),
\end{eqnarray*}
where the notation $(_{||i})$ stands for the $h$-covariant derivative in the Riemannian space $(M, h)$.\\
\textbf{Remark 1}\hspace{0.7in}
\textit{We can introduce a Finsler connection $\Gamma^*=(^h{{\gamma_j}^i}_k(x), {N_j}^i:=$
$^h{{\gamma_j}^i}_k(x)y^k, {{C_j}^i}_k)$
associated with the linear connection $^h{{\gamma_j}^i}_k(x)$ of the Riemannian space $(M, h)$.
The \textit{$h$-covariant derivative} are defined as follows ([2]):}
\textit{For a vector field $W^i(x)$ on $M$,
\begin{eqnarray*}
1,\hspace{0.2in} W^i(x)_{||j}&=&\frac{\partial W^i}{\partial x^j}-\frac{\partial W^i}{\partial y^s}{N_j}^s
+^h{{\gamma_j}^i}_sW^s\\
&=&\frac{\partial W^i}{\partial x^j}+^h{{\gamma_j}^i}_sW^s.
\end{eqnarray*}}
\textit{For a reference vector $y^i$,
\begin{eqnarray*}
2,\hspace{0.2in} {y^i}_{||j}&=& \frac{\partial y^i}{\partial x^j}-\frac{\partial y^i}{\partial y^s}{N_j}^s
+^h{{\gamma_j}^i}_sy^s\\
&=&-{N_j}^i+{N_j}^i\\
&=&0.
\end{eqnarray*}}
We put
\begin{eqnarray*}
\texttt{R}_{ij}:=\frac{W_{i||j}+W_{j||i}}{2}, \hspace{0.1in}
\texttt{S}_{ij}:=\frac{W_{i||j}-W_{j||i}}{2}, \hspace{0.1in}
&&{\texttt{R}^i}_j:=h^{ir}\texttt{R}_{rj},\hspace{0.1in}
{\texttt{S}^i}_j:=h^{ir}\texttt{S}_{rj}\\
\texttt{R}_{i}:=W^r\texttt{R}_{ri}, \hspace{0.1in}
\texttt{S}_{i}:=W^r\texttt{S}_{ri}, \hspace{0.1in}
&&\texttt{R}^{i}:=h^{ir}\texttt{R}_{r}, \hspace{0.1in}
\texttt{S}^{i}:=h^{ir}\texttt{S}_{r}.
\end{eqnarray*}
It follows
\begin{eqnarray*}
r_{ij}=2e^{-\kappa} \bigg(\texttt{R}_{ij}-\frac{1}{2}W_r\overline{\kappa}^rh_{ij}\bigg),\hspace{0.1in}
s_{ij}=2e^{-\kappa} \bigg(\texttt{S}_{ij}+\frac{\kappa_iW_j-\kappa_jW_i}{2}\bigg).
\end{eqnarray*}
Furthermore, we get
\begin{eqnarray*}
{s^i}_j=2{\texttt{S}^i}_j+\overline{\kappa}^iW_j-\kappa_jW^i,&&\hspace{0.1in}
{s^i}_0=2{\texttt{S}^i}_0+W_0\overline{\kappa}^i-\kappa_0W^i\\
s_i=2e^{-\kappa} \bigg(2\texttt{S}_i+W_r\overline{\kappa}^rW_i-\kappa_i\bigg),&&\hspace{0.1in}
s_0=2e^{-\kappa} \bigg(2\texttt{S}_0+W_r\overline{\kappa}^rW_0-\kappa_0\bigg) ,\\
r_{00}=2e^{-\kappa} \bigg(\texttt{R}_{00}-\frac{1}{2}W_r\overline{\kappa}^rh_{00}\bigg) &&\hspace{0.1in}
b^i=a^{ir}b_r=e^\kappa h^{ir}\frac{2W_r}{e^\kappa}=2W^i.
\end{eqnarray*}
Substituting the above equality in (2.5), we have
\begin{eqnarray}
2\Phi^i=\frac{h_{00}}{W_0}(\texttt{S}_0W^i-{\texttt{S}^i}_0)
+(\texttt{R}_{00}W^i-2\texttt{S}_0y^i)
-\frac{2W_0}{h_{00}}\texttt{R}_{00}y^i.
\end{eqnarray}
Multiplying now the above equality by $2h_{00}W_0$, we get
\begin{eqnarray*}
&&4h_{00}W_0 \Phi^i
=2(h_{00})^2(\texttt{S}_0W^i-{\texttt{S}^i}_0)+2h_{00}W_0(\texttt{R}_{00}W^i-2\texttt{S}_0y^i)
-4(W_0)^2\texttt{R}_{00}y^i
\end{eqnarray*}
and by putting
\begin{eqnarray*}
A_{(1)}^i:=2(\texttt{S}_0W^i-{\texttt{S}^i}_0),\hspace{0.1in}
A_{(2)}^i:=2(\texttt{R}_{00}W^i-2\texttt{S}_0y^i),\hspace{0.1in}
A_{(3)}^i:=-4\texttt{R}_{00}y^i,
\end{eqnarray*}
it follows
\begin{eqnarray}
4h_{00}W_0 \Phi^i =(h_{00})^2A_{(1)}^i +h_{00}W_0A_{(2)}^i +(W_0)^2A_{(3)}^i.
\end{eqnarray}
\section{The necessary and sufficient conditions for a Kropina space to be of constant curvature.}\label{}
\hspace{0.2in}
In this section, we consider a Kropina space $(M, F=\alpha^2/\beta)$ of constant curvatue $K$, where
$\alpha=\sqrt{a_{ij}(x)y^iy^j}$ and $\beta=b_i(x)y^i$. Furthermore, we suppose that the matrix $(a_{ij})$ is always
positive definite and that the dimension $n$ is more than or equal two.
Hence, it follows that $\alpha^2$ is not divisible by $\beta$. This is an important relation and
is equivalent to that $h_{00}$ is not divisible by $W_0$.
Using these, we will obtain the necessary and sufficient conditions for a Kropina space to be of
constant curvature.
\subsection{The curvature tensor of a Kropina space.}\label{}
\hspace{0.2in}
Let ${{R_j}^i}_{kl}$ be the $h$-curvature tensor of Cartan connection in Finsler space.
The Berwald's spray curvature tensor is
\begin{eqnarray}
^{(b)}{{R_j}^i}_{kl}&=&A_{(kl)}\bigg\{\frac{\partial {{G_j}^i}_k}{\partial x^l}+{{G_j}^r}_k{{G_r}^i}_l\bigg\}.
\end{eqnarray}
It is well-known that the equality ${{R_0}^i}_{kl}=^{(b)}{{R_0}^i}_{kl}$ holds good ([2]).
From $2G^i={{^h\gamma_0}^i}_0+2\Phi^i$, it follows
\begin{eqnarray*}
{G^i}_j={{^h\gamma_0}^i}_j+{\Phi^i}_j \hspace{0.2in} and \hspace{0.2in}
{{G_j}^i}_k={{^h\gamma_j}^i}_k+{{\Phi_j}^i}_k.
\end{eqnarray*}
Substituting the above equalities in (3.1), we get
\begin{eqnarray*}
{{R_j}^i}_{kl}={{^hR_j}^i}_{kl}+ A_{(kl)}\bigg\{ {{\Phi_j}^i}_{k||l}+{{\Phi_j}^r}_k{{\Phi_r}^i}_l\bigg\}.
\end{eqnarray*}
The following Proposition is well-known ([2]):\\
\textbf{Proposition 1}\hspace{0.5in}
\textit{The necessary and sufficient condition for a Finsler space $(M, F)$ to be of scalar curvature $K$ is that the equality
\begin{eqnarray}
{{R_0}^i}_{0l}=KF^2 ({\delta^i}_l-l^il_l),
\end{eqnarray}
where $l^i=y^i/F$ and $l_l=\partial F/\partial y^l$, holds.} \\
If the equality (3.2) holds good and $K$ is constant, then the Finsler space is called of {\it constant curvature} $K$.
For a Kropina space of constant curvatue $K$, the equality (3.2) holds and $K$ is constant.
Since $F=\alpha^2/\beta$ is written as $F=h_{00}/(2W_0)$, we have
\begin{eqnarray*}
{\delta^i}_l-l^il_l={\delta^i}_l-\frac{2W_0h_{0l}-h_{00}W_l}{h_{00}W_0}y^i.
\end{eqnarray*}
Using the curvature we obtained above, we have
\begin{eqnarray*}
{{R_0}^i}_{0l}={{^hR_0}^i}_{0l}+ 2{\Phi^i}_{||l}- {\Phi^i}_{l||0}
+2\Phi^r{{\Phi_r}^i}_l -{\Phi^r}_l{\Phi^i}_r.
\end{eqnarray*}
Substituting the above equalities in (3.2), we get
\begin{eqnarray}
KF^2\bigg({\delta^i}_l-\frac{2W_0h_{0l}-h_{00}W_l}{h_{00}W_0}y^i\bigg)={{^hR_0}^i}_{0l}+ 2{\Phi^i}_{||l}- {\Phi^i}_{l||0}
+2\Phi^r{{\Phi_r}^i}_l -{\Phi^r}_l{\Phi^i}_r.
\end{eqnarray}
\subsection{Rewriting the equation (3.3) using $h_{00}$ and $W_0$. }\label{}
\textbf{(1)The calculations for ${\Phi^i}_{||l}$.}
First, applying the $h$-covariant derivative $_{||l}$ to (2.7), it follows
\begin{eqnarray*}
&&4h_{00}W_{0||l} \Phi^i+4h_{00}W_0 {\Phi^i}_{||l}\\
&=&(h_{00})^2{A_{(1)}^i}_{||l} +h_{00}W_0{A_{(2)}^i}_{||l}
+h_{00}W_{0||l}A_{(2)}^i+(W_0)^2{A_{(3)}^i}_{||l} +2W_0W_{0||l}A_{(3)}^i \nonumber
\end{eqnarray*}
and using again (2.7), we get
\begin{eqnarray*}
&&4h_{00}(W_0)^2 {\Phi^i}_{||l}\\
&=&(h_{00})^2W_0{A_{(1)}^i}_{||l}
-(h_{00})^2A_{(1)}^iW_{0||l}
+h_{00}(W_0)^2{A_{(2)}^i}_{||l} \\
&&\hspace{2in} +(W_0)^3{A_{(3)}^i}_{||l}
+(W_0)^2A_{(3)}^iW_{0||l}.
\end{eqnarray*}
Putting now
\begin{eqnarray*}
&&B_{(1)l}^i={A_{(1)}^i}_{||l} ,\hspace{0.1in}
B_{(21)l}^i=-A_{(1)}^iW_{0||l}, \hspace{0.1in}
B_{(22)l}^i={A_{(2)}^i}_{||l} ,\hspace{0.1in}
B_{(3)l}^i={A_{(3)}^i}_{||l},\hspace{0.1in}
B_{(4)l}^i=A_{(3)}^iW_{0||l},
\end{eqnarray*}
we have
\begin{eqnarray}
&&4h_{00}(W_0)^2 {\Phi^i}_{||l}
=(h_{00})^2W_0B_{(1)l}^i
+(h_{00})^2B_{(21)l}^i \\
&&\hspace{2in} +h_{00}(W_0)^2B_{(22)l}^i +(W_0)^3B_{(3)l}^i
+(W_0)^2B_{(4)l}^i.\nonumber
\end{eqnarray}
\textbf{(2)The calculations for ${\Phi^i}_l$.}
Secondly, derivating (2.7) by $y^l$, we get
\begin{eqnarray*}
&&8h_{0l}W_0 \Phi^i+4h_{00}W_l \Phi^i+4h_{00}W_0 {\Phi^i}_l\\
&=& (h_{00})^2A_{(1)}^i._l +h_{00}W_0A_{(2).l}^i
+h_{00}(4h_{0l}A_{(1)}^i+W_lA_{(2)}^i)\\
&&\hspace{2in} +(W_0)^2A_{(3)}^i._l +2W_0(W_lA_{(3)}^i+h_{0l}A_{(2)}^i),
\end{eqnarray*}
where the notation $(_{.l})$ stands for the derivation by $y^l$.
Using the above equality and (2.7), we obtain
\begin{eqnarray}
&&4(h_{00})^2(W_0)^2 {\Phi^i}_l\\
&=&(h_{00})^3W_0C_{(0)l}^i
+(h_{00})^3C_{(11)l}^i
+(h_{00})^2(W_0)^2C_{(12)l}^i \nonumber \\
&&\hspace{1in} +(h_{00})^2W_0C_{(21)l}^i
+h_{00}(W_0)^3C_{(22)l}^i
+h_{00}(W_0)^2C_{(3)l}^i
+(W_0)^3C_{(4)l}^i,\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
&&C_{(0)l}^i:=A_{(1)}^i._l=2(\texttt{S}_lW^i-{\texttt{S}^i}_l), \hspace{0.1in}
C_{(11)l}^i:=-A_{(1)}^iW_l=-2(\texttt{S}_0W^i-{\texttt{S}^i}_0)W_l,\\
&&C_{(12)l}^i:=A_{(2).l}^i
=4( \texttt{R}_{0l}W^i-\texttt{S}_ly^i-\texttt{S}_0{\delta^i}_l),\hspace{0.1in}
C_{(21)l}^i:=2A_{(1)}^ih_{0l}=4(\texttt{S}_0W^i-{\texttt{S}^i}_0)h_{0l},\\
&&C_{(22)l}^i:=A_{(3)}^i._l=-4(2\texttt{R}_{0l}y^i+\texttt{R}_{00}{\delta^i}_l),\hspace{0.1in}
C_{(3)l}^i:=W_lA_{(3)}^i=-4\texttt{R}_{00}W_ly^i,\\
&&C_{(4)l}^i:=-2A_{(3)}^ih_{0l}=8\texttt{R}_{00}h_{0l}y^i.
\end{eqnarray*}
\textbf{(3)The calculations for ${\Phi^i}_{l||0}$.}
Applying the $h$-covariant derivaive ${_{||0}}$ to (3.5), we get
\begin{eqnarray*}
&&8(h_{00})^2W_0 W_{0||0}{\Phi^i}_l+4(h_{00})^2(W_0)^2 {\Phi^i}_{l||0}\\
&=&(h_{00})^3W_0C_{(0)l||0}^i
+(h_{00})^3(W_{0||0}C_{(0)l}^i+C_{(11)l||0}^i) \\
&& +(h_{00})^2(W_0)^2C_{(12)l||0}^i
+(h_{00})^2W_0(2W_{0||0}C_{(12)l}^i+C_{(21)l||0}^i)\\
&&+(h_{00})^2W_{0||0}C_{(21)l}^i
+h_{00}(W_0)^3C_{(22)l||0}^i
+h_{00}(W_0)^2(3W_{0||0}C_{(22)l}^i+C_{(3)l||0}^i)\\
&&+2h_{00}W_0W_{0||0}C_{(3)l}^i
+(W_0)^3C_{(4)l||0}^i
+3(W_0)^2W_{0||0}C_{(4)l}^i.
\end{eqnarray*}
Using the above equality and (3.5), we obtain
\begin{eqnarray}
&&4(h_{00})^2(W_0)^3 {\Phi^i}_{l||0}\\
&=&(h_{00})^3(W_0)^2D_{(1)l}^i
+(h_{00})^3W_0D_{(21)l}^i
+(h_{00})^3D_{(31)l}^i\nonumber\\
&&\hspace{0.5in} +(h_{00})^2(W_0)^3D_{(22)l}^i
+(h_{00})^2(W_0)^2D_{(32)l}^i
+(h_{00})^2W_0D_{(41)l}^i\nonumber\\
&&\hspace{0.5in} +h_{00}(W_0)^4D_{(33)l}^i
+h_{00}(W_0)^3 D_{(42)l}^i
+(W_0)^4D_{(5)l}^i
+(W_0)^3D_{(6)l}^i,\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
&&D_{(1)l}^i=C_{(0)l||0}^i,\hspace{0.2in}
D_{(21)l}^i=C_{(11)l||0}^i-W_{0||0}C_{(0)l}^i,\hspace{0.2in}
D_{(31)l}^i=-2W_{0||0}C_{(11)l}^i, \\
&& D_{(22)l}^i=C_{(12)l||0}^i,\hspace{0.2in}
D_{(32)l}^i=C_{(21)l||0}^i , \hspace{0.2in}
D_{(41)l}^i=-W_{0|0}C_{(21)l}^i, \\
&&D_{(33)l}^i=C_{(22)l||0}^i , \hspace{0.2in}
D_{(42)l}^i=W_{0||0}C_{(22)l}^i+C_{(3)l||0}^i, \hspace{0.2in}
D_{(5)l}^i= C_{(4)l||0}^i , \hspace{0.2in}
D_{(6)l}^i= W_{0||0}C_{(4)l}^i.
\end{eqnarray*}
\textbf{(4)The calculations for ${\Phi^r}_l{\Phi^i}_r$.}
Using (3.5), we have
\begin{eqnarray}
&&16(h_{00})^4(W_0)^4 {\Phi^r}_l{\Phi^i}_r\\
&=&(h_{00})^6(W_0)^2E_{(01)l}^i
+(h_{00})^6W_0E_{(11)l}^i
+(h_{00})^6E_{(21)l}^i
+(h_{00})^5(W_0)^3E_{(12)l}^i \nonumber\\
&&+(h_{00})^5(W_0)^2E_{(22)l}^i
+(h_{00})^5W_0E_{(31)l}^i
+(h_{00})^4(W_0)^4E_{(23)l}^i
+(h_{00})^4(W_0)^3E_{(32)l}^i \nonumber\\
&&+(h_{00})^4(W_0)^2E_{(41)l}^i
+(h_{00})^3(W_0)^5E_{(33)l}^i
+(h_{00})^3(W_0)^4E_{(42)l}^i
+(h_{00})^3(W_0)^3E_{(51)l}^i \nonumber\\
&&+(h_{00})^2(W_0)^6E_{(43)l}^i
+(h_{00})^2(W_0)^5E_{(52)l}^i
+(h_{00})^2(W_0)^4E_{(61)l}^i
+h_{00}(W_0)^6E_{(62)l}^i \nonumber\\
&&+h_{00}(W_0)^5E_{(7)l}^i
+(W_0)^6E_{(8)l}^i, \nonumber
\end{eqnarray}
where
\begin{eqnarray*}
E_{(0)l}^i:&=&C_{(0)r}^iC_{(0)l}^r,\hspace{0.1in}
E_{(11)l}^i:=C_{(11)r}^iC_{(0)l}^r+C_{(0)r}^iC_{(11)l}^r,\hspace{0.1in}
E_{(21)l}^i:=C_{(11)r}^iC_{(11)l}^r,\\
E_{(12)l}^i:&=&C_{(0)r}^iC_{(12)l}^r +C_{(12)r}^iC_{(0)l}^r ,\\
E_{(22)l}^i:&=&C_{(12)r}^iC_{(11)l}^r +C_{(11)r}^iC_{(12)l}^r+C_{(21)r}^iC_{(0)l}^r +C_{(0)r}^iC_{(21)l}^r ,\\
E_{(31)l}^i:&=&C_{(21)r}^iC_{(11)l}^r +C_{(11)r}^iC_{(21)l}^r, \hspace{0.1in}
E_{(23)l}^i:=C_{(12)r}^iC_{(12)l}^r+C_{(22)r}^iC_{(0)l}^r +C_{(0)r}^iC_{(22)l}^r, \\
E_{(32)l}^i:&=&C_{(21)r}^iC_{(12)l}^r +C_{(12)r}^iC_{(21)l}^r+C_{(3)r}^iC_{(0)l}^r+C_{(22)r}^iC_{(11)l}^r
+C_{(11)r}^iC_{(22)l}^r +C_{(0)r}^iC_{(3)l}^r \\
E_{(41)l}^i:&=&C_{(3)r}^iC_{(11)l}^r+C_{(11)r}^iC_{(3)l}^r +C_{(21)r}^iC_{(21)l}^r, \hspace{0.1in}
E_{(33)l}^i:=C_{(22)r}^iC_{(12)l}^r+C_{(12)r}^iC_{(22)l}^r , \\
E_{(42)l}^i:&=&C_{(4)r}^iC_{(0)l}^r+C_{(3)r}^iC_{(12)l}^r+C_{(22)r}^iC_{(21)l}^r
+C_{(21)r}^iC_{(22)l}^r+C_{(12)r}^iC_{(3)l}^r +C_{(0)r}^iC_{(4)l}^r,\\
E_{(51)l}^i:&=&C_{(3)r}^iC_{(21)l}^r+C_{(21)r}^iC_{(3)l}^r+C_{(4)r}^iC_{(11)l}^r +C_{(11)r}^iC_{(4)l}^r,\hspace{0.1in}
E_{(43)l}^i:=C_{(22)r}^iC_{(22)l}^r , \\
E_{(52)l}^i:&=&C_{(4)r}^iC_{(12)l}^r+C_{(12)r}^iC_{(4)l}^r
+C_{(3)r}^iC_{(22)l}^r +C_{(22)r}^iC_{(3)l}^r, \\
E_{(61)l}^i:&=&C_{(3)r}^iC_{(3)l}^r +C_{(4)r}^iC_{(21)l}^r +C_{(21)r}^iC_{(4)l}^r,\hspace{0.1in}
E_{(62)l}^i:=C_{(4)r}^iC_{(22)l}^r +C_{(22)r}^iC_{(4)l}^r,\\
E_{(7)l}^i:&=&C_{(4)r}^iC_{(3)l}^r+C_{(3)r}^iC_{(4)l}^r,\hspace{0.1in}
E_{(8)l}^i:=C_{(4)r}^iC_{(4)l}^r.
\end{eqnarray*}
\textbf{(5)The calculations for ${\Phi^r}{{\Phi_r}^i}_l$.}
Derivating (3.5) by $y^r$, we get
\begin{eqnarray*}
&&16h_{00}h_{0r}(W_0)^2 {\Phi^i}_l+8(h_{00})^2W_0W_r {\Phi^i}_l+4(h_{00})^2(W_0)^2 {{\Phi_r}^i}_l\\
&=&6(h_{00})^2h_{0r}W_0C_{(0)l}^i+(h_{00})^3W_rC_{(0)l}^i+(h_{00})^3W_0C_{(0)l}^i._r\\
&&+6(h_{00})^2h_{0r}C_{(11)l}^i+(h_{00})^3C_{(11)l}^i._r\\
&&+4h_{00}h_{0r}(W_0)^2C_{(12)l}^i+2(h_{00})^2W_0W_rC_{(12)l}^i+(h_{00})^2(W_0)^2C_{(12)l}^i._r\\
&&+4h_{00}h_{0r}W_0C_{(21)l}^i+(h_{00})^2W_rC_{(21)l}^i+(h_{00})^2W_0C_{(21)l}^i._r\\
&& +2h_{0r}(W_0)^3C_{(22)l}^i+3h_{00}(W_0)^2W_rC_{(22)l}^i+h_{00}(W_0)^3C_{(22)l}^i._r\\
&&+2h_{0r}(W_0)^2C_{(3)l}^i+2h_{00}W_0W_rC_{(3)l}^i+h_{00}(W_0)^2C_{(3)l}^i._r\\
&&+3(W_0)^2W_rC_{(4)l}^i+(W_0)^3C_{(4)l}^i._r.
\end{eqnarray*}
Using the above equality, (3.5) and $C_{(0)l}^i._r=0$, we have
\begin{eqnarray*}
&&4(h_{00})^3(W_0)^3 {{\Phi_r}^i}_l\\
&=&(h_{00})^4W_0H_{(01)rl}^i
+(h_{00})^4H_{(11)rl}^i
+(h_{00})^3(W_0)^3H_{(02)rl}^i
+(h_{00})^3(W_0)^2H_{(12)rl}^i \\
&&+(h_{00})^3W_0H_{(21)rl}^i
+(h_{00})^2(W_0)^4H_{(13)rl}^i
+(h_{00})^2(W_0)^3H_{(22)rl}^i\\
&& +h_{00}(W_0)^4H_{(3)rl}^i
+h_{00}(W_0)^3H_{(4)rl}^i
+(W_0)^4H_{(5)rl}^i,
\end{eqnarray*}
where
\begin{eqnarray*}
&&H_{(01)rl}^i=C_{(11)l}^i._r -W_rC_{(0)l}^i, \hspace{0.1in}
H_{(11)rl}^i=-2W_rC_{(11)l}^i,\hspace{0.1in}
H_{(02)rl}^i=C_{(12)l.r}^i,\\
&&H_{(12)rl}^i=2h_{0r}C_{(0)l}^i+C_{(21)l}^i._r, \hspace{0.1in}
H_{(21)rl}^i=-W_rC_{(21)l}^i+2h_{0r}C_{(11)l}^i, \\
&&H_{(13)rl}^i=C_{(22)l}^i._r, \hspace{0.1in}
H_{(22)rl}^i=W_rC_{(22)l}^i+C_{(3)l}^i._r, \hspace{0.1in}
H_{(3)rl}^i=C_{(4)l}^i._r-2h_{0r}C_{(22)l}^i,\\
&&H_{(4)rl}^i=W_rC_{(4)l}^i-2h_{0r}C_{(3)l}^i,\hspace{0.1in}
H_{(5)rl}^i=-4h_{0r}C_{(4)l}^i.
\end{eqnarray*}
Using the above equality and (2.7), we get
\begin{eqnarray}
&&16(h_{00})^4(W_0)^4 \Phi^r{{\Phi_r}^i}_l\\
&=&(h_{00})^6W_0J_{(11)l}^i
+(h_{00})^6J_{(21)l}^i
+(h_{00})^5(W_0)^3J_{(12)l}^i
+(h_{00})^5(W_0)^2J_{(22)l}^i\nonumber\\
&&+(h_{00})^5W_0J_{(31)l}^i
+(h_{00})^4(W_0)^4J_{(23)l}^i
+(h_{00})^4(W_0)^3J_{(32)l}^i
+(h_{00})^4(W_0)^2J_{(41)l}^i \nonumber\\
&&+(h_{00})^3(W_0)^5J_{(33)l}^i
+ (h_{00})^3(W_0)^4J_{(42)l}^i
+ (h_{00})^3(W_0)^3J_{(51)l}^i
+(h_{00})^2(W_0)^6J_{(43)l}^i\nonumber\\
&&+(h_{00})^2(W_0)^5J_{(52)l}^i
+ (h_{00})^2(W_0)^4J_{(61)l}^i
+h_{00}(W_0)^6J_{(62)l}^i
+h_{00}(W_0)^5J_{(71)l}^i
+(W_0)^6J_{(8)l}^i, \nonumber
\end{eqnarray}
where
\begin{eqnarray*}
&&J_{(11)l}^i=A_{(1)}^rH_{(01)rl}^i,\hspace{0.1in}
J_{(21)l}^i=A_{(1)}^rH_{(11)rl}^i,\hspace{0.1in}
J_{(12)l}^i=A_{(1)}^rH_{(02)rl}^i,\\
&&J_{(22)l}^i=A_{(1)}^rH_{(12)rl}^i+A_{(2)}^rH_{(01)rl}^i,\hspace{0.1in}
J_{(31)l}^i=A_{(1)}^rH_{(21)rl}^i+A_{(2)}^rH_{(11)rl}^i,\\
&&J_{(23)l}^i=A_{(1)}^rH_{(13)rl}^i+A_{(2)}^rH_{(02)rl}^i,\hspace{0.1in}
J_{(32)l}^i=A_{(1)}^rH_{(22)rl}^i +A_{(2)}^rH_{(12)rl}^i+A_{(3)}^rH_{(01)rl}^i,\\
&&J_{(41)l}^i=A_{(2)}^rH_{(21)rl}^i+A_{(3)}^rH_{(11)rl}^i,\hspace{0.1in}
J_{(33)l}^i=A_{(2)}^rH_{(13)rl}^i+A_{(3)}^rH_{(02)rl}^i,\\
&&J_{(42)l}^i=A_{(1)}^rH_{(3)rl}^i
+A_{(2)}^rH_{(22)rl}^i+A_{(3)}^rH_{(12)rl}^i,\\
&&J_{(51)l}^i=A_{(1)}^rH_{(4)rl}^i+A_{(3)}^rH_{(21)rl}^i,\hspace{0.1in}
J_{(43)l}^i=A_{(3)}^rH_{(13)rl}^i,\\
&&J_{(52)l}^i=A_{(2)}^rH_{(3)rl}^i+A_{(3)}^rH_{(22)rl}^i,\hspace{0.1in}
J_{(61)l}^i=A_{(2)}^rH_{(4)rl}^i+A_{(1)}^rH_{(5)rl}^i,\\
&&J_{(62)l}^i=A_{(3)}^rH_{(3)rl}^i,\hspace{0.1in}
J_{(7)l}^i=A_{(2)}^rH_{(5)rl}^i+ A_{(3)}^rH_{(4)rl}^i,\hspace{0.1in}
J_{(8)l}^i=A_{(3)}^rH_{(5)rl}^i.
\end{eqnarray*}
\textbf{(6)The main relation.}
Multiplying (3.3) by $16(h_{00})^4(W_0)^4$ and using $F^2=(h_{00})^2/\{4(W_0)^2\}$, we have the equality
\begin{eqnarray*}
&&4K(h_{00})^6(W_0)^2 {h^i}_l
= 16(h_{00})^4(W_0)^4 \cdot {{^hR_0}^i}_{0l}+8(h_{00})^3(W_0)^2 \cdot 4h_{00}(W_0)^2{\Phi^i}_{||l} \\
&&- 4(h_{00})^2W_0\cdot 4(h_{00})^2(W_0)^3{\Phi^i}_{l||0}
+2 \cdot 16(h_{00})^4(W_0)^4\Phi^r{{\Phi_r}^i}_l -16(h_{00})^4(W_0)^4{\Phi^r}_l{\Phi^i}_r.
\end{eqnarray*}
Substituting (3.4), (3.5), (3.6), (3.7) and (3.8) in the above equality, we get
\begin{eqnarray*}
&&4K(h_{00})^6(W_0)^2{\delta^i}_l-8K(h_{00})^5(W_0)^2h_{0l}y^i+4K(h_{00})^6W_0W_ly^i\\
&=&-(h_{00})^6(W_0)^2E_{(0)l}^i
+(h_{00})^6W_0(2J_{(11)l}^i -E_{(11)l}^i)\\
&&+(h_{00})^6(2J_{(21)l}^i-E_{(21)l}^i )
+(h_{00})^5(W_0)^3(8B_{(1)l}^i-4D_{(1)l}^i+2J_{(12)l}^i-E_{(12)l}^i) \\
&&+(h_{00})^5(W_0)^2(-4D_{(21)l}^i+8B_{(21)l}^i+2J_{(22)l}^i-E_{(22)l}^i)\\
&&+(h_{00})^5W_0(2J_{(31)l}^i -4D_{(31)l}^i-E_{(31)l}^i)\\
&&+(h_{00})^4(W_0)^4(16 {{^hR_0}^i}_{0l}+8B_{(22)l}^i-4D_{(22)l}^i+2J_{(23)l}^i-E_{(23)l}^i)\\
&& +(h_{00})^4(W_0)^3(2J_{(32)l}^i-4D_{(32)l}^i -E_{(32)l}^i )
+(h_{00})^4(W_0)^2(2J_{(41)l}^i -E_{(41)l}^i-4D_{(41)l}^i)\\
&&+(h_{00})^3(W_0)^5(8B_{(3)l}^i-4D_{(33)l}^i+2J_{(33)l}^i-E_{(33)l}^i) \\
&&+ (h_{00})^3(W_0)^4(2J_{(42)l}^i+8B_{(4)l}^i-4 D_{(42)l}^i-E_{(42)l}^i)\\
&&+ (h_{00})^3(W_0)^3(2J_{(51)l}^i-E_{(51)l}^i)
+(h_{00})^2(W_0)^6(2J_{(43)l}^i-E_{(43)l}^i)\\
&&+(h_{00})^2(W_0)^5(2J_{(52)l}^i-4D_{(5)l}^i-E_{(52)l}^i)
+ (h_{00})^2(W_0)^4(2J_{(61)l}^i-4D_{(6)l}^i-E_{(61)l}^i)\\
&&+h_{00}(W_0)^6(2J_{(62)l}^i-E_{(62)l}^i )
+h_{00}(W_0)^5(2J_{(71)l}^i-E_{(7)l}^i)
+(W_0)^6(2J_{(8)l}^i-E_{(8)l}^i).
\end{eqnarray*}
Taking into account the equalities $J_{(21)l}^i=0$ and $E_{(21)l}^i=0$, we have
\begin{eqnarray}
(h_{00})^4{P_{(5)}^i}_l+(h_{00})^2{Q_{(9)}^i}_l+(W_0)^4{R_{(9)}^i}_l=0,
\end{eqnarray}
where
\begin{eqnarray*}
{P_{(5)}^i}_l&=&(h_{00})^2W_0(-E_{(0)l}^i-4K{\delta^i}_l)\\
&&\hspace{0.2in}+(h_{00})^2(2J_{(11)l}^i -E_{(11)l}^i-4KW_ly^i)\nonumber\\
&&+h_{00}(W_0)^2(8B_{(1)l}^i-4D_{(1)l}^i+2J_{(12)l}^i-E_{(12)l}^i) \nonumber\\
&&\hspace{0.2in}+h_{00}W_0(-4D_{(21)l}^i+8B_{(21)l}^i+2J_{(22)l}^i-E_{(22)l}^i+8Kh_{0l}y^i)\nonumber\\
&&+(W_0)^3(16 {{^hR_0}^i}_{0l}+8B_{(22)l}^i-4D_{(22)l}^i+2J_{(23)l}^i-E_{(23)l}^i)\nonumber\\
&&\hspace{0.2in} +(W_0)^2(2J_{(32)l}^i-4D_{(32)l}^i -E_{(32)l}^i ),\nonumber\\
{Q_{(9)}^i}_l&=&(h_{00})^3(2J_{(31)l}^i -4D_{(31)l}^i-E_{(31)l}^i)\\
&&\hspace{0.4in}+(h_{00})^2W_0(2J_{(41)l}^i -E_{(41)l}^i-4D_{(41)l}^i)\nonumber\\
&&+h_{00}(W_0)^4(8B_{(3)l}^i-4D_{(33)l}^i+2J_{(33)l}^i-E_{(33)l}^i) \nonumber\\
&&\hspace{0.2in}+ h_{00}(W_0)^3(2J_{(42)l}^i+8B_{(4)l}^i-4 D_{(42)l}^i-E_{(42)l}^i)\nonumber\\
&&\hspace{0.4in}+ h_{00}(W_0)^2(2J_{(51)l}^i-E_{(51)l}^i)\nonumber\\
&&+(W_0)^5(2J_{(43)l}^i-E_{(43)l}^i)\nonumber\\
&&\hspace{0.2in}+(W_0)^4(2J_{(52)l}^i-4D_{(5)l}^i-E_{(52)l}^i) \nonumber\\
&&\hspace{0.4in}+ (W_0)^3(2J_{(61)l}^i-4D_{(6)l}^i-E_{(61)l}^i)\nonumber\\
{R_{(9)}^i}_l&=&h_{00}W_0(2J_{(62)l}^i-E_{(62)l}^i )\\
&&\hspace{0.6in}+h_{00}(2J_{(7)l}^i-E_{(7)l}^i)\nonumber\\
&&+W_0(2J_{(8)l}^i-E_{(8)l}^i) . \nonumber
\end{eqnarray*}
We call ${P_{(5)}^i}_l$, ${Q_{(9)}^i}_l$ and ${R_{(9)}^i}_l$ the \textit{curvature part},
the \textit{vanishing part} and the \textit{Killing part}, respectively.\\
\textbf{Proposition 2}\hspace{0.5in}
\textit{The necessary and sufficient condition for a Kropina space $(M, F=\alpha^2/\beta=h_{00}/2W_0)$ to be of constant curvature
$K$ is that (3.9) holds good.}
\subsection{The Killing part.}\label{}
We consider the Killing part ${R_{(9)}^i}_l$ and obtain the conclusion that the vector field $W$ is Killing.
First, we have
\begin{eqnarray*}
J_{(62)l}^i &=&-64\texttt{R}_{00}(2\texttt{R}_{00}h_{0l}+h_{00}\texttt{R}_{0l})y^i
-32(\texttt{R}_{00})^2h_{00}{\delta^i}_l,\\
E_{(62)l}^i&=&-64\texttt{R}_{00} h_{00}\texttt{R}_{0l}y^i
-128 (\texttt{R}_{00})^2h_{0l}y^i,\\
J_{(7)l}^i &=&-32\texttt{R}_{00}\{(3\texttt{R}_{00}W_0-4\texttt{S}_0h_{00}) h_{0l}
+ h_{00}\texttt{R}_{00}W_l\}y^i,\\
E_{(7)l}^i &=&-32(\texttt{R}_{00})^2(h_{00}W_ly^i+W_0h_{0l}y^i),\\
2J_{(8)l}^i-E_{(8)l}^i&=&192(\texttt{R}_{00})^2h_{00}h_{0l}y^i.\nonumber
\end{eqnarray*}
Using the above equalities, we get
\begin{eqnarray*}
{R_{(9)}^i}_l =-32h_{00}\texttt{R}_{00}\bigg(W_0(2\texttt{R}_{00}h_{00}{\delta^i}_l
+ 2h_{00}\texttt{R}_{0l}y^i
+7\texttt{R}_{00}h_{0l}y^i)
-8\texttt{S}_0h_{00} h_{0l} y^i
+\texttt{R}_{00}h_{00}W_ly^i\bigg).
\end{eqnarray*}
Substituting the above equality in (3.9) and dividing it by $W_0h_{00}$, we get
\begin{eqnarray}
&&(h_{00})^3{P_{(5)}^i}_l+h_{00}{Q_{(9)}^i}_l\\
&&\hspace{0.5in} -32(W_0)^4\texttt{R}_{00}\bigg(W_0(2\texttt{R}_{00}h_{00}{\delta^i}_l
+ 2h_{00}\texttt{R}_{0l}y^i
+7\texttt{R}_{00}h_{0l}y^i) \nonumber\\
&&\hspace{2.5in} -8\texttt{S}_0h_{00} h_{0l} y^i
+\texttt{R}_{00}h_{00}W_ly^i\bigg)=0.\nonumber
\end{eqnarray}\\
\textbf{Lemma 1}\hspace{0.5in}
\textit{In the equation (3.10), it follows that $\texttt{R}_{00}$ is divisible by $h_{00}$.}\\
($Proof$.)\hspace{0.5in}
Suppose that $\texttt{R}_{00}$ is not divisible by $h_{00}$ and since $(h_{ij})$ is positive definite, $(R_{00})^2$
is not divisible by $h_{00}$.
Taking into account that ${P_{(5)}^i}_l$ and ${Q_{(9)}^i}_l$ are homogeneous polynomials of $y^i$
and that $(W_0)^2$ is not divisible by $h_{00}$, it follows that the equation
\begin{eqnarray*}
h_{0l}y^i=h_{00}{\eta^i}_l,
\end{eqnarray*}
where ${\eta_l}^i(x)$ is a function of $(x^i)$ alone, holds good. Transvecting the above equation by $W^l$, we get
\begin{eqnarray*}
W_0y^i=h_{00}{\eta_l}^i(x)W^l.
\end{eqnarray*}
Since $h_{00}$ is not divisible by $W_0$, the above equation is impossible. \hspace{0.5in} Q.E.D.\\
Therefore, it follows that
$\texttt{R}_{00}$ is divisible by $h_{00}$ and the following equation holds:
\begin{eqnarray*}
\texttt{R}_{00}=c(x)h_{00},
\end{eqnarray*}
where $c(x)$ is a function of $(x^i)$ alone.
Derivating the above equation by $y^i$ and $y^j$, we get
\begin{eqnarray}
W_{i||j}+W_{j||i}=2c(x)h_{ij}.
\end{eqnarray}
Transvecting (3.11) by $W^iW^j$, we get $W_{i||j}W^iW^j=c(x)h_{ij}W^iW^j$ and using $h_{ij}W^iW^j=|W|^2=1$ and
$W_{i||r}W^i=0$, we obtain $c(x)=0$.
Therefore, it follows that the equality
\begin{eqnarray}
\texttt{R}_{ij}=0
\end{eqnarray}
holds good. Hence, we have that $W$ is a Killing vector field.
Therefore, we can state\\
\textbf{Lemma 2}\hspace{0.5in}
\textit{If a Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$, then\\
\hspace{0.5in} (1) \hspace{0.1in} $W(x)$ is a Killing vector field,\\
and then\\
\hspace{0.5in}(2) \hspace{0.1in} Killing part $R^i_{(9)l}=0$.}\\
Using (3.12), the equation (3.10) reduces to
\begin{eqnarray}
(h_{00})^2{P_{(5)}^i}_l+{Q_{(9)}^i}_l=0
\end{eqnarray}
and we have the following equalities:
\begin{eqnarray}
W_{i||j}=\texttt{S}_{ij}, \hspace{0.1in}
\texttt{S}_j=W_{i||j}W^i=0,\hspace{0.1in}
W_{0||j}=\texttt{S}_{0j},\hspace{0.1in}
W_{i||0}=\texttt{S}_{i0},\hspace{0.1in}
W_{0||0}=0.
\end{eqnarray}
\subsection{The vanishing part.}\label{}
In this subsection, we will show that the equality $Q_{(9)l}^i=0$ holds from the conclusion $\texttt{R}_{00}=0$
in the previous subsection.
Using (3.12) and (3.14), the $A's$, $B's$, $C's$, $D's$, $E's$, $H's$ and $J's$ reduce to
\begin{eqnarray*}
&&A_{(1)}^i:=-2{W^i}_{||0},\hspace{0.1in}
B_{(1)l}^i=-2{W^i}_{||0||l},\hspace{0.1in}
B_{(21)l}^i=2{W^i}_{||0}W_{0||l}, \hspace{0.1in}
C_{(0)l}^i=-2{W^i}_{||l},\\
&& C_{(11)l}^i=2{W^i}_{||0}W_l,\hspace{0.1in}
C_{(21)l}^i=-4{W^i}_{||0}h_{0l},\hspace{0.1in}
D_{(1)l}^i=-2{W^i}_{||l||0},\hspace{0.1in} \\
&&D_{(21)l}^i =2{W^i}_{||0||0}W_l+2{W^i}_{||0}W_{l||0},\hspace{0.1in}
D_{(32)l}^i=-4{W^i}_{||0||0}h_{0l},\hspace{0.1in}
E_{(0)l}^i =4{W^i}_{||r}{W^r}_{||l},\\
&&E_{(11)l}^i=-4{W^i}_{||r}{W^r}_{||0}W_l,\hspace{0.1in}
E_{(22)l}^i =8{W^i}_{||0}W_{0||l}
+8{W^i}_{||r} {W^r}_{||0}h_{0l},\\
&&H_{(01)rl}^i =2{W^i}_{||r}W_l+2W_r{W^i}_{||l},\hspace{0.1in}
H_{(11)rl}^i =-4W_r{W^i}_{||0}W_l,\hspace{0.1in}\\
&&H_{(12)rl}^i =-4(h_{0r}{W^i}_{||l}+{W^i}_{||r}h_{0l}+{W^i}_{||0}h_{rl}),\hspace{0.1in}
H_{(21)rl}^i=4(W_r{W^i}_{||0}h_{0l}+h_{0r}{W^i}_{||0}W_l),\\
&&J_{(11)l}^i =-4{W^i}_{||r}{W^r}_{||0}W_l,\hspace{0.1in}
J_{(22)l}^i=8{W^i}_{||r}{W^r}_{||0}h_{0l}+8 {W^i}_{||0}W_{l||0},
\end{eqnarray*}
and the others are zero.
Using the above equalities, we get
\begin{eqnarray*}
&&2J_{(31)l}^i -4D_{(31)l}^i-E_{(31)l}^i=0,\hspace{0.2in}
2J_{(41)l}^i -E_{(41)l}^i-4{D^i}_{(41)l}=0\\
&&8B_{(3)}^i-4D_{(33)l}^i+2J_{(33)l}^i-E_{(33)l}^i=0,\hspace{0.2in}
2J_{(42)l}^i+8B_{(4)}^i-4 D_{(42)l}^i-E_{(42)l}^i=0,\\
&&2J_{(51)l}^i-E_{(51)l}^i=0,\hspace{0.2in}
2J_{(43)l}^i-E_{(43)l}^i=0,\\
&&2J_{(52)l}^i-4D_{(5)l}^i-E_{(52)l}^i=0,\hspace{0.2in}
2J_{(61)l}^i-4D_{(6)l}^i-E_{(61)l}^i=0.
\end{eqnarray*}
Therefore, from Lemma 2, it follows \\
\textbf{Lemma 3}\hspace{0.5in}
\textit{If a Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$, then\\
\hspace{0.5in} (1) \hspace{0.1in}vanishing part $Q^i_{(9)l}=0$,\\
and then\\
\hspace{0.5in}(2) \hspace{0.1in} curvature part $P^i_{(5)l}=0$.}\\
\subsection{The curvature part.}\label{}
\hspace{0.2in}
In this subsection, we will show that Lemma 3 implies that $(M, h)$ is a Riemannian space of constant curvature $K$.
Using the resuls given at the beginning of the previous subsection, we have
\begin{eqnarray*}
-E_{(0)l}^i-4K{\delta^i}_l&=&-4{W^i}_{||r}{W^r}_{||l}-4K{\delta^i}_l,\\
2J_{(11)l}^i -E_{(11)l}^i-4KW_ly^i &=&-4{W^i}_{||r}{W^r}_{||0}W_l-4KW_ly^i, \\
8B_{(1)l}^i-4D_{(1)l}^i+2J_{(12)l}^i-E_{(12)l}^i&=&-16{W^i}_{||0||l}+8{W^i}_{||l||0},\\
-4D_{(21)l}^i+8B_{(21)l}^i+2J_{(22)l}^i-E_{(22)l}^i+8Kh_{0l}y^i
&=& -8{W^i}_{||0||0}W_l +8{W^i}_{||r}{W^r}_{||0}h_{0l} +8Kh_{0l}y^i,\\
16 {{^hR_0}^i}_{0l}+8B_{(22)l}^i-4D_{(22)l}^i+2J_{(23)l}^i-E_{(23)l}^i
&=&16 {{^hR_0}^i}_{0l},\\
2J_{(32)l}^i-4D_{(32)l}^i -E_{(32)l}^i&=&16{W^i}_{||0||0}h_{0l}.
\end{eqnarray*}
Therefore, from (2) of Lemma 3 and the above equalities we have
\begin{eqnarray}
&&-\frac{1}{4}{P_{(5)}^i}_l=(h_{00})^2W_0({W^i}_{||r}{W^r}_{||l}+K{\delta^i}_l )
+(h_{00})^2({W^i}_{||r}{W^r}_{||0}W_l+KW_ly^i )\\
&& +2h_{00}(W_0)^2(2{W^i}_{||0||l}-{W^i}_{||l||0})
+2h_{00}W_0({W^i}_{||0||0}W_l -{W^i}_{||r}{W^r}_{||0}h_{0l} -Kh_{0l}y^i)\nonumber\\
&& -4(W_0)^3 \cdot {{^hR_0}^i}_{0l}
-4(W_0)^2{W^i}_{||0||0}h_{0l} =0,\nonumber
\end{eqnarray}
First, we consider the term $(h_{00})^2({W^i}_{||r}{W^r}_{||0}W_l+KW_ly^i )$ which does not contain $W_0$.
Taking into account that $(h_{00})^2$ is not divisible by $W_0$, the following equality must hold good
\begin{eqnarray}
{W^i}_{||r}{W^r}_{||0}W_l+KW_ly^i=W_0{c_l}^i(x),
\end{eqnarray}
where ${c_l}^i(x)$ are functions of $(x^i)$ alone, must hold.
Transvecting (3.16) by $y^l$ and dividing it by $W_0$, we get
\begin{eqnarray*}
{W^i}_{||r}{W^r}_{||0}+Ky^i={c_l}^i(x)y^l.
\end{eqnarray*}
Derivating it by $y^l$, we have
\begin{eqnarray*}
{W^i}_{||r}{W^r}_{||l}+K{\delta^i}_l={c_l}^i(x).
\end{eqnarray*}
Substituting the above equality in (3.16), we get
\begin{eqnarray*}
{W^i}_{||r}{W^r}_{||0}W_l+KW_ly^i=W_0({W^i}_{||r}{W^r}_{||l}+K{\delta^i}_l).
\end{eqnarray*}
Transvecting the above equality by $W^l$, we get
\begin{eqnarray*}
{W^i}_{||r}{W^r}_{||0}+Ky^i=KW_0W^i,
\end{eqnarray*}
where we have used ${W^i}_{||r}{W^r}_{||l}W^l=0$.
Derivating the above equality by $y^l$, we get
\begin{eqnarray}
{W^i}_{||r}{W^r}_{||l}=KW_lW^i-K{\delta^i}_l.
\end{eqnarray}
Substituting (3.17) in (3.15) and dividing it by $2W_0$, it follows
\begin{eqnarray}
&&K(h_{00})^2W_lW^i +h_{00}W_0(2{W^i}_{||0||l}-{W^i}_{||l||0}) \\
&&\hspace{0.6in} +h_{00}({W^i}_{||0||0}W_l -KW_0W^ih_{0l})
-2(W_0)^2 \cdot {{^hR_0}^i}_{0l}
-2W_0{W^i}_{||0||0}h_{0l} =0.\nonumber
\end{eqnarray}
Transvecting the above equality by $h_{0i}$, we get
\begin{eqnarray}
{W_0}_{||l||0}=K(h_{00}W_l - W_0h_{0l}).
\end{eqnarray}
Using $W_{i||j}+W_{j||i}=0$ and (3.19), we have
\begin{eqnarray}
{W_l}_{||0||0}=-K(h_{00}W_l - W_0h_{0l}).
\end{eqnarray}
Derivating (3.19) by $y^i$, we have
\begin{eqnarray*}
{W_i}_{||l||0}+{W_0}_{||l||i} =K(2h_{0i}W_l - W_ih_{0l}- W_0h_{il}).
\end{eqnarray*}
From the above equality, we have
\begin{eqnarray}
{W_i}_{||0||l} =-{W_i}_{||l||0}-K(2h_{0l}W_i - W_lh_{0i}- W_0h_{li}).
\end{eqnarray}
Using (3.20) and (3.21), the equality (3.18) reduces to
\begin{eqnarray*}
3h_{00}(KW_ly^i -Kh_{0l}W^i -{W^i}_{||l||0} )
-2W_0( {{^hR_0}^i}_{0l} + K y^i h_{0l}- K h_{00}{\delta^i}_l)=0.
\end{eqnarray*}
Since $h_{00}$ is not divisible by $W_0$, it follows that the equality
\begin{eqnarray}
{{^hR_0}^i}_{0l}+K h_{0l}y^i-Kh_{00}{\delta^i}_l=h_{00}{d^i}_l(x),
\end{eqnarray}
where ${d^i}_l(x)$ are functions of $(x^i)$ alone, must hold. Transvecting the above equality by $y^l$, we get
${d^i}_l(x)y^l=0$.
Derivating the above equality by $y^l$, we get ${d^i}_l(x)=0$. Substituting it in (3.22), we get
\begin{eqnarray}
{{^hR_0}^i}_{0l}+K h_{0l}y^i-Kh_{00}{\delta^i}_l=0.
\end{eqnarray}
We can rewrite the above equality as
\begin{eqnarray*}
{{^hR_0}^i}_{0l}&=&K(h_{00}{\delta^i}_l- h_{0l}y^i).
\end{eqnarray*}
Derivating (3.23) by $y^j$ and $y^k$, we get
\begin{eqnarray}
{{^hR_j}^i}_{kl}+{{^hR_k}^i}_{jl}=K(2h_{jk}{\delta^i}_l-h_{jl}{\delta^i}_k-h_{kl}{\delta^i}_j),
\end{eqnarray}
and interchanging $j$ and $l$, we obtain
\begin{eqnarray}
{{^hR_l}^i}_{kj}+{{^hR_k}^i}_{lj}=K(2h_{lk}{\delta^i}_j-h_{lj}{\delta^i}_k-h_{kj}{\delta^i}_l).
\end{eqnarray}
Subtracting (3.25) from (3.24), we get
\begin{eqnarray*}
{{^hR_j}^i}_{kl}+2{{^hR_k}^i}_{jl}-{{^hR_l}^i}_{kj}=3K(h_{jk}{\delta^i}_l-h_{kl}{\delta^i}_j).
\end{eqnarray*}
Since the left-hand side of the above equality can be changed as follows:
\begin{eqnarray*}
{{^hR_j}^i}_{kl}+2{{^hR_k}^i}_{jl}-{{^hR_l}^i}_{kj}
=2{{^hR_k}^i}_{jl}-{{^hR_j}^i}_{lk}-{{^hR_l}^i}_{kj}
=2{{^hR_k}^i}_{jl}+{{^hR_k}^i}_{jl}
=3{{^hR_k}^i}_{jl},
\end{eqnarray*}
we obtain
\begin{eqnarray*}
{{^hR_k}^i}_{jl}=K(h_{jk}{\delta^i}_l-h_{kl}{\delta^i}_j).
\end{eqnarray*}
This means that the Riemannian space $(M, h)$ is of constant curvature $K$.\\
Therefore, we obtain \\
\textbf{Theorem 2}\hspace{0.5in}
\textit{Let $M$ be an $n(\ge 2)$-dimensional Riemannian manifold. Put $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ and
$\beta=b_i(x)y^i$.
Let $(M, \alpha^2/\beta)$ be a Kropina space and define a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and
a vector field $W$ with $|W|=1$ by (1.2) and (1.3).}
\textit{If the Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$, then the vector field $W$ is a
Killing one and the Riemannian space $(M, h)$ is of constant curvature $K$ .}
\subsection{The converse of Theorem 2.}\label{}
\hspace{0.2in}
Let $(M, \alpha^2/\beta)$ be a Kropina space and define a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and
a vector field $W$ with $|W|=1$ by (1.2) and (1.3).
Suppose that the vector field $W$ is a Killing one and that the Riemannian space $(M, h)$ is of constant curvature $K$.
To prove that the Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$, we have only to show that
the equality (3.9) holds.
Since the vector field $W$ is a Killing one, we have $\texttt{R}_{00}=0$.
Taking into account (2) of Lemma 2 and (1) of Lemma 3, the Killing part $R$ and the vanishing part $Q$ vanishes
respectively.
Therefore, we have only to show that the curvature part $P^i_{(5)l}$ vanishes. At the rest of this subsection, we will prove it.
The curvature part $P^i_{(5)l}$ is defined by (3.15).
First, we give the following Lemma 4:\\
\textbf{Lemma 4}\hspace{0.5in}
\textit{ For a Killing vector field $W=W^i(\partial/\partial x^i)$ of constant length $|W|=1$, the equality
\begin{eqnarray}
W_{i||j||k}={W_r}\hspace{0.05in}{^h{{R_k}^r}_{ij}}
\end{eqnarray}
holds good.}\\
($Proof$.)\hspace{0.5in}
From the Ricci's formula, it follows
\begin{eqnarray}
W_{i||j||k}-W_{i||k||j}=-{W_r}\hspace{0.05in} {^h{{R_i}^r}_{jk}}.
\end{eqnarray}
On the other hand, since $W$ is a Killing vector field, we have
\begin{eqnarray*}
W_{i||j}+W_{j||i}=0.
\end{eqnarray*}
Applying the $h$-covariant derivative ${_{||k}}$ to the above equality, we get
\begin{eqnarray*}
W_{i||j||k}+W_{j||i||k}=0.
\end{eqnarray*}
Replacing $i$, $j$, $k$ by $j$, $k$, $i$ and $k$, $i$, $j$ respectively, we obtain the following equalities:
\begin{eqnarray*}
W_{j||k||i}+W_{k||j||i}&=&0,\\
W_{k||i||j}+W_{i||k||j}&=&0.
\end{eqnarray*}
Subtracting the second equality from the first equality and adding the third equality to it, we get
\begin{eqnarray}
W_{i||j||k}+W_{i||k||j}-{W_r}\hspace{0.05in} {^h{{R_j}^r}_{ik}}-{W_r}\hspace{0.05in} {^h{{R_k}^r}_{ij}}=0.
\end{eqnarray}
From (3.27) and (3.28), we get
\begin{eqnarray*}
2W_{i||j||k}=-{W_r}\hspace{0.05in} {^h{{R_i}^r}_{jk}}
+{W_r}\hspace{0.05in} {^h{{R_j}^r}_{ik}}
+{W_r}\hspace{0.05in} {^h{{R_k}^r}_{ij}}.
\end{eqnarray*}
Using the formula
\begin{eqnarray*}
{^h{{R_i}^r}_{jk}}+{^h{{R_j}^r}_{ki}}+{^h{{R_k}^r}_{ij}}=0,
\end{eqnarray*}
we have
\begin{eqnarray*}
2W_{i||j||k}&=&{W_r}\hspace{0.05in} {^h{{R_j}^r}_{ki}}
+{W_r}\hspace{0.05in}{^h{{R_k}^r}_{ij}}
+{W_r}\hspace{0.05in} {^h{{R_j}^r}_{ik}}
+{W_r}\hspace{0.05in} {^h{{R_k}^r}_{ij}}\\
&=&2 {W_r}\hspace{0.05in}{^h{{R_k}^r}_{ij}}
\end{eqnarray*}
that is, (3.26) holds good. \hspace{1in} Q.E.D.\\
From the assumption that the Riemannian space $(M, h)$ is of constant curvature, we have
\begin{eqnarray}
{^h{{R_k}^r}_{ji}}=K(h_{kj}{\delta^r}_i-h_{ki}{\delta^r}_j).
\end{eqnarray}
Using the above equality and (3.26), we get
\begin{eqnarray}
{W^i}_{||j||k} =K({\delta^i}_kW_j-h_{kj}W^i)
\end{eqnarray}
and from here and ${y^i}_{||j}=0$ (See, Remark 1), it follows
\begin{eqnarray}
{W^i}_{||0||l}&=&K({\delta^i}_lW_0-h_{l0}W^i),\nonumber\\
{W^i}_{||0||0}&=&K(y^iW_0-h_{00}W^i),\\
{W^i}_{||l||0}&=&K(y^iW_l-h_{0l}W^i).\nonumber
\end{eqnarray}
From (3.29), we have
\begin{eqnarray}
{^h{{R_0}^i}_{0l}}=K(h_{00}{\delta^i}_l-h_{0l}y^i)
\end{eqnarray}
and applying the $h$-covariant derivative ${_{||i}}$ to the equality $|W|^2=W_rW_sh^{rs}=1$, we get
\begin{eqnarray*}
W_{r||i}W^r=-W_{i||r}W^r=0.
\end{eqnarray*}
Furthermore, applying the $h$-covariant derivative ${_{||l}}$ to the above equality, we obtain
\begin{eqnarray*}
W_{i||r}{W^r}_{||l}+W_{i||r||l}W^r=0.
\end{eqnarray*}
From the above equality and (3.30), we have
\begin{eqnarray}
W_{i||r}{W^r}_{||l}&=&-W_{i||r||l}W^r\\
&=&K(h_{lr}W_i-h_{li}W_r)W^r \nonumber\\
&=&K(W_lW_i-h_{li}).\nonumber
\end{eqnarray}
Substituting the equalities (3.31)-(3.33) in (3.15), we can easily recognize the curvature part ${P_{(5)}^i}_l=0$.
Therefore, (3.9) holds good. Hence, from Proposition 2, we get\\
\textbf{Theorem 3}\hspace{0.5in}
\textit{Let $M$ be an $n(\ge 2)$-dimensional Riemannian space. Put $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ and $\beta=b_i(x)y^i$.
Let $(M, \alpha^2/\beta)$ be a Kropina space and define a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and
a vector field $W=W^i(\partial/\partial x^i)$ of constant length $|W|=1$ by (1.2) and (1.3).}
\textit{If the vector field $W=W^i(\partial/\partial x^i)$ is a Killing one and the Riemannian space $(M, h)$ is of
constant curvature $K$, the Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$}.\\
From Theorem 2 and Theorem 3, we have\\
\textbf{Theorem 4}\hspace{0.5in}
\textit{Let $(M, \alpha^2/\beta)$ be an $n(\ge 2)$-dimensional Kropina space, where $\alpha^2=a_{ij}(x)y^iy^j$,
$\beta=b_i(x)y^i$ and the matrix $(a_{ij})$ is positive definite.
For the Kropina space, we define a new Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector field
$W=W^i(\partial/\partial x^i)$ of constant length $|W|=1$ on $M$ by (1.2) and (1.3).}
\textit{Then, the Kropina space $(M, \alpha^2/\beta)$ is of constant curvature $K$ if and only if the following conditions
hold:}
\textit{(1)$W_{i||j}+W_{j||i}=0$, that is, $W=W^i(\partial/\partial x^i)$ is a Killing vector field.}
\textit{(2)The Riemannian space $(M, h)$ is of constant curvature $K$.}\\
Let $(M, F=\alpha^2/\beta)$ be an $n(\ge 2)$-dimensional Kropina space. From Theorem 1, for this Kropina metric $F=\alpha^2/\beta$, we can
define a Riemannian metric $h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector field $W=W^i(\partial/\partial x^i)$ of constant
length 1 on $M$
by (1.2) and (1.3). We suppose that the vector field $W$ is a Killing one. Then, we have ${\texttt{R}}_{00}=0$.
From this assumption, we get the second equation of (3.14), that is, ${\texttt{S}}_0=0$.
Substituting ${\texttt{R}}_{00}=0$, ${\texttt{S}}_0=0$ and $F=h_{00}/(2W_0)$ in (2.6), we obtain the equation $\Phi^i=-F{\texttt{S}^i}_0$.
Substituing this in (2.4), we get\\
\textbf{Theorem 5}\hspace{0.5in}
\textit{Let $(M, F=\alpha^2/\beta)$ be an $n(\ge 2)$-dimensional Kropina space. We define a Riemannian metric
$h=\sqrt{h_{ij}(x)y^iy^j}$ and a vector field $W=W^i(\partial/\partial x^i)$ of constant length $|W|=1$ on $(M,h)$
by (1.2) and (1.3).}
\textit{Suppose that the vector field $W$ is a Killing one, then the coefficients $G^i$ of the geodesic spray of the
Kropina space $(M, \alpha^2/\beta)$ is written as follows:
\begin{eqnarray}
2G^i&=&{{^h\gamma_0}^i}_0-2F{\texttt{S}^i}_0,
\end{eqnarray}
where ${{^h\gamma_j}^i}_k$ are Christoffel symbols of the Riemannian space $(M, h)$.}\\
\textbf{Remark 2}\hspace{0.5in}
\textit{The geodesic spray of the Randers space $(M, \alpha+\beta)$ is given in the subsection 2.3 ([5] p.5).
Comparing to this, the geodesic spray of the Kropina space $(M, F=\alpha^2/\beta)$ is in a very simple form (3.33).} \\
\textit{Acknowledgments.}\hspace{0.5in}
The authors would like to express their sincere thanks to Professor Dr. S. V. Sabau and Professor Dr. H. Shimada
for the valuable advices, the efforts to check all the calculation and the continuous encouragements.
\vspace {0.5in}
\begin{center}
References
\end{center}
\begin{description}
\item[[1]] S. B\'acs\'o, X. Cheng and Z. Shen : Curvature properties of $(\alpha, \beta)$-metrics,
Advanced Studies in Pure Mathematics 48, 2007, Finsler Geometry, Sappro 2005 -
In Memory of Makoto Matsumoto, 73-110.
\item[[2]] M. Matsumoto : Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Saikawa, Otsu,
Japan, 1986.
\item[[3]] M. Matsumoto: Finsler spaces of constant curvature with Kropina metric, Tensor N.S., 50(1991), 194-201.
\item[[4]] M. Matsumoto: Theory of Finsler spaces with $(\alpha, \beta)-$metric, Rep. Math. Phys., 31(1992), 43-83.
\item[[5]] C. Robles : Geodesic in Randers spaces of constant curvature, arXiv:math/0501358v2 [math.DG] 8 July 2005.
\item [[6]] S. Tachibana : Riemannian geometry (Japanese), Modern mathematical lecture 15, Asakura Shoten,
Tokyo, 1967.
\item[[7]] R. Yoshikawa and K. Okubo: Kropina spaces of constant curvature, Tensor N.S., 68(2007), 190-203.
\end{description}
\vspace{0.1in}
\begin{center}
\hspace{0.7in} Hachiman technical \hspace{1in} Faculty of Education \\
\hspace{0.8in} High School \hspace{1.4in} Shiga University \\
\hspace{0.2in} 5 Nishinosho-cho Hachiman \hspace{0.8in} 2-5-1 Hiratsu Otsu \\
\hspace{0.6in} 523-0816 Japan \hspace{1.3in} 520-0862 Japan \\
\hspace{0.7in} E-mail: ryozo@e-omi.ne.jp \hspace{0.4in} E-mail: okubo@edu.shiga-u.ac.jp
\end{center}
\end{document}
\end{document} |
1,314,259,993,025 | arxiv | \section{Introduction}
\label{sec1}
The study of the decays of $B$ mesons is important and interesting
for the determination of the flavor parameters of the Standard Model
(SM), the exploration of $CP$ violation, the search of new physics
beyond SM, etc. In recent years, theoretical studies of $B_{u,d}$
mesons have been investigated widely in the literatures. They are
tested and supported by the experimental data collected by the detectors
at the $e^{+}e^{-}$ colliders, such as the CLEO, Babar, and Belle.
With the bright hope arising from the startup of the CERN
Large Hadron Collider (LHC) \cite{epjc65p111}, the heavier $B_{s}$ and
$B_{c}$ mesons could be produced abundantly and studied in detail at
the hadron colliders.
It is estimated that one could expect around $5$ ${\times}$ $10^{10}$
``self-tagging'' $B_{c}$ events per year at the LHC \cite{0412158} due to
a relatively large production cross section \cite{prd71p074012} plus the
huge luminosity ${\cal L}$ $=$ $10^{34}$ $\hbox{cm}^{-2}\hbox{s}^{-1}$ \cite{lhc}.
There seems to exist a real possibility to study not only some $B_{c}$
rare decays, but also $CP$ violation and polarization asymmetries.
The study of the $B_{c}$ mesons will highlight the advantages of $B$
physics.
The $B_{c}$ mesons are the ``double heavy-flavored'' binding systems
and share many features with the heavy quarkonia.
The first observation of the $B_{c}$ mesons at the Tevatron \cite{CDF1998}
provokes the physicist's particular interest in them. Many studies
and investigation of the properties of the $B_{c}$ mesons have been
made, and will be further scrutinized by the LHC experiments.
Because the $B_{c}$ mesons lie below the $BD$ threshold (here we only
discuss the lightest $1^{1}S_{0}$ ground state pseudoscalar $B_{c}$
mesons, excluding their excited states) and carry flavors, they
cannot annihilate into gluon and/or photon so are stable for the
strong and/or electromagnetic interaction.
Because of the flavor quantum numbers $B$ $=$ $-C$ $=$ ${\pm}1$,
the $B_{c}$ mesons can decay through the weak interaction only.
The $B_{c}$ mesons have more decay modes than the $B_{u,d,s}$ mesons
due to several reasons.
One is that many decay modes, such as $B_{c}$ ${\to}$
$B_{u,d,s}$ $+$ $X$, are only accessible by $B_{c}$ mesons because of
their sufficiently large masses. Another is that the $B_{c}$ mesons carry
open flavors, so either $b$ or $c$ quarks can decay individually.
The potential decays of the $B_{c}$ mesons permit us to over-constrain
quantities determined by the $B_{u,d,s}$ meson decays.
The decays of $B_{c}$ mesons can be divided into three classes:
(1) the $b$-quark decay (i.e. $b$ ${\to}$ $q_{_{U}}$, where the up-type quark
$q_{_{U}}$ $=$ $u$, $c$) accompanied with the spectator $c$-quark,
(2) the $c$-quark decay (i.e. $c$ ${\to}$ $q_{_{D}}$, where the down-type quark
$q_{_{D}}$ $=$ $d$, $s$) accompanied with the spectator $b$-quark, and
(3) the annihilation channel (i.e. $B_{c}^{-}$ ${\to}$ ${\ell}^{-}\tilde{\nu}_{\ell}$,
$q_{_{D}}\bar{q}_{_{U}}$, where the lepton ${\ell}^{-}$ $=$ $e^{-}$,
${\mu}^{-}$, ${\tau}^{-}$).
Among the multitudinous $B_{c}$ decays, the weak annihilation channels are
expected to take ${\sim}$ $10\%$ shares according to the estimates in
\cite{0412158,0211021} for which a major part comes from the tree weak
annihilation process $B_{c}^{-}$ ${\to}$ $s\bar{c}$ which is not
helicity-suppressed because of the large charm quark mass and
produces a large weak annihilation branching ratio with charm in
the final state, while the charmless pure weak annihilation decay
$B_{c}$ ${\to}$ $KK$ is helicity-suppressed like the $B_{d}$ ${\to}$
$KK$ decay and would have a very small branching ratio.
It is highly expected that the LHC experiments might shed light on
a better understanding of weak annihilation processes for $B_{c}$ mesons.
In recent years, several attractive methods have been proposed to
study the nonleptonic $B$ decays,
such as the QCD factorization (QCDF) \cite{0006124},
perturbative QCD method (pQCD) \cite{9607214,9701233,0004004},
soft and collinear effective theory \cite{prd63p114020,prd65p054022}, etc.
Here, we would like to investigate the charmless pure weak
annihilation $B_{c}$ ${\to}$ $KK$ decay with the pQCD approach
due to several reasons.
(1) One reason is that fits of nonleptonic charmless decays $B_{u,d}$
${\to}$ $PP$, $PV$ without taking into account weak annihilation
contributions are generally of poor quality \cite{npb675p333}
(here $P$ and $V$ denote the lightest ground pseudoscalar and vector
mesons, respectively).
Our present understanding of the weak annihilation contributions
remains limited and unclear.
So the pure weak annihilation processes, such as $B_{c}$ ${\to}$
$KK$ decays, are interesting and worthy of study, which will
certainly help us to improve our understanding of the weak
annihilation contributions.
(2) Another is that due to both kinematic improvement from the large
phase spaces and dynamic enhancement of the CKM factor
${\vert}V_{cb}V_{ud}^{\ast}{\vert}$, the $B_{c}$ ${\to}$ $KK$ decay is
expected to have a large branching ratio among two-body nonleptonic charmless
$W$-annihilation $B_{c}$ ${\to}$ $PP$ processes. In addition to the absence
of penguin operators for the tree annihilation process $B_{c}$ ${\to}$ $KK$,
final state interactions arising from soft gluon exchanges are expected
to be extremely small because of the large momenta of the final $K$
mesons. Therefore a relatively accurate estimation of annihilation
contributions could be obtained effectively from the charmless
$B_{c}$ ${\to}$ $KK$ decay.
(3) Still another is that Ref.\cite{prd65p114007} obtains
a very large $B_{c}$ ${\to}$ $KK$ branching ratio, about $1.6\%$, at
$4$ orders of magnitude bigger than the estimate ${\cal O}(10^{-6})$
of Ref.\cite{prd80p114031}, but this estimate is not valid because
Ref.\cite{prd65p114007} in their calculation incorrectly uses the
measured penguin-dominated $B_{u}^{\pm}{\to}{\pi}^{\pm}K$ branching
ratio while the decay $B_{c}$ ${\to}$ $KK$ is a pure tree weak
annihilation and should be related to $B_{d}$ ${\to}$ $KK$.
In addition, the branching ratio of the charmless decay $B_{c}$
${\to}$ $KK$ is estimated to be ${\cal O}(10^{-8})$ with the QCDF
approach \cite{prd80p114031}.
Recently, this charmless decay is also studied with the pQCD
approach and its branching ratio is ${\cal O}(10^{-7})$ with the
off-mass-shell final states \cite{09121163}, which is the same
order of magnitude as ours obtained in this paper with the on-shell
final states.
This paper is organized as follows : In Section \ref{sec2},
we will discuss the theoretical framework and give the decay amplitudes
for $B_{c}$ ${\to}$ $KK$ with the perturbative QCD approach.
In our calculation, we shall ignore the final state interactions
because the final states have very large momenta and move far away
before soft gluon exchange.
Section \ref{sec3} is devoted to the numerical result of the branching
ratio. Finally, we summarize in Section \ref{sec4}.
\section{Theoretical framework and the decay amplitudes}
\label{sec2}
\subsection{The effective Hamiltonian}
\label{sec21}
Using the Operator Product Expansion approach and renormalization
group (RG) equation, the low-energy effective Hamiltonian for
$B_{c}$ ${\to}$ $KK$ decay can be written as
\begin{equation}
{\cal H}_{eff}\,=\,\frac{G_{F}}{\sqrt{2}} V_{cb}V_{ud}^{\ast} \Big\{
C_{1}({\mu})Q_{1}+ C_{2}({\mu})Q_{2} \Big\} + \hbox{H.c.},
\label{eq:hamiltonian}
\end{equation}
where $G_{F}$ is the Fermi coupling constant for electroweak interactions.
$V_{cb}V_{ud}^{\ast}$ is the CKM factor accounting for the strengths of
the nonleptonic $B_{c}$ decays.
$C_{i}({\mu})$ are Wilson coefficients at the renormalization scale
${\mu}$ which have been evaluated to the next-to-leading order with
the perturbation theory. The local tree operators are process dependent.
Their expressions are defined as
\begin{equation}
Q_{1}=\big[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}\big]
\big[\bar{d}_{\beta} {\gamma}^{\mu}(1-{\gamma}_{5})u_{\beta} \big],~~~~~
Q_{2}=\big[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta} \big]
\big[\bar{d}_{\beta} {\gamma}^{\mu}(1-{\gamma}_{5})u_{\alpha}\big],
\label{eq:operator}
\end{equation}
where ${\alpha}$, ${\beta}$ are $SU(3)$ color indices.
The most difficult problem in theoretical calculation of nonleptonic
charmless decay $B_{c}$ ${\to}$ $KK$ is how to evaluate the hadronic
matrix elements ${\langle}KK{\vert}Q_{1,2}{\vert}B_{c}{\rangle}$
properly and accurately.
\subsection{Hadronic matrix elements}
\label{sec22}
For convenience, the kinematics variables are described in the terms of
the light cone coordinate. The momenta of the valence quarks and hadrons
in the rest frame of the $B_{c}$ meson are defined by
\[ \begin{array}{lclcl}
p_{B_{c}^{-}}\,=\,p_{1}\,=\,\frac{m_{_{B_{c}}}}{\sqrt{2}}(1,1,\vec{0}_{\perp}),
&~~& k_{1}=x_{1}p_{1}+(0,0,\vec{k}_{1{\perp}}), &~~& \\
p_{K^{-}}\,=\,p_{2}\,=\,\frac{m_{_{B_{c}}}}{\sqrt{2}}(1,0,\vec{0}_{\perp}), &&
k_{2}=x_{2}p_{2}+(0,0,\vec{k}_{2{\perp}}), & & n_{2}=(1,0,0), \\
p_{K^{0}}\,=\,p_{3}\,=\,\frac{m_{_{B_{c}}}}{\sqrt{2}}(0,1,\vec{0}_{\perp}), &&
k_{3}=x_{3}p_{3}+(0,0,\vec{k}_{3{\perp}}), & & n_{3}=(0,1,0),
\end{array} \]
where $n_{2}{\cdot}n_{3}$ $=$ $1$. The null vectors $n_{2}$ and $n_{3}$ are
the plus and minus directions, respectively. $k_{1}$ is the momentum of $c$
quark in the $B_{c}$ meson. $k_{2}$ and $k_{3}$ are the momenta of the light
non-strange quark in the $K^{-}$ and $K^{0}$ mesons, respectively.
$\vec{k}_{i{\perp}}$ denotes the transverse momentum. $x_{i}$ denotes the
longitudinal momentum fraction of the valence quark.
The calculation of the hadronic matrix elements is difficult due to the
nonperturbative effects arising from the strong interactions.
Phenomenologically, using the Brodsky-Lepage approach \cite{prd22p2157},
a modified perturbative QCD formalism has been proposed recently under
the $k_{T}$ factorization framework \cite{9607214,9701233,0004004}.
Taking into account the transverse momentum of the valence quarks in
the hadrons, the Sudakov factors are introduced to modify the endpoint
behavior of the hadronic matrix elements. The amplitudes are factorized
into three convolution parts : the ``harder'' functions, the heavy quark
decay subamplitudes, and the nonperturbative meson wave functions,
which are characterized by the $W^{\pm}$ boson mass $m_{W}$, the
typical scale $t$ of the decay processes, and the hadronic scale
${\Lambda}_{QCD}$, respectively. The pQCD approach has been
extensively applied to study semileptonic and nonleptonic $B$ decays
with phenomenological results. More information about pQCD approach
can be found in \cite{9607214,9701233,0004004}.
The final decay amplitudes can be expressed as
\begin{equation}
{\cal A}(B_{c}^{-}{\to}K^{-}K^{0})\, {\propto}\,
C(t){\otimes}H(t){\otimes}{\Phi}_{B_{c}^{-}}(x_{1},b_{1}) {\otimes}
{\Phi}_{K^{-}}(x_{2},b_{2}){\otimes}{\Phi}_{K^{0}}(x_{3},b_{3}),
\label{eq:am01}
\end{equation}
where the Wilson coefficient $C(t)$ is calculated in perturbative theory
at the scale of $m_{W}$ and evolved down to the typical scale $t$ using the
RG equations. ${\otimes}$ denotes the convolution over parton kinematic
variables. $H(t)$ is the hard-scattering subamplitude which is dominated
by hard gluon exchange and can be factorized. The universal wave functions
${\Phi}(x,b)$ absorb nonperturbative long-distance dynamics, which can be
extracted from experiments or constrained by lattice calculation and QCD
sum rules. $b$ is the conjugate variable of the transverse momentum of
the valence quark of the meson. According to the arguments in
\cite{9607214,9701233,0004004}, the amplitude of Eq.(\ref{eq:am01}) is
free from the renormalization scale dependence.
\subsection{Bilinear operator matrix elements}
\label{sec23}
Within the pQCD framework, the long-distance hadronic information is
contained by the the so-called light-cone distribution amplitudes (LCDAs)
which are defined from hadron-to-vacuum matrix elements of nonlocal
bilinear operators. Although LCDAs are not calculable in QCD perturbation
theory, some of their properties are well understood for both light
and heavy mesons. For example, the LCDAs for the $K$ meson including higher-twist
contributions are systematically presented in \cite{jhep0605004}.
In our calculation, we only consider two-particle (valence quarks) twist-2
and twist-3 LCDAs for $K$ mesons, and neglect contributions from higher
Fock states. The LCDAs for $K$ mesons are written as
\begin{equation}
{\langle}K(p){\vert}\bar{s}_{\alpha}(0)q_{\beta}(z){\vert}0{\rangle}=
\!\frac{i}{\sqrt{2N_{c}}}{\int}_{0}^{1}\!{\bf d}x\,e^{ixp{\cdot}z}
\Big\{\!{\gamma}_{5} \not{\!\!p}\,{\phi}_{K}^{a}(x)
+\!{\gamma}_{5}{\mu}_{K}\! \Big[ {\phi}_{K}^{p}(x)
-(\not{\!n}_{_{+}}\!\!\not{\!n}_{_{\!-}}\!-1)
{\phi}_{K}^{t}(x)\Big]\Big\}_{{\beta}{\alpha}}
\end{equation}
where $N_{c}$ is the color number.
The parameter ${\mu}_{K}$ is the chiral factor ${\mu}_{K}$ $=$
$m_{K}^{2}/(m_{s}+m_{q})$. The null vector $n_{_{+}}$ and $n_{_{-}}$
are parallel to $p$ and $z$, respectively. The expressions of the
twist-2 LCDAs ${\phi}_{K}^{a}$ and the twist-3 LCDAs ${\phi}_{K}^{p}$,
${\phi}_{K}^{t}$ are collected in Appendix \ref{app01}.
Unlike the ${\pi}$ and $K$ mesons, our knowledge of the LCDAs for
$B_{c}$ mesons has been relatively poor until recently (for a recent view,
see \cite{jhep0804061}), but we know that the $B_{c}$ mesons are
composed of heavy valence quark both $b$ and $c$. Given $m_{B_{c}}$
${\approx}$ $m_{b}$ $+$ $m_{c}$, the $B_{c}$ mesons can be described
approximately by nonrelativistic dynamics. In this paper, we
will take
\begin{equation}
{\langle}0{\vert}\bar{c}_{\alpha}(z)b_{\beta}(0){\vert}B_{c}^{-}(p_{1}){\rangle}
=\frac{if_{B_{c}}}{4N_{c}}{\int}{\bf d}x_{1}\,{\rm e}^{-ix_{1}p_{1}{\cdot}z}
\Big[ \Big(\!\not{\!\!p}_{1}\!+\!m_{B_{c}} \Big) {\gamma}_{5} {\phi}_{B_{c}}(x_{1})
\Big]_{{\beta}{\alpha}},
\label{eq:wf-bc-01}
\end{equation}
where $f_{B_{c}}$ is the decay constant of the $B_{c}$ meson.
As the arguments in \cite{jhep0804061}, this simplest form,
${\phi}_{B_{c}}(x)$ $=$ ${\delta}(x-m_{c}/m_{B_{c}})$, is the
two-particle nonrelativistic LCDAs at the tree level where both heavy
valence quarks just share the total momentum of the $B_{c}$ mesons
according to their masses. For a rough estimation of the branching
ratio for $B_{c}$ ${\to}$ $KK$ decay, we will take the simplest
form as an approximation, and neglect the relativistic corrections
and contributions from higher Fock states.
\subsection{The decay amplitudes}
\label{sec24}
The $B_{c}$ ${\to}$ $KK$ decay is the pure annihilation process.
According to the effective Hamiltonian Eq.(\ref{eq:hamiltonian}),
the lowest order Feynman diagrams are shown in FIG.\ref{fig1},
where (a) and (b) are nonfactorizable topologies, (c) and (d) are
factorizable topologies. After a straightforward calculation using
the modified perturbative QCD formalism Eq.(\ref{eq:am01}), we
find that the contributions of factorizable topologies are zero,
which is a result of exact isospin symmetry. The decay amplitude
comes only from the nonfactorizable topologies, and can be
written as
\begin{eqnarray}
& & {\cal A}(B_{c}^{-}{\to}K^{-}K^{0}) \nonumber \\
&=&-i\frac{G_{F}8{\pi}C_{F}f_{B_{c}}m_{B_{c}}^{4}}{\sqrt{2}N_{c}}
V_{cb}V_{ud}^{\ast}{\int}_{0}^{1}{\bf d}x_{1}{\bf d}x_{2}{\bf d}x_{3}
{\int}_{0}^{\infty}b_{1}{\bf d}b_{1}{\int}_{0}^{\infty}b_{2}{\bf d}b_{2}\,
{\phi}_{B_{c}}(x_{1}) \nonumber \\ &{\times}& \Big\{
{\alpha}_{s}(t_{a})C_{2}(t_{a})E(t_{a})H({\Delta},{\alpha},b_{1},b_{2})
\Big[ {\phi}_{K^{-}}^{a}{\phi}_{K^{0}}^{a}\left(r_{b}+x_{1}-x_{3}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}}{\phi}_{K^{-}}^{p}{\phi}_{K^{0}}^{p}
\left( 4r_{b}+2x_{1}-x_{2}-x_{3}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}} \left( {\phi}_{K^{-}}^{t}{\phi}_{K^{0}}^{p}
+{\phi}_{K^{-}}^{p}{\phi}_{K^{0}}^{t} \right) \left(x_{3}-x_{2}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}} {\phi}_{K^{-}}^{t}{\phi}_{K^{0}}^{t}
\left(2x_{1}-x_{2}-x_{3}\right) \Big]
\nonumber \\ &+&
{\alpha}_{s}(t_{b})C_{2}(t_{b})E(t_{b})H({\Delta},{\beta},b_{1},b_{2})
\Big[ {\phi}_{K^{-}}^{a}{\phi}_{K^{0}}^{a}\left(x_{2}-\bar{x}_{1}-r_{c}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}}{\phi}_{K^{-}}^{p}{\phi}_{K^{0}}^{p}
\left(x_{2}+x_{3}-2\bar{x}_{1}-4r_{c}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}} \left( {\phi}_{K^{-}}^{t}{\phi}_{K^{0}}^{p}
+{\phi}_{K^{-}}^{p}{\phi}_{K^{0}}^{t} \right) \left(x_{3}-x_{2}\right)
\nonumber \\ & &~~~~~~~~~~~~~~~~~~~~~~+
r_{K^{-}}r_{K^{0}} {\phi}_{K^{-}}^{t}{\phi}_{K^{0}}^{t}
\left(x_{2}+x_{3}-2\bar{x}_{1}\right) \Big] \Big\}_{b_{2}=b_{3}}
\label{eq:amplitude}
\end{eqnarray}
where the CKM matrix elements $V_{cb}V_{ud}^{\ast}$ $=$
$A{\lambda}^{2}(1-{\lambda}^{2}/2-{\lambda}^{4}/8)$ $+$
${\cal O}({\lambda}^{8})$ with the phenomenological Wolfenstein
parameterization.
$r_{b}$ $=$ $m_{b}/m_{B_{c}}$ and $r_{c}$ $=$ $m_{c}/m_{B_{c}}$
are the ratios of the mass of $b$ and $c$ quark to the mass of $B_{c}$
mesons, respectively.
$r_{K}$ $=$ ${\mu}_{K}/m_{B_{c}}$ $=$ $m_{K}^{2}/[m_{B_{c}}(m_{s}+m_{q})]$.
$C_{F}$ $=$ $4/3$ is the $SU(3)$ color factor. $t_{a(b)}$ is the
characteristic scale. ${\Delta}$ is the virtualities of internal
gluons, which is a timelike variable for the pure annihilation $B_{c}$
${\to}$ $KK$ decay concerned. ${\alpha}$ and ${\beta}$ are the
virtualities of internal quarks. $E$ and $H$ are the Sudakov factor
and the hard kernel functions, respectively. Their expressions are
listed in Appendix \ref{app02}.
\section{Numerical results and discussions}
\label{sec3}
The branching ratio in the $B_{c}$ meson rest frame can be written as:
\begin{equation}
{\cal BR}(B_{c}{\to}KK)=\frac{{\tau}_{B_{c}}}{8{\pi}}
\frac{p}{m_{B_{c}}^{2}}
{\vert}{\cal A}(B_{c}{\to}KK){\vert}^{2}
\label{eq:br-01},
\end{equation}
where $p$ is the center-of-mass momentum of $K$ mesons.
The lifetime and mass of the $B_{c}$ meson are
$m_{B_{c}}$ $=$ $6.276$ ${\pm}$ $0.004$ GeV and
${\tau}_{B_{c}}$ $=$ $0.46{\pm}0.07$ ps \cite{pdg2008},
respectively.
Other input parameters are
\[ \begin{array}{lll}
m_{c}=1.27^{+0.07}_{-0.11}~\hbox{\rm GeV}~\hbox{\cite{pdg2008}}, &
{\lambda}=0.2257^{+0.0009}_{-0.0010}~\hbox{\cite{pdg2008}}, &
f_{B_{c}}=489{\pm}4~\hbox{\rm MeV~\cite{pos180}}, \\
m_{b}=4.20^{+0.17}_{-0.07}~\hbox{\rm GeV~\cite{pdg2008}}, &
A=0.814^{+0.021}_{-0.022}~\hbox{\cite{pdg2008}}, &
f_{K}=159.8{\pm}1.4{\pm}0.44~\hbox{\rm MeV~\cite{pdg2006}}.
\end{array} \]
If not specified explicitly, we shall take their central values as
the default input. The numerical result of the branching ratio is
\[ {\cal BR}(B_{c}^{-}{\to}K^{-}K^{0}){\approx}
[1.63^{+0.67}_{-0.17}(m_{b})^{+0.35}_{-0.10}(m_{c})]{\times}
[1{\pm}0.3\%(\hbox{CKM}){\pm}1.6\%(f_{B_{c}}){\pm}3.7\%(f_{K})]
{\times}10^{-7}, \]
where the errors come from the uncertainties of quark masses $m_{b}$
and $m_{c}$, the CKM factor $V_{cb}V_{ud}^{\ast}$, and the decay
constants $f_{B_{c}}$ and $f_{K}$. The largest error arises from
the parameter of $m_{b}$, which can reach $40\%$. The errors
arising from both the CKM factor and the decay constants are
relatively small. Of course, there are some other uncertainties
not considered here, such as the radiative corrections to the LCDAs
of $B_{c}$ mesons, the final states interactions, etc. So
the results might just be an estimation of the pQCD approach.
Our estimation of the branching ratio ${\cal BR}(B_{c}{\to}KK)$
is slightly different with the result in \cite{09121163},
although they are calculated with the same pQCD approach
resulting in the same order of magnitude ${\cal O}(10^{-7})$.
Besides the input parameters, the reasons may be
(1) whether the final states are on-mass-shell or not, and
(2) whether the contributions of factorizable topologies are
zero or not\footnotemark[3].
\footnotetext[3]{Because of almost equal masses of the final states,
the same two-particle LCDAs for the charged and neutral
$K$ mesons are taken in our calculation. With this approximation,
a similar conclusion, that the contributions of factorizable topologies
cancel each other because of the isospin symmetry, can be found
in $B_{s}$ ${\to}$ ${\pi}{\pi}$ decays with the pQCD approach
\cite{prd70p034009}.}
With appropriate input parameters, the results in \cite{09121163}
and ours are in agreement with each other within an error range.
As the arguments in \cite{prd80p114031}, the inconsistencies
among various estimations of the branching ratio
${\cal BR}(B_{c}{\to}KK)$, such as ${\cal O}(10^{-6})$
based on $B_{d}$ annihilation by using the relations among the
charmless weak annihilation $B_{c}$ decay channels relying on
the $SU(3)$ flavor symmetry \cite{prd80p114031},
${\cal O}(10^{-7})$ (or ${\cal O}(10^{-8})$ \cite{prd80p114031})
based on perturbative one-gluon exchange with the pQCD (or QCDF)
approach, arise from conceptually different methods.
Anyway, for weak annihilation to light quarks in the final state, the
tree annihilation $B_{c}^{-}$ ${\to}$ $d\bar{u}$ process is helicity
suppressed because of small light quark masses, so that gluon
emission either from the initial or final state must occur in
this annihilation and the decay amplitude is then $O({\alpha}_{s})$
as given in pQCD. Both the estimations in \cite{prd80p114031,09121163}
and our result are in accordance with an intuitive expectation for
nonleptonic charmless $W$-annihilation of heavy meson decays
which are usually suppressed.
There are some additional factors for the tiny
estimation of ${\cal BR}(B_{c}{\to}KK)$. One the is that although the
$B_{c}{\to}KK$ decay is a tree weak annihilation process,
its amplitude is color suppressed and associated with $C_{2}/N_{c}$.
Another is that there is a large destructive interference between the
nonfactorizable topologies due to the near equal final state particle
masses. This can be clearly found in Eq.(\ref{eq:amplitude}).
The numerical results also confirm the cancellation between the
nonfactorizable topologies, and give the strong phases ${\sim}$
$-31^{\circ}$ and ${\sim}$ $+127^{\circ}$ for FIG.\ref{fig1} (a)
and (b), respectively.
If the pQCD prediction is right, then there should be some $10^{3}$
events for $B_{c}{\to}KK$ decay per year at the LHC.
Considering the detection efficiency and selection efficiency, there
would be just a few events per year. The signal of the pure weak
annihilation $B_{c}{\to}KK$ decay would be very tiny at the LHC.
As $B$ nonleptonic charmless decays, the charmless pure weak
annihilation is expected to be small in $B_{c}$ nonleptonic decays,
so the LHC measurement could confirm our understanding of the
annihilation terms in weak decays based on perturbative QCD.
\section{Summary}
\label{sec4}
In this paper, we study the $B_{c}$ ${\to}$ $KK$ decay with the pQCD
approach, which would call for another reassessment of the weak annihilation
processes and might provide some valuable hints of our understanding on
perturbative QCD and long-distance contributions.
It is found that the contributions
of factorizable annihilation topologies are zero, and that there is a large
cancellation between the nonfactorizable topologies, which result
in the branching ratio ${\cal BR}(B_{c}{\to}KK)$ ${\sim}$
${\cal O}(10^{-7})$.
The branching ratio with the pQCD approach is so tiny that the
$B_{c}$ ${\to}$ $KK$ decay might not be measured at the LHC experiments.
\section*{Acknowledgments}
This work is supported by both National Natural Science Foundation
of China (under Grant No. 10805014) and the program for Science \&
Technology Innovation Talents in Universities of Henan Province,
China (under Grant No. 2010HASTIT001). We would like to thank the
referees for their helpful comments.
\begin{appendix}
\section{Distribution amplitude of the $K$ meson}
\label{app01}
The expression of the LCDAs of the $K$ meson incluing higher-twist
contributions can be found in \cite{jhep0605004}. In our calculation,
the twist-2 distribution amplitude ${\phi}_{K}^{a}$ and the twist-3
distribution amplitude ${\phi}_{K}^{p}$ and ${\phi}_{K}^{t}$
are \cite{klcda}
\begin{eqnarray}
\hbox{twist-2} &&
{\phi}_{K}^{a}(x)\,=\,\frac{f_{K}}{2\sqrt{2N_{c}}}6x\bar{x}
\Big\{1+0.17C_{1}^{3/2}(t)+0.115C_{2}^{3/2}(t)\Big\} \\
\hbox{twist-3} &&
{\phi}_{K}^{p}(x)\,=\,\frac{f_{K}}{2\sqrt{2N_{c}}}
\Big\{1+0.24C_{2}^{1/2}(t)-0.12C_{4}^{1/2}(t)\Big\}, \\
\hbox{twist-3} &&
{\phi}_{K}^{t}(x)\,=\,\frac{-f_{K}}{2\sqrt{2N_{c}}}
\Big\{C_{1}^{1/2}(t)+0.35C_{3}^{1/2}(t)\Big\}
\end{eqnarray}
where the decay constant $f_{K}$ $=$ $160$ MeV.
$t$ $=$ $x$ $-$ $\bar{x}$ $=$ $2x$ $-$ $1$.
The Gegenbauer polynomials are
\[ \begin{array}{lll}
\displaystyle C_{1}^{3/2}(z)=3z, &~~
& \displaystyle C_{2}^{3/2}(z)=\frac{3}{2}(5z^{2}-1), \\
\displaystyle C_{1}^{1/2}(z)=z, &
& \displaystyle C_{2}^{1/2}(z)=\frac{1}{2}(3z^{2}-1), \\
\displaystyle C_{3}^{1/2}(z)=\frac{1}{2}(5z^{3}-3z), &
& \displaystyle C_{4}^{1/2}(z)=\frac{1}{8}(35z^{4}-30z^{2}+3)
\end{array} \]
\section{Some parameters and formulas}
\label{app02}
The expression of the Sudakov factors $E$ is
\begin{equation}
E(t)\,=\,{\exp}\left(-S_{B_{c}}(t)-S_{K^{-}}(t)-S_{K^{0}}(t)\right)
\end{equation}
where
\begin{eqnarray}
S_{B_{c}}(t)&=&s(x_{1}p_{1}^{+},b_{1})
+2{\int}_{1/b_{1}}^{t}\frac{{\bf d}{\mu}}{\mu}{\gamma}_{q} \\
S_{K^{-}}(t)&=&s(x_{2}p_{2}^{+},b_{2})+s(\bar{x}_{2}p_{2}^{+},b_{2})
+2{\int}_{1/b_{2}}^{t}\frac{{\bf d}{\mu}}{\mu}{\gamma}_{q} \\
S_{K^{0}}(t)&=&s(x_{3}p_{3}^{-},b_{3})+s(\bar{x}_{3}p_{3}^{-},b_{3})
+2{\int}_{1/b_{3}}^{t}\frac{{\bf d}{\mu}}{\mu}{\gamma}_{q}
\end{eqnarray}
The anomalous dimension of the quark is ${\gamma}_{q}$ $=$
$-{\alpha}_{s}/{\pi}$. The explicit expression of $s(Q,b)$
can be found in \cite{npb642p263}.
The hard kernel function $H$ is defined as follows
\begin{eqnarray}
H({\Delta},Z,b_{1},b_{2})&=&\left\{{\theta}(b_{1}-b_{2})
\frac{i{\pi}}{2}H_{0}^{(1)}(\sqrt{\Delta}b_{1})J_{0}(\sqrt{\Delta}b_{2})
+\left(b_{1}{\leftrightarrow}b_{2}\right)\right\}
\nonumber \\ &{\times}&
\left\{ {\theta}(Z)K_{0}(\sqrt{Z}b_{1})+{\theta}(-Z)
\frac{i{\pi}}{2}H_{0}^{(1)}(\sqrt{{\vert}Z{\vert}}b_{1}) \right\}
\end{eqnarray}
where the hard scales are
\begin{eqnarray}
{\Delta}&=&m^{2}(1-x_{2})(1-x_{3}) \\
-{\alpha}&=&m^{2}(x_{1}-x_{2})(x_{1}-x_{3})-m_{b}^{2} \\
-{\beta} &=&m^{2}(x_{1}+x_{2}-1)(x_{1}+x_{3}-1)-m_{c}^{2} \\
t_{a}&=&\max\left(\sqrt{\Delta},\sqrt{\alpha},1/b_{1},1/b_{2}\right) \\
t_{b}&=&\max\left(\sqrt{\Delta},\sqrt{\beta},1/b_{1},1/b_{2}\right)
\end{eqnarray}
\end{appendix}
|
1,314,259,993,026 | arxiv | \section{Introduction}
\label{intro}
Being able to automatically generate a description from an image is a fundamental problem in artificial intelligence, connecting computer vision and natural language processing. The problem is particularly challenging because it requires to correctly recognize different objects in images and how they interact. Another challenge is that an image description generator needs to express these interactions in a natural language (\emph{e.g.} English). Therefore, a language model is implicitly required in addition to visual understanding.
Recently, this problem has been studied by many different authors. Most of the attempts are based on recurrent neural networks to generate sentences. These models leverage the power of neural networks to transform image and sentence representations into a common space~\citep{MaoXYWY14,Andrej2014,VinyalsTBE14,DonahueHGRVSD14}.
In this paper, we propose a different approach to the problem that does not rely on complex recurrent neural networks.
An exploratory analysis of two large datasets of image descriptions reveals that their syntax is quite simple.
The ground-truth descriptions can be represented as a collection of noun, verb and prepositional phrases.
The different objects in a given image are described by the noun phrases, while the interactions between these objects are encoded by both the verb and the prepositional phrases.
We thus train a model that predicts the set of phrases present in the sentences used to describe the images.
By leveraging previous works on word vector representations, each phrase can be represented by the mean of the representations of the words that compose the phrase. Vector representations for images can also be easily obtained from some pre-trained convolutional neural networks.
The model then learns a common embedding between phrase and image representations (see Figure~\ref{fig:schema}).
Given a test image, a bilinear model is trained to predict a set of top-ranked phrases that best describe it. Several noun phrases, verb phrases and prepositional phrases are in this set. The objective is therefore to generate syntactically correct sentences from (possibly different) subsets of these phrases.
We introduce a trigram constrained language model based on our knowledge about how the sentence descriptions are structured in the training set.
With a very constrained decoding scheme, sentences are inferred with a beam search.
Because these sentences are not conditioned to the given image (apart with the initial phrases selection), a re-ranking is used to pick the sentence that is closest to the sample image (according to the learned metric).
The quality of our sentence generation is evaluated on two very popular datasets for the task: Flickr30k~\citep{HodoshYH13} and the recently published COCO~\citep{mscoco2014}. Using the popular BLEU score~\citep{Papineni:2002}, our results are competitive with other recent works. Our generated sentences also achieve a similar performance as humans on the BLEU metric.
The paper is organized as follows. Section~\ref{related-works} presents related works. Section~\ref{dataanalysis} presents the analysis we conducted to better understand the syntax of image descriptions. Section~\ref{model} describes the proposed phrase-based model. Section~\ref{sec:generation} introduces the sentence generation from the predicted phrases. Section~\ref{exp-results} describes our experimental setup and the results on the two datasets. Section~\ref{conclusion} concludes.
\section{Related Works}
\label{related-works}
The classical approach to sentence generation is to pose the problem as a retrieval problem: a given test image will be described with the highest ranked annotation in the training set \citep{HodoshYH13,SocherKLMN14,srivastava14b}.
These matching methods may not generate proper descriptions for a new combination of objects.
Due to this limitation, several generative approaches have been proposed.
Many of them use syntactic and semantic constraints in the generation process~\citep{YaoYLLZ10,Mitchell:2012,Kulkarni:2011,Kuznetsova:2012}. These approaches benefit from visual recognition systems to infer words or phrases, but in contrast to our work they do not leverage a multimodal metric between images and phrases.
More recently, automatic image sentence description approaches based on deep neural networks have emerged with the release of new large datasets. As starting point, these solutions use the rich representation of images generated by Convolutional Neural Networks \citep{lecun1998gradient} (CNN) that were previously trained for object recognition tasks.
These CNN are generally followed by recurrent neural networks (RNN) in order to generate full sentence descriptions~\cite{VinyalsTBE14,Andrej2014,DonahueHGRVSD14,Chen14,MaoXYWY14,VenugopalanXDRMS14,KirosSZ14}. Among these recent works, long short-term memory (LSTM) is often chosen as RNN.
In such approaches, the key point is to learn a common space between images and words or between images and sentences, i.e. a multimodal embedding.
\citet{VinyalsTBE14} consider the problem in a similar way as a machine translation problem. The authors propose an encoder/decoder (CNN/LSTM networks) system that is trained to maximize the likelihood of the target description sentence given a training image. \citet{Andrej2014} propose an approach that is a combination of CNN, bidirectional RNN over sentences and a structured objective responsible for a multimodal embedding. They then propose a second RNN architecture to generate new sentences.
Similarly, \citet{MaoXYWY14} and \citet{DonahueHGRVSD14} propose a system that uses a CNN to extract image features and a RNN for sentences. The two networks interact with each other in a multimodal common layer.
Our model shares some similarities with these recent proposed approaches. We also use a pre-trained CNN to extract image features. However, thanks to the phrase-based approach, our model does not rely on complex recurrent networks for sentence generation, and we do not fine-tune the image features.
As our approach, \citet{FangGISDDGHMPZZ14} proposes to not use recurrent networks for generating the sentences.
Their solution can be divided into three steps: (i) a visual detector for words that commonly occur are trained using multiple instance learning, (ii) a set of sentences are generated using a Maximum-Entropy language model and (iii) the set of sentences is re-ranked using sentence-level features and a proposed deep multimodal similarity model. Our work differs from this approach in two different important ways: our model infers phrases present in the sentences instead of words and we use a considerably simpler language model.
\section{Syntax Analysis of Image Descriptions}
\label{dataanalysis}
The art of writing sentences can vary a lot according to the domain.
When reporting news or reviewing an item, not only the choice of the words might vary, but also the general structure of the sentence.
In this section, we wish to analyze the syntax of image descriptions to identify whether images have their own structures.
We therefore proceed to an exploratory analysis of two recent datasets containing a large amount of images with descriptions: Flickr30k \citep{HodoshYH13} and COCO \citep{mscoco2014}.
\subsection{Datasets}
The Flickr30k dataset contains 31,014 images where 1,014 images are for validation, 1,000 for testing and the rest for training (i.e. 29,000 images).
The COCO dataset contains 123,287 images, 82,783 training images and 40,504 validation images. The testing images has not yet been released.
We thus use two sets of 5,000 images from the validation images for validation and test, as in \citet{Andrej2014}\footnote{Available at \small\url{http://cs.stanford.edu/people/karpathy/deepimagesent/}}.
In both datasets, images are given with five (or six) sentence descriptions annotated using Amazon Mechanical Turk.
This results in 559,113 sentences when combining both training datasets.
\subsection{Chunking-based Approach}
A quick overview over these sentence descriptions reveals that they all share a common structure, usually describing the different objects present in the image and how they interact between each other.
This interaction among objects is described as actions or relative position between different objects.
The sentence can be short or long, but it generally respects this process.
To confirm this claim and better understand the description structures, we used a chunking (also called shallow parsing) approach which identifies the constituents of a sentence.
These constituents are usually noun phrases (NP), verb phrases (VP) and prepositional phrases (PP).
We extract them from the training sentences with the SENNA software\footnote{Available at \small\url{http://ml.nec-labs.com/senna/}}.
Pre-verbal and post-verbal adverb phrases are merged with verb phrases to limit the number of phrase types.
\begin{figure}
\includegraphics[width=\columnwidth]{phrase-counts-crop.pdf}
\caption{Statistics on the number of phrases (NP, VP, PP) per ground-truth descriptions in Flickr30k and COCO training datasets.}
\label{fig:stats1}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{sentence-stats-crop.pdf}
\caption{The 20 most frequent sentence structures in Flickr30k and COCO training datasets. The black line is the appearance frequency for each structure, the red line is the cumulative distribution.}
\label{fig:stats2}
\end{figure}
Statistics reported in Figure~\ref{fig:stats1} and Figure~\ref{fig:stats2} confirm that image descriptions possess a simple and distinct structure.
These sentences do not have much variability.
All the key elements in a given image are usually described with a noun phrase (NP).
Interactions between these elements can then be explained using prepositional phrases (PP) or verb phrases (VP).
A large majority of sentences contain from two to four noun phrases. Two noun phrases then interact using a verb or prepositional phrase.
Describing an image is therefore just a matter of identifying these constituents.
We thus propose to train a model which can predict the phrases which are likely to be in a given image.
\section{Phrase-based Model for Image Descriptions}
\label{model}
\begin{figure*}[ht]
\centering
\includegraphics[height=7cm]{model-schematic-remi-crop.pdf}
\caption{Schematic illustration of our phrase-based model for image descriptions.}
\label{fig:schema}
\end{figure*}
By leveraging previous works on word and image representations, we propose a simple model which can predict the phrases that best describe a given image. For this purpose, a metric between images and phrases is trained, as illustrated in Figure~\ref{fig:schema}. The proposed architecture is then just a low-rank bilinear model $U^TV$.
\subsection{Image Representations}
For the representation of images, we choose to use a Convolutional Neural Network. CNN have been widely used in different vision domains and are currently the state-of-the-art in many object recognition tasks. We consider a CNN that has been pre-trained for the task of object classification~\cite{Chatfield14}. We use a CNN solely to the purpose of feature extraction, that is, no learning is done in the CNN layers.
\subsection{Learning a Common Space for Image and Phrase Representations}
Let $\mathcal{I}$ be the set of training images, $\mathcal{C}$ the set of all phrases used to describe $\mathcal{I}$, and $\theta$ the trainable parameters of the model.
By representing each image $i \in \mathcal{I}$ with a vector $\mathrm{z}_i \in \mathbb{R}^n$ thanks to the pre-trained CNN, we define a metric between the image $i$ and a phrase $c$ as a bilinear operation:
\begin{equation}\label{eq:score}
f_\theta(c,i) = \mathrm{u}_c^T V \mathrm{z}_i \,,
\end{equation}
with $U= (\mathrm{u}_{c_1}, \ldots, \mathrm{u}_{c_{|\mathcal{C}|}}) \in \mathbb{R}^{m \times |\mathcal{C}|}$ and $V\in \mathbb{R}^{ m \times n}$ being the trainable parameters $\theta$.
Note that $U^TV$ could be a full matrix, but a low-rank setting eases the capacity control.
\subsection{Phrase Representations Initialization}
Noun phrases or verb phrases are often a combination of several words.
Good word vector representations can be obtained very efficiently with many different recent approaches~\citep{Mikolov2013,Mnih2013,pennington2014glove,Lebret14b}.
\citet{MikolovICLR2013} also showed that simple vector addition can often produce meaningful results, such as \emph{king - man + woman $\approx$ queen}.
By leveraging the ability of these word vector representations to compose by simple summation, representations for phrases are easily computed with an element-wise addition.
Each phrase $c$ composed of $K$ words $w_k$ is therefore represented by a vector $\mathrm{x}_{w_k} \in \mathbb{R}^m$ thanks to a word representation model pre-trained on large unlabeled text corpora.
A vector representation $\mathrm{u}_c$ for a phrase $c = \{w_1,\ldots,w_K\}$ is then calculated by averaging its word vector representations:
\begin{equation}
\mathrm{u}_{c} = \frac{1}{K} \sum_{k=1}^K \mathrm{x}_{w_k}\,.
\end{equation}
Vector representations for all phrases $c \in \mathcal{C}$ can thus be obtained to initialized the matrix $U \in \mathbb{R}^{m \times |\mathcal{C}|}$.
$V \in \mathbb{R}^{ m \times n}$ is initialized randomly and trained to encode images in the same vector space than the phrases used for their descriptions.
\subsection{Training with Negative Sampling}
Each image $i$ is described by a multitude of possible phrases $\mathcal{C}^{i}\subseteq \mathcal{C}$.
We consider $|\mathcal{C}|$ classifiers attributing a score for each phrase.
We train our model to discriminate a target phrase $c_j$ from a set of negative phrases $c_k \in \mathcal{C}^{-} \subseteq \mathcal{C}$, with $c_k \neq c_j$.
With $\theta=\{U,V\}$, we minimize the following logistic loss function with respect to $\theta$:
\begin{multline}
\theta \mapsto \sum_{i \in \mathcal{I}} \sum_{c_{j} \in \mathcal{C}^{i}} \Big( \log \Big(1 +e^{-\mathrm{u}_{c_j}^T V \mathrm{z}_i} \Big) \\+ \sum_{c_k \in \mathcal{C}^{-}} \log \Big(1 +e^{+\mathrm{u}_{c_k}^T V \mathrm{z}_i}\Big) \Big)\,.
\end{multline}
The model is trained using stochastic gradient descent. A new set of negative phrases $\mathcal{C}^{-}$ is randomly picked from the training set at each iteration.
\section{From Phrases to Sentence}
\label{sec:generation}
After identifying the $L$ most likely constituents $c_j$ in the image $i$, we propose to generate sentences out of them.
From this set, $l \in \{1,\ldots,L\}$ phrases are used to compose a syntactically correct description.
\subsection{Sentence Generation}
\label{sec:markov}
Using a statistical language framework, the likelihood of a certain sentence is given by:
\begin{equation}\label{eq:ngram}
P(c_1,c_2,\ldots,c_l) = \prod_{j=1}^l P(c_j | c_1,\ldots,c_{j-1})
\end{equation}
Keeping this system as simple as possible and using the second order Markov property, we approximate Equation~\ref{eq:ngram} with a trigram language model:
\begin{equation}\label{eq:trigram}
P(c_1,c_2,\ldots,c_l) \approx \prod_{j=1}^l P(c_j | c_{j-2}, c_{j-1})\,.
\end{equation}
The best candidate corresponds to the sentence $P(c_1,c_2,\ldots,c_l)$ which maximizes the likelihood of Equation~\ref{eq:trigram} over all the possible sizes of sentence.
Because we want to constrain the decoding algorithm to include prior knowledge on chunking tags $t \in \{NP,VP,PP\}$, we rewrite Equation~\ref{eq:trigram} as:
\begin{align}
&\prod_{j=1}^l \sum_{t} P(c_j | t_j = t, c_{j-2}, c_{j-1})P(t_j = t | c_{j-2}, c_{j-1})\nonumber\\
&= \prod_{j=1}^l P(c_j | t_j, c_{j-2}, c_{j-1})P(t_j| c_{j-2}, c_{j-1})\,.
\end{align}
Both conditions $P(c_j | t_j, c_{j-2}, c_{j-1})$ and $P(t_j| c_{j-2}, c_{j-1})$ are probabilities estimated by counting trigrams in the training datasets.
\subsection{Sentence Decoding}
At decoding time, we prune the graph of all possible sentences made out of the top $L$ phrases with a beam search, according to three heuristics:
(i) we consider only the transitions which are likely to happen (we discard any sentence which would have a trigram transition probability inferior to 0.01). This thresholding helps to discard sentences that are semantically incorrect;
(ii) each predicted phrases $c_j$ may appear only once\footnote{This is easy to implement with a beam search, but intractable with a full search.};
(iii) we add syntactic constraints which are illustrated in Figure~\ref{fig:bayes}.
The last heuristic is based on the analysis of syntax in Section~\ref{dataanalysis}.
In Figure~\ref{fig:stats2}, we see that a noun phrase is, in general, always followed by a verb phrase or a prepositional phrase, and both are then followed by another noun phrase. A large majority of the sentences contain three noun phrases interleaved with verb phrases or prepositional phrases.
According the statistics reported in Figure~\ref{fig:stats1}, sentences with two or four noun phrases are also common, but sentences with more than four noun phrases are marginal.
We thus repeat this process $N=\{2,3,4\}$ times until reaching the end of a sentence (characterized by a period).
\begin{figure}[h!]
\begin{center}
\input{lm}
\end{center}
\caption{The constrained language model for generating description given the predicted phrases for an image.}
\label{fig:bayes}
\end{figure}
\subsection{Sentence Re-ranking}
For each test image $i$, the proposed model will generate a set of $M$ sentences.
Sentence generation is not conditioned on the image, apart from phrases which are selected beforehand.
Some phrase sequences might be syntactically good, but have low match with the image.
Consider, for instance, an image with a cat and a dog. Both sentences \emph{``a cat sitting on a mat and a dog eating a bone''} and \emph{``a cat sitting on a mat''} are correct, but the second is missing an important part of the image.
A ranking of the generated sentences is therefore necessary to choose the one that has the best match with the image.
Because a generated sentence is composed from $l$ phrases predicted by our system, we simply average the phrase scores given by Equation~\ref{eq:score}.
For a generated sentence $s$ composed of $l$ phrases $c_j$, a score between $s$ and $i$ is calculated as:
\begin{equation}
\frac{1}{l} \sum_{c_j \in s} f_{\theta}(c_j,i)\,.
\end{equation}
The best candidate is the sentence which has the highest score out of the $M$ generated sentences.
This ranking helps the system to chose the sentence which is closer to the sample image.
\section{Experiments}
\label{exp-results}
\subsection{Experimental Setup}
\subsubsection{Feature Selection}
Following \citet{Andrej2014}, the image features are extracted using VGG CNN \citep{Chatfield14}. This model generates image representations of dimension 4096 from RGB input images.
\begin{table}[h!]
\ra{1.3}
\begin{center}
\begin{tabular}{@{}lcccc@{}}
\hline\toprule
& \phantom{a} & {\bf Flickr30k} & \phantom{a} & {\bf COCO} \\
\bottomrule
Noun Phrase (NP) & & 4818 & & 8982 \\
Verb Phrase (VP) & & 2109 & & 3083\\
Prepositional Phrase (PP) & & 128 & & 189\\
\midrule
Total $\mathcal{|C|}$ & & 7055 & & 12254 \\
\bottomrule
\hline
\end{tabular}
\end{center}
\caption{Statistics of phrases appearing at least ten times.}
\label{tab:stats}
\end{table}
For each training set, only phrases occurring at least ten times are considered.
This threshold is chosen to fulfil two objectives: (i) limit the number of phrases $\mathcal{C}$ and therefore the size of the matrix $U$ and (ii) exclude rare phrases to better generalize the descriptions. Statistics on the number of phrases are reported in Table~\ref{tab:stats}.
For Flickr30k, this threshold covers about 81\% of NP, 83\% of VP and 99\% of PP. For COCO, it covers about 73\% of NP, 75\% of VP and 99\% of PP.
Phrase representations are then computed by averaging vector representations of their words.
We obtained word representations from the Hellinger PCA of a word co-occurrence matrix, following the method described in~\citet{Lebret14b}.
The word co-occurrence matrix is built over the entire English Wikipedia\footnote{Available at \small\url{http://download.wikimedia.org}. We took the January 2014 version.}, with a symmetric context window of ten words coming from the 10,000 most frequent words. Words, and therefore also phrases, are represented in 400-dimensional vectors.
\subsubsection{Learning the Multimodal Metric}
The parameters $\theta$ are $V \in \mathbb{R}^{400 \times 4096 }$ (initialized randomly) and $U \in \mathbb{R}^{400 \times |\mathcal{C}|}$ (initialized with the phrase representations) which are tuned on the validation datasets. They are trained with $15$ randomly chosen negative samples and a learning rate set to 0.00025.
\subsubsection{Generating Sentences from the Predicted Phrases}
\label{sec:generatingSentences}
Transition probabilities for our constrained language model (see Figure~\ref{fig:bayes}) are calculated independently for each training set.
No smoothing has been used in the experiments.
Concerning the set of top-ranked phrases for a given test image, we select only the top five predicted verb phrases and the top five predicted prepositional phrases.
Since the average number of noun phrases is higher than for the two other types of phrases (see Figure~\ref{fig:stats1}), more noun phrases are needed. The top twenty predicted noun phrases are thus selected.
\subsection{Experimental Results}
As a first evaluation, we consider the task of retrieving the ground-truth phrases from test image descriptions.
Results reported in Table~\ref{tab:recall} show that our system achieves a recall of around 50\% on this task on the test set of both datasets, assuming the threshold considered for each type of phrase (see \ref{sec:generatingSentences}). Note that this task is extremely difficult, as semantically similar phrases (\emph{the women} / \emph{women} / \emph{the little girls}) are classified separately.
Despite the possible number of noun phrases being higher, results in Table~\ref{tab:recall} reveal that noun phrases are better retrieved than verb phrases.
This shows that our system is able to detect different objects in the image. However, finding the right verb phrase seems to be more difficult.
A possible explanation could be that there exists a wide choice of verb phrases to describe interactions between the noun phrases.
For instance, we see in Figure~\ref{fig:schema} that two annotators have used the same noun phrases (\emph{a man}, \emph{a skateboard} and \emph{a (wooden) ramp}) to describe the scene, but they have then chosen a different verb phrase to link them (\emph{riding} versus \emph{is grinding}).
Therefore, we suspect that a low recall for verb phrases does not necessarily mean that the predictions are wrong.
Finding the right prepositional phrase seems, on the contrary, much easier.
The high recall for prepositional phrase can be explained by much lower variability of this type of phrase compared to the two others (see Table~\ref{tab:stats}).
\begin{table}[h!]
\ra{1.3}
\begin{center}
\begin{tabular}{@{}lcccc@{}}
\hline\toprule
& \phantom{a} & {\bf Flickr30k} & \phantom{a} & {\bf COCO} \\
\bottomrule
Noun Phrase (NP) & & 38.14 & & 45.44 \\
Verb Phrase (VP) & & 20.61 & & 27.83 \\
Prepostional Phrase (PP) & & 81.70 & & 84.49\\
\midrule
Total & & 44.92 & & 52.49 \\
\bottomrule
\hline
\end{tabular}
\end{center}
\caption{Recall on phrase retrieval. For each test image, we take the top 20 predicted NP, the top 5 predicted VP, and the top 5 predicted PP.}
\label{tab:recall}
\end{table}
\begin{table*}[ht!]
\ra{1.3}
\begin{center}
\begin{tabular}{@{}lccccccccccc@{}}
\hline\toprule
& \phantom{abc} & \multicolumn{4}{c}{\bf Flickr30K} & \phantom{abc} & \multicolumn{4}{c}{\bf COCO} \\
\cmidrule{3-6} \cmidrule{8-11}
& & B-1 & B-2 & B-3 & B-4 & & B-1 & B-2 & B-3 & B-4\\
\bottomrule
Human agreement & & $0.55$ & $0.35$ & $0.23$ & $0.15$ & & $0.68$ & $0.45$ & $0.30$ & $0.20$\\
\midrule
\citet{MaoXYWY14} & & $0.55$ & $0.24$ & $0.19$ & - & & - & - & - & - \\
\citet{Andrej2014} & & $0.50$ & $0.30$ & $0.15$ & - & & $0.57$ & $0.37$ & $0.19$ & -\\
\citet{VinyalsTBE14} & & $0.66$ & - & - & - & & $0.67$ & - & - & - \\
\citet{DonahueHGRVSD14} & & $0.59$ & $0.39$ & $0.25$ & $0.16$ & & $0.63$ & $0.44$ & $0.30$ & $0.21$ \\
\citet{FangGISDDGHMPZZ14} & & - & - & - & - & & - & - & - & $0.21$ \\
Our model & & $0.59$ & $0.35$ & $0.20$ & $0.12$ & & $0.70$ & $0.46$ & $0.30$ & $0.20$ \\
\bottomrule
\hline
\end{tabular}
\end{center}
\caption{Comparison between human agreement scores, state of the art models and our model on both datasets. Note that there are slight variations between the test sets chosen in each paper.}
\label{tab:results}
\end{table*}
As a second evaluation, we consider the task of generating full descriptions. We measure the quality of the generated sentences using the popular, yet controversial, BLEU score~\citep{Papineni:2002}.
Table~\ref{tab:results} shows our sentence generation results on the two datasets considered.
BLEU scores are reported up to 4-gram. Human agreement scores are computed by comparing the first ground-truth description against the four others\footnote{For all models, BLEU scores are computed against five reference sentences which give a slight advantage compared to human scores.}.
For comparison, we include results from recently proposed models.
Our model, despite being simpler, achieves similar results to state of the art results. It is interesting to note that our results are very close to the human agreement scores.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=\textwidth]{results2.pdf}
\end{center}
\caption{Quantitative results for images on the COCO dataset. Ground-truth annotation (in blue), the NP, VP and PP predicted from the model and generated annotation (in black) are shown for each image. The last row are failure samples.}
\label{fig:results}
\end{figure*}
We show examples of full automatic generated sentences in Figure~\ref{fig:results}. The simple language model used is able to generate sentences that are in general syntactically correct. Our model produces sensible descriptions with variable complexity for different test samples. Due to the generative aspect of the model, it can occur that the sentence generated is very different from the ground-truth and still provides a descent description. The last row of Figure~\ref{fig:results} illustrates failure samples. We can see in these failure samples that our system has however outputted relevant phrases.
There is still room for improvement for generating the final description. We deliberately choose a simple language model to show that competitive results can be achieved with a simple approach. A more complex language model could probably avoid these failure samples by considering a larger context. The probability for \emph{a dog} to stand on top of \emph{a wave} is obviously very low, but this kind of mistake cannot be detected with a simple trigram language model.
\subsection{Diversity of Image Descriptions}
In contrast to RNN-based models, our model is not trained to match a given image $i$ with its ground-truth descriptions $s$, i.e., to give $P(s|i)$.
Because our model outputs instead a set of phrases, this is not really surprising that only 1\% of our generated descriptions are in the training set for Flickr30k, and 9.7\% for COCO.
While a RNN-based model is generative, it might easily overfit a small training data.
\citet{VinyalsTBE14} report, for instance, that the generated sentence is present in the training set 80\% of the time.
Our model therefore offers a good alternative with the possibility of producing unseen descriptions with a combination of phrases from the training set.
\subsection{Phrase Representation Fine-Tuning}
\begin{table}[h]
\vspace{-.8cm}
\resizebox{\columnwidth}{!}{
\begin{tabular}{@{}lcccc@{}}\hline\toprule
{\bf PHRASES} & & \multicolumn{2}{c}{\bf NEAREST NEIGHBORS} \\
\cmidrule{3-4}
& \# & \sc before & \sc after \\\bottomrule
\\
\multirow{5}{1.95cm}{\sc a grey cat} & \small 1 & \sc\small a grey dog & \sc\small a gray cat\\
& \small 2 & \sc\small a grey and black cat & \sc\small a grey and black cat \\
& \small 3 & \sc\small a gray cat & \sc\small a brown cat \\
& \small 4 & \sc\small a grey elephant & \sc\small a grey and white cat \\
& \small 10 & \sc\small a yellow cat & \sc\small grey and white cat \\\bottomrule
\\
\multirow{5}{1.95cm}{\sc home plate} & \small 1 & \sc\small a home plate & \sc\small a home plate \\
& \small 4 & \sc\small a plate & \sc\small home base\\
& \small 6 &\sc\small another plate & \sc\small the pitch\\
& \small 9 &\sc\small a red plate & \sc\small the batter \\
& \small 10 &\sc\small a dinner plate & \sc\small a baseball pitch \\\bottomrule
\\
\multirow{5}{1.95cm}{\sc a half pipe} & \small 1 & \sc\small a pipe & \sc\small a pipe \\
& \small 2 &\sc\small a half & \sc\small the ramp \\
& \small 5 &\sc\small a small clock & \sc\small a hand rail \\
& \small 9 &\sc\small a large clock & \sc\small a skate board ramp \\
& \small 10 &\sc\small a small plate & \sc\small an empty pool \\\bottomrule
\end{tabular}}
\caption{Examples of three noun phrases from the COCO dataset with five of their nearest neighbors before and after learning.}
\label{tab:neigh}
\vspace{-.8cm}
\end{table}
Before training the model, the matrix $U$ is initialized with phrase representations obtained from the whole English Wikipedia.
This corpus of unlabeled text is well structured and large enough to provide good word vector representations, which can then produce good phrase representations.
However, the content of Wikipedia is clearly different from the content of the image descriptions.
Some words used for describing images might be used in different contexts in Wikipedia, which can lead to out-of-domain representations for certain phrases.
This becomes thus crucial to adapt these phrase representations by fine-tuning the matrix $U$ during the training\footnote{Experiments with a fixed $U$ phrase representations matrix significantly hurt the general performance. We observe about a 50\% decrease in both datasets with the BLEU metric. Since the number of trainable parameters is reduced, the capacity of $V$ should be increased to guarantee a fair comparison.}.
Some examples of noun phrases are reported in Table~\ref{tab:neigh} with their nearest neighbors before and after the training.
These confirm the importance of fine-tuning to incorporate visual features.
In Wikipedia, \emph{cat} seems to occur in the same context than \emph{dog} or other animals. When looking at the nearest neighbors of a phrase such as \emph{a grey cat}, other \emph{grey} animals arise. After training on images, the word \emph{cat} becomes the important feature of that phrase. And we see that the nearest neighbors are now cats with different colours.
In some cases, averaging word vectors to represent phrases is not enough to capture the semantic meaning.
Fine-tuning is thus also important to better learn specific phrases.
Images related to baseball games, for example, have enabled the phrase \emph{home plate} to be better defined.
This is also true for the phrase \emph{a half pipe} with images about skateboarding.
This leads to interesting phrase representations, grounded in the visual world, which could be possibly used in natural language applications in future work.
\vspace{-.2cm}
\section{Conclusion}
\label{conclusion}
In this paper, we propose a simple model that is able to infer different phrases from image samples. From the phrases predicted, our model is able to automatically generate sentences using a statistical language model. We show that the problem of sentence generation can be effectively achieved without the use of complex recurrent networks.
Our algorithm, despite being simpler than state-of-the-art models, achieves similar results on this task. Also, our model generate new sentences which are not generally present in training set.
Future research directions will go towards leveraging unsupervised data and more complex language models to improve sentence generation.
Another interest is assessing the impact of visually grounded phrase representations into existing natural language processing systems.
\section*{Acknowledgements}
This work was supported by the HASLER foundation through the grant ``Information and Communication Technology for a Better World 2020'' (SmartWorld).
|
1,314,259,993,027 | arxiv | \section{Introduction}
Next-to-leading (NLO) order event generators interfaced to Parton Showers (\NLOPS{} from now on) have, in the past
decades, become the state of the art for the simulation of hard collider processes. The \MCatNLO{} algorithm was the first one
to be proposed~\cite{Frixione:2002ik}. Soon after, the \POWHEG{} method was proposed~\cite{Nason:2004rx}
in order to overcome the problem of negative weights inherent to \MCatNLO{}.
Novel methods were proposed later~\cite{Platzer:2012bs,Lonnblad:2012ix,Jadach:2015mza}. Among them,
the \KrK{} method, like the \POWHEG method, has the characteristic of being free from negative weights, although its applicability
is at the moment restricted to relatively simple processes.
Both the \MCatNLO{} and the \KrK{} methods are intimately intertwined with the parton shower generator that is
adopted. In fact, the two methods provide the
corrections one must apply to the parton shower result in order to achieve
next-to-leading order (NLO) accuracy.
This is at variance with \POWHEG{}, which is largely independent from
the shower generator. In fact, \POWHEG{} takes care of the generation of the hardest event
in such a way that NLO accuracy is preserved, while the PS takes care of the remaining (less hard) radiation.
This feature is convenient, since one can generate events
in the Les Houches format~\cite{Boos:2001cv} and then shower them with any available PS generator that complies
with the Les Houches format specifications.
In view of recent developments aimed at the improvement of the
logarithmic accuracy of the
shower (see Refs.~\cite{Dasgupta:2018nvj,Dasgupta:2020fwr,Hamilton:2020rcu,Karlberg:2021kwr},
\cite{Forshaw:2019ver,Forshaw:2020wrq,Holguin:2020joq}
and \cite{Nagy:2020rmk,Nagy:2020dvz})
it is legitimate to ask
whether \POWHEG{} will maintain the same advantage, i.e. whether minor
modifications to the \POWHEG{} algorithm will be sufficient to
guarantee that, when interfaced to an already NLL accurate shower, NLL
accuracy will be maintained, or whether it will be more
feasible to correct an already NLL accurate shower to achieve also
NLO accuracy, or rather whether both alternatives will be feasible, and
will productively compete among each other.
It is therefore important to explore the full range of options for matching
NLO calculations and parton showers.
In this note, we will address certain limitations of \MCatNLO{} and \KrK{},
and show how they can be overcome.
The \MCatNLO{} approach is widely applicable, and widely used, but
has the undesirable
feature of negative weights.\footnote{Recent proposals for the
reduction of the
negative-weight fraction include
Refs.~\cite{Frederix:2020trv,Andersen:2021mvw,Danziger:2021xvr}.}
On the other hand, the \KrK{} method has positive
weights, but it is difficult to extend it to generic processes.
We will show that combining the two methods one could achieve both
positive weights and unrestricted applicability.
The paper is organised as follows, in section~\ref{sec:review} we will compare
the \POWHEG{}, \MCatNLO{} and \KrK{} methods by formulating them in a common language.
We will do this by extending the formulation of the \POWHEG{} and \MCatNLO{} methods of ref.~\cite{Nason:2012pr}
also to the \KrK{} method.
In section~\ref{sec:newmethod} we will present the combined method,
and in section~\ref{sec:variant} we will present some of its possible variants.
Finally, in section~\ref{sec:conc} we present our conclusions.
\section{\POWHEG{}, \MCatNLO{} and \KrK{}}\label{sec:review}
We assume that the phase space with radiation $\Phi$ can be written in terms of
an underlying Born phase space $\Phi_{\rm B}$ and three radiation variables, indicated collectively as
$\Phi_{\rm rad}$. We also assume that
the mapping from $\Phi$ to $\Phi_{\rm B}$ is such that in the singular (collinear or soft) limit,
the Born configuration matches the full phase space with
the collinear pair merged into a single parton or
with the soft particle removed. Such mappings are easy to realise for processes with a single
singular region, while for more complex processes one must separate the real cross section into
contributions having a single singular region. In the following illustrative discussion we ignore these
complications.
We define the following quantity, a function of the underlying Born kinematics,
\begin{equation}\label{eq:barb}
\BBs(\Phi_{\rm B})= B_0(\Phi_{\rm B})+V(\Phi_{\rm B})+\int \Rs(\Phi) \mathrm{d} \Phi_{\rm rad},
\end{equation}
where the full phase space $\Phi$ is defined as function of $\Phi_{\rm B}$ and $\Phi_{\rm rad}$,
$B_0$ is the Born cross section, $V$ comprises
the virtual corrections and, for hadron initiated processes, the collinear counterterms
integrated at fixed underlying Born,
and $\Rs{}$ is a part of the real cross section that includes all soft and collinear singularities.
In other words, $R-\Rs{}$ is non-singular. Notice that if $\Rs{}$ is taken equal to $R$,
$\BBs{}$ is the inclusive cross section at fixed underlying Born.
We imagine that renormalisation has been carried out, and that the infrared divergences
arising in $V$ and in the $\mathrm{d} \Phi_{\rm rad}$ integral of $R_s$ are regularised in some way.
Notice also that $\BBs{}$ is finite, since the singularities present
in $V$
cancel those arising integrating the $\Rs$ term.
We also define the Sudakov form factor associated with $\Rs{}$
\begin{align}
S(t,\Phi_{\rm B}) &= \exp\left[-\int_{t_{\Phi}>t} \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \mathrm{d} \Phi_{\rm rad}\right],
\end{align}
where $\Phi$ is defined in terms of the variables $\Phi_{\rm B}$ and $\Phi_{\rm rad}$,
and $t$ is some definition of hardness, depending upon the full phase space with radiation.
One may think, for example, that $t$ is the relative transverse momentum of the splitting pair.
If $\Rs$ was taken equal to $R$, the Sudakov form factor would represent the probability for
not radiating anything harder than $t$.
The hardest event cross section can be represented in both \POWHEG{}
and \MCatNLO{} as
\begin{multline}
\mathrm{d} \sigma =\BBs{}(\Phi_{\rm B}) S(\tcut,\Phi_{\rm B}) \mathrm{d} \Phi_{\rm B}
+
\BBs{}(\Phi_{\rm B})\, S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \,\theta(t_\Phi-\tcut) \mathrm{d} \Phi
\,+\\+ \left[R(\Phi)-\Rs{}(\Phi)\right] \mathrm{d} \Phi, \label{eq:showerForm1}
\end{multline}
where $\mathrm{d} \Phi=\mathrm{d} \Phi_{\rm B}\,\mathrm{d} \Phi_{\rm rad}$, and
$\tcut$ represents a lower limit for radiation, needed to avoid the
Landau-pole singularities.
Events are
generated with a probability proportional to each term of the cross section formula.
The first two terms are generated with a Monte Carlo technique. In fact, they satisfy the
shower unitarity equation
\begin{equation} \label{eq:unitarity}
S(\tcut,\Phi_{\rm B}) + \int_{t_\Phi>\tcut} S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \mathrm{d} \Phi_{\rm rad} = 1,
\end{equation}
which follows from the fact that the expression under the integral sign
is an exact differential. In \POWHEG, the generation of the hardness
$t$ is uniform in the Sudakov form factor, and one can generate events
with the standard shower algorithm by equating a random number
with the Sudakov form factor, and solving for $t$.
If the $t$ value so obtained
is above $t_{\rm cut}$, the radiation kinematics is generated, and the event with the
hardest radiation is fed to a shower generator, which takes care of
adding subsequent (less hard) radiation.
In the case of \MCatNLO{}, the implementation of the first term is generally
more involved. If the shower is ordered in transverse momentum,
the hardest emission is the first, and the radiation mechanism
is the same as in \POWHEG{}, except that it is implemented within the shower
generator, rather than in the NLO program.
In case of an angular ordered shower, large angle soft radiation is generated first,
and the hardest radiation occurs somewhere down the shower.\footnote{Taking $t$ to be the angular
scale of the shower means that $t$ does not represent a hardness.
%
As a result, in the absence of an infrared cutoff,
the first emission is
dominated by the infinitely soft region (where $\Rs=R$) rather than
by the hard region.} It was shown in ref.~\cite{Nason:2004rx}
that, in this case,
by suitable rearrangement of the Sudakov factors for each emission, one reconstructs
the transverse momentum Sudakov form factor in formula~(\ref{eq:showerForm1}).
The last term in
Eq.~(\ref{eq:showerForm1}) is non-singular, and thus is dominated by
hard radiation. It is handled essentially in the same way in \MCatNLO and \POWHEG.
The NLO accuracy of Eq.~(\ref{eq:showerForm1}) can be demonstrated
by computing the expectation value of a generic infrared safe observable $O(\Phi)$ as
follows
\begin{subequations}
\begin{align}
\langle O \rangle &=\int \mathrm{d} \Phi_{\rm B} \, \BBs{}(\Phi_{\rm B}) \Bigg\{ S(\tcut,\Phi_{\rm B})\,O(\Phi_{\rm B})
+\int_{t_\Phi>\tcut} S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} O(\Phi) \mathrm{d} \Phi_{\rm rad} \Bigg\}\,+\nonumber \\
&\hspace{16em} \qquad\qquad + \int \left[R(\Phi)-\Rs{}(\Phi)\right] O(\Phi) \mathrm{d} \Phi, \\
\intertext{or equivalently,}\label{eq:NLO-accuracy-check-b}
\langle O \rangle &=\int \mathrm{d} \Phi_{\rm B} \BBs{}(\Phi_{\rm B}) \Bigg\{ S(\tcut,\Phi_{\rm B})\,O(\Phi_{\rm B})
+ \int_{t_\Phi>\tcut} S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} O(\Phi_{\rm B}) \mathrm{d} \Phi_{\rm rad} \,+\nonumber \\
&\hspace{12em}+\int_{t_\Phi>\tcut} S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})}
[O(\Phi)-O(\Phi_{\rm B})] \mathrm{d} \Phi_{\rm rad} \Bigg\}\,+
\nonumber \\
&\hspace{16em}\qquad\qquad
+ \int \left[R(\Phi)-\Rs{}(\Phi)\right] O(\Phi) \mathrm{d} \Phi\,.
\end{align}
\end{subequations}
In view of the unitarity equation~(\ref{eq:unitarity}), the first two
terms in the curly bracket of equation (\ref{eq:NLO-accuracy-check-b})
collapse into $O(\Phi_{\rm B})$. Furthermore, the factor $[O(\Phi)-O(\Phi_{\rm B})]$ in the third
term in the curly bracket suppresses the singular region, so that up to higher order terms
the Sudakov form factor can be omitted, and the lower integration limit can be set to zero
up to a power-suppressed
correction. We thus get
\begin{subequations}
\begin{align}
\langle O \rangle &=\int \mathrm{d} \Phi_{\rm B} \BBs{}(\Phi_{\rm B}) \Bigg\{O(\Phi_{\rm B})
+\int \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} [O(\Phi)-O(\Phi_{\rm B})] \mathrm{d} \Phi_{\rm rad} \Bigg\}\,+\nonumber \\
&\hspace{12em}+ \int
\left[R(\Phi)-\Rs{}(\Phi)\right] O(\Phi) \mathrm{d}
\Phi + {\cal O}({\rm NNLO})\,, \\
&=\int \mathrm{d} \Phi_{\rm B} \left\{ \BBs{}(\Phi_{\rm B}) O(\Phi_{\rm B})
+\int \Rs(\Phi) [O(\Phi)-O(\Phi_{\rm B})] \mathrm{d} \Phi_{\rm rad} \right\}\,+\nonumber \\
&\hspace{12em}+ \int \left[R(\Phi)-\Rs{}(\Phi)\right] O(\Phi) \mathrm{d} \Phi + {\cal O}({\rm NNLO})\,.
\end{align}
\end{subequations}
Inserting into the above equation the expression for $\BBs$, Eq.~(\ref{eq:barb}), we get
\begin{align}
\langle O \rangle =\int \mathrm{d} \Phi_{\rm B} [B(\Phi_{\rm B})+V(\Phi_{\rm B})] O(\Phi_{\rm B})
+ \int R(\Phi) O(\Phi) \mathrm{d} \Phi+ {\cal O}({\rm NNLO}),
\end{align}
which is the correct NLO expression for the observable.
\POWHEG{} implements Eq.~(\ref{eq:showerForm1}) directly. $\Rs{}$ is defined as
\begin{equation}
\Rs{}(\Phi)=F(\Phi) R\,,
\end{equation}
where $F(\Phi)\leq 1$, and approaches 1 as $\Phi$ approaches the singular region,
so that $\Rs{}$ carries all the singularity structure of $R$.
In \MCatNLO{}, $\Rs{}$ is the shower approximation to $R$, typically given by the Born term
times a DGLAP splitting function. Thus, in \MCatNLO{} one computes a Born
configuration using the $\BBs{}$ function, and passes it directly to the Shower Monte Carlo.
The $R-\Rs{}$ terms are generated separately, and fed directly to the shower.
Negative weights can arise at this stage, since there is no guarantee
that $R-\Rs{}$ is positive.
Using the same language adopted here for \POWHEG{}
and \MCatNLO{}, the key formula for the \KrK{} method can be written
as follows
\begin{equation} \label{eq:KrK}
\mathrm{d} \sigma =\BBs{}(\Phi_{\rm B}) S(\tcut,\Phi_{\rm B}) \mathrm{d} \Phi_{\rm B}
+ \BBs{}(\Phi_{\rm B}) \left\{S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})}\right\} \times \left[ \frac{R(\Phi)}{\Rs(\Phi)} \right] \mathrm{d} \Phi\,,
\end{equation}
(with an implicit $\theta(t-\tcut)$ in $d\Phi$).
In the literature dealing with the \KrK{}
method~\cite{Jadach:2017ujd,Jadach:2016qti,Jadach:2016viv,Jadach:2016acv,Jadach:2015mza,Jadach:2015zsq,Jadach:2020xfl},
particular emphasis has been put on the
use of a specific scheme for the parton densities, which considerably simplifies the expression
for the $\BBs{}$ function.
Here we are instead interested in a simpler aspect of the method,
which is that to generate NLO accurate radiation, it uses a
multiplicative correction, rather than the additive correction of the
\MCatNLO{} method.
The NLO accuracy of the \KrK{}
formula can be simply demonstrated by showing that \KrK{} is equivalent to \MCatNLO{} at the NLO level.
In fact, we can rewrite formula~(\ref{eq:KrK}) as
\begin{multline} \label{KrK}
\mathrm{d} \sigma =\BBs{}(\Phi_{\rm B}) S(\tcut,\Phi_{\rm B}) \mathrm{d} \Phi_{\rm B}
+
\BBs{}(\Phi_{\rm B}) S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \times \left[ \frac{R(\Phi)}{\Rs(\Phi)}-1 \right] \mathrm{d} \Phi
+\\+
\BBs{}(\Phi_{\rm B}) S(t_\Phi,\Phi_{\rm B}) \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \mathrm{d} \Phi\,.
\end{multline}
The middle term is now insensitive to the soft region, because the factor in square brackets vanishes there,
so we can drop the Sudakov form factor and, neglecting terms of NNLO size, set $\BBs{}/B_0=1$. By
doing this we recover exactly Eq.~(\ref{eq:showerForm1}).
The \KrK{} method leads to positive weighted events. On the other hand, unlike \MCatNLO{} and \POWHEG{},
its cross section at fixed underlying Born does not exactly match the
corresponding fixed order result, but differs from it by NNLO terms.\footnote{Although some authors consider this to
be an undesirable feature, it does not constitute a real problem, since choices for
uncontrolled NNLO terms are made, for example, when choosing the scales, and there is no preferred
way to define an NLO result as far as the neglected NNLO terms are concerned.}
The \KrK{} method, however, generates weighted events, so, the unweighting efficiency may constitute a problem
if one wants to apply the method to generic processes without having to do process-by-process adjustments.
As a related problem, the shower generator may not cover the
full radiation phase space. This is the same as saying that $\Rs{}$ can become zero in certain regions, in which case
the method is not applicable. For these
reasons, it seems difficult to apply the method to generic processes in automated framework, something
that has been available in \POWHEG{} and \MCatNLO{} since a long time.
\section{The new method}\label{sec:newmethod}
By comparing the \MCatNLO{} and \KrK{} method using a common language,
it becomes clear that the two methods
have much in common, and can in fact be merged in such a way that the \KrK{} positivity is maintained, unweighted
events can be generated on the fly, and no issues arise from the
limited coverage of the phase space by the parton shower code.
The merged method is defined by the following formula
\begin{equation}\label{eq:hybrid}
\mathrm{d} \sigma = \BBs{}(\Phi_{\rm B}) \, S(t_\Phi,\Phi_{\rm B})
\times \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})}
\times \left\{1+ \frac{R-\Rs{}}{\Rs} \theta(\Rs-R)\right\} \mathrm{d} \Phi
+\theta(R-\Rs) \left[R-\Rs\right] \mathrm{d} \Phi\,.
\end{equation}
It is easy to see by inspection that this formula has the same NLO accuracy as \MCatNLO{}. In fact, the $\theta$ function
in the first term of eq.~(\ref{eq:hybrid}) is regular,
and as such the Sudakov form factor multiplying it and the $\BBs{}/B_0$ ratio
can both be set to 1 up to NNLO corrections. Having done this, the contribution proportional
to the $\theta$ function in the first term becomes identical to the
last term, except
for the theta function, which has the opposite sign in the argument. The two terms thus combine, and the theta function
disappears, yielding the \MCatNLO{} formula.
The method of Eq.~(\ref{eq:hybrid}) is immediately seen to have
positive weights, since the second term
in eq.~(\ref{eq:hybrid}) is forced to be positive by the
theta function, while the factor in the curly bracket of the
first term
is forced
by the theta function to be less than 1. Notice that regions of phase
space not populated by the Monte Carlo only contribute to the additive
term in the square bracket, since they have $\Rs=0$.
Let us consider how the method of Eq.~(\ref{eq:hybrid}) might be
implemented in a scenario where it is the parton shower that is
responsible for generating $\Rs$.\footnote{Alternatively, one might
also implement the approach in a \POWHEG-style scheme, where the NLO
program takes responsibility for the first emission, and $\Rs$ is not required
to be smaller than $R$.}
We first consider the case of a parton shower ordered in a genuine
hardness variable (for example, transverse momentum).
As in the \MCatNLO approach, the NLO program is responsible for
generating a sample of Born events, with a distribution corresponding
to the correct $\BBs{}(\Phi_{\rm B})\mathrm{d}\Phi_{\rm B}$ weight.
The event is communicated to the parton shower, which generates a
first emission, i.e.\ accounting for a remaining factor
$S(t_\Phi,\Phi_{\rm B}) \frac{\Rs(\Phi)}{B_0(\Phi_{\rm B})} \mathrm{d}\Phi/\mathrm{d}\Phi_{\rm B}$
in Eq.~(\ref{eq:hybrid}).
Next, the parton shower communicates the event, with its first
emission, back to the NLO program, which evaluates the contents of the
curly bracket in Eq.~(\ref{eq:hybrid}).
The result of that evaluation, which is bounded to be between zero and
one, is to be used as an acceptance probability for the event.
If the event is accepted, the parton shower then continues from that
point, generating the remaining emissions.
In addition, the NLO program is responsible for generating a second
(positive definite) sample of events, corresponding to the
$\theta(R - \Rs$) term of Eq.~(\ref{eq:hybrid}), covering the part of
the real phase space where the shower $\Rs$ underestimates the true
real matrix element.
These events are passed to the parton shower code for normal
showering.
The above scheme requires a more connected workflow between the parton
shower and the NLO code than either of the \MCatNLO or \POWHEG
approaches.
Recall that those approaches pass a Les Houches Event file to the
shower code, and then let the shower take over from there.
In contrast, the scheme outlined above requires additional action from
the NLO code \emph{after} the first parton shower emission.
However, this is simply a technical consideration, which in our view
is a small price to pay for an approach that, like the \MCatNLO
scheme, leaves responsibility for the first emission with the parton
shower, while eliminating negative weights.
A further remark concerns angular-ordered parton showers.
For such showers, the application of the acceptance probability
(factor in curly brackets in Eq.~(\ref{eq:hybrid})) after the first
emission would not be correct.
Instead, one might envisage an approach in which one runs the complete
shower, uses a jet algorithm to map the full event to the real phase
space and then applies the acceptance probability.
\section{Variants of the method}\label{sec:variant}
Eq.~(\ref{eq:hybrid}) can be viewed as a special case of a family of
approaches, parameterised by a constant $c \ge 1$,
\begin{multline}\label{eq:hybridvar}
\mathrm{d} \sigma = \BBs{}(\Phi_{\rm B}) \, S(t_\Phi,\Phi_{\rm B})
\times \frac{c\,\Rs(\Phi)}{B_0(\Phi_{\rm B})}
\times \left\{1+ \frac{R-c\,\Rs{}}{c\,\Rs}
\theta(c\,\Rs-R)\right\} \mathrm{d} \Phi \,+ \\
+\theta(R-c\,\Rs) \left[R-c\,\Rs\right] \mathrm{d} \Phi\,.
\end{multline}
This formula was obtained by replacing $\Rs \to c\,\Rs$ in all
occurrences where it appears in Eq.~(\ref{eq:hybrid}), except for the
Sudakov form factor.
First of all, we should convince ourselves
that this formula has the correct behaviour near the soft limit, and
that it is NLO accurate. This is seen immediately
by writing~(\ref{eq:hybridvar}) as
\begin{multline}
\mathrm{d} \sigma = \BBs{}(\Phi_{\rm B}) \, S(t_\Phi,\Phi_{\rm B})
\times \frac{c\,\Rs(\Phi)}{B_0(\Phi_{\rm B})}
\times \left\{1+ \frac{R-c\,\Rs{}}{c\,\Rs}
\right \} \mathrm{d} \Phi\,, \\
+ \theta(R-c\,\Rs) (R-c\,\Rs)
\left[1-\frac{\BBs{}(\Phi_{\rm B})}{B_0(\Phi_{\rm B})} \, S(t_\Phi,\Phi_{\rm B})
\right] \mathrm{d} \Phi\,.
\label{eq:hybridvar1}
\end{multline}
The first line is equal to the \KrK{} formula, eq.~(\ref{eq:KrK})
(leaving aside the $\tcut$, for simplicity).
In the second line, the factor in square brackets is of order $\as$
and it multiplies a term dominated by the hard region (and thus also of order $\as$), since
\begin{displaymath}
\theta(R-c\,\Rs) (R-c\,\Rs) \le \theta(R-\Rs)(R-\Rs)\,.
\end{displaymath}
Accordingly, the second line is of NNLO order, and we can conclude
that formula~(\ref{eq:hybridvar}) is equivalent to the \KrK{} formula
up to NNLO corrections.
For a shower where $\Rs / R$ is always larger than some value $r$,
taking $c \ge 1/r$ causes Eq.~(\ref{eq:hybridvar}) to reduces to the
\KrK approach over all of phase space.
The implementation of formula~(\ref{eq:hybridvar}) goes as follows. The soft
events are oversampled by a factor of $c$, and accepted with a probability
proportional to the expression in the curly bracket, while the hard events
are generated in a standard way.
There are several reasons why it may be of use to have such a family of
approaches parameterised by $c$.
One is that it is useful to have a parameter to help gauge systematic
uncertainties associated with terms that are beyond the accuracy of
the method. In fact, the $c$ parameter also gauges the amount of radiation
that is multiplied by the inclusive $K$-factor, with respect to the amount
that is added in as hard radiation. In this sense, it would play a similar
role to the {\tt hdamp} parameter in
\POWHEG{}~\cite{Nason:2004rx,Alioli:2008tz}.
Another reason is that the $\theta$-functions in
Eqs.~(\ref{eq:hybrid}), (\ref{eq:hybridvar}) can, conceivably, introduce
non-smoothness in phase-space coverage even when the underlying $\Rs$
and $R$ functions are smooth.
Sampling over a range of $c$ values would allow one to address that
issue.\footnote{It is not clear to us that this would be a genuine
problem.
%
A worse issue is potentially present in the
\MCatNLO approach (and also in Eqs.~(\ref{eq:hybrid}) and
(\ref{eq:hybridvar})) if a shower has a discontinuity in the
distribution of the hardness variable.
We are not aware of this having caused significant problems in
practice, possibly because subsequent showering smoothens any
discontinuities.
%
Were it to be a problem, one solution could be to sample over a
range of shower starting scales.
%
}
So far, the variants that we have discussed involve the rejection of
events.
For showers with an ordering variable that involve a genuine hardness
scale, such as transverse momentum, it is also possible to envisage a
variant where instead of rejecting the event with probability
$\frac{\Rs-R}{\Rs} \theta(\Rs - R)$, one rejects the shower's first
emission with that same probability.
If that first emission is rejected, at a value of the ordering
variable that we label $t_1$, the shower then continues from that
scale $t_1$, based on an event without the first emission.
As the shower continues, each time the shower again attempts to create
a new first emission, that emission continues to be rejected with
probability $\frac{\Rs-R}{\Rs} \theta(\Rs - R)$.
Once the shower has generated a first emission that is accepted, the
shower continues as normal.
This ensures that the shower (with first emission rejection) remains
unitary.\footnote{As a matching algorithm to ensure the correct matrix
element for the first emission, it bears strong similarities to the
algorithm of Refs.~\cite{Bengtsson:1986et,Seymour:1994df}.
%
That algorithm works within the assumption that one can find some
constant $c$ such $c \Rs > R$ over the full phase space, evading the
need for an additive term.}
The Born normalisation factor that multiplies the shower generation
then needs to be modified to read
\begin{equation}\label{eq:tildeb}
\BTs(\Phi_{\rm B})= B_0(\Phi_{\rm B})+V(\Phi_{\rm B})+\int \min[R(\Phi),\Rs(\Phi)] \mathrm{d} \Phi_{\rm rad}\,.
\end{equation}
With this variant, the distribution of the hardest radiation is given by
\begin{equation}
\mathrm{d} \sigma = \BTs{}(\Phi_{\rm B}) \, \tilde S(t_\Phi,\Phi_{\rm B})
\times \frac{\min(R(\Phi),\Rs(\Phi))}{B_0(\Phi_{\rm B})}
\mathrm{d} \Phi
+\theta(R-\Rs) \left[R-\Rs\right] \mathrm{d} \Phi\,,
\end{equation}
where $\tilde S(t_\Phi,\Phi_{\rm B})$ is the Sudakov form factor that is
effectively obtained as a result of the emission rejection procedure,
\begin{align}
\tilde S(t,\Phi_{\rm B}) = \exp\left[-\int_{t_{\Phi}>t}
\frac{\min[R(\Phi),\Rs(\Phi)]}{B_0(\Phi_{\rm B})} \mathrm{d} \Phi_{\rm rad}\right].
\end{align}
\section{Conclusion}\label{sec:conc}
In this article we have proposed an algorithm for NLOPS generators
that, like the \MCatNLO approach, relies on the parton shower code for
driving the showering steps, but avoids the negative-weight events
that complicate the practical use of the \MCatNLO approach.
The approach can be viewed as a hybrid version of the \MCatNLO{} and
\KrK{} methods.
The method should be straightforward to implement, requiring no more
information than is already used for \MCatNLO codes, though it should
be noted that it will probably require a closer integration of the
showering and fixed-order codes.
Following the tradition of associating an acronym to
NLOPS methods, a suitable one for this method could be {\tt MAcNLOPS},
reflecting the combination of \underline{M}ultiplicative and
\underline{Ac}cumulative (additive) steps.
|
1,314,259,993,028 | arxiv | \section{Introduction}
\noindent The global crisis of 2008 had the most devastating consequences in the world economy \cite{imf}.
One of the main causes for the beginning and aggravation of the crisis was the strengthening
of international economic interdependence. Primarily, the crisis hit the financial system
and the debtor countries. As a result, systemic financial risks occurred,
and the crisis spread to other countries.
The global financial system is a kind of configuration of numerous interrelations
between national economies, and every day the world economy becomes more like
a unified space with a network nature. Failure of one subject of the financial system
generates a chain reaction through interconnections and causes shocks and systemic risk.
This risk is associated with the incapability of one of the participants to perform
their obligations (or to accomplish them properly), which leads to the interruption
in the functioning of other participants and, thus, of the entire system.
The Bank for International Settlements (BIS) provides the following definition
of systemic financial risks: ``the risk that the failure of a participant to meet
its contractual obligations may in turn cause other participants to default,
with the chain reaction leading to broader financial difficulties'' \cite{bisreport}.
Therefore, the systemic risk is the likelihood of negative changes in the financial
system and the economy of a particular country that affect the financial stability
of the global market \cite{investopedia,Kaufman}.
Crises continue to occur at different economic levels, both at micro and macro
levels \cite{kosnaz}, which make the economy an interesting object of study.
The interest of scientists to this field is increasing after some collapse in the economy.
The contribution of scientists in the study of the world economy is huge and has increased
rapidly over the past decade. In \cite{inci}, the authors investigated contagion between
international equity markets using the local correlation. The contagion effects among
the stock markets were investigated in \cite{alexakis} using the asymmetric dynamic
conditional correlation dynamics. Authors in \cite{giudici} investigated Corporate
Default Swap spreads using the vector autoregressive regression with correlation
networks in their model. Also, one of the interesting types of research in this area
is the work \cite{ahnert}, where the authors studied information contagion due to
the counterparty risk and examined its effects on banks \emph{ex-ante} choices and systemic risk.
Mathematical epidemiology is widely developed, as described in \cite{brauer},
and has wide application in various fields of science \cite{rodrigues2,ozturka,cannarella,rodrigues,wu}.
However, the use of epidemiological models in the economy is scanty and the economy
has not been studied yet completely. Thus, the economy needs to be investigated in order
to prevent possible negative consequences, since the systemic risks accumulate in
the world financial system and become a general threat to the new global crisis \cite{rockinger}.
The study of the systemic financial risks allows to characterize comprehensively
the current picture of the global financial world, and also to develop new methods
of protection against global threats. The significance of global systemic financial
risks is increased by their complexity in the identification, estimation, and developing
methods for their calculation and minimization \cite{cerutti}.
A key feature of global systemic financial risks is the potential
infection of the world economy with a financial virus \cite{paltalidis}.
For example, if some European Union countries are a source of global systemic risk,
as they experience a debt crisis, then they threaten the stability
of a larger system, which is a global threat.
For this reason, it is necessary to study the spread of financial
viruses in the world economic network. The complex study of country
interrelations shows which national banking systems are most exposed
to a particular country, both on an immediate counterparty
basis and on an ultimate risk basis \cite{cerutti}.
Our research focuses on total foreign claims on an ultimate risk basis,
which captures lending to a borrower in any country that is guaranteed
by an entity that resides in the counterparty country. The object of study
is the process of infection spreading through network interconnections.
Moreover, we investigate economic relations between the subjects of the
global financial system, which arise in the process of managing systemic
financial risks. The aim is to study the process of spreading the infection
through network interconnections, identify regularities, and whenever
possible give recommendations for minimization risks in global scale management.
The scientific novelty of our study consists in modelling and investigating
the process of contagion in the network using epidemiological models.
The research was done with statistical data from BIS \cite{bis}
on the volumes of consolidated foreign claims on ultimate risk bases
in a number of countries, and data of countries credit rating
from the Guardian Datablog \cite{guardata}.
The paper is structured as follows.
In Section~\ref{sec:02} of ``Methodology'',
the basic concepts of network and epidemiological models
are introduced as well as the data used for the considered models.
The results of modelling and various scenarios
of contagion spreading are presented in Section~\ref{sec:03}
of ``Results''. We end our work
with Section~\ref{sec:04} of ``Conclusions''.
\section{Methodology}
\label{sec:02}
\noindent Based on the network nature of the global economy, described above,
the systemic risk can be considered as a network risk, which causes infection of networks.
Our method for investigating the spread of a virus
in the financial system consists of six steps:
$1)$ to build the network;
$2)$ to define the virus transmitting rate and recovery rate;
$3)$ to visualise the process of virus transmission in the network
by implementing a multi-agent programmable modelling environment
in NetLogo \cite{netlogo};
$4)$ to run the spreading process in a closed population by solving
the Kermack--McKendrick SIR model \cite{kermack} in the multi-paradigm
numerical computing environment MATLAB \cite{matlab};
$5)$ to compare results between dynamics of infection in the network
and dynamics obtained by solving the SIR system of differential equations;
$6)$ to confirm or disprove the economic reasonableness of the results.
\subsection{Network}
\noindent Network analysis is well used in various fields of science \cite{bartlett}:
in computer science, to describe the internet topology \cite{alderson};
in social sciences, to describe the evolution and spread of ideas
and innovations in societies \cite{hufnagel}; in ecology,
to model networks of ecological interactions \cite{rezende};
in biology, to investigate the neurovascular structure
of the human brain \cite{david}; in biochemistry, to infer
how selection acts on metabolic pathways \cite{proulx,albert};
as well as in economics, to study financial contagion
in the banking system \cite{paltalidis,garas}.
Many mechanisms and quantitative tools for describing networks
have been provided by research in graph theory.
Networks are mathematically described as graphs.
There are different types of graphs: random graphs,
small-world graphs, scale-free graphs, and others.
A network consists of multiple nodes connected to each other.
In this research we construct a fully connected network, which includes $n$ nodes.
This network is also known as a complete graph, denoted by $K_{n}$.
The complete graph is a regular graph, where each vertex has the same
degree $n-1$, and $K_{n}$ always has $n(n-1)/2$ links.
It means that all nodes are interconnected, i.e.,
each vertex has the same number of neighbours.
In graph theory, a finite graph is often represented as an adjacency matrix:
\begin{equation}
\label{amatrx}
A =
\left[
\begin{array}{cccc}
a_{11}& a_{12} &\ldots & a_{1n}\\
a_{21}& a_{22} &\ldots & a_{2n}\\
\vdots& \vdots &\ddots & \vdots\\
a_{n1}& a_{n2} &\ldots & a_{nn}
\end{array}\right],
\end{equation}
where elements $a_{ij}$ equal to zero or one,
respectively for disconnected and connected vertices.
Such matrix is the basis for network building.
A network construction provides a good visualization of its structure,
knowledge and understanding, which allows us to compute
the epidemic dynamics and to predict a spreading phenomena.
\subsection{Epidemiological model}
\noindent The epidemic spreading can be described by many models.
Epidemiological models, in their majority, are based on dividing
the population according to the disease status of their individuals.
The main models describe the proportion of population that is infected,
susceptible to infection, and recovered after a disease \cite{MyID:417}.
In our study, we use the classical Kermack--McKendrick SIR model \cite{kermack},
which considers such factors as infection spreading and recovery \cite{kosroddelf}:
\begin{eqnarray}
\label{eqSIR}
\begin{cases}
\displaystyle \frac{dS(t)}{dt} = - \beta S(t) I(t),\\[0.3cm]
\displaystyle \frac{dI(t)}{dt} = \beta S(t) I(t) - \gamma I(t),\\[0.3cm]
\displaystyle \frac{dR(t)}{dt} = \gamma I(t),
\end{cases}
\end{eqnarray}
$t \in [0,T]$, subject to the initial conditions
\begin{equation}
\label{eqIC}
S(0) = S_0, \quad I(0) = I_0, \quad R(0) = R_0.
\end{equation}
The SIR model \eqref{eqSIR}--\eqref{eqIC} expresses the spread among
the population compartments as a system of differential equations,
where $S$, $I$ and $R$ refer to the number of susceptible, infectious
and recovered individuals, respectively, in a constant population
of $N$ individuals for all time $t$:
\begin{equation}
\label{eqN}
S(t)+I(t)+R(t) = N, \quad t\in [0,T], \quad T > 0.
\end{equation}
System \eqref{eqSIR} describes the relationship between the three compartments:
a susceptible individual changes its state to infected with probability $\beta$
(the contagion spreading rate), while an infected changes its state to recovered
with probability $\gamma$ (the speed of recovery).
These parameters are assumed constant for the entire sample.
\subsection{Data}
\noindent Sixteen European and Non-European developed countries
were chosen based on statistical data from the Bank for International
Settlements (BIS) for the end of the year $2012$ \cite{bis}.
The number of countries $N = 16$ is fixed throughout the contamination time.
They connected to the network (see Figure~\ref{fnetwork}) according
to the adjacency matrix \eqref{amatrx} as follows:
\begin{equation}
\label{matrx}
A =
\left[
\begin{array}{cccc}
0& 1 &\ldots & 1\\
1& 0 &\ldots & 1\\
\vdots& \vdots &\ddots & \vdots\\
1& 1 &\ldots & 0
\end{array}\right],
\end{equation}
where the elements $a_{ij} $ are equal to one for connected vertices,
and zero for disconnected. The connection is provided by the presence
of bilateral foreign claims on an ultimate risks basis.
The diagonal elements are all zero, since loops are not determined by
statistical data of amounts outstanding from BIS \cite{bis}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.25]{results_network_0}
\caption{Fully connected network, where each country is represented as a node
and edges indicate the existence of a link between countries.}
\label{fnetwork}
\end{figure}
In our work, we assume that only one country is contagious at the initial time.
Thus, the values of initial conditions \eqref{eqIC} for the SIR model are as follows:
$S(0) = 15$, $I(0) = 1$, and $R(0) = 0$.
We also assume that the initially infected country $I$ cannot
fulfil all of its obligations to other countries (for example, by domestic reasons).
This means that all foreign claims $\alpha_{ij}$ of a counterparty country $i$ are infected.
A contribution of infected debts to the total amount of claims from all countries,
defines the value of $\beta$ parameter:
\begin{equation}
\label{eqbeta}
\beta_{i} = \frac{\sum\limits_{j=1}^{16} \alpha_{ij}}
{\sum\limits_{i=1}^{16}\sum\limits_{j=1}^{16} \alpha_{ij}},
\quad i \in \left\{1, \ldots, 16\right\}.
\end{equation}
The values of the infection spreading rate $\beta_i \in [0, 1]$ and $\sum_{i=1}^{16} \beta_{i}=1$ or $100\%$.
Thus, the more outstanding debts in the total amount of debts, imply the higher possibility of infection.
Statistical information was taken from the BIS consolidated international banking statistics
on an ultimate risk basis \cite{bis}. It is the most appropriate source for measuring
the aggregate exposures of a banking system to a given country \cite{avdjiev}.
The recovery rate was calculated according to country's credit rating:
\begin{equation}
\label{eqgamma}
\gamma_{i} = \frac{1} {101-C_{i}},
\quad i \in \left\{1, \ldots, 16\right\}.
\end{equation}
Here, $\gamma_i$ implies that it takes $\frac{1}{\gamma_i} = 101-C_{i}$ time steps to recover.
The credit rating $C_{i}$ takes into account not only countries' debt,
but also assets. This measures the ability to fulfil their obligations as borrowers
-- the probability of recovery. The data of countries' credit rating is taken from
the Guardian Datablog \cite{guardata} and converted by ourselves to the numerical
representation based on the rating table from \cite{TE}, where the credit rating
is shown by country's credit worthiness between $100$ (riskless) and $0$ (likely to default).
Thus, a susceptible country $S$ can obtain contagion if it has a relationship
with an infected country $I$, and if it has not enough money in reserve
to cover possible risk losses.
The values of contagion spreading rate and the speed
of recovery are given in Figure~\ref{fdata}.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.53]{data}
\vspace*{0.3cm}
\caption{Summary statistics of $\gamma$ and $\beta$ parameters for
the 16 European and Non-European developed countries considered in our study.}
\label{fdata}
\end{figure}
\section{Results}
\label{sec:03}
\noindent We now present the obtained results.
For comparison, all countries are grouped according to the value
of the recovery parameter $\gamma$ (see Table~\ref{tabular:gamma}).
\begin{table}[ht!]
\doublerulesep 0.1pt
\tabcolsep 7.8mm
\centering
\caption{\rm Grouping of countries depending on $\gamma$ parameter.}
\label{tabular:gamma}
\vspace*{2mm}
\renewcommand{\arraystretch}{1.3}
\setlength{\tabcolsep}{22pt}
\begin{center}
\footnotesize{
\begin{tabular*}{12.5cm}{ccc} \hline\hline\hline
\raisebox{-2ex}[0pt][0pt]{Group 1: $\gamma \leq 0.1$} &
\raisebox{-2ex}[0pt][0pt]{Group 2: $0.1 < \gamma \leq 0.5$} &
\raisebox{-2ex}[0pt][0pt]{Group 3: $0.5 < \gamma \leq 1$} \\ \\ \hline
BE & AT & AU \\
ES & FR & CH \\
GR & US & DE \\
IE & & GB \\
IT & & NL \\
JP & & SE \\
PT & & \\ \hline\hline\hline
\end{tabular*}}
\end{center}
\renewcommand{\arraystretch}{1}
\end{table}
We compare the results depending on the belonging of the initially
infected country to a particular group. For illustrative purposes, we consider
the dynamics of the chain propagation reaction for three cases: when the contagion
process starts 1) from Portugal (PT, Group~1); 2) from United States
(US, Group~2); and 3) from Switzerland (CH, Group~3).
Both network and SIR model simulation results are consistent.
\subsection{Network}
\noindent To investigate the dynamics of infection spread in the network,
we use the NetLogo agent-based programming language and integrated
modelling environment \cite{netlogo}. It is well-recognized that its
visualization makes it easy to understand chain reaction processes
\cite{netlogprog}.
Figure~\ref{fnet} demonstrates the spread of the financial virus through the network,
where each node represents a random country from the considered list represented
in Figure~\ref{fdata}. At the initial moment, all nodes are susceptible (white colour)
except one infected node (black colour). In each time step (``days''), the ``nodes''
check whether they have an infection, and an infected node attempts
to infect all of its neighbours. ``Days'' is an arbitrary unit
during which the ``nodes'' check and change their status.
If an infection has been detected, then there is a probability
of $\beta$ that susceptible neighbours will get an infection and change
their colour to black, and there is a probability $\gamma$ that an infection
will be removed and the nodes will be recovered.
Recovered nodes (grey colour) cannot be infected.
When a node becomes recovered, the links between it and its neighbours are darkened,
since they are no longer possible vectors for contagion spreading \cite{netlogprog}.
It is important to notice that, in reality, the country's financial system cannot recover evermore.
Therefore, applying this model, we consider that recovered countries are resistant for some
short period of time, and then they again become susceptible to the virus.
\begin{figure}[hp]
\centering
\begin{subfigure}
[$T_{PT}=0$]{\label{fnetpt0}
\includegraphics[width=0.23\textwidth]{results_PT2_0}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{PT}=9$]{\label{fnetpt9}
\includegraphics[width=0.23\textwidth]{results_PT2_9}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{PT}=15$]{\label{fnetpt15}
\includegraphics[width=0.23\textwidth]{results_PT2_15}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{PT}=48$]{\label{fnetpt48}
\includegraphics[width=0.23\textwidth]{results_PT2_48}}
\end{subfigure}
\vfill
\begin{subfigure}[$T_{PT}=280$ ]{\label{fnetpt280}
\includegraphics[width=0.23\textwidth]{results_PT2_280}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{PT}=281$ ]{\label{fnetpt281}
\includegraphics[width=0.23\textwidth]{results_PT2_281}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{USA}=0$]{\label{fnetusa0}
\includegraphics[width=0.23\textwidth]{results_USA4_0}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{USA}=1$]{\label{fnetusa1}
\includegraphics[width=0.23\textwidth]{results_USA4_1}}
\end{subfigure}
\vfill
\begin{subfigure}[$T_{USA}=2$]{\label{fnetusa2}
\includegraphics[width=0.23\textwidth]{results_USA4_2}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{USA}=5$]{\label{fnetusa5}
\includegraphics[width=0.23\textwidth]{results_USA4_5}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{USA}=14$]{\label{fnetusa14}
\includegraphics[width=0.23\textwidth]{results_USA4_14}}
\end{subfigure}
\hfill
\begin{subfigure}[$T_{USA}=15$]{\label{fnetusa15}
\includegraphics[width=0.23\textwidth]{results_USA4_15}}
\end{subfigure}
\vfill
\begin{subfigure}[$T_{CH}=0$]{\label{fnetch0}
\includegraphics[width=0.23\textwidth]{results_CH1_0}}
\end{subfigure}
\begin{subfigure}[$T_{CH}=1$]{\label{fnetch1}
\includegraphics[width=0.23\textwidth]{results_CH1_1}}
\end{subfigure}
\caption{Virus spreading in the network of countries with parameters
$\beta$ and $\gamma$ taken from Figure~\ref{fdata};
\eqref{fnetpt0}--\eqref{fnetpt281} -- initially infected country is Portugal (PT);
\eqref{fnetusa0}--\eqref{fnetusa15} -- initially infected country is United States (USA);
\eqref{fnetch0}--\eqref{fnetch1} -- initially infected country is Switzerland (CH).
Nodes in white mean ``Susceptible''; nodes in black mean ``Infected''; nodes in grey mean ``Recovered''.}
\label{fnet}
\end{figure}
For the case where the infection begins from Portugal
(Figures~\ref{fnetpt0}--\ref{fnetpt281}),
the first infected node (Figure~\ref{fnetpt0}) spreads
the virus to one of its neighbours at time $T=9$ (Figure~\ref{fnetpt9}).
At time $T=15$ (Figure~\ref{fnetpt15}) the contagion process slowly continues,
and the last infected node (Figure~\ref{fnetpt280}) changes its state
to recovered at time $T=281$ (Figure~\ref{fnetpt281}).
It is easy to see that the chain reaction of infection and recovery of nodes
occurs much faster when it starts from United States of America
(Figures~\ref{fnetusa0}--\ref{fnetusa15}).
The initially infected node (Figure~\ref{fnetusa0}) spreads
the infection to neighbouring nodes in the next time step $T=1$ (Figure~\ref{fnetusa1}).
The maximum number of infected nodes is reached at time $T=2$ (Figure~\ref{fnetusa2}).
At time $T=5$ (Figure~\ref{fnetusa5}) most of the infected nodes have already been recovered
and the last infected node (Figure~\ref{fnetusa14}) changed its state to recovered
at time $T=15$ (Figure~\ref{fnetusa15}).
In the case when Switzerland is initially infected, the virus is not transmitted
to the neighbours and the infected node is immediately recovered
(Figures~\ref{fnetch0}--\ref{fnetch1}).
\subsection{Epidemiological SIR model}
\noindent The initial value problem \eqref{eqSIR}--\eqref{eqIC}
can be solved using a numerical approach. In practice, the solution
can be obtained in the form of a time-series function of each compartment.
In our work we solve the system of differential equations in MATLAB.
The obtained results are consistent with those that were obtained
with the network simulations.
The behaviour of the epidemiological model for Portugal,
United States of America, and Switzerland parameters,
are shown in Figure~\ref{fsir}.
\begin{figure}[ht!]
\centering
\begin{subfigure}[PT: Portugal]{\label{fsirpt}
\includegraphics[width=0.32\textwidth]{PT}}
\end{subfigure}
\begin{subfigure}[USA: United States]{\label{fsirusa}
\includegraphics[width=0.32\textwidth]{USA}}
\end{subfigure}
\begin{subfigure}[CH: Switzerland]{\label{fsirch}
\includegraphics[width=0.32\textwidth]{CH}}
\end{subfigure}
\caption{The SIR contagion risk model \eqref{eqSIR}--\eqref{eqIC} with
parameters $\beta$ and $\gamma$ for Portugal, United States, and Switzerland,
taken from Figure~\ref{fdata}; the initial conditions are $S(0)=15$, $I(0)=1$, $R(0)=0$.}
\label{fsir}
\end{figure}
When infection spreading begins from Portugal (Figure~\ref{fsirpt}),
contagion has almost reached the contagion-free equilibrium ($I(T)=0$)
after 281 time steps. The spread of contagion occurs over a long period
of time and the recovery process goes slowly too.
If the United States is the starting point for virus spreading (Figure~\ref{fsirusa}),
the contagion spreads rapidly and affects a large number of countries
in a short period of time, and then swiftly decreases as the recovery process takes fast.
If the initially infected country is Switzerland, the virus immediately dies out (Figure~\ref{fsirch}).
The results in Figure~\ref{fsir} coincide with those that were obtained earlier in Figure~\ref{fnet}.
It means that both methods of modelling of contagion spreading are in agreement with each other.
Figures~\ref{fnet}--\ref{fsir} show that the contagion spreading processes take place
in different ways, depending on the country where it begins.
The countries that are in Group 3 of Table~\ref{tabular:gamma}
have the highest recovery rate.
Within a short period of time, the infected will recover
(Figure~\ref{fsirch} and Figure~\ref{fsircde}--~\ref{fsircse}).
If the infection begins from a country listed in Group~2 of
Table~\ref{tabular:gamma}, then the contagion ceases to spread
and all infected become recovered after 10 to 25 time steps
(Figure~\ref{fsirusa} and Figure~\ref{fsircat}--~\ref{fsircfr}).
The situation is completely opposite for the countries in Group~1
of Table~\ref{tabular:gamma}. For them, the virus infect the highest number
of countries and takes much more time, and the recovery process
is slower too (Figure~\ref{fsirpt} and Figure~\ref{fsircbe}--~\ref{fsircjp}).
\begin{figure}[hp]
\centering
\begin{subfigure}[BE: Belgium]{\label{fsircbe}
\includegraphics[scale=0.30]{BE}}
\end{subfigure}
\begin{subfigure}[ES: Spain]{\label{fsirces}
\includegraphics[scale=0.30]{ES}}
\end{subfigure}
\begin{subfigure}[GR: Greece]{\label{fsircgr}
\includegraphics[scale=0.30]{GR}}
\end{subfigure}
\begin{subfigure}[IE: Ireland]{\label{fsircie}
\includegraphics[scale=0.30]{IE}}
\end{subfigure}
\begin{subfigure}[IT: Italy]{\label{fsircit}
\includegraphics[scale=0.30]{IT}}
\end{subfigure}
\begin{subfigure}[JP: Japan]{\label{fsircjp}
\includegraphics[scale=0.30]{JP}}
\end{subfigure}
\begin{subfigure}[AT: Austria]{\label{fsircat}
\includegraphics[scale=0.30]{AT}}
\end{subfigure}
\begin{subfigure}[FR: France]{\label{fsircfr}
\includegraphics[scale=0.30]{FR}}
\end{subfigure}
\begin{subfigure}[DE: Germany]{\label{fsircde}
\includegraphics[scale=0.30]{DE}}
\end{subfigure}
\begin{subfigure}[GB: United Kingdom]{\label{fsircgb}
\includegraphics[scale=0.30]{GB}}
\end{subfigure}
\begin{subfigure}[NL: Netherlands]{\label{fsircnl}
\includegraphics[scale=0.30]{NL}}
\end{subfigure}
\begin{subfigure}[AU: Australia]{\label{fsircau}
\includegraphics[scale=0.30]{AU}}
\end{subfigure}
\begin{subfigure}[SE: Sweden]{\label{fsircse}
\includegraphics[scale=0.30]{SE}}
\end{subfigure}
\caption{The SIR contagion risk model with parameters $\beta$
and $\gamma$ taken from Figure~\ref{fdata}.}
\label{fsirc}
\end{figure}
The reason for the identified differences lies in the
different economic state of the country where contagion begins,
especially in the adequacy of country's reserve capital.
If a country has a big reserve capital and, consequently,
a high credit rating position, then a high recovery rate indicates its ability
to cover possible risks in the shortest time period.
The situation is completely opposite for countries with low recovery rate.
If any of these countries will be forced to fulfil their obligations,
it will be difficult for their economies and, therefore,
the recovery process will take longer.
\section{Conclusions}
\label{sec:04}
\noindent The recent global crisis of 2008 placed
the economic analysis as one of the most relevant
political and social concerns of the most indebted countries.
Here we considered some of the western countries
in these conditions. Precisely, we investigated
and modelled the process of contagion spreading
in a global inter-country network, revealing
the degree of interconnection of national financial systems,
identifying the potential systemic financial risks and their
effects. Our research was done with real data from the Bank of International
Settlements on the volumes of consolidated foreign claims on ultimate risk bases in
several countries, and data of credit rating from the Guardian Datablog \cite{guardata}.
The dynamics of infection spreading of a virus in the financial
system on the given network of countries was simulated
with NetLogo, an agent-based programming language, and
integrated modelling environment, and confirmed by an
epidemiological SIR model. The infection process was shown
to depend on the parameter value of the recovery rate,
as well as on the country, which initially begins the process of infection.
We found out that if one of the financially unstable countries will be the starting
point in the spread of contagion, and will be forced to fulfil its obligations
as a counterparty, then the global financial system will have serious problems,
the negative effects of which will continue during a long period of time.
Therefore, the countries with a powerful economy and good credit rating position
are more reliable counterparties, since if necessary
they will be able to fulfil their obligations.
According to the standard SIR methodology, both parameters $\beta$ and $\gamma$
are constant for the entire sample. However, in reality, these data are unique
for each country and depend on the infection force of the affected country
and the financial stability of the susceptible. Another line of research,
motivated by the fact that the level of exposure and interference between
countries and financial institutions is not the same, consists
to consider a more realistic representation as a graph with varying edge weights.
These and other issues are under investigation and will be addressed elsewhere.
\section*{Acknowledgements}
\noindent This research was supported by the Portuguese Foundation
for Science and Technology (FCT -- Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia),
through CIDMA -- Center for Research and Development in Mathematics and Applications,
within project UID/MAT/04106/2019. Kostylenko is also supported
by the FCT Ph.D. fellowship PD/BD/114188/2016.
We are very grateful to Professor Yuriy Petrushenko, Doctor of Economics,
for providing us with a consultation regarding the data used to calculate
the beta and gamma parameters in our work; and to four anonymous Reviewers,
for valuable remarks and comments, which significantly contributed
to the quality of the paper.
\section*{Conflict of interest}
The authors declare that there is no conflicts of interest in this paper.
|
1,314,259,993,029 | arxiv |
\section{Introduction}
Throughout this paper $G$ is an $n$-vertex, simple graph with vertex set $V(G)$ and edge set $E(G)$. A \textit{matching} $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. The cardinality of $M$ is called the \textit{size} of the matching. A matching $M$ is a \textit{maximum matching} if there is no matching in $G$ with greater size. The \textit{matching number} $\nu (G)$ of $G$ is the cardinality of any maximum matching in $G$. Since each vertex can be incident to at most one edge of a matching, it follows that $\nu (G) \le \lfloor n/2 \rfloor$ for any graph $G$. If every vertex of $G$ is incident with an edge in $M$, then $M$ is called a \textit{perfect} matching and such graphs have $\nu (G)=n/2$. It is clear that perfect matchings are also maximum matchings but the converse is not generally true.
Matchings serve as models of many phenomena across the sciences. An important motivation for their study arose from chemistry, when it was observed that the stability of benzenoid compounds is related to the number of perfect matchings, also known as \textit{Kekul\'{e} structures}, in the corresponding chemical graphs. For a survey of these results, see \cite{gutman1}. With the discovery of fullerenes in 1985 \cite{kroto1}, the desire to identify properties characteristic for stable fullerenes led to the enumeration of perfect matchings \cite{doslic4,zhang1,kardos1,doslic3} in these corresponding graphs.
Maximum matchings give one way to quantify the largeness of a matching. Both the enumerative and structural properties of maximum matchings are well studied and well understood, see \cite{lovasz1} for a general background on such matching theory.
There is yet another way to quantify the largeness of a matching. A \textit{maximal matching} in $G$ is a matching that cannot be extended to a larger matching in $G$. Clearly, every maximum matching is also maximal but the opposite is usually not true. Chemically, maximal matchings model the adsorption of dimers to a molecule, where each dimer bonds to a pair of adjacent atoms in the molecule. Any such adsorption pattern corresponds to a matching in the graph of the molecule, and once no further adsorption is possible, such a matching must be maximal. The best case of adsorption can be viewed as a maximum matching, while the worst case concerns the smallest possible maximal matching. This idea gives rise to the study of the \textit{saturation number} of a graph $G$, which is the cardinality of any smallest maximal matching in $G$. Thus the saturation number is a measure of how inefficient the adsorption process can be. Aside from chemistry, the saturation number has a number of interesting applications in networks, engineering, etc. The saturation number of a graph is equal to the cardinality of an independent edge dominating set. Finding an independent edge dominating set in a graph is an NP-Hard problem \cite{Yannakakis1980}.
Maximal matchings are much less understood than their maximum counterparts. Some work has been done on enumerating maximal matchings in certain chemical graphs \cite{doslic5,short1} but this area remains largely unexplored. Structural properties, such as the saturation number, have been studied for benzenoid graphs \cite{doslic2}, fullerenes \cite{andova,doslic7,doslic6}, and nanotubes \cite{tratnik1}. The paper \cite{doslic2} mentions the saturation number of nanocones as an interesting, unexplored avenue of study.
This paper considers both nanocones and nanotubes, which are carbon networks situated between graphene and fullerene in terms of structure. New upper and lower bounds on the saturation number of nanocones are established, which are asymptotically equal. In addition, lower bounds for the saturation number of two classes of nanotubes are presented, which improve recent results \cite{tratnik1}.
\section{Statement of Results}
A \textit{hexagonal patch}, or \textit{patch} for short, is a planar graph where all faces are hexagons except one \textit{outer} or \textit{boundary} face. All internal vertices have degree 3 and all vertices on the outer face have degree 2 or 3. For a face $F$ in a planar graph $G$, let $n_2(F)$ be the number of degree 2 vertices incident to $F$ and let $n_2=n_2(G)$ be the total number of degree 2 vertices in a graph $G$.
Next it will be useful to introduce some definitions utilized in \cite{graves1,graves2,graver1,brinkmann1,bornhoft1}. The \textit{boundary code} of a patch is described by a sequence of 2's and 3's corresponding to the degree of the vertices on the boundary of the patch in cyclic order. A \textit{break edge} is an edge on the boundary whose endpoints are both degree 2. A \textit{bend edge} is an edge on the boundary whose endpoints are both degree 3.
This paper limits itself to patches with nice boundaries. A patch is \textit{pseudoconvex} if it does not contain any bend edges. A \textit{side} of a patch is a path on the boundary between a consecutive pair of break edges, including the break edges, and the \textit{length} of a side is the number of degree 3 vertices on the side.
A \textit{defect} in a patch is a non-hexagonal face. A defect is \textit{internal} if all vertices incident to the face are degree 3. A defect is \textit{external} if there are degree 2 vertices incident to the face. Using this terminology, the outer face of a patch can also be called an external defect. Patches can have more than one external defect. In such a graph, any face incident to degree 2 vertices could the outer face in a planar drawing.
The following theorem is the main tool in proving lower bounds on the saturation number. This theorem is a generalization of the theorem proven for fullerenes in \cite{andova}, in that a fullerene graph can be viewed as a patch containing exactly 12 pentagonal defects and no vertices of degree 2 (i.e. fullerenes have no external defects). For the sake of consistency, the proof in Section \ref{seclowerbound} uses similar terminology and structure to what was presented in \cite{andova}.
\begin{theorem} \label{lowerbound}
Let $G$ be a pseudoconvex patch with $n$ vertices, $o_k$ internal defects which are $k$-gonal where $k$ is odd, $e_k$ internal defects which are $k$-gonal where $k\neq 6$ is even, and $n_2$ vertices of degree 2. Then
$$
s(G) \ge \frac{n}{3}-\frac{1}{18}\left( \sum _{k \text{ odd}}(k-2)o_k+ \sum _{k \text{ even}} ke_k \right)-\frac{n_2}{6}.
$$
\end{theorem}
\subsection{Nanocones}
Generally speaking, nanocones are planar graphs where the majority of faces are hexagons, along with some non-hexagonal faces, most commonly pentagons, in addition to the outer face. This paper considers these pentagonal defect nanocones as well as nanocones with a single $k$-gonal defect.
A \textit{single-defect $k$-gonal nanocone with $\ell$ layers}, $CNC_k(\ell)$, is obtained by taking a cycle on $k \ge 3$ vertices, $C_k$, and surrounding it with $\ell$ concentric layers of hexagons. Using previous terminology, a single-defect $k$-gonal nanocone is a pseudoconvex patch with a single $k$-gonal defect at its apex. By induction, it follows that there are $k{\ell+1\choose 2}$ hexagonal faces, $k(\ell+1)^2$ total vertices, and $(2\ell+1)k$ external vertices. There are $k\ell$ external vertices of degree 3 and $k(\ell+1)$ vertices of degree 2. The following Corollary is an immediate consequence of Theorem \ref{lowerbound}.
\begin{corollary} \label{lowersd}
$$
s(CNC_k(\ell)) \ge \begin{cases} \frac{k(\ell+1)^2}{3}-\frac{k-2}{18} - \frac{k(\ell+1)}{6} & \text{if } k \text{ is odd} \\ \frac{k(\ell+1)^2}{3}-\frac{k}{18} - \frac{k(\ell+1)}{6} & \text{if } k \text{ is even}\end{cases}
$$
\end{corollary}
\textit{Pentagonal defect nanocones} are pseudoconvex patches with $p$ pentagons, where $1\le p \le 5$. While many arrangements of pentagons and hexagons are possible, a classification result first from \cite{klein1,klein2} and then independently in \cite{justus1} shows that it suffices to consider the 8 configurations of pentagons and or hexagons in Figure \ref{cones}. Note that the single pentagon configuration is merely $CNC_5(0)$.
\begin{figure}[h]
\begin{center}
\input{cone1} \input{cone2s.tex} \input{cone2n} \input{cone3s} \input{cone3n} \input{cone4s} \input{cone4n} \input{cone5s}
\end{center}
\caption{The 8 configurations of pentagons and or hexagons for pentagonal defect nanocones.} \label{cones}
\end{figure}
A \textit{pentagonal defect nanocone with $i$ pentagons and $\ell$ layers}, $CN_i^j(\ell)$, $i\in \{2,3,4,5\}$ and $j \in \{s,a\}$, is defined to be the configuration $CN_i^j$ in Figure \ref{cones} surrounded by $\ell$ concentric layers of hexagons. For reference, the use of $s$ in the superscript designates a symmetric configuration as drawn in Figure \ref{cones} and the use of $a$ represents asymmetric. The configurations $CNC_5(0)$ and $CN_i^j$ in Figure \ref{cones} are called the \textit{caps} of the nanocone.
The following Corollary is another consequence of Theorem \ref{lowerbound}, and its proof, containing additional details, is provided in Section \ref{secnanocone}.
\begin{corollary} \label{lowernanocone}\.\\
\begin{enumerate}
\item[(a)] $s(CN_2^s(\ell)) \ge \frac{14+4\ell(\ell+4)}{3} - \frac{1}{3} - \frac{4(\ell+2)}{6}$
\item[(b)] $s(CN_2^a(\ell)) \ge \frac{11+2\ell(2\ell+7)}{3} - \frac{1}{3 }-\frac{4\ell+7}{6}$
\item[(c)] $s(CN_3^s(\ell)) \ge \frac{10+3\ell(\ell+4)}{3} - \frac{1}{2 }-\frac{3(\ell+2)}{6}$
\item[(d)] $s(CN_3^a(\ell)) \ge \frac{16+\ell(3\ell+16)}{3} - \frac{1}{2 }-\frac{3\ell+8}{6}$
\item[(e)] $s(CN_4^s(\ell)) \ge \frac{12+2\ell(\ell+6)}{3} - \frac{2}{3 }-\frac{2(\ell+3)}{6}$
\item[(f)] $s(CN_4^a(\ell)) \ge \frac{15+2\ell(\ell+7)}{3} - \frac{2}{3 }-\frac{2\ell+7}{6}$
\item[(g)] $s(CN_5^a(\ell)) \ge \frac{16+\ell(\ell+12)}{3} - \frac{5}{6 }-\frac{\ell+6}{6}$
\end{enumerate}
\end{corollary}
The upper bound on the saturation number of nanocones relies on splitting the nanocone in subgraphs, where the number of subgraphs depends on the number of break edges. The following Lemma was proven in \cite{graves1}.
\begin{lemma} \label{break} \cite{graves1}
In a nanocone, the number of pentagons $p$ and the number of break edges $s$ are related by
$$
s+p=6.
$$
\end{lemma}
\begin{comment}
A \textit{benzenoid parallelograms }, $P_{p,q}$, consists of a configuration of $p \times q$ congruent regular hexagons arranged in $p$ rows, each row containing $q$ hexagons, and each row shifted for half a hexagon to the right from the row immediately below.
\begin{proposition}\cite{doslic2} \label{benzparallel}
$$
s(P_{p,q}) \le \left \lceil \frac{2p+1}{3} \right \rceil q+p
$$
\end{proposition}
\end{comment}
\noindent The subgraph used is a \textit{benzenoid triangle}, $T_p$, which is a patch that can be constructed by arranging ${p+1 \choose 2}$ hexagonal faces in the shape of an equilateral triangle, so that each side of the triangle has $p$ hexagons. For an example of a benzenoid triangle, see Figure \ref{benztriangleT_5}. Note that the saturation number of similar graphs, such as benzenoid parallelograms, was studied in \cite{doslic2}. The upper bound on the saturation number of benzenoid triangles is presented in Lemma \ref{benztriangle} which is used to deduce the upper bounds on the saturation number of nanocones in Theorem \ref{uppernanocone}. The proofs of these results are presented in Section \ref{secnanocone}.
\begin{figure}[h]
\begin{center}
\include{triangle}
\end{center}
\caption{The benzenoid triangle, $T_5$.} \label{benztriangleT_5}
\end{figure}
\begin{lemma} \label{benztriangle}
$$
s(T_p) \le \left \lfloor \frac{(p+1)(p+3)}{3} \right \rfloor
$$
\end{lemma}
\noindent The upper bound presented in Lemma \ref{benztriangle} is believed to be the exact value of $s(T_p)$ and is the sequence A032765 in the OEIS \cite{oeis}.
\begin{theorem} \label{uppernanocone} \.\\
\begin{enumerate}
\item[(a)] $s(CNC_k(\ell)) \le k \left \lfloor \frac{\ell(\ell+2)}{3} \right \rfloor + k(\ell+1)$
\item[(b)] $s(CN_2^s(\ell)) \le 4 \left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor + 4(\ell + 1) + 1$
\item[(c)] $s(CN_2^a(\ell)) \le 3 \left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor + \left \lfloor \frac{\ell (\ell+2)}{3} \right \rfloor + 4(\ell + 1) + 1$
\item[(d)] $s(CN_3^s(\ell)) \le 3 \left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor + 3(\ell + 1) + 1$
\item[(e)] $s(CN_3^a(\ell)) \le 2 \left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor + \left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor + 3(\ell + 1) + 2$
\item[(f)] $s(CN_4^s(\ell)) \le 2 \left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor + 2(\ell + 1) + 3$
\item[(g)] $s(CN_4^a(\ell)) \le \left \lfloor \frac{(\ell+3)(\ell+5)}{3} \right \rfloor + \left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor + 2(\ell + 1) + 4$
\item[(h)] $s(CN_5^a(\ell)) \le \left \lfloor \frac{(\ell+5)(\ell+7)}{3} \right \rfloor + (\ell + 1) + 6$
\end{enumerate}
\end{theorem}
Combining Corollaries \ref{lowersd} and \ref{lowernanocone} along with Theorem \ref{uppernanocone} shows that if $G$ is any nanocone graph with $n$ vertices, then $s(G)\sim n/3$. Hence, in a smallest maximal matching, as $n$ gets large there are roughly 2 matched edges per hexagon. These findings are consistent with the work done on the saturation number of fullerenes \cite{andova} and benzenoid graphs \cite{doslic2}.
\begin{comment}
\begin{corollary} \label{oldupsatcone}
$$
s(CNC_k[\ell]) \le \begin{cases} \frac{k}{2}\left ( \left \lceil \frac{2\ell+1}{3} \right \rceil \cdot \ell+2\ell+1 \right ) & \text{if } k \text{ is even}\\
\left \lfloor \frac{k}{2} \right \rfloor \left ( \left \lceil \frac{2\ell+1}{3} \right \rceil \cdot \ell + \ell \right ) + \frac{1}{2} \left( \left \lceil \frac{2\ell-1}{3} \right \rceil \cdot \ell +(\ell-1) \right) + \left \lceil \frac{k}{2} \right \rceil(\ell+1) & \text{if } k \text{ is odd} \end{cases}
$$
\end{corollary}
\end{comment}
\subsection{Nanotubes}
\textit{Open ended nanotubes}, also called \textit{tubulenes}, can be obtained in the following way. Starting with a hexagonal tessellation of a cylinder, take the finite graph induced by all hexagons that lie between two vertex disjoint cycles, where each cycle encircles the axis of the cylinder. This paper considers two types of tubulenes having particularly nice structure, namely zig-zag and arm chair tubulenes, shown in Figures \ref{zigzag} and \ref{armchair}.
Zig-zag tubulenes, $ZT(\ell,m)$, have $\ell$ horizontal layers of hexagons, each containing $m$ hexagons. Bounds on the saturation number of such zig-zag were first established in \cite{tratnik1}, shown in Corollary \ref{tratnikzigzag}.
\begin{figure}[h]
\begin{center}
\input{zigzag}
\end{center}
\caption{The zig-zag tubulene, $ZT(6,5)$, is obtained from the figure above by gluing the lines $L_1$ and $L_2$ together.} \label{zigzag}
\end{figure}
\begin{corollary} \cite{tratnik1} \label{tratnikzigzag}
$$
\frac{m(\ell+1)}{2} \le s(ZT(\ell,m)) \le \begin{cases} \frac{m(2\ell+3)}{3} & \text{if } 3|\ell \\ \frac{m(2\ell+1)}{3} & \text{if } 3|(\ell-1) \\ \frac{m(2\ell+2)}{3} & \text{if } 3|(\ell-2) \end{cases}
$$
\end{corollary}
\noindent Corollary \ref{lowerzigzag} follows as an application of Theorem \ref{lowerbound} and improves the lower bound for the saturation number of zig-zag tubulenes. The proof is contained in Section \ref{sectubes}.
\begin{corollary} \label{lowerzigzag}
$$
s(ZT(\ell,m)) \ge \frac{m(2\ell+1)}{3}
$$
\end{corollary}
Combining the new lower bound from Corollary \ref{lowerzigzag} and the upper bounds presented in Corollary \ref{tratnikzigzag}, it follows that $s(ZT(\ell,m)) = \frac{m(2\ell+1)}{3}$ whenever $3|(\ell-1)$.
Armchair tubulenes, $AT(m,\ell)$, have $\ell$ vertical layers of hexagons, each containing $m$ hexagons. The saturation number of armchair tubulenes was also studied in \cite{tratnik1}, as seen in Corollary \ref{tratnikarmchair}.
\begin{figure}[h]
\begin{center}
\input{armchair}
\end{center}
\caption{The armchair tubulene, $AT(4,6)$, is obtained from the figure above by gluing the curves $L_1$ and $L_2$ together.} \label{armchair}
\end{figure}
\begin{corollary} \cite{tratnik1} \label{tratnikarmchair}
$$
\frac{\ell(m+1)}{2} \le s(AT(m,\ell)) \le \begin{cases} \frac{2\ell(m+1)}{3} & \text{if } 3|\ell \\ \frac{(2\ell+1)(m+1)}{3} & \text{if } 3|(\ell-1) \\ \frac{2(\ell+2)(m+1)}{3} & \text{if } 3|(\ell-2) \end{cases}
$$
\end{corollary}
\noindent Another application of Theorem \ref{lowerbound}, Corollary \ref{lowerarmchair} improves the lower bound for the saturation number of armchair tubulenes and its proof is found in Section \ref{sectubes}.
\begin{corollary} \label{lowerarmchair}
$$
s(AT(m,\ell)) \ge \frac{\ell(2m+1)}{3}
$$
\end{corollary}
From the work above, it follows that the saturation number of zigzag and armchair tubulenes with $n$ vertices is essentially $n/3$. This is now consistent with the findings for nanocones, fullerenes, and benzenoid graphs.
\section{Proof of the main tool}\label{seclowerbound}
\begin{proof}[Proof (of Theorem \ref{lowerbound})]
Let $M$ be a maximal matching in $G$. Let the edges in $M$ and the vertices saturated by $M$ be called black, and let the remaining edges and vertices be called white. Let $B$ and $W$ be the set of all black and white vertices, respectively.
The proof using the discharging method, setting the initial charges as follows:
\begin{itemize}
\item Let the initial charge of each black vertex be $3$;
\item Let the initial charge of each white vertex be $-6$;
\item Let the initial charge of each $k$-gonal, internal defect be equal to $\begin{cases}$k-2$ & \text{ if } k \text{ is odd} ,\\ k & \text{ if } k \text{ is even}\end{cases}$; and
\item Let the initial charge of each external defect, $E$, be $3n_2(E)$.
\end{itemize}
It remains to show that the total sum of the charge in the graph $3|B|-6|W|+\sum _{k \text{ odd}}(k-2)o_k+ \sum _{k \text{ even}} ke_k+3 n_2$ is non-negative. From this it follows that
$$
3|B|\ge 2|B|+2|W|-\frac{1}{3}\left( \sum _{k \text{ odd}}(k-2)o_k+ \sum _{k \text{ even}} ke_k \right)-n_2
$$
implying that
\begin{align*}
|M|=\frac{|B|}{2} &\ge \frac{|B|+|W|}{3}-\frac{1}{18} \left( \sum _{k \text{ odd}}(k-2)o_k+ \sum _{k \text{ even}} ke_k \right)-\frac{n_2}{6} \\
& = \frac{n}{3}-\frac{1}{18} \left( \sum _{k \text{ odd}}(k-2)o_k+ \sum _{k \text{ even}} ke_k \right)-\frac{n_2}{6}.
\end{align*}
\noindent The initial charge is distributed as follows:
\begin{enumerate}
\item[(R1)] Each external defect sends +3 charge to each incident, degree 2 white vertex.
\end{enumerate}
First note that all vertices of degree 2 in $G$ are incident to an external defect. There are a total of $n_2(E)$ vertices of degree 2 incident to an external defect $E$, not all of them white vertices, so after applying (R1) all white vertices of degree 2 in $G$ now have charge -3.
\begin{enumerate}
\item[(R2)] Each white vertex distributes its negative charge evenly among the adjacent black vertices.
\end{enumerate}
Since $M$ is a maximal matching, $W$ is an independent set in $G$, so no 2 white vertices are adjacent. The white vertices of degree 2 are adjacent to exactly 2 black vertices and sends -1.5 charge to each adjacent black vertex. All other white vertices are adjacent to 3 black vertices and sends -2 charge to each adjacent black vertex. After applying (R2), all white vertices have charge 0.
Let $v$ be a black vertex. Since $v$ is saturated by $M$, $v$ is adjacent to at least one black vertex and hence, $v$ is adjacent to at most 2 white vertices. After receiving charge 0, -1.5, -2, -3, -3.5, or -4 from (R1) according to the number and type of white neighbors, $v$ now has charge 3, 1.5, 1, 0, -0.5, or -1.
Next, let $e_v$ be the black edge incident with $v$, and let $f_v$ be the face incident to $v$ but not $e_v$, if such a face exists. Note that it's possible $f_v$ is an external defect. If such a face does not exists, then both $v$ and $e_v$ must be incident to an external defect, in which case set $u_v$ to be the incident external defect.
\begin{enumerate}
\item[(R3)] Each black vertex sends all of its remaining charge to $f_v$ or $u_v$.
\end{enumerate}
Note that all charge that was initially present at the vertices of $G$ is now at its faces. Due to the face that $G$ is pseudoconvex, it is straightforward to check that if $v$ is a black vertex that sent charge to an external defect according to (R3), then $v$ previously had charge 0, 1, 1.5, or 3. So external defects receive no negative charge after applying (R3), and hence, their total charge is non-negative.
Now the only case when a face receives negative charge from (R3) is when a black vertex $v$ with 2 white neighbors sends charge $-1$ or $-\frac{1}{2}$ (depending on the degrees of the white neighbors) to $f_v$. So if a face is incident with at most 1 white vertex, then its charge is certainly non-negative. It turns out that if a face is incident to at most two white vertices, then its charge is non-negative.
An internal $k$-gonal defect is incident to at most $k/2$ white vertices if $k$ is even and $(k-1)/2$ white vertices if $k$ is odd. Hence, the negative charge an internal $k$-gonal defect receives after applying (R3) is at most $k/2$ in either case. Therefore, the charge of each such defect is at least $\lfloor k/2 \rfloor$ after applying (R3).
All hexagonal faces have nonnegative charge except those incident to three white vertices. For the hexagons incident to at least two white vertices, the different cases for hexagons are split into figures depending on the number of incident vertices of degree 2. The cases for hexagons incident to 0, 1, 2, and 3 vertices of degree 2 can be seen in Figures \ref{internalhex}, \ref{externalhex1}, \ref{externalhex2}, and \ref{externalhex3}, respectively.
\begin{figure}[h]
\begin{center}
\input{h-bad} \hspace{1in} \input{h-transition} \hspace{1in} \input{h-neutral}
\vspace{0.25in}
\input{h-good1} \input{h-good2} \input{h-good3}
\end{center}
\caption{Hexagons adjacent to at least 2 white vertices and 0 vertices of degree 2.} \label{internalhex}
\end{figure}
\begin{figure}[h]
\begin{center}
\input{he-bad} \hspace{1in} \input{he-transition1} \hspace{-0.25in} \input{he-transition-sep} \hspace{-0.25in} \input{he-transition2}
\vspace{0.25in}
\input{he-neutral1} \hspace{-0.25in} \input{he-neutral-sep} \hspace{-0.25in} \input{he-neutral2} \hspace{1in} \input{he-nuetral3}
\vspace{0.25in}
\input{he-good1} \input{he-good2} \input{he-good3} \input{he-good4} \input{he-good5}
\end{center}
\caption{Hexagons adjacent to at least 2 white vertices and 1 vertex of degree 2.} \label{externalhex1}
\end{figure}
\begin{figure}[h]
\begin{center}
\input{he2-transition} \hspace{1in} \input{he2-neutral1} \hspace{-0.25in} \input{he2-neutral-sep} \hspace{-0.25in} \input{he2-neutral2}
\vspace{0.25in}
\input{he2-neutral3} \hspace{1in} \input{he2-good1} \hspace{-0.25in} \input{he2-good-sep} \hspace{-0.25in} \input{he2-good2}
\end{center}
\caption{Hexagons adjacent to at least 2 white vertices and 2 vertices of degree 2.} \label{externalhex2}
\end{figure}
\begin{figure}[h]
\begin{center}
\input{he3-neutral1} \hspace{-0.25in} \input{he3-neutral-sep} \hspace{-0.25in} \input{he3-neutral2} \hspace{1in} \input{he3-neutral3} \hspace{1in} \input{he3-good1}
\end{center}
\caption{Hexagons adjacent to at least 2 white vertices and 3 vertices of degree 2.} \label{externalhex3}
\end{figure}
If a hexagon has negative charge as in Figure \ref{internalhex} (a) or Figure \ref{externalhex1} (a), then these hexagons are called \textit{bad}. If a hexagon has zero charge as in Figure \ref{internalhex} (b), Figure \ref{externalhex1} (b), or Figure \ref{externalhex2} (a), then these hexagons are called \textit{transitional}. If a hexagon has zero charge as in Figure \ref{internalhex} (c), Figure \ref{externalhex1} (c), Figure \ref{externalhex2} (b), or Figure \ref{externalhex3} (a), then these hexagons are called \textit{neutral}. Those hexagons with charge $1/2$ as in Figure \ref{externalhex1} (d), Figure \ref{externalhex2} (b), or Figure \ref{externalhex3} (b) are called \textit{almost neutral}. All other hexagons have a positive charge, and the value of the positive charge is at least the number of incident white vertices. These hexagons with positive charge are called \textit{good}.
Let $f$ be a transitional hexagon. Then $f$ is incident to one black edge, two white vertices, and two black vertices that are incident to black edges not incident to $f$. Let the white vertex adjacent to the black edge incident to $f$ be called \textit{outgoing}. Let the other white vertex, between the black edges that are not incident to $f$, be called \textit{incoming}.
The last steps of the discharging are given by the following rules:
\begin{enumerate}
\item[(R4)] Each good face sends charge 1 to each incident, degree 3 white vertex.
\item[(R5)] Each bad hexagonal face sends charge -1 to each incident, degree 3 white vertex.
\item[(R6)] Each transitional hexagonal face sends charge -1 to the incoming degree 3 white vertex, and it sends charge 1 to the outoing degree 3 white vertex.
\end{enumerate}
After applying (R4)-(R6), there is no negative charge left at the faces of $G$, and the only possible negative charge resides at white vertices of degree 3 in $G$.
Let $v$ be a vertex that was sent charge $-1$ by either (R5) or (R6), let $h$ be the hexagon that sent the negative charge to $v$, and let $u_i$, $i=1,2$, be the black vertices adjacent to $v$ and incident to $h$. Since $h$ is either a bad hexagon or transitional hexagon, then the black edges incident to the $u_i$ are not incident to $h$.
Now let $x$ be the black vertex adjacent to $v$ but not incident to $h$. Since $G$ is pseudoconvex, there exists two faces $f_i$, $=1,2$, incident with $v$ different from $h$, where $f_i$ is incident to $u_i$, $i=1,2$. Without loss of generality, assume the black edge incident with $x$ is incident with $f_1$. Since the black edges incident to both $u_1$ and $x$ are incident to $f_1$, it follows that $f_1$ is either good or neutral, so it does not send negative charge to $v$.
Now consider $f_2$. Since the black edge incident to $u_2$ is incident to $f_2$, then $f_2$ cannot be a bad hexagon, nor a neutral hexagon due to the black edge incident to $x$. Furthermore, $f_2$ cannot be almost-neutral since $v$ is a degree 3 white vertex. If $f_2$ is not incident to any other white vertex other than $v$, then $f_2$ is a good hexagon. If $f_2$ is incident to another white vertex at distance 3 from $v$, then it is a good hexagon as well. If $f_2$ is incident to another white vertex at distance 2 from $v$, then $f_2$ is a transitional hexagon. In the case that $f_2$ is transitional, then $v$ is the outgoing white vertex for $f_2$. Hence in all considered cases, $f_2$ has sent positive 1 charge to $v$ by (R4) or (R5).
\begin{figure}[h]
\begin{center}
\input{f2bad} \hspace{1in} \input{f2transition}
\end{center}
\caption{On the left shows the case when $h$ is a bad hexagon and $f_2$ is transitional. On the right shows when both $h$ and $f_2$ are transitional.}\label{f2}
\end{figure}
Repeating the above argument, it's possible $G$ has a chain of adjacent, transitional hexagons, which in turn, would move charge between adjacent hexagons. If such a chain starts with a bad hexagon, then the chain cannot close on itself forming a cycle of hexagons. Such a cycle would have to close at the bad hexagon, implying a white vertex receives negative charge from both a bad hexagon and transitional hexagon and this cannot happen according to the above argument. A chain beginning with a transitional hexagon could close to form a cycle of transitional hexagons, in which case, since transitional hexagons have zero charge, the discharging simply moved zero charge around in a cycle.
Thus after these last steps of discharging, there is no negative charge in the graph. So the total sum of charge is non-negative, which finishes the proof.
\end{proof}
\section{Proofs for nanocones} \label{secnanocone}
\begin{proof}[Proof (of Lemma \ref{benztriangle})]
First a construction of a maximal matching $M$ is given and then below it is shown this matching yields the desired bound. To construct $M$, $T_p$ is drawn in the plane so that the hexagons appear in columns and the number of hexagons in columns decreases moving to the right, as in Figure \ref{benztrianglematch}. Moving left to right, the following pattern of matched edges is iterated every 3 columns of hexagons: the first column of $k$ hexagons requires $k+1$ matched edges, the second column of $k-1$ hexagons requires $k$ matched edges, and the third column is skipped, since edges from the second column partially matches the third column. See the bold edges in Figure \ref{benztrianglematch} for examples of these matchings. This pattern is continued so long as there are at least 3 columns of hexagons remaining, at which point the pattern breaks. This process yields a maximal matching of size
$$
(p+1)+p+(p-2)+(p-3)+ \cdots + k_2 + k_1
$$
where the end values $k_i$, $i=1,2$, fall into 3 cases depending on the value of $p+1$ modulo 3, which can be seen in the matchings of $T_3$, $T_4$, and $T_5$ in Figure \ref{benztrianglematch}.
\begin{figure}[h]
\input{triangle2} \hfill \input{triangle3} \hfill \input{triangle4} \hfill \input{triangle5}
\caption{Maximal matchings of $T_2$, $T_3$, $T_4$, and $T_5$ as described in Lemma \ref{benztriangle}.} \label{benztrianglematch}
\end{figure}
The remaining argument is broken into these 3 cases:
\begin{enumerate}
\item[(Case 1)] $p+1 \equiv 0 \pmod 3$
\end{enumerate}
In this case, the construction gives a matching of size
$$
((p+1)+p) + ((p-2)+(p-3)) + \cdots + (6+5) + (3+2)
$$
and then iteratively rearranging terms so that the next largest term is now paired with the next smallest term yields
$$
((p+1)+2) + (p+3) + ((p-2)+5) + ((p-3)+6) + \cdots ((p-k)+(k+3))
$$
for some value $k$. This new sum consists of $(p+1)/3)$ pairs each summing to $(p+3)$, so the matching has size exactly $(p+1)(p+3)/3$.
\begin{enumerate}
\item[(Case 2)] $p+1 \equiv 1 \pmod 3$
\end{enumerate}
In this case, it also follows that $(p+3) \equiv 0 \pmod 3$. The construction gives a matching of size
$$
((p+1)+p) + ((p-2)+(p-3)) + \cdots + (4+3) + 1
$$
which can be rearranged to
$$
((p+1)+0) + (p+1) + ((p-2)+3) + ((p-3)+4) + \cdots ((p-k)+(k+1))
$$
for some value $k$. This new sum consists of $(n+3)/3)$ pairs each summing to $(p+1)$, so the matching has size exactly $(p+1)(p+3)/3$.
\begin{enumerate}
\item[(Case 3)] $p+1 \equiv 2 \pmod 3$
\end{enumerate}
This case has $(p+1) = 3q + 2$ for some integer $q$. The construction gives a matching of size
$$
((p+1)+p) + ((p-2)+(p-3)) + \cdots + (5+4) + 2.
$$
First, the smallest and largest terms are paired together, $((p+1)+2)$, and the next largest term, $p$, is reserved. The remaining terms are iteratively rearranged so that the next largest term is now paired with the next smallest term to obtain
$$
((p-2)+4) + ((p-3)+5) + \cdots ((p-k)+(k+2))
$$
for some value $k$. This last sum results in $(q-1)$ pairs each summing to $(p+2)$. Hence the matching has size $(p+3)+p+(q-1)(p+2)$. Now
\begin{align*}
(p+3)+p+(q-1)(p+2) &= (p+3)+p+(q-1)(p+3)-(q-1) \\
&= p+q(p+3)-(q-1) \\
&= p-\frac{p-4}{3} + q(p+3) \\
&< \frac{2}{3}(p+3) + q(p+3) \\
&=\frac{(p+1)(p+3)}{3}
\end{align*}
which proves the desired bound.
\end{proof}
\begin{proof}[Proof (of Corollary \ref{lowernanocone})]
The claimed lower bounds are a straight forward application of Theorem \ref{lowerbound} depending on the total number of vertices, the number of pentagons, and the number of vertices of degree 2. The counts of these values are provided below, where both counts of vertices follow by induction on $\ell$.
(a) $CN_2^s(\ell)$ has $14+4\ell(\ell+4)$ total vertices, 2 pentagons, and $4(\ell +2)$ vertices of degree 2.
(b) $CN_2^a(\ell)$ has $11+2\ell(2\ell+7)$ total vertices, 2 pentagons, and $4\ell +7$ vertices of degree 2.
(c) $CN_3^s(\ell)$ has $10+3\ell(\ell+4)$ total vertices, 3 pentagons, and $3(\ell +2)$ vertices of degree 2.
(d) $CN_3^a(\ell)$ has $16+\ell(3\ell+16)$ total vertices, 3 pentagons, and $3\ell +8$ vertices of degree 2.
(e) $CN_4^s(\ell)$ has $12+\ell(2\ell+12)$ total vertices, 4 pentagons, and $2(\ell+3)$ vertices of degree 2.
(f) $CN_4^a(\ell)$ has $15+\ell(2\ell+14)$ total vertices, 4 pentagons, and $2\ell+7$ vertices of degree 2.
(g) $CN_5^a(\ell)$ has $16+\ell(\ell+12)$ total vertices, 5 pentagons, and $\ell+6$ vertices of degree 2.
\end{proof}
\begin{figure}
\centering
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CNC54-shaded.png}
\caption*{$CNC_5(4)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN2s3-shaded}
\caption*{$CN_2^s(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN2a3-shaded}
\caption*{$CN_2^a(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN3s3-shaded}
\caption*{$CN_3^s(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN3a3-shaded}
\caption*{$CN_3^a(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN4s3-shaded}
\caption*{$CN_4^s(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN4a3-shaded}
\caption*{$CN_4^a(3)$}
\end{subfigure}
\begin{subfigure}[h]{.3\textwidth}
\includegraphics[width=\textwidth]{CN5a3-shaded}
\caption*{$CN_5^a(3)$}
\end{subfigure}
\caption{Nanocones split into benzenoid triangles, or subgraphs thereof, which are represented by the shaded regions.} \label{nanoconetriangle}
\end{figure}
\begin{proof}[Proof (of Theorem \ref{uppernanocone})]
Observe that a nanocone with $s$ break edges can be split into $s$ benzenoid triangles, or subgraphs of benzenoid triangles. Each such benzenoid triangle or subgraph resides between successive hexagons containing the break edges each layer of hexagons, see Figure \ref{nanoconetriangle}. The sizes of the triangles depends on the lengths of the sides of the nanocone. Lemma \ref{benztriangle} can be used to find a maximal matching of the benzenoid triangles of the indicated size. The union of these matchings augmented by a matching of size at most $s(\ell+1)$ along the break edges from each layer, and potentially an additional matching of the cap, gives an upper bound on the size of a maximal matching of the nanocone. Additional details for each case are provided below.
(a) $CNC_k(\ell)$ has $k$ break edges and therefore can be split into $k$ benzenoid triangles $T_{\ell-1}$, each triangle with a matching of size $\left \lfloor \frac{\ell(\ell+2)}{3} \right \rfloor$. The union of these matching augmented by a matching of size of size $k(\ell+1)$ along the break edges from each layer provides an upper bound for a maximal matching of $CNC_k(\ell)$.
(b) By Lemma \ref{break}, $CN_2^s(\ell)$ has 4 break edges and can be split into 4 benzenoid triangles, $T_\ell$, each with a matching of size $\left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor$. Their union augmented by a matching along the break edges of size at most $4(\ell +1)$ plus an addition edge needed for the remaining edges on the cap yields the desired upper bound.
(c) By Lemma \ref{break}, $CN_2^a(\ell)$ has 4 break edges and can be split into 3 $T_\ell$'s each with a matching of size $\left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor$ and an additional $T_{\ell-1}$ with a matching of size $\left \lfloor \frac{\ell(\ell+2)}{3} \right \rfloor$. The break edges require at most $4(\ell +1)$ edges after which the cap requires 1 additional edge.
(d) By Lemma \ref{break}, $CN_3^s(\ell)$ has 3 break edges and can be split into 3 $T_\ell$'s each with a matching of size $\left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor$. The break edges union the cap of the nanocone require at most an additional $3(\ell + 1)+1$ edges.
(e) Again using Lemma \ref{break}, $CN_3^a(\ell)$ has 3 break edges and can be split into $T_{\ell + 1}$, $T_\ell$, and a subgraph of $T_{\ell+1}$, which in total require at most
$$
2 \left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor + \left \lfloor \frac{(\ell+1)(\ell+3)}{3} \right \rfloor
$$
matched edges. The break edges need at most $3(\ell + 1)$ edges and the cap requires at most 2 edges, proving the desired bound.
(f) Lemma \ref{break} gives that $CN_4^s(\ell)$ has 2 break edges. So $CN_4^s(\ell)$ can be split into two pieces which turn out to be subgraphs of $T_{\ell+1}$, and each subgraph has a maximal matching of size at most $\left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor$. The union of these matchings augmented by a matching of the break edges of size at most $2(\ell+1)$ along with a matching of size 3 for the remaining edges of the cap provides the desired maximal matching.
(g) Similar to the case in (f), $CN_4^a(\ell)$ can be split into subgraphs of $T_{\ell+1}$ and $T_{\ell+2}$ requiring at most
$$
\left \lfloor \frac{(\ell+3)(\ell+5)}{3} \right \rfloor + \left \lfloor \frac{(\ell+2)(\ell+4)}{3} \right \rfloor
$$
matched edges. The break edges again require at most $2(\ell + 1)$ matched edges, after which the cap needs at most 4 edges.
(h) By Lemma \ref{break}, $CN_5^a(\ell)$ has 1 break edge and contains a subgraph of $T_{\ell+4}$, which according to Lemma \ref{benztriangle} has a maximal matching of size at most $\left \lfloor \frac{(\ell+5)(\ell+8)}{3} \right \rfloor$. The break edges require at most $(\ell+1)$ matched edges and the cap needing an additional 6 matched edges.
\end{proof}
\begin{comment}
\begin{proof}[Proof (of Corollary \ref{upsatcone})]
If $k$ is even, then $CNC_k[\ell]$ can be split into $k/2$ benzenoid parallelograms each with dimension $\ell \times \ell$. See Figure *** for an example of ***.
\begin{figure}[h]
\input{CNC_6[4]}\input{CNC_5[4]}
\caption{On the left, $CNC_6[3]$, and on the right, $CNC_5[4]$.}
\end{figure}
\noindent By Proposition \ref{benzparallel}, each parallelogram has a maximal matching of size
$$
\left \lceil \frac{2\ell+1}{3} \right \rceil \cdot \ell+\ell.
$$
Their union augmented by a matching of size at most $k/2\cdot (\ell+1)$ along the hexagons joining the parallelograms gives the claimed bound for $s(CNC_k[\ell])$.
If $k$ is odd, then then $CNC_k[\ell]$ can be split into $\lfloor k/2 \rfloor + 1$ pieces: $\lfloor k/2 \rfloor$ benzenoid parallelograms each with dimension $\ell\times \ell$ with one remaining piece which is half of a benzenoid parallelogram with dimension $\ell\times (\ell-1)$. The $\ell\times \ell$ parallelograms have maximal matchings of the size mentioned above, and by Proposition \ref{benzparallel} the half benzenoid parallelogram has a maximal matching of size
$$
\frac{1}{2}\left( \left \lceil \frac{2\ell-1}{3} \right \rceil \cdot \ell +(\ell-1) \right).
$$
The union of these matchings augmented by a matching of size at most $\lceil k/2 \rceil (\ell+1)$ along the hexagons joining the pieces gives the claimed bound for $k$ odd.
\end{proof}
It should be noted the bounds in Corollary \ref{upsatcone} could be tightended, however, these bounds show that the saturation number is at most $n/3$ which is asymptotically tight with our lower bounds that follow.
\end{comment}
\section{Proofs for nanotubes} \label{sectubes}
\begin{proof}[Proof (of Corollary \ref{lowerzigzag})]
It follows that $ZT(\ell, m)$ has $2m\ell + 2m$ total vertices and two external defects at the ends of the cylinder. The external defects each have $m$ vertices of degree 2, for a total of $2m$ degree 2 vertices. Now Theorem \ref{lowerbound} gives that
\begin{align*}
s(ZT(\ell, m)) &\ge \frac{2m\ell+2m}{3} - \frac{2m}{6} \\
&= \frac{m(2\ell+1)}{3}.
\end{align*}
\end{proof}
\begin{proof}[Proof (of Corollary \ref{lowerarmchair})]
The armchair tubulene $AT(m, \ell)$ has $2m\ell + 2\ell$ total vertices and $2\ell$ vertices of degree 2. Theorem \ref{lowerbound} gives that
\begin{align*}
s(AT(m, \ell)) &\ge \frac{2m\ell+2\ell}{3} - \frac{2\ell}{6} \\
&= \frac{\ell(2m+1)}{3}.
\end{align*}
\end{proof}
\section{Acknowledgements}
This work was partially supported by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina. I am extremely grateful to Tomislav Do\v{s}li\'{c}, whose mentoring and hosting during the Summer 2015 eventually led to the completion of this work. I would also like acknowledge the software CaGe \cite{cage} which helped draw Figures \ref{cones} and \ref{nanoconetriangle} in this paper, and I would like to thank Nico Van Cleemput, who helped export graphs generated with CaGe for my initial experimentation. I would like to thank Michael Santana for a helpful discussion concerning Theorem \ref{lowerbound}.
\bibliographystyle{amsplain}
|
1,314,259,993,030 | arxiv | \section{Introduction}
\label{sec:Introduction}
Ant Colony Optimization (ACO) is a population-based metaheuristic
inspired by the social behavior of ants~\cite{Dorigo2004}.
It has been successfully applied in solving many NP-hard problems,
including the Traveling Salesman Problem (TSP),
the Quadratic Assignment Problem,
and the Sequential Ordering Problem~\cite{Skinderowicz2017,Stutzle2000,Talbi2001}.
Being metaheuristic, the ACO does not guarantee
finding an optimum solution, however it is often able to offer
satisfactory \emph{approximate} solutions within an acceptable
time compared to exact methods~\cite{Dorigo2018}.
However, even metaheuristics can be prohibitively time consuming
if faced with a large enough problem instance.
For this reason, a lot of research attention has been devoted both to
improving the effectiveness of the ACO search process,
and to speeding up its execution~\cite{Pedemonte2011}.
The idea of applying GPU-based computing to the ACO
is an example of the latter.
In recent years, the use of graphics processing units (GPUs)
to speed up scientific computations has become commonplace.
This has been, in part, dictated by the slow-down in Moore's law,
and the progress made in the GPU architecture development, which has resulted
in more computing capacity, flexibility, and ease-of-use~\cite{Thompson2018}.
In fact, currently a significant proportion of the world's fastest supercomputers
is equipped with GPUs to accelerate their computations~\cite{top500}.
Still, the efficient use of the computing capacity offered by GPUs
remains a difficult task~\cite{Jia2018}.
In an ACO, a population of agents (ants) construct, in parallel,
a set of solutions to the optimization problem begin tackled.
Unfortunately, the inherent parallel nature of the ACO
does not translate easily into an efficient GPU-based parallel
implementation~\cite{Cecilia2012,Dawson2013b,Skinderowicz2016}.
The difficulties arise partly from the fact that not all of the ACO
computations are independent, e.g., pheromone trail updates; as well as from the
computing restrictions inflicted by GPU architectures.
Although considerable research attention has been devoted to using GPUs to
speed up the ACO-based algorithms, both the improved computational capabilities
of successive generations of GPUs as well as novel algorithmic ideas, offer new
opportunities for greater progress.
In this paper, we present a GPU-based parallel implementation of the MAX-MIN
Ant System (MMAS), which is one of the best-performing ACO
variants for solving various optimization problems including
the production-distribution scheduling problem~\cite{Jia2019},
the blocks relocation problem~\cite{Jovanovic2019},
the routing and scheduling of home health care caregivers
problem~\cite{Decerle2019}, and the traveling purchaser
problem~\cite{Skinderowicz2018}.
Building on existing research we show how each of
the essential MMAS components can be parallelized to allow the efficient use of
the significant computing power offered by the current generation of GPUs.
The main contributions presented in this paper can be summarized as follows:
\begin{itemize}
\item
We present a novel parallel implementation of the next node (proportional) selection
procedure used in the MMAS and other ACO algorithms.
The implementation is based on the weighted reservoir sampling algorithm and
fits well within the parallel computing model of contemporary GPU architectures.
\item
We present a novel, memory-efficient implementation of the tabu list structure
used by the ACO solution construction process. The implementation
allows for the better utilization of fast, but very size limited, shared memory
of GPUs.
\item
Combining these novel ideas and the solutions from the literature, we
present a total of six MMAS variants and evaluate
their computational efficiency based on two subsequent generations
of Nvidia GPUs, namely Pascal and Volta.
\item
The computational evaluation based on a set of TSP instances ranging from 198
to 3,795 cities
shows that the proposed GPU-based MMAS is competitive with
state-of-the-art GPU-based~\cite{Dawson2013,Dawson2013b,Cecilia2018}
and multi-core CPU-based~\cite{Zhou2018} parallel ACO implementations.
The times obtained were up to 7.18x and 21.79x smaller, respectively.
\item
Acknowledging, that the ACO algorithms are typically paired with
an efficient, problem-specific local search (LS) method, we
combine the proposed GPU-based MMAS with a parallelized 2-opt heuristic.
The computational experiments that consider the TSP instances of up to 18,512
nodes show that the proposed implementation is able to generate
high-quality solutions, i.e., within 1\% from an optimum, in a relatively
short time.
\end{itemize}
The remainder of this paper is organized as follows. In Section~\ref{sec:background}
we provide a brief description of the MAX-MIN Ant System and short piece on the
characteristics of
general-purpose GPU computations. Section~\ref{sec:related-work} summarizes the existing
work on applying GPUs in speeding up ACO computations, including the MMAS.
Our main ideas on the efficient implementation of the GPU-based MMAS are presented
in Section~\ref{sec:implementing-gpu-based-mmas}; while
Section~\label{sec:experimental-results} presents an analysis of the computational
experiments we conducted in order to evaluate the proposals, and compare them with the work
described in the literature.
Finally, we summarize our findings and provide a few ideas for future work
in Section~\ref{sec:conclusions}.
\section{Background}
\label{sec:Background}
\label{sec:background}
\subsection{MAX-MIN Ant System}
\label{sec:MAX_MIN_Ant_System}
The ACO metaheuristic belongs to a group of swarm-based metaheuristics (SBMs)
in which the problem-solving abilities are a result of the interactions of
simple information-processing units (agents)~\cite{Kennedy2006}.
The inspirations for the SBMs often come from biological systems including ant
colonies, swarms of bees, flocks of birds, and schools of fish, among
others~\cite{Engelbrecht2005}. Typically, the agents in the SBMs follow simple
rules and are given a certain degree of autonomy, e.g., in selecting the next
action to perform. The agents may also interact with each other, e.g., by
transferring data about the solutions found, or with the environment, e.g., by
depositing artificial pheromone trails that can be read by other agents
(indirect communication).
In addition to ACO, one of the most-successful SBMs are particle swarm
optimization (PSO) and artificial bee colony algorithms.
In the MMAS, a number of ants (agents) iteratively construct solutions to a
combinatorial optimization problem (COP)~\cite{Stutzle2000}. In this paper, we
focus on the TSP, following the existing research on parallel
ACO~\cite{Bai2009,Dawson2013,Dawson2013b,Delevacq2013,Zhou2017,Cecilia2018},
although, in principle, the ACO algorithms can be applied to any
COP~\cite{Dorigo2004}.
The TSP can be defined using a complete graph $G = (V, A)$, where $V$ is a set
of nodes numbered from 0 to $n-1$ ($n$ being the number of cities), and $A$ as
the set of edges (arcs) between the nodes, i.e., $E = \{ (i, j): i, j \in V, \, i
\ne j \}$. The set of nodes represents a set of cities to be visited by a
salesman, while each edge in $E$ corresponds to a road between a pair of
cities. Additionally, for every edge, $(i, j)$, a positive value, $d_{ij}$, is
given, which represents the distance (weight) between the cities, $i$ and $j$. If the TSP
is \emph{symmetric}, then $d_{ij} = d_{ji}$, otherwise the instance is
asymmetric (ATSP) and the distance from $i$ to $j$ might not equal the distance
from $j$ to $i$.
Typically, the distances between the nodes in graph $G$ satisfy the triangle
inequality but, in general, they may be arbitrary.
Solving the TSP is equivalent to finding the minimum weight Hamiltonian cycle
in graph $G$. In general, solving the Hamiltonian cycle is an NP-hard problem
and finding the optimum to the TSP is at least as difficult.
A comprehensive overview of both exact and approximate approaches to solving the TSP
can be found in the work of Applegate et al.~\cite{Applegate2007}.
In the ACO, the edges of graph $G$ correspond to the \emph{solution
components} from which the ants' solutions are being constructed. Additionally,
for every edge, $(i, j) \in E$, there is an associated \emph{pheromone trail},
$\tau_{ij}(t)$, where $t$ denotes a (discrete) time.
In nature, the pheromones are chemical substances that some ant species
use as an indirect
medium of communication between individuals~\cite{Dorigo2004}.
In ACOs, including
the MMAS, the artificial pheromone trails are stored as positive real values,
and influence the probability of including the corresponding solution
components into the solutions being constructed by the ants. In the MMAS,
in contrast to other ACO algorithms, the values of the pheromone trails are
\emph{bounded} by limits: $\tau_{\rm min}$ and $\tau_{\rm
max}$~\cite{Stutzle2000}.
In the MMAS, an ant starts its solution construction process from an
initial node and in each of the subsequent steps it selects an edge (a solution
component) that connect its current node with one of the neighboring, as yet
unvisited, nodes. The choice of an edge is probabilistic and depends on
so-called \emph{heuristic information} (available \emph{a priori}) and
the values of the pheromone trails.
Specifically, an ant, $k$, positioned at node $i$ selects an edge, $(i, j)$,
leading to node $j$ with the probability:\\
\begin{equation}
\label{eq:prob}
p_{ij}^k(t) = \frac{ [ \tau_{ij}(t) ]^\alpha [ \eta_{ij} ]^\beta }{ \sum_{l \in \mathcal{N}_i^k} [\tau_{il}(t)]^\alpha [ \eta_{il} ]^\beta } \; \quad \textrm{if} \; j \in \mathcal{N}_i^k \, ,
\end{equation}\\
where $\tau_{ij}(t)$ is the value of the pheromone trail deposited on the edge,
$(i,j)$; $\eta_{ij}$ is the value of the heuristic information for the edge,
$(i, j)$; $\alpha$ and $\beta$ are parameters that control the relative
influence of the pheromone values and the heuristic information on the
probability; and, finally, $\mathcal{N}_i^k$, denotes the set of nodes
that neighbor $i$ to be visited the ant, $k$.
The heuristic information, $\eta_{ij}$, specifies how \emph{attractive} a
particular edge $(i,j)$ is, and in the case of the TSP, $\eta_{ij} = 1 /
d_{ij}$, makes an edge more attractive the shorter it is.
This is based on the assumption that good quality solutions consist of edges
connecting nodes located near to each other~\cite{Helsgaun2009}.
Each ant stores the previously visited cities in a \emph{tabu list}, which
allows $\mathcal{N}_i^k$ to be computed, guaranteeing that only valid
Hamiltonian cycles are constructed.
\begin{algorithm}[h]
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\def\ant[#1]{\textrm{Ant}(#1)}
\def\route[#1]{\textit{route}_{\textrm{Ant}(#1)}}
\def\tabu[#1]{\textit{tabu}_{\textrm{Ant}(#1)}}
\def\textit{iter\_best}{\textit{iter\_best}}
\def\textit{global\_best}{\textit{global\_best}}
Calculate pheromone trails limits: $\tau_{\rm{min}}$ and
$\tau_{\rm{max}}$ \\
Set pheromone trails values to $\tau_{\rm{max}}$ \label{alg:mmas:pherinit} \\
$\textit{global\_best} \leftarrow \emptyset$
\For{ $i \leftarrow 1$ \KwTo $\textit{\#iterations} $ }{
\label{alg:mmas:main.loop.start}
\For{ $j \leftarrow 0$ \KwTo $\textit{\#ants}-1$ }{
$u \leftarrow \mathcal{U}\{0, n-1\}$ \quad \tcp{Select the first node randomly}
$\route[j] \left[0\right] \leftarrow u$ \\
Add $u$ to $\tabu[j]$
\For(\tcp*[h]{Complete the solution (route)}){ $k \leftarrow 1$ \KwTo $n - 1$ }{
$u \leftarrow \textrm{select\_next\_node}( \route[j] \left[k-1\right], \tabu[j] )$ \\
$\route[j] \left[k\right] \leftarrow u$ \\
Add $u$ to $\tabu[j]$
}
}
$\textit{iter\_best} \leftarrow \textrm{select\_shortest} \left(\route[0], \ldots, \route[\#ants-1] \right)$
\label{alg:mmas:iter.best} \\
\If{ $\textit{global\_best} = \emptyset$ {\bf or} $\textit{iter\_best}$ {\rm is shorter than} $\textit{global\_best}$ }{
$\textit{global\_best} \leftarrow \textit{iter\_best}$ \\
Update pheromone trails limits
$\tau_{\rm{min}}$ and $\tau_{\rm{max}}$ using $\textit{global\_best}$
}
Evaporate pheromone according to $\rho$ parameter \\
Deposit pheromone based on $\textit{iter\_best}$ \label{alg:mmas:pher.deposition} \\
\label{alg:mmas:main.loop.end}
}
\caption{The MAX-MIN Ant System.}
\label{alg:mmas}
\end{algorithm}
The pseudocode for the MMAS is shown in Fig.~\ref{alg:mmas}. At first, the initial
pheromone trail limits, $\tau_{\rm min}$ and $\tau_{\rm max}$, are computed
based on a solution constructed using the nearest neighbor heuristic.
Next, the pheromone trails values are set to $\tau_{\rm max}$
(line~\ref{alg:mmas:pherinit}).
In the main loop of the algorithm
(lines~\ref{alg:mmas:main.loop.start}--\ref{alg:mmas:main.loop.end}), each ant
constructs a complete solution to the problem starting from a randomly chosen
node.
After the solutions have been constructed, the \emph{iteration best}
solution is selected (line \ref{alg:mmas:iter.best}). If it is shorter than
the current \emph{global best} solution, it becomes the new global best, and
the trail limits are updated accordingly.
Finally, a pheromone update is performed.
This means lowering (evaporating) the values of the pheromone trails:
$\tau_{ij} \leftarrow {\rm max}\left( \rho \tau_{ij}, \tau_{\rm min} \right)$,
where $\rho$ is a parameter that controls the evaporation speed.
The values of the pheromone trails never drop below the minimum value,
$\tau_{\rm min}$, which ensures that all edges have a non-zero probability of
being selected even in the late stages of the algorithm's execution.
The pheromone trails' values only increase if they correspond to the components
of the current iteration's best solution (line~\ref{alg:mmas:pher.deposition}).
The values are increased according to:
$\tau_{ij} \leftarrow {\rm min}\left( \tau_{ij} + \Delta_{ij}, \tau_{\rm max} \right)$, where
\[
\Delta_{ij} =
\begin{cases}
{\rm cost}( {\it iter\_best } )^{-1} \, , & {\rm if~~} (i, j) \in {\it iter\_best} \, , \\
0 \, , & {\rm otherwise} \, .
\end{cases}
\]
Increasing pheromone levels for the trails corresponding to the edges (solution
components) of good quality solutions, increases the probability that, in
subsequent iterations, the ants will choose these edges more often. This process
allows the algorithm to learn and construct higher quality solutions over
time~\cite{Dorigo2004}.
It is possible to use the current \emph{global best} value instead of the
iteration best~\cite{Stutzle2000}.
It is worth noting, that in contrast to the Ant Colony System (ACS),
parallelization is made simpler because the MMAS \emph{lacks} a local pheromone
update~\cite{Skinderowicz2016}.
In fact, the pheromone trail values remain constant during the solution
construction phase, allowing a \emph{beforehand} computation of the product of
the pheromone trails and the heuristic values required by~Eq.~(\ref{eq:prob}).
This optimization is in common use as it reduces both the computation time,
and more importantly, the number of loads from the
memory~\cite{Cecilia2012,Dawson2013b}.
We also apply it in our work, storing the product in a matrix called
\emph{choice\_info}.
A single iteration of the MMAS has a $O(mn^2)$ time complexity, as each of the
$m$ ants constructs a complete solution to the problem in $n-1$ steps (assuming
that the starting node is chosen arbitrarily), $n$ being the size of the problem
instance.
Each step has a complexity of $O(n)$ as an ant has to move from the
current node to the next, chosen from up to $n-1$ unvisited nodes.
If candidate lists are used, the average complexity of the solution
construction process falls to $O(n \cdot \textit{cl}) = O(n)$, where
$\textit{cl}$ is a constant that denotes the size of the list.
\subsection{General-purpose computing using GPUs}
\label{sec:General_purpose_computing_on_GPUs}
Architectural differences between the CPUs and the GPUs allow the latter to
offer a higher computing power but often at the cost of reorganizing the
structure of the calculations to enable parallel execution~\cite{Kirk2016}.
For the sake of clarity of further discussion,
it is worth clarifying the distinction between \emph{parallelism} and
\emph{concurrency}. Assuming that the required computations were divided
into independent portions or tasks, parallel execution refers to the case in which
the available processing elements (cores) execute the tasks
\emph{at the same time}. Concurrency, on the other hand, is a more general
term also including the cases in which some of the computations
may not overlap in time. In the simplest case, concurrent computations
require only a single computing unit working in a time-shared manner.
\R{
In other words, concurrency allows to handle multiple computing tasks \emph{at
once}, while parallelism emphasises doing multiple computations \emph{at the same
time}.
}
GPUs allow for a high degree of parallelism as they typically contain several replicated
\emph{streaming multiprocessors} (SMs), each comprising a number of
\emph{processing elements} that share control units, a register file,
caches, and shared memory.
However, the number of computational tasks should typically exceed the number
of available processing elements. This is helpful in situations in which the
computations get stalled, e.g., while waiting for the data to be read from
memory. In such cases, it is possible to switch to
another task for which the necessary data are available.
Somewhat related is the description of GPUs as being \emph{throughput-oriented}
meaning that a large number of computations can be performed in a given
period of time, however, the speed of execution of individual computations could be low
compared with that of CPUs~\cite{Kirk2016}.
SMs schedule and execute hundreds of parallel threads in groups of 32 called
\emph{warps}.
The warps employ a model called a \emph{single-instruction, multiple-thread}
(SIMT), in which all threads start at the same program address and typically
execute the same instructions over different data (data-parallelism).
However, each thread has its own program counter and register state so its
execution may diverge from the other threads in the warp.
From the performance point of view, it is best to keep the
number of diverging executions as low as possible; although, in the newer
Nvidia architectures (Volta, Turing) the penalty paid is lower than previous
versions~\cite{Jia2018}.
It is also worth adding that the threads within a warp have access to
primitives allowing them to access each other's registers directly, i.e.,
without the need for accessing slower, shared memory.
Nvidia's Compute Unified Device Architecture (CUDA) provides a
\emph{programming model} that forms an abstract layer over the hardware
architecture~\cite{CUDA2018}.
The CUDA divides programs into CPU (host) and GPU (device) parts.
The host and the device have separate memory spaces, but a unified memory
extension exists in CUDA 6.0 (and newer versions) that allows the CPU and GPU
threads to store data in a shared address space.
A programmer can define functions, called \emph{kernels}, which are executed by
the GPU.
Each kernel is executed by a specified number of concurrent threads that are
divided into several \emph{blocks}, which in turn are organized into a
\emph{grid}.
Each thread-block is assigned to a single SM and can communicate through the
shared memory with the other threads within the same block.
The blocks are scheduled independently; hence, threads belonging to separate
blocks can only communicate using large but high-latency global memory.
Summarizing, each thread executing on the GPU has access to a memory hierarchy,
with the privately-accessed registers being the fastest, the shared memory
being a bit slower, a small but cached \emph{constant memory} also offering
relatively fast reads, and, finally, the global memory being the slowest. L1
and L2 caches are also present but not directly accessible to the programmer.
The CUDA programming model assumes that a large number of threads (tens of
thousands) is executed concurrently to allow memory-related latencies to be
hidden.
\section{Related work}
\label{sec:Related_work}
\label{sec:related-work}
Being a population-based metaheuristic, the ACO naturally exhibits some degree
of parallelism~\cite{Dorigo2004}.
For example, there is no direct communication between the ants.
In fact, the ants cooperate \emph{indirectly} (stigmergy) by modifying the
values of the pheromone trails that correspond to the components of the problem
they select during the solution construction phase.
If the solutions are constructed quickly, as they are in the case of the TSP,
the frequent updates of the pheromone trails become problematic from the
parallelization point of view.
Overall, a lot of research has been devoted to the parallelization of the ACO,
especially for execution on multi-CPU systems aimed at both improving the
quality of the generated solutions, and shortening the execution
time~\cite{Randall2002, Chu2004, Manfrin2006}.
A good summary of the research was done by Pedemonte et
al.~\cite{Pedemonte2011}.
Although valuable, the CPU-based parallelization of the ACO is difficult to
transfer directly to the GPUs due to differences in the hardware architectures.
In most of the approaches to CPU-based ACO parallelization, a coarse-grained
organization of computations is favored, with the multi-colony ACO being one of
the most efficient.
The GPUs on the other hand, are \emph{throughput oriented} and contain
thousands of relatively simple processing elements.
Only recently has the
increasing number of CPU cores and the availability of wide vector
instructions (e.g. AVX2) allowed for a more efficient, fine-grained
approach~\cite{Zhou2017}. For these reasons, the rest of the section will
focus on research that targets the GPU-based parallelization of the ACO.
The first attempts at using GPUs to speedup the ACO predate the CUDA
programming framework.
Catala et al.~\cite{Catala2007} presented a parallel ACO for solving the
Orienteering Problem, although some speed increases were reported, the
implementation was complicated as the authors had to use graphics generation
primitives to perform computations.
A similar programming approach was used by Wang et al.~\cite{Wang2009}, who
proposed a GPU-based MMAS.
The authors reported a modest speedup compared to a sequential, CPU-based MMAS
implementation.
\subsection{Task-based vs data-parallel approaches}
One of the first attempts to speed up the MMAS using the first generation of
general purpose GPUs was made by Bai et al.~\cite{Bai2009}.
This approach used multiple ant colonies with a single colony assigned to a single thread block, and each thread within the block assigned to an ant.
The distance matrix was stored in texture memory to facilitate cache memory.
Computational experiments showed the execution was around 2x
faster than a reference CPU implementation when solving the TSP.
This is an example of \emph{task-based} parallelism, as the threads are directly
mapped to the ants.
The problem with this approach is that it leads to \emph{warp-branching}, i.e.,
different threads within a thread-warp (using the Nvidia CUDA-based
terminology) are likely to take different execution paths as the ants follow
divergent paths, causing the remaining threads to wait.
A more efficient, \emph{data-parallel} approach was proposed by Cecilia et
al.~\cite{Cecilia2012}.
In that implementation, a single thread block is mapped to a single ant in the
Ant System (AS), that is, all threads within a thread block work on a single
solution to the problem.
This avoids the warp-branching present
in the task-based approach as the threads execute the same instructions but for
different data, i.e., nodes.
The authors considered block sizes of 16 to 1,024 threads.
The computational experiments done on the Nvidia Tesla C2050 GPU showed that
the best performance was obtained for 128 threads per block.
The work's most notable contribution was the introduction of the so-called
\emph{I-Roulette} (independent roulette) method for selecting, in parallel, the next
city to be visited by an ant.
This was the alternative to the proportional selection method, also known as
the Roulette Wheel Method (RWM), which was used originally.
In the I-Roulette method, the probabilities of selecting each of the unvisited
nodes (assigned to separate threads) are multiplied by random numbers, and the
node that has the maximum product is selected through a parallel reduction.
Although, the I-Roulette method did not produce the same results as the
sequential RWM, it was up to 2.36x faster.
The authors also considered two parallel pheromone update methods, in which the
simpler one used atomic instructions to allow the safe simultaneous
modification of memory by multiple threads.
Overall, the reported speedups were up to 20x faster compared to the sequential
implementation.
A valuable comparison between the task-based and data-parallel approaches can be
found in the work of Del{\'e}vacq et al.~\cite{Delevacq2013} who presented a
GPU-based parallel implementation of the MMAS for the TSP.
In the task based approach, each ant was assigned to a CUDA thread.
In the data-parallel approach, a whole thread-block was assigned to a single ant.
Moreover, the 3-opt local search for improving the ants' solutions was also
included in both approaches.
The data-parallel approach was significantly faster than the task-based one,
and up to 19.47x faster than the reference sequential implementation.
The inclusion of the 3-opt resulted in more modest speedups of up to 8.03x for
the data-parallel implementation.
The authors concluded that the 3-opt is not well suited to GPU architecture
as it has a low computation to memory access (reads and writes) ratio.
\subsection{Alternative Implementations of the RWM}
The I-Roulette method used by Cecilia et al.~\cite{Cecilia2012} was analyzed,
both experimentally and analytically, by Lloyd and Amos~\cite{Lloyd2017} who
concluded that it behaves in a \emph{qualitatively} different way to the RWM.
Specifically, it tends to increase the probability of selecting an edge with a
high pheromone value in cases where there are a large number of edges to choose
from and the majority of the pheromone is concentrated on one edge.
This results in a slight degradation in the quality of MMAS solutions for
TSP instances with more than 1,000 nodes.
On the other hand, there is also a slight improvement in the quality of the
solutions produced by the parallel ACS.
Another approach to speeding up the AS on GPUs was proposed by Uchida et
al.~\cite{Uchida2012}.
In this algorithm, the RWM was replaced by a method called the
\emph{stochastic trial}.
The stochastic trial utilizes a matrix that has its rows assigned to the nodes,
each containing the prefix sums of the selection probabilities for the
corresponding node.
During the solution construction phase, an ant located at the node, $i$, draws a
uniform number from the range $[0, 1]$ and checks if the cell from the $i$-th row
of the matrix corresponds to an unvisited node.
If it does, it is selected, otherwise the process is repeated a specified
number of times.
In the case of a failure, the next node is selected using a (slower) parallel RWM.
Together with a parallel pheromone
update method, the proposed algorithm was up to 43.47x faster than a sequential
AS executed on a CPU.
The data-parallel approach was also adopted by Dawson and
Stewart~\cite{Dawson2013b} who applied a GPU-based AS to the TSP.
The authors proposed a new, efficient parallel implementation of the RWM -- the
Double-Spin Roulette (DS-Roulette) method.
The DS-Roulette method consists of three stages.
In the first stage, all nodes are divided between four thread warps
(128 threads).
Within a warp, the yet to be visited nodes (cities) are determined,
and the threads perform a warp-level reduction of the selection probabilities
that correspond to the nodes.
In the second stage, the reduced values are used by the RWM to select a winning
warp.
In the third stage, the winning warp draws a second random number and performs
a node selection from the assigned nodes.
The selected node becomes the final result of the DS-Roulette execution.
The DS-Roulette method avoids the block-level reduction, and its results are
closer to the results of the sequential RWM when compared to the proposals of
Uchida et al.~\cite{Uchida2012} and Cecilia et al.~\cite{Cecilia2012}.
Combined with a parallel pheromone update, the resulting algorithm was up to
82.3x faster than the CPU-based implementation when tested on the Nvidia GTX 580
GPU.
In subsequent work, Dawson and Stewart~\cite{Dawson2013} presented a parallel
AS in which candidate lists were used to speed up the node selection process.
By limiting the length of the candidate list to 32 they were able to exploit
the warp-level communication primitives provided by the CUDA to efficiently
implement the RWM.
Along with the tabu list compression
method by Uchida et al.~\cite{Uchida2012}, the resulting implementation was up
to 18x faster than its sequential counterpart.
\subsection{Recent Advancements}
In recent work, Cecilia et al.~\cite{Cecilia2018} discussed several aspects
of an efficient GPU-based AS implementation.
Specifically, they introduced a parallel implementation of the RWM that uses
scan and stencil patterns to efficiently select an unvisited node.
To further speed up the calculations, the authors applied a partial
synchronization between the warps within thread-blocks to create a
\emph{super-warp} comprised of two warps (64 threads).
Combined with the previous parallel pheromone update
methods~\cite{Cecilia2012}, the resulting implementation, being
\emph{state-of-the-art}, was up to 8x faster than the baseline version proposed
earlier.
The most recent proposals include work by Borisenko and
Gorlatch~\cite{Borisenko2018} who presented a GPU-based parallel implementation
of the ACO, combined with Simulated Annealing (SA) for the
optimization of the multi-product batch plants used, e.g., in the chemical
industry.
The proposed metaheuristic was able to quickly find near-optimal solutions,
making it a viable alternative to the exact, but very time-consuming,
branch-and-bound approach.
Another work, by Rey et al.~\cite{Rey2018}, discusses
an interesting hybrid-parallel ACO for solving the Vehicle Routing Problem
(VRP).
The first stage of the algorithm consists of the MMAS being executed on the GPU
and generating TSP routes which are then combined into the VRP solutions and
improved using LS procedures during the second stage being executed
on the CPU.
The GPU-based MMAS is also one of components in the recently proposed parallel
framework for the Multi-population Cultural Algorithm by Unold and
Tarnawski~\cite{Unold2017}.
One should also be aware that a lot of work has been done on efficient parallel
implementations of other SBMs including the PSO~\cite{Mussi2011} and bees
algorithm~\cite{Luo2014}. A more general summary can be found in the work of
Tan and Ding~\cite{Tan2016}.
\section{Implementing a GPU-based MMAS}
\label{sec:Implementing_GPU_based_MMAS}
\label{sec:implementing-gpu-based-mmas}
In this section we discuss how the MMAS can be parallelized in order to achieve
efficient execution on GPUs.
We devote most of our attention to the solution construction phase of the
algorithm which is its most time-consuming part.
\subsection{Tabu implementation}
\label{sec:Tabu_implementation}
\label{sec:tabu}
In order to calculate the probabilities defined by Eq.~(\ref{eq:prob}) it is
necessary to determine the set of nodes yet to be visited by an ant.
In the Ant System, the MMAS, and other ACO algorithms, the nodes already
visited by the ant are typically stored in a data structure called a \emph{tabu
list}~\cite{Dorigo1996}.
If the nodes are added in the order in which they are
visited by the ant, then the tabu list comprises a partial solution to the
problem, which, at the end of the construction phase, becomes the complete
solution.
It is
worth noting, that in order to calculate the probabilities given by
Eq.~(\ref{eq:prob}) it is necessary to determine the, as yet, unvisited nodes,
however, the relative order in which they are considered is not important.
In fact, we want the tabu list to fit into the fast but small shared memory of
the GPUs' SMs.
Hence, we can generalize the notion of the tabu list (or \emph{tabu} for short
-- to avoid confusion) to any data structure that provides the following
operations:
\begin{itemize}
\item \texttt{mark}($v$) -- marks node $v$ as \emph{visited};
\item \texttt{is\_visited}($v$) -- returns \texttt{true} if the node $v$ has
already been visited by an ant, or otherwise \texttt{false};
\item \texttt{length}() -- returns a number that is equal to or \emph{greater}
than the number of yet unvisited nodes;
\item \texttt{get\_candidate}($i$) -- where $i \in \{0, 1, \ldots,
\texttt{length}() - 1\}$, returns either an unvisited node $u, \; u \in V$, or
a special \emph{sentinel} value $s, \; s \notin V$; we assume that if
\texttt{get\_candidate}() is executed for every $i \in \{0, 1, \ldots,
\texttt{length}() - 1\}$ then the returned set of values contains all the nodes
to be visited by an ant.
\end{itemize}
The non-obvious definition of \texttt{get\_candidate}() allows the tabu list to
be implemented using a \emph{bitmask}.
The general scheme for accessing the set of nodes to be visited using the
presented operations is shown in Fig.~\ref{alg:tabu}.
\begin{algorithm}[h]
l $\leftarrow$ tabu.length()\;
\For{$i \leftarrow 0$ \KwTo $l-1$}{
$v \leftarrow $ tabu.get\_candidate(i)\;
\If{ $v \ne $ sentinel }{
$v$ can be processed\;
}
}
\caption{
General scheme for processing the tabu using the generalized scheme (see
Sec.~\ref{sec:Tabu_implementation})
}
\label{alg:tabu}
\end{algorithm}
It is worth emphasising that the tabu is at the core of the MMAS and other ACO
algorithms, and its implementation is important for the efficiency of the whole
algorithm.
In case of GPUs, the tabu should be stored in the \emph{shared}
(local) memory so that it can be accessed quickly~\cite{Cecilia2012}.
Unfortunately, the size of the shared memory available to each thread block is
very limited -- usually only 48kB to 96kB in the last few generations of Nvidia
GPUs~\cite{Jia2018}.
Hence, both the time and space complexity of the tabu are important.
A simple linked list is sufficient to implement all of the tabu operations,
however accessing an arbitrary element in the list has
$O(n)$ time complexity, with $n$ being the number of nodes.
A more efficient implementation, known as the \emph{tabu with list compression}
(LC), has been described by Dawson and Stewart~\cite{Dawson2013}, who applied
the \emph{list compression method} first proposed by~Uchida et
al.~\cite{Uchida2012}.
The data structure consists of two single
dimensional arrays of integers of size $n$, and a variable $L$.
For the sake of simplicity, lets denote them by $\textit{unvisited}$ and
$\textit{indices}$.
The first one, stores the list of $L$ nodes to be visited by an ant, while the
second stores the indices of every node in the first list.
For example, if $\textit{unvisited}[i] = v$ then $\textit{indices}[v] = i$ (we
assume that the nodes are denoted by numbers from 0 to $n-1$).
At the start of the construction process, both arrays contain a sequence of $n$
consecutive integers from $0$ to $n-1$, where $n$ is the number of nodes and
$L=n$.
In
subsequent steps, if a node, $v = \textit{unvisited}[i]$, is being visited, then
the $\texttt{mark}(v)$ operation involves the updates: $\textit{unvisited}[i]
\leftarrow \textit{unvisited}[L-1]$, i.e., the visited node is replaced by the
last one, and $\textit{indices}\big[ \textit{unvisited}[i] \big] \leftarrow i$,
is followed by $L \leftarrow L-1$.
It can be seen that, if $\textit{indices}[u] \ge L$ then the node, $u$, has
already been visited by the ant.
Figure~\ref{fig:compressed-tabu-example} shows an example of how the LC works.
\begin{figure}[h]
\centering
\includegraphics[]{tabu-compressed-list.pdf}
\caption{An example showing the subsequent removal of nodes 2, 5, and 7 from the
LC tabu, which contains nodes 0 to 7. The dashed line marks the end of the nodes
list.}
\label{fig:compressed-tabu-example}
\end{figure}
The LC allows all the tabu operations to be performed in constant
time.
The major disadvantage is the necessity for storing two $n$-element arrays in
the memory.
On the other hand, the \textit{indices} array allows the
order in which the nodes were visited (in reverse order) to be recovered.
For example, if we consider the case shown in
Fig~\ref{fig:compressed-tabu-example}, nodes 2, 5 and 7 were already
visited, and the corresponding values in the \emph{indices} array are equal to 7, 6
and 5. By subtracting each value from $n-1$ we get $n-1-7=0$, $n-1-6=1$ and
$n-1-5=2$, respectively, which is exactly the order in which the nodes were
visited.
By analyzing the LC tabu, we notice that it is possible to implement the
tabu using only one list, \textit{entries}, of length $n$, and a variable,
$L$.
The trick is to divide the $\textit{entries}$ (logically) into two parts.
The first (left) part consists of the first $L \; (L \le n)$ entries and
contains the list of $L$ distinct nodes to be visited, i.e., $\textit{entries} \left[ i \right] = u \; ,
i < L$ if, and only if, node $u$ is as yet unvisited.
The second (right) part comprises entries at positions from $L$ up to $n-1$.
It is used to store indices for the nodes \emph{yet to be visited} that were
relocated to the left part, or the \emph{sentinel} value of $n$ for the nodes
that have already been \emph{visited}, i.e., are not in the left part.
Initially, the \textit{entries} array contains consecutive numbers from 0 to
$n-1$, denoting the unvisited nodes.
In subsequent steps, if a node, $u$, is visited, one of two cases is possible:
either $u < L$ or $u \ge L$.
In the first case, the node $u$ is at its initial position, i.e., $\textit{entries}
\left[ u \right] = u$.
In the second case, the node has been relocated to the left
part, and the value $i_u = \textit{entries} \left[ u \right]$ denotes the
\emph{index} at which the node is currently located, i.e., $\textit{entries}
\left[ i_u \right] = u$.
If $i_u = L-1$ then $u$ is at the end of the list,
and it is enough to set $\textit{entries} \left[ i_u \right] \leftarrow n$ to
mark that node $u$ has been visited.
Otherwise, $i_u < L-1$ and the last element, $t = \textit{entries}\left[ L-1
\right]$, of the list replaces it:
$\textit{entries} \left[ i_u \right] \leftarrow t$.
The new position of $t$ is
saved: $\textit{entries} \left[ t \right] \leftarrow i_u$.
It is worth noting that this scheme allows for checking in $O(1)$ time whether
node $u$ was visited,
simply by checking whether $\textit{entries} \left[ u \right] > u$.
We will refer to this tabu implementation as the \emph{compact tabu} (CT).
An example showing the removal of three nodes from a CT that contains eight
nodes is shown in Fig.~\ref{fig:compact-tabu-example}.
\begin{figure}[h]
\centering
\includegraphics[]{tabu-compact.pdf}
\caption{
An example showing the subsequent removal of nodes 2, 5, and 7 from a CT
containing nodes 0 to 7. The dashed line marks the end of the nodes
list. The entries to the right of the dashed line are used to store
positions (indices) for the nodes relocated to the left.
}
\label{fig:compact-tabu-example}
\end{figure}
Even more memory efficient implementation can be achieved if a \emph{bitmask}
of length $n$ is used to mark the visited /unvisited state of each node.
In that case, the \texttt{mark} and \texttt{is\_visited} tabu operations can be
performed in $O(1)$ time.
However, accessing the unvisited nodes (Fig.~\ref{alg:tabu}) requires checking
the state of $n$ bits, while the two previous tabu implementations provide
\emph{direct} accesses to the \emph{unvisited} nodes.
For the sake of completeness, we will refer to this
tabu as the \emph{bitmask tabu} (BT).
A summary of the presented tabu implementations is shown in
Tab.~\ref{tab:tabu-cmp}.
\begin{table}[]
\footnotesize
\centering
\caption{
A summary of the considered tabu implementations. We assume that the nodes
are denoted as numbers 0 to $n-1$, where $n < 2^{16}$, i.e. a node can
fit into an unsigned 16-bit variable.
}
\label{tab:tabu-cmp}
\begin{tabular}{@{}lrrr@{}}
\toprule
\multirow{2}{*}{Tabu characteristic} & \multicolumn{3}{c}{Tabu implementation} \\ \cmidrule(l){2-4}
& \multicolumn{1}{c}{TLC} & \multicolumn{1}{c}{CT} & \multicolumn{1}{c}{BT} \\ \midrule
Testing if node was visited (\texttt{is\_visited}) & $O(1)$ & $O(1)$ & $O(1)$ \\ \cmidrule(r){1-1} \cmidrule(l){2-4}
Marking node as visited (\texttt{mark}) & $O(1)$ & $O(1)$ & $O(1)$ \\ \cmidrule(r){1-1} \cmidrule(l){2-4}
\begin{tabular}[c]{@{}l@{}}Number of calls to \texttt{get\_candidate} \\ to get all $k \; (k \le n)$ unvisited nodes\end{tabular} & $k$ & $k$ & $n$ \\ \cmidrule(r){1-1} \cmidrule(l){2-4}
Required memory in bytes & $4n$ & $2n$ & $\left\lceil n/8 \right\rceil$ \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Next node selection}
\label{sec:Next_node_selection}
\label{sec:next.node.selection}
The procedure for the selection of the next node by an ant during the solution
construction phase has the biggest impact on the performance of the MMAS and other
ACO algorithms~\cite{Cecilia2012,Uchida2012}.
Equation~(\ref{eq:prob}) defines the probabilities for selecting each of the
unvisited nodes.
The sequential version of the procedure has a simple and efficient
implementation, often referred to as the \emph{roulette wheel method} (RWM).
The RWM takes $O(n)$ time, where $n$ is the number of nodes to choose from.
\subsubsection{Parallel RWM}
\label{sec:Parallel_RWM}
The RWM is a typical example of an algorithm that has a simple and efficient
sequential implementation but is difficult to parallelize
effectively~\cite{Cecilia2012}. The difficulties reside in the dependencies
between the subsequent computations of the RWM (e.g., the summation of the
pheromone and heuristic information products [weights in short]), searching for
a "winning" node based on a randomly drawn value.
It is not surprising that
multiple alternative RWM implementations have been proposed in the literature,
including the I-Roulette method by Cecilia et. al~\cite{Cecilia2012}, the DS-Roulette
by Dawson and Stewart~\cite{Dawson2013b}, and the stochastic trial by Uchida et.
al~\cite{Uchida2012}.
Although these methods allow for an efficient parallel
execution, they are \emph{qualitatively} different from the sequential
version~\cite{Lloyd2017}.
Only recently has Cecilia et. al~\cite{Cecilia2018}
proposed the parallel \emph{SS-Roulette} method, which is essentially a parallel
version of the RWM, i.e., it offers the same quality of results as the
sequential implementation.
This was possible mainly due to the increasing
computational capacity of GPU architectures, and also improvements
on the software side, e.g., the CUDA toolkit.
\begin{figure}[h]
\centering
\includegraphics[]{par-rwm.pdf}
\caption{An example showing the execution of the parallel RWM.}
\label{fig:par-rwm}
\end{figure}
Following the description of the SS-Roulette method (as the source code is not
available), we have implemented a parallel version of the RWM.
Figure~\ref{fig:par-rwm} shows how our implementation of the parallel RWM
(PRWM) works.
First, each thread that is executing the PRWM is assigned a chunk of the
unvisited nodes, for which it computes a sum of the corresponding weights.
Next, a prefix sum of the chunks' sums is computed in parallel.
Following this, the last thread draws a uniform random number and multiplies it
by the total.
The resulting value is broadcast to all threads so that the \emph{winning}
chunk of nodes can be selected.
If the chunk contains more than one node, then it is necessary to locate the
selected node within the chunk.
This process again involves the calculation of the prefix sums of the nodes'
weights but this time only for the nodes within the chunk.
This can also be done in parallel by splitting the chunk's nodes across all
threads.
In general, this process may require up to $\left \lceil \log_p n
\right \rceil$ stages, where $n$ is the number of nodes and $p$ is the number
of processors.
For example, if $n = 1024$ and $p=32$, then in the first stage
each of the 32 threads processes 32-node chunks, and in the second (final)
stage each thread is assigned one out of the 32 nodes from the chunk selected
during the first stage.
It is worth noting that the weights belonging to the chunks
selected in a single stage are read from the memory again in the subsequent stage,
and so on.
Fortunately, the total number of times the weights are read from the memory
equals
$\sum_{k = 0}^{ \left \lceil log_p n \right \rceil } \frac{n}{p^k}
\le n \sum_{k = 0}^{\infty} \frac{1}{p^k} = n \frac{p}{p - 1}$,
which is $O(n)$ because the number of threads, $p$, is constant.
\subsubsection{Weighted Reservoir Sampling}
\label{sec:Weighted_Reservoir_Sampling}
The problem of the selection of the next node in the MMAS can also be seen as an
instance of random weighted sampling without replacement, or, more briefly,
weighted reservoir sampling (WRS).
In WRS, one has to select $m$ distinct items randomly out of a population of size
$n$, while the probability of choosing an item is proportional to its
\emph{weight}~\cite{Efraimidis2006}.
In the case of the MMAS, only one item (node) needs to be selected, and the
weights are products of the heuristic information values and the pheromone trails
values (see Eq.~(\ref{eq:prob})) that are stored in the \emph{choice\_info} matrix.
\begin{algorithm}[h]
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\Input{ A population $V$ of $n$ weighted items}
\Output{A reservoir (sample) $R$ with the WRS of size $m$}
Insert first $m$ items of $V$ into $R$ \;
\For{ $i \leftarrow 1$ \KwTo $m$ }{
$k_i \leftarrow u_i^{(1/w_i)}$ where $u_i = \textrm{random}(0,1)$\;
}
\For{$i \leftarrow m+1$ \KwTo $n$}{
$T \leftarrow $ the smallest key in $R$ \;
$k_i \leftarrow u_i^{(1/w_i)}$ where $u_i = \textrm{random}(0,1)$\;
\If{ $k_i > T$ }{
The item with the minimum key in $R$ is replaced by item $v_i$ \;
}
}
\caption{Algorithm A-Res for computing WRS~\cite{Efraimidis2006}.}
\label{alg:wrs}
\end{algorithm}
Efraimidis and Spirakis proposed an efficient algorithm, named \emph{A-Res}
(Fig.~\ref{alg:wrs}), for computing WRS~\cite{Efraimidis2006}.
The A-Res algorithm assigns each item a key, $u_i ^ {(1/w_i)}$, where $u_i$ is
a uniformly chosen number from the range $(0, 1)$, and then selects $m$ items
with the \emph{largest keys}.
The most important property of the algorithm is that it
selects the sample in \emph{one pass}, i.e., it considers each item and its
weight \emph{once} and, in contrast to the RWM, does not require the summation of
all the weights.
It also worth noting that the relative order in which the items are processed
can be arbitrary, what makes the algorithm easier to parallelize.
Therefore, we propose to adapt this algorithm to implement a parallel
equivalent of the RWM for application in the MMAS.
\begin{algorithm}[h]
\SetKw{KwBy}{by}
$t \leftarrow \textrm{threadIdx.x}$ \quad \tcp{CUDA-based thread id}
$p \leftarrow \textrm{blockDim.x}$ \quad \tcp{Number of threads in a block}
$R_t \leftarrow \emptyset $ \quad \tcp{No element was selected} \label{alg:par-wrs:init1}
$T_t \leftarrow 0$ \quad \tcp{Initial key for the thread $t$} \label{alg:par-wrs:init2}
$l \leftarrow$ tabu.length()
\ForPar(\tcp*[f]{Thread $t$ processes indices: $t, t + p, \ldots, \lfloor \frac{n - t}{p} \rfloor p + t$}){$i \leftarrow t$ \KwTo $l-1$ \KwBy $p$ }{
\label{alg:par-wrs:for-beg}
$v \leftarrow $ tabu.get\_candidate(i)
\If(\quad \tcp*[h]{$v$ is an unvisited node}){ $v \ne $ sentinel }{
$r \leftarrow random(0, 1)$ \\
$w \leftarrow \left[ \tau_{uv} \right]^\alpha \left[ \eta_{uv} \right ]^\beta$ \label{alg:par-wrs:weight} \\
$k \leftarrow r^{(1 / w)}$ \label{alg:par-wrs:key}
\If{ $k > T_t$ }{ \label{alg:par-wrs:cmp}
$T_t \leftarrow k$ \\
$R_t \leftarrow v$
}
}
\label{alg:par-wrs:for-end}
}
$k \leftarrow \argmax_{i \in \{ 0, 1, \ldots, p-1 \}} T_i$ \quad \tcp{Parallel
reduction of $(T_0, T_1, \ldots, T_{p-1})$}
\label{alg:par-wrs:reduction}
\Return{$R_k$}
\caption{The WRS-based parallel pseudo-random proportional selection of the next node in the MMAS.}
\label{alg:par-wrs}
\end{algorithm}
Figure~\ref{alg:par-wrs} presents a pseudocode of the parallel version
of the WRS (a single thread-block is assumed) adapted to perform the
pseudo-random proportional selection of the next node in the MMAS.
Each thread starts with its own reservoir that has a size of one (as only one
node has to be selected) and the corresponding key (lines
\ref{alg:par-wrs:init1} and \ref{alg:par-wrs:init2}).
Next, the elements of the tabu are processed by $p$ threads in parallel
(loop in lines \ref{alg:par-wrs:for-beg}--\ref{alg:par-wrs:for-end}).
A thread, $t$, processes every $p$-th element and selects its own (locally) maximum
key, $T_t$, and the corresponding element (node) $R_t$.
After this, a parallel reduction of the keys selected by the threads, $(T_0,
T_1, \ldots, T_p)$, is performed and the (final) maximum key is elected (line
\ref{alg:par-wrs:reduction}).
The corresponding element becomes the result of the WRS.
The algorithm selects a node in $O(\frac{l}{p} + \log p)$ time.
Figure~\ref{fig:par-wrs} shows an example of the WRS-based node selection.
\begin{figure}[h]
\centering
\includegraphics[]{par-wrs.pdf}
\caption{
An example showing the execution of the parallel, WRS-based pseudo-random
proportional selection of the next node in the MMAS.
}
\label{fig:par-wrs}
\end{figure}
The algorithm has a few potential drawbacks that can negatively affect its
runtime.
Firstly, a single random number is required per element of the tabu, hence a
separate pseudo-random number generator's state needs to be stored for
each thread.
Secondly, the calculation of the keys (line \ref{alg:par-wrs:key} in
Fig.~\ref{alg:par-wrs}) involves costly operations on float numbers: division
and exponentiation.
The former can be alleviated if the reciprocal of each weight (line
\ref{alg:par-wrs:weight}) is calculated in advance.
This is possible because the weights depend on the heuristic information
values that are constant, while the values of the pheromone trails in the MMAS
are updated only once per iteration -- after each ant has completed
construction of its solution.
The latter can be computed using the \texttt{powf()} CUDA function but another
important problem arises -- as the weights are often very small, their
reciprocals become large, and therefore so do their exponents.
This in turn causes problems, as the 32-bit floating point type does
not provide enough precision for the calculations.
Fortunately, it is possible to replace the exponentiation with a logarithm,
i.e., the calculation in line~\ref{alg:par-wrs:key} can be replaced by
$k \leftarrow \frac{1}{w} \log_2 r$, where $r$ is a uniform random number in
the range $(0, 1)$
(Also, the initial $T_t$ value needs to be set carefully as $\lim_{x \to 0^+}
\log_2 x = -\infty$).
In fact, it is now possible to use the fast approximate logarithm calculation
provided by the CUDA \texttt{\_\_log2f()} (intrinsic) function.
\subsection{Candidate lists}
\label{sec:Candidate_lists}
During the solution construction process an ant located at a given node selects
the next node from the set of (yet) unvisited neighboring nodes.
If the set of nodes the ant can choose from is limited to the so called
\emph{candidate list}, the computation time of the MMAS can be greatly reduced,
often without sacrificing quality~\cite{Stutzle2000}.
The candidate list for each node consists of a number, \emph{cl}, of the
closest neighboring nodes. Assuming that $\textit{cl}$ is constant, the time
complexity of the solution construction process for a single ant is reduced
from $O(n^2)$ to $O(n)$.
In the original (CPU-based) MMAS implementation for solving the TSP,
$\textit{cl} = 20$~\cite{Stutzle2000}.
Even smaller numbers are possible, however it requires a more complex
definition of the \emph{closeness} between nodes, e.g., the
$\alpha$-measure~\cite{Helsgaun2009}.
In the case of the GPU-based computations, setting $\textit{cl}$ to a multiple
of the warp size (32 in case of the modern Nvidia GPUs) seems an obvious
choice~\cite{Dawson2013}.
\subsection{Final details}
\label{sec:Final_details}
The final structure of the proposed MMAS implementation is shown in
Fig.~\ref{fig:flowchart}. The main component of the parallel MMAS is the ants'
solution construction process computed using a single CUDA kernel. We apply the
proven data-parallel approach~\cite{Cecilia2012,Dawson2013,Uchida2012} in which
a single thread-block computes the ant's solution, and the size of each block
is a multiple of the warp size, which is 32 in the Nvidia GPUs, so that all
the processing elements (CUDA cores) within a warp are used efficiently.
\begin{figure}[h]
\centering
\includegraphics[]{flowchart.pdf}
\caption{
Flowchart of the GPU-based MMAS ($n$ is the size of the problem, $p$ is the
number of threads per ant). The pairs of numbers in the angle brackets
denote the number of thread blocks and the number of threads per block that
are executing a kernel, respectively.
}
\label{fig:flowchart}
\end{figure}
After the solutions construction kernel is executed, the iteration best
solution is selected.
If its cost is lower than the cost of the current global best solution, it
becomes the new global best solution and the pheromone trail limits, $\tau_{\rm
min}$ and $\tau_{\rm max}$, are updated.
The new limits are then used by the pheromone evaporation kernel.
This kernel is executed with one thread-block per row of the pheromone memory
matrix, and with 256 threads in every block.
Finally, a pheromone deposition kernel is called.
It is responsible for increasing the values of the pheromone trails that
correspond to the edges of the iteration best solution.
This kernel is executed by a single block of 256 threads.
The presented pheromone evaporation-deposition scheme is simpler than in the
AS, in which the pheromone is updated for every ant and where the changes are
often conflicting, i.e., they concern the same pheromone trails~\cite{Cecilia2012}.
In the MMAS, both the evaporation and deposition of the pheromone can be split
into \emph{independent} and non-conflicting parts that do not require atomic
operations as are used in the GPU-based AS
implementations~\cite{Cecilia2012,Cecilia2018}.
\section{Experimental results}
\label{sec:Experimental_results}
\label{sec:experimental-results}
In this section we present the results of the computational experiments
conducted in order to evaluate the efficiency of the proposed MMAS implementations.
The computations were performed on a set of symmetric TSP instances from the
well-known TSPLIB~\cite{Reinelt1991} repository. The instances were selected so
that the results could be compared to the reports available in the literature.
Following the works by Cecilia et. al~\cite{Cecilia2012,Cecilia2018}, we set the
MMAS parameters as follows:
\begin{itemize}
\item
the number of ants was equal to the problem size, i.e., $m = n$, where $n$ denotes the size of the TSP instance;
\item
$\rho = 0.5$ -- the parameter controlling the pheromone evaporation rate;
\item
$\alpha = 1$ and $\beta = 2$ -- the parameters controlling the influence of the pheromone and the heuristic information on the next node selection probability~(Eq.~\ref{eq:prob});
\item
the number of iterations equaled 100.
\end{itemize}
The pheromone trail limits, $\tau_{\rm min}$ and $\tau_{\rm max}$, were
calculated following the work of St{\"{u}}tzle and Hoos~\cite{Stutzle2000} with
$p_{\rm best} = 0.01$. The initial values of the limits were set based on the
value of a solution constructed using the nearest neighbor heuristic. If not
stated otherwise, the presented numbers were averaged over 30 repeated
executions of a given algorithm. The presented time measurements were obtained
using CUDA provided timers and refer to the \emph{kernels} executing respective
parts of the MMAS.
Almost all of the computation time in the AS and other ACO algorithms is spent
in the solutions construction phase and relatively little on the pheromone
updates~\cite{Cecilia2012}. This proportion is even higher in the MMAS in which
only a single ant deposits pheromone, i.e., the current global or iteration
best, depending on the chosen strategy~\cite{Dawson2013}. For this reason, in
the computational experiments, we mainly focused on the impact of the tabu
and the implementations of the node selection procedure.
Combining the three tabu implementations presented in Section~\ref{sec:tabu} and
the two node selection procedures in Section~\ref{sec:next.node.selection} results
in a total of six MMAS variants under consideration.
For convenience, they are denoted as MMAS--\emph{node selection
procedure}--\emph{tabu implementation}, where the \emph{node selection procedure}
is either denoted by RWM or WRS; and \emph{tabu implementation} is denoted by one of the
following three: LC, BT and CT tabu implementations.
\subsection{Computing environment}
\label{sec:Computing_environment}
\begin{table}[]
\footnotesize
\centering
\caption{Characteristics of the GPUs used in the computational experiments.}
\label{tab:gpu-spec}
\begin{tabular}{@{}lrr@{}}
\toprule
Characteristic & \multicolumn{1}{l}{Nvidia Tesla P100} & \multicolumn{1}{l}{Nvidia Tesla V100} \\ \midrule
Architecture & Pascal & Volta \\
Multiprocessors (SM) & 56 & 80 \\
Streaming processors (CUDA cores) & 3584 & 5120 \\
FP32 processing power & 9,340 GFLOPS & 14,028 GFLOPS \\
Memory size & 16 GB & 16 GB \\
Memory bandwidth & 732 GB/s & 900 GB/s \\
L2 cache size (per die) & 4,096 KB & 6,144 KB \\
Shared mem. size (per SM) & 64 KB & Up to 96 KB \\
Transistors count & 15 Billion & 21.1 Billion \\
Maximum TDP & 300W & 300W \\ \bottomrule
\end{tabular}
\end{table}
The implementation of the proposed algorithms was done in C++ using Nvidia CUDA version 10.
Sources~\footnote{Sources are available at
https://github.com/RSkinderowicz/GPU-based-MMAS} were compiled using GCC v6.3
with a \emph{-O3} switch for the CPU-side code, while the GPU-side
code was compiled with a \emph{-gencode arch=compute\_60,code=sm\_60} switch for the Nvidia Tesla P100 GPU,
and \emph{-gencode arch=compute\_70,code=sm\_70} switch for the Nvidia Tesla V100 GPU.
Table~\ref{tab:gpu-spec} presents the characteristics of the two GPU architectures used.
The computations were conducted on servers running under the Debian 9 Linux OS and
equipped with a 20-core Intel Xeon 6138 (Skylake) CPU clocked at 2 GHz (a single
core was used in the computations).
\subsection{Solution construction phase}
\label{sec:Solution_construction_phase}
The efficient use of the computing power offered by GPUs equipped with
thousands of processing elements (CUDA cores), requires the proper
organization of the computations, i.e., the computations should also be split
into multiple, mostly independent portions~\cite{Kirk2016}.
In the proposed MMAS implementations we adopted the data-parallel approach in
which each ant is assigned a thread-block, and the threads within the
thread-blocks are responsible for computing the solution.
This leaves one crucial decision: the number of threads within a thread-block.
To best adapt to the Nvidia GPUs used in the computations, the size of each
thread block was a multiple of the warp size, that is, 32.
Although the Volta architecture allows the threads within a warp to follow
divergent paths simultaneously~\cite{Jia2018}, setting the number of threads to
a multiple of the warp size simplifies the implementation and allows for
efficient execution on the pre-Volta generations of GPUs, i.e., Pascal.
\begin{figure}[h]
\centering
\includegraphics{fig/warps-vs-time-pcb1173-p100.pdf}
\caption{
The mean time required by the ants to construct solutions in the MMAS
vs. the number of thread-warps per ant for the \emph{pcb1173} instance
($n$ is the size of the problem). The lowest value for each
MMAS variant is shown above the respective bar.
The results are for the Nvidia Pascal P100 GPU.
}
\label{fig:warps-vs-time-pcb1173-p100}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics{fig/warps-vs-time-pr2392-p100.pdf}
\caption{
The mean time required by the ants to construct solutions in the MMAS
vs. the number of thread-warps per ant for the \emph{pr2392} instance
($n$ is the size of the problem). The lowest value for each
MMAS variant is shown above the respective bar.
The results are for the \emph{Nvidia P100 GPU}.
}
\label{fig:warps-vs-time-pr2392-p100}
\end{figure}
\begin{figure}
\centering
\includegraphics{fig/p100-occupancy-pr2392.pdf}
\caption{
SM occupancy vs. the number of thread warps for the \emph{pr2392} instance and
the \emph{Nvidia P100 GPU}.
}
\label{fig:p100-occupancy-pr2392}
\end{figure}
Figures~\ref{fig:warps-vs-time-pcb1173-p100}
and~\ref{fig:warps-vs-time-pr2392-p100} show the mean solution construction
time for the proposed MMAS variants that executed on the Nvidia P100 GPU vs the
number of threads per ant obtained for the \emph{pcb1173} and \emph{pr2392} TSP
instances, respectively.
A few observations can be made here.
First, the MMAS variants differ significantly in speed depending both on the
node selection and the tabu implementation.
The fastest is the MMAS-WRS-CT variant followed by
the MMAS-WRS-LC.
The slowest are the variants with the BT, which can be explained by the fact
that the BT does not provide direct access to the list of nodes to visit, but
checks the status (and the corresponding bit) of each node repeatedly, hence
significantly increasing the number of computations.
Second, the parallel WRS-based node selection procedure is faster and \emph{scales}
better than the parallel RWM implementation.
Even though the WRS method involves costly computations (i.e., the random
number generation and logarithm calculation), it reduces the number of data
exchanges and synchronizations between the threads.
Third, the results indicate that the number of threads should be adjusted to the
size of the problem.
Specifically, the bigger the problem is, the higher the number of threads per
ant are required.
This can be observed when comparing the results for
the 1,173-city instance (Fig.~\ref{fig:warps-vs-time-pcb1173-p100}) to the
results for the 2,392-city instance (Fig.~\ref{fig:warps-vs-time-pr2392-p100}).
For the former, the fastest computations were observed for 32 and 64 threads (1
and 2 warps, respectively) per ant; while for the latter, the fastest execution
was observed when the number of threads per ant was at least 128 (4 warps).
Generally, the efficient use of the GPU's computing power is related to its
\emph{occupancy}: understood as the ratio of the number of \emph{active}
thread warps to the maximum number of warps per SM~\cite{Kirk2016}.
In most cases, this ratio should be high.
Often, having more active threads than processing
elements (CUDA cores) is beneficial as it allows latencies generated by
global memory accesses and synchronization operations to be hidden.
The number of thread blocks (each consisting of thread warps) that can execute
simultaneously on the same SM is limited by the size of the shared memory and
the number of registers used.
By increasing the number of ants, we are increasing the number of thread
blocks.
The LC tabu takes twice as much shared memory as the CT.
Thus the latter allows more thread blocks per SM to be executed at the same time.
Another, complementary, solution that increases the occupancy is to increase the
number of threads per ant.
A comparison of occupancy for the proposed MMAS
variants with LC and CT tabu implementations solving the \emph{pr2392} TSP
instance, is shown in Fig.~\ref{fig:p100-occupancy-pr2392}.
As can be seen, the CT allows for a higher occupancy, with the MMAS-WRS-CT
achieving values exceeding 90\% for 7 or 8 thread warps per ant; while for the
MMAS with the LC tabu the occupancy does not exceed 80\%.
\begin{figure}
\centering
\includegraphics{fig/warps-vs-time-pr2392-v100.pdf}
\caption{
The mean time required for the ants to construct solutions in the MMAS
vs. the number of thread-warps per ant for the \emph{pr2392} instance
($n$ is the size of the problem). The lowest value for each
MMAS variant is shown above the respective bar.
The results are for the \emph{Nvidia V100 GPU}.
}
\label{fig:warps-vs-time-pr2392-v100}
\end{figure}
Executing the same MMAS variants on the \emph{Nvidia Volta V100} GPU reveals a
significant advantage in computing power over the Pascal architecture.
This can be seen in Fig.~\ref{fig:warps-vs-time-pr2392-v100}, which shows a
comparison of the execution times of the solution construction phase for the
\emph{pr2392} instance.
The fastest variant is again the MMAS-WRS-CT taking about 39.2~ms to
complete, vs. 89~ms for the previous generation of GPU.
This difference can be somewhat surprising as it cannot be explained simply by
the higher processing power, which increased by only about 50\%, i.e., 9,340 to
14,028 GFLOP/s.
Rather, the
advantage can be attributed to \emph{a combination} of multiple factors
including an increased number of the processing elements (cores) (see
Tab.~\ref{tab:gpu-spec}), a larger shared memory per SM, a larger L2 cache, a
higher global memory bandwidth and an improved L1 cache replacement policy,
among others~\cite{Jia2018}.
\begin{table}[]
\footnotesize
\centering
\caption{
A comparison of the mean solution construction phase times (in ms) for the
proposed MMAS variants executed on the \emph{Nvidia P100} GPU.
The numbers in parentheses denote the number of thread warps per
block (ant). The smallest time for each instance was marked in bold.
}
\label{tab:mmas-cmp-p100}
\begin{tabular}{@{}lrrrrrrrr@{}}
\toprule
\multirow{2}{*}{Algorithm} & \multicolumn{8}{c}{Instance} \\
& \multicolumn{1}{c}{\textit{d198}} & \multicolumn{1}{c}{\textit{pcb442}} & \multicolumn{1}{c}{\textit{rat783}} & \multicolumn{1}{c}{\textit{pr1002}} & \multicolumn{1}{c}{\textit{pcb1173}} & \multicolumn{1}{c}{\textit{rl1889}} & \multicolumn{1}{c}{\textit{pr2392}} & \multicolumn{1}{c}{\textit{fl3795}} \\ \cmidrule(r){1-1} \cmidrule(l){2-9}
\#1: AS~(CPU)~\cite{Zhou2018} & 0.87 & 5.66 & 25.25 & 47.17 & NA & NA & 822.54 & NA \\
\#2: AS~(GPU)~\cite{Cecilia2018} & 0.40 & 1.59 & 6.81 & 15.54 & 23.53 & 78.87 & 185.13 & 726.25 \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
MMAS-RWM-LC & 0.33 (1) & 1.05 (1) & \textbf{2.75 (1)} & 7.77 (1) & 12.92 (1) & 62.26 (4) & 122.46 (5) & 482.60 (8) \\
MMAS-RWM-BT & 0.76 (3) & 1.37 (1) & 3.67 (1) & 8.59 (1) & 15.25 (1) & 83.50 (5) & 165.11 (6) & 656.51 (10) \\
MMAS-RWM-CT & 0.47 (1) & 1.15 (1) & 2.80 (1) & 6.45 (1) & 14.00 (2) & 65.05 (4) & 126.63 (6) & 521.57 (10) \\
MMAS-WRS-LC & \textbf{0.30 (2)} & \textbf{0.92 (2)} & 3.06 (2) & 7.68 (3) & 11.46 (3) & 48.59 (6) & 95.96 (8) & 383.67 (10) \\
MMAS-WRS-BT & 0.42 (4) & 1.58 (2) & 6.33 (2) & 12.17 (2) & 19.37 (2) & 94.87 (5) & 164.54 (4) & 652.62 (6) \\
MMAS-WRS-CT & 0.31 (3) & 0.95 (2) & 3.18 (2) & \textbf{6.09 (2)} & \textbf{9.54 (2)} & \textbf{46.02 (5)} & \textbf{88.99 (5)} & \textbf{354.43 (10)} \\ \cmidrule(r){1-1} \cmidrule(l){2-9}
Speedup vs. \#1 & 2.90x & 6.14x & 9.19x & 7.74x & - & - & 9.24x & - \\
Speedup vs. \#2 & 1.33x & 1.73x & 2.48x & 2.55x & 2.47x & 1.71x & 2.08x & 2.05x \\ \bottomrule
\end{tabular}
\end{table}
Table~\ref{tab:mmas-cmp-p100} presents a summary of the mean times needed to
execute the MMAS solution construction kernel for the proposed MMAS variants
for the Nvidia Tesla P100 GPU. Generally, the MMAS with the WRS-based node
selection implementation was faster in all cases but one, while the memory
savings resulting from the CT had an impact only for sufficiently large
instances, i.e., those with at least 1,000 nodes.
For reference point, we have also provided the results from the two recent works by
Zhou et al.~\cite{Zhou2018} and Cecilia et al.~\cite{Cecilia2018}.
Although in both cases the authors proposed parallel implementations of the AS,
the main differences between the AS and the MMAS relate to the pheromone
update, while the solution construction phase is analogous, assuming that no
candidate lists are used~\cite{Stutzle2000}.
Specifically, the numbers from Zhou et
al.~\cite{Zhou2018} refer to a parallel, multicore-SIMD CPU based AS
implementation named VETPAM-CPU-AS, executed on a six-core Intel i7-5820
(Haswell) CPU.
The numbers from Cecilia et al.~\cite{Cecilia2018} are
the execution times of the solution construction kernel for the CUDA-based AS
implementation running on an Nvidia GTX TITAN X GPU with 3,072 CUDA cores and
6.69 GFLOP/s of computing power (Maxwell architecture).
In comparison, the slower of our GPUs, the Nvidia P100, has about 52\% more
computing power (single precision), 25\% larger L2 cache and more than double
the global memory bandwidth (732 GB/s vs.~336 GB/s).
The results show that the proposed
MMAS-WRS-CT is up to 2.55x faster, which confirms that our implementation
utilizes the additional computing resources efficiently.
In comparison, the highly optimized VETPAM-CPU-AS is up to 9.24x slower, though
still, being very fast if we account for the relatively low computing power of
the CPU used for the computations (316 GFLOP/s).
\begin{table}[]
\footnotesize
\centering
\caption{
The comparison of duration (in ms) of the mean solution construction phase
for the proposed MMAS variants executed on the \emph{Nvidia V100} GPU. The
numbers in parentheses denote the number of thread warps per block (ant).
The smallest time for each instance is marked in bold.
}
\label{tab:mmas-cmp-v100}
\begin{tabular}{@{}lrrrrrrrr@{}}
\toprule
\multirow{2}{*}{Algorithm} & \multicolumn{8}{c}{Instance} \\
& \multicolumn{1}{c}{\textit{d198}} & \multicolumn{1}{c}{\textit{pcb442}} & \multicolumn{1}{c}{\textit{rat783}} & \multicolumn{1}{c}{\textit{pr1002}} & \multicolumn{1}{c}{\textit{pcb1173}} & \multicolumn{1}{c}{\textit{rl1889}} & \multicolumn{1}{c}{\textit{pr2392}} & \multicolumn{1}{c}{\textit{fl3795}} \\ \cmidrule(r){1-1} \cmidrule(l){2-9}
\#1: AS~(CPU)~\cite{Zhou2018} & 0.87 & 5.66 & 25.25 & 47.17 & NA & NA & 822.54 & NA \\
\#2: AS~(GPU)~\cite{Cecilia2018} & 0.40 & 1.59 & 6.81 & 15.54 & 23.53 & 78.87 & 185.13 & 726.25 \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
MMAS-RWM-LC & 0.28 (6) & 0.77 (1) & 1.87 (1) & 3.08 (1) & 5.21 (1) & 32.75 (4) & 65.50 (5) & 285.72 (6) \\
MMAS-RWM-BT & 0.42 (1) & 1.22 (1) & 2.84 (2) & 5.04 (2) & 7.66 (1) & 31.20 (2) & 68.90 (3) & 301.75 (6) \\
MMAS-RWM-CT & 0.29 (6) & 0.81 (1) & 1.85 (1) & 3.23 (1) & 5.55 (1) & 24.06 (2) & 53.26 (3) & 250.62 (6) \\
MMAS-WRS-LC & \textbf{0.18 (3)} & \textbf{0.50 (5)} & \textbf{1.26 (4)} & \textbf{2.16 (3)} & 3.36 (3) & 20.39 (5) & 40.03 (8) & 196.00 (10) \\
MMAS-WRS-BT & 0.23 (4) & 0.70 (5) & 2.19 (4) & 3.85 (4) & 5.97 (4) & 28.36 (5) & 52.54 (2) & 236.14 (8) \\
MMAS-WRS-CT & 0.19 (6) & 0.52 (5) & 1.30 (4) & 2.20 (3) & \textbf{3.31 (3)} & \textbf{19.12 (5)} & \textbf{39.24 (6)} & \textbf{193.05 (10)} \\ \cmidrule(r){1-1} \cmidrule(l){2-9}
Speedup vs. \#1 & 4.77x & 11.42x & 20.07x & 21.79x & - & - & 20.96x & - \\
Speedup vs. \#2 & 2.19x & 3.21x & 5.41x & 7.18x & 7.10x & 4.12x & 4.72x & 3.76x \\ \bottomrule
\end{tabular}
\end{table}
Table~\ref{tab:mmas-cmp-v100} shows the results for the Nvidia Volta GPU
architecture.
In a similar way to the results for the Pascal architecture, the MMASs with the
WRS-based implementation were faster than their counterparts with the RWM-based
implementation.
Specifically, the MMAS-WRS-CT was about 2.88x faster when solving the
\emph{pcb1173} instance, but for the larger instances, the speedup dropped to
only 2x.
This difference can be explained (partially) by the larger
L2 cache present in the V100 GPU, 6MB vs. 4MB in the P100 GPU.
Storing the \emph{choice\_info} data for the \emph{pcb1173} instance takes
up about 5.25MB which fits entirely into the larger cache of the V100 GPU, but
only partially into the L2 of the P100 GPU.
Generally, the newer GPU offers advantages over the older architecture in all
cases, but the differences are larger for medium and large instances, as can be
seen in Figure~\ref{fig:gpu-cmp}.
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{fig/times-cmp-p100-v100.pdf}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{fig/times-cmp-p100-v100-log-scale.pdf}
\end{subfigure}
\caption{
A comparison of the solution construction times for
the TSP instances under consideration, obtained for the P100 and V100 GPUs.
}
\label{fig:gpu-cmp}
\end{figure}
Compared with the times reported by Cecilia et al.~\cite{Cecilia2018}, the
proposed MMAS implementation is up to 7.18x faster.
At the same time, the advantage over the VETPAM-CPU-AS exceeds
20x for the following instances: \emph{rat783, pr1002} and \emph{pr2392}.
\subsection{Candidate Lists}
\label{sec:Candidate_Lists}
Using candidate lists can have a significant impact on the MMAS convergence
and also on the execution speed~\cite{Stutzle2000}.
In our experiments, the length of the candidate lists was set to 32, following
Dawson and Stewart~\cite{Dawson2013} who proposed a GPU-based AS using the LC
tabu implementation.
This value results in a single thread warp per ant.
It is
worth noting that the V100 GPU has 5,120 CUDA cores, thus we need at least 160
ants to utilize the available computing power.
In fact, this number should be even higher to allow for effective memory
latency hiding through context switching between the active thread
warps~\cite{Kirk2016}.
\begin{table}[]
\footnotesize
\centering
\caption{
The mean solution construction phase duration (in ms) for the
proposed MMAS variants with a \emph{candidate list} length of 32,
executed on the \emph{Nvidia P100 (Pascal)} GPU.
}
\label{tab:mmas-cl-p100}
\begin{tabular}{lrrrrrrrr}
\hline
\multirow{2}{*}{Algorithm} & \multicolumn{8}{c}{Instance} \\
& \multicolumn{1}{l}{\textit{d198}} & \multicolumn{1}{l}{\textit{pcb442}} & \multicolumn{1}{l}{\textit{rat783}} & \multicolumn{1}{l}{\textit{pr1002}} & \multicolumn{1}{l}{\textit{pcb1173}} & \multicolumn{1}{l}{\textit{rl1889}} & \multicolumn{1}{l}{\textit{pr2392}} & \multicolumn{1}{c}{\textit{fl3795}} \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
AS~\cite{Dawson2013} & 0.77 & 3.67 & 12.13 & 19.76 & - & - & 131.85 & - \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
MMAS-RWM-LC & 0.34 & 0.64 & 1.11 & 2.34 & 2.77 & 10.09 & 19.84 & 88.19 \\
MMAS-RWM-BT & 0.35 & 0.67 & 1.20 & 1.70 & 2.09 & \textbf{6.69} & \textbf{9.33} & 28.78 \\
MMAS-RWM-CT & 0.35 & 0.67 & 1.19 & 1.63 & \textbf{2.01} & 7.34 & 12.04 & 52.32 \\
MMAS-WRS-LC & \textbf{0.33} & \textbf{0.62} & \textbf{1.09} & 2.25 & 2.67 & 9.70 & 18.91 & 85.71 \\
MMAS-WRS-BT & 0.34 & 0.64 & 1.17 & 1.69 & 2.13 & 6.75 & 9.44 & \textbf{28.75} \\
MMAS-WRS-CT & 0.35 & 0.65 & 1.16 & \textbf{1.62} & 2.07 & 7.23 & 11.77 & 51.08 \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
Speedup vs. AS~\cite{Dawson2013} & 2.32x & 5.91x & 11.18x & 12.18x & - & - & 14.12x & - \\ \hline
\end{tabular}
\end{table}
Table~\ref{tab:mmas-cl-p100} presents the mean duration of the solution
construction phase for the proposed MMAS implementations.
Generally, for the smallest instances, the differences between the algorithms
are also small.
Only when
starting with the \emph{pr1002} instance do the differences increase with
performance, mainly depending on the shared memory's efficiency for the
given tabu implementation, with the BT offering the best performance, and the
LC the worst (Tab.~\ref{tab:tabu-cmp}).
For example, for the largest instance,
\emph{fl3795}, and the MMAS with the WRS node selection implementation; the BT
results in a 1.78x and 2.98x speedup over the CT and LC tabu implementations,
respectively.
The implementation is also up to 14.12x faster than the GPU-based
AS by Dawson and Stewart~\cite{Dawson2013} who used the Nvidia GTX 580 GPU
(Fermi architecture) with 1,581 GFLOP/s computing power and 192 GB/s global
memory bandwidth.
As the source code is not available, it is difficult to
repeat the computations on the same GPU. However, the observed speedup suggests
that our implementation efficiently utilizes the newer GPU's computing
capacity.
\begin{table}[]
\footnotesize
\centering
\caption{
The mean solution construction phase duration (in ms) for the
proposed MMAS variants with a \emph{candidate list} length of 32,
executed on the \emph{Nvidia V100 (Volta)} GPU.
}
\label{tab:mmas-cl-v100}
\begin{tabular}{@{}lrrrrrrrr@{}}
\toprule
\multirow{2}{*}{Algorithm} & \multicolumn{8}{c}{Instance} \\
& \multicolumn{1}{c}{\textit{d198}} & \multicolumn{1}{c}{\textit{pcb442}} & \multicolumn{1}{c}{\textit{rat783}} & \multicolumn{1}{c}{\textit{pr1002}} & \multicolumn{1}{c}{\textit{pcb1173}} & \multicolumn{1}{c}{\textit{rl1889}} & \multicolumn{1}{c}{\textit{pr2392}} & \multicolumn{1}{c}{\textit{fl3795}} \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
AS~\cite{Dawson2013} & 0.77 & 3.67 & 12.13 & 19.76 & - & - & 131.85 & - \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
MMAS-RWM-LC & \textbf{0.19} & 0.40 & 0.72 & 0.95 & 1.13 & 3.54 & 8.18 & 32.16 \\
MMAS-RWM-BT & 0.20 & 0.37 & 0.71 & 0.97 & 1.16 & 2.49 & 5.18 & 12.83 \\
MMAS-RWM-CT & 0.20 & 0.41 & 0.74 & 1.00 & 1.20 & 2.32 & 5.04 & 19.32 \\
MMAS-WRS-LC & \textbf{0.19} & 0.38 & 0.69 & \textbf{0.92} & \textbf{1.08} & 3.40 & 7.74 & 30.77 \\
MMAS-WRS-BT & \textbf{0.19} & \textbf{0.36} & \textbf{0.68} & 0.93 & 1.12 & 2.44 & \textbf{3.26} & \textbf{12.65} \\
MMAS-WRS-CT & \textbf{0.19} & 0.40 & 0.71 & 0.97 & 1.15 & \textbf{2.26} & 4.89 & 18.66 \\
\cmidrule(r){1-1} \cmidrule(l){2-9}
Speedup vs. AS~\cite{Dawson2013} & 4.14x & 10.26x & 17.84x & 21.51x & - & - & 40.39x & - \\ \bottomrule
\end{tabular}
\end{table}
Repeating the computations on the newer, Volta V100, GPU results in speedups up
to 2.86x for the \emph{pr2392} instance compared to the Pascal P100 GPU.
However, for the largest instance, \emph{fl3795}, the speedup falls to 2.27x suggesting
that the larger computing capacity of the newer GPU is not able to compensate
for the significantly increased number of high-latency memory accesses associated
with solving the bigger TSP instance.
Compared to the GPU-based AS by Dawson and Stewart~\cite{Dawson2013}, the
proposed MMAS implementations are up to 40.39x faster.
Considering the impact of the node selection procedure and the tabu
implementations, the latter is more important, especially for the largest TSP
instances.
\subsection{Varying the number of ants}
\label{sec:Varying_the_number_of_ants}
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{fig/variable-numer-of-ants-v100-build-time.pdf}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{fig/variable-numer-of-ants-v100-quality.pdf}
\end{subfigure}
\caption{
The influence of the number of ants in the MMAS-RWM-CL on the mean solution
build time. The results are mean values obtained over 30 runs of the
algorithm solving the \emph{pr1002} instance on the Nvidia V100 GPU.
}
\label{fig:var-number-of-ants}
\end{figure}
In the experiments presented above, the number of ants was equal to the size of
the instance to enable a comparison with the existing results from the
literature~\cite{Cecilia2018,Dawson2013,Zhou2017}. However nothing prevents
the number of ants being set to a different value.
Figure~\ref{fig:var-number-of-ants} shows how the number of ants affects the
runtime and the quality of the solutions generated by the MMAS-RWM-LC (with and
without the CL) solving the \emph{pr1002} instance on the Nvidia V100 GPU. The
number of iterations was set to 1,000 and the $\rho$ parameter governing the
pheromone evaporation speed was set to $0.9$.
As can be seen, the computation time rises steadily with the increasing number
of ants but not as fast as it might be expected based on the numbers alone. This can
be explained by the fact that the efficient utilization of the 5,120 computing
cores of the Nvidia Tesla V100 GPU (see Tab.~\ref{tab:gpu-spec}) requires
running a large number of calculations in parallel, i.e., the ants constructing
the solutions. On the other hand, each kernel executing on the GPU requires a
number of the SM's registers and a portion of the shared memory to run.
Combined with other architecture-related restrictions, the number of thread
warps resulting in the best occupancy of the GPU's SMs can be calculated.
For example, the \texttt{cudaOccupancyMaxActiveBlocksPerMultiprocessor}
function offered by the CUDA framework reports that the best occupancy for the
solution construction kernel of the MMAS-RWM-LC (assuming 1 warp per thread
block) can be achieved for 1,680 ants. This is consistent with the significant
increase in the execution time as observed when going from 1,600 to 1,800 ants.
Focusing on the quality of the solutions, we can observe a positive effect of
increasing the number of ants. However, the benefit is much more visible when
the number of ants goes from 200 to 1,000, than when it goes from 1,000 to 2,000.
In other words, assuming that the number of ants is high enough, it could be
more reasonable to increase the number of iterations than to further increase
the number of ants.
\subsection{Pheromone Update and Remaining Operations}
\label{sec:Pheromone_Update_and_Remaining_Operations}
The solution construction process is definitely the most time consuming part of the MMAS.
The remaining operations, including the pheromone evaporation and deposition,
take significantly less time.
Table~\ref{tab:remaining-times} shows the times needed to complete all the
kernels except for the solution construction time for the MMAS-RWM-LC
algorithm, as executed on the Pascal and Volta GPUs.
The times for the remaining MMAS variants are very close which stems from the
fact that most of the differences between the algorithms affect only the solution
construction process.
As can be seen, the times for the newer-generation Volta GPU are close to two
times slower than for the older Pascal GPU.
If we compare the numbers to the solution construction kernel times
(Tab.~\ref{tab:mmas-cmp-p100} and Tab.~\ref{tab:mmas-cmp-v100}), it can be seen
that they are significantly lower and the relative differences increase with
the size of the problem.
For example, for the \emph{pcb442} instance the solution construction kernel
time of the MMAS-WRS-CT executed on the P100 GPU was close to $7.92x$ greater
than the total time needed by all the remaining kernels (0.95 ms vs 0.12 ms).
For the largest instance, \emph{fl3795}, the ratio was close to 246.13x, meaning
that the solution construction process accounted for over 99.5\% of the
computation time.
The ratio remains high ($\sim 35.47x$) even if the construction process is sped
up by using the candidate lists.
It is be possible that a faster execution of the remaining kernels could be
observed if some of them were joined (fused) into one; for example, the
pheromone evaporation and deposition kernels.
However, as the above analysis has shown, it would have little impact on the
overall MMAS execution time while, at the same time, making the implementation
more complicated.
\begin{table}[]
\footnotesize
\centering
\caption{
A comparison of the mean times (in ms) necessary to complete the execution of
all kernels except the solution construction for the MMAS-RWM-LC, with a
candidate list of 32, executed on the P100 and V100 GPUs.
}
\label{tab:remaining-times}
\begin{tabular}{@{}lcrrrrrrr@{}}
\toprule
\multirow{2}{*}{GPU} & \multicolumn{8}{c}{Instance} \\
& \textit{d198} & \multicolumn{1}{c}{\textit{pcb442}} & \multicolumn{1}{c}{\textit{rat783}} & \multicolumn{1}{c}{\textit{pr1002}} & \multicolumn{1}{c}{\textit{pcb1173}} & \multicolumn{1}{c}{\textit{rl1889}} & \multicolumn{1}{c}{\textit{pr2392}} & \multicolumn{1}{c}{\textit{fl3795}} \\ \midrule
P100 (Pascal) & \multicolumn{1}{r}{0.08} & 0.12 & 0.21 & 0.28 & 0.35 & 0.61 & 0.81 & 1.44 \\
V100 (Volta) & \multicolumn{1}{r}{0.05} & 0.07 & 0.12 & 0.15 & 0.18 & 0.34 & 0.44 & 0.89 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Solution Quality}
\label{sec:Solution_Quality}
Our work focused on an efficient parallel GPU-based implementation of the
MMAS, hence no quality-related modifications of the MMAS were investigated.
In fact, to confirm that the proposed MMAS variants differ only in terms of
the computation speed, the samples of the final solutions' quality obtained
using the proposed MMAS implementations were checked for statistically significant
differences based on the non-parametric Friedman test~\cite{Hollander2013}.
\begin{table}[]
\footnotesize
\centering
\caption{
The results of the non-parametric Friedman test with
the null hypothesis $H_0$ stating that the six proposed MMAS
variants produced solutions with the same median quality.
The results are provided for each TSP instance separately.
Assumed significance level of $\alpha = 0.05$.
}
\label{tab:stat-test}
\begin{tabular}{@{}lrrrrrrrr@{}}
\toprule
Instance & \multicolumn{1}{l}{\textit{d198}} & \multicolumn{1}{l}{\textit{pcb442}} & \multicolumn{1}{l}{\textit{rat783}} & \multicolumn{1}{l}{\textit{pr1002}} & \multicolumn{1}{l}{\textit{pcb1173}} & \multicolumn{1}{l}{\textit{rl1889}} & \multicolumn{1}{l}{\textit{pr2392}} & \multicolumn{1}{l}{\textit{fl3795}} \\ \midrule
Test statistic & 2.13 & 2.98 & 3.72 & 8.76 & 3.32 & 6.14 & 6.41 & 4.34 \\
$p$-value & 0.83 & 0.70 & 0.59 & 0.12 & 0.65 & 0.29 & 0.27 & 0.50 \\
$H_0$ rejected & No & No & No & No & No & No & No & No \\ \bottomrule
\end{tabular}
\end{table}
The null hypothesis evaluated with this test checks if at least two of the
samples represent populations with different median values, in a set of $k$
samples (where $k \ge 2$).
If the null hypothesis is rejected, then the quality of the results generated
by the proposed MMAS variants is not equivalent.
Table~\ref{tab:stat-test} shows the results of the Friedman test for the six
proposed MMAS variants: MMAS-RWM-LC, MMAS-RWM-BT, MMAS-RWM-CT, MMAS-WRS-LC,
MMAS-WRS-BT and MMAS-WRS-CT.
The test was calculated separately for each of the eight TSP instances
investigated.
As can be seen from Table~\ref{tab:stat-test}, the null hypothesis was not
rejected in any of the cases, hence no statistically significant differences in
the quality of the results generated by the MMAS variants were found.
\subsection{Local Search}
\label{sec:Local_Search}
The ACO algorithms are typically combined with problem-specific LS
heuristics~\cite{Dorigo2004}. In this combination, the ACO performs
a \emph{coarse-grained} search throughout the space of possible solutions,
while the LS is responsible for a fine-grained exploitation of the neighborhood
of a solution built by an ant.
For the sake of completeness of the current work, we have implemented a parallel version of the
2-opt heuristic as described by Bentley~\cite{Bentley1992}.
The 2-opt works by searching for a pair of edges to remove so that
the resulting parts of the route can be reconnected using a new pair
of edges that has a combined length smaller than the length of the removed pair.
If such an improvement is found and applied, the
search is restarted until no further improvements can be found.
Although there are only $O(n^2)$ possible pairs to consider while searching for
the improvement, it is still too time-consuming if the number of nodes is on the
order of thousands or more. For this reason, in our implementation
the performance-oriented heuristics described by Bentley are applied, i.e. the search for a new
edge to insert is limited only to the nearest neighbors of the considered node (32 in our
case) and an array of so-called \emph{don't look bits} is used to further limit the
search. In the proposed parallel version of the 2-opt, a number of threads divided into
groups of 32 threads each (warps) search in parallel for the pair of edges to
remove, and, if any group succeeds, the improvement is applied (in
parallel) by all groups of threads reversing the respective part of the route.
The 2-opt is applied to every solution constructed by the ants.
It is worth mentioning, that the GPU-based implementation of the 2-opt
was first proposed by Rocki and Suda~\cite{Rocki2013} but without any additional
heuristics that could limit the $O(n^2)$ search space size. The approach was refined by
Zhou et al.~\cite{Zhou2016} who improved its execution speed and,
combined it with the Iterated Local Search. The authors solved TSP instances of up
to 4,461 nodes obtaining for the largest instance solutions with the mean
distance from the optimum equal to 3.67\%.
Another proposal to improve the 2-opt implementation by Rocki and Suda was made
by Robinson et al.~\cite{Robinson2018}, who allowed multiple non-conflicting
changes to a route to be performed simultaneously. The resulting 2-opt
hill-climbing with random restarts was used to solve the 7,397-city
instance with an error of around 8\% relative to the optimum.
The GPU-computing power was also utilized to solve the TSP in the work of Wang et
al.~\cite{Wang2017}, who proposed a parallel computation model for the self-organizing
map neural network. The authors considered instances up to 85,900 cities but the
quality of obtained solutions was typically at least 5\% above the respective optimum.
To assess the performance of the proposed parallel MMAS with the 2-opt, we
conducted a number of experiments using the Nvidia V100 GPU and several
instances from the TSPLIB repository with up to 18,512 nodes.
Specifically, the MMAS-RWM-BT with the candidate lists of length 32 was run
with the number of ants equal to 800 (multiple of the number of 80 SMs in the V100).
The values of the remaining parameters were set based on the size of the
instance. For the instances with up to $10^5$ nodes,
the number of iterations was 2,000 and the pheromone was evaporated slowly
($\rho = 0.9$), while for the larger
instances the number of iterations was increased to 3,000 and the speed of
the pheromone evaporation was higher \R{($\rho = 0.7$)}.
These values were selected with the aim of allowing the algorithm to find good
quality solutions within a relatively short time.
The computations were repeated 20 times.
\begin{table}[]
\footnotesize
\centering
\caption{\R{
A comparison of the results generated by the ESACO~\cite{Ismkhan2017} and MMAS-RWM-BT
(with the 2-opt LS) metaheuristics for the TSP instances from
the TSPLIB repository. The best and mean solution costs (route lengths)
are given along with the relative difference from the optimum reported in
round brackets. The values in the \emph{Time} column refer to the CPU runtime (in
seconds) as reported in the respective work and the Nvidia Tesla V100 GPU
runtime in the case of the MMAS-RWM-BT. The lowest mean cost for
every instance is marked in bold.
}}
\R{
\label{tab:cmp-with-esaco}
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}lrrrrrrr@{}}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{Instance}}
& \multicolumn{1}{c}{\multirow{2}{*}{Optimum}}
& \multicolumn{3}{c}{ESACO\cite{Ismkhan2017}}
& \multicolumn{3}{c}{MMAS-RWM-BT} \\ \cmidrule(lr){3-5} \cmidrule(l){6-8}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Best solution} & \multicolumn{1}{c}{Mean solution} & \multicolumn{1}{c}{Time} & \multicolumn{1}{c}{Best solution} & \multicolumn{1}{c}{Mean solution} & \multicolumn{1}{c}{Time} \\ \midrule
\textit{eil51} & 426 & 426 (0\%) & \textbf{426.0 (0\%)} & 1.12 & 426 (0\%) & \textbf{426.0 (0\%)} & 0.43 \\
\textit{eil76} & 538 & 538 (0\%) & \textbf{538.0 (0\%)} & 1.39 & 538 (0\%) & \textbf{538.0 (0\%)} & 0.49 \\
\textit{kroA100} & 21282 & 21282 (0\%) & \textbf{21282.0 (0\%)} & 2.61 & 21282 (0\%) & \textbf{21282.0 (0\%)} & 0.58 \\
\textit{lin105} & 14379 & 14379 (0\%) & \textbf{14379.0 (0\%)} & 2.0 & 14379 (0\%) & \textbf{14379.0 (0\%)} & 0.6 \\
\textit{d198} & 15780 & 15780 (0\%) & \textbf{15780.0 (0\%)} & 6.5 & 15780 (0\%) & \textbf{15780.0 (0\%)} & 0.9 \\
\textit{kroA200} & 29368 & 29368 (0\%) & \textbf{29368.0 (0\%)} & 4.7 & 29368 (0\%) & \textbf{29368.0 (0\%)} & 0.8 \\
\textit{a280} & 2579 & 2579 (0\%) & 2579.1 (0\%) & 4.5 & 2579 (0\%) & \textbf{2579.0 (0\%)} & 1.0 \\
\textit{lin318} & 42029 & 42029 (0\%) & \textbf{42053.9 (0.06\%)} & 10.2 & 42029 (0\%) & 42069.6 (0.1\%) & 1.7 \\
\textit{pcb442} & 50778 & 50778 (0\%) & \textbf{50803.6 (0.05\%)} & 11.5 & 50809 (0.06\%) & 50950.7 (0.34\%) & 2.0 \\
\textit{att532} & 27686 & 27686 (0\%) & \textbf{27701.2 (0\%)} & 23.1 & 27686 (0\%) & 27708.9 (0.08\%) & 1.8 \\
\textit{rat783} & 8806 & 8806 (0\%) & \textbf{8809.8 (0.04\%)} & 22.6 & 8810 (0.05\%) & 8825.5 (0.22\%) & 3.1 \\
\textit{pr1002} & 259045 & 259045 (0\%) & \textbf{259509.0 (0.18\%)} & 35.8 & 259415 (0.14\%) & 259712.7 (0.26\%) & 4.0 \\
\textit{fl3795} & 28772 & 28787 (0.05\%) & 28883.5 (0.39\%) & 119.3 & 28793 (0.07\%) & \textbf{28819.3 (0.16\%)} & 19.5 \\
\textit{fnl4461} & 182566 & 183254 (0.38\%) & \textbf{183446.0 (0.48\%)} & 192.6 & 183361 (0.44\%) & 183627.6 (0.58\%) & 34.9 \\
\textit{rl5915} & 565530 & 567177 (0.29\%) & 568935.0 (0.60\%) & 216.9 & 566123 (0.11\%) & \textbf{567699.9 (0.38\%)} & 49.3 \\
\textit{pla7397} & 23260728 & 23345479 (0.36\%) & 23389341.0 (0.55\%) & 213.9 & 23365046 (0.45\%) & \textbf{23386240.5 (0.54\%)} & 58.0 \\
\textit{rl11849} & 923288 & 928876 (0.61\%) & 930338.0 (0.76\%) & 575.8 & 926840 (0.38\%) & \textbf{928618.83 (0.58\%)} & 247.4 \\
\textit{usa13509} & 19982859 & 20172735 (0.95\%) & 20195089.0 (1.06\%) & 914.2 & 20128078 (0.73\%) & \textbf{20168030.2 (0.93\%)} & 224.2 \\
\textit{brd14051} & 469385 & 473718 (0.92\%) & \textbf{474087.0 (1.00\%)} & 682.5 & 473389 (0.85\%) & 474715.65 (1.14\%) & 228.4 \\
\textit{d15112} & 1573084 & 1587150 (0.89\%) & 1589288.0 (1.03\%) & 776.7 & 1584054 (0.70\%) & \textbf{1586604.05 (0.86\%)} & 404.2 \\
\textit{d18512} & 645238 & 652516 (1.13\%) & 653154.0 (1.23\%) & 684.4 & 650784 (0.86\%) & \textbf{651413.58 (0.96\%)} & 335.5 \\ \bottomrule
\end{tabular}%
}
}
\end{table}
\begin{table}[]
\footnotesize
\centering
\R{
\caption{\R{
A comparison of the results generated by the MFC-ABC and MMAS-RWM-BT
(with the 2-opt LS) metaheuristics. The meaning of the columns is the same
as in Tab.~\ref{tab:cmp-with-esaco}.
}}
\label{tab:cmp-with-mfc-abc}
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
\multicolumn{1}{c}{\multirow{2}{*}{Instance}}
& \multicolumn{1}{c}{\multirow{2}{*}{Optimum}}
& \multicolumn{2}{c}{MCF-ABC\cite{Choong2019}}
& \multicolumn{3}{c}{MMAS-RWM-BT} \\ \cmidrule(lr){3-4} \cmidrule(l){5-7}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{Mean solution} & \multicolumn{1}{c}{Time} & \multicolumn{1}{c}{Best solution} & \multicolumn{1}{c}{Mean solution} & \multicolumn{1}{c}{Time} \\ \midrule
\textit{eil101} & 629 & \textbf{629.0 (0\%)} & 0.0 & 629 (0\%) & \textbf{629.0 (0\%)} & 0.6 \\
\textit{lin105} & 14379 & \textbf{14379.0 (0\%)} & 0.0 & 14379 (0\%) & \textbf{14379.0 (0\%)} & 0.6 \\
\textit{pr107} & 44303 & \textbf{44303.0 (0\%)} & 0.0 & 44303 (0\%) & \textbf{44303.0 (0\%)} & 0.6 \\
\textit{gr120} & 6942 & \textbf{6942.0 (0\%)} & 0.0 & 6942 (0\%) & \textbf{6942.0 (0\%)} & 0.6 \\
\textit{pr124} & 59030 & \textbf{59030.0 (0\%)} & 0.1 & 59030 (0\%) & \textbf{59030.0 (0\%)} & 0.6 \\
\textit{bier127} & 118282 & \textbf{118282.0 (0\%)} & 0.1 & 118282 (0\%) & \textbf{118282.0 (0\%)} & 0.7 \\
\textit{ch130} & 6110 & \textbf{6110.0 (0\%)} & 0.0 & 6110 (0\%) & \textbf{6110.0 (0\%)} & 0.6 \\
\textit{pr136} & 96772 & \textbf{96772.0 (0\%)} & 0.2 & 96772 (0\%) & \textbf{96772.0 (0\%)} & 0.6 \\
\textit{gr137} & 69853 & \textbf{69853.0 (0\%)} & 0.1 & 69853 (0\%) & \textbf{69853.0 (0\%)} & 0.7 \\
\textit{pr144} & 58537 & \textbf{58537.0 (0\%)} & 1.8 & 58537 (0\%) & \textbf{58537.0 (0\%)} & 0.6 \\
\textit{ch150} & 6528 & \textbf{6528.0 (0\%)} & 0.1 & 6528 (0\%) & \textbf{6528.0 (0\%)} & 0.7 \\
\textit{kroA150} & 26524 & \textbf{26524.0 (0\%)} & 0.1 & 26524 (0\%) & \textbf{26524.0 (0\%)} & 0.7 \\
\textit{kroB150} & 26130 & \textbf{26130.0 (0\%)} & 0.1 & 26130 (0\%) & \textbf{26130.0 (0\%)} & 0.7 \\
\textit{pr152} & 73682 & \textbf{73682.0 (0\%)} & 1.8 & 73682 (0\%) & \textbf{73682.0 (0\%)} & 0.8 \\
\textit{u159} & 42080 & \textbf{42080.0 (0\%)} & 0.1 & 42080 (0\%) & \textbf{42080.0 (0\%)} & 0.7 \\
\textit{si175} & 21407 & \textbf{21407.0 (0\%)} & 0.2 & 21407 (0\%) & 21407.5 (0\%) & 0.8 \\
\textit{brg180} & 1950 & \textbf{1950.0 (0\%)} & 0.0 & 1950 (0\%) & \textbf{1950.0 (0\%)} & 0.7 \\
\textit{rat195} & 2323 & \textbf{2323.0 (0\%)} & 0.2 & 2323 (0\%) & 2325.2 (0.2\%) & 0.8 \\
\textit{d198} & 15780 & \textbf{15780.0 (0\%)} & 1.3 & 15780 (0\%) & \textbf{15780.0 (0\%)} & 0.9 \\
\textit{kroA200} & 29368 & \textbf{29368.0 (0\%)} & 0.1 & 29368 (0\%) & \textbf{29368.0 (0\%)} & 0.8 \\
\textit{kroB200} & 29437 & \textbf{29437.0 (0\%)} & 0.0 & 29437 (0\%) & \textbf{29437.0 (0\%)} & 0.8 \\
\textit{gr202} & 40160 & \textbf{40160.0 (0\%)} & 0.6 & 40160(0\%) & 40161.3(0\%) & 0.9 \\
\textit{tsp225} & 3916 & \textbf{3916.0 (0\%)} & 0.0 & 3916 (0\%) & \textbf{3916.0 (0\%)} & 0.8 \\
\textit{ts225} & 126643 & \textbf{126643.0 (0\%)} & 0.1 & 126643 (0\%) & \textbf{126643.0 (0\%)} & 0.9 \\
\textit{pr226} & 80369 & \textbf{80369.0 (0\%)} & 1.4 & 80369 (0\%) & \textbf{80369.0 (0\%)} & 0.9 \\
\textit{gr229} & 134602 & \textbf{134602.0 (0\%)} & 0.8 & 134602 (0\%) & 134608.4 (0\%) & 0.9 \\
\textit{gil262} & 2378 & \textbf{2378.0 (0\%)} & 0.1 & 2378 (0\%) & 2378.2 (0.01\%) & 1.0 \\
\textit{pr264} & 49135 & \textbf{49135.0 (0\%)} & 0.1 & 49135 (0\%) & \textbf{49135.0 (0\%)} & 1.0 \\
\textit{a280} & 2579 & \textbf{2579.0 (0\%)} & 0.0 & 2579 (0\%) & \textbf{2579.0 (0\%)} & 1.0 \\
\textit{pr299} & 48191 & \textbf{48191.0 (0\%)} & 0.2 & 48191 (0\%) & 48197.1 (0.01\%) & 1.1 \\
\textit{lin318} & 42029 & \textbf{42029.0 (0\%)} & 1.5 & 42029 (0\%) & 42069.6 (0.1\%) & 1.7 \\
\textit{rd400} & 15281 & \textbf{15281.0 (0\%)} & 2.4 & 15281 (0\%) & 15282.9 (0.01\%) & 1.4 \\
\textit{fl417} & 11861 & \textbf{11861.0 (0\%)} & 5.7 & 11861 (0\%) & \textbf{11861.0 (0\%)} & 1.6 \\
\textit{gr431} & 171414 & \textbf{171414.0 (0\%)} & 13.0 & 171414 (0\%) & 171419.7 (0\%) & 1.5 \\
\textit{pr439} & 107217 & \textbf{107217.0 (0\%)} & 2.3 & 107217 (0\%) & \textbf{107217.0 (0\%)} & 1.4 \\
\textit{pcb442} & 50778 & \textbf{50778.0 (0\%)} & 1.6 & 50809 (0.06\%) & 50950.7 (0.34\%) & 2.0 \\
\textit{d493} & 35002 & \textbf{35002.7 (0\%)} & 20.9 & 35004 (0.01\%) & 35046.9 (0.13\%) & 1.7 \\
\textit{att532} & 27686 & \textbf{27686.5 (0\%)} & 11.4 & 27686 (0\%) & 27708.9 (0.08\%) & 1.8 \\
\textit{ali535} & 202339 & \textbf{202339.0 (0\%)} & 10.6 & 202339 (0\%) & 202378.1 (0.02\%) & 2.0 \\
\textit{si535} & 48450 & 48498.3 (0.1\%) & 53.3 & 48450 (0\%) & \textbf{48455.7 (0.01\%)} & 1.9 \\
\textit{pa561} & 2763 & \textbf{2763.1 (0\%)} & 7.1 & 2763 (0\%) & 2768.4 (0.20\%) & 1.8 \\
\textit{u574} & 36905 & \textbf{36905.0 (0\%)} & 2.1 & 36905 (0\%) & 36923.2 (0.05\%) & 2.1 \\
\textit{rat575} & 6773 & \textbf{6774.3 (0.02\%)} & 6.3 & 6774 (0.01\%) & 6781.5 (0.13\%) & 1.9 \\
\textit{p654} & 34643 & \textbf{34643.0 (0\%)} & 21.5 & 34643 (0\%) & 34653.6 (0.03\%) & 2.6 \\
\textit{d657} & 48912 & \textbf{48915.1 (0.01\%)} & 15.5 & 48913 (0\%) & 48993.2 (0.17\%) & 2.3 \\
\textit{gr666} & 294358 & \textbf{294404.8 (0.02\%)} & 32.1 & 294358 (0\%) & 294544.3 (0.06\%) & 2.2 \\
\textit{u724} & 41910 & \textbf{41916.5 (0.02\%)} & 12.9 & 41929 (0.05\%) & 41973.8 (0.15\%) & 2.3 \\
\textit{rat783} & 8806 & \textbf{8806.0 (0\%)} & 4.2 & 8810 (0.05\%) & 8825.5 (0.22\%) & 3.1 \\
\textit{dsj1000} & 18659688 & \textbf{18661580.6 (0.01\%)} & 53.2 & 18665058 (0.03\%) & 18677500.3 (0.10\%) & 4.4 \\
\textit{pr1002} & 259045 & \textbf{259073.0 (0.01\%)} & 16.6 & 259415 (0.14\%) & 259712.7 (0.26\%) & 4.0 \\
\textit{si1032} & 92650 & \textbf{92650.0 (0\%)} & 4.4 & 92650 (0\%) & \textbf{92650.0 (0\%)} & 3.3 \\
\textit{vm1084} & 239297 & \textbf{239322.3 (0.01\%)} & 24.3 & 239297 (0\%) & 239480.0 (0.08\%) & 3.9 \\
\textit{pcb1173} & 56892 & \textbf{56897.9 (0.01\%)} & 15.4 & 56892 (0\%) & 56994.7 (0.18\%) & 3.8 \\
\textit{d1291} & 50801 & \textbf{50843.7 (0.08\%)} & 15.7 & 50803 (0\%) & 50855.5 (0.11\%) & 4.5 \\
\textit{d1655} & 62128 & \textbf{62221.6 (0.15\%)} & 27.0 & 62192 (0.10\%) & 62321.1 (0.31\%) & 5.7 \\
\textit{u1817} & 57201 & \textbf{57354.2 (0.27\%)} & 24.6 & 57209 (0.01\%) & 57428.2 (0.4\%) & 7.0 \\
\textit{u2152} & 64253 & \textbf{64426.5 (0.27\%)} & 28.3 & 64366 (0.18\%) & 64520.1 (0.42\%) & 7.1 \\
\textit{pr2392} & 378032 & \textbf{378549.6 (0.14\%)} & 28.9 & 378390 (0.1\%) & 379872.0 (0.49\%) & 10.3 \\
\textit{fl3795} & 28772 & 28825.1 (0.18\%) & 131.7 & 28793 (0.07\%) & \textbf{28819.3 (0.16\%)} & 19.5 \\
\textit{fnl4461} & 182566 & \textbf{183002.8 (0.24\%)} & 66.6 & 183361 (0.44\%) & 183627.6 (0.58\%) & 34.9 \\
\textit{rl5915} & 565530 & 567990.2 (0.44\%) & 94.2 & 566123 (0.11\%) & \textbf{567699.9 (0.38\%)} & 49.3 \\
\textit{pla7397} & 23260728 & \textbf{23324321.9 (0.24\%)} & 247.6 & 23365046 (0.45\%) & 23386240.5 (0.54\%) & 58.0 \\
\textit{rl11849} & 923288 & \textbf{928015.1 (0.51\%)} & 308.9 & 926840 (0.38\%) & 928618.83 (0.58\%) & 247.4 \\ \bottomrule
\end{tabular}
\end{table}
\R{
Table~\ref{tab:cmp-with-esaco} and Tab.~\ref{tab:cmp-with-mfc-abc} present the
obtained results and compare them with the
results of two high-performing metaheuristics, namely the Effective Strategies+ACO
(ESACO)~\cite{Ismkhan2017} and the Artificial Bee Colony with a Modified Choice
Function (MFC-ABC)~\cite{Choong2019} (respectively). The ESACO was chosen as it is the
current state-of-the-art ACO-based TSP solver. Specifically, it is based on the
ACS combined with an efficient LS comprising the 2-opt, 3-opt and so-called
\emph{double-bridge moves}. The second algorithm, the MFC-ABC, combines a modified ABC
metaheuristic with the well-known Lin-Kernighan heuristic. Extensive comparisons with
the existing state-of-the-art metaheuristic TSP solvers showed that the MFC-ABC is
very competitive in terms of both computation time and quality of the
results~\cite{Choong2019}.
Analysis of the results presented in Tab.~\ref{tab:cmp-with-esaco} and
Tab.~\ref{tab:cmp-with-mfc-abc} reveals
several facts.
Firstly, the proposed MMAS with the 2-opt LS is able to obtain
good quality results, i.e. within 1\% from the optima, for all but one
instance, \emph{brd14051}, for which the mean cost was 1.14\% above the optimum.
As expected, the error increases with the instance size.
Secondly, the 2-opt LS is too simplistic to allow the optima to be found except
for small instances. On the other hand, it proved to be effective
enough to allow the proposed implementation to obtain better results than the
ESACO in 14 out of 21 cases, including the largest instances with up to
18,512 cities.
A comparison of the proposed GPU-based MMAS with the MFC-ABC (Tab.~\ref{tab:cmp-with-mfc-abc}) confirms the effectiveness of the latter as it
generated (on average) better results for 33 out of 63 (52\%) instances while
worse only for 3. Still, the results obtained for the
GPU-based MMAS can be considered satisfactory with the relative error being
below 0.5\% for all but three (\emph{fnl4461, pla7397} and
\emph{rl11849}) instances for which it reached 0.58\%.
Unfortunately, the $O(n^2)$ memory complexity of the MMAS prevents solving
the largest TSP instance (with 85,900 cities) considered in~\cite{Choong2019}
as it would require more than 16 GB of RAM offered by the V100 GPU.
It is also worth adding that the GPU-based MMAS is also
competitive in terms of computation time.
The average runtime for the instances with less than 10,000 cities was
below 1 minute, whereas solving each of the remaining (larger) instances took less than
7 minutes.
Summarizing, the results show that the high computational power of the GPU
is enough to allow the relatively simple method (MMAS with the 2-opt) to compete
with sophisticated but sequential algorithms. This suggests that more effort
should be put into the development of parallel versions of the more advanced
LS methods, e.g., the Lin-Kernighan heuristic, allowing
the available computing power of multi-core CPUs and the increasingly
popular GPUs to be fully utilized.
}
\section{Conclusions}
\label{sec:Conclusions}
\label{sec:conclusions}
The efficient parallel execution of the ACO on GPUs requires a careful
organization of the computations, which must harmonize with their capabilities
and limitations.
A large number of processing elements (cores) allow a high
degree of parallelism, and thus require/benefit-from high concurrency.
In this paper, we have presented a novel parallel
implementation of the solution construction procedure, which is, in general, the most
important and the most time-consuming part of the MMAS and ACO.
Specifically, we have proposed a novel implementation of the node selection
procedure based on the WRS algorithm~\cite{Efraimidis2006}. It requires little
cooperation between parallel threads, resulting in a high degree of
scalability, yet it is easy to implement and offers the same quality of
generated solutions as does the original RWM-based method.
The efficient use of the fast, but size limited, shared memory offered by the
SMs that comprise the GPU is also essential from the performance point of view.
We have discussed three tabu implementations, namely, LC, CT, and BT, which
differ in memory overhead and the time complexity of the offered operations.
A total of six MMAS
variants have been evaluated empirically on a set of TSP instances ranging
from 198 to 3,795 cities.
The computations were conducted for two subsequent Nvidia
GPU architectures, namely, Pascal (P100) and Volta (V100).
In general, the MMAS with the parallel WRS-based node selection implementation
is faster than the MMAS with a parallel version of the RWM-based method.
The impact of the tabu implementation depends on the size of the problem
instance and, more importantly, on the use of the candidate lists.
If the candidate lists are
used, the BT is preferred as the slow enumerations of its contents are
executed rarely.
On the other hand, if the lists are not used, then the proposed CT
offers better speed at the cost of increased shared memory usage.
Overall, the MMAS-WRS-BT and MMAS-WRS-CT are the recommended choices,
depending on whether the candidate lists are used or not, respectively.
Both GPUs offer excellent performance, with the newer V100 GPU being up to 2.88x
faster than the previous-generation P100.
For example, the solution construction phase for the \emph{pr1002} instance
takes about 6.09 ms on the P100 GPU and only 2.16 on the V100 GPU.
If candidate lists with a size of 32 are used, the MMAS-WRS-BT executing on the
V100 needs only 0.93 ms, i.e., it generates solutions at the impressive rate of
over 1 million solutions per second.
The analysis of the results presented in the literature on speeding
up the ACO's execution using the GPU-based computing shows that
huge progress has been made since the emergence of GPUs as a platform
for general-purpose computing.
For example, it took approximately 392 ms per iteration of the ACO to
build solutions to the \emph{pr1002} TSP instance when running
on the C2050 (Fermi) GPU~\cite{Cecilia2012}.
Five GPU generations forward, and the time has decreased to about 2 ms
for the V100 (Volta) GPU, that is, 196x faster.
Overall, advances in the design of GPU architecture,
progress in the development of software tools (compilers, libraries),
and algorithmic refinements seem to be a promising answer to the slowing-down
of Moore's law.
Additionally, the computational experiments considering the MMAS combined with
the 2-opt LS, show that the high computing speed of GPUs may be sufficient to
compete with more sophisticated but sequential algorithms, especially
if the results are to be generated quickly.
\subsection*{Future work}
The presented work may be extended in multiple directions. In this paper, we
have considered TSP instances for which the tabu fits entirely into the shared
(local) memory of a group of threads. However, its size is very limited, up to
64kB or 96kB even for newer Nvidia GPU architectures (Pascal, Volta). If part
of the shared memory is reserved for other purposes, for example, a local
search procedure, this leaves even less space for the tabu. It may be
interesting to consider tabu implementations in which the shared memory is used
along with global memory, i.e., variants of the described BT. It is worth
adding, that the proposed implementations of the tabu data structure and the
WRS-based node selection procedure can be applied to other ACO-based
algorithms, for example, the ACS~\cite{Skinderowicz2016}, and even to other
metaheuristics.
In this present work, we have mostly focused on the algorithmic refinements of
the GPU-based ACO implementation, but we think that further improvements are
possible. We agree with Cecilia et al.~\cite{Cecilia2018} that the new
mechanism of cooperative groups introduced in CUDA 9 could prove useful. Also,
improvements to the SIMT model introduced in the Volta architecture are worth
investigating as they allow for more divergence in the computations performed
by the threads belonging to the same warp~\cite{V1002018}.
Additionally, solving large problem instances would require significant changes
to the MMAS to be made, including a replacement of the pheromone memory with a more
space-efficient alternative and exploring ideas for alternative, faster
proportional selection method implementations.
\noindent
\textbf{Acknowledgments:} This research was supported in part by PL-Grid Infrastructure.
\bibliographystyle{plainnat}
\footnotesize
|
1,314,259,993,031 | arxiv | \section*{Introduction}
Throughout the paper $X$ will be an algebraic
variety of dimension $\ge 2$ over an algebraically closed field ${\Bbbk}$ of characteristic 0.
The {\em special automorphism group}
$\mathop{\rm SAut}(X)$ of such a variety $X$ is the subgroup of the full automorphism group
$ \operatorname{{\rm Aut}}(X)$ generated by all one-parameter unipotent subgroups of
$ \operatorname{{\rm Aut}}(X)$.\footnote{I.e. by subgroups isomorphic to $\ensuremath{\mathbb{G}}_a$. By abuse of language we do
not distinguish between one-parameter
unipotent subgroups of the group $ \operatorname{{\rm Aut}}(X)$ and effective
$\ensuremath{\mathbb{G}}_a$-actions on $X$. } Let
${\ensuremath{\mathcal{U}}}(X)$ denote the set of all these subgroups. A quasi-affine variety $X$ is
called {\em flexible}, if the tangent space $T_xX$ in
any smooth point $x\in X_{\rm reg}$ is spanned by the tangent
vectors at $x$ to the orbits $U.x$, where $U$ runs over ${\ensuremath{\mathcal{U}}}(X)$.
If $X$ is affine then this amounts to the notion of flexibility as introduced in \cite{AKZ, AFKKZ}.
For such varieties the flexibility is equivalent to the transitivity,
and even to infinite transitivity of the group
$\mathop{\rm SAut}(X)$ acting on the smooth locus
$X_{\mathop{\rm reg}}$ of $X$ (see \cite[Theorem 0.1]{AFKKZ}).
(We say that a group action is {\em infinitely transitive} if it is $m$-transitive for any $m\ge 1$.)
These characterizations of flexibility can be extended to any quasi-affine variety
(see Remarks \ref{1.7} and Theorem \ref{1.10} in Sect.\ 1).
It is worthwhile mentioning that the class of flexible varieties
is rather wide. It includes in particular
\begin{enumerate}[-]
\item homogeneous spaces of semi-simple groups (and even homogeneous
spaces of extensions of semi-simple groups by unipotent radicals);
\item non-degenerate toric varieties (i.e.\ toric varieties without nonconstant
invertible regular functions);
\item cones over flag varieties and anti-canonical cones over Del Pezzo surfaces of degree at least 4;
\item normal hypersurfaces of the form $uv=p (\bar x )$ in $\ensuremath{\mathbb{C}}^{n+2}_{u,v, {\bar x}}$;
\item homogeneous Gizatullin surfaces;
\end{enumerate}
see \cite{AKZ}, \cite{AFKKZ}, \cite{Kov}.
If on a quasi-affine variety $X$ the group $\mathop{\rm SAut}(X)$ has an open
orbit, then this open orbit is a flexible quasi-affine variety. A
normal quasi-affine variety $X$ is flexible if and only if so is
$X_{\mathop{\rm reg}}$. In its simplest form the main result of this paper is
the following theorem; see Sect.\ 1 for generalizations and
refinements.
\begin{thm}\label{mthm}
Let $X$ be a smooth quasi-affine
variety of dimension $\ge 2$ and $Y\subseteq X$ a closed subscheme of codimension $\ge 2$. If $X$
is flexible then so is $X \backslash Y$.
\end{thm}
That is, if $\mathop{\rm SAut}(X)$ acts transitively on $X$ then $\mathop{\rm SAut}(X\backslash Y)$
acts transitively on $X\backslash Y$.
We note that in the setup of the Theorem any action of a unipotent group on $X\backslash Y$
extends to an action on $X$ preserving $Y$; see Proposition \ref{1.7a} for a more general statement.
Moreover, our main result (see Theorem \ref{mthm1}) yields that the pointwise stabilizer $\mathop{\rm SAut}_Y(X)$
acts transitively on $X\backslash Y$.
This answers in affirmative a question posed in
\cite[4.22(2)]{AFKKZ}. Partial results in this direction were
obtained in Theorem 2.5 and Proposition 4.19 in \cite{AFKKZ}, see
also Proposition \ref{1.10} below.
Let us note that Theorem \ref{mthm} does not hold for subsets $Y$ of $X$ of codimension 1,
in general; see \cite[Proposition 4.13]{AFKKZ}. In this sense the result above is optimal.
For an affine space $X={\mathbb A}^n$, $n\ge 2$,
the flexibility of $X\backslash Y$
was first observed by M.\ Gromov in \cite[\S 2.1.5, p.\ 72,
Exercise (b$'$)]{Gr1}, cf.\ also 4.6(b) and 5.3(c) in \cite{Gr2}.
The transitivity of $\mathop{\rm SAut}_Y(X)$ in $X\backslash Y$ was proven in
this particular case by J.\ Winkelmann \cite[\S 2, Proposition
1]{Wi}.
The paper is organized as follows. In Section \ref{sec1} we recall
some useful facts from \cite{AFKKZ} and formulate, after
introducing necessary definitions, a stronger version of Theorem
\ref{mthm}, see Theorem \ref{mthm1}. As an important ingredient of the proof we show that for
any flexible variety $X$ one can find a subgroup
of $\mathop{\rm SAut}(X)$ acting with an open orbit on $X$, which is
generated by two locally nilpotent derivations $\delta_0, \delta_1$ along with their replicas $f_0\delta_0$, $f_1\delta_1$, where $f_0\in \ker \delta_0$ and $f_1\in\ker\delta_1$; see Proposition \ref{1.14}. In
Sections \ref{sec2} and \ref{sec3} we prepare the setup for the
proof of Theorem \ref{mthm1}.
The proof is then contained in Section \ref{sec4}.
It should be possible, after reading Section
\ref{sec1}, to go directly to Section \ref{sec4}
addressing results in Sections \ref{sec2} and \ref{sec3} when necessary.
Let us sketch the scheme of the proof of Theorem \ref{mthm}.
By a result in \cite{AFKKZ} the pointwise stabilizer $\mathop{\rm SAut}_Y(X)$ of $Y$ in
$\mathop{\rm SAut}(X)$ has an open orbit, say, $O$ in $X$. We consider a
completion $\bar X$ of $X$ compatible with partial quotients by
the two $\ensuremath{\mathbb{G}}_a$-subgroups $U_0=\exp({\Bbbk}\delta_0)$ and
$U_1=\exp({\Bbbk}\delta_1)$, where $\delta_0$ and $\delta_1$ are as in Proposition
\ref{1.14}. These quotients define on $\bar X$ two
${\mathbb P}^1$-fibrations $\bar\rho_0,\bar\rho_1$ with privileged
sections $D_0,D_1$, which lie on the boundary of $X$ in $\bar X$. Acting with
a suitable replica of $U_0$ one can move the part of the boundary $\partial Y\cap D_1$ to a fixed proper subset of $D_1$, and symmetrically for $U_1$ and $\partial
Y\cap D_0$, see Proposition \ref{prop-one}. Up to a controllable (and so negligible) proper subset
of $D_0\cup D_1$, this property is preserved when we iterate
subsequently actions by suitable replicas of $U_0$ and $U_1$, see Proposition \ref{prop-two}.
Using the transitivity property of the subgroup
$H\subseteq\mathop{\rm SAut}(X)$ generated by $U_0,U_1$ and their replicas, we
can move a given codimension $\ge 2$ subset $Y$ as in Theorem \ref{mthm} and, simultaneously, a given point $x\in X\backslash Y$ to a
generic fiber, say, $F$ of the ${\mathbb P}^1$-fibration $\bar\rho_0$ so
that $F$ does not meet $\partial Y\cap D_0$. Using the Transversality
Theorem from \cite{AFKKZ} we can achieve that $F$ does not meet $Y$ hence in total $F$ and
$\bar Y$ are disjoint. This enables us to find a $U_0$-invariant function
$f\in{\ensuremath{\mathcal{O}}}_X(X)$, which vanishes on $Y$ and not in $x$. The corresponding replica $U_0'$ of $U_0$ fixes $Y$ and moves $x$ along $F$. Since the
fiber $F$ is generic it meets the open orbit $O$ of $\mathop{\rm SAut}_Y(X)$, hence so
does $U_0'.x$. Thus $x$ belongs to $O$, and so $O=X\backslash Y$, as stated.
In order to prove Propositions \ref{prop-one} and \ref{prop-two} we develop in Sections \ref{sec2} and \ref{sec3} a machinery, which allows to reduce the proof to the model case of a standard birational transformation of a ruled surface induced by a $\ensuremath{\mathbb{G}}_a$-action. This reduction is the most lengthly part of the proof.
We thank M.\ Gizatullin for his interest in our work and in particular for his suggestion to treat in Theorem \ref{mthm1} also non-reduced subschemes $Y$ of $X$.
\section{Main theorem}\label{sec1}
\subsection{Basic notions and the main result} We let ${\mathbb A}^n={\mathbb A}^n_{\Bbbk}$
and $\ensuremath{\mathbb{G}}_a=\ensuremath{\mathbb{G}}_a({\Bbbk})$.
In the sequel $X$ denotes a quasi-affine variety over ${\Bbbk}$.
Thus $X$ can be embedded into an affine variety $X'= \operatorname{{\rm Spec}} B$ as an open subset.
We let $A={\ensuremath{\mathcal{O}}}_X(X)$ so that $B$ is a finitely generated ${\Bbbk}$-subalgebra of $A$. The embedding
$X\hookrightarrow X'$ factors as $X\to \operatorname{{\rm Spec}} A\to \operatorname{{\rm Spec}} B$. Furthermore $X\hookrightarrow \operatorname{{\rm Spec}} A$ is an open embedding.
We note that $A$ is in general not a finitely generated algebra over ${\Bbbk}$.
\begin{lem}\label{1.1}
With the notation
as above the following hold.
\begin{enumerate}[(a)]
\item Every action of an algebraic group on $X$ extends in a canonical way to $ \operatorname{{\rm Spec}} A$.
\item
Every subgroup $U\in {\ensuremath{\mathcal{U}}}(X)$ with infinitesimal generator $\delta$ yields a locally nilpotent ${\Bbbk}$-derivation on $A$.
\end{enumerate}
\end{lem}
\begin{proof}
(a) is standard, and (b) is a consequence of (a).
\end{proof}
Let us recall some notions and useful facts from \cite{AFKKZ}.
Given a subgroup $U\in{\ensuremath{\mathcal{U}}}(X)$ we let $\delta$
denote an infinitesimal generator of $U$; the latter is uniquely determined up to a nonzero constant factor.
Thus $\delta$ is a
locally nilpotent derivation of the algebra $A={\ensuremath{\mathcal{O}}}_X(X)$ such that
$U=\exp({\Bbbk} \delta)$. Geometrically $\delta$ can be viewed as a complete
vector field on $X$
with phase flow $u_t=\exp(t\delta)$, $t\in {\Bbbk}$. The tangent vector at the
point $x\in X$ given by this vector field
is denoted $\delta_x$.
\begin{lem}\label{1.2}
Let $Y$ be a closed (not necessarily reduced)
subscheme of the quasi-affine variety $X$ with ideal sheaf
${\ensuremath{\mathcal{I}}}\subseteq {\ensuremath{\mathcal{O}}}_X$, and consider the ideal of global sections $I={\ensuremath{\mathcal{I}}}(X)\subseteq A={\ensuremath{\mathcal{O}}}_X(X)$. Given
$U\in {\ensuremath{\mathcal{U}}}(X)$ with an infinitesimal generator $\delta$ the following hold.
\begin{enumerate}[(a)]
\item $\delta(A)\subseteq I$ if and only if $u|Y={\rm id}_Y$ for any $u\in U$.
\item $\delta(I)\subseteq I$ if and only if $u.Y\subseteq Y$ for any $u\in U$.\footnote{In the terminology
of \cite[p.\ 10]{Fr} this means that $I$ is an integral ideal.}
\end{enumerate}
\end{lem}
Let us fix the following notation.
\begin{nota}\label{1.3}\label{nota1.3}
(a) Let as before $X$ be a quasi-affine variety and $A={\ensuremath{\mathcal{O}}}_X(X)$ be its ring of regular functions.
If ${\mathfrak a}\subseteq A$ is the ideal of the complement $ \operatorname{{\rm Spec}} (A)\backslash X$, then the set of nonzero locally
nilpotent derivations $\delta$ of $A$ with $\delta({\mathfrak a})\subseteq {\mathfrak a}$ is denoted by
$$
\operatorname{{\rm LND}}(X).
$$
In view of Lemmas \ref{1.1} and \ref{1.2}(b) any element
$\delta\in \operatorname{{\rm LND}}(X)$ gives rise to a one-parameter subgroup
$U=\exp({\Bbbk} \delta)$ in ${\ensuremath{\mathcal{U}}}(X)$ and vice versa.
(b)
In order to deal with quasi-affine varieties we choose a
${\Bbbk}$-subalgebra $\Lambda$ of $A$ such that
the induced map $X\to \operatorname{{\rm Spec}} \Lambda$ is an open embedding.
Letting ${\mathfrak b}$ be the ideal of the complement $ \operatorname{{\rm Spec}}(\Lambda)\backslash X$ we let $ \operatorname{{\rm LND}}_\Lambda(X)$ denote the set
of all locally nilpotent derivations $\delta$ on
$\Lambda$ with $\delta({\mathfrak b})\subseteq {\mathfrak b}$. Every such derivation induces as before a one-parameter subgroup
$U\in {\ensuremath{\mathcal{U}}}(X)$ and consequently extends to an element in $ \operatorname{{\rm LND}}(X)$.
Thus $ \operatorname{{\rm LND}}_\Lambda(X)$ can be considered as a subset of $ \operatorname{{\rm LND}}(X)$.
(c) Given a collection ${\ensuremath{\mathcal{N}}}\subseteq \operatorname{{\rm LND}}_\Lambda(X)$ of nonzero
locally nilpotent derivations we let $G=G_{\ensuremath{\mathcal{N}}}=\langle{\ensuremath{\mathcal{N}}}\rangle$
be the subgroup of the group $\mathop{\rm SAut}(X)$
generated by the corresponding one-parameter unipotent subgroups $U=\exp({\Bbbk}\delta)$, $\delta\in {\ensuremath{\mathcal{N}}}$.
\end{nota}
\begin{rems}\label{emph} 1.
We emphasize that the subring $\Lambda$ of $A$ is not supposed to be finitely generated over ${\Bbbk}$
so that the choice $\Lambda=A$ is also possible.
In other words, we consider $X$ as
an open subset of an affine ${\Bbbk}$-scheme $ \operatorname{{\rm Spec}} \Lambda$,
which is not necessarily an algebraic variety, in contrast with \cite{AFKKZ};
see also
Remark \ref{1.7} below.
2. We observe as well that the $G$-action on $X$ as in \ref{nota1.3}(c)
extends to a $G$-action on the affine scheme $ \operatorname{{\rm Spec}}\Lambda$.\end{rems}
Let us recall some notation and standard facts.
\begin{sit} \label{1.4}
(1) Given a group $G=G_{\ensuremath{\mathcal{N}}}$ as before, the set of all one-parameter
unipotent subgroups of $G$ will be denoted by ${\ensuremath{\mathcal{U}}}(G)$, and the
set of all nonzero locally nilpotent derivations on $\Lambda$ generating
one-parameter subgroups of $G$ by $ \operatorname{{\rm LND}}_\Lambda(G)$ or simply $ \operatorname{{\rm LND}}(G)$.
(2) A {\em $\Lambda$-replica} of a
subgroup $U=\exp({\Bbbk}\delta)\in{\ensuremath{\mathcal{U}}}(G)$ is a subgroup $U_f=\exp({\Bbbk}
f\delta)\in \operatorname{{\rm LND}}_\Lambda(G)$, where $f\in \Lambda$ is in the kernel of $\delta$ (\cite{AFKKZ}).
(3) We say that ${\ensuremath{\mathcal{N}}}$
is {\em $\Lambda$-saturated} if ${\ensuremath{\mathcal{N}}}$ is closed under conjugation by
elements in $G$ and taking $\Lambda$-replicas i.e.,
$$
f\delta\in{\ensuremath{\mathcal{N}}}\qquad\forall\delta\in {\ensuremath{\mathcal{N}}}
\quad\text{and}\quad\forall f\in\ker_\Lambda\delta\,.
$$ Hereafter $\Lambda$ will be fixed, hence in
most cases we omit the symbol $\Lambda$ and say simply `replica' or `saturated'.
(4) A point $x\in X$ is
called {\em $G$-flexible} if $T_xX= \operatorname{{\rm Span}}({\ensuremath{\mathcal{N}}}(x))$, where ${\ensuremath{\mathcal{N}}}(x)$ denotes the set of tangent vectors $\delta_x$
with $\delta\in {\ensuremath{\mathcal{N}}}$. We say that $X$
is {\em $G$-flexible} if $X_{\rm reg}$ consists of $G$-flexible
points.
(5) Given a (not necessarily reduced) closed subscheme $Y$ in $X$ we let $G_{{\ensuremath{\mathcal{N}}},Y}$ denote
the subgroup of $G$ generated by all replicas $f\delta$ in ${\ensuremath{\mathcal{N}}}$ vanishing on $Y$ in the ideal theoretic sense,
see Lemma \ref{1.2}(a). Therefore
$G_{{\ensuremath{\mathcal{N}}},Y}\subseteq G_Y$, where $G_Y=\{g\in G:g|Y={\rm id}_Y\}$ stands for the
`pointwise' stabilizer of $Y$ in $G$ in the scheme theoretic sense.
\end{sit}
The following result is our main theorem.
\begin{thm}\label{mthm1} Let $X$ be a quasi-affine
variety of dimension $\ge 2$ and $X\hookrightarrow \operatorname{{\rm Spec}}\Lambda$ be an open embedding into an affine ${\Bbbk}$-scheme, see \ref{nota1.3}(b).
Let $G=\langle{\ensuremath{\mathcal{N}}}\rangle$
be a subgroup of the group $\mathop{\rm SAut}(X)$ generated by
a $\Lambda$-saturated set ${\ensuremath{\mathcal{N}}}$ of locally nilpotent
derivations as in \ref{1.4}. Suppose that $X$ is $G$-flexible.
If $Y$ is a closed (possibly non-reduced\footnote{The authors are grateful to M.~Gizatullin for the suggestion
to take also into account non-reduced subschemas $Y$ of $X$.}) subscheme
of $X$ of codimension $\ge 2$, then the
complement $X \backslash Y$ is $G_{{\ensuremath{\mathcal{N}}},Y}$-flexible.
\end{thm}
In the case of a smooth variety $X$ applying Theorem \ref{mthm1} to the group $G=\mathop{\rm SAut}(X)$ we get
Theorem \ref{mthm} from the Introduction.
\begin{rems}\label{1.7}
1. Since $G\subseteq \operatorname{{\rm Aut}} ( \operatorname{{\rm Spec}}\Lambda)$ the variety $X$ satisfies the requirements of Theorem \ref{mthm1} whenever so does its ($G$-stable) regular locus $X_{\rm reg}$. Therefore
it suffices to prove Theorem \ref{mthm1} under the assumption that $X$ is smooth.
This explains the necessity to fix a subring $\Lambda \subseteq A$ as in \ref{nota1.3}(b). Indeed, $A$ can be properly contained in $A'={\ensuremath{\mathcal{O}}}_{X_{\rm reg}}(X_{\rm reg})$. If instead of fixing $\Lambda$ we consider always LND's and their replicas with respect to the ring $A={\ensuremath{\mathcal{O}}}_X(X)$, then an $A'$-replica
is possibly not an $A$-replica and so the notion of saturated set
of derivations could change when passing from $X$
to $X_{\rm reg}$.
2. The viewpoint of the paper \cite{AFKKZ} is slightly different as it deals
with open subsets $X$ of affine algebraic varieties $Z= \operatorname{{\rm Spec}} B$,
and with subgroups $G$ of $\mathop{\rm SAut} (Z)$ stabilizing $X$.
It might happen in principle that although $ \operatorname{{\rm Aut}}(X)$
acts transitively on $X_{\rm reg}$ there is no subgroup $G$ of $ \operatorname{{\rm Aut}}(Z)$
acting transitively on $X_{\rm reg}$, whatever is the choice of an embedding of $X$ into an affine variety $Z$;
cf.\ Question \ref{1.9}
below. Thus {\em a priori} our viewpoint here is more general.
3. Working with quasi-affine varieties has yet another advantage: given a subgroup $G\subseteq \mathop{\rm SAut}(X)$,
in the subsequent proofs we may at any step replace $X$ by an open orbit of $G$. This considerably
simplifies our notation.
\end{rems}
It is worthwhile to note that if $X$ as in Theorem \ref{mthm1} is normal then the group
$\mathop{\rm SAut}(X\backslash Y)$ is in a natural way a subgroup of $\mathop{\rm SAut}(X)$.
This is a consequence of the following proposition.
\begin{prop}\label{1.7a}
Let $X$ be a normal quasi-affine variety and $Y\subseteq X$ a subset of codimension $\ge 2$.
Then every $\ensuremath{\mathbb{G}}_a$-action on $X\backslash Y$
extends to a $\ensuremath{\mathbb{G}}_a$-action on $X$ that stabilizes $Y$.
\end{prop}
\begin{proof}
A $\ensuremath{\mathbb{G}}_a$-action on $X\backslash Y$ corresponds to a locally nilpotent derivation on $A={\ensuremath{\mathcal{O}}}_X(X\backslash Y)$
such that the ideal, say, ${\mathfrak c}$ of the complement $Z\backslash (X\backslash Y)$ is stabilized by $\delta$,
where $Z= \operatorname{{\rm Spec}} A$. Because of $\mathop{\rm codim}_XY\ge 2$
the ${\Bbbk}$-algebras $A$ and ${\ensuremath{\mathcal{O}}}_X(X)$ coincide.
Consider the ideal ${\mathfrak a}\subseteq A$ of the complement $X^c=Z\backslash X$ and the ideal
${\mathfrak b}\subseteq A$ of the closure $\bar Y$ so that ${\mathfrak c}={\mathfrak a}\cap{\mathfrak b}$ is the ideal of the complement of
$X\backslash Y$ in $Z$. We have to show that ${\mathfrak a}$ is stabilized by $\delta$.
This is easy in the case that $A$ is finitely generated, thus $Z$ is an affine algebraic variety.
Indeed, if $U$ stabilizes $X^c\cup {\bar Y}$
then it stabilizes all irreducible components of that set
(see e.g.\ \cite[Proposition 1.14(b)]{Fr}), thus also $X^c$ and ${\bar Y}$ and
consequently their respective ideals.
In the general case, by Lemma \ref{1.8} below $A$ is a direct limit of its $\partial$-stable
finitely generated subalgebras $A_i$ such that $X$ embeds as an open subset into $ \operatorname{{\rm Spec}} A_i$.
Applying the first case to every $A_i$ the result follows easily.
\end{proof}
The following fact is an easy consequence of the Lemma of Cartier \cite[Chapt.\ I, \S 1]{Mu}.
\begin{lem}\label{1.8}
Given $\delta\in \operatorname{{\rm LND}}_\Lambda(X)$ and a finite dimensional ${\Bbbk}$-subspace $E\subseteq \Lambda$ there is
a finitely generated $\delta$-stable ${\Bbbk}$-subalgebra $\Lambda'\subseteq \Lambda$ containing $E$ such that $X$
embeds as on open subset of the affine variety $ \operatorname{{\rm Spec}} \Lambda'$.
\end{lem}
Since $X$ is quasi-affine there is a finitely generated subalgebra
$C$ of $B$ such that $X$ embeds as an open subset in $ \operatorname{{\rm Spec}} C$. We may suppose that $E$ contains a finite set of generators of $C$.
Since $\partial$ is locally nilpotent, the set $E'=\bigcup_{i\ge 1}\partial^i(E)$ is finite. Since it is also $\partial$-stable,
it generates a subalgebra $\Lambda'$ of $C$ with the desired properties.
We do not know whether this result remains true for any finite collection of locally nilpotent derivations.
More precisely:
\smallskip
\begin{ques}\label{1.9}
Suppose that ${\ensuremath{\mathcal{N}}}\subseteq \operatorname{{\rm LND}}(X)$ is a finite subset. Does there exist a finitely generated
${\ensuremath{\mathcal{N}}}$-stable ${\Bbbk}$-subalgebra $\Lambda'$ of $A={\ensuremath{\mathcal{O}}}_X(X)$ such that $X$ embeds into $ \operatorname{{\rm Spec}} \Lambda'$
as an open subset?
\end{ques}
\subsection{Transitivity versus flexibility on quasi-affine varieties}
Let $X= \operatorname{{\rm Spec}} A$ be an affine variety. By the main result in \cite{AFKKZ} the flexibility of $X$
is equivalent to the transitivity of $\mathop{\rm SAut}(X)$ on $X_{\mathop{\rm reg}}$, which in turn is equivalent to infinite
transitivity. In the sequel we need this and related facts in the more general setting
of quasi-affine varieties.
We will state the necessary results in the generality that we need below. The proofs
in \cite{AFKKZ} can be carried over to our more general quasi-affine setup without any difficulty.
Let us start with the main result of \cite{AFKKZ}, see 1.11 and 2.2 in {\em loc.cit.}
\begin{thm}\label{1.10}
Let $X$ be a smooth, quasi-affine variety of dimension $\ge 2$,
and let $G=\langle{\ensuremath{\mathcal{N}}}\rangle$ be a subgroup of $\mathop{\rm SAut}(X)$ generated by
a $\Lambda$-saturated set ${\ensuremath{\mathcal{N}}}\subseteq \operatorname{{\rm LND}}_\Lambda(X)$ as in Notation \ref{1.3} and \ref{1.4}.
Then the following are equivalent.
(i) $X$ is $G_{\ensuremath{\mathcal{N}}}$-flexible.
(ii) $G_{{\ensuremath{\mathcal{N}}}}$ acts transitively on $X$.
(iii) $G_{\ensuremath{\mathcal{N}}}$ acts infinitely transitively on $X$.
\end{thm}
In the proof of Theorem \ref{mthm1}
we use the following auxiliary results. They are established
in 2.5, 4.19, and 4.2 in \cite{AFKKZ} in the case of affine schemes $X$ and reduced subvarieties $Y$ of $X$.
The proofs given there carry immediately over to our more general situation.
\begin{prop}\label{1.11}
Let $X$ and $G_{\ensuremath{\mathcal{N}}}$ be as in Theorem \ref{1.10}, and let $Y$ be a closed subscheme of $X$.
If $X$ is $G_{\ensuremath{\mathcal{N}}}$-flexible\footnote{Equivalently, if $G_{\ensuremath{\mathcal{N}}}$ acts transitively on $X$.} then the following hold.
(1) The group $G_{{\ensuremath{\mathcal{N}}},Y}$
acts on $X\backslash Y$ with a dense open orbit, say, $O_{Y}$, which
consists of all $G_{{\ensuremath{\mathcal{N}}},Y}$-flexible points of $X\backslash Y$.
Consequently, the $G_{{\ensuremath{\mathcal{N}}},Y}$-action on $O_{Y}$ is
infinitely transitive.
(2) If $Y$ is finite then $O_{Y}=X\backslash Y$.
(3) If $x\in X$ then
the image of the tangent representation $G_{{\ensuremath{\mathcal{N}}},x}\to \operatorname{{\rm GL}}(T_xX)$
given by the differential coincides with the special linear group $ \operatorname{{\bf SL}}(T_xX)$.
\end{prop}
Finally we need the following interpolation result, see \cite[Theorem 4.14 and Remark 4.16]{AFKKZ}.
\begin{prop}\label{1.12}
Let $X$ and $G_{\ensuremath{\mathcal{N}}}$ be as in Theorem \ref{1.10}. If $G$ acts transitively on $X$
then for any
finite subset $Z\subseteq X$
there exists an automorphism $g \in G$ with $g(x)=x$ for $x\in Z$ and prescribed tangent map $d_xg\in \operatorname{{\bf SL}}(T_xX)$ at the points $x\in Z$.\footnote{In fact this proposition holds more generally for any finite collection of
$m$-jets provided these jets fix the corresponding points and
preserve local volume forms on $X$ at these points; see \cite[Remark 4.16]{AFKKZ}.}
\end{prop}
\subsection{ Generation of subgroups by LND's. }
Let as before $X$ be a quasi-affine algebraic variety of dimension $n\ge 2$
equipped with an open
embedding into an affine ${\Bbbk}$-scheme $ \operatorname{{\rm Spec}}\Lambda$, where $\Lambda\subseteq{\ensuremath{\mathcal{O}}}_X(X)$.
Given a set of locally nilpotent derivations ${\ensuremath{\mathcal{N}}}\subseteq
\operatorname{{\rm LND}}_\Lambda(X)$ we enrich it by adding all the $\Lambda$-replicas
of derivations in ${\ensuremath{\mathcal{N}}}$. Letting $\tilde {\ensuremath{\mathcal{N}}}$ be this enlarged set we consider
the subgroup $\lan\lan{\ensuremath{\mathcal{N}}}\ran\ran:=\langle\tilde{\ensuremath{\mathcal{N}}}\rangle$ of the group
$ \operatorname{{\rm Aut}}(X)$ generated by $\tilde{\ensuremath{\mathcal{N}}}$.
In this section we prove the following result.
\begin{prop}\label{1.14} Let $G=\langle{\ensuremath{\mathcal{N}}}\rangle\subseteq\mathop{\rm SAut}(X)$
be a subgroup generated by a
$\Lambda$-saturated set ${\ensuremath{\mathcal{N}}}$ of locally nilpotent
derivations. Suppose that $G$ acts transitively on $X$.
Then for any locally nilpotent derivation
$\delta_0\in {\ensuremath{\mathcal{N}}}$ one
can find another one
$\delta_1\in {\ensuremath{\mathcal{N}}}$ such that the subgroup
\begin{equation}\label{3.2a} H=\langle\langle \delta_0,\delta_1\rangle\rangle
\end{equation}
generated by $\delta_0$, $\delta_1$ and all their replicas acts with an open orbit on $X$.
\end{prop}
To deduce this result let us recall a few facts.
Let $U$ be a one-parameter unipotent subgroup
with an infinitesimal generator $\delta\in \operatorname{{\rm LND}}_\Lambda(X)$ (see Notation \ref{1.3}).
By assumption $X$ is contained as an open subset in $ \operatorname{{\rm Spec}} \Lambda$ and by Lemma \ref{1.8} even in
$ \operatorname{{\rm Spec}} \Lambda'$ for some $\delta$-stable finitely generated subalgebra $\Lambda'$ of $\Lambda$.
By the Rosenlicht Theorem (see \cite[Theorem
2.3]{PV})
one can find a finite set of $U$-invariant functions $f_1,\ldots,f_m\in \Lambda'^U$,
which separate general $U$-orbits. Let $B$ be the integral closure of the finitely generated ${\Bbbk}$-algebra
${\Bbbk}[f_1,\ldots, f_m]$. It is a standard result that $B$ is again finitely generated,
see e.g.\ \cite[Theorem 4.14]{Ei}.
\begin{defi}\label{partial quotient}
The normal affine variety $Q_U= \operatorname{{\rm Spec}} B$ will be called a
{\em partial quotient} of $X$ by $U$. In general it depends on the choice of the functions
$f_1,\ldots, f_m$.\footnote{Alternatively, one could use the {\em Winkelmann quotient} \cite{Wi2}.
This quasi-affine quotient is canonically defined, but has the disadvantage to be non-affine, in general.}
The inclusion $B\hookrightarrow {\ensuremath{\mathcal{O}}}_X(X)$
defines a dominant morphism $\rho_U:X\to Q_U$ such that the general fibers of $\rho_U$
are general orbits of $U$.
\end{defi}
\begin{proof}[Proof of Proposition \ref{1.14}]
Let as before $\rho_0:X\to Q_0$ be a partial quotient of $X$ by
$U^0$, where $\dim Q_0=n-1$. Since $n\ge 2$ there exists
$\sigma\in{\ensuremath{\mathcal{N}}}$ such that $\ker\sigma\neq\ker\delta_0$ and so $U^0$ and $U=\exp(\ensuremath{\mathbb{C}}\sigma)$
have different general orbits.
We can choose $x\in X$ such that the tangent vector $\delta_{0,x}$ of $\delta_0$ at $x$ is
nonzero, hence $\dim
U^0.x=1$. Choosing $x$ in an appropriate way there are points $x_1,\ldots,x_{n-1}$ on
the orbit $U^0.x$ such that the vectors
$v_i'=\sigma_{x_i}\in T_{x_i}X$ are all nonzero. Letting $q=\rho_0(x)\in Q_0$ we fix for
each $i=1\ldots, n-1$ a tangent vector $v_i\in T_{x_i}X$ in such a way
that the vectors $d\rho_0(v_i)\in T_qQ_0$, $i=1,\ldots,n-1$, generate the tangent space
$T_qQ_0$ to $Q_0$ at $q$.
For every $i=1,\ldots,n-1$ we can choose a $1$-jet
of a local automorphism at the point $x_i$ that fixes $x_i$ and
sends $v_i'$ to $v_i$. This amounts to choosing
$\alpha_i\in \operatorname{{\bf SL}}(T_{x_i}X)$ such that $\alpha_i(v_i')=v_i$.
According to Proposition \ref{1.12} one can
interpolate these jets by an automorphism, say, $\alpha\in G$ such
that $\alpha(x_i)=x_i$ and $d\alpha(v_i')=v_i$ for $
i=1,\ldots,n-1$. Replacing $U$ by
$$
U^1=\alpha\circ U\circ \alpha^{-1}=\exp(\ensuremath{\mathbb{C}}\delta_1)\in{\ensuremath{\mathcal{U}}}(G)\,.
$$
we obtain a one-parameter unipotent subgroup with tangent vector $v_i$ at $x_i$, $i=1,\ldots, n-1$.
We claim that the locally nilpotent derivation
$\delta_1$ satisfies our requirement. Indeed, $\delta_1\in{\ensuremath{\mathcal{N}}}$ since
${\ensuremath{\mathcal{N}}}$ is saturated and so, in particular, is closed under conjugation in $G$.
Consider the conjugated one-parameter subgroups
$$
U^1_i=\alpha_i^{-1}\circ U^1\circ \alpha_i=\exp(\ensuremath{\mathbb{C}}\sigma_i)\in{\ensuremath{\mathcal{U}}}(H),\quad
i=1,\ldots,n-1\,,
$$
where $\alpha_i\in U^0$ is an element which maps $x$ to $x_i$. Here $H$ is as in (\ref{3.2a}) and
$\sigma_i$ is a conjugate of $\delta_1$ under the action of $H$ for $ i=1,\ldots,n-1$. For any
$i$ in this range the vector $u_i=d\alpha_i(v_i)$ is tangent to the
orbit $U^1_i.x$ at the point $x\in X$. Furthermore, the
vectors $d\rho_0(u_i)=d\rho_0(v_i)\in T_q Q_0$, $i=1,\ldots,n-1$
still generate $T_qQ_0$. Hence the vectors
$$
u_0=\delta_{0,x},\,\,u_1=\sigma_{1,x},\,\,\ldots,\,\,u_{n-1}
=\sigma_{n-1,x}\in T_xX
$$
span $T_xX$ as well.
Consequently, $x$ is an $H$-flexible point and so the $H$-orbit
$H.x$ is open and dense in $X$ (see \cite[Corollary 1.11(a)]{AFKKZ}).
\end{proof}
\section{$m$-blowups, tangency, and $m$-contractions}\label{sec2}
This section is technical; we use its results and notions (see especially Definitions \ref{x.5} and \ref{x.8} and Proposition \ref{x.13})
in the proof of Proposition \ref{prop-one} in the next section.
\begin{sit}\label{x.1}
In the sequel we deal with rational maps $\begin{diagram} g:X&\rDotsto& Y\end{diagram}$ which
fit into a diagram
\begin{diagram}
&& \hat X\\
&\ldTo<{h}&&\rdTo>{g'}\\
X&&\rDotsto^g && Y
\end{diagram}
where $h$ is a sequence of blowups and $g'$ is a proper morphism. This somewhat restricted class of
rational maps is suitable for our purposes. Given subsets $A\subseteq X$ and $B\subseteq Y$ we let
$$
g(A)=g'(h^{-1}(A))\quad\mbox{and}\quad g^{-1}(B)=h(g'^{-1}(B))
$$
denote the total image and preimage, respectively.\footnote{These notions should be treated with
caution, because they are not compatible with composition of rational maps.} Since any two resolutions
of the indeterminacy set are dominated by a third one, the total image
and the total preimage are well defined.
\end{sit}
\subsection{$m$-blowups and tangency}
In the next Definition we introduce a setup which is used repeatedly in this and the next section.
\begin{defi}\label{x.2}
Let $X$ be an algebraic variety and $C$, $D$ be divisors in $X$, which are Cartier near $C\cap D$.
The {\em $m$-blowup $\sigma_m:X_m\to X$ of $D$ along $C$} is defined recursively as follows.
With $X_0=X$ we let $X_1$ be the blowup of $X$ along the subscheme $C\cap D$.
If $X_{m-1}$ is already defined for some $m\ge 2$,
then we let $X_{m}\to X_{m-1}$ be the blowup along $D^{(m-1)}\cap
E_{m-1}$, where $D^{(m-1)}$ is the proper transform of $D$ in $X_{m-1}$ and $E_{m-1}$ the exceptional
set of the previous blowup $X_{m-1}\to X_{m-2}$.
In the following we call the proper transforms
$$
E'_1,\ldots, E_m'\subseteq X'= X_m
$$
of the exceptional sets $E_i$ of $X_i\to X_{i-1}$ the {\em exceptional sets of the $m$-blowup} of $D$
along $C$. The proper transforms of $C$ and $D$ will always be denoted $C',D'$, respectively.
\end{defi}
\begin{exa}\label{x.3}
Suppose that $S$ is a complete smooth surface and $C\cap D=\{p\}$, where the intersection
is transversal. Then the dual graph of
$C'\cup E_1'\cup\ldots\cup E_m'\cup D'$ is a linear chain:
\begin{equation}\label{eqegraph} \cou{C^2-1}{C'}\vlin{11.5}
\cou{-2}{E'_1}\vlin{8.5}\ldots\vlin{8.5}
\cou{-1}{E'_m}\vlin{14}\cou{D^2-m}{D'}
\qquad.\end{equation}
\end{exa}
Let us consider next the effect of an $m$-blowup as in Definition \ref{x.2} on the boundary of
a closed subset of $X$.
\begin{prop}\label{x.4}
We keep the notation and assumptions as in Definition \ref{x.2}.
Given a closed subset $Y\subseteq X$ we let $Y'$ denote its proper transform in $X'$ and $\partial Y'$
its boundary $\partial Y'= Y'\cap \sigma_m^{-1}(C\cup D)$.
Then with $P= \overline{Y\cap D\backslash C}$, for $m\gg 0$
$$
\partial Y'\subseteq E_1'\cup\ldots\cup E_{m-1}' \cup \sigma_m^{-1}(P)\,.
$$
\end{prop}
\begin{proof}
The assertion is local around points in $C\cap D\backslash P$. Thus we may assume
that $P=\emptyset$, $X= \operatorname{{\rm Spec}} A$ is affine, and that $D=V(x)$, $C=V(y)$
with functions $x,y\in A$. The subset
$$
U'=X'\backslash \bigcup_{i=0}^{m-1} E'_i
$$
of $X'$ is affine with coordinate ring
$$
A'= A[u]\,,\quad\mbox{where }u=x/y^m,
$$
cf.\ Lemma \ref{x.10} below for the special case of surfaces. Furthermore
\begin{equation}\label{aux6a}
U'\cap E'_m=\{y=0\}\quad\mbox{and}\quad U'\cap D'=\{u=0\}.
\end{equation}
If $I\subseteq A$ is the ideal of $Y$ then $B=A/I$ is the affine coordinate ring of $Y$.
Since $\overline{Y\cap D\backslash C}=\emptyset$ the set $Y\cap D$ is contained in $C\cap D$
and so the localization $(B/xB)_y$ is zero. Hence there exists a natural number $m$
such that $y^{m-1}\in xB$. In other words, we can find $a\in A$ such that
\begin{equation}\label{aux6b}
y^{m-1} -a\cdot x \in I \,.
\end{equation}
In the blowup ring $A'$
the ideal $I'$ of $ Y'$ is given by
$$
I'=\{g\in A'\mid \exists k\in {\mathbb N}: y^kg\in IA'\}.
$$
Since $u=x/y^m$ condition \eqref{aux6b} can be rewritten in the form
$$
y^{m-1}\cdot\left(1- yau\right)\in IA'.
$$
Hence $1- yau\in I'$. This shows that in the affine coordinate ring $B'=A'/I'$ of $U'\cap Y'$
the residue classes of $y$ and $u$ are units. In view of \eqref{aux6a} this implies that
$$
U'\cap Y'\cap E'_m=\emptyset\quad\mbox{and}\quad
U'\cap Y'\cap D'=\emptyset,
$$
which immediately yields the required result.
\end{proof}
\begin{defi}\label{x.5}
We say that a closed subset $Y$ of $X$ is {\em at most $m$-tangent} to $D$ along $C$,
if the conclusion of Proposition \ref{x.4} holds with this particular value of $m$.
The subset $N=C\cap \overline{Y\cap D\backslash C}$ of $C\cap D$ will be called the {\em defect set}.
\end{defi}
We note that if $Y$ is at most $m$-tangent to $C$ along $D$ then it is also at most $m'$-tangent
to $C$ along $D$ for all $m'\ge m$.
The following observation is important.
\begin{lem}\label{x.6}
If $\mathop{\rm codim}_XY\ge 1$ and $Y\backslash D$ is dense in $Y$ then the defect set $N$ is nowhere
dense in $C\cap D$.
\end{lem}
\begin{proof}
If $\mathop{\rm codim}_XY\ge 1$ then the set $Y\cap D$ has
codimension $\ge 1$ in $D$. Hence its closure
cannot contain any component of $C\cap D$.
\end{proof}
\begin{rem}\label{x.7}
In the setup of Proposition \ref{x.4} suppose that $(Y_s)_{s\in S}$ is a family of proper
closed subsets of $X$. Then there is a natural $m$ such that $Y_s$ is at most $m$-tangent
to $D$ along $C$ for any $s\in S$.
This follows easily from the fact that the construction of Proposition \ref{x.4} can be
done al least generically in the given family and that it is then compatible with restriction
to the general fiber. More precisely, one can find an open dense subset $U\subseteq S$
so that all fibers $Y_s$ are at most $m$-tangent to $D$ along $C$ with $m$ independent
of $s\in U$, and with a defect set $N_s=C\cap\overline{Y_s\cap D\backslash C}$.
Restricting the family to $S'=S\backslash U$ and applying induction on $\dim S$,
we may assume that $Y_s$ is at most $m$-tangent to $D$ along $C$ for any $s\in S'$.
Hence the assertion follows.
\end{rem}
\subsection{$m$-contractions}
\begin{defi}\label{x.8}
Let $C$, $D$ be divisors on an the algebraic variety $X$,
which are Cartier near $C\cap D$. Consider a birational map $\begin{diagram} g:X&\rDotsto & X\end{diagram}$
and a resolution of the indeterminacy set of $g$ which factors through the $m$-blowup
$\sigma_m:X'=X_m\to X$ of $D$ along $C$, see Definition \ref{x.2}:
\begin{diagram}
&& \hat X\\
&\ldTo<{h_m}&\dTo>h &\rdTo>{g'}\\
X'=X_m& \rTo^{\sigma_m} &X&\rDotsto^g & X
\end{diagram}
$g$ is called an {\em $m$-contraction for $C$ along $D$} if the following hold.
\begin{enumerate}
\item $g$ is biregular in the points of $X\backslash C$;
\item with $g_m=g\circ\sigma_m$, the total image\footnote{See \ref{x.1}}
$g_m(C'+E_1'+\ldots+ E_{m-1}')$ is a subset of $D$, where $E_1', \ldots, E_m'$ are
as in Definition \ref{x.2}.
\end{enumerate}
\end{defi}
Clearly, an $m$-contraction for $C$ along $D$ is also an $m'$-contraction for $C$ along
$D$ for any $m'\le m$.
The following example is important and serves as a model case.
\begin{nota}\label{x.9}
Let $\Gamma=(\Gamma,o)$ be a germ of a smooth affine curve with a uniformizing parameter $u$ such that $u(o)=0$,
and let $d(u)$ denote a nowhere vanishing function on $\Gamma$. We consider homogeneous coordinates $(\zeta_1:\zeta_2)$ on ${\mathbb P}^1$ and an affine coordinate
$v=\zeta_1/\zeta_2$ on ${\mathbb A}^1={\mathbb P}^1\backslash \{(1:0)\}$. The product $S:=\Gamma\times{\mathbb P}^1$ is a ${\mathbb P}^1$-fibered surface over
$\Gamma$. Its fiber, say, $C$ over $o\in \Gamma$ and the section
$D=\Gamma\times \{(0:1)\}\subseteq \Gamma\times{\mathbb A}^1$ can be described in coordinates by
$$
C=\{u=0\}\quad\mbox{and}\quad D=\{v=0\}.
$$
Let us study the rational map $g_m: S\dashrightarrow S$, where $m\in{\mathbb N}$,
given in affine coordinates by
\begin{equation}\label{hm}
g_m(u,v)=
\left( u,\,\,\frac{u^mv}{d(u) v+u^m} \right)\,.
\end{equation}
Its indeterminacy set consists of the intersection point $C\cap D=\{u=v=0\}$, which will be
denoted by $\bar 0$.
\end{nota}
\begin{lem}\label{x.10}
Let
\begin{diagram}[small]
& &{S'} & &\\
&\ldTo^{\sigma_m} & &
\rdTo>{{g'}_m}& \\
S &&\rDashto^{g_m} && S\\
\end{diagram}
\noindent
be the minimal resolution of indeterminacies of $g_m$, where $\sigma_m$ is a sequence of blowups
and $g'_m$ is a morphism.
Then the total transform of $C+D$ on $S'$ under $\sigma_m$ has weighted dual graph
\medskip
\begin{equation}\label{aux8a} \cou{-1}{C'}\vlin{8.5}
\cou{-2}{E'_1}\vlin{8.5}\ldots\vlin{8.5}
\cou{\!\!\!\!\!\!\!\!\!\!\!\!-2}{E'_m}\nlin\cshiftup{D'}{-m}
\vlin{8.5}\ldots\vlin{8.5}\cou{-2}{E'_{2m-1}}\vlin{8.5}\cou{-1}{E'_{2m}} \quad,\end{equation}
\noindent
where $C'$ and $D'$
are the proper transforms of $C$
and $D$, respectively.
The map $\sigma_m$ contracts the components
$E'_1,\ldots,E'_{2m}$ to the origin
$\bar 0\in S$, while $g'_m$ contracts the curves
$C',E'_1,\ldots,E'_{2m-1}$ to $\bar 0\in S$. Furthermore $g'_m(D')=D$ and $g'_m(E'_{2m})=C$.
\end{lem}
\begin{proof}
Letting $v_0=v$ we define a sequence of coordinates charts $(u,v_i)$
on $S'$, $i=0,\ldots,2m$, so that the
$2m$ blowing-downs over the origin
with exceptional curves $E'_1,\ldots,E'_{2m}$ that constitute the map
$$
\sigma:(u,v_{2m})\mapsto (u,v_{2m-1})\mapsto\ldots
\mapsto(u,v_{1})\mapsto (u,v)\,
$$
can be described by the formulae
\begin{equation}\label{x.10a}
v_1=v/u,\quad v_2=v_1/u=v/u^2,\quad\ldots,\quad
v_m=v_{m-1}/u=v/u^m\,,
\end{equation}
and
\begin{equation}\label{x.10b}
v_{m+1}=(1 +d(u) v_m)/u,\quad v_{m+2}=v_{m+1}/u,\quad
\ldots,\quad v_{2m}=v_{2m-1}/u=(1+d(u)v_m)/u^m\,.
\end{equation}
The map $g_m$ can be written in these coordinate charts as
$$
(u,v)\mapsto \left(u,\,\frac{u^mv}{d(u)v+u^m}\right)=
\left(u,\, \frac{u^mv_1}{d(u)v_1+u^{m-1}}\right)=\ldots
$$
$$
\ldots =
\left(u,\, \frac{u^m v_m}{1+d(u) v_m}\right)
=\left(u,\, \frac{d(u) u^mv_{m+1}- u^{m-1}}{v_{m+1}}\right)=\ldots=
\left(u,\,\frac{d(u) u^m v_{2m}-1}{v_{2m}}\right)\,.
$$
Hence the curve $E'_i$ given in the chart $(u,v_i)$ by equation $u=0$ is contracted under
$g'_m$ for every $i=0,\ldots,2m-1$,
while the curve $E'_{2m}$
given by the same equation in the chart $(u,\, v_{2m})$
maps birationally onto the curve $C$ in $S$. Now the assertion
follows.\end{proof}
An immediate consequence is the following corollary.
\begin{cor}\label{x.11}
The birational map $g_m$ in \eqref{hm} is an $m$-contraction of $C$ along $D$.
\end{cor}
Let us note that $g_m$ is not an ($m+1$)-contraction of $C$ along $D$.
This example can be generalized to higher dimensions as follows.
\begin{nota}\label{x.11a}
Instead of a curve $\Gamma$ in \ref{x.9} we consider now a smooth affine algebraic variety $Q$
and a smooth divisor $T\subseteq Q$ given by the equation $\{u=0\}$, where $u\in {\ensuremath{\mathcal{O}}}_Q(Q)$.
The product $X=Q\times {\mathbb P}^1 $ is ${\mathbb P}^1$-fibered over $Q$ and contains the divisors
$$
C=T\times {\mathbb P}^1\quad\mbox{and}\quad
D=Q\times \{(0:1)\}\subseteq Q\times{\mathbb A}^1,
$$
where we equip ${\mathbb P}^1$ with homogeneous coordinates $(\zeta_1:\zeta_2)$. As before
$v=\zeta_1/\zeta_2$ stands for an affine coordinate on ${\mathbb A}^1={\mathbb P}^1\backslash \{(1:0)\}$.
Thus we have
$$
C=\{u=0\}\quad\mbox{and}\quad D=\{v=0\}.
$$
\end{nota}
\begin{lem}\label{x.11b}
Given a nowhere vanishing function $d(q)$ on $Q$ and $m\in{\mathbb N}$ the rational map
\begin{equation}\label{x.11aa}
g_m: X\dashrightarrow X\,,\quad \mbox{where}\quad
g_m(q,v)=
\left( q,\,\,\frac{u(q)^mv}{d(q) v+u(q)^m} \right)\,,
\end{equation}
is an $m$-contraction of $C$ along $D$.
\end{lem}
\begin{proof}
A resolution
\begin{diagram}[small]
& &{X'} & &\\
&\ldTo^{\sigma_m} & &
\rdTo>{{g'}_m}& \\
X &&\rDashto^{g_m} && X\\
\end{diagram}
of the indeterminacy points of $g_m$ can be obtained (with obvious changes) by the same sequence
of blowups as in the proof of Lemma \ref{x.10}. Letting $v_0=v$ we define a sequence of coordinates
charts $(q,v_i)\in U_i=Q\times {\mathbb A}^1$
on $X'$, $i=0,\ldots,2m$, so that the
$2m$ blowdowns over $C\cap D$
with exceptional divisors $E'_1,\ldots,E'_{2m}$ that constitute the map
$$
\sigma:(q,v_{2m})\mapsto (q,v_{2m-1})\mapsto\ldots
\mapsto(q,v_{1})\mapsto (q,v)\,
$$
can be described by the formulae in \eqref{x.10a} and \eqref{x.10b}, where $u$ is now the function $u(q)$.
With the same calculation as before the map $g_m$ can be written in these coordinate charts as
$$
(q,v)\mapsto
\left(q,\,\frac{d(q) u(q)^m v_{2m}-1}{v_{2m}}\right)\,.
$$
As in the proof of \ref{x.10} the exceptional set $E'_i$ is given in the chart $U_i$ by the equation $u=0$,
and it is contracted under $g'_m$ to the subset $C\cap D$ for every $i=0,\ldots,2m-1$.
Finally, the exceptional set $E'_{2m}$
given by $\{u=0\}$ in the chart $U_{2m}$
maps under $g_m'$ isomorphically onto the divisor $C$ in $X$. Since
the divisors $C', E_1',\ldots, E_{m-1}'$ in $X'$ are contracted under $g_m'$ to $C\cap D$, the result follows.
\end{proof}
Next we show that $m$-contractions are compatible with certain blowups.
\begin{prop}\label{x.12}
Let $X$ be an algebraic variety and $C$, $D$ be connected divisors on $X$, which are
Cartier near $C\cap D$.
Let $\begin{diagram} g: X&\rDotsto &X\end{diagram} $ be an $m$-contraction of $C$ along $D$ and $p:Z\to X$ be
a modification, which is an isomorphism over $D\cup (X\backslash C)$. Then the rational map
$\begin{diagram} f: Z&\rDotsto &Z\end{diagram} $ induced by $g$ is an $m$-contraction of $C_Z=p^{-1}(C)$ along
$D_Z=p^{-1}(D)\cong D$.
\end{prop}
\begin{proof}
Let $X_m\to X$ and $Z_m\to Z$ be the $m$-blowups of $X$ and $Z$, respectively. Since $p$ is an
isomorphism in the points near $D$, the exceptional sets $E_1',\ldots, E_m'$
of $X_m\to X$ can be identified in a natural way with the exceptional sets, say,
$E_{1,Z}',\ldots, E_{m,Z}'$
of $Z_m\to Z$. Consider the composed rational maps
\begin{diagram}
Z'_m &&\rDotsto^{f_m} && Z\quad\mbox{and}\quad X'_m &&\rDotsto^{g_m} && Z\,.
\end{diagram}
and a diagram
\begin{diagram}
&& \hat Z\\
&\ldTo<{h_m}&&\rdTo>{f_m'}\\
Z'_m &&\rDotsto^{f_m} && Z\\
\dTo<{p'} &&&&\dTo>p\\
X'_m &&\rDotsto^{g_m} && X
\end{diagram}
where $\hat Z$ is a resolution of the indeterminacy locus of $f_m$
and then also of $g_m$. By our assumption the set
$$
(p'\circ h_m)^{-1}(C'\cup E_1'\cup\ldots\cup E'_{m-1})
=h_m^{-1}(C_Z'\cup E_{1,Z}'\cup\ldots\cup E'_{m-1,Z})
$$
is contracted under $p\circ f_m'$ to a subset of $D$. Since $p$ is an isomorphism near $D$
the latter set is already contracted under $f_m'$ to a subset of $D$. This proves the assertion.
\end{proof}
Let us now study the effect of an $m$-contraction of $C$ along $D$ on the boundary
of a closed subset $Y$ of $X$.
\begin{prop}\label{x.13}
Let $X$ be an algebraic variety and $C$, $D$ divisors on $X$, which are Cartier
near $C\cap D$.
Assume that $\begin{diagram} g:X&\rDotsto& X\end{diagram}$ is an $m$-contraction of $C$ along $D$ and
that $Y\subseteq X$
is a closed subset, which is at most $m$-tangent to $C$ along $D$ with defect set
$N=C\cap \overline{Y\cap D\backslash C}$.
Then the proper image $\hat Y$ of $Y$ under $g$ satisfies
$$
\partial\hat Y\subseteq D\cup g(N)\,,
$$
where $g(N)$ is the total image of $N$ and $\partial \hat Y$ denotes the intersection of $\hat Y$
with $D\cup C$.
\end{prop}
\begin{proof}
Let $\sigma_m:X'=X_m\to X$ be the $m$-blowup of $C$ along $D$ with exceptional sets
$E'_1,\ldots, E'_m$ and consider the composition $\begin{diagram} g_m=g\circ\sigma_m:X'&\rDotsto&X\end{diagram}$.
We can find a resolution of the indeterminacy locus of $g_m$
\begin{diagram}
&& \hat X\\
&\ldTo<{h_m}&&\rdTo>{g_m'}\\
X'&&\rDotsto^{g_m} && X\,.
\end{diagram}
Since $Y$ is at most $m$-tangent to $C$ along $D$, the boundary $\partial Y'$ of the proper transform $Y'$
of $Y$ in $X'$ satisfies
$$
\partial Y'\subseteq E_1'\cup\ldots\cup E_{m-1}'\cup \sigma_m^{-1}(P)\,,
$$
where $P=\overline{Y\cap D\backslash C}$, see Proposition \ref{x.4}.
By condition (2) in Definition \ref{x.8}
$$
h_m^{-1}(C'\cup E_1'\cup\ldots\cup E_{m-1}')
$$
is contracted under $g_m'$ to a subset of $D$. Hence
$$
g_m'(h_m^{-1}(\partial Y'))\subseteq D\cup g_m'(h_m^{-1}(\sigma_m^{-1}(P))))
=D\cup g(P).
$$
Since $g_m'$ is proper the set on the right is easily seen to contain $\partial \hat Y$, as stated.
\end{proof}
\section{Replicas as $m$-contractions}\label{sec3}
\begin{nota}\label{5.1}
(a) Let $X$ be a smooth quasi-affine algebraic variety and $G_{\ensuremath{\mathcal{N}}}$ a group of automorphisms on $X$
generated by a set of $\Lambda$-saturated locally nilpotent derivations
${\ensuremath{\mathcal{N}}}\subseteq \operatorname{{\rm LND}}_\Lambda(X)$,
see Notation \ref{1.3} and \ref{1.4}. Suppose that $G_{\ensuremath{\mathcal{N}}}$ acts transitively on $X$.
(b) We choose two locally nilpotent derivations $\delta$, $\delta_0\in \operatorname{{\rm LND}}_\Lambda(X)$ such that
$$
\ker\delta\ne \ker\delta_0 .
$$
Let $U$, $U^0$ denote the associated one-parameter subgroups and choose partial quotients
$$
\rho:X\to Q \quad\mbox{and}\quad \rho_0:X\to Q_0
$$
as introduced in \ref{partial quotient}.
(c) We can embed $Q$ and $Q_0$ into normal projective varieties $\bar Q$ and $\bar Q_0$,
respectively. Let ${\bar X}$ be a smooth projective completion of $X$. After blowing up ${\bar X}$ in the boundary $\partial X={\bar X}\backslash X$, if necessary, we may extend $\rho$ and $\rho_0$ to morphisms
\begin{diagram}
{\bar X}&\rTo^{\bar\varrho_0}& {\bar Q}_0\\
\dTo>{\bar\varrho}\\
{\bar Q}
\end{diagram}
The general fiber of $\rho$ is an orbit of $U$
isomorphic to ${\mathbb A}^1$. Clearly
$$
\bar\varrho^{-1}(q)\cong {\mathbb P}^1
$$
for a general point $q\in Q$. Hence there is a unique divisor $D\subseteq \bar X\backslash X$
which maps birationally onto $\bar Q$.
Similarly there is a unique divisor $D_0$ in ${\bar X}\backslash X$
mapping birationally
onto ${\bar Q}_0$. Thus both $D$ and $D_0$ are contained in the boundary $\partial X={\bar X}\backslash X$. \end{nota}
The following observations will be important.
\begin{lem}\label{5.2}
\begin{enumerate}[(1)]
\item Let $\varphi\in \ker \delta \backslash \ker \delta_0$ be a regular function on $X$.
Then $\varphi$ is a rational function on ${\bar X}$ with poles at general points of $D_0$.
\item We have
$$
\bar\varrho(D_0)\subseteq {\bar Q}\backslash Q
\quad\mbox{and}\quad
\bar\varrho_0(D)\subseteq {\bar Q}_0\backslash Q_0\,.
$$
In particular, $D\ne D_0$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) Since $D_0\to {\bar Q}_0$ is dominant, an orbit closure $\overline{H_0 .x}$ of a general point $x\in X$
meets $D_0$ at a general point $\bar x\in D_0$. Let us consider
$\varphi$ as a rational map
$\begin{diagram} {\bar X}&\rDashto & {\mathbb P}^1\end{diagram}$. Since the indeterminacy set of $\varphi$ on ${\bar X}$ is of codimension
at least 2, $\varphi$ is regular on the orbit closure $\overline{H_0 .x}\cong {\mathbb P}^1$ for a general $x\in X$.
Since $\varphi\not\in \ker \delta_0$ this map is not constant on general orbits of $H_0$.
In particular it restricts to a dominant morphism $\varphi:\overline{H_0 .x}\to{\mathbb P}^1$ such that
$\varphi(\bar x) = \infty$.
(2) It is sufficient to prove the first part. If $\bar\varrho(D_0)\cap Q\ne\emptyset$ then a function
$\varphi\in {\ensuremath{\mathcal{O}}}(Q) \backslash \ker \delta_0$ would be holomorphic in a general point of $D_0$ contradicting (1).
\end{proof}
\begin{lem}\label{5.3}
After blowing up the boundaries $\partial X={\bar X}\backslash X$ and $\partial Q={\bar Q}\backslash Q$
suitably we can achieve that
\begin{enumerate}[(a)]
\item $T=\bar\varrho(D_0)$ is a divisor in ${\bar Q}$, and
\item ${\bar X}$, $D$ and $D_0$ are smooth.
\end{enumerate}
\end{lem}
\begin{proof}
(a) By Lemma \ref{5.2}(2) $T$ sits in the boundary of ${\bar Q}$.
According to a theorem of Zariski, see \cite{Za} and \cite[Theorem 1.3]{Kol}, there is a blowup $\bar Q'\to\bar Q$ with a center in
$\bar\varrho(D_0)$ such that the proper transform of $D_0$ in ${\bar X}_{{\bar Q}'}$ maps onto a divisor in ${\bar Q}'$.
Thus replacing $\bar Q$ by $\bar Q'$ we can achieve that $T$ is a divisor.
Since $X$ is smooth and does not meet $D\cup D_0$, by a suitable blowup of the boundary ${\bar X}\backslash X$
we can achieve that (b) holds.
\end{proof}
\begin{lem}\label{5.3a} \label{}
There is a closed subset $B_0$ of ${\bar Q}$ with $\mathop{\rm codim}_{\bar Q} B_0\ge 2$ such that the following hold.
\begin{enumerate}[(a)]
\item $ \operatorname{{\rm Sing}} {\bar Q}\cup \operatorname{{\rm Sing}} T\subseteq B_0$.
\item $D\to {\bar Q}$ is an isomorphism in the points $D\backslash \bar\varrho^{-1}(B_0)$.
\item ${\bar X}\to {\bar Q}$ is flat in the points over ${\bar Q}\backslash B_0$.
\end{enumerate}
\end{lem}
\begin{proof}
(a) can be satisfied as ${\bar Q}$ is normal and $T$ is reduced. Since $D\to {\bar Q}$ is a birational map, also (b) can be achieved.
(c)
By the theorem on generic flatness \cite[Theorem 14.4]{Ei}
there is a proper closed subset $E$ in ${\bar Q}$ such that $\bar\varrho$
is flat in the points over ${\bar Q}\backslash E$. Applying the theorem on generic flatness again gives that
the restricted map $\bar\varrho|E:\bar\varrho^{-1}(E)\to E$ is flat over a subset $E\backslash B'$ of $E$, where $B'$
is a nowhere dense closed subset of $E$. Using Corollary 6.9 in \cite{Ei} it follows that $f$ is flat over
the set ${\bar Q}\backslash B''$, where
$$
B''=B'\cup \{s\in E: E \mbox{ is not a Cartier divisor in ${\bar Q}$ at }x \}
$$
Since ${\bar Q}$ is normal this set has codimension $\ge 2$ in ${\bar Q}$.
Adding $B''$ to $B_0$, also (c) is satisfied.
\end{proof}
The following facts should be well known; in lack of a reference
we provide a brief argument.
\begin{lem}\label{5.4}
Let $p:S\to \Gamma$ be a ${\mathbb P}^1$-fibration of a smooth surface $S$ over a smooth affine curve $\Gamma$
admitting a smooth section $D\subseteq S$ so that $D\cong \Gamma$. Then for any point $t\in\Gamma$
the fiber $F=p^{-1}(t)$ over $t$ is a tree of rational curves. Furthermore the following hold.
\begin{enumerate}[(a)]
\item
If $\{x\}=F\cap D$ then $h^0(F,{\ensuremath{\mathcal{O}}}_F(x))=2$ and $H^i(F,{\ensuremath{\mathcal{O}}}_F(x))=0$ for $i\ge 1$.
\item The sheaf ${\ensuremath{\mathcal{O}}}_F(x)$ is generated by its global sections.
\item If $s_0, s_1\in H^0(F,{\ensuremath{\mathcal{O}}}_F(x))$ is a basis, then the map $(s_0:s_1): F\to {\mathbb P}^1$ is an isomorphism near $x$.
\end{enumerate}
\end{lem}
\begin{proof}
Blowing down successively $(-1)$-curves in the fibers of $p$ not meeting $D$ we obtain a locally trivial ${\mathbb P}^1$-bundle
${\ensuremath{\mathcal{V}}}\to \Gamma$. The curve $D$ can as well be considered as a section of ${\ensuremath{\mathcal{V}}}\to \Gamma$ and so we have an isomorphism
${\ensuremath{\mathcal{V}}}\cong \operatorname{{\rm Proj}}_\Gamma(p_*({\ensuremath{\mathcal{O}}}_{\ensuremath{\mathcal{V}}}(D)))$. If $S={\ensuremath{\mathcal{V}}}$ then the assertions (a)-(c) are trivial.
Blowing up subsequently points in the fibers these assertions also follow for $p:S\to \Gamma$.
\end{proof}
In what follows we may assume that the conditions (a), (b) in Lemma \ref{5.3} are satisfied.
\begin{lem}\label{5.5}
Letting ${\bar X}_q=\bar\varrho^{-1}(q)$ and $D_q=D\cap {\bar X}_q$
there is a closed subset $B$ of codimension $\ge 2$ in ${\bar Q}$ such that for $q\in {\bar Q}\backslash B$ the following assertions hold.
\begin{enumerate}[(a)$_q$]
\item $h^0({\bar X}_q,{\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))=2$ and $H^i({\bar X}_q,{\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))=0$ for $i\ge 1$.
\item The sheaf ${\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))$ is generated by its global sections.
\item If $s_0, s_1\in H^0({\bar X}_q,{\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))$ is a basis, then the map $(s_0:s_1)): {\bar X}_q\to {\mathbb P}^1$
is an isomorphism near $D_q$.
\item The map $\bar\varrho_*({\ensuremath{\mathcal{O}}}_{{\bar X}}(D))_q\to
H^0({\bar X}_q,{\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))$ is surjective, and $\bar\varrho_*({\ensuremath{\mathcal{O}}}_{{\bar X}}(D))_q$ is free of rank 2.
\end{enumerate}
\end{lem}
\begin{proof}
Let $B_0\subseteq {\bar Q}$ be a set as in Lemma \ref{5.3a}.
We choose a proper closed subset $P$ of ${\bar Q}$ such that any fiber over ${\bar Q}\backslash P$ is isomorphic to ${\mathbb P}^1$.
For any $q\in {\bar Q}\backslash P$ the assertions (a)$_q$-(d)$_q$ follow easily.
Let a curve $\Gamma$ in ${\bar Q}$ be an intersection of $n-1$ general ample divisors in ${\bar Q}$.
Since ${\bar Q}$ is normal and $\mathop{\rm codim} B_0\ge 2$, $\Gamma$ meets neither $ \operatorname{{\rm Sing}} {\bar Q}$ nor $B$.
By Bertini's theorem both $\Gamma$ and the surface $S=\bar\varrho^{-1}(\Gamma)$ are smooth.
The restriction $\bar\varrho|S:S\to{\mathbb P}^1$ is a ${\mathbb P}^1$-fibration. This ${\mathbb P}^1$-fibration admits a section,
namely $D\cap S$. The intersection $D\cap S$ is smooth in view of Bertini's theorem and Lemma
\ref{5.3}(b). The fiber of $S\to \Gamma$ over $q\in \Gamma\subseteq {\bar Q}$ coincides with ${\bar X}_q$.
By Lemma \ref{5.4} such a fiber ${\bar X}_q$ is a tree of rational curves satisfying (a)$_q$-(c)$_q$. Since
$\Gamma$ meets every component, say, $P_i$ of $P$ of codimension 1 and does not meet $B_0$,
for some $q_i\in P_i\backslash B_0$ the conditions (a)$_{q_i}$-(c)$_{q_i}$ are satisfied.
By semicontinuity (see\cite[III, 12.8]{Ha}) we obtain the inequalities
$$
h^j({\bar X}_p,{\ensuremath{\mathcal{O}}}_{{\bar X}_p}(D_p))\le
h^j({\bar X}_q,{\ensuremath{\mathcal{O}}}_{{\bar X}_q}(D_q))\le h^j({\bar X}_{q_i},{\ensuremath{\mathcal{O}}}_{{\bar X}_{q_i}}(D_{q_i})), \quad j\ge 0,
$$
where $q\in P_i$ is a point near $q_i$ and $p\in {\bar Q}\backslash P$ is a point near $q$.
Since the outer terms are equal, condition
(a)$_q$ holds for $q$ in some open dense subset $P^o_i$ of $P_i$.
By Grauert's criterion (see \cite[III, 12.9]{Ha}) now also (d)$_q$ is satisfied. Since (b)$_q$ and (c)$_q$
are open conditions on $P_i^o$, which are satisfied for some $q\in P_i^o$, they are satisfied generically
on $P_i$. Now the lemma follows.
\end{proof}
\begin{cor} \label{5.6}
There is a proper closed subset $B\subseteq {\bar Q}$ containing $ \operatorname{{\rm Sing}} T$ and $ \operatorname{{\rm Sing}} {\bar Q}$ with
$\mathop{\rm codim}_T(T\cap B)\ge 1$ such that, letting
$$
X^o={\bar X}\backslash \bar\varrho^{-1}(B)\,,
\quad Q^o={\bar Q}\backslash B\,,\quad T^o=T\backslash B\quad\mbox{and}\quad
C=\bar\varrho^{-1}(T)\,,
$$
there is a birational morphism
\begin{equation}\label{5.6a}
\varphi: X^o\longrightarrow {\ensuremath{\mathcal{X}}}= Q^o\times {\mathbb P}^1
\end{equation}
compatible with the projection to $Q^o$,
which restricts to a biregular morphism
\begin{equation}\label{5.6b}
X^o\backslash C\longrightarrow {\ensuremath{\mathcal{X}}}\backslash {\ensuremath{\mathcal{C}}}=(Q^o\backslash T)\times {\mathbb P}^1\,,
\end{equation}
where ${\ensuremath{\mathcal{C}}}=T^o\times {\mathbb P}^1$. Furthermore $\varphi$ is biregular in a neighborhood of $D^o=D\cap X^o$.
\end{cor}
\begin{proof}
Let $B\subseteq {\bar Q}$ be the subset constructed in Lemma \ref{5.5}. Enlarging it in a suitable
way we may assume that it contains $ \operatorname{{\rm Sing}} T\cup \operatorname{{\rm Sing}} {\bar Q}$. According to Lemma \ref{5.5}(c)
the sheaf ${\ensuremath{\mathcal{E}}}=\bar\varrho_*({\ensuremath{\mathcal{O}}}_{X^o}(D))$ is locally free of rank 2 on $Q^o$. Thus enlarging $B$
we may suppose that $\bar\varrho_*({\ensuremath{\mathcal{O}}}_{X^o}(D))$ is free. Choose two sections $s_0, s_1$ which
form a basis of this bundle. They provide a morphism
$$
\varphi= (\bar\varrho, (s_0:s_1)): X^o\to Q^o\times {\mathbb P}^1.
$$
Restricting to a fiber over $q\in Q^o$, in view of Lemma \ref{5.5}(c)$_q$ this yields an isomorphism
near $D_q$.
Hence $\varphi$ is an isomorphism near $D^o$.
Enlarging $B$ further we may also assume that all fibers in $Q^o\backslash T$ are isomorphic to ${\mathbb P}^1$.
This implies that the restricted morphism \eqref{5.6b} is an isomorphism.
\end{proof}
\begin{nota}\label{5.7}
Consider the restriction of the locally nilpotent vector field $\delta$ to $X^o\cap X$.
The associated action of $U=\exp({\Bbbk}\delta)$ has no fixed points in this set and extends to an action on
$X^o\backslash C$, where as before $C=\bar\varrho^{-1}(T)$. The fibers of $X^o \backslash C\to Q^o\backslash T$
are preserved under $U$.
Under the isomorphism $X^o\backslash C \simeq {\ensuremath{\mathcal{X}}}=(Q^o\backslash T) \times {\mathbb P}^1$
the second factor can be equipped with
a homogeneous coordinate system $(\zeta_1: \zeta_2)$ such that the image, say, ${\ensuremath{\mathcal{D}}}$
of $D^o=D\cap X^o$ in $X^o$ is defined by the equation $\zeta_1=0$. We treat
$$
v=\zeta_1/\zeta_2
$$
as a coordinate in the neighborhood ${\ensuremath{\mathcal{X}}} \backslash \{ \zeta_2 =0 \}$
of ${\ensuremath{\mathcal{D}}}$ in ${\ensuremath{\mathcal{X}}}$.
We fix a function $f \in {\Bbbk} [Q]$ such that its pullback on $X$ belongs to $\ker \delta
\backslash\ker \delta_0$.
This pullback induces rational functions on $X^o$ and on ${\ensuremath{\mathcal{X}}}$ denoted by the same symbol $f$.
By Lemma \ref{5.2}(1)
$f$ has poles along $D_0\cap X^o$.
By our choice of $B$ in Corollary \ref{5.6} $T^o$ is a submanifold of $Q^o$. Thus
locally the ideal of $T^o$ is generated by some function, say, $u$ on $Q^o$. On $Q^o$ the function $f$
is of form $a/u^s$. Here $s\ge 1$ is the pole order of $f$ along $T^o$, so $a$ is a rational function on $Q^o$,
which is nonzero in the general point of $T^o$.
Later on we will replace $f$ by a sufficiently large power $f^k$. By this we can achieve that the
pole order $s$ is arbitrary large.
Recall that $U_f$ stands for the replica of $U$ associated with the locally nilpotent vector field $f\delta$.
We note that $U_f$ is well defined on the set
$$
X^o\backslash C \cong (Q^o\backslash T^o)\times {\mathbb P}^1,
$$
cf.\ Corollary \ref{5.6}. Its element at moment $\tau \in {\Bbbk}$ will be denoted by $h_{f, \tau}$.
Considered as an automorphism of $(Q^o\backslash T)\times {\mathbb P}^1$ it
preserves the first factor but not the second one. The action of $h_{f, \tau}$ on $v$
is described
by the following Lemma.
\end{nota}
\begin{lem}\label{5.8} There exist a regular function $d = d(f)$ on $Q^o$, which does not vanish at
general points of $T$, and an
integer $l$ such that the automorphism of $(Q^o\backslash T)\times {\mathbb P}^1$ defined by
$h_{f, \tau}$
is given in the coordinates
$(q,v)$ by
the formula
$$
h_{f,\tau}:\, (q,v)\mapsto \left( q\,\,,\, {\frac{u(q)^{m}v}{u(q)^{m} +\tau d(q) v}}\right)\,,
$$
where $m=s-l$. In particular ${\ensuremath{\mathcal{D}}} \cap {\ensuremath{\mathcal{C}}}= \{ u=v=0 \}$
is the set of indeterminacy points of $h_{f,\tau}$.
\end{lem}
\begin{proof} In homogeneous coordinates $(\zeta_1 : \zeta_2)$ the action of $U=\exp({\Bbbk}\delta)$ on
$(Q^o\backslash T)\times{\mathbb P}^1$
is of form $(\zeta_1 : \zeta_2) \to (\zeta_1, \zeta_2 + \tau c \zeta_1)$ where $c$
is non-vanishing function
on $Q^o\backslash T$.
That is, $c=c_0u^l$ where $c_0$ is a non-vanishing function on $Q^o$ and $l\in {\mathbb Z}$.
Hence $h_{f,\tau }$
is of form
$(\zeta_1 : \zeta_2) \mapsto (\zeta_1: \zeta_2 + {\frac{\tau d}{u^{s-l}}} \zeta_1)$,
where $d$ does not
vanish at general points of $T^o$. Note that $m>0$ since $f\delta$ has a pole along $D_0$. Passing to the
affine coordinate $v=\zeta_1/\zeta_2$ this yields
the desired conclusion.
\end{proof}
Letting $s$ be the pole order of $f$ along $T$ we consider the set
\begin{equation}\label{Pf}
P_f=\{q\in T:\mbox{ locally } f=a/u^s\mbox{ with } a(q)=0\mbox{ or }a\not\in{\ensuremath{\mathcal{O}}}_{{\bar Q},q}\}\,,
\end{equation}
where $u$ is as before (i.e.\ $u=0$ is a local equation of $T$ near $q$) and $a$ is a rational function.
This set is a proper closed subset of $T$.
The next proposition is the main result of this section.
\begin{prop}\label{5.9}
Given $m$ and a function $f\in {\Bbbk}[Q]\cap \ker\delta\backslash \ker \delta_0$ there exists a positive integer $k_0$ such
that any transformation
$$
h\in U_{f^k}, \quad h\ne {\rm id}, \; k\ge k_0,
$$
is an $m$-contractions of $C$ along $D$ over the points of $Q^o\backslash P_f$.
\end{prop}
\begin{proof}
Let $s,l$ be as in Notation \ref{5.7} and Lemma \ref{5.8}.
If we chose $k_0$ in such a way that $m'=k_0s-l\ge m$ then by Lemma \ref{x.11b} the map $h=h_{f^k,\tau}$
is indeed an $m$-contraction for any $\tau\ne 0$.
\end{proof}
Let now $Y\subseteq X$ be a closed subset. Consider the partial boundary
$$
\partial_0Y={\bar Y}\cap D_0\,.
$$
For $U\in {\ensuremath{\mathcal{U}}}(X)$ we let $U^*=U\backslash \{{\rm id}\}$.
With this notation the following result holds.
\begin{prop}\label{prop-one}
Let the notation and conventions be as in Notation \ref{5.1} and assume that (a), (b) in Lemma \ref{5.3} are satisfied.
Let $(Y_{\alpha,\beta})_{(\alpha,\beta)\in A\times B}$ be a flat family of proper closed subsets of $X$. Suppose
that there is a flat family $(E_\alpha)_{\alpha\in A}$ of proper, closed subset of $D$ such that
$$
\partial Y_{\alpha,\beta}\cap D\subseteq E_\alpha \quad \mbox{for all}
\quad (\alpha,\beta)\in A\times B.
$$
Given an invariant function $f\in\ker\delta\backslash\ker \delta_0$, there is a dense open subset $A^o$ of $A$ and
a flat family $(E'_\alpha)_{\alpha\in A^o}$ of proper closed subset of $D_0$ satisfying
$$
\partial_0 h.Y_{(\alpha,\beta)}\subseteq E'_\alpha
\quad
\forall\, (\alpha,\beta)\in A^o\times B,\,\,\forall\, h\in U^*_{f^k},\,\,\forall \, k\ge k_0\,.
$$
\end{prop}
\begin{proof}
According to Proposition \ref{x.4} and Remark \ref{x.7} the closure ${\bar Y}_{\alpha\beta}$ of $Y_{\alpha\beta}$
in ${\bar X}$ is at most $m$-tangent to $D$ along $C$ for $m\gg 0$ and for all $(\alpha,\beta)\in A\times B$ simultaneously.
Let $N_{\alpha\beta}=C\cap \overline{D\cap Y_{\alpha\beta}\backslash C}$ denote the defect set. By Proposition \ref{5.9}
for $k\gg 0$ any map $h\in U_{f^k}^*$ is an $m$-contraction of $C$ along $D$ over the points of $Q^o\backslash P_f$.
Applying Proposition \ref{x.13} the image $h.Y_{\alpha\beta}$ satisfies
\begin{equation}\label{mmm}
\overline{ h.Y_{\alpha\beta}}\cap (D^o\cup C^o)\subseteq D\cup h(N_{\alpha\beta})\cup \bar\varrho^{-1}(P_f)
\end{equation}
where $h(N_{\alpha\beta})$ stands for the total transform of $N_{\alpha\beta}$ under $h$.
By our assumption the defect set $N_{\alpha\beta}$ is contained in $N_\alpha =C\cap\overline{E_\alpha\backslash C}$.
Since our birational transformation $h$ is compatible with the fibration $\bar\varrho$, the total image $h(N_{\alpha\beta})$
is contained in $\bar\varrho^{-1}(\bar\varrho(N_\alpha))$. Taking in \eqref{mmm} the intersection with $D_0$ gives
$$
\partial_0 (h.Y_{\alpha\beta})\subseteq E'_\alpha=(D\cup \bar\varrho^{-1}(B\cup \bar\varrho(N_\alpha)\cup P_f)) \cap D_0,
$$
where $B={\bar Q}\backslash {\bar Q}^o$ is as in Corollary \ref{5.6}.
Using the theorem on generic flatness it is easily seen that over an open dense subset $A^o$ of $A$ the sets
$E'_\alpha$ form a flat family of closed subsets of $D_0$. This yields the assertion.
\end{proof}
\section{Proof of the main theorem}\label{sec4}
\subsection{Algebraic families of automorphisms}\label{transitivity results}
Following Ramanujam \cite{Ram} let us introduce the following notion.
\begin{defi}\label{algebraic family} Given irreducible algebraic varieties $X$ and $A$ and
a map $\varphi:A\to \operatorname{{\rm Aut}}(X)$ we say that $(A,\phi)$
is an {\em algebraic family of automorphisms on $X$} if the induced map
$A\times X\to X$, $(\alpha,x)\mapsto \varphi(\alpha).x$, is a morphism.
\end{defi}
By abuse of notation, we do not distinguish in the sequel $A$ and its image $\varphi(A)$, and we identify
$\alpha\in A$ with its image $\varphi(\alpha)$ in $ \operatorname{{\rm Aut}}(X)$. As in the case of group action, given a point $x\in X$
the set $A.x$ will be called the {\em $A$-orbit} of $x$,
and the set $A_x=\{\alpha\in A\,|\,\alpha(x)=x\}$ the {\em stabilizer} of $x$ in $A$. The stabilizer admits
a natural linear representation
$d_x: A_x\to \operatorname{{\rm GL}}(T_xX)$, $\alpha\mapsto d\alpha|T_xX$, called the tangent representation.
The following result allows to work with finite dimensional algebraic
families instead of dealing with infinite dimensional groups of automorphisms.
\begin{lem}\label{uuuuyo}
Let $X$ be a smooth quasi-affine variety and $G=G_{{\ensuremath{\mathcal{N}}}}$ a group of automorphisms generated by a saturated
set of locally nilpotent derivations so that $G$ acts transitively on $X$. Then there exists
an algebraic family of automorphisms
$A\subseteq G$
such that for any $x\in X$ we have
(a) $A.x=X$ and
(b) $d_x(A_x)= \operatorname{{\bf SL}}(T_xX)$.
\end{lem}
\begin{proof}
According to Proposition 1.5 in \cite{AFKKZ} there exist one-parameter unipotent
subgroups $H_1,\ldots,H_s$ of $G$ such that with $H=H_1\cdot\ldots\cdot H_s\subseteq G$ we have
$H.x=G.x$ for any $x\in X$.
In particular, (a) holds with the algebraic family $A=H$.
By Theorem 4.2 \cite{AFKKZ} and its proof, for a fixed point $x\in
X$ the group $ \operatorname{{\bf SL}}(T_xX)$ is equal to the image in
$d_x(H')\subseteq \operatorname{{\rm GL}}(T_xX)$ for an algebraic family
$H'=H'_1\cdot\ldots\cdot H'_r$, where $H_1',\ldots ,H_r'$ are
suitable one-parameter subgroups of $G_{{\ensuremath{\mathcal{N}}},x}$. Taking the
product $A=HH'H^{-1}$, where $H$ is as in (a) and
$H^{-1}=H_s\cdot\ldots \cdot H_1$, we thus achieve that both (a)
and (b) are satisfied at every point $x\in X$. \end{proof}
\begin{nota}\label{44a} (a) As before we let $X$ be a smooth
quasi-affine variety and $G=G_{{\ensuremath{\mathcal{N}}}}$ a group of automorphisms
generated by a saturated set of locally nilpotent derivations as
in Notation \ref{5.1}(a). We suppose that $G$ acts transitively on
$X$. According to Theorem \ref{1.14} there are derivations $\delta_0,\delta_1\in {\ensuremath{\mathcal{N}}}$ such that the group
$$
H=\lan\lan \delta_0,\delta_1\ran\ran\subseteq G
$$
generated by $\delta_0,\delta_1$ and their replicas acts with an open orbit on $X$.\footnote{In
contrast to Notation \ref{5.1}(a)
in this section the role of $\delta_0$ and $\delta_1$ will be symmetric
so that it is convenient to replace the former
$\delta$ by $\delta_1$.}
These locally nilpotent vector fields generate one-parameter unipotent subgroups
$U^0, U^1\in{\ensuremath{\mathcal{U}}}(G)$.
Any function $f\in \ker\delta_0\backslash \ker \delta_1$ yields a replica $U^0_f$, and similarly
$g\in \ker\delta_1\backslash \ker \delta_0$ yields a replica $U^1_g$.
(b) To any sequence of invariant functions
\begin{equation}\label{seq}
{\ensuremath{\mathcal{F}}}=\{f_1,\ldots,f_s, g_1, \ldots,g_s\},\,\,\,\,\mbox{where}
\,\,\,\, f_i\in\ker\delta_1\backslash\ker
\delta_0\,\,\,\,\mbox{and}\,\,\,\, g_i\in\ker\delta_0\backslash\ker
\delta_1\,,
\end{equation}
we associate an algebraic family
of automorphisms ${\mathbb A}^{2s}\to \operatorname{{\rm Aut}}(X)$ defined by the product
\begin{equation}\label{00121}
U^{\ensuremath{\mathcal{F}}}=U^1_{f_s}\cdot
U^0_{g_s}\cdot\ldots\cdot U^1_{f_1}\cdot
U^0_{g_1}\subseteq H\,.
\end{equation}
More generally, given a tuple
$\kappa=(k_i,l_i)_{i=1,\ldots,s}\in\ensuremath{\mathbb{N}}^{2s}$ the product
\begin{equation}\label{001210}
U_\kappa=U_\kappa^{\ensuremath{\mathcal{F}}}=
U^1_{f_s^{k_s}}\cdot
U^0_{g_s^{l_s}}\cdot\ldots\cdot U^1_{f_1^{k_1}}\cdot
U^0_{g_1^{l_1}}
\subseteq H\,
\end{equation}
is as well an algebraic family of automorphisms.
\end{nota}
\begin{cor}\label{cor-0012} There is a finite collection of invariant functions ${\ensuremath{\mathcal{F}}}$ as in (\ref{seq})
such that for any sequence
$\kappa=(k_i,l_i)_{i=1,\ldots,s}\in\ensuremath{\mathbb{N}}^{2s}$ the algebraic family of automorphisms
$U_\kappa$ as in (\ref{001210})
has a dense open orbit in $X$.
This orbit $O(U_\kappa)$ coincides with $O(H)$ and so
does not depend on the choice of $\kappa\in\ensuremath{\mathbb{N}}^{2s}$.\end{cor}
\begin{proof} According to Proposition 1.5 in \cite{AFKKZ} there is a
sequence ${\ensuremath{\mathcal{F}}}$ as in (\ref{seq}) such that
$$
H.x= U^{\ensuremath{\mathcal{F}}}.x\quad\forall x\in X
\,.
$$
In
particular, for $x\in O(H)$ the orbit $ U^{\ensuremath{\mathcal{F}}}.x=O(H)$ is open in
$X$. It is easily seen that for any $\kappa\in\ensuremath{\mathbb{N}}^{2s}$ we have
$O(U_\kappa)=O( U^{\ensuremath{\mathcal{F}}})=O(H)$. Indeed, $O(H)$ consists of all the $U^{\ensuremath{\mathcal{F}}}$-flexible points in $X$.
Now the assertions follow. \end{proof}
\subsection{Proof of the main theorem}
\begin{nota}\label{4.5}
We keep the notation and assumptions from \ref{44a}(a).
(a) Let $\rho_0:X\to Q_0$ and $\rho_1:X\to Q_1$ be partial quotients with respect
to the unipotent subgroups $U^0$ and $U^1$, respectively.
Let us choose open embeddings $X\hookrightarrow {\bar X}$, $Q_0\hookrightarrow{\bar Q}_0$, and $Q_1\hookrightarrow{\bar Q}_1$
into normal projective varieties, see
Notation \ref{5.1}. We can assume that the following conditions
are satisfied.
\begin{enumerate}[(i)]
\item $\rho_0$ and $\rho_1$ extend to morphisms
$\bar\varrho_0:{\bar X}\to{\bar Q}_0$ and $\bar\varrho_1:{\bar X}\to {\bar Q}_1$.
Let $D_0$ and $D_1$ as in \ref{5.1} be the unique horizontal divisors that map birationally onto
${\bar Q}_0$ and ${\bar Q}_1$, respectively.
\item ${\bar X}$, $D_0$ and $D_1$ are smooth, see Lemma \ref{5.3}(b).
\item $T_0=\bar\varrho(D_0)$ and $T_1=\bar\varrho(D_1$ are divisors in ${\bar Q}_0$ and ${\bar Q}_1$, respectively;
see Lemma \ref{5.3}(a).
\end{enumerate}
(b) Given a closed subscheme $Y\subseteq X$ of codimension $\ge 2$ we call
$$
\partial_0Y={\bar Y}\cap D_0\quad\mbox{and}\quad \partial_1Y={\bar Y}\cap D_1
$$
the {\em partial boundaries.} Furthermore $O_Y$ will denote the open orbit of $G_{{\ensuremath{\mathcal{N}}},Y}$ in $X\backslash Y$.
\end{nota}
\begin{sit}\label{cvnt}
In the course of the proof of the main Theorem we move the given pair $(Y,x)$ to
another one $(Y_\alpha, x_\alpha)$ by means of an automorphism $\alpha \in G_{\ensuremath{\mathcal{N}}}$, where
$Y_\alpha=\alpha. Y$ and $x_\alpha=\alpha.x$. In this way we can adopt
the position of our pair with respect to the ${\mathbb P}^1$-fibration
$\bar\varrho_0:{\bar X}\to{\bar Q}_0$
so that the conditions (i)-(iii) below hold.
\begin{enumerate}\label{condis}
\item[(i)] $U^0.x_\alpha\cap O_{Y_\alpha}\neq\emptyset\,;$
\item[(ii)] $U^0.x_\alpha\cap Y_\alpha=\emptyset\,;$
\item[(iii)] $\partial_0(U^0.x_\alpha)\notin\partial_0 (Y_\alpha)$.\end{enumerate}
\end{sit}
The following lemma allows to deduce Theorem \ref{mthm1}
provided that (i)-(iii) hold for any $x\in X\backslash Y$ with some $\alpha\in G$ depending on $x$.
\begin{lem}\label{777} If for a point $x\in X\backslash Y$ and for some $\alpha\in G$ conditions (i)-(iii) in \ref{condis}
are fulfilled then $x\in O_{Y}$.
If these conditions are fulfilled for any $x\in X\backslash Y$ with some $\alpha\in G$ depending on $x$,
then the conclusion of Theorem \ref{mthm1} holds.
\end{lem}
\begin{proof}
Since $O_{Y\alpha}=\alpha.O_{Y}$ we have
$$
x\in O_{Y}\Longleftrightarrow x_\alpha\in O_{Y_\alpha}\,.$$
Replacing $(Y,x)$ by $(Y_\alpha,x_\alpha)$ we will assume that (i)-(iii) hold for the pair
$(Y,x)$ and $\alpha={\rm id}$. We need to show that then $x\in O_Y$. Conditions (ii) and (iii) yield that
$$
\rho_0(x)\in\rho_0(O_{Y})\backslash\overline{\rho_0(Y)}\,.
$$
Therefore there exists a regular function
$h\in{\ensuremath{\mathcal{O}}}(Q_0)$ such that $h(\rho_0(x))=1$ and $h$ vanishes
on $\rho_0 (Y)$. Replacing $h$ by a suitable power of $h$
we may suppose that the $\delta_0$-invariant function $f=h\circ\rho_0$ on $X$ vanishes on $Y$.
Thus the replica $U^0_f=\exp({\Bbbk} f\delta_0)$ of $U^0$ fixes $Y$ pointwise i.e.
$U^0_f\in {\ensuremath{\mathcal{U}}}(G_{{\ensuremath{\mathcal{N}}},Y})$.
By (i) one can find $u\in U^0_f$ such that $u.x\in
O_{Y}$. Hence also $x\in O_{Y}$, as stated.
\end{proof}
Thus to prove Theorem \ref{mthm1} it is enough to show that (i)-(iii) hold for every point
$x\in X\backslash Y$ with a suitable $\alpha\in G$ depending on $x$.
\begin{lem}\label{constr} Given a point $x\in X\backslash Y$ and
an algebraic family of automorphisms $\varphi:A\to \operatorname{{\rm Aut}}(X)$ the following hold.
(a) The set of all $\alpha\in A$ satisfying (i) is open in $A$.
(b) The set of all $\alpha\in A$ satisfying (ii) is constructible in $A$.
\end{lem}
\begin{proof}
(a) The subset $B\subseteq A$ where (i) does not hold is the set of $\alpha\in A$ satisfying
$$
U^0.x_\alpha\subseteq Y_\alpha
\quad\mbox{or, equivalently,}\quad \alpha^{-1}U^0\alpha. x\subseteq Y.
$$
Thus $B=\bigcap_{u\in U^0} B_u$, where
$B_u=\{\alpha\in A: \alpha^{-1}u\alpha.x\in Y\}$
is the preimage of $Y$ under the morphism $A\to X$, $\alpha\mapsto
\alpha^{-1}u\alpha.x$. Hence $B$ is closed in $A$. This proves (a).
(b) Similarly, the subset $C\subseteq A$ where (ii) does not hold is the set of $\alpha\in A$ with
$\alpha^{-1}U^0\alpha\cap Y\ne \emptyset.$
Consider the set
$$
C'=\{(\alpha, u)\in A\times U^0: \alpha^{-1}u\alpha.x\in Y\}\,.
$$
This set is closed in $A\times U^0$ since it is the preimage of $Y$
under the morphism $A\times U^0\to X$, $(\alpha,u)\mapsto
\alpha^{-1}u\alpha.x$. Since $C$ is the image of $C'$ under the projection to $A$, (b) follows.
\end{proof}
The next proposition allows to verify conditions (i) and (ii).
\begin{prop}\label{555} Let as before $x\in X\backslash Y$.
\begin{enumerate}[(a)]
\item If $A$ is an algebraic family of automorphisms of $X$ with $d_x(A_x)\supseteq \operatorname{{\bf SL}}(T_xX)$,
then the set of all $\alpha\in
A$ satisfying (i) is a dense open subset of $A$.
\item
There exists an algebraic family $A^*\subseteq G_x$
transitive in $X^*=X\backslash\{x\}$ such that for any subgroup $U^0\in {\ensuremath{\mathcal{U}}}(X)$ condition
(ii) holds for a general $\alpha\in A^*$.
\item
Given an algebraic family $B\subseteq \operatorname{{\rm Aut}}(X)$
we let $\tilde A=B\cdot A^*\subseteq \operatorname{{\rm Aut}}(X)$,
where $A^*\subseteq G_x$
is as in (b).
Then (ii) holds for a general $\tilde\alpha\in \tilde A$.
\end{enumerate}\end{prop}
\begin{proof} (a) By Lemma \ref{constr} it suffices to find $\alpha\in A$ satisfying (i),
or, equivalently, such that $\alpha^{-1}U^0\alpha.x\cap O_{Y}\neq\emptyset$.
By our assumptions in (a) for any nonzero vector $v\in T_xX$ there is an element $\alpha\in A_x$ such that
$v$ is tangent to the orbit through $x$ of the one-parameter group $\alpha^{-1}U^0\alpha\subseteq \operatorname{{\rm Aut}}(X)$.
These orbits form
an algebraic family of smooth rational curves in $X$ through the point $x$ that dominates $X$ and so
meets the open orbit $O_{Y}$, as required.
(b) By the Transversality Theorem \cite[1.16]{AFKKZ} there exists an algebraic family $A^*\subseteq G_x$
transitive in $X^*$
such that for any two subvarieties $Y,Z\subseteq X$ there is a dense open subset
$A_0\subseteq A^*$ with the property that
for any $\alpha\in A_0$ the varieties $\alpha.Y$ and $Z$ are transversal. Applying this to $Z=U^0.x$
the varieties $U^0.x$ and $\alpha.Y$ are disjoint, because under our assumptions
$$\dim U^0.x +\dim Y< \dim X\,.$$
Since $x_\alpha=x$, (b) follows.
To deduce (c) we note that the set, say $C$ of points $\tilde \alpha\in \tilde A$, where (ii) fails is the set of
$\tilde\alpha=(\beta,\alpha)$ with $\alpha^{-1}\beta^{-1}U^0\beta\alpha.x\cap Y\ne\emptyset$. Consider
similarly as in the proof of Lemma \ref{constr}(b) the closed subset of $B\times A^*\times U^0$
$$
C'=\{(\beta,\alpha,u)\in B\times A^*\times U^0: \alpha^{-1}\beta^{-1}u\beta\alpha.x
\in Y\}\,,
$$
where $A^*$ satisfies the conclusion of (b).
According to (b) for any $\beta\in B$ the set
$$
C'_\beta=C'\cap (\{\beta\}\times A^*\times U^0)
$$
maps under the projection to $A^*$ to a nowhere dense subset.
Hence also the image $C$ of $C'$ under the projection to $\tilde A=B\times A^*$ will be nowhere dense.
Thus its complement contains an open dense subset proving (c).
\end{proof}
\begin{nota}\label{nota4.10}
Given a one-parameter group $U\in {\ensuremath{\mathcal{U}}}(X)$ we let as before $U^*=U\backslash\{{\rm id}\}$.
Given a collection ${\ensuremath{\mathcal{F}}}$ of invariant functions
$$
f_1,\ldots,f_s\in\ker\delta_1\backslash\ker \delta_0\quad\mbox{and}\quad g_1,
\ldots,g_s\in\ker\delta_0\backslash\ker \delta_1
$$
and $U_\kappa=
U^1_{f_s^{k_s}}\cdot
U^0_{g_s^{l_s}}\cdot\ldots\cdot U^1_{f_1^{k_1}}\cdot
U^0_{g_1^{l_1}}$ as in \eqref{00121}, we let
$$
U_\kappa^*=
U^{1*}_{f_s^{k_s}}\cdot
U^{0*}_{g_s^{l_s}}\cdot\ldots\cdot U^{1*}_{f_1^{k_1}}\cdot
U^{0*}_{g_1^{l_1}}\,.
$$
\end{nota}
Using Proposition \ref{prop-one} we can deduce the following result.
\begin{prop}\label{prop-two}
Let $(Y_\alpha)_{\alpha\in A }$ be a flat family of proper closed subsets of $X$.
Assume that the partial boundaries
$\partial_i Y_\alpha$ (see Notation \ref{4.5}) are contained in $E_{\alpha,i}$,
where the $(E_{\alpha,i})_{\alpha\in A}$, $i=0,1$,
form flat families of proper closed subsets of $D_i$.
Then one can find an open dense subset $A^o$ of $A$, flat families of proper, closed subsets
$(E^o_{\alpha, i})_{\alpha\in A^o}$ of $ D_i$ ($i=0,1$), and a sequence
$\kappa=(k_1,l_1,\ldots,k_s,l_s)\in\ensuremath{\mathbb{N}}^{2s}$ such that for any element $h\in U_\kappa^*$ we have
$$
\partial_i (h.Y_\alpha)\subseteq E^o_{\alpha ,i}\, ,\qquad i=0,1\,,\; \forall\,\alpha\in A^o\,.
$$
\end{prop}
\begin{proof}
The proof proceeds by induction on $s$. For $s=0$
the assertion clearly holds with $A^o=A$ and
$E_{\alpha, i}=\partial_i Y_\alpha$, $i=0,1$. Assume that it holds at step $s-1$, i.e. we can find
$\kappa'=(k_j,\, l_j)_{j=1,\ldots,s-1}\in\ensuremath{\mathbb{N}}^{2s-2}$, a dense open subset $A'\subseteq A$
and flat families of proper closed subsets $(E_{\alpha,i})_{\alpha\in A'}$ of $ D_i$ such that for $\alpha\in A'$
$$
\partial_i(h.Y_\alpha)\subseteq E_{\alpha,i}\,,\quad i=0,1,\quad \forall\,
h\in U_{\kappa'}^*\,.
$$
The varieties $(h.Y_\alpha)_{(h,\alpha)\in U_{\kappa'}^*\times A'}$ form a flat algebraic family.
By Proposition \ref{prop-one}
one can find an open dense subset $A''\subseteq A'$ and flat families $(E'_{\alpha,i})_{\alpha\in A''}$, $i=0,1$,
of proper closed subsets of $D_i$ such that
$$
\partial_i(h'h.Y_\alpha)\subseteq E'_{\alpha,i}\,\,\, (i=0,1)\quad
\forall \,\, l_s\gg 0,\,\, \forall\alpha\in A'',\,\,
\,\,\forall \, (h',h)\in U^{0*}_{g_s^{l_s}}\times U_{\kappa'}^*\,.
$$
Fixing a sufficiently large $l_s$ and applying the same argument again
one can find an open dense subset $A^o\subseteq A''$ and flat families $(E^o_{\alpha,i})_{\alpha\in A^o}$, $i=0,1$,
of proper closed subsets of $D_i$ such that
$$
\partial_i(h''h'h.Y)\subseteq E^o_{\alpha,i}\,\,\, (i=0,1)\quad
\forall k_1\gg 0, \,\, \forall\alpha\in A^o\,,\,
\forall \, (h'', h',h)\in U^{1*}_{f_s^{k_s}}\times U^{0*}_{g_s^{l_s}}\times U_{\kappa'}^*\,.
$$
This concludes the induction.
\end{proof}
Using Proposition \ref{prop-two} and
Corollary \ref{cor-0012} we can now
deduce Theorem \ref{mthm1}.
\begin{proof}[Proof of Theorem \ref{mthm1}.]
Let $x\in X\backslash Y$ be a fixed point.
We show that for a suitable choice of an algebraic family $A$ of automorphisms conditions (i)-(iii) are satisfied for
the pair $(Y_\alpha, x_\alpha)$, if $\alpha\in A$ is generic. Then our theorem follows by applying Lemma \ref{cvnt}.
{\em Step 1.} Consider an algebraic family $A\subseteq G$ satisfying conditions (a)
and (b) of Lemma \ref{uuuuyo}.
Applying Proposition \ref{555}(a) condition (i) holds
when $\alpha$ varies in a dense open subset of $A$.
Replacing the original pair $(Y,x)$ by a suitable new one
$(Y_\alpha,x_\alpha)=(\alpha.Y,\alpha.x)$ we may suppose that $(Y,x)$ satisfies (i).
{\em Step 2.}
In the following we construct an algebraic family $B$ of automorphisms such that for a generic choice of $\beta\in B$
the translates $(Y_\beta, x_\beta)$ satisfy (ii), (iii). Since by Proposition \ref{555}(a) condition (i) is open then the pair
$(Y_\beta, x_\beta)$ also satisfies (i).
Let $A^*$ be a family of automorphisms as in Proposition \ref{555}(b). The translates $Y_\alpha=\alpha. Y$,
$\alpha\in A^*$, form a flat family of proper closed subsets of $X$. Using the theorem of generic flatness it is
easily seen that over an open dense subset $A'\subseteq A^*$ also the partial boundaries $E_{\alpha,i}=\partial_i Y_\alpha$,
$\alpha\in A'$, form flat families of proper closed subsets of $D_i$, $i=0,1$.
Let now ${\ensuremath{\mathcal{F}}}$, $U_\kappa$,
and $U_\kappa^*$ be as in Notation \ref{nota4.10}.
By Proposition \ref{prop-two} we can find $\kappa=(k_1,l_1,\ldots,k_s,l_s)\in\ensuremath{\mathbb{N}}^{2s}$,
a dense open subset $A^o\subseteq A'$,
and families $(E^o_{\alpha,i})_{\alpha\in A^o}$, $i=0,1$, of proper closed subsets of $D_i$ such that
$\partial_i(h.Y_\alpha)\subseteq E^o_{\alpha, i}$ for $i=0,1$, $\alpha\in A^o$ and all $h\in U_\kappa^*$.
We claim that for a generic choice of $(h,\alpha)\in B=U^*_\kappa\times A^*$ conditions (ii) and (iii) are
satisfied for $h.Y_\alpha$. To check (ii) we note that $h.Y_\alpha=h\alpha.Y$. Thus applying Proposition \ref{555}(c)
to the family $B=U^*_\kappa\times A^*$ condition (ii) is indeed satisfied for a generic choice of $(h,\alpha)$.
It remains to show that (iii) is satisfied for a generic choice of $(h,\alpha)$.
Condition (iii) is equivalent to $\bar\varrho_0(h.x_\alpha)\not \in \partial_0(h.Y_\alpha)$. By construction
$\partial_0(h.Y_\alpha)\subseteq E_{\alpha,0}\subseteq D_0$ for any $h\in U_\kappa^*$,
while for a fixed $\alpha\in A^o$ the points
$h.x_\alpha$, $h\in U_\kappa^*$, fill in a dense subset of $X$,
and so their images $\bar\varrho_0(h.x)$ fill in a dense subset of $Q_0\subseteq \bar Q_0\backslash \bar\varrho_0(D_0)$.
Thus (iii) holds for a generic choice of $(h,\alpha)\in U^*_\kappa\times A^o$.
This concludes the proof of Theorem \ref{mthm1}.
\end{proof}
|
1,314,259,993,032 | arxiv | \section*{Introduction}
The transmission of human respiratory diseases such as influenza, tuberculosis, and COVID-19 is directly driven by the rate of close social contact between individuals. Social contact studies such as the pivotal POLYMOD study~\cite{mossong_social_2008} have been widely acknowledged as an effective method of obtaining social contact estimates to assess infection risk and to parameterise mathematical infectious disease models~\cite{goeyvaerts_estimating_2010, eichner_4flu_2014, schmidt-ott_influence_2016}. Consequently, they provide critical epidemiological insights which inform the implementation and evaluation of non-pharmaceutical interventions~\cite{leung_transmissibility_2021} as well as public health policies such as vaccination schedules~\cite{wallinga_optimizing_2010}.
Since the outbreak of COVID-19, numerous contact studies have been conducted in Europe and around the world, providing indispensable information on the evolving patterns of human mixing behaviour during the pandemic \cite{verelst_socrates-comix_2021, feehan_quantifying_2021}. In Germany, the COVIMOD study collected social contact data for close to two years, and initial analyses~\cite{tomori_individual_2021} focused on data from April to June 2020 during the first partial lockdown in Germany to quantify the scale of social contact reductions relative to pre-pandemic contact patterns. High-resolution estimates of age- and gender-specific human contact patterns and their trends over time are substantially more difficult to obtain, due to limitations in available inference methods~\cite{goeyvaerts_estimating_2010, van_de_kassteele_efficient_2017, funk_socialmixr_2020}.
First, COVIMOD and other COVID-era social contact studies record the age of contacts by large age categories of 5 to 10 years, reflecting that often it is challenging for study participants to know the exact age of their contacts. In contrast, the pre-pandemic POLYMOD surveys collected data on the exact age of contacts and subsequently developed methods including thin-plate regressions~\cite{goeyvaerts_estimating_2010} and Gaussian Markov Random Field model approaches~\cite{van_de_kassteele_efficient_2017} relied on such data to estimate high-resolution contact patterns. The bootstrap approach implemented in the \texttt{socialmixr} library~\cite{funk_socialmixr_2020} provides a convenient and speedy way to estimate contact matrices. But, it only provides contact estimates in discrete large age categories, which is unsatisfactory as it may mask subtle but important age effects~\cite{farrington_contact_2005}.
Second, most COVID-era studies adopted retrospective web-based survey protocols and conducted longitudinal repeat surveys~\cite{gimma_changes_2022, coletti_comix_2020, backer_impact_2021, verelst_socrates-comix_2021}. Importantly, the survey waves are typically inter-dependent because a varying number of participants were surveyed in multiple waves, and additional participants were recruited to replenish the cohort size. While this approach provides valuable longitudinal data, it also introduces the issue of reporting fatigue, where participants tend to report fewer contacts in subsequent participation due to becoming tired of filling out the survey. It follows that directly applying existing methods, which do not incorporate adjustments to counter the confounding reporting fatigue effects, is bound to lead to incorrect estimates. Additionally, participants in the COVID-era surveys sometimes found it difficult to recall specific age and gender information for all of their contacts. Instead, they were allowed to report an estimate for the total number of contacts on that occasion~\cite{tomori_individual_2021}, which again may result in under-ascertainment of contact intensities if these data are not accounted for in inference approaches of contact patterns.
In this work, we present a temporal Bayesian model to infer age- and gender-specific contact patterns and trends at high 1-year resolution from longitudinal survey data. The primary innovation of the model is the ability to infer contact patterns by 1-year age bands even when the age contacts are reported in broad age categories. We call this model-based approach the Bayesian rate consistency model for reasons that will be clear soon. In addition, we use recently developed Hilbert Space Gaussian Process approximations~\cite{solin_hilbert_2020} to gain substantial advances in computational efficiency, which in turn enable us to make full Bayesian inferences over time and uncover the dynamics in social contact structure. We demonstrate that it is crucial to model contact patterns over time to account for reporting fatigue effects in inter-dependent longitudinal survey waves. The primary purpose behind developing the Bayesian rate consistency model is its application to contemporary COVID19-era survey data, which we present for data spanning the first five survey waves of the COVIMOD study in Germany. We present high-resolution estimates of age- and gender-specific social contacts for each survey wave and describe their time evolution. We also place the inferred contact dynamics into a pre-pandemic context and quantify the differences in contact intensity change by the age of contacts.
\section*{Methods}
\subsection*{The COVIMOD study}
The COVIMOD study was launched in April 2020 and continued until December 2021, constituting 33 survey waves. Participants were recruited through email invitations to existing panel members of the online market research platform IPSOS i-say~\cite{ipsos_ipsos_2022}. To ensure the sample's broad representativeness of the German population, quota sampling was conducted based on age, gender, and region. Participants were invited to participate in multiple waves to track changes in social behaviour and attitudes toward COVID-19. When the participant size did not meet the sampling quota due to study withdrawals, new participants were recruited into the study. This approach enabled COVIMOD study to obtain longitudinal samples, but it also introduced the issue of response fatigue, where the number of detailed contacts reported decreased compared to previous participation, irrespective of the survey wave. To procure information on children, a subgroup of adult participants living with children under the age of 18 were selected to be proxies. This procedure meant that middle-aged adults were under-sampled as they completed the survey on behalf of their children.
The COVIMOD questionnaire was based on the CoMix study and includes questions on demographics, the presence of a household member belonging to a high-risk group, attitudes towards COVID-19 as well as related government measures, and current preventative behaviors~\cite{jarvis_quantifying_2020, tomori_individual_2021}. Participants were also asked to provide information about their social contacts between 5 a.m. the preceding day to 5 a.m. the day of answering the survey. Following the pre-pandemic POLYMOD study, a contact is defined as either a skin-to-skin contact such as a kiss or a handshake (physical contact) or an exchange of words in the presence of another person (non-physical contacts)~\cite{mossong_social_2008}. Participants were asked to report the age group, gender, relation, the contact setting (e.g. home, school, workplace, place of entertainment, etc.), and whether the contact was a household member. For survey waves 1 and 2, participants were asked to provide each contact's information separately. However, some participants reported contacts to groups of individuals (e.g., customers, clients) for which a specific number of contacts was assumed (Additional file 2 of~\cite{tomori_individual_2021}). From wave 3 onwards, participants were given the possibility to record group contacts in addition to the recording of individual contacts.
Additionally, some participants could not recall or preferred not to answer the age or gender information of some individual contacts. We treat these three types of entries with missing age or gender equally and refer to them as \emph{missing} \& \emph{aggregate contact reports}. A copy of the COVIMOD questionnaire may be found in Additional file 1 of~\cite{tomori_individual_2021}. COVIMOD was approved by the ethics committee of the Medical Board Westfalen-Lippe and the University of Münster, reference number 2020-473-f-s.
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-1.png}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Timeline and participants of the longitudinal COVIMOD study} A. Daily COVID-19 case counts in Germany (red bars), the OxCGRT Stringency Index (blue line), and COVIMOD survey administration periods (grey ribbons). B. Cumulative COVID-19-related deaths in Germany (red line), the OxCGRT Stringency Index (blue line), and COVIMOD survey administration periods (grey ribbons). C. Sample sizes and the proportion of people repeatedly sampled in the COVIMOD survey for which zero repeats indicate first-time participants.}
\label{fig:1}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
This current work concerns the first five survey waves of the COVIMOD study. In Fig~\ref{fig:1}A and B, we show the sampling periods with the number of daily COVID-19 cases, cumulative COVID-19-related deaths, and the OxCGRT Stringency Index~\cite{hale_global_2021}. The following COVID-19 policy timeline is obtained from the ACAPS COVID-19 Government Measures dataset~\cite{acaps_covid}. The first COVIMOD survey was administered from April 30\textsuperscript{th} to May 6\textsuperscript{th} in year 2020, towards the end of the first partial lockdown and the first wave of cases. Before the beginning of the first survey (April 20\textsuperscript{th}), small stores, auto dealers, and bookstores were allowed to reopen under strict hygiene regulations. During the final few days of the survey period (May 4\textsuperscript{th} to 6\textsuperscript{th}), phase-out measures were announced by the government, including the step-wise uptake of schools, the reopening of hairdressers under strict hygiene regulations, lifting of the ban on public gatherings of 30 people indoors and 50 outdoors, resumption of religious services, and reopening of public services such as museums, botanical gardens, zoos, and playgrounds. The second wave of the COVIMOD survey was administered from May 14\textsuperscript{th} to May 21\textsuperscript{st}. During this period, additional phase-out measures were announced, including the resumption of all cross-country transport and the reopening of hotels and restaurants. International travel to neighbouring countries was also slightly relaxed during this period. The third, fourth, and fifth waves of COVIMOD surveys were taken from May 28\textsuperscript{th} to July 4\textsuperscript{th}, June 11\textsuperscript{th}-22\textsuperscript{nd}, and June 26\textsuperscript{th} to July 1\textsuperscript{st}, respectively. There was no notable introduction or reduction of social contact restriction measures during this time, but international travel restrictions were relaxed primarily for Schengen and EU countries. COVID-19 cases and deaths remained stable during this period (Fig~\ref{fig:1}).
After excluding participants who prefer not to provide age or gender information and 25 participants above the age of 84, there were 1549, 1345, 1076, 1881, and 1603 participants for waves 1 to 5. We observed 3244, 4852, 6344, 13471, and 8353 total contacts for each wave. In Fig~\ref{fig:1}C, we show the proportion of participants who consented to the survey multiple times. Most participants in waves 2 and 3 had participated in wave 1, with only 6.8\% and 16\% of participants being new to the survey. The proportion dropped sharply in wave 4, where only 35.1\% of initial participants remained. Hence the majority (57.7\%) of wave 4 participants were first-time participants. On the contrary, no new participants were enrolled for wave 5, and individuals who participated for the second, third, and fifth time took up approximately 45\%, 10.7\%, 9.6\%, and 34.6\% of the sample.
\subsection*{Data processing}
Following ethical guidelines, the participant age information for children is reported in discrete categories, i.e., $\text{0−4, 5−9, 10−14, 15−18}$. To obtain fine-age information for participants
under 18, we imputed their age by drawing from a discrete uniform distribution with
bounds set as the minimum and maximum age of the participant’s age category. We
excluded 20 participants (0.3\% of the total) without age or gender information as we
could not estimate the information accurately. For total contact counts $Y^{gh}_{ab}$ between all participants aged $a \in \{0, 1, \ldots, 84\}$ of gender $g \in \{M, F\}$ and contacted individuals aged $b \in \{0, 1, \ldots, 84\}$ of gender $h \in \{M, F\}$, we filled in
missing entries with zeroes if a participant of age $a$ and gender $g$ is present. If not, we treated the entries as missing. We truncated group contacts at 60 (90\textsuperscript{th} percentile of group contacts) to remove the effects of extreme outliers. We show the distribution of observed contacts in \nameref{S1_Fig}.
\subsection*{Estimating contact patterns}
Throughout, we focus on estimating contact patterns and dynamics between men and women (superscripts $g, h \in \{M, F\}$) and high resolution one year age groups (subscripts $a, b \in \mathcal{B} = \{0, 1, 2, \cdots\, 84\}$). To introduce basic notations, let us momentarily consider male-to-female contacts at some fixed time $t$ and suppress the time index. We start by considering the total number of contacts $\ccntMF$ that are reported by men of age $a$ who participate in the survey to all women of age $b$ in the population. The number of male survey participants of age $a$ is $\PartM$, and the number of women of age $b$ in the population is $\PopF$. When $\PartM>0$ for age group $a$, the contact counts $\ccntMF$ are defined for all $b \in \mathcal{B}$, and are either zero or positive. When there are no participants for some age group $a$, there is no corresponding contact data, and we denote the participant age groups in the survey by $\AgesPartM = \{ a \in \mathcal{B} \colon \PartM>0\}$. Finally, we denote the number of age groups in $\AgesPartM$ by $A^M$ and the number of age groups in $\mathcal{B}$ by $B$. We also consider analogous notations for all other gender combinations using the superscripts $g$ and $h$.
From the observed data, we seek to estimate the \emph{contact intensity} $\cintMF$, the average number of contacts from one male participant of age $a$ to all women aged $b$ in the population. We follow~\cite{mossong_social_2008, hens_mining_2009, goeyvaerts_estimating_2010, van_de_kassteele_efficient_2017, vandendijck_cohort-based_2022, wallinga_using_2006, feehan_quantifying_2021} and model the count data with an overdispersion adjusting Negative Binomial observation model, in shape-scale form,
\begin{subequations}\label{lkl-contact-patterns}
\begin{align}
\ccnt & \sim \text{NegBinomial}\left(\alpha_{ab}^{gh}, \frac{\odisp}{1+\odisp}\right)\\
\eccnt & = \alpha_{ab}^{gh}\odisp\\
\log \eccnt & = \log \cint + \log \Part \label{lkl-contact-patterns-b}
\end{align}
\end{subequations}
where $g,h\in\{M,F\}$, $a\in\AgesPartG$, $b\in\mathcal{B}$, and $\eccnt$ are the expected contacts that are expressed of the target quantity of interest, $\cint$, and the known $\Part$. The overdispersion parameter $\odisp>0$ such that $\text{Var}[Y_{ab}^{gh}] = \mu_{ab}^{gh}(1 + \nu)$ is allowed to be greater than the mean.
The \emph{contact rate} is defined as the probability of contact between one male aged $a$ and one female aged $b$, i.e.
\begin{equation}\label{contact-rate}
\crateMF = \cintMF / \PopF.
\end{equation}
Crucially, in a closed population, the (unknown) number of total contacts must be self-consistent,~i.~e.
\begin{equation}\label{contact-rate-symmetry}
\PopMa \PopF \crateMF = \PopFa \PopM \crateFM,
\end{equation}
from which we find that contact rates are symmetric in the sense that $\crateMF = \crateFM$ for all $a$, $b$, and similarly $\crateMM = \crateMMb$ and $\crateFF = \crateFFb$ for all $a < b$. A similar symmetry property does not hold for contact intensities.
Property~\eqref{contact-rate-symmetry} implies that data on age group $a$ informs contact rates in both age dimensions, which we will exploit heavily below. If the survey captures participants for all possible age groups, i.~e. $\AgesPartG = \mathcal{B}$, the estimation problem reduces to $B\times B + B \times (B+1)/2\times 2 = B(2B+1)$ free contact rate parameters rather than $4B^2$ free parameters, an almost 50\% reduction. To take advantage of these self-consistency constraints, we follow~\cite{van_de_kassteele_efficient_2017} and expand Eq~\eqref{lkl-contact-patterns-b} to
\begin{subequations}\label{random-functions}
\begin{align}
\log \cint & = \beta_0 + \bmf^{gh}(a,b) + \log(\Pop[b]{h}), \quad g=M, \: h=F, \: a, b\in\mathcal{B},\\
\log \cinthg & = \beta_0 + \bmf^{gh}(b,a) + \log(\Pop[b]{g}),\quad g=M, \: h=F, \: a, b\in\mathcal{B},\\
\log \cintgg & = \beta_0 + \bmf^{gg}(a,b) + \log(\Pop[b]{g}),\quad g\in \{M,F\}, \: a\leq b, \\
\log \cintgg & = \beta_0 + \bmf^{gg}(b,a) + \log(\Pop[b]{g}),\quad g\in \{M,F\}, \: a>b,
\end{align}
\end{subequations}
where $\beta_0 \in \mathbb{R}$ is a real-valued baseline parameter, and $\bmf^{MF}$, $\bmf^{MM}$, $\bmf^{FF}$ are three real-valued, random functions of two-dimensional continuous inputs on the compact domain $[0,84]\times [0,84]$. Specifically, we model the $\bmf^{gh}$ through computationally efficient Gaussian process approximations as described below, and for ease of notation, simply write $\bmf^{gh}(a,b)$. The random functions act as age-age-specific offsets to the baseline parameter and thus capture the age structure in human contact intensities. Using random functions, we can estimate arbitrary age-specific contact patterns, which is important because human contact patterns have changed substantially since the COVID-19 pandemic with school closures and other non-pharmaceutical interventions.
\subsection*{Recovering fine age structure from coarse data}
For COVIMOD and similar contact surveys, participants were asked to report their contacts in the coarse age groups
\begin{equation}
\begin{split}
c\in\mathcal{C} = \{& \text{0-4, 5-9, 10-14, 15-19, 20-24, 25-34, 35-44,} \\ &\text{45-54, 55-64, 65-69, 70-74, 75-79, 80-84} \}
\end{split}
\end{equation}
to facilitate reporting because participants often do not know or remember the exact age of their contacts. Importantly, the exact age of the participant is known, and we can leverage this information through the symmetry property in Eq~\eqref{contact-rate-symmetry} to estimate contact intensities at a much finer resolution. Because of this fundamental property, we call our resulting model the ``Bayesian rate consistency model". Using the shape-scale parameterisation in Eq~\eqref{lkl-contact-patterns}, it follows that
\begin{subequations}\label{coarse-lkl-contact-patterns}
\begin{align}
Y_{ac}^{gh} & = \sum_{b \in c} \ccnt \sim \text{NegBinomial}\left(\sum_{b \in c} \alpha_{ab}^{gh}, \frac{\odisp}{1 + \odisp}\right)\\
\eccnt & = \alpha_{ab}^{gh} \odisp\\
\log \eccnt & = \log \cint + \log \Part, \label{coarse-lkl-contact-patterns-b}
\end{align}
\end{subequations}
where $g,h\in\{M,F\}$, $a\in \AgesPartG$, $b\in \mathcal{B}$, and $c\in\mathcal{C}$. We will demonstrate below that the high-resolution contact intensities $\cint$ are identifiable from coarse contact data.
\subsection*{Estimating dynamics in contact patterns}
It is in principle, straightforward to extend the model (\ref{lkl-contact-patterns}-\ref{coarse-lkl-contact-patterns}) to capture time trends in contact patterns, but a particular challenge arises in the context of repeat surveillance. Across COVIMOD survey waves, many participants agreed to report data in multiple rounds, and analyses indicate that participants tend to repeat fewer contacts in subsequent surveys regardless of the time they were initially surveyed, a phenomenon we call reporting fatigue. The time trends in the primary data are thus confounded by longitudinal reporting behaviour. To control for reporting fatigue, we denote by $Y_{trac}^{gh}$ the number of contacts to individuals of age group $c$ and gender $h$ that are reported at survey time $t$ by all participants of age $a$ and gender $g$ who have participated $r$ time(s). All other notation extends analogously.
In the simplest case, we introduce age-homogeneous reporting fatigue effects $\rho_r \in \mathbb{R}$ at repeat response times $r= 0,1,2,\dotsc$, and jointly model the longitudinal data with
\begin{subequations}\label{coarse-lkl-contact-dynamics}
\begin{align}
Y_{trac}^{gh} & \sim \text{NegBinomial}\left(\sum_{b \in c} \alpha_{trab}^{gh}, \frac{\odisp}{1 + \odisp}\right)\\
\mu_{trab}^{gh} & = \alpha_{trab}^{gh} \odisp\\
\log \mu_{trab}^{gh} & = \log m_{tab}^{gh} + \rho_r + \log(\Parttr),
\end{align}
\end{subequations}
where $t=1,2,\dotsc$ indicates the survey waves, $r= 0,1,2,\dotsc$ repeat surveillance, and $\rho_0 = 0$ for ease of notation. Here, reporting fatigue is captured by negative $\rho_r$ (that decreases with $r$), which in turn will adjust the contact intensities $\log m_{tac}^{gh}$ in follow-up survey rounds to higher estimates than the primary data suggest.
New participants entered the COVIMOD survey in each survey wave, so we have data to provide independent information on the contact dynamics and reporting fatigue, with the model borrowing strength across all the data available.
A second challenge is estimating the dynamics in contact patterns while accounting for missing \& aggregated contact reports. We denote the number of missing \& aggregate contact reports by participants of age $a$ and gender $g$ in survey wave $t$ by $T_{ta}^g$, and in addition, consider the number of total contacts with detailed age information by participants of age $a$ and gender $g$ in survey wave $t$ by $Y_{ta}^g = \sum_{r,c, h} Y_{trac}^{gh}$. Thus, we can calculate the proportion of contacts that are reported with detailed age information,
\begin{equation}
S_{ta}^g = Y_{ta}^g / (Y_{ta}^g + T_{ta}^g),
\end{equation}
and use it as an additional offset term in the linear predictor,
\begin{equation}\label{eq_group-contacts}
\log \mu_{trab}^{gh} = \log m_{tab}^{gh} + \rho_r + \log(\Parttr) + \log(S_{ta}^g).
\end{equation}
In practice, if increasingly many participants only provide aggregated contact reports, we have that $S_{ta}^g < 1$, and in turn, this will adjust the contact intensities $\log m_{tab}^{gh}$ in later survey waves to higher estimates than the data with full age-specific details suggest.
\subsection*{Non-parametric modelling of contact dynamics}
We regularise our inferences in high-dimensional parameter space by associating the random functions $\bmf_t^{MF}$, $\bmf_t^{MM}$, $\bmf_t^{FF}$ in Eq~\eqref{random-functions} with computationally efficient, zero-mean, two-dimensional Hilbert Space Gaussian Process approximation priors~\cite{xi_inferring_2022, riutort-mayol_practical_2022}. In the next sections, we will drop the time and gender sub- and superscripts to ease notation and present our modelling of a generic random function $\bmf$ that represents age structure in contact patterns.
Zero-mean two-dimensional Gaussian Processes (GPs) are powerful prior models for random functions. For any finite collection of two-dimensional inputs, the function values are multivariate normal with mean zero. We always have $AB$ count observations on the grid $x_1 = (a_1,b_1), \dotsc, x_{AB} = (a_A,b_B)$ defined by $A$ participant age groups in $\mathcal{A}$ and all possible $B$ population age groups in $\mathcal{B}$. The multivariate normal has then a covariance matrix $\bK\in\mathbb{R}^{AB\times AB}$ whose $i$, $j$\textsuperscript{th} entries are specified by a covariance kernel function $k(x_i, x_j)$. Here, we decompose the 2D kernel function for computational efficiency and model each component through squared exponential or Mat\'ern class kernels. Here, we have used squared exponential kernels as an example:
\begin{subequations}\label{squared-exp-kernels}
\begin{align}
& k\left( (a, b), (a', b') \right) = k^1(a, a')k^2(b, b') \label{kernel-kronecker-decomposition} \\
& k^1(a, a') = \mgnt_a^2\exp \left( -\frac{(a-a')^2}{2\lnsc_a^2} \right)\label{eq_kernel-eq} \\
& k^2(b, b') = \mgnt_b^2\exp \left( -\frac{(b-b')^2}{2\lnsc_b^2} \right),
\end{align}
\end{subequations}
where the scaling parameters $\mgnt_a$, $\mgnt_b$ control the magnitude of the random function in the corresponding dimension, and the lengthscale parameters $\lnsc_a$, $\lnsc_b$ control the bandwidth. The product in Eq~\eqref{kernel-kronecker-decomposition} is also known as Kronecker decomposition, because the covariance matrix $\bK$ equals the Kronecker product of the covariance matrices of the kernels with one-dimensional inputs, $\bK = \bK^2 \otimes \bK^1$, where the $i$, $j$th entry in $\bK^1\in\mathbb{R}^{A\times A}$ is given by $k^1(a_i, a_j)$ and the $i$, $j$th entry in $\bK^2\in\mathbb{R}^{B\times B}$ is given by $k^2(b_i, b_j)$. For computing purposes, we exploit that $\bK^1$, $\bK^2$ are positive semi-definite and decompose the covariance matrices as $\bK^1= \bL^1 {\bL^1}^\top$, $\bK^2= \bL^2 {\bL^2}^\top$, where the superscript $\top$ denotes transposition. Using the mixed product property of Kronecker operations, we obtain
\begin{equation}
\bK = \left( \bL^2 \otimes \bL^1 \right)\left( \bL^2 \otimes \bL^1 \right)^{\top}.
\end{equation}
This shows that the zero-mean two-dimensional GP prior attached to the random function $\bmf$ on the $AB$ inputs $\bx=(x_1, \dotsc, x_{AB})$ can be obtained by linear transformation of $(AB)^2$ i.~i.~d. standard Gaussian random variables $\bz \sim \mathcal{N}(0,1)$,
\begin{equation}\label{GP-representation}
\bmf(\bx) = \left( \bL^2 \otimes \bL^1 \right) \bz = \text{vec}\left( \left( \bL^2 \Big(\bL^1 \: \text{reshape}(\bz, A, B) \Big)^{\top} \right)^{\top}\right).
\end{equation}
In Eq~\eqref{GP-representation}, the left-hand side denotes the $AB$-dimensional column vector of the random function evaluated at the inputs, and the right-hand side shows how the Kronecker product is calculated by a series of basic arithmetic operations. The $\text{reshape}$ operation transforms the $AB$ dimensional column vector $\bz$ column-wise into a $A\times B$ dimensional matrix, and the $\text{vec}$ operation flattens $A\times B$ dimensional matrices column-wise into an $AB$ dimensional column vector.
Eq~\eqref{GP-representation} also shows that the computational cost of two-dimensional GPs is entirely determined by calculating, first, $\bL^1$, $\bL^2$ for each new set of GPs hyperparameters, and then, second, performing the arithmetic operations associated with $( \bL^2 \otimes \bL^1 )\bz$. We further use Hilbert Space Gaussian Process (HSGP) approximations\cite{solin_hilbert_2020} to each of the kernels $k^1$ and $k^2$ in Eq~\eqref{squared-exp-kernels} to substantially reduce the cost associated with the first step. For brevity, we refer readers to the excellent introductions to HSGPs in~\cite{xi_inferring_2022, riutort-mayol_practical_2022}, and here merely note that the stationary isotropic kernels can be expressed as an infinite sum that involves the spectral density $S$, eigenfunctions $\phi_i$ and eigenvalues $\lambda_i$, $i=1,\dotsc,\infty$, associated with a certain Laplacian eigenvalue problem on a compact domain $\Omega$ that is strictly larger than $\mathcal{B}$. For convenience, the input domain $\mathcal{B}$ is shifted with the midpoint at zero, and then $\Omega$ is written as $[-L, L]$ for some $L>0$. To ease notation, we continue to write the shifted inputs as $a_i$, and $b_i$ in what follows. The HSGP approximation $\HSGPk^1$ to $k^1$ on the domain $[-L^1,L^1]$ is then obtained by truncating the infinite sum to the first $M^1$ terms,
\begin{subequations}\label{hsgp-series}
\begin{align}
k^1(a, a') \approx \HSGPk^1(a, a') & = \sum_{j=1}^{M^1}S^1(\sqrt{\lambda^1_j})\phi^1_j(a)\phi^1_j(a'),\label{hsgp-series-sum}\\
S^1(\omega) & = \mgnt_a^2(2\pi\lnsc_a)\exp\left(- \lnsc_a^2\omega^2/2\right) \label{hsgp-series-S},\\
\sqrt{\lambda^1_j} &= (j\pi)/(2L^1), \label{hsgp-series-ev}\\
\phi^1_j(x) &= \sqrt{1/L^1}\sin\left( \sqrt{\lambda^1_j}(x + L^1) \right).\label{hsgp-series-ef}
\end{align}
\end{subequations}
Note that the spectral density above is specific to squared exponential kernels and that expressions for Mat\'ern class kernels may be found in~\cite{riutort-mayol_practical_2022, carl_edward_rasmussen_gaussian_2006}. Crucially, the GP hyperparameters $\mgnt_a$, $\lnsc_a$ enter only in Eq~\eqref{hsgp-series-S}, and the eigenvalues and eigenfunctions are the same regardless of the GP hyperparameters and depend only on the domain boundary value $L^1$ together with the observed inputs $a$, $a'$. This speeds up Bayesian computations significantly because the eigenvalues in Eq~\eqref{hsgp-series-ev} and eigenfunctions in Eq~\eqref{hsgp-series-ef} can be precomputed once and for all. Rewriting Eq~\eqref{hsgp-series-sum} in matrix notation, we see that $\bL^1$ is approximated by
\begin{equation}\label{hsgp-L-approx}
\bL^1 \approx \tilde{\bL}^1 = \bm{\Phi}^1\sqrt{\Delta^1},
\end{equation}
where the $A\times M^1$ matrix $\bm{\Phi}^1$ has the $i,j$ entries $\phi^1_j(a_i)$, and the $M^1\times M^1$ matrix $\bm{\Delta}^1$ is diagonal with $j,j$ entries $S^1(\sqrt{\lambda^1_j})$. Again, $\bm{\Phi}^1$ does not depend on the GP hyperparameters and can be precomputed. The arithmetic operations in Eq~\eqref{hsgp-L-approx} can harness computationally efficient diagonal-post-multiply functions in many linear algebra libraries. The HSGP approximation to the $k^2$ kernel is analogous. The tuning parameters of the HSGP approximations are the integers $M^1$, $M^2$ and the boundary values $L^1$, $L^2$, and we determine these using established diagnostics~\cite{riutort-mayol_practical_2022}. The zero-mean Kronecker-decomposed HSGP prior associated with our random functions $\bmf$ on the input grid $\bx$ is then
\begin{equation}\label{HSGP-representation}
\bmf(\bx) = \left( \tilde{\bL}^2 \otimes \tilde{\bL}^1 \right) \tilde{\bz} = \text{vec}\left( \left( \tilde{\bL}^2 \Big(\tilde{\bL}^1 \: \text{reshape}(\tilde{\bz}, M^1, M^2) \Big)^{\top} \right)^{\top}\right),
\end{equation}
where $\tilde{\bz}$ is a $M^1M^2$ dimensional column vector of i.~i.~d. standard normal random variables, and the non-negative hyperparameters are $\theta= (\mgnt_a,\lnsc_a,\mgnt_b,\lnsc_b)$.
\subsection*{Difference-in-age parameterisation}
Human contact patterns tend to concentrate among individuals of similar age and individuals with similar age gaps (parent-child, grandparent-child and grandparent-parent). To capture this diagonal structure in the simple Kronecker decomposed priors in Eq~\eqref{squared-exp-kernels} for our 2D random functions $\bmf$, we follow~\cite{vandendijck_cohort-based_2022} and define $\bmf$ on an age by difference-in-age space rather than an age by age space. This amounts to rotating the age-by-age space by 45 degrees so that the peer-peer, parent-child, grandparent-child, and grandparent-parent contacts correspond to horizontal lines in the re-parameterised space and match the structure of our Kronecker decomposed priors~\eqref{squared-exp-kernels} (See Fig 1 of Vandendijck et al.~\cite{vandendijck_cohort-based_2022}).
Specifically, we consider age differences $d\in\mathcal{D}=\{ -84, -83, \ldots, 83, 84 \}$, and re-parameterise the points $(a,b) \in \mathcal{A}\times\mathcal{B}$ to $(a,d) = d(a,b) = (a, b-a) \in \mathcal{A}\times\mathcal{D}$. The number of age differences $D = 169$ in $\mathcal{D}$ is larger than the number of one-year age groups $B = 85$, and we are only interested in the random functions evaluated on the original points, which we write as $\bmf(d(a,b))$ for all $(a,b)\in \mathcal{A}\times\mathcal{B}$. We will show below that the difference-in-age parameterisation is associated with higher estimation accuracy of typical diagonal and off-diagonal human contact patterns at higher computational costs that arise from the larger input space with an additional $A^2 - A$ elements.
\subsection*{Full Bayesian model and numerical inference}
To complete our model for inferring contact dynamics from longitudinal survey data, we specified commonly used priors on all remaining model parameters, leading for survey waves $t=1,\dotsc,5$, reporting repeats $r=0,\dotsc,4$, gender $g,h \in \{M,F\}$, participant age groups $a\in\mathcal{A}^{trg}$ and population age groups $b\in\mathcal{B}$ to
\begin{subequations} \label{eq:full-model}
\begin{align}
Y_{trac}^{gh} & \sim \text{NegBinomial}\left(\sum_{b \in c} \alpha_{trab}^{gh}, \frac{\odisp}{1+\odisp}\right)\\
\mu_{trab}^{gh} & = \alpha_{trab}^{gh} \odisp\\
\log \mu_{trab}^{gh} & = \log m_{tab}^{gh} + \rho_r + \log(\Parttr) + \log(S^g_{ta})\\
\log m_{tab}^{gh} & = \beta_0 + \tau_t + \bmf_t^{gh}\big(d(a,b)\big) + \log(\Pop[b]{h}), \quad g=M, \: h=F, \: a, b\in\mathcal{B}\\
\log m_{tab}^{hg} & = \beta_0 + \tau_t + \bmf_t^{gh}\big(d(b,a)\big) + \log(\Pop[b]{g}),\quad g=M, \: h=F, \: a, b\in\mathcal{B}\\
\log m_{tab}^{gg} & = \beta_0 + \tau_t +\bmf_t^{gg}\big(d(a,b)\big) + \log(\Pop[b]{g}),\quad g\in \{M,F\}, \: a\leq b, \\
\log m_{tab}^{gg} & = \beta_0 + \tau_t + \bmf_t^{gg}\big(d(b,a)\big) + \log(\Pop[b]{g}),\quad g\in \{M,F\}, \: a>b,
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\beta_0 & \sim \mathcal{N}(0, 10) \\
\rho_r & \sim \mathcal{N}(0, 1) \\
\tau_t & \sim \mathcal{N}(0, 1) \\
\odisp & \sim \text{Exponential}(1) \\
\bmf_{t}^{gh}\big(d(\bx)\big)|\mgnt_{ti}, \lnsc_{ti} & \sim \text{HSGP}(\bm{0}, \tilde{\bL}_t^{gh,2} \otimes \tilde{\bL}_t^{gh,1} ), \quad gh\in \{ MF, MM, FF\}, i = 1, 2 \\
\mgnt_{ti} & \sim \text{Cauchy}^+(0,1),\quad i = 1,2 \\
\lnsc_{ti} & \sim \text{InvGamma}(5, 5),\quad i = 1,2.
\end{align}
\end{subequations}
Monte Carlo draws from the joint posterior distribution of all parameters were obtained with the probabilistic computing language \texttt{Stan}~\cite{carpenter_stan_2017} via the \texttt{cmdstanr} interface version 0.5.2. Eight chains were run in parallel for 500 warmup iterations and 1000 iterations thereafter. Initial sampling was facilitated by adding the nugget $10^{-13}$ to $\alpha_{trab}^{gh}$. We typically observed a small number of divergences in the NUTS algorithm, but these accounted for less than 0.005\% of samples and were considered to be of no concern. The typical minimum effective sample sizes were 1627, and the $\hat{R}$ convergence diagnostics were below $1.01$, indicating that the Markov chains converged and mixed well~\cite{vehtari_rank-normalization_2021, betancourt_conceptual_2018}. The corresponding trace plots are shown in the Supp.
\subsection*{Simulated social contact data}
To validate the Bayesian models, we created synthetic datasets that mimic the social contact patterns with some simplifications before the COVID-19 pandemic (pre-COVID-19) and during the pandemic (in-COVID-19). We also varied the participant sample size in our experiments to assess its effect on estimation accuracy. To reduce experiment run time, we limited participants and contacts ages 6 to 49 and assumed that contact intensity patterns do not vary by gender. We generated contact intensity patterns based on the crude estimated marginal contact intensities of targeted age groups, which we obtained from the POLYMOD (pre-COVID-19)~\cite{mossong_social_2008} and CoMix studies (in-COVID-19)~\cite{jarvis_quantifying_2020} studies.
Contact intensities were set to be highest among individuals of similar age, mimicking age-assortative contact behaviour. To simulate parent-children contact dynamics, we define individuals between 6-18 as children and individuals between 30-39 as parents and increase the contact intensities between these two age categories. Similarly, individuals between 19-29 are defined as children of individuals from 40 to 49. The resulting patterns are shown in the top left panel of Fig~\ref{fig:sim-exp-pre} and Fig~\ref{fig:sim-exp-in}. For full details, we refer the readers to~\nameref{S2_Text}.
From the stylised contact intensity scenarios, we next randomly generated age- and gender-specific contact counts for five different participant size configurations, $N = 250, 500, 1000, 2000, 5000$, by sampling from a Poisson distribution such that $Y_{ab}^{gh} \sim \text{Poisson}(\lambda_{ab}^{gh})$ where $\lambda_{ab}^{gh} = \tilde{m}_{ab}^{gh}N_{a}^{g}$. We set $N_{a}^{g}$ such that the age-gender counts of the participants is representative of the 2011 German census population~\cite{statistische_amter_des_bundes_und_der_lander_zensus2011_2011}. To mimic the age reporting scheme in the COVIMOD surveys, we aggregate the simulated contact counts by $Y^{gh}_{ac} = \sum_{b \in c} Y_{ab}^{gh}$ where,
$$
c \in \mathcal{C}^{\text{sim}} = \{ \text{6-9, 10-14, 15-19, 20-24, 25-34, 35-44, 45-49}\},
$$
as illustrated in the top right panels of Fig~\ref{fig:sim-exp-pre}-\ref{fig:sim-exp-in}. In this fashion, we generated 10 replicate datasets for each experiment configuration (pre-COVID-19/in-COVID-19 and sample size) to obtain representative accuracy and runtime estimates. As we were interested in the performance of our GP-based models for the age by age and difference-in-age by age parameterisations, we ran cross-sectional versions of our model in Eq~\eqref{eq:full-model} where we dropped the time and repeat terms.
\section*{Results}
\subsection*{Contact patterns by 1-year age band can be estimated}
\begin{figure}[!t]
\includegraphics[width=\textwidth]{figures/figure-2.jpeg}
\caption{{\bf Pre-COVID19 scenario simulation experiments.}
(Top left) Simulated social contact intensities for Male-Male contacts. (Top right) Simulated social contact counts for Male-Male contacts with a COVIMOD-like age aggregation scheme. (Bottom left) Estimated social contact intensities from the age-age parameterisation HSGP model. (Bottom right) Estimated social contact intensities from the difference-in-age parameterisation HSGP model.}
\label{fig:sim-exp-pre}
\end{figure}
Fig~\ref{fig:sim-exp-pre} illustrates the fits of the Bayesian rate consistency model using the age-age and difference-in-age parameterisations for the pre-COVID-19 scenario, and Fig~\ref{fig:sim-exp-in} for the in-COVID-19 scenario, both with a sample size of 2,000. The age-age parameterisation performed poorly for both simulation scenarios, especially in regions of the contact matrix where the degree of age aggregation is large, i.e., 10-year age intervals as opposed to 5-year age intervals. For such large reporting intervals of age groups, the contact intensity patterns that we could estimate with the age-age parameterisation showed idiosyncratic bimodal patterns along the main diagonal of the contact intensity matrices (bottom left panels in Fig~\ref{fig:sim-exp-pre}-\ref{fig:sim-exp-in}). In comparison, the difference-in-age parameterisation captured age-assortative contact patterns and the sub-diagonal parent-children contact patterns with much better accuracy. The estimated contact intensity patterns for other gender combinations and other simulation scenarios were qualitatively very similar and are reported in~\nameref{S3_Fig} and~\nameref{S4_Fig}.
\begin{figure}[!t]
\includegraphics[width=\textwidth]{figures/figure-3.jpeg}
\caption{{\bf In-COVID19 scenario simulation experiments.}
(Top left) Simulated social contact intensities for Male-Male contacts. (Top right) Simulated social contact counts for Male-Male contacts with a COVIMOD-like age aggregation scheme. (Bottom left) Estimated social contact intensities from the age-age parameterisation HSGP model. (Bottom right) Estimated social contact intensities from the difference-in-age parameterisation HSGP model.}
\label{fig:sim-exp-in}
\end{figure}
\subsection*{Gaussian process approximations enable fast Bayesian inference}
In Table~\ref{table:sim-exp}, we numerically compare the performance of the Bayesian rate consistency model with various parameterisations for different sample sizes and scenarios in terms of estimation accuracy and computing runtimes. Specifically, to assess how well HSGP models can approximate full-rank 2DGP models, we ran simulations for both scenarios and parameterisations with a sample size fixed at 2000. We compared the model fits to the simulation truth in terms of the mean absolute error (MAE) based on the age-age-specific inferred contact intensities, expected log posterior density (ELPD), the percentage that the predicted values are inside the 95\% prediction intervals according to posterior predictive check (PPC) and the median running time.
\begin{table}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\centering
\caption{
{\bf Comparison of performance on simulated data for different scenarios, models, sample sizes, and parameterisations.}}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\bf Scenario & \bf N & \bf Model & \bf Parameterisation & \bf MAE$^a$ & \bf ELPD$^b$ & \bf PPC$^c$ & \bf Runtime$^d$ \\ \thickhline
pre$^e$ & 2000 & 2DGP & age-age & $9.17 \times 10^{-2}$ & -3224.1 & 99.9\% & 1.3 hours \\ \hline
pre & 2000 & HSGP & age-age & $9.04 \times 10^{-2}$ & -3175.8 & 99.8\% & 0.5 hours \\ \hline
pre & 2000 & 2DGP & difference-in-age & $4.24 \times 10^{-2}$ & -3017.7 & 98.2\% & 27.5 hours\\ \hline
pre & 2000 & HSGP & difference-in-age & $4.44 \times 10^{-2}$ & -3027.6 & 98.5\% & 2.1 hours\\ \hline
in$^f$ & 2000 & 2DGP & age-age & $4.91 \times 10^{-2}$ & -2745.4 & 99.9\% & 1.5 hours \\ \hline
in & 2000 & HSGP & age-age & $4.93 \times 10^{-2}$ & -2720.4 & 99.7\% & 0.4 hours \\ \hline
in & 2000 & 2DGP & difference-in-age & $2.77 \times 10^{-2}$ & -2639.1 & 98.5\% & 16.7 hours\\ \hline
in & 2000 & HSGP & difference-in-age & $2.85 \times 10^{-2}$ & -2674.5 & 98.7\% & 1.4 hours \\ \thickhline
pre & 5000 & HSGP & difference-in-age & $4.17 \times 10^{-2}$ & -3865.2 & 98.2\% & 2.7 hours \\ \hline
pre & 2000 & HSGP & difference-in-age & $4.44 \times 10^{-2}$ & -3027.6 & 98.5\% & 2.1 hours\\ \hline
pre & 1000 & HSGP & difference-in-age & $4.79 \times 10^{-2}$ & -2587.8 & 98.7\% & 1.5 hours \\ \hline
pre & 500 & HSGP & difference-in-age & $5.20 \times 10^{-2}$ & -2256.5 & 98.7\% & 1.7 hours \\ \hline
pre & 250 & HSGP & difference-in-age & $5.73 \times 10^{-2}$ & -1765.6 & 99.0\% & 0.7 hours \\ \hline
in & 5000 & HSGP & difference-in-age & $2.60 \times 10^{-2}$ & -3357.8 & 98.4\% & 2.6 hours \\ \hline
in & 2000 & HSGP & difference-in-age & $2.85 \times 10^{-2}$ & -2674.5 & 98.7\% & 1.4 hours \\ \hline
in & 1000 & HSGP & difference-in-age & $3.23 \times 10^{-2}$ & -2245.7 & 99.0\% & 1.4 hours \\ \hline
in & 500 & HSGP & difference-in-age & $3.27 \times 10^{-2}$ & -1716.6 & 99.1\% & 1.3 hours \\ \hline
in & 250 & HSGP & difference-in-age & $4.94 \times 10^{-2}$ & -1327.2 & 99.5\% & 1.3 hours \\ \hline
\end{tabular}
\begin{flushleft}
$^a$Mean absolute error, $^b$Expected log posterior density, $^c$Posterior predictive check, $^d$Median runtime, $^e$pre-COVID19 scenario, $^f$in-COVID19 scenario.
\end{flushleft}
\label{table:sim-exp}
\end{adjustwidth}
\end{table}
Next, the difference-in-age parameterisation achieved significantly better accuracy than the age-age parameterisation but also required more time to fit due to the introduction of additional $A^2 - A$ nuisance parameters. The computational toll of the difference-in-age parameterisation was most strongly reflected in the median runtimes under the full-rank 2DGPs, which could take more than a day to fit. Using HSGPs, we reduced median runtimes by more than 20-fold.
However, the HSGP models resulted in a slight decrease in accuracy (MAE and ELPD) than full-rank 2DGP models.
Finally, the bottom section of Table~\ref{table:sim-exp} compares the accuracy of the difference-in-age HSGP models across survey sample sizes. In general, a larger sample size entailed longer computations. Smaller sample sizes led to less accurate estimates, with a 5-fold reduction in sample size resulting in approximately a 10-15\% increase in MAE for the pre-COVID-19 scenario and a 20-35\% increase in MAE for the in-COVID-19 scenario.
\subsection*{Modelling marked structures in age-specific contact patterns}
Social contacts are strongly structured by age, reflecting common behaviour and social norms around family size, reproductive age, schooling, and other factors ~\cite{vandendijck_cohort-based_2022}. In turn, smooth process kernels such as the squared exponential (Eq~\eqref{eq_kernel-eq}) may not be well suited to describe marked changes in contact intensities. On our simulated contact scenarios, we find indeed that Mat\'ern $\frac{5}{2}$ and Mat\'ern $\frac{3}{2}$ kernels performed better in comparison to squared exponential kernels in terms of accuracy (Table~\ref{table:sim-exp}, \nameref{S5_Table}). The difference in accuracy between Mat\'ern $\frac{3}{2}$ and Mat\'ern $\frac{5}{2}$ was small and qualitatively indistinguishable (\nameref{S3_Fig}, \nameref{S4_Fig}), and in the following results we considered the Mat\'ern $\frac{5}{2}$ kernel.
\subsection*{Model-based estimates of contact patterns in Germany by 1-year age groups}
The age- and gender-specific crude empirical contact intensities, contact intensity estimates from the \texttt{socialmixr} package~\cite{funk_socialmixr_2020} estimated via bootstrapping, and contact intensity estimates from the Bayesian rate consistency model for the first wave of COVIMOD are shown in Fig~\ref{fig:4}. Here, the ``crude" contact intensities were calculated from the data without any statistical modelling via
\begin{equation*}
\hat{m}_{ac}^{gh} = \frac{Y_{ac}^{gh}}{N_a^g} \frac{1}{S_{a}^{g}},
\end{equation*}
where $Y_{ac}^{gh}$ denote the total number of contacts from participants of gender $g$ and age group $a$ to individuals of gender $h$ and age category $c$, $N_a^g$ are age- and gender-specific sample size, and $S_{a}^{g}$ are age- and gender-specific proportion of reports with complete age and gender information. The exact runtime arguments for this comparison are given in script \texttt{figure-4.R} on our accompanying GitHub repository. The crude estimates are sparse and fluctuate greatly, even between neighbouring age groups. They are also not symmetric in contact rates. In particular, some parents answered the survey on behalf of their children resulting in a larger asymmetry between parent-to-children versus children-to-parent contacts.
The estimates from \texttt{socialmixr} aggregate the observations by the large age categories in which the contacts were reported and did not borrow available information through the exact age of participants to obtain higher resolution estimates. The \texttt{socialmixr} estimates are adjusted for symmetry in contact rates but not reporting fatigue or missing \& aggregate contact reports (see respectively Eq~\eqref{coarse-lkl-contact-dynamics} and Eq~\eqref{eq_group-contacts}). Furthermore, contacts with missing age information are imputed by sampling the missing age only from all contacts of the participants of the same age group~\cite{funk_socialmixr_2020}.
The estimated contact patterns from the Bayesian rate consistency model align with those estimated by \texttt{socialmixr} but provide much higher age resolution. Importantly, we achieve this higher resolution not via imputation but by logical constraints on who contacts whom in a closed population (recall Eq~\eqref{coarse-lkl-contact-patterns}). These constraints imply that data on the exact age of survey participants provides information on the exact contacts even though they are reported in coarse age brackets. The patterns reveal strong age-assortativeness in mixing patterns indicated by the high intensities on the main diagonals of the contact intensity matrices shown in the bottom row of Fig~\ref{fig:4}. Lying approximately 30 years away from the main diagonal, two strips of high contact intensity fade with increasing age and correspond to inter-generational contacts between parents and children. This age-dependent pattern persists over time (\nameref{S6_Fig} to \nameref{S9_Fig}). However, we show below that the increases in social contact intensities in subsequent waves were far from uniform.
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-4.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Empirical and estimated contact intensity patterns for COVIMOD wave 1}. (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-4.R} on our accompanying GitHub repository.}
\label{fig:4}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-5.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Marginal social contact intensities}. Dark blue dashed lines represent estimates without adjustments. Green dashed lines represent estimates adjusted for reporting fatigue but not missing \& aggregated contact reports. Turquoise dashed lines represent estimates adjusted for missing \& aggregated contact reports but not for reporting fatigue. Red solid lines represent estimates that are adjusted for both. Red bands represent 95\% credible intervals for adjusted estimates.}
\label{fig:5}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\subsection*{Controlling for time-varying reporting effects}
We can sum the contacts' age dimension of estimated contact intensities to obtain the average number of contacts from one person of age $a$ per day. For brevity, we call these ``marginal" contact intensities. In Fig~\ref{fig:5}, we show the estimated marginal contact intensities under the Bayesian rate consistency model, which simultaneously accounts for the time-varying reporting effects that emerge through reporting fatigue (repeat participation in the longitudinal COVIMOD survey) and missing and aggregate contact reports (participants unable to list contacts individually), and which are clearly present in the data as shown in Fig~\ref{fig:1} and \nameref{S1_Fig}. We compare these (adjusted) marginal contact intensities in Fig~\ref{fig:5} to those obtained without adjusting for missing \& aggregate contact reports (but adjusting for reporting fatigue), those obtained without adjusting for reporting fatigue (but adjusting for missing and aggregate contact reports), and those obtained without any adjustments for time-varying, missing \& aggregate contact reports or reporting fatigue. Importantly, the reporting effect sizes can be estimated simultaneously and thus borrow strength across data from neighbouring age groups and are mathematically consistent with the symmetry constraints in contact rates in closed populations. The results clearly show that adjusting for missing and aggregate contact reports and reporting fatigue significantly increased the estimated marginal contact intensities in a non-trivial manner that depends on the contribution of repeat survey participants to each survey wave. As contact reduction measures were progressively eased between wave 1 to wave 5 of the COVIMOD survey, the adjusted estimates are more consistent with the timeline of non-pharmaceutical interventions in Germany.
\subsection*{Overall time trends in contact intensities from May to July 2020 in Germany}
Fig~\ref{fig:5} shows that, overall, the contact intensities increased consecutively from wave 1 to wave 5, although the increases were most substantial from wave 1 to wave 2. Fig~\ref{fig:6} illustrates the relative percentage increase in the marginal contact intensities relative to those in wave 1. Although there were marked differences in the contact patterns between men and women (Fig~\ref{fig:5}), the relative increases showed no significant differences between the two genders, which suggests that the gender differences in contact patterns may arise from underlying gender-dependent contact dynamics rather than non-pharmaceutical interventions. For waves 2 and 3, increases in contact intensities were higher in adults over 30 and were generally age-homogeneous. In wave 4, we observe a sharp increase in contacts among men approximately 20 years of age, but we find this pattern is sensitive to data pre-processing criteria as we explain below. In wave 5, the increase in contact intensities showed a rising pattern with age, where contact intensities increased most in older individuals.
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-6.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Relative percentage change of marginal contact intensities from wave 1}. The Red and blue lines represent posterior median estimates of the relative percentage change for adjusted marginal contact intensity estimates in females and males, respectively. Shaded ribbons represent 95\% credible intervals.}
\label{fig:6}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\subsection*{Social contact intensities remained largely below pre-pandemic levels}
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-7.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Marginal social contact intensities.} Solid lines represent COVIMOD estimates after adjusting for aggregate contacts and reporting fatigue. Dashed lines represent estimates from the pre-pandemic POLYMOD study. Shaded ribbons represent 95\% credible intervals.}
\label{fig:7}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
To gain more insights into the difference between social contact patterns before and during the COVID-19 pandemic, we compared the contact intensities seen during the first 5 waves of the COVIMOD study to those observed in the pre-pandemic POLYMOD study conducted between 2006 and 2008~\cite{mossong_social_2008} (Fig~\ref{fig:7}). We find that by wave 5, the contact intensities of individuals aged 70 were very similar to pre-pandemic levels. In contrast, the social contact intensities of all younger age groups in women and men remained substantially below pre-pandemic levels. Furthermore, similar to POLYMOD, we find that women between age 20 and 50 had more contacts than men of the same age range (\nameref{S10_Fig}).
\subsection*{Differential rebound of age-specific social contacts}
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/figure-8.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption{{\bf Time evolution in age-specific social contact intensities.} The conditional contact intensities (top row) and relative change in the conditional contact intensities in waves 2 to 5 relative to those in wave 1 (bottom row) for individuals aged 10, 20, 35 and 70, respectively. Conditional contact intensities were aggregated across men and women. The colours represent different COVIMOD survey waves, and the shaded ribbons represent 95\% credible intervals. We only show credible intervals for wave 5 to reduce overlaps and ease interpretation.}
\label{fig:8}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
We next focus on characterising the dynamics in contact intensities for specific age groups. Intuitively, this corresponds to slicing the contact intensity matrices reported in Fig~\ref{fig:4}, \nameref{S6_Fig} to \nameref{S9_Fig} across rows for a fixed column, and for brevity, call these the ``conditional" contact intensities. In Fig~\ref{fig:8} (top), we illustrate the conditional contact intensities for individuals aged 10, 20, 35, and 70 years to represent the contact intensities of school children, young adults, the working population, and the ageing population. For participants aged 10 and 20, we observe two peaks: a larger sharp peak corresponding to contacts between peers and a shorter rounded peak with individuals approximately 45-50 years older, representing contacts with their parents.
For participants aged 35, we observe an additional third peak with individuals aged 60 to 70, predominantly corresponding to contacts with their parents. Participants aged 70 generally mixed with individuals of a similar age, with some contact between individuals aged approximately 40 but almost no contact with individuals under 20. These core patterns of social contacts are present across all survey waves.
Fig~\ref{fig:8} (bottom) shows the ratio of the conditional contact intensities in survey waves 2, 3, 4 and 5 relative to those in wave 1. Ratios above 1 thus indicate increases in social contacts of one individual with age shown in column facets to individuals with age indicated on the x-axis in Fig~\ref{fig:8}. We find that the increases in social contacts were not homogeneous by the age of contacted individuals. Focusing on wave 5 relative to wave 1, we find that for children aged 10, the conditional contact intensities in individuals of the same age and roughly aged 70 rose particularly strongly. For individuals aged 20, increases in their social contacts were more homogeneous, except for those with young children, which remained similar to those seen in wave 1. For individuals aged 35, increases in their social contacts were concentrated in slighter younger and all older individuals, but not their peers, reflecting that individuals in the 35-year age group retained social contact with their peers during intense non-pharmaceutical interventions in wave 1. Individuals aged 70 homogeneously increased their conditional contact intensities with younger individuals.
\section*{Discussion}
We developed the Bayesian rate consistency model in order to regain the ability to quantify and characterise social contact patterns at high age resolution from contemporary, longitudinal survey data.
The key contributions of our model-based approach to estimating the trends in human contact patterns are as follows. Our approach can accurately estimate high-resolution social contact patterns from data that aggregate the age of contacts into large age bands. We also provide a unified framework to adjust for the confounding effects due to the aggregated reporting of contacts and reporting fatigue. These advancements are particularly important for COVID-era studies from which fine-age contact reporting is unavailable and in which participants contributed to multiple survey waves~\cite{coletti_comix_2020, backer_impact_2021, verelst_socrates-comix_2021, gimma_changes_2022}. We draw from methodologies which serve as principal workhorses in spatial statistics to map the landscape of social contacts~\cite{diggle_model-based_2007, ton_spatial_2017} and incorporate a recently developed GP approximation technique to alleviate the computational bottleneck which often plagues such spatial models~\cite{solin_hilbert_2020, riutort-mayol_practical_2022}. For more complex and high-dimensional models, e.g. those involving more survey waves and other participant demographics, recent advances in approximating GP priors with variational autoencoders such as priorVAE\cite{semenova_priorvae_2022} and $\pi$VAE~\cite{mishra_pivae_2022} may be incorporated within our fully Bayesian framework.
In applying our model to the first five waves of the COVIMOD study, we gained insights into the age- and gender-specific social contact dynamics and their evolution over time at high age resolution. Despite a rebound in contact intensities after the lifting of contact reduction measures, we found that the estimates were still considerably lower than the pre-pandemic estimates obtained through POLYMOD (Fig~\ref{fig:6}), indicating a sustained behavioural change. These results are consistent with from the CoMix arms in England~\cite{gimma_changes_2022}, Belgium~\cite{coletti_comix_2020}, and the Netherlands~\cite{backer_impact_2021}. This suggests that the sustained shifts in contact patterns might apply more generally beyond the individual sampling frames of each study. Relative to pre-pandemic estimates, the largest contact reductions occurred among children and young adults (Fig~\ref{fig:6}), possibly due to school closures and the transition to remote work. Such results may indicate that non-pharmaceutical interventions successfully reduced risk in groups with the highest infection risk in pre-pandemic times.
In general, our results can aid epidemiologists and policymakers by providing a clearer picture of how COVID-19 and other infectious respiratory illnesses are propagated through the population. They can be used to parameterise infectious disease models to obtain more realistic estimates for epidemiological quantities such as the reproduction number $R$~\cite{van_de_kassteele_efficient_2017, wallinga_optimizing_2010}. Contact estimates from the first wave of COVIMOD may be of particular interest as they represent the patterns during a time of stay-at-home orders across Germany and subsequent relaxation of non-pharmaceutical interventions. As such, they could be used as template social contact patterns for modelling and forecasting future pandemics~\cite{flaxman_estimating_2020, monod_age_2021}, and pandemic preparedness building~\cite{wallinga_optimizing_2010}. This point is particularly important because the dynamics in social contact patterns were highly non-homogeneous.
Further, despite a continued increase in contact intensity, COVID-19 cases remained stable during the analysis period (Fig~\ref{fig:1}). This may be because contact counts remained much lower than pre-pandemic estimates despite the ease in non-pharmaceutical interventions~\cite{tomori_individual_2021}. Other protective measures such as face masks, hygiene regulations including surface disinfection, remote work, and warmer summer weather may also have contributed to keeping infections at bay~\cite{leung_transmissibility_2021}.
Our work is not without limitations. First, some issues arise from the sampling methodology of the COVIMOD survey. Participants were recruited from an online panel for market research with email invitations, and although quota sampling was performed, the final samples were not fully representative of the population. Previous work proposed using post-stratification weights to re-scale the data, but a sensitivity analysis did not reveal large differences between weighted and unweighted estimates~\cite{tomori_individual_2021}. Participants consisted only of those with internet access who possibly adhered more to social distancing rules as such a demographic is more likely to respond to health surveys. Participants received guidance only through the text within the questionnaires, which may have been misinterpreted, and participants may report more contacts in paper-based surveys than in an online survey~\cite{beutels_social_2006}. Additionally, we truncate aggregate contact reports at 60, but different thresholds may lead to slight changes in the inference results.
Secondly, the difference-in-age parameterisation may not be appropriate if social contacts do not follow the pattern where high intensities lie on the contact matrix's main diagonal and sub-diagonals. This is relevant when investigators wish to conduct analyses for various contexts, e.g., work and transport, where contact patterns may not depend on age and age difference. However, it is easy for investigators to revert to the classical age-age parameterisation if they deem it more appropriate. We provide template \texttt{Stan} model files in the accompanying GitHub repository. Third, our fully Bayesian modelling framework is currently limited to analysing approximately 10 longitudinal survey rounds. While the recently proposed Hilbert Space Gaussian Process priors enable fast Bayesian inferences on cross-sectional data~\cite{xi_inferring_2022, solin_hilbert_2020, riutort-mayol_practical_2022}, additional research is needed to scale up the approach to survey data from 30 waves or more. Fourth, our model requires participant age information to be reported by 1-year age bands. The exact age of participants is usually recorded without error but not necessarily made publicly available~\cite{verelst_socrates-comix_2021}, which limits the applicability of the model-based approach developed here.
\section*{Conclusion}
In summary, we propose a novel model-based Bayesian framework for high-resolution estimation of age- and gender-specific social contact patterns. We validate our model on simulated social contact data for different scenarios and demonstrate the effectiveness of our approach. In applying our model to the COVIMOD contact survey, we provide a detailed picture of how social contact patterns evolved during Germany's first wave of the COVID-19 pandemic. This work promises to aid the understanding of contact behaviour, more realistic parameterisations of infectious disease models, and a deeper understanding of how infectious respiratory diseases are propagated through populations.
\section*{Supporting information}
\paragraph*{S1 Fig.}
\label{S1_Fig}
{\bf The number of complete contact reports and aggregated contact reports by age, gender, and COVIMOD wave.} Pink bars represent aggregated contact reports, light blue bars represent non-household contacts, and dark blue bars represent household contacts. Aggregate contact reports were truncated at 60 (90\textsuperscript{th} percentile of aggregate contact reports) to remove the effects of extreme outliers.
\paragraph{S2 Text}
\label{S2_Text}
{\bf The construction of simulated social contact patterns.} A detailed description of how the contact intensity patterns used in the simulation experiments are generated.
\paragraph*{S3 Fig.}
\label{S3_Fig}
{\bf Simulation experiment results for the pre-COVID19 scenario with different covariance kernels.}. From top to bottom: results for the squared exponential kernel, results for the Mat\'ern $\frac{5}{2}$ kernel, and results for the Mat\'ern $\frac{3}{2}$ kernel. All experiments were run with HSGP using the difference-in-age parameterisation models with $M^1=40$ (Number eigenfunctions on the difference-in-age dimension) and $M^2=20$ (Number of eigenfunctions on the contacts' age dimension). The sample size was fixed at $N=2000$.
\paragraph{S4 Fig.}
\label{S4_Fig}
{\bf Simulation experiment results for the in-COVID19 scenario with different covariance kernels.}. From top to bottom: results for the squared exponential kernel, results for the Mat\'ern $\frac{5}{2}$ kernel, and results for the Mat\'ern $\frac{3}{2}$ kernel. All experiments were run with HSGP using the difference-in-age parameterisation models with $M^1=40$ (Number eigenfunctions on the difference-in-age dimension) and $M^2=20$ (Number of eigenfunctions on the contacts' age dimension). The sample size was fixed at $N=2000$.
\paragraph{S5 Table.}
\label{S5_Table}
{\bf Comparison of different covariance kernels and number of basis functions for HSGP models.} Results were obtained with models using the difference-in-age parameterisation. The sample size was fixed at $N=2000$ throughout. $M^1$: The number of HSGP basis functions on the difference-in-age dimension. $M^2$: The number of HSGP basis functions on the contacts' age dimension. $^a$Mean absolute error, $^b$Expected log posterior density, $^c$Posterior predictive check, $^d$Median runtime, $^e$pre-COVID19, $^f$in-COVID19 scenario.
\paragraph*{S6 Fig.}
\label{S6_Fig}
{\bf Empirical and estimated social contact intensity patterns for COVIMOD wave 2.} (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.
\paragraph*{S7 Fig.}
\label{S7_Fig}
{\bf Empirical and estimated social contact intensity patterns for COVIMOD wave 3.} (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.
\paragraph*{S8 Fig.}
\label{S8_Fig}
{\bf Empirical and estimated social contact intensity patterns for COVIMOD wave 4.} (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.
\paragraph*{S9 Fig.}
\label{S9_Fig}
{\bf Empirical and estimated social contact intensity patterns for COVIMOD wave 5.} (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.
\paragraph*{S10 Fig.}
\label{S10_Fig}
{\bf Ratio of female to male marginal contact intensities for POLYMOD and COVIMOD}. Lines represent posterior median estimates of the ratio of female to male marginal contact intensities, i.e., $m_a^F / m_a^M$. A ratio of 1 (dashed lines) indicates no difference in contact intensities between genders. Shaded ribbons represent 95\% credible intervals.
\section*{Acknowledgments}
YuC gratefully acknowledges funding from the Imperial President’s PhD Scholarship program, and the EPSRC Centre for Doctoral Training in Modern Statistics and Statistical Machine Learning at Imperial and Oxford (EP/S023151/1); OR from the Bill \& Melinda Gates Foundation (OPP1175094) and
the Medical Research Council (MR/V038109/1); MM from the EPSRC Centre for Doctoral Training in Modern Statistics and Statistical Machine Learning at Imperial and Oxford (EP/S023151/1) and the Bill \& Melinda Gates Foundation (OPP1175094); SB acknowledge support from the MRC Centre for Global Infectious Disease Analysis (MR/R015600/1), jointly funded by the UK Medical Research Council (MRC) and the UK Foreign, Commonwealth \& Development Office (FCDO), under the MRC/FCDO Concordat agreement, and also part of the EDCTP2 programme supported by the European Union. S.B. acknowledges support from the Novo Nordisk Foundation via The Novo Nordisk Young Investigator Award (NNF20OC0059309). S.B.\ acknowledges support from the Danish National Research Foundation via a chair position. S.B.\ acknowledges support from The Eric and Wendy Schmidt Fund For Strategic Innovation via the Schmidt Polymath Award (G-22-63345). S.B.\ acknowledges support from the National Institute for Health Research (NIHR) via the Health Protection Research Unit in Modelling and Health Economics.
This work was further supported by the Imperial College Research Computing Service, DOI: 10.14469/hpc/2232. COVIMOD is funded by intramural funds of the Institute of Epidemiology and Social Medicine, University of M\"{u}nster, and of the Institute of Medical Epidemiology, Biometry and Informatics, Martin Luther University Halle-Wittenberg, as well as by funds provided by the Robert Koch Institute, Berlin, the Helmholtz-Gemeinschaft Deutscher Forschungszentren e.V. via the HZEpiAdHoc ``The Helmholtz Epidemiologic Response against the COVID-19 Pandemic” project, the Saxonian COVID-19 Research Consortium SaxoCOV (co-financed with tax funds on the basis of the budget passed by the Saxon state parliament), the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, via the project SpaceImpact project number 458526380) and the Federal Ministry of Education and Research (BMBF) via the projects Respinow (project number 031L0298F) and OptimAgent (project number 031L0299J) and as part of the Network University Medicine (NUM) via the egePan Unimed project (funding code: 01KX2021). The authors would like to thank Christopher Jarvis, Kevin Van Zandvoort, Amy Gimma, John Edmunds and the entire CoMix team for allowing the COVIMOD team to use an adapted version of the CoMix questionnaire for COVIMOD and for their great cooperation. The authors would also like to thank the team at IPSOS-Mori for their work on implementing the COVIMOD survey.
\section*{S2 Text. The construction of simulated social contact patterns}
Existing social contact studies showed that human social contacts are highly dependent on age and age-difference~\cite{mossong_social_2008, van_de_kassteele_efficient_2017, jarvis_quantifying_2020, monod_age_2021, feehan_quantifying_2021}. Specifically, they reveal that contact intensities are strong between people of the same age and between parents and children. We aimed to mimic these real-world social contact dynamics in creating our simulation datasets. For both the pre-COVID-19 and in-COVID-19 scenarios, we limited the age range to 6 to 49 for simplicity and to reduce the run time of our experiments. We designate individuals in age bands 6-18, 19-29, 30-39 and 40-49 as children, young adults, adults, and middle-aged adults, respectively. A subset of people aged 30-39 and 40-49 was assumed to be the parents of children and young adults.
The following text describes the exact process by which the simulated contact intensities were generated. The R code may be found in script \texttt{R/sim-intensity-utility.R}, and the resulting contact intensity matrices can be found under \texttt{data/simulations/intensity} in the accompanying GitHub repository.
\subsection*{Pre-COVID-19 scenario}
Based on findings from POLYMOD study~\cite{mossong_social_2008}, we set the number of social contacts per day in children to roughly 32, which includes the 2 contacts with their parents. We assumed young adults have approximately 25 contacts per day with peers (i.e., people in the same age category) and 2 contacts with their parents. Adults were assumed to have approximately 20 daily contacts with peers and 4 with children. Middle-aged adults were assumed to have approximately 15 contacts per day with peers and 4 with young adults. We set contact intensities to be the highest for age-assortative contacts (i.e., between the same age groups) and decreased the intensity as the absolute age difference (AAD) increased. Additionally, we increased contact intensity for parent-child contacts (the AAD is 24) and decreased the intensity as the ADD increased.
The following text describes the full data generation set-up for the pre-COVID-19 scenario. The contact intensity of age pairs not included in the list was set to 0.
\begin{enumerate}
\item Participants aged 6-18: \\set the highest intensity as 2.5 ADD is zero, and decrease the intensity by 0.2 as the AAD increases to 8; \\
set the intensity as 0.1 when AAD $\in[9:11]$; \\
set the intensity as 0.03 when the AAD $\in[12:13]$;\\
set the intensity as 0.01 when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the ADD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\item Participants aged 19-29:\\ the highest intensity is 2.5 with 0.3 decrease by AAD until AAD = 5;\\
set the intensity as 0.8 when AAD $\in[6:9]$; \\
set the intensity as 0.04 when the AAD $\in[10:13]$;\\
set the intensity as 0.01 when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\item Participants aged 25-29, to generate the contacts with their children: \\set the intensity as 0.2 when the AAD-24 is from $\in[2:3]$;\\
set the intensity as 0.02 when the AAD-24 is from $\in[4:5]$; \\
set the intensity as 0.6 when the AAD-24 is 1 and participants aged 29.
\item Participants aged 30-39:\\ the highest intensity is 2 with 0.24 decrease by AAD until AAD = 5;\\
set the intensity as 0.64 when AAD $\in[6:9]$; \\
set the intensity as 0.03 when the AAD $\in[10:13]$;\\
set the intensity as 0.01 when the AAD $\in[14:15]$;\\
set the second highest intensity as 1.6 for AAD is 24, and decrease the intensity by 1 as the AAD-24 increases to 1; \\
set the intensity as 0.2 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.02 when the AAD-24 $\in[4:5]$.
\item Participants aged 40-49:\\ the highest intensity is 1.5 with 0.18 decrease by AAD until AAD = 5;\\
set the intensity as 0.5 when AAD $\in[6:9]$; \\
set the intensity as 0.02 when the AAD $\in[10:13]$;\\
set the intensity as 0.007 when the AAD $\in[14:15]$;\\
set the second highest intensity as 1.6 for AAD is 24, and decrease the intensity by 1 as the AAD-24 increases to 1; \\
set the intensity as 0.2 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.02 when the AAD-24 $\in[4:5]$.
\end{enumerate}
\subsection*{In-COVID-19 scenario}
Using the pre-COVID-19 scenario as a baseline, we incorporated the effects of contact reduction measures such as school closures and remote work for the in-COVID-19 scenario based on findings from the CoMix study~\cite{jarvis_quantifying_2020}. In children, the number of social contacts per day was set to roughly 3, among which 2 are with their parents. Young adults were assumed to have around 15 contacts per day with people of the same age and 2 with their parents. We assumed that adults have about 12 contacts with people of the same age and 2 with their children. Middle-aged adults were assumed to have approximately 10 contacts per day with peers and 2 with their children (young adults). Following the pre-COVID-19 scenario, contact intensities were set to be the highest for age-assortative contacts and decreased as the absolute age difference increased. Additionally, we increased contact intensity for parent-child contacts (the AAD is 24) and reduced the intensity as the ADD increased.
The following text describes the full data generation setup for the in-COVID-19 scenario. The contact intensity of age pairs not included in the list was set to 0.
\begin{enumerate}
\item Participants aged 6-10: \\set the highest intensity as 0.08 ADD is zero, and decrease the intensity by 0.007 as the AAD increases to 8; \\
set the intensity as 0.003 when AAD $\in[9:11]$; \\
set the intensity as 0.001 when the AAD is $\in[12:13]$;\\
set the intensity as 3 $\times 10^{-4}$ when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the ADD-24 is $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 is $\in[4:5]$.
\item Participants aged 11-18: \\set the highest intensity as 0.4 ADD is zero, and decrease the intensity by 0.035 as the AAD increases to 8; \\
set the intensity as 0.015 when AAD $\in[9:11]$; \\
set the intensity as 0.005 when the AAD $\in[12:13]$;\\
set the intensity as 15 $\times 10^{-4}$ when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the ADD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\item Participants aged 19-29:\\ the highest intensity is 1.5 with 0.18 decrease by AAD until AAD = 5;\\
set the intensity as 0.48 when AAD $\in[6:9]$; \\
set the intensity as 0.024 when the AAD $\in[10:13]$;\\
set the intensity as 0.006 when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\item Participants aged 25-29, to generate the contacts with their children: \\set the intensity as 0.1 when the AAD-24 is from $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 is from $\in[4:5]$; \\
set the intensity as 0.6 when the AAD-24 is 1 and participants aged 29.
\item Participants aged 30-39:\\ the highest intensity is 1.25 with 0.15 decrease by AAD until AAD = 5;\\
set the intensity as 0.4 when AAD $\in[6:9]$; \\
set the intensity as 0.01875 when the AAD is $\in[10:13]$;\\
set the intensity as 0.00625 when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\item Participants aged 40-49:\\ the highest intensity is 1 with 0.12 decrease by AAD until AAD = 5;\\
set the intensity as 0.33 when AAD $\in[6:9]$; \\
set the intensity as 0.01375 when the AAD $\in[10:13]$;\\
set the intensity as 0.00475 when the AAD $\in[14:15]$;\\
set the second highest intensity as 0.8 for AAD is 24, and decrease the intensity by 0.5 as the AAD-24 increases to 1; \\
set the intensity as 0.1 when the AAD-24 $\in[2:3]$;\\
set the intensity as 0.01 when the AAD-24 $\in[4:5]$.
\end{enumerate}
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-2.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S3 Fig. Simulation experiment results for the pre-COVID19 scenario with different covariance kernels.}. From top to bottom: results for the squared exponential kernel, results for the Mat\'ern $\frac{5}{2}$ kernel, and results for the Mat\'ern $\frac{3}{2}$ kernel. All experiments were run with HSGP using the difference-in-age parameterisation models with $M^1=40$ (Number eigenfunctions on the difference-in-age dimension) and $M^2=20$ (Number of eigenfunctions on the contacts' age dimension). The sample size was fixed at $N=2000$.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\clearpage
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-3.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S4 Fig. Simulation experiment results for the in-COVID19 scenario with different covariance kernels.}. From top to bottom: results for the squared exponential kernel, results for the Mat\'ern $\frac{5}{2}$ kernel, and results for the Mat\'ern $\frac{3}{2}$ kernel. All experiments were run with HSGP using the difference-in-age parameterisation models with $M^1=40$ (Number eigenfunctions on the difference-in-age dimension) and $M^2=20$ (Number of eigenfunctions on the contacts' age dimension). The sample size was fixed at $N=2000$.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\clearpage
\begin{table}[!ht]
\begin{adjustwidth}{-2.25in}{0in}
\centering
\caption*{
{\bf S5 Table. Comparison of performance on simulated data for different scenarios, covariance kernels, and number of basis functions.}}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\bf Scenario & \bf Kernel & \bf $M^1$ & \bf $M^2$ & \bf MAE$^a$ & \bf ELPD$^b$ & \bf PPC$^c$ & \bf Runtime$^d$ \\ \thickhline
pre$^e$ & SE & 20 & 20 & $5.62 \times 10^{-2}$ & $-3120.4$ & 98.4\% & 1.6 hours \\ \hline
pre & SE & 30 & 20 & $5.00 \times 10^{-2}$ & $-3065.7$ & 98.3\% & 1.1 hours\\ \hline
pre & SE & 40 & 20 & $4.97 \times 10^{-2}$ & $-3064.2$ & 98.3\% & 1.2 hours\\ \hline
pre & SE & 40 & 30 & $4.97 \times 10^{-2}$ & $-3064.4$ & 98.3\% & 1.6 hours\\ \hline
pre & Mat\'ern $5/2$ & 20 & 20 & $5.38 \times 10^{-2}$ & $-3099.4$ & 98.6\% & 1.4 hours \\ \hline
pre & Mat\'ern $5/2$ & 30 & 20 & $4.79 \times 10^{-2}$ & $-3040.5$ & 98.5\% & 2.0 hours\\ \hline
pre & Mat\'ern $5/2$ & 40 & 20 & $4.44 \times 10^{-2}$ & $-3027.6$ & 98.5\% & 2.1 hours\\ \hline
pre & Mat\'ern $5/2$ & 40 & 30 & $4.41 \times 10^{-2}$ & $-3027.9$ & 98.5\% & 3.1 hours\\ \hline
pre & Mat\'ern $3/2$ & 20 & 20 & $5.63 \times 10^{-2}$ & $-3094.6$ & 99.0\% & 2.1 hours \\ \hline
pre & Mat\'ern $3/2$ & 30 & 20 & $4.81 \times 10^{-2}$ & $-3037.4$ & 98.8\% & 1.4 hours\\ \hline
pre & Mat\'ern $3/2$ & 40 & 20 & $4.37 \times 10^{-2}$ & $-3024.2$ & 98.9\% & 1.2 hours\\ \hline
pre & Mat\'ern $3/2$ & 40 & 30 & $4.36 \times 10^{-2}$ & $-3023.7$ & 98.9\% & 1.5 hours\\ \thickhline
in$^f$ & SE & 20 & 20 & $3.73 \times 10^{-2}$ & $-2723.2$ & 98.3\% & 2.3 hours \\ \hline
in & SE & 30 & 20 & $3.27 \times 10^{-2}$ & $-2691.4$ & 98.4\% & 1.1 hours \\ \hline
in & SE & 40 & 20 & $3.27 \times 10^{-2}$ & $-2691.3$ & 98.4\% & 0.7 hours \\ \hline
in & SE & 40 & 30 & $3.27 \times 10^{-2}$ & $-2691.3$ & 98.4\% & 0.9 hours \\ \hline
in & Mat\'ern $5/2$ & 20 & 20 & $3.56 \times 10^{-2}$ & $-2675.7$ & 98.6\% & 1.3 hours \\ \hline
in & Mat\'ern $5/2$ & 30 & 20 & $2.97 \times 10^{-2}$ & $-2639.4$ & 98.6\% & 1.1 hours \\ \hline
in & Mat\'ern $5/2$ & 40 & 20 & $2.85 \times 10^{-2}$ & $-2674.5$ & 98.7\% & 1.4 hours \\ \hline
in & Mat\'ern $5/2$ & 40 & 30 & $2.85 \times 10^{-2}$ & $-2635.2$ & 98.7\% & 1.8 hours \\ \hline
in & Mat\'ern $3/2$ & 20 & 20 & $3.41 \times 10^{-2}$ & $-2652.0$ & 98.9\% & 2.7 hours \\ \hline
in & Mat\'ern $3/2$ & 30 & 20 & $2.86 \times 10^{-2}$ & $-2619.5$ & 99.0\% & 1.2 hours \\ \hline
in & Mat\'ern $3/2$ & 40 & 20 & $2.70 \times 10^{-2}$ & $-2614.1$ & 99.0\% & 1.4 hours \\ \hline
in & Mat\'ern $3/2$ & 40 & 30 & $2.69 \times 10^{-2}$ & $-2612.6$ & 99.0\% & 1.6 hours \\ \hline
\end{tabular}
\begin{flushleft}
Results were obtained with models using the difference-in-age parameterisation. The sample size was fixed at $N=2000$ throughout. $M^1$: The number of HSGP basis functions on the difference-in-age dimension. $M^2$: The number of HSGP basis functions on the contacts' age dimension. $^a$Mean absolute error, $^b$Expected log posterior density, $^c$Posterior predictive check, $^d$Median runtime, $^e$pre-COVID19, $^f$in-COVID19 scenario.
\end{flushleft}
\end{adjustwidth}
\end{table}
\clearpage
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-6.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S6 Fig. Empirical and estimated contact intensity patterns for COVIMOD wave 2}. (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\clearpage
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-7.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S7 Fig. Empirical and estimated contact intensity patterns for COVIMOD wave 3}. (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{sup-figure-6-9.R} on our accompanying GitHub repository.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\clearpage
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-8.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S8 Fig. Empirical and estimated contact intensity patterns for COVIMOD wave 4}. (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\clearpage
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-9.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S9 Fig. Empirical and estimated contact intensity patterns for COVIMOD wave 5}. (Top row) Crude empirical social contact intensity patterns, with crude contact intensities above a value of 3 truncated for visualisation purposes. There are some age groups with no participants, and they are represented by white vertical columns. (Middle row) Contact intensity patterns as estimated by the \texttt{socialmixr} R package~\cite{funk_socialmixr_2020}. (Bottom row) Contact intensity patterns are given by our Bayesian model. The exact runtime arguments for this comparison are given in script \texttt{figure-6-9.R} on our accompanying GitHub repository.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\begin{figure}[!t]
\begin{adjustwidth}{-2.25in}{0in}
\includegraphics[width=\linewidth]{figures/sup-figure-10.jpeg}
\begin{adjustwidth}{0.75in}{0in}
\captionsetup{width=\linewidth}
\caption*{{\bf S10 Fig. Ratio of female to male marginal contact intensities for POLYMOD and COVIMOD}. Lines represent posterior median estimates of the ratio of female to male marginal contact intensities, i.e., $m_a^F / m_a^M$. A ratio of 1 (dashed lines) indicates no difference in contact intensities between genders. Shaded ribbons represent 95\% credible intervals.}
\end{adjustwidth}
\end{adjustwidth}
\end{figure}
\section*{Abstract}
Since the emergence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), many contact surveys have been conducted to measure the fundamental changes in human interactions that occurred in the face of the pandemic and non-pharmaceutical interventions. These surveys were typically conducted longitudinally, using protocols that have important differences from those used in the pre-pandemic era.
Here, we present a model-based statistical approach that can reconstruct contact patterns at 1-year resolution even when the age of the contacts is reported coarsely by 5 or 10-year age bands. This innovation is rooted in population-level consistency constraints in how contacts between groups must add up, which prompts us to call the approach presented here the Bayesian rate consistency model. The model also incorporates computationally efficient Hilbert Space Gaussian process priors to infer the dynamics in age- and gender-structured social contacts, and is designed to adjust for reporting fatigue emerging in longitudinal surveys.
On simulations, we show that social contact patterns by gender and 1-year age interval can indeed be reconstructed with adequate accuracy from coarsely reported data and within a fully Bayesian framework to quantify uncertainty. We then investigate the patterns and dynamics of social contact data collected in Germany from April to June 2020 across five longitudinal survey waves. We estimate the fine age structure in social contacts during the early stages of the pandemic and demonstrate that social contact intensities rebounded in a structured, non-homogeneous manner. We also show that by July 2020, social contact intensities remained well below pre-pandemic values despite a considerable easing of non-pharmaceutical interventions.
The Bayesian rate consistency model provides a modern, non-parametric, computationally tractable approach for estimating the fine structure and longitudinal trends in social contacts, and is readily applicable to contemporary survey data as long as the exact age of survey participants is reported.
\section*{Author summary}
The transmission of respiratory infectious diseases occurs during close social contacts. Hence, characterising social contact patterns within a population, encoded in contact matrices, leads to a better understanding of disease spread. Contact matrices also parameterise mathematical models, which played a pivotal role in informing health policy during the coronavirus disease 2019 (COVID-19) pandemic. Unlike pre-pandemic surveys, which recorded contacts' age in one-year age intervals, COVID-era studies recorded contacts' age in discrete large age categories to facilitate reporting. Some studies allowed participants to report an estimate for the total number of contacts for which they could not remember age and gender information. Many studies were partially longitudinal, which introduced the issue of reporting fatigue. Thus, directly applying existing statistical methods for estimating social contact matrices may result in a loss of age detail and confounded estimates. To this end, we develop a longitudinal model-based approach which estimates fine-age contact patterns from coarse-age data that also adjusts for the confounding effects of aggregate contact reporting and reporting fatigue in a unified manner. We apply our approach to the COVIMOD study to provide a detailed picture of how social contact dynamics evolved during the first wave of COVID-19 in Germany.
\nolinenumbers
\input{C1_introduction}
\input{C2_methods}
\input{C3_results}
\input{C4_discussion}
\input{C5_conclusion}
\input{S1_appendix}
\nolinenumbers
|
1,314,259,993,033 | arxiv | \section{Introduction}
The suggestion was made in work of Crane and Frenkel \cite{cf} \cite{c}
that the
inverse relation to the Grothendieck rig (fusion rig) construction
should shed light on the relation between topological quantum field theories
(TQFT's) in various dimensions, and, as well, should provide constructions
for TQFT's in dimension 4.
It is the purpose of this note to consider several simple cases of this
relation.
\begin{defin} A {\bf rig} $R$ is a set equipped with a monoid structure,
$(R, +, 0)$ and a semi-group structure $(R, \cdot)$ satisfying, moreover,
\begin{center}
\begin{tabular}{rl}
commutativity & $a + b = b + a$ \\
right-distributivity & $(a + b)\cdot c = a\cdot c + b\cdot c$ \\
left-distributivity & $a\cdot (b + c) = a\cdot b + a\cdot c$. \\
\end{tabular}
\end{center}
\noindent A rig is {\bf unital} if the multiplicative semi-group is a monoid
with unit $1$. A rig is {\bf of finite rank} if the additive monoid $(R,+,0)$
is finitely generated.
\end{defin}
Observe that in analogy to the coproduct of rings---tensor product of rings
over ${\Bbb Z}$---rigs admit a coproduct, ``tensor product over ${\Bbb N}$''.
The one other general property of rigs (of finite rank) which we will need
in the sequel is
\begin{propo}
If $R$ is a rig of finite rank, then $R$ has a unique minimal set of additive
generators.
\end{propo}
\noindent{\bf proof:} Suppose $S$ and $T$ are minimal sets of generators.
Then there exist expressions
\[ s = \sum_{t \in T} n^s_t t \]
\noindent and
\[ t = \sum_{s \in S} m^t_s s \]
\noindent Observe then that $[n^s_t]$ and $[m^t_s]$ are mutually inverse
matrices of natural numbers. The only such pairs are pairs of inverse
permutation matrices. $\Box$
\begin{defin}
If $\cal C$ is an (essentially small) tensor category, that is an
abelian category equipped with a (bi-)exact monoidal
productnot necessarily with unit object),
then the set of isomorphism classes of
objects in $\cal C$ equipped
with the operations induced by direct sum and tensor product is called
the {\bf Grothendieck rig} of $\cal C$, and denoted $Groth({\cal C})$.
\end{defin}
Notice that if $\cal C$ has a unit object,
then $Groth({\cal C})$ is unital. Similarly if $\cal C$ is Artinian
semi-simple, then $Groth({\cal C})$ is of finite rank.
Observe that the universal property of direct sum imposes some constraints
on the structure of a Grothendieck rig, the most salient of which is
the condition that all 1-generator sub-additive-monoids are free.
\begin{defin} An {\bf abstract fusion-rule algebra} is a rig, in which
all 1-generator sub-additive-monoids are free.
An {\bf abstract fusion-rule bialgebra}
is an abstract fusion rule algebra $A$ equipped with a co-operation
$\Delta:A\rightarrow A\otimes_{\Bbb N} A$
which is a rig homomorphism (or, equivalently, which
satisfies the usual compatibility relations for bialgebras). It is counital
if it is equipped with a co-operation $\epsilon:A\rightarrow {\Bbb N}$ which
is a rig homomorphism. In the case where $A$ is unital, we require
$\Delta$ and $\epsilon$ to preserve the unit.
\end{defin}
This latter, the notion of an abstract fusion-rule bialgebra, is important
because of the construction proposed by Crane and Frenkel \cite{cf}
of state-sum invariants of 4-manifolds using as initial data
categorical analogues of Hopf-algebras, and the result of Crane and Yetter
\cite{cy.alg.tqfts} showing that any 4D TQFT with factorization
at corners has as part of its structure a formal ``bialgebra category.''
In order to explain the structure of these analogues to Hopf-algebras,
and to fix the context in which our reversal of the Grothendieck rig
construction will take place, first fix an algebraically closed field
$K$. We will briefly mention the difficulties which non-algebraically
closed fields present later. Let $VECT_K$ or simply $VECT$ denote the
category of finite-dimensional vector-spaces over $K$ with its usual
monoidal structure.
\begin{defin} Let $VECT_K-mod$ denote the 2-category of all (small)
Artinian semi-simple
$K$-linear categories, that is $K$-linear categories equivalent
to finite powers of $VECT_K$ with exact functors and natural transformations
as 1- and 2-arrows. We refer to the objects of this 2-category as
$VECT$-modules. Let $\boxtimes$ denote a chosen bifunctor
from $VECT_K-mod^2$ to $VECT_K-mod$ which selects an object
${\cal C}\boxtimes{\cal D}$ equipped with a functor from
${\cal C} \times {\cal D}$ exact in each variable separately and universal
among such.
\end{defin}
The existence of $\boxtimes$ has been shown in Yetter \cite{catlinalg},
and can be readily verified
by mimicking the construction of the tensor product of vector-spaces,
except that instead of identifying objects as one identified elements, one
must adjoin an isomorphism. It follows readily from the universality
properties that $\boxtimes$ makes $VECT_K-mod$ into
a monoidal bicategory (cf. Gordon/Power/Street \cite{gps} and
Kapranov/Voevodsky \cite{kv}), whose underlying bicategory is a
2-category.
In this setting if a $VECT$-module $\cal C$ is equipped with a monoidal
structure (with unit object), exact in each variable, we may regard this as
being given
by (exact) functors $\otimes: {\cal C}\boxtimes{\cal C}\rightarrow
{\cal C}$ (and $I: VECT\rightarrow {\cal C}$, sending $K$ to the monoidal
identity object), equipped with natural transformation(s) $\alpha$ (and
$\rho$ and $\lambda$) satisfying the usual pentagon (and triangle)
coherence condition(s). Similarly the structural transformations of
exact monoidal functors between $VECT$-modules $\cal C$ and $\cal D$
may be understood as
natural transformations between functors from ${\cal C}\boxtimes
{\cal C}$ to $\cal D$ (and $VECT$ to $\cal D$). We will refer to monoidal
categories of this sort as {\em (Artinian semi-simple) tensor
categories.}\footnote{More generally, we
advocate the use of ``tensor category'' to
refer to a monoidal abelian category with $\otimes$ exact in each variable.}
We call exact monoidal functors {\em tensor functors}.
It is now sensible to consider in this setting duals to the notions of tensor
category and tensor functor:
\begin{defin}
An Artinian semi-simple {\bf (counital) cotensor category}
over $K$ is a $VECT$-module
$\cal C$ equipped with functors $\Delta:{\cal C}\rightarrow {\cal C}
\boxtimes {\cal C}$ (and $\epsilon:{\cal C}\rightarrow VECT$), together
with natural transformations $\beta$ (and $r$ and $l$)
satisfying the obvious pentagon
(and triangle) relation(s).
A {\bf strong cotensor functor} $F:{\cal C}\rightarrow
{\cal D}$ is a functor equipped with natural isomorphisms $F_{\sim}:
(F\boxtimes F)(\Delta)\rightarrow \Delta(F)$
(and $F^0:\epsilon \rightarrow \epsilon(F)$)
satisfying coherence conditions formally dual
to those for strong monoidal functors.
\end{defin}
We can now succinctly define the categorical analogue of a bialgebra
as given by Crane and Frenkel \cite{cf} including the conditions on
the unit and counit functors which were omitted in \cite{cf}:
\begin{defin}
An Artinian semi-simple {\bf bitensor category} over $K$ is an
Artinian semi-simple category equipped with both a tensor category
structure and a cotensor category structure for which the structure
functors for the cotensor category strucure are (strong)
tensor functors, the structure functors for the tensor
structure are cotensor functors, and the structural transformations
for the tensor functor and cotensor functor structures coincide
whenever their sources and targets coincide. It is {\bf biunital} if the
tensor structure is unital and the cotensor structure is counital.
\end{defin}
In particular, we fix notation for these ``compatibility transformations''
as follows:
\[ \Phi = \tilde{\Delta} = \otimes_{\sim}^{-1} \]
\[ \eta = \epsilon^0 = I_0^{-1} \]
\[ \tau = \tilde{\epsilon} = \otimes_0^{-1} \]
\[ \delta = \Delta^0 = I_{\sim}^{-1} \]
The other two ``coherence cubes'' of \cite{cf} (besides the pentagon
and dual pentagon) are simply the coherence condition for a tensor
functor and its dual.
Thus the complete structure of a biunital bitensor
category $\cal C$ is given by
four functors $\otimes, I, \Delta,$ and $\epsilon$ and
ten natural isomorphisms $\alpha, \rho, \lambda, \beta, r, l, \Phi, \eta,
\tau,$ and $\delta$ satisfying coherence conditions which can be read off
from the definitions of tensor and cotensor categories and tensor and
cotensor functors.
The appropriate notion of structure preserving functors between
bitensor categories is given by
\begin{defin}
A {\bf (strong) bitensor functor} is a 5-tuple
$(F,\tilde{F},F_{\sim},F^0,F_0)$, where $F:{\cal C}\rightarrow {\cal D}$
is a functor between bitensor categories, and
such that $(F,\tilde{F},F_0)$ (resp.
$(F,F_{\sim},F^0)$) is a tensor (resp. cotensor) functor and which moreover
satisfies the following, in which primes ($^\prime$) indicate structural
functors or natural transformations belonging to $\cal D$:
\[ [ F\boxtimes F (\Phi_{A,B})][ \oplus \tilde{F}_{A_{(1)},A_{(2)}}\boxtimes
\tilde{F}_{B_{(1)},B_{(2)}}] [ F_{\sim A}\boxtimes F_{\sim B}] =
F_{\sim A\otimes B} \; \Delta^\prime(\tilde{F}_{A,B}) \;
\Phi^\prime_{F(A),F(B)} \]
\[ F^0_{A\otimes B} \epsilon^\prime (\tilde{F}_{A,B}) \tau^\prime_{F(A),F(B)}
= \tau_{A,B}
(F^0_A\otimes F^0_B) \]
\[ F_{\sim I} \Delta^\prime (F_0) \delta^\prime
= (F\boxtimes F)(\delta) (F_0\boxtimes F_0) \]
\noindent and
\[ F^0_n F_0 \eta^\prime = \eta \]
If either category
is not unital or counital, the appropriate functor and attendant
natural isomorphisms are omitted.
\end{defin}
Likewise, we can define cotensor and bitensor natural transformations:
a natural transformation is a cotensor transformation if it satisfies
the dual condition to monoidal naturality, and is a bitensor transformation
if it is both a monoidal natural transformation and a cotensor transformation.
Now, observe that if $\cal C$ is a bitensor category,
$Groth({\cal C})$ has the
structure of an abstract fusion rule bialgebra with the co-operations
induced by the cotensor and counit functors on the category. We will refer
to this as the {\bf Grothendieck birig} of the bialgebra category.
\begin{defin} A {\bf $K$-categorification} of an abstract fusion-rule
algebra (resp. bialgebra) $A$ is a $K$-linear tensor category
(resp. bialgebra category) whose Grothendieck rig (resp. birig) is $A$.
If $A$ is of finite rank, we call a categorification {\bf semi-simple} if it
is semi-simple as a tensor category, and, moreover, the objects whose images
under the Grothendieck rig construction are the additive generators of
$A$ are simple objects.
\end{defin}
Notice in general,
in the non-finite rank case, one must specify a set of additive
generators to make sense of the notion of semi-simple categorification. We
will have no need of this more general notion in this paper.
In the remainder of the paper, we consider examples of categorifications
and their relevance to the construction of TQFT's. We restrict our
attention to the case of $K$ algebraically closed because this
covers the most interesting case of $K = {\Bbb C}$ and removes the
possibility of having objects in semi-simple categories whose endomorphism
algebras are division-algebra extensions of the ground field.
\section{Categorifying ${\Bbb N}[G]$, Dijkgraaf-Witten Theory, and the
Turaev-Viro Construction}
Our first example sheds light on the relationship between two well-known
constructions of (2+1)-dimensional TQFT's: Dijkgraaf-Witten theory
\cite{dw}, in particular its simplicial construction as in Wakui \cite{wak},
(cf. also Yetter \cite{yetgrps}), and the generalized Turaev-Viro
construction, cf. Barrett and Westbury \cite{bw}. As examples of
tensor categories, these are reasonably well-known (cf. \cite{cp}, \cite{dpr})
and are included here merely as the simplest examples of categorification.
Fix a finite group $G$. Observe that the group rig ${\Bbb N}[G]$ is
an abstract fusion-rule bialgebra with the operations induced on the basis
$G$ by $g\cdot h = gh$ (the null-infix denoting the group-law) and
$\Delta(g) = g\otimes g$. The following shows that the
categorifications of ${\Bbb N}[G]$ are essentially classified by the
3-cocyles on $G$ with coefficients in $K^\times$ when
$K$ is algebraically closed.
In this and all subsequent proofs, it first should be observed
that each equivalence class of $K$-linear semi-simple categories contains
skeletal categories (i.e. categories with only one object in each
isomorphism class). We begin by restricting our attention to the structure of
these skeletal categories up to isomorphism, then consider the question
of monoidal equivalence. Notice, we are taking an approach orthogonal
to that of the Mac Lane Coherence Theorem \cite{mac}: we will not
be able to strictify the structure maps and retain skeletalness.
\begin{thm}
Let $G$ be a finite group, and $K$ an algebraically closed field, then the
isomorphism classes of skeletal semi-simple
$K$-categorifications of ${\Bbb N}[G]$ as an abstract fusion-rule
algebra (resp. unital abstract fusion-rule algebra;
abstract fusion-rule bialgebra;
biunital abstract fusion-rule bialgebra)
are in one-to-one correspondence with
the set of $K^\times$-valued 3-cocycles on $G$ (resp.
pairs $(\alpha, \rho)$, where $\alpha$ is a $K^\times$-valued 3-cocycle
on $G$ , and $\rho$ is an element of $K^\times$; $K^\times$-valued 2-cochains
on $G$; triples $(\phi, \rho, r)$, where $\phi$ is a $K^\times$-valued
2-cochain on $G$, $\rho$ is an element of $K^\times$, and $r$ is a
$K^\times$-valued 1-cochain on $G$).
\end{thm}
\noindent {\bf proof:} We begin by proving the first two statements, that
skeletal categorifications as an algebra (resp. unital algebra) are given by
3-cocycles (resp. pairs of a 3-cocycle and a scalar).
Now, in any skeletal categorification of ${\Bbb N}[G]$, we
may identify the object whose image under the Grothendieck rig
construction is $g \in G$ with $g$. Since the tensor product (resp.
identity object) must be carried to the multiplication (resp. unit)
in the fusion ring, the functors of the monoidal structure are given
by
\[ g \otimes h = gh \;\;\; I = e \]
To specify a categorification (as an algebra), it remains only
to describe the rest of the monoidal structure: in this case the
structure maps become families of maps $\alpha_{g,h,k}:ghk\rightarrow ghk$,
$\rho_g:g\rightarrow g$ and $\lambda_g:g\rightarrow g$. Observe, moreover,
that these ``maps'' are just elements of $K$, that
the semi-simplicity condition implies that any such families of
maps will satisfy the required naturality conditions, and that invertibility
consists in restricting the choices to elements of $K^\times$.
The pentagon condition for a 4-tuple $g,h,k,l$ can then be written:
\[ \alpha_{g,h,k} \alpha_{g,hk,l} \alpha_{h,k,l} =
\alpha_{gh,k,l} \alpha_{g,h,kl} \]
\noindent which is precisely the condition that $\alpha_{-,-,-}$ be a
$K^\times$-valued 3-cocycle on $G$.
Upon including a unit, the triangle condition relating
$\rho, \lambda,$ and $\alpha$ becomes
\[ \rho_g = \alpha_{g,e,h} \lambda_h \]
\noindent from which it follows that all components of $\rho$ and $\lambda$
are completely determined by $\alpha$ and the choice of a number
$\rho$ such that $\rho_e = \rho \cdot Id_e$.
The exercise of verifying that the choice of $\lambda_h$'s
determined by $\rho_e$, and the choice of $\rho_g$'s determined by
$\lambda_e$ satisfy the triangle condition for all pairs $g$ and $h$ is
left to the reader.
(Hint: use the cocycle condition with two indices equal to $e$.)
To specify a categorification as a bialgebra, notice first that
the dual pentagon condition on the coassociator reduces to
\[ \beta_g^3 = \beta_g^2 \]
\noindent for all $g \in G$, and thus since $\beta_g$ must be invertible,
$\beta_g = 1$.
Similarly the compatibility condition between the connecting
transformation $\phi$ and the coassociator $\beta$ gives no restriction
on the components of $\phi$.
On the other hand, the compatibility between $\phi$ and $\alpha$
is given by
\[ \phi_{g,h} \phi_{gh,k} \alpha_{g,h,k} = \alpha_{g,h,k} \alpha_{g,h,k}
\phi_{h,k} \phi_{g,hk} \]
\noindent which reduces to
\[ \phi_{g,h} \phi_{g,hk}^{-1} \phi_{gh,k} \phi_{h,k}^{-1} = \alpha_{g,h,k} \]
Thus, in this case the structure is completely determined by
the connecting transformation $\phi$, which is simply a 2-cochain on $G$.
Finally in the case of a biunital bialgebra categorification,
the remaining structure maps are similarly completely determined by the
component maps at at simple objects, which are again given by scalars
in $K^\times$,
$r_g, l_g, \tau_{g,h}$; and since
$e = I$
is simple, scalars $\delta$ and $\eta$. The coherence conditions
then become:
\bigskip
\begin{center}
\begin{tabular}[h]{ll}
triangle for tensor structure & $\rho_g =
\alpha_{g,e,k}\lambda_k$ \\
triangle for cotensor structure & $r_g =
\beta_g l_g$ \\
$\Delta$ respects right unit &
$\phi_{g,e}\delta\rho_g^2
= \rho_g $ \\
$\Delta$ respects left unit &
$\phi_{g,e}\delta\lambda_g^2
= \lambda_g $ \\
$\otimes$ respects right counit & $
\phi_{g,k}\tau_{g,k} r_g
r_k = r_{gk} $\\
$\otimes$ respects left counit & $
\phi_{g,k}\tau_{g,k} l_g
l_k = l_{gk} $\\
$\epsilon$ preserves $\otimes$ & $ \alpha_{g,k,m}\tau_{g,km}\tau_{k,m} =
\tau_{g,k}\tau_{gk,m} $\\
$I$ preserves $\Delta$ & $
\beta_e \delta^2
= \delta^2 $ \\
$\epsilon$ respects right unit & $\tau_{g,e}\eta = \rho_g$ \\
$\epsilon$ respects left unit & $\tau_{e,g}\eta = \lambda_g$ \\
$I$ respects right counit & $\delta \eta = r_e $\\
$I$ respects left counit & $\delta \eta = l_e $\\
\end{tabular}
\end{center}
\bigskip
We can then analyse these equations to determine a minimal set of
data and conditions for specifying the unital and counital structures
on a categorification of ${\Bbb N}[G]$. We assume that $\alpha$ and
$\phi$ have been chosen so as
to specify a bitensor categorification without unit or counit.
First, observe that it follows from the two triangle conditions that
$\rho, \lambda, r$ and $l$ are completely determined by the values
of $\rho = \rho_e$ and $r_g$ $(g \in G)$ by
the formulas
\[ \rho_g = \alpha_{g,e,e}\rho \]
\[ \lambda_k = \alpha^{-1}_{e,e,k}\rho \]
\[ l_g = r_g \]
\noindent The only restriction on these needed to ensure that they
unambiguously determine cochains satisfying the first two equations
has already been imposed by the condition that $\alpha$ be a
coboundary in the suitable sense.
In a similar way, $\otimes$ respecting
both the right and left counit conditions is equivalent to
\[ \tau(g,k) = d(r^{-1})(g,k) \phi^{-1}_{g,k}. \]
Similarly, $\Delta$ respecting both the right and left
unit conditions is equivalent to
\[ \delta = \rho^{-1} \phi^{-1}_{e,e} \]
Given these last two equations, the conditions that $\epsilon$ preserve
$\otimes$ and $I$ preserve $\Delta$ follow from the
cocycle conditions and the fact that
$d$ distributes over multiplication of cochains and (remember we write
the operation on cochains multiplicatively since the coefficients
are in $K^\times$).
The four remaining conditions are all equivalent to
\[ \eta = \rho r_e \phi_{e,e} \]
\noindent and we are done.$\Box$
\smallskip
Turning to the more interesting question of monoidal equivalence
classes of categorifications, we have:
\begin{defin}
Two algebra (resp. bialgebra) categorifications are {\bf equivalent} if
there exists a monoidal equivalence (resp. bitensor
equivalence) between them which induces the identity on Grothendieck
rigs.
\end{defin}
\begin{thm}
The equivalence classes of categorifications of ${\Bbb N}[G]$ as
an abstract fusion-rule algebra (whether unital or not)
are in 1-1 correspondence with
$H^3(G,K^\times)$. The bialgebra categorification of ${\Bbb N}[G]$ is
unique up to equivalence.
\end{thm}
\noindent {\bf proof:} Now, observe that the structure of a monoidal
equivalence which induces the identity on the Grothendieck rig
is given entirely by the structure transformations of the monoidal
equivalence. In this case, these are given by a choice of units
$\psi_{a,b}$ for pair of group elements, and satisfying
\[ \alpha_{g,h,k}\psi_{h,k}\psi_{g,hk} =
\psi_{g,h}\psi_{gh,k}\alpha^\prime_{g,h,k} \]
\noindent and
\[ \rho_g\psi_{g,e} = \rho^\prime_g \]
\noindent where $\alpha_{g,h,k}$ and $\alpha^\prime_{g,h,k}$
(resp. $\rho_g$ and $\rho^\prime_g$)\
are the components of
the associativity (resp. unit) transformations on the two categorifications.
Solving gives the condition that $\alpha$ and $a$ be cohomologous, while
the condition on the unit transformations is just a normalization condition
which can be trivially satisfied.
The statement for bialgebra categorifications follows immediately from this
and the condition that the associativity constraint be the coboundary of the
connecting constraint: $\psi$ must be the ratio of the two connecting
constraints.
The additional structure of unital-counital categorifications
is effaced by equivalence.
A similar analysis to that
in the proof of the previous theorem shows that the structural transformations
for an equivalence of unital and counital categorifications are determined
by the 2-cochain $\psi$ which determines the non-unital non-counital
equivalence, together with a scalar $f_0$ and a 1-cochain $f^0$.
The conditions besides the cobounding conditions
reduce to
\[ f_0 =
\frac{\rho}{\rho^\prime\psi_{e,e}} \]
\[ f^0(g) = \frac{r_g}{r^\prime_g}
\]
\noindent which can always be solved for any choice of $\rho$ and $\rho^\prime$
(resp. $r$ and $r^\prime$).
$\Box$
The algebra case of this result in fact shows that the Dijkgraaf-Witten
invariants of 3-manifolds are examples of the generalized Turaev/Viro
construction of Barrett and Westbury \cite{bw}. It is not hard to show
that the categorifications of a finite group rig are spherical
categories in the sense of \cite{bw}. The construction as given by
Wakui \cite{wak} is then immediately seen to be a special case of their
general construction.
\section{Categorifying $D({\Bbb N}[G])$}
We now turn to the question of categorifying the simplest really
non-trivial Hopf algebra: the Drinfel'd double of a finite (non-commutative)
group algebra (here taken with ${\Bbb N}$-coefficients to produce
a birig).
The same techniques may be used to classify the semi-simple
categorifications of arbitary bicrossproducts of a finite group algebra with
a dual finite group algebra (cf. Majid \cite{maj}).
Recall that the Drinfel'd double of a finite group algebra can
be constructed by taking as a basis pairs $(g,\hat{h})$, where
$g$ and $h$ are elements of the group, and $\hat{h}$ indicates the
element in the dual basis corresponding to $h$ as a basis element
in ${\Bbb N}[G]$, with structure maps given by
\begin{center}
\begin{tabular}{rl}
multiplication & $(g,\hat{h})\cdot (k,\hat{l}) = \delta_{k^{-1}hk,l}
(gk, \hat{l})$ \\
unit & $1 = \sum_h (e,\hat{h})$ \\
comultiplication & $\Delta (g,\hat{h}) = \sum_{kl = h} (g,\hat{k})\otimes
(g,\hat{l})$ \\
counit & $\epsilon (g,\hat{h}) = \delta_{h,e}$ \\
\end{tabular}
\end{center}
To categorify this as an abstract fusion-algebra (resp. -bialgebra) we
must first consider what are the source and target data for a component of
the associator (resp. components of the associator, coassociator and
compatiblity transformation).
The typical component of an associator is a map
\[ \alpha_{(g,\hat{h}),(k,\hat{l}),(m,\hat{n})}:((g,\hat{h})\otimes
(k,\hat{l}))\otimes (m,\hat{n}) \rightarrow (g,\hat{h})\otimes ((k,\hat{l})
\otimes (m,\hat{n})) \]
\noindent Observe that the source and target objects are both $0$ unless
$l = k^{-1}hk$ and $n = m^{-1}lm$, in which case both are the object
$(gkm,\hat{n})$.
Thus the associator may be regarded as a family of non-zero scalars
$\alpha(g,k,m;\hat{n})$ giving the non-zero components as multiples
of the identity on $(gkm,\hat{n})$. (Note: a choice
of $g,k,m,$ and $\hat{n}$ contains enough information to recover
the source and target data for a non-zero component of the associator.)
Similarly, the coassociator has components given by maps (in the
tensor cube of the category)
\[ \beta_{(g,\hat{h})}: \oplus_{pk = h} \oplus_{ij = p} (g,\hat{i})\boxtimes
(g,\hat{j})\boxtimes (g,\hat{k}) \rightarrow \oplus_{iq = h} \oplus_{jk = q}
(g,\hat{i})\boxtimes (g,\hat{j})\boxtimes (g,\hat{k}) \]
Each such map is determined by its components on the various
\[ (g,\hat{i})\boxtimes (g,\hat{j})\boxtimes (g,\hat{k}) \]
\noindent and thus by a family of scalars $\beta(g; \hat{i}, \hat{j}, \hat{k})$
(As above, the given indices contain enough information to recover to which
summand of which component of the coassociator this scalar belongs.)
Finally, the compatibility transformation or ``coherer''
has as components maps (in the
tensor square of the category)
\[ \phi_{(g,\hat{h}),(k,\hat{l})}: \oplus_{mn = l} (gk,\hat{m})\boxtimes
(gk,\hat{n})
\rightarrow \oplus_{ab = h} (gk, \widehat{k^{-1}ak})\boxtimes
(gk, \widehat{k^{-1}bk})
\]
\noindent In this case, the sources and targets of components are
given only in the case
where the source and target are non-zero, which happens precisely when
$k^{-1}hk = l$.
Thus the coherer is determined by a family of scalars
$\phi(g,k;\hat{m},\hat{n})$.
Simply writing down the pentagon coherence condition on the associator
in terms of the scalars $\alpha(g,k,m;\hat{n})0$ gives
\[ \alpha(k,m,p;\hat{q}) \alpha(g,km,p;\hat{q})
\alpha(g,k,m;\widehat{pqp^{-1}}) = \alpha(gk,m,p;\hat{q})
\alpha(g,k,mp;\hat{q}) \]
Similarly, the dual pentagon in terms of the scalars becomes
\[ \beta(g;\hat{j},\hat{k},\hat{l}) \beta(g;\hat{i},\hat{jk},\hat{l})
\beta(g;\hat{i},\hat{j},\hat{k}) = \beta(g;\hat{ij},\hat{k},\hat{l})
\beta(g;\hat{i},\hat{j},\hat{kl}) \]
\noindent (A choice of 3-cocycle for each group element.)
The coherence condition provided by the compatibility transformation
as the structure transformation for $\Delta$ as a monoidal
functor becomes:
\[ \alpha(g,k,m;\hat{p})\alpha(g,k,m;\hat{q})\phi(k,m;\hat{p},\hat{q})
\phi(g,km;\hat{p},\hat{q}) = \phi(g,k;\widehat{mpm^{-1}},\widehat{mqm^{-1}})
\phi(gk,m;\hat{p},\hat{q})\alpha(g,k,m;\widehat{pq}) \]
(The two occurences of $\alpha$ on the left come from the associator
for ${\cal C}\boxtimes{\cal C}$.)
Similarly, the coherence condition provided by the compatibility transformation
as the structure transformation for $\otimes$ as a cotensor functor
becomes
\[ \phi(g,k;\hat{p};\hat{r})\phi(g,k;\widehat{pr},\hat{s})
\beta(gk;\hat{p},\hat{r},\hat{s}) =
\beta(g;\widehat{kpk^{-1}},\widehat{krk^{-1}},\widehat{ksk^{-1}})
\beta(k;\hat{p},\hat{r},\hat{s})\phi(g,k;\hat{r},\hat{s})
\phi(g,k;\hat{p},\widehat{rs}) \]
These conditions and those involving the other structural transformations
become more intelligible if we introduce a general setting for such scalar
valued functions:
For any finite group, $G$, let $\hat{G}$ denote the set of characteristic
functions of 1-element subsets of $G$. Let $C_{n,m}(G,K^\times)$ (or
$C_{n,m}$ when $G$ and $K$ are clear from context) denote the
abelian group of all functions from $G^n\times \hat{G}^m$ to $K^\times$.
The groups $C_{n,m}$ then form a double complex
(written multiplicatively) with differentials
$d_2:C_{n,m}\rightarrow C_{n,m+1}$, given by the Hochschild coboundary
in the ``hatted indices'', and $\tilde{d}_1:C_{n,m}\rightarrow C_{n+1,m}$
given by the same formula as Hochschild coboundary in the ``unhatted indices''
except that when the last index is dropped, all hatted indices are
left-conjugated by the dropped index.
In terms of these coboundary operations, the coherence conditions already
interpreted for categorifications of $D({\Bbb N}[G])$ become
\[ \tilde{d}_1(\alpha ) = 1 \]
\[ d_2(\beta) = 1 \]
\[ d_2(\alpha) = \tilde{d}_1(\phi) \]
\[ d_2(\phi) = \tilde{d}_1(\beta) \]
\noindent that is, the triple $(\alpha, \phi, \beta)$ forms a coboundary in
the total complex of the double complex
$(C_{n,m},\tilde{d}_1,d_2$ $n,m \geq 1)$. We will index the cohomology of
the total complex of $(C_{n,m},...)$ by $n+m-1$, and denote the groups
by ${\Bbb H}_\bullet (G,K^\times)$
By identical methods, one can verify that the structure transformations
for the preservation of the tensor product and cotensor product by a
bitensor functor with the identity as underlying functor are determined
by families of scalars $\tilde{f}(g,k;\hat{l})$ and $f_\sim(g;\hat{m},\hat{n})$
such that the ratios of corresponding structure maps satisfy
\[ \alpha \alpha^{\prime -1} = \tilde{d}_1 (\tilde{f}) \]
\[ \phi \phi^{\prime -1} = d_2 (\tilde{f}) \tilde{d}_1(f_\sim) \]
\noindent and
\[ \beta \beta^{\prime -1} = d_2(f_\sim). \]
Thus we have
\begin{thm} \label{cohom.thm}
The skeletal semi-simple $K$-categorifications
of $D({\Bbb N}[G])$ as an abstract fusion-rule
algebra (resp. abstract fusion-rule bialgebra)
are in one-to-one correspondence with the 3-coboundaries in
$(C_{n,1},\tilde{d}_1)$ (resp. the 3-coboundaries in the total
complex of the double complex $(C_{n,m},\tilde{d}_1,d_2)$). Moreover,
the equivalence classes of categorifications are in natural one-to-one
correspondence with the elements of the cohomology group $H_{3,1}$
(resp. ${\Bbb H}_3(G,K^\times)$).
\end{thm}
As was done explicitly for the associator, coherer, and coassociator above,
we can examine the components of each of the other structural transformations
at a simple object. In this way we find that $\rho$ (resp. $\lambda$,
$r$, $l$) is determined by a $1,1$-cochain $\rho(g;\hat{h})$ (resp.
$\lambda(g;\hat{h}) $, $r(g;\hat{h}) $, $l(g;\hat{h}) $).
For example, the typical component of $\rho$ at a simple object is a map
from $(g;\hat{h})\otimes \oplus_k (e;\hat{k})$ to $(g;\hat{h})$, but
the source is just $(g;\hat{h})$, so the map is a scalar multiple of
$1_{(g;\hat{h})}$.
Likewise, $\delta$ is determined by $0,2$-cochain $\delta(\hat{k},\hat{l})$,
$\tau$ by a $2,0$-cochain, and $\eta$ by a single element of $K^\times$.
(Note: although the double complex actually used in defining categorifications
does not have a 0-row or 0-column, it is helpful here and in what
follows to consider the larger complex which does, since much of what
is needed to handle the unital and counital structures is conveniently
phrased in terms of cochains and coboundaries in the larger complex.)
Again, by way of example, $\delta$ is a map from $\Delta(I)$ to $I\boxtimes I$
(the latter being the unit object in ${\cal C}\boxtimes {\cal C}$), that is
a map from $\oplus_{h \; kl=h} (e;\hat{k})\boxtimes (e;\hat{l})$ to
$\oplus_{k \; l} (e;\hat{k})\boxtimes (e;\hat{l})$, and is thus determined by
a choice of scalar for each summand (the simple summands of the two sides
being the same, and each occurring with multiplicity one), that is a
$0,2$-cochain.
Of course, these functions satisfy conditions equivalent to the coherence
conditions for the natural transformations they define. It is an easy exercise
to write out each of the coherence conditions in turn, instantiate the
objects with simple objects (or simple summands of $I$, as appropriate), and
write out the corresponding equation on the cochains.
The table below summarizes the resulting equations:
\bigskip
\begin{center}
\begin{tabular}[h]{ll}
triangle for tensor structure & $\rho(g;\widehat{klk^{-1}}) =
\alpha(g,e,k;\hat{l})\lambda(k;\hat{l})$ \\
triangle for cotensor structure & $r(g;\hat{k}) =
\beta(g;\hat{k},\hat{e},\hat{l}) l(g;\hat{l})$ \\
$\Delta$ respects right unit &
$\phi(g,e;\hat{k},\hat{l})\delta(\hat{k},\hat{l})\rho(g;\hat{k})
\rho(g;\hat{l}) = \rho(g;\widehat{kl}) $ \\
$\Delta$ respects left unit &
$\phi(e,g;\hat{k},\hat{l})\delta(\hat{k},\hat{l})\lambda(g;\hat{k})
\lambda(g;\hat{l}) = \lambda(g;\widehat{kl}) $ \\
$\otimes$ respects right counit & $
\phi(g,k;\hat{l},\hat{e})\tau(g,k) r(g;\widehat{klk^{-1}})
r(g;\hat{l}) = r(gk;\hat{l}) $\\
$\otimes$ respects left counit & $
\phi(g,k;\hat{l},\hat{e})\tau(g,k) l(g;\widehat{klk^{-1}})
l(g;\hat{l}) = l(gk;\hat{l}) $\\
$\epsilon$ preserves $\otimes$ & $ \alpha(g,k,m;\hat{e})\tau(g,km)\tau(k,m) =
\tau(g,k)\tau(gk,m) $\\
$I$ preserves $\Delta$ & $
\beta(e;\hat{k},\hat{l},\hat{m})\delta(\hat{k},\widehat{lm})
\delta(\hat{l},\hat{m})
= \delta(\hat{k},\hat{l})\delta(\widehat{kl},\hat{m}) $ \\
$\epsilon$ respects right unit & $\tau(g,e)\eta = \rho(g;\hat{e})$ \\
$\epsilon$ respects left unit & $\tau(e,g)\eta = \lambda(g;\hat{e})$ \\
$I$ respects right counit & $\delta(\hat{h},\hat{e})\eta = r(e;\hat{h}) $\\
$I$ respects left counit & $\delta(\hat{e},\hat{h})\eta = l(e;\hat{h}) $\\
\end{tabular}
\end{center}
\bigskip
By way of example, the condition that $\Delta$ be a monoidal functor
includes the condition that $\Delta$ respect the right unit transformation.
Written out this become the equation
\[ \Phi_{A,I} (Id \otimes_2 \delta) \rho_2 = \Delta(\rho) \]
\noindent where the subscripts $_2$ indicate the corresponding structure
in ${\cal C}\boxtimes {\cal C}$. Now, $\rho_2 = \rho \boxtimes \rho$,
and if $A = (g;\hat{h})$ then all of the sources and targets of the
maps in the equation are $\oplus_{kl = h} (g;\hat{k})\boxtimes (g;\hat{l})$,
and each map is thus determined by a scalar for each triple
$(g;\hat{k},\hat{l})$. Writing out a component of the equation above
gives
\[ \phi(g,e;\hat{k},\hat{l})\delta(\hat{k},\hat{l})\rho(g;\hat{k})
\rho(g;\hat{l}) = \rho(g;\widehat{kl}) \]
\noindent(Notice: on the right hand side, we use the fact that
$\Delta$ preserves identities
and scalar multiples.)
We can then analyse these equations to determine a minimal set of
data and conditions for specifying the unital and counital structures
on a categorification of $D({\Bbb N}[G])$. We assume that $\alpha,
\phi$ and $\beta$ have been chosen as in the previous theorem
to specify a bitensor categorification without unit or counit.
First, observe that it follows from the two triangle conditions that
$\rho, \lambda, r$ and $l$ are completely determined by the values
of $\rho(e;\hat{m})$ $(m \in G)$ and $r(g;\hat{e})$ $(g \in G)$ by
the formulas
\[ \rho(g;\hat{m}) = \alpha(g,e,e;\hat{m})\rho(e;\hat{m}) \]
\[ \lambda(k;\hat{l}) = \alpha^{-1}(e,e,k;\hat{l})\rho(e;\widehat{klk^{-1}}) \]
\[ r(g;\hat{k}) = \beta(g;\hat{k},\hat{e},\hat{e})r(g;\hat{e}) \]
\[ l(g;\hat{k}) = \beta(g;\hat{e},\hat{e},\hat{k})r(g;\hat{e}) \]
\noindent The only restriction on these needed to ensure that they
unambiguously determine cochains satisfying the first two equations
has already been imposed by the condition that $\alpha$ and $\beta$ be
coboundaries in the suitable sense.
In a similar way, $\otimes$ respecting
both the right and left counit conditions is equivalent to
\[ \tau(g,k) = \tilde{d}_1(r^{-1})(g,k;\hat{e})
\phi^{-1}(g,k;\hat{e},\hat{e}) \]
\noindent with no further conditions imposed on $r$ or $\phi$.
On the other hand, $\Delta$ respecting both the right and left
unit conditions is equivalent to
\[ \delta(\hat{k},\hat{l}) = d_2(\rho^{-1})(e;\hat{k},\hat{l})
\phi^{-1}(e,e;\hat{k},\hat{l}) \]
\noindent together with the condition that $\delta$ be invariant under
simultaneous conjugation of both indices by elements of $G$.
Given these last two equations, the conditions that $\epsilon$ preserve
$\otimes$ and $I$ preserve $\Delta$ follow from the condition that
$(\alpha,\phi,\beta)$ be a cocycle and the fact that $\tilde{d}_1$ and
$d_2$ distribute over multiplication of cochains and
$\tilde{d}_1^2 = 1$ and $d_2^2 =1$ (remember we write
the operation on cochains multiplicatively since the coefficients
are in $K^\times$).
The four remaining conditions are all equivalent to
\[ \eta = \rho(e;\hat{e})r(e;\hat{e})\phi(e,e;\hat{e},\hat{e}) \]
A similar analysis shows that the structural transformations
for an equivalence of unital and counital categorifications are determined
by the 1,2-cochain and 2,1-cochain which determine the non-unital non-counital
equivalence, together with a 0,1-cochain $f_0$ and a 1,0-cochain $f^0$.
The conditions besides the cobounding conditions of Theorem
\ref{cohom.thm} reduce to
\[ f_0(\hat{k}) =
\frac{\rho(e;\hat{k})}{\rho^\prime(e;\hat{k})\tilde{f}(e,e;\hat{k})} \]
\[ f^0(g) = \frac{r(g,\hat{e})}{r^\prime(g,\hat{e})f_\sim(g;\hat{e},\hat{e})}
\]
\noindent Thus, it follows that within any equivalence class of
categorifications, the {\em only} constraint upon the choice of
the functions $\rho(e;\hat{h})$ and $r(g;\hat{e})$ is the condition
that $\delta(\hat{k},\hat{l})$ be invariant under simultaneous
conjugation of both indices. We have thus almost shown
\begin{thm}
Every skeletal biunital semi-simple bitensor $K$-categorification
of $D({\Bbb N}[G])$ is determined by a choice
of a 3-cocycle $(\alpha,\phi,\beta)$
in the total complex
of the double complex $C_{i,j}$, together with a choice of
functions $\rho(e;\hat{k}):\hat{G}\rightarrow K^\times$ and
$r(g;\hat{e}):G\rightarrow K^\times$ subject to
the condition that $\phi(e,e;\hat{k},\hat{l})d_2(\rho)(e,\hat{k},\hat{l})$
be invariant under simultaneous conjugation of the hatted indices. Conversely,
every such choice determines such a categorification up to isomorphism.
The equivalence classes of
unital counital bitensor categorifications of $D({\Bbb N}[G])$ are
in natural one-to-one correspondence with ${\Bbb H}_3(G,K^\times)$.
\end{thm}
\noindent{\bf proof:} For the two statements, it remains only to
observe that the condition on $\delta$ is equivalent to the given
condition on $\phi$ and $\rho$.
The final statement requires a little more work. By the preceding
remark, it suffices to show that every cohomology class admits a
representative for which the $\rho(e;\hat{k})$ can be chosen so the
invariance condition holds. Let $(\alpha,\phi,\beta)$ be an
arbitrary 3-cocycle. Now, consider the 2-cochain $(1,f)$, where
$1$ is the constant 2,1-cochain, and $f(g;\hat{k},\hat{l}) =
\phi^{-1}(e,e;\hat{k},\hat{l})$. Multiplying $(\alpha,\phi,\beta)$ by
the (total) coboundary of $(1,f)$ give a cohomologous 3-cocycle
such that $\phi(e,e;\hat{k},\hat{l}) = 1$. Thus any constant
$\rho(e;\hat{k})$ suffices. $\Box$
Now observe that for any group $G$ there is at least one solution
to the required equations: if we choose {\em all} of the families of
scalars to be identically $1$, we obtain a solution. We will refer to
this and any equivalent bitensor categorifications
as {\bf trivial categorifications} of $D({\Bbb N}[G])$.
Of course, it behooves us to exhibit a non-trivial bitensor
categorification, since we as yet have no examples. The simplest family
of such may be described as follows: let all of the families of scalars
be identically $1$ except $\beta(g;\hat{i},\hat{j},\hat{k})$.
Observe that all conditions not involving $\beta$ are trivially satisfied,
and that the conditions involving $\beta$ then reduce to those defining
other quantities in terms of $\beta$ and the other scalars, and
the conditions
\[ \beta(g;\widehat{hih^{-1}},\widehat{hjh^{-1}},\widehat{hkh^{-1}})
\beta(h;\hat{i},\hat{j},\hat{k}) = \beta(gh;\hat{i},\hat{j},\hat{k}) \]
\noindent and
\[ \beta(g;\hat{i},\hat{j},\hat{k})\beta(g;\hat{i},\hat{jk},\hat{l})
\beta(g;\hat{j},\hat{k},\hat{l}) = \beta(g;\hat{ij},\hat{k},\hat{l})
\beta(g;\hat{i},\hat{j},\hat{kl}) \]
Thus, we may regard $\beta(g;-,-,-)$ as a function from $G$
to 3-cocycles on $G$ (written with hatted indices), satisfying the first
equation. In particular, any group homomorphism from $G$ to the (abelian)
group of 3-cocyles invariant under simultaneous conjugation gives such
a function.
For a specific example, let $G$ be any group with $C_2$, the
cyclic group of order 2, as a quotient (e.g. $G = {\frak S}_n$). Call an
element of $G$ even when its image in $C_2$ is the identity; odd otherwise.
In this case $\beta$ given by
\begin{eqnarray*}
\beta(g;\hat{i},\hat{j},\hat{k}) & = & -1 \;\;
\parbox{2in}{\rm if $g,i,j,k$ are all odd} \\
& = & 1 \;\;\; \parbox{2in}{\rm otherwise} \\
\end{eqnarray*}
\noindent has all of the desired properties. In particular, $(1,1,\beta)$
represents a non-trivial element of ${\Bbb H}_3(G,K^\times)$ (provided
$char(K) \neq 2$).
\section{Conclusions}
The cohomological setting which provided a natural setting for these
constructions and classification theorems suggests that the process of
categorification should, at least in the semi-simple case, be viewed as
a deformation process for tensor or bitensor categories.
The authors, in work in preparation \cite{cy.prep} have constructed
a general framework for the infinitesimal deformation of general
(semi-simple) bitensor categories in terms of a similar
double complex, and have isolated the cohomological obstructions to the
existence of formal power-series deformations as classes in the total
cohomology of the double complex.
It still remains to use the examples presented herein to provide
explicit examples of $4$-manifold invariants of Crane/Frenkel type \cite{cf}
and to
construct the monoidal bicategory of representations of the bitensor
categories constructed herein, thereby giving initial data for a
fully bicategorical version of the Crane-Yetter construction (cf. \cite{cy},
\cite{cky}).
|
1,314,259,993,034 | arxiv | \section{Introduction}
Alzheimer's disease (AD), one of the most prevalent degenerative conditions, is a degenerative dementia that begins with minor memory loss and gradually escalates to a complete loss of mental and physical capacities. For the patient's health, a diagnosis should be made as soon as possible so that treatment and preventative measures can be commenced. A thorough and comprehensive medical evaluation involving an array of psychological and physical testing is necessary to diagnose AD. The ability and expertise of the clinician have a direct impact on the precision of the psychological and cognitive tests, and a magnetic resonance imaging (MRI) medical data analysis is necessary for a conclusive medical diagnosis of AD. Medical specialists are in responsibility of evaluating and interpreting medical data, however due to the data subjectivity and complexity, this procedure is particularly complex and constrained for a medical specialist. Consequently, there is a need for the development of quick and efficient diagnostic tools, and machine learning (ML) can be employed for this.
The ability of the computer to learn from experience without explicit programming has improved thanks to machine learning approaches. For classification, regression, clustering, and dimensionality reduction in a variety of applications like image processing, predictive analytics, and data mining, machine learning approaches are applied. In the field of medical imaging research, machine learning (ML) is frequently used and produces outstanding results in tasks like segmentation, regression, classification, etc. Medical image analysis tasks seldom have access to such expansive datasets, unlike some visual image datasets like ImageNet, which offer large-scale annotated examples for developing models for tasks like object detection. Using pre-trained models for some related domains using the transfer learning method is a practical and effective solution.
This study deals with a specific type of transfer learning approach called domain adaptation. Domain adaptation is a very effective learning technique that has been used for analyzing medical image data \cite{2021arXiv210209508G}. In this project, we will be focusing on studying 3D structural MRI brain scans in the context of Alzheimer’s disease classification. Real-world medical datasets are prone to being filled with unlabeled and ill-labeled samples, which makes working with them very challenging. The process of labeling medical images is typically expensive, time-consuming, and labor-intensive, necessitating the involvement of doctors, radiologists, and other experts. Where transfer learning strategies like domain adaptation may be useful is in this situation. While there has been a lot of work related to the development of novel classification algorithms for Alzheimer’s detection and classification, the study of interoperability and reusability of such models is often ignored, and this is very important since a model that performs well on one dataset might not replicate that performance when tested on a different dataset, and the samples themselves can be very different with a large domain gap between them, and domain adaptation could be useful in mitigating problems associated with domain shift.
\section{Background}
Here we discuss some of the prior work done in the domain of AD detection using machine learning. Detecting Alzheimer’s disease in its early stages can have a positive impact on the patient. To accomplish this, machine learning models have been at the forefront. Authors in \cite{Muhammed_Raees_2021} use the well-known ADNI (Alzheimer’s Disease Neuroimaging Initiative) dataset \cite{Jack2008-ia} to train, test, and compare SVM and different Deep Neural Networks (DNN), showcasing how the newer DNN models (VGG19) without lesser data preprocessing can achieve better accuracy of 90\% and more when compared to SVMs. The authors in \cite{Liu2022} employed 3D CNNs in conjunction with structural MRIs to accurately distinguish between those with moderate cognitive impairment, those with cognitive health, and those with mild dementia brought on by Alzheimer's disease. The authors of \cite{Odusami2021-np} trained ResNet18, an 18-layer CNN architecture, to predict various stages of cognitive impairment, and the model was able to achieve classification accuracy of near unity.
These texts mostly deal with using the entire image to train and predict. On the other hand, texture and shape features of the hippocampus region specifically are extracted, and a neural network acts as a multiclass classifier in \cite{Raut2017AML}. This approach is believed by the authors to yield better results. Using more than one model, as in ensemble learning (EL), has been shown to improve results and learning system performance. The authors of \cite{10.3389/fnins.2020.00259} combine CNN and EL to identify subjects with mild cognitive impairment (MCI) or AD. The authors in \cite{OROUSKHANI2022100066} employ a deep triplet network and a few-shot learning method, which surpass state-of-the-art models in terms of accuracy scores. This model addresses the issue of overfitting due to the limited image samples being used to train. After comparing many models on the OASIS dataset, the authors \cite{Battineni2021-rt} conclude that the gradient boosting technique may be a better classifier than others.
\section{Overview}
We will be studying AD detection in the context of both binary classification and anomaly detection. Binary classification is a supervised learning problem where the goal is to categorize new observations into one of the two presented classes. Anomaly detection, on the other hand, tries to solve the problem of identifying rare events or observations in data that don't fit the normal patterns. Both tasks require two distinct classes of samples, and here we use the Clinical Dementia Rating (CDR) scores to divide the samples into ``cognitive normal" or ``AD/Demented."
For binary classification, we will be training various supervised convolutional neural network (CNN)-based models to classify between the two classes. We first perform hyperparameter tuning and record scores for various architectures. We will then use the model with the best performance for domain adaptation. For domain adaptation, we will consider one dataset to be the source distribution and a different dataset to be the target distribution. The model will be trained in an unsupervised domain adaptation setting where the source dataset is labeled and the target data is unlabeled. This will give us good performance metrics for scenarios where we want to use pre-trained models on a set of unlabeled MRI samples. The datasets are publicly available and are sourced from the OASIS database.
Then we look into the problem of anomaly detection, where the samples classified as demented are considered anomalies. before training the models for anomaly detection. We first test various generative autoencoder-based models for image reconstruction and image synthesis. This will give us the performance benchmarks to help pick the best model for both anomaly detection and domain adaptation for anomaly detection. Image reconstruction is an image-to-image operation that deals with the problem of recovering observations from latent space. On the other hand, image synthesis tries to learn the distribution of the data in order to efficiently generate new data samples while maintaining the best distribution and feature variety. We will be studying domain adaptation for anomaly detection in both supervised and unsupervised settings. The unsupervised approach is similar to the previous discussion of binary classification, where the target samples are unlabeled. For the supervised approach, we assume that we have knowledge of the fact that all the available samples are from non-anomalous samples, which is the cognitive normal class. The specific approach used for domain adaptation is ADDA, or adversarial discriminative domain adaptation \cite{tzeng_adversarial_2017}, which will be discussed in detail in the coming sections.
\section{Methodology and Results}
\subsection{Experimental Data}
While there are various publicly available MRI datasets, In particular, OASIS-1 and OASIS-2 datasets from the OASIS database will be used. Neuroimaging datasets are now freely accessible for research thanks to OASIS, also known as the Open Access Series of Imaging Studies. The OASIS-1 dataset is a cross-sectional compilation of 416 participants' T1-weighted MRI scans, ranging in age from 18 to 96. And OASIS-2 is a longitudinal set of 150 individuals' T1-weighted MRI images from 60 to 96 years old. The term "T1-weighted" refers to the timing of radiofrequency pulse sequences used in scanning. Hence, considering the range of ages of the subjects, OASIS-2 is a more balanced dataset when it comes to the distribution of the two classes. We also have access to the demographic data of the subjects for both of the datasets.
The demographic data contains information about the Clinical Dementia Rating (CDR) scores. We use the CDR values to first differentiate the samples into "cognitive normal" or "AD/Demented". The CDR is an alternate semi-structured interview that uses scores from six different cognitive and behavioral areas to calculate a number between 0 and 3 for evaluating dementia. We classify samples with a CDR score of 0 as cognitively normal, and samples with a CDR score greater than zero as demented. The samples are normalized based on their mean and standard deviation. The normalized samples are then sliced along the sagittal plane to obtain ten distinct slices of each sample. We use sliced samples due to the computational complexity associated with training the models on whole-brain 3D MRI scans. Figure ~\ref{fig:MRI} shows samples from both the OASIS-1 and the OASIS-2 datasets along the coronal plane, sagittal plane, and axial plane.
\onecolumngrid
\begin{figure}[h]
\begin{tabular}{ccc}
\includegraphics[width=1.0in]{OAS2_sample2.png} & \includegraphics[width=2.0in]{OAS2_sample.png} & \includegraphics[width=1.0in]{OAS2_sample3.png}\\
(a) coronal plane & (b) sagittal plane & (c) axial plane\\[6pt]
\includegraphics[width=1.0in]{OAS1_sample2.png} & \includegraphics[width=2.0in]{OAS1_sample.png} & \includegraphics[width=1.0in]{OAS1_sample3.png}\\
(a) coronal plane & (b) sagittal plane & (c) axial plane\\[6pt]
\end{tabular}
\caption{Data samples of OASIS-2 (top) and OASIS-1 (bottom)}
\label{fig:MRI}
\end{figure}
\twocolumngrid
\subsection{Binary Classification}
We have explored several models for binary classification in order to get a baseline performance on the source data and compare the models so that we can pick the one with the best performance that can be used for domain adaptation. We used grid search based on the learning rate and the number of epochs. Each data sample contains 10 slices along the sagittal view, so the dimension of each sample is 10x256x256. The models tested are ResNet-18, EfficientNet-B3, ResNeXt-50, and a 3D ResNet-18 model.
The \textbf{ResNet} model \cite{2015arXiv151203385H} was the first one to be tested. To address complexities, deep neural networks are often built with additional layers, which increase accuracy and performance. When more layers are added, the theory behind layering is that they will eventually learn features that are more complex. For example, the first layer might learn to distinguish edges in the case of image analysis, the second layer might learn to recognize textures, the third layer might learn to recognize objects, and so on. Nonetheless, adding too many layers might result in overfitting, vanishing gradients, and other issues. The development of ResNet, or residual networks, which are composed of residual blocks, has solved the complexity associated with training very deep networks. Layers with skipped connections form residual blocks, which are stacked together. The skip connections in ResNet provide a different shortcut way for the gradient to flow through, resolving the issue of vanishing gradients in deep neural networks. The identity functions are learned by the model as a result of these linkages, which helps ensure that the layers that are stacked higher will perform at least as well as the ones present lower. Here, we specifically test a network with 18 layers, which is the ResNet-18 model.
Next, we explored the \textbf{EfficientNet} model \cite{2019arXiv190511946T}. Using a compound coefficient, the convolutional neural network formulation and scaling method EfficientNet uniformly scales all depth, breadth, and resolution dimensions. The EfficientNet scaling method increases network width, depth, and resolution consistently using a set of preset scaling coefficients, in contrast to traditional methods that scale these variables arbitrarily. Network width, depth, and resolution are all uniformly scaled by EfficientNet with the help of a compound coefficient. The rationale behind the compound scaling method is that larger input images require more layers in order to expand the network's receptive field and more channels in order to acquire more fine-grained features on the larger image. We use the EfficientNet-B3 architecture in this instance. In order to determine the relationship between the several fundamental network scaling dimensions while adhering to a set resource limitation, this architecture employs the grid search approach. This could help determine the proper scaling factors for each of the scaled dimensions. The basic network was scaled to the required size based on these computed factors.
Another architecture we tested is the \textbf{ResNeXt-50} model, a ResNeXt model \cite{2016arXiv161105431X} with 50 layers. A ResNeXt replicates a structural component that combines a number of transformations with the same topology. It reveals an additional dimension that is cardinality, or the magnitude of the set of transformations, which is in contrast to a standard ResNet architecture. Thus, the ResNeXt model introduces a new hyperparameter called cardinality to adjust the model's capacity. Additionally, ResNeXt also uses the split-transform-merge paradigm from the Inception model. The ResneXt block's input is projected into a number of lower-dimensional representations instead of applying convolutions over the entire input feature map. We then independently apply only a few convolutional filters to each representation before combining the results. Group convolutions, which were suggested in the AlexNet paper \cite{10.5555/2999134.2999257} as a technique to distribute the convolution computation across numerous GPUs, are pretty similar to this notion. The input is divided channel-wise into groups rather than building filters with the entire channel depth of the input. \textbf{Since the MRI scans are sliced and the slices are passed as channels, each channel of the input is thus uncorrelated with each other, and since ResNeXt divides the input channel-wise into groups, we hypothesize that it could be the ideal architecture for our task.}
Lastly, we evaluate a \textbf{3D-ResNet-18} model. which is essentially a ResNet-18-based model that employs 3D convolution layers. With the earlier architectures, we were passing the slices along the sagittal view as channels, but here we pass the slices as a spatial dimension using 3D convolution to check whether passing the data in this fashion provides any improvement in the performance.
The results of the hyperparameter tuning, i.e., the best model and set of hyperparameters, are presented in the Table ~\ref{table:HTResults} below, and the complete scores are compiled in the appendix. As hypothesized earlier, the ResNeXt model performed the best. Since ResNeXt is the best scoring model, we will be using it as our base architecture for domain adaptation. All models have been built using the PyTorch framework \cite{paszke_pytorch_2019} and trained on an NVIDIA A100 GPU.
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Model & ResNeXt-50 \\ \hline
Training Epochs & 50\\ \hline
Learning Rate & 2e-4\\ \hline
\end{tabular}
\caption{Best Modela and Hyperparameters}
\label{table:HTResults}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
Model & ResNeXt-50 \\ \hline
AUC & 0.91157\\ \hline
Accuracy & 0.84897\\ \hline
Sensitivity/Recall & 0.88489\\ \hline
Specificity & 0.80188\\ \hline
Precision & 0.85416\\ \hline
F1 & 0.86925 \\ \hline
\end{tabular}
\caption{Supervised Binary Classification on Source (OASIS-2)}
\label{table:BCResults}
\end{table}
We will now train the ResNeXt model with this set of hyperparameters on the OASIS-2 dataset, which will be used as our source dataset for domain adaptation. The model is trained with the cross-entropy loss. A classification model's performance is measured by cross-entropy loss, also known as log loss, whose output is a probability value between 0 and 1. For learning, we use a gradient descent optimization algorithm called Adam, or Adaptive Moment Estimation \cite{2014arXiv1412.6980K}. It is an adaptive learning rate method that combines momentum-based stochastic gradient descent with RMSprop. It scales the learning rate using squared gradients, similar to RMSprop, and leverages momentum by using the gradient's moving average rather than the gradient itself, akin to SGD with momentum. The results are presented in the Table ~\ref{table:BCResults}, and Figure ~\ref{fig:Plot1} shows the training loss. The metrics calculated are AUC, accuracy, sensitivity/recall, specificity, precision, and the F1 score.
\begin{figure}[H]
\centering
\includegraphics[width = 3.0in]{ResNeXT_OAS2_Loss.png}
\caption{Training Loss of ResNeXt Model}
\label{fig:Plot1}
\end{figure}
\subsection{Domain Adaptation}
We will now discuss domain adaptation for binary classification. The ADDA approach is the specific method we employed for domain adaptation. A supervised model will be trained on a source data set with the intention of applying it to a target data set. We employ the ResNeXt-50 as our base architecture for this endeavor, specifically for the encoder, as was previously stated. Referring back to the discussion of the ADDA method, Figure ~\ref{fig:ADDA} gives a description of this method.
ADDA, or Adversarial Discriminative Domain Adaptation, is an adversarial adaptation method with the goal of minimizing the domain discrepancy distance through an adversarial objective with respect to a discriminator. Ideally, the discriminator will be unable to distinguish between the
source and the target distributions. We consider that we have access to source images and labels that come from a source distribution, but when it comes to the target distribution, we have the target images but not the labels. Our objective is to train a target encoder and classifier that can classify the target samples into the source classes. That is done by training the source and target encoders adversarially, and the discriminator tries to distinguish between the mapped latent vectors of the source and target images. The results of domain adaptation are presented in the table ~\ref{table:DAResults}. Surprisingly, the accuracy score obtained for the OASIS-1 dataset via unsupervised domain adaptation is actually higher than some of the state-of-the-art supervised classification results.
\onecolumngrid
\begin{figure}[h]
\centering
\includegraphics[width = 6.0in]{ADDA_Overview.png}
\caption{Overview of the ADDA domain adaptation method.}
\label{fig:ADDA}
\end{figure}
\twocolumngrid
\hspace{1cm}
\begin{table}[H]
\centering
\begin{tabular}{|l|l|}
\hline
Model & ResNeXt-50 \\ \hline
Accuracy without DA & 0.74624 \\ \hline
Accuracy ADDA & \textbf{0.8312} \\ \hline
SOTA Accuracy (Saratxaga et al. \cite{Saratxaga2021-fm}) & \textbf{0.81} \\ \hline
Sensitivity/Recall & 0.82926 \\ \hline
Specificity & 0.83783 \\ \hline
Precision & 0.94444 \\ \hline
F1 & 0.88311 \\ \hline
\end{tabular}
\caption{Domain Adaptation, Source: OASIS-2, Target: OASIS-1}
\label{table:DAResults}
\end{table}
\subsection{Anomaly Detection}
In this section, we will be discussing image reconstruction, image synthesis, anomaly detection, and domain adaptation for anomaly detection. We use autoencoder-based generative models such as the adversarial autoencoder \cite{2015arXiv151105644M} and variational autoencoder \cite{2013arXiv1312.6114K} for image reconstruction and image synthesis. This is in order to get a baseline performance of the models so that we can use the best-performing model for anomaly detection and also explore domain adaptation in an anomaly detection setting. The auto-encoder models used consist of an encoder and a decoder network. The decoder network learns to reconstruct the input observations from the latent space after the encoder learns to map the input observations to a latent space with a lower dimension than the input samples. The variational autoencoder imposes an additional constraint on the standard autoencoder in the form of KL divergenceloss , and the adversarial autoencoder replaced this with adversarial learning. The loss function is the mean squared error (MSE) in the case of the adversarial auto-encoder and MSE plus KL divergence for the variational auto-encoder. The optimizer used for training is the Adam optimizer in both cases. The architecture of the autoencoder model has been presented in Figure ~\ref{fig:AE}. We have used Tanh as the output activation.
\onecolumngrid
\begin{figure}[H]
\centering
\includegraphics[width = 6.0in]{Autoencoder_Architecture.png}
\caption{Architecture of the Autoencoder model}
\label{fig:AE}
\end{figure}
\twocolumngrid
Figures ~\ref{fig:plt2}, and ~\ref{fig:plt3} present the training loss of the autoencoder models and the results obtained for image reconstruction and image synthesis are presented in the below Tables ~\ref{table:EMDResults1} and ~\ref{table:EMDResults2}. To test and compare the models quantitatively, we have used the earthmovers' distance (EMD) as the metric. On a high level, EMD measures the distance between two probability distributions. Both models perform similarly, and the OASIS-1 dataset looks to be much more complex than the OASIS-2 dataset.
\onecolumngrid
\hspace{1cm}
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|}
\hline
Model & Generated Samples & Reconstructed Samples \\ \hline
Adversarial Autoencoder & 27.04159& 28.40938 \\ \hline
Variational Autoencoder & 25.15828 &23.93365 \\ \hline
\end{tabular}
\caption{EMD values (OASIS-2)}
\label{table:EMDResults1}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|l|l|l|}
\hline
Model & Generated Samples & Reconstructed Samples \\ \hline
Adversarial Autoencoder & 64.21179 & 63.64729 \\ \hline
Variational Autoencoder & 66.7877 & 66.27392 \\ \hline
\end{tabular}
\caption{EMD values (OASIS-1)}
\label{table:EMDResults2}
\end{table}
\twocolumngrid
\begin{figure}[H]
\centering
\includegraphics[width = 3.0in]{Autoencoder_OAS1_Loss.png}
\caption{Training Loss of Autoencoder Models on OASIS-1 dataset}
\label{fig:plt2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width = 3.0in]{Autoencoder_OAS2_Loss.png}
\caption{Training Loss of Autoencoder Models on OASIS-2 dataset}
\label{fig:plt3}
\end{figure}
We now use the trained autoencoder models for anomaly detection. We are considering the demented samples to be anomalies. The basic premise is that the autoencoder models are trained specifically on the cognitively normal samples for image reconstruction, and when the models are tested on the test data samples, they should produce a higher reconstruction loss for the demented samples. We will first test the models on their respective datasets and then use domain adaptation to transfer the model trained on the source dataset, which is OASIS-2, to the target dataset, which is OASIS-1.
The AUC scores calculated based on the reconstruction loss are shown in the Table ~\ref{table:ADResults}, and while the score for the OASIS-1 dataset is poor, we get a decent score for the OASIS-2 dataset, implying that the model can distinguish between cognitively normal and demented samples to some extent. These results should also give us a baseline for domain adaptation.
\begin{table}[H]
\small
\centering
\caption{AUC scores for Anomaly Detection}
\begin{tabular}{| >{\centering\arraybackslash}m{1.2in} | >{\centering\arraybackslash}m{0.8in} |>{\centering\arraybackslash}m{0.8in} |}
\hline
Model & OASIS-1 & OASIS-2
\\
\hline
Adversarial Autoencoder & 0.60727& 0.71692\\
Variational Autoencoder & 0.67645& 0.68715\\
\hline
\end{tabular}
\label{table:ADResults}
\end{table}
Similar to how we have experimented in the case of binary classification, we will be using the ADDA approach of domain adaptation for anomaly detection. This has been done in two parts: the supervised approach, where we only use the cognitively normal samples of the target dataset, and the unsupervised approach, where we use all the samples of the target dataset from both classes. During our testing, we observed that the adversarial autoencoder model was frequently outperforming the variational autoencoder model, which is why we specifically use the adversarial autoencoder for domain adaptation.
\onecolumngrid
\hspace{1cm}
\begin{table}[H]
\small
\centering
\caption{AUC scores for Domain Adaptation, Source: OASIS-2, Target: OASIS-1}
\begin{tabular}{| >{\centering\arraybackslash}m{1.2in} | >{\centering\arraybackslash}m{0.8in} |>{\centering\arraybackslash}m{1.8in} |>{\centering\arraybackslash}m{1.8in} |}
\hline
Model &Without DA& ADDA (For Supervised Anomaly Detection) & ADDA (For Unsupervised Anomaly Detection)
\\
\hline
Adversarial Autoencoder & 0.78162 &0.81097 &0.73742\\
Variational Autoencoder & 0.77341& 0.79150 & 0.71228\\
\hline
\end{tabular}
\label{table:DAADResults}
\end{table}
\twocolumngrid
Examining the results in Table ~\ref{table:DAADResults}, even simply using a naive transfer learning approach that is performing inference using a pre-trained model without domain adaptation improves the AUC score for OASIS-1, but the ADDA approach gives us an improvement in the score, and while using ADDA for anomaly detection in an unsupervised setting resulted in degraded performance, it still outscores the result we got for anomaly detection without any transfer learning or domain adaptation.
\section{Discussion and Future Work}
Although we were able to achieve encouraging results, we have only used the sagittal plane for slicing and training the models, and training the models using samples along other planes could be explored. The anomaly detection results are currently not on par with those obtained for binary classification. However, this approach still has a lot of potential since anomaly detection only requires samples from cognitively normal subjects i.e. we could train the anomaly detection model with a larger set of samples since it’s easier to obtain more samples of cognitively normal subjects as opposed to demented subjects.
Despite the fact that the motivating anomaly detection results show that the auto-encoder models are able to capture the features of the MRI scans, the images generated by them don’t look visually appealing, which is why we chose a more quantitative measure such as EMD to compare the models. Alternative deep generative models, such as generative adversarial networks (GAN), could be explored to improve image synthesis performance.
\section{Conclusion}
In conclusion, we have performed various studies on the OASIS-1 and OASIS-2 datasets. Domain adaptation in the case of binary classification resulted in a significant improvement to the results of the OASIS-1 dataset and outperformed some of the state-of-the-art supervised classification approaches despite the model being trained in an unsupervised setting. In addition, we then explored an alternative way of Alzheimer’s disease detection called anomaly detection and noticed that domain adaptation improved the scores for this application as well. Hence, domain adaptation is a useful technique for studying MRI brain scans and can be used for developing competent methods for Alzheimer’s disease classification.
\section{Acknowledgements}
We would like to thank Dr. Corey Toler-Franklin for her guidance and the University of Florida for providing access to computational resources.
\nocite{*}
|
1,314,259,993,035 | arxiv | \section{Introduction}
\label{sec:intro}
Consider Polish spaces $\ALP A$, $\ALP B$ and $\ALP C$, with generic elements in the spaces denoted by $a$, $b$ and $c$, respectively. Let $\wp(\cdot)$ denote the space of probability measures over the space $(\cdot)$. If $\mu\in\wp(\ALP A\times\ALP B)$, then $\mu(\cdot|a)$ denotes the conditional measure on the space $\ALP B$ given an element $a\in\ALP A$.
Suppose $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp(\ALP A\times\ALP B\times\ALP C)$ be a converging sequence of measures such that $\mu_n$ converges to $\mu_0$ in some topology as $n\rightarrow\infty$, and for all $n\in\mathbb{N}$, the conditional measure on the space $\ALP B$ given elements $a$ and $c$ is independent of $c$, that is, $\mu_n(db|a,c) =\mu_n(db|a)$ for $a\in\ALP A,c\in\ALP C$ $\mu_n$-almost everywhere (see Subsection \ref{subsec:condI} for a formal definition of conditional independence). A natural question that arises is whether the limit $\mu_0$ also satisfy this conditional independence condition, that is, does $\mu_0(db|a,c) =\mu_0(db|a)$ $\mu_0$-almost every $a\in\ALP A,c\in\ALP C$ hold? As we will soon see (by an example) in Section \ref{sec:toi}, if we endow the space of measures with the usual weak-* topology, then this conditional independence condition may not be satisfied in the limit. Thus, we must endow the space of measures with a stronger topology such that if we take a convergence sequence (or
net) of measures in that topology, then the conditional independence condition is maintained in the limit.
To see why a stronger topology is essential in certain problems, consider an optimization, a team, or a game problem, in which the actions of decision makers depend on their information. If a sequence of measures induced by the strategies of a decision maker is taken, then in the weak-* limit, the actions of the decision maker may become independent of the information of the decision maker, or may become dependent on some other random variables that are not observed by the decision maker. This may be unacceptable in many circumstances\footnote{The action of a decision maker becoming independent of the information in the limit is not necessarily troublesome; the decision maker can decide not to use the information while making a decision.}, as it may violate information or causality constraints of the problem. Due to this issue, several authors studying game or optimization problems have assumed specific structures on sequences of measures or corresponding conditional measures in order to ensure that the
causality or the information constraints are not violated by the limiting measure (see, for example, \cite{jordan1977, milgrom1985, engl1995, jackson2012}, among several others).
The failure of preserving causality or information constraints in the limit under usual topologies on measure spaces led \citet{hellwig1996} to define the {\it topology of convergence in information}\footnote{We prefer to use ``topology of information'' instead of {\it topology of convergence in information} for brevity.} on measure spaces, which, generally speaking, is stronger than the usual weak-* topology on the measure spaces. Under this stronger topology, a convergent sequence of measures preserves the conditional independence property in the limit. The purpose of this paper is twofold: (i) to study the structure of this new topology over measure spaces, and (ii) to identify sufficient conditions when convergence of a sequence of measures under any of the other well-known topologies on measure spaces imply convergence in the topology of information.
\subsection{Previous Work}
One of the first papers to make assumptions on a sequence of measures over general Polish spaces in order to preserve informational constraints in the limit is \citet{jordan1977}. The author considered an infinite horizon one-person discrete-time optimization problem in which the state of the nature evolved as time progressed, and the decision maker, at any time step, observed the realizations of the state until that time step and actions taken until the previous time step. The author studied the continuity properties of the value functions as a function of the distribution of the states of nature. In order to maintain the information and causality constraint, that is, the action at a time step must be a function of the past actions, and realizations of the past and the current states of the world, the author assumed that the conditional measure of the future states given the past and the current states is a continuous function of the realizations of the past and the current states.
The continuity assumption on the conditional measures is restrictive, as pointed out by \citet{hellwig1996}. This motivated \citet{hellwig1996} to define the topology of information on the measure spaces, under which a convergent sequence of measures maintain the informational constraints in the limit. Further, using this topology, he obtained the continuity properties of the value function as a function of the distribution of exogenous states of nature variables and proved the existence and continuity of optimal strategies in infinite horizon optimization problems. There is a mistake in the crucial steps in the proof of one of the main results, Lemma 4, in \citet{hellwig1996}, which we also address and correct in this paper.
Independently, \citet{milgrom1985} considered a game of incomplete information where the type spaces and action spaces of the decision makers, respectively, are Polish spaces and compact metric spaces. They assumed certain absolute continuity condition on the joint measures over the product space of the type spaces of the decision makers. This absolute continuity assumption was crucial in showing that the limit of a weak-* convergent sequence of distributional strategies\footnote{A distributional strategy of a decision maker is the joint measure over the action and type spaces of a decision maker induced by an equivalence class of behavioral strategies of the decision maker. For a precise definition and details, the reader is referred to \citet{milgrom1985}.} retain informational constraints in the limit. \citet{engl1995} studied the continuity properties of Nash equilibrium correspondence, as a function of the joint measures over the type spaces (also known as beliefs), in games with incomplete information.
The author assumed that the type space of the decision makers is countable and the beliefs on the type space converge in the topology of setwise convergence. Similar setups have been studied in \citet{kajii1998} and \citet{jackson2012} later on.
\subsection{Outline of the Paper}
The paper is organized as follows. We discuss some preliminary results in Section \ref{sec:prelim}. In Section \ref{sec:toi}, we motivate the discussions on why the topology of information is important, and then define the topology of information on the space of measures over the product of two Polish spaces. Section \ref{sec:main} is the main section of this paper, where we prove that a convergent sequence of measures in the topology of information maintains the conditional independence property in the limit and discuss how this result fixes the mistake in the proof of \citet[Lemma 4]{hellwig1996}. We study some topological properties of the topology of information in Section \ref{sec:top} and present an example that applies the concept of topology of information to show existence of an optimal solution to an optimization problem described in that section. Thereafter, we study the relation between the topology of information and other well-known topologies like weak-* topology, topology of setwise
convergence and convergence assumptions made in the literature in Section \ref{sec:relation}. In particular, we identify certain sufficient conditions for a sequence of measures under which convergence in either topology imply convergence of that sequence in the topology of information. Finally, we conclude our discussion in Section \ref{sec:conclusion}.
\subsection{Notation}
Throughout the paper, we use the following notation. Let $\ALP X$ be a set and $X\subset\ALP X$ be a subset. Then, $X^\complement$ denotes the complement of the set $X$. Now, let $\ALP X$ be a topological space. The vector space of all bounded continuous functions on the topological space $\ALP X$ endowed with supremum norm $\infnorm{\cdot}$ is denoted by $C_b(\ALP X)$, that is, $C_b(\ALP X):=\{f:\ALP X\rightarrow \Re: f \text{ is continuous and }\infnorm{f}<\infty\}$. For a metric space $\ALP X$, we let $U_b(\ALP X)$ denote the vector space of all bounded uniformly continuous functions with the supremum norm.
We use $\FLD B(\ALP X)$ and $\FLD P(\ALP X)$ to denote, respectively, the Borel $\sigma$-algebra and the power set of a topological space $\ALP X$. For two topological spaces $\ALP X_1$ and $\ALP X_2$, $\FLD B(\ALP X_1)\otimes\FLD B(\ALP X_2)$ denotes the Borel $\sigma$-algebra generated by the set of sets $\{X_1\times X_2:X_1\in\FLD B(\ALP X_1), X_2\in\FLD B(\ALP X_2)\}$.
The space of probability measures over the measurable space $(\ALP X,\FLD B(\ALP X))$ is denoted by $\wp(\ALP X)$. We let $\ind{\cdot}$ denote the Dirac measure over a point $\{\cdot\}$. Henceforth, we use $\wp_w(\ALP X)$ to denote the space of probability measures over the space $\ALP X$ endowed with the weak-* topology, which is defined to be the weakest topology such that the map $\mu\mapsto \int_{\ALP X} f\:d\mu$ is a continuous map for every $f\in C_b(\ALP X)$. If $\{\mu_{\alpha}\}\subset\wp(\ALP X)$ is a convergent net of measures converging to $\mu_0$ in weak-* topology, then we denote it by $\mu_\alpha\overset{w^*}{\rightharpoonup}\mu_0$. For a measure $\mu\in\wp(\ALP X\times\ALP Y)$, $\mu^{\ALP X}\in\wp(\ALP X)$ denotes the marginal of $\mu$ onto the space $\ALP X$.
A Polish space is defined as a separable topological space which is completely metrizable. It is well known that the space of measures over Polish spaces with weak-* topology is also a Polish space (see \citet[p. 505]{ali2006} or \citet[Theorem 8.9.4, p. 213]{bogachev2006b}). Thus, if $\ALP A$ and $\ALP B$ are Polish spaces, then $\ALP A\times\ALP B$, $\wp_w(\ALP A)$, $\wp_w(\wp_w(\ALP A))$, $\wp_w(\ALP A\times\wp_w(\ALP B))$ are all Polish spaces.
\section{Preliminaries}\label{sec:prelim}
Before we discuss the topology of information, we recall disintegration theorem for measures \citet[Theorem 5.3.1, p. 121]{amb2008}. The main statement of the theorem is that if $\ALP A$ and $\ALP B$ are Polish spaces and $\mu\in\wp(\ALP A\times\ALP B)$, then there exists a conditional measure on the space $\ALP B$ given an element $a\in\ALP A$. Thus, one can disintegrate a joint measure into a product of a conditional measure and a marginal. We state the following lemma without proof, the proof of which relies on the disintegration theorem.
\begin{lemma}[\citet{amb2008}]\label{lem:glue}
Let $\ALP A,\ALP B$ and $\ALP C$ be Polish spaces. Consider $\mu_1\in\wp(\ALP A\times\ALP B)$ and $\mu_2\in\wp(\ALP B\times\ALP C)$ such that $\mu^{\ALP B}_1= \mu^{\ALP B}_2$. Then, there exists a measure $\mu_0\in\wp(\ALP A\times\ALP B\times\ALP C)$ such that
\beq{\label{eqn:glue}\mu_0^{\ALP A\times \ALP B} = \mu_1,\quad \mu_0^{\ALP B\times\ALP C} = \mu_2.}
Moreover, $\mu_0$ is unique if either there exists a Borel measurable function $h_1:\ALP B\rightarrow\ALP A$ such that $\mu_1(da,db) = \ind{h_1(b)}(da)\mu^{\ALP B}_1(db)$, that is, $\mu_1(A\times B)=\int_B \delta_{\lbrace h_1(b)\rbrace} (A) \mu_1^{\ALP B}(db)$ for any $A\in \FLD B(\ALP A)$ and $B\in \FLD B(\ALP B)$, or there exists a Borel measurable function $h_2:\ALP B\rightarrow\ALP C$ such that $\mu_2(db,dc) = \ind{h_2(b)}(dc)$ $\mu^{\ALP B}_2(db)$.
\end{lemma}
\begin{proof}
For proof, the reader is referred to \citet[Lemma 5.3.2, pp. 122]{amb2008}. The first part is also proved in \citet[Theorem 1.1.10, p. 7]{dudley1999}.
\end{proof}
We use the following result on the convergence of marginals of measures.
\begin{lemma}[Convergence of Marginals]\label{lem:marg}
Let $\ALP X$ and $\ALP Y$ be Polish spaces. Suppose $\{\nu_n\}_{n\in\mathbb{N}}\subset\wp_w(\ALP X\times\ALP Y)$ is a converging sequence of measures that converges to $\nu_0$ as $n\rightarrow\infty$ in weak-* topology. Define $\zeta_n(B) := \nu^{\ALP Y}_n(B) = \nu_n(\ALP X\times B)$ for all Borel sets $B\subset\ALP Y$ and $n\in\mathbb{N}\cup\{0\}$. Then, $\lf{n}\zeta_n\overset{w^*}{\rightharpoonup}\zeta_0$, that is, $\nu^{\ALP Y}_n$ converges to $\nu^{\ALP Y}_0$ in the weak-* topology.
\end{lemma}
\begin{proof}
This is a simple consequence of Lemma 5.2.1 in \citet[p. 118]{amb2008}.
\end{proof}
We now formally define conditional independence property of a measure in the next subsection.
\subsection{Conditional Independence}\label{subsec:condI}
We recall here the formal definition of conditional independence. Let $\mu\in \wp(\ALP A\times\ALP B\times\ALP C)$ be a probability measure. Let $\mu(\cdot\vert a)$ denote the regular conditional distribution of $\mu$ on $\ALP B\times \ALP C$ given $a\in \ALP A$. Then, the distribution $\mu$ is said to be conditionally independent given a point $a\in \ALP A$ if
\begin{align}\label{eqn:condI1}
\mu(B\times C\vert a)=\mu(B\vert a) \mu(C\vert a)
\end{align}
for all $B\in \mathcal{B}(\ALP B)$, $C\in \mathcal{B}(\ALP C)$ and $\mu$-almost every $a\in \ALP A$. If $\mu(\cdot\vert a,c)$ denotes the regular conditional distribution of $\mu$ on $\ALP B$ given $a\in \ALP A$ and $c\in \ALP C$. Then, conditional independence of $\mu$ given $a$ is equivalent to
\begin{align}\label{eqn:condI2}
\mu(B\vert a,c)=\mu(B\vert a)
\end{align}
for all $B\in \mathcal{B}(\ALP B)$, $\mu$-almost every $c\in \ALP C$ and $a\in \ALP A$ (see Lemma 2.7 in \citet{jordan1977} or any other probability textbook)\footnote{Jordan (1977) states things in terms of conditional independence from $\sigma$-algebras generated by random variables. These properties translate straightforwardly into the properties fro regular conditional distributions}. We will use the characterization of conditional given by (\ref{eqn:condI2}) in this paper. For condition (\ref{eqn:condI2}), we use the short-hand notation $\mu(db\vert a,c)=\mu(db\vert a)$.
\section{The Topology of Information}\label{sec:toi}
In this section, we present the definition of the topology of information on the space of measures over a product of two Polish spaces. This topology has been defined in \citet{hellwig1996}. For this topology, we answer the following question: If $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp(\ALP A\times\ALP B\times\ALP C)$ is a sequence of measures that converges to $\mu_0$ in some topology and each $\mu_n$ satisfies (\ref{eqn:condI1}) (or equivalently (\ref{eqn:condI2})), does (\ref{eqn:condI1}) also hold for $\mu_0$?
First, an example is presented that demonstrates that the usual weak-* topology does not retain conditional independence property in the limit if we consider a weak-* convergent sequence of measures.
\subsection{Motivation}
We first take a look at the following example.
\begin{example}\label{exm:ex2}
Let $\ALP A = \Re$, $\ALP B = \{-1,1\}$, $\ALP C = \Re$. In this example, $c$ is a bijective function of $b$, whereas $a$ is a noise corrupted version of $c$. Define $h_n:\ALP B\rightarrow\ALP C$ as $h_n(b) = b\left( 1+\frac{1}{n} \right)$. Let us define $\{\mu_n\}_{n\in\mathbb{N}\cup\{0\}}\subset\wp(\ALP A\times\ALP B\times\ALP C)$ as
\beqq{\mu_n(da|b) &=& \frac{1}{2}\ind{h_n(b)+1}(da)+\frac{1}{2}\ind{h_n(b)-1}(da)\qquad \mu_n(dc|b) =\ind{h_n(b)}(dc), \\
\mu_0(da|b) &=& \frac{1}{2}\ind{b+1}(da)+\frac{1}{2}\ind{b-1}(da)\qquad\qquad\quad \mu_0(dc|b) =\ind{b}(dc),\\
\mu_n^{\ALP B} &=& \mu_0^{\ALP B} = \frac{1}{2}\ind{-1}+\frac{1}{2}\ind{1}.}
Since $h_n(b)\rightarrow b$ as $n\rightarrow\infty$, we conclude that $\mu_n\overset{w^*}{\rightharpoonup} \mu_0$. For every $n\in\mathbb{N}$, we have
\beqq{\mu_n(db|a) &=& \ind{\text{sgn}(a)}(db) = \left\{\begin{array}{ll}
\ind{-1}(db) &\text{if } a\in\{-\frac{1}{n},-2-\frac{1}{n}\}\\
\ind{1}(db) &\text{if } a\in\{\frac{1}{n},2+\frac{1}{n}\}\\\end{array}\right.,\\
\mu_n(dc|a) &=& \ind{(1+\frac{1}{n})\text{sgn}(a)}(dc).}
This implies, given $a$, conditional independence holds. On the other hand, $\mu_0(db|a,c) = \ind{c}(db)\neq \mu_0(db|a)$ for $a=0$ as $\mu_0(db\vert a)=\frac{1}{2}\delta_{\lbrace -1\rbrace}+\frac{1}{2}\delta_{\lbrace 1\rbrace}$. In other words, given $a$, $b$ and $c$ are completely determined if the three-tuple $(a,b,c)$ are distributed according to measure $\mu_n,\: n\in\mathbb{N}$, but the same does not hold if the they are distributed according to the measure $\mu_0$. Thus, the conditional independence property is lost in the limit. {\hfill $\Box$}
\end{example}
The conditional independence property is crucial to show the existence of optimal strategies in the problems of optimization under uncertainty, where the decision makers have informational or causality constraints, such as the ones considered in \citet{jordan1977, hellwig1996, milgrom1985} and others. In such problems, if we take a convergent sequence of measures, then in the limit, we have to avoid situations where the control actions (lying in the space $\ALP C$) of the decision makers become independent of their information (lying in the space $\ALP A$) or dependent on some other random variables (lying in the space $\ALP B$) that they do not observe.
In the next subsection, we define the topology of information on probability measures over a product of two Polish spaces. The definition of this topology over measure spaces comes from \citet[p. 448]{hellwig1996}.
\subsection{Topology of Information on Probability Measure Space}
We let $\wp_I(\ALP A\times\ALP B)$ denote the space of probability measures over the space $\ALP A\times\ALP B$ endowed with the topology of information. This is defined as follows: Let $\ALP N\subset \wp_w(\ALP A\times\wp_w(\ALP B))$ be the set of measures\footnote{We reserve this notation for the rest of this paper.} that are induced by measurable functions, that is, for every $\nu\in\ALP N$, there exists a measurable function $h_{\nu}:\ALP A\rightarrow\wp(\ALP B)$ such that $\nu(da,d\zeta) = \ind{h_{\nu}(a)}(d\zeta)\nu^{\ALP A}(da)$. Assume that $\ALP N$ is endowed with the subspace topology of $\wp_w(\ALP A\times\wp_w(\ALP B))$. There is a bijection, say $\psi:\wp(\ALP A\times\ALP B)\rightarrow\ALP N$, between the space of measures $\wp(\ALP A\times \ALP B)$ and $\ALP N$ which can be seen as follows: If $\mu\in\wp_I(\ALP A\times\ALP B)$, then there exists a unique measure $\nu\in\ALP N$ such that $\nu(da,d\zeta) = \ind{\mu(\cdot|a)}(d\zeta)\mu^{\ALP A}(da)$. Conversely, if $\nu\in\ALP N$, then there
exists a unique $\mu\in\wp_I(\ALP A\times\ALP B)$ defined by $\mu(A\times B) = \int_A \int_{\wp(\ALP B)} \zeta(B)\nu(da,d\zeta)$. Also recall that by the definition of conditional measure, the function that maps $a\mapsto\mu(\cdot|a)$ is a Borel measurable function from $\ALP A$ to $\wp_w(\ALP B)$. We now define the topology of information.
\begin{definition}[Topology of Information (\citet{hellwig1996})]\label{def:topinfo}
The topology of information is defined to be the coarsest topology on $\wp(\ALP A\times\ALP B)$, denoted by $\wp_I(\ALP A\times\ALP B)$, that makes the function $\psi:\wp_I(\ALP A\times\ALP B)\rightarrow\ALP N$ continuous. Thus, if a set $U\subset\wp_I(\ALP A\times\ALP B)$ is open, then there exists an open set $V\subset\ALP N$ such that $U=\psi^{-1}(V)$.
\end{definition}
The topology of information is stronger than the weak-* topology on the space of measures over the space $\ALP A\times\ALP B$, and we prove this later in Corollary \ref{cor:toiweak}. Since $\psi$ is one-to-one mapping, the space $\wp_I(\ALP A\times\ALP B)$ is homeomorphic to $\ALP N\subset\wp_w(\ALP A\times\wp_w(\ALP B))$ with $\psi$ and $\psi^{-1}$ being the homeomorphism between the spaces. This fact is a consequence of the following result.
\begin{lemma}
$\psi^{-1}:\ALP N\rightarrow\wp_I(\ALP A\times\ALP B)$ is continuous.
\end{lemma}
\begin{proof}
Let $U\subset\wp_I(\ALP A\times\ALP B)$ be open. Then, $\psi(U)$ is open in $\ALP N$ by Definition \ref{def:topinfo}. Thus, $\psi^{-1}$ is continuous.
\end{proof}
The above result is also stated in \citet[p. 449]{hellwig1996}. In the next section, we show that the a convergent sequence of measures in the topology of information retains the conditional independence property in the limit.
\section{Limit of Convergent Sequences in Topology of Information}\label{sec:main}
This section is devoted to prove the main result of this paper, that is, the topology of information retains the conditional independence condition in the limit. In order to show this, we first prove a few auxiliary results, that are used in the main result of this section, Theorem \ref{thm:main}.
\subsection{Auxiliary Results}
We need the following definition to prove a few results later.
\begin{definition}\label{def:consistent}
Let $\ALP X$, $\ALP Y$ and $\ALP Z$ be Polish spaces and let $\mu_1\in\wp(\ALP X\times\ALP Y)$ and $\mu_2\in\wp(\ALP Y\times\ALP Z)$. We say that the measures $\mu_1$ and $\mu_2$ are consistent if and only if $\mu^{\ALP Y}_1 = \mu^{\ALP Y}_2$. {\hfill$\Box$}
\end{definition}
The next lemma discusses some properties of the function that glues two consistent probability measures in a specific manner.
\begin{lemma}\label{lem:aux1}
Let $\ALP X$, $\ALP Y$ and $\ALP Z$ be Polish spaces. Let $\ALP M\subset\wp_w(\ALP X\times\ALP Y)\times\wp_w(\ALP Y\times\ALP Z)$ be the set of all consistent measure pairs, and let $\tilde{\ALP M}\subset\ALP M$ be a tight set of consistent measure pairs. Define $\chi_1:\ALP M\rightarrow \wp_w(\ALP X\times\ALP Y\times\ALP Z)$ as
\beq{\label{eqn:chi1}\chi_1(\mu,\nu)(X\times Y\times Z) = \int_{Y} \mu(X|b)\nu(Z|b)\mu^{\ALP Y}(db) = \int_{Y\times Z} \mu(X|b)\nu(db,dc).}
for all Borel sets $X\subset\ALP X,Y\subset\ALP Y,Z\subset\ALP Z$. Then,
\begin{enumerate}
\item $(\chi_1(\mu,\nu))^{\ALP X\times\ALP Y} = \mu $ and $(\chi_1(\mu,\nu))^{\ALP Y\times\ALP Z} = \nu$.
\item $\chi_1(\tilde{\ALP M})$ is a tight set of measures.
\item Let there exists a Borel measurable function $h_0:\ALP Y\rightarrow\ALP X$ such that $\mu_0(dx,dy) = \ind{h_0(y)}(dx)\mu^{\ALP Y}_0(dy)$. If $\{(\mu_n,\nu_n)\}_{n\in\mathbb{N}}\subset\ALP M$ is a convergent sequence with the limit $(\mu_0,\nu_0)\in\ALP M$, then $\lf{n}\chi_1(\mu_{n},\nu_{n}) = \chi_1(\mu_{0},\nu_{0})$.
\end{enumerate}
\end{lemma}
\begin{proof}
See \ref{app:aux1}.
\end{proof}
This leads us to the following result, which is a corollary of the Lemma \ref{lem:aux1}.
\begin{corollary}\label{cor:cor1}
Let $\ALP M \subset \ALP N\times\wp_w(\ALP A\times\ALP C)$ be a set of consistent measure pairs. Then, $\varphi_1:\ALP M\rightarrow\wp_w(\ALP A\times\wp_w(\ALP B)\times\ALP C)$, defined in an identical fashion as $\chi_1$ in \eqref{eqn:chi1}, is a continuous function.
\end{corollary}
\begin{proof}
The proof follows from the third part of the result in Lemma \ref{lem:aux1}.
\end{proof}
The next lemma is also an important result.
\begin{lemma}\label{lem:aux2}
Let $\ALP X$ and $\ALP Y$ be Polish spaces, and $\chi_2:\wp_w(\ALP X\times\wp_w(\ALP Y))\rightarrow\wp_w(\ALP X\times\ALP Y)$ be defined as
\beq{\label{eqn:chi2}\chi_2(\nu)(X\times Y) = \int_{X}\int_{\wp_w(\ALP Y)} \zeta(Y)\nu(dx,d\zeta)}
for all Borel sets $X\subset\ALP X,Y\subset\ALP Y$. Then, the following holds:
\begin{enumerate}
\item For any bounded measurable function $g:\ALP X\times\ALP Y\rightarrow\Re$, define $\bar{g}:\ALP X\times\wp_w(\ALP Y)\rightarrow\Re$ as $\bar{g}(x,\zeta) = \int_{\ALP Y} g(x,y)\zeta(dy)$. We have
\beq{\label{eqn:bargxy}\int_{\ALP X\times\ALP Y}g(x,y)\chi_2(\nu)(dx,dy) = \int_{\ALP X\times\wp_w(\ALP Y)} \bar{g}(x,\zeta)\nu(dx,d\zeta).}
\item If $g$ is a bounded continuous function on its domain, then $\bar{g}$ is a bounded continuous function on its domain.
\item $\chi_2$ is a continuous function.
\end{enumerate}
\end{lemma}
\begin{proof}
See \ref{app:aux2}.
\end{proof}
\begin{corollary}\label{cor:cor2}
$\varphi_2:\wp_w(\ALP A\times\wp_w(\ALP B)\times\ALP C)\rightarrow\wp_w(\ALP A\times\ALP B\times\ALP C)$, defined in an identical fashion as $\chi_2$ in \eqref{eqn:chi2}, is a continuous function.
\end{corollary}
\begin{proof}
This is a direct application of Lemma \ref{lem:aux2}.
\end{proof}
We have now proved all the auxiliary results to prove the main result of this paper in the next subsection.
\subsection{Main Result}
We show that when we take a convergent sequence of measures in the topology of information, then conditional independence property holds.
\begin{theorem}
\label{thm:main}
Let $\ALP P\subset \wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)$ be a set of consistent measure pairs. Let $\varphi:\ALP P\rightarrow\wp_w(\ALP A\times\ALP B\times\ALP C)$ be defined as
\beqq{\varphi(\mu,\nu)(A\times B\times C) = \int_{A\times C} \mu(db|a)\nu(da,dc).}
Then, $\varphi$ is a continuous function on $\ALP P$ and $\varphi(\mu,\nu)(db|a,c) = \varphi(\mu,\nu)(db|a)=\mu(db|a)$.
\end{theorem}
\begin{proof}
Note that $\varphi(\mu,\nu) = \varphi_2(\varphi_1(\psi(\mu),\nu))$, where $\varphi_1$ and $\varphi_2$ are defined in Corollaries \ref{cor:cor1} and \ref{cor:cor2}, respectively. It is clear that $\varphi$ is continuous since $\varphi_1,\varphi_2$ and $\psi$ are continuous functions on their domain. We now show that conditional independence property of $\varphi$.
Let $g\in C_b(\ALP A\times\ALP B\times\ALP C)$ and $\bar{g}(a,\zeta,c):=\int_{\ALP B} g(a,b,c)\zeta(db)$. Then, $\bar{g}\in C_b(\ALP A\times\wp_w(\ALP B)\times\ALP C)$ by Lemma \ref{lem:aux2} Part 2, which further implies
\beqq{\int_{\ALP A\times\ALP B\times\ALP C}g(a,b,c)\varphi(\mu,\nu)(da,db,dc) &=& \int_{\ALP A\times\wp_w(\ALP B)\times\ALP C}\bar{g}(a,\zeta,c)\:\varphi_1(\psi(\mu),\nu)(da,d\zeta,dc),\\
&=& \int_{\ALP A\times\wp_w(\ALP B)\times\ALP C}\bar{g}(a,\zeta,c)\ind{\mu(\cdot|a)}(d\zeta)\nu(da,dc),\\
&=& \int_{\ALP A\times\ALP B\times\ALP C}g(a,b,c)\mu(db|a)\nu(da,dc).}
In the expressions above, the first equality follows from the definition of $\varphi_2$, the second equality follows from the definition of $\varphi_1(\psi(\cdot),\cdot)$ and the third equality follows from the definition of $\bar{g}$. On the other hand
\beqq{\int_{\ALP A\times\ALP B\times\ALP C}g(a,b,c)\varphi(\mu,\nu)(da,db,dc) &=& \int_{\ALP A\times\ALP B\times\ALP C}g(a,b,c)\varphi(\mu,\nu)(db|a,c)\nu(da,dc),}
which follows from the fact that $(\varphi(\mu,\nu))^{\ALP A\times\ALP C}=\nu$ by the definition of $\varphi_1$ and $\varphi_2$ in Corollaries \ref{cor:cor1} and \ref{cor:cor2}. Since the above equality holds for all continuous bounded functions on space $\ALP A\times\ALP B\times\ALP C$, we conclude that $\varphi(\mu,\nu)(db|a,c) = \varphi(\mu,\nu)(db|a)=\mu(db|a)$, and the proof of the theorem is complete.
\end{proof}
Now, if we take a sequence of measures $\{(\mu_n,\nu_n)\}_{n\in\mathbb{N}}\subset \ALP P$ which converges to $(\mu_0,\nu_0)\in\wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)$, then $\varphi(\mu_0) = \lf{n}\varphi(\mu_n)$. Moreover, we also conclude that \[\varphi(\mu_0)(db|a,c) = \varphi(\mu_0)(db|a)=\mu_0(db|a).\]
Thus, the conditional independence is retained. We also have the following corollary.
\begin{corollary}\label{cor:toiweak}
The topology of information is a stronger topology than the usual weak-* topology on the space of measures. Thus, the space of probability measure endowed with the topology of information is a Hausdorff space.
\end{corollary}
\begin{proof}
The statement follows from taking $\ALP C$ to be a one-point space in the result of Theorem \ref{thm:main}. Second statement follows immediately from the first statement of the corollary.
\end{proof}
\begin{corollary}\label{cor:main2}
Using the same notation as in Theorem \ref{thm:main}, consider $\mu\in\wp_I(\ALP A\times\ALP B)$ and $\{\nu_n\}_{n\in\mathbb{N}}\subset\wp_w(\ALP A\times\ALP C)$ such that $\mu^{\ALP A} = \nu^{\ALP A}_n$ for all $n\in\mathbb{N}$. If $\nu_n\overset{w^*}{\rightharpoonup}\nu$ for some $\nu\in\wp_w(\ALP A\times\ALP C)$, then $\varphi(\mu,\nu_n)\overset{w^*}{\rightharpoonup}\varphi(\mu,\nu)$.
\end{corollary}
\begin{proof}
First note that by Lemma \ref{lem:marg}, we know that $\nu^{\ALP A}= \mu^{\ALP A}$. Consequently, the tuple $(\mu,\nu)$ is a consistent pair of measures. Since $\wp_I(\ALP A\times\ALP B)$ is a Hausdorff space by Corollary \ref{cor:toiweak}, the statement follows.
\end{proof}
Next example illustrates that if we use $\wp_w(\ALP A\times\ALP B)$ instead of $\wp_I(\ALP A\times\ALP B)$ in the statement of Theorem \ref{thm:main}, then the function $\varphi$ may not be continuous.
\begin{example}
Let $\ALP A=\left\lbrace 0,1,\frac{1}{2},\frac{1}{3},...\right\rbrace , \ALP B=\left\lbrace \underline{b},\overline{b}\right\rbrace, \ALP C=\left\lbrace \underline{c},\overline{c}\right\rbrace $, where $\ALP A$ is endowed with the subspace topology of the real line, and $\ALP B$ and $\ALP C$ is endowed with discrete topology. Consider two sequences $\{\mu_n\}_{n\in\mathbb{N}} \subset \wp(\ALP A\times \ALP B)$ and $\{\nu_n\}_{n\in\mathbb{N}}\subset \wp(\ALP A\times \ALP C)$ given by
\beqq{\mu_n:=\frac{1}{2}\ind{(\frac{1}{n},\underline{b})}+\frac{1}{2}\ind{(0,\overline{b})}\qquad \nu_n = \frac{1}{2}\ind{(\frac{1}{n},\underline{c})}+\frac{1}{2}\ind{(0,\overline{c})}\qquad n\in\mathbb{N}.}
Clearly, $\nu_n\overset{w^*}{\rightharpoonup}\nu$ and $\mu_n\overset{w^*}{\rightharpoonup}\mu$ in the weak-* topology, where $\mu:=\frac{1}{2}\ind{(0,\underline{b})}+\frac{1}{2}\ind{(0,\overline{b})}$ and $\nu := \frac{1}{2}\ind{(0,\underline{c})}+\frac{1}{2}\ind{(0,\overline{c})}$. Also, note that $\mu_n$ and $\nu_n$ have identical marginal distributions on $\ALP A$ for all $n\in\mathbb{N}$. Let $f\in C_b(\ALP A\times \ALP B\times \ALP C)$ with $f(0,\overline{b},\underline{c})<f(0,\overline{b},\overline{c})$ and $f(0,\underline{b},\overline{c})
<f(0,\underline{b},\underline{c})$. Then we have
\beqq{\int f(a,b,c)d\varphi(\mu_n,\nu_n)=\int_{\ALP A\times \ALP C} \int_\ALP B f(a,b,c)\mu_n\left(db\vert a\right)\nu_n(da,dc) =\frac{1}{2} f\left(\frac{1}{n},\underline{b},\underline{c}\right)+\frac{1}{2} f(0,\overline{b},\overline{c})}
so that this converges to $\frac{1}{2} f(0,\underline{b},\underline{c})+\frac{1}{2} f(0,\overline{b},\overline{c})$ as $n\rightarrow\infty$. On the other hand, we have
\beqq{ \int f(a,b,c)d\varphi(\mu,\nu) &=&\int_{\ALP A\times \ALP C} \int_\ALP B f(a,b,c)\mu\left(db\vert a\right) \nu(da,dc) \\
&=&\frac{1}{2} \left( \frac{1}{2}f(0,\underline{b},\underline{c})+\frac{1}{2}f(0,\overline{b},\underline{c})\right) +\frac{1}{2}\left( \frac{1}{2}f(0,\overline{b},\overline{c})+\frac{1}{2}f(0,\underline{b},\overline{c})\right).}
Given the assumptions on $f$, we have no equality. So, $\varphi(\nu_n,\mu_n)$ does not converge to $\varphi(\nu,\mu)$ as $n\rightarrow\infty$ in the weak-* topology.{\hfill$\Box$}
\end{example}
\subsection{Revisiting Example \ref{exm:ex2}}
Recall Example \ref{exm:ex2}. We show that the sequence of measures $\{\mu_n^{\ALP A\times\ALP B}\}_{n\in\mathbb{N}}$ does not converge in the topology of information. We use here the same notation as in the example. First, note that
\beqq{\mu_n(db|a) = \ind{\text{sgn}(a)}(db),\quad
\mu_n^{\ALP A} = \sum_{i\in\{-1,1\}} \frac{1}{4} \Big(\ind{h_n(i)+1}+\ind{h_n(i)-1}\Big).}
We now show that the sequence $\{\mu_n^{\ALP A\times\ALP B}\}_{n\in\mathbb{N}}$ does not converge to $\mu_0^{\ALP A\times\ALP B}$ in the topology of information. For any continuous function $f\in C_b(\ALP A\times\wp_w(\ALP B))$ such that $f(a,\ind{1})\neq f(a,\ind{-1})$, we get
\beqq{\int f(a,\zeta) \ind{\ind{\text{sgn}(a)}}\mu_n^{\ALP A}(da)
= \sum_{i\in\{-1,1\}} \frac{1}{4} \Big(f(h_n(i)+1,\ind{h_n(i)+1}) +f(h_n(i)-1,\ind{h_n(i)-1})\Big).}
The right side of the equation above does not converge as $n\rightarrow\infty$, because $\text{sgn}(h_n(1)-1) = \text{sgn}(\frac{1}{n})$ and $\text{sgn}(h_n(-1)+1) = -\text{sgn}(\frac{1}{n})$ do not converge as $n\rightarrow\infty$. Furthermore, since $\ALP B = \{-1,1\}$, $f(a,\ind{0})$ is not well-defined. Using a similar approach as above, one can show that the sequence $\{\mu_n^{\ALP A\times\ALP C}\}_{n\in\mathbb{N}}$ does not converge in the topology of information to $\mu_0^{\ALP A\times\ALP C}$.
\begin{remark}
In the above setting, if either sequence $\{\mu_n^{\ALP A\times\ALP B}\}_{n\in\mathbb{N}}$ or sequence $\{\mu_n^{\ALP A\times\ALP C}\}_{n\in\mathbb{N}}$ converges in the topology of information, then the conditional independence property ($b$ and $c$ are conditionally independent given $a$) would hold. However, for the example we constructed, both sequences fail to converge in the topology of information, which implied that the conditional independence property failed to hold for the limit, $\mu_0$.{\hfill $\Box$}
\end{remark}
\subsection{Relation to \texorpdfstring{\citet{hellwig1996}}{TEXT}}
Our main result Theorem \ref{thm:main} can be used to give a correct proof of Lemma 4 in \citet{hellwig1996}. Hellwig considers a infinite-horizon sequential optimization problem. We present here a simpler version of the problem of Hellwig, and we refer the reader to \citet{hellwig1996} for a detailed description of the original model.
The basic structure of Hellwig's problem can be set up as follows: Let $\ALP A$ denote the set of exogenous states today, $\ALP B$ is the set of states tomorrow and $\ALP C$ is the set of actions chosen today. There is an exogenous probability distribution on the states of the world today and tomorrow given by $\nu\in \wp(\ALP A\times \ALP B)$. Any choice of the decision maker is represented by a probability measure on action and both the states, $\mu\in \wp(\ALP A\times \ALP B \times \ALP C)$. Note that $\mu$ is consistent with $\nu$ in the sense that $\mu^{\ALP A\times \ALP B}=\nu$. The choice of actions that the decision maker can take is restricted by a correspondence $\beta(\nu)$, which assigns the possible choices of joint distributions of states and action for each exogenous measure $\nu$. Besides the technological constraints in $\beta$, it captures an informational constraint that the action chosen today cannot depend on the information (the state of the world) revealed tomorrow. Formally, this
requires that conditioned on the state of the world today $a$, the action today $c$ and the state of the world
tomorrow $b$ have to be conditionally independent. This means that for each $\mu\in \beta(\nu)$, we must have $\varphi(\nu,\mu^{\ALP A\times \ALP C})=\mu$, where $\varphi$ is defined in Theorem \ref{thm:main}. Lemma 4 in \citet{hellwig1996} aims to show that if the set of exogenous states is endowed with the topology of information, then the correspondence $\beta$ is upper-hemicontinuous. Since this topology is metrizable (see Lemma \ref{lem:met} below), the proof relies on the sequential characterization of upper-hemicontinuous correspondences.
If $\nu_n$ converges to $\nu$ in $\wp_I(\ALP A\times \ALP B)$, $\mu_n\in \beta(\nu_n)$, and $\mu_n\overset{w^*}{\rightharpoonup} \mu$ to some $\mu\in\wp_w(\ALP A\times \ALP B\times \ALP C)$, it must be true that $\mu\in\beta(\nu)$. It is easy to verify the technological constraints imposed by $\beta$. However, in order to belong to $\beta(\nu)$,$\mu$ has to satisfy the conditional independence property. This follows now immediately from Theorem \ref{thm:main}, since $\mu_n=\varphi(\nu_n,\mu_n^{\ALP A\times \ALP C})\overset{w^*}{\rightharpoonup}\varphi(\nu,\mu^{\ALP A\times \ALP C})$, so by the uniqueness of limits in the weak-* topology, $\mu=\varphi(\nu,\mu^{\ALP A\times \ALP C})$.
\citet{hellwig1996} instead tries to prove the preservation of conditional independence in the limit as follows: He considers $\psi(\nu)\in \ALP N\subseteq \wp_w(\ALP A\times \wp_w(\ALP B))$ and $\mu^{\ALP A\times \ALP C}$ on page 452 of his paper, applies the mapping $\varphi_1$ to these measures. He claims that (on p. 452) that
\beqq{\varphi_1(\psi(\nu),\mu^{\ALP A\times \ALP C})(A\times B\times C) = \frac{\psi(\nu)(A\times B)\mu^{\ALP A\times \ALP C}(A\times C)}{\mu^{\ALP A}(A)}.}
for any $A\in \FLD B(\ALP A)$ with $\mu^{\ALP A}(A)>0$ and $B\in \FLD B(\wp_w(\ALP B))$ and $C\in \FLD B(\ALP C)$, and applies this equality to prove the conditional independence. However, the equality in the equation above is not true in general. We provide a counterexample to this claim now.
Consider $\ALP A = \{a_1,a_2\}, \:\ALP B =\{b_1,b_2\}$ and $\ALP C = \{c_1,c_2\}$. Now, define $h_1:\ALP A\rightarrow\ALP B$ and $h_2:\ALP A\rightarrow\ALP C$ as
\beqq{h_1(a_i) = b_i,\qquad h_2(a_i) = c_i,\quad i=1,2.}
Let $\nu\in\wp(\ALP A\times\ALP B)$ and $\mu\in\wp(\ALP A\times\ALP C)$ be probability measures, respectively, induced by functions $h_1$ and $h_2$, with the marginal on $\ALP A$ as the uniform distribution:
\beqq{\nu^{\ALP A}(da) = \mu^{\ALP A}(da) = \frac{1}{2}\ind{a_1}(da)+\frac{1}{2}\ind{a_2}(da).}
Consider $\psi(\nu)$, which assigns probabilities $\frac{1}{2}$ to $(a_1,\ind{b_1})$ and $(a_2,\ind{b_2})$. Now $\varphi_1(\psi(\nu),\mu)\in\wp_w(\ALP A\times\wp_w(\ALP B)\times\ALP C)$ is given by
\beqq{ \varphi_1(\psi(\nu),\mu)= \frac{1}{2} \ind{(a_1,\ind{b_1},c_1)}+\frac{1}{2} \ind{(a_2,\ind{b_2},c_2)}.}
Now, let $A = \ALP A,\: B = \lbrace\ind{b_1}\rbrace$ and $C = \{c_2\}$, we get
\beqq{\frac{\psi(\nu)(A\times B)\mu(A\times C)}{\nu^{\ALP A}(A)}=\frac{1}{2}\times\frac{1}{2} = \frac{1}{4},\\
\text{but}\qquad\varphi_1(\psi(\nu),\mu)(A\times B\times C) &=& 0,}
which is what we wanted to show. This completes the counterexample.
\section{Topological Properties of \texorpdfstring{$\wp_I(\ALP A\times\ALP B)$}{TEXT}}\label{sec:top}
In this section, we study a few topological properties of the space of measures endowed with topology of information. Since $\wp_I(\ALP A\times\ALP B)$ and $\ALP N$ are homeomorphic, we can show that $\wp_I(\ALP A\times\ALP B)$ is a metrizable separable space, which is also proved in \citet[Lemma 2, p. 449]{hellwig1996}.
\begin{lemma}\label{lem:met}
$\wp_I(\ALP A\times\ALP B)$ is a metrizable and separable space.
\end{lemma}
\begin{proof}
Let $\rho_{(\cdot)}$ denote the metric on a metric space $(\cdot)$. For any $\mu_1,\mu_2\in \wp_I(\ALP A\times\ALP B)$, one can just the take metric
\beqq{\rho_{\wp_I(\ALP A\times\ALP B)}(\mu_1,\mu_2) = \rho_{\ALP N}(\psi(\mu_1),\psi(\mu_2)).}
It is easy to verify that the above definition is indeed a metric on space $\wp_I(\ALP A\times\ALP B)$. Since $\ALP N$ is a subset of a separable space, we conclude that $\wp_I(\ALP A\times\ALP B)$ is also separable under this metric. This completes the proof of this lemma.
\end{proof}
However, it is easy to show that $\ALP N$ is not a closed subset of $\wp_w(\ALP A\times\wp_w(\ALP B))$\footnote{One can use a variation of Example \ref{exm:set} to show this fact.}. Thus, we cannot conclude that $\wp_I(\ALP A\times\ALP B)$ is complete under the metric defined in the proof above.
Corollary \ref{cor:toiweak} allows us to state the following result.
\begin{lemma}\label{lem:funcweak}
Let $\ALP T$ be a topological space. We use the same notation as in Theorem \ref{thm:main}.
\begin{enumerate}
\item Let $f:\wp_w(\ALP A\times\ALP B\times\ALP C)\rightarrow \ALP T$ be any function, and define $\bar{f}:\wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)\rightarrow \ALP T$ as $\bar{f}(\mu,\nu) = f(\varphi(\mu,\nu))$ for any $\mu\in\wp(\ALP A\times\ALP B)$ and $\nu\in\wp(\ALP A\times\ALP C)$. If $f$ is lower (resp. upper) semi-continuous, then $\bar{f}$ is lower (resp. upper) semi-continuous. Thus, if $f$ is continuous, then so is $\bar{f}$.
\item Let $c:\ALP A\times\ALP B\times\ALP C\rightarrow\Re\cup\{-\infty,\infty\}$ be a measurable function. Define $\bar{f}_c:\wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)\rightarrow\Re$ as $\bar{f}_c(\mu,\nu) = \int_{\ALP A\times\ALP B\times\ALP C} c\:d\varphi(\mu,\nu)$ for $\mu\in\wp(\ALP A\times\ALP B)$ and $\nu\in\wp(\ALP A\times\ALP C)$. If $c$ is lower semi-continuous and bounded from below, then $\bar{f}_c$ is a lower semi-continuous function. If $c$ is upper semi-continuous and bounded from above, then $\bar{f}_c$ is an upper semi-continuous function. Thus, if $c$ is continuous and bounded, then so is $\bar{f}_c$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof of Part 1 follows from Theorem \ref{thm:main} and Corollary \ref{cor:toiweak}. Part 2 follows immediately from Part 1 of the result along with Lemma 4.3 of \citet[p. 43]{villani2009}.
\end{proof}
We now present an example below that applies the result of Lemma \ref{lem:funcweak} to prove the existence of an optimal solution to an optimization problem. The proof technique adopted in the following example can be extended to a game of incomplete information or any sequential optimization with perfect recall.
\begin{example}\label{exm:opt}
Let $\ALP B$ be the state space of the world, $\ALP A$ be the observation space of a decision maker and $\ALP C$ be the decision space of the decision maker. Assume that $\ALP C$ is a compact space and $c:\ALP A\times\ALP B\times\ALP C\rightarrow[0,\infty)$ is a continuous function. Further, assume that observation $a$ and state $b$ are correlated with each other and let $\mu\in\wp(\ALP A\times\ALP B)$ denote the joint probability measure over the observation space and the state space. Let $\Gamma$ denote the space of all measurable functions $\gamma:\ALP A\rightarrow\ALP C$. The question now is that under what conditions, there exists a measurable function $\gamma^\star\in\Gamma$ such that the following holds
\beq{\label{eqn:cabc} \ex{c(a,b,\gamma^\star(a))} = \inf_{\gamma\in\Gamma} \ex{c(a,b,\gamma(a))} := \inf_{\gamma\in\Gamma}\int_{\ALP A\times\ALP B} c(a,b,\gamma(a))\mu(da,db).}
We now show that the aforementioned optimization problem admits an optimal solution, thereby showing the existence of an optimal $\gamma^\star$. We show the existence result in four steps.
{\it Step 1:} Let us first expand the strategy space of the decision maker to include all randomized strategies as well. Thus, each decision maker decides on a conditional measure $\nu(dc|a)$ such that the measure on $\ALP A\times\ALP C$ is given by $\nu(da,dc):=\nu(dc|a)\mu^{\ALP A}(da)$, and let $\ALP P\subset\wp_w(\ALP A\times\ALP C)$ denote the set of all such $\nu$. It is immediate that $\ALP P$ is a tight and weak-* closed set of measures, thus weak-* compact. Since the space $\ALP P$ subsumes the measures induced by strategies in $\Gamma$, we have
\beq{\label{eqn:infNG}\inf_{\tilde \nu\in\ALP P} \int c(a,b,c)\tilde \nu(dc|a)\mu(da,db)\leq\inf_{\gamma\in\Gamma}\int_{\ALP A\times\ALP B} c(a,b,\gamma(a))\mu(da,db).}
{\it Step 2:} Let us endow the space of measures over $\ALP A\times\ALP B$ with the topology of information. Consider the sequence $\{\mu_n\}_{n\in\mathbb{N}}$, defined by $\mu_n=\mu$ for all $n\in\mathbb{N}$. Since $\wp_I(\ALP A\times\ALP B)$ is a metric space, the sequence $\{\mu_n\}_{n\in\mathbb{N}}$ converges to $\mu$ in the topology of information.
{\it Step 3:} Now, consider a sequence $\{\nu_n\}_{n\in\mathbb{N}}\subset\ALP P$ satisfying
\beqq{\int c(a,b,c)\nu_n(dc|a)\mu(da,db)< \inf_{\tilde \nu\in\ALP P} \int c(a,b,c)\tilde \nu(dc|a)\mu(da,db)+\frac{1}{n}.}
Since $\ALP P$ is weak-* compact, there exists a convergent subsequence, say $\{\nu_{n_k}\}_{k\in\mathbb{N}}\subset \{\nu_n\}_{n\in\mathbb{N}}$, such that it converges to some $\nu^\star\in\ALP P$. A consequence of this result is that $\varphi(\mu_{n_k},\nu_{n_k})\overset{w^*}{\rightharpoonup} \varphi(\mu,\nu^\star)$ as $k\rightarrow\infty$. Since $c$ is continuous and bounded from below, applying the result of Lemma \ref{lem:funcweak} Part 2, we conclude that
\beqq{\int c(a,b,c)\nu^\star(dc|a)\mu(da,db)\leq \lif{n}\int c(a,b,c) \nu_n(dc|a)\mu(da,db),}
which further implies
\beqq{\int c(a,b,c)\nu^\star(dc|a)\mu(da,db)= \inf_{\tilde \nu\in\ALP P} \int c(a,b,c)\tilde \nu(dc|a)\mu(da,db).}
Hence, we know that there exists an optimal randomized strategy of the decision maker.
{\it Step 4:} Now, we can apply {\it Blackwell's principle of irrelevant information} (see \citet{blackwell1964}, \citet[p. 457]{yukselbook} for details) to conclude that there exists a measurable function, say $\gamma^\star:\ALP A\rightarrow\ALP C$, such that
\beqq{\int c(a,b,\gamma^\star(a))\mu(da,db)=\int c(a,b,c)\nu^\star(dc|a)\mu(da,db),}
which, together with \eqref{eqn:infNG}, completes the proof of existence of an optimal solution to the optimization problem posed in \eqref{eqn:cabc}.
{\hfill$\Box$}
\end{example}
\begin{remark}
Instead of formulating the optimization problem over the space $\wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)$ in the example above, if we had formulated it over the space $\wp_w(\ALP A\times\ALP B\times\ALP C)$, then we could show that there exists a $\lambda^\star\in \wp_w(\ALP A\times\ALP B\times\ALP C)$ such that
\beqq{\int c\: d\lambda^\star= \inf_{\tilde \lambda\in\wp_w(\ALP A\times\ALP B\times\ALP C)} \int c\:d\tilde \lambda}
using similar arguments as above, but to show the conditional independence property (state and action are independent given the observation) of the limiting measure, we will have to use a similar approach as used in Lemmas \ref{lem:aux1} and \ref{lem:aux2}. Thus, formulating the optimization problem over a product space $\wp_I(\ALP A\times\ALP B)\times\wp_w(\ALP A\times\ALP C)$ that uses topology of information makes it easier to show the conditional independence property of the limiting decision strategy.{\hfill$\Box$}
\end{remark}
The example stated above also shows how to apply topology of information to solve optimization problems. A similar approach, with certain modifications, can be used to analyze game problems. In the next section, we study the relation between other well-known topologies over measure spaces and the topology of information.
\section{Relation to Other Topologies on Measure Spaces}\label{sec:relation}
In this section, we show that under some conditions, convergence of a sequence of measures in some well-known topologies on the space of measures - weak-* topology, topology of setwise convergence, and the norm topology (the topology induced by total variation norm), implies convergence of that sequence in the topology of information. First, we state definitions of setwise convergence and convergence in total variation of a sequence of measures for a Borel space $(\ALP X,\FLD B(\ALP X))$.
\begin{definition}[Setwise Convergence of measures]
A sequence of measures $\{\nu_n\}_{n\in\mathbb{N}}$ over the space $\ALP X$ is said to converge setwise to a measure $\nu_0$ if
\beqq{\lf{n}\nu_n(X) = \nu_0(X)}
for every measurable set $X\subset\ALP X$.{\hfill$\Box$}
\end{definition}
\begin{definition}[Convergence of measures in total variation norm]
A sequence of measures $\{\nu_n\}_{n\in\mathbb{N}}$ over the space $\ALP X$ is said to converge in total variation to a measure $\nu_0$ if
\beqq{\lf{n}\|\nu_n-\nu_0\|_{TV} = 0,}
where the total variation norm of any countably additive signed measure $\nu$ is defined to be
\beqq{\|\nu\|_{TV} := \sup_{f:\ALP X\rightarrow[-1,1]} \int_{\ALP X} f\:d\nu,}
where the supremum is taken over all functions $f$ that are Borel measurable.{\hfill$\Box$}
\end{definition}
In the next few subsections, we identify certain sufficient conditions on the sequences of measures, such that if the sequence converges under some topology, then it implies that the sequence also converges in the topology of information.
\subsection{Relation to the Weak-* Topology}
Example \ref{exm:ex2} is an example of the case where weak-* convergence of a sequence of measures does not imply convergence of that sequence in the topology of information. Thus, it is clear that the notion of weak-* convergence is not sufficient to guarantee conditional independence property of limiting measures. However, under some restrictive assumptions, weak-* convergence implies convergence in the topology of information. Our next two results identify two such sets of conditions.
\begin{theorem}\label{thm:wstoi}
Let $\ALP X$ be a Polish space and $\ALP Y$ be a locally compact Polish space. Consider a sequence $\lbrace \mu_n\rbrace_{n\in\mathbb{N}}$ of probability measures on $\ALP X\times \ALP Y$ such that each $\mu_n$ has a (measurable) density $f_n$ with respect to $\mu^{\ALP X}_n\otimes\mu^{\ALP Y}_n$. Further, assume that (i) $\{\mu_n\}_{n\in\mathbb{N}}$ converges to $\mu$ in the weak-* topology, (ii) $\mu$ has a continuous density $f$ with respect to $\mu^{\ALP X}\otimes\mu^{\ALP Y}$, and (iii) $f_n$ converges uniformly to $f$ on each compact subset of $\ALP X\times \ALP Y$ as $n\rightarrow\infty$. Then, $\lbrace \mu_n\rbrace_{n\in\mathbb{N}}$ converges to $\mu$ as $n\rightarrow\infty$ in $\wp_I(\ALP X\times\ALP Y)$.
\end{theorem}
\begin{proof}
See \ref{app:wstoi}.
\end{proof}
The conditions above is motivated from the convergence condition in \citet{milgrom1985}, which considers convergence results for Bayesian games\footnote{In fact, they require that the limit density be a.s.-continuous with respect to the product measure. They also do not require the spaces to be locally compact. We impose a slightly stronger assumption in order to obtain a general result.}. We have the following corollary of the theorem above.
\begin{corollary}\label{cor:wstoi2}
Let $\ALP X,\ALP Y_1, \ALP Y_2$ and $\ALP Z$ be locally compact Polish spaces. Consider weak-* sequences of measures $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp_w(\ALP X\times\ALP Y_1\times\ALP Y_2)$ and $\{\nu_n\}_{n\in\mathbb{N}}\subset\wp_w(\ALP Y_1\times\ALP Z)$ such that $\mu_n^{\ALP Y_1} = \nu_n^{\ALP Y_1}$, $\mu_n\overset{w^*}{\rightharpoonup} \mu$ and $\nu_n\overset{w^*}{\rightharpoonup}\nu$ for some $\mu\in\wp_w(\ALP X\times\ALP Y_1\times\ALP Y_2)$ and $\nu\in\wp_w(\ALP Y_1\times\ALP Z)$. Assume that (i) $\mu_n$ has a (measurable) density $f_n$ with respect to $\mu^{\ALP X}_n\otimes\mu^{\ALP Y_1}_n\otimes\mu^{\ALP Y_2}_n$ for every $n\in\mathbb{N}$, (ii) $\mu$ has a continuous density $f$ with respect to $\mu^{\ALP X}\otimes\mu^{\ALP Y_1}\otimes\mu^{\ALP Y_2}$, and (iii) $f_n$ converges uniformly to $f$ on each compact subset of $\ALP X\times \ALP Y_1\times\ALP Y_2$ as $n\rightarrow\infty$. Define a sequence of measures $\{\lambda_n\}_{n\in\mathbb{N}}\subset\wp(\ALP X\times\ALP Y_1\times\ALP Y_2\times\ALP Z)$ and $\lambda\in\wp(\ALP X\times\ALP Y_1\times\ALP Y_2\times\ALP Z)$ as
\beq{\label{eqn:lambdan}\lambda_n(dx,dy_1,dy_2,dz) = \nu_n(dz|y_1)\mu_n(dx,dy_1,dy_2)\;\; n\in\mathbb{N},\quad \lambda(dx,dy_1,dy_2,dz) = \nu(dz|y_1)\mu(dx,dy_1,dy_2).}
Then, the following holds:
\begin{enumerate}
\item The sequence $\lbrace \mu_n\rbrace_{n\in\mathbb{N}}$ converges to $\mu$ as $n\rightarrow\infty$ in three topological spaces: $\wp_I(\ALP X\times(\ALP Y_1\times\ALP Y_2))$ and $\wp_I(\ALP Y_i\times(\ALP X\times\ALP Y_j))$, where $i,j\in\{1,2\}, i\neq j$.
\item The sequence $\{\lambda_n\}_{n\in\mathbb{N}}$ converges to $\lambda$ as $n\rightarrow\infty$ in the weak-* topology.
\item The sequence $\{\lambda_n\}_{n\in\mathbb{N}}$ converges to $\lambda$ as $n\rightarrow\infty$ in the space $\wp_I(\ALP Y_2\times(\ALP X\times\ALP Y_1\times\ALP Z))$.
\end{enumerate}
\end{corollary}
\begin{proof}
Theorem \ref{thm:wstoi} implies the first part of the corollary. The second part then follows from Theorem \ref{thm:main}. We now prove Part 3 of the corollary.
Let $\tilde{\ALP Y}_1 = \ALP Y_1\times\ALP Z$. Also note that, by the definition of $\lambda_n$ and $\lambda$ in \eqref{eqn:lambdan}, $\lambda^{\tilde{\ALP Y}_1}_n = \nu_n$ and $\lambda^{\tilde{\ALP Y}_1} = \nu$. Define $\tilde f_n,\tilde f:\ALP X\times\ALP Y_2\times\tilde{\ALP Y}_1\rightarrow\Re$ as $\tilde f_n(x,y_2,(y_1,z)) = f_n(x,y_1,y_2)$ for $n\in\mathbb{N}$ and $\tilde f(x,y_2,(y_1,z)) = f(x,y_1,y_2)$. Then, for any $n\in\mathbb{N}$, we have $\mu_n(dx,dy_1,dy_2) = f(x,y_1,y_2) \lambda^{\ALP X}_n(dx)\lambda^{\ALP Y_1}_n(dy_1)\lambda^{\ALP Y_2}_n(dy_2) $, which further implies
\beqq{\lambda_n(dx,dy_1,d_2,dz) &=& \nu_n(dz|y_1)\mu_n(dx,dy_1,dy_2) = f_n(x,y_1,y_2) \lambda^{\ALP X}_n(dx)\lambda^{\ALP Y_2}_n(dy_2) \Big(\nu_n(dz|y_1)\lambda^{\ALP Y_1}_n(dy_1)\Big),\\
& =& \tilde f_n(x,y_2,(y_1,z)) \lambda^{\ALP X}_n(dx)\lambda^{\ALP Y_2}_n(dy_2)\lambda^{\tilde{\ALP Y}_1}_n(d y_1,dz).}
A similar result holds for $\lambda$. Thus, the following statements follow immediately from the definitions above, hypotheses of the corollary, and the above equations:
\begin{enumerate}
\item[(i)] $\lambda_n$ is absolutely continuous with respect to the measure $\lambda^{\ALP X}_n\otimes\lambda^{\ALP Y_2}_n\otimes\lambda^{\tilde{\ALP Y}_1}_n$ for every $n\in\mathbb{N}$ with the Radon-Nikodym derivative as $\tilde f_n$.
\item[(ii)] $\lambda$ has a continuous density $\tilde f$ with respect to $\lambda^{\ALP X}\otimes\lambda^{\ALP Y_2}\otimes\lambda^{\tilde{\ALP Y}_1}$. \item[(iii)] $\tilde f_n$ converges uniformly to $\tilde f$ on each compact subset of $\ALP X\times \ALP Y_2\times\tilde{\ALP Y}_1$ as $n\rightarrow\infty$.
\end{enumerate}
The above statements, Part 2 of the corollary, together with the result of Theorem \ref{thm:wstoi}, imply that the sequence $\{\lambda_n\}_{n\in\mathbb{N}}$ converges to $\lambda$ as $n\rightarrow\infty$ in the space $\wp_I(\ALP Y_2\times(\ALP X\times\tilde{\ALP Y}_1))=\wp_I(\ALP Y_2\times(\ALP X\times\ALP Y_1\times\ALP Z))$. This completes the proof of the corollary.
\end{proof}
The above corollary is useful in optimization or game problems in which multiple decision makers act simultaneously based on their observations. To see this, consider a game or an optimization problem with $N\in\mathbb{N}$ decision makers. Let $\ALP X$, $\ALP Y_i$ and $\ALP Z_i$ denote the state space of the nature, observation space of decision maker $i$ and the decision space of decision maker $i$, respectively, for $i\in\{1,\ldots,N\}$. Assume that all the spaces are locally compact Polish spaces, and the joint distribution of the state and the observations of the decision makers admits a continuous density function with respect to the product measure of their marginals. Then, the result of Corollary \ref{cor:wstoi2} can be used iteratively to conclude that a weak-* convergent sequence of joint measures over state, observation and action spaces of the decision makers, induced by appropriate strategies of the decision makers, maintains conditional independence properties\footnote{In this setup, the number of
conditional independence properties to check are the same as the number of decision makers.} in the limit.
Our proof of Theorem \ref{thm:wstoi} relies on a property of measure spaces over a locally compact Polish space with weak-* topology. Therefore, it is not clear as of now if the restriction of locally compact Polish spaces can be weakened in the hypotheses of Theorem \ref{thm:wstoi} and its corollary above.
A somewhat different condition was considered in \cite{jordan1977}, in which the conditional measure on $\ALP Y$ given $x$ is assumed to be continuous in $x$. In our next theorem, we show that under such an assumption with another condition, weak-* convergence of a sequence of measures imply convergence in the topology of information.
\begin{theorem}\label{thm:wstoi3}
Let $\ALP X$ and $\ALP Y$ be Polish spaces. Let $\{\mu_n\}_{n\in\mathbb{N}} \subset \wp_w(\ALP X\times\ALP Y)$ be a weak-* convergent sequence of measures, converging to $\mu$ as $n\rightarrow\infty$. For each $n\in\mathbb{N}$, let $f_n:\ALP X\rightarrow\wp_w(\ALP Y)$ be the measurable function defined as $f_n(x)(\cdot) = \mu_n(\cdot|x)$. Similarly, define measurable function $f:\ALP X\rightarrow\wp_w(\ALP Y)$ as $f(x)(\cdot) = \mu(\cdot|x)$. Let $E\subset\ALP X$ be the set of all $x\in\ALP X$ such that there exists a sequence $\{x_n\}_{n\in\mathbb{N}}\subset\ALP X$ such that $x_n\rightarrow x$, but the sequence $\{f_n(x_n)\}_{n\in\mathbb{N}}$ does not converge to $f(x)$. If $\mu^{\ALP X}(E) = 0$, then $\lbrace \mu_n\rbrace_{n\in\mathbb{N}}$ converges to $\mu$ as $n\rightarrow\infty$ in $\wp_I(\ALP X\times\ALP Y)$.
\end{theorem}
\begin{proof}
Define $h_n:\ALP X\rightarrow\ALP X\times\wp_w(\ALP Y)$ to be the function as $h_n(x) = (x,f_n(x))$, and similarly define $h(x) = (x,f(x))$. Then, for any $x\in E^\complement$, if $\{x_n\}_{n\in\mathbb{N}}\subset\ALP X$ is a sequence converging to $x$, then the sequence $\{h_n(x_n)\}_{n\in\mathbb{N}}$ converges to $h(x)$. Since $\mu^{\ALP X}_n\overset{w^*}{\rightharpoonup}\mu^{\ALP X}$, we know from \citet[Theorem 5.5, p. 34]{billing1968} or \citet[Theorem 8.4.1 (iii), p. 195]{bogachev2006b} that for any continuous function $g\in C_b(\ALP X\times\wp_w(\ALP Y))$, we have
\beqq{\lf{n}\int_{\ALP X\times\wp_w(\ALP Y)} g(h_n(x))d\mu^{\ALP X}_n = \int_{\ALP X\times\wp_w(\ALP Y)}g(h(x))d\mu^{\ALP X}.}
This implies that $\mu_n$ converges to $\mu_0$ as $n\rightarrow\infty$ in $\wp_I(\ALP X\times\ALP Y)$, which completes the proof of the theorem.
\end{proof}
\citet{jackson2012} also make similar assumptions as mentioned in Theorems \ref{thm:wstoi} and \ref{thm:wstoi3} above. For one part, their assumption on page 207 of ther paper is similar to the one in \citet{milgrom1985}, which we discuss in Theorem \ref{thm:wstoi}. For other parts, on page 206 of their paper, they assume a sort of uniform continuity version of the original continuity assumption of \citet{jordan1977}. Theorem \ref{thm:wstoi3} can be used to show that both of these assumptions imply convergence in the topology of information. We refer the reader to \ref{app:jordan} to see how the result of Theorem \ref{thm:wstoi3} can be applied to the setting considered in \citet{jordan1977}.
To see how the assumptions of page 206 in \citet{jackson2012} are a special case of Theorem \ref{thm:wstoi3}, recall that their assumption requires that $\mu_n$ converges weakly to $\mu_\infty$. Further, if $d_{\ALP X}$ denotes some fixed metric generating the topology on $\ALP X$, they require that for each $\varepsilon>0$ and each continuous function $f:\ALP X\times \ALP Y\rightarrow \left[ 0,1\right]$, there exists $N\in \mathbb{N}$ and $\delta>0$ such that for all $m,n>N$ (including $n=\infty$) and for all $x,x'\in \ALP X$ with $d_{\ALP X}(x,x')<\delta$
\begin{align}\label{eqn:Jackson}
\left\vert \int_{\ALP Y} f(x,y)d\mu_n(y\vert x)-\int_{\ALP Y} f(x',y)d\mu_m(y\vert x') \right\vert<\varepsilon.
\end{align}
Now, let $g:\ALP Y\rightarrow \left[ 0,1\right]$ be a continuous function. Fix some $x_\infty\in \ALP X$ and consider a sequence $\{x_m\}_{m\in\mathbb{N}}$ such that $x_m\rightarrow x_\infty$ as $m\rightarrow\infty$. In \eqref{eqn:Jackson} above, take $f=g$, $n=\infty$, $x=x_\infty$ and $x'=x_m$. Then, (\ref{eqn:Jackson}) implies that $\mu_m(\cdot\vert x_m)$ converges to $\mu_\infty(\cdot\vert x_\infty)$ as $m\rightarrow\infty$ in weak-* topology\footnote{Weak-* convergence for conditional distributions in Theorem \ref{thm:wstoi3} requires one to consider convergence for all bounded continuous functions, not just those mapping to $\left[ 0,1\right]$. However, by adding a constant and rescaling, it suffices to show convergence for functions mapping to $\left[ 0,1\right]$.}. From Theorem \ref{thm:wstoi3}, it follows that $\mu_n$ converges to $\mu_\infty$ as $n\rightarrow\infty$ in $\wp_I(\ALP X\times\ALP Y)$.
\subsection{Relation to the Topology of Setwise Convergence}
In this subsection, we state sufficient conditions when setwise convergence of measures imply convergence in topology of information. Before we state the conditions, let us first consider an example where setwise convergence of a sequence of measures {\it does not} imply convergence in the topology of information.
\begin{example}\label{exm:set}
This example uses Rademacher functions, which were used in \citet[p. 445]{hellwig1996} to construct an example for discontinuous behavior of conditional distributions under weak convergence. Let $\left[0,1\right[$ with Borel $\sigma$-algebra $\FLD B_{\left[ 0,1\right[}$ be given. Let $\lambda$ denote the Lebesgue measure restricted to $\left[ 0,1 \right[$. Recall that the $n$-th Rademacher function is defined as $F_n(\omega)=\sum_{k=0}^{2^{n-1}-1} \mathbb{I}_{\left[ \frac{2k}{2^n},\frac{2k+1}{2^n}\right[ }(\omega)$ for any $\omega\in \left[ 0,1\right[$. We have the following result.
\begin{lemma}\label{lem:cl1}
Define a sequence of measures $\{\lambda_n\}_{n\in\mathbb{N}}$ such that $\lambda_n(A) := \lambda\left( A\cap F_n^{-1}(1)\right)$ for any $A\in\FLD B_{\left[ 0,1\right[}$ and $n\in\mathbb{N}$. For any $A\in \FLD B_{\left[ 0,1\right[}$,
\begin{align}\label{rademacher}
\lambda_n(A)=\lambda\left( A\cap F_n^{-1}(1)\right) \longrightarrow \frac{1}{2}\lambda(A) \quad\text{ as }\quad n\rightarrow\infty.
\end{align}
In other words, $\lambda_n\rightarrow\frac{1}{2}\lambda$ as $n\rightarrow\infty$ in the setwise topology over the space of measures over $[0,1[$.
\end{lemma}
\begin{proof}
See \ref{app:cl1}.
\end{proof}
Note that then also $\lambda\left( A\cap F_n^{-1}(0)\right) \longrightarrow \frac{1}{2}\lambda(A)$ as $n\rightarrow\infty$. Let $\ALP X := [0,1[$ and $\ALP Y:=\{1,2\}$. We consider now the set $\ALP X\times\ALP Y$ with $\sigma$-algebra $\FLD B(\ALP X)\otimes\mathcal{P}(\lbrace 1,2\rbrace)$. Let $\mu$ be the measure given by
\begin{align*}
\mu(A)=\int_{\left[ 0,1\right[} \mu(A_x\vert x) d\lambda
\end{align*}
where $A_x$ is the $x-$section of the set $A\in \FLD B(\ALP X)\otimes\FLD P(\ALP Y)$ and for each $x\in \left[ 0,1\right[$ we set $\mu(\lbrace 1\rbrace\vert x)=\mu(\lbrace 2\rbrace\vert x)= \frac{1}{2}$. For each $n$, we set $\mu_n(\lbrace 1\rbrace\vert x)=1$ if $F_n(x)=0$ and $\mu_n(\lbrace 1\rbrace\vert x)=0$ if $F_n(x)=1$. The measure $\mu_n\in\wp(\ALP X\times\ALP Y)$ is defined for each $A\in \FLD B(\ALP X)\otimes\FLD P(\ALP Y)$ as
\beqq{\mu_n(A)=\int_{\left[ 0,1\right[} \mu_n(A_x\vert x) d\lambda}
For each $A\in \FLD B(\ALP X)\otimes\FLD P(\ALP Y)$, let
$A_1:=\left\lbrace x\in \ALP X \vert A_x=\lbrace 1\rbrace \right\rbrace$, $A_2:=\left\lbrace x\in \ALP X \vert A_x=\lbrace 2\rbrace \right\rbrace$ and $A_{12}:=\left\lbrace x\in \ALP X \vert A_x=\lbrace 1,2\rbrace \right\rbrace$. Note that $A_1,A_2$ and $A_{12}$ are all measurable sets. We have
\beqq{\mu_n(A)&=&\int_{A_1} (1-F_n) d\lambda+\int_{A_2} F_n d\lambda+\int_{A_{12}} \mu_n(\lbrace 1,2\rbrace\vert x) d\lambda\\
&=& \lambda(A_1\cap F_n^{-1}(0))+\lambda(A_2\cap F_n^{-1}(1))+\lambda(A_{12}),\\
\mu(A)&=&\int_{A_1} \mu(\lbrace 1\rbrace\vert x) d\lambda+\int_{A_2} \mu(\lbrace 2\rbrace\vert x) d\lambda+\lambda(A_{12})=\frac{1}{2}\lambda(A_1)+\frac{1}{2}\lambda(A_2)+\lambda(A_{12}).}
Lemma \ref{lem:cl1} implies that $\mu_n(A)\rightarrow \mu(A)$ for each $A\in \FLD B(\ALP X)\otimes\FLD P(\ALP Y)$, that is, $\mu_n$ converges to $\mu$ setwise. On the other hand, $\mu_n$ does not converge to $\mu$ in the topology of information, as $\mu(\cdot|x)=\frac{1}{2}\ind{1}+\frac{1}{2}\ind{2}$ for all $x\in\ALP X$ and $\mu_n(\cdot|x)$ is either equal to $\ind{1}$ or $\ind{2}$.{\hfill$\Box$}
\end{example}
It turns out that if $\ALP X$ is countable with discrete metric and $\ALP Y$ is a Polish space, then setwise convergence of a sequence of measures over $\ALP X\times\ALP Y$ implies convergence of that sequence of measures in the topology of information. This assumption corresponds to the convergence assumption in \citet{engl1995}.
\begin{theorem}\label{thm:settoi}
Let $\ALP X$ be a countable space with discrete metric and $\ALP Y$ be a Polish space. If a sequence $\{\mu_n\}_{n\in\mathbb{N}}$ of probability measures on $\ALP X\times \ALP Y$ converges setwise to a probability measure $\mu$, then the sequence $\lbrace \mu_n\rbrace_{n\in\mathbb{N}}$ converges to $\mu$ as $n\rightarrow\infty$ in $\wp_I(\ALP X\times\ALP Y)$.
\end{theorem}
\begin{proof}
See \ref{app:settoi}.
\end{proof}
We also have an immediate corollary to the above theorem.
\begin{corollary}\label{cor:settoi}
Under the same assumptions of Theorem \ref{thm:settoi}, the result of Theorem \ref{thm:settoi} holds if $\mu_n\rightarrow\mu$ as $n\rightarrow\infty$ in the metric induced by the total variation norm.
\end{corollary}
\begin{proof}
For a sequence of measures, convergence in total variation norm implies setwise convergence (see the discussion on page 291 of \cite{bogachev2006a}). This fact implies the result of the corollary.
\end{proof}
We can use Theorem \ref{thm:settoi} to conclude the following result for measures over countable discrete spaces.
\begin{theorem}
Let $\ALP X$ and $\ALP Y$ be countable spaces, each of which is endowed with the discrete metric. Let $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp_w(\ALP X\times\ALP Y)$ be a weak-* convergent sequence of measures, converging to $\mu$. Then, the sequence $\{\mu_n\}_{n\in\mathbb{N}}$ converges to $\mu$ in $\wp_I(\ALP X\times\ALP Y)$.
\end{theorem}
\begin{proof}
Note that both $\ALP X$ and $\ALP Y$ are Polish spaces. Furthermore, the space of measurable functions and the space of continuous functions over discrete countable spaces are the same. Since $\mu_n\overset{w^*}{\rightharpoonup}\mu$, we know from that $\mu_n$ converges to $\mu$ in total variation norm, which further implies that $\mu_n$ converges to $\mu$ setwise, as $n\rightarrow\infty$. The result then follows from Theorem \ref{thm:settoi}.
\end{proof}
The assumption of countable spaces in the above theorem is part of the conditions on a convergent sequence of measures considered in \cite{kajii1998}. Besides this assumption, they require additional assumptions motivated by game theoretic considerations, which makes their convergence concept stronger than weak-* convergence (or in this case equivalently setwise or norm convergence) on a discrete countable space. Their assumptions on the convergent sequence of measures imply that the sequence converges in the topology of information.
\subsection{Discussion}
The theorems we proved in this section show that topology of information is weaker that other well-known topologies under certain conditions on the Polish spaces or underlying distributions of random variables. However, for general Polish spaces, it is not clear as of now if convergence of a sequence of measures in total variation norm implies convergence of that sequence in the topology of information or vice-versa. It could be possible that the topology of information is stronger than the topology of convergence in total variation metric under certain conditions, but we have been unable to prove this or construct a counterexample. Thus, we leave the following question as a topic for further research:
{\bf Open Problem 1:} Let $\ALP X$ and $\ALP Y$ be Polish spaces, and let $\wp_{TV}(\ALP X\times\ALP Y)$ denote the space of probability measures over $\ALP X\times\ALP Y$ with the topology induced by total variation norm. What is the relation between the topological spaces $\wp_I(\ALP X\times\ALP Y)$ and $\wp_{TV}(\ALP X\times\ALP Y)$? Does there exists a sequence $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp(\ALP X\times\ALP Y)$ that converges to $\mu$ in the metric induced by the total variation norm, but does not converge to $\mu$ in the topology of information?{\hfill $\Box$}
We showed that a sequence of measures converging in the topology of information preserves conditional independence property in the limit. A natural question to ask would be if this is also necessary, that is, if a sequence of measures converging under some topology preserves conditional independence property in the limit, then does it also converge in the topology of information? In other words, the following problem is also interesting in its own right:
{\bf Open Problem 2:} Let $\ALP X$, $\ALP Y$, $\ALP Z$ be Polish spaces and let $\{\mu_n\}_{n\in\mathbb{N}}\subset\wp(\ALP X\times\ALP Y\times\ALP Z)$ be a sequence of measures. Assume that (i) $\mu_n\rightarrow\mu$ for some measure $\mu$ as $n\rightarrow\infty$ in some topology, (ii) $\mu_n(dy,dz|x) = \mu_n(dy|x)\mu_n(dz|x)$ for $\mu_n$ almost every $x$, and (iii) $\mu(dy,dz|x) = \mu(dy|x)\mu(dz|x)$ for $\mu$ almost every $x$. Does either of the following holds: (i) $\{\mu_n^{\ALP X\times\ALP Y}\}_{n\in\mathbb{N}}$ converges to $\mu^{\ALP X\times\ALP Y}$ in the space $\wp_I(\ALP X\times\ALP Y)$, or (ii) $\{\mu_n^{\ALP X\times\ALP Z}\}_{n\in\mathbb{N}}$ converges to $\mu^{\ALP X\times\ALP Z}$ in the space $\wp_I(\ALP X\times\ALP Z)$?{\hfill$\Box$}
\section{Conclusion}\label{sec:conclusion}
In this paper, we studied the topology of information on the space of measures over Polish spaces. We showed, through examples, that the weak-* topology and the topology of setwise convergence are weaker notions of topology than the topology of information for probability measures over general Polish spaces. We also determined conditions under which weak-* convergence or setwise convergence of a sequence of measures implied convergence in the topology of information.
This topology is useful in game or optimization problems that feature informational constraints among the decision makers or causality constraints in the decision making process. For applications, we refer the reader to the papers listed in the introduction. In particular, \citet{hellwig1996} uses topology of information explicitly in showing the existence of optimal solution in an infinite horizon decision problem, while \cite{milgrom1985, jordan1977, kajii1998} assume specific conditions on the underlying spaces, sequences of measures, and/or the topology over measure spaces, to ensure that the limits satisfy information and causality constraints in the problem.
Our results show that many notions of convergence used in this literature are a special case of convergence in the topology of information and we believe that the results of this paper will be helpful in unifying and extending conditions for existence and continuity of optimal solution or Nash equilibrium strategies in such problems. Recently, \citet{yuksel2012} considered one-person optimization of observation channels in dynamic decision problems. The technical difficulty in \citet{yuksel2012} arose partly due to the fact that the actions of a decision maker affected future states of the world, but the decision maker does not recall the past observations. It will be interesting to consider such dynamic decision making problems (for example, Markov decision problems) with or without memory, and use topology of information to identify conditions that guarantee existence of optimal decision rules.
|
1,314,259,993,036 | arxiv | \section{Introduction}
\subsection{Planet formation: bottom-up or top-down?}
Core Accretion (CA) theory \citep{PollackEtal96,AlibertEtal05} posits that
planet formation starts when some of the dust grains in the protoplanetary
disc grow and sediment to the disc midplane. This step is then followed by a
less well understood one in which much larger solid objects of a few km or
more are made \citep[cf.][]{Youdin02,JohansenEtal07}. These rocky objects,
called ``planetesimals'', then co-coagulate into terrestrial-like planetary
cores \citep{Safronov72}. These cores continue to gain mass by mergers and
accretion of planetesimals. If the cores are still embedded in the parent
gaseous disc, then a gas atmosphere builds up around the cores. When the solid
core mass exceeds a few to a few tens Earth masses, the atmosphere becomes
self-gravitating and collapses hydrodynamically to much higher densities,
forming a proto-giant gas planet \citep{Mizuno80,Stevenson82,Rafikov06}. This
model may be termed a bottom-up scenario for planet formation.
An alternative top-down scenario for the origin of planets is also physically
plausible, although it is currently much less developed
\citep{BoleyEtal10,Nayakshin10b}. The very first step in this ``Tidal
Downsizing'' (TD) scheme is reminiscent of formation of stars, that is the
Jeans self-gravitation instability of a gas cloud/clump \citep[for example,
see][]{Larson69}. The cloud is however much less massive, e.g., its mass is
around the opacity fragmentation limit of $\sim 10 M_J$
\citep{Rees76,Nayakshin10a,ForganRice11}, and is born inside a gas disc
orbiting the parent star \citep[e.g.,][]{Boss97}. The second step in TD
scenario is similar to the first one in CA model, e.g., grains grow and
sediment. However, this process occurs inside the gas clumps rather than the
whole of the disc, where the densities are much lower \citep[cf. simulations
of][]{ChaNayakshin11a}. A massive solid core therefore forms inside the gas
clump \citep{Boss98,HS08,Nayakshin10b}. Rapid inward radial migration of the
gas clump \citep{VB05,VB06,MachidaEtal11,MichaelEtal11,BaruteauEtal11} then
expose the clump to the ever increasing tidal force of the parent star.
Removal of all or a part of the original gas cloud by tidal forces of the star
may hypothetically leave behind both terrestrial-like and gas giant planets
\citep{BoleyEtal10,Nayakshin10c}. Note that while the combination of these
physical steps is a very recent development, the TD hypothesis may be viewed
as a physics-upgraded version of the gravitational disc instability model
\citep[GI; e.g.,][]{Boss97,Rice05,Rafikov05}. The complicated fate of gas
clumps in a self-gravitating disc was also discussed by \cite{MayerEtal04},
who also noted that these clumps may be tidally disrupted if they migrate
inward. Furthermore, except for the crucial step of the radial
migration\footnote{\cite{DW75} showed that the gas clump with properties
envisaged by \cite{McCreaWilliams65}, sitting at the present location of the
Earth, would be tidally disrupted well before grains could have grown and
sedimented to its centre. Making the clump at $\sim 100$ AU, where the clump
may be much cooler, and having grains sediment into a massive solid core
before bringing the clump into the inner Solar System by migration is the
only physically plausible way for the model to work.} of the gas clump, the
TD scheme is similar to ideas of \cite{McCreaWilliams65}.
The current status of TD planet formation hypothesis is best described as
``work in progress''. Progress is being made in terms of building more
self-consistent time-dependent models of the protoplanetary discs with
embedded massive gas clumps \citep[e.g.,][]{BoleyEtal11a,NayakshinLodato11};
addressing the surprising high temperature content of Solar System comets
\citep{Vorobyov11a,NayakshinEtal11a}; chemistry of self-gravitating discs
\citep{IleeEtal11}; the ``hot'' super-Earth planets \citep{Nayakshin11b}; and
rotation of the Solar System terrestrial planets \citep{Nayakshin11a}.
Despite this, no detailed statistical predictions in the spirit of population
synthesis models of \cite{IdaLin08,IdaLin10} have yet been made. The TD
hypothesis thus cannot yet be compared with exoplanetary data in detail.
One promising way to constrain the TD hypothesis observationally is via
observations of the early ``embedded'' phase of star (and potentially planet)
formation \citep{DunhamVorobyov12}, especially with the advent of the {\em
ALMA} telescope when resolving individual massive gas clumps in large $R
\lower.5ex\hbox{\gtsima} 50$ AU discs around young protostars may become possible.
\subsection{Can solid debris constrain planet formation theories?}\label{sec:debris_intro}
Planets are not the only bodies orbiting their host stars: solid bodies from
microscopic dust to sub-terrestrial planet size objects such as Pluto also
form during planet assembly both in the Solar System and around other nearby
stars. In particular, Solar System contains several repositories of $\lower.5ex\hbox{\gtsima} 1$
km sized solid bodies -- the asteroid belt, the Kuiper belt, the scattered
disc and the Oort cloud of long-period comets. Starting from the discovery of
the infrared excess beyond 12$\mu$m around Vega by IRAS \citep{Aumann84},
circum-stellar dust disc are commonly found around pre- and main sequence stars
\citep{Zuckerman01,Wyatt08}. Since microscopic dust in the discs around main
sequence dust should be blown away rapidly due to radiation pressure from the
parent stars, a continuous replenishment source for the dust is required
\citep{WyattEtal07}. The universally accepted picture is that the
circum-stellar dust results from a fragmentation "cascade"
\citep[e.g.,][]{Hellyer70,WyattEtal07b,Wyatt08} of larger solid bodies such as
comets and asteroids. This view is reinforced by observations of dust
resulting from such fragmentation cascades in our own Solar System
\citep{NesvornyEtal03,NesvornyEtal10}, and also by the signatures of
collisional sculpting of the asteroid belt \citep[e.g.,][]{BottkeEtal05}.
It is obvious that the solid {\em debris} around stars has a rather natural
explanation in the context of the CA theory: these bodies are the
planetesimals and ``half-grown'' planetary embryos that did not get consumed
or ejected completely from the system by the growing planets. The question we
pose in this paper is this: Can TD hypothesis produce solid debris too, and if
so, how do the properties of that debris differ from the CA model?
We shall argue below that it is economical and logical to have all smaller
solid objects to be born inside the same ``parent'' gas clumps where the
terrestrial-like solid cores grow. We arrive at these conclusions based on
the following ideas. In the 1D spherically symmetric models of Helled and
co-authors, and in \cite{Nayakshin10b}, only one central core is
formed. However, \citep{Nayakshin11b} has shown that specific angular momentum
of the grains may be too large to allow gravitational collapse into just one
massive body. Gravitational collapse of a rapidly rotating cloud may result in
formation of not only the central body but also a number of smaller objects
possibly orbiting the larger one in a centrifugally supporting disc. The
objects may fragment or grow further due to subsequent collisions. In
addition, the formation of the solid core releases a significant amount of
energy into the surrounding gas \citep{Nayakshin10b}, which stirs up strong
convective motions. This second effect may also promote formation of
additional solid bodies by gravitational collapse of smaller grain-dominated
regions \citep[cf. \S 3.6.2 and 3.6.3 of][]{Nayakshin10a} on chaotically
oriented orbits.
The next logical step in our picture is the disruption of the parent gas
clump, as required by the TD hypothesis. We show below that the solid debris
population within the clump can be divided into two groups: one bound to the
planetary core (be it terrestrial-like or more massive by that point), and the
other to the host gas clump. Upon disruption of the clump the first group
``survives'' and becomes planet's satellites; the other group gets disrupted
and dispersed with the gas. Generically, the second population of debris forms
a ``disruption ring'' centred on the location of the host gas clump
disruption.
Below we study analytically and numerically (for three representative cases)
the properties of the post-disruption orbits of the planetesimal debris. We
start in \S 2 by discussing the likely structure of the gas clump before the
disruption. We show that there are two possible ways of the host clump
disruption -- either tidally or due to an internal energy release by the
growing protoplanetary core itself. We estimate the minimum mass of the core
being able to disrupt the host clump to be $\sim 10$ Earth masses.
In \S \ref{sec:drag} we consider the aerodynamic drag acting on the solid
bodies within the host clump. We find that bodies smaller than $\sim$ 1~cm and
larger than $\sim 1$~km stand a good chance of remaining ``independent'' after
the clump disruption compared with the bodies between these size limits: these
suffer strong drag from the gas and must end up joining the protoplanetary
core in the centre of the gas clump. In \S \ref{sec:theory} we consider
analytically the kind of orbits that the large solid bodies obtain after the
gas clump disruption. \S \ref{sec:numerics} describes the set up of our
numerical experiments. Several following sections present the numerical
results, with \S \ref{sec:gas} showing the gas flow, \S\S \ref{sec:unbound}
and \ref{sec:bound_orbits} focusing on the population of unbound and bound
solid bodies, respectively. \S \ref{sec:discussion} contains a discussion of
the main results of our paper.
\section{Host clump structure and disruption}\label{sec:host}
\subsection{Setup and terminology}\label{sec:defs}
For definitiveness, we consider a host gas clump of $M_{\rm hc} = 5 M_J$ mass
in the numerical part of the paper. We assume that the initial density and
temperature profiles are that of a polytropic sphere with the polytropic index
$n=5/2$ as appropriate for a molecular hydrogen-dominated gas in the
temperature range from a few hundred K to about 1500 K
\citep{BoleyEtal07}. The clump is initially located at $R=40$ AU on a circular
orbit around the star of mass $M_* = 1 M_\odot$.
We shall refer to the host gas clump as such to distinguish it from the solid
planetary core, which we frequently call ``planet''. The planet (of mass
$M_{\rm p} = 10 \mearth$) is treated as a point mass in this paper. We assume
that the density of the planet is much higher than that of the host
clump. Therefore, the internal structure of the planet is not affected by the
host gas clump disruption. We note that this setting does not require the
planet to consist of high-Z material only; volatiles could be present also as
long as their densities are much higher than the tidal density, $M_*/2\pi
R^3$.
\subsection{Clump disruption by tidal forces}\label{sec:tidal}
We study two limiting cases of clump disruption. In the case of disruption due
to tidal forces from the star, the host clump fills its Roche lobe at the
start of the simulation, time $t=0$. The size of the Roche lobe (Hill's
radius) at this location is
\begin{equation}
\label{eq:hillsr}
r_h = R\left(\frac{M_{\rm hc}}{3M_*}\right)^{1/3}\approx 4.7\;\mbox{AU}\;.
\end{equation}
A polytropic cloud with the clump mass $M_{\rm hc} = 5 M_J$ and the clump
radius, $r_{\rm hc}$, satisfying $r_{\rm hc} = r_h$ has central temperature $T
= 195$ K, for reference.
The mass-radius relation of a polytropic cloud is given by
\begin{equation}
r_{\rm hc} \propto M_{\rm hc}^\frac{1-n}{3-n} \propto M_{\rm hc}^{-3} \mbox{
for } n = 5/2\;.
\label{mass_radius}
\end{equation}
This implies that the host clump expands rapidly as mass is lost. We therefore
expect a prompt tidal disruption of such a host clump by Roche lobe overflow:
$r_{\rm hc}$ increases while $r_h$ decreases as $M_{\rm hc}$ drops.
We shall add to this that the stabilising effect of the host clump outward
migration due to the mass exchange between the gas clump and the star,
discovered by \cite{NayakshinLodato11}, does not occur in our simulations
because the clump is destroyed rather rapidly in all of the simulations below.
In addition, we find that disruption proceeds via both L1 and L2 points, in
contrast to what is found for much more gentle ``hot disruptions'' at $R \sim
0.1$ AU separations by \cite{NayakshinLodato11}. For both of these reasons
there is no outward torque on the host clump; the tidal disruption has a
runaway character.
\subsection{Disruption by an internal energy release}\label{sec:explosion}
In the second class of models we consider here, the clump's initial radius is
smaller than the Hill's radius $r_h$. In particular, for the two simulations
presented below, we set the initial central temperature to $T=500$ K, which
corresponds to the clump's initial radius of $r_{\rm hc}=1.84$ AU. We then
assume that a sudden burst of energy release occurs in the centre of the
clump. Physically, we relate the burst to the assembly of the planetary core,
as is found in 1D simulations of \cite{Nayakshin10b}. In detail, the energy
released by the solid core is passed to the surrounding gas by radiative
diffusion. If the radiative diffusion time scale is longer than the planetary
core's assembly time and the injection energy is large enough then even an
isolated gas clump may be disrupted. An example of this is simulation
M0$\alpha$3, Figures 5 to 7 in \cite{Nayakshin10b}.
We estimate the energy
released by the planetary core to be about its binding energy,
\begin{equation}
E_{\rm bind, c} \approx\frac{G M_p^2}{2 r_p} = 5 \times 10^{40} \;
\hbox{erg}\; \left(\frac{M_p}{10 M_\oplus}\right)^{5/3} \rho_p^{1/3} \;,
\label{ebind_p}
\end{equation}
where $G$ is the gravitational constant, and $\rho_p$ is the solid core
density in g cm$^{-3}$.
Let us now compare the core's binding energy with the minimum amount of energy
needed to disrupt the host clump. The total energy of a polytropic gas clump,
$E_{\rm hc}$, is given by
\begin{equation}
\label{eq:tot_poly}
E_{\rm hc} = -\frac{3-n}{5-n}\frac{GM_{\rm hc}^2}{r_{\rm hc}}\;.
\end{equation}
For simplicity we assume that the clump remains polytropic after the energy
injection (which is not obvious at all; we will come back to this later) with
same $n$ but a different adiabatic constant. The energy input sufficient to
disrupt the clump, $\Delta U_0$ is the energy needed to inflate the clump to
the point of its tidal disruption, i.e., when $r_{\rm hc}$ increases and
becomes equal to the Hill's radius, $r_h$. Thus,
\begin{equation}
\label{eq:ene_dis}
\Delta U_0 = \frac{3-n}{5-n} \left(1-\frac{r_{\rm hc}}{r_h}\right)\;
\frac{GM_{\rm hc}^2}{r_{\rm hc}}\;.
\end{equation}
For reference, $GM_{\rm hc}^2/r_{\rm hc}\approx 2.4\times 10^{41} $erg for
clump mass $M_{\rm hc} = 5 M_J$ and clump radius $r_{\rm hc} = 1.84$ AU.
As energy released by the core heats the surrounding gas, it is instructive to
compare the required energy injection $\Delta U_0$ with the initial total
internal energy of the host clump, $U_0$:
\begin{equation}
\label{deltau_disr}
\frac{\Delta U_0}{U_0} =
-\left(1-\frac{3}{n}\right)\left(1-\frac{r_{\rm hc}}{r_h}\right)\;.
\end{equation}
In the limit of the initially small host clump, $r_{\rm hc} \ll r_h$, the
disruption energy is $\Delta U_0 = 0.2 U_0$ for $n=5/2$. In the case under
consideration, $r_{\rm hc} = 1.84$ AU, and $r_h=4.7$ AU, and thus $\Delta U_0
= 0.123 U_0$.
From the following we conclude that the energy required to unbind the host gas
clump is of the order of $3\times 10^{40}$ erg. This is of the same order as
the binding energy of the planetary core of mass $M_p=10 M_\oplus$. Therefore,
we see that, to disrupt the host clump with the energy released by the
planetary core, the mass of the core must be at least $M_p \sim 10
M_\oplus$. We also note in passing that the similarity of this mass to the
critical solid core mass in the CA theory
\citep[e.g.,][]{Mizuno80,Stevenson82} seems to be a pure coincidence. While
the minimum mass capable of disrupting the gas host clump is the function of
the conditions in that clump (mass, age, opacity, etc.), the CA critical mass
is a function of the core's surroundings -- the location from the star,
opacity and the planetesimal accretion rate from the surrounding
protoplanetary disc \citep[e.g.,][]{Rafikov11}.
\section{Solids within the clump}\label{sec:solids_within}
\cite{Nayakshin10a,Nayakshin10b} finds that when grains contained in the host
clump grow by the hit-and-stick mechanism to sizes of the order of $s\sim 10$
cm, their sedimentation within $\sim 1000$ years becomes possible \citep[see
also][]{McCreaWilliams65,Boss98}. Accumulation of these to the centre of the
clump creates a region dominated by grains rather than by gas. This region,
called ``grain cluster'' by \cite{Nayakshin10a} then becomes gravitationally
unstable and collapses to form a massive core composed of high-Z materials.
As argued in \S \ref{sec:debris_intro}, due to rotation and chaotic convective
gas motions in the centre of the host clump, one may expect numerous smaller
bodies to form in the centre of the clump as well. It is the fate of these
bodies that interests us here.
\subsection{Aerodynamic drag in the host clump}\label{sec:drag}
We shall now consider what happens to solids of different sizes, $s$, from
microscopic grains to asteroid-sized bodies, if they are placed within the
host clump. As we shall see, due to the aerodynamic drag that these solids
suffer, small grains ($s \lower.5ex\hbox{\ltsima} 0.1 - 1$ cm) are nearly frozen in with the
gas, so have to follow its motion, whereas bodies larger than $s \sim 1$ km
experience negligible friction with the gas. Bodies in the intermediate size
range are most likely to end up joining the solid core. The smaller and the
larger objects may survive the gas clump disruption and become either planet
satellites or independent bodies in heliocentric orbits.
To quantify this discussion, we model the host gas clump in this section only
as in the analytical model of \cite{Nayakshin10c}, which shows that the outer
radius of the host clump is approximately independent of the clump's mass,
$r_{\rm hc} = 0.8$~AU~$k_*^{1/2} (10^4 \mbox{yr}/t_{\rm hc})^{1/2}$, where
$t_{\rm hc}$ is the host clump's age, and $k_*$ is dimensionless opacity
\citep[cf. \S 2.1 of][]{Nayakshin10c}. The age of the host clump chosen for
the representative calculation below is $10^4$ years, the mass is $10 M_J$,
and opacity $k_* = 1$. Note that in detail the structure of the host clump is
different from that of the gas clumps we consider in the numerical of the
paper, but the main conclusions that we reach in this section are
qualitatively unaffected by this.
For simplicity, clump rotation is neglected in this section, but we note that
results for rotating gas clumps are actually quite similar as long as gas
pressure is the main means of support against gravity for the host clump
\citep[in the opposite case the gas clump would be unstable to various fluid
instabilities; see, e.g., chapter 7 of][]{ShapiroTeukolsky83}.
Our goal here is to calculate how long it takes for a grain of size $s$ and a
given initial condition for the grain position and velocity to settle to the
centre of the clump due to aerodynamic friction with the gas. To achieve this
goal, we solve for the grain's motion within the host clump using the formulae
of \cite{Weiden77} for the aerodynamic forces acting on the grains. This
allows us to calculate the radial sedimentation velocity of the grain, $v_r$,
and define the sedimentation time as
\begin{equation}
t_{\rm sed} = {r \over |v_r|}\;.
\label{tsed}
\end{equation}
where $r$ is distance to the clump's centre. For solids released from rest
results of such calculations are presented in fig. 2 of
\cite{NayakshinEtal11a}. Here we present a very similar calculation for dust
grains that are on initially circular orbits around the host gas clump's
centre.
Figure \ref{fig:tsed} shows the sedimentation time scale for grains released
on circular orbits with initial radii $r$ of 0.02, 0.1 and 0.5 times $r_{\rm
hc}$ for the blue dot-dashed, brown dashed and black solid curves,
respectively. For comparison, the red dash-triple-dot curve shows the grain
sedimentation time for a grain released from rest, as in
\cite{NayakshinEtal11a}.
Without rotation, grains larger than a few cm manage to sediment to the centre
of the gas clump within its age. Smaller grains are tightly bound to the gas,
and if the host is disrupted, these grains are released into the disc
around the protoplanet. The grains closest to the solid core could have been
thermally reprocessed into crystalline materials; mixing these with surrounding
disc's ices and incorporating into comets may explain their puzzling
compositions \citep{Vorobyov11a,NayakshinEtal11a}.
In a purely spherical geometry with no rotational support for grains, large
grains ($s\lower.5ex\hbox{\gtsima} 1$ m) fall into the centre and join the solid core on the
free-fall time, which for a constant density model clump is constant with
radius (cf. the horizontal part of the red dash-triple-dot curve).
For grains on circular orbits, however, centrifugal force balances gravity,
and the grains sediment only because of a gradual angular momentum loss due to
aerodynamic forces. These forces become progressively less important for
larger grains, and therefore the grain sedimentation time increases with $s$
again (cf. the black, the brown and the blue curves in
Fig. \ref{fig:tsed}). As a result, grains larger than 1-10 km may orbit the
centre of the clump for times comparable with the clump's lifetime (which we
assume of the same order as the clump's age here, as presumably the clump
continues to migrate inward at roughly same migration speed).
Concluding, we see that solid objects larger than a few km in size do not
necessarily contribute to growth of the solid planetary core as it may take
too long for these bodies to spiral into the core: the host clump is likely to
be disrupted before such an inspiral occurs.
\begin{figure}
\centerline{
\psfig{file=tsed_omega0_vrot1.pdf,width=0.49\textwidth,angle=0}}
\caption{Sedimentation times versus grain size for grains set initially on
circular orbits, except for the red dash-triple-dot curve which is for a
grain released from rest.}
\label{fig:tsed}
\end{figure}
\subsection{On planetesimal birth and sizes}\label{sec:birth}
In this paper we do not simulate the phase of planetesimal formation due to
numerical challenges. Previous simulations of the whole proto-planetary disc
by \cite{ChaNayakshin11a} lacked numerical resolution (to resolve smaller
objects) and resulted in formation of single self-bound massive ($M \lower.5ex\hbox{\gtsima}$ a
few $\mearth$) clusters of dust particles inside the gas clumps. These were
held from a further self-gravitational collapse artificially by employing a
finite gravitational softening length in the simulations. Increasing
numerical resolution further (that is, the number of SPH and dust particles)
by up to several orders of magnitude is needed to resolve small, e.g., $\lower.5ex\hbox{\ltsima}
0.01$ AU scales, on which planetesimals we are interested here could
form. This is beyond our present numerical capabilities.
Figure \ref{fig:tsed} shows a difficulty for formation of planetesimals in the
TD scheme, very much similar in nature to the well known ``metre-size
barrier'' for the planetesimal growth in the protoplanetary discs
\citep{Weiden77}. In the CA theory, aerodynamic coupling with the gas makes
$\sim$~metre-sized boulders migrate radially inward in the disc, so that they
are probably lost to the protostar before they become planetesimals.
Similarly, Figure \ref{fig:tsed} shows that solids of intermediate sizes
migrate radially inward very rapidly even if they are initially set on
circular orbits around the centre of the host gas clump. Such objects thus
must join the massive solid core there. To arrive at km-sized and larger
bodies that suffer much weaker aerodynamic drag, smaller grains must thus
somehow ``jump'' from being small to being large without suffering the
aerodynamic drag.
We leave a detailed study of this issue to a future paper, but make some
suggestions on how this difficulty may be avoided. First of all, again
reminiscent of the well known ideas in the CA theory context, channels for a
rapid growth of solids may be available: via self-gravitational instability
\citep{Safronov72,GoldreichWard73} of dust-dominated regions (although not in
necessarily in the shape of a disc), or via turbulence
\citep[e.g.,][]{YoudinGoodman05,JohansenEtal07,CuzziEtal08}.
Secondly, and unlike the CA ``metre-size barrier'' problem, here it is much
less clear that aerodynamic drag does make all the medium-sized solids to
accrete onto the central object. In the case of a proto-planetary disc (the CA
model), the protostar contains almost all the mass and is thus the natural
central (nearly) point-mass around which the disc rotates. Grains migrating
inward must therefore end up in the star. In the TD case, however, the mass of
the ``massive'' solid core we are thinking about is measured in Earth masses,
which is $\sim 10^{-3}$ of the total host gas clump mass. If there are
turbulent or convective motions in the inner region of the clump (which is
almost a certainty), then the solid core itself will be affected by the gas
motions through its coupling to the gas via gravity of the latter. Thus it may
not ``sit'' in the centre of the clump accreting all the medium-sized solids
sedimenting there. Furthermore, motion of the grains would also be affected,
with turbulent and convective motions of gas dragging the grains around. There
is clearly a limit here; too much of turbulence would prevent grains from
sedimenting into the centre in the first place, but moderate turbulent motions
may be expected to delay grain accretion onto the solid core and lead to
formation of new condensation centres which perhaps would lead to formation of
the larger planetesimals that we study in the rest of the paper.
It would also be desirable to understand just how large the planetesimals
formed inside the host gas clump are likely to be. Following \S 3.6.2 and
3.6.3 of \cite{Nayakshin10a}, we find that the linear extent of the region of
the gas cloud within which the first gravitational collapse of grain-dominated
regions could occur is about $\sim 0.1$ times the size of the gas clump. For
the total grain mass in the collapsing region of $10\mearth$, the expected
velocity dispersion of fragments formed by collapse is then $\sim 0.5$ km
s$^{_1}$.
Now, collisions of large planetesimals of equal size, $a$, would lead to their
fragmentation if the collision velocity ($\sim$ the velocity dispersion
calculated above) is larger than the escape velocity from the surface of the
planetesimal \citep[e.g.,][]{StewartL09}. This would then suggest that
collisions of equal size bodies inside the ``grain sphere'' would split
planetesimals smaller than about 1000 km, while larger bodies would
``stick''. One however need to estimate the frequency of such collisions.
Assuming that the grain sphere fragments into bodies of approximately equal
size, we find that all the planetesimals would have had at least one
catastrophic (shattering) collision within a thousand years inside the grain
sphere for $a\lower.5ex\hbox{\ltsima} 100$ km. Collisions with smaller bodies -- fragments of
the ``original'' planetesimals -- increase this collision rate estimate
further. We thus conclude that expected sizes of planetesimals surviving the
dense environment of the inner region of the clump are at least a few hundred
km. Presumably both smaller and larger solid objects could then be obtained
over much longer time scales by collisional evolution of the disrupted
population.
\subsection{Planet's influence radius: bound and unbound debris}\label{sec:influence}
Having understood the range of body sizes ($s\lower.5ex\hbox{\gtsima} 1$km) that may be
considered as independent for the duration of the host clump, we now shift out
attention to what may happen when the host clump is disrupted. In detail the
answer on this question is quite complicated as it depends on the nature of
planetesimal orbits within the clump and the exact way the clump is
disrupted. However, for not too eccentric orbits, we can divide the
planetesimal population on those that are ``close'' and those that are ``far''
from the solid planetary core. The former have a fair chance of remaining
bound to the solid core when the host clump is disrupted, and the latter
become unbound independent objects.
To quantify this, consider the simplest setting, placing a solid core of mass
$M_p$ in the centre of the gas clump and setting planetesimals on circular
orbits around the planet. The rotation of the planetesimal's disc is assumed
prograde with respect to the orbit of the host clump around the star. The
circular speed of the planetesimals around the planet is given by
\begin{equation}
v_{\rm circ}^2 = G {M_p + M(r)\over r}\;,
\label{vcirc}
\end{equation}
where $M(r)$ is the gas mass enclosed inside radius $r$ (distance from the
centre of the host clump).
If planet's mass is low enough to build up a massive self-gravitating
atmosphere around it \citep[which is expected to be the case for an adiabatic
young gas clump where the critical core mass is above 100
$\mearth$; see][]{PerriCameron74}, then the gas density is nearly constant in the
clump's centre. Let us call that central density $\rho_0$; thus $M(r)\approx
(4\pi /3) \rho_0 r^3$. We can now define the planet influence radius, $r_i$,
such that $M(r_i) = M_p$. Apparently,
\begin{equation}
r_i = \left[{3 M_p\over 4\pi \rho_0}\right]^{1/3}\;.
\label{ri}
\end{equation}
Note that since $r_{\rm hc} = (3 M_{\rm hc}/4\pi \rho_{\rm mean})^{1/3}$,
where $\rho_{\rm mean}$ is the mean density of the host clump, then we have
\begin{equation}
r_i \approx r_{\rm hc} \left({M_p \over M_{hc}}\right)^{1/3} \left({\rho_{\rm
mean}\over \rho_0}\right)^{1/3} = 0.18\; r_{\rm hc}
\left({\rho_{\rm mean}\over \rho_0}\right)^{1/3}
\label{ri_app}
\end{equation}
Solids inside $r_i$ are gravitationally bound mainly to the planet, whereas
those outside are bound by the enclosed gas. We expect that when the host
clump is disrupted, objects within $r_i$ remain bound to the solid planet
whereas objects outside this radius become unbound from the planet.
\subsection{Post-disruption orbits for unbound debris}\label{sec:theory}
We now build an approximate analytical theory for the orbits of the
planetesimals after the host clump tidal disruption. To this end we assume
that gas clump disruption is instantaneous. We also assume that solids within
$r_i$ remain bound to the planet and continue to follow its motion around the
central star, and we concentrate here only on solids outside the influence
radius.
Before the disruption, the position of a planetesimal with respect to the star
is given by $\mathbf R+\mathbf r$, where $\mathbf R$ is the vector connecting
the star with the centre of the gas clump (solid planet's location), and
$\mathbf r$ is the vector connecting the centre of the host clump with the
given planetesimal. The instantaneous velocity of the planetesimal is also
comprised of two parts, one due to the heliocentric Keplerian motion of the
gas clump, $\mathbf V$, and the other given by the circular prograde rotation
velocity, $v_{\rm circ}$, around the centre of the host clump.
We now call $\mathbf{V + \Delta v}$ the planetesimal's velocity right after
the host clump disruption. Physically, $\mathbf{\Delta v}$ is the velocity
with which planetesimal becomes unbound, and may be called a velocity
``kick''. We expect that $0 < \Delta v < v_{\rm circ}$ for the energy
conservation reasons.
We shall now calculate the new heliocentric orbit of the planetesimal by
considering its specific energy and angular momentum. The specific energy and
specific angular momentum of the host gas clump before the disruption are $E_0 =
-GM_*/2R$ and $\mathbf{L_0 = R\times V}$, respectively. The post-disruption
specific energy, $E$, and specific angular momentum, $ \mathbf L$, of the
planetesimal are
\begin{eqnarray}
E &=& -\frac{GM_*}{|\mathbf R+\mathbf r|} +
\frac{1}{2}\left|\mathbf V + \mathbf{\Delta v}\right|^2,\\
\mathbf L &=& (\mathbf R+\mathbf r) \times (\mathbf V + \mathbf{\Delta v}),
\end{eqnarray}
where, $M_*$ is the mass of the central protostar. Decomposing these
expressions into Taylor series with respect to small parameters $r/R$ and
$\Delta v/V$, up to the linear terms only, and writing $E \approx E_0 + \delta
E$, $L^2 \approx (L_0 + \delta L)^2$ (the latter is possible since
$\mathbf{\Delta v}$ is in the clump's orbital plane by setup), we have
\begin{equation}
\delta E = V^2 \left({\mathbf{R \cdot r} \over R^2} + {\mathbf{V \cdot \Delta
v} \over V^2}\right)\;.
\label{deltae_app}
\end{equation}
From this we see that the semi-major axis of the planetesimal's orbit after
disruption is
\begin{equation}
a = \frac{GM_*}{2|E|}\; = R \left(1 + {\delta E\over |E_0|}\right) = R
\left[ 1 + 2 \left({\mathbf{R \cdot r} \over R^2} + {\mathbf{V \cdot \Delta v}
\over V^2}\right)\right]\;.
\label{a_linear}
\end{equation}
Similarly, we can write
\begin{equation}
\mathbf{\delta L \; \approx \; r \times V + R\times \Delta v}\;.
\label{l_linear}
\label{deltaL1}
\end{equation}
Now, using vector identity
\begin{equation}
\mathbf{[A\times B]\cdot [C\times D] = (A\cdot C)(B \cdot D) - (A \cdot D) (B
\cdot C)}\;,
\end{equation}
we find that
\begin{equation}
\mathbf{L_0 \delta L =} V^2 \mathbf{R \cdot r} + R^2 \mathbf{V \cdot \Delta v}
\end{equation}
Comparing this expression with equation \ref{deltae_app}, we observe that
\begin{equation}
{\delta E\over E_0} = - 2 {\mathbf{\delta L\cdot L_0}\over L_0^2} = -2 {\delta
L \over L_0}\;,
\label{de_dl}
\end{equation}
where we also recalled that the orbital plane of the planetesimals does not
change since the proto-disc of planetesimals rotates in the prograde direction
in the orbital plane of the host clump by assumption.
The eccentricity of the planetesimal's orbit is
\begin{equation}
e^2 = 1+\frac{2E L^2}{G^2M_*^2}\;,
\label{eq:ana_e}
\end{equation}
which can be decomposed to become
\begin{equation}
e^2 \approx - {\delta E\over E_0} - 2 {\delta
L \over L_0} - 2 {\delta E\over E_0} {\delta
L \over L_0} - \left({\delta
L \over L_0}\right)^2\;.
\end{equation}
Using equation \ref{de_dl}, we find that the linear terms cancel, and the
always positive result is
\begin{equation}
e^2 = {3 \over 4} \left({\delta E \over E_0}\right)^2\;.
\label{ecc1}
\end{equation}
\subsubsection{Eccentricity -- semi-major axis correlation}\label{Sec:e_a}
We can eliminate $\delta E$ and $E_0$ in favour of orbital elements. Since $E_0 =
-GM_*/2R$, and $a = GM_*/2|E|$, equation \ref{ecc1} actually shows that
\begin{equation}
e\approx \left({3 \over 4}\right)^{1/2} {|\delta a| \over a_0}\;,
\label{ecc_a}
\end{equation}
where $a_0 = R$, and $\delta a \equiv a - R$ is the difference in the
semi-major axis of the planetesimal's post-disruption orbit and that of the
host gas clump before disruption.
Equation \ref{ecc_a} predicts a correlation in the orbital elements of
planetesimals after disruption: the belt of planetesimal debris left after the
host gas clump disruption should have circular orbits at the centre, $a=a_0$,
and increasingly eccentric orbits towards the belt's edges.
We note one rather interesting feature of equation \ref{ecc_a}: it does not
depend on the properties of the host gas clump {\em or} the solid
protoplanetary core inside. The eccentricity -- semi-major axis correlation
should thus be a general property of the debris rings left after disruption of
the host gas clumps (as long as the pre-disruption orbit of the clump is
nearly circular).
We also note that for the case of the proto-disc of planetesimals {\em not
coinciding} with the orbital plane, the inclination of the planetesimal
orbits, $i$, after the disruption is non zero but can be shown to be small.
\subsubsection{Rings versus discs}\label{Sec:rings}
Despite the universal correlation we just discussed, the properties of the
host gas clump are still imprinted on the {\em range} of possible planetesimal
orbits through the fact that the pre-disruption protoplanet is finite in
spatial extent, and that the maximum velocity kick to a planetesimal released
by the clump disruption, $\Delta v$, is also finite. Referring to equation
\ref{a_linear}, we see that the width of the ``disruption ring'' -- the ring
occupied by the planetesimals left over after the clump disruption -- is
\begin{equation}
w\equiv {a_{\rm max} - a_{\rm min} \over a_0} = 4 \max \left| {\mathbf{R \cdot r} \over
R^2} + {\mathbf{V \cdot \Delta v} \over V^2} \right|
\label{a_width}
\end{equation}
Noting that $\Delta v \sim v_{\rm circ} \propto r$ within a constant density
host clump, and that the maximum possible value of $r$ is the Hill's radius of
the host clump, $r_h$, we can summarise this prediction by writing
\begin{equation}
w \approx 4 \zeta {r_h \over a_0}\;,
\label{zeta}
\end{equation}
where $\zeta$ is a dimensionless number probably smaller than but comparable
to unity. For the setup of this paper, in particular, where $r_h = 4.7$ AU and
$a_0 = 40$ AU, we have
\begin{equation}
w \approx 0.5 \zeta\;.
\label{w_predict}
\end{equation}
We shall see below that $\zeta \approx 0.7-1$ for the three simulations
performed in the paper. In general this parameter will depend on how
concentrated the distribution of planetesimals is within the host protoplanet
before the disruption, with more compact distributions leading to smaller
$\zeta$. It would also vary, likely increase, if planetesimal orbits in the
pre-disruption clump are eccentric.
\section{Numerics}\label{sec:numerics}
\subsection{Method} \label{sec:method}
We now turn to hydro/N-body simulations for a more detailed investigation of
the problem. SPH \citep{Gingold77,Lucy77} is a Lagrangian simulation algorithm
well suited for irregular and self-gravitating systems. SPH has been applied
to a variety of astrophysical contexts \citep{Monaghan92,Springel10}. In this
paper, we use \textsc{gadget-3}, an updated version of the SPH/N-body code
presented by \cite{Springel05}.
In both of the disruption scenarios that we study in this paper, host clump
disruption occurs on a time scale shorter than the Kelvin-Helmholtz time of the
clump. An adiabatic equation of state for the gas is thus sufficient for our
purposes. As hydrogen is molecular for temperatures smaller than $\sim 2000$
K, we chose the polytropic index, $n$, to be $n=5/2$, which corresponds to the
ratio of specific heats of $\gamma=1.4$. The number of SPH particles in each
of the simulations presented is $N_{sph} = 10^{6}$, so that the mass of each
particle is $5\times 10^{-6} M_J$.
The protostar, the planet and the planetesimals are all modelled as N-body
particles of appropriate masses. Accretion onto the planet is not allowed, but
accretion onto the protostar is simulated with the sink particle approach
\citep[as in][]{ChaNayakshin11a}. This is necessary to prevent the build up of
very short time step SPH particles near the star due to a non-negligible
artificial viscosity of the code in this low density region, which is
additionally of little interest to our study.
The N-body particles interact with gas only through gravity (except for gas
accretion as described above). We set the total mass of the planetesimal disc
to $M_d = 0.1$M$_\oplus$. This mass is shared equally between $N_{\rm pl} =
2\times10^4$ particles. The mass of each is thus $\sim 3\times 10^{22}$ g, and
the corresponding radius is about 200 km. For such low masses, gravitational
interactions between planetesimal particles and their effect onto the planet
and the gas are negligible, even though these interactions are included in the
calculation. The large linear size of the planetesimals allows us to neglect
the aerodynamic coupling between them and the gas completely.
\subsection{Initial conditions and disruption method} \label{sec:init_cond}
Below we present three numerical simulations. The first of these, labelled U0,
uses the initial condition for the polytropic gas cloud described in \S
\ref{sec:tidal}. In this case the host gas clump exactly fills its Roche lobe
at the beginning of the simulation. The clump is thus disrupted due to tidal
forces of the star only.
In the other two cases, the host clump radius, $r_{\rm hc} = 1.84$ AU, is
initially smaller than the Hills radius, $r_h=4.7$ AU. The host clump central
temperature is $500$ K for this initial condition. As explained in \S
\ref{sec:explosion}, we assume that at $t=0$ the central solid core releases a
given amount of energy which is then instantaneously added to the internal
energy of the gas particles within radius $r_{\rm ej} = 0.2$ AU (chosen
somewhat arbitrarily). The inner region is then over-pressured compared to its
surroundings, which drives an expansion that puffs up the whole gas clump.
In \S \ref{sec:explosion} we found that injecting energy $\Delta U_0 = 0.123
U_0$, where $U_0$ is the total internal energy of the gas protoplanet, should
increase $r_{\rm hc}$ to $r_h$. This derivation assumed that the clump keeps
its polytropic structure even after the energy injection, which is clearly not
actually correct for the equation of state we use. We therefore performed two
simulations with energy injection bracketing the critical injection energy
$\Delta U_0$. These are labelled U10 and U15, so that $\Delta U = $ 10\% and
\%15 of $U_0$ for the two simulations, respectively. We find that both of
these simulations resulted in the disruption of the host gas clump, although
the process was much faster in U15 than in U10. There are interesting
differences between the orbital parameters of the planetesimals in U10 and
U15, making both simulations worth presenting here.
The exact outcome of the gas protoplanet disruption depends on a number of
assumptions about the pre-disruption stage, such as (i) the exact location of
the planet at the moment of the gas protoplanet disruption (which needs not be
the centre of the gas clump in general); (ii) the duration of the disruption
process (rapid or slow); (iii) the distributions and orbits of solids within
the clumps, and their total mass, which may start influence the dynamics of
planetesimals if their total mass is comparable or larger than that of the
planet, and (iv) the orbit of the gas clump around the protostar.
The parameter space of the problem is too large to cover in this first
study. Therefore, we only aim here at learning the most roburst features of
the results, which we hope will be generally correct within a factor of a few
despite all the uncertainties pointed out above.
The planetesimal particles are placed in a geometrically thin disc on circular
orbits around the centre of the gas clump with prograde velocity given by
equation \ref{vcirc}. The outer radius of the planetesimal disc, $r_d$, is
$0.4$ AU for the simulations U10,U15 and $1$ AU for simulation U0,
respectively (to scale approximately as $0.2 r_{\rm hc}$ for all the
cases). The inner radius of the planetesimal disc, $r_{in}$, is set to 10$\%$
of $r_d$. Introduction of the inner disc cutoff at $r_d$ does not change our
results at all but speeds up simulations considerably. Particles at $r \lower.5ex\hbox{\ltsima}
r_{in}$ are bound very strongly to the solid core (planet), so that disruption
of the host gas clump hardly changes their orbits. These ``uninteresting''
planetesimals have short time steps and are expensive to simulate.
The planetesimal disc has the intitial disc surface density profile of
$\Sigma_{\rm pl} (r) \propto r^{-2}$. As partciles in the disc do not
interact significantly due to their low masses, behaving as test particles,
such a choice for $\Sigma_{\rm pl} (r)$ has no consequences for the results
but allows an approximately uniform parameter sampling in terms of the initial
distance $r$ to the solid planet.
Initially, the gas host clump rotates around the protostar with the Keplerian
velocity, $\sqrt{GM_*/R}$. The orbital period of the initial orbit, $P_0$, is
$\approx 253$ yrs. We use $P_0$ as a unit of time in presenting the results
below. All of the simulations were run until time $t=10$, i.e., 10 initial
orbits of the clump.
\section{Dynamics during host protoplanet disruption}\label{sec:gas}
\subsection{Simulation U15} \label{sec:u15}
In this simulation, the energy injected into the gas exceeds the critical
energy needed to disrupt the host clump
(cf. eq. \ref{deltau_disr}). Therefore, a rapid disintegration of the gas
clump is expected. Fig. \ref{fig:OFF00EB015VY_early} shows the projected gas
column density at two early stages of the simulation, $t = 0.06$ (left panel)
and $t = 0.25$ (right panel). The projection is done along the $z$-axis, e.g.,
along the direction normal to the host clump's orbital plane. The figures are
centered on the position of the solid core (planet), shown with the thick
green symbol at the centre of each panel. Vectors show the gas velocity field,
likelywise centered on the solid planet's velocity. The black dots are
individual planetesimals.
Initially (left panel), clump expansion appears nearly spherically symmetric
even though the outer gas shells did overflow the Roche lobe by that
time. However, the flow beyond $r_h$ is controlled more and more by the star's
tidal field. This is evident in the right panel, where the flow becomes highly
assymetric at large $r$. The star is positioned exactly North of the clump in
the right panel. One observes a flow of gas towards the star at the upper
left corner of the panel, and away from the star in the lower part of the
panel. The planetesimal disc is also becoming affected, visually following
the motion pattern of the gas.
The host clump disruption is indeed dynamic as the right panel corresponds to
time just slightly later than one dynamical time, defined as $P_0/2\pi$.
\begin{figure*}
\centerline{
\psfig{file=OFF00EB015VY_50_0_063P_ZI.pdf,width=0.49\textwidth,angle=0}
\psfig{file=OFF00EB015VY_200_0_252P_ZI.pdf,width=0.49\textwidth,angle=0}}
\caption{The top view of the gas surface density map for simulation U15 at
time t=0.06 (left panel) and $t=0.25$ (right panel). The map is centered on
the position of the solid planet (core), which is shown with the thick green
symbol. Vectors show the velocity field, also centered on the velocity of
the planet. The black dots show locations of individual planetesimals.}
\label{fig:OFF00EB015VY_early}
\end{figure*}
Fig. \ref{fig:OFF00EB015VY_late} shows the same as
Fig. \ref{fig:OFF00EB015VY_early}, but now at times $t=1$ and $10$ in the left
and the right panels, respectively. The figure presents a larger field of view
centered onto the star now. The velocity field is also centered onto the star
rather than the planet. Note in the left panel that the planetesimals
disrupted off the initial gas clump continue to follow orbits similar to that
of the densest regions of gas. This is hardly surprising given that potential
due to the host clump initially dominates the planetesimals' orbits (until the
host clump is completely disrupted). At late times (see the right panel), the
host clump is disrupted into a ring centered about the initial clump's
separation from the star. The planetesimals are also spread into a ring-like
feature which shows several streams. The streams appear to have some eccentricity to
them. We also note that the fine structure of the streams is now not
correlated to the gas component. This is probably caused by the gas
gravitational potential being smoothed out due to circularisation of the gas
flow, leading to a reduced gas gravitational force on the planetesimals.
\begin{figure*}
\centerline{
\psfig{file=OFF00EB015VY_795.pdf,width=0.49\textwidth,angle=0}
\psfig{file=OFF00EB015VY_7945.pdf,width=0.49\textwidth,angle=0}}
\caption{Same as Fig. \ref{fig:OFF00EB015VY_early} but at later time; $t=1$
(left panel), and $t=10$ (right panel), except that the center of the
coordinate system is now the protostar (the red symbol at the center of the
panels). Note that planetesimals initially continue to follow the orbital
motion of the densest parts of the gas streams at $t=1$ but separate out at
$t=10$ when the gas streams are dispersed and circularised into a ring.}
\label{fig:OFF00EB015VY_late}
\end{figure*}
\subsection{Simulation U10}\label{sec:u10_gas}
As derived in \S \ref{sec:explosion}, the critical injection energy sufficient
to inflate the clump to the point of its tidal disruption is $\Delta U_0 =
0.123 U_0$ for the parameters of the clumps we consider. Therefore, if the
approximate theory of \S \ref{sec:explosion} were correct, the clump would not
be disrupted at all in this particular simulation as $\Delta U = 0.1 U_0 <
\Delta U_0$. However, the argument given by equation \ref{deltau_disr} assumes
that the energy injected into the central regions of the gas is shared by the
whole clump, so that gas in the clump effectively finds itself on a new
(single) polytropic relation. In the simulations, however, the central regions
are shifted to a different polytropic relation by the energy injection,
whereas the rest of the clump remains at the initial one. One may thus expect
the simulation results to differ from the simple analytical prediction.
Figure \ref{fig:OFF00EB010VY_early_internal} shows the radial density
profile of the host clump (red curve) in simulation U10 at time $t=0.05$
and the theoretical polytropic density profile (black curve) with $n=5/2$
inflated by the energy input from the solid core as assumed in \S
\ref{sec:explosion}. One notices significant differences between the two
curves, with a tail of gas material extending further out in the red curve. It
is this tail that allows some material to siphon out of the Roche lobe and
for the gas clump's radius to continue swelling (cf. equation
\ref{mass_radius} on this point) as mass is lost.
The left panel of Figure \ref{fig:OFF00EB010VY_3P_snap} illustrates this
initially slow expansion. The host clump is largely intact at $t=3$ in
U10. This is in stark contrast to simulation U15 in which the host gas clump
was completely obliterated by $t=1$ already (left panel of
Fig. \ref{fig:OFF00EB015VY_late}). However, the host gas clump is eventually
destroyed in simulation U10 also. The gas column density and the distribution
of planetesimals appear to be quite similar at late times in U10 and U15
($t=10$, right panels of Figs. \ref{fig:OFF00EB010VY_3P_snap} and
\ref{fig:OFF00EB015VY_late}, respectively). We shall make a more detailed
analysis of planetesimals' orbits in \S \ref{sec:unbound} below.
\begin{figure}
\centerline{\psfig{file=poly_int.pdf,width=0.49\textwidth,angle=0}}
\caption{The internal density structure of the expanding protoplanetary clump
for simulation U10 ($\Delta U/U_0=0.1$) at time $t=0.05$. The red solid line
is the actual gas density from the simulation, while the black solid line is
a polytrope of 5M$_J$ mass and $4.7$AU radius. The latter curve is expected
based on the approximate theory shown in \S \ref{sec:explosion}. The figure
shows that a single polytropic relation approximation breaks down after the
energy injection, although the approximation is still useful in roughly
determining the critical injection energy $\Delta U_0$.}
\label{fig:OFF00EB010VY_early_internal}
\end{figure}
\begin{figure*}
\centerline{
\psfig{file=OFF00EB010VY_RESUM_284_3P.pdf,width=0.49\textwidth,angle=0}
\psfig{file=OFF03ZEB010VY_RESUM_5846.pdf,width=0.49\textwidth,angle=0}}
\caption{Gas column density projections similar to Fig.
\ref{fig:OFF00EB010VY_early_internal} but for run U10 at time $t=3$ (left
panel) and $t=10$ (right panel). As the planet is less extended than in test
U15, the disruption process is much more gradual. At $t=3$ (3 whole orbits
of the host clump around the star), only a small fraction of gas mass is
lost through the Roche lobe overflow. No planetesimals have yet been unbound
from the clump. However, at late time (the right panel), the structure of
the disrupted gas and planetesimal population is quite similar to that in
U15.}
\label{fig:OFF00EB010VY_3P_snap}
\end{figure*}
\subsection{Simualtion U0}\label{sec:u0_gas}
In this simulation, the gas protoplanet is more extended initially, so that
$r_{\rm hc} = r_h$ at $t=0$. No energy input from the solid core is
assumed. Figure \ref{fig:BIG_snap} shows two snapshots for simulation U0 at
times $t=4$ (left panel) and $t=10$ (right panel) in the same format as
Fig. \ref{fig:OFF00EB010VY_3P_snap}. The disruption process is not as rapid as
in simulation U15 but is a little faster than in U10. The morphology of the
gas flow is different: the disrupted gas spiral at $t=4$ is wider in U0 than
it is in Fig. \ref{fig:OFF00EB010VY_3P_snap}.
Since the initial disc or planetesimals inside the gas protoplanet is large to
begin with ($r_d = 1$ AU for U0) than for simulations U10 and U15 ($r_d = 0.4$
AU in both), the flow of disrupted planetesimals is initially wider in U0 (see
the banana-shaped feature in the left panel of Fig. \ref{fig:BIG_snap}.) The
end result is at least visually not too dissimilar from runs U10 and U15,
however.
\begin{figure*}
\centerline{
\psfig{file=BIG_1_796.pdf,width=0.49\textwidth,angle=0}
\psfig{file=BIG_3_681.pdf,width=0.49\textwidth,angle=0}}
\caption{Same as Fig. \ref{fig:OFF00EB010VY_3P_snap} but for simulation U0,
the case where the initial cloud is cooler and already fills its Roche
lobe. The corresponding times are $t=4$ (left panel) and $t=10$ (right
panel).}
\label{fig:BIG_snap}
\end{figure*}
\section{Orbits of the unbound planetesimals}\label{sec:unbound}
We now turn to analysis of the unbound population of planetesimals in terms of
their orbital elements at the end of the simulations. To avoid confusion, tt
must be stressed that, all of our planetesimals remain bound to the central
star at the end of the simulations, but some are also bound to the planet, as
planetary satellites. The ``unbound'' population of planetesimals are those
particles moving on their own independent heliocentric orbits.
In the simulations, the bound and unbound populations are differentiated
according to two conditions. We define the specific energy of a planetesimal
with respect to the planet, $E_{\rm rel}$,
\begin{equation}
E_{\rm rel} = - {GM_p \over r} + {1 \over 2}\;\left(\mathbf{v}-\mathbf{V_p}\right)^2\;,
\label{erel}
\end{equation}
where $r$ is the distance between the planet and the planetesimal,
$\mathbf{v}$ and $\mathbf{V_p}$ are the planetesimal and the planet
velocities, respectively (note that these quantities are calculated here at
$t=10$ whereas in \S \ref{sec:explosion} we measured relative positions and
velocities of planetesimals and the planet at $t=0$). The planetesimal is
considered unbound if
\begin{equation}
E_{\rm rel} > 0\;,
\label{ebound}
\end{equation}
and bound if
\begin{equation}
E_{\rm rel} < 0\quad {\rm and} \quad r \le r_h'\;,
\label{unbound}
\end{equation}
where $r_h' = R_0 (M_p/3M_*)^{1/3}$ is the Hill's radius of the solid
planet.
The two populations should be analysed differently, of course. The orbits of
the unbound population should be defined with respect to the central star,
whereas the orbits of the bound planetesimals are best defined with respect to
the planet. In the rest of this section we consider the unbound part of the
planetesimals only.
\subsection{Simulation U15}\label{sec:u15_unbind}
Fig. \ref{fig:EB015_orbit_LIVE} shows the orbital eccentricity (bottom panel)
and the inclination (top panel) of the unbound planetesimals versus the
semi-major axis of their orbits ($a$) at $t=10$. As explained above, their
orbital parameters are calculated with respect to the star as their orbits are
heliocentric.
The top panel of figure \ref{fig:EB015_orbit_LIVE} shows that planetesimals
continue to have very small orbital inclinations after the disruption, with
the mean inclination angle of only $i\approx 1^\circ$. This is natural since
in our setup tidal disruption of the host gas protoplanet is symmetric around
its orbital plane. In fact, the approximate analytical theory of \S
\ref{sec:explosion} predicts that inclination of planetesimal orbits should
remain exactly zero. Therefore, the inclination found in the simulation must
result either from a small but finite asymmetry with respect to $z \rightarrow
-z$ inversion developing during the disruption, or due to effects not taken
into account in the analytical theory.
The central $a\approx 32$ to $a\approx 40$ AU region of this plot shows a
higher dispersion in the values of $i$ than do the more distant regions. To
understand the origin of this result, we note that the central region is the
one where most of the gas ends up after being disrupted, e.g., see the right
panel of figure \ref{fig:OFF00EB015VY_late}. We also note that gas from the
disrupted host clump is initially arranged in a high density spiral,
gravitational potential of which may well be significant enough to scatter
planetesimals about and to pump their inclinations.
This explanation is consistent with the fact that planetesimals on wider
orbits, e.g., $a \lower.5ex\hbox{\gtsima} 42$ AU, which should be affected by the interactions
with the gas spiral less, indeed have smaller dispersion in their inclination
angles. The non-zero mean value of $i$ for this population, more distant from
the planet, should be due to asymmetries developing during the disruption
process. Comparison with simulation U0 below, which produces far smaller
inclinations, demonstrates that the asymmetries are probably related to the
injection of energy in the centre of the host in U15 (and U10). This energy
injection increases the entropy of the gas within $r < r_{\rm ej}$ around the
planet but not at larger radii. The initial entropy profile is thus strongly
unstable to convection in U10 and U15, most likely leading to an asymmetric
(with respect to the initial orbital plane of the clump) expansion of the host
clump.
\begin{figure}
\centerline{
\psfig{file=OFF00EB015VY_7945_10P_kbo_orbit.pdf,width=0.49\textwidth,angle=0}}
\caption{The inclination (top panel) and eccentricity (bottom panel) of the
unbound planetesimals versus semi-major axis, $a$, in the simulation U15 at
the end of the simulation, $t=10$. Individual planetesimals' orbital
elements are shown with black dots. The coloured chevron-shaped curves are
theoretical predictions (see \S \ref{sec:u15_unbind}). The nearly empty
vertical contains many planetesimals still bound to the planet, which are
not shown here but analysed in \S \ref{sec:bound_orbits}.}
\label{fig:EB015_orbit_LIVE}
\end{figure}
The bottom panel of figure \ref{fig:EB015_orbit_LIVE} shows the planetesimals
in the eccentricity -- semi-major axis ($e$--$a$) plane. The coloured curves
show the theoretically calculated correlations of $e$ and $a$ with slightly
different assumptions (explained below) on the basis of the approximate theory
of \S \ref{sec:theory}. All of these curves were shifted to $a_0 = 38$ AU to
fit the simulations better. The shift may be empirically justified as
following. Our approximate theory is based on the assumption of an
instantaneous disruption of the protoplanet. However, in the simulations the
protoplanet is not disrupted instantaneously, and it does migrate inwards
during the disruption, presumably due to gas gravitational torques. Therefore,
in terms of our approximate theory, the ``effective'' location of protoplanet
disruption should be smaller than the initial radius of the host clump's
circular orbit.
Amongst the theoretically computed curves, the green dots show the simplest
one -- equation \ref{ecc_a}. We recall that this equation was obtained with
the help of a Taylor decomposition of the specific energy and the specific
angular momentum in powers of relatively small parameters $r/R$ and $\delta
v/V$. In contrast, the red and the blue curves show the predicted orbits of
the planetesimals that did not use the decomposition. To draw the curves, we
consider a ring of planetesimals of radius $r=0.3$ and $0.4$ AU for the red
and blue curves, respectively. We also assume that the kick velocity of the
planetesimals after the disruption, $\mathbf{\Delta v}$ is equal to $\delta
\mathbf{v}_{\rm circ}$, where $\delta = 0.5$ and $\mathbf{v}_{\rm circ}$ is
the circular velocity of the planetesimal before the disruption (cf. equation
\ref{vcirc}). This simple (but somewhat more accurate than the green dots)
theory also predicts the ``V'' shaped $e-a$ correlation.
We also note that our neglect of the second order terms in equations
\ref{deltae_app} and \ref{deltaL1} lead to the equation \ref{ecc_a} being
symmetrical with respect to the sign of $a-a_0$. The more exact calculations
given with the red and the blue curves are not quite symmetrical and also show
non-zero eccentricity for all the particles. On the other hand, SPH/N-body
simulations show a wider distribution of eccentricities at a given $a$, with
some particles reaching $e\approx 0$. Nevertheless, it appears that by
varying the parameter $\delta$ (which is only constrained to be less than
unity) and by considering the different rings of planetesimals $r$ within the
host before its disruption, we can qualitatively explain the observed range of
$e$ and $a$ as well as their correlation.
The approximate analytical theory of \S \ref{sec:theory} also makes a
prediction on the width of the disruption ring, as $w = 0.5 \zeta$, where
$\zeta$ is expected to be of order unity. The debris ring in simulation U15
has width of $\sim 20$ AU, commensurate with $\zeta =1 $. The simple
``instantaneous'' disruption model is thus reasonably accurate in predicting
the radial extent of the debris disc in this simulation.
\subsection{Unbound debris in U10}\label{sec:unbound_u10}
\begin{figure}
\centerline{
\psfig{file=OFF00EB010VY_RESUM_5846_10P_kbo_orbit.pdf,width=0.49\textwidth,angle=0}}
\caption{Same as Figure \ref{fig:EB015_orbit_LIVE} but at the end of
simulation U10.}
\label{fig:OFF00EB010VY_RESUM_LIVE}
\end{figure}
Figure \ref{fig:OFF00EB010VY_RESUM_LIVE} shows the inclination and the
eccentricity of the unbound planetesimals in simulation U10, along with
theoretical curves, in the same format as in
fig. \ref{fig:EB015_orbit_LIVE}. Simulation U10 differs from U15 by the
smaller amount of internal energy injected into the host clump by the solid
core (planet). As the result, the disruption process is much more gradual and
protracted in U10 than in U15 (see \S \ref{sec:gas}).
The $e$--$a$ diagram shows a great deal of similarity between simulation U10
and U15, but also some interesting differences. For example, the radial width
of the disruption ring is narrower in the former. This makes intuitive sense
as there is less internal energy input into the gas clump in U10, and
therefore the velocity ``kicks'' $\Delta v$ after the disruption are probably
smaller than they are in U15.
The inclination angle $i$ versus $a$ plot shows somewhat larger mean value and
larger dispersion of $i$ in U10 than in U15. In addition, there are pronounced
vertical bands in the top panel of Figure
\ref{fig:OFF00EB010VY_RESUM_LIVE}. Finally, if we break the pattern of
fig. \ref{fig:EB015_orbit_LIVE} on the ``gas-dominated'' central and the
``far-out'' population at $a > 43$ AU, then its fair to say that in simulation
U10 the far-out population is all turned into the central gas-dominated one.
All of these differences are explainable by the fact that the disruption
process is more gradual in U10, which means that the tidal tails (arms) of the
host clump in the process of disruption (cf. the left panel of
fig. \ref{fig:OFF00EB010VY_3P_snap}) survive for longer, before being
circularised into a diffuse gaseous ring (the right panel of
fig. \ref{fig:OFF00EB010VY_3P_snap}). This implies that the planetesimals are
influenced stronger by these tidal tails, explaining a higher dispersion in
inclination $i$. In addition to that, the tidal tails ``shepherd'' the
planetesimals out of the cloud, as in seen in the left panels of figures
\ref{fig:OFF00EB015VY_late} and \ref{fig:BIG_snap}, causing bunching of the
orbital parameters in the vertical bands evident in the top panel of figure
\ref{fig:OFF00EB010VY_RESUM_LIVE}. As the spiral arms survive for longer in
U10 than they do in U15, the bunching effect is far stronger in the former than
in the latter.
\subsection{Unbound planetesimals in simulation U0}\label{sec:unbound_u0}
Figure \ref{fig:BIG_ANA} shows the orbital parameters of the unbound
planetesimals at the end of simulation U0. This figure shows, yet again, the
familiar $e$--$a$ correlation diagram but a significantly different
distribution of planetesimal orbital inclinations. The red and the blue curves
in this case were computed assuming two rings of planetesimals with $r=0.75$
and $1$ AU, respectively, because the planetesimal disc is larger in U0 than
in U10 and U15, and we used $\delta = 0.9$ (see \S \ref{sec:u15_unbind}). The
latter is chosen by trial and error to find a visually reasonable match to the
distribution of orbits in the simulation.
The much stronger bunching of planetesimals towards the $i=0$ plane is a
testament to the different nature of the host clump disruption in U0 as
compared to U10 and U15. While in the latter two the inner region of the clump
was the actual source of the disruption (as the core accretion energy was
dumped there), in U0 the innermost region is ``passive''. Therefore the
disruption process, as experienced by the planetesimals initially located at
$r \le 1$ AU for this simulation, is much less abrupt, leading to even less
inclined orbits for the unbound particles. In addition, as mentioned in
\ref{sec:u15_unbind}, the injection of energy stirs up strong convective
motions in those simulations, pumping up asymmetries and thus orbital
inclinations.
On the other hand, the larger dispersion in values of $i$ in the centre of the
planetesimal ring, and the presence of the vertical bands in
fig. \ref{fig:BIG_ANA} confirms that these features are formed by the tidal
arms (tails) of the host clump before they are wound up and completely erased.
\begin{figure}
\centerline{
\psfig{file=BIG_681_10P_kbo_orbit.pdf,width=0.49\textwidth,angle=0}}
\caption{Same as Figs. \ref{fig:EB015_orbit_LIVE} and
\ref{fig:OFF00EB010VY_RESUM_LIVE}, except for the red and blue curves, as
explained in the text (see \S \ref{sec:unbound_u0}). Note the much larger
$i\approx 0$ population of planetesimals in this simulation compared to U10
and U15}
\label{fig:BIG_ANA}
\end{figure}
\section{Bound population: planet satellites}\label{sec:bound_orbits}
We now switch to discussing planetesimals bound to the solid planet, e.g.,
those that have a negative energy with respect to the planet and that are
within the planet's Hill's radius (see \S \ref{sec:unbound}).
Figure \ref{fig:U15_U10_sat} presents the orbital parameters of the
planetesimal orbits around the planet for the simulation U15 (left panel) and
U10 (right panel). The red vertical lines show the location of the planet's
influence radius, $r_i$, defined by equation \ref{ri}. This radius divides the
region where gravitational potential is dominated by the solid planet (inside
$r_i$) and the gas (outside $r_i$). As stated in \S \ref{sec:influence}, the
orbits of planetesimals are expected to be only mildly perturbed within $r_i$
and suffer strong perturbations outside $r_i$, in fact being completely
unbound from the planet at $r \gg r_i$.
Figure \ref{fig:U15_U10_sat} confirms this expectation qualitatively,
including the fact that there are very few particles with semi-major axis $a$
larger than 0.4 AU. The particles inside $r_i$ tend to have mild eccentricity
$0 \le e \le 0.4$, whereas planetesimals outside $r_i$ have higher values of
$e$ on average. The orbital inclination $i$ also increases with $a$, reaching
$10^\circ$ to $20^\circ$ outside $r_i$ in simulation U15, and slightly higher
values in U10. Note that a larger mean value of $i$ in U10 compared to that in
U15 found here for the bound population is consistent with a similar trend
that we found for the unbound population (cf. figures
\ref{fig:EB015_orbit_LIVE} and \ref{fig:OFF00EB010VY_RESUM_LIVE}).
Figure \ref{fig:U0_sat} shows the orbital parameters of the bound population
of planetesimals in simulation U0. There is a marked difference in these
distributions compared with that for U15 and U10. Instead of a monotonic
increase in eccentricities and inclinations with increasing $a$, evident in
Figure \ref{fig:U15_U10_sat}, here there is an almost a step-function change
in the nature of the orbits. Orbits at $a \lower.5ex\hbox{\ltsima} 0.3$ have small eccentricity,
$0 \le e \le 0.2$, whereas orbits at $a \lower.5ex\hbox{\gtsima} 0.3$ have a large spread in
eccentricities, with some approaching unity. The planetesimals with the
largest values of $e$ may become unbound later since their orbits are
comparable to $r_h$. There is also a group of particles with large
inclinations, $i\lower.5ex\hbox{\gtsima} 90^\circ$ at $a \approx 0.35$ AU.
The significant differences in the orbits of the bound populations of
planetesimals between Figures \ref{fig:U15_U10_sat} and \ref{fig:U0_sat}
demonstrate that the exact way in which the innermost region of the host clump
is disrupted influences the orbits of the ``satellites'' remaining after the
disruption. By extension it also means that the results would also be somewhat
different if there was a massive gas atmosphere around the solid
protoplanetary core.
\begin{figure*}
\centerline{
\psfig{file=OFF00EB015VY_7945_10P_sat_orbit.pdf,width=0.49\textwidth,angle=0}
\psfig{file=OFF00EB010VY_RESUM_5846_10P_sat_orbit.pdf,width=0.49\textwidth,angle=0}}
\caption{The bound population of planetesimals around the solid core
(planet). The orbital characteristic are calculated with respect to the
planet rather than the star. Left: Simulation U15, Right:
simulation U10.}
\label{fig:U15_U10_sat}
\end{figure*}
\begin{figure}
\centerline{
\psfig{file=BIG_681_10P_sat_orbit.pdf,width=0.49\textwidth,angle=0}}
\caption{Same as Fig. \ref{fig:U15_U10_sat} but for simulation U0.}
\label{fig:U0_sat}
\end{figure}
\section{Discussion}\label{sec:discussion}
\subsection{General conclusions}\label{sec:general}
In this paper we considered the predictions of the TD hypothesis for the
``planetesimal'' debris left over after the disruption of a single gas host
clump. These gas clumps are the parent bodies within which all (or at least
most) of planet formation action takes place in the TD hypothesis. The most
important results of our paper can be summarised as following:
1. In the context of TD hypothesis, large solids form only inside the massive
gas host clumps \citep[because the densities of solids initially reflect that
of gas, and gas densities are orders of magnitude higher inside the host
clumps than in the ``ambient disc''; see,
e.g.,][]{Nayakshin10b,ChaNayakshin11a} that are then disrupted. The solid
debris is then spread around in a ring around the disruption location. TD
hypothesis thus generically predicts rings of planetesimals remaining
after planet formation rather than continuous discs.
2. The role of planetesimals in the formation of planets may be very
different in the CA and TD scenarios. Whereas planets could not have formed
without planetesimals forming first in the CA picture, in the TD hypothesis
this is less clear. In the latter case, once the initial proto solid core
is born by gravitational collapse of the grain-dominated inner region of the
host clump\citep{Nayakshin10a,Nayakshin10b}, further growth may be dominated
by the proto solid core accreting $\sim$ cm to tens of cm grains. Solids in
this size range experience aerodynamic drag that is strong enough to dump
possible centrifugal support and yet small enough to sediment to the centre of
the host clump within its lifetime (see Fig. 1). Bodies in the planetesimal
size range, i.e., km-sized and larger, on the other hand, experience too weak
an aerodynamic drag. They are not expected to join the solid core en masse if
they have some orbital support due to an excess angular momentum or chaotic
convective motions of the gas in the centre of the host clump. In this case
planetesimals are no parent bodies of planets in the strict sense.
On the other hand, if turbulence, convection or angular momentum prevents
small grains from reaching the proto solid core directly, turning most of
these grains into planetesimals as discussed in \S \ref{sec:birth}, then the
situation is less clear. If this happens sufficiently close to the proto
solid core and planetesimal densities are high, most of that material can be
subsequently accreted by the proto core somewhat like in the CA theory. In
this case planetesimals are also building blocks of planets, with only those
on larger less bound orbits managing to avoid being accreted onto the core.
3. Related to the point above, the solid planetary cores and the planetesimal
debris form at approximately same time in our picture. After host gas clump
disruption, some of the solid debris around the planetary core could fall on
it and be accreted in direct collisions, so that planetary growth could
actually continue in a manner analogous to that of the later oligarchic
stages of solid core assembly in the traditional scenario
\citep[e.g.,][]{Safronov72}. In particular, as velocity dispersion of the
disrupted population is high, e.g. a fraction of a km~s$^{-1}$, only large
planetesimals of size $\lower.5ex\hbox{\gtsima} 1000$ km would continue their growth, whereas
smaller objects would accrete on large bodies or fragment in collisions with
smaller bodies. If they fragment to sizes as small as a few metres then
their inward radial migration inside the gas disc becomes important.
4. Also related to point (2) above, the mass budget of planetesimals can be
potentially very different in the two planet formation theories. In the CA
scenario, the initial population of planetesimals must have had a total mass
larger than the total high-Z element (other than H and He) mass of the
resulting planets. This does not {\em have to be} the case in the TD
scenario. In particular, there does not seem anything wrong with assuming that
the debris population may be far less massive than the protoplanetary core.
5. In the CA scenario, the properties of the protoplanetary disc are expected
to be a rather smoothly varying function of radius, except for the regions
near the ice line \citep[e.g.,][]{Armitage10}. Thus one expects that
planetesimal properties are also a smoothly varying function of $R$. In
contrast, in the TD hypothesis, it is the host clump that determines the
outcome of the planet formation process. The internal structure of the clump
is a strong function of its mass \citep{Nayakshin10a,Nayakshin10b}; that structure is
non-uniquely coupled to radius $R$ where the clump is disrupted. It is
possible that clumps of different history, mass, internal structure are
disrupted at roughly same location. Further, we envisage a number of host
clumps forming and being destroyed during the early gas-rich phase as required
to explain the FU Ori outbursts of young protostars
\citep{NayakshinLodato11}. This implies a far greater diversity in the
properties of the solid debris at the same spatial location than is possible
in the CA model.
6. We found in this paper that disruption of a host gas clump generically
produces a ``V''-shaped pattern in the eccentricity-semi-major axis space for
the planetesimal population released by the disruption. The planetesimals in
the centre of the pattern have nearly circular orbits while whose at the edges
of it are much more eccentric. The maximum eccentricities possible can be
estimated based on the simple analytical theory; from equations \ref{ecc_a} to
\ref{w_predict}, it follows that, within a factor of $\sim 2$, it is $e_{\rm
max} \sim r_h/R \approx 0.2$ for host clump mass of $\sim 10$ Jupiter
masses.
7. The eccentricities and inclinations of the disrupted planetesimal
population are generally much too high to allow planetesimals to stick to one
another. The typical dispersion velocity of the disrupted population is
$\delta v \sim v_K r_h/R \sim 0.1 v_K$. At the distance of the Kuiper belt,
for example, this yields dispersion velocity of the order of $300$ m/s. Only
Pluto-sized bodies could survive {\em equal-size} collisions at these
velocities. Accretion on massive solid cores, however, is allowed as already
noted in (3) above, so the post-disruption evolution may be somewhat similar
to the final phases of the run-away growth of planetary embryos in the
traditional scenario \citep[e.g.,][]{Safronov72}.
8. We also find that solid bodies on orbits tightly bound to the planetary
core could survive the disruption of the gas host clump, remaining bound to
the planet. These bodies are satellites of the planetary cores.
Satellites on orbits more strongly bound to the planet are affected by the
gas envelope destruction less than those on less bound orbits. Some of the
outermost satellites may even find themselves in eccentric counter-rotating
orbits (e.g., see fig. \ref{fig:U15_U10_sat}). This is qualitatively similar
to the satellites of the giant planets in the Solar System, although we note
that our simulations are not designed to study these issues; one also would
need to model composition, size differences and long-term survivability of
the bound objects.
\subsection{Potential relevance to the Solar System}\label{sec:ss}
Even though we did not specifically attempt to reproduce the structure of the
outer Solar System here (which would require an additional study of how the
system evolves on the 4.5 Giga year time scale), our simulations do have
potentially interesting implications for it. We mention these implications
only briefly here, with the intent of quantifying them in a future
publication:
\begin{itemize}
\item One of the key properties of the trans-Neptunian region of the Solar
System, including the Kuiper belt, is the ``mass deficit problem''. The
current mass of the Kuiper belt is estimated at $\sim 0.01 - 0.1 \mearth$
\citep{BernsteinEtal04}. On the other hand, to grow the observed populations
of solid bodies in the context of the \cite{Safronov72} model for formation
of solids, 10 to 100 $\mearth$ of solids is required
\citep[e.g.,][]{KL99}. Thus, removal of $\lower.5ex\hbox{\gtsima} 99.9$\% of solid material is
required. In the TD hypothesis, solids are made within the central fractions
of AU of the host gas clumps \citep{Nayakshin10a}. The host clumps contain
as much as $\sim 60\mearth$ of high-Z elements \citep{Nayakshin10b} which
are then partially used to make the massive solid cores, the
``planetesimals'', and partially remain bound to gas in small grains. If it
is possible to build solid cores as massive as $\sim 10 \mearth$ by direct
gravitational collapse and accretion of $\sim$cm size grains
\citep{Nayakshin10b}, it should also be possible to build Pluto-sized
objects {\em without} having to put tens of $\mearth$ of material into the
planetesimals. There is thus no mass deficit problem for the Kuiper belt in
the TD scenario.
\item There is a sharp outer edge to the Kuiper belt at around $R\approx 50$
AU which is currently not well understood \citep[e.g.,][]{Morbidelli08}. The
disruption of a gas host clump naturally produces a ring with sharp outer
and inner edges (point 1 above) because the range of orbits available to
planetesimals after disruption is limited by the parameters of the initial
host clump. This could naturally account for the outer edge of the Kuiper
belt objects.
\item The structure of the classical Kuiper belt is best explained by our
model if we assume that the left hand side (lower $a$) of the disruption
ring (e.g., Fig. \ref{fig:BIG_ANA}) has been destroyed by interactions with
planets. Realistically, the planet, left at $a\sim 35-39$ AU at the end of
our simulations, could have continued to migrate inward via type I migration
in the gas disc: the disrupted gas ring is still quite massive (5 Jupiter
masses for our simulations here, and there would be more at larger $R$ to
effect the migration of the host clump in the first place). Using the
standard type-I gas migration \citep[e.g.,][]{Tanaka02} for Neptune
initially located at $\approx 40$ AU, assuming the disc mass being $\sim 10
$ Jupiter masses, and the gas disc aspect ratio of $H/R \sim 0.1$ yields
type I migration rate time scale of $\sim 10^6$ yrs. This is sufficiently
fast to allow Neptune to migrate inward significantly to end up, for
example, in the configuration proposed by the NICE model of the outer Solar
System \citep{GomesEtal05}. While migrating inward in the gas disc, Neptune
would have scattered the left part of the ``V''-pattern of the
planetesimals, but leaving behind the right hand side of the pattern as a
belt reminiscent of the Kuiper belt.
\item Kuiper belt contains the hot and the cold populations. These are best
accounted for by two different gas clump disruptions in our model. The
hot population in our model would have most naturally resulted if the host
gas protoplanet rotation axis were highly inclined to the orbital plane. In
this case one could end up with higher inclinations than we obtained here.
\item \cite{NesvornyYR10} show that if planetesimals form by a local
gravitational collapse in a high density environment, then one can naturally
explain the surprisingly large fraction of binaries in the $\sim $100 km
class low-inclination objects in the classical Kuiper Belt. We note that our
model produces ``planetesimals'' in a similar fashion (gravitational
collapse in a high density environment) albeit inside of the host clumps
rather than the disc. Therefore by extension one might expect to see a high
fraction of massive objects locked in binaries after the dissipation of the
host clump. This idea however needs to be checked with a longer time
$N$-body calculation as there are non trivial physical constraints on
collisional destruction of binaries in the Kuiper Belt
\citep{NesvornyEtal11}.
\end{itemize}
\section{Conclusions}
We have considered the origin of solid debris, such as comets, asteroids and
large bodies such as Pluto in the context of a recent planet formation
hypothesis (Tidal Downsizing). We assumed (see \S \ref{sec:birth}) that
the solid ``debris'' is formed in a way very similar to the massive solid
protoplanetary cores themselves, e.g., inside the $\sim 10$ Jupiter mass host
gas clouds. The latter are eventually disrupted either tidally or due to an
internal energy release during the solid core accretion. The release of these
solid bodies into the field forms rings potentially reminiscent of the Kuiper
belt and the debris discs around nearby main sequence stars. While much work
remains to be done to detail predictions of the TD hypothesis further, it is
already clear that these predictions are sufficiently different from the
standard planetesimal-based paradigm for planet formation \citep{Safronov72}
to be critically tested by observations in the near future.
As an astrophysical aside, we note that the rapid accretion of large solid
bodies (Pluto-like and even Neptune-like) in the TD scheme suggests that
planet and solid debris formation is a very robust process and may even occur
in very crowded environments such as the inner parsecs of galaxies
\citep{NayakshinSS12,ZubovasNM12}; this is an untenable proposition in the CA
model.
\section*{Acknowledgments}
Theoretical Astrophysics research in Leicester is supported by an STFC rolling
grant. This research used the ALICE High Performance Computing Facility at the
University of Leicester. Some resources on ALICE form part of the DiRAC
Facility jointly funded by STFC and the Large Facilities Capital Fund of
BIS. We acknowledge constructive report by the anonymous referee whose
suggestions helped to improve the clarity of this paper significantly.
|
1,314,259,993,037 | arxiv | \section{Introduction}
\subsection{}
The Vorono\"i summation formula is an equality between a weighted sum of Fourier coefficients of an automorphic form twisted by an additive character, and a dual weighted sum of Fourier coefficients of the dual form twisted by a Kloosterman sum. The Vorono\"i formula for $\GL_2$ is a basic tool in the study of automorphic forms, while more general applications have followed with the more general formulas for $\GL_N$ proved by Goldfeld-Li \cite{GL} and Miller-Schmid \cite{MS} over $\QQ$, and Ichino-Templier \cite{IT} over any number field $F$ which, importantly, removes any ramification assumptions in the previous cases.
A balanced formula on $\GL_N$ was first obtained by Zhou \cite{Zhou} under certain restrictions, then later in general by Miller-Zhou \cite{MZ}, in which the lengths of the hyper-Kloosterman sums on either side of the formula can be chosen in a `balanced' manner. That is, the dimensions of the hyper-Kloosterman sums on either side of the summation formula can be taken to be $M'$ and $N'$ respectively, where $M'+L'+2 = N$.
This was applied in the recent work of Blomer-Li-Miller \cite{BLM} to prove a spectral reciprocity law via a so-called `Kuznetsov-Vorono\"i-Kuznetsov triad' for a spectral sum of automorphic $L$-functions on $\GL_4\times \GL_2$ as follows: a Kuznetsov trace formula on $\GL_2$ is applied, and then the balanced Vorono\"i formula for $\GL_4$ is used on the geometric side, and the Kuznetsov formula is applied again to the dual geometric side, to give a dual spectral sum. As an application, the authors prove a non-vanishing result for automorphic $L$-functions on $\GL_4\times \GL_2$. A modified version has also been developed in Blomer-Khan \cite{Bkhan}, and is used to bound moments of twisted $L$-functions on $\GL_4$. The mechanics of the spectral reciprocity law suggest that a general formula may exist for $\GL_{2N}\times \GL_{N}$. Unfortunately, even for $N=3$ one finds that the Kuznetsov formula involves Kloosterman sums of varying lengths, which prevents a direct application of the balanced Vorono\"i formula as in the $N=2$ case.
\subsection{}
In this paper, we generalise the balanced Vorono\"i to a general number field. Besides allowing for extensions of the results on spectral reciprocity laws to number fields in special cases, another feature of our work is that rather than Kloosterman sums, more general Kloosterman integrals appear on either side of the balanced formula, which allows for the possibility of a more flexible relative trace formula, which involves Kloosterman integrals, to be used in place of the Kuznetsov trace formula.
A second motivation for our study comes from a somewhat different source. Recent developments with regards to the conjectures of Braverman and Khazdan \cite{BK1} such as \cite{N1, N3, BNS} developing geometric methods to generalise the theory of Godement and Jacquet \cite{GJ}, which proves the functional equation of standard automorphic $L$-functions on $\GL_N$ using Poisson summation. In particular, \cite{N3} proposes a construction of the conjectural $\rho$-Fourier transform $\mathscr F^\rho$, which generalizes the Hankel transform that occurs in the Vorono\"i formula for $\GL_2$. The existence of balanced Vorono\"i formulas then suggests that the $\rho$-Poisson summation formula of the form
\[
\sum_{\gamma\in G(F)} \phi(\gamma) = \sum_{\gamma\in G(F)}\mathscr F^\rho(\phi)(\gamma),
\]
where $\phi$ belongs to a certain $\rho$-Schwartz space $\mathscr S^\rho(G(\A_F))$, can be again `balanced' in a similar manner, and it would be interesting to explore potential applications to the analytic theory of $L$-functions.
\subsection{Main result} Our method essentially follows that of Miller-Zhou, where instead of starting with the Vorono\"i formula of Miller-Schmid \cite{MS} over $\QQ$ we use the more general formula of Ichino-Templier \cite{IT}, and avoid the use of multiple Dirichlet series. The key observation is that the proof of the balanced Vorono\"i formula reduces to the usual Vorono\"i formula through a series of character sums, parallel to the repeated use of the crucial identity \cite[Lemma 3.2]{MZ}.
Let $N+2=L+M$. Let $T$ be the maximal torus of diagonal matrices in GL$_N$, and $T^L, T^M$ disjoint sub-tori of dimensions $L-1$ and $M-1$ respectively, so that $T \simeq T^L\times T^M$. The case $L=1$ then reduces to the ordinary Vorono\"i summation formula. Then referring to Section \ref{defs} below for the definitions and notations, our balanced Vorono\"i formula is as follows.
\begin{thm}
\label{balancedvoronoi}
Let $F$ be a number field. Let $\pi=\otimes_v\pi_v$ be an irreducible cuspidal automorphic representation of $GL_N(\A_F)$, and let $S$ be the set of places of $F$ over which $\pi_v$ is ramified. For any $\zeta\in \A^S_F$ and $\omega_S \in C_c^\infty(F_S^\times)$, we have
\begin{align}
\label{bvformula}
&\sum_{t\in T^M_\zeta/T^M_\circ} \sum_{\gamma\in F^\times} Kl_M(\gamma\zeta,t) W_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}\Big) w_{S}(\gamma)\\
&=
\sum_{s\in T^L_\zeta/T^L_\circ} c(\zeta,s) \det(s) \sum_{\gamma\in F^\times} Kl_L(\gamma\zeta^{-1},s) \tilde{W}_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}a(s)\Big) \tilde{w}_{S}(\gamma),\notag
\end{align}
where $c(\zeta,s)$ is defined in \eqref{const}
\end{thm}
\noindent In principle, one should be able to apply this formula to obtain generalizations spectral reciprocity formulae, for example of \cite[Theorem 3]{BLM} in the case $N=2$ to totally real number fields, using the relevant Kuznetsov formula of Bruggeman and Miatello or relative trace formula. We will discuss this briefly at the end in Remark \ref{blm}.
\begin{rem}
We briefly describe how the notation in \cite[Theorem 1.1]{MZ} can be compared to ours. First, for the same $N$, their parameters are chosen such that $M'+L'+2 = N$. Our choice of $M+L-2=N$ here differs from theirs due to the convention on hyper-Kloosterman sums used in \cite{IT}. Their balanced Vorono\"i formula takes the form:
\begin{align*}
&\sum_{\bf D|Q} D^{M}_1\dots D_M^1 \sum_{n=1}^\infty Kl_M(\bar a,n,c;{\bf Q}, {\bf D}) A({\bf q},{\bf D}, n)\omega\Big(\frac{nD_1^{M+1}\dots D_M^2}{q_1^L\dots q_L^1}\Big) \\
&= \sum_{\bf d|q}\frac{d_1^L\dots d_L^1}{c^{L+1}} \sum_{n=1}^\infty\sum_{\epsilon = \pm} Kl_L(a,\epsilon n,c; {\bf q},{\bf d}) \tilde{A}({\bf Q},{\bf d},n)\Omega\Big(\frac{(-1)^M\epsilon n d_1^{L+1}\dots d_L^2}{c^NQ_1^M\dots Q_M^1}\Big).
\end{align*}
Letting $F=\QQ$, we specialise $\psi(x_v)$ to be $e^{-2\pi i x_\infty}$ for $x_\infty\in \RR$, and $e^{2 \pi i x_p}$ for $x_p\in {\QQ}_p$. Our $\zeta$ corresponds to $\frac{\bar a}{c}$, and the set of places $S$ are the prime divisors of $c$. Our $\gamma\in F^\times$ correspond to the arguments of $\omega$ and $\Omega$ above. Our $t \in T^M_\zeta/T^M_\circ$ corresponds to their sequence of positive integers $d_1,\dots,d_M$, where up to units we have $(t_2,\dots,t_{M-2},t_{M-1})$ equal to $\frac1 c(d_1d_2\dots d_{M}, \dots, d_1d_2, d_1)$, and similarly $s \in T^L_\zeta/T^L_\circ$ corresponds to $D_1,\dots, D_L$. Their hyper-Kloosterman sum $Kl_N(a,n,c;{\bf q,d})$ corresponds to $Kl_N(\gamma\zeta^{-1},t)$ as outlined in \cite[p.72]{IT}. Finally, the Fourier coefficients $A$ correspond to $W_{\circ f}$ and $\tilde W_{\circ f}$ up to normalization as in \eqref{FC}, and our functions $\omega,\tilde\omega$ correspond to their $\omega,\Omega$ respectively, though their test function $\omega$ is compactly supported on $(0,\infty)$.
\end{rem}
\section{The Vorono\"i formula of Ichino-Templier}
\label{defs}
\subsection{}
Let $F$ be a number field, and $\A=\A_F$ the ring of adeles. Also let $F_v$ be a completion of $F$ at a prime $v$, with ring of integers $\oo_v$. Fix a non-trivial additive character $\psi=\otimes_v\psi_v$ of $F\backslash \A$. Let $\pi=\otimes_v\pi_v$ be an irreducible cuspidal automorphic representation of $GL_N(\A_F)$, $n\ge2$, and let $S$ be the set of places of $F$ over which $\pi_v$ is ramified. Let $\A^S$ be the adeles with trivial component above $S$. Define the unramified Whittaker function of $\pi^S$ to be $W_\circ^S = \prod_{v\not\in S}W_{\circ v}$, and similarly for the contragredient representation $\tilde{\pi}^S$ we write $\tilde{W}_{\circ}^S$, where
\[
\tilde{W}_{\circ}^S(g) = W_{\circ}^S(w^tg^{-1})
\]
for all $g\in GL_N(\A^S)$, and $w$ is the long Weyl element of GL$_N$. Over $\QQ$, they are related to the Fourier coefficients $A(m_1,m_2,\dots,m_{N-1})$ of $\pi$ by the following relation:
\be
\label{FC}
\prod_{p<\infty} W_p(\Delta_m)= \prod_{i=1}^{N-1}|m_i|^{-i(n-i)/2}A(m_1,m_2,\dots,m_{N-1}),
\ee
where
\[
\Delta_m= \text{diag}(m_1\dots m_{N-1}, m_2\dots m_{N-1},\dots, m_{N-1},1)
\]
is a diagonal matrix in $GL_N(\QQ)$.
\subsubsection{Measures}
Throughout, we make the following choices of measures. The measure $dx_v$ on the local field $F_v$ is chosen to be self-dual with respect to the fixed additive character $\psi_v$. Fix a non-zero differential form $\omega$ in Hom$_F(\wedge^\text{top}\text{Lie}(U),F)$ and also for $Y$, so that $\omega_v$ and $dx_v$ determine a measure on Lie$(U)(F_v)$, hence an invariant measure on $U(F_v)$. The product of these measures gives the Tamagawa measure.
\subsubsection{Generalised Bessel transforms}
Define for each $w_v \in C_c^\infty(F_v^\times)$ a dual function $\tilde{\omega}_v$ such that
\begin{align*}
&\int_{F^\times_v}\tilde{\omega}_v(y) \chi(y)^{-1}|y|^{s-\frac{N-1}{2}} dy \notag\\
&= \chi(-1)^{N-1}\gamma(1-s,\pi_v\times\chi,\psi_v)\int_{F^\times_v} w_v(y)\chi(y)|y|^{1-s-\frac{N-1}{2}}dy
\end{align*}
for all Re$(s)$ large enough and any unitary character $\chi$ of $F^\times_v$. This defines $\tilde{\omega}_v$ uniquely in terms of $\pi_v,\psi_v,$ and $\omega_v$, independent of the choice of Haar measure $dy$. Note that $\tilde{\omega}_v(x)$ is smooth of rapid decay, but not necessarily compactly supported, as $|x|\to\infty$, which is important for the convergence of the dual sum.
\subsubsection{Kloosterman integrals}
Define for any $\gamma_v,\zeta_v \in F^\times_v$, the hyper-Koosterman integral,
\[
K_v(\gamma_v,\zeta_v,\tilde{W}_{\circ v}) :=
|\zeta_v|^{N-2} \int_{U_\tau^-(F_v)}\overline{\psi}_v(u_{N-2,N-1})\tilde{W}_{\circ v}(\tau u) du
\]
where
\[
\tau = \begin{pmatrix}&1& \\ 1_{N-2}&&\\ &&1 \end{pmatrix} \begin{pmatrix}1_{N-2}&&\\ &-\gamma_v\zeta_v^{-1}&\\ &&-\zeta\end{pmatrix},
\]
and set
\[
K_R(\gamma,\zeta,\tilde{W}_{\circ R})=\prod_{v\in R}K_v(\gamma_v,\zeta_v,\tilde{W}_{\circ v})
\]
for $\gamma,\zeta \in \A^\times_R$. It relates to hyper-Kloosterman sums as follows. Let $T$ be the maximal torus of diagonal matrices in GL$_N$, then
\[
K_v(\gamma_v,\zeta_v,\tilde{W}_{\circ v})=
|\zeta_v|^{N-2} \sum_{T(F)^+/T(\oo_v)} \tilde{W}(t)Kl_N(\gamma_v\zeta_v^{-1},t)
\]
where the sum is taken over elements $t=(t_1,\dots,t_N)$ in $T(F_v)^+/T(\oo_v)$ such that
\[
1\leq |t_2|\leq \dots \leq |t_N|=|\zeta_v|, \text{ and }|t_1t_2\dots t_{N-1}|=|\zeta_v|.
\]
Here $Kl_N(\gamma\zeta^{-1},t)$ is the hyper-Kloosterman sum of dimension $N-1$ and can be expressed as
\[
\sum_{v_{N-1}\in t_{N-1}\oo^\times_v/\oo_v} \cdots \sum_{v_{2}\in t_2\oo^\times_v/\oo_v}
\psi(v_{N-1}+\dots + v_2)\psi((-1)^n\gamma\zeta_v^{-1}v_2^{-1}\dots v^{-1}_{N-1})
\]
by \cite[Corollary 6.7]{IT}.
\subsubsection{Vorono\"i formula}
We can now state the main result of Ichino and Templier \cite[Theorem 1]{IT}, which will be the basis for our balanced Vorono\"i formula.
\begin{thm}[Ichino-Templier]
\label{IchinoTemplier}
Let $\zeta\in \A^S_F$, and $R$ the set of places $v$ such that $|\zeta_v|>1$. Then with notation as above, we have
\begin{align*}
&\sum_{\gamma \in F^\times} \psi(\gamma\zeta)W_\circ^S\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}\Big) w_S(\gamma)\\
&=
\sum_{\gamma \in F^\times}K_R(\gamma, \zeta,\tilde{W}_{\circ R})\tilde{W}_\circ^{R\cup S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}\Big) \tilde{w}_S(\gamma),
\end{align*}
for any $\omega_S \in C_c^\infty(F_S^\times)$.
\end{thm}
\noindent From the preceding discussion, we can expand the right-hand side along the maximal torus $T$ to obtain an expression in terms of Kloosterman sums
\[
\sum_{t\in T_\zeta/T_\circ} \sum_{\gamma\in F^\times} Kl_N(\gamma\zeta^{-1},t) \tilde{W}_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}a(t)\Big) \tilde{w}_S(\gamma),
\]
here $T_\zeta$ denotes the set of $(t_2,\dots,t_{N-1})$ in $F_R^{N-2}$ such that
\[
1\leq |t_{2}|_v\leq \dots \leq |t_{N-1}|_v \leq |\zeta|_v
\]
for all $v\in R$. Here $T_\circ = (\oo^\times_R)^{N-2}$ and $a(t)$ is the diagonal matrix $(t_1,\dots,t_N)$ in $T(\A_R)/T(\oo_R)$ uniquely completed such that $|t_{N}|_v = |\zeta|_v$ and $|t_1\cdots t_N|_v=1$ for all $v\in R$. Taking $F=\QQ$, and $\pi$ to be unramified at every finite prime, this recovers the main result of \cite{MS} (see \cite[Theorem 2]{IT}).
\section{Proof of Theorem \ref{balancedvoronoi}}
We are now ready to prove a balanced Vorono\"i formula over an arbitrary number field, which specialises to Theorem \ref{IchinoTemplier} at $M=0$. First, we open up the hyper-Kloosterman sum on the left-hand side of \eqref{bvformula}, and then bring in the $\gamma$ sum,
\begin{align}
\sum_{t\in T^M_\zeta/T^M_\circ} \sum_{\substack{v_{M-1}\in t_{M-1}\oo^\times_R/\oo_R\\ \dots \\ v_{2}\in t_{2}\oo^\times_R/\oo_R}}& \psi(v_{M-1}+\dots + v_2) \ \times \notag\\
&\sum_{\gamma\in F^\times} \psi((-1)^M\gamma\zeta^{-1}v_2^{-1}\dots v^{-1}_{M-1})W_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}\Big) w_{S}(\gamma).
\end{align}
Note that interchanging the summation is justified by the compact support of the test function $\omega_S$. Then applying the Vorono\"i summation of Theorem \ref{IchinoTemplier} to the inner sum, we obtain the dual expression
\begin{align}
\label{innervoronoi}
\sum_{t\in T^M_\zeta/T^M_\circ} \sum_{\substack{v_{M-1}\in t_{M-1}\oo^\times_R/\oo_R\\ \dots \\ v_{2}\in t_{2}\oo^\times_R/\oo_R}}& \psi(v_{M-1}+\dots + v_2) \ \times \notag\\
&\sum_{s\in T_\zeta/T_\circ} \sum_{\gamma\in F^\times} Kl_N(\gamma\breve\zeta^{-1},s) \tilde{W}_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}a(s)\Big) \tilde{w}_S(\gamma),
\end{align}
where we have denoted $\breve\zeta:=(-1)^M\zeta^{-1} v_2^{-1}\dots v_{M-1}^{-1}$. Recall that here $T$ is the maximal split torus in $G$, so that relabelling indices if necessary, we may decompose any $s\in T_\zeta/T_\circ$ into $s=s_1s_2$ where
\begin{align*}
s_1&=(t_1,\dots, t_{L-1}) \in T^L_\zeta/T^L_\circ, \\
s_2&=(t_L,\dots, t_{N-1}) \in T^M_\zeta/T^M_\circ,
\end{align*}
such that
\begin{align*}
&1\leq |t_{1}|_v\leq \dots \leq |t_{L-1}|_v \leq |\breve\zeta|_v,\\
& 1\leq |t_{L}|_v\leq \dots \leq |t_{N-1}|_v \leq |\breve\zeta|_v
\end{align*}
for all $v\in R$. Note that $s_2$ is an $(M-2)$-tuple.
Now on the dual side, opening up the $(N-1)$-dimensional hyper-Kloosterman sum along $t_2$, down to $(L-1)$ dimension, we have
\[
Kl_N(\gamma\breve\zeta^{-1},s_1s_2)=
\sum_{u_{N-1}\in t_{N-1}\oo^\times_R/\oo_R} \cdots \sum_{u_L\in t_L\oo^\times_R/\oo_R}
\psi(u_{N-1}+\dots + u_L)Kl_L(\gamma\zeta^{-1}_L,s_1)
\]
where
\[
\zeta_L = (-1)^{N-L}\breve\zeta u_{L}\dots u_{N-1} = \zeta^{-1} v_2^{-1}\dots v_{M-1}^{-1}u_{L}\dots u_{N-1}.
\]
Only the innermost sum over $\gamma$ is infinite, so we may rearrange the order of summation by pairing the $v_{M-1}$ sum with the $u_L$ sum, the $v_{M-2}$ sum with the $u_{L+1}$ sum, and so on. Separating the $s_1$ and $s_2$ sums, we write \eqref{innervoronoi} as
\begin{align}
\label{pairs}
\sum_{t\in T^M_\zeta/T^M_\circ} &\sum_{\substack{v_{M-1}\in t_{M-1}\oo^\times_R/\oo_R\\ \dots \\ v_{2}\in t_{2}\oo^\times_R/\oo_R}}
\sum_{s_2\in T^M_\zeta/T^M_\circ} \sum_{\substack{u_{N-1}\in t_{N-1}\oo^\times_R/\oo_R\\ \dots \\ u_{L}\in t_{L}\oo^\times_R/\oo_R}}
\psi(v_{M-1}+u_{L})\dots \psi(v_{2}+u_{N-1}) \notag\\
& \times\sum_{s_1\in T^L_\zeta/T^L_\circ} \sum_{\gamma\in F^\times} Kl_L(\gamma\zeta^{-1}_L,s_1) \tilde{W}_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}a(s_1)\Big) \tilde{w}_S(\gamma),
\end{align}
where the third sum is over $s_2=(t_L,\dots,t_{N-1})$ as above. It remains then to evaluate the first line, noting that it is independent of the second line except for $\zeta_L$. To treat the first four sums, we separate also the sum on $t$ in $T^M_\zeta/T^M_\circ$ into its $(M-2)$ components $(t_2,\dots,t_{M-1})$ such that
$1\le |t_{1}|_v \le \dots \le |t_{M-1}|_v\le |\zeta|_v$
for all $v\in R$. We will omit the subscript on $|\cdot|_v$ when the context is clear. So the first two sums of \eqref{pairs} then reads for every fixed $s_2, u_L, u_{L+1},\dots, u_{N-1}$ as follows:
\be
\label{pair}
\sum_{\substack{ t_{M-1}\in F_R \\ |t_{M-1}|\le |\zeta_R|}} \sum_{v_{M-1}\in t_{M-1}\oo^\times_R/\oo_R} \psi(v_{M-1}+u_{L})
\dots
\sum_{\substack{ t_{2}\in F_R \\ |t_{2}|\le |t_3|}} \sum_{v_{2}\in t_{2}\oo^\times_R/\oo_R} \psi(v_{2}+u_{N-1}).
\ee
Consider then the first pair. We observe that for each fixed $s_2\in T^M_\zeta/T^M_\circ$ and $u_{L}\in t_{L}\oo^\times_R/\oo_R$, the sum:
\[
\label{lemma}
\sum_{ |t_{M-1}|\le |\zeta_R|}
\sum_{v_{M-1}\in t_{M-1}\oo^\times_R/\oo_R}
\psi(v_{M-1}+u_{L}) \\
\]
is nonzero only if $t_{M-1} = t_L, u_L \equiv - v_{M-1}\text{ (mod } \oo_R)$.
To see this, simply observe that
\[
\sum_{v_{M-1} \in t_{M-1} \oo_R^\times / \oo_R } \psi(v_{M-1}+u_L) = |t_{M-1}|_R
\]
if $t_{M-1} = t_L$ and $u_L \equiv - v_{M-1}$ modulo $\oo_R$, and is zero otherwise. We note that this is the analogue of Lemma 3.2 of \cite{MZ}. This implies that $ v_{M-1}^{-1}u_{L}\equiv-1$ mod $\oo_R$ in $\zeta_L$. Moving on to the second pair, for fixed $t_{M-1}$ and $u_{L+1}$,
\[
\sum_{ |t_{M-2}|\le |t_{M-1}|} \sum_{v_{M-2}\in t_{M-2}\oo^\times_R/\oo_R} \psi(v_{M-2}+u_{L+1})
\]
we see that the sum is again nonzero only if $t_{M-2} = t_{L+1}$ and $u_{L+1}\equiv - v_{M-2}$ mod $\oo_R$, and zero otherwise. Applying this $M-2$ times, we collect the evaluated sums \eqref{pairs} into a constant equal to the product of
\[
|t_{M-1} \dots t_2|_R = \det(s_1)
\]
and the cardinality
\be
\label{const}
c(\zeta,s_1)= \#\{t\in T^M_\zeta/T^M_\circ: 1\le |t_{1}|_v \le \dots \le |t_{M-1}|_v\le |\zeta|_v, v\in R\}.
\ee
And finally the sum reduces to
\[
\sum_{s_1\in T^L_\zeta/T^L_\circ} c(\zeta,s_1) \det(s_1)\sum_{\gamma\in F^\times} Kl_L(\gamma\zeta_L^{-1},s_1) \tilde{W}_\circ^{S}\Big( \begin{pmatrix}\gamma & \\ & 1_{N-1}\end{pmatrix}a(s_1)\Big) \tilde{w}_{S}(\gamma)
\]
as desired.
\begin{rem}
\label{blm}
The spectral reciprocity formula obtained in \cite{BLM} corresponds to the case $N=2$ and $L=M=3$. Our balanced formula in Theorem \ref{bvformula} then consists of Kloosterman sums on either side, attached to tori $T^L$ and $T^M$ in GL$_3$ over $F$. Fixing a cuspidal automorphic representation $\Pi$ on GL$_4(\A)$ and a suitable test function $h$, we consider the spectral mean value roughly of the form
\[
\sum_{\pi}\frac{L(\frac12,\Pi\times \pi)}{L(1,\pi, \text{Sym}^2)}h(t_\pi) +\frac{1}{2\pi}\int_{-\infty}^\infty\frac{L(\frac12+it,\Pi)L(\frac12-it,\Pi)}{|\zeta(1+2it)|^2}h(t) dt
\]
where $\pi$ runs over cuspidal automorphic representations of GL$_2(\A)$. (In \cite{BLM} the sum is twisted by the parity of the eigenvalue of $\pi_\infty$ under the reflection $z\mapsto -\bar{z}$.) We expand the $L$-functions as usual to obtain
\[
L(s,\Pi\times \pi) = \sum_{n,m>1}\frac{a_\Pi(n,m)a_\pi(n)}{n^sm^{2s}}
\]
and similarly for $L(s+it,\Pi)L(s-it,\Pi)$, and applying known convexity the original expression is seen to be absolutely convergent.
To apply the so-called Kuznetsov-Voronoi-Kuznetsov triad over $F$, one may use either the relative or Kuznetsov trace formula. In the former case, one has Kloosterman integrals as in \cite[\S3.3]{KL} roughly of the form
\[
I_\delta(h) = \int_{H_\delta (F)\backslash H(\A)} h(n_1^{-1}\delta n_2)\overline{\theta_{m_1}(n_1)}\theta_{m_2}(n_2) dn_1dn_2
\]
where $dn_1$ and $dn_2$ are quotient measures obtained from $H(\A) = N(\A) \times N(\A)$, $H_\delta$ is the stabilizer of $\delta\in N(F) \backslash \mathrm{GL}_2(F)/N(F)Z(F)$, and $\theta_{m_1}$ and $\theta_{m_1}$ are additive characters on $N(F)\backslash N(\A)$. (See also \cite[\S7]{KL2}.)Up to a constant, this is equal to
\[
\sum_{c\in N{\bf Z}^+}\frac{S(m_1,m_2;\mathfrak n,c)}{c}J_{k-1}\left(\frac{4\pi \sqrt{\mathfrak n m_1m_2}}{c}\right).
\]
If we decompose the adelic integral into
\[
I_\delta(h) = I_{\delta_S}(h) I_{\delta^S}(h),
\]
then we see that $I_{\delta_S}(h)$, which specializes to $J_{k-1}$, corresponds to our generalized Bessel transform $\tilde{\omega}_S(\gamma)$, whereas $I_{\delta_S}(h)$ which specializes to the Kloosterman sum corresponds to the Kloosterman integral $Kl_3(\gamma\zeta,t)$. To execute the spectral reciprocity will require more careful estimates as in \cite{BLM}, and a proper generalization of the parity condition which allows the Kuznetsov formula to be inverted as Motohashi's work is over ${\bf Q}$ \cite[Theorems 2.3 and 2.5]{Mo}.
\end{rem}
\noindent\emph{Acknowledgments.} The author thanks Giacomo Cherubini for helpful discussions and comments on a preliminary version of this paper, and the referee for comments improving the content and exposition of the paper.
\begin{comment}
\section{Braverman-Khazdan}
\subsection{} Fix $T$ a split torus of $G^\lambda$. Let $M_{T,\rho}$ be the monoid defined by $\rho|_{\hat{T}}$, or by the strictly convex cone generated by the weights $\mu_1,\dots\mu_r$ of $\rho$ in Hom$(\G_m,T)\otimes\RR$. Consider the projection of toric varieties $p_\rho:\A^r\to M_{T,\rho}$ given by
\be
p_\rho(x_1,\dots,x_r)=\mu_1(x_1)\dots\mu_r(x_r).
\ee
This gives a map from $\G_m\to T$ by the open embeddings of the units of $\A_r$ and $M_{T,\rho}$. Let $S$ be its kernel. The $\rho$-Schwartz space $\mathscr{S}_\rho(T(F))$ is defined to be the coinvariants of $\mathscr{S}(F^r)$ under the action of $S(F)$, which is given by $(sf)(x):=f(s^{-1}x)$. Then the $\rho$-Fourier transform, or generalised Hankel transform is formally defined by $\mathscr{J}_\rho(p_{\rho!}f)=p_{\rho!}\mathscr{F}(f)$ where the pushforward $p_{\rho!}:\mathscr{S}(F^r)\to\mathscr{S}_\rho(T(F))$is defined by integration along fibers
\be
p_{\rho!}(f)(x)=\int_{p_{\rho!}^{-1}(x)}f(y)dy = \int_{S(F)}f(y_0s)ds
\ee
where $x=p_\rho(y_0)$ and $ds$ is the Haar measure on $S(F)$. Here $\mathscr F$ is the usual Euclidean Fourier transform.
\end{comment}
\bibliographystyle{alpha}
|
1,314,259,993,038 | arxiv | \section{Introduction}
\label{sec:intro}
In real-world point cloud generation applications, the controllability to generated shape is a fundamental requirement.
Structures are the ideal intermidiate representation to control the generation as they are the brief abstraction of 3D objects and have advantages of conciseness and intuitiveness.
Therefore, structure-based controllable point cloud generation is attracting rising research interests.
Different structure representations are proposed by previous works.
Some works model objects at the part level.
Mitra et al. \cite{Mitra2014} proposed structured objects which contain part-level connectivity information and inter-part relationship information.
The information can be naturally formed as a part tree \cite{Mo_2019_CVPR} named N-ary part hierarchies.
Mo et al. \cite{mo2019structurenet} proposed the StructureNet to generate point clouds via hierarchical graph networks, and later they proposed PT2PC \cite{mo2020pt2pc} to generate shapes directly from the part tree.
Unlike above methods, Yang et al. \cite{Yang_Wu_Zhang_Jin_2021} used a simple sparse point cloud to represent structures and introduced point-level semantic labels into structure extraction.
They proposed CPCGAN to generate a 3D shape from a structure point cloud by breeding 64 points from each structure point.
Previous works have achieved great performances in controllable point cloud generation from structures.
But there are two major limitations in their works:
1) Lack of control in details.
Part-tree-based methods cannot control the generation inside a part.
CPCGAN represents the structure by only 32 points, leading to the lack of small-scale details.
Controllabilities of works above are mainly available on large-scale structures, while the generation of details is learned by neural networks which cannot be controlled.
2) Lack of generalization ability.
Some kinds of category-specific semantic information have been introduced into generation by above works, which limits the ability to generate in more categories.
Besides, absolute positions of points are used in their generation processes, causing those methods unable to work well when structures are rotated, scaled, or translated.
To tackle the challenge of controlling details, we propose a novel representation of structures.
It can be observed that definitions of details and structures are subjective.
Details have no differences with structures other than the spatial scale.
With this assumption, details can be modeled in the same way as the structures.
Besides, different parts of an object require structure representations in different scales, i.e. delicate parts require relatively dense representations while sparse representations are enough for rough parts.
This means an ideal structure representation needs to be spatially uneven.
However, most previous methods model structures in a spatially even manner.
Thus we propose a graph-based representation that can model different scales of structures in one graph, namely Multiscale Structure Graph (MSG).
As shown in Fig.~\ref{fig:ExampleofPatterns}(a), the MSG represents structures continuously in scale by connecting vertices with different capacities, which are the numbers of points they represent.
To model large-scale structures, fewer vertices with larger capacities are enough.
And for small-scale details, more vertices with smaller capacities will better model the local shape.
Without considering physical or chemical influences, all 3D objects and parts can be scaled to any size without losing its rationality.
Complex real-world 3D objects are often assembled by multiple simple shapes in different scales.
Thus, as shown in Fig.~\ref{fig:ExampleofPatterns}(b), plenty of similar patterns in local structures can be found in 3D objects at different positions, scales, angles, densities, and categories.
The knowledge learned from a pattern can be transferred to other similar patterns no matter the differences of those spatial properties.
Previous methods cannot associate those similar patterns as they use absolute positions of points which are sensitive to scaling and rotation.
In this paper, we propose a novel Multiscale Structure-based Point Cloud Generator (MSPCG) to generate point clouds from local structures in a similarity-transformation-invariant (invariant to scale,rotation and translation) manner.
The invariances make MSPCG is able to jointly learn similar patterns in different scales, which largely extend the scope of training data.
As local structures are category-irrelevant and even dataset-irrelevant, learning the generation from local structures also helps the proposed method has better generalization ability.
The main contributions of our work are as follows:
\begin{enumerate}
\item A concise and intuitive graph representation of structure called Multiscale Structure Graph (MSG) aiming to model a 3D object in different scales simultaneously. It enables the controlling to details during generation.
\item A graph encoding and point cloud generation method called Multiscale Structure-based Point Cloud Generator (MSPCG). It generates point clouds from local structures and strongly improves the generalization ability towards scaling and rotation.
\item A framework to simultaneously learn from different scales, positions, and angles. It extends the scope of training data and helps the MSPCG to perform competitively in point cloud reconstructions compared to SOTA methods with complete point clouds as input.
\item The combination of above contributions has great generalization capabilities crossing categories and datasets. Trained on ShapeNet, our method is able to generate point clouds from given structures for unseen categories, objects from ModelNet, and indoor scenes from ScanNet.
\end{enumerate}
\section{Related works}
\label{sec:related works}
After various researches have been done in 1D signal and 2D image\cite{goodfellow2020generative,kingma2018glow,kingma2013auto,oord2016conditional,prenger2019waveglow,shen2018natural}, methods of using a neural network to generate 3D point clouds have been explored in recent years.
Achlioptas et al. \cite{achlioptas2017learning} firstly adopted simple fully connected layers as the generator to encode and generate point clouds.
Valsesia et al. \cite{valsesia2018learning} used dynamic graph convolution networks to enhance the generation performance.
Shu et al. \cite{shu20193d} proposed TreeGAN, which generates point clouds from a tree-based network.
Yang et al. \cite{yang2018foldingnet} proposed FoldingNet deforming 2D squares into 3D surfaces to generate point clouds.
AtlasNet\cite{groueix2018} followed the idea of FoldingNet and further expanded the deforming operation into multiple branches.
TearingNet\cite{pang2021tearingnet} researched on modeling shapes with more complex topology.
PointFlow \cite{pointflow} modeled the point cloud generation as a distribution transformation by introducing free-form normalizing flows\cite{chen2018neural,grathwohl2019ffjord}.
Above methods aim to generate point clouds from random latent codes.
For more controllable and detailed generation, some researchers focused on generating point clouds from different kinds of intermediate representations.
Some works \cite{Mitra2014,mo2020pt2pc,Mo_2019_CVPR} controllably generate point clouds from part-level structure representations.
But the part-level structure representation is expensive to annotate.
Yang et al. \cite{Yang_Wu_Zhang_Jin_2021} constructed structure point clouds using the point-level semantic annotations.
The CPCGAN they presented is also controllable and is able to generate semantic labels for points.
A popular way to explicitly construct rotation-insensitive neural networks is using capsule networks\cite{sabour2017dynamic}.
Researchers adopted capsule networks in 3D point cloud reconstruction task\cite{sun2021canonical,zhao20193d}.
Unlike capsule networks, which force the network to estimate the poses while learning embeddings of objects, our method explicitly extracts pose information.
Apart from the invariances of rotation, our method is also able to generate reasonable shapes when inputs are scaled.
\section{Approach}
\label{sec:methods}
\subsection{Overview}
The overview of our MSPCG is shown in Fig.~\ref{fig:Network}.
The Multiscale Structure Graph (MSG) is presented as $ G_{MS}(V,E) $, where each vertex $V_i$ contains a $C_i$ standing for the capacity and an $L_i$ meaning the location coordinates.
Given a $ G_{MS} $ with $K$ vertices, we aim to generate a dense 3D point cloud $P_{ge} = \{g_j\}_{j=1}^{N}$ where $g_j \in \mathbb{R}^3$ and $N = \sum_{i=1}^{K} C_i$ is the number of points we want to generate.
In training, each $ G_{MS} $ has a corresponding ground truth point cloud $P_{gt} = \{gt_j\}_{j=1}^{N}$.
The goal of MSPCG is to generate $P_{ge}$ closer to $P_{gt}$.
There are mainly four parts in the MSPCG.
1) The graph extractor samples a $G_{MS}$ from the $P_{gt}$.
2) The rotation and scale invariant sorter (RS-Inv sorter) takes a $G_{MS}$ as input and then extracts multiple properties on each vertex.
3) The latent space sampler first encodes the $G_{MS}$ in a similarity-transformation-invariant manner via a modified graph attention network.
Then an expanding operation is performed to sample $N$ features where each feature represents one point in the final point cloud.
4) The RS-Inv generator projects the features from latent space into 3D space, getting the $P_{ge}$ finally by generating a bunch of points around each vertex in MSG.
Besides, to balance the influences caused by the different capacities of vertices, a weighted Chamfer Distance is proposed as the loss function of our MSPCG.
\begin{figure}[t]
\centering
\includegraphics[height = 6cm]{pics//Model.pdf}
\caption{Overview framework of proposed MSPCG. K is the number of vertices and N is the number of points we finally generate. $ ^* $Relative Capacity ratios are calculated in 3-ring neighbors because the depth of GAT is 3.}
\label{fig:Network}
\end{figure}
\subsection{Multiscale Structure Graph}
To model details of 3D objects in different scales, we propose a graph-based structure representation $ G_{MS} $ as shown in Fig.~\ref{fig:ExampleofPatterns}(a).
Each vertex $V_i$ is an abstraction of a point cluster in the original point cloud.
$L_i \in \mathbb{R}^3$ represents the center of the cluster and the capacity $ C_i \in \mathbb{N}^+$ is the size of the cluster.
Edges $E$ between vertices represent the connectivities of the clusters in 3D space.
By this definition, we treat structures in large scale as under-described details and treat the low-level details as over-described structures.
\subsection{Graph Extractor}
To enhance the generalization ability, graphs containing multiscale structures are required in training.
Yang et al. \cite{Yang_Wu_Zhang_Jin_2021} utilize the K-Means algorithm to get point clusters and then extract a structure point from every cluster.
Following this idea, we propose a simple and efficient algorithm called Mixed-Precision Random K-Means to extract point clusters within different sizes.
Pseudo-code of the algorithm is given in Algorithm ~\ref{alg1}.
The gravity center of cluster $i$ will be the $L_i$ in $G_{MS}$, and the $C_i$ is assigned by the number of points in the cluster $i$.
Vertices are connected by an edge if the spatial distance between the clusters they represent is closer than a threshold.
With the mixed-precision K-means, we are able to automatically extract MSGs that contain structures in various scales.
And the randomness in the algorithm also improves the robustness of MSPCG to different graph distributions.
\begin{algorithm}
\caption{Mixed-Precision Random K-Means}
\label{alg1}
\begin{algorithmic}
\STATE \textbf{Input:} $P_{gt} = \{gt_j\}_{j=1}^{N}\ where\ gt_j \in \mathbb{R}^3$
\STATE $N_c \gets RandInt(4,16),\ N_f \gets RandInt(64,128)$
\STATE $\mathit{Centroids}_{coarse} \gets KMeans(P_{gt}, N_c),\ \mathit{Centroids}_{fine} \gets KMeans(P_{gt}, N_f)$
\STATE $N_{f2c} \gets RandInt(12,32)$
\STATE $\mathit{Centroids}_{f2c} \gets RandChoice(\mathit{Centroids}_{fine}, N_{f2c})$
\STATE $\mathit{Clusters} \gets GetClustersFromCentroids(P_{gt}, \mathit{Centroids}_{f2c})$
\STATE \textbf{Output:} $\mathit{Clusters}$
\end{algorithmic}
\end{algorithm}
\subsection{RS-Inv Sorter}
To encode the graph in a similarity-transformation-invariant manner, a rotation matrix $R_i$ and a scale factor $\mathit{SF}_i$ are needed for every vertex $V_i$ in the graph.
As the capacity of each vertex is also scale-related, directly feeding capacities into the network will also damage the scale-invariances of the model.
Thus we extract another property called relative capacity ratio $\mathit{RCr}_i$ for each vertex $ V_i$ in the RS-Inv sorter.
Assume $EC_i = \{E_j\},\ E_j = (V_i, V_n), V_n \in \mathcal{N}_i$ is the set of edges connected to $V_i$.
We define the edge vector $\overrightarrow {E_j} = L_n - L_i$ for $E_j$ where $\overrightarrow {E_j} \in \mathbb{R}^3$.
Without loss of generality, we suppose the two longest non-collinear edge vectors in $EC_i$ are $E_1$ and $E_2$.
$R_i$ is a rotation matrix that meets the following conditions:
\begin{equation}
\begin{aligned}
&R_{i} \times \frac{\overrightarrow {E_1}}{|\overrightarrow {E_1}|} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}^T\ &if\ |EC_i| > 0 \\
and\ &R_{i} \times \overrightarrow {E_2} \times \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} = 0 \ &if\ |EC_i| > 1 \\
and\ &R_{i} \times \overrightarrow {E_2} \times \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} > 0 \ &if\ |EC_i| > 1
\end{aligned}
\end{equation}
If $|EC_i| = 0$, the $R_i$ is an identity matrix.
The scale factor for $V_i$ is calculated as:
\begin{equation}
\mathit{SF}_i = \frac{\sum_{E_j \in EC_i}|\overrightarrow {E_j}|}{|EC_i|} \ \ \ \ \ if\ |EC_i| > 0
\end{equation}
if $|EC_i| = 0$, $\mathit{SF}_i$ is the average length of all edges.
The relative capacity ratio for $V_i$ comes from:
\begin{equation}
\mathit{RCr}_i = \frac{C_i}{\frac{1}{|N(3)_i|}\sum_{V_j \in N(3)_i}C_j}
\end{equation}
Where $N(3)_i$ is the set of vertices whose graph distance to $V_i$ is smaller than 3.
\subsection{Latent Space Sampler}
With a $G_{MS}$ as input, the latent space sampler aims to encode the graph with $K$ vertices and sample $N$ features in latent space.
Graph encoders have been widely researched in recent years\cite{perozzi2014deepwalk,bruna2013spectral,hamilton2017inductive,velivckovic2017graph}.
The Graph Attention Network\cite{velivckovic2017graph}(GAT) is one of the most popular message-passing-based graph neural networks.
GAT can efficiently aggregate features from neighbors and avoid the over-smooth problem.
Thus we choose GAT as the backbone of our graph encoder.
To avoid the usage of absolute positions, we propose a relative graph attention layer (relative GAT) as the first layer of the graph encoder.
The attention weight $\alpha$ between $V_s$ and $V_t$ is expressed as:
\begin{equation}
\alpha_{st} = \frac{e^{\mathit{LR}(\overrightarrow {a}^T[R_s\times \frac{L_t - L_s}{SF_s} \parallel RCr_s \parallel RCr_t])}}{\sum_{k \in \mathcal{N}_s}e^{\mathit{LR}(\overrightarrow {a}^T[R_s\times \frac{L_k - L_s}{SF_s} \parallel RCr_s \parallel RCr_k])}}
\end{equation}
where $\overrightarrow {a}\in \mathbb{R}^5$ is the weight vector, $\parallel$ is the concatenation operation, and $\mathit{LR}$ stands for LeakyReLU nonlinearity.
The feature vectors $F_{st}$ for $V_t$ that will be aggregated by the attention weights of $V_s$ is expressed as:
\begin{equation}
F_{st} = [R_s\times \frac{L_t - L_s}{SF_s} \parallel RCr_s \parallel RCr_t]
\end{equation}
The introduction of the relative capacity ratio helps the network to recognize the differences between vertices representing different scales.
At the same time, the relative capacity ratio only considers vertices in a certain range, which is also robust to local density changes.
After relative GAT, two layers of ordinary GAT are followed to further encode the graph.
The encoder outputs noise variances $\mathit{NV}_i \in \mathbb{R} ^{c}$ and encoded features $F_i \in \mathbb{R}^{c}$ for each vertex $V_i$, where $c$ stands for the number of feature channels.
To generate a point cloud for $N = \sum_{i=1}^{K} C_i$ points, each vertex $V_i$ needs to breed $C_i$ points.
Most multi-branch generation methods in previous works can only generate the same number of points in every branch\cite{groueix2018,mo2020pt2pc,Yang_Wu_Zhang_Jin_2021}.
FoldingNet\cite{yang2018foldingnet} proposed a sample-transform method, which can generate a shape in different number of points.
With the inspiration FoldingNet gave to us, we propose a method to sample different numbers of features for different vertices.
A binary expanding matrix $\mathit{EP} \in \mathbb{N}^{N*K}$ is presented:
\begin{equation}
\mathit{EP}_{ij} = \left\{
\begin{array}{rcl}
1 & & i \ge \sum_{k = 0}^{j-1}C_k\ and\ i < \sum_{k=0}^j C_k\\
0 & & otherwise\\
\end{array}
\right.
\end{equation}
By pre-multiplying $EP$ to the features $F \in \mathbb{R}^{K \times c}$, we can expand the $K$ features into $N$ features $F \in \mathbb{R}^{N \times c}$, where features from vertex $V_i$ will be repeated $C_i$ times.
Pre-multiplying other properties matrices (e.g. $R,\mathit{SF},L$) by $EP$ can also expand other properties in the same way.
Similar to FoldingNet's sampling results, each feature is concatenated by a unique noise and a common global feature.
Points expanded from $V_i$ share the global feature.
Noises for these points are sampled by a reparameterization module from a gaussian distribution whose variances are $\mathit{NV}_i$.
\subsection{RS-Inv Generator}
The RS-Inv generator firstly uses a shared MLP to get offsets $O \in \mathbb{R}^{N \times 3}$ by projecting features and sampled noise into 3D space.
The generated point cloud $P_{ge} = \{g_j\}_{j=1}^{N}, g_j \in \mathbb{R}^{3}$ can be expressed as:
\begin{equation}
g_j = R_j^{-1} \times O_j \cdot \mathit{SF}_j + L_j
\end{equation}
\subsection{Weighted Chamfer Distance Loss}
To learn in multiple scales simultaneously, each vertex should get the same attention no matter which scale it locates.
But obviously, the vertices with bigger capacities will get more attention as the Chamfer Distance\cite{fan2017point} treats equally to every point rather than to every vertex.
Thus, we propose the weighted Chamfer Distance $\mathcal{L}_{wCD}$ as the loss function to rebalance the influences of different capacities.
The weighted Chamfer Distance $\mathcal{L}_{wCD}$ between $P_{ge} = \{g_i\}_{i=1}^{N}$ and $P_{gt} = \{gt_i\}_{i=1}^{N}$ is defined as:
\begin{equation}
\begin{aligned}
\mathcal{L}_{wCD}(P_{ge}, P_{gt}) &= \frac{1}{|P_{gt}|} \sum_{gt_j\in P_{gt}} \mathop{min}_{g_i \in P_{ge}}\parallel gt_j - g_i\parallel_2\\
&+ \frac{1}{K} \sum_{g_i\in P_{ge}} \mathop{min}_{gt_j \in P_{gt}}(\frac{\parallel gt_j - g_i\parallel_2}{C_i})
\end{aligned}
\end{equation}
As each point in $P_{ge}$ may not only be matched once, weighting them may cause the summation of weights not equal to 1.
Therefore the first term in $\mathcal{L}_{wCD}$ is not balanced by the capacity.
Thus weighted Chamfer Distance can only alleviate the problem but still cannot solve it completely.
\subsection{Invariance for similarity transformation and Learning from Multiple Scales}
Similarity transformation includes translation, rotation, and scaling.
MSPCG gets the invariances for similarity transformation by
1) Giving the neural network a consistent input after a similarity transformation is adopted to the MSG.
2) Applying inverse transformations to the output of the neural network to match the ground truth for calculating losses.
From the perspective of one vertex in the relative GAT, the information from its neighbors is the rotated, normalized relative positions, and the relative capacity ratio.
Vertices will get different information from the same neighboring vertex.
RS-Inv sorter extracts a unique similarity transformation for each vertex, which will transform the vertex and its neighbors to a canonical pose.
Therefore, the input of the attention module will be consistent no matter what similarity transformation is applied to the input MSG.
As the transformation for a vertex is explicitly calculated, the RS-Inv generator directly applies a inverse transformation to the offsets generated from this vertex.
After the relative GAT wipes similarity transformation, other GATs will only focus on the topological and feature-wise relationship between vertices.
Thus, information from different scales can be learned simultaneously by the shared network in our method.
\begin{table}[]
\begin{tabular}{c|c|cccccc}
\toprule[2pt]
Methods & \begin{tabular}[c]{@{}c@{}}Input\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Seen\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen +\\ R-rotate\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen +\\ R-scale\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen +\\ R-rotate +\\R-scale\end{tabular} & ModelNet \\ \midrule[1pt]
\begin{tabular}[c]{@{}c@{}}AtlasNet\\ V2\cite{deprelle2019learning}\end{tabular} & \multirow{4}{*}[-2em]{\begin{tabular}[c]{@{}c@{}}Point\\ Cloud\end{tabular}} & 6.107 & 6.988 & 7.096 & 7.602 & 7.714 & 2049 \\ \cline{1-1} \cline{3-8}
\begin{tabular}[c]{@{}c@{}}TearingNet\\ \cite{pang2021tearingnet}\end{tabular} & & 6.924 & 6.767 & 6.819 & 7.322 & 7.397 & 36.25 \\ \cline{1-1} \cline{3-8}
\begin{tabular}[c]{@{}c@{}}3D Point\\ Capsules\cite{zhao20193d}\end{tabular} & & 6.947 & 7.626 & 7.718 & 8.317 & 8.464 & 88.72 \\ \cline{1-1} \cline{3-8}
\begin{tabular}[c]{@{}c@{}}Canonical\\ Capsules\cite{sun2021canonical}\end{tabular} & & \textbf{4.386} & \textbf{5.307} & \textbf{5.317} & \textbf{5.729} & \textbf{5.713} & 46.50 \\ \midrule[1pt]
\begin{tabular}[c]{@{}c@{}}MSPCG\\ w/o rot\&scale\end{tabular} & \multirow{2}{*}{MSG} & 6.354 & 7.281 & 7.263 & 8.037 & 7.993 & 49.30 \\ \cline{1-1} \cline{3-8}
MSPCG(ours) & & 5.177 & 6.323 & 6.312 & 6.757 & 6.781 & \textbf{27.06} \\ \bottomrule[2pt]
\end{tabular}
\caption{Average reconstruction performance in Chamfer Distance, multiplied by $10^4$. "R-" represents random.}
\label{table:CompareToSOTA}
\end{table}
\section{Experiments}
\label{sec:experiments}
\subsection{Dataset and Implementation Details}
We train our MSPCG on the ShapeNetCore-v2-PC15K\cite{shapenet2015,pointflow} dataset.
In training, we down-sample the 15,000 points into 2048 points using the farthest points sampling method\cite{qi2017pointnetplusplus} to be the $P_{gt}$.
Categories with more than 300 samples in the training split of ShapeNet are collected to be the training set.
Thus some categories are unavailable for the model in the training period.
We sample 5 MSGs for every point cloud to overcome the randomization of Mixed-Precision Random K-Means.
On implementation of the MSPCG, Adam optimizer\cite{kingma2014adam} is used with $\alpha = 1e-4,\ \beta_1 = 0$ and $\beta_2 = 0.99$.
The model is trained for 90 epochs parallelly on 3 GPUs with batch size set to 32 on each card.
The bottleneck for increasing batch size is the inference time.
As each input data may have a different number of vertices, the time complexity of multiplying expanding matrix to features is $\mathcal{O} ((\sum K)\times(\sum N)\times C)$ where C is the number of feature channels.
Increasing batch size will cause the inference time to squarely increase.
As there is no relationship between different data samples, we split the big batch to different processes on different GPUs to better use the compute resources.
Each process with a batch size of 32 will cost about 5 GB of GPU memory in training.
\subsection{Comparisons in Point Clouds Reconstruction}
As the MSPCG generates point clouds from given structures, it is unfair to compare MSPCG with point cloud generation methods that generate from random latent codes.
MSPCG can be regarded as a reconstruction model that reconstructs complete point clouds from spare structures.
Therefore, we compare our MSPCG mainly on point cloud reconstruction task.
Six testing sets of ShapeNet are constructed in experiments.
The average Chamfer Distances between the generated point clouds and the ground truths are reported.
All the testing sets come from the official testing split of ShapeNet.
Without any other declaration, the input MSG in those testing sets are extracted by a K-Means algorithm with K randomly chosen from 16 to 64.
The ShapeNet Seen includes the categories that are used in the training period, and the ShapeNet Unseen only contains the other categories.
The suffix of "32 vertices" means the K in K-Means algorithm is restricted to 32.
The R-rotate stands for randomly rotating each object and the R-scale stands for randomly scaling each object with a ratio between 0.8 and 1.25.
\begin{figure}[t]
\centering
\includegraphics[width = 12cm]{pics/Scale-CD_ver1.png}
\caption{Reconstruction Chamfer Distances in ShapeNet Unseen when all point clouds are scaled in a certain ratio.}
\label{fig:Scale-CD}
\end{figure}
We first compared our MSPCG to some state-of-the-art point cloud reconstruction methods in Tab.~\ref{table:CompareToSOTA}.
Augmented data (randomly rotated and scaled) are used to train those methods.
It can be found that although the input of MSPCG contains less information, our method can provide competitive or even better reconstruction results.
We believe that it can be attributed to the strong representation ability of MSG and the simultaneous learning of local point patterns at multiple scales.
When scale changes more, it is more difficult to reconstruct a reasonable shape.
Fig.~\ref{fig:Scale-CD} shows the reconstruction Chamfer Distances when all point clouds are scaled at some specific ratios.
We re-sampled 2048 points from the original 15,000 points for all point clouds and reported the Chamfer Distances between re-sampled point clouds and ground truth point clouds as the lower bound of the distance.
As revealed by the figure, most methods other than MSPCG cannot provide consistent reconstruction performance when the inputs are zoomed.
The Chamfer Distance growth trend of MSPCG is the same as the re-sampled point clouds, which illustrates the MSPCG is insensitive to scale changes.
To verify the cross-generalization ability, we adopt those models trained on ShapeNet into the reconstruction task on ModelNet.
The testing set of ModelNet-40 is used to compare the methods, and all objects are randomly rotated.
The experimental results show that the proposed MSPCG has much better cross-generalization ability than other methods.
\begin{figure}[t]
\centering
\includegraphics[width = 12cm]{pics//ShowResults2.pdf}
\caption{Reconstruction results on ShapeNet and ModelNet. Note that our MSPCG takes structure graphs as input and other methods use point clouds.}
\label{fig:ShowResults}
\end{figure}
Besides quantitative comparisons, we also qualitatively compared our method with SOTA methods.
Some reconstruction results are shown in Fig.~\ref{fig:ShowResults}.
It can be found that other methods tend to reconstruct the large-scale shapes of the input and lack the ability to generate sharp details.
When testing on a different dataset, the gaps are more significant.
MSPCG can maintain the performance which other methods can hardly produce visually similar outputs.
The distributions of point coordinates in ModelNet are far different from those in ShapeNet.
Neural networks without special designs for scale changes have less knowledge about these new distributions as they have never seen them.
With the help of similarity-transformation-invariant designs and generating point clouds from local structures, MSPCG can associate the new distributions with the similar distributions in ShapeNet and outputs reasonable results.
To investigate the influences of each part in MSPCG, ablation studies have been down in Tab.~\ref{table:Ablation}.
Graph Interpolation is a baseline method that evenly samples points on the edges of MSG.
Graph + Gaussian method samples a Gaussian distribution around each vertex in MSG with the variance of its scale factor.
FC-based method uses the decoder of CPCGAN\cite{Yang_Wu_Zhang_Jin_2021}, which transforms each vertex's feature into 64 offsets.
As the decoder of CPCGAN cannot generate points with different numbers for different vertices, the FC-based method can only take the MSG with 32 vertices as input.
To get an overall view of the advantages brought by the proposed framework, augmented data are used to train an MSPCG without any rotation or scale invariant module.
\begin{table}[t]
\small
\begin{center}
\begin{tabular}{@{}cccccccc@{}}
\toprule[2pt]
& \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Seen\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen\\ 32 vertices\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen +\\ R-rotate\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen + \\ R-scale\end{tabular} & \begin{tabular}[c]{@{}c@{}}ShapeNet\\ Unseen + \\ R-rotate \\R-scale\end{tabular} & ModelNet \\\midrule[1pt]
\begin{tabular}[c]{@{}c@{}}Graph \\ Interp.\end{tabular} &14.45 &17.07 &18.23 &17.07 &18.39 &18.36 &72.51 \\
\hline
\begin{tabular}[c]{@{}c@{}}Graph \\ + Gaussian\end{tabular} &8.926 &11.78 &12.39 &11.82 &12.72 &12.73 &41.15 \\
\hline
FC-based & - & - &\bf{6.365} & - & - & - & - \\
\hline
\begin{tabular}[c]{@{}c@{}}MSPCG \\ w/o R\&S \\ w/ aug.\end{tabular} &6.354 &7.281 &7.250 &7.263 &8.037 &7.993 &49.30 \\
\hline
\begin{tabular}[c]{@{}c@{}}MSPCG \\ w/o scale\end{tabular} &5.790 &6.854 &6.850 &6.850 &7.563 &7.574 &47.73 \\
\hline
\begin{tabular}[c]{@{}c@{}}MSPCG \\ w/o rotate\end{tabular} &\bf{4.818} &\bf{6.180} &6.443 &7.996 &\bf{6.618} &8.719 &35.31 \\
\hline
\begin{tabular}[c]{@{}c@{}}MSPCG \\ w/o wCD\end{tabular} &5.201 &6.322 &6.582 &6.315 &6.771 &6.795 &27.10 \\
\hline
MSPCG &5.177 &6.323 &6.571 &\bf{6.312} &6.757 &\bf{6.781} &\bf{27.06} \\
\bottomrule[2pt]
\end{tabular}
\end{center}
\caption{Average reconstruction performance in Chamfer Distance, multiplied by $10^4$. Note that our MSPCG achieves best results in the last two columns, which have the most challenging settings.}
\label{table:Ablation}
\end{table}
From Tab.~\ref{table:Ablation}, we can find that:
1) The learning-based methods have significantly better performance.
2) The FC-based method performs slightly better than MSPCG in the 32 vertices' test set, which is because the FC-based method only focuses on one scale in both training and testing.
The proposed MSPCG also has a competitive performance in the 32 vertices' test set and have better generalization ability.
3) Adopting rotation-invariant and scale-invariant mechanisms is much more efficient than training on augmented data.
4) Removing scale factors causes a performance decline even when objects are not scaled.
It proves the positive influences of jointly learning the structures in different scales.
5) Without the rotation-invariant module the MSPCG can generate better in non-rotated cases, which shows the rotation-invariant mechanism of our MSPCG is not perfect enough.
Rotating edges to a canonical pose will cause a huge information loss.
The full MSPCG performs considerably better when the objects are rotated and have competitive performances in non-rotated cases.
Thus for generalization consideration, we still add the rotation-invariant mechanism.
6) The weighted Chamfer Distance will slightly help the performance.
\subsection{Multiscale Editing}
\begin{figure}[t]
\centering
\includegraphics[width = 12cm]{pics//Edit_3.pdf}
\caption{Examples of multiscale editing.}
\label{fig:Edit}
\end{figure}
By adding or deleting vertices with different capacities in MSG, the MSPCG can control the generation in multiple scales.
Fig.~\ref{fig:Edit}(a) shows some examples of multiscale editing.
Adding one vertex above the barrel built a sight on a relatively small scale, and deleting a bunch of vertices modified the point cloud on a large scale, e.g. removing the magazine.
Fig.~\ref{fig:Edit}(b) shows how the reconstruction Chamfer Distances changes when the number of vertices increases on another object.
The reconstruction Chamfer Distance will gradually decrease when more vertices are added to the graph, and the generated shape will be closer to the ground truth.
The Chamfer Distances between ground truth and $N$ points sampled from ground truth by FPS method are also shown in Fig.~\ref{fig:Edit}(b) as $\mathit{CD}\ \mathit{FPS-N}$.
With vertices less than the sampled points, the point cloud generated by MSPCG can have more information than the down-sampled point clouds.
\subsection{Scene Generation}
The MSPCG can be easily adopted to larger structure graphs, such as MSGs for indoor scenes.
A scene generated by MSPCG is shown in Fig.~\ref{fig:SceneGen}.
It is noticeable that the model used in this experiment is the same as the one above, which is trained on ShapeNet.
The structures are extracted within each instance to show the differences between even structure and uneven structure.
To extract uneven structure, each instance will produce an MSG with the number of vertices equal to 1/16 the number of points.
If an instance has more than 1024 points, it will produce an MSG with only 64 vertices.
Objects with more points will be represented more sparse in this setting.
The even structure is extracted with a consistent down-sampling ratio in all instances.
The down-sampling ratio is specially set to make the even structure has a similar number of vertices with the uneven structure.
Both even and uneven structure have about 2400 vertices, and the generated point clouds have about 210k points.
It can be observed that details generated from the uneven structure are sharper, cleaner, and closer to the ground truth.
The results show that the proposed MSG can be considered as a possible intermediate representation for scene generation in the future.
\begin{figure}[t]
\centering
\includegraphics[height = 4.7cm]{pics//SceneGen.png}
\caption{Generation performances of MSPCG on ScanNet dataset. The colors are copied from the closest points in ground truth.}
\label{fig:SceneGen}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we focused on controlling the generation of 3D details.
A Multiscale Structure Graph(MSG) is proposed to represent multiscale structures in a continuous way, and a Multiscale Structure-based Point Cloud Generator(MSPCG) is presented to generate high-quality point clouds from MSG.
Various experiments demonstrate that the special designs of MSG and MSPCG help the proposed methods to jointly learn the structures in different scales, perform competitive in point cloud reconstruction and achieve great generalization ability.
\clearpage
\bibliographystyle{splncs04}
|
1,314,259,993,039 | arxiv | \section{Introduction}
\label{s:intro}
The formation and evolution of the Milky Way is one of the outstanding questions facing astrophysics today.
The study of stars and stellar populations is a pillar of Galactic Archaeology \citep{2002ARA&A..40..487F}, as they contain the chemical imprint of the gas from which they formed and serve as birth tags.
This allows stars to be used as a fossil record to unravel the history of the Milky Way.
During the past decade, traditional fiber-fed spectrographs employed by most large-scale spectroscopic surveys like RAVE \citep{2006AJ....132.1645S}, LAMOST \citep{2012RAA....12..735D, 2012RAA....12..723Z}, Gaia-ESO \citep{2012Msngr.147...25G}, GALAH \citep{2015MNRAS.449.2604D}, and APOGEE \citep{2017AJ....154...94M}
combined with astrometric and photometric information from Gaia \citep{2018A&A...616A...1G, 2021A&A...649A...1G} provided detailed chemodynamics for millions of stars from the solar neighborhood to even several kilo-parsecs away \citep{2016ARA&A..54..529B}.
The large volume of spectroscopic observations is posing great challenges for data analysis, particularly in deriving stellar labels (atmospheric parameters and chemical abundances) precisely and efficiently from spectra.
In the past few years, with the use of radiation transfer codes (e.g. SME \citealt{1996A&AS..118..595V}, Turbospectrum \citealt{1998A&A...330.1109A, 2012ascl.soft05004P}, MOOG \citealt{2012ascl.soft02009S}), many model-driven (e.g. \emph{The Payne}{} \citealt{2019ApJ...879...69T} and \emph{StarNet} \citealt{2018MNRAS.475.2978F, 2020MNRAS.498.3817B}) and data-driven methods (e.g. \emph{The Cannon}{} \citealt{2015ApJ...808...16N} and \emph{Astro-NN}{} \citealt{2019MNRAS.483.3255L}) and models in between (e.g. \emph{DD-Payne}{} \citealt{2019ApJS..245...34X} and \emph{Cycle-StarNet} \citealt{2021ApJ...906..130O}) have been developed for this purpose.
One benefit of the data-driven approach is it can generate a model from a training set sample, rather than traditional physics-based approaches using theoretical stellar atmospheres; this
is much easier to do and in general leads to more precise stellar parameters as it can potentially make use of the full wavelength region of the spectra.
However, one limitation of the data-driven methods is that some chemical abundances could be determined by astrophysical correlations in the training set, instead of physically motivated measurements from spectral features.
This has a non-negligible impact on the precision of some stellar labels, especially those derived from low-resolution spectra, where many of the absorption features are blended.
To overcome this issue, \citealt{2017ApJ...849L...9T, 2018ApJ...860..159T} used the methods outlined in \emph{The Payne}{} \citep{2019ApJ...879...69T} to build a neural network to determine stellar labels. They imported a new term in the loss function which takes into account the similarity of gradient spectra from the data-driven model and Kurucz model atmospheres \citep{1970SAOSR.309.....K, 1993KurCD..18.....K, 2005MSAIS...8...14K}, to regularize the training process and force the model to measure labels from real absorption features. They verified that more than 10 elements can be extracted even from the low-resolution spectra as in LAMOST.
Based on this approach \citep{2017ApJ...849L...9T}, \citealt{2019ApJS..245...34X} further developed the Data-Driven Payne (hereafter \emph{DD-Payne}{}) which used input training labels from the high-resolution spectroscopic surveys APOGEE \citep{2019ApJ...879...69T} and GALAH \citep{2018MNRAS.478.4513B} to determine stellar labels for LAMOST DR5 spectra.
These authors found that they could precisely estimate close to 20 stellar labels for more than 6 million spectra. These studies highlight
that if one has a reliable training set then one can obtain precise elemental abundances even from low-resolution spectra.
The methods outlined above demonstrate how one can measure stellar labels for millions of spectra using machine learning approaches. However, all of the surveys described previously are conducted using fiber-fed spectrographs, which take spectra by plugging a series of optical fibers onto a plate based on the positions of stars on the sky.
Fibers are typically set to have a minimum separation (e.g. $30^{\prime \prime}$ for GALAH; $56-71.5^{\prime \prime}$ for APOGEE; $268.2^{\prime \prime}$ for LAMOST) during each observation. And each fiber has a typical diameter of $2\sim3^{\prime \prime}$.
This means that these instruments can not observe dense fields such as cores of globular clusters and the very dense regions of Galactic bulge, where the separation of stars is less than $1^{\prime \prime}$.
However, stars in these fields also play an essential role in the Galactic evolution, it is vital to obtain more complete spectroscopic observations in these regions to unravel the history of the Milky Way.
The Multi-Unit Spectroscopic Explorer (MUSE) \citep{2010SPIE.7735E..08B} on the Very Large Telescope is an integral field spectrograph, which is regularly used to observe external galaxies and can be an ideal solution to deal with high stellar density fields.
With the advanced development of image slicer and spectrometer, MUSE can perform both imaging and spectroscopy simultaneously.
The benefit is that each MUSE spectrum is obtained for a spatial pixel ($0.2^{\prime \prime} \times 0.2^{\prime \prime}$) on the sky rather than the area of the fiber, so it is not restricted by the fiber size or collision effects.
The data format of MUSE is a 3-D data-cube, with 2D in spatial on the sky and 1D in wavelength. Combining with the PSF-fitting spectral extraction software PampelMUSE{} \citep{2013A&A...549A..71K}, MUSE becomes the best instrument to obtain spectra in fields with high stellar density.
There have been several studies on stars in globular clusters using spectra from MUSE observations \citep{2016A&A...588A.148H, 2016A&A...588A.149K}.
Near the core of these clusters, more than 6000 spectra (e.g., 47 Tuc at a distance of 4 kpc analyzed by \citealt{2016A&A...588A.149K}) can be extracted on average per MUSE cube (in wide-field mode with a field of view of $1^{\prime} \times 1^{\prime}$). Currently, most studies focused exclusively on kinematics (e.g. \citealt{2016A&A...588A.149K, 2018MNRAS.473.5591K, 2018A&A...616A..83V}).
Some studies explored specific spectral region or metallicity (e.g. \citealt{2019A&A...631A.118G, 2020A&A...635A.114H}).
However, these spectra contain a wealth of chemical information that has not yet been fully explored.
This is because currently there is no robust and easy-to-use method to automatically derive chemical abundances for stars in these fields.
The success of data-driven models in
deriving about $20$ abundances from LAMOST spectra highlights the potential of other
low-resolution surveys, for example MUSE,
having wavelength coverage and resolution
similar to LAMOST to also derive abundances.
Inspired by this, in this paper, we develop an approach based on the \emph{DD-Payne}{} \citep{2017ApJ...849L...9T, 2019ApJS..245...34X}, to measure stellar labels from MUSE observations.
Using this approach we derive robust stellar labels, including effective temperature (\ensuremath{T_{\mathrm{eff}}}{}), surface gravity (\mbox{$\log g$}{}), metallicity \ensuremath{[\mathrm{Fe/H}]}{} and abundances of four $\alpha$ elements, \ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{Si/Fe}]}{}, \ensuremath{[\mathrm{Ca/Fe}]}{} and \ensuremath{[\mathrm{Ti/Fe}]}{}.
In Section~\ref{s:data}, we introduce the data-sets used in our analysis and the spectral extraction method used by us. In Section~\ref{s:meth}, we outline the procedures used to measure stellar labels for MUSE spectra.
We do two validation tests in Section~\ref{s:resu}. First, we verify the bias and dispersion of our approach by applying it to common stars between LAMOST and MUSE. Second, for dense fields, we explore the \ensuremath{[\mathrm{Fe/H}]}{}-\ensuremath{[\mathrm{Mg/Fe}]}{} distribution of stars towards the bulge region using MUSE observations and compare it with that obtained from the high-resolution APOGEE survey. In Section~\ref{s:predict}, we explore and discuss the dependence of label precision on magnitude and exposure time. In Section~\ref{s:discuss}, we discuss the exciting prospects of using MUSE to observe dense stellar fields in the Milky Way and other galaxies. Finally, a summary of our results and conclusions is presented in Section~\ref{s:summary}.
\section{Data}
\label{s:data}
\subsection{MUSE}
\label{s:data-muse}
\subsubsection{Overview}
\label{ss:data-muse-ow}
MUSE (the Multi-Unit Spectroscopic Explorer) \citep{2010SPIE.7735E..08B, 2014Msngr.157...13B} is an optical wide-field integral field spectrograph using the image slicing technique mounted on the UT4 of the Very Large Telescope at the Paranal Observatory in Chile.
MUSE operates in two spatial modes, the wide-field mode (WFM) and the narrow-field mode (NFM).
The wide-field mode covers a FoV of $1^{\prime} \times 1^{\prime}$ with $0.2^{\prime \prime} \times 0.2^{\prime \prime}$ spatial bins.
The narrow-field mode covers a FoV of $7.5^{\prime \prime} \times 7.5^{\prime \prime}$ with $0.025^{\prime \prime} \times 0.025^{\prime \prime}$ spatial bins.
It operates in two wavelength modes, the nominal mode covering 4800-9300~$\Angstrom$ and the extended mode covering 4650-9300~$\Angstrom$. It samples the wavelength in 1.25$\Angstrom$ bins at a spectral resolution of R$\sim$3000.
To date, more than 13000 data-cubes have been observed by MUSE and they are publicly available via the ESO Science Portal Website\footnote{\url{http://archive.eso.org/scienceportal/home}}. MUSE is typically used to conduct studies on external galaxies but it has also been used to study a wide variety of other things, e.g., exoplanets \citep{2018Icar..302..426I, 2019NatAs...3..749H}, Galactic regions \citep{2015A&A...582A.114W, 2015MNRAS.450.1057M} and resolved stellar populations of Globular clusters \citep{2016A&A...588A.148H, 2018MNRAS.473.5591K}.
\subsubsection{Reductions}
\label{ss:data-muse-rd}
The data format of MUSE observations follows the ESO science data product standard for data-cubes. Each data-cube has three dimensions, two in spatial and one in spectral.
The data processing pipeline transforms the raw CCD-based data into fully calibrated and corrected data-cubes, which can be directly used for studies. The transformation processes include bias, dark and flat-field reduction, wavelength and flux calibration, line spread function and illumination correction, sky creation, and astrometrically correlation and multiple exposures combination, which are discussed in detail in \citealt{2012SPIE.8451E..0BW, 2014ASPC..485..451W, 2020A&A...641A..28W}.
For this study, all the data-cubes we use are published online and have been processed and calibrated using the MUSE Data Processing Pipeline\footnote{\url{https://www.eso.org/sci/software/pipelines/muse/}}.
\subsubsection{Stellar Spectral Extraction}
\label{ss:data-muse-se}
To extract stellar spectra, we use the PampelMUSE{} package \citep{2013A&A...549A..71K} which was specifically developed for 3D data-cubes in dense fields. It assumes the data-cube to be a sum of many overlapping PSFs and sky background.
First, a single object is represented by a PSF function such as Moffat and Gaussian, and assuming parameters in the PSF function change smoothly with wavelengths.
Next, the PSF photometry is employed to fit each star in each wavelength layer and then all the layers are combined to generate a single stellar spectrum.
One great advantage of this method is that it can de-blend the spectra of different stars.
Therefore, MUSE observations combined with PampelMUSE{} become an ideal way to study dense fields.
To get spectra in the field using PampelMUSE{}, a reference catalog with stellar positions and magnitude is needed as input.
Various reference catalogs can be used, such as all-sky surveys (e.g. Gaia \citealt{2018A&A...616A...1G, 2021A&A...649A...1G}, Sky-Mapper \citealt{2007PASA...24....1K} or HST) or smaller surveys for specific purposes. In this study, we mainly use LAMOST DR5 and VVV DR2 \citep{2017yCat.2348....0M} as reference catalogs for the two validation test data, respectively.
Moreover, a transformation function of the filter band is needed when aligning the MUSE data-cube slice with reference catalogs.
Here we refer to the transformation function from SVO Filter Profile Service\footnote{\url{https://svo.cab.inta-csic.es/main/index.php}} and use the Moffat function as the PSF profile, which has better performance than the Gaussian profile when fitting flux distribution of stars (see Figure 3 in \citealt{2013A&A...549A..71K}).
After going through all processes in PampelMUSE{}, including source position transformation, PSF fitting, sky background estimation and subtraction, and unresolved star reduction, the output is a list of stellar spectra which are identified as resolvable by the algorithm.
A few additional cleaning steps (removal of emission lines and non-stellar features) are performed on the spectra and will be discussed in Section~\ref{sss:resu-bulge-data}.
\subsection{LAMOST}
\label{s:data-lamost}
\subsubsection{Overview}
\label{ss:data-lamost-ow}
The LAMOST Galactic survey \citep{2012RAA....12..735D, 2012RAA....12..723Z, Liu_2013, 2015RAA....15.1089L} is the first dedicated spectroscopic survey to obtain spectra of millions of stars covering most of the sky.
LAMOST DR5 has released more than 8 million stellar spectra covering the optical wavelength regime 3700-9000 $\Angstrom$ with low-resolution R$\sim$1800.
For about 5 million spectra, LAMOST DR5 also provides the basic stellar labels such as \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{} and \ensuremath{[\mathrm{Fe/H}]}{} which were derived with the LAMOST stellar parameter pipeline (LASP; \citealt{2011RAA....11..924W, 2015RAA....15.1095L}). There are also several value-added catalogs providing stellar labels with some chemical abundances and extinction via different methods (e.g., \citealt{2015MNRAS.448..822X, 2016RAA....16..110L, 2017MNRAS.464.3657X}). Despite LAMOST's low spectral resolution, \citealt{2017ApJ...849L...9T, 2018ApJ...860..159T} has verified that $\geq$10 chemical abundances should be derivable with precision on the level of $0.1\sim0.2$ dex or better.
\subsubsection{Data-Driven Payne}
\label{ss:data-lamost-ddp}
Given the success of machine learning and data-driven models in stellar parameter determination, an effective way to estimate stellar labels from MUSE spectra would be to develop a data-driven model for the MUSE spectra based on stellar labels from other high-resolution spectroscopic surveys.
Presently, we do not have enough common stars (ideally $\sim10^4$) covering a wide range in the parameter space between MUSE and other surveys for a robust training set.
However, with LAMOST spectra having similar resolution and wavelength range as MUSE (see Table~\ref{tab:tab1}), we can apply a model developed for LAMOST to MUSE spectra.
\citealt{2017ApJ...849L...9T, 2019ApJS..245...34X} (hereafter T17 \& X19) had developed a \emph{DD-Payne}{} model for LAMOST, which we discuss below.
It was shown to have good performance and hence we adopted it for our analysis.
\emph{DD-Payne}{} utilizes the neural network interpolator and the fitting technique from \emph{The Payne}{} \citep{2019ApJ...879...69T} combined with physical gradient spectra from Kurucz spectral model \citep{1970SAOSR.309.....K, 1993KurCD..18.....K, 2005MSAIS...8...14K} to regularize the training process.
This model can derive stellar labels of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, micro-turbulence velocity ($V_{\rm mic}$) and 16 reliable chemical abundances.
Compared with other similar models (\citealt{2017ApJ...836....5H, 2020ApJ...898...58W}), the biggest advantage of \emph{DD-Payne}{} is it can enforce these stellar labels to be measured physically from the spectral features.
For $\rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>50$ $\text{pix}^{-1}${}, the typical theoretical precision of the \emph{DD-Payne}{} abundances is $\sim$0.05 dex for Fe, Mg, Ca, Ti, Cr and Ni, $\sim$0.1 dex for C, N, O, Na, Al, Si, Mn, and Co. This study demonstrates that while obtaining reliable elemental abundances remains challenging for low-resolution spectra, precise abundances are still derivable for spectra with SNR$\sim$50 $\text{pix}^{-1}${}.
\begin{table}
\caption{Comparison of MUSE and LAMOST survey.}
\label{tab:tab1}
\begin{tabular}{lcc}
\hline
Survey & Wavelength range ($\Angstrom$) & Resolution\\
\hline
MUSE & 4750-9300 & R$\sim$3000\\
LAMOST & 3700-9000 & R$\sim$1800\\
\hline
\end{tabular}
\end{table}
\subsection{Validation data and the motivations}
\label{s:data-vbd}
We do two different validation tests.
Since the model used by us is designed for LAMOST spectra, the question is how precisely does it work when applied to MUSE spectra? Therefore, for the first validation test, we run the model on both the LAMOST spectra and the MUSE spectra of the same star and compare the results.
We cross-matched the LAMOST DR5 catalog from X19\footnote{\url{http://dr5.lamost.org/doc/vac}} with all the MUSE observations so far published on the ESO website, and found 79 stars across 162 MUSE data-cubes.
\begin{figure*}
\includegraphics[width=1.48\columnwidth]{figs_new/lb_distribution_zoccali.png}
\caption{
Distribution of 9 fields observed by MUSE (0101.B-0381(A), PI: Zoccali) in the Galactic coordinate with the background taken by DSS2. The nine square fields are marked in cyan and are divided into three areas. The purple cross in the middle represents the Galactic center where $(l, b)=(0, 0)$. On the right-hand side are zoomed-in figures of these three areas guided by yellow arrows. The white light picture of the MUSE data-cube is shown in each square field.
}
\label{f:bulge_cube}
\end{figure*}
The second validation is done by comparing
the abundance distribution of stars in the inner Galaxy (bulge region) with those obtained from the high-resolution APOGEE survey.
The Galactic bulge region has a very high stellar density and its stars suffer from strong extinction.
Hence, validation of bulge stars is important to test the performance of our method in challenging conditions.
Therefore, for the second test, we selected 29 data-cubes (0101.B-0381(A), PI: Zoccali) covering nine fields that are less than 3 degrees to the Galactic center.
They are dense fields and provide a large sample of stars for which to verify our spectroscopic abundances. Figure \ref{f:bulge_cube} shows the distribution of 9 fields in the Galactic coordinate with the background taken by DSS2. The nine square fields are marked in cyan and are divided into three areas.
The purple cross in the middle represents the Galactic center where $(l,b)=(0,0)$.
On the right-hand side are zoom-in figures of these three areas guided by yellow arrows. The white light picture of the MUSE data-cube is shown in each square field.
For each cube field, there are three or four repetitive exposures.
Galactic bulge fields present some unique challenges and need some additional reprocessing, such as low SNR wavelength cut-off, emission line masking, etc, which will be discussed in detail in Section~\ref{sss:resu-bulge-data}.
\section{Methods}
\label{s:meth}
The feasibility of applying the \emph{DD-Payne}{} on MUSE observations is due to the similar wavelength range and spectral resolution of the MUSE and LAMOST spectrographs.
The processes of deriving stellar labels from MUSE observations include five steps:
\begin{itemize}
\item Stellar spectra are extracted from MUSE data-cubes by running the automatic spectra extraction algorithm PampelMUSE{} \citep{2013A&A...549A..71K} to obtain spectra at R$\sim$3000. This step was talked about in Section~\ref{ss:data-muse-se}.
\item The R$\sim$3000 MUSE spectra are degraded by using the line spread function (LSF) of these two instruments to R$\sim$1800.
\item The spectra are normalized by a pseudo-continuum derived in the same way as in T17 \& X19, to make
it compatible with the LAMOST \emph{DD-Payne}{} model.
\item The degraded and normalized spectra are then fitted by three \emph{DD-Payne}{} models of X19 to derive stellar labels.
\item For each label, the cross-correlation of gradient spectra between \emph{DD-Payne}{} and Kurucz spectral model \citep{1970SAOSR.309.....K, 1993KurCD..18.....K, 2005MSAIS...8...14K} are calculated to check if the label is estimated from real physical features rather than astrophysical correlations.
\item All the stellar information is generated and combined to create an output catalog.
\end{itemize}
The detailed process of each step and the calculating procedure is described in the following sub-sections.
\subsection{Degrading and normalizing the MUSE spectra}
\label{ss:meth-degrade}
To make \emph{DD-Payne}{} executable on MUSE spectra, we need to make the MUSE spectra as similar to LAMOST spectra in its wavelength grid and spectral resolution as possible, which leads to the process of degrading and interpolating MUSE spectra.
This is the procedure that determines whether we can derive precise stellar labels from MUSE spectra successfully.
To minimize the difference between LAMOST and MUSE, we employ the line spread function (LSF) of both instruments from X19 and \citealt{2017A&A...608A...1B}. The line spread function is FWHM in each wavelength grid. The process of degrading is performed by Gaussian kernel smoothing over the whole wavelength with $\sigma\mathrm{(\lambda_{i})}$, which is calculated from the equation given by
\begin{equation}
\sigma\mathrm{(\lambda_{i})}=\sqrt{\left(\frac{\mathrm{FWHM_{L}(\lambda_{i})}}{2.355}\right)^{2}-\left(\frac{\mathrm{FWHM_{M}(\lambda_{i})}}{2.355}\right)^{2}},
\label{e:eqn1}
\end{equation}
where $\mathrm{FWHM_{L}(\lambda_{i})}$ and $\mathrm{FWHM_{M}(\lambda_{i})}$ are the LAMOST and MUSE line-spread function, respectively.
After degrading, the spectra will be normalized by a pseudo-continuum which is derived by smoothing the spectra with a Gaussian kernel of 50$\Angstrom$ in width, which is the same method employed by T17 \& X19 for \emph{DD-Payne}{} training set spectra normalization. This is also an important procedure to ensure the similarity between these two sets of spectra.
Figure \ref{f:lm_muse_comparison} shows the normalized spectra of one single star observed by both LAMOST (in orange) and MUSE (in blue) and their residuals. The MUSE spectra were degraded using the above method.
The gray areas are the masked areas due to telluric lines and other interference in the LAMOST wavelength grid, which will not be processed by \emph{DD-Payne}{}.
This figure demonstrates that for the same star, the difference between the LAMOST and the MUSE spectra is less than 0.03 even in regions with strong absorption lines. This is encouraging and is essential for reliably estimating chemical abundances.
For the LAMOST-MUSE cross-validation data-cubes, the spectral SNR $\text{pix}^{-1}${} is in general higher than 200, which is much more than the average SNR of spectra in LAMOST ($\sim$40). After degrading (binning), the SNR is expected to increase even further.
\subsection{Estimating stellar labels and errors}
\label{ss:meth-ddp}
\begin{figure*}
\includegraphics[width=1.86\columnwidth]{figs_new/compare_lamostvmuse.png}
\caption{
Comparison of LAMOST spectra (in yellow) with MUSE spectra (in blue) of the same star. The MUSE spectra have been degraded to the same resolution. Then both spectra were normalized in the same way as X19 with a pseudo-continuum derived by smoothing the spectra with a Gaussian kernel of 50$\Angstrom$ in width. Gray areas are the masked pixels (due to telluric lines and other sources of interference) which will not be used when fitting with \emph{DD-Payne}{}. The bottom plot shows the residuals of these two spectra. From the residuals, we can see that LAMOST spectra and MUSE spectra are similar with residuals less than 0.03. The blue shaded area represents the uncertainty of the residual, which is calculated using the flux error of the LAMOST and MUSE spectra. The flux error of MUSE spectra is re-estimated by multiplying $\sigma$ in the distribution of flux residuals of the fitting divided by the flux error, see the right panel of Figure~\ref{f:muse_ddp_comparison}}.
\label{f:lm_muse_comparison}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.86\columnwidth]{figs_new/compare_musevmodel.png}
\caption{
Comparison of MUSE spectra (in black) with model predicted spectra (in red) from \emph{DD-Payne-G}{} on the same star as Figure \ref{f:lm_muse_comparison}. The left bottom plot shows their residuals. The masked regions in grey in Figure \ref{f:lm_muse_comparison} are not shown in this plot. From the residuals we can see the difference between MUSE spectra and predicted spectra is small, with the residuals less than 0.03 even in absorption regions, demonstrating that \emph{DD-Payne}{} training models can fit the MUSE spectra very well. The right panel shows the distribution of flux residuals divided by the flux error. A Gaussian fit to the distribution yields a standard deviation of 3.75, which demonstrates the underestimation of flux error of MUSE spectra.
}
\label{f:muse_ddp_comparison}
\end{figure*}
After degrading and normalizing the MUSE spectra to the LAMOST wavelength grid and spectral resolution, the next step is to estimate the stellar labels using the \emph{DD-Payne}{} models.
There are three models (so-called \emph{DD-Payne-G}{}, \emph{DD-Payne-A}{} and \emph{DD-Payne-A-mp}{} below) trained by X19 with different training sets.
The \emph{DD-Payne-G}{} model used spectra from LAMOST ($\rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>50$ $\text{pix}^{-1}${} and $\rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>30$ $\text{pix}^{-1}${} for stars with $\ensuremath{[\mathrm{Fe/H}]}{}<-0.6$) and stellar labels from GALAH DR2 \citep{2018MNRAS.478.4513B} while the \emph{DD-Payne-A}{} model adopted LAMOST spectra ($\rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>50$ $\text{pix}^{-1}${}) and stellar labels from APOGEE-\emph{Payne} \citep{2019ApJ...879...69T} for their respective cross-matched common samples.
The \emph{DD-Payne-G}{} derives stellar labels for \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{} and abundance ratios [X/Fe] for 20 elements: Li, C, O, Na, Mg, Al, Si, Ca, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Y, Ba, and Eu. \emph{DD-Payne-A}{} derive stellar labels of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, micro-turbulence velocity ($V_\mathrm{mic}$), \ensuremath{[\mathrm{Fe/H}]}{} and abundance ratios for 15 elements: C, N, O, Na, Mg, Al, Si, Ca, Ti, Cr, Mn, Co, Ni, Cu, and Ba.
Since X19 described these models in great detail, we direct the readers to this paper for more information about the selection criteria, validation and comparisons of the performances of different models.
However, we plot the parameter ranges of these training sets in \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{} and \ensuremath{[\mathrm{Fe/H}]}{} in Figure~\ref{f:ddp_vali_parameter} as square markers for later comparison with the validation data in Section~\ref{ss:resu-veri}.
In this work, we run all these models to test their performances because behind the training sets are GALAH and APOGEE spectra, which have different wavelength regions. The use of all these models will build a complete set of elements and also test the non-negligible systematics among the high-resolution surveys' labels. In addition, for some overlapped elements, we also compare their bias and dispersion, which can provide a flexible choice of models for the users in the future.
Based on results from Figure 2 in X19, Li, Sc, V, Zn, Y, Eu were removed due to the weak correlations of the gradients, and will not be considered in this study.
The process of applying the \emph{DD-Payne}{} to MUSE spectra is as follows: After loading the \emph{DD-Payne}{} models, same as X19, several wavelength regions are masked including areas with telluric bands, the very red wavelength region ($>8750\Angstrom$), and a region overlapped by the dichroic $(5720-6060\Angstrom)$ where the LAMOST spectral performance is low \citep{2015MNRAS.448..822X}. Each MUSE spectra is then fitted with both the \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{} model.
Following the procedure outlined in X19, we also run an additional model for stars being identified as metal-poor in \emph{DD-Payne-A}{} ($\ensuremath{[\mathrm{Fe/H}]}{}<-0.6$) to improve the precision for metal-poor stars.
This \emph{DD-Payne}{} APOGEE metal-poor model (\emph{DD-Payne-A-mp}{}) is trained by X19 based on common metal-poor stars in APOGEE-\emph{Payne} and LAMOST DR5. For stars with $\ensuremath{[\mathrm{Fe/H}]}{}<−1.0$ dex, only labels from \emph{DD-Payne-A-mp}{} model are adopted.
For stars with metallicity in $−1.0<\ensuremath{[\mathrm{Fe/H}]}{}<−0.6$ dex, we take the weighted mean value of results from \emph{DD-Payne-A}{} and \emph{DD-Payne-A-mp}{} models, with weighting given below by Equation~\ref{apogeemetal}. For stars with $\ensuremath{[\mathrm{Fe/H}]}{}>-0.6$ dex, we adopt the the \emph{DD-Payne-A}{} measurements.
\begin{equation}
\begin{split}
&\omega_{\rm MP} = (\ensuremath{[\mathrm{Fe/H}]}_{\rm MR} + 0.6) / (-0.4), \\
&\omega_{\rm MR} = 1 - \omega_{\rm MP}, \\
&\text{[X/Fe]} = \omega_{\rm MR} \times \text{[X/Fe]}_{\rm MR} + \omega_{\rm MP} \times \text{[X/Fe]}_{\rm MP}.
\label{apogeemetal}
\end{split}
\end{equation}
Here MR is the label from \emph{DD-Payne-A}{}, and MP is the label from \emph{DD-Payne-A-mp}{}, $\omega$ is the relative weight for each label [X/Fe]. In the following sections, when we refer to the labels derived from \emph{DD-Payne-A}{} we mean this combined set of results.
In addition, we determine the radial velocity and its error using the Doppler equation at the same time during the fit, which is the same as \emph{The Payne}{} \citep{2019ApJ...879...69T}.
We also determine two values to represent the quality of fitting. One is the reduced $\chi^2$, calculated given by
\begin{equation}
\chi^{2}= \frac{1}{n} \sum_{i=1}^{n} \frac{\left(P_{i}-O_{i}\right)^{2}}{e_{i}^2},
\end{equation}
where $n$ is the number of wavelength pixels, $P_{i}$, $O_{i}$ and $e_{i}$ are the best-fit spectra, observed spectra and the flux error in the $i$-th wavelength pixel.
The other is the Pearson correlation coefficient between the pixel values of the observed and the best-fit \emph{DD-Payne}{} spectra.
Figure \ref{f:muse_ddp_comparison} illustrates the comparison of observed MUSE spectra (in black) with the best fit by \emph{DD-Payne}{} (in red) for the same star as that in Figure \ref{f:lm_muse_comparison}. The residual spectrum is shown in the bottom panel.
The difference is less than 0.03 over most of the spectrum, including areas with strong absorption lines. This demonstrates that \emph{DD-Payne}{} training models can reproduce the MUSE spectra with high fidelity.
\subsection{Assessing the precision of labels}
Even though our best-fit model spectra are quite similar to observed MUSE spectra, it is not guaranteed that all stellar labels from \emph{DD-Payne}{} are estimated precisely.
Many spectral features suffer from blending due to the low spectral resolution. Since \emph{DD-Payne}{} is developed to avoid astrophysical correlations, the imprecise stellar labels are due to weak or no absorption features in the fitting wavelength window.
When compared to LAMOST spectra, MUSE loses information in the blue regions $3700-4750\Angstrom$, which contains absorption features of many elements.
Therefore, we cannot expect the performance of \emph{DD-Payne}{} on the MUSE spectra to be as good as that on the LAMOST spectra, even if MUSE spectra have higher SNR. To check the precision of estimated stellar labels, we follow T17 \& X19's work, and for each stellar label, we compare the gradient spectra of \emph{DD-Payne}{} with Kurucz spectral model \citep{1970SAOSR.309.....K, 1993KurCD..18.....K, 2005MSAIS...8...14K} and calculate their correlation coefficient.
If a label is physically measured from the spectral absorption features, the gradient spectra predicted by \emph{DD-Payne}{} should be very similar to the Kurucz spectral model, with a correlation coefficient close to 1.
Otherwise, if a weak or no absorption feature is identified for the label the correlation coefficient will be very low.
Due to the computational expense of calculating the theoretical model, we use a template of Kurucz model gradient spectra from X19 (see Table 1) which is calculated based on a list of reference stars covering a wide range
in the parameter space, from −2.5 to 0.5 dex in \ensuremath{[\mathrm{Fe/H}]}{} and 4000
to 7000 K in \ensuremath{T_{\mathrm{eff}}}{}. Therefore, for each star, we compare the gradient spectra from \emph{DD-Payne}{} with one of the template spectra having the closest parameter distance to the target star. The parameter distance is defined by X19 as
\begin{equation}
D=\sqrt{\left(\Delta \ensuremath{T_{\mathrm{eff}}} / 100 \mathrm{K}\right)^{2}+(\Delta \log g / 0.2)^{2}+(\Delta \ensuremath{[\mathrm{Fe/H}]} / 0.1)^{2}}.
\end{equation}
Finally, the correlation coefficient of each label will be provided in the final catalog as well as a flag based on the correlation values to represent whether the label is precise or not. Here we assign the flag of 1 if the correlation coefficient is larger than 0.5, otherwise, the flag is set to 0.
In this way, we will know which elements are determined precisely for each star.
\subsection{Building the catalog}
\label{ss:meth-catalog}
\begin{table*}
\caption{Descriptions of all the column names and information for the DD-Payne stellar label catalog.}
\label{tab:tab2}
\begin{tabular}{lll}
\hline
Column & Desceiption & Unit\\
\hline
stellar$_{-}$id & Stellar ID of stars & String\\
ra & Right ascension from the input catalog (or RAJ2000) & deg\\
dec & Declination from the input catalog (or DEJ2000) & deg\\
snr$_{-}$muse & Spectra signal-to-noise ratio calculated by median of all pixels & $\text{pix}^{-1}${}\\
\ensuremath{T_{\mathrm{eff}}} & Effective temperature & K\\
\ensuremath{T_{\mathrm{eff}}}$_{-}$err & Uncertainty in \ensuremath{T_{\mathrm{eff}}} & K\\
\ensuremath{T_{\mathrm{eff}}}$_{-}$flag & Quality for the value based on the examination of the DD-Payne gradient spectra $\partial f_{\mathrm{\lambda}} / \partial T_{\mathrm{eff}}$ \\
\ensuremath{T_{\mathrm{eff}}}$_{-}$gradcorr & Correlation coefficients of $\partial f_{\mathrm{\lambda}} / \partial T_{\mathrm{eff}}$ between the DD–Payne and the Kurucz model \\
\mbox{$\log g$} & Surface gravity & in cgs \\
\mbox{$\log g$}$_{-}$err & Uncertainty in \mbox{$\log g$} & in cgs \\
\mbox{$\log g$}$_{-}$flag & Quality for the value based on the examination of the DD-Payne gradient spectra $\partial f_{\mathrm{\lambda}} / \partial \mbox{$\log g$}$ \\
\mbox{$\log g$}$_{-}$gradcorr & Correlation coefficients of $\partial f_{\mathrm{\lambda}} / \partial \mbox{$\log g$}$ between the DD–Payne and the Kurucz model \\
V$_\mathrm{mic}$ & Micro-turbulent velocity & {\rm km s}$^{-1}$ \\
V$_\mathrm{mic}$$_{-}$err & Uncertainty of micro-turbulent velocity & {\rm km s}$^{-1}$ \\
$V_\mathrm{mic}$$_{-}$flag & Quality for the value based on the examination of the DD-Payne gradient spectra $\partial f_{\mathrm{\lambda}} / \partial V_\mathrm{mic}$ \\
$V_\mathrm{mic}$$_{-}$gradcorr & Correlation coefficients of $\partial f_{\mathrm{\lambda}} / \partial V_\mathrm{mic}$ between the DD–Payne and the Kurucz model \\
\ensuremath{[\mathrm{Fe/H}]} & Metallicity & \ensuremath{\,\mathrm{dex}} \\
\ensuremath{[\mathrm{Fe/H}]}$_{-}$err & Uncertainty of metallicity & \ensuremath{\,\mathrm{dex}} \\
\ensuremath{[\mathrm{Fe/H}]}$_{-}$flag & Quality for the value based on the examination of the DD-Payne gradient spectra $\partial f_{\mathrm{\lambda}} / \partial \ensuremath{[\mathrm{Fe/H}]}$ \\
\ensuremath{[\mathrm{Fe/H}]}$_{-}$gradcorr & Correlation coefficients of $\partial f_{\mathrm{\lambda}} / \partial \ensuremath{[\mathrm{Fe/H}]}$ between the DD–Payne and the Kurucz model \\
\ensuremath{[\mathrm{X/Fe}]} & Chemical abundance fraction to iron, sorted by Mg, Si, Ca, Ti & \ensuremath{\,\mathrm{dex}} \\
\ensuremath{[\mathrm{X/Fe}]}$_{-}$err & Uncertainty of chemical abundance & \ensuremath{\,\mathrm{dex}} \\
\ensuremath{[\mathrm{X/Fe}]}$_{-}$flag & Quality for the value based on the examination of the DD-Payne gradient spectra $\partial f_{\mathrm{\lambda}} / \partial \mathrm{[X/H]}$ \\
\ensuremath{[\mathrm{X/Fe}]}$_{-}$gradcorr & Correlation coefficients of $\partial f_{\mathrm{\lambda}} / \partial \mathrm{[X/H]}$ between the DD–Payne and the Kurucz model \\
red$_{-}$chi2 & Reduced $\chi^2$ of the spectral fit (hereafter $\chi^{2}$)\\
corr$_{-}$flux & Correlation coefficient of the muse and best-fitting model spectra \\
\hline
\end{tabular}
\end{table*}
After deriving two series of stellar labels for each star by applying \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{} models and assessing the precision of each label value, we compile two catalogs.
Table \ref{tab:tab2} presents a description of the columns in the catalogs. Due to different science goals, wavelength coverage, and data quality, the relative performances of the two models are different for different stellar labels.
Some elements are estimated better in one model as compared to the other.
In normal circumstances, the version of the label with a larger gradient spectra correlation coefficient should be selected.
In the following Section \ref{ss:resu-veri}, we discuss in detail the bias and precision of stellar labels from these training models.
\section{Results}
\label{s:resu}
In Section \ref{s:meth} we gave detailed steps for applying the \emph{DD-Payne}{} on MUSE spectra, measuring their stellar labels and building the \emph{DD-Payne-A}{} and \emph{DD-Payne-G}{} catalogs.
To test whether our method can measure stellar labels from MUSE observations with high precision, we now do two validations: First, we compare the \emph{DD-Payne}-estimated stellar labels from common stars observed by both MUSE and LAMOST; Second, we use our method to analyze the \ensuremath{[\mathrm{Fe/H}]}{} and \ensuremath{[\mathrm{Mg/Fe}]}{} distributions of stars in the bulge and compare our results with that of the previous high-resolution spectroscopic studies. The motivation for each validation has been discussed in detail in Section~\ref{s:data-vbd}.
\subsection{Validation 1: LAMOST-MUSE cross-validation samples}
\label{ss:resu-veri}
\begin{figure*}
\includegraphics[width=1.88\columnwidth]{figs_new/ddp_validation_parameter_range.png}
\caption{
Distributions of the \emph{DD-Payne}{} training set (Figure 5 in X19, in square markers) and LAMOST-MUSE cross-validation samples (in star markers) in the \ensuremath{T_{\mathrm{eff}}}{}-\mbox{$\log g$}{} plane color-coded by \ensuremath{[\mathrm{Fe/H}]}{}. Left panel: Distribution of the \emph{DD-Payne-G}{} training stars, the \ensuremath{T_{\mathrm{eff}}}{} and \mbox{$\log g$}{} are from a corrected version GALAH DR2 values (see Section 3 and Appendix A of X19) The number of stars for each sample is noted in each subplot. The LAMOST-MUSE cross-validation samples are selected following the criteria of Section~\ref{ss:resu-veri} and span reasonable parameter ranges than those from \emph{DD-Payne}{} training sets. No stellar parameter is measured by extrapolation.
}
\label{f:ddp_vali_parameter}
\end{figure*}
We follow the steps described in Section \ref{s:meth} and derive stellar labels of common stars in LAMOST and MUSE using both \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{} models.
To ensure the precision of labels, we use the selection criteria given by
\begin{equation}
\left\{\begin{array}{l}
$\ensuremath{\mathrm{SNR_{\mathrm{LAMOST}}}}${}>35 \text{pix}^{-1} (\sim \rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>23 \text{pix}^{-1}) \\
$\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${}>80 \text{pix}^{-1} \\
\end{array}\right.
\label{eqn:thresholdval1}
\end{equation}
Here $\ensuremath{\mathrm{SNR_{\mathrm{LAMOST}}}}${} and $\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${} is the median signal-to-noise ratio of the LAMOST and MUSE spectra, respectively.
We made a cut-off of $\ensuremath{\mathrm{SNR_{\mathrm{LAMOST}}}}${} to ensure the correction of stellar parameters measured from LAMOST spectra, this threshold is looser than the criteria of \emph{DD-Payne}{} training set ($\rm{SNR}_{\rm{LAMOST}}^{\rm{g-band}}>30$ $\text{pix}^{-1}${}) to keep more common stars. The threshold of $\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${} cut-off is more strict than that of $\ensuremath{\mathrm{SNR_{\mathrm{LAMOST}}}}${} because the SNR of MUSE spectra is somehow overestimated after data reduction pipeline, which is confirmed by looking at the spectra of the two validation data and the distribution of flux residuals of the fitting divided by the flux error (the right panel of Figure~\ref{f:muse_ddp_comparison}).
Nevertheless, most MUSE spectra have SNR $\text{pix}^{-1}${} greater than 200.
This is because these observations typically have long exposure time, as they were targeting external galaxies which are much fainter than the stars we are interested in.
Finally, we end up with 27 common stars in \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{} results, during which the $\chi^2$ is mostly concentrated in 4$\sim$25, meaning the $\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${} is overestimated by a factor of 2$\sim$5.
Star markers/diamonds in Figure~\ref{f:ddp_vali_parameter} represent this cross-validation sample. These stars span a reasonable range of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{} and \ensuremath{[\mathrm{Fe/H}]}{} than those from \emph{DD-Payne}{} training sets. One star is measured by extrapolation, which will be discussed in Section~\ref{sss:resu_veri_some}.
\subsubsection{Bias and precision check for main labels}
\label{sss:resu_veri_some}
\begin{figure*}
\includegraphics[width=1.53\columnwidth]{figs_new/comparison_labels_selected.png}
\caption{
Differences of stellar labels derived from MUSE spectra with those from LAMOST spectra using \emph{DD-Payne-G}{} (left column) and \emph{DD-Payne-A}{} (right column) models as a function of \ensuremath{[\mathrm{Fe/H}]} from MUSE spectra, respectively. All the stars in these plots have LAMOST spectra with SNR higher than 35 $\text{pix}^{-1}${}. The median and dispersion of all points are marked in the upper left corner of each panel, the dispersion is calculated as half of the difference between 15.87 and 84.13 percentile values.
}
\label{f:com_ddp}
\end{figure*}
In this section, we focus on \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{} and several chemical abundances which we reported with confidence in the abstract.
Figure \ref{f:com_ddp} shows the differences of \emph{DD-Payne}-estimated stellar labels from MUSE and LAMOST spectra, as a function of \ensuremath{[\mathrm{Fe/H}]}{}.
The median and dispersion of all points are marked in the upper left corner of each panel, where the dispersion is calculated as half of the difference between 15.87 and 84.13 percentile values. The median and dispersion are indicative of the bias {$\mu$} and precision {$\sigma$} of our label estimates, respectively.
The left and right panels show results for the \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{} models, respectively.
This figure demonstrates that the dispersion of $\Delta$\ensuremath{T_{\mathrm{eff}}}{} is about 75K, $\Delta$\mbox{$\log g$}{} is around 0.15 dex for both the \emph{DD-Payne-G}{} and \emph{DD-Payne-A}{}. As for chemical abundances, the dispersion is less than 0.1 dex in general and we consider this to be precise, as this is roughly the uncertainty in the LAMOST labels themselves.
Therefore, most of the stellar labels from MUSE spectra and LAMOST spectra are in good agreement in Figure~\ref{f:com_ddp}.
Even though the dispersion of some elements (e.g. \ensuremath{[\mathrm{Si/Fe}]}{} from \emph{DD-Payne-G}{} and \ensuremath{[\mathrm{Cr/Fe}]}{} from \emph{DD-Payne-A}{}) is slightly higher, it is still close to 0.1 dex.
The $\alpha$-elements listed in this figure are some of the most useful ones as they can be used to constrain intrinsic stellar properties such as evolutionary stage, distance, birth radius, extinction, and even age \citep{2019ApJ...883..177N, 2020arXiv201113745H, 2022MNRAS.510..734S}.
Therefore, this agreement shows the usefulness of \emph{DD-Payne}{} based estimates from MUSE observations for Galactic studies.
The bias in this figure reflects the differences between LAMOST and MUSE instruments and the different performances when losing wavelength in $3700-4750\Angstrom$. However, it can be seen that all of them are smaller than the dispersion. Therefore, in this work we do not perform the bias correction on any label. Whether to do it will be re-investigated when we have a larger sample.
In addition, by comparing the dispersion of the same elements under different model estimates, we also found that except \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Ti/Fe}]}{} and \ensuremath{[\mathrm{Cr/Fe}]}{}, all the labels calculated using the \emph{DD-Payne-A}{} model have lower dispersion than that from the \emph{DD-Payne-G}{} model.
The dispersion of \ensuremath{[\mathrm{Mg/Fe}]}{} from the \emph{DD-Payne-A}{} model is even less than 0.05 dex.
Since we derived labels from the same spectra, the reason must be the relative performance of the different models. Additionally, when comparing the derived gradient spectra with Kurucz spectral model for these labels, \emph{DD-Payne-G}{} with lower correlation coefficients, which indicates \emph{DD-Payne-A}{} model has better performance. This is because the training set of \emph{DD-Payne-A}{} is larger than that of \emph{DD-Payne-G}{}, so the measured stellar labels will be more precise.
There is one star in the panel of \emph{DD-Payne-A}{} measured by extrapolation since the \ensuremath{[\mathrm{Fe/H}]}{} is less than -1.5 dex, which is shown in the right panel of Figure 4 as a "diamond". By looking at the parameter differences in each panel, we can see from Figure 5 that \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Mg/Fe}]}{} and \ensuremath{[\mathrm{Si/Fe}]}{} measured from MUSE spectra of this star have good agreements with those measured from LAMOST spectra. But for the other parameters, the differences are larger. Note that the LAMOST spectra of this star was also measured by extrapolation, whether this indicates anything about extrapolation will be investigated when we have more stars.
\subsubsection{Bias and precision check for all labels}
\label{sss:resu-accuracy}
\begin{figure*}
\includegraphics[width=1.75\columnwidth]{figs_new/scatter_precision_galah.png}
\caption{
The bias (top panel) and precision (bottom panel) of all stellar labels derived from \emph{DD-Payne-G}{} model, respectively. The bias and precision are calculated in the same way as Figure \ref{f:com_ddp} by taking the median and half the difference between 15.87 and 84.13 percentile values. We list all measured labels and arrange them from small to large according to the precision of the elements. We also plotted results in the Bulge case which loses pixels between $(4750-5991\Angstrom)$. From these two plots, for the \emph{DD-Payne-G}{} model, Fe, Mg, Ni, Si are precisely estimated with $\sigma$ close to or better than 0.1 dex when fitting MUSE spectra between both $(4750-8750\Angstrom)$ and $(5991-8750\Angstrom)$.
As for bias, most of the elements are within 0.1 dex except Co, Mn, Na, and Al.
}
\label{f:acc_galah}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.75\columnwidth]{figs_new/scatter_precision_apogee.png}
\caption{
The same plot as Figure \ref{f:acc_galah} but now using \emph{DD-Payne-A}{} model. From these two plots, for \emph{DD-Payne-A}{} model, Fe, C, Si, Mg, Ni, and Ca are well estimated with $\sigma$ less than 0.1 dex when fitting MUSE spectra between both $(4750-8750\Angstrom)$ and $(5991-8750\Angstrom)$. Besides, MUSE $(5991-8750\Angstrom)$ shows a larger bias and dispersion. As for bias, most of the elements are within 0.1 dex except Mn, O, N, and Cu. Comparing with results from \emph{DD-Payne-G}{} model, for the same element, \emph{DD-Payne-A}{} model has better precision.}
\label{f:acc_apogee}
\end{figure*}
Other than the parameters discussed in the previous section, we also estimated abundances for additional elements, with 23 stellar labels for \emph{DD-Payne-G}{} and 16 labels for \emph{DD-Payne-A}{}. However, the dispersion for many of the abundances is not shown as they are much larger than those analyzed in Figure~\ref{f:com_ddp}.
This is because, unlike LAMOST, MUSE does not have any pixels in the wavelength range $3700-4750\Angstrom$, which contains a large number of absorption features.
In this section, we will further discuss the precision of these labels and compare them with the results of cross-validation samples between LAMOST and GALAH or APOGEE.
We perform the same analysis in Figure~\ref{f:com_ddp}, and derive bias ($\mu$) and the dispersion ($\sigma$) for all labels in Figure \ref{f:acc_galah} and Figure \ref{f:acc_apogee}.
In each plot stellar labels from MUSE are determined in two regions: $\lambda=(4750-8750\Angstrom)$ (in blue) and $\lambda=(5991-8750\Angstrom)$ (in yellow), corresponding to normal spectra and spectra in high extinction fields such as the Galactic bulge.
We also over-plot the dispersion of label differences between those from LAMOST spectra and GALAH DR2 \citep{2018MNRAS.478.4513B} or APOGEE-\emph{Payne} \citep{2019ApJ...879...69T} in red.
For the LAMOST-MUSE cross-validation sample (the blue line), elements such as Na, Al, O, and Ba in \emph{DD-Payne-G}{} model and Mn, Al, O, N, and Cu in \emph{DD-Payne-A}{} have high median and dispersion (larger than 0.1 dex).
This can be due to two reasons.
We find that the bias and dispersion of Na, Al, O, and Ba in cross-validation samples are also high.
For these elements, the high dispersion is the inheritance from the \emph{DD-Payne}{} models.
However, for Mn and N, the high dispersion is due to the loss of the blue wavelength region ($3700-4750\Angstrom$) from the MUSE spectra compared to the larger wavelength region covered by LAMOST.
This can be seen in Figure~\ref{f:acc_apogee}, where $\mu$ and $\sigma$ show a large discrepancy between the LAMOST-GALAH/APOGEE cross-validation estimates and the LAMOST-MUSE cross-validation estimates.
This is further confirmed by the gradient spectra of these two elements (See Figure 22 in X19), where the dominant absorption features of Mn and N are in $3700-4750\Angstrom$, which the MUSE spectra are lacking.
In general, for most of the labels in these two figures, the dispersion of the cross-validation estimates and MUSE estimates are in agreement.
Therefore, it can be verified that the models designed for LAMOST spectra can be applied to MUSE spectra, and measure some of labels (\ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{}, [Ni/Fe], [Cr/Fe], [C/Fe] and $\alpha$ abundances) with similar precision.
It is worth noting that there are several elements in these figures in which MUSE has higher precision than the LAMOST cross-validations, for example, Fe, Cr, Mg in \emph{DD-Payne-G}{} and Mg in \emph{DD-Payne-A}{}, this is due to the small number of stars in the common sample of LAMOST and MUSE.
For the LAMOST-MUSE cross-validation estimates in the wavelength region of $\lambda=(5991-8750\Angstrom)$ (the yellow line). Both bias and dispersion are larger than when it is fitted in $4750-8750\Angstrom$ (the blue line).
This is because more absorption features in the region of $4750-5991\Angstrom$ are lost.
Nevertheless, some elements such as \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{Si/Fe}]}{} in \emph{DD-Payne-G}{} and \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{Ca/Fe}]}{}, \ensuremath{[\mathrm{C/Fe}]}{}, \ensuremath{[\mathrm{Ni/Fe}]}{}, \ensuremath{[\mathrm{Si/Fe}]}{} in \emph{DD-Payne-A}{} still have dispersion lower than 0.1 dex.
This indicates that after further losing pixels in the wavelength range of $4750-5991 \Angstrom$ due to high extinction from the bulge, we still can recover some stellar labels for these fields.
\subsubsection{Theoretical precision prediction}
\label{sss:resu-veri-internal}
\begin{figure*}
\includegraphics[width=1.64\columnwidth]{figs_new/intrin_precision_predic.png}
\caption{
Estimation of the theoretical precision for a typical solar-metallicity K-Giant with a g-band SNR of 50 $\text{pix}^{-1}${} in three different scenarios. The LAMOST spectra cover the wavelength range between $3700-8750 \Angstrom$. Both MUSE $(4750-8750 \Angstrom)$ and MUSE $(5991-8750 \Angstrom)$ have the same resolution as LAMOST. The theoretical precision is predicted by using the \emph{Chem-I-Calc}{} method which is an efficient method for computing the expected precision of stellar labels determined via full spectral fitting \citep{2020ApJS..249...24S}. When losing wavelength pixels in wavelength $3700-4750 \Angstrom$ (LAMOST versus MUSE $(4750-8750 \Angstrom)$, the precision of all the labels is getting slightly worse by an average factor of 1.17, but there is no label having a large change. When further losing wavelength pixels in the range of $4750-5991\Angstrom$ (i.e., for MUSE cubes in high extinction fields), the precision of all the labels is worse by an average factor of 2.16. However, there are some labels such as Cr, N, Na, and C which have much larger shifts towards larger uncertainties by a factor of 2.66, 4.30, 4.02, 3.53, respectively due to the loss in wavelength coverage.}
\label{f:inter_preci}
\end{figure*}
In Section~\ref{sss:resu_veri_some} and Section~\ref{sss:resu-accuracy} we discussed the bias and precision of stellar labels from MUSE spectra in comparison with results from LAMOST spectra and discussed the influence of losing wavelength pixels in $3700-4750\Angstrom$ and $4750-5991\Angstrom$.
The theoretical precision, which is the standard deviation of each label X (written as $\mathrm{X_{-}err}$ in Table~\ref{tab:tab2}) as a by-product of \emph{DD-Payne}{}, will also be affected by the changes in wavelength coverage.
Since we don't have enough LAMOST and MUSE common stars with a given SNR, we use the method \emph{Chem-I-Calc}{} \citep{2020ApJS..249...24S} to forecast the theoretical precision, which uses \emph{ab initio}{} spectral model and the Cramér-Rao Lower Bound (CRLB)
to quantify the chemical information content of stellar spectra in terms of theoretical precision.
Figure~\ref{f:inter_preci} shows the predicted theoretical precision of stellar labels for a typical solar-metallicity K-Giant with a g-band SNR of 50 $\text{pix}^{-1}${} in three different scenarios.
By comparing these scenarios, we find that when losing wavelength range in $3700-4750\Angstrom$ (cyan versus blue), the theoretical precision of all the labels is getting slightly worse by an average factor of 1.17, but there is no label having a large change.
When further losing wavelength pixels in wavelength $4750-5991\Angstrom$ (blue versus pink), the precision of all the labels is getting worse by an average factor of 2.16.
However, there are some labels such as Cr, N, Na, and C which have a much greater theoretical precision change with a factor of 2.66, 4.30, 4.02, 3.53, respectively, as we lose critical absorption lines in a more restricted wavelength range.
This is shown in the gradient spectra of these labels from Figure 22 of X19, where they show that there are no significant features for these abundances in the remaining wavelength range.
Fortunately, precision of labels like \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{}, and some $\alpha$ elements are not significantly affected.
\subsection{Validation 2: observations towards the Galactic bulge}
\label{ss:resu-bulge}
From the above sections we have shown that we can derive stellar labels precisely for individual stars in MUSE observations via the utilization of the \emph{DD-Payne}{}. Here we want to demonstrate that our method is also useful to determine stellar labels for stars in dense fields. The Milky Way bulge is one of the densest regions of the Galaxy.
In the past decades, there have been many fiber-fed spectroscopic surveys focusing on the bulge to study its chemical distribution such as Gaia-ESO \citep{2012Msngr.147...25G}, ARGOS \citep{2013MNRAS.430..836N}, and GIBS survey \citep{2002Msngr.110....1P}, which have revealed multiple stellar components with different mean metallicity.
A highlight of this study is from \citealt{2019A&A...626A..16R} who revealed the bimodality in \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution using the bulge stars from APOGEE DR14 \citep{2018ApJS..235...42A}.
In terms of the integral-field spectroscopic survey, \citealt{2018A&A...616A..83V} observed four fields with MUSE towards the inner bulge and measure radial velocity based on the CaT region of spectra and reached $\sigma V_{\mathrm{gc}}\sim$ 140 {\rm km s}$^{-1}$. They also found that the velocity dispersion peak is symmetric with respect to the distance $z$ from the Galactic plane.
To summarize previous bulge studies, chemical element distributions have been studied based on high-resolution fiber-fed spectroscopic surveys, and kinematics has been studied using IFS data.
Since our method can precisely determine stellar labels from MUSE spectra, this should enable us to study the chemodynamics of the Milky Way bulge using IFS data.
\subsubsection{Data refinement}
\label{sss:resu-bulge-data}
\begin{figure*}
\includegraphics[width=1.89\columnwidth]{figs_new/bulge_star_spectra_example.png}
\caption{
The detailed process of spectral processing of one bulge star in the MUSE data-cube. (a) Original MUSE spectra after degrading which demonstrates the high extinction in the Galactic center causing almost no flux in short wavelength. (b) Spectra after normalization, which shows the extremely high extinction makes the short-wavelength spectra between $(4750-5991\Angstrom)$ impossible to fit. (c) The predicted spectra and original MUSE spectra of this star. (d) The residuals of predicted spectra and the original MUSE spectra.
}
\label{f:bulge_emp}
\end{figure*}
\begin{figure*}
\includegraphics[width=1.89\columnwidth]{figs_new/H-R-color-rgc_av.png}
\caption{
H-R diagrams of stars with good fitting to the model defined by Equation~\ref{eqn:thresholdval21}. The left-hand side panel is color-coded with $R_{\mathrm{gc}}${} (distance to the Galactic center) in cylindrical Galactocentric coordinates, where distance from the Sun to the Galactic Center is adopted as $R_{\odot}${}$=8.2$ kpc, $z_{\odot}=25$ pc \citep{2016ARA&A..54..529B}. The right-hand side panel is color-coded with extinction $A_V$. Both $R_{\mathrm{gc}}${} and $A_V$ are predicted by BSTEP \citep{2018MNRAS.473.2004S} taking parameters of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Mg/Fe}]}{} from DD-Payne and $m_{J}$, $m_{Ks}$, $m_{H}$ magnitude from VVV DR2.
}
\label{f:hrbulge}
\end{figure*}
The bulge data processing begins with the steps outlined in Section~\ref{s:meth}.
The spectra are extracted from the 29 bulge data-cubes using PampelMUSE{} with VVV DR2 \citep{2017yCat.2348....0M} as the input photometry catalog. Next, the spectra are degraded to LAMOST spectral resolution of R$\sim$1800.
Figure \ref{f:bulge_emp} shows the degraded spectra, both raw and normalized, of one bulge star (VVV{-}J175141.74{-}282207.85) extracted from the data-cube.
The bulge stars pose several challenges for data reduction.
They have large heliocentric distances, which means they are typically faint and have low SNR.
These stars also typically lie along lines of sight with high extinction, making SNR quite low in the shorter wavelength regions.
Finally, subtracting the night sky emission lines from faint stellar spectra is challenging.
This highlights the challenges associated with reducing the spectra in the bulge region and reinforces the need to adopt additional data refinement procedures.
To remove emission lines, we attempted to apply the Zurich Atmosphere Purge (ZAP) \citep{2016MNRAS.458.3210S}, which is an approach based on principal component analysis (PCA) to reduce emission lines for faint spectra in data-cubes.
However, using ZAP has some undesirable side effects.
By adjusting some tunable parameters in ZAP, we can increase the efficiency of subtracting the emission lines, but then ZAP has a significant impact on the absorption lines which are desirable for estimating chemical abundances. Strangely, the spectra of bright stars were also found to become noisy.
In the end, we find the most effective way to deal with emissions is to replace emission pixels with the average value of adjacent non-emission pixels
In this way, we can retain more pixels in the spectra.
In addition, due to extinction in the bulge, the flux and hence the SNR in short-wavelength ranges is not enough to extract useful information (See panel (a) and (b) in Figure~\ref{f:bulge_emp}).
The NaD doublet line $(5803-5966\Angstrom)$ is also masked in data-cubes that are observed using the adaptive optics module.
Therefore, we decided to discard information for wavelengths less than $5991\Angstrom$.
We also mask the following telluric bands; $6270-6305\Angstrom$, $6850-6966\Angstrom$, $7160-7320\Angstrom$, $7585-7715\Angstrom$, $8130-8310\Angstrom$.
The widths of the telluric bands are narrower than those used by X19, this was done to increase the number of useful wavelength pixels.
This range was determined after many tests to ensure that the unmasked pixels provide useful information about the chemical elements. Figure~\ref{f:acc_galah} and Figure~\ref{f:acc_apogee} can indicate that when losing pixels in $4750-5991\Angstrom$, the dispersion of some elements is still below 0.1 dex.
In panel (c) of Figure \ref{f:bulge_emp} shows the best-fit model spectra in red, alongside the original spectra in black. It is clear from the comparison between LAMOST and MUSE samples presented in Figure~\ref{f:muse_ddp_comparison}, that the spectra of bulge stars have more noise, due to a combination of these stars having higher extinction and lower average SNR.
In addition, the distribution of residuals between the observed and the best-fit model spectra for a bulge star (Figure~\ref{f:bulge_emp} panel (d)) is larger than that for the star in Figure~\ref{f:muse_ddp_comparison}, which shows the difficulty of fitting spectra in the bulge fields.
After these refinement and masking steps, stellar labels are derived from two \emph{DD-Payne}{} models. We adopt the labels from the \emph{DD-Payne-A}{} model and elect not to use the \emph{DD-Payne-G}{} model to compare directly with high-resolution APOGEE data which has ample stars in the bulge.
Given the loss of precision due to the omission of the blue wavelength region, we only adopt the \ensuremath{[\mathrm{Mg/Fe}]}{} when studying $\alpha$ abundances for the bulge fields.
Our final MUSE bulge catalog is comprised of 8720 individual spectra containing 2721 unique stars, with the following columns
\ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{} and \ensuremath{[\mathrm{Mg/Fe}]}{}.
On average, from each cube about 300 spectra were extracted and 40\% of them have $$\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${}>80$ $\text{pix}^{-1}${}.
Compared with typical MUSE wavelength coverage, the bulge spectra lose the bluer regions of the spectrum, between $4750-5991\Angstrom$, so we adopt more strict selection criteria compared to our LAMOST verification sample with
\begin{equation}
\left\{\begin{array}{l}
$\ensuremath{\mathrm{SNR_{\mathrm{MUSE}}}}${}>80 \text{pix}^{-1} \\
\rm{corr_{-}flux}>0.85 \\
\end{array}\right.
\label{eqn:thresholdval21}
\end{equation}
Here the SNR$_{\mathrm{MUSE}}$ is the SNR $\text{pix}^{-1}${} in wavelength higher than $5991\Angstrom$. Since there are multiple spectra for each star, we selected the observation having the highest corr$_{-}$flux between the MUSE and model-predicted spectra, to represent the most physical fitting.
\begin{figure}
\includegraphics[width=0.95\columnwidth]{figs_new/av_rgc_z_galaxia.png}
\caption{
\textbf{Top:} Spatial distribution in cylindrical Galactocentric coordinates ($R_{\mathrm{gc}}${},z) of stars observed in three nine bulge fields. The distance to the Galactic center is adopted as $R_{\odot}${}$=8.2$ kpc, $z_{\odot}=25$ pc, same as Figure~\ref{f:hrbulge}. The red dots are stars with $R_{\mathrm{gc}}${}$<3.5$kpc.
\textbf{Middle:} Extinction-$R_{\mathrm{gc}}${} distribution of all-stars with good fitting to the model defined by Equation~\ref{eqn:thresholdval21} in dots color-coded by absolute J mag. The red curve shows the median value of each distance bin and the blue fill shows the standard deviation of each bin.
\textbf{Bottom:} Extinction $A_V$ as a function of Galactocentric radius $R$ for various lines of sight from the Sun. The extinction is based on an analytical model from \citealt{2011ApJ...730....3S} consisting of an exponential disc with warp and flare; scale length $R_{\rm d}=4.2$ kpc and scale height $0.88$ kpc. From lines towards the center, we could see that with the larger line of sight latitude, the extinction will become a constant more quickly, which is in agreement with the left bottom plot.
}
\label{f:avrgczgal}
\end{figure}
\subsubsection{Distance estimation and its relationship with extinction}
\label{sss:resu-bulge-dist-extin}
To discriminate bulge stars from other foreground stars, we make use of stellar distance. The distance was estimated by BSTEP (Bayesian stellar labels estimator) \citep{2018MNRAS.473.2004S}, a Bayesian scheme that is used for predicting stellar ages, distances, and extinction from a set of observables given by
\begin{equation}
\mathbf{y}=\left(l, b, m_{J}, m_{K s}, T_{\mathrm{eff}}, \log g,[\mathrm{Fe/H}]_{\rm eff}\right),
\end{equation}
Briefly speaking, for each star, if we know its position, \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{} and magnitude, its intrinsic parameters such as distance, age, and extinction can be derived by making use of isochrones \citep{2017ARA&A..55..213S}. The isochrones employed in this study are from \citealt{2012MNRAS.427..127B}.
For our catalog, parameters of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{} are determined by \emph{DD-Payne}{}. As for $m_J$, $m_{Ks}$, they are derived from VVV DR2 catalog.
Figure \ref{f:hrbulge} shows the H-R diagram of stars with model spectra with good fitting to the model defined by Equation~\ref{eqn:thresholdval21} and color-coded by Galactocentric cylindrical radial distance $R_{\mathrm{gc}}${} and extinction $A_V$.
We identify bulge stars with $R_{\mathrm{gc}}$$<3.5$kpc.
The left panel shows that bulge stars are predominantly giants. The right panel shows that bulge stars have extinction $A_V$ in the range 4-6 mag. Note, there is a selection bias in the sense that low extinction stars are more likely to be detected.
In fact, extinction towards the Galactic center can be as high as $A_V=50$ mag \citep{2016MNRAS.456.2692N}.
In Figure~\ref{f:avrgczgal} we explore the extinction in more detail.
The top panel shows the $(R_{\mathrm{gc}} ,z)$ distribution of stars in nine MUSE cubes. The line of sight of three fields can be seen.
Stars with distances beyond the Galactic center have low temperature ($\ensuremath{T_{\mathrm{eff}}}{}<4300$ K) and surface gravity ($\mbox{$\log g$}{}<1$), which fall outside of the label ranges for which we can reliably determine $\alpha$ abundances, so these stars were removed.
The middle panel of Figure~\ref{f:avrgczgal} shows a scatter plot of extinction and $R_{\mathrm{gc}}${} color-coded by absolute magnitude in $J$ band.
The median value of extinction is shown by the red line, along with 25 and 75 percentile dispersion as a shaded blue region.
This figure demonstrates that MUSE can successfully extract bulge giant stars with extinction up to $A_V=6$ mag.
To understand the trend of extinction with $R_{\mathrm{gc}}${}, in the bottom panel of Figure~\ref{f:avrgczgal} we show extinction $A_V$ as a function of Galactocentric radius $R_{\mathrm{gc}}${} for various lines of sight from the Sun as predicted by a theoretical model in \textit{Galaxia} \citep{2011ApJ...730....3S} consisting of an exponential disc with warp and flare; scale length $R_{\rm d}=4.2$ kpc and scale height $0.88$ kpc.
It can be seen that if the latitude $|b|$ is greater than 5 degrees, the extinction first increases as we go towards smaller $R$, but then flattens and reaches a plateau at distances 1-2 kpc away from the sun.
A similar trend is seen in the observed data in the middle panel. This flattening is due to the finite height of the dusty disc responsible for the extinction.
For a given line of sight as long as we are in the dusty disc, the extinction increases with distance, but once we are out of the dusty disc there is no further increase in extinction.
The model qualitatively explains the trend seen in the observed data, but there are differences. The extinction for $b=5$ degrees is not high enough compared to our observations. For lower $|b|$ extinction can be high but the rise with distance is much slower.
These differences suggest that the scale height of the disc in the model is too high.
\subsubsection{\ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution in the Galactic bulge}
\label{sss:resu-bulge-mgfe}
\begin{figure*}
\includegraphics[width=1.80\columnwidth]{figs_new/mg-fe-bulge.png}
\caption{
Distribution of stars satisfying $|z|<0.5$ kpc in the (\ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{}) plane as a function of $R_{\mathrm{gc}}${}, with $R_{\mathrm{gc}}${} increasing from left to right.
\textbf{Top}: The \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution of stars in APOGEE-\emph{Payne} \citep{2019ApJ...879...69T}. \textbf{Bottom}: The same distribution of stars in MUSE data-cubes towards the Galactic bulge with abundances measured by \emph{DD-Payne-A}{} model (hereafter MUSE-\emph{DD-Payne-A}{}). The labels are calibrated by adding an offset for \ensuremath{[\mathrm{Mg/Fe}]}{}, which is due the \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{T_{\mathrm{eff}}}{} trend inherited from APOGEE-\emph{Payne} (see details in Appendix~\ref{app:mg_teff_trend}), respectively.
In each panel, the black dots mark stars in this location and the distribution is smoothed by a Gaussian kernel function and then used to plot the color maps.
For each $R_{\mathrm{gc}}${} bin, the APOGEE-\emph{Payne} and MUSE-\emph{DD-Payne-A}{} distributions are in agreement with the densest peak at the same location. This validates the precision of the MUSE results in dense regions.
}
\label{f:mgfe_bulge}
\end{figure*}
We next analyze the abundance distribution of stars towards the bulge. We make some additional quality selections to guarantee all the stars have precisely estimated parameters:
\begin{equation}
\left\{\begin{array}{l}
\ensuremath{[\mathrm{Fe/H}]}>-1.5 \mathrm{dex} \\
\ensuremath{T_{\mathrm{eff}}}>4300 \mathrm{K} \\
\mathrm{\ensuremath{[\mathrm{Mg/Fe}]}{}_{-}gradcorr}>0.65 \\
\chi^2<20 \\
\end{array}\right.
\label{eqn:thresholdval22}
\end{equation}
Here stars with \ensuremath{[\mathrm{Fe/H}]}{}$<-1.5$~dex
and \ensuremath{T_{\mathrm{eff}}}{}$<4300$K are removed as it has been demonstrated in X19 that \ensuremath{[\mathrm{Mg/Fe}]}{} for these giants are erroneous with low gradient spectra correlation coefficients (see their Figure 2).
Moreover, we apply a cut in the $\chi^2<20$ to remove stars with poor fittings from \emph{DD-Payne}{} according to the $\chi^2$ distribution of the LAMOST-MUSE cross-validation sample. The removed stars are primarily giants with low \ensuremath{T_{\mathrm{eff}}}{} ($<4300$ K) and \mbox{$\log g$}{} ($<1.5$ dex), which shows low correlation in gradient spectra of \emph{DD-Payne}{} (see Figure 2 of X19).
We also limit our analysis to stars with correlation coefficient gradient spectra for
\ensuremath{[\mathrm{Mg/Fe}]}{} to larger than 0.65.
After applying all the selection criteria, we end up with 75 stars physically located in the Galactic bulge.
Panel (a) and (d) of Figure \ref{f:mgfe_bulge} show the \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution of stars in MUSE data-cubes towards the Galactic bulge with values measured by \emph{DD-Payne-A}{} model (hereafter MUSE-\emph{DD-Payne-A}{}) with comparison of the same plot using stars from APOGEE-\emph{Payne} \citep{2019ApJ...879...69T}.
We use the Gaussian kernel estimation and plot the density color map.
The selection criteria of APOGEE stars is the same as that in \citealt{2019A&A...626A..16R} and the distances were estimated by \citealt{2016A&A...585A..42S}.
\ensuremath{[\mathrm{Mg/Fe}]}{} in MUSE-\emph{DD-Payne-A}{} are calibrated by adding an offset. This is because there is a non-negligible \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{T_{\mathrm{eff}}}{} trend in \emph{DD-Payne-A}{} model, which is inherited from the APOGEE-\emph{Payne} values in the training set (See Figure 14 in X19).
The average temperature differences between stars in APOGEE-\emph{Payne} and MUSE-\emph{DD-Payne-A}{} in the bulge will lead to roughly 0.08~dex deviation in \ensuremath{[\mathrm{Mg/Fe}]}{}. So we calibrate this deviation here.
This trend is beyond the scope of this study, and more details will be discussed in the Appendix~\ref{app:mg_teff_trend}.
In Figure \ref{f:mgfe_bulge}, both panel (a) and (d) demonstrate two sequences of stars in the bulge: Metal-poor/$\alpha$-rich sequence with center in (\ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{Fe/H}]}{}) space at $(0.36,-0.43)$ and metal-rich/$\alpha$-poor sequence at $(0.19, 0.18)$.
These two centers are in agreement in MUSE-\emph{DD-Payne-A}{} and APOGEE-\emph{Payne} samples. In addition, the two sequences merge at \ensuremath{[\mathrm{Fe/H}]}{}$\sim-0.07$ dex in both panels.
All these agreements reassure and validate the precision of the MUSE results, and the ability of \emph{DD-Payne}{} to measure stellar labels in dense fields.
However, The two plots also show differences.
The $\alpha$-rich stars in the MUSE-\emph{DD-Payne-A}{} are less dense than those in APOGEE-\emph{Payne} and the distribution is also broader.
Therefore, the small sample size of MUSE is certainly a limiting factor in comparing the two panels.
Another important factor that is different in the two panels is the distribution of stars in the $(R_{\mathrm{gc}},z)$ plane (See the top panel of Figure~\ref{f:avrgczgal}), neither is the apparent magnitude selection. MUSE data is restricted to 3 lines of sight, which are all close to the plane.
\subsubsection{\ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution of other regions}
\label{sss:resu-bulge-35}
In the top panel of Figure~\ref{f:avrgczgal}, we can see that in addition to the bulge stars, there are many foreground stars observed in these nine fields.
We can also study the \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distribution of these stars and compare them with distributions in APOGEE-\emph{Payne}.
Here, we select the stars with $R_{\mathrm{gc}}${} between $3\sim5$ and $5\sim7$ kpc, and plot the \ensuremath{[\mathrm{Mg/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distributions in the panel (b), (e), (c), (f) of Figure~\ref{f:mgfe_bulge}, respectively.
This is similar to \citealt{2015ApJ...808..132H, 2021MNRAS.507.5882S}, which used stars in APOGEE DR12 and plotted \ensuremath{[\mathrm{\alpha/Fe}]}{}-\ensuremath{[\mathrm{Fe/H}]}{} distributions in different Galactic locations.
Here the \ensuremath{[\mathrm{Fe/H}]}{} and \ensuremath{[\mathrm{Mg/Fe}]}{} values in MUSE-\emph{DD-Payne-A}{} are calibrated for the same reason as before.
In this figure, for the same $R_{\mathrm{gc}}${}, the distributions of stars in MUSE-\emph{DD-Payne-A}{} and APOGEE-\emph{Payne} samples are in good agreement, with the centers at the same (\ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{Fe/H}]}{}) locations.
All the panels are dominated by low-$\alpha$ stars, and the overall trends are consistent.
Despite the small number of MUSE stars (52 in panel (e) and 103 in panel (f)), the good agreement with APOGEE-\emph{Payne} is once again reassuring and validating the precision of the MUSE results.
\section{Estimating exposure times to achieve a given uncertainty}
\label{s:predict}
The uncertainty $\sigma_{\rm X}$ of a stellar label $X$ derived with MUSE using the \emph{DD-Payne}{} depends on the signal to noise ratio (SNR hereafter) of the spectra, which in turn depends on the exposure time $T_{\rm exp}$ and the magnitude of a star.
The brighter the star and longer exposure time, the higher the SNR and thus the more precise the stellar labels will be.
Therefore, when doing a survey or planning future observations, one should ensure that the exposure time is long enough to reach the expected uncertainty.
In the next sections, we provide formulas to estimate the $\sigma_{\rm X}$ as a function of $V$ band magnitude and exposure time, to provide a reference for future MUSE observations.
To do this, we first estimate the $\sigma_{\rm X}$ as a function of SNR. Next, we estimate the SNR as a function of magnitude $V$ and $T_{\rm exp}$.
\subsection{Uncertainty of stellar labels as a function of SNR}
\begin{figure}
\includegraphics[width=0.98\columnwidth]{figs_new/predict_sigma_snr_new.png}
\caption{
Prediction of precision of stellar labels as a function of SNR. It is derived by estimating the theoretical precision using \emph{Chem-I-Calc}{} from typical K-giant spectra from MUSE Exposure Time Calculator in different SNRs. Then a factor is applied to transfer the theoretical precision to real precision for each label which is calculated by Equation~\ref{e:facpair}.}
\label{f:predict_sigma}
\end{figure}
\begin{table}
\caption{Parameters for the equation~\ref{eq:sigmasnr} of uncertainty as a function of SNR.}
\begin{tabular}{lcclcc}
\hline
stellar label & $A_X$ & $B_X$ & stellar label & $A_X$ & $B_X$\\
\hline
\ensuremath{T_{\mathrm{eff}}}{} & -0.9875 & 1.652 & \ensuremath{[\mathrm{Ca/Fe}]}{} & -0.9834 & 0.868\\
\mbox{$\log g$}{} & -0.9922 & 1.052 & \ensuremath{[\mathrm{Ti/Fe}]}{} & -0.991 & 0.856\\
\ensuremath{[\mathrm{Fe/H}]}{} & -0.9746 & 0.5823 & \ensuremath{[\mathrm{C/Fe}]}{} & -0.9891 & 0.5737\\
\ensuremath{[\mathrm{Mg/Fe}]}{} & -0.9862 & 0.4486 & \ensuremath{[\mathrm{Ni/Fe}]}{} & -0.9925 & 0.8681\\
\ensuremath{[\mathrm{Si/Fe}]}{} & -0.9886 & 0.7546 & \ensuremath{[\mathrm{Cr/Fe}]}{} & -0.9835 & 0.7855\\
\hline
\end{tabular}
\label{tab:tab3}
\end{table}
Figure~\ref{f:predict_sigma}
shows the uncertainty $\sigma_{X}$ of a stellar label $X$ as a function of SNR.
We firstly derive theoretical uncertainty $\sigma_{ X,{\rm Theoretical}}$ using \emph{Chem-I-Calc}{} \citep{2020ApJS..249...24S} from a typical K-giant spectra.
SNR is estimated from MUSE Exposure Time Calculator\footnote{\url{https://www.eso.org/observing/etc/bin/gen/form?INS.NAME=MUSE+INS.MODE=swspectr}}.
The air-mass and seeing are set as 1.5 and $0.8^{\prime \prime}$.
Then $\sigma_{X}$ is calculated by
\begin{equation}
\sigma_{X}=\sigma_{ X,{\rm Theoretical}}({\rm SNR})f_{X,{\rm corr}},
\end{equation}
where $f_{X,\rm corr}$ is the scale factor applied to transform the theoretical uncertainty to the observed uncertainty for each label $X$.
The factor is calculated from MUSE repeat observations by computing the dispersion of pairwise differences as follows.
\begin{equation}
f_{X,\rm corr} = \text{std-dev} \left( \frac{X_{m}-X_{n}}{
\sigma^{2}_{X,{\rm Theoretical},m}
+\sigma^{2}_{X,{\rm Theoretical},n}} \right),
\label{e:facpair}
\end{equation}
where $m$ and $n$ represent the repeated observations of a star. $\sigma_{X,{\rm Theoretical}}$ is the theoretical uncertainty of a spectra as given by \emph{Chem-I-Calc}{}.
Comparing with the error from \emph{DD-Payne}{}, \emph{Chem-I-Calc}{} estimates are normally 1.5 times smaller (see Fig. 21 in \citealt{2020ApJS..249...24S}), so we multiply this factor.
Then $\sigma_{ X,{\rm Theoretical}}$ mentioned in this section means the theoretical uncertainty of multiplying 1.5.
Figure~\ref{f:predict_sigma} indicates that the uncertainty will decrease as the SNR increases.
Generally, \ensuremath{[\mathrm{Mg/Fe}]}{} has the lowest uncertainty with $\sigma_{\ensuremath{[\mathrm{Fe/H}]}}<0.1$ dex for SNR higher than 29.44 $\text{pix}^{-1}${}.
As for other elements, to make the uncertainty less than 0.1 dex, the SNR needs to be higher than $\sim$39.00, 42.02, 59.54, 79.35, 74.62, 76.24, 65.39 $\text{pix}^{-1}${} for \ensuremath{[\mathrm{C/Fe}]}{}, \ensuremath{[\mathrm{Fe/H}]}{}, \ensuremath{[\mathrm{Si/Fe}]}{}, \ensuremath{[\mathrm{Ca/Fe}]}{}, \ensuremath{[\mathrm{Ti/Fe}]}{}, \ensuremath{[\mathrm{Ni/Fe}]}{} and \ensuremath{[\mathrm{Cr/Fe}]}{}, respectively.
To facilitate the calculation of the SNR required to achieve a given uncertainty, we provide analytical fits to the uncertainty curves shown in Figure~\ref{f:predict_sigma}.
The curves are fitted with the functional form of
\begin{equation}
\log \sigma_{X} = A_X \log ({\rm SNR}) + B_X,
\label{eq:sigmasnr}
\end{equation}
The coefficients $A_X$ and $B_X$ for each label are listed in Table~\ref{tab:tab3}.
\subsection{SNR as a function of magnitude and exposure time}
\begin{figure}
\includegraphics[width=0.98\columnwidth]{figs_new/mag_snr_new.png}
\caption{
SNR of MUSE spectra at $R\sim3000$ as a function of $V$ magnitude (purple line) for an exposure time of one hour. Each colored line represents the threshold where the uncertainty of a label reaches 0.1 dex (or 10 K),
which is derived using equation~\ref{eq:sigmasnr}.
All chemical abundances can be derived with uncertainty less than 0.1 dex till $V=18.5$ mag, for \ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{C/Fe}]}{} and \ensuremath{[\mathrm{Fe/H}]}{} even till $V=19.7$ mag.
}
\label{f:mag_snr}
\end{figure}
Using equation~\ref{eq:sigmasnr}, with the known SNR we can obtain the uncertainty of stellar labels from its spectra.
We know SNR depends on magnitude $V$, exposure time $T_{\exp}$, and air-mass.
Therefore, when making science goals, what we care about is how long the exposure time is needed to achieve the required label uncertainty for stars with different magnitudes.
Here we provide an analytical function to calculate SNR with given $V$ magnitude and exposure time $T_{\exp}$ as
\begin{equation}
{\rm SNR}={\rm SNR}_{\rm 1 hr}(V) \sqrt{\left(T_{\exp } / 3600 \mathrm{s}\right)}.
\label{eq:snrsnr1rexp}
\end{equation}
Here ${\rm SNR}_{\rm 1 hr}(V)$ is the SNR estimated from the K-giant spectra by the MUSE Exposure Time Calculator for an exposure of one hour.
The analytical function is given by
\begin{equation}
{\rm SNR}_{\rm 1 hr}(V)=\frac{462.34 \times 10^{0.4(14.83-V)}}{\sqrt{0.00223+10^{0.4(14.83-V)}}},
\label{eq:snr1hv}
\end{equation}
This idea is taken from \citealt{2018MNRAS.473.2004S}. For the above equations, we set the air-mass as 1.5 and seeing as $0.8^{\prime \prime}$. Therefore, by combining equations~\ref{eq:sigmasnr}-\ref{eq:snr1hv}, with a given magnitude $V$ and exposure time $T_{\exp}$, one can predict the uncertainty of each label.
Figure~\ref{f:mag_snr} provides the SNR as a function of $V$ magnitude for an one-hour exposure (purple line).
We also plot the threshold of maximum magnitude for each label where the uncertainty reaches 0.1 dex or 10K, which are derived by Figure~\ref{f:predict_sigma}.
As we can see, all chemical abundances can be derived with uncertainty less than 0.1 dex till $V=18.5$ mag, for \ensuremath{[\mathrm{Mg/Fe}]}{}, \ensuremath{[\mathrm{C/Fe}]}{} and \ensuremath{[\mathrm{Fe/H}]}{} even till $V=19.7$ mag.
This figure combining with equations~\ref{eq:sigmasnr}-\ref{eq:snr1hv} can be used as a guide for future research, providing an indication of the exposure time for different magnitude stars to achieve the expected uncertainty.
\section{Discussions}
\label{s:discuss}
\subsection{Performance of MUSE stellar label extraction under different scenarios}
Ideally, the methods outlined in this paper to determine stellar labels would be applicable to all stars in MUSE data-cubes.
However, the label ranges of the \emph{DD-Payne}{} training set do not cover the space for all potential observations: for stars outside of this parameter space, for example having very low metallicity $(\ensuremath{[\mathrm{Fe/H}]}<-1.5)$ or very high effective temperatures $(\ensuremath{T_{\mathrm{eff}}}>7000K)$, labels are determined through extrapolation of the model.
Despite this, some studies have shown \emph{DD-Payne}{} can do some moderate extrapolation in the linear region of the gradient spectra varying with labels.
This is because \emph{DD-Payne}{} imposes the gradient spectra to be similar to that of the Kurucz models \citep{1970SAOSR.309.....K, 1993KurCD..18.....K, 2005MSAIS...8...14K}, which is an advantage of such a hybrid method compared to pure data-driven models.
For example, the LAMOST \ensuremath{[\mathrm{Fe/H}]}{} estimates can be robust down to $-2.5$ dex (see Fig. 10 of \citealt{2019ApJ...887..237C}); \emph{DD-Payne}{} can also generate reasonable [Ba/Fe] stars with $1<{\rm [Ba/Fe]}<3$ dex and $\ensuremath{T_{\mathrm{eff}}}{}>7000$K \citep{2020ApJ...898...28X}.
Typically in MUSE data, stellar spectra come from the following types of fields: random individual stars near a galaxy, high extinction fields, and globular clusters.
In each of these scenarios the
performance of the spectroscopic analysis is different and some of them pose unique challenges for analysis.
For individual stars near a galaxy, they typically have high SNR, as observations in galaxy fields generally have long exposure times. Therefore, as long as the PSF does not overlap with the nearby galaxy, it is relatively simple to extract and estimate the stellar labels. All of the common stars between MUSE and LAMOST belong to this case; one typical spectrum is shown in Figure~\ref{f:muse_ddp_comparison}.
For these stars, stellar labels will have high precision due to their high SNR, as well as the fact that most of them belong to the nearby disk, which are stellar populations that are well covered by the \emph{DD-Payne}{} training sets.
For stars in high extinction fields, such as the bulge, the spectra are restricted to a narrower range in wavelength and the SNR of the extracted spectra is in general low (due to a combination of large distance and dust). For low SNR, in addition to SNR directly affecting precision, the emission lines are comparatively stronger and more difficult to remove. The above-listed factors lead to lower precision in stellar labels.
Hence, in high extinction regions, one can only study giants which are intrinsically bright.
Therefore, we can only get reliable label estimates for stars with relatively low extinction ($A_V<6$ mag) along the bulge lines of sight, which was shown in the middle plot of Figure~\ref{f:avrgczgal}.
As for stars of $A_V>6$ mag in the Galactic center, the dust will make the giants difficult to observe with typical MUSE exposure times.
For globular clusters, which have high stellar density, our work will be of particular importance.
It will provide an automated way to determine high precision stellar labels for these dense fields down to the center of the cluster; previous studies using fiber-fed spectrographs were often limited to the outskirts of these objects.
Additionally, it is possible to increase the precision of stellar parameters and several abundance estimations by making use of full wavelength fitting, which has not been done so far.
Previously, the MUSE GC survey \citep{2016A&A...588A.148H, 2016A&A...588A.149K} observed 26 globular star clusters, with most of these studies focusing on specific features of spectra or just metallicity, e.g. emission-line studies \citep{2019A&A...631A.118G}, the dynamics using CaT absorption lines \citep{2018MNRAS.473.5591K}, the relative abundance variations using the equivalent widths \citep{2019A&A...631A..14L}, metallicity distributions \citep{2020A&A...635A.114H, 2021A&A...653L...8L}.
Our method has the potential to enable more chemical abundances measured automatically by the full-wavelength fitting frameworks, rather than applying any spectral library to execute the fittings.
Even though globular clusters typically have $\ensuremath{[\mathrm{Fe/H}]}{}<-1.0$~dex, some of them are still in the parameter range of \emph{DD-Payne}{}, and the moderate linear extrapolation ability of \emph{DD-Payne}{} still enables this method to measure robust stellar labels. We will perform the test of it in the future work.
\subsection{Computational challenges in spectral extraction}
In this work, the spectral extraction processes are executed using PampelMUSE \citep{2013A&A...549A..71K}. The typical time spent on extracting spectra from each data-cube with a 24-core CPU and 32GB RAM is $\sim30$ minutes and it is independent of the number of stars in the field. Therefore, for the verification data analyzed in this paper (the cross-match of LAMOST and MUSE), we spent more time on spectral extraction for these cubes compared to the data-cubes in the bulge, even though there were two orders of magnitude more spectra extracted from the bulge fields. Hence, to perform a survey of all individual stars in publicly available MUSE data-cubes, the most time-consuming step will be the extraction of spectra.
For individual stars in sparse fields or more limited studies focused purely on kinematics in dense fields, it is also possible to use other stellar extraction strategies such as aperture photometry in IRAF for each layer subtracted by the sky estimated locally (e.g. \citealt{2018A&A...616A..83V}).
This approach has the advantage of retaining as many stars as possible and has a more accurate estimate of the sky background, which will lead to reduced artifacts in the final spectra \citep{2018A&A...616A..83V}.
For regions with modest crowding (i.e., $<200$ stars per field), aperture photometry and PSF-fitting could yield similar results. However, PSF-fitting is still necessary when dealing with more crowded fields ($>200$ stars per field) as there will be significant blending for stars in these dense regions.
\citealt{2013A&A...549A..71K} tested the performance of PampelMUSE{} PSF-fitting using a simulated MUSE data-cube towards the center of a globular cluster 47 Tuc. Under the average seeing of 0.8 arcsec, $\sim$5000 useful spectra could be extracted. In good conditions, this value can be 3 or 4 times larger. With severe crowding, aperture photometry will suffer from significant blending between stellar spectra. Therefore, for globular clusters, we still recommend PampelMUSE.
\subsection{Future research}
There are several aspects in the data pipeline that can be improved in the future.
In the methods outlined in this paper, stellar labels are estimated from the \emph{DD-Payne}{}, which depends sensitively on the precision of labels in GALAH DR2 \citep{2018MNRAS.478.4513B} and APOGEE-\emph{Payne} \citep{2019ApJ...879...69T}.
Given that improved versions of both these catalogs are now available, GALAH DR3 \citep{2021MNRAS.506..150B} and APOGEE DR16 \citep{2020ApJS..249....3A}, one needs to update the \emph{DD-Payne}{} models.
In addition, we can train a new \emph{DD-Payne}{} model if a sufficient number of common stars in MUSE and other high-resolution spectroscopic surveys for a training set is available.
The advantage is that there will be no need to degrade the MUSE spectra to a lower resolution, which will help extraction of blended features.
In addition, some absorption lines such as [Na/Fe] and [O/Fe] are masked in the LAMOST spectra due to dichroic issues in the wavelength range $(5803-5966\Angstrom)$.
This range can now be used when working in MUSE non-AO mode.
It will enable to estimate precise sodium and oxygen abundances, which is essential in identifying different stellar populations in globular clusters \citep{2018ARA&A..56...83B}. The primary difficulty for retraining is having a training set that is both large and covers a wide range of parameter space.
In the future, the method outlined here can be utilized for many purposes. A survey can be executed based on stars from publicly available MUSE data-cubes using Gaia eDR3 \citep{2021A&A...649A...1G} and SkyMapper \citep{2007PASA...24....1K} as photometric input catalogs. Our method can be applied to estimate their stellar labels and compile a catalog that will be useful for Galactic archaeology studies.
For the Galactic bulge, our method has the opportunity to study the chemistry and dynamics of the Galactic center in greater detail.
However, more observational data is needed, such as fields in Baade's window, which are less affected by the dust.
In addition, this method can be applied to globular clusters observed by \citealt{2016A&A...588A.148H, 2016A&A...588A.149K} to study their chemistry, and identify multiple stellar populations if any, and their connection to kinematics.
Moreover, the interest in using MUSE for spectroscopy of resolved stellar populations has been growing quite strongly, even beyond the limits of the Galaxy (e.g. \citealt{2019AN....340..989R}), our method can also provide parameter estimations for these targets.
With the launch of the BlueMUSE project \citep{2019arXiv190601657R}, we will obtain spectra in the wavelength range between $3500-5800\Angstrom$, which is also covered by LAMOST. As mentioned previously, the loss of the blue portion of the spectrum in MUSE is the primary reason why we are not able to estimate as many precise chemical elements as those determined from LAMOST spectra.
BlueMUSE will provide the opportunity to measure these abundances with higher precision, and for more elements that are currently not available; critically the s-process elements such as [Y/Fe] and [Ba/Fe] which provide key age diagnostics for stellar targets \citep{2020arXiv201113745H}.
The blue portion will also bring strong absorption features for determining precise abundances of Na, C, N, and O, which are essential for identifying multiple populations in globular clusters.
In addition, MAVIS \citep{2020arXiv200909242M}, which is the new IFS instrument being built for ESO's VLT AOF (Adaptive Optics Facility, UT4 Yepun) also covers a wavelength range $3700-10000\Angstrom$ similar to BlueMUSE and MUSE. With the lower spatial sampling of $0.02^{\prime \prime} - 0.05^{\prime \prime}$, we have the opportunity to study the core of globular clusters with more stars providing an insight into its chemistry and dynamics. Because of the similar wavelength coverage of \emph{DD-Payne}{} and these instruments, our method can be directly applied to them.
Finally, based on the fact that the study of resolved stellar populations represents one of the major science cases for the European Extremely Large Telescope (ELT) \citep{2007Msngr.127...11G, 2009ASSP....9..225H}, it is reassuring that our method is paving the way for future studies in multi-dimensional chemical space.
For the Galactic bulge, ELT will allow us to measure chemical abundances all the way down to the main sequence turn-off stars \citep{Minniti_2007}. For these targets, we can use BSTEP \citep{2018MNRAS.473.2004S} to map the age distribution and reveal the detailed formation and evolution history of the innermost bulge, the peanut bar, and the long bar.
For Local Group galaxies, the ELT can also derive more stellar spectra of bright main-sequence (potentially even turn-off) stars and giants with necessary SNR than VLT (e.g. \citealt{2019MNRAS.486.5263M, 2020AJ....159..152C}); this includes the SMC and LMC, as well as more distant galaxies such as Andromeda and M33, extending Galactic Archaeology beyond the Milky Way to the Local Group.
With the unprecedented detection sensitivity of ELT, our method can provide a way for future stellar label estimation and science goals for the ELT-related IFS instruments, to measure stellar labels of stars beyond the Local Group \citep{Wyse_2005, 2008RMxAC..33...23E, 2012PASP..124..653G}.
\section{Summary}
\label{s:summary}
In this study, we applied \emph{DD-Payne}{} \citep{2019ApJS..245...34X}, which was developed for LAMOST spectra, on MUSE data-cubes and estimated \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{}, \ensuremath{[\mathrm{Fe/H}]}{} and chemical abundances of Mg, Si, Ti, Ca for individual stars in MUSE data-cubes.
By comparing our MUSE results with that of LAMOST using common stars, we show that despite instrumental differences, it is possible to extract precise stellar parameters and abundances with typical dispersion of \ensuremath{T_{\mathrm{eff}}}{}, \mbox{$\log g$}{} and other chemical abundances less than 75K and 0.15 dex and 0.1 dex, respectively.
In dense fields, which is the unique advantage of the MUSE instrument, we selected 29 data-cubes towards the Galactic center.
Based on previous studies \citep{2019A&A...626A..16R, 2015ApJ...808..132H, 2021MNRAS.507.5882S}, by comparing the \ensuremath{[\mathrm{Fe/H}]}{}-\ensuremath{[\mathrm{Mg/Fe}]}{} distribution estimated by our method in the bulge and inner disk region with stars from APOGEE-\emph{Payne}, we found excellent agreements in two sequences with similar center values and overall trends.
This indicates the ability of our method to measure chemical abundances in regions with high stellar density and extinction.
In addition, we also studies the extinction as a function of $R_{\mathrm{gc}}${} in the direction of the Galactic center and compared these results with the prediction from \textit{Galaxia} \citep{2011ApJ...730....3S}.
We found there is a qualitative agreement with the overall trends, but the scale height of the dust predicted by \textit{Galaxia} is higher than observations.
In addition, we provided analytical formulas to predict the precision of each label as a function of $V$ magnitude and exposure time. This can be used for observational proposal designing before a survey to achieve the expected label uncertainty.
In the future, this method can be applied to do a survey for all individual stars in public MUSE cubes and estimate their stellar labels. The result can be a supplementary catalog for all-sky spectroscopic surveys.
For the Galactic bulge, more observations are needed to expand the sample size. By combining the chemical and kinematic information, there is a great opportunity to study the evolution and assembly history of the Galactic center.
This method can also be applied to MUSE observations on globular clusters, to study the connection between kinematics and multiple populations.
This is the first time to measure stellar labels using full wavelength fitting in dense fields observed by the MUSE instrument. We demonstrate that despite PSF/LSF being different, it is possible to apply a model trained on spectra from one instrument to another. Given a large number of spectrographs having wavelength similar to LAMOST, our results suggest that one can easily do detailed spectroscopic analysis with them. In the future, this method can be used directly for BlueMUSE \citep{2019arXiv190601657R} and MAVIS \citep{2020arXiv200909242M}. It will also help with the label measurement development for IFS instruments on ELT \citep{2007Msngr.127...11G}, to study Galactic archaeology from the Milky Way to somewhere more than 10 Mpc away.
\section*{Acknowledgements}
We thank Xu Zhang, David M. Nataf, Thorsten Tepper Garcia, Sven Buder, and Jesse Van de Sande for their very useful comments.
This research has made use of the services of the ESO Science Archive Facility. The results are based on public data released from the MUSE commissioning observations at the VLT Yepun (UT4) telescope.
This work has also made use of the data from LAMOST. The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
This work was also based on data products from VVV Survey observations made with the VISTA telescope at the ESO Paranal Observatory under programme ID 179.B-2002.
This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093/cds/vizier). The original description of the VizieR service was published in \citealt{2000A&AS..143...33B}; Astropy\footnote{\url{http://www.astropy.org}}, a community-developed core Python package for Astronomy \citep{2013A&A...558A..33A, 2018AJ....156..123A}, numpy \citep{2020Natur.585..357H}, scipy \citep{2020NatMe..17..261V}, matplotlib \citep{2007CSE.....9...90H}, ZAP \citep{2016MNRAS.458.3210S}, PampelMUSE \citep{2013A&A...549A..71K}, DD-Payne \citep{2019ApJS..245...34X} and MPDAF \citep{2017arXiv171003554P}.
We acknowledge the University of Sydney HPC service at The University of Sydney for providing HPC resources that have contributed to the research results reported in this paper.
ZW is supported by the China Scholarship Council and Australian Research Council Centre of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D) through project number CE170100013.
YST is grateful to be supported by the NASA Hubble Fellowship grant HST-HF2-51425.001 awarded by the Space Telescope Science Institute. YST acknowledges financial support from the Australian Research Council through DECRA Fellowship DE220101520.
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
1,314,259,993,040 | arxiv |
\section{Introduction and preliminaries}{}
Associated with every continued fraction expansion is its Space of Jager Pairs, used in assessing the approximation quality for its convergents \cite{JK}. In this paper, we will examine the one parameter family of continued fractions
\[ [a_1,a_2,...]_k := \frac{k}{k+a_1+\frac{k}{k+a_2+ ...}}, \hspace{1pc} a_n \in \IZ_{\ge 0}, \hspace{1pc} 0 < k \in \mathbb{R},\]
introduced in \cite{HM1}, was also studied extensively in \cite{Avi1, Avi2}. Letting $x_0 :=[a_1,a_2,...]_k$ leads to the definitions of the convergents $\frac{p_0}{q_0} = \frac{0}{1}$ and
\[ \frac{p_n}{q_n} := [a_1,a_2,...,a_n]_k = \frac{k}{k+a_1+\frac{k}{k+a_2+ ... + \frac{k}{k+a_n}}}, \hspace{.1pc} n\ge1.\]
Define the future and past of $x_0$ at time $n \ge 1$ to be $x_n := [a_{n+1},a_{n+2},...]_k \in (0,1)$ and $y_n := -k-a_n - [a_{n-1},a_{n-2},...,a_1]_k \in (-\infty, -k]$ (we take $y_1 = -k-a_1$) and call the pairs $(x_n,y_n)$ the dynamic pairs of $x_0$ at time $n$. The pair of consecutive terms $(\theta_{n-1}(x_0),\theta_n(x_0))$ in the sequence of approximation coefficients $\{\theta_n\}_0^\infty := \left\{\abs{x_0 - \frac{p_n}{q_n}}q_n^2\right\}_0^\infty$ is called the approximation pair of $x_0$ at time $n$. There is a correspondence between these pairs, which was established by Haas and Molnar in \cite[Theorem 4]{HM1}, who proved that $(\theta_{n-1},\theta_n)$ is the image of $(x_n,y_n)$ under the bijective map
\begin{equation}\label{Psi}
\Psi_k(x,y) := \left(\frac{1}{x-y}, -\frac{x{y}}{k(x-y)}\right).
\end{equation}
We will prove that there is no such bijection when $0<k<1$ and, consequently, that the formula for the Space of Jager Pairs for these continued fraction expansions does not follow suit with the $k\ge 1$ cases.
\section{The space of Jager Pairs when k is less than one}{}
\noindent For all $k \in (0,\infty)$ and $a \in \IZ_{\ge 0}$ define the region $P_{(k,a)} := (0,1) \times (-k-a-1, -k-a]$ and $P_{(k,a)}^\# := \Psi_k(P_{(k,a)})$. The continuity of the map $\Psi$ provides us with the partition $\Gamma_k = \Psi_k(\bigcup{P_a}) = \bigcup{\Psi(P_a)} = \bigcup{P_a^\#}$. To find $P_a^\#$, we will use the following propositions:
\begin{proposition}{\cite[Proposition 3.1]{Avi1}}\label{P_a}\\
If $\Psi$ is injective on $P_{(k,a)}$, then $P_{(k,a)}^\#$ is the quadrangle with vertices $\left(\frac{1}{k+a},0\right), \hspace{.1pc} \left(\frac{1}{k+a+1},\frac{k+a}{k(k+a+1)}\right),\\ \hspace{.1pc} \left(\frac{1}{k+a+2},\frac{k+a+1}{k(k+a+2)}\right)$ and $\left(\frac{1}{k+a+2},0\right)$.
\end{proposition}
\begin{proposition}\label{Psi_fold}
$\Psi$ is injective on the region $\{(x,y) \in \mathbb{R}^2 : x + y < 0\}$ and is invariant under the reflection about the line $x+y=0$
\end{proposition}
\begin{proof}
The equality $\Psi(x, y) = \big(\frac{1}{x-y}, -\frac{xy}{k(x-y)}\big) = \Psi(-y,-x)$ proves the first statement. Let $(x_1,y_1)$ and $(x_2,y_2)$ be points which are on or below the line $x + y = 0$, so that $x_1 + y_1 \le 0, \hspace{.1pc} x_2 + y_2 \le 0$ and let $u_1,v_1,u_2,v_2 \in \mathbb{R}$ be such that $(u_1,v_1) = \Psi(x_1,y_1)= \Psi(x_2,y_2) = (u_2,v_2)$. Then the definition of $\Psi$ \eqref{Psi} implies that $\frac{1}{x_1-y_1} = u_1 = u_2 = \frac{1}{x_2-y_2}$, hence
\begin{equation}\label{x-y_G}
x_1 - y_1 = x_2 - y_2 \ne 0.
\end{equation}
Also
\[-\dfrac{u_1}{k}x_1y_1 = v_1 = v_2 = -\dfrac{u_2}{k}x_2y_2 = -\dfrac{u_1}{k}x_2y_2\]
and $u_1,u_2 > 0$ imply that $x_1y_1 = x_2y_2$, so that
\[(x_1 +y_1)^2 = (x_1 - y_1)^2 + 4x_1y_1 = (x_2 - y_2)^2 + 4x_2y_2 = (x_2 + y_2)^2.\]
Since $x_1 + y_1 \le 0$ and $x_2 + y_2 \le 0$, this last equation proves $x_1 + y_1 = x_2 + y_2$, which in tandem with condition \eqref{x-y_G}, proves $x_1=x_2$ and $y_1 = y_2$, hence $\Psi$ is injective on or below the line $x+y=0$.
\end{proof}
\noindent These propositions proves that when $k \ge 1$, the Space of Jager Pairs
\[\Gamma_k := \{(\theta_{n-1}(x_0),\theta_n(x_0))\}_{x_0 \in (0,1)}\]
is the convex quadrangle with vertices $\left(0,0\right), \hspace{.1pc} \left(\frac{1}{k},0\right),\hspace{.1pc}\left(\frac{1}{k+1},\frac{1}{k+1}\right)$ and $\left(0,\frac{1}{k+1}\right)$.
\begin{center}
\includegraphics[scale=.5]{gauss_larger_than_one.pdf}\\
$\Gamma_k \text{ when $k\ge 1$}$\\
\end{center}
\noindent When $0<k<1$, $\Psi$ is injective on $P_{(k,a)}$ precisely when $a>0$. In order to finish the characterization of the space of approximation coefficients for these cases, we prove:
\begin{theorem}\label{P_0_k<1}
When $0<k<1$, $P_0^\#$ is the intersection of the unbounded regions $u{k}+v < 1, \hspace{.1pc} v > 0, \hspace{.1pc} (k+1)^2u+k{v} > k+1, \hspace{.1pc} u+k{v} < 1$ and $4k{u}{v} \le 1$.
\end{theorem}
\begin{proof}
Let $(u,v) := \Psi(x,-k-a)$. From the definition \eqref{Psi} of $\Psi$, we obtain
\begin{equation}\label{u=f(x,y)}
u = \dfrac{1}{x-y} = \dfrac{1}{x+k+a}
\end{equation}
and $v = -\frac{x{y}}{x-y}$ so that
\begin{equation}\label{v=f(u)}
v = -\dfrac{x{y}}{k(x-y)} = \dfrac{u}{k}x(-y) = \dfrac{u}{k}\bigg(\dfrac{1}{u}- (a+k)\bigg)(a+k) = \dfrac{a+k}{k}\big(1-(a+k)u\big).
\end{equation}
After restricting $P_0$ to its part on or below $x+y=0$, we are left with the region whose boundary is the 5-gon with vertices $(k,-k), (1,-1), (1,-k), (1,-k-1), (0,-k-1)$ and $(0,-k)$, which includes the part of its perimeter, which is the open line segment connecting the first vertex to the second vertex. We will use these formulas to evaluate the image of each open line segment in this 5-gon under $\Psi$.
\begin{enumerate}
\item When $y=-k$, we obtain that $u$ ranges between $\frac{1}{k}$ and $\frac{1}{2k}$ as $x$ ranges between $0$ and $k$. Plugging $a=0$ to formula \eqref{v=f(u)} yields that the line segment $(0,k) \times \{-k\}$ in the $x{y}$ plane maps to the open segment in the line $uk+v=1$ between the points $\left(\frac{1}{2k},\frac{1}{2}\right)$ and $\left(\frac{1}{k},0\right)$ on the $u{v}$ plane\\
\item Plugging $x=a=0$ yields that the line segment $\{0\} \times (-k-1,-k)$ in the $x{y}$ plane maps to $\left(\frac{1}{k+1},\frac{1}{k}\right)\times\{0\}$ on the $u{v}$ plane.
\item The line segment $(0,1) \times \{-(k+1)\}$ in the $x{y}$ plane maps to the open segment of the line $(k+1)^2u+k{v}=k+1$ between $\left(\frac{1}{k+1},0\right)$ and $\left(\frac{1}{k+2},\frac{k+1}{k(k+2)}\right)$ on the $u{v}$ plane.
\item Plugging $x=1$ and $a=0$ yields that the line segment $\{1\} \times \left(-1,-(k+1)\right)$ in the $x{y}$ plane maps to the open segment of the line $u+k{v}=1$ between the points $\left(\frac{1}{k+2},\frac{k+1}{k(k+2)}\right)$ and $\left(\frac{1}{2},\frac{1}{2k}\right)$ on the $u{v}$ plane.
\item Finally, if $y=-x$ then $u = \frac{1}{x-y}= \frac{1}{2x}$ hence $x = \frac{1}{2u}$ and $v= -\frac{u}{k}x{y} = \frac{1}{4k{u}}$. Thus the line $x+y=0$ in the $x{y}$ plane is mapped under $\Psi$ to the hyperbola $4k{u}v=1$ in the $u{v}$ plane. The points $(1,-1)$ and $(k,-k)$ map to $\big(\frac{1}{2},\frac{1}{2k}\big)$ and $\big(\frac{1}{2k}, \frac{1}{2}\big)$ under $\Psi$, so that the open segment of the line $x+y=0$ from $(1,-1)$ to $(k,-k)$ maps to the open arc on this hyperbola from $\big(\frac{1}{2},\frac{1}{2k}\big)$ to $\big(\frac{1}{2k}, \frac{1}{2}\big)$.
\end{enumerate}
\noindent Since $\Psi$ is a continuous bijection on and below the line $x+y=0$, it maps the interior and boundary of this 5-gon bijectively into the interior and boundary of the region in the $u{v}$ plane whose boundary we have just determined and which coincides with the hypothesis. Using proposition \ref{Psi_fold}, we know that the part of $P_0$ which is above the line $x+y=0$ has the same image under $\Psi$ as its reflection about this line. Since this reflection is also contained in $P_0$, we conclude that $P_0$ is mapped in its entirety onto this region, thus concluding the result.
\end{proof}
\begin{corollary}
When $0<k<1$, the Space of Jager Pairs is the region in the $u{v}$ plane which is the union of the quadrangle vertices $\left(0,0\right), \hspace{.1pc} \left(\frac{1}{k},0\right),\hspace{.1pc}\left(\frac{1}{k+1},\frac{1}{k+1}\right)$ and $\left(0,\frac{1}{k+1}\right)$ and the part of the hyperbola $4k{u}{v}=1$ in the $u{v}$ plane between $u=\frac{1}{2}$ and $u=\frac{1}{2k}$.
\end{corollary}
\begin{center}
\includegraphics[scale=.5]{gauss_less_than_one.pdf}\\
$\Gamma_k \text{ when $0 < k < 1$}$
\end{center}
|
1,314,259,993,041 | arxiv | \section{Introduction}
Interfaces of water are the most important subjects not only
because water is widely involved in physical, chemical,
environmental as well as biological processes, but also because
water is so far the most mysteries molecule in the
universe.\cite{FranksBook,science-Luecke,TobiasPRL2002,RicePNAS1999}
Among them, air/water interface has been intensively investigated
theoretically or experimentally over the last decades.
Spectroscopy, molecular structure and dynamics at air/water
interface is studied with theoretical analysis such as \textit{ab
initio} calculation or molecular dynamics
simulation,\cite{Mundy-science,
BenjaminPRL1994,HynesCP2000,HynesJPCB2002,MooreJCP2003,ChandraCPL2003,
ChandraCPL2004,RiceJCP1991,BenjaminCR1996} or experimental
techniques such as X-ray
reflection,\cite{PershanPRL1985,PershanPRA1988} Stimulated Raman
Scattering (SRS),\cite{SawadaCPL2002} Near-edge X-Ray Adsorption
Fine Structure (NEXAFS),\cite{SaykallyJPCM2002} Second Harmonic
Generation (SHG),\cite{EisenthalJPC1988,FreyMP2001} as well as Sum
Frequency Generation, etc.\cite{ShultzIRPC2000,
RichmondARPC2001,Richmond:cr102:2693,Shen-science,richmond:science,Shen-prl1994,
DuQuanPRL1993,Richmond-jpca2000,WeiXingPRL2001} Among these
experimental techniques, Second Harmonic Generation and Sum
Frequency Generation are the most important methods for molecular
interface studies because of their surface sensitivity and
specificity.\cite{RichmondSHGReview,ShenANRP1989,CornHigginsReview1994,
ShenMirandaJPCBReview,shen:nature:review,eisenthal:review,Shen-5CT,somorjai:ap:review}
With these investigations, the properties of the water molecules
at the interface, such as the surface density, surface structure,
surface potential as well as surface dynamics, have been
intensively discussed.
However, conclusions on the surface molecule species at air/water
interface are still under discussion.\cite{DuQuanPRL1993,
Richmond-jpca2000,WeiXingPRL2001,SaykallyJPCM2002} With SFG-VS
experimental studies, the following interfacial water species have
been reported in literatures, namely, water molecules straddle at
the interface with one OH bond hydrogen bonded to neighboring
molecules in liquid phase (singly bonded OH) and another OH bond
free from hydrogen bonding (free OH) in gas
phase;\cite{Shen-prl1994,ShultzIRPC2000,Richmond-jpca2000,richmond:jpcb1998}
water molecules with both OH bonds symmetrically hydrogen bonded
in a tetrahedral network (ice-like and liquid-like
structures);\cite{Shen-prl1994,
ShultzIRPC2000,Richmond-jpca2000,richmond:jpcb1998} and water
molecules in gas phase with both OH bonds not hydrogen bonded
pointing into the liquid
phase.\cite{Richmond-jpca2000,richmond:science} With NEXAFS
measurement and \textit{ab initio} molecular dynamics simulation,
water molecules with both OH bonds not hydrogen bonded pointing
out of the interface was also
proposed.\cite{SaykallyJPCM2002,Mundy-science} The latter case is
particularly controversial because NEXAFS is not strictly a
surface specific technique.\cite{SaykallyJPCM2002} With
polarization SFG-VS measurement, Wei \textit{et al.} discussed the
absence of SFG spectra in some polarization combinations and
proposed an explanation through fast orientational motion in a
broad range of about $102^{\circ}$ in a time scale comparable or
less than 0.5 \textit{ps}.\cite{WeiXingPRL2001} However, puzzle
still remains because some of the experimental studies suggests
ordered and slow dynamics for interfaces of hydrogen bonding
liquids, while some experimental investigations suggested a more
dynamic and less ordered picture for the liquid interfaces,
air/water interface included.\cite{ShenMirandaJPCBReview} In
addition, whether the surface orientation relaxation is fast or
slow than the bulk water molecules is also an issue under
discussion in the recent
literatures.\cite{BakkerScience,ChandraCPL2004,BenjaminJPCBASAP}
Besides SHG and SFG-VS experimental
studies,\cite{Eisenthal1992ACR,ShenMirandaJPCBReview} Structure
and dynamics of water molecules at the air/water interface have
also been intensively discussed with theoretical
simulations.\cite{Mundy-science,BenjaminPRL1994,HynesCP2000,HynesJPCB2002,
ChandraCPL2003,ChandraCPL2004,MooreJCP2003,RiceJCP1991,BenjaminCR1996}
Even though with so much efforts and progresses both by
experimentalists and theoreticians, our detailed understanding of
air/water interface is still limited. Just as indicated by B. C.
Garrett recently,\cite{GarrettScience} `...(direct) experiments
are difficult to perform because the liquid interface is
disordered, dynamic, and small (typically only a few molecules
wide) relative to the bulk'.
Actually, direct measurement of the liquid interface is not as
difficult as suggested as above. It has been known that along with
SHG, SFG-VS can provide direct measurement on liquid interface no
other technique can
match.\cite{Eisenthal1992ACR,ShenMirandaJPCBReview} As pointed out
by Miranda and Shen, `SFG is currently the only technique that can
yield a vibrational spectrum for a neat liquid
interface'.\cite{ShenMirandaJPCBReview} In fact, with the advances
of ultrafast laser and detection technology in the past decade and
especially recent few
years,\cite{RichterOL1998,AllenAnaSci2001BroadBandSFG,RichmondApplySpec2004,
JohnsonPCCP2005} particularly with commercial systems designed for
SFG-VS measurement,\cite{EKSPLA&EROSACAN} SFG-VS, as well as SHG,
experiments have come from easier to routine.\cite{ShenApplyPhys}
The real difficulty lies on the fact that quantitative analysis
and interpretation of the SFG-VS, as well as SHG, data had been
not as well developed and widely performed until
recently.\cite{Shen-5CT,WeiXingPRL2001,weixing:pre2000,WHFRaoJCP2003,Lurong1,Lurong2,
HongfeiCJCPPaper,Lurong3,ChenhuaJPCBacetone,
ChenhuaJPCBmethanol,ChenhuaCPLacetone,GanweiCPLNull,HongfeiIRPCreview}
Therefore, conclusions in many previous reports on the
investigations of air/water interface, as well as other liquid
interfaces, with SFG-VS are subjected to different
interpretations.
As we have demonstrated in a series of recent publications,
systematically quantitative treatment to SFG-VS data is not only
possible, but also very effective for obtaining detailed
spectroscopic, structural and thermodynamic properties of liquid
interfaces.\cite{WHFRaoJCP2003,Lurong1,Lurong2,
HongfeiCJCPPaper,Lurong3,ChenhuaJPCBacetone,ChenhuaJPCBmethanol,
ChenhuaCPLacetone,GanweiCPLNull,HongfeiIRPCreview} In these works,
we not only developed methodology for quantitative polarization
and experimental configuration analysis in SFG-VS and SHG, we also
tested accuracy and sensitivity of some of the methodology. We
have applied them to elucidated the anti-parallel double layered
structure and thermodynamics of some organic liquid aqueous
solution interfaces. In addition, we also demonstrated that a set
of polarization selection rules (or guidelines) in SFG-VS can be
developed for vibrational spectrum assignment through symmetry
analysis of the SFG-VS spectral features.\cite{Lurong2,Lurong3}
This latter approach is extremely useful for discerning complex
SFG-VS spectrum with unidentified or controversial assignments.
Recently, based on polarization analysis, Ostroverkhov \textit{et
al.} demonstrated a phase-sensitive interference analysis of SFG
polarization spectra of water/quartz
interface.\cite{Shen2005PRLWaterQuartz} With these development, in
this report we intend to apply these analysis methodologies to the
study of air/water interface.
In this work, we examined SFG-VS spectra at air/water interface
measured in different polarizations under four experimental
configurations with polarization analysis method and experimental
configuration analysis. With these analysis, detailed new
information are obtained for understanding of the spectroscopy,
structure and dynamics of the air/water interface. In the
following sections, after a brief introduction of the theoretical
background and experimental conditions, we first discuss the
motion of the interfacial water molecules at the air/water
interface, which was previously suggested experiencing rapidly
motion over a broad angular range in the vibrational relaxation
time; then we use polarization and symmetry analysis of the SFG-VS
spectral features for assignment of the SFG-VS spectra peaks; in
the end, we shall discuss the structure and orientation of the
water molecules at the air/water interface.
\section{Polarization and Experimental Configuration Analysis in SFG-VS}
Quantitative polarization analysis and experimental configuration
analysis can provide rich and detailed information of
spectroscopy, structure and dynamics of molecular
interfaces.\cite{Shen-5CT,WeiXingPRL2001,WHFRaoJCP2003,Lurong2,Lurong3,GanweiCPLNull}
Generally, the SFG intensity in the reflective direction
is,\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
I(\omega)&=&\frac{{8\pi ^3 \omega ^2sec^2\beta
}}{{c^{3}n_{1}(\omega)n_{1}(\omega_{1}})n_{1}(\omega_{2})}\left|\chi
_{eff}^{(2)}\right|^2 I(\omega_{1})I(\omega_{2})\label{all}
\end{eqnarray}
\noindent
\noindent in which $\omega$, $\omega_{1}$ and $\omega_{2}$ are the
frequencies of the SFG signal, visible and IR laser beam,
respectively. $n_{i}(\omega_{i})$ is the refractive index of bulk
medium $i$ at frequency $\omega_{i}$, and $n'(\omega_{i})$ is the
effective refractive index of the interface layer at $\omega_{i}$.
$\beta_{i}$ is the incident or reflection angle from interface
normal of the $i$th light beams; $I(\omega_{i})$ is the intensity
of the SFG signal or the input laser beam. $\chi_{eff}^{(2)}$ is
the effective second order susceptibility for an interface. The
notations and the experiment geometry have been described in
detail previously.\cite{Lurong2,Shen-5CT}
$\chi_{eff}^{(2)}$ for the four generally used independent
polarization combinations can be deduced from the 7 nonzero
macroscopic susceptibility tensors for an achiral rotationally
isotropic interface ($C_{\infty v}$).\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
\chi_{eff}^{(2),ssp}&=&
L_{yy}(\omega)L_{yy}(\omega_{1})L_{zz}(\omega_{2})sin\beta_{2}\chi_{yyz}\label{ssp}\nonumber
\\
\chi_{eff}^{(2),sps}&=&L_{yy}(\omega)L_{zz}(\omega_{1})L_{yy}(\omega_{2})sin\beta_{1}\chi_{yzy}\label{sps}\nonumber
\\
\chi_{eff}^{(2),pss}&=&L_{zz}(\omega)L_{yy}(\omega_{1})L_{yy}(\omega_{2})sin\beta\chi_{zyy}\label{pss}\nonumber
\\
\chi_{eff}^{(2),ppp}&=&
-L_{xx}(\omega)L_{xx}(\omega_{1})L_{zz}(\omega_{2})
cos\beta{cos\beta_{1}}sin\beta_{2}\chi_{xxz}\nonumber\\
&&-L_{xx}(\omega)L_{zz}(\omega_{1})L_{xx}(\omega_{2})cos\beta{sin\beta_{1}}cos\beta_{2}\chi_{xzx}\nonumber\\
&&+L_{zz}(\omega)L_{xx}(\omega_{1})L_{xx}(\omega_{2})sin\beta{cos\beta_{1}}cos\beta_{2}\chi_{zxx}\nonumber\\
&&+L_{zz}(\omega)L_{zz}(\omega_{1})L_{zz}(\omega_{2})sin\beta{sin\beta_{1}}sin\beta_{2}\chi_{zzz}\nonumber\\
\label{ppp}
\end{eqnarray}
\noindent It is so defined that the $xy$ plane in the laboratory
coordinates system $\lambda(x,y,z)$ is the plane of interface; all
the light beams propagate in the $xz$ plane; $\textit{p}$ denotes
the polarization of the optical field in the $xz$ plane, with $z$
as the surface normal, while $\textit{s}$ the polarization
perpendicular to the $xz$ plane. The consecutive superscript, such
as \textit{ssp}, represents the following polarization
combinations: SFG signal \textit{s} polarized, visible beam
\textit{s} polarized, IR beam \textit{p} polarized, and so forth.
$L_{ii}$ ($i=x,y,z$) is the Fresnel coefficient determined by the
refractive indexes of the two bulk phase and the interface layer,
and the incident and reflected angles.\cite{Lurong2,Shen-5CT}
$\chi_{ijk}^{(2)}$ tensors are related to the microscopic
hyperpolarizability tensor $\beta_{i'j'k'}^{(2)}$ of the molecules
in the molecular coordinates system $\lambda'(a,b,c)$ through the
ensemble average over all possible molecular orientations.
\cite{Lurong2,Shen-5CT}
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{i'j'k'}\langle{R_{ii'}R_{jj'}R_{kk'}\rangle}\beta_{i'j'k'}^{(2)}
\label{hyper}
\end{eqnarray}
\noindent where $R_{\lambda\lambda'}(\theta,\phi,\psi)$ is the
matrix element of the Euler rotational transformation matrix from
the molecular coordination $\lambda'(a,b,c)$ to the laboratory
coordination $\lambda$(\textit{x,y,z}); $\beta_{i'j'k'}^{(2)}$ is
the microscopic (molecular) hyperpolarizability
tensor.\cite{GoldsteinBook,HongfeiCJCPPaper,HongfeiIRPCreview}
Here $N_{s}$ is the molecular number density at the interface.
$\langle A \rangle$ represents orientational average of property
$A(\theta,\phi,\psi)$ over the orientational distribution function
$f(\theta,\phi,\varphi)$.
\begin{eqnarray}
\langle A
\rangle=\frac{\int^{\pi}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}A(\theta,\phi,\psi)
f(\theta,\phi,\psi)\sin\theta d\theta\ d\phi d\psi}
{\int^{\pi}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}f(\theta,\phi,\psi)\sin\theta
d\theta d\phi d\psi}\label{OAverage}
\end{eqnarray}
For SFG-VS, $\beta^{(2)}$ is IR frequency ($\omega_{2}$)
dependent,
\begin{eqnarray}
\beta^{(2)}_{i'j'k'}&=&\beta_{NR,i'j'k'}^{(2)}+\sum_{q}\frac{\beta_{q,i'j'k'}}
{\omega_{2}-\omega_{q}+i\Gamma_{q}}\label{spectrum}
\end{eqnarray}
Thus, $\chi_{ijk}^{(2)}$ can be expressed into,
\begin{eqnarray}
\chi_{ijk}^{(2)}&=&\chi_{NR,ijk}^{(2)}+\sum_{q}\frac{\chi_{q,ijk}}
{\omega_{2}-\omega_{q}+i\Gamma_{q}}\label{spectrum1}
\end{eqnarray}
Therefore, SFG-VS measures the vibrational spectroscopy of
molecular interfaces. For dielectric interfaces, such as liquid
interfaces, the non-resonant term $\beta_{NR,i'j'k'}^{(2)}$ or
$\chi_{NR,ijk}^{(2)}$ is generally negligible compare with the
resonant terms.
Recently, we have found that the following formulation is very
effective in quantitative polarization and orientation analysis of
SFG and SHG data. It can be generally shown that in surface SFG
and SHG for an interface with orientational order, the effective
second order susceptibility $\chi_{eff}^{(2)}$ can be simplified
into the following form.\cite{WHFRaoJCP2003}
\begin{equation}
\chi_{eff}^{(2)} = N_{s}\ast\textit{d}\ast(\langle \cos \theta
\rangle - \textit{c}\ast \langle \cos ^3\theta \rangle
)=N_{s}\ast\textit{d}\ast \textit{r}(\theta) \label{chi}
\end{equation}
\noindent $r(\theta)$ is called the \textit{orientational field
functional}, which contains all molecular orientational
information at a given SFG experimental configuration; while the
dimensionless parameter \textit{c} is called the \textit{general
orientational parameter}, which determines the orientational
response $r(\theta)$ to the molecular orientation angle $\theta$;
and $\textit{d}$ is the susceptibility strength factor, which is a
constant in a certain experimental polarization configuration with
a given molecular system. The $d$ and $c$ values are both
functions of the related Fresnel coefficients including the
refractive index of the interface and the bulk phases, and the
experimental geometry.
The key for quantitative analysis is that both \textit{d} and
\textit{c} can be explicitly derived from the expressions of the
$\chi_{eff}^{(2)}$ in relationship to the macroscopic
susceptibility and microscopic (molecular) hyperpolarizability
tensors for a particular molecular vibrational
modes,\cite{Lurong2,HongfeiIRPCreview} as shown for the water
molecules with $C_{2v}$ symmetry in the appendix. With the
parameters $c$ and $d$, the polarization dependence and the
orientation dependence of the SFG/SHG signal for a certain
interface at certain experimental configuration can be analyzed
and calculated with clear physical picture on molecular
orientation and orientational distribution.\cite{WHFRaoJCP2003}
Reciprocally, information on the molecular symmetry, molecular
orientation and dynamics can be obtained from the analysis on the
SFG intensity relationships measured in different polarization
combinations and experimental
configurations.\cite{Lurong2,Lurong3,HongfeiIRPCreview,GanweiCPLNull}
The orientational average in Eq.\ref{hyper} is only the static
average on molecular orientations, without considering fast
molecular motion effects. Recently Wei \textit{et al.} discussed
the fast and slow limit of the time average over orientational
motion for $\chi_{eff}^{(2)}$, and they also applied this
treatment to analysis the polarization dependence of SFG
measurement of the OH stretching vibrational spectra for the
air/water interface.\cite{WeiXingPRL2001} In the fast motion
limit, the orientational motion is faster than the vibrational
relaxation time scale $1/\Gamma_{q}$ of the $q$th vibrational
mode; while in the slow motion limit, the orientational motion is
much slower than $1/\Gamma_{q}$.
According to Wei \textit{et al.},\cite{WeiXingPRL2001} the slow
motion limit gives,
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{q}\sum_{i'j'k'}
\frac{\beta^{(2)}_{q,i'j'k'}}{\omega_{2}-\omega_{q}+i\Gamma_{q}}
\langle R_{ii'}R_{jj'}R_{kk'}\rangle \label{SlowAverage}
\end{eqnarray}
\noindent while the fast motion gives,
\begin{eqnarray}
\chi^{(2)}_{ijk}&=&N_{s}\sum_{q}\sum_{i'j'k'}
\frac{\beta^{(2)}_{q,i'j'k'}}{\omega_{2}-\omega_{q}+i\Gamma_{q}}
\langle R_{ii'}R_{jj'}\rangle \langle R_{kk'}
\rangle\label{FastAverage}
\end{eqnarray}
\noindent in which
$R_{\lambda\lambda'}(t)=\hat{\lambda}\cdot\hat{\lambda'}(t)$ is
the time-dependent direction Euler transformation matrix from
$\lambda'(a,b,c)$ to $\lambda(x,y,z)$ coordinates system. Because
of the molecular orientational motion, the molecular coordinates
$\lambda'(a,b,c)$ is time-dependent. Eq.\ref{SlowAverage} is
equivalent to Eq.\ref{spectrum1}, which is obtained by insertion
of Eq.\ref{spectrum} into Eq.\ref{hyper}.
\section{Experiment}
The details of the laser system has been described in our previous
reports.\cite{Lurong2,ChenhuaJPCBacetone,ChenhuaJPCBmethanol}
Briefly, the 10Hz and 23 picosecond SFG spectrometer laser system
(EKSPLA) is in a co-propagating configuration. The efficiency of
the detection system has been improved for the weak SFG signal of
air/water interface. A high-gain low-noise photomultiplier
(Hamamatsu, PMT-R585) and a two channel Boxcar average system
(Stanford Research Systems) are integrated into the EKSPLA system.
The voltage of R585 was 1300V in the measurement for air/water
interface, and 900V for the Z-cut quartz surface. The wavelength
of the visible is fixed at 532nm and the full range of the IR
tunability is $1000cm^{-1}$ to $4300cm^{-1}$. The specified
spectral resolution of this SFG spectrometer is $<6cm^{-1}$ in the
whole IR range, and about $2cm^{-1}$ around $3000cm^{-1}$. Each
scan was with a $5cm^{-1}$ increment and was averaged over 300
laser pulses per point. Each spectrum has been repeated for at
least several times. Moreover, for \textit{sps} polarization, each
spectrum has been repeated for more than a dozen times and
averaged. The energy of visible beam is typically less than
300$\mu J$ and that of IR beam less than 150$\mu J$ around
$3000cm^{-1}$ and $3700cm^{-1}$, and less than 100$\mu J$ in the
region in between. These are comparable to literature reported
values for measurement of air/water interface.\cite{DuQuanPRL1993}
All measurements were carried out at controlled room temperature
($22.0\pm0.5^{\circ}C$) and humidity (40$\%$) . The sample used
was ultrapure water from standard Millipore treatment (18.2
M$\Omega \cdot cm$). The whole experimental setup on the optical
table was covered in a plastic housing to reduce the air flow. No
detectable evaporation effect was observed for SFG spectrum during
each scan.
The normalization procedure of the SFG signal in different
experimental configurations need to be specifically discussed. The
detail of the normalization procedure for a single experimental
configuration was presented in Xing Wei's Ph.D.
dissertation.\cite{Wei:Thesis} However, the difference of coherent
length and Fresnel factors with different incident angles in the
quartz SFG signal measurements has to be corrected when comparing
SFG signal in different experimental configurations. Therefore,
the measured spectrum is firstly normalized with the energy of the
incident laser beams, and then normalized to the SFG signal of
Z-cut quartz (also normalized by the energy of the incident
lasers). Then it times with a converting factor between different
experimental configurations. This factor contains the influence of
the coherent length of Z-cut quartz,\cite{Wei:Thesis} the Fresnel
coefficients,\cite{Wei:Thesis} the $\chi_{ijk}$ value for Z-cut
quartz, and the factor $sec^{2}\beta$ for each experimental
configuration. Therefore, the end result is directly proportional
to the SFG intensity in Eq.\ref{all}. If the spectrum in Fig.
\ref{allSpectra} is divided by the factor $sec^{2}\beta$ and the
factor of the PMT efficiency between 1300V and 900V, which is
determined as 24.1 in our detection system, and then times the
unit factor $1\times 10^{-40}V^{4}m^{-2}$ which we left out for
simplicity of graph presentation, it will give the value for
$|\chi^{(2)}_{eff}|^{2}$. For example, the peak at about
$3700cm^{-1}$ in the \textit{ssp} spectra of Config.2 in
Fig.\ref{allSpectra} is about 0.23 unit. After above conversion it
gives $|\chi^{(2)}_{eff}|^{2}=4.7\times 10^{-40}V^{4}m^{-2}$,
matching satisfactorily with the reported value for less than
$10\%$ difference.\cite{WeiXingPRL2001}
Even though the normalized intensities are generally consistent
with each other, there can be possibly other sources of errors
when intensities in different experimental configurations need to
be compared. Because the visible and IR beams have different
coherent lengthes in the Z-cut crystal, and because these coherent
lengthes vary with different experimental incident angles, one of
the most likely error might come from the different focusing
parameters with different beam overlapping quality of the visible
and IR beams in the Z-cut quartz crystal with different
experimental configurations. Therefore, quantitative comparison of
the SFG spectral intensities in different polarizations with the
same experimental configuration can be more accurate than
comparison intensities between different experimental
configurations. Even though the latter is a good solution to
reduce such relative error associated with different experimental
configurations need to be developed.
\section{Results and Discussion}
\subsection{Polarization SFG Spectra of the air/water interface}
Firstly we would like to present the polarization SFG spectra of
the air/water interface measured in four different experimental
configurations.
We have demonstrated recently that the change of the SFG spectra
in different polarizations by varying the experimental
configurations can be used for quantitative polarization analysis
and orientational analysis.\cite{HongfeiIRPCreview,GanweiCPLNull}
Here we present in Fig.\ref{allSpectra} the SFG spectra in the
\textit{ssp}, \textit{ppp} and \textit{sps} polarizations on the
air/water interface at four experimental configurations with
different incident angles for the visible and IR laser beams. They
are, Config.1: Visible=39$^{\circ}$, IR=55$^{\circ}$; Config.2:
Visible=45$^{\circ}$, IR=55$^{\circ}$; Config.3:
Visible=48$^{\circ}$, IR=57$^{\circ}$; Config.4:
Visible=63$^{\circ}$, IR=55$^{\circ}$.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=15cm,width=15cm]{allSpectra.eps}
\caption{SFG spectra of air/water interface in different
polarization combination and experimental configurations. All
spectra are normalized to the same scale. The solid lines are
globally fitted curves with Lorentzian line shape function in Eq.
\ref{spectrum1}. Note the different error bars for graphs in
different scales.}\label{allSpectra}
\end{center}
\end{figure*}
\begin{table}[h!]
\caption{The fitting results of the SFG spectra at air/watrer
interface. The spectra are fitted with Lorentzian line shape
function as Eq.\ref{spectrum1}. The peak position of the
vibrational modes $\omega_{q}$, the peak width $\Gamma_{q}$ and
the oscillator strength factor $\chi_{eff,q,ijk}$ of the
vibrational modes are listed. The first column is the fitted value
for $\chi_{NR,eff,ijk}$. The relative error in fitting of
\textit{sps} is larger because of the small signal strength for
\textit{sps} spectra.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline $\omega_{q}(cm^{-1})$ & & & 3281$\pm$5 & 3446$\pm$3 & 3536$\pm$6 & 3693$\pm$1& $$ \\
$\Gamma_{q}(cm^{-1})$ & & & 89$\pm$9 & 103$\pm$7 & 77$\pm$11 & 17$\pm$1 \\
\hline
& ssp & 0.17 & -6.7$\pm$0.6 & -20.1$\pm$1.3 & -5.2$\pm$1.2 & 6.8$\pm$0.2 \\
Config.1 & ppp &-0.04 & 3.2$\pm$0.5 & 2.6$\pm$0.7 & 5.0$\pm$0.4 & 1.1$\pm$0.1 \\
& sps & -0.01 & -0.1$\pm$0.1 & -0.2$\pm$0.1 & -3.5$\pm$0.5 & 0.9$\pm$0.2 \\
\hline
& ssp & 0.19 & -8.3$\pm$0.6 & -24.0$\pm$3.5 & -5.0$\pm$3.5 & 8.5$\pm$0.1 \\
Config.2 & ppp &-0.02 & 1.1$\pm$0.5 & 0.0$\pm$0.8 & 6.9$\pm$0.3 & 2.4$\pm$0.6 \\
& sps & 0.02 & -0.1$\pm$0.1 & -0.3$\pm$0.1 & -4.5$\pm$0.5 & 1.6$\pm$0.1 \\
\hline
& ssp & 0.22 & -10.1$\pm$0.7 & -22.8$\pm$1.5 & -7.0$\pm$2.0 & 8.8$\pm$0.2 \\
Config.3 & ppp &-0.01 & 2.4$\pm$0.6 & 0.9$\pm$0.7 & 6.6$\pm$0.3 & 2.8$\pm$0.1 \\
& sps & 0.01 & -0.2$\pm$0.1 & -0.3$\pm$0.1 & -3.4$\pm$0.7 & 1.4$\pm$0.2 \\
\hline
& ssp& 0.21 & -8.8$\pm$0.7 & -23.3$\pm$1.4 & -5.0$\pm$1.5 & 9.2$\pm$0.2 \\
Config.4 & ppp & 0.15 & -1.0$\pm$0.8 & -3.0$\pm$1.3 & -9.0$\pm$0.7 & 9.3$\pm$0.2 \\
& sps & 0.01 & -0.2$\pm$0.1 & -0.4$\pm$0.1 & -6.6$\pm$0.8 & 3.1$\pm$0.2 \\
\end{tabular}\label{fittingResults}
\end{center}
\end{table}
There are four apparent peaks can be identified in the SFG spectra
in Fig.\ref{allSpectra}. They are around $3700cm^{-1}$,
$3550cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$, respectively. The
$3700cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$ peaks has been
extensively discussed in the SFG
literature.\cite{Shen-prl1994,ShultzIRPC2000,
Richmond-jpca2000,richmond:jpcb1998,Shen-science} However, the
$3550cm^{-1}$ peak has been observed, but not yet clearly
identified or assigned.\cite{WeiXingPRL2001} The results of global
fit of these spectra with four Lorentzian peaks in
Eq.\ref{spectrum1} are listed in Table \ref{fittingResults}. From
the fitting results we can see that the peak bandwidths of the
$3550cm^{-1}$, $3450cm^{-1}$ and $3250cm^{-1}$ peaks are
$77\pm11cm^{-1}$, $103\pm 7cm^{-1}$ and $89\pm 9cm^{-1}$,
respectively. Such broad bandwidths indicate that they all belong
to different hydrogen bonded O-H stretching vibrational modes.
However, the bandwidth of the $3693cm^{-1}$ peak width is only
$17cm^{-1}$, consistent with the symmetric stretching (ss)
vibrational mode of the free O-H bond.\cite{DuQuanPRL1993} The
signs in Table \ref{fittingResults} contain the information of the
relative phase and interference effects of the different
vibrational modes. Here the phase of the $3693cm^{-1}$ peak is
held positive in each fit. Altering the relative phases of the
peaks on the same spectrum can not give a reasonable fit. Because
we used global fitting with all the spectra, these relative phases
can be determined accurately. They can be used to determine the
symmetry properties of each vibrational mode in Section IV.C.
According to Eq.\ref{ppp}, the \textit{ssp} spectra in different
experimental configurations should have the same features from the
$\chi_{yyz}$ term. As shown in Fig.\ref{TryOverlap}, all
\textit{ssp} curves overlap quit well when normalized to the
$3693cm^{-1}$ peak. Calculation of the Fresnel factors with
different incident angles can quantitatively explain the relative
intensities in all four configurations.\cite{JohnsonJPCBpaper}
Because the SFG spectral intensity from the air/water interface in
the OH region is usually several times smaller than that of the
C-H region from other air/liquid interfaces, the air/water
interface SFG spectra are usually very hard to measure
experimentally. Therefore, the well overlapping of the
\textit{ssp} spectra in different experimental configurations is a
proof for the quality of our SFG-VS data. Furthermore, the spectra
we obtained agree very well with these in the
literatures.\cite{WeiXingPRL2001,AllenJPCB2004}
In principle, the \textit{sps} spectra in different experimental
configurations should also overlap with each other when
normalized. However, consistent with the calculations of the
corresponding Fresnel factors, the \textit{sps} signal level for
Config. 1, 2 and 3 are very close to the noise level, and features
in the \textit{sps} spectra can not be clearly identified except
for the spectra of Config.4. Therefore, such normalization and
comparison for \textit{sps} spectra is not as meaningful as the
\textit{ssp} spectra.
Different from the \textit{ssp} and \textit{sps} spectra, the
features in the \textit{ppp} spectra in Fig.\ref{allSpectra}
changed drastically with different experimental configurations.
This is because that the \textit{ppp} spectra is determined by
combination of four different $\chi_{ijk}$ tensors. Detailed
polarization analysis and experimental configuration analysis of
these changes in the \textit{ppp} spectra can provide symmetry
properties for each spectral features, as well as orientation and
structure information of the interfacial molecular groups, as
shall be shown later.\cite{HongfeiIRPCreview,Lurong2,Lurong3} We
shall show that analysis of the \textit{ppp} spectra in different
experimental configurations is very informative. However, this
advantage of \textit{ppp} spectra analysis has not been well
utilized in the previous literatures.
\begin{figure}[h!]
\begin{center}
\includegraphics[height=5cm,width=6cm]{TryOverlap.eps}
\caption{Overlap of the normalized \textit{ssp} spectra of the
air/water interface in different experimental
configurations.}\label{TryOverlap}
\end{center}
\end{figure}
\subsection{Orientation and Motion of the Free OH Bond}\label{IVA}
Now with the knowledge of the SFG vibrational spectra of the
air/water interface, we can discuss the orientation and motion of
the free O-H bond at the air/water
interface.\cite{watershortpaper}
The sharp peak around $3700cm^{-1}$ was generally accepted as the
free OH bond protruding out of the liquid
water,\cite{DuQuanPRL1993,Richmond-jpca2000,WeiXingPRL2001,ShenLeePaper}
and it has been treated with $C_{\infty v}$ symmetry in
polarization analysis.\cite{DuQuanPRL1993,WeiXingPRL2001} Wei
\textit{et al.} studied the polarization dependence of the
intensity of this peak in the \textit{ssp}, \textit{ppp}, and
\textit{sps} polarizations measured with experimental
configuration of Visible$=45^{\circ}$,
IR=$57^{\circ}$.\cite{WeiXingPRL2001} Their SFG-VS data are
quantitatively very close to our data with Config.2 as expected.
Therefore, the \textit{ssp} intensity of the $3693cm^{-1}$ peak is
about 10 times of that of \textit{ppp}, and the \textit{sps}
intensity is essentially close zero. Wei \textit{et al.} realized
that using the step orientational distribution function in
Eq.\ref{StepAverage}, as well as other distribution functions,
such as Gaussian, centered at the surface normal, can not explain
such \textit{ssp}, \textit{ppp} and \textit{sps} intensity
relationships with the slow motion average in
Eq.\ref{SlowAverage}. On the other hand, the fast motion average
centered at the interface normal, as shown in Eq.\ref{FastAverage}
with $\theta_{M}=51^{\circ}$, can fairly well explain the observed
intensity relationships. Thus, it was concluded that the
orientation of the free OH bond of the interfacial water molecule
varies over a very broad angular range ($\theta_{M}=51^{\circ}$)
within the vibrational relaxation time as short as
$0.5ps$.\cite{WeiXingPRL2001}
\begin{eqnarray}
f(\theta)&=&cost\ \ \ \ for \ \ \ 0\leq \theta
\leq\theta_{M}\nonumber\\
f(\theta)&=&0\ \ \ \ \ \ \ \ for \ \ \ \theta \geq
\theta_{M}\label{StepAverage}
\end{eqnarray}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8cm]{FourMotionSimulation.eps}
\caption{SFG intensity of the free OH bond simulated with both
slow motion limit (solid curves) and fast motion limit (dotted
curves) following the procedure and parameters as Wei \textit{et
al.} \cite{WeiXingPRL2001}. $\theta_{M}$ is the range of
orientational motion of the free OH bond. All the curves presented
include the factor of $\sec^{2}\beta$, and all intensities are
normalized to the \textit{ssp} intensity in Config.4 with
$\theta_{M}=0^{\circ}$. The vertical lines indicate the
distribution width suggested by Wei \textit{et
al.}}\label{FourMotionSimulation}
\end{center}
\end{figure}
As shown in Fig.\ref{FourMotionSimulation}, Wei \textit{et al.}'s
treatment predicts clearly zero intensity for the \textit{sps}
spectra at $3693cm^{-1}$ with the assumption of fast orientational
motion centered at the surface normal. Using exactly the same
parameters, our calculation of Config.2 gives the same results as
that by Wei \textit{et al.} as it should have
been.\cite{WeiXingPRL2001} It is clear that the simulation results
in Fig.\ref{FourMotionSimulation} can fairly well explain the data
in Config. 1, 2 and 3, because all of them have relatively very
small \textit{sps} spectral intensity at $3693cm^{-1}$. However,
even though the fast orientational motion picture can explain the
relative intensity between the \textit{ssp} and \textit{ppp}
polarization in Config.4, it is clear that it can not explain the
apparently non-zero \textit{sps} intensity at $3693cm^{-1}$ with
Config.4. As long as the orientation distribution or orientational
motion is assumed to be centered to the interface
normal,\cite{ShenPrivate} orientational distribution functions
other than the step function in Eq.\ref{StepAverage} give the same
conclusion. Since the slow motion limit is already not an
option,\cite{WeiXingPRL2001} alternative description of the motion
and orientation at the air/water interface has to be invoked.
Because the air/water interface is rotationally isotropic around
the interface normal, now we assume that the molecular orientation
is centered around the tilt angle $\theta_{0}\neq 0$, instead of
the interface normal ($\theta_{0}=0$). If the Gaussian
distribution function is assumed, we have
\begin{eqnarray}
f(\theta)&=&\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(\theta-\theta_{0})^{2}/2\sigma^{2}}\label{Gaussian}
\end{eqnarray}
\noindent in which $\sigma$ is the standard deviation of the
angular distribution. We shall show in the followings that by
using this distribution function, the $3693cm^{-1}$ peak in
different polarization and experimental configurations in Fig.
\ref{allSpectra} can be quantitatively analyzed.
\begin{table}[h!]
\caption{The general orientational parameter \textit{c} and the
strength factor \textit{d} for the vibrational stretching mode of
free OH bond in different experimental configurations. The
\textit{d} value bear the unit $\beta_{ccc}$ of single OH bond.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline & & d-ssp & c-ssp & d-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & 0.274 & 0.515 & 0.112 & 1 & -0.154 & 1.53 \\
Config.2 & & 0.256 & 0.515 & 0.118 & 1 & -0.120 & 2.05 \\
Config.3 & & 0.248 & 0.515 & 0.118 & 1 & -0.104 & 2.43 \\
Config.4 & & 0.176 & 0.515 & 0.107 & 1 & -0.035 & 6.55 \\
\end{tabular}\label{CandDvalueForC3v}
\end{center}
\end{table}
Because the $3693cm^{-1}$ peak belongs to the \textit{ss} mode of
the free O-H bond at the air/water interface, it has been treated
with $C_{}\infty v$ symmetry. Now we calculate the general
orientational parameter \textit{c} and the strength factor
\textit{d} for the \textit{ssp}, \textit{sps} and \textit{ppp}
polarizations in all four experimental configurations with the
same parameters of the air/water interface as those used by Wei
\textit{et al.}\cite{WeiXingPRL2001} The details of the
calculation of $c$ and $d$ can be found
elsewhere.\cite{Lurong2,Lurong3,HongfeiIRPCreview} It is clear
from Table \ref{CandDvalueForC3v} that the $c$ values for the
\textit{ssp} and \textit{sps} polarizations are the same for all
four experimental configurations; whereas the $c$ values of the
\textit{ppp} polarization differ significantly for different
experimental configurations.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8cm]{FourC3vSimulation.eps}
\caption{The simulated SFG intensity of vibrational stretch mode
for free OH bond at different orientation angle $\theta$ assuming
$\sigma=0^{\circ}$. The factor $sec^{2}\beta$ in Eq.\ref{all} is
also included for comparison of SFG intensity in different
experimental configurations. All curves are normalized to the
\textit{ssp} intensity in Config.4 with $\theta_{0}=0^{\circ}$.
The vertical lines indicate the orientation which quantitatively
explains the observed SFG data.}\label{FourC3vSimulation}
\end{center}
\end{figure}
As we have demonstrated
previously,\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
the $[d\ast r(\theta)]^{2}$ vs. $\theta$ plot with $\sigma=0$ in
different polarizations can provide direct first look of the
physical picture for polarization analysis of SFG-VS data. Here we
plot $[d\ast r(\theta)\ast \sec\beta]^{2}$ vs. $\theta$ in
Fig.\ref{FourC3vSimulation} in order to compare data in different
experimental configurations. Thus, the relative intensity for the
$3693cm^{-1}$ peak in experimental Config.1, 2, 3 and 4 can be
used to calculate the orientation angle of the free O-H bond.
Using the known
procedures\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
and parameters,\cite{WeiXingPRL2001} they give the following four
values, i.e. $28.7\pm1.2^{\circ}$, $32.6\pm0.5^{\circ}$,
$34.6\pm0.7^{\circ}$ and $35.8\pm1.0^{\circ}$, respectively. These
values agree with each other quite well. However, the value from
Config.1, whose \textit{ppp} and \textit{sps} intensities are both
very weak, is not as reliable as the other three configurations.
Averaging over these values gives $\theta=33^{\circ}\pm
1^{\circ}$.
It is clear that $\sigma=0^{\circ}$ is not physically possible for
the liquid interface. However, the apparent success of the
quantitative explanation of the observed SFG spectra of the free
O-H bond in different experimental configurations using
$\sigma=0^{\circ}$ indicates that the actual $\sigma$ value can
not be very broad. Simulation of the $3693cm^{-1}$ peak in
different polarizations in each of the four experimental
configurations using the Gaussian distribution function in
Eq.\ref{Gaussian} concludes that $\sigma$ has to be smaller than
$15^{\circ}$ to satisfy the measured $3693cm^{-1}$ peak
intensities in all four experimental configurations.
$\sigma=15^{\circ}$ is the largest distribution width allowed by
the SFG experiment data for a Gaussian orientational distribution
function. With $\sigma=15^{\circ}$, we have
$\theta_{0}=30^{\circ}$. This indeed confirms that the orientation
of the free O-H bond is within a relatively narrow range (between
$30^{\circ}$ to $33^{\circ}$), with a relatively small
distribution width ($\sigma\leq 15^{\circ}$). Calculation with
both Eq.\ref{SlowAverage}, i.e. slow average limit, and
Eq.\ref{FastAverage}, i.e. fast average limit, gives
indistinguishable results with $\sigma$ as small as $\leq
15^{\circ}$ if $\theta_{0}$ is around $30^{\circ}$. This is
because that with a small distribution width, fast and slow motion
average should be the same according to Eq.\ref{FastAverage} and
Eq.\ref{SlowAverage}. Using a step distribution function around
$\theta_{0}\neq 0^{\circ}$ also give very close orientation angle
and distribution width.
Thus, our conclusion of the free O-H orientation and distribution
at the air/water interface is drastically different from the
conclusion given by Wei \textit{et al.}, which concluded that the
free O-H bond orientation varies in a broad range as big as
$102^{\circ}$ and as fast as $0.5$ picosecond, which is the
relaxation time for the O-H stretching
vibration.\cite{WeiXingPRL2001} It is clear that our conclusion is
based on the successful explanation of the observed polarized SFG
spectral intensities in different experimental configurations,
especially the relatively small but clearly non-vanishing SFG
spectral intensity at $3693cm^{-1}$ in the \textit{sps}
polarization. Our conclusion explicitly supports ultrafast
libratory motions with a relatively narrow angular range. As we
have known, the dynamics libratory motion of the hydrogen bonding
can be as fast as 0.1
picosecond.\cite{FeckoScience2003,Chandler1996PRL} Even with such
ultrafast dynamics, the air/water interface is nevertheless well
ordered. This is consistent with the generally well ordered
picture of the air/liquid and air/liquid mixture interfaces.
Recent quantitative analysis of data in SFG vibrational
spectroscopy have suggested that vapor/liquid interface are
generally well ordered, and sometimes even with anti-parallel
double-layered structures
\cite{ShenMirandaJPCBReview,ShenLinJCPAcetone2001,JohnsonJPCBpaper,ChenhuaJPCBacetone,
ChenhuaJPCBmethanol,ChenhuaCPLacetone}.
It has been generally accepted that liquid interface with strong
hydrogen bonding between molecules should be well ordered
\cite{ShenMirandaJPCBReview}. Our analysis here not only confirmed
this conclusion, but also provided solid and direct experimental
measurement of the orientation and motion at the air/water
interface.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=7cm]{ssC2vsimulation.eps}
\hspace{1.5cm}%
\includegraphics[width=7cm]{asC2vsimulation.eps}
\caption{The simulated SFG intensity for symmetric stretching
(\textit{ss}) mode (left) and asymmetric stretching (\textit{as})
mode (right) of water molecule with $C_{2v}$ symmetry. All curves
presented include the factor of $\sec^{2}\beta$. The intensity of
\textit{ss} mode is normalized to the \textit{ssp} intensity in
Config.4 with $\theta=0^{\circ}$. The intensity for \textit{as}
mode is normalized to the \textit{sps} intensity in Config.4 with
$\theta_{0}=0^{\circ}$. The units between plots of the \textit{ss}
and \textit{as} modes differ by 9.11 times according to the
$\beta_{ccc}$ and $\beta_{aca}$ values in the
appendix.}\label{C2vsimulation}
\end{center}
\end{figure*}
\subsection{Polarization Analysis and Determination of Spectral Symmetry Property}
Here we try to apply polarization analysis for identifying the
symmetry property and for assignment of the SFG vibrational
spectra of the air/water interface.
The assignment of the SFG spectra of air/water interface in the
range of 3000 to 3800$cm^{-1}$ has been discussed
intensively.\cite{Shen-prl1994,RamanSpectroscopyBook,ShultzIRPC2000,
Richmond-jpca2000,richmond:science,RichmondCPL2004,
richmond:jpcb1998,Shen-science,richmondJPCB2003p546paper,RichmondARPC2001,Richmond:cr102:2693}
Richmond recently reviewed the current understanding of the
bonding and energetics, as well as the SFG spectra assignment, of
various aqueous interfaces, including the air/water
interface.\cite{RichmondARPC2001,Richmond:cr102:2693} The SFG
spectral assignments heavily relied on band fitting of IR and
Raman peak positions of bulk water or water cluster
spectra,\cite{Richmond:cr102:2693} as well as based on theoretical
calculations.\cite{TobiasJPCB2005,JPCA2000Buch,HynesCP2000,HynesJPCB2002}
The sharp peak at about 3700$cm^{-1}$ has been unanimously
assigned to the free O-H stretching vibration mode. The broad
peaks around 3250$cm^{-1}$ and 3450$cm^{-1}$ undoubtedly belong to
the hydrogen bonded O-H stretching modes, but their assignments
are not as unanimous as the 3700$cm^{-1}$ peak. The spectrum
around 3250$cm^{-1}$ was assigned to a continuum of O-H symmetric
stretches(ss), $\nu_{1}$ of water molecules in a symmetric
environment (ss-s), and was generally referred as "ice-like"
region because of its similarity in energy to O-H bonds in bulk
ice. The broad band around 3450$cm^{-1}$ was assigned to more
weakly correlated hydrogen bonded stretching modes, and was called
the "liquid-like" hydrogen-bonded region, where water molecules
reside in a more asymmetrically bonded (as) water
environment.\cite{RichmondARPC2001,Richmond:cr102:2693} The broad
peak around 3550$cm^{-1}$ appeared clearly in the \textit{ppp} SFG
spectra has been identified once and it has not been clearly
assigned so far.\cite{WeiXingPRL2001} Shultz \textit{et al.}
pointed out that these broad peak should also include the
asymmetric stretching mode of water molecules in a symmetry
environment and the bending overtone.\cite{ShultzIRPC2000}
Richmond \textit{et al.} also suggested that the intensity at
about 3450$cm^{-1}$ include the contribution of donor O-H
bond.\cite{richmond:science}
Recent progresses on SFG-VS have made it possible to determine the
symmetry properties of SFG-VS vibrational spectral features
through comparison of SFG spectra in different polarizations and
experimental
configurations.\cite{Lurong2,Lurong3,GanweiCPLNull,HongfeiIRPCreview}
The key idea of this development is from the commonsense of
molecular spectroscopy that vibrational modes of molecular groups
with different symmetry properties have different polarization
dependence on the interacting optical
fields.\cite{MichlBook,SymmetrySpectroscopyBook} Applying these
ideas to polarization analysis of SFG spectra has led to a set of
polarization selection rules for different stretching vibrational
modes of molecular groups with different molecular symmetry
properties, such as stretching vibrational modes of the $CH_{3}$
($C_{3v}$), $CH_{2}$ ($C_{2v}$) and $CH$ ($C_{\infty v}$)
groups.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Many of these
selection rules are independent from molecular orientation and
orientational distribution. Therefore, they can be directly used
to identify symmetry property of SFG stretching vibrational band.
These progresses make it possible to analyze SFG vibrational
spectra \textit{in situ}, instead of rely only on the assignments
from Raman and IR studies of the bulk phases, which can be called
\textit{ex situ}. Because SFG spectra usually has more features
than those from IR and Raman measurement, some confusions and
errors in the previous spectral assignments have also been
clarified.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Even though SFG
is naturally a polarized spectroscopy and the interfacial
molecular groups are usually ordered, this idea has not been
systematically explored until very
recently.\cite{WHFRaoJCP2003,Lurong2,Lurong3,HongfeiIRPCreview}
Water molecule possesses $C_{2v}$ symmetry. If the two O-H bond of
a water molecule are asymmetrically bonded, both O-H bond has to
be treated separately with $C_{\infty v}$ symmetry. This
classification of the water molecule symmetry is generally true no
matter it is hydrogen-bonded or not, in cluster, in bulk or at the
interface. Therefore, the symmetry property of the SFG vibrational
spectra features of the air/water interface can all be classified
accordingly. Thus, there are three kind of stretching vibrational
modes for us to deal with, namely, the symmetric (ss) and
asymmetric (as) stretching modes for $C_{2v}$ symmetry, and the
stretching mode for $C_{\infty v}$ symmetry. It is fairly easy to
distinguish these three stretching vibrational modes from the
polarization selection rules for SFG spectra of $CH_{2}$ and $CH$
groups.\cite{Lurong2,Lurong3,HongfeiIRPCreview} Because the bond
angle of the $CH_{2}$ group is slightly larger than that of water
molecule, the polarization dependence of the $C_{2v}$ water
molecule are slightly different. The key difference is that even
when the water molecule at the interface rotate freely around its
symmetry axis, the \textit{sps} spectral intensity of its
\textit{ss} mode does not vanish as that for $CH_{2}$. However,
this fact does not make the polarization selection rules different
for the interfacial $CH_{2}$ group and the $C_{2v}$ water
molecule.
Two of the major selection rules for the $C_{2v}$ group at a
dielectric interface are: \textit{(a) $\textit{ssp}$ intensity is
always many times of that of $\textit{ppp}$ for \textit{ss} mode.}
and \textit{(b) $\textit{ppp}$ intensity for \textit{as} mode is
always several times of that of $\textit{ssp}$. That is to say, if
there is any peak which is stronger in the $\textit{ssp}$ than
$\textit{ppp}$ spectra, it can not be from the $\textit{as}$
mode.}\cite{Lurong2,Lurong3} These two rules are independent from
molecular orientation and orientational distributions at a
rotationally isotropic interface.
It is clear from these selection rules, the sharp peak around
$3693cm^{-1}$ in Fig.\ref{allSpectra} does not belong to the
$C_{2v}$ symmetry, because its intensity in \textit{ppp}
polarization in Config. 1,2,3 is smaller than that in \textit{ssp}
polarization; while larger in Config.4. On the other hand, these
all fits well with the simulations in Fig.\ref{FourC3vSimulation}.
Therefore, $3693cm^{-1}$ peak is with $C_{\infty v}$ symmetry as
the free O-H stretching mode. Dissimilarly, both the broad peaks
around $3250cm^{-1}$ and $3450cm^{-1}$ in Fig.\ref{allSpectra} are
very strong only in the \textit{ssp} spectra in all experimental
configurations. They fit well with the \textit{ss} mode of the
$C_{2v}$ symmetry, and can not belong to the \textit{as} mode of
the $C_{2v}$ symmetry, or the $C_{\infty v}$ symmetry.
It is not so easy to determine the symmetry property of the broad
$3550cm^{-1}$ peak, because it appears to be buried in the high
frequency tail of the broad $3450cm^{-1}$ peak. It is not so
straight forward to read its relative intensity in \textit{ssp}
and \textit{ppp} polarizations from Fig.\ref{allSpectra}. However,
it appears significantly bigger in \textit{sps} polarization in
Config.4 than that in Config.1,2,3. Therefore, it appears to fit
with the simulations in Fig.\ref{FourC3vSimulation}. In order to
exclude the possibility that it may belong to a $C_{2v}$ mode (ss
or as), detailed simulation of the $C_{2v}$ modes in different
polarizations and experimental configurations is now called upon.
As describe in the appendix, the parameter \textit{c} and
\textit{d} of the $C_{2v}$ vibrational stretching modes are
calculated for different polarizations and experimental
configurations. Plots of $[d\ast r(\theta)\ast \sec\beta]^{2}$ vs.
tilted angle $\theta$ of the water molecule $c$ axis from the
interface normal using these $c$ and $d$ values are presented in
Fig.\ref{C2vsimulation}. These plots again confirm that the
$3693cm^{-1}$ can not belong to any $C_{2v}$ mode, especially with
the one order of magnitude increase of this peak in \textit{ppp}
polarization.
Clearly, the polarization dependence of the broad $3550cm^{-1}$
peak does not fit to the \textit{ss} mode of the $C_{2v}$
symmetry. Otherwise, according to Fig.\ref{C2vsimulation}, its
\textit{ppp} intensity in Config.3 has to be about one order of
magnitude weaker than that in the observed spectra. This peak can
not be the \textit{as} mode of the $C_{2v}$ symmetry either.
According to the \textit{c} and \textit{d} values for the
\textit{as} mode of the $C_{2v}$ symmetry, the phases in
\textit{ssp} and \textit{ppp} polarizations have to be with
opposite signs in all four experimental configurations. However,
fitting of the \textit{ssp} and \textit{ppp} spectra indicates
that in Config.4, the oscillator strengths had the same signs for
\textit{ssp} and \textit{ppp} polarizations, even though in
Congfig. 1,2,3, the oscillator strengths of this peak do possess
opposite phases these two polarizations. This indicates that the
$3550cm^{-1}$ peak is not $C_{2v}$ symmetry, and it appears to
have $C_{\infty v}$ symmetry.
Because in the bulk phase there is no observation of free O-H
bond, and because this broad peak $3550cm^{-1}$ appears to be
hydrogen-bonded, there is only one possibility that it is the
other O-H bond of the interfacial water molecule which has a free
O-H bond extruding away from the liquid bulk phase. According to
Fig.\ref{FourC3vSimulation}, the $C_{\infty v}$ O-H stretching
mode in \textit{sps} polarization is about twice as large in
Cogfig.4 as that in the other configurations. This is fully
consistent with the SFG spectra data in Fig.\ref{allSpectra} and
the fitting results in Table \ref{fittingResults}. Furthermore, in
Table \ref{fittingResults}, the phase of the broad $3550cm^{-1}$
peak is just opposite to that of the $3693cm^{-1}$ peak in both
\textit{ssp} and \textit{sps} polarizations, indicating these two
O-H pointing to opposite directions. The phase of the \textit{ppp}
polarization of the broad $3550cm^{-1}$ peak changes signs with
different experimental configuration. This is because the
orientational angle of the two O-H bonds are some times on the
same side of the minimum on the \textit{ppp} curves in
Fig.\ref{FourC3vSimulation}, and sometime on the different side of
the minimum, just as predicted with the experimental configuration
analysis. These detail features indicated the ability to
understand very subtle dependence of the SFG spectra on
experimental configurations and the parameters used for the
spectra calculations. Further study shall be reported elsewhere.
Further support for the assignment of the broad $3550cm^{-1}$ peak
came from the IR spectra measurement of the water dimer clusters,
where the stretching frequency for the donor O-H bond is just at
about $3550cm^{-1}$.\cite{RamanSpectroscopyBook,
PimentalWaterDimerPaper,NixonWaterDimerPaper,ShenLeePaper} This
assignment is a good support for our assignment of the peak at
3550$cm^{-1}$ in \textit{ppp} spectra to the single
hydrogen-bonded water molecule at the interface. Furthermore, the
two O-H stretching vibrations for the methanol dimer are at
3574$cm^{-1}$ and 3684$cm^{-1}$.\cite{MethanolJCP1991} The donor
O-H stretching mode is also in the same region of 3550$cm^{-1}$.
There is no observable spectra features in Fig.\ref{allSpectra}
for the \textit{as} mode of the $C_{2v}$ water molecules, neither
hydrogen bonded nor non-hydrogen bonded. According to
Fig.\ref{C2vsimulation}, for the \textit{as} modes corresponding
to the ss mode around $3250cm^{-1}$ and $3450cm^{-1}$, their
intensities have to be at least one order of magnitude weaker than
that of their corresponding \textit{ss} modes. It is
understandable that we do not observe them. Above discussion also
throw doubts on the existence of interfacial water molecules with
two free O-H bonds, as suggested somewhat less convincingly by
some recent studies.\cite{Richmond-jpca2000,richmond:science,
RichmondCPL2004,SaykallyJPCM2002,Mundy-science} According to the
polarization selection rules and the calculation for the
polarization dependence of the $C_{2v}$ water molecules, no
detectable spectral features satisfying the $C_{2v}$ symmetry in
the 3600$cm^{-1}$ and 3800$cm^{-1}$ has been observed in the SFG
spectra.
Here we clearly see that how polarization selection rules,
quantitative polarization and experimental configuration analysis
can help determine the symmetry property of the observed spectra
features. The importance of studying of spectral interference has
been demonstrated in recent
reports.\cite{Richmond-jpca2000,Shen2005PRLWaterQuartz,DaviesInterferenceJPCB2004}
Analysis in this work also demonstrated that, in order to discern
spectral details, it is useful and effective to analyze the
spectral interference of different spectral features through
global fitting of SFG spectra in different polarizations and
experimental configurations, and to compare fitting results with
the prediction from the calculated $c$ and $d$ values. This also
indicates the usefulness of the formulation of total SFG signal
with functions of $c$ and $d$ parameters in Eq.\ref{chi}.
\subsection{Molecular Structure at Air/Water Interface}
With the analysis of the orientation and motion, vibrational
spectral symmetry of the water molecules at the air/water
interface in above sections, we can have more understanding of the
molecular structure of the air/water interface.
In Section IV.B, we have determined that the free O-H oriented
around 30$^{\circ}$ away from the interface normal with a
orientational distribution narrower than $\sigma=15^{\circ}$, and
in Section IV.C, we have identified the spectral feature around
$3550cm^{-1}$ of the other hydrogen bonded O-H bond of this water
molecule. If the plane of this interfacial water molecule is close
to perpendicular to the interface, the orientation of the
hydrogen-bonded O-H should point into the liquid phase with a
orientation around 135$^{\circ}$ away from the interface normal.
This orientation is fully consistent with the calculation of the
polarization and experimental configuration dependence of the
broad $3550cm^{-1}$ peak with a $C_{\infty v}$ symmetry with the
observed SFG intensities, detail to be reported elsewhere. Such
orientation makes the dipole of this water molecule points around
97$^{\circ}$ from the interface normal. This picture is fully
consistent with conclusions in many previous experimental and
theoretical studies,
\cite{DuQuanPRL1993,Mundy-science,JaqamanJCP2004,LAAKSONENMolecularPhysics,
LAAKSONENJCP1997,TILDESLEYMolecularPhysics,HynesCP2000,GarrettJPC1996,RiceJCP1991,
BerkowitzCPL1991,WilsonJPC1987,TownsendJCP1985} but certainly
different from some.\cite{WeiXingPRL2001}
From Section IV.C, the broad spectral features between
3100$cm^{-1}$ to 3500$cm^{-1}$ are determined to be symmetric
stretching modes of the $C_{2v}$ symmetry. Because the peaks are
broad, and their energies is in the range of hydrogen-bonded O-H
stretching range, they can only come from the water molecules with
two donor O-H bonds, whose oxygen atom can accept either two, one
or zero hydrogen atom from other water molecules as hydrogen
donors. Certainly, the water molecule with the oxygen atom forming
two hydrogen-bonds is tetrahedral in shape and is "ice-like". This
is consistent with the previous assignment of the broad
$3250cm^{-1}$ peak.
The water molecule with no hydrogen bond for the oxygen atom is
obviously with $C_{2v}$ symmetry. However, the water molecule with
only one hydrogen bond for the oxygen atom may or may not preserve
the $C_{2v}$ symmetry. However, if this hydrogen bond perturbation
to the water structure is limited, this water molecule can still
be treated as with $C_{2v}$ symmetry. The last two kinds of water
molecules are certainly not "ice-like", but "liquid-like". The two
"liquid-like" species may have slightly different O-H vibrational
frequencies. However, only two apparently broad peaks in the
3100$cm^{-1}$ to 3500$cm^{-1}$ region have been identified in the
literatures.\cite{Shen-prl1994,RamanSpectroscopyBook,ShultzIRPC2000,
Richmond-jpca2000,richmond:science,RichmondCPL2004,
richmond:jpcb1998,Shen-science,richmondJPCB2003p546paper,RichmondARPC2001,Richmond:cr102:2693}
More studies on the possible hydrogen-bonded species are certainly
warranted in the future.
Here we confirm the conclusion by Brown \textit{et al.} that these
$C_{2v}$ water species all have their dipole vector point out of
the bulk liquid phase, i.e. with both hydrogen atoms point into
the bulk liquid phase.\cite{Richmond-jpca2000} It is clear in
Table \ref{fittingResults}, the signs of the \textit{ssp}
polarization oscillator strength factors of the $C_{2v}$ water
species are all in opposite phase to that of the free O-H peak at
$3693cm^{-1}$ in all experimental configurations. The signs and
values of the $c$ and $d$ parameters of the $C_{2v}$ and
$C_{\infty v}$ in Table \ref{CandDforC2vWater} and Table
\ref{CandDvalueForC3v}, respectively, indicate that the $c$ axis
of the $C_{2v}$ species has to be in opposite direction to the $c$
axis of the free O-H bond at the interface. Therefore, as defined
as in the appendix, the $C_{2v}$ species have to have their dipole
vector point out of the bulk liquid phase. The calculation of the
phase of the \textit{ppp} as well as \textit{sps} spectral
features are all consistent with this picture.
However, because the SFG spectral intensities of the \textit{ppp}
and \textit{sps} polarizations are generally in the noise level in
the 3100$cm^{-1}$ to 3500$cm^{-1}$ region (Fig.\ref{allSpectra}),
it is difficult to determine the range of the orientation angle
$\theta$ of these hydrogen-bonded species relative to the
interface normal. The orientational distribution of these $C_{2v}$
species can be quite broad, different from that for the
interfacial water molecules with the free O-H bond. From our
simulations, it appears to us that SFG measurement may not be very
effective to determine the orientational angle of the $C_{2v}$
species at the air/water interface, even though it can do very
well with the $C_{\infty v}$ O-H bonds as shown above. However,
our recent analysis of the SHG measurement of the neat air/water
interface showed that SHG measurement might be able to help
determine the orientational angle of the $C_{2v}$ species, but not
the $C_{\infty v}$ O-H bonds. Recent SHG results indicated that
the average orientation of the interfacial $C_{2v}$ water
molecules is about 40$^{\circ}$ to 50$^{\circ}$ from the surface
normal.\cite{wkzhangSHGwaterPaper}
The molecular structure, orientation and dynamics at nonpolar
material/water interfaces have been studied by \textit{ab initio}
calculation, MD simulation, or them
combined.\cite{Mundy-science,BenjaminPRL1994,HynesCP2000,
HynesJPCB2002,MooreJCP2003,ChandraCPL2003,ChandraCPL2004,RiceJCP1991,BenjaminCR1996,
JedlovskyWaterDCE,JedlovskyWaterCCl4} It appears that some
different conclusions were drawn on the molecular orientation and
structure of the air/water interface in different
studies.\cite{JedlovskyWaterCCl4,BenjaminPRL1994,MatsumotoJCP1987}
Nevertheless, many of these studies concluded that the dipole
vector of the interfacial water molecules prefers lying parallel
to the interface and have one of the O-H bond protrude out of the
liquid phase. The majority of the conclusions from theoretical
calculations agree satisfactorily with the experimental analysis
of ours and previous studies, but all the simulation results were
with significantly broader orientational
distributions.\cite{Mundy-science,JaqamanJCP2004,LAAKSONENMolecularPhysics,
LAAKSONENJCP1997,TILDESLEYMolecularPhysics,HynesCP2000,GarrettJPC1996,RiceJCP1991,
BerkowitzCPL1991,WilsonJPC1987,TownsendJCP1985} There were reports
concluded that some interfacial water molecules have their two O-H
bonds projecting into the vapor phase and with oxygen atoms in the
liquid phase.\cite{Mundy-science,GrayCondencedMatter1994,
RobinsonJPC1991,CroctonPhysica1981} However, we have not found
explicit spectroscopic evidence for such species at the air/water
interface. These all indicate that detailed comparison of the
theoretical calculations and the experimental analysis is
certainly an important subject in the future studies.
\section{Conclusion}
Detailed understanding of the air/water interface is important,
and can be used for the general understanding of the liquid water
structure. In this work, we presented detailed analysis of the SFG
vibrational spectra of the air/water interface taken in different
polarizations and experimental configurations. Polarization and
experimental configuration analysis have provided detailed
information on the orientation, structure and dynamics of the
water molecules at the air/water interface. The success of these
analysis indicated the effectiveness and ability of SFG-VS as a
uniquely interface specific spectroscopic probe of liquid
interfaces and other molecular interfaces. It also indicates that
for the neat air/water interface, as has been studied in the
literature for some other simple air/liquid interfaces, the
contribution from the interface region dominates the SFG
spectra.\cite{ShenApplyPhys,WeiXinJPCBBulkVsSurface,WeiPRBBulkvsSurface,Shen-methanolPRL1991}
Here are major conclusions we have reached for the air/water
interface. Firstly, we concluded that the motion of the
interfacial water molecules can only be in a limited angular
range, instead rapidly varying over a broad angular range in the
vibrational relaxation time suggested previously. Secondly,
because different vibrational modes of different molecular species
at the interface has different symmetry properties, polarization
and symmetry analysis of the SFG-VS spectral features can help
assignment of the SFG-VS spectra peaks to different interfacial
species. These analysis concluded that the narrow $3693cm^{-1}$
and broad $3550cm^{-1}$ peaks belong to $C_{\infty v}$ symmetry,
while the broad $3250cm^{-1}$ and $3450cm^{-1}$ peaks belong to
the symmetric stretching modes with $C_{2v}$ symmetry. Thus, the
$3693cm^{-1}$ peak is assigned to the free OH, the $3550cm^{-1}$
peak is assigned to the single hydrogen bonded OH stretching mode,
and the $3250cm^{-1}$ and $3450cm^{-1}$ peaks are assigned to
interfacial water molecules as two hydrogen donors for hydrogen
bonding (with $C_{2v}$ symmetry), respectively. Thirdly, analysis
of the SFG-VS spectra concluded that the singly hydrogen bonded
water molecules at the air/water interface have their dipole
vector direct almost parallel to the interface, and is with a very
narrow orientational distribution. The doubly hydrogen bond donor
water molecules have their dipole vector point away from the
liquid phase. Finally, we did not find any observable evidence for
interfacial water molecules with doubly free O-H bonds at the
air/water interface.
Many of the conclusions in this work agree well with previous
reports, with much more detailed understandings. The conclusion of
the narrow range motion of the free O-H bond is different from the
literature. The explicit assignment of the broad $3550cm^{-1}$
peak and determination of the symmetry property of the
hydrogen-bonded O-H stretching modes in the 3100$cm^{-1}$ to
3500$cm^{-1}$ region are based on firm evidences. These
conclusions as a whole provided a detailed and general picture of
the spectroscopy, structure and dynamics of the air/water
interface, which can be used for understanding chemical and
biological problems related to the ubiquitous water molecule in
general. The concepts and approaches used in the analysis in this
report can be applied to studying on more complex molecular
interfaces.
Recently, extensive efforts with SFG-VS, as well as SHG,
experimental studies and theoretical simulations have been devoted
to the renewed interests on ion adsorption and the Jones-Ray
effect at the air/aqueous solution
interfaces.\cite{TobiasJPCB2005,ShultzIRPC2000,RichmondJPCB2004,ShultzJPCB2002,
AllenJPCB2004,SaykallyCPL2004,SaykallyCPL2004-2,SaykallyJPCB2005,MuchaJPCB2005}
We suggest that detailed polarization and experimental
configuration analysis of the SFG vibrational spectra be applied
to these interfaces.
\vspace{0.8cm}
\noindent \textbf{Acknowledgment.} This work was supported by the
Chinese Academy of Sciences (CAS, No.CMS-CX200305), the Natural
Science Foundation of China (NSFC, No.20425309) and the Chinese
Ministry of Science and Technology (MOST, No.G1999075305). We
thank Bao-hua Wu for help derive the bond polarizability
derivative model expressions. H.F.W. acknowledges Y. R. Shen for
helpful discussions.
\section*{Appendix: Calculation of \textit{d} and \textit{c} Parameters for $C_{2v}$ Molecule}
Here we present the expressions to calculate the parameter
\textit{c} and \textit{d} for water molecules with C$_{2v}$
symmetry using the bond polarizability derivative model first used
by Hirose \textit{et al}.\cite{hirose:jcp1992,hirose:jpc1993} The
detailed re-derivation of the complete expressions and the
effectiveness of the model can be found in a recent
review.\cite{HongfeiIRPCreview}
The relationship between the Raman depolarization ratio $\rho$ and
the bond polarizability \textit{r} for a molecule group with
C$_{2v}$ symmetry was:\cite{HongfeiIRPCreview}
\begin{eqnarray}
\rho&=\frac{3}{4+20\frac{(1+2r)^2}{(1-r)^2(1+3\cos^2\tau)}}\label{rCH2definition}
\end{eqnarray}
\noindent in which $\tau$ is the H-O-H bond angle between the two
OH bonds of a water molecule. With the Raman depolarization ratio
measured as about 0.03,\cite{Murphy-r} the bond polarizability
\textit{r} for OH bond in water molecule can be deduced to be
0.32, as used by Du \textit{et al.}\cite{DuQuanPRL1993}
The 7 hyperpolarizability tensor elements of water molecule with
$C_{2v}$ symmetry are as the followings.
\begin{eqnarray}
\nonumber\beta_{aac}&=&\frac{G_{a}\beta_{OH}^{0}}{\omega_{a1}}\left[(1+r)-(1-r)\cos\tau\right]\cos(\frac{\tau}{2})\\
\nonumber\beta_{bbc}&=&\frac{2G_{a}\beta_{OH}^{0}}{\omega_{a1}}r\cos(\frac{\tau}{2})\\
\nonumber\beta_{ccc}&=&\frac{G_{a}\beta_{OH}^{0}}{\omega_{a1}}\left[(1+r)+(1-r)\cos\tau\right]\cos(\frac{\tau}{2})\\
\nonumber\beta_{aca}&=&\beta_{caa}=\frac{G_{b}\beta_{OH}^{0}}{\omega_{b1}}\left[(1-r)\sin\tau\right]\sin(\frac{\tau}{2})\\
\beta_{bcb}&=&\beta_{cbb}=0\label{c2vbond}
\end{eqnarray}
\noindent Where G$_{a}=(1+\cos\tau)/M_{O}+1/M_{H}$ and
G$_{b}=(1-\cos\tau)/M_{O}+1/M_{H}$ are the inverse effective mass
for the symmetric ($a_{1}$) and asymmetric ($b_{1}$) normal modes,
with $M_{O}$ and $M_{H}$ as the atomic mass of O and H atoms,
respectively. $\omega_{a1}$ and $\omega_{b1}$ are the vibrational
frequencies of the respective modes.
$\beta_{OH}^{0}=\frac{1}{2\varepsilon_{0}}\alpha'_{\zeta\zeta}\mu'_{\zeta}$,
as defined by Wei \textit{et al.}\cite{weixing:pre2000} The water
molecule are fixed in the molecular coordination $\lambda'(a,b,c)$
with the O atom at the coordination center, the molecule plane in
\textit{ac} plane, and the bisector from the oxygen to the two
hydrogen atoms side is the \textit{c} axis.
For the achiral rotationally isotropic ($C_{\infty v}$) liquid
interface, the symmetric stretching (\textit{ss}, $a_{1}$)
vibrational modes have,\cite{HongfeiIRPCreview}
\begin{eqnarray}
\chi_{xxz}^{(2),ss}&=&\chi_{yyz}^{(2),ss}\nonumber\\
&=&\frac{1}{2}N_{s}[\langle\cos^{2}\psi\rangle\beta_{aac}+
\langle\sin^{2}\psi\rangle\beta_{bbc}+\beta_{ccc}]\langle\cos\theta\rangle\nonumber\\
&+&\frac{1}{2}N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}-
\beta_{ccc}]\langle\cos^{3}\theta\rangle\nonumber\\
\chi_{xzx}^{(2),ss}&=&\chi_{zxx}^{(2),ss}=\chi_{yzy}^{(2),ss}=\chi_{zyy}^{(2),ss}\nonumber\\
&=&-\frac{1}{2}N_{s}[\langle\cos\theta\rangle\nonumber-\langle\cos^{3}\theta\rangle]\nonumber\\
&&[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}
-\beta_{ccc}]\nonumber\\
\chi_{zzz}^{(2),ss}&=&N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}]
\langle\cos\theta\rangle\nonumber\\
&-&N_{s}[\langle\sin^{2}\psi\rangle\beta_{aac}+\langle\cos^{2}\psi\rangle\beta_{bbc}-
\beta_{ccc}]\langle\cos^{3}\theta\rangle\nonumber\\
\label{ssofC2Vpsi}
\end{eqnarray}
\noindent And the asymmetric stretching (\textit{as}, $b_{1}$)
vibrational modes have,
\begin{eqnarray}
\chi_{xxz}^{(2),as}&=&\chi_{yyz}^{(2),as}=-N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle
[\langle\cos\theta\rangle-\langle\cos^{3}\theta\rangle]\nonumber\\
\chi_{xzx}^{(2),as}&=&\chi_{zxx}^{(2),as}=\chi_{yzy}^{(2),as}=\chi_{zyy}^{(2),as}\nonumber\\
&=&\frac{1}{2}N_{s}\beta_{aca}[\langle\cos^{2}\psi\rangle-\langle\sin^{2}\psi\rangle]
\langle\cos\theta\rangle\nonumber\\
&+&N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle\langle\cos^{3}\theta\rangle\nonumber\\
\chi_{zzz}^{(2),as}&=&2N_{s}\beta_{aca}\langle\sin^{2}\psi\rangle
[\langle\cos\theta\rangle-\langle\cos^{3}\theta\rangle]\label{asofC2Vpsi}
\end{eqnarray}
The $b_{2}$ asymmetric mode are SFG inactive since the
hyperpolarizability tensors $\beta_{bcb}$ and $\beta_{cbb}$ are
zero.
The Euler angel $\psi$ can be integrated if the H-X-H plane of the
XH$_{2}$ group can rotate freely around its symmetry axis $c$. For
water molecules with both OH bond hydrogen bonded to neighboring
molecules in liquid phase, the Euler angel $\psi$ should not be a
fixed value. Assuming a random $\psi$ distribution we have the
following non-vanishing tensor elements for the
symmetric-stretching mode.,\cite{Lurong2,HongfeiIRPCreview}
\begin{eqnarray}
\chi_{xxz}^{(2),ss}&=&\chi_{yyz}^{(2),ss}=\frac{1}{4}N_{s}(\beta_{aac}
+\beta_{bbc}+2\beta_{ccc})\langle{cos\theta}\rangle\nonumber\\
&+&\frac{1}{4}N_{s}(\beta_{aac}+\beta_{bbc}-2\beta_{ccc})
\langle{cos^3\theta}\rangle\nonumber\\
\chi_{xzx}^{(2),ss}&=&\chi_{zxx}^{(2),ss}=\chi_{yzy}^{(2),ss}=\chi_{zyy}^{(2),ss}\nonumber\\
&=&-\frac{1}{4}N_{s}(\beta_{aac}+\beta_{bbc}-2\beta_{ccc})
(\langle{cos\theta}\rangle-\langle{cos^3\theta}\rangle)\nonumber
\\\chi_{zzz}^{(2),ss}&=&\frac{1}{2}N_{s}(\beta_{aac}+\beta_{bbc})
\langle{cos\theta}\rangle\nonumber\\
&-&\frac{1}{2}N_{s}(\beta_{aac}+\beta_{bbc}
-2\beta_{ccc})\langle{cos^3\theta}\rangle\label{ssofC2V}
\end{eqnarray}
\noindent And the non-vanishing tensor elements for water
asymmetric-stretching modes are,
\begin{eqnarray}
\chi_{xxz}^{(2),as}&=&\chi_{yyz}^{(2),as}=-\frac{1}{2}N_{s}\beta_{aca}
(\langle{cos\theta}\rangle-\langle{cos^3\theta}\rangle)\nonumber\\
\chi_{xzx}^{(2),as}&=&\chi_{zxx}^{(2),as}=\chi_{yzy}^{(2),as}=\chi_{zyy}^{(2),as}
=\frac{1}{2}N_{s}\beta_{aca}\langle{cos^3\theta\rangle}\nonumber\\
\chi_{zzz}^{(2),as}&=&
N_{s}\beta_{aca}(\langle{cos\theta}\rangle-\langle{cos^3\theta\rangle})\label{asofC2V}
\end{eqnarray}
\begin{table}[t]
\caption{The general orientational parameter \textit{c} and the
strength factor \textit{d} for \textit{ss} mode and \textit{as}
mode of water molecule with $C_{2v}$ symmetry in different
polarization combinations. The \textit{d} values bear the unit
$\beta_{ccc}$.}
\begin{center}
\begin{tabular}{lcccccccccccccc}
\hline
\textit{ss} mode& & d-ssp & c-ssp & d-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & 0.400 & 0.038 & 0.012 & 1 & -0.146 & 0.174 \\
Config.2 & & 0.374 & 0.038 & 0.013 & 1 & -0.079 & 0.338 \\
Config.3 & & 0.362 & 0.038 & 0.013 & 1 & -0.046 & 0.589 \\
Config.4 & & 0.257 & 0.038 & 0.012 & 1 & 0.066 & -0.378 \\
\hline
\textit{as} mode& & d-ssp & c-ssp & d$\ast$c-sps & c-sps & d-ppp & c-ppp \\
\hline
Config.1 & & -0.154 & 1 & -0.122 & $\infty$ & 0.262 & 0.98 \\
Config.2 & & -0.144 & 1 & -0.129 & $\infty$ & 0.272 & 0.99 \\
Config.3 & & -0.139 & 1 & -0.128 & $\infty$ & 0.277 & 0.99 \\
Config.4 & & -0.099 & 1 & -0.117 & $\infty$ & 0.250 & 1.01 \\
\end{tabular}\label{CandDforC2vWater}
\end{center}
\end{table}
For CH$_{2}$ group, there is a general relationship
$\beta_{aac}+\beta_{bbc}-2\beta_{ccc}\cong 0$, because $\tau
=109.5^{\circ}$.\cite{Lurong2,HongfeiIRPCreview} This relationship
makes $\chi_{xzx}^{(2),ss}$=$\chi_{zxx}^{(2),ss}$=
$\chi_{yzy}^{(2),ss}$=$\chi_{zyy}^{(2),ss}\cong 0$, which means
that the \textit{ss} vibrational mode should vanish in the
\textit{sps} and \textit{pss} polarizations according to Eq.
\ref{ssofC2V}. For water molecule, $\tau=104.5^{\circ}$. Then
hyperpolarizability tensors of the water molecule are as the
followings: $\beta_{aac}=1.296$; $\beta_{bbc}=0.557$;
$\beta_{ccc}=1$; $\beta_{aca}=\beta_{caa}=0.741$;
$\beta_{bcb}=\beta_{cbb}=0$. Here all value are normalized to
$\beta_{ccc}=1$. Then,
$\beta_{aac}+\beta_{bbc}-2\beta_{ccc}=-0.147$. This value is not
$0$, but is very small. So the \textit{ss} vibrational mode
spectra in the \textit{sps} and \textit{pss} polarizations should
vanish as the $CH_{2}$ group mentioned above. However, they have
to be very small comparing with in \textit{ssp} spectra. This is
fully consistent with the small intensities in the \textit{sps}
SFG spectra for the $C_{2v}$ water modes in Fig.\ref{allSpectra}.
With above deduction, and following the procedure in previous
report,\cite{Lurong3} the general orientational parameter
\textit{c} and strength factor \textit{d} for the symmetric
stretching (\textit{ss}) mode and asymmetric stretching
(\textit{as}) mode of water molecule in different polarizations
and experimental configurations can be calculated (see Table
\ref{CandDforC2vWater}). The parameters used in the calculation
are $n_{1}(\omega)$=$n_{1}(\omega_{1})$=$n_{1}(\omega_{2})$=1;
$n_{2}(\omega)$=$n_{2}(\omega_{1})$=1.34;
$n_{2}(\omega_{2})$=1.18; $n'(\omega)$=$n'(\omega_{1})$=1.15;
$n'(\omega_{2})$=1.09, respectively. These parameters are the same
as the dielectric constants used for calculation of the air/water
interface by Wei \textit{et al.}\cite{WeiXingPRL2001} As we have
discussed in our reports,\cite{Lurong2,Lurong3} polarization
analysis with the co-propagating experimental geometry is
insensitive to the value of the dielectric constants of the IR
frequency.\cite{GanweiCPLNull,HongfeiIRPCreview} Therefore, we
used the same refractive constants for the IR frequencies across
the whole $3100cm^{-1}$ to $3800cm^{-1}$ region, and this does not
appear to affect our analysis. These \textit{c} and \textit{d}
values are used to calculate the polarization and orientation
dependence of the SFG intensity, as well as the interference
(phase) of different spectral features in different experimental
configurations. These calculations can satisfactorily explain the
detailed changes of the observed spectral features, as discussed
in the main text.
It is to be noticed that in the above discussion we only used
single water molecule parameters. When there is association and
clustering of water molecules, as long as the $C_{2v}$ symmetry
preserves, and the H-O-H bond angle does not change significantly,
above expressions dictated by symmetry properties should still be
valid.
|
1,314,259,993,042 | arxiv | \section{Introduction}
An irrational number is a real number that cannot be expressed as a fraction with the numerator as integers and denominator as nonzero integers. One of the most famous irrational number is $\sqrt{2}$, sometimes called Pythagoras's constant. Proof of the irrationality of $\sqrt{2}$ can be obtained in the following way: assume $\sqrt{2}$ is rational, that is, it can be expressed as a fraction of the form $\frac{w}{y}$, where $w$ and $y$ are two relatively prime positive integers. Now, since $\sqrt{2} = \frac{w}{y}$, we have $2 = \frac{w^{2}}{y^{2}}$, or $w^{2} = 2y^{2}$. Since $2y^{2}$ is even, $w^{2}$ must be even and since $w^{2}$ is even, so is $w$. Let $w = 2z$. We have $4z^{2} = 2y^{2}$ and thus $y^{2} = 2z^{2}$. Since $2z^{2}$ is even, $y^{2}$ is even, and since $y^{2}$ is even, so is y. However, two even numbers cannot be relatively prime, so $\sqrt{2}$ cannot be expressed as a rational fraction; hence $\sqrt{2}$ is irrational. Similarly, proving that the number $\sqrt{2} + \sqrt{3}$ is irrational can be done in the following manner: let $\sqrt{2} + \sqrt{3}$ be a rational number, say $x$. $x = \sqrt{2} + \sqrt{3}$ implies $x - \sqrt{2} = \sqrt{3}$. Squaring on both sides, we obtain, $x^{2} + 2 - 2x\sqrt{2} = 3$, or $\frac{x^{2} - 1}{2x} = \sqrt{2}$. Now, $\frac{x^{2} - 1}{2x}$ is a rational number. But this contradicts the fact that $\sqrt{2}$ is an irrational number. So, our supposition is false. Therefore, $\sqrt{2} + \sqrt{3}$ is an irrational number. The techniques which we have used to establish the irrationality of the above two numbers, if we use the same techniques on other rational linear combination of radicals, for example $\pm \sqrt[8]{12} \pm \frac{2}{3}\sqrt[27]{108} \pm \frac{1}{2}\sqrt{12}$, then it would become cumbersome.
Let $U$ denotes the set of all radicals which are irrationals, that is, $$U = \{\sqrt[m]{b} \mid b \in \mathbb{Q}^{+}, m \in \mathbb{N} - \{1\}, \sqrt[m]{b} \notin \mathbb{Q}\}.$$
Further, if $\mathcal{S}$ denotes the set of all finite rational linear combination of radicals in $U$ such that if $\alpha \in \mathcal{S}$ then the terms in the expression of $\alpha$ do not trivially cancel out by simplifying the radicals in the expression of $\alpha$ (for example, we do not consider $3\sqrt{12} - 5\sqrt{3} - \sqrt[4]{9}$ to be an element of $\mathcal{S}$ as $3\sqrt{12} - 5\sqrt{3} - \sqrt[4]{9} = 6\sqrt{3} - 5\sqrt{3} - \sqrt{3} = 0$), then we prove the following.
\begin{theorem}
If $\alpha \in \mathcal{S}$ then $\alpha$ cannot be expressed as $\frac{p}{q}$, where $p \in \mathbb{Z}$ and $q \in \mathbb{N}$.
\end{theorem}
An equivalent form of the above theorem has already been proved\cite{1940}\cite{1974}, which says that finite rational linear combination of radicals, under certain constraint, are linearly independent. The main motivation behind this note is to provide an elegant alternative proof of Theorem 1.
\begin{mydef}
An element $\sqrt[m]{b} \in U$ is said to be a reduced irrational if it can not be written of the form $e\sqrt[n]{d}$ where $n \in \mathbb{N}$, $e, d \in \mathbb{Q}^{+}$ and $n < m$.
\end{mydef}
For example, $\sqrt[3]{\frac{81}{5}}, \sqrt[4]{9}$ are not a reduced irrational numbers as $\sqrt[3]{\frac{81}{5}} = 3\sqrt[3]{\frac{3}{5}}$ and $\sqrt[4]{9} = \sqrt{3}$, whereas $\sqrt[3]{\frac{9}{5}}, \sqrt[4]{3^{3}}$ is a reduced irrational number.
\begin{lemma}
Let $\sqrt[m]{b}$ be a reduced irrational number and $l_{0}, l_{1}, \ldots, l_{t} \in \mathbb{Q}$ be such that $l_{t} \ne 0$. If $t < m$, then $l_{0} + l_{1}\sqrt[m]{b} + \cdots + l_{t}\sqrt[m]{b^{t}}$ is an irrational number.
\end{lemma}
\begin{proof}
Suppose $l_{0} + l_{1}\sqrt[m]{b} + \cdots + l_{t}\sqrt[m]{b^{t}} = \frac{p}{q}$ for some $p \in \mathbb{Z}$ and $q \in \mathbb{N}$. If we consider the polynomials $u(X), v(X) \in \mathbb{Q}[X]$ given by, $u(X) = ql_{t}X^{t} + \cdots + ql_{1}X + ql_{0} - p$ and $v(X) = X^m - b$, then observe that $\sqrt[m]{b}$ is a common zero for both of the polynomials. Therefore $gcd(u(X), v(X)) = r(X)$ exists in $\mathbb{Q}[X]$. Since $r(X)$ divides $u(X)$ and $v(X)$, so we have $deg(r) < m$ and the zeros of $r(X)$ are also the zeros of $v(X)$. If $ r_{0} \in \mathbb{Q} - \{0\}$ be the constant term of the polynomial $r(X)$, then $\prod{(\omega\sqrt[m]{b})} = r_{0}$, where the product is taken over all zeros of $r(X)$ and $\omega$ is some $m$-th root of unity. Now taking modulus on both sides, we obtain that $|r_{0}| = (\sqrt[m]{b})^{deg(r)}$, that is, $\sqrt[deg(r)]{|r_{0}|} = \sqrt[m]{b}$. But then this contradicts the fact that $\sqrt[m]{b}$ is a reduced irrational number. Therefore it must be that our supposition is false. This completes the proof.
\end{proof}
Observe that from the above lemma it is easy to understand that what can be the minimal polynomial for a reduced irrational number $\sqrt[m]{b}$. That is, if we take $p = 0$ and $q = 1$ in the above proof, then it tells us that the degree of the minimal polynomial cannot be less than $m$, and it is exactly $m$, namely, $X^{m} - b$. Further consider the finite product $c = \prod_{i}(\sqrt[m_{i}]{b_{i}})^{\epsilon_{i}}$ with $0 \le \epsilon_{i} \le (m_{i} - 1)$ such that $c \notin \mathbb{Q}$ and each $\sqrt[m_{i}]{b_{i}}$ is a reduced irrational number. Since $c^{\prod_{i}m_{i}} \in \mathbb{N}$, so by well ordering principle we can have a smallest positive integer $s$, such that $c^{s} \in \mathbb{Q}$. If we set $c^{s} = k$, then observe that $\sqrt[s]{k}$ is the reduced irrational number and that $X^{s} - k$ is the minimal polynomial for $c$ over $\mathbb{Q}$.
Now, let $\mathcal{I_{L}}$ denotes the set of all reduced irrational numbers.
\begin{mydef}
For $k \in \mathbb{N}$, a subset $S = \{\sqrt[m_{i}]b_{i} \mid 1 \le i \le k\}$ of $\mathcal{I_{L}}$ is said to be a reduced set if and only if $\prod_{i = 1}^{k}(\sqrt[m_{i}]{b_{i}})^{\epsilon_{i}} \notin \mathbb{Q}$, $\forall$ $(\epsilon_{1}, \epsilon_{2}, \ldots, \epsilon_{k}) \in V_{1} \times V_{2} \times \cdots \times V_{k} - \{0\}$, where $V_{i} = \{0, 1, 2, \ldots, (m_{i} - 1)\}$ and $0 = (0, 0, \ldots, 0)$.
\end{mydef}
Observe that every nonempty subset of a reduced set is again a reduced set.
\begin{lemma}
If $\alpha \in \mathcal{S}$, then there exist a reduced set $S$ such that $\alpha \in \mathbb{Q}(S)$.
\end{lemma}
\begin{proof}
Let $\alpha = r_{1}\sqrt[m_1]{c_{1}} + r_{2}\sqrt[m_2]{c_{2}} + \cdots + r_{n}\sqrt[m_n]{c_{n}}$. Assume, without loss of generality, that $\sqrt[m_{i}]{c_{i}} \ne \sqrt[m_{j}]{c_{j}}$ whenever $i \ne j$ where $1 \le i, j \le n$. let $c_{1} = \frac{f_{1}}{s_{1}}, c_{2} = \frac{f_{2}}{s_{2}}, \ldots, c_{n} = \frac{f_{n}}{s_{n}}$. For each $i$ with $1 \le i \le n$, let $f_{i}s_{i}^{m_{i} - 1} = p_{i1}^{\delta_{i1}} \cdots p_{ir_{i}}^{\delta_{ir_{i}}}$ be the factorisation into primes and $R_{i}$ denotes the set containing the reduced irrational numbers reduced form the numbers $\sqrt[m_i]{p_{i1}^{\delta_{i1}}}, \ldots, \sqrt[m_i]{p_{ir_{i}}^{\delta_{ir_{i}}}}$. Let $R = \bigcup_{i = 1}^{n}R_{i}$. If $R$ is singleton set $R = S$. Otherwise, let $\sqrt[\beta_{1}]{p_{1}^{a_{1}}}, \sqrt[\beta_{2}]{p_{2}^{a_{2}}}, \ldots, \sqrt[\beta_{z}]{p_{z}^{a_{z}}}$ be the distinct elements of $R$ where $p_{i}$'s are primes. Clearly $gcd(\beta_{i}, a_{i}) = 1$ for all $i$ with $1 \le i \le z$. Now for each $i$ with $1 \le i \le z$, choose $u_{ij} \in \mathbb{N}$ such that $a_{i} = u_{i1} + u_{i2} + \cdots + u_{iv_{i}}$ and $\frac{\beta_{i}}{u_{ij}} = \theta_{j} \in \mathbb{N}$, $\forall j$ with $1 \le j \le v_{i}$. Further, for $1 \le i \le z$ let $L_{i} = \{\sqrt[\theta_{j}]{p_{i}} \mid 1 \le j \le v_{i}\}$ and $L = \bigcup_{i = 1}^{z}L_{i}$. Now, if $L$ is singleton then set $L = S$. Otherwise, let $\sqrt[\mu_{1}]{q_{1}}, \sqrt[\mu_{2}]{q_{2}}, \ldots, \sqrt[\mu_{y}]{q_{y}}$ be the distinct elements of $L$ where $q_{i}$'s are primes for $1 \le i \le y$. For $1 \le k \le t$, let $q_{kl_{k}}$ be the distinct primes that appear $l_{k}$ times inside the radical signs, where $l_{1} + l_{2} + \cdots + l_{t} = y$. Now for each $q_{kl_{k}}$ with $1 \le k \le t$, let $\sqrt[\mu^{\prime}_{1}]{q_{kl_{k}}}, \sqrt[\mu^{\prime}_{2}]{q_{kl_{k}}}, \ldots, \sqrt[\mu_{l_{k}}^{\prime}]{q_{kl_{k}}}$ be the corresponding radicals and $\eta_{k} = lcm(\mu^{\prime}_{1}, \mu^{\prime}_{2}, \ldots, \mu^{\prime}_{l_{k}})$. Set $S = \{\sqrt[\eta_{k}]{q_{kl_{k}}} \mid 1 \le k \le t\}$. We claim that $S$ is a reduced set. To see this, observe that if the number
$$\prod_{k = 1}^{t}(\sqrt[\eta_{k}]{q_{kl_{k}}})^{\epsilon_{k}} = \sqrt[(\eta_{1}\eta_{2} \cdots \eta_{t})]{\prod_{k = 1}^{t}q_{kl_{k}}^{(\epsilon_{k}\eta_{1} \cdots \eta_{k - 1}\eta_{k + 1} \cdots \eta_{t})}}$$
would be a rational number, where not all $\epsilon_{k}$'s are zero and $0 \le \epsilon_{k} \le \eta_{k} - 1$, then since $q_{kl_{k}}$'s are the distinct primes so it must be that $(\eta_{1}\eta_{2} \cdots \eta_{t}) \mid (\epsilon_{k}\eta_{1} \cdots \eta_{k - 1}\eta_{k + 1} \cdots \eta_{t})$ for each $k$ such that $\epsilon_{k} \ne 0$. But this means $\eta_{k} \mid \epsilon_{k}$ which is absurd. This completes the proof.
\end{proof}
Now we give an example to find the reduced set for the number $\sqrt[8]{12} - \frac{2}{3}\sqrt[27]{108} + \frac{1}{2}\sqrt{12}$ by applying the above lemma(3). Here, $c_{1} = 12$, $c_{2} = 108$, $c_{3} = 12$ and $m_{1} = 8$, $m_{2} = 27$, $m_{3} = 2$. Also, $f_{1} = 12$, $f_{2} = 108$, $f_{3} = 12$ and $s_{1} = 1$, $s_{2} = 1$, $s_{3} = 1$. Let $f_{1}s_{1}^{7} = f_{3}s_{3} = 12 = 2^{2}3$ and $f_{2}s_{2}^{26} = 108 = 2^{2}3^{3}$ be the factorisation into primes. Further, $R_{1} = \{\sqrt[4]{2}, \sqrt[8]{3}\}$, $R_{2} = \{\sqrt[27]{4}, \sqrt[9]{3}\}$, $R_{3} = \{\sqrt{3}\}$ and $R = \{\sqrt[4]{2}, \sqrt[8]{3}, \sqrt[27]{4}, \sqrt[9]{3}, \sqrt{3}\}$. Now observe that we have $\beta_{1} = 4$, $\beta_{2} = 8$, $\beta_{3} = 27$, $\beta_{4} = 9$, $\beta_{5} = 2$, $a_{1} = 1$, $a_{2} = 1$, $a_{3} = 2$, $a_{4} = 1$, $a_{5} = 1$ and $p_{1} = 2$, $p_{2} = 3$, $p_{3} = 2$, $p_{4} = 3$, $p_{5} = 3$. So we only have to look at the element $a_{3} = 2$ as rest all of $a_{i}$'s are 1. Since, $2 = 1 + 1$, so $L_{3} = \{\sqrt[27]{2}\}$ and $L_{1} = \{\sqrt[4]{2}\}$, $L_{2} = \{\sqrt[8]{3}\}$, $L_{4} = \{\sqrt[9]{3}\}$, $L_{5} = \{\sqrt{3}\}$. So we get $L = \{\sqrt[4]{2}, \sqrt[8]{3}, \sqrt[27]{2}, \sqrt[9]{3}, \sqrt{3}\}$. As, $lcm(4, 27) = 108$ and $lcm(8, 9, 2) = 72$, so we get that $S = \{\sqrt[108]{2}, \sqrt[72]{3}\}$ and $\sqrt[8]{12} - \frac{2}{3}\sqrt[27]{108} + \frac{1}{2}\sqrt{12} \in \mathbb{Q}(\sqrt[108]{2}, \sqrt[72]{3})$.
$$\textbf{Proof of the Theorem 1}$$
\begin{proof}
Let $\alpha \in \mathcal{S}$. Then the lemma(3) guarantees that there exists natural number $k$ and a reduced set $S = \{\sqrt[m_{i}]b_{i} \mid 1 \le i \le k\}$ such that we have, $\alpha = \gamma_{0} + \gamma_{1}\sqrt[m_{k}]{b_{k}^{i_{1}}} + \cdots + \gamma_{M}\sqrt[m_{k}]{b_{k}^{i_{M}}}$, where $\gamma_{j} \in K - \{0\}$ for $0 \le j \le M$, $0 \le i_{1} \le i_{2} \ldots \le i_{M - 1} < i_{M} < m_{k}$, $K = \mathbb{Q}(S - \{\sqrt[m_{k}]b_{k}\})$.
Now, suppose on the contrary we have $\alpha = \frac{p}{q}$ for some $p \in \mathbb{Z}$ and $q \in \mathbb{N}$. Consider the polynomials $f(X) = q\gamma_{M}X^{i_{M}} + \ldots + q\gamma_{1}X^{i_{1}} + q\gamma_{0} - p$ and $g(X) = X^{m_{k}} - b_{k}$. Since, $\sqrt[m_{k}]{b_{k}}$ is a common zero for both of the polynomials, so $gcd(f(X), g(X)) = h(X)$ exists in $K[X]$. Since $h(X)$ divides $g(X)$ thus, the zeros of $h(X)$ are of the form $\omega\sqrt[m_{k}]{b_{k}}$ where $\omega$ is some $m_{k}$-th root of unity. If $h_{0} \in K - \{0\}$ be the constant term of the polynomial $h(X)$, then we have $\prod{(\omega\sqrt[m_{k}]{b_{k}})} = h_{0}$, where the product is taken over all zeros of $h(X)$. Now taking modulus on both sides, we obtain that $|h_{0}| = \sqrt[m_{k}]{b_{k}^{deg(h)}}$, that is, $\sqrt[m_{k}]{b_{k}^{deg(h)}} \in K$.
Let $\mathcal{B} \subseteq H$ be a basis for the $\mathbb{Q}$ vector space $K$, where $H$ denotes the set $$\{\prod_{i = 1}^{k - 1}(\sqrt[m_{i}]{b_{i}})^{\epsilon_{i}} \mid 0 \le \epsilon_{i} \le (m_{i} - 1)\}.$$ We claim that $Tr_{K/\mathbb{Q}}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta)$ is nonzero for at least one $\beta$ in $\mathcal{B}$, where $Tr_{K/\mathbb{Q}} : K \rightarrow \mathbb{Q}$ is the well known trace function. It is because, if $Tr_{K/\mathbb{Q}}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta) = 0$ $\forall \beta \in \mathcal{B}$, then since $\{\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta \mid \beta \in \mathcal{B}\}$ forms a basis for the $\mathbb{Q}$ vector space $K$, so for any $u \in K$ with $u = \sum_{\beta \in \mathcal{B}}c_{\beta}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta)$ for some $c_{\beta} \in \mathbb{Q}$, we get that $$Tr_{K/\mathbb{Q}}(u) = Tr_{K/\mathbb{Q}}(\sum_{\beta \in \mathcal{B}}c_{\beta}\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta) = \sum_{\beta \in \mathcal{B}}c_{\beta}Tr_{K/\mathbb{Q}}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta) = 0,$$ whereas $Tr_{K/\mathbb{Q}}(1) = |\mathcal{B}|$. Let $\beta^{*} \in \mathcal{B}$ be such that $Tr_{K/\mathbb{Q}}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta^{*}) \ne 0$. Now if $m(X) = X^{d} + a_{d - 1}X^{d - 1} + \cdots + a_{1}X + a_{0}$ be the minimal polynomial for $\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta^{*}$ over $\mathbb{Q}$, then certainly $a_{d - 1} = -\frac{d}{|\mathcal{B}|}Tr_{K/\mathbb{Q}}(\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta^{*})$ is non-zero. But from the very next discussion on the lemma(2) we know that the minimal polynomial for $\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta^{*}$ over $\mathbb{Q}$ is of the form $X^{s} - (\sqrt[m_{k}]{b_{k}^{deg(h)}}\beta^{*})^{s}$ for some $s \in \mathbb{N}$. Since the monic minimal polynomial is unique, therefore we must have $s = d$ and $a_{d - 1} = a_{0}$, that is $d = 1$. Hence we get that $(\sqrt[m_{k}]{b_{k}})^{deg(h)}\beta^{*} \in \mathbb{Q}$. Since $deg(h) < m_{k}$ and $\beta^{*} \in H$, so this contradicts the fact that $S$ is a reduced set, completing the proof of the theorem.
\end{proof}
|
1,314,259,993,043 | arxiv |
\section{Introduction}
Real-time and high-accuracy positioning is a crucial component for a large variety of applications, such as autonomous driving, logistics tracking, search-and-rescue, emergency response, Internet-of-Things (IoT), UAV sensing, and emerging integrated sensing and communication (ISAC)\cite{8025618,7426565,meyer2018scalable,9282206,liu2022survey}. The $5^\mathrm{th}$ generation (5G) and beyond networks require the ability of precise positioning, since ubiquitous real-time position information can be extracted by using node-to-node communication capabilities of the network that consists of anchor nodes and agent nodes \cite{win2018theoretical,del2017survey,ca}.
Traditional positioning technologies in the wireless network have been studied extensively. Most of the works exploit either time of arrival (ToA) or direction of arrival (DoA)\cite{win2011network,8240645,han2015performance} measured at the receiver equipped with a single antenna or an antenna array. For time-based estimation, such as ToA or joint DoA and ToA, extremely precise synchronization between the terminal and the receiver must be ensured\cite{guerra2018single}. As for positioning algorithms, ESPRIT and MUSIC approaches have been widely proposed to estimate terminal position by using antenna array to observe the channel's array manifold vector, which is only characterized by DoA. Note that the aforementioned positioning techniques all consider a terminal located in the Fraunhofer (far-field) region\footnote{
In the far-field region, the transceiver distance is larger than the Fraunhofer distance $d_{F}=2D^{2}/\lambda$\cite{sherman1962properties}, where $D$ is the maximum dimension of the receiving antenna (array), and $\lambda$ is the wavelength.}, where the wavefront of an electromagnetic (EM) wave transmitted by the terminal can be approximated as a plane wave.
Envisioned as the key features of the beyond 5G networks, adoption of larger antenna arrays or surfaces\cite{larsson2014massive,lu2014overview,bjornson2019massive,tang2020wireless} and exploitation of higher frequency bands\cite{ghosh2014millimeter,rappaport2019wireless,akyildiz2018combating} will push the electromagnetic diffraction field from the far-field region towards the near-field region\footnote{In this paper, the term “near-field” refers to the “radiative near-field” and “Fresnel region”, where the transceiver distance is smaller than the Fraunhofer distance, but larger than the Fresnel distance $d_{f}=0.62\sqrt{D^{3}/\lambda}$\cite{selvan2017fraunhofer}.},
in which the wavefront tends to be spherical and the uniform plane wave assumption will no longer hold\cite{cui2022near,zhang2022beam}. Wireless communication taking place in the near-field region provides both new opportunities
and challenges for positioning. In particular, the near-field channel’s array manifold vectors contain more information related to the terminal position, as both distance information and DoA information can be inferred from the receiving array. Since traditional positioning technologies are not suitable for near-field positioning, it is critical to develop new
architectures and approaches to achieve high-accuracy and high-resolution near-field positioning.
The study of near-field positioning has attracted extensive attention. They can be classified into positioning model design, signal processing algorithm, and performance evaluation. For the model design, reference \cite{1165222} proposed a model with an imperfectly calibrated array for near-field positioning and studied a calibration method. To simplify the near-field model, many works applied the Fresnel approximation to the antenna arrays with special geometries, e.g., uniform linear arrays (ULAs)\cite{grosicki2005weighted,chen2004new,deng2007closed}, and considered the model mismatch while analyzing the achievable positioning precision. This mismatch inevitably reduced the estimation accuracy \cite{hsu2011mismatch}. To solve this problem and to characterize the incident waves emitted from a near-field terminal as accurately as possible, the \textit{spherical wavefront model} was developed. An antenna array was utilized to extract the distance and DoA information based on the \textit{spherical wavefront model} and it was revealed that the spherical wavefront provided an underlying generic parametric model for near-field positioning\cite{haneda2006parametric}.
In \cite{yin2017scatterer}, the \textit{spherical wavefront model} was extended to a practical scenario with large-scale antenna arrays. The results indicated that terminals in the near-field region could be roughly identified through employing large-scale antenna arrays to estimate the wavefront curvature, i.e., curvature of arrival (CoA). In order
to reduce the complexity and implementation cost of large-scale antenna arrays, the authors in \cite{9335528} introduced the electromagnetic (EM) lens to the \textit{spherical wavefront model}.
Based on the aforementioned \textit{spherical wavefront model}, some works investigated signal processing algorithms for near-field positioning. Reference \cite{huang1991near} estimated DoA by using a modified two-dimensional (2-D) MUSIC algorithm and a global-optimum maximum likelihood (ML) searching approach. In \cite{661337}, a high-order ESPRIT-like algorithm formulated for observations collected from a ULA was proposed. An overlapping symmetric sub-arrays algorithm was proposed in \cite{4217610} to estimate terminal position with low complexity that did not require computation of high-order statistics in contrast to the traditional near-field ESPRIT algorithm. In \cite{liang2009passive}, a two-stage MUSIC algorithm was proposed to estimate the position of a mixed near-field and far-field terminal. The result indicated that the curvature information should be exploited when the moving terminal approaches the receiver.
A subspace-based algorithm without eigendecomposition was proposed in \cite{8410024}, which could provide remarkable and satisfactory estimation performance compared with some existing near-field positioning algorithms. To further reduce the algorithm complexity, the authors in \cite{8509135} proposed a CoA algorithm. For positioning model utilizing large-scale antenna
arrays equipped with EM-lens, an effective parameterized estimation algorithm was proposed in \cite{9314267}, which could directly reuse receiving signals to extract position parameters.
In addition to signal processing, many works have studied the performance evaluation of near-field positioning. In practical scenarios, as the transceived EM waves encounter non-ideal phenomena such as noise, shadowing, and fading, the estimation accuracy of positioning is subject to uncertainty.
In the interest of system design and operation, it is momentous to obtain achievable accuracy in positioning operations to provide benchmarks for evaluating performance of the actual positioning systems. The most commonly used tool is the Cramér-Rao bound (CRB), which describes the fundamental lower limits for estimation accuracy. For instance, in \cite{9314267,6705635,9145059,delmas2016crb}, the \textit{spherical wavefront model} was employed to derive the CRBs for the near-field estimator with ULA, planar arrays, or large-scale antenna arrays.
All of the aforementioned works\cite{haneda2006parametric,9145059,yin2017scatterer,9335528,huang1991near,661337,4217610,liang2009passive,8410024,8509135,9314267,6705635,delmas2016crb} adopted the \textit{spherical wavefront model}, which has been proved to be inaccurate in \cite{8736783}. Specifically, the \textit{spherical wavefront model} does not correspond to the equations governing the EM field near an antenna or array, and often disregards the physical characteristics of the near-field source. This could have a profound impact on the generated electromagnetic fields and the observations collected by the receiving antenna. The \textit{analytic model} (\textit{true model}) is by far the most accurate electromagnetic theory-based model for describing signals in the near-field region. The authors in
\cite{de2021cramer} first evaluated the performance of the near-field positioning system using electromagnetic theory. They computed the CRBs for a terminal located on the central perpendicular line (CPL) of the receiving antenna surface by using the \textit{vector electric field}. However, in addition to \textit{vector electric field} observation, \textit{scalar electric field} and \textit{overall scalar electric field} observations are also possible due to the different observation capabilities of various receiving antenna paradigms. Moreover, it is more common for the terminal not to be located on the CPL. A more comprehensive study of positioning arbitrary terminal positions by utilizing different electric field observations is necessary. Consequently, it remains unclear how to evaluate the performance of near-field positioning in such a study using the electromagnetic propagation theory and estimation theory.
In this paper,
we develop a generic model for near-field positioning. It is also referred to as the general scenario. In this scenario, the receiving antennas\footnote{In the remainder of our paper, the receiving antenna is a broad concept referring to various antenna paradigms with different observation capabilities, such as a conventional surface antenna and intelligent surfaces with a large number of finely customizable antennas.} with different observation capabilities are employed. This results in the extraction of various electric field observations that require distinct CRB computation methods. In addition, unlike \cite{de2021cramer}, the position of the terminal in front of the receiving antenna is unrestricted such that it can be placed anywhere. A special case when the terminal is on the CPL of the receiving antenna surface is considered. The generality and
validity of the generic model are illustrated to obtain further simplifications and insights. Additionally, to show the scaling behavior of the CRBs, two further simplified scenarios are investigated: 1) The system is operating at frequencies in the range of GHz or above; 2) the surface diagonal length of the receiving antenna is significantly greater than the distance from the terminal to the receiver. Finally, the impact of multiple distributed receiving antennas is extensively discussed. The main contributions of this paper are summarized as follows.
\begin{itemize}
\item \textbf{Utilize \textit{analytic model}.} Unlike traditional near-field positioning methods following the inaccurate \textit{spherical wavefront model}, an \textit{analytic model} without any approximation is used based on the
electromagnetic propagation theory. The CRBs for estimating the terminal position are derived by combining the \textit{analytic model} with the estimation theory to provide fundamental limits for estimation accuracy of the actual near-field positioning system.
\item \textbf{Generic CRB expressions.} A generic near-field positioning model considering the diversity of observations and the universality of the terminal position is designed. Specifically, three electric fields (\textit{vector}, \textit{scalar} and \textit{overall scalar}) are extracted by receiving antennas with different observation capabilities to derive the generic expressions of CRBs for the terminal with an arbitrary position. This generalizes the existing results in \cite{de2021cramer}. In the CPL case, the precise closed-form expressions or upper and
lower bounds of the CRBs using the \textit{vector}
or \textit{scalar electric field} are provided to make it possible to
compute and analyze the CRBs in the asymptotic regime.
\item \textbf{SIMO positioning system.} To investigate the impact of the multiple receiving antennas on the positioning performance, the generic positioning model is extended to the system with multiple distributed receiving antennas under the CPL assumption, i.e., the single-input multiple-output (SIMO) system and the expressions of CRBs are derived. The results reveal that multiple receiving antennas can significantly
improve the estimation accuracy of dimensions parallel to the
receiving antenna surface.
\end{itemize}
The remaining of this paper is organized as follows. Section \ref{SEC:2} describes the generic system model, provides the CRB computation methods using the three electric field observations, and derives the specific CRB expressions. In Section \ref{sectionCPL}, the CPL case and two further simplified scenarios are discussed. In Section \ref{sec:simo}, SIMO positioning system is proposed. Numerical results and discussions are presented in Section \ref{sectionV}, and the conclusions are given in Section \ref{sec:con}.
The following notation is used throughout the paper. Vectors and matrices are denoted in bold lowercase and uppercase respectively, e.g., $\mathbf{a}$ and $\mathbf{A}$. We use $[\mathbf{A}]_{ij}$ to
denote the $(i,j)$th entry of $\mathbf{A}$ and $\mathbf{a}_{i}$ to denote the $i$th entry of $\mathbf{a}$. The superscripts $(\cdot)^{\mathrm{H}}$, $(\cdot)^{-1}$, and $(\cdot)^{\mathrm{T}}$ represent the matrix hermitian-transpose, inverse, and transpose, respectively. $(\cdot)^{*}$ and $\operatorname{Re}\{\cdot\}$ denote the complex conjugate and real part of the input operations. The operator $\|\cdot\|$ means to obtain $\mathcal{L}_{2}$-norm of the input and $|\cdot|$ stands for the modulo operator. The notations $\mathbb{C}$ and $\mathbb{R}$ represent sets of complex numbers and of real numbers, respectively. $n\sim\mathcal{C}\mathcal{N}\left(0, \sigma^{2}\right)$ stands for a circularly-symmetric complex-Gaussian random variable with variance $\sigma^{2}$. The notation $\jmath$ denotes the imaginary unit. $\mathbf{I}_{N}$ is the $N\times N$ identity matrix, $\mathbf{0}_{N}$ is the $N$-dimensional zero vector, and the suffix $\kappa = x, y, z$ represents the
$X$-, $Y$- and $Z$-dimension in the Cartesian
coordinate system, respectively.
\section{System Model and CRB Computation}
\label{SEC:2}
This section first introduces a generic near-field positioning system aiming to estimate the position of a point source terminal based on the electric field observed over the receiving antenna surface area. Since different receiving antenna settings have different observation capabilities, which are embodied in obtaining different observations, i.e., \textit{vector}, \textit{scalar} and \textit{overall scalar electric field}, we will consider these observations for the near-field positioning system. Finally, we will derive and analyze the CRBs for estimating the terminal position by using the above three electric fields and combining electromagnetic propagation theory with estimation theory.
\subsection{Generic System Model of Near-field Positioning}\label{subsection_SMNP}
Consider the near-field positioning system depicted in Fig. \ref{system1}. The terminal is a point source equipped with
a monochromatic single-antenna located at $\mathbf{p}_{t}$ inside a three-dimensional source region $\mathcal{R}_{t}$, and it generates the vector electric field $\mathbf{e}\left(\mathbf{p}_{r}\right) \in \mathbb{C}^{3}$ at an arbitrary point $\mathbf{p}_{r}$ on the surface $\mathcal{R}_{r}$ of the receiving antenna through a homogeneous and isotropic medium with neither scatterers nor reflectors.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.59]{TSP+WCL.pdf}
\caption{The generic near-field positioning system.}
\label{system1}
\end{figure}
In order to quantitatively describe the positional relationship between the terminal and receiving antenna, we create two Cartesian coordinate systems, $OXYZ$ and $PX^{\prime}Y^{\prime}Z^{\prime}$, with $O$ (the center of $\mathcal{R}_{r}$) and $\mathbf{p}_{t}$ (the centroid of $\mathcal{R}_{t}$) as the origins of the coordinates that have a pure translational relationship. In the $OXYZ$ system, we denote $\mathbf{p}_{t}=\left(x_{t},y_{t},z_{t}\right)$, $\mathbf{p}_{r}=(x_{r},y_{r},0)$ and $\mathcal{R}_{r}=\left \{(x_{r},y_{r},0):|x_{r}|\leq D/\sqrt{8},|y_{r}|\leq D/\sqrt{8} \right \}$, where $D$ is the maximum geometric dimension of the receiving antenna, i.e., the diagonal length of the square surface. Since the wavefront is spherical, we establish a spherical coordinate system $\left(r,\theta,\phi\right)$ at point $\mathbf{p}_{t}$ to facilitate the description of the spherical wave model. $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ are unit vectors along the $X$-, $Y$-, and $Z$-dimension in the $OXYZ$ system while $\hat{\bm{\theta}}$ and $\hat{\bm{\phi}}$ are unit vectors along the $\theta$ and $\phi$ coordinate curves. $\hat{\mathbf{r}}$ is a unit vector denoting the direction of $\mathbf{r}=\mathbf{p}_{r}-\mathbf{p}_{t}$.
The near-field positioning system can estimate the position of the terminal by using the electric field observations obtained over the receiving antenna area. Note that depending on the actual communication requirements, cost or technical limitations, the types of receiving antennas may be different, leading to extraction of varying types of observed electric fields and thus affecting the positioning performance. Next, we will introduce three different cases of the electric field observations.
\textbf{\textit{1) Vector Electric Field (VEF):}} The most ideal case is that the \textit{vector electric field} at each point on the whole contiguous surface of the receiving antenna can be observed. To obtain the \textit{VEF}, we apply Holographic MIMO (H-MIMO)\cite{huang2020holographic,dardari2021holographic,pizzo2020spatially} or large intelligent surfaces (LIS)\cite{hu2018beyond1,hu2018beyond2,dardari2020communicating} as the receiving antenna, which is a spatially-contiguous electronically active surface with a
vast amount of tiny antenna-elements. In the $OXYZ$ system, the \textit{vector electric field} $\mathbf{e}\left(\mathbf{p}_{r} \right)$ can be written as
\begin{equation}
\mathbf{e}\left(\mathbf{p}_{r} \right)= e_{x}\left(\mathbf{p}_{r} \right)\hat{\mathbf{x}}+e_{y}\left(\mathbf{p}_{r} \right)\hat{\mathbf{y}}+e_{z}\left(\mathbf{p}_{r} \right)\hat{\mathbf{z}}.
\end{equation}
Then, the observation equation using \textit{VEF} is
\begin{equation}
\hat{\mathbf{e}}\left(\mathbf{p}_{r} \right)=\mathbf{e}\left(\mathbf{p}_{r} \right)+\mathbf{n}\left( \mathbf{p}_{r}\right),
\label{observation_equation1}
\end{equation}
where $\hat{\mathbf{e}}\left(\mathbf{p}_{r} \right)$ is the noisy \textit{VEF} and $\mathbf{n}\left( \mathbf{p}_{r}\right)\in \mathbb{C}^{3}$ accounts for thermal noise that is distributed as $\mathbf{n}\left( \mathbf{p}_{r}\right)\sim \mathcal{C}\mathcal{N}\left(\mathbf{0}_{3}, \sigma^{2}\mathbf{I}_{3}\right)$.
\textbf{\textit{2) Scalar Electric Field (SEF):}} If the observation capability of the receiving antenna decreases, there will be a different approach for estimating the position of $\mathbf{p}_{t}$. This method uses a \textit{scalar electric field} that is a component of the Poynting vector perpendicular to each point of the the whole contiguous receiving surface $\mathcal{R}_{r}$. In fact, the \textit{SEF} can be regarded as a scalar approximation to the \textit{VEF} and provide an intermediate step to understand the \textit{analytic model}. In the $OXYZ$ system, the \textit{scalar electric field} $e\left( \mathbf{p}_{r}\right)$ can be written as
\begin{equation}
e\left(\mathbf{p}_{r}\right)=\sqrt{\|\mathbf{e}\left(\mathbf{p}_{r} \right)\|^{2}\left(- {\hat{\mathbf{r}}}^{\mathrm{T}}\cdot{\hat{\mathbf{z}}}\right)}\mathrm{e}^{-\jmath k_{0}r},
\label{eq:Epr_definition}
\end{equation}
where $k_{0}=\omega/c=2\pi/\lambda$ is the wave number, $\omega$ is the angular frequency, $\lambda$ is the wavelength, $c$ is the speed-of-light, $\cdot$ indicates inner product of vectors, and $r=\|\mathbf{r} \|$. Then, the observation equation using \textit{SEF} is
\begin{equation}
\hat{e}\left(\mathbf{p}_{r} \right)=e\left(\mathbf{p}_{r}\right)+n\left( \mathbf{p}_{r}\right),
\label{observation_equation2}
\end{equation}
where $\hat{e}\left(\mathbf{p}_{r} \right)$ is the observation of the \textit{SEF} with noise.
\textbf{\textit{3) Overall Scalar Electric Field (OSEF):}} With a further decline in the observation capability of the receiving antenna, we assume that only the \textit{overall scalar electric field} can be obtained, which is the integral of the \textit{scalar electric field} $e\left(\mathbf{p}_{r} \right)$ over the receiving antenna surface. In this case, the receiving antenna is a conventional surface antenna\cite{bjornson2021primer}. According to \eqref{eq:Epr_definition}, the \textit{overall scalar electric field} $e$ is
\begin{equation}
e=\sqrt{\frac{2}{D^{2}}}\iint_{\mathcal{R}_{r}}e\left(\mathbf{p}_{r} \right)d\mathbf{p}_{r},
\label{eq:overall}
\end{equation}
where $D^{2}/2$ is the area of the receiving surface antenna. Then, the observation equation using the \textit{OSEF} is
\begin{equation}
\hat{e}={e}+n,
\label{observation_equation3}
\end{equation}
where $\hat{e}$ is the noisy \textit{OSEF}.
Based on the estimation theory, the computation methods of the CRBs for estimating the position of $\mathbf{p}_{t}$ using the above three observation equations are given in the following proposition and corollaries.
\begin{prop}[CRB computation method using \textit{VEF}]\label{prop:CRBvector}
{Denote the real vector to be estimated as ${{\bm{\xi}}}\in \mathbb{R}^{3}=\left(x_{t},y_{t},z_{t} \right)$, which collects the unknown coordinates of $\mathbf{p}_{t}$ with respect to the Cartesian system $OXYZ$. The Fisher's Information Matrix (FIM), denoted as $\mathbf{I}(\bm{\xi})$, is a $3\times 3 $ matrix, whose elements are given by the following double integral:
\begin{equation}
\begin{aligned}
\left[\mathbf{I}(\bm{\xi})\right]_{mn} &=\frac{2}{\sigma^{2}} \iint_{\mathcal{R}_{r}} \operatorname{Re}\left\{\frac{\partial e_{x}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{n}} \frac{\partial e_{x}^{*}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{m}}+\right.\\
&\left.\frac{\partial e_{y}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{n}} \frac{\partial e_{y}^{*}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{m}}+\frac{\partial e_{z}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{n}} \frac{\partial e_{z}^{*}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{m}}\right\} d x_{r} d y_{r},
\label{eq:CRBvector}
\end{aligned}
\end{equation}
where $m,n=1,2,3$. The CRB for estimating the $i$th entry of ${\bm{\xi}}$ is
\begin{equation}
\mathrm{CRB}\left(\bm{\xi}_{i} \right)=\left[\mathbf{I}(\bm{\xi})^{-1}\right]_{ii}.
\label{eq:CRB}
\end{equation}}
\end{prop}
\begin{IEEEproof}
The results can be derived from \cite[Appendix 15C]{kay1993fundamentals} by replacing the noisy observation and the parameter to be estimated with the complex vector $\hat{\mathbf{e}}\left(\mathbf{p}_{r} \right)$ and the real vector ${{\bm{\xi}}}$, respectively.
\end{IEEEproof}
From Proposition \ref{prop:CRBvector}, the CRBs using \textit{SEF} and \textit{OSEF} can be computed by Corollary \ref{coro:CRBscalar} and Corollary \ref{coro:CRBoverall}.
\begin{coro}[CRB computation method using \textit{SEF}]\label{coro:CRBscalar}
{Using the \textit{scalar electric field}, the elements of FIM can be computed as:
\begin{equation}
[\mathbf{I}(\bm{\xi})]_{m n}=\frac{2}{\sigma^{2}}\iint_{\mathcal{R}_{r}} \operatorname{Re}\left\{\frac{\partial e\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{n}} \frac{\partial e^{*}\left(\mathbf{p}_{r}\right)}{\partial \bm{\xi}_{m}}\right\}dx_{r}dy_{r}.
\label{eq:FIMscalar}
\end{equation}
By substituting \eqref{eq:FIMscalar} into \eqref{eq:CRB}, CRBs in this case can be derived.}
\end{coro}
\begin{IEEEproof}
According to Proposition \ref{prop:CRBvector}, FIM is additive since $e_{x}\left(\mathbf{p}_{r}\right)$, $e_{y}\left(\mathbf{p}_{r}\right)$, and $e_{z}\left(\mathbf{p}_{r}\right)$ are independent observations. So if we only have one noisy observation $\hat{e}\left(\mathbf{p}_{r} \right)$, expression \eqref{eq:FIMscalar} can be derived.
\end{IEEEproof}
\begin{coro}[CRB computation method using \textit{OSEF}]\label{coro:CRBoverall}
Similar to Corollary \ref{coro:CRBscalar}, the elements of FIM can be derived as:
\begin{equation}
\left[\mathbf{I}(\bm{\xi})\right]_{mn} =\frac{2}{\sigma^{2}}\operatorname{Re}\left\{\frac{\partial e}{\partial \bm{\xi}_{n}} \frac{\partial e^{*}}{\partial \bm{\xi}_{m}} \right\}.
\label{eq:FIMoverall}
\end{equation}
By substituting \eqref{eq:FIMoverall} into \eqref{eq:CRB}, CRBs in this case are obtained.
\end{coro}
\begin{IEEEproof}
The only difference between \eqref{eq:FIMscalar} and \eqref{eq:FIMoverall} is that $e\left(\mathbf{p}_{r}\right)$ has already been integrated in \eqref{eq:overall}.
\end{IEEEproof}
\subsection{Electric Field Expressions}\label{subsection_EFE}
From the Maxwell equations, the vector electric field $\mathbf{e}\left(\mathbf{p}_{r}\right)$ generated in the point $\mathbf{p}_{r}$ from the isoptropic point antenna $\mathbf{p}_{t}$ is due to the
current density ${\mathbf{J}}\left({\mathbf{p}}_{t}\right)$ and satisfies \cite{poon2005degrees,harrington1961time}
\begin{equation}
{\mathbf{E}}\left({\mathbf{p}}_{r}\right)={\mathbf{G}}\left(\mathbf{r}\right){\mathbf{J}}\left({\mathbf{p}}_{t}\right),
\label{eq:E}
\end{equation}
where ${\mathbf{J}}\left({\mathbf{p}}_{t}\right)$ is Fourier representation ${\mathbf{J}}\left({\mathbf{p}}_{t},\omega \right)$ of the current $\mathbf{j}{\left(\mathbf{p}_{t} \right)}$ at point $\mathbf{p}_{t}$. ${\mathbf{G}}\left({\mathbf{r}}\right) \in \mathbb{C}^{3\times 3}$ is referred as the tensor Green function in electromagnetic theory and can be expressed as
\begin{equation}
\mathbf{G}(\mathbf{r}) \simeq-\frac{\jmath \eta \mathrm{e}^{-\jmath k_{0}r} }{2 \lambda r}\left(\mathbf{I}-\hat{\mathbf{r}} \cdot \hat{\mathbf{r}}^{\mathrm{T}}\right),
\label{eq:Gr}
\end{equation}
where $\eta=\sqrt{\mu/\epsilon}$, $\mu$, and $\epsilon$ are the permeability, permittivity, and impedance of free-space, respectively. The approximation in \eqref{eq:Gr} is tight when $r \geq \lambda$, which always holds when the terminal is in the near-field region (between the reactive near-field and the far-field region) of the receiving antenna\cite{bjornson2021primer}. Without loss of generality, we assume that the electromagnetic wave emitted from the terminal $\mathbf{p}_{t}$ is polarized in the $Y$-dimension, which means that $\mathbf{J}\left(\mathbf{p}_{t}\right)=J_{y}\left(\mathbf{p}_{t}\right)\mathbf{e}_{y}$.
Using \eqref{eq:E} and \eqref{eq:Gr}, the specific expressions of the three electric fields \textit{VEF}, \textit{SEF}, and \textit{OSEF} in the near-field region can be obtained by Proposition \ref{prop:electric_vec}, Corollary \ref{coro:electric_sca} and \ref{coro:electric_overall}.
\begin{prop} [\textit{Vector electric field}]\label{prop:electric_vec}
In the coordinate system $OXYZ$, the three components of the \textit{vector electric field} can be expressed as
\begin{align}
&e_{x}(\mathbf{p}_{r})=\jmath E_{0}\frac{\left(x_{r}-x_{t} \right)\left(y_{r}-y_{t} \right)}{r^{3}}\mathrm{e}^{-\jmath k_{0}r}\\
&e_{y}(\mathbf{p}_{r})=-\jmath E_{0}\frac{1}{r}\Big[1-\frac{\left(y_{r}-y_{t}\right)^{2}}{r^{2}}\Big]\mathrm{e}^{-\jmath k_{0}r}\\
&e_{z}(\mathbf{p}_{r})=-\jmath E_{0}\frac{z_{t}\left(y_{r}-y_{t}\right)}{r^{3}}\mathrm{e}^{-\jmath k_{0}r},
\end{align}
where $E_{0}=\frac{\eta J_{y}\left(\mathbf{p}_{t}\right)}{2 \lambda}$ is initial electric intensity and is measured in volts ($V$).
\end{prop}
\begin{IEEEproof}
Please see Appendix \ref{proof:VEF}.
\end{IEEEproof}
\begin{coro}[\textit{Scalar electric field}]\label{coro:electric_sca}
In the coordinate system $OXYZ$, the \textit{scalar electric field} can be derived as
\begin{equation}
e\left(\mathbf{p}_{r}\right)=E_{0}\frac{\sqrt{z_{t}[\left(x_{r}-x_{t} \right)^{2}+z_{t}^{2}]}}{r^{5/2}}\mathrm{e}^{-\jmath k_{0}r}.
\end{equation}
\end{coro}
\begin{IEEEproof}
Substituting \eqref{eq:er} in Appendix \ref{proof:VEF} into \eqref{eq:Epr_definition} yields
\begin{equation}
\begin{aligned}
e\left(\mathbf{p}_{r}\right)=\sqrt{\|\mathbf{G}\left(\mathbf{r}\right)\mathbf{J}\left(\mathbf{p}_{t}\right)\|^{2}\sin{\theta}\sin{\phi}}\mathrm{e}^{-\jmath k_{0}r}.
\label{eq:Epr_GJsinsine}
\end{aligned}
\end{equation}
By using \eqref{eq:Ex_simply} -- \eqref{eq:Ez_simply} in Appendix \ref{proof:VEF} into \eqref{eq:Epr_GJsinsine}, the \textit{scalar electric field} with respect to $\left(r,\theta,\phi \right)$ can be expressed as
\begin{equation}
e\left(\mathbf{p}_{r}\right)=|G(r)|J_{y}\left(\mathbf{p}_{t} \right)\sqrt{\sin^{3}\theta\sin{\phi}}\mathrm{e}^{-\jmath k_{0}r}.
\label{eq:Epr_GJsinsine1}
\end{equation}
By substituting \eqref{eq:r} -- \eqref{eq:tan} in Appendix \ref{proof:VEF} into \eqref{eq:Epr_GJsinsine1} yields Corollary \ref{coro:electric_sca}.
\end{IEEEproof}
\begin{coro}[\textit{Overall scalar electric field}]\label{coro:electric_overall}
In the coordinate system $OXYZ$, the \textit{overall scalar electric field} is
\begin{equation}
e=E_{0}\sqrt{\frac{2}{D^{2}}}\iint_{\mathcal{R}_{r}}\frac{\sqrt{z_{t}[\left(x_{r}-x_{t} \right)^{2}+z_{t}^{2}]}}{r^{5/2}}\mathrm{e}^{-\jmath k_{0}r}dx_{r}dy_{r}.
\end{equation}
\end{coro}
\begin{IEEEproof}
From \eqref{eq:overall} and \eqref{eq:Epr_GJsinsine1}, the overall scalar observation is derived in Corollary \ref{coro:electric_overall}.
\end{IEEEproof}
\subsection{CRB Computation and Analysis} \label{CRB_NCPL}
Using results in Sec. \ref{subsection_SMNP} and \ref{subsection_EFE}, the expressions of the CRBs for estimating the position of $\mathbf{p}_{t}$ in Fig. \ref{system1} are provided.
\begin{prop}[CRB expressions, $\mathbf{e}\left(\mathbf{p}_{r} \right)$]\label{prop:CRB_vec_specific}
Using the \textit{vector electric field}, the CRBs can be computed as
\begin{align}
&\mathrm{CRB}_{1}\left(x_{t} \right)=\frac{\rm{SNR}^{-1}}{2}\cdot\frac{-I_{23}^{2}+I_{22}I_{33}}{I_{\rm{s}}}\label{eq:CRB1x}\\
&\mathrm{CRB}_{1}\left(y_{t} \right)=\frac{\rm{SNR}^{-1}}{2}\cdot\frac{-I_{13}^{2}+I_{11}I_{33}}{I_{\rm{s}}}\\
&\mathrm{CRB}_{1}\left(z_{t} \right)=\frac{\rm{SNR}^{-1}}{2}\cdot\frac{-I_{12}^{2}+I_{11}I_{22}}{I_{\rm{s}}}\label{eq:CRB1z},
\end{align}
where $\rm{SNR}={|E_{0}|^{2}}/{\sigma^{2}}$, $I_{mn}=\rho_{11}^{mn}+\rho_{12}^{mn}$, $m\leq n$, $\rho^{mn}_{11}$ and $\rho^{mn}_{12}$ are computed in \eqref{eq:rho_11^11} -- \eqref{eq:rho_12^23} in Appendix \ref{experssion}, and
\begin{equation*}
I_{\rm{s}}=2I_{12}I_{13}I_{23}+I_{11}I_{22}I_{33}-I_{13}^{2}I_{22}-I_{11}I_{23}^{2}-I_{12}^{2}I_{33}.
\end{equation*}
\end{prop}
\begin{IEEEproof}
According to Proposition \ref{prop:CRBvector} and Proposition \ref{prop:electric_vec}, the first-order derivatives ${\partial h_{x}\left(\mathbf{p}_{r}\right)}/{\partial x_{t}}$, $\cdots$, ${\partial h_{z}\left(\mathbf{p}_{r}\right)}/{\partial z_{t}}$ in FIM, where $h_{\kappa}\left(\mathbf{p}_{r}\right)\triangleq e_{\kappa}\left(\mathbf{p}_{r}\right)/E_{0}$, are first computed. For their specific expressions, please see \eqref{eq:hx_xt} -- \eqref{eq:hz_zt} in Appendix \ref{experssion}. Then by substituting these expressions into \eqref{eq:CRBvector}, we can derive the elements of FIM as follows.
\begin{equation}
[\mathbf{I}(\bm{\xi})]_{mn}
=2{\rm{SNR}}\left(\rho_{11}^{mn}+\rho_{12}^{mn}\right).
\end{equation}
Since FIM is a symmetric matrix, we have $[\mathbf{I}(\bm{\xi})]_{mn,m\neq n}=[\mathbf{I}(\bm{\xi})]_{nm,m\neq n}$. By applying the matrix inversion lemma, we obtain the inverse of $\mathbf{I}(\bm{\xi})$, denoted as $\mathbf{I}(\bm{\xi})^{-1}$, whose diagonal elements are the CRBs for estimating $x_{t}$, $y_{t}$, and $z_{t}$.
\end{IEEEproof}
Based on the above expressions of the CRBs for \textit{VEF}, the CRBs for \textit{SEF} and \textit{OSEF} are provided in the following Corollary \ref{coro:CRB_sca_specific} and Corollary \ref{coro:CRB_overall_specific}.
\begin{coro}[CRB expressions, $e\left(\mathbf{p}_{r}\right)$]\label{coro:CRB_sca_specific} If using the \textit{scalar electric field} observation, the specific expressions of the CRBs can also be computed by \eqref{eq:CRB1x} -- \eqref{eq:CRB1z}, and we denote them as $\mathrm{CRB}_{2}\left(\kappa_{t} \right)$. The only difference from Proposition \ref{prop:CRB_vec_specific} is the computation of $I_{mn}$, where $I_{mn}=\rho_{21}^{mn}+\rho_{22}^{mn}$. $\rho_{21}^{mn}$ and $\rho_{22}^{mn}$ are given in \eqref{eq:rho_21^11} -- \eqref{eq:rho_22^23} in Appendix \ref{experssion}.
\end{coro}
\begin{IEEEproof}
According to Corollary \ref{coro:electric_sca}, the first-order derivatives ${\partial h\left(\mathbf{p}_{t}\right)}/{\partial \kappa_{t}}$ involved in FIM $\mathbf{I}(\bm{\xi})$, where $h\left(\mathbf{p}_{t}\right)\triangleq e\left(\mathbf{p}_{t}\right)/E_{0}$, are computed in \eqref{eq:h_xt} -- \eqref{eq:h_zt} in Appendix \ref{experssion}. According to Corollary \ref{coro:CRBscalar}, $\mathrm{CRB}_{2}\left(\kappa_{t} \right)$ can be derived.
\end{IEEEproof}
\begin{coro}[CRB expressions, $e$]\label{coro:CRB_overall_specific} If we can only obtain the \textit{overall scalar electric field} observation. The CRBs, denoted as $\mathrm{CRB}_{3}\left(\kappa_{t} \right)$, can also be computed by \eqref{eq:CRB1x} -- \eqref{eq:CRB1z}, but the expression of $I_{mn}$ is different. Specifically, $I_{mn}=\rho_{3}^{mn}$,
\begin{equation}
\rho_{3}^{mn}=\frac{2}{D^{2}}\operatorname{Re}\left\{\frac{\partial {h}}{\partial \xi_{n}}\frac{\partial {h}^{*}}{\partial \xi_{m}}\right\},
\label{eq:rho3mn}
\end{equation}
and $h\triangleq \frac{D}{\sqrt{2}}e/E_{0}$.
\end{coro}
\begin{IEEEproof}
The results can be derived based on Corollary \ref{coro:CRBoverall} and Corollary \ref{coro:electric_overall}.
\end{IEEEproof}
Note that it might be difficult to compute the value of $\rho_{3}^{mn}$ due to the double integral in the molecule of partial derivative ${\partial h}/{\partial \kappa_{t}}$ in \eqref{eq:rho3mn}. By approximating the integral as a summation, a simpler expression of $\rho_{3}^{mn}$ can be obtained. In particular, we divide the receiving surface $\mathcal{R}_{r}$ into $\alpha$ parts, where $\sqrt{\alpha}$ is a positive integer and an odd number for simplicity. Denote the coordinate of each small part as $(x_{i},y_{j})$, where $x_{i}$ is the arithmetic sequence $(x_{1},x_{2},\ldots,x_{\sqrt{\alpha}})$, the common difference is $\frac{D}{\sqrt{2\alpha}}$, and the first item is $x_{1}=\frac{D}{2\sqrt{2\alpha}}-\frac{D}{2\sqrt{2}}$. Similarly, the arithmetic sequence $y_{j}$ has the same common difference and the first item as $x_{i}$. So ${h}$ can be written approximately as ${h}_{d}$,
\begin{equation}
{h}_{d}=\frac{D^{2}}{2\alpha}\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}} \frac{\sqrt{z_{t}[\left(x_{i}-x_{t}\right)^{2}+z_{t}^{2}]}}{r_{i,j}^{5/2}}{\mathrm{e}}^{-\jmath k_{0}r_{i,j}},
\label{eq:hd}
\end{equation}
where $r_{i,j}=\sqrt{\left(x_{i}-x_{t}\right)^{2}+\left(y_{j}-y_{t}\right)^{2}+z_{t}^{2}}$. Therefore, $\rho_{3}^{mn}$ can be computed by $\frac{2}{D^{2}}\operatorname{Re}\left\{\frac{\partial {h}_{d}}{\partial \xi_{n}}\frac{\partial {h}_{d}^{*}}{\partial \xi_{m}}\right\}$. Further, the specific expressions of $\rho_{3}^{mn}$ are given in \eqref{eq:rho_3^11} -- \eqref{eq:rho_3^23} in Appendix \ref{experssion}.
\section{CRB for a Transmitter on the CPL}\label{sectionCPL}
To validate the results derived in Sec. \ref{CRB_NCPL}, a simplified case of the generic near-field positioning system is considered, where the terminal is on the CPL of the receiving antenna surface. Specifically, the CPL is the line perpendicular to the receiving antenna surface $\mathcal{R}_{r}$ passing through the centre point $O$ and the three-dimensional source region $\mathcal{R}_{t}$ degenerates into the one-dimensional region $\mathcal{L}_{t}$, as shown in Fig. \ref{system2}.
\subsection{CRB Computation and Analysis for CPL Case}\label{subsection:CRB_CPL}
In CPL case, we have $x_{t}=y_{t}=0$ (but they are unknown), and $r=\sqrt{x_{r}^{2}+y_{r}^{2}+z_{t}^{2}}$. Since $r$ is an even function with respect to $x_{r}$ and $y_{r}$, and the integration domain $\mathcal{R}_{r}$ is symmetric, the cross-terms of different dimensions in the FIM $\mathbf{I}(\bm{\xi})$ are zero, meaning that the FIM $\mathbf{I}(\bm{\xi})$ is a diagonal matrix. Using the properties of the diagonal matrix inversion, the process of computing CRBs will be greatly simplified.
We denote a parameter $\tau\triangleq{D}/{z_{t}}$, which measures the diagonal length of the receiving antenna surface normalized by the distance from the considered terminal position to the receiver. For a terminal in the near-field region, the value of $\tau$ is large, and for a terminal far away from the receiving antenna, $\tau$ becomes small. Then we define a new integration domain $\mathcal{R}_{\tau}=\left \{(u,v):|u|\leq \tau/\sqrt{8},|v|\leq \tau/\sqrt{8} \right \}$.
Based on Proposition \ref{prop:CRB_vec_specific}, Corollary \ref{coro:CRB_sca_specific}, and Corollary \ref{coro:CRB_overall_specific}, the following results can be obtained.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.59]{system2.pdf}
\caption{The near-field positioning system for CPL case.}
\label{system2}
\end{figure}
\begin{coro}[CRB, \textit{VEF}, CPL]\label{coro:CRB_vec_CPL}
If the terminal is on the CPL, the CRBs for the estimation of $x_{t}$, $y_{t}$, and $z_{t}$ using the \textit{vector electric field}, denoted as $\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)$, are
\begin{equation}
\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)=\frac{{\rm{SNR}}^{-1}}{2\left(k_{0}^{2}\rho_{11\kappa}+z_{t}^{-2}\rho_{12\kappa}\right)},
\end{equation}
where
\begin{align}
&\rho_{11x}\triangleq \iint_{\mathcal{R}_{\tau}} \frac{u^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{3}}dudv\label{eq:11x}\\
&\rho_{12x}\triangleq \iint_{\mathcal{R}_{\tau}}\frac{u^{4}+v^{4}+u^{2}+v^{2}-u^{2}v^{2}}{(u^{2}+v^{2}+1)^{4}}dudv\label{eq:12x}\\
&\rho_{11y}\triangleq \iint_{\mathcal{R}_{\tau}} \frac{v^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{3}}dudv\label{eq:11y}\\
&\rho_{12y}\triangleq \iint_{\mathcal{R}_{\tau}}\frac{(u^{2}+1)(u^{2}+4v^{2}+1)}{(u^{2}+v^{2}+1)^{4}}dudv\label{eq:12y}\\
&\rho_{11z}\triangleq \iint_{\mathcal{R}_{\tau}} \frac{u^{2}+1}{(u^{2}+v^{2}+1)^{3}}dudv\label{eq:11z}\\
&\rho_{12z}\triangleq \iint_{\mathcal{R}_{\tau}}\frac{v^{4}+u^{2}v^{2}+1}{(u^{2}+v^{2}+1)^{4}}dudv\label{eq:12z}.
\end{align}
\end{coro}
\begin{IEEEproof}
Since FIM $\mathbf{I}(\bm{\xi})$ is a diagonal matrix, \eqref{eq:CRB} can be rewritten as
\begin{equation}
\mathrm{CRB}\left(\bm{\xi}_{i} \right)=\left[\mathbf{I}(\bm{\xi})\right]_{ii}^{-1}=I_{ii}^{-1},
\label{eq:CRB_CPL}
\end{equation}
where $I_{ii}=\rho_{11}^{ii}+\rho_{12}^{ii}$, $\rho_{11}^{ii}$ and $\rho_{12}^{ii}$ can be computed by replacing $x_{r,t}$ and $y_{r,t}$ in \eqref{eq:rho_11^11} -- \eqref{eq:rho_12^33} with $x_{t}$ and $y_{t}$.
\end{IEEEproof}
\begin{rem}[The generalizability of proposition \ref{prop:CRB_vec_specific}]\label{rem:related} Proposition \ref{prop:CRB_vec_specific} can be simplified to Corollary \ref{coro:CRB_vec_CPL} by utilizing diagonal matrix inversion and simplification of $\rho_{11}^{11}$ -- $\rho_{12}^{33}$ when the terminal is on the CPL. Besides, the expressions of $\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)$ are consistent with the results in \cite[Eqs. (28)--(36)]{de2021cramer}. The only difference is that we have replaced the integration variables $x_{t}$ and $y_{t}$ with $u$ and $v$ for a more intuitive analysis of the effect of $\lambda$ and $z_{t}$ on the CRBs. Consequently, the expressions of the CRBs (using the \textit{vector electric field}) in proposition \ref{prop:CRB_vec_specific} are more general than \cite{de2021cramer}. In fact, compared with the CPL case, Sec. \ref{subsection_SMNP} provides a generic positioning model.
\end{rem}
\begin{rem}[Closed-form expressions of $\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)$]\label{rem:CRB_vec_CPL} Different from \cite[Eqs. (39)--(46)]{de2021cramer}, the more precise closed-form expressions for $\rho_{12x}$, $\rho_{12y}$, $\rho_{11z}$, and $\rho_{12z}$ are given in \eqref{eq:12x_close} -- \eqref{eq:12z_close} in Appendix \ref{ap:remark2}. Since the closed-form expressions of $\rho_{11x}$ and $\rho_{11y}$ are hard to obtain, their closed-form upper and lower bounds are provided in \eqref{eq:11x+} -- \eqref{eq:11y-}.
\end{rem}
\begin{coro}[CRB, $\textit{SEF}$, CPL]\label{coro:CRB_sca_CPL}
For the CPL case, the CRBs for estimating $x_{t}$, $y_{t}$, and $z_{t}$ using the \textit{scalar electric field}, denoted as $\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)$, are given by
\begin{equation}
\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)=\frac{{\rm{SNR}}^{-1}}{2\left(k_{0}^{2}\rho_{21\kappa}+z_{t}^{-2}\rho_{22\kappa}\right)},
\end{equation}
where
\begin{align}
&\rho_{21x}\triangleq\iint_{\mathcal{R}_{\tau}} \frac{u^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{7/2}}dudv\label{eq:21x}\\
&\rho_{22x}\triangleq\iint_{\mathcal{R}_{\tau}}\frac{u^{2}(3u^{2}-2v^{2}+3)^{2}}{4(u^{2}+1)(u^{2}+v^{2}+1)^{9/2}}dudv\\
&\rho_{21y}\triangleq\iint_{\mathcal{R}_{\tau}} \frac{v^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{7/2}}dudv\\
&\rho_{22y}\triangleq\iint_{\mathcal{R}_{\tau}}\frac{25v^{2}(u^{2}+1)}{4(u^{2}+v^{2}+1)^{9/2}}dudv\\
&\rho_{21z}\triangleq\iint_{\mathcal{R}_{\tau}} \frac{u^{2}+1}{(u^{2}+v^{2}+1)^{7/2}}dudv\\
&\rho_{22z}\triangleq\iint_{\mathcal{R}_{\tau}}\frac{(u^{4}+u^{2}v^{2}+3v^{2}-u^{2}-2)^{2}}{4(u^{2}+1)(u^{2}+v^{2}+1)^{9/2}}dudv\label{eq:22z}.
\end{align}
\end{coro}
\begin{IEEEproof}
The diagonal elements of FIM $\mathbf{I}(\bm{\xi})$ in \eqref{eq:CRB_CPL} can be written as $I_{ii}=\rho_{21}^{ii}+\rho_{22}^{ii}$, $\rho_{21}^{ii}$ and $\rho_{22}^{ii}$ can be computed by replacing $x_{r,t}$ and $y_{r,t}$ in \eqref{eq:rho_21^11} -- \eqref{eq:rho_22^33} with $x_{t}$ and $y_{t}$.
\end{IEEEproof}
The closed-form expressions of $\rho_{21\kappa}$ and $\rho_{22\kappa}$ are complicated and lengthy, so we provided their closed-form upper and lower bounds in \eqref{eq:21xu} -- \eqref{eq:22zl}.
Corollary \ref{coro:CRB_vec_CPL} and Corollary \ref{coro:CRB_sca_CPL} clearly demonstrate the effects of the wavelength $\lambda = 2\pi/k_{0}$ and the propagation distance $d=z_{t}$ on the CRBs for fixed values of $\tau$ and $\mathrm{SNR}$ in the near-field positioning
system (using the \textit{vector} or \textit{scalar electric field}). In particular, the CRBs for all dimensions decrease as $\lambda$ or $z_{t}$ decreases. In other words, the estimation accuracy of the positioning
system increases as the carrier frequency ($f_{c}$) becomes higher or as the propagation distance becomes closer.
\begin{coro}[CRB, $\textit{OSEF}$, CPL]\label{coro:CRB_overall_CPL}
When we use the \textit{overall scalar electric field}, the CRBs for the CPL case, denoted as $\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right)$, can be computed as follows.
\begin{align}
\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right)=\frac{{\rm{SNR}}^{-1}}{\frac{4}{D^{2}}\left|\rho_{3\kappa} \right|^{2}},
\end{align}
where $\rho_{3\kappa}\triangleq\frac{\partial {h}}{\partial \kappa_{t}}$. Utilize ${h}_{d}$ to discretize ${h}$, we have
\begin{equation}
\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right)\approx\frac{\frac{\alpha^{2}}{D^{2}}{\rm{SNR}}^{-1}}{\left| \sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}f_{iz}\rho_{3\kappa}^{i,j}\mathrm{e}^{-\jmath k_{0} r_{i,j}}\right|^{2}},\label{eq:CRB3c}
\end{equation}
where $f_{iz}\triangleq\sqrt{z_{t}\left(x_{i}^{2}+z_{t}^{2}\right)}$ and
\begin{align}
&\rho_{3x}^{i,j}\triangleq x_{i}\big({\jmath k_{0}} r_{i,j}^{-\frac{7}{2}}+\frac{5}{2} r_{i,j}^{-\frac{9}{2}}-\frac{1}{z_{t}^{2}+x_{i}^{2}} r_{i,j}^{-\frac{5}{2}}\big)\label{eq:rho3xij} \\
&\rho_{3y}^{i,j}\triangleq y_{j}\big({\jmath k_{0}} r_{i,j}^{-\frac{7}{2}}+\frac{5}{2} r_{i,j}^{-\frac{9}{2}}\big)\\
&\rho_{3z}^{i,j}\triangleq-\jmath k_{0} z_{t} r_{i,j}^{-\frac{7}{2}}-\frac{5}{2}z_{t} r_{i,j}^{-\frac{9}{2}}+\frac{3 z_{t}^{2}+x_{i}^{2}}{2 f_{iz}^{2}}r_{i,j}^{-\frac{5}{2}}\label{eq:rho3zij}.
\end{align}
\end{coro}
\begin{IEEEproof}
The results can be derived utilizing Corollary \ref{coro:CRB_overall_specific} and equation \eqref{eq:hd} following the property of the inverse of a diagonal matrix $\mathbf{I}(\bm{\xi})$.
\end{IEEEproof}
\begin{rem}[$\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)<\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right)$]\label{rem:CRB_sca_overall} We can either compute \eqref{eq:CRB3c} numerically or use the Cauchy-Schwarz inequality to show
\begin{equation}
\begin{aligned}
\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right)&> \frac{\frac{\alpha^{2}}{D^{2}}{\rm{SNR}}^{-1}}{ \sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}\alpha\left|f_{iz}\rho_{3\kappa}^{i,j}\mathrm{e}^{-\jmath k_{0} r_{i,j}}\right|^{2}}\\
&=\frac{{\rm{SNR}}^{-1}}{\frac{D^{2}}{\alpha} \sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}\left(k_{0}^{2}\rho_{21\kappa}^{i,j}+z_{t}^{-2}\rho_{22\kappa}^{i,j}\right)}\\
&=\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right),
\end{aligned}
\end{equation}
where $\rho_{21\kappa}^{i,j}$ and $\rho_{22\kappa}^{i,j}$ are the discretized sampling of the integrand in \eqref{eq:21x} -- \eqref{eq:22z}. It can be seen that, under the same condition, the CRBs using \textit{SEF} are the lower bounds of the CRBs using \textit{OSEF}. Using \textit{OSEF} can significantly reduce the complexity of the near-field positioning system, but at the cost of reduced estimation accuracy.
\end{rem}
\subsection{Two Further Simplified Scenarios}\label{CRB further simplified}
\subsubsection{CRB analysis for \texorpdfstring{$z_{t}\gg \lambda$}{zt}} \label{CRB analysis for} Consider a scenario where the distance from the terminal located on the CPL to the receiver is much larger than the wavelength, namely $z_{t}\gg \lambda$\footnote{$z_{t}\gg \lambda$ corresponds to the near-field region when the size of the receiving antenna is on the order of meters, because $z_{t}\ll 2D^{2}/\lambda$ when $z_{t}\gg \lambda$ and $D$ is not very small.}. It generally holds in wireless communication systems with frequencies in the range of GHz ($10^{9}$ Hz) or above. Expressions of the CRBs in Corollary \ref{coro:CRB_vec_CPL} and \ref{coro:CRB_sca_CPL} can be further simplified as follows.
\begin{coro}[CRB, CPL, $z_{t}\gg \lambda$]\label{coro:CRB_CPL_zt}
If $z_{t}\gg \lambda$, the CRBs for the CPL case can be further simplified as
a) {Using the \textit{vector electric field}, $\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)$ reduces to
\begin{equation}
\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)\approx \frac{{\rm{SNR}}^{-1}}{2k_{0}^{2}\rho_{11\kappa}}.
\end{equation}}
b) {Using the \textit{scalar electric field}, $\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)$ reduces to
\begin{equation}
\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)\approx \frac{{\rm{SNR}}^{-1}}{2k_{0}^{2}\rho_{21\kappa}}.
\end{equation}}
\end{coro}
\begin{IEEEproof}
Please refer to Appendix \ref{ap:prop4}.
\end{IEEEproof}
Corollary \ref{coro:CRB_CPL_zt} shows that, when $z_{t}\gg\lambda$, the CRBs for all dimensions are
solely determined by the values of $\lambda$ and
$\tau$. Particularly, if we keep $\tau$ and $\mathrm{SNR}$ fixed, $\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)$ and $\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)$ are proportional to the square of $\lambda$. Furthermore, for a fixed value of
$\tau$, if $z_{t}$ increases
by a factor $\alpha$, the surface diagonal length $D$ needs to be scaled by the same
factor $\alpha$ (the surface area of the receiving antenna increases by the factor $\alpha^{2}$) to keep the CRBs unchanged.
\begin{rem}[Comparison of estimation accuracy]\label{rem:CRB_vec_sca_overall} From Corollary \ref{coro:CRB_vec_CPL} and Corollary \ref{coro:CRB_sca_CPL}, we find that $\rho_{11\kappa}>\rho_{12\kappa}$. Accordingly, based on Corollary \ref{coro:CRB_CPL_zt} and Remark \ref{rem:CRB_sca_overall}, we derive that
\begin{equation}
\mathrm{CRB}_{1}^{C}\left(\kappa_{t} \right)<\mathrm{CRB}_{2}^{C}\left(\kappa_{t} \right)<\mathrm{CRB}_{3}^{C}\left(\kappa_{t} \right).\label{eq:CRB_vec_sca_overall}
\end{equation}
Inequality \eqref{eq:CRB_vec_sca_overall} shows that using the \textit{vector electric field} at each point on the receiving surface renders lower CRBs, i.e., higher estimation accuracy. Using the \textit{scalar electric field} will reduce the complexity of the observations, but the CRBs will increase.
If the conventional surface antenna is employed as the receiver, the near-field positioning system can only obtain the \textit{overall scalar electric field}, which will further reduce the complexity of the system but the accuracy decreases too.
\end{rem}
\subsubsection{Asymptotic CRB analysis for \texorpdfstring{$\tau \to \infty$}{zt}}\label{Asymptotic CRB analysis for}
Based on the above analysis, it is interesting to analyze the behaviour of the asymptotic CRBs when the surface diagonal length $D$ is much larger than the distance $z_{t}$ from the terminal to the receiver. Corollary \ref{prop:CRB_CPL_tau} gives the CRBs in the asymptotic regime $\tau \to \infty$.
\begin{coro}[CRB, CPL, $\tau \to \infty$]\label{prop:CRB_CPL_tau}
For the CPL case and $z_{t}\gg \lambda$, in the asymptotic regime $\tau \to \infty$, the CRBs for the estimation of $x_{t}$, $y_{t}$, and $z_{t}$ are given by
a) Using the \textit{vector electric field}, we have
\begin{align}
&\lim_{\tau\to \infty}\mathrm{CRB}_{1}^{C}\left(x_{t} \right)=\frac{{\rm{SNR}}^{-1}}{6\pi^{3}}\frac{\lambda^{2}}{\ln{\tau}}\label{eq:CRB1xttau}\\
&\lim_{\tau\to \infty}\mathrm{CRB}_{1}^{C}\left(y_{t} \right)=\frac{{\rm{SNR}}^{-1}}{2\pi^{3}}\frac{\lambda^{2}}{\ln{\tau}}\label{eq:CRB1yttau}\\
&\lim_{\tau\to \infty}\mathrm{CRB}_{1}^{C}\left(z_{t} \right)=\frac{{\rm{SNR}}^{-1}}{6\pi^{3}}{\lambda^{2}}.\label{eq:CRB1zttau}
\end{align}
b) Using the \textit{scalar electric field}, we have
\begin{align}
&\lim_{\tau\to \infty}\mathrm{CRB}_{2}^{C}\left(x_{t} \right)=\frac{15}{64}\frac{{\rm{SNR}}^{-1}}{\pi^{3}}{\lambda^{2}}\label{eq:CRB2xttau}\\
&\lim_{\tau\to \infty}\mathrm{CRB}_{2}^{C}\left(y_{t} \right)=\frac{15}{32}\frac{{\rm{SNR}}^{-1}}{\pi^{3}}\lambda^{2}\label{eq:CRB2yttau}\\
&\lim_{\tau\to \infty}\mathrm{CRB}_{2}^{C}\left(z_{t} \right)=\lim_{\tau\to \infty}\mathrm{CRB}_{2}^{C}\left(x_{t} \right)\label{eq:CRB2zttau}.
\end{align}
\end{coro}
\begin{IEEEproof}
We have provided the closed-form expressions or upper and lower bounds in Appendix \ref{ap:remark2}, making it possible to compute and analyze the asymptotic CRBs. By computing the limit values of \eqref{eq:11x+} and \eqref{eq:11x-}, we have that $\rho_{11x}\sim\frac{3\pi}{4}\ln{\tau}$ for $\tau\to\infty$. According to \eqref{eq:11y+} and \eqref{eq:11y-}, we have that $\rho_{11y}\sim\frac{\pi}{4}\ln{\tau}$ for $\tau\to\infty$. Similarly, according to \eqref{eq:11z_close} and \eqref{eq:21xu} -- \eqref{eq:21zl}, we have $\lim \rho_{11z}=\frac{3\pi}{4}$, $\lim\rho_{21x}=\frac{8\pi}{15}$, $\lim\rho_{21y}=\frac{4\pi}{15}$ and $\lim\rho_{21z}=\frac{8\pi}{15}$, where we use $\lim$ to represent $\lim_{\tau\to \infty}$. Thus, Corollary \ref{prop:CRB_CPL_tau} holds.
\end{IEEEproof}
From Corollary \ref{prop:CRB_CPL_tau}, the following observations can be made. Firstly, for the near-field positioning system, if we use the \textit{vector electric field}, the CRBs for estimating $x_{t}$ and $y_{t}$ will decrease as a $\ln^{-1}$ function of $\tau$ and go to zero as $\tau$ increases infinitely. But $\mathrm{CRB}_{1}^{C}\left(z_{t} \right)$ tends to a fixed value which depends uniquely on the $\lambda$ and $\mathrm{SNR}$, and does not change with $\tau$. In the CPL case, $z_{t}$ represents the propagation distance, so equation \eqref{eq:CRB1zttau} provides a fundamental lower limit to the near-field ranging precision. Secondly, if we use the \textit{scalar electric field}, the CRBs for the estimation of $x_{t}$ and $z_{t}$
are identical and the three CRBs are solely determined by $\lambda$ and $\mathrm{SNR}$ as $\tau$ increases. Finally, in order to get more insights on the difference of fundamental limit of the estimation accuracy between \textit{VEF} and \textit{SEF} as $\tau$ increases, we denote their difference as $\Delta C_{\kappa}=p_{\kappa} \mathrm{SNR}^{-1}\lambda^{2}$ with $p_{x}=15/(64\pi^{3})\approx7.56\times10^{-3}$, $p_{y}=15/(32\pi^{3})\approx1.512\times10^{-2}$, and $p_{z}=13/(192\pi^{3})\approx6.77\times10^{-5}$. This indicates that using \textit{SEF} has a smaller performance penalty for the estimation of $z_{t}$ than $x_{t}$ and $y_{t}$ compared to using \textit{VEF}.
\section{CRB of the SIMO Positioning System}
\label{sec:simo}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.59]{system_3.pdf}
\caption{The SIMO near-field positioning system.}
\label{fig3}
\end{figure}
The receiving antenna in the previous sections is a single antenna or intelligent surface\footnote{The single intelligent surface represents a \textit{centralized-deployment} LIS/H-MIMO, which can observe $\textit{VEF}$/$\textit{SEF}$. Besides, the single antenna corresponds to a conventional surface antenna and it can only obtain $\textit{OSEF}$. For simplicity, we define both of them as “single-output”.}, where the positioning system can be defined as the single-input single-output (SISO) system. In this section, the system with multiple distributed receiving antennas will be discussed, referred to as the single-input multiple-output (SIMO) system depicted in Fig. \ref{fig3}. This SIMO system is specifically interpreted as follows.
\begin{itemize}
\item \textbf{Space constraints:} Each of the small receiving antenna is an intelligent surface or conventional surface antenna as previously described and they are distributed on a large rectangular surface $\mathcal{R}_{s}$ with size $\frac{R}{\sqrt{2}}\times\frac{R}{\sqrt{2}}$, where $R$ is usually a fixed value (a few meters to tens of meters) due to space constraints\footnote{The receiving antenna, such as LIS, can be easily embedded in daily life objects with limited size such as buildings, walls, cars, etc.} of the positioning system.
\item \textbf{Total surface area:} The total surface area is the same for different numbers of the small receiving antennas and each of them has the same surface area and properties. In particular, we consider that the total surface area is $\frac{D^{2}}{2}$ and the number of the receiving antennas is $N^{2}$. Therefore, the size of each receiving antenna is $\frac{D}{N\sqrt{2}}\times\frac{D}{N\sqrt{2}}$.
\item \textbf{Terminal position:} For simplicity, the terminal is located on the CPL with coordinates $(0, 0, z_{t})$, which makes the FIM matrix diagonalize as will be shown in Lemma \ref{lem:fim}.
\end{itemize}
Note
that if $N=1$, the SIMO
system degenerates into the SISO system, where the CRBs for all three dimensions using the three electric fields have been computed and analyzed in Sec. \ref{subsection_EFE} and Sec. \ref{sectionCPL}. In this section, we assume $N\geq2$. To derive the CRBs of the SIMO system, we provide Lemma \ref{lem:fim}.
\begin{lem}[Properties of the Fisher's information]\label{lem:fim}
The FIM of the SIMO system is a diagonal matrix, and the Fisher's information is identical for every four small receiving antennas rotationally symmetric about the origin (rotation angle is $90^{\circ}$).
\end{lem}
\begin{IEEEproof}
Since $\rho_{11}^{12}$ -- $\rho_{12}^{23}$ in \eqref{eq:rho_11^12} -- \eqref{eq:rho_12^23} and $\rho_{21}^{12}$ -- $\rho_{22}^{23}$ in \eqref{eq:rho_21^12} -- \eqref{eq:rho_22^23} (items in FIM off-diagonal elements) contain at least an odd power term of either $x_{r}$ or $y_{r}$, and $r$ is an even function with respect to $x_{r}$ and $y_{r}$, we can prove that even though the integral domains of $\rho_{11}^{12}$ -- $\rho_{12}^{23}$ and $\rho_{21}^{12}$ -- $\rho_{22}^{23}$ are no longer symmetric about the origin, due to the additivity of the Fisher's information, there can be a symmetric integral of each integral whose sum is zero. Consequently, the off-diagonal elements of the FIM matrix are canceled. Similarly, $\rho_{11}^{11}$ -- $\rho_{12}^{33}$ in \eqref{eq:rho_11^11} -- \eqref{eq:rho_12^33} and $\rho_{21}^{11}$ -- $\rho_{22}^{33}$ in \eqref{eq:rho_21^11} -- \eqref{eq:rho_22^33} (items in FIM diagonal elements) contain even power terms of $x_{r}$ and/or $y_{r}$,
so the diagonal elements are non-zero, and the values of $\rho_{11}^{11}$ -- $\rho_{12}^{33}$ and $\rho_{21}^{11}$ -- $\rho_{22}^{33}$ remain unchanged if $x_{r}$ becomes $-x_{r}$ and/or $y_{r}$ becomes $-y_{r}$. Therefore, Lemma \ref{lem:fim} holds.
\end{IEEEproof}
Based on Lemma \ref{lem:fim}, we divide the large
rectangular surface into four equal parts using the $X$ and $Y$ axes as their boundaries. Then, we only need to study one of the four parts, which contains $\frac{N^{2}}{4}$ small receiving antennas with index $({i},{j}),i,j=1,\cdots,\frac{N}{2} $. The integral domain of the small receiving antenna with index $({i},{j})$ is denoted as $\mathcal{R}_{i,j}=\left(x_{r},y_{r},0\right)$, where $x_{r}\in [\frac{\left(2i-1\right)R-D}{2\sqrt{2}N},\frac{\left(2i-1\right)R+D}{2\sqrt{2}N}]$, $y_{r}\in[\frac{\left(2j-1\right)R-D}{2\sqrt{2}N},\frac{\left(2j-1\right)R+D}{2\sqrt{2}N}]$. Additionally, $\mathcal{R}_{i,j}$ can be rewritten as $\mathcal{R}^{\tau}_{i,j}=\left(u,v,0\right)$, where $u\in \frac{1}{2\sqrt{2}N}[\frac{\left(2i-1\right)R}{z_{t}}-\tau,\frac{\left(2i-1\right)R}{z_{t}}+\tau]$, $v\in \frac{1}{2\sqrt{2}N}[\frac{\left(2j-1\right)R}{z_{t}}-\tau,\frac{\left(2j-1\right)R}{z_{t}}+\tau]$. The CRBs of the SIMO positioning system using \textit{VEF}, \textit{SEF} and \textit{OSEF} are derived as follows.
\begin{prop}[CRB, SIMO]\label{prop:CRB_CPL_SIMO}
For the defined SIMO positioning system depicted in Fig. \ref{fig3}, we have that:
a) {Using the \textit{vector electric field}, the CRBs can be given by
\begin{equation}
\mathrm{CRB}_{1}^{\rm{M}}\left(\kappa_{t}\right)=\frac{{\rm{SNR}}^{-1}}{8\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}\left(k_{0}^{2}\rho_{11\kappa}^{i,j}+z_{t}^{-2}\rho_{12\kappa}^{i,j}\right)},
\label{eq:CRB_SIMO_1}
\end{equation}
where $\rho_{11\kappa}^{i,j}$, $\rho_{12\kappa}^{i,j}$ have the same integrand as $\rho_{11\kappa}$, $\rho_{12\kappa}$ in \eqref{eq:11x} -- \eqref{eq:12z}, but their integral domain is $\mathcal{R}_{i,j}^{\tau}$.}
b) {Using the \textit{scalar electric field}, the CRBs are given by
\begin{equation}
\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t}\right)=\frac{{\rm{SNR}}^{-1}}{8\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}\left(k_{0}^{2}\rho_{21\kappa}^{i,j}+z_{t}^{-2}\rho_{22\kappa}^{i,j}\right)},
\label{eq:CRB_SIMO_2}
\end{equation}
where $\rho_{21\kappa}^{i,j}$, $\rho_{22\kappa}^{i,j}$ have the same integrand as $\rho_{21\kappa}$, $\rho_{22\kappa}$ in \eqref{eq:21x} -- \eqref{eq:22z}, but their integral domain is $\mathcal{R}_{i,j}^{\tau}$.}
c) {Using the \textit{overall scalar electric field}, the CRBs are
\begin{align}
\mathrm{CRB}_{3}^{\rm{M}}\left(\kappa_{t}\right)=\frac{{\rm{SNR}}^{-1}}{\frac{16}{D^{2}}\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}\Big|\rho_{3\kappa}^{ij} \Big|^{2}},\label{eq:CRB3M}
\end{align}
where $\rho_{3\kappa}^{ij}=\frac{\partial {h}_{ij}}{\partial \kappa_{t}}$, ${h}_{ij}$ has the same integrand as $h$ in Corollary \ref{coro:CRB_overall_specific} while its integral domain is $\mathcal{R}_{i,j}$. According to \eqref{eq:CRB3c}, the more feasible expression of \eqref{eq:CRB3M} is given. Similarly, we divide each small receiving surface $\mathcal{R}_{i,j}$ into $\alpha$ parts, and denote that $x_{m,i}=\frac{\left(2i-1\right)R-D}{2\sqrt{2}N}+\frac{(2m-1)D}{2\sqrt{2\alpha}N}$, $y_{n,j}=\frac{\left(2j-1\right)R-D}{2\sqrt{2}N}+\frac{(2n-1)D}{2\sqrt{2\alpha}N}$, $r_{mn,ij}=\sqrt{x_{m,i}^{2}+y_{n,j}^{2}+z_{t}^{2}}$, then $\mathrm{CRB}_{3}^{\rm{M}}\left(\kappa_{t}\right)$ can be further written as follows.
\begin{equation}
\mathrm{CRB}_{3}^{\rm{M}}\left(\kappa_{t}\right)\approx\frac{\frac{\alpha^{2}}{4D^{2}}{\rm{SNR}}^{-1}}{\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}\left| \sum_{m=1}^{\sqrt{\alpha}} \sum_{n=1}^{\sqrt{\alpha}}\varrho_{3\kappa}^{mn,ij}\right|^{2}},
\end{equation}
where $\varrho_{3\kappa}^{mn,ij}=\sqrt{z_{t}(x_{m,i}^{2}+z_{t}^{2})}\rho_{3\kappa}^{mn,ij}\mathrm{e}^{-\jmath k_{0} r_{mn,ij}}$ and $\rho_{3\kappa}^{mn,ij}$ is given in \eqref{eq:rho3xij} -- \eqref{eq:rho3zij}, but $x_{i},y_{j}$ and $r_{i,j}$ need to be modified to $x_{m,i},y_{n,j}$ and $r_{mn,ij}$, respectively. }
\end{prop}
\begin{IEEEproof}
Corollary \ref{coro:CRB_vec_CPL} -- \ref{coro:CRB_overall_CPL}
have computed the CRBs for all three
dimensions using the three electric fields in the SISO positioning system and the crux of the computation is to give the values of double integrals $\rho_{11\kappa}$, $\rho_{12\kappa}$, $\rho_{21\kappa}$, $\rho_{22\kappa}$ and $\rho_{3\kappa}$, whose domain is $\mathcal{R}_{\tau}$ or $\mathcal{R}_{r}$. In the SIMO positioning system, since the domain of each small receiving antenna is different and discontinuous, we change the domains from $\mathcal{R}_{\tau}$/$\mathcal{R}_{r}$ to $\mathcal{R}_{i,j}^{\tau}$/$\mathcal{R}_{i,j}$. Moreover, the electric field observations of each small receiving antennas are independent. So the Fisher's
information is additive. Hence, Proposition \ref{prop:CRB_CPL_SIMO} holds.
\end{IEEEproof}
Proposition \ref{prop:CRB_CPL_SIMO} indicates that $\mathrm{CRB}_{1}^{\rm{M}}\left(\kappa_{t}\right)$ and $\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t}\right)$ decrease as $\lambda$ or $z_{t}$ decreases for fixed values of $N$ and $\tau$ or, equivalently,
of the functions $\rho_{ab\kappa}^{i,j},a,b=1,2$. The impact of the number $N^{2}$ of small receiving antennas on the CRBs will be investigated in Sec. \ref{subsection:CRB_simo_numer}. Note that, same as Remark \ref{rem:CRB_sca_overall}, $\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t}\right)$ can be verified as the lower bounds of the $\mathrm{CRB}_{3}^{\rm{M}}\left(\kappa_{t}\right)$ by using the Cauchy-Schwarz inequality.
The authors in \cite{hu2018beyond2} use the scalar field observed from LIS to derive the CRBs in the SIMO positioning system. They consider a simple and idealized radiation model, which overlooks the physical characteristics of the source. Besides, they assume that the terminal is in the far-field to simplify the
computation of CRBs. In Proposition \ref{prop:CRB_CPL_SIMO}, we derive CRBs of the SIMO near-field positioning system while considering the characteristics of the near-field source and using different electric fields.
Next, we analyze
the behaviour of the CRBs in the SIMO positioning system when $z_{t}\gg\tau$ and $\tau\to\infty$. The main results are summarized in the
following two corollaries.
\begin{coro}[SIMO, $z_{t}\gg\tau$]\label{prop:CRB_CPL_SIMO_zt}
If $z_{t}\gg \lambda$, the CRBs of the
SIMO positioning system can be simplified as
\begin{equation}
\mathrm{CRB}_{1}^{\rm{M}}\left(\kappa_{t}\right)\approx\frac{{\rm{SNR}}^{-1}}{8\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}k_{0}^{2}\rho_{11\kappa}^{i,j}}\label{CRB1Mzt}
\end{equation}
\begin{equation}
\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t}\right)\approx\frac{{\rm{SNR}}^{-1}}{8\sum_{j=1}^{\frac{N}{2}}\sum_{i=1}^{\frac{N}{2}}k_{0}^{2}\rho_{21\kappa}^{i,j}}\label{CRB2Mzt}.
\end{equation}
\end{coro}
\begin{IEEEproof}
We can show that: 1) $\rho_{11\kappa}^{i,j}>\rho_{12\kappa}^{i,j}$ or $\rho_{11\kappa}^{i,j}$ has the same order of magnitude as $\rho_{12\kappa}^{i,j}$; 2) both of them are positive. Then, $k_{0}^{2}\rho_{11\kappa}^{i,j}\gg z_{t}^{2}\rho_{12\kappa}^{i,j}$ for $z_{t}\gg \lambda$. Similarly, we can show that $k_{0}^{2}\rho_{21\kappa}^{i,j}\gg z_{t}^{2}\rho_{22\kappa}^{i,j}$ for $z_{t}\gg \lambda$. Thus, expression \eqref{eq:CRB_SIMO_1} and \eqref{eq:CRB_SIMO_2} can be simplified to \eqref{CRB1Mzt} and \eqref{CRB2Mzt}.
\end{IEEEproof}
Note that $\mathrm{CRB}_{1}^{\rm{M}}\left(\kappa_{t} \right)<\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t} \right)<\mathrm{CRB}_{3}^{\rm{M}}\left(\kappa_{t} \right)$ can be derived based on Corollary \ref{prop:CRB_CPL_SIMO_zt}, which is similar to inequality \eqref{eq:CRB_vec_sca_overall}. It clearly indicates that using multiple
distributed receiving antennas does not affect the order of estimation accuracy of using different electric field observations.
\begin{coro}[SIMO, $\tau \to \infty$ ]\label{prop:CRB_CPL_SIMO_tau}
{If $z_{t}\gg \lambda$ and $\tau \to \infty$, the CRBs of the SIMO positioning system can be given by}
\begin{align}
&\lim_{\tau\to \infty}\mathrm{CRB}_{1}^{\rm{M}}\left(\kappa_{t}\right)=\lim_{\tau\to \infty}{\mathrm{CRB}_{1}^{C}\left(\kappa_{t}\right)}/{N^{2}}\\
&\lim_{\tau\to \infty}\mathrm{CRB}_{2}^{\rm{M}}\left(\kappa_{t}\right)=\lim_{\tau\to \infty}{\mathrm{CRB}_{2}^{C}\left(\kappa_{t}\right)}/{N^{2}}.
\end{align}
\end{coro}
\begin{IEEEproof}
The results can be derived based on Corollary \ref{prop:CRB_CPL_tau} and Corollary \ref{prop:CRB_CPL_SIMO_zt}. Particularly, we have that $\lim\rho_{11\kappa}^{i,j}=\lim\rho_{11\kappa}$ and $\lim\rho_{21\kappa}^{i,j}=\lim\rho_{21\kappa}$, where we use $\lim$ to represent $\lim_{\tau\to \infty}$. Thus, Corollary \ref{prop:CRB_CPL_SIMO_tau} holds.
\end{IEEEproof}
It can be seen from Corollary \ref{prop:CRB_CPL_SIMO_tau} that the CRBs of the SIMO positioning system will be one-$N^{2}$th of the SISO system as $\tau$ increases unboundedly. On the large surface with fixed size, different small receiving antennas will be stacked on top of each other with $\tau$ increasing, resulting in multiplexing benefits and lower CRBs. Besides, the total area of the small receiving antennas will be larger than $\mathcal{R}_{s}$ as $\tau\to \infty$, which ignores the space constraints. In fact, the more practical and meaningful case is $\tau \leq R/z_{t}$, which will be analyzed in Sec. \ref{subsection:CRB_simo_numer}.
\section{Numerical Results and Discussions}\label{sectionV}
In this section, we will provide numerical results to illustrate the propositions and corollaries that we have derived in previous sections.
We set the signal-to-noise ratio as
$\mathrm{SNR}={|E_{0}|^{2}}/{\sigma^{2}} = -10 \mathrm{dB}$ and the wavelength as $\lambda= 0.01m$ (corresponding to $f_{c} = 30 \mathrm{GHz}$) unless otherwise specified.
\subsection{CRB Evaluation for CPL Case}\label{subsection:CRB_CPL_numer}
We first show the CRBs for a terminal on the CPL computed in Sec. \ref{sectionCPL}. In order to illustrate the influence of the system carrier frequency on the CRBs, we consider two different values of the wavelength, i.e., $\lambda=0.01m$ and $\lambda=0.001m$ (corresponding to $f_{c} = 300 \mathrm{GHz}$). Fig. \ref{fig:cpl111} and Fig. \ref{fig:cpl222} demonstrate the CRBs, measured in square meters [$m^{2}$], versus the surface diagonal length $D$ or the distance
from the terminal to the receiving antenna (terminal-surface distance) $d=z_{t}$ when $z_{t}=6m$ or $D=9m$, respectively.
It can be seen from Fig. \ref{fig:cpl111} that all the CRBs decrease dramatically with the surface diagonal length in the range $1m\leq D \leq 10m$, which contains the values of $D$ commonly used in the practical system. In addition, the CRBs for $z_{t}$ are much lower than those for $x_{t}$ and $y_{t}$ in the above range. More interestingly, the CRBs using \textit{SEF} are greater than CRBs using \textit{VEF} for all values of $D$, which agrees with Remark \ref{rem:CRB_vec_sca_overall}. The difference between $\mathrm{CRB}_{1}\left(\kappa_{t}\right)$ and $\mathrm{CRB}_{2}\left(\kappa_{t}\right)$ is negligible when $D\leq10m$, but it will increase gradually with the increase of $D$. As for the CRBs in the asymptotic regime, we find that: 1) $\mathrm{CRB}_{1}\left(x_{t}\right)$
and $\mathrm{CRB}_{1}\left(y_{t}\right)$ decrease unboundedly with the trend of the $\ln^{-1}$ function in \eqref{eq:CRB1xttau} and \eqref{eq:CRB1yttau}; 2) $\mathrm{CRB}_{1}\left(z_{t}\right)$ and $\mathrm{CRB}_{2}\left(z_{t}\right)$ approach the asymptotic limit in \eqref{eq:CRB1zttau} and \eqref{eq:CRB2zttau} from $D\approx20m$; 3) $\mathrm{CRB}_{2}\left(x_{t}\right)$ and $\mathrm{CRB}_{2}\left(y_{t}\right)$ converge to the asymptotic limit in \eqref{eq:CRB2xttau} and \eqref{eq:CRB2yttau} when $D>10^{3}m$. These phenomena are consistent with Corollary \ref{prop:CRB_CPL_tau}.
Fig. \ref{fig:cpl222} shows that the CRBs for all the
dimensions increase very slowly with the terminal-surface distance in the range $0.1m\leq z_{t}\leq 1m$, but they increase considerably (among them, $\mathrm{CRB}_{1}^{C}\left(z_{t}\right)$ and $\mathrm{CRB}^{C}_{2}\left(z_{t}\right)$ are much lower than the CRBs for $x_{t}$ and $y_{t}$) when $z_{t}> 1m$. It is worth noting that all the CRBs depend linearly on $\lambda^{2}$ regardless of \textit{VEF} or \textit{SEF}, as in Corollary \ref{coro:CRB_CPL_zt}.
\begin{figure}
\centering
\includegraphics[scale=0.45]{cplD.pdf}
\caption{CRBs versus surface diagonal length $D$, with $\lambda= 0.01m$ or $0.001m$, $z_{t}=6m$, when $\mathbf{p}_{t}$ is on the CPL and using $\textit{VEF}$ or $\textit{SEF}$.}
\label{fig:cpl111}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.45]{cplz.pdf}
\caption{CRBs versus terminal-surface distance $z_{t}$, with $D=9m$, $\lambda= 0.01m$ or $0.001m$, when $\mathbf{p}_{t}$ is on the CPL and using $\textit{VEF}$ or $\textit{SEF}$.}
\label{fig:cpl222}
\end{figure}
Table. \ref{tab:performance} provides the square root of the CRBs (RCRB, denoted as $\mathrm{R}\left(\kappa_{t}\right)$), measured
in centimeters [$\mathrm{c}m$], for the three components $x_{t}$, $y_{t}$, and $z_{t}$, for terminals located on the CPL. $D_{1}$, $D_{2}$, $D_{3}$, and $D_{4}$ represent that the surface diagonal length is $0.5m$, $1m$, $2m$, and $3m$ when $z_{t}=6m$, respectively. To evaluate the average positioning performance, we use the receiving antenna with $D=3m$ to compute the average RCRB of $1000$ terminals with coordinates of $z_{t}$ dimension uniformly distributed in $[1m, 20m]$, which is denoted as $Ave$. It can be seen that using \textit{VEF} or \textit{SEF} can guarantee
a centimeter-level accuracy (within a few centimeters) for estimating all three dimensions in the mmWave or sub-THz bands. This is in contrast with the results in \cite{de2021cramer}. Unfortunately, we find that, even though an accuracy on the order of tens of centimeters in $Z$-dimension can be achieved by using \textit{OSEF}, we are unable to estimate $x_{t}$ and $y_{t}$ with an acceptable accuracy. This reveals that a single conventional surface antenna possesses only the near-field ranging function, which can be considered a one-dimensional special case of near-field positioning.
\begin{table}[!t]
\center
\caption{Comparison of estimation accuracy between using \textit{VEF}, \textit{SEF}, and \textit{OSEF} (- means the value is too large).}
\label{tab:performance}
\begin{tabular}{c|c|c|c|c|c||c}
\toprule
\multicolumn{2}{c|}{} & \multicolumn{5}{c}{RCRB [$\mathrm{c}m$]} \\
\cline{3-7}
\multicolumn{2}{c|}{} & ${D}_{1}$& $D_{2}$& $D_{3}$ & $D_{4}$ & $Ave$ \\
\midrule
\multirow{3}{*}{\textit{VEF}}& $\mathrm{R}\left(x_{t}\right)$&35.5 &8.91 &2.25 & 1.02& \textbf{3.88} \\
&$\mathrm{R}\left(y_{t}\right)$& 35.5& 8.91& 2.26&1.02 & \textbf{3.88}\\
&$\mathrm{R}\left(z_{t}\right)$& 0.604& 0.303& 0.153&0.103 & \textbf{0.179}\\
\midrule
\multirow{3}{*}{\textit{SEF}}& $\mathrm{R}\left(x_{t}\right)$&35.5 &8.92 &2.26 &1.03 & \textbf{3.89} \\
&$\mathrm{R}\left(y_{t}\right)$& 35.6& 8.92& 2.26&1.03 & \textbf{3.89}\\
&$\mathrm{R}\left(z_{t}\right)$& 0.605& 0.303& 0.153&0.104 & \textbf{0.179}\\
\midrule
\multirow{3}{*}{\textit{OSEF}}& $\mathrm{R}\left(x_{t}\right)$&- &- &- &- & \textbf{-} \\
&$\mathrm{R}\left(y_{t}\right)$& -& -& -&- & \textbf{-}\\
&$\mathrm{R}\left(z_{t}\right)$& 11.8& 21.1& 20.4&23.7 & \textbf{18.0}\\
\bottomrule
\end{tabular}
\end{table}
\subsection{CRB Evaluation for the General Scenario}\label{subsection:CRB_general_numer}
Next, we will evaluate the CRBs for a terminal not on the CPL as discussed in Proposition \ref{prop:CRB_vec_specific}, Corollary \ref{coro:CRB_sca_specific} and \ref{coro:CRB_overall_specific}. Fig. \ref{fig:6} illustrates the CRBs as a function of the distance $d=\sqrt{x_{t}^{2}+y_{t}^{2}+z_{t}^{2}}$ for a ternimal at $(2,3,z_{t})$ when $D=9m$. It can be found that the estimation accuracy reduces as the terminal-surface distance increases, which is consistent with our intuition. Particularly, the CRBs for estimating $x_{t}$ and $y_{t}$ increase faster than $z_{t}$ regardless of \textit{VEF} or \textit{SEF}. Furthermore, all the CRBs increase rapidly when the the terminal is close to the receiving antenna ($0<z_{t}\leq \sqrt{3}m$). This occurs because the estimation for all dimensions is nearly perfect (CRB is approaching $0$) when the terminal approaches the receiving antenna ($z_{t}\to0$, $|x_{t}|$ and $|y_{t}|$ are less than $\frac{D}{2\sqrt{2}}$), and as $z_{t}$ increases from $0$, CRBs will rapidly increase to greater orders of magnitude. Besides, we find that $\mathrm{CRB}_{2}\left(\kappa_{t}\right)$ is greater than $\mathrm{CRB}_{1}\left(\kappa_{t}\right)$ when the terminal-surface distance is less than $10m$, otherwise they are equal. This indicates that for a receiving antenna with fixed size, there is a considerable performance gap between \textit{VEF} and \textit{SEF}, only when the terminal is close to the receiving antenna.
\begin{figure}
\centering
\includegraphics[scale=0.45]{3.pdf}
\caption{CRBs as a function of the terminal-surface distance for a terminal at $\left(2,3,z_{t}\right)$ when using \textit{VEF} or \textit{SEF}, and $D=9m$.}
\label{fig:6}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.45]{multixtyt.pdf}
\caption{CRBs as a function of the terminal-surface distance for terminals with different $x_{t}$ and $y_{t}$ when using \textit{SEF}, $D=9m$ and $\lambda=0.001m$. Since using \textit{VEF} or \textit{SEF} has the same rules, we take the use of \textit{SEF} as an example.}
\label{fig:7}
\end{figure}
Fig. \ref{fig:7} illustrates the CRBs for terminals with different $x_{t}$ and $y_{t}$ versus the terminal-surface distance when using \textit{SEF}, $D=9m$ and $\lambda=0.001m$. It shows that the CRBs have different trends and the curve shapes vary from each other for different $x_{t}$ and $y_{t}$ when the terminal is close to the receiving antenna. For instance, if the terminal is on the CPL ($x_{t}=y_{t}=0$, $d=z_{t}$), the CRBs for all dimensions are almost unchanged in the range $0.1m<d<1m$. However, if $x_{t}$ or $y_{t}$ are greater than $\frac{D}{2\sqrt{2}}$ and $z_{t}$ is small, which means the vertical projection of the terminal along the $Z$-dimension is not on the receiving antenna surface and the distance from the terminal to the CPL is much larger than $z_{t}$, the CRBs sharply decrease from infinity. We refer this phenomenon as the near-field positioning \textit{blocking zone} effect, which always exists for a fixed-size receiving surface antenna. Notably, extensive numerical simulations in Fig. \ref{fig:cpl111} for terminals not on the CPL demonstrate that results obtained in the analysis of the CPL case in Sec. \ref{sectionCPL} are also applicable to the generic near-field positioning system proposed in Sec. \ref{SEC:2}, which provides support for the generalizability of our insights and results.
Fig. \ref{fig:simu1} demonstrates the normalized CRBs for the terminal not on the CPL, versus $x_{t}$ and $y_{t}$ when $z_{t}=6m$, $D = 3m$, and using \textit{VEF} or \textit{SEF}. These normalized CRBs, measured in [$\mathrm{dB}$] and denoted as $\mathrm{CRB}_{1}^{N}\left(\kappa_{t}\right)$ and $\mathrm{CRB}_{2}^{N}\left(\kappa_{t}\right)$, are defined as the values of CRBs normalized by their minimum, which can be achieved when the terminal is on the CPL $(x_{t} = y_{t} = 0)$. In order to clearly illustrate the different behaviours of the CRBs when the terminal moves away from the CPL, the color of the point $(x_{t}, y_{t})$ is used to measure the normalized CRB values corresponding to that point. In particular, the normalized CRB values are mapped to the color gamut, where warmer colors represent higher values and lower values are associated to cooler colors. It can be seen that the CRB for estimating $z_{t}$ increases faster than those for $x_{t}$ and $y_{t}$ regardless of using \textit{VEF} or \textit{SEF}. In addition, the maximum normalized values of $\mathrm{CRB}_{1}\left(\kappa_{t}\right)$ (as shown in Fig.\ref{fig:CRBxa}, \ref{fig:CRBya} and \ref{fig:CRBza}) and $\mathrm{CRB}_{2}\left(\kappa_{t}\right)$ (as shown in Fig. \ref{fig:CRBxb}, \ref{fig:CRByb} and \ref{fig:CRBzb}) are $18.40\mathrm{dB}$,
$18.41\mathrm{dB}$, $45.68\mathrm{dB}$, $22.41\mathrm{dB}$, $22.42\mathrm{dB}$, and $49.69\mathrm{dB}$, respectively. This result indicates that the CRBs using \textit{SEF} have a more significant increase than those using \textit{VEF}, and the difference is about $4\mathrm{dB}$ for all dimensions.
\begin{figure}
\centering
\vspace{-1em}
\subfloat[$\mathrm{CRB}_{1}^{N}\left(x_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_x1_1.pdf}
\label{fig:CRBxa}
}
\subfloat[$\mathrm{CRB}_{2}^{N}\left(x_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_x2_1.pdf}
\label{fig:CRBxb}
}
\vfill
\subfloat[$\mathrm{CRB}_{1}^{N}\left(y_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_y1_1.pdf}
\label{fig:CRBya}
}
\subfloat[$\mathrm{CRB}_{2}^{N}\left(y_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_y2_1.pdf}
\label{fig:CRByb}
}
\vfill
\subfloat[$\mathrm{CRB}_{1}^{N}\left(z_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_z1_1.pdf}
\label{fig:CRBza}
}
\subfloat[$\mathrm{CRB}_{2}^{N}\left(z_{t} \right)$ in $\mathrm{dB}$.]{
\includegraphics[scale=0.243]{simu1_z2.pdf}
\label{fig:CRBzb}
}
\caption{Normalized CRBs, measured in [$\mathrm{dB}$], as a function of $x_{t}$ and $y_{t}$ for the terminal not on the CPL when using $\textit{VEF}$/$\textit{SEF}$, $z_{t} = 6 m$ and $D=3m$. We have that $\mathrm{CRB}_{1}^{N}\left(\kappa_{t}\right)=10\log_{10}[\mathrm{CRB}_{1}\left(\kappa_{t}\right)/\mathrm{CRB}_{1}^{C}\left(\kappa_{t}\right)]$ and $\mathrm{CRB}_{2}^{N}\left(\kappa_{t}\right)=10\log_{10}[\mathrm{CRB}_{2}\left(\kappa_{t}\right)/\mathrm{CRB}_{2}^{C}\left(\kappa_{t}\right)]$.}
\label{fig:simu1}
\end{figure}
\subsection{CRB Evaluation for the SIMO Positioning System}\label{subsection:CRB_simo_numer}
Finally, we will evaluate the CRBs for the SIMO positioning system as
discussed in Sec. \ref{sec:simo}. We set $R=30m$, $z_{t}=6m$ and $\lambda=0.001m$. Based on Proposition \ref{prop:CRB_CPL_SIMO}, we compare the CRBs for a terminal on the CPL with different number of small receiving antennas, i.e., $N^{2}=1,4,16,64,256$.
\begin{figure}[!t]
\centering
\subfloat[$\mathrm{CRB}_{1}^{\mathrm{M}}\left(x_{t} \right)$ and $\mathrm{CRB}_{1}^{C}\left(x_{t} \right)$ versus $D$.]{
\includegraphics[scale=0.45]{simox.pdf}
\label{fig:simo1}
}
\vfill
\subfloat[$\mathrm{CRB}_{1}^{\mathrm{M}}\left(y_{t} \right)$ and $\mathrm{CRB}_{1}^{C}\left(y_{t} \right)$ versus $D$.]{
\includegraphics[scale=0.45]{simoy.pdf}
\label{fig:simo2}
}
\vfill
\subfloat[$\mathrm{CRB}_{1}^{\mathrm{M}}\left(z_{t} \right)$ and $\mathrm{CRB}_{1}^{C}\left(z_{t} \right)$ versus $D$.]{
\includegraphics[scale=0.45]{simoz.pdf}
\label{fig:simo3}
}
\caption{CRBs with different number of small receiving antennas ($N^{2}=1, 4, 16, 64, 256$, equivalently expressed as $\mathrm{CRB}_{1}^{C}\left(\kappa_{t}\right)$, $1\times4, 1\times16,1\times64,1\times256$) with $R=30m$, $z_{t}=6m$, and $\lambda=0.001m$ when using \textit{VEF}.}
\label{fig:simo}
\end{figure}
As shown in Fig. \ref{fig:simo}, when $D>R$, the SIMO positioning system renders lower CRBs than the SISO positioning system for all three dimensions. More precisely, $\mathrm{CRB}_{1}^{\mathrm{M}}\left(\kappa_{t}\right)$ will be one-$N^{2}$th of $\mathrm{CRB}_{1}^{C}\left(\kappa_{t}\right)$ as $D$ increases infinitely, as in Corollary \ref{prop:CRB_CPL_SIMO_tau}. Due to the space constraints, we are more interested in the range $D\leq R$, where the surface area covered by the small receiving antennas will be smaller than the large rectangular surface $\mathcal{R}_{s}$. It can be seen that the CRBs for estimating $x_{t}$ and $y_{t}$ are significantly improved when using the SIMO system in the above range of practical interest, although the CRBs for $z_{t}$ become worse. For instance, the CRBs for $x_{t}$ and $y_{t}$ with $4$ small receiving antennas, each antenna has a surface diagonal length $0.025m$, can achieve the same CRBs for a single receiving antenna with $D \approx 0.9m$, that is, the antenna surface area needed for estimating $X$- and $Y$-dimension by the SIMO positioning system is only $1.23\%$ of that by the SISO system when $D$ is small. The CRB for estimating $z_{t}$ with $4$ small receiving antennas is around $10\mathrm{dB}$ larger than $\mathrm{CRB}^{C}_{1}\left(z_{t}\right)$ when $D$ is the same and less than $10m$. Besides, we find that $\mathrm{CRB}_{1}^{\mathrm{M}}\left(x_{t}\right)$ remains the same when the number of small receiving antennas changes, whereas $\mathrm{CRB}_{1}^{\mathrm{M}}\left(y_{t}\right)$ is slightly lower when $N^{2}=4$ compared to $N^{2}=16,64,256$, and $\mathrm{CRB}_{1}^{\mathrm{M}}\left(z_{t}\right)$ is slightly larger when $N^{2}=4$. In fact, to achieve cooperation and coupling calibration among the small receiving antennas, more stringent hardware is required as the number of small antennas rises. Therefore, in light of the performance of the positioning system and the cost of hardware, the SIMO positioning system with $4$ small receiving antennas is the superior option for estimating $x_{t}$ and $y_{t}$, whereas the SISO system is the better choice for estimating $z_{t}$, i.e., ranging. It is worth noting that using \textit{SEF} in the SIMO system has the same rules as using \textit{VEF}. Using \textit{OSEF} in the SIMO system with $4$ small antennas still fails to estimate the three coordinates of the terminal, but when the number of small receiving antennas is large enough, using \textit{OSEF} can be approximated as using \textit{SEF}.
\section{Conclusions}\label{sec:con}
In this paper, a generic near-field positioning model considering different electric field observations and the universality of the terminal position has been proposed. With the purpose of evaluating
the estimation accuracy of this system, we have combined electromagnetic propagation theory with estimation theory to develop generic CRB expressions for three Cartesian coordinates of the terminal. Three electric fields (\textit{vector}, \textit{scalar} and \textit{overall scalar electric field}) have been studied for various antenna paradigms with varying observation capabilities. The derived CRB expressions generalize the existing results in \cite{de2021cramer}, where the terminal is located on the CPL of the receiving antenna surface and the \textit{vector electric
field} is utilized. As a result of the CPL assumption, simplifications and insights have been obtained, as well as closed-form CRB expressions. The correlation between estimation precision and observation capability has been discovered. Additionally, the generic CPL model has been expanded to account for systems with multiple distributed receiving antennas, and its optimal estimation precision has been thoroughly discussed. Asymptotic
expressions of the CRBs have been provided to illustrate their scaling behaviors in relation to carrier frequency and surface diagonal length. Numerical results have shown that centimeter-level accuracy can be achieved in the near-field of a receiving antenna of a practical size in the mmWave or sub-THz bands by using the \textit{vector} or \textit{scalar electric field}. The \textit{overall scalar electric field} observed by a conventional surface antenna can only be used for ranging. Furthermore, the multiple receiving antennas enhance the estimation accuracy of dimensions parallel to the receiving antenna surface.
\begin{appendices}
\section{Proof of Proposition \ref{prop:electric_vec}}\label{proof:VEF}
The transformation relationship between the basis vectors $\hat{\mathbf{r}}$, $\hat{\bm{\theta}}$, $\hat{\bm{\phi}}$ of the spherical coordinate system $\left(r,\theta,\phi \right)$ and the basis vectors $\hat{\mathbf{x}}$, $\hat{\mathbf{y}}$, $\hat{\mathbf{z}}$ of the Cartesian coordinate system $OXYZ$ is
\begin{align}
&\hat{\mathbf{r}} =\sin{\theta}\cos{\phi}\hat{\mathbf{x}}+\cos{\theta}\hat{\mathbf{y}}-\sin{\theta}\sin{\phi}\hat{\mathbf{z}}
\label{eq:er}\\
&\hat{\bm{\theta}} =\cos{\theta}\cos{\phi}\hat{\mathbf{x}}-\sin{\theta}\hat{\mathbf{y}}-\cos{\theta}\sin{\phi}\hat{\mathbf{z}}\\
&\hat{\bm{\phi}} =-\sin{\phi}\hat{\mathbf{x}}-\cos{\phi}\hat{\mathbf{z}}.
\end{align}
Plugging \eqref{eq:er} into \eqref{eq:E} and \eqref{eq:Gr} yields
\begin{align}
&\notag e_{x}(\mathbf{p}_{r})=G(r)\left[\left(1-\sin^{2}{\theta}\cos^{2}{\phi}\right)J_{x}\left(\mathbf{p}_{t} \right)-\left(\sin{\theta}\right.\right.\\
&\left.\left.\quad\cos{\phi}\cos{\theta}\right)J_{y}\left(\mathbf{p}_{t} \right)+\left(\sin^{2}{\theta}\cos{\phi}\sin{\phi}\right)J_{z}\left(\mathbf{p}_{t} \right)\right]\label{eq:Ex}\\
&\notag e_{y}(\mathbf{p}_{r})=G(r)\left[\left(-\sin{\theta}\cos{\theta}\cos{\phi}\right)J_{x}\left(\mathbf{p}_{t}\right)+\sin^{2}{\theta}\right.\\
&\left.\quad J_{y}\left(\mathbf{p}_{t} \right)+\left(\sin{\theta}\cos{\theta}\sin{\phi}\right)J_{z}\left(\mathbf{p}_{t} \right)\right]\label{eq:Ey}\\
&\notag e_{z}(\mathbf{p}_{r})=G(r)\left[\left(\sin^{2}{\theta}\sin{\phi}\cos{\phi}\right)J_{x}\left(\mathbf{p}_{t}\right)+\left(\sin{\theta}\right.\right.\\
&\left.\left.\quad\cos{\theta}\sin{\phi}\right)J_{y}\left(\mathbf{p}_{t} \right)+\left(1-\sin^{2}{\theta}\sin^{2}{\phi}\right)J_{z}\left(\mathbf{p}_{t} \right)\right]\label{eq:Ez},
\end{align}
where $\mathbf{J}\left(\mathbf{p}_{t}\right)=J_{x}\left(\mathbf{p}_{t}\right)\hat{\mathbf{x}}+J_{y}\left(\mathbf{p}_{t}\right)\hat{\mathbf{y}}+J_{z}\left(\mathbf{p}_{t}\right)\hat{\mathbf{z}}$, and $G(r)$ is the scalar Green function
\begin{equation}
G(r)=-\frac{\jmath \eta \mathrm{e}^{-\jmath k_{0}r}}{2 \lambda r}.
\label{eq:Gr_sca}
\end{equation}
Since we assume that the electromagnetic wave is polarized in the $Y$-dimension, \eqref{eq:Ex} -- \eqref{eq:Ez} can be simplified as
\begin{align}
&e_{x}(\mathbf{p}_{r})=G(r)\left(-\sin{\theta}\cos{\phi}\cos{\theta}\right)J_{y}\left(\mathbf{p}_{t} \right)\label{eq:Ex_simply}\\
&e_{y}(\mathbf{p}_{r})=G(r)\sin^{2}{\theta}
J_{y}\left(\mathbf{p}_{t} \right)\label{eq:Ey_simply}\\
&e_{z}(\mathbf{p}_{r})=G(r)\left(\sin{\theta}
\cos{\theta}\sin{\phi}\right)J_{y}\left(\mathbf{p}_{t} \right)\label{eq:Ez_simply}.
\end{align}
The dependence of $e_{x}(\mathbf{p}_{r})$, $e_{y}(\mathbf{p}_{r})$ and $e_{z}(\mathbf{p}_{r})$ on the position $\left(x_{t},y_{t},z_{t}\right)$ is hidden in $\left(r,\theta,\phi \right)$. We have
\begin{align}
&r=\sqrt{\left(x_{r}-x_{t}\right)^{2}+\left(y_{r}-y_{t}\right)^{2}+z_{t}^{2}}, \label{eq:r}\\
&\cos \theta=\frac{y_{r}-y_{t}}{r},\label{eq:cos}\\
&\tan \phi=\frac{z_{t}}{x_{r}-x_{t}}\label{eq:tan}.
\end{align}
from which it follows that
\begin{align}
&\sin \theta \cos \theta \cos \phi=\frac{\left(x_{r}-x_{t}\right)\left(y_{r}-y_{t}\right)}{r^{2}} ,\label{eq:sin1}\\
&\sin ^{2} \theta=1-\frac{\left(y_{r}-y_{t}\right)^{2}}{r^{2}},\label{eq:sin2} \\
&\sin \theta \cos \theta \sin \phi=\frac{z _{t}\left(y_{r}-y_{t}\right)}{r^{2}}\label{eq:sin3}.
\end{align}
By substituting \eqref{eq:Gr_sca} and \eqref{eq:sin1} -- \eqref{eq:sin3} into \eqref{eq:Ex_simply} -- \eqref{eq:Ez_simply} yields Proposition \ref{prop:electric_vec}.
\section{Some Complex Expressions}\label{experssion}
In the proof of Proposition \ref{prop:CRB_vec_specific}, we should compute some first-order derivatives to derive the elements of FIM $\mathbf{I}(\bm{\xi})$, and the specific expressions are as follows.
\begin{subequations}
\begin{align}
&\frac{\partial h_{x}\left(\mathbf{p}_{r}\right)}{\partial x_{t}}=x_{r,t}^{2}y_{r,t}\Big(\frac{3\jmath}{r^{5}}-\frac{{\jmath}}{{x_{r,t}^{2}r^{3}}}-\frac{k_{0}} {r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r}\label{eq:hx_xt} \\
&\frac{\partial h_{x}\left(\mathbf{p}_{r}\right)}{\partial y_{t}}=x_{r,t}y_{r,t}^{2}\Big(\frac{3\jmath }{r^{5}}-\frac{{\jmath}}{{y_{r,t}^{2}r^{3}}}-\frac{k_{0}} {r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r} \\
&\frac{\partial h_{x}\left(\mathbf{p}_{r}\right)}{\partial z_{t}}=x_{r,t}y_{r,t}z_{t}\Big(-\frac{3\jmath}{r^{5}}+\frac{k_{0}}{r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r}\\
&\frac{\partial h_{y}\left(\mathbf{p}_{r}\right)}{\partial x_{t}}=x_{r,t}\Big(\jmath\frac{3y_{r,t}^{2}-r^{2}} {r^{5}}-k_{0}\frac{y_{r,t}^{2}-r^{2}} {r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r} \\
&\frac{\partial h_{y}\left(\mathbf{p}_{r}\right)}{\partial y_{t}}=y_{r,t}\Big(\jmath\frac{3y_{r,t}^{2}-3r^{2}} {r^{5}}-k_{0}\frac{y_{r,t}^{2}-r^{2}} {r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r} \\
&\frac{\partial h_{y}\left(\mathbf{p}_{r}\right)}{\partial z_{t}}=z_{t}\Big(\jmath\frac{-3 y_{r,t}^{2}+r^{2}}{r^{5}}+k_{0}\frac{y_{r,t}^{2}-r^{2}}{r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r} \\
&\frac{\partial h_{z}\left(\mathbf{p}_{r}\right)}{\partial x_{t}}=x_{r,t}y_{r,t}z_{t}\Big(-\frac{3\jmath}{r^{5}}+\frac{k_{0}}{r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r} \\
&\frac{\partial h_{z}\left(\mathbf{p}_{r}\right)}{\partial y_{t}}=y_{r,t}^{2}z_{t}\Big(-\frac{3\jmath}{r^{5}}+\frac{\jmath}{y_{r,t}^{2}r^{3}}+\frac{k_{0}}{r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r}\\
&\frac{\partial h_{z}\left(\mathbf{p}_{r}\right)}{\partial z_{t}}=y_{r,t}z_{t}^{2}\Big(\frac{3\jmath}{r^{5}}-\frac{\jmath}{z_{t}^{2}r^{3}}-\frac{k_{0}}{r^{4}}\Big) \mathrm{e}^{-\jmath k_{0} r}, \label{eq:hz_zt}
\end{align}
\end{subequations}
where we have set $x_{r,t}=x_{r}-x_{t}$ and $y_{r,t}=y_{r}-y_{t}$.
The specific expressions of $\rho^{mn}_{11}$ and $\rho^{mn}_{12}$ are as follows.
\begin{align}
&\rho_{11}^{11}=k_{0}^{2}\iint_{\mathcal{R}_{r}} \frac{x_{r,t}^{2}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{6}} d x_{r} d y_{r}\label{eq:rho_11^11}\\
&\rho_{12}^{11}=\iint_{\mathcal{R}_{r}} \frac{\left(x_{r,t}^{2}+y_{r,t}^{2}\right)r^{2}-3x_{r,t}^{2}y_{r,t}^{2}}{r^{8}} d x_{r} d y_{r} \\
&\rho_{11}^{22}=k_{0}^{2}\iint_{\mathcal{R}_{r}}\frac{y_{r,t}^{2}(x_{r,t}^{2}+z_{t}^{2})}{r^{6}} d x_{r} d y_{r}\\
&\rho_{12}^{22}=\iint_{\mathcal{R}_{r}}\frac{(x_{r,t}^{2}+z_{t}^{2})(x_{r,t}^{2}+z_{t}^{2}+4y_{r,t}^{2})}{r^{8}}d x_{r} d y_{r}\\
&\rho_{11}^{33}=k_{0}^{2}z_{t}^{2}\iint_{\mathcal{R}_{r}}\frac{x_{r,t}^{2}+z_{t}^{2}}{r^{6}} d x_{r} d y_{r}\\
&\rho_{12}^{33}=\iint_{\mathcal{R}_{r}}\frac{y_{r,t}^{2}\left(r^{2}-2z_{t}^{2}\right)+z_{t}^{2}\left(z_{t}^{2}+x_{r,t}^{2}\right)}{r^{8}}d x_{r} d y_{r}\label{eq:rho_12^33}\\
&\rho_{11}^{12}=k_{0}^{2}\iint_{\mathcal{R}_{r}} \frac{x_{r,t}y_{r,t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{6}} d x_{r} d y_{r}\label{eq:rho_11^12}\\
&\rho_{12}^{12}=\iint_{\mathcal{R}_{r}} \frac{x_{r,t}y_{r,t}\left(x_{r,t}^{2}-2y_{r,t}^{2}+z_{t}^{2}\right)}{r^{8}} d x_{r} d y_{r} \\
&\rho_{11}^{13}=k_{0}^{2}\iint_{\mathcal{R}_{r}} \frac{-x_{r,t}z_{t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{6}} d x_{r} d y_{r}\\
&\rho_{12}^{13}=\iint_{\mathcal{R}_{r}} \frac{x_{r,t}z_{t}\left(2y_{r,t}^{2}-x_{r,t}^{2}-z_{t}^{2}\right)}{r^{8}} d x_{r} d y_{r} \\
&\rho_{11}^{23}=k_{0}^{2}\iint_{\mathcal{R}_{r}} \frac{-y_{r,t}z_{t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{6}} d x_{r} d y_{r}\\
&\rho_{12}^{23}=\iint_{\mathcal{R}_{r}} \frac{y_{r,t}z_{t}\left(2y_{r,t}^{2}-x_{r,t}^{2}-z_{t}^{2}\right)}{r^{8}} d x_{r} d y_{r}.
\label{eq:rho_12^23}
\end{align}
In the proof of Corollary \ref{coro:CRB_sca_specific}, some first-order derivatives should be computed and the expressions are as follows.
\begin{subequations}
\begin{align}
&\frac{\partial h\left(\mathbf{p}_{r}\right)}{\partial x_{t}}=x_{r,t}\Big({\jmath k_{0}} r^{-\frac{7}{2}}+\frac{5}{2} r^{-\frac{9}{2}}-\frac{ r^{-\frac{5}{2}}}{f_{xz}}\Big) f_{sz}\label{eq:h_xt}\\
&\frac{\partial h\left(\mathbf{p}_{r}\right)}{\partial y_{t}}=y_{r,t}\Big(\frac{5}{2} r^{-\frac{9}{2}}+{\jmath k_{0}} r^{-\frac{7}{2}}\Big) f_{sz}\\
&\frac{\partial h\left(\mathbf{p}_{r}\right)}{\partial z_{t}}=\Big(\frac{3 z_{t}^{2}+x_{r,t}^{2}}{2 z_{t}f_{xz}} r^{-\frac{5}{2}}-\jmath k_{0} z_{t} r^{-\frac{7}{2}}-\frac{5}{2}z_{t} r^{-\frac{9}{2}}\Big) f_{sz}.\label{eq:h_zt}
\end{align}
\end{subequations}
where $f_{xz}=x_{r,t}^{2}+z_{t}^{2}$ and $f_{sz}=\sqrt{z_{t}f_{xz}}\mathrm{e}^{-\jmath k_{0} r}$.
The specific expressions of $\rho^{mn}_{21}$ and $\rho^{mn}_{22}$ are as follows.
\begin{align}
&\rho_{21}^{11}=k_{0}^{2}z_{t}\iint_{\mathcal{R}_{r}} \frac{x_{r,t}^{2}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{7}} d x_{r} d y_{r}\label{eq:rho_21^11}\\
&\rho_{22}^{11}=z_{t}\iint_{\mathcal{R}_{r}}\frac{x_{r,t}^{2}\left({25}f_{xz}/4-5r^{2}+f_{xz}^{-1}r^{4}\right)}{r^{9}}d x_{r} d y_{r}\\
&\rho_{21}^{22}=k_{0}^{2}z_{t}\iint_{\mathcal{R}_{r}} \frac{y_{r,t}^{2}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{7}} d x_{r} d y_{r}\\
&\rho_{22}^{22}=z_{t}\iint_{\mathcal{R}_{r}} \frac{{25}y_{r,t}^{2}\left(x_{r,t}^{2}+z_{t}^{2}\right)/4}{r^{9}} d x_{r} d y_{r}\\
&\rho_{21}^{33}=k_{0}^{2}z_{t}^{3}\iint_{\mathcal{R}_{r}} \frac{x_{r,t}^{2}+z_{t}^{2}}{r^{7}} d x_{r} d y_{r}\\
&\rho_{22}^{33}=\iint_{\mathcal{R}_{r}}
\frac{\big[x_{r,t}^{2}\left(r^{2}-2z_{t}^{2}\right)+z_{t}^{2}f_{yz}\big]^{2}}{4z_{t}\left(x_{r,t}^{2}+z_{t}^{2}\right)r^{9}}d x_{r} d y_{r}\label{eq:rho_22^33}\\
&\rho_{21}^{12}=k_{0}^{2}z_{t}\iint_{\mathcal{R}_{r}}\frac{x_{r,t}y_{r,t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{7}} d x_{r} d y_{r}\label{eq:rho_21^12}\\
&\rho_{22}^{12}=z_{t}\iint_{\mathcal{R}_{r}}\frac{x_{r,t}y_{r,t}\left(25f_{xz}/4-5r^{2}/2\right)}{r^{9}} d x_{r} d y_{r}\\
&\rho_{21}^{13}=k_{0}^{2}z_{t}^{2}\iint_{\mathcal{R}_{r}} \frac{-x_{r,t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{7}} d x_{r} d y_{r}\\
&\rho_{22}^{13}=\iint_{\mathcal{R}_{r}} \frac{x_{r,t}\big(f_{5z}f_{xz}-f_{3z}-25z_{t}^{2}f_{xz}^{2}/2\big)}{2\left(x_{r,t}^{2}+z_{t}^{2}\right)r^{9}} d x_{r} d y_{r}\\
&\rho_{21}^{23}=k_{0}^{2}z_{t}^{2}\iint_{\mathcal{R}_{r}}\frac{-y_{r,t}\left(x_{r,t}^{2}+z_{t}^{2}\right)}{r^{7}} d x_{r} d y_{r}\\
&\rho_{22}^{23}=\iint_{\mathcal{R}_{r}} \frac{5y_{r,t}\left(f_{3z}/r^{2}-5z_{t}^{2}f_{xz} \right)}{4r^{9}} d x_{r} d y_{r},\label{eq:rho_22^23}
\end{align}
where $f_{3z}=\left(x_{r,t}^{2}+3z_{t}^{2}\right)r^{4}$, $f_{5z}=5\left(x_{r,t}^{2}+5z_{t}^{2}\right)r^{2}/2$, and $f_{yz}=3y_{r,t}^{2}-2z_{t}^{2}$.
In Corollary \ref{coro:CRB_overall_specific}, $\mathrm{CRB}_{3}\left(\kappa_{t} \right)$ can be computed by \eqref{eq:CRB1x} -- \eqref{eq:CRB1z} and $I_{mn}=\rho_{3}^{mn}$. The expressions of $\rho^{mn}_{3}$ are as follows.
\begin{align}
&\rho_{3}^{11}=\frac{D^{2}}{2\alpha^{2}}\bigg|\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}x_{i,t} g_{zx}\Big(g_{r}-\frac{z_{t}}{|g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}\Big)\bigg|^{2}\label{eq:rho_3^11}\\
&\rho_{3}^{22}=\frac{D^{2}}{2\alpha^{2}}\bigg|\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}y_{j,t} g_{zx}g_{r}\bigg|^{2}\\
&\rho_{3}^{33}=\frac{D^{2}}{2\alpha^{2}}\bigg|\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}g_{zx}\Big(\frac{3 z_{t}^{2}+x_{i,t}^{2}}{2 |g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}-z_{t}g_{r}\Big)\bigg|^{2}\\
&\notag\rho_{3}^{12}=\frac{D^{2}}{2\alpha^{2}}\operatorname{Re}\Big\{\Big(\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}y_{j,t}g_{zx}g_{r}\Big)\\
&\quad\quad\cdot\Big[\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}x_{i,t} g_{zx}\Big(g_{r}-\frac{z_{t}}{|g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}\Big)\Big]^{*}\Big\}\\
&\notag\rho_{3}^{13}=\frac{D^{2}}{2\alpha^{2}}\operatorname{Re}\Big\{\Big[\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}g_{zx}\Big(\frac{3 z_{t}^{2}+x_{i,t}^{2}}{2 |g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}-z_{t}g_{r}\Big)\Big]\\
&\quad\quad\cdot\Big[\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}x_{i,t} g_{zx}\Big(g_{r}-\frac{z_{t}}{|g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}\Big)\Big]^{*}\Big\}\\
&\notag\rho_{3}^{23}=\frac{D^{2}}{2\alpha^{2}}\operatorname{Re}\Big\{\Big[\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}g_{zx}\Big(\frac{3 z_{t}^{2}+x_{i,t}^{2}}{2 |g_{zx}|^{2}} r_{i,j}^{-\frac{5}{2}}-z_{t}g_{r}\Big)\Big]\\
&\quad\quad\cdot\Big(\sum_{i=1}^{\sqrt{\alpha}} \sum_{j=1}^{\sqrt{\alpha}}y_{j,t} g_{zx}g_{r}\Big)^{*}\Big\},\label{eq:rho_3^23}
\end{align}
where $x_{i,t}=x_{i}-x_{t}$, $y_{j,t}=y_{j}-x_{t}$, $g_{r}=\frac{5}{2}r_{i,j}^{-\frac{9}{2}}+\jmath k_{0}r_{i,j}^{-\frac{7}{2}}$, and $g_{zx}=\sqrt{z_{t}\left(z_{t}^{2}+x_{i,t}^{2}\right)}\mathrm{e}^{-\jmath k_{0} r_{i,j}}$.
\section{The closed-form expressions}\label{ap:remark2}
The double integral formulas \eqref{eq:12x}, \eqref{eq:12y}, \eqref{eq:11z} and \eqref{eq:12z} can be computed in the following closed-form expressions.
\begin{align}
&\rho_{12x}=\frac{1}{\tau^{2}+8}\Big[\frac{f_{t}}{2\sqrt{\tau^{2}+8}}-\frac{\tau^{2}(3\tau^{2}+16)}{(\tau^{2}+4)^{2}}\Big]\label{eq:12x_close}\\
&\rho_{12y}=\frac{(9\tau^{4}+152\tau^{2}+544)}{2(\tau^{2}+8)^{5/2}\tau^{-1}f_{t}^{-1}}+\frac{\tau^{2}(3\tau^{4}+8\tau^{2}-32)}{(\tau^{2}+8)^{2}(\tau^{2}+4)^{2}}\label{eq:12y_close}\\
&\rho_{11z}=\frac{\tau}{\tau^{2}+8}\Big[\frac{(3\tau^{2}+28)}{\sqrt{\tau^{2}+8}}f_{t}+\frac{2\tau}{\tau^{2}+4}\Big]\label{eq:11z_close}\\
&\rho_{12z}=\frac{2\tau}{(\tau^{2}+8)^{2}}\Big[\frac{\tau^{4}+16\tau^{2}+88}{\sqrt{\tau^{2}+8}f_{t}^{-1}}+\frac{16\tau(\tau^{2}+5)}{(\tau^{2}+4)^{2}}\Big],\label{eq:12z_close}
\end{align}
where $f_{t}=\arctan{\frac{\tau}{\sqrt{\tau^{2}+8}}}$.
To provide the closed-form upper and lower bounds of $\rho_{11x}$ and $\rho_{11y}$, we denote two circular domains $\mathcal{C}^{-}=\big \{(u,v):u^{2}+v^{2}\leq (\frac{\tau}{\sqrt{8}})^{2}\big\}$, $\mathcal{C}^{+}=\big \{(u,v):u^{2}+v^{2}\leq (\frac{\tau}{2})^{2}\big\}$ and two non-negative function $g_{11x}={u^{2}(u^{2}+1)}/{(u^{2}+v^{2}+1)^{3}}$, $g_{11y}={v^{2}(u^{2}+1)}/{(u^{2}+v^{2}+1)^{3}}$, then we have
\begin{equation}
\iint_{\mathcal{C}^{-}}g_{11x}dudv<\rho_{11x}<\iint_{\mathcal{C}^{+}}g_{11x}dudv
\end{equation}
\begin{equation}
\iint_{\mathcal{C}^{-}}g_{11y}dudv<\rho_{11y}<\iint_{\mathcal{C}^{+}}g_{11y}dudv.
\end{equation}
Therefore, the closed-form upper and lower bounds of \eqref{eq:11x} and \eqref{eq:11y} can be derived as follows.
\begin{align}
&\iint_{\mathcal{C}^{+}}g_{11x}dudv=\frac{3\pi}{8}\ln{(1+\frac{\tau^{2}}{4})}-\frac{\pi\tau^{2}(5\tau^{2}+24)}{16(\tau^{2}+4)^{2}}\label{eq:11x+}\\
&\iint_{\mathcal{C}^{-}}g_{11x}dudv=\frac{3\pi}{8}\ln{(1+\frac{\tau^{2}}{8})}-\frac{\pi\tau^{2}(5\tau^{2}+48)}{16(\tau^{2}+8)^{2}}\label{eq:11x-}\\
&\iint_{\mathcal{C}^{+}}g_{11y}dudv=\frac{\pi}{8}\ln{(1+\frac{\tau^{2}}{4})}+\frac{\pi\tau^{2}(\tau^{2}-8)}{16(\tau^{2}+4)^{2}}\label{eq:11y+}\\
&\iint_{\mathcal{C}^{-}}g_{11y}dudv=\frac{\pi}{8}\ln{(1+\frac{\tau^{2}}{8})}+\frac{\pi\tau^{2}(\tau^{2}-16)}{16(\tau^{2}+8)^{2}}\label{eq:11y-}.
\end{align}
Similarly, we denote $g_{2i\kappa}$, $i=1,2$ as the integrand functions of \eqref{eq:21x} -- \eqref{eq:22z}, then we have
\begin{equation}
\rho_{2i\kappa}^{(l)}=\iint_{\mathcal{C}^{-}}g_{2i\kappa}dudv<\rho_{2i\kappa}<\iint_{\mathcal{C}^{+}}g_{2i\kappa}dudv=\rho_{2i\kappa}^{(u)}.
\end{equation}
The closed-form upper and lower bounds of $\rho_{21\kappa}$ and $\rho_{22\kappa}$ can be computed as follows.
\begin{align}
&\rho_{21x}^{(u)}=\frac{8\pi}{15}-\frac{\pi(45\tau^{4}+320\tau^{2}+512)}{30(\tau^{2}+4)^{5/2}\label{eq:21xu}}\\
&\rho_{21x}^{(l)}=\frac{8\pi}{15}-\frac{\sqrt{2}\pi(45\tau^{4}+640\tau^{2}+2048)}{30(\tau^{2}+8)^{5/2}}\label{eq:21xl}\\
&\notag\rho_{22x}^{(u)}=\frac{3\pi}{14}-\frac{\pi(63\tau^{4}-112\tau^{2}\sqrt{\tau^{2}+4})}{14(\tau^{2}+4)^{7/2}}\label{eq:22xu}\\
&\quad\quad-\frac{\pi(672\tau^{2}-448\sqrt{\tau^{2}+4}+1280)}{14(\tau^{2}+4)^{7/2}}\\
&\notag\rho_{22x}^{(l)}=\frac{3\pi}{14}-\frac{\pi(63\sqrt{2}\tau^{4}+1344\sqrt{2}\tau^{2})}{7(\tau^{2}+8)^{7/2}}\\
&\quad\quad+\frac{\pi[224(\tau^{2}+8)^{3/2}-5120\sqrt{2}]}{7(\tau^{2}+8)^{7/2}}\\
&\rho_{21y}^{(u)}=\frac{4\pi}{15}-\frac{\pi(15\tau^{4}+160\tau^{2}+256)}{30(\tau^{2}+4)^{5/2}}\\
&\rho_{21y}^{(l)}=\frac{4\pi}{15}-\frac{\sqrt{2}\pi(15\tau^{4}+320\tau^{2}+1024)}{30(\tau^{2}+8)^{5/2}}\\
&\rho_{22y}^{(u)}=\frac{10\pi}{21}-\frac{5\pi(35\tau^{4}+448\tau^{2}+512)}{42(\tau^{2}+4)^{7/2}}\\
&\rho_{22y}^{(l)}=\frac{10\pi}{21}-\frac{5\sqrt{2}\pi(35\tau^{4}+896\tau^{2}+2048)}{21(\tau^{2}+8)^{7/2}}\label{eq:22yl}\\
&\rho_{21z}^{(u)}=\frac{8\pi}{15}-\frac{8\pi(5\tau^{2}+32)}{15(\tau^{2}+4)^{5/2}}\\
&\rho_{21z}^{(l)}=\frac{8\pi}{15}\Big[1-\frac{10\sqrt{2}}{(\tau^{2}+8)^{3/2}}-\frac{48\sqrt{2}}{(\tau^{2}+8)^{5/2}}\Big]\label{eq:21zl}\\
&\notag\rho_{22z}^{(u)}=\frac{13\pi}{42}-\frac{\pi(21\tau^{6}+224\tau^{4}-896\tau^{2})}{42(\tau^{2}+4)^{7/2}}\label{eq:22zu}\\
&\quad\quad+\frac{512\pi}{21(\tau^{2}+4)^{7/2}}-\frac{\pi(168\tau^{2}+672)}{21(\tau^{2}+4)^{3}}\\
&\notag\rho_{22z}^{(l)}=\frac{13\pi}{42}-\frac{\pi(21\sqrt{2}\tau^{6}+448\sqrt{2}\tau^{4})}{42(\tau^{2}+8)^{7/2}}\\
&\quad\quad-\frac{\pi[1344(\tau^{2}+8)^{3/2}-\sqrt{2}]}{42(\tau^{2}+8)^{7/2}}.\label{eq:22zl}
\end{align}
\section{Proof of Corollary \ref{coro:CRB_CPL_zt}}\label{ap:prop4}
We only need to prove that $k_{0}^{2}\rho_{11\kappa}\gg z_{t}^{2}\rho_{12\kappa}$ and $k_{0}^{2}\rho_{21\kappa}\gg z_{t}^{2}\rho_{22\kappa}$ for $z_{t}\gg \lambda$, then the approximation in Corollary \ref{coro:CRB_CPL_zt} can be proved immediately.
When $z_{t}\gg \lambda$, we have $k_{0}^{2}\gg z_{t}^{-2}$. Observe that
\begin{equation}
2\rho_{11x}>\iint_{\mathcal{R}_{\tau}}\frac{2u^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{4}}dudv>\rho_{12x}>0,
\end{equation}
from which we obtain
\begin{equation}
k_{0}^{2}\rho_{11x}\gg z_{t}^{-2}\rho_{12x}.
\end{equation}
Similarly, we have
\begin{equation}
\frac{u^{2}+1}{(u^{2}+v^{2}+1)^{3}}\geq \frac{v^{4}+u^{2}v^{2}+1}{(u^{2}+v^{2}+1)^{4}}>0.
\end{equation}
Then, we have that $\rho_{11z}>\rho_{12z}$. Accordingly, we have that
\begin{equation}
k_{0}^{2}\rho_{11z}\gg z_{t}^{-2}\rho_{12z}.
\end{equation}
Observe that
\begin{equation}
4\pi^{2}\rho_{21y}>\iint_{\mathcal{R}_{\tau}} \frac{4\pi^{2}v^{2}(u^{2}+1)}{(u^{2}+v^{2}+1)^{9/2}}dudv>\rho_{22y}>0.
\end{equation}
So the following inequality can be proved.
\begin{equation}
k_{0}^{2}\rho_{21y}\gg z_{t}^{-2}\rho_{22y}.
\end{equation}
Note that the remaining inequalities are challenging to demonstrate analytically, therefore we provide numerical proofs. If we define the difference function $f_{dy1}(\tau)=\rho_{11y}-\rho_{12y}$, we can deduce from \eqref{eq:12y_close} and \eqref{eq:11y-} that the minimum value of the function $f_{dy1}(\tau)$ is greater than $-2.34$, which testifies that
\begin{equation}
k_{0}^{2}\rho_{11y}\gg z_{t}^{-2}\rho_{12y}.
\end{equation}
Define the difference function $f_{dx2}(\tau)=\rho_{21x}-\rho_{22x}$, from \eqref{eq:21xl} and \eqref{eq:22xu}, we have that $f_{dx2}(\tau)>\rho_{21x}^{(l)}-\rho_{22x}^{(u)}$, and we can derive that the minimum value of the function $f_{dx2}(\tau)$ is greater than $67\pi/210-1.23\approx-0.23$, which indicates that
\begin{equation}
k_{0}^{2}\rho_{21x}\gg z_{t}^{-2}\rho_{21x}.
\end{equation}
Similarly, we define the difference function $f_{dz2}(\tau)=\rho_{21z}-\rho_{22z}$, then we have that $f_{dz2}(\tau)>\rho_{21z}^{(l)}-\rho_{22z}^{(u)}$ based on \eqref{eq:21zl} and \eqref{eq:22zu}. Next, we can deduce that the minimum value of the function $f_{dz2}(\tau)$ is greater than $47\pi/210-0.80\approx-0.10$, which verifies that
\begin{equation}
k_{0}^{2}\rho_{21z}\gg z_{t}^{-2}\rho_{22z}.
\end{equation}
Therefore, Corollary \ref{coro:CRB_CPL_zt} holds.
\end{appendices}
\bibliographystyle{IEEEtran}
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1,314,259,993,044 | arxiv | \section*{\refname
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\begin{document}
\title{\textbf{Compactness of conformal metrics with constant $Q$-curvature. I}
\bigskip}
\author{\medskip YanYan Li\footnote{Supported in part by NSF grants DMS-1065971 and DMS-1203961.} \ \ and \ \
Jingang Xiong\footnote{Supported in part by Beijing Municipal Commission of Education for the Supervisor of Excellent Doctoral Dissertation (20131002701).}}
\date{}
\fancyhead{}
\fancyhead[CO]{Compactness of conformal metrics with constant $Q$-curvature}
\fancyhead[CE]{Y. Y. Li \& J. Xiong}
\fancyfoot{}
\fancyfoot[CO, CE]{\thepage}
\renewcommand{\headrule}{}
\maketitle
\begin{abstract}
We establish compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are
universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is trivial and its Green function is positive, we establish universal energy bounds on manifolds which are either locally conformally flat (LCF) or of dimension $\le 9$. By assuming a positive mass type theorem for the Paneitz operator, we prove compactness in $C^4$. Positive mass type theorems have been verified recently on LCF manifolds or manifolds of dimension $\le 7$, when the Yamabe invariant is positive. We also prove that, for dimension $\ge 8$, the Weyl tensor has to vanish at possible blow up points of a sequence of solutions. This implies the compactness result in dimension $\ge 8$ when the Weyl tensor does not vanish anywhere.
To overcome difficulties stemming from fourth order elliptic equations, we develop a blow up analysis procedure via integral equations.
\end{abstract}
\tableofcontents
\section{Introduction}
On a compact smooth Riemannian manifold $(M,g)$ of dimension $\ge 3$, the Yamabe problem, which concerns the existence of constant scalar curvature metrics in the conformal class of $g$, was solved through the works of Yamabe \cite{Y}, Trudinger \cite{Tr}, Aubin \cite{Aubin} and Schoen \cite{Schoen84}. Different proofs of the Yamabe problem in the case $n\le 5$ and in the case $(M,g)$ is locally conformally flat are given by Bahri and Brezis \cite{BB} and Bahri \cite{B}.
The problem is equivalent to solving the Yamabe equation
\begin{equation}} \newcommand{\ee}{\end{equation} \label{Yamabe equation}
-L_g u= Sign(\lda_1) u^{\frac{n+2}{n-2}}, \quad u>0 \quad \mbox{on }M,
\ee
where $L_g:=\Delta_g -\frac{(n-2)}{4
(n-1)}R_g$, $\Delta_g$ is the Laplace-Beltrami operator associated with $g$, $R_g$ is the scalar curvature, and $Sign(\lda_1)$ denotes the sign of the first eigenvalue $\lda_1$ of the conformal Laplacian $-L_g$. The sign of $\lda_1$ is conformally invariant, i.e., it is the same for every metric in the conformal class of $g$.
If $\lda_1<0$, there exists a unique solution of \eqref{Yamabe equation}. If $\lda_1=0$, the equation is linear and solutions are unique up to multiplication by a positive constant. If $\lda_1>0$, non-uniqueness has been established; see Schoen \cite{Schoen89} and Pollack \cite{P}. If $(M,g)$ is the standard unit sphere, all solutions are classified by Obata \cite{Obata} and there is no uniform $L^\infty$ bound for them. Schoen \cite{Schoen91} established a uniform $C^2$ bound for all solutions if $M$ is locally conformally flat but not conformal to the sphere. The uniform $C^2$ bound was established in dimensions $n\le 7$ by Li-Zhang \cite{Li-Zhang05} and Marques \cite{Marques} independently. For $n = 3, 4, 5$, see works of Li-Zhu \cite{Li-Zhu99}, Druet \cite{Druet03, Druet04} and Li-Zhang \cite{Li-Zhang04}.
For $8\le n\le 24$, the answer is positive provided
that the positive mass theorem holds in these dimensions; see Li-Zhang \cite{Li-Zhang05, Li-Zhang06}
for $8\le n\le 11$, and Khuri-Marques-Schoen \cite{KMS} for $12\le n\le 24$. On the other
hand, the answer is negative in dimension $n \ge 25$; see Brendle \cite{Brendle} for
$n \ge 52$, and Brendle-Marques \cite{BM} for $25\le n \le 51$.
In this paper, we are interested in a fourth order analogue of the Yamabe problem. Namely, the constant $Q$-curvature problem. Let us recall the conformally invariant Paneitz operator and the corresponding \emph{$Q$-curvature}, which are defined as \footnote{If $n=4$, $\frac12 Q_g$ is defined as the $Q$-curvature in some papers.}
\begin{align}
\label{Paneitz operator}
P_g&= \Delta_g^2 -\mathrm{div}_g(a_n R_g g+b_nRic_g)d+\frac{n-4}{2}Q_g \\
\label{Q-curvature}
Q_g&=-\frac{1}{2(n-1)} \Delta_g R_g+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2} R_g^2-\frac{2}{(n-2)^2} |Ric_g|^2,
\end{align} where $R_g$ and $Ric_g$ denote the scalar curvature and Ricci tensor of $g$ respectively, and $
a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}.$ The self-adjoint operator $P_g$ was discovered by Paneitz \cite{Pan83} in 1983, and $Q_g$ was introduced later by Branson \cite{Bra85}. Paneitz operator is conformally invariant in the sense that
\begin{itemize}
\item If $n=4$, for any conformal metric $\hat g=e^{2w} g$, $w\in C^\infty(M)$, there holds
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:conformal change2}
P_{\hat g}=e^{-4w}P_g \quad \mbox{and} \quad P_g w+Q_g=Q_{\hat g} e^{4w}.
\ee
\item If $n=3$ or $n\ge 5$, for any conformal metric $\hat g=u^{\frac{4}{n-4}} g$, $0<u\in C^\infty(M)$, there holds
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:conformal change1}
P_{\hat g}(\phi)=u^{-\frac{n+4}{n-4}}P_g(u\phi)\quad \forall~ \phi\in C^\infty(M).
\ee
\end{itemize}
Hence, finding constant $Q$-curvature in the conformal class of $g$ is equivalent to solving
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Q-4d}
P_g w+Q_g=\lda e^{4w} \quad \mbox{on }M
\ee if $n=4$, and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:4th Yamabe}
P_g u=\lda u^{\frac{n+4}{n-4}}, \quad u>0 \quad \mbox{on }M,
\ee
if $n=3$ or $n\ge 5$, where $\lda$ is a constant.
When $n=4$, there is a Chern-Gauss-Bonnet type formula involving the $Q$-curvature; see Chang-Yang \cite{CY99}. The constant $Q$-curvature problem has been studied by Chang-Yang \cite{CY95}, Djadli-Malchiodi \cite{DM}, Li-Li-Liu \cite{LLL} and references therein. Bubbling analysis and compactness for solutions have been studied by Druet-Robert \cite{DR}, Malchiodi \cite{Mal}, Weinstein-Zhang \cite{WZhang} among others.
When $n\ge 5$, the constant $Q$-curvature problem is a natural extension of the Yamabe problem. However, the lack of maximum principle for fourth order elliptic equations makes the problem much harder. The first eigenvalues of fourth order self-adjoint elliptic operators are not necessarily simple and the associated eigenfunctions may change signs. We might not be able to divide the study of \eqref{eq:4th Yamabe} into three mutually exclusive cases by linking the constant $\lda$ to the sign of the first eigenvalue of the Paneitz operator. Up to now, the existence of solutions has been obtained with $\lda=1$, roughly speaking, under the following three types of assumptions. The first one is on the equation. Assuming, among others, the coefficients of the Paneitz operator are constants, Djadli-Hebey-Ledoux \cite{DHL} proved some existence results, where they decompose the operator as a product of two second order elliptic operators and use the maximum principle of second order elliptic equations. This assumption is fulfilled, for instance, when the background metric is Einstein. The second one is on the geometry and topology of the manifolds. Assuming that the Poincar\'e exponent is less than $(n-4)/2$, Qing-Raske \cite{QR0, QR} proved the existence result on locally conformally flat manifolds of positive scalar curvature. The last one is purely geometric. Assuming that there exists a conformal metric of nonnegative scalar curvature and semi-positive $Q$-curvature, Gursky-Malchiodi \cite{GM} recently proved the existence result for $n\ge 5$. By their condition, the scalar curvature was proved to be positive. In a very recent preprint, Hang-Yang \cite{HY14b} replaced the positive scalar curvature condition by the positive Yamabe invariant (which is equivalent to $\lda_1>0$). More precisely, \eqref{eq:4th Yamabe} admits a solution with $\lda=1$ if
\begin{equation}} \newcommand{\ee}{\end{equation}\label{condition:main}
\lda_1(-L_{g})>0, \quad Q_{g}\ge 0 \mbox{ and } Q_{g}>0 \mbox{ somewhere on }M,
\ee
where $\lda_1(-L_{g})$ is the first eigenvalue of $-L_g$ defined above. See also Hang-Yang \cite{HY14a} for $n=3$.
Each of the above three types of assumptions implies that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{condition:main2} \mathrm{Ker} P_g=\{0\} \mbox{ and the Green's function }G_g \mbox{ of $P_g$ is positive}.
\ee
In fact, $P_g$ is coercive in Djadli-Hebey-Ledoux \cite{DHL}, Qing-Raske \cite{QR0, QR} and Gursky-Malchiodi \cite{GM}. We refer to the latest paper Gursky-Hang-Lin \cite{GHL} for further discussions on these conditions.
If \eqref{condition:main2} holds and $\lda_1>0$, there exists a positive mass type theorem for $G_g$, provided $M$ is locally conformally flat or $n=5,6,7$, but not conformal to the standard sphere; see Humbert-Raulot \cite{HuR}, Gursky-Malchiodi \cite{GM} and Hang-Yang \cite{HY14c}.
Starting from this paper, we study the compactness of solutions of the constant $Q$-curvature equation for $n\ge 5$.
For positive constant $Q$-curvature problem, there are non-compact examples. If $(M,g)$ is a sphere, the constant $Q$-curvature metrics are not compact in $C^4$ due to the non-compactness of the conformal diffeomorphism group of the sphere. Recently, Wei-Zhao \cite{WZ} produced non-compact examples on manifolds of dimension $n\ge 25$ not conformal to the standard sphere.
\begin{thm} \label{thm:main theorem} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, but not conformal to the standard sphere.
Assume (\ref{condition:main2}).
For $1< p\le \frac{n+4}{n-4}$, let $0<u\in C^4(M)$ be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:main1}
P_g u=c(n)u^{p} \quad \mbox{on }M,
\ee where $c(n)=n(n+2)(n-2)(n-4)$. Suppose that one of the following conditions is also satisfied:
\begin{itemize}
\item[i)] $\lda_1(-L_g)>0$ and $(M,g)$ is locally conformally flat,
\item[ii)] $\lda_1(-L_g)>0$ and $n=5,6,7$,
\item[iii)] $(M,g)$ is locally conformally flat or $5\le n\le 9$, and the positive mass type theorem holds for the Paneitz operator,
\item[iv)] The Weyl tensor of $g$ does not vanish anywhere, i.e., $|W_g|^2>0$ on $M$.
\end{itemize} Then there exists a constant $C>0$, depending only on $M,g$, and
a lower bound of $p-1$, such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:C4 estimate}
\|u\|_{C^4(M)} +\|1/u\|_{C^4(M)}\le C.
\ee
\end{thm}
The assumption (\ref{condition:main2})
in the theorem can be
replaced by \eqref{condition:main},
as explained above. The positive mass type theorem for Paneitz operator in dimension $8,9$ is understood as in Remark \ref{rem:positive mass}. The case $5\le n\le 9$ for positive constant $Q$-curvature equation shows some similarity to $3\le n\le 7$ for the Yamabe equation with positive scalar curvature.
The following situations, included in
Theorem \ref{thm:main theorem}, were proved before.
If $M$ is locally conformally flat and $p=\frac{n+4}{n-4}$, \eqref{eq:C4 estimate} was established
by Qing-Raske \cite{QR0, QR} with the assumptions that $\lda_1>0$ and the Poincar\'e exponent
is less than $(n-4)/2$, and by Hebey-Robert \cite{HR, HR11} with $C$ depending on the $H^2$ norm of $u$,
where they assumed that $P_g$ is coercive.
Neither $\lda_1(-L_g)>0$ nor the positive mass type theorem for Paneitz operator is assumed, we have an energy bound of solutions.
\begin{thm}\label{thm:energy} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$.
Assume (\ref{condition:main2}), and assume that either $n\le 9$ or $(M,g)$ is
locally conformally flat.
Let $0<u\in C^4(M)$ be a solution of \eqref{eq:main1}. Then
\[
\|u\|_{H^2(M)}\le C,
\]
where $C>0$ depends only on $M,g$, and
a lower bound of $p-1$.
\end{thm}
Next, we establish Weyl tensor vanishing results.
\begin{thm}\label{thm:main-b} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 8$.
Assume (\ref{condition:main2}).
Let $u_i$ be a sequence of positive solutions of
\[
P_g u_i=c(n)u_i^{p_i},
\] where $p_i\le \frac{n+4}{n-4}$,
$p_i\to \frac{n+4}{n-4}$ as $i\to \infty$. Suppose that there is a sequence of $X_i\to \bar X\in M$ such that $u_i(X_i)\to \infty$. Then the Weyl tensor has to vanish at $\bar X$, i.e., $W_g(\bar X)=0$.
Furthermore, if $n=8,9$, there exists
$X_i'\to \bar X$
such that, for all $i$,
\[
|W_g(X_i')|^2\le C\begin{cases} (\log u_i(X_i'))^{-1},& \quad \mbox{if }n=8,\\
u_i(X_i')^{-\frac{2}{n-4}}, &\quad \mbox{if }n=9,
\end{cases}
\]
where $C>0$ depends only on $M$ and $g$.
\end{thm}
\begin{thm}\label{thm:main-c}
In addition to the assumptions in Theorem \ref{thm:main-b} with $n\ge 10$, we assume that
there exist a neighborhood $\Omega} \newcommand{\pa}{\partial$ of $\bar X$ and a constant $\bar b>0$
such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:technical condition 12}
u_i(X)\le \bar b \cdot dist_g(X, X_i)^{ -\frac 4{p_i-1}} \quad \forall~ X\in \Omega} \newcommand{\pa}{\partial,
\ee
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:technical condition}
X_i\ \mbox{is a local maximum point of}\
u_i,
\quad \sup_{\Omega} \newcommand{\pa}{\partial}u_i\le \bar bu_i(X_i).
\ee
Then, for sufficiently large $i$,
\[
|W_g(X_i)|^2\le C\begin{cases} u_i(X_i)^{-\frac{4}{n-4}}\log u_i(X_i), &\quad \mbox{if }n=10,\\
u_i(X_i)^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11,
\end{cases}
\]
where $C>0$ depends only on $M,g, dist_g(\bar X,\pa \Omega} \newcommand{\pa}{\partial)$ and $\bar b$.
\end{thm}
The rates of decay of $|W_g(X_i)|$ in
Theorem \ref{thm:main-b} and
Theorem \ref{thm:main-c} correspond to the Yamabe
problem case $n=6,7$ and $n\ge 8$ respectively;
see theorem 1.3 and theorem 1.2 in
\cite{Li-Zhang05}.
Condition (\ref{eq:technical condition 12}) and (\ref{eq:technical condition})
can often be reduced to, by some elementary consideration,
in applications.
In a subsequent paper, we will establish compactness results analogous to those established in $8\le n\le 24$ for the Yamabe equation by Li-Zhang \cite{Li-Zhang05,Li-Zhang06} and Khuri-Marques-Schoen \cite{KMS}. The present paper provides analysis foundations.
For the negative constant $Q$-curvature equation, we have
\begin{thm}\label{thm:compact1} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. Then for any $1<p<\infty$, there exists a positive constant $C$, depending only on $M,g$ and $p$, such that every nonnegative $C^4$ solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:-Q}
P_g(u)=-u^{p} \quad \mbox{on }M
\ee
satisfies
\[
\|u\|_{C^4(M)} \le C.
\]
\end{thm}
The proofs of Theorems \ref{thm:main theorem}, Theorem \ref{thm:main-b} and Theorem \ref{thm:main-c}
make use of important ideas for the proof of compactness of positive solutions of the Yamabe equation, which were outlined first by Schoen \cite{Schoen89, Schoen89b, Schoen91}, as well as methods developed through the work Li \cite{Li95}, Li-Zhu \cite{Li-Zhu99}, Li-Zhang \cite{Li-Zhang04,Li-Zhang05,Li-Zhang06},
and Marques \cite{Marques}.
Our main difficulty now stems from the fourth order equation, which we explain in details. To understand the profile of possible blow up solutions, it is natural to scale the solutions in local coordinates centered at local maximum points. By the Liouville theorem in Lin \cite{Lin}, one can conclude that these solutions are close to some standard bubbles in small geodesic balls, whose sizes become smaller and smaller as solutions blowing up; see e.g., Proposition \ref{prop:reduction}. Then we need to answer two questions:
\begin{itemize}
\item[(i)] Do these blow up points accumulate?
\item [(ii)] If not, how do these solutions behave in geodesic balls with some fixed size?
\end{itemize}
For the first one, we may scale possible blow up points apart and look at them individually. It turns out that we end up with the situation of question (ii). After scaling we need to carry out local analysis.
In the Yamabe case, properties of second order elliptic equations, which include the maximum principle, comparison principle, Harnack inequality and B\^ocher theorem for isolated singularity, were used crucially. Now we don't have these properties for fourth order elliptic equations.
This leads to an obstruction to using fourth order equations to develop local analysis.
We observe that along scalings the bounds of Green's function are preserved. In view of Green's representation, we develop a blow up analysis procedure for integral equations and answer the above two questions completely in dimensions less than $10$. This is inspired by our recent joint work with Jin \cite{JLX3} for a unified treatment of the Nirenberg problem and its generalizations, which in turn was stimulated by our previous work on a fractional Nirenberg problem \cite{JLX, JLX2}. The approach of the latter two papers were based on the Caffarelli-Silvestre extension developed in \cite{CaffS}. Our analysis is very flexible and can easily be adapted to deal with higher order and fractional order conformally invariant elliptic equations. The organization of the paper is shown in the table of Contents.
\medskip
\textbf{Notations.} Letters $x,y,z$ denote points in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$, and capital letters $X,Y,Z$ denote points on Riemannian manifolds. Denote by $B_r(x)\subset \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ the ball centered at $x$ with radius $r>0$. We may write $B_r$ in replace of $B_r(0)$ for brevity. For $X\in M$, $\B_{\delta}(X)$ denotes the geodesic ball centered at $X$ with radius $\delta$. Throughout the paper, constants $C>0$ in inequalities may vary from line to line and are universal, which means they depend on given quantities but not on solutions. $f=O^{(k)}(r^m)$ denotes any quantity satisfying $|\nabla^j f(r)|\le C r^{m-j}$ for all integers $1\le j\le k$, where $k$ is a positive integer and $m$ is a real number. $|\mathbb{S}^{n-1}|$ denotes the area of the standard $n-1$-sphere. Here are specified constants used throughout the paper:
\begin{itemize}
\item $c(n)=n(n+2)(n-2)(n-4)$ appears in constant $Q$-curvature equation,
\item $\alpha} \newcommand{\lda}{\lambda_n=\frac{1}{2(n-2)(n-4)|\mathbb{S}^{n-1}|}$ appears in the expansion of Green's functions,
\item $c_n=c(n)\cdot \alpha} \newcommand{\lda}{\lambda_n =\frac{n(n+2)}{2|\mathbb{S}^{n-1}|}$.
\end{itemize}
\noindent\textbf{Added note on June 1, 2015:} Theorem \ref{thm:main theorem}
was announced by the first named author in his talk at the
International Conference on Local and Nonlocal Partial Differential
Equations, NYU Shanghai, China, April 24-26, 2015; while
the part of the theorem
for general manifolds of dimension $n=5,6,7$ and
for locally conformally flat manifolds of dimension $n\ge 5$
was announced in his talk at the
Conference on Partial Differential Equations,
University of Sussex, UK, September 15-17, 2014.
We noticed that two days ago an article
was posted on the arXiv, [Gang Li, A compactness theorem on Branson$'$s
$Q$-curvature equation,
arXiv:1505.07692v1 [math.DG] 28 May 2015], where
a compactness result in dimension $n=5$, under the assumption that $R_g>0$ and $Q_g\ge 0$ but not identically equal to zero, was proved independently.
\bigskip
\noindent\textbf{Acknowledgments:}
J. Xiong is grateful to Professor Jiguang Bao and Professor Gang Tian for their supports.
\medskip
\section{Preliminaries}
\label{section:pre}
\subsection{Paneitz operator in conformal normal coordinates}
Let $(M,g)$ be a smooth Riemannian manifold (with or without boundary) of dimension $n\ge 5$, and $P_g$ be the Paneitz operator on $M$.
For any point $X\in M$, it was proved in \cite{LP}, together with some improvement in \cite{Cao} and \cite{Guther}, that there exists a positive smooth function $\kappa $ (with control) on $M$ such that the conformal metric $\tilde g=\kappa^{\frac{-4}{n-4}}g$ satisfies, in $\tilde g$-normal coordinates $\{x_1,\dots,x_n\}$ centered at $X$,
\[
\det \tilde g=1 \quad \mbox{in }B_{\delta}
\] for some $\delta>0$. We refer such coordinates as conformal normal coordinates. Notice that $\det \tilde g=1+O(|x|^N)$ will be enough for our use if $N$ is sufficiently large. Since one can view $x$ as a tangent vector of $M$ at $X$, thus $
\det g(x)=1+O(|x|^2)$. It follows that $\kappa(x)=1+O(|x|^2)$. In particular,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:cfn-1}
\kappa(0)=1, \quad \nabla \kappa(0)=0.
\ee
In the $\tilde g$-normal coordinates,
\begin{eqnarray*}
R_{ij}(0)=0, & & Sym_{ijk} R_{ij,k}(0)=0, \\
R_{,i}(0)=0, & & \Delta_{ \tilde g} R(0)=-\frac{1}{6}|W_{\tilde g}(0)|^2,
\end{eqnarray*}
where the Ricci tensor $R_{ij}$, scalar curvature $R$, and Weyl tensor $W$ are with respect to $\tilde g$.
We also have \[
\Delta_{\tilde g} =\Delta+\pa_l \tilde g^{kl}\pa_k+(\tilde g^{kl}-\delta^{kl})\pa_{kl}=:\Delta+d^{(1)}_k\pa_k+d^{(2)}_{kl}\pa_{kl},
\]
and
\[
\Delta_{\tilde g}^2=\Delta^2+f^{(1)}_{k}\pa_k +f^{(2)}_{kl}\pa_{kl}+f^{(3)}_{kls}\pa_{kls}+f^{(4)}_{klst}\pa_{klst},
\]
where
\begin{align*}
f^{(1)}_{k}:&=\Delta d^{(1)}_k+d^{(1)}_s\pa_s d^{(1)}_k+d^{(2)}_{st}\pa_{st}d^{(1)}_k=O(1), \\
f^{(2)}_{kl}:&=\pa_k d^{(1)}_l+\Delta d^{(2)}_{kl}+d^{(1)}_kd^{(1)}_l+d^{(1)}_s\pa_sd^{(2)}_{kl}+d^{(2)}_{sl}\pa_s d^{(1)}_k+d^{(2)}_{st}\pa_{st}d^{(2)}_{kl}=O(1), \\
f^{(3)}_{kls}:&=2d^{(1)}_s\delta^{kl}+\pa_s d^{(2)}_{kl}+2d^{(1)}_s d^{(2)}_{kl}+d^{(2)}_{st}\pa_t d^{(2)}_{kl}=O(|x|),\\
f^{(4)}_{klst}:&=2d^{(2)}_{kl}\delta^{st}+d^{(2)}_{kl}d^{(2)}_{st}=O(|x|^2).
\end{align*}
Now the second term of the Paneitz operator $P_{\tilde g}$ can be expressed as
\[
-\mathrm{div}_{\tilde g}(a_n R_{\tilde g} \tilde g+b_nRic_{\tilde g})d=-\pa_{l}((a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}\pa_k)=:f^{(5)}_k\pa_k+f^{(6)}_{kl}\pa_{kl},
\]
where
\begin{align*}
f^{(5)}_{k}:&=-\pa_{l}\big((a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}\big)=O(1),\\
f^{(6)}_{kl}:&=-(a_nR{\tilde g}_{st}+b_nR_{st}){\tilde g}^{sk}{\tilde g}^{tl}=O(|x|).
\end{align*}
By abusing notations, we relabel $f^{(1)}_k$ as $f^{(1)}_{k}+f^{(5)}_k$, and $f^{(2)}_{kl}$ as $f^{(1)}_{kl}+f^{(6)}_{kl}$. Hence,
\begin{align}
E(u):&=P_{\tilde g}u-\Delta^2 u\nonumber\\&
=\frac{n-4}{2}Q_{\tilde g}u+ f^{(1)}_{k}\pa_ku +f^{(2)}_{kl}\pa_{kl}u+f^{(3)}_{kls}\pa_{kls}u+f^{(4)}_{klst}\pa_{klst}u,
\label{eq:Eu}
\end{align}
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:co-Eu}
f^{(1)}_{k}(x)=O(1), \quad f^{(2)}_{kl}(x)=O(1),\quad f^{(3)}_{kls}(x)=O(|x|), \quad f^{(4)}_{klst}(x)=O(|x|^2).
\ee
We point out that each term of $f^{(1)}_k$ takes up to three times derivatives of $\tilde g$ totally, each term of $f^{(2)}_{kl}(x)$ takes twice, each term of $f^{(3)}_{kls}(x)$ takes once, and no derivative of $\tilde g$ is taken in any term of $f^{(4)}_{klst}$. Hence, we see that
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:r1}
\begin{split}
\|f^{(1)}_{k}\|_{L^\infty(B_\delta)}+ \|f^{(2)}_{kl}\|_{L^\infty(B_\delta)}& +\|\nabla f^{(3)}_{kls} \|_{L^\infty(B_\delta)}+\|\nabla^2 f^{(4)}_{klst}\|_{L^\infty(B_\delta)} \\&
\le C\sum_{k\ge 1, 2\le k+1\le 4} \|\nabla^k g\|_{L^\infty(B_\delta)}^l
\end{split}
\ee
\begin{lem}\label{lem:GM2.8} In the $\tilde g$-normal coordinates, we have, for any smooth radial function $u$,
\begin{align*}
P_{\tilde g}u=&\Delta^2 u+\frac{1}{2(n-1)}R_{,kl}(0)x^kx^l (c_1^*\frac{u'}{r} +c_2^*u'')-\frac{4}{9(n-2)r^2}\sum_{kl}(W_{ikjl}(0)x^i x^j)^2(u''-\frac{u'}{r})\\&+\frac{n-4}{24(n-1)}|W_g(0)|^2 u+(\frac{\psi_5(x)}{r^2}+\psi_3(x))u'' -(\frac{\psi_5(x)}{r^3} +\frac{\psi_3'(x)}{r}) u'+\psi_1(x)u \\&+O(r^4)u''+O(r^3) u'+O(r^2) u,
\end{align*}
where $r=|x|$, $\psi_k(x),\psi_k'(x)$ are homogeneous polynomials of degree $k$, and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:c-star}
c_1^*=\frac{2(n-1)}{(n-2)}-\frac{(n-1)(n-2)}{2}+6-n,\quad c_2^*=-\frac{n-2}{2}-\frac{2}{n-2}.
\ee
\end{lem}
\begin{proof} Since $\det \tilde g=1$ and $u$ is radial, we have $\Delta_{\tilde g}^2 u =\Delta^2 u$. The rest of the proof is same as that of Lemma 2.8 of \cite{GM}. It suffices to expand the coefficients of lower order terms of $P_{\tilde g}$ in Taylor series to a higher order so that $(\frac{\psi_5(x)}{r^2}+\psi_3(x))u'' -(\frac{\psi_5(x)}{r^3} +\frac{\psi_3'(x)}{r}) u'+\psi_1(x)u$ appears.
\end{proof}
If $\mathrm{Ker} P_{g}=\{0\}$, then $P_g$ has unique Green function $G_g$, i.e., $P_gG_g(X,\cdot)=\delta_{X}(\cdot)$ for every $X\in M$, where $\delta_X(\cdot)$ is the Dirac measure at $X$ on manifolds $(M,g)$. It is easy to check that $\mathrm{Ker} P_{g}=\{0\}$ is conformally invariant.
\begin{prop}[\cite{GM}, \cite{HY14b}]\label{prop:GM} Let $(M,\tilde g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, on which $\mathrm{Ker} P_{\tilde g}=\{0\}$. Then there exists a small constant $\delta>0$, depending only on $(M,\tilde g)$, such that if $\det \tilde g=1$ in the normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$, the Green's function $G(\bar X, \exp_{\bar X}x)$ of $P_{\tilde g}$ has the expansion, for $x\in B_\delta(0)$,
\begin{itemize}
\item If $n=5,6,7$, or $M$ is flat in a neighborhood of $\bar X$,
\[
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}+A+O^{(4)}(|x|),
\]
\item If $n=8$,
\[
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}-\frac{\alpha} \newcommand{\lda}{\lambda_n}{1440}|W(\bar X)|^2 \log |x|+O^{(4)}(1),
\]
\item If $n\ge 9$, \begin{align*}
G(\bar X, \exp_{\bar X}x)=\frac{\alpha} \newcommand{\lda}{\lambda_n}{|x|^{n-4}}\Big(1+\psi_4(x)\Big)+O^{(4)}(|x|^{9-n}),
\end{align*}
\end{itemize}
where $\alpha} \newcommand{\lda}{\lambda_n=\frac{1}{2(n-2)(n-4)|\mathbb{S}^{n-1}|}$, $A$ is a constant, $W(\bar X)$ is the Weyl tensor at $\bar X$, and $\psi_4(x)$ a homogeneous polynomial of degree $4$.
\end{prop}
\begin{cor}\label{cor:Q-gf-expansion} Suppose the assumptions in Proposition \ref{prop:GM}. Then in the normal coordinate centered at $\bar X$ we have
\[
G(\exp_{\bar X}x, \exp_{\bar X}y)=\frac{\alpha} \newcommand{\lda}{\lambda_n(1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2))}{|x-y|^{n-4}}+\bar a+O^{(4)}(|x-y|^{6-n}),
\]
where $x,y\in B_\delta$, $x-y=(x_1-y_1, \dots, x_n-y_n)$, $|x-y|=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$, $\bar a$ is a constant and $\bar a=0$ if $n\ge 6$.
\end{cor}
\begin{proof} We only prove the case that $M$ is non-locally formally flat. Denote $X=\exp_{\bar X}x$ and $Y= \exp_{\bar X}y$ for $x,y\in B_\delta$, where $\delta>0$ depends only on $(M,\tilde g)$. For $X\neq \bar X$, we can find $g_{X}=v^{\frac{4}{n-4}} \tilde g$ such that in the $g_{X}$-normal coordinate centered at $X$ there hold $\det g_{X}=1$ and $v(Y)=1+O^{(4)}(dist_{g_{X}}(X,Y)^2)$. Let $G_{g_{X}}$ be the Green's function of $P_{g_{X}}$. By Proposition \ref{prop:GM},
\[
G_{g_{X}}(X,Y)=\alpha} \newcommand{\lda}{\lambda_{n} dist_{g_{X}}(X,Y)^{4-n}+A+O^{(4)}(dist_{g_{X}}(X,Y)^{6-n}),
\] where $A$ is a constant and $A=0$ if $n\ge 6$. By the conformal invariance of the Paneitz operator, we have the transformation law
\[
G(X,Y) =G_{g_{X} }(X,Y) v(X)v(Y)=G_{g_{X} }(X,Y) v(Y).
\]
Since $g_{X}=v^{\frac{4}{n-4}} \tilde g$ and $v(Y)=1+O^{(4)}(dist_{g_{X}}(X,Y)^2)$, we obtain
\[
\begin{split}
dist_{g_{X}}(X,Y)&=dist_{g_{X}}(\exp_{\bar X}x,\exp_{\bar X}y)\\&
=(1+O^{(4)}(|x-y|^2))dist_{\tilde g}(\exp_{\bar X}x,\exp_{\bar X}y)\\&
=(1+O^{(4)}(|x-y|^2))(1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2))|x-y|,
\end{split}
\]
where $\tilde g$ is viewed as a Riemannian metric on $B_\delta$ because of the exponential map $\exp_{\bar X}$.
Therefore, we get
\[
G(\exp_{\bar X}x, \exp_{\bar X}y)=\alpha} \newcommand{\lda}{\lambda_{n}\frac{1+O^{(4)}(|\bar x|^2)+O^{(4)}(|y|^2)}{|\bar x-y|^{n-4}}+O^{(4)}(|\bar x-y|^{6-n}).
\]
If $X=\bar X$, it follows Proposition \ref{prop:GM}.
We complete the proof.
\end{proof}
The following positive mass type theorem for Paneitz operator was proved through \cite{HuR}, \cite{GM} and \cite{HY14b}.
\begin{thm}\label{thm:positive mass} Let $(M,g)$ be a compact manifold of dimension $n\ge 5$, and $\bar X\in M$ be a point. Let $g$ be a conformal metric of $g$ such that $\det \tilde g=1$ in the $\tilde g$-normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$. Suppose also that $\lda_1(-L_{g})>0$ and \eqref{condition:main2} holds. If $n=5,6,7$, or $(M,g)$ is locally conformally flat, then the constant
$A$ in Proposition \ref{prop:GM} is nonnegative, and $A=0$ if and only if $(M, g)$ is conformal to the standard $n$-sphere.
\end{thm}
\begin{rem}\label{rem:positive mass} Suppose the assumptions in Theorem \ref{thm:positive mass}. If $W(\bar X)=0$, it follows from Proposition 2.1 of \cite{HY14b} that, in the $\tilde g$-normal coordinates centered at $\bar X$, the Green's function $G$ of $P_{\tilde g}$ has the expansion
\[
G(\bar X, \exp_{\bar X}x)= \begin{cases} \alpha} \newcommand{\lda}{\lambda_8|x|^{-4}+\psi(\theta) +\log |x| O^{(4)}(|x|),& \quad n=8,\\[2mm]
\alpha} \newcommand{\lda}{\lambda_9|x|^{-5}(1+\frac{R_{,ij}(\bar X) x^ix^j |x|^2}{384}) +A+ O^{(4)}(|x|), &\quad n=9,
\end{cases}
\]
where $x=|x|\theta$, $\psi$ is a smooth function of $\theta$, and $A$ is constant. In dimension $n=8,9$, we say the positive mass type theorem holds for Paneitz operator if $\int_{\mathbb{S}^{n-1}} \psi(\theta)\,\ud \theta>0 $ and $A>0$ respectively.
\end{rem}
Let
\[
U_\lda(x):=\left(\frac{\lda}{1+\lda^2|x|^2}\right)^{\frac{n-4}{2}}, \quad \lda>0,
\]
which is the unique positive solution of $\Delta^2 u=c(n)u^{\frac{n+4}{n-4}}$ in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$, $n\ge 5$, up to translations by Lin \cite{Lin}. By Lemma \ref{lem:GM2.8}, in the $\tilde g$-normal coordinates we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:cor-GM}
P_{\tilde g} U_{\lda}= c(n)U_{\lda}^{\frac{n+4}{n-4}}+ f_{\lda}U_{\lda},
\ee
where $f_{\lda}(x)$ is a smooth function satisfying that $\lda^{-k}|\nabla_x^kf_\lda(x)| $, $k=0,1,\dots, 5$, is uniformly bounded in $B_\delta$ independent of $\lda \ge 1$.
Indeed,
by direct computations
\begin{align*}
\pa_rU_\lda&=(4-n)\lda^{\frac{n}{2}}(1+\lda^2r^2)^{\frac{2-n}{2}}r\\
\pa_{rr}^2 U_\lda &=(4-n)(2-n)\lda^{\frac{n+4}{2}}(1+\lda^2r^2)^{\frac{-n}{2}}r^2+(4-n)\lda^{\frac{n}{2}}(1+\lda^2r^2)^{\frac{2-n}{2}}.
\end{align*}
Inserting them to the expression in Lemma \ref{lem:GM2.8}, \eqref{eq:cor-GM} follows.
\begin{cor}\label{cor:GM2.8} Let $(M,\tilde g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$, on which $\mathrm{Ker} P_{\tilde g}=\{0\}$. Then there exists a small constant $\delta>0$, depending only on $(M,\tilde g)$, such that if $\det \tilde g=1$ in the normal coordinate $\{x_1,\dots, x_n\}$ centered at $\bar X$, then
\[
U_{\lda}(x)=c(n)\int_{B_{\delta}}G(\exp_{\bar X} x, \exp_{\bar X} y) \{U_{\lda}(y)^{\frac{n+4}{n-4}}+c_\lda'(x) U_{\lda}(y)\}\,\ud y+c_\lda''(x),
\]
where $\delta>0$ depends only on $M,\tilde g$, and $c_\lda', c_\lda''$ are smooth functions satisfying
\[
\lda^{-k}|\nabla^kc_\lda'(x)| \le C, \quad |\nabla ^kc_\lda''(x)|\le C\lda^{\frac{4-n}{2}},
\]
for $k=0,1,\dots, 5$ and some $C>0$ independent of $\lda\ge 1$.
\end{cor}
\begin{proof} Let $\eta(x)=\eta(|x|)$ be a smooth cutoff function satisfying
\[
\eta(t)=1 ~ \mbox{for }t<\delta/2, \quad \eta(t)=0 ~ \mbox{for }t>\delta.
\] By the Green's representation formula, we have
\[
(U_\lda\eta)(x)=\int_{B_\delta} G(\exp_{\bar X} x, \exp_{\bar X} y) P_{\tilde g}(U_\lda\eta)(y)\,\ud y.
\]
Making use of \eqref{eq:cor-GM} and Lemma \ref{lem:GM2.8}, we see that $c_\lda'=\frac{f_\lda}{c(n)}$ and proof is finished.
\end{proof}
\subsection{Two Pohozaev type identities}
For $r>0$, define in Euclidean space
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Poho-a}\begin{split}
\mathcal{P}(r,u):=\int_{\pa B_r}&\frac{n-4}{2}\Big(\Delta u \frac{\pa u}{\pa \nu}-u\frac{\pa }{\pa \nu}(\Delta u)\Big) -\frac{r}{2}|\Delta u|^2
\\&-x^k \pa_k u \frac{\pa }{\pa \nu}(\Delta u)+\Delta u \frac{\pa }{\pa \nu}(x^k\pa_ku)\,\ud S,
\end{split}
\ee
where $\nu=\frac{x}{r}$ is the outward normal to $\pa B_r$.
\begin{prop}\label{prop:4-pohozaev} Let $0<u\in C^4(\bar B_r)$ satisfy
\[
\Delta^2 u+E(u)=Ku^{p}\quad \mbox{in }B_r,
\]
where $E:C^4(\bar B_r)\to C^0(\bar B_{r})$ is an operator, $p>0, r>0$ and $K\in C^1(\bar B_{r})$. Then
\begin{align*}
\mathcal{P}(r,u)=& \int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) E(u)\,\ud x+\mathcal{N}(r,u),
\end{align*}
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:right-pohozaev}
\begin{split}
\mathcal{N}(r,u):=& (\frac{n}{p+1}-\frac{n-4}{2}) \int_{B_r} Ku^{p+1}\,\ud x +\frac{1}{p+1}\int_{B_r} x^k\pa_k K u^{p+1}\,\ud x \\& -\frac{r}{p+1}\int_{\pa B_{r}} K u^{p+1}\,\ud S.
\end{split}
\ee
\end{prop}
\begin{proof} A similar Pohozaev identity without $E(u)$ was derived in \cite{DMO}.
We present the proof for completeness.
For any $u\in C^4(\bar B_r)$, by Green's second identity we have
\[
\int_{B_r} u \Delta^2 u\,\ud x= \int_{B_r} (\Delta u)^2\,\ud x+ \int_{\pa B_r} u\frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa u}{\pa \nu}\Delta u\,\ud S
\]
and
\[
\int_{B_r} x^k\pa_k u \Delta^2 u\,\ud x =\int_{B_r} \Delta(x^k\pa_k u) \Delta u\,\ud x + \int_{\pa B_r} x^k\pa_k u \frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa }{\pa \nu}(x^k\pa_ku)\Delta u\,\ud S.
\]
Using Green's first identity, we have
\begin{align*}
\int_{B_r} \Delta(x^k\pa_k u) \Delta u\,\ud x &=2\int_{B_r} (\Delta u)^2\,\ud x+\frac12\int_{B_r} x^k\pa_k (\Delta u)^2\,\ud x \\&
=\frac{4-n}{2}\int_{B_r} (\Delta u)^2\,\ud x+ \int_{\pa B_r} \frac{r}{2}(\Delta u)^2\,\ud S.
\end{align*}
Therefore, we obtain
\begin{align*}
&\frac{n-4}{2} \int_{B_r} u \Delta^2 u\,\ud x+\int_{B_r} x^k\pa_k u \Delta^2 u\,\ud x\\
&=\frac{n-4}{2} \int_{\pa B_r} u\frac{\pa }{\pa \nu}( \Delta u)- \frac{\pa u}{\pa \nu}\Delta u\,\ud S
\\&\quad +
\int_{\pa B_r} \frac{r}{2}|\Delta u|^2 +x^k \pa_k u \frac{\pa }{\pa \nu}(\Delta u)- \frac{\pa }{\pa \nu}(x^k\pa_ku)\Delta u\,\ud S.
\end{align*}
By the equation of $u$ we get
\[
\mathcal{P}(r,u)=\int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) E(u)\,\ud x-\int_{B_r} (x^k\pa_k u +\frac{n-4}{2} u) Ku^p\,\ud x.
\]
Since
\begin{align*}
\int_{B_r} x^k\pa_k u Ku^p\,\ud x&=\frac{1}{p+1}\int_{B_r} K x^k\pa_k u^{p+1}\,\ud x\\
&=-\frac{n}{p+1}\int_{B_r} Ku^{p+1}\, \ud x-\frac{1}{p+1}\int_{B_r}x^k \pa_k K u^{p+1}\\& \quad +\frac{r}{p+1}\int_{B_r} K u^{p+1}\,\ud S,
\end{align*}
we complete the proof.
\end{proof}
\begin{lem}\label{lem:test-poho} For $G(x)=|x|^{4-n}+A+O^{(4)}(|x|)$, where $A$ is constant. Then
\[
\lim_{r\to 0}\mathcal{P}(r,G)=-(n-4)^2(n-2)A|\mathbb{S}^{n-1}|.
\]
\end{lem}
The following proposition is a special case of Proposition 2.15 of \cite{JLX3}.
\begin{prop} \label{prop:pohozaev} For $R>0$, let $0\le u\in C^1(\bar B_R)$ be a solution of
\[
u(x)= \int_{B_R} \frac{K(y)u(y)^{p}}{|x-y|^{n-4}}\,\ud y+ h_R(x),
\]
where $p>0$, and $h_R(x)\in C^1(B_R)$, $\nabla h_R\in L^1(B_R)$.
Then
\begin{align*}
&\left(\frac{n-4}{2}-\frac{n}{p+1}\right) \int_{B_R} K(x)u(x)^{p+1}\,\ud x-\frac{1}{p+1} \int_{B_R} x\nabla K(x) u(x)^{p+1}\,\ud x \\ &
=\frac{n-4}{2} \int_{B_R} K(x) u(x)^p h_R(x)\,\ud x+ \int_{B_R} x\nabla h_R(x) K(x)u(x)^p \,\ud x
\\& \quad - \frac{R}{p+1} \int_{\pa B_R} K(x) u(x)^{p+1}\,\ud S.
\end{align*}
\end{prop}
\section{Blow up analysis for integral equations}
\label{s:blowup}
In the section, the idea of dealing with integral equation is inspired by \cite{JLX3}, but we have to consider general integral kernels and remainder terms. We will use $A_1, A_2,A_3 $ to denote positive constants, and $\{\tau_i\}_{i=1}^\infty$ to denote a sequence of nonnegative constants satisfying $\lim_{i\to \infty}\tau_i=0$. Set
\begin{equation}} \newcommand{\ee}{\end{equation} \label{p}
p_i=\frac{n+4}{n-4}-\tau_i.
\ee
Let $\{G_i(x,y)\}_{i=1}^\infty$ be a sequence of functions on $B_3\times B_3$ satisfying
\begin{equation}} \newcommand{\ee}{\end{equation} \label{G}
\begin{aligned}
&G_i(x,y)=G_i(y,x),\qquad G_i(x,y)\ge A_1^{-1}|x-y|^{4-n},\\[2mm]
& |\nabla^l_x G_i(x,y)|\le A_1|x-y|^{4-n-l}, \quad l=0,1,\dots, 5 \\[2mm]
&G_i(x,y)=c_{n}\frac{1+O^{(4)}(|x|^2)+O^{(4)}(|y|^2)}{|x-y|^{n-4}}+\bar a_i+O^{(4)}(\frac{1}{|x-y|^{n-6}})
\end{aligned}
\ee
for all $x,y\in \bar B_3$, where $
c_n=\frac{n(n+2)}{2|\mathbb{S}^{n-1}|}
$ is the constant given towards the end of the introduction, $f=O^{(4)}(r^m)$ denotes any quantity satisfying $|\nabla^j f(r)|\le A_1 r^{m-j}$ for all integers $1\le j\le 4$, and $\bar a_i$ is a constant and $\bar a_i=0$ if $n\ge 6$.
Let $\{K_i\}_{i=1}^\infty \in C^\infty(\bar B_3)$ satisfy
\begin{equation}} \newcommand{\ee}{\end{equation} \label{K}
\lim_{i\to \infty}K_i(0)=1, \quad K_i\ge A_2^{-1} , \quad
\| K_i\|_{C^5(B_3)} \le A_2.
\ee
Let $\{ h_i\}_{i=1}^\infty$ be a sequence of nonnegative functions in $C^\infty( B_3)$ satisfying
\begin{equation}} \newcommand{\ee}{\end{equation} \label{H}
\begin{aligned}
\max_{\bar B_{r}(x)} h_i &\le A_2 \min_{\bar B_r(x)} h_i \\
\sum_{j=1}^5r^j|\nabla^j h_i(x)| &\le A_2 \|h_i\|_{L^\infty(B_r(x))}
\end{aligned}
\ee
for all $x\in B_{2}$ and $0<r<1/2$.
Given $p_i, G_i, K_i, $ and $ h_i$ satisfying \eqref{p}-\eqref{H}, let $0\le u_i\in L^{\frac{2n}{n-4}}(B_3)$
be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:s1}
u_i(x)=\int_{B_3 }G_i(x,y) K_i(y) u_i^{p_i}(y)\,\ud y +h_i(x) \quad \mbox{in } B_3.
\ee
It follows from \cite{Li04} and Proposition \ref{prop:local estimates} that $u_i\in C^{4}(B_3)$. In the following we will always assume $u_i\in C^{4}(B_3)$.
We say that $\{u_i\}$ blows up if $\|u_i\|_{L^\infty(B_3)}\to \infty$ as $i\to \infty$.
\begin{defn}\label{def4.1}
We say a point $\bar x\in B_3$ is an isolated blow up point of $\{u_i\}$ if there exist
$0<\overline r<dist(\overline x,\pa B_3)$, $\overline C>0$, and a sequence $x_i$ tending to $\overline x$, such that,
$x_i$ is a local maximum of $u_i$, $u_i(x_i)\to \infty$ and
\[
u_i(x)\leq \overline C |x-x_i|^{-4/(p_i-1)} \quad \mbox{for all } x\in B_{\overline r}(x_i).
\]
\end{defn}
Let $x_i\to \overline x$ be an isolated blow up of $u_i$. Define
\begin{equation}} \newcommand{\ee}{\end{equation}\label{def:average}
\overline u_i(r)=\frac{1}{|\pa B_r|} \int_{\pa B_r(x_i)}u_i\,\ud S,\quad r>0,
\ee
and
\[
\overline w_i(r)=r^{4/(p_i-1)}\overline u_i(r), \quad r>0.
\]
\begin{defn}\label{def4.2}
We say $x_i \to \overline x\in B_3$ is an isolated simple blow up point, if $x_i \to \overline x$ is an isolated blow up point, such that, for some
$\rho>0$ (independent of $i$) $\overline w_i$ has precisely one critical point in $(0,\rho)$ for large $i$.
\end{defn}
\begin{lem}\label{lem:harnack} Given $p_i, G_i, K_i$ and $h_i$ satisfying \eqref{p}-\eqref{H}, let $0\le u_i\in C^{4}(B_3)$ be a solution of \eqref{eq:s1}. Suppose that $0$ is an isolated blow up point of $\{u_i\}$ with $\bar r=2$, i.e., for some positive constant $A_3$ independent of $i$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{4.7}
u_i(x)\leq A_3|x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_{2}.
\ee
Then for any $0<r<1/3 $ we have
\[
\sup_{B_{2r}\setminus B_{r/2}} u_i\leq C \inf_{B_{2r}\setminus B_{r/2}} u_i,
\]
where $C$ is a positive constant depending only on $n, A_1, A_2, A_3$.
\end{lem}
\begin{proof} For every $0<r<1/3$, set
\[
w_i(x)=r^{4/(p_i-1)}u_i(rx).
\]
By the equation of $u_i$, we have
\[
w_i(x)= \int_{B_{3/r}} G_{i,r}(x,y)K_i(ry)w_i(y)^{p_i}\,\ud y + \tilde h_i(x)\quad x\in B_{3/r},
\]
where
\[
G_{i,r}(x,y)=r^{n-4}G_i(rx,ry) \quad \mbox{for }r>0
\]
and $\tilde h_i(x):=r^{4/(p_i-1)} h_i(rx)$.
Since $0$ is an isolated blow up point of $u_i$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:out}
w_i(x)\leq A_3 |x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_3.
\ee
Set $\Omega} \newcommand{\pa}{\partial_1=B_{5/2}\setminus B_{1/4}$, $\Omega} \newcommand{\pa}{\partial_2=B_{2}\setminus B_{1/2}$ and $
V_i(y)=K_i(ry)w_i(y)^{p_i-1}$. Thus $w_i$ satisfies the linear equation
\[
w_i(x)= \int_{\Omega} \newcommand{\pa}{\partial_1} G_{i,r}(x,y)V_i(y)w_i(y)\,\ud y + \bar h_i(x)\quad \mbox{for } x\in B_{5/2}\setminus B_{1/4},
\]
where
\[
\bar h_i(x)=\tilde h_i(x)+\int_{B_{3/r}\setminus \Omega} \newcommand{\pa}{\partial_1} G_{i,r}(x,y)K_i(ry)w_i(y)^{p_i}\,\ud y.
\]
By \eqref{eq:out} and \eqref{K}, $\|V_i\|_{L^\infty(\Omega} \newcommand{\pa}{\partial_1)} \le C(n,A_1,A_2,A_3)<\infty$.
Since $K_i$ and $w_i$ are nonnegative, by \eqref{G} on $G_i$ and \eqref{H} on $h_i$ we have $\max_{\bar \Omega} \newcommand{\pa}{\partial_2}\bar h_i \le C(n,A_1,A_2) \min_{\bar \Omega} \newcommand{\pa}{\partial_2} \bar h$. Applying Proposition \ref{prop:har} to $w_i$ gives
\[
\max_{\bar \Omega} \newcommand{\pa}{\partial_2} w_i \le C \min_{\bar \Omega} \newcommand{\pa}{\partial_2} w_i,
\]
where $C>0$ depends only on $n, A_1,A_2 $ and $A_3$. Rescaling back to $u_i$, the lemma follows.
\end{proof}
\begin{prop}\label{prop:blow up a bubble}
Suppose that $0\le u_i\in C^{4}(B_3)$ is a solution of \eqref{eq:s1} and all assumptions in Lemma \ref{lem:harnack} hold.
Let $R_i\rightarrow \infty$ with $R_i^{\tau_i}=1+o(1)$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}_i\rightarrow 0^+$, where $o(1)$ denotes some quantity tending to $0$ as $i\to \infty$. Then we have,
after passing to a subsequence (still denoted as $\{u_i\}$, $\tau_i$ and etc . . .),
\[
\|m_i^{-1}u_i(m_i^{-(p_i-1)/4} \cdot)-(1+ |\cdot|^2)^{(4-n)/2}\|_{C^3(B_{2R_i}(0))}\leq \varepsilon} \newcommand{\ud}{\mathrm{d}_i,
\]
\[
r_i:=R_im_i^{-(p_i-1)/4}\rightarrow 0\quad \mbox{as}\quad i\rightarrow \infty,
\]
where $m_i=u_i(0)$.
\end{prop}
\begin{proof} Let
\[
\varphi_i(x)=m_i^{-1} u_i(m_i^{-(p_i-1)/4} x) \quad \mbox{for }|x|<3 m_i^{(p_i-1)/4}.
\]
By the equation of $u_i$, we have,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scal}
\varphi_i(x)= \int_{B_{3 m_i^{(p_i-1)/4}}} \tilde G_i (x,y) \tilde K_i(y)\varphi_i(y)^{p_i}\,\ud y+\tilde h_i(x),
\ee
where $\tilde G_i (x,y)=G_{i,m_i^{-\frac{p_i-1}{4}} }(x,y)$, $\tilde K_i(y)=K_i(m_i^{-\frac{p_i-1}{4}} y )$ and $\tilde h_i(x)= m_i^{-1} h_i(m_i^{-\frac{p_i-1}{4}} x)$.
First of all, $\max_{\pa B_1 } h_i\le \max_{\pa B_1 } u_i\le A_3$, by \eqref{H} we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq: h goes}
\tilde h_i \to 0\quad \mbox{in } C^5_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n) \quad \mbox{as } i\to \infty.
\ee
Secondly, since $0$ is an isolated blow up point of $u_i$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scalbound}
\varphi_i(0)=1, \quad \nabla \varphi_i(0)=0, \quad 0<\varphi_i(x)\leq A_3 |x|^{-4/(p_i-1)}.
\ee
For any $R>0$, we claim that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:scalbound2}
\|\varphi_i\|_{C^{4}(B_R)}\leq C(R)
\ee
for sufficiently large $i$.
Indeed, by Proposition \ref{prop:local estimates} and \eqref{eq:scalbound}, it suffices to prove that $\varphi_i \le C$ in $B_1$.
If $\varphi_i(\bar x_i)=\sup_{B_{1}} \varphi_i \to \infty $, set
\[
\tilde \varphi_i(z)=\varphi_i(\bar x_i)^{-1}\varphi_i(\varphi_i(\bar x_i)^{-(p_i-1)/4}z+\bar x_i)\leq 1\quad \mbox{for }|z| \le \frac12 \varphi_i(\bar x_i)^{(p_i-1)/4}.
\]
By \eqref{eq:scalbound},
\[
\tilde \varphi_i(z_i)= \varphi_i(\bar x_i)^{-1} \varphi_i(0)\to 0
\] for $z_i=-\varphi_i(\bar x_i)^{(p_i-1)/4}\bar x_i$. Since $\varphi_i(\bar x_i) \leq A_3 |\bar x_i|^{-4/(p_i-1)}$, we have $|z_i|\leq A_3^{4/(p_1-1)}$. Hence, we can find $t>0$ independent of $i$ such that such that $z_i\in B_t$.
Applying Proposition \ref{prop:har} to $\tilde \varphi_i$ in $B_{2t}$ (since $\tilde\varphi_i$ satisfies a similar equation to \eqref{eq:scal}), we have
\[
1=\tilde \varphi_i(0)\le C \tilde \varphi_i(z_i)\to 0,
\]
which is impossible. Hence, $\varphi_i \le C$ in $B_1$.
It follows from \eqref{eq:scalbound2} that there exists a function $\varphi\in C^4(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ such that, after passing subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:phigeos}
\varphi_i(x)\to \varphi \quad \mbox{in }C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n) \quad \mbox{as }i\to \infty.
\ee
Thirdly, for every $R>0$, let
\[
g_i(R,x):= \int_{B_{3 m_i^{(p_i-1)/4}}\setminus B_{R}} \tilde G_i (x,y) \tilde K_i(y)\varphi_i(y)^{p_i}\,\ud y.
\]
Since $K_i$ and $\varphi_i$ are nonnegative, a simple computation using \eqref{G} gives that, for any $x\in B_{R-1}$,
\[
|\nabla^k g_i(R,x)| \le C g_i(R,x), \quad k=1,\dots,5.
\]
Note that $g_{i}(R,x)\le \varphi_i(x) \le C(R)$. It follows that, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:ggoes}
g_i(R,x) \to g(R,x)\ge 0 \quad \mbox{in } C^{4}(B_{R-1}) \quad \mbox{as }i\to \infty.
\ee
By \eqref{G} and \eqref{K}, we have
\[
\tilde G_i(x,y)\to c_n \frac{1}{|x-y|^{n-4}}\quad \forall~ x\neq y
\]
and $\tilde K_i(y)\to K_i(0)=1.$
Combining \eqref{eq: h goes}, \eqref{eq:phigeos} and \eqref{eq:ggoes} together, by \eqref{eq:scal} we have that for any fixed $R>0$ and $x\in B_{R-1}$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:limit1}
g(R,x)=\varphi(x)-c_{n}\int_{B_R}\frac{ \varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}\,\ud y.
\ee
By \eqref{eq:limit1}$, g(R,x)$ is non-increasing in $R$. For any fixed $x$ and $|y|\ge R>>|x|$, by \eqref{G} we have
\begin{align*}
\frac{G_{i,m_i^{-(p_i-1)/4} } (x,y)}{G_{i,m_i^{-(p_i-1)/4} } (0,y)}=\frac{G_{i,|y|m_i^{-(p_i-1)/4} } (\frac{x}{|y|},\frac{y}{|y|})}{G_{i,|y| m_i^{-(p_i-1)/4} } (0,\frac{y}{|y|})}=1+O(\frac{|x|}{|y|}).
\end{align*}
Hence, $g_i(R,x)=(1+O(\frac{|x|}{R}))g_i(R,0)$, which implies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:remainder1}
\lim_{R\to \infty}g(R,x)= \lim_{R\to \infty} g(R,0):=c_0\ge 0.
\ee
Sending $R$ to $\infty$ in \eqref{eq:limit1}, it follows from Lebesgue's monotone convergence theorem that
\[
\varphi(x)=c_{n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{\varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}} \ud y+c_0 \quad x\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n.
\]
We claim that $c_0=0$. If not,
\[
\varphi(x)-c_0= c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{\varphi(y)^{\frac{n+4}{n-4}}}{|y|^{n-4}}\,\ud y >0,
\]
which implies that
\[
1=\varphi(0)\geq c_{n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{c_0^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}=\infty.
\]
This is impossible.
The use of monotonicity in the above argument is taken from \cite{JLX3}.
It follows from the classification theorem in \cite{CLO} or \cite{Li04} that
\[
\varphi(x)=\left(1+|x|^2\right)^{-\frac{n-4}{2}},
\]
where we have used that $\varphi(0)=1$ and $\nabla \varphi(0)=0$.
The proposition follows immediately.
\end{proof}
Since passing to subsequences does not affect our proofs, in the rest of the paper we will always choose $R_i\to\infty$ with $R_i^{\tau_i}=1+o(1) $ first, and then $\varepsilon} \newcommand{\ud}{\mathrm{d}_i\to 0^+$ as small as we wish (depending on $R_i$) and then choose our subsequence $\{u_{j_i}\}$ to work with. Since $i\le j_i$ and $\lim_{i\to \infty}\tau_i=0$, one can ensure that $R_i^{\tau_{j_i}}=1+o(1) $ as $i\to \infty$. In the sequel, we will still denote the subsequences as $u_i, \tau_i$ and etc.
\begin{rem}\label{rem:blow} By checking the proof of Proposition \ref{prop:blow up a bubble}, together with the fact $\nabla^2 (1+|x|^2)^{-\frac{n-4}{2}}$ is negatively definite near zero and the $C^2$ convergence in a fixed neighborhood of zero, the following statement holds. Let $0\le u_i\in C^{4}(B_3)$ be a solution of \eqref{eq:s1} and satisfy \eqref{4.7}. Suppose that $u_i(0)\to \infty$ as $i\to \infty$, $\nabla u_i(0)=0$ and $\max_{B_3}u_i\le b u_i(0)$ for some constant $b\ge 1$ independent of $i$. Then, after passing to a subsequence, $0$ must be a local maximum point of $u_i$ for $i$ large. Namely, $0$ is an isolated blow up point of $u_i$ after passing to a subsequence.
\end{rem}
\begin{prop}\label{prop:lower bounded by bubble} Under the hypotheses of Proposition \ref{prop:blow up a bubble}, there exists a constant $C>0$, depending only on $n, A_1,A_2$ and $A_3$, such that,
\[
u_i(x)\geq C^{-1}m_i(1+m_i^{(p_i-1)/2}|x|^2)^{(4-n)/2}, \quad |x|\leq 1.
\]
In particular, for any $e\in \mathbb{R}^n$, $|e|=1$, we have
\[
u_i(e)\geq C^{-1}m_i^{-1+((n-4)/4)\tau_i}.
\]
\end{prop}
\begin{proof}
By change of variables and using Proposition \ref{prop:blow up a bubble}, we have for $r_i\le |x|\le 1$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:lower bound bubble}
\begin{split}
u_i(x)&\ge C^{-1}\int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y\\&
\ge C^{-1} m_i\int_{|z|\le R_i} \frac{\big(m_i^{-1}u_{i}(m_i^{-(p_i-1)/4}z)\big)^{p_i}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z
\\&
\ge C^{-1} m_i \int_{|z|\le R_i} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z\\&
\ge \frac 12 C^{-1} m_i \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(z)^{\frac{n+4}{n-4}}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z\\&
=\frac 12 C^{-1}m_i U_1(m_i^{(p_i-1)/4}x).
\end{split}
\ee
Recall that
\[
U_\lda(z)=\left(\frac{\lda}{1+\lda^2|z|^2}\right)^{(n-4)/2}, \quad \lda>0.
\]
The proposition follows immediately.
\end{proof}
\begin{lem} \label{lem:upbound1} Suppose the hypotheses of Proposition \ref{prop:blow up a bubble} and in addition that $ 0$ is also an isolated simple blow up point with the constant $\rho>0$. Then there exist $\delta_i>0$, $\delta_i=O(R_i^{-4})$,
such that
\[
u_i(x)\leq C u_i(0)^{-\lda_i}|x|^{4-n+\delta_i},\quad \mbox{for all }r_i\leq |x|\leq 1,
\]
where $\lda_i=(n-4-\delta_i)(p_i-1)/4-1$ and $C>0$ depends only on $n, A_1,A_3$ and $\rho$.
\end{lem}
\begin{proof} We divide the proof into several steps.
Step 1. From Proposition \ref{prop:blow up a bubble}, we see that
\begin{align}
u_i(x)&\le C m_i \left(\frac{1}{1+|m_i^{(p_i-1)/4}x|^2}\right)^{\frac{n-4}{2}} \nonumber \\ &
\le C m_iR_i^{4-n} \quad \mbox{for all } |x|=r_i=R_i m_i^{-(p_i-1)/4}.
\label{4.8}
\end{align}
Let $\overline u_i(r)$ be the average of $u_i$ over the sphere of radius $r$ centered at $0$.
It follows from the assumption of isolated simple
blow up points and Proposition \ref{prop:blow up a bubble} that
\begin{equation}} \newcommand{\ee}{\end{equation}\label{4.9}
r^{4/(p_i-1)}\overline u_i(r) \quad \mbox{is strictly decreasing for $r_i<r<\rho$}.
\ee
By Lemma \ref{lem:harnack}, \eqref{4.9} and \eqref{4.8}, we have, for all $r_i<|x|<\rho$,
\[
\begin{split}
|x|^{4/(p_i-1)}u_i(x)&\leq C|x|^{4/(p_i-1)}\overline u_i(|x|)\\&
\leq C r_i^{4/(p_i-1)}\overline u_i(r_i)\\&
\leq CR_i^{\frac{4-n}{2}},
\end{split}
\]
where we used $R_i^{\tau_i}=1+o(1)$. Thus,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:coeff}
u_i(x)^{p_i-1}\leq C R_i^{-4}|x|^{-4} \quad \mbox{for all } r_i\leq |x|\le \rho.
\ee
\medskip
Step 2. Let
\[
\mathcal{L}_i\phi(y):= \int_{B_3} G_i(x,z)K_i(z) u_i(z)^{p_i-1}\phi(z)\,\ud z.
\]
Thus
\[
u_i=\mathcal{L}_i u_i+h_i.
\]
Note that for $4<\mu<n$ and $0<|x|<2$,
\begin{align*}
\int_{B_3} G_i(x,y)|y|^{-\mu}\,\ud y&\le A_1\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{1}{|x-y|^{n-4}|y|^{\mu}}\,\ud y
\\&=A_1 |x|^{4-n} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}
\frac{1}{||x|^{-1}x-|x|^{-1}y|^{n-4}|y|^{\mu}}\,\ud y \\&
= A_1|x|^{-\mu+4} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{1}{||x|^{-1}x-z|^{n-4}|z|^{\mu}}\,\ud z \\&
\le C\Big( \frac{1}{n-\mu}+\frac{1}{\mu- 4} \Big)|x|^{-\mu+4},
\end{align*}
where we did the change of variables $y=|x|z$.
By \eqref{eq:coeff}, one can properly choose $0<\delta_i=O(R_i^{-4})$ such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a1}
\int_{r_i<|y|<\rho} G_i(x,y)K_i(y)u_i(y)^{p_i-1} |y|^{-\delta_i} \,\ud y\leq \frac{1}{4} |x|^{-\delta_i},
\ee
and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a2}
\int_{r_i<|y|<\rho} G_i(x,y)K_i(y)u_i(y)^{p_i-1} |y|^{4-n+\delta_i} \,\ud y\leq \frac{1}{4} |x|^{4-n+\delta_i},
\ee
for all $r_i< |x|<\rho$.
Set $M_i:=4^n A_1^2\max_{\pa B_\rho} u_i+2\max_{\bar B_\rho}h_i$,
\[
f_i(x):=M_i \rho^{\delta_i} |x|^{-\delta_i}+A m_i^{-\lda_i} |x|^{4-n+\delta_i},
\] and
\[
\phi_i(x)=\begin{cases}
f_i(x), & \quad r_i< |x|< \rho,\\
u_i(x),&\quad \mbox{otherwise} ,
\end{cases}
\]
where $A>1$ will be chosen later.
By \eqref{eq:a1} and \eqref{eq:a2}, we have for $r_i<|x|< \rho$.
\begin{align*}
\mathcal{L}_i \phi_i (x)&= \int_{B_3} G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y\\&
=\left(\int_{|y|\le r_i} + \int_{r_i<|y|<\rho} + \int_{\rho\le |y|<3} \right)G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y
\\&
\leq A_1 \int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y+ \frac{f_i}{4}+ \frac{M_i}{2^{n-4}},
\end{align*}
where we used, in view of \eqref{G},
\[
\begin{split}
\int_{\rho\le |y|<3}& G_i(x,y)K_i(y)u_i(y)^{p_i-1} \phi_i(y)\,\ud y\\&
=\int_{\rho\le |y|<3} G_i(x,y)K_i(y)u_i(y)^{p_i} \,\ud y\\ &\le A_1^2 2^{n+4}\int_{\rho\le |y|<3} G_i(\frac{\rho x}{|x|},y)K_i(y)u_i(y)^{p_i}\,\ud y \\ &\le A_1^2 2^{n+4} \max_{\pa B_{\rho}} u_i \le 2^{4-n} M_i.
\end{split}
\]
By change of variables and using Proposition \ref{prop:blow up a bubble}, we have, similar to \eqref{eq:lower bound bubble},
\begin{align*}
\int_{|y|\le r_i} \frac{u_{i}(y)^{p_i}}{|x-y|^{n-4}} \,\ud y&
=m_i\int_{|z|\le R_i} \frac{\big(m_i^{-1}u_{i}(m_i^{-(p_i-1)/4}z)\big)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z
\\&
\le 2m_i \int_{|z|\le R_i} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-4}} \,\ud z\\&
\le C m_i \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(z)^{\frac{n+4}{n-4}}}{|m_i^{(p_i-1)/4}x-z|^{n-4}} \,\ud z\\&
=Cm_i U_1(m_i^{(p_i-1)/4}x),
\end{align*}
where we used $R^{(n-4)\tau_i}=1+o(1)$.
Since $|x|>r_i$, we see
\begin{align*}
m_i U_1(m_i^{(p_i-1)/4} x) &\le Cm_i^{1-(p_i-1)(n-4)/4}|x|^{4-n} \\&
\le Cm_i^{-\lda_i}|x|^{4-n+\delta_i}.
\end{align*}
Therefore, we conclude that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:intineq}
\mathcal{L}_i \phi_i(x)+h_i(x) \leq \phi_i(x)\quad \mbox{for all } r_i\le |x| \leq \rho,
\ee
provided $A$ is large independent of $i$.
\medskip
Step 3. Note that
\[
\liminf_{|x|\to r_i ^+}f_i(x) >A m_i^{-\lda_i}R_i^{4-n+\delta_i} m_i^{(p_i-1)(n-4-\delta_i)/4}=A R_i^{4-n+\delta_i} m_i.
\] In view of \eqref{4.8}, we may choose $A$ large such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:strict}
\liminf_{|x|\to r_i ^+}(f_i(x)-u_i(x))>0 .
\ee We claim that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:upper bounded}
u_i(x)\le \phi_i(x).
\ee Indeed, if not, let
\[
1<t_i:=\inf\{t>1, t\phi_i(x)\ge u_i(x) \quad \mbox{for all } r_i\le |x|\le \rho \}<\infty.
\] By \eqref{eq:strict}, $t_i>1$, together with $f_i>u_i$ on $\pa B_{\rho}$, we can find a sufficient small open neighborhood of $\pa B_{r_i}\cup \pa B_\rho$ in which $t_i \phi_i> u_i $. By the continuity there exists $y_i\in B_\rho \setminus \bar B_{r_i}$ such that
\[
0=t_i\phi_i(y_i)-u_i(y_i)\ge \mathcal{L}_i(t_i\phi_i-u_i)(y_i)+(t_i-1)h_i(y_i)>0.
\]
We derived a contradiction and thus \eqref{eq:upper bounded} is valid.
\medskip
Step 4. By \eqref{H}, we have $\max_{\bar B_{\rho}} h_i \le A_2\max_{\pa \bar B_{\rho}}h_i \le A_2 \max_{\pa \bar B_{\rho}}u_i$.
Hence,
\[
M_i\le C\max_{\pa B_{\rho}} u_i .
\]
For $r_i<\theta<\rho$,
\[
\begin{split}
\rho^{4/(p_i-1)}M_i&\leq C \rho^{4/(p_i-1)}\overline u_i(\rho)\\
&\leq C\theta^{4/(p_i-1)}\overline u_i(\theta)\\
&\leq C\theta^{4/(p_i-1)}\{M_i\rho^{\delta_i}\theta^{-\delta_i}+Am_i^{-\lda_i}\theta^{4-n+\delta_i}\}.
\end{split}
\]
Choose $\theta=\theta(n,\rho,A_1,A_2, A_3)$ sufficiently small so that
\[
C\theta^{4/(p_i-1)}\rho^{\delta_i}\theta^{-\delta_i}\leq \frac12 \rho^{4/(p_i-1)}.
\]
Hence, we have
\[
M_i\le C m_i^{-\lda_i}.
\]
It follows from \eqref{eq:upper bounded} that
\[
u_i(x)\le \phi_i(x) \le Cm_i^{-\lda_i}|x|^{-\delta_i}+A m_i^{-\lda_i} |x|^{4-n+\delta_i} \le Cm_i^{-\lda_i} |x|^{4-n+\delta_i}.
\]
We complete the proof of the lemma.
\end{proof}
\begin{lem} \label{lem:aux1} Under the assumptions in Lemma \ref{lem:upbound1}, for $k<n$ we have
\[
I_k[u_i](x) \le C\begin{cases}
m_i^{\frac{n-2k+4}{n-4}+o(1)}, &\quad \mbox{if } |x|<r_i, \\
m_i^{-1+o(1)}|x|^{k-n}, &\quad \mbox{if }r_i \le |x|<1,
\end{cases}
\]
where
\[
I_k[u_i](x) =\int_{B_1}|x-y|^{k-n} u_i(y)^{p_i}\,\ud y.
\]
\end{lem}
\begin{proof} Making use of Proposition \ref{prop:blow up a bubble} and Lemma \ref{lem:upbound1}, we have
\begin{align*}
I_k[u_i](x) & = \int_{B_{r_i}}\frac{u_i(y)^{p_i}}{ |x-y|^{n-k}} \,\ud y+ \int_{B_1\setminus B_{ r_i}} \frac{u_i(y)^{p_i}}{ |x-y|^{n-k}}\,\ud y\\&
\le Cm_i^{\frac{n-2k+4}{n-4}+o(1)} \int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z\\& \quad +Cm_i^{-\frac{n+4}{n-4}+o(1)}\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y.
\end{align*}
If $|x|<r_i$, we see that
\[
\int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z \le C
\]
by Lemma \ref{lem:aux2},
and
\begin{align*}
\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y \le \int_{B_1\setminus B_{ r_i}}\frac{1}{|y|^{2n-k+4}}\,\ud y
\le C(n) R_i^{-(n-k+4)}m_i^{\frac{2(n-k+4)}{n-4}+o(1)}.
\end{align*}
Hence, $I_k[u_i](x) \le Cm_i^{\frac{n-2k+4}{n-4}+o(1)}$.
If $r_i<|x|<1 $, then $|m_i^{(p_i-1)/4} x|\ge 1$. It follows from Lemma \ref{lem:aux2} that
\begin{align*}
\int_{B_{R_i}} \frac{U_1(z)^{p_i}}{|m_i^{(p_i-1)/4} x-z|^{n-k}}\,\ud z & \le \int_{B_{R_i}} \frac{1}{|m_i^{(p_i-1)/4} x-z|^{n-k}(1+|z|)^{n+4+o(1)}}\,\ud z \\&
\le C|m_i^{(p_i-1)/4} x|^{k-n}.
\end{align*}
By change of variables $z=m_i^{(p_i-1)/4} y$,
\begin{align*}
\int_{B_1\setminus B_{ r_i}}\frac{1}{|x-y|^{n-k}|y|^{n+4}}\,\ud y&=m_i^{\frac{2(n-k+4)}{n-4}+o(1)} \int_{B_{m_i^{(p_i-1)/4}}\setminus B_{R_i}} \frac{1}{|m_i^{(p_i-1)/4}x-z|^{n-k}|z|^{n+4}}\,\ud z\\&
\le Cm_i^{\frac{2(n-k+4)}{n-4}+o(1)} |m_i^{(p_i-1)/4} x|^{k-n}.
\end{align*}
Thus
\[
I_k[u_i](x) \le Cm_i^{\frac{n-2k+4}{n-4}+o(1)} |m_i^{(p_i-1)/4} x|^{k-n} =m_i^{-1+o(1)} |x|^{k-n}.
\]
Therefore, the proof of the lemma is completed.
\end{proof}
\begin{lem}\label{lem:error}
Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
\tau_i=O(u_{i}(0)^{-2/(n-4)+o(1)}).
\]
Consequently, $m_i^{\tau_i}=1+o(1)$.
\end{lem}
\begin{proof} For $x\in B_1$, we write equation \eqref{eq:s1} as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:flat}
u_i(x)= c_{n}\int_{B_1} \frac{K_i(y)u_i(y)^{p_i}}{|x-y|^{n-4}}\,\ud y +b_i(x),
\ee
where $b_i(x):=Q'_i(x)+Q''_i(x) +h_i(x)$,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{F}
Q_i' (x):=\int_{B_1}(G_i(x,y)-c_{n}|x-y|^{4-n}) K_i(y) u_i(y)^{p_i}\,\ud y
\ee
and
\[
Q_i''(x):= \int_{B_3\setminus B_1} G_i(x,y)K_i(y) u_i(y)^{p_i}\,\ud y.
\]
Notice that
\[
|G_i(x,y)-c_{n}|x-y|^{4-n}| \le \frac{C|x|^2 }{|x-y|^{n-4}}+|\bar a_i|+C|x-y|^{6-n}.
\]
\[
|\nabla_x(
G_i(x,y)-c_{n}|x-y|^{4-n})| \le \frac{C|x|^2 }{|x-y|^{n-3}}+\frac{C|x| }{|x-y|^{n-4}}+C|x-y|^{5-n}.
\]
Hence,
\begin{align*}
|Q_i'(x)| & \le C(|x|^2 u_i(x)+|a_i| \|u_i^{p_i}\|_{L^1(B_1)}+I_{6} [u_i^{p_i}](x)), \\
|\nabla Q_i'(x)| &\le C ( |x|^2 I_{3}+|x|I_{4} +I_{5}) [u_i^{p_i}](x),
\end{align*}
where $I_k [u_i^{p_i}](x)=\int_{B_1}|x-y|^{k-n} u_i(y)^{p_i}\,\ud y $.
By Lemma \ref{lem:upbound1}, we have $u_i(x)\le C m_i^{-\lda_i}$ for all $x\in B_{3/2}\setminus B_{1/2}$. Hence, $Q_i''(x) +h_i(x)\le u_i(x)\le C m_i^{-1+o(1)}$ for any $x\in \pa B_1$ . It follows from \eqref{H} that
\[
\max_{\bar B_2}h_i(x) \leq C \min_{\pa B_1} h_i(x)\le C m_i^{-1+o(1)}.
\]
and
\[
|\nabla h_i(x)|\le C\max_{\bar B_2}h_i(x) \le C m_i^{-1+o(1)} \quad \mbox{for all }x\in B_1.
\]
Since $u_i$ is nonnegative, by \eqref{G} it is easy to check that
\[
|Q_i''(x)|+|\nabla Q_i''(x)|\le Cm_i^{-1+o(1)} \quad \mbox{for all }x\in B_1.
\]
Applying Proposition \ref{prop:pohozaev} to \eqref{eq:flat}, we have
\begin{align}
\tau_i & \int_{B_1} u_i(x)^{p_i+1}-A_2\int_{B_1} |x| u_i(x)^{p_i+1} \,\ud x \nonumber\\& \leq C\Big(\int_{B_1}(|Q'_i(x)|+|x||\nabla Q_i'(x)) u_i(x)^{p_i}+ m_i^{-1+o(1)}\int_{ B_1} u_i^{p_i}+ \int_{\pa B_1} u_i^{p_i+1}\, \ud s\Big).
\label{eq:a6}
\end{align}
By Proposition \ref{prop:blow up a bubble} and change of variables,
\begin{align*}
\int_{B_1} u_i(x)^{p_i+1} \,\ud x &\ge C^{-1}\int_{B_{r_i}} \frac{m_i^{p_i+1}}{(1+|m_i^{(p_i-1)/4}y|^2)^{(n-4)(p_i+1)/2}}\,\ud y \\&
\ge C^{-1} m_i^{\tau_i(1-n/4)} \int_{R_i} \frac{1}{(1+|z|^2)^{(n-4)(p_i+1)/2}}\,\ud z \\&
\ge C^{-1} m_i^{\tau_i(1-n/4)} ,
\end{align*}
By Proposition \ref{prop:blow up a bubble}, Lemma \ref{lem:upbound1}, we have
\[
\int_{B_{1}} u_i^{p_i} \le C m_i^{-1+o(1)},
\]
\[
\int_{B_{1}} |x|^su_i^{p_i+1} \le C m_i^{-2s/(n-4)+o(1)}, \quad \mbox{for }-n<s<n,
\]
and
\[
\int_{\pa B_1} u_i^{p_i+1}\, \ud s \le Cm_i^{-2n/(n-4) +o(1)}.
\]
It follows from Lemma \ref{lem:aux1} that
\[
\int_{B_{1}} (|Q'_i(x)|+|x||\nabla Q_i'(x)|) u_i(x)^{p_i+1}\,\ud x\le Cm_i^{-2/(n-4)+o(1)}.
\]
Therefore, we complete the proof.
\end{proof}
\begin{lem}\label{lem:aux3} For $-4<s<4$, we have, as $i\to \infty$,
\[
m_i^{1+\frac{2s}{n-4}} \int_{B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y \to \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}|z|^s (1+|z|^2)^{-\frac{n+4}{2}}\,\ud z\]
and
\[
m_i^{1+\frac{2s}{n-4}} \int_{B_1\setminus B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y \to 0.
\]
\end{lem}
\begin{proof} By a change of variables $y=m_i^{-(p_i-1)/4} z$, we have
\[
\int_{B_{r_i}}|y|^s u_i(y)^{p_i}\,\ud y =m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i} \int_{B_{R_i}}|z|^s (m_i^{-1}u_i(m_i^{-(p_i-1)/4} z))^{p_i}\,\ud z
\]
By Lemma \ref{lem:error}, $m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i}=(1+o(1)) m_i^{-1-\frac{2s}{n-4}}$. In view of Proposition \ref{prop:blow up a bubble} and $-4<s<4$, it follows from Lebesgue's dominated convergence theorem that
\[ \int_{B_{R_i}}|z|^s (m_i^{-1}u_i(m_i^{-(p_i-1)/4} z))^{p_i}\,\ud z \to \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}|z|^s (1+|z|^2)^{-\frac{n+4}{2}}\,\ud z.
\]
Hence, the first convergence result in the lemma follows.
By Lemma \ref{lem:upbound1},
\begin{align*}
\int_{r_i\le |y|<1}|y|^s u_i(y)^{p_i}\,\ud y&\le Cm_i^{-\lda_i p_i} \int_{r_i\le |y|<1}|y|^s |y|^{(4-n+\delta_i)p_i}\,\ud y\\&
\le C m_i^{-\lda_i p_i}m_i^{\frac{(p_i-1)((n-4-\delta_i)p_i-s-n)}{4}} R_i^{n+s-(n-4-\delta_i)p_i}\\&
= Cm_i^{-\frac{(p_i-1)(s+n)}{4}+p_i} R_i^{n+s-(n-4-\delta_i)p_i},
\end{align*}
where $0<\delta_i=O(R_i^{-4})$ and $\lda_i=(n-4-\delta_i)(p_i-1)/4-1$. Since $m_i^{-\frac{(p_i-1)(s+n)}{4}+p_i}=(1+o(1)) m_i^{-1-\frac{2s}{n-4}}$ and $n+s-(n-4-\delta_i)p_i \to s-4<0$ as $i\to \infty$, we have
the second convergence result in the lemma.
In conclusion, the lemma is proved.
\end{proof}
\begin{prop}\label{prop:upbound2} Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
u_i(x)\leq Cu_i^{-1}(0)|x|^{4-n},\quad \mbox{for all } |x|\leq 1.
\]
\end{prop}
\begin{proof} For $|x|\le r_i$, the proposition follows immediately from Proposition \ref{prop:blow up a bubble} and Lemma \ref{lem:error}.
We shall show first that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a7'}
\sup_{|e|=1}u_i( \rho e) u_i(0)\le C.
\ee If not, then along a subsequence we have, for some unit vectors $\{e_i\}$,
\[
\lim_{i\to\infty}u_i(\rho e_i) u_i(0)=+\infty.
\]
Since $u_i(x)\le A_3 |x|^{-4/(p_i-1)}$ in $B_2$, it follows from Proposition \ref{prop:har} that for any $0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1$ there exists a positive constant
$C(\varepsilon} \newcommand{\ud}{\mathrm{d})$, depending only on $n, A_1, A_2, A_3$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$, such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:extra1}
\sup_{B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d}} u_i \le C(\varepsilon} \newcommand{\ud}{\mathrm{d})\inf_{B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d}} u_i.
\ee
Let $\varphi_i (x)=u_i(\rho e_i) ^{-1} u_i(x)$. Then for $|x|\le 1$,
\[
\varphi_i(x)= \int_{B_3} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y+\tilde h_i(x),
\]
where $\tilde h_i(x)=u_i(\rho e_i) ^{-1} h_i(x)$.
Since $\varphi_i(\rho e_i)=1$, by \eqref{eq:extra1}
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a8}
\|\varphi_i\|_{L^\infty(B_{3/2}\setminus B_\varepsilon} \newcommand{\ud}{\mathrm{d})} \le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) \quad \mbox{for }0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1.
\ee
By \eqref{H}, we have that for any $x\in B_1$
\[
\tilde h_i(x) \le A_2 \tilde h_i(\rho e_i) \le A_2.
\]
Besides,
by Lemma \ref{lem:upbound1},
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:a8'}
u_i(\rho e_i) ^{p_i-1} \to 0
\ee
as $i\to \infty$.
Because of \eqref{G}-\eqref{H}, by applying Proposition \ref{prop:local estimates} to $\varphi_i$ we conclude that there exists $\varphi\in C^3(B_1\setminus \{0\})$ such that $\varphi_i\to \varphi$ in $C^3_{loc}(B_1\setminus \{0\})$ after passing to a subsequence.
Let us write the equation of $\varphi_i$ as
\begin{equation}} \newcommand{\ee}{\end{equation}
\varphi_i(x)= \int_{B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y+b_i(x),
\ee
where $b_i(x):= \int_{B_3\setminus B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y +\tilde h_i(x)$.
By \eqref{eq:a8}, there exists $b\in C^3(B_1)$ such that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:a9}
b_i(x) \to b(x)\ge 0\quad \mbox{in }C_{loc}^3(B_1)
\ee
after passing to a subsequence. Therefore,
\[
\int_{B_1} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y= \varphi_i(x)-b_i(x) \to \varphi(x) -b(x)
\]
in $C^3_{loc}(B_1\setminus \{0\})$. Denote $\Gamma(x):=\varphi(x)-b(x)$.
For any $|x|>0$ and $0< \varepsilon} \newcommand{\ud}{\mathrm{d}<\frac{1}{2}|x|$, in view of \eqref{eq:a8} and \eqref{eq:a8'} we have
\begin{align}
\Gamma(x)&= \lim_{i\to \infty} \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} G_i(x,y)K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y \nonumber\\ &
=\lim_{i\to \infty} \left(G_i(x,0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y +O(m_i)\int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}}|y| u_i(y)^{p_i}\,\ud y\right)\nonumber \\&
= \lim_{i\to \infty} (G_i(x,0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y + O(m_i^{-\frac{2}{n-4}}))\nonumber\\&
=:G_\infty(x,0) a(\varepsilon} \newcommand{\ud}{\mathrm{d}),
\label{eq:the constant a}
\end{align}
where we used Lemma \ref{lem:aux3} in the third identity, $a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ is a bounded nonnegative function of $\varepsilon} \newcommand{\ud}{\mathrm{d}$,
\[
G_\infty(x,0)=c_{n} |x|^{4-n}+\bar a+O'(|x|^{6-n})
\] by \eqref{G}, $\bar a\ge 0$ and $\bar a=0$ if $n\ge 6$.
Clearly, $a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ is nondecreasing in $\varepsilon} \newcommand{\ud}{\mathrm{d}$, so $\lim_{\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0} a(\varepsilon} \newcommand{\ud}{\mathrm{d})$ exists which we denote as $a$.
Sending $\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0$, we obtain
\[
\Gamma(x)= a G_\infty(x,0).
\]
Since $ 0$ is an isolated simple blow point of $\{u_i\}_{i=1}^\infty$, we have $r^{\frac{n-4}{2}}\bar \varphi(r) \ge \rho^{\frac{n-4}{2}}\bar \varphi(\rho)$ for $0<r<\rho$. It follows that $\varphi$ is singular at $0$, and thus, $a>0$. Hence,
\[
\lim_{i\to \infty} \int_{B_{1/8}}K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y\ge a(\varepsilon} \newcommand{\ud}{\mathrm{d})\ge a>0.
\]
However,
\[
\begin{split}
&\int_{B_{1/8}}K_i(y) u_i(\rho e_i)^{p_i-1} \varphi_i(y)^{p_i}\,\ud y\\
&\quad\le C u_i(\rho e_i)^{-1} \int_{B_{1/8}} u_i(y)^{p_i}\,\ud y\\
& \quad\le \frac{C}{u_i(\rho e_i) u_i(0)} \to 0 \quad\mbox{as } i \to \infty,
\end{split}
\]
where we used Lemma \ref{lem:aux3} in the last inequality. This is a contradiction.
Without loss of generality, we may assume that $\rho\le 1/2$.
It follows from Proposition \ref{prop:har} and \eqref{eq:a7'} that Proposition \ref{prop:upbound2} holds for $\rho\le |x| \le 1$.
To establish the inequality in the Proposition for $r_i\le |x|\le \rho$, we only need to rescale and reduce it to the case of $|x|=1$. Suppose the contrary that there exists a subsequence $ x_i$ satisfying $|x_i|\le \rho$
and $\lim_{i\to \infty}u_i( x_i) u_i(0)|x_i|^{n-4}= +\infty$.
Set $\tilde r_i:=| x_i|$, $\tilde u_i(x)= \tilde r_i^{4/(p_i-1)}u_i(\tilde r_i x)$. Then $\tilde u_i$ satisfies
\[
\tilde u_i(x)= \int_{B_3} G_{i,\tilde r_i}(x,y)K_i(\tilde r_i y)\tilde u_i(y)^{p_i}\,\ud y +\tilde h_i(x) \quad \mbox{for } x\in B_2,
\]
where $\tilde h_i(x)=\int_{B_{3/\tilde r_i}\setminus B_3} G_{i,\tilde r_i}(x,y)K_i(\tilde r_i y)\tilde u_i(y)^{p_i}\,\ud y + \tilde r_i^{4/(p_i-1)}h_i(\tilde r_i x)$. One can easily check that $\tilde u_i$ and the above equation
satisfy all hypotheses of Proposition \ref{prop:upbound2} for $u_i$ and its equation. It follows from
\eqref{eq:a7'} that
\[
\tilde u_i(0) \tilde u_i(\frac{x_i}{\tilde r_i}) \le C.
\]
It follows (using Lemma \ref{lem:error}) that
\[
\lim_{i\to \infty} u_i(x_i) u_i(0) |x_i|^{n-4} <\infty.
\]
This is again a contradiction.
Therefore, the proposition is proved.
\end{proof}
\begin{prop}\label{prop:upbound3} Under the assumptions in Lemma \ref{lem:upbound1}, we have
\[
|\nabla^k u_i(x)|\leq Cu_i^{-1}(0)|x|^{4-n-k},\quad \mbox{for all } r_i\le |x|\leq 1,
\]
where $k=1,\dots, 4$.
\end{prop}
\begin{proof} Since $0$ is an isolated blow up point in $B_2$, by Proposition \ref{prop:har} we see that Proposition \ref{prop:upbound2} holds for all $|x|\le \frac32$. For any $r_i\le |x|<1$, let
\[
\varphi_i(z)=\Big( \frac{|x|}{4}\Big)^{\frac{4}{p_i-1}} u_i(x+\frac{|x|}{4} z).
\]
By the equation of $u_i$, we have
\[
\varphi_i(z)=\int_{\{y:|x+\frac{|x|}{4} y|\le 3\}} \tilde G_{i}(z,y) \tilde K_i(y) \varphi_i(y)^{p_i-1} \varphi_i(y)\,\ud y+\tilde h_i(z),
\]
where $\tilde G_{i}(z,y)=(\frac{|x|}{4})^{n-4} G_i(x+\frac{|x|}{4} z, x+\frac{|x|}{4} y)$, $\tilde K_i(y)=K_i(x+\frac{|x|}{4} y)$, and $\tilde h_i(z) =( \frac{|x|}{4})^{\frac{4}{p_i-1}} h_i(x+\frac{|x|}{4} z)$. Since $0$ is an isolated blow up point of $u_i$, we have $\varphi_i(z)^{p_i-1}\le A_2^{p_i-1}$ for all $|z|\le 1$. Since $\varphi_i, \tilde G_i, \tilde K_i$ and $\tilde h_i$ are nonnegative, by Proposition \ref{prop:local estimates} we have
\[
|\nabla^k \varphi_i(0)|\le C(\|\varphi_i\|_{L^\infty(B_1)} +\| \tilde h_i\|_{C^4(B_1)}).
\]
This gives
\begin{align*}
(\frac{|x|}{4})^k |\nabla^k u_i(x)| &\le C\|u_i\|_{L^\infty(B_{\frac{|x|}{4}}(x))}+Cm_i^{-1}\\
& \le C u_i(0)^{-1}|x|^{4-n}.
\end{align*}
Therefore, the proposition follows.
\end{proof}
\begin{cor}\label{cor:energy} Under the hypotheses of Lemma \ref{lem:upbound1}, we have
\[
\int_{B_1}|x|^{s}u_i(x)^{p_i+1}\le C u_i(0)^{-2s/(n-4)}, \quad\mbox{for } -n< s<n,
\]
\end{cor}
\begin{proof} Making use of Proposition \ref{prop:blow up a bubble}, Lemma \ref{lem:error} and Proposition
\ref{prop:upbound2}, the corollary follows immediately.
\end{proof}
By Proposition \ref{prop:upbound2} and its proof, we have the following corollary.
\begin{cor} \label{cor:convergence} Under the assumptions in Lemma \ref{lem:upbound1}, if we let $T_i(x)=T'_i(x)+T''_i(x)$, where
\[
T'_i(x):=u_i(0)\int_{B_1} G_{i}(x,y)K_i(y) u_i(y)^{p_i}\,\ud y
\]
and
\[
T''_i(x):= u_i(0) \int_{B_3\setminus B_1} G_{i}(x,y) K_i(y) u_i(y)^{p_i}\,\ud y+u_i(0) h_i(x).
\] then, after passing a subsequence,
\[
T'_i(x) \to aG_\infty(x,0) \quad \mbox{in } C^3_{loc}(B_1\setminus \{0\})
\]
and
\[
T''_i(x) \to h(x) \quad \mbox{in } C_{loc}^3(B_1)
\] for some $h(x)\in C^3(B_2)$,
where $G_\infty$ is the limit of a subsequence of $G_i$ in $C^3_{loc}(B_1\setminus \{0\})$,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:number a}
a=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\left(\frac{1}{1+|y|^2}\right)^\frac{n+4}{2}\ud y.
\ee
Consequently, we have
\[u_i(0) u_i(x)\to aG_\infty(x,0) +h(x) \quad \mbox{in }C^3_{loc}(B_1\setminus \{0\}).
\]
\end{cor}
\begin{proof} Similar to that in the proof of Proposition \ref{prop:upbound2}, we set
$\varphi_i(x)=u_i(0) u_i(x)$, which satisfies
\begin{align*}
\varphi_i(x)&=\int_{B_3} G_{i}(x,y)K_i(y) u_i(0)^{1-p_i} \varphi_i(y)^{p_i}\,\ud y+u_i(0)h_i(x)\\&
= : \int_{B_1} G_{i}(x,y)K_i(y) u_i(0)^{1-p_i} \varphi_i(y)^{p_i}\,\ud y +T''_i(x) =T'_i(x) +T''_i(x) .
\end{align*}
We have all the ingredients as in the proof of Proposition \ref{prop:upbound2}. Hence, we only need to evaluate the positive constant $a$. By \eqref{eq:the constant a} and Lemma \ref{lem:aux3}, we have
\[
a= \lim_{\varepsilon} \newcommand{\ud}{\mathrm{d}\to 0}\lim_{i\to \infty} u_i(0) \int_{B_{\varepsilon} \newcommand{\ud}{\mathrm{d}}} K_i (y)u_i(y)^{p_i}\,\ud y=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\left(\frac{1}{1+|y|^2}\right)^\frac{n+4}{2}\ud y.
\]
\end{proof}
\section{Expansions of blow up solutions of integral equations}
\label{section:bubble-expansion}
In this section, we are interested in stronger estimates than that in Proposition \ref{prop:upbound2}. To make statements closer to the main goal of the paper, we restrict our attention to a special $K_i$. Namely, given $p_i, G_i$, and $ h_i$ satisfying \eqref{p}, \eqref{G} and \eqref{H} respectively, $\kappa_i$ satisfying \eqref{K} with $K_i$ replaced by $\kappa_i$, let $0\le u_i\in C^4(B_3)$
be a solution of
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:s1'}
u_i(x)=\int_{B_3 }G_i(x,y) \kappa_i(y)^{\tau_i} u_i^{p_i}(y)\,\ud y +h_i(x) \quad \mbox{in } B_3.
\ee
We also assume
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:56}
\nabla \kappa_i(0)=0.
\ee Suppose that $0$ is an isolated simple blow up point of $\{u_i\}$ with $\rho=1$, i.e.,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:A_3}
u_i(x)\leq A_3 |x|^{-4/(p_i-1)}\quad \mbox{for all } x\in B_2.
\ee
and $
r^{4/(p_i-1)} \bar u_i(r)$ has precisely one critical point in $(0,1)$.
Let us first introduce a non-degeneracy result.
\begin{lem}\label{lem:non-degeneracy} Let $v\in L^\infty(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ be a solution of
\[
v(x)=c_{n}\frac{n+4}{n-4}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(y)^{\frac{8}{n-4}}v(y)}{|x-y|^{n-4}}\,\ud y.
\]
Then
\[
v(z)=a_0 \left(\frac{n-4}{2}U_1(z)+z\cdot \nabla U_1(z)\right)+\sum_{j=1}^n a_j \pa_j U_1(z),
\]
where $a_0,\dots,a_n$ are constants.
\end{lem}
\begin{proof} Since $v\in L^\infty(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$, by using Lemma \ref{lem:aux2} iteratively a finite number of times we obtain
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:d1}|v(x)|\le C (1+|x|)^{4-n}.
\ee Let $F:\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n\to \mathbb{S}^n\setminus\{N\}$,
\[
F(x)=\left(\frac{2x}{1+|x|^2}, \frac{1-|x|^2}{1+|x|^2}\right)
\]
denote the inverse of the inverse of the stereographic projection and $ h(F(x)):=v(x) J_{F}(x)^{-\frac{n-4}{2n}} $, where $J_F=(\frac{2}{1+|x|^2})^{n}$ is the Jacobian determinant of $F$ and $N$ is the north pole. It follows from \eqref{eq:d1} that $h\in L^\infty(\Sn)$. Let $\xi=F(x)$ and $\eta=F(y)$.
Then
\[
|\xi-\eta| = \frac{2|x-y|}{\sqrt{(1+|x|^2)(1+|y|^2)}} \quad\mbox{and} \quad
\ud \eta =\left(\frac{2}{1+|y|^2}\right)^{n}\ud y
\]
are respectively the distance between $\xi$ and $\eta$ in $\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^{n+1}$ and the surface measure of $\Sn$.
It follows that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{inte eq}
h(\xi)=2^{-4}n(n-2)(n+2)(n+4) \alpha} \newcommand{\lda}{\lambda_n\int_{\Sn} |\xi-\eta|^{4-n} h(\eta)\,\ud \eta.
\ee
By the regularity theory for Riesz potentials, $h\in C^\infty(\Sn)$. Note that the Paneitz operator
\[
P_{g_{\Sn}}=\Delta^2_{g_{\Sn}}-\frac{n^2-2n-4}{2}\Delta_{g_{\Sn}} +\frac{n(n-2)(n+2)(n-4)}{16}
\]
with respect to the standard metric $g_{\Sn}$ on $\Sn$ satisfies $P_{g_{\Sn}} \phi=|J_{F}|^{-\frac{n+4}{2n}} \Delta^2 (|J_{F}|^{\frac{n-4}{2n}} \phi \circ F)$ for any $\phi\in C^\infty(\Sn)$. By the integral equation of $v$ we have
\begin{align*}
P_{g_{\Sn}} h= 2^{-4}n(n-2)(n+2)(n+4)h.
\end{align*}
Let $Y^{(k)}$ be a spherical harmonics of degree $k\ge 0$. We have
\[
P_{g_{\Sn}}Y^{(k)}=2^{-4}(2k+n+2)(2k+n)(2k+n-2)(2k+n-4) Y^{(k)}.
\]
Hence, $h$ must be a spherical harmonics of degree one. Transforming $h$ back, we complete the proof.
\end{proof}
In view of Corollary \ref{cor:GM2.8}, we assume in this and next section that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:IE-cond}
U_{\lda}(x)=\int_{B_3} G_i(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+c_{\lda,i}'(y) U_{\lda}(y)\}\,\ud y+c_{\lda,i}''(x) \quad \forall ~\lda\ge 1 ,~x\in B_3
\ee
where $c_{\lda,i}',c_{\lda,i}''\in C^5(B_3)$ satisfy
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:non-flat}
\Theta_i:=\sum_{k=0}^5\|\lda^{-k}\nabla^kc_{\lda,i}'\|_{L^\infty(B_2)}\le A_2
,\ee and $\|c_{\lda,i}''\|_{C^5(B_2)}\le A_2 \lda^{\frac{4-n}{2}}$, respectively.
\begin{lem}
\label{lem:expansion-a}
Let $0\le u_i\in C^4(B_3)$ be a solution of \eqref{eq:s1'} and $0$ is an isolated simple blow up point of $\{u_i\}$ with some constant
$\rho$, say $\rho=1$. Suppose \eqref{eq:IE-cond} holds and let $\Theta_i$ be defined in \eqref{eq:non-flat}. Then we have
\[
|\varphi_i(z)-U_1(z)| \le C \begin{cases}
\max\{\tau_i, m_i^{-2}\},& \quad \mbox{if } 5\le n\le 7, \\
\max\{ \tau_i, \Theta_i m_i^{-2}\log m_i, m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \tau_i, \Theta_i m_i^{-\frac{8}{n-4}}, m_i^{-2}\},& \quad\mbox{if } n\ge 9,
\end{cases}
\quad \forall~ |z|\le m_i^{\frac{p_i-1}{4}},
\]where $\varphi_i(z)=\frac{1}{m_i}u_i(m_i^{-\frac{p_i-1}{4}}z)$, $m_i=u_i(0)$, and $C>0$ depends only on $n,A_1,A_2$ and $A_3$.
\end{lem}
\begin{proof}
For brevity, set $\ell_i= m_i^{\frac{p_i-1}{4}}$. By the equation of $u_i$, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:varphi}
\varphi_i(z)=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y)\tilde \kappa_i(y)^{\tau_i}\varphi_i(y)^{p_i}\,\ud y +\bar h_i(z),
\ee
where $G_{i,\ell_i^{-1}}(z,y)=\ell_i^{4-n}G_i(\ell_i^{-1}x,\ell_i^{-1} y)$, $\tilde \kappa_i(z)=\kappa_i(\ell_i^{-1} z)$, and
$
\bar h_i(z)=m_i^{-1}\tilde h_i(\ell_i^{-1}z)
$ with
\[
\tilde h_i (x)= \int_{B_3\setminus B_1} G_i(x,y) \kappa_i(y)^{\tau_i} u_i(y)^{p_i} \,\ud y+h_i(x).
\]
Since $0$ is an isolated simple blow up point of $u_i$, by Proposition \ref{prop:upbound2} we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:exp-1}
u_i(x) \le C m_i^{-1}|x|^{4-n} \quad \mbox{for }|x|<1.
\ee It follows that $\tilde h_i(x)\le C m_i^{-1}$ for $x\in B_1$ and $\bar h_i(z) \le Cm_i^{-2}$ for $z\in B_{\ell_i}$.
Notice that $U_{\ell_i}(x)\le Cm_i^{-1}$ for $1\le |x|\le 3$. Let $z=\ell_i x$. By \eqref{eq:IE-cond} with $\lda=\ell_i$ we have for $|z|\le \ell_i$
\begin{align}
U_1(z)&=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y)(U_1(y)^{\frac{n+4}{n-4}}+m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1}y)U_1(y))\,\ud y+\mathcal{O}(m_{i}^{-2})\nonumber \\&
=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i}+T_i(y))\,\ud y+\mathcal{O}(m_i^{-2}),
\label{eq:U1}
\end{align}
where we used $m_i^{\tau_i}=1+o(1)$, and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
\emph{Here and throughout this section, $\mathcal{O}(m_i^{-2})$ denotes some function $f_i$ satisfying $\|\nabla ^k f_i\|_{B_{(1-\varepsilon} \newcommand{\ud}{\mathrm{d})\ell_i}} \le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) m_i^{-2-\frac{2k}{n-4}}$ for small $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $k=0,\dots, 5$.}
In the following, we adapt some arguments from Marques \cite{Marques} for the Yamabe equation; see also the proof of Proposition 2.2 of Li-Zhang \cite{Li-Zhang05}.
Let
\[
\Lambda} \newcommand{\B}{\mathcal{B}_i=\max_{|z|\le \ell_i} |\varphi_i-U_1|.
\]
By \eqref{eq:exp-1}, for any $0<\varepsilon} \newcommand{\ud}{\mathrm{d}<1$ and $\varepsilon} \newcommand{\ud}{\mathrm{d} \ell_i\le |z|\le \ell_i$, we have $|\varphi_i (z)-U_1(z)|\le C(\varepsilon} \newcommand{\ud}{\mathrm{d}) m_i^{-2}$, where we used $m_i^{\tau_i}=1+o(1)$. Hence, we may assume that $\Lambda} \newcommand{\B}{\mathcal{B}_i$ is achieved at some point $|z_i|\le \frac12 \ell_i$, otherwise the proof is finished.
Set
\[
v_i(z)= \frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}(\varphi_i(z)-U_1(z)).
\]
It follows from \eqref{eq:varphi} and \eqref{eq:U1} that $v_i$ satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:newscale1}
v_i(z)= \int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}T_i(y))\,\ud y+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}\mathcal{O}(m_i^{-2}),
\ee
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1}.
\]
Since
\[
G_{i,\ell_i^{-1}}(z,y)\le A_1|z-y|^{4-n}
\] and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Ti-estimate}
|T_i(y)|\le C\tau_i (|\log U_i|+|\log \tilde \kappa_i|)(1+|y|)^{-p_i(n-4)}+\Theta_i m_i^{-\frac{8}{n-4}} (1+|y|)^{4-n},
\ee we obtain
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y\le C (\tau_i+\Theta_i \alpha} \newcommand{\lda}{\lambda_i)\quad \mbox{for }|z|\le \frac{\ell_i}{2},
\]
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:ali}
\alpha} \newcommand{\lda}{\lambda_i= \begin{cases}
m_i^{-2},& \quad \mbox{if }5\le n\le 7, \\
m_i^{-2}\log m_i,& \quad \mbox{if }n=8,\\
m_i^{-\frac{8}{n-4}},& \quad \mbox{if }n\ge 9.
\end{cases}
\ee
Since $\kappa_i(x)$ is bounded and $\varphi_i\le C U_1$, we see
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:bi-estimate}
|b_i(y)|\le CU_1(y)^{p_i-1}\le C (1+|y|)^{-7.5}, \quad y\in B_{\ell_i}.
\ee
Noticing that $\|v_i\|_{L^\infty(B_{\ell_i})}\le 1$, by Lemma \ref{lem:aux2} we have
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |b_i(y) v_i(y)|\,\ud y\le C(1+|z|)^{-\min\{n-4,3.5\}}.
\] Hence, we get
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:b-expansion-1}
v_i (z) \le C((1+|z|)^{-\min\{n-4,3.5\}}+\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}(\tau_i+\Theta_i\alpha} \newcommand{\lda}{\lambda_i+m_i^{-2})) \quad \mbox{for }|z|\le \frac{\ell_i}{2}.
\ee
Suppose the contrary that $\frac{1}{\Lambda} \newcommand{\B}{\mathcal{B}_i}\max\{\tau_i,\Theta_i\alpha} \newcommand{\lda}{\lambda_i,m_i^{-2} \}\to 0$ as $i \to \infty$. Since $v(z_i)=1$, by \eqref{eq:b-expansion-1} we see that
\[
|z_i|\le C.
\]
Differentiating the integral equation \eqref{eq:newscale1} up to three times, together with \eqref{eq:Ti-estimate} and \eqref{eq:bi-estimate}, we see that the $C^3$ norm of $v_i$ on any compact set is uniformly bounded. By Arzel\`a-Ascoli theorem let $v:=\lim_{i\to \infty}v_i$ after passing to a subsequence. Using Lebesgue's dominated convergence theorem, we obtain
\[
v(z)=c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n}\frac{U_1(y)^{\frac{8}{n-4}} v(y)}{|z-y|^{n-4}}\,\ud y.
\]
It follows from Lemma \ref{lem:non-degeneracy} that
\[
v(z)=a_0 (\frac{n-4}{2}U_1(z)+z\cdot \nabla U_1(z))+\sum_{j=1}^n a_j \pa_j U_1(z),
\]
where $a_0,\dots,a_n$ are constants.
Since $v(0)=0$ and $\nabla v(0)=0$, $v$ has to be zero. However, $v(z_i)=1$. We obtain a contradiction.
Therefore, $\Lambda} \newcommand{\B}{\mathcal{B}_i\le C (\tau_i+\alpha} \newcommand{\lda}{\lambda_i)$ and the proof is completed.
\end{proof}
\begin{lem}\label{lem:expansion-b} Under the same assumptions in Lemma \ref{lem:expansion-a}, we have
\[
\tau_i \le C \begin{cases}
m_i^{-2},& \quad \mbox{if }5\le n\le 7, \\
\max\{ \Theta_i m_i^{-2}\log m_i,m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\},& \quad \mbox{if }n\ge 9.
\end{cases}
\]
\end{lem}
\begin{proof} The proof is also by contradiction. Recall the definition of $\alpha} \newcommand{\lda}{\lambda_i$ in \eqref{eq:ali}. Suppose the contrary that $\frac{1}{\tau_i}\max\{\Theta_i\alpha} \newcommand{\lda}{\lambda_i,m_i^{-2} \}\to 0$ as $i\to \infty$.
Set
\[
v_i(z)=\frac{\varphi_i(z)-U_1(z)}{\tau_i}.
\]
It follows from Lemma \ref{lem:expansion-a} that $|v_i(z)|\le C$ in $B_{\ell_i}$, where $\ell_i= m_i^{\frac{p_i-1}{4}}$. As \eqref{eq:newscale1}, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:newscale2}
v_i(z)= \int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\tau_i}T_i(y))\,\ud y+\frac{1}{\tau_i}\mathcal{O}(m_i^{-2}),
\ee
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1},
\]
and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
By the estimates \eqref{eq:bi-estimate} and \eqref{eq:Ti-estimate} for $b_i$ and $T_i$ respectively, we conclude from the integral equation that $\|v_i\|_{C^3}$ is uniformly bounded over any compact set. It follows that $v_i\to v$ in $C^2_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ for some $v\in C^3(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$.
Multiplying both sides of \eqref{eq:newscale2} by $b_i(z)\phi(z)$, where $\phi(z)=\frac{n-4}{2} U_1(z)+z\cdot \nabla U_1(z)$, and integrating over $B_{\ell_i}$, we have, using the symmetry of $G_{i,\ell_i^{-1}}$ in $y$ and $z$,
\begin{align*}
&\int_{B_{\ell_i}}b_i(z)v_i(z)\left(\phi(z)-\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\right)\,\ud z\\&
=\frac{1}{\tau_i}\int_{B_{\ell_i}}T_i(z)\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\,\ud z+\frac{1}{\tau_i}\mathcal{O}(m_i^{-2})\int_{B_{\ell_i}} b_i(z)\phi(z)\,\ud z.
\end{align*}
As $i\to \infty$, we have
\[
\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y \to c_{n}\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{U_1(y)^{\frac{8}{n-4}}\phi(y)}{|z-y|^{n-4}}\,\ud z=\phi(z),
\]
\[
\frac{1}{\tau_i}\mathcal{O}(m_i^{-2})\int_{B_{\ell_i}} b_i(z)\phi(z)\,\ud z \to 0\mbox{ by the contradiction hypothesis},
\]
and
\[
\frac{T_i(z)}{\tau_i} \to (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}.
\]
Hence, by Lebesgue's dominated convergence theorem we obtain
\[
\begin{split}
\lim_{i\to \infty}&\frac{1}{\tau_i}\int_{B_{\ell_i}}T_i(z)\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) b_i(y) \phi(y)\,\ud y\,\ud z\\&=\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \phi(z) (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}\ud z=0.
\end{split}
\]
This is impossible, because
\begin{align*}
&\int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \phi(z) (\log U_1(z))U_1(z)^{\frac{n+4}{n-4}}\ud z\\&=\frac{(n-4)^2|\mathbb{S}^{n-1}|}{4} \int_{0}^\infty \frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log (1+r^2)\,\ud r\\&
=\frac{(n-4)^2|\mathbb{S}^{n-1}|}{2} \int_{1}^\infty \frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log r\,\ud r>0,
\end{align*} where we used
\[
\int_{0}^1\frac{(r^2-1)r^{n-1}}{(1+r^2)^{n+1}} \log (1+r^2)\,\ud r= -\int_{1}^\infty \frac{(s^2-1)s^{n-1}}{(1+s^2)^{n+1}} (\log (1+s^2)-\log s^2)\,\ud s
\]
by the change of variable $r=\frac{1}{s}$.
We obtain a contradiction and thus $\tau_i\le \alpha} \newcommand{\lda}{\lambda_i$. Therefore, the lemma is proved.
\end{proof}
\begin{prop}\label{prop:expansion} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have
\[
|\varphi_i (z)-U_1(z)| \le C \begin{cases}
m_i^{-2},& \quad \mbox{if } 5\le n\le 7, \\
\max\{ \Theta_i m_i^{-2}\log m_i,m_i^{-2}\},& \quad \mbox{if }n=8,\\
\max\{ \Theta_i m_i^{-\frac{8}{n-4}},m_i^{-2}\},& \quad\mbox{if } n\ge 9,
\end{cases}
\quad \forall~ |z|\le m_i^{\frac{p_i-1}{4}}.
\]
\end{prop}
\begin{proof} It follows immediately from Lemma \ref{lem:expansion-a} and Lemma \ref{lem:expansion-b}.
\end{proof}
\begin{prop}\label{prop:expansion8+} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have, for every $|z|\le m_i^{\frac{p_i-1}{4}}$,
\[
|\varphi_i (z)-U_1(z)| \le C \begin{cases}
\max\{ \Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\},& \quad \mbox{if } n=8, \\
\max\{ \Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\} ,& \quad\mbox{if } n\ge 9.
\end{cases}
\]
\end{prop}
\begin{proof} Let $\alpha} \newcommand{\lda}{\lambda_i$ be defined in \eqref{eq:ali}. We may assume that $\frac{m_i^{-2}}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i}\to 0$ as $i\to \infty$ for $n\ge 8$; otherwise the proposition follows immediately from Proposition \ref{prop:expansion}. Set
\[
\alpha} \newcommand{\lda}{\lambda'_i=\begin{cases}
m_i^{-2}m_i^{\frac{2}{n-4}},& \quad \mbox{if } n=8, \\
m_i^{-2}m_i^{\frac{2(n-8)}{n-4}},& \quad\mbox{if } n\ge 9,
\end{cases}
\]
and
\[
v_i(z)=\frac{\varphi_i(z)-U_1(z)}{\Theta_i \alpha} \newcommand{\lda}{\lambda'_i}, \quad |z|\le m_i^{\frac{p_i-1}{4}}.
\]
Since $\frac{m_i^{-2}}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i}\to 0$, it follows from Proposition \ref{prop:expansion} that $|v_i|\le C$. Since $0<\varphi_i\le CU_1$, we only need to prove the proposition when $|z|\le \frac{1}{2}\ell_i$, where $\ell_i=m_i^{\frac{p_i-1}{4}}$. Similar to \eqref{eq:newscale1}, $v_i$ now satisfies
\[
v_i(z)=\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) (b_i(y) v_i(y)+\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda'_i}T_i(y))\,\ud y+\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda_i'}\mathcal{O}(m_i^{-2}),
\]
where
\[
b_i=\tilde \kappa_i^{\tau_i}\frac{\varphi_i^{p_i}-U_1^{p_i}}{\varphi_i-U_1}
\]
and
\[
T_i(y):=U_1(y)^{\frac{n+4}{n-4}}-\tilde \kappa_i(y)^{\tau_i}U_1(y)^{p_i} +m_i^{-\frac{8}{n-4}}c'_{\ell_i,i}(\ell_i^{-1} y)U_1(y).
\]
Noticing that
\[
|T_i(y)|\le C\tau_i (|\log U_1|+|\log \tilde \kappa_i|)(1+|y|)^{-4-n}+m_i^{-\frac{8}{n-4}} \Theta_i (1+|y|)^{4-n},
\]
we have
\begin{align*}
\frac{1}{\Theta_i \alpha} \newcommand{\lda}{\lambda_i'}\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y&\le C\int_{B_{\ell_i}} \frac{1}{|z-y|^{n-4}(1+|y|)^4 m_i^{\frac{2}{n-4}}} \,\ud y\\&
\le C \int_{B_{\ell_i}} \frac{1}{|z-y|^{n-4}(1+|y|)^5} \,\ud y\\&
\le C(1+|z|)^{-1},
\end{align*}
if $n=8$, and
\[
\frac{1}{\Theta_i\alpha} \newcommand{\lda}{\lambda_i'}\int_{B_{\ell_i}} G_{i,\ell_i^{-1}}(z,y) |T_i(y)|\,\ud y \le C(1+|z|)^{8-n}
\]
if $n\ge 9$, where we used Lemma \ref{lem:aux2}.
Thus
\[
|v_i(z)|\le C((1+|z|)^{-3.5}+(1+|z|)^{-1})
\]
for $n=8$, and
\[
|v_i(z)|\le C((1+|z|)^{-3.5}+(1+|z|)^{8-n})
\] for $n\ge 9$. If $n=8,9,10,11$, the conclusion follows immediately from multiplying both sides of the above inequalities by $\alpha} \newcommand{\lda}{\lambda'_i$. If $n\ge 12$, the above estimate gives $|v_i(z)|\le C(1+|z|)^{-3.5}$. Plugging this estimate to the term $\int G_{i,\ell_i^{-1}}(z,y)b_i(y)v_i(y)\,\ud y$ yields $|v_i(z)|\le C(1+|z|)^{8-n}$ as long as $n\le 14$. Repeating this process, we complete the proof.
\end{proof}
\begin{cor}\label{cor:hot-expansion} Under the hypotheses in Lemma \ref{lem:expansion-a}, we have, for very $|z|\le m_i^{\frac{p_i-1}{4}}$,
\begin{align*}
&|\nabla^k(\varphi_i-U_1)(z)|\\& \le C(1+|z|)^{-k} \begin{cases}
m_i^{-2},& \quad \mbox{if } 5\le n\le 7,\\
\max\{ \Theta_im_i^{-2}m_i^{\frac{2}{n-4}}(1+|z|)^{-1}, m_i^{-2}\},& \quad \mbox{if } n=8, \\
\max\{ \Theta_i m_i^{-2}m_i^{\frac{2(n-8)}{n-4}}(1+|z|)^{8-n}, m_i^{-2}\} ,& \quad\mbox{if } n\ge 9.
\end{cases}
\end{align*}
where $k=1,2,3,4$.
\end{cor}
\begin{proof} Considering the integral equation of $v_i=\varphi_i-U_1$, the conclusion follows from Lemma \ref{lem:aux2}.
Indeed, if $k<4$, we can differentiate the integral equation of $v_i$ directly and then use Lemma \ref{lem:aux2}. If $k=4$, we can use a standard technique (see the proof of Proposition \ref{prop:local estimates}) for proving the higher order regularity of Riesz potential since $v_i$ and the coefficients are of $C^1$.
\end{proof}
\section{Blow up local solutions of fourth order equations}
\label{section:Q equation blow}
In the previous two sections, we have analyzed the blow up profiles of the blow up local solutions of integral equations. In this section, we will assume that those blow up solutions also satisfy differential equations, which is only used to check the Pohozaev identity in Proposition \ref{prop:4-pohozaev}. It should be possible to completely avoid using differential equations after improving Corollary \ref{cor:Q-gf-expansion}. This is the case on the sphere; see our joint work with Jin \cite{JLX3}. On the other hand, as mentioned in the Introduction, without extra information fourth order differential equations themselves are not enough to do blow up analysis for positive local solutions.
\begin{prop}\label{prop:one-side} In addition to the hypotheses in Lemma \ref{lem:expansion-a}, assume that $u_i$ also satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:Q-sub}
P_{ g_i} u_i=c(n) \kappa_i^{\tau_i} u_i^{p_i}\quad \mbox{in }B_3,
\ee
where $\det g_i=1$, $B_3$ is a normal coordinates chart of $g_i$ at $0$ and $\|g_i\|_{C^{10}(B_3)}\le A_1$.
\begin{itemize}
\item[(i)] If either $n\le 9$ or $g_i$ is flat, then
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sign restrict}
\liminf_{r\to 0}\mathcal{P}(r,\Gamma)\ge 0,
\ee
where $\Gamma$ is a limit of $u_i(0)u_i(x)$ along a subsequence.
\item[(ii)] If $n\ge 8$, then $|W_{g_i}(0)|^2 \le C^* \mathcal{G}_i \beta_i$ with $C^*>0$ depending only on $n,A_1,A_2, A_3$,
where
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:gi}
\mathcal{G}_i:= \sum_{k\ge 1,~ 2\le k+l\le 4}\Theta_i \|\nabla^k g_i\|_{L^\infty(B_3)}^l+\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l ,
\ee
and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:beta_i}
\beta_i:=\begin{cases} (\log m_i)^{-1},& \quad \mbox{if }n=8,\\
m_i^{-\frac{2}{n-4}}, &\quad \mbox{if }n=9,\\
m_i^{-\frac{4}{n-4}}\log m_i, &\quad \mbox{if }n=10,\\
m_i^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11.\\
\end{cases}
\ee
\item[(iii)] \eqref{eq:sign restrict} holds if $n\ge 10$ and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:r2}
|W_{g_i}(0)|^2 > C^* \mathcal{G}_i \beta_i.
\ee
\end{itemize}
\end{prop}
\begin{proof} It follows from Corollary \ref{cor:convergence} that after passing a subsequence
\[
\lim u_i(0)u_i(x)=:\Gamma(x),
\] where $\Gamma(x)$ is in $C^3(B_{1}\setminus \{0\})$. We will still denote the subsequence as $u_i$.
Notice that for every $0<r<1$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-0}
m_i^{2}\mathcal{P}(r,u_i)\to \mathcal{P}(r,\Gamma)\quad \mbox{as }i\to \infty.
\ee
By Proposition \ref{prop:4-pohozaev},
\begin{align*}
\mathcal{P}(r,u_i)= \int_{B_r} (x^k \pa_k u_i +\frac{n-4}{2} u_i) E(u_i)+\mathcal{N}(r,u_i),
\end{align*}
where $E(u_i)$ is as in \eqref{eq:Eu} with $\tilde g$ and $u$ replaced by $g_i$ and $u_i$ respectively, i.e.,
\begin{align*}
E(u_i):&=P_{g_i}u_i-\Delta^2 u_i\\&
=\frac{n-4}{2}Q_{g_i}u_i+ f^{(1)}_{i,k}\pa_ku_i +f^{(2)}_{i,kl}\pa_{kl}u_i+f^{(3)}_{i,kls}\pa_{kls}u_i+f^{(4)}_{i,klst}\pa_{klst}u_i,
\end{align*}
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:fi}
f^{(1)}_{i,k}(x)=O(1), \quad f^{(2)}_{i,kl}(x)=O(1),\quad f^{(3)}_{i,kls}(x)=O(|x|), \quad f^{(4)}_{i,klst}(x)=O(|x|^2),
\ee
and
\begin{align*}
\mathcal{N}(r,u_i)=\frac{c(n)\tau_i}{p_i+1} \int_{B_r} (\frac{n-4}{2}\kappa_i^{\tau_i}+ x^k\pa_k \kappa_i \kappa_i^{\tau_i-1}) u_i^{p_i+1}
-\frac{r}{p_i+1}\int_{\pa B_{r}} c(n)\kappa_i^{\tau_i} u_i^{p_i+1}.
\end{align*}
By Proposition \ref{prop:upbound2}, for $0<r<1$ we have, for some $C>0$ independent of $i$ and $r$,
\begin{align*}
m_i^{2}\mathcal{N}(r,u_i) \ge
-\frac{m_i^{2}r}{p_i+1}\int_{\pa B_{r}} c(n)\kappa_i^{\tau_i} u_i^{p_i+1}\ge -Cr^{-n}m_i^{1-p_i},
\end{align*}
where we used the facts that $\kappa_i(x)=1+O(|x|^2)$ and $|\nabla \kappa_i(x)|=O(|x|)$ with $O(\cdot)$ independent of $i$. Hence, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-1}
\liminf_{i\to \infty}m_i^{2}\mathcal{N}(r,u_i)\ge 0.
\ee
Throughout this section, without otherwise stated, we use $C$ to denote some constants independent of $i$ and $r$.
If $g_i$ is flat, then we complete the proof because $E(u_i)=0$.
Now we assume $g_i$ is not flat. By a change of variables $z=\ell_i x$ with $\ell_i=m_i^{\frac{p_i-1}{4}}$, we have
\begin{align*}
\mathcal{ E}_i(r):&= m_i^{2}\int_{B_r} (x^k \pa_k u_i +\frac{n-4}{2} u_i) E(u_i) \,\ud x\\&
=m_i^{2} m_i^{2+(4-n)\frac{p_i-1}{4}} \int_{B_{\ell_i r}}\Big( z^k\pa_k\varphi_i+\frac{n-4}{2}\varphi_i\Big)\cdot\\& \quad
\Big(\frac{n-4}{2}\ell_i^{-4}Q_{g_i}(\ell_i^{-1}z)\varphi_i+ \sum_{j=1}^4\ell_i^{-4+j}f_i^{(j)}(\ell_i^{-1}z)\nabla^j \varphi_i\Big)\,\ud z,
\end{align*}
where $\varphi_i(z)=m_i^{-1}u_i(m_i^{-\frac{p_i-1}{4}}z)$, $
f_i^{(1)}(\ell_i^{-1}z)\nabla^1 \varphi_i=f^{(1)}_{i,k}(\ell_i^{-1}z)\pa_k\varphi_i$
and $f_i^{(j)}(\ell_i^{-1}z)\nabla^j \varphi_i$ is defined in the same fashion for $j\neq 1$. Define
\begin{align*}
\mathcal{\hat E}_i(r):&=m_i^{2} m_i^{2+(4-n)\frac{p_i-1}{4}} \int_{B_{\ell_i r}}\Big( z^k\pa_kU_1+\frac{n-4}{2}U_1\Big)\cdot\\& \quad
\Big(\frac{n-4}{2}\ell_i^{-4}Q_{g_i}(\ell_i^{-1}z)U_1+ \sum_{j=1}^4\ell_i^{-4+j}f_i^{(j)}(\ell_i^{-1}z)\nabla^j U_1\Big)\,\ud z.
\end{align*}
Notice that $m_i^{2+(4-n)\frac{p_i-1}{4}}=1+o(1)$, and $Q_{\tilde g}=O(1)$. By Proposition \ref{prop:expansion}, Proposition \ref{prop:expansion8+}, Corollary \ref{cor:hot-expansion}, \eqref{eq:fi} and \eqref{eq:r1}, we have
\begin{align}
&|\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r)|\\&\le Cm_i^{2}m_i^{-\frac{4}{n-4}}\int_{B_{\ell_i r}} \sum_{j=0}^4|\nabla^j(\varphi_i-U_1)|(z)(1+|z|)^{2-n+j}\,\ud z\nonumber \\&
\le C \sum_{k\ge 1,~ 2\le k+l\le 4}\|\nabla^k g_i\|_{L^\infty(B_3)}^l
\begin{cases}
r^2, & \quad \mbox{if }n=5,6,7,\\
\max\{\Theta_i r, r^2\}, & \quad \mbox{if }n=8,9,\\
\max\{\Theta_i \log (rm_i), r^2\},& \quad \mbox{if }n=10,\\
\max\{\Theta_i m_i^{\frac{2(n-10)}{n-4}}, r^2\},& \quad \mbox{if }n\ge 11.
\end{cases}
\label{eq:tos-2}
\end{align}
Now we estimate $\mathcal{\hat E}_i(r)$.
If $n=5,6,7$, we have
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:why 8d}
\begin{split}
\mathcal{\hat E}_i(r)&= m_i^{2+(n-4)\tau_i}\int_{B_r} (x^k \pa_k U_{\ell_i} +\frac{n-4}{2} U_{\ell_i}) E(U_{\ell_i}) \,\ud x\\
&= m_i^{2+(n-4)\tau_i}\int_{B_r} (x^k \pa_k U_{\ell_i} +\frac{n-4}{2} U_{\ell_i}) (P_{ g_i}-\Delta^2)U_{\ell_i}\,\ud x\\&
=O(1)m_i^{2}\int_{B_r} |x^k \pa_k U_{\ell_i}+\frac{n-4}{2} U_{\ell_i}| U_{\ell_i},
\end{split}
\ee
where we have used $(P_{ g_i}-\Delta^2)U_{\ell_i}=O(1) U_{\ell_i}$ because of \eqref{eq:cor-GM}, and
\[
||x|^k\nabla_x^k U_{\ell_i}(x)|\le C(n,k) U_{\ell_i}(x)\quad \mbox{for } k\in \mathbb{N}.
\]
Hence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-3}
|\mathcal{\hat E}_i(r)|\le Cr^{8-n}\le Cr.
\ee
Therefore, \eqref{eq:sign restrict} follows from \eqref{eq:tos-1}, \eqref{eq:tos-2} and \eqref{eq:tos-3} when $n=5,6,7$.
If $n\ge 8$, by Lemma \ref{lem:GM2.8} we have
\begin{align*}
\mathcal{\hat E}_i(r) =& -\frac{2}{n}\gamma_i\int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)(c_1^*s\pa_s U_1+c_2^*s^2 \pa_{ss}U_1)\,\ud z\\&
-\frac{32(n-1)\gamma_i}{3(n-2)n^2} \int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)s^2(\pa_{ss}U_1-\frac{\pa_s U_1}{s})\,\ud z \\&
+(n-4)\gamma_i\int_{B_{\ell_i r}}(s\pa_s U_1+\frac{n-4}{2}U_1)U_1\,\ud z+O(\alpha} \newcommand{\lda}{\lambda_i'')\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l,
\end{align*}
where we used the symmetry so that those terms involving homogeneous polynomials of odd degrees are gone, $s=|z|$, $\gamma_i=\frac{m_i^{\frac{2(n-8)}{n-4}+(n-4)\tau_i}|W_{g_i}(0)|^2}{24(n-1)}\ge 0$,
\[
\alpha} \newcommand{\lda}{\lambda_i''=\int_{B_r} |x|^2 U_{\ell_i}(x)^2\,\ud x=O(1)\begin{cases}
r^{10-n}, & \quad \mbox{if }n=8,9,\\
\log rm_i,& \quad \mbox{if }n=10,\\
m_i^{\frac{2(n-10)}{n-4}},& \quad \mbox{if }n\ge 11.
\end{cases}
\]
and $c_1^*, c_2^*$ are given in \eqref{eq:c-star}.
By direct computations,
\[
r\pa_r U_1+\frac{n-4}{2}U_1=\frac{n-4}{2}\frac{1-r^2}{(1+r^2)^{\frac{n-2}{2}}},
\]
\begin{align*}
c_1^*r\pa_r U_1+c_2^*r^2 \pa_{rr}U_1&=(4-n)\frac{(c_1^*+c_2^*)r^2}{(1+r^2)^{\frac{n-2}{2}}}+(4-n)(2-n)\frac{c_2^*r^4}{(1+r^2)^{\frac{n}{2}}}\\
&=(4-n)\frac{(c_1^*+c_2^*)r^2+(c_1^*+(3-n)c_2^*)r^4}{(1+r^2)^{\frac{n}{2}}},
\end{align*}
\[
\pa_{rr} U_1-\frac{\pa_r U_1}{r}=(n-4)(n-2)(1+r^2)^\frac{-n}{2} r^2.
\]
Thus
\begin{align*}
\mathcal{\hat E}_i(r) = \frac{(n-4)^2}{n}\gamma_i |\mathbb{S}^{n-1}| J_i+O(\alpha} \newcommand{\lda}{\lambda_i'')\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l,
\end{align*}
where
\[
J_i:=\int^{\ell_i r}_0 \frac{(1-s^2)[\frac{n}{2}+(c_1^*+c_2^*+n)s^2+(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2})s^4]s^{n-1}}{(1+s^2)^{n-1}}\,\ud s.
\]
If $n=8$, we have $-(c_1^*+(3-n)c_2^*+\frac{n}{2})= (2n-12)+\frac{14}{3}-4=\frac{14}{3}$. Since $\int^{\ell_i r}_0 \frac{s^{13}}{(1+s^2)^7}\,\ud s\to \infty$ as $\ell_i\to \infty$, Hence, $J_i\to \infty$ as $i\to \infty$.
For $n\ge 9$, we notice that for positive integers $2< m+1<2k$,
\[
\int_0^\infty \frac{t^m}{(1+t^2)^k}\,\ud t=\frac{m-1}{2k-m-1}\int_{0}^\infty \frac{t^{m-2}}{(1+t^2)^k}\,\ud t.
\]
If $\ell_i r =\infty$, we have
\begin{align*}
J_i=&\Big\{-\frac{2n}{n-4}-(c_1^*+c_2^*+n)\frac{8n}{(n-6)(n-4)}\\&-(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2}) \frac{12n(n+2)}{(n-8)(n-6)(n-4)}\Big\} \int_0^\infty \frac{s^{n-1}}{(1+s^2)^{n-1}}\,\ud s.
\end{align*}
We compute the coefficients of the integral, \begin{align*}
&-\frac{2n}{n-4}-(c_1^*+c_2^*+n)\frac{8n}{(n-6)(n-4)}\\&\quad -(c_1^*+(3-n)c_2^*-\frac{16(n-1)}{3n}+\frac{n}{2}) \frac{12n(n+2)}{(n-8)(n-6)(n-4)}\\&
=\frac{2n}{n-4}\Big\{-1+(\frac{n(n-2)}{2}-8)\frac{4}{(n-6)}+(\frac{3n}{2}+\frac{16(n-1)}{3n}-12) \frac{6(n+2)}{(n-8)(n-6)}\Big\}\\&
= \frac{2n}{n-4}\Big\{\frac{2n^2 -5n-26}{n-6}+\frac{9(n+2)}{(n-6)}+\frac{32(n-1)(n+2)}{n(n-8)(n-6)}\Big\}\\&
\ge \frac{4n(n^2+2n-4)}{(n-4)(n-6)}>0.
\end{align*}
Therefore, for any $0<r<1$ and sufficiently large $i$ (the largeness of $i$ may depend on $r$), we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-5}
J_i\ge 1/C(n)>0.
\ee In conclusion, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:tos-6}
\mathcal{\hat E}_i(r) \ge \begin{cases} \frac{1}{C}|W_{g_i}(0)|^2 \log m_i+O(\alpha} \newcommand{\lda}{\lambda''_i)\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l, &\quad \mbox{if }n=8,\\
\frac{1}{C} |W_{g_i}(0)|^2 m_i^{\frac{2(n-8)}{n-4}}+O(\alpha} \newcommand{\lda}{\lambda''_i)\sum_{k\ge 1,~ 6\le k+l\le 8}\|\nabla^k g_i\|_{L^\infty(B_3)}^l, &\quad \mbox{if }n\ge 9. \end{cases}
\ee
Combing \eqref{eq:tos-1}, \eqref{eq:tos-2} and \eqref{eq:tos-6} together, we see that, for $n\ge 8$,
\begin{align}
m_i^2 \mathcal{P}(r,u_i)&=m_i^2 \mathcal{N}(r, u_i)+(\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r))+\mathcal{\hat E}_i(r)
\nonumber\\& \ge m_i^2 \mathcal{N}(r, u_i) +\frac{1}{2} \mathcal{\hat E}_i(r) -Cr
\label{eq:tos-7}
\end{align}
where $Cr$ can be set to zero when $n\ge 9$. If $n=8,9$, by sending $i\to \infty$ in \eqref{eq:tos-7} we have $ \mathcal{P}(r,\Gamma) \ge -Cr$. Thus \eqref{eq:sign restrict} follows and the conclusion (i) is proved.
If $n\ge 10$ and $|W_{g_i}(0)|^2 $ satisfies \eqref{eq:r2} for large $C^*>0$, by \eqref{eq:tos-6} we see that $(\mathcal{ E}_i(r)-\mathcal{\hat E}_i(r))+\mathcal{\hat E}_i(r) \ge 0$. Hence, the conclusion (iii) follows.
Since $|\mathcal{P}(r,\Gamma)|\le C$, it follows from \eqref{eq:tos-7} that for large $i$, $
\mathcal{\hat E}_i(r) \le C$. In view of \eqref{eq:tos-6} and the definition of $\alpha} \newcommand{\lda}{\lambda''$, the conclusion (ii) follows.
\end{proof}
\begin{prop} \label{prop:isolated to isolated simple} Given $p_i, G_i$, and $ h_i$ satisfying \eqref{p}, \eqref{G} and \eqref{H} respectively, $\kappa_i$ satisfying \eqref{K} with $K_i$ replaced by $\kappa_i$, let $0\le u_i\in C^4(B_3)$ solve both \eqref{eq:s1'} and \eqref{eq:Q-sub}, and assume \eqref{eq:IE-cond} holds. Suppose that $0$ is an isolated blow up point of $\{u_i\}$ with \eqref{eq:A_3} holds. Then $0$ is an isolated simple blow up point, if one of the three cases happens:
\begin{itemize}
\item $g_i$ is flat;
\item $n\le 9$;
\item $n\ge 10$ and \eqref{eq:r2} holds.
\end{itemize}
\end{prop}
\begin{proof} By Proposition \ref{prop:blow up a bubble}, $r^{4/(p_i-1)}\overline u_i(r)$ has precisely
one critical point in the interval $0<r<r_i$,
where $R_i\to \infty$ $r_i=R_iu_i(0)^{-\frac{p_i-1}{4}}$ as in Proposition \ref{prop:blow up a bubble}.
Suppose the contrary that $0$ is not an isolated simple blow up point and let $\mu_i$ be the second critical point of $r^{4/(p_i-1)}\overline u_i(r)$. Then we must have
\begin{equation}} \newcommand{\ee}{\end{equation}\label{5.2}
\mu_i\geq r_i,\quad \displaystyle\lim_{i\to \infty}\mu_i=0.
\ee
Set
\[
v_i(x)=\mu_i^{4/(p_i-1)}u_i(\mu_i x),\quad x\in B_{3/\mu_i}.
\]
By the assumptions of Proposition \ref{prop:blow up a bubble}, $v_i$ satisfies
\begin{align*}
v_i(x)&=\int_{B_{3/\mu_i}}\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i} v_i(y)^{p_i}\,\ud y+\tilde h_i(x)
\\[2mm]
|x|^{4/(p_i-1)}v_i(x)&\leq A_3,\quad |x|<2/\mu_i \to \infty,
\\[2mm]
\lim_{i\to \infty}v_i(0)&=\infty,
\end{align*}
\[
r^{4/(p_i-1)}\overline v_i(r)\mbox{ has precisely one critical point in } 0<r<1,
\]
and
\[
\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}\overline v_i(r)\right\}\Big|_{r=1}=0,
\]
where $\tilde G_i=G_{i,\mu_i}$, $\tilde \kappa_i(y)=\kappa_i(\mu_i y)$,
$\tilde h_i(x)=\mu_i^{4/(p_i-1)}h_i(\mu_i x)
$ and $\overline v_i(r)=|\pa B_r|^{-1}\int_{\pa B_r}v_i$.
Therefore, $0$ is an isolated simple blow up point of $\{v_i\}$.
\textbf{Claim.} We have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:converg2}
v_i(0) v_i (x) \to \frac{ac_{n}}{|x|^{n-4}} + ac_{n} \quad \mbox{in }C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n \setminus \{0\}).
\ee
where $a>0
$ is given in \eqref{eq:number a}.
First of all, by Proposition \ref{prop:upbound2} we have $\tilde h_i(e)\le v_i(e)\le C v_i(0)^{-1}$ for any $e\in \mathbb{S}^{n-1}$, where $C>0$ is independent of $i$. It follows from the assumption \eqref{H} on $h_i$ that
\[
v_i(0)\tilde h_i(x)\le C \quad \mbox{for all } |x|\le 2/\mu_i
\] and
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos0a}
\|\nabla (v_i(0)\tilde h_i)\|_{L^\infty(B_{\frac{1}{9\mu_i}})} \le \mu_i \|v_i(0)\tilde h_i\|_{L^\infty(B_{\frac{1}{4\mu_i}})} \le C\mu_i.
\ee Hence, for some constant $c_0\ge 0$, we have, along a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:iso to isos0b}
\lim_{i\to \infty}\|v_i(0)\tilde h_i(x)-c_0\|_{L^\infty(B_t)} =0, \quad \forall~t>0.
\ee
Secondly, by Corollary \ref{cor:convergence} and Proposition \ref{prop:upbound2} we have, up to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos1}
v_i(0)\int_{B_t}\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i} v_i(y)^{p_i}\,\ud y \to \frac{a c_n}{|x|^{n-4}} \quad \mbox{in }C^3_{loc}(B_t \setminus \{0\}) \mbox{ for any } t>0,
\ee
where we used that $\tilde G_i(x,0)\to c_{n}|x|^{4-n}$.
Notice that for any $x\in B_{t/2}$
\[
Q''_i(x):= \int_{B_{3/\mu_i}\setminus B_t}
\tilde G_{i}(x,y)\tilde \kappa_i(y) ^{\tau_i}v_i(y)^{p_i}\,\ud y \le C(n, A_1)\max_{\pa B_t} v_i.
\]
Since $\max_{\pa B_t} v_i\le Ct^{4-n} v_i(0)^{-1}$, we have as in the proof of \eqref{eq:phigeos}, after passing to a subsequence,
\[
v_i(0) Q''_i(x)\to q(x) \quad \mbox{in }C^3_{loc}(B_t) \quad \mbox{as }i\to \infty
\] for some $q\in C^3(B_t)$. For any fixed large $R>t+1$, it follows from \eqref{eq:iso to isos1} that \[
v_i(0)\int_{t \le |y|\le R}
\tilde G_{i}(x,y)\tilde \kappa_i(y)^{\tau_i}v_i(y)^{p_i}\,\ud y \to 0
\]
as $i\to \infty$, since the constant $a$ is independent of $t$. By the assumption \eqref{G} on $G_i$, for any $x\in B_t$ and $|y|>R$, we have
\[
|\nabla_x \tilde G_{i}(x,y)| \le A_1 |x-y|^{3-n} \le \frac{A_1}{R-t} |x-y|^{4-n} \le \frac{A_1^2}{R-t} \tilde G_{i}(x,y).
\]
Therefore, we have $
|\nabla q(x)| \le \frac{A_1^2}{R-t} q(x).$
By sending $R\to \infty$, we have $|\nabla q(x)| \equiv 0$ for any $x\in B_t$. Thus,
\[
q(x)\equiv q(0)\quad \mbox{for all } x\in B_t.
\]
Since
\[
\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}v_i(0)\overline v_i(r)\right\}\Big|_{r=1}=
v_i(0)\frac{\mathrm{d}}{\mathrm{d}r}\left\{ r^{4/(p_i-1)}\overline v_i(r)\right\} \Big|_{r=1}=0,
\]
we have, by choosing, for example, $t=2$ and sending $i$ to $\infty$, that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:iso to isos3}
q(0)+c_0=ac_{n}>0.
\ee
Therefore, \eqref{eq:converg2} is proved.
It follows from \eqref{eq:converg2} and Lemma \ref{lem:test-poho} that
\[
\liminf_{i\to \infty} v_i(0)^2\mathcal{P}(r,v_i)=-(n-4)^2(n-2)a^2c_n^2 |\mathbb{S}^{n-1}|<0\quad \mbox{for all }0<r<1.
\]
On the other hand, by \eqref{eq:Q-sub} $v_i$ satisfies
\[
P_{\tilde g_i} v_i=c(n)\tilde \kappa_i^{\tau_i}v_i^{p_i} \quad \mbox{in }B_{3/\mu_i},
\]where $\tilde g_i(z)=g_i(\mu_i z)$. It is easy to see that \eqref{eq:IE-cond} is still correct with $G_i$ replaced by $\tilde G_i$. If $n\le 9$ or $g_i$ is flat, it follows from Proposition \ref{prop:one-side} that
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:another-side}
\liminf_{r\to 0}\liminf_{i\to \infty} v_i(0)^2\mathcal{P}(r,v_i)\ge 0.
\ee
If $n \ge 10$, by \eqref{eq:IE-cond}, we have
\begin{align*}
U_{\lda}(x)&=\int_{B_{3/\mu_i}} G_{i,\mu_i}(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) U_{\lda}(y)\}\,\ud y+\mu_i^{\frac{n-4}{2}}c_{\lda/\mu_i,i}''(\mu_ix) \\& = \int_{B_3} \tilde G_i(x,y) \{U_\lda(y)^{\frac{n+4}{n-4}}+ \tilde c_{\lda,i}'(y) U_{\lda}(y)\}\,\ud y+\tilde c_{\lda,i}''(x) \quad \forall ~\lda\ge 1 ,~x\in B_3,
\end{align*}
where $c_{\lda,i}'(y):=\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) $ and
\[
\tilde c_{\lda,i}''(x) = \int_{B_{3/\mu_i}\setminus B_3} G_{i,\mu_i}(x,y)\{U_\lda(y)^{\frac{n+4}{n-4}}+\mu_i^4 c_{\lda/\mu_i,i}'(\mu_i y) U_{\lda}(y)\}\,\ud y+\mu_i^{\frac{n-4}{2}}c_{\lda/\mu_i,i}''(\mu_ix).
\]
By the assumptions for $c'_{\lda,i}$ and $c''_{\lda,i}$, we have
\[
\tilde \Theta_i:=\sum_{i=0}^5 \|\lda^{-k} \nabla^k \tilde c'_{\lda, i}\|_{L^\infty(B_3)} \le \mu_i^4\Theta_i,
\]
and $\|\tilde c_{\lda,i}''\|_{C^5(B_2)}\le CA_2 \lda^{\frac{4-n}{2}}$, where $C>0$ depends only on $n,A_1,A_2$. Clearly, we have $|W_{\tilde g_i}(0)|^2=\mu_i^4 |W_{ g_i}(0)|^2$. Hence \eqref{eq:r2} is satisfied. By Proposition \ref{prop:one-side}, we also have \eqref{eq:another-side}. We obtain a contradiction.
Therefore, $0$ must be an isolated simple blow up point of $u_i$
and the proof is completed.
\end{proof}
\begin{lem}\label{lem:isolated to isolated simple} Let $0\le u_i\in C^4(B_3)$ solve both \eqref{eq:s1'} and \eqref{eq:Q-sub} with $n\ge 10$, and assume \eqref{eq:IE-cond} holds. For $\mu_i\to 0$, let
\[
v_i(x)=\mu_i^{\frac{4}{p_i-1}} u_i(\mu_i x).
\]
Suppose that $0$ is an isolated blow up point of $\{v_i\}$ and \eqref{eq:r2} holds. Then $0$ is also an isolated simple blow up point.
\end{lem}
\begin{proof} From the end of proof of Proposition \ref{prop:isolated to isolated simple}, we see that the condition \eqref{eq:r2} is preserved under the scaling $v_i(x)=\mu_i^{\frac{4}{p_i-1}} u_i(\mu_i x)$. Hence, the lemma follows from Proposition \ref{prop:isolated to isolated simple}.
\end{proof}
\section{Global analysis, and proof of Theorem \ref{thm:energy}}
\label{section:thm1.1}
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. Suppose that $\mathrm{Ker} P_g=\{0\}$ and the Green's function $G_g$ of $P_g$ is positive. Consider the equation
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE}
P_gu= c(n)u^p, \quad u \ge 0 \quad \mbox{on } M,
\ee
where $1<p\le \frac{n+4}{n-4}$.
\begin{prop} \label{prop:reduction} Assume the above. For any given $R>0$ and $0\le \varepsilon} \newcommand{\ud}{\mathrm{d}<\frac{1}{n-4}$, there exist positive constants $C_0=C_0(M,g,R,\varepsilon} \newcommand{\ud}{\mathrm{d})$, $C_1=C_1(M, g,R,\varepsilon} \newcommand{\ud}{\mathrm{d})$ such that, for any smooth positive solution of \eqref{eq:sub DE} with
\[
\max_{M} u(X)\ge C_0,
\]
then $\frac{n+4}{n-4}-p<\varepsilon} \newcommand{\ud}{\mathrm{d}$ and there exists a set of finite distinct points
\[
\mathscr{S}(u):=\{Z_1,\dots,Z_N\}\subset M
\] such that the following statements are true.
(i) Each $Z_i$ is a local maximum point of $u$ and
\[
\overline{\B_{\bar r_i}(Z_i)} \cap \overline{\B_{\bar r_j}(Z_j)}=\emptyset \quad \mbox{for }i\neq j,
\]
where $\bar r_i=R u(Z_i)^{(1-p)/4}$, and $\B_{r_i}(Z_i)$ denotes the geodesic ball in $B_2$ centered at $Z_i$ with radius $\bar r_i$
(ii) For each $Z_i$,
\[
\left\| \frac{1}{u(Z_i)}u\left(\exp_{Z_i}\left(\frac{y}{u(Z_i)^{(p-1)/4}}\right)\right)-\left(\frac{1}{1+ |y|^2}\right)^{\frac{n-4}{2}}\right\|_{C^4(B_{2R})} <\varepsilon} \newcommand{\ud}{\mathrm{d}.
\]
(iii) $u(X)\le C_1 dist_g(x,\{Z_1,\dots,Z_N\})^{-4/(p-1)}$ for all $X\in M$.
\end{prop}
\begin{proof}
The proof is standard by now.
\end{proof}
\begin{prop}\label{prop:ruling out accumulation} If either $n\le 9$ or $(M,g)$ is locally conformally flat, then, for $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$, $R>1$ and any solution \eqref{eq:sub DE} with $\max_M u>C_0$, we have
\[
|Z_1-Z_2|\ge \delta^*>0 \quad \mbox{for any }Z_1,Z_2\in \mathscr{S}(u), ~Z_1\neq Z_2,
\]
where $\delta^*$ depends only on $M,g$.
\end{prop}
\begin{proof} Suppose the contrary, then there exist a sequence $0\le \frac{n+4}{n-4}-p_i<\varepsilon} \newcommand{\ud}{\mathrm{d}$ and $u_i$ satisfying \eqref{eq:sub DE} with $p=p_i$,
\[
\max_{M}u_i(X)> C_0,
\] and
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:e1}
dist_g(Z_{1i},Z_{2i})=\min_{1\le k,l\le N_i, ~k\neq l} dist_g(Z_{ki},Z_{li})\to 0
\ee
as $i\to \infty$, where $\mathscr{S}(u_i)=(Z_{1i},\dots, Z_{N_ii} )$ be the local maximum points of $u_i$ as selected by Proposition \ref{prop:reduction}. Without loss of generality, we may assume
\[
u_i(Z_{1i})\ge u_i(Z_{2i}).
\]
Since $\B_{Ru_i(Z_{1i})^{-(p_i-1)/4}}(Z_{1i})$ and $\B_{Ru_i(Z_{2i})^{-(p_i-1)/4}}(Z_{2i})$ have to be disjoint,
we have, because of \eqref{eq:e1}, that
$ u_i(Z_{1i})\to \infty$ and $ u_i(Z_{2i})\to \infty$.
Let $\{x_1,\dots, x_n\}$ be the conformal normal coordinates centered at $Z_{1i}$. We write \eqref{eq:sub DE} as
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE-a}
P_{ g_i}\tilde u_i=c(n) \kappa_i^{\tau_i}\tilde u_i^{p_i} \quad \mbox{on }M,
\ee
where $g_i=\kappa_i^{\frac{-4}{n-4}} g$, $\tilde u_i=\kappa_i u_i$, $\kappa_i>0$, $\kappa_i(Z_{1i})=1$, $\nabla_g \kappa_i(Z_{1i})=0$, and $\tau_i=\frac{n+4}{n-4}-p_i$.
Since $dist_g(Z_{1i},Z_{2i})\to 0$, for large $i$ we let $z_{2i} \in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ such that $\exp_{Z_{1i}} z_{2i}=Z_{2i}$, and let
\[
\vartheta_i:=|z_{2i}|\to 0.
\]
We will sit in the conformal normal coordinates chart $B_t$ at $Z_{1i}$, where $t>0$ is independent of $i$, and write $f(\exp_{Z_{1,i}}x)$ simply as $f(x)$.
Set
\[
\varphi_i(x)=\vartheta_i^{4/(p_i-1)}\tilde u_i(\vartheta_i x) \quad \mbox{for } |x|\le t/\vartheta_i.
\]
By the equation \eqref{eq:sub DE-a}, we have
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:sub DE-b}
P_{\tilde g_i} \varphi_i(x)=c(n)\tilde \kappa_i(x)^{\tau_i} \varphi_i(x)^{p_i} \quad \mbox{in }B_{t/\vartheta_i},
\ee
where $\tilde \kappa_i(x)=\kappa_i(\vartheta_i x)$, $\tilde g_i(x)=g_i(\vartheta_i x)$. Using the Green representation for \eqref{eq:sub DE-a},
\[
\tilde u_i(x)=c(n)\int_{B_t}G_i(x,y)\kappa_i(y)^{\tau_i} \tilde u_i(y)^{p_i}\,\ud y+h_i(x),
\]
where $G_i(x,y)=G_{g_i}(\exp_{Z_{1,i}}x,\exp_{Z_{1,i}}y)$ and
\[
h_i(x)=\int_{M\setminus \exp_{Z_{1,i}} B_t} G_{g_i}(\exp_{Z_{1,i}} x, Y)\kappa_i(Y)^{\tau_i}u_i(Y)\,\ud vol_{g_i}(Y).
\]
Hence, $\varphi_i$ also satisfies
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:e5}
\varphi_i(x)=\int_{B_{t/\vartheta_i}}G_{i,\vartheta_i}(x,y)K_i(\vartheta_iy)\varphi_i(y)^{p_i} \,\ud y+\tilde h_i(x)
\quad \mbox{for all } x\in B_{t/\vartheta_i},
\ee
where $G_{i,\vartheta_i}(x,y) =\vartheta_i^{n-4}G(\vartheta_ix,\vartheta_iy)$ and $\tilde h_i=
\vartheta_i^{4/(p_i-1)}h_i(\vartheta_i y)$.
By proposition \ref{prop:reduction},
we have
\begin{align*}
\tilde u_i(x)&\leq C_1^*|x|^{-4/(p_i-1)}\quad \mbox{for all }|x|\leq 3\vartheta_i/4,\\
\tilde u_i(x) &\le C_1^* |x-z_{2i}|^{-4/(p_i-1)} \quad \mbox{for all }|x-z_{2i}|\leq 3\vartheta_i/4,
\end{align*}
where $C_1^*$ depending only on $C_1$, $M$ and $g$.
Hence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{9-4}
\begin{split}
\varphi_i(x)&\leq C_1^*|x|^{-4/(p_i-1)}\quad \mbox{for all }|x|\leq 3/4,\\
\varphi_i(x) &\le C_1^* |x-\vartheta_i^{-1}z_{2i}|^{-4/(p_i-1)} \quad \mbox{for all }|x-\vartheta_i^{-1}z_{2i}|\leq 3/4.
\end{split}
\ee
Set $\xi_i=\vartheta_i^{-1}z_{2i}$. We claim that, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:9-5}
\varphi_i(0), \varphi_i(\xi_i)\to \infty \quad \mbox{as }i\to \infty.
\ee
It is clear that $\varphi_i(0)$ and $\varphi_i(\xi_i)$ are bounded from below by
some positive constant independent of $i$.
If there exists a subsequence (still denoted as $\varphi_i$) such that $\displaystyle\lim_{i\to \infty}\varphi_i(0)=\infty $ but $\varphi_i(\xi_i)$ stays bounded, we have that $0$ is an isolated blow up point for $\varphi_i$ in $B_{3/4}$ when $i$ is large; see Remark \ref{rem:blow}. Using equation \eqref{eq:e5} and \eqref{9-4}, by the same proof of \eqref{eq:scalbound2} we have $\sup_{B_{1/2}(\xi_i)} \varphi_i<\infty$. It follows from Proposition \ref{prop:upbound2} and Proposition \ref{prop:har} that $\displaystyle\lim_{i\to \infty}\varphi_i(\xi_i) = 0$, but this is impossible since $\vartheta_i>Ru_i(Z_{2i})^{-(p_i-1)/4} $ and thus $
\varphi_i(\xi_i)\ge \frac{1}{C}R$ for some $C>0$ depending only on $M $, $g$, $R$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$. On the other hand, if there exists a subsequence (still denoted as $\varphi_i$) such that $\varphi_i(0)$ and $\varphi_i(\xi_i)$ remain bounded, we know from a similar argument as above that $\varphi_i$ is locally bounded. The same proof of Proposition \ref{prop:blow up a bubble} yields that after passing to a subsequence $\varphi_i\to \varphi$ in $C^3_{loc}(\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n)$ for some $\varphi$ satisfying
\[
\varphi(x)=c_n \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} \frac{\varphi(y)^{\frac{n+4}{n-4}}}{|x-y|^{n-4}}\,\ud y \quad \mbox{for all }x\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n,
\]
$\nabla \varphi(0)=0$, $\nabla \varphi(\bar z)=\lim_{i\to \infty} \nabla \varphi_i(\xi_i)= \lim_{i\to \infty} \frac{\vartheta_i \varphi_{i}(\xi_i) \nabla \kappa_i(z_{2i})}{\kappa_i(z_{2i})} =0$, where $|\bar z|=1$ is the limit of $\xi_i$ up to passing a subsequence. This contradicts to the Liouville theorem in \cite{CLO} or Li \cite{Li04}. Hence, \eqref{eq:9-5} is proved.
Since $\nabla \varphi_i(0)=0$, it follows from the first inequality of \eqref{9-4} and \eqref{eq:9-5} that $0$ is an isolated blow up point of $\{\varphi_i\}$. Since $n\le 9$ or $(M,g)$ is locally conformally flat, by Proposition \ref{prop:isolated to isolated simple} we conclude that $0$ is an isolated simple blow up point of $\{\varphi_i\}$. It follows from Corollary \ref{cor:convergence} that for all $x\in B_{1/2}$
\begin{equation}} \newcommand{\ee}{\end{equation} \label{eq:e6}
\varphi_i(0)\int_{B_{1/2}}G_{i,\vartheta_i}(x,y)\varphi_i(y)^{p_i} K_i(\vartheta_i y)\,\ud y \to a c_{n}|x|^{4-n}
\ee
and
\[
\varphi_i(0) (Q_i''(x)+\tilde h_i(x))\to h(x)\ge 0 \quad \mbox{in } C^3_{loc}(B_{1/2}),
\] where $a>0$ is given in \eqref{eq:number a}, $h(x)\in C^5(B_{1/2})$ and
\[
Q''_i(x)=\int_{B_{t/\vartheta}\setminus B_{1/2}}G_{i,\vartheta_i}(x,y)\varphi_i(y)^{p_i}K_i(\vartheta_i y)\,\ud y \quad x\in B_{1/2}.
\]
Note that \[
\varphi_i(0)Q_i''(x) \ge \frac{1}{C} \varphi_i(0) \int_{B_{1/2}(\xi_i)}\varphi_i(y)^{p_i} \,\ud y.
\]
It follows from \eqref{9-4}, \eqref{eq:9-5} and the proof of
\eqref{eq:scalbound2} that there exists a constant $C>0$, depending only on $M,g,R $ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$ such that $\varphi_i(x)\le C \varphi_i(\xi_i)$ for all $|x-\xi_i|\le \frac{1}{2}$. It follows from the proof of Proposition \ref{prop:blow up a bubble} that there exist a constant $\lda$ and an point $x_0\in \mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n$ with $1\le \lda\le C$ and $|x_0|\le C$ such that for any fixed $\bar R>0$ we have
\[
\lim_{i\to \infty} \left \|\frac{1}{\varphi_i(\xi_i)} \varphi_i(\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x)-U_{\lda}(x-x_0) \right\|_{C^4(\bar B_{R})}=0
\]
By changing of variables $y=\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x$, we have
\begin{align*}
&\varphi_i(0) \int_{B_{1/2}(\xi_i)}\varphi_i(y)^{p_i} \,\ud y\\&=\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{B_{\varphi_i(\xi_i)/2}(0)}\left( \frac{1}{\varphi_i(\xi_i)} \varphi_i(\xi_i+\varphi_i(\xi_i)^{-(p_i-1)/4} x)\right)^{p_i} \,\ud x\\ &
\ge \frac{1}{C}\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} U_{\lda}(x-x_0)^{p_i}\,\ud x\\&
\ge \frac{1}{C}\varphi_i(0) \varphi_i(\xi_i)^{p_i-\frac{(p_i-1)n}{4}} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} (1+|x|^2)^{\frac{n+4}{2}}\,\ud x,
\end{align*}
where we used $1\le \lda\le C$ and $|x_0|\le C$. Since $u_i(Z_{1i})\ge u_i(Z_{2i})$, we have $ \varphi_i(0) \ge \frac{1}{C} \varphi_i(\xi_i)$ for some $C$ depending only on $M,g$. By Lemma \ref{lem:error}, we have $\varphi_i(\xi_i)^{1+p_i-\frac{(p_i-1)n}{4}}=1+o(1)$. Therefore, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}\label{eq:e7}
\lim_{i\to \infty} \varphi_i(0)Q_i''(x) \ge \frac{1}{C} \int_{\mathbb{R}} \newcommand{\Sn}{\mathbb{S}^n^n} (1+|\xi|^2)^{\frac{n+4}{2}}\,\ud \xi=:a_0>0.
\ee
In conclusion,
\[
\varphi_i(0)\varphi_i(x)\to ac_n|x|^{4-n}+h(x) \quad \mbox{in }C^3_{loc} (B_{1/2} \setminus \{0\})
\]
for some nonnegative bounded function in $C^3(B_{1/2})$ with $h(0)\ge a_0$.
It follows from Lemma \ref{lem:test-poho} that
\[
\liminf_{r\to 0}\liminf_{i\to \infty} \varphi_i^2\mathcal{P}(r,\varphi_i)<-(n-4)^2(n-2)aa_0c_n|\mathbb{S}^{n-1}|.
\]
On the other hand, notice that $\varphi_i$ satisfies \eqref{eq:sub DE-a}. We also have Corollary \ref{cor:GM2.8}. It follows from Proposition \ref{prop:one-side} that
\[
\liminf_{r\to 0}\liminf_{i\to \infty} \varphi_i(0)^2\mathcal{P}(r,\varphi_i)\ge 0.
\]
We arrive at a contradiction. Therefore, \eqref{eq:e1} is not valid and the proposition follows.
\end{proof}
Theorem \ref{thm:energy} is a part of the following theorem.
\begin{thm}\label{thm:final-a} Let $u_i\in C^4(M)$ be a sequences of positive solutions of $P_g u_i=c(n)u_i^{p_i}$ on $M$, where $0\le (n+4)/(n-4)-p_i \to 0$ as $i\to \infty$. Assume \eqref{condition:main2}. If either $n\le 9$ or $(M,g)$ is locally conformally flat, then
\[
\|u_i\|_{H^2(M)} \le C,
\]
where $C>0$ depending only on $M,g$. Furthermore, after passing to a subsequence, $\{u_i\}$ is uniformly bounded or has only isolated simple blow up points and the distance between any two blow up points is bounded below by some positive constant depending only on $M,g$.
\end{thm}
\begin{rem}\label{rem:isolated on manifolds} On $(M,g)$, we say a point $\bar X\in M$ is an isolated blow up point for $\{u_i\}$ if there exists a sequence $X_i\in M$, where each $X_i$ is a local maximum point for $u_i$ and $X_i\to \bar X$, such that $u_i(X_i)\to \infty$ as $i\to \infty$ and $u_i(X) \le C dist_g(X,X_i)^{-\frac{4}{p_i-1}} $ in $\B_{\delta}(X_i)$ for some constants $C,\delta>0$ independent of $i$. Under the assumptions that $u_i$ is a positive solution of $P_{g}u_i=c(n)u_i^{p_i}$ with $0\le \frac{n+4}{n-4}-p_i\to 0$, $\mathrm{Ker} P_g=\{0\}$ and that the Green's function $G_g$ of $P_g$ is positive, it is easy to see that if $X_i\to \bar X\in M$ is an isolated blow up point of $\{u_i\}$, then in the conformal normal coordinates centered at $X_i$, $0$ is an isolated blow up point of $\{\tilde u_i(\exp_{X_i} x)\}$, where the exponential map is with respect to conformal metric $g_i=\kappa_i^{\frac{-4}{n-4}}g$, $\kappa_i>0$ is under control on $M$, and $\tilde u_i=\kappa_i u_i$; see Remark \ref{rem:blow}.
Since in Theorem \ref{thm:final-a} and the sequel those assumptions will always be assumed, the notation of isolated simple blow up points on manifolds is understood in conformal normal coordinates.
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:final-a}] The last statement follows immediately from Proposition \ref{prop:reduction}, Proposition \ref{prop:ruling out accumulation} and Proposition \ref{prop:isolated to isolated simple}. Consequently, it follows from Corollary \ref{cor:energy} and Proposition \ref{prop:upbound2} and Proposition \ref{prop:har} that $\int_{M} u_i^{\frac{2n}{n-4}}\,\ud vol_g\le C$. By the Green's representation and standard estimates for Riesz potential, we have the $H^2$ estimates.
\end{proof}
Now we consider that $n\ge 10$.
\begin{prop}\label{prop:r} Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 10$.
Assume (\ref{condition:main2}).
Let $u_i$ be a sequence of positive solutions of $P_g u=c(n)u^{p_i}$,
where $p_i\le \frac{n+4}{n-4}$,
$p_i\to \frac{n+4}{n-4}$ as $i\to \infty$. Suppose that there is a sequence $X_i\to \bar X\in M$ such that $u_i(X_i)\to \infty$. For any small $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $R>1$, let $\mathscr{S}(u_i)$ denote the set selected as in Proposition \ref{prop:reduction} for $u_i$. If $|W_g(\bar X)|^2\ge \varepsilon} \newcommand{\ud}{\mathrm{d}_0>0$ on $M$ for some constant $\varepsilon} \newcommand{\ud}{\mathrm{d}_0$,
then there exists $\delta^*>0$ depending only on $M,g$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}_0$ such that $\B_{\delta^*}(\bar X)\cap \mathscr{S}(u_i)$ contains precisely one point.
\end{prop}
\begin{proof}
Let $\bar\delta>0$ such that $|W_g(X)|^2\ge \varepsilon} \newcommand{\ud}{\mathrm{d}_0/2$
for $X\in \B_{\bar \delta}(\bar X)$.
Assume the contrary of the proposition, then for a
subsequence
of $\{u_i\}$ (still denoted as $\{u_i\}$)
there exist distinct points $X_{1i},\hat X_{1i}\in \mathscr{S}(u_i)$ such that $X_{1i}, \hat X_{1i}\to \bar X$.
Define $f_i:\mathscr{S}(u_i)\to (0,\infty)$ by
\[
f_i(X):=\min_{X'\in \mathscr{S}(u_i)\setminus \{X\}} dist_g(X',X).
\]Let $R_i\to \infty$ satisfying $R_i f_i(X_{1i})\to 0$.
\textbf{Claim.} There exists a subsequence of $i\to \infty$ such that one
can find $X_{i}'\in \mathscr{S}(u_i)\cap \B_{\bar \delta/9}(\bar X)$ satisfying
\[
f_i(X_i') \le (2R_i+1)f_i(X_{1i})
\]
and
\[
\min_{X\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_i')} (X_i')} f_i(X) \ge \frac12 f_i(X_i').
\]
Indeed, suppose the contrary, then there exists $I\in \mathbb{N}$ such that for any $i\ge I$, $X_i'$ in the claim can not been selected. Since $f_i(X_{1i})\le (2R_i+1) f_i(X_{1i})$, by the contradiction hypothesis, there must exist $X_{2i}\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_{1i})} (X_{1i})$ such that $f_{i}(X_{2i})<\frac12 f_{i}(X_{1i})$. We can define $X_{li}\in \mathscr{S}(u_i)$, $l=3\dots$, satisfying
$f_i(X_{li})<\frac{1}{2} f_i(X_{(l-1)i})$
and $0<dist_g(X_{li},X_{(l-1)i})< R_i f_i(X_{(l-1)i})$
inductively as follows. Once $X_{li}$, $l\ge 2
$, is defined, we have, for $2\le m\le l$, that
\[
dist_g(X_{mi},X_{(m-1)i})<R_i f_i(X_{(m-1)i})<R_i 2^{-1} f_i(X_{(m-2)i})<\cdots <R_i 2^{2-m} f_i(X_{1i}),
\]
which implies
$$
dist_g(X_{li},X_{1i})\le \sum_{m=2}^l dist_g(X_{mi},X_{(m-1)i})
< R_i f_i(X_{1i}) \sum_{m=2}^l 2^{2-m}
< 2R_i f_i(X_{1i}),
$$
and
$$
f_i(X_{li})\le dist_g(X_{li}, X_{1i})+f_i(X_{1i})\le (2R_i+1) f_i(X_{1i}).
$$
so $X_i':=X_{li}$ satisfies
$X_{i}'\in \mathscr{S}(u_i)\cap \B_{\bar \delta/9}(\bar X)$
and the first inequality
of the claim.
By the contradiction hypothesis, there must exist $X_{(l+1)i}\in \mathscr{S}(u_i)\cap \B_{R_i f_i(X_{li})} (X_{li})$ such that $f_{i}(X_{(l+1)i})<\frac12 f_{i}(X_{li})$.
But $\mathscr{S}(u_i)$ is a finite set and we can not work for all $l\ge 2$. Therefore, the claim follows.
By the claim, we can follow the proof of Proposition \ref{prop:ruling out accumulation} with $Z_{1i}$ replaced by $X_i'$. We then derive a contradiction to Proposition \ref{prop:one-side}. Therefore, we complete the proof.
\end{proof}
\section{Proof of Theorems \ref{thm:main theorem}, Theorem \ref{thm:main-b}, Theorem \ref{thm:main-c}}
\label{section:thm1.2}
\begin{proof}[Proof of Theorem \ref{thm:main-b}] For $n=8,9$, since $u_i(X_i)\to \infty$, it follows from Proposition \ref{prop:ruling out accumulation} and Theorem \ref{thm:final-a} that
$\bar X$ is an
isolated simple blow up points of $\{u_i\}$. Then Theorem \ref{thm:main-b} follows from item (ii) of Proposition \ref{prop:one-side}. When $n\ge 10$, we may argue by contradiction. Suppose that $|W_{g}(\bar X)|^2>0$.
By Proposition \ref{prop:r},
Proposition \ref{prop:isolated to isolated simple},
and in view of the proof of
(\ref{eq:scalbound2}) under (\ref{eq:scalbound}),
that there exists a sequence of $X_i'\to \bar X$ which is an isolated simple blow up point of $\{u_i\}$; see Remark \ref{rem:isolated on manifolds}. By item (ii) of Proposition \ref{prop:one-side}, we have $|W_g(X_i')|^2 \to 0$. It gives
$|W_{g}(\bar X)|^2=0$. We obtain a contradiction. Hence, we complete the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main-c}] Suppose the contrary,
then, after passing to a subsequence,
\begin{equation}} \newcommand{\ee}{\end{equation} \label{c-hy}
|W_g(X_i)|^2> \frac 1{ |o(1)| } \begin{cases} u_i(X_i)^{-\frac{4}{n-4}}\log u_i(X_i), &\quad \mbox{if }n=10,\\
u_i(X_i)^{-\frac{4}{n-4}}, &\quad \mbox{if }n\ge 11.
\end{cases}
\ee
For any $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ and $R>1$, let $\mathscr{S}(u_i)=\{Z_{1i},\dots, Z_{N_i i}\}$ be the set selected as in Proposition \ref{prop:reduction} for $u_i$, where $N_i\in \mathbb{N}^+$. Let, without loss of generality,
\[
dist_g(X_i, Z_{2i}) =\inf_{Z_{ji}\in \mathscr{S}(u_i), Z_{ji}\neq X_i}dist_g(X_i, Z_{ji}).
\] If there exists a constant $\delta^*>0$
independent of $i$
such that $dist_g(X_i, Z_{2i})\ge \delta^*$, then $X_i\in \mathscr{S}(u_i)$
for large $i$. It follows from item (iii) of Proposition \ref{prop:reduction} and Proposition \ref{prop:isolated to isolated simple} that $X_i\to \bar X$ has to be an isolated simple blow up point of $\{u_i\}$,
using the fact that
\eqref{c-hy} guarantees
\eqref{eq:r2}. By item (ii) of Proposition \ref{prop:one-side}, we obtain an opposite side inequality of
\eqref{c-hy}. Contradiction.
If $dist_g(X_i, Z_{2i}) \to 0$ as $i\to \infty$. Let $\{x_1,\dots, x_n\}$ be the conformal normal coordinates centered at $X_i$. Define $\varphi_i$ as that in the proof of Proposition \ref{prop:ruling out accumulation} with $Z_{1i}$ replaced by $X_i$. Since $\sup_{\Omega} \newcommand{\pa}{\partial}u_i\le \bar bu_i(X_i)$, we must have $\varphi_i(0) \to \infty$ by the Liouville theorem in \cite{CLO} or \cite{Li04}; see the proof of \eqref{eq:9-5}.
Because of \eqref{eq:technical condition 12} and
\eqref{eq:technical condition},
$0$ has to be an isolated blow up point of $\{\varphi_i\}$; see Remark \ref{rem:blow}. It follows from the contradiction hypothesis
\eqref{c-hy}, which guarantees \eqref{eq:r2},
as in the proof of
Lemma \ref{lem:isolated to isolated simple} that $0$ has to be an isolated simple blow up point of $\{\varphi_i\}$ for $i$ large. Then we we arrive at a contradiction by item (ii) of Proposition \ref{prop:one-side} again.
Therefore, we complete the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main theorem}]
If $n\ge 8$ and $|W_g|^2>0$ on $M$, Theorem \ref{thm:main theorem} is a direct corollary of Theorem \ref{thm:main-b}. Hence, we only need to consider $n\le 9$ or $(M,g)$ is locally conformally flat.
By Proposition \ref{prop:reduction}, it suffices to consider that $p$ is close to $\frac{n+4}{n-4}$. Suppose the contrary that there exists a sequences of positive solutions $u_i\in C^4(M)$ of $P_g u_i=c(n)u_i^{p_i}$ on $M$, where $p_i\to (n+4)/(n-4)$ as $i\to \infty$, such that $\max_{M}u_i\to \infty$. By Theorem \ref{thm:final-a}, let $X_i\to \bar X\in M$ be an isolated simple blow up point of $\{u_i\}$; see Remark \ref{rem:isolated on manifolds}. It follows from Proposition \ref{prop:one-side} that, in the $g_{\bar X}$-normal coordinates centered at $\bar X$,
\[
\liminf_{r\to 0} \mathcal{P}(r, c(n)G)\ge 0,
\]
where $g_{\bar X}$ a conformal metric of $g$ with $\det g_{\bar X}=1$ in an open ball $B_{\delta}$ of the $g_{\bar X}$-normal coordinates, $G(x)=G_{g_{\bar X}}(\bar X,\exp_{\bar X} x)$ and $G_{g_{\bar X}}$ is the Green's function of $P_{g_{\bar X}}$.
On the other hand, if $n=5,6,7$ or $(M,g)$ is locally conformally flat, by Theorem \ref{thm:positive mass} and Lemma \ref{lem:test-poho} we have
\[
\mathcal{P}(r, c(n)G) <-A \quad \mbox{for small }r,
\]
where $A>0$ depends only on $M,g$. We obtain a contradiction.
If $n=8,9$, by Theorem \ref{thm:main-b} we have $W_g(\bar X)=0$. In view of Remark \ref{rem:positive mass}, we have
\[
\lim_{r\to 0}\mathcal{P}(r,G)=\begin{cases} -2 \Xint-_{\mathbb{S}^{n-1}} \psi(\theta), &\quad n=8, \\
-\frac{5}{2}A, &\quad n=9,
\end{cases}
\]
where $\psi(\theta)$ and $A$ are as in Remark \ref{rem:positive mass}.
If the positive mass type theorem holds for Paneitz operator in dimension $n=8,9$, we obtain $\lim_{r\to 0}\mathcal{P}(r,G)<0$. Again, we derived a contradiction.
Therefore, $u_i$ must be uniformly bounded and the proof is completed.
\end{proof}
\section{Proof of Theorem \ref{thm:compact1}}
This section will not use previous analysis and thus is independent. The proof of Theorem \ref{thm:compact1} is divided into two steps.
\emph{Step 1}. $L^p$ estimate. Let $u\ge 0$ be a solution of \eqref{eq:-Q}. Integrating both sides of \eqref{eq:-Q} and using H\"older inequality, we have
\[
\int_{M}u^p \ud vol_g =\left| -\int_{M} uP_g(1) \ud vol_g\right| \le \frac{n-4}{2} \|u\|_{L^p(M)} \|Q_g\|_{L^{p'}(M)},
\]
where $\frac{1}{p'}+\frac1p=1$. It follows that $\|u\|_{L^p(M)} ^{p-1} \le \frac{n-4}{2} \|Q_g\|_{L^{p'}(M)}$.
\emph{Step 2.} If $\mathrm{Ker} P_{g}=\{0\}$, there exist a unique Green function of $P_g$. If the kernel of $P_{g}$ is non-trivial, since the spectrum of Paneitz operator is discrete, there exists a small constant $\varepsilon} \newcommand{\ud}{\mathrm{d}>0$ such that the kernel of $P_g-\varepsilon} \newcommand{\ud}{\mathrm{d}$ is trivial. Let $G_g$ be the Green function of the operator $P_g-\varepsilon} \newcommand{\ud}{\mathrm{d}$, where $\varepsilon} \newcommand{\ud}{\mathrm{d}\ge 0$. Then there exists a constant $\delta>0$, depending only $M,g$ and $\varepsilon} \newcommand{\ud}{\mathrm{d}$, such that, for every $X\in M$, we have $G(X,Y)>0$ for $Y\in \mathcal{B}_{\delta}(X)$ and $|G_g(X,Y)|\le C(\delta,\varepsilon} \newcommand{\ud}{\mathrm{d})$ for $Y\in M\setminus \mathcal{B}_{\delta}(X)$. Rewrite the equation of $u$ as
\[
P_g u-\varepsilon} \newcommand{\ud}{\mathrm{d} u=-(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u).
\]
It follows from the Green representation theorem that
\begin{align*}
u(X)& =-\int_{M}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\&
=-\int_{\mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\& \quad -\int_{M\setminus \mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y)\\&
\le -\int_{M\setminus \mathcal{B}_{\delta}(X)}G_g(X,Y)(u^p+\varepsilon} \newcommand{\ud}{\mathrm{d} u)(Y)\ud vol_g(Y) \\&
\le C\max\{\|u\|_{L^p(M)}^p, \|u\|_{L^p(M)}\} \le C.
\end{align*}
By the arbitrary choice of $X$, we have $\|u\|_{L^\infty}\le C$. The higher order estimate follows from the standard linear elliptic partial differential equation theory; see Agmon-Douglis-Nirenberg \cite{ADN}.
Therefore, we complete the proof.
|
1,314,259,993,045 | arxiv | \section{Introduction}
Inspired by Lawvere's pioneering work \cite{Lawvere1973} which presents metric spaces as enriched categories, during the past decades category theory has been playing an important role in the study of metric spaces and their generalizations. Lawvere's construction, in modern terms, is based on the \emph{quantale}
$$[0,\infty]_+=([0,\infty],+,0)$$
whose underlying complete lattice is the extended non-negative real line $[0,\infty]$ equipped with the order ``$\geq$''. Categories enriched in the quantale $[0,\infty]_+$, i.e., sets $X$ equipped with a map $\al:X\times X\to[0,\infty]$ such that
\begin{enumerate}[label=(M\arabic*),leftmargin=2.8em]
\item \label{I-M:d}
$\al(x,x)=0$, and
\item \label{I-M:t}
$\al(x,z)\leq\al(y,z)+\al(x,y)$
\end{enumerate}
for all $x,y,z\in X$, are precisely \emph{(generalized) metric spaces}; that is, (classical) metric spaces dropping the requirements of \emph{symmetry} ($\al(x,y)=\al(y,x)$), \emph{finiteness} ($\al(x,y)<\infty$) and \emph{separatedness} ($\al(x,y)=\al(y,x)=0\iff x=y$).
The motivation of this paper comes from two important generalizations of metric spaces, i.e., \emph{probabilistic metric spaces} \cite{Menger1942,Schweizer1983} (also known as \emph{fuzzy metric spaces} \cite{George1994,Kramosil1975}) and \emph{partial metric spaces} \cite{Bukatin2009,Matthews1994}, both of which can be understood as enriched categories.
Probabilistic metric spaces are metric spaces in which the distance is defined as a map $\phi:[0,\infty]\to[0,1]$ with
$$\phi(t)=\bv_{s<t}\phi(s)$$
for all $t\in[0,\infty]$, called a \emph{distance distribution}, rather than a non-negative real number. As discovered by Chai \cite{Chai2009} and investigated by Hofmann-Reis \cite{Hofmann2013a}, (generalized) probabilistic metric spaces are categories enriched in the quantale
$$\De_*=(\De,\otimes_*,\ka_{0,1})$$
of distance distributions w.r.t. a continuous t-norm $*$ on the unit interval $[0,1]$ (see Proposition \ref{De-quantale} for further explanations) which, in elementary words, are precisely sets $X$ equipped with a map $\al:X\times X\times[0,\infty]\to[0,1]$ such that
\begin{enumerate}[label=(ProbM\arabic*),leftmargin=4.6em,start=0]
\item \label{I-ProbM:d}
$\al(x,y,-):[0,\infty]\to[0,1]$ is a distance distribution,
\item \label{I-ProbM:r}
$\al(x,x,t)=1$ for all $t>0$, and
\item \label{I-ProbM:t}
$\al(y,z,r)*\al(x,y,s)\leq\al(x,z,r+s)$
\end{enumerate}
for all $x,y,z\in X$ and $r,s\in[0,\infty]$.
Partial metric spaces are metric spaces in which the distance from a point to itself may not be zero. Explicitly, (generalized) partial metric spaces are sets $X$ equipped a with a map $\al:X\times X\to[0,\infty]$ such that
\begin{enumerate}[label=(PM\arabic*),leftmargin=3.3em]
\item \label{I-PM:d}
$\al(x,x)\vee \al(y,y)\leq \al(x,y)$, and
\item \label{I-PM:t}
$\al(x,z)\leq\al(y,z)-\al(y,y)+\al(x,y)$
\end{enumerate}
for all $x,y,z\in X$. As discovered by H{\"o}hle-Kubiak \cite{Hoehle2011a} and Pu-Zhang \cite{Pu2012} and formalized later by Stubbe \cite{Stubbe2014} in a more general setting, although partial metric spaces are not categories enriched in the quantale $[0,\infty]_+$, one may construct a \emph{quantaloid of diagonals} of the quantale $[0,\infty]_+$, usually denoted by $\sD[0,\infty]_+$, such that partial metric spaces are precisely categories enriched in the quantaloid $\sD[0,\infty]_+$.
It is now natural to ask whether it is possible to consider the probabilistic version of partial metric spaces or, equivalently, the partial version of probabilistic metric spaces. As far as we know, this topic has been considered recently by several authors under the name \emph{probabilistic partial metric spaces} or \emph{fuzzy partial metric spaces}; see, e.g. \cite{Amer2016,Sedghi2015,Wu2017,Yue2015,Yue2014}. Following the categorical interpretation of partial metric spaces, this paper aims at a full-scale investigation of the quantaloid
$$\sD\De_*$$
of diagonals of the quantale $\De_*$ of distance distributions w.r.t. a continuous t-norm $*$ on $[0,1]$, so that a categorical foundation of probabilistic partial metric spaces can be established, which was more or less neglected in the existing references.
In general, let $\sQ=(\sQ,\with,1)$ be a commutative and integral quantale (i.e., complete residuated lattice), in which
$$p\with q\leq r\iff p\leq q\ra r$$
for all $p,q,r\in\sQ$. With $\DQ$ denoting the quantaloid of diagonals of $\sQ$ (see Proposition \ref{DQ-def}), a \emph{$\DQ$-category} (also called \emph{partial $\sQ$-category}, see \cite{Hofmann2016}) consists of a set $X$ and a map $\al:X\times X\to\sQ$ such that
\begin{enumerate}[label=(\arabic*)]
\item $\al(x,y)$ is a diagonal between $\al(x,x)$ and $\al(y,y)$, and
\item $(\al(y,y)\ra\al(y,z))\with\al(x,y)=\al(y,z)\with(\al(y,y)\ra\al(x,y))\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$.
By taking $\sQ=[0,\infty]_+$ we will see that $\sD[0,\infty]_+$-categories are exactly partial metric spaces, as pointed out in \cite{Hoehle2011a,Pu2012}. In this case, it is easy to determine diagonals between non-negative real numbers; indeed, $\al(x,y)$ is a diagonal between $\al(x,x)$ and $\al(y,y)$ if, and only if, $\al(x,x)\vee \al(y,y)\leq \al(x,y)$, which is precisely the condition \ref{I-PM:d} for partial metric spaces $(X,\al)$. However, the elegant form of diagonals between non-negative real numbers relies heavily on the fact that the quantale $[0,\infty]_+$ is \emph{divisible}. In fact, in every divisible quantale $\sQ$ it holds that
\begin{equation} \label{I-DQpq-div}
\DQ(p,q)=\,\da\!(p\wedge q)
\end{equation}
for all $p,q\in\sQ$; that is, diagonals between $p$ and $q$ are exactly the down set generated by $p\wedge q$. Unfortunately, Proposition \ref{De-non-div} tells us that the quantale $\De_*$ of distance distributions w.r.t. any continuous t-norm $*$ is non-divisible, and therefore the following question becomes a tough one for the quantale $\De_*$:
\begin{ques}
What are the diagonals between distance distributions?
\end{ques}
With necessary preparations in Section \ref{Preliminaries}, we investigate the quantale $\De_*$ and the quantaloid $\sD\De_*$ thoroughly in Sections \ref{Delta} and \ref{DDelta}, and the main results of this paper answer the above question from two aspects:
\begin{itemize}
\item Theorem \ref{dd-div} characterizes diagonals between any pair of distance distributions in the quantale $\De_*$ w.r.t. an arbitrary continuous t-norm $*$.
\item Theorem \ref{step-only-divisible-lower} shows that the characterization of diagonals obtained from Theorem \ref{dd-div} cannot be simplified to Equation \eqref{I-DQpq-div} unless the involved distance distributions intersect to yield one-step functions, whose proof is the most challenging one in this paper.
\end{itemize}
As an application of these results, in Definition \ref{ProbPM} we propose a rigorous definition of probabilistic partial metric spaces w.r.t. a continuous t-norm $*$ through $\sD\De_*$-categories. Finally, we attach an appendix with some interesting results about the categorical connections between $\sQ$-categories and $\DQ$-categories which, in particular, reveal the interactions between (probabilistic) metric spaces and their partial version.
\section{Diagonals between non-negative real numbers: Partial metric spaces as enriched categories} \label{Preliminaries}
A \emph{(generalized) partial metric space} \cite{Bukatin2009,Matthews1994,Pu2012} is a set $X$ that comes equipped with a map $\al:X\times X\to[0,\infty]$ such that
\begin{enumerate}[label=(PM\arabic*),leftmargin=3.3em]
\item \label{PM:d}
$\al(x,x)\vee \al(y,y)\leq \al(x,y)$, and
\item \label{PM:t}
$\al(x,z)\leq\al(y,z)-\al(y,y)+\al(x,y)$
\end{enumerate}
for all $x,y,z\in X$. In particular, a partial metric space $(X,\al)$ satisfying $\al(x,x)=0$ for all $x\in X$ is exactly a \emph{(generalized) metric space} in the sense of Lawvere \cite{Lawvere1973}; that is, a set $X$ equipped with a map $\al:X\times X\to[0,\infty]$ such that
\begin{enumerate}[label=(M\arabic*),leftmargin=2.7em]
\item \label{M:r}
$\al(x,x)=0$, and
\item \label{M:t}
$\al(x,z)\leq\al(y,z)+\al(x,y)$
\end{enumerate}
for all $x,y,z\in X$.
\begin{rem} \label{ParMet-classical}
As noted above, our terminology of partial metric spaces here naturally extends Lawvere's notion of generalized metric spaces; so, they are more precisely \emph{generalized} partial metric spaces in the sense of Pu-Zhang \cite{Pu2012}. Besides the conditions \ref{PM:d} and \ref{PM:t}, the notion of ``partial metric'' originally introduced by Matthews \cite{Matthews1994} additionally requires $\al$ to be \emph{symmetric} ($\al(x,y)=\al(y,x)$), \emph{finitary} ($\al(x,y)<\infty$) and \emph{separated} ($\al(x,x)=\al(y,y)=\al(x,y)=\al(y,x)\iff x=y$).
\end{rem}
Although it has been well known for decades that metric spaces can be studied as enriched categories \cite{Lawvere1973} or, more precisely, \emph{quantale}-enriched categories, it is only until recently that partial metric spaces are also understood as enriched categories in a more general framework, where a \emph{quantaloid} is considered as the base for enrichment \cite{Hoehle2011a,Pu2012}.
Explicitly, a \emph{quantaloid} $\CQ$ is a locally ordered category whose hom-sets are complete lattices, such that the composition $\circ$ of $\CQ$-arrows preserves suprema in each variable, i.e.,
$$v\circ\Big(\bv_{i\in I} u_i\Big)=\bv_{i\in I}v\circ u_i\quad\text{and}\quad\Big(\bv_{i\in I} v_i\Big)\circ u=\bv_{i\in I}v_i\circ u$$
for all $\CQ$-arrows $u,u_i:p\to q$, $v,v_i:q\to r$ $(i\in I)$. Hence, the corresponding right adjoints induced by the composition maps
$$-\circ u\dv -\lda u:\ \CQ(p,r)\to\CQ(q,r)\quad\text{and}\quad v\circ -\dv v\rda -:\ \CQ(p,r)\to\CQ(p,q)$$
satisfy
$$v\circ u\leq w\iff v\leq w\lda u\iff u\leq v\rda w$$
for all $\CQ$-arrows $u:p\to q$, $v:q\to r$, $w:p\to r$, where the operations $\lda$ and $\rda$ are called \emph{left} and \emph{right implications} in $\CQ$, respectively.
Given a \emph{small} quantaloid $\CQ$ (i.e., $\CQ$ has a set $\CQ_0$ of objects), a \emph{$\CQ$-category} (or, a \emph{category enriched in $\CQ$}) consists of a set $X$, a map $|\text{-}|:X\to\CQ_0$, and a family of $\CQ$-arrows $\al(x,y)\in\CQ(|x|,|y|)$ $(x,y\in X)$, subject to
$$1_{|x|}\leq \al(x,x)\quad\text{and}\quad \al(y,z)\circ \al(x,y)\leq \al(x,z)$$
for all $x,y,z\in X$. A \emph{$\CQ$-functor} $f:(X,\al)\to(Y,\be)$ between $\CQ$-categories $(X,\al)$, $(Y,\be)$ is a map $f:X\to Y$ such that
$$|x|=|fx|\quad\text{and}\quad \al(x,y)\leq \be(fx,fy)$$
for all $x,y\in X$. The category of $\CQ$-categories and $\CQ$-functors is denoted by $\QCat$.
A one-object quantaloid is precisely a \emph{(unital) quantale}. Let $\with$ denote the multiplication in a quantale $\sQ$, i.e., the composition of arrows in the unique hom-set. We say that
\begin{itemize}
\item $\sQ$ is \emph{commutative} if $p\with q=q\with p$ for all $p,q\in\sQ$;
\item $\sQ$ is \emph{integral} if the unit $1$ of $\sQ$ is also the top element of the complete lattice $\sQ$.
\end{itemize}
Throughout this paper, we let $\sQ=(\sQ,\with,1)$ denote a commutative and integral quantale, which is also known as a \emph{complete residuated lattice}, and we write
$$p\ra q:=q\lda p=p\rda q$$
for implications in $\sQ$. In what follows we are particularly interested in the quantaloid of \emph{diagonals} of $\sQ$:
\begin{prop} \label{DQ-def} (See \cite{Hoehle2011a,Pu2012,Stubbe2014}.)
The following data defines a quantaloid $\DQ$ of ``diagonals of $\sQ$'':
\begin{itemize}
\item objects of $\DQ$ are elements $p,q,r,\dots$ in $\sQ$;
\item for $p,q\in\sQ$, a morphism $d:p\rqa q$ in $\DQ$, called a \emph{diagonal} from $p$ to $q$, is an element $d\in\sQ$ with
\begin{equation} \label{diagonal-def}
(p\ra d)\with p=d=q\with(q\ra d);
\end{equation}
\item for diagonals $d:p\rqa q$, $e:q\rqa r$, the composition $e\diamond d:p\rqa r$ is given by
\begin{equation} \label{diagonal-comp}
e\diamond d=(q\ra e)\with d=e\with(q\ra d);
\end{equation}
\item $q:q\rqa q$ is the identity diagonal on $q$.
\end{itemize}
\end{prop}
Given $q\in\sQ$, following the terminology in \cite{Hoehle2015}, an element $d\in\sQ$ is said to be \emph{divisible by $q$} if there exists $p\in\sQ$ such that $q\with p=d$. Note that
\begin{equation} \label{diagonal-divisible}
q\with(q\ra d)=d\iff\exists p\in\sQ:\ q\with p=d.
\end{equation}
Hence, from \eqref{diagonal-def} we see that $d:p\rqa q$ is a diagonal if, and only if, $d$ is divisible by $p$ and $q$. In particular, $d$ is divisible by $q$ if, and only if, $d:q\rqa q$ is a diagonal on $q$.
We say that the quantale $\sQ$ is \emph{divisible} \cite{Hajek1998,Hoehle1995a} if $d$ is a diagonal on $q$ (or equivalently, $d$ is divisible by $q$) whenever $d\leq q$ in $\sQ$. It is easy to check that divisible quantales are necessarily integral (see, e.g., the proof of \cite[Proposition 2.1]{Pu2012}).
\begin{rem} \label{diagonal}
The construction of $\DQ$ does not rely on the commutativity or the integrality of $\sQ$. In fact, one may construct the quantaloid $\sD\CQ$ of ``diagonals of $\CQ$'' for any quantaloid $\CQ$ as considered in \cite{Stubbe2014}. Explicitly, a $\CQ$-arrow $d:p_1\to q_2$ with
$$(d\lda u)\circ u=d=v\circ(v\rda d),$$
$$\bfig
\square<700,500>[p_1`p_2`q_1`q_2;v\rda d`u`v`d\lda u]
\morphism(0,500)/-->/<700,-500>[p_1`q_2;d]
\efig$$
denoted by $d:u\rqa v$, is called a \emph{diagonal} between $\CQ$-arrows $u$ and $v$, which is precisely the \emph{diagonal} of the above commutative square. The composition of diagonals $d:u\to v$, $e:v\to w$ is given by
\begin{align*}
e\diamond d&=(e\lda v)\circ d=(e\lda v)\circ(d\lda u)\circ u\\
&=e\circ(v\rda d)=w\circ(w\rda e)\circ(v\rda d).
\end{align*}
$$\bfig
\iiixii<700,500>[p_1`p_2`p_3`q_1`q_2`q_3;v\rda d`w\rda e`u`v`w`d\lda u`e\lda v]
\morphism(0,500)/-->/<700,-500>[p_1`q_2;d]
\morphism(700,500)/-->/<700,-500>[p_2`q_3;e]
\iiixii(2000,0)/->`->`->``->`->`->/<700,500>[p_1`p_2`p_3`q_1`q_2`q_3;v\rda d`w\rda e`u``w`d\lda u`e\lda v]
\morphism(2000,500)/-->/<1400,-500>[p_1`q_3;e\diamond d]
\morphism(2000,500)|b|/-->/<700,-500>[p_1`q_2;d]
\morphism(2700,500)/-->/<700,-500>[p_2`q_3;e]
\place(1700,250)[\mapsto]
\efig$$
It is straightforward to check that $\CQ$-arrows and diagonals constitute a quantaloid $\sD\CQ$. In particular for a commutative and integral quantale $\sQ$, if we denote by $\star$ the single object of the one-object quantaloid $\sQ$, then the commutative square
$$\bfig
\square<700,500>[\star`\star`\star`\star;q\ra d`p`q`p\ra d]
\morphism(0,500)/-->/<700,-500>[\star`\star;d]
\efig$$
illustrates a diagonal $d:p\rqa q$ in $\DQ$ defined by Equation \eqref{diagonal-def}.
\end{rem}
Applying the definition of $\CQ$-categories to the special case of $\CQ=\DQ$, one sees that a $\DQ$-category consists of a set $X$, a map $|\text{-}|:X\to\sQ$, and a map $\al:X\times X\to\sQ$, such that
\begin{enumerate}[label=(\arabic*)]
\item \label{DQ-Cat:d}
$\al(x,y)=|x|\with(|x|\ra\al(x,y))=|y|\with(|y|\ra\al(x,y))$, i.e., $\al(x,y)\in\DQ(|x|,|y|)$,
\item \label{DQ-Cat:r}
$|x|\leq\al(x,x)$,
\item \label{DQ-Cat:t}
$(|y|\ra\al(y,z))\with\al(x,y)=\al(y,z)\with(|y|\ra\al(x,y))\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$. Since $\sQ$ is integral, it is easy to deduce that
\begin{equation} \label{DQpq}
\DQ(p,q)\subseteq\,\da\!(p\wedge q)
\end{equation}
for all $p,q\in\sQ$, where $\da\!(p\wedge q)$ is the down set generated by $p\wedge q$. Hence, the condition \ref{DQ-Cat:d} together with \eqref{DQpq} implies $\al(x,x)\leq|x|$ for all $x\in X$ and, moreover, the condition \ref{DQ-Cat:r} forces $|x|=\al(x,x)$. Therefore:
\begin{prop} \label{integral-DQCat}
A map $\al:X\times X\to\sQ$ defines a $\DQ$-category structure on a set $X$ if, and only if,
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{DQ-Cat:s}
$\al(x,y)=\al(x,x)\with(\al(x,x)\ra\al(x,y))=\al(y,y)\with(\al(y,y)\ra\al(x,y))$, i.e., $\al(x,y)$ is a diagonal between $\al(x,x)$ and $\al(y,y)$,
\item \label{DQ-Cat:tt}
$(\al(y,y)\ra\al(y,z))\with\al(x,y)=\al(y,z)\with(\al(y,y)\ra\al(x,y))\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$. In this case, one has
$$\al(x,y)\leq\al(x,x)\wedge\al(y,y)$$
for all $x,y\in X$.
\end{prop}
If the quantale $\sQ$ is divisible, then every $d\leq p\wedge q$ is a diagonal between $p,q\in\sQ$, which in conjunction with \eqref{DQpq} forces
\begin{equation} \label{DQpq-div}
\DQ(p,q)=\,\da\!(p\wedge q)
\end{equation}
for all $p,q\in\sQ$; that is, $d:p\rqa q$ is a diagonal if, and only if, $d\leq p\wedge q$. In this case, one may further simplifies Proposition \ref{integral-DQCat} to the following:
\begin{prop} \label{divisible-DQCat} (See \cite{Pu2012}.)
If $\sQ$ is divisible, then a map $\al:X\times X\to\sQ$ defines a $\DQ$-category structure on a set $X$ if, and only if,
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{DQ-Cat:ss}
$\al(x,y)\leq\al(x,x)\wedge\al(y,y)$, i.e., $\al(x,y)$ is a diagonal between $\al(x,x)$ and $\al(y,y)$,
\item \label{DQ-Cat:ttt}
$(\al(y,y)\ra\al(y,z))\with\al(x,y)=\al(y,z)\with(\al(y,y)\ra\al(x,y))\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$.
\end{prop}
In order to exhibit partial metric spaces as $\DQ$-categories, let us now look at the well-known Lawvere's quantale (see \cite{Lawvere1973})
$$[0,\infty]_+=([0,\infty],+,0),$$
where $[0,\infty]$ is the extended non-negative real line equipped with the order ``$\geq$'' (so that $0$ becomes the top element and $\infty$ the bottom element), and ``$+$'' is the usual addition extended via
\begin{equation} \label{infty-add}
p+\infty=\infty+p=\infty
\end{equation}
to $[0,\infty]$. Note that the implication in $[0,\infty]_+$ is given by
$$p\ra q=\begin{cases}
0 & \text{if}\ p\geq q,\\
q-p & \text{if}\ p<q
\end{cases}$$
for all $p,q\in[0,\infty]$, where the subtraction ``$-$'' is extended via
\begin{equation} \label{infty-minus}
\infty-p=\begin{cases}
\infty & \text{if}\ p<\infty,\\
0 & \text{if}\ p=\infty
\end{cases}
\end{equation}
to $[0,\infty]$. Hence, it is easy to see that $[0,\infty]_+$ is a commutative and divisible quantale, and thus a diagonal $d:p\rqa q$ between non-negative real numbers $p,q\in[0,\infty]$ is precisely a real number $d\in[0,\infty]$ with $p\vee q\leq d$. Consequently, the quantaloid
$$\sD[0,\infty]_+$$
of diagonals in $[0,\infty]_+$ has $[0,\infty]$ as the object-set, and its hom-sets are given by
$$\sD[0,\infty]_+(p,q)=\{d\in[0,\infty]\mid p\vee q\leq d\}.$$
By Proposition \ref{divisible-DQCat}, a $\sD[0,\infty]_+$-category structure $\al:X\times X\to[0,\infty]$ on a set $X$ is exactly a partial metric on $X$ described by \ref{PM:d} and \ref{PM:t}. With non-expanding maps $f:(X,\al)\to(Y,\be)$ between partial metric spaces as morphisms, i.e., maps $f:X\to Y$ with
$$\al(x,x)=\be(fx,fx)\quad\text{and}\quad\be(fx,fy)\leq\al(x,y)$$
for all $x,y\in X$, one obtains the category
$$\ParMet:=\sD[0,\infty]_+\text{-}\Cat,$$
which contains the category
$$\Met:=[0,\infty]_+\text{-}\Cat$$
of metric spaces as a full subcategory.
\begin{rem}
In general, $\sQ$-categories are studied as \emph{$\sQ$-preordered sets} in the fuzzy community \cite{Bvelohlavek2004,Denniston2014,Hoehle2015,Lai2006,Lai2009,Shen2013}, while $\DQ$-categories are also referred to as \emph{$\sQ$-preordered $\sQ$-subsets} (or \emph{preordered fuzzy sets}) in the literatures \cite{GutierrezGarcia2017,Pu2012,Shen2016b,Shen2013b}. Therefore, metric spaces are a special kind of $\sQ$-preordered sets, while partial metric spaces can be regarded as examples of $\sQ$-preordered $\sQ$-subsets.
\end{rem}
\section{The quantale of distance distributions w.r.t. a continuous t-norm} \label{Delta}
Probabilistic metric spaces, as defined by \cite{Menger1942,Schweizer1983}, are a generalization of metric spaces in which the distance is defined as a \emph{distance distribution} rather than a non-negative real number, and it has been known in \cite{Chai2009,Clementino2017,Hofmann2013a} that probabilistic metric spaces can be considered as categories enriched in the quantale of distance distributions.
Explicitly, a map $\phi:[0,\infty]\to[0,1]$ is called a \emph{distance distribution} if
$$\phi(t)=\bv_{s<t}\phi(s)$$
for all $t\in[0,\infty]$. In other words, $\phi:[0,\infty]\to[0,1]$ is a distance distribution if, and only if,
\begin{enumerate}[label=(\arabic*)]
\item $\phi(0)=0$,
\item $\phi$ is monotone\footnote{The monotonicity here refers to the usual order ``$\leq$'' on $[0,\infty]$; that is, $s\leq t$ in $[0,\infty]$ implies $\phi(s)\leq\phi(t)$ in $[0,1]$.}, and
\item $\phi$ is left-continuous on $(0,\infty]$.
\end{enumerate}
The set $\De$ of all distance distributions becomes a complete lattice under the pointwise order given by
$$\phi\leq\psi\iff\forall t\in[0,\infty]:\ \phi(t)\leq\psi(t)$$
for all $\phi,\psi\in\De$. The following lemma introduces a canonical procedure of generating distance distributions:
\begin{lem} \label{monotone-dd}
For every map $\phi:[0,\infty]\to[0,1]$, the map
$$\phi^-:[0,\infty]\to[0,1],\quad\phi^-(t)=\bv_{s<t}\bw_{r>s}\phi(r)$$
is a distance distribution. In particular, if $\phi:[0,\infty]\to[0,1]$ is monotone, then
$$\phi^-(t)=\bv_{s<t}\phi(s)$$
for all $t\in[0,\infty]$, and $\phi^-$ is the largest distance distribution that does not exceed the monotone map $\phi$.
\end{lem}
It is often useful to consider a special family of distance distributions, called \emph{one-step functions}, defined by
$$\ka_{p,a}:[0,\infty]\to[0,1],\quad\ka_{p,a}(t)=\begin{cases}
0 & \text{if}\ t\leq p,\\
a & \text{if}\ t>p
\end{cases}$$
for $p\in[0,\infty)$, $a\in[0,1]$.\footnote{We exclude the case of $p=\infty$ in the definition of one-step functions to avoid inconsistency in future discussions; for example, Lemma \ref{De-calc}\ref{De-calc:monotone} may fail to be true if $p=\infty$. Indeed, even if $p=\infty$ is taken into consideration, it is easy to see that $\ka_{\infty,a}=\ka_{0,0}$ for all $a\in[0,1]$.} In particular, $\ka_{0,1}$ and $\ka_{0,0}$ are respectively the top and the bottom elements of $\De$. Moreover, distance distributions can always be written as suprema of one-step functions:
\begin{lem} \label{De-step-join}
For every $\phi\in\De$,
$$\phi=\bv_{a<\phi(p)}\ka_{p,a}=\bv_{p\in[0,\infty)}\ka_{p,\phi(p)}.$$
\end{lem}
Recall that the unit interval $[0,1]$ equipped with a continuous t-norm $*$ \cite{Klement2000,Klement2004} is a commutative and divisible quantale
$$[0,1]_*=([0,1],*,1).$$
It is well known \cite{Faucett1955,Klement2000,Klement2004a,Klement2004b,Mostert1957} that every continuous t-norm $*$ on $[0,1]$ can be written as an ordinal sum of three basic t-norms, i.e., the minimum, the product, and the {\L}ukasiewicz t-norm:
\begin{itemize}
\item (Minimum t-norm) $[0,1]_{\wedge}=([0,1],\wedge,1)$, in which $p\ra_{\wedge} q=\begin{cases}
1 & \text{if}\ p\leq q\\
q & \text{if}\ p>q
\end{cases}$ for all $p,q\in[0,1]$.
\item (Product t-norm) $[0,1]_{\times}=([0,1],\times,1)$, where $\times$ is the usual multiplication on $[0,1]$, and $p\ra_{\times} q=\begin{cases}
1 & \text{if}\ p\leq q\\
\dfrac{q}{p} & \text{if}\ p>q
\end{cases}$ for all $p,q\in[0,1]$. The quantales $[0,1]_{\times}$ and $[0,\infty]_+$ are clearly isomorphic.
\item ({\L}ukasiewicz t-norm) $[0,1]_{\opl}=([0,1],\opl,1)$, in which $p\opl q=\max\{0,p+q-1\}$, and $p\ra_{\opl} q=\min\{1,1-p+q\}$ for all $p,q\in[0,1]$.
\end{itemize}
The \emph{convolution} of monotone maps $\phi,\psi:[0,\infty]\to[0,1]$ w.r.t. a continuous t-norm $*$ is defined as a (necessarily monotone) map $\phi\otimes_*\psi:[0,\infty]\to[0,1]$ with
$$(\phi\otimes_*\psi)(t)=\bv_{r+s=t}\phi(r)*\psi(s).$$
\begin{rem} \label{monotone-convolution-rep}
It is easy to see that
\begin{equation} \label{monotone-convolution-finite}
(\phi\otimes_*\psi)(t)=\bv_{s\leq t}\phi(s)*\psi(t-s)=\bv_{s\leq t}\phi(t-s)*\psi(s)
\end{equation}
for all $t\in[0,\infty)$, but
$$(\phi\otimes_*\psi)(\infty)=\phi(\infty)*\psi(\infty)$$
may not satisfy Equation \eqref{monotone-convolution-finite} since $\infty-\infty=0$; see \eqref{infty-add} and \eqref{infty-minus} for the addition and subtraction involving $\infty$.
\end{rem}
\begin{lem} \label{convolution-dd}
For monotone maps $\phi,\psi:[0,\infty]\to[0,1]$, it holds that
\begin{equation} \label{monotone-convolution-dd}
\phi^-\otimes_*\psi^-=(\phi\otimes_*\psi)^-.
\end{equation}
In particular, if $\phi$ is a distance distribution, then
\begin{equation} \label{monotone-convolution-dd-finite}
(\phi\otimes_*\psi^-)(t)=(\phi\otimes_*\psi)(t)
\end{equation}
for all $t\in[0,\infty)$.
\end{lem}
\begin{proof}
It is obvious that $(\phi^-\otimes_*\psi^-)(0)=(\phi\otimes_*\psi)^-(0)=0$, and
$$(\phi^-\otimes_*\psi^-)(\infty)=\bv_{r<\infty}\phi(r)*\bv_{s<\infty}\psi(s)=\bv_{r+s<\infty}\phi(r)*\psi(s)=\bv_{t<\infty}\bv_{r+s=t}\phi(r)*\psi(s)
=\bv_{t<\infty}(\phi\otimes_*\psi)(t)=(\phi\otimes_*\psi)^-(\infty).$$
It remains to prove the case of $t\in(0,\infty)$. Since $\phi^-\otimes_*\psi^-\leq\phi\otimes_*\psi$, from Lemma \ref{monotone-dd} one immediately obtains $\phi^-\otimes_*\psi^-\leq(\phi\otimes_*\psi)^-$. For the reverse inequality, one needs to show that
$$\bv_{s<t}\bv_{r\leq s}\phi(r)*\psi(s-r)\leq\bv_{p\leq t}\phi^-(p)*\psi^-(t-p)$$
by Equation \eqref{monotone-convolution-finite}. Indeed, whenever $r\leq s<t<\infty$, let $q=r+\dfrac{t-s}{2}$, then $r<q<t$ and $s-r<t-q$. It follows that
$$\phi(r)*\psi(s-r)\leq\Big(\bv_{p<q}\phi(p)\Big)*\Big(\bv_{p<t-q}\psi(p)\Big)=\phi^-(q)*\psi^-(t-q)\leq\bv_{p\leq t}\phi^-(p)*\psi^-(t-p).$$
In particular, if $\phi\in\De$, then $(\phi\otimes_*\psi^-)(0)=(\phi\otimes_*\psi)(0)=0$, and
\begin{align*}
(\phi\otimes_*\psi)(t)&=\bv_{s\leq t}\phi(t-s)*\psi(s)&(\text{Equation \eqref{monotone-convolution-finite}})\\
&=\bv_{s<t}\phi(t-s)*\psi(s)&(\phi\in\De\implies\phi(0)=0)\\
&=\bv_{s<r<t}\phi(r-s)*\psi(s)&(\phi\in\De\implies\phi\ \text{is left-continuous})\\
&=\bv_{r<t}(\phi\otimes_*\psi)(r)\\
&=(\phi\otimes_*\psi)^-(t)\\
&=(\phi\otimes_*\psi^-)(t)&(\text{Equation \eqref{monotone-convolution-dd}})
\end{align*}
for all $t\in(0,\infty)$.
\end{proof}
\begin{rem}
As pointed out by an anonymous referee, Equation \eqref{monotone-convolution-dd-finite} in Lemma \ref{convolution-dd} may not be true for $t=\infty$. For example, let $\phi=\ka_{0,1}$, and let $\psi:[0,\infty]\to[0,1]$ be the monotone map given by
$$\psi(t)=\begin{cases}
0 & \text{if}\ t\in[0,\infty),\\
1 & \text{if}\ t=\infty.
\end{cases}$$
Then $(\phi\otimes_*\psi^-)(\infty)=(\phi\otimes_*\ka_{0,0})(\infty)=\ka_{0,0}(\infty)=0$, but $(\phi\otimes_*\psi)(\infty)=\phi(\infty)*\psi(\infty)=1$.
\end{rem}
If $\phi,\psi\in\De$, it is clear that
\begin{align*}
(\phi\otimes_*\psi)(t)&=\bv_{r+s=t}\phi(r)*\psi(s)=\bv_{s\leq t}\phi(s)*\psi(t-s)=\bv_{s\leq t}\phi(t-s)*\psi(s)\\
&=\bv_{r+s<t}\phi(r)*\psi(s)=\bv_{s<t}\phi(s)*\psi(t-s)=\bv_{s<t}\phi(t-s)*\psi(s)
\end{align*}
for all $t\in[0,\infty]$, and thus $\phi\otimes_*\psi\in\De$. With $\ka_{0,1}$ being the neutral element for $\otimes_*$ one actually defines a quantale structure on $\De$:
\begin{prop} \label{De-quantale}
$\De_*=(\De,\otimes_*,\ka_{0,1})$ is a commutative and integral quantale.
\end{prop}
\begin{rem}
As revealed by \cite{GutierrezGarcia2017a}, the quantale $\De_*$ of distance distributions is the \emph{tensor product} of Lawvere's quantale $[0,\infty]_+$ and the quantale $[0,1]_*$ of continuous t-norm $*$ on $[0,1]$ in the category $\Sup$ of complete lattices and $\sup$-preserving maps.
\end{rem}
A \emph{(generalized) probabilistic metric space} \cite{Chai2009,Hofmann2013a,Hofmann2014} w.r.t. a continuous t-norm $*$ or, equivalently, a $\De_*$-category, is then defined as a set $X$ equipped with a map
$$\al:X\times X\to\De,$$
called the \emph{probabilistic distance function}, subject to the following conditions, for all $x,y,z\in X$ and $r,s\in[0,\infty]$:
\begin{enumerate}[label=(ProbM\arabic*),leftmargin=4.6em]
\item \label{ProbM:r}
$\al(x,x)(t)=1$ for all $t>0$; that is, $\al(x,x)=\ka_{0,1}$.
\item \label{ProbM:t}
$\al(y,z)(r)*\al(x,y)(s)\leq\al(x,z)(r+s)$; that is, $\al(y,z)\otimes_*\al(x,y)\leq\al(x,z)$.
\end{enumerate}
With $\De_*$-functors $f:(X,\al)\to(Y,\be)$ as morphisms, i.e., maps $f:X\to Y$ with
$$\al(x,y)(t)\leq\be(fx,fy)(t)$$
for all $x,y\in X$, $t\in[0,\infty]$, we obtain the category
$$\ProbMet_*:=\De_*\text{-}\Cat.$$
\begin{rem} \label{ProbMet-classical}
Similarly as we remarked in \ref{ParMet-classical}, a probabilistic metric space is classically defined as a pair $(X,\al)$ satisfying \ref{ProbM:r}--\ref{ProbM:t} and, additionally, the following conditions \cite{Menger1942,Schweizer1983} for all $x,y\in X$:
\begin{enumerate}[label=(ProbM\arabic*),start=3,leftmargin=4.6em]
\item \label{ProbM:sym}
(symmetry) $\al(x,y)=\al(y,x)$;
\item \label{ProbM:f}
(finiteness) $\al(x,y)(\infty)=1$;
\item \label{ProbM:sep}
(separatedness) $\al(x,y)(t)=\al(y,x)(t)=1$ for all $t>0$ implies $x=y$; that is, $\al(x,y)=\al(y,x)=\ka_{0,1}$ implies $x=y$.
\end{enumerate}
\end{rem}
\section{Diagonals between distance distributions: Probabilistic partial metric spaces as enriched categories} \label{DDelta}
It is now natural to define the partial version of probabilistic metric spaces, i.e., probabilistic partial metric spaces, as categories enriched in the quantaloid $\sD\De_*$ of ``diagonals of $\De_*$''. However, the following fact indicates that the structure of $\sD\De_*$ would be far more complicated than what was described by \eqref{DQpq-div}:
\begin{prop} \label{De-non-div}
The quantale $\De_*=(\De,\otimes_*,\ka_{0,1})$ is non-divisible.
\end{prop}
\begin{proof}
Let $\phi,\xi\in\De$ be given by
$$\phi(t):=\begin{cases}
t & \text{if}\ 0\leq t\leq 1,\\
1 & \text{if}\ t>1
\end{cases}\quad\text{and}\quad
\xi(t):=\begin{cases}
0 & \text{if}\ 0\leq t\leq 1,\\
1 & \text{if}\ t>1.
\end{cases}$$
Then $\xi<\phi$, but there is no $\psi\in\De$ with $\xi=\phi\otimes_*\psi$. Indeed, if such $\psi$ exists, then
$$\xi\Big(\dfrac{4}{3}\Big)=\bv_{s<\frac{4}{3}}\phi\Big(\dfrac{4}{3}-s\Big)*\psi(s)=1.$$
Hence, for each $p\in\Big(0,\dfrac{1}{3}\Big)$ one may find $s_p<\dfrac{4}{3}$ such that
\begin{align*}
\phi\Big(\dfrac{4}{3}-s_p\Big)*\psi(s_p)>1-p&\implies\dfrac{4}{3}-s_p\geq\phi\Big(\dfrac{4}{3}-s_p\Big)>1-p\quad\text{and}\quad\psi(s_p)>1-p\\
&\implies s_p<\dfrac{1}{3}+p<\dfrac{2}{3}\quad\text{and}\quad\psi(s_p)>1-p,
\end{align*}
and consequently
$$\psi\Big(\dfrac{2}{3}\Big)\geq\psi(s_p)>1-p$$
for all $p\in\Big(0,\dfrac{1}{3}\Big)$, which forces $\psi\Big(\dfrac{2}{3}\Big)=1$. It follows that
$$0=\xi(1)=\bv_{s<1}\phi(1-s)*\psi(s)\geq\phi\Big(\dfrac{1}{3}\Big)*\psi\Big(\dfrac{2}{3}\Big)=\dfrac{1}{3}*1=\dfrac{1}{3},$$
giving a contradiction.
\end{proof}
Therefore, although we know from \eqref{DQpq} and the integrality of the quantale $\De_*$ that $\xi:\phi\rqa_*\psi$ is a diagonal of $\De_*$ only if
\begin{equation} \label{xi-leq-phi-psi}
\xi\leq\phi\wedge\psi,
\end{equation}
it requires more efforts to determine which distance distributions below $\phi\wedge\psi$ are actually diagonals between $\phi$ and $\psi$. To achieve this, let us first describe implications in the quantale $\De_*$. As a preparation we list here some rules that are needed in the calculations later on:
\begin{lem} \phantomsection \label{De-calc}
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{De-calc:step-comp} (See \cite{Hofmann2013a} for the case of $*=\times$)
$\ka_{p,a}\otimes_*\ka_{q,b}=\ka_{p+q,a*b}$ for all $p,q\in[0,\infty)$, $a,b\in[0,1]$.
\item \label{De-calc:monotone}
If $\phi:[0,\infty]\to[0,1]$ is a monotone map, then
$$a\leq\bw_{r>0}\phi(p+r)\iff\ka_{p,a}\leq\phi^-$$
for all $p\in[0,\infty)$, $a\in[0,1]$.
\item \label{De-calc:imp}
Let $\phi,\xi,\theta\in\De$. Then $\phi\otimes_*\psi\leq\xi\iff\psi\leq\theta$ for all $\psi\in\De$ if, and only if,
$$\phi\otimes_*\ka_{p,a}\leq\xi\iff\ka_{p,a}\leq\theta$$
for all $p\in[0,\infty)$, $a\in[0,1]$.
\end{enumerate}
\end{lem}
\begin{proof}
\ref{De-calc:step-comp} Follows immediately from the definition of one-step functions and convolutions.
\ref{De-calc:monotone} From the definition of one-step functions one soon has $a\leq\dbw_{r>0}\phi(p+r)\iff\ka_{p,a}\leq\phi$, and $\ka_{p,a}\leq\phi\iff\ka_{p,a}\leq\phi^-$ since $\phi^-$ is the largest distance distribution less than or equal to $\phi$ (see Lemma \ref{monotone-dd}).
\ref{De-calc:imp} Since $\phi\otimes_* -$ preserves suprema, one derives the non-trivial direction immediately by Lemma \ref{De-step-join}.
\end{proof}
Let $\ra_*$ denote the implication in the quantale $[0,1]_*$. We start by investigating implications of one-step functions in the quantale $\De_*$:
\begin{prop} \label{step-implication} (See \cite{Hofmann2013a} for the case of $*=\times$)
For any $\xi\in\De$ and $p\in[0,\infty)$, $a\in[0,1]$, the implication $\ka_{p,a}\Ra_*\xi$ in the quantale $\De_*$ is given by
$$(\ka_{p,a}\Ra_*\xi)(t)=\bv_{s<t}a\ra_*\xi(p+s).$$
In particular,
$$\ka_{p,a}\Ra_*\ka_{r,c}=\ka_{\max\{0,r-p\},\,a\ra_* c}$$
for all $r\in[0,\infty)$, $c\in[0,1]$.
\end{prop}
\begin{proof}
The map $\theta:[0,\infty]\to[0,1]$ with $\theta(t)=a\ra_*\xi(p+t)$ is clearly monotone. By Lemma \ref{De-calc}\ref{De-calc:imp} it suffices to prove
$$\ka_{p,a}\otimes_*\ka_{q,b}\leq\xi\iff\ka_{q,b}\leq\theta^-$$
for all $q\in[0,\infty)$, $b\in[0,1]$. Indeed,
\begin{align*}
\ka_{p,a}\otimes_*\ka_{q,b}\leq\xi&\iff\ka_{p+q,a*b}\leq\xi&(\text{Lemma \ref{De-calc}\ref{De-calc:step-comp}})\\
&\iff a*b\leq\bw_{r>0}\xi(p+q+r)&(\text{Lemma \ref{De-calc}\ref{De-calc:monotone}})\\
&\iff b\leq a\ra_*\bw_{r>0}\xi(p+q+r)=\bw_{r>0}\theta(q+r)\\
&\iff\ka_{q,b}\leq\theta^-,&(\text{Lemma \ref{De-calc}\ref{De-calc:monotone}})
\end{align*}
as desired. In particular, if $\xi=\ka_{r,c}$, then for all $q\in[0,\infty)$, $b\in[0,1]$ one has
\begin{align*}
\ka_{p,a}\otimes_*\ka_{q,b}\leq\ka_{r,c}&\iff\ka_{p+q,a*b}\leq\ka_{r,c}&(\text{Lemma \ref{De-calc}\ref{De-calc:step-comp}})\\
&\iff r\leq p+q\ \text{and}\ a*b\leq c\\
&\iff\max\{0,r-p\}\leq q\ \text{and}\ b\leq a\ra_*c\\
&\iff\ka_{q,b}\leq\ka_{\max\{0,r-p\},\,a\ra_* c}.
\end{align*}
Hence $\ka_{p,a}\Ra_*\ka_{r,c}=\ka_{\max\{0,r-p\},\,a\ra_* c}$, again by Lemma \ref{De-calc}\ref{De-calc:imp}.
\end{proof}
Now we turn to the general case. For $\phi,\xi\in\De$, define a map
\begin{equation} \label{rho-def}
\rho_*(\phi,\xi):[0,\infty]\to[0,1],\quad\rho_*(\phi,\xi)(t)=\bw_{q>0}\phi(q)\ra_*\xi(q+t).
\end{equation}
If we interpret $a\ra_* b$ as the ``distance'' of $a,b\in[0,1]$ w.r.t. the t-norm $*$, then $\rho_*(\phi,\xi)(t)$ gives the ``smallest vertical distance'' between the graphs of $\phi$ and the monotone map obtained by shifting $\xi$ to the left for $t$ units. Then the implication in $\De_*$ will be given by
$$\phi\Ra_*\xi=\rho_*(\phi,\xi)^-:$$
\begin{thm} \label{De-implication}
Let $\phi,\xi\in\De$. Then the implication $\phi\Ra_*\xi$ in the quantale $\De_*$ is given by
$$(\phi\Ra_*\xi)(t)=\rho_*(\phi,\xi)^-(t)=\bv_{s<t}\bw_{q>0}\phi(q)\ra_*\xi(q+s)$$
for all $t\in[0,\infty]$.
\end{thm}
\begin{proof}
By Lemma \ref{De-calc}\ref{De-calc:imp} it suffices to prove
$$\phi\otimes_*\ka_{p,a}\leq\xi\iff\ka_{p,a}\leq\rho_*(\phi,\xi)^-$$
for all $p\in[0,\infty)$, $a\in[0,1]$. Indeed,
\begin{align*}
\phi\otimes_*\ka_{p,a}\leq\xi&\iff\forall q\in(0,\infty):\ \ka_{q,\phi(q)}\otimes_*\ka_{p,a}\leq\xi&(\text{Lemma \ref{De-step-join}})\\
&\iff\forall q\in(0,\infty):\ \ka_{p+q,a*\phi(q)}\leq\xi&(\text{Lemma \ref{De-calc}\ref{De-calc:step-comp}})\\
&\iff\forall q\in(0,\infty):\ a*\phi(q)\leq\bw_{r>0}\xi(p+q+r)&(\text{Lemma \ref{De-calc}\ref{De-calc:monotone}})\\
&\iff\forall q\in(0,\infty):\ a\leq\phi(q)\ra_*\bw_{r>0}\xi(p+q+r)\\
&\iff a\leq\bw_{\substack{q>0\\r>0}}\phi(q)\ra_*\xi(p+q+r)=\bw_{r>0}\rho_*(\phi,\xi)(p+r)\\
&\iff\ka_{p,a}\leq\rho_*(\phi,\xi)^-,&(\text{Lemma \ref{De-calc}\ref{De-calc:monotone}})
\end{align*}
which completes the proof.
\end{proof}
Now we are ready to characterize diagonals of the quantale $\De_*$. Recall from \eqref{diagonal-def} that $\xi:\phi\rqa_*\psi$ is a diagonal if, and only if, $\xi:\phi\rqa_*\phi$ is a diagonal on $\phi$ and $\xi:\psi\rqa_*\psi$ is a diagonal on $\psi$; that is, $\xi$ is simultaneously divisible by $\phi$ and $\psi$. Hence, along with \eqref{xi-leq-phi-psi} it suffices to find those distance distributions below a given $\phi\in\De$ that are diagonals on $\phi$. First, let us look at the case when $\phi$ is a one-step function:
\begin{prop} \label{step-divisible}
Let $p\in[0,\infty)$, $a\in[0,1]$. Then $\xi\in\De$ is a diagonal on $\ka_{p,a}$ in the quantale $\De_*$ if, and only if, $\xi\leq\ka_{p,a}$.
\end{prop}
\begin{proof}
It suffices to show that every $\xi\leq\ka_{p,a}$ is divisible by $\ka_{p,a}$. Define $\theta:[0,\infty]\to[0,1]$ with $\theta(t)=a\ra_*\xi(p+t)$ as in the proof of Proposition \ref{step-implication}. Then, by Equation \eqref{monotone-convolution-finite}, for any $t\in[0,\infty)$ one has
$$(\ka_{p,a}\otimes_*\theta)(t)=\bv_{s\leq t}\ka_{p,a}(s)*(a\ra_*\xi(p+t-s))=\bv_{p<s\leq t}a*(a\ra_*\xi(p+t-s))=\bv_{p<s\leq t}\xi(p+t-s)=\xi(t),$$
where the penultimate equality follows from $\xi(p+t-s)\leq a$ and the divisibility of the quantale $[0,1]_*$. Thus
$$(\ka_{p,a}\otimes_*\theta^-)(t)=(\ka_{p,a}\otimes_*\theta)(t)=\xi(t)$$
for all $t\in[0,\infty)$ by Lemma \ref{convolution-dd}. Since $\ka_{p,a}\otimes_*\theta^-$ and $\xi$, as distance distributions, are both left-continuous at $\infty$,
$$(\ka_{p,a}\otimes_*\theta^-)(\infty)=\xi(\infty)$$
follows immediately. Hence $\ka_{p,a}\otimes_*\theta^-=\xi$, showing that $\xi$ is divisible by $\ka_{p,a}$.
\end{proof}
For a general $\phi\in\De$, diagonals on $\phi$ in $\De_*$ usually constitute a proper subset of $\da\phi$ (as we will see in Theorem \ref{step-only-divisible-lower} below), and they are characterized as follows:
\begin{thm} \label{dd-div}
Let $\phi\in\De$. Then $\xi\in\De$ is a diagonal on $\phi$ in the quantale $\De_*$ if, and only if,
$$\xi(t)=\bv_{s<t}\bw_{q>0}\phi(t-s)*(\phi(q)\ra_*\xi(q+s))$$
for all $t\in[0,\infty)$.
\end{thm}
Note that $\xi\in\De$ is a diagonal on $\phi$ if, and only if, $\xi=\phi\otimes_*(\phi\Ra_*\xi)$, which is equivalent to say that
$$\xi(t)=(\phi\otimes_*(\phi\Ra_*\xi))(t)$$
for all $t\in[0,\infty)$ for the same argument as in the last part of the proof of Proposition \ref{step-divisible}. Therefore, Theorem \ref{dd-div} is an immediate consequence of the following lemma:
\begin{lem} \label{dd-diagonal-comp}
For $\phi,\psi,\xi\in\De$,
$$(\psi\otimes_*(\phi\Ra_*\xi))(t)=\bv_{s<t}\bw_{q>0}\psi(t-s)*(\phi(q)\ra_*\xi(q+s))$$
for all $t\in[0,\infty)$.
\end{lem}
\begin{proof}
From Lemma \ref{convolution-dd} one sees that
$$(\psi\otimes_*(\phi\Ra_*\xi))(t)=(\psi\otimes_*\rho_*(\phi,\xi)^-)(t)=(\psi\otimes_*\rho_*(\phi,\xi))(t)$$
for all $t\in[0,\infty)$, where $\rho_*(\phi,\xi)$ is defined by Equation \eqref{rho-def}. Since
$$(\psi\otimes_*\rho_*(\phi,\xi))(t)=\bv_{s\leq t}\psi(t-s)*\rho_*(\phi,\xi)(s)=\bv_{s<t}\bw_{q>0}\psi(t-s)*(\phi(q)\ra_*\xi(q+s))$$
for all $t\in[0,\infty)$ by Equation \eqref{monotone-convolution-finite}, the conclusion thus follows.
\end{proof}
\begin{exmp}
It follows soon from \eqref{diagonal-divisible} that the convolution
$$\phi\otimes_*\psi$$
is a diagonal between $\phi,\psi\in\De$ in the quantale $\De_*$. In particular,
$$\{\phi\otimes_*\psi\mid\psi\in\De\}$$
gives a set of diagonals on $\phi$. Hence, $\phi\otimes_*\psi$ must satisfy the condition given in Theorem \ref{dd-div}; indeed, for any $t\in[0,\infty)$,
$$\bv_{s<t}\phi(t-s)*\psi(s)=(\phi\otimes_*\psi)(t)\leq\bv_{s<t}\bw_{q>0}\phi(t-s)*(\phi(q)\ra_*(\phi\otimes_*\psi)(q+s))$$
since
$$\forall q,s\in[0,\infty]:\ \phi(q)*\psi(s)\leq(\phi\otimes_*\psi)(q+s)$$
implies
$$\forall s\in[0,\infty]: \psi(s)\leq\bw_{q>0}\phi(q)\ra_*(\phi\otimes_*\psi)(q+s).$$
\end{exmp}
Here we present an alternative characterization for diagonals of the quantale $\De_{\wedge}$. Note that each distance distribution $\phi:[0,\infty]\to[0,1]$ is $\sup$-preserving, and thus admits a right adjoint
$$\phi^{\flat}:[0,1]\to[0,\infty],\quad\phi^{\flat}(a):=\bv\{p\in[0,\infty]\mid\phi(p)\leq a\}.$$
\begin{thm} \label{dd-div-minimum}
Let $\phi\in\De$. Then $\xi\in\De$ is a diagonal on $\phi$ in the quantale $\De_{\wedge}$ if, and only if, there exists $\psi\in\De$ with
$$\xi^{\flat}=\phi^{\flat}+\psi^{\flat}.$$
\end{thm}
\begin{proof}
It suffices to show that $(\phi\otimes_{\wedge}\psi)^{\flat}=\phi^{\flat}+\psi^{\flat}$ for all $\phi,\psi\in\De$. This is true since
\begin{align*}
(\phi\otimes_{\wedge}\psi)(t)\leq a&\iff\forall s\leq t:\ \phi(s)\wedge\psi(t-s)\leq a\\
&\iff\forall s\leq t:\ \phi(s)\leq a\ \ \text{or}\ \ \psi(t-s)\leq a\\
&\iff\forall s\leq t:\ s\leq\phi^{\flat}(a)\ \ \text{or}\ \ t-s\leq\psi^{\flat}(a)\\
&\iff t\leq\phi^{\flat}(a)+\psi^{\flat}(a)
\end{align*}
for all $t\in[0,\infty]$, $a\in[0,1]$, where the last equivalence is valid since
$$\exists s\leq t:\ s>\phi^{\flat}(a)\ \ \text{and}\ \ t-s>\psi^{\flat}(a)\iff t>\phi^{\flat}(a)+\psi^{\flat}(a)$$
trivially holds.
\end{proof}
In fact, as the following theorem shows, Theorem \ref{dd-div} cannot be reduced to Proposition \ref{step-divisible} unless $\phi$ is a one-step function, which also gives a stronger proof for Proposition \ref{De-non-div}:
\begin{thm} \label{step-only-divisible-lower}
Every $\xi\leq\phi$ is a diagonal on $\phi$ in the quantale $\De_*$ if, and only if, $\phi$ is a one-step function.
\end{thm}
To prove this theorem we need the following consequence of the well-known representation theorem of continuous t-norms:
\begin{lem} \label{t-norm-rep} (See \cite{Klement2000,Klement2004b,Mostert1957}.)
Let $*$ be a continuous t-norm on $[0,1]$. Then the set of non-idempotent elements of $*$ in $[0,1]$ is a union of countably many pairwise disjoint open intervals $$\{(b_i,c_i)\mid 0<b_i<c_i<1,\ i\in I,\ I\ \text{is countable}\},$$
and for each $i\in I$, the quantale $([a_i,b_i],*,b_i)$ obtained by restricting $*$ on $[a_i,b_i]$ is either isomorphic to the product t-norm $[0,1]_{\times}$ or isomorphic to the {\L}ukasiewicz t-norm $[0,1]_{\oplus}$.
\end{lem}
\begin{proof}[The proof of Theorem \ref{step-only-divisible-lower}]
Suppose that $\phi$ is not a one-step function, then there exists $p\in(0,\infty)$ with $0<\phi(p)<\phi(\infty)$. We proceed with two cases.
{\bf Case 1.} There exists a strictly increasing sequence $\{a_n\}$ in $[0,1]$ such that each $a_n$ is an idempotent element of $*$ and that $\lim\limits_{n\ra\infty}a_n=\phi(\infty)$. In this case, one may find a positive integer $N$ with $a_N\in(\phi(p),\phi(\infty))$. Note that the set
$$\{t\in(0,\infty)\mid\phi(t)>a_N\}$$
is non-empty by applying the left-continuity of $\phi$ to the point $\infty$, and it has $p$ as a lower bound since $\phi$ is monotone. Thus it makes sense to define
$$q:=\inf\{t\in(0,\infty)\mid\phi(t)>a_N\}.$$
Then $q\in[p,\infty)$, and the left-continuity of $\phi$ guarantees that $\phi(q)\leq a_N$. Hence, $\phi(t)\leq a_N$ for all $t\leq q$ and $a_N<\phi(t)\leq\phi(\infty)$ for all $t>q$. Now define $\xi\in\De$ with
$$\xi(t):=\begin{cases}
0 & \text{if}\ t\leq q,\\
\phi(t) & \text{if}\ t>q.
\end{cases}$$
Then $\xi<\phi$, but there is no $\psi\in\De$ with $\xi=\phi\otimes_*\psi$. Indeed, if such $\psi$ exists, then for every $t>q$,
$$\bv_{s<t}\phi(t-s)*\psi(s)=\xi(t)=\phi(t)>a_N.$$
But $\phi(t-s)\leq a_N$ if $t-s\leq q$, the above inequality then implies
$$a_N<\bv_{s<t-q}\phi(t-s)*\psi(s)\leq\bv_{s<t-q}\psi(s)=\psi(t-q)$$
for all $t>q$; that is, $\psi(t)>a_N$ for all $t>0$. It follows that
$$0=\xi(q)=\bv_{s<q}\phi(q-s)*\psi(s)\geq\bv_{0<s<q}\phi(q-s)*a_N=\phi(q)*a_N\geq\phi(p)*a_N=\phi(p)\wedge a_N=\phi(p)>0,$$
where the penultimate equality follows from the idempotency of $a_N$, giving a contradiction.
{\bf Case 2.} There exists no strictly increasing sequence in $[0,1]$ consisting of idempotent elements of $*$ that approaches to $\phi(\infty)$. In this case, by Lemma \ref{t-norm-rep} one may find idempotent elements $b,c$ of $*$ such that $\phi(\infty)\in(b,c]\subseteq[0,1]$ and that the quantale $([b,c],*,c)$ is isomorphic to $[0,1]_{\times}$ or to $[0,1]_{\oplus}$. Let
$$a:=\dfrac{(b\vee\phi(p))+\phi(\infty)}{2}\in(b\vee\phi(p),\ \phi(\infty)),$$
and let
$$q:=\inf\{t\in(0,\infty)\mid\phi(t)>a\}\in[p,\infty)$$
similarly as in Case 1. Then $\phi(t)\leq a$ for all $t\leq q$ and $a<\phi(t)\leq\phi(\infty)$ for all $t>q$. Now define $\xi\in\De$ with
$$\xi(t):=\begin{cases}
0 & \text{if}\ t\leq q,\\
\phi(t) & \text{if}\ t>q.
\end{cases}$$
Then $\xi<\phi$, but there is no $\psi\in\De$ with $\xi=\phi\otimes_*\psi$. Indeed, if such $\psi$ exists, write
$$a_0:=\bw_{s>q}\phi(s),$$
then for every $t>q$,
$$\bv_{s<t}\phi(t-s)*\psi(s)=\xi(t)=\phi(t)\geq a_0\geq a,$$
where at least one of the last two inequalities is strict. But $\phi(t-s)\leq a$ if $t-s\leq q$, the above inequality then implies
$$a_0\leq\bv_{s<t-q}\phi(t-s)*\psi(s)\leq\bv_{s<t-q}\phi(t)*\psi(s)=\phi(t)*\Big(\bv_{s<t-q}\psi(s)\Big)=\phi(t)*\psi(t-q)$$
for all $t>q$, and consequently
$$a_0\leq\bw_{t>q}\phi(t)*\psi(t-q)=\Big(\bw_{t>q}\phi(t)\Big)*\Big(\bw_{t>q}\psi(t-q)\Big)=a_0*\Big(\bw_{t>q}\psi(t-q)\Big),$$
where the first equality follows from the continuity of $*$. Since $a_0\in[a,c]\subseteq(b,c]$ and the quantale $([b,c],*,c)$ is either isomorphic to $[0,1]_{\times}$ or isomorphic to $[0,1]_{\oplus}$, the above inequality holds only if
$$\bw_{t>0}\psi(t)=\bw_{t>q}\psi(t-q)\geq c;$$
that is, $\psi(t)\geq c$ for all $t>0$. Since $c$ is idempotent, a contradiction arises from
$$0=\xi(q)=\bv_{s<q}\phi(q-s)*\psi(s)\geq\bv_{0<s<q}\phi(q-s)*c =\phi(q)*c\geq\phi(p)*c=\phi(p)\wedge c=\phi(p)>0,$$
which completes the proof.
\end{proof}
For $\phi,\psi\in\De$, since it is easy to extract the condition for $\phi\wedge\psi$ to be a one-step function, the following corollary is an immediate consequence of Theorem \ref{step-only-divisible-lower}:
\begin{cor}
Every $\xi\leq\phi\wedge\psi$ is a diagonal between $\phi$ and $\psi$ in the quantale $\De_*$ if, and only if, there exists $p\in[0,\infty)$, $a\in[0,1]$ such that
\begin{enumerate}[label={\rm(\arabic*)}]
\item either $\phi$ or $\psi$ is constant on $[0,p]$ with value $0$, and
\item either $\phi$ or $\psi$ is constant on $(p,\infty]$ with value $a$.
\end{enumerate}
\end{cor}
With Theorem \ref{dd-div} and Lemma \ref{dd-diagonal-comp} it is now possible to characterize $\sD\De_*$-categories through Proposition \ref{integral-DQCat}, which are precisely probabilistic partial metric spaces:
\begin{defn} \label{ProbPM}
A \emph{(generalized) probabilistic partial metric space} w.r.t. a continuous t-norm $*$ is a set $X$ equipped with a map
$$\al:X\times X\to\De,$$
called the \emph{probabilistic partial distance function}, satisfying the following conditions, for all $x,y,z\in X$:
\begin{enumerate}[label=(ProbPM\arabic*),leftmargin=5.2em]
\item \label{ProbPM:r}
$\al(x,y)$ is a diagonal between $\al(x,x)$ and $\al(y,y)$ in the quantale $\De_*$; that is,
\begin{align*}
\al(x,y)(t)&=\bv_{s<t}\bw_{q>0}\al(x,x)(t-s)*(\al(x,x)(q)\ra_*\al(x,y)(q+s))\\
&=\bv_{s<t}\bw_{q>0}\al(y,y)(t-s)*(\al(y,y)(q)\ra_*\al(x,y)(q+s))
\end{align*}
for all $t\in[0,\infty)$.
\item \label{ProbPM:t}
$\al(y,z)\otimes_*(\al(y,y)\Ra_*\al(x,y))\leq\al(x,z)$; that is,
$$\bw_{q>0}\al(y,z)(r)*(\al(y,y)(q)\ra_*\al(x,y)(q+s))\leq\al(x,z)(r+s)$$
for all $r,s\in[0,\infty)$.
\end{enumerate}
\end{defn}
With $\sD\De_*$-functors $f:(X,\al)\to(Y,\be)$ as morphisms, i.e., maps $f:X\to Y$ with
$$\al(x,x)(t)=\be(fx,fx)(t)\quad\text{and}\quad\al(x,y)(t)\leq\be(fx,fy)(t)$$
for all $x,y\in X$, $t\in[0,\infty]$, one obtains the category
$$\ProbParMet_*:=\sD\De_*\text{-}\Cat,$$
which contains the category $\ProbMet_*=\De_*\text{-}\Cat$ of probabilistic metric spaces as a full subcategory.
\begin{rem}
Similar to Remarks \ref{ParMet-classical} and \ref{ProbMet-classical}, it makes sense to say that a probabilistic partial metric space $(X,\al)$ is
\begin{enumerate}[label=(ProbPM\arabic*),start=3,leftmargin=5.2em]
\item \label{ProbPM:sym}
\emph{symmetric}, if $\al(x,y)=\al(y,x)$ for all $x,y\in X$;
\item \label{ProbPM:f}
\emph{finitary}, if $\al(x,y)(\infty)=1$ for all $x,y\in X$;
\item \label{ProbPM:sep}
\emph{separated}, if $\al(x,x)=\al(y,y)=\al(x,y)=\al(y,x)$ implies $x=y$.
\end{enumerate}
\end{rem}
\appendices
\section{Appendix: $\sQ$-categories vs. $\DQ$-categories}
It is easy to observe the following fact, where the coreflector sends each $\DQ$-category $(X,\al)$ to the set
$$X_1:=\{x\in X\mid\al(x,x)=1\}$$
equipped with the $\sQ$-category structure inherited from $(X,\al)$:
\begin{prop} \label{QCat-coref-DQCat}
$\sQCat$ is a full coreflective subcategory of $\DQCat$. In particular, $\Met$ (resp. $\ProbMet_*$) is a full coreflective subcategory of $\ParMet$ (resp. $\ProbParMet_*$).
\end{prop}
The interaction between $\sQ$-categories and $\DQ$-categories is far more profound than Proposition \ref{QCat-coref-DQCat}. First note that there are \emph{lax homomorphisms} of quantaloids\footnote{A lax homomorphism $f:\CQ\to\CQ'$ of small quantaloids consists of a map $f:\CQ_0\to\CQ'_0$ between the object sets and monotone maps $f:\CQ(p,q)\to\CQ'(fp,fq)$ $(p,q\in\CQ_0)$ between hom-sets, such that
$$1_{fq}\leq f1_q\quad\text{and}\quad fv\circ fu\leq f(v\circ u)$$
for all $q\in\CQ_0$ and composable morphisms $u,v$ of $\CQ$.}
$$\Ff:\DQ\to\sQ,\quad(u:p\rqa q)\mapsto(p\ra u)\quad\text{and}\quad\Fb:\DQ\to\sQ,\quad(u:p\rqa q)\mapsto(q\ra u),$$
which induce two functors
$$\Gf:\DQCat\to\sQCat\quad\text{and}\quad\Gb:\DQCat\to\sQCat,$$
called respectively the \emph{forward globalization} and the \emph{backward globalization} functors \cite{Pu2012,Tao2014}. Explicitly, the forward globalization of a $\DQ$-category $(X,\al)$ is the $\sQ$-category $(X,\Gf\al)$ with
$$\Gf\al(x,y)=\al(x,x)\ra\al(x,y),$$
and the backward globalization of $(X,\al)$ is the $\sQ$-category $(X,\Gb\al)$ with
$$\Gb\al(x,y)=\al(y,y)\ra\al(x,y).$$
While considering $\sQ$ as a $\sQ$-category $(\sQ,\pi)$ with $\pi(p,q)=p\ra q$, one obtains a slice category $\sQCat/\sQ$ whose objects are $\sQ$-functors $f:(X,\al)\to(\sQ,\pi)$, and whose morphisms from $f:(X,\al)\to(\sQ,\pi)$ to $g:(Y,\be)\to(\sQ,\pi)$ are $\sQ$-functors $h:(X,\al)\to(Y,\be)$ making the diagram
\begin{equation} \label{QCat-over-Q}
\bfig
\Vtriangle<500,400>[(X,\al)`(Y,\be)`(\sQ,\pi);h`f`g]
\efig
\end{equation}
commutative. Note that each $\DQ$-category $(X,\al)$ induces a $\sQ$-functor $t_{\al}:(X,\Gf\al)\to(\sQ,\pi)$ with $t_{\al}x=\al(x,x)$; indeed, $t_{\al}$ is a $\sQ$-functor since Proposition \ref{integral-DQCat} implies
$$\Gf\al(x,y)=\al(x,x)\ra\al(x,y)\leq\al(x,x)\ra\al(y,y)=\pi(t_{\al}x,t_{\al}y)$$
for all $x,y\in X$. Moreover:
\begin{prop} \label{Gf-fully-faithful}
Let $(X,\al)$, $(Y,\be)$ be $\DQ$-categories.
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{Gf-fully-faithful:injective}
$(X,\al)=(Y,\be)$ if, and only if, $t_{\al}=t_{\be}$.
\item \label{Gf-fully-faithful:ff}
$f:(X,\al)\to(Y,\be)$ is a $\DQ$-functor if, and only if, $f$ is a morphism from $t_{\al}:(X,\Gf\al)\to(\sQ,\pi)$ to $t_{\be}:(Y,\Gf\be)\to(\sQ,\pi)$ in $\sQCat/\sQ$.
\end{enumerate}
\end{prop}
\begin{proof}
\ref{Gf-fully-faithful:injective} If $t_{\al}:(X,\Gf\al)\to(\sQ,\pi)$ and $t_{\be}:(Y,\Gf\be)\to(\sQ,\pi)$ coincide, then $X=Y$, $\al(x,x)=\be(x,x)$ and $\al(x,x)\ra\al(x,y)=\be(x,x)\ra\be(x,y)$ for all $x,y\in X$. It then follows from Proposition \ref{integral-DQCat}\ref{DQ-Cat:s} that
$$\al(x,y)=\al(x,x)\with(\al(x,x)\ra\al(x,y))=\be(x,x)\with(\be(x,x)\ra\be(x,y))=\be(x,y)$$
for all $x,y\in X$. Hence $\al=\be$.
\ref{Gf-fully-faithful:ff} Note that
\begin{align*}
f\ \text{is a}\ \DQ\text{-functor}&\iff\forall x,y\in X:\ \al(x,x)=\be(fx,fx)\ \text{and}\ \al(x,y)\leq\be(fx,fy)\\
&\iff\forall x,y\in X:\ t_{\be}f=t_{\al}\ \text{and}\ \al(x,x)\ra\al(x,y)\leq\be(fx,fx)\ra\be(fx,fy)\\
&\iff\forall x,y\in X:\ t_{\be}f=t_{\al}\ \text{and}\ \Gf\al(x,y)\leq\Gf\be(fx,fy)\\
&\iff f:t_{\al}\to t_{\be}\ \text{is a morphism from}\ \text{in}\ \sQCat/\sQ,
\end{align*}
where we have applied the same method as in \ref{Gf-fully-faithful:injective} to the second equivalence, and thus the conclusion holds.
\end{proof}
From Proposition \ref{Gf-fully-faithful} we obtain a fully faithful functor
$$\Gf^{\dag}:\DQCat\to\sQCat/\sQ,\quad (X,\al)\mapsto(t_{\al}:(X,\Gf\al)\to(\sQ,\pi))$$
that embeds $\DQCat$ in $\sQCat/\sQ$ as a full subcategory. Furthermore, this embedding is reflective if $\sQ$ is divisible:
\begin{thm} \label{DQCat-ref-QCatQ}
If $\sQ$ is divisible, then $\DQCat$ is a full reflective subcategory of $\sQCat/\sQ$.
\end{thm}
\begin{proof}
{\bf Step 1.} For each $\sQ$-category $(X,\al)$ equipped with a $\sQ$-functor $f:(X,\al)\to(\sQ,\pi)$,
$$\al_f(x,y)=\al(x,y)\with fx$$
defines a $\DQ$-category $(X,\al_f)$. To see this, it suffices to verify that $(X,\al_f)$ satisfies the conditions given in Proposition \ref{divisible-DQCat}.
First, since $f$ is a $\sQ$-functor, $\al(x,y)\leq\pi(fx,fy)=fx\ra fy$, and thus $\al_f(x,y)=\al(x,y)\with fx\leq fy$ for all $x,y\in X$. It follows that $\al_f(x,y)\leq fx\wedge fy=\al_f(x,x)\wedge\al_f(y,y)$ for all $x,y\in X$. Here
\begin{equation} \label{alf=f}
\al_f(x,x)=fx
\end{equation}
for all $x\in X$ since, with $(X,\al)$ being a $\sQ$-category, one always has $\al(x,x)=1$.
Second, since $(X,\al)$ is a $\sQ$-category, one also has $\al(y,z)\with\al(x,y)\leq\al(x,z)$ for all $x,y,z\in X$, and hence
\begin{align*}
\al_f(y,z)\with(\al_f(y,y)\ra\al_f(x,y))&=\al(y,z)\with fy\with(fy\ra(\al(x,y)\with fx))\\
&\leq\al(y,z)\with\al(x,y)\with fx\leq\al(x,z)\with fx=\al_f(x,z).
\end{align*}
{\bf Step 2.} For each commutative diagram \eqref{QCat-over-Q} in $\sQCat$, $h:(X,\al_f)\to(Y,\be_g)$ is a $\DQ$-functor, hence the above assignment defines a functor $\FT:\sQCat/\sQ\to\DQCat$. Indeed, for all $x\in X$, $y\in Y$,
$$\be_g(hx,hx)=ghx=fx=\al_f(x,x)$$
follows from \eqref{alf=f}, and
$$\al_f(x,y)=\al(x,y)\with fx\leq\be(hx,hy)\with ghx=\be_g(hx,hy)$$
since $h:(X,\al)\to(Y,\be)$ is a $\sQ$-functor.
{\bf Step 3.} $\FT:\sQCat/\sQ\to\DQCat$ is a left adjoint of $\Gf^{\dag}:\DQCat\to\sQCat/\sQ$. To this end, we establish a bijection
\begin{equation} \label{DQCat-cong-sQCatQ}
\DQCat((X,\al_f),(Y,\be))\cong\sQCat/\sQ(f,t_{\be})
\end{equation}
for each $\sQ$-functor $f:(X,\al)\to(\sQ,\pi)$ and $\DQ$-category $(Y,\be)$. Indeed,
\begin{align*}
&h:(X,\al_f)\to(Y,\be)\ \text{is a}\ \DQ\text{-functor}\\
\iff{}&\forall x,y\in X:\ \al_f(x,x)=\be(hx,hx)\ \text{and}\ \al_f(x,y)\leq\be(hx,hy)\\
\iff{}&\forall x,y\in X:\ fx=t_{\be}hx\ \text{and}\ \al(x,y)\with fx\leq\be(hx,hy)&(\text{definition of}\ t_{\be},\ \al_f\ \text{and}\ \eqref{alf=f})\\
\iff{}&\forall x,y\in X:\ fx=t_{\be}hx\ \text{and}\ \al(x,y)\leq\be(hx,hx)\ra\be(hx,hy)&(\text{definition of}\ t_{\be})\\
\iff{}&\forall x,y\in X:\ fx=t_{\be}hx\ \text{and}\ \al(x,y)\leq\Gf\be(hx,hy)\\
\iff{}&h:f\to t_{\be}\ \text{is a morphism in}\ \sQCat/\sQ.
\end{align*}
Finally, it is straightforward to check that the bijection \eqref{DQCat-cong-sQCatQ} is natural in $f:(X,\al)\to(\sQ,\pi)$ and $(Y,\be)$, and thus the proof is completed.
\end{proof}
Let $\sQ^{\op}=(\sQ,\pi^{\op})$ be the $\sQ$-category with the underlying set $\sQ$ and the $\sQ$-category structure $\pi^{\op}(p,q)=q\ra p$. Then similarly one obtains another full embedding
$$\Gb^{\dag}:\DQCat\to\sQCat/\sQ^{\op}$$
that sends each $\DQ$-category $(X,\al)$ to the $\sQ$-category $(X,\Gb\al)$ equipped with the $\sQ$-functor $s_{\al}:(X,\Gb\al)\to(\sQ,\pi^{\op})$ with $s_{\al}x=\al(x,x)$ for all $x\in X$, and this embedding is also reflective when $\sQ$ is divisible:
\begin{thm} \label{DQCat-ref-QCatQop}
If $\sQ$ is divisible, then $\DQCat$ is a full reflective subcategory of $\sQCat/\sQ^{\op}$.
\end{thm}
In particular, if $\sQ=[0,\infty]_+$, then $[0,\infty]=([0,\infty],\pi)$ and $[0,\infty]^{\op}=([0,\infty],\pi^{\op})$ are both metric spaces with
$$\pi(p,q)=\begin{cases}
0 & \text{if}\ p\geq q,\\
q-p & \text{if}\ p<q
\end{cases}\quad\text{and}\quad\pi^{\op}(p,q)=\begin{cases}
0 & \text{if}\ q\geq p,\\
p-q & \text{if}\ q<p
\end{cases}$$
for all $p,q\in[0,\infty]$. In this case, it is easy to see that the functors $\Gf^{\dag}$ and $\FT$ are inverses to each other, and thus
$$\Gf^{\dag}:\ParMet\to\Met/[0,\infty]$$
gives an isomorphism of categories; and so is the functor $\Gb^{\dag}:\ParMet\to\Met/[0,\infty]^{\op}$:
\begin{thm} \label{ParMet-Met-iso}
$\ParMet\cong\Met/[0,\infty]\cong\Met/[0,\infty]^{\op}$.
\end{thm}
If $\sQ=\De_*$, then $\De_*=(\De,\pi_*)$ and $\De_*^{\op}=(\De,\pi_*^{\op})$ are both probabilistic metric spaces with
$$\pi_*(\phi,\psi)(t)=\bv_{s<t}\bw_{q>0}\phi(q)\ra_*\psi(q+s)\quad\text{and}\quad\pi_*^{\op}(\phi,\psi)(t)=\bv_{s<t}\bw_{q>0}\psi(q)\ra_*\phi(q+s)$$
for all $\phi,\psi\in\De$, $t\in[0,\infty]$, and in this case:
\begin{cor}
$\ProbParMet_*$ is a full subcategory of both $\ProbMet_*/\De_*$ and $\ProbMet_*/\De_*^{\op}$.
\end{cor}
\section*{Acknowledgement}
The first author acknowledges the support of National Natural Science Foundation of China (No. 11771311). The second author acknowledges the support of National Natural Science Foundation of China (No. 11771310). The third author acknowledges the support of National Natural Science Foundation of China (No. 11701396) and the Fundamental Research Funds for the Central Universities (No. YJ201644).
The authors are grateful to the anonymous referees for their helpful remarks and suggestions.
|
1,314,259,993,046 | arxiv | \section{Introduction and notations}
For a number field $k$, the classical wild kernel $W\!K_{2}(k)$ is the kernel of all local power norm residue symbols. It fits in the Moore exact sequence
\begin{equation}\label{sequence0}
0\to W\!K_{2}(k)\to K_2(k)\to \underset{v}{\oplus} \mu(k_v)\to \mu(k)\to 0,
\end{equation}
where $v$ runs through all non complex places of $k$, $k_v$ is the completion of $k$ at $v$ and $\mu(k_v)$ (resp. $\mu(k)$) is the group of roots of unity of $k_v$ (resp. $k$). It is known (see \cite{Ta1} and also \cite{Hu}) that, for an odd prime $p$, the $p$-primary part $W\!K_{2}(k)\{p\}$ of the wild kernel consists of the elements of infinite height of $K_2(k)\{p\}$. In particular, the inclusion $W\!K_{2}(k)\{p\}\subset K_2(k)\{p\}$ never splits, unless $W\!K_{2}(k)\{p\}=0$.
The wild kernel is actually contained in the tame kernel of $k$, denoted $K_2(o_k)$, and there is an exact sequence
$$0\to W\!K_{2}(k)\{p\}\to K_2(o_k)\{p\}\to \underset{v\mid p}{\oplus} \mu(k_v)\{p\}\to \mu(k)\{p\}\to 0.$$
In the study of the structure of $K_2(o_k)$, it is interesting to know when the inclusion $W\!K_{2}(k)\{p\}\subseteq K_2(o_k)\{p\}$ splits. Strangely enough, it seems that this question has never been studied so far. The aim of this paper is to give conditions for the inclusion $W\!K_{2}(k)\{p\}\subseteq K_2(o_k)\{p\}$ to split. Together with some additional results (e.g. the Birch-Tate formula), this sometimes allows one to completely determine the structure of $K_2(o_k)\{p\}$ (see Section \ref{examples}).
Actually, using the results of Tate (\cite{Ta2}), $W\!K_{2}(k)\{p\}$ (resp. $K_2(o_k)\{p\}$) can be interpreted as a Tate-Shafarevic (resp. cohomology) group. This leads directly to a generalization of the inclusion $W\!K_{2}(k)\{p\}\subseteq K_2(o_K)\{p\}$, as we shall now explain.
Let $p$ be an odd prime and let $i\in\mathbb{Z}$ be an integer. Let $o'_k$ be the ring of $p$-integers of $k$, which embeds into $k_v$ for every $v\mid p$. Accordingly we have a homomorphism in cohomology
$$\lambda_i:\hdet{o'_k}{(i)}\to \underset{v\mid p}{\oplus} \hdet{k_v}{(i)}.$$
Here $\hdet{o'_k}{(i)}=\lim\limits_{\longleftarrow}H^2_{\acute{e}t}(o'_k,\mu_{p^n}^{\otimes i})$, where the limits are taken over the natural maps $\mu_{p^{n+m}}^{\otimes i}\to \mu_{p^n}^{\otimes i}$ of $Spec(o'_k)$-\'etale sheaves (and similarly for $\hdet{k_v}{(i)}$).
One can define the $i$-th \'etale wild kernel of $K$ as
$$\wket{k}{(i)}=\mathrm{ker}(\lambda_i)$$
(see \cite{Sc}, \cite{N4}, \cite{Kol1}, \cite{Ba}).
Note that $\wket{k}{(i)}$ (resp. $\hdet{o'_k}{(i)}$) is finite if $i\geq 1$ (resp. $i\geq 2$), see \cite{Sc}, \cite{So}. In fact $\wket{k}{(i)}$ is also conjectured to be finite when $i\leq 0$, but for the rest of this introduction we will suppose $i\geq 1$. Part of the (dual) exact sequence of Poitou-Tate (see \cite[\S 4, Satz 8]{Sc}) then reads
\begin{equation}\label{sequence}
0\to \wket{k}{(i)}\stackrel{\iota_{k,i}}{\rightarrow} \hdet{o'_k}{(i)}\stackrel{\lambda_i}{\longrightarrow} \underset{v\mid p}{\oplus}\hdet{k_v}{(i)} \stackrel{res^*}{\longrightarrow} \hzet{k}{(1-i)}^*\to 0,
\end{equation}
where $(\--)^*$ denotes the Pontryagin dual and, by local duality,
$$\hdet{k_v}{(i)}\cong \hzet{k_v}{(1-i)}^*.$$
For $i=2$, there is a commutative diagram
$$
\begin{CD}
W\!K_{2}(k)\{p\}@>>> K_2(o_k)\{p\}\\
@V\wr VV@V\wr VV\\
\wket{k}{(2)}@>\iota_{k,2}>> \hdet{o'_k}{(2)},\\
\end{CD}
$$
where the left isomorphism is due to Schneider (\cite{Sc}) and the right one is due to Soul\'e (\cite{So}). Therefore our splitting problem is equivalent to that of $\iota_{k,2}$. As it is well-known, there are analogues of the above diagrams for any $i\geq 2$: the vertical arrows of these ``higher'' diagrams are still isomorphisms in view of the Quillen-Lichtenbaum conjecture, which is a consequence of the Bloch-Kato conjecture (see for example \cite[Theorem 2.7]{Kol2}) proven by Rost-Voevodsky (see \cite{We} for more details). In fact we will study the splitting of the inclusion $\iota_{k,i}$ for general $i\geq 2$ (it is easy to see that $\iota_{k,1}$ always splits, see Section \ref{code}).
To this end, we look at this problem along the cyclotomic tower of the base field and apply Iwasawa theoretical tools to derive our results.
Let $\wkin{k}{(i)}$ (resp. $\hdin{o'_k}{(i)}$) be the direct limit of $\wket{k_n}{(i)}$ (resp. $\hdet{o'_{k_n}}{(i)}$) with respect to the restriction homomorphisms along the cyclotomic $\mathbb{Z}_p$-extension $k_\infty$ of $k$, whose $n$-th layer is denoted $k_n$. Set $K=k(\mu_p)$ and $\Delta=\mathrm{Gal}(K/k)$. Then the map $\iota_{K,\infty}=\iota_{K,\infty,i}=\lim\limits_{\longrightarrow}\iota_{K_n,i}$ fits in a commutative diagram (see Proposition \ref{nqd})
$$
\begin{CD}
\wkin{K}{(i)}@>\iota_{K,\infty}>>\hdin{o'_K}{(i)}\\
@A\wr AA@A\wr AA\\
t_{\Lambda}(BP_{K_\infty})(-i)^*@>>>t_{\Lambda}(\mathfrak{X}_{K_\infty})(-i)^*
\end{CD}
$$
assuming Leopoldt's conjecture. Here $\mathfrak{X}_{K_\infty}$ (resp. $BP_{K_\infty}$) is the Galois group of the maximal abelian pro-$p$-extension of $K_\infty$ which is unramified outside $p$ (resp. the Galois group of the union of the fields of Bertrandias-Payan over $K_n$ for all $n$, see Definition \ref{BP}). Both $\mathfrak{X}_{K_\infty}$ and $BP_{K_\infty}$ are finitely generated $\Lambda$-modules (where $\Lambda=\mathbb{Z}_p[\![\mathrm{Gal}(K_\infty/K)]\!]$ is the Iwasawa algebra) and $t_{\Lambda}(\cdot)$ denotes the $\Lambda$-torsion submodule.
The advantage of considering limits lies mainly in the fact that, using the results of \cite{LMN}, one can easily spot a sufficient condition for the surjection
\begin{equation}\label{surje}
t_{\Lambda}(\mathfrak{X}_{K_\infty})\to t_{\Lambda}(BP_{K_\infty})
\end{equation}
to split (as a morphism of $\Lambda$-modules), namely the triviality of the invariant $\Psi(K_\infty)$. The latter is a finite group (under Gross's conjecture for every layer $K_n$) defined as the direct limit of the groups
$$\Psi(K_n):=\ker \left(\left(X'_{K_\infty}\right)_{\mathrm{Gal}(K_\infty/K_n)}\to A'_{K_n}\right).$$
Here $X'_{K_\infty}=\varprojlim A'_{K_n}$, where $A'_{K_n}$ is the $p$-Sylow subgroup of the class group of $o'_{K_n}$ and the limit is taken with respect to norms.
Once a condition for the splitting of the inclusion $\iota_{K,\infty}$ has been found, one may transfer it to finite layers. The most direct way is to perform Galois descent from $K_\infty$ to $K$, namely consider the commutative diagram
\begin{equation}\label{desquare}
\begin{CD}
\wket{k}{(i)} @>\iota_k>>\hdet{o'_k}{(i)}\\
@VVV@VVV\\
\qquad\wkin{K}{(i)}^{\mathrm{Gal}(K_\infty/k)}@>\iota_{K,\infty}>>\qquad\hdin{o'_K}{(i)}^{\mathrm{Gal}(K_\infty/k)}.
\end{CD}
\end{equation}
The vertical arrows of the above diagram are surjective and have the same kernel $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathrm{Gal}(K_\infty/k)}$ (see Theorem \ref{descent}). Here $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$ denotes the maximal finite $\Lambda$-submodule of $X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$. Therefore $\iota_k$ splits as soon as $\Psi(K_\infty)$ and $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$ vanish. It turns out that something stronger is also true (see Proposition \ref{main} and Theorem \ref{randonnee}). For instance $\iota_k$ splits, provided that $\Psi(K_\infty)(i-1)^{\Delta}=0$ and the right vertical arrow of (\ref{desquare}) splits (as abelian groups). The latter condition holds, for example, for $k_n$, if $n$ is sufficiently large and the Iwasawa $\mu$-invariant of $X'_{K_\infty}$ is trivial (see \cite{Va2}).
We have also developed another approach which uses codescent instead of descent. Let $\wkiw{K}{(i)}$ (resp. $\hdiw{o'_K}{(i)}$) be the inverse limit of $\wket{K_n}{(i)}$ (resp. $\hdet{o'_{K_n}}{(i)}$) over the corestriction homomorphisms. Then we have an injection $\iota_{K,Iw}=\iota_{K,Iw,i}=\lim\limits_{\longleftarrow}\iota_{K_n,i}$ and a commutative diagram
\begin{equation}\label{cosquare}
\begin{CD}
\wket{k}{(i)} @>\iota_k>>\hdet{o'_k}{(i)}\\
@A\wr AA@A\wr AA\\
\qquad\wkiw{K}{(i)}_{\mathrm{Gal}(K_\infty/k)}@>\iota_{K,Iw}>>\qquad\hdiw{o'_K}{(i)}_{\mathrm{Gal}(K_\infty/k)}
\end{CD}
\end{equation}
whose vertical arrows are isomorphisms. Hence, for $\iota_k$ to split it is sufficient that $\iota_{K,Iw}$ splits in the category of $\Lambda[\Delta]$-modules and we shall give a condition for that in Theorem \ref{main2} (see also Lemma \ref{pavia}). When the $p$-adic primes are totally ramified in $K_\infty/K$, the above condition simply reduces to $\Psi(K)(i-1)^\Delta=0$, see Corollary \ref{simplecase}.
One can also follow a quite different approach, based on the following remark. The inclusion $\iota_k:\wket{k}{(i)}\to\hdet{o'_k}{(i)}$ splits if and only if, for every $n\in\mathbb{N}$, the induced homomorphism $\wket{k}{(i)}/p^n\to \hdet{o'_k}{(i)}/p^n$ stays injective. In Section \ref{crit}, we express the above criterion in terms of class groups, following the strategy of \cite{Ca} and using in particular Schneider's isomorphism (\cite{Sc})
$$X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathrm{Gal}(K_\infty/k)}\cong \wket{k}{(i)} \quad (i\ne 1).$$
This turns out to be particularly useful to give examples of number fields $k$ for which $\iota_k$ does not split. We also use our criterion to give a different proof of Corollary \ref{simplecase}.
We conclude by applying our results to the inclusion
$$W\!K_{2}(k)\{3\}\subseteq K_2(o_k)\{3\},$$
for families of quadratic fields $k$, giving examples for both split and non-split cases. In particular we are able to confirm some of the predictions of \cite{BG} on the exact structure of wild and tame kernels of imaginary quadratic fields.
\subsection*{Notations}
Let $k$ be a number field and let $p$ be an odd prime, we now give the most relevant notations used in the paper:
\noindent
\begin{tabular}{lcl}
$\mu_{p^n}$, $\mu_{p^\infty}$&&group of $p^n$-th (resp. $p$-power order) roots of unity in an algebraic closure $\overline{k}$ of $k$;\\
$K$&& $k(\mu_p)$;\\
$k_\infty$&&cyclotomic $\mathbb{Z}_p$-extension of $k$ inside $\overline{k}$;\\
$k_n$&&$n$-th layer of $k_\infty$;\\
$\Gamma_n$&&Galois group of $k_\infty/k_n$, which is identified with $\mathrm{Gal}(K_\infty/K_n)$ ($\Gamma=\Gamma_0$);\\
$\mathcal{G}_n$&&Galois group of $K_\infty/k_n$ ($\mathcal{G}=\mathcal{G}_0$);\\
$\Delta$&&Galois group of $K_\infty/k_\infty$, which is identified with $\mathrm{Gal}(K/k)$;\\
$A'_{k_n}$&&$p$-Sylow subgroup of the ideal class group of the ring of $(p)$-integers of $k_n$;\\
$A'_{k_\infty}$ &&direct limit of $A'_{k_n}$ with respect to the maps induced by the inclusions $k_n\subset k_{n+1}$;\\
$X'_{k_\infty}$&&projective limit of $A'_{k_n}$ with respect to the maps induced by the norms $k_{n+1}\to k_n$;\\
$\mathfrak{X}_{k_n}$, $\mathfrak{X}_{k_\infty}$&&Galois group of the maximal pro-$p$-abelian extension of $k_n$ (resp. $k_\infty$) unramified outside $p$.\\
\end{tabular}
\linebreak
\\
Let $M$ be a (topological) $\mathbb{Z}_p[\mathrm{Gal}(\overline{k}/k)]$-module. The Pontryagin dual $M^*=\mathrm{Hom}_{\mathbb{Z}_p}(M,\mathbb{Q}_p/\mathbb{Z}_p)$ of $M$ is a $\mathbb{Z}_p[\mathrm{Gal}(\overline{k}/k)]$-module with the action defined by $(g\phi)(m)=\phi(g^{-1}m)$, for any $g\in \mathrm{Gal}(\overline{k}/k)$, $m\in M$ and $\phi\in M^*$. We also consider, for any $i\in \mathbb{Z}$, the $i$-th Tate twist of $M$, denoted $M(i)=M\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(i)$, as a $\mathbb{Z}_p[\mathrm{Gal}(\overline{k}/k)]$ with the action $g(m\otimes x)=g(m)\otimes g(x)$.
If $A$ is a ring and $M$ is an $A$-module, we will denote by $t_A(M)$ the torsion submodule of $M$. For $a\in A$, $M\{a\}$ denotes the submodule of elements of $M$ which are killed by a power of $a$. If $G$ is a group acting on $M$, we denote by $M^G$ (resp. $M_G$) the submodule of invariants (resp. the quotient of coinvariants) of $M$, \textit{i.e.} the maximal submodule (resp. the maximal quotient) of $M$ on which $G$ acts trivially.
We also recall three classical conjectures we shall often refer to throughout the paper.
\begin{leoconj*}
The $\mathbb{Z}_p$-module $\mathfrak{X}_{k}$ has rank $r_2(k)+1$, where $r_2(k)$ is the number of complex places of $k$.
\end{leoconj*}
\begin{groconj*}
The group $(X'_{\!k_{_{\!\scriptscriptstyle{\infty}}}})_\Gamma$ (or equivalently $(X'_{\!k_{_{\!\scriptscriptstyle{\infty}}}})^\Gamma$) is finite.
\end{groconj*}
\begin{greconj*}
If $k$ is totally real, then $X'_{\!k_{_{\!\scriptscriptstyle{\infty}}}}$ is finite.
\end{greconj*}
If $E/k$ is a finite extension and Leopoldt's conjecture (resp. Gross's conjecture) holds for $E$, then it holds for $k$ too. Moreover, both Leopoldt's and Gross's conjectures are known to hold if $k/\mathbb{Q}$ is abelian.
\section{Background}
This section is devoted to the description of some Iwasawa theoretical objects which will allow us to formulate our main results.
Let $p$ be an odd prime. For a number field $k$, set $K=k(\mu_p)$, where $\mu_p$ is the group of $p$-th roots of unity in an algebraic closure $\overline{k}$ of $k$, and $\Delta=\Delta_k=\mathrm{Gal}(K/k)$. Let $k_\infty$ denote the cyclotomic $\mathbb{Z}_p$-extension of $k$ and set $\Gamma=\Gamma_k=\mathrm{Gal}(k_\infty/k)$. As usual, we shall denote by $\Lambda=\Lambda_k=\mathbb{Z}_p[\![\Gamma_k]\!]$ the completed group algebra of $\Gamma_k$. For every $n\in\mathbb{N}$, we set $\Gamma_n=\Gamma^{p^n}$ and denote by $k_n$ the subfield of $k_\infty$ fixed by $\Gamma_n$, so that $k_\infty=\cup_{n\in\mathbb{N}} k_n$.
Let $o'_k$ be the ring of $p$-integers of $k$ and, for any $i\in\mathbb{Z}$, let $\wket{k}{(i)}$ and $\hdet{o'_k}{(i)}$ denote the higher wild and tame kernels of $k$, respectively, as defined in the introduction. We will use the following notation:
$$\wkiw{k}{(i)}=\lim_{\longleftarrow}\wket{k_n}{(i)} \quad\textrm{and}\quad\hdiw{o'_k}{(i)}=\lim_{\longleftarrow}\hdet{o'_{k_n}}{(i)},$$
where the limits are taken over corestriction maps, and
$$\wkin{k}{(i)}=\lim_{\longrightarrow}\wket{k_n}{(i)} \quad\textrm{and}\quad\hdin{o'_k}{(i)}=\lim_{\longrightarrow}\hdet{o'_{k_n}}{(i)},$$
where the limits are taken over restriction maps. Note that the inclusions $\iota_{k_n}:\wket{k_n}{(i)}\to \hdet{o'_{k_n}}{(i)}$ induce inclusions
$$\iota_{k,Iw,i}=\iota_{k,Iw}:\wkiw{k}{(i)}\to \hdiw{o'_k}{(i)}\quad\textrm{and}\quad\iota_{k,\infty,i}=\iota_{k,\infty}:\wkin{k}{(i)}\to \hdin{o'_k}{(i)}.$$
It is well known that both higher tame and wild kernels behave well under Galois co-descent along the cyclotomic $\mathbb{Z}_p$-extension. More precisely, for any $n\in \mathbb{N}$ and $i\geq 2$, the natural projection homomorphisms induce isomorphisms
\begin{equation}\label{codescentwh}
\wkiw{k}{(i)}_{\Gamma_n}\cong \wket{k_n}{(i)} \quad\textrm{and}\quad \hdiw{o'_k}{(i)}_{\Gamma_n}\cong \hdet{o'_{k_n}}{(i)},
\end{equation}
(see \cite[Corollary 2.7]{KM}).
On the other hand the natural descent homomorphisms
$$\wket{k_n}{(i)}\to\wkin{k}{(i)}^{\Gamma_n}\quad\textrm{and}\quad \hdin{k_n}{(i)}\to \hdin{o'_k}{(i)}^{\Gamma_n}$$
are not in general injective but still surjective, as we shall recall in Theorem \ref{descent} below.
First we need some notation. For any number field $k$, let $A'_k$ denote the $p$-Sylow subgroup of the ideal class group of $o'_{k}$. For any $n\in\mathbb{N}$, set $\mathcal{G}_{k_n}=\mathcal{G}_{n}=\mathrm{Gal}(K_\infty/k_n)$ and we shall also write $\mathcal{G}=\mathcal{G}_k$ for $\mathcal{G}_0$. Let $X'_{k_\infty}$ denote the Galois group of the maximal abelian pro-$p$-extension of $k_\infty$ which is completely decomposed everywhere. Then, by class field theory, $X'_{k_\infty}$ is isomorphic to the inverse limit $\lim\limits_{\longleftarrow}A'_{k_n}$ with respect to the norm homomorphisms. Finally, for a finitely generated $\Lambda$-module $M$, we denote by $M^\circ$ the maximal finite $\Lambda$-submodule of $M$. Then the following theorem (more or less known) gives the description of Galois descent for higher wild and tame kernels.
\begin{teo}\label{descent}
Let $i\geq 2$ be an integer. There exists a commutative diagram of $\mathrm{Gal}(k_n/k)$-modules
\begin{equation}\label{ladiscesa}
\begin{CD}
0@>>>X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathcal{G}_n}@>>>\wket{k_n}{(i)}@>>>\wkin{k}{(i)}^{\Gamma_n}@>>>0\\
@.@|@VVV@VVV\\
0@>>>X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathcal{G}_n}@>>>\hdet{o'_{k_n}}{(i)}@>>>\hdin{o'_k}{(i)}^{\Gamma_n}@>>>0,
\end{CD}
\end{equation}
whose rows are exact and vertical arrows are injective. Moreover, if the Iwasawa $\mu$-invariant of the $\Lambda$-module $X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$ is trivial, then both the top and the bottom exact sequences split (as abelian groups) for $n$ large.
\end{teo}
\begin{proof}
The bottom sequence is defined and proved to be exact in \cite[Theorem 7]{Co} for $i=2$ in terms of $K$-theory and in \cite[Section 4]{N3} for general $i$ (see also \cite[Proposition 3.2, Corollary 3.3, Theorem 3.6]{KM}).
The surjectivity of the map $\wket{k_n}{(i)}\to\wkin{k}{(i)}^{\Gamma_n}$ is noticed in \cite[page 854]{LMN} (it is an easy consequence of \cite[Lemma 1.1]{LMN} together with Schneider's isomorphism $X'_{K_\infty}(i-1)_{\mathcal{G}_n}\cong \wket{k_n}{(i)}$, see \cite[\S6, Lemma 1]{Sc}) and its kernel is indeed isomorphic to $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}$ as shown in \cite[Proposition 3.8]{KM}. Using the injectivity of $\wket{k_n}{(i)}\to\hdet{o'_{k_n}}{(i)}$ and the finiteness of $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathcal{G}_n}$, we deduce that the left-hand square of the diagram of the theorem is indeed commutative. The last assertion of the theorem is due to Validire (see \cite[Proposition 3.3 and Theorem 4.1]{Va2}).
\end{proof}
\begin{remark}
One can actually show that the last assertion of the above theorem holds under the weaker assumption that the Iwasawa $\mu$-invariant of the $\Lambda$-module $X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}$ is trivial (see \cite[Th\'eor\`eme 3.1.8]{Va1}).
\end{remark}
The module $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$ has a relevant role in our approach. Of course $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}$ has other arithmetic interpretations, discovered mainly by Iwasawa, which we shall now briefly recall.
When $i=1$, $\wket{k}{(1)}$ is isomorphic to $A'_k$ (see \cite[\S 4, Satz 8]{Sc}). A classical result of Iwasawa (whose proof is essentially contained in \cite{Iw}) states that there is an exact sequence
\begin{equation}\label{seclass}
0\to (X'^{\,\circ}_{\!k_{_{\!\scriptscriptstyle{\infty}}}})_{\Gamma_n}\to A'_{k_n}\to A'^{\,\Gamma_n}_{k_\infty},
\end{equation}
where $A'_{k_\infty}=\lim\limits_{\longrightarrow}A'_{k_n}$ denotes the direct limit of class groups with respect to the maps induced by extension of ideals. The above sequence may therefore be considered as the classical analogue of the top sequence of the diagram of Theorem \ref{descent}. The splitting (as abelian groups) of the inclusion $(X'^{\,\circ}_{\!k_{_{\!\scriptscriptstyle{\infty}}}})_{\Gamma_n}\to A'_{k_n}$ in (\ref{seclass}) for $n$ large was proved by Grandet and Jaulent (see \cite[Th\'eor\`eme]{GJ}).
Another interpretation of $X'^{\,\circ}_{\!k_{_{\!\scriptscriptstyle{\infty}}}}$ can be given in the following context. Let $\mathfrak{X}_{k}$ (resp. $\mathfrak{X}_{k_\infty}$) be the Galois group of the maximal pro-$p$-abelian extension of $k$ (resp. $k_\infty$) unramified outside $p$ and the archimedean primes. In fact we have $\mathfrak{X}_{k_\infty}=\lim\limits_{\longleftarrow}\mathfrak{X}_{k_n}$, the limit being taken over the restriction maps. Moreover $\mathfrak{X}_{k}$ (resp. $\mathfrak{X}_{k_\infty}$) can be shown to a be finitely generated $\mathbb{Z}_p$-module (resp. $\Lambda$-module). More precisely, a well-known theorem of Iwasawa asserts that there is a pseudo-isomorphism $f:\mathfrak{X}_{k_\infty}/t_{\Lambda}(\mathfrak{X}_{k_\infty})\to \Lambda^{r_2(k)}$ where $r_2(k)$ is the number of complex places of $k$ (see \cite[Theorem 13.31]{Wa}) and $H_k=\mathrm{coker}(f)$ is independent of $f$ up to isomorphism (see \cite[\S 3]{Gr}, \cite{Ja}). Then, as explained by Greenberg in \cite[\S 5]{Gr}, one can deduce from \cite{Iw} an isomorphism of $\Lambda[\Delta]$-modules
$$X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}\cong H_{K}^*(1),$$
where, for a $\Lambda[\Delta]$-module $M$, we denote by $M^*=\mathrm{Hom}_{\mathbb{Z}_p}(M,\mathbb{Q}_p/\mathbb{Z}_p)$ its Pontryagin dual. Note that, if $k$ is totally real, then the plus part of $H_{K}$ is clearly trivial. In particular the above isomorphism can be used, for instance, to prove the following well-known result.
\begin{prop}\label{coates}
If $k$ is totally real and $i\in\mathbb{Z}$ is even, then
$$X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\mathcal{G}_n}=0\quad\textrm{for every $n\in\mathbb{N}$.}$$
\end{prop}
We now come to another object which is fundamental for our approach.
\begin{defi}
For a number field $k$, let $\Psi(k)$ denote the kernel of the natural map $(X'_{k_\infty})_{\Gamma}\to A'_{k}$.
\end{defi}
Note that, if Gross's conjecture holds for $k$, $\Psi(k)$ is a finite group. If $m\geq n$ it is easy to see that the map $(X'_{k_\infty})_{\Gamma_n}\to (X'_{k_\infty})_{\Gamma_m}\,,$ given by multiplication by $\omega_m/\omega_n$ induces a map $\Psi(k_n)\to \Psi(k_m)$.
\begin{defi}
For a number field $k$, set
$$\Psi(k_\infty):=\lim_{\longrightarrow} \Psi(k_n),$$
where the limits are taken over the maps described above.
\end{defi}
Let $n_0=n_0(k)$ denote the smallest integer for which all $p$-adic primes are totally ramified in $k_\infty/k_{n_0}$ (in particular the map $X'_{k_\infty}\to A'_{k_n}$ is surjective for $n\geq n_0$).
\begin{lemma}\label{CPsi}
For any $j\in\mathbb{Z}$, if $\Psi(K_n)(j)_\Delta=0$ for some $n\geq n_0(K)$, then $\Psi(K_m)(j)_\Delta=0$ for every $m\geq n$ and therefore $\Psi(K_\infty)(j)_\Delta=0$.
Assume moreover that the Gross conjecture holds for all the layers $k_n$. Then for all $n$ large enough the groups $\Psi(k_n)$ stabilize and the natural maps $\Psi(k_n)\to \Psi(k_\infty)$ become isomorphisms (in particular $\Psi(k_\infty)$ is finite).
\end{lemma}
\begin{proof}
The proof of the first assertion is an easy generalization of \cite[proof of Corollary 1.6]{LMN}. The last assertion is \cite[Lemma 1.3]{LMN}.
\end{proof}
\begin{remark}\label{oujda}
If there is only one prime above $p$ in $k_\infty$ then $\Psi(k_{n})=0$ for every $n\geq n_0(k)$ (this can be easily deduced for example from \cite[Lemma 13.15]{Wa}) and therefore $\Psi(k_\infty)=0$. Note also that if $k$ is a totally real field satisfying Greenberg's conjecture, then one can easily show that the norm induces isomorphisms $A'_{k_m}\cong A'_{k_n}$ for $m\geq n$ large enough, implying that $\Psi(k_\infty)$ is trivial.
\end{remark}
The invariant $\Psi(k_\infty)$ has at least two interesting interpretations, described in \cite{LMN}. To explain the first one, set, for any $n\in \mathbb{N}$,
$$C_{k_n}:=\mathrm{coker}(A'_{k_n}\to A'^{\,\Gamma_n}_{k_\infty}).$$
Then, if Gross's conjecture holds for $k_n$, $C_{k_n}$ is a finite group (actually $(X'_{\!k_{_{\!\scriptscriptstyle{\infty}}}})_{\Gamma_n}$ is finite if and only if $A'^{\,\Gamma_n}_{k_\infty}$ is, as is easily seen using \cite[Theorem 11]{Iw}). As is shown in \cite[Theorem 1.4]{LMN}, if $n$ is large enough, $C_{k_n}$ is independent of $n$ (up to isomorphism) and indeed isomorphic to $\Psi(k_\infty)$.
The second interpretation of $\Psi(k_\infty)$, which will be the most relevant for us, is in terms of Galois theory.
\begin{defi}\label{BP}
Let $K$ be a number field containing $\mu_p$. The field of Bertrandias-Payan $K^{BP}$ over $K$ is the compositum of all the (cyclic) $p$-extensions of $K$ which are embeddable in cyclic extensions of degree $p^m$, for all $m\geq 1$. Note that $K^{BP}/K$ is a Galois extension and we set $BP_K=\mathrm{Gal}(K^{BP}/K)$.
\end{defi}
In fact $K^{BP}$ coincides with the compositum of all extensions of $K$ which are locally $\mathbb{Z}_p$-embeddable for any finite place (see \cite[Lemme 4.1]{N1}). Using this observation, it is not difficult to show that, if $M_K$ denotes the maximal abelian pro-$p$-extension of $K$ unramified outside $p$ (thus $\mathfrak{X}_K=\mathrm{Gal}(M_K/K)$), then $K^{BP}\subseteq M_K$ and accordingly the restriction induces a surjection
\begin{equation}\label{XBP}
\mathfrak{X}_{K}\twoheadrightarrow BP_K.
\end{equation}
Moreover, if $E$ is a number field containing $K$, then one easily sees that $K^{BP}\subseteq E^{BP}$ and therefore the restriction defines a map $BP_E\to BP_K$. In particular we can consider $BP_{K_\infty}=\lim\limits_{\longleftarrow}BP_{K_n}$, the limit being taken over the restriction maps. Set also $K^{BP}_\infty=\cup_{n\in \mathbb{N}}K^{BP}_n$, so that $\mathrm{Gal}(K^{BP}_\infty/K_\infty)=BP_{K_\infty}$ (note, however, that in general $K^{BP}_\infty$ is strictly contained in the compositum of all the $p$-extensions of $K_\infty$ which are embeddable in cyclic extensions of degree $p^m$ for all $m\geq 1$).
Now $K_\infty^{BP}$ is contained in $M_{K_\infty}=\cup M_{K_n}$ (thus $M_{K_\infty}$ is the maximal abelian pro-$p$-extension of $K_\infty$ which is unramified outside $p$ and $\mathfrak{X}_{K_\infty}=\mathrm{Gal}(M_{K_\infty}/K_\infty)$) and the inverse limits of the surjections (\ref{XBP}) gives a surjection of $\Lambda$-modules
$$\mathfrak{X}_{K_\infty}\twoheadrightarrow BP_{K_\infty}.$$
Using the arguments of \cite[proof of Theorem 4.2]{N1}, one can show that the kernel of the above map is a $\Lambda$-torsion module, giving a surjection between the torsion submodules
\begin{equation}\label{slt}
t_\Lambda(\mathfrak{X}_{K_\infty})\twoheadrightarrow t_\Lambda(BP_{K_\infty}).
\end{equation}
Let $T_{K_\infty}$ be the subextension of $M_{K_\infty}$ which is fixed by the torsion submodule $t_\Lambda(\mathfrak{X}_{K_\infty})$ of $\mathfrak{X}_{K_\infty}$ and $N'_{K_\infty}$ be the (Galois) extension of $K_\infty$ which is obtained by adjoining to $K_\infty$ the $p^m$-th roots of all the $p$-units of $K_\infty$, for all $m\in\mathbb{N}$. We have $T_{K_{\infty}}\subseteq N'_{K_{\infty}}$, thanks to \cite[Theorem 15]{Iw}, and $T_{K_\infty}\subseteq K_\infty^{BP}$, by (\ref{slt}). If we set $N''_{K_\infty}=N'_{K_\infty}\cap K^{BP}_\infty$, then the promised Galois theoretical interpretation of $\Psi(K_\infty)$ is given by the following result.
\begin{teo}
Assume that the Gross conjecture holds for all layers $K_n$. Then there is an isomorphism of $\Lambda[\Delta]$-modules
$$\Psi(K_\infty)^*(1)\cong\mathrm{Gal}(N''_{K_\infty}/T_{K_\infty}).$$
\end{teo}
\begin{proof}
See \cite[Th\'eor\`eme 2.5]{LMN}.
\end{proof}
The above setting can therefore be summarized by the following diagram of fields which are all Galois extensions of $k$:
\begin{displaymath}
\xymatrix{
&M_{K_\infty}\ar@{-}[2,1]\ar@{-}[2,-1]\ar@/^8pc/@{-}^{t_\Lambda(\mathfrak{X}_{K_\infty})}[6,0]\\
\\
N'_{K_\infty}\ar@{-}[2,1]&&K^{BP}_\infty\ar@{-}[2,-1]\ar@/^/@{-}^{t_\Lambda(BP_{K_\infty})}[4,-1]\\
\\
&N''_{K_\infty}\ar@{-}_{\Psi(K_\infty)^*(1)}[2,0]\\
\\
&T_{K_\infty}
}
\end{displaymath}
\begin{remark}\label{idea}
As is apparent in the above diagram, if $\Psi(K_\infty)=0$, the restriction sends $\mathrm{Gal}(M_{K_\infty}/N'_{K_\infty})$ isomorphically onto $t_\Lambda(BP_{K_\infty})$. Therefore the triviality of $\Psi(K_{\infty})$ implies that the surjection $t_\Lambda(\mathfrak{X}_{K_\infty})\to t_\Lambda(BP_{K_\infty})$ splits in the category of $\Lambda$-modules.
\end{remark}
\section{Conditions for the splitting via Galois descent}
Now that the Iwasawa theoretical setting has been described, we give conditions for the splitting of the inclusion $\iota_{k,\infty}:\wkin{k}{(i)} \to \hdin{o'_k}{(i)}$. This map is intimately related to the projection $\pi:t_\Lambda(\mathfrak{X}_{K_\infty})\to t_\Lambda(BP_{K_\infty})$ that appeared in the preceding section, as we shall explain in this section.
We begin with a refined version of Remark \ref{idea}. We need the following lemma that will also be used in the next section.
\begin{lemma}\label{casa}
Let $R$ be a ring and let
$$
\begin{CD}
@.@.0@.0\\
@.@.@VVV@VVV\\
0@>>>A_1@>\alpha_1>>A_2@>\alpha_2>>A_3@>>>0\\
@.@V\iota_1VV@V\iota_2 VV@V \iota_3 VV\\
0@>>>B_1@>\beta_1>>B_2@>\beta_2>>B_3@>>>0\\
@.@.@V\pi_2VV@V\pi_3VV\\
@.@.T_2@>\tau_2>>T_3\\
@.@.@VVV@VVV\\
@.@.0@.0\\
\end{CD}
$$
be a commutative diagram of $R$-modules with exact rows and columns. Suppose that $\tau_2$ (or equivalently $\iota_1$) is an isomorphism. Then the following are equivalent:
\begin{enumerate}
\item[(i)] $\alpha_2$ and $\pi_2$ split;
\item[(ii)] $\beta_2$ and $\pi_3$ split.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\pi_2$ has a section, say $\lambda_2$, then $\beta_2\circ \lambda_2\circ \tau_2^{-1}$ is a section of $\pi_3$. If moreover $\alpha_2$ splits, then $\alpha_1$ has a section, say $\rho_1$. Then $\iota_1\circ\rho_1\circ\nu_2$ is a section of $\beta_1$, where $\nu_2$ is a section of $\iota_2$. In particular $\beta_2$ splits.\\
If $\beta_2$ splits, then $\beta_1$ has a section, say $\sigma_1$. Then $\iota_{1}^{-1}\circ\sigma_1\circ \iota_2$ is a section of $\alpha_1$ (in particular $\alpha_2$ splits).
If moreover $\pi_3$ has a section, say $\lambda_3$, then $\sigma_2\circ\lambda_3\circ\tau_2$ is a section of $\pi_2$, where $\sigma_2$ is a section of $\beta_2$.
\end{proof}
\begin{prop}\label{lmn2}
Suppose that Leopoldt's and Gross's conjectures hold for all the layers $K_n$. Then the following conditions are equivalent:
\begin{itemize}[leftmargin=12pt]
\item the surjective $\Lambda$-morphism $t_\Lambda(\mathfrak{X}_{K_\infty})(-i)_\Delta\to \Psi(K_\infty)^*(1-i)_\Delta$ splits;
\item the surjective $\Lambda$-morphisms $t_{\Lambda}(BP_{K_\infty})(-i)_\Delta\to \Psi(K_\infty)^*(1-i)_\Delta$ and $t_\Lambda(\mathfrak{X}_{K_\infty})(-i)_\Delta\to t_{\Lambda}(BP_{K_\infty})(-i)_\Delta$ split.
\end{itemize}
In particular, if $\Psi(K_\infty)(i-1)^\Delta=0$, then the surjective $\Lambda$-homomorphism
$$t_\Lambda(\mathfrak{X}_{K_\infty})(-i)_\Delta\to t_{\Lambda}(BP_{K_\infty})(-i)_\Delta$$
splits.
\end{prop}
\begin{proof}
Note that $N'_{K_\infty}/k$ and $K^{BP}_\infty/k$ are Galois extensions. This implies that $\mathrm{Gal}(M_{K_\infty}/N'_{K_\infty})$ and $\mathrm{Gal}(K^{BP}_\infty/N''_{K_\infty})$ are $\Lambda[\Delta]$-submodules of $t_\Lambda(\mathfrak{X}_{K_\infty})$ and $t_\Lambda(BP_{K_\infty})$, respectively. Therefore we have a commutative diagram of $\Lambda[\Delta]$-modules with exact rows and columns
$$
\begin{CD}
@.@.0@.0\\
@.@.@VVV@VVV\\
0@>>>\mathrm{Gal}(M_{K_\infty}/K^{BP}_\infty)@>>>\mathrm{Gal}(M_{K_\infty}/N''_{K_\infty})@>>>\mathrm{Gal}(K^{BP}_\infty/N''_{K_\infty})@>>>0\\
@.@|@VVV@VVV\\
0@>>>\mathrm{Gal}(M_{K_\infty}/K^{BP}_\infty)@>>>t_\Lambda(\mathfrak{X}_{K_\infty})@>>>t_\Lambda(BP_{K_\infty})@>>>0\\
@.@.@VVV@VVV\\
@.@.\Psi(K_\infty)^*(1)@=\Psi(K_\infty)^*(1)@>>>0\\
@.@.@VVV@VVV\\
@.@.0@.0
\end{CD}
$$
Furthermore the surjective homomorphism $\mathrm{Gal}(M_{K_\infty}/N''_{K_\infty})\to\mathrm{Gal}(K^{BP}_\infty/N''_{K_\infty})$ of $\Lambda[\Delta]$-modules splits (since $\mathrm{Gal}(M_{K_\infty}/N'_{K_\infty})$ is an isomorphic preimage of $\mathrm{Gal}(K^{BP}_\infty/N''_{K_\infty})$). Now tensor the above diagram with $\mathbb{Z}_p(-i)$ and take $\Delta$-coinvariants. The resulting diagram satisfies the hypotheses of Lemma \ref{casa}, which implies the equi\-va\-lence between the two assertions of the proposition.
\end{proof}
Before the description of the relation between $\iota_{k,\infty}$ and the projection $t_\Lambda(\mathfrak{X}_{K_\infty})\to t_\Lambda(BP_{K_\infty})$, we quote the following well-known lemma.
\begin{lemma}\label{ltzpt}
Let $M$ be a $\Lambda[\Delta]$-module which is a finitely generated $\Lambda$-module. Suppose that $(t_\Lambda(M))_{\Gamma_n}$ is finite for all $n\in \mathbb{N}$. Then the natural map $M\to \lim\limits_{\longleftarrow}M_{\Gamma_n}$ induces an isomorphism of $\Lambda[\Delta]$-modules
$$t_\Lambda(M)\cong \lim\limits_{\longleftarrow}t_{\mathbb{Z}_p}(M_{\Gamma_n}).$$
\end{lemma}
\begin{proof}
See \cite[proof of Proposition 2.1]{N3}.
\end{proof}
The proof of the following proposition is inspired by the papers \cite{N1, N2}.
\begin{prop}\label{nqd}
Suppose that Leopoldt's conjecture holds for all the layers $K_n$. Then, for every $i\geq 0$ and $i\ne 1$, there are natural isomorphisms of $\Lambda[\Delta]$-modules
$$t_{\Lambda}(\mathfrak{X}_{K_\infty})(-i)\cong \hdin{o'_K}{(i)}^*\quad \textrm{and} \quad t_{\Lambda}(BP_{K_\infty})(-i)\cong \wkin{K}{(i)}^*,$$
fitting into a commutative diagram
$$
\begin{CD}
\hdin{o'_K}{(i)}^*@>\iota_{K,\infty}^*>>\wkin{K}{(i)}^*\\
@V\wr VV@V\wr VV\\
t_{\Lambda}(\mathfrak{X}_{K_\infty})(-i)@>\pi>>t_{\Lambda}(BP_{K_\infty})(-i),
\end{CD}
$$
where $\pi$ is induced by the natural projection $\mathfrak{X}_{K_\infty}\to BP_{K_\infty}$.
\end{prop}
\begin{proof}
We shall prove the statement supposing that $K_{n_0}=K$, \textit{i.e.} all $p$-adic primes are totally ramified in $K_\infty/K$. The general case easily follows since no object appearing in the statement changes if $K$ is replaced by a higher layer of the cyclotomic tower. Furthermore every morphism we shall consider in the proof can be easily checked to be invariant with respect to the action of the Galois group of any Galois extension $K/E$.
Recall that, for every $i\in\mathbb{Z}$, there is an isomorphism of $\mathbb{Z}_p[\Delta]$-modules
\begin{equation}\label{h2t}
\hdet{o'_K}{(i)}^*\cong t_{\mathbb{Z}_p}(\mathfrak{X}_{K_\infty}(-i)_{\Gamma})
\end{equation}
(see \cite[Lemme 4.1]{N3}). Similarly one shows that for every prime $v$ above $p$ in $K$ there is an isomorphism of $\mathbb{Z}_p[\Delta_v]$-modules
$$\hdet{K_v}{(i)}^*\cong t_{\mathbb{Z}_p}(\mathfrak{X}_{K_{v,_\infty}}(-i)_{\Gamma_v}),$$
where $\mathfrak{X}_{K_{v,\infty}}$ is the Galois group of the maximal abelian pro-$p$-extension of the cyclotomic $\mathbb{Z}_p$-extension $K_{v,\infty}/K_v$, $\Gamma_v=\mathrm{Gal}(K_{v,\infty}/K_v)$ and $\Delta_v=\mathrm{Gal}(K_v/k_{v_0})$ where $v_0$ is the prime of $k$ below $v$.
The above isomorphisms yield a commutative diagram of $\mathbb{Z}_p[\Delta]$-modules
\begin{equation}\label{diag}
\begin{CD}
\underset{v\mid p}{\oplus}\hdet{K_v}{(i)}^*@>>>\hdet{o'_K}{(i)}^*\\
@V\wr VV@V\wr VV\\
\underset{v\mid p}{\oplus}t_{\mathbb{Z}_p}(\mathfrak{X}_{K_{v,\infty}}(-i)_{\Gamma_v})@>>>t_{\mathbb{Z}_p}(\mathfrak{X}_{K_\infty}(-i)_{\Gamma}).
\end{CD}
\end{equation}
If $i\ne 1$, then the group $\left(t_{\Lambda_v}(\mathfrak{X}_{K_{v,\infty}}(-i)\right)_{\Gamma_v}$ is finite for every $v\mid p$ in $K$, where $\Lambda_v=\mathbb{Z}_p[\![\Gamma_v]\!]$ (this follows from \cite[Theorem 25]{Iw}). The same holds for $\left(t_{\Lambda}(\mathfrak{X}_{K_\infty})(-i)\right)_{\Gamma}$ provided $i\geq 2$ (this follows easily from \cite[Lemme 4.3]{N3}) or $i=0$ and Leopoldt's conjecture holds for $K$ (see \cite[Lemma 21]{Iw}). Moreover, an easy check shows the diagram in (\ref{diag}) to be compatible with the restriction maps on both sides. Therefore, for every $i\geq 0$ and $i\ne 1$, taking direct limits over $K_\infty/K$ with respect to those maps and using Lemma \ref{ltzpt}, we get a commutative diagram (under Leopoldt's conjecture for all the layers $K_n$ when $i=0$)
$$
\begin{CD}
\underset{v\mid p}{\oplus}\hdin{K_v}{(i)}^*@>>>\hdin{o'_{K}}{(i)}^*\\
@V\wr VV@V\wr VV\\
\underset{v\mid p}{\oplus}t_{\Lambda_v}(\mathfrak{X}_{K_{v,\infty}}(-i))@>>>t_{\Lambda}(\mathfrak{X}_{K_\infty}(-i)),
\end{CD}
$$
where $\hdin{K_v}{(i)}=\lim\limits_{\longrightarrow}\hdet{(K_v)_n}{(i)}$. Since clearly
\begin{equation}\label{tinv}
t_{\Lambda}(\mathfrak{X}_{K_\infty}(-i))=t_{\Lambda}(\mathfrak{X}_{K_\infty})(-i)\quad\textrm{and} \quad t_{\Lambda_v}(\mathfrak{X}_{K_{v,\infty}}(-i))=t_{\Lambda_v}(\mathfrak{X}_{K_{v,\infty}})(-i),
\end{equation}
we deduce that, for every $i\geq 2$, there is a commutative diagram
\begin{equation}\label{h2inv}\begin{CD}
\hdin{o'_{K}}{(i)}@>>>\underset{v\mid p}{\oplus}\hdin{K_v}{(i)}\\
@V\wr VV@V\wr VV\\
\hdin{o'_{K}}{}(i)@>>>\underset{v\mid p}{\oplus}\hdin{K_v}{}(i)\\
\end{CD}
\end{equation}
which implies in particular that
\begin{equation}\label{wkinv}
\wkin{o'_K}{(i)}\cong \wkin{o'_K}{}(i).
\end{equation}
Now, the field of Bertrandias-Payan over $K$ certainly contains the compositum of all the $\mathbb{Z}_p$-extensions of $K$. Therefore the map $t_{\mathbb{Z}_p}(\mathfrak{X}_{K})\to t_{\mathbb{Z}_p}(BP_{K})$, induced by (\ref{XBP}), is surjective. In fact, it follows easily from the arguments of \cite[Th\'eor\`eme 4.2]{N1} that there is a commutative diagram of $\mathbb{Z}_p[\mathrm{Gal}(K/k)]$-modules with surjective rows
$$
\begin{CD}
\hdet{o'_{K}}{}^*@>>>\wket{K}{}^*\\
@V\wr VV@V\wr VV\\
t_{\mathbb{Z}_p}(\mathfrak{X}_{K}) @>>> t_{\mathbb{Z}_p}(BP_{K})\\
\end{CD}
$$
where the left-hand isomorphism is (\ref{h2t}), while the right-hand one is induced by the diagram. As before, the above diagram is compatible with restriction maps and taking inverse limits with respect to those maps over $K_\infty/K$ yields a commutative diagram
\begin{equation}\label{coomgalois}
\begin{CD}
\hdin{o'_K}{}^*@>>> \wkin{o'_K}{}^*\\
@V\wr VV@V\wr VV\\
\lim\limits_{\longleftarrow} t_{\mathbb{Z}_p}(\mathfrak{X}_{K_n})@>>>\lim\limits_{\longleftarrow} t_{\mathbb{Z}_p}(BP_{K_n}).\\
\end{CD}
\end{equation}
If Leopolt's conjecture holds for $K_n$ for every $n\in\mathbb{N}$, using Lemma \ref{ltzpt} we can replace the bottom line of diagram (\ref{coomgalois}) by the surjection $t_{\Lambda}(\mathfrak{X}_{K_\infty})\to t_{\Lambda}(BP_{K_\infty})$. Then the result follows by applying the functor $\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(-i)$ to diagram (\ref{coomgalois}) and using (\ref{tinv}), (\ref{h2inv}) and (\ref{wkinv}).
\end{proof}
\begin{remark}\label{chazadinfty}
From the proof of the above proposition, we deduce that, if $i\geq 2$ and Leopoldt's conjecture holds for $K_n$ for all $n\in\mathbb{N}$, then there is a commutative diagram of $\Lambda[\Delta]$-modules
$$
\begin{CD}
\wkin{K}{(i)} @>\iota_{K,\infty,i}>> \hdin{o'_K}{(i)}\\
@V\wr VV@V\wr VV\\
\wkin{K}{}(i) @>\iota_{K,\infty,0}(i)>> \hdin{o'_K}{}(i)\\
\end{CD}
$$
whose vertical arrows are natural isomorphisms.
\end{remark}
\begin{cor}\label{ideaNQD}
Suppose that Leopoldt's conjecture holds for all the layers $K_n$. Then for every $i\geq 2$ the following conditions are equivalent:
\begin{enumerate}
\item the injective $\Lambda$-homomorphism $\wkin{k}{(i)}\to \hdin{o'_k}{(i)}$ splits;
\item the surjective $\Lambda$-homomorphism $t_\Lambda(\mathfrak{X}_{K_\infty})(-i)_\Delta\to t_{\Lambda}(BP_{K_\infty})(-i)_\Delta$ splits.
\end{enumerate}
In particular, if $\Psi(K_\infty)(i-1)^\Delta=0$, the inclusion $\wkin{k}{(i)}\to \hdin{o'_k}{(i)}$ splits as soon as Gross's conjecture holds for all the layers $K_n$.
\end{cor}
\begin{proof}
Using standard properties of Pontryagin duality and the equality $\hdin{o'_K}{(i)}^\Delta=\hdin{o'_k}{(i)}$, it is easy to show that the first condition is equivalent to the splitting of the surjective map
$$\left(\hdin{o'_K}{(i)}^*\right)_\Delta\to \left(\wkin{K}{(i)})^*\right)_\Delta.$$
Therefore the equivalence between the two statements of the corollary follows by Proposition \ref{nqd}. The last assertion follows by Proposition \ref{lmn2}.
\end{proof}
Once we have given conditions for the inclusion $\wkin{k}{(i)}\subseteq \hdin{o'_k}{(i)}$ to split, we aim to see when the splitting at infinite level implies the splitting at finite levels. It is natural to look at this problem using Galois descent.
\begin{prop}\label{main}
Let $k$ be a number field and $i\geq 2$ be an integer. Suppose that
\begin{enumerate}
\item[(i)] the surjective homomorphism $\hdet{o'_k}{(i)}\to \hdin{o'_k}{(i)}^{\Gamma}$ splits;
\item[(ii)] the injective $\Lambda$-homomorphism $\wkin{k}{(i)}\to \hdin{o'_k}{(i)}$ splits.
\end{enumerate}
Then the inclusion $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits.
\end{prop}
\begin{proof}
If $\wkin{k}{(i)} \to \hdin{o'_k}{(i)}$ splits in the category of $\Lambda$-modules, then
$\wkin{k}{(i)}^{\Gamma}\to \hdin{o'_k}{(i)}^{\Gamma}$ splits. The latter condition implies the splitting of $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$, thanks to (i), Theorem \ref{descent} and Lemma \ref{casa}.
\end{proof}
Combining Proposition \ref{main} and Corollary \ref{ideaNQD} gives the main result of this section.
\begin{teo}\label{randonnee}
Let $k$ be a number field and $i\geq 2$ be an integer. Suppose that
\begin{enumerate}
\item[(a)] the surjective homomorphism $\hdet{o'_k}{(i)}\to \hdin{o'_k}{(i)}^{\Gamma}$ splits;
\item[(b)] $\Psi(K_\infty)(i-1)^{\Delta}=0$;
\item[(c)] Gross's and Leopoldt's conjectures hold for all the layers $K_n$.
\end{enumerate}
Then the injective homomorphism $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits.
\end{teo}
\begin{remarks}
\begin{enumerate}
\item[1)] Suppose that the Iwasawa $\mu$ invariant of $X'_{K_\infty}$ is trivial (this holds for example if $k$ is abelian by a classical result of Ferrero and Washington). Then, by Theorem \ref{descent}, Condition (a) of Theorem \ref{randonnee} holds asymptotically, \textit{i.e.} the surjection $\hdet{o'_{k_n}}{(i)}\to \hdin{o'_k}{(i)}^{\Gamma_n}$ splits for $n$ large enough.
\item[2)] Suppose that $k$ is totally real (hence $K$ is CM).
\begin{itemize}
\item If $i$ is even, then $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta=0$ by Proposition \ref{coates}. In particular Condition (a) of Theorem \ref{randonnee} holds, thanks to Theorem \ref{descent}.
\item If $i$ is odd, then we have
$$\Psi(K_{\infty})(i-1)^{\Delta}\subseteq \Psi(K_{\infty})(i-1)^{\Delta_0}\cong \Psi(K_{\infty})^{\Delta_0}(i-1),$$
where $K^+$ denotes the maximal totally real subfield of $K$ and $\Delta_0=\mathrm{Gal}(K/K^+)\subseteq \Delta$. Now, under Greenberg's conjecture $\Psi(K_\infty)^{\Delta_0}=\Psi(K^+_\infty)=0$ by Remark \ref{oujda}. In particular, under Greenberg's conjecture, Condition (b) of Theorem \ref{randonnee} holds.
\end{itemize}
\item[3)] Of course, if $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}=0$, then, by Theorem \ref{descent}, Condition (a) of Theorem \ref{randonnee} is satisfied. However the condition $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}=0$ could in general be too strong for our purposes. In fact, by Schneider's isomorphism
$$\wket{k}{(i)} \cong X'_{K_\infty}(i-1)_{\mathcal{G}}=(X'_{K_\infty}(i-1)_{\Delta})_\Gamma.$$
In particular, by Nakayama's lemma, $\wket{k}{(i)}$ is trivial precisely when $X'_{K_\infty}(i-1)_{\Delta}$ is. Now, if $k$ is a totally real field satisfying Greenberg's conjecture and $i\equiv 1 \pmod{\#\Delta}$, then
$$X'_{K_\infty}(i-1)_{\Delta}\cong X'_{k_\infty}=X'^{\,\circ}_{\!k_{_{\!\scriptscriptstyle{\infty}}}}\congX'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}.$$
Therefore, in this situation, the triviality of $X'^{\,\circ}_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_{\Delta}$ is equivalent to that of $\wket{k}{(i)}$.
\end{enumerate}
\end{remarks}
\section{Conditions for the splitting via Galois codescent}\label{code}
In this section we shall follow a different approach to find a condition for the inclusion $\wket{k}{(i)}\to\hdet{k}{(i)}$ to split. We study the splitting of the inclusion of $\Lambda$-modules $\wkiw{k}{(i)}\to\hdiw{k}{(i)}$ and then use the good codescent properties of higher wild and tame kernels.
Taking inverse limits of the exact sequence (\ref{sequence}) for $i\geq 1$ along the cyclotomic tower, we obtain the following exact sequence of $\Lambda$-modules
\begin{equation}\label{seiw}
0\to\wkiw{k}{(i)}\stackrel{\iota_{k,Iw}}{\longrightarrow} \hdiw{o'_k}{(i)} \stackrel{\pi_{k,Iw}}{\longrightarrow} \underset{v\mid p}{\oplus}\underset{w\mid v}{\oplus}\hdiw{k_v}{(i)}\to \hzin{k}{(1-i)}^*\to 0
\end{equation}
where, in the second sum, $w$ runs over the primes of $k_\infty$ dividing a $p$-adic prime $v$ of $k$. Set
$$\oplus_{k,Iw}=\underset{v\mid p}{\oplus}\underset{w\mid v}{\oplus}\hdiw{k_v}{(i)},\qquad\tilde\oplus_{k,Iw}=\mathrm{ker}\left(\oplus_{k,Iw}\to \hzin{k}{(1-i)}^*\right),$$
so that we have an exact sequence
\begin{equation}\label{seiwsimp}
0\to\wkiw{k}{(i)}\stackrel{\iota_{k,Iw}}{\longrightarrow}\hdiw{o'_k}{(i)} \stackrel{\pi_{k,Iw}}{\longrightarrow} \tilde\oplus_{k,Iw}\to 0.
\end{equation}
Note that the above exact sequence splits in the category of $\mathbb{Z}_p$-modules since $\tilde\oplus_{k,Iw}$ is either trivial or a free $\mathbb{Z}_p$-module ($\hdiw{k_v}{(i)}\cong\hzin{k_v}{(1-i)}^*$ is either trivial or $\mathbb{Z}_p$-free of rank $1$).
Let $M,N$ be two finitely generated compact $\Lambda$-modules, then we let $\mathrm{Hom}(M, N)$ (resp. $\mathrm{Hom}_\Lambda(M, N)$) denote the group of continuous $\mathbb{Z}_p$-homomorphisms (resp. $\Lambda$-homomorphisms). Then $\mathrm{Hom}(M, N)$ has a structure of $\Gamma$-module such that $(\gamma f)(m)=\gamma f(\gamma^{-1}m)$, for every $f\in \mathrm{Hom}(M, N)$, $\gamma\in\Gamma$ and $m\in M$. In particular $\mathrm{Hom}(M, N)^{\Gamma}=\mathrm{Hom}_\Gamma(M, N)=\mathrm{Hom}_\Lambda(M, N)$.
\begin{lemma}\label{pavia}
Let
\begin{equation}\label{m123}
0\to M_1 \stackrel{\iota}{\longrightarrow} M_2 \stackrel{\pi}{\longrightarrow} M_3\to 0
\end{equation}
be an exact sequence of finitely generated compact $\Lambda$-modules which splits in the category of $\mathbb{Z}_p$-modules. Then (\ref{m123}) splits in the category of $\Lambda$-modules if and only if
\begin{equation}\label{condition123}
\mathrm{ker}\Big(\mathrm{Hom}(M_3, M_1)_\Gamma\stackrel{\iota\scriptscriptstyle{\circ}}{\longrightarrow}\mathrm{Hom}(M_3, M_2)_\Gamma\Big)=0.
\end{equation}
\end{lemma}
\begin{proof}
Since (\ref{m123}) splits as $\mathbb{Z}_p$-modules, we have an exact sequence of $\Gamma$-modules
$$0\to \mathrm{Hom}(M_3, M_1)\stackrel{\iota\scriptscriptstyle{\circ}}{\longrightarrow}\mathrm{Hom}(M_3, M_2)\stackrel{\pi\scriptscriptstyle{\circ}}{\longrightarrow}\mathrm{Hom}(M_3, M_3)\to 0.$$
Note that if (\ref{m123}) splits as $\Gamma$-modules, then clearly the same holds for the above sequence and in particular (\ref{condition123}) holds.
Suppose conversely that (\ref{condition123}) holds. A standard application of the snake lemma to the above exact sequence gives an exact sequence
$$\mathrm{Hom}_\Gamma(M_3, M_2)\stackrel{\pi\scriptscriptstyle{\circ}}{\longrightarrow}\mathrm{Hom}_\Gamma(M_3, M_3) \to \mathrm{Hom}(M_3, M_1)_\Gamma\stackrel{\iota\scriptscriptstyle{\circ}}{\longrightarrow}\mathrm{Hom}(M_3, M_2)_\Gamma.$$
We deduce that $\pi{\scriptscriptstyle\circ}:\mathrm{Hom}_\Gamma(M_3, M_2)\to \mathrm{Hom}_\Gamma(M_3, M_3)$ is surjective. In particular, there exists $\sigma\in \mathrm{Hom}_\Gamma(M_3, M_2)=\mathrm{Hom}_\Lambda(M_3, M_2)$ such that $\pi{\scriptscriptstyle\circ\,}\sigma=\mathrm{id}_{M_3}$.
\end{proof}
When $i=1$ we have $\wket{K}{(1)}\cong A'_K$ and therefore (\ref{sequence}) reads
$$0\to A'_K\to \hdet{o'_K}{(1)}\to \underset{v\mid p}{\oplus} \hdet{K_v}{(1)}\to \hzet{K}{}^*\to 0.$$
Note that
$$\hdet{K_v}{(1)}\cong \hzet{K_v}{}^*\cong \mathbb{Z}_p$$
by local duality. Hence the inclusion $A'_K\to \hdet{o'_K}{(1)}$ splits as $\mathbb{Z}_p[\Delta]$-modules and therefore also as $\Lambda[\Delta]$-modules (with trivial $\Gamma$-action), since it maps $A'_K$ isomorphically onto $t_{\mathbb{Z}_p}(\hdet{o'_K}{(1)})$. In particular the inclusion $A'_K(i-1)_\Delta\to \hdet{o'_K}{(1)}(i-1)_\Delta$ splits as $\Lambda$-modules.
\begin{lemma}\label{chazad}
For every $i\in\mathbb{Z}$, there is a commutative diagram of $\Lambda[\Delta]$-modules
$$
\begin{CD}
\wkiw{K}{(i)} @>\iota_{K,Iw,i}>> \hdiw{o'_K}{(i)}\\
@V\wr VV@V\wr VV\\
\wkiw{K}{}(i) @>\iota_{K,Iw,0}(i)>> \hdiw{o'_K}{}(i)\\
\end{CD}
$$
whose vertical arrows are natural isomorphisms.
\end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\hdiw{o'_K}{(i)}&=&\lim\limits_{\underset{n}{\longleftarrow}} \lim\limits_{\underset{m}{\longleftarrow}}\hdetp{o'_{K_n}}{m}{(i)}\\
&=&\lim\limits_{\underset{m}{\longleftarrow}} \lim\limits_{\underset{n\geq m}{\longleftarrow}}\hdetp{o'_{K_n}}{m}{(i)}\\
&\cong&\lim\limits_{\underset{m}{\longleftarrow}} \lim\limits_{\underset{n\geq m}{\longleftarrow}}\hdetp{o'_{K_n}}{m}{}(i)\quad\textrm{(since $\mu_{p^m}\subseteq K_n$ if $n\geq m$)}\\
&=&\lim\limits_{\underset{n}{\longleftarrow}} \lim\limits_{\underset{m}{\longleftarrow}}\hdetp{o'_{K_n}}{m}{}(i)=\hdiw{o'_K}{}(i).
\end{eqnarray*}
A similar argument shows that, for any $p$-adic prime $w$ of $K$, there is an isomorphism $\hdiw{K_w}{(i)}\cong \hdiw{K_w}{}(i)$ which is compatible with the one above. Hence the statement of the lemma follows.
\end{proof}
\begin{remark}
If the Iwasawa $\mu$-invariant of $X'_{K_\infty}$ is trivial and $i\geq 2$, there exist isomorphisms of $\Lambda[\Delta]$-mo\-du\-les
$$\wkin{K}{(i)}\cong \wkiw{K}{(i)}\otimes \mathbb{Q}_p/\mathbb{Z}_p,\quad\hdin{o'_K}{(i)}\cong \hdiw{o'_K}{(i)}\otimes \mathbb{Q}_p/\mathbb{Z}_p$$
(see \cite[Lemma 2.4]{KM2}). Thus Lemma \ref{chazad} can be used to prove the result of Remark \ref{chazadinfty}.
\end{remark}
We are now ready for the main result of this section.
\begin{teo}\label{main2}
Let $k$ be a number field and $i\geq 2$ be an integer. Suppose that the map
\begin{equation}\label{mappa}
\alpha:\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)_\Gamma \to \mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)_\Gamma,
\end{equation}
induced by the natural map $X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta\to A'_K(i-1)_\Delta$, is injective.
Then the injective homomorphism $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits.
\end{teo}
\begin{proof}
By (\ref{codescentwh}), it suffices to show that the exact sequence (\ref{seiwsimp}) splits in the category of $\Lambda$-modules.
Note that
$$\wkiw{k}{(i)}\cong X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta,$$
by Schneider's isomorphism. Then, according to Lemma \ref{pavia}, the exact sequence (\ref{seiwsimp}) splits if and only if the map
\begin{equation}\label{remplace}
\beta:\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)_\Gamma\to\mathrm{Hom}\left(\tilde\oplus_{k,Iw}, \hdiw{o'_k}{(i)}\right)_\Gamma,
\end{equation}
induced by $\iota_{K,Iw}$ via the above isomorphism, is injective. Hence it is sufficient to show that $\mathrm{ker}\left(\beta\right)\subseteq \mathrm{ker}\left(\alpha\right)$.
Using Lemma \ref{chazad}, we get an isomorphism $\hdiw{o'_k}{(i)}\cong\hdiw{o'_K}{(1)}(i-1)_\Delta$ and therefore a morphism $\hdiw{o'_k}{(i)}\to \hdet{o'_K}{(1)}(i-1)_{\Delta}$ fitting into a commutative diagram of $\Lambda$-modules
$$
\begin{CD}
X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta @>>> A'_K(i-1)_\Delta\\
@VVV@VVV\\
\hdiw{o'_k}{(i)} @>>>\hdet{o'_K}{(1)}(i-1)_{\Delta}
\end{CD}
$$
whose vertical arrows are injective. Applying the functor $\mathrm{Hom}(\tilde\oplus_{k,Iw},\--)_\Gamma$ to the above diagram, we get a commutative diagram
$$
\begin{CD}
\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)_\Gamma @>\alpha>> \mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)_\Gamma\\
@V\beta VV@VVV\\
\mathrm{Hom}(\tilde\oplus_{k,Iw},\hdiw{o'_k}{(i)})_\Gamma @>>>\mathrm{Hom}(\tilde\oplus_{k,Iw},\hdet{o'_K}{(1)}(i-1)_{\Delta})_\Gamma.
\end{CD}
$$
According to the discussion prior to Lemma \ref{pavia}, the injection $A'_K(i-1)_\Delta\to \hdet{o'_K}{(1)}(i-1)_\Delta$ splits as a homomorphism of $\Lambda$-modules. Therefore the same holds for the injection $\mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)\to\mathrm{Hom}(\tilde\oplus_{k,Iw},\hdet{o'_K}{(1)}(i-1)_{\Delta})$. Hence the right vertical arrow of the above diagram is injective and we have $\mathrm{ker}\left(\beta\right)\subseteq \mathrm{ker}\left(\alpha\right)$, as desired.
\end{proof}
The condition of Theorem \ref{main2}, despite its aspect, can be simplified in several concrete situations, as the following result shows.
\begin{cor}\label{simplecase}
Suppose that $n_0(K)=0$ (\textit{i.e.} all $p$-adic places are totally ramified in $K_\infty/K$). Then, if $\Psi(K)(i-1)^\Delta=0$, the inclusion $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits.
\end{cor}
\begin{proof}
Since $Q:=\hzin{k}{(1-i)}^*$ is either trivial or a free $\mathbb{Z}_p$-module, the exact sequence of $\Lambda$-modules
$$0\to \tilde\oplus_{k,Iw}\to\oplus_{k,Iw}\to Q\to 0$$
splits in the category of $\mathbb{Z}_p$-modules. We thus get a commutative diagram of $\Lambda$-modules with exact rows
$$
\xymatrix{
0\ar[r]&\mathrm{Hom}(Q,X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta) \ar[d] \ar[r] & \mathrm{Hom}(\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)\ar[d] \ar[r] &\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)\ar[d]\ar[r]&0\,\,\\
0\ar[r]&\mathrm{Hom}(Q,A'_K(i-1)_\Delta)\ar[r] & \mathrm{Hom}(\oplus_{k,Iw},A'_K(i-1)_\Delta) \ar[r] &\mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)\ar[r]&0.\\}
$$
Now note that
\begin{eqnarray*}
Q=\hzin{k}{(1-i)}^*&\cong&\left\{\begin{array}{ll}\mathbb{Z}_p(i-1)&\textrm{if $i\equiv 1 \pmod{[K:k]}$}\\0&\textrm{otherwise,}\end{array}\right. \quad \textrm{as $\Gamma$-modules,}\\
\hdiw{k_v}{(i)}&\cong&\left\{\begin{array}{ll}\mathbb{Z}_p(i-1)&\textrm{if $i\equiv 1 \pmod{[k_v(\mu_p):k_v]}$}\\0&\textrm{otherwise,}\end{array}\right. \quad \textrm{as $\Gamma_v$-modules,}
\end{eqnarray*}
where $\Gamma_v=\mathrm{Gal}(k_{v,\infty}/k_v)$ is the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $k_v$. Since, by hypothesis, all $p$-adic places are totally ramified in $K_\infty/K$ (and hence in $k_\infty/k$), we have $\Gamma=\Gamma_v$ and an isomorphism of $\Lambda$-modules
$$\oplus_{k,Iw}\cong \underset{v\mid p}{\oplus^\prime}\mathbb{Z}_p(i-1),$$
where the module on the right-hand side is the sum of the $\Lambda$-module $\mathbb{Z}_p(i-1)$ over the $p$-adic places $v$ of $k$ such that $i\equiv 1 \pmod{[k_v(\mu_p):k_v]}$.
For every $\Lambda[\Delta]$-module $M$ and every $j\in \mathbb{Z}$, there is a natural isomorphism of $\Lambda[\Delta]$-modules $M(-j)\cong \mathrm{Hom}(\mathbb{Z}_p(j),M)$. Moreover, $M(j)_\Delta(-j)\cong M^{[-j]}$ as $\Lambda[\Delta]$-modules. As usual, here $M^{[-j]}$ denotes the eigenspace of $M$ where $\Delta$ acts as multiplication by $\omega^{-j}$, $\omega:\Delta\to \mathbb{Z}_p^\times$ being the Teichm\"uller character. In particular, if $i\equiv 1 \pmod{[K:k]}$, then obviously $i\equiv 1 \pmod{[k_v(\mu_p):k_v]}$ for every $p$-adic place $v$ and from the above diagram we deduce a commutative diagram of $\Lambda$-modules with exact rows
$$
\begin{CD}
0@>>>\xKj{1-i} @>>> \underset{v\mid p}{\oplus^\prime}\xKj{1-i} @>>>\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)@>>>0\,\,\\
@.@VVV@VVV@VVV\\
0@>>>\aKj{1-i} @>>> \underset{v\mid p}{\oplus^\prime}\aKj{1-i}@>>>\mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)@>>>0.
\end{CD}
$$
If instead $i\not \equiv 1 \pmod{[K:k]}$, the terms in the left-hand column are trivial. We shall continue the proof in the case where $i\equiv 1 \pmod{[K:k]}$, the case $i\not\equiv 1 \pmod{[K:k]}$ follows by similar arguments. Of course if $i\equiv 1 \pmod{[K:k]}$, then $\oplus^\prime_{v\mid p}\mathbb{Z}_p(i-1)=\oplus_{v\mid p}\mathbb{Z}_p(i-1)$.
Then taking $\Gamma$-coinvariants in the above diagram we get a commutative diagram of $\mathbb{Z}_p$-modules with exact rows
$$
\begin{CD}
@.(\xKj{1-i})_\Gamma @>>> \underset{v\mid p}{\oplus}(\xKj{1-i})_\Gamma @>>>\mathrm{Hom}(\tilde\oplus_{k,Iw},X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(i-1)_\Delta)_\Gamma @>>>0\,\,\\
@.@VVV@VVV@VV\alpha V\\
0@>>>\aKj{1-i} @>>> \underset{v\mid p}{\oplus}\aKj{1-i} @>>>\mathrm{Hom}(\tilde\oplus_{k,Iw},A'_K(i-1)_\Delta)_\Gamma @>>>0,
\end{CD}
$$
where $\alpha$ is the map of Theorem \ref{main2}.
Note that $\ker\left((\xKj{1-i})_\Gamma\to \aKj{1-i}\right)=\Psi(K)^{[1-i]}=0$ by hypothesis.
Hence the snake lemma gives an inclusion
$$\ker\left(\alpha
\right)\hookrightarrow \mathrm{coker}\left((\xKj{1-i})_{\Gamma}\to \aKj{1-i} \right).$$
Since $n_0(K)=0$, the map $X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}\to A'_K$ is surjective so the corollary follows by Theorem \ref{main2}.
\end{proof}
\begin{remark}
It is interesting to compare the hypotheses of Theorem \ref{randonnee} with those of Corollary \ref{simplecase} (and hence of Theorem \ref{main2}). If $\Psi(K)(i-1)^\Delta=0$, then of course $\Psi(K_\infty)(i-1)^\Delta=0$ by Lemma \ref{CPsi}. On the other hand, even if $\Psi(K_\infty)(i-1)^\Delta=0$ and $\hdet{o'_k}{(i)}\to \hdin{o'_k}{(i)}^\Gamma$ splits, $\Psi(K)(i-1)^\Delta$ may be nontrivial. For example, consider the case where $p=3$, $i=2$ and $k=\mathbb{Q}(\sqrt{-3\cdot 3739})$ (for which $n_0(K)=0$). Then the PARI program developed by Browkin and Gangl (see \cite{BG}) gives $\#H^2_{\acute{e}t}(o'_k,\mathbb{Z}_3(2))=3^2$. Moreover $k$ has one prime $v$ above $3$ and $k_v=\mathbb{Q}_3(\mu_3)$. In particular there is an exact sequence
$$0\to \left(A'_K/3(1)\right)_\Delta\to H^2_{\acute{e}t}(o'_k,\mathbb{Z}_3(2))/3\to \mathbb{Z}/3\mathbb{Z}\to 0$$
(see \cite[proof of Theorem 6.6]{Ke} or \cite[Theorem 6.2]{Ta2}). Using PARI, we get $\#\left(A'_K/3(1)\right)_\Delta=3$ and therefore $H^2_{\acute{e}t}(o'_k,\mathbb{Z}_3(2))$ has $3$-rank $2$, giving $H^2_{\acute{e}t}(o'_k,\mathbb{Z}_3(2))\cong \mathbb{Z}/3\times\mathbb{Z}/3\mathbb{Z}$. This immediately implies that the surjection $H^2_{\acute{e}t}(o'_k,\mathbb{Z}_3(2))\to H^2_{\infty}(o'_k,\mathbb{Z}_3(2))^\Gamma$ splits. Furthermore $\Psi(K_\infty)(i-1)^\Delta\cong\Psi(k'_\infty)(i-1)$, where $k'=\mathbb{Q}(\sqrt{3739})$ is the totally real subfield of $K=k(\mu_3)$. In particular $\Psi(K_\infty)(i-1)^\Delta=0$ by Remark \ref{oujda}. On the other hand, using again PARI and the methods of \cite[Section 4.3]{LMN}, we get $\Psi(K)(i-1)^\Delta\cong\Psi(k')(i-1)\ne 0$. Note that this also shows that the converse of Corollary \ref{simplecase} does not hold in general.
\end{remark}
\section{A general criterion using class groups}\label{crit}
In this section we will prove a criterion for the inclusion $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ to split in terms of the triviality of some codescent kernels which are somehow reminiscent of the $\Psi(K_n)$. In fact we shall use this criterion to give a different proof of Corollary \ref{simplecase}.
For every $i\geq 1$ and every $m\in \mathbb{N}$, let $k(\mu_{p^m}^{\otimes i})$ be the subfield of $k(\mu_{p^m})$ which is fixed by the kernel of the homomorphism $\mathrm{Gal}(k(\mu_{p^m})/k)\to \mathrm{Aut}(\mu_{p^m}^{\otimes i})$, induced by the action of $\mathrm{Gal}(k(\mu_{p^m})/k)$ on $\mu_{p^m}^{\otimes i}$.
For this section, we shall set $j=i-1$. We also recall the notation $\mathcal{G}_n=\mathrm{Gal}(K_\infty/k_n)$ (and $\mathcal{G}_0=\mathcal{G}$) and we set
$$\wketp{k}{m}{(i)}=\ker\left(\hdetp{o'_k}{m}{(i)}\to \underset{v\mid p}{\oplus} \hdetp{k_v}{m}{(i)}\right).$$
The following well-known lemma indicates the strategy of our criterion.
\begin{lemma}\label{splitpure}
The following conditions are equivalent:
\begin{itemize}
\item $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits;
\item for every $m\in\mathbb{N}$, the map $\wket{k}{(i)}/p^m\to \hdet{o'_k}{(i)}/p^m$ is injective.
\end{itemize}
\end{lemma}
\begin{proof}
The second condition can be rephrased by saying that $\wket{k}{(i)}$ is pure in $\hdet{o'_k}{(i)}$ (see \cite[Chapter 7]{Ka}). Then the lemma follows by \cite[Theorem 5]{Ka} and the (obvious) fact that a direct summand of an abelian group is a pure subgroup.
\end{proof}
For any $m\in \mathbb{N}$ and any $i\geq 1$, set
$$\Psi(k,p^m,i):=\mathrm{ker}\left(\wket{k}{(i)}/p^m\to \hdet{o'_k}{(i)}/p^m\right).$$
This notation is reminiscent of $\Psi(k)=\mathrm{ker}\left((X'_{\!k_{_{\!\scriptscriptstyle{\infty}}}})_\Gamma\to A'_k\right)$ and the next proposition partially explains this relation. We will need the following observation: for every $i\ne 1$, Schneider's isomorphism
$\wket{k}{(i)}\cong \left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}(j)\right)_{\mathcal{G}}$ can also be obtained taking projective limits over $m>0$ of the isomorphisms
\begin{equation}\label{modpn}
\wketp{k}{m}{(i)}\cong \left(A_{k(\mu_{p^m}^{\otimes j})}'/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)}.
\end{equation}
Using the definition of $k(\mu_{p^m}^{\otimes j})$, the isomorphism (\ref{modpn}) follows from class field theory and Poitou-Tate duality (see \cite[Section 3]{Ca}).
\begin{prop}
For every integer $m$ and $j+1=i\geq 2$, there is an exact sequence
$$0\to \Psi(k,p^m,i)\to \left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/p^m(j)\right)_{\mathcal{G}}\to \left(A'_{k(\mu_{p^m}^{\otimes j})}/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)}\to D_{m,i}\to 0$$
where $D_{m,i}$ is the Pontryagin dual of the kernel of $\hzet{k}{(-j)}/p^m\to\underset{v\mid p}{\oplus}\hzet{k_v}{(-j)}/p^m$ and is trivial if $i\not\equiv1 \pmod{[K:k]}$.
\end{prop}
\begin{proof}
Taking cohomology of the exact sequence
$$0\to \mathbb{Z}_p(i)\stackrel{p^m}{\longrightarrow}\mathbb{Z}_p(i)\to\mathbb{Z}/p^m\mathbb{Z}(i)\to 0$$
and using the long exact sequence of Poitou-Tate, we get the following commutative diagram with exact rows
$$
\begin{CD}
0@>>>\Psi(k,p^m,i)@>>>\wket{k}{(i)}/p^m @>>> \hdet{o'_k}{(i)}/p^m @>>> \tilde\oplus_k/p^m @>>>0\,\,\\
@.@.@VVV@VVV@VVV\\
@.0@>>> \wketp{k}{m}{(i)} @>>> \hdetp{o'_k}{m}{(i)}@>>>\tilde\oplus_{k,p^m}@>>>0,
\end{CD}
$$
where
\begin{eqnarray*}
\tilde\oplus_{k}=&\!\!\!\!\!\!\!&\mathrm{ker}\left(\underset{v\mid p}{\oplus}\hdet{k_v}{(i)}\to \hzet{k}{(-j)}^*\right),\\
\tilde\oplus_{k,p^m}=&\!\!\!\!\!\!\!&\mathrm{ker}\left(\underset{v\mid p}{\oplus}\hdetp{k_v}{m}{(i)}\to \hzetp{k}{m}{(-j)}^*\right).
\end{eqnarray*}
The vertical arrow in the middle is an isomorphism, since $cd_p(Spec(o'_k))\leq 2$. Then the snake lemma and the discussion prior to Lemma \ref{splitpure} give the exact sequence
$$0\to \Psi(k,p^m,i)\to \left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/p^m(j)\right)_{\mathcal{G}}\to \left(A'_{k(\mu_{p^m}^{\otimes j})}/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)}\to D_{m,i}\to 0,$$
where $D_{m,i}=\ker\left(\tilde\oplus_k/p^m\to \tilde\oplus_{k,p^m}\right)$. We also have a commutative diagram with exact rows and columns
$$
\begin{CD}
@.0\\
@.@VVV\\
@.D_{m,i}\\
@.@VVV\\
@.\tilde \oplus_k/p^m @>>> \underset{v\mid p}{\oplus}\hdet{k_v}{(i)}/ p^m @>>> \hzet{k}{(-j)}^*/p^m @>>>0\,\,\\
@.@VVV@VVV@VVV\\
0@>>> \tilde\oplus_{k,p^m} @>>> \underset{v\mid p}{\oplus}\hdetp{k_v}{m}{(i)} @>>>\hzetp{k}{m}{(-j)}^*@>>>0 .
\end{CD}
$$
Since $cd_p(Spec(k_v))\leq 2$ for any $v\mid p$, we deduce that the vertical arrow in the middle is an isomorphism. Hence, using the snake lemma and local duality,
\begin{eqnarray*}
D_{m,i}&=&\ker\left(\tilde\oplus_k/p^m \to \underset{v\mid p}{\oplus}\hdet{k_v}{(i)}/p^m\right)\\
&=&\mathrm{coker}\left( \underset{v\mid p}{\oplus}\hdet{k_v}{(i)}[p^m]\to \hzet{k}{(-j)}^*[p^m]\right)\\
&=&\mathrm{coker}\left( \underset{v\mid p}{\oplus}\hzet{k_v}{(-j)}^*[p^m]\to \hzet{k}{(-j)}^*[p^m]\right)\\
&=&\mathrm{coker}\left( \underset{v\mid p}{\oplus}\left(\hzet{k_v}{(-j)}/p^m\right)^*\to \left(\hzet{k}{(-j)}/p^m\right)^*\right)\\
&=&\left(\ker\left(\hzet{k}{(-j)}/p^m\to\underset{v\mid p}{\oplus}\hzet{k_v}{(-j)}/p^m \right)\right)^*,
\end{eqnarray*}
where for an abelian group $M$, we denote by $M[p^m]$ the elements of $M$ annihilated by $p^m$. Note that, if $i\not\equiv1 \pmod{[K:k]}$, then $\hzet{k}{(-j)}=0$ and hence $D_{m,i}$ is trivial for every $m\in \mathbb{N}$.
\end{proof}
The above proposition and Lemma \ref{splitpure} give the main result of this section.
\begin{teo}\label{criterion}
Let $k$ be a number field. For any integer $m$ and $j+1=i\geq 2$, the inclusion $\wket{k}{(i)}\to \hdet{o'_k}{(i)}$ splits if and only if
\begin{equation}\label{comeinca}
\mathrm{ker}\left(\left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/p^m(j)\right)_{\mathcal{G}}\to \left(A'_{k(\mu_{p^m}^{\otimes j})}/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)}\right)=0,\quad \textrm{for every $m\in\mathbb{N}$.}
\end{equation}
\end{teo}
\begin{remark}
It is interesting to compare (\ref{comeinca}) with the condition
$$\left(A'_{k(\mu_{p^m}^{\otimes j})}/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)}=0,\quad \textrm{for every $m\in\mathbb{N}$,}
$$
which is equivalent to the splitting of the inclusion $\hdet{o'_k}{(i)}\to H^2_{cont}(k,\mathbb{Z}_p(i))$ by \cite[Theorem 1]{Ca}.
\end{remark}
Using Theorem \ref{criterion}, we can give a different proof of Corollary \ref{simplecase}.
\begin{proof}[Alternate proof of Corollary \ref{simplecase}.]
Fix $m\in \mathbb{N}$ and write $[k(\mu_{p^m}^{\otimes j}):k]=d_jp^n$, with $(d_j,p)=1$. Then $n$ is the smallest integer such that $k(\mu_{p^m}^{\otimes j})\subseteq K_n$ (in particular $\mathrm{Gal}(K_\infty/k(\mu_{p^m}^{\otimes i}))$ acts trivially on $\mathbb{Z}/p^m\mathbb{Z}(j)$).
Since $n_0(K)=0$, we have an exact sequence
$$0\to \Psi(K_n)\to (X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}})_{\Gamma_n}\to A'_{K_n}\to 0.$$
Taking $(-j)$-components with respect to the action of $\Delta$, we get an exact sequence
$$0\to \Psi(K_n)^{[-j]}\to (\xKj{-j})_{\Gamma_n}\to \aKnj{-j}\to 0$$
(note that the actions of $\Gamma_n$ and $\Delta$ commute since $\mathcal{G}_n$ is abelian). Since $\Psi(K)(j)^{\Delta}=0$, we deduce that $\Psi(K_n)(j)^{\Delta}=\Psi(K_n)^{[-j]}=0$ by Lemma \ref{CPsi}. Therefore the above exact sequence reduces to an isomorphism
$$(\xKj{-j})_{\Gamma_n}\cong \aKnj{-j}.$$
Now applying the functor $\otimes \mathbb{Z}/p^m\mathbb{Z}(j)$ we get an isomorphism
$$\left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/p^m(j)\right)_{\mathcal{G}_n}\to \left(A'_{K_n}/p^m(j)\right)_{\Delta},$$
since $\Gamma_n$ acts trivially on $\mathbb{Z}/p^m\mathbb{Z}(j)$.
Finally, taking $\mathrm{Gal}(K_n/K)$-coinvariants we get an isomorphism
$$\left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/p^m(j)\right)_{\mathcal{G}}\cong \left(A'_{K_n}/p^m(j)\right)_{\mathrm{Gal}(K_n/k)}=\left(A'_{k(\mu_{p^m}^{\otimes j})}/p^m(j)\right)_{\mathrm{Gal}(k(\mu_{p^m}^{\otimes j})/k)},$$
where the last equality comes from the fact that $\mathrm{Gal}(K_n/k(\mu_{p^m}^{\otimes j}))$ acts trivially on $\mathbb{Z}/p^m\mathbb{Z}(j)$ and has order coprime with $p$.
\end{proof}
\section{Examples}\label{examples}
In this section we will illustrate our results in the case of quadratic number fields and $p=3$. We further assume $i=2$ to recover the classical context. In other words (see the introduction for more details), we shall consider the exact sequence
\begin{equation}\label{sekt}
0\to W\!K_2(k)\{3\}\to K_2(o_{k})\{3\}\to \underset{v\mid 3}{\tilde\oplus}\mu(k_v)\{3\}\to 0,
\end{equation}
where
$$\underset{v\mid 3}{\tilde\oplus}\mu(k_v)\{3\}=\mathrm{ker}\left(\underset{v\mid 3}{\oplus}\mu(k_v)\{3\}\to \mu(k)\{3\}\right).$$
Using PARI \cite{PARI}, we will provide examples for both split and non-split cases of the above exact sequence.
Let $k=\mathbb{Q}(\sqrt{\delta})$ be a quadratic field. As usual we set $K=k(\mu_3)$ and let $k'=\mathbb{Q}(\sqrt{-3\delta})$ denote the quadratic subfield of $K=k(\mu_3)$ which is different from $k$ and $\mathbb{Q}(\mu_3)$. Note that in this situation $n_0(K)=0$, \textit{i.e.} all the primes above $3$ in $K_\infty/K$ are totally ramified. Moreover, we have an isomorphism
$$\Psi(K)(1)^\Delta\cong\Psi(k').$$
\begin{remark}\label{comune}
For a quadratic number field $k$, Schneider's isomorphism $W\!K_2(k)\{3\}\congX'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}\!\!(1)_\mathcal{G}=\xKj{-1}(1)_\Gamma,$ together with Nakayama's lemma, implies that $W\!K_2(k)\{3\}=0$ if and only if $X'_{k'_\infty}=0$, since $\xKj{-1}\cong X'_{k'_\infty}$. Moreover, if $\Psi(k')=0$, then there is an isomorphism $(X'_{k'_\infty})_\Gamma\cong A'_{k'}$. In particular, when $\Psi(k')=0$, $W\!K_2(k)\{3\}=0$ if and only if $A'_{k'}=0$.
\end{remark}
It is easy to see that, if $k=\mathbb{Q}(\sqrt{\delta})$ with $\delta\in \mathbb{Z}$ square-free, then $\tilde\oplus_{v\mid 3}\mu(k_v)\{3\}$ is non trivial if and only if $\delta\equiv -3 \,\,(3^2)$ and $\delta\ne -3$. In fact in these circumstances there is only one prime $v$ above $3$ and $\mu(k_v)\{3\}$ has order $3$ while $\mu(k)\{3\}$ is trivial, so that $[K_2(o_{k})\{3\}:W\!K_2(k)\{3\}]=3$.
Thus for the rest of this section we set $k=\mathbb{Q}(\sqrt{\delta})$, with $\delta\in D$ where
$$D=\{x\in\mathbb{Z}\setminus\{-3\}\,|\,\textrm{$x$ square-free, $x\equiv -3 \!\!\pmod{3^2}$}\}.$$
\begin{example}\label{splitcase}
Suppose that
\begin{equation}\label{reqsplit}
\Psi(k')=\mathrm{ker}((X'_{k'_\infty})_\Gamma\to A'_{k'})=0\quad \textrm{and}\quad A'_{k'}\ne 0.
\end{equation}
The first of the above conditions can be easily verified using the methods of \cite[Section 4.3]{LMN}. Then (\ref{sekt}) splits with $W\!K_2(k)\{3\}\ne0$, thanks to Corollary \ref{simplecase} and Remark \ref{comune}, and, since $\delta\in D$, we have
$$K_2(o_k)\{3\}\cong W\!K_2(k)\{3\}\oplus\mathbb{Z}/3\mathbb{Z}$$
as abelian groups (an easy argument shows that the same holds with $k$ replaced by $K$).
For positive $\delta$, combined with the Birch-Tate formula $\zeta_k(-1)=(\#K_2(o_k))/24$, the splitting of (\ref{sekt}) sometimes allows to completely determine the structures of $K_2(o_k)\{3\}$ and $W\!K_2(k)\{3\}$. For example, if $k=\mathbb{Q}(\sqrt{3\cdot 239})$, the Birch-Tate formula gives $\#K_2(o_k)\{3\}=3^2$. We also have $A'_{k'}\cong \mathbb{Z}/3\mathbb{Z}$, which implies that the $3$-rank of $K_2(o_k)$ is $2$, thanks to the Keune-Tate exact sequence (\cite[proof of Theorem 6.6]{Ke} or \cite[Theorem 6.2]{Ta2}). Finally $\Psi(k')=0$ and therefore we get
$$W\!K_2(k)\{3\}\cong\mathbb{Z}/3\mathbb{Z}\quad\textrm{and}\quad K_2(o_{k})\{3\}\cong \mathbb{Z}/3\mathbb{Z}\oplus \mathbb{Z}/3\mathbb{Z}.$$
Based on the computation of the regulator $R_2(k)$ in Lichtenbaum's generalization of the Birch-Tate formula $\zeta_k(-1)=R_2(k)(\#K_2(o_k))/24$, Browkin and Gangl (\cite{BG}) predicted the structure of $K_2(o_k)$ when $k$ is an imaginary quadratic field whose discriminant has absolute value less than $5000$. We have applied our methods to the cases in the list of \cite{BG} and, as far as the $3$-parts are concerned, our results agree with those in their list.
\end{example}
\begin{example}
If $\Psi(k')\ne 0$, we cannot apply Corollary \ref{simplecase}. Nevertheless, if \begin{equation}\label{reqnsplit}
\Psi(k')=\mathrm{ker}((X'_{k'_\infty})_\Gamma\to A'_{k'})\ne0\quad \textrm{and}\quad A'_{k'}= 0,
\end{equation}
then (\ref{sekt}) does not split. Indeed, when $A'_{k'}=0$, the condition $\Psi(k')\ne 0$ is equivalent to $X'_{k'_\infty}\ne 0$. Therefore
\begin{equation}\label{wknt}
\mathrm{ker}\left(\left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/3(1)\right)_{\mathcal{G}}\to \left(A'_{K}/3(1)\right)_{\Delta}\right)=\left(X'_{\!K_{_{\!\scriptscriptstyle{\infty}}}}/3(1)\right)_{\mathcal{G}}\cong X'_{k'_\infty}/3(1)\ne 0,
\end{equation}
which implies that (\ref{sekt}) does not split by Theorem \ref{criterion}. In fact, this can also be proved in the following way: if $A'_{k'}=0$, then the $3$-rank of $K_2(o_k)$ is $1$ (by the Keune-Tate exact sequence) and, since $W\!K_2(k)\{3\}\ne 0$ by (\ref{wknt}), this implies that (\ref{sekt}) does not split. Note that the first condition in (\ref{reqnsplit}) can be verified using the recipe of \cite[Section 4.3]{LMN}. If (\ref{reqnsplit}) holds, then, since $\delta\in D$, we have
$$W\!K_2(k)\{3\}\cong\mathbb{Z}/3^a\mathbb{Z}\quad\textrm{and}\quad K_2(o_{k})\{3\}\cong \mathbb{Z}/3^{a+1}\mathbb{Z}$$
for some $a\in\mathbb{N}$.
As in the previous example in the totally real case we can use the information about the non splitting of (\ref{sekt}) to completely determine the structures of $K_2(o_k)\{3\}$ and $W\!K_2(k)\{3\}$. For example, when $k=\mathbb{Q}(\sqrt{3\cdot14})$, the Birch-Tate formula gives $\#K_2(o_k)\{3\}=3^3$. We also have $A'_{k'}=0$, which implies that $K_2(o_k)\{3\}$ is cyclic. Finally $\Psi(k')\cong\mathbb{Z}/3\mathbb{Z}\ne 0$ and therefore we get
$$W\!K_2(k)\{3\}\cong\mathbb{Z}/3^2\mathbb{Z}\quad\textrm{and}\quad K_2(o_{k})\{3\}\cong \mathbb{Z}/3^3\mathbb{Z}.$$
In the imaginary case, the results given by our methods agree once more with the predictions of \cite{BG}, as far as the $3$-parts are concerned.
\end{example}
\begin{remark}
If $\Psi(k')\ne0$ and $A'_{k'}\ne0$, then the inclusion $W\!K_2(k)\{3\}\subseteq K_2(o_{k})\{3\}$ may or may not split, indeed both cases may occur. For instance, for $\delta=3\cdot 1409$ or $3\cdot 1658$, we have $\#\Psi(k')=\#A'_{k'}=3$. Moreover the Birch-Tate formula gives $\#K_2(o_{k})\{3\}=3^3$ and the Keune-Tate exact sequence implies that $K_2(o_{k})\{3\}$ has $3$-rank $2$ and its exponent must therefore be $3^2$.
In particular (\cite[Theorem 6.6]{Ke}), we have
\begin{equation}\label{last}
W\!K_2(k)\{3\}\cong \left(A'_{K_1}\otimes\mu_9\right)_{\mathrm{Gal}(K_1/k)}.
\end{equation}
Now PARI tells us that the right-hand term of (\ref{last}) is isomorphic to $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$ (resp. $\mathbb{Z}/3^2\mathbb{Z}$) when $\delta=3\cdot 1409$ (resp. $\delta=3\cdot 1658$). This implies that the inclusion $W\!K_2(k)\{3\}\subseteq K_2(o_{k})\{3\}$ does not split (resp. splits) when $\delta=3\cdot 1409$ (resp. $\delta=3\cdot 1658$).
\end{remark}
\bigskip
\noindent
\textbf{Acknowledgments}
We are very grateful to Thong Nguyen Quang Do for several helpful exchanges during the preparation of this article. We would also like to thank Herbert Gangl for providing us with the program used to compute the table of \cite{BG}, along with useful suggestions on how to use it. Finally, we thank Filippo Nuccio for his remarks on a preliminary version.
|
1,314,259,993,047 | arxiv | \section{Introduction}
Cosmic strings \cite{original} are one-dimensional topological defects
which arise during phase transitions in the very early universe. Since
they carry energy, they will lead to density fluctuations and
cosmic microwave background (CMB) anisotropies (see e.g.
\cite{CSreviews} for reviews on cosmic strings and structure formation).
Causality implies that the network of strings which forms during the
phase transition contains infinite strings.
Once formed in the early universe, the network of strings will approach a
``scaling solution'' which is characterized by of the order one infinite
string segment in each Hubble volume, and a distribution of cosmic string
loops which are the remnants of the previous evolution. In particular,
this implies that in any theory which admits cosmic strings, a network of
strings will be present during the time period relevant to the CMB,
namely between the time of recombination $t_{rec}$ and the present time
$t_0$.
A network of cosmic strings will generate a scale-invariant spectrum of
cosmological perturbations \cite{CSstructure}. As first discussed
by Kaiser and Stebbins (KS) \cite{KS},
the non-Gaussian nature of the density field
produced by strings will lead to a distinctive signature
(which we will call KS signature) in CMB anisotropy
maps, namely line discontinuities. These line discontinuities are a
consequence of the non-trivial nature of the metric produced by a
cosmic string: space perpendicular to a cosmic string is a cone with
deficit angle given by
\be
\alpha \, = \, 8 \pi G \mu \, ,
\ee
where $\mu$ is the mass per unit length of the string and $G$ is Newton's
gravitational constant \cite{deficit}. Since cosmic strings have a
tension comparable to their mass per unit length, they will typically
be moving with a relativistic transverse velocity $v$. As illustrated in
Figure 1, if we are looking at the CMB in direction of the string, we
will see the photons passing on different sides of the string with a
Doppler shift
\be \label{KSsig}
{{\delta T} \over T} \, = \, 8 \pi \gamma(v) v G \mu \, ,
\ee
where $\gamma(v)$ is the relativistic gamma factor. Looking in direction
of the string, we will see a line in the sky across which the CMB
temperature jumps by the above amount.
\begin{figure}
\includegraphics[height=6cm]{deficit.eps}
\caption{Geometry of the Kaiser-Stebbins effect: Photons passing
on the two sides of the moving cosmic string obtain a relative Doppler
shift for the observer who is at rest.} \label{fig:1}
\end{figure}
Any cosmic string which photons
reaching us today pass on their way from the last scattering surface will
produce an effect. The most numerous strings, however, are those present
close to the time of last scattering $t_{rec}$. At that time, the
typical curvature radius of a long string segment is of the order of
the Hubble radius at $t_{rec}$. The corresponding angular scale today
is about $1^{0}$. In order to be able to identify the cosmic signature
as a line discontinuity, as good an angular resolution as possible is
required. If probed with an angular resolution not significantly
smaller than $1^{0}$, the KS signature will be washed out.
In this work we will study the potential of surveys with
an angular of several minutes of arc to detect the KS signature for
strings. Less important than the angular resolution of the survey is
the total sky coverage, as long as it includes a sufficiently
large number of Hubble patches at $t_{rec}$. Obviously, increasing the
sky coverage will reduce the standard deviation of the results and
thus lead to some better discriminating power.
There has been surprisingly little work devoted to detecting the KS
signature. An early study \cite{Moessner} showed that the angular
resolution of the WMAP satellite would not be adequate to pick out
the KS signature, even for values of $G \mu$ for which the cosmic
strings would dominate the power spectrum of density fluctuations.
In order to detect the KS signature, new statistical methods must
be employed. In \cite{Lo}, a matched filtering method was applied
to the WMAP data to search for cosmic string signatures. In
\cite{Smoot}, new statistics (such as a statistic measuring
the connectedness of neighboring temperature steps or
a decomposition of the temperature map into constant background
temperature, Gaussian noise plus straight string step
discontinuities) were introduced
and applied to the WMAP
data to search for strings. Based on the null results of both studies,
an upper bound on $G \mu$ of $G \mu < 10^{-6}$ could be set.
However, this bound is not competitive with other existing bounds on
the string tension coming from the observed acoustic peak structure
of the CMB angular power spectrum \cite{CMBlimit}
(which yields $G \mu < 10^{-7}$) and from pulsar timing measurements
\cite{Pulsarlimit} (which yield limits of between
$G \mu < 2 \times 10^{-8}$ and $G \mu < 10^{-5.5}$, the difference being
due to conflicting ways \cite{timinglimits}
in which the pulsar timing data is analyzed to obtain limits on
the amplitude of the stochastic background of gravitational waves, and due
to different assumptions about the distribution of cosmic string loops,
the main source of gravitational radiation from a string network -
see \cite{Pol} for a recent discussion). Note that an advantage
of using the KS signature over pulsar bounds is that the KS signature
is more robust (less dependent on the unknown distribution of cosmic
string loops) than the pulsar bounds (which depend sensitively on
the details of the cosmic string loop distribution)
\footnote{The statistical
properties of the long strings are much better established from
numerical string evolution simulations than is the distribution of
string loops. Different numerical codes all agree that the
distribution of long strings scales. Even the number of long
strings crossing each Hubble volume is known to
within one order of magnitude. The distribution of cosmic string loops,
however, much less well known. Even the form of the scaling distribution,
not to mention the parameters of such a distribution, are unknown.}
As discussed above, we expect to be able to set much stronger limits
on $G \mu$ from searches for the KS signature by looking at small-scale
anisotropy maps, provided we find a good statistic to identify line
discontinuities. The goal of this paper is to explore the potential
of one specific edge detection technique, the Canny algorithm \cite{Canny}
to pick out these line discontinuities in CMB anisotropy maps.
One reason for the relative lack of previous work on detecting the
KS signature of cosmic strings is the fact that interest in cosmic
strings as a source for structure in the universe \footnote{Initially
\cite{CSstructure}, cosmic strings were studied as an alternative to
inflation as a mechanism for structure formation.} decreased
dramatically after the discovery of the sharpness of the acoustic peaks
in the CMB angular power spectrum \cite{Boomerang,WMAP}. Since
perturbations seeded by strings are not coherent \cite{incoherent},
a scenario with only cosmic strings as the source of density perturbations
would only produce one broad Doppler peak rather than the observed
narrow acoustic oscillations \cite{Periv,Albrecht,Turok}.
However, recent years have seen a resurgence of interest in cosmic strings.
This was fueled by two developments. Firstly, it was realized \cite{Rachel}
that in many supersymmetric particle physics models, cosmic strings are
formed after inflation, and thus contribute to but do not completely
replace inflationary perturbations as the seeds for structure formation.
Secondly, it has recently been realized that models with cosmic superstrings
\cite{Witten} may well be viable \cite{CMP}. They could, for example,
be generated as the remnant of brane annihilation processes in
brane inflation models \cite{Tye}, or they may play an important roles
in inflationary models in warped backgrounds \cite{stringinflation}.
Cosmic superstrings may also be left behind after the initial Hagedorn
phase in string gas cosmology \cite{BV}, where they would add an additional
component to the spectrum of fluctuations produced by thermal string
gas fluctuations \cite{NBV}. Thus, it is of great interest to
find new ways to search for signatures of strings in cosmological data.
The outline of this paper is as follows: In the following
section we review the Canny algorithm and describe its application
to our problem. Section 3 discusses how the temperature maps
with and without strings are produced, introduces the parameters
chosen in the specific simulations, and presents our results.
We conclude with a summary and a discussion of the caveats
of the current analysis and prospects for future work.
\section{The Canny Algorithm}
The Canny algorithm was developed in 1986 as a technique to detect
edges in images \cite{Canny} such as the
two-dimensional images considered here. When applied to a map of raw data,
the algorithm is intended to produce a map tracing the edges in the
map, the lines across which the intensity contrast is largest.
The first step in the algorithm is to filter the data to eliminate
point source noise. The filtering is achieved using a convolution of
the map with a Gaussian filter. The filtering length is a free parameter
in the algorithm. It must be chosen sufficiently large to eliminate
unwanted point source noise, but must be smaller than the characteristic
size of the structure in the maps which one is trying to identify. Let us
denote the original map by $M(i, j)$, and the filter by $F(i, j)$. The
filtered map $FM$ is then given by
\be
FM(i, j) \, = \, \sum_{k,l} M(i - k, j - l) F(k, l) \, ,
\ee
where the pixel points of the map are denoted by the labels $(i, j)$.
In our simulations, the filtering length is taken to be $1.5$ grid
units.
The second step of the algorithm is to find the gradient of the
filtered image, a vector obtained by taking the discrete derivative
of the map in each of the two directions. The gradient vector at
a grid point $(i, j)$ is denoted by $(G_x, G_y)$, where $G_x$ and $G_y$
are the derivatives in the respective directions. The {\it edge
strength} $|G|$ at any given grid point is defined by
\be
|G| \, = \,\sqrt{|G_x|^2 + |G_y|^2} \, ,
\ee
and the {\it edge direction} is given by the angle
\be
\theta \, = \, arctan(G_y / G_x) \, .
\ee
Since we are working on a grid, it makes sense to replace the edge
direction by one of the eight distinguished directions on a quadratic
grid, namely the four directions along the coordinate axes and the
four diagonal directions.
The average maximal gradient in maps without cosmic strings (the
average taken over all of our runs) is denoted by $G_{m}$.
For a specific run, the maximal gradient is obtained by scanning
the entire map and searching for the maximal value of the grid
strength. The value of $G_{m}$ is
used to set the thresholds discussed below.
The next step of the Canny algorithm is to find grid points with
gradients which are maximal when we vary the point
in direction of the gradient. Grid
points which are not local maxima of the gradient are assigned
a number $0$.
At this stage, two thresholds must be set, an upper and a lower
threshold $t_u$ and $t_l$. The algorithm first finds local maxima in
the gradient along one of the four directional axes on the grid.
Next, it goes through each of the list of local maxima and
determines whether the gradient is greater than the upper threshold fraction of $G_m$, i.e.
\be
|G| \, > \, t_u G_m \, .
\ee
Grid point thus selected are marked with a number $1$. If the
gradient is larger than the lower threshold fraction of $G_m$,
\be
|G| \, > \, t_l G_m \, ,
\ee
the grid point is assigned a value of $1/2$. For
points marked with $1$ or $1/2$, the direction of the gradient
is also stored. For each grid point with a value of $1/2$, the
algorithm next checks whether there is a grid point with
value of $1$ or $1/2$ which is in a direction perpendicular
to the gradient and
whose gradient is parallel or next to parallel (thus three
possible directions in total) to the initial gradient. If such a
grid point is found, the algorithm continues until it either does
not find another point satisfying the criteria or else finds a
point satisfying the criteria marked with a $1$. If it finds such
a point, then all of the points (initially marked $1/2$) found along
the way are marked as $1$. Neighboring (in all eight directions)
grid points marked by $1$ are then said to belong to the same edge.
These are edges across which the gradient of the map is large.
The reason for using two thresholds is as follows: we want the algorithm
to find lines with a large gradient. At the same time, we do not want
noise to lead to points along a line of large gradients to be missed
(and thus the lines cut) just because noise has reduced the amplitude
of the edge strength at one point along the line.
In this way, the algorithm produces a list of grid points belonging to
edges, and this list can be read out as a map of edges.
In our simulations, the upper and lower thresholds are $t_u = 0.5$
and $t_l = 0.4$.
It is important to quantify the edge maps. A simple way to do this is
to produce a histogram of edge lengths. Thus, the implementation of
the Canny algorithm is arranged such as to output a list of edge
lengths which can then be used to produce a histogram of edge
lengths. The histogram contains useful information about the
presence of edges in the map.
\section{Applying the Canny Algorithm to CMB Anisotropy Maps}
Eventually, we would like to apply the Canny algorithm to actual
CMB maps. For this initial feasibility study, however, we will
apply the algorithm to theoretical maps produced in numerical
simulations. We wish to compare temperature maps for the
``standard'' Gaussian $\Lambda$CDM model with maps in which a network
of strings contributes a fraction $f$ of the total power.
The maps are characterized by the angular scale of the survey
(we take the survey area to be square) and by the angular scale
of the grid, i.e. the angular resolution of the survey.
The Gaussian maps are produced in the following way: We start with
the angular power spectrum $C_l$ of cosmic microwave anisotropies
taken from the CMBFAST simulation \cite{CMBFAST}
with the appropriate cosmological parameters
\footnote{Since the range of $l$ values for which the CMBFAST
program generates $C_l$ values is limited to $3000$, we
are currently unable to explore angular scales relevant to
upcoming experiments. We are at the moment completing an improved
code which will allow us to obtain an improved angular resolution.}.
Since we have
in mind applications of our algorithm to small angular scale surveys,
we perform a ``flat sky'' approximation \cite{White}.
We introduce a two-dimensional
Cartesian coordinate system covering the survey area which we take
to be rectangular, choosing the upper right corner of the survey area
to be the origin of the coordinates. We need to compute the temperature
field $T_G(x, y)$
of the map at the coordinate values $(x, y)$ corresponding
to the grid points. This map is determined by an inverse fast Fourier
transform from the temperature values $\tilde{T}({\vec{k}})$ in
Fourier space.
For each vector ${\vec{k}}$ in Fourier space, we find the integers
$l_1(k)$ and $l_2(k)$ which bracket the $l$ value $l(k)$ corresponding
to ${\vec{k}}$ and take a linear interpolation of the values of $C_l$.
Let us denote the result of this linear interpolation by $C_{l(k)}$,
where $k \equiv |{\vec{k}}|$. The
value of ${\tilde{T}}({\vec{k}})$ is then given as follows:
\be
{\tilde{T}}({\vec{k}}) \, = \, g({\vec{k}}) {\sqrt{C_{l(k)}}} \, ,
\ee
where $g$ is a random variable drawn from a probability distribution
with variance $1$.
In the above analysis, we are singling out one grid point to be
special, namely the origin. This introduces unwanted phase
correlations which we overcome by superimposing the results from
four separate simulations, indicated by $T_i, i = 1 , .., 4$:
\bea
T_G(x, y) \, &=& \, {1 \over 2} \bigl( T_1(x, y) + T_2(x_m - x, y_m - y) \\
&& + T_3(x_m - x, y) + T_4(x, y_m - y) \bigr) \, , \nonumber
\eea
where $x_m$ and $y_m$ are the maximal $x$ and $y$ values of the survey
volume (the pre-factor of $1/2$ is required in order to maintain the
original standard deviation).
A network of cosmic strings will produce a temperature map
$T_{CS}(x, y)$. In the following, we take a toy model for a
temperature map produced by strings introduced by Perivolaropoulos
\cite{Leandros,Moessner}. The toy model takes into
account that at all times between $t_{rec}$ (the time of
recombination) and the present time $t_0$, the network of
strings is described by a scaling solution in which there are
a fixed number of long string segments (strings which are
not loops with radius smaller than the Hubble radius) crossing
each Hubble volume. Each such string gives rise to a
Kaiser-Stebbins line discontinuity (\ref{KSsig})
in the temperature map. The cosmic string network is
continuously evolving via motion of the strings and string
interactions which lead to the production of string loops.
Thus, in each Hubble time step the string network can be
taken to be uncorrelated.
In the algorithm, we divide the time interval $t_{rec} < t < t_0$
into 15 Hubble time steps. In each time step $t$ we lay down a network
of strings at random, uncorrelated with the network at the previous
time step. We take the network to consist of straight string
segments of length $\alpha_1 t$. In order to avoid missing strings
at the edges of the survey area, the survey area is extended
in each direction by a Hubble distance. The program runs through
all points of the simulation volume and picks points to be
midpoints of a string segment with
a probability chosen such that the average number of strings in the
Hubble volume equals the number $N$ of the scaling solution. The
directions of the strings are chosen at random, as are the string
transverse velocities. To take into account also the projection onto
the last scattering surface, we add a temperature
\be
{{\delta T} \over T} \, = \, {1 \over 2} {\tilde v} r 8 \pi G \mu \, ,
\ee
to one side of the string projection onto the last scattering surface,
and subtract it from the other (so as to maintain the average temperature
of the CMB). Here, ${\tilde v}$ stands for the maximal value of
$v \gamma(v)$, where $v$ is the transverse velocity of the
string, and $\gamma(v)$ is the relativistic $\gamma$ factor associated
with $v$. Also, $r$ is a random number between $0$ and $1$, to take
into account both the distribution of velocities, and also projection
effects (the formula (\ref{KSsig}) for the line
discontinuity in temperature is modulated by an angular factor if
the string velocity is not perpendicular to our line of sight
to the string). The regions affected by the temperature fluctuation are
rectangles on either side of the string. The depth of these
rectangles in direction transverse to the direction of the string
is taken to be a fraction $f$ of the Hubble radius. The length
of the string segments is taken to be $\alpha_1$ times the
Hubble radius. The survey area
is always taken to correspond to the central region of the
simulation area.
The free parameters of the string simulation are $G \mu$, the
number of strings $N$ per Hubble volume, the length coefficients
$\alpha_1$ and $f$, as well as ${\tilde v}$. In our work, we
will fix $N = 10$, $\alpha_1 = 2$, $f = 1$ and ${\tilde v} = 0.15$.
These values are representative of the results of numerical
simulations of cosmic string evolution (see \cite{CSreviews} for reviews).
However, it should be kept in mind that the results for
these parameters obtained using different numerical codes differ significantly.
Let us make a couple of comments on the expected
dependence of the results on the value of $N$. As $N$ initially
increases from zero, the signature of the strings should increase since
there are more strings. However, once $N$ rises above a certain
critical value, then further increasing the number $N$ will render the
string distribution more Gaussian by the Central Limit Theorem and
will hence decrease the sensitivity of the Canny algorithm. We
expect that the critical value of $N$ will be the one for which
the regions of the sky effected by individual string segments begin
to overlap. If the strings have length comparable to the Hubble radius,
the the critical value of $N$ is expected to be less than $N = 10$.
Increasing the string velocity ${\tilde v}$ will boost the line
discontinuites in the temerature maps and will make our algorithm
more effective. Since the weight in our statistical analysis of
the edge maps comes from fairly short segments, our results are
not very sensitive to the specific values of $f$ and $\alpha_1$.
We are interested in using the Canny algorithm to find signatures
for cosmic strings in maps which contain both Gaussian fluctuations
from inflation and a cosmic string component (which is sub-dominant
in terms of its contribution to the power spectrum). The total
map $T_T(x, y)$ is given by the superposition of a pure Gaussian
noise map and the map produced by a distribution of strings:
\be
T_T(x, y) \, = \, T_G(x, y) + T_{CS}(x, y) \, ,
\ee
and where the amplitude of the Gaussian term is adjusted
such that the total angular power spectrum agrees with the
COBE results \cite{COBE}.
The output of the Canny algorithm is a map of edges which have
been picked out. In the presence of cosmic strings, we would
expect more longer edges than in the absence of strings. In
order to test for this effect, we produced histograms of the
distribution of edge lengths for each simulation. Since both
the Gaussian maps and the maps with cosmic strings are produced
by a Gaussian random process (the phases of the Fourier modes for
the Gaussian maps are picked at random, and for the string maps
the locations of the string centers, their directions and their
transverse velocity vectors are all random), we ran many
(${\tilde N} = 50$)
simulations for both the Gaussian map and the maps with strings.
This gave us statistical error bars on the histograms of edge
lengths. In turn, this allows us to assign statistical weight to
the difference in the histograms. Our results are based on this
analysis.
In our simulations, we consider the ``Gaussian noise'' maps to be
given by a standard $\Lambda$CDM model with adiabatic
fluctuations with a spectral index of $n_s = 0.99$, and
with parameters
$\Omega_B = 0.046$ (baryon fraction), $\Omega_{CDM} = 0.224$ (cold
dark matter fraction), $\Omega_{\Lambda} = 0.730$ (cosmological
constant contribution), and $H_0 = 72$ (Hubble expansion rate). No
massless neutrinos, standard recombination history and a Helium
fraction of $Y = 0.24$ were assumed (in the $C_l$ spectra which
were used to construct the maps).
The parameters of the string simulation are indicated
above. Our maps have angular extent $15$ degrees by $15$ degrees.
We fix the angular resolution and search for the minimal value
of the cosmic string mass parameter $G \mu$ for which the maps
with the strings can be distinguished from the pure Gaussian
maps at a statistically significant level. We demand
significance at the $3 \sigma$ level.
Figure 2 shows a CMB temperature fluctuation map of a simulation
without strings, and Figure 3 is the corresponding map of a simulation
which includes strings according to the prescription described above.
\begin{figure}
\includegraphics[height=9cm]{Gaussian_Image.ps}
\caption{Map of the CMB temperature in a $15^2$ square of the sky
for a $\Lambda$CDM simulation with the parameters described in the text.}
\label{fig:2}
\end{figure}
\begin{figure}
\includegraphics[height=9cm]{5e-7_Image.ps}
\caption{Corresponding map of the CMB temperature in a simulation which
includes cosmic strings with a mass per unit length parameter given
by $G \mu = 10^{-7}$ (and other parameters as described in the text.}
\label{fig:3}
\end{figure}
The output of the Canny algorithm applied to the above maps is
shown in Figures 4 and 5, respectively.
\begin{figure}
\includegraphics[height=9cm]{Gaussian_Image_output.ps}
\caption{Output map of the Canny algorithm showing the edges in the
$\lambda$CDM map of Figure 2.}
\label{fig:4}
\end{figure}
\begin{figure}
\includegraphics[height=9cm]{5e-7_Image_output.ps}
\caption{Corresponding edge distribution in the map of Figure 3.} \label{fig:5}
\end{figure}
Based on these results, our algorithm evaluates the distribution of
edge lengths and checks whether the distributions of the histograms
with and without strings are statistically different.
Maps with cosmic strings showed, as expected,
a slight excess of edges of all lengths. This excess is more
pronounced for edge lengths which lie in the non-Gaussian tail.
The results for an
angular resolution of $8^{'}$ are presented in Table 1. In this
table, the rows indicate the length $L$ of the edge. The entries
in the columns are the mean number [standard deviation of the mean]
of edges of the maps which have the respective length. The
first column of numbers is for simulations without strings, the
next two are for simulations including a network of strings with the
indicated mass per unit length parameter. Each histogram
was produced from 50 runs.
From the histogram it follows that the difference between
the distributions is significant (at the $3 \sigma$ level)
if $G \mu = 3.5 \times 10^{-8}$,
but not if $G \mu = 3 \times 10^{-8}$.
For a given angular resolution, we varied the string mass parameter
$G \mu$ to find the limiting value below
which the differences in the histograms ceased to be statistically
significant (using Fisher's combined probability test).
Our results are summarized in Table 2.
Based on the above results, prospects for being able to use the
Canny algorithm to significantly improve the limits on the
cosmic string mass parameter $G \mu$ (or detect cosmic strings)
are excellent. The angular resolution of the Planck satellite
experiment will be about $5^{'}$, the South Pole Telescope is
aiming for a resolution of $1^{'}$ with a field of view of
$4,000$ square degrees, and the resolution of the ACT telescope
in Chile will be $1.5^{'}$.
\begin{table}
\caption{Histograms of the distribution of edge lengths $L$.}
\label{table:1}
\begin{center}
\begin{tabular}{cccc} \hline \hline
& no strings & $G \mu = 3 \times 10^{-8}$ & $G \mu = 3.5 \times 10^{-8}$
\\ \hline
L = 2 & 165.0 [2.84] & 169.9 [3.60] & 178.0 [3.84] \\
L = 3 & 40.0 [0.86] & 42.3 [1.19] & 43.6 [1.18] \\
L = 4 & 11.8 [0.51] & 13.4 [0.58] & 14.0 [0.57] \\
L = 5 & 4.5 [0.31] & 4.8 [0.32] & 5.5 [0.34] \\
L = 6 & 1.7 [0.23] & 2.1 [0.21] & 2.2 [0.21] \\
L = 7 & 0.66 [0.13] & 0.74 [0.11] & 0.84 [0.12] \\
L = 8 & 0.18 [0.07] & 0.38 [0.09] & 0.28 [0.07] \\
L = 9 & 0.12 [0.06] & 0.1 [0.04] & 0.08 [0.05] \\
L = 10 & 0.04 [0.03] & 0.08 [0.04] & 0.04 [0.03] \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\caption{Critical values of the string mass parameter $G \mu$.}
\label{table:2}
\begin{center}
\begin{tabular}{ccc} \hline \hline
angular resolution & $G \mu$ \\ \hline
$10^{'}$ & $4.3 \times 10^{-8}$ \\
$9^{'}$ & $4.0 \times 10^{-8}$ \\
$8^{'}$ & $3.1 \times 10^{-8}$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusions}
We have suggested a new way of looking for the specific signature
of cosmic strings, namely the Kaiser-Stebbins
line discontinuities, in small-scale
cosmic microwave anisotropy maps. Our
method makes use of the Canny algorithm, an edge-detection technique
used often in pattern recognition.
Since the cosmic microwave temperature anisotropy maps induced by
a network of cosmic strings are dominated by the strings present at
the time of last scattering (when the Hubble radius, which is
comparable to the correlation length of the string network at that
time, is of the order of one degree), good small-scale angular resolution
is essential in order to be able to detect strings. An angular resolution
substantially less than $1^{o}$ is required, otherwise the signals
from the line discontinuities are washed out.
We have constructed CMB anisotropy maps which correspond to having
both Gaussian "noise" with a nearly scale-invariant power spectrum
from inflation and anisotropies produced by a distribution of
straight string segments. Based on our numerical simulations,
we find that for an angular resolution
of CMB maps of $8^{'}$ the Canny algorithm has the potential
to detect strings with a mass per unit length $\mu$ above a value
of $G \mu \simeq 4 \times 10^{-8}$, close to an order of magnitude
better than current limits based on the CMB.
A drawback of our work is that it is based on toy model cosmic string
CMB maps which are obtained by superimposing idealized line
discontinuities of straight string segments, not from actual
string networks. Actual string networks contain both infinite strings
and string loops. The infinite strings are not completely straight,
but have a curvature radius comparable to the Hubble radius.
An improved analysis should start from a numerical simulation of the
distribution of cosmic strings, calculate the induced temperature
anisotropies taking into account of all effects following the formalism
set out in \cite{Turok},
and then apply the Canny algorithm to the resulting maps. After
completion of this work, a paper appeared \cite{Fraisse} in which
CMB temperature maps on scales similar to the ones we are considering
were constructed based on full cosmic string simulations. This
work confirmed that the Kaiser-Stebbins from long string segments
is the dominating visible effect, and suggested, in agreement with
the message of our work, that strings should be visible in high
resolution small-scale CMB anisotropy maps. However, no string-specific
statistical analyses of the maps like the one we are proposing were
performed in \cite{Fraisse}.
Another important issue which remains to be analyzed is the effect
of instrumental noise. We have done preliminary work on this
topic and modelled instrumental noise by a component to the $C_l$
spectrum which rises rapidly as a function of $l$ at the scale of
the angular resolution of the survey \footnote{We thank Gil Holder
for suggesting this method.}, becoming dominant at the
angular resolution scale. Initial results show that the efficiency
of the Canny algorithm is not reduced. We plan to study this question
in more detail.
\begin{acknowledgments}
This work is supported in part by a NSERC Discovery Grant, by funds from
the CRC Program, and by a FQRNT Team Grant. We wish to thank Matt Dobbs,
Christophe Ringeval, and in particular Gil Holder for useful discussions.
\end{acknowledgments}
|
1,314,259,993,048 | arxiv | \section{Introduction}
\label{Introduction}
This paper involves an algebraically defined $\mathfrak{S}_n$-module, and is concerned with modelling the $\mathfrak{S}_n$ action on this module via combinatorially defined objects. In particular, we will give a basis indexed by a certain type of noncrossing set partition for which the $\mathfrak{S}_n$ action has a nice combinatorial interpretation.
The module in question was introduced by Jongwon Kim and Rhoades \cite{KR}, and is defined as follows. Let $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots, \xi_n)$ be two sets of $n$ anticommuting variables, and let
\begin{equation}
\wedge \{\Theta_n, \Xi_n \} := \wedge \{\theta_1, \dots, \theta_n , \xi_1, \dots, \xi_n \}
\end{equation}
be the exterior algebra generated by these symbols over $\mathbb{C}$. The symmetric group $\mathfrak{S}_n$ acts on this exterior algebra via a diagonal action given by
\begin{equation}
w \cdot \theta_i := \theta_{w(i)} \qquad w \cdot \xi'_i := \xi'_{w(i)}.
\end{equation}
for any permutation $w\in \mathfrak{S}_n$ and $1 \leq i \leq n$. Let $\wedge \{\Theta_n, \Xi_n \}^{\mathfrak{S}_n}_+$ denote the subspace of $\mathfrak{S}_n$-invariants with vanishing constant term. Then the fermionic diagonal coinvariant ring is defined as
\begin{equation}
FDR_n := \wedge \{\Theta_n, \Xi_n \}/\langle\wedge \{\Theta_n, \Xi_n \}^{\mathfrak{S}_n}_+ \rangle.
\end{equation}
The ring $FDR_n$ is a variant of the Garsia-Haiman diagonal coinvariant ring \cite{Haiman}, which is defined analogously but with the anticommuting variables replaced with commuting ones. Several other variants involving more sets of variables or mixtures of anticommuting and commuting variables have been studied by other authors \cite{Bergeron, BRT, DIW, KR, OZ, PRR, RW, RW2, Swanson, SW, ZabrockiDelta, ZabrockiFermion}.
The ring $\wedge \{\Theta_n, \Xi_n \}$ has a bigrading given by
\begin{equation}
(\wedge \{\Theta_n, \Xi_n \})_{i,j} := \wedge^i \{\theta_1, \dots, \theta_n \} \otimes \wedge^j \{\xi_1, \dots, \xi_n \} .
\end{equation}
The invariant ideal $\langle\wedge \{\Theta_n, \Xi_n \}^{\mathfrak{S}_n}_+ \rangle$ is homogeneous, so $FDR_n$ inherits the bigrading. In \cite{KR}, Kim and Rhoades calculated the frobenius image of $FDR_n$ to be given by
\begin{equation}
\textrm{Frob}(FDR_n)_{i,j} = s_{(n-i, 1^i)} * s_{(n-j, 1^j)} - s_{(n-i-1, 1^{i+1})} * s_{(n-j-1, 1^{j+1})}
\end{equation}
where $*$ denotes the Kronecker product of Schur functions. They remark that in the case when $i+j = n-1$, the above shows that the dimension of $(FDR_n)_{n-k,k-1}$ is given by the Narayana number $\textrm{Nar}(n,k)$. Narayana numbers count noncrossing set partitions of $[n]$ into $k$ blocks, and in joint work with Rhoades \cite{me} we gave a combinatorial basis of $(FDR_n)_{n-k,k-1}$ indexed by set partitions for which the $\mathfrak{S}_n$-action was given by a skein action on noncrossing partitions first described by Rhoades in \cite{Rhoades}.
In this paper we will give a similar result for all bidegrees, although our results will not give a combinatorial description for the full $\mathfrak{S}_n$-action. Instead, we will focus on the subgroup of $\mathfrak{S}_n$ consisting of permutations which leave $n$ fixed (which we will abusively refer to as $\mathfrak{S}_{n-1}$). We will define a basis of $(FDR_n)_{i,j}$ indexed by a certain class of noncrossing set partitions defined in Section 3 for which the action of $\mathfrak{S}_{n-1}$ can be described via combinatorial manipulations of the indexing partitions and use this basis to give an expression for the Frobenius image
\begin{equation}
\textrm{Frob}(\textrm{Res}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}}(FDR_n)_{i,j}).
\end{equation}
The rest of the paper is organized as follows. Section 2 will give relevant background information on set partitions, exterior algebras, and $\mathfrak{S}_n$ representation theory. Section 3 will describe an action of $\symm_{n-1}$ on certain set partitions and map this action into $FDR_n$. Section 4 will show that a restriction of this map is an isomorphism and use it to obtain a combinatorial basis of $FDR_n$. Section 5 will use the basis developed to calculate the bigraded $\symm_n$-structure of $FDR_n$. Section 6 will connect this basis to a cyclic sieving result of Thiel and address some avenues of further inquiry.
\section{Background}
\label{Background}
\subsection{Combinatorics}
A \textit{noncrossing set partition} of $[n]$ is a set partition of $[n]$ in which for any $1 \leq a<b<c<d \leq n$ if $a$ and $c$ are in the same block, and $b$ and $d$ are in the same block, then $a,b,c,d$ are all in the same block.
An \textit{integer partition} $\lambda \vdash n$ of length $k$ is a sequence of integers $(\lambda_1, \lambda_2, \dots, \lambda_k)$ where $\lambda_1 + \cdots + \lambda_k = n$ and $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 1$. \textit{Dominance order}, denoted by $\mu \preceq \lambda$, is a partial order on set partitions defined by $\mu \preceq \lambda$ if and only if $\mu_1 + \cdots + \mu_i \leq \lambda_1 + \cdots + \lambda_i$ for all $i$, taking $\mu_i$ or $\lambda_i$ to be 0 whenever $i$ exceeds the length of $\mu$ or $\lambda$ respectively. The conjugate of an integer partition $\lambda$ denoted $\lambda'$.
\subsection{Exterior Algebras}
As in the introduction, we will use $\wedge \{\Theta_n, \Xi_n\}$ to denote the exterior algebra generated by the $2n$ symbols $\theta_1, \dots, \theta_n, \xi_1, \dots, \xi_n$. There is an isomorphism of graded vector spaces (see e.g. \cite{KR})
\begin{equation}
FDR_n \cong \wedge \{\theta_1, \dots, \theta_{n-1}, \xi'_1 , \dots, \xi'_{n-1}\} / \langle \theta_1\xi'_1 + \cdots + \theta_{n-1}\xi'_{n-1} \rangle
\end{equation}
given by
\begin{align*}
&\theta_i \rightarrow \theta_i & 1\leq i \leq n-1\\
&\theta_n \rightarrow -(\theta_1 + \cdots + \theta_{n-1})&\\
&\xi_i \rightarrow \xi'_i - \frac{1}{n}(\xi'_1 + \cdots + \xi'_{n-1}) & 1 \leq i \leq n-1 \\
&\xi_n \rightarrow - \frac{1}{n}(\xi'_1 + \cdots + \xi'_{n-1})
\end{align*}
As this paper will focus on the action of $\mathfrak{S}_{n-1}$, we will extensively use this alternate formulation of $FDR_n$, and use $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ to denote $\wedge \{\theta_1, \dots, \theta_{n-1}, \xi'_1 , \dots, \xi'_{n-1}\}$. The ring $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ inherits the action of $\mathfrak{S}_n$ and $\mathfrak{S}_{n-1}$ from $FDR_n$, and the action of $\symm_{n-1}$ simply permutes indices of variables.
Given subsets $S, T \subseteq [n-1]$, let $\theta_S, \xi'_T$ with $S = \{s_1 < \cdots < s_a\}$, $T = \{t_1 < \cdots < t_b\}$ denote the monomial
\begin{equation}
\theta_{s_1} \cdots \theta_{s_a}\cdot \xi'_{t_1} \cdots \xi'_{t_b}
\end{equation}
The set $\{\theta_S \cdot \xi'_T : S,T \subseteq [n-1]\}$ is a basis of $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$. Define an inner product $\langle -,-\rangle$ on $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ by declaring this basis to be orthogonal.
An exterior algebra $\wedge \{ \omega_1, \dots, \omega_n\}$ acts on itself via {\em exterior differentiaiton}, denoted by $\odot$. The action $\odot: \wedge \{ \omega_1, \dots, \omega_n\} \times \wedge \{ \omega_1, \dots, \omega_n\} \rightarrow \wedge \{ \omega_1, \dots, \omega_n\}$ is defined by
\[
\omega_i \odot (\omega_{s_1}\cdots \omega_{s_k}) = \begin{cases} \omega_{s_1}\cdots\widehat{\omega_{s_j}}\cdots \omega_{s_k} & i = s_j \\ 0 & i \neq s_j \textrm{ for all } 1\leq j \leq k\end{cases}.
\]
\subsection{$\mathfrak{S}_n$-Representation Theory}
Irreducible representations of $\mathfrak{S}_n$ are in one-to-one correspondence with partitions $\lambda \vdash n$. Given $\lambda \vdash n$ let $S^\lambda$ denote the corresponding $\mathfrak{S}_n$-irreducible. Any finite dimensional $\mathfrak{S}_n$-module $V$ can be decomposed uniquely as $V \cong \bigoplus_{\lambda \vdash n} c_\lambda S_\lambda$ for some multiplicities $c_\lambda$. The \textit{Frobenius image} of $V$ is the symmetric function given by
\begin{equation}
\textrm{Frob } V := \sum_{\lambda \vdash n} c_\lambda s_\lambda
\end{equation}
where $s_\lambda$ is the Schur function corresponding to $\lambda$.
If $V$ is an $\mathfrak{S}_n$-module and $W$ is an $\mathfrak{S}_m$-module, their \textit{induction product} $V\circ W$ is given by
\begin{equation}
V \circ W := \textrm{Ind}_{\mathfrak{S}_n \times \mathfrak{S}_m}^{\mathfrak{S}_{n+m}} (V \otimes W)
\end{equation}
where the action of $\mathfrak{S}_m \times \mathfrak{S}_n$ on $V\otimes W$ is given by $(\sigma, \sigma') \cdot (v \otimes w) := (\sigma \cdot v) \otimes (\sigma' \cdot w)$. We have
\begin{equation}
\label{indprod}
\textrm{Frob } V\circ W = \textrm{Frob } V \cdot \textrm{Frob }W
\end{equation}
so induction product of modules corresponds to multiplication of Frobenius image.
Given a partition $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_k) \vdash n$, let $\mathfrak{S}_\lambda \subseteq \mathfrak{S}_n$ denote the Young subgroup $\mathfrak{S}_\lambda := \mathfrak{S}_{\{1,\dots,\lambda_1\}} \times \mathfrak{S}_{\{\lambda_1+1,\dots,\lambda_1+\lambda_2\}} \times \cdots \times \mathfrak{S}_{\{n-\lambda_k,\dots,n\}}$. To any subgroup $X \subseteq \mathfrak{S}_n$ we associate two group algebra elements $[X]_+$ and $[X]_-$ defined by $[X]_+ = \sum_{w\in X} w$ and $[X]_+ = \sum_{w\in X} \textrm{sign}(w)w$. We will need the following standard lemma.
\begin{lemma}
\label{symmetrizationkills}
Let $\lambda, \mu \vdash n$. Then $[S_\lambda]_+$ kills $S^\mu$ unless $\lambda \preceq \mu$ and $[S_{\lambda'}]_-$ kills $S^\mu$ unless $\mu \preceq \lambda$.
\end{lemma}
\subsection{Cyclic Sieving}
Cyclic sieving was introduced by Reiner, Stanton and White \cite{RSW} as a way to express various related results about the enumeration of fixed points of a cyclic action. If $X$ is a finite set, $C$ is a cyclic group of order $n$ generated by $c$ acting on $X$, and $P(q)$ is a polynomial in $\mathbb{N}[q]$, then we say that
\begin{defn}
The triple $(X,C, P(q))$ exhibits the cyclic sieving phenomenon if for all nonnegative integers $d$,
\[
|\{x\in X \mid c^d \cdot x = x \}| = P(\zeta^d)
\]
where $\zeta$ is a primitive $n^{th}$ root of unity.
\end{defn}
The polynomial $P(q)$ is often given in terms of {\em $q$-analogs}. The $q$-analog of a positive integer $n$ is denoted $[n]_q$ and is defined to be $1+q+q^2+\cdots + q^{n-1}$. The $q$-analogs of $n!$, $\binom{n}{k}$ and the multinomial coefficient $\binom{n}{k_1,k_2, \dots, k_l}$ , are denoted and defined as follows:
\[
[n]_q! = [n]_q[n-1]_q\cdots[1]_q,
\]
\[
\begin{bmatrix} n\\k \end{bmatrix}_q = \frac{[n]_q!}{[n-k]_q![k]_q!},
\]
\[
\begin{bmatrix} n\\k_1, k_2, \dots k_l \end{bmatrix}_q = \frac{[n]_q!}{[k_1]_q!\cdots[k_l]_q!}.
\]
The {\em fake degree polynomial} of a representation is defined by
\[
\textbf{fd}(S^{\lambda}) = q^{b(\lambda)}\frac{[r]_q!}{\prod_{(i,j \in \lambda)} [h(i,j)]_q}
\]
where the product is over all cells $(i,j)$ of $\lambda$, $h(i,j)$ is its hook length and $b(\lambda) = \lambda_2 + 2\lambda_3 + 3\lambda_4 + \cdots$ and the fake degree of a general representation is the sum of fake degrees of the irreducibles it contains, with multiplicity.
Cyclic sieving results are often proven via representation theory. In particular Reiner, Stanton and White \cite{RSW} realized the following was implied by a result of Springer's \cite{Springer}.
\begin{theorem} [Springer, 1974]
\label{fakedegree}
Let $V$ be a representation of $\mathfrak{S}_n$ with a basis $X$ which is preserved by the long cycle, $c$. Let $P(q) = \textbf{fd}(V)$. Then $(X, \langle c \rangle, P(q))$ exhibits the cyclic sieving phenomenon.
\end{theorem}
\section{Set partitions and the action of $\symm_{n-1}$}
\label{basis-section}
The indexing set for our combinatorial basis will be a certain partially labelled subset $\Phi(n)$ of noncrossing set partitions of $[n]$.
\begin{defn}
Let $n,k,x,t$ be nonnegative integers. We define the following sets of set partitions:
\begin{itemize}
\item Let $\Psi(n)$ denote the set of all set partitions of $n$ for which all blocks not containing $n$ are size 1 or size 2, and blocks of size 1 not containing $n$ are labelled with either a $\theta$ or a $\xi'$. \\
\item
Let $\Psi(n,k)$ be the set of partitions in $\Psi(n)$ in which the block containing $n$ is size $k$. \\
\item Let $\Psi(n,k,t,x)$ denote the set of partitions in $\Psi(n,k)$ which have exactly $t$ singletons labelled $\theta$ and exactly $x$ singletons labelled $\xi'$. \\
\item Let $\Phi(n)$, $\Phi(n,k)$, and $\Phi(n,k,t,x)$ be the subsets of $\Psi(n)$, $\Psi(n,k)$, or $\Psi(n,k,t,x)$ respectively which consist only of the those set partitions which are noncrossing.
\end{itemize}
\end{defn}
For the rest of this paper, when we refer to the singleton blocks of a partition $\pi \in \Psi(n)$, we only refer to those blocks of size 1 that do not contain $n$, even if the block containing $n$ happens to be size 1. Similarly when we refer to the blocks of size two we refer to only the blocks of size two that do not contain $n$.
There is a natural action of $\symm_{n-1}$ on $\Psi(n)$, given by simply permuting elements between blocks and preserving labels of blocks. The sets $\Psi(n,k)$ and $\Psi(n,k,x,t)$ are closed under this action, but $\Phi(n)$ is not, as permuting the elements of a noncrossing permutation may introduce crossings. However, we can define an action of $\symm_{n-1}$ on the linearization $\mathbb{C}\Phi(n)$ by mapping $\mathbb{C}\Psi(n)$ into $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ in such a way that $\mathbb{C}\Phi(n)$ is $\symm_{n-1}$-invariant and pulling back the $\symm_{n-1}$-action.
Towards this goal, to each element $\pi \in \Psi(n)$ we will associate an element $G_\pi$ of $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$. To define $G_\pi$ we will make use of a tool we will call \emph{block operators}. Let $B$ be a block of a set partition $\pi \in \Psi(n)$, i.e. $B$ is a nonempty subset of $[n]$ that either contains $n$ or is size at most two. Define the \emph{block operator} $\tau_B : \wedge \{\Theta_{n-1}, \Xi'_{n-1}\} \rightarrow \wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ by
\begin{equation}
\tau_B(f) = \begin{cases} (\prod_{i \in B\setminus \{n\}} \theta_i ) \odot f & n \in B \\ \xi'_i \cdot(\theta_j \odot f) + \xi'_j \cdot (\theta_i \odot f) & n \not\in B, B = \{i,j\}\\ f & B = \{i_\theta\} \\ \xi'_i \cdot(\theta_i \odot f) & B = \{i_\xi'\} \end{cases}
\end{equation}
It will be important for what follows to note that block operators corresponding to blocks not containing $n$ commute
\begin{lemma}
\label{block-operator-commute}
Let $A$ and $B$ be two nonempty subsets of $[n-1]$ of size at most two. Then $\tau_A$ and $\tau_B$ commute.
\end{lemma}
\begin{proof}
The lemma reduces to the fact that the family of operators $\{\xi'_1 \cdot, \dots, \xi'_{n-1} \cdot, \theta_1 \odot, \dots, \theta_{n-1} \odot \}$ all anticommute, and that each block operator is a degree two polynomial in these.
\end{proof}
Block operators also interact nicely with the action of $\mathfrak{S}_{n-1}$.
\begin{lemma}
\label{blockoperatorequiv}
Let $A$ be a subset of $[n-1]$ and let $\sigma \in \mathfrak{S}_{n-1}$. Then for any $f\in \wedge\{\Theta,_{n-1}, \Xi'_{n-1}\}$
\[
\sigma \cdot \tau_A (f) = \tau_{\sigma \cdot A}( \sigma \circ f)
\]
where the action of $\mathfrak{S}_n$ on subsets is given by $\sigma \cdot \{a_1, \dots, a_k\} = \{\sigma(a_1), \dots, \sigma(a_k)\}.$
\end{lemma}
We can now define $G_\pi$.
\begin{defn}
Let $\pi \in \Psi(n)$ with blocks $B_1, \dots ,B_k$ and $n \in B_k$. Then
\begin{equation}
G_\pi := \tau_{B_1} \cdots \tau_{B_k} (\theta_1\theta_2 \cdots \theta_{n-1}).
\end{equation}
\end{defn}
We can also give a description of the $G_\pi$ not involving block operators as follows.
\begin{proposition}
Let $\pi \in \Psi(n)$. Take the product of $\theta_i\xi'_i - \theta_j\xi'_j$ for every size two block $\{i,j\}$ of $\pi$ with $i<j$. For each singleton block $\{i\}$ of $\pi$, multiply by $\theta_i$ or $\xi'_i$ according to its label in increasing order. Then $G_\pi$ is equal to the result multiplied by $(-1)^\textrm{inv}(\pi')$ where $\pi'$ is the word formed by listing all size two blocks not containing $n$ increasing within each block and by order of increasing minimal element, then listing all size one blocks not containing n in increasing order.
\end{proposition}
For example, if $\pi = 1_\theta / 2,5 / 3,4 / 5,6,8 / 7_\xi'$, then
\begin{equation}
G_\pi = (-1)^{\inv(253417)}(\theta_2\xi'_2 - \theta_5\xi'_5)(\theta_3\xi'_3-\theta_4\xi'_4)\theta_1\xi'_7
\end{equation}
\begin{proof}
By Lemma~\ref{block-operator-commute} we can assume that all of the block operators corresponding to size two blocks appear before block operators according to singletons. Applying $\tau_{B_k}$ and any block operators corresponding to singletons to $ (\theta_1\theta_2 \cdots \theta_{n-1})$ removes all $\theta_i$ indexed by elements of $B_k$ and replaces $\theta_i$ indexed by $\xi'$-labelled singletons with $\xi'_i$. Note that $\tau_{\{i,j\}} \theta_i \theta_j = \theta_i\xi'_i - \theta_j\xi'_j$, and the proof follows.
\end{proof}
The $\symm_{n-1}$ action on these $G_\pi$ matches the natural $\symm_{n-1}$ action on $\Psi(n)$, up to sign.
\begin{proposition}
Let $\sigma \in \mathfrak{S}_{n-1}$ and $\pi \in \Psi(n)$. Then $\sigma \circ G_\pi = \mathrm{sign}(\sigma) G_{\sigma \circ \pi}$.
\end{proposition}
\begin{proof}
Using the block operator definition of $G_\pi$ and Lemma~\ref{blockoperatorequiv} we have,
\begin{align}
\sigma \circ G_\pi &= \sigma \circ (\tau_{B_1} \cdots \tau_{B_k} (\theta_1\theta_2 \cdots \theta_{n-1}))\\ & = \tau_{\sigma(B_1)} \cdots \tau_{\sigma(B_k)}( \sigma \circ (\theta_1\theta_2 \cdots \theta_{n-1})) \\ &= \tau_{\sigma(B_1)} \cdots \tau_{\sigma(B_k)}( \textrm{sign} (\sigma) \theta_1\theta_2 \cdots \theta_{n-1}) \\ &= \textrm{sign}(\sigma) G_{\sigma \circ \pi}
\end{align}
\end{proof}
The goal of the remainder of this section is to show that $\textrm{span} (\{ G_\pi \mid \pi \in \Phi(n)\})$ is $\symm_{n-1}$ invariant. For this end we will need the following two relations of block operators.
\begin{lemma}
\label{skein}
Let $a,b,c,d \in [n-1]$. Then
\begin{equation}
\tau_{\{a,b\}}\tau_{\{c,d\}} + \tau_{\{a,c\}}\tau_{\{b,d\}} +\tau_{\{a,d\}}\tau_{\{b,c\}} = 0
\end{equation}
as operators on the ring $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$
\end{lemma}
\begin{proof}
This is a straightforward calculation from the definition of $\tau$.
\end{proof}
\begin{lemma}
\label{2-cross-more-than-2}
Let $A = \{a_1 < a_2\} \subset [n-1]$ and $B \subset [n]$ be two disjoint sets with $n \in B$. Let $b_1<b_2<\cdots < b_m$ be the elements of $B$ that lie between $a_1$ and $a_2$, and suppose at least one such element exists. Then
\begin{equation}
\tau_A \tau_B + \tau_{\{a_1,b_1\}} \tau_{B + a_2 - b_1} + \sum_{i=1}^{m-1} \tau_{\{b_i, b_{i+1}\}} \tau_{B+a_1+a_2- b_i-b_{i+1}} +\tau_{\{b_m, a_2\}} \tau_{B+a_1-b_m} = 0
\end{equation}
as operators on the ring $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$
\end{lemma}
\begin{proof}
For any two cyclically consecutive elements $c_1, c_2$ of $a_1, b_2, \dots, b_m ,a_2$, and any third element $c_3 \in B \cup \{a_1, a_2\}$, the terms $\xi'_{c_1} \cdot \theta_{c_2} \odot \theta_{c_3} \odot$ and $\xi'_{c_1} \cdot \theta_{c_3} \odot \theta_{c_2} \odot$ will appear in the expansion of left hand side both exactly once or both exactly twice, depending on whether $c_3$ is also cyclically consecutive with $c_1$. In either case, anticommutativity results in the sum being 0.
\end{proof}
Together these lemmas allow us to demonstrate the $\symm_{n-1}$ invariance via a combinatorial algorithm.
\begin{corollary}
\label{algorithm}
Let $\sigma \in \mathfrak{S}_{n-1}$ and let $\pi \in \Phi(n)$. Then $\sigma \cdot G_\pi$ can be expressed as a linear combination of $\{G_\pi \mid \pi \in \Phi(n)\}$ via the following algorithm:
\begin{enumerate}
\item Apply $\sigma$ to $\pi$, resulting in a set partition $\pi'$ not necessarily in $\Phi(n)$.\\
\item If $\pi'$ is contains any crossing two element blocks $\{a,c\}$, $\{b,d\}$, neither of which contain $n$, replace $\pi'$ with minus the sum of the partitions obtained by replacing $\{a,c\}, \{b,d\}$ with $\{a,b\},\{c,d\}$ and $\{a,d\}, \{b,c\}$. Repeat on each new term of the sum until all terms of the sum do not contain crossing two element blocks.\\
\item For each term of the sum obtained in step 2, replace any two element set that crosses the block containing $n$ as described by Lemma~\ref{2-cross-more-than-2}.\\
\item Replace each partition $\pi''$ in the sum obtained from step 3 with its corresponding $G_{\pi''}$ to express $\sigma \cdot G_\pi$ as a linear combination.
\end{enumerate}
\end{corollary}
\begin{example}
Let $n=8$ and let $\sigma \in \symm_{n-1}$ be the cycle $(3576)$. Let $\pi \in \Phi(n)$ be the set partition $\{23/45/7_\theta/186\}$. An example of applying Corollary~\ref{algorithm} to this situation is given in Figure 1.
\begin{center}
\begin{figure}[h]
\caption{Applying Corollary~\ref{algorithm}}
\begin{tikzpicture}
\equicc[1 cm]{8}{0}{0}
\draw (N2)--(N3);
\draw (N4)--(N5);
\draw (N1)--(N8)--(N6)--(N1);
\draw[->] (1.5,0)--(2,0);
\node at (1.05,-1) {\scriptsize $\theta$};
\equicc[1 cm]{8}{4}{0}
\draw (N2)--(N5);
\draw (N4)--(N7);
\draw (N1)--(N3)--(N8)--(N1);
\node at (2.3,0) {$-$};
\node at (4.15,-1.4) {\scriptsize $\theta$};
\draw[->] (5.5,0)--(6,0);
\equicc[1 cm]{8}{7.5}{0}
\draw (N2)--(N4);
\draw (N1)--(N3)--(N8)--(N1);
\draw (N5)--(N7);
\node at (7.65,-1.4) {\scriptsize $\theta$};
\equicc[1 cm]{8}{11}{0}
\draw (N4)--(N5);
\draw (N2)--(N7);
\draw (N1)--(N3)--(N8)--(N1);
\node at (9.3,0) {$+$};
\node at (11.15,-1.4) {\scriptsize $\theta$};
\draw[->] (-2,-4)--(-1.5,-4);
\equicc[1 cm]{8}{0.5}{-4}
\draw (N7)--(N5);
\draw (N2)--(N3);
\draw (N1)--(N4)--(N8)--(N1);
\node at (-1.2,-4) {$-$};
\node at (.65,-5.4) {\scriptsize $\theta$};
\equicc[1 cm]{8}{4}{-4}
\draw (N2)--(N3);
\draw (N4)--(N5);
\draw (N1)--(N7)--(N8)--(N1);
\node at (2.3,-4) {$-$};
\node at (4.15,-5.4) {\scriptsize $\theta$};
\equicc[1 cm]{8}{7.5}{-4}
\draw (N7)--(N5);
\draw (N2)--(N3);
\draw (N1)--(N4)--(N8)--(N1);
\node at (5.8,-4) {$-$};
\node at (7.65,-5.4) {\scriptsize $\theta$};
\equicc[1 cm]{8}{11}{-4}
\draw (N4)--(N5);
\draw (N7)--(N3);
\draw (N1)--(N2)--(N8)--(N1);
\node at (9.3,-4) {$-$};
\node at (11.15,-5.4) {\scriptsize $\theta$};
\end{tikzpicture}
\end{figure}
\end{center}
\end{example}
\section{A combinatorial basis}
We have shown that there is a mapping of $\symm_{n-1}$-modules $\mathbb{C}\Psi(n) \rightarrow \wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$. In this section we will show that the restriction of this mapping to $\mathbb{C}\Phi(n)$ is injective and becomes an isomorphism when composed with the quotient map $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\} \rightarrow FDR_n$, thereby proving the following.
\begin{theorem}
\label{basis}
The set $\{ [G_\pi] \mid \pi \in \Psi(n) \}$ forms a basis for $FDR_n$, where $[f]$ denotes the equivalence class in $FDR_n$ of $f \in \wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$.
\end{theorem}
\begin{proof}
We begin with a dimension count; Kim and Rhoades \cite{KR} gave a basis of $FDR_n$ indexed by a set $\Pi(n)_{>0}$ of Motzkin-like lattice paths defined as follows.
\begin{defn}
Let $\Pi(n)_{>0}$ be the set of all lattice paths which
start at $(0,0)$, take steps $(1,0), (1,1)$ or $(1,-1)$, only touch the $x$-axis at $(0,0)$ and have all $(1,0)$ steps labelled by $\theta$ or $\xi'$.
\end{defn}
The two indexing sets are in bijection.
\begin{lemma}
\label{path-partition-bijection}
There is a bijection between $\Pi(n)_{>0}$ and $\Phi(n)$.
\end{lemma}
\begin{proof}
Given a Motzkin path in $\Pi(n)_{>0}$, draw a horizontal line extending to the right of each up step until it first intersects the path again. Label each step after the first $1$ to $n-1$. Construct a set partition by placing every up step in a block with the down step it is connected to if such a down step exists, or in the block containing $n$ otherwise. Place every horizontal step in a singleton block with the same label. The process can be reversed, and is therefore a bijection.
\end{proof}
The bijection is best described with a picture example as in Figure 2.
\begin{center}
\begin{figure}[h]
\caption{An example of Lemma~\ref{path-partition-bijection}}
\begin{tikzpicture}[scale = .8]
\equicc[2 cm]{8}{-5}{2.1}
\draw (N1) -- (N3) ;
\draw (N8) -- (N5) -- (N4) -- (N8);
\draw (N6) -- (N7);
\node at (-4.8,4.2) {\scriptsize $\theta$};
\draw [to-to] (-2,2.1) -- (0,2.1);
\draw [thick] (0,0) -- (1,1) -- (2,2) -- (3,2) -- (4,1) -- (5,2) -- (6,3) -- (7,4) -- (8,3);
\draw [dotted] (0,0) -- (8,0);
\draw (.5, .5) -- (8,.5);
\draw (1.5,1.5) -- (3.5,1.5);
\draw (4.5,1.5) -- (8,1.5);
\draw (5.5, 2.5) -- (8,2.5);
\draw (6.5, 3.5) -- (7.5, 3.5);
\node at (1.5,2) {1};
\node at (2.5,2.5) {2};
\node at (2.7,2.3) {\scriptsize $\theta$};
\node at (3.5,2) {3};
\node at (4.5,2) {4};
\node at (5.5,3) {5};
\node at (6.5,4) {6};
\node at (7.5,4) {7};
\end{tikzpicture}
\end{figure}
\end{center}
Therefore it suffices to show that $\{ [G_\pi] \mid \pi \in \Psi(n) \}$ spans. By Corollary~\ref{algorithm} and since $FDR_n$ is defined as a quotient, it suffices to show that together, the sets
\[
\beta := \{ G_\pi \mid \pi \in \Psi(n) \}
\] and
\[ \beta' := \{m(\theta_1\xi'_1 + \cdots + \theta_{n-1}\xi'_{n-1}) \mid m \textrm{ a monomial in } \wedge \{\Theta_{n-1}, \Xi'_{n-1}\}\}
\] span
\[
\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}.
\]
To show that $\beta \cup \beta'$ spans, we will break $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ into many subspaces and show that each subspace is spanned.
Let $S$ be a subset of $[n-1]$ of size $2k$ for some integer $k$. Let $m$ denote a fixed monomial of $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ such that the following conditions hold for all $i \in [n-1]$:
\begin{enumerate}
\item If $i \in S$, then neither $\xi'_i$ nor $\theta_i$ appears in $m$.
\item If $\xi'_i$ appears in $m$, then $\theta_i$ does not appear in $m$.
\end{enumerate}
Let $V_{S,m}$ denote the subspace of $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ spanned by monomials of the form $\theta_{s_1}\xi'_{s_1} \cdots \theta_{s_k}\xi'_{s_k}m$, where $s_1, \dots, s_k \in S$. There are $\binom{2k}{k}$ such monomials, so
\begin{equation}
\textrm{dim}(V_{S,m}) = \binom{2k}{k}.
\end{equation} Consider the set of $\beta \cap V_{S,m}$. These will consist of all $\pi \in \Psi(n)'$ such that the size 1 parts of $\pi$ and their labels correspond exactly with the monomial $m$, and the size two parts partition $S$. This set is therefore in bijection with noncrossing perfect matchings of $S$, so we have
\begin{equation}
| \beta \cap V_{S,m}| = \textrm{Cat}(k)
\end{equation}
where $\textrm{Cat}(k)$ is the $k$th Catalan number. Consider as well the set $\beta' \cap V_{S,m}$. If $m'$ is a degree $n-3$ monomial such that $(\theta_1\xi'_1 + \cdots + \theta_{n-1}\xi'_{n-1})m' \in V_{S,m}$, then it must be the case that $m' = \theta_{s_1}\xi'_{s_1}\cdots\theta_{s_{k-1}}\xi'_{s_{k-1}}m$ for some choice of $s_1, \dots, s_{k-1} \in S$. So we have
\begin{equation}
| \beta' \cap V_{S,m}| = \binom{2k}{k-1}.
\end{equation}
Putting the above equations together we have
\begin{equation}
|(\beta \cup \beta') \cap V_{S,m} | = \textrm{Cat}(k) + \binom{2k}{k-1} = \binom{2k}{k} = \textrm{dim}(V_{S,m})
\end{equation}
and so it suffices to show that $(\beta \cup \beta') \cap V_{S,m}$ is a linearly independent set.
Let $d$ be the degree of $m$, and let $M$ be the set of monomials of degree $d+2k$ in $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ whose variables are in increasing numerical order with $\theta_1 < \xi_1 < \cdots < \theta_n < \xi_n$. Define an inner product $ \langle - , - \rangle $ on the degree $d+2k$ part of $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ such that $M$ is an orthonormal set. With respect to this inner product,
\begin{equation}
\beta \cap V_{S,m} \subseteq (\beta' \cap V_{S,m})^\perp
\end{equation} To see this, suppose that $f_\pi \in V_{S,m}$ and $(\theta_1\xi'_1 + \cdots + \theta_{n-1}\xi'_{n-1})m' \in V_{S,m}$ have monomials in common. Then $m'$ must be equal to $ \theta_{s_1}\xi'_{s_1}\cdots\theta_{s_{k-1}}\xi'_{s_{k-1}}m'$ where each $s_i$ is in a distinct size 2 part of $\pi$. If this is the case, then $ \theta_{s_1}\xi'_{s_1}\cdots\theta_{s_{k-1}}\xi'_{s_{k-1}}m'$ and $f_\pi$ share exactly two monomials, corresponding to the two elements in the last size 2 part of $\pi$. These monomials will have coefficients of opposite sign in $f_\pi$ and the same sign in $ \theta_{s_1}\xi'_{s_1}\cdots\theta_{s_{k-1}}\xi'_{s_{k-1}}m'$, so the inner product will be 0. Therefore it suffices to show that $\beta \cap V_{S,m}$ and $\beta' \cap V_{S,m}$ are both individually linearly independent sets.
To see that $\beta \cap V_{S,m}$ is linearly independent, consider the lexicographic term order on monomials with respect to the variable order $\theta_1, \xi'_1, \theta_2, \xi'_2, \dots$. With respect to this order, the leading term of $f_\pi$ is $\theta_{s_1}\xi'_{s_1}\cdots\theta_{s_k}\xi'_{s_k}m$, where $s_1, \dots, s_k$ are the numerically smaller elements of each size two block of $\pi$. Since $\pi$ is noncrossing and $m$ and $S$ determine the singletons and block containing $n$, specifying the set of elements that are the smaller of their part uniquely determines $\pi$. Therefore the $f_\pi$ contained in $V_{S,m}$ all have unique leading terms and are therefore linearly independent.
Kim and Rhoades proved \cite{KR} that in $FDR_n$, multiplication by $\theta_1\xi'_1 + \cdots + \theta_{n-1}\xi'_{n-1}$ is an injection, so $\beta' \cap V_{S,m}$ is also a linearly independent set and $V_{S,m}$ is spanned by $(\beta \cup \beta') \cap V_{S,m}$. Since every monomial is contained in some $V_{S,m}$, we therefore have that $\wedge \{\Theta_{n-1}, \Xi'_{n-1}\}$ is spanned by $\beta \cup \beta'$ and therefore $\{ [G_\pi] \mid \pi \in \Psi(n) \}$ is a basis for $FDR_n$ as desired.
\end{proof}
\section{$\mathfrak{S}_{n-1}$ module structure}
\label{structure}
In this section we will describe the Frobenius image of each bigraded piece of $FDR_n$ as an $\mathfrak{S}_{n-1}$ module. Consider the family of subspaces:
\begin{equation}
V(n,k,x,y) := \textrm{span} \{ [G_\pi] \mid \pi \in \Phi(n, k , x ,y ) \} \subseteq FDR_n
\end{equation}
These subspaces are in fact submodules of $\textrm{Res}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}}(FDR_n)$, since they are closed under the action of $\mathfrak{S}_{n-1}$. To see this, note that no step of the algorithm described in Corollary~\ref{algorithm} replaces a set partition with one with a different number of size two blocks, $\xi'$-labelled elements, or $\theta$-labelled elements. Since $\Phi(n) = \oplus_{k,x,y} \Phi(n,k,x,y)$ the subspaces $V(n,k,x,y)$ make up all of $FDR_n$:
\begin{proposition}
\label{fdrsum}
The $i,j$-graded piece of $DR_n$ is a direct sum of $V(n,k,x,y)$:
\[
(FDR_n)_{i,j} = \bigoplus_{\substack{k,x,y\\k+x = i \\ k+y = j}} V(n,k,x,y)
\]
\end{proposition}
\begin{proof}
From the definition of $G_\pi$ it is clear that if $\pi \in \Phi(n,k,x,y)$ then $G_\pi$ has bidgree $(k+x,k+y)$. The result follows.
\end{proof}
To determine the structure of these modules we begin with $V(n,k,0,0)$. We first need a lemma
\begin{lemma}
\label{bijection}
There exists a bijection from $\Phi(n,k,0,0)$ to $SYT(n-k-1,k)$, the set of standard Young tableau of shape $\lambda = (n-k-1, k)$.
\end{lemma}
\begin{proof}
Define a function $g: \Phi(n,k,0,0) \rightarrow \binom{[n-1]}{[k]}$ by
\begin{equation}
g(\pi) = \{i \in [n-1] \mid i \textrm{ is in a block of size 2, and is the larger element in its block}.\}
\end{equation}
For example, $g(14/23/78/569) = \{3,4,8\}$. Then $g$ is injective, it is possible to recover the preimage of a set $S$ under $g$ by starting with the smallest $i$ element of $S$, if $g(\pi) = S$, then for $\pi$ to satisfy the noncrossing condition, $\{i-1, i\}$ must be a block of $\pi$. Then the next smallest element of $S$ must be paired with the largest element smaller than it that is not already paired, and so on. This algorithm will produce a unique preimage iff $S$ satisfies the condition that for any $k \in [n-1]$, $|S \cap [k]| \leq k/2$.
Define another function $h: SYT(n-k-1, k) \rightarrow \binom{[n-1]}{k}$ by
\begin{equation}
h(T) = \{ i \in [n-1] \mid i \textrm{ is in the second row of } T\}
\end{equation}
Then $h$ is also injective, and $S \in h(SYT(n-k-1,k))$ iff $S$ satisfies the condition that for any $k \in [n-1]$, $|S \cap [k]| \leq k/2$. So the image of $h$ and $g$ are the same and the result follows.
\end{proof}
\begin{proposition}
We have that $V(n,k,0,0) \cong_{\mathfrak{S}_{n-1}} S^{(n-k-1, k)}$.
\end{proposition}
\begin{proof}
Let $\lambda = (n-k-1,k)$. By Theorem~\ref{basis} and Lemma~\ref{bijection}, the dimensions of the modules agree, so by Lemma~\ref{symmetrizationkills} it suffices to show that $[\mathfrak{S}_\lambda]_+$ does not kill $V(k,0,0)$, but $ [\mathfrak{S}_\mu]_+$ does kill $V(n,k,0,0)$ for all partitions $\mu \succ \lambda$.
We begin by showing that $[\mathfrak{S}_\lambda]_+$ does not kill $V(k,0,0)$. Let $\pi_0 \in \Phi(n,k,0,0)$ be the parition whose blocks are
\[
\{n-1,n-2k\}, \{n-2,n-2k+1\} \{n-3,n-2k+2\}, \dots, \{n-k, n-k-1\}, \{1,2,3, \dots, n-2k-1, n\}
\]
Then using the block operator definition of $F_{\pi_0}$ we have
\begin{equation}
\label{idk}
[\mathfrak{S}_\lambda]_+ F_{\pi_0} = \sum_{\sigma \in \mathfrak{S}_\lambda} \sigma \cdot \tau_{\{n-1, n-2k\}} \cdots \tau_{\{n-k, n-k-1\}} \tau_{\{1,2,3,\dots, n-2k-1, n\}} \theta_{1}\cdots \theta_{n-1}
\end{equation}
Consider the coefficient of $\theta_{n-1} \cdots \theta_{n-k}\xi'_{n-1} \cdots \xi'_{n-k}$ in the above expression. For a term to contribute to this coefficient, it must be the case that $\sigma \cdot \{1,2,3, \dots, n-2k-1, n\} = \{1,2,3, \dots, n-2k-1, n\}$. If this is the case, then the summand corresponding to $\sigma$ can be written as
\begin{equation}
\label{sigmaprime}
\tau_{\{n-1, \sigma'(n-2k)\}} \cdots \tau_{\{n-k, \sigma'(n-k-1)\}} \theta_{n-2k}\cdots \theta_{n-1}
\end{equation}
for some permutation $\sigma'$ of $\{n-k-1, n-2k\}$. The coefficient of $\theta_{n-1} \cdots \theta_{n-k}\xi'_{n-1} \cdots \xi'_{n-k}$ in equation~\ref{sigmaprime} above does not depend on $\sigma'$, so all terms of the sum in equation~\ref{idk} which contribute to the coefficient of $\theta_{n-1} \cdots \theta_{n-k}\xi'_{n-1} \cdots \xi'_{n-k}$ contribute the same sign, and thus the coefficient of $\theta_{n-1} \cdots \theta_{n-k}\xi'_{n-1} \cdots \xi'_{n-k}$ in $[\mathfrak{S}_\lambda]_+ F_{\pi_0}$ is nonzero. Thus $V(k,0,0)$ is not killed by $[\mathfrak{S}_\lambda]_+$.
Now let $\mu$ be any partition of $n-1$ such that $\lambda \succ \mu$, i.e. $\mu = (n-m, m-1)$ for any $m\leq k$. Let $\pi \in \Phi(n,k,0,0)$. Since $m-1 < k$, there must be at least two elements of $i$ and $j$ of $[n-m]$ in the same block in $\pi$. Then the transposition $(i, j)$ acts on $G_\pi$ via multiplication by $-1$, so $(1 + (i, j)) G_\pi = 0$. But $ [\mathfrak{S}_\lambda]_+ = A(1+(i,j)$ for some symmteric group algebra element $A$, so indeed $[\mathfrak{S}_\lambda]_+ G_\pi = 0$, and the result follows.
\end{proof}
We can use $V(n,k,0,0)$ to determine the structure of $V(n,k,x,y)$ for any $x,y$.
\begin{proposition}
\label{vfrob}
We have that
\[
V(n,k,x,y) \cong_{\mathfrak{S}_{n-1}} \mathrm{Ind}_{\mathfrak{S}_{n-x-y-1} \otimes \mathfrak{S}_{x} \otimes \mathfrak{S}_{y}}^{\mathfrak{S}_{n-1}} S^{(n-x-y-k-1, k)} \otimes \mathrm{sign}_{\mathfrak{S}_x} \otimes \mathrm{sign}_{\mathfrak{S}_y}.
\]
\end{proposition}
\begin{proof}
We can represent an element $\pi$ of $\Phi(n,k,x,y)$ by the triple $(X,Y,\pi')$, where $X$ is the set of singletons labelled by $\xi'$, $Y$ is the set of singletons labelled by $\theta$, and $\pi'$ is the set partition obtained by removing all singletons from $\pi$ and decrementing indices. Let $F_{(X,Y,\pi')}$ denote $G_\pi$ for the corresponding $\pi$. The action of a transposition $(i,j)$ on $F_{(X,Y, \pi')}$ is then given by
\begin{equation}
(i,j) \circ F_{(X,Y, \pi')} = \begin{cases} -F_{(X,Y,\pi')} & \{i,j\} \subset X \textrm{ or } \{i,j\} \subset Y \\ F_{(X,Y, (i,j) \circ \pi')} & \{i,j\} \subset (X \cup Y)^c \\ F_{(i,j) \circ X, (i,j) \circ Y, \pi'} \textrm{ otherwise }\end{cases}
\end{equation}
The proposition follows from the definition of induced representation.
\end{proof}
\begin{corollary}
\label{frobimg}
The Frobenius image of $V(n,k,x,y)$ is given by $s_{(n-x-y-k-1, k)}s_{(1^x)}s_{(1^y)}$. The Frobenius image of $(FDR_n)_{i,j}$ is
\[
\sum_{\substack{k,x,y \\ k+x = i \\ k+y = j}} s_{(n-x-y-k-1, k)}s_{(1^x)}s_{(1^y)}
\]
\end{corollary}
\begin{proof}
This follows directly from Proposition~\ref{vfrob}, Proposition~\ref{fdrsum}, and equation~\ref{indprod}.
\end{proof}
\begin{corollary}
The bigraded Frobenius image of $\textrm{Res}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}}(FDR_n)$ is given by
\[
\mathrm{grFrob} (\mathrm{Res}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}}(FDR_n); q,t) = (1-qt)\prod_{i=1}^\infty \frac{(1+x_iqz)(1+x_itz)}{(1-x_iz)(1-x_iqtz)} \bigg|_{z^{n-1}}
\]
where the operator $(\cdots)\mid_{z^{n-1}}$ extracts the coefficient of $z^{n-1}$.
\end{corollary}
By Proposition~\ref{frobimg} we have
\begin{equation}
\textrm{grFrob} (\textrm{Res}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}}(FDR_n); q,t) = \sum_{i} \sum_j \sum_{\substack{k,x,y \\ k+x = i \\ k+y = j}} s_{(n-x-y-k-1, k)}s_{(1^x)}s_{(1^y)} q^i t^j.
\end{equation}
Applying Jacobi-Trudi \cite{Sagan} to the $s_{(n-x-y-k-1, k)}$ terms on the right gives
\begin{equation}
\sum_{i} \sum_j \sum_{\substack{k,x,y \\ k+x = i \\ k+y= j}} s_{(n-x-y-k-1, k)}s_{(1^x)}s_{(1^y)} q^i t^j = \sum_{i} \sum_j \sum_{\substack{k,x,y \\ k+x = i \\ k+y= j}} (h_{n-x-y-k-1}h_{k}- h_{n-x-y-k}h_{k-1})e_{x}e_{y} q^i t^j
\end{equation}
and reindexing sums gives
\begin{equation}
\sum_{i} \sum_j \sum_{\substack{k,x,y \\ k+x = i \\ k+y= j}} h_{n-x-y-k-1}h_{k}e_{x}e_{y} q^i t^j = \sum_{k} h_kq^kt^kz^k\sum_{x}e_xq^xz^x \sum_{y}e_yq^yz^y \sum_m h_mz^m \bigg|_{z^{n-1}}
\end{equation}
and
\begin{equation}
\sum_{i} \sum_j \sum_{\substack{k,x,y \\ k+x = i \\ k+y= j}} h_{n-x-y-k}h_{k-1}e_{x}e_{y} q^i t^j = \sum_{k} h_kq^{k+1}t^{k+1}z^k\sum_{x}e_xq^xz^x \sum_{y}e_yq^yz^y \sum_m h_mz^m \bigg|_{z^{n-1}}
\end{equation}
from which the result follows.
\section{Maximal bidegrees, cyclic sieving and further directions}
Let $X_n$ denote the subset of $\Phi(n)$ corresponding to bidegrees $(i,j)$ where $i+j= n-1$, in other words,
\begin{equation}
X_n = \bigcup_{2k+x+y = n-1} \Phi(n,k,x,y).
\end{equation}
This set consists of noncrossing set partitions set partitions of $[n]$ in which $n$ is in a block by itself, all other blocks are size 1 or 2, and singleton blocks other than $n$ are labelled by $\theta$ or $\xi'$. The set $\{G_\pi \mid \pi \in X_n\}$ is invariant (up to sign changes) under the action of the cycle $(1,2, \dots, n-1)$, since $n$ is in a block by itself and rotating all elements except $n$ cannot introduce any new crossings. We therefore have the setup for a cyclic sieving result using Springer's theorem of regular elements (Theorem~\ref{fakedegree}).
\begin{theorem}
\label{sieving}
The triple $(X_n, C_{n-1}, q^{\binom{n}{2}}\mathbf{fd}(FDR_n)_{i+j = n-1})$ exhibits the cyclic sieving phenomenon where $C_{n-1}$ is the cyclic group generated by $(1,2,\dots, n-1)$.
\end{theorem}
\begin{proof}
This follows directly from Theorem~\ref{fakedegree}.
\end{proof}
Thiel \cite{Thiel} studied a version of this cyclic action in which rotation does not introduce a sign change, while in our setup it introduces a sign when $n$ is odd. Thiel proved the following cyclic sieving.
\begin{theorem}[Thiel, 2016]
\label{thielsieving}
The triple $(X_n, C_{n-1}, C_n(q))$ exhibits the cyclic sieving phenomenon, where $C_{n-1}$ is the cyclic group generated by $(1,2,\dots, n-1)$ and $C_n(q)$ is the Mac-Mahon $q$-Catalan number, defined by
\[
C_n(q) := \frac{1}{[n+1]_q} \begin{bmatrix}2n\\q \end{bmatrix}_q.
\]
\end{theorem}
Thiel proved his result via direct computation of $C_n(q)$ and enumeration of fixed points instead of using representation theory, so one might wonder if our basis could give an altenate algebraic proof of his result. The expression for Frobenius image given in Corollary~\ref{frobimg} allows for the computation of the fake degree as
\begin{equation}
\textbf{fd}((FDR_n)_{i+j = n-1}) = \sum_{\substack{k,x,y\\2k+x+y = n-1}} \begin{bmatrix}{n-1}\\{2k,x,y}\end{bmatrix}_q C_k(q) q^{k+\binom{x}{2}+\binom{y}{2}}
\end{equation}
Combining the two cyclic sieving results it must follow that $q^{\binom{n}{2}}\textbf{fd}((FDR_n)_{i+j = n-1})$ is equivalent to $C_n(q)$ modulo $q^{n-1} - 1$. We have had difficulty in determining this equivalence directly, however, so we propose the following problem:
\begin{problem}
Is there a direct computational proof that $q^{\binom{n}{2}}\textbf{fd}((FDR_n)_{i+j=n-1})$ and $C_n(q)$ are equivalent modulo $q^n-1$?
\end{problem}
Such a proof would complete an alternative representation theoretic proof of Thiel's result.
In joint work with Rhoades \cite{me} we developed a similar combinatorial model for the maximal bidegree components of $FDR_n$, with a basis indexed by all noncrossing set partitions. The action of $S_n$ on that basis could be understood in terms of skein-like relations described by Rhoades \cite{Rhoades}. Patrias, Pechenik, and Striker \cite{PPS} independently discovered an alternate algebraic/geometric model for the irreducible submodule of this action generated by singleton-free noncrossing set partitions sitting in the coordinate ring of a certain algebraic variety. They associated to each partition a polynomial in this coordinate ring defined in terms of matrix minors, and showed that these polynomials satisfied the skein relations described in \cite{Rhoades}. This suggests the following problem:
\begin{problem}
Can our basis for $S^{(n-k-1, k)}$ be realized as a set of polynomials, similarly to the methods of Patrias, Pechenik, and Striker \cite{PPS}?
\end{problem}
One reason for thinking an analogous model might exist is that the relation of block operators described in Lemma~\ref{skein} also appears in the maximal bidegree model and corresponds to a certain identity of two-by-two matrix minors in the work of Patrias, Pechenik and Striker. Their construction therefore extends to give a model for the submodule generated by partitions in $\Phi(n)$ for which the block containing $n$ is at most size two, but we have as yet been unable to discover a treatment of larger blocks satisfying our other relations.
\section{Acknowledgements}
We are very grateful to Brendon Rhoades for many helpful discussions and comments on this project.
|
1,314,259,993,049 | arxiv | \section{Introduction}
Exploiting additional structure has always been central to the success of compressed sensing, ever since it was introduced by Cand\`es, Romberg \& Tao \cite{CandesRombergTao} and Donoho \cite{donohoCS}. Sparsity and incoherence has allowed us to recover signals and images from uniformly subsampled measurements. Recently \cite{AHPRBreaking} the notions of asymptotic sparsity in levels and asymptotic incoherence were introduced to provide enough flexibility to recover signals in a larger variety of inverse problems using subsampling in levels. The key is that optimal subsampling strategies will depend both on the signal structure (asymptotic sparsity) and the asymptotic incoherence structure.
There is a wide variety of problems that lack incoherence, a fact that has been widely recognized \cite{AHPRBreaking, discrete, VanderEtAlSpreadSpectrum, VanderEtAlVariable, ChauffertGradientwaveform, ChauffertVDS, BoyerBlockStructured, PoonFrames, Siemens, Gitta_Fourier, PoonTV, WardFourierAndPolys}, however, they instead posses asymptotic incoherence. Examples include Magnetic Resonance Imaging (MRI) \cite{Unser,Lustig3}, X-ray Computed Tomography \cite{Stanford_CT, quinto2006xrayradon}, Electron Tomography \cite{lawrence2012et,leary2013etcs}, Fluorescence microscopy \cite{Candes_PNAS, Roman} and Surface scattering \cite{JonesTamtoglHAS}, to name a few. This phenomena often originates from the inverse problems being based upon integral transforms, for example, reconstructing a function $f$ from pointwise evaluations of its Fourier transform. In compressed sensing, such a transform is combined with an appropriate sparsifying transformation associated to a basis or frame, giving rise to an infinite measurement matrix $U$. The `coherence' of $U \in \bbC^{\bbN \times \bbN} $ or $U' \in C^{N \times N}$ is defined by
\bes{
\mu(U) = \sup_{i,j \in \bbN} | U_{ij} |^2, \qquad \mu(U') = \max_{i,j=1,\ldots,N} | U'_{ij} |^2.
}
Small coherence is refered to as `incoherence'. Asymptotic incoherence is the phenomena of when
\be{ \label{asympinco}
\mu(P^\perp_N U), \mu(U P^\perp_N) \to 0, \qquad N \to \infty,
}
where $P^\perp_N$ denotes the projection onto the indices $N+1,N+2,...$. As a general rule, the faster asymptotic incoherence decays the more we are able to subsample (see (\ref{conditions31_levels})). The study of more precise notions of coherence has also been considered for the one and two dimensional discrete Fourier sampling, separable Haar sparsity problems in \cite{discrete}. This paper focuses on studying the structure of (\ref{asympinco}) in continuous multidimensional inverse problems and the impact this has on the ability to effectively subsample.
In previous work \cite{onedimpaper}, the structure of incoherence was analyzed as a general problem and theoretical limits on how fast it can decay over all such inverse problems were established. Furthermore, the notion of optimal decay was introduced, which describes the fastest asymptotic incoherence decay possible for a given inverse problem. The notion of an optimal ordering was also introduced, which acted as a set of instructions on how to actually attain this optimal incoherence decay rate by ordering the sampling basis. Optimal decay rates and optimal orderings were determined for the one-dimensional Fourier-wavelet and Fourier-polynomial cases and the former was found to attain the theoretically optimal incoherence decay rate of $N^{-1}$. By `optimal' here we mean in the sense of over all inverse problems that has $U$ an isometry. Furthermore, it is the fastest decay as a power of $N$. This paper extends the basic findings in \cite{onedimpaper} to general $d$-dimensional problems.
\begin{figure}[t]
\begin{center}
\begin{subfigure}[t]{0.43\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{haarmat}
\caption{\footnotesize Coherence Matrix for the 1D case}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.43\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{haar1}
\caption{\footnotesize 1D Column Coherences}
\end{center}
\end{subfigure}\\
\begin{subfigure}[t]{0.43\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{haar2}
\caption{\footnotesize 2D analogue of (b)}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.43\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{haar3}
\caption{\footnotesize Isosurface of 3D Case}
\end{center}
\end{subfigure}
\end{center}
\caption{Fourier - Separable Haar Cases: Incoherence Structures in Different Dimensions. In (b), the coherences are calculated by taking the maxima over the columns in (a), demonstrating decay that scales with frequency. In 2D this decay roughly matches that of the norm of the frequency as seen in (c). However in 3D there are hyperbolic spikes around the coordinate axes that lead to poor incoherence decay (see (d)) when using sampling patterns with rotational invariance or linear scaling. In the black and white plots, white indicates larger absolute value.}
\label{haarimages}
\end{figure}
The optimal orderings in these one dimensional cases matched the leveled schemes that were already used for subsampling. For example, when sampling from the 1D Fourier basis, the sampling levels are usually ordered according to increasing frequency. In multiple dimensions there is no such consensus, instead many different sampling patterns are used, especially when it comes to 2D sampling patterns where radial lines \cite{radial}, spirals \cite{spiralmed} or other k-space trajectories are used. There are also a variety of other sampling techniques used in even higher dimensional (3-10D) problems, such as in the field of NMR spectroscopy \cite{highdimnmr}. If one desires to exploit asymptotic incoherence to its fullest it must be understood whether the coherence structure is consistent with the sampling pattern that one intends to use.
This paper determines optimal orderings for the case of Fourier sampling and (separable) wavelet sparsity in any dimension. It is shown that the optimal decay is always that of the one-dimensional case, and moreover in two dimensions the optimal orderings are compatible with the structure of the 2D sampling patterns mentioned above. However, in higher dimensions problems with poor wavelet smoothness, such as the three dimensional separable Haar case, the class of optimal orderings\footnote{Technically we mean \emph{strongly} optimal here (see Definition \ref{strongoptimality}).} are no longer rotationally invariant (as in Figure \ref{haarimages}), hindering the ability to subsample with traditional sampling schemes. It is also shown that using a pair of tensor bases in general leads to a best possible incoherence decay that is always anisotropic and suboptimal.
We should mention here that for many inverse problems in higher dimensions, using separable wavelets as a reconstruction basis fairs poorly against other bases such as shearlets \cite{shearlet} and curvelets \cite{candes2004new} for approximating images with curve-like features. However, it is not our goal to focus on a particular reconstruction basis in this paper, instead we wish to demonstrate how the incoherence structure can vary for different bases and the impact this has on its application in compressed sensing problems, for good or for worse.
\subsection{Setup \& Key Concepts : Incoherence, Sparsity \& Orderings}
Throughout this paper we shall work in an infinite dimensional separable Hilbert space $ \mathcal{H}$, typically $\mathcal{H}=L^2(\bbR^d)$, with two closed infinite dimensional subspaces $V_1, V_2$ spanned by orthonormal bases $B_1,B_2$ respectively,
\[ V_1 = \overline{ \text{Span} \{ f \in B_1 \}}, \qquad V_2 = \overline{ \text{Span} \{ f \in B_2 \} }.\]
We call $(B_1,B_2)$ a `basis pair'. If we are to form the change of basis matrix $U=(U_{i,j})_{i,j \in \bbN}$ we must list the two bases, which leads to following definitions:
\begin{definition}[Ordering]
Let $S$ be a set. Say that a function $\rho: \mathbb{N} \to S$ is an `ordering' of $S$ if it is bijective.
\end{definition}
\begin{definition}[Change of Basis Matrix]
For a basis pair $(B_1,B_2)$, with corresponding orderings $\rho:\mathbb{N} \to B_1$ and $\tau:\mathbb{N} \to B_2$, form a matrix $U$ by the equation
\begin{equation}\label{U}
U_{m,n} := \langle \tau(n) , \rho(m) \rangle.
\end{equation}
Whenever a matrix $U$ is formed in this way we write `$U:=[(B_1,\rho),(B_2,\tau)]$'.
\end{definition}
Standard compressed sensing theory says that if $x \in \mathbb{C}^N$ is $s$-sparse, i.e.\ $x$ has at most $s$ nonzero components, then, with probability exceeding $1-\epsilon$, $x$ is the unique minimiser to the problem
\bes{
\min_{\eta \in \bbC^N} \| \eta \|_{l^1} \quad \mbox{subject to} \quad P_{\Omega} U \eta
= P_{\Omega} Ux,
}
where $P_{\Omega}$ is the projection onto $\mathrm{span}\{e_j:j\in \Omega\}$, $\{e_j\}$ is the canonical basis, $\Omega$ is chosen uniformly at random with $|\Omega| = m$ and
\be{
\label{m_est_Candes_Plan}
m \ge C \cdot \mu(U) \cdot N \cdot s \cdot \log (\epsilon^{-1}) \cdot \log (N),
}
for some universal constant $C>0$ (see \cite{Candes_Plan} and
\cite{BAACHGSCS}). In \cite{AHPRBreaking} a new theory of compressed sensing was introduced based on the following three key concepts:
\defn{[Sparsity in Levels]
\label{d:Asy_Sparse}
Let $x$ be an element of either $\bbC^N$ or $l^2(\bbN)$. For $r \in \bbN$ let $\mathbf{M} = (M_1,\ldots,M_r) \in \bbN^r$ with $1 \leq M_1 < \ldots < M_r$ and $\mathbf{s} = (s_1,\ldots,s_r) \in \bbN^r$, with $s_k \leq M_k - M_{k-1}$, $k=1,\ldots,r$, where $M_0 = 0$. We say that $x$ is $(\mathbf{s},\mathbf{M})$-sparse if, for each $k=1,\ldots,r$,
\bes{
\Delta_k : = \mathrm{supp}(x) \cap \{ M_{k-1}+1,\ldots,M_{k} \},
}
satisfies $| \Delta_k | \leq s_k$. We denote the set of $(\mathbf{s},\mathbf{M})$-sparse vectors by $\Sigma_{\mathbf{s},\mathbf{M}}$.
}
\defn{[Multi-level sampling scheme]
\label{multi_level_dfn}
Let $r \in \bbN$, $\mathbf{N} = (N_1,\ldots,N_r) \in \bbN^r$ with $1 \leq N_1 < \ldots < N_r$, $\mathbf{m} = (m_1,\ldots,m_r) \in \bbN^r$, with $m_k \leq N_k-N_{k-1}$, $k=1,\ldots,r$, and suppose that
\bes{
\Omega_k \subseteq \{ N_{k-1}+1,\ldots,N_{k} \},\quad | \Omega_k | = m_k,\quad k=1,\ldots,r,
}
are chosen uniformly at random, where $N_0 = 0$. We refer to the set
\bes{
\Omega = \Omega_{\mathbf{N},\mathbf{m}} := \Omega_1 \cup \ldots \cup \Omega_r
}
as an $(\mathbf{N},\mathbf{m})$-multilevel sampling scheme.
}
\defn{[Local coherence]\label{loc_coherence}
Let $U$ be an isometry of either $\bbC^{N}$ or $l^2(\bbN)$.
If $\mathbf{N} = (N_1,\ldots,N_r) \in \bbN^r$ and $\mathbf{M} = (M_1,\ldots,M_r) \in \bbN^r$ with $1 \leq N_1 < \ldots N_r $ and $1 \leq M_1 < \ldots < M_r $ the $(k,l)^{\rth}$ local coherence of $U$ with respect to $\mathbf{N}$ and $\mathbf{M}$ is given by
\ea{ \label{localincoherence}
\mu_{\mathbf{N},\mathbf{M}}(k,l) &= \sqrt{\mu(P^{N_{k-1}}_{N_{k}}UP^{M_{l-1}}_{M_{l}}) \cdot \mu(P^{N_{k-1}}_{N_{k}}U)},\qquad k,l=1,\ldots,r,
}
where $N_0 = M_0 = 0$ and $P^{a}_{b}$ denotes the projection matrix corresponding to indices $\{a+1,\hdots, b\}$.
}
The paper \cite{AHPRBreaking} provided the following estimate (with $C>0$ a universal constant) regarding the local number of measurements $m_k$ in the $k^{\rth}$ level in order to obtain a good reconstruction with probability $\ge 1-\epsilon$:
\be{
\label{conditions31_levels}
\frac{m_k}{N_k-N_{k-1}} \ge C \cdot \log(\epsilon^{-1}) \cdot \left(
\sum_{l=1}^r \mu_{\mathbf{N},\mathbf{M}}(k,l) \cdot s_l\right) \cdot \log\left(N\right),\quad k=1,\ldots,r.
}
In particular, the sampling strategy (i.e.\ the parameters $\mathbf{N}$ and $\mathbf{m}$) is now determined through the local sparsities and coherences. Since the local coherence (\ref{localincoherence}) is rather difficult to analyze in its current form, we bound it above by the following:
\ea{ \label{local2asymp}
\mu_{\mathbf{N},\mathbf{M}}(k,l) &= \sqrt{\mu(P^{N_{k-1}}_{N_{k}}UP^{M_{l-1}}_{M_{l}}) \cdot \mu(P^{N_{k-1}}_{N_{k}}U)}
\\ \label{local2asymp2} & \le \sqrt{\min(\mu(P^{N_{k-1}}_{N_{k}}U), \mu(U P^{M_{l-1}}_{M_{l}})) \cdot \mu(P^{N_{k-1}}_{N_{k}}U)}
\\ \label{local2asymp3} & \le \sqrt{\min(\mu(P^\perp_{N_{k-1}}U), \mu(U P^\perp_{M_{l-1}})) \cdot \mu(P^\perp_{N_{k-1}}U)}
}
It is arguably (\ref{local2asymp3}) rather than (\ref{local2asymp2}) that is the roughest bound here, however we shall see that this becomes effectively an equality in what follows. The crucial improvement of (\ref{local2asymp3}) over (\ref{local2asymp}) is that it is completely in terms of the asymptotic incoherences $\mu(P_N^\perp U), \mu(U P_N^\perp)$, which depend only on the orderings of $B_1,B_2$ respectively, rather than both of them. Furthermore, we can treat the two problems of maximizing the decay of
$\mu(P_N^\perp U), \mu(U P_N^\perp)$ separately and then combine the two resulting orderings together at the end.
Next we describe how one determines the fastest decay of $\mu(P_N^\perp U)$. In \cite{onedimpaper} this was done via the notion of optimality up to constants:
\begin{definition}[Optimal Orderings] \label{fasterdecay}
Let $\rho_1, \rho_2 : \mathbb{N} \to B_1$ be any two orderings of a basis $B_1$ and $\tau$ any ordering of a basis $B_2$. Let $U_1:=[(B_1,\rho_1) , (B_2,\tau)], \ U_2:=[(B_1,\rho_2) , (B_2,\tau)]$ as in (\ref{U}). Also let $Q_N:=P_{N-1}^\perp$. If there is a constant $C>0$ such that
\[ \mu(Q_NU_1) \le C \cdot \mu(Q_NU_2), \qquad \forall N \in \mathbb{N}, \]
then we write $\rho_1 \prec \rho_2$ and say that `$\rho_1$ has a faster decay rate than $\rho_2$ for the basis pair $(B_1, B_2)$'. $\rho_1$ is said to be an `optimal ordering of $(B_1,B_2)$' if $\rho_1 \prec \rho_2$ for all other orderings $\rho_2$ of $B_1$. The relation $\prec$, defined on the set of orderings of $B_1$, is independent of the ordering $\tau$ since the values of $\mu(Q_N U_1), \mu(Q_N U_2)$ are invariant under permuation of the columns of $U_1, U_2$.
\end{definition}
It was shown in \cite{onedimpaper} that optimal orderings always exist. Optimal orderings are used to give us the optimal decay rate:
\begin{definition}[Optimal Decay Rate] Let $f,g : \mathbb{N} \to \mathbb{R}_{>0}$ be decreasing functions.
We write $f \lesssim g$ to mean there is a constant $C>0$ such that
\[ f(N) \le C \cdot g(N), \qquad \forall N \in \mathbb{N}. \]
If both $f \lesssim g$ and $g \lesssim f$ holds, we write `$f \approx g$'.
Now suppose that $\rho: \mathbb{N} \to B_1$ is an optimal ordering for the basis pair $(B_1,B_2)$ and we let $U=[(B_1,\rho),(B_2,\tau)]$ be a corresponding incoherence matrix (with some ordering $\tau$ of $B_2$). Then any decreasing function $f: \mathbb{N} \to \mathbb{R}_{>0}$ which satisfies $f \approx g$, where $g$ is defined by $g(N) = \mu(Q_N U)$, $\forall N \in \bbN$, is said to `represent the optimal decay rate' of the basis pair $(B_1,B_2)$.
\end{definition}
Notice that the optimal decay rate is unique up to the equivalence relation $\approx$ defined on the set of decreasing functions $f: \mathbb{N} \to \mathbb{R}_{>0}$.
We also have a stronger notion of optimality, which gives us finer details on the exact decay:
\begin{definition} [Strong Optimality] \label{strongoptimality}
Let $U=[(B_1,\rho),(B_2,\tau)]$ and $\pi_N$ denote the projection onto the single index $N$. If $f$ represents the optimal decay rate of the basis pair $(B_1,B_2)$ then $\rho$ is said to be `strongly optimal' if the function $g(N):= \mu(\pi_N U)$ satisfies $f \approx g$.
\end{definition}
Estimates in terms of the row incoherence $\mu(\pi_N U)$ have used before in \cite{discrete}, where it was called the `local coherence'. If $\rho$ is a strongly optimal ordering, $U=[(B_1,\rho),(B_2,\tau)]$ and $f$ represents the optimal decay of $(B_1,B_2)$ then
\[
\mu(Q_N U) \le C_1 \cdot f(N) \le C_2 \cdot \mu(\pi_N U) \le C_2 \cdot \mu(P_{N-1}^{N-1+M} U), \qquad N, M \in \bbN,
\]
for some constants $C_1(\rho), C_2(\rho)>0$, which can then be used to show the $\le$ in (\ref{local2asymp3}) can be replaced by $\approx$.
We shall introduce the Fourier basis here as it is used in all of the examples discussed in this paper:
\begin{definition}[Fourier Basis] \label{fourier}
If we define
\[ \chi_k(x) = \sqrt{\epsilon} \exp(2 \pi \mathrm{i} \epsilon k x)\cdot \ \mathds{1}_{[(- 2 \epsilon)^{-1},(2 \epsilon)^{-1}]} (x), \qquad k \in \mathbb{Z}, \]
then the $(\chi_k)_{k \in \bbZ}$ form a basis\footnote{The little $\mathrm{f}$ here stands for `Fourier'.} $B_\mathrm{f}(\epsilon)$ of $L^2([-(2 \epsilon)^{-1},(2 \epsilon)^{-1}])$ . We can form a $d$-dimensional basis of $L^2([-(2 \epsilon)^{-1},(2 \epsilon)^{-1}]^d)$ by taking tensor products (see Section \ref{tensors})
\[ \chi_{k} := \bigotimes_{j=1}^d \chi_{k_j} ,
\qquad k \in \mathbb{Z}^d, \]
and setting $B^d_\mathrm{f}(\epsilon)= \{ \chi_{k} \ : \ k \in \mathbb{Z}^d \} $.
It shall be convenient to identify $B^d_\mathrm{f}(\epsilon)$ with $\mathbb{Z}^d$ using the function
\be{ \label{multidimlambda}
\lambda_d:B_\mathrm{f}^d \to \mathbb{Z}^d, \quad \lambda_d(\chi_k):=(\lambda(\chi_{k_1}), ..., \lambda(\chi_{k_d}))= (k_1,...,k_d)=k.
}
\end{definition}
\section{Main Results}
It turns out that the task of determining the asymptotic incoherence for general $d$-dimensional cases is substantially more difficult and subtle than the $1$-dimensional problems. However, we are able to present sharp results on the decay as well as optimal orderings of the bases.
The main results can be broken down into two groups: one for tensor cases in general and one for the Fourier-Separable wavelet case. In what follows $ d \in \bbN$ denotes dimension.
\subsubsection{Fourier to Tensor Wavelets}
\begin{theorem} \label{tensormainwavelet}
Let $B^d_\mathrm{w}$ be a tensor wavelet basis. The optimal decay rate of both $(B^d_\mathrm{f}, B^d_\mathrm{w})$ and $(B^d_\mathrm{w},B^d_\mathrm{f})$ is represented by $f(N)= \log^{d-1}(N) \cdot N^{-1}$.
\end{theorem}
This theorem is a user friendly and easy-to-read restatement of Theorem \ref{TensorResultsWavelet}. The latter theorem contains the more subtle and technical statements of the results.
\subsubsection{Fourier to Legendre polynomials}
\begin{theorem} \label{tensormainpoly}
Let $B^d_\mathrm{p}$ be a (tensor) Legendre polynomial basis. The optimal decay rate of both $(B^d_\mathrm{f}, B^d_\mathrm{p})$ and $(B^d_\mathrm{p},B^d_\mathrm{f})$ is represented by $f(N)= \big( \log^{(d-1)}(N) \cdot N^{-1} \big)^{2/3}$.
\end{theorem}
This theorem is a restatement of Theorem \ref{TensorResultsPoly} for the purpose of an easy-to-read exposition.
The additional logarithmic factors in the tensor cases here demonstrates the typical problems associated with dimensionality. In general the optimal orderings for all tensor problems are constructed using the hyperbolic cross on the original one-dimensional optimal orderings.
\subsubsection{Fourier to Separable Wavelets}
The definition of a separable wavelet basis $B_\text{sep}^d$ is provided in Section \ref{separable}. The main results on these cases are summarized below:
\begin{theorem} \label{separablesummary}
Consider the Fourier basis $B^d_\mathrm{f}$ and the wavelets basis $B^d_\text{sep}$. Then the following is true.
\begin{itemize}
\item[(i)]
The optimal decay rate of $(B^d_\text{sep},B^d_\mathrm{f})$ is represented by $f(N)= N^{-1}$. The optimal decay rate of $(B^d_\text{sep},B^d_\mathrm{f})$ is obtained by using an ordering $\tau$ that is consistent with the wavelet levels (see definitions \ref{consistent_ordering} and \ref{leveled}) and this ordering is strongly optimal.
\item[(ii)]
In 2D ($d=2$) the optimal decay rate of $(B^d_\mathrm{f},B^d_\text{sep})$ is represented by $f(N)= N^{-1}$. This optimal decay rate is obtained by using an ordering $\rho$ of $B^d_\mathrm{f}$ that satisfies, for some constants $C_1, C_2>0$ and some norm $\| \cdot \|$ on $\bbR^d$,
\be{ \label{linearrough}
\max(\| \lambda_d(\rho(N)) \|,1) \approx N^{1/d}, \qquad N \in \mathbb{N}.
}
In fact $\rho$ is strongly optimal in 2D if and only if (\ref{linearrough}) holds.
\item[(iii)]
In higher dimensions ($d \ge 3$) the optimal decay rate of $(B^d_\mathrm{f},B^d_\text{sep})$ is still represented by $f(N)= N^{-1}$. However the optimal ordering used to obtain this decay rate is dependent on the wavelet used to generate the basis $B^d_\text{sep}$.
\end{itemize}
\end{theorem}
Part (i) is the subject of Section \ref{separablewaveletordering} and is proven in Corollary \ref{leveledresults}. Part (ii), tackled in Section \ref{linearproof}, is the same as Corollary \ref{twodimresults}. Part (iii), covered in Section \ref{semihypsection}, is proven in Theorem \ref{semihyperbolicthm}.
An ordering satisfying (\ref{linearrough}) is called a `linear ordering'. The class of linear orderings are rotation invariant and compatible with sampling schemes based on linearly scaling a fixed shape from the origin (see Section \ref{linearsection}).
Optimal orderings in the case of high dimensions and poor wavelet smoothness can be found by
interpolating between the case of (\ref{linearrough}) and the hyperbolic cross, which generates semi-hyperbolic orderings (see Definition \ref{semihyperbolic}). If the wavelet is sufficiently smooth relative to the dimension then linear orderings are optimal. It is also shown that if a linear ordering is optimal then the wavelet used must have some degree of smoothness proportional to the dimension; in 3D it is $C^0$, 5D it is $C^1$, 7D it is $C^2$, etc. (see Section \ref{orderingsandsmoothness}).
The differences between the two incoherence structures of the Fourier-Tensor wavelet and Fourier-Separable wavelet cases are tested in 2D in Section \ref{numericalsection}.
\subsection{Outline for the Remainder}
Some key tools that we use to find optimal orderings are given in Section \ref{orderings}. Those familiar with \cite{onedimpaper} can skip the majority of this section except for the concept of characterization. We then cover the general tensor case and introduce hyperbolic orderings in Section \ref{tensors} and prove Theorem \ref{tensormainwavelet} and Theorem \ref{tensormainpoly}. In Section \ref{separable} we discuss the separable cases, first covering how to optimally order the wavelet basis before quickly moving on to the central problem of finding optimal orderings of the Fourier basis. Linear orderings are introduced first, then we justify the need for semihyperbolic orderings. Finally we move onto some simple compressed sensing experiments, one demonstrating the benefits of multilevel subsampling and one showing the impact of differing incoherence structures between the 2D tensor and separable cases.
\section{Tools for Finding Optimal Orderings \& Theoretical Limits on Optimal Decay} \label{orderings}
The first tool is perhaps the most important, as it is a very easy way to identify a strongly optimal ordering:
\begin{lemma} \label{StrongOptimalEquivalence}
\textbf{1):} Let $(B_1,B_2)$ be a basis pair and $\tau$ any ordering of $B_2$. Furthermore, let $ B_1$ have an ordering $\rho_1 : \mathbb{N} \to B_1$, and define $U_1:=[(B_1,\rho_1),(B_2,\tau)]$. Suppose that that there is a decreasing function $f_1: \mathbb{N} \to \mathbb{R}_{>0}$ such that
\[ f_1(N) \le \mu(\pi_N U_1), \qquad \forall N \in \mathbb{N}. \]
Then if $\rho_2: \mathbb{N} \to B_1$ is an ordering, $U_2=[(B_1,\rho_2),(B_2,\tau)]$ and $f_2: \mathbb{N} \to \mathbb{R}_{>0}$ is a function with
\[ \mu( Q_N U_2) \le f_2(N), \qquad \forall N \in \mathbb{N}, \]
then $f_1(N) \le f_2(N)$ for every $N \in \mathbb{N}$.
\textbf{2):} Let $\rho$ be an ordering of $B_1$ with $U:=[(B_1,\rho),(B_2,\tau)]$ and $f: \mathbb{N} \to \mathbb{R}_{\ge 0}$ be a decreasing function with $f(N) \to 0$ as $N \to \infty$. If, for some constants $C_1, C_2 > 0$, we have
\begin{equation} \label{rowordercond}
C_1 f(N) \le \mu( \pi_N U) \le C_2 f(N), \qquad \forall N \in \mathbb{N},
\end{equation}
then $\rho$ is a strongly optimal ordering and $f$ is a representative of the optimal decay rate.
\end{lemma}
\begin{proof}
See Lemma 2.11 in \cite{onedimpaper}.
\end{proof}
\begin{definition}[Best ordering]
Let $(B_1,B_2)$ be a basis pair. Then any ordering $\rho: \mathbb{N} \to B_1$ is said to be a `best ordering' if for any ordering $\tau$ of $B_2$ and $U=[(B_1,\rho),(B_2,\tau)]$ we have that the function $g(N):= \mu(\pi_N U)$ is decreasing.
\end{definition}
Notice that any best ordering is also a strongly optimal ordering. We shall need the notion of a best ordering briefly to prove Lemma \ref{characterisationlemma}.
\begin{lemma} \label{bestexistence}
Suppose that we have a basis pair $(B_1, B_2)$ with two orderings $\rho: \bbN \to B_1$, $\tau: \bbN \to B_2$ of $B_1, B_2$ respectively. If $U=[(B_1,\rho),(B_2,\tau)]$ satisfies
\[ \mu(\pi_N U) \to 0 \quad \text{as} \quad N \to \infty, \]
then a best ordering exists.
\end{lemma}
\begin{proof}
See Lemma 2.10 in \cite{onedimpaper}.
\end{proof}
Throughout this paper we would like to define an ordering according to a particular property of the basis but this property may not be enough to specify a unique ordering. To deal with this issue we introduce the notion of consistency:
\begin{definition}[Consistent ordering]\label{consistent_ordering}
Let $F: S \to \mathbb{R}$ where $S$ is a set. We say that an ordering $\rho: \mathbb{N} \to S$ is `consistent with F' if
\[ F(f) < F(g) \quad \Rightarrow \quad \rho^{-1}(f) < \rho^{-1}(g), \qquad \forall f,g \in S. \]
\end{definition}
The notion of consistency becomes important if we want to convert bounds on the coherence into optimal orderings:
\begin{definition} \label{Characterisation}
\begin{itemize}
\item[1.)]
Suppose $F:S \to \bbR_{> 0} $ satisfies $| \{ x \in S : 1/F(x) \le K \}| < \infty$ for all $K>0$, $\sigma: \bbN \to S$ is consistent with $1/F$ and $F(\sigma(N)) \to 0$ as $N \to \infty$. Then any decreasing function $f: \bbN \to \bbR_{>0}$ such that $f \approx F \circ \sigma$ is said to `represent the fastest decay of $F$'.
\item[2.)]
Suppose $(B_1,B_2)$ is a basis pair and $\iota: S \to B_1$ a bijection. If there exists a function $F:S \to \bbR_{>0}$ and a constant $C_1>0$ such that
\be{ \label{dominate}
\sup_{g \in B_2} | \langle \iota(s) , g \rangle |^2 \le C_1 \cdot F(s), \quad \forall s \in S,
}
then $F$ is said to `dominate the optimal decay of $(B_1,B_2)$'. If the inequality is reversed we say $F$ is `dominated by the optimal decay of $(B_1,B_2)$'. Furthermore, if there is a constant $C_2>0$ such that
\be{ \label{characterise}
C_2 \cdot F(s) \le \sup_{g \in B_2} | \langle \iota(s) , g \rangle |^2 \le C_1 \cdot F(s), \quad \forall s \in S,
}
then $F$ is said to `characterize the optimal decay of $(B_1,B_2)$'.
\end{itemize}
\end{definition}
\begin{lemma} \label{characterisationlemma}
1): \ \ Suppose $f$ is a representative of the optimal decay rate for the basis pair $(B_1,B_2)$, $\iota: S \to B_1$ is a bijection, $F: S \to \bbR$ dominates the optimal decay of $(B_1,B_2)$, $\sigma : \bbN \to S$ is consistent with $1/F$ and $U=[(B_1, \iota \circ \sigma), (B_2, \tau)]$ . Then if $g$ represents the fastest decay of $F$ then $f, \mu(\pi_{\cdot}U) \lesssim g$.
2): \ \ If $F$ is instead is dominated by the optimal decay of $(B_1,B_2)$ then $f, \mu(\pi_{\cdot}U) \gtrsim g$.
3): \ \ If $F$ now characterizes the optimal decay of $(B_1,B_2)$ then $f, \mu(\pi_{\cdot}U) \approx g$ and therefore $\rho$ is a strongly optimal ordering for the basis pair $(B_1,B_2)$ if and only if $F(\iota^{-1} \circ \rho(\cdot)) \approx g $.
\end{lemma}
\begin{proof} 1.) \ \ We may assume, without loss of generality, that $g(N) \to 0$ as $N \to \infty$ else there is nothing to prove as $f(N), \mu(\pi_N U)$ are bounded functions of $N$. Therefore, a best ordering exists by Lemma \ref{bestexistence}. (\ref{dominate}) becomes (for $C_1'>0$ a constant),
\[
\mu(\pi_N U) \le C_1 \cdot F(\sigma(N)) \le C_1' \cdot g(N), \quad \forall N \in \bbN.
\]
Since $g$ is decreasing we have $\mu(Q_N U) \le C_1'\cdot g(N)$ and therefore we can apply part 1) of Lemma \ref{StrongOptimalEquivalence} to $f_1=f$ and $f_2=g$ (using a best ordering as $\rho_1$ and $\rho_2 = \iota \circ \sigma$) to deduce that $f \lesssim g$.
2.) \ \ (\ref{dominate}) reversed becomes
\[
\mu(\pi_N U) \ge C \cdot F(\sigma(N)) \ge C_1' \cdot g(N), \quad \forall N \in \bbN.
\]
Therefore we can apply part 1) of Lemma \ref{StrongOptimalEquivalence} to $f_1=g$ and $f_2=f$ (using $\rho_1 = \iota \circ \sigma$, $\rho_2$ an optimal ordering) to deduce that $f \lesssim g$.
3.) \ \ Notice that if
$F$ characterizes the optimal decay of $(B_1,B_2)$ then (\ref{characterise}) becomes
\[
C_2' \cdot g(N) \le C_2 \cdot F(\sigma(N)) \le \mu(\pi_N U) \le C_1 \cdot F(\sigma(N)) \le C'_1 \cdot g(N), \quad \forall N \in \bbN,
\]
and we can then apply part 2) of Lemma \ref{StrongOptimalEquivalence} to show $f, \mu(\pi_{\cdot}U) \approx g$. If we let $U':=[(B_1, \rho), (B_2, \tau)]$ then (\ref{characterise}) becomes
\[
C_2 \cdot F(\iota^{-1} \circ \rho(N)) \le \mu(\pi_N U') \le C_1 \cdot F(\iota^{-1} \circ \rho(N)) , \quad \forall N \in \bbN,
\]
and the result follows from Definition \ref{strongoptimality}.
\end{proof}
Before moving on, we recall from \cite{onedimpaper} some results on the fastest optimal decay rate for a basis pair:
\begin{theorem} \label{isometrydecaylowerbound}
Let $U \in \cB(l^2(\bbN))$ be an isometry. Then
$
\sum_{N} \mu(Q_N U)
$
diverges.
\end{theorem}
\begin{proof}
See Theorem 2.14 in \cite{onedimpaper}.
\end{proof}
\begin{corollary}
Let $U \in \cB(l^2(\bbN))$ be any isometry. Then there does not exist an $\epsilon > 0$ such that
$$
\mu(Q_N U) = \mathcal{O}(N^{-1-\epsilon}) , \qquad \ N \to \infty.
$$
\end{corollary}
It turns out that Theorem \ref{isometrydecaylowerbound} cannot be improved without imposing additional conditions on $U$:
\begin{lemma} \label{incoherencecounter}
Let $f,g: \bbN \to \bbR$ be any two strictly positive decreasing functions and suppose that $\sum_N f(N)$ diverges. Then there exists $U \in \cB(l^2(\bbN))$ an isometry with
\be{ \label{strongerthanoptimal}
\mu(Q_N U) \le f(N), \quad \mu(U Q_N) \le g(N), \qquad N \in \bbN .
}
\end{lemma}
\begin{proof}
See Lemma 2.16 in \cite{onedimpaper}.
\end{proof}
If we restrict our decay function to be a power law, i.e. $f(N):= CN^{- \alpha}$ for some constants $\alpha, C >0$ then the largest possible value of $\alpha>0$ such that (\ref{strongerthanoptimal}) holds for an isometry $U$ is $\alpha=1$. This gives us a notion of the fastest optimal decay rate as a power of $N$ over all pairs of bases where the span of $B_2$ lies in the span of $B_1$.
\section{One-dimensional Bases and Incoherence Results} \label{onedim}
Before we begin our review of the one-dimensional cases by quickly going the one-dimensional bases and orderings that we shall be working with to construct multi-dimensional bases and orderings in Section \ref{tensors}.
\subsection{Fourier Basis}
We recall the one-dimensional Fourier basis $B_\mathrm{f}(\epsilon)=(\chi_k)_{k \in \bbZ}$ from Definition \ref{fourier}.
\begin{definition}[Standard ordering]\label{standard_ordering}
We define $F_\mathrm{f}:B_\mathrm{f} \to \mathbb{N} \cup \{0\}$ by $F_\mathrm{f}(\chi_k)=|k|$ and say that an ordering $\rho: \mathbb{N} \to B_\mathrm{f}$ is a `standard ordering' if it is consistent with $F_\mathrm{f}$ (recall Definition \ref{consistent_ordering}).
\end{definition}
\subsection{Standard Wavelet Basis} \label{waveletbasis}
Take a Daubechies wavelet $\psi$ and corresponding scaling function $\phi$ in $L^2(\mathbb{R})$ with \[\text{Supp} (\phi) = \text{Supp} (\psi) = [-p+1,p]. \]
We write
\[
\begin{aligned}
\phi_{j,k}(x) =2^{j/2} \phi(2^j x-k) , \qquad \psi_{j,k} (x) = 2^{j/2} \psi(2^j x - k),
\\ V_j := \overline{\text{Span} \{ \phi_{j,k}: k \in \bbZ \}}, \quad W_j := \overline{\text{Span} \{ \psi_{j,k}: k \in \bbZ \}}.
\end{aligned}
\]
With the above notation, $(V_j)_{j \in \mathbb{Z}}$ is the multiresolution analysis for $\phi$, and therefore
\[V_j \subset V_{j+1} , \qquad V_{j+1} = V_j \oplus W_j, \qquad L^2(\bbR) = \overline{\bigcup_{j \in \bbZ} V_j}, \]
where $W_j$ here is the orthogonal complement of $V_j$ in $V_{j+1}$. For a fixed $J \in \bbN$ we define the set\footnote{`$\mathrm{w}$' here stands for `wavelet'.}
\begin{align} \label{waveletbasisdefine}
B_\mathrm{w} := \left\{ \begin{array}{cc} & \mathrm{Supp}(\phi_{J,k}) \cap (-1,1) \neq \emptyset , \\ \phi_{J,k} , \ \psi_{j,k} : & \mathrm{Supp}(\psi_{j,k}) \cap (-1,1) \neq \emptyset, \\ & j \in \mathbb{N}, j \ge J , \ k \in \mathbb{Z} \end{array} \right \} ,
\end{align}
Let $\rho$ be an ordering of $B_\mathrm{w}$. Notice that since $L^2(\mathbb{R})= \overline{ V_J \oplus \bigoplus^{\infty}_{j=J} W_j}$ for all
$f \in L^2(\mathbb{R})$ with $\mathrm{supp}(f) \subseteq [-1,1]$ we have
\[ f = \sum_{n=1}^\infty c_{n} \rho(n) \quad \text{for some} \quad (c_n)_{n \in \bbN} \in \ell^2(\bbN) .\]
\begin{definition} [Leveled ordering (standard wavelets)]\label{leveled}
Define $F_\mathrm{w}:B_\mathrm{w} \to \mathbb{R}$ by
\[ F_\mathrm{w}( f) \ = \ \begin{cases}
\ j, \ & \mbox{if } f \in W_j\\
\ -1, \ & \mbox{if }f \in V_J
\end{cases} , \]
and say that any ordering $\tau: \mathbb{N} \to B_\mathrm{w}$ is a `leveled ordering' if it is consistent with $F_\mathrm{w}$.
\end{definition}
Notice that $F_\mathrm{w}(\psi_{j,k})=j$. We use the name ``leveled'' here since requiring an ordering to be leveled means that you can order however you like within the individual wavelet levels themselves, as long as you correctly order the sequence of wavelet levels according to scale.
Suppose that $U=[(B_\mathrm{f}(\epsilon), \rho), (B_\mathrm{w},\tau)]$ for orderings $\rho, \tau$. If we require $U$ to be an isometry we must impose the constraint $(2\epsilon)^{-1} \ge 1+2^{-J+1}(p-1)$ otherwise the elements in $B_\mathrm{w}$ do not lie in the span of $B_\mathrm{f}(\epsilon)$. For convenience we rewrite this as $\epsilon \in I_{J,p}$ where
\[I_{J,p}:=(0,(2+2^{-J+2}(p-1))^{-1}]. \]
\subsection{Boundary Wavelet Basis} \label{BoundaryWavelets}
We now look at an alternative way of decomposing a function $f \in L^2([-1,1])$ in terms of a wavelet basis, which involves using boundary wavelets \cite[Section 7.5.3]{dDwav}. The basis functions all have support contained within $[-1,1]$, while still spanning $L^2[-1,1]$. Furthermore, the new multiresolution analysis retains the ability to reconstruct polynomials of order up to $p-1$ from the corresponding original multiresolution analysis. We shall not go into great detail here but we will outline the construction; we take, along with a Daubechies wavelet $\psi$ and corresponding scaling function $ \phi$ with $ \mathrm{Supp} (\psi) = \mathrm{Supp} (\phi)=[-p+1,p]$, boundary scaling functions and wavelets (using the same notation as in \cite{dDwav} except that we use $[-1,1]$ instead of $[0,1]$ as our reconstruction interval)
\[ \phi^{\text{left}}_n, \ \phi^{\text{right}}_n, \ \psi^{\text{left}}_n , \ \psi^{\text{right}}_n , \qquad n =0,\cdots,p-1 .\]
Like in the standard wavelet case we shift and scale these functions,
\[ \phi^{\text{left}}_{j,n}(x) = 2^{j/2} \phi^{\text{left}}_{n}(2^j (x+1)),
\qquad \phi^{\text{right}}_{j,n}(x)= 2^{j/2} \phi^{\text{right}}_{n}(2^j (x-1)). \]
We are then able to construct nested spaces ,
$ (V^{\text{int}}_j)_{j \ge J}$, $ (W^{\text{int}}_j)_{j \ge J}$ for a fixed base level $J \ge \lceil \log_2 (p) \rceil $, such that \mbox{$L^2([-1,1])=\overline{ \bigoplus^{\infty}_{j=J} V^{\text{int}}_j}$}, $V^{\text{int}}_{j+1}=V^{\text{int}}_j \oplus W^{\text{int}}_j$ with $W^\text{int}_j$ the orthogonal complement of $V^\text{int}_j$ in $V^\text{int}_{j+1}$ by defining
\begin{equation*}
V^{\text{int}}_j = \overline{ \text{Span}
\left \{ \begin{aligned}
\phi^{\text{left}}_{j,n} & , \phi^{\text{right}}_{j,n} \\ & \phi_{j,k}
\end{aligned}
:
\begin{aligned}
& n =0 , \cdots , p-1 \ \\ & k \in \mathbb{Z} \ s.t. \ \mathrm{Supp}( \phi_{j,k} ) \subset (-1,1)
\end{aligned}
\right \} } ,
\end{equation*}
\begin{equation*}
W^{\text{int}}_j = \overline{ \text{Span}
\left \{ \begin{aligned}
\psi^{\text{left}}_{j,n} & , \psi^{\text{right}}_{j,n} \\ & \psi_{j,k}
\end{aligned}
:
\begin{aligned}
& n =0 , \cdots , p-1 \ \\ & k \in \mathbb{Z} \ s.t. \ \mathrm{Supp}( \psi_{j,k} ) \subset (-1,1)
\end{aligned}
\right \} } .
\end{equation*}
We then take the spanning elements of $V^{ \text{int}}_J$ and the spanning elements of $W^{\text{int}}_j$ for every $j \ge J$ to form the basis $B_{\mathrm{b} \mathrm{w}}$ ($\mathrm{b} \mathrm{w}$ for 'boundary wavelets').
\begin{definition}[Leveled ordering (boundary wavelets)]
Define $F_w: B_{\mathrm{b} \mathrm{w}} \to \mathbb{R}$ by the formula
\[ F_{\mathrm{b} \mathrm{w}}( f) \ = \ \begin{cases}
\ j, \ & \mbox{if } f \in W^{\text{int}}_j\\
\ -1, \ & \mbox{if }f \in V^{\text{int}}_J
\end{cases}. \]
Then we say that an ordering $\tau: \mathbb{N} \to B_{\mathrm{b} \mathrm{w}}$ of this basis is a `leveled ordering' if it is consistent with $F_{\mathrm{b} \mathrm{w}}$.
\end{definition}
\subsection{Legendre Polynomial Basis} \label{polynomialbasis}
If $(p_n)_{n \in \mathbb{N}}$ denotes the standard Legendre polynomials on $[-1,1]$ (so $p_n(1)=1$ and $p_1(x)=1$ for $x \in [-1,1]$) then the $L^2$-normalised Legendre polynomials are defined by $\tilde{p}_n=\sqrt{n-1/2} \cdot p_n$ and we write $B_\mathrm{p} := (\tilde{p}_n )_{n=1}^\infty$ (the $\mathrm{p}$ here stands for ``polynomial'' ). $B_\mathrm{p}$ is already ordered; call this the \emph{natural ordering} .
\subsection{Incoherence Results for One-dimensional Bases} \label{1DResults}
Next we recall the one-dimensional incoherence results proved in \cite{onedimpaper}, which shall be used to prove the corresponding multi-dimensional tensor results in Section \ref{tensors}:
\begin{theorem} \label{FourierWaveletResults}
Let $\rho$ be a standard ordering of $B_\mathrm{f}(\epsilon)$ with $\epsilon \in I_{J,p}$, $\tau$ a leveled ordering of $B_\mathrm{w}$ and $U=[(B_\mathrm{f}(\epsilon),\rho),(B_\mathrm{w},\tau)]$. Then we have, for some constants $C_1, C_2>0$ the decay
\be{ \label{FourierWaveletOptimalBounds}
\frac{C_1}{N} \le \mu(\pi_N U), \ \mu(U \pi_N) \le \frac{C_2}{N}, \qquad \forall N \in \mathbb{N} ,
}
The same conclusions also hold if the basis $B_\mathrm{w}$ is replaced by $B_{\mathrm{b} \mathrm{w}}$ and the condition $\epsilon \in I_{J,p}$ by $\epsilon \in (0,1/2]$.
\end{theorem}
\begin{theorem} \label{FourierPolynomialResults}
Let $\rho$ be a standard ordering of $B_\mathrm{f}(\epsilon)$ with $\epsilon \in (0,0.45], $ $\tau$ a natural ordering of $B_\mathrm{p}$ and $U=[(B_\mathrm{f}(\epsilon),\rho),(B_\mathrm{p},\tau)]$. Then we have, for some constants $C_1, C_2>0$ the decay
\be{ \label{FourierPolynomialOptimalBounds}
\frac{C_1}{N^{2/3}} \le \mu(\pi_N U), \ \mu(U \pi_N), \le \frac{C_2}{N^{2/3}}, \qquad \forall N \in \mathbb{N} .
}
\end{theorem}
\section{Multidimensional Tensor Cases: Proof of Theorem \ref{tensormainwavelet} and Theorem \ref{tensormainpoly} } \label{tensors}
In this section we prove Theorem \ref{tensormainwavelet} and Theorem \ref{tensormainpoly}. In fact, we state and prove their slightly more involved generalisations: Theorems \ref{TensorResultsWavelet} and \ref{TensorResultsPoly}. We also provide examples of hyperbolic orderings.
\subsection{General Estimates}
\begin{definition}[Tensor basis]
Suppose that $B$ is an orthonormal basis of some space $T \le L^2 (\mathbb{R})$ (i.e. $T$ is a subspace $L^2 (\mathbb{R})$) and we already have an ordering $\rho: \mathbb{N} \to B$. Define $\rho^d: \mathbb{N}^d \to \bigotimes_{j=1}^d T \le L^2 (\mathbb{R}^d)$ by the formula ($m \in \mathbb{N}^d$)
\[ \rho^d(m)(x):= \Big( \bigotimes_{j=1}^d \rho(m_j) \Big) (x) = \prod_{j=1}^d \rho(m_j)(x_j). \]
This gives a basis of $\bigotimes_{j=1}^d T \le L^2 (\mathbb{R}^d)$ because of the formula
\begin{equation} \label{prodsplit}
\langle \rho^d (m),\rho^d(n) \rangle_{L^2(\mathbb{R}^d)} = \prod_{j=1}^d \langle \rho(m_j), \rho(n_j) \rangle_{L^2(\mathbb{R})}.
\end{equation}
We call $B^d:=(\rho^d(m))_{m \in \mathbb{N}^d}$ a `tensor basis'. The function $\rho^d$ is said to be the `d-dimensional indexing induced by $\rho$'. Notice that $\rho^d$ is not an ordering unless $d=1$.
\end{definition}
Now suppose that we have two one-dimensional bases $B_1$, $B_2$ with corresponding optimal orderings $\rho_1, \rho_2$. Let $\rho^d_1, \rho^d_2$ be the d-dimensional indexings induced by $\rho_1,\rho_2$ of the bases $B^d_1,B^d_2$. What are optimal orderings of the basis pair $(B^d_1,B^d_2)$ and what is the resulting optimal decay rate? Some insight is given by the following Lemma:
\begin{lemma} \label{generaltensor}
Let $(B_1,B_2)$ be a pair of bases with corresponding tensor bases $B_1^d, B_2^d$.
Let $\rho_1$ be a strongly optimal ordering of $B_1$ and $\rho_1^d$ be the $d$-dimensional indexing induced by $\rho_1$. Finally, for some ordering $\tau$ of $B_2$, let $U=[(B_1, \rho_1), (B_2, \tau)]$ . Then if $f$ represents the optimal decay rate corresponding to the basis pair $(B_1,B_2)$ we have, for some constants $C_1, C_2>0$,
\be{ \label{generaltensorequation}
\prod_{i=1}^d C_1^d \cdot f(n_i) \le \sup_{g \in B^d_2} | \langle \rho_1^d (n) , g \rangle |^2 = \prod_{i=1}^d \mu(\pi_{n_i} U) \le \prod_{i=1}^d C_2^d \cdot f(n_i), \quad n \in \bbN^d.
}
Consequently, if we let $\iota := \rho_1^d$ then $F(n):=\prod_{i=1}^d f(n_i)$ characterizes the optimal decay of $(B_1,B_2)$.
\end{lemma}
\begin{proof}
Let $\tau^d$ denote the $d$-dimensional indexing induced by $\tau$. Then by breaking the down the tensor product into terms and using the bijectivity of $\tau^d$ we have
\[
\begin{aligned}
\sup_{g \in B^d_2} | \langle \rho_1^d (n) , g \rangle |^2 & = \sup_{m \in \bbN^d} | \langle \rho_1^d (n) , \tau^d(m) \rangle |^2 = \sup_{m \in \bbN^d} \prod_{i=1}^d | \langle \rho_1 (n_i) , \tau(m_i) \rangle |^2
\\ & = \prod_{i=1}^d \sup_{m \in \bbN} | \langle \rho_1 (n_i) , \tau(m) \rangle |^2 = \prod_{i=1}^d \mu(\pi_{n_i} U).
\end{aligned}
\]
Therefore (\ref{generaltensorequation}) follows from applying the definition of a strongly optimal ordering to each term in the product.
\end{proof}
Lemma \ref{generaltensor} says that if we have a strongly optimal ordering for the basis pair $(B_1,B_2)$ then we can use Lemma \ref{characterisationlemma} to find all strongly optimal orderings for the corresponding tensor basis pair $(B_1^d,B_2^d)$. In particular, we have
\begin{corollary} \label{generaltensorcorollary}
We use the framework of the previous Lemma. Let $\sigma:\bbN \to \bbN^d$ be consistent with $1/F$. Then an ordering $\rho$ is strongly optimal for the basis pair $(B^d_1,B^d_2)$ if and only if there are constants $C_1,C_2>0$ such that
\[
C_1 F(\sigma(N)) \le F((\rho_1^d)^{-1} \circ \rho(N)) \le C_2 F(\sigma(N)), \quad N \in \bbN.
\]
\end{corollary}
Suppose that we have a strongly optimal ordering $\rho_1$ of $B_1$ such that the optimal decay rate is a power of $N$, namely that $f(n)=n^{-\alpha}$ for some $\alpha>0$, which is the case for the one dimensional examples we covered in Section \ref{onedim}. The above Lemma tells us that to find the optimal decay rate we should take an ordering $\sigma : \bbN \to \bbN^d$ that is consistent with $1/F(n):= \prod_{i=1}^d 1/f(n_i)= \prod_{i=1}^d n_i^{\alpha}$ which is equivalent to being consistent with $1/F^{1/\alpha}(n)=\prod_{i=1}^d n_i$. This motivates the following:
\begin{definition}[Corresponding to the hyperbolic cross]
Define $F_H: \mathbb{N}^d \to \mathbb{R}$ by $ F_{H}(n)= \prod_{i=1}^d n_i$.
Then we say a bijective function $\sigma: \mathbb{N} \to \mathbb{N}^d$ `corresponds to the hyperbolic cross' if it is consistent with $F_H$.
\end{definition}
The name `hyperbolic cross' originates from its use in approximation theory \cite{crossorig,hypcross}. We now claim that if $\sigma$ corresponds to the hyperbolic cross and $d \ge 2$, then
\be{ \label{hyperbolicdecayrate}
\prod_{i=1}^d \sigma(N)_i \sim \frac{(d-1)! N}{\log^{d-1}(N+1)} \quad \text{as } \quad N \to \infty.
}
Next we proceed to prove this claim.
\begin{definition}
For $d \in \mathbb{N}$ let $f_d(x) = x \log^{d-1} x$ be defined on $[1,\infty)$. We define
$g_d$ as the inverse function of $f_d$ on $[1,\infty)$, and so $g_d: [ 0, \infty) \to [1,\infty)$. Furthermore, we define
\be{ \label{hyperbolicdecay}
h_d(x):= \frac{x}{\log^{d-1}(x+1)}
, \qquad x \in [1, \infty).}
\end{definition}
\begin{lemma} \label{ordersimplify}
The following holds:
\\
1.) \ \
$g_d(x)/h_{d}(x) \to 1 \quad \text{as} \quad x \to \infty.$
\\
2.) \ \ Let $\tilde{f}(x) = x \log^{d-1} x + x p( \log(x) ) + \beta$ with $p$ a polynomial of degree at most $d-2$, $ \beta \in \mathbb{R}$ and let $\tilde{g}$ be its inverse function defined for large $x \in \bbR_+$. Then we also have $\tilde{g}(x)/h_{d}(x) \to 1 \quad \text{as} \quad x \to \infty.$
\end{lemma}
\begin{proof}
1.) \ \ For notational convenience we shall prove the equivalent result
\[ \frac{g_d(x) \log^{d-1}(x)}{x} \to 1 \quad \text{as} \quad x \to \infty. \]
By taking logarithms we change the problem from studying the asymptotics of a fraction to the asymptotics of the difference
\be{ \label{equivalentasymp}
\log(g_d(x))-\log(h_d(x)) = \log(g_d(x)) - \log x + (d-1) \log \log x \to 0 \quad \text{as} \quad x \to \infty.
}
With this in mind we notice that the function $\log(g_d)$ (defined on $[0,\infty)$) is the inverse function of $e_d(x):= f_d(\exp(x)) = x^{d-1} \exp x$ (defined on $[0,\infty)$).
Notice that for $x$ large we have $e_d(x- (d-1) \log x)= \frac{(x- (d-1) \log x)^{d-1}}{x^{d-1}} \exp(x) \le \exp(x)$ which implies that $ x - (d-1) \log x \le \log(g_d( \exp(x))) $. Now if we let $\epsilon>0$ then we deduce that
\[e_d(x- (d-1) \log x+ \epsilon)= \frac{(x- (d-1) \log x+\epsilon)^{d-1}}{x^{d-1}} \exp(x+\epsilon) \ge \exp(x) \quad \text{for} \quad x \quad \text{large.} \]
This implies that $ x - (d-1) \log x + \epsilon \ge \log(g_d( \exp(x))) $ for $x$ large. We therefore conclude that for all $x$ sufficiently large we have
\[ x - (d-1) \log x \le \log(g_d( \exp(x))) \le x - (d-1) \log x + \epsilon,\]
from which (\ref{equivalentasymp}) follows since $\epsilon>0$ is arbitrary.
2.) \ \ Notice that by part 1. it suffices to show that $\tilde{g}(x)/g_{d}(x) \to 1 \ $ as $\ x \to \infty.$ Again, we shall show this by taking logarithms, reducing the proof to showing
\[ \log( \tilde{g}(x)) - \log( g_d(x)) \to 0 \quad \text{as} \quad x \to \infty. \]
Notice that $\log( \tilde{g}(x))$ is the inverse function, defined for large $x$, of
\[\tilde{e}(x):=\tilde{f}( \exp(x)) = x^{d-1} \exp(x) + p(x) \cdot \exp(x) + \beta, \]
Then since
\[\tilde{e}'(x)= x^{d-1} \exp(x) + ((d-1) \cdot x^{d-2} +p'(x)+p(x)) \cdot \exp(x), \]
we can use the hypothesis that $p$ is of a lower order than $x^{d-1}$ to show that for every $\epsilon>0$, there is an $L(\epsilon)>0$ such that for all $x \ge L(\epsilon)$ we have $\epsilon \cdot \tilde{e}'(x -\epsilon) \ge |\tilde{e}(x) - e_d(x)|=|p(x) \cdot \exp(x) + \beta|$. We therefore deduce from the mean value theorem that for $x \ge \exp(L(\epsilon))$ we have
\begin{align*}
\tilde{e}(\log(g_d(x))-\epsilon) \le e_d(\log(g_d(x)))= & x \le \tilde{e}( \log(g_d(x))+ \epsilon)
\\ & \Rightarrow \log(g_d(x)) - \epsilon \le \log(\tilde{g}(x)) \le \log(g_d(x)) + \epsilon,
\end{align*}
where we applied $\log(\tilde{g})$ to the inequality in the last step (this preserves the inequality since $\log(\tilde{g})$ is an increasing function of $x$ for $x$ large).
\end{proof}
\begin{lemma} \label{orderset}
1). \ \ For every $d \in \mathbb{N}$ we have
\be{ \label{hyplogestimate}
R_N:=\sum_{i=1}^N \frac{1}{i} (\log(N) - \log(i))^d \ = \ \frac{1}{d+1} \log^{d+1} N + \mathcal{O}(\log^d N) \qquad N \to \infty.
}
2). \ \ Let $S_d(N)$ for $d, N \in \mathbb{N}$ be defined by
\begin{equation} \label{simplehyperbolic}
S_d(N):= \# \Big\{ m \in \mathbb{N}^d : \prod_{i=1}^d m_i \le N \Big\}.
\end{equation}
Then for every $d \in \mathbb{N}$, there exists polynomials $ \underline{p}_d, \overline{p}_d$ both of degree $d-1$ with identical leading coefficient $1/(d-1)!$ such that
\begin{equation} \label{hyperbolic_count}
N \underline{p}_d( \log(N)) \le S_d(N) \le N \overline{p}_d( \log(N)) .
\end{equation}
3). \ \ If we let $\sigma: \mathbb{N} \to \mathbb{N}^d$ correspond to the hyperbolic cross then (\ref{hyperbolicdecayrate}) holds.
\end{lemma}
\begin{proof}
1). \ \ Let $I_N:=\int_1^N \frac{1}{x} (\log(N) - \log(x))^d \, dx$. Since the integrand is a decreasing function of $x$ (with $N$ fixed) we find that by the Maclaurin integral test that
$0 \le R_N -I_N \le \log^d(N)$. This means that showing (\ref{hyplogestimate}) is equivalent to showing that
\[ \int_1^N \frac{1}{x} (\log(N) - \log(x))^d \, dx = \frac{1}{d+1} \log^{d+1} N + \mathcal{O}(\log^d N). \]
Now, by expanding out the factors of the integrand and integrating (recall that the integral of $x^{-1} \log^k x$ is $\frac{1}{k+1} \cdot \log^{k+1}x$) the integral becomes
\[ \log^{d+1}(N) \cdot \sum_{i=0}^d \frac{1}{i+1} \binom{d}{i} (-1)^{i}. \]
Since $\frac{1}{i+1} \binom{d}{i} = \frac{1}{d+1} \binom{d+1}{i+1}$ we see that the sum simplifies to $ \frac{1}{d+1}$ and we are done.
2). \ \ We use induction on the dimension $d$. The case $d=1$ is immediate since $\underline{p}_1(x)=\overline{p}_1(x)=1$ satisfies inequality (\ref{hyperbolic_count}). Therefore suppose that inequality (\ref{hyperbolic_count}) holds for dimension $d=k$. We shall extend the result to $d=k+1$ using the equality:
\begin{equation} \label{hyperbolicdimreduce}
S_{k+1}(N)= \sum_{i=1}^N S_{k}\Big( \left\lfloor \frac{N}{i} \right\rfloor \Big).
\end{equation}
This equality follows from rewriting the set defining $S_{k+1}$ as the following disjoint union:
\[
\Big\{ m \in \bbN^{k+1} : \prod_{i=1}^{k+1} m_i \le N \Big\} = \coprod_{j=1}^N \Bigg\{ m \in \bbN^{k+1} : m_{k+1}=j, \prod_{i=1}^k m_i \le \left\lfloor \frac{N}{i} \right\rfloor \Bigg\} .
\]
\textbf{Upper Bound:} We may assume without loss of generality that $\overline{p}_k$ has all coefficients positive. Therefore, by replacing $ \lfloor \frac{N}{i} \rfloor $ with $\frac{N}{i}$ and using the upper bound in (\ref{hyperbolic_count}), we can upper bound equation (\ref{hyperbolicdimreduce}) by
\[ \sum_{i=1}^N \frac{N}{i} \cdot \overline{p}_k \Big( \log \Big( \frac{N}{i} \Big) \Big) \ \le \ N \sum_{i=1}^N \frac{1}{i} \cdot \overline{p}_k ( \log(N) - \log(i)). \]
We can then get the required upper bound by applying part 1) of the lemma to each term in the polynomial; for example the highest order term becomes
\[
\begin{aligned}
\sum_{i=1}^N \frac{N}{i} \cdot \frac{1}{(k-1)!} ( \log(N) - \log(i))^{k-1} \le \frac{N }{k!} \log^{k}N + C N \log^{k-1} N, \qquad \forall N \in \mathbb{N},
\end{aligned}
\]
for some constant $C>0$ sufficiently large. The other terms in $\overline{p}_k$ are handled similarly.
\textbf{Lower Bound:} Notice that without loss of generality we can assume all the coefficients of $\underline{p}_k$ apart from the leading coefficient are negative. Using the lower bound in (\ref{hyperbolic_count}), we can lower bound equation (\ref{hyperbolicdimreduce}) by
\[ \sum_{i=1}^N \left\lfloor \frac{N}{i} \right\rfloor \cdot \overline{p}_k \Big( \log \Big( \left\lfloor \frac{N}{i} \right\rfloor \Big) \Big). \]
This means we can tackle the $<k-1$ order terms in the same way as in the upper bound since we can replace $ \left\lfloor \frac{N}{i} \right\rfloor $ with $\frac{N}{i}$ (recall we have assumed these terms are negative). Now we are left with bounding the highest order term:
\begin{equation}
\begin{aligned}
\sum_{i=1}^N \left\lfloor \frac{N}{i} \right\rfloor \frac{1}{(k-1)!} (\log \Big( \left\lfloor \frac{N}{i} \right\rfloor \Big) )^k = \sum_{i=1}^N \left\lfloor \frac{N}{i} \right\rfloor \frac{1}{(k-1)!} \cdot \Big[ \log \Big( \frac{N}{i} \Big) - \big( \log \Big( \frac{N}{i} \Big) - \log \Big( \left\lfloor \frac{N}{i} \right\rfloor \Big) \big) \Big]^k.
\end{aligned}
\end{equation}
Therefore expanding out the binomial term, setting the sign of all terms except the first to be negative, and noticing $\log \Big( \frac{N}{i} \Big) - \log \Big( \left\lfloor \frac{N}{i} \right\rfloor \Big) \le 1$ for every $i,N$ we get the lower bound
\[
\begin{aligned}
\sum_{i=1}^N & \left\lfloor \frac{N}{i} \right\rfloor \frac{1}{(k-1)!} \log^{k} \Big( \frac{N}{i} \Big) - \sum_{i=1}^N \sum_{j=0}^{k-1} \left\lfloor \frac{N}{i} \right\rfloor \binom{k}{j} \frac{1}{(k-1)!} \log^{j} \Big( \frac{N}{i} \Big).
\end{aligned}
\]
From here we can replace $\left\lfloor \frac{N}{i} \right\rfloor$ by $ \frac{N}{i} $ for the right term, $\left\lfloor \frac{N}{i} \right\rfloor$ by $\frac{N}{i} -1 $ on the left term and use part 1) of the lemma again to prove the lower bound.
3.) \ \ From the second part of the lemma we know that for some degree $d-1$ polynomials $\underline{p}_d , \overline{p}_d$ with leading coefficient $1/(d-1)!$ we have
$ N \underline{p}_d (\log(N)) \le S_d(N) \le N \overline{p}_d ( \log(N)). $
Now notice that if $m \in \mathbb{N}$ then because of consistency we must have $ S_d( F_H(\sigma(m))-1) \le m$ since $\sigma$ must first list all the terms $n$ in $\bbN^d$ with $F_H(n) \le F_H(\sigma(m))-1$ before listing $\sigma(m)$. Likewise we must have $m \le S_d( F_H(\sigma(m)))$ since the $S_d( F_H(\sigma(m)))$ terms with $F_H(n) \le F_H(\sigma(m)), n \in \bbN^d$ must be listed by $\sigma$ first, including $m$ , before any others. Consequently we deduce
\begin{equation} \label{productbound}
\begin{aligned}
(F_H(\sigma(m)) -1) \underline{p}_d (\log(F_H(\sigma(m)) & -1)) \le m \le F_H(\sigma(m)) \overline{p}_d ( \log( F_H( \sigma(m)) )).
\end{aligned}
\end{equation}
We now treat both sides separately. Looking at the LHS we get the estimate
$F_H(\sigma(m)) -1 \le \tilde{g}_d(m),$
where $\tilde{g}_d(m)$ is the inverse function (defined for large $m$) of
\[
\begin{aligned}
\tilde{f}_d(x):= & \frac{1}{(d-1)!} x \log^{d-1}(x) + \mbox{(degree $d-2$ poly)}(\log(x)),
\end{aligned}
\]
and so we may apply part 2. of Lemma \ref{ordersimplify} to deduce
$ F_H(\sigma(m)) \le h_d( (d-1)! m ) \cdot (1 + \epsilon(m)),$
where $\epsilon(m) \to 0$ as $m \to \infty$. The right hand side is handled similarly to get the same asymptotic lower bound on $F_H(\sigma(m))$, namely
$ F_H(\sigma(m)) \ge h_d( (d-1)! m ) \cdot (1 + \epsilon(m)),$ where $\epsilon(m) \to 0$ as $m \to \infty$. Since $\frac{h_d((d-1)! x)}{(d-1)! h_d(x)} \to 1$ as $x \to \infty$ the proof is complete.
\end{proof}
(\ref{hyperbolicdecayrate}) allows us to determine the optimal decay rate for when the optimal one dimensional decay rate is a power of $N$.
\begin{theorem} \label{generaltensortheorem}
Returning to the framework of Corollary \ref{generaltensorcorollary}, if $f(n)=n^{- \alpha}$ for $n \in \bbN$, $F(n)= \prod_{i=1}^d f(n)$ for $n \in \bbN^d$ and $\sigma : \bbN \to \bbN^d$ corresponds to the hyperbolic cross then
\be{ \label{finaltensordecay}
F(\sigma(N)) = \Bigg( \prod_{i=1}^d \sigma(N)_i \Bigg)^{- \alpha} \sim \big((d-1)! \cdot h_d(N) \big) ^{-\alpha}, \quad N \to \infty.
}
Consequently $h_d^{-\alpha}$ is representative of the optimal decay rate for the basis pair $(B_1^d,B_2^d)$. Furthermore, an ordering $\rho$ is strongly optimal for the basis pair $(B_1^d,B_2^d)$ if and only if there are constants $C_1, C_2>0$ such that
\begin{equation} \label{hyperbolicdef}
C_1 \cdot h_d(N) \le \prod_{i=1}^d \Big((\rho_1^d)^{-1} \circ \rho(N) \Big)_i \le C_2 \cdot h_d(N), \quad N \in \bbN.
\end{equation}
\end{theorem}
\begin{proof}
(\ref{finaltensordecay}) follows immediately from (\ref{hyperbolicdecayrate}). This implies that $F \circ \sigma \approx h_d^{-\alpha}$. The statement on the optimal decay rate then follows from the characterization result from Lemma \ref{generaltensor} applied to Lemma \ref{characterisationlemma}. The statement on strongly optimal orderings follows from Corollary \ref{generaltensorcorollary}.
\end{proof}
\begin{definition} \label{hyperbolicordering}
Using the framework of Lemma \ref{generaltensor}, any ordering $\rho: \bbN \to B_1^d$ such that (\ref{hyperbolicdef}) holds is called a 'hyperbolic' ordering. with respect to $\rho_1$. Notice that by (\ref{finaltensordecay}) that if $\sigma: \bbN \to \bbN^d$ corresponds to the hyperbolic cross then $\rho_1^d \circ \sigma$ is hyperbolic with respect to $\rho_1$.
\end{definition}
We now apply Theorem \ref{generaltensortheorem} to the one-dimensional cases we have already covered:
\subsection{Fourier-Wavelet Case}
\begin{theorem} \label{TensorResultsWavelet}
We use the setup of Lemma \ref{generaltensor}.
Suppose $B_1=B_\mathrm{f}(\epsilon)$, $B_2=B_\mathrm{w}$ for some fixed $\epsilon \in I_{J,p}$, $\rho_1$ is a standard ordering of $B_1$ and $\tau_1$ is a leveled ordering of $B_2$. Let $U_d=[(B_1^d, \rho), (B_2^d, \tau)]$ where $\rho, \tau$ is hyperbolic with respect to $\rho_1, \tau_1$ respectively. Then we have, for some constants $C_1,C_2>0$,
\begin{equation}\label{FourWave1}
\frac{C_1 \log^{d-1}(N+1)}{N} \le \mu( \pi_N U_d ), \ \mu(U_d \pi_N) \le \frac{C_2 \log^{d-1}(N+1)}{N}, \qquad N \in \bbN.
\end{equation}
The above also holds if the basis $B_\mathrm{w}$ is replaced by $B_{\mathrm{b} \mathrm{w}}$ and the condition $\epsilon \in I_{J,p}$ by $\epsilon \in (0,1/2]$.
\end{theorem}
\begin{proof}
Inequality (\ref{FourWave1}) follows from applying Theorem \ref{FourierWaveletOptimalBounds} to Theorem \ref{generaltensortheorem}.
\end{proof}
\subsection{Fourier-Polynomial Case}
\begin{theorem} \label{TensorResultsPoly}
We use the setup of Lemma \ref{generaltensor}. Suppose $B_1=B_\mathrm{f}(\epsilon)$, $B_2=B_\mathrm{p}$ for some fixed $\epsilon \in (0,0.45]$, $\rho_1$ is a standard ordering of the Fourier basis and $\tau_1$ is the natural ordering of the polynomial basis. Let $U_d=[(B_1^d, \rho), (B_2^d, \tau)]$ where $\rho, \tau$ is hyperbolic with respect to $\rho_1, \tau_1$ respectively. Then we have, for some constants $C_1,C_2>0$, that
\begin{equation}\label{FourLeg1}
\frac{C_1 (\log^{d-1}(N+1))^{2/3}}{N^{2/3}} \le \mu( \pi_N U_d ), \ \mu(U_d \pi_N) \le \frac{ C_2 (\log^{d-1}(N+1))^{2/3}}{N^{2/3}}, \qquad N \in \bbN.
\end{equation}
\end{theorem}
\begin{proof}
Inequality (\ref{FourLeg1}) follows from applying Theorem \ref{FourierPolynomialOptimalBounds} to Proposition \ref{generaltensortheorem}.
\end{proof}
\subsection{Examples of Hyperbolic Orderings} \label{hyperbolicexamples}
The generalisation introduced by Definition \ref{hyperbolicordering}, apart from allowing us to characterise all orderings that are strongly optimal, may seem to fulfil little other purpose. However, as we shall see in this section, this definition admits orderings which in specific cases are very natural and appear a little less abstract than an ordering derived from the hyperbolic cross.
\begin{example} \label{hypcrossZd} (Hyperbolic Cross in $\mathbb{Z}^d$)
Our first example is unremarkable but nonetheless important. In $d$ dimensions, take $B_1^d:=B_\mathrm{f}^d$ as a d-dimensional tensor Fourier basis. Recall we can identify this basis with $\mathbb{Z}^d$ using the function $\lambda_d$. Suppose that we define a function $H_d: \mathbb{Z}^d \to \mathbb{R}$ by
\be{ \label{hddef}
H_d(m) = \prod_{i=1}^d | \max(|m_i|,1)|,
}
and say that a bijective function $\sigma: \mathbb{N} \to \mathbb{Z}^d$ `corresponds to the hyperbolic cross in $\mathbb{Z}^d$' if it is consistent with $H_d$. Figure \ref{hyperbolic} shows the first few contour lines of $H_d$ in two dimensions.
With this definition we can then prove the analogous result of Lemma \ref{orderset}:
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{cross}
\caption{Hyperbolic Fourier Ordering in Two Dimensions: A Contour Plot of $H_2$}
\label{hyperbolic}
\end{figure}
\begin{lemma} \label{orderset2}
Let $\sigma: \mathbb{N} \to \mathbb{Z}^d$ correspond to the hyperbolic cross and let $h_d$ be as in (\ref{hyperbolicdecayrate}). Then we have
\be{ \label{hyperboliccrossZdecay}
\prod_{i=1}^d | \max(| \sigma(m)_i|,1)| \sim \frac{(d-1)!}{2^d} \cdot h_d(m) \quad \mbox{as} \quad m \to \infty.
}
Moreover, if $\rho_1$ is a standard ordering of $B_\mathrm{f}$ and $\sigma: \mathbb{N} \to \mathbb{Z}^d$ corresponds to the hyperbolic cross. Then $\lambda_d^{-1} \circ \sigma$ is a hyperbolic ordering with respect to $\rho_1$.
\end{lemma}
\begin{proof}
Let $R_d(n)$ denote the number of lattice points in the hyperbolic cross of size $n$ in $\bbZ^d$, namely
\[ R_d(n) := \# \{ m \in \mathbb{Z}^d : \prod_{i=1}^d \max(| m_i|,1) \le n \}. \]
Call the set in the above definition $ \mathcal{H}_d(n)$. If we remove the hyperplanes $\{m_i=0 \}$ for every $i$ from $\mathcal{H}_d(n)$, we are left with $2^d$ quadrants in $\mathbb{Z}^d$ which are congruent to set in equation (\ref{simplehyperbolic}). From the second part of Lemma \ref{orderset} we therefore have
\[ R_d(n) \ge 2^d n \underline{p}_d( \log(n)). \]
Next notice that the intersection of $\mathcal{H}_d(n)$ with each hyperplane $\{ m_i=0 \}$ can be identified with $\mathcal{H}_{d-1}(n)$ and so we also have the upper bound
\[
\begin{aligned}
R_d(n) & \le 2^d n \overline{p}_d( \log(n)) + d \cdot R_{d-1}(n) \quad \Rightarrow \quad R_{d}(n) \le n \overline{r}_d( \log (n)),
\end{aligned}
\]
for some degree $d-1$ polynomial $\overline{r}_d$ with leading coefficient $\frac{2^{d}}{(d-1)!}$. Combining the upper and lower bounds we see that for some polynomials $\underline{r}_d, \overline{r}_d$ of degree $d-1$ with leading coefficient $ \frac{2^{d}}{(d-1)!}$ we have
\[ n \underline{r}_d (\log(n)) \le R_d(n) \le n \overline{r}_d( \log(n) ). \]
Therefore for $m \in \mathbb{N}$ since
\[R_d(H_d(\sigma(m))-1) \le m \le R_d(H_d(\sigma(m))),\]
we have
\[
\begin{aligned}
(H_d(\sigma(m))-1) & \underline{r}_d (\log(H_d(\sigma(m))-1)) \le m \le H_d(\sigma(m)) \overline{r}_d (\log(H_d(\sigma(m)))).
\end{aligned}
\]
Consequently we can apply Lemma \ref{ordersimplify} to both sides to derive (\ref{hyperboliccrossZdecay}) like in the proof of Lemma \ref{orderset}.
For the last part of the Lemma notice that since $\rho_1$ is a standard ordering then $\max(|\lambda_1 \circ \rho_1(N)|,1) \approx N$. This means that the bounds on $\mu(\pi_N U)$ in Theorem \ref{FourierWaveletResults} can be rephrased as (for some constants $C_1,C_2>0$)
\[
C_1 \cdot (\max(|n|,1))^{-1} \le \sup_{g \in B_\mathrm{w}} | \langle \lambda_1^{-1}(n), g \rangle |^2 \le C_2 \cdot (\max(|n|,1))^{-1}, \quad n \in \bbZ,
\]
and by Lemma \ref{generaltensor} this extends to the dD tensor case:
\be{ \label{zhypcrosscharacterise}
C^d_1 \cdot \prod_{i=1}^d (\max(|n_i|,1))^{-1} \le \sup_{g \in B^d_\mathrm{w}} | \langle \lambda_d^{-1}(n), g \rangle |^2 \le C^d_2 \cdot \prod_{i=1}^d (\max(|n_i|,1))^{-1}, \quad n \in \bbZ^d.
}
This describes a characterization of the optimal decay of $(B_\mathrm{f}^d(\epsilon),B^d_\mathrm{w})$. Lemma \ref{characterisationlemma} tells us that $\lambda_d^{-1} \circ \sigma$ is strongly optimal for $(B_\mathrm{f}^d(\epsilon),B^d_\mathrm{w})$, which by Theorem \ref{generaltensortheorem} is hyperbolic with respect to $\rho_1$.
\end{proof}
\end{example}
\begin{example} (Tensor Wavelet Ordering)
Now we look at an example of a less obvious hyperbolic ordering. We first introduce some notation to describe a tensor wavelet basis:
For $j \in \mathbb{N}, \ k \in \mathbb{Z}$ let $\phi^0_{j,k}:= \phi_k, \ \phi^1_{j,k}:= \psi_{j,k}$. Now for $s \in \{ 0,1 \}^d , \ j \in \mathbb{N}^d, \ k \in \mathbb{Z}^d$ define
\[ \Psi^s_{j,k} := \bigotimes_{i=1}^d \phi^{s_i}_{j_i,k_i}.\]
Then it follows that for $J \in \bbN$ fixed, we have
\begin{align} \label{tensorwaveletbasisdefine}
B^d_\mathrm{w} := \left\{ \begin{array}{cc} & \mathrm{Supp}(\phi^{s_i}_{j_i,k_i}) \cap (-1,1) \neq \emptyset \quad \forall i , \\ \Psi^s_{j,k} : & s_i=0 \Rightarrow j_i=J, \quad s_i=1 \Rightarrow j_i\ge J \\ & j \in \mathbb{N}^d, s \in \{ 0,1 \}^d , \ k \in \mathbb{Z}^d \end{array} \right \} ,
\end{align}
The same approach can be applied to the boundary wavelet basis $B_{\mathrm{b} \mathrm{w}}$ to generate a boundary tensor wavelet basis $B^d_{\mathrm{b} \mathrm{w}}$, although we must include the extra boundary terms, which can be done by letting $s \in \{ 0,1,2,3 \}^d$ where $\phi^2_{J,n}$ would be a boundary scaling function term and $\phi^3_{j,n}$ a boundary wavelet term.
\begin{lemma} \label{tensorwavelethyp}
Let $\rho_1$ be any leveled ordering of a one-dimensional Haar wavelet basis $B^d_{\mathrm{w}}$. Setting $\overline{j}=\sum_{i=1}^d j_i$ define $F_\text{hyp}:B^d_{\mathrm{w}} \to \mathbb{R}$ by the formula
\[ F_\text{hyp}( f) = \overline{j} \quad \mbox{if } \quad f= \Psi^s_{j,k}.
\]
Then any ordering $\rho: \mathbb{N} \to B^d_{\mathrm{w}}$ that is consistent with $F_\text{hyp}$ is a hyperbolic ordering with respect to $\rho_1$.
\end{lemma}
\begin{remark}
Such an ordering $\rho$ is used to implement a tensor wavelet basis in Section \ref{numericalsection}.
\end{remark}
\begin{remark}
For the sake of simplicity we only work with the Haar wavelet case, although we could cover the boundary wavelet case with the same argument.
\end{remark}
\begin{proof}
By recalling inequality (3.10) in \cite{onedimpaper} or by using Lemma \ref{levelgrowth} in the case $d=1$ we know that there are constants $C_1, C_2>0$ such that for $\rho_1(N) = \phi^{s}_{j,k}, s \in \{0,1\}, j \in \bbN, k \in \bbZ$,
\[
C_1 2^{ j} \le N \le C_2 2^{j}, \qquad N \in \bbN.
\]
Therefore, writing $\rho_1^d(m) = \Psi^{s(m)}_{j(m),k(m)}$,
\[ C^d_1 2^{ \overline{j(m)}} \le \prod_{i=1}^d m_i \le C^d_2 2^{ \overline{j(m)}}, \qquad m \in \bbN^d. \]
Consequently if we rewrite this with an actual ordering $\rho(N)= \Psi^{s(N)}_{j(N),k(N)}$ for $N \in \mathbb{N}$ we deduce
\begin{equation} \label{down2j}
C^d_1 2^{ \overline{j(N)}} \le \prod_{i=1}^d \Big( (\rho_1^d)^{-1} \circ \rho(N) \Big)_i \le C^d_2 2^{ \overline{j(N)}} ,
\end{equation}
and so we have reduced the problem to determining how $\overline{j(N)}$ scales with $N$.
Notice that from our ordering of the wavelet basis that $\overline{j(N)}$ is a monotonically increasing function in $N$ and moreover, for every value of $\overline{j(N)}$ there are $r_d(\overline{j(N)})2^{ \overline{j(N)}}$ terms in $B_w^d$ with this value of $\overline{j(N)}$ in the wavelet basis, where
\[
\begin{aligned}
r_d(N):= \# \Bigg\{ (j,s) \in \mathbb{N}^d \times \{0,1 \}^d : \quad \overline{j} = N , \quad j_i \ge J, \quad (s_i-1)(j_i-J)=0 \quad \forall i=1,...,d \Bigg\},
\end{aligned}
\]
This is where we are using that the support of the Haar wavelet is $[0,1]$ and so there are $2^j$ shifts of $\phi_{j,0}, \psi_{j,0}$ in $B_\mathrm{w}$. Notice that $r_d(N)$ is a polynomial of degree $d-1$. With this in mind notice we can define, consistent for $n \in \mathbb{N}, \ n \ge J$,
\begin{align*}
T_d(x) & ``=" \sum_{i=J}^x r_d(i)2^{i} := \ p_d(x) 2^{x} + \alpha_d ,
\end{align*}
for some degree $d-1$ polynomial $p_d$ and constant $\alpha_d$. This is possible by taking the formula for the geometric series expansion and differentiating repeatedly. By the consistency property of $\rho$ we deduce the inequality
\begin{align*}
T_d( \overline{j(N)}-1) \le N \le T_d(\overline{j(N)}) \quad &\Rightarrow \quad \overline{j(N)}-1 \le T_d^{-1}(N) \le \overline{j(N)} \\
&\Rightarrow \quad 2^{\overline{j(N)}-1} \le 2^{T_d^{-1}(N)} \le 2^{\overline{j(N)}}.
\end{align*}
Notice that $2^{T_d^{-1}(x)}$ is the inverse function of $T_d( \log_2 x)$ which is of the form
$ x \cdot p_d( \log_2x) + \alpha,$
Therefore, applying parts 2. \& 3. of Lemma \ref{ordersimplify} gives, for some constants $D_1,D_2 >0$ and $N$ large,
\begin{equation} \label{gasymp}
\quad (1+ \epsilon_1(N)) \cdot D_1 \cdot 2^{\overline{j(N)}} \le g_d(N) \le (1+\epsilon_2(N)) \cdot D_2 \cdot 2^{\overline{j(N)}} ,
\end{equation}
where $\epsilon_1(N), \epsilon_2(N) \to 0$ as $N \to \infty$. Combining this with (\ref{down2j}) shows that we have a hyperbolic ordering.
\end{proof}
\end{example}
\subsection{Plotting Tensor Coherences}
Let us consider a simple illustration of this theory applied to a 2D tensor Fourier-Wavelet case $(B^2_\mathrm{f},B^2_\mathrm{w})$. We can identify the 2D Fourier Basis $B^2_\mathrm{f}$ with $\bbZ^2$ using the function $\lambda_2$, so the row incoherences can also be identified with $\bbZ^2$ and therefore they can be imaged directly in 2D, as in Figure \ref{tensorincoherences}.
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensororig}
\caption{\footnotesize Original Coherences}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorscaling}
\caption{\footnotesize Hyperbolic Scaling }
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorscaled}
\caption{\footnotesize Scaled Coherences \\ (Product of (a) \& (b)) }
\end{center}
\end{subfigure}
\end{center}
\caption{2D Tensor Fourier - Tensor Haar Incoherences. We show the subset $\{-250,-249,...,249,250\}^2 \subset \bbZ^2$. Notice that the scaled coherences have no vanishing values (no pure black) and no values that blow up (no pure white, baring the center value which = 1) indicating that we have characterised the coherence in terms of the hyperbolic scaling used. Formally this is shown by equation (\ref{zhypcrosscharacterise}). The coherences shown in the Figure are square rooted to reduce contrast (i.e. we image $ \sqrt{\mu(\pi_N U)}$ instead of $\mu(\pi_N U)$).}
\label{tensorincoherences}
\end{figure}
\section{Multidimensional Fourier - Separable Wavelet Case: Proof of Theorem \ref{separablesummary}} \label{separable}
We repeat the notation of the one-dimensional case, with scaling function $\phi$ (in one dimension) \& Daubechies wavelet $\psi$:
\[ \phi_{j,k}(x) =2^{j/2}\phi(2^j x - k) , \quad \psi_{j,k} (x) = 2^{j/2} \psi(2^j x - k). \]
We can construct a d-dimensional scaling function $\Phi$ by taking the tensor product of $\phi$ with itself, namely
\[ \Phi(x) \ := \ \Big( \bigotimes_{j=1}^d \phi \Big)(x) = \prod_{j=1}^d \phi(x_j), \qquad x \in \mathbb{R}^d, \]
which has corresponding multiresolution analysis $(\tilde{V}_j)_{j \in \mathbb{Z}}$ with diagonal scaling matrix $A \in \mathbb{R}^{d \times d}$ with $A_{i,j}=2 \delta_{i,j}$.
Let $\phi^0:=\phi, \ \phi^1:=\psi$ and for $s \in \{ 0,1 \}^d, \ j \ge J, \ k \in \mathbb{Z}^{d}$ where $J \in \bbN$ is fixed define the functions
\be{ \label{separabledefine}
\Psi^s_{j,d}:= \bigotimes_{i=1}^d \phi^{s_i}_{j,k_i}.
}
If we write (for $s \in \{ 0,1 \}^d \setminus \{0\}, \ j \ge J$)
\[W^s_j := \overline{ \text{Span} \{ \Psi^s_{j,k} : k \in \mathbb{Z}^d \} }.\] Then it follows that
\[ \tilde{V}_{j+1} = \tilde{V}_j \oplus \bigoplus_{s \in \{ 0,1 \}^d \setminus \{0\}} W^s_j, \quad L^2(\mathbb{R}^d) = \overline{\tilde{V}_J \oplus \bigoplus_{\substack{s \in \{ 0,1 \}^d \setminus \{0\} \\ j \ge J}} W^s_j}.\]
This corresponds to taking $2^d-1$ wavelets for our basis in d dimensions (see \cite{dDwav}). As before we take the spanning functions from the above whose support has non-zero intersection with $[-1,1]^d$ as our basis $B_2$ (called a `separable wavelet basis'):
\begin{align} \label{separablewaveletbasisdefine}
B^d_{\text{sep}} := \left\{ \begin{array}{cc} & \mathrm{Supp}(\phi^{s_i}_{j,k_i}) \cap (-1,1) \neq \emptyset \quad \forall i , \\ \Psi^s_{j,k} : & s=0 \Rightarrow j=J, \\ & j \in \mathbb{N}, s \in \{ 0,1 \}^d , \ k \in \mathbb{Z}^d \end{array} \right \} ,
\end{align}
\begin{remark}
We can also construct a separable boundary wavelet basis in the same manner like in the one-dimensional case however, for the sake of simplicity, we stick to the above relatively simple construction throughout (although all the coherence results we cover here also hold for the separable boundary wavelet case as well).
\end{remark}
\subsection{Ordering the Separable Wavelet Basis: Proving Theorem \ref{separablesummary} Part (i)}
\label{separablewaveletordering}
We note a few key equalities from the one-dimensional case that will come in handy:
\be{ \label{1Dequalities}
\mathcal{F}\phi_{j,k}( \omega ) = e^{-2\pi \mathrm{i} 2^{-j} k \omega} 2^{-j/2} \mathcal{F} \phi(2^{-j} \omega), \qquad
\mathcal{F}\psi_{j,k}( \omega ) = e^{-2\pi \mathrm{i} 2^{-j} k \omega} 2^{-j/2} \mathcal{F} \psi(2^{-j} \omega),
}
where $\mathcal{F}$ here denotes the Fourier Transform, i.e. for $f \in L^2(\bbR^d)$ we define
$$
\mathcal{F}f(\omega) = \int_{\mathbb{R}^d} f(x) e^{-2\pi i \omega \cdot x } \, dx, \qquad \omega \in \bbR^d.
$$
Recall $\chi_k$ from Definition \ref{fourier}. We observe that by (\ref{separabledefine})
\begin{equation} \label{fourierprod}
\begin{aligned}
\langle \Psi^s_{j,k} , \chi_n \rangle & = \epsilon^{d/2} \cdot \mathcal{F} \Psi^s_{j,k}(\epsilon n)
= \epsilon^{d/2} \prod_{i=1}^d \mathcal{F} \phi^{s_i}_{j,k_i} (\epsilon n_i), \qquad n \in \bbZ^d,
\\ \Rightarrow & \sup_{n \in \bbN^d} |\langle \Psi^s_{j,k} , \chi_n \rangle|^2 = \epsilon^{d} 2^{-dj} \cdot \prod_{i=1}^d \sup_{n \in \bbN} |\mathcal{F} \phi^{s_i} (\epsilon 2^{-j} n)|^2.
\end{aligned}
\end{equation}
By careful treatment of the product term we can determine the optimal decay of $(B^d_\text{sep}, B_\mathrm{f}^d (\epsilon))$, using the following result:
\begin{proposition} \label{dDSeparableWaveletFourierCharacterisation}
There are constants $C_1,C_2>0$ such that for all $\epsilon \in I_{J,p}, \Psi^s_{j,k} \in B^d_\text{sep}$ we have
\[
C_1 \cdot \epsilon^{d} 2^{-dj} \le \sup_{n \in \bbN^d} |\langle \Psi^s_{j,k} , \chi_n \rangle|^2 \le C_2 \cdot \epsilon^{d} 2^{-dj}.
\]
Consequently, fixing $\epsilon$, the function $F_\text{power}: B^d_\text{sep} \to \bbR$ defined by $F_\text{power}(\Psi^s_{j,k})=2^{-dj}$ characterizes the optimal decay of $(B_\text{sep}^d, B_\mathrm{f}^d(\epsilon))$.
\end{proposition}
\begin{proof}
Let $A=\max(\sup_{\omega \in \bbR} |\mathcal{F} \phi (\omega)|^2, \sup_{\omega \in \bbR} |\mathcal{F} \psi (\omega)|^2)$. Then (\ref{fourierprod}) gives us the upper bound
\[
\sup_{n \in \bbN^d} |\langle \Psi^s_{j,k} , \chi_n \rangle|^2 \le \epsilon^{d} 2^{-dj} \cdot A^d.
\]
This leaves the lower bound. This can be achieved if we can show that there exists constants $D_1, D_2>0$ such that for all $\epsilon \in I_{J,p}$\footnote{Notice that replacing $J$ with $j \ge J$ below would have been redundant.}
\be{ \label{fourierinfimum}
\mathcal{F}_1(\epsilon):=\sup_{n \in \bbN} |\mathcal{F} \phi (\epsilon 2^{-J} n)| \ge D_1, \quad \mathcal{F}_2(\epsilon):= \sup_{n \in \bbN} |\mathcal{F} \psi (\epsilon 2^{-J} n)| \ge D_2.
}
By the Riemann-Lebesgue Lemma the functions $\mathcal{F}_1, \mathcal{F}_2$ are continuous on $I_{J,p}$ and
\[
\mathcal{F}_1(\epsilon) \to \sup_{\omega \in \bbR} |\mathcal{F} \phi (\omega)|>0 \quad \text{as} \quad \epsilon \to 0.
\]
Likewise for $\mathcal{F}_2$. Therefore $\mathcal{F}_1, \mathcal{F}_2$ can be extended to continuous functions over the closed interval $I_{J,p} \cup \{0\}$. Finally we notice that $\mathcal{F}_1(\epsilon)>0, \mathcal{F}_2(\epsilon)>0$ for every $\epsilon \in I_{J,p}$ otherwise we would deduce that $\phi$ or $\psi$ has no support in $[-1,1]$ since the span of $B_\mathrm{f}^d(\epsilon)$ covers $L^2[-1,1]$. This means that the infimums over $I_{J,p} \cup \{0\}$ are attained and are strictly positive, proving (\ref{fourierinfimum}) and the lower bound.
\end{proof}
Let $F_\text{level}:B^d_\text{sep} \to \bbR$ be defined by $F_\text{level}(\Psi^s_{j,k})=j$. Lemma \ref{characterisationlemma} tells that an ordering that is consistent with $1/F_\text{power}$, i.e. consistent with $F_\text{level}$ will be strongly optimal.
\begin{definition} \label{sepleveled}
We say that an ordering $\rho: \mathbb{N} \to B^d_{\text{sep}}$ is `leveled' if it is consistent with $F_\text{level}$.
\end{definition}
\begin{lemma} \label{levelgrowth}
Let $\rho: \mathbb{N} \to B^d_{\text{sep}}$ be leveled. Then there are constants $D_1, D_2>0$ such
\be{ \label{levelgrowthequation}
D_1 \cdot N \le 2^{d F_\text{level}(\rho(N))} \le D_2 \cdot N.
}
\end{lemma}
\begin{proof}
Let $a \in \mathbb{N}$ denote the length of the support of $\phi, \psi$. Notice that for each $j \in \mathbb{N}$ and $s \in \{ 0,1 \}^d$, there are $(2^{j+1} +a-1)^d $ shifts of $\Psi^{s}_{j,0}$ whose support lies in $[-1,1]^d$. For convenience we use the notation $j(N):= F_\text{level}(\rho(N))$ and shall also be using the simple bounds $ 2^{j(N)+1} \le 2^{j(N)+1}+a-1 \le 2^{j(N)+a}$. Now for every $N \in \bbN$ with $j(N)>J$, we must have had all the terms of the form $f \in B^d_{\text{sep}}, F_\text{level}(f)=j(N)-1$ come before $N$ in the leveled ordering and there are at least $(2^d-1) \cdot 2^{dj(N)}$ of these terms, implying that
\[ (2^d-1) \cdot 2^{dj(N)} \le N.\]
This completes the upper bound for $j(N)>J$. Likewise for every $N \in \bbN$ with $j(N)\ge J$ there can be no more than
\[
2^d \cdot \sum_{i=J}^{j(N)} 2^{d(i+a)} \le 2^d \cdot 2^{d(j(N)+a+1)}= 2^{d(a+2)} \cdot 2^{dj(N)},
\]
terms such that $F_\text{level}(f) \le j(N)$. This shows that $N \le 2^{d(a+2)} \cdot 2^{dj(N)}$, completing the upper bound for $j(N)>J$. Extending (\ref{levelgrowthequation}) to all $N \in \bbN$ (i.e. $j(N) \ge J$) is trivial since we have only omitted finitely many terms so a change of constants will suffice.
\end{proof}
\begin{corollary} \label{leveledresults}
Any ordering $\rho$ of $B^d_\text{sep}$ that is leveled is strongly optimal for the basis pair $(B^d_\text{sep},B^d_\mathrm{f}(\epsilon))$. Furthermore, the optimal decay rate of $(B^d_\text{sep},B_\mathrm{f}^d(\epsilon))$ is represented by the function $f(N)=N^{-1}$.
\end{corollary}
\begin{proof}
Lemma \ref{characterisationlemma} applied to Proposition \ref{dDSeparableWaveletFourierCharacterisation} tells us that $\rho$ is strongly optimal and moreover the optimal decay rate is represented by $F_\text{power}(\rho(N))$ which by Lemma \ref{levelgrowth} is of order $N^{-1}$.
\end{proof}
\subsection{Ordering the Fourier Basis: Proving Theorem \ref{separablesummary} Part (ii)} \label{linearproof}
We now want to find the optimal decay rate of $(B^d_\mathrm{f}(\epsilon), B^d_{\text{sep}})$ which means looking at orderings of the Fourier basis. It might be tempting to try and extend the standard ordering definition from the one dimensional Fourier basis. Recall as well that, using the function $\lambda_d$ defined in (\ref{multidimlambda}), ordering $B^d_\mathrm{f}(\epsilon)$ is equivalent to ordering $\bbZ^d$.
If we let $s \in \{ 0,1 \}^d , \ j \in \bbN, \ k \in \bbZ^d$, then in order to bound the coherence $\mu(\pi_NU)$ we need to be bounding terms of the form
\be{ \label{sepprodexample2}
| \langle \Psi^{s}_{j,k} , \lambda_d^{-1} (n) \rangle |^2
= \epsilon^d 2^{-dj} \prod_{i=1}^d | \mathcal{F} \phi^{s_i}(2^{-j} \epsilon n_i) |^2.
}
In the one-dimensional case in \cite{onedimpaper} the following decay property of the Fourier transform of the scaling function $\phi$ was used:
\begin{lemma} \label{FTdecayLemma}
If $\phi$ is any Daubechies scaling function with corresponding mother wavelet $\psi$ then there exists a constant $K>0$ such that for all $\omega \in \bbR \setminus \{0 \}$,
\be{ \label{FTdecay}
| \mathcal{F} \phi(\omega)|, |\mathcal{F} \psi (\omega)| \le \frac{K}{|\omega|}.
}
Furthermore, suppose that for some $\alpha>0$ we have, for some constant $K>0$, the decay $| \mathcal{F} \phi(\omega)| \le K | \omega|^{-\alpha}$ for all $\omega \in \bbR \setminus \{0\}$. Then, for a larger constant $K>0$, $| \mathcal{F} \psi(\omega)| \le K | \omega|^{-\alpha}$ for all $\omega \in \bbR \setminus \{0\}$.
\end{lemma}
\begin{proof}
The first result is a direct result of Lemma 3.5 in \cite{onedimpaper}. The last statement follows immediately from the equality (taken from equation (3.14) in \cite{onedimpaper}):
\begin{equation} \label{fourierscalingwavelet}
|\mathcal{F}\psi(2 \omega)|= |m_0(\omega + 1/2) \cdot \mathcal{F}\phi(\omega)|,
\end{equation}
where $m_0$ is the low pass filter corresponding to $\phi$ which satisfies $|m_0(\omega)| \le 1$ for all $\omega \in \bbR$.
\end{proof}
Therefore let us first consider the case where we use (\ref{FTdecay}) to bound every term in the product, giving us ($n \in \bbZ^d, n_i \neq 0, i=1,...,d$)
\be{ \label{hyperbolicbound}
| \langle \Psi^{s}_{j,k} , \lambda_d^{-1} (n) \rangle |^2 \le \epsilon^d 2^{-dj} \prod_{i=1}^d \frac{K^{2}}{|\epsilon 2^j n_i|} =\frac{K^{2d}}{ \prod_{i=1}^d | n_i|}.
}
Making adjustments to prevent dividing by zero by using $\sup_{\omega \in \bbR} \max(| \mathcal{F} \phi(\omega)|, | \mathcal{F} \psi(\omega)|)\le 1$ (for $\phi$ this follows from Proposition 1.11 in \cite{wav}. We extend this to $\psi$ using equation (\ref{fourierscalingwavelet})), this can then be rephrased as
\be{ \label{hyperbolicbound2}
\sup_{g \in B_\text{sep}^d} | \langle g , \lambda_d^{-1} (n) \rangle |^2 \le \frac{\max (K^{2d},1)}{ \prod_{i=1}^d \max(| n_i|,1)}, \qquad n \in \bbZ^d.
}
This tells us that the function $F_\text{hyp}: \bbZ^d \to \bbR, F_\text{hyp}(n)= (\prod_{i=1}^d \max(|n_i|,1))^{-1}$ dominates the optimal decay of $(B_\mathrm{f}^d, B_\text{sep}^d)$ (see Definition \ref{Characterisation} for the definition of domination).
Therefore if we want to maximise the utility of this bound then we should use an ordering $\sigma$ of $\bbZ^d$ so that $\prod_{i=1}^d \max(|\sigma(N)_i|,1)$ is increasing, namely an ordering corresponding to the hyperbolic cross in $\bbZ^d$ (see Example \ref{hypcrossZd}). However, using such an ordering will not give us the $N^{-1}$ decay rate that we got from the one dimensional case:
\begin{proposition} \label{Hyperbolic4Separable}
Let $\sigma: \bbN \to \bbZ^d$ correspond to the hyperbolic cross in $\bbZ^d$ and define an ordering $\rho$ of $B^d_\mathrm{f}(\epsilon)$ by $\rho:=\lambda_d^{-1} \circ \sigma$, where $\epsilon \in I_{J,p}$. Next let $U=[(B^d_\mathrm{f}(\epsilon),\rho),(B^d_{\text{sep}},\tau)]$ for any ordering $\tau$ and fix $\epsilon$. Then there are constants $C_1, C_2 > 0$
\[ \frac{C_1 \log^{d-1}(N+1)}{N} \le \mu( Q_N U ) \le \frac{C_2 \log^{d-1}(N+1)}{N}, \qquad N \in \bbN. \]
\end{proposition}
As this result is primarily for motivation, its proof is left to the appendix.
Since this approach gives us suboptimal results, we return to our bound of (\ref{sepprodexample2}). Instead of using (\ref{FTdecay}) on every term in the product, why not just use it once on the term that give us the best decay instead? To bound the remaining terms we can simply use $\sup_{\omega \in \bbR} \max(| \mathcal{F} \phi(\omega)|, | \mathcal{F} \psi(\omega)|)\le 1$ . This approach gives us the following bound
\be{ \label{linearbound1}
| \langle \Psi^{s}_{j,k} , \lambda_d^{-1} (n) \rangle |^2 \le \epsilon^d 2^{-dj} \cdot \min_{i=1,...d} \frac{K^{2}}{|\epsilon 2^j n_i|}
=\epsilon^{d-1} 2^{-(d-1)j} \cdot \frac{K^{2}}{ \max_{i=1,...,d} | n_i|}, \qquad n \in \bbZ^d.
}
As we shall see in Lemma \ref{normest}, choosing $\rho$ so that we maximise the growth of the $\max_{i=1,...,d} | n_i|$ leads to $ \max_{i=1,...,d} | n_i| \ge E \cdot N^{1/d}$ for some constant $E>0$ and so (\ref{linearbound1}) is bounded above by $\text{constant} \cdot N^{-1/d}$, which is very poor decay. However, if we instead replace (\ref{FTdecay}) by the stronger condition
\begin{equation} \label{dDFTdecay}
|\mathcal{F} \phi( \omega )| \le \frac{K}{| \omega |^{d/2}} , \qquad \omega \in \bbR \setminus \{0 \} .
\end{equation}
then we can obtain the following upper bound\footnote{noting that (\ref{dDFTdecay}) also holds for $\psi$ by (\ref{fourierscalingwavelet}).}
\be{ \label{linearbound2}
| \langle \Psi^{s}_{j,k} , \rho(N) \rangle |^2 \le \epsilon^d 2^{-dj} \cdot \min_{i=1,...d} \frac{K^{2d}}{|\epsilon 2^j n_i|^d}
= \frac{K^{2d}}{ \max_{i=1,...,d} | n_i|^d}.
}
Let us write $\|n\|_\infty:=\max_{i=1,...,d} | n_i |$. The above can be rephrased as
\be{ \label{linearbound3}
\sup_{g \in B_\text{sep}^d} | \langle g , \lambda_d^{-1}(n) \rangle |^2 \le \frac{\max(K^{2d},1)}{ \max(\|n\|_\infty^d,1)}, \qquad n \in \bbZ^d .
}
Therefore we deduce that $F_\text{lin}: \bbZ^d \to \bbR, F_\text{lin}(n)=(\max(\|n\|_\infty^d,1))^{-1}$ dominates the optimal decay of $(B_\mathrm{f}^d (\epsilon), B_\text{sep}^d)$. In fact in can be shown that $F_\text{lin}$ also \textit{characterizes} the optimal decay (i.e. a lower bound of the same form is possible) by using the following preliminary Lemma:
\begin{lemma} \label{wavelower}
For any compactly supported wavelet $\psi$ there exists an $R \in \mathbb{N}$ such that for all $q \ge R, \ (q \in \mathbb{N})$ we have
\be{ \label{wavelowerequation}
L_q := \inf_{\omega \in [2^{-(q+1)},2^{-q}] } | \mathcal{F}\psi(\omega)| \ > \ 0.
}
\end{lemma}
\begin{proof}
See Lemma 3.6 in \cite{onedimpaper}.
\end{proof}
\begin{proposition} \label{dDSeparableFourierWavelet}
We fix the choice of wavelet basis $B^d_\text{sep}$ and recall the function $\lambda_d : B^d_\mathrm{f} (\epsilon) \to \bbZ^d$ from (\ref{multidimlambda}).
1.) \ \ Then there are constants $C_1(\phi)>0, D(J)>0$ such that for all $\epsilon \in I_{J,p}$ and $n \in \bbZ^d$ with $\| n \|_\infty \ge D \epsilon^{-1}$ we have
\be{ \label{linearbound4}
\sup_{g \in B_\text{sep}^d} | \langle g , \lambda_d^{-1}(n) \rangle |^2 \ge \frac{C_1}{ \|n\|_\infty^d}.
}
Therefore (by fixing $\epsilon$) the function $F_\text{lin}$ is dominated by the optimal decay of $(B^d_\mathrm{f}(\epsilon), B^d_\text{sep})$.
2.) \ \ Suppose that $\phi$ satisfies (\ref{dDFTdecay}). Then there is a constant $C_2(\phi)>0$ such that for all $\epsilon \in I_{J,p}$ and $n \in \bbZ^d$,
\be{ \label{linearbound5}
\sup_{g \in B_\text{sep}^d} | \langle g , \lambda_d^{-1}(n) \rangle |^2 \le \frac{C_2}{ \max(\|n\|_\infty^d,1)}.
}
Therefore (by fixing $\epsilon$) the function $F_\text{lin}$ characterizes the optimal decay of $(B^d_\mathrm{f}(\epsilon), B^d_\text{sep})$.
\end{proposition}
\begin{proof}
2.) \ \ Follows from (\ref{linearbound3}).
1.) \ \ If we set $j= \lceil \log_2 \epsilon \| n \|_\infty \rceil +q$ for some $q \in \bbN$ fixed we observe that $| \epsilon 2^{-j} n_i | \in [0,2^{-q}]$ for every $i=1,...,d$ and, since we are using the max norm, $| \epsilon 2^{-j} n_i | \in [2^{-q-1},2^{-q}]$ for at least one $i$, say $i'$ . Set $s_i=0$ for $i \neq i'$ and $s_{i'}=1$. Then, assuming $j \ge J$, by (\ref{sepprodexample2}) we have the lower bound.
\begin{equation} \label{lastcall}
\begin{aligned}
&| \langle \Psi^s_{j,0} , \lambda_d^{-1}(n) \rangle |^2 \ge \frac{2^{-d(q+1)}}{ \|n\|_\infty^d} \prod_{i=1}^d | \mathcal{F} \phi^{s_i} (\epsilon 2^{-j} n_i)|^2
\\ & \qquad \ge \frac{ 2^{-d(q+1)}}{\|n\|_\infty^d} \cdot \inf_{ \omega \in (2^{-q-1},2^{-q}]}| \mathcal{F} \psi(\omega)|^2
\cdot
\inf_{ \omega \in [0,2^{-q}]}| \mathcal{F} \phi (\omega)|^{2(d-1)}.
\end{aligned}
\end{equation}
Recall that by Lemma \ref{wavelower} there exists a $q \in \mathbb{N}$ such that $L_q>0$ and $\inf_{ \omega \in [0,2^{-q}]}| \mathcal{F} \phi (\omega)| >0 $\footnote{We are using the fact that $|\mathcal{F} \phi(0)|=1$ and continuity of $\mathcal{F} \phi$ here which follows from $\phi \in L^1(\bbR)$.} and therefore (\ref{linearbound5}) follows as long as $j \ge J$.
To ensure that $j= \lceil \log_2 ( \epsilon \|n\|_\infty) \rceil + q$ satisfies $j \ge J$ we must therefore impose the constraint that $n$ is sufficiently large. $j \ge J$ is satisfied if
\[ J \le \log_2 ( \epsilon \|n\|_\infty) \quad \Rightarrow \quad \|n\|_\infty \ge 2^{J} \epsilon^{-1}. \]
\end{proof}
\begin{remark} \label{2doptimal}
If $d=2$ then (\ref{dDFTdecay}) always holds by Lemma \ref{FTdecayLemma}. This means we have characterized every 2D Separable wavelet case (for Daubechies Wavelets).
\end{remark}
\begin{remark}
A similar upper bound in two dimensions based on the norm of $n \in \mathbb{Z}^2$ has already been considered in a discrete framework for separable Haar wavelets \cite{discrete}.
\end{remark}
Let $F_\text{norm}(n):=\max(\|n\|_\infty,1)$ . By \ref{characterisationlemma} we know that if (\ref{dDFTdecay}) holds then the optimal decay of $(B_\mathrm{f}^d,B^d_\text{sep})$ is determined by the fastest growth of $F_\text{norm}$. This motivates the following:
\begin{lemma} \label{normest}
Let $\sigma : \mathbb{N} \rightarrow \mathbb{Z}^d$ be consistent with $F_\text{norm}$. Then there are constants $E_1 , E_2 >0$ such that
\begin{equation} \label{normestresult}
E_1 \cdot N^{1/d} \le \max(\| \sigma(N) \|_\infty,1) \le E_2 \cdot N^{1/d}, \qquad \forall N \in \mathbb{N}.
\end{equation}
\end{lemma}
\begin{proof}
If $ \| \sigma(N) \|_\infty = L \ge 2$, then $\sigma$ must have enumerated beforehand all points $m$ in $ \mathbb{Z}^d$ with $\|m\|_\infty \le L-1$ and there are $(2L -1)^d$ of such points. This means that
\[
N \ge (2 L -1)^d \quad \Rightarrow \quad \| \sigma(N) \|_\infty \le \frac{N^{1/d}+1}{2}, \qquad N \in \bbN.
\]
which proves the upper bound when $ \| \sigma(N) \|_\infty = L \ge 2$. The lower bound is tackled similarly by noting $\sigma$ must first list all $m \in \bbZ^d$ with $\| m \|_\infty \le L$, including $\sigma(N)$ which shows
\[
N \le (2 L +1)^d \quad \Rightarrow \quad \| \sigma(N) \|_\infty \ge \frac{N^{1/d}-1}{2}, \qquad N \in \bbN.
\]
This proves (\ref{normest}) for $ \| \sigma(N) \|_\infty = L \ge 2$. Extending this to all $ N \in \bbN$ is trivial since we have only omitted finitely many terms, so changing the constants will suffice since all terms are strictly positive.
\end{proof}
\begin{definition}[Linear Ordering]
Any ordering $\rho: \bbN \to B_\mathrm{f}^d(\epsilon)$ such that $\sigma=\lambda_d \circ \rho$ satisfies (\ref{normestresult}) is called a `linear ordering'.
\end{definition}
\begin{corollary} \label{linearresults}
Assuming (\ref{dDFTdecay}) holds for the scaling function corresponding to $B^d_\text{sep}$, an ordering $\rho$ of $B_\mathrm{f}^d(\epsilon)$ is strongly optimal for the basis pair $(B_\mathrm{f}^d(\epsilon),B^d_\text{sep})$ if and only if it is linear. Furthermore, the optimal decay rate of $(B_\mathrm{f}^d(\epsilon),B^d_\text{sep})$ is represented by the function $f(N)=N^{-1}$.
\end{corollary}
\begin{proof}
If we apply part 2.) of Proposition \ref{dDSeparableFourierWavelet} to Lemma \ref{characterisationlemma} we kow that if $\sigma: \bbN \to \bbZ^d$ is consistent with $1/F_\text{lin}=F_\text{norm}^d$, i.e. consistent with $F_\text{norm}$, then $F_\text{lin}(\sigma(\cdot))=1/F_\text{norm}^d(\sigma(\cdot))$ represents the optimal decay rate. Lemma \ref{normest} tells us that this optimal decay is $1/(N^{1/d})^d=1/N$. Furthermore, Lemma \ref{characterisationlemma} says that an ordering $\rho$ is strongly optimal for $(B_\mathrm{f}^d(\epsilon),B^d_\text{sep})$ if and only if $F_\text{lin}(\lambda_d \circ \rho(\cdot)) \approx F_\text{lin}(\sigma(\cdot))$ which holds if and only if $F_\text{norm}(\lambda_d \circ \rho(\cdot)) \approx F_\text{norm}(\sigma(\cdot))$, namely $\rho$ is linear.
\end{proof}
Corollary \ref{linearresults} gives us the same optimal decay as in one dimension, which is in contrast to the multidimensional tensor case, where the best we can do is have $d-1$ extra log factors. We can use this result to cover the two dimensional case in full:
\begin{corollary} \label{twodimresults}
In 2D the optimal decay rate of $(B^2_\mathrm{f},B^2_\text{sep})$ is represented by $f(N)= N^{-1}$. This optimal decay rate is obtained by using a linear ordering. In fact an ordering $\rho$ of $B^2_\mathrm{f}$ is strongly optimal in 2D if and only if it is linear.
\end{corollary}
\begin{proof}
Using Lemma \ref{FTdecayLemma} we observe that the decay condition (\ref{dDFTdecay}) holds automatically if $d=2$. Therefore we may apply Corollary \ref{linearresults} directly.
\end{proof}
This result \emph{does not extend to higher dimensions:}
\begin{example} \label{3DHaar}
If we do not have condition (\ref{dDFTdecay}) then our argument can break down very badly:
For Haar wavelets we have an explicit formula for the Fourier transform of the one-dimensional mother wavelet,
\[ \mathcal{F} \phi( \omega) = \frac{ \exp(2 \pi i \omega)-1}{2 \pi i \omega }. \]
Therefore we have that (\ref{dDFTdecay}) is not satisfied for $d \ge 3$ and furthermore we have (for $\epsilon<1$ and $J \in \bbN$ fixed)
\begin{equation} \label{FTdecayfail}
| \mathcal{F} \phi( \epsilon 2^{-J} k ) | \ge \frac{1}{2\pi \epsilon k} ,
\end{equation}
for infinitely many $k \in \mathbb{N}$.
Now consider the case of $d$D separable Haar wavelets with a linear ordering $\rho$ of the Fourier Basis. Then, for $m \in \mathbb{N}$ such that $\lambda_d \circ \rho(m)=(\lambda_d \circ \rho(m)_1,0,\cdots,0)$ we know that by (\ref{FTdecayfail}) there are infinitely many $m$ such that
\be{ \label{poorlowerbound}
\begin{aligned}
| \langle \Phi , \rho(m) \rangle |^2 & = \epsilon^d |\mathcal{F} \phi(\epsilon 2^{-J} \lambda_d \circ \rho(m)_1)|^2 \cdot |\mathcal{F} \phi(0)|^2(d-1)
\\ & \ge \epsilon^d \cdot \frac{1}{(2\pi \epsilon |\lambda_d \circ \rho(m)_1|)^2} \ge \frac{\epsilon^{d-2} E}{4 \pi^2 m^{2/d}},
\end{aligned}
}
for some constant $E$ using Lemma \ref{normest}. Therefore an upper bound of the form $\text{Constant} \cdot N^{-1}$ is not possible for a linear scaling scheme if $d \ge 3$.
This can be rectified by applying a semi-hyperbolic scaling scheme, as in the next subsection.
\end{example}
\subsection{Examples of Linear Orderings - Linear Scaling Schemes} \label{linearsection}
A wide variety of sampling schemes that are commonly used happen to be linear. In particular we demonstrate that sampling according to how a shape scales linearly from the origin always corresponds to a linear ordering (see Figure \ref{linearscalingimages}):
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{shape}
\caption{\footnotesize Scaling shape}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{shapelevels}
\caption{\footnotesize Sampling according to a linear scaling scheme with scaling shape (a)}
\end{center}
\end{subfigure}
\end{center}
\caption{A Simple Linear Scaling Scheme}
\label{linearscalingimages}
\end{figure}
\begin{definition} \label{lineardef}
Let $D \subset \mathbb{R}^d$ be bounded with $0$ in its interior and define $S_{D}: \mathbb{Z}^d: \to \mathbb{R}$
\[ S_{D}(x) := \inf \big\{ \kappa >0 : x \in \kappa D \ \big\}. \]
An ordering $\sigma: \bbN \to \bbZ^d$ is said to `correspond to a linear scaling scheme with scaling shape $D$' if it is consistent with $S_D$. Furthermore, an ordering $\rho : \mathbb{N} \rightarrow B^d_\mathrm{f}(\epsilon)$ is said to `correspond to a linear scaling scheme with scaling shape $D$' if it is consistent with $S_D \circ \lambda_d$.
\end{definition}
\begin{remark} \label{linearnorm}
If we put a norm $\| \cdot \|$ on $\mathbb{Z}^d$ and take an ordering consistent with this norm then this ordering corresponds to a linear scaling scheme with scaling shape $\{ x \in \mathbb{R}^d \ : \ \|x\|=1 \}.$
\end{remark}
\begin{lemma} \label{normestold}
Let $\rho: \bbN \to B^d_\mathrm{f}(\epsilon)$ corresponds to a linear scaling scheme with scaling shape $D$. Then $\rho$ is linear.
\end{lemma}
\begin{proof}
Let $\sigma=\lambda_d \circ \rho$. Because the scaling shape $D$ is bounded and contains $0$ in its interior we have that there exists constants $C_1,C_2>0$ such that $C_1 \mathcal{S} \subset D \subset C_2 \mathcal{S}$ where $\mathcal{S}$ is defined to be the unit hypercube, i.e.
$ \mathcal{S} := \{ x \in \mathbb{R}^d : \|x\|_\infty=1 \}.$
Therefore if $ \| \sigma(N) \|_\infty = L$, then since $D \subset C_2 \mathcal{S}$ we have that $S_D(\sigma(N))\ge L C_2^{-1}$. Applying this to $C_1 \mathcal{S} \subset D$ we deduce that $\sigma$ must have enumerated beforehand all points $m$ in $ \mathbb{Z}^d$ with $\|m\|_\infty<LC_1 C_2^{-1}$ and there are at least $(2 ( L C_1 C_2^{-1} -1 )+1)^d$ of such points. This means that
\[
N \ge (2 ( \| \sigma(N) \|_\infty C_1 C_2^{-1} -1 ) +1)^d \quad \Rightarrow \quad \| \sigma(N) \|_\infty \le \frac{N^{1/d}+1}{2 C_1 C_2^{_1}} \le \frac{N^{1/d}}{C_1 C_2^{-1}}, \qquad N \in \bbN.
\]
which proves the upper bound. The lower bound is tackled similarly to prove (\ref{normestresult}).
\end{proof}
\subsection{2D Separable Incoherences}
By Remark \ref{2doptimal} we have shown that linear orderings are strongly optimal for all 2D Fourier - separable wavelet cases, so this is a good point to have a quick look at a few of these in Figure \ref{separableincoherences}.
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{separablehaarorig}
\caption{\footnotesize Haar Coherences}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{separablehaarscaled}
\caption{\footnotesize Scaled Haar Coherences}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{separablescaling}
\caption{\footnotesize Linear Scaling used for Both Bases}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{separabledauborig}
\caption{\footnotesize Daubechies16 Coherences }
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{separabledaubscaled}
\caption{\footnotesize Scaled \\ Daubechies16 Coherences }
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\end{center}
\end{subfigure}
\end{center}
\caption{2D Fourier - Separable Wavelet Coherences. We show the subset $\{-250,-249,...,249,250\}^2 \subset \bbZ^2$. Notice again that the scaled coherences are bounded above zero and below 1 indicating that we have characterised the incoherence in terms of the linear scaling used, as shown in Proposition \ref{dDSeparableFourierWavelet}. The incoherences shown in the Figure are square rooted to reduce contrast.}
\label{separableincoherences}
\end{figure}
\subsection{Semi-Hyperbolic Orderings: Proof of Theorem \ref{separablesummary} Part (iii)} \label{semihypsection}
By Example \ref{3DHaar} we know that if (\ref{dDFTdecay}) does not hold then our approach of using a linear ordering can fail. We therefore return once more to (\ref{sepprodexample2}). Let us now try to use an approach that is halfway between our two previous linear/hyperbolic approaches. Let $r \in \{1,...,d-1\}$ be fixed. We shall first impose a decay condition that is stronger than (\ref{FTdecay}) but weaker than (\ref{dDFTdecay}):
\be{ \label{semiFTdecay}
|\mathcal{F}\phi (\omega)| \le \frac{K}{| \omega|^{d/2r} }, \qquad \omega \in \mathbb{R} \setminus \{ 0 \} .
}
Instead of just taking out the dominant term of the product in (\ref{sepprodexample2}), let us take out the $r$ smallest terms:
\be{ \label{semihypbound}
\begin{aligned}
| \langle \Psi^{s}_{j,r} , \lambda_d^{-1}(n) \rangle |^2 & \le \epsilon^d 2^{-dj} \cdot \min_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}} \prod_{r=1}^r \frac{K^{2}}{|\epsilon 2^j n_{i_r}|}^{d/r}
\\ & = K^{2r} \cdot \Big( \max_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}} \prod_{r=1}^r | n_{i_r}| \Big)^{-d/r}, \qquad n \in \bbZ^d, n_i \neq 0, i=1,...,d.
\end{aligned}
}
Again we can extend this bound to all $n \in \bbZ^d$:
\be{ \label{semihypbound2}
\sup_{g \in B_\text{sep}^d} | \langle g , \lambda_d^{-1}(n) \rangle |^2 \le \max(K^{2r},1) \cdot \Big( \max_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}} \prod_{r=1}^r \max(| n_{i_r}|,1) \Big)^{-d/r}, \qquad n \in \bbZ^d.
}
We deduce that the function $F_{\text{hyp},r}: \bbZ^d \to \bbR, F_{\text{hyp},r}(n)=\Big( \max_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}} \prod_{r=1}^r \max(| n_{i_r}|,1) \Big)^{-d/r}$ dominates the optimal decay of of $(B_\mathrm{f}^d, B_\text{sep}^d)$.
\begin{definition} \label{semihyperbolic}
Let us define, for $r,d \in \bbN, r \le d$ the function
\[ H_{d,r}(n):= \max_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}}
\prod_{j=1}^r \max(|n_{i_j}|,1) , \qquad n \in \bbZ^d.
\]
Then we say an ordering $\sigma: \bbN \to \bbZ^d$ is `semi-hyperbolic of order $r$ in $d$ dimensions' if it is consistent with $H_{d,r}$.
\end{definition}
Figure \ref{Consistent3D} presents some isosurface plots of $H_{3,r}$ for the various values of $r$
Notice that a semi-hyperbolic ordering of order d in d dimensions corresponds to the hyperbolic cross in $\bbZ^d$ (see Example \ref{hypcrossZd}). Furthermore, if $\sigma: \bbN \to \bbZ^d$ is a semi-hyperbolic ordering of order $1$ in $d$ dimensions then, by Remark \ref{linearnorm}, $\sigma$ corresponds to a linear scaling scheme because $H_{d,1}(n)= \|n\|_\infty$ for the componentwise max norm $\| \cdot \|_\infty$ on $\bbR^d$. Like in the linear and hyperbolic cases discussed in the previous sections, we want to determine how $H_{d,r}(\sigma(n))$ scales with $n \in \bbN$.
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Linear3DConsistent}
\caption{\footnotesize Case $r=1$ (Linear) ; \\ Isosurface value=10. }
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{SemiHyp3DConsistent}
\caption{\footnotesize Case $r=2$ (Semi-Hyperbolic) ; \\ Isosurface value=20. }
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Hyp3DConsistent}
\caption{\footnotesize Case $r=3$ (Hyperbolic) ; \\ Isosurface value=20. }
\end{center}
\end{subfigure}
\end{center}
\caption{Isosurfaces of $H_{3,r}$, $r=1,2,3$ describing the three types of ordering available in 3D}
\label{Consistent3D}
\end{figure}
\begin{lemma} \label{semihypbehaviour}
1). \ \ Let $r,d\in \bbN, r \le d-1$ be fixed. Let us define
\[ S_{d,r}(n):= \# \{ m \in \bbZ^d : H_{d,r}(m) \le n \}, \qquad n \in \bbN.\]
Then there is a constant $C>0$ such that
\[ n^{d/r} \le S_{d,r}(n) \le C \cdot n^{d/r}, \qquad n \in \mathbb{N}. \]
2). \ \ If $\sigma: \bbN \to \bbZ^d$ is semi-hyperbolic of order $r$ with $r \le d-1$ then there are constants $C_1,C_2>0$ such that
\[ C_1 \cdot n^{r/d} \le H_{d,r}(\sigma(n)) \le C_2 \cdot n^{r/d}, \qquad n \in \bbN. \]
\end{lemma}
\begin{proof}
1). \ \ For notational simplicity we prove the same bounds but with $S_{d,r}$ replaced by the smaller set
\[ \tilde{S}_{d,r}(n):= \# \{ m \in \bbN^d : H_{d,r}(m) \le n \}, \qquad n \in \bbN. \]
The same bounds for $S_{d,r}$ then follows immediately, albeit with a larger constant $C>0$. The lower bound is straightforward since the set defining $\tilde{S}_{d,r}(n)$ contains the set $\{ m \in \bbN^d : m_i \le n^{1/r}, \ i=1,..,d \}$.
We prove the upper bound by induction on $r$. The case $r=1$ is clear because $\tilde{S}_{d,1}(n)$
is simply the number of points inside a $d$-dimensional hypercube with side length $n$. Suppose the result holds for $r=r'-1$. We use the following set inclusion:
\be{
\begin{aligned}
\{ m \in \bbZ^d : H_{d,r'}(m) \le n \} \subset & \{ m \in \bbN^d : m_i \le n^{1/r'}, \ i=1,..,d \}
\\ & \cup \bigcup_{i=1}^d \{m \in \bbN^d : n^{1/r'} \le m_i \le n, \ H_{d-1,r'-1}(\tilde{m}_i) \le n/m_i \},
\end{aligned}
}
where $\tilde{m}_i$ here refers to $m$ with the $i$th entry removed. The cardinality of the first set on the right is just $n^{d/r'}$ and so we are done if we can show that for some constant $C>0$,
\[ \# \{m \in \bbN^d : n^{1/r'} \le m_1 \le n, \ H_{d-1,r'-1}((m_2,...,m_d)) \le n/m_1 \} \le C n^{d/r'}, \qquad n \in \bbN \]
We achieve this by applying our inductive hypothesis:
\be{
\begin{aligned}
\# \{m & \in \bbN^d : n^{1/r'} \le m_1 \le n, \ H_{d-1,r'-1}((m_2,...,m_d)) \le n/m_1 \} \\
& \le \sum_{ i= \lfloor n^{1/r'} \rfloor} ^n S_{d-1,r'-1}(\lfloor n/i \rfloor) \le C' \cdot \sum_{ i= \lfloor n^{1/r'} \rfloor}^n \big( n/i \big)^{(d-1)/(r'-1)}
\\ & \le C'n^{(d-1)/(r'-1)} \cdot \int_{n^{1/r'}-2}^n x^{-(d-1)/(r'-1)} \, dx
\\ & \le C'n^{(d-1)/(r'-1)} \cdot (n^{1/r'}-2)^{(1-(d-1)/(r'-1))}, \qquad ( \text{noting } r' \le d-1)
\end{aligned}
}
where $C'>0$ is some constant. We can replace $(n^{1/r'}-2)$ by $n^{1/r'}$ in the above by changing the constant $C'$ and assuming $n>2^{r'}$. Finally, we notice that the exponents add to the desired expression:
\[ \frac{d-1}{r-1} + \frac{1}{r'}\Big( 1- \frac{d-1}{r'-1} \Big)= \frac{d-1}{r'-1} - \frac{ d-r'}{r'(r'-1)} = \frac{d}{r'}.\]
This gives the required upper bound for $n>2^{r'}$. Since the terms involved are all positive, we can just increase the constant $C'$ to include the cases $n \le 2^{r'}$. This shows that the result holds for $r=r'$ and the induction argument is complete.
2.) \ \ By consistency we know that
\[ S_{d,r}(H_{d,r}(\sigma(n))-1) \le n \le S_{d,r}(H_{d,r}(\sigma(n))), \qquad n \in \bbN \]
and therefore we can directly apply part 1 to deduce
\[ ( H_{d,r}(\sigma(n))-1)^{d/r} \le n \le C \cdot ( H_{d,r}(\sigma(n)))^{d/r}, \qquad n \in \bbN, \]
from which the result follows.
\end{proof}
Armed with this result, we can now completely tackle the separable wavelet case.
\begin{theorem} \label{semihyperbolicthm}
Suppose that the scaling function $\phi$ corresponding to the separable wavelet basis $B^d_{\text{sep}}$, satisfies (\ref{semiFTdecay}) for some constant $K\ge 0$ and $r \in \{1,...,d-1\}$.
Next let $\sigma: \bbN \to \bbZ^d$ be semi-hyperbolic of order $r$ in $d$ dimensions and $\rho:=\lambda_d^{-1} \circ \sigma$. Finally, we let $U=[(B^d_\mathrm{f}(\epsilon),\rho),(B^d_{\text{sep}}, \tau)]$, where $\tau$ is an ordering of $B^d_{\text{sep}}$. Let us also fix $\epsilon \in I_{J,p}$. Then there are constants $C_1, C_2$ such that
\[ \frac{C_1}{N} \le \mu(Q_N U) \le \frac{C_2}{N}, \quad N \in \bbN. \]
Furthermore it follows that the ordering $\rho$ is optimal for the basis pair $(B^d_\mathrm{f}(\epsilon),B^d_\text{sep})$.
\end{theorem}
\begin{proof}
Applying part 1.) from Proposition \ref{dDSeparableFourierWavelet} (with $\epsilon$ fixed) to part 2.) of Lemma \ref{characterisationlemma} immediately gives us the lower bound for the semihyperberbolic ordering since this bound also holds for the optimal decay rate. Furthermore this lower bound holds for any other ordering and therefore if we have the upper bound then the ordering $\rho$ is automatically optimal. We now focus on the upper bound.
By (\ref{semihypbound2}) we know that the optimal decay of $(B_\mathrm{f}^d(\epsilon),B^d_\text{sep})$ is dominated by $F_\text{hyp,r}$. Therefore by part 1.) of Lemma \ref{characterisationlemma} if $\sigma: \bbN \to \bbZ^d$ is consistent with $1/F_\text{hyp,r}$, i.e. $\sigma$ is semihyperbolic of order $r$ then we can bound the row incoherence $\mu(\pi_N U)$ by $F_\text{hyp,r}(\sigma(N))= H_{d,r}^{-d/r}(\sigma(N)) \approx (N^{r/d})^{-d/r}=N^{-1}$ by Lemma \ref{semihypbehaviour}. Since $N^{-1}$ is decreasing this bound extends to $\mu(Q_N U)$.
\end{proof}
Finally we can summarise our results on the $(B_\mathrm{f}^d (\epsilon),B^d_{\text{sep}})$ case as follows:
\begin{theorem} \label{SeparableResults}
Let $\rho$ be a Linear ordering of the d-dimensional Fourier basis $B_\mathrm{f}^d(\epsilon)$ with $\epsilon \in I_{J,p}$, $\tau$ a leveled ordering of the d-dimensional separable wavelet basis $B^d_{\text{sep}}$ and $U=[(B_\mathrm{f}^d(\epsilon),\rho),(B^d_{\text{sep}},\tau)]$. Furthermore, suppose that the decay condition (\ref{FTdecay}) holds for the wavelet basis. Then, keeping $\epsilon>0$ fixed, we have, for some constants $C_1, C_2>0$ the decay
\be{ \label{dDSeparableFourierPolynomialLinearBounds}
\frac{C_1}{N} \le \mu(\pi_N U), \ \mu(U \pi_N), \le \frac{C_2}{N} \qquad \forall N \in \mathbb{N}.
}
Let us now instead replace $\rho$ by a semi-hyperbolic ordering of order $r$ in $d$ dimensions with $r \in \{1,....,d-1\}$ and assume the weaker decay condition (\ref{semiFTdecay}). Then, keeping $\epsilon>0$ fixed, we have, for some constants $C_1, C_2>0$ the decay
\be{ \label{dDSeparableFourierPolynomialSemiHypBounds}
\frac{C_1}{N} \le \mu(Q_N U), \ \mu(U \pi_N), \le \frac{C_2}{N} \qquad \forall N \in \mathbb{N},
}
and furthermore $\rho$ is optimal for the basis pair $(B_\mathrm{f}^d(\epsilon),B^d_{\text{sep}})$. Since, for any separable Daubechies wavelet basis, (\ref{semiFTdecay}) always holds for $r=d-1$ any semi-hyperbolic ordering of order $d-1$ in $d$ dimensions will produce (\ref{dDSeparableFourierPolynomialSemiHypBounds}).
\end{theorem}
\begin{proof}
(\ref{dDSeparableFourierPolynomialLinearBounds}) follows from Corollaries \ref{leveledresults} and \ref{linearresults}. (\ref{dDSeparableFourierPolynomialSemiHypBounds}) follows from Corollary \ref{leveledresults} and Theorem \ref{semihyperbolicthm}. To show that (\ref{dDSeparableFourierPolynomialSemiHypBounds}) always holds for a $d-1$ degree semi-hyperbolic ordering in $d$ dimensions, we note that the weakest decay on the scaling function $\phi$ is $| \mathcal{F} \phi (\omega)| \le K \cdot | \omega |^{-1}$ (see Lemma \ref{FTdecayLemma}) and therefore (\ref{semiFTdecay}) is automatically satisfied for $r=d-1$.
\end{proof}
\subsection{Optimal Orderings \& Wavelet Smoothness} \label{orderingsandsmoothness}
Theorem \ref{SeparableResults} demonstrates how certain degrees of smoothness, in terms of decay of the Fourier transform, allows us to show certain orderings are optimal and this smoothness requirement becomes increasingly more demanding as the dimension increases. But if a certain ordering is optimal for the basis pair $(B_\mathrm{f}^d, B^d_\text{sep})$, does this mean that the wavelet must also have some degree of smoothness as well? The answer to this question turns out to be yes, and it is the goal of this section to prove this result.
We shall rely heavily on the following simple result from \cite[Thm. 9.4]{korner}:
\begin{theorem} \label{kornertheorem}
Let\footnote{$\bbR/\bbZ$ denotes the unit circle which we write as $[0,1)$ with the quotient topology induced by $M: \bbR \to [0,1), M(x)=x (\text{mod} \ 1)$.} $f: \bbR/\bbZ \to \bbC$ be continuous and for $k \in \bbZ$ define $\hat{f}(k)= \int_0^1 f(x) \exp(2 \pi \mathrm{i} k x) \, dx$. If $\sum_{k=-\infty}^\infty |k||\hat{f}(k)| < \infty$ then $f \in C^1$. Consequently, using $\widehat{f'}(k)=(2 \pi \mathrm{i} k)^{-1} \cdot \hat{f}(k)$, if $\sum_{k=-\infty}^\infty |k|^n|\hat{f}(k)| < \infty$ then $f \in C^n$.
\end{theorem}
Now the main result itself:
\begin{theorem} \label{ordering2smooth}
Let $\sigma : \bbN \to \bbZ^d$ be semihyperbolic of order $r<d$ in $d$ dimensions and let $\rho:= \lambda_d^{-1} \circ \sigma : \bbN \to B_\mathrm{f}^d(\epsilon)$ where $\epsilon \in I_{J,p}$. Then if $\rho$ is optimal for the basis pair $(B_\mathrm{f}^d(\epsilon),B_\text{sep}^d)$ then $\phi \in C^l$ for any $l \in \bbN \cup \{0\}$ with $l+1 < d/2r$.
\end{theorem}
\begin{proof}
By Theorem \ref{SeparableResults} we know that the optimal decay rate for the basis pair is $N^{-1}$, therefore if $\rho$ is optimal we must have, for some constant $C_1>0$, the bound
\[
\sup_{g \in B^d_\text{sep}} | \langle g, \lambda_d^{-1} \circ \sigma(N) \rangle |^2 \le C_1 \cdot N^{-1}, \qquad N \in \bbN.
\]
Next since $\sigma$ is semihyperbolic we also know that, by Lemma \ref{semihypbehaviour}, there is a constant $C_2>0$ such that
\[
H_{d,r}(\sigma(N)) \le C_2 \cdot N^{r/d}, \qquad N \in \bbN.
\]
Consequently we deduce,
\be{ \label{conversion}
\begin{aligned}
\sup_{g \in B^d_\text{sep}} | \langle g, \lambda_d^{-1} \circ \sigma(N) \rangle |^2 \le C_1 \cdot N^{-1} \le C_1 C_2^{-d/r} \cdot H^{-d/r}_{d,r}(\sigma(N)), \qquad N \in \bbN.
\\
\Rightarrow \sup_{g \in B^d_\text{sep}} | \langle g, \lambda_d^{-1}(n) \rangle |^2 \le \frac{C_1 C_2^{-d/r}}{ \big( \max_{\substack{i_1,...,i_r \in \{1,...,d\} \\ i_1<...<i_r}}
\prod_{j=1}^r \max(n_{i_j},1) \big)^{d/r}}, \qquad n \in \bbZ^d.
\end{aligned}
}
Letting $g=\Psi^s_{J,0}$ where $s=\{0,...,0\}$ and $n=(k,0,...,0)$ for $k \in \bbZ$ we see that (\ref{conversion}) becomes
\be{ \label{phiprior}
\epsilon^d 2^{-dJ} | \mathcal{F}\phi(2^{-J} \epsilon k) |^2 \le \frac{C_1 C_2^{-d/r}}{\max(|k|,1)^{d/r}}, \qquad k \in \bbZ.
}
Since the scaling function $\phi$ has compact support in $[-p+1,p]$ and $\epsilon \in I_{J,p}$, $\phi_{J,0}$ can be viewed as a function on $\bbR/\bbZ$ and (\ref{phiprior}) describes a bound on the Fourier coefficients of $\phi$. Formally, if we write $\varphi(x):=\phi(2^J \epsilon^{-1}(x-1/2))$, then since $\epsilon \in I_{J,p}$ we have that $\varphi$ is supported in $[0,1]$ and (\ref{phiprior}) becomes, for some constant $D(\epsilon,J,p)>0$:
\[
| \mathcal{F}\varphi(k) |^2 = | \widehat{\varphi}(k)|^2 \le \frac{D}{\max(|k|,1)^{d/r}}, \qquad k \in \bbZ.
\]
If $\phi \in C^{0}$ then the result follows from Theorem \ref{kornertheorem}. If $\phi \notin C^0$, i.e. $\phi$ corresponds to a Haar wavelet basis, then (\ref{phiprior}) cannot hold with $d/2r > 1$ as this would contradict (\ref{FTdecayfail}).
\end{proof}
\begin{corollary} \label{hyperbolictendency}
Let the scaling function $\phi$ corresponding to the Daubechies wavelet basis $B^d_\text{sep}$ be fixed. Then for every order $r \in \bbN$, there exists a dimension $d' \in \bbN, d'>r$ such that for all $d \ge d'$, we have that a semihyperbolic ordering $\sigma $ of order $r$ in $d$ dimensions is such that $\rho= \lambda_d^{-1} \circ \sigma$ is not optimal for the basis pair $(B_\mathrm{f}^d(\epsilon),B^d_\text{sep})$.
\end{corollary}
\begin{proof}
If this result was not true then we would deduce by Theorem \ref{ordering2smooth} that the wavelet $\phi$ satisfies $\phi \in C^{\infty}$, which is a contradiction because no compactly supported wavelet can be infinitely smooth \cite[Thm. 3.8]{wav}.
\end{proof}
\subsection{Hierarchy of Semihyperbolic Orderings}
One other notable point from Theorem \ref{SeparableResults} is that we can have multiple values of $r$ such that if $\sigma$ is semi-hyperbolic of order $r$ in $d$ dimensions then $\rho= \lambda_d^{-1} \circ \sigma$ is optimal for the basis pair $(B_\mathrm{f}^d, B^d_\text{sep})$, so which one should we choose? We know that in the case of sufficient smoothness linear orderings are strongly optimal and therefore this suggests that the lower the order $r$ the stronger the optimality result. We now seek to prove this conjecture.
\begin{lemma} \label{semihypinequality}
Let $r,r',d \in \bbN, r \le r' \le d$. Then for all $n \in \bbZ^d$ we have that $H_{d,r'}^r(n) \le H_{d,r}^{r'}(n)$.
\end{lemma}
\begin{proof}
Let $n \in \bbZ^d$ be fixed. For each $j=1,..d$ let $i_j$ denote the $j$th largest terms of the form $\max(|n_{i_j}|,1)$. Observe that
\[
H_{d,r}(n)= \prod_{j=1}^r \max(|n_{i_j}|,1), \quad H_{d,r'}(n)= \prod_{j=1}^{r'} \max(|n_{i_j}|,1),
\]
\[
\Rightarrow \frac{ H_{d,r}^{r'}(n)}{H_{d,r'}^r(n)} = \frac{\prod_{j=1}^r \max(|n_{i_j}|,1)^{r'-r}}{\prod_{j=r+1}^{r'} \max(|n_{i_j}|,1)^r}.
\]
Finally we observe that the numerator and denominator have the same number ($r(r'-r)$) of terms in the product and that each term in the numerator is greater than each term in the denominator, proving the inequality.
\end{proof}
\begin{corollary} \label{hierarchycorollary}
Let $r,r',d \in \bbN, r \le r' < d$ and $\sigma, \sigma'$ be semihyperbolic of orders $r,r'$ in $d$ dimensions respectively. If $\rho=\lambda_d^{-1} \circ \sigma$ is optimal for the basis pair $(B_\mathrm{f}^d(\epsilon), B_\text{sep}^d)$ then so is $\rho'=\lambda_d^{-1} \circ \sigma'$.
\end{corollary}
\begin{proof}
Recalling (\ref{conversion}) we know that there is a constant $C>0$ such that
\[
\sup_{g \in B^d_\text{sep}} | \langle g, \lambda_d^{-1} \circ \sigma(N) \rangle |^2 \le C \cdot H^{-d/r}_{d,r}(\sigma(N)), \qquad N \in \bbN,
\]
\[
\Rightarrow \sup_{g \in B^d_\text{sep}} | \langle g, \lambda_d^{-1} \circ \sigma'(N) \rangle |^2 \le C' \cdot H^{-d/r'}_{d,r'}(\sigma'(N)), \qquad N \in \bbN,
\]
where we have used Lemma \ref{semihypinequality} on the second line. We then apply Lemma \ref{semihypbehaviour} to deduce the result.
\end{proof}
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.49\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Daub43d}
\caption{\footnotesize Daubechies4 - Isosurface Value $ = 5 \cdot 10^{-3}$}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Daub83d}
\caption{\footnotesize Daubechies8 - Isosurface Value $ = 5 \cdot 10^{-3}$}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Haar3d}
\caption{\footnotesize Haar - Isosurface Value $ = 5 \cdot 10^{-3}$}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Daub45d}
\caption{\footnotesize 3D slice of 5D Daubechies4 - Isosurface Value $ = 5 \cdot 10^{-4}$ }
\end{center}
\end{subfigure}
\end{center}
\caption{3D Fourier - Separable Wavelet Incoherence Isosurface Plots. We draw the isosurface plots over the subset $\{-50,-49,...,49,50\}^3 \subset \bbZ^3$. These pictures should be compared with the ordering plots in Figure \ref{Consistent3D}. Notice that for the smoother wavelets in (a) \& (b), the growth matches that of a linear ordering however the 3D Haar case lacks this smoothness, resulting in semi-hyperbolic scaling in (c). If we keep the wavelet basis fixed and let the dimension increase, the scaling becomes increasingly hyperbolic, as seen in (d) and proved in Corollary \ref{hyperbolictendency}. }
\label{3dseparableincoherences}
\end{figure}
\begin{remark}
Corollary \ref{hierarchycorollary} tells us that if there are several orders $r$ that give us optimality then the smallest $r$ possible, say $r^*$, is the strongest result.
\end{remark}
\subsection{3D Separable Incoherences}
We have found optimal orderings for every multidimensional Fourier- separable wavelet case however, we have not shown that (apart from in the linear case with sufficient Fourier decay) that the ordering is strongly optimal and we have not characterized the decay. Therefore it is of interest to see how the incoherence scales in further detail by directly imaging them in 3D. We do this by drawing levels sets in $\bbZ^3$, as seen in Figure \ref{3dseparableincoherences}.
\section{Asymptotic Incoherence and Compressed Sensing in Levels} \label{numericalsection}
We now return to the original compressed sensing problem which was described in the introduction of this paper and aim to study how asymptotic incoherence can influence the ability to subsample effectively. We shall be working exclusively in 2D for this section.
Consider the problem of reconstructing a function $f \in L^2([-1,1]^2)$ from its samples $\{ \langle f, g \rangle : g \in B^2_\mathrm{f} (2^{-1}) \} $. The function $f$ is reconstructed as follows: Let $U:=[(B^2_\mathrm{f}(2^{-1}),\rho), (B_2, \tau)]$ for some orderings $\rho, \tau$ and a reconstruction basis $B_2$. The number $2^{-1}$ is present here to ensure the span of $B_\mathrm{f}$ contains $L^2([-1,1]^2)$. Next let $\Omega \subset \bbN$ denote the set of subsamples from $B^2_\mathrm{f}(2^{-1})$ (indexed by $\rho$), $P_\Omega$ the projection operator onto $\Omega$ and $\hat{f}:=( \langle f , \rho(m) \rangle )_{m \in \bbN}$. We then attempt to approximate $f$ by $\sum_{n=1}^\infty \tilde{x}_n \tau(n)$ where $\tilde{x} \in \ell^1(\bbN)$ solves the optimisation problem
\be{ \label{basicl1full}
\min_{x \in \ell^1(\bbN)} \| x \|_1 \quad \text{subject to} \quad P_\Omega Ux= P_\Omega \hat{f} .
}
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphant}
\caption{\footnotesize Rasterized Phantom \\ Resolution = $2^{12} \times 2^{12}$}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantbasicrecon}
\caption{\footnotesize Reconstruction from pattern A \\ $L^1$ error = 0.0735}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantleveledrecon}
\caption{\footnotesize Reconstruction from pattern B \\ $L^1$ error = 0.0620}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantzoom}
\caption{\footnotesize Rasterized Phantom - Closeup}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantbasicreconzoom}
\caption{\footnotesize Closeup of (b)}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantleveledreconzoom}
\caption{\footnotesize Closeup of (c)}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantbasiclist}
\caption{\footnotesize Sampling Pattern A \\ Number of Samples: 40401}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{resphantleveledlist}
\caption{\footnotesize Sampling Pattern B \\ Number of Samples: 39341}
\end{center}
\end{subfigure}
\end{center}
\caption{Simple Resolution Phantom Experiment. Samples are from the subset $\{-200,-199,...,199,200\}^2 \subset \bbZ^2$. Notice that the checkerboard feature are captured by the leveled sampling pattern but not by pattern (a), even though it uses fewer samples. Reconstructions are at a resolution of $2^{10} \times 2^{10}$.}
\label{resphantimages}
\end{figure}
Since the optimisation problem is infinite dimensional we cannot solve it numerically so instead we proceed as in \cite{BAACHGSCS} and truncate the problem, approximating $f$ by
$\sum_{n=1}^R \tilde{x}_n \tau(n)$ (for $R \in \bbN$ large) where $\tilde{x} = (\tilde{x}_n)_{n=1}^R$ now solves the optimisation problem
\be{ \label{basicl1}
\min_{x \in \bbC^R} \| x \|_1 \quad \text{subject to} \quad P_\Omega U P_R x= P_\Omega \hat{f} .
}
We shall be using the SPGL1 package \cite{SPGL} to solve (\ref{basicl1}) numerically.
\subsection{Demonstrating the Benefits of Multilevel Subsampling}
We shall first demonstrate directly how subsampling in levels is beneficial in situations with asymptotic incoherence ($\mu(Q_N U) \to 0$) but poor global incoherence ($\mu(U)$ is relatively large). The image $f$ that we will attempt to reconstruct is made up of regions defined by Bezier curves with one degree of smoothness, as in \cite{GLPU}. This image is intended is model a resolution phantom\footnote{`resolution' here refers to `resolving' a signal from a MRI device.} which is often used to calibrate MRI devices \cite{resphantom}. A rasterization of this phantom is provided in image (a) of Figure \ref{resphantimages}.
We reconstruct with 2D separable Haar wavelets, ordered according to its resolution levels, from a base level of 0 up to a highest resolution level of 8. The Fourier basis is ordered by the linear consistency function $H_{2,1}$, which gives us a square leveling structure when viewed in $\bbZ^2$. We choose these orderings because we know that they are both strongly optimal for the corresponding bases, and therefore should allow reasonable degrees of subsampling when given an (asymptotically) sparse problem.
By looking at Figure \ref{resphantimages}, we observe that subsampling in levels (pattern (b)) allows to pick up features that would be otherwise impossible from a direct linear reconstruction from the first number of samples (pattern (a)) and moreover the $L^1$ error is smaller.
\subsection{Tensor vs Separable - Finding a Fair Comparison}
We would like to study how different asymptotic incoherence behaviours can impact how well one can subsample. In 2D it would be unwise to compare 2 different separable wavelet bases, since we know that they have the same optimal orderings and decay rates in 2D (see Corollary \ref{linearresults}). Therefore we are left with comparing a separable wavelet basis to a tensor basis. The incoherence decay rates for the 2D Haar cases are shown in the table below for Linear and Hyperbolic orderings of the Fourier basis $B^2_\mathrm{f}$:
\begin{table}[H] \label{incoherencetable2D}
\begin{center}
\large
\begin{tabular}{c|cc}
\hline
\multicolumn{3}{c}{2D Haar Basis Incoherence Decay Rates} \\
\hline
Ordering & Tensor & Separable \\
\hline
Linear & $ N^{-1/2} $ & $ N^{-1} $ \\
Hyperbolic & $\log(N+1) \cdot N^{-1}$ & $\log(N+1) \cdot N^{-1} $ \\
\hline
\end{tabular}
\end{center}
\caption{ \footnotesize The decay rates for the hyperbolic case comes from Theorem \ref{TensorResultsWavelet} and Proposition \ref{Hyperbolic4Separable}. For the linear case, the separable result comes from Theorem \ref{SeparableResults} and the tensor result can be deduced from Lemma \ref{normest} applied to (\ref{zhypcrosscharacterise}), although we do not provide the details here.
}
\end{table}
Observe that for linear orderings, there is a large discrepancy between the decay rates, however they are the same for hyperbolic orderings. Therefore, comparing separable and tensor reconstructions appears to be a good method for testing the behaviour of differing speeds of asymptotic incoherence.
However, there is one serious problem, namely the choice of image $f$ that we would like to reconstruct. Recall from (\ref{conditions31_levels}) that the ability to subsample depends on both the coherence structure of the pair of bases and the sparsity structure of the function $f$ we are trying to reconstruct. Ideally, to isolate the effect of asymptotic incoherence we would like to choose an $f$ that has the same sparsity structure in both a tensor and separable wavelet basis. If $f$ was chosen to be the resolution phantom like before then the tensor wavelet approximation would be a poor comparison to that of the separable wavelet reconstruction (due to a poor resolution structure). Therefore we need to choose a function that we expect to reconstruct well in tensor wavelets, for example a tensor product of one dimensional functions.
\begin{figure}[!t]
\begin{center}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{spectrumorig}
\caption{\footnotesize Rasterized Spectrum}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{spectrumfullsep}
\caption{\footnotesize Separable Reconstruction \\ $L^1$ error = 0.0157}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{spectrumfulltensor}
\caption{\footnotesize Tensor Reconstruction \\ $L^1$ error = 0.0159}
\end{center}
\end{subfigure}
\end{center}
\caption{Spectrum Model and `Full Sampling' Reconstructions. Reconstructions uses all samples from the subset $\{-200,-199,...,199,200\}^2 \subset \bbZ^2$. Images are at a resolution of $2^{10} \times 2^{10}$. Haar wavelets are used for tensor and separable cases. Observe that both reconstructions match the original very closely and have similar $L^1$ approximation errors.}
\label{spectrumsetup}
\end{figure}
Such an example is provided by NMR spectroscopy \cite[Eqn. (5.24)]{spindynamics}. A 2D spectrum is sometimes modelled as a product of 1D Lorentzian functions:
\be{ \label{spectrumdefine}
\begin{aligned}
f(x) & = \sum_{i=1}^r L_{2,p(i),s(i)}(x), \quad x,p(i),s(i) \in \bbR^2,
\\
L_{2,p,s}(x) & = L_{p_1,s_1}(x_1) \cdot L_{p_2,s_2}(x_2), \quad x,p,s \in \bbR^2
\\
L_{p,s} & = \frac{s}{s^2 + (x -p)^2}, \quad x,p,s \in \bbR.
\end{aligned}
}
We consider a specific spectrum $f$ of the above form. By looking at Figure \ref{spectrumsetup} we observe that, without any subsampling from the subset $\{-200,-199,...,199,200\}^2 \subset \bbZ^2$, the tensor and separable Haar wavelet reconstructions have almost identical $L^1$ errors, suggesting that this problem does not bias either reconstruction basis. We order the tensor and separable reconstruction bases using their corresponding level based orderings, which are defined in Lemma \ref{tensorwavelethyp} and Definition \ref{sepleveled} respectively. For separable wavelets we start at a base level of $J=0$ and stop at level 8 (so we truncate at the first $2^{10} \times 2^{10}$ wavelet coefficients) and for tensor wavelets we start at level
$J=0$ and stop at level 10 (when the problem was truncated at higher wavelet resolutions the improvement in reconstruction quality was negligible).
\begin{figure}[t]
\begin{center}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{linearsampling2d}
\caption{\footnotesize Linear Sampling Pattern}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{hypsampling2d}
\caption{Hyperbolic Sampling Pattern \\ (Boxed in)}
\end{center}
\end{subfigure}
\end{center}
\caption{Sampling Patterns. Samples are from the subset $\{-200,-199,...,199,200\}^2 \subset \bbZ^2$. White indicates sample is taken.}
\label{spectrumsamples}
\end{figure}
\begin{figure}[!h]
\begin{center}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{seplinear2d}
\caption{\footnotesize Separable Reconstruction \\ $L^1$ error = 0.0367}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{seplinear2dcloseup}
\caption{\footnotesize Separable Closeup}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorlinear2d}
\caption{\footnotesize Tensor Reconstruction \\ $L^1$ error = 0.0592}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorlinear2dcloseup}
\caption{\footnotesize Tensor Closeup}
\end{center}
\end{subfigure}
\end{center}
\caption{Reconstructions from Linear Sampling Pattern}
\label{linearrecons2d}
\end{figure}
We are now going to test how well these two bases perform under subsampling with different orderings of $B^2_\mathrm{f}$. Two subampling patterns, one based on a linear ordering and another on a hyperbolic ordering, are presented in Figure \ref{spectrumsamples}. Ideally the hyperbolic subsampling pattern would not be restricted by the $\{-200,-199,...,199,200\}^2$ but this is numerically unfeasible.
Let us first consider what happens when using pattern (a) (see Figure \ref{linearrecons2d}). Notice that the separable reconstruction performs far better than the tensor reconstruction and therefore is more tolerant to subsampling with a linear ordering than the tensor case. This is unsurprising as the tensor problem suffers from noticeably large $1/\sqrt{N}$ incoherence when using a linear ordering when compared to the $1/N$ separable decay rate.
Of course we should have fully considered the sparsity of these two problems which also factor into the ability to subsample, however $f$ was specifically chosen because it was sparse in the tensor basis and moreover we have seen that it provides a comparable reconstruction to the separable case when taking a full set of $\{-200,-199,...,199,200\}^2$ samples.
Next we observe what happens when using the pattern (b) (Figure \ref{hyprecons2d}). There is now a stark contrast to the linear case, in that both separable and tensor cases provide very similar reconstructions and furthermore the $L^1$ errors are very close. This suggests that both problems have similar susceptibility to subsampling when using hyperbolic sampling, which is reflected by their identical rates of incoherence decay with hyperbolic orderings.
\begin{figure}[h]
\begin{center}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{sephyp2d}
\caption{\footnotesize Separable Reconstruction \\ $L^1$ error = 0.0263}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{sephyp2dcloseup}
\caption{\footnotesize Separable Closeup}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorhyp2d}
\caption{\footnotesize Tensor Reconstruction \\ $L^1$ error = 0.0277}
\end{center}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{tensorhyp2dcloseup}
\caption{\footnotesize Tensor Closeup}
\end{center}
\end{subfigure}
\end{center}
\caption{Reconstructions from Hyperbolic Sampling Pattern}
\label{hyprecons2d}
\end{figure}
\section{Appendix}
\textit{Proof of Proposition \ref{Hyperbolic4Separable}:} (\ref{hyperbolicbound2}) applied to part 1.) of Lemma \ref{characterisationlemma} shows that the decay of $\mu(\pi_N U)$ is bounded above by\footnote{for the definitions of $H_d, h_d$ see (\ref{hyperbolicdecay}) and (\ref{hddef}).} $F_\text{hyp}(\sigma(N))=1/H_d(\sigma(N)) \approx 1/h_d(N)$, which gives us the upper bound for $\mu(Q_N U)$ since $1/h_d(N)$ is decreasing.
For the lower bound, we focus on terms of the form $\lambda_d \circ \rho(m)=(t,...,t)$ for some $t \in \bbN$ and we set, for a fixed $q \in \bbN$
\[s=(1,...,1), \quad j:= \lceil \epsilon \log_2 t \rceil + q , \]
where we assume for now that $j \ge J$ is satisfied. This gives us
\be{ \label{specificlower}
\begin{aligned}
| \langle \Psi^s_{j,0}, \rho(m) \rangle |^2 & = \epsilon^d 2^{-dj} \prod_{i=1}^d | \mathcal{F} \psi(\epsilon 2^{-j} t) |^2
\\ & \ge \frac{1}{2^{d(1+q)} t^d} \cdot | \mathcal{F} \psi(\epsilon 2^{-(\lceil \epsilon \log_2 t \rceil + q)} t) |^2
\\ & \ge \frac{1}{2^{d(1+q)} t^d} \cdot L_q^{2d} \quad ( \text{using (\ref{wavelower})}) .
\end{aligned}
}
Let $m$ now be arbitrary with $\prod_{i=1}^d \max( | \lambda_d \circ \rho(m)_i|,1)=M \ge 1$ and let $t = \lceil M^{1/d} \rceil +1$. Because $\rho$ corresponds to the hyperbolic cross there exists an $m'>m$ such that $\prod_{i=1}^d \max( | \lambda_d \circ \rho(m')_i|,1)=t^d$ where $\lambda_d \circ \rho(m')=(t,...,t)$. Notice that $t^d \le E(d)M$ for some constant dependent on the dimension $d$. Furthermore, (\ref{specificlower}) holds for $m=m'$ if we have that $j \ge J$, which is satisfied if $m$ is sufficiently large. Therefore we deduce by (\ref{specificlower}) that
\[
\begin{aligned}
\mu( Q_m U) \ge | \langle \Psi^s_{j,0}, \rho(m') \rangle |^2 & \ge \frac{1}{2^{d(1+q)} t^d} \cdot L_q^{2d}
\\ & \ge \frac{1}{E 2^{d(1+q)+1} M} \cdot L_q^{2d}
\\ & = \frac{1}{E 2^{d(1+q)} \prod_{i=1}^d \max( | \lambda_d \circ \rho(m)_i|,1) } \cdot L_q^{2d}
\\ & \ge \frac{C}{E 2^{d(1+q)} h_d(m)} \cdot L^{2d}_q \quad (\text{using (\ref{hyperboliccrossZdecay}), $C>0$ some constant}).
\end{aligned}
\]
This proves the lower bound.
\bibliographystyle{abbrv}
|
1,314,259,993,050 | arxiv | \section{Introduction} \goodbreak
\label{sec-intro}
\subsection{Background and motivation}
\label{subs-back}
We consider a many-server queueing system in which customers with independent,
identically\vadjust{\goodbreak} distributed (henceforth, i.i.d.) service requirements
chosen from
a general distribution are processed in the order of arrival.
In addition, a customer is assumed to abandon the queue if his/her time
spent waiting in queue reaches his/her patience time. The patience times
of customers are also assumed to be i.i.d. and drawn from a general
distribution.
When there are $N$ servers and the
cumulative customer arrival process is assumed to be a renewal process,
this reduces to the
so-called G/GI/N${}+{}$GI model.
Over the last couple of decades, several applications have spurred the
study of
many-server models with abandonment \cite{bachet81,boxwaal94,gansetal}.
Specifically, in applications to telephone contact
centers and (more generally) customer contact centers,
the effect of customers' impatience has been shown to
have a substantial impact on the performance of the system
\cite{gansetal}.
For example, customer abandonment can stabilize
a system that was formerly unstable.
Under the assumption that the interarrival,
service and abandonment time distributions are (possibly time-varying)
exponential, process-level
fluid and diffusion approximations
were obtained by Mandelbaum, Massey and Reiman \cite{manmasrei}
for the total number in system
in networks of multiserver queues
with abandonments and retrials.
On the other hand, for the case of Poisson arrivals, exponential
service times and
general abandonment distributions (the M/M/N${}+{}$GI queue),
explicit formulae for the steady state distributions of the queue
length and
virtual waiting time were obtained by Baccelli and Hebuterne
\cite{bachet81} (see Sections IV and V.2 therein),
whereas several other steady state performance measures and their
asymptotic approximations, in the limit as the arrival rates
and servers go to infinity, were derived by Mandelbaum and Zeltyn \cite
{manzel05}.
In addition, approximations for performance measures
suggested by these limit theorems were
used by Garnett et al. \cite{garmanrei02} and Mandelbaum and Zeltyn
\cite{manzel08} for the
case of exponential and general abandonment distributions, respectively,
to provide insight into the design of large call centers.
In all the previously mentioned works, the service times
were assumed to be exponential.
However, statistical analysis of real call centers has shown that both
service times and abandon times
are typically not exponentially distributed \cite{brownetal,manzel05},
thus providing strong motivation for considering many-server systems with
general service and abandonment distributions.
A step toward incorporating
more realistic general service distributions was taken in
the insightful paper by Whitt~\cite{whifluid06}, where a
deterministic fluid approximation for a G/GI/N${}+{}$GI queue with general
service and abandonment distributions was proposed.
However, the convergence of the discrete system starting empty
to this fluid approximation was left as a conjecture (see Conjecture
2.1 in
\cite{whifluid06}).
In this work,
we rigorously identify the functional law of large numbers or mean-field limit, as the number of servers goes to
infinity, of a many-server queueing system with general service and abandonment
distributions starting from general initial conditions. In a recent
work, Mandelbaum and Momcilovic
\cite{MM09} have established diffusion approximations for the queue-length and virtual waiting time processes in
a G/GI/N${}+{}$GI queue.
With a view to providing a Markovian representation of the dynamics
with a state space that is independent of the number of servers,
we introduce a pair of measure-valued processes to describe the
evolution of
the system. One measure-valued process keeps track of the waiting
times of customers in queue and the other keeps track of the
amounts of time each customer present in the system has been in service.
Under rather general assumptions (specified in Sections~\ref{subs-modyn}
and~\ref{subs-scaling}),
we establish an asymptotic limit theorem for the scaled (divided by $N$)
pair of measure-valued processes, as the number of servers $N$ and the mean
arrival rate into the system simultaneously go to infinity. In a recent
independent study,
Zhang \cite{Zhang} also considered the fluid limit for the same
G/GI/N${}+{}$GI system by using a measure-valued representation. His
approach is based on tracking the ``residual'' service and patience
times rather than tracking the ``ages'' in system and service as
considered in this work.
As in \cite{kasram07} and \cite{kasram08}, an advantage of the particular measure-valued
representation used here, in terms of ages in system and service,
rather than residual service and residual patience times, is that it
facilitates the
application of martingale techniques, which streamlines the analysis
and also
allows for a more intuitive representation of the dynamics of
the limiting process. In addition, the measure-valued approach also
simultaneously allows for the characterization of asymptotic limits of
several other functionals of interest.
In order to illustrate this point,
we also derive a limit theorem for the virtual waiting time of
a customer, defined to be the time before entry to service of a
(virtual) customer with infinite patience.
This work generalizes the framework of Kaspi and Ramanan
\cite{kasram07}, in which the corresponding model without abandonments
was considered. The presence of two coupled measure-valued processes,
rather than just one as in \cite{kasram07}, makes the analysis here
significantly more involved. In addition, an important step is the
identification of an explicit expression for the cumulative reneging
process. This paper also forms the basis of subsequent work in which we establish,
under suitable conditions, the convergence of the stationary distributions of the
fluid-scaled $N$-server systems to the invariant state of the
fluid limit, as $N$ tends to infinity \cite{kanram08c}.
It is worthwhile to mention that the models discussed above are
relevant when the mean demand
of customers is known (or can be accurately learned from an initial
period of measurements), which is a realistic assumption
in many applications.
In other scenarios, it may be more natural to model the demand
as being doubly stochastic. This approach was adopted
by Harrison and Zeevi \cite{harzee05} (see also \cite{basharzee05}),
who proposed optimal staffing and design of multi-class call centers
with several agent pools in the presence of abandonment
under the assumption that the dominant variability arises from the
randomness in the mean demand, rather than fluctuations around the
mean demand.
\subsection{Outline of the paper}
The outline of the paper is as follows.
We provide a more precise description of the model and the
measure-valued representation of the state, and describe the dynamical
equations governing the evolution of the system
in Section \ref{sec-mode} (the explicit
construction of the state process
is relegated to Appendix~\ref{ap-markov} and the strong Markov
property of the state process is established in Appendix \ref{sec:SMF}).
A key result here is
Theorem \ref{th-prelimit}, which provides a succinct
characterization of the state dynamics.
An analog of this characterization for continuous state
processes leads to
the fluid equations, which are
introduced in Section \ref{subs-fleqs}
(see Definition~\ref{def-fleqns}).
Next, the main results of the paper are summarized in
Section \ref{subs-mainres}. The first (Theorem \ref{thm:1})
is a uniqueness result that states that
(under the assumption that the service
and abandonment distributions have densities and finite first moments)
there exists at most one solution to the fluid equations.
The proof of this result, which is considerably more
involved than in the case without abandonment,
is the subject of Section \ref{sec-uniq}.
The second and main result of the paper (Theorem \ref{thm:2})
states that
under mild additional assumptions (namely,
Assumptions \ref{ass-init}--\ref{ass-h} introduced in
Section \ref{subs-scaling}),
the scaled sequence of state processes converges weakly to
the (unique) solution of the fluid equations, and provides
a fairly explicit representation for the solution.
The proof of this result consists of two main steps.
First, in Section \ref{Sec:relcom},
the sequence of scaled state processes is shown to be tight and
then, in Section \ref{sec:CSL},
it is shown that every subsequential limit is a solution to the
fluid equations.
Both of these results make use of properties of a family of
martingales that are established in Section \ref{subs-prelim}.
Finally, the last result (Theorem \ref{thm:3})
formulates the asymptotic limit theorem for
the virtual waiting time process, which is proved
in Section \ref{subs-prf3}.
To start with, in Section \ref{subs-notat},
we first collect some basic notation and
terminology used throughout the paper.
\subsection{Notation and terminology}
\label{subs-notat}
The following notation will be used throughout the paper.
${\mathbb Z}$ is the set of integers, ${\mathbb N}$ is the set of
strictly positive
integers, ${\mathbb R}$ is set of real numbers,
${\mathbb R}_+$ the set of nonnegative real numbers and ${\mathbb Z}_+$
is the set of
nonnegative integers.
For $a, b \in{\mathbb R}$, $a \vee b$ denotes the maximum of $a$ and $b$,
$a \wedge b$ the minimum of $a$ and $b$ and the short-hand $a^+$ is
used for $a \vee0$.
Given $A \subset{\mathbb R}$ and $a \in{\mathbb R}$, $A - a$ equals
the set
$\{x \in{\mathbb R}\dvtx x + a \in A\}$ and $\mathbh{1}_B$ denotes
the indicator
function of the set $B$
[i.e., $\mathbh{1}_B (x) = 1$ if $x \in B$ and $\mathbh{1}_B(x) = 0$
otherwise].
\subsubsection{Function and measure spaces}
\label{subsub-funmeas}
Given any metric space $E$, $\mathcal{C}_b(E)$ and $\mathcal{C}_c
(E)$ are, respectively,
the space of bounded, continuous functions and
the space of continuous real-valued functions with compact support
defined on
$E$, while $\mathcal{C}^1(E)$ is the space of real-valued,
once continuously differentiable functions on~$E$, and $\mathcal{C}^1_c(E)$
is the subspace of functions in $\mathcal{C}^1(E)$ that have compact support.
The subspace of functions in $\mathcal{C}^1(E)$ that, together with their
first derivatives, are bounded, will be denoted by $\mathcal
{C}^1_b(E)$. For
$H\le\infty$, let
$\mathcal{L}^1[0,H)$ and $\mathcal{L}^1_{\mathrm{loc}}[0,H)$, respectively,
represent the
spaces of
integrable and locally integrable functions on $[0,H)$, where a locally
integrable function $f$ on $[0,H)$
is a measurable function on $[0,H)$ that satisfies
$\int_{[0,a]}f(x)\,dx<\infty$ for all $a<H$. The constant functions
$f \equiv1$ and $f \equiv0$ will be represented by the symbols
${\mathbf{1}}$ and $\mathbf{0}$, respectively. Given any c\`{a}dl\`{a}g,
real-valued function $\varphi$ defined on $[0,\infty)$, we define
$\Vert\varphi\Vert_T \doteq\sup_{s \in[0,T]} |\varphi(s)|$ for
every $T < \infty$,
and let $\Vert\varphi\Vert_\infty\doteq\sup_{s \in[0,\infty)}
|\varphi(s)|$, which could possibly take the value $\infty$.
In addition,
the support of a function $\varphi$ is denoted by $\operatorname
{supp}(\varphi)$.
Given a
nondecreasing function $f$ on $[0,\infty)$, $f^{-1}$ denotes the inverse
function of $f$ in the sense that
\begin{equation}\label{inverse} f^{-1}(y)=\inf\{x\geq0\dvtx f(x)\geq
y\}.
\end{equation}
For each differentiable function $f$ defined on ${\mathbb R}$, $f'$
denotes the
first derivative of~$f$. For each function $f(t,x)$ defined on
${\mathbb R}
\times{\mathbb R}^n$, $f_t$ denotes the partial derivative of $f$ with respect
to $t$, and
$f_x$ denotes the partial derivative of $f$ with respect to $x$.
The space of Radon measures on a metric space $E$, endowed with the
Borel $\sigma$-algebra,
is denoted by $\mathcal{M}(E)$, while
$\mathcal{M}_F(E)$, $\mathcal{M}_1(E)$ and $\mathcal{M}_{\leq1}
(E)$ are, respectively, the subspaces of finite,
probability and sub-probability measures in $\mathcal{M}(E)$. Also,
given $B <
\infty$,
$\mathcal{M}_{\leq B} (E)\subset\mathcal{M}_F(E)$ denotes the space
of measures $\mu$ in $\mathcal{M}_F(E)$
such that $|\mu(E)| \leq B$.
Recall that a Radon measure is one that assigns finite measure to every
relatively
compact subset of ${\mathbb R}_+$.
The space $\mathcal{M}(E)$ is equipped with the vague topology, that
is, a
sequence of measures
$\{\mu_n\}$ in $\mathcal{M}(E)$ is said to converge to $\mu$ in the vague
topology (denoted
$\mu_n \stackrel{v}{\rightarrow}\mu$) if and only if for every
$\varphi\in\mathcal{C}_c (E)$,
\begin{equation}
\label{w-limit}
\int_{E} \varphi(x) \mu_n(dx) \rightarrow\int_E \varphi(x) \mu(dx)
\qquad\mbox{as } n \rightarrow\infty.
\end{equation}
By identifying a Radon measure $\mu\in\mathcal{M}(E)$ with the
mapping on
$\mathcal{C}_c (E)$ defined by
\[
\varphi\mapsto\int_{E} \varphi(x) \mu(dx),
\]
one can equivalently define a Radon measure on $E$ as a linear mapping
from $\mathcal{C}_c (E)$ into ${\mathbb R}$ such that for every
compact set $\mathcal{K}
\subset E$, there exists
$L_{\mathcal{K}} < \infty$ such that
\[
\biggl| \int_{E} \varphi(x) \mu(dx) \biggr| \leq L_{\mathcal{K}}
\Vert\varphi\Vert_\infty\qquad\forall
\varphi\in\mathcal{C}_c (E)\mbox{ with } \operatorname
{supp}(\varphi) \subset{\mathcal{K}}.
\]
On $\mathcal{M}_F(E)$, we will also consider the
weak topology, that is, a sequence $\{\mu_n\}$ in $\mathcal{M}_F(E)$
is said to
converge weakly to $\mu$ (denoted $\mu_n \stackrel{w}{\rightarrow
}\mu$) if and only
if (\ref{w-limit}) holds for every $\varphi\in\mathcal{C}_b(E)$.
As is
well known, $\mathcal{M}(E)$ and $\mathcal{M}_F(E)$, endowed with the
vague and weak
topologies, respectively,
are Polish spaces.
The symbol $\delta_x$ will be used to denote the measure with unit
mass at the point $x$, and,
by some abuse of notation, we will use $\mathbf{0}$ to denote the
identically zero
Radon measure on $E$. When $E$ is an interval, say $[0,H)$, for
notational conciseness,
we will often write $\mathcal{M}[0,H)$ instead of $\mathcal
{M}([0,H))$. For
any finite measure $\mu$ on $[0,H)$,
we define
\begin{equation}
\label{def-cdfmu}
F^\mu(x) \doteq\mu[0,x],\qquad x \in[0,H).
\end{equation}
We say a measure $\mu$ is continuous at $x$ if and only if $\mu(\{x\})=0$.
We will mostly be interested in the case when
$E = [0,H)$ and $E = [0,H) \times{\mathbb R}_+$, for some $H \in
(0,\infty]$.
To distinguish these cases, we will usually use $f$ to denote generic functions
on $[0,H)$ and $\varphi$ to denote generic
functions on $[0,H) \times{\mathbb R}_+$. By some abuse of notation,
given $f$ on $[0,H)$, we will sometimes
also treat it as a function on $[0,H) \times{\mathbb R}_+$ that is
constant in
the second
variable. For any
Borel measurable function $f\dvtx[0,H) \rightarrow{\mathbb R}$ that
is integrable
with respect to $\xi\in\mathcal{M}[0,H)$, we often use the
short-hand notation
\[
\langle f, \xi\rangle\doteq\int_{[0,H)} f(x) \xi(dx).
\]
Also, for ease of notation, given $\xi\in\mathcal{M}[0,H)$ and an
interval $(a,b)
\subset[0,H)$, we will
use $\xi(a,b)$ and $\xi(a)$ to denote $\xi((a,b))$ and $\xi(\{a\}
)$, respectively.
\subsubsection{Measure-valued stochastic processes}
Given a Polish space $\mathcal{H}$, we denote by $\mathcal
{D}_{\mathcal{H}}[0,T]$
(resp., $\mathcal{D}_{\mathcal{H}}[0,\infty)$) the space of
$\mathcal{H}$-valued,
c\`{a}dl\`{a}g functions on $[0,T]$ (resp., $[0,\infty)$),
and we endow this space with the usual Skorokhod $J_1$-topology
\cite{parbook}. Then $\mathcal{D}_{\mathcal{H}}[0,T]$ and $\mathcal
{D}_{\mathcal{H}}[0,\infty)$ are also Polish
spaces (see \cite{parbook}). In this work, we will be interested
in $\mathcal{H}$-valued stochastic processes, where $\mathcal{H} =
\mathcal{M}_F[0,H)$
for some $H\le\infty$.
These are random elements that are defined on a probability space
$(\Omega, \mathcal{F}, \mathbb{P})$ and take values in $\mathcal
{D}_{\mathcal{H}}[0,\infty)$,
equipped with the Borel $\sigma$-algebra (generated by open sets under the
Skorokhod $J_1$-topology).
A~sequence $\{X_n\}$ of c\`{a}dl\`{a}g, $\mathcal{H}$-valued processes,
with $X_n$ defined on the probability space $(\Omega_n, \mathcal
{F}_n, \mathbb{P}_n)$,
is said to converge in distribution
to a c\`{a}dl\`{a}g $\mathcal{H}$-valued process $X$ defined on
$(\Omega,
\mathcal{F}, \mathbb{P})$ if, for every bounded, continuous functional
$F\dvtx\mathcal{D}_{\mathcal{H}}[0,\infty)\rightarrow{\mathbb R}$,
we have
\[
\lim_{n \rightarrow\infty} \mathbb{E}_n [ F(X_n) ] = \mathbb{E}[
F(X) ],
\]
where $\mathbb{E}_n$ and $\mathbb{E}$ are the expectation operators
with respect to
the probability measures
$\mathbb{P}_n$ and $\mathbb{P}$, respectively.
Convergence in distribution of $X_n$ to $X$ will be denoted by $X_n
\Rightarrow X$. Let $\mathcal{I}_{{\mathbb R}_+}[0,\infty)$ be the
subset of nondecreasing functions
$f \in\mathcal{D}_{{\mathbb R}_+}[0,\infty)$ with $f(0) = 0$.
\section{Description of model and state dynamics}
\label{sec-mode}
In Section \ref{subs-modyn} we describe the basic model and the
model primitives.
In Section \ref{sec:repdyn} we introduce the state descriptor and some
auxiliary processes, and derive some equations that describe the
dynamics of
the state. Finally, in Section \ref{subs-prelimit} (see Theorem
\ref{th-prelimit}), we provide a succinct characterization of the
state dynamics.
This characterization motivates the
form of the fluid equations, which are introduced in Section
\ref{subs-fleqs}.
\subsection{Model description and primitive data}
\label{subs-modyn}
Consider a system with $N$ servers, in which arriving
customers are served in a nonidling, First-Come-First-Serve (FCFS)
manner, that is, a newly
arriving customer immediately enters service if there are any idle
servers or,
if all servers are busy, then the customer joins the back of
the queue, and the customer at the head of the queue (if one is
present) enters
service as soon as a server becomes free.
Our results are not sensitive to the exact mechanism used to assign an
arriving customer to an idle server, as long as the nonidling condition,
that there cannot simultaneously be a positive queue and an idle server,
is satisfied.
It is assumed that customers are impatient, and that a customer reneges
from the queue as soon
as the amount of time he/she has spent in queue reaches his/her
patience time.
Customers do not renege once they have entered service.
The patience times of customers are given by an i.i.d. sequence,
$\{r_i, i\in{\mathbb Z}\}$, with common cumulative distribution
function $G^r$ on $[0,\infty]$,
while the service requirements of customers are given by another
i.i.d.
sequence, $\{v_i, i\in{\mathbb Z}\}$, with common cumulative
distribution function
$G^s$ on $[0,\infty)$. For $i\in{\mathbb N}$, $r_i$ and $v_i$
represent, respectively, the
patience time and the service requirement of the $i$th customer to
enter the
system after time zero, while $\{r_i, i\in-{\mathbb N}\cup\{0\}\}$
and $\{v_i,
i\in-{\mathbb N}\cup\{0\}\}$ represent, respectively, the patience
times and the
service requirements of customers that arrived prior to time zero (if such
customers exist), ordered according to their arrival times
(prior to time zero). We assume that $G^s$ has density $g^s$ and $G^r$,
restricted on $[0,\infty)$, has density $g^r$.
This implies, in particular, that
$G^r(0+) = G^s(0+) = 0$. Let
\begin{eqnarray*}
H^r & \doteq& \sup\{x \in[0,\infty)\dvtx g^r(x) >0 \}, \\
H^s & \doteq
& \sup\{x \in[0,\infty)\dvtx g^s(x) >0 \}
\end{eqnarray*}
denote the right
ends of the supports of $g^r$ and $g^s$, respectively. The superscript
$(N)$ will be used to refer to quantities associated with the system
with $N$ servers.
Let $E^{(N)}$ denote the cumulative arrival process, with $E^{(N)}(t)$
representing the total number of customers that arrive into the system
with $N$ servers in the time interval $[0,t]$. Also, consider the c\`
{a}dl\`{a}g, real-valued process $\alpha_E^{(N)}$ defined by $\alpha
_E^{(N)}(s)=s$ if
$E^{(N)}(s)=0$ and, if $E^{(N)}(s)>0$, then
\begin{equation}
\label{def-ren}
\alpha_E^{(N)}(s) \doteq s- \sup\bigl\{u<s\dvtx E^{(N)}(u)<E^{(N)}(s) \bigr\},
\end{equation}
which denotes the time elapsed since the last arrival. If $E^{(N)}$ is
a renewal
process, then $\alpha_E^{(N)}$ is simply the backward recurrence time process.
Also, let $\mathcal{E}^{(N)}_0$ be an a.s. ${\mathbb Z}_+$-valued
random variable that
represents the
number of customers that entered the system prior to time zero.
This random variable does not play an important role in the analysis,
but is
used for bookkeeping purposes to keep track of the indices of customers.
The following mild assumptions on $E^{(N)}$ will be imposed throughout,
without explicit
mention:
\begin{itemize}
\item$E^{(N)}$ is a nondecreasing, pure jump process with $E^{(N)}(0)
= 0$
and a.s.,
for $t\in[0, \infty), E^{(N)}(t) < \infty$ and
$E^{(N)}(t)-E^{(N)}(t-) \in\{
0, 1\}$;
\item
the process $\alpha_E^{(N)}$ is Markovian with respect to its own natural
filtration (this holds, e.g., when $E^{(N)}$ is a renewal process);
\item
the cumulative arrival process $E^{(N)}$, the sequence of service requirements
$\{v_j, j \in{\mathbb Z}\}$ and the sequence of patience times
$\{r_j, j \in{\mathbb Z}\}$ are independent.
\end{itemize}
The assumption on the jump size of $E^{(N)}$ is not crucial and is imposed
mainly for convenience. On the other hand, the assumed
independence of the service and patience times is a genuine
restriction. It would be of interest to consider the case of correlated
service and patience times.
\subsection{State descriptor and dynamical equations}
\label{sec:repdyn}
As mentioned in Section \ref{subs-back}, our representation of the
state of the system with $N$ servers involves a pair of measure-valued
processes, the ``potential queue measure'' process, $\eta^{(N)}$,
which keeps track of the waiting times of customers
in queue and the ``age measure'' process, $\nu^{(N)}$,
which encodes the amounts of time
that customers currently receiving service have been in service.
In fact, the potential queue measure process
keeps track not only of the waiting times of customers in
queue, but also of the potential waiting times (equivalently,
times since entry into system) of those customers
who may have already entered service (and possibly departed the system),
but for whom the time since entry into the system has not yet exceeded the
patience time. In order to determine which subset of these customers
is actually in queue, the process $X^{(N)}$, which
represents the total number of
customers in system with $N$ servers (including those in service and
those in queue), is also incorporated into the state descriptor.
Thus the state of the system is represented by the
vector of processes $(\alpha_E^{(N)}, X^{(N)}, \nu^{(N)}, \eta
^{(N)})$, where
$\alpha_E^{(N)}$ determines the cumulative arrival process via (\ref
{def-ren}).
The reason for introducing the process
$\eta^{(N)}$ into the state (rather than working directly with a
restricted measure that only encodes the waiting times
of customers in queue) is that its dynamics is decoupled
from the service dynamics. It is governed
purely by the primitive data $E^{(N)}$ and $G^r$, and is thus
more amenable to analysis
(see Remark \ref{rem-compdyn} for further elaboration of this point).
Indeed, the queue measure process $\eta^{(N)}$ can also be viewed as
describing the ages of customers in an infinite server queue that
has cumulative arrivals $E^{(N)}$ and i.i.d. service requirements
distributed according to $G^r$. Thus the dynamics of the process
$\eta^{(N)}$ is also of independent interest.
Precise mathematical descriptions of $\eta^{(N)}$ and $\nu^{(N)}$
are given in Sections \ref{subsub-queue} and \ref{subsub-service},
respectively. Some auxiliary processes that are useful for describing the
evolution of the state are introduced in Section \ref{subsub-aux}.
Finally, in Section \ref{subsub-filt},
a~filtration $\{\mathcal{F}_t^{(N)}\}$ corresponding to the system
with $N$ servers is introduced, and it is shown that the state
processes and
auxiliary processes are all adapted to this filtration.
In fact, it is shown in Appendix \ref{sec:SMF} that
the state process is a strong Markov process with respect to this filtration.
\subsubsection{Description of queue dynamics}
\label{subsub-queue}
The potential waiting time process $w^{(N)}_j$ of customer $j$ is
(for every realization) defined to be the piecewise linear function
on $[0,\infty)$ that is identically zero till the customer enters the system,
then increases linearly, representing the amount of time elapsed since
entering the system,
and then remains constant (equal to the patience time)
once the time elapsed exceeds the patience time. More precisely,
for $j\in{\mathbb N}$, if $\zeta_j^{(N)}$ is the time at which the $j$th
customer arrives into the system after time $0$, then for $j \in
{\mathbb N}$
$\zeta_j^{(N)} = (E^{(N)})^{-1} (j) \doteq\inf\{ t > 0\dvtx E^{(N)} (t)
= j \}$
and
\begin{equation}
\label{def-waitjn}
w^{(N)}_j (t) = \cases{
\bigl[t - \zeta_j^{(N)} \bigr] \vee0, &\quad if $t - \zeta
{}^{(N)}_j < r_j$, \vspace*{2pt}\cr
r_j, &\quad otherwise.}
\end{equation}
For $j\in-{\mathbb N}\cup\{0\}$, $w^{(N)}_j$ represents the potential
waiting time
process of the $j$th customer who entered the system before time zero (if
such a customer exists). Observe that the potential waiting time
$w^{(N)}_j(t)$
of a customer at time $t$ equals its actual waiting time (equivalently,
time spent in queue) if and only if the customer has neither entered service
nor reneged by time $t$. For $t\in[0,\infty)$, let $\eta^{(N)}_t$ be the
nonnegative Borel measure on $[0,H^r)$ that has a unit mass at the potential
waiting time of each customer that has entered the system by time $t$ and
whose potential waiting time has not yet reached its patience time. Recall
that $\delta_x$ represents the Dirac mass at $x$. The potential queue
measure $\eta^{(N)}_t$ can be written in the form
\begin{eqnarray}
\label{def-etan} \eta^{(N)}_t &=& \sum_{j = -\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)}
\delta_{w^{(N)}_j(t)} \mathbh{1}_{\{w^{(N)}_j (t) <
r_j\}}\nonumber\\[-8pt]\\[-8pt]
&=&\sum_{j = -\mathcal{E}^{(N)}_0+ 1}^{E^{(N)}(t)} \delta_{w^{(N)}_j(t)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(t+) >0 \}},\nonumber
\end{eqnarray}
where the last equality holds because at any time $t$, the potential waiting
time process of any customer has a right derivative that is positive if and
only if the customer has entered the system and the customer's potential
waiting time has not yet reached its patience time.
For $t\in[0,\infty)$, let $Q^{(N)}(t)$ be the number of customers
waiting in queue
at time~$t$.
Due to the nonidling condition, the queue length process is then given by
\begin{equation}
\label{def-qnt}Q^{(N)}(t)=\bigl[X^{(N)}(t)-N\bigr]^+.
\end{equation}
Moreover, since the head-of-the-line customer is the customer in queue
with the
longest waiting time, the quantity
\begin{equation}\label{def-chi} \chi^{(N)}(t)\doteq
\inf\bigl\{x>0\dvtx\eta^{(N)}_t[0,x]\geq Q^{(N)}(t) \bigr\} =
\bigl(F^{\eta^{(N)}_t} \bigr)^{-1}
\bigl(Q^{(N)}(t)\bigr)
\end{equation}
represents the waiting time of the head-of-the-line customer in the
queue at
time $t$.
Here, recall from (\ref{def-cdfmu}) that $F^{\eta^{(N)}_t}$ is the
c.d.f. of the
measure $\eta^{(N)}_t$ and $(F^{\eta^{(N)}_t})^{-1}$ represents its
inverse, as defined in (\ref{inverse}).
Since this is an FCFS system,
any mass in $\eta^{(N)}_t$ that lies to the right of $\chi^{(N)}(t)$
represents
a customer that has already entered service by time $t$.
Therefore, the queue length process $Q^{(N)}$ admits the following
alternative representation in terms of $\chi^{(N)}$ and $\eta^{(N)}$:
\begin{eqnarray} \label{qn}
Q^{(N)}(t)&=& \sum_{j=-\mathcal{E}^{(N)}_0+ 1}^{E^{(N)}(t)}
\mathbh{1}_{\{w^{(N)}_j (t)\leq\chi^{(N)}(t), w^{(N)}_j (t)<r_j \}
}
\nonumber\\[-8pt]\\[-8pt]
&=&
\eta^{(N)}_t\bigl[0,\chi^{(N)}(t)\bigr]. \nonumber
\end{eqnarray}
\subsubsection{Description of service dynamics}
\label{subsub-service}
Analogous to the potential waiting process $w_j^{(N)}$,
the age process $a_j^{(N)}$ associated with customer $j$ is (for every
realization)
defined to be the piecewise linear function on $[0,\infty)$ that
equals $0$ till the customer enters service, then increases linearly
while the customer is in service (representing the amount of time
elapsed since entering service) and is then constant (equal to the
total service requirement)
after the customer completes service and departs the system.
For\vspace*{-2pt} $j=-\mathcal{E}^{(N)}_0+1, \ldots,0$, let $a^{(N)}_j(0)$
represent the age of the $j$th
customer in service at time 0 and for $j\in{\mathbb N}$, we set
$a^{(N)}_j(0)=0$.
Due to the First-Come-First-Serve (FCFS) nature of the system,
customers in service at
time $t$ are those that did not renege, that
have been in the system longer than
the head-of-the-line customer at time $t$, but
have not yet completed service and departed.
Therefore, a.s., for $j=-\mathcal{E}^{(N)}_0+1, \ldots,0,\ldots
,E^{(N)}(t)$, $t\geq0$,
\begin{equation}\label{adif}
\frac{d a^{(N)}_j(t+)}{dt}=\cases{
0, &\quad if $a^{(N)}_j(t)=0, w^{(N)}_j(t)=r
_j$, \vspace*{2pt}\cr
&\quad or $a^{(N)}_j(t)=0, w^{(N)}_j(t)\leq\chi^{(N)}(t)$, \vspace*{2pt}\cr
&\quad or $a^{(N)}_j(t) = v_j$, \vspace*{2pt}\cr
1, &\quad if $a^{(N)}_j(t)=0, \chi^{(N)}(t)< w^{(N)}_j(t) < r_j$, \vspace*{2pt}\cr
&\quad or $0< a^{(N)}_j(t) < v_j$.}
\end{equation}
Note that the condition in the penultimate line of the right-hand side
above represents the scenario in which a
customer enters service precisely at time $t$, which causes
$\chi^{(N)}$ to have a downward jump at time $t$ since the
condition that the arrival process increases only in unit jumps
ensures that there is at most one customer with a given potential
waiting time.
Now, for $t\in[0,\infty)$, let $\nu^{(N)}_t$ be the discrete nonnegative
Borel measure on $[0,H^s)$ that has a unit mass at the age of each of the
customers in service at time~$t$. Then, in a fashion analogous to
(\ref{def-etan}), the age measure $\nu^{(N)}_t$ can be
explicitly represented as
\begin{equation}
\label{def-nun}
\nu^{(N)}_t = \sum_{j = -\mathcal{E}^{(N)}_0+ 1}^{E^{(N)}(t)} \delta
_{a^{(N)}_j(t)}
\mathbh{1}_{ \{{da^{(N)}_j }/{dt}(t+) >0 \}}.
\end{equation}
\subsubsection{Auxiliary processes}
\label{subsub-aux}
We now introduce certain auxiliary processes that will be useful for
the study of the evolution
of the system.
\begin{itemize}
\item
The cumulative reneging process $R^{(N)}$, where $R^{(N)}(t)$ is the
cumulative number
of customers that have reneged from the system in the time interval $[0,t]$;
\item
the cumulative potential reneging process $S^{(N)}$, where $S^{(N)}(t)$
represents the cumulative number of customers whose potential waiting times
have reached their patience times in the interval $[0,t]$;
\item
the cumulative departure process $D^{(N)}$, where $D^{(N)}(t)$ is the cumulative
number of customers that have departed the system after completion of service
in the interval $[0,t]$;
\item
the process $K^{(N)}$, where $K^{(N)}(t)$ represents the cumulative
number of
customers that
have entered service in the interval $[0,t]$.
\end{itemize}
Now, a customer $j$ completes service (and therefore departs the
system) at
time $s$ if and only if, at time $s$, the left derivative of
$a_j^{(N)}$ is
positive and the right derivative of $a_j^{(N)}$ is zero. Therefore, we
can write
\begin{equation}
\label{def-depart} D^{(N)}(t) = \sum_{j = -\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)}
\sum_{s\in[0,t]} \mathbh{1}_{ \{{da^{(N)}_j }/{dt}(s-) >0,
{da^{(N)}_j }/{dt}(s+)=0 \}}.
\end{equation}
Note that the second sum in (\ref{def-depart}) is well defined since
for each $t\geq0$ and each $j$ between $-\mathcal{E}^{(N)}_0+ 1$ and
$E^{(N)}(t)$, the
piecewise linear structure of\vspace*{1pt}
$a^{(N)}_j$ ensures that the indicator function in the sum is nonzero
for at most one $s\in[0,t]$, that is, there exists at most one $s\in
[0,t]$ such that the customer $j$ completes service at time $s$. A
similar logic shows that the cumulative potential reneging
process $S^{(N)}$ admits the representation
\begin{equation}
\label{def-cvrp} S^{(N)}(t) = \sum_{j = -\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)} \sum
_{s\in[0,t]} \mathbh{1}_{ \{{dw^{(N)}_j}/{dt}(s-) >0,
{dw^{(N)}_j }/{dt}(s+)=0 \}},
\end{equation}
and the cumulative reneging process $R^{(N)}$ admits the representation
\begin{eqnarray}
\label{def-crp}
&&
R^{(N)}(t) \nonumber\\[-10pt]\\[-10pt]
&&\qquad= \sum_{j = -\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)} \sum
_{s\in[0,t]} \mathbh{1}_{ \{w^{(N)}_j(s)\leq\chi^{(N)}(s-),
{dw^{(N)}_j}/{dt}(s-) >0, {dw^{(N)}_j }/{dt}(s+)=0 \}},\nonumber\hspace*{-28pt}
\end{eqnarray}
where the additional restriction $w^{(N)}_j(s)\leq\chi^{(N)}(s-)$ is imposed
so as to only count the reneging of customers actually in queue (and
not the
reneging of all
customers in the potential queue, which is captured by $S^{(N)}$).
Here, one considers the left limit $\chi^{(N)}(s-)$ of $\chi^{(N)}$
at time
$s$ to capture the situation in which $\chi^{(N)}$ jumps down at time $s$
due to the head-of-the-line customer reneging from the queue or
entering service.
Now, $\langle{\mathbf{1}}, \nu^{(N)}_t\rangle= \nu
^{(N)}_t[0,\infty)$ represents
the total number of customers in service at time $t$.
Therefore, mass balances on the total number of customers in the
system, the
number of customers waiting in the ``potential queue'' and the number
of customers in service show that
\begin{eqnarray}
\label{def-dn}
X^{(N)}(0) + E^{(N)}&=& X^{(N)}+ D^{(N)}+ R^{(N)},
\\
\label{def-sn}
\bigl\langle{\mathbf{1}}, \eta^{(N)}_0 \bigr\rangle+ E^{(N)}&=& \bigl\langle
{\mathbf{1}}, \eta^{(N)}
\bigr\rangle+ S^{(N)}
\end{eqnarray}
and
\begin{equation}
\label{def-kn}
\bigl\langle{\mathbf{1}}, \nu^{(N)}_0 \bigr\rangle+ K^{(N)}= \bigl\langle{\mathbf
{1}}, \nu^{(N)}
\bigr\rangle+ D^{(N)}.
\end{equation}
In addition, it is also clear that
\begin{equation}
\label{def-xn}
X^{(N)}= \bigl\langle{\mathbf{1}}, \nu^{(N)}\bigr\rangle+ Q^{(N)}.
\end{equation}
Combining (\ref{def-dn}), (\ref{def-kn}) and (\ref{def-xn}), we
obtain the following mass balance for the number of customers in queue:
\begin{equation}
\label{mass-queue}
Q^{(N)}(0) + E^{(N)}= Q^{(N)}+ R^{(N)}+ K^{(N)}.
\end{equation}
Furthermore, the nonidling condition takes the form
\[
N-\bigl\langle{\mathbf{1}}, \nu^{(N)}\bigr\rangle= \bigl[N - X^{(N)}\bigr]^+.
\]
Indeed, note that this ensures that when $X^{(N)}(t) < N$, the number in
the system is equal to the
number in service, and so there is no queue, while if $X^{(N)}(t) > N$,
there is a positive queue and
$\langle{\mathbf{1}}, \nu^{(N)}_t \rangle=N$, indicating that there
are no
idle servers.
An advantage of the measure-valued state representation that we adopt is
that it allows us to simultaneously
study several other functionals of interest. As an example,
we consider the so-called virtual waiting time process, which
is important for applications.
For each $t\geq0$, the virtual waiting time $W^{(N)}(t)$
is defined to be the amount of time a (virtual) customer with infinite
patience would have to wait before entering service if he were to
arrive at time $t$. For a more precise definition of $W^{(N)}$,
let $t\in[0,\infty)$ and
for each $s\in
[0,\infty)$, define
\begin{eqnarray} \label{dis:Tn}
\mathcal T_t^{(N)}(s)
&\doteq& \sum_{u \in[t,t+s]} \sum
_{j=-\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(u-) >0, {dw^{(N)}_j }/{dt}(u
+)=0 \}}\nonumber\\[-8pt]\\[-8pt]
&&\hspace*{79.28pt}{}\times
\mathbh{1}_{\{w^{(N)}_j (u)\leq\chi^{(N)}(u-)\}}. \nonumber
\end{eqnarray}
Observe that $\mathcal T_t^{(N)}(s)$ equals the cumulative number of
customers
who arrived before time $t$ and reneged from the queue (before entering
service) in the time interval $[t,t+s]$. Once again, for each $j$ there
is at most one $u\in[t,t+s]$ for which both indicator functions in the
summation are nonzero, and so
the sum is well defined. The virtual waiting time
$W^{(N)}(t)$ of a customer at time $t$ is the least amount of
time $s$ that elapses
after time $t$ such that the cumulative departure from the system of
customers that arrived prior to time $t$
strictly exceeds the queue length at time $t$.
Observing that this cumulative departure in the interval
$[t,t+s]$ can be
due to either departure from service or
reneging of customers that arrived prior to time $t$,
we can express the virtual waiting time as
\begin{equation} \label{T}\qquad
W^{(N)}(t)
\doteq \inf\bigl\{s\geq0\dvtx D^{(N)}(t+s)- D^{(N)}(t)+ {\mathcal
T}_t^{(N)}(s)> Q^{(N)}(t)\bigr\}.
\end{equation}
Here, we have used the fact that for all $s$ such that $D^{(N)}(t+s)- D^{(N)}
(t)+ {\mathcal T}_t^{(N)}(s)\leq Q^{(N)}(t)$, every customer that departed
in the time interval $[t,t+s]$ must have arrived prior to time $t$.
\subsubsection{Filtration}
\label{subsub-filt}
The total number of customers
in service at time $t$ is given by $\langle{\mathbf{1}}, \nu^{(N)}_t
\rangle
=\nu^{(N)}_t [0,H^s)$
and is bounded above by $N$. In addition, from (\ref{def-sn}) it
follows that
\[
\bigl\langle{\mathbf{1}}, \eta^{(N)}_t \bigr\rangle= \eta^{(N)}_t[0,H^r)
\leq E^{(N)}(t) + \bigl\langle{\mathbf{1}}, \eta^{(N)}_0 \bigr\rangle\leq
E^{(N)}(t) + \mathcal{E}_0^{(N)},
\]
which is a.s. finite by assumption. Therefore,
for every $t\in[0,\infty)$, a.s., $\nu^{(N)}_t \in\mathcal{M}_F
[0,H^s)$ and
$\eta^{(N)}_t \in\mathcal{M}_F [0,H^r)$.
Hence, the state descriptor
$(\alpha_E^{(N)},X^{(N)},\break\nu^{(N)},\eta^{(N)})$ takes values in
${\mathbb R}_+ \times{\mathbb Z}_+
\times\mathcal{M}_F
[0,H^s) \times\mathcal{M}_F [0,H^r)$.
For purely technical purposes we will find it
convenient to also introduce the additional ``station process''
$s^{(N)} \doteq(s_j^{(N)}, j \in{\mathbb Z})$, defined on the same
probability space $(\Omega,\mathcal F,\mathbb{P})$. For each $t \in
[0,\infty)$,
if customer $j$ has already
entered service by time $t$, then
$s_j^{(N)} (t)$ is equal to the index $i \in\{1, \ldots, N\}$ of the
station at which
customer $j$ receives/received service and $s_j^{(N)} (t) \doteq
0$ otherwise.
For $t\in[0,\infty)$, let $\tilde{\mathcal{F}}_t^{(N)}$ be the
$\sigma$-algebra generated by
\begin{eqnarray*}
&&\bigl\{\mathcal{E}^{(N)}_0,X^{(N)}(0),\alpha_E^{(N)}(s), w^{(N)}_j(s),
a^{(N)}_j(s), s_j^{(N)},\\
&&\hspace*{45pt} j \in\{-\mathcal{E}^{(N)}_0+1, \ldots, 0\}
\cup{\mathbb N}, s \in
[0,t] \bigr\},
\end{eqnarray*}
and let $\{\mathcal{F}_t^{(N)}\}$ denote the associated
right-continuous filtration, completed with respect to $\mathbb{P}$. In
Appendix \ref{ap-markov}, an explicit construction of
the state descriptor and auxiliary
processes is provided, which shows in particular
that the state descriptor $(\alpha_E^{(N)},X^{(N)},\nu^{(N)},\eta
^{(N)})$ and
auxiliary processes are c\`{a}dl\`{a}g.
Moreover, in Lemma~\ref{app:adapt}, it is proved that the state process
$V^{(N)}\doteq(\alpha_E^{(N)},X^{(N)},\nu^{(N)},\eta^{(N)})$ and
the processes
$E^{(N)}$, $Q^{(N)}$, $S^{(N)}$, $R^{(N)}$, $D^{(N)}$ and $K^{(N)}$ are
all $\mathcal
{F}_t^{(N)}$-adapted, and in Lemma \ref{lem:Mark}, it is shown that
$(V^{(N)}, \mathcal{F}_t^{(N)})$ is a strong Markov process.
\subsection{A succinct characterization of the dynamics}
\label{subs-prelimit}
The main result of this section is Theorem \ref{th-prelimit}, which provides
equations that more succinctly characterize the dynamics of the state
$(\alpha_E^{(N)},X^{(N)},\nu^{(N)},\eta^{(N)})$ described in Section
\ref{sec:repdyn}.
First, we introduce some notation that is required to state the result.
For any measurable function $\varphi$ on $[0,H^s)\times{\mathbb
R}_+$, consider
the process $D^{(N)}_\varphi$ that takes
values in ${\mathbb R}$, and is given by
\begin{equation}
\label{def-baren}
D^{(N)}_\varphi(t) \doteq\sum_{s \in[0,t]} \sum_{j=-\mathcal
{E}^{(N)}_0+ 1}^{E^{(N)}(t)}
\mathbh{1}_{ \{{da^{(N)}_j }/{dt}(s-) >0, {da^{(N)}_j
}/{dt}(s+)=0 \}}
\varphi\bigl(a^{(N)}_j (s),s\bigr)\hspace*{-32pt}
\end{equation}
for $t \in[0,\infty)$.
It follows immediately from (\ref{def-baren}) and the right continuity
of the filtration $\{\mathcal{F}_t^{(N)}\}$ that
$D^{(N)}_\varphi$ is $\{\mathcal{F}_t^{(N)}\}$-adapted.
Also, comparing (\ref{def-baren}) with (\ref{def-depart}), it is
clear that when $\varphi$ is the constant function ${\mathbf{1}}$,
$D^{(N)}_{\mathbf{1}}$
is exactly the cumulative departure process $D^{(N)}$, that is,
\begin{equation}D^{(N)}_{\mathbf{1}}
=D^{(N)}. \label{equivD}
\end{equation}
In an exactly analogous fashion, for any measurable function $\psi$ on
$[0,H^r)\times{\mathbb R}_+$, consider the
process $S^{(N)}_\psi$ that takes
values in ${\mathbb R}$, and is given by
\begin{equation}
\label{def-barwn}
S^{(N)}_\psi(t) \doteq\sum_{s \in[0,t]} \sum_{j=-\mathcal
{E}^{(N)}_0+ 1}^{E^{(N)}(t)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(s-) >0, {dw^{(N)}_j
}/{dt}(s+)=0 \}}
\psi\bigl(w^{(N)}_j (s),s\bigr).\hspace*{-37pt}
\end{equation}
It follows immediately from (\ref{def-barwn}) and the right continuity
of the filtration $\{\mathcal{F}_t^{(N)}\}$ that for $t\in[0,\infty)$,
$S^{(N)}_\psi$ is $\{\mathcal{F}_t^{(N)}\}$-adapted.
Moreover, $S^{(N)}_{\mathbf{1}}$ is clearly equal to the cumulative potential
reneging process $S^{(N)}$, that is,
\begin{equation}S^{(N)}_{\mathbf{1}}=S^{(N)}. \label{equivS}
\end{equation}
In addition, using
(\ref{def-dn}), (\ref{def-xn}) and the nonnegativity of $Q^{(N)}$, $R^{(N)}$
and $\langle{\mathbf{1}}, \nu^{(N)}\rangle$,
it follows that for any $t \in[0,\infty)$ and bounded,
measurable $\varphi$,
\begin{equation}
\label{bd-1}
\mathbb{E}\bigl[ \bigl|D^{(N)}_\varphi(t) \bigr| \bigr]
\leq\Vert\varphi\Vert_\infty\mathbb{E}\bigl[X^{(N)}(0) + E^{(N)}(t)
\bigr]<\infty
\end{equation}
and likewise, for each $t \in
[0,\infty)$ and bounded measurable $\psi$, (\ref{def-sn}) shows that
\begin{equation}
\label{bd-2}
\mathbb{E}\bigl[ \bigl|S^{(N)}_\psi(t) \bigr| \bigr] \leq
\Vert\psi\Vert_\infty\mathbb{E}\bigl[\bigl\langle{\mathbf{1}}, \eta
^{(N)}_0\bigr\rangle+
E^{(N)}
(t) \bigr] <
\infty.
\end{equation}
Next, comparing (\ref{def-crp}) with (\ref{def-barwn}), it is clear
that the cumulative reneging process $R^{(N)}$ satisfies
\begin{equation} \label{RQ}
R^{(N)}(t)=S^{(N)}_{\theta^{(N)}}(t),\qquad t\geq0,
\end{equation}
where $\theta^{(N)}$ is given by
\begin{equation}\label{ps}\theta^{(N)}(x,s)=\mathbh{1}_{[x,\infty
)}\bigl(\chi
^{(N)}(s-)\bigr),\qquad
x\in{\mathbb R}, s\geq0.
\end{equation}
We now state the main result of this section.
For $s, r \in[0,\infty)$, recall that
$\langle\varphi(\cdot+ r,s), \nu^{(N)}_s \rangle$ is used as a
short form for
$\int_{[0, H^s)} \varphi(x+ r,s) \nu^{(N)}_s (dx)$, and likewise
for~$\eta^{(N)}$.
\begin{theorem}
\label{th-prelimit}
The processes $(E^{(N)}, X^{(N)},
\nu^{(N)},\eta^{(N)})$ a.s. satisfy the following coupled set of
equations: for
$\varphi\in\mathcal{C}^1_c([0,H^s)\times{\mathbb R}_+)$ and $t \in
[0,\infty)$,
\begin{eqnarray}
\label{eqn-prelimit1}
\bigl\langle\varphi(\cdot,t), \nu^{(N)}_{t} \bigr\rangle
& = & \bigl\langle\varphi(\cdot, 0), \nu^{(N)}_{0} \bigr\rangle+ \int
_{0}^t \bigl\langle\varphi_x(\cdot,s) + \varphi_s(\cdot,s), \nu^{(N)}_s
\bigr\rangle \,ds \nonumber\\[-8pt]\\[-8pt]
& &{} - D^{(N)}_\varphi(t) + \int_{[0,t]} \varphi(0,s)\, dK^{(N)}(s)
,\nonumber
\end{eqnarray}
for
$\psi\in\mathcal{C}^1_c([0,H^r)\times{\mathbb R}_+)$ and $t \in
[0,\infty)$,
\begin{eqnarray}\quad
\label{eqn-prelimit3} \bigl\langle\psi(\cdot, t), \eta^{(N)}_{t}
\bigr\rangle
& = & \bigl\langle\psi(\cdot, 0), \eta^{(N)}_{0} \bigr\rangle+ \int
_{0}^t \bigl\langle\psi_x(\cdot,s) + \psi_s(\cdot,s), \eta^{(N)}_s
\bigr\rangle \,ds \nonumber\\[-8pt]\\[-8pt]
& &{} - S^{(N)}_\psi(t) + \int_{[0,t]} \psi(0,s) \,dE^{(N)}(s)
,\nonumber\\
\label{eqn-prelimit2}
X^{(N)}(t) & = & X^{(N)}(0) + E^{(N)}(t) - D^{(N)}_{\mathbf{1}}(t)- R^{(N)}
(t), \\
\label{comp-prelimit}
N - \bigl\langle{\mathbf{1}}, \nu^{(N)}_t \bigr\rangle &=& \bigl[N - X^{(N)}(t) \bigr]^+,
\end{eqnarray}
where $K^{(N)}$ satisfies (\ref{def-kn}), $R^{(N)}$ satisfies
(\ref{RQ}) and $D^{(N)}_\varphi$ and $S^{(N)}_\psi$ are the processes
defined in (\ref{def-baren}) and (\ref{def-barwn}), respectively.
\end{theorem}
\begin{remark}
\label{rem-compdyn}
In the service dynamics, customer arrivals into
service are governed by the process $K^{(N)}$, the random duration in
service is determined by the distribution $G^s$ and departures are
represented by $D^{(N)}$. As captured by
(\ref{eqn-prelimit1}) and (\ref{eqn-prelimit3}), the dynamics of the
\textit{potential queue} is exactly analogous, with the customer
arrivals now governed by the process $E^{(N)}$, the random duration of stay
in the potential queue determined by $G^r$ and potential departures due
to reneging represented by $S^{(N)}$. Moreover, \textit{given
$K^{(N)}$}, the
dynamics of $\nu^{(N)}$ are exactly the same as in the case without
abandonment, which was well studied in~\cite{kasram07}. However, in the
presence of reneging, there is a significantly more complicated
coupling between $\nu^{(N)}$ and $K^{(N)}$, as captured by
(\ref{eqn-prelimit2}) and~(\ref{comp-prelimit}). In particular, this
involves the cumulative reneging process $R^{(N)}$, which has no analogy
with any quantity in the system without abandonments. Instead, as shown
in the sequel [see Lemma \ref{cor:1}, (\ref{rep-rcomp2}) and
Proposition \ref{prop:3}], we will exploit the representation
(\ref{RQ}) of $R^{(N)}$ in terms of the ``known'' quantity $S^{(N)}$
in order
to characterize the limit of the scaled sequence of reneging processes.
\end{remark}
\begin{pf*}{Proof of Theorem \ref{th-prelimit}}
The proof of (\ref{eqn-prelimit1}) can be carried out in exactly the
same way as the proof of (5.2) in Theorem 5.1 of \cite{kasram07}, since
the definition of $\nu^{(N)}$ in \cite{kasram07} is equivalent to the
definition given in (\ref{def-nun}) here since $da^{(N)}_j(t+)/dt = 0$
for all $j > K^{(N)}(t)$ in \cite{kasram07}. For the reasons mentioned in
Remark \ref{rem-compdyn}, the proof of (\ref{eqn-prelimit3}) is also
analogous except that the condition that each $\nu^{(N)}_t$ has total mass
no greater than $N$ is replaced by the argument below, which shows that
each $\eta^{(N)}_t$ has finite mass. We know that for $k = 0, \ldots,
\lfloor nt \rfloor$,
\[
\bigl\langle{\mathbf{1}},\eta^{(N)}_{({k+1})/{n}}\bigr\rangle \leq
E^{(N)}\biggl(\frac{k+1}{n} \biggr)+ \bigl\langle{\mathbf{1}},\eta^{(N)}_0
\bigr\rangle\leq
E^{(N)}(t+1)+\bigl\langle
{\mathbf{1}}, \eta^{(N)}_0\bigr\rangle.
\]
Thus, by taking the supremum over $k = 0, \ldots, \lfloor nt \rfloor$,
we have a.s.
\begin{equation} \label{renegnf}
\sup_{k = 0, \ldots, \lfloor nt
\rfloor} \bigl\langle{\mathbf{1}},\eta^{(N)}_{({k+1})/{n}} \bigr\rangle\leq
E^{(N)}(t+1)+\mathcal{E}^{(N)}_0<\infty.
\end{equation}
Equation (\ref{eqn-prelimit2})
follows from (\ref{def-dn}), (\ref{equivD}) and (\ref{RQ}), while
(\ref{comp-prelimit}) is just the nonidling condition
formulated in Section \ref{subsub-aux}.
\end{pf*}
\section{Main results}
\label{sec-flconj}
In this section we summarize our main results.
First, in Section \ref{subs-scaling}, we introduce the
fluid-scaled quantities and state our basic assumptions.
Then, in Section \ref{subs-fleqs}, we introduce the so-called
fluid equations, which provide a continuous analog of the characterization
of the discrete model given in
Theorem~\ref{th-prelimit}.
In Section \ref{subs-mainres} we present our main theorems. In
particular, we show that the fluid equations uniquely characterize the
strong law of large numbers or ``fluid''
limit of the many-server system, as the number of
servers goes to infinity.
\subsection{Fluid scaling and basic assumptions}
\label{subs-scaling}
Consider the following scaled versions of the basic processes
described in Section \ref{sec-mode}. For each $N \in{\mathbb N}$, the scaled
version of the state descriptor
$(\overline{\alpha}{}^{(N)}_E,\overline{X}{}^{(N)},\overline{\nu
}^{(N)},\overline{\eta}^{(N)})$ is given by
\begin{eqnarray} \label{fl-scaling}
\overline{\alpha}_E^{(N)}(t)& \doteq&\alpha_E^{(N)}(t),\qquad \overline
{X}{}^{(N)}(t)
\doteq\frac{X^{(N)}(t)}{N}, \\ \overline{\nu}{}^{(N)}_t (B) &\doteq&
\frac{\nu^{(N)}_t (B)}{N}, \qquad\overline{\eta}^{(N)}_t (B) \doteq
\frac{\eta^{(N)}_t (B)}{N},
\end{eqnarray}
for $t \in[0,\infty)$ and any Borel subset $B$ of ${\mathbb R}_+$.
Analogously, define
\begin{equation}
\label{fl-scaling2}
\overline{I}^{(N)}\doteq\frac{I^{(N)}}{N} \qquad\mbox{for } I=E,D,K,Q,R,S,
\mathcal{T}_t.
\end{equation}
Recall that $\mathcal{I}_{{\mathbb R}_+}[0,\infty)$ is the subset of
nondecreasing functions
$f \in\mathcal{D}_{{\mathbb R}_+}[0,\infty)$ with $f(0) = 0$,
$H^s=\sup\{x \in[0,\infty)\dvtx g^s(x) > 0\}$ and $H^r=\sup\{x \in
[0,\infty)\dvtx\break g^r(x) > 0\}$. Define
\begin{equation}\qquad
\label{def-newspace}
\mathcal{S}_0\doteq\left\{\matrix{
(e,x,\nu, \eta) \in\mathcal{I}_{{\mathbb R}_+}[0,\infty)\times
{\mathbb R}_+ \times
\mathcal{M}_F[0,H^s)\times\mathcal{M}_F[0,H^r)\mbox{:}\cr
1 - \langle{\mathbf{1}}, \nu\rangle= [1-x]^+}\right\}.
\end{equation}
$\mathcal{S}_0$ serves as the space of possible input data for the
fluid equations.
Our goal is to identify
the limit in distribution of the quantities
$(\overline{X}{}^{(N)}, \overline{\nu}{}^{(N)},\overline{\eta
}^{(N)})$, as \mbox{$N \rightarrow\infty$}. To this end,
we impose some natural assumptions on the sequence of
initial conditions $(\overline{E}{}^{(N)}, \overline{X}{}^{(N)}(0),
\overline{\nu}{}^{(N)}_0,\overline{\eta}{}^{(N)}_0)$.
\begin{ass}[(Initial conditions)]
\label{ass-init}
There exists an $\mathcal{S}_0$-valued random variable $(\overline
{E}, \overline{X}(0),
\overline{\nu}_0,\overline{\eta}_0)$
such that,
as $N \rightarrow\infty$, the following limits hold:
\begin{enumerate}
\item
$\overline{E}{}^{(N)}\rightarrow\overline{E}$ in $\mathcal
{D}_{{\mathbb R}_+}[0,\infty)$ $\mathbb{P}$-a.s., and
$\mathbb{E}[\overline{E}{}^{(N)}(t) ] \rightarrow\mathbb{E}[\overline
{E}(t) ]<\infty$ for
every $t \in[0,\infty)$;
\item
$\overline{X}{}^{(N)}(0) \rightarrow\overline{X}(0)$ in ${\mathbb
R}_+$ $\mathbb{P}$-a.s.;
\item
$\overline{\nu}{}^{(N)}_0 \stackrel{w}{\rightarrow}\overline{\nu
}_0$ in $\mathcal{M}_F[0,H^s)$;
\item
$\overline{\eta}^{(N)}_0 \stackrel{w}{\rightarrow}\overline{\eta
}_0$ in $\mathcal{M}_F[0,H^r)$, and $\mathbb{E}[\langle
1, \overline{\eta}^{(N)}_0 \rangle] \rightarrow\mathbb{E}[\langle
1, \overline{\eta}_0 \rangle] < \infty$.
\end{enumerate}
\end{ass}
\begin{remark}
\label{rem-skorep}
If the limits in (1) and (2) of Assumption \ref{ass-init} hold only in
distribution rather than
almost surely, then using the Skorokhod representation theorem in the
standard way,
it can be shown that all the stochastic process convergence results in the
paper continue to hold. Also, (1) and (4) of Assumption \ref{ass-init} and
(\ref{comp-prelimit}) imply that, for every $T\in[0,\infty)$,
\begin{equation}\label{init-bd}\qquad
\sup_{t\in[0,T]}\sup_N\mathbb
{E}\bigl[\overline{X}{}^{(N)}
(0)+\overline{E}{}^{(N)}
(t) \bigr]\leq\mathbb{E}\bigl[1+\bigl\langle1, \overline{\eta}^{(N)}_0 \bigr\rangle
+\overline{E}{}^{(N)}(T)
\bigr]<\infty.
\end{equation}
\end{remark}
The next assumption imposes some regularity conditions on $\overline
{\eta}_0$ and
$\overline{E}$.
\begin{ass} \label{ass-jump}
For each $t\geq0$, if $\overline{\eta}_0(\{t\})>0$, then $\overline
{\eta}
_0(t,t+\varepsilon)>0$ for every $\varepsilon>0$ and if $\overline
{E}(t)-\overline{E}
(t-)>0$, then $\overline{E}(t-)-\overline{E}(t-\varepsilon)>0$ for
every $\varepsilon>0$.
\end{ass}
\begin{remark} Assumption \ref{ass-jump} is trivially satisfied if
$\overline{\eta}_0$ and $\overline{E}$ are continuous, that is,
$\overline{\eta}_0(\{t\})=0$
for all $t\geq0$ and the function $\overline{E}$ is continuous.
\end{remark}
In order to state our last assumption,
define the hazard rate functions of $G^r$ and $G^s$ in the usual manner
\begin{eqnarray}
\label{def-h}
h^r(x) &\doteq&\frac{g^r(x)}{1 - G^r(x)},\qquad x\in[0,H^r),\\
h^s(x) &\doteq&\frac{g^s(x)}{1 - G^s(x)},\qquad x \in[0,H^s).
\end{eqnarray}
It is easy to verify that $h^r$ and $h^s$ are locally integrable on
$[0,H^r)$ and $[0,H^s)$, respectively.
\begin{ass} \label{ass-h}
There exists $L^s<H^s$ such that $h^s$ is either bounded or
lower-semicontinuous on $(L^s,H^s)$, and, likewise, there exists
$L^r<H^r$ such that $h^r$ is either bounded or lower-semicontinuous on
$(L^r,H^r)$.
\end{ass}
\subsection{Fluid equations}
\label{subs-fleqs}
We now introduce the so-called fluid
equations and provide some intuition as to why the
limit of any sequence $(\overline{X}{}^{(N)},\overline{\nu
}^{(N)},\overline{\eta}^{(N)})$ should be expected
to be a solution to these equations.
In Section \ref{sec:CSL}, we provide a rigorous proof of this fact.
\begin{defn}[(Fluid equations)]
\label{def-fleqns}
The c\`{a}dl\`{a}g function $(\overline{X},
\overline{\nu},\overline{\eta})$ defined on $[0,\infty)$ and
taking values in ${\mathbb R}_+
\times
\mathcal{M}_F[0,H^s)\times\mathcal{M}_F[0,H^r)$ is said to solve the
\textit{fluid equations} associated with
$(\overline{E}, \overline{X}(0), \overline{\nu}_0,\overline{\eta
}_0) \in\mathcal{S}_0$ and the hazard
rate functions $h^r$ and $h^s$
if and only if
for every $t \in[0,\infty)$,
\begin{equation}
\label{cond-radon}
\int_0^t \langle h^r, \overline{\eta}_s \rangle \,ds < \infty,\qquad \int_0^t
\langle h^s, \overline{\nu}_s \rangle \,ds < \infty,
\end{equation}
and the following relations are satisfied: for every $\varphi\in
\mathcal{C}^1_c([0,H^s)\times{\mathbb R}_+)$,
\begin{eqnarray}
\label{eq-ftmeas}\quad
\langle\varphi(\cdot,t), \overline{\nu}_t \rangle& = & \langle
\varphi(\cdot,0),
\overline{\nu}_0 \rangle+
\int_0^t \langle\varphi_s(\cdot,s), \overline{\nu}_s \rangle \,ds +
\int_0^t \langle\varphi_x(\cdot,s), \overline{\nu}_s \rangle \,ds
\nonumber\\[-8pt]\\[-8pt]
& &{}
- \int_0^t \langle h^s(\cdot) \varphi(\cdot,s), \overline{\nu}_s
\rangle \,ds+
\int_0^t \varphi(0,s) \,d\overline{K}(s),\nonumber
\end{eqnarray}
where
\begin{equation}
\label{eq-fk}
\overline{K}(t) = \langle{\mathbf{1}}, \overline{\nu}_t \rangle-
\langle{\mathbf{1}},
\overline{\nu}_0 \rangle+
\int_0^t \langle h^s, \overline{\nu}_s \rangle \,ds;
\end{equation}
for every $\psi\in\mathcal{C}^1_c([0,H^r)\times{\mathbb R}_+)$
\begin{eqnarray}
\label{eq-ftreneg}\qquad\quad
\langle\psi(\cdot,t), \overline{\eta}_t \rangle& = & \langle\psi
(\cdot
,0), \overline{\eta}_0 \rangle+
\int_0^t \langle\psi_s(\cdot,s), \overline{\eta}_s \rangle \,ds +
\int_0^t \langle\psi_x(\cdot,s), \overline{\eta}_s \rangle \,ds
\nonumber\\[-8pt]\\[-8pt]
& &{}
- \int_0^t \langle h^r(\cdot) \psi(\cdot,s), \overline{\eta}_s
\rangle \,ds+ \int_0^t \psi(0,s)\, d\overline{E}(s);\nonumber
\\
\label{fq}
\overline{Q}(t)&=& \overline{X}(t) - \langle{\mathbf{1}}, \overline
{\nu}_t \rangle; \\
\label{fqfreneg} \overline{Q}(t)&\leq&\langle{\mathbf{1}},
\overline{\eta}_t \rangle; \\
\label{fr}
\overline{R}(t) &=& \int_0^t \biggl(\int_0^{\overline
{Q}(s)}h^r((F^{\overline\eta_s}
)^{-1}(y))\,dy \biggr) \,ds,
\end{eqnarray}
where we recall that $F^{\overline\eta_t}(x)=\overline{\eta}_t[0,x];$
\begin{equation}
\label{eq-fx}
\overline{X}(t) = \overline{X}(0) + \overline{E}(t) - \int_0^t
\langle h^s, \overline{\nu}_s
\rangle \,ds - \overline{R}(t)
\end{equation}
and
\begin{equation}
\label{eq-fnonidling}
1 - \langle{\mathbf{1}}, \overline{\nu}_t \rangle= [1 - \overline
{X}(t)]^+.
\end{equation}
\end{defn}
It immediately follows from (\ref{fq}) and (\ref{eq-fnonidling}) that
for each $t\in[0,\infty)$,
\begin{equation}
\label{fqfx}\overline{Q}(t)=[\overline{X}(t)-1]^+.
\end{equation}
For future use, we also observe that (\ref{eq-fk}), (\ref{fq})
and (\ref{eq-fx}), when combined, show that for every $t\in[0,\infty)$,
\begin{equation}\label{qt-conserve} \overline{Q}(0)+\overline
{E}(t)=\overline{Q}(t)+\overline{K}
(t)+\overline{R}(t).
\end{equation}
We now provide an informal, intuitive explanation for the form of the fluid
equations. Equations (\ref{eq-fk}), (\ref{fq}) and (\ref{eq-fx}) are simply
mass conservation equations, that are fluid analogs of (\ref{def-kn}),
(\ref{def-xn}) and (\ref{eqn-prelimit2}), respectively, while (\ref
{fqfreneg})
expresses a bound, whose analog clearly holds in the pre-limit, as can
be seen
from (\ref{qn}). The relation (\ref{eq-fnonidling}) is simply the
fluid analog
of the nonidling condition (\ref{comp-prelimit}). Equations (\ref{eq-ftmeas})
and (\ref{eq-ftreneg}), which govern the evolution of the fluid age measure
$\overline{\nu}$ and queue measure $\overline{\eta}$, respectively,
are natural
analogs of the
pre-limit equations (\ref{eqn-prelimit1}) and (\ref{eqn-prelimit3}),
respectively. It is worthwhile to comment further on~the fourth terms
on the
right-hand sides of (\ref{eq-ftmeas}) and (\ref{eq-ftreneg}), which
characterize the fluid departure rate and potential reneging rate,
respectively, as integrals
of the corresponding hazard rate with respect to the age and queue
measures. Note that $\overline{\nu}_s(dx)$ represents the amount of
mass (limiting
fraction of customers) whose age lies in the range $[x,x+dx)$ at time
$s$, and
$h^s(x)$ represents the fraction of mass with age $x$ (i.e., with
service time
no less than $x$) that would depart from the system while having age in
$[x,x+dx)$. Hence, it is natural to expect $\langle h^s,\overline{\nu
}_s \rangle$
to represent the
departure rate of mass from the fluid system at time $s$. This was rigorously
proved in the case without abandonments in \cite{kasram07} (see Proposition
5.17 therein). By exploiting the exact analogy between $(\overline{\nu
},\overline{K},
\overline{D})$ and
$(\overline{\eta},\overline{E}, \overline{S})$ (see Remark \ref
{rem-compdyn}), it is clear
that the
potential reneging rate at time $s$ can be similarly represented as
$\langle
h^r,\overline{\eta}_s \rangle$. Thus the fluid potential reneging
process $\overline{S}
$, defined by
\begin{equation}
\label{def-fs}
\overline{S}(t) \doteq\int_0^t \langle h^r, \overline{\eta}_s
\rangle \,ds,\qquad t
\in[0,\infty),
\end{equation}
represents the cumulative amount of potential reneging from the fluid system
in the interval $[0,t]$. Due to the FCFS nature of the system, the
fluid queue at
time $s$ contains all the mass in $\overline{\eta}$ that is to the
left of
$(F^{\overline\eta_s})^{-1}(\overline{Q}(s))$, where recall
$F^{\overline\eta_s}$ is the
c.d.f. of $\overline{\eta}_s$.
Moreover, roughly speaking, given any $y\in[0,\overline{Q}(s)]$,
there is a
mass of
$dy$ customers in the queue whose waiting time at $s$ is
$(F^{\overline\eta_s})^{-1}(y)$ and the mean reneging rate of
customers with this
waiting time is $h^r((F^{\overline\eta_s})^{-1}(y))$. Thus the total
actual reneging
that has occurred in the interval $[0,t]$, is represented by the
integral, as specified in (\ref{fr}).
We close the section with a simple result on the action of time-shifts
on solutions to the fluid equations. For this, we need the following
notation: for any $t\in[0,\infty)$,
\begin{eqnarray*}
\overline{E}^{[t]}&\doteq&\overline{E}(t+\cdot)-\overline{E}(t),\qquad
\overline{K}{}^{[t]}\doteq\overline{K}
(t+\cdot)-\overline{K}(t),\\
\overline{X}{}^{[t]}&\doteq&\overline{X}(t+\cdot),\qquad
\overline{\nu}^{[t]}\doteq\overline{\nu}_{t+\cdot},
\\
\overline{R}{}^{[t]}&\doteq&\overline{R}(t+\cdot)-\overline{R}(t),\qquad
\overline{\eta}^{[t]}\doteq
\overline{\eta}_{t+\cdot},\qquad \overline{Q}{}^{[t]}\doteq\overline
{Q}(t+\cdot).
\end{eqnarray*}
\begin{lemma} \label{lem:shift}
Suppose the c\`{a}dl\`{a}g function $(\overline{X},
\overline{\nu},\overline{\eta})$ defined on $[0,\infty)$ and
taking values in ${\mathbb R}_+
\times\mathcal{M}_F[0,H^s)\times\mathcal{M}_F[0,H^r)$ solves the
fluid equations associated with
$(\overline{E}, \overline{X}(0), \overline{\nu}_0,\overline{\eta
}_0) \in\mathcal{S}_0$, then $(\overline{X}{}^{[t]},
\overline{\nu}^{[t]},\overline{\eta}^{[t]})$ solves the fluid
equations associated with
$(\overline{E}^{[t]}, \overline{X}(t), \overline{\nu}_t,\overline
{\eta}_t) \in\mathcal{S}_0$, where
$\overline{K}{}^{[t]}, \overline{R}{}^{[t]}, \overline{Q}{}^{[t]}$ are the
corresponding processes
that satisfy (\ref{eq-fk}), (\ref{fr}), (\ref{fq}) with $\overline
{\nu}
^{[t]}$, $\overline{\eta}^{[t]}$ and $\overline{X}{}^{[t]}$ in place
of $\overline{\nu}$,
$\overline{\eta}$ and $\overline{X}$.
\end{lemma}
The proof of the lemma just involves a rewriting of the fluid
equations, and is thus omitted.
\subsection{Summary of main results}
\label{subs-mainres}
Our first result establishes uniqueness of solutions to the fluid equations.
\begin{theorem} \label{thm:1}
Given any $(\overline{E}, \overline{X}(0), \overline{\nu}_0,
\overline{\eta}_0) \in\mathcal{S}_0$, there
exists at
most one solution $(\overline{X},\overline{\nu},\overline{\eta})$
to the associated fluid equations
(\ref{cond-radon})--(\ref{eq-fnonidling}). Moreover, if $\overline
{\nu}$ and
$\overline{\eta}$
satisfy (\ref{cond-radon}), then $(\overline{X},\overline{\nu
},\overline{\eta})$ is a
solution to the
fluid equations if and only if for every $f \in\mathcal
{C}_b({\mathbb R}_+)$,
\begin{eqnarray} \label{eq-freneg2}
\int_{[0,H^r)} f (x) \overline{\eta}_t (dx) & = & \int_{[0,H^r)} f(x+t)
\frac{1 - G^r(x+t)}{1 - G^r(x)} \overline{\eta}_0 (dx)
\nonumber\\[-8pt]\\[-8pt]
& &{}
+ \int_{[0,t]} f(t-s) \bigl(1
- G^r(t-s)\bigr)\, d \overline{E}(s), \nonumber\\
\label{eq-fmeas2}
\int_{[0,H^s)} f (x) \overline{\nu}_t (dx) & = & \int_{[0,H^s)} f(x+t)
\frac{1 - G^s(x+t)}{1 - G^s(x)} \overline{\nu}_0 (dx) \nonumber\\[-8pt]\\[-8pt]
& &{}
+ \int_{[0,t]} f(t-s) \bigl(1
- G^s(t-s)\bigr)\, d \overline{K}(s), \nonumber
\end{eqnarray}
where
\begin{eqnarray}\label{dis:fk}
\overline{K}(t) &=& [ \overline
{X}(0)-1]^+ - [\overline{X}(t) -
1]^+ + \overline{E}
(t)\nonumber\\[-8pt]\\[-8pt]
&&{} - \int_0^t \biggl( \int_0^{[\overline{X}(s) - 1]^+} h^r ( (
F^{\overline{\eta}_s} )^{-1} (y)
) \,dy \biggr) \,ds\nonumber
\end{eqnarray}
and for all $t\in[0,\infty)$, $\overline{X}$ satisfies $[\overline
{X}(t)-1]^+\leq
\langle{\mathbf{1}},\overline{\eta}_t \rangle$, the nonidling
condition (\ref
{eq-fnonidling}) and
\begin{eqnarray}\label{dis:fx}
\overline{X}(t) &=& \overline{X}(0) +
\overline{E}(t) - \int_0^t
\langle h^s,
\overline{\nu}_s \rangle \,ds\nonumber\\[-8pt]\\[-8pt]
&&{} -
\int_0^t \biggl( \int_0^{[\overline{X}(s) - 1]^+} h^r ( (
F^{\overline{\eta}_s} )^{-1} (y)
) \,dy \biggr) \,ds.\nonumber
\end{eqnarray}
Moreover, $\overline{K}$ also satisfies
\begin{eqnarray}
\label{eq-fk2}\qquad
\overline{K}(t) &=& \langle{\mathbf{1}}, \overline{\nu}_{t-s}
\rangle- \langle{\mathbf{1}},
\overline{\nu}_0
\rangle+\int_{[0,H^s )}
\frac{G^s(x+t-s)- G^s(x)}{1 - G^s(x)} \overline{\nu}_0 (dx)
\nonumber\\
&&{} + \int_0^t \biggl( \langle{\mathbf{1}}, \overline{\nu}_{t-s} \rangle-
\langle
\mathbf{1}, \overline{\nu}_0 \rangle\\
&&\hspace*{31.4pt}{} +\int_{[0,H^s )}
\frac{G^s(x+t-s)- G^s(x)}{1 - G^s(x)} \overline{\nu}_0 (dx) \biggr)
u^s(s) \,ds,\nonumber
\end{eqnarray}
where $u^s$ is the density of the renewal function $U^s$ associated with
$G^s$
($u^s$ exists since $G^s$ is assumed to have a density).
\end{theorem}
Next, we state the main result of the paper, which shows that, under
fairly general conditions, a solution to the fluid equations exists and
is the functional law of large numbers limit, as $N\rightarrow\infty
$, of the $N$-server system with abandonment.
\begin{theorem} \label{thm:2}
Suppose that Assumptions \ref{ass-init}--\ref{ass-h} hold, and let
$(\overline{E}, \overline{X}(0)$,\break $\overline{\nu}_0,\overline{\eta
}_0)\in\mathcal{S}_0$ be the limiting
initial condition. Then there exists a unique solution $(\overline
{X},\overline{\nu}
,\overline{\eta})$ to the associated fluid equations, and the
sequence $(\overline{X}{}^{(N)}
,\overline{\nu}{}^{(N)},\break\overline{\eta}^{(N)})$ converges weakly, as
$N\rightarrow\infty$, to
$(\overline{X},\overline{\nu},\overline{\eta})$.
\end{theorem}
Theorem \ref{thm:2} follows from Theorem \ref{th-tight}, which establishes
tightness of the sequence $\{\overline{X}{}^{(N)},\overline{\nu
}^{(N)},\overline{\eta}^{(N)}\}$, Theorem \ref{thm:FE},
which shows that any subsequential limit of the sequence
$\{\overline{X}{}^{(N)},\overline{\nu}{}^{(N)},\overline{\eta}^{(N)}\}
$ satisfies the fluid equations, and the uniqueness
of solutions to the fluid equations stated in Theorem \ref{thm:1}.
\begin{cor}
Suppose that Assumptions \ref{ass-init}--\ref{ass-h} hold. Given any
$(\overline{E}$, $\overline{X}(0), \overline{\nu}_0,\overline{\eta
}_0) \in\mathcal{S}_0$, let $(\overline{X}
,\overline{\nu},\overline{\eta})$ be the unique solution to the
associated fluid
equations (\ref{cond-radon})--(\ref{eq-fnonidling}) specified in
Theorem \ref{thm:1}. If the function $\overline{E}$ is absolutely continuous
and $\overline{\nu}_0$ and $\overline{\eta}_0$ are absolutely
continuous measures,
then the function $\overline{X}$ is also absolutely continuous and for every
$t\in[0,\infty)$, the measures $\overline{\nu}_t$ and $\overline
{\eta}_t$ are also
absolutely continuous.
\end{cor}
\begin{pf}
Since $\overline{E}$ is absolutely continuous, (\ref{dis:fx}) allows
us to
deduce that $\overline{X}$ is absolutely continuous. In turn, (\ref{dis:fk})
shows that $\overline{K}$ is also absolutely continuous. Then the
argument used
in proving Lemma 5.18 of \cite{kasram07} can be adapted, together with
(\ref{eq-freneg2}) and (\ref{eq-fmeas2}), to show that $\overline
{\nu}_t$ and
$\overline{\eta}_t$ are absolutely continuous for every $t\in
[0,\infty)$. This
proves the corollary.
\end{pf}
We now state the fluid limit result for the virtual waiting time process
$W^{(N)}$. This result is of particular interest in the context of call
centers.
Note that in the fluid system, for any $u > t$
the total mass of customers in queue at time
$u$ that arrived before time $t$ equals $\overline{Q}(u) -
\overline{\eta}_{u}[0,u-t]$, and the ages of these (fluid) customers
lie in the interval $(u-t, \overline\chi(u)]$, where
\begin{equation}\overline\chi(u) \doteq(F^{\overline{\eta
}_{u}})^{-1}(\overline{Q}
(u)).
\end{equation}
Observe that this definition is analogous to the definition of
$\chi^{(N)}$ given in (\ref{def-chi}). Therefore, by the same logic
that was used to justify the
expression (\ref{fr}) for $\overline{R}$ in Definition \ref{def-fleqns},
it is natural to conjecture that, for each
$t \in[0,\infty)$, the fluid limit of the sequence $\{\overline
{\mathcal
T}{}^{(N)}_t\}$ equals $\overline
{\mathcal T}_t$, where for $s\in[0,\infty)$,
\begin{eqnarray}
\label{dis:T}
\overline
{\mathcal T}_t(s) & \doteq&
\int_t^{t+s} \biggl( \int_{\overline{\eta}_{u}[0,u-t]}^{\overline{Q}(u)}
h^r( (F^{\overline\eta_{u}})^{-1}(y)) \,dy \biggr) \,du \nonumber\\[-8pt]\\[-8pt]
& = & \int_0^s \biggl( \int_{\overline{\eta}_{t+u}[0,u]}^{\overline{Q}(t
+u)}h^r( (F^{\overline\eta_{t+u}})^{-1}(y)) \,dy \biggr) \,du.
\nonumber
\end{eqnarray}
Also, define
\begin{equation}\label{Wbar} \overline W(t) \doteq\inf\biggl\{s\geq
0\dvtx\int
_t^{t+s} \langle h^s, \overline{\nu}_u \rangle \,du + \overline
{\mathcal T}_t(s)
\geq
\overline{Q}(t) \biggr\}.
\end{equation}
We will say a function $f\in\mathcal{D}[0,\infty)$ is uniformly strictly
increasing if it is absolutely continuous and there exists $a>0$ such
that the derivative of $f$ is bigger than and equal to $a$ for a.e.
$t\in[0,\infty)$. Note that for
any such
function, $f^{-1}(f(t))=t$ and $f^{-1}$ is continuous and strictly
increasing on $[0,\infty)$. We now characterize the fluid limit of the
(scaled) virtual waiting time in the system.
\begin{theorem} \label{thm:3}
Suppose that the conditions of Theorem \ref{thm:2} hold and that the
function $\int_0^{\cdot} \langle h^s, \overline{\nu}_u \rangle \,du$
is uniformly
strictly increasing. For each $t \geq0$, if $\overline{Q}$ is
continuous at
$t$, then $\overline{\mathcal
T}{}^{(N)}_t
\Rightarrow\overline{\mathcal T}_t$ and
$W^{(N)}(t) \Rightarrow\overline W(t)$, as
$N\rightarrow\infty$.
\end{theorem}
\section{Uniqueness of solutions to the fluid equations}
\label{sec-uniq}
In Section \ref{subs-cont}, we show that if $(\overline{X}, \overline
{\nu}, \overline{\eta})$
solve the fluid equations associated with a given initial condition
$(\overline{E},\overline{X}(0),\overline{\nu}_{0},\overline{\eta
}_{0})\in\mathcal{S}_0$,
then $\overline{\nu}$ (resp., $\overline{\eta}$) can be written
explicitly in terms
of the auxiliary fluid process $\overline{K}$ (resp., cumulative
arrival process $\overline{E}$). In
Section \ref{sec-flsol}, these representations are
used, along with the nonidling condition and the remaining fluid equations,
to show that there is at most one solution to the fluid
equations for a given initial condition.
\subsection{Integral equations for $(\overline{\nu},\overline{K})$
and $(\overline{\eta},\overline{E})$}
\label{subs-cont}
We begin by recalling Theorem~4.1 and Remark 4.3 of \cite{kasram07},
which we state here as Proposition \ref{th-pde}. This proposition
identifies an implicit relation that must be satisfied by the processes
$(\overline{\nu},\overline{K})$ and $(\overline{\eta},\overline
{E})$ that solve (\ref{eq-ftmeas}) and
(\ref{eq-ftreneg}), respectively.
\begin{prop}[\cite{kasram07}]
\label{th-pde} Let $G$ be the cumulative distribution function of a
probability distribution with density $g$ and hazard rate function
$h=g/(1-G)$, let $H\doteq\sup\{x\in[0,\infty)\dvtx g(x)>0\}$. Suppose
$\overline\pi\in\mathcal{D}_{\mathcal{M}_F[0,H)}[0,\infty)$ has
the property
that for every $m\in[0,H)$ and
$T\in[0,\infty)$, there exists $C(m,T)<\infty$ such that
\begin{equation}
\label{dis:hcontr} \int_0^\infty\langle\varphi(\cdot,s)h(\cdot
),\overline
\pi_s \rangle\,ds<C(m,T)\Vert\varphi\Vert_\infty
\end{equation}
for every $\varphi\in\mathcal{C}_c({\mathbb R}^2)$ with
$\operatorname{supp}(\varphi)\subset
[0,m]\times[0,T]$. Then given
any $\overline\pi_0 \in
\mathcal{M}_F[0,H)$ and $\overline{Z}\in\mathcal{I}_{{\mathbb
R}_+}[0,\infty)$, $\overline\pi$ satisfies the
integral equation
\begin{eqnarray}\label{eq-pde}\qquad
\langle
\varphi(\cdot,t), \overline\pi_t \rangle& = & \langle\varphi
(\cdot,0),
\overline\pi_0
\rangle+ \int_0^t \langle\varphi_s(\cdot,s), \overline\pi_s
\rangle \,ds +
\int_0^t \langle\varphi_x(\cdot,s), \overline\pi_s \rangle \,ds
\nonumber\\[-8pt]\\[-8pt]
& &{} - \int_0^t \langle\varphi(\cdot,s)h(\cdot), \overline\pi_s
\rangle \,ds + \int_{[0,t]} \varphi(0,s) \,d \overline{Z}(s)\nonumber
\end{eqnarray}
for
every $\varphi\in\mathcal{C}_c((-\infty,H)\times{\mathbb R})$ and
$t \in
[0,\infty)$, if and only if $\overline\pi$ satisfies
\begin{eqnarray}\label{eq2-fmeas}
\int_{[0,M)} f (x) \overline\pi_t (dx) &=& \int_{[0,M )} f(x+t)
\frac{1 - G(x+t)}{1 - G(x)} \overline\pi_0 (dx)\nonumber\\[-8pt]\\[-8pt]
&&{} + \int_{[0,t]}
f(t-s) \bigl(1
- G(t-s)\bigr) \,d \overline{Z}(s),\nonumber
\end{eqnarray}
for every $f \in\mathcal{C}_b({\mathbb R}_+)$ and $t\in
(0,\infty)$. Moreover, for every $f \in\mathcal{C}^1_b({\mathbb
R}_+)$ and $t\in(0,\infty
)$,
\begin{eqnarray}
\label{eq-ibp}
&&\int_0^t f(t-s) \bigl(1 - G(t-s)\bigr) \,d \overline{Z}(s) \nonumber\\
&&\qquad=
f(0) \overline{Z}(t) + \int_{[0,t]} f^\prime(t-s) \bigl(1 -
G(t-s)\bigr)\overline{Z}(s)
\,ds \\
&&\qquad\quad{}- \int_{[0,t]} f(t-s)
g(t-s)\overline{Z}(s) \,ds.\nonumber
\end{eqnarray}
\end{prop}
Fluid equations (\ref{cond-radon})--(\ref{eq-ftreneg}) show that
(\ref{dis:hcontr}) and (\ref{eq-pde}) are satisfied with
$(h,\overline
\pi,\overline Z)$ replaced by $(h^s,\overline{\nu},\overline{K})$
and $(h^r,\overline{\eta}
,\overline{E})$,
respectively. Therefore, the next result
follows from Proposition \ref{th-pde}.
\begin{cor} \label{cor:nueta} Processes $(\overline{\eta},\overline
{E})$ and $(\overline{\nu}
,\overline{K})$ satisfy (\ref{eq-freneg2}) and (\ref{eq-fmeas2}) for every
bounded Borel measurable function $f$ and $t\in[0,\infty)$.
Moreover, $\overline{K}$ satisfies the renewal equation
\begin{eqnarray}\label{dis:fkl}
\overline{K}(t)&=&\langle{\mathbf{1}}, \overline{\nu}_t \rangle-
\langle{\mathbf{1}}, \overline{\nu}
_0 \rangle
+\int_{[0,H^s )}
\frac{G^s(x+t)- G^s(x)}{1 - G^s(x)} \overline{\nu}_0 (dx)\nonumber\\[-8pt]\\[-8pt]
&&{} + \int_0^t
g^s(t-s)\overline{K}(s) \,ds\nonumber
\end{eqnarray}
for each $t\geq0$ and admits the representation
\begin{eqnarray*}
\overline{K}(t) &=& \int_{[0,t]} (\langle{\mathbf{1}}, \overline
{\nu}_{t-s} \rangle
- \langle
{\mathbf{1}}, \overline{\nu}_0 \rangle) \,dU^s(s) \\
& &{} +\int
_{[0,t]} \biggl(\int_{[0,H^s )}
\frac{G^s(x+t-s)- G^s(x)}{1 - G^s(x)} \overline{\nu}_0 (dx) \biggr) \,dU^s(s),
\end{eqnarray*}
where $dU^s$ is the renewal measure associated with the distribution $G^s$.
\end{cor}
\begin{remark}
Strictly speaking, in \cite{kasram07} the cumulative distribution
function $G$ was
assumed to be absolutely continuous and supported on $[0,\infty)$. However,
the proofs given there only use the local integrability
of the hazard rate function $h$ on $[0,H)$ and so continue
to hold for $G^r$ here, which may possibly have a positive mass at
$\infty$.
In fact, in the case that $G^r$ has a positive mass at $\infty$ the
hazard rate function $h^r$ is globally integrable on $[0,H^r)$.
\end{remark}
\subsection{Uniqueness of solutions}
\label{sec-flsol}
Let $(\overline{X},
\overline{\nu},\overline{\eta})$ be a solution to the fluid
equations associated with
$(\overline{E},
\overline{X}(0), \overline{\nu}_0,\overline{\eta}_0)$. Recall the
definitions of $\overline{Q}$ and
$\overline{R}$ that
are given in (\ref{fq}) and (\ref{fr}).
As an immediate consequence of (\ref{fr}), we have the following elementary
property.
\begin{lemma}
\label{fq0}
For any $0 \leq a \leq b < \infty$, if $\overline{Q}(t)=0$ for all
$t\in
[a,b]$, then $\overline{R}(b)-\overline{R}(a)=0$.
\end{lemma}
Next, we establish the intuitive result that the process $\overline
{K}$ that
represents the cumulative entry of ``fluid'' into service is nondecreasing.
\begin{lemma}
The function $\overline{K}$ is nondecreasing.
\end{lemma}
\begin{pf}
Fix $t\in[0,\infty)$ and $0\leq s<t$. If $\overline{X}(t)\geq1$,
then $\langle
{\mathbf{1}}, \overline{\nu}_t \rangle=1\geq\langle{\mathbf
{1}},\overline{\nu}_s \rangle
$ by
(\ref{eq-fnonidling}). Hence, by (\ref{eq-fk}), it follows that
\begin{equation}
\label{eq-kinc1} \overline{K}(t)-\overline{K}(s)=\langle{\mathbf
{1}}, \overline{\nu}_t \rangle-
\langle
{\mathbf{1}},
\overline{\nu}_s \rangle+ \int_s^t \langle h^s, \overline{\nu}_l
\rangle \,dl \geq0.
\end{equation}
If
$\overline{X}(t)< 1$, we consider two cases.
\textit{Case} 1. $\overline{X}(v)<1$ for all $v\in(s,t]$. In this case,
by (\ref{fq}) and (\ref{eq-fnonidling}), $\overline{Q}(v)=0$ for all
$v\in
(s,t]$. Hence, by Lemma \ref{fq0} and the right continuity of
$\overline{R}$,
$\overline{R}(t)-\overline{R}(s)=0$. By (\ref{qt-conserve}), it then
follows that
\begin{eqnarray*}
\overline{K}(t)-\overline{K}(s)&=& \overline{K}(t)-\overline{K}(s)
+ \overline{R}(t)-\overline{R}(s) +\overline{Q}(t)-\overline{Q}(s)\\
&=& \overline{E}(t)-\overline{E}(s)\\ &\geq& 0.
\end{eqnarray*}
\textit{Case} 2. There exists $v\in(s,t]$ such that $\overline{X}
(v)\geq
1$. Define $l\doteq\sup\{v\leq t\dvtx\overline{X}(v)\geq1\}$.
Then, clearly
$l\in(s,t]$
and $\overline{X}(l-)\geq1$. Now, (\ref{fr}) implies that $\overline
{R}$ is
continuous and
hence, by (\ref{eq-fx}), $\overline{X}(v)-\overline{X}(v-)\geq0$
for every $v\in
(0,\infty)$. Therefore, $\overline{X}(l)\geq1=\langle{\mathbf
{1}},\overline{\nu}_l
\rangle$,
with the latter equality being a consequence of the nonidling
condition (\ref{eq-fnonidling}). Due to the case assumption $\overline
{X}(t)<1$,
we must have $l<t$. Then (\ref{eq-kinc1}), with $t$
replaced by $l$, shows that $\overline{K}(l)-\overline{K}(s)\geq0$.
On the other hand, since
$\overline{X}(v)<1$ for all $v\in(l,t]$, the argument in case 1 above
shows that
$\overline{K}(t)-\overline{K}(l)\geq0$. Thus, in this case too, we
have $\overline{K}(t)-\overline{K}
(s)\geq0$.
\end{pf}
We now state the main result of this section.
\begin{theorem}\label{th-unique2}
For $i=1,2$, let $(\overline{X}{}^i,\overline{\nu}^i,\overline{\eta
}^i)$ be a solution to the
fluid equations associated with $(\overline{E},\overline
{X}(0),\overline{\nu}_0,\overline{\eta}_0)
\in\mathcal{S}_0$. Then $\overline{X}{}^1=\overline{X}{}^2, \overline
{\nu}^1=\overline{\nu}^2$ and
$\overline{\eta}^1=\overline{\eta}{}^2$.
\end{theorem}
\begin{pf}
For each $i=1,2$, let $\overline{Q}{}^i, \overline{K}{}^i, \overline
{D}{}^i, \overline{R}{}^i$ be the processes
associated with the solution $(\overline{X}{}^i,\overline{\nu
}^i,\overline{\eta}^i)$ to the fluid
equations for $(\overline{E},\overline{X}(0),\overline{\nu
}_0,\overline{\eta}_0)\in\mathcal{S}_0$. It
follows directly from Corollary \ref{cor:nueta} that
$\overline{\eta}^1=\overline{\eta}^2$. Let $\triangle A$ denote
$A^2-A^1$ for $A=\overline{Q},
\overline{K}, \overline{D}, \overline{R}$ and $\overline{\nu}$.
For each $t\geq0$, let $\triangle
\overline{\nu}_t$ be the measure that satisfies $\triangle
\overline{\nu}_t(\Xi)=\overline{\nu}_t^2(\Xi)-\overline{\nu
}_t^1(\Xi)$ for every
measurable set
$\Xi\subset[0,\infty)$. Choose $\delta>0$ and define
\[
\tau=\tau_\delta\doteq\inf\{t\geq0\dvtx\triangle\overline
{K}(t)\vee
\triangle
\overline{K}(t-)\geq\delta\}.
\]
We shall argue by contradiction to show that
$\tau=\infty$. Suppose that $\tau<\infty$.
We first claim that for each $t\in[0,\tau]$,
\begin{equation}\label{claim1}
\triangle\overline{K}(t)< \delta\qquad\mbox{if } \langle{\mathbf{1}},
\overline{\nu}^1_t
\rangle
=1.
\end{equation}
To see why this is true, suppose that $\langle{\mathbf{1}}, \overline
{\nu}
^1_t \rangle=1$ for some $t\in[0,\tau]$. Since $\langle{\mathbf{1}},
\overline{\nu}
^2_t \rangle\leq1$, we have $\langle{\mathbf{1}}, \triangle
\overline{\nu}_t
\rangle\leq
0$. When combined with (\ref{dis:fkl}) and the identity $\triangle
\overline{\nu}_0=0$, this shows that
\begin{equation} \label{dis:Kest}\qquad
\triangle\overline{K}(t)= \langle{\mathbf{1}}, \triangle\overline
{\nu}_t \rangle+ \int_0^t
g^s(t-s)\triangle\overline{K}(s) \,ds \leq\int_0^t g^s(t-s)\triangle
\overline{K}(s) \,ds.
\end{equation}
If $G^s(t)>0$ then, along with the fact that $\triangle\overline
{K}(s)<\delta
$ for all
$s\in[0,t)$, this implies $\triangle\overline{K}(t) <\delta
G^s(t)\leq\delta
$. On
the other hand, if $G^s(t)=0$, it must be that $g^s(s)=0$ for a.e.
$s\in
[0,t]$ and so (\ref{dis:Kest}) implies that $\triangle\overline
{K}(t)=0\leq
\delta$. Thus (\ref{claim1}) follows in either case. In addition, the
right-continuity of $\overline{K}{}^1$ and $\overline{K}{}^2$ implies
that $\triangle\overline{K}
(\tau)\geq\delta$. When combined with (\ref{claim1}), (\ref{fq})
and (\ref{eq-fnonidling}), this shows that
\begin{equation}\label{claim2} \overline{X}{}^1(\tau) = \langle
{\mathbf{1}},
\overline{\nu}^1_\tau\rangle
<1 \quad\mbox{and}\quad \overline{Q}{}^1(\tau)=0.
\end{equation}
Now, define
\[
r\doteq\sup\{t<\tau\dvtx\overline{Q}{}^2(t)<\overline{Q}{}^1(t) \}
\vee0.
\]
Then for every $t\in[r,\tau]$, $\overline{Q}{}^2(t)\geq\overline{Q}{}^1(t)$.
If $r=0$, then $\triangle\overline{K}(r)=\triangle\overline
{K}(0)=0<\delta$. On the
other hand, if $r>0$, there exists a sequence of $\{t_n\}_{n=1}^\infty
$ such that $t_n<r$ and $t_n\rightarrow r$ as $n\rightarrow\infty$
and $0\leq\overline{Q}{}^2(t_n)<\overline{Q}{}^1(t_n)$ for each $n\in
{\mathbb N}$. Since $\overline{Q}{}^1$
and $\overline{Q}{}^2$ are c\`{a}dl\`{a}g, this implies that
\begin{equation}
\label{uniq-qineq}\overline{Q}{}^2(r-)\leq\overline{Q}{}^1(r-),
\end{equation}
and, due to (\ref{fq}) and
(\ref{eq-fnonidling}), it also follows that $\overline{X}{}^1(t_n)>
\langle\mathbf{1}, \overline{\nu}^1_{t_n} \rangle=1$ for every
$n\in{\mathbb N}$. When combined with
(\ref{dis:Kest}), this shows that for $n\in{\mathbb N}$,
\[
\triangle\overline{K}(t_n)\leq\int_0^{t_n} g^s(t_n-s)\triangle
\overline{K}(s)
\,ds=\int_0^{t_n} g^s(s)\triangle\overline{K}(t_n-s) \,ds.
\]
Since $\overline{K}{}^1$ and $\overline{K}{}^2$ are c\`{a}dl\`{a}g, this
implies that
\[
\triangle\overline{K}(r-)\leq\int_0^{r} g^s(s)\triangle\overline
{K}\bigl((r-s)-\bigr) \,ds.
\]
Using the fact that $\triangle\overline{K}((r-s)-)<\delta$ for all
$s\in
(0,r)$, it is easy to see [once again, as in the analysis of (\ref
{dis:Kest}), by considering the cases $G^s(r)>0$ and $G^s(r)=0$
separately] that this implies
\begin{equation}\label{uniq-kineq} \triangle\overline{K}
(r-)<\delta.
\end{equation}
On the other hand, since (\ref{qt-conserve}) is satisfied with
$(\overline{K},\overline{R},\overline{Q})$ replaced by $(\overline{K}{}^i,\overline
{R}{}^i$, $\overline{Q}{}^i)$ for $i=1,2$, and
$\triangle\overline{Q}(0) + \triangle\overline{E}(t)=0$ for each
$t\geq0$, it
follows that
\[
\triangle\overline{K}(\tau)+\triangle\overline{R}(\tau)+\triangle
\overline{Q}(\tau
)=\triangle\overline{K}(r-)+\triangle\overline{R}(r-)+\triangle
\overline{Q}(r-)=0.
\]
Hence,
\[
\triangle\overline{K}(\tau) - \triangle\overline{K}(r-) =
-\bigl(\triangle\overline{R}(\tau) -
\triangle\overline{R}(r-) \bigr) -\triangle\overline{Q}(\tau)+\triangle
\overline{Q}(r-).
\]
Since
$-\triangle\overline{Q}(\tau)=\overline{Q}{}^1(\tau)-\overline
{Q}^2(\tau)=-\overline{Q}{}^2(\tau) \leq
0$ due to
(\ref{claim2}) and $\triangle\overline{Q}(r-)\leq0$ by (\ref{uniq-qineq}),
we obtain
\begin{equation}\label{KR}
\triangle\overline{K}(\tau) - \triangle\overline{K}(r-) \leq
-\bigl(\triangle
\overline{R}(\tau) - \triangle\overline{R}(r-) \bigr).
\end{equation}
We now show that the right-hand side of the above display is
nonpositive. For each $t\geq0$, by (\ref{fr}), we see that
\begin{eqnarray*}
\triangle\overline{R}(t)&=& \overline{R}{}^2(t)-\overline{R}{}^1(t) \\
&=& \int_{0}^{t}
\biggl(\int_0^{\overline{Q}{}^2(s)} h^r((\overline F^{\overline\eta
^2_s})^{-1}(y))\,dy \biggr) \,ds\\
&&{} -\int_{0}^{t} \biggl(\int_0^{\overline{Q}{}^1(s)}
h^r((\overline F^{\overline\eta^1_s})^{-1}(y))\,dy \biggr) \,ds.
\end{eqnarray*}
Since $\overline{\eta}^1=\overline{\eta}^2$, it follows that
$\overline F^{\overline
\eta^1_{\cdot}}=\overline F^{\overline\eta^2_{\cdot}}$. Together
with the continuity of $\overline{R}{}^1$ and~$\overline{R}{}^2$, this
yields the equation
\begin{eqnarray}
\label{R12}
&& \triangle\overline{R}(\tau)-\triangle\overline
{R}(r-)\nonumber\\
&&\qquad=
\triangle\overline{R}(\tau)-\triangle\overline{R}(r)\nonumber\\[-8pt]\\[-8pt]
&&\qquad= \int_{r}^{\tau
} \biggl(\int_0^{\overline{Q}{}^2(s)} h^r((\overline F^{\overline\eta
^1_s})^{-1}(y))\,dy \biggr) \,ds\nonumber\\
&&\qquad\quad{} -\int_{r}^{\tau} \biggl(\int_0^{\overline{Q}
^1(s)} h^r((\overline F^{\overline\eta^1_s})^{-1}(y))\,dy \biggr) \,ds.
\nonumber
\end{eqnarray}
However, by the definition of $r$, for each $t\in[r,\tau]$,
$\overline{Q}
^2(t)\geq\overline{Q}{}^1(t)$, and so $\triangle\overline{R}(\tau
)-\triangle\overline{R}
(r-)\geq0$. Together with (\ref{KR}) and (\ref{uniq-kineq}), this implies
\[
\triangle\overline{K}(\tau)\leq\triangle\overline{K}(r-) <\delta.
\]
Essentially the same argument can be used to also show that $\triangle
\overline{K}(\tau-)\leq\triangle\overline{K}(r-)<\delta$. Hence
$\triangle\overline{K}
(\tau)\vee\triangle\overline{K}(\tau-)<\delta$, which contradicts the
definition of $\tau$.
Thus we have proved that $\tau=\infty$ and $\overline
{K}^2(t)-\overline{K}{}^1(t)\leq
\delta$ for each $\delta>0$ and $t\geq0$. By letting $\delta
\rightarrow0$, we have $\overline{K}{}^2(t)\leq\overline{K}{}^1(t)$ for
all $t\geq0$. An
exactly analogous argument yields the reverse inequality $\overline
{K}^1(t)\leq
\overline{K}{}^2(t)$ for each $t\geq0$, and so it must be that
$\overline{K}{}^2= \overline{K}{}^1$.
By Corollary \ref{cor:nueta}, it follows that $\overline{\nu
}^1=\overline{\nu}^2$.
Also, by (\ref{qt-conserve}), we obtain
\begin{equation}\label{frq}\overline{R}{}^1+\overline{Q}
^1=\overline{R}{}^2+\overline{Q}{}^2.
\end{equation}
We now show that, in fact $\overline{Q}{}^1=\overline{Q}{}^2$ and
$\overline{R}{}^1=\overline{R}{}^2$. If there exists $t\in(0,\infty)$
such that $\overline{Q}
^1(t)>\overline{Q}{}^2(t)$, let
\[
s\doteq\sup\{v<t\dvtx\overline{Q}{}^1(v)\leq\overline{Q}{}^2(v)\}\vee0.
\]
Then $\overline{Q}{}^1(s-)\leq\overline{Q}{}^2(s-)$ and $\overline
{Q}^1(v)>\overline{Q}{}^2(v)$ for each
$v\in(s,t]$. Due to the fact that $\overline\eta^1=\overline\eta
^2$, we have
\begin{eqnarray*}
\overline{R}{}^1(t)-\overline{R}{}^1(s)&=&\int_{s}^{t} \biggl(\int
_0^{\overline{Q}{}^1(v)}
h^r((\overline F^{\overline\eta^1_v})^{-1}(y))\,dy \biggr) \,dv
\\
&\geq&
\int_{s}^{t} \biggl(\int_0^{\overline{Q}{}^2(v)} h^r((\overline
F^{\overline
\eta^2_v})^{-1}(y))\,dy \biggr) \,dv\\
&=& \overline{R}{}^2(t)-\overline{R}{}^2(s).
\end{eqnarray*}
From (\ref{frq}) and the continuity of $\overline{R}{}^i, i=1,2$, we deduce
that $\overline{Q}{}^1(t)-\overline{Q}{}^1(s-)\leq\overline
{Q}{}^2(t)-\overline{Q}{}^2(s-)$. Combining this
with the inequality $\overline{Q}{}^1(s-)\leq\overline{Q}{}^2(s-)$
proved above, we obtain
$\overline{Q}{}^1(t)\leq\overline{Q}{}^2(t)$, which leads to a
contradiction. Hence $\overline{Q}
^1(v)\leq\overline{Q}{}^2(v)$ for all $v\in(0,\infty)$. By symmetry,
we can
also argue that $\overline{Q}{}^1(v)\geq\overline{Q}{}^2(v)$ for all
$v\in(0,\infty)$.
This shows $\overline{Q}{}^1= \overline{Q}{}^2$ and, hence, $\overline
{R}{}^1= \overline{R}{}^2$. Finally, by
(\ref{fq}), we have $\overline{X}{}^1=\overline{X}{}^2$.
\end{pf}
\begin{pf*}{Proof of Theorem \ref{thm:1}}
The first statement in Theorem \ref{thm:1} follows from Theorem
\ref{th-unique2}. The second statement follows directly from Corollary
\ref{cor:nueta} and the fluid equations (\ref{fq}), (\ref{fr}),
(\ref{eq-fx}) and (\ref{fqfx}). The alternative representation
(\ref{eq-fk2}) for $\overline{K}$ is a direct consequence of the renewal
equation (\ref{dis:fkl}) and the fact that the first three terms on the
right-hand side of (\ref{dis:fkl}) are bounded by one.
\end{pf*}
\begin{remark}
\label{rem-compen0}
For future use, we observe here that
the result of Lemma 5.16 in \cite{kasram07} (and the analog with
$\overline{\nu}$ replaced by $\overline{\eta}$), which was
obtained for the model without abandonments,
is also valid in the present context.
This is because equations (\ref{eq-freneg2}) and (\ref{eq-fmeas2}) of
Theorem \ref{thm:1} and Corollary 4.14 of \cite{kasram07} show that,
in the terminology of \cite{kasram07}, $\{\overline{\eta}_s\}$
(resp., $\{\overline{\nu}_s\}$) satisfies
the simplified age equation associated with a certain Radon measure
$\xi(\overline{\eta}_0$, $\overline{E})$ and $h^r$
[resp.,
$\xi(\overline{\nu}_0$, $\overline{K})$ and $h^s$]. Therefore, by
Proposition 4.15
of \cite{kasram07}, it follows that the result of Lemma 5.16 of
\cite{kasram07} is also valid in the present context.
\end{remark}
\section{A family of martingales}
\label{subs-prelim}
In Section \ref{subs-comp}, we identify the compensators (with respect
to the
filtration $\mathcal{F}_t^{(N)}$) of the cumulative
departure, potential reneging and (actual) reneging processes. Then,
in Section \ref{subs-altcomp}, we establish a more convenient representation
for the compensator of the reneging process.
\subsection{Compensators}
\label{subs-comp}
For any bounded measurable function $\varphi$ on $[0,H^s)\times
{\mathbb R}
_+$, consider
the sequence $\{A^{(N)}_{\varphi,\nu}\}$ of processes given by
\begin{equation}
\label{def-dcompns}
A^{(N)}_{\varphi,\nu} (t) \doteq\int_0^t \biggl(
\int_{[0, H^s)} \varphi(x,s) h^s(x) \nu^{(N)}_s (dx) \biggr)
\,ds,\qquad
t \in[0,\infty).
\end{equation}
Likewise, for any bounded measurable function $\varphi$ on
$[0,H^r)\times{\mathbb R}_+$
and $N \in{\mathbb N}$, let
\begin{equation}
\label{def-dcompnr}
A^{(N)}_{\varphi,\eta} (t) \doteq\int_0^t \biggl(
\int_{[0, H^r)} \varphi(x,s) h^r(x) \eta^{(N)}_s (dx) \biggr)
\,ds,\qquad
t \in[0,\infty).
\end{equation}
In Proposition \ref{cor-compensatormeasn}, we show that
$A^{(N)}_{\varphi,\nu}$ (resp., $A^{(N)}_{\varphi,\eta}$) is
the $\mathcal{F}_t^{(N)}$-compensator
of the associated ``$\varphi$-weighted'' cumulative departure process
$D^{(N)}_\varphi$
(resp., $S^{(N)}_\varphi$).
A~similar result was established
in \cite{kasram07} for the model without abandonments. However,
the filtration $\{\mathcal{F}_t^{(N)}\}$ considered here
is larger than the one considered in \cite{kasram07}, and
so Proposition \ref{cor-compensatormeasn} does not directly
follow from the results in \cite{kasram07}.
\begin{prop}
\label{cor-compensatormeasn} The following properties hold:
\begin{enumerate}
\item For every bounded measurable function $\varphi$ on
$[0,H^s)\times
{\mathbb R}_+$ such that the function $s \mapsto\varphi
(a^{(N)}_j(s),s)$ is left
continuous on $[0,\infty)$ for each $j$, the process
$M^{(N)}_{\varphi,\nu}$ defined by
\begin{equation}\label{def-martnmeasn}
M^{(N)}_{\varphi,\nu} \doteq D^{(N)}_\varphi- A^{(N)}_{\varphi
,\nu}
\end{equation}
is a local $\mathcal{F}_t^{(N)}$-martingale.
Moreover, for every $N \in{\mathbb N}$, $t \in[0,\infty)$ and
$m \in[0,H^s)$,
\begin{equation}
\label{bd-3}
\bigl|A^{(N)}_{\varphi,\nu} (t)\bigr| \leq\Vert\varphi\Vert_{\infty}
\bigl( X^{(N)}(0) + E^{(N)}(t) \bigr) \biggl( \int_0^m h^s(x) \,dx
\biggr) < \infty
\end{equation}
for every $\varphi\in\mathcal{C}_{c} ([0,H^s) \times
{\mathbb R}_+)$ with $\operatorname{supp}(\varphi) \subset[0,m]
\times{\mathbb R}_+$.
In addition, the quadratic variation process $\langle\overline
{M}{}^{(N)}_{\varphi
,\nu} \rangle$ of the scaled process $\overline{M}{}^{(N)}_{\varphi
,\nu}\doteq
M^{(N)}_{\varphi,\nu}/N$ satisfies
\begin{equation}\label{lim-qv}
\lim_{N\rightarrow\infty}\mathbb{E}\bigl[ \bigl\langle\overline
{M}{}^{(N)}_{\varphi,\nu} \bigr\rangle
(t) \bigr] =
0;\qquad \overline{M}{}^{(N)}_{\varphi,\nu}\Rightarrow\mathbf{0}\qquad\mbox{as
} N \rightarrow\infty.
\end{equation}
\item
Furthermore, properties (\ref{def-martnmeasn})--(\ref{lim-qv}) also hold
with $D, a_j, \nu, H^s$ and $h^s$, respectively, replaced
by $S, w_j, \eta, H^r$ and $h^r$.
\end{enumerate}
\end{prop}
\begin{pf}
In Lemma 5.4 and Corollary 5.5 of \cite{kasram07}, it was shown that
$A^{(N)}_{\varphi,\nu}$ is the compensator of $D^{(N)}_\varphi$
with respect
to a
certain filtration. The filtration $\{\mathcal{F}_t^{(N)}\}$ that we
consider here is larger than the filtration used in \cite{kasram07}
since it also includes the $\sigma$-algebra generated by the potential
waiting times $\{\eta^{(N)}_j(s), s\leq t, j=-\mathcal
{E}^{(N)}_0+1,\ldots,E^{(N)}(t)\}$.
Thus the results of \cite{kasram07} do not directly apply here.
Nevertheless, as we prove below, the result continues to hold due to
the assumed independence of the patience and service times.
We first claim that for every $\mathcal{F}_t^{(N)}$-stopping time
$\Upsilon$,
\begin{eqnarray}\label{eqn-KR}
&&
\mathbb{E}\bigl[\mathbh{1}_{\{\theta_n^k\leq
{j}/{2^m}<\Upsilon, \zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta
_n^k\leq
({j+1})/{2^m}\}}|\mathcal{F}^{(N)}_{{j}/{2^m}}
\bigr]\nonumber\\[-8pt]\\[-8pt]
&&\qquad
=\mathbh{1}_{\{\theta_n^k\leq{j}/{2^m}<\Upsilon, \zeta
_n^k>{j}/{2^m}\}}\int_{j/2^m}^{(j+1)/2^m}\frac{g^s(u-\theta
_n^k)}{1-G^s({j}/{2^m}-\theta_n^k)}\,du,\nonumber
\end{eqnarray}
where $\theta_n^k$ (resp., $\zeta_n^k$) is the time at which the $n$th
customer to be served at station $k$ starts (resp., completes)
service. Then $\zeta_n^k-\theta_n^k$ is the service time of the $n$th
customer to be served at station $k$, which has cumulative distribution
function $G^s$. In order to show the equality in (\ref{eqn-KR}), it
suffices to show that for every bounded $\mathcal{F}^{(N)}_{
{j}/{2^m}}$-adapted random variable $H$,
\begin{eqnarray}\label{eqn-KR2}\quad
&&
\mathbb{E}\bigl[H\mathbh{1}_{\{\theta_n^k\leq
{j}/{2^m}<\Upsilon, \zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta
_n^k\leq
({j+1})/{2^m}\}} \bigr] \nonumber\\[-8pt]\\[-8pt]
&&\qquad
= \mathbb{E}\biggl[H\mathbh{1}_{\{\theta
_n^k\leq{j}/{2^m}<\Upsilon, \zeta_n^k>{j}/{2^m}\}}\int
_{j/2^m}^{(j+1)/2^m}\frac{g^s(u-\theta_n^k)}{1-G^s(
{j}/{2^m}-\theta_n^k)}\,du \biggr].\nonumber
\end{eqnarray}
For $j\in{\mathbb N}$, $m\in{\mathbb N}$, define $\mathcal{G}_
{j/2^m}^{(N)}$
be the
$\sigma$-algebra to be generated by the events $\{(\theta_n^k\leq
x)\cap(\theta_n^k\leq\frac{j}{2^m},\zeta_n^k>\frac{j}{2^m}),
x\geq0\}$. In particular, $\mathcal{G}_{j/2^m}^{(N)}$ contains the
information of the ages of all customers in service at time $\frac
{j}{2^m}$. Recall that the patience times and the service times of
customers are assumed to be independent. Therefore, given $\mathcal
{G}_{j/2^m}^{(N)}$, $\zeta_n^k-\theta_n^k$ and $\mathcal
{F}^{(N)}_{{j/2^m}}$ are conditionally\vspace*{1pt} independent. Hence, it
follows from the left-hand side of (\ref{eqn-KR2}) that
\begin{eqnarray*}
&&\hspace*{-4pt}
\mathbb{E}\bigl[H\mathbh{1}_{\{\theta_n^k\leq
{j}/{2^m}<\Upsilon,
\zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta_n^k\leq(
{j+1})/{2^m}\}}
\bigr] \\
&&\hspace*{-4pt}\qquad=
\mathbb{E}\bigl[\mathbb{E}\bigl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon\}
}\mathbh{1}_{\{
\theta_n^k\leq
{j}/{2^m},
\zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta_n^k-\theta_n^k\leq
({j+1})/{2^m}-\theta_n^k\}}|\mathcal{G}_{j/2^m}^{(N)}
\bigr] \bigr] \\
&&\hspace*{-4pt}\qquad=
\mathbb{E}\bigl[\mathbb{E}\bigl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon\}
}|\mathcal
{G}_{j/2^m}^{(N)} \bigr]\\
&&\qquad\hspace*{15pt}{}\times\mathbb{E}\bigl[\mathbh{1}_{\{\theta_n^k\leq
{j}/{2^m},
\zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta_n^k-\theta_n^k\leq
({j+1})/{2^m}-\theta_n^k\}}|\mathcal{G}_{j/2^m}^{(N)}
\bigr] \bigr]
\end{eqnarray*}
and
\begin{eqnarray*}
&&\mathbb{E}\bigl[\mathbh{1}_{\{\theta_n^k\leq{j}/{2^m},
\zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{\zeta_n^k-\theta_n^k\leq
({j+1})/{2^m}-\theta_n^k\}}|\mathcal{G}_{j/2^m}^{(N)}
\bigr] \\
&&\qquad=\mathbh{1}_{\{\theta_n^k\leq{j/2^m},
\zeta_n^k>{j/2^m}\}}\int_{j/2^m}^{(j+1)/2^m}\frac
{g^s(u-\theta_n^k)}{1-G^s({j}/{2^m}-\theta_n^k)}\,du.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&&\mathbb{E}\bigl[\mathbb{E}\bigl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon
\}}|\mathcal{G}_{j/2^m}^{(N)} \bigr]\mathbb{E}\bigl[\mathbh{1}_{\{
\theta
_n^k\leq{j}/{2^m}, \zeta_n^k>{j}/{2^m}\}}\mathbh{1}_{\{
\zeta
_n^k-\theta_n^k\leq({j+1})/{2^m}-\theta_n^k\}}|\mathcal{G}_
{j/2^m}^{(N)} \bigr] \bigr]
\\
&&\qquad= \mathbb{E}\biggl[\mathbb{E}
\bigl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon\}}|\mathcal{G}_
{j/2^m}^{(N)} \bigr]\mathbh{1}_{\{\theta_n^k\leq{j}/{2^m}, \zeta
_n^k>{j}/{2^m}\}}\\
&&\qquad\quad\hspace*{28.6pt}{}\times\int_{j/2^m}^{(j+1)/2^m}\frac{g^s(u-\theta
_n^k)}{1-G^s({j}/{2^m}-\theta_n^k)}\,du \biggr]
\\
&&\qquad= \mathbb{E}
\biggl[\mathbb{E}\biggl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon\}}\mathbh{1}_{\{
\theta
_n^k\leq{j}/{2^m}, \zeta_n^k>{j}/{2^m}\}}\\
&&\qquad\quad\hspace*{9.6pt}{}\times\int
_{j/2^m}^{(j+1)/2^m}\frac{g^s(u-\theta_n^k)}{1-G^s(
{j}/{2^m}-\theta_n^k)}\,du\Big|\mathcal{G}_{j/2^m}^{(N)}
\biggr] \biggr]
\\
&&\qquad= \mathbb{E}\biggl[H\mathbh{1}_{\{{j}/{2^m}<\Upsilon\}}\mathbh
{1}_{\{
\theta_n^k\leq{j}/{2^m}, \zeta_n^k>{j}/{2^m}\}}\int
_{j/2^m}^{(j+1)/2^m}\frac{g^s(u-\theta_n^k)}{1-G^s(
{j}/{2^m}-\theta_n^k)}\,du \biggr].
\end{eqnarray*}
This shows that (\ref{eqn-KR2}), and therefore (\ref{eqn-KR}), holds.
If $\varphi$ is bounded, measurable and such that the function
$s\mapsto
\varphi(a^{(N)}_j(s),s)$ is left
continuous for each $j$, then the process $\{\varphi(a^{(N)}_j(s),s),
s\geq0\}$ is $\mathcal{F}_t^{(N)}$-predictable. Therefore, it follows
from the standard theory
(cf. Theorem 3.18 of \cite{jacshibook}) that $M^{(N)}_{\varphi,\nu}$
is a local
$\mathcal{F}_t^{(N)}$-martingale. Inequality (\ref{bd-3}) can be established
exactly as in Proposition 5.7 of \cite{kasram07} and assertions
(\ref{lim-qv}) can be proved using the same argument as in Lemma 5.9 of
\cite{kasram07}, thus establishing property (1).
Due to the analogy between the service dynamics
and the potential queue dynamics (see Remark \ref{rem-compdyn}),
property (2) is a direct consequence of
property (1).
\end{pf}
\begin{remark}
\label{rem-compen}
It is easy to see that Lemmas 5.6 and 5.8 of \cite{kasram07}
continue to be valid in the presence of abandonments. Indeed, the proofs
of Lemmas 5.6 and 5.8 of \cite{kasram07} only depend on Assumption 1
and Corollary
5.5 therein (since, as shown in Lemma 5.12 of \cite{kasram07},
the additional conditions (5.32) and (5.33) of Lemma 5.8 of \cite
{kasram07} can be derived from
Assumption 1), which correspond to Assumption \ref{ass-init} and Proposition
\ref{cor-compensatormeasn} of this paper. In addition, due to the parallels
between the dynamics of $\nu^{(N)}$ and $\eta^{(N)}$ (see Remark \ref
{rem-compdyn}),
the analogs of the
results in Lemmas 5.6 and 5.8, with $D^{(N)}, \nu^{(N)}, G^s$ and $H^s$,
respectively, replaced by
$S^{(N)}, \eta^{(N)}, G^r$ and $H^r$, also hold. In this case, even though
$\eta^{(N)}_0$ is (unlike $\nu^{(N)}_0$) not necessarily a
sub-probability measure,
the verification of the conditions analogous to (5.32) and
(5.33) of Lemma 5.8 in \cite{kasram07} can still be carried out in
the same manner since
Assumption \ref{ass-init} implies that the sequence
$\{\langle{\mathbf{1}}, \eta^{(N)}_0\rangle\}$ is tight. Moreover,
even though
$G^r$ is
allowed to have a mass at $\infty$, the proofs of
Lemmas 5.6 and 5.8 are still valid, with the renewal function $U^s$ now
replaced by the function
$U^r(\cdot) = \int_0^\cdot\sum_{n=1}^\infty(g^r)^{*n} (s) \,ds$, where
$(g^r)^{*n}$ is the $n$th convolution of $g^r$ on $[0,\infty)$.
\end{remark}
Now, note from (\ref{RQ}) that $R^{(N)}=S^{(N)}_{\theta^{(N)}}$, where
$\theta^{(N)}$ is defined by (\ref{ps}). In view of the fact that
$A^{(N)}_{\varphi,\eta}$ is the compensator for
$S^{(N)}_{\varphi}$, it is natural to conjecture that the compensator of
$R^{(N)}$ is equal to $A^{(N)}_{\theta^{(N)},\eta}$, where
\begin{eqnarray}
\label{rep-rcomp1}
A^{(N)}_{\theta^{(N)},\eta} (t) \doteq\int_0^t \biggl(
\int_{[0, H^r)} \mathbh{1}_{[0,\chi^{(N)}(s-)]}(x) h^r(x) \eta^{(N)}_s
(dx) \biggr) \,ds,\nonumber\\[-8pt]\\[-8pt]
\eqntext{ t \in[0,\infty).}
\end{eqnarray}
However, this is not immediate from Proposition \ref
{cor-compensatormeasn}(2) since $\theta^{(N)}(w^{(N)}_j(\cdot),\cdot
)$ is not left continuous for any $j$. Instead, we approximate $\theta
^{(N)}$ by a sequence $\{\theta^{(N)}_m\}_{N\in{\mathbb N}}$ defined by
\begin{equation}
\label{def-thetam}\theta^{(N)}_m(x,s)\doteq\mathbh{1}_{(x-
{1}/{m},\infty)}\bigl(\chi^{(N)}(s-)\bigr),
\end{equation}
which is shown to be left
continuous in Lemma \ref{psim}. Then in Lemma \ref{cor:1}, we use an
approximation argument to show that $A^{(N)}_{\theta^{(N)},\eta}$
is indeed the compensator of $R^{(N)}$.
\begin{lemma} \label{psim}
For each $m\geq1$, $x\in{\mathbb R}$ and $s\in{\mathbb R}_+$, the
sequence $\{\theta
^{(N)}_m\}_{N\in{\mathbb N}}$ defined by (\ref{def-thetam}) satisfies the
following two properties:
\begin{enumerate}
\item For every $N\in{\mathbb N}, x\in{\mathbb R}, s\in{\mathbb R}$,
$\theta
^{(N)}_m(x,s)$ is nonincreasing in $m$ and converges, as $m\rightarrow
\infty$, to $\theta^{(N)}(x,s)$ for every sample point in $\Omega$.
\item For each $N,m\in{\mathbb R}$, $j\in{\mathbb Z}$, the process
$\theta
^{(N)}_m(w^{(N)}_j(\cdot),\cdot)$ has left continuous paths on
$(0,\infty)$.
\end{enumerate}
\end{lemma}
\begin{pf}
The first property is immediate from the definition of
$\theta^{(N)}_m$. For the second property, fix $N,m\in{\mathbb N}, s>0,
j\in
{\mathbb Z}$ and $\omega\in\Omega$. To ease the notation, we shall suppress
$\omega$ from the notation. Let $\{s_n\}$ be a sequence in $(0,\infty)$
such that $s_n\uparrow s$ as $n\rightarrow\infty$. We now consider two
mutually exclusive cases.
\textit{Case} 1. $\theta^{(N)}_m(w^{(N)}_j(s),s)=1$. Then
$w^{(N)}_j(s)< \chi^{(N)}(s-)+1/m$. Since $w^{(N)}_j$ is
nondecreasing, $w^{(N)}_j(s_n)\leq w^{(N)}_j(s)$ and since the process
$\{\chi^{(N)}(s-), s\geq0\}$ is left continuous, we have, for all
$n$ large enough, $w^{(N)}_j(s_n)<\chi^{(N)}(s_n-)+1/m$. Hence,
$\theta^{(N)}_m(w^{(N)}_j(s_n),s_n)=1$ for all $n\in{\mathbb N}$.
Thus, in
this case, $\theta^{(N)}_m(w^{(N)}_j(\cdot),\cdot)$ is left
continuous at $s$.
\textit{Case} 2. $\theta^{(N)}_m(w^{(N)}_j(s),s)=0$. Then
$w^{(N)}_j(s)\geq\chi^{(N)}(s-)+1/m$. It follows from Lemma \ref
{lem-chi} that for all sufficiently large $n$, $\chi^{(N)}(s-)-\chi
^{(N)}(s_n-)=s-s_n>0$. Since (\ref{def-waitjn}) implies
$w_j^{(N)}(s)-w_j^{(N)}(s_n)\leq s-s_n$ for all $n\in{\mathbb N}$, this
implies $w^{(N)}_j(s_n)\geq\chi^{(N)}(s_n-)+1/m$ for all $n$ large
enough. Hence, $\theta^{(N)}_m(w^{(N)}_j(s_n),s_n)=0$ and $\theta
^{(N)}_m(w^{(N)}_j(\cdot),\cdot)$ is again left continuous at $s$.
\end{pf}
\begin{lemma} \label{cor:1}
For every $N\in{\mathbb N}$,
the process $M^{(N)}_{\theta^{(N)},\eta}$ defined by
\begin{equation}
M^{(N)}_{\theta^{(N)},\eta} \doteq R^{(N)}- A^{(N)}_{\theta
^{(N)},\eta}
\end{equation}
is a local $\mathcal{F}_t^{(N)}$-martingale.
In addition, as $N
\rightarrow\infty$,
\begin{equation}\label{lim-psi}
\mathbb{E}\bigl[ \bigl\langle\overline
{M}{}^{(N)}_{\theta^{(N)},\eta}
\bigr\rangle(t) \bigr] \rightarrow 0,\qquad \overline{M}{}^{(N)}_{\psi,\eta} \Rightarrow
\mathbf{0}
\quad\mbox{and}\quad \overline{M}{}^{(N)}_{\theta^{(N)},\eta} \Rightarrow
\mathbf{0}.
\end{equation}
\end{lemma}
\begin{pf}
Fix $N\in{\mathbb N}$, and let $A^{(N)}_{\theta^{(N)}_m,\eta}$,
$m\in{\mathbb N}
$, be defined in the obvious way
\begin{equation}
\label{def-dcompnpsim}
A^{(N)}_{\theta^{(N)}_m,\eta} (t) \doteq\int_0^t \biggl(
\int_{[0, H^r)} \theta_m^{(N)}(x,s) h^r(x) \eta^{(N)}_s (dx)
\biggr) \,ds.
\end{equation}
By Proposition \ref{cor-compensatormeasn}(2) and Lemma \ref{psim}(2), the
process $A^{(N)}_{\theta^{(N)}_m,\eta}$ is the $\mathcal
{F}_t^{(N)}$-compen\-sator of
the process $S^{(N)}_{\theta_m^{(N)}}$, and the process
$M^{(N)}_{\theta^{(N)}_m,\eta}$ defined by
\begin{equation}\label{def-martnpsim}
M^{(N)}_{\theta^{(N)}_m,\eta} \doteq S^{(N)}_{\theta_m^{(N)}}-
A^{(N)}_{\theta
^{(N)}_m,\eta}
\end{equation}
is a local $\mathcal{F}_t^{(N)}$-martingale. Now, by Lemma \ref{psim}(1),
$\theta^{(N)}_m\rightarrow\theta^{(N)}$ pointwise on ${\mathbb R}^2_+$,
$|\theta^{(N)}_m(x,s)-\theta^{(N)}(x,s)|\leq1$ for all $(x,s)\in
{\mathbb R}^2_+$,
and $\mathbb{E}[S^{(N)}_{\mathbf{1}}(t) ]<\infty$,
$\mathbb{E}[A^{(N)}_{{\mathbf{1}},\eta}(t) ]<\infty$ for all
$t\in
(0,\infty)$. Hence, an application of the dominated convergence
theorem shows
that for all $t\in(0,\infty)$, as $m\rightarrow\infty$,
\[
\mathbb{E}\Bigl[\sup_{0\leq s\leq t} \bigl|A^{(N)}_{\theta^{(N)}_m,\eta}(s)-
A^{(N)}_{\theta^{(N)},\eta}(s) \bigr| \Bigr]\rightarrow0
\]
and
\[
\mathbb{E}\Bigl[\sup_{0\leq s\leq t} \bigl|S^{(N)}_{\theta_m^{(N)}}(s)-
S^{(N)}_{\theta^{(N)}}(s) \bigr|
\Bigr]\rightarrow
0,
\]
and hence\vspace*{-1pt} $M^{(N)}_{\theta^{(N)}_m,\eta}$ converges in distribution to
$M^{(N)}_{\theta^{(N)},\eta}$. Since $ |S^{(N)}_{\theta
_m^{(N)}}(t)-\break S^{(N)}_{\theta_m^{(N)}}
(t-) |\leq1$
for all $t\in[0,\infty)$ and $m\in{\mathbb N}$, we conclude that
$M^{(N)}_{\theta^{(N)},\eta}$ is a local $\mathcal
{F}_t^{(N)}$-martingale by
Corollary 1.19 of Chapter IX of \cite{jacshibook}.
Given that $M^{(N)}_{\theta^{(N)}, \eta}$ is a martingale,
the proof of the limits (\ref{lim-psi}) is exactly analogous to
the proof of~(\ref{lim-qv}), as carried out in
Lemma 5.9 of \cite{kasram07}.
\end{pf}
\subsection{An alternative representation for the compensator of $R^{(N)}$}
\label{subs-altcomp}
We now derive an alternative, more convenient representation for
$A^{(N)}_{\theta^{(N)},\eta}$, or more generally, for processes of
the form $A^{(N)}_{\theta^{(N)},\eta}$, but with $h^r$ replaced by
an arbitrary measurable function $h$.
In what follows, recall that $F^{\eta_t^{(N)}}(x)=\eta^{(N)}_t[0,x]$
and its inverse
$(F^{\eta_t^{(N)}})^{-1}$ is as defined in (\ref{inverse}).
\begin{prop} \label{lem:uni5}
For each $N\in{\mathbb N}$, $t\geq0$ and measurable function $h$ on
$[0,H^r)$,
\begin{eqnarray}\label{lem:ext1}
&&\int_{[0,H^r)}\mathbh{1}_{[0,\chi
^{(N)}(t-)]}(x)h(x)\eta^{(N)}_t(dx)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad=\int_0^{Q^{(N)}(t)+\iota^{(N)}(t)}h((F^{\eta_t^{(N)}})^{-1}(y))\,dy,\nonumber
\end{eqnarray}
where
\begin{equation}
\label{iota}\iota^{(N)}(t)\doteq\cases{
0, &\quad if $\bigl(\chi^{(N)}(t-)-\chi^{(N)}(t)\bigr)\bigl(K^{(N)}
(t)-K^{(N)}(t-)\bigr)=0$,\vspace*{2pt}\cr
1, &\quad if
$\bigl(\chi^{(N)}(t-)-\chi^{(N)}(t)\bigr)\bigl(K^{(N)}(t)-K^{(N)}(t-)\bigr)>0$.}\hspace*{-30pt}
\end{equation}
\end{prop}
\begin{pf}
Fix $N\in{\mathbb N}$, $t\geq0$ and a measurable function $h$ on $[0,H^r)$.
By the representation (\ref{def-etan}) for $\eta^{(N)}_t$, we have
\begin{eqnarray}
\label{dis:2}
&&\int_{[0,H^r)}\mathbh{1}_{[0,\chi
^{(N)}(t-)]}(x)h(x)\eta^{(N)}_t(dx) \nonumber\\[-8pt]\\[-8pt]
&&\qquad= \sum_{j=-\mathcal{E}^{(N)}_0+
1}^{E^{(N)}(t)}
h \bigl(w^{(N)}_j (t) \bigr) \mathbh{1}_{\{w^{(N)}_j (t)\leq\chi
^{(N)}(t-)\}} \mathbh{1}_{\{ w^{(N)}_j (t) < r_j \}}.\nonumber
\end{eqnarray}
Moreover, by (\ref{qn}),
\[
Q^{(N)}(t)=\eta^{(N)}_t\bigl[0,\chi^{(N)}(t)\bigr]=\sum_{j=-\mathcal
{E}^{(N)}_0+ 1}^{E^{(N)}(t)}
\mathbh{1}_{\{w^{(N)}_j (t)\leq\chi^{(N)}(t)\}} \mathbh{1}_{\{
w^{(N)}_j (t) <
r_j \}}.
\]
Thus $Q^{(N)}(t)$ is the total number of customers who have arrived to
the system and have not reneged by $t$ and whose potential waiting
times at $t$ are less than or equal to $\chi^{(N)}(t)$. If we arrange
those customers in increasing order of their potential waiting times at
$t$, then for $i=1,2,\ldots, Q^{(N)}(t)$, $(F^{\eta
_t^{(N)}})^{-1}(i)$ is exactly
the potential waiting time at $t$ of the $i$th customer from the back
of the queue.
Suppose that $(\chi^{(N)}(t-)-\chi^{(N)}(t))(K^{(N)}(t)-K^{(N)}(t-))=0$.
This implies that either $\chi^{(N)}(t-)=\chi^{(N)}(t)$ holds or both
$\chi^{(N)}(t-)>\chi^{(N)}(t)$ and $K^{(N)}(t)=K^{(N)}(t-)$ hold. The
latter condition indicates that the head-of-the-line customer right
before time $t$ reneged at time $t$. In this case,
the right-hand side of (\ref{dis:2}) admits the alternative representation
\[
\int_0^{Q^{(N)}(t)}h((F^{\eta_t^{(N)}})^{-1}(y))\,dy.
\]
On the other hand, suppose that
$(\chi^{(N)}(t-)-\chi^{(N)}(t))(K^{(N)}(t)-K^{(N)}(t-))>0$. In this
case, the
head-of-the-line customer right before time $t$ departs for service at time
$t$ and this customer is counted in the right-hand side of (\ref
{dis:2}) but
not in $Q^{(N)}(t)$. Since $E^{(N)}(t)-E^{(N)}(t-)\leq1$, there is exactly
one such
customer, that is, $K^{(N)}(t)-K^{(N)}(t-)=1$. Hence the right-hand
side of
(\ref{dis:2})
can be rewritten as
\[
\int_0^{Q^{(N)}(t)+1}h((F^{\eta_t^{(N)}})^{-1}(y))\,dy.
\]
\upqed\end{pf}
As an immediate consequence of (\ref{rep-rcomp1}), Lemma \ref{cor:1},
and Proposition \ref{lem:uni5}, we obtain the following alternative
representation for the compensator $A^{(N)}_{\theta^{(N)},\eta}$
of $R^{(N)}$:
\begin{eqnarray}\label{rep-rcomp2}
A^{(N)}_{\theta^{(N)},\eta} (t) &\doteq& \int_0^t \biggl(\int
_0^{Q^{(N)}(t)+\iota^{(N)}(t)}h^r((F^{\eta_s^{(N)}})^{-1}(y))\,dy
\biggr)\,ds,\nonumber\\
\eqntext{t\in[0,\infty),}
\end{eqnarray}
where $\iota^{(N)}$ is given by (\ref{iota}).
\section{Tightness of pre-limit sequence}
\label{Sec:relcom}
The main objective of this section is to show that, under
suitable assumptions, the sequence of scaled state processes
$\{(\overline{X}{}^{(N)},\overline{\nu}{}^{(N)},\overline{\eta
}^{(N)})\}$ and the sequences of auxiliary
processes are tight.
Specifically, from (\ref{bd-1}) and (\ref{bd-3}) it is clear that for every
$t$, the linear functionals $\overline{D}{}^{(N)}_{\cdot}(t)\dvtx
\varphi\mapsto\overline{D}{}^{(N)}
_{\varphi}(t)$ and
$\overline{A}{}^{(N)}_{\cdot, \nu}(t)\dvtx\varphi\mapsto\overline
{A}^{(N)}_{\varphi, \nu}(t)$
are finite Radon measures on $[0,H^s)\times{\mathbb R}_+$. Likewise, from
(\ref{bd-2}) and the fact that
(\ref{bd-3}) holds with $\nu, h^s$, respectively, replaced by $\eta,
h^r$ by property (2) of Proposition \ref{cor-compensatormeasn}, it
follows that the linear functionals $\overline{S}{}^{(N)}_{\cdot
}(t)\dvtx\psi\mapsto
\overline{S}{}^{(N)}_{\psi}(t)$ and
$\overline{A}{}^{(N)}_{\cdot, \eta}(t)\dvtx\psi\mapsto\overline
{A}^{(N)}_{\psi, \eta}(t)$
define finite Radon measures on $[0,H^r)\times{\mathbb R}_+$. Thus $\{
\overline{D}{}^{(N)}
_{\cdot}(t)\dvtx t\in[0,\infty)\}$ and $\{\overline{A}{}^{(N)}_{\cdot
, \nu
}(t)\dvtx t\in[0,\infty)\}$
can be viewed as $\mathcal{M}_F([0,H^s)\times{\mathbb R}_+)$-valued
c\`{a}dl\`{a}g processes, and $\{\overline{S}{}^{(N)}
_{\cdot}(t)\dvtx t\in[0,\infty)\}$ and $\{\overline{A}{}^{(N)}_{\cdot
, \eta
}(t)\dvtx t\in[0,\infty)\}$
can be viewed as $\mathcal{M}_F([0,H^r)\times{\mathbb R}_+)$-valued
c\`{a}dl\`{a}g processes.
Now, for $N\in{\mathbb N}$, let
\begin{eqnarray}\label{Y}\overline Y{}^{(N)}
& \doteq &
\bigl(\overline{X}{}^{(N)}(0),\overline{E}{}^{(N)},\overline
{X}{}^{(N)},\overline{R}{}^{(N)}, \overline{\nu}{}^{(N)}_0,\nonumber\\[-8pt]\\[-8pt]
& &\hspace*{4.8pt} \overline{\nu}{}^{(N)},
\overline{\eta}^{(N)}_0,\overline{\eta}^{(N)},
\overline{A}{}^{(N)}_{\cdot,\nu}, \overline{D}{}^{(N)}_{\cdot
},\overline{A}{}^{(N)}_{\cdot,\eta},
\overline{S}{}^{(N)}_{\cdot} \bigr). \nonumber
\end{eqnarray}
Then each $\overline Y{}^{(N)}$ is a $\mathcal Y$-valued process, where
$\mathcal
Y$ is the space
\begin{eqnarray*} \mathcal{Y}&\doteq&{\mathbb R}_+ \times(\mathcal
{D}_{{\mathbb R}
_+}[0,\infty))^3 \times\mathcal{M}_F[0,H^s)
\times\mathcal D_{\mathcal{M}_F[0,H^s)}[0,\infty) \times\mathcal
{M}_F[0,H^r)\\
&&{} \times
\mathcal
D_{\mathcal{M}_F[0,H^r)}[0,\infty) \times\bigl(\mathcal{D}_{\mathcal
{M}_F([0,H^s)\times{\mathbb R}_+)}[0,\infty)\bigr)^2 \\
&&{}\times\bigl(\mathcal
{D}_{\mathcal{M}_F([0,H^r){\mathbb R}_+)}[0,\infty)\bigr)^2
\end{eqnarray*}
equipped with the product metric. Clearly, $\mathcal Y$ is a Polish space.
Now we state
the main result of this section.
\begin{theorem} \label{th-tight} Suppose Assumption \ref{ass-init} is
satisfied. Then the sequence $\{\overline Y{}^{(N)}\}$ defined in (\ref
{Y}) is
relatively compact in the Polish space $\mathcal{Y}$, and is therefore tight.
\end{theorem}
The relative compactness of $\{\overline{Y}{}^{(N)}\}$ follows
from Assumption \ref{ass-init} and Lemmas \ref{lem:rc}, \ref{lem:fnueta},
\ref{lem-tight1} and \ref{lem-tight2} below.
Since $\mathcal{Y}$ is a Polish space, tightness is then a direct
consequence of Prohorov's theorem.
We start by recalling Kurtz's criteria (see Theorem 3.8.6 of \cite
{ethkurbook} for details)
for the relative compactness of a sequence $\{\overline{F}{}^{(N)}\}$ of
processes in $\mathcal{D}_{{\mathbb R}_+}[0,\infty)$.
\begin{prop}[(Kurtz's criteria)]\label{Kurtz}
The sequence of processes $\{\overline{\mathcal Z}{}^{(N)}\}$ is
relatively compact if and only if the following two properties hold:
\begin{enumerate}[K2.]
\item[K1.] For every rational $t\geq0$,
\[
\lim_{R\rightarrow\infty}\sup_N\mathbb{P}\bigl(\overline{\mathcal
Z}{}^{(N)}(t)>R\bigr)=0.
\]
\item[K2.] For each $t>0$, there exists $\beta>0$ such that
\begin{equation}
\label{newk2}
\lim_{\delta\rightarrow
0}\sup_N\mathbb{E}\bigl[ \bigl|\overline{\mathcal Z}{}^{(N)}(t+\delta
)-\overline{\mathcal Z}{}^{(N)}(t) \bigr|^\beta\bigr]=0.
\end{equation}
\end{enumerate}
\end{prop}
\begin{lemma} \label{lem:rc} Suppose Assumption \ref{ass-init} holds.
Then the sequences $\{\overline{X}{}^{(N)}\}$,
$\{\overline{K}{}^{(N)}\}$, $ \{\overline{R}{}^{(N)}\}$,
$\{\langle{\mathbf{1}},\overline{\nu}{}^{(N)}\rangle\}, \{\langle
{\mathbf{1}},\overline{\eta}^{(N)}
\rangle\}$, the
sequences $\{\overline D{}^{(N)}_\varphi\}, \{\overline
{A}{}^{(N)}_{\varphi,\nu}\}$, for every
$\varphi\in\mathcal{C}_b([0,H^s)\times{\mathbb R}_+)$ and the
sequences $\{S^{(N)}_\psi\}, \{\overline{A}{}^{(N)}_{\psi
,\eta}\}$, for every $\psi\in\mathcal{C}_b([0, H^r)\times{\mathbb
R}_+)$, are relatively compact.
\end{lemma}
\begin{pf}
Fix $T\in(0,\infty)$. It follows from Proposition \ref
{cor-compensatormeasn}(1), (\ref{bd-1}) and (\ref{init-bd}) that for
$\varphi\in\mathcal{C}_b([0,H^s)\times{\mathbb R}_+)$,
\[
\sup_{N}\mathbb{E}\bigl[\overline{A}{}^{(N)}_{\varphi,\nu}(T) \bigr]= \sup
_{N}\mathbb{E}
\bigl[\overline D{}^{(N)}_\varphi(T)\bigr] \leq\Vert\varphi\Vert_\infty\sup
_N\mathbb{E}\bigl[\overline{X}{}^{(N)}(0)+\overline{E}{}^{(N)}(T)\bigr]
<\infty.
\]
Similarly, by Proposition \ref{cor-compensatormeasn}(2), (\ref{bd-2})
and (\ref{init-bd}), we have for every $\psi\in\mathcal
{C}_b([0,\break H^r)\times{\mathbb R}_+)$,
\[
\sup_{N}\mathbb{E}\bigl[\overline{A}{}^{(N)}_{\psi,\eta}(T)\bigr]= \sup
_{N}\mathbb{E}\bigl[\overline S^{(N)}_\psi(T)\bigr]
\leq
\Vert\psi\Vert_\infty\sup_N\mathbb{E}\bigl[\overline
{X}^{(N)}(0)+\overline{E}{}^{(N)}(T)\bigr] <\infty,
\]
which verifies condition K1 for $\mathcal Z = A^{(N)}_{\varphi,\nu},
D^{(N)}_{\varphi},
\varphi\in\mathcal{C}_b([0,H^s) \times{\mathbb R}_+)$ and
$\mathcal Z =
A^{(N)}
_{\psi,\eta},
S^{(N)}_{\psi}, \psi\in\mathcal{C}_b([0,H^r) \times{\mathbb R}_+)$.
The same argument that was used to prove
Lemma 5.8(2) in \cite{kasram07} can then be used to
show that (\ref{newk2}) is also satisfied by the same collection of
$\mathcal Z$ (see Remark \ref{rem-compen}).
The fact that $\overline{R}{}^{(N)}$ and its increments are dominated,
respectively,
by $\overline{S}{}^{(N)}$ and its increments shows that the sequence $\{
\overline{R}{}^{(N)}\}$ also
satisfies conditions K1 and K2, and is thus relatively compact.
Since $\overline{D}{}^{(N)}=\overline D{}^{(N)}_{\mathbf{1}}$ and
$\overline{S}{}^{(N)}=\overline S{}^{(N)}_{\mathbf{1}}$, it follows that
the sequences
$\{\overline{D}{}^{(N)}\}$ and $\{\overline{S}{}^{(N)}\}$ are also
relatively compact. By Assumption
\ref{ass-init}, the sequences $\{\overline{E}{}^{(N)}\}$ and $\{
\overline{X}{}^{(N)}(0)\}$ are relatively
compact.
Since for every $t\geq0$, $\langle{\mathbf{1}},\overline{\nu
}{}^{(N)}_t\rangle\leq\overline{X}{}^{(N)}
(t)\leq\overline{X}{}^{(N)}(0)+\overline{E}{}^{(N)}(t)$ by (\ref
{comp-prelimit}) and (\ref
{def-dn}), it follows from Markov's inequality that
$\langle\mathbf{1},\overline{\nu}{}^{(N)}_t\rangle$ and $\overline
{X}{}^{(N)}$ satisfy K1 of Proposition \ref{Kurtz}.
In addition, (\ref{def-dn}) also shows that
\begin{eqnarray*}
\bigl|\overline{X}{}^{(N)}(t)-\overline{X}{}^{(N)}(s) \bigr| &\leq& \bigl|\overline
{E}{}^{(N)}(t)-\overline{E}{}^{(N)}(s)
\bigr|+ \bigl|\overline{D}{}^{(N)}(t)-\overline{D}{}^{(N)}(s) \bigr|\\
& &{} + \bigl|\overline
{R}{}^{(N)}(t)-\overline{R}{}^{(N)}(s) \bigr|,
\end{eqnarray*}
and by (\ref{comp-prelimit}) and the Lipschitz continuity of the
function $x\mapsto[1-x]^+$ with Lipschitz constant $1$, we have
\[
\bigl|\bigl\langle{\mathbf{1}},\overline{\nu}^{(N)}_t\bigr\rangle-\bigl\langle
{\mathbf{1}},\overline{\nu}{}^{(N)}
_s\bigr\rangle
\bigr|= \bigl|\bigl[1-\overline{X}{}^{(N)}(t)\bigr]^+-\bigl[1-\overline{X}{}^{(N)}(s)\bigr]^+ \bigr|\leq
\bigl|\overline{X}{}^{(N)}(t)-\overline{X}{}^{(N)}
(s) \bigr|.
\]
When combined with the properties of $\overline{E}{}^{(N)}$, $\overline
{D}{}^{(N)}$ and $\overline{R}{}^{(N)}$ established
above, this shows that $\{\overline{X}{}^{(N)}\}$ and $\{\langle
{\mathbf{1}},\overline{\nu}{}^{(N)}
\rangle\}$
satisfy K2
of Proposition \ref{Kurtz} and, are relatively compact. In turn, by
(\ref{mass-queue}), the relative compactness of $\{\overline
{D}{}^{(N)}\}$ and $\{
\langle
{\mathbf{1}}, \overline{\nu}{}^{(N)}\rangle\}$ implies that of $\{
\overline{K}{}^{(N)}\}$. Moreover,
due to
(\ref{def-sn}), for every $s, t \in[0,\infty)$, we have that
\begin{eqnarray}\quad \bigl|\bigl\langle{\mathbf{1}},\overline{\eta
}^{(N)}_t\bigr\rangle-\bigl\langle
\mathbf{1},\overline{\eta}^{(N)}_s\bigr\rangle\bigr|& \leq& \bigl|\overline
{E}{}^{(N)}(t)-\overline{E}{}^{(N)}(s)
\bigr|+ \bigl|\overline{S}{}^{(N)}(t)-\overline{S}{}^{(N)}(s) \bigr|, \\
\label{dis:etaxe}
\bigl\langle{\mathbf{1}},\overline{\eta}{}^{(N)}
_t\bigr\rangle
&\leq& \bigl\langle{\mathbf{1}},\overline{\eta}^{(N)}_0\bigr\rangle
+\overline{E}{}^{(N)}(t).
\end{eqnarray}
Thus $\langle{\mathbf{1}},\overline{\eta}^{(N)}\rangle$ is also
relatively compact,
and the
proof is complete.
\end{pf}
\begin{lemma} \label{lem:fnueta}
Suppose Assumption \ref{ass-init} holds.
For every $f\in\mathcal{C}^1_c({\mathbb R}_+)$, the sequences $\{
\langle f,\overline{\nu}{}^{(N)}\rangle\}$ and
$\{\langle f,\overline{\eta}^{(N)}\rangle\}$ of $\mathcal
{D}_{{\mathbb R}}[0,\infty)$-valued random variables are
relatively compact.
\end{lemma}
\begin{pf}
Fix $t\in[0,\infty)$. By (\ref{eqn-prelimit1}) and (\ref
{eqn-prelimit3}), for every $f \in\mathcal{C}_c^1({\mathbb R}_+)$, we have
\[
\bigl\langle f, \overline{\nu}{}^{(N)}_t \bigr\rangle- \bigl\langle f, \overline{\nu
}^{(N)}_0\bigr\rangle
= \int_0^t \bigl\langle f^\prime, \overline{\nu}{}^{(N)}_s \bigr\rangle \,ds -
\overline{D}{}^{(N)}_f (t) + f(0) \overline{K}{}^{(N)}(t)
\]
and
\[
\bigl\langle f,\overline{\eta}^{(N)}_t\bigr\rangle-\bigl\langle f,\overline{\eta
}^{(N)}_0\bigr\rangle= \int_0^t\bigl\langle
f',\overline{\eta}^{(N)}_s\bigr\rangle \,ds -\overline
{S}{}^{(N)}_{f}(t)+f(0)\overline{E}{}^{(N)}(t).
\]
Since $\{\overline{D}_f^{(N)}\}$, $\{\overline{K}{}^{(N)}\}$,
$\{\overline{S}_f^{(N)}\}$ and
$\{ \overline{E}{}^{(N)}\}$ are relatively
compact due to Lem\-ma~\ref{lem:rc} and property 1 of Assumption \ref{ass-init},
it suffices to show that the sequences
$\{\int_0^\cdot\langle f^\prime, \overline{\nu}{}^{(N)}_s \rangle \,ds
\}$ and
$\{\int_0^\cdot\langle f^\prime, \overline{\eta}_s \rangle \,ds \}$
are tight.
It follows from (\ref{dis:etaxe}) that for $\delta\in(0,1)$,
\begin{eqnarray*}
\biggl|\int_t^{t+\delta}\bigl\langle f',\overline{\eta}^{(N)}_s\bigr\rangle \,ds \biggr|
&\leq&
\Vert f'\Vert_\infty\int_t^{t+\delta}\bigl|\bigl\langle{\mathbf
{1}},\overline{\eta}^{(N)}_s\bigr\rangle\bigr| \,ds
\\
&\leq&
\Vert f'\Vert_\infty\delta\bigl(\bigl\langle{\mathbf{1}},\overline
{\eta}^{(N)}_0\bigr\rangle
+\overline{E}{}^{(N)}
(t+1) \bigr).
\end{eqnarray*}
Hence, we have
\begin{equation} \label{dis:preeta}
\mathbb{E}\biggl[ \biggl|\int_t^{t+\delta}\bigl\langle f',\overline{\eta
}^{(N)}_s\bigr\rangle \,ds
\biggr| \biggr]\leq\Vert f'\Vert_\infty\delta\sup_N\mathbb{E}\bigl[\bigl\langle
{\mathbf
{1}},\overline{\eta}^{(N)}
_0\bigr\rangle+\overline{E}{}^{(N)}(t+1)\bigr].
\end{equation}
For each $t\in[0,\infty)$, by (\ref{def-etan}) and Assumption \ref
{ass-init}, it follows that
\begin{equation}\label{bound-eta} \sup_N\mathbb{E}
\bigl[\bigl\langle{\mathbf{1}},\overline{\eta}^{(N)}_t\bigr\rangle\bigr] \leq\sup
_N\mathbb{E}\bigl[\bigl\langle
\mathbf{1},\overline{\eta}^{(N)}_0\bigr\rangle+\overline{E}{}^{(N)}(t) \bigr]
<\infty.
\end{equation}
Therefore, taking the limit, as $\delta\rightarrow0$, in (\ref{dis:preeta})
and using the last inequality in (\ref{bound-eta}), we have
\[
\lim_{\delta\rightarrow0} \sup_N\mathbb{E}\biggl[ \biggl|\int_t^{t+\delta
}\bigl\langle f',\overline{\eta}^{(N)}_s\bigr\rangle \,ds \biggr| \biggr]=0.
\]
Similarly, since $\langle{\mathbf{1}}, \overline{\nu}{}^{(N)}_s
\rangle\leq1$ for every
$s\in
[0,\infty)$ and $N\in{\mathbb N}$,
\[
\lim_{\delta\rightarrow0} \sup_N\mathbb{E}\biggl[ \biggl|\int_t^{t+\delta
}\bigl\langle f',\overline{\nu}{}^{(N)}_s\bigr\rangle \,ds \biggr| \biggr]\leq\lim_{\delta
\rightarrow0}\Vert f'\Vert_\infty\delta=0.
\]
Moreover, by (\ref{bound-eta}), we also have, for every $t\in
[0,\infty)$,
\[
\sup_N \mathbb{E}\biggl[ \biggl|\int_0^{t}\bigl\langle f',\overline{\eta}^{(N)}
_s\bigr\rangle \,ds \biggr| \biggr] \leq\Vert f'\Vert_\infty t \sup_N\mathbb{E}
\bigl[\bigl\langle{\mathbf{1}},\overline{\eta}^{(N)}_0\bigr\rangle+\overline
{E}{}^{(N)}(t) \bigr]<\infty.
\]
Similarly, we have
\[
\sup_N\mathbb{E}\biggl[ \biggl|\int_0^{t}\bigl\langle f',\overline{\nu}{}^{(N)}
_s\bigr\rangle \,ds \biggr| \biggr] \leq\sup_N\mathbb{E}\biggl[\int_0^{t}\bigl|\bigl\langle
f',\overline{\nu}{}^{(N)}_s\bigr\rangle\bigr| \,ds \biggr] \leq\Vert f'\Vert_\infty
t<\infty.
\]
This implies that $ \{\int_0^\cdot\langle f',\overline{\eta
}^{(N)}_s\rangle
\,ds \}$ and $ \{\int_0^\cdot\langle f',\overline{\nu}{}^{(N)}_s\rangle \,ds
\}$ both satisfy criteria K1 and K2 of Proposition \ref{Kurtz} and
hence are relatively compact. This completes the proof of the lemma.
\end{pf}
Next, we show that $\{\overline{\nu}{}^{(N)}\}$ and $\{\overline{\eta
}^{(N)}\}$ are tight, and
hence are relatively compact with respect to the topology on $\mathcal
D_{\mathcal{M}_F[0,H^s)}[0,\infty)$ and $\mathcal D_{\mathcal
{M}_F[0,H^r)}[0,\infty)$, respectively.
Since, as mentioned in Section \ref{subsub-funmeas}, $\mathcal
{M}_F[0,H^s)$ and
$\mathcal{M}_F[0,H^r)$, equipped with the topology of weak
convergence, are Polish
spaces, we can apply Jakubowski's criteria to establish the tightness
of $\{\overline{\nu}{}^{(N)}\}$ and $\{\overline{\eta}^{(N)}\}$. For
convenience, we recall
Jakubowski's criteria.
\begin{prop}[(Jakubowski)]\label{prop:JC} A sequence $\{\overline\pi
^{(N)}\}$ of $\mathcal D_{\mathcal{M}_F[0,H)}[0,\infty)$-valued
random elements
defined on $(\Omega,\mathcal F,\mathbb{P})$ is tight if and only if the
following two conditions hold:
\begin{enumerate}[J2.]
\item[J1.] For each $T>0$ and $0<\delta<1$, there are compact subsets
$\tilde C_{T,\delta}$ of $\mathcal{M}_F[0,H)$ such that
\[
\liminf_{N\rightarrow\infty} \mathbb{P}\bigl(\overline{\nu}{}^{(N)}_t\in
\tilde
C_{T,\delta} \mbox{ for all }t\in[0,T] \bigr)>1-\delta.
\]
\item[J2.] There exists a family $\mathbb{F}$ of real continuous
functions $F$
on $\mathcal{M}_F[0,H)$ that separates points in $\mathcal{M}_F[0,H)$
and is closed under
addition, and $\{\overline\pi^{(N)}\}$ is $\mathbb{F}$-weakly tight, that
is, for every $F\in\mathbb{F}$, the sequence $\{F(\overline\pi
^{(N)}),s\in
[0,\infty)\}$ is tight in $\mathcal D_{{\mathbb R}}[0,\infty)$.
\end{enumerate}
\end{prop}
\begin{lemma} \label{lem-tight1}
Suppose Assumption \ref{ass-init} holds. The sequences $\{\overline
{\nu}^{(N)}\}$ and
$\{\overline{\eta}^{(N)}\}$ are relatively compact.
\end{lemma}
\begin{pf}
By Remark 5.11 of \cite{kasram07} and Lemma \ref{lem:fnueta}, it
follows that $\{\overline{\nu}{}^{(N)}\}$ and $\{\overline{\eta
}^{(N)}\}$ satisfy Jakubowski's J2
criterion. Therefore, it suffices to show that they also satisfy Jakubowski's
J1 criterion. By (2) and (3) of Assumption \ref{ass-init}, for almost every
$\omega\in\Omega$, $\sup_N \overline{\nu}{}^{(N)}_0(\omega
)[0,H^s)<\infty$. By
Lemma A 7.5
of \cite{Kal}, for every $\varepsilon>0$, there exists
$k(\omega,\varepsilon)<\infty$ such that $\sup_N
\overline{\nu}{}^{(N)}_0(\omega)(k(\omega,\varepsilon
),\break H^s)<\varepsilon$.
The argument for tightness of $\{\overline{\nu}{}^{(N)}\}$ (in the
absence of reneging)
presented in Lemma 5.12 of \cite{kasram07} can be directly applied to show
that $\{\overline{\nu}{}^{(N)}\}$ satisfies Jakubowski's J1 criterion,
and hence
$\{\overline{\nu}{}^{(N)}\}$ is tight in the presence of reneging as well.
Similarly, due to (2) and (4) of Assumption \ref{ass-init}, for almost every
$\omega\in\Omega$, $\sup_N \overline{\eta}^{(N)}_0(\omega
)[0,H^r)<\infty$.
Once again, by
Lemma A 7.5 of \cite{Kal}, we infer that for every $\varepsilon>0$, there
exists $l(\omega,\varepsilon)<\infty$ such that $\sup_N
\overline{\eta}^{(N)}_0(\omega)(l(\omega,\varepsilon
),H^r)<\varepsilon$. Since
$\{\langle{\mathbf{1}},
\overline{\eta}^{(N)}\rangle\}$ is tight by Lemma \ref{lem:fnueta}, the argument
for tightness of
$\{\overline{\nu}{}^{(N)}\}$ presented in Lemma 5.12 of \cite
{kasram07} can also be adapted
to show that the sequence $\{\overline{\eta}^{(N)}\}$ satisfies
Jakubowski's J1
criterion, and is therefore tight. We omit the details.
\end{pf}
We end this section by
establishing the relative compactness of the measure-valued
processes associated with the cumulative departure and reneging
functionals and their compensators.
\begin{lemma}\label{lem-tight2}
Suppose Assumption \ref{ass-init} holds. Then the sequences $\{
\overline{D}{}^{(N)}
_{\cdot}\}$ and $\{\overline{A}{}^{(N)}_{\cdot,\nu}\}$ are
relatively compact
in $\mathcal{D}_{\mathcal{M}_F([0,H^s)\times{\mathbb R}_+)}[0,\infty
)$. Similarly, the sequences $\{\overline{S}{}^{(N)}_{\cdot}\}$ and $\{
\overline{A}{}^{(N)}_{\cdot,\eta}\}$ are relatively compact in
$\mathcal{D}_{\mathcal{M}_F([0,H^r)\times{\mathbb R}_+)}[0,\infty)$.
\end{lemma}
\begin{pf}
This can be proved by combining Lemma \ref{lem:rc} and Proposition
\ref{cor-compensatormeasn}
with the argument that was used in Lemma 5.13 of \cite{kasram07} to
establish the tightness of the sequences
$\{\overline\mathcal{Q}{}^{(N)}\}$ and $\{\overline\mathcal{A}{}^{(N)}\}$
therein. Since the adaptation of
the argument in \cite{kasram07} is
straightforward, we omit the details.
\end{pf}
\section{Strong law of large numbers limits}
\label{sec:CSL}
\subsection{Characterization of subsequential limits}
\label{subs-csl}
The focus of this section is the following theorem which, in particular,
establishes existence of a solution to the fluid equations.
\begin{theorem}\label{thm:FE}
Suppose that Assumptions \ref{ass-init}--\ref{ass-h} hold. Let
$(\overline{X}
,\overline{\nu},\overline{\eta})$ be the limit of any subsequence
of $\{\overline{X}{}^{(N)},\overline{\nu}{}^{(N)}
,\overline{\eta}^{(N)}\}$. Then $(\overline{X},\overline{\nu
},\overline{\eta})$ solves the fluid equations.
\end{theorem}
The rest of the section is devoted to the proof of this theorem. Let
$(\overline{E},
\overline{X}(0)$, $\overline{\nu}_0,\overline{\eta}_0)$ be the
$\mathcal{S}_0$-valued random
variable\vspace*{1pt} that
satisfies Assumption~\ref{ass-init}, and let $\{\overline Y{}^{(N)}\}
_{N\in{\mathbb N}}$
be the sequence of processes defined in (\ref{Y}). Then, by Assumption
\ref{ass-init}, Theorem \ref{th-tight} and the limits $\overline{M}{}^{(N)}
_{\cdot,\nu}=
\overline{D}{}^{(N)}_{\cdot} - \overline{A}{}^{(N)}_{\cdot,\nu
}\Rightarrow0$ and
$\overline{M}{}^{(N)}_{\cdot,\eta}= \overline{S}{}^{(N)}_{\cdot} -
\overline{A}{}^{(N)}_{\cdot,\eta
}\Rightarrow0$
established in Proposition \ref{cor-compensatormeasn},
there exist processes
$\overline{X}\in\mathcal{D}_{{\mathbb R}_+}[0,\infty), \overline
{R}\in\mathcal{D}_{{\mathbb R}
_+}[0,\infty), \overline{\nu}\in\mathcal
D_{\mathcal{M}_F[0,H^s)}[0,\infty)$, $\overline{\eta}\in\break \mathcal
D_{\mathcal{M}_F[0,H^r)}[0,\infty)$,
$\overline{A}_{\cdot,\nu}\in\mathcal{D}_{\mathcal
{M}_F([0,H^s)\times{\mathbb R}_+)}[0,\infty)$, $\overline{D}_{\cdot
}\in$ $\mathcal{D}_{\mathcal{M}_F([0,H^s)\times{\mathbb
R}_+)}[0,\infty)$,
$\overline{A}_{\cdot,\eta}\in\mathcal{D}_{\mathcal
{M}_F([0,H^r)\times{\mathbb R}_+)}[0,\infty)$, $\overline{S}_{\cdot
}\in\mathcal{D}_{\mathcal{M}_F([0,H^r)\times{\mathbb
R}_+)}[0,\infty)$
such that $\overline Y{}^{(N)}$ converges weakly (along a suitable
subsequence) to
\[
\overline Y\doteq(\overline{X}(0),\overline{E},\overline
{X},\overline{R}, \overline{\nu}_0, \overline{\nu},
\overline{\eta}_0,\overline{\eta},\overline{A}_{\cdot,\nu},
\overline{A}_{\cdot,\nu
},\overline{A}_{\cdot,\eta}, \overline{A}_{\cdot,\eta} ) \in
\mathcal{Y}.
\]
Denoting this subsequence again by $\overline Y{}^{(N)}$ and invoking the
Skorokhod representation theorem, with a slight abuse of notation, we
can assume that, $\mathbb{P}$
a.s., $\overline Y{}^{(N)}\rightarrow\overline Y$ as $N\rightarrow
\infty$.
Without loss of generality, we may further assume that the above
convergence holds everywhere.
We now identify some properties of the limit that will be used to prove
Theorem \ref{thm:FE}. From Proposition \ref{cor-compensatormeasn}(1),
it follows that, as $N\rightarrow\infty$, $(\overline Y{}^{(N)},\break
\overline{D}{}^{(N)}_{\cdot
}) \rightarrow(\overline Y,\overline{A}_{\cdot,\nu})$. Together
with (\ref
{def-dn}), this implies that
\begin{equation}\label{limX}
\overline{X}= \overline{X}(0)+\overline{E}-\overline{A}_{{\mathbf
{1}},\nu}-\overline{R}.
\end{equation}
Moreover, we claim that
\begin{equation}\label{dis:13}
\overline{A}_{\varphi,\nu}=\int_0^\cdot\langle\varphi
h^s,\overline{\nu}_s\rangle \,ds.
\end{equation}
This
corresponds to relation (5.48) established in Proposition 5.17 of \cite
{kasram07} for the
model without abandonments. However, essentially the same argument can
be used
here as well. Specifically, the proof of (5.48) in \cite{kasram07}
relies on
Lemmas 5.8(1) and~5.16 of \cite{kasram07},
which continue to be valid in the presence of abandonments due to Remarks
\ref{rem-compen} and
\ref{rem-compen0}.
On substituting (\ref{dis:13}) into (\ref{limX}), we see that the
fluid equation (\ref{eq-fx}) is satisfied.
Next, in Proposition \ref{prop:3}, we establish representation (\ref{fr})
for $\overline{R}$ given in the fluid equations. The proof of this result
relies on the alternative representation for the compensator $A^{(N)}
_{\theta^{(N)},\eta}$ of $R^{(N)}$ given in
(\ref{rep-rcomp2}).
\begin{prop}
\label{prop:3}
For every $T\in[0,\infty)$, as $N\rightarrow\infty$,
\begin{equation}\label{rep-Aconv}
\mathbb{E}\biggl[\sup_{t\in[0,T]} \biggl|\overline{A}{}^{(N)}_{\theta
^{(N)},\eta}(t) - \int_0^t \biggl(\int_0^{\overline
{Q}(s)}h^r((F^{\overline\eta_s}
)^{-1}(y))\,dy \biggr)\,ds \biggr| \biggr]\rightarrow0.
\end{equation}
Moreover, almost surely,
\begin{equation}\label{rep-R}\overline{R}(t)=\int_0^t \biggl(\int
_0^{\overline{Q}(s)}h^r((F^{\overline\eta_s})^{-1}(y))\,dy \biggr) \,ds,\qquad
t\in [0,\infty).
\end{equation}
\end{prop}
The proof of Proposition \ref{prop:3} is given near the end of this
section and relies on the following preliminary
observations.
Let $\tilde R(t)$ be defined by the right-hand side of (\ref{rep-R})
for $t\in[0,\infty)$. We first show how (\ref{rep-R}) can be deduced
from (\ref{rep-Aconv}). From (\ref{rep-Aconv}), it follows that
$\overline{A}{}^{(N)}_{\theta^{(N)},\eta} \Rightarrow\tilde R$ as
$N\rightarrow
\infty$. Since $\tilde{R}$ is continuous, $\overline
{R}{}^{(N)}=\overline{M}{}^{(N)}_{\theta
^{(N)},\eta} + \overline{A}{}^{(N)}_{\theta^{(N)},\eta}$ and
$\overline{M}{}^{(N)}
_{\theta^{(N)},\eta} \Rightarrow0$ by
Lemma \ref{cor:1}, it follows that $\overline{R}{}^{(N)} \Rightarrow
\tilde{R}$.
This implies, a.s., $\tilde{R} = \overline{R}$, and thus the second statement
of Proposition \ref{prop:3} follows from the first statement.
The proof of\vspace*{1pt} (\ref{rep-Aconv}) relies on Lemmas
\ref{lem:uni4}--\ref{lem:2} below and the following observations.
Using (\ref{rep-rcomp2}) and the elementary relation $(F^{\eta
^{(N)}_s})^{-1}(N\cdot)=(F^{\overline\eta^{(N)}_s})^{-1}(\cdot)$,
simple algebraic manipulations show
that
\begin{eqnarray}\label{rep-rcomp3} \overline{A}{}^{(N)}_{\theta
^{(N)},\eta} (t)
\doteq\int_0^t \biggl(\int_0^{\overline{Q}{}^{(N)}(t)+\overline\iota
^{(N)}(t)}h^r((F^{\overline\eta^{(N)}_s})^{-1}(y))\,dy
\biggr)\,ds,\nonumber\\[-8pt]\\[-8pt]
\eqntext{t\in[0,\infty),}
\end{eqnarray}
where, as usual, $\overline\iota
^{(N)}\doteq\iota^{(N)}/N$ and $\iota^{(N)}$ is given by (\ref
{iota}). Next, observe that for all $t\in[0,T]$ and $L\in[0,H^r)$,
\begin{equation}\label{dcomp1} \bigl|\overline{A}{}^{(N)}_{\theta
^{(N)},\eta
}(t)-\tilde
R(t) \bigr|\leq\overline C{}^{(N)}_1(t,L)+\overline C{}^{(N)}_2(t,L)+\overline C_3(t,L),
\end{equation}
where $\overline C{}^{(N)}_i(t,L), i=1,2$, and $\overline C_3(t,L)$ are defined, for
$t\in[0,\infty)$, by
\begin{eqnarray} \label{def-c1} \qquad\overline C{}^{(N)}_1(t,L)
&\doteq& \biggl|\int_0^t \biggl(\int_0^{(\overline{Q}{}^{(N)}(s)+\overline\iota
^{(N)}(s))\wedge F^{\overline\eta_s^{(N)}}(L)} h^r((F^{\overline\eta
_s^{(N)}})^{-1}(y))\,dy \biggr)\,ds\\
& &\hspace*{52.4pt}{} - \int_0^t \biggl(\int_0^{\overline{Q}
(s)\wedge F^{\overline\eta_s}(L)}h^r((F^{\overline\eta_s})^{-1}(y))\,dy
\biggr)\,ds \biggr|, \nonumber
\\
\label{def-c2} \overline C{}^{(N)}_2(t,L) &\doteq&
\biggl|\int
_0^t \biggl( \int_{(\overline{Q}{}^{(N)}(s)+\overline\iota^{(N)}(s))\wedge
F^{\overline\eta_s^{(N)}}
(L)}^{\overline{Q}{}^{(N)}(s)+\overline\iota
^{(N)}(s)}h^r((F^{\overline\eta_s^{(N)}})^{-1}(y))\,dy
\biggr)\,ds \biggr|
\end{eqnarray}
and
\begin{equation}\label{def-c3} \overline C_3(t,L) \doteq\int
_0^t \biggl( \int
_{\overline{Q}(s)\wedge F^{\overline\eta_s}(L)}^{\overline
{Q}(s)}h^r((F^{\overline\eta_s})^{-1}(y))\,dy
\biggr)\,ds.
\end{equation}
As a precursor to the proof of (\ref{rep-Aconv}) of Proposition \ref
{prop:3}, we first establish
some path properties of the limiting queue measure $\overline{\eta}$
in Lemma\ref{lem:uni4}
and some estimates in Lemma \ref{lem-est}. These two preliminary
results will be used in Lemma \ref{lem:1} to show that for any $L\in
[0,H^r)$, $\lim_{N\rightarrow\infty} \sup_{t\in[0,T]}
|\overline C{}^{(N)}_1(t,L) |=0$ in the case when $h^r$ is
continuous. Next, Lemma \ref{lem:2} extends this to include general
$h^r$ that is locally integrable in $[0,H^r)$. All these results are
then combined to prove Proposition \ref{prop:3}.
\begin{lemma} \label{lem:uni4}
For every $L\in[0,H^r)$, $\overline{\eta}_t$ is continuous at $L$
for almost
every \mbox{$t\geq0$}. Moreover, for $t\in(0,\infty)$ and $L\in[0,H^r)$,
if $\overline{\eta}_t(\{L\})>0$, then $\overline{\eta
}_t(L,L+\varepsilon)>0$ for all
sufficiently small $\varepsilon$.
\end{lemma}
\begin{pf}
It was shown in Corollary \ref{cor:nueta} that $(\overline{\eta
},\overline{E})$
satisfies (\ref{eq-freneg2}) for every bounded Borel measurable
function $f$.
For every $L \in[0,H^r)$, substituting $f = \mathbh{1}_{L}$ in (\ref
{cor:nueta}),
we obtain
\begin{eqnarray} \label{atL} \overline{\eta}_t (\{L\}) & = &\int
_{[0,H^r )}
\mathbh{1}_{\{L\}} (x+t)
\frac{1 - G^r(x+t)}{1 - G^r(x)} \overline{\eta}_0
(dx)\nonumber\\[-8pt]\\[-8pt]
& &{} + \int
_{[0,t]} \mathbh{1}_{\{L\}} (t-s) \bigl(1
- G^r(t-s)\bigr) \,d \overline{E}(s).\nonumber
\end{eqnarray}
It is easy to see that the right-hand side of the above display is zero
except when $\overline{\eta}_0(\{L-t\})>0$ if $t\leq L$ or when
$\overline{E}(t-L)-\overline{E}
((t-L)-)>0$ if $t>L$. Since the jump times of both $\overline{\eta
}_0$ and
$\overline{E}$ are at most countable, (\ref{atL}) shows that
$\overline{\eta}_t$ is
continuous at $L$ for almost every $t\geq0$.
Next, suppose $\overline{\eta}_t(\{L\})>0$. Then by (\ref{atL}), at
least one
of the following two inequalities must hold:
\begin{equation}\label{atL1}\int
_{[0,H^r )} \mathbh{1}_{\{L\}} (x+t)
\frac{1 - G^r(x+t)}{1 - G^r(x)} \overline{\eta}_0 (dx)>0
\end{equation}
or
\begin{equation}\label{atL2}\int_{[0,t]} \mathbh{1}_{\{L\}} (t-s) \bigl(1-
G^r(t-s)\bigr)\, d
\overline{E}(s)>0.
\end{equation}
If (\ref{atL1}) holds, then it must be that $L-t\in[0,H^r)$, $(1 -
G^r(L))/(1 - G^r(L-t))>0$ and $\overline{\eta}_0(\{L-t\})>0$. By Assumption
\ref{ass-jump} and the continuity of $(1 - G^r(\cdot+t))/(1 -
G^r(\cdot))$, it then follows that for all sufficient small
$\varepsilon>0$,
\begin{equation}\label{dis:lle}\int_{[0,H^r )} \mathbh{1}
_{(L,L+\varepsilon)} (x+t)
\frac{1 - G^r(x+t)}{1 - G^r(x)} \overline{\eta}_0 (dx)>0.
\end{equation}
Substituting
$f=\mathbh{1}_{(L,L+\varepsilon)}$ into (\ref{eq-freneg2}) in Corollary
4.2 shows that $\overline{\eta}_t(L,L+\varepsilon)$ is greater than
or equal
to the left-hand side of (\ref{dis:lle}), and so the lemma is
established in this case. On the other hand, suppose (\ref{atL2})
holds. In this case, $t-L>0$, $1-G^r(t-L)>0$ and $\overline
{E}(t-L)-\overline{E}
((t-L)-)>0$. By Assumption \ref{ass-jump} and the continuity of
$1-G^r(t-\cdot)$, for all sufficiently small $\varepsilon>0$,
$1-G^r(t-\cdot)$ is strictly positive on $(L,L+\varepsilon)$ and
$\overline{E}
((t-L)-)-\overline{E}(t-L-\varepsilon)>0$. Another application of
(\ref
{eq-freneg2}) of Corollary \ref{cor:nueta}, with $f=\mathbh{1}
_{(L,L+\varepsilon)}$, shows that
\[
\overline{\eta}_t(L,L+\varepsilon) \geq\int_0^t \mathbh
{1}_{(L,L+\varepsilon)}
(t-s) \bigl(1
- G^r(t-s)\bigr) \,d \overline{E}(s)>0,
\]
and the proof of the lemma is complete.
\end{pf}
\begin{lemma}
\label{lem-est} Let $T\in[0,\infty)$ and $L\in[0,H^r)$. The
following estimates hold:
\begin{enumerate}
\item For $m \in[0, H^r)$ and every $\ell\in L^1_{\mathrm{loc}}[0, H^r)$ with
support in $[0,m]$, there exists $\tilde L(m,T) < \infty$ such that
\begin{equation}\label{lem-est-1} \biggl|\int_0^T\langle\ell, \overline
{\eta}
_s\rangle
\,ds \biggr|\leq
{\tilde L(m,T) \int_{[0,H^r)}}|\ell(x)|\,dx.
\end{equation}
\item Suppose $h$ is a measurable function such that $\tilde
C_L^h\doteq\sup_{x\in[0,L]}|h(x)|<\infty$. Then, $\mathbb{P}$-a.s.,
\begin{equation}
\label{lem-est-2}\sup_N\sup_{s\in[0,T]}\int_0^Lh(x)\overline{\eta}^{(N)}
_s(dx)\leq\tilde C_L^h \sup_N \bigl(\bigl\langle{\mathbf{1}},\overline{\eta}^{(N)}
_0\bigr\rangle
+\overline{E}{}^{(N)}(T) \bigr)<\infty.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{pf}
It was established in Lemma 5.16 of \cite{kasram07} that inequality
(\ref{lem-est-1}) holds with $\overline{\eta}$ replaced by the fluid age
measure $\overline{\nu}$
associated with a many-server queue without abandonments. The proof follows
directly from Proposition 4.15 and the estimate (5.46) of
\cite{kasram07}. Since the dynamic equations (\ref{eqn-prelimit3}) and
(\ref{eq-freneg2}) for $\eta^{(N)}$ and~$\overline{\eta}$,
respectively, are exactly
analogous to the dynamic equations for $\nu^{(N)}$ and~$\overline{\nu}$.
Estimate (5.46) of \cite{kasram07} can be shown to hold for $\overline
{\eta}$
using the same
argument as in~\cite{kasram07}. When combined with Proposition 4.15 of
\cite{kasram07}, this shows that (\ref{lem-est-1}) holds.
Estimate (\ref{lem-est-2}) follows directly from (\ref{def-sn}) and
Assumption \ref{ass-init}.
\end{pf}
\begin{lemma}\label{lem:1} For $T\geq0$ and all but countably many
$L\in[0,H^r)$, given any continuous function $h$ on $[0,\infty)$, as
$N\rightarrow\infty$, for every realization,
\begin{eqnarray} \label{dis:ext2}\qquad& & \sup_{t\in[0,T]} \biggl|\int
_0^t \biggl(\int_0^{(\overline{Q}{}^{(N)}(s)+\overline\iota^{(N)}(s))\wedge
F^{\overline\eta_s^{(N)}}
(L)} h((F^{\overline\eta_s^{(N)}})^{-1}(y))\,dy \biggr)\,ds
\nonumber\\[-8pt]\\[-8pt]
& &\hspace*{80pt}{}
- \int_0^t \biggl(\int_0^{\overline{Q}(s)\wedge F^{\overline\eta_s}
(L)}h((F^{\overline\eta_s})^{-1}(y))\,dy \biggr)\,ds \biggr|\rightarrow
0.\nonumber
\end{eqnarray}
\end{lemma}
\begin{pf}
Fix $\omega\in\Omega$. To ease the notation, we shall suppress
$\omega$ from the notation.
From the convergence of $\overline{\eta}^{(N)}$ to $\overline{\eta
}$ and $\overline{Q}{}^{(N)}$ to $\overline{Q}
$, it follows that, as $N\rightarrow\infty$, $\overline{\eta
}^{(N)}_s \stackrel{w}{\rightarrow}
\overline{\eta}_s$ and $\overline{Q}{}^{(N)}(s) \rightarrow\overline
{Q}(s)$ for almost every $s\geq
0$. Also, by Lem\-ma~\ref{lem:uni4}, $\overline{\eta}_s$ is continuous
at $L$
for almost every $s\geq0$. Let $s\geq0$ be a time at which $\overline
{\eta}^{(N)}
_s \stackrel{w}{\rightarrow}\overline{\eta}_s$ and $\overline
{Q}^{(N)}(s) \rightarrow\overline{Q}(s)$ as
$N\rightarrow\infty$ and $\overline{\eta}_s$ is continuous at $L$.
Then, as
$N\rightarrow\infty$, $F^{\overline\eta_s^{(N)}}(x)\rightarrow
F^{\overline\eta_s}(x)$ for $x=L$ and all but
a countable number of $x\in[0,H^r)$. Therefore, by Theorem 13.6.3 of
\cite{whi-SPL}, we have $(F^{\overline\eta_s^{(N)}})^{-1}
\rightarrow(F^{\overline\eta_s})^{-1}$ on
$[0,F^{\overline\eta_s}(H^r-))$ in the $M_1$ topology. For $s\in
[0,T]$, we now show
that, as $N\rightarrow\infty$,
\begin{eqnarray}\label{dis:4}
&&
\int_0^{(\overline{Q}{}^{(N)}
(s)+\overline\iota^{(N)}(s))\wedge F^{\overline\eta
_s^{(N)}}(L)}h((F^{\overline\eta_s^{(N)}}
)^{-1}(y))\,dy\nonumber\\[-8pt]\\[-8pt]
&&\qquad\rightarrow\int_0^{\overline{Q}(s)\wedge F^{\overline
\eta_s}
(L)}h((F^{\overline\eta_s})^{-1}(y))\,dy.\nonumber
\end{eqnarray}
From the inequality $ |\overline\iota^{(N)} |\leq1/N$, we
immediately see that
\begin{equation} \label{dis:Mconv}\qquad
\bigl(\overline{Q}{}^{(N)}(s)+\overline\iota^{(N)}(s)\bigr)\wedge
F^{\overline\eta_s^{(N)}}(L) \rightarrow\overline{Q}(s)\wedge
F^{\overline\eta_s}(L)\qquad\mbox{as
}N\rightarrow\infty.
\end{equation}
We now consider the
following two cases:
\textit{Case} 1. $\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)<F^{\overline\eta_s}(H^r-)$. In this
case, due
to (\ref{dis:Mconv}), for all sufficiently large $N$, $(\overline{Q}{}^{(N)}
(s)+\overline
\iota^{(N)}(s))\wedge F^{\overline\eta_s^{(N)}}(L) <F^{\overline
\eta_s}(H^r-)$. For each $n\in{\mathbb N}$,
by Theorem
11.5.1 of \cite{whi-SPL} and the continuity of $h$, we obtain for each
$t<F^{\overline\eta_s}(H^r-)$,
\[
\lim_{N \rightarrow\infty}\sup_{u\in
[0,t]} \biggl|\int_0^{u}h((F^{\overline\eta_s^{(N)}})^{-1}(y))\,dy-\int
_0^{u}h((F^{\overline
\eta_s})^{-1}(y))\,dy \biggr| = 0.
\]
By the case assumption, this implies, in particular, that
\[
\lim_{N \rightarrow
\infty} \biggl|\int_0^{\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)}h((F^{\overline\eta_s^{(N)}})^{-1}(y))\,dy -
\int_0^{\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)}h((F^{\overline\eta
_s})^{-1}(y))\,dy \biggr| =
0.
\]
On the other hand, (\ref{dis:Mconv}) and the continuity of $h$ show that
\[
\lim_{N \rightarrow\infty}\int_{(\overline{Q}{}^{(N)}(s)+\overline
\iota^{(N)}(s))\wedge
F^{\overline\eta_s^{(N)}}(L)}^{\overline{Q}(s)\wedge F^{\overline
\eta_s}(L)}h((F^{\overline\eta_s^{(N)}})^{-1}(y))\,dy = 0.
\]
Together, the last two assertions imply (\ref{dis:4}).
\textit{Case} 2. $\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)=F^{\overline\eta_s}(H^r-)$. We
first claim that in this case
\begin{equation}\label{dis:qlh} \overline{Q}(s)=F^{\overline\eta_s}
(L)=F^{\overline\eta_s}(H^r-).
\end{equation}
Indeed, $F^{\overline\eta_s}(L)\leq F^{\overline\eta_s}(H^r-)$ because
$F^{\overline\eta_s}$ is nondecreasing and $L<H^r$, while $\overline
{Q}(s)\leq\overline{\eta}
_s[0,H^r)=F^{\overline\eta_s}(H^r-)$ by (\ref{fqfreneg}). On the
other hand, the
reverse inequalities $\overline{Q}(s)\geq F^{\overline\eta_s}(H^r-)$
and $F^{\overline\eta_s}(L)\geq
F^{\overline\eta_s}(H^r-)$ hold by the case assumption, and so the
claim follows.
Now, define $\overline{L}\doteq(F^{\overline\eta_s})^{-1}(F^{\overline
\eta_s}(H^r-))$.
Then $\overline L=( F^{\overline\eta_s})^{-1}(F^{\overline\eta
_s}(L))$ by (\ref
{dis:qlh}). Hence, $\overline L\leq L$ and
\begin{equation}\label{feta-eq} F^{\overline\eta_s}
(\overline L)=F^{\overline\eta_s}(L)=F^{\overline\eta_s}(H^r-).
\end{equation}
This implies $\overline{\eta}_s(\overline
L,H^r)=0$, and from the second assertion of Lemma \ref{lem:uni4}, it
follows that
\begin{equation}\label{dis:reneg0}
\overline{\eta}_s(\{\overline L\})=0.
\end{equation}
The change of variables formula and (\ref{feta-eq}) then yield
\begin{eqnarray}
\label{eq-cvf1} \int_0^{\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)}h((F^{\overline\eta
_s})^{-1}(y))\,dy&=&\int_{[0,H^r)}h(x)\overline{\eta}_s(dx)\nonumber\\[-8pt]\\[-8pt]
&=&\int
_{[0,\overline
L]}h(x)\overline{\eta}_s(dx).\nonumber
\end{eqnarray}
Also, by Proposition \ref{lem:uni5} and
another application of the change of variables formula, we have
\begin{eqnarray}
\label{eq-cvf2}
&&
\int_0^{(\overline{Q}{}^{(N)}(s)+\overline\iota^{(N)}(s))\wedge
F^{\overline\eta_s^{(N)}}(L)}h((F^{\overline\eta
_s^{(N)}})^{-1}(y))\,dy \nonumber\\[-8pt]\\[-8pt]
&&\qquad=\int_{[0,\chi
^{(N)}(s-)]}\mathbh{1}_{[0,L]}(x)h(x)\overline{\eta
}^{(N)}_s(dx).\nonumber
\end{eqnarray}
Expanding the term on the right-hand side of (\ref{eq-cvf2}) and using
the inequality \mbox{$\overline L \leq L$}, we obtain
\begin{eqnarray}
\label{dis:3}
& &\int_{[0,\chi^{(N)}(s-)]}\mathbh{1}
_{[0,L]}(x)h(x)\overline{\eta}^{(N)}_s(dx) \nonumber\\
&&\qquad=\int_{[0,\overline
L]}\mathbh{1}
_{[0,L]}(x)h(x)\overline{\eta}^{(N)}_s(dx)\nonumber\\[-8pt]\\[-8pt]
&&\qquad\quad{}+\int_{(\chi
^{(N)}(s-)\wedge\overline
L,\chi^{(N)}(s-)]}\mathbh{1}_{[0,L]}(x)h(x)\overline{\eta
}^{(N)}_s(dx) \nonumber\\
&&\qquad\quad{} - \int_{(\chi^{(N)}(s-)\wedge\overline L,\overline L]}\mathbh{1}
_{[0,L]}(x)h(x)\overline{\eta}^{(N)}_s(dx). \nonumber
\end{eqnarray}
By (\ref{eq-cvf1}) and (\ref{eq-cvf2}), the left-hand side and the
first term
on the right-hand side of (\ref{dis:3}), respectively, equal the left-hand
side and right-hand side of (\ref{dis:4}). Therefore, to prove (\ref{dis:4})
it suffices to show that the second and the third terms on the
right-hand side
of (\ref{dis:3}) converge to zero, as $N\rightarrow\infty$. Recall the
constant $\tilde C_L^h$ defined in Lemma \ref{lem-est}. Note that
$\tilde
C_L^h<\infty$ since $h$ is continuous. Therefore, the second term on the
right-hand side of (\ref{dis:3}) is bounded above by $\tilde C_L^h
\overline{\eta}^{(N)}_s(\chi^{(N)}(s-)\wedge\overline L,\chi
^{(N)}(s-)]$. By
(\ref{dis:reneg0}), Portmanteau's theorem and (\ref{feta-eq}), it
follows that
\[
\lim_{N\rightarrow\infty}\overline{\eta}^{(N)}_s\bigl(\chi
^{(N)}(s-)\wedge\overline
L,\chi^{(N)}(s-)\bigr] \leq\lim_{N\rightarrow\infty}\overline{\eta
}^{(N)}_s(\overline
L,H^r)=\overline{\eta}[\overline L,H^r)=0.
\]
On the other hand, the absolute value of the third term on the
right-hand side
of (\ref{dis:3}) is bounded above by $\tilde C_L^h
\overline{\eta}^{(N)}_s(\chi^{(N)}(s-)\wedge\overline L,\overline L]$. We now
argue by contradiction
to show that $\liminf_{N\rightarrow\infty}\chi^{(N)}(s-)\geq\overline
L$ and,
consequently, that $\overline{\eta}^{(N)}_s(\chi^{(N)}(s-)\wedge
\overline L,\overline L]$
converges to zero as $N\rightarrow\infty$. Indeed, suppose this
assertion were false. Then there must exist a subsequence $\{N_k\}
_{k\in{\mathbb N}}$ such that $\lim_{k\rightarrow\infty}\chi^{(N_k)}(s-)=
\overline L-\delta$ for some $\delta>0$. Hence, for $k$ large enough,
$\chi^{(N_k)}(s-)<\overline L-\delta/2$. By Lemma \ref{lem-chi}, we have
$\chi^{(N_k)}(s-)\geq\chi^{(N_k)}(s)$. Hence $\overline{\eta}
_s^{(N_k)}[0,\overline L-\delta/2]\geq\overline{Q}{}^{(N_k)}(s)$ by (\ref{qn}).
Sending $k\rightarrow\infty$ and using the convergence $\overline
{\eta}
_s^{(N_k)}\Rightarrow\overline{\eta}_s$, the fact that $[0,\overline
L-\delta/2]$
is closed and Portmanteau's theorem, we obtain $\overline{\eta
}_s[0,\overline
L-\delta/2]\geq\overline{Q}(s)$. This contradicts the definition of
$\overline L$,
and hence completes the proof of (\ref{dis:4}).
Finally, we deduce (\ref{dis:ext2}) from (\ref{dis:4}) using the
bounded convergence theorem, whose application is justified by the
bounds (\ref{eq-cvf1}), (\ref{eq-cvf2}) and the estimate (\ref{lem-est-2}).
\end{pf}
We now generalize Lemma \ref{lem:1} to allow for a general locally
integrable (not necessarily continuous) function $h^r$ on $[0,H^r)$.
\begin{lemma} \label{lem:2}
Let $L<H^r$, and let $\overline C{}^{(N)}_1(t,L), t\in[0,\infty),
N\in{\mathbb N}$ be defined as in (\ref{def-c1}). Then for every
$T\in
[0,\infty)$, almost surely for $L<H^r$,
\begin{equation}\label{dis:5}\lim_{N\rightarrow
\infty}\sup_{t\in[0,T]}\overline C{}^{(N)}_1(t,L)=0.
\end{equation}
\end{lemma}
\begin{pf}
Fix $L<H^r$. Since $h^r$ lies in $\mathcal{L}^1_{\mathrm{loc}}[0,H^r)$ and is
nonnegative,
there exists a sequence of nonnegative continuous functions $\{h^r_n\}
_{n\geq
1}$ on $[0,H^r)$ such that $\int_0^L|h^r(x)-h^r_n(x)|\,dx\rightarrow0$ as
$n\rightarrow\infty$ and $h^r_n$ has common compact support in
$[0,H^r)$. For
each $n\in{\mathbb N}$, (\ref{dis:5}) holds with $h^r_n$ in place of $h^r$
due to
Lemma \ref{lem:1}.
Let $l^r_n=|h^r_n-h^r|$ for each $n\geq1$.
Then, in order to prove (\ref{dis:5}), it clearly suffices to show
that the following two limits hold: almost everywhere,
\begin{equation}\label
{dis:11}\lim_{N\rightarrow\infty}\sup_N\int_0^T \biggl(\int
_0^{(\overline{Q}{}^{(N)}
(s)+\overline\iota^{(N)}(s))\wedge F^{\overline\eta
_s^{(N)}}(L)}l^r_n((F^{\overline\eta_s^{(N)}}
)^{-1}(y))\,dy \biggr)\,ds= 0\hspace*{-28pt}
\end{equation}
and
\begin{equation}\label{dis:12}\lim_{N\rightarrow\infty
}\int_0^T \biggl(\int_0^{\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)}l^r_n((F^{\overline\eta_s}
)^{-1}(y))\,dy \biggr)\,ds= 0.
\end{equation}
We first consider (\ref{dis:11}). By Proposition \ref{lem:uni5},
applied to
$h=l^r_n$, and the same scaling argument that was used to obtain (\ref
{rep-rcomp3}), for every $N,n\in{\mathbb N}$,
\begin{eqnarray*}
&&
\int_0^T \biggl(\int_0^{(\overline{Q}{}^{(N)}(s)+\overline\iota
^{(N)}(s))\wedge
F^{\overline\eta_s^{(N)}}(L)}l^r_n((F^{\overline\eta
_s^{(N)}})^{-1}(y))\,dy \biggr)\,ds \\
&&\qquad= \int
_0^T \biggl(\int_{[0,\chi^{(N)}(s-)\wedge L]}l^r_n(x)\overline{\eta}^{(N)}
_s(dx) \biggr)\,ds \leq\int_0^T \biggl(\int_{[0,L]}l^r_n(x)\overline{\eta}^{(N)}
_s(dx) \biggr)\,ds.
\end{eqnarray*}
By (\ref{def-waitjn}) and the representation of $\eta^{(N)}$ in (\ref
{def-etan}), we have
\begin{eqnarray*}
&&
\int_0^T \biggl(\int_{[0,L]}l^r_n(x)\overline{\eta}^{(N)}_s(dx) \biggr) \,ds \\
&&\qquad\leq\frac{1}{N} \sum_{j = -\mathcal{E}^{(N)}_0+ 1}^{0} \int_0^T
l^r_n\bigl(w^{(N)}_j(0)+s\bigr)
\mathbh{1}_{\{w^{(N)}_j (0)+s <L\wedge r_j\}} \,ds \\
&&\qquad\quad{} +
\frac{1}{N}\sum_{j = 1}^{E^{(N)}(T)} \int_{\zeta_j^{(N)}}^T
l^r_n\bigl(s-\zeta_j^{(N)}\bigr) \mathbh{1}_{\{s-\zeta_j^{(N)} <L\}} \,ds \\
&&\qquad\leq\sup_N \bigl( \bigl\langle1,\overline{\eta}^{(N)}_0 \bigr\rangle+\overline{E}{}^{(N)}(T)
\bigr)\int_0^Ll^r_n(x) \,dx.
\end{eqnarray*}
Since $\sup_N ( \langle 1,\overline{\eta}{}^{(N)}_0 \rangle +\overline{E}{}^{(N)}(t)
)<\infty$ almost surely, due to Assumption \ref{ass-init}, and
$h_n^r$ converges in $\mathcal{L}^1_{\mathrm{loc}}[0,H^r)$ to $h^r$, we obtain
(\ref{dis:11}).
On the other hand, observe that, by (\ref{lem-est-1}) of Lemma \ref
{lem-est} applied to $l=l^r_n$,
\begin{eqnarray*} \int_0^T \biggl(\int_0^{\overline{Q}(s)\wedge
F^{\overline\eta_s}
(L)}l^r_n((F^{\overline\eta_s})^{-1}(y))\,dy \biggr)\,ds &\leq& \int_0^T
\biggl(\int
_{[0,L]}l^r_n(x)\overline{\eta}_s(dx) \biggr)\,ds \\ &\leq& \tilde
L(L,T)\int
_0^Ll^r_n(x)\,dx.
\end{eqnarray*}
By the convergence of $h_n^r$ to $h^r$ in $\mathcal
{L}^1_{\mathrm{loc}}[0,H^r)$, the last
term on the right-hand side of the above display converges to $0$, as
$n\rightarrow\infty$, and (\ref{dis:12}) follows.
\end{pf}
\begin{pf*}{Proof of Proposition \ref{prop:3}}
Given the
discussion prior
to Lemma \ref{lem:uni4} and, in particular, (\ref{dcomp1}), to
complete the
proof of the proposition, it only remains to show that
\begin{equation}
\label{dis:6} \lim_{L\rightarrow H^r}\limsup_{N\rightarrow\infty
}\mathbb{E}\Bigl[\sup
_{t\in
[0,T]}\overline C{}^{(N)}_i(t,L) \Bigr]=0,\qquad i=1,2,
\end{equation}
and
\begin{equation}\label{dis:6'}\lim_{L\rightarrow H^r}\mathbb{E}
[\overline
C_3(T,L) ]=
0.
\end{equation}
For the case $i=1$ in (\ref{dis:6}), this follows from Lemma
\ref{lem:2}
and the dominated convergence theorem, whose application is justified
because, by (\ref{eq-cvf1}), (\ref{eq-cvf2}) and the fact that $\overline
{L} \leq
L$,
\begin{eqnarray*}\mathbb{E}\Bigl[\sup_{t\in[0,T]}\overline
C_1^{(N)}(t,L) \Bigr]
&\leq&\mathbb{E}\biggl[\int_0^T \biggl(\int_{[0,L]}h^r(x)\overline{\eta}^{(N)}
_s(dx) \biggr)\,ds \biggr]\\
& &{} +\mathbb{E}\biggl[\int_0^T \biggl(\int
_{[0,L]}h^r(x)\overline{\eta}_s(dx) \biggr)\,ds \biggr],
\end{eqnarray*}
which is bounded uniformly in $N$ by (\ref{lem-est-2}) and Assumption
\ref{ass-init}.
Now, by Remark \ref{rem-compen}, an application of Lemma 5.8(1) of
\cite{kasram07} (with $\nu$, $h^s$ and $H^s$, resp., replaced by
$\eta$, $h^r$ and $H^r$, resp.), shows that
\begin{equation}
\label{use-temp}
\lim_{L \rightarrow H^r} \sup_{N} \mathbb{E}\biggl[ \int_0^t \biggl( \int
_{[L,H^r)} h^r(x)
\overline{\eta}^{(N)}_s (dx) \biggr) \,ds \biggr] = 0.
\end{equation}
On the other hand, the definition of $\overline C{}^{(N)}_2(T,L)$ in
(\ref{def-c2}), when combined
with Proposition \ref{lem:uni5} and (\ref{eq-cvf2}), shows that
\[
\sup_N\mathbb{E}\bigl[\overline C{}^{(N)}_2(T,L) \bigr]\leq\sup_N
\mathbb{E}\biggl[\int_0^T \biggl(\int_{[L,H^r)} h^r(x) \overline{\eta}^{(N)}
_s(dx) \biggr) \,ds \biggr].
\]
Taking the limit, as $L\rightarrow H^r$, and invoking (\ref
{use-temp}), it follows that (\ref{dis:6}) holds for $i=2$. Finally,
to show (\ref{dis:6'}), we see that, by the definition of $\overline
C_3(T,L)$ in (\ref{def-c3}) and the change of variables formula,
\begin{eqnarray*} \mathbb{E}[\overline C_3(T,L) ] &= &\mathbb{E}
\biggl[\int_0^t \biggl( \int_{\overline{Q}(s)\wedge F^{\overline\eta
_s}(L)}^{\overline{Q}(s)}h^r((F^{\overline\eta_s}
)^{-1}(y))\,dy \biggr)\,ds \biggr]\\
&\leq& \int_0^t \biggl( \int
_{[L,H^r)}h^r(x)\overline{\eta}_s(dx) \biggr)\,ds.
\end{eqnarray*}
If $h^r$ is bounded, then (\ref{dis:6'}) holds by simply applying the
bounded convergence theorem on the right-hand side of the equality in
the above display. On the other hand, suppose $h^r$ is
lower-semicontinuous on $(L^r,H^r)$ for some $L^r<H^r$. Then, by
Theorem A.3.12 of
\cite{dupellbook} and the
fact that $\mathbb{P}$ a.s., $\overline{\eta}^{(N)}_s \stackrel
{w}{\rightarrow}\overline{\eta}_s$, as $N\rightarrow\infty$,
for a.e. $s\in[0,T]$, this implies that for any
such $s$ and $L>L^r$,
\[
\int_0^t \biggl( \int_{[L,H^r)}h^r(x)\overline{\eta}_s(dx) \biggr)\,ds \leq
\liminf_{N\rightarrow\infty} \int_0^t \biggl( \int
_{[L,H^r)}h^r(x)\overline{\eta}^{(N)}
_s(dx) \biggr)\,ds.
\]
Integrating both sides over $s\in[0,T]$ and taking expectations, an
application of
Fatou's lemma yields
\[
\mathbb{E}[\overline C_3(T,L) ]\leq\liminf_{N\rightarrow\infty
}\mathbb{E}
\biggl[\int_0^t \biggl( \int_{[L,H^r)}h^r(x)\overline{\eta}^{(N)}_s(dx)
\biggr)\,ds \biggr].
\]
Taking the limit as $L\rightarrow H^r$, an application of (\ref{use-temp})
shows that
(\ref{dis:6'}) holds.
\end{pf*}
We now prove the main limit result.
\begin{pf*}{Proof of Theorem \ref{thm:FE}}
Fix $t\in[0,\infty)$ such that $\overline{\nu}{}^{(N)}_t \stackrel
{w}{\rightarrow}\overline{\nu}_t$, $\overline{\eta}^{(N)}_t
\stackrel{w}{\rightarrow}\overline{\eta}_t$, $\overline
{E}{}^{(N)}(t)\rightarrow\overline{E}(t)$, $\overline
{X}{}^{(N)}(t)\rightarrow
\overline{X}(t)$, $\overline{R}{}^{(N)}(t)\rightarrow\overline
{R}(t)$, $\overline{A}{}^{(N)}_{\cdot,\nu
}(t)\stackrel{w}{\rightarrow}\overline{A}_{\cdot,\nu}(t)$,
$\overline{D}{}^{(N)}_{\cdot}(t) \stackrel{w}{\rightarrow}
\overline{A}_{\cdot,\nu}(t)$, $\overline{A}{}^{(N)}_{\cdot,\eta
}(t)\stackrel{w}{\rightarrow}
\overline{A}_{\cdot,\eta}(t)$, $\overline{S}{}^{(N)}_{\cdot
}(t)\stackrel{w}{\rightarrow}\overline{A}_{\cdot
,\eta}(t)$ as $N\rightarrow\infty$. Since $\overline Y{}^{(N)}\rightarrow\overline Y$ a.s., this occurs for $t$ outside a
countable set. By (\ref{dis:13}), this implies that as $N\rightarrow
\infty$,
\begin{eqnarray}\overline{D}{}^{(N)}_{\varphi}(t)\rightarrow\overline
{A}_{\varphi,\nu
}(t)=\int
_0^t\langle\varphi(\cdot,s)h^s(\cdot,s),\overline{\nu}_s\rangle
\,ds,\nonumber\\[-8pt]\\[-8pt]
\eqntext{\varphi\in\mathcal{C}_b\bigl([0,H^s)\times{\mathbb
R}_+\bigr).}
\end{eqnarray}
An analogous argument also implies that, as $N\rightarrow\infty$,
\begin{eqnarray}\label{dis:fsn}\overline{S}{}^{(N)}_{\psi
}(t)\rightarrow\overline{A}
_{\psi,\eta
}(t)=\int_0^t\langle\psi(\cdot,s)h^r(\cdot,s),\overline{\eta
}_s\rangle \,ds,\nonumber\\[-8pt]\\[-8pt]
\eqntext{\psi\in\mathcal{C}_b\bigl([0,H^r)\times{\mathbb R}_+\bigr).}
\end{eqnarray}
In particular, when $\varphi=\psi={\mathbf{1}}$, the above two displays
imply that (\ref{cond-radon}) holds.
Also, we immediately obtain that, as $N\rightarrow\infty$, $\langle
\mathbf{1}, \overline{\nu}{}^{(N)}_t\rangle\rightarrow\langle
{\mathbf{1}}, \overline{\nu}_t\rangle$ and
$\langle
{\mathbf{1}}, \overline{\eta}^{(N)}_t\rangle\rightarrow\langle
{\mathbf{1}}, \overline{\eta}
_t\rangle$.
When combining with (\ref{def-xn}), (\ref{comp-prelimit}), (\ref
{def-kn}), (\ref{equivD}), (\ref{def-dn}), (\ref{qn}), (\ref
{rep-R}), this implies that all the equations in Definition \ref
{def-fleqns} are satisfied at time $t$ except (\ref{eq-ftmeas}) and
(\ref{eq-ftreneg}).
It only remains to show that (\ref{eq-ftmeas}) and (\ref{eq-ftreneg})
are also satisfied at time $t$. We shall just prove (\ref
{eq-ftreneg}). The same argument will also show that (\ref{eq-ftmeas})
holds. Dividing (\ref{eqn-prelimit3}) by $N$, we have
\begin{eqnarray*}
\bigl\langle\psi(\cdot, t), \overline{\eta}^{(N)}_{t} \bigr\rangle
& = & \bigl\langle\psi(\cdot, 0), \overline{\eta}^{(N)}_{0} \bigr\rangle+
\int_{0}^t \bigl\langle\psi_x(\cdot,s) + \psi_s(\cdot,s), \overline
{\eta}^{(N)}
_s \bigr\rangle \,ds \\
& &{} - \overline S{}^{(N)}_\psi(t) + \int_{[0,t]} \psi(0,s) \,d\overline
{E}{}^{(N)}(s).\nonumber
\end{eqnarray*}
Since $\overline{\eta}^{(N)}_{0} \stackrel{w}{\rightarrow}\overline
{\eta}_0$ by Assumption \ref
{ass-init}(4), $\overline{\eta}^{(N)}_s \stackrel{w}{\rightarrow
}\overline{\eta}_s$ for a.e. $s \in[0, t]$,
$\overline{\eta}^{(N)}_t \stackrel{w}{\rightarrow}\overline{\eta
}_t$ by our
choice of $t$ and $\psi(\cdot, t)$ and $\psi_x(\cdot, s)+\psi
_s(\cdot, s), s\in[0, t]$, are bounded and continuous,
as $N\rightarrow\infty$, we have
\[
\bigl\langle\psi(\cdot, t), \overline{\eta}^{(N)}_{t} \bigr\rangle
\rightarrow\langle
\psi(\cdot, t), \overline{\eta}_{t} \rangle\quad\mbox{and}\quad
\bigl\langle\psi(\cdot, 0), \overline{\eta}^{(N)}_{0} \bigr\rangle
\rightarrow\langle
\psi(\cdot, 0), \overline{\eta}_{0} \rangle,
\]
and, by the bounded convergence theorem,
\[
\int_{0}^t \bigl\langle\psi_x(\cdot,s) + \psi_s(\cdot,s), \overline
{\eta}^{(N)}
_s \bigr\rangle \,ds \rightarrow\int_{0}^t \langle\psi_x(\cdot,s) + \psi
_s(\cdot,s), \overline{\eta}_s \rangle \,ds.
\]
On the other hand, using an integration-by-parts argument, the facts
that $\overline{E}{}^{(N)}(0)=0$, $\overline{E}{}^{(N)}\rightarrow
\overline{E}$,
$\overline{E}$ is nondecreasing and $\psi_s(0, \cdot)$ is bounded and
continuous on $[0,t]$,
along with the bounded convergence theorem, we see that, as
$N\rightarrow
\infty$,
\[
\int_{[0,t]} \psi(0,s) \,d\overline{E}{}^{(N)}(s) \rightarrow\int
_{[0,t]} \psi(0,s) \,d\overline{E}(s).
\]
Combining the last four displays with (\ref{dis:fsn}), it follows that
(\ref{eq-ftreneg}) holds.
Then it follows that all fluid equations are satisfied for all but
countably many $t$. By right-continuity (with respect to $t$) of each
of the terms in all fluid equations, we conclude that all fluid
equations are a.s. satisfied for all $t\in[0,\infty)$. This completes
the proof of the desired result that $(\overline{X},\overline{\nu
},\overline{\eta})$
satisfies the fluid equations.
\end{pf*}
\subsection{\texorpdfstring{Proof of Theorem \protect\ref{thm:3}}{Proof of Theorem 3.8}}
\label{subs-prf3}
This section is devoted to the proof of Theorem~\ref{thm:3}.
Recall ${\mathcal T}_t^{(N)}(s)$ in (\ref{dis:Tn}) and its fluid
scaled version defined in (\ref{fl-scaling2}).
Observe that the virtual waiting time defined in (\ref{T}) can be
rewritten in terms of the fluid-scaled quantities as
\begin{equation} \label{T2}\qquad
W^{(N)}(t) \doteq\inf\bigl\{s\geq0\dvtx\overline{D}{}^{(N)}(t+s)
-\overline{D}{}^{(N)}(t)+\overline
{\mathcal T}{}^{(N)}_t(s) > \overline{Q}{}^{(N)}(t) \bigr\}.
\end{equation}
We first show that for each $t\in[0,\infty)$, $\overline
{\mathcal T}{}^{(N)}_t
\Rightarrow\overline{\mathcal T}_t$ as $N \rightarrow\infty$, where
$\overline{\mathcal T}_t$ is defined in (\ref{dis:T}). Notice that
a customer $j$ who arrived into the system before time $t$ and has
not reneged by time $t$ must have a potential waiting time
$w^{(N)}_j(u) > u-t$ for all $u > t$ sufficiently small.
In addition, for that customer to have reneged from the queue
(before entering service) in the period $[t,t+s]$, there must
exist a time $u \in[t,t+s]$ such that the customer is still in queue
(i.e., has not yet entered service) or, equivalently, such that
$w^{(N)}_j(u) < \chi^{(N)}(u-)$,
the waiting time of the head-of-the-line customer just prior to $u$,
and the customer reneges, so that the slope of her
potential waiting time changes
from one to zero.
Therefore, for each $s\in
[0,\infty)$, $\mathcal T_t^{(N)}(s)$ can be alternatively expressed as
\begin{eqnarray*}
\mathcal T_t^{(N)}(s) &=& \sum_{u \in[t,t+s]} \sum_{j=-\mathcal{E}^{(N)}_0+
1}^{E^{(N)}
(u)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(u-) >0, {dw^{(N)}_j }/{dt}(u
+)=0 \}}\\
&&\hspace*{78.85pt}{}\times
\mathbh{1}_{\{u-t<w^{(N)}_j (u)\leq\chi^{(N)}(u-)\}}.
\end{eqnarray*}
Let
\[
\mathcal T_t^{(N),1}(s) \doteq\sum_{u \in[t,t+s]} \sum_{j=-\mathcal
{E}^{(N)}_0+
1}^{E^{(N)}
(u)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(u-) >0, {dw^{(N)}_j }/{dt}(u
+)=0 \}}
\mathbh{1}_{\{w^{(N)}_j (u)\leq\chi^{(N)}(u-)\}}
\]
and
\[
\mathcal T_t^{(N),2}(s) \doteq\sum_{u \in[t,t+s]} \sum_{j=-\mathcal
{E}^{(N)}_0+
1}^{E^{(N)}
(u)}
\mathbh{1}_{ \{{dw^{(N)}_j }/{dt}(u-) >0, {dw^{(N)}_j }/{dt}(u
+)=0 \}}
\mathbh{1}_{\{w^{(N)}_j (u)\leq u-t\}}.
\]
It is easy to see that $\mathcal T_t^{(N)}(s)=\mathcal T_t^{(N),1}
(s)-\mathcal T_t^{(N),2}(s)$, $\mathcal T_t^{(N),1}(s)=R^{(N)}(t+s)-R^{(N)}
(t)$, $\mathcal T_t^{(N),2}(s) \leq S^{(N)}(t+s)-S^{(N)}(t)$
and $\mathcal T_t^{(N),2}(s+\delta)-\mathcal T_t^{(N),2}(s) \leq S^{(N)}
(t+s+\delta)-S^{(N)}(t+s)$.
Therefore, an application of Kurtz's criteria in Proposition \ref{Kurtz}
shows
that the relative compactness of
the fluid scaled versions
$\overline{\mathcal T}{}^{(N),1}_t$ and $\overline{\mathcal T}{}^{(N),2}_t$
of
${\mathcal T}_t^{(N),1}$ and ${\mathcal T}_t^{(N),2}$, respectively,
follows from that of $\overline{R}{}^{(N)}$ and $\overline{S}{}^{(N)}$
established in Lemma \ref{lem:rc}.
By a straightforward adaption of the argument used in Proposition \ref
{prop:3} to
show the convergence of $\overline{R}{}^{(N)}$ to $\overline{R}$, we can
conclude that
$\overline{\mathcal T}{}^{(N)}_t\Rightarrow\overline{\mathcal T}_t$
as $N\rightarrow\infty$.
Recall the application of the Skorokhod representation theorem in Theorem
\ref{thm:FE} to assume, without loss of generality, that $\overline Y{}^{(N)}$
converges a.s. to $\overline Y$. Here, we can also assume, in
addition, that
$\overline{\mathcal T}{}^{(N)}_t(s)\rightarrow\overline{\mathcal T}_t$ a.s.,
as $N\rightarrow\infty$. Since $\overline{Q}$ is continuous at $t$
and, by
(\ref{dis:13}), $\overline{A}_{{\mathbf{1}},\nu}=\int_0^\cdot
\langle
h^s,\overline{\nu}_s\rangle \,ds$ is
continuous by the integral representation,
and $\overline{\mathcal T}_t$ has continuous paths by definition,
it follows that, almost surely, $\overline{Q}{}^{(N)}(t)\rightarrow
\overline{Q}(t)$ and for
each $T\in[0,\infty)$, as $N\rightarrow\infty$,
\[
\sup_{s\in[0,T]}\bigl|\overline{D}{}^{(N)}(t+s)-\overline{A}_{{\mathbf
{1}},\nu
}(t+s)\bigr|\rightarrow0 \quad\mbox{and}\quad \sup_{s\in
[0,T]}\bigl|\overline{\mathcal T}{}^{(N)}_t(s)- \overline{\mathcal
T}_t\bigr|\rightarrow
0.
\]
From (\ref{T2}), it is easy to see that
$W^{(N)}(t)\leq(\overline{D}{}^{(N)})^{-1}(\overline
{D}{}^{(N)}(t)+\overline{Q}{}^{(N)}(t))-t$ for each $N$. By the
tightness result established in
Theorem \ref{th-tight}, we know that
$\overline{D}{}^{(N)}(t)+\overline{Q}{}^{(N)}(t)$ is bounded uniformly in
$N$, and due to Lemma 4.10 of
\cite{ramrei03} and the assumption that $\overline{A}_{{\mathbf
{1}},\nu
}$ is
uniformly strictly increasing, we also know that
$(\overline{D}{}^{(N)})^{-1}\rightarrow
(\overline{A}_{{\mathbf{1}},\nu} )^{-1}$ uniformly on compact
sets, as $N\rightarrow\infty$.
Hence,
$W^{(N)}(t)$ is bounded uniformly in $N$. Therefore, there exists a
subsequence, $W^{(N_n)}(t)$, $n \in{\mathbb N}$, that converges to a
limit in
$[0,\infty)$,
which we denote by $W^*$. From (\ref{T2}) and
the right-continuity of $\overline{D}{}^{(N)}, \overline{Q}{}^{(N)}$ and
$\overline{\mathcal T}{}^{(N)}_t$, we then have
$\overline D{}^{(N_n)}(t+\overline W^{(N_n)}(t))-\overline
D{}^{(N_n)}(t)+\overline{\mathcal
T}{}^{(N_n)}_t(\overline W{}^{(N_n)}(t))\geq\overline Q{}^{(N_n)}(t)$.
Sending $n \rightarrow\infty$, we
obtain
\begin{equation}
\label{ineq-Wstar}
\overline{A}_{{\mathbf{1}},\nu}(t+W^*)- \overline{A}_{{\mathbf
{1}},\nu
}(t)+\overline{\mathcal T}_t(W^*)\geq\overline{Q}(t).
\end{equation}
Together with (\ref{Wbar}),
this shows that $\overline W(t)\leq W^*$. Now, suppose that $\overline
W(t)< W^*$,
and
fix $w$ such that $\overline W(t)<w< W^*$.
Since $\overline{A}_{{\mathbf{1}},\nu}$ is uniformly strictly
increasing and $\overline{\mathcal T}_t$ is nondecreasing, the
inequality $\overline W(t)<w$ implies
that $\overline{A}_{{\mathbf{1}},\nu}(t+w)-\overline{A}_{{\mathbf
{1}},\nu
}(t)+\overline{\mathcal T}_t(w) > \overline{Q}(t)$.
Therefore, for sufficiently large $N$,
we have $\overline{D}{}^{(N)}(t+w)-\overline{D}{}^{(N)}(t)+\overline
{\mathcal T}{}^{(N)}_t(w) > \overline{Q}{}^{(N)}(t)$ and hence $W^{(N)}(t)
\leq w$. In
turn, this implies
that $W^{(N_n)}(t) \leq w$ for sufficiently large $n \in{\mathbb N}$.
Sending $n\rightarrow\infty$ and using the convergence of
$W^{(N_n)}(t)$ to $W^*$, we then obtain $W^*\leq w$.
This contradicts the choice of $w$. Hence $\overline W(t)= W^*$, and
this proves the
desired result.
\begin{appendix}
\section{Explicit construction of the state processes}
\label{ap-markov}
In this section, we construct all state processes and auxiliary
processes described in Section \ref{sec:repdyn} from the initial data
$\{\mathcal{E}^{(N)}_0, X^{(N)}(0),w^{(N)}_j(0), a^{(N)}_j(0),
j=-\mathcal{E}^{(N)}_0+1,\ldots,0\}$, $\{
\alpha_E^{(N)}(t),t\in[0,\infty)\}$, $\{v_j, j\in{\mathbb Z}\}$ and
$\{r_j,
j\in{\mathbb Z}\}$.
Fix $N$ and, for simplicity, we omit the dependence on $N$ in notation.
Let \mbox{$E(0)=0$}. The process $E$ on $[0,\infty)$ can be obtained from
$\alpha_E$ using the relation (\ref{def-ren}). Let $\ell=0$, $\tau
_0=0$, and let $R(\tau_\ell)=D(\tau_\ell)=K(\tau_\ell)=0$,
\begin{equation}
\label{app-q}
Q(\tau_\ell) \doteq[X(\tau_\ell) - N]^+,
\end{equation}
and for $j>E(\tau_\ell)$, let $w_j(\tau_\ell)=a_j(\tau_\ell)=0$.
Now, for $t \in[\tau_\ell, \infty)$, define
\begin{equation}
\label{app-chi}
\chi^\ell(t) \doteq
\inf\{ x > 0\dvtx\eta_{\tau_\ell} [0,x] \geq Q(\tau_\ell)\} + t -
\tau_{\ell}.
\end{equation}
Also, for $j=-\mathcal{E}_0+1,\ldots,0,\ldots,E(\tau_\ell)$ and
$t\in[\tau_\ell,\infty)$, let
\begin{eqnarray*}
w_j^\ell(t) &\doteq& \bigl(w_j(\tau_\ell)+t-\tau_\ell\bigr)\wedge r
_j, \\
a_j^\ell(t) &\doteq& \cases{
0, &\quad if $w_j(\tau_\ell)=r_j$
or $w_j(\tau_\ell)\leq\chi^\ell(\tau_\ell)$, \vspace*{2pt}\cr
\bigl(a_j(\tau_\ell)+t-\tau_\ell\bigr)\wedge v_j, &\quad if $\chi^\ell(\tau
_\ell
)<w_j(\tau_\ell)<r_j$,}
\\
\eta^\ell_t &\doteq& \sum_{j=-\mathcal{E}_0+1}^{E(\tau
_\ell)}\delta_{w_j(t)}\mathbh{1}_{ \{{d w_j}/{dt}(t+) >0 \}
}, \\
\nu^\ell_t &\doteq& \sum_{j=-\mathcal{E}_0+1}^{E(\tau_\ell
)}\delta_{a_j(t)}\mathbh{1}_{ \{{d a_j}/{dt}(t+) >0 \}}, \\
R^\ell(t) &\doteq& \sum_{j=-\mathcal{E}_0+1}^{E(\tau_\ell)}\sum
_{s\in[0,t]}\mathbh{1}_{ \{w_j(s)\leq\chi^l(s-), {d
w_j}/{dt}(s-) >0, {d w_j }/{dt}(s+)=0 \}}, \\
D^\ell(t)
&\doteq& \sum_{j=-\mathcal{E}_0+1}^{E(\tau_\ell)}\sum_{s\in
[0,t]}\mathbh{1}_{ \{{d a_j}/{dt}(s-) >0, {d a_j
}/{dt}(s+)=0 \}}.
\end{eqnarray*}
Next, define
\[
\tau_{\ell+ 1} \doteq\inf\bigl\{t > 0\dvtx\bigl(D^{\ell}(t) -
D(\tau_{\ell})\bigr) \wedge
\bigl(R^{\ell}(t) - R(\tau_{\ell})\bigr) \wedge\bigl(E(t) - E(\tau_{\ell})\bigr) > 0
\bigr\}.
\]
For $t \in[\tau_{\ell},\tau_{\ell+1})$, let $Y(t) = Y^\ell(t)$ for $Y =
w_j, a_j, j \in-\mathcal{E}_0+1, \ldots, E(\tau_\ell)$, $R, D, \eta,
\nu$ and $\chi$ and set $Y(t) = Y(\tau_\ell)$ for $Y = X, Q, w_j, a_j,
j>E(\tau_\ell)$. Moreover, define
\begin{eqnarray*}
X(\tau_{\ell+1}) & \doteq& X(\tau_{\ell}) + E(\tau_{\ell+1}) -
E(\tau_{\ell})
- D(\tau_{\ell+1}) + D(\tau_{\ell}) \\
& &{} - R(\tau_{\ell+1}) + R(\tau_{\ell}), \\
\eta_{\tau_{\ell+1}} & \doteq& \eta^{\ell}_{\tau_{\ell+1}} +
\bigl(E(\tau_{\ell+1}) - E(\tau_{\ell})\bigr) \delta_0,
\end{eqnarray*}
and, if $E(\tau_{\ell+ 1}) > E(\tau_{\ell})$, then $E(\tau_{\ell+
1}) = E(\tau_{\ell})+1$, and then let
$w_j (\tau_{\ell+1}) \doteq0$ for $j \in\{ E(\tau_{\ell})+1,
\ldots,
E(\tau_{\ell+1})\}$.
In this case, $Q(\tau_{\ell+1})$ and $\chi(\tau_{\ell+1})$ can be
defined via equations (\ref{app-q}) and (\ref{app-chi}), but with
$\ell$
replaced by $\ell+ 1$, and
the procedure can be reiterated.
Now, $\max\{\ell\dvtx\tau_\ell\leq t\}$ is
bounded by $\mathcal{E}_0 + E(t)$, and is therefore
a.s. finite. Therefore, $\tau_{\ell} \rightarrow\infty$ as $\ell
\rightarrow
\infty$,
and so the above procedure constructs the above processes on
$[0,\infty)$. $K$ and $S$ can then be defined, respectively,
via equations (\ref{def-kn}) and (\ref{def-sn}).
For each $j\geq-\mathcal{E}^{(N)}_0$, by the construction, we have
\begin{eqnarray*}w_j(t)&=&\sum_{E(\ell)\geq j}\mathbh{1}_{[\tau
_\ell
,\tau_\ell+1)}(t)\bigl(w_j(\tau_\ell)+t-\tau_\ell\bigr)\wedge r_j \\
&=& \cases{
t\wedge r_j, &\quad if $j=-\mathcal{E}^{(N)}_0,\ldots,
0$,\cr
(t-\zeta_j)\wedge r_j, &\quad otherwise,}
\end{eqnarray*}
where $\zeta_j=\inf\{t>0\dvtx E(t)=j\}$. Hence the process $w_j$ defined
above is
indeed the potential waiting time process of customer $j$. It is also
not to
hard to see that the process $a_j$ defined above is the age process of
customer $j$ and satisfies (\ref{adif}). We next show that the process
$\chi$ constructed above satisfies (\ref{def-chi}). It is easy to see
that $\chi(0)=\chi^0(0)$ by (\ref{app-chi}) with $t=0$ and $\ell=
0$. The $\chi(0)$ satisfies (\ref{def-chi}) for $t=0$. When $t\in
[\tau_0, \tau_1)$, $Q(t)=Q(0)$, $\eta_t=\eta^0_t$ and $\chi
(t)=\chi^0(t)$. Then we have
\[
\chi^0(t)=\inf\{ x > 0\dvtx\eta_{\tau_0} [0,x] \geq Q(\tau_0)\} + t
- \tau_{0}=\inf\{ x > 0\dvtx\eta_{t} [0,x] \geq Q(t)\}.
\]
Hence $\chi$ satisfies (\ref{def-chi}) on the interval $[\tau_0,\tau
_1)$. By the standard induction argument, we can see that $\chi$
satisfies (\ref{def-chi}) for all $t\geq0$.
For each $t\geq0$, by the construction, we have
\begin{eqnarray*}\eta_t &=&\sum_{\ell= 0}^\infty\mathbh{1}_{[\tau
_\ell
,\tau_\ell+1)}(t)\sum_{j=-\mathcal{E}_0+1}^{E(\tau_\ell)}\delta
_{w_j(t)}\mathbh{1}_{ \{{d w_j}/{dt}(t+) >0 \}} \\
&=& \sum
_{\ell= 0}^\infty\mathbh{1}_{[\tau_\ell,\tau_\ell+1)}(t)\sum
_{j=-\mathcal{E}_0+1}^{E(t)}\delta_{w_j(t)}\mathbh{1}_{ \{{d
w_j}/{dt}(t+) >0 \}} \\
&=& \sum_{j=-\mathcal
{E}_0+1}^{E(t)}\delta_{w_j(t)}\mathbh{1}_{ \{{d w_j}/{dt}(t+)
>0 \}} .
\end{eqnarray*}
This shows that the $\eta$ constructed satisfies (\ref{def-etan}). A
similar argument shows that the processes $\nu$, $D$ and $R$
constructed satisfy (\ref{def-nun}), (\ref{def-depart}) and (\ref
{def-crp}), respectively. Finally, $K$ and $S$ satisfy (\ref{def-kn})
and (\ref{def-sn}) by construction.
Recall that, for $t\in[0,\infty)$, $\tilde{\mathcal{F}}_t$ is the
$\sigma$-algebra generated by
\[
\bigl(\mathcal{E}_0,X(0),\alpha_E(s), w_j(s), a_j(s), j \in\{-\mathcal
{E}_0+1, \ldots, 0\}\cup{\mathbb N}, s \in[0,t]\bigr\}
\]
and $\{\mathcal{F}_t\}$ is the associated completed,
right-continuous filtration.
\begin{lemma} \label{app:adapt}
The processes $w_j, a_j, j \geq-\mathcal{E}_0+1$ and $E, R, D,
\eta
, \nu, \chi, X, Q$, $K, S$ are c\`{a}dl\`{a}g and $\{\mathcal
{F}_t\}
$-adapted.
\end{lemma}
\begin{pf}
The c\`{a}dl\`{a}g property of those processes follows from the
construction. Now we show that all the processes are $\{\mathcal{F}_t\}
$-adapted. Indeed, it follows immediately from (\ref{def-ren}),
(\ref{def-etan}), (\ref{def-nun}), (\ref{def-depart}) and (\ref{def-cvrp})
that $E, \eta, \nu, D$ and $S$ are $\mathcal{F}_t$-adapted. We next
show that
$\chi$ is $\mathcal{F}_t$-adapted. By equations (\ref{def-qnt}) and
(\ref{def-chi}) evaluated at time~$0$, it follows that $\chi(0)$ is a function
of $X(0)$ and $\eta_0$ and hence $\mathcal{F}_0$-adapted. Now, let $t>
0$. For
each $\ell\geq0$, by the induction argument, $\chi^\ell(t)$ is
$\mathcal{F}_t$-adapted, and $\tau_\ell$ is an $\mathcal
{F}_t$-stopping time. Since
$\chi_t=\chi^\ell_t$ if $t\in[\tau_\ell,\tau_{\ell+1})$, $\chi
$ is
$\mathcal{F}_t$-adapted.~Equations (\ref{def-crp}) and (\ref{def-dn})
show that $R$
and $X$ are $\mathcal{F}_t$-adapted,
and it follows from (\ref{def-qnt}) and (\ref{def-kn}) that $Q$ and
$K$ are
$\mathcal{F}_t$-adapted.
\end{pf}
The next lemma establishes some basic properties of $\chi(t)$, the
waiting time of the head-of-the-line customer at time $t$, defined in
(\ref{def-chi}).
\begin{lemma} \label{lem-chi} $\chi$ is piecewise linear with
downward jumps
that occur when the head-of-the-line customer either enters service
(due to
a departure from service) or reneges from the queue. Hence, $\chi
(t-)\geq\chi(t)$ for every $t\in(0,\infty)$. Moreover, for every
$t>0$, there exists $\varepsilon_t(\omega)\in(0,t)$ such that for
all $\tilde t\in(t-\varepsilon_t(\omega),t)$, $\chi(t-)-\chi
(\tilde t-)=t-\tilde t>0$.
\end{lemma}
\begin{pf}
By the construction, $\chi_t=\chi^\ell_t$ if $t\in
[\tau_\ell,\tau_{\ell+1})$. Since $\chi^\ell$ is linear on
$[\tau_\ell,\tau_{\ell+1})$, $\chi$ is piecewise linear. Also
$\chi$
can only jump at $\tau_{\ell+1}$, $\ell\geq0$. Based on the definition
of $\tau_{\ell+1}$, it is not hard to see that $\chi$ can only have a
downward jump at $\tau_{\ell+1}$ when the head-of-the-line customer
either enters service [$D^{\ell}(\tau_{\ell+1}) - D(\tau_{\ell
})>0$] or
reneges from the queue [$R^{\ell}(\tau_{\ell+1}) - R(\tau_{\ell})>0$].
Then we have $\chi(t-)\geq\chi(t)$ for every $t\in(0,\infty)$. The
last statement of the lemma follows from the fact that $\chi$ is
c\`{a}dl\`{a}g and piecewise linear.
\end{pf}\vspace*{-14pt}
\section{Strong Markov property} \label{sec:SMF}
In this section we show that the state descriptor $V^{(N)} =
(\alpha_E^{(N)}, X^{(N)}, \nu^{(N)},\break \eta^{(N)})$ is a strong Markov
process with
respect to the filtration $\{\mathcal{F}_t^{(N)}, t\geq0\}$ defined in
Section \ref{subsub-filt}. To ease the notation, we shall suppress the
superscript $(N)$ from the notation.
Let $\mathcal{M}_D[0,H^s)$ and $\mathcal{M}_D[0,H^r)$ be the subsets
of $\mathcal{M}_F[0,H^s)$ and $\mathcal{M}_F[0,H^r)$,
respectively, such that each measure in $\mathcal{M}_D[0,H^s)$ and
$\mathcal{M}_D[0,H^r)$ takes the
form $\sum_{i=1}^k \delta_{x_i}$. Define
\begin{equation}
\mathcal{V}\doteq\left\{
\matrix{(\alpha, x, \mu, \pi) \in{\mathbb R}_+\times{\mathbb Z}
_+\times\mathcal{M}_D[0,H^s)\times\mathcal{M}_D[0,H^r)\mbox{:}\cr
x\leq\langle{\mathbf{1}}, \mu\rangle+\langle{\mathbf{1}}, \pi
\rangle,
\langle{\mathbf{1}},
\mu\rangle\leq N}
\right\},
\end{equation}
where ${\mathbb R}_+$ is endowed with the Euclidean topology $d$,
${\mathbb Z}_+$ is
endowed with the discrete topology $\rho$ and $\mathcal{M}_D[0,H^s)$
and $\mathcal{M}_D[0,H^r)$
are endowed with the weak topology, respectively. The space
$\mathcal{V}$ is a closed subset of ${\mathbb R}_+\times{\mathbb
Z}_+\times\mathcal{M}_F[0,H^s)
\times\mathcal{M}_F[0,H^r)$ and is endowed with the usual product
topology. Since
${\mathbb R}_+\times{\mathbb Z}_+\times\mathcal{M}_F[0,H^s)\times
\mathcal{M}_F[0,H^r)$ is a Polish space, then
the closed subset $\mathcal{V}$ is also a Polish space. Now, denote
\[
V(t)\doteq(\alpha_E(t), X(t), \nu_t,\eta_t),\qquad t\geq0.
\]
It is obvious that $V$ is a $\mathcal{V}$-valued process adapted
to the filtration $\{\mathcal{F}_t^V, t\geq0\}$, the natural
filtration generated by $V$.
For each $y, z\in\mathcal{V}$ and $t\geq0$, let
\begin{equation}\label{dis:semiG}
P_t(y,z)=\mathbb{P}\bigl(V(t)=z|V(0)=y\bigr).
\end{equation}
For any measurable function
$\psi$ defined on
$\mathcal{V}$ and $t\geq0$, define the function $P_t\psi$ on
$\mathcal{V}$ as
\begin{equation}
\label{def:Ppsi} P_t\psi(y)=\mathbb{E}[\psi(V(t))|V(0)=y],\qquad
y\in\mathcal{V}.
\end{equation}
\begin{lemma} \label{lem:Mark}
The state descriptor $V$ is strong Markov with respect to $\{
\mathcal{F}_t$, $t\geq0\}$, and hence is strong Markov with respect to
$\{
\mathcal{F}_t^V, t\geq0\}$. Moreover, $\{P_t, t\geq0\}$ in
(\ref
{dis:semiG}) is the Markov semigroup of $V$.
\end{lemma}
\begin{pf}
To establish the strong Markov property, we shall identify $V$ as
a, so-called, piecewise deterministic Markov process (cf. \cite
{jacobsen}). From the explicit pathwise construction of $V$ in
Appendix \ref{ap-markov}, it follows that $V$ is a piecewise deterministic
process with jump times $\{\tau_1,\tau_2,\ldots\}$. Each jump time
is either the arrival time of a new customers or the time of a service
completion or the time to the end of a patience time. Note that, due to
the nonidling condition, the time of entry into service of a customer
must coincide with either the arrival time of that customer or the time
of service completion of another customer. Let $\tau_0=0$. For each
integer $n\geq0$, let $P_n=V(\tau_n)$. Then $\{(\tau_n,P_n),
n\geq0\}$ forms a marked point process. For each $n\geq0$, $V$
evolves in a deterministic fashion on $[\tau_n,\tau_{n+1})$. For each
$t\geq0$ and $y\in\mathcal{V}$ with $y=(\alpha, x, \sum_{i=1}^k
\delta_{u_i}, \sum_{j=1}^l \delta_{z_i})$ and $k\leq N$, define
{\renewcommand{\theequation}{$*$}
\begin{equation}\label{zv1}
\phi_t(y)\doteq \Biggl(\alpha+t, x, \sum_{i=1}^k \delta_{u_i+t}, \sum
_{j=1}^l \delta_{z_i+t} \Biggr).
\end{equation}}
\noindent It is easy to see that
\[
\phi_{t+s}(y)=\phi_s(\phi_t(y)),\qquad \phi_0(y)=y,
\]
and the map $t\mapsto\phi_t(y)$ is continuous in the interval
$[0,\infty)$. For each $t\geq0$, let
\[
\langle t\rangle=\max\{n\geq1\dvtx\tau_n \leq t\}
\]
with the convention that $\max\varnothing= 0$. We can see that
\setcounter{equation}{3}
\begin{equation}
V(t)=\phi_{t-\tau_{\langle t\rangle}}(V_{\tau_{\langle t\rangle
}}).
\end{equation}
The jump dynamics are captured by $\{r_t(y,C), t\geq0, y\in
\mathcal{V}, C\subset\mathcal{V}\}$. For each $t\geq0, y\in
\mathcal{V}, C\subset\mathcal{V}$, $r_t(y,C)$ is the conditional
probability that a jump leads to a state in\vspace*{1pt} $C$, given that the jump
occurs at time $t$ from state $y$. Let $y=(\alpha, x, \sum_{i=1}^k
\delta_{u_i}, \sum_{j=1}^l \delta_{z_i})$. Recall that there are only
three types of jump times for the process $V$. Given that $V$
jumps at time $t$ from state $y$, if we know which type the jump time
$t$ is, then we know to which state the process $V$ jumps to. For
example, suppose that the number $k$ in the expression of $y$ is less
than $N$, then, at state $y$, there is at least one idle server. If the
jump\vspace*{1pt} is due to the new arrival, then the process $V$ will jump\vspace*{1pt} to
state $(0, x+1, \sum_{i=1}^k \delta_{u_i}+\delta_0, \sum_{j=1}^l
\delta_{z_i}+\delta_0)$. Let $p_1, p_2, p_3$, respectively, be the
conditional probability that the jump at time $t$ is due to the arrival
of a new customer, service completion of a customer in service, the end
of patience time for some customer in the system, respectively, given
that the jump occurs at time $t$ from state $y$. Then the probability
measure $r_t(y,\cdot)$ can be easily written from $y$ and $p_i$,
$i=1,2,3$.
The jump time dynamics are captured by the survivor functions
$\{\overline H_{s,y}(t)\dvtx\break 0\leq s\leq t, y\in\mathcal{V}\}$, where
$\overline H_{s,y}(t)$ is the conditional probability that the time for
the next jump is more than time $t$ given the state being at $y$ at
time $s$, in other words, for $y=(\alpha, x, \sum_{i=1}^k
\delta_{u_i}, \sum_{j=1}^l \delta_{z_i})$,
{\renewcommand{\theequation}{$**$}
\begin{eqnarray}\label{zv2}
\overline
H_{s,y}(t)&=&\frac{1- F(\alpha+t-s)}{1-
F(\alpha)}\prod_{i=1}^k\frac{1-G^s(u_i+t-s)}{1-G^s(u_i)}\nonumber\\[-8pt]\\[-8pt]
&&\hspace*{0pt}{}\times\prod
_{j=1}^l\frac{1-G^r(z_j+t-s)}{1-G^r(z_j)}.\nonumber
\end{eqnarray}}
\noindent It is easy to see that $\overline H_{s,y}(t)$ satisfies
\[
\overline
H_{s,y}(u)=\overline H_{s,y}(t)\overline H_{t,\phi_{t-s}(y)}(u),\qquad
s\leq t\leq u.
\]
Then by Theorem 7.3.2 of \cite{jacobsen}, $V$ is
a piecewise deterministic Markov process constructed from
$\{(\tau_n,P_n), n\geq0\}$ using functions $\phi_t$ for the
deterministic part, survivor functions $\overline H_{s,y}$ for jump
time distributions and transition probabilities $r_t$ for the jumps.
Thus it follows from Theorem 7.5.1 of \cite{jacobsen} that $V$ is
a strong Markov process. The second part of the lemma follows directly
from the definition of the $\{P_t, t\geq0\}$ in
(\ref{dis:semiG}).
\end{pf}
\begin{remark}
For future purposes, we note that
the results of this paper including, in particular, the strong Markov
property established above, continue to be valid if the state component
$\alpha_E^{(N)}$ introduced in Section \ref{subs-modyn} is, instead, defined as follows:
\[
\alpha_E^{(N)}(s) \doteq \cases{
s, &\quad if $E^{(N)}(s) = 0$, \cr
\inf\bigl\{u>s\dvtx E^{(N)}(u)>E^{(N)}(s)\bigr\}-s, &\quad if $E^{(N)}(s) >
0$.}
\]
Observe that\vspace*{-1pt} when $E^{(N)}(s) > 0$, $\alpha_E^{(N)}(s)$ represents
the time from $s$ until the next arrival, and if $E^{(N)}$ is a
renewal process, then $\alpha_E^{(N)}$ is simply the forward recurrence time process.
A minor variation of the proof of Lemma \ref{lem:Mark} given above shows that
the strong Markov property holds in this case as well. First, the
definition of $\phi_t(y)$ should be modified by replacing $\alpha + t$ by
$\alpha - t$ in (\ref{zv1}). With $V$, $r_t$ and
$p_1$, $p_2$, $p_3$ defined as before,
in this case,
the probability measure $r_t(y,\cdot)$ can be easily determined from $y$,
the distribution of the remaining time from $t$ to the next arrival
and $p_i$, $i=1,2,3$. Note that if $\alpha > 0$ at time $t$,
then $p_1=0$. On the other hand, if $\alpha = 0$ at time $t$,
then $V$ jumps at time $t$ due to the arrival of a
new customer, and, hence, $p_1=1$.
Moreover, given that $V$
jumps at time $t$ from state~$y$, if the type of the jump at time
$t$ is known, then it is possible to determine the state to which
the process $V$ jumps.
For example,
suppose that the number $k$ in the expression for $y$ is less than $N$.
Then, at state $y$, there is at least one idle server. If the jump is
due to a new arrival, then the state $V$ will jump to the
region $\{c\in [0,\infty)\dvtx (c, x+1, \sum_{i=1}^k
\delta_{u_i}+\delta_0, \sum_{j=1}^l \delta_{z_i}+\delta_0)\}$
according\vspace*{1pt} to the distribution of the time to the next arrival
(which is determined by the current state $\alpha$ due to the assumption that
$\alpha_E^{(N)}$ is Markov with respect to its own filtration).
Once again, the jump time dynamics are captured by the
survivor functions, with the only difference that now
the ratio $(1-F(\alpha+t-s))/(1-F(\alpha))$
on the right-hand side of (\ref{zv2}) should be replaced by
$\mathbh{1}_{\{\alpha\geq t-s\}}$. The rest of the proof then follows
as before.
\end{remark}
\end{appendix}
|
1,314,259,993,051 | arxiv | \section{Problem statement}
Consider a common resource (e.g. an actuator, a camera, etc), where only one out of $N$ processes may be simultaneously using this resource. This requirement implies that these $N$ processes coordinate their access and that access is only granted to one and only one process at any given time. The block of code, where exclusive access is to be guaranteed, is called a critical section. This problem is known as the mutual exclusion problem, which, in the case of a standalone application, is solved using operating system primitives (mutex, semaphores, etc.). The problem we now tackle is distributed mutual exclusion, where the competing processes only communicate through message passing. The used failure model assumes: (i)~a reliable channel between communicating processes, (ii)~processes do not crash, (iii)~processes eventually leave the critical section (processes are well behaving).
\section{Evaluation metrics}
Next to the number of exchanged messages, there are two important metrics to characterize the performance of algorithms for distributed mutual exclusion:
\begin{itemize}
\item Client delay: denotes how much time does it take for a client to enter/leave the critical section, in case the critical section is not used by another client, where the system is assumed to be unloaded. The client delay is most influenced by the network delay (the one-way network delay is denoted as $\delta$) and the processing delay.
\item Synchronization delay: this metric gives an indication how much time the algorithm wastes between subsequent accesses to the critical section. The synchronization delay is the time (typically expressed as a function of $\delta$) needed between one client leaving the critical section, and the next client accessing the critical section. Here, we suppose that the system is heavily loaded (to allow fair comparison). Obviously, this synchronization delay puts an upper bound on the number of clients that can be served per unit of time.
\end{itemize}
\section{Three considered algorithms}
Several algorithms exist for distributed mutual exclusion, three of them are detailed below.
\subsection{Central Server Algorithm}
In the Central Server algorithm, one process takes the role of coordinator, receiving requests to access the critical section from all other processes, and granting access. Often, centralized approaches are simple in terms of algorithmic complexity, but exhibit poor scaling behaviour.
\subsection{Ring Algorithm}
In a ring based approach, the communicating processes are organized in a logical ring, and only communicate with one neighbour (Figure 3): in case the ring consists of N processes ${p_0, . . . , p_{N-1}}$, process $p_i$ has a single, unidirectional communication channel to process $p_{(i+1)mod N}$. The token, granting access now travels along the ring. There is only one message type exchanged, i.e. the message Token, so we only have to specify what happens when this message is received. Also note that all processes implement the same logic (there is no central server), making this algorithm suitable to dynamic adaptations.
\subsection{Raymond's Algorithm}
This algorithm is a token-passing algorithm for tree-based networks. It is assumed that processes are connected through a tree topology, and a token is passed between processes. A node that holds the token, serves a root of the tree, and child nodes maintain a pointer to the root node. Each node has two variables: (i)~a parent pointer, this pointer is empty in the root node and non-empty in the child nodes, and (ii) a local queue Q to store pending requests.
A node that wants access to the critical, sends a request towards the root via its parent. In each visited node, pointers to the root node are followed. Only the first request in Q is forwarded to the parent node. When a token moves from one process to another, the following two steps are taken: (i)~the root changes, by swapping the parent variables between the processes, and (ii)~the node with the token becomes the new root.
\section{Simulation approach}
Assume that the communication delay $\delta$ always follows a Gaussian distribution with a mean of 30 milliseconds and standard deviation of 5 milliseconds.
Consider the following three algorithms for distributed mutual exclusion:
\begin{enumerate}
\item the central server algorithm, where the processing time in the central server equals $40 \times e^{N/10}$ milliseconds (for both request- and leave-messages), where $N$ denotes the number of clients involved and $e^x$ denotes the exponential function.
\item the ring-based algorithm with $N$ different nodes. The processing delay in each node (to receive and send the token, or receive and keep the token) follows a gaussian distribution with a mean of 15 milliseconds and standard deviation of 2 milliseconds.
\item the Raymond's algorithm for distributed mutual exclusion, with N different nodes. The processing delay in each node (to receive and send the token, or receive and keep the token) follows a gaussian distribution with a mean of 15 milliseconds and standard deviation of 2 milliseconds. Each node has a pointer to the parent node (value=NULL for the root node), and two pointers to the left child node and the right child node, respectively.
\end{enumerate}
Assume $N$ equals 100, implement the three algorithms above in the C++ programming language and keep track of the client delay and the synchronization delay for 50 different requests. Determine the average client delay and synchronization delay, together with the standard deviation for the three algorithms.\\
\section{HPC infrastructure}
All examples are supposed to be compiled and executed using the UGent HPC (High Performance Computing) infrastructure. Once connected to the HPC infrastructure, students are logged on to one of the interface nodes. These nodes can be used to compile software and submit jobs to the different clusters, but running software on these interface nodes is generally considered bad practice. For this assignment however, it is fine to run lightweight examples on the interface node. Hence, you may simply execute the commands in the assignments on the interface nodes. The project should be compiled and run on a interface nodes. In order to compile all source files, please use the provided compile.sh script (that relies on CMake internally to generate a Makefile).
\section{Discussion}
The comparison study of the three algorithms for distributed mutual exclusion gives insight in the following elements:
\begin{enumerate}
\item the internal algorithmic details of the algorithms (centralized, ring-based and tree-based),
\ item the typical simulation-based modelling of the communication delays and processing delays,
\item the evaluation setup and statistical processing of the obtained numerical results, and
\item typical numerical values for the two important evaluation metrics: client delay and synchronization delay.
\end{enumerate}
An important guideline is to clearly guide the simulation study, and emphasize the difference with a typical emulation study.\\
Distributed mutual exclusion algorithms are beneficial in several application domains. Important use cases where distributed mutual exclusion algorithms can be applied are: mobile augmented reality applications~\cite{JSSVerbelen}, hierarchical network management systems~\cite{hierMoens, hiermgmtsystem}, scalable and adaptive video delivery~\cite{scalablevideolaga, 6dofadaptive, reviewMariaQoE}, efficient resource management for virtual desktop cloud computing~\cite{SupercomputingDeboosere}, smart city applications~\cite{CNSM2017Santos, NOMS2018Santos}, replica placement in ring based content delivery networks~\cite{ComComWauters}, and softwarized networks~\cite{vnfp, tnsmmoens}.
\section{GitHub repository}
\texttt{https://github.ugent.be/fdeturck/PDS/tree/main/HWA3}
|
1,314,259,993,052 | arxiv | \section{Introduction}
Cold, thick snow and atmospheric icing are three well known winter phenomenon in Nordic countries. Such climate puts a strain on infrastructure, companies and people, which affects the normal operations of the whole society. It is therefore central to gather knowledge and methods to counter problems caused by the extreme winter climate. In this study, the effect of harsh winter climate on railway system is investigated.
The railway system is an important part of the infrastructure and a large number of people as well as companies depend on it for transportation on sorts of purposes every day. Thus, punctuality becomes one key criterion for the railway industry in order to minimise the society costs and increase its reliability of the railway system. To achieve this goal, an important task is therefore to investigate and figure out how train delays are affected by winter climate factors.
The performance of the train operations is commonly measured by cumulative delay and current delay. Cumulative delay measures the increment in delay within two consecutive measuring spots in terms of running time, and current delay is the delay in terms of arrival time at each measuring spot compared to the schedule.
Earlier studies about train performance and the effect of weather have been conducted. \citet{Xia2013} fitted a linear model and showed that weather factors like snow, temperature, precipitation and wind had significant effects on the punctuality of trains in Netherlands. In a more recent study by \citet{Wang2019}, a machine learning approach was used to create a predictive model to predict the current delay at each station for a train line in China with help of weather observations. \citet{Ottosson} used count data models such as negative binomial regression and zero-inflated model and showed that weather variables, such as snow depth, temperature and wind direction, had significant effects on the train performance.
The cumulative delay information of a train trip collected from the measuring spots in time could also be seen as time-to-event (i.e. cumulative delay) data, and the transitions between the (current) delayed and non-delayed states in a train trip can be treated as a Markov chain. To the best of our knowledge, very few applications of the survival analysis to the railway field can be found in literature, and they do not concern the delay issues \citep{Jardine,Annelise,Andersson}; there are also a few applications of Markov chain to the train delay investigation, but none of them considered the weather impact on the transition probability \citep{Sahin,Kecman,Gaurav}.
Therefore, novelty of this study is that survival analysis is used to investigate how the winter climate affects the occurrence of cumulative delays, and Markov chain model is applied to study the effect of winter climate on the transitions between delayed and non-delayed states.
The weather data is simulated from the Weather Research and Forecasting (WRF) model instead of using real meteorological observations, since the distance between the nearest meteorological station and measuring spot along the train line ranges from 17 to 24 km \citep{Ottosson}. Thus, using the meteorological data is not an ideal choice in the analysis. However, WRF with high spatial resolution is a good alternative under this situation, which is one of the most commonly used numerical weather prediction models for both atmospheric research and operational forecasting needs. Its reliable performance has been assessed in a number of studies \citep{wangbayesian, wangdownscal,Mohan,Cassano}.
The paper is organized as follows. In Section 2, we introduce the statistical models in details. Data processing and model diagnostic methods are described in Section 3. Section 4 is reserved for results. Section 5 is devoted to the conclusion and discussion.
\section{Statistical modelling}
\subsection{Cox proportional hazard model with time dependent covariates for recurrent event}
\citet{Andersen1982} proposed a regression model used to analyse the relationship between survival times (or hazard rates) of subjects with recurrent event and covariates, which is probably the most often applied model for recurrent event survival analysis. It is a simple extension of the original Cox model \citep{Cox}. Formally, the extended Cox model with time dependent covariates for recurrent event is an expression of the hazard function $h(t)$, which gives the risk of an event at time \emph{t}, and covariates
\begin{equation}
\label{coxmodel}
h_{ij}(t) = h_0(t)\exp{(\boldsymbol{\beta}^T \mathbf{x}_{ij}(t))},
\end{equation}
where
\begin{itemize}
\item $h_{ij}(t)$ represents the hazard function for the $j$th event of
the $i$th subject at time $t$.
\item $h_0(t)$ is the baseline hazard which is the hazard rate when all the covariates are equal to zero.
\item $\mathbf{x}_{ij}(t)$ represents a covariate vector for the $i$th subject and the $j$th event at time $t$.
\item $\boldsymbol{\beta}$ is an unknown coefficient vector to be estimated.
\end{itemize}
The coefficients can be estimated with partial likelihood by taking into account the conditional probabilities for the events that occur for subjects, which is given by
\begin{equation}
\label{partialLikli}
L(\boldsymbol{\beta}) = \prod_{i = 1}^n \prod_{j = 1}^{k_i} \left(\frac{\exp{( \boldsymbol{\beta}^T \mathbf{x}_{i}(t_{ij})))}}{\sum_{l \in R(t_{ij})} \exp{( \boldsymbol{\beta}^T \mathbf{x}_{l}(t_{ij})})}\right)^{\delta_{ij}},
\end{equation}
where $j$ is the event index with $k_i$ being the subject-specific maximum number of events, $\mathbf{x}_{i}(t_{ij})$ denotes the covariate vector for the $i$th subject at the $j$th event time $t_{ij},$ $\delta_{ij}$ is an event indicator which equals $1$ for the $j$th event of the $i$th subject and $0$ for censoring, $R(t) = \{l, l=1,\cdots,n: \exists j\in\{1,\cdots,k_l\} \text{ with } t_{lj}\ge t\}$ is a group of subjects that are at risk for an event at time $t$.
The proportional hazard assumption is a vital assumption to the use and interpretation of a Cox model, and it states that the hazard ratio of two subjects, $i$ and $l$, at time $t$
\begin{equation}
\label{hr}
HR(t)=\frac{h_{i}(t)}{h_{l}(t)}=\exp{\left(\boldsymbol{\beta}^T\left(\mathbf{x}_{i}(t)-\mathbf{x}_{l}(t)\right)\right)}
\end{equation}
is constant. An obvious case that (\ref{hr}) is constant when the covariate vectors $\mathbf{x}_i(t)$ and $\mathbf{x}_l(t)$ are independent of $t$. In general, the proportionality assumption may be invalid for time dependent covariates. A likelihood ratio test or Schoenfeld residuals can be used to examine whether a covariate satisfies the assumption or not \citep{Abeysekera}. If not, a heaviside function $g(t)$ can be chosen to fix the proportionality violation \citep{Miftahuddin}. The purpose of a heaviside function is to partition the observational time into intervals and make the proportionality assumption be valid within each interval. Then (\ref{coxmodel}) could be rewritten element-wise as
\begin{equation}
\label{heaviside}
h_{ij}(t) = h_0(t)\exp{\left(\sum_{m=1}^p\beta_m x_{ijm}+\sum_{m=1}^p\theta_m x_{ijm}g_{m}(t)\right)},
\end{equation}
where $x_{ijm}$ is the $m$th covariate of the $i$th subject and the $j$th event, $\theta_m$ is a new introduced coefficient for the $m$th covariate in an interval defined by $g_m(t)$, and $g_m(t)$ is a function of time for the $m$th covariate. For the case, $g_m(t)=0$ for all $m$ implies no time dependent covariates, then (\ref{heaviside}) is reduced to a simpler version of (\ref{coxmodel}) with only time independent covariates
\begin{equation}
\label{timeinde}
h_{ij}(t) = h_0(t)\exp{(\boldsymbol\beta^T\mathbf{x}_{ij})}.
\end{equation}
In this study, a simple step function is chosen for covariates that do not satisfy the proportionality assumption
\begin{equation}
\label{heav}
g_m(t) = \Bigg\{
\begin{array}{l c }
0, \ t \ \leq t_0,\\
1, \ t \ >t_0,
\end{array}
\end{equation}
where $t_0$ is chosen through Schoenfeld residuals to make the proportionality assumption hold within each interval.
\subsection{Markov chain model}
A multi-state model describes how an subject moves among a number of states in continuous time \citep{Jackson}. The movement from state $r$ to $s$ at the time $t$ is governed by transition intensity
\begin{equation}
\label{inte}
q_{rs}(t) = \lim_{\Delta t \to 0} P( S(t+\Delta t) = s |S(t)=r)/\Delta t,
\end{equation}
which may depend also potentially on a time dependent explanatory vector $\mathbf{x}(t)$, i.e. $q_{rs}(t)\to q_{rs}(t,\mathbf{x}(t))$.
The $q_{rs}$ of a $q$ states process forms a $q\times q$ transition intensity matrix $Q$, whose rows sum to zero, so that the diagonal entries are defined by $q_{rr} = -\sum_{s\neq r}q_{rs}$. An example of transition intensity matrix $Q$ with two states can be seen below
\begin{equation}
Q = \begin{bmatrix}
q_{11} & q_{12} \ \\
q_{21} \ & q_{22} \end{bmatrix},
\end{equation}
where $q_{11}=-q_{12}$ and $q_{22}=-q_{21}$.
A multi-state model generally relies on the Markov assumption that next transition only depends on the current state, i.e. $q_{rs}(t, \mathbf{x}(t),\mathcal{H}_t)$ is independent of $\mathcal{H}_t$ which is the observation history of the process up to the time preceding $t$.
At the early investigate of the study, we limit our focus on a homogeneous process in time meaning that the transition intensity $Q$ is constant and the transition probability to move from a state to another depends solely on the time difference between two time points, i.e.
\begin{equation}
\label{mark2}
P( S(u+t) = s |S(u) = r) = P(S(t) = s | S(0) = r).
\end{equation}
Corresponding to the transition intensity matrix $Q$, the entry in a transition probability matrix $P(u,u+t)$ is $p_{rs}(u,u+t)$ representing the probability of being in state $s$ at a time $u+t$ given the state at time $u$ is $r$. The relationship between transition intensity matrix and transition probability matrix is defined through the Kolmogorov differential equations \citep{Cox1977}. Specially, when a process is homogeneous, the transition probability matrix can be calculated by taking the matrix exponential of the transition intensity
matrix
\begin{equation}
\label{pro}
P(u,u+t)=P(t) = \text{Exp}(tQ).
\end{equation}
In a Markov chain model, to take account of the effects of explanatory variables, a proportional hazard like model was proposed by \citet{Marshall1995}
\begin{equation}
\label{explo}
q_{rs}(t,\mathbf{x}(t)) = q_{rs}^{(0)}(t) \exp{(\boldsymbol{\beta}_{rs}^T \mathbf{x}(t))},
\end{equation}
where $q_{rs}^{(0)}(t)$ is baseline transition intensity when all covariates are zero.
The coefficient vectors $\boldsymbol{\beta}_{rs}$ as well as the transition intensity matrix $Q$ and the transition probability matrix $P(t)$ can be estimated through maximising the likelihood
\begin{equation}
\label{logts}
L(Q) = \prod_{i,j} p_{S(t_{i,j})S(t_{i,j+1})}(t_{i,j + 1}- t_{i,j}),
\end{equation}
where the transition probability is evaluated at time $t_{i,j + 1}- t_{i,j}$ and $S(t_{i,j})$ represents the $j$th observed state of the $i$th subject at time $t_{i,j}$.
\section{Method}
\subsection{Train data}
The investigation focuses on the high speed passenger train, which is a type of trains with top speed of between 200 to 250 km/h. This type of train often travels longer distances and is therefore more prone to experience disturbances caused by climate factors. A train line comprises of a number of measuring spots where the operational times are recorded such as departure and arrival times.
In the study, the train line between Umeå and Stockholm is selected, which includes 116 measuring spots in total. The total length of the train line is 711 km and the planned drive time for a high speed passenger train is approximately 6.5 hours. The lengths of any two consecutive measuring spots vary from 0.3 km to 15 km. The train operation data in the year 2017 is provided by the Swedish Transport Administration. The key variables are listed in Table \ref{table:1}.
\begin{table}[H]
\centering
\caption{List of variables in the train operation data}
\begin{tabular}{ | m{11em} | m{9cm}| } \hline
\textbf{Variables} & \textbf{Description} \\ \hline
Train Number & An identification number for train used in the trip \\ \hline
Arrival location & Name of arrival measuring spot \\ \hline
Departure location & Name of departure measuring spot \\ \hline
Departure date & The departure date (yyyy-mm-dd) for a train at a location. \\ \hline
Arrival date & The arrival date (yyyy-mm-dd) for a train at a location. \\ \hline
Train type & Type of train, for example: high speed, commute train and regional \\ \hline
Section Length & Length (km) between two consecutive measuring spots \\ \hline
Planned departure time & The planned departure time (hh:mm) at a measuring spot \\ \hline
Planned arrival time & The planned arrival time (hh:mm) at a measuring spot \\ \hline
Actual departure time & The Actual departure time (hh:mm) at a measuring spot\\ \hline
Actual arrival time & The Actual arrival time (hh:mm) at a measuring spot \\ \hline
\end{tabular}
\label{table:1}
\end{table}
\subsection{Weather data}
A WRF model is a numerical weather prediction system that is used for research and operational purposes. The model can be set up with different configurations and over different regions. Actual atmospheric conditions and idealised conditions can be used in the model. The WRF model simulates desired weather variables estimations over grids. Higher spatial resolution gives smaller square of each grid. Temporal resolution decides the time interval between each simulation. The extreme winter weather data from January to February in the year 2017 is of special interest in the study. The spatial resolution is $3\times3$ km and the temporal resolution is 1 hour. The simulation region as well as the train line in blue of interest are shown in \textit{Figure \ref{pic:weather}}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.7]{map.PNG}
\caption{Train line in the region with simulated WRF data}
\label{pic:weather}
\end{figure}
The weather variables of interest are shown in \textit{Table \ref{table:2}}. These variables are chosen because they are believed to have impacts on the train operation in winter and we want to investigate how these variables affect the train operation.
\begin{table}[H]
\centering
\caption{The weather variables of interest}
\begin{tabular}{ | m{8em} | m{10cm}| }
\hline
\textbf{Variables} & \textbf{Description} \\
\hline
Temperature & The temperature ($^\circ$C) at 2 meters above the ground \\ \hline
Humidity & Relative Humidity (\%) at 2-meters \\ \hline
Snow depth & The snow depth in meters (m) \\ \hline
Atmospheric icing & Accumulated snow and ice in millimeter (mm) \\ \hline
\hline
\end{tabular}
\label{table:2}
\end{table}
Every measuring spot on the train line is matched with the closest grid point by date and time. The measuring time in train operation data has to be rounded to the closest hour.
The mean of the weather variables within any two consecutive spots is calculated and used in the analysis. Since a large number of the atmospheric icing values is zero along the train line, a categorical variable is used instead of the continuous atmospheric icing variable, i.e. 0 if atmospheric icing is zero, 1 otherwise.
\subsection{Missing values in the train operation data}
A section between two consecutive measuring spots for a train trip often has missing departure/arrival times that can be classified into three different classes which are defined in \textit{Table \ref{table:miss}}.
\begin{table}[H]
\centering
\caption{Classes of missing times}
\begin{tabular}{ | m{8em} | m{5cm}| m{5cm}| }
\hline
\textbf{Class} & \textbf{Departure time missing}& \textbf{Arrival time missing} \\
\hline
1& True & False\\ \hline
2& False & True \\ \hline
3& True & True \\ \hline
\end{tabular}
\label{table:miss}
\end{table}
A common method to impute missing values in such longitudinal data is called last observation carried forward (LOCF), i.e. the latest recorded value is used to impute the missing value. The advantages of using LOCF are that the number of observations removed from the study decreases and make it possible to study all subjects over the whole time period. A disadvantage with the method is the introduction of bias of the estimates if the values changes considerably large with time or the time period between the most recent value and the missing value is long.
Because the intervals with missing values are short in the dataset which decreases the risk of bias, thus it is reasonable to apply this approach. Based on the LOCF, the imputation procedure is explained further below.
\begin{enumerate}
\item Start from the beginning of the trip and save the latest arrival and departure time until a missing time is occurring.
\item If the missing time is arrival time then (a); if departure time is missing then (b):
\begin{enumerate}
\item Replace the missing arrival time with the latest departure time + the planned driving time for the previous section
\item Replace the missing departure time with the latest arrival time + the planned dwell time.
\end{enumerate}
\item Save the imputed time as the new latest time.
\item If the section is not the last section of the trip, go back to step 1.
\end{enumerate}
\subsection{Proportionality test}
It is necessary to verify whether the proportionality assumption holds for a fitted Cox model. A Schoenfeld residuals based test can be used to test the assumption. Schoenfeld residuals has the form \textit{Observed - Expected}. The Schoenfeld residual $r$ for the $j$th event is defined as
\begin{equation}
\label{Schoenfeld}
r_j(\hat{\boldsymbol{\beta}})=\mathbf{x}_i(t_{ij})-E(\mathbf{x}(t_{ij})|R(t_{ij}))=
\mathbf{x}_i(t_{ij})-\sum_{l\in R(t_{ij})}\mathbf{x}_{l}(t_{ij})w_l,
\end{equation}
where $w_l=\frac{\exp{\left( \hat{\boldsymbol{\beta}}^T \mathbf{x}_{l}(t_{ij}))\right)}}{\sum_{l' \in R(t_{ij})} \exp{\left( \hat{\boldsymbol{\beta}}^T \mathbf{x}_{l'}(t_{ij})\right)}}$ is a weight of $\mathbf{x}_{l}(t_{ij})$ over those observations still at risk at time $t_{ij}$. If the proportional hazard assumption holds, $E(r_j(\hat{\boldsymbol{\beta}}))\approx 0$ \citep{Xue,Abeysekera}. \citet{harrell1979phglm} proposed a test based on the Schoenfeld residual. It is a test of correlation between the Schoenfeld residuals and time, for example, a correlation of zero indicates that the model met the proportional hazard assumption.
\subsection{Analysis tools}
R is the programming language that used throughout the study. Therein, the package \textit{dplyr} is used for data processing, the package \textit{survival} and the package \textit{msm} are used for the Cox model and Markov chain model, respectively.
\section{Results}
\subsection{Cox model}
By using the proportionality test, it shows that the temperature variable does not satisfy the proportionality assumption. After inspecting the Schoenfeld residuals plotted against travel distance (not shown), the hazard ratios are different between 0 - 150km and 150 - the end. Therefore, a heaviside function
\begin{equation}
\label{heavyy}
g(t) = \Bigg\{
\begin{array}{l c }
0, \ t \ \leq 150,\\
1, \ t \ >150,
\end{array}
\end{equation}
is chosen for temperature.
The results from the fitted extended Cox proportional hazard model with (\ref{heavyy}) can be found in \textit{Table \ref{table:solocox}}. Two predictors have significant effects on the occurrence of the cumulative delay which are temperature and humidity. To be specific, as temperature increases 1 $^\circ$C, the hazard rates decrease 17\% for the first 150 km and 6\% for the remaining trip, respectively. As humidity increase 1\%, the hazard rate increases 2.6\%.
\begin{table}[H]
\caption{Estimates from the fitted extended Cox model }
\centering
\begin{tabular}{ | m{6em} | m{2cm}| m{2cm}| m{2cm}| m{2cm}| m{2cm}| }
\hline
\textbf{Predictor} & \textbf{Coefficient} & \textbf{Hazard ratio} & \textbf{Robust standard error} & \textbf{z-value} & \textbf{p($>$z)} \\ \hline
Temperature (0 - 150 km) & -0.19& 0.83& 0.055 & -3.39 &0.00070* \\\hline
Temperature (150 km - the end)& -0.062& 0.94& 0.025 &-2.46952& 0.014* \\\hline
Humidity & 0.025& 1.026 & 0.0081 & 3.14 &0.0017* \\\hline
Snow depth & 0.0073 & 1.0074 & 0.015 &0.48 &0.63 \\\hline
Categorical icing & -0.16& 0.85& 0.15 & -1.061 &0.29 \\\hline
\end{tabular}
\label{table:solocox}
\end{table}
The proportionality test for the model with the heaviside function for each predictor can be seen in \textit{Table \ref{proportionality}}. It shows the proportionality assumption holds for extended Cox model with the heaviside function.
\begin{table}[ht]
\centering
\caption{P-values from proportionality test}
\begin{tabular}{ | m{10em} | m{5cm}| }
\hline
\textbf{Predictor} & \textbf{P-value} \\ \hline
Temperature & 0.45 \\\hline
Humidity &0.54 \\ \hline
Snow depth &0.51 \\ \hline
Categorical icing &0.48 \\ \hline
\textbf{Global} &0.78 \\ \hline
\end{tabular}
\label{proportionality}
\end{table}
\subsection{Markov chain model}
\textit{Table \ref{table:markpre}} shows temperature and atmospheric icing have significant impacts on the transition rate from non-delayed to delayed states in the model. The hazard ratios indicate that as the temperature increases 1 $^\circ$C, the transition rate from non-delay to delayed states decreases 3\%, and the transition rate from non-delay to delayed states increases 46\% with the occurrence of atmospheric icing.
\begin{table}[H]
\centering
\caption{Hazard ratios from non-delayed to delayed states}
\begin{tabular}{ | m{6em} | m{2cm}| m{2cm}|m{2cm}| }
\hline
\textbf{Predictor} & \textbf{Hazard Ratio} & \textbf{CI: Lower}& \textbf{CI: Upper} \\ \hline
Temperature & 0.97 &0.94&0.99\\\hline
Humidity & 0.99 &0.98 &1.002 \\\hline
Snow depth & 1 &0.97&1.024 \\\hline
Categorical icing & 1.460 & 1.163& 1.83 \\\hline
\end{tabular}
\label{table:markpre}
\end{table}
\textit{Table \ref{table:markov}} shows that the estimates of hazard ratios from delayed to non-delayed states. Temperature, humidity and snow depth are significant in the model. It indicates that as the temperature increases 1 $^\circ$C, the transition rate from delayed to non-delay states increases 3.3\%, as the humidity increases 1\%, the transition rate decreases 2\%, and as the snow depth increases 1 m, the transition rate decreases 5\%.
\begin{table}[H]
\centering
\caption{Hazard ratios from delayed to non-delayed states}
\begin{tabular}{ | m{6em} | m{2cm}| m{2cm}| m{2cm}|}
\hline
\textbf{Predictor} & \textbf{Hazard Ratio} & \textbf{CI: Lower}& \textbf{CI: Upper} \\ \hline
Temperature & 1.033&1.0010&1.065 \\\hline
Humidity & 0.98 &0.96&0.99 \\\hline
Snow depth & 0.95& 0.92&0.99 \\\hline
Categorical icing & 1.19&0.945&1.50 \\\hline
\end{tabular}
\label{table:markov}
\end{table}
\section{Conclusion and discussion}
The purpose of this study was to investigate how the winter weather as well as the atmospheric icing affect the occurrence of cumulative delays and the transitions between delayed and non-delayed states. The cumulative delay was investigated with the Cox proportional hazard model with time dependent covariates for recurrence event, which showed that the temperature and humidity have significant effects on the occurrence of cumulative delays. Lower temperature and higher humidity increase the probability of the occurrence of cumulative delays.
The transitions between delayed and non-delayed states were investigated with the two-states Markov model. This showed that the temperature and the atmospheric icing have significant effects on the transition rate from non-delayed to delayed states. More specifically, lower temperatures and the presence of icing increase the transition rate from non-delayed to delayed states. In the other side, humidity, temperature and snow depth have significant effects on the transition rate from delayed to non-delayed states. To be specific, higher temperatures, lower snow depth and lower humidity increase the transition rate from delayed to non-delayed states.
In summary, both models show that the temperature and humidity have impacts on the performance of a train in the winter climate, i.e. lower temperature and higher humidity are against the train to be punctual.
Much could be done in terms of statistical modelling in the further investigation. For instance, 1) the stratified Cox model can be used, which takes the order of event into account and could also avoid to violate the proportionality assumption \citep{Ozga}; 2) fitting an inhomogeneous Markov chain model to the train operation data is more reasonable, since it is strict to assume the transition rate does not change over time in reality; 3) a more than two states' Markov chain model can be used to acquire a deeper understanding about the climate effects. Besides, more train operation data as well as weather data could to be included in the model construction and verification procedure.
\section*{Acknowledgements}
We acknowledge EU Intereg Botnia-Atlantica Programme and Regional Council of V\"{a}sterbotten and Ostrobothnia for their support of this work through the noICE project. We would like to thank the Swedish Transport Administration for providing the train operation data. We would like to thank the Atmospheric Science Group (ASG) at Lule\r{a} University of Technology for simulating and providing the WRF data.
\printbibliography
\end{document}
|
1,314,259,993,053 | arxiv |
\section{Introduction}
A pulsar is a rapidly spinning magnetized compact star.
The highly collimated beam of radiation of the star is rotated into and out of the
line of sight of a distant observer as the star itself rotates around a
fixed axis, manifesting a periodic sequence of pulsations.
Hence, pulsars are thought to be the accurate clocks in the Universe.
During pulsar observations, a sudden increase in pulsar's spin frequency $\nu$ (i.e. glitch) is discovered. The
observed fractions $\Delta\nu/\nu$ range between $10^{-10}$ and $10^{-5}$ (Yu et al. 2013).
In neutron star models, pulsars are thought to be a fluid star with a thin
solid shell. The physical mechanism of glitch is believed to be the coupling
and decoupling between outer crust (rotates slower) and the inner superfluid
(rotates faster) (Anderson $\&$ Itoh 1975; Alpar et al. 1988). However, the absence of evident energy
release during even the largest glitches ($\Delta \nu/\nu \sim 10^{-6}$)
of Vela pulsar is a great challenge to this glitch scenario (G$\mathrm{\ddot{u}}$rkan et al. 2000; Helfand et al. 2001).
Recently, the glitches detected from AXP/SGRs (anomalous X-ray
pulsars/soft gamma repeaters), which are usually accompanied with energy release
(Kaspi et al. 2003; Tong $\&$ Xu 2011; Dib $\&$ Kaspi 2014) become another challenge to the previous glitch models.
\section{Two types of glitches}
In our model, glitches can be divided into two types depending on the energy released.
According to our calculations, these two types of glitches might be induced by two
types of starquakes of solid stars, i.e. the bulk variable starquake and bulk invariable starquake.
The bulk variable starquake is induced by the accretion.
Due to the accretion, the mass of the star increases, and the radius will change as well.
However, as a solid star, the pulsar will gain elastic energy to resist this change in shape.
A sudden collapse will happen when this energy is too large for the solid structure to stand.
With a typical pulsar and glitch parameter, the estimated energy released in this way coincides with the observed value during the glitches of AXP/SGRs.
The other type of starquake occurs without a bulk variation, and it's induced by the spinning down of the pulsar.
The pulsars will spin down due to the magnetic dipole radiation. Stars made of perfect fluid will change its ellipticity
and deform like a Maclaurin ellipsoid when it spins down. However, for a solid star, the shearing force will resist this change
in ellipticity and gain additional elastic energy. A starquake will happen when the shearing force exceeds a critical value.
The moment of inertia will decrease and the pulsar will spin up. With some available parameters, we can reproduce the glitch properties,
i.e. the glitch amplitude and the intervals between glitches, with very little energy released.
Fig. 1 show our calculating result of the two types of starquakes.
For same glitch amplitude, the energy released during a bulk variable
will be about $10^7$ times larger than that of bulk invariable starquake.
Besides the energy release estimation, there are also some other hints showing that the glitch
on AXP/SGRs and the glitch on Vela might be bulk variable starquake and bulk invariable starquake, respectively.
Some observation hints the possibility of accretion on AXP/SGRs, which is the key to induce a
bulk varible starquake. And on the other hand, as a younger pulsar,
the spin of Vela is much faster than AXP/SGRs, so that the role of stellar ellipticity is much more
important for the evolution of Vela-like pulsars.
In neutron star models, a pulsar is thought to be a fluid star with a thin solid shell.
In fact, so far only quark star and quark cluster star model develop a solid star model.
Then the two types of glitches may be an implication that the pulsar
is composed by quark matter or quark cluster matter.
\begin{figure}[htb]
\includegraphics[width=0.6\textwidth]{glitchenergyfinal.eps}
\caption{The total energy release during the bulk-variable glitches and bulk-invariable
glitches with amplitudes of $10^{-6}$ and $10^{-9}$.
The Lennard-Jones interaction is applied as an approximation to work out
the mass-radius relation (Lai $\&$ Xu 2009).
There are two main factors in this approximation: the number of quarks in one
cluster ($N_{\mathrm{q}}$) and the depth of the potential($U_{0}$).
The case of 3-quark clusters with potential of 100MeV (solid lines) and 18-quark
clusters with potential of 50MeV (dashed lines) are considered.
It's also worth noting that the energy release during a bulk-invariable
glitch is related to the time intervals between two glitches.
In this calculation the glitch is thought to happen once per month and the
spin down power is calculated according to the observational data of Vela.}
\label{fig:Mott}
\end{figure}
\subsection*{Acknowledgement}
We express our thanks to the organizers of the CSQCD IV conference for providing an
excellent atmosphere which was the basis for inspiring discussions with all participants.
We have greatly benefitted from this.
|
1,314,259,993,054 | arxiv | \section{Conclusions and Future Work}
We formulated a novel method to decompose registration based trackers into sub modules and tested several different combinations of methods for each sub module to gain interesting insights into the strengths and weaknesses of these methods. We also obtained some rather surprising results that proved previously published theoretical analysis to be somewhat inaccurate in practice, thus demonstrating the usefulness of our framework in testing out new ideas in the domain of registration based tracking. We also make publicly available the open source modular tracking framework so all results can be reproduced. This framework, with its highly efficient and ROS compatible C++ implementations for several well established trackers, will hopefully address practical tracking needs of the wider robotics community too.
\section{Introduction}
\label{introduction}
Since its inception, research in object tracking has focused on presenting new tracking algorithms to address specific challenges in a wide variety of application domains like surveillance, targeting systems, augmented reality and medical analysis. However, before an algorithm can be adopted in a real life application, it needs to be extensively tested so that both its advantages and limitations can be determined. Recent studies in tracking evaluation \cite{Wu13benchmark, Kristan2015_vot15} show increasing efforts to standardize this crucial process. However, though such studies assign a global rank to each tracker, they often provide little feedback to improve these trackers since they treat them as black boxes predicting the trajectory of the object. A more useful evaluation methodology would be to have empirical validation of the tracker's design or point out its shortcomings.
An exhaustive analysis of learning based trackers is admittedly a daunting and impracticable task as these often use widely varying techniques that have little in common. This, however, is not true for registration based trackers \cite{Lucas81lucasKanade, Baker04lucasKanade_paper} which - as we show in this work - can be decomposed into three well defined modules, thus making their systematic analysis feasible. These trackers are generally faster and more precise than learning based trackers \cite{Roy2015_tmt} which makes them more suitable for applications such as robotic manipulations, visual servoing and SLAM, where multiple trackers are used in parallel. On the other hand, lacking an online learning component, they are known to be non robust to changes in the object's appearance and prone to failure in the presence of motion blur, occlusion, lighting variations or viewpoint changes. As a result, they are less popular in the vision community and often underrepresented in the aforementioned studies, thus making such an evaluation particularly useful for applications where learning based trackers are unsuitable.
A detailed analysis, with a test framework in registration based tracking, to the best of our knowledge. has never been attempted before.
Many reported studies in this domain \cite{Lucas81lucasKanade, Baker04lucasKanade_paper, Benhimane07_esm_journal} have introduced new methods for only one of the three submodules without exploring the full extent of their contributions. For instance, Baker et. al \cite{Baker04lucasKanade_paper} reported a compositional update scheme for the state parameters $\mathbf{p}$ (Eq. \ref{reg_tracking}) instead of the additive scheme used in \cite{Hager98parametricModels}, but never experimented with different AMs.
Conversely, Richa et. al \cite{Richa11_scv_original} showed an improvement over the existing efficient second order minimization \cite{Benhimane07_esm_journal} approach by using the sum of conditional variance as the similarity metric instead of the sum of squared differences. Similarly, Dame et. al \cite{Dame10_mi_ict} used mutual information while Scandaroli et. al \cite{Briechle2001_ncc_template_matching} used normalized cross correlation with the inverse compositional method of \cite{Baker04lucasKanade_paper}. However, neither of them tested their similarity measures with other search methods even though the latter had previously been shown to be a good metric when used with the standard Lucas Kanade type tracker \cite{Briechle2001_ncc_template_matching}.
Finding the optimal combination of methods for any tracking algorithm is a two step process. First, the sub module where the algorithm's main contribution lies needs to be determined, using, for instance, the method employed in \cite{Wu13benchmark}. Second, all possible combinations for the other sub modules that are compatible with this algorithm (since not all methods for different sub modules work with each other) need to be enumerated and evaluated. A generic framework would thus be useful to avoid such fragmentation.
To summarize, following are the main contributions of this work:
\begin{itemize}
\item Empirically test different combinations of submodules leading to several interesting observations and insights that were missing in the original papers. Experiments are done using two large datasets with over 77,000 frames in all to ensure their statistical significance.
\item Report for the first time, to the best of our knowledge, results comparing robust similarity metrics \cite{Richa12_robust_similarity_measures}, with traditional SSD type measures.
\item Compare formulations against popular online learning based trackers to validate their usability in precise tracking applications.
\item Provide an open source tracking framework \footnote{available online at \url{http://webdocs.cs.ualberta.ca/~vis/mtf/}} using which all results can be reproduced and which, owing to its efficient C++ implementation, can also be used to address practical tracking requirements.
\end{itemize}
\section{Experimental Results and Analysis}
\label{experiments}
\subsection{Dataset and Error Metric}
Two publicly available datasets have been used to analyze the trackers:
\begin{enumerate}
\item Tracking for Manipulation Tasks (\textbf{TMT}) dataset \cite{Roy2015_tmt} that contains videos of some common tasks performed at several speeds and under varying lighting conditions. It has 109 sequences with a total of 70592 frames.
\item Visual Tracking Dataset provided by \textbf{UCSB} \cite{Gauglitz2011_ucsb} that has 96 short sequences of different challenges in object tracking with a total of 6889 frames. The sequences here are more challenging but also rather artificial since they were created specifically to address various challenges rather than represent realistic scenarios.
\end{enumerate}
Both these datasets have full pose (8 DOF) ground truth data which makes them suitable for evaluating high precision trackers that are the subject of this study. In addition, we use \textbf{Alignment Error} ($E_{AL}$) \cite{Dick13nn} as metric to compare tracking result with the ground truth since it accounts for fine misalignments of pose better than other common measures like center location error and Jaccard index.
\subsection{Evaluation Measure}
We measure a tracker's overall accuracy through its \textbf{success rate} (SR) which is defined as the fraction of total frames where the tracking error $E_{AL}$ is less than a threshold of $t_p$ pixels. Formally,
$ SR = \frac{|S|}{|F|} $
where $ S = \{f^{i} \in F : E_{AL}^{i} < t_p \}$, $F$ is the set of all frames and $E_{AL}^{i}$ is the error in the $i^{th}$ frame $f^{i}$. Since we have far too many sequences to present results for each, we instead report an overall summary of performance by averaging the success rates over all the sequences in both datasets, i.e. $ F $ is treated as the set of all frames in TMT and UCSB with $ |F| = 70592 + 6889 - 205 = 77276 $ - we do not consider the first frame in each sequence, where the tracker is initialized, for computing the SR. Finally, we evaluate SR for several values of $ t_p $ ranging from 0 to 20 and study the resulting SR vs. $ t_p $ plot to get an overall idea of how precise and robust a tracker is.
\subsection{Parameters Used}
All results have been generated using a fixed sampling resolution of $ 50{\times}50 $ irrespective of the tracked object's size. The input images were smoothed using a Gaussian filter with a $ 5{\times}5 $ kernel before being fed to the trackers. Iterative SMs were allowed to perform a maximum of $ 30 $ iterations per frame but only as long as the L2 norm of the change in bounding box corners in each iteration remained greater than $ 0.001 $. For the NN tracker, a standard deviation of $ 0.05 $ was used for generating the random warps. The learning based trackers whose results are reported in Sec. \ref{res_ssm} were run using default settings provided by their respective authors. All speed tests were run on a 2.66 GHz Intel Core 2 Quad Q9450 machine with 4 GB of RAM. No multi threading was used.
\subsection{Results}
The results presented in this section are organized into three sections corresponding to the three sub modules. In each of these, we present and analyze results comparing different methods for implementing the respective sub module with one or more combinations of methods for the other sub modules. SSM is fixed to homography for the first two sections.
\label{results}
\subsubsection{Search Methods}
\label{res_sm}
\setlength{\floatsep}{0pt}
\begin{figure*}[!htbp]
\begin{subfigure}{\textwidth}
\centering
\includegraphics[height=0.23\textheight]{sm_ssd_zncc_rscv}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[height=0.23\textheight]{sm_lscv_mi_ncc}
\end{subfigure}
\caption{Success rates for SMs using Homography SSM and different AMs. Best viewed on a high resolution screen.}
\label{fig_sm}
\end{figure*}
\begin{figure*}[!htbp]
\begin{subfigure}{\textwidth}
\centering
\includegraphics[height=0.23\textheight]{am_fc_ic_esm}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[height=0.23\textheight]{am_fa_ia_nnic}
\end{subfigure}
\caption{Success rates for AMs using Homography SSM and different SMs. Best viewed on a high resolution screen.}
\label{fig_am}
\end{figure*}
Fig. \ref{fig_sm} presents the results for all SMs except NN1K and NN10K which are presented separately in Fig. \ref{fig_am_nn}. This separation was needed because NN, due to its stochastic nature, tends to have significantly lower SR for smaller thresholds than other SMs. In order to maximize the visibility of individual curves in the various plots within Fig. \ref{fig_sm}, the y axis in each has been limited to the range where the curves in that plot actually lie. Inclusion of NN results here would have caused this range to increase significantly, thus decreasing the separation between these curves and making analysis more difficult. SCV and CCRE results are excluded here too, the former because they are very similar to LSCV while the latter are presented separately in Fig. \ref{fig_sm_ccre} for the same reason as NN but now pertaining to Fig. \ref{fig_am}.
Several interesting observations can be made from Figs. \ref{fig_sm} and \ref{fig_sm_ccre}. Firstly, we see that the four variants of LK do not perform identically - FCLK is the best for all AMs and is significantly better than FALK especially for smaller thresholds. ICLK with IALK, on the other hand, are more contentious, being very similar for three AMs - SSD, RSCV and LSCV - but ICLK being appreciably better for the other four. This is especially true for CCRE where it is almost equivalent to FCLK for larger $ t_p $ and much better than both the additive variants. This finding contradicts the equivalence between these variants that was reported in \cite{Baker04lucasKanade_paper} and justified there using both theoretical analysis and experimental results. The latter, however, were only performed on synthetic images and even the former used several approximations. So, it is perhaps not surprising that this supposed equivalence does not hold under real world conditions.
Secondly, we note that ESM fails to outperform FCLK for any AM except MI and even there it does not lead by much. This fact too emerges in contradiction to the theoretical analysis in \cite{Benhimane07_esm_journal} where ESM was shown to have second order convergence and so should be better than first order methods including FCLK. It might be argued that the extended version of ESM \cite{Brooks10_esm_ic,Scandaroli2012_ncc_tracking} used here might not possess the characteristics of the formulation described in \cite{Benhimane07_esm_journal} but we conducted extensive experiments with that exact formulation too and can confirm that the version reported here performs identically to that one.
Thirdly, we see that NNIC does not perform better than ICLK on any of the AMs and is in fact significantly poorer with ZNCC. This yet again does not agree with the results reported in \cite{Dick13nn} using both static experiments and the Metaio dataset \cite{Lieberknecht2009_metaio}. We have already seen in our first observation that static experiments may not always agree with real world tests and it must be admitted that sequences in the Metaio benchmark are highly artificial in nature as they neither represent real tasks nor include an actual background around the tracked patch. We did try to perform experiments on this dataset to check for possible bugs in our implementation but unfortunately the Metaio evaluation service is no longer available. However, to the best of our belief, there is no such bug and the discrepancy does indeed arise from the differences between artificial and real world benchmarks.
Fourthly, we can note that both additive LK variants and especially IALK perform much poorer compared to the compositional variants with the robust AMs than with the SSD like AMs. This is probably to be expected since the Hessian after convergence approach used for extending the Gauss Newton method to these AMs does not make as much sense for additive formulations \cite{Dame10_mi_thesis}.
We conclude this section by examining the effect of number of samples on NN as well as its relative performance to gradient descent SMs from Figs. \ref{fig_sm_ccre} and \ref{fig_am_nn}. We can see by comparing these plots to Fig. \ref{fig_am} that NN performs better relative to the latter with the robust AMs and in fact CCRE actually fares best with NN10K for larger $ t_p $. This might indicate that the poor performance of CCRE, and to an extent MI, with LK type trackers has more to do with gradient descent optimization itself rather than some limitation of these AMs as good similarity metrics.
The gain in performance between NN10K and NN1K though seems to be similar for all AMs as it is caused by an improved coverage of the SSM search space and so should depend only on that.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sm_ccre}
\caption{Success rates for SMs using CCRE with Homography}
\label{fig_sm_ccre}
\end{figure}
\subsubsection{Appearance Models}
\label{res_am}
\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{am_nn1k_nn10k_lone_ccre}
\caption{Success rates for AMs using NN10K and NN1K with Homography represented with \textbf{solid and dashed lines} respectively. SCV, being almost identical to LSCV, has been ommitted for clarity.}
\label{fig_am_nn}
\end{figure}
Fig. \ref{fig_am} shows the SR curves for all AMs except CCRE whose results are in Fig. \ref{fig_sm_ccre} for reasons already mentioned in the previous section. This reason itself is the most obvious point to be noted by comparing Figs. \ref{fig_am} and \ref{fig_sm_ccre} - that CCRE, even though it is the most sophisticated and computationally expensive AM, performs much poorer than other AMs with all SMs except those based on NN. Another interesting fact is that it actually performs far worse with NNIC than it does with either NN1K or ICLK which is very unexpected as the composite tracker uses inputs from both and so should perform at least as well as the best of these. A similar phenomenon can be observed with ZNCC too. We repeated these experiments several times but these discrepancies remained.
Further, even MI is only slightly better than SSD on average, except with NN where it is among the best, being almost at par with NCC. It is much better than CCRE, however, in spite of the two AMs differing only in the latter using a cumulative joint histogram. It seems likely that the additional complexity of CCRE along with the resultant invariance to appearance changes significantly \textit{reduces} its basin of convergence \cite{Dame10_mi_ict}. This leads to poor performance with gradient descent type SMs but, as expected, does not affect the efficacy of stochastic SMs.
The next fact to note is that NCC is the best performer with all SMs except IALK (which performs poorly with all robust AMs anyway as noted in the previous section).
We also note that, though ZNCC is supposedly equivalent to NCC \cite{Ruthotto2010_thes_ncc_equivalence} and also has a wider basin of convergence due to its SSD like formulation, it usually does \textit{not} perform as well as NCC.
However both ZNCC and NCC are almost always better than SCV and its extensions LSCV/RSCV.
This last observation is rather contrary to expectations since SCV is supposedly more robust against lighting changes due to its use of joint probability distributions while ZNCC is merely the L2 norm between the pixel values normalized to have zero mean and unit variance. We can note too that LSCV, notwithstanding, its reported \cite{Richa14_scv_constraints} increased invariance to localized intensity changes, fails to offer any improvement over either SCV or RSCV even though several of the tested sequences do exhibit such lighting changes. Considering that SCV and its variants are significantly more expensive than ZNCC to compute, there seems little reason to use these instead as the computational savings from ZNCC can be used to employ other ways (i.e. higher sampling resolution or more iterations) to improve performance.
\subsubsection{State Space Models}
\label{res_ssm}
\begin{figure*}[!htbp]
\begin{center}
\includegraphics[width=\textwidth]{sm_translation_learning_speed_num_log}
\caption{Success Rates for SMs using ZNCC and Translation as well as for 5 learning based trackers. The former are shown with \textbf{solid} lines and the latter in \textbf{dashed} lines. 2 DOF ground truth was used for all evaluations. Note that the speed plot on the right uses \textbf{logarithmic scaling} on the x axis to increase visibility of the latter though the actual figures are mentioned too.}
\label{fig_sm_rscv_2_gt2_speed}
\end{center}
\end{figure*}
The results presented in this section follow a slightly different format from the other two sections due to the difference in the motivations for using low DOF SSMs - the principle one being that reducing the dimensionality of the search space of warp parameters decreases the likelihood of the search process getting stuck in a local optimum, thus making the tracker more robust. The other less important motivation is that lower DOF SSMs tend to be faster since Jacobians are typically less expensive to compute.
Limiting the DOF also makes registration based trackers directly comparable to learning based trackers as these too work in low DOF search spaces. As a result, in this section, we also present results for five state of the art learning based trackers \cite{Kristan2015_vot15} - discriminative scale space tracker (\textbf{DSST}), kernelized correlation filter tracker (\textbf{KCF}), tracking-learning-detection (\textbf{TLD}), real time compressive tracker (\textbf{RCT}) and consensus-based matching of keypoints tracker (\textbf{CMT}). We have used C++ implementations of all these trackers that are fully integrated into our framework. This not only makes it easy to reproduce the results presented here and but also makes it reasonable to compare the speeds of these trackers with the faster registration based trackers since slower speed is one of the main reasons why learning trackers are often not used in robotics applications.
Lastly, in order to make the evaluations fair, we have used \textit{lower DOF ground truths} for all accuracy results in this section. These were generated for each SSM using least squares optimization to find the warp parameters that, when applied to the initial bounding box, will produce a warped box whose alignment error ($E_{AL}$) with respect to the full 8 DOF ground truth is as small as it is possible to achieve given the constraints of that SSM. In most cases, the ground truth corners thus generated represent the best possible performance that can theoretically be achieved by any tracker that follows the constraints of that SSM. In some rare cases, however, the resulting corners can be quite unexpected so we also visually inspected all lower DOF corners and corrected any that appeared unreasonable.
Fig. \ref{fig_sm_rscv_2_gt2_speed} shows the performance of all SMs with translation SSM in terms of both accuracy, evaluated against 2 DOF ground truth, and speed, measured in terms of the average number of frames processed by the tracker per second (FPS). In addition to the SMs described in Sec. \ref{searchMethod}, results from another SM based on particle filter \cite{Isard98condensation}, generated using 1000 particles (\textbf{PF1K}), are also reported here. This is another stochastic SM like NN that, though present in our framework, only works well with translation at the time of this writing and is thus not mentioned in the previous sections.
As expected, all the learning based trackers have low SR for smaller $ t_p $ since they are less precise in general \cite{Kristan2015_vot15}. What is more interesting, however, is that none of these trackers, with the exception of DSST, managed to surpass the best registration based trackers even for larger $ t_p $ though they did close the gap. Even DSST only managed it at the extreme tail end of the plot and by a small margin. The superiority of DSST over other learning based trackers is at least consistent with results published elsewhere \cite{Kristan2015_vot15}.
The speed comparisons in Fig. \ref{fig_sm_rscv_2_gt2_speed} clearly show the main reason why learning trackers are not suitable for high speed tracking scenarios - they are $ 10 $ to $ 30 $ times slower than their registration based counterparts.
It is not surprising that tracking based SLAM systems like SVO \cite{Forster2014_svo} use registration based trackers as they need to track hundreds to thousands of patches per frame.
It may be noted that the speeds of the former depend on the size of the initial bounding box and so varied widely between sequences unlike the latter where a fixed sampling resolution was used. However, the mean figures reported here do provide a good idea of the general performance that can be expected from these trackers.
Some interesting observations can be made by comparing the different SMs too. Firstly, we see that FALK and FCLK show perfect overlap which is to be expected as the two formulations are identical for translation. Secondly, we note that NN1K and NN10K have practically identical performance in terms of both accuracy and speed. The latter is to be expected since the KD Tree index used by FLANN library \cite{Muja2009_flann} is largely independent of the number of samples - only the initialization time increases when a larger index is to be built. The former, however, is a bit more difficult to explain since NN10K does perform significantly better than NN1K with homography (Fig. \ref{fig_am_nn}). It seems, however, that more samples do not help much with low DOF search spaces as 1000 samples is already enough to cover it well and it is the \textit{quality} of samples that forms the bottleneck now. It may be noted too that PF performs at par with the best registration trackers. This is unsurprising since PF is known to perform well with low DOF when large number of particles are available - an asset that comes at the cost of much slower speed.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{ssm_esm_zncc}
\caption{Success Rates for all SSMs using ESM with ZNCC. Note that homography has 3 parameterizations that overlap perfectly. These plots were generated using corresponding low DOF ground truth for each SSM.}
\label{fig_ssm}
\end{center}
\end{figure}
To conclude the analysis in this section, we tested the performance of different SSMs against each other and the results are reported in Fig. \ref{fig_ssm} using ESM with ZNCC. The plots for each SSM were generated by using the corresponding low DOF ground truth.
As stated before, we were expecting lower DOF trackers to perform better here but this is not the case since higher DOF trackers seem to perform better with the exception of affine which is better than homography for larger values of $ t_p $. However, the increased robustness of low DOF SSMs is at least partially apparent in the fact that their curves approach those of homography as $ t_p $ increases with several surpassing it too. Thus, though they may not be as precise as homography, they do tend to be more resistant to complete drift.
In fact, a general trend noticeable from the SR plots for high DOF SSMs, not only in Fig. \ref{fig_ssm} but also others analyzed earlier, is that, unlike low DOF SSMs and learning based trackers, their SR does not continue to increase through the entire range of $ t_p $ but instead flattens out after a certain point (often for $ t_p<10 $). This results from the fact that, as long as these trackers work, they track the object very precisely but once they diverge, they do not drift off gradually but rather lose track quite abruptly.
Finally, it can be noted that all three parameterizations of homography have exactly identical performance with their plots showing perfect overlap. This indicates that the theoretical justification given in \cite{Benhimane07_esm_journal} for parameterizing ESM with $\mathcal{SL}3$ has little practical significance.
This, in turn, may also suggest that, contrary to the assumption in \cite{Benhimane07_esm_journal}, the reason for ESM's superior performance has more to do with its use of the information from both $ I_0 $ and $ I_t $ rather than with it providing a pseudo second order convergence (opposed to LK's first order convergence).
\section{Descriptions of submodules}
A registration based tracker can be decomposed into three sub modules: appearance model (\textbf{AM}), state space model (\textbf{SSM}) and search method (\textbf{SM}). Figure \ref{flow} shows how these modules work together in a complete tracking system assuming there is no model update wherein the appearance model of the object is updated as tracking progresses.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2.8in]{submodules_ood}
\caption{Modular breakdown of a registration based tracker assuming there is no dynamic update to the appearance model. This shows how different components work, as formulated in Eq \ref{reg_tracking}}
\label{flow}
\end{center}
\end{figure}
When a geometric transform $\mathbf{w}$ with parameters $\mathbf{p}=(p_1, p_2, ..., p_S)$ is applied to an image patch $\mathbf{x}$, the transformed patch is denoted by $\mathbf{x}'=\mathbf{w}(\mathbf{x}, \mathbf{p})$ and the corresponding pixel values in image $ I $ as $\mathbf{I}(\mathbf{w}(\mathbf{x}, \mathbf{p}))$. Tracking can then be formulated (Eq \ref{reg_tracking}) as a search problem where we need to find the optimal transform parameters $\mathbf{p_t}$ for an image $I_t$ that maximize the similarity, measured by a suitable metric $f$, between the target patch $\mathbf{I^*} = \mathbf{I_0}(\mathbf{w}(\mathbf{x},\mathbf{p_0}))$ and the warped image patch $\mathbf{I_t}(\mathbf{w}(\mathbf{x},\mathbf{p_t}))$.
\begin{equation}
\begin{aligned}
\label{reg_tracking}
\mathbf{p_t} = \underset{\mathbf{p}} {\mathrm{argmax}} ~f(\mathbf{I^*},\mathbf{I_t}(\mathbf{w}(\mathbf{x},\mathbf{p})))
\end{aligned}
\end{equation}
We refer to the similarity metric $ f $, the warp function $ \mathbf{w} $ and the algorithm that maximizes Eq \ref{reg_tracking} respectively as AM, SSM and SM.
A more detailed description of these submodules follows.
\subsection{Search Method}
\label{searchMethod}
This is the optimization procedure that searches for the warped patch in the the current image that best matches the original template. Gradient descent is the most popular optimization approach used in tracking due to its speed and simplicity and is the basis of the classic Lucas Kanade (LK) tracker \cite{Lucas81lucasKanade}. This algorithm can be formulated in four different ways \cite{Baker04lucasKanade_paper} depending on which image is searched for the warped patch - $ I_t $ or $ I_0 $ - and how the parameters of the warping function are updated in each iteration - additive or compositional. The four resulting variants - forward additive (\textbf{FALK}) \cite{Lucas81lucasKanade}, inverse additive (\textbf{IALK}) \cite{Hager98parametricModels}, forward compositional (\textbf{FCLK}) \cite{Szeliski06_fclk_extended} and inverse compositional (\textbf{ICLK}) \cite{Baker04lucasKanade_paper} - were analyzed mathematically and shown to be equivalent to first order terms in \cite{Baker04lucasKanade_paper}. Here, however, we show experimental results proving that their performance on real video benchmarks is quite different (Sec. \ref{res_sm}).
A relatively recent update to this approach was in the form of Efficient Second order Minimization (\textbf{ESM}) \cite{Benhimane07_esm_journal} technique that tries to make the best of both inverse and forward formulations by using the mean of the initial and current Jacobians.
We would like to mention here that, even though the authors of \cite{Benhimane07_esm_journal} used $\mathcal{SL}3$ parameterization for their ESM formulation and gave theoretical proofs as to why it is essential for this SM, we have used standard parameterization (i.e. using matrix entries \cite{Szeliski06_fclk_extended, Baker04lucasKanade_paper}) for all our experiments since, as we show later (Sec. \ref{res_ssm}), ESM actually performs identically with several different parameterizations.
Further, since the standard formulations for these SMs using the Gauss Newton Hessian \cite{Lucas81lucasKanade, Baker04lucasKanade_paper, Benhimane07_esm_journal} do not work with any AMs besides SSD \cite{Dame10_mi_ict,Scandaroli2012_ncc_tracking}, a modified version with the so called \textit{Hessian after convergence} \cite{Dame10_mi_ict,Scandaroli2012_ncc_tracking} has been used for our experiments. Also, the extended formulation for ESM reported in \cite{Brooks10_esm_ic,Scandaroli2012_ncc_tracking} has been used instead of the original one in \cite{Benhimane07_esm_journal}. The exact formulations used can be found in \cite{singh16_mtf}.
Nearest neighbor search (NN) is another SM that has recently been used for tracking \cite{Dick13nn} thanks to the FLANN library \cite{Muja2009_flann} that makes real time search feasible. Since the performance of stochastic SMs like NN depends largely on the number of random samples used, we have reported results with 1000 and 10000 samples, with the respective SMs named as \textbf{NN1K} and \textbf{NN10K}. Further, this method tends to give jittery and unstable results when used by itself due to the very limited search space and so was used in conjunction with a gradient descent type SM in \cite{Dick13nn} to create a composite tracker that performs better than either of its constituents. As in \cite{Dick13nn}, we have used ICLK as this second tracker due to its speed and the resultant composite SM is named \textbf{NNIC}. Unlike NN, results for NNIC are only reported using 1000 samples for NN as NN10K is too slow to be combined with ICLK.
\subsection{Appearance Model}
\label{appearanceModel}
This is the similarity metric defined by the function $f$ in Eq. \ref{reg_tracking} using which the SM compares different warped patches from the current image to get the closest match with the original template.
The sum of squared differences (\textbf{SSD}) \cite{Lucas81lucasKanade,Baker04lucasKanade_paper,Benhimane07_esm_journal} or the L2 norm of pixel differences is the AM used most often used in literature especially with SMs based on gradient descent search due to its simplicity and the ease of computing its derivatives. However, the same simplicity also makes it vulnerable to providing false matches when the object's appearance changes due to factors like lighting variations, motion blur and occlusion.
To address these issues, more robust AMs have been proposed including Sum of Conditional Variance (\textbf{SCV}) \cite{Richa11_scv_original}, Normalized Cross Correlation (\textbf{NCC}) \cite{Scandaroli2012_ncc_tracking}, Mutual Information (\textbf{MI}) \cite{Dowson08_mi_ict,Dame10_mi_ict} and Cross Cumulative Residual Entropy (\textbf{CCRE}) \cite{Wang2007_ccre_registration,Richa12_robust_similarity_measures}, all of which supposedly provide a degree of invariance to changes in illumination. There also exists a slightly different formulation of the former known as Reversed SCV (\textbf{RSCV}) \cite{Dick13nn} where $ \mathbf{I_t} $ is updated rather than $ \mathbf{I_0} $. There has also been a recent extension to it called \textbf{LSCV} \cite{Richa14_scv_constraints} that uses multiple joint histograms from corresponding sub regions within the target patch to achieve greater robustness to localized intensity changes.
It has further been shown \cite{Ruthotto2010_thes_ncc_equivalence} that maximizing NCC between two images is equivalent to minimizing the SSD between two z-score \cite{Jain2005_ncc_z_score} normalized images. We consider the resultant formulation as a different AM called Zero Mean NCC (\textbf{ZNCC}).
It may ne noted that these AMs can be divided into 2 distinct categories - those that use some form of the L2 norm as the similarity function - SSD, SCV, RSCV, LSCV and ZNCC - and those that do not - MI, CCRE and NCC. The latter are henceforth called robust models after \cite{Richa12_robust_similarity_measures}.
\subsection{State Space Model}
\label{stateSpace}
The SSM represents the set of allowable image motions of the tracked object and thus embodies any constraints that are placed on the search space of warp parameters to make the optimization more efficient. This includes both the degrees of freedom (DOF) of allowed motion, as well as the actual parameterization of the warping function. For instance the ESM tracker, as presented in \cite{Benhimane07_esm_journal}, can be considered to have a different SSM than conventional LK type trackers \cite{Lucas81lucasKanade,Baker04lucasKanade_paper} even though both involve 8 DOF homography, since it uses the $\mathcal{SL}3$ parameterization rather than the actual entries of the corresponding matrix.
We model 7 different SSMs - translation, isometry, similitude, affine and homography \cite{Szeliski06_fclk_extended} along with two extra parameterizations of homography - $ \mathcal{SL}3 $ and corner based (using x,y coordinates of the four corners of the bounding box).
The advantage of using higher DOF SSM is achieving greater precision in the aligned warp since transforms that are higher up in the hierarchy \cite{Hartley04MVGCV} can better approximate the projective transformation process that captures the relative motion between the camera and the object in the 3D world into the 2D images.
However, there are two issues with having to estimate more parameters - the iterative search takes longer to converge making the tracker slower and the search process becomes more likely to either diverge or end up in a local optimum causing the tracker to be less stable and more likely to lose track. The latter is a well known phenomenon with LK type trackers \cite{Bouguet01} whose higher DOF variants are usually less robust.
It may be noted that this sub module differs from the other two in that it does not admit new methods in the conventional sense and may even be viewed as a part of the SM with the two being often closely intertwined in practical implementations.
However, though the SSMs used in this work are limited to the standard hierarchy of geometric transformations, more complex models like piecewise projective transforms do exist and it is also theoretically possible to impose novel constraints on the search space that can significantly decrease the search time while still producing sufficiently accurate results. The fact that such a constraint will be an important contribution in its own right justifies the use of SSM as a sub module in this work to motivate further research in this direction.
|
1,314,259,993,055 | arxiv | \section{Introduction}
Deep neural networks (DNN) produce excellent image classification results and are the current state-of-the-art, but have been shown to be vulnerable to attacks by adversarial examples: Images altered by the introduction of small perturbations that cause the neural network to misclassify the image \cite{carlini2017adversarial}.
\par
Many researchers have proposed defenses against adversarial image attacks on neural network classification systems.
In image preprocessing defenses, the images are altered in some way before being classified (blurring, cropping, noise) in order to disrupt any adversarial perturbations \cite{graese2016assessing} \cite{guo2017countering}.
In dropout randomization defenses, as the name suggests, the neural network adds randomization which supports the use of Bayesian uncertainty measurements to assess the likelihood of an image being adversarial \cite{feinman2017detecting} \cite{papernot2017extending}.
\par
Our model uses a combination defense: We base our model on Bayesian uncertainty in a dropout neural network \cite{srivastava2014dropout}, but use a secondary defense, preprocessing, as a double-check for \enquote{edge} cases.
The existence of the secondary defense allows us to tune the uncertainty aspect of our defense in favor of declaring \enquote{edge} images (images close to the threshold uncertainty) adversarial, with less sacrifice of clean-image accuracy than would otherwise be the case.
\section{Related Work}
Feinman et al. \cite{feinman2017detecting} proposed a defense based on Bayesian uncertainty with dropout, combined with kernel density estimation.
They tested it with good results against adversarial examples generated via the Fast Gradient Sign Method (FGSM) \cite{goodfellow6572explaining}, Basic Iterative Method (BIM) \cite{kurakin2016adversarial}, Jacobian-based Saliency Map Attack (JSMA) \cite{papernot2016limitations}, and C\&W \cite{carlini2017adversarial} attacks.
They achieved ROC-AUC scores on sets of adversarial images generated from the MNIST dataset ranging from 90.57\% to 98.13\% depending on attack method.
This approach took advantage of the uncertainty estimates possible with dropout networks by assuming that the Bayesian uncertainty will be greater for adversarial examples than for clean data, because of the effect of the randomization on the necessarily precise perturbations.
In addition, they used a Gaussian Mixture Model to analyze the outputs of the last hidden layer of their neural network, arguing that adversarial images will have a different distribution than clean ones.
They also incorporated a kernel density estimate defense and evaluated their approach on MNIST \cite{lecun1998mnist}, CIFAR-10 \cite{krizhevsky2009learning}, and SVHN \cite{netzer2011reading} datasets.
\par
Papernot and McDaniel \cite{papernot2017extending} also presented an uncertainty based defense: A model-agnostic system in which the uncertainty is based on the predictions of a second, separate neural network which is used to train the classification network.
This defense showed good results for the MNIST dataset, tested with adversarial images generated by FGSM, JSMA, and AdaDelta \cite{carlini2017towards} attacks.
\par
Several researchers have considered preprocessing based defenses against adversarial examples.
Graese, Rozsa, and Boult \cite{graese2016assessing} explored several preprocessing techniques with the MNIST dataset, tested with FGSM and Fast Gradient Value (FGV) \cite{rozsa2016adversarial} attacks, and found the best results with cropping and resizing: 76\% and 78\% accuracy for FGV and FGSM samples respectively, compared to 65\% and 68\% for the next best \enquote{translation} technique on their raw model.
Guo et al. \cite{guo2017countering} tested cropping-resizing as well as other image transformations (image quilting, JPEG compression, etc.) and found that cropping-resizing was \enquote{very efficient} with accuracy up to 76\% depending on the strength and method of attack.
\section{Methodology}
Our method uses Bayesian uncertainty \cite{gal2016dropout} \cite{feinman2017detecting} with a relatively low \enquote{adversarial} threshold for initial assessments and corrects for false negatives using image pre-processing.
Our method involves a Convolutional Neural Network (CNN) with dropout which reports the Bayesian uncertainty of its classifications, but which then takes multiple crops of images which fall near our threshold uncertainty level and classifies each crop again using our CNN.
The final classification or adversarial image indication is based on both the level of Bayesian uncertainty and the agreement or lack thereof among the crops.
Algorithm \ref{Algo} illustrates the procedure.
\begin{algorithm}[t]
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{an image $img$, a high threshold $H$, a low threshold $L$, and a count of agreed predictions $C$.}
\Output{image label (clean or adversarial).}
calculate $img$ prediction\;
calculate $img$ uncertainty\;
\eIf{$uncertainty > H$}
{declare adversarial\;}
{
\eIf{$uncertainty < L$}
{declare clean\;}
{
\For{1:5}{
crop \& resize $img$\;
classify $img$\;
\If{class = prediction}
{increment counter\;}
}
\eIf{$counter > C $}
{declare clean\;}
{declare adversarial\;}
}
}
\caption{Detecting adversarial images during inferring time.}
\label{Algo}
\end{algorithm}
\subsection{Computing Model Bayesian Uncertainty}
Dropout was first introduced as a means of avoiding overfitting in deep neural network training \cite{srivastava2014dropout}.
Dropout layers mean that some weights are randomly zeroed out, that is, some links between neurons are cut.
Gal and Ghahramani \cite{gal2016dropout} noted that including dropout layers in a neural network provides information about uncertainty for a wide variety of DNN architectures, sometimes without modification.
Srivastava et al. \cite{srivastava2014dropout} used dropout at training time, however Gal and Ghahramani applied dropout to the inference stage.
\par
Our model uses dropout layers in the inference stage: For each image, we made $N$ stochastic passes through the network.
Each pass produced a probability for each available class by applying Softmax to the resulting logit vectors $z_{1}(x), ..., z_{N}(x)$.
\par
To obtain the stochastic prediction, we take the mean of the logit vector $z(x)$ for each class.
The image is provisionally classified as the class with the highest mean.
We also obtain the stochastic uncertainty by computing the standard deviation of the predictions over the $N$ stochastic passes.
\subsection{Image Preprocessing}
At this point, if the stochastic uncertainty is sufficiently high, our model declares the image adversarial without reference to the secondary method.
If the uncertainty is sufficiently low, our model outputs the provisional classification as the final classification, again without reference to the secondary method.
However, if the uncertainty lies within a certain middle range, the model employs the crop-resize secondary method.
\par
In the cropping stage, the network takes any image which has been classified with an uncertainty laying in the \enquote{edge} range, splits it into five overlapping crops (top left, bottom left, top right, bottom right, and center), in the manner of Graese et al. \cite{graese2016assessing}, and resizes each crop back to the original size.
Each of these crops is reclassified by the network, and if four out of the five new classifications agree with the original, the image is not declared adversarial and the original classification is output.
This allows us to set the \enquote{threshold} Bayesian uncertainty level low enough to risk a noticeable number of false negatives (clean images declared adversarial) at the uncertainty stage of the process, while providing a mechanism to prevent these \enquote{extra} false negatives from making a large impact on our model's final accuracy.
The lower uncertainty threshold level, in turn, gives a greater chance of detecting borderline adversarial images.
\section{Experiments}
We implemented our model using Python 3.6 with CleverHans 2.0, TensorFlow, Keras 2.0, Scikit-learn, OpenCV, and Pillow 5.1.0.
CleverHans facilitates the generation of adversarial examples using a variety of techniques.
Using multiple techniques to generate a set of adversarial images for testing allows us to assess the general applicability of our defense.
We trained and tested our model using the MNIST dataset: a dataset of handwritten digits consisting of 60,000 training examples and 10,000 test images.
We tested our model with 50/50 mixed clean and adversarial test sets for each of three attacks.
\subsection{Neural Networks}
For MNIST, we used the LeNet \cite{lecun1989backpropagation} convnet architecture.
We used a dropout rate of 0.5 after the last pooling layer and after the inner-product layer, as in \cite{feinman2017detecting}.
On normal, non-adversarial samples, this network shows an accuracy of 98\%.
\subsection{Adversarial Attacks}
We evaluated our model using three different methods of generating adversarial images: FSGM \cite{goodfellow6572explaining}, BIM \cite{kurakin2016adversarial}, and JSMA \cite{papernot2016limitations}.
\begin{center}
\begin{table}[h!]
\center
\caption{Classification accuracy of undefended model.}
\begin{tabular}{ m{3.5cm} | m{2cm}| m{2cm} | m{2cm}}
\hline
& FGSM & JSMA & BIM \\
\hline
Classification accuracy & 59.0\% & 52.0\% & 50.0\% \\
\hline
\end{tabular}
\label{undefended}
\end{table}
\end{center}
\begin{center}
\begin{table}[h!]
\center
\caption{Results for adversarial image detection.}
\begin{tabular}{ m{3.5cm} | m{2cm}| m{2cm} | m{2cm}}
\hline
& FGSM & JSMA & BIM \\
\hline
False negative & 252 & 85 & 249 \\
\hline
False positive & 134 & 74 &104 \\
\hline
True negative & 4796 & 4856 & 4826 \\
\hline
True positive & 4725 & 4892 & 4728\\
\hline
Detection accuracy & 96.1\% & 98.3\% & 96.4\% \\
\hline
\end{tabular}
\label{detection}
\end{table}
\end{center}
\begin{center}
\begin{table}[h!]
\center
\caption{Results of applying the defense with different attacks using MNIST dataset.}
\begin{tabular}{ m{1.5cm} | m{1.7cm} | m{1.7cm} | m{1.7cm} | m{1.7cm} | m{2cm} }
\hline
\multirow{2}{1.1cm}{Attack}
&
\multicolumn{2}{|c|}{Clean images reported clean}
&
\multicolumn{2}{|c|}{Adv. images reported clean}
&
\multirow{2}{1.1cm}{Classification Accuracy}
\\ \cline{2-5}
& Classified correctly & Classified incorrectly & Classified correctly & Classified incorrectly & \\
\hline
FGSM & 97.3\% & 0.0\% & 1.7\% & 1.0\% & 99.0\% \\
\hline
JSMA & 98.5\% & 0.0\% & 1.4\% & 0.1\% & 99.9\% \\
\hline
BIM & 97.8\% & 0.0\% & 1.2\% & 1.0\% & 99.0\% \\
\hline
\end{tabular}
\label{result}
\end{table}
\end{center}
\section{Results}
For our model, we adjusted the uncertainty levels based on the information gain to improve accuracy and used these thresholds to calculate high and low levels of Bayesian uncertainty.
Our experiments also showed that requiring four out of five crops be in agreement produced the greatest ultimate accuracy.
We generated adversarial images using FGSM, BIM, and JSMA, with separate test runs for each attack.
\par
We tested our model with a set of 10,000 images comprised of 50\% clean and 50\% adversarial images.
After testing with the basic, undefended CNN, we removed clean images which were misclassified; in testing the defended model, we used only adversarial images generated from images which were correctly classified by the basic CNN when clean.
\par
For the FGSM attack, as seen in Table \ref{undefended}, our basic CNN without defense had an accuracy of 59\% on the 50/50 mixed test set; most of adversarial images were indeed misclassified.
By adding our defense to the network, we were able to detect adversarial images with an accuracy of 96.1\% which increased classification accuracy, for the images flagged as clean, to 99.0\% (see Table \ref{detection} and \ref{result}).
\par
Adversarial images produced using the BIM attack were even more successful against the unprotected CNN, which was only 50\% accurate on the mixed test set.
With our defense added, our model handled a BIM mixed test set with an accuracy detecting adversarial images of 96.4\% and a classification accuracy on clean-flagged images of 99.0\%.
\par
On a mixed test set produced with the JSMA attack, our basic CNN had an accuracy of 52\%.
With our defense, our model achieved 99.9\% classification accuracy with 98.3\% detection accuracy.
\par
In all cases above, accuracy was calculated as \textless images correctly classified\textgreater/ \textless total remaining (flagged as clean) images\textgreater, and did not include images discarded as adversarial.
Detection accuracy was calculated using the \textless true positive + true negative\textgreater/\textless total images in set\textgreater. Total images now are less than 10,000 due to removal of misclassified clean images from consideration.
\par
Feinman et al. \cite{feinman2017detecting} used a Bayesian uncertainty model, but combined it with kernel density rather than preprocessing.
With MNIST, they achieved mixed test set adversarial sample detection ROC-AUC scores of 90.57\% for FGSM, 97.23\% for BIM-A, and 98.13\% for JSMA, slightly lower than the detection accuracy results for our model.
On average, Feinman et al. achieved an MNIST detection ROC-AUC of 95.31\%, somewhat lower than our average MNIST detection accuracy of \textbf{97\%}.
\par
Graese, Rozsa, and Boult \cite{graese2016assessing} tested their techniques with the FGSM attack, but did not consider BIM nor JSMA attacks.
For their 5-crops-and-resize method they reported a classification accuracy of 90.94\% for FGSM, noticeably lower than our combined model results of \textbf{99.0\%} for that attack, however their binarization defense had a comparable accuracy of 99.21\%.
\par
Wang et al. \cite{wang2018detecting} applied a set of mutations, obtained by imposing a minor random but realistic perturbation on the image.
Based on their observation, clean images preserved their labels while adversarial did not.
They scored, on average, 88.0\% for MNIST detection. Ma et al. \cite{ma2018characterizing} used Local Intrinsic Dimensionality (LID) to characterize the dimensional properties of adversarial subspaces or adversarial regions which facilitates the distinction of adversarial examples. For MNIST, on average, they scored 96.20\% accuracy, slightly lower than our average.
\section{Conclusion}
Our combined Bayesian uncertainty and image pre-processing proved effective, with accuracy in the high 90s, at detecting adversarial examples in a mixed MNIST test set.
This allowed for considerably higher accuracy in classifying the remaining, primarily clean, images.
Our model has shown results which are for the most part similar to those of other recent work, with slightly greater accuracy in adversarial example detection for the test sets and attacks we considered.
\par
Future work in those areas could be promising, as our results suggest that a combined defense has some advantages.
In particular, it would be interesting to explore a combination method that involves binarization in the manner of Graese et al.
Also, future work in defenses against additional attack methods could shed light on the general applicability of this combined method.
\bibliographystyle{splncs03}
|
1,314,259,993,056 | arxiv | \section{Introduction}
Noncommutative distributions are at the heart of noncommutative probability theory and of free probability theory in particular. These, in general, purely combinatorial objects allow some very elegant translation of various questions arising for instance in operator algebra or random matrix theory into the unifying language of noncommutative probability theory; in this way, they can build bridges between originally unrelated fields and often also make available tools from free probability theory in those areas.
Within the algebraic frame of of a \emph{noncommutative probability space} $(\mathcal{A},\phi)$, i.e., a unital complex algebra $\mathcal{A}$ with a distinguished unital linear functional $\phi:\mathcal{A}\to\mathbb{C}$, the \emph{(joint) noncommutative distribution} of a tuple $X=(X_1,\dots,X_n)$ of finitely many noncommutative random variables $X_1,\dots,X_n\in\mathcal{A}$ is given as the linear functional
$$\mu_X:\ \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C},\qquad P \mapsto \phi(P(X))$$
that is defined on the algebra $\mathbb{C}\langle x_1,\dots,x_n\rangle$ of noncommutative polynomials in $n$ formal noncommuting variables $x_1,\dots,x_n$.
In practice, one often works -- as we will do in the following -- in the more analytic setting of a \emph{tracial $W^\ast$-probability space} $(\mathcal{M},\tau)$, i.e., a von Neumann algebra $\mathcal{M}$ that is endowed with some faithful normal tracial state $\tau: \mathcal{M}\to\mathbb{C}$.
If tuples $X=(X_1,\dots,X_n)$ of noncommutative random variables $X_1,\dots,X_n$ living in $\mathcal{M}$ are considered, then their joint noncommutative distribution $\mu_X$ determines the generated von Neumann algebra $\operatorname{vN}(X_1,\dots,X_n)$ up to isomorphism. Thus, $\mu_X$ provides a kind of combinatorial ``barcode'' for $\operatorname{vN}(X_1,\dots,X_n)$ and consequently contains all spectral properties of $X_1,\dots,X_n$; the challenging question, however, is how to read off those information from a given $\mu_X$.
Here, we are concerned with regularity properties of noncommutative distributions.
In a groundbreaking series of papers \cite{Voi93,Voi94,Voi96,Voi97,Voi98,Voi99}, Voiculescu developed free probability analogues of the classical notions of Fisher information and entropy; see \cite{Voi02} for a survey. Here, we follow the non-microstates approach that Voiculescu presented in \cite{Voi98,Voi99}. To tuples $X=(X_1,\dots,X_n)$ of noncommutative random variables living in a tracial $W^\ast$-probability space $(\mathcal{M},\tau)$, he associates the \emph{non-microstates free Fisher information $\Phi^\ast(X)$} and the \emph{non-microstates free entropy $\chi^\ast(X)$}; each of those numerical quantities, if finite, gives some rich structure to the joint noncommutative distribution $\mu_X$, however, without determining it completely.
It is the common viewpoint that both $\Phi^\ast(X) < \infty$ and $\chi^\ast(X) > - \infty$ imply in particular some strong regularity of $\mu_X$, making this guess precise, however, turns out to be quite intricate.
One of the major drawbacks in that respect is the lack of an effective analytic machinery to handle noncommutative distributions in a way similar to the measure theoretic description of distributions in classical probability theory.
Such tools, however, are available only in very limited situations. Even in the strong analytic framework of a tracial $W^\ast$-probability space $(\mathcal{M},\tau)$, we typically must restrict ourselves to the case of a single noncommutative random variable $X\in\mathcal{M}$ in order to gain such an analytic description. For instance, if the considered operator $X$ is selfadjoint, then its combinatorial noncommutative distribution can be encoded by some compactly supported Borel probability measure $\mu_X$ on the real line $\mathbb{R}$, called the \emph{analytic distribution of $X$}; more precisely, the analytic distribution $\mu_X$ is uniquely determined among all Borel measures on $\mathbb{R}$ by the requirement that
$$\tau(X^k) = \int_\mathbb{R} t^k\, d\mu_X(t) \qquad\text{for all integers $k\geq 0$}.$$
For the sake of completeness, we note that this notion can be generalized to normal operators $X$, resulting in a compactly supported Borel probability measure on the complex plane $\mathbb{C}$; on the other hand, for operators that fail to be normal, on can study instead its so-called Brown measure.
Accordingly, it is not even clear what ``regularity'' should mean for general noncommutative distributions.
In recent years, evaluations of ``noncommutative test functions'' such as noncommutative polynomials or noncommutative rational functions were successfully developed as a kind of substitute for the measure theoretic description in order to overcome those difficulties. In fact, each such evaluation produces a single noncommutative random variable whose analytic distribution can be studied by measure theoretic means. The guiding idea is that the larger the considered class of test functions is, the more information we gain about the underlying multivariate noncommutative distribution.
In this way, also the aforementioned problem becomes treatable: ``regularity'' of noncommutative distributions $\mu_X$, imposed by conditions such as $\Phi^\ast(X) < \infty$ and $\chi^\ast(X) > - \infty$, is understood as being reflected in properties of the analytic distributions $\mu_{f(X)}$ that arise from evaluations $f(X)$ of noncommutative test functions $f$.
Several results have already been obtained in that direction; see, for instance, \cite{SS15,CS16,MSW17,MSY18}. Inspired by \cite{CS16}, we elaborate here on the H\"older continuity of cumulative distribution functions of analytic distributions associated to noncommutative polynomial evaluations in variables having finite Fisher information.
Recall that the \emph{cumulative distribution function $\mathcal{F}_\mu$} of a probability measure $\mu$ on $\mathbb{R}$ is the function $\mathcal{F}_\mu: \mathbb{R} \to [0,1]$ that is defined by $\mathcal{F}_\mu(t) := \mu((-\infty,t])$; note that in the case of the analytic distribution $\mu_Y$ of $Y$, we will abbreviate $\mathcal{F}_{\mu_Y}$ by $\mathcal{F}_Y$.
Our first main result reads then as follows.
\begin{theorem}\label{thm:Hoelder_continuity}
Let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space and let $X_1,\dots,X_n$ be selfadjoint noncommutative random variables living in $\mathcal{M}$ satisfying
$$\Phi^\ast(X_1,\dots,X_n)<\infty.$$
Moreover, let $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be any selfadjoint noncommutative polynomial of degree $d\geq 1$ and consider the associated selfadjoint noncommutative random variable $$Y:=P(X_1,\dots,X_n)$$ in $\mathcal{M}$. Then the cumulative distribution function $\mathcal{F}_Y$ of the analytic distribution $\mu_Y$ of $Y$ is H\"older continuous with exponent $\frac{2}{2^{d+2}-5}$, i.e., there is some constant $C>0$ (depending only on $P$ and $X_1,\dots,X_n$) such that
$$|\mathcal{F}_Y(t)-\mathcal{F}_Y(s)| \leq C |t-s|^{\frac{2}{2^{d+2}-5}} \qquad\text{for all $s,t\in\mathbb{R}$}.$$
\end{theorem}
Theorem \ref{thm:Hoelder_continuity} has some important consequences.
It was shown in \cite{Jam15} that Borel probability measures with H\"older continuous cumulative distribution functions have finite logarithmic energy. The latter quantity, if the analytic distribution $\mu_Y$ of a selfadjoint noncommutative random variable $Y\in\mathcal{M}$ is considered, is known to be closely related to the non-microstates free entropy $\chi^\ast(Y)$, which coincides in that case with the microstates free entropy $\chi(Y)$; see \cite{Voi98}.
Thus, in summary, we obtain the following result.
\begin{theorem}\label{thm:finite_entropy}
Let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space and let $X_1,\dots,X_n$ be selfadjoint noncommutative random variables living in $\mathcal{M}$ satisfying
$$\Phi^\ast(X_1,\dots,X_n)<\infty.$$
Then, for every selfadjoint noncommutative polynomial $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ which is non-constant, we have that
$$\chi^\ast(P(X_1,\dots,X_n)) > -\infty.$$
\end{theorem}
This provides a partial and conceptual answer to a question formulated in \cite{CS16}. There, it is conjectured that the conclusion of Theorem \ref{thm:finite_entropy}, i.e., that $\chi^\ast(P(X))>-\infty$ holds for every non-constant noncommutative polynomial $P$, remains true under the weaker condition $\chi^\ast(X)>-\infty$.
At first sight, as we have strengthened that condition to $\Phi^\ast(X)<\infty$, it might be tempting to guess that this should even enforce $\Phi^\ast(P(X)) < \infty$. This guess, however, is much too optimistic, as one already sees in the case of a single variable: for a standard semicircular variable $S$, we have that $\Phi^\ast(S) < \infty$ while $S^2$ is a free Poisson distribution, for which we know that $\Phi^\ast(S^2)=\infty$.
Thus, also under the stronger assumption $\Phi^\ast(X)<\infty$, one cannot hope in general for more than $\chi^\ast(P(X))>-\infty$.
The rest of this article is organized as follows.
In Section \ref{sec:free_differential_operators}, we recall some basic facts from the $L^2$-theory for free differential operators as initiated by Voiculescu.
The proof of Theorem \ref{thm:Hoelder_continuity} will be given in Section \ref{sec:Hoelder_continuity}; for that purpose, we will first collect and extend there some of the more recent results on which the proof builds.
Of independent interest is Section \ref{sec:Kolmogorov}, which is devoted to the well-known phenomenon that convergence in distribution of Borel probability measures on $\mathbb{R}$ to a limit measure with H\"older continuous cumulative distribution function automatically improves itself to convergence in Kolmogorov distance. With Theorems \ref{thm:Hoelder_criterion} and \ref{thm:Hoelder_criterion_compact}, we prove quantified versions thereof that provide explicit rates of convergence for the Kolmogorov distance.
In the last Section \ref{sec:random_matrices}, we combine our previously obtained results to a wide class of random matrix models. In particular, we consider a tuple $(X_1^{(N)},\dots,X_n^{(N)})$ of $N\times N$ selfadjoint random matrices following some Gibbs law and whose asymptotic behavior as $N\to\infty$ is described by a tuple $(X_1,\dots,X_n)$ of selfadjoint noncommutative random variables with the property $\Phi^\ast(X_1,\dots,X_n) < \infty$. We then prove, in Corollaries \ref{cor:random_matrices_polynomial} and \ref{cor:random_matrices_block}, that the limiting eigenvalue distribution of a random matrix of the form $Y^{(N)}=f(X_1^{(N)},\dots,X_n^{(N)})$, for certain ``noncommutative functions'' $f$, has a H\"older continuous cumulative distribution function and that this convergence holds with respect to the Kolmogorov distance. Finally, we provide, in Corollaries \ref{cor:block-GUE} and \ref{cor:p-GUE}, rates of convergence of the Kolmogorov distance for the particular cases where $(X_1^{(N)},\dots,X_n^{(N)})$ is a tuple of independent GUE random matrices.
\tableofcontents
\section{A glimpse on the $L^2$-theory for free differential operators}\label{sec:free_differential_operators}
This section is devoted to the $L^2$-theory for free differential operators, which underlies the non-microstates approach to free entropy as developed by Voiculescu in \cite{Voi98,Voi99}. For reader's convenience, we recall here the needed terminology and some fundamental results.
\subsection{Noncommutative polynomials and noncommutative derivatives}
As usual, we will denote by $\mathbb{C}\langle x_1,\dots,x_n\rangle$ the unital complex algebra of \emph{noncommutative polynomials} in $n$ formal noncommuting variables $x_1,\dots,x_n$. Let us recall that any noncommutative polynomial $P\in \mathbb{C}\langle x_1,\dots,x_n\rangle$ can be written in the form
\begin{equation}\label{eq:ncpoly}
P = \sum^d_{k=0} \sum_{1\leq i_1,\dots,i_k\leq n} a_{i_1,\dots,i_k}\, x_{i_1} \cdots x_{i_k}.
\end{equation}
for some integer $d\geq 0$ and coefficients $a_{i_1,\dots,i_k} \in \mathbb{C}$; if there exist $1\leq i_1,\dots,i_d\leq n$ such that $a_{i_1,\dots,i_d} \neq 0$, then we say that \emph{$P$ has degree $d$} and we put $\deg(P) := d$.
Note that $\mathbb{C}\langle x_1,\dots,x_n\rangle$ becomes a $\ast$-algebra if it is endowed with the involution defined by
$$P^\ast = \sum^d_{k=0} \sum_{1\leq i_1,\dots,i_k\leq n} \overline{a_{i_1,\dots,i_k}}\, x_{i_k} \cdots x_{i_1}$$
for every noncommutative polynomial $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ which is written in the form \eqref{eq:ncpoly}.
Elements in the algebraic tensor product $\mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ will be called \emph{bi-polynomials} in the following. By definition, $\mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ forms a unital complex algebra; it moreover forms a $\mathbb{C}\langle x_1,\dots,x_n\rangle$-bimodule with the natural left and right action determined by $P_1 \cdot (Q_1 \otimes Q_2) \cdot P_2 := (P_1 Q_1) \otimes (Q_2 P_2)$.
Therefore, we may introduce on $\mathbb{C}\langle x_1,\dots,x_n\rangle$ the so-called \emph{non-commutative derivatives} $\partial_1,\dots,\partial_n$ as the unique derivations
$$\partial_j:\ \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle,\qquad j=1,\dots,n,$$
with values in $\mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ that satisfy $\partial_j x_i = \delta_{i,j} 1 \otimes 1$ for $i,j=1,\dots,n$.
\subsection{Conjugate systems and non-microstates free Fisher information}
Let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space (i.e., a von Neumann algebra $\mathcal{M}$ that is endowed with a faithful normal tracial state $\tau: \mathcal{M} \to \mathbb{C}$) and consider $n$ selfadjoint noncommutative random variables $X_1,\dots,X_n\in \mathcal{M}$.
Throughout the following, we will denote in such cases by $\mathcal{M}_0\subseteq \mathcal{M}$ the von Neumann subalgebra that is generated by $X_1,\dots,X_n$; in order to simplify the notation, the restriction of $\tau$ to $\mathcal{M}_0$ will be denoted again by $\tau$.
In \cite{Voi98}, Voiculescu associated to the tuple $(X_1,\dots,X_n)$ the so-called \emph{non-microstates free Fisher information $\Phi^\ast(X_1,\dots,X_n)$}; note that, while he assumed for technical reasons in addition that $X_1,\dots,X_n$ do not satisfy any non-trivial algebraic relation over $\mathbb{C}$, it was shown in \cite{MSW17} that this constraint is not needed as an a priori assumption on $(X_1,\dots,X_n)$ but is nonetheless enforced a posteriori by some general arguments. We call $(\xi_1,\dots,\xi_n) \in L^2(\mathcal{M}_0,\tau)^n$ a \emph{conjugate system for $(X_1,\dots,X_n)$}, if the \emph{conjugate relation}
$$\tau\big(\xi_j P(X_1,\dots,X_n)\big) = (\tau \mathbin{\overline{\otimes}} \tau)\big((\partial_j P)(X_1,\dots,X_n)\big)$$
holds for each $j=1,\dots,n$ and for all noncommutative polynomials $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$, where $\tau \mathbin{\overline{\otimes}}\tau$ denotes the faithful normal tracial state that is induced by $\tau$ on the von Neumann algebra tensor product $\mathcal{M} \mathbin{\overline{\otimes}} \mathcal{M}$. The conjugate relation implies that such a conjugate system, in case of its existence, is automatically unique; thus, one can define
$$\Phi^\ast(X_1,\dots,X_n) := \sum^n_{j=1} \|\xi_j\|_2^2$$
if a conjugate system $(\xi_1,\dots,\xi_n)$ for $(X_1,\dots,X_n)$ exists and $\Phi^\ast(X_1,\dots,X_n) := \infty$ if there is no conjugate system for $(X_1,\dots,X_n)$.
\subsubsection{Free differential operators}
Suppose now that $\Phi^\ast(X_1,\dots,X_n) < \infty$ holds and let $(\xi_1,\dots,\xi_n)$ be the conjugate system for $(X_1,\dots,X_n)$. It was shown in \cite{MSW17} that $\operatorname{ev}_X: \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C}\langle X_1,\dots,X_n\rangle$ constitutes under this hypothesis an isomorphism, so that the noncommutative derivatives induce unbounded linear operators
$$\partial_j:\ L^2(\mathcal{M}_0,\tau) \supseteq D(\partial_j) \to L^2(\mathcal{M}_0 \mathbin{\overline{\otimes}} \mathcal{M}_0, \tau \mathbin{\overline{\otimes}} \tau)$$
with domain $D(\partial_j) := \mathbb{C}\langle X_1,\dots,X_n\rangle$. Since $\partial_j$ is densely defined, we may consider the adjoint operators
$$\partial_j^\ast:\ L^2(\mathcal{M}_0 \mathbin{\overline{\otimes}} \mathcal{M}_0, \tau \mathbin{\overline{\otimes}} \tau) \supseteq D(\partial_j^\ast) \to L^2(\mathcal{M}_0,\tau)$$
and we conclude from the conjugate relations that $1\otimes 1 \in D(\partial_j^\ast)$ with $\partial_j^\ast(1\otimes 1) = \xi_j$.
If restricted to its domain, each of the unbounded linear operator $\partial_j$ gives a $\mathbb{C}\langle X_1,\dots,X_n\rangle \otimes \mathbb{C}\langle X_1,\dots,X_n\rangle$-valued derivation on $\mathbb{C}\langle X_1,\dots,X_n\rangle$.
\subsection{Non-microstates free entropy}
It was shown in \cite{Voi98} that arbitrarily small perturbations of any tuple $(X_1,\dots,X_n)$ of selfadjoint operators in $\mathcal{M}$ by freely independent semicircular elements lead to finite non-microstates free Fisher information. Indeed, if $S_1,\dots,S_n$ are semicircular elements in $\mathcal{M}$ which are freely independent among themselves and also free from $\{X_1,\dots,X_n\}$, then \cite[Corollary 6.14]{Voi98} tells us that $(X_1+\sqrt{t}S_n,\dots,X_n+\sqrt{t}S_n)$ admits a conjugate system for each $t>0$ and we have the estimates
\begin{equation}\label{eq:Fisher_perturbation}
\frac{n^2}{C^2 + nt} \leq \Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n) \leq \frac{n}{t} \quad\text{for all $t>0$},
\end{equation}
where $C^2 := \tau(X_1^2 + \dots + X_n^2)$; moreover, the function $t\mapsto \Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n)$, which is defined on $[0,\infty)$ and takes its values in $(0,\infty)$, is decreasing and right continuous.
Based on this observation, Voiculescu introduced in \cite{Voi98} the \emph{non-microstates free entropy} $\chi^\ast(X_1,\dots,X_n)$ of $X_1,\dots,X_n$ by
$$\chi^\ast(X_1,\dots,X_n) := \frac{1}{2} \int^\infty_0\Big(\frac{n}{1+t}-\Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n)\Big)\, dt + \frac{n}{2}\log(2\pi e).$$
Note that the left inequality in \eqref{eq:Fisher_perturbation} implies in particular that (cf. \cite[Proposition 7.2]{Voi98})
$$\chi^\ast(X_1,\dots,X_n) \leq \frac{n}{2}\log(2\pi e n^{-1} C^2).$$
Of particular interest is the case $n=1$ of a single noncommutative random variable $X=X^\ast \in M$. It was shown in \cite[Proposition 7.6]{Voi98} that $\chi^\ast(X)$ coincides then with the microstates free entropy $\chi(X)$; for the latter quantity, it was found in \cite[Proposition 4.5]{Voi94} that
\begin{equation}\label{eq:entropy-log_energy}
\chi(X) = -I(\mu_X) + \frac{3}{4} + \frac{1}{2}\log(2\pi)
\end{equation}
holds, where $I(\mu_X)$ denotes the logarithmic energy of the analytic distribution $\mu_X$ of $X$.
Recall that the \emph{logarithmic energy} of a Borel probability measure $\mu$ on $\mathbb{R}$ is defined as
\begin{equation}\label{eq:log_energy}
I(\mu) := \int_\mathbb{R} \int_\mathbb{R} \log\frac{1}{|s-t|} \, d\mu(s)\, d\mu(t).
\end{equation}
\section{H\"older continuity under the assumption of finite free Fisher information}\label{sec:Hoelder_continuity}
Throughout the following, let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space and let $X_1,\dots,X_n$ be selfadjoint noncommutative random variables living in $\mathcal{M}$ that satisfy the regularity condition $\Phi^\ast(X_1,\dots,X_n)<\infty$. The goal of this section is the proof of Theorem \ref{thm:Hoelder_continuity}. In doing so, we will follow ideas of \cite{CS16}, but with simplified arguments similar to \cite{MSY18}. In fact, Theorem \ref{thm:Hoelder_continuity}, in the case $d=1$ of an affine linear polynomial, overlaps with the corresponding result of \cite{MSY18}, if applied to the scalar-valued case; both of them yield the same exponent $\beta=\frac{2}{3}$.
The proof of Theorem \ref{thm:Hoelder_continuity} will be given below, in Subsection \ref{subsec:Hoelder_continuity_proof}. This builds on several previous results, which we collect in Subsection \ref{subsec:Hoelder_continuity_ingredients}.
\subsection{Ingredients for the proof of Theorem \ref{thm:Hoelder_continuity}}\label{subsec:Hoelder_continuity_ingredients}
In this subsection, we lay the groundwork for the proof of Theorem \ref{thm:Hoelder_continuity} in Subsection \ref{subsec:Hoelder_continuity_proof}. We will remind the reader of some facts from free analysis. Most of the material presented here is well-known, but some of these results are slightly modified or extended in order to meet our needs.
\subsubsection{H\"older continuity via spectral projections}
The easy but crucial observation that underlies our approach is the following lemma which is \cite[Lemma 8.3]{MSY18} and which was inspired by \cite{CS16}.
\begin{lemma}\label{lem:Hoelder_criterion}
Let $Y$ be a selfadjoint noncommutative random variable in $(\mathcal{M},\tau)$. If there exist $c>0$ and $\alpha>1$ such that
$$c \|(Y-s)p\|_2 \geq \|p\|_2^\alpha$$
holds for all $s\in\mathbb{R}$ and each spectral projection $p$ of $Y$, then the cumulative distribution function $\mathcal{F}_Y$ of the analytic distribution $\mu_Y$ of $Y$ is H\"older continuous with exponent $\beta := \frac{2}{\alpha-1}$; more precisely, we have that
$$|\mathcal{F}_Y(t)-\mathcal{F}_Y(s)| \leq c^\beta |t-s|^\beta \qquad\text{for all $s,t\in\mathbb{R}$}.$$
\end{lemma}
For a detailed proof we refer the interested reader to \cite{MSY18}.
\subsubsection{$L^2$-comparison of left- and right restrictions}
Another ingredient is a nice argument taken from \cite{CS16}; a streamlined version thereof is recorded in the following lemma. Because this is not stated explicitly in \cite{CS16} and since our situation is moreover slightly different, we provide here also the short proof of that statement.
\begin{lemma}\label{lem:comparison}
Let $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be a noncommutative polynomial of degree $d\geq1$. Then, for every non-zero projection $p$ in $\mathcal{M}$, there exists a non-zero projection $q$ in $\mathcal{M}$ such that
$$\tau(q) = \tau(p) \qquad\text{and}\qquad \| P(X_1,\dots,X_n)^\ast q \|_2 = \| P(X_1,\dots,X_n) p\|_2.$$
\end{lemma}
\begin{proof}
Put $Y:=P(X_1,\dots,X_n)$ and consider its polar decomposition $Y = u |Y|$ with a partial isometry $u\in\mathcal{M}$. As $P$ has degree $d\geq1$ and hence is non-constant, we conclude with the results that were obtained in \cite{CS16,MSW17} that $Y$ has no kernel, which finally yields that $u$ is in fact a unitary.
We define $q := upu^\ast$, which is clearly a non-zero projection in $\mathcal{M}$ satisfying $\tau(q) = \tau(p)$. Furthermore, we may check that
$$\|Y^\ast q\|_2 = \| |Y| u^\ast q \|_2 = \| |Y| p u^\ast \|_2 = \| u |Y| p\|_2 = \|Y p\|_2,$$
which concludes the proof.
\end{proof}
We note that the proof given above actually verifies the claim of Lemma \ref{lem:comparison} under the much weaker assumption $\delta^\star(X_1,\dots,X_n)=n$ where $\delta^\star$ is a variant of the non-microstates free entropy dimension defined in \cite[Section 4.1.1]{CS05}; this fact, however, is not needed in the following.
\subsubsection{A quantitative reduction argument}
Next, we recall \cite[Proposition 3.7]{MSW17}. It is this result which allows us to weaken the assumptions that in \cite{CS16} were imposed on $X_1,\dots,X_n$ to finiteness of free Fisher information.
In the sequel, we denote by $(\xi_1,\dots,\xi_n)$ the conjugate system for $X=(X_1,\dots,X_n)$. Furthermore, $\mathcal{M}_0$ will stand for the von Neumann subalgebra of $\mathcal{M}$ that is generated by $X_1,\dots,X_n$, i.e., $\mathcal{M}_0 := \operatorname{vN}(X_1,\dots,X_n)$.
Moreover, let us introduce on the subalgebra $\mathbb{C}\langle X_1,\dots,X_n\rangle \otimes \mathbb{C}\langle X_1,\dots,X_n\rangle$ of the algebraic tensor product $\mathcal{M}_0 \otimes \mathcal{M}_0$ the projective norm $\|\cdot\|_\pi$ by
$$\|w\|_\pi = \inf\bigg\{\sum^m_{k=1} \|w_{1,k}\| \|w_{2,k}\| \mathrel{\bigg|} w = \sum^m_{k=1} w_{1,k} \otimes w_{2,k},\ w_{1,k},w_{2,k} \in \mathbb{C}\langle X_1,\dots,X_n\rangle\bigg\}$$
for every $w\in \mathbb{C}\langle X_1,\dots,X_n\rangle \otimes \mathbb{C}\langle X_1,\dots,X_n\rangle$.
\begin{proposition}\label{prop:Fisher-bound}
For all noncommutative polynomials $P\in\mathbb{C}\langle x_1,\dots,x_n \rangle$ (not necessarily selfadjoint) and for all $u,v\in \mathcal{M}_0$, we have that
\begin{equation}\label{eq:Fisher-bound}
|\langle v^\ast (\partial_i P)(X) u, Q(X)\rangle| \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|Q(X)\|_\pi
\end{equation}
for all noncommutative bi-polynomials $Q\in \mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ and $i=1,\dots,n$.
\end{proposition}
The proof of Proposition \ref{prop:Fisher-bound} can be found in \cite{MSW17}; a matrix-valued variant thereof was proven in \cite{MSY18}. In either case, the proof makes heavily use of results from \cite{Voi98} and \cite{Dab10}. An alternative approach building on \cite{CS05} was presented in \cite{CS16}. An extension to the case of more general derivations, with an eye towards free stochastic calculus, is provided in \cite{Mai15}.
We show next an important consequence of Proposition \ref{prop:Fisher-bound}, which will be used in the sequel. For that purpose, let us introduce
\begin{itemize}
\item for every $v\in\mathcal{M}$ the linear functional $$\phi_v:\ \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C},\quad P \mapsto \tau(v^\ast P(X)),$$
\item and for every $v\in\mathcal{M}$ and $i=1,\dots,n$ the linear map $$\Delta_{v,i}:\ \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C}\langle x_1,\dots,x_n\rangle,\quad P \mapsto (\phi_v \otimes \operatorname{id})(\partial_i P).$$
\end{itemize}
Note that both $\phi_v$ and $\Delta_{v,i}$ depend implicitly on $X$, but in order to keep the notation as simple as possible, we prefer not to indicate that dependency as $X$ is fixed throughout our discussion.
\begin{corollary}\label{cor:Fisher-bound_reduced}
For all noncommutative polynomials $P\in\mathbb{C}\langle x_1,\dots,x_n \rangle$ (not necessarily selfadjoint) and for all $u,v\in\mathcal{M}_0$, we have for $i=1,\dots,n$ that
\begin{equation}\label{eq:Fisher-bound_reduced}
\|(\Delta_{v,i} P)(X) u\|_2^2 \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|u\| \|v\| \|(\partial_i P)(X)\|_\pi,
\end{equation}
and
\begin{equation}\label{eq:Fisher-bound_reduced_trace}
|\tau((\Delta_{v,i} P)(X) u)| \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr).
\end{equation}
\end{corollary}
\begin{proof}
Take any noncommutative polynomial $Q_2 \in \mathbb{C}\langle x_1,\dots,x_n\rangle$. We apply Proposition \ref{prop:Fisher-bound} to the noncommutative bi-polynomial $Q := 1 \otimes Q_2$ and so we may derive from \eqref{eq:Fisher-bound} that
$$|\langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, Q_2(X)\rangle| \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|Q_2(X)\|,$$
or in other words, that
$$|\langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, w\rangle| \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|w\|$$
for every $w\in \mathbb{C}\langle X_1,\dots,X_n\rangle$. Now, by Kaplansky's density theorem, as $\mathbb{C}\langle X_1,\dots,X_n\rangle$ is strongly dense in $\mathcal{M}_0$, the latter inequality extends to all $w\in\mathcal{M}_0$. Indeed, if $w\in\mathcal{M}_0$ is given, we find a net $(w_\lambda)_{\lambda\in\Lambda}$ in $\mathbb{C}\langle X_1,\dots,X_n\rangle$ that converges strongly to $w$ and satisfies $\|w_\lambda\| \leq \|w\|$ for all $\lambda\in\Lambda$; thus
\begin{align*}
|\langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, w_\lambda\rangle|
&\leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|w_\lambda\|\\
&\leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|w\|
\end{align*}
and since
$$\lim_{\lambda\in\Lambda} \langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, w_\lambda \rangle = \langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, w\rangle,$$
we may conclude that
$$|\langle (\tau \otimes \operatorname{id})(v^\ast (\partial_i P)(X)) u, w\rangle| \leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|w\|.$$
Fix now any $i=1,\dots,n$; we apply the latter inequality to
$$w := (\tau\otimes\operatorname{id})\big(v^\ast (\partial_i P)(X)\big) u = (\Delta_{v,i} P)(X) u,$$
which moreover satisfies that
\begin{align*}
\lefteqn{\|(\tau\otimes\operatorname{id})(v^\ast (\partial_i P)(X)) u\|}\\
&\qquad \leq \|(\tau\otimes\operatorname{id})(v^\ast (\partial_i P)(X))\| \|u\| \leq \|v^\ast (\partial_i P)(X)\| \|u\| \leq \|v^\ast\| \|(\partial_i P)(X)\|_\pi \|u\|,
\end{align*}
and hence we obtain that
\begin{align*}
\|(\Delta_{v,i} P)(X) u\|_2^2
&\leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|(\tau\otimes\operatorname{id})(v^\ast (\partial_i P)(X)) u\|\\
&\leq 4 \|\xi_i\|_2 \bigl(\|P(X)u\|_2 \|v\| + \|u\| \|P(X)^\ast v\|_2\bigr) \|u\| \|v\| \|(\partial_i P)(X)\|_\pi,
\end{align*}
which is the inequality asserted in \eqref{eq:Fisher-bound_reduced}.
The second inequality \eqref{eq:Fisher-bound_reduced_trace} follows directly from the inequality \eqref{eq:Fisher-bound} given in Proposition \ref{prop:Fisher-bound} if the latter is applied to the noncommutative bi-polynomial $Q := 1 \otimes 1$.
\end{proof}
\subsubsection{A Bernstein type inequality for noncommutative derivatives}
Finally, we address the question of how to control the projective norm of evaluations $(\partial_i P)(X)$ in terms of the noncommutative polynomial $P$; in other words, we are asking for an analogue of Bernstein's inequality for noncommutative polynomials.
On $\mathbb{C}\langle x_1,\dots,x_n\rangle$, we may define, for any fixed $R>0$, a norm $\|\cdot\|_R$ by putting
$$\|P\|_R := \sum^d_{k=0} \sum_{1\leq i_1,\dots,i_k\leq n} |a_{i_1,\dots,i_k}| R^k$$
for each $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ that is written in the form \eqref{eq:ncpoly}. Correspondingly, on the space $\mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ of all noncommutative bi-polynomials, we may introduce the associated projective norm $\|\cdot\|_{R,\pi}$ for every $Q\in \mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$ by
$$\|Q\|_{R,\pi} = \inf\bigg\{\sum^m_{k=1} \|Q_{1,k}\|_R \|Q_{2,k}\|_R \mathrel{\bigg|} Q = \sum^m_{k=1} Q_{1,k} \otimes Q_{2,k},\ Q_{1,k},Q_{2,k} \in \mathbb{C}\langle x_1,\dots,x_n\rangle\bigg\}.$$
\begin{lemma}\label{lem:projective_norm_bound_1}
Let $q\in\mathcal{M}_0$ be any projection and suppose that $R>0$ is chosen such that
\begin{equation}\label{eq:radius_constraint}
R \geq \max_{i=1,\dots,n} \|X_i\|.
\end{equation}
Then the following holds true:
\begin{itemize}
\item[(i)] The associated linear functional $\phi_q$ is positive and satisfies for all $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$
$$|\phi_q(P)| \leq \tau(q) \|P\|_R.$$
\item[(ii)] For each noncommutative bi-polynomial $Q\in\mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$, we have
$$\|(\phi_q \otimes \operatorname{id})(Q)\|_R \leq \tau(q) \|Q\|_{R,\pi}.$$
\item[(iii)] For every $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ and $i=1,\dots,n$, we have that
$$\|\Delta_{q,i} P\|_R \leq \tau(q) \|\partial_i P\|_{R,\pi}.$$
\end{itemize}
\end{lemma}
\begin{proof}
(i) Let $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be any noncommutative polynomial. Suppose that $P$ is written in the form \eqref{eq:ncpoly}, so that
$$\phi_q(P) = \sum^d_{k=0} \sum_{1\leq i_1,\dots,i_k\leq n} a_{i_1,\dots,i_k} \tau(q X_{i_1} \cdots X_{i_k}).$$
By H\"older's inequality we see that $|\tau(q X_{i_1} \cdots X_{i_k})| \leq \|q\|_1 \|X_{i_1} \cdots X_{i_k}\| \leq \|q\|_1 R^k$, where $\|q\|_1 = \tau(q)$ as $q$ is a projection. Thus, in summary, we obtain as claimed that
$$|\phi_q(P)| \leq \tau(q) \bigg(\sum^d_{k=0} \sum_{1\leq i_1,\dots,i_k\leq n} |a_{i_1,\dots,i_k}| R^k\bigg) = \tau(q) \|P\|_R.$$
(ii) Take any noncommutative bi-polynomial $Q$ and write $Q = \sum^m_{k=1} Q_{1,k} \otimes Q_{2,k}$. Then
$$\|(\phi_q \otimes \operatorname{id})(Q)\|_R \leq \sum^m_{k=1} |\phi_q(Q_{1,k})| \|Q_{2,k}\|_R \leq \tau(q) \sum^m_{k=1} \|Q_{1,k}\|_R \|Q_{2,k}\|_R,$$
and by passing to the infimum over all possible representations of $Q$, we finally arrive at the assertion.
(iii) Since $\Delta_{q,i} P = (\phi_q \otimes \operatorname{id})(\partial_i P)$, applying (ii) to $Q=\partial_i P$ directly yields the claim.
\end{proof}
It is easily seen that it holds under the assumption \eqref{eq:radius_constraint}
$$\|P(X)\| \leq \|P\|_R \qquad\text{for all $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$}$$
and accordingly
$$\|Q(X)\|_\pi \leq \|Q\|_{R,\pi} \qquad\text{for all $Q\in \mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle$}.$$
Thus, we see that in order to control $\|(\partial_i P)(X)\|_\pi$, it suffices to provide bounds for $\|\partial_i P\|_{R,\pi}$.
For that purpose, we have to restrict attention to subspaces of $\mathbb{C}\langle x_1,\dots,x_n\rangle$ consisting of all noncommutative polynomials with degree below a given threshold; more precisely, for every $d\geq 0$, we work with the subspace of $\mathbb{C}\langle x_1,\dots,x_n\rangle$ that is given by
$$\P_d := \Big\{P\in \mathbb{C}\langle x_1,\dots,x_n\rangle \mathrel{\Big|} \deg(P) \leq d\Big\}.$$
Now, we may formulate the desired estimate, which is a variant of a result that can be found in \cite[Section 4]{Voi98}.
\begin{lemma}\label{lem:projective_norm_bound_2}
Take any $R>0$ that satisfies \eqref{eq:radius_constraint}. Then, for each $P\in\P_d$ and $i=1,\dots,n$, it holds true that
$$\|\partial_i P\|_{R,\pi} \leq \frac{d}{R} \|P\|_R.$$
\end{lemma}
\begin{proof}
Take any $P\in\P_d$ that is written in the form \eqref{eq:ncpoly}. Then, for $i=1,\dots,n$, we have by definition of the noncommutative derivatives that
$$\partial_i P = \sum^d_{k=1} \sum_{1\leq i_1,\dots,i_k\leq n} \sum^k_{j=1} \delta_{i,i_j} a_{i_1,\dots,i_k} x_{i_1} \cdots x_{i_{j-1}} \otimes x_{i_{j+1}} \cdots x_{i_k}.$$
Therefore, we may conclude that
\begin{align*}
\|\partial_i P\|_{R,\pi}
&\leq \sum^d_{k=1} \sum_{1\leq i_1,\dots,i_k\leq n} \sum^k_{j=1} \delta_{i,i_j} |a_{i_1,\dots,i_k}| \|x_{i_1} \cdots x_{i_{j-1}}\|_R \|x_{i_{j+1}} \cdots x_{i_k}\|_R\\
&\leq \sum^d_{k=1} \sum_{1\leq i_1,\dots,i_k\leq n} k |a_{i_1,\dots,i_k}| R^{k-1}\\
&= \frac{d}{R} \|P\|_R,
\end{align*}
which is the asserted inequality.
\end{proof}
\begin{corollary}\label{cor:projective_norm_bound}
For a $k\in\mathbb{N}$, let $q_1,\dots,q_{k-1}\in\mathcal{M}_0$ be arbitrary projections and let $1\leq i_1,\dots,i_k \leq n$ be any collection of indices. Moreover, suppose that $R>0$ is given which satisfies the condition \eqref{eq:radius_constraint}. Then
$$\|\partial_{i_k} \Delta_{q_{k-1},i_{k-1}} \cdots \Delta_{q_1,i_1} P\|_{R,\pi} \leq \frac{d!}{(d-k)!} \frac{\tau(q_1) \cdots \tau(q_{k-1})}{R^k} \|P\|_R$$
holds for every noncommutative polynomial $P\in\P_d$.
\end{corollary}
\begin{proof}
We proceed by mathematical induction on $k$. In the case $k=1$, the asserted estimate reduces to $\|\partial_i P\|_{R,\pi} \leq \frac{d}{R} \|P\|_R$, which was shown in Lemma \ref{lem:projective_norm_bound_2}. Suppose now that the assertion is already proven for some $k\in\mathbb{N}$; then, because $\Delta_{q_1,i_1} P$ must belong to $\P_{d-1}$, we may conclude that
$$\|\partial_{i_{k+1}} \Delta_{q_k,i_k} \cdots \Delta_{q_2,i_2} (\Delta_{q_1,i_1} P)\|_{R,\pi} \leq \frac{(d-1)!}{((d-1)-k)!} \frac{\tau(q_2) \cdots \tau(q_k)}{R^k} \|\Delta_{q_1,i_1} P\|_R.$$
Now, with the help of Lemma \ref{lem:projective_norm_bound_1} Item (iii), we get that $\|\Delta_{q_1,i_1} P\|_R \leq \tau(q_1) \|\partial_{i_1} P\|_{R,\pi}$, and using Lemma \ref{lem:projective_norm_bound_2}, we see that $\|\partial_{i_1} P\|_{R,\pi} \leq \frac{d}{R} \|P\|_R$; this together yields that $\|\Delta_{q_1,i_1} P\|_R \leq d \frac{\tau(q_1)}{R}\|P\|_R$. Hence, in summary, we obtain
$$\|\partial_{i_{k+1}} \Delta_{q_k,i_k} \cdots \Delta_{q_2,i_2} (\Delta_{q_1,i_1} P)\|_{R,\pi} \leq \frac{d!}{(d-(k+1))!} \frac{\tau(q_1) \tau(q_2) \cdots \tau(q_k)}{R^{k+1}} \|P\|_R,$$
which verifies the assertion in the case $k+1$.
\end{proof}
\subsection{Proof of Theorem \ref{thm:Hoelder_continuity}}\label{subsec:Hoelder_continuity_proof}
Let us fix any selfadjoint noncommutative polynomial $P\in \mathbb{C}\langle x_1,\dots,x_n\rangle$ that has degree $d := \deg(P) \geq 1$. Accordingly, $P$ belongs to the space $\P_d$; we suppose that $P$ is written in the form \eqref{eq:ncpoly}. Let us fix some leading coefficient $a_{i_1,\dots,i_d}$ of $P$ that is non-zero. Further, we choose $R>0$ such that $R \geq \max_{i=1,\dots,n} \|X_i\|$.
The H\"older continuity of $\mu_Y$ for the noncommutative random variable $Y=P(X)$ will follow from Lemma \ref{lem:Hoelder_criterion}; for that purpose, we are going to prove that there are $\alpha>1$ and $c>0$ such that $Y$ satisfies
\begin{equation}\label{eq:Hoelder_continuity_proof}
c \|(Y-s) p\|_2 \geq \|p\|_2^\alpha
\end{equation}
for every $s\in\mathbb{R}$ and every projection $p\in\mathcal{M}_0$; note that it clearly suffices to consider the case $p\neq 0$.
Correspondingly, let us take now any $s\in\mathbb{R}$ and any non-zero projection $p\in\mathcal{M}_0$. Put $P_0 := P-s$; note that $\deg(P_0) = \deg(P) \geq 1$. We construct then recursively, for every $1\leq k\leq d$, a non-zero projection $q_k\in\mathcal{M}_0$ and a non-constant noncommutative polynomial $P_k\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ according to the following rules:
\begin{enumerate}
\item With the help of Lemma \ref{lem:comparison}, applied to the non-constant polynomial $P_{k-1}$, we construct the projection $q_k\in\mathcal{M}_0$ so that $$\tau(q_k) = \tau(p) \qquad\text{and}\qquad \|P_{k-1}(X)^\ast q_k\|_2 = \|P_{k-1}(X) p\|_2.$$
\item Subsequently, we put $P_k := \Delta_{q_k,i_k} P_{k-1} = \Delta_{q_k,i_k} \dots \Delta_{q_1,i_1} P$, which is again a non-constant polynomial.
In fact, we have $\deg(P_k) = d-k$, since necessarily $\deg(P_k) \leq d-k$, due to the iterative application of noncommutative derivatives, and since the monomial $x_{i_{k+1}} \dots x_{i_d}$ of degree $d-k$ shows up in $P_k$ with the non-zero coefficient $\tau(p)^k a_{i_1,\dots,i_d}$.
\end{enumerate}
Involving now the inequality \eqref{eq:Fisher-bound_reduced} provided in Corollary \ref{cor:Fisher-bound_reduced}, we infer that for $k=1,\dots,d$
$$\|P_k(X) p\|_2^2 \leq 8 \|\xi_{i_k}\|_2 \|(\partial_{i_k} P_{k-1})(X)\|_\pi \|P_{k-1}(X) p\|_2.$$
We estimate $\|\xi_{i_k}\|_2 \leq \Phi^\ast(X)^{1/2}$; moreover, by using Lemma \ref{lem:projective_norm_bound_2} and Corollary \ref{cor:projective_norm_bound}, respectively, we get that
$$\|(\partial_{i_k} P_{k-1})(X)\|_\pi \leq \|\partial_{i_k} P_{k-1}\|_{R,\pi} \leq \frac{d!}{(d-k)!} \frac{\tau(q_1) \cdots \tau(q_{k-1})}{R^k} \|P\|_R .$$
This, in summary, yields that for every $k=1,\dots,d-1$
$$\|P_k(X) p\|_2^2 \leq c_k \|P_{k-1}(X) p\|_2 \qquad\text{with}\qquad c_k := 8 \Phi^\ast(X)^{1/2} \frac{d!}{(d-k)!} \frac{\tau(p)^{k-1}}{R^k} \|P\|_R.$$
Moreover, by the second inequality \eqref{eq:Fisher-bound_reduced_trace} of Corollary \ref{cor:Fisher-bound_reduced}
$$|\tau(P_d(X) p)| \leq c_d \|P_{d-1}(X) p\|_2 \qquad\text{with}\qquad c_d := 8 \Phi^\ast(X)^{1/2}.$$
By iterating the latter inequalities, we obtain that
$$|\tau(P_d(X) p)|^{2^{d-1}} \leq \bigg(\prod^{d}_{k=1} c_k^{2^{k-1}}\bigg) \|P_0(X) p\|_2.$$
By using $\tau(p) = \|p\|_2^2$ and use the formulas
$$\sum^d_{k=1} 2^{k-1} = 2^d-1,\quad \sum^{d-1}_{k=1}(k-1)2^k = (d-3)2^d + 4, \quad\text{and}\quad \sum^{d-1}_{k=1} k2^{k-1} = (d-2)2^{d-1}+1,$$
then the involved product simplifies to
$$\prod^{d}_{k=1} c_k^{2^{k-1}}= \bigg(8 \Phi^\ast(X)^{1/2}\bigg)^{2^d-1} \|P\|_R^{2^{d-1}-1} \frac{\|p\|_2^{(d-3)2^d+4}}{R^{(d-2)2^{d-1}+1}}\prod^{d-1}_{k=1} \Big(\frac{d!}{(d-k)!}\Big)^{2^{k-1}}$$
Note that $P_0 = P-s$ and $P_d = \Delta_{q_d,i_d} \dots \Delta_{q_1,i_1} P = \tau(p)^d a_{i_1,\dots,i_d}$ as $P$ has degree $d$; thus
$$\|P_0(X) p\|_2 = \|(Y-s)p\|_2 \qquad\text{and}\qquad|\tau(P_d(X) p)|^{2^{d-1}} = |a_{i_1,\dots,i_d}|^{2^{d-1}} \|p\|_2^{(d+1)2^d}.$$
we conclude now that \eqref{eq:Hoelder_continuity_proof} holds with $\alpha = (d+1)2^d - (d-3)2^d - 4 = 2^{d+2}-4$ and
\begin{equation}\label{eq:Hoelder_constant-1}
c = \frac{1}{|a_{i_1,\dots,i_d}|^{2^{d-1}}} \big(8 \Phi^\ast(X)^{1/2}\big)^{2^d-1} \|P\|_R^{2^{d-1}-1} \frac{1}{R^{(d-2)2^{d-1}+1}} \prod^{d-1}_{k=1} \Big(\frac{d!}{(d-k)!}\Big)^{2^{k-1}}.
\end{equation}
Now, using Lemma \ref{lem:Hoelder_criterion}, we see that $\mathcal{F}_Y$ is H\"older continuous with exponent $\beta = \frac{2}{\alpha-1} = \frac{2}{2^{d+2}-5}$ and the associated constant $C= c^\beta$; this concludes the proof of Theorem \ref{thm:Hoelder_continuity}.
\subsection{More about the H\"older constant}
We take now a closer look at the constant $c$ given in \eqref{eq:Hoelder_constant-1}. Besides $\|P\|_R$, we can extract from there another quantity that solely depends on $R$ and the algebraic structure of $P$.
More precisely, for any given $R>0$, we define for every noncommutative polynomial $0\neq P\in \mathbb{C}\langle x_1,\dots,x_n\rangle$ its \emph{leading weight} $\rho_R(P) \in (0,1]$ by
$$\rho_R(P) := \max_{1\leq i_1,\dots,i_d \leq n} \frac{|a_{i_1,\dots,i_d}| R^d}{\|P\|_R}, \qquad\text{where $d:=\deg(P)$}.$$
Using this quantity, we can rearrange the terms appearing in \eqref{eq:Hoelder_constant-1} as
$$c = \rho_R(P)^{-2^{d-1}} \big(8 R \Phi^\ast(X)^{1/2}\big)^{2^d-1} \frac{1}{\|P\|_R} \prod^{d-1}_{k=1} \Big(\frac{d!}{(d-k)!}\Big)^{2^{k-1}}.$$
Since the explicit value for the H\"older constant $C>0$ of $\mathcal{F}_Y$ that we found in the proof of Theorem \ref{thm:Hoelder_continuity} is $C = c^\beta$ with $\beta = \frac{2}{2^{d+2}-5}$, we infer from the latter that
\begin{equation}\label{eq:Hoelder_constant-2}
C = C_d \rho_R(P)^{-\frac{2^d}{2^{d+2}-5}} \big(8 R \Phi^\ast(X)^{1/2}\big)^{\frac{2(2^d-1)}{2^{d+2}-5}} \|P\|_R^{-\frac{2}{2^{d+2}-5}},
\end{equation}
where $C_d$ is a numerical quantity depending only on $d$ which is given by
\begin{equation}\label{eq:constant}
C_d := \bigg(\prod^{d-1}_{k=1} \Big(\frac{d!}{(d-k)!}\Big)^{2^{k-1}}\bigg)^{\frac{2}{2^{d+2}-5}}.
\end{equation}
It is natural to ask for the order by which $C_d$ grows with $d$; this question is addressed in the next lemma.
\begin{lemma}\label{lem:constant_bound}
For every $d\in\mathbb{N}$, the constant $C_d$ from \eqref{eq:constant} satisfies $(d!)^{1/8} \leq C_d \leq (d!)^{1/4}$.
\end{lemma}
\begin{proof}
Since $C_1=1$, the assertion is trivially true in the case $d=1$. Thus, assume from now on that $d\geq 2$. It is straightforward then to check that
$$\log(C_d) = \frac{1}{2^{d+2}-5} \sum^{d-1}_{k=1} \sum_{l=d-k}^{d-1} 2^k \log(l+1) = \frac{2^d}{2^{d+2}-5} \Bigg(\log(d!) - \sum^{d-1}_{l=1} 2^{-l} \log(l+1)\Bigg).$$
From the latter, we easily deduce that
$$\log(C_d) \leq \frac{2(2^{d-1}-1)}{2^{d+2}-5} \log(d!) \leq \frac{1}{4} \log(d!) \quad\text{and}\quad \log(C_d) \geq \frac{2^{d-1}}{2^{d+2}-5} \log(d!) \geq \frac{1}{8} \log(d!),$$
which proves the assertion.
\end{proof}
\begin{remark}
Depending on the situation, it might be useful to have a simplified upper bound for the H\"older constant $C>0$ given in \eqref{eq:Hoelder_constant-2}.
For that purpose, we note that $\Phi^\ast(X)\tau(X_1^2 + \dots + X_n^2) \geq n^2$ according to the free Cramer-Rao inequality \cite[Proposition 6.9]{Voi98}, from which we infer that $R \Phi^\ast(X)^{1/2} \geq \sqrt{n}$ as $\tau(X_j^2) \leq R^2$ for $j=1,\dots,n$.
Furthermore, by definition of the leading weight of $P$, we have that $\rho_R(P)^{-1} \geq 1$.
Thus, since $\frac{2^d}{2^{d+2}-5} \leq \frac{2}{3}$ and $\frac{2(2^d-1)}{2^{d+2}-5} \leq \frac{2}{3}$, we can conclude that
$$C \leq 4 (d!)^{1/4} \rho_R(P)^{-2/3} R^{2/3} \Phi^\ast(X)^{1/3} \|P\|_R^{-\frac{2}{2^{d+2}-5}},$$
where we have used also the upper bound of $C_d$ that was found in Lemma \ref{lem:constant_bound}.
\end{remark}
\section{H\"older continuity and finite free entropy}
This section is devoted to the proof of Theorem \ref{thm:finite_entropy}. In fact, we will prove the following theorem, which provides an explicit upper bound for the logarithmic energy (as defined in \eqref{eq:log_energy}) of the analytic distribution of the considered polynomial evaluation. Thanks to \eqref{eq:entropy-log_energy}, the latter results directly in a lower bound for both the microstates and the non-microstates free entropy; this, in particular, verifies the assertion of Theorem \ref{thm:finite_entropy}.
\begin{theorem}\label{thm:finite_entropy_bound}
Let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space and let $X_1,\dots,X_n$ be selfadjoint noncommutative random variables living in $\mathcal{M}$ satisfying
$$\Phi^\ast(X_1,\dots,X_n)<\infty.$$
Furthermore, let $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be any selfadjoint noncommutative polynomial of degree $d\geq 1$ and consider the associated selfadjoint noncommutative random variable $$Y:=P(X_1,\dots,X_n)$$ in $\mathcal{M}$. Then the analytic distribution $\mu_Y$ of $Y$ has finite logarithmic energy $I(\mu_Y)$ that can be bounded from above by
\begin{equation}\label{eq:log_energy_bound}
I(\mu_Y) \leq C_d (2^{d+2} -5) \rho_R(P)^{-\frac{2^d}{2^{d+2}-5}} \big(8 R \Phi^\ast(X)^{1/2}\big)^{\frac{2(2^d-1)}{2^{d+2}-5}} \|P\|_R^{-\frac{2}{2^{d+2}-5}}.
\end{equation}
where $C_d>0$ is the constant introduced in \eqref{eq:constant}.
\end{theorem}
Using \cite{Jam15}, Theorem \ref{thm:finite_entropy_bound} follows rather immediately from Theorem \ref{thm:Hoelder_continuity}. To be more precise, it was shown in \cite{Jam15} that for every Borel probability measure $\mu$ on $\mathbb{R}$ that has a cumulative distribution function $\mathcal{F}_\mu$ which satisfies
\begin{equation}\label{eq:Jam_condition}
|\mathcal{F}_\mu(t)-\mathcal{F}_\mu(s)| \leq C |t-s|^\beta \qquad\text{for all $s,t\in\mathbb{R}$}
\end{equation}
with some constants $\beta>0$ and $C>0$, the logarithmic energy of $\mu$ can be bounded by
\begin{equation}\label{eq:Jam_bound}
I(\mu) \leq 2\frac{C}{\beta}.
\end{equation}
\begin{proof}[Proof of Theorem \ref{thm:finite_entropy_bound}]
Using Theorem \ref{thm:Hoelder_continuity}, we see that $\mathcal{F}_Y$ satisfies the \eqref{eq:Jam_condition} with the constant $C$ given by \eqref{eq:Hoelder_constant-2} and $\beta = \frac{2}{2^{d+2} -5}$. Thus, the asserted bound \eqref{eq:log_energy_bound} follows from \eqref{eq:Jam_bound}.
\end{proof}
\section{Convergence in distribution and the Kolmogorov distance}\label{sec:Kolmogorov}
Among the strongest metrics that are usually studied on the space of all Borel probability measures on the real line $\mathbb{R}$ is the so-called \emph{Kolmogorov distance}; this metric $\Delta$ is defined for any two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$ by
$$\Delta(\mu,\nu) := \sup_{t\in\mathbb{R}} |\mathcal{F}_\mu(t) - \mathcal{F}_\nu(t)|.$$
Though its definition is quite appealing, convergence with respect to the Kolmogorov distance is much more rigid than, for instance, convergence with respect to the so-called \emph{L\'evy distance}. The latter is defined by
$$L(\mu,\nu) := \inf\{\epsilon>0 \mid \forall t\in\mathbb{R}:\ \mathcal{F}_\mu(t-\epsilon) - \epsilon \leq \mathcal{F}_\nu(t) \leq \mathcal{F}_\mu(t+\epsilon) + \epsilon\}$$
and is known to provide a metrization of convergence in distribution.
It is accordingly a challenging task to control the Kolmogorov distance in concrete situations. In view of our regularity results, some known ``self-improvement'' phenomenon is worth mentioning: if convergence towards a measure with H\"older continuous cumulative distribution function is considered, then convergence in distribution automatically implies convergence in L\'evy distance; see Theorem \ref{thm:convergence_in_distribution} below.
The drawback of this approach, however, is that it does not give rates of convergence for the Kolmogorov distance if the convergence is measured only in terms of the associated Cauchy-Stieltjes transforms. Based on estimates derived in \cite{Bai93a,Bai93b} (see also \cite{BS10}), we provide here with Theorem \ref{thm:Hoelder_criterion} a criterion that gives explicitly such rates in general situations.
\subsection{Convergence in Kolmogorov distance}
Let us denote by $\mathbb{C}^\pm$ the complex upper respectively lower half-plane, i.e., $\mathbb{C}^\pm := \{z\in\mathbb{C} \mid \pm\Im(z)>0\}$. To each Borel probability measure $\mu$ on the real line $\mathbb{R}$, we may associate its \emph{Cauchy transform}, i.e., the holomorphic function $G_\mu: \mathbb{C}^+ \to \mathbb{C}^-$ that is given by
$$G_\mu(z) := \int_\mathbb{R} \frac{1}{z-t}\, d\mu(t) \qquad\text{for all $z\in\mathbb{C}^+$}.$$
Let us first recall the following well-known facts that are well surveyed in \cite{GH03}.
\begin{theorem}\label{thm:convergence_in_distribution}
Let $(\mu_n)_{n=1}^\infty$ be a sequence of Borel probability measures on $\mathbb{R}$ and let $\nu$ be another Borel probability measure on $\mathbb{R}$. Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $(\mu_n)_{n=1}^\infty$ converges in distribution to $\nu$.
\item[(ii)] We have that $(G_{\mu_n})_{n=1}^\infty$ converges uniformly on compact subsets of $\mathbb{C}^+$ to $G_\nu$.
\item[(iii)] There is an infinite subset $K \subseteq \mathbb{C}^+$ with an accumulation point in the complex upper half-plane $\mathbb{C}^+$ such that $G_{\mu_n}(z) \to G_\nu(z)$ as $n\to\infty$ for each $z\in K$.
\end{itemize}
If we assume in addition that the target measure $\nu$ has a cumulative distribution function $\mathcal{F}_\nu$ that is H\"older continuous with exponent $\beta>0$, i.e., there is some constant $C>0$ such that
\begin{equation}\label{eq:Hoelder_criterion}
|\mathcal{F}_\nu(t)-\mathcal{F}_\nu(s)| \leq C |t-s|^\beta \qquad\text{for all $s,t\in\mathbb{R}$},
\end{equation}
then, the above statements (i), (ii), and (iii) are equivalent also to
\begin{itemize}
\item[(iv)] We have $\Delta(\mu_n,\nu) \to 0$ as $n\to\infty$.
\end{itemize}
\end{theorem}
If we require the target measure $\nu$ to have a cumulative distribution function $\mathcal{F}_\nu$ that satisfies \eqref{eq:Hoelder_criterion}, then \cite[Lemma 12.18]{BS10} says that
$$L(\mu_n,\nu) \leq \Delta(\mu_n,\nu) \leq (C+1) L(\mu_n,\nu)^\beta,$$
from which the equivalence of (i) and (iv), since we have that $L(\mu_n,\nu) \to 0$ as $n\to\infty$ if and only if (i) holds.
Here, we will prove the following quantitative version of Theorem \ref{thm:convergence_in_distribution}. We will denote by $\mathbb{S}_\rho$ for any $0<\rho\leq \infty$ the strip $\{z\in\mathbb{C} \mid 0 < \Im(z) < \rho\}$ in $\mathbb{C}^+$; clearly $\mathbb{S}_\infty = \mathbb{C}^+$.
\begin{theorem}\label{thm:Hoelder_criterion}
Let $(\mu_n)_{n=1}^\infty$ be a sequence of Borel probability measures on $\mathbb{R}$ and let $\nu$ be any other Borel probability measure on $\mathbb{R}$. Suppose the following:
\begin{enumerate}
\item\label{it:cond-1} The cumulative distribution function $\mathcal{F}_\nu$ of the measure $\nu$ is H\"older continuous with exponent $\beta\in(0,1]$, i.e., condition \eqref{eq:Hoelder_criterion} is satisfied.
\item\label{it:cond-2} There are continuous functions $\Theta: \mathbb{S}_\rho \to [0,\infty)$ for some $0<\rho \leq \infty$ and $\Theta_0: [0,\infty) \to [0,\infty)$ that satisfy the growth conditions $$\limsup_{R\to\infty} R^{-l} \max_{r\in[0,R]} \Theta_0(r) < \infty$$ for some $l\geq 0$ and $$\Theta(z) \leq \frac{\Theta_0(|z|)}{\Im(z)^k} \qquad\text{for all $z\in\mathbb{S}_\rho$}$$ for some $k\geq 0$, and a sequence $(\epsilon_n)_{n=1}^\infty$ in $(0,\infty)$ converging to $0$ such that the estimate $$|G_{\mu_n}(z) - G_\mu(z)| \leq \Theta(z) \epsilon_n$$ holds for every $n\in\mathbb{N}$ and all $z\in \mathbb{S}_\rho$.
\item\label{it:cond-3} We have that $\sup_{n\in\mathbb{N}} \int_\mathbb{R} t^2\, d\mu_n(t) < \infty$.
\end{enumerate}
Then, $(\mu_n)_{n=1}^\infty$ converges in Kolmogorov distance to $\nu$; in fact, there is $D>0$, such that
$$\Delta(\mu_n,\nu) \leq D \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}} \qquad\text{for all $n\in\mathbb{N}$}.$$
\end{theorem}
The proof will be given in Subsection \ref{subsec:Hoelder_criterion_proof}. If one replaces \ref{it:cond-3} by the much stronger condition that all $\mu_n$ have support contained in a fixed compact interval, one can establish with similar but significantly simplified arguments a better rate for the Kolmogorov distance; see Remark \ref{rem:Hoelder_criterion_compact}. We present the precise statement in the next theorem, but the details of its straightforward proof are left to the reader.
\begin{theorem}\label{thm:Hoelder_criterion_compact}
Let $(\mu_n)_{n=1}^\infty$ be a sequence of compactly supported Borel probability measures on $\mathbb{R}$ and let $\nu$ be any other Borel probability measure on $\mathbb{R}$. Suppose the following:
\begin{enumerate}
\item The cumulative distribution function $\mathcal{F}_\nu$ of the measure $\nu$ is H\"older continuous with exponent $\beta\in(0,1]$, i.e., condition \eqref{eq:Hoelder_criterion} is satisfied.
\item There are continuous functions $\Theta: \mathbb{S}_\rho \to [0,\infty)$ and $\Theta_0: \overline{\mathbb{S}_\rho} \to [0,\infty)$ for some $0<\rho\leq\infty$ that satisfy $$\Theta(z) \leq \frac{\Theta_0(|z|)}{\Im(z)^k} \qquad\text{for all $z\in\mathbb{S}_\rho$}$$ for some $k\geq 0$, and a sequence $(\epsilon_n)_{n=1}^\infty$ in $(0,\infty)$ converging to $0$ such that the estimate $$|G_{\mu_n}(z) - G_\mu(z)| \leq \Theta(z) \epsilon_n$$ holds for every $n\in\mathbb{N}$ and all $z\in \mathbb{S}_\rho$.
\item There exists $M>0$ such that $\operatorname{supp}(\mu_n) \subseteq [-M,M]$ for all $n\in\mathbb{N}$.
\end{enumerate}
Then, $(\mu_n)_{n=1}^\infty$ converges in Kolmogorov distance to $\nu$; in fact, there is $D>0$, such that
$$\Delta(\mu_n,\nu) \leq D \epsilon_n^{\frac{\beta}{k+\beta}} \qquad\text{for all $n\in\mathbb{N}$}.$$
\end{theorem}
\subsection{Bai's inequalities}
The proof of Theorem \ref{thm:Hoelder_criterion} relies crucially on the following result, which is \cite[Theorem 2.2]{Bai93a}; see also \cite[Theorem 2.2]{Bai93b}.
\begin{theorem}\label{thm:Kolmogorov}
Let $\mu$ and $\nu$ be two Borel probability measures such that
\begin{equation}\label{eq:BS_condition-0}
\int_\mathbb{R} |\mathcal{F}_\mu(t) - \mathcal{F}_\nu(t)|\, dt < \infty.
\end{equation}
Then, for every $y>0$,
\begin{align*}
\Delta(\mu,\nu) &\leq \frac{1}{\pi(1-\kappa)(2\gamma-1)}\Bigg[\int^A_{-A} |G_\mu(x+iy)-G_\nu(x+iy)|\, dx\\
&\qquad\quad + \frac{2\pi}{y} \int_{|t|>B} |\mathcal{F}_\mu(t) - \mathcal{F}_\nu(t)|\, dt + \frac{1}{y} \sup_{t\in\mathbb{R}} \int_{|s|\leq 2ya} |\mathcal{F}_\nu(t+s)-\mathcal{F}_\nu(t)|\, ds\Bigg],
\end{align*}
where $a$ and $\gamma$ are constants related to each other by
\begin{equation}\label{eq:BS_condition-1}
\gamma = \frac{1}{\pi} \int_{|x|<a} \frac{1}{x^2+1}\, dx > \frac{1}{2}
\end{equation}
and $A$, $B$, and $\kappa$ are positive constants such that $A>B$ and
\begin{equation}\label{eq:BS_condition-2}
\kappa = \frac{4B}{\pi(A-B)(2\gamma-1)} < 1.
\end{equation}
\end{theorem}
This useful methodology to control the Kolmogorov distance in terms of the corresponding Cauchy transforms is surveyed nicely in the book \cite{BS10}.
\subsection{Bounding integrals of Cauchy transforms}
In order to apply Theorem \ref{thm:Kolmogorov}, we will have to control integrals of the form
$$\int_{|x|\geq A} |G_\mu(x+iy)-G_\nu(x+iy)|\, dx$$
as $A\to\infty$, uniformly over a large class of measures. Providing such bounds is the purpose of this subsection.
For a Borel probability measure $\mu$ on $\mathbb{R}$ having finite first and second moments, we denote by
$$m(\mu) := \int_\mathbb{R} t\, d\mu(t) \qquad\text{and}\qquad \sigma^2(\mu) := \int_\mathbb{R} (t-m(\mu))^2\, d\mu(t)$$
its \emph{mean} and \emph{variance}, respectively. Furthermore, in preparation of the next lemma, we define another quantity that is associated to $\mu$ and any real number $y>0$ by
$$W_y(\mu) := \bigg(1 + \frac{1}{2y} \int_\mathbb{R} |t|\, d\mu(t) + \frac{1}{2y^2} \int_\mathbb{R} t^2\, d\mu(t)\bigg)^{1/2}.$$
Moreover, if two such measures $\mu$ and $\nu$ are given, we put
$$c(\mu,\nu) := \Big(\sigma^2(\mu) + \sigma^2(\nu) + (m(\mu)-m(\nu))^2\Big)^{1/2}.$$
Using that notation, we are ready to formulate with the next lemma the desired integral bounds.
\begin{lemma}\label{lem:integral_bound}
Let $\mu$ and $\nu$ be any two Borel probability measures on $\mathbb{R}$ having finite first and second moments. Then, for each $y>0$ and for all $A>0$, it holds true that
\begin{equation}\label{eq:integral_bound-2}
\int_{|x|\geq A} |G_\mu(x+iy)-G_\nu(x+iy)|\, dx \leq c(\mu,\nu) W_y(\mu) W_y(\nu) \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx.
\end{equation}
\end{lemma}
\begin{proof}
Let us first take any $z\in\mathbb{C}^+$. We may write
\[
G_\mu(z) - G_\nu(z) = \int_\mathbb{R} \int_\mathbb{R} \frac{t-s}{(z-t)(z-s)}\, d\mu(t)\, d\nu(s),
\]
which yields after an application of the Cauchy Schwarz inequality
\begin{align}\label{eq:integral_bound-3}
|G_\mu(z) - G_\nu(z)| &\leq \bigg(\int_\mathbb{R} \int_\mathbb{R} (t-s)^2\, d\mu(t)\, d\nu(s)\bigg)^{1/2} \bigg(\int_\mathbb{R} \int_\mathbb{R} \frac{1}{|z-t|^2|z-s|^2}\, d\mu(t)\, d\nu(s)\bigg)^{1/2} \nonumber
\\&= c(\mu,\nu) \bigg(\int_\mathbb{R} \frac{1}{|z-t|^2}\, d\mu(t)\bigg)^{1/2} \bigg(\int_\mathbb{R} \frac{1}{|z-s|^2}\, d\nu(s)\bigg)^{1/2}
\end{align}
Now, let us fix any $y>0$. In order to establish \eqref{eq:integral_bound-2}, we use \eqref{eq:integral_bound-3} and again the Cauchy Schwarz inequality; this gives for every $A>0$
\begin{multline}\label{eq:integral_bound-4}
\lefteqn{\int_{|x|\geq A} |G_\mu(x+iy) - G_\nu(x+iy)|\, dx}\\
\leq c(\mu,\nu) \bigg[\int_{|x|\geq A} \int_\mathbb{R} \frac{1}{(x-t)^2+y^2}\, d\mu(t)\, dx\bigg]^{1/2} \bigg[\int_{|x|\geq A} \int_\mathbb{R} \frac{1}{(x-s)^2+y^2}\, d\nu(s)\, dx\bigg]^{1/2}.
\end{multline}
Then, using Fubini's theorem and in turn a substitution, we may compute that
\begin{equation}\label{eq:integral_bound-5}
\begin{aligned}
\int_{|x|\geq A} \int_\mathbb{R} \frac{1}{(x-t)^2+y^2}\, d\mu(t)\, dx
&= \int_\mathbb{R} \int_{|x+t|\geq A} \frac{1}{x^2+y^2}\, dx\, d\mu(t).
\end{aligned}
\end{equation}
We want to control the integrand $\int_{|x+t|\geq A} \frac{1}{x^2+y^2}\, dx$ for every fixed $t\in\mathbb{R}$. We consider the case $t\geq0$ first. To begin with, we observe that
$$\{ x\in\mathbb{R} \mid |x+t|\geq A \} \cup (-A-t,-A] = \{ x\in\mathbb{R} \mid |x|\geq A\} \cup [A-t,A),$$
where, in the case $t<2A$, the sets on both sides are disjoint, and otherwise
$$\{ x\in\mathbb{R} \mid |x+t|\geq A \} \cap (-A-t,-A] = \{ x\in\mathbb{R} \mid A-t \leq x \leq -A\} = \{ x\in\mathbb{R} \mid |x|\geq A\} \cap [A-t,A).$$
Thus, with respect to the measure $\rho_y$ that is given by $d\rho_y(x) = \frac{1}{x^2+y^2}\, dx$, we have in either case that
$$\rho_y\big(\{ x\in\mathbb{R} \mid |x+t|\geq A \}\big) + \rho_y\big((-A-t,-A]\big) = \rho_y\big(\{ x\in\mathbb{R} \mid |x|\geq A\}\big) + \rho_y\big([A-t,A)\big),$$
which gives us that
\begin{align*}
\int_{|x+t|\geq A} \frac{1}{x^2+y^2}\, dx
&= \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx + \int^A_{A-t} \frac{1}{x^2+y^2}\, dx - \int^{-A}_{-A-t} \frac{1}{x^2+y^2}\, dx\\
&= \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx + \int^{A+t}_A \frac{1}{(x-t)^2+y^2}\, dx - \int^{A+t}_A \frac{1}{x^2+y^2}\, dx\\
&= \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx + \frac{t}{y} \int^{A+t}_A \frac{2(x-t)y}{(x-t)^2+y^2} \frac{1}{x^2+y^2}\, dx\\
&\qquad\qquad + t^2 \int^{A+t}_A \frac{1}{(x-t)^2+y^2} \frac{1}{x^2+y^2}\, dx.
\end{align*}
The second integral in the last line above can be estimated by the inequality of arithmetic and geometric means as
$$\bigg| \int^{A+t}_A \frac{2(x-t)y}{(x-t)^2+y^2} \frac{1}{x^2+y^2}\, dx \bigg|
\leq \int^\infty_A \frac{1}{x^2+y^2}\, dx.$$
For the third integral, which has a positive integrand, we see that
$$\int^{A+t}_A \frac{1}{(x-t)^2+y^2} \frac{1}{x^2+y^2}\, dx \leq \frac{1}{y^2} \int^\infty_A \frac{1}{x^2+y^2}\, dx.$$
Thus, in summary, we have that
\begin{equation}\label{eq:integral_bound-6}
\int_{|x+t|\geq A} \frac{1}{x^2+y^2}\, dx \leq \Big(1 + \frac{|t|}{2y} + \frac{t^2}{2y^2}\Big) \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx.
\end{equation}
So far, we have established \eqref{eq:integral_bound-6} only in the case $t\geq0$, we claim, however, that it also holds for every $t\leq 0$. To see that, we note that the integral on the left hand side is taken over a mirror symmetric function, which gives that $\int_{|x+t|\geq A} \frac{1}{x^2+y^2}\, dx = \int_{|x+(-t)|\geq A} \frac{1}{x^2+y^2}\, dx$, and since the right hand side of \eqref{eq:integral_bound-6} remains the same if $t$ is replaced by $-t$, we infer that \eqref{eq:integral_bound-6} holds verbatim also for $t\leq 0$.
Inserting the bound \eqref{eq:integral_bound-6} into the formula \eqref{eq:integral_bound-5}, we obtain the inequality
\begin{equation}\label{eq:integral_bound-7}
\int_{|x|\geq A} \int_\mathbb{R} \frac{1}{(x-t)^2+y^2}\, d\mu(t)\, dx \leq W_y(\mu)^2 \int_{|x|\geq A} \frac{1}{x^2+y^2}\, dx.
\end{equation}
Note that \eqref{eq:integral_bound-7} holds, of course, also for the measure $\nu$ instead of $\mu$; thus, using \eqref{eq:integral_bound-7}, we can infer from \eqref{eq:integral_bound-4} the validity of \eqref{eq:integral_bound-2}.
\end{proof}
\begin{remark}
Another interesting estimating which is however not sufficient for our purposes is the following:
\begin{equation}\label{eq:integral_bound-1}
\int_\mathbb{R} |G_\mu(x+iy)-G_\nu(x+iy)|\, dx \leq \frac{\pi}{y} c(\mu,\nu).
\end{equation}
It can be simply proved following the strategy of the proof of Lemma \ref{lem:integral_bound}.
\end{remark}
\subsection{Convergence in distribution and absolute moments}
Let us remind ourselves of the following well-known fact.
\begin{lemma}\label{lem:absolute_moments}
Let $(\mu_n)_{n=1}^\infty$ a sequence of Borel probability measures on $\mathbb{R}$ which converges in distribution to a Borel probability measure $\nu$ on $\mathbb{R}$. Suppose that, for some $p\geq 1$,
$$\sup_{n\in\mathbb{N}} \int_\mathbb{R} |t|^p\, d\mu_n(t) < \infty$$
holds. Then
$$\int_\mathbb{R} |t|^p\, d\nu(t) \leq \sup_{n\in\mathbb{N}} \int_\mathbb{R} |t|^p\, d\mu_n(t).$$
\end{lemma}
\subsection{The proof of Theorem \ref{thm:Hoelder_criterion}}\label{subsec:Hoelder_criterion_proof}
Now, we are prepared to give the proof of Theorem \ref{thm:Hoelder_criterion}. In doing so, we will follow the strategy of Theorem \ref{thm:Kolmogorov}, for which we will need the bounds that were derived in Lemma \ref{lem:integral_bound}.
\begin{proof}[Proof of Theorem \ref{thm:Hoelder_criterion}]
First, we fix $a$ and $\gamma$ according to the condition \eqref{eq:BS_condition-1} in Theorem \ref{thm:Kolmogorov} and we choose any $\kappa \in(0,1)$. We then define sequences $(y_n)_{n=1}^\infty$ and $(K_n)_{n=1}^\infty$ in $(0,\infty)$ by
$$y_n := \epsilon_n^{\frac{1}{2+k+(2-\beta)l}} \qquad\text{and}\qquad K_n := \frac{1}{y_n^{2-\beta}}$$
for every $n\in\mathbb{N}$; note that we clearly have $y_n \to 0$ and $K_n \to \infty$ as $n\to\infty$. We proceed now as follows:
\begin{itemize}
\item The H\"older continuity condition in Item \ref{it:cond-1} yields for every $n\in\mathbb{N}$ that
$$\int_{|s|\leq 2y_na} |\mathcal{F}_\nu(t+s)-\mathcal{F}_\nu(t)|\, ds \leq 2C \int^{2y_na}_0 s^\beta\, ds = \frac{2C(2y_na)^{1+\beta}}{1+\beta}$$
and therefore, with $C_1 := \frac{2C(2a)^{1+\beta}}{1+\beta} > 0$, that
\begin{equation}\label{eq:Kolmogorov_cond-1}
\frac{1}{y_n} \sup_{t\in\mathbb{R}} \int_{|s|\leq 2y_na} |\mathcal{F}_\nu(t+s)-\mathcal{F}_\nu(t)|\, ds \leq C_1 y_n^\beta = C_1 \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}.
\end{equation}
\item The condition formulated in Item \ref{it:cond-3} of the theorem guarantees that there are $m_1,m_2>0$ such that
$$\sup_{n\in\mathbb{N}} \int_\mathbb{R} |t|\, d\mu_n(t) \leq m_1 \qquad\text{and}\qquad \sup_{n\in\mathbb{N}} \int_\mathbb{R} t^2\, d\mu_n(t) \leq m_2.$$
Since the assumption made in Item \ref{it:cond-2} of the theorem guarantees due to Theorem \ref{thm:convergence_in_distribution} that $\mu_n \to \nu$ in distribution as $n\to\infty$, Lemma \ref{lem:absolute_moments} tells us that both
\begin{equation}\label{eq:absolute_moments}
\int_\mathbb{R} |t|\, d\nu(t) \leq m_1 \qquad\text{and}\qquad \int_\mathbb{R} t^2\, d\nu(t) \leq m_2.
\end{equation}
Consequently, we also have that
$$c := \sup_{n\in\mathbb{N}} c(\mu_n,\nu) < \infty$$
Using Lemma \ref{lem:integral_bound}, we get that
$$\int_{|x|\geq K_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx \leq c W_{y_n}(\mu_n) W_{y_n}(\nu) \int_{|x|\geq K_n} \frac{1}{x^2+y_n^2}\, dx.$$
We have then for every $n\in\mathbb{N}$
$$\int_{|x|\geq K_n} \frac{1}{x^2+y_n^2}\, dx \leq \int_{|x|\geq K_n} \frac{1}{x^2}\, dx = \frac{2}{K_n}$$
and furthermore, if $n$ is large enough,
$$W_{y_n}(\mu_n) W_{y_n}(\nu) \leq \frac{m_2}{y_n^2}.$$
In combination, this shows that for sufficiently large $n\in\mathbb{N}$
$$\int_{|x|\geq K_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx \leq \frac{2cm_2}{K_ny_n^2} = 2cm_2 y_n^\beta = 2cm_2 \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}.$$
We conclude that, with some suitably chosen constant $C_2>0$, for all $n\in\mathbb{N}$
\begin{equation}\label{eq:Kolmogorov_cond-2.1}
\int_{|x|\geq K_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx \leq C_2 \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}.
\end{equation}
\item Now, we invoke the estimates given in Item \ref{it:cond-2}. We put $R_n := (K_n^2+y_n^2)^{1/2}$ and we note first that for all sufficiently large $n\in\mathbb{N}$
\begin{itemize}
\item $R_n < 2^{1/l} K_n$,
\item $\{x + iy_n \mid x\in [-K_n,K_n]\} \subset \mathbb{S}_\rho$,
\item $\max_{r\in[0,R_n]} \Theta_0(r) \leq \theta R_n^l$ for some $\theta>0$.
\end{itemize}
Thus, the bound on $\Theta$ yields that
$$\max_{x\in [-K_n,K_n]} \Theta(x+iy_n) \leq \frac{1}{y_n^k} \max_{r\in[0,R_n]} \Theta_0(r) \leq \theta \frac{R_n^l}{y_n^k} \leq 2 \theta \frac{K_n^l}{y_n^k},$$
and with the bound for the Cauchy transforms we conclude that
$$\max_{x\in[-K_n,K_n]} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)| \leq 2\theta \epsilon_n \frac{K_n^l}{y_n^k}.$$
Using this, we can now verify that for all such $n\in\mathbb{N}$
\begin{align*}
\lefteqn{\int^{K_n}_{-K_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx}\\
& \qquad \leq 2 K_n \max_{x\in[-K_n,K_n]} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\\
& \qquad \leq 4 \theta \epsilon_n \frac{K_n^{l+1}}{y_n^k} = 4 \theta y_n^\beta = 4 \theta \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}
\end{align*}
Hence, we conclude that for all $n\in\mathbb{N}$, with some suitably chosen constant $C_3>0$,
\begin{equation}\label{eq:Kolmogorov_cond-2.2}
\int^{K_n}_{-K_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx \leq C_3 \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}.
\end{equation}
\item By the fact that $\int_\mathbb{R} t^2\, d\mu_n(t) < \infty$ for every $n\in\mathbb{N}$, \eqref{eq:absolute_moments}, and the Chebyshev inequality, we get for every $n\in\mathbb{N}$
$$\int_\mathbb{R} |\mathcal{F}_{\mu_n}(t) - \mathcal{F}_\nu(t)|\, dt < \infty,$$
so that $\mu_n$ and $\nu$ satisfy condition \eqref{eq:BS_condition-0} of Theorem \ref{thm:Kolmogorov}; furthermore, this guarantees that we can choose $B_n>0$ such that
\begin{equation}\label{eq:Kolmogorov_cond-3}
\frac{1}{y_n} \int_{|t|>B_n} |\mathcal{F}_{\mu_n}(t) - \mathcal{F}_\nu(t)|\, dt \leq \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}.
\end{equation}
\item Now, we associate to the so found sequence $(B_n)_{n=1}^\infty$ another sequence $(A_n)_{n=1}^\infty$ by
$$A_n := B_n \bigg(1 + \frac{4}{\kappa\pi(2\gamma-1)}\bigg) \qquad\text{for all $n\in\mathbb{N}$}.$$
Then, for each $n\in\mathbb{N}$, we have that $A_n > B_n$ and \eqref{eq:BS_condition-2} is satisfied with the $\kappa$ that we have chosen above.
\item Finally, since by construction $A_n > K_n$ for every $n\in\mathbb{N}$, we may check that
\begin{align*}
\lefteqn{\int^{A_n}_{-A_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx}\\
& \quad \leq \int^{K_n}_{-K_n} |G_\mu(x+iy_n)-G_\nu(x+iy_n)|\, dx + \int_{K_n \leq |x| < A_n} |G_\mu(x+iy_n)-G_\nu(x+iy_n)|\, dx\\
& \quad \leq \int^{K_n}_{-K_n} |G_\mu(x+iy_n)-G_\nu(x+iy_n)|\, dx + \int_{|x| \geq K_n} |G_\mu(x+iy_n)-G_\nu(x+iy_n)|\, dx.
\end{align*}
Due to \eqref{eq:Kolmogorov_cond-2.1} and \eqref{eq:Kolmogorov_cond-2.2}, the latter yields that for all $n\in\mathbb{N}$
\begin{equation}\label{eq:Kolmogorov_cond-2}
\int^{A_n}_{-A_n} |G_{\mu_n}(x+iy_n)-G_\nu(x+iy_n)|\, dx < (C_2 + C_3) \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}}
\end{equation}
\end{itemize}
Putting these pieces together, we see that for every $n\in\mathbb{N}$, the conditions \eqref{eq:BS_condition-0}, \eqref{eq:BS_condition-1}, and \eqref{eq:BS_condition-2} are satisfied for $A_n$ and $B_n$; therefore, we may apply Theorem \ref{thm:Kolmogorov}, which yields, in combination with \eqref{eq:Kolmogorov_cond-1}, \eqref{eq:Kolmogorov_cond-2}, and \eqref{eq:Kolmogorov_cond-3}, that
$$\Delta(\mu_n,\nu) < D \epsilon_n^{\frac{\beta}{2+k+(2-\beta)l}} \qquad\text{with}\qquad D := \frac{1+C_1+C_2+C_3}{\pi(1-\kappa)(2\gamma-1)}$$
for all $n\in\mathbb{N}$, as claimed.
\end{proof}
\begin{remark}\label{rem:Hoelder_criterion_compact}
We point out that also the proof of Theorem \ref{thm:Hoelder_criterion_compact} relies on Theorem \ref{thm:Kolmogorov}. Indeed, if we choose $A>B>M$ such that condition \eqref{eq:BS_condition-2} is satisfied, then Theorem \ref{thm:Kolmogorov} yields for $y_n := \epsilon_n^{\frac{1}{k+\beta}}$ the bound asserted in Theorem \ref{thm:Hoelder_criterion_compact}.
\end{remark}
\section{Random matrix applications}\label{sec:random_matrices}
The aim of this section is to discuss some applications of our results in the context of random matrix theory. The simple idea is roughly the following: let $(X_1^{(N)},\dots,X_n^{(N)})$, for every $N\in\mathbb{N}$, be a tuple of selfadjoint random matrices of size $N\times N$ and suppose that their asymptotic behavior as $N\to\infty$ is described by a tuple $(X_1,\dots,X_n)$ of selfadjoint noncommutative random variables living in some tracial $W^\ast$-probability space $(\mathcal{M},\tau)$ with the property that $\Phi^\ast(X_1,\dots,X_n) < \infty$. For many types of ``noncommutative functions'' $f$, the limiting eigenvalue distribution of the random matrices $Y^{(N)}=f(X_1^{(N)},\dots,X_n^{(N)})$ as $N\to\infty$ is given by the analytic distribution of the operator $Y=f(X_1,\dots,X_n)$.
We shall see how our results in Theorems \ref{thm:Hoelder_continuity}, \ref{thm:convergence_in_distribution} and \ref{thm:Hoelder_criterion} could be combined to obtain H\"older continuity and provide rates of convergence with respect to the Kolmogorov distance for such matrix models.
As concrete instances of such ``composed'' random matrices we will consider here
\begin{itemize}
\item for fixed (deterministic) selfadjoint matrices $a_0,a_1,\dots,a_n\in M_d(\mathbb{C})$, the \emph{generalized block matrices}
\begin{equation}\label{eq:block_matrix}
Y^{(N)} := a_0 \otimes 1_N + \sum^n_{j=1} a_j \otimes X_j^{(N)};
\end{equation}
\item for a non-constant selfadjoint noncommutative polynomial $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ the random matrices
\begin{equation}\label{eq:polynomial_evaluation}
Y^{(N)} := P(X^{(N)}_1,\dots,X^{(N)}_n).
\end{equation}
\end{itemize}
In Section \ref{subsec:Gibbs_laws}, we will work with tuples $(X_1^{(N)},\dots,X_n^{(N)})$ of random matrices that follow general Gibbs laws; this includes the important case of GUEs, which is addressed separately in Section \ref{subsec:GUEs}. In Section \ref{subsec:random_matrices_basics}, we first recall some basic terminology.
\subsection{Random matrices and noncommutative probability theory}\label{subsec:random_matrices_basics}
Many types of random matrices fit nicely into the frame of noncommutative $\ast$-probability spaces. In fact, one can often treat them as noncommutative random variables living in the $\ast$-probability space $(\mathcal{M}_N,\tau_N)$ given by the $\ast$-algebra $\mathcal{M}_N := M_N(\mathbb{C}) \otimes L^{\infty -}(\Omega,\mathbb{P})$ that is endowed with the tracial state $\tau_N := \operatorname{tr}_N \otimes \mathbb{E}$ for some classical probability space $(\Omega,\mathcal{F},\mathbb{P})$ with the associated expectation $\mathbb{E}$.
Let a selfadjoint random matrix $X^{(N)} \in \mathcal{M}_N$ be given. We will be interested in the random eigenvalues $\lambda_1(X^{(N)}),\dots,\lambda_N(X^{(N)})$ of $X^{(N)}$, to which we associate a random probability measure $\mu_{X^{(N)}}$ on $\mathbb{R}$ by
$$\mu_{X^{(N)}} := \frac{1}{N} \sum_{j=1}^N \delta_{\lambda_j(X^{(N)})},$$
called the \emph{empirical eigenvalue distribution of $X^{(N)}$}. By $\overline{\mu}_X$, we will denote the \emph{mean eigenvalue distribution of $X^{(N)}$} which is the probability measure on $\mathbb{R}$ that is defined as $\overline{\mu}_X := \mathbb{E}[\mu_X]$. We point out that the Cauchy transform of $\overline{\mu}_{X^{(N)}}$ agrees with the Cauchy transform of the noncommutative random variable $X^{(N)}$ in $(\mathcal{M}_N,\tau_N)$, i.e., we have
$$G_{\overline{\mu}_{X^{(N)}}}(z) = \tau_N\big((z 1_N - X^{(N)})^{-1}\big) \qquad\text{for all $z\in\mathbb{C}^+$}.$$
In the following, we shall see random matrices as elements in $M_N(\mathbb{C})_{\operatorname{sa}}$ chosen randomly according to some probability measure on this space.
\subsection{Gibbs laws}\label{subsec:Gibbs_laws}
Consider a selfadjoint noncommutative polynomial $V\in\mathbb{C}\langle x_1,\dots,x_n\rangle$; in the following, we will refer to $V$ as a \emph{potential}. Following \cite{GS09}, we say that the potential $V$ is \emph{selfadjoint $(c,M)$-convex} if
$$(DV(X) - DV(Y)) . (X-Y) \geq c (X-Y) . (X-Y)$$
for any $n$-tuples $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ of selfadjoint operators in some $C^\ast$-algebra $\mathcal{A}$ that are bounded in norm by $M$, where $X . Y := \frac{1}{2} \sum^n_{j=1} (X_j Y_j + Y_j X_j)$.
Suppose now that $V$ is selfadjoint $(c,\infty)$-convex for some $c>0$. We will use $V$ to introduce a probability measure on $M_N(\mathbb{C})^n_{\operatorname{sa}}$. For that purpose, let us first define the Lebesgue measure on $M_N(\mathbb{C})_{\operatorname{sa}}$ by
$$dX^{(N)} := \prod_{k=1}^N dX_{kk} \prod_{1\leq k < l \leq N} d\Re(X_{kl})\, d\Im(X_{kl}).$$
Further, let $\operatorname{Tr}$ denote the unnormalized trace on $M_N(\mathbb{C})$. On the space $M_N(\mathbb{C})^n_{\operatorname{sa}}$, we then define the probability measure
$$\mathbb{P}_V^N(X_1^{(N)},\dots,X_n^{(N)}) = \frac{1}{Z_N(V)} \exp\big(-N \operatorname{Tr}(V(X^{(N)}_1,\dots,X^{(N)}_n))\big)\ dX_1^{(N)}\, \dots\, dX_n^{(N)},$$
where $Z_N(V)$ is the normalizing constant that is given by
$$Z_N(V) := \int_{M_N(\mathbb{C})^n_{\operatorname{sa}}} \exp\big(-N \operatorname{Tr}(V(X^{(N)}_1,\dots,X^{(N)}_n))\big)\ dX_1^{(N)}\, \dots\, dX_n^{(N)}.$$
We call $\mathbb{P}^N_V$ the \emph{Gibbs measure with potential $V$}.
The Brascamp-Lieb inequality \cite{BL76} guarantees that those measures are well-defined (i.e., that $Z_N(V)$ is finite) for potentials $V$ that are selfadjoint $(c,\infty)$-convex for some $c>0$. Those measures are extensively studied for instance in \cite{GMS06,GMS07,GS09}; see also the surveys \cite{Gui06,Gui14,Gui16}.
It was shown in \cite{GS09} that $n$-tuples $(X^{(N)}_1,\dots,X^{(N)}_n)$ of selfadjoint random matrices of size $N\times N$ following the Gibbs law $\mathbb{P}^N_V$ can be described in the limit $N\to\infty$ by an $n$-tuple $(X_1,\dots,X_n)$ of selfadjoint operators in some tracial $W^\ast$-probability space with the property that $\Phi^\ast(X_1,\dots,X_n) <\infty$. Before we can state their result, we need to introduce some further notation: for every noncommutative polynomial $V\in\mathbb{C}\langle x_1,\dots,x_n\rangle$, we denote by $DV = (D_1 V, \dots, D_n V)$ the \emph{cyclic gradient} of $V$; the \emph{cyclic derivatives} $D_1 V, \dots, D_n V$ of $V$ are given by $D_j V = \tilde{m}(\partial_j V)$ for $j=1,\dots,n$, where $\tilde{m}: \mathbb{C}\langle x_1,\dots,x_n\rangle \otimes \mathbb{C}\langle x_1,\dots,x_n\rangle \to \mathbb{C}\langle x_1,\dots,x_n\rangle$ denotes the flipped multiplication that is determined by $\tilde{m}(P_1 \otimes P_2) := P_2 P_1$.
\begin{theorem}[{\cite[Theorem 1.6]{GS09}}]\label{thm:Gibbs}
Let $V$ be selfadjoint $(c,\infty)$-convex for some $c>0$. For every $N\in\mathbb{N}$, let $X^{(N)}=(X^{(N)}_1,\dots,X^{(N)}_n)$ be an $n$-tuple of selfadjoint random matrices of size $N\times N$ with law $\mathbb{P}^N_V$. Then there is an $n$-tuple $X=(X_1,\dots,X_n)$ of selfadjoint operators in some tracial $W^\ast$-probability space $(\mathcal{M},\tau)$ (whose joint distribution $\mu_X$ is then in fact uniquely determined) which satisfy the \emph{Schwinger-Dyson equation with respect to the potential $V$}, i.e.,
$$(\tau\otimes\tau)\big((\partial_j P)(X)\big) = \tau\big(P(X) (D_j V)(X)\big)$$
for every $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ and all $j=1,\dots,n$, and for each $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$, we have
$$\lim_{N\to\infty} \operatorname{tr}_N(P(X^{(N)})) = \tau(P(X)) \qquad\text{almost surely}.$$
\end{theorem}
In the situation of Theorem \ref{thm:Gibbs}, the Schwinger-Dyson equation yields that $(\xi_1,\dots,\xi_n)$ with $\xi_j := (D_j V)(X)$ for $j=1,\dots,n$ are the conjugate variables of $X=(X_1,\dots,X_n)$; thus, we infer that $\Phi^\ast(X_1,\dots,X_n) <\infty$.
With the result obtained in the previous subsection, we conclude the following about matrix models of the type \eqref{eq:polynomial_evaluation}.
\begin{corollary}\label{cor:random_matrices_polynomial}
In the situation of Theorem \ref{thm:Gibbs}, the following holds for each selfadjoint noncommutative polynomial $P\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ of degree $d\geq 1$:
\begin{enumerate}
\item The empirical eigenvalue distribution $\mu_{Y^{(N)}}$ of $$Y^{(N)} = P(X^{(N)}_1,\dots,X^{(N)}_n)$$ converges in distribution almost surely to a compactly supported Borel probability measure $\nu$ on $\mathbb{R}$ whose cumulative distribution function is H\"older continuous with exponent $\frac{2}{2^{d+2}-5}$.
\item We have that $$\lim_{N\to\infty} \Delta(\mu_{Y^{(N)}},\nu) = 0 \quad \text{almost surely} \qquad\text{and}\qquad \lim_{N\to\infty} \Delta(\overline{\mu}_{Y^{(N)}},\nu) = 0.$$
\end{enumerate}
\end{corollary}
\begin{proof}
Theorem \ref{thm:Gibbs} tells us that $\mu_{Y^{(N)}}$ converges in distribution almost surely as $N\to\infty$ to the analytic distribution $\nu:=\mu_Y$ of $Y:=P(X_1,\dots,X_n)$. Since $X_1\dots,X_n$ satisfy the Schwinger-Dyson equation with potential $V$, we infer that $\Phi^\ast(X_1,\dots,X_n) < \infty$ as outlined above. Therefore, with the help of Theorem \ref{thm:Hoelder_continuity}, we see that the cumulative distribution function of $\nu$ is H\"older continuous with exponent $\frac{2}{2^{d+2}-5}$.
As a consequence of Theorem \ref{thm:convergence_in_distribution}, we obtain that $\Delta(\mu_{Y^{(N)}},\nu) \to 0$ almost surely as $N\to \infty$ and in particular $\Delta(\overline{\mu}_{Y^{(N)}},\nu) \leq \mathbb{E}[\Delta(\mu_{Y^{(N)}},\nu)] \to 0$ as $N\to \infty$.
\end{proof}
We point out that an analogous statement holds true for certain random matrices of the form \eqref{eq:block_matrix}. For that purpose, we need the following terminology: if $a_1,\dots,a_n\in M_d(\mathbb{C})$ are selfadjoint matrices, we call
\begin{equation}\label{eq:quantum_operator}
\mathcal{L}:\ M_d(\mathbb{C}) \to M_d(\mathbb{C}),\quad b\mapsto \sum^n_{j=1} a_j b a_j
\end{equation}
the \emph{quantum operator (associated to $a_1,\dots,a_n$)}; we say that $\mathcal{L}$ is \emph{semi-flat}, if there is some constant $c>0$ such that $\mathcal{L}(b) \geq c \operatorname{tr}_d(b) 1_d$ for all positive semidefinite matrices $b\in M_d(\mathbb{C})$.
In \cite[Theorem 8.1]{MSY18}, it is stated that whenever $a_0,a_1,\dots,a_n\in M_d(\mathbb{C})$ are selfadjoint matrices such that the quantum operator $\mathcal{L}: M_d(\mathbb{C}) \to M_d(\mathbb{C})$ associated to $a_1,\dots,a_n$ is semi-flat and $X_1,\dots,X_n$ are selfadjoint operators in a tracial $W^\ast$-probability space $(\mathcal{M},\tau)$ that satisfy $\Phi^\ast(X_1,\dots,X_n) < \infty$, then $\mathcal{F}_Y$ is H\"older continuous with exponent $\beta=\frac{2}{3}$ for the selfadjoint operator in the tracial $W^\ast$-probability space $(M_d(\mathbb{C}) \otimes \mathcal{M}, \operatorname{tr}_d \otimes \tau)$ given by
$$Y:=a_0 \otimes 1 + \sum^n_{j=1} a_j \otimes X_j.$$
This approach was inspired by \cite{AjEK18,AEK18}, where a very detailed analysis of such operators in the special case for freely independent semicircular operators $X_1,\dots,X_n$ is carried out.
\begin{corollary}\label{cor:random_matrices_block}
In the situation of Theorem \ref{thm:Gibbs}, for every choice of selfadjoint matrices $a_0,a_1,\dots,a_n\in M_d(\mathbb{C})$ for which the quantum operator $\mathcal{L}: M_d(\mathbb{C}) \to M_d(\mathbb{C})$ associated to $a_1,\dots,a_n$ is semi-flat, the following statements holds true:
\begin{enumerate}
\item The empirical eigenvalue distribution $\mu_{Y^{(N)}}$ of the random matrix $$Y^{(N)} = a_0 \otimes 1_N + \sum^n_{j=1} a_j \otimes X^{(N)}_j$$ converges in distribution almost surely to a compactly supported Borel probability measure $\nu$ on $\mathbb{R}$ whose cumulative distribution function is H\"older continuous with exponent $\frac{2}{3}$.
\item We have that $$\lim_{N\to\infty} \Delta(\mu_{Y^{(N)}},\nu) = 0 \quad \text{almost surely} \qquad\text{and}\qquad \lim_{N\to\infty} \Delta(\overline{\mu}_{Y^{(N)}},\nu) = 0.$$
\end{enumerate}
\end{corollary}
\begin{proof}
It follows from Theorem \ref{thm:Gibbs} that $\mu_{Y^{(N)}}$ converges in distribution almost surely as $N\to\infty$ to the analytic distribution $\nu:=\mu_Y$ of the operator $Y:=a_0 \otimes 1 + \sum^n_{j=1} a_j \otimes X_j$ living in $(M_d(\mathbb{C}) \otimes \mathcal{M}, \operatorname{tr}_d \otimes \tau)$. Since $X_1\dots,X_n$ satisfy $\Phi^\ast(X_1,\dots,X_n) < \infty$, we can use \cite[Theorem 8.1]{MSY18} which tells us that the cumulative distribution function of $\nu$ is H\"older continuous with exponent $\frac{2}{3}$. The rest is shown like in the proof of Corollary \ref{cor:random_matrices_polynomial}.
\end{proof}
\subsection{Gaussian random matrices and rates of convergence}\label{subsec:GUEs}
A \emph{(standard) selfadjoint Gaussian random matrix} (or \emph{GUE}) of size $N\times N$ is a selfadjoint complex random matrix $X=(X_{kl})_{k,l=1}^N$ in $\mathcal{M}_N$ for which
$$\{X_{kk} \mid 1\leq k \leq n\} \cup \{\Re(X_{kl}) \mid 1\leq k < l \leq N\} \cup \{\Im(X_{kl}) \mid 1\leq k < l \leq N\}$$
are independent real Gaussian random variables such that
$$\mathbb{E}[X_{kl}] = 0 \quad\text{and}\quad \mathbb{E}[|X_{kl}|^2] = \frac{1}{N} \quad \text{for $1\leq k \leq l \leq N$}.$$
Those fall into the general class of Gibbs measures considered in the previous section with the particular potential $V = \frac{1}{2}(x_1^2 + \dots + x_n^2)$.
Our goal is to strengthen Corollary \ref{cor:random_matrices_polynomial} by proving explicit rates for the Kolmogorov distance. This improvement crucially depends on the results of \cite{HT05} about random matrices of the form \eqref{eq:block_matrix}, which we are going to recall now.
Note that each random matrix like in \eqref{eq:block_matrix} is an element in $\mathcal{M}_{dN} \cong M_d(\mathbb{C}) \otimes \mathcal{M}_N$. For each $X = X^\ast \in M_d(\mathbb{C}) \otimes \mathcal{M}_N$, we define its \emph{matrix-valued Cauchy transform} by
$$\mathbf{G}_X:\ \mathbb{H}^+(M_d(\mathbb{C})) \to \mathbb{H}^-(M_d(\mathbb{C})), \quad b \mapsto (\operatorname{id}_{M_d(\mathbb{C})} \otimes \tau_N)\big((b \otimes 1_N - X)^{-1}\big),$$
where $\mathbb{H}^+(M_d(\mathbb{C}))$ and $\mathbb{H}^-(M_d(\mathbb{C}))$ denotes the upper and lower half-plane in $M_d(\mathbb{C})$, respectively, that is the set of all $b\in M_d(\mathbb{C})$ with positive and negative imaginary part $\Im(b) := \frac{1}{2i}(b-b^\ast)$, respectively. Note that $G_{\overline{\mu}_{X}}(z) = \operatorname{tr}_d(\mathbf{G}_{X}(z 1_d))$ for all $z\in\mathbb{C}^+$.
The limit of those random matrices will be described accordingly by some selfadjoint operator in the tracial $W^\ast$-probability space $(M_d(\mathbb{C}) \otimes \mathcal{M}, \operatorname{tr}_d \otimes \tau)$. Note that $(M_d(\mathbb{C}) \otimes \mathcal{M}, \operatorname{tr}_d \otimes \tau)$ is again a tracial $W^\ast$-probability space, which can further be regarded as an operator-valued probability spaces over $M_d(\mathbb{C})$ with the conditional expectation that is given by $\operatorname{id}_{M_d(\mathbb{C})} \otimes \tau$. Accordingly, we can consider the matrix-valued Cauchy transform of any $X = X^\ast \in M_d(\mathbb{C}) \otimes \mathcal{M}$; it is defined by
$$\mathbf{G}_X:\ \mathbb{H}^+(M_d(\mathbb{C})) \to \mathbb{H}^-(M_d(\mathbb{C})), \quad b \mapsto (\operatorname{id}_{M_d(\mathbb{C})} \otimes \tau)\big((b \otimes 1 - X)^{-1}\big).$$
Now, we can formulate the precise convergence result, which is \cite[Theorem 5.7]{HT05}.
\begin{theorem}\label{thm:HT}
Let $a_0,a_1,\dots,a_n \in M_d(\mathbb{C})$ be selfadjoint matrices. We consider, for each $N\in\mathbb{N}$, a tuple $(X^{(N)}_1,\dots,X^{(N)}_n)$ of $n$ independent GUEs. Let further $(S_1,\dots,S_n)$ be a tuple of freely independent semicircular elements in some tracial $W^\ast$-probability space $(\mathcal{M},\tau)$. Consider
$$X^{(N)} := a_0 \otimes 1_N + \sum^n_{j=1} a_j \otimes X_j^{(N)} \qquad\text{and}\qquad S := a_0 \otimes 1 + \sum^n_{j=1} a_j \otimes S_j.$$
Then the matrix-valued Cauchy transforms $\mathbf{G}_{X^{(N)}}, \mathbf{G}_S: \mathbb{H}^+(M_d(\mathbb{C})) \to \mathbb{H}^-(M_d(\mathbb{C}))$ satisfy
$$\|\mathbf{G}_{X^{(N)}}(b) - \mathbf{G}_S(b)\| \leq \frac{4C}{N^2} (K + \|b\|)^2 \|\Im(b)^{-1}\|^7$$
for all $b\in\mathbb{H}^+(M_d(\mathbb{C}))$, with the constants $C>0$ and $K>0$ that are given by
$$C = d^3 \bigg\|\sum^n_{j=1} a_j^2\bigg\|^2 \qquad\text{and}\qquad K = \|a_0\| + 4 \sum^n_{j=1} \|a_j\|.$$
\end{theorem}
Accordingly (see \cite[Lemma 6.1]{HT05}), the associated scalar-valued Cauchy transforms $G_{X^{(N)}}$ and $G_S$, which are related to the respective matrix-valued Cauchy transforms by $G_{\overline{\mu}_{X^{(N)}}}(z) = \operatorname{tr}_d(\mathbf{G}_{X^{(N)}}(z 1_d))$ and $G_{\mu_S}(z) = \operatorname{tr}_d(\mathbf{G}_S(z 1_d))$ for every $z\in\mathbb{C}^+$, satisfy
\begin{equation}\label{eq:HT}
|G_{X^{(N)}}(z) - G_S(z)| \leq \frac{4C}{N^2} \frac{(K + |z|)^2}{\Im(z)^7}.
\end{equation}
Putting these facts together, we conclude now the following.
\begin{corollary}\label{cor:block-GUE}
Let $a_0,a_1,\dots,a_n \in M_d(\mathbb{C})$ be selfadjoint such that the quantum operator $\mathcal{L}: M_d(\mathbb{C}) \to M_d(\mathbb{C})$ associated to $a_1,\dots,a_n$ by \eqref{eq:quantum_operator} is semi-flat.
For each $N\in\mathbb{N}$, let $(X^{(N)}_1,\dots,X^{(N)}_n)$ be a tuple of $n$ independent GUEs. Further, let $(S_1,\dots,S_n)$ be a tuple of freely independent semicircular elements in some tracial $W^\ast$-probability space $(\mathcal{M},\tau)$. Set
$$X^{(N)} := a_0 \otimes 1_N + \sum^n_{j=1} a_j \otimes X_j^{(N)} \qquad\text{and}\qquad S := a_0 \otimes 1 + \sum^n_{j=1} a_j \otimes S_j.$$
Then the averaged empirical eigenvalue distribution $\overline{\mu}_{X^{(N)}}$ of $X^{(N)}$ satisfies
$$\Delta(\overline{\mu}_{X^{(N)}},\mu_S) \leq D N^{-4/35}.$$
\end{corollary}
\begin{proof}
We want to apply Theorem \ref{thm:Hoelder_criterion}. Therefore, we check that $\mu_N := \overline{\mu}_{X^{(N)}}$ and $\nu := \mu _S$ have the required properties:
\begin{itemize}
\item Since $\mathcal{L}$ is semi-flat, the cumulative distribution function of $\mu_S$ is H\"older continuous with exponent $\beta=\frac{2}{3}$, as it follows from \cite[Theorem 8.1]{MSY18}.
\item Let us define $\epsilon_N := N^{-2}$. Then, due to \eqref{eq:HT}, we have that
$$|G_{X^{(N)}}(z) - G_S(z)| \leq \Theta(z) \epsilon_N \qquad\text{for all $z\in\mathbb{C}+$}$$
with a continuous function $\Theta: \mathbb{C}^+ \to [0,\infty)$ that satisfies the growth condition $\Theta(z) \leq \frac{\Theta_0(|z|)}{\Im(z)^7}$ on $\mathbb{S}_\infty = \mathbb{C}^+$ with the continuous function $\Theta_0: [0,\infty) \to [0,\infty)$ that is given by $\Theta_0(r) := (K+r)^2$; the latter satisfies $\lim_{R\to\infty} R^{-2} \max_{r\in[0,R]} \Theta_0(r) = 1$.
\item For each $N\in\mathbb{N}$, the measure $\overline{\mu}_{X^{(N)}}$ satisfies $$\int_\mathbb{R} t^2\, d\overline{\mu}_{X^{(N)}}(t) = \mathbb{E}\big[(\operatorname{tr}_d\otimes\operatorname{tr}_N)\big( (X^{(N)})^2 \big)\big] = \operatorname{tr}_d(\mathcal{L}(1_d)).$$
\end{itemize}
Therefore, Theorem \ref{thm:Hoelder_criterion} guarantees the existence of some numerical constant $D>0$ for which $\Delta(\overline{\mu}_{X^{(N)}},\mu_S) \leq D N^{-4/35}$ holds, as claimed.
\end{proof}
\begin{remark}
In the proof of Theorem \ref{thm:Hoelder_criterion}, on which the previous corollary relies substantially, the behavior of the cumulative distribution functions near $\infty$ was controlled with the help of Chebyshev's inequality. For the sake of completeness, we note that in the case of the mean empirical eigenvalue distribution $\overline{\mu}_{X^{(N)}}$ of $X^{(N)}$ much stronger statements are possible -- although this does not improve the conclusion of Theorem \ref{thm:Hoelder_criterion}. More precisely, we have
\begin{equation}\label{eq:cumulative_decay}
1-\mathcal{F}_{\overline{\mu}_{X^{(N)}}}(2+t) \leq 2N \exp\Big(-\frac{N t^2}{2}\Big) \quad\text{and}\quad \mathcal{F}_{\overline{\mu}_{X^{(N)}}}(2-t) \leq 2N \exp\Big(-\frac{N t^2}{2}\Big).
\end{equation}
This follows from \cite[Proof of Lemma 3.3]{HT03}, \cite[Proof of Lemma 6.4]{S05}, and \cite[Proof of Proposition 6.4]{HST06}.
\end{remark}
With the help of linearization techniques that we outline in Section \ref{sec:linearization} of the appendix, we can give rates for the Kolmogorov distance in the case of polynomial evaluations.
\begin{corollary}\label{cor:p-GUE}
Let $p\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be a selfadjoint noncommutative polynomial of degree $d\geq 1$. For each $N\in\mathbb{N}$, we consider a tuple $X^{(N)} = (X^{(N)}_1,\dots,X^{(N)}_n)$ of $n$ independent GUEs. Further, let $S=(S_1,\dots,S_n)$ be a tuple of freely independent semicircular elements in some tracial $W^\ast$-probability space $(\mathcal{M},\tau)$. We define
$$Y^{(N)} := p(X^{(N)}_1,\dots,X^{(N)}_n) \qquad\text{and}\qquad Y := p(S_1,\dots,S_n).$$
Then there is a constant $D>0$ such that for all $N\in\mathbb{N}$
$$\Delta(\overline{\mu}_{Y^{(N)}},\mu_Y) \leq D N^{-\frac{1}{13 \cdot 2^{d+3} - 138}}.$$
\end{corollary}
For the particular case $p(x)=x$ of a GUE matrix, the rate of convergence to the semicirular distribution with respect to the Kolmogorov distance was studied by G\"otze and Tikhomirov in \cite{GT02} and then in \cite{GT05} where they obtain the optimal rate, conjectured by Bai \cite{Bai93a} for the more general Wigner matrices. Even for $d=1$ or $n=1$, our result still covers a larger class of matrices than a single GUE.
\begin{proof}[Proof of Corollary \ref{cor:p-GUE}]
This will follow from Theorem \ref{thm:Hoelder_criterion}. Note that the convergence in distribution of $(\overline{\mu}_{Y^{(N)}})_{N=1}^\infty$ to $\mu_Y$ can be taken for granted as by the results of \cite{Voi91} on asymptotic freeness, the tuple $X^{(N)}$ is known to converge in distribution to $S$ as $N\to \infty$.
First of all, it follows from Theorem \ref{thm:Hoelder_continuity} that the cumulative distribution function of the analytic distribution of $Y$ is H\"older continuous with exponent $\frac{2}{2^{d+2}-5}$.
In order to verify condition \ref{it:cond-2} of Theorem \ref{thm:Hoelder_criterion}, we choose a selfadjoint linear representation $\rho=(u,Q,v)$ of $p$ and we consider the associated selfadjoint linearization $\hat{p}$.
For a moment, let us fix $z\in\mathbb{C}^+$ and $N\in\mathbb{N}$; we define $\epsilon>0$ by $\epsilon := N^{-1/4} \Im(z)$.
Since in particular $\epsilon \leq \Im(z) \leq |z|$, we see that $\|\Lambda_\epsilon(z)\| = |z|$ and $\|\Im(\Lambda_\epsilon(z))^{-1}\| = \frac{1}{\epsilon} = N^{1/4} \frac{1}{\Im(z)}$. Thus, involving Theorem \ref{thm:HT}, we get that
$$\|\mathbf{G}_{\hat{p}(X^{(N)})}(\Lambda_\epsilon(z)) - \mathbf{G}_{\hat{p}(S)}(\Lambda_\epsilon(z))\| \leq 4C N^{-1/4} \frac{(K + |z|)^2}{\Im(z)^7}.$$
Furthermore, applying Theorem \ref{thm:approximation_improved}, we find noncommutative polynomials $p=p_1,p_2,\dots,p_d\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ such that
\begin{align*}
\big| G_{p(X^{(N)})}(z) - \big[ \mathbf{G}_{\hat{p}(X^{(N)})}(\Lambda_\epsilon(z)) \big]_{1,1} \big| &\leq N^{-1/4} \frac{2}{\Im(z)} \sum^d_{j=1} \mathbb{E}\big[\operatorname{tr}_N\big(p_j(X^{(N)})^\ast p_j(X^{(N)})\big)\big],\\
\big| G_{p(X)}(z) - \big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1} \big| &\leq N^{-1/4} \frac{2}{\Im(z)} \sum^d_{j=1} \|p_j(S)\|_2^2.
\end{align*}
Since $|[A]_{1,1}| \leq \|A\|$ for every matrix $A\in M_N(\mathbb{C})$, we obtain by putting these pieces together that
$$|G_{\overline{\mu}_{Y^{(N)}}}(z) - G_{\mu_Y}(z)| \leq N^{-1/4} \Big(\frac{2C'}{\Im(z)} + 4C \frac{(K + |z|)^2}{\Im(z)^7}\Big)$$
with the constant
$$C' := \sum^d_{j=1} \|p_j(S)\|_2^2 + \sup_{N\in\mathbb{N}} \sum^d_{j=1} \mathbb{E}\big[\operatorname{tr}_N\big(p_j(X^{(N)})^\ast p_j(X^{(N)})\big)\big],$$
which is finite because $X^{(N)}$ converges in distribution to $S$ as $N\to \infty$ and therefore
$$\lim_{N\to\infty} \sum^d_{j=1} \mathbb{E}\big[\operatorname{tr}_N\big(p_j(X^{(N)})^\ast p_j(X^{(N)})\big)\big] = \sum^d_{j=1} \|p_j(S)\|_2^2.$$
Thus, in summary, we see that with the continuous function
$$\Theta:\ \mathbb{C}^+ \to [0,\infty),\quad z\mapsto \frac{2C'}{\Im(z)} + 4C \frac{(K + |z|)^2}{\Im(z)^7}$$
we have for all $z\in\mathbb{C}^+$ and for all $N\in\mathbb{N}$ that
$$|G_{\overline{\mu}_{Y^{(N)}}}(z) - G_{\mu_Y}(z)| \leq \Theta(z) N^{-1/4}.$$
Taking now a closer look at $\Theta$, we see that it can be bounded on the strip $\mathbb{S}_1$ as
$$\Theta(z) \leq \frac{\Theta_0(|z|)}{\Im(z)^7} \qquad\text{for all $z\in\mathbb{S}_1$},$$
where the function $\Theta_0: [0,\infty) \to [0,\infty)$ is defined by $\Theta_0(r) := 2 C' + (K + r)^2$ for $r>0$ and thus satisfies the growth condition $\limsup_{R\to\infty} R^{-2} \max_{r\in[0,R]} \Theta_0(r) = 1$.
This means that condition \ref{it:cond-1} of Theorem \ref{thm:Hoelder_criterion} is fulfilled with $l=2$, $k=7$, and the sequence $(\epsilon_N)_{N=1}^\infty$ defined by $\epsilon_N := N^{-1/4}$.
It remains to check condition \ref{it:cond-3} of Theorem \ref{thm:Hoelder_criterion}. This, however, is clear as
$$\lim_{N\to\infty} \int_\mathbb{R} t^2\, d\overline{\mu}_{Y^{(N)}} = \lim_{N\to\infty} \mathbb{E}\big[\operatorname{tr}_N(p(X^{(N)})^2)\big] = \tau\big(p(S)^2\big),$$
since $X^{(N)}$ converges in distribution to $S$ as $N\to \infty$.
Thus, Theorem \ref{thm:Hoelder_criterion} guarantees the existence of a constant $D>0$ such that
$$\Delta(\overline{\mu}_{Y^{(N)}},\mu_Y) \leq D N^{-\frac{1}{13 \cdot 2^{d+3} - 138}} \qquad\text{for all $N\in\mathbb{N}$},$$
which proves the assertion.
\end{proof}
\begin{appendix}
\section{Approximation of Cauchy transforms by linearizations}\label{sec:linearization}
Linearization techniques have turned out to be very useful when dealing with evaluations of noncommutative polynomials or noncommutative rational functions. Especially when evaluations in freely independent selfadjoint operators are considered, these methods allow an algorithmic computation of the corresponding analytic distributions and Brown measures, respectively; see \cite{BMS17,HMS18}.
Here, we focus on the case of noncommutative polynomials $p\in\mathbb{C}\langle x_1,\dots,x_n\rangle$. We can associate to $p$ by purely algebraic techniques a \emph{linear representation} $\rho=(u,Q,v)$, i.e., a triple that consists of a row vector $u$ and a column vector $v$, both of the size, say $d\in\mathbb{N}$, and an invertible matrix $Q \in M_d(\mathbb{C}\langle x_1,\dots,x_n\rangle)$ of the form
$$Q = Q_0 + Q_1 x_1 + \dots + Q_n x_n$$
with scalar matrices $Q_0,Q_1,\dots,Q_n\in M_d(\mathbb{C})$ which enjoys the crucial property that
$$p(X_1,\dots,X_n) = -u Q(X_1,\dots,X_n)^{-1} v$$
for every tuple $(X_1,\dots,X_n)$ of elements in any unital complex algebra $\mathcal{A}$.
Moreover, if $p$ is selfadjoint, we may find a particular linear representation $\rho$ which is additionally selfadjoint in the sense that $v=u^\ast$ holds and all $Q_0,Q_1,\dots,Q_n$ are selfadjoint.
With the help of the well-known Schur complement formula, one easily sees that the scalar-valued Cauchy transform of $p(X_1,\dots,X_n)$ can be obtained from the matrix-valued Cauchy transform of the selfadjoint operator $\hat{p}(X_1,\dots,X_n) \in M_{d+1}(\mathcal{A})$, where
\begin{equation}\label{eq:linearization}
\hat{p} := \begin{bmatrix} 0 & u\\ v & Q\end{bmatrix} \in M_{d+1}(\mathbb{C}\langle x_1,\dots,x_n\rangle),
\end{equation}
which is some matrix-valued but linear polynomial, called the \emph{selfadjoint linearization of $p$ associated to $\rho$}; in fact, we have for every point $z\in\mathbb{C}^+$ that
\begin{equation}\label{eq:Cauchy-linearization}
G_{p(X_1,\dots,X_n)}(z) = \lim_{\epsilon \searrow 0} \big[\mathbf{G}_{\hat{p}(X_1,\dots,X_n)}(\Lambda_\epsilon(z))\big]_{1,1},
\end{equation}
where $[A]_{1,1}:= A_{11}$ for any matrix $A$ with entries $A_{ij}$ and $\Lambda_\epsilon(z)$ is a matrix in $\mathbb{H}^+(M_{d+1}(\mathbb{C}))$ that is given by
$$\Lambda_\epsilon(z) := \begin{bmatrix} z & 0 & \hdots & 0\\ 0 & i\epsilon & \hdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \hdots & i\epsilon\end{bmatrix}.$$
Notably, $\mathbf{G}_{\hat{p}(X_1,\dots,X_n)}$ at any point in $\mathbb{H}^+(M_{d+1}(\mathbb{C}))$ can be computed efficiently by means of operator-valued free probability theory.
Our goal is the following quantitative version of \eqref{eq:Cauchy-linearization}.
\begin{theorem}\label{thm:approximation_improved}
Let $p\in\mathbb{C}\langle x_1,\dots,x_n\rangle$ be a selfadjoint noncommutative polynomial. Consider the selfadjoint linearization $\hat{p} \in M_{d+1}(\mathbb{C}\langle x_1,\dots,x_n\rangle)$ of $p$ associated to a given selfadjoint linear representation $\rho=(u,Q,v)$ of $p$ with $u\neq 0$. Then there are (not necessarily selfadjoint) polynomials $p_1,\dots,p_d\in\mathbb{C}\langle x_1,\dots,x_n\rangle$, where $p_1$ can be chosen to be $p$, such that the following statements hold true:
\begin{enumerate}
\item\label{it:approximation_improved} If $X=(X_1,\dots,X_n)$ is a tuple of selfadjoint operators in any tracial $W^\ast$-probability space $(\mathcal{M},\tau)$, then for all $z\in\mathbb{C}^+$ and all $\epsilon>0$
\begin{equation}\label{eq:approximation_improved}
\big| G_{p(X)}(z) - \big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1} \big| \leq \frac{2\epsilon}{\Im(z)^2} \sum^d_{j=1} \|p_j(X)\|_2^2 .
\end{equation}
\item\label{it:approximation_improved_random_matrices} If $X^{(N)} = (X^{(N)}_1,\dots,X_n^{(N)})$ is a tuple of selfadjoint matrices in $M_N(L^{\infty -}(\Omega,\mathbb{P}))$ for any classical probability space $(\Omega,\mathcal{F},\mathbb{P})$ and arbitrary $N\in\mathbb{N}$, then for all $z\in\mathbb{C}^+$ and all $\epsilon>0$
\begin{equation}\label{eq:approximation_improved_random_matrices}
\big| G_{p(X^{(N)})}(z) - \big[ \mathbf{G}_{\hat{p}(X^{(N)})}(\Lambda_\epsilon(z)) \big]_{1,1} \big| \leq \frac{2\epsilon}{\Im(z)^2} \sum^d_{j=1} \mathbb{E}\big[\operatorname{tr}_N\big(p_j(X^{(N)})^\ast p_j(X^{(N)})\big)\big].
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
It is easily seen that with $\rho=(u,Q,v)$ also $\rho_\lambda = (\lambda^{1/2} u, \lambda Q, \lambda^{1/2} v)$, for every $\lambda>0$, yields a selfadjoint linear representation of $p$; thus, since $u\neq 0$ by assumption, we may assume with no loss of generality that $u$ is normalized such that $uu^\ast = 1$. Basic linear algebra tells us that we may find then an orthonormal basis $\{u_1,\dots,u_d\}$ of $\mathbb{C}^d$ with $u_1 = u$. We use these row vectors to define the wanted noncommutative polynomials $p_1,\dots,p_d$ by $p_j := - u_j Q^{-1} v$ for $j=1,\dots,d$; by construction, we clearly have that $p_1 = p$.
We shall show that these polynomials $p_1,\dots,p_d$ have the required properties. We will only prove the validity of Item \ref{it:approximation_improved}; the details of the proof of Item \ref{it:approximation_improved_random_matrices} are left to the reader.
Let us take any selfadjoint operators $X_1,\dots,X_n$ living in an arbitrary tracial $W^\ast$-probability space $(\mathcal{M},\tau)$. Further, let us choose $z\in\mathbb{C}^+$ and $\epsilon>0$.
We begin with the observation that the operator $z - u(i\epsilon 1_d - Q(X))^{-1} u^\ast$ is invertible in $\mathcal{M}$ with
\begin{equation}\label{eq:inverse-bound}
\|(z - u(i\epsilon 1_d - Q(X))^{-1} u^\ast)^{-1}\| \leq \frac{1}{\Im(z)}.
\end{equation}
In order to verify this, let us abbreviate $h:=z-u(i\epsilon 1_d - Q(X))^{-1}u^\ast$; we observe that
\begin{align*}
\Im(h) &= \Im(z) - \epsilon u (i\epsilon 1_d + Q(X))^{-1} (i\epsilon 1_d - Q(X))^{-1} u^\ast\\
&= \Im(z) + \epsilon u (\epsilon^2 1_d + Q(X)^2)^{-1} u^\ast\\
&\geq \Im(z),
\end{align*}
since $(\epsilon^2 1_d + Q(X)^2)^{-1} \geq 0$. This implies, as desired, that $h$ is invertible with $\|h^{-1}\| \leq \frac{1}{\Im(z)}$.
Next, we note that according to the Schur complement formula
$$\big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1} = \tau\big( \big(z-u(i\epsilon 1_d - Q(X))^{-1} u^\ast \big)^{-1} \big).$$
Thus, we obtain with the help of the resolvent identity
\begin{align*}
\lefteqn{G_{p(X)}(z) - \big[\mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1}}\\
&\qquad = \tau\big((z-p(X))^{-1}\big) - \tau\big( \big(z-u(i\epsilon 1_d - Q(X))^{-1} u^\ast \big)^{-1} \big)\\
&\qquad = \tau\big((z-p(X))^{-1} \big( p(X) - u(i\epsilon 1_d-Q(X))^{-1}u^\ast \big) \big(z-u(i\epsilon 1_d - Q(X))^{-1}u^\ast\big)^{-1}\big)\\
&\qquad = -i\epsilon\, \tau\big((z-p(X))^{-1} u Q(X)^{-1} (i\epsilon 1_d - Q(X))^{-1} u^\ast \big(z-u(i\epsilon 1_d-Q(X))^{-1}u^\ast\big)^{-1}\big).
\end{align*}
Let us consider now the elements in $\mathcal{M}^d \subset L^2(\mathcal{M},\tau)^d$ that are given by
\begin{align*}
\Psi_1 &:= (i\epsilon 1_d-Q(X))^{-1} u^\ast \big(z-u(i\epsilon 1_d-Q(X))^{-1}u^\ast\big)^{-1},\\
\Psi_2 &:= Q(X)^{-1} u^\ast (\overline{z}-p(X))^{-1}.
\end{align*}
With these abbreviations, we can rewrite the previous result as
$$G_{p(X)}(z) - \big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1} = -i\epsilon\, \langle \Psi_1, \Psi_2 \rangle_{L^2(\mathcal{M},\tau)^d}.$$
Thus, by the Cauchy-Schwarz inequality on $L^2(\mathcal{M},\tau)^d$, we get that
$$\big|G_{p(X)}(z) - \big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1}\big| \leq \epsilon\, \|\Psi_1\|_{L^2(\mathcal{M},\tau)^d} \|\Psi_2\|_{L^2(\mathcal{M},\tau)^d}.$$
One easily sees that
$$\|\Psi_2\|_{L^2(\mathcal{M},\tau)^d} \leq \frac{1}{\Im(z)}\, \|Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}$$
and similarly, by \eqref{eq:inverse-bound}, we get that
$$\|\Psi_1\|_{L^2(\mathcal{M},\tau)^d} \leq \frac{1}{\Im(z)}\, \|(i\epsilon 1_d-Q(X))^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}.$$
Combining these observations leads us to
$$\big|G_{p(X)}(z) - \big[ \mathbf{G}_{\hat{p}(X)}(\Lambda_\epsilon(z)) \big]_{1,1}\big| \leq \frac{\epsilon}{\Im(z)^2} \|Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d} \|(i\epsilon 1_d-Q(X))^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}.$$
Finally, we involve $1_d = u_1^\ast u_1 + \dots + u_d^\ast u_d$ in order to obtain
$$\|Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}^2 = \sum^d_{j=1} \tau(u Q(X)^{-1} u_j^\ast u_j Q(X)^{-1} u^\ast) = \sum^d_{j=1} \|p_j(X)\|_2^2.$$
Furthermore, by the resolvent identity,
\begin{align*}
\lefteqn{\|(i\epsilon 1_d-Q(X))^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}}\\
&\qquad \leq \|Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d} + \epsilon\, \|(i\epsilon 1_d-Q(X))^{-1} Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}\\
&\qquad \leq 2 \|Q(X)^{-1} u^\ast\|_{L^2(\mathcal{M},\tau)^d}.
\end{align*}
Thus, in summary, we arrive at \eqref{eq:approximation_improved}, which concludes the proof of Item \ref{it:approximation_improved}.
\end{proof}
\end{appendix}
\bibliographystyle{amsalpha}
|
1,314,259,993,057 | arxiv | \section{}
Whether a supersolid state, where both of solidity and superfluidity coexists, realizes in matters or not. The possibility of the supersolid in ${\rm ^4He}$ have been discussed theoretically\cite{Andreev,Chester,Leggett,Matsuda,Liu}. Recently, characteristic behaviors of superfluidity have been reported in torsional oscillator experiments on the solid ${\rm ^4He}$ by E. Kim and M. H. W. Chan(KC)\cite{Kim}, while a number of experiments failed to detect the superfluidity in the solid ${\rm ^4He}$. In their measurements, a sudden drop in the resonant period was observed around $T \sim$ 0.2K. The drop implies emergence of non-classical rotational inertia in the solid ${\rm ^4He}$. They concluded that this was a signature of a transition into the supersolid phase. However, some experimental and theoretical studies suggested different interpretations of KC's observation. For instance, the signal of the superfluidity in the solid ${\rm ^4He}$ became weaker as annealing cycles were repeated\cite{Ritter} and the superflow was blocked by the solid ${\rm ^4He}$ with no grain boundaries\cite{sasaki}. Theoretically, the possibility of the superflow induced by vacancies in the commensurate solid ${\rm ^4He}$, in which the total number of atoms equals a multiple of the number of lattice sites, was ruled out\cite{Clark,Boninsegni}, and the superglass behavior appeared in a quenched system\cite{Boninsegni2}. Although several mechanisms of the superflow in the solid ${\rm ^4He}$ have been proposed, the satisfactory interpretation of KC's results is still controversial.
Bosonic lattice model was introduced as a reasonable model of liquid ${\rm ^4He}$\cite{MM}. Recently, the possibility of the supersolid on the lattice model in triangular lattice\cite{Wessel,Heidarian,Melko,Boninsegni3} and kagome lattice\cite{kagome} cases was studied by quantum Monte Carlo simulations. From these studies it became clear that the frustrated interactions on the triangular lattice stabilize the super current induced by vacancies in the crystalline ordering with the wave vector ${\bf Q}$=$(4\pi/3,0)$ or $(2\pi/3,0)$. In contrast, the supersolid is not stabilized on the kagome lattice where the frustrated interactions exist. In the three dimensional lattice cases, the phase diagrams of the system on the body-centered-cubic (BCC) lattice was obtained by a mean-field approximation and concluded that the supersolid state appears if the next-nearest-neighbor interactions are present\cite{Matsuda,Liu}. However, the reason for the stabilization of the supersolid state on the BCC lattice and the microscopic picture was not cleared from the mean-field results. (Note that the BCC lattice is bipartite and has no frustration if one does not take into account the next-nearest-neighbor interactions.)
Theoretical study beyond the mean-field theory for three dimensional systems is still missing. In this letter, the supersolid state in a three-dimensional bosonic-lattice model is studied by a quantum Monte Carlo method based on the directed loop algorithm\cite{Sandvik,Harada}. We wish to address a generic question what ingredient is necessary to realize the supersolidity. From the study of the two-dimensional case mentioned above, it is presumable that the geometrical frustration plays an essential role in the supersolidity in the bosonic lattice model. We therefore focus on a hardcore-bosonic model on the face-centered-cubic (FCC) lattice, which does not have a direct connection to the lattice structure of the real solid helium, because the FCC lattice is one of the simplest lattices with geometric frustration.
More specifically, we consider bosonic lattice model with the positive hopping amplitude $t>0$ and the nearest-neighbor repulsion $V>0$ on the FCC lattice. The model Hamiltonian is defined by
\begin{eqnarray}
{\mathcal H}=-t\sum_{\langle ij \rangle} \left( {b_{i}}^{\dagger}b_{j} + h.c. \right) + V\sum_{\langle ij \rangle}\hat{n}_{i}\hat{n}_{j} -\mu\sum_{i}\hat{n}_{i},
\label{Ham1}
\end{eqnarray}
where $\mu$ is the chemical potential, ${b_{i}}^{\dagger}(b_{i})$ is the bosonic creation (annihilation) operator, and $\hat{n}_{i}={b_{i}}^{\dagger}b_{i}$. The summation $\langle ij \rangle$ is over the nearest neighbor pairs and the system size is defined by $N=L^3$. The periodic boundary condition is applied. Under the hardcore condition, the original bosonic-lattice model is identically mapped onto the $S$=$1/2$ XXZ model,
\begin{eqnarray}
{\mathcal H}&=&-J_{\perp}\sum_{\langle ij \rangle}({S_{i}}^{x}{S_{j}}^{x}+{S_{i}}^{y}{S_{j}}^{y})-J_{z}\sum_{\langle ij \rangle}{S_{i}}^{z}{S_{j}}^{z}\nonumber\\
& &-H\sum_{i}{S_{i}}^{z},
\label{Ham2}
\end{eqnarray}
where $J_{\perp}=2t$, $J_z=-V$ and $H=\mu-6V$. Note that $J_\perp$ and $J_z <(>)0$ mean the antiferromagnetic (ferromagnetic) interactions. In the spin language, the supersolidity is characterized by the following two properties: sublattice-dependent expectation values of the longitudinal spin components (broken translational symmetry, or crystallization), and non-vanishing transverse spin components (off-diagonal long range order, or superfluidity).
In the limit $J_{\perp}/|J_{z}|\rightarrow0$ (Ising model), the ordered states were investigated and the $H$-$T$ phase diagram was obtained\cite{Binder,Hagai,Kammerer}. At the magnetization $m$=$\langle\sum_i {S_{i}}^z/N \rangle$=$0$ and $1/4$, there appear two solid phases conventionally referred to as ${\rm AB}$ and ${\rm A_3B}$. Representative spin configurations of the two phases are shown in Fig. \ref{spin-arr}. The phase transition from the ${\rm AB}$ phase to the ${\rm A_3B}$ phase occurs at $H_{\rm Ising}=2|J_z|$ at absolute zero temperature.
\vspace{-5 mm}
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=5mm 0mm 100mm 0mm,angle=270, scale=0.33]{fig1.eps}
\vspace{-5 mm}
\caption{The perfectly ordered spin configurations. (a) ${\rm AB}$ state and (b) ${\rm A_3B}$ state. Note that the dashed lines connecting next-nearest neighbors are mere guide lines to the eye and there is no direct coupling corresponding to these lines. (The direct couplings exist only for nearest neighbor pairs.) The gray lines denote the superflow paths in the supersolid state (see text).}
\label{spin-arr}
\end{center}
\end{figure}
In order to investigate the crystalline order and the off-diagonal long-range order for $J_{\perp}> 0$, we calculate the static structure factor $S({\bf Q})$ (SSF) and the superfluid density $\rho_s$\cite{Pollack}, defined by
\begin{eqnarray}
S({\bf Q})&=&\left\langle \left| \sum_{i} \exp [i{\bf Q}\cdot{\bf r}_{i}] S_{i}^{z} \right|^{2}\right\rangle
\label{sq}
\end{eqnarray}
and
\begin{eqnarray}
\rho_s&=&\frac{k_BT\left\langle {\bf W}^2 \right\rangle}{3J_{\perp}L},
\end{eqnarray}
where ${\bf W}=(W_x,W_y,W_z)$ denotes the winding number of the world-lines. In what follows, we express the wave vector ${\bf Q}$ by the conventional choice of the unit reciprocal vectors.
When the system is in the ${\rm AB}$ or ${\rm A_3B}$ ordered state, the SSF is proportional to $N$ and strong system-size dependence is expected at ${\bf Q_{sol}}$=$(\pi,\pi,0)$, $(\pi,0,\pi)$, and $(0,\pi,\pi)$. For other ${\bf Q}$s, the SSF should be system-size independent. It was reported for the classical case ($J_{\perp}$=$0$)\cite{Kammerer,Ackermann} that a perfect solid state, either ${\rm AB}$ or ${\rm A_3B}$, can hardly be observed in a system of computationally accessible size due to antiphase domain boundaries (APB). While the states with domain boundaries have negligible weight in the thermodynamic limit, they have non-negligible contributions for small systems because the domain-wall free-energy is small due to the frustrated nature of the interactions. The APBs reduce $S({\bf Q_{sol}})$ since contributions from different phases have opposite signs. However, the magnitude of the reduction depends on the locations of the APBs and the cancellation does not in general make $S({\bf Q_{sol}})$ completely vanishing. Therefore, the average $S({\bf Q_{sol}})$ is still proportional to the system size even if the APBs are present. As for the effect of the APBs on the superfluid density, we have confirmed that it is relatively minor compared to that on $S({\bf Q_{sol}})$. To see this, we evaluated $\rho_s$ in two ways (see Fig. 4 (b)). One is a long equilibrium simulation starting from random initial configurations, in which APBs are observed. The other is relatively short Monte Carlo simulations starting from the perfect ${\rm AB}$ or ${\rm A_3B}$ configuration. In the latter, the length of the simulation is chosen such that APBs do not appear. In both cases, the superfluid density yielded the same value within the statistical error.
In Fig. \ref{sq-res}, we show the results of the field dependence of $S({\bf Q_{sol}})$ and $\rho_s$ at $(J_{\perp},J_z)=(0.2,-1.0)J$ and $k_BT=0.1J$, where $J\equiv|J_z|$ is our unit of energy. The $H$-axis can be divided into four regions according to the behaviors of $S({\bf Q_{sol}})$ and $\rho_s$; (I) the low-field region $0<H<H_{\rm solid1}\sim1.15J$, (II) the lower-intermediate region $H_{\rm solid1}<H<H^*\sim2.2J$, (III) the upper-intermediate region $H^*<H<H_{\rm solid2}\sim3.1J$, and (IV) the high-field region $H_{\rm solid2}<H$, where $H_{\rm solid1}$, $H^*$, and $H_{\rm solid2}$ are temperature-dependent transition fields.
In the regions I, III and IV, the crystalline order exists. This is evident from the fact that $S({\bf Q_{sol}})$ increases in proportion to the system size. In the region II, on the other hand, $S({\bf Q_{sol}})$ does not show a system-size dependence indicating no crystalline ordering in this region. The superfluid density $\rho_s$ is almost zero in the regions I and IV whereas in the intermediate regions II and III, it stays finite. Judging from these results, we conclude that the ground state is the solid state in the regions I and IV. As shown in Fig. \ref{sz-res}, the magnetization plateaus at $m$=$0$ and $1/4$ appear in the corresponding fields. Therefore, these solid phases are the ${\rm AB}$ and ${\rm A_3B}$ ordered phases, respectively. The region II is the superfluid phase. Finally in the region III, since the crystalline order and the superfluidity coexist, there appears the supersolid phase. Hence, we conclude that the supersolid state is stable in three dimensions.
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=0mm 0mm 0mm 0mm ,scale=0.38]{fig2.eps}
\vspace{-3 mm}
\caption{The field dependence of $S({\bf Q_{sol}})$ and $\rho_s$ at $(J_{\perp},J_z)=(0.2,-1.0)J$ and $k_BT=0.1J$. The open circles, the solid circles, and the inverted triangles denote the results of $L=6$, $12$, and $18$, respectively. Note that the results of $S({\bf Q_{sol}})$ are averaged values of $S(0,\pi,\pi)$, $S(\pi,0,\pi)$ and $S(\pi,\pi,0)$. The inset shows the system size dependence of $S({\bf Q_{sol}})/N$ at $H/J_z=3.0$.}
\label{sq-res}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=0mm 0mm 0mm 0mm ,scale=0.40]{fig3.eps}
\vspace{-5 mm}
\caption{The field dependence of the magnetization at $(J_{\perp},J_z)$=$(0.2, -1.0)J$ and $k_BT$=$0.1J$. The value of $m$=$1/4$ corresponds to a half of the saturation magnetization.}
\label{sz-res}
\end{center}
\end{figure}
Next we study the temperature dependence. The results at $H=2.7J$ (in the supersolid region III) are shown in Fig. \ref{scaling}. As we decrease the temperature with fixed magnetic field, $S({\bf Q_{sol}})$ almost discontinuously increases at $k_BT$=$k_BT_{{\rm solid}}\sim0.32J$ and the system-size dependence appears for $T<T_{{\rm solid}}$. However, the superfluid density remains very small; $\rho_s<5\times 10^{-3}$ near $T_{{\rm solid}}$. This is the transition from the normal fluid to the solid phase. Since $S({\bf Q_{sol}})$ takes the same values of those in the ${\rm A_3B}$ solid region IV and scarcely shows the temperature dependence in $T<T_{\rm solid}$, we identify this solid phase as the ${\rm A_3B}$ ordered phase. At even lower temperature, the system undergoes another transition from the solid phase to the supersolid phase. This is marked by the increase in the superfluid density $\rho_s$ that starts at $k_BT_{{\rm super}}\sim0.22J$. To estimate $T_{{\rm super}}$, we analyze the finite-size-scaling behavior of the superfluid density $\rho_s$ by the scaling form $\rho_s L$=$f(L^{1/\nu}(T-T_{\rm super}))$ using the exponents of the three dimensional $XY$ model $\nu$=$0.6723$\cite{Hasenbusch}. As shown in the inset of Fig. \ref{scaling}(b), the data collapse is obtained with the critical temperature, $k_BT_{{\rm super}}$=$0.221(2)J$. Thus, we conclude that the phase transition from the solid phase to the supersolid phase is of the second order and its universality class is that of the three dimensional $XY$ model as expected. In this way, we estimate the critical temperatures $T_{{\rm solid}}$ and $T_{{\rm super}}$ for various other values of $H$ to obtain the phase boundary. Here, $T_{{\rm solid}}$ and $T_{{\rm super}}$ denote the transition temperatures where the crystalline order and the superfluid order, respectively, emerge. The results are shown in Fig. \ref{phased}.
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=0mm 0mm 0mm 0mm ,scale=0.38]{fig4.eps}
\vspace{-3 mm}
\caption{The temperature dependence of $S({\bf Q_{sol}})$ and $\rho_s$ for $(J_{\perp},J_z)$=$(0.2, -1.0)J$ and $H$=$2.7J$. In (b), the cross symbols denote the results starting from the perfect ${\rm A_3B}$ configuration in $L$=$18$ and the others are those starting from the random initial configurations. The inset in (b) is the finite-size scaling of the superfluid density.}
\label{scaling}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=20mm 140mm 50mm 15mm,scale=0.45]{fig5.eps}
\caption{The $H$-$T$ phase diagram for $(J_{\perp},J_z)$=$(0.2,-1.0)J$. The open circles and the solid circles indicate the first-order and the second-order transitions, respectively. The solid lines are mere guide lines to the eye. The labels, "N.F.", "S.F.", and "S.S." stand for the normal fluid phase, the superfluid phase, and the supersolid phase, respectively.}
\label{phased}
\end{center}
\end{figure}
From the phase diagram in Fig.\ref{phased}, we find that the supersolid phase exists between the ${\rm AB}$ and ${\rm A_3B}$ solid phases. This region locates slightly above $H_{\rm Ising}$=$2.0J$ at which the phase transition occurs in the classical case ($J_{\perp}$=$0$) from the ${\rm AB}$ to ${\rm A_3B}$ phase. To discuss the mechanism of the supersolid state on the FCC lattice, we consider the spin configuration above $H_{\rm Ising}$. In the classical case, at the critical field, the spins at the centers of faces of the cubic lattice (the up spins in Fig. \ref{spin-arr}(b)) become {\it dangling} spins; they can be reversed without changing the energy. Let us regard down (up) spins at these locations as hard-core particles (holes). Then, $H-H_{\rm Ising}$ can be interpreted as the excitation gap for creating a particle and the ground state is the empty state at $H$ larger than the critical value. However, once the hopping ($J_{\perp}$) is turned on, the excited particles may move along the gray lines in Fig.\ref{spin-arr}(b) and can in general condense. If the magnetic field is far larger than $H_{\rm Ising}$, the classical gap is larger than the scale of the hopping constant $J_{\perp}$ (i.e., the band width of particle excitation), the gap remains open even if the quantum hopping is present. As we decrease the magnetic field, however, at some point the classical gap becomes smaller than the scale of the hopping constant. Accordingly the actual gap closes and the ground state starts exhibiting superfluidity. At this point, in contrast to the spins on the faces, the spins at the corners can hardly be affected by the hopping term, because the energy cost of reversing one of these spins is $\Delta E$$\sim\frac{3}{4}|J_z|$ and is still too large. Therefore, they stay in a solid crystalline order. As we further decrease the magnetic field, the density of condensed particles at the dangling spin locations gradually increases. This generates positive molecular fields at the corners, destabilizing the crystalline order. This destabilizing effect finally melts the crystal at $H$=$H^{\ast}$. This latter transition point must be larger than $H_{\rm Ising}$ because the transition must take place before the classical excitation gap closes and therefore the density of the excited particles diverges. This is the microscopic scenario of the two transitions, the solid to the supersolid transition and the supersolid to the super fluid transition. Indeed, we successfully confirmed the existence of the ${\rm A_3B}$-type supersolid states in the corresponding parameter region in the present simulation.
This scenario predicts a supersolid phase of another type in the region $H<H_{\rm Ising}$, which we could call ${\rm AB}$-type. The mechanism of the ${\rm AB}$-type supersolid is again understood by the {\it dangling} spins. This time, the dangling spins appear at the sites occupied by down spins in Fig.\ref{spin-arr}(a), and we should regard the up spins on these sites as excited particles, which condense in the AB-type supersolid phase that locates below $H_{\rm Ising}$. The excited particles hop along the gray lines in Fig.\ref{spin-arr}(a), while rigid spins (those pointed up in Fig.\ref{spin-arr}(a)) stay in the crystalline order. The ${\rm AB}$-type supersolid has the characteristic two-dimensional paths of the superfluid due to the alternatively stacks of the superfluid and solid layers, while the ${\rm A_3B}$-type supersolid has the three dimensional superfluid connections. Perturbatively, the effective interactions between these superfluid layers may arise in the second order of $J_{\perp}$. In Fig. 6, we show some results of the superfluid density and the structure factor at $(J_{\perp},J_z)$=$(0.15,-1.0)J$ and $H$=$1.65J<H_{\rm Ising}$. In this case, while an anomaly, which is cleared by the calculations in the larger system size, appears in $S({\bf Q_sol})$ at $k_BT\sim 1.5$ due to the APB's, the ${\rm AB}$-type supersolid realizes in the region $k_{B}T < k_{B}T_{\rm super} \sim 0.104(2)$.
\begin{figure}[htb]
\begin{center}
\vspace{3 mm}
\includegraphics[trim=0mm 0mm 0mm 0mm ,scale=0.35, angle=270]{fig6.eps}
\vspace{-3 mm}
\caption{(i)The temperature dependence of $S({\bf Q_{sol}})$ and $\rho_s$ in $(J_{\perp},J_z)$=$(0.15, -1.0)J$ and $H$=$1.65J$. (ii)The cross-section of a spin configuration in the AB-type supersolid at $k_BT=0.1$. The gray circles denote up spins and the unoccupied sites correspond to down spins.}
\label{ABss}
\end{center}
\end{figure}
To summarize, we have calculated the SSF and $\rho_s$ for $S$=$1/2$ XXZ model on the FCC lattice and obtained a phase diagram at fixed $J_{\perp}/J_z$. We have also discussed the microscopic mechanism of the supersolidity in the present model and pointed out that {\it the connections of dangling spins} resulting from the geometrical frustration play a key role in the formation of the supersolid state.
We would like thank Y. S. Wu, M. Kohmoto, K. Harada, and C. Batista for useful comments and fruitful discussions. This work is supported by Next Generation Supercomputing Project, Nanoscience Program, MEXT, Japan. Numerical computations were carried out at the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo.
|
1,314,259,993,058 | arxiv | \section{Introduction}
Whole genome sequences are readily available and affordable like never before \cite{NGS} due to the advent of high-throughput Next Generation Sequencing (NGS) which has provided researchers with vast amounts of genomic sequencing data that has transformed the landscape of understanding of genomes.
The field of phylogenetics, which discovers the evolutionary relationship between taxa, has been no exception to this transformation.
Phylogenetics has responded to the copious amounts of high throughput data with novel alignment-free and assembly-free methods \cite{AAF,CVTree} that are better suited \cite{NextGenPhylo} to handle the large amounts of data more efficiently than the traditional alignment-based phylogenetic methods.
The traditional approach to phylogenetic tree reconstruction requires a homology search throughout the genomes of the taxa, a Multiple Sequence Alignment (MSA) of the homologs, and a tree construction from the resulting matrix.
Each of these steps can be computationally expensive and may introduce many unnecessary assumptions that can be avoided by using an alignment-free and assembly-free method.
Alignment-free and assembly-free methods \cite{AlignFreeReview,AlignFreeReview2,NoMSA,Cophylog} don't come without their disadvantages, one of which being that many of these methods abstract away the source of the phylogenetic signal to a method akin to shared kmer-counting.
We propose an assembly-free whole genome phylogenetic tree reconstruction method using the Colored de Bruijn Graph (CdBG) \cite{CdBG}, a data structure that is commonly used for detecting variation and comparing genomes.
The CdBG is similar to a traditional de Bruijn Graph (dBG) in that the substrings of a certain length, referred to as kmers, of a sequence represent the vertices of the dBG and an edge exists between two vertices if the suffix of the first vertex is the prefix of the second vertex.
The CdBG differs from the traditional dBG in that each vertex is associated to an unique color (or set of colors) which could be a differing sample, species, or taxon.
We introduce the \texttt{kleuren} (Dutch for "colors" in tribute of Nicolaas Govert de Bruijn, the de Bruijn graph's namesake) software package which implements our methods.
\texttt{kleuren} works by finding \textit{bubble} regions \cite{CdBG,Bubbles} of the CdBG, which are where one or more colors diverge at a node, which act as pseudo-homologous regions between the taxa.
The sequence for each taxon in each bubble is then extracted and a MSA is performed, then the MSA's from each bubble are concatenated to form a supermatrix in which a phylogenetic tree of evolution is constructed.
\section{Methods}
\subsection{Definitions}
Given the alphabet $\Sigma = \{A, C, G, T\}$ which are nucleotide codes, let a dBG $\mathbf{G}$, be defined as $\mathbf{G} = (V, E)$ where $V = \{v_1, v_2, \ldots, v_i, \ldots, v_s\}$ is the set of vertices and where $v_i$ is the $i^{th}$ unique sequence of length $k$ of $\mathbf{G}$ and where $E = \{e_1, e_2, \ldots, e_i, \ldots, e_t\}$ is the set of edges and where $e_i = \left(v_i, v_{i+1}\right)$ is an edge connecting two vertices such that the sequence of $v_i$ and $v_{i+1}$ overlap by $(k-1)$ characters.
Let a CdBG, $\mathbf{CG}$, be defined as $\mathbf{CG} = \{G_1, G_2, \ldots, G_i, \ldots, G_u\}$ for $u$ taxa where $G_i = \left(V_i, E_i\right)$ is the dBG of the $i^{th}$ taxon.
We refer to each $G \in \mathbf{CG}$ as a distinct color or taxon.
Furthermore, let a \textit{path}, $P = \left(v_1,\ldots, v_w\right)$ in $G_i$ be defined as a sequence of vertices from $V_i$ such that for all subsequences $\left(v_j,v_{j+1}\right)$ of $P$, the edge $\left(v_j, v_{j+1}\right) \in E_i$.
Let a \textit{bubble}, $B$, in $\mathbf{CG}$ be defined as $B = \{P_1, \ldots, P_z\}$ such that each $P \in B$ is associated with one or more colors, that the first and last vertices of $\forall P \in B$ are identical, and that $2 \leq z \leq u$ (see Figure~\ref{fig:bubble}).
Finally, let $\mathbf{K}$ be defined as $\mathbf{K} = \{V_1 \cup V_2 \cup \ldots \cup V_i \cup \ldots \cup V_u\}$ where $V_i$ is the vertices (or the unique kmers) of the $i^{th}$ dBG, $\mathbf{G_i}$.
\begin{figure}
\centering
\includegraphics[scale=0.55]{bubble-graph}
\begin{flushleft}
\textbf{A.} Bubble in a Colored de Bruijn Graph \\
\end{flushleft}
\begin{tabular}{cl}
{\color{yellow}Color 1} & Path: ACTGTG \\
{\color{red}Color 2} & Path: ACTAGGTG \\
{\color{blue}Color 3} & Path: ACTAGTG \\
\end{tabular}
\begin{flushleft}
\textbf{B.} Paths in the Bubble of Each Color \\
\end{flushleft}
\caption{\textbf{A.} An example of a bubble in a Colored de Bruijn Graph with $3$ colors (i.e. $3$ taxa), and where $k=3$.
The colors of the vertices represent the following: gray- all colors contain the vertex, purple- Color 2 and Color 3 contain the vertex, yellow- Color 1 contains the vertex, red- Color 2 contains the vertex, and blue- Color 3 contains the vertex.
In this example \textit{ACT} is the \textit{startVertex} and \textit{GTG} is the \textit{endVertex} which are both contained in all of the colors.
\textbf{B.} The extended paths of each color between the \textit{startVertex} and \textit{endVertex}.
\label{fig:bubble}}
\end{figure}
\subsection{Software Architecture}
We use the \texttt{dbgfm} software package \cite{dbgfm} to construct and represent the dBG's of the individual taxa.
\texttt{kleuren} provides an interface to interact with the individual dBG's to create a CdBG, where each taxon is considered a color.
The \texttt{dbgfm} package uses the FM-Index \cite{FM-Index}, as a space efficient representation of the dBG.
\subsection{\texttt{kleuren} Algorithms}
\subsubsection{Overall Algorithm}\label{overall}
\begin{algorithm}
\caption{kleuren Algorithm}\label{overallAlg}
\begin{algorithmic}[1]
\Function{kleuren}{$\mathbf{K}, \mathbf{CG}$}
\State $bubbles \gets \left[ \ \right]$ \Comment{$bubbles$ is initialized to an empty list}
\ForAll{$k \in \mathbf{K}$}
\If{$k$ is in $c$ or more colors of $\mathbf{CG}$}
\State $endVertex \gets$ \Call{findEndVertex}{$k, \mathbf{CG}$}
\ForAll{$color \in \mathbf{CG}$}
\State $path \gets$ \Call{extendPath}{$k, endVertex,$ $color$}
\State add $path$ to $bubble$
\EndFor
\State append $bubble$ to $bubbles$
\EndIf
\EndFor
\State $alignments \gets \left[ \ \right]$
\ForAll{$bubble \in bubbles$}
\State $alignment \gets$ multiple sequence alignment of each $path$ in $bubble$
\State append $alignment$ to $alignments$
\EndFor
\State $supermatrix \gets$ concatenation of $alignments$
\EndFunction
\end{algorithmic}
\end{algorithm}
\texttt{kleuren} works by iterating over the superset of vertices, $\mathbf{K}$, and discovering vertices that could form a \textit{bubble}.
A vertex, $s$, could form a \textit{bubble} if $s$ is present in $c$ or more colors of $\mathbf{CG}$, where $c$ is set by the user as a command line parameter.
Note that the lower that $c$ is, the more potential bubbles that may be found, but \texttt{kleuren} will take longer to run because more vertices will be considered as the starting vertex of a \textit{bubble}.
Let $s$ be considered as the starting vertex of the \textit{bubble}, $b$; then the end vertex, $e$, of $b$ is found (see Section~\ref{findEndVertex}).
After the end vertex is found, the path, $p$, between $s$ and $e$ is found for each color in $\mathbf{CG}$ (see Section~\ref{extendPath}).
This process is repeated until each vertex in $\mathbf{K}$ has been either considered as a starting vertex of a \textit{bubble}, or has been visited while extending the path between a starting and ending vertex.
\subsubsection{Finding the End Vertex}\label{findEndVertex}
\begin{algorithm}
\caption{Find End Vertex Function}\label{findEndVertexAlg}
\begin{algorithmic}[1]
\Function{findEndVertex}{$startVertex, \mathbf{CG}$}
\State $endVertex \gets ``~"$ \Comment{$endVertex$ is initialized to an empty string}
\State $neighbors \gets \Call{getNeighbors}{startVertex}$
\While{$!\Call{isEmpty}{neighbors}\ and\ $ $\Call{isEmpty}{endVertex}$}
\ForAll{$neighbor \in neighbors$}
\If{$k$ is in $c$ or more colors of $\mathbf{CG}$}
\State $endVertex \gets neighbor$
\EndIf
\EndFor
\EndWhile
\State \Return{$endVertex$}
\EndFunction
\end{algorithmic}
\end{algorithm}
The end vertex is found by traversing the path from the $startVertex$ until a vertex is found that is in at least $c$ colors.
The $endVertex$ is then used in the function to extend the path (see Section~\ref{extendPath}).
\subsubsection{Extending the Path}\label{extendPath}
\begin{algorithm}
\caption{Extend the Path Functions}\label{extendPathAlg}
\begin{algorithmic}[1]
\Function{extendPath}{$startVertex, endVertex, color$, $maxDepth$}
\State $path \gets ``~"$
\State $visited \gets \{\}$ \Comment{$visited$ is initialized to the empty set}
\If{\Call{recursivePath}{$startVertex, endVertex, path,$ $color,visited, 0, maxDepth$}}
\State \Return{$path$}
\EndIf
\EndFunction
\\
\Function{recursivePath}{$currentVertex, endVertex,$ $path, color, visited, depth, maxDepth$}
\State add $currentVertex$ to $visited$
\If{$depth >= maxDepth$}
\State \Return $false$
\EndIf
\If{$currentKmer == endKmer$}
\State \Return $true$
\EndIf
\State $neighbors \gets \Call{neighbors}{currentVertex, color}$
\ForAll{$neighbor \in neighbors$}
\If{$neighbor$ is in $visited$}
\State continue
\EndIf
\State $oldPath \gets path$
\State append suffix of $currentKmer$ to $path$
\State $depth \gets depth + 1$
\If{!\Call{recursivePath}{$neighbor, endVertex, path,$ $color, visited, depth, maxDepth$}}
\State $path \gets oldPath$
\Else
\State \Return{$true$}
\EndIf
\EndFor
\EndFunction
\end{algorithmic}
\end{algorithm}
The main functions that discover the sequences found in a bubble are the Extend the Path Functions (see Section~\ref{extendPath}).
To extend the $path$ between the $startVertex$ and $endVertex$ we use a recursive function that traverses the dBG for a color in which every possible path between the $startVertex$ and $endVertex$ is explored up to the $maxDepth$ (provided as a command line parameter by the user).
The $maxDepth$ parameter allows the user to specify how thorough \texttt{kleuren} will search for a \textit{bubble}; the higher the $maxDepth$ the more \textit{bubbles} that \texttt{kleuren} will potentially find, but the longer \texttt{kleuren} will take because at each depth there are exponentially more potential paths to traverse.
\subsection{Data Acquisition}
To measure the effectiveness of our method we used 12 \textit{Drosophila} species, obtained from FlyBase \cite{FlyBase}.
We chose this group of species because there is a thoroughly researched and established phylogenetic tree \cite{Hahn-true-tree}.
\subsection{Tree Construction and Parameters}\label{sec:tree-construction}
We used the DSK software package \cite{DSK} to count the kmers present in all of the \textit{Drosophila} species.
To find the bubbles, we used the following parameters: $k = 17$ (kmer size of $17$) and $c = 12$ (all colors in the $\mathbf{CG}$ were required to contain a vertex in order to search for a bubble starting at that vertex) and ran $32$ instances of \texttt{kleuren} concurrently for $4$ days to find $3,277$ bubbles.
When all of the \textit{bubbles} in the CdBG had been identified, we used MAFFT \cite{MAFFT} to perform a MSA for each sequence in every \textit{bubble} that \texttt{kleuren} identified (see Figure~\ref{fig:overall} A.).
Then each MSA was concatenated to form a supermatrix (see Figure~\ref{fig:overall} B.) using Biopython \cite{Biopython}.
The phylogenetic tree was then inferred from the supermatrix by Maximum Likelihood using IQ-TREE \cite{iqtree} (see Figure~\ref{fig:overall} C.).
Once the tree was constructed, we used the ETE 3 software package \cite{ETE3} to compare the tree to the established one and Phylo.io \cite{phylo.io} to visualize the trees.
\subsection{Bubble Assumptions}
Our method is based on the assumption that \textit{bubbles} are representative of homologous regions of the taxa genomes.
We propose that this assumption is reliable because it has been shown that dBG's are a suitable method to align sequences \cite{MultipleAlignment,Sibelia,TwoPaCo}, and by identifying the \textit{bubbles} in the CdBG we find the sections of the graph that contain the most phylogenetic signal.
\begin{figure}
\centering
\begin{tabular}{cl}
{\color{yellow}Color 1} & Path: ACT\texttt{-}\texttt{-}GTG \\
{\color{red}Color 2} & Path: ACTAGGTG \\
{\color{blue}Color 3} & Path: ACTA\texttt{-}GTG \\
\end{tabular}
\begin{flushleft}
\textbf{A.} Multiple Sequence Alignment of the Sequences in Bubble (Figure~\ref{fig:bubble}) \\
\end{flushleft}
\medskip
\begin{tabular}{cl}
{\color{yellow}Color 1} & Path: ACT\texttt{-}\texttt{-}GTGATT\texttt{-}A... \\
{\color{red}Color 2} & Path: ACTAGGTGATTC\texttt{-}... \\
{\color{blue}Color 3} & Path: ACTA\texttt{-}GTGATTCA... \\
\end{tabular}
\begin{flushleft}
\textbf{B.} Supermatrix of Multiple Sequence Alignments concatenated \\
\end{flushleft}
\medskip
\begin{forest}
forked edges,
/tikz/every pin edge/.append style={Latex-, shorten <=2.5pt, darkgray},
/tikz/every pin/.append style={darkgray},
/tikz/every label/.append style={darkgray},
before typesetting nodes={
delay={
where content={}{coordinate}{},
},
where n children=0{tier=terminus, label/.wrap pgfmath arg={right:#1}{content()}, content=}{},
},
for tree={
grow'=0,
s sep'+=10pt,
l sep'+=15pt,
},
l sep'+=10pt,
[,
[
[\color{blue} Color 3]
[\color{red} Color 2]
]
[\color{yellow} Color 1]
]
\end{forest}
\begin{flushleft}
\textbf{C.} Phylogenetic Tree \\
\end{flushleft}
\caption{\textbf{A.} The Multiple Sequence Alignment (MSA) of the sequences from the bubble presented in Figure~\ref{fig:bubble}.
\textbf{B.} The MSA's from each bubble are concatenated into a supermatrix, from which a phylogenetic tree is constructed.
\textbf{C.} The resulting tree from the supermatrix inferred by Maximum Likelihood.
\label{fig:overall}}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.45]{Tree_mod}
\caption{The phylogenetic tree of 12 \textit{Drosophila} species constructed using \texttt{kleuren}.
This tree resulted from using a kmer size of 17 and required all species to contain a vertex in order for the algorithm to search for a bubble starting at that vertex; and this tree is consistent with the established tree for these 12 species.
\label{fig:tree}}
\end{figure*}
\section{Results}\label{results}
\texttt{kleuren} constructed a tree (see Figure~\ref{fig:tree}) consistent with the established tree found in \cite{Hahn-true-tree} (the Robinson-Foulds distance \cite{Robinson1981} between the two trees is 0).
Even though we ran many concurrent instances of \texttt{kleuren} for multiple days (see Section~\ref{sec:tree-construction}), not all of the kmers in $\mathbf{K}$ were explored for potential bubbles; meaning that many more bubbles could be found in this CdBG which would only make the phylogeny more concrete.
Before this final successful run, there were a number of unsuccessful attempts made to construct the tree.
Initial attempts were unsuccessful because $\mathbf{K}$ (the super-set of kmers) that \texttt{kleuren} uses to find bubbles was semi-sorted (segments of the file were sorted, but all of the kmers in the file were not in lexicographic order) so the vertices that \texttt{kleuren} used to search for bubbles were skewed towards vertices that were lexicographically first.
We remedied this issue by shuffling the order of the kmer file so that there was no lexicographic bias towards the bubbles that \texttt{kleuren} finds.
A previous attempt resulted in a tree that had a $0.44$ normalized Robinson-Fould's distance from the established tree occurred because there were too few bubbles, and therefore there was not enough phylogenetic signal for the correct tree to be constructed.
To find more bubbles, we split up the kmer file into parts so that multiple instances of \texttt{kleuren} could find bubbles concurrently.
We also discovered that there was a high frequency of adenines (A) (a frequency around $40\%$ in comparison to the other nucleotides) in the final supermatrix that could skew the final tree because nucleotides have differing mutation rates.
We thought this bias towards A was due to the fact that in the $recursivePath$ function (see Algorithm~\ref{extendPathAlg}) the $neighbors$ may be sorted, so the function would traverse the $neighbor$ that started with an A before traversing the other $neighbors$ (see Algorithm~\ref{extendPathAlg}, line: 18).
Similar to the previous sorting problem, we shuffled the order of the $neighbors$ so that the first $neighbor$ that was traversed would not always be lexicographically first.
Despite this change, the final supermatrix that produced the true tree still had a bias towards A (see Section~\ref{futureWork}).
\section{Conclusion}
We introduced a novel method of constructing accurate phylogenetic trees using a CdBG.
Our method, \texttt{kleuren}, uses whole genome sequences to construct a CdBG representation, then it traverses the CdBG to discover bubble structures which become the basis for phylogenetic signal between taxa and eventually produces a phylogenetic tree.
As the NGS era progresses, whole genome sequences are becoming more prevalent for more non-model organisms, in which phylogenies of these organisms have never been constructed.
\texttt{kleuren} is a viable method to relatively quickly and accurately construct the phylogenies for these newly sequenced organisms.
\section{Future Work} \label{futureWork}
We plan to optimize \texttt{kleuren} so that it can find more bubbles in a shorter amount of time.
We will do this by replacing the underlying data structure for how the CdBG is represented.
\texttt{dbgfm}, the current method used to represent the dBG in \texttt{kleuren}, sacrifices time efficiency for memory efficiency by storing the FM-Index entirely on disk, thus slowing down queries into the dBG.
When \texttt{kleuren} runs faster, more bubbles will be found, and more phylogenetic signal will be present so that a more accurate tree can be constructed.
We also plan to investigate the reasons for the high abundance of A's in the supermatrix (see Section~\ref{results}) further, and balance the frequency of nucleotides in the supermatrix.
Furthermore, we would like to look into how \texttt{kleuren} performs when the CdBG is constructed using read sequencing data rather than assembled genomes.
\section*{Acknowledgment}
This work was funded through the Utah NASA Space Grant Consortium and EPSCoR and through the BYU Graduate Research Fellowship.
The authors would like to thank Kristi Bresciano, Michael Cormier, Justin B. Miller, Brandon Pickett, Nathan Schulzke, and Sage Wright for their thoughts concerning the project.
The authors would also like to thank the Fulton Supercomputing Laboratory at Brigham Young University for their work to maintain the super-computer on which these experiments were run.
\newpage
\bibliographystyle{IEEEtran}
|
1,314,259,993,059 | arxiv | \section{INTRODUCTION}
The recent direct imaging of the companion to the brown dwarf 2M1207A, and the estimates of its mass (2--5 Jupiter masses) open a new era in extrasolar planet research (Schneider et~al.\ 2005; Chauvin et~al.\ 2005). At the same time, \emph{Spitzer} observations are now revealing detailed properties of dusty disks around young stellar objects (e.g., Meyer et~al.\ 2004).
In the present letter we propose that disks and planets may be found in an unexpected place---around massive white dwarfs. Our model is presented in Section~2 and some observational tests are discussed in Section~3.
\section{MERGING WHITE DWARFS, PLANETS AND DUST DISKS}
The merger of two white dwarfs is an inevitable outcome of binary star evolution in some binary systems (see, e.g., Iben \& Livio 1993 for a review). Liebert, Bergeron, and Holberg (2005) examined a sample of 348 DA white dwarfs (WDs). They found that the mass distribution has three components: (i)~A peak centered around 0.6~$M_{\odot}$, (ii)~a low-mass component around 0.4~$M_{\odot}$, and (iii)~a high-mass component extending above 0.8~$M_{\odot}$. The high-mass component was estimated to contribute about
15\% of the WD formation rate. Liebert et~al.\ (2005) also concluded that most ($\gtrsim80$\%) of the high-mass objects formed via mergers of two lower-mass WDs. The merger process itself has been studied in some detail (e.g., Benz et~al.\ 1990; Mochkovitch \& Livio 1990). It leads to the total dissipation of the lighter of the two WDs (which fills its Roche lobe first) within a few orbital periods. The dissipated WD material forms a disk around the more massive component, most of which is eventually accreted onto the heavier member to form the massive WDs found by Liebert et~al. However, in order for this disk material to be accreted at all, some fraction of the matter in the disk needs to take up the angular momentum. The matter into which the angular momentum is deposited spreads out to form a disk that is much lower in mass, but much larger in radius. Note, however, that the remaining disk is essentially 100\% metals, since it typically represents the composition of a CO WD.
To model this process we start with two identical white dwarfs, each of mass $M_{WD} = 0.5~M_\odot$ in a circular orbit around each other, and filling their Roche lobes. These merge to give a new white dwarf whose mass is essentially equal to
$M_{\rm NWD} = 2~M_{WD} = 1~M_\odot$. This implies that if the semi-major axis of the binary is $a$, the mean white dwarf radii are $R_{\rm WD} = 0.38 a$. In this case the orbital angular momentum is
\begin{equation}
J_{\rm orb} = \frac{1}{4} M_{\rm NWD} (G M_{\rm NWD} a)^{1/2}~~.
\end{equation}
It is straightforward to show that this angular momentum cannot be taken into the spin of the newly formed white dwarf (since the breakup speed would be exceeded by a factor of around 3). We therefore model the ensuing dynamics of the merger as an accretion disk with fixed angular momentum $J_{\rm orb}$ around a central point mass $M_{\rm NWD}$. The time evolution of such disks is well known (e.g., Pringle 1981; Pringle 1991). The exact behavior depends on the details of the dissipative processes within the disk, but at late times the outer radius of the disk expands to take up the angular momentum, and the inner regions behave as a steady accretion disk, but with gradually decreasing accretion rate. In this sense, the disk dynamics can be expected to be akin to that of the usual protostellar disks, in which all the accretion comes from large radii, but at a steadily declining rate. For simplicity we assume that the surface density profile of the disk is of the form
\begin{equation}
\Sigma \propto R^{-\alpha}~~,
\end{equation}
where we require $\alpha < 2$. For example, the usual solar nebula profile is assumed to have $\alpha \approx 3/2$ (Armitage et~al.\ 2002), whereas if we adopt the Lin \& Papaloizou (1985) prescription for the viscosity in cool protostellar disks of $\nu\propto\Sigma^2$ we would obtain $\alpha \approx 0$. With this profile, angular momentum of the disk is given in terms of the disk radius $R_d$ and the disk mass $M_d$ by
\begin{equation}
J_d = \xi M_d (G M_{NWD} R_d)^{1/2}~~,
\end{equation}
where $\xi = (2-\alpha)/(\frac{5}{2} -\alpha)$. We shall adopt $\alpha = 3/2$ and $\xi = 1/2$ as typical parameters.
If we equate the angular momentum of the disk to that of the original binary we find that the disk mass is given in terms of the disk radius by
\begin{equation}
M_d = 0.81 \left( \frac{\xi}{1/2} \right)^{-1} M_{\rm NWD}
\left(\frac{R_{\rm WD}}{R_d} \right)^{1/2}~~.
\end{equation}
We now need to ask at what stage planets might start to form in such a disk. We have already argued that a disk of this sort does not differ in its dynamics greatly from a standard protostellar disk. If this were a disk with standard cosmic abundances then it might seem
reasonable to assume that planetesimal formation sets in when the disk radius exceeds the snow-line, which is the radius outside which the disk temperature equals the ice condensation temperature of $T_c \approx 170$~K. But this disk does not have solar composition, and in particular lacks the hydrogen to produce the water for the `snow.' In this situation it seems sensible to adopt the dust grain condensation temperature of $T_c \approx 1600$~K as the equivalent of the snow line. Adopting the temperature profile for an optically thin disk given by
\begin{equation}
T_d = 2.8 \times 10^2 \left( \frac{R}{1~\mathrm{AU}} \right)^{-1/2}
\left(
\frac{L}{L_\odot} \right)^{1/4}~\mathrm{K}~~,
\end{equation}
gives the condensation radius as
\begin{equation}
R_\mathrm{dust} = 0.0306 \left( \frac{L}{L_\odot} \right)^{1/2} \left(
\frac{T_c}{1600~\mathrm{K}} \right)^{-2}~\mathrm{AU}~~.
\end{equation}
Taking the radius of the newly formed white dwarf to be $R_{\rm NWD} = 6 \times 10^8$~cm, and its temperature to be $T_{\rm NWD} = 50,\!000$~K, we find that the dust condensation line occurs at radius
\begin{equation}
R_\mathrm{dust} = 0.02 \left( \frac{R_\mathrm{NWD}}{6\times10^8~\mathrm{cm}} \right)
\left( \frac{T_\mathrm{NWD}}{50,000~\mathrm{K}} \right)^2 \left(
\frac{T_c}{1600~\mathrm{K}} \right)^{-2}~\mathrm{AU}~~.
\end{equation}
When this radius is reached, the mass in the disk is
\begin{equation}
M_d = 0.047 \left( \frac{\xi}{1/2} \right)^{-1} \frac{M_\mathrm{NWD}}{M_\odot}
\left( \frac{0.6 R_\mathrm{WD}}{ R_\mathrm{NWD}} \right)^{1/2}
\left( \frac{T_\mathrm{NWD}}{50,000~\mathrm{K}} \right)^{-1}
\left( \frac{T_c}{1600~\mathrm{K}} \right)^{-1}~M_\odot~~.
\end{equation}
Since the mass is concentrated at large radii, we may take this mass as an estimate of the mass available for forming planets. As the radius expands to $\sim$1~AU, the mass in the disk is $\sim\!0.007~M_{\odot}$.
In a recent, important work, Fischer and Valenti (2005) have shown that the probability to host a planet increases quite dramatically
with the metallicity of the host star (approximately as [Fe/H]$^{1.8}$, where [Fe/H] represents the iron abundance). If this result can be extrapolated to the metallicities under discussion here, it implies that planets formation would be highly efficient in disks of the kind we are considering.
The only other situation we are aware of which is analogous to the planet formation picture we have described above is the formation of planets around the pulsar PSR1257+12 (Wolszcan \& Frail 1992). The planets in this system, presumably around a neutron star of mass around $M_\mathrm{NS} \approx 1.4~M_\odot$, move in almost circular orbits, and so most likely formed in a disk. They have masses of 2.8~$M_\mathrm{Earth}$ at 0.47~AU and 3.4~$M_\mathrm{Earth}$ at 0.36~AU. The disk in which the planets have formed is thought to have come from the disruption of a low mass companion (Stevens, Rees \& Podsiadlowski 1992) of mass $M_2 \approx
0.016~M_\odot$ (King et~al.\ 2005). In this case, we calculate that the fraction of the original orbital angular momentum which ended up in these two planets is $f \sim 1$\%. If the planets, being Earth-mass, are also Earth-like, then this implies a high efficiency of converting
the initial angular momentum stored in elements heavier that H and He into planets.
\section{OBSERVATIONAL CONSEQUENCES}
One of the predictions of the proposed scenario is the potential existence of dusty disks around massive WDs. Our calculations suggest that a typical circumstellar dust disk will have a mass and radius of $M_d\sim 0.007~M_\odot$ and $R_d\sim 1$~AU, respectively. The disk will be gas poor and the dust will reprocess radiation from the white dwarf to produce an infrared excess spectral energy distribution (SED). Figure~1 shows a simulated SED for a dust disk around a white dwarf (assumed to be at a distance of 12~pc, typical of nearby WDs; Holberg, Oswalt \& Sion 2002), demonstrating that the thermal emission from the disk will be detectable above the white dwarf photosphere at infrared wavelengths (the different plots are evenly spaced in the cosine of the inclination). In this simulation, the white dwarf (after merger) has $M_\mathrm{NWD}=M_\odot$, $T_\mathrm{NWD}=5\times10^4$~K, $R_\mathrm{NWD}=6\times10^8$~cm, and a Kurucz model atmosphere was used. Using a $5\times10^4$~K black body did not result in any significant changes. The disk has a dust mass $M_d = 0.007~M_\odot$, outer radius $R_d = 1$~AU, and inner radius corresponding to a dust destruction temperature of 1600~K. The dust opacity and scattering properties are assumed to be those for interstellar dust (e.g., Kim, Martin, \& Henry 1994), which gives the prominent $10~\mu$m silicate feature in the simulated SED. If the dust in the disk has grown to sizes larger than typical ISM grains, the opacity will be grayer and the silicate features less prominent (e.g., Wood et~al.\ 2002). We assume the dust is in vertical hydrostatic equilibrium and the disk has a radial surface density $\Sigma\sim r^{-3/2}$. The disk temperature, vertical hydrostatic density structure, and emergent SED are computed using the Monte Carlo radiation transfer code described in Walker et~al.\ (2004). If the dust is not in vertical hydrostatic equilibrium and has settled to the midplane, the infrared excess emission will be less than shown in Figure~1, but will still be detectable above the photosphere and resemble that from a flat reprocessing disk (e.g., Adams, Lada, \& Shu 1987). Note that massive WDs are considerably smaller than their low-mass counterparts, and hence, at a given temperature they are less luminous.
Disks that are extremely metal-rich may also be expected to form planets. Following the pioneering work of Becklin \& Zuckerman (1988), a few more extensive searches for planets around WDs have already started. The search methods include IR imaging, IR spectroscopy, and the use of WD pulsations. Clarke \& Burleigh (2004; see also Burleigh, Clarke \& Hodgkin 2002) describe the first results from a deep infrared imaging ($J\sim24$) campaign of twenty-four WDs. They found two objects that appear to have the same proper motion as the WDs (at least at the 1$\sigma$ level), and whose luminosities are consistent with masses in the range 7--10~M$_\mathrm{Jup}$. The Clarke \& Burleigh search aims at planets that existed around the WD progenitor, and which have survived stellar evolution. The current paper identifies \emph{massive WDs} as high-probability hosts of dusty disks or planets, with these disks and planets having formed during a late evolutionary phase, and having unusual compositions.
A second existing search for planets around WDs uses the pulsations of the WDs as intrinsic clocks (Mullally et~al.\ 2003). The idea here is that hot, DAV-type WDs near 12,000~K pulsate with an extraordinarily stable pulsation period. The reflex orbital motion causes a rate of change $\dot P$ in the period $P$ of order (e.g., Kepler et~al.\ 1991)
\begin{equation}
\frac{\dot P}{P}=\frac{G}{c} \frac{PM_p \sin i}{r^2_p}~~,
\end{equation}
where $G$ and $c$ are the gravitational constant and speed of light, respectively, and $i$ is the orbital inclination. Again, in the present paper we have identified massive WDs as promising targets. While WD cooling also produces a period change, this change is expected to be monotonic, while the change caused by a planet is periodic.
Finally, Dobbie et~al.\ (2005) have used a near-infrared spectroscopic analysis of eight white dwarfs, in an attempt to search for low-mass companions. They were able to place upper limits on putative companions at substellar masses ($\sim$0.07~$M_{\odot}$). We have also examined the WDs with cool companions reported by Wachter et~al.\ (2003), who used 2MASS photometry. However, none of the confirmed high-mass WDs appears to be on that list.
We should also note that the probability of even a Jovian-mass planet (most likely of CO composition) at 0.01~AU eclipsing its host WD is rather small, $\sim$2\%. Note, however, that a Jovian planet of CO composition will be larger (by more than a factor 6) than a 1.2~$M_{\odot}$ WD (e.g., Zapolsky \& Salpeter 1969), thus enabling it to produce a total eclipse.
If future observations fail to detect any disks or planets around massive WDs, this may mean that the formation of dust is somehow suppressed in these somewhat unusual, hydrogen-poor environments.
\begin{acknowledgments}
We are grateful to Jim Liebert and Keith Horne for helpful discussions. M.L.\ thanks the Dept.\ of Astronomy at the University of St.\ Andrews for its hospitality during a Carnegie Centenary Professorship.
\end{acknowledgments}
|
1,314,259,993,060 | arxiv | \section{Introduction}
Multi-user networks with relays, sensors and Internet of Things (IoT) in the 5G and beyond networks will generate enormous amount of data and consume large amount of energy for a wide range of services in different domain, e.g., \cite{Andrews2014jsac,Atapattu2019twc} and references therein.
One of the key challenges in such wireless networks is energizing the remote devices for successful communication. Although natural energy resources such as wind and solar can be used, they are often hindered by inconsistent availability, implementation overhead or the requirement of large infrastructure. Thus, energy harvesting (EH) using radio frequency (RF) signals, is motivated as existing communication circuitry can be used with low cost modifications \cite{Zhou}. Since such low power communication interfaces make the seamless connectivity more challenging, relaying or cooperative communication has been promoted as a viable solution, especially for the Internet of Things (IoT) \cite{Wang}. Thus, RF energy harvesting in relay networks has gained much attention recently.
\subsection{Related Work}
Since energy at the EH node is not automatically replenished as in a traditional node with fixed power supply, the performance of an EH network depends on the EH protocol and the usage scheme of the harvested energy. For simultaneous information and power transfer (SWIPT), two basic EH protocols, i) time-switching (TS) and ii) power-splitting (PS), are introduced for amplify-and-forward (AF) and decode-and-forward (DF) relay networks in \cite{Nasir1,Saman1,Saman2}. An optimal hybrid EH protocol, which is a combination of PS and TS protocols is introduced in \cite{Saman3,bhathiya1} and it outperforms both TS and PS protocols. An improved receiver architecture for PS protocol is introduced in \cite{FTChien} and \cite{WChoi}, which makes use of the level of the harvested energy as side information to assist the decoding of the source transmitted message. The common assumption of most of these work is that the total harvested energy is used for data transmission and thus a battery for long term energy storage is not required at the EH node. However, a long term energy storage enables a PS energy harvesting node to manage two basic resources i) PS ratio and ii) transmit energy. Thus, an efficient resource allocation scheme, which store excess amount of harvested energy for future use, can achieve a better performance compared to a network without a battery in the EH node. Due to the battery energy dependency on the resource allocation decisions made earlier, the analysis of the system performance needs more attention.
For EH relaying with a battery, several resource allocation methods are discussed in literature. An AF relaying network with TS energy harvesting is considered in \cite{Krikidis}, where data relaying is realized when sufficient energy is collected through EH. An AF relaying network with PS energy harvesting is considered in \cite{Blum}, where the remaining energy after data transmission is stored in the battery. The optimal resource allocation that maximizes the energy efficiency in a WSN with DF relaying is considered in \cite{LZheng}. A sum-throughput maximization problem is formulated for DF relay \cite{WTu}, where the relay node opportunistically switch between modes of total EH and PS based information processing. Resource allocation schemes for EH nodes which harvest energy from renewable sources such as wind or solar are investigated in \cite{FYuan, Amirnavaei}. All these work assume full CSI at the decision node. The outage performance is analyzed in \cite{HJiang} for a sub-optimal resource allocation scheme based on incremental DF relay protocol.
\subsection{Problem Statement and Contribution}
In contrast to previous work \cite{Nasir1,Saman1,Saman2,Saman3,Blum,Krikidis,LZheng,WTu}, this paper thus considers a dual hop DF relaying network with the PS energy harvesting protocol assuming that no CSI of forward channels is available at any node. The system performance is evaluated by the average success probability of the source to destination communication. To efficiently use the harvested energy, the relay is equipped with a battery, which consists of a finite capacity. In contrast to \cite{HJiang}, we focus our attention to find the maximum average success probability over the set of resource allocation policies. The evaluation of maximum is important to assess the feasibility of the network for a practical set of system parameters. Due to the intractability of the problem, we develop a mathematical framework to find an upper bound for the maximum average success probability by formulating a discrete state Markov decision problem (MDP).
\section{System Model}\label{system_model}
\begin{table}[]
\centering
\caption{Notations}
\label{notations}
\renewcommand{\arraystretch}{1}
\begin{tabular}{| >{\centering\arraybackslash}m{2.3cm}| >{\centering\arraybackslash}m{5.5cm}|}
\hline
Notation & Remark \\
\hline\hline
$P_s$ & Source transmit power \\
\hline
$\sigma^2$ & Noise power \\
\hline
$T$ & Block duration \\
\hline
\small{$m$} & \small{Block index} \\
\hline
\small{$h_m$} & \small{S-R channel power gain in the $m$th~block} \\
\hline
\small{$g_m$} & \small{R-D channel power gain in the $m$th~block} \\
\hline
\small{$E_m$} & \small{Battery energy at the beginning of the $m$th~block} \\
\hline
\small{$\lambda_m$} & \small{PS ratio used in the $m$th~block} \\
\hline
\small{$u_m$} & \small{Relay transmit energy used in the $m$th~block} \\
\hline
\small{$S_m$} & \small{State of the relay in the $m$th~block - $\left(E_m,h_m\right)$ pair } \\
\hline
\small{$A_m$} & \small{Relay action in the $m$th~block - $\left(\lambda_m,u_m\right)$ pair} \\
\hline
\small{$\mathcal{S}$} & \small{State space - set of all possible $S_m$} \\
\hline
\small{$\mathcal{A}_s$} & \small{Action space - set of all possible $A_m$} \\
\hline
\small{$d_m\left(\cdot\right)$} & \small{Decision rule in the $m$th~block, which gives an action for each state - $A_m=d_m\left(S_m\right)$}\\
\hline
\small{$\pi$} & \small{Resource allocation policy - the sequence of decision rules $d_1,d_2,\cdots$}\\
\hline
\small{$\widetilde{\text{P}}_\pi\left(s\right)$} & \small{Average success probability of policy $\pi$ for the initial state $S_1=s$} \\
\hline
\small{$\text{P}_\pi$} & \small{Average success probability of policy $\pi$} \\
\hline
\end{tabular}
\end{table}
In this section, we discuss main assumptions and the operation of the network.
\subsection{Network Model}
We consider a wireless relay network in which a source node (S) communicates with a destination node (D) via a single relay node (R). The relay operates in the DF mode. We assume that the direct link between S and D is not available due to a blockage. The communication takes place in half-duplex mode. Each node has a single antenna.
The network operates block by block, where each block has a duration $T$ and is indexed by $m \in \{1,2,\cdots\}$. The fading coefficients of S to R channel (S-R) and R to D channel (R-D) in the $m$th~block are denoted by $\tilde{h}_m$ and $\tilde{g}_m$, respectively, which are independent. Since an unbounded flat-fading channel may be modeled by a finite number of channel states with an arbitrary low error \cite{Parastoo, Blum}, both channel coefficients are drawn from finite sets. We assume that there is no feedback from D to R or from R to S. Thus, no CSI is available on the forward channel, i.e., S does not have any channel knowledge, R has knowledge on $\tilde{h}_m$, and D has knowledge on $\tilde{g}_m$. The source transmits with constant power $P_s$ and information rate $\tau$. The relay harvests energy from source transmitted information signal and uses that energy for information transmission to the destination. The PS protocol is used in R. The source transmits the message during the first half of the block. The relay uses $\sqrt{\lambda_m}$ portion of the received signal for the EH, and the remaining $\sqrt{1-\lambda_m}$ portion of the received signal is utilized for the information decoding. During the second half of the block, the relay transmits the decoded message to the destination using $u_m$ amount of energy.
\subsection{Analytical Model}
\subsubsection{S-R and R-D Transmission}
The discrete time received signal at the information decoder of $R$ in $k$th symbol index of $m$th block is
\begin{equation*}
\hat{y}_{r,m}^{(k)} = \sqrt{1-\lambda_m}\bigg(\sqrt{P_s} \tilde{h}_{m} s^{(k)}_m + n_{r,a}^{(k)}\bigg)+n_{r,c}^{(k)} \, ,
\end{equation*}
where $s^{(k)}_m$ is the $k$th symbol transmitted by S, $n_{r,a}^{(k)}$ and $n_{r,c}^{(k)}$ are AWGN at the antenna and the information decoder of $R$, respectively with variance $\sigma^2$. Therefore, the signal-to-noise-ratio (SNR) of $S-R$ channel in the $m$th block is
\begin{equation}\label{gamSR}
\gamma_1\left(h_m,\lambda_m\right)= \frac{\left(1-\lambda_m\right) h_m P_s}{\left(2-\lambda_m\right) \sigma^2} \, ,
\end{equation}
where $h_m= | \tilde{h}_m|^2$ and $h_m \in \mathcal{H}$ for all $m$. Since fading coefficients are drawn from a finite set, $\mathcal{H}$ is also finite. Thus, we have $\mathcal{H}=\left\{h^{(i)}| \ i=1,2,\cdots,N_c \right\}$, where $N_c$ is the total number of elements in $\mathcal{H}$. To omit the use of the index $i$ when not necessary, we may denote a general element of $\mathcal{H}$ by $h$. The probability mass functions for $\mathcal{H}$ is $f_\mathcal{H}\left(h\right)$.
If the Relay uses $u_m$ energy to transmit information, the discrete time received signal at $D$ in the $k$th symbol index of the $m$th block is
\begin{equation*}
{y}_{d,m}^{(k)} = \sqrt{\frac{2 u_m}{T}} \tilde{g}_{m} \hat{s}^{(k)}_m + n_{d,a}^{(k)} + n_{d,c}^{(k)}\, ,
\end{equation*}
where $\hat{s}^{(k)}_m$ is the $k$th symbol transmitted by R. Therefore the SNR at D in the $m$th~block is
\begin{equation}\label{gamRD}
\gamma_2\left(g_m,u_m\right) = \frac{u_m g_m}{T \sigma^2} \, ,
\end{equation}
where $g_m= | \tilde{g}_m|^2$ and $g_m \in \mathcal{G}$ for all $m$. Since fading coefficients are drawn from a finite set, $\mathcal{G}$ is also finite. We denote the largest element of $\mathcal{G}$ by $g_{max}$.
\subsection{Relay Operations and Battery Behavior}\label{relay_operations}
The total harvested energy during the $m$th~block by neglecting the noise energy, is $\eta P_s h_m \lambda_m \frac{T}{2}$ where $\eta\in (0,1)$ is the conversion efficiency \cite{Zhou}. This energy is directly transfered to the battery. Thus, the battery energy at $t=\left(m+\frac{1}{2}\right)T$ is
\begin{equation}\label{diff_eqn_sol}
E_{m+\frac{1}{2}} = \text{min}\Bigg[ \frac{\eta P_s h_m \lambda_m T}{2} + E_m , B\Bigg] \, ,
\end{equation}
where $B<\infty$ is the battery capacity and $E_m$ is the residual battery energy at the beginning of the $m$th~block.
For information transmission from R to D, the relay uses $u_m$ amount energy. The residual battery energy for the next block, is
\begin{equation}\label{end_block_energy}
E_{m+1} = \left[E_{m+\frac{1}{2}} - u_m\right] \, .
\end{equation}
If Shannon channel capacity is larger than the information rate $\tau$, the receiving node may decode the received signal with arbitrary small error probability. This is defined as a successful decoding. Thus, to achieve a successful decoding with a minimum received SNR $\gamma_\tau$, we have $\tau = \frac{1}{2}\mathrm{log}_2\left(1+\gamma_\tau\right)$\,bits/s/Hz, in which the factor $\frac{1}{2}$ is due to each S-R and R-D links are used only half of the total time. This satisfies $\gamma_\tau = 4^\tau -1$. Thus, for a successful decoding at the relay and the destination, we have $\gamma_1 \left(h_m,\lambda_m\right) \geqslant \gamma_\tau$ and $\gamma_2 \left(g_m,u_m\right) \geqslant \gamma_\tau$, respectively. The PS ratio $\lambda_m$ and relay transmit energy $u_m$ used, impact the SNRs $\gamma_1\left(h_m,\lambda_m\right)$ and $\gamma_2\left(g_m,u_m\right)$. Subsequently, they effect the probability of successful transmission from the source to the destination. In the next section, we discuss the calculation of the average success probability.
\section{The Average Success Probability}\label{succ_cal_section}
We first define the \emph{state} $S_m$ in the $m$th~block to be the pair $S_m=\left(E_m, h_m\right)$. The state $S_m$ for each $m$, takes an element from the the \emph{state space} defined as\break $\mathcal{S}=\left\{s=\left(E,h\right)|\ h\in \mathcal{H}, E\in [0,B] \right\}$, where a general element of $\mathcal{S}$ is denoted by $s=\left(E,h\right)$. The \emph{action}, $A_m$, taken by the relay in the $m$th~block is defined as the pair $A_m=\left(\lambda_m, u_m\right)$. For the brevity, we then define two functions related to \eqref{diff_eqn_sol} and \eqref{end_block_energy} as
\begin{equation}\label{diff_eqn_sol2}\begin{split}
\mathcal{E}_{\frac{T}{2}}\left(\lambda_m,E_m,h_m\right) &= \text{min}\Bigg[ \frac{\eta P_s h_m \lambda_m T}{2} + E_m , B\Bigg] \, , \\
\mathcal{E}_{T}\left(\lambda_m,u_m,E_m,h_m\right) &= \left[\mathcal{E}_{\frac{T}{2}}\left(\lambda_m,E_m,h_m\right) - u_m\right]\, ,
\end{split}
\end{equation}
which are used to represent $E_{m+\frac{1}{2}}= \mathcal{E}_{\frac{T}{2}}\left(\lambda_m,E_m,h_m\right)$ and $E_{m+1}=\mathcal{E}_{T}\left(\lambda_m,u_m,E_m,h_m\right)$, respectively. The PS ratio $\lambda_m$ may take any value in $[0,1]$. The transmit energy $u_m$ and the residual battery energy for the next block $E_{m+1}$ are non-negative. By considering these constraints, the action $A_m$ at each $m$ takes an element from the \emph{action space}, $\mathcal{A}_s$, which is defined as the set of all actions for state $s$ and it can be given as
\begin{equation}\label{As}
\mathcal{A}_s=\left\{a=\left(\lambda, u\right) |\ \lambda\in [0,1] , \ 0\leqslant u \ , \ 0\leqslant \mathcal{E}_{T}\left(\lambda,u,s\right)\right\} \, ,
\end{equation}
where a general element of $\mathcal{A}_s$ is denoted by $a=\left(\lambda,u\right)$.
The knowledge of $S_m=\left(E_m, h_m\right)$ is available in the relay at the beginning of each $m$th~block. We thus consider each action $A_m$ as a function of the current state denoted by $d:\mathcal{S}\to \mathcal{A}_s$, i.e. $A_m = d\left(S_m\right)$, where this function is termed as the \emph{decision rule}. Since each action is an element of $\mathcal{A}_s$, the \emph{decision rule space}, $\mathcal{D}$, which is the set of all possible decision rules can be given as
\begin{equation}\label{D_space}
\mathcal{D}= \left\{d \ | \ d\left(s\right) \in \mathcal{A}_s \forall s \in \mathcal{S} \right\} \ .
\end{equation}
The relay can be configured to have a sequence of decision rules $\pi=\left\{d_1, d_2,\cdots \right\}$, which is termed as \emph{policy}. For each $S_m$, the action $A_m$ is chosen according to $d_m$. The \emph{policy space} is thus given by $\Pi = \mathcal{D}\times\mathcal{D}\times\mathcal{D}\times\cdots$. A stationary policy employs the same decision rule $d$ at all blocks, i.e., $d^\infty$. Without loss of generality, we may denote a stationary policy by $d$.
For a given state $S_m=\left(E_m, h_m\right)$ and action $A_m=\left(\lambda_m,u_m\right)$, the success probability of S-R link can be given as
\begin{align}\label{suc_prob_SR_for_given_Et_gSRt}
\Pr\left(\text{S-R} \ \text{success} \, \big|\, S_m,A_m\right) &\overset{(a)}{=}\mathds{1}_{\left[\gamma_1\left(h_m,\lambda_m\right) \geqslant \gamma_\tau\right]} \nonumber \\
&\overset{(b)}{=} \mathds{1}_{\left[\lambda_m \leqslant \frac{h_m P_s - 2 \sigma^2 \gamma_\tau}{h_m P_s - \sigma^2 \gamma_\tau}\right]} \, ,
\end{align}
where $\mathds{1}_{\left[\gamma_1\left(h_m,\lambda_m\right) \geqslant \gamma_\tau\right]}=1$ when $\gamma_1\left(h_m,\lambda_m\right) \geqslant \gamma_\tau$, and $0$ otherwise. The equation $(a)$ follows as the requirements for the successful decoding at the relay, and $(b)$ comes from (\ref{gamSR}). For a given state $S_m=\left(E_m, h_m\right)$, and action $A_m=\left(\lambda_m,u_m\right)$, the success probability in R-D link can be given with the aid of \eqref{gamRD} as
\begin{align}\label{suc_prob_RD_for_given_Et_gSRt}
\Pr\left(\text{R-D} \ \text{success} \, \big|\, S_m,A_m \right) &= \Pr\left(g_m\geqslant\frac{T \sigma^2 \gamma_\tau}{u_m}\right) \, .
\end{align}
For state $S_m$ and action $A_m$, we define the \emph{reward}, $p\left(S_m, A_m\right)$, as the end-to-end success probability, which is evaluated as
\begin{equation}\label{suc_prob_endtoend_blockn}
\begin{split}
p\left(S_m, A_m\right) =\Pr\left(g_m\geqslant\frac{T \sigma^2 \gamma_\tau}{u_m}\right) \mathds{1}_{\left[\lambda_m \leqslant \frac{h_m P_s - 2 \sigma^2 \gamma_\tau}{h_m P_s - \sigma^2 \gamma_\tau}\right]} \, .
\end{split}
\end{equation}
For the policy $\pi=\{d_1,d_2\cdots\}$ and the initial state $S_1=s$, the time average success probability over $M$ blocks is given as
\begin{equation}\label{lim_avg_suc_prob}
\bar{p}_{\pi,M}\left(s\right) = \frac{1}{M} \mathbb{E}\left[\sum_{m=1}^{M} p\left(S_m, d_m\left(S_m\right)\right) \, \bigg| \,S_1=s\right] \, ,
\end{equation}
where $\mathbb{E}\left[\cdot\right]$ denotes the
expectation operator. The long term average success probability for initial state $S_1=s$, is thus given by $\widetilde{\text{P}}_\pi\left(s\right)=\lim_{M \to \infty} \bar{p}_{\pi,M}\left(s\right)$. We consider all policies for which the limit exists. Without loss of generality, we assume that the initial battery energy $E_1=0$. The channel fading is independant from the battery energy in the relay. Therefore, the long term average success probability is given by
\begin{equation}\label{lim_inf_avg_suc_prob_vector}
\text{P}_\pi = \mathbb{E}\left[\widetilde{\text{P}}_\pi\big(\left(0,h\right)\big)\right] \, .
\end{equation}
It is important to find the maximum $\text{P}_\pi$ in order to assess the feasibility of the system. Since the state space $\mathcal{S}$ and the action space $\mathcal{A}_s$ is uncountably infinite, maximization of $\text{P}_\pi$ with respect to policy $\pi$, is intractable. Therefore, the main objective of this paper is to find an upper bound for the maximum $\text{P}_\pi$, denoted by $P_u $, by making use of a suitable discretization of $\mathcal{S}$ and $\mathcal{A}_s$. For comparison purposes we also provide a heuristic resource allocation policy. These will be discussed in the next section
\section{A Heuristic Policy and the Upper bound}\label{Upper_Lower_bound_calculation}
We notice that in some states $s\in \mathcal{S}$ any action $a\in \mathcal{A}_s$ taken results in $p\left(s, a\right)=0$. Therefore, when deriving the heuristic policy and the upper bound $P_u$, these states can be treated differently to other states. To this end, we categories each state $s=\left(E,h\right)$ in to two subsets depending on the resulting reward $p\left(s, a\right)$ for action $a=\left(\lambda, u\right)$;
\begin{itemize}
\item \emph{Subset-1} : $\mathcal{C}_1 = \Big\{\left(h, E\right)\in \mathcal{S} \, | \, \mathcal{E}_{\frac{T}{2}}\left(\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau},h,E\right) <\frac{T \sigma^2 \gamma_\tau}{g_{max}} \ \text{or} \ h < \frac{2 \sigma^2 \gamma_\tau}{P_s} \Big\}$
\vspace{0.4cm}
As given in \eqref{suc_prob_SR_for_given_Et_gSRt}, when $\lambda > \frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau}$, the relay cannot decode the source message. The maximum $\lambda$, which helps successful decoding is $\lambda = \frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau}$. The condition $h < \frac{2 \sigma^2 \gamma_\tau}{P_s}$ describes the situation where no $\lambda \in [0,1]$ satisfies $\lambda \leqslant \frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau}$, which causes $p\left(s, a\right)=0$ for all $a \in \mathcal{A}_s$.
\quad On the other hand, it can be seen from (\ref{As}) that selection of $\lambda$ restricts the selection of $u$. A lager value for $\lambda$ allows the relay to harvest more energy, which results in more energy in the battery. This enable the relay to use a larger $u$. Therefore, with the aid of \eqref{end_block_energy}, the maximum value $u$ can take, while allowing the relay to decode the source message is $u= \mathcal{E}_{\frac{T}{2}}\left(\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau},h,E\right)$. When the relay uses this energy to transmit to the destination, the largest SNR at the destination is achieved when $g=g_{max}$ in (\ref{gamRD}). The condition $\mathcal{E}_{\frac{T}{2}}\left(\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau},h,E\right)<\frac{T \sigma^2 \gamma_\tau}{g_{max}}$ describes the situation when the largest achievable SNR falls below $\gamma_\tau$. This causes $p\left(s, a\right)=0$ for all $a \in \mathcal{A}_s$.
Therefore, $p\left(s, a\right)=0$ for all $a \in \mathcal{A}_s$ whenever $s\in \mathcal{C}_1$.
\vspace{0.4cm}
\item \emph{Subset-2} : $\mathcal{C}_2=\mathcal{S}\backslash \mathcal{C}_1$
\vspace{0.2cm}
When the state $s$ does not belong to $\mathcal{C}_1$, we have $\mathcal{E}_{\frac{T}{2}}\left(\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau},h,E\right) >0$, which makes \break $\lambda=\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau}$ and $u> 0$ feasible. Therefore, whenever $s\in \mathcal{C}_2$, there exists an action $a \in \mathcal{A}_s$, which gives $p\left(s, a\right)>0$.
\end{itemize}
\subsection{Heuristic Policy}
If the conditional distribution of the state $S_{m+1}$ given $S_m=s=\left(h,E\right)$ is known, the evaluation of expectation operation in \eqref{lim_avg_suc_prob} is straight forward. A simple way this can be achieved is by driving the energy level of the battery to zero by using the total amount of the battery energy for $u_m$. Thus, for any $S_m$, the residual battery energy $E_{m+1}=0$ and the $h_{m+1}$ is independent from $S_m$. With the aid of \eqref{As}, a heuristic decision rule, which always drives the battery energy to zero can be given as
\begin{equation}\label{decision_rule_IRM}
d_{l}\left(s\right) = \begin{cases}
\lambda=1, \qquad\qquad\qquad\qquad\quad \text{if} \ s \in \mathcal{C}_1\\ \quad u=\mathcal{E}_{\frac{T}{2}}\left(\lambda,h,E\right) \\ \lambda=\frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau}, \qquad\qquad\quad \text{otherwise \, .}\\ \quad u= \left[\mathcal{E}_{\frac{T}{2}}\left(\lambda,h,E\right) \right]
\end{cases}
\end{equation}
The stationary policy generated by the above decision rule is $\pi_l=d_l^\infty$. If $\pi_l$ is used, the states $S_m$ for all $m>1$ is known to be an element from the set $\left\{\left(0,h\right)|\ h\in \mathcal{H} \right\}$. Therefore, the average success probability for initial state $S_1=s$ can be written as
{\small
\begin{multline*}
\widetilde{\text{P}}_{\pi_l}(s)=\lim_{M \to \infty} \frac{1}{M} \Bigg[ p\left(s,d_l\left(s\right)\right)
+ \sum_{m=2}^{M} \mathbb{E}\left[p\left(\left(0,h\right), d_l\left(0,h\right)\right)\right]\Bigg] \ .
\end{multline*}
}
By taking the limit in the above equation and noting that $\widetilde{\text{P}}_{\pi_l}(s)$ is constant with respect to $s$, with the aid of \eqref{lim_inf_avg_suc_prob_vector} we have
\begin{equation}\label{lower_bound_eq}
\text{P}_{\pi_l}=\mathbb{E}
\left[p\left(\left(0,h\right), d_l\left(0,h\right)\right)\right] \ .
\end{equation}
This can be evaluated using \eqref{suc_prob_endtoend_blockn} and \eqref{decision_rule_IRM} for each state $\left(0,h\right)$ with $h \in \mathcal{H}$ and taking the average using the probability mass function $f_\mathcal{H}$.
\subsection{Upper Bound Calculation}
\begin{figure}
\centering
\includegraphics[width=8cm]{Modification.eps}
\caption{Discretization of the battery energy levels.}
\label{Modification}
\end{figure}
Although, the state transition of any policy can be modeled by a Markov chain, finding an upper bound using a MDP is involved due to the state space $\mathcal{S}$ is uncountably infinite. Therefore, instead of formulating a MDP for the original system model, we first appropriately modify the system to have a finite state space. We prove that the maximum of the average success probability of the finite state space system gives an upper bound for the maximum of the average success probability of the original system. To this end, we discretize the battery energy assuming that there exists a hypothetical energy source in the relay, which injects energy to the battery at the beginning of each block, such that battery energy occupy only predefined $N_b$ number of levels. For the current state $S_m$ and action $A_m$ the residual battery energy for the next block given in \eqref{end_block_energy} is modified by the hypothetical energy source according to
\begin{equation}\label{upper_modi}
E_{m+1} = \begin{cases}
e_{i+1}=\frac{i B}{N_b-1}, \ \text{if} \ \mathcal{E}_{T}\left(A_m,S_m\right) \in \left[\frac{\left(i-1\right)B}{N_b-1} , \frac{ i B}{N_b-1} \right)\\ \quad\quad\quad\quad\quad\quad\quad \text{for each} \ i=1,2,\cdots,N_b-1 \\ e_{N_b}=B, \quad\quad \text{otherwise} \ .
\end{cases}
\end{equation}
Each $e_i=\frac{ \left(i-1\right) B}{N_b-1}$ for all $i=1,\cdots,N_b$ denotes the finite battery levels in the battery. According to \eqref{upper_modi}, the hypothetical energy source drives the battery energy to the nearest upper level defined by each $e_i$. This is shown in Fig.~\ref{Modification}b. Thus, the state space has finite number of elements and we denote it by $\mathcal{S}'=\left\{e_1,\cdots,e_{N_b}\right\}\times \mathcal{H}$. We denote a general element of $\mathcal{S}'$ by $s_i$, which are indexed in such a way, that states $\left(e_j, h^{(1)}\right)$ to $\left(e_j, h^{(N_c)}\right)$ map with $s_{(j N_c -N_c +1)}$ to $s_{(j N_c)}$, respectively. Due to the finite nature of the state space, one-step transition probability from the state $S_m$ to state $S_{m+1}$ for any decision rule $d$ can be given in a matrix form according to
\begin{equation}\label{Theta_d}
\Theta_d^{\left(i,j\right)}=\Theta_d\left(s_i,s_j\right) = \Pr\left(S_{m+1}=s_j\ \big| \ S_{m}=s_i \right) \ .
\end{equation}
If the current state is $s_i$ and the residual battery energy determined by the action is $e_j$, the $i$th row of the transition matrix $\Theta_d$ consists of the channel probability values $f_{\mathcal{H}}\left(h^{(1)}\right)$ to $f_{\mathcal{H}}\left(h^{(N_c)}\right)$ from column $N_c\left(j-1\right)+1$ to column $N_c j$.
Since the state space is finite, for any decision rule $d$, we can define a reward vector $p_d$ in which, each element gives the reward for each state and action defined by the decision rule for the state, i.e. $p_d\left(s_i\right) = p\big(s_i,d(s_i)\big)$ for all $s_i \in \mathcal{S}' , \ d\in \mathcal{D}$. Using the transition matrix $\Theta_d$ and the reward vector $p_d$ we can write the average success probability of the modified system, in a vector form as \cite{Puterman}
\begin{equation}\label{lim_inf_avg_suc_prob_vector_modi}
\widetilde{\text{P}}'_\pi =\lim_{M \to \infty} \frac{1}{M} \left[p_{d_1}+\Theta_{d_1} p_{d_2}+\cdots + \prod_{m=1}^{M-1}\Theta_{d_m} p_{d_M}\right] \ .
\end{equation}
The average success probability for the initial state $S_1=s_i$ is given by $\widetilde{\text{P}}'_\pi\left(s_i\right)$, which is the $i$th element of the vector $\widetilde{\text{P}}'_\pi$. Although the state space $\mathcal{S}'$ is finite, the action space $\mathcal{A}_{s_i}$ for each $s_i\in \mathcal{S}'$ is uncountably infinite for each $s_i$. However, the number of levels of residual battery energy is finite with the modification \eqref{upper_modi}. Thus, we have groups of actions for which the resulting residual battery energy is the same. In fact, it is sufficient to consider a finite action space to find $\underset{\pi \in \Pi}{\text{max}} \ \widetilde{\text{P}}'_\pi\left(s_i\right)$. This is proved in the next lemma and the proposition.
\begin{lemma}
\label{Adash}
For any decision rule $d \in \mathcal{D}$ there exists\break $d' \in \left\{d \ | \ d\left(s\right) \in \mathcal{A}'_s \ \forall s \in \mathcal{S}' \right\}$ such that $\Theta_d = \Theta_{d'}$, where
\begin{multline}\label{A_dash}
\mathcal{A}'_s = \big\{\lambda, u \ | \lambda\in [0,1], \ 0\leqslant u, \ \mathcal{E}_{T}\left(\lambda,u,s\right)=e_i,\\ i=1,2,\cdots,N_b \big\} \ .
\end{multline}
\end{lemma}
\begin{IEEEproof}
Channel fading is independent from the decision rule use and we denote $h_{m+1}=h$. Let $E_{m+1}=e_j$ with $j\in \left\{2,\cdots,N_b\right\}$ be the level of residual battery energy resulted from the action $d\left(S_m\right)$ for the state $S_m$. State of the next block is $S_{m+1}=\left(e_j,h\right)$ and we have $\Theta_d\left(S_m,S_{m+1}\right)=f_{\mathcal{H}}\left(h\right)$. In addition, with the aid of (\ref{upper_modi}) it can be seen that the action $d'\left(S_m\right)=\left(\lambda',E'_t\right)$ such that $\mathcal{E}_{T}\left(\lambda',E'_t,S_m\right)=e_{j-1}$ results in the same $E_{m+1}=e_j$. Therefore, we define $\mathcal{A}'_s$ as given in the lemma and thus $d'\left(S_m\right) \in \mathcal{A}'_s$ with $\Theta_{d'}\left(S_m,S_{m+1}\right)=f_{\mathcal{H}}\left(h\right)$, which concludes the proof.
\end{IEEEproof}
Using the following proposition we can further reduce the dimension of $\mathcal{A}_s$ to be finite.
\begin{proposition}
\label{finite_decision_space}
For any policy $\pi = \{d_1,d_2,\cdots,\}$ with $d_m\in \mathcal{D}$ for all $m$, there exists a policy $\pi' = \{d_1',d_2',\cdots\}$ with $d_m'\in \tilde{\mathcal{D}}$ for all $m$, such that $\text{P}'_{\pi'}\geqslant \text{P}'_{\pi}$, where
\begin{equation*}
\tilde{\mathcal{D}} = \left\{d \ | \ d\left(s\right) \in \mathcal{A}^*_s \ \forall s \in \mathcal{S}' \right\} \subset \mathcal{D} \ ,
\end{equation*}
\begin{equation*}
\mathcal{A}^*_s = \mathcal{A}'_{s,1}\cup \mathcal{A}'_{s,2} \ ,
\end{equation*}
\begin{equation*}
\mathcal{A}'_{s,1} = \left\{\lambda, u \ | \ \left(\lambda, u\right) \in \mathcal{A}'_s, \ \lambda=1 \right\} \ ,
\end{equation*}
\begin{equation}\label{A2}
\mathcal{A}'_{s,2} = \begin{cases}
\phi \qquad \ \text{if} \ s \in \mathcal{C}_1 \\ \quad\quad\ \ \ \text{otherwise,}\\
\left\{\lambda, u \ | \ \left(\lambda, u\right) \in \mathcal{A}'_s, \ \lambda = \frac{h P_s - 2 \sigma^2 \gamma_\tau}{h P_s - \sigma^2 \gamma_\tau} \right\}
\end{cases} \, ,
\end{equation}
where $\phi$ denotes the empty set.
\end{proposition}
\begin{IEEEproof}
See Appendix A.
\end{IEEEproof}
The operation of $\mathcal{A}^*_s$ is shown in Fig.~\ref{Modification}a.
With proposition~\ref{finite_decision_space}, we can claim, that for any policy $\pi \in \Pi$, there exists a policy in $\tilde{\Pi}=\tilde{\mathcal{D}}\times\tilde{\mathcal{D}}\times\tilde{\mathcal{D}}\times\cdots$, which has an average success probability, larger or equal to that of policy $\pi$. Therefore, it is sufficient to restrict our attention to the reduced policy space $\tilde{\Pi}$, when we search for a solution to $\underset{\pi \in \Pi}{\text{max}} \ \text{P}'_\pi\left(s_i\right)$, which is useful to calculate the upper bound $P_u$ as per the following proposition.
\begin{proposition}\label{supremum_modifed}
Average success probability in the modified system $\text{P}'_{\pi}$ satisfies, $\underset{\pi}{\text{max}} \ \widetilde{\text{P}}'_{\pi}\left(s_i\right)\geqslant \underset{\pi}{\text{max}}\ \widetilde{\text{P}}_{\pi}\left(s_i\right)$ for all $s_i \in \mathcal{S}'$
\end{proposition}
\begin{IEEEproof}
See Appendix B.
\end{IEEEproof}
Therefore, the upper bound $P_u$ can be calculated using
\begin{equation}\label{op_prob_reduced}
P_u = \mathbb{E}\left[ \underset{\pi \in \tilde{\Pi}}{\text{max}} \ \widetilde{\text{P}}'_\pi\big(\left(0,h_1\right)\big)\right] \ .
\end{equation}
Since the state space $\mathcal{S}'$ and the set $\tilde{\mathcal{D}}$ are both finite, the existence of $\underset{\pi \in \tilde{\Pi}}{\text{max}} \ \text{P}'_\pi\big(s\big)$ for all $s\in \mathcal{S}'$, is guaranteed \cite[Chapter 9]{Puterman}. To evaluate $\underset{\pi \in \tilde{\Pi}}{\text{max}} \ \text{P}'_\pi\big(s\big)$, we can use a standard average reward policy iteration algorithm, which consists of iterations of following two steps,
\begin{itemize}
\item At iteration $n$ ; $\pi_n \leftarrow d_n^\infty$
\begin{itemize}
\item \emph{Step-1} ; $\widetilde{\text{P}}'_{\pi_n} \leftarrow Evaluate\_Policy\left(\pi_n\right)$ \ ,
\item \emph{Step-2} ; $d_{n+1} \leftarrow Improve\_Policy\left(\widetilde{\text{P}}'_{\pi_n}\right)$ \ .
\end{itemize}
\end{itemize}
The policy iteration algorithm can be initiated with any resource allocation policy $\pi_1 = d_1^\infty$. For the details of the functions $Evaluate\_Policy\left(\pi_n\right)$, $Improve\_Policy\left(\widetilde{\text{P}}'_{\pi_n}\right)$ and the stopping criterion, the reader is referred to \cite[Algorithm 9.2.1]{Puterman}.
\section{Numerical Results}\label{Numerical}
Although our analysis is valid for any finite fading distributions of $\mathcal{H}$ and $\mathcal{G}$, in this section we consider a equiprobable quantization of a unit mean Rayleigh fading \cite{Parastoo} with $N_c=200$ channel states. Simulation results for $\text{P}_{\pi_l}$ in \eqref{lower_bound_eq} are generated by simulating the system with the stationary policy $\pi_l=d_l^\infty$.
\begin{figure}
\centering
\includegraphics[width=8cm]{Vs_B.eps}
\caption{The variation average success probability $\text{P}_\pi$ with the relay battery capacity $B$.}
\label{Vs_B}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{Vs_Q.eps}
\caption{The variation average success probability $\text{P}_\pi$ with the source transmit power $P_s$.}
\label{Vs_Q}
\end{figure}
Fig.~\ref{Vs_B} shows the variation of $\text{P}_{\pi_l}$ in \eqref{lower_bound_eq} and $P_u$ in \eqref{op_prob_reduced} for difference values of $N_b$ and, with the relay battery capacity $B$, where the source transmit power $P_s=0.5$\,mW and $2$\,mW. Simulation results match with analytical results in \eqref{lower_bound_eq}. As shown in the figure, smaller upper bounds can be obtained with a larger values for $N_b$. The gain of the upper bound from battery capacity $B=B_1$ compared to $B=B_2$ is $100 \times \frac{P_u|_{B=B_1}-P_u|_{B=B_2}}{P_u|_{B=B_2}} \% $. When source transmit power $P_s=2$\,mW, the gain is $29.8\%$ from battery capacity $10$\,$\mu$J compared to $4$\,$\mu$J, whereas the gain is $4.9\%$ from $16$\,$\mu$J compared to $10$\,$\mu$J. For the same increase in the battery capacity, the gain is small. This is also true for $P_s=0.5$\,W. Although a larger battery capacity results in more battery states, occupying a higher battery state is improbable, which explains the diminishing returns in average success probability with battery capacity. The performance gain of $\text{P}_{u}$ compared to $\text{P}_{\pi_l}$ is $100 \times \frac{\text{P}_{\pi^*}-\text{P}_{\pi_l}}{\text{P}_{\pi_l}} \% $. When the source transmit power $P_s=2$\,mW and $B=10$\,$\mu$J the performance gain of $\text{P}_{u}$ is $31\%$ and when the source transmit power $P_s=0.5$\,mW and $B=10$\,$\mu$J the gain is $107.8\%$.
Fig.~\ref{Vs_Q} shows the variation of $P_u$ and $\text{P}_{\pi_l}$ with the source transmit power $P_s$, for $B=2$\,$\mu$J and $2$\,$\mu$J. Average success probability achieved by the heuristic policy $\pi_l$ gets closer to the upper bound $P_u$ as the source transmit power is increased. This is more noticeable when the battery capacity is small. When the source transmit power $P_s$ is large such that for all $s \in \mathcal{S}$ and $\left(\lambda,u\right) \in \mathcal{A}^*_s$ the half block battery energy is $\mathcal{E}_{\frac{T}{2}}\left(\lambda,s\right)=B$, then for it is optimal to use total battery energy for data transmission to the destination. This makes heuristic policy optimal in this situation, which explains $\text{P}_{\pi_l}$ gets closer to $P_u$ for large $P_s$ or small $B$.
\section{Conclusion}\label{conclusion}
This paper considers SWIPT over a DF relay network with the power-splitting (PS) energy harvesting protocol at the relay. A mathematical framework is presented to investigate the feasibility of the network by evaluating an upper bound of the performance. Numerical results show that performance gain has diminishing returns with battery capacity and the proposed heuristic resource allocation policy achieves a performance close to the upper bound when the source power is large or the relay battery is small. Mathematical framework can be changed to include battery imperfections and power consumption by the information processing circuits and we intend to investigate these in a future work.
|
1,314,259,993,061 | arxiv | \section{Introduction}
In the last few years, information theory has seen vibrant developments
in the study of the non-vanishing error probability regime,
and in particular, the successes in applying normal approximations
to gauge the back-off from the asymptotic limits as a function of delay.
Extending the achievements for point-to-point communication systems in \cite{hayashi2009information}\cite{polyanskiy2010channel}\cite{kostina2012fixed}
to network information theory problems
usually requires new ideas for proving tight non-asymptotic bounds. For achievability, single-shot covering lemmas and packing lemmas \cite{verdu2012non}\cite{liu_marton} supply convenient tools for distilling single-shot achievability bounds from the classical asymptotic achievability proofs.
These single-shot bounds are easy to analyze in the stationary memoryless case by choosing the auxiliary random variables to be i.i.d.~and applying the law of large numbers or the central limit theorem.
In contrast, there are few examples of single-shot converse bounds in the network setting. Indeed, unlike their achievability counterparts, single-shot converses are often non-trivial to single-letterize to a strong converse.
In fact, there are few methods for obtaining strong converses for network information theory problems whose single-letter solutions involve auxiliaries; see e.g.~\cite[Section~9.2 ``Open problems and challenges ahead'']{CIT-086}.
Exceptions include the strong converses for select source networks \cite{csiszar2011information} where the method of types plays a pivotal role.
In this paper, through the example of a common randomness (CR) generation problem \cite[Theorem~4.2]{ahlswede1998common}, we demonstrate the power
of
a functional inequality, the \emph{Generalized Brascamp-Lieb-like (GBLL) inequality} \cite{lccv2015}:
\begin{align}
\int\exp\left(\sum_{j=1}^m\mathbb{E}[\log f_j(Y_j)|X=\cdot]-d\right){\rm d}\mu
&\le \prod_{j=1}^m\|f_j\|_{\frac{1}{c_j}},
\label{e_func}
\end{align}
in proving single-shot converses for problems involving multiple sources.
Here $\mu$, $(Q_{Y_j|X})$, $(\nu_j)$, $(c_j)$, $d$ are given and
$\|f_j\|_{\frac{1}{c_j}}:=\left(\int f_j^{1/c_j}{\rm d}\nu_j\right)^{c_j}$.
The key tool for single-letterizing such single-shot converses to strong converses is the ``achievability'' of the following problem: infimize the best constant $d$ in \eqref{e_func}
with the substitutions $\mu\leftarrow \mu_n$, $\nu_j\leftarrow\nu_j^{\otimes n}$ and $Q_{Y_j|X}\leftarrow Q_{Y_j|X}^{\otimes n}$,
where
the auxiliary measure $\mu_n$ is within a neighborhood (say in total variation) of $\mu^{\otimes n}$.
Interestingly, a product $\mu_n$ is generally not a good choice.
On the surface, this is reminiscent of the smooth R\'{e}nyi entropy \cite{renner2005simple}, who showed that the infimum (resp.~supremum) of the R\'{e}nyi entropy of order $\alpha<1$ (resp.~$\alpha>1$) of an auxiliary measure with a neighborhood of a product distribution behaves like the Shannon entropy. In reality, the smooth version of GBLL appears to be a much deeper problem, since structure at a finer resolution than weak typicality is involved.
The general philosophy appears to be that under certain regularity conditions,
$\frac{d}{n}$ (where $d$ is the best constant in the setting of product measures and smoothing above) converges to the best constant in a mutual information inequality.
We provide a general approach for verifying this principle, and apply it to the discrete memoryless and the Gaussian source.
When this principle holds, our single-shot converse proves the strong converse for the CR generation problem.
The proposed approach to strong converses has two main advantages compared with the method of types approach in \cite{csiszar2011information},
which are nicely illustrated by the example of CR generation:
1) The argument covers possibly stochastic decoders.
2) As illustrated by the Gaussian example, the approach is applicable to some non-discrete sources where the method of types is futile.
This is perhaps the first instance of a strong converse for a continuous source when the rate region involves auxiliaries.
We also refine the analysis to bound the second order rate.
In addition, we discuss the ``converse'' part of smooth BLL, which
generally follows from the achievability of CR generation problems. In fact, smooth BLL and CR generation may be considered as dual problems where the achievability of one implies the converse of the other, and vice versa.\footnote{Another example of such ``dual problems'' is channel resolvability and identification coding \cite{han1993approximation}.}
It is also interesting to note that for hypercontractivity, which is a special case of the BLL inequality with the best constant being zero, Anantharam et~al.~\cite{anan_13} showed the equivalence between a relative entropy inequality and a mutual information inequality. This equivalence is lost for positive best constants. Thus smooth BLL is a conceptually satisfying way to regain the connection between these two inequalities.
Omitted proofs are given in the appendices of \cite{lccv_smooth2016}.
\section{Preliminaries}
\begin{defn}\label{defn1}
Given a nonnegative $\mu$ on $\mathcal{X}$, $\nu_j$ on $\mathcal{Y}_j$, and random transformations $Q_{Y_j|X}$, and $c_j\in(0,\infty)$, $j\in\{1,\dots,m\}$, define
\begin{align}
\rmd(\mu,(Q_{Y_j|X}),(\nu_j),c^m)
:=\sup\left\{\sum_{l=1}^m c_l D(P_{Y_l}\|\nu_j)-D(P_X\|\mu)\right\}
\nonumber
\end{align}
where the sup is over $P_X\ll \mu$ and $P_X\to Q_{Y_j|X}\to P_{Y_j}$.
\end{defn}
We shall abbreviate the notation in Definition~\ref{defn1} as $\rmd(\mu,\nu_j,c^m)$ when there is no confusion.
Note that $\mu$ and $\nu_j$ are not necessarily probability measures, and $\mu\to Q_{Y_j|X}\to \nu_j$ need not hold. These liberties are useful, e.g. in the proof of Theorem~\ref{thm_gauss}.
Generalizing an approach in \cite{carlen2009subadditivity}, we established the following \cite{lccv2015}:
\begin{prop}\label{prop_func}
Under the assumptions of Definition~\ref{defn1}, $\rmd(\cdot)$ is the minimum $d$ such that
\eqref{e_func} holds
for all nonnegative measurable functions $f_j$.
\end{prop}
We call \eqref{e_func} a \emph{generalized Brascamp-Lieb-like inequality} (GBLL).
The case of deterministic $Q_{Y_j|X}$ was considered in \cite{carlen2009subadditivity}, which we shall call a \emph{Brascamp-Lieb-like inequality} (BLL).
In the special case where $Q_{Y_j|X}$'s are a linear projections and $\mu$ and $\nu_j$ are Gaussian or Lebesgue, \eqref{e_func} is called a Brascamp-Lieb inequality; it is well-known that a Brascamp-Lieb inequality holds for a specific value of $d$ if and only if it holds for all Gaussian functions $(f_j)$ \cite{brascamp1976best}.
\begin{defn}
For nonnegative measures $\nu$ and $\mu$ on the same measurable space $(\mathcal{X},\mathscr{F})$ and $\gamma\in[1,\infty)$,
the $E_{\gamma}$ divergence is defined as
\begin{align}
E_{\gamma}(\nu\|\mu):=\sup_{\mathcal{A}\in\mathscr{F}}
\{\nu(\mathcal{A})-\gamma \mu(\mathcal{A})\}.
\end{align}
\end{defn}
Note that under this definition $E_1(P\|\mu)$ does not equal $\frac{1}{2}|P-\mu|$ if $\mu$ is not a probability measure. Properties of $E_{\gamma}$ used in this paper can be found in \cite{liu2015_egamma_arxiv}.
\begin{defn}
For $\delta\in[0,1)$, $Q_X$, $(Q_{Y_j|X})$ and $(\nu_j)$, define
\begin{align}
{\rm d}_{\delta}(Q_X,\nu_j,c^m):= \inf_{\mu\colon E_1(Q_X\|\mu)\le \delta}{\rm d}(\mu,\nu_j,c^m).
\label{e_smoothconst}
\end{align}
In the stationary memoryless case, define the \emph{$\delta$-smooth GBLL rate}\footnote{As is clear from the context, the random transformations implicit on the right side of \eqref{e6} are $(Q_{Y_j|X}^{\otimes n})$.}
\begin{align}
{\rm D}_{\delta}(Q_X,\nu_j,c^m):=
\limsup_{n\to\infty}\frac{1}{n}{\rm d}_{\delta}(Q_X^{\otimes n},
\nu_j^{\otimes n},c^m),
\label{e6}
\end{align}
and the \emph{smooth GBLL rate} is the limit
\begin{align}
{\rm D}_{0^+}(Q_X,\nu_j,c^m):=\lim_{\delta\downarrow0}{\rm D}_{\delta}(Q_X,\nu_j,c^m).
\end{align}
\end{defn}
\begin{rem}
Allowing unnormalized measures avoids the unnecessary step of normalization in the proof, and is in accordance with the literature on smooth R\'{e}nyi entropy, where such a relaxation generally gives rise to nicer properties and tighter non-asymptotic bounds, cf.~\cite{renner2005simple}\cite{liu2015_egamma_arxiv}.
\end{rem}
\begin{defn}\label{defn_dstar}
Given $Q_X$, $(Q_{Y_j|X})$ and $c^m\in(0,\infty)^m$, define
\begin{align}
\rmds(Q_X,c^m):=\sup_{P_{U|X}}\left\{\sum_{l=1}^m c_l I(U;Y_l)-I(U;X)\right\}.
\end{align}
We say $Q_X$, $(Q_{Y_j|X})$ and $(c_j)$ satisfy the \emph{$\delta$-smooth property} if
\begin{align}
{\rm D}_{\delta}(Q_X,Q_{Y_j},c^m)
=\rmds(Q_X,c^m),
\label{e_smooth}
\end{align}
\emph{(weak) smooth property} if ${\rm D}_{0^+}(Q_X,Q_{Y_j},c^m)
=\rmds(Q_X,c^m)$,
and \emph{strong smooth property} if \eqref{e_smooth} holds for all $\delta\in(0,1)$.
\end{defn}
From these definitions and a tensorization property of $\rmd(\cdot)$ \cite{lccv2015} we clearly have
\begin{align}
\rmd(Q_X,Q_{Y_j},c^m)
={\rm D}_0(Q_X,Q_{Y_j},c^m)
&\ge {\rm D}_{\delta}(Q_X,Q_{Y_j},c^m).
\end{align}
The goal is to explore conditions for ${\rm D}_{\delta}(Q_X,Q_{Y_j},c^m)=\rmds(Q_X,c^m)$.
\section{Achievabilities for Smooth GBLL}
Under various conditions, we provide upper bounds on ${\rm D}_{\delta}(Q_X,Q_{Y_j},c^m)$, establishing the achievability part of the strong smooth property.
\subsection{Hypercontractivity
\label{sec_hypercontractivity}
If $\rmds(Q_X,c^m)=0$,
by an extension of the proof of equivalent formulations of hypercontractivity \cite{anan_13} we also have
$
\rmd(Q_X,Q_{Y_j},c^m)=0
$,
establishing that ${\rm D}_{0}(Q_X,Q_{Y_j},c^m)=\rmds(Q_X,c^m)$.
\subsection{Finite $|\mathcal{X}|$, and Beyond}
The main objective of this section is to show that
\begin{thm}\label{thm_discrete}
${\rm D}_{0^+}(Q_X,Q_{Y_j},c^m)\le\rmds(Q_X,c^m)$ if $\mathcal{X}$ is finite.
\end{thm}
We present a general method of proving achievability of smooth GBLL which, although not intuitive at the first sight, turns out to be successful for the distinct cases of the discrete and Gaussian sources. The following tensorization result is useful:
\begin{lem}\label{lem_singleletter}
Suppose $\tau_{\alpha}\colon \mathcal{X}\to\mathbb{R}$ is measurable for each (abstract) index $\alpha\in\mathcal{A}$. Fix any $\epsilon\in(0,1)$, and for each $n\in\{1,\dots\}$ define $g(n)$ as the supremum of
\begin{align}
\frac{1}{n} \left[
\sum_j c_j D(P_{Y^n|U}\|\nu_j^{\otimes n}|P_U)-D(P_{X^n|U}\|\mu^{\otimes n}|P_U)\right]
\label{e_f}
\end{align}
over $P_{UX^n}$ such that
$
\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n \tau_{\alpha}(\hat{X}_i)\right]\le\epsilon
$,
where
$\hat{X}^n\sim P_{X^n}$
and
$P_{UX^nY^n}:=P_{UX^n}Q_{Y|X}^{\otimes n}$. Then $g(n)\le g(1)$.
\end{lem}
The functions $\tau_{\alpha}(\cdot)$ can be thought of as (possibly negative) cost functions that enforce the $P_{UX}$
maximizing \eqref{e_f} to satisfy $P_X\approx Q_X$. If the probability that an i.i.d.~sequence induces a small cost is large, then one can choose the $\mu$ in the definition of the smooth property to be the restriction\footnote{In this paper, by restriction of a measure on a set we mean the result of cutting off the measure outside that set (without renormalizing).} of $Q_X^{\otimes n}$ on such a set. Therefore the following lemma will be the key to our proofs of the smooth property:
\begin{lem}\label{lem_key}
Suppose $\tau_{\alpha}$ is as in Lemma~\ref{lem_singleletter} and
define
\begin{align}
\mathcal{S}_{\epsilon}^n
:=\left\{x^n\colon\frac{1}{n}\sum_{i=1}^n \tau_{\alpha}(x_i)\le \epsilon\right\}.
\label{e_s}
\end{align}
If $P_{X^n}$ is supported on $\mathcal{S}_{\epsilon}^n$ for each $n$, then
\begin{align}
&\quad\limsup_{n\to\infty}\frac{1}{n}
\left[\sum_j c_jD(P_{Y_j^n}\|\nu_j^{\otimes n})
-D(P_{X^n}\|\mu^{\otimes n})\right]
\nonumber\\
&\le
\sup
\left\{\sum c_j D(P_{Y_j|U}\|\nu_j|P_U)
-D(P_{X|U}\|\mu|P_U)\right\}
\label{e_14}
\end{align}
where the sup on the right is over $P_{UX}$ such that $\mathbb{E}[\tau_{\alpha}(\hat{X})]\le\epsilon$.
\end{lem}
A remarkable aspect of Lemma~\ref{lem_key} is that the left side of \eqref{e_14}, which is a multi-letter quantity from the definition of $\rmd(\cdot)$, is upper bounded by a single-letter quantity.
\begin{lem}\label{lem_weakcont}
Suppose $(\mathcal{X},\mathscr{F})$ is a second countable topological space and $Q_X$ is a Borel measure. Define
\begin{align}
\sigma\colon P_X\mapsto\sum c_j D(P_{Y_j}\|Q_{Y_j})
-D(P_{X}\|Q_X).
\label{e_semi}
\end{align}
If $\phi$, the concave envelope of $\sigma$, is upper semicontinuous at $Q_X$,
then ${\rm D}_{0^+}(Q_X,Q_{Y_j},c^m)\le\rmds(Q_X,c^m)$.
\end{lem}
\begin{rem}
If $c_1=\dots =c_m=0$, then $\phi(P_X)=-D(P_{X}\|Q_X)$ always satisfies the upper semicontinuity in Lemma~\ref{lem_weakcont} because of the weak semicontinuity of the relative entropy. On the other hand, taking $m=1$, $c_1=2$, $Q_X$ any distribution on a countably infinite alphabet with $H(Q_X)<\infty$,
and $Q_{Y_1|X}$ the identity transformation,
we see $\sigma(P_X)=H(P_X)+D(P_X\|Q_X)$ and the upper semicontinuity condition in Lemma~\ref{lem_weakcont} fails.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm_discrete}]
Assume w.l.o.g.~that $Q_X(x)>0,\,\forall x$ since otherwise we can delete $x$ from $\mathcal{X}$. Then $Q_X$ is in the interior of the probability simplex. Moreover $\phi(\cdot)$ in Lemma~\ref{lem_weakcont} is clearly bounded.
Thus by \cite[Corollary~7.4.1]{rockafellar2015convex},
the weak semicontinuity in Lemma~\ref{lem_weakcont} is fulfilled.
\end{proof}
\begin{rem}
For general $\mathcal{X}$, one cannot use the property of convex functions to conclude the semicontinuity as in the proof of Theorem~\ref{thm_discrete}. In fact, whenever $|\mathcal{X}|=\infty$, there are points in $\mathcal{X}$ with arbitrarily small probability, thus $Q_X$ cannot be in the interior of the probability simplex even under the stronger topology of total variation.
\end{rem}
\subsection{Gaussian Case}
The semicontinuity assumption in
Lemma~\ref{lem_weakcont}
appears too strong
for the case of the Gaussian distribution, which has a non-compact support. Nevertheless, we can proceed by picking a different $\tau_{\alpha}(\cdot)$ in Lemma~\ref{lem_key}.
\begin{thm}\label{thm_gauss}
${\rm D}_{0^+}(Q_X,Q_{Y_j},c^m)\le\rmds(Q_X,c^m)$ if $Q_{\bf X}$ and $(Q_{{\bf Y}_j|\bf X})$ are Gaussian.
\end{thm}
The proof hinges on our prior result \cite{lccv2015} about the Gaussian optimality in an optimization under a covariance constraint: suppose $\mu$ and $\nu_j$ are the Lebesgue measures. Define
\begin{align}
F({\bf M})
&:=\sup\left\{-\sum c_j h({\bf Y}_j|U)+h({\bf X}|U)\right\}
\label{e14}
\\
=&\sup
\left\{\sum c_j D(P_{{\bf Y}_j|U}\|\nu_j|P_U)
-D(P_{\mathbf{X}|U}\|\mu|P_U)\right\}
\label{e15}
\end{align}
where the supremums are over $P_{U\bf X}$ such that ${\bf \Sigma}_{\bf X}\preceq {\bf M}$.
Also suppose w.l.o.g.~that ${\bf X}\sim \mathcal{N}({\bf 0},{\bf \Sigma})$ under $Q_{\bf X}$.
\begin{prop}[\cite{lccv2015}]
\label{prop14}
$F(\mathbf{M})$ equals the sup in \eqref{e15} restricted to constant $U$ and Gaussian $\mathbf{X}$, which implies that
\begin{align}
F({\bf \Sigma})+C=
\rmds(Q_{\bf X},Q_{{\bf Y}_j},c^m)
\label{e16}
\end{align}
where
\begin{align}
C:=\sum_j c_j h({\bf Y}_j)-h({\bf X}_j).
\label{e_c}
\end{align}
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm_gauss}]
Put $\mathcal{A}$ as the set of unit length vectors in $\mathcal{X}$ (a Euclidean space), and for each $\alpha\in\mathcal{A}$ define
$
\tau_{\alpha}(\mathbf{x}):=(\alpha^{\top}{\bf \Sigma}^{-\frac{1}{2}}{\bf x})^2-1
$.
Now, observe that for ${\bf x}^n\in\mathcal{X}^n$,
\begin{align}
\frac{1}{n}\sum_i\tau_{\alpha}(\mathbf{x}_i)
:=\alpha^{\top}{\bf \Sigma}^{-\frac{1}{2}}\left(\frac{1}{n}\sum_i{\bf x}
{\bf x}^{\top}\right)
{\bf \Sigma}^{-\frac{1}{2}}\alpha
-1,
\end{align}
so $\frac{1}{n}\sum_i\tau_{\alpha}(\mathbf{x}_i)\le\epsilon_1$ for all $\alpha\in\mathcal{A}$ is equivalent to the bound on the empirical covariance:
$
\frac{1}{n}\sum_i{\bf x}
{\bf x}^{\top}
\preceq (1+\epsilon_1){\bf \Sigma}
$.
Consider also the ``weakly typical set'' $\mathcal{T}_{\epsilon_2}^n$, defined as the set of sequences $\mathbf{x}^n$ such that
\begin{align}
&\frac{1}{n}\sum_i\left[\imath_{Q_{\bf X}\|\mu}(\mathbf{x}_i)-\sum_j c_j\mathbb{E}[\imath_{Q_{\mathbf{Y}_j}\|\nu_j}(\mathbf{Y}_j)|\mathbf{X}=\mathbf{x}_i]
\right]
\le
C+\epsilon_2
\end{align}
where $C$ was defined in \eqref{e_c}.
Now set $\mu_n$ as the restriction of $Q_{\bf X}^{\otimes n}$ on $\mathcal{S}_{\epsilon_1}^n\cap \mathcal{T}_{\epsilon_2}^n$. If $P_{{\bf X}^n}\ll \mu_n$, by Lemma~\ref{lem_key} we have
\begin{align}
\limsup_{n\to\infty}\frac{1}{n}
\left[\sum_j c_jD(P_{{\bf Y}_j^n}\|\nu_j^{\otimes n})
-D(P_{{\bf X}^n}\|\mu^{\otimes n})\right]
\le F((1+\epsilon_1){\bf \Sigma}).
\label{e_opt}
\end{align}
Since $P_{{\bf X}^n}$ is supported on $\mathcal{T}_{\epsilon_2}^n$, we also have
\begin{align}
&\frac{1}{n}
\left[\sum_j c_jD(P_{{\bf Y}_j^n}\|\nu_j^{\otimes n})
-D(P_{{\bf X}^n}\|\mu^{\otimes n})\right]\nonumber
+C
\nonumber
\\
&\ge\frac{1}{n}
\left[\sum_j c_jD(P_{{\bf Y}_j^n}\|Q_{\mathbf{Y}_j}^{\otimes n})
-D(P_{{\bf X}^n}\|Q_{\bf X}^{\otimes n})\right]-\epsilon_2
\label{e33}
\end{align}
Hence from \eqref{e_opt}-\eqref{e33} we conclude
\begin{align}
&\limsup_{n\to\infty}\frac{1}{n}
\left[\sum_j c_jD(P_{{\bf Y}_j^n}\|Q_{\mathbf{Y}_j}^{\otimes n})
-D(P_{{\bf X}^n}\|\mu_n)\right]
\nonumber
\\
&\le
F((1+\epsilon_1){\bf \Sigma})+C+\epsilon_2
\label{e34}
\end{align}
where we used $D(P_{{\bf X}^n}\|Q_{\bf X}^{\otimes n})=D(P_{{\bf X}^n}\|\mu_n)$.
Also, by the law of large numbers, $\lim_{n\to\infty} Q_{\bf X}^{\otimes n}(\mathcal{S}_{\epsilon_1}^n\cap \mathcal{T}_{\epsilon_2}^n)=1$ so
$
\lim_{n\to\infty} E_1(Q_{\bf X}^{\otimes n}\|\mu_n)=1
$.
Thus \eqref{e34}, Proposition~\ref{prop14} and the continuity of $F$ (which can be verified since \eqref{e14} is essentially a matrix optimization problem) imply the desired result.
\end{proof}
\section{Converse for the One-Communicator Problem}
\label{sec_onecommunicator}
We prove a single-shot bound connecting smooth GBLL and one-communicator CR generation~\cite[Theorem~4.2]{ahlswede1998common}, allowing us to prove the converse of one using the achievability of the other.
Let $Q_{XY^m}$ be the joint distribution of sources $X$, $Y_1$, \dots, $Y_m$, observed by terminals ${\sf T}_0$, \dots, ${\sf T}_m$ as shown in Figure~\ref{f_1com}.
The communicator ${\sf T}_0$ computes the integers $W_1(X)$, \dots, $W_m(X)$ and sends them to ${\sf T}_1$, \dots, ${\sf T}_m$, respectively. Then, terminals ${\sf T}_0$, \dots, ${\sf T}_m$ compute integers $K(X)$, $K_1(Y_1,W_1)$,\dots, $K_m(Y_m,W_m)$.
The goal is to produce $K=K_1=\dots=K_m$ with high probability
with $K$ almost equiprobable.
\begin{figure}[h!]
\centering
\begin{tikzpicture}
[scale=2,
dot/.style={draw,fill=black,circle,minimum size=0.7mm,inner sep=0pt},arw/.style={->,>=stealth}]
\node[rectangle,draw,rounded corners] (A) {${\sf T}_1$};
\node[rectangle,draw,rounded corners] (B) [right= 1.4cm of A] {${\sf T}_2$};
\node[rectangle] (C) [right =of B] {$\dots$};
\node[rectangle,draw,rounded corners] (D) [right =of C] {${\sf T}_m$};
\node[rectangle,draw,rounded corners] (T) [above right=of B, xshift=-13mm, yshift=10mm] {${\sf T}_0$};
\node[rectangle] (Z) [left=0.4cm of T] {$X$};
\node[rectangle] (K1) [below =0.4cm of A] {$K_1$};
\node[rectangle] (K2) [below =0.4cm of B] {$K_2$};
\node[rectangle] (Km) [below =0.4cm of D] {$K_m$};
\node[rectangle] (K) [above =0.4cm of T] {$K$};
\node[rectangle] (X1) [left =0.4cm of A] {$Y_1$};
\node[rectangle] (X2) [left =0.4cm of B] {$Y_2$};
\node[rectangle] (Xm) [left =0.4cm of D] {$Y_m$};
\draw[arw] (Z) to node[]{} (T);
\draw[arw] (X1) to node[]{} (A);
\draw[arw] (X2) to node[]{} (B);
\draw[arw] (Xm) to node[]{} (D);
\draw [arw] (A) to node[midway,above]{} (K1);
\draw [arw] (B) to node[midway,above]{} (K2);
\draw [arw] (D) to node[midway,above]{} (Km);
\draw [arw] (T) to node[]{} (K);
\draw [arw,line width=1.5pt] (T) to node[midway,left]{$W_1$} (A.north);
\draw [arw,line width=1.5pt] (T) to node[midway,left]{$W_2$} (B.north);
\draw [arw,line width=1.5pt] (T) to node[midway,left]{$W_m$} (D.north);
\end{tikzpicture}
\caption{CR generation with one-communicator}
\label{f_1com}
\end{figure}
In the stationary memoryless case, put $X\leftarrow X^n$, $Y_j\leftarrow Y_j^n$. Denote by $R$ and $R_j$ the rates of $K$ and $W_j$, respectively.
Under various performance metrics (cf.~\cite{ahlswede1998common}\cite{liu2015key}), the achievable region is the set of $(R,R_1,\dots,R_m)$ such that
\begin{align}
\rmds(Q_X,c^m)+\sum_j c_jR_j\ge \left(\sum_jc_j-1\right)R
\label{e36}
\end{align}
for all $c^m\in(0,\infty)^m$. \footnote{Remark in passing that the corresponding \emph{key} generation problem, which places the additional constraint that $W_j\perp K$ asymptotically for each $j$, is solved in \cite{liu2015key} with a different rate region involving $m+1$ auxiliaries.}
\begin{thm}[Strong converse for one-communicator CR generation]
\label{thm_onecommunicator}
For finite $|\mathcal{X}|$, $|\mathcal{Y}_1|,\dots,|\mathcal{Y}_m|$,
suppose $(R,R_1,\dots,R_m)$ fails \eqref{e36} for some $c^m$.
If $(\delta_1,\delta_2)$ is such that
\begin{align}
\mathbb{P}[K=K_1=\dots=K_m]
&\ge 1-\delta_1;
\label{e_38}
\\
\frac{1}{2}|Q_K-T_K|
&\le\delta_2
\label{e_39}
\end{align}
can hold for some CR generation scheme at rates $(R,R_1,\dots,R_m)$ for sufficiently large $n$
where $T_K$ is the equiprobable distribution on $\mathcal{K}$,
then
$
\delta_1+\delta_2\ge 1
$.
\end{thm}
The following lemma establishes a \emph{single-shot} connection between one-communicator CR generation and smooth GBLL, which allows us to prove the converse of one problem from the achievability of the other.
For simplicity of presentation, we state it in the case of $m=1$.\footnote{Note that this problem is unlike the usual ``image-size characterization'' \cite[Chapter~15]{csiszar2011information} which is difficult to generalize to $m\ge 3$ case.}
\begin{lem}\label{lem_connect}
Suppose that there exist $\delta_1,\delta_2\in(0,1)$,
a stochastic encoder $Q_{W|X}$, and deterministic decoders $Q_{K|X}$ and $Q_{\hat{K}|WY}$, such that \eqref{e_38} and \eqref{e_39} hold.
Also, suppose that there exist $\mu_X$, $\delta,\epsilon,\epsilon'\in(0,1)$ and $c,d\in(0,\infty)$ such that
\begin{align}
E_1(Q_X\|\mu_X)
&\le \delta;
\label{e_neighbor}
\\
\mu_X\left(x\colon Q_{Y|X=x}(\mathcal{A})\ge 1-\epsilon'\right)
&\le
2^c\exp(d) Q_Y^{c(1-\epsilon)}(\mathcal{A})
\label{e39}
\end{align}
for any $\mathcal{A}\subseteq \mathcal{Y}$.
Then, for any $\delta_3,\delta_4\in(0,1)$ such that $\delta_3\delta_4=\delta_1+\delta$, we have
\begin{align}
\delta_2\ge 1-\delta-\delta_3-\frac{1}{|\mathcal{K}|}
-\frac{2^{\frac{1}{1-\epsilon}}\exp\left(\frac{d}{c(1-\epsilon)}\right)
|\mathcal{W}|}{(\epsilon'-\delta_4)
^{\frac{1}{c(1-\epsilon)}}|\mathcal{K}|^{1-\frac{1}{c(1-\epsilon)}}}.
\label{e_41}
\end{align}
\end{lem}
\begin{rem}\label{rem15}
The relevance of the Lemma~\ref{lem_connect} to smooth GBLL is seen by setting
$
f(y):=(1_{\mathcal{A}}(y)+ Q_Y(\mathcal{A}) 1_{\mathcal{\bar{A}}}(y))^c
$
in \eqref{e_func}. We then see \eqref{e39} holds for any $\epsilon=\epsilon'\in(0,1)$.
\end{rem}
\begin{rem}\label{rem_16}
In the stationary memoryless case $Q_X\leftarrow Q_X^{\otimes n}$, $Q_{Y|X}\leftarrow Q_{Y|X}^{\otimes n}$, suppose $|\mathcal{X}|,|\mathcal{Y}|<\infty$.
Using the blowing-up lemma \cite{ahlswede1976bounds}, we can show that
for any $\delta,\epsilon,\epsilon'\in(0,1)$ and $d>\rmds(Q_X,c)$, there exists $n$ large enough such that \eqref{e39} is satisfied with $d\leftarrow nd$ for some $\mu_X$ (more precisely, the restriction of $Q_X^{\otimes n}$ on a strongly typical set) satisfying \eqref{e_neighbor}.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm_onecommunicator}]
Again consider $m=1$ case for simplicity.
Suppose that $(R,R_1)$ is such that
\eqref{e36} fails for some $c>0$. Then, there is $\epsilon\in(0,1)$ and $d>\rmds(Q_X,c)$ such that
\eqref{e_54} does not hold.
If we choose $\delta>0$ arbitrarily small, then $\delta_3$ can be made arbitrarily close to $\delta_1$, in which case $\delta_4$ is forced to be close to $1$. Pick $\epsilon'>\delta_4$. These choices combined with Remark~\ref{rem_16}, Theorem~\ref{thm_discrete} and \eqref{e_41}, show that
$
\delta_1+\delta_2\ge 1
$.
\end{proof}
Another application of Lemma~\ref{lem_connect} is the following:
\begin{thm}[Weak converse for smooth GBLL]
\label{thm_weak}
\begin{align}
{\rm D}_{0^+}(Q_X,Q_{Y_j},c^m)
\ge\rmds(Q_X,c^m)
\label{e52}
\end{align}
\end{thm}
\begin{proof}
For simplicity, we prove for the case of $m=1$. For any $d>{\rm D}_{0^+}(Q_X,Q_Y,c)$ (achievable rate for smooth GBLL)
and any $(R,R_1)$ achievable for one-communicator CR generation,
we show that
\begin{align}
\frac{d}{c(1-\epsilon)}+R_1>R\left(1-\frac{1}{c(1-\epsilon)}\right)
\label{e_54}
\end{align}
for any $\epsilon\in(0,1)$, which will establish \eqref{e52} because of the achievable region formula \eqref{e36}.
We can choose $\delta,\delta_1,\delta_2,\delta_3,\delta_4$ such that
$\delta_2< 1-\delta-\delta_3$
and $\delta_4<\epsilon$.
For large $n$, \eqref{e_38} and \eqref{e_39} can be satisfied, and by Remark~\ref{rem15}, for $\epsilon'=\epsilon$, we can find $\mu_X$ satisfying \eqref{e_neighbor} and \eqref{e39} with $Q_X\leftarrow Q_X^{\otimes n}$, $Q_{Y|X}\leftarrow Q_{Y|X}^{\otimes n}$ and $d\leftarrow nd$.
Thus \eqref{e_54} holds because the last term in \eqref{e_41} must vanish as $n\to\infty$.
\end{proof}
\section{Converse for the Omniscient Helper Problem}
Note that Theorem~\ref{thm_weak} only establishes a weak converse for smooth GBLL and Theorem~\ref{thm_onecommunicator} is only for finite alphabets and deterministic decoders, because of the use of the blowing-up lemma. In this section we improve these results in a special case where $X=(Y_1,\dots,Y_m)$, that is, in the special case of smooth BLL and \emph{omniscient helper} CR generation.
To see why the problem becomes simpler in this special case, note that the set $\{x\colon Q_{Y|X=x}(\mathcal{A})\ge 1-\epsilon'\}$ in \eqref{e39} can be regarded as the ``preimage''
of the set $\mathcal{A}$ under the random transformation. In the case of deterministic $Q_{Y_j|X}$, there is no difference regarding the choice of $\epsilon'\in(0,1)$. However, in general a large $\epsilon'$ may imply a large $\epsilon$ on the right side of \eqref{e39}. Nevertheless, under the conditions for the blowing-up lemma, $\epsilon'$ and $\epsilon$ can be chosen independently (Remark~\ref{rem_16}).
In our prior work \cite{liu2015key}, a single-shot bound was derived via hypercontractivity which shows the strong converse property of the secret key (or CR) per unit cost.
From the current perspective, no smoothing is needed for that particular $c^m$ (which can be viewed as the orientation of the supporting hyperplane) for the reason explained in Section~\ref{sec_hypercontractivity}.
Straightforward extensions of the analysis from hypercontractivity to BLL inequality yields only a loose outer bound for the rate region when $\rmd(Q_X,Q_{Y_j},c^m)>\rmds(Q_X,c^m)$.
However, following the philosophy in the present paper, we may choose $\mu$ which is $E_1$-close to $Q_X$ and expect that $\rmd(\mu,Q_{Y_j},c^m)\approx\rmds(Q_X,c^m)$.
Thus by a slight change of the analysis in \cite{liu2015key}, we can show the following.
\begin{thm}[single-shot converse for omniscient helper CR generation]
\label{thm_oneshot}
If $d\ge \rmd(\mu, Q_{Y_j},c^m)$ for some $\mu$ satisfying
$
E_1(Q_{Y^m}\|\mu)\le\delta
$,
then
{\small
\begin{align}
\frac{1}{2}|Q_{K^m}-T_{K^m}|
\ge
1-\frac{1}{|\mathcal{K}|}
-\frac{\prod_{l=1}^m |\mathcal{W}_l|^{\frac{c_l}{\sum c_i}}}
{|\mathcal{K}|^{1-\frac{1}{\sum c_i}}}
\exp\left(\frac{d}{\sum c_i}\right)
-\delta.
\label{e38}
\end{align}
}
where $T_{K^m}(k^m):=\frac{1}{|\mathcal{K}|}1\{k_1=\dots=k_m\}$.
\end{thm}
Note that Theorem~\ref{thm_oneshot} applies for stochastic encoders and decoders, and in its proof, the function $f_j(\cdot)$ in \eqref{e_func} will take the role of $\max_w Q_{K_j|W_jY_j}(k|w,\cdot)$. However, the intuition is best explained in the case of deterministic decoders: let $\mathcal{A}_{kw_j}^j$ be the decoding set for $K_j=k$ upon receiving $w_j$ by ${\sf T}_j$. Then
\begin{align}
\mu(K_1=\dots=K_m=k)
&\le
\mu\left(\cap_j\cup_{w_j}\mathcal{A}_{kw_j}^j\right)
\\
&\le
\exp(d)\prod_j Q_{Y_j}^{c_j}\left(\cup_{w_j}\mathcal{A}_{kw_j}^j\right)
\label{e_35}
\end{align}
where the crucial step \eqref{e_35}, which may be viewed as a change-of-measure from a joint distribution to uncorrelated distributions (with powers), follows by choosing indicator functions in the BLL inequality. After some manipulations, one can bound the total variation between $\mu_{K^m}$ (consequently $Q_{K^m}$) and $T_{K^m}$.
\begin{cor}[Strong converse for omniscient helper CR generation]
Suppose $(R,R_1,\dots,R_m)$ fails \eqref{e36} for some $c^m$, and there exist a coding scheme at rates $(R,R_1,\dots,R_m)$
\begin{align}
\frac{1}{2}|Q_{K_1\dots K_m}-T_{K_1\dots K_m}|\le \delta
\label{e_62}
\end{align}
for sufficiently large $n$.
Then $\delta\ge1$ if $Q_{Y^m}$, $(Q_{Y_j|Y^m})$ and $c^m$ satisfy the smooth property (as in the case of discrete/Gaussian $Q_{Y^m}$).
\end{cor}
In the Gaussian case, refining the analysis in Theorem~\ref{thm_gauss}, we can derive a second order achievability bound for smooth BLL, which, in view of Theorem~\ref{thm_oneshot},
implies a second order converse bound for CR generation: for any sequence of CR generation schemes with non-vanishing error probability, we have
{\small
\begin{align}
\liminf_{n\to\infty}\sqrt{n}\left[\left(\sum c_j-1\right)R_n-\sum c_j R_{jn}-\rmds(Q_{Y^m},c^m)\right]
\le D\nonumber
\end{align}
}
for some constant $D$ (explicit formula given in \cite{lccv_smooth2016}), where $R_n$, $R_{1n}$, \dots, $R_{mn}$ are rates at blocklength $n$.
\begin{rem}
We used slightly different performance measures for the one-communicator problem and the omniscient helper problem.
If $\delta_1$ and $\delta_2$ satisfy \eqref{e_38}-\eqref{e_39}
then $\delta\leftarrow\delta_1+\delta_2$ satisfies \eqref{e_62},
so a strong converse measured by \eqref{e_62} implies a strong converse measured by \eqref{e_38}-\eqref{e_39}.
On the other hand,
if $\delta$ satisfies \eqref{e_62} then
$\delta_1\leftarrow\delta$ and $\delta_2\leftarrow\delta$ satisfy \eqref{e_38}-\eqref{e_39}.
Thus the strong converse in the sense of \eqref{e_38}-\eqref{e_39} only implies a ``$\frac{1}{2}$-converse'' in the sense of \eqref{e_62}.
\end{rem}
Unlike the more general one-communicator case, the rate region for omniscient helper \emph{key} generation can be obtained as the intersection of the region for omniscient helper CR generation and $\{R\le \min_j H(Y_j)\}$ \cite{liu2015key}.
(Though, the misleading similarities between the rate regions for the omniscient helper CR and key generation is only a coincidence from optimizing of the rate regions.)
As a consequence, the strong converse for the omniscient helper key generation is also proved, since the key generation counterpart obviously places more constraints, and the strong converse property of the outer-bound $\{R\le \min_j H(Y_j)\}$ is comparatively trivial.
As alluded before, the achievability for the omniscient helper CR generation implies the strong converse for smooth BLL:
\begin{cor}
\label{cor22}
For any $Q_{Y^m}$, $c^m$, and $\delta\in(0,1)$,
\begin{align}
{\rm D}_{\delta}(Q_{Y^m},Q_{Y_j},c^m)
\ge\rmds(Q_{Y^m},c^m).
\end{align}
\end{cor}
Theorem~\ref{thm_oneshot}
essentially establishes a single-shot connection between the smooth BLL and omniscient helper CR generation.
Thus the proof of Corollary~\ref{cor22} follows easily by a similar reasoning as the proof of Theorem~\ref{thm_weak}.
In fact, for a general sequence (not necessarily stationary memoryless) of sources, if the $\delta$-smooth BLL rate is strictly smaller than the supremum of $(\sum_j c_j-1)R-\sum_j c_jR_j$ over achievable rates, then the second and third terms on the right side of \eqref{e38} can be made to vanish exponentially in the blocklength. Thus $(1-\delta)$-achievability of CR generation implies $\delta$-converse for smooth BLL.
\bibliographystyle{ieeetr}
|
1,314,259,993,062 | arxiv | \section{Introduction}
The associative dialgebras (also known as diassociative algebras) has been introduced by Loday in 1990 (see [6] and references therein) as a
generalization of associative algebras.
They are a generalization of associative algebras in the sens that they possess two associative multiplications and obey to three other conditions;
when the two associative low are equal we recover associative algebra.
One of his motivation were to find an algebra whose commutator give rises to Leibniz algebra as it is the case in the relation between Lie and
associative algebra. Another motivation come from the research of an obstruction to the periodicity in algebraic
K-theory. Now, these algebras found their applications in classical geometry, non-commutative geometry and physics.
The centroid plays an important role in understanding forms of an algebra. It is an element in the classification of associative and diassociative
algebras. They occurs naturally is in the study of derivations of an algebra. The centroid and averaging operators are used in the deformation of
algebra in order to
generate another algebraic structure. The Nijenhuis operator on an associative algebra was introduced in \cite{CJ} to study quantum bi-Hamiltonian
systems while the notion Nijenhuis operator on a Lie algebra originated from the concept of Nijenhuis tensor that was introduced by Nijenhuis
in the study of pseudo-complex manifolds and was related to the well known concepts of Schouten-Nijenhuis bracket , the Frolicher-Nijenhuis
bracket \cite{FN}, and the Nijenhuis-Richardson bracket. The associative analog of the Nijenhuis relation may be regaded as the homogeneous version of
Rota-Baxter relation\cite{PL}.
BiHom-algebraic structures were introduced in 2015 by G. Graziani, A. Makhlouf, C. Menini and F. Panaite
in \cite{GACF} from a categorical approach as an extension of the class of Hom-algebras.
Since then, other interesting BiHom-type algebraic structures of many Hom-algebraic structures has been intensively studied as
BiHom-Lie colour algebras structures \cite{ABA}, Representations of BiHom-Lie algebras \cite{YH},
BiHom-Lie superalgebra structures \cite{SS},
$\{\sigma, \tau\}$-Rota-Baxter operators, infinitesimal
Hom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation \cite{MFP}, The construction and deformation of BiHom-Novikov algebras \cite{SZW},
On n-ary Generalization of BiHom-Lie algebras
and BiHom-Associative Algebras \cite{KMS}, Rota-Baxter operators on BiHom-associative
algebras and related structures \cite{LAC}.
The goal of this paper is to introduce, classify and study structures, central extensions and derivations of BiHom-associative algebras. The paper
is organized as follows. In section 2, we define BiHom-associative dialgebras, give some constructions using twisting,
direct sum, elements of centroid, averaging operator, Nijenhuis operator
and Rota-Baxter relation. We give a connection between BiHom-associative dialgebras and BiHom-Leibniz algebras.
We introduce action of a BiHom-Leibniz algebra onto another and give a Leibniz structure on the semidirect structure. Then, we show that
the semidirect sum of BiHom-Leibniz algebras associated to BiHom-associative dialgebras is the same that the BiHom-Leibniz algebra associated
to the semidirect of BiHom-associative dialgebras. Finally, we introduce BiHom-associative dialgebras and show that any BiHom-associative
dialgebra carries a structure of BiHom-Poisson dialgebra.
In section 3, we introduce the notion of central extension of BiHom-associative dialgebras and define $2$-cocycles and $2$-coboundaries
of BiHom-associative dialgebras with coefficients in a trivial BiHom-module. Then we establish relationship between $2$-cocycles and central
extensions. Section 4, is devoted to the classification of $n$-dimensional BiHom-associative dialgebras for $n\leq 4$. We dedicated Section 5 to the
derivations of BiHom-associative dialgebras.
\section{Structure of BiHom-associative dialgebras}
\begin{definition}\label{dia}
A BiHom-associative dialgebras is a $5$-truple $(A, \dashv, \vdash, \alpha, \beta)$ consisting of a linear space $A$ linear maps
$\dashv, \vdash,: A\times A \longrightarrow A$ and $\alpha, \beta : A\longrightarrow A$ satisfying, for all $x, y, z\in A$ the following
conditions :
\begin{eqnarray}
\alpha\circ\beta&=&\beta\circ\alpha,\\
(x\dashv y)\dashv\beta(z)&=&\alpha(x)\dashv(y\dashv z),\label{eq4}\\
(x\dashv y)\dashv\beta(z)&=&\alpha(x)\dashv(y\vdash z),\label{eq5}\\
(x\vdash y)\dashv\beta(z)&=&\alpha(x)\vdash(y\dashv z),\label{eq6}\\
(x\dashv y)\vdash\beta(z)&=&\alpha(x)\vdash(y\vdash z),\label{eq7}\\
(x\vdash y)\vdash\beta(z)&=&\alpha(x)\vdash(y\vdash z).\label{eq8}
\end{eqnarray}
\end{definition}
We called $\alpha$ and $\beta$ ( in this order ) the structure maps of A.
\begin{example}
Any Hom-associative dialgebra \cite{BB2} or any associative dialgebra is a BiHom-associative dialgebra by
setting $\beta = \alpha$ or $\alpha=\beta=id$.
\end{example}
\begin{example}
Let $(A, \dashv, \vdash, \alpha, \beta)$ a BiHom-associative dialgebra. Consider the module of $n\times n$-matrices
$\mathcal{M}_n(D)=\mathcal{M}_n(\mathbb{K})\otimes D$ with the linear maps ${\bf \alpha} (A)=(\alpha(a_{ij}))$, ${\bf\beta}(A)=(\beta(a_{ij})$ for all
$A\in \mathcal{M}_n(D)$ and the products $(a\triangleleft b)_{ij}=\sum_{k}a_{ik}\dashv b_{kj}$ and $(a\triangleright b)_{ij}=\sum_{k}a_{ik}\vdash b_{kj}$.
Then, $(\mathcal{M}_n(D), \triangleleft, \triangleright, {\bf \alpha}, {\bf \beta})$ is a BiHom-associative dialgebra.
\end{example}
\begin{definition}
A morphism $ f : ({D}, \dashv, \vdash, \alpha, \beta)$ and $({D}', \dashv',\vdash', \alpha', \beta')$ be a BiHom-associative dialgebras is a linear map
$f : {D}\rightarrow {D}'$ such that $\alpha'\circ f=f\circ\alpha,\, \beta'\circ f=f\circ\beta$ and
$f(x\dashv y)=f(x)\dashv'f(y),\quad f(x\vdash y)=f(x)\vdash'f(y)$, for all
$x, y \in {D}.$
\end{definition}
\begin{definition}
A BiHom-associative dialgebra $(A, \dashv, \vdash, \alpha, \beta)$ in which $\alpha$ and $\beta$ are morphism is said to be a multiplicative
BiHom-associative dialgebra.\\
If moreover, $\alpha$ and $\beta$ are bijective (i.e. automorphisms), then $(A, \dashv, \vdash, \alpha, \beta)$ is said to be a
regular BiHom-associative dialgebra.
\end{definition}
We prove in the following proposition that any BiHom-associative dialgebra turn to another one via morphisms.
\begin{theorem}\label{tw}
Let $(D, \dashv, \vdash, \alpha, \beta )$ be a BiHom-associative dialgebra and $\alpha', \beta' : D\rightarrow D$ two morphisms of
BiHom-associative dialgebras such that the maps $\alpha, \alpha', \beta, \beta'$ commute pairewise. Then
$$D_{(\alpha', \beta')}=(D, \triangleleft:=\dashv(\alpha'\otimes\beta'), \triangleright:=\vdash(\alpha'\otimes\beta'), \alpha\alpha', \beta\beta')$$
is a BiHom-associative dialgebra.
\end{theorem}
\begin{proof}
We prove only one axiom and leave the rest to the reader. For any $x, y, z\in D$,
\begin{eqnarray}
(x\triangleleft y)\triangleleft\beta\beta'(z)- \alpha\alpha'(x)\triangleleft(y\triangleright z)&=&
\alpha'(\alpha'(x)\dashv\beta'(y))\dashv\beta'\beta\beta'(z)-\alpha'\alpha\alpha'(x)\dashv\beta'(\alpha'(y)\vdash\beta'(z))\nonumber\\
&=&(\alpha'\alpha'(x)\dashv\alpha'\beta'(y))\dashv\beta\beta'\beta'(z)-\alpha\alpha'\alpha'(x)\dashv(\alpha'\beta'(y)\vdash\beta'\beta'(z)).\nonumber
\end{eqnarray}
The left hand side vanishes by (\ref{eq5}). And, this ends the proof.
\end{proof}
\begin{corollary}
Let $(D, \dashv, \vdash, \alpha, \beta )$ be a multiplicative BiHom-associative dialgebra. Then
$$(D, \dashv\circ(\alpha^n\otimes\beta^n), \vdash\circ(\alpha^n\otimes\beta^n), \alpha^{n+1}, \beta^{n+1})$$
is also a multiplicative BiHom-associative dialgebra.
\end{corollary}
\begin{proof}
It suffises to take $\alpha'=\alpha^n$ and $\beta'=\beta^n$ in Theorem \ref{tw}.
\end{proof}
\begin{corollary}
Let $(D, \dashv, \vdash, \alpha)$ be a multiplicative Hom-associative dialgebra and $\beta : D\rightarrow D$ an endomorphism of $D$. Then
$$(D, \dashv\circ(\alpha\otimes\beta), \vdash\circ(\alpha\otimes\beta), \alpha^{2}, \beta)$$
is also a Hom-associative dialgebra.
\end{corollary}
\begin{proof}
It suffises to take $\alpha'=\alpha$ and replace $\beta$ by $Id_D$, and $\beta'$ by $\beta$ in Theorem \ref{tw}.
\end{proof}
Any regular Hom-associative dialgebra give rises to associative dialgebra as stated in the next corollary.
\begin{corollary}
If $(D, \dashv, \vdash, \alpha, \beta)$ is a regular BiHom-associative dialgebra, then
$$(D, \dashv\circ(\alpha^{-1}\otimes\beta^{-1}), \vdash\circ(\alpha^{-1}\otimes\beta^{-1}))$$
is an associative dialgebra.
\end{corollary}
\begin{proof}
We have to take $\alpha'=\alpha^{-1}$ and $\beta'=\beta^{-1}$ in Theorem \ref{tw}.
\end{proof}
\begin{corollary}
Let $(D, \dashv, \vdash)$ be an associative dialgebra and $\alpha : D\rightarrow D$ and $\beta : D\rightarrow D$ a pair of
commuting endomorphisms of $D$. Then
$$(D, \dashv\circ(\alpha\otimes\beta), \vdash\circ(\alpha\otimes\beta), \alpha, \beta)$$
is a BiHom-associative dialgebra.
\end{corollary}
\begin{proof}
We have to take $\alpha=\beta=Id_D$ and replace $\alpha'$ by $\alpha$, and $\beta'$ by $\beta$ in Theorem \ref{tw}.
\end{proof}
\begin{definition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra. For any integers $k, l$, an even linear map $\theta : D\rightarrow D$
is called an element of $(\alpha^k, \beta^l)$-centroid on $D$ if
\begin{eqnarray}
\alpha\circ\theta&=&\theta\circ\alpha,\quad \beta\circ\theta=\theta\circ\beta,\\
\theta(x)\dashv \alpha^k\beta^l(y)&=&\theta(x)\dashv\theta(y)=\alpha^k\beta^l(x)\dashv \theta(y),\\
\theta(x)\vdash \alpha^k\beta^l(y)&=&\theta(x)\vdash \theta(y)=\alpha^k\beta^l(x)\vdash \theta(y),
\end{eqnarray}
for all $x, y\in D$.
\end{definition}
The set of elements of centroid is called centroid.
\begin{proposition}\label{t1}
Let $(A, \dashv, \vdash, \alpha, \beta)$ a BiHom-associative dialgebra and $\phi : A\rightarrow A$ and $\psi : A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined
$$x\triangleleft y:=\phi(x)\dashv y \quad\mbox{and}\quad x\triangleright y:=\psi(x)\vdash y.$$
Then, $(A, \triangleleft, \triangleright, \alpha, \beta)$ is a BiHom-associative dialgebra if and only if\\
$Im(\phi-\psi)\in Z_{\dashv}(A):=\{x\in A/x\dashv y=0, \forall y\in A\}
\;\mbox{and}\;
Im(\phi-\psi)\in Z_{\vdash}(A):=\{x\in A/y\vdash x=0, \forall y\in A\}$.
\end{proposition}
\begin{proof}
We only prove axioms (\ref{eq5}) and (\ref{eq7}), the three other comes from BiHom-associativity.
So for any $x, y, z\in A$,
\begin{eqnarray}
(x\triangleleft y)\triangleleft\beta(z)-\alpha(x)\triangleleft(y\triangleright z)
&=&(\phi(x)\dashv y)\dashv\phi\beta(z)-\phi\alpha(x)\dashv(y\vdash\psi(z))\nonumber\\
&=&(\phi(x)\dashv y)\dashv\beta\phi(z)-\alpha\phi(x)\dashv(y\vdash\psi(z))\nonumber\\
&=&(\phi(x)\dashv y)\vdash\beta\phi(z)-(\phi(x)\dashv y)\vdash\beta\psi(z))\nonumber\\
&=&(\phi(x)\dashv y)\vdash\beta(\phi-\psi)(z)\nonumber\\
&=&\alpha\phi(x)\dashv (y\vdash(\phi-\psi)(z))\nonumber.
\end{eqnarray}
and
\begin{eqnarray}
(x\triangleleft y)\triangleright \beta(z)-\alpha(x)\triangleright(y\triangleright z)
&=&(\phi(x)\dashv y)\vdash\beta\psi(z)-\psi\alpha(x)\vdash(y\vdash\psi(z))\nonumber\\
&=&(\phi(x)\dashv y)\vdash\beta\psi(z)-\alpha\psi(x)\vdash(y\psi(z))\nonumber\\
&=&(\phi(x)\dashv y)\vdash\beta\psi(z)-(\psi(x)\dashv y)\vdash\beta\psi(z)\nonumber\\
&=&[(\phi(x)-\psi(x))\dashv y]\vdash\beta\psi(z)\nonumber.
\end{eqnarray}
A study of cancellation of the two equalities allows to conclude.
\end{proof}
\begin{proposition}\label{t2}
Let $(A, \cdot, \alpha, \beta)$ be a BiHom-associative algebra, and $\phi : A\rightarrow A$ and $\psi : A\rightarrow A$ be a paire of commuting
elements of cenroid. Let us defined
$$x\dashv y:=\phi(x)\cdot y \quad\mbox{and}\quad x\vdash y:=\psi(x)\cdot y.$$
Then, $(A, \dashv, \vdash, \alpha, \beta)$ is a BiHom-associative dialgebra if and only if $Im(\phi-\psi)$ is contained in the
set of isotropic vectors.
\end{proposition}
\begin{proof}
We only prove axioms (\ref{eq5}) and (\ref{eq7}), the three other comes from BiHom-associativity.
So for any $x, y, z\in A$,
\begin{eqnarray}
(x\dashv y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)
&=&(\phi(x)y)\phi\beta(z)-\phi\alpha(x)(y\psi(z))\nonumber\\
&=&(\phi(x)y)\beta\phi(z)-\alpha\phi(x)(y\psi(z))\nonumber\\
&=&(\phi(x)y)\beta\phi(z)-(\phi(x)y)\beta\psi(z))\nonumber\\
&=&(\phi(x)y)\beta(\phi-\psi)(z)\nonumber\\
&=&\alpha\phi(x)(y(\phi-\psi)(z))\nonumber.
\end{eqnarray}
and
\begin{eqnarray}
(x\dashv y)\vdash \beta(z)-\alpha(x)\vdash(y\vdash z)
&=&(\phi(x)y)\beta\psi(z)-\psi\alpha(x)(y\psi(z))\nonumber\\
&=&(\phi(x)y)\beta\psi(z)-\alpha\psi(x)(y\psi(z))\nonumber\\
&=&(\phi(x)y)\beta\psi(z)-(\psi(x)y)\beta\psi(z)\nonumber\\
&=&[(\phi(x)-\psi(x))y]\beta\psi(z)\nonumber.
\end{eqnarray}
A study of cancellation of the two equalities allow to conclude.
\end{proof}
\begin{remark}
Proposition \ref{t2} may be seen as a consequence of Proposition \ref{t1}.
\end{remark}
\begin{proposition}
Let $(A, \cdot, \alpha, \beta)$ be a BiHom-associative algebra and $(M, \ast_L, \ast_R, \alpha_M, \beta_M)$ an $A$-BiHom-bimodule i.e. $M$ is a
vector space, $\alpha_M :M\rightarrow M$ and $\beta_M : M\rightarrow M$ are two linear maps, and $\ast_L : A\rightarrow M$ and
$\ast_R : M\rightarrow A$ two bilinear maps such that
\begin{eqnarray}
\alpha(x)\ast_L(y\ast_L m)&=&(x\cdot y)\ast_L\beta_M(m)\\
\alpha(x)\ast_L(m\ast_R y)&=&(x\ast_L m)\ast_R\beta(y)\label{m2}\\
\alpha_M(m)\ast_R(x\cdot y)&=&(m\ast_R x)\ast_R\beta(y).
\end{eqnarray}
Suppose that $f :M\rightarrow A$ is a morphism of $A$-BiHom-bimodule i.e. $f$ is linear such that $\alpha\circ f=f\circ\alpha_M$,
$\beta\circ f=f\circ\beta_M$ and
\begin{eqnarray}
f(x\ast_L m)&=&x\cdot f(m)\\
f(m\ast_R x)&=&f(m)\cdot x.\\
\end{eqnarray}
Then, $(M, \triangleleft, \triangleright, \alpha_M, \beta_M)$ is a BiHom-associative dialgebra with
$$m\triangleleft n=f(m)\ast_R n\quad\mbox{and}\quad m\triangleright n=m\ast_Rf(n),$$
for all $m, n\in M$.
\end{proposition}
\begin{proof}
We only prove axiom (\ref{eq8}), the other being proved similarly. For any $x, y, z\in A$,
\begin{eqnarray}
(m\triangleleft n)\triangleright \beta_M(p)
&=&(f(m)\ast_L n)\ast_R f\beta_M(p)\nonumber\\
&=&(f(m)\ast_L n)\ast_R \beta f(p)\nonumber.
\end{eqnarray}
By (\ref{m2}),
\begin{eqnarray}
(m\triangleleft n)\triangleright \beta_M(p)
&=&\alpha f(m)\ast_L (n\ast_R f(p))\nonumber\\
&=& f\alpha_M(m)\ast_L (n\triangleright p)\nonumber\\
&=& \alpha_M(m)\triangleleft(n\triangleright p)\nonumber.
\end{eqnarray}
This completes the proof.
\end{proof}
\begin{remark}
Any $(\alpha^0, \beta^0)$-element of centroid of a BiHom-associative algebra is a morphism of BiHom-bimodule.
\end{remark}
Thanks to the above remark, we have what follows :
\begin{corollary}
Let $(A, \cdot, \alpha, \beta)$ be a BiHom-associative algebra and let $\theta$ be an element of cenroid on $A$. Then,
$(A, \triangleleft, \triangleright, \alpha, \beta)$ is a BiHom-associative dialgebra with
$$x\triangleleft y=\theta(x)\cdot y\quad\mbox{and}\quad x\triangleright y=x\cdot \theta(y),$$
for any $x, y\in A.$
\end{corollary}
\begin{proposition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $R: D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$ i.e.
$R$ is linear and $\alpha\circ R=R\circ\alpha$ , $\beta\circ R=R\circ\beta$, and
\begin{eqnarray}
R(x)\dashv R(y)&=&R(R(x)\dashv y+x\dashv R(y))\\
R(x)\vdash R(y)&=&R(R(x)\vdash y+x\vdash R(y))
\end{eqnarray}
Then, $(D, \triangleleft, \triangleright, \alpha, \beta)$ is also a BiHom-associative algebra with
\begin{eqnarray}
x\triangleleft y=R(x)\dashv y+x\dashv R(y),\\
x\triangleright y=R(x)\vdash y+x\vdash R(y),
\end{eqnarray}
for all $x, y\in D$.
\end{proposition}
\begin{proof}
We only prove axiom (\ref{eq8}), the other being proved in a similar way. Thus, For any $x, y, z\in A$,
\begin{eqnarray}
&&\qquad (x\triangleleft y)\triangleleft \beta(z)-\alpha(x)\triangleleft(y\triangleright z)=\nonumber\\
&&=(x\dashv R(y)+R(x)\dashv y)\dashv R\beta(z)+R(R(x)\dashv y+x\dashv R(y))\dashv\beta(z)\nonumber\\
&&\quad-\alpha(x)\dashv R(R(y)\dashv z+y\dashv R(z))-R\alpha(x)\dashv(R(y)\vdash z+y\vdash R(z))\nonumber\\
&&=(x\dashv R(y))\dashv \beta R(z)+(R(x)\dashv y)\dashv\beta R(z)+(R(x)\dashv R(y))\dashv \beta(z)\nonumber\\
&&\quad-\alpha(x)\dashv(R(y)\vdash R(z))-\alpha R(x)\dashv(y\vdash R(z))\alpha R(x)\dashv(R(y)\vdash z).\nonumber
\end{eqnarray}
The left hand side vanishes by axiom (\ref{eq8}). This ends the proof.
\end{proof}
\begin{corollary}
Let $(D, \dashv, \vdash, \alpha, \beta)$ BiHom-associative dialgebra and $R: D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D, \ast, \alpha, \beta)$ is a BiHom-associative algebra with $x\ast y=x\triangleleft y+x\triangleright y$.
\end{corollary}
\begin{corollary}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $R: D\rightarrow D$ a Rota-Baxter operator of weight $0$ on $D$. Then,
$(D, [-, -], \alpha, \beta)$ is a BiHom-Lie algebra with $$[x, y]=x\ast y-\alpha^{-1}\beta(y)\ast\alpha\beta^{-1}(x),$$
with $x\ast y=x\triangleleft y+x\triangleright y$.
\end{corollary}
As in the previous proposition, it is well known that a Nijenhuis operator on an associative algebra allows to define another associative algebra.
In the next result, we try to establish an analoq of this result for BiHom-associative dialgebras.
\begin{proposition}\label{tns}
Let $(D, \dashv, \vdash, \alpha, \beta)$ BiHom-associative dialgebra and $N: D\rightarrow D$ a Nijenhuis operator on $D$ i.e.
$N$ is linear and $\alpha\circ N=N\circ\alpha$ , $\beta\circ N=N\circ\beta$, and
\begin{eqnarray}
N(x)\dashv N(y)&=&N(N(x)\dashv y+x\dashv N(y)-N(x\cdot y))\label{n1}\\
N(x)\vdash N(y)&=&N(N(x)\vdash y+x\vdash N(y)-N(x\cdot y))\label{n2}
\end{eqnarray}
Then, $(D, \triangleleft, \triangleright, \alpha, \beta)$ is also a BiHom-associative algebra with
\begin{eqnarray}
x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),\\
x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),
\end{eqnarray}
for all $x, y\in D$.
\end{proposition}
\begin{proof}
We only prove axiom (\ref{eq6}) for the products $\triangleleft$ and $\triangleright$. The others are leave to the reader.
\begin{eqnarray}
&&\qquad (x\triangleright y)\triangleleft \beta(z)-\alpha(x)\triangleright(y\triangleleft z)=\nonumber\\
&&=N\Big(N(x)\vdash y+x\dashv y-N(x\dashv y)\Big)\dashv \beta(z)+\Big(N(x)\dashv y+x\vdash N(y)-N(x\vdash y)\Big)\dashv N\beta(z)\nonumber\\
&&\quad-N\Big((N(x)\vdash y+x\vdash N(y)-N(x\vdash y))\dashv\beta(z)\Big)
-N\alpha(x)\vdash\Big(N(y)\dashv z+y\dashv N(z)-N(y\dashv z)\Big)\nonumber\\
&&-\alpha(x)\vdash N\Big(N(y)\dashv z+y\dashv N(z)-N(y\dashv z)\Big)
+N\Big(\alpha(x)\vdash(N(y)\dashv z+y\dashv N(z)-N(y\dashv z))\Big)\nonumber.
\end{eqnarray}
By (\ref{n1}) and (\ref{n2}), we have
\begin{eqnarray}
&&\qquad (x\triangleright y)\triangleleft \beta(z)-\alpha(x)\triangleright(y\triangleleft z)=\nonumber\\
&&=(N(x)\vdash N(y))\dashv \beta(z)+(N(x)\dashv y)\dashv \beta N(z)+(x\vdash N(y))\dashv \beta N(z)-N(x\vdash y))\dashv N\beta(z)\nonumber\\
&&\quad-N\Big((N(x)\vdash y)\dashv\beta(z)\Big)-N\Big((x\vdash N(y))\dashv\beta(z)\Big)-N\Big(N(x\vdash y)\dashv\beta(z)\Big)\nonumber\\
&&-\alpha N(x)\vdash(N(y)\dashv z)-\alpha N(x)\vdash(y\dashv N(z))+N\alpha(x)\vdash N(y\dashv z))\nonumber\\
&&-\alpha(x)\vdash (N(y)\dashv N(z))
+N\Big(\alpha(x)\vdash(N(y)\dashv z\Big)+N\Big(\alpha(x)\vdash(y\dashv N(z))\Big)-N\Big(\alpha(x)\vdash N(y\dashv z))\Big)\nonumber.
\end{eqnarray}
By (\ref{eq6}), we have
\begin{eqnarray}
&&\qquad (x\triangleright y)\triangleleft \beta(z)-\alpha(x)\triangleright(y\triangleleft z)=\nonumber\\
&&=-N(x\vdash y))\dashv N\beta(z)-N\Big((N(x)\vdash y)\dashv\beta(z)\Big)-N\Big((x\vdash N(y))\dashv\beta(z)\Big)-N\Big(N(x\vdash y)\dashv\beta(z)\Big)\nonumber\\
&&\quad+N\alpha(x)\vdash N(y\dashv z))
+N\Big(\alpha(x)\vdash(N(y)\dashv z\Big)+N\Big(\alpha(x)\vdash(y\dashv N(z))\Big)-N\Big(\alpha(x)\vdash N(y\dashv z))\Big)\nonumber.
\end{eqnarray}
Using again (\ref{n1}) and (\ref{n2}), it comes
\begin{eqnarray}
&&\qquad (x\triangleright y)\triangleleft \beta(z)-\alpha(x)\triangleright(y\triangleleft z)=\nonumber\\
&&=-N\Big(N(x\vdash y)\dashv\beta(z)+ (x\vdash y)\dashv\beta N(z)-N((x\vdash y)\dashv\beta(z))\Big)\nonumber\\
&&-N\Big((N(x)\vdash y)\dashv\beta(z)\Big)-N\Big((x\vdash N(y))\dashv\beta(z)\Big)-N\Big(N(x\vdash y)\dashv\beta(z)\Big)\nonumber\\
&&\quad+N\Big(\alpha(x)\vdash (N(y)\dashv z)+\alpha(x)\vdash N(y\dashv z)-N(\alpha(x)\vdash (y\dashv z))\Big)\nonumber\\
&&+N\Big(\alpha(x)\vdash(N(y)\dashv z\Big)+N\Big(\alpha(x)\vdash(y\dashv N(z))\Big)-N\Big(\alpha(x)\vdash N(y\dashv z))\Big)\nonumber.
\end{eqnarray}
The left hand side vanishes by (\ref{eq6}).
\end{proof}
\begin{corollary}
If $(D, \dashv, \vdash, \alpha)$ is a Hom-associative dialgebra and $N: D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D, \triangleleft, \triangleright, \alpha)$ is also a Hom-associative algebra with
\begin{eqnarray}
x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),\nonumber\\
x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),\nonumber
\end{eqnarray}
for all $x, y\in D$.
\end{corollary}
\begin{corollary}
If $(D, \dashv, \vdash, \alpha, \beta)$ is an associative dialgebra and $N: D\rightarrow D$ a Nijenhuis operator on $D$, then
$(D, \triangleleft, \triangleright, \alpha, \beta)$ is also an associative algebra with
\begin{eqnarray}
x\triangleleft y=N(x)\dashv y+x\dashv N(y)-N(x\dashv y),\nonumber\\
x\triangleright y=N(x)\vdash y+x\vdash N(y)-N(x\vdash y),\nonumber
\end{eqnarray}
for all $x, y\in D$.
\end{corollary}
The next proposition asserts that the twist of the products of any BiHom-associative dialgebra by an averaging operator gives rise to another
BiHom-associative dialgebra.
\begin{proposition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $\theta: D\rightarrow D$ an injective averaging operator on $D$ i.e.
$\theta$ is an injective linear map such that $\alpha\circ \theta=\theta\circ\alpha$ , $\beta\circ \theta=\theta\circ\beta$, and
\begin{eqnarray}
\theta(x)\dashv\theta(y)&=&\theta(\alpha^k\beta^l(x)\dashv\theta(y))=\theta(\theta(x)\dashv\alpha^k\beta^l(y)),\label{c1}\\
\theta(x)\vdash\theta(y)&=&\theta(\alpha^k\beta^l(x)\vdash\theta(y))=\theta(\theta(x)\vdash\alpha^k\beta^l(y)),\label{c2}
\end{eqnarray}
for any $x, y\in D$.
Then, $(D, \triangleleft, \triangleright, \alpha, \beta)$ is also a BiHom-associative algebra with
\begin{eqnarray}
x\triangleleft y=\theta(x)\dashv\alpha^k\beta^l(y))\\
x\triangleright y=\alpha^k\beta^l(x)\vdash\theta(y),
\end{eqnarray}
for all $x, y\in D$.
\end{proposition}
\begin{proof}
We only prove one identity, the others have a similar proof. For any $x, y, z\in D$, one has :
\begin{eqnarray}
&&\qquad \theta[(x\triangleleft y)\triangleright\beta(z)-\alpha(x)\triangleright(y\triangleleft z)]=\nonumber\\
&&=\theta[\theta(\theta(x)\dashv\alpha^k\beta^l(y)))\vdash\alpha^k\beta^{l+1}(z))
-\theta\alpha(x)\vdash(\theta(y)\dashv\alpha^k\beta^l(z)))]\nonumber\\
&&=\theta[(\theta(x)\dashv\theta(y)\vdash\alpha^k\beta^{l+1}(z)]
-\theta\alpha(x)\vdash\theta(\theta(y)\dashv\alpha^k\beta^l(z)))]\nonumber\\
&&=(\theta(x)\vdash\theta(y))\dashv\beta\theta(z)-\alpha\theta(x)\vdash(\theta(y)\dashv\theta(z))\nonumber.
\end{eqnarray}
Which vanishes by axiom (\ref{eq6}), and the conclusion holds by injectivity.
\end{proof}
At this moment, we introduce ideals for BiHom-associative dialgebra in order to give another construction of BiHom-associative dialgebras.
\begin{definition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $D_o$ a subset of $D$. We say that $D_o$ is a BiHom-subalgebra of $D$ if
$D_o$ is stable under $\alpha$ and $\beta$, and $x\dashv y, x\vdash y\in D_o$, for any $x, y\in D_o$.
\end{definition}
\begin{example}
If $\varphi : D_1\rightarrow D_2$ is a homomorphism of BiHom-associative dialgebras, the image $Im\varphi$ is a BiHom-subalgebra of $D_2$.
\end{example}
\begin{definition}
A two side BiHom-ideal of a BiHom-associative dialgebra $(D, \dashv, \vdash, \alpha, \beta)$ is subspace $I$ such that $\alpha(I)\subset I,
x\ast y, y\ast x\in I$ for all $x\in D, y\in I$ with $\ast=\dashv$ and $\vdash$. Note that $I$ is called the left and right BiHom-ideal if $x\dashv y, x\vdash y$ and
$y\dashv x, y\vdash x$ are in $I$, respectively, for all $x\in D. y\in I$.
\end{definition}
\begin{example}
i) Obviously $I=\{0\}$ and $I=D$ are two-sided ideals.\\
ii) If $\varphi : D_1\rightarrow D_2$ is a homomorphism of BiHom-associative dialgebras, the kernel $Ker\varphi$ is a two sided ideal in $D_1$.\\
iii) If $I_1$ and $I_2$ are two sided ideals of $D$, then so is $I_1+I_2$.
\end{example}
In the below proposition, we prove that BiHom-associative dialgebras are closed under direct summation, and give a condition for which
a linear map becomes a morphism.
\begin{proposition}
Let $({A}, \dashv_{A}, \vdash_{A}, \alpha_{A}, \beta_{A})$ and $({B}, \dashv_{B}, \vdash_{B}, \alpha_{B}, \beta_{B})$ be two
BiHom-associative dialgebras. Then there exists a BiHom-associative dialgebra structure
on ${A}\oplus{B}$ with the bilinear maps $\triangleleft, \triangleright : ({A}\oplus{B})^{\otimes 2}\rightarrow {A}\oplus{B}$
given by
$$(a_1+b_2)\dashv(a_2+b_2)=a_1\dashv_{A}a_2+b_2\dashv_{B}b_2,$$ $$(a_1+b_1)\vdash(a_2+b_2)=a_1\vdash_{A}a_2+b_{A}\vdash_{B}b_2$$ and the linear maps
$\alpha=\alpha_{A}+\alpha_{B},\, \beta=\beta_{A}+\beta_{B} : {A}\oplus{B}\rightarrow {A}\oplus{B}$ given by
$$(\alpha_{A}+\alpha_{B})(a+b)=\alpha_{A}(a)+\alpha_{B}(b),\, (\beta_{A}+\beta_{B})(a+b)=\beta_{A}(a)+\beta_{B}(b),\,
\forall(a,b)\in({A}\times{B}).
$$
Moreover, if $\xi : {A}\rightarrow {B}$ is a linear map.
Then $ \xi : ({A}, \dashv_{A}, \vdash_{A}, \alpha_A, \beta_A)$ to
$({B}, \dashv_{B}, \vdash_{B}, \alpha_B, \beta_B)$ is a morphism if and only if its graph $\Gamma_\xi=\{(x, \xi(x)), x\in A\}$
is a BiHom-subalgebra of $({A}\oplus{B}, \triangleleft, \triangleright, \alpha, \beta)$.
\end{proposition}
\begin{proof}
The proof of the first part of the proposition comes from a simple computation.\\
Let us suppose that $\xi : ({A}, \dashv_{A}, \vdash_{A},\alpha_A, \beta_A)\rightarrow({B}, \dashv_{B}, \vdash_{B},\alpha_B, \beta_B)$ is a
morphism of BiHom-associative dialgebras.
Then $$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))=(u\dashv_{A}v+\xi(u\dashv_{A}v)$$
$$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))=(u\vdash_{A}v+\xi(u\vdash_{A}v).$$
Thus the graph $\Gamma_\xi$ is closed under the operations $\dashv$ and $\vdash$.\\
Furthermore since $\xi\circ\alpha_A=\alpha_B\circ\xi,$ and $\xi\circ\beta_A=\beta_B\circ\xi,$ we have
$$
(\alpha_A\oplus\alpha_B)(u, \xi(u))=(\alpha_A(u), \alpha_B\circ\xi(u))=(\alpha_A(u), \xi\circ\alpha_A(u)).
$$
and
$$
(\beta_A\oplus\beta_B)(u, \xi(u))=(\beta_A(u), \beta_B\circ\xi(u))=(\beta_A(u), \xi\circ\beta_A(u)),
$$
implies that $\Gamma_\xi$ is closed $\alpha_A\oplus\alpha_B$ and $\beta_A\oplus\beta_B.$
Thus, $\Gamma_\xi$ is a BiHom-subalgebra of
$({A}\otimes{B}, \dashv, \vdash, \alpha, \beta).$ \\
Conversely, if the graph $\Gamma_\xi\subset{A}\oplus{B}$ is a BiHom-subalgebra of
$({A}\oplus{B}, \dashv, \vdash, \alpha, \beta)$ then we
$$(u+\xi(u))\dashv(v+\xi(v))=(u\dashv_{A}v+\xi(u)\dashv_{B}\xi(v))\in \Gamma_\xi $$
$$(u+\xi(u))\vdash(v+\xi(v))=(u\vdash_{A}v+\xi(u)\vdash_{B}\xi(v))\in \Gamma_\xi.$$
Furthermore, $(\alpha_A\oplus\alpha_B)(\Gamma_\xi)\subset \Gamma_\xi,\, (\beta_A\oplus\beta_B)(\Gamma_\xi)\subset \Gamma_\xi,$ implies
$$
(\alpha_A\oplus\alpha_B)(u, \xi(u))=(\alpha_A(u),\alpha_B\circ\xi(u))\in \Gamma_\xi,\,(\beta_A\oplus\beta_B)(u, \xi(u))
=(\beta_A(u),\beta_B\circ\xi(u))\in \Gamma_\xi,
$$
which is equivalent to the condition $\alpha_B\circ\xi(u)=\xi\circ\alpha_A(u),$ i.e $\alpha_B\circ\xi=\xi\circ\alpha_A.$ Similary,
$\beta_B\circ\xi=\xi\circ\beta_A$. Therefore, $\xi$ is a
morphism BiHom-associative dialgebras.
\end{proof}
\begin{proposition}\label{P3}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $I$ be a two sided BiHom-ideal of
$(D,\dashv, \vdash, \alpha, \beta)$. Then, $(D/I, \left[\cdot, \cdot\right],\overline{\dashv}, \overline{\vdash},
\overline{\alpha}, \overline{\beta})$ is a
BiHom-associative dialgebra where
$$\overline{x}\;\overline{\dashv}\;\overline{y}:=\overline{x\dashv y},\;\;
\overline{x}\;\overline{\vdash}\;\overline{y}:=\overline{x\vdash y},\;\;
\overline{\alpha}(\overline{x}):=\overline{\alpha(x)},\;\;
\overline{\beta}(\overline{x}):=\overline{\beta(x)},$$
for all $\overline{x}, \overline{y}\in A/I.$
\end{proposition}
\begin{proof}
We only prove left associativity, the other being proved similarly. For all $\overline{x}, \overline{y},
\overline{z}\in D/I$,
we have
\begin{eqnarray}
(\overline{x}\overline{\vdash}\overline{y})\overline{\vdash}\overline{\beta}(\overline{z})
-\overline{\alpha}(\overline{x})\overline{\vdash}(\overline{y}\overline{\vdash}\overline{z})
=\overline{(x\vdash y)\vdash\beta(z)-\alpha(x)\vdash(y\vdash z)}
=0.\nonumber
\end{eqnarray}
Then, $(D/I, \overline{\dashv}, \overline{\vdash}, \overline{\alpha}, \overline{\beta})$ is BiHom-associative dialgebra.
\end{proof}
Now, let us recall the definition of BiHom-Lie algebra.
\begin{definition}\cite{GACF}
$A$ BiHom-Lie algebra $(L, \left[\cdot, \cdot\right], \alpha,\beta)$ is a $4$-tuple in where L is linear space, $\alpha,
\beta : A\rightarrow A $,are linear maps and
$\left[\cdot, \cdot\right] : L\otimes L\rightarrow L$ is a bilinear maps, such that, for all $x, y, z\in L$ :
\begin{equation}
\alpha\circ\beta=\beta\circ\alpha,
\end{equation}
\begin{equation}
\alpha(\left[x, y\right])=\left[\alpha(x), \alpha(y)\right],\,\text{and},\,\beta(\left[x, y\right])=\left[\beta(x), \beta(y)\right],
\end{equation}
\begin{equation}
\left[\beta(x),\alpha(y)\right])=-\left[\beta(y), \alpha(x)\right],\, (\text{BiHom-skew-symetry}),
\end{equation}
\begin{equation}
\left[\beta^2(x),\left[\beta(y),\alpha(z)\right]\right]+\left[\beta^2(y),\left[\beta(z),\alpha(x)\right]\right]
+\left[\beta^2(z),\left[\beta(x),\alpha(y)\right]\right]=0,
\end{equation}
\begin{center}
(\text{BiHom-Jacobi identity}).
\end{center}
\end{definition}
The maps $\alpha$ and $\beta$ (in this order) are called the structure maps of L.
\begin{definition}
A morphism between two BiHom-Lie algebras
$f : (L, [-, -], \alpha, \beta)\rightarrow(L', [-, -]', \alpha', \beta')$ is a linear map
$f : L\rightarrow L'$ such that $\alpha'\circ f=f\circ\alpha,\, \beta'\circ f=f\circ\beta$ and
$f(\left[x, y\right])=\left[f(x), f(y)\right]'$, for all $x, y \in L.$
\end{definition}
The following lemma asserts that the commutator of any BiHom-associative algebra gives rise to BiHom-Lie.
\begin{lemma}\cite{GACF}\label{cll}
Let $(A, \cdot, \alpha, \beta )$ be a regular BiHom-associative algebra. Then
$$L(A)=(A, [-, -], \alpha, \beta)$$
is a regular BiHom-Lie algebra, where
$$[x, y]=x\cdot y-\alpha^{-1}\beta(y)\cdot\alpha\beta^{-1}(x),$$
for any $x, y\in A.$
\end{lemma}
\begin{proposition}
Let $(L, [-, -], \alpha,\beta)$ be a BiHom-Lie algebra and $N : L\rightarrow L$ be a Nijenhuis operator on $L$ i.e.
$\alpha\circ N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and
\begin{eqnarray}
[N(x), N(y)]=N([N(x), y]+[x, N(y)]-N([x, y]))\nonumber
\end{eqnarray}
for any $x, y\in L$.
Then, $(L, [-, -]_N, \alpha,\beta)$ is a BiHom-Lie algebra with
\begin{eqnarray}
[x, y]_N=[N(x), y]+[x, N(y)]-N([x, y])\nonumber
\end{eqnarray}
for all $x, y\in L$.
\end{proposition}
\begin{proof}
It follows from direct computation.
\end{proof}
\begin{corollary}\label{aln1}
Let $(A, \cdot, \alpha,\beta)$ be a BiHom-associative algebra and $N : A\rightarrow A$ be a Nijenhuis operator on $A$ i.e.
$\alpha\circ N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and
\begin{eqnarray}
N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))\nonumber
\end{eqnarray}
for any $x, y\in A$.
Let us denote by $L(A)$ the BiHom-Lie algebra associated with $A$ as in Proposition \ref{cll}. Then,
$(A, [-, -]_N, \alpha,\beta)$ is a BiHom-Lie algebra.
\end{corollary}
\begin{corollary}\label{aln2}
Let $(A, \cdot, \alpha,\beta)$ be a BiHom-associative algebra and $N : A\rightarrow A$ be a Nijenhuis operator on $A$ i.e.
$\alpha\circ N=N\circ\alpha$, $\beta\circ N=N\circ\beta$ and
\begin{eqnarray}
N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot y))\nonumber
\end{eqnarray}
for any $x, y\in A$.
Then, $(A, \{-, -\}, \alpha,\beta)$ is a BiHom-Lie algebra with
\begin{eqnarray}
\{x, y\}= x\ast_N y-\alpha^{-1}\beta(y)\ast_N\alpha\beta^{-1}(x)\nonumber
\end{eqnarray}
and
\begin{eqnarray}
x\ast_Ny= N(x)\cdot y+x\cdot N(y)-N(x\cdot y)\nonumber
\end{eqnarray}
for all $x, y\in A$.
\end{corollary}
\begin{proof}
It is similar to the one of Proposition \ref{tns}. And the Lemma \ref{cll} will end the proof.
\end{proof}
\begin{remark}
The BiHom-Lie algebra generated by Corollary \ref{aln1} and Corollary \ref{aln2} are equal.
\end{remark}
\begin{proposition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra. Then,for all $x, y\in D$, the bracket
$$[x, y]=[x, y]_L+[x, y]_R,$$
where
\begin{eqnarray}
[x, y]_L&=&x\dashv y-\alpha^{-1}\beta(y)\dashv\alpha\beta^{-1}(x),\nonumber\\
{[x, y]_R}&=&x\vdash y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x),\nonumber
\end{eqnarray}
is a BiHom-Lie bracket if and only if
\begin{eqnarray}
\alpha(x)\dashv(y\vdash z)=(x\dashv y)\vdash\beta(z),\label{dil1}\\
\alpha(x)\dashv(y\dashv z)=(x\vdash y)\vdash\beta(z).\label{dil2}
\end{eqnarray}
\end{proposition}
\begin{proof} It is essentialy based on Lemma \ref{cll}. That is,
an expansion of BiHom-Jacobi identity leads to $48$ terms including $8$ terms which cancel pairewise by axiom (\ref{eq4}), $4$ terms
cancel pairewise by axiom (\ref{eq5}), $12$ terms cancel pairewise by axiom (\ref{eq6}), $6$ terms cancel pairewise by axiom (\ref{eq7}) and
$6$ terms cancel pairewise by axiom (\ref{eq8}).\\
For the of the $12$ terms, $8$ terms cancel pairewise by axiom (\ref{dil1}) and $4$ terms cancel pairewise by axiom (\ref{dil2}).
\end{proof}
\begin{definition}\label{Leib}
A (right ) BiHom-Leibniz algebra is a $4$-tuple $(L, \left[\cdot, \cdot\right], \alpha, \beta)$, where L is a linear space, $\left[\cdot, \cdot\right] : L \times L\rightarrow L$
is a bilinear map and $\alpha, \beta : L\rightarrow L$ are linear maps satisfying
\begin{equation}
\left[\left[x, y\right], \alpha\beta(z),\right]=\left[\left[x, \beta(z)\right], \alpha(y)\right]+\left[\alpha(x),\left[y, \alpha(z)\right]\right],
\end{equation}
for all $x, y, z\in L$.
\end{definition}
\begin{example}
Let $L$ be a two-dimensional vector space and $\left\{e_1, e_2\right\}$ be a basis of $L$. Then, $(L, [-, -], \alpha, \beta)$ is a
BiHom-Leibniz algebra with
$$\left[e_1, e_2\right]=ae_1,
\left[e_2, e_2\right]=be_1,\;
\alpha(e_2)=\beta(e_2)=e_1,
a, b\in \mathbb{R}.$$
\end{example}
Now, we introduce BiHom-Poisson dialgebras and study its connection with BiHom-associative dialgebras.
\begin{definition}\label{pois}
A BiHom-Poisson dialgebra is a BiHom-associative dialgebra $({P}, \dashv, \vdash, \alpha, \beta )$ and a BiHom-Leibniz algebra
$({P}, [-, -], \alpha, \beta )$ such that
\begin{eqnarray}
{[x\dashv y, \alpha\beta(z)]}&=&\alpha(x)\dashv[y, \alpha(z)]+[x, \beta(z)]\dashv\alpha(y),\nonumber\\
{[x\vdash y, \alpha\beta(z)]}&=&\alpha(x)\vdash[y, \alpha(z)]+[x, \beta(z)]\vdash\alpha(y),\nonumber\\
\{\alpha\beta(x), y\dashv z\}&=&\beta(y)\vdash[\alpha(x), z]+[\beta(x), y]\dashv\beta(z)=[\alpha\beta(x), y\vdash z],\nonumber
\end{eqnarray}
are satisfied for $x, y, z\in{P}$.
\end{definition}
\begin{theorem}
Let $({D}, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra. Then,
$$P(D)=(D,[-, -],\dashv, \vdash, \alpha, \beta)$$
is a BiHom-Poisson dialgebra, where $[x, y]=x \dashv y-y\vdash x$, for any $x, y\in {D}$.
\end{theorem}
\begin{proof}
By Theorem \ref{dilei} $P(D)$ is a BiHom-Leibniz algebra. Moreover, for any $x, y, z\in D$,
\begin{eqnarray}
&& [x\dashv y, \alpha\beta(z)]-\alpha(x)\dashv[y, \alpha(z)]-[x, \beta(z)]\dashv\alpha(y)=\nonumber\\
&&=(x\dashv y)\dashv\alpha\beta(z)-\alpha^{-1}\beta\alpha\beta(z)\vdash\alpha\beta^{-1}(x\dashv y)
-\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))\nonumber\\
&&-(x\dashv\beta(z)-\alpha^{-1}\beta\beta(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)\nonumber\\
&&=(x\dashv y)\dashv\alpha\beta(z)-\beta^2(z)\vdash(\alpha\beta^{-1}(x)\dashv \alpha\beta^{-1}(y))-\alpha(x)\dashv(y\dashv\alpha(z)\nonumber\\
&&\quad+\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))-
(x\dashv\beta(z))\dashv\alpha(y)+(\alpha^{-1}\beta^2(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)\nonumber.
\end{eqnarray}
The last three axioms are proved analagously. This completes the proof.
\end{proof}
\begin{theorem}
Let $(P, \dashv, \vdash, [-, -], \alpha, \beta )$ be a BiHom-Poisson dialgebra and $\alpha', \beta' : D\rightarrow D$ two morphisms of
BiHom-Poisson dialgebras such that the maps $\alpha, \alpha', \beta, \beta'$ commute pairewise. Then
$$P_{(\alpha', \beta')}=(D, \, \triangleleft:=\dashv(\alpha'\otimes\beta'),\; \triangleright:=\vdash(\alpha'\otimes\beta'),\;
\{-, -\}:=[-, -](\alpha'\otimes\beta'),\; \alpha\alpha',\; \beta\beta'),$$
is a BiHom-Poisson dialgebra.
\end{theorem}
\begin{proof}
It is essentialy based on the one of Theorem \ref{tw}.
\end{proof}
Now, we introduce action of BiHom-Leibniz algebra on another one.
\begin{definition}
Let $D$ and $L$ be two BiHom-Leibniz algebras. An action of $D$ on $L$ consists of a pair of bilinear maps,
$D\times L\rightarrow L, (x, a)\mapsto [x, a]$ and $L\times D\rightarrow [x, a]$, such that
\begin{eqnarray}\label{le}
\left[\alpha(x),\left[a, \alpha(b)\right]\right]&=&\left[\left[x, a\right], \alpha\beta(b),\right]
-\left[\left[x, \beta(b)\right], \alpha(a)\right]\label{la1}\\
\left[\alpha(a),\left[x, \alpha(b)\right]\right]&=&\left[\left[a, x\right], \alpha\beta(b),\right]
-\left[\left[a, \beta(b)\right], \alpha(x)\right]\\
\left[\alpha(a),\left[b, \alpha(x)\right]\right]&=&\left[\left[a, b\right], \alpha\beta(x),\right]
-\left[\left[a, \beta(x)\right], \alpha(b)\right]\\
\left[\alpha(a),\left[x, \alpha(y)\right]\right]&=&\left[\left[a, x\right], \alpha\beta(y),\right]
-\left[\left[a, \beta(y)\right], \alpha(x)\right]\\
\left[\alpha(x),\left[a, \alpha(y)\right]\right]&=&\left[\left[x, a\right], \alpha\beta(y),\right]
-\left[\left[x, \beta(y)\right], \alpha(a)\right]\\
\left[\alpha(x),\left[y, \alpha(a)\right]\right]&=&\left[\left[x, y\right], \alpha\beta(a),\right]
-\left[\left[x, \beta(a)\right], \alpha(y)\right]
\end{eqnarray}
for all $x, y\in D, a, b\in L$.
\end{definition}
\begin{lemma}
Given a BiHom-Leibniz action of $D$ on $L$, we can consider the {\it semidirect product} Leibniz algebra $L\rJoin D$, which consists of
vector space $D\oplus L$ together with the Leibniz bracket given by
\begin{eqnarray}
[(x, a), (y, b)]=([x, y]+[x, b]+[a, y], [a, b])
\end{eqnarray}
for all $(x, a), (x, b)\in D\times L$.
\end{lemma}
\begin{proof}
\begin{eqnarray}
[\alpha(x, a), [(y, b), \alpha(z, c)]]
&=&[(\alpha(x), \alpha(a)), ([y, \alpha(z)]+[y, \alpha(c)]+[b, \alpha(z)], [b, \alpha(c)])]\nonumber\\
&=&\Big([\alpha(x), [y, \alpha(z)]]+[\alpha(x), [y, \alpha(c)]]+[\alpha(x), [b, \alpha(z)]]+[\alpha(x), [b, \alpha(c)]]\nonumber\\
&&+[\alpha(a), [y, \alpha(z)]]+[\alpha(a), [y, \alpha(z)]]+[\alpha(a), [y, \alpha(c)]]
+[\alpha(a), [b, \alpha(z)],\nonumber\\
&& [\alpha(a), [b, \alpha(c)]]\Big).\nonumber\\
{[[(x, a), (y, b)], \alpha\beta(z, c)]}&=&[([x, y]+[x, b]+[a, y]), [a, b]), (\alpha\beta(z), \alpha\beta(c))]\nonumber\\
&=&([[x, y], \alpha\beta(z)]+[[x, b], \alpha\beta(z)]+[[a, y], \alpha\beta(z)]+[[x, y], \alpha\beta(c)]\nonumber\\
&&+[[x, b], \alpha\beta(c)]
+[[a, y], \alpha\beta(c)]+[[a, b], \alpha\beta(c)], [[a, b], \alpha\beta(c)].\nonumber\\
{[[(x, a), \beta(z, c)], \alpha(y, b)]}
&=&[([x, \beta(z)]+[x, \beta(c)]+[a, \beta(z)], [a, \beta(c)]), (\alpha(y), \alpha(b))]\nonumber\\
&=&([[x, \beta(z)], \alpha(y)]+[[x, \beta(c)], \alpha(y)]+[[a, \beta(z)], \alpha(y)]+[[x, \beta(z)], \alpha(b)]\nonumber\\
&&+[[x, \beta(c)], \alpha(b)]+[[a, \beta(z)], \alpha(b)]+[[a, \beta(c)], \alpha(y)], [[a, \beta(c)], \alpha(b)])\nonumber.
\end{eqnarray}
Using axioms in Definition \ref{le}, it follows that
$${[[(x, a), (y, b)], \alpha\beta(z, c)]}={[[(x, a), \beta(z, c)], \alpha(y, b)]}+[\alpha(x, a), [(y, b), \alpha(z, c)]].$$
Which proves the proposition.
\end{proof}
\begin{theorem}\label{dilei}
Let $({D}, \dashv, \vdash, \alpha, \beta)$ be a regular BiHom-associative dialgebra. Then the bracket defined by
$\left[x, y\right]=x \dashv y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x)$, defines a structure of BiHom-Leibniz algebra on ${D}$, and denoted
${\bf Lb}(D)$.
\end{theorem}
\begin{proof}
For any $x, y, z\in {D}$, we have
\begin{eqnarray}
[[x, y], \alpha\beta(z)]
&=&(x\dashv y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)\nonumber\\
&&-\alpha^{-1}\beta\alpha\beta(z)\vdash(x\dashv y-\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})\nonumber\\
&=&(x\dashv y)\dashv\alpha\beta(z)-(\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(z)\nonumber\\
&&-\beta^2(z)\vdash(\alpha\beta^{-1}(x)\dashv\alpha\beta^{-1}(y)
+\beta^2(z)\vdash(y\vdash\alpha^2\beta^{-2}(x)).\nonumber\\
{[[x, \beta(z)], \alpha(y)]}
&=&(x\dashv\beta(z)-\alpha^{-1}\beta^2(z)\vdash\alpha\beta^{-1}(x)\dashv\alpha(y)\nonumber\\
&&-\alpha^{-1}\beta\alpha(z)\vdash(\alpha\beta^{-1}(x\dashv y+\alpha^{-1}\beta(y)\vdash\alpha\beta^{-1})\nonumber\\
&=&(x\dashv\beta(z))\dashv\alpha(y)-(\alpha^{-1}\beta^2(z)\vdash\alpha\beta^{-1}(x))\dashv\alpha(y)\nonumber\\
&&-\beta(y)\vdash(\alpha\beta^{-1}(x)\dashv\alpha(z)) +\beta(y)\vdash(\beta(z)\vdash\alpha^2\beta^{-2}(x))\nonumber.\\
{[\alpha(x), [y, \alpha(z)]]}
&=&\alpha(x)\dashv(y\dashv\alpha(z)-\alpha^{-1}\beta\alpha(z)\vdash\alpha\beta^{-1}(y))
-\alpha^{-1}\beta(y\dashv\alpha(z)\nonumber\\
&&-\beta(z)\vdash\alpha\beta^{-1}(y))\vdash\alpha\beta^{-1}\alpha(x)\nonumber\\
&=&\alpha(x)\dashv(y\dashv\alpha(z))-\alpha(x)\dashv(\beta(z)\vdash\alpha\beta^{-1}(y))\nonumber\\
&&-(\alpha^{-1}\beta(y)\dashv\beta(z))\vdash\alpha^2\beta^{-1}(x) -(\alpha^{-1}\beta^2(z)\vdash y)\vdash\alpha^2\beta^{-1}(x)\nonumber.
\end{eqnarray}
By axioms in Definition \ref{dia}, the conclusion holds.
\end{proof}
In the relations contained in the below definition, we omitted the subsript for simplifying the typography.
\begin{definition}\label{act1}
Let $D$ and $L$ be dialgebras. An action of $D$ on $L$ consists of four linear maps, two of them denoted by the symbol $\dashv$ and other two by
$\vdash$,
\begin{eqnarray}
\dashv : D\otimes L\rightarrow L, & & \dashv : L\otimes D\rightarrow L,\nonumber\\
\dashv : D\otimes L\rightarrow L, & & \dashv : L\otimes D\rightarrow L\nonumber
\end{eqnarray}
such that the following $30$ equalities hold :\\
\begin{tabular}{lr}
${(01)}\quad (x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\dashv b) $,
&$\qquad\qquad{(16)}\quad (a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\dashv y)$,\\
${(02)}\quad (x\dashv a)\dashv\beta(b)=\alpha(x)\dashv(a\vdash b)$,
&$\qquad\qquad {(17)}\quad (a\dashv x)\dashv\beta(y)=\alpha(a)\dashv(x\vdash y)$,\\
${(03)}\quad (x\vdash a)\dashv\beta(b)=\alpha(x)\vdash(a\dashv b)$,
&$\qquad\qquad {(18)}\quad (a\vdash x)\dashv\beta(y)=\alpha(a)\vdash(x\dashv y)$,\\
${(04)}\quad (x\dashv a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$,
&$\qquad\qquad {(19)}\quad (a\dashv x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,\\
${(05)}\quad (x\vdash a)\vdash\beta(b)=\alpha(x)\vdash(a\vdash b)$,
&$\qquad\qquad {(20)}\quad (a\vdash x)\vdash\beta(y)=\alpha(a)\vdash(x\vdash y)$,\\
\\
${(06)}\quad (a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\dashv b)$,
&$\qquad\qquad {(21)}\quad (x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\dashv y)$,\\
${(07)}\quad (a\dashv x)\dashv\beta(b)=\alpha(a)\dashv(x\vdash b)$,
&$\qquad\qquad{(22)}\quad (x\dashv a)\dashv\beta(y)=\alpha(x)\dashv(a\vdash y)$,\\
${(08)}\quad (a\vdash x)\dashv\beta(b)=\alpha(a)\vdash(x\dashv b)$,
&$\qquad\qquad {(23)}\quad (x\vdash a)\dashv\beta(y)=\alpha(x)\vdash(a\dashv y)$,\\
${(09)}\quad (a\dashv x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$,
&$\qquad\qquad{(24)}\quad (x\dashv a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,\\
${(10)}\quad (a\vdash x)\vdash\beta(b)=\alpha(a)\vdash(x\vdash b)$,
&$\qquad\qquad {(25)}\quad (x\vdash a)\vdash\beta(y)=\alpha(x)\vdash(a\vdash y)$,\\
\\
\end{tabular}
\begin{tabular}{lr}
${(11)}\quad (a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\dashv x)$,
&$\qquad\qquad {(26)}\quad (x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\dashv a)$,\\
${(12)}\quad (a\dashv b)\dashv\beta(x)=\alpha(a)\dashv(b\vdash x)$,
&$\qquad\qquad {(27)}\quad (x\dashv y)\dashv\beta(a)=\alpha(x)\dashv(y\vdash a)$,\\
${(13)}\quad (a\vdash b)\dashv\beta(x)=\alpha(a)\vdash(b\dashv x)$,
&$\qquad\qquad {(28)}\quad (x\vdash y)\dashv\beta(a)=\alpha(x)\vdash(y\dashv a)$,\\
${(14)}\quad (a\dashv b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$,
&$\qquad\qquad{(29)}\quad (x\dashv y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,\\
${(15)}\quad (a\vdash b)\vdash\beta(x)=\alpha(a)\vdash(b\vdash x)$,
&$\qquad\qquad{(30)}\quad (x\vdash y)\vdash\beta(a)=\alpha(x)\vdash(y\vdash a)$,
\end{tabular}
\\
\\
for all $x, y\in D, a, b\in L$. The action is called trivial if these four maps are trivial.
\end{definition}
\begin{example}
i) Any BiHom-associative dialgebra may be seen as acting on itself
ii)Given a homomorphism $\varphi : D\rightarrow L$ of BiHom-associative dialgebras, then there is an action of $D$ on $L$ via the maps
$x\triangleleft a:=\varphi(x)\dashv a,\; x\triangleright a:=\varphi(x)\triangleright, a\;
a\triangleleft x:=a\vdash\varphi(x)\;\;\mbox{and}\;\; a\triangleright x:=a\vdash\varphi(x)$.\\
iii)If $\psi : L\rightarrow D$ is an isomorphism of BiHom-associative dialgebras, then there is an action of $D$ on $L$ via the maps
$x\triangleleft a:=\psi^{-1}(x)\dashv a,\; x\triangleright a:=\psi^{-1}(x)\triangleright a,\;
a\triangleleft x:=a\vdash\psi^{-1}(x)\;\;\mbox{and}\;\; a\triangleright x:=a\vdash\psi^{-1}(x)$.\\
iv) If $I$ is an ideal of $D$, then the left and the right product yield an action of $D$ on I.
\end{example}
\begin{lemma}\label{lbs}
Given two regular BiHom-associative dialgebras $D$ and $L$ together with an action of $D$ on $L$, there is an action an action
${\bf Lb} (D)$ on ${\bf Lb} (L)$ given by
\begin{eqnarray}
[x, a]&=&x\dashv a-\alpha^{-1}\beta(a)\dashv\alpha\beta^{-1}(x),\nonumber\\
{[a, x]}&=&a\vdash x-\alpha^{-1}\beta(x)\vdash\alpha\beta^{-1}(a),\nonumber
\end{eqnarray}
for all $x\in {\bf Lb} (D)$, $a\in {\bf Lb} (L)$.
\end{lemma}
\begin{proof}
For all $x\in {\bf Lb} (D)$, $a\in {\bf Lb} (L)$,
\begin{eqnarray}
[[x, a], \alpha\beta(b)]
&=&(x\dashv a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\alpha\beta(b)\nonumber\\
&&-\alpha^{-1}\beta\alpha\beta(b)\vdash\alpha\beta^{-1}(x\dashv a-\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\nonumber\\
&=&(x\dashv a)\dashv\beta\alpha(b)-(\alpha^{-1}\beta(a)\vdash\alpha\beta^{-1}(x))\dashv\beta\alpha(b)\nonumber\\
&&-\beta^2(b)\vdash(\alpha\beta^{-1}(x)\dashv\beta^{-1}\alpha(a))
+\beta^2(b)\vdash(a\vdash\alpha^2\beta^{-2}(x))\nonumber.
\end{eqnarray}
On the other hand,
\begin{eqnarray}
&&\qquad [[x, \beta(b)], \alpha(a)]+[\alpha(x), [a, \alpha(b)]]=\nonumber\\
&&=\Big(x\dashv\beta(b)-\alpha^{-1}\beta^2(b)\vdash\alpha\beta^{-1}(x)\Big)\dashv\alpha(a)
-\alpha^{-1}\beta\alpha(a)\vdash\alpha\beta^{-1}\Big(x\dashv\beta(b)-\alpha^{-1}\beta^2(b)\vdash\alpha\beta^{-1}(x)\Big)\nonumber\\
&&\quad+\alpha(x)\dashv\Big(a\dashv \alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big)
-\alpha^{-1}\beta\Big(a\dashv \alpha(b)-\alpha^{-1}\beta\alpha(b)\vdash\alpha\beta^{-1}(a)\Big)\vdash\alpha\beta^{-1}\alpha(x)\nonumber\\
&&=(x\dashv\beta(b))\dashv\alpha(a)-(\alpha^{-1}\beta^2(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)
-\beta(a)\vdash (\alpha\beta^{-1}(x)\dashv\alpha(b))\nonumber\\
&&\quad+\beta(a)\vdash(\beta(b)\vdash\alpha^2\beta^{-2}(x))+\alpha(x)\dashv(a\dashv\alpha(b))-\alpha(x)\dashv(\beta(b)\vdash\alpha\beta^{-1}(a))\nonumber\\
&&\quad-(\alpha^{-1}\beta(a)\dashv\beta(b))\vdash\beta^{-1}\alpha^2(x)+(\alpha^{-1}\beta^2(b)\vdash a)\vdash\beta^{-1}\alpha^2(x)\nonumber.
\end{eqnarray}
Using axioms (\ref{eq5}), (\ref{eq7}), it comes
\begin{eqnarray}
&&\qquad [[x, \beta(b)], \alpha(a)]+[\alpha(x), [a, \alpha(b)]]=\nonumber\\
&&=-(\alpha^{-1}\beta^2(b)\vdash\alpha\beta^{-1}(x))\dashv\alpha(a)-\beta(a)\vdash (\alpha\beta^{-1}(x)\dashv\alpha(b))\nonumber\\
&&\quad+\alpha(x)\dashv(a\dashv\alpha(b)) +(\alpha^{-1}\beta^2(b)\vdash a)\vdash\alpha^2\beta^{-1}(x)\nonumber.
\end{eqnarray}
By comparing, we get the attended result. The five other axioms are proved in the same way.
\end{proof}
\begin{lemma}\label{lbsa}
Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an action of $D$ on $L$. There is a BiHom-associative dialgebra structure on
$L\rJoin D$ which consists with vector space $L\oplus D$ and
\begin{eqnarray}
(a, x)\triangleleft (b, y)&=&(a\dashv b+a\dashv y+x\dashv b, x\dashv y)\nonumber,\\
(a, x)\triangleright (b, y)&=&(a\vdash b+a\vdash y+x\vdash b, x\vdash y)\nonumber,
\end{eqnarray}
for any $(a, x), (b, y)\in L\times D$
\end{lemma}
\begin{proof}
For any $a, b, c\in L, x, y, z\in D$, one has
\begin{eqnarray}
&&\qquad \Big((a, x)\triangleleft(b, y)\Big)\triangleleft\beta(c, z)-\alpha(a, x)\triangleleft\Big((b, y)\triangleright(c, z)\Big)=\nonumber\\
&&=(a\dashv b+a\dashv y+x\dashv b, x\dashv y)\triangleleft(\beta(c), \beta(z))-(\alpha(a), \alpha(x))\triangleleft
(b\vdash c+b\vdash z+y\vdash c, y\vdash z)\nonumber\\
&&=\Big((a\dashv b+a\dashv y+x\dashv b)\dashv\beta(c)+(a\dashv b+a\dashv y+x\dashv b)\dashv\beta(z)
+(x\dashv y)\dashv\beta(c),\;\; (x\dashv y)\dashv\beta(z)\Big)\nonumber\\
&&\quad-\Big(\alpha(a)\dashv(b\vdash c+b\vdash z+y\vdash c)+\alpha(a)\dashv(y\vdash z)
+\alpha(x)\dashv(b\vdash c+b\vdash z+y\vdash c),\;\; \alpha(x)\dashv(y\vdash z)\Big)\nonumber\\
&&=\Big((a\dashv b)\dashv\beta(c)+(a\dashv y)\dashv\beta(c)
+(x\dashv b)\dashv\beta(c)+(a\dashv b)\dashv\beta(z)+(a\dashv y)\dashv\beta(z)+(x\dashv b)\dashv\beta(z)\nonumber\\
&&\quad+(x\dashv y)\dashv\beta(c)-\alpha(a)\dashv(b\vdash c)-\alpha(a)\dashv(b\vdash z)-\alpha(a)\dashv(y\vdash c)-\alpha(a)\dashv(y\vdash z)
-\alpha(x)\dashv(b\vdash c)\nonumber\\
&&\quad-\alpha(x)\dashv(b\vdash z)-\alpha(x)\dashv(y\vdash c),\;\; (x\dashv y)\dashv\beta(z)-\alpha(x)\dashv(y\vdash z)\Big)\nonumber.
\end{eqnarray}
The left hand side vanishes by axiom (\ref{eq5}) and axioms $(02), (07), (12), (17), (22), (27)$ in Definition \ref{act1}.
The other axioms are proved in the same way.
\end{proof}
\begin{theorem}
Let $D$ and $L$ be two regular BiHom-associative dialgebras together with an action of $D$ on $L$.
Then, ${\bf Lb}(L\rJoin D)={\bf Lb}(L)\rJoin{\bf Lb}(D)$.
\end{theorem}
\begin{proof}
By lemma $\ref{lbs}$, ${\bf Lb}(D)$ acts on ${\bf Lb}(L)$, so it makes sense to consider the semidirect product Leibniz algebra
${\bf Lb}(L)\rJoin{\bf Lb}(D)$. It is clear that ${\bf Lb}(L\rJoin D)$ and ${\bf Lb}(L)\rJoin{\bf Lb}(D)$ are egal as vector space, so we only need
to verify that they share the same bracket. Let $(a, x), (b, y)\in L\times D$. If we use the bracket in ${\bf Lb}(L)\rJoin{\bf Lb}(D)$, we get :
\begin{eqnarray}
[(a, x), (b, y)]
&=&([a, b]+[x, b]+[a, y], [x, y])\nonumber\\
&=&(a\dashv y-b\vdash+x\dashv b-b\vdash x+a\dashv y-y\vdash a, x\dashv y-y\vdash x).\nonumber
\end{eqnarray}
On the other hand, if we use the Leibniz bracket in ${\bf Lb}(L\rJoin D)$ (Lemma \ref{lbsa}), we get
\begin{eqnarray}
\{(a, x), (b, y)\}
&=&(a, x)\triangleleft(b, y)-(b, y)\triangleright (a, x)\nonumber\\
&=&(a\dashv b+x\dashv b+a\dashv y, x\dashv y)-(b\vdash a+y\vdash a+b\vdash x, y\vdash x)\nonumber,
\end{eqnarray}
So the brackets are equal.
\end{proof}
\section{Central extensions}
This section concerns the central extension of BiHom-associative dialgebras in relation with cocycles.
\begin{definition}
Let $(D_i, \dashv_i, \vdash_i, \alpha_i, \beta_i), i=1,2,3$ be three BiHom-associative dialgebras. The BiHom-associative dialgebra $D_2$ is called
the extension of $D_3$ by $D_1$ if there are homomorphisms $\phi : D_1\rightarrow D_2$ and $\psi : D_2\rightarrow D_3$ such that the following sequence
$$0\rightarrow D_1\stackrel{\phi}{\longrightarrow}D_2\stackrel{\psi}{\rightarrow}D_3\rightarrow0$$
is exact.
\end{definition}
\begin{definition}
An extension is called trivial if there exists a BiHom-ideal $I$ of $D_2$ complementary to $Ker\psi$ i.e.
$$D_2=Ker\psi\oplus I$$
\end{definition}
It may happen that there exist several extensions of $D_3$ by $D_1$. To classify extensions the notion of equivalent extensions is defined.
\begin{definition}
Two sequences
$$0\rightarrow D_1\stackrel{\phi}{\longrightarrow}D_2\stackrel{\psi}{\rightarrow}D_3\rightarrow0$$
and
$$0\rightarrow D_1\stackrel{\phi'}{\longrightarrow}D_2\stackrel{\psi'}{\rightarrow}D_3\rightarrow0$$
are equivalent extensions if there exists a associative dialgebra isomorphism $f : D_2\rightarrow D'_2$ such that
$$f\circ\phi=\phi'\quad\mbox{and}\quad \psi'\circ f=\psi.$$
\end{definition}
\begin{definition}
An extension
$$0\rightarrow D_1\stackrel{\phi}{\longrightarrow}D_2\stackrel{\psi}{\rightarrow}D_3\rightarrow0$$
is called central if the kernel of $\psi$ is contained in the center $Z(D_2)$ of $D_2$, i.e. $Ker\psi\subset Z(D)$.
\end{definition}
Now, we introduce $2$-cocycle on BiHom-associative dialgebra with values in a BiHom-module.
\begin{definition}
Let $(D, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra and $(M, \alpha_M, \beta_M)$ a BiHom-module over the same field that $D$. A pair
$\Theta=(\theta_1, \theta_2)$ of bilinear maps $\theta_1 : D\times D\rightarrow V$ and $\theta_2 : D\times D\rightarrow V$ is called a $2$-cocycle on
$D$ with values in $V$ if $\theta_1$ and $\theta_2$ satisfy
\begin{eqnarray}
\theta_1(x\dashv y, \beta(z))&=&\theta_1(\alpha(x), y\dashv z), \label{cc1}\\
\theta_1(x\dashv y, \beta(z))&=&\theta_1(\alpha(x), y\vdash z), \\
\theta_2(x\vdash y, \beta(z))&=&\theta_2(\alpha(x), y\vdash z), \\
\theta_2(x\dashv y, \beta(z)) &=& \theta_2(\alpha(x), y\vdash z), \\
\theta_1(x\vdash y, \beta(z))&=&\theta_2(\alpha(x), y\dashv z),
\end{eqnarray}
for all $x, y, z\in D$.
\end{definition}
The set of all $2$-cocycles on $D$ with values in $M$ is denoted $Z^2(D, M)$, which a vector space. \\
In the below lemma, we give a special type of $2$-cocycles which are called $2$-coboundaries.
\begin{lemma}
Let $\nu : D\rightarrow V$ be a linear map, and define $\varphi_1(x, y)=\nu(x\dashv y)$ and $\varphi_2(x, y)=\nu(x\vdash y)$. Then,
$\Phi=(\varphi_1, \varphi_2)$ is a $2$-cocycle on $D$.
\end{lemma}
\begin{proof}
We will prove one equality, the others being proved in the same way. For any $x, y, z\in D$, one has
\begin{eqnarray}
\varphi_1(\alpha(x), y\dashv z)&=&\nu(\alpha(x)\dashv(y\dashv z))=\nu((x\dashv y)\dashv\beta(z))\nonumber\\
&=&\nu(\alpha(x)\dashv(y\vdash z))=\varphi_1(\alpha(x), y\vdash z).\nonumber
\end{eqnarray}
This finishes the proof.
\end{proof}
The set of all $2$-coboundaries is denoted by $B^2(D, M)$ and it is a subgroup of $Z^2(D, M)$.
The group $H^2(D, M)=Z^2(D, M)/B^2(D, M)$ is said to be a second cohomology group of $D$ with values in $M$.
Two cocycles $\Theta_1$ and $\Theta_2$ are said to be cohomologous cocycles if $\Theta_1-\Theta_2$ is a coboundary.
\begin{theorem}
Let $(D, \dashv, \vdash, \alpha_D, \beta_D)$ be a BiHom-associative dialgebra, $(M, \alpha_M, \beta_M)$ a BiHom-module,
$$\theta_1 : D\times D\rightarrow M\quad\mbox{and}\quad \theta_2 : D\times D\rightarrow M$$
be bilinear maps. Let us set $D_\Theta=D\oplus M$, where $\Theta=(\theta_1, \theta_2)$. For any $x, y\in D$, $v, w\in M$, let us define
\begin{eqnarray}
(x+u)\triangleleft(y+v)=x\dashv y+\theta_1(x, y)\quad\mbox{and}\quad (x+u)\triangleright(y+v)=x\vdash y+\theta_2(x, y).\nonumber
\end{eqnarray}
Then, $(D_\Theta, \triangleleft, \triangleright, \alpha_A\otimes \alpha_M, \beta_A\otimes \beta_M)$ is a BiHom-associative dialgebra if and only
if $\Theta$ is a $2$-cocycle.
\end{theorem}
\begin{proof}
For any $x, y, z\in D, u, v, w\in M$, we have
\begin{eqnarray}
&& ((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+v)\triangleleft((y+w)\triangleleft(z+w))=\nonumber\\
&=&((x+v)\triangleleft(y+w))\triangleleft(\beta(z)+w)-(\alpha(x)+ v)\triangleleft((y\dashv z)+\theta_1(y, z))\nonumber\\
&=&((x\dashv y)\dashv \beta(z))+\theta_1(x\dashv y, \beta(z))-(\alpha(x)\dashv (y\dashv z))-\theta_1(\alpha(x), y\dashv z).\nonumber
\end{eqnarray}
The left hand vanishes by axioms (\ref{eq4}) and (\ref{cc1}). The other axioms are proved analagously.
\end{proof}
\begin{lemma}
Let $\Theta$ be a $2$-cocycle and $\Phi$ a $2$-coboundary. Then, $D_{\Theta+\Phi}$ is a BiHom-associative dialgebra with
\begin{eqnarray}
(x+u)\unlhd(y+v)=x\dashv y+\varphi_1(x, y)+\theta_1(x, y),\nonumber\\
(x+u)\unrhd(y+v)=x\vdash y+\varphi_2(x, y)+\theta_2(x, y)\nonumber.
\end{eqnarray}
Moreover, $D_\Theta\cong D_{\Theta+\Phi}$.
\end{lemma}
\begin{proof}
First, we have to shown that $D_{\Theta+\Phi}$ is a BiHom-associative dialgebra. So, for any $x+u,\\ y+v, z+w\in D\oplus M$,
\begin{eqnarray}
&&\qquad((x+u)\unlhd(y+v))\unlhd\beta(z+w)-\alpha(x+u)\unlhd((y+v))\unlhd(z+w))=\nonumber\\
&&=(x\dashv y+\varphi_1(x, y)+\theta_1(x, y))\unlhd(\beta(z)+\beta(w))
-(\alpha(x)+\alpha(u))\unlhd(y\dashv z+\varphi_1(y, z)+\theta_1(y, z))\nonumber\\
&&=(x\dashv y)\dashv\beta(z)+\varphi_1(x\dashv y,\beta(z))+\theta_1(x\dashv y,\beta(z))
-\alpha(x)\dashv y\dashv z-\varphi_1(\alpha(x), y\dashv z)+\theta_1(\alpha(x), y\dashv z)\nonumber
\end{eqnarray}
The left hand side vanishes by (\ref{eq4}) and (\ref{cc1}). The proofs of the rest of axioms are leaved to the reader.\\
Next, the isomorphism $f : D_\Theta\rightarrow D_{\Theta+\Phi}$ is given by
$f(x+v)=x+\nu(x)+v$. In fact, it is clear that $f$ is a bijective linear map and
\begin{eqnarray}
f(\alpha_D+\alpha_M)(x+v)&=&f(\alpha_D(x)+\alpha_M(v))\nonumber\\
&=&\alpha_D(x)+\nu\alpha_D(x)+\alpha_M(v)\nonumber\\
&=&\alpha_D(x)+\alpha_M\nu(x)+\alpha_M(v)\nonumber\\
&=&(\alpha_D+\alpha_M)(x+\nu(x)+v)\nonumber\\
&=&(\alpha_D+\alpha_M)\circ f(x+v).\nonumber
\end{eqnarray}
Thus, $f$ commutes $\alpha_D+\alpha_M$, and similarly with $\beta_D+\beta_M$.\\
Then,
\begin{eqnarray}
f((x+v)\triangleleft (y+w))
&=&f(x\dashv y+\theta_1(x, y))\nonumber\\
&=&f(x\dashv y)+f(\theta_1(x, y))\nonumber\\
&=&x\dashv y+\nu(x\dashv y)+\theta_1(x, y)\nonumber\\
&=&x\dashv y+\varphi_1(x, y)+\theta_1(x, y)\nonumber.
\end{eqnarray}
and
\begin{eqnarray}
f(x+v)\unlhd f(y+w)
&=&(x+\nu(x)+v)\unlhd(y+\nu(y)+w)\nonumber\\
&=&(x\dashv y)+\varphi_1(x, y)+\theta_1(x, y).\nonumber
\end{eqnarray}
\end{proof}
\begin{corollary}
Let $\Theta_1, \Theta_2$ be two cohomologous $2$-cocycles on a BiHom-associative dialgebra $D$, and $D_1, D_2$ be the central extensions constructed
with these $2$-cocycles, respectively. The the central extensions $D_1$ and $D_2$ are equivalent extensions. In particular a central extension
defined by a coboundary is equivalent with a trivial central extension.
\end{corollary}
The following theorem is proved Mutatis Mutandis as (\cite{ISR}, Theorem 4.1). So we omitted the proof.
\begin{theorem}
There exists one to one correspondence between elements of $H^2(D, M)$ and nonequivalents central extensions of associative dialgebra $D$ by $M$.
\end{theorem}
\section{Classification }
In this section, we give classification of BiHom-associative dialgebras in low dimension.
Let $(D, \dashv, \vdash, \alpha, \beta)$ be an $n$-dimensional BiHom-associative dialgebra, $\{e_i\}$ be a basis of $D$. For any $i, j\in \mathbb{N}, 1\leq i, j\leq n$, let us put
$$e_i\dashv e_j=\sum_{k=1}^{n}\gamma_{ij}^ke_k,\quad e_i\vdash e_j\sum_{k=1}^{n}\delta_{ij}^ke_k,\quad\alpha(e_j)=\sum_{k=1}^{n}\alpha_{kj}e_k,\quad
\beta(e_j)=\sum_{k=1}^{n}\beta_{kj}e_k.$$
The axioms in Definition \ref{dia} are respectively equivalent to
\begin{eqnarray}
\beta_{kj}\alpha_{pk}-\alpha_{ji}\beta_{pj}&=&0,\\
\gamma_{ij}^p\beta_{qk}\gamma_{pq}^r-\alpha_{pi}\gamma_{jk}^q\gamma_{pq}^r&=&0,\\
\gamma_{ij}^p\beta_{qk}\gamma_{pq}^r-\alpha_{pi}\delta_{jk}^q\gamma_{pq}^r&=&0,\\
\gamma_{ij}^p\beta_{qk}\delta_{pq}^r-\alpha_{pi}\delta_{jk}^q\delta_{pq}^r&=&0,\\
\delta_{ij}^p\beta_{qk}\gamma_{pq}^r-\alpha_{pi}\gamma_{jk}^q\delta_{pq}^r&=&0,\\
\delta_{ij}^p\beta_{qk}\gamma_{pq}^r-\alpha_{pi}\delta_{jk}^q\delta_{pq}^r&=&0.
\end{eqnarray}
\subsection{One dimensional}
There is only one $1$-dimensional BiHom-associative dialgebra ; the nul (or trivial) BiHom-associative dialgebra.
\subsection{Two dimensional}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications &Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_1$&
$\begin{array}{ll}
e_1\dashv e_2=ae_1,\\
e_2\dashv e_1=be_1,\\
e_1\vdash e_2=ce_1,\\
e_2\vdash e_1=de_1,\\
e_2\vdash e_2=fe_1.
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1,\\
\beta(e_2)=e_1
\end{array}$
\\ \hline
$\mathcal{A}lg_2$
&
$\begin{array}{ll}
e_1\dashv e_2=ae_1,\\
e_2\dashv e_1=ae_1,\\
e_2\dashv e_2=e_1,\\
e_1\vdash e_2=e_1,\\
e_2\vdash e_1=e_1,
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1,\\
\beta(e_2)=e_1
\end{array}$
\\ \hline
$\mathcal{A}lg_3$
&
$\begin{array}{ll}
e_1\dashv e_2=ae_1,\\
e_1\vdash e_2=be_1,\\
e_2\vdash e_1=ce_1,\\
e_2\vdash e_2=de_1
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1,\\
\beta(e_2)=e_1
\end{array}$
\\ \hline
$\mathcal{A}lg_4$
&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_1,\\
e_2\dashv e_2=ae_1,\\
e_1\vdash e_2=be_1,\\
e_2\vdash e_1=ce_1,\\
e_2\vdash e_2=de_1,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1,\\
\beta(e_2)=e_1
\end{array}$
\\ \hline
\end{tabular}
\begin{remark}
In two dimensional, all of the BiHom-associative dialgebras are Hom-associative dialgebras i.e. $\alpha=\beta$.
\end{remark}
\subsection{Three dimensional}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications&Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_1$&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_1,\\
e_2\dashv e_2=ae_1,\\
e_2\dashv e_3=be_1,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_2=ce_1,\\
e_2\vdash e_1=e_1,\\
e_2\vdash e_2=de_1,\\
e_3\vdash e_2=fe_1,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\beta(e_2)=e_1,\\
\beta(e_3)=be_3
\end{array}$
\\ \hline
\end{tabular}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications&Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_2$&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_1,\\
e_2\dashv e_2=e_1,\\
e_2\dashv e_3=e_1,\\
e_3\dashv e_2=e_1,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_1,\\
e_2\vdash e_1=e_1,\\
e_2\vdash e_2=e_1,\\
e_3\vdash e_2=e_1,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\beta(e_2)=e_1,\\
\beta(e_3)=be_3
\end{array}$
\\ \hline
$\mathcal{A}lg_3$&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_,\\
e_2\dashv e_2=e_1,\\
e_2\dashv e_3=e_1,\\
e_3\dashv e_2=e_1,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_1,\\
e_2\vdash e_2=e_1,\\
e_2\vdash e_3=e_1,\\
e_3\vdash e_2=e_1,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\beta(e_2)=e_1,\\
\beta(e_3)=be_3
\end{array}$
\\ \hline
$\mathcal{A}lg_4$&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_1,\\
e_2\dashv e_2=e_1,\\
e_2\dashv e_3=e_1,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_1,\\
e_2\vdash e_1=e_1,\\
e_2\vdash e_2=e_1,\\
e_2\vdash e_3=e_1,\\
e_3\vdash e_2=e_1,
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\beta(e_2)=e_1,\\
\beta(e_3)=be_3
\end{array}$
\\ \hline
$\mathcal{A}lg_5$&
$\begin{array}{ll}
e_1\dashv e_2=e_1,\\
e_2\dashv e_1=e_1,\\
e_2\dashv e_2=e_1,\\
e_2\dashv e_3=e_1,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_1,\\
e_2\vdash e_1=e_1,\\
e_2\vdash e_3=e_1,\\
e_3\vdash e_2=e_1,
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\beta(e_2)=e_1,\\
\beta(e_3)=be_3
\end{array}$
\\ \hline
\end{tabular}
\subsection{Four dimensional}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications&Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_1$&
$\begin{array}{ll}
e_2\dashv e_1=e_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_1=e_4,\\
e_3\dashv e_2=e_4,\\
e_4\dashv e_4=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=ce_4,\\
e_3\vdash e_3=e_4,\\
e_3\vdash e_4=de_3,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=be_2\\
\beta(e_2)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_2$&
$\begin{array}{ll}
e_1\dashv e_2=e_4,\\
e_1\dashv e_4=e_4,\\
e_2\dashv e_1=ae_4,\\
e_2\dashv e_3=be_4,\\
e_3\dashv e_1=-ce_4,\\
e_3\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=de_4,\\
e_3\vdash e_3=fe_4,\\
e_3\vdash e_4=e_4,\\
e_4\vdash e_4=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\beta(e_2)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_3$&
$\begin{array}{ll}
e_1\dashv e_4=e_4,\\
e_2\dashv e_1=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=be_4,\\
e_3\dashv e_1=ce_4,\\
e_3\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_4,\\
e_3\vdash e_2=ce_4,\\
e_3\vdash e_3=de_4,\\
e_4\vdash e_4=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\alpha(e_3)=e_3\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_4$&
$\begin{array}{ll}
e_1\dashv e_4=e_4,\\
e_2\dashv e_2=ae_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_1=e_4,\\
e_3\dashv e_2=ce_4,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_3=e_4,\\
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_4,\\
e_3\vdash e_3=e_4,\\
e_4\vdash e_4=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\alpha(e_3)=e_3\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
\end{tabular}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications&Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_5$&
$\begin{array}{ll}
e_1\dashv e_4=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_4=e_4,\\
e_3\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_3=e_4,\\
e_3\dashv e_4=e_4,\\
e_1\vdash e_3=e_4,\\
e_2\vdash e_2=e_4,\\
e_3\vdash e_3=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_6$&
$\begin{array}{ll}
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_2=e_4,\\
e_3\dashv e_3=e_4,\\
e_3\dashv e_4=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_3=e_4,\\
e_1\vdash e_4=e_4,\\
e_2\vdash e_2=e_4,\\
e_3\vdash e_1=e_4,\\
e_3\vdash e_3=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\alpha(e_4)=e_4
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_7$&
$\begin{array}{ll}
e_1\dashv e_2=e_4,\\
e_1\dashv e_4=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_4=fe_4,\\
e_3\dashv e_3=-ge_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_4=e_4,\\
e_2\vdash e_2=e_4,\\
e_2\vdash e_3=e_4,\\
e_3\vdash e_1=e_4,\\
e_3\vdash e_2=-he_4,\\
e_3\vdash e_3=ke_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_3)=e_3,\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_8$&
$\begin{array}{ll}
e_1\dashv e_3=e_4,\\
e_1\dashv e_4=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_4=e_4,\\
e_3\dashv e_3=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_3=e_4,\\
e_3\vdash e_1=e_4,\\
e_3\vdash e_2=e_4,\\
e_3\vdash e_3=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_3)=e_3\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1,\\
\beta(e_3)=e_2,\\
\beta(e_4)=e_3,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_9$&
$\begin{array}{ll}
e_2\dashv e_2=e_1+e_4,\\
e_2\dashv e_3=e_1+e_4,\\
e_3\dashv e_2=e_1+e_4,\\
e_4\dashv e_2=e_1+e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=-e_1+e_4,\\
e_2\vdash e_2=e_1,\\
e_3\vdash e_3=e_1+e_4,\\
e_4\vdash e_2=e_1+e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\alpha(e_3)=e_2\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_3)=e_3,\\
\beta(e_4)=e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{10}$&
$\begin{array}{ll}
e_1\dashv e_2=e_4,\\
e_2\dashv e_2=e_1+e_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_2=e_1,\\
e_3\dashv e_3=e_4,\\
\end{array}$
$\begin{array}{ll}
e_4\dashv e_2=e_4,\\
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_1,\\
e_3\vdash e_3=e_1+e_4,\\
e_4\vdash e_2=e_1+e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\alpha(e_3)=e_2\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_3)=e_3,\\
\beta(e_4)=e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{11}$&
$\begin{array}{ll}
e_2\dashv e_2=fe_1+ge_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_2=e_1+e_4,\\
e_3\dashv e_3=e_4,\\
e_4\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=he_1-ke_4,\\
e_3\vdash e_3=e_1+e_4,\\
e_4\vdash e_2=e_1+e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_1\\
\alpha(e_3)=e_2\\
\alpha(e_4)=e_4\\
\end{array}$
$\begin{array}{ll}
\beta(e_3)=e_3,\\
\beta(e_4)=e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{12}$&
$\begin{array}{ll}
e_1\dashv e_4=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=ae_4,\\
e_2\dashv e_4=e_4,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_3=e_4,\\
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_4,\\
e_2\vdash e_3=-be_4,\\
e_3\vdash e_2=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_3)=e_3\\
\alpha(e_4)=e_4\\
\beta(e_1)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1+e_2,\\
\beta(e_3)=e_2+e_3,\\
\beta(e_4)=e_3+e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{13}$&
$\begin{array}{ll}
e_1\dashv e_2=e_4,\\
e_1\dashv e_3=e_4,\\
e_2\dashv e_1=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=e_4,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_1=e_4,\\
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_4,\\
e_2\vdash e_3=e_4,\\
e_3\vdash e_3=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\alpha(e_3)=e_3\\
\beta(e_1)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1+e_2,\\
\beta(e_3)=e_2+e_3,\\
\beta(e_4)=e_3+e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{14}$&
$\begin{array}{ll}
e_1\dashv e_1=e_4,\\
e_1\dashv e_3=-ce_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=e_4,\\
e_3\dashv e_1=e_4,\\
e_3\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_3\dashv e_3=-2ae_4,\\
e_1\vdash e_2=e_4,\\
e_2\vdash e_2=e_4,\\
e_2\vdash e_3=e_4,\\
e_3\vdash e_2=e_4,\\
e_3\vdash e_3=be_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_1)=e_1\\
\alpha(e_2)=e_2\\
\beta(e_1)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1+e_2,\\
\beta(e_3)=e_2+e_3,\\
\beta(e_4)=e_3+e_4,\\
\end{array}$
\\ \hline
\end{tabular}
\begin{tabular}{||c||c||c||c||c||c||c||}
\hline
$Algebras$&Multiplications&Morphisms $\alpha,\beta$.
\\ \hline
$\mathcal{A}lg_{15}$&
$\begin{array}{ll}
e_1\dashv e_1=-e_4,\\
e_1\dashv e_2=ae_4,\\
e_2\dashv e_3=be_4,\\
e_3\dashv e_1=ce_4,\\
e_3\dashv e_2=de_4,\\
e_3\dashv e_3=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=fe_4,\\
e_1\vdash e_4=e_4,\\
e_2\vdash e_2=e_4,\\
e_2\vdash e_3=e_4,\\
e_3\vdash e_2=ge_4,\\
e_3\vdash e_3=e_4,\\
e_3\vdash e_4=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_2)=e_2\\
\beta(e_1)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1+e_2,\\
\beta(e_3)=e_2+e_3,\\
\beta(e_4)=e_3+e_4,\\
\end{array}$
\\ \hline
$\mathcal{A}lg_{16}$&
$\begin{array}{ll}
e_1\dashv e_2=e_4,\\
e_2\dashv e_1=e_4,\\
e_2\dashv e_2=e_4,\\
e_2\dashv e_3=ae_4,\\
e_2\dashv e_4=e_4,\\
e_3\dashv e_2=e_4,\\
\end{array}$
$\begin{array}{ll}
e_1\vdash e_2=be_4,\\
e_2\vdash e_2=ce_4,\\
e_3\vdash e_2=de_4,\\
e_3\vdash e_3=e_4,\\
e_3\vdash e_4=e_4,\\
\end{array}$
&
$\begin{array}{ll}
\alpha(e_1)=ae_1\\
\beta(e_1)=e_1,\\
\end{array}$
$\begin{array}{ll}
\beta(e_2)=e_1+e_2,\\
\beta(e_3)=e_2+e_3,\\
\beta(e_4)=e_3+e_4,\\
\end{array}$
\\ \hline
\end{tabular}
\section{Derivation of BiHom-associative dialgebras}
In this section, we introduce and study derivations of BiHom-dendrifom, BiHom-dialgebras.
\begin{definition}
Let $(A,\mu, \alpha, \beta)$ be a BiHom-associative algebra. A linear map ${D} : A\longrightarrow A$
is called an $(\alpha^s, \beta^r)$-derivation of
$(A,\mu, \alpha, \beta)$, if it satisfies
$$
{D}\circ\alpha=\alpha\circ {D}\quad \text{and}\quad{D}\circ\beta=\beta\circ {D}
$$
$$
{D}\circ\mu(x, y)=\mu({D}(x), \alpha^s \beta^r(y))+\mu(\alpha^s \beta^r(x), {D}(y))
$$
\end{definition}
\begin{example}
We consider the $2$-dimensional BiHom-associative with a basis $\left\{e_1, e_2\right\}$.
For $\mu(e_1,e_1)=-e_1,\quad \mu(e_1, e_2)=-e_2,\quad \mu(e_2, e_1)=0,\quad \mu(e_2, e_2)=e_2$ and \\
$\alpha(e_1)=e_1,\quad\alpha(e_2)=-e_2,\quad \beta(e_1)=e_1,\quad \beta(e_2)=e_2.$ A direct computation gives that :
${D}(e_1)=d_{22}e_1,\quad {D}(e_2)=d_{22}e_2,$\\
$\alpha^s(e_1)=\frac{\alpha_{21}\beta_{22}}{\beta_{21}}e_1+\frac{e_2}{2\beta_{21}},\quad\alpha^s(e_2)=\alpha_{21}e_1,\quad
\beta^r(e_1)=\frac{e_2}{2\beta_{21}}e_2,\quad\beta^r(e_2)=\beta_{21}e_1+\beta_{22}e_2$.
\end{example}
\begin{definition}
Let $({D}, \dashv, \vdash, \alpha, \beta)$ be a BiHom-associative dialgebra. A linear map ${D} : {D}\rightarrow {D}$ is called an
$(\alpha^k, \beta^l)$-derivation of ${D}$ if it satisfies
\begin{enumerate}
\item [$1.$] ${D}\circ\alpha=\alpha\circ{D},\,{D}\circ\beta=\beta\circ{D}$;
\item [$2.$] ${D}(x\dashv y)=\alpha^k\beta^l(x)\dashv{D}(y)+{D}(x)\dashv\alpha^k\beta^l(y);$
\item [$3.$] ${D}(x\vdash y)=\alpha^k\beta^l(x)\vdash {D}(y)+{D}(x)\vdash\alpha^k\beta^l(y),$
\end{enumerate}
for $x, y\in {D}.$
We denote by $Der({D}):=\displaystyle\bigoplus_{k\geq 0}\displaystyle\bigoplus_{l\geq 0}Der_{(\alpha^k, \beta^l)}({D})$, where
$Der_{(\alpha^k, \beta^l)}({D})$ is the set of all $(\alpha^k,\beta^l)$-derivations of ${D}$.
\end{definition}
\begin{proposition}
For any ${D}\in Der_{(\alpha^s, \beta^r)}(A)$ and ${D}'\in Der_{(\alpha^{s'},\beta^{r'})}(A)$, we have
$\left[{D},{D'}\right]\in Der_{(\alpha^{s+s'}, \beta^{r+r'})}(A)$.
\end{proposition}
\begin{proof}
For $x, y\in A$, we have
$$\begin{array}{ll}
\left[{D}, {D'}\right]\circ\mu(x, y)
&={D}\circ{D'}\circ\mu(x, y)-{D'}\circ{D}\circ\mu(x, y)\\
&={D}(\mu({D}'(x), \alpha^s\beta^r(y))+\mu(\alpha^s\beta^r(x),{D}'(y)))\\
&-{D}'(\mu({D}(x), \alpha^s\beta^r(y))+\mu(\alpha^s\beta^r(x),{D}(y)))\\
&=\mu({D}\circ{D'}(x),\alpha^{s+s'}\beta^{r+r'}(y))+\mu(\alpha^s\beta^r\circ{D'}(x),{D}\circ\alpha^s\beta^r(y))\\
&+\mu({D}\circ\alpha^s\beta^r(x),\alpha^s\beta^r\circ{D}'(y))+\mu(\alpha^{s+s'}\beta^{r+r'}(x),{D}\circ{D'}(y))\\
&-\mu({D'}\circ{D}(x),\alpha^{s+s'}\beta^{r+r'}(y))-\mu(\alpha^s\beta^r\circ{D}(x),{D'}\circ\alpha^s\beta^r(y))\\
&-\mu({D'}\circ\alpha^s\beta^r(x),\alpha^s\beta^r{D}(y))-\mu(\alpha^{s+s'}\beta^{r+r'}(x),{D'}\circ{D}(y)).
\end{array}$$
Since ${D}$ and ${D}'$ satisfy
${D}\circ\alpha=\alpha\circ{D},\,{D}'\circ\alpha=\alpha\circ{D}'$,\,
${D}\circ\beta=\beta\circ{D},\,{D}'\circ\beta=\beta\circ{D}'$.\\ We obtain
$\alpha^s\beta^r\circ{D'}={D'}\circ\alpha^s\beta^r,\, {D}\circ\alpha^{s'}\beta^{r'}=\alpha^{s'}\beta^{r'}\circ{D}.$
Therefore, we arrive at\\
$\left[{D},{D'}\right]\circ\mu(x, y)=\mu(\alpha^{s+s'}\beta^{r+r'}(x),\left[{D},{D'}\right](y))+
\mu(\left[{D},{D'}\right](x),\alpha^{s+s'}\beta^{r+r'}(y)).$
Furthermore, it is straightforward to see that
$$\begin{array}{ll}
\left[{D},{D'}\right]\circ\alpha
&={D}\circ{D'}\circ\alpha-{D'}\circ{D}\circ\alpha\\
&=\alpha\circ{D}\circ{D}'-\alpha\circ{D}'\circ{D}=\alpha\circ\left[{D},{D'}\right].
\end{array}$$
$$\begin{array}{ll}
\left[{D},{D'}\right]\circ\beta
&={D}\circ{D'}\circ\beta-{D'}\circ{D}\circ\beta\\
&=\beta\circ{D}\circ{D}'-\beta\circ{D}'\circ{D}=\beta\circ\left[{D},{D'}\right]
\end{array}$$
which yields that $\left[{D},{D'}\right]\in Der_{(\alpha^{s+s'}, \beta^{r+r'})}(A)$ with $\mu=\dashv=\vdash.$
\end{proof}
\begin{proposition}
The space $Der_{(\alpha^{s}, \beta^{r})}(A)$ is an invariant of the triple BiHom-associative algebra A.
\end{proposition}
\begin{proof}
Let $\sigma : (A, \dashv_A, \vdash_A, \alpha^s, \beta^r)\longrightarrow (B, \dashv_B, \vdash_B, \alpha^s, \beta^r)$ be a triple BiHom-associative algebra isomorphism
and let ${D}$ be a $(\alpha^s, \beta^r)$-derivation of A. Then for any $x, y, z\in B$. We have :
$$\begin{array}{ll}
\sigma{D}\sigma^{-1}\circ(((x)\dashv_B(y))\dashv_B (z))
&=\sigma{D}\circ((\sigma^{-1}(x)\dashv_A\sigma^{-1}(y))\dashv_A\sigma^{-1}(z))\\
&=\sigma({D}\circ\sigma^{-1}(x)\vdash_A\sigma^{-1}\circ\alpha^s\beta^r(y))\vdash_A\sigma^{-1}\circ\alpha^s\beta^r(z))\\
&+\sigma(\sigma^{-1}\circ\alpha^s\beta^r(x)\vdash_A{D}\circ\sigma^{-1}(y))\vdash_A\sigma^{-1}\circ\alpha^s\beta^r(z))\\
&+\sigma(\sigma^{-1}\circ\alpha^s\beta^r(x)\vdash_A\sigma^{-1}\circ\alpha^s\beta^r(y)\vdash_A{D}\circ\sigma^{-1}(z))\\
&=({D}\circ\sigma^{-1}(x)\dashv_B\alpha^s\beta^r(y))\dashv_B\alpha^s\beta^r(z))\\
&+(\alpha^s\beta^r(x)\dashv_B\sigma\circ{D}\circ\sigma^{-1}(y))\dashv_B\alpha^s\beta^r(z))\\
&+(\alpha^s\beta^r(x)\dashv_B\alpha^s\beta^r(y))\dashv_B{D}\circ\sigma^{-1}(z)).
\end{array}$$
Thus $\sigma\circ{D}\circ\sigma^{-1}$ is a $(\alpha^s, \beta^r)$-derivation of $B$, hence the mapping.
$\psi : Der_{(\alpha^{s},\beta^{r})}(A)\longrightarrow Der_{(\alpha^{s}, \beta^{r})}(B)$,\\ ${D}\longmapsto \sigma{D}\sigma^{-1}$
is an isomorphism of triple BiHom-associative algebras.
In fact, it is easy to see that $\psi$ is linear. Moreover let
${D}_1, {D}_2, {D}_3$ be derivations of A :
$$\begin{array}{ll}
&\alpha^s\beta^r\circ\psi({D}_1\dashv_{tr}{D}_2)\dashv_{tr}{D}_3)=\\
&=\alpha^s\beta^r\psi(tr({D}_1)({D}_2\dashv{D}_3))+\alpha^s\beta^r\psi(tr({D}_3)({D}_1{D}_2))
+\alpha^s\beta^r\psi(tr({D}_2)({D}_3\dashv {D}_1))\\
&=\alpha^s\beta^r tr({D}_1)\psi({D}_2\dashv{D}_3)
+\alpha^s\beta^r tr({D}_3)\psi({D}_1\dashv{D}_2)+\alpha^s\beta^r tr({D}_2)\psi({D}_3\dashv{D}_1)\\
&=\alpha^s\beta^r tr(\psi({D}_1))(\psi({D}_2)\dashv\psi({D}_3))+\alpha^s\beta^r tr(\psi({D}_3))(\psi({D}_1)\dashv\psi({D}_2))\\
&+\alpha^s\beta^r tr(\psi({D}_2))\psi((\psi({D}_3)\dashv\psi({D}_1)),
\end{array}$$
since $\psi$ is a morphism of the $Der_{(\alpha^{s},\beta^{r})}(A)$ and $Der_{(\alpha^{s},\beta^{r})}(B)$, and
$tr({D})=tr(\sigma\circ{D}\circ\sigma^{-1}).$\\
Then $\alpha^s\beta^r\psi(({D}_1\dashv_{tr}{D}_2)\dashv_{tr}{D}_3))=\alpha^s\beta^r((\psi({D}_1)\dashv_{tr}\psi({D}_2))\dashv_{tr}\psi({D}_3)).$
\end{proof}
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