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+{"text":"\\section{Introduction} \\label{sec:intro}\nSupernova (SN) feedback plays a critical role in galaxy formation by regulating the phase structure of the interstellar medium (ISM; \\citealt{MO77, MacLow2004, Joung2006}), launching galactic winds \\citep{Strickland2009, Heckman2017, Zhang2018}, accelerating cosmic rays \\citep{Drury1994, Socrates2008, Caprioli2011, Girichidis2016}, and enriching the intergalactic and circumgalactic medium with metals \\citep{Andrews2017, Weinberg2017, Telford2018}. Through a combination of these phenomena, feedback regulates the global star-formation efficiency of galaxies \\citep{Ostriker2011, Hopkins2012}. Simulations imply that, without feedback, galaxies would rapidly convert cold gas into stars, resulting in up to a factor of 100 overproduction of stars compared to what is observed \\citep{Navarro1991, Hopkins2011}. \n\nUnfortunately, current state-of-the-art cosmological simulations that study the evolution of galaxy population over cosmic time cannot resolve the spatial scales on which supernova remnants (SNRs) interact with the ISM. Even modern `zoom-in' simulation of isolated galaxies can only marginally resolve SNRs \\citep{Hopkins2014, Hopkins2018}, and properly resolved SNRs can only be obtained in simulations of smaller regions of the ISM disk \\citep{Gatto2017, Kim2017,2020ApJ...896...66K}. This limitation motivated development of subgrid models of SN feedback at the resolution limit of simulations.\nInitial efforts to implement SN feedback in the form of thermal energy deposition were ineffective due to efficient radiative cooling \nin high-density star-forming regions \\citep{Katz1992}. The quest to limit plasma cooling and runaway star-formation spawned a variety of subgrid models which employed techniques like delayed gas-cooling \\citep{Stinson2006, Governato2007, Governato2010}, stochastic thermal feedback \\citep{Dalla2012}, an effective equation of state for a pressure-supported multi-phase ISM, with hydrodynamically decoupled wind particles \\citep{Springel2000, Springel2003, Oppenheimer2006, Vogelsberger2014} These techniques ranged from being unphysical in nature to being inaccurate in the details of the SN-ISM coupling \\citep{Martizzi2015, Rosdahl2017, Smith2018}.\n\nMore recent cosmological simulations \\citep[e.g., FIRE,][]{Hopkins2018, Hopkins2018b} have explored subgrid models that deposit momentum, which unlike thermal energy, cannot be radiated away before impacting ambient gas \\citep{Murray2005, Socrates2008, Agertz2013}. During the Sedov-Taylor phase of SNRs, the blast wave increases its momentum yield by a factor of 10--30 as it sweeps up ambient ISM. Later, it transitions into a cold, dense, momentum-conserving shell that ultimately merges with the ISM \\citep{Chevalier1974, Cioffi1998, Thornton1998, Martizzi2015, Kim2015, 2020ApJ...896...66K}. This momentum budget per SN has been quantified by several realistic models of the ISM \\citep[e.g.,][]{Martizzi2015, Kim2015, Li2015, Walch2015,2020ApJ...896...66K}. It has been shown to effectively drive turbulence and winds, and reproduce key features of galaxies such as the Kennicutt-Schmidt relation and galactic morphologies \\citep{Martizzi2016, Smith2018, Hopkins2018}. \nMore recent studies however have shown that the momentum deposition per SN depends sensitively on effects like SN clustering \\citep{Sharma2014, Gentry2017}, entrainment of cold clouds \\citep{Pittard2019}, the abundance pattern of the ISM \\citep{2020ApJ...896...66K}, thermal conduction \\citep{Badry2019}, enhanced cooling due to fluid-instability-driven mixing across the contact discontinuity \\citep{Gentry2019}, the SN delay-time distribution (DTD) model \\citep{Gentry2017, Keller2020}, and pre-SN feedback via winds, photoionization and radiation pressure \\citep{Fierlinger2016, Smith2020}\n\nObservations that are specifically sensitive to SN feedback can identify a reliable subgrid model. Generally, cosmological simulations calibrate subgrid models to reproduce bulk properties of the galaxy population such as the stellar mass function and stellar mass to halo mass relation, but this necessarily limits the predictive power of the simulations \\citep{Schaye2015}. Extragalactic multi-wavelength have served as useful references for setting subgrid model components such as SN rates \\citep[e.g.][]{Mannucci2006}, the efficiency of SN energy driving ISM turbulence \\citep[e.g.][]{Tamburro2009, Stilp2013} and mass-loading in supersonic winds \\citep[e.g.][]{Martin1999, Veilleux2005, Strickland2009}. However, the main source of uncertainty in modern subgrid models stem from a poor understanding of the SN-ISM interaction physics that originates on scales of 10-100 pc, which is beyond the reach for most distant surveys. In this respect, the resolved environments of Local Group galaxies provide detailed information available on stellar populations, ISM distribution and kinematics, and SNRs at the highest affordable spatial resolution. They are the ideal testing grounds for SN feedback models.\n\nIn this work, we test models of mass-weighted ISM turbulence predicted by SN momentum feedback models against observations of turbulence in M31, with a focus on the long-lived, prominent 10-kpc star-forming ring \\citep{Lewis2015, Williams2015}. The proximity of M31 pushes the frontier of  turbulence studies to $<100$ pc, where the effects of feedback are spatially-resolved, complete with detailed maps of the atomic ISM distribution \\citep[e.g.][]{Braun1991,Nieten2006,Braun2009}, and spatially-resolved stellar age distribution (SAD) measurements with sensitivity down to masses $\\approx 1.5$ M$_{\\odot}$ obtained by the Panchromatic Hubble Andromeda Treasury survey \\citep[PHAT;][]{Dalcanton2012, Williams2017}. We can use these SADs to estimate SN rates by taking into account currently known constraints on the efficiencies of the different progenitor channels of core-collapse and Type Ia SNe, expressed in the form of SN delay-time distributions (DTDs) \\citep{Maoz2014, Zapartas17, Eldridge2017}. The SADs of the older stellar populations allow us to quantify the Type Ia SN rate as a function of location, which is important for a `green-valley' galaxy like M31 \\citep{Mutch2011, Davidge2012}, and lacks correlation with conventional star-formation rate tracers.\n\nThis paper is organized as follows. In Section \\ref{sec:model} we describe our analytical momentum-driven ISM turbulence model, and how we use stellar population and ISM data to constrain ISM densities, SN rates and velocity dispersion in the M31 ring. Section \\ref{sec:results} describes the results of our analysis and checks on potential systematics, and in Section \\ref{sec:disc}, we discuss the implications of these results on the assumed subgrid models of feedback used in cosmological simulations.\n\n\\section{Modeling ISM Velocity Dispersion in M31} \\label{sec:model}\nHere, we compare the observed non-thermal velocity dispersion in M31's neutral ({\\sc H\\,i}\\xspace) and cold ISM \nwith the predicted turbulent velocity dispersion from the SN momentum-driven ISM turbulence model of \\cite{Martizzi2016}.\nOur calculations are supplemented by measured SN rates from the SAD of the PHAT survey and known forms of the SN DTD.\nWe describe these efforts below. For all measurements, we assume that the distance to M31 is 785 kpc \\citep{McConnachie2005}, and 1$^{\\prime\\prime}$ = 3.78 pc at the distance of M31.\n\nWe restrict our analysis to the 10 kpc star-forming ring of M31 (Figure \\ref{fig:maps}). We expect the main source of turbulence here to be star-formation, which we are mainly interested in testing, as opposed to other sources of turbulence observed in galaxies such as galactic spiral arms and magnetorotational instabilities \\citep{Tamburro2009,Koch2018,utomo2019}. Both atomic and molecular gas are most abundantly located and detected at high signal-to-noise along the ring \\citep{Braun2009, Nieten2006}. Additionally, since the gas scale height in pressure-supported star-forming disks can vary with radius \\citep{utomo2019}, staying within the ring helps justify the use of a constant scale height in Eq.\\ \\eqref{eq:nh1} and \\eqref{eq:nh2}. We do however assess the impact of variable scale heights later on in Section \\ref{sec:modelvsobsveldisp}. \n\nIn the following sub-sections, we describe our methodology for modeling the ISM velocity dispersion using momentum-driven turbulence from SNe, and the observations we use for comparison\n\n\\subsection{Momentum Injection by Supernovae} \\label{sec:predsigma}\n\nFollowing \\cite{Martizzi2015} and \\cite{Martizzi2016} -- hereafter, M15 and M16 respectively -- we assume that the non-thermal velocity dispersion in {\\sc H\\,i}\\xspace is a result of SNR momentum-driven turbulence driven on spatial scales comparable to the radius at which the SNR merges with the ISM, i.e. the shock velocity becomes of the order of velocity dispersion in the ISM. M15 quantified the final momentum ($p_{\\rm fin}$\\xspace) driven by an isolated SNR well past its shell formation stage in a turbulent ISM as\n\n\\begin{equation} \\label{eq:pfin_nh}\n\\frac{p_{\\rm fin}}{m_*} = 1110\\ \\mathrm{km\/s} \\left(\\frac{Z}{Z_{\\odot}}\\right)^{-0.114} \\left(\\frac{n_h}{100\\ \\mathrm{cm^{-3}}}\\right)^{-0.19}\n\\end{equation}\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{PaperFig1_HIDens.png}\n\\caption{Map of {\\sc H\\,i}\\xspace column density derived from 21 cm observations by \\cite{Braun2009}. Overlaid is the footprint of the PHAT survey, with observation bricks outlined as large red rectangles \\citep{Dalcanton2012}. Stellar age distributions are measured in 83$^{\\prime\\prime}$ cells within the PHAT area \\citep{Williams2017}. The shaded red squares show the locations of cells located between deprojected radii of 10--13 kpc from M31's center; they roughly cover the main star-forming ring of M31 (see Section \\ref{subsec:ring}). We will compare our model (Section \\ref{sec:model}) with the observed velocity dispersion in this ring.}\n\\label{fig:maps}\n\\end{figure}\n\nwhere $p_{\\rm fin}$\\xspace\/$m_*$ is the momentum deposited per mass of stellar population (we set $m_* = 100$ M$_{\\odot}$ per M15), $Z$ is the metallicity and $n_h$ is the ISM density\\footnote{The reader is refer to \\citet{2020ApJ...896...66K} for revised prescriptions at low $Z$}. We note that this form of $p_{\\rm fin}$\\xspace is similar to other independent high-resolution studies of momentum deposition by SNRs in an inhomogenous ISM \\citep[e.g.][]{Kim2015}.\n\nM16 used this subgrid model of momentum feedback to simulate the SN-driven ISM at 2--4 pc spatial resolution, and showed that the resulting velocity dispersion in a steady-state Milky Way-like ISM can be described by an analytical equation where the energy injection rate of SN momentum-driven turbulence is balanced by its corresponding rate of decay. The resulting mass-weighted velocity dispersion ($\\sigma_p$) is given by the Eq 22. in M15, which we repeat here for convenience, \n\n\\begin{equation}\n\\label{eq:sigma}\n \\sigma_{p} = \\dfrac{3}{4\\pi} \\left(\\dfrac{32\\pi^2}{9}\\right)^{3\/7} \\left(\\dfrac{p_{\\rm fin}}{\\rho}\\right)^{4\/7} \\left(f\\, \\dot{n}_{SN}\\right)^{3\/7}\n \\end{equation}\n\t\nwhere $\\rho$ is the density of gas, $\\dot{n}_{SN}$ is the SN rate per unit volume, and $f$ is a factor that accounts for momentum cancellation when multiple blast waves interact. We set $f=1$ for our fiducial runs, then revisit the issue of $f$ in Section \\ref{sec:disc}. We will use this predicted $\\sigma_p$ in different regions of M31's ring, as a function of the measured $\\rho$ and $\\dot{n_{SN}}$, for comparison with observations in the subsequent sections.\n    \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{CurrentDTDIllustration.pdf}\n    \\caption{Form of the SN DTD for core-collapse and Type Ia SNe. The solid blue line shows the DTD for core-collapse \\citep{Zapartas17} assuming single stellar evolution, while the dashed line model shows the extended delay-time tail due to binary evolution. Dashed red line shows the form of the Type Ia DTD used in this paper, in comparison with observations from SN Ia surveys \\citep{Totani2008,Maoz2010a, Maoz2011, Maoz2012b} and SNR surveys \\citep{Maoz2010}. See Section \\ref{sec:ratesmodel} for details.}\n    \\label{fig:dtd}\n\\end{figure}\n\\subsection{Measurement of SN rate density ($\\dot{n}_{SN}$)} \\label{sec:ratesmodel}\n\nWe set the SN rate per unit volume, $\\dot{n}_{SN}$ using the detailed SAD maps from the PHAT survey and the known properties of SN DTD.  We use the \\cite{Williams2017} SAD map of the northern third of the disk of M31, spanning a total area of about 0.8 deg$^2$ (Figure \\ref{fig:maps}). An SAD is measured in each of 826 spatial cells, each 83$^{\\prime\\prime}$ wide. Each cell contains the stellar mass formed per look-back time ($M_{ij}$, where $i$ is the cell and $j$ is the age bin), estimated by comparing resolved color-magnitude diagrams of the stars in the region with stellar isochrone models. \n\nWe then convert these SAD maps into maps of the SN rate per cell using observationally constrained DTDs. The DTD is defined as the SN rate versus time elapsed after a hypothetical brief burst of star formation. When convolved with the SAD maps described in the previous section, the DTD provides the current SN rate in each region of the galaxy \\citep{MaozMan2012, Maoz2014} in the following way:\n\n\\begin{equation}\n\\label{eq:r_i}\nR_i = \\sum_{j=1}^{N} M_{ij} \\Psi_j\n\\end{equation}\n\nwhere M$_{ij}$ is the stellar mass formed in cell $i$ in the age-interval $j$ given by the SAD map, and $\\Psi_j$ is the DTD value in the age bin $j$. We use the form of the core-collapse DTD given Eq A.2 of \\cite{Zapartas17}, which accounts for the effects of binary stellar interactions at $Z_{\\odot}$. For Type Ia SNe, we assume the parametric form of the Type Ia DTD from Maoz et al (2012) based on the compilation of all observational constraints to date,\n\n\\begin{equation}\n\\label{eq:psiIa}\n\\Psi_{Ia}(t) = (4 \\times 10^{-13} \\mathrm{\\ SN\\ yr^{-1}\\ M_{\\odot}^{-1}}) \\left(\\frac{t}{1 \\ \\mathrm{Gyr}}\\right)^{-1}\n\\end{equation}\n\nwhere $t$ is the delay-time between star-formation and SN. The form of the Type Ia and core-collapse SN DTDs are shown in Figure \\ref{fig:dtd}. The volumetric SN rate in cell $i$ ($\\dot{n}_{SN, i}$ for use in Eq.\\ \\eqref{eq:sigma})\ncan then be estimated from R$_i$ as\n\n\\begin{equation}\n\\label{eq:dotsn}\n\\dot{n}_{SN, i} = \\frac{R_{i} \\mathrm{cos}(i)}{2 A_{i} z_{sn}}\n\\end{equation}\n\nwhere $A_i$ is the cell size of each SAD region ($\\approx 83^{\\prime\\prime} \\times83^{\\prime\\prime}$ or $310 \\times 310$ pc$^2$) and $z_{sn}$ is scale height of the vertical distribution of SNe. The factor $\\mathrm{cos}(i)$ accounts for the extended line of sight through the disk as a result of the galaxy's inclination angle $i = 77\\degree$ \\citep{Corbelli2010}, so $z_{sn} \\rightarrow z_{sn}\/\\mathrm{cos}(i)$. We assume $z_{sn}=150$ pc for core-collapse SNe and $z_{sn}=600$ pc for Type Ia SNe, as explained in Section \\ref{sec:scaleheights}.\n\n\\subsection{Measurements of ISM density and velocity dispersion} \\label{sec:measurementofismdensity}\nMost of the ISM mass in star-forming regions is in the atomic ({\\sc H\\,i}\\xspace) and molecular phases, so we use maps of the 21 cm line of {\\sc H\\,i}\\xspace \\citep{Braun2009} and the 115 GHz line  $^{12}$CO(J=1--0) \\citep{Nieten2006} in M31. The data cubes of \\cite{Braun2009} were obtained using the Westerbork Synthesis Radio Telescope (WSRT) and the Green Bank Telescope (GBT), with a spatial resolution of 30$^{\\prime\\prime}$ (or 113 pc at the distance of M31). The {\\sc H\\,i}\\xspace column density ($N_{HI}$) \nand non-thermal velocity dispersion ($\\sigma_{HI}$) were measured by \\cite{Braun2009} from the 21 cm emission along each line of sight assuming a model of an isothermal, turbulence-broadened line profile. We note that evidence for opacity-corrected {\\sc H\\,i}\\xspace features in 21 cm is somewhat inconclusive in more recent observations in M31 and M33 \\citep{Koch2018, Koch2021}, so we use the opacity-uncorrected map of \\cite{Braun2009} (their Fig. 15). The difference in the predicted velocity dispersion from the two different version of the density maps is about $\\approx 12\\%$, which does not affect our conclusions later. Molecular hydrogen column densities were obtained from the $^{12}$CO(J=1--0) emission  map of \\cite{Nieten2006} using the single-dish IRAM 30m telescope. The survey covered $2^{\\degree} \\times 0.5^{\\degree}$ of the M31 disk, yielding a map of CO-line intensity at a final angular resolution of 23$^{\\prime\\prime}$ (spatial resolution $\\approx 87$ pc at the distance of M31).\n\nThe CO-line intensities were converted into H$_2$ column densities ($N_{H2}$)\nusing the conversion factor $\\mathrm{X_{CO}} = 1.9 \\times 10^{20}$ mol cm$^{-2}$ (K km s$^{-1}$)$^{-1}$ assumed by \\cite{Nieten2006}. The total mass of H$_2$ in M31 is about 14$\\%$ that of {\\sc H\\,i}\\xspace in the M31 ring. For convenience, we use the {\\sc H\\,i}\\xspace velocity dispersion as a proxy for H$_2$ velocity dispersion using the radius-independent ratio of $\\sigma_{HI}\/\\sigma_{H2} = 1.4$ measured by the HERACLES CO and THINGS {\\sc H\\,i}\\xspace surveys of nearby galaxies \\citep{Mogotsi2016}, as well as in M33 \\citep{Koch2019}. We note here that our inclusion of H$_2$ measurements is done to account for the velocity dispersion of the `mass-weighted' ISM, in order to be consistent with the M15 and M16 models, which also predicts the mass-weighted turbulent velocity dispersion.\n\nWe combine the density and velocity dispersion of the atomic and molecular phases into an effective mass-weighted ISM. The total mass-weighted non-thermal velocity dispersion in the {\\sc H\\,i}\\xspace and molecular phases is then  $\\sigma_{obs} = \\sqrt{(N_{HI}\/N_{tot})\\sigma_{HI}^2 + (N_{H2}\/N_{tot})\\sigma_{H2}^2}$, where $N_{tot}=N_{HI}+N_{H2}$.\n\nWe assume the vertical distribution of ISM in M31 is centered on the disk midplane, and approximately Gaussian for the molecular phase and exponential for {\\sc H\\,i}\\xspace, consistent with observations of our Galaxy \\citep{Dickey1990, Ferriere2001}. Each phase is characterized by an `effective' scale height, which we discuss further in Section \\ref{sec:scaleheights}.\n\n\nFor each SAD cell with {\\sc H\\,i}\\xspace column density N$_{HI}$, the {\\sc H\\,i}\\xspace density along the line of sight $z$ is,\n\n\\begin{equation} \\label{eq:nh1}\n    n_{HI}(z) = \\frac{N_{HI}\\, \\mathrm{cos}(i)}{2 z_{HI}}\\mathrm{exp}\\left(-\n    \\left|\\frac{z}{z_{HI}}\\right|\\right)\n\\end{equation}\n\nAs in Eq.\\ \\ref{eq:dotsn}, the scale height has been corrected for the inclination of M31 with the factor $\\mathrm{cos}(i)$. Similarly for each SAD cell with H$_2$ column density N$_{H2}$, the corresponding H$_2$ volume density is,\n\n\\begin{equation} \\label{eq:nh2}\n    n_{H2}(z) = \\frac{N_{H2}\\, \\mathrm{cos}(i)}{\\sqrt{2\\pi z_{H2}^2}} \\mathrm{exp}\\left(-\\frac{z^2}{2\n    z_{H2}^2}\\right)\n\\end{equation}\n\nFor ease of interpretation (given our simplified ISM model), we will compare the observed velocity dispersions with the \\emph{minimum} velocity dispersion predicted by models. We enforce this by assuming all SNe explode at the mid-plane density, i.e. $n_h = n_h(z=0)$. This is approximately a lower-limit on $\\sigma_p$ (we call this $\\sigma^{min}_p$) per SAD cell since SNe exploding away from the midplane but still within the scale height of gas would interact with lower densities than at the midplane, deposit greater momenta (Eq.\\ \\ref{eq:pfin_nh}), and contribute to an effectively higher $\\sigma_p$ per SAD cell. This lower limit is also a good assumption since we neglect all other sources of stellar feedback (e.g. winds, cosmic rays) that could add to the momentum budget per SAD cell depending on the environment. Comparing this minimum feedback from SNe with observations leads to some interesting insight as we show in Section \\ref{sec:disc}. For each value of ($N_{HI}$, $N_{H2}$), we derive ($n_{HI}$, $n_{H2}$) using Eq \\eqref{eq:nh1} and \\eqref{eq:nh2}, convert to a total hydrogen mass density $\\rho = m_p (n_{HI} + 2n_{H2})\/X_H$ in units of g\/cm$^{3}$ (where $m_p = 1.67 \\times 10^{-24}$ g and $X_H = 0.76$ is the mass fraction of hydrogen), and feed it into Eq.\\ \\eqref{eq:sigma}. We also take the total number density of hydrogen, in units of atoms cm$^{-3}$ as $n_h = n_{HI} + 2 n_{H2}$, for use in Eq \\ref{eq:pfin_nh}.\n\n\\subsection{Galactocentric radii of SAD cells} \\label{subsec:ring}\n\nWe first calculate the deprojected distance of each cell from the center of M31 using the method in \\cite{Hakobyan2009}. Let $\\left(\\alpha,\\delta \\right)$ be the sky-projected location of each cell centroid, and $\\left(\\Delta \\alpha, \\Delta \\delta\\right)$ be the sky-projected angular offset from M31 center (located at $\\alpha_{M31} = 00^h42^m44.3^s$, $\\delta_{M31} = +41\\degree16^{\\prime}9^{\\prime\\prime}$)\\footnote{\\url{http:\/\/ned.ipac.caltech.edu\/}}. Assuming a position angle of M31's disk, $\\theta_p$= 38$\\degree$ \\citep{Corbelli2010}, the location ($u,v$) of each SAD cell in M31's coordinate system is\n\n\\begin{align*}\n\tu &= \\Delta \\alpha\\ \\mathrm{sin} \\theta_p + \\Delta \\delta\\ \\mathrm{cos} \\theta_p \\\\\n\tv &= \\Delta \\alpha\\ \\mathrm{cos} \\theta_p - \\Delta \\delta\\ \\mathrm{sin} \\theta_p\n\\end{align*}\n\nThe radial distance of each cell in the plane of M31 from the M31 center, corrected for M31's inclination ($i = 77\\degree$; \\citealt{Corbelli2010}), is, \n\n\\begin{equation}\n\td^2 = u^2 + \\left(\\frac{v}{\\mathrm{cos}\\, i}\\right)^2\n\\end{equation}\n\nwhere $d$ is the angular distance from the center in arcseconds. We identify the ``ring'' as SAD cells with 10-13 kpc, as shown by the shaded region in Figure \\ref{fig:maps}\n\n\n\\subsection{Assumptions about SN and ISM scale heights} \\label{sec:scaleheights}\nIn this section, we describe plausible ranges and fiducial values for our free parameters: the atomic and molecular scale heights ($z_{HI}$ and $z_{H2}$) and SN scale heights $z_{sn}$ (here on, we will specify the separate scale heights of SNe Ia and CC as $z_{Ia}$ and $z_{cc}$ respectively).\n\nCore-collapse SNe generally occur at lower effective scale heights than SNe Ia \\citep{Hakobyan2017}.\n\n\\begin{figure*}\n\\subfigure[]{\\includegraphics[width=0.5\\textwidth]{total_sn_rate_ring.pdf}\\label{fig:snrate}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{sn_ia_cc_fraction.pdf}\\label{fig:snfrac}}\n\\caption{(a) The distribution of SN rate (log $R_i$; see Eq.\\ \\ref{eq:r_i}) in the portion of M31's 10 kpc ring covered by the PHAT survey. Greyscale is the {\\sc H\\,i}\\xspace column density map of \\cite{Braun2009} (see Section \\ref{sec:ratesmodel} for details). The red shading denotes total SN rate (Type Ia + core-collapse) in units of SNe yr$^{-1}$. Black circles show locations of optically-selected SNRs from the \\cite{Lee2014} survey, and the blue dashed elliptical contours show the region of the 10--13 kpc ring demarcated for analysis in this paper (see Section \\ref{subsec:ring}). (b) The fraction of Type Ia to core-collapse SNe. The figure is the same as in panel (a), except red shading denotes this ratio.}\n\\end{figure*}\n\nIn the Milky Way, open clusters younger than 100 Myrs are all situated within 200 pc of the midplane, with an effective scale height of 60--80 pc \\citep{Joshi2016, Soubiran2018}. Since their age and velocity distribution nicely follows that of field stars \\citep{Baumgardt2013, Soubiran2018}, we can assume that the general population of core-collapse SN progenitors in the Milky Way also has a scale height of 60--80 pc. However, the disk of M31 is kinematically hotter and more extended than the Milky Way \\citep{Ivezic2008, Collins2011}. Based on the ratio of scale heights to scale lengths observed in edge-on disk galaxies \\citep{Yoachim2006, Yoachim2008}, \\cite{Collins2011} proposed that the M31 disk could be 2-3 times thicker than the Milky Way (although this may be an over-estimate as the galaxies in the \\citealt{Yoachim2006} sample are different and less massive than M31). We therefore assume that in M31, 60 pc $<$ $z_{cc}$ $< 200$ pc is a plausible range for the scale height of core-collapse SNe. \n\nOlder ($\\sim$Gyr) stars are mostly concentrated in the thin and thick disk with measured scale heights in the range of 140-300 pc and 500-1100 pc respectively in the Milky Way \\citep{Li2018, Mateu2018}. The thin disk is slightly younger, with ages in the range of $7-9$ Gyrs compared to the thick disk's age of $\\sim 10$ Gyrs \\citep{Kilic2017}. The measured shape of the Type Ia SN DTD suggests that progenitors younger than 10 Gyrs will produce the majority of Type Ia SNe \\citep{Maoz2010}, so we assume Type Ia SN progenitors are roughly distributed at the same scale height as the thin disk, $\\sim$300 pc. This is also consistent with the scale height of SDSS white dwarfs \\citep{deGennaro2008, Kepler2017, Gentile2019} and about 4 times the scale height of young core-collapse progenitors, so for simplicity we assume that in M31, $z_{Ia} \\approx 4 z_{cc}$, and $z_{cc}$ is in the range mentioned previously.\n\n{\\sc H\\,i}\\xspace scale heights in M31 were measured by \\cite{Braun1991} in the range of $z_{HI} = 275-470$ pc between radii of 10--13 kpc in M31. We are not aware of any scale height measurements of the molecular phase in M31, but the Milky Way can provide some supplementary information. Studies of the H$_2$ profiles traced by CO in the Milky Way have measured a half-width at half-maximum scale height of 50--80 pc (consistent with being a bit smaller than $z_{cc}$), which is about a factor of 3 lower than the scale height of {\\sc H\\,i}\\xspace in the Milky Way \\citep{Marasco2017}.\n\nGiven these constraints, we can assume that $z_{cc}$ is always less than $z_{Ia}$, $z_{H2}$ is always less than $z_{HI}$, $z_{Ia} \\gtrsim 4 z_{cc}$ and $z_{H2} \\approx z_{HI}\/3$ . Given the range of values allowed by observations, we first analyze our results for a fiducial model where $z_{cc} = 150$ pc, and $z_{HI}=350$ pc, giving $z_{Ia} = 600$ pc, and $z_{H2}=117$ pc. We then change the values of these parameters and their ratios within the plausible ranges discussed previously to assess the impact of assumptions in Section \\ref{sec:modelvsobsveldisp}.\n\n\\section{Results} \\label{sec:results}\nIn this section, we show the distribution of SN rates across the M31 ring as measured from our SAD map and DTDs, and a comparison of our predicted velocity dispersions predicted by these rates with the observed values along the ring.\n\\subsection{Distribution of SN Rates} \\label{subsec:snrates}\n\nFigure \\ref{fig:snrate} shows our SN rate distribution \nestimated from the DTDs and SADs as described in in Section \\ref{sec:ratesmodel} (Eq \\ref{eq:r_i}). The integrated SN rate in the region we identify as M31's 10 kpc ring is $1.74 \\times 10^{-3}$ SN yr$^{-1}$\\xspace, with roughly 39$\\%$ contribution from SN Ia and $61\\%$ from core-collapse SNe. \n\nThe fraction of this SN rate of Type Ia versus core-collapse is shown in Figure \\ref{fig:snfrac}. About 75$\\%$ of the ring has a higher core-collapse rate than Type Ia. These regions coincide well with young star-forming regions identified in UV and IR images of M31 \\citep{Lewis2017}, and are mostly concentrated in the inner parts of the ring. Regions with the highest core-collapse rates, exceeding that of Type Ia by more than a factor of 3, coincide with the well-known star-forming region OB54, with\nnearly $3.8 \\times 10^{5}$ M$_{\\odot}$ of stars younger than 300 Myrs \\citep[][also see Figure \\ref{fig:overpredictedcells} in this paper]{Johnson2016}. SNe Ia generally dominate the total SN rate near the edges of our ring region, coinciding with the inter-arm region as seen in Figure \\ref{fig:snfrac}, and exceeding the core-collapse rate by up to a factor of 3 in some SAD cells. \n\nAs evidence of the high characteristic SN rate of the ring, we also show the distribution of optically-selected SNRs in M31 by \\cite{Lee2014} in Figure \\ref{fig:snfrac}. The majority of SNRs are concentrated along the M31 ring, and particularly associated with regions of higher core-collapse fraction. A more quantitative test of whether the observed SNR distribution is consistent with the SN rates will be the subject of a future paper, since it requires a more rigorous analysis of the poorly understood completeness of SNR catalogs (particularly at optical wavelengths).\n\n\n\\subsection{Comparison of model and observed velocity dispersion}\n\\label{sec:modelvsobsveldisp}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{sigma_obs_v_predict.pdf}\n    \\caption{The observed atomic+molecular velocity dispersion in the M31 ring (Section \\ref{sec:measurementofismdensity}) compared with our predictions of the minimum velocity dispersion from the fiducial SN momentum-driven turbulence model in Section \\ref{sec:predsigma}. The dashed lines indicate values where $\\sigma^{min}_p$ is twice, equal and half the $\\sigma_{obs}$ values.}\n    \\label{fig:modelvsobsvel}\n\\end{figure}\nWe compare the observed ($\\sigma_{obs}$) versus mininum predicted velocity dispersion ($\\sigma^{min}_p$) in the mass-weighted ISM in Figure \\ref{fig:modelvsobsvel}. The observed velocity dispersion exhibits a range of values spanning 4-12 km\/s, whereas the predicted values extend up to 20 km\/s or higher. On average, we find that for our fiducial model described in Section \\ref{sec:scaleheights}, the $\\sigma^{min}_p$ mostly exceed the observed values $\\sigma_{obs}$, but within a factor of two for 84$\\%$ of the SAD pixels in the ring. To understand why our velocity dispersion model over-estimates the observed values in Figure 4., we checked the ratio of observed to predicted velocity dispersion values, i.e. $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace against the column density and SN rate, the two fundamental parameters in our model, in Figure \\ref{fig:sigrationh}. We find hint of a negative correlation in $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace with $N_H$ and a positive correlation with SN rate.\nIn particular, SAD pixels with log $(N_{tot}\/\\mathrm{cm}^{-2})<21.3$ and SN rate or Log $(R_i\/\\mathrm{yr}^{-1})>-4.7$ mostly have $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace$>1$. We examine this more closely in Section \\ref{sec:disc}\n\n\n\n\\begin{figure*}\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.5\\textwidth]{sigmas_NH_fiducial.pdf}}\n    \\subfigure[]{\\hspace{-0.2cm}\\includegraphics[width=0.5\\textwidth]{sigmas_rIa_rCC_fiducial.pdf}}\n    \\caption{The ratio of velocity dispersion predicted by our model ($\\sigma_{p}^{min}$) and to that measured from 21 cm + CO-line observations ($\\sigma_{obs}$) in each SAD cell in the M31 ring (see Figure \\ref{fig:maps}) is compared with the observed column density (atomic + molecular; panel a) and the estimated SN rate (panel b) in those cells. Both panels provide the same information, but N$_H$ and SN Rate switches between the x-axis and the colorbar. The points represent the lower limit to the ratio as it corresponds to SNe exploding in M31's midplane (see Section \\ref{sec:scaleheights}).}\n    \\label{fig:sigrationh}\n\\end{figure*}\nWe briefly discuss the impact of model uncertainties which are certain to alter the $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace measurements. The SN rates can vary by an average of 15$\\%$ (maximum of 50$\\%$), depending on the isochrone model used for constructing SADs \\citep{Williams2017}, but this has a relatively small impact on our result. For example, using the MIST SAD solutions (which we have been using), we have about 82$\\%$ SAD pixels with $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace$>1$, whereas using the PARSEC SAD solutions results in 74$\\%$ of SAD cells having $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace $>1$, and the correlations with density and SN rates remain. Assumptions about the scale height of gas and SNe directly affect the midplane densities and SN rate densities, which has a larger effect on $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace measurements. We therefore assess the impact of varying scale heights on the $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace values as as shown in Figure \\ref{fig:checkz}. For smaller gas scale heights and larger core-collapse scale heights,\n$\\sigma^{min}_p\/\\sigma_{obs}$\\xspace decreases. This is because smaller gas scale heights imply a higher volume density of ISM for a given column density (Eq.\\ \\ref{eq:nh1}, \\ref{eq:nh2}), which reduces the momentum deposition and turbulence driving based on Eq.\\ \\ref{eq:pfin_nh}. For larger \nSN scale heights, \nthe SN rate per unit volume is smaller (Eq.\\ \\ref{eq:dotsn}), which likewise reduces the momentum deposition rate (Eq.\\ \\ref{eq:pfin_nh}). Within the plausible range of scale heights  discussed in Section \\ref{sec:scaleheights} and marked as a box in Figure \\ref{fig:checkz}, about $60\\%$ of SAD pixels still have over-predicted velocity dispersion. The part of the parameter space in Figure \\ref{fig:checkz} where the fraction of over-predicted cells are below $10-20\\%$ involves $z_{cc}>z_{HI}$, which is unlikely given the close association of core-collapse SNe with gas-rich star-forming regions.\n\n\\begin{figure}\n    \\hspace{-0.3in}\n    \\includegraphics[width=1.2\\columnwidth]{pspace_zcc_zhi.pdf}\n    \\caption{Effect of scale heights on the velocity dispersion calculated from our model. Here, we assume $z_{H2} =  z_{HI}\/3$ and $z_{Ia} = 4 z_{cc}$. The plausible values for $z_{cc}$ and $z_{HI}$ in M31, as laid out in Section \\ref{sec:scaleheights}, are outlined by the white box. The fiducial model used for predicting values of velocity dispersion in Figure \\ref{fig:sigrationh} are shown as a white cross. Colorbar indicates the fraction of SAD pixels with over-predicted mass-weighted velocity dispersion.}\n    \\label{fig:checkz}\n\\end{figure}\n\n\\section{DISCUSSION} \\label{sec:disc}\n\\begin{figure}\n    \\includegraphics[width=\\columnwidth]{overpredicted_cells.pdf}\n    \\caption{Zoomed-in section of the M31 ring (demarcated by black dashed ellipses) on the grayscale \\cite{Braun2009} {\\sc H\\,i}\\xspace map, showing \n    the low density (light red) and high SN rate cells (bold red) where the ratio of predicted-to-observed velocity dispersion is mainly above 1 (See Section \\ref{sec:disc} for details). Blue circles show locations of star clusters younger than 50 Myr and more massive than $10^3$ M$_{\\odot}$ from \\cite{Johnson2016}.\n    \\label{fig:overpredictedcells}\n\\end{figure}\n\\subsection{Insight Into Momentum feedback efficiency}\nOur analysis has shown that simple models of ISM turbulence driven by isolated non-overlapping SNRs are consistent within a factor of 2 of observations for most of the star-forming\/ISM environment of the M31 ring. Some of the discrepancy can be explained by variation in model parameters (e.g. scale heights of stars and gas) as explained in Section \\ref{sec:modelvsobsveldisp}, but in this discussion, we particularly focus on cells with Log $(N_{tot}\/\\mathrm{cm}^{-2})<21.3$ and Log $(R_i\/\\mathrm{yr}^{-1})>-4.7$, since all the cells in this range have $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace$\\gtrsim 1$ even after accounting for plausible variation in SN rates and scale heights in Section \\ref{sec:modelvsobsveldisp}.\nRegions with $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace$>1$ values are interesting in the sense that there are multiple sources of stellar feedback such as stellar winds, radiation pressure, photoionization and cosmic rays, but here the hydrodynamical momentum from SN blast-waves alone over-predict the observed ISM turbulence.\n\nOne reason behind these over-predicted cells could be that $\\sigma_{obs}$ were underestimated in our maps, but this is unlikely. More recent, sensitive VLA-based {\\sc H\\,i}\\xspace surveys \\citep[e.g.][]{Koch2018, Koch2021} suggest that a clean separation of thermal and non-thermal components of the 21 cm line is non-trivial. It is likely that the assumption of \\cite{Braun2009} of an isothermal {\\sc H\\,i}\\xspace component along the line of sight results in some residual thermal contribution to the non-thermal velocity dispersion. Thus the non-thermal velocity dispersion in M31 we are using from \\cite{Braun2009} may be an upper-limit to the actual turbulence contribution.\n\nThe regions with $\\sigma^{min}_p\/\\sigma_{obs}$\\xspace$>1$, especially in the cells with Log $(N_{tot}\/\\mathrm{cm}^{-2})<21.3$ and Log $(R_i\/\\mathrm{yr}^{-1})>-4.7$, may therefore indicate a drawback in the SN momentum-driven turbulence model, so we investigate these regions visually in Figure \\ref{fig:overpredictedcells}.\nFrom here on, we couch the column density and SN rate cutoff in terms of a volume density and SN rate surface density cutoff, to be consistent with values used in simulations. The low column density cutoff corresponds to $n_h < 0.2$ cm$^{-3}$ for our fiducial scale heights, and the SN rate cutoff corresponds to a SN rate surface density $\\Sigma_{SN}>2.1\\times10^{-4}$ SN yr$^{-1}$ kpc$^{-2}$\\xspace.\n\n\nThe low-density cells are situated at the upper and lower edges of the star-forming ring as shown in Figure \\ref{fig:overpredictedcells}. Comparison with Figure \\ref{fig:snfrac} shows that these regions also have a higher rate of SN Ia than CC, with an average SN Ia\/CC ratio $\\approx 1.67$ in these cells. M16 noted that at low densities, SNRs have longer cooling timescales, and may come into pressure equilibrium with the ISM before cooling or depositing a significant amount of momentum \\citep{MO77}. This missing physics in our model is likely the reason for the over-predicted $\\sigma^{min}_p$.\n\nSAD cells with Log $(R_i\/\\mathrm{yr}^{-1})>-4.7$ are spatially correlated with the prominent star-forming region OB54 as mentioned in Section \\ref{subsec:snrates}. One possibility, as raised by M15 and M16, is that in high star-forming regions, overlapping shocks from close-proximity SNRs might cancel some of the outgoing momentum (parameterized by $f$ in Eq \\ref{eq:sigma}, which we had set to 1). For example, a reduction of more than a factor of 2 in  $\\sigma^{min}_p$ is achieved with $f<0.2$, which is consistent, though slightly less than $f=0.3-0.4$ assumed in the SN-driven turbulent ISM simulations of M16. Other possibilities include a non-negligible fraction of the cold ISM mass is driven out by clustered SNe driving a hot, over-pressurized outflow \\citep{Sharma2014, Gentry2017}. This explanation is plausible given the detection of X-ray emission in this region by \\cite{Kavanagh2020}, and was also given as an explanation by previous energy-balance studies that similarly observed an excess of SN energy over the measured ISM turbulent energy in the central high star-forming regions of galaxies \\citep{Tamburro2009, Stilp2013, Koch2018, utomo2019}. A remaining possibility is that a non-negligible ISM mass in these regions is sustained at warmer phases, invisible to 21 cm or CO-line maps, due to the cumulative heating by SNe and pre-SN processes like winds and photoionization. \n\n\nThe results above indicate that fiducial models of momentum feedback from SNe used by most cosmological simulations, which generally assume non-overlapping, non-clustered SNRs, may require adjustment at low densities and at high SN rates due to aforementioned non-linear effects of clustering and SNR evolution at low-densities. This can be quantified by a suppression factor $f(n_h,\\, \\Sigma_{SN})<1$ for $n_h \\lesssim 0.2$ cm$^{-3}$ and $\\Sigma_{SN}>2.1\\times10^{-4}$ SN yr$^{-1}$ kpc$^{-2}$\\xspace, although a more precise form of this relation will be explored in a subsequent paper where we account for the energy and momentum carried away by any high-velocity outflows or warm diffuse gas from these regions. \n\nThe result also highlights the role of Type Ia SN feedback in low-density regions of the ISM, where it can exceed core-collapse SN rates \\citep{Li2020a, Li2020b}. The energetics of Type Ia SNe are particularly pronounced in the central few kpc of M31 (though not explored in this paper), where it is likely responsible for the bright X-ray halo emission and depleted metal abundances in the region \\citep{Tang2009, Telford2018}.\n\n\\subsection{Comparison with previous studies of ISM energy balance}\nAs the molecular ISM in our data is only $\\sim 14\\%$ of the atomic ISM, our results primarily explore feedback in the atomic ISM, and therefore it is interesting to compare our work with previous studies of energy balance in the ISM traced by atomic hydrogen. \\cite{Tamburro2009} showed that SN energy alone can drive turbulence in atomic gas within the optical radius of nearby galaxies, with an approximate coupling efficiency of $\\sim 10\\%$ \\citep{Thornton1998, MacLow2004}. Similar results were also obtained by \\cite{Stilp2013} with globally-averaged {\\sc H\\,i}\\xspace observations. More recently, \\cite{Koch2018} and \\cite{utomo2019} extended these techniques to M33, with the latter study allowing coupling efficiency to vary with radius.\n\nA key difference between our work and previous ones is that we examine spatially-resolved {\\sc H\\,i}\\xspace line profiles along different lines of sight, as opposed to globally or radially-stacked {\\sc H\\,i}\\xspace profiles. This allows us to compare the observed ISM turbulence with the local properties of the star-forming and ISM environment.\n\nPhenomenologically, there are a few key differences between our work and previous studies also worth mentioning. Similar to \\cite{utomo2019}, we account for turbulence driven by momentum-conserving phase of isolated SNRs, and constrain the efficiency of this momentum-feedback driving the observed turbulence, as opposed to previous studies that considered the efficiency of initial SN energy (= $10^{51}$ ergs) going into the ISM turbulence (the majority of which will be radiated away without impacting the gas). The M15 and M16 models also assume that turbulence is driven at the radius where SNR dissolves into the ISM, which strongly depends on the ISM density (i.e. when $v_s \\sim \\sigma$). This is different from the assumption of constant driving scale (equal to the scale height) or decay timescale in \\cite{Tamburro2009} and \\cite{Koch2018}. These assumptions affect the predicted SN feedback. For example, \\cite{utomo2019} showed that a spatially-varying decay timescale allowed SNe to drive turbulence in M33 out to 7 kpc instead of 4 kpc, by which point the star-formation rate and gas densities decrease by an order of magnitude compared to the central region. \\cite{Bacchini2020} similarly showed that a variable decay timescale makes SNe efficient enough to drive turbulence in the THINGS galaxies throughout, as opposed to just within the optical radius \\citep{Tamburro2009}. \n\nDespite the differences in methodology, our work agrees with previous studies that SN energy driving is inefficient, particularly in regions characterized by high star-formation rates. The higher spatial resolution offered by a more nearby galaxy like M31 reveals that regions where our models disagree with observations also correlate with regions of clustered star-formation, signifying the importance of taking into account clustering effects in SN feedback models. A direct comparison with previous studies is complicated given the differences in methodology, but our work highlights the importance of spatially-resolved observations in Local Group galaxies in the study of SN feedback.\n\n\n\\section{Conclusion}\nIn this paper, we have tested the paradigm of SN momentum-driven ISM turbulence developed by recent high-resolution vertical disk simulations. We compare model prescriptions with resolved observations of stellar populations and ISM in the prominent 10-kpc star-forming of M31, where stellar feedback is expected to be the main source of turbulence. The spatially-resolved PHAT stellar photometry in the northern third of the disk provides detailed stellar-age distributions (SADs) in $\\approx 310$ pc$^2$ cells, which we convolved with known forms of the SN delay-time distribution to predict the core-collapse and Type Ia rates across M31's ring. We used ISM densities of the neutral atomic gas (traced by 21 cm {\\sc H\\,i}\\xspace line maps) and molecular gas (traced by $^{12}$CO(1-0) line maps) alongside the SN rates to predict the steady-state mass-weighted turbulent velocity dispersion, using the feedback prescriptions of \\cite{Martizzi2015} and \\cite{Martizzi2016}. We compared these model estimates against the scaled turbulent velocity dispersion obtained from {\\sc H\\,i}\\xspace and CO maps of M31. We assumed all SNe explode in the galaxy midplane where the line-of-sight density is highest, effectively providing a lower-limit on the predicted velocity dispersion. We summarize the following key results from our work :-\n\n\\begin{enumerate}\n\\item We find an integrated rate of $\\approx 1.7 \\times 10^{-3}$ SN yr$^{-1}$ in the ring covered by PHAT, with 61$\\%$ contribution from core-collapse SNe. Regions with dominant core-collapse contribution coincide with known star-forming regions as expected, while regions with dominant Type Ia contribution fall near the edges of the ring.\n\n\\item We found that the minimum predicted velocity dispersion exceed observed values in 84$\\%$ of the ring covered by PHAT for our fiducial model within a factor of 2. Some of the discrepancy can be explained by varying the assumptions regarding SADs and ISM\/SN scale heights within plausible limits, but for densities $\\lesssim 0.2$ cm$^{-3}$ and SN rates $>2.1 \\times 10^{-4}$ SN yr$^{-1}$ kpc$^{-2}$\\xspace, the discrepancy appears to increase.\n\\item SAD cells with SN rates $>2.1 \\times 10^{-4}$ SN yr$^{-1}$ kpc$^{-2}$\\xspace where velocity dispersion is over-predicted are spatially correlated with dense concentration of young clusters embedded in a bright thermal X-ray region. This supports the possibility of clustering of SNe in this regime, which is not captured in our momentum feedback model. Clustering of SNe can lower the momentum deposited per SN and mass-weighted turbulence in the ISM as a result of converging shocks from adjacent explosions, mass-loaded outflows, or higher mass fraction in warmer ISM phases due to cumulative action of stellar winds and SNe.\n\n\\item The low density ($\\lesssim 0.2$ cm$^{-3}$) regions where velocity dispersion is over-predicted coincide with the edges of our ring region where Type Ia SNe dominate the injection rate by nearly a factor of 2. However given the overall low SN rate in these regions, it is likely that the discrepancy could be due to isolated SNRs coming into pressure equilibrium with the ISM before significant amount of cooling and momentum deposition takes place---another effect not included in our models.\n\\end{enumerate}\nOur results provide observational support for including adjustments in fiducial subgrid models of momentum feedback, to account for SNR evolution in clustered and in low-density environments. The work underscores the importance of resolved stellar photometry and cloud-scale atomic and molecular ISM observations for assessing feedback models and ISM turbulence. Newer, more sensitive observations at high spectral resolution, such as the VLA maps of \\cite{Koch2021} can provide more detailed characterization of turbulence in atomic clouds in M31. Preliminary comparison have shown that the 2nd moment line-widths of the VLA maps are within a factor of 2 of \\cite{Braun2009} non-thermal values, although the former do not yet cover the full PHAT area or the M31 ring. We will expand our present analysis in future papers with data from the ongoing Local Group L-Band Survey\\footnote{\\url{https:\/\/www.lglbs.org\/home}} which will cover all of M31, as well as M33 and four Local Group dwarfs, providing {\\sc H\\,i}\\xspace maps of unprecedented sensitivity at a wide range of spatial resolution. This extension would allow us to test feedback models with spatially resolved observations across a wide range of star-forming conditions, and empirically obtain corrections to the fiducial models to be included in cosmological simulations.\n\n\\acknowledgements\nS.K.S is grateful to Robert Braun for sharing the WSRT 21 cm maps of {\\sc H\\,i}\\xspace density and non-thermal velocity dispersion in M31. SKS and LC are grateful for support from NSF grants AST-1412549, AST-1412980 and AST-1907790. Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. We acknowledge support from the Packard Foundation.\nE.R-R is supported by the  Heising-Simons Foundation and  the Danish National Research Foundation (DNRF132).\n\n\\software{numpy \\citep{numpy}, scipy \\citep{scipy}, matplotlib \\citep{matplotlib}, astropy \\citep{astropy1, astropy2}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Quark Model}\n\nThe quark model describes mesons and baryons in terms of constituent quarks.\nMesons are $q\\bar q$ states and baryons are $qqq$, while all the other combinations of\nquarks, such as $qq\\bar q\\bar q$, $qqqq\\bar q$, are called exotic.\nThe constituent quarks must be effective degrees of freedom which are valid only \nin low energy regime. \nThey must have the same conserved charges as the QCD quarks: \nbaryon number 1\/3, spin 1\/2, color 3 and flavor 3.\nThey acquire dynamical masses of order $\\sim 300-500$ MeV\ninduced by chiral symmetry breaking of the QCD vacuum.\n\nThe ground-state mesons and baryons have been classified \nbased on $SU(3)\\times SU(2) \\to SU(6)$ symmetry, i.e., \nthe pseudoscalar and vector meson nonets, and\nthe octet and decuplet baryons.\nTheir mass spectra, in particular, the pattern of SU(3) breaking,\nand the electro-magnetic properties\nare well reproduced by minimum dynamics of quarks,\nquark confinement and color-magnetic type interactions,\nexcept for pseudoscalar mesons, i.e., the octet pseudoscalars, $\\pi, K, \\eta$,\nare very light according to chiral symmetry breaking,\nwhile the $\\eta'$ mass is large due to the $U_A(1)$ anomaly.\n\nThe quark model based on $SU(6) \\times O(3)$ symmetry, however, encounters some difficulties when it is applied to\nexcited hadron states.\nAn example is the scalar meson nonet; ($\\sigma(600), a_0(980), f_0(980), K_0^*(800)$).\n($K_0^*$ has been indicated in $K\\pi$ final states in $J\/\\psi$ and $D$ meson decays, \nbut not yet established\\cite{kappa}.)\nTheir mass ordering as $q\\bar q$ states is expected to be like\n$m(\\sigma) \\sim m(a_0) < m(f_0)$, assuming the ideal mixing, i.e.,\n\\begin{eqnarray}\n&& \\sigma\\sim \\frac{u\\bar u + d\\bar d}{\\sqrt{2}} , \\qquad\na_0\\sim \\frac{u\\bar u - d\\bar d}{\\sqrt{2}} , \\qquad\nf_0 \\sim s\\bar s \\nonumber\n\\end{eqnarray}\nFurthermore, while they are classified as $^3P_0$ states, \ntheir spin-orbit partners $J= 1$ and 2 states are not observed in their vicinity.\n\nA possible solution of the difficulty is to consider four-quark exotic states for the scalar mesons\\cite{4q}.\nSuppose that diquarks with flavor 3, color 3 and spin 0, i.e.,\n\\begin{eqnarray}\n&& U=(\\bar d\\bar s)_{S=0,C=3,f=3}\\qquad D=(\\bar s\\bar u)_{S=0,C=3,f=3}\n\\qquad S=(\\bar u\\bar d)_{S=0,C=3,f=3} . \n\\label{diquark}\n\\end{eqnarray}\nare building blocks of the scalar mesons.\nThen the scalar nonets in the ideal mixing may appear as\n\\begin{eqnarray}\n&& \\sigma\\sim S\\bar S \\sim (ud)(\\bar u\\bar d) \\nonumber\\\\\n&& a_0\\sim \\frac{1}{\\sqrt{2}} (U\\bar U - D\\bar D) \\sim \\frac{1}{\\sqrt{2}} ((ds)(\\bar d \\bar s)-(su)(\\bar s\\bar u))\n\\nonumber\\\\\n&& f_0 \\sim \\frac{1}{\\sqrt{2}} (U\\bar U + D\\bar D) \\sim \\frac{1}{\\sqrt{2}} ((ds)(\\bar d \\bar s)+(su)(\\bar s\\bar u))\n\\nonumber\n\\end{eqnarray}\nThen one sees that the strange quark counting predicts the observed mass pattern,\n$m(\\sigma) < m(a_0) \\sim m(f_0)$.\nIt also explains that the $J=0$ state is isolated without $J=1, 2$ partners.\n\nThus one sees that multi-quark components may help to explain anomalies in the scalar meson \nnonets.  There are other hadrons which are suspected to contain exotic multi-quark components, such as \n$D_s^*$, $X(3872)$ and $\\Lambda(1405)$, \nmainly because they do not fit well in the spectra of the ordinary mesons or baryons.\n\nThe next question is whether the QCD dynamics allows such states?\nWe actually have a simple reason,  for instance, why $\\Lambda (1405)$ \nis possible to be a 5-quark state.\n$\\Lambda(1405)$ is a $J^{\\pi} = 1\/2^-$ flavor singlet baryon.\nThe three quark ($uds$) configuration has to contain orbital excitation $L=1$ with spin 1\/2,\nleading to $J=1\/2^-$ and $3\/2^-$ states. \nThen the candidate of the $3\/2^-$ partner is $\\Lambda(1520)$, but the spin-orbit splitting is\nunusually large compared to the nonstrange baryons with $L=1$.\nOn the other hand, the 5-quark content, $udsu\\bar u+udsd\\bar d$ may be \nrealized in $L=0$ without orbital excitation,\nand thus has advantage over the $L=1$ excited states.\nIn terms of the diquark language, one may assign the $L=0$ and $S=1\/2$ configuration.\nThis gives an isolated $J=1\/2$ state and has no difficulty of large LS splitting.\nTherefore it is quite interesting and important to answer whether realistic dynamical calculation\nindeed gives multi-quark states a lower energy.\n\n\\section{Pentaquark $\\Theta^+$}\nWe first look at the status of the pentaquark $\\Theta^+$ in the quark picture.\nThe pentaquark $\\Theta^+$ is a baryon with strangeness $+1$, whose minimal quark content\nis $uudd\\bar s$.  It may be affiliated to a member of flavor antidecuplet\\cite{Theta-quark}.\nHow does the known dynamics of the quark model work for $\\Theta^+$?\nIt has been clarified that the ``standard'' quark model, which explains the ground state mesons and baryons very well, does not easily yield the small mass and the tiny width of $\\Theta^+$ simultaneously.\nRecently variational techniques for solving multi-body bound states have been applied to the 5-quark systems. The results\\cite{Theta-variational} show that \nthe typical masses of spin $1\/2^{\\pm}$ and $3\/2^{\\pm}$ pentaquarks are\nmore than 500 MeV above the threshold of $NK$, and thus the masses are 1.9 GeV or higher.\nFurthermore, if they have such large masses, various decay channels are open and their widths\nmust be very large.  It was pointed out that even at the mass 1.54 GeV, the decay widths of\n$1\/2^-$ 5-quark states\\cite{Hosaka-width}, \nwhich decay into S wave $NK$  states, must be a few hundred MeV or larger.\n \nIn order to bring such high-mass resonances down to the observed mass\\cite{Nakano}, \none may require a very strong correlation. \nDiquark correlation is a possibility\\cite{Theta-diquark}. \nMost dynamical models of quarks, such as one-gluon exchange, instanton induced interaction, and so on,\ngive attraction to color $\\bar 3$, flavor $\\bar 3$, spin 0 diquarks (Eq.(\\ref{diquark})).\nIt is, however, noted that the same interaction causes (often stronger) attraction to $q \\bar q$ color singlet, flavor 8, spin 0 system (i.e., pseudoscalar mesons). Then the ground state of the pentaquark may well be \na state of a baryon and a meson far separated.\nThus, the di-quark scenario may not work unless the spin (or other quantum numbers) is chosen so that\nit hinders a pseudoscalar subsystem. One such case is that the spin of $\\Theta^+$ is \n3\/2\\cite{Hosaka-width,spin3_2}.\n\nEven if the observed $\\Theta^+$ might not be what we expected originally,\nthe techniques developed in the study of the pentaquark $\\Theta^+$ are useful\nin studying other exotic multiquark resonances as well as multiquark components of ordinary hadrons.\nIn particular, the variational method is powerful and useful to judge \nwhether a certain quark model allows multiquark hadrons as a ground state.\n\n\n\\section{Pentaquarks in Lattice QCD}\n\nAs the model calculations have various ambiguity in treating multiquark systems with color singlet sub-systems,\ndirect applications of QCD to exotics are desperately needed.\nIn fact, QCD does not a priori exclude exotic multi-quark bound (resonance) states as far as they are color-singlet.\nWe here show some of the results from the lattice QCD simulations and the approach using QCD sum rules.\n\nLattice QCD is powerful in understanding non-perturbative physics of QCD: \nvacuum structure, phase diagrams, mass spectra of ground state mesons and baryons, interactions of quarks, . . . \nHowever, at this moment the lattice QCD has two rather severe restrictions:\n(1) Light quarks are too expensive. The simulations are made at a large quark mass and require (often drastic) extrapolation to physical quark masses for $u$ and $d$ quarks.\n(2) No direct access to resonance poles is possible. \nIt is hard to distinguish resonances from hadron scattering states. Real number simulations can not access complex poles.\nThus applications to exotic hadrons are yet limited.\n\nWe  have developed a new method, called hybrid boundary condition (HBC) method, to extract the 5-quark resonances out of the meson-baryon background\\cite{Ishii}.\nThe basic idea of HBC is to apply anti-periodic boundary conditions to certain quarks,\nwhich effectively raise the energy of the lowest-energy meson-baryon scattering state.\nComparing the results with the standard boundary condition with the hybrid one, one can determine\nwhether a state seen in the lattice simulation is a compact resonance state or a hadronic continuum state.\nThis technique has been applied to the pentaquark $\\Theta^+$ first and later to \n$\\Lambda(1405)$ and the other possible exotics.\n\nThe results for $\\Theta^+$ pentaquark are summarized as follows.\\\\\n(1) The negative parity $1\/2^-$ state appears at $m\\sim 1.75$ GeV,\nwhich seems consistent with $NK$ ($L=0$) scattering state on the lattice.\nThe HBC analysis confirms that the observed state is not a compact 5-quark state.\\\\\n(2) The positive parity $1\/2^+$ state appears at 2.25 GeV, which is too heavy for $\\Theta^+$.\\\\\n(3) The $J =3\/2$ states are generated by three different operators: di-quark type, \n$NK^*$, and color-twisted $NK^* $ operators.\nThe results show that the mass of $3\/2^-$ state is around 2.11 GeV, \nthat is consistent with $NK^*$ ($L=0$) threshold, while the positive parity $3\/2^+$ is around 2.42 GeV, consistent with $NK^*$ ($L=1$) threshold.\n\nThus no candidate for compact 5-quark state is found. \nMost lattice QCD parties agree with these results with a few exceptions.\nThe QCD sum rules also give the consistent results.\n\nA similar method has been applied to study the penta-quark nature of\nthe flavor singlet $\\Lambda (1\/2^-)$ state\\cite{LQCD-Lambda}.  \nIn this case, one may twist the spatial boundary condition \nonly of quarks in the quenched approximation,\nwhile antiquarks satisfy the standard boundary condition.\nThis ``HBC'' will enable us to isolate $\\Lambda^*$ resonance \nfrom $N\\bar K$ and $\\Sigma\\pi$ scattering states. \nThe results are satisfactory, showing that the 5-quark operator \ngives $m_{5Q}$ = 1.63(7) GeV, which is around the $m(N)+m(K)$ threshold \non the current choice of the lattice parameters.\nIt is further interesting that the HBC analysis indicates that this 5-quark state \nseems a compact resonance and thus it is a strong candidate for $\\Lambda(1405)$.\nIn contrast, the 3-quark operator gives a higher mass, $m_{3Q}$ = 1.79(8) GeV.\n\n\n\\section{Exotic Multi-quark States in QCD Sum Rule}\n\nIn nature, it is expected that exotic multi-quark components mix with the standard Fock\nstate in hadrons. In such mixed states, one may ask how large the mixing probability is\nof the exotic multi-quark components.\n\nIt turns out that such a mixing is not easily quantified.\nIt would be natural to consider the strengths of the couplings of \nthe 3-quark and 5-quark \noperators to the physical state and then evaluate the mixing angle.\nHowever, such a procedure is largely dependent on the definition\nand normalization of the local operators.  Indeed, a numerical factor\ncan be easily hidden in the local operators and thus the magnitudes\nof the coupling strengths are ambiguous.\n\nThis problem happens to be more fundamental than just \nthe definition of the operators. \nIn the field theory one may not be able to ``measure'' the number of quarks without\nambiguity because no conserved charge corresponding the number\nof quarks, $N(q)+N(\\bar q)$, is available.\nNamely, the Fock space separation may not be unique.\nThus we have to consider the concept of the ``number of quarks''\nin the context of the quantum mechanical interpretation of the \nfield theoretical state.\n\nIn recent study\\cite{SINNO}, we propose two ways to ``define'' the ratio of the Fock space\nprobability.\nIn the first approach, we define local operators\nin the context of a 5-quark operator $J_5$.\nAs the 5-quark operator contains $qqq$ component, \none can write the operator into $J_5= \\tilde J_5 + \\tilde J_3$.\nThen the mixing parameter may be defined as the ratio of the couplings \nto $\\tilde J_5$ and $\\tilde J_3$.\nThis mixing angle can be evaluated from the correlation functions\nwith an assumption that the poles are at the same position.\nThe result is model independent, but it depends on the choice of the\noperators.\nTherefore it does not necessarily have a direct relation to the mixing parameters\nemployed in the quark models.\n\nIn order to define a mixing angle more appropriate to the quark models, \none must determine the normalization of the operators using a quark model\nwave function.\nOne may use the MIT bag model wave functions for the normalization,\nassuming that the bag model states (with define number of quarks) are normalized\nproperly.\n\nThe above two methods have been applied to the $a_0$ scalar meson and we have\nfound that the physical state for $a_0$ is indeed given by the mixings of \n$q\\bar q$ and $qq\\bar q\\bar q$ states.\nIt turns out that the 4-quark mixing probability is about 90\\%, which does not \nstrongly depend on the choice of the definition.\n\nIn contrast, for the flavor singlet $\\Lambda^* (1\/2^-)$ state, we find that the 3-quark operator gives \na higher mass and thus the lowest energy state is predominantly 5-quark state.  In such a case,\nthe mixing is not properly defined in our method.\n\n\\section{Conclusion}\n\nA technique using the hybrid boundary conditions seems to work in lattice QCD to distinguish compact states from scattering states.\nThe quenched lattice QCD suggests that $\\Lambda(1405)$ is predominantly a 5-quark state.\nThe mixing of the multi-quark components is not quantified from the field theoretical viewpoint.\nHowever, one can define a set of useful mixing amplitudes using the well-defined matrix elements of local operators. Whether this definition of the mixing is relevant in the quark model is yet an open problem. \nThe QCD sum rule indicates a large 4-quark components in scalar mesons. \n\n\\bigskip\nThe contents of this paper come from various collaborations.\nI acknowledge Drs. N. Ishii, H. Suganuma, T. Doi, Y. Nemoto, H. Iida, F. Okiharu, S. Takeuchi, A. Hosaka,\nT. Shinozaki, T. Nishikawa, J. Sugiyama,  and T. Nakamura for productive collaborations and discussions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe latest observations from the Planck satellite \n\\cite{planck1,planck2,planck3} confirm the\nvanilla predictions of cosmic inflation for the primordial curvature\nperturbation in that the latter is predominantly Gaussian (non-Gaussianities \nhave not been observed, with upper bound $f_{\\rm NL}^{\\rm local}=0.9\\pm 5.7$), \nadiabatic (no isocurvature contribution has been observed, with upper bound to \nless that 3\\%), statistically isotropic (no statistical anisotropy has been \nobserved, with upper bound to less than 2\\%) and almost scale-invariant, but \nwith a significant red tilt ($n_s=0.968\\pm0.006$). Moreover, the Planck data \nfavour canonically normalised, single-field, slow-roll inflation \\cite{planck1}.\nIn fact, in conjunction with other data, Planck seems to favour an inflationary\nplateau \\cite{bestinf}. \n\nThere have been many examples of such inflationary\nmodes, such as the original $R^2$-inflation \\cite{staro}, Higgs inflation \n\\cite{higgs} or T-model inflation \\cite{Tmodel}. However, most of these attempts\nconsider an exponential approach to the inflationary plateau. Here we design a \nmodel, which approaches the inflationary plateau in a power-law manner, \noffering distinct observational signatures.\n\n\\section{Bottom-up versus top-down approach}\n\nIn inflationary model-building one can identify two broad strategies.\nThe top-down scenario corresponds to designing models based on ``realistic'' \nconstructions, for example inspired by string theory, supergravity etc. Then,\none looks for specific signatures in the data (e.g. non-Gaussianity). Since the\nlatest Planck data favour single-field, slow-roll inflation, they seem to \nsupport such relatively straightforward constructions. \n\nIn contrast, the bottom-up scenario amounts to inflationary model constructions,\nwhich are ``suggested'' by the data, i.e. they are data-inspire \n``guess-stimates''. As such, this approach uses the Early Universe as \nlaboratory to investigate fundamental physics, in the best tradition of \nparticle cosmology. We adopt this strategy (see also Ref.~\\cite{shaft}). \nOur model proposes a power-law approach to the inflationary plateau in the \ncontext of global supersymmetry.\n\n\\section{The scalar potential for Shaft Inflation}\n\nConsider a toy-model superpotential of the form: \n\\mbox{$W=M^2\\Phi^{nq+1}\/(\\Phi^n+m^n)^q$}, where $n,q$ are real numbers and\n$M,m$ are mass-scales. For \\mbox{$|\\Phi|\\gg m$}, this superpotential approaches\nan O' Raifearteagh form \\mbox{$W\\simeq M^2\\Phi$} leading to de~Sitter inflation.\nFor \\mbox{$|\\Phi|\\ll m$}, the superpotential becomes \\mbox{$W\\propto\\Phi^{nq+1}$}\nleading to chaotic inflation. To simplify it even further, we choose to\neliminate the numerator, and take \\mbox{$q=-1\/n$}. We end up with the \nsuperpotential for Shaft Inflation \\cite{shaft}:\n\\begin{equation}\nW=M^2\\left(\\Phi^n+m^n\\right)^{1\/n}.\n\\label{W}\n\\end{equation}\nTo obtain the scalar potential, we consider \\mbox{$\\Phi=\\phi e^{i\\theta}$},\nwhere $\\phi,\\theta$ are real scalar fields with $\\phi>0$.%\n\\footnote{A normalisation factor of \n$1\/\\sqrt 2$ has been absorbed in the mass scales.}\nThen the scalar potential is:\n\\begin{equation}\nV=M^4|\\Phi|^{2(n-1)}|\\Phi^n+m^n|^{2(\\frac{1}{n}-1)}=\n\\frac{M^4\\phi^{2(n-1)}}{[\\phi^{2n}+m^{2n}+2\\cos(n\\theta)m^n\\phi^n]^{\\frac{n-1}{n}}}.\n\\end{equation}\nThe potential is minimised when \\mbox{$n\\theta=2\\ell\\pi$}, with $\\ell$ being an\ninteger. Further, noting that \\mbox{$-\\phi=\\phi e^{i\\pi}$}, we can make the\npotential symmetric over the origin [\\mbox{$V(\\phi)=V(-\\phi)$}] if \n\\mbox{$n=2\\ell$}, i.e. even. In this case,\n\\begin{equation}\nV(\\phi)=M^4\\phi^{2n-2}(\\phi^n+m^n)^{\\frac{2}{n}-2},\n\\label{V}\n\\end{equation}\nfor all real values of $\\phi$.\nFrom the above we see that the scalar potential has the desired behaviour,\nfor \\mbox{$n\\geq 2$}, i.e. it approaches a constant \\mbox{$V\\approx M^4$}\nfor \\mbox{$\\phi\\gg m$}, while for \\mbox{$\\phi\\ll m$} the potential becomes \nmonomial, with \\mbox{$V\\propto\\phi^{2(n-1)}$}, see Fig.~\\ref{fig0}. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.6\\textwidth]{shaftinffig1.eps}\n\\caption{%\nThe scalar potential in shaft inflation for \\mbox{$n=2,4,8$} and $16$. The \nshaft becomes sharper as $n$ grows. Far from the origin the potential \napproximates the inflationary plateau with \\mbox{$V\\approx M^4$}. Near the \norigin the potential becomes monomial, as in chaotic inflation. \n}\n\\label{fig0}\n\\end{center}\n\\end{figure}\n\n\\vspace{-1cm}\n\n\\section{The spectral index and the tensor to scalar ratio}\n\n\\subsection{\\boldmath Slow-roll parameters, $n_s$ and $r$}\n\nFrom Eq.~(\\ref{V}), we readily obtain the slow-roll parameters as\n\\begin{eqnarray}\n& & \\epsilon\\equiv\\frac12 m_P^2\\left(\\frac{V'}{V}\\right)^2=\n2(n-1)^2\\left(\\frac{m_P}{\\phi}\\right)^2\\left(\\frac{m^n}{\\phi^n+m^n}\\right)^2\n\\label{eps}\\\\\n& & \\eta\\equiv m_P^2\\frac{V''}{V}= \n2(n-1)\\left(\\frac{m_P}{\\phi}\\right)^2\\left(\\frac{m^n}{\\phi^n+m^n}\\right)\n\\frac{(2n-3)m^n-(n+1)\\phi^n}{\\phi^n+m^m},\n\\label{eta}\n\\end{eqnarray}\nwhere the prime denotes derivative with respect to the inflaton field and\n$m_P=2.4\\times 10^{18}\\,$GeV is the reduced Planck mass. Hence, \nthe spectral index of the curvature perturbation is\n\\begin{equation}\nn_s=1+2\\eta-6\\epsilon=1-4(n-1)\\left(\\frac{m_P}{\\phi}\\right)^2\n\\frac{m^n[(n+1)\\phi^n+nm^n]}{(\\phi^n+m^n)^2}.\n\\label{ns}\n\\end{equation}\nTo rewrite the above as functions of the remaining e-folds of inflation $N$ \nwe have to investigate the end of inflation.\nIt is straightforward to see that inflation is terminated when \n\\mbox{$|\\eta|\\simeq 1$} so that, for the end of inflation, we find\n\\begin{equation}\n\\phi_{\\rm end}\\simeq m_P\\left[2(n^2-1)\\alpha^n\\right]^{1\/(n+2)},\n\\label{fend}\n\\end{equation}\nwhere we assumed that \\mbox{$\\phi>m$} (so that the potential deviates\nfrom a chaotic monomial) and we defined\n\\begin{equation}\n\\alpha\\equiv\\frac{m}{m_P}\\,.\n\\label{alpha}\n\\end{equation}\nUsing this, we obtain $\\phi(N)$ \n\\begin{eqnarray}\nN=\\frac{1}{m_P^2}\\int_{\\phi_{\\rm end}}^\\phi\\frac{V}{V'}{\\rm d}\\phi & \\simeq &\n\\frac{1}{2(n-1)(n+2)\\alpha^n}\\left[\\left(\\frac{\\phi}{m_P}\\right)^{n+2}-\n\\left(\\frac{\\phi_{\\rm end}}{m_P}\\right)^{n+2}\\right]\n\\label{N}\\\\\n& \\Rightarrow & \\phi(N)\\simeq m_P\\left[2(n-1)(n+2)\\alpha^n\n\\left(N+\\frac{n+1}{n+2}\\right)\\right]^{1\/(n+2)}.\n\\label{fN}\n\\end{eqnarray}\nInserting the above into Eqs.~(\\ref{eps}) and (\\ref{ns})\nrespectively we obtain the tensor to scalar ratio $r$ and the spectral index \n$n_s$ as functions of $N$:\n\\begin{eqnarray}\n& & r=16\\epsilon=32(n-1)^2\\alpha^{\\frac{2n}{n+2}}\\left[2(n-1)(n+2)\n\\left(N+\\frac{n+1}{n+2}\\right)\\right]^{-2(\\frac{n+1}{n+2})}\n\\label{r}\\\\\n& & n_s=1-2\\,\\frac{n+1}{n+2}\\left(N+\\frac{n+1}{n+2}\\right)^{-1}.\n\\label{nsN}\n\\end{eqnarray}\nNotice that only $r$ is dependent on $m$ (through $\\alpha$), which means that\n$r$ can be affected by changing $m$ without disturbing $n_s$. We will return to\nthis possibility later.\n\n\\subsection{Examples}\n\nTo investigate the performance of the model, we consider the two extreme cases\nfor the values of $n$, namely \\mbox{$n=2$} and \\mbox{$n\\gg 1$}. For \nillustrative purposes we take \\mbox{$\\alpha=1$}, i.e. \\mbox{$m=m_P$}.\n\n\\subsubsection{\\boldmath $n=2$}\n\nIn this case the scalar potential becomes\n\\begin{equation}\nV(\\phi)=M^4\\frac{\\phi^2}{\\phi^2+m^2}.\n\\label{Vquad}\n\\end{equation}\nWe see that the above can be thought of as a modification of quadratic \nchaotic inflation, because after the end of inflation, the inflaton field \noscillates in a quadratic potential. However, for large values of the inflaton \nthe potential approaches a constant.\nThis potential has been obtained also in S-dual superstring inflation \n\\cite{Sdualinf} with \\mbox{$\\alpha=1\/4$} and also in radion assisted gauge \ninflation \\cite{RAGI} with \\mbox{$\\alpha\\sim 10^{-3\/2}$}. In this case,\nEqs.~(\\ref{r}) and (\\ref{nsN}) become\n\\begin{equation}\nr=\\frac{32\\alpha}{\\left[8\\left(N+\\frac34\\right)\\right]^{3\/2}}\n\\qquad{\\rm and}\\qquad\nn_s=1-\\frac32\\left(N+\\frac34\\right)^{-1}.\n\\end{equation}\nFrom the above, we find the values for $n_s$ and $r$, as shown in \nTable~\\ref{tab1}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\\hline\n$N$ & $n_s$ & $r$\\\\\\hline\\hline\n50 & 0.970 & 0.0039\\\\\n60 & 0.975 & 0.0030\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Values of $n_s(N)$ and $r$ in the case $n=2$.}\n\\label{tab1}\n\\end{table}\n\n\\subsubsection{\\boldmath $n\\gg 1$}\n\nIn the opposite extreme \\mbox{$n\\gg 1$}, Eqs.~(\\ref{r}) and (\\ref{nsN}) become\n\\begin{equation}\nr=\\frac{8\\alpha^2}{n^2(N+1)^2}\\rightarrow 0\n\\quad{\\rm and}\\quad\nn_s=1-\\frac{2}{N+1}\\,.\n\\label{nbig}\n\\end{equation}\nThe spectral index is now the same as in the original $R^2$~inflation model\n\\cite{staro} (also in Higgs inflation \\cite{higgs}), which is not surprising \nsince we expect power-law behaviour to approach the exponential when \n\\mbox{$n\\rightarrow\\infty$}. The values of $n_s$, in this case, are shown in \nTable~\\ref{tab2}.\n\n\\begin{figure}\n\\begin{center}\n\\vspace{-12cm}\n\n\\mbox{\\hspace{-3.4cm}\n\\includegraphics[width=1.4\\textwidth]{shafttalk1.ps}}\n\\vspace{-9.5cm}\n\n\\caption{%\nShaft inflation for \\mbox{$n=2$} is depicted with the large \\{small\\} red cross\nfor \\mbox{$N\\simeq 60$} \\{\\mbox{$N\\simeq 50$}\\}. Shaft inflation for \n\\mbox{$n\\gg 1$} is depicted with the large \\{small\\} black cross for \n\\mbox{$N\\simeq 60$} \\{\\mbox{$N\\simeq 50$}\\}.\nIntermediate values of $n$ lie in-between the depicted points.\nAs evident, there is excellent agreement with the Planck observations.%\n}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|}\\hline\n$N$ & $n_s$ \\\\\\hline\\hline\n50 & 0.961 \\\\\n60 & 0.967 \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Values of $n_s(N)$ in the case $n\\gg 1$ (where $r\\approx 0$).}\n\\label{tab2}\n\\end{table}\nFrom the above, we find that the values for $n_s$ and $r$ are very close to the\nbest fit point for the Planck data for all values of $n$, \nas shown in Fig.~\\ref{fig1}.\\footnote{%\nThe crosses are superimposed to the original Planck paper image, taken from\nRef.~\\cite{planck1}, which includes also the original caption.}\n\n\n\n\n\\section{Gravitational waves}\n\nPlanck observations, in conjunction with BICEP2 and Keck Array data suggest\n\\mbox{$r\\leq 0.1$} \\cite{PKB}. As we have seen, in Shaft Inflation, \n\\mbox{$r\\propto\\alpha^{2n\/(n+2)}$}, while there is no $\\alpha$-dependence of \n$n_s$. Thus, by changing $m$, $r$ can vary without affecting the spectral index \n(c.f. Eq.~(\\ref{alpha})). Therefore, sizeable tensors can be attained by \nwidening the shaft in field space. Indeed, rendering $m$ mildly super-Planckian\ncan produce potentially observable values of $r$ as shown in Table~\\ref{table3},\nwhere Eq.~(\\ref{alpha}) suggests that \n\\mbox{$m=\\alpha\\,m_P=\\frac{\\alpha}{\\sqrt{8\\pi}}M_P$}, with \n$M_P=1.2\\times 10^{19}\\,$GeV being the Planck mass. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|}\\hline\n$n$ & $n_s$ & $r\\;(\\alpha=1)$ \n& $r\\;(\\alpha=2\\sqrt{8\\pi}\\approx 10)$ \n& $r\\;(\\alpha=5\\sqrt{8\\pi}\\approx 25)$ \\\\\\hline\\hline\n2 & 0.975 & 0.0030 & 0.0299 & 0.0747\\\\\n4 & 0.973 & 0.0008 & 0.0168 & 0.0570\\\\\n6 & 0.971 & 0.0003 & 0.0089 & 0.0352\\\\\n8 & 0.970 & 0.0001 & 0.0052 & 0.0227\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Values of $n_s$ and $r$ for $N=60$ and $n=2,4,6,8$. Three choices of \n$\\alpha=m\/m_P$ are depicted, which correspond to $m=m_P$, $m=2M_P$ and $m=5M_P$,\nwhere \\mbox{$M_P=\\sqrt{8\\pi}\\,m_P$}.\nIt is shown that, with $m$ mildly super-Planckian, $r$ can approach the \nobservational bound $r<0.1$ without affecting $n_s$.}\n\\label{table3}\n\\end{table}\nThus, we see that with \n$m\\simeq 5\\,M_P$ we can have $r\\simeq 0.07$, which is on the verge of \nobservability. This is shown clearly in Fig,~\\ref{fig2}.\n\n\n\\begin{figure}\n\\begin{center}\n\\vspace{-4cm}\n\\includegraphics[width=.9\\textwidth]{shafttalk2.ps}\n\\vspace{-6cm}\n\\caption{%\nShaft inflation predictions for \\mbox{$N=60$}. The crosses in the image \ncorrespond to \\mbox{$n=2,4,6,8$} as depicted from right to left. \nBlack crosses correspond to \\mbox{$\\alpha=1$} (\\mbox{$m=m_P$}),\nred crosses correspond to \\mbox{$\\alpha=2\\sqrt{8\\pi}\\approx 10$} \n(\\mbox{$m=2M_P$}) and yellow crosses correspond to \n\\mbox{$\\alpha=5\\sqrt{8\\pi}\\approx 25$} (\\mbox{$m=5M_P$}).\nIt is evident that, for mildly super-Planckian values of $m$ the model \npredictions lie at the verge of observability.}\n\\label{fig2}\n\\vspace{-1cm}\n\\end{center}\n\\end{figure}\n\n\\section{More on Shaft Inflation}\n\nThe running of the spectral index is easily obtained as\n\\begin{equation}\n\\frac{{\\rm d}n_s}{{\\rm d}\\ln k}=\n-\\frac{2\\left(\\frac{n+1}{n+2}\\right)}{\\left(N+\\frac{n+1}{n+2}\\right)^2}.\n\\end{equation}\nIn the two extreme cases, this gives\n\\begin{eqnarray}\nn=2: & & \n\\frac{{\\rm d}n_s}{{\\rm d}\\ln k}=-\\frac{3}{2\\left(N+\\frac34\\right)^2}=\n-4.064\\times 10^{-4}\\\\\nn\\gg 1: & &\n\\frac{{\\rm d}n_s}{{\\rm d}\\ln k}=-\\frac{2}{\\left(N+1\\right)^2}=\n-5.375\\times 10^{-4} \n\\end{eqnarray}\nwhere the numerical values correspond to $N=60$. Thus, for all values of $n$,\nwe the above suggests:\n\\mbox{$\\frac{{\\rm d}n_s}{{\\rm d}\\ln k}\\approx-(4-5)\\times 10^{-4}$},\nwhich is in agreement with the Planck findings:\n\\mbox{$\\frac{{\\rm d}n_s}{{\\rm d}\\ln k}=-0.003\\pm0.007$}.\n\nFinally, the inflationary scale is determined by the COBE constraint\n\\begin{equation}\n\\sqrt{{\\cal P}_\\zeta}=\\frac{1}{2\\sqrt 3\\pi}\\frac{V^{3\/2}}{m_P^3|V'|},\n\\end{equation}\nwhere \\mbox{${\\cal P}_\\zeta=(2.208\\pm 0.075)\\times 10^{-9}$} is the spectrum of\nthe curvature perturbation \\cite{planck2}. This provides an estimate for the \nrequired value of $M$\n\\begin{equation}\n\\left(\\frac{M}{m_P}\\right)^2=4\\sqrt 3(n-1)\\alpha^{-\\frac{n}{n+2}}\n\\pi\\sqrt{{\\cal P}_\\zeta}\\left[2(n\\!-\\!1)(n\\!+\\!2)\\!\\left(N+\\frac{n+1}{n+2}\\right)\n\\right]^{-\\frac{n+1}{n+2}}.\n\\label{M}\n\\end{equation}\nFor illustrative purposes, using \n\\mbox{$\\sqrt{{\\cal P}_\\zeta}\\simeq 4.7\\times 10^{-5}$}, \\mbox{$n=2$}, \n\\mbox{$\\alpha=1$} (i.e. \\mbox{$m=m_P$}) and \\mbox{$N=60$} we find\n\\mbox{$M=7.7\\times 10^{15}\\,$GeV}, which is very near the scale of grand \nunification, as expected.\n\n\\section{Conclusions}\n\nPlanck data favour single-field, slow-roll inflation, characterised by a \nscalar potential which approaches an inflationary plateau. In contrast to many \nother successful models, Shaft Inflation approaches this plateau in a power-law\nmanner. Shaft Inflation is based on a simple superpotential: \n\\mbox{$W=M^2\\left(\\Phi^n+m^n\\right)^{1\/n}$}. Without any fine tuning \n(\\mbox{$m\\sim m_P$} and \\mbox{$M\\sim 10^{16}\\,$}GeV, i.e. the scale of grand \nunification) Shaft Inflation produces a scalar spectral index very close to the\nPlanck sweet spot with very small (negative) running, in agreement with Planck.\nRendering $m$ mildly super-Planckian one can easily obtain potentially \nobservable tensors without affecting the spectral index. The challenge in now \nto obtain realistic setups which can realise the (deceptively) simple Shaft \nInflation superpotential.\n\n\\section*{Acknowledgements}\nKD is supported (in part) by the Lancaster-Manchester-Sheffield Consortium for \nFundamental Physics under STFC grant ST\/L000520\/1.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMore than 100 cases of strong gravitational lensing are now known in which quasars are\nmultiply lensed by foreground galaxies, about the same quantity as the\nnumber of galaxy-galaxy lensing systems. The two types of system have\ndifferent advantages. Systems with lensed galaxies are usually extended\nand therefore typically provide more constraints on the first derivative\nof the gravitational potential, as has been shown by the large survey of\nsuch systems from the Sloan Digital Sky Survey, SLACS (Bolton et al.\n2006; Koopmans et al. 2006; Bolton et al. 2008). On the other hand,\ntime delay measurements of variations in the images of lensed quasars \nprovide a measurement of the combination of the Hubble constant $H_0$ \n(Refsdal 1964) and the average surface density of the lens in the annulus \nbetween the images used to determine the delay (Kochanek 2002).\nMoreover, the selection effects are often different; galaxy-galaxy\nsystems such as the SLACS survey are usually selected based on the\nlenses, whereas lensed quasars are usually selected based on the\nsources. This has important implications for statistical studies.\n\nIn many cases, the statistics of a well-selected set of gravitational\nlenses can provide important cosmological information. The original\napplication of source-selected lens samples, the determination of \ncombinations of the cosmic matter density $\\Omega_m$ and cosmological \nconstant density $\\Omega_{\\Lambda}$ in units of the critical density \n(Fukugita et al. 1992, Maoz \\& Rix 1993, Kochanek 1996) has now been largely superseded \nby other methods such as studies of the cosmic microwave background,\nsupernova brightness, and baryon acoustic oscillations. However, once\nthe global cosmological model is known, the statistics of gravitational\nlensing can provide important information about the evolution of\ngalaxies. Early studies used the radio sample CLASS (Myers et al. 2003,\nBrowne et al. 2003) which contained 13 quasar lenses in a statistically\ncomplete sample (22 lenses overall) of radio sources with 5-GHz flux\ndensity $\\geq$30~mJy. One major use of such samples is the\n``lens--redshift'' test (Kochanek 1992) in which knowledge of the lens\nand source redshifts and image separations can be used to make\ninferences about galaxy evolution, given a global cosmology. \nThis was used by Ofek, Rix \\& Maoz (2003) and most recently Matsumoto \n\\& Futamase (2008) to derive limits on the evolution of the galaxy number\ndensity and velocity dispersion, in terms of the redshift evolution of \na fiducial number density and velocity dispersion from a Schechter-like \nfunction. In surveys to date, the available sample of lenses is consistent\nwith no evolution up to $z\\sim1$ and a standard\n$\\Lambda$CDM cosmology, but expansion of the sample is desirable in\norder to enable a more stringent test. Capelo \\& Natarajan (2007) study\nthe robustness of this test, concluding that larger and more\nuniform samples of lenses, with complete redshift information and good\ncoverage of separation distributions, are required.\n\nIn recent years, larger samples have become available\nby investigation of quasars from the Sloan Digital Sky Survey quasar\nlist (Schneider et al. 2007). These have been used by Inada and\ncollaborators (e.g. Inada et al. 2003; Inada et al. 2008) to discover 30 lensed\nquasars to date, which form the SQLS (SDSS Quasar Lens Search, Oguri et\nal. 2006). Optical\nsurveys are somewhat more difficult to carry out, in that the high\nresolution needed to separate the components of the lens system is less\neasily available in the optical; the CLASS survey, which had a limiting\nlens separation of 0\\farcs3, showed that the median lens separation is of the\norder 0\\farcs8. \n\nAlthough the SDSS covers a large fraction of the sky to a relatively\nfaint ($r\\sim 22$) limiting magnitude, with the Legacy DR7 spectroscopy\nnow totalling 9380 square degrees, the PSF width of the images is\ntypically 1\\farcs4. More recently the UKIRT Deep Sky Survey (UKIDSS,\nLawrence et al. 2007) has become available; the UKIDSS Large Area Survey\n(ULAS) now covers just over 1000 square degrees to a depth of K=18.4\n(corresponding to $R\\sim24$ for a typical elliptical galaxy at $z=0.3$)\nand, importantly, has a median seeing of 0\\farcs8. UKIDSS uses the\nUKIRT Wide Field Camera (WFCAM; Casali et al, 2007); the photometric\nsystem is described in Hewett et al (2006), and the calibration is\ndescribed in  Hodgkin et al. (2009). The pipeline processing and science\narchive are described in Irwin et al (2009, in prep) and Hambly et al\n(2008). \n\n\nWe are therefore\nconducting a programme (Major UKIDSS-SDSS Cosmic Lens Survey, or\nMUSCLES) which aims to discover lenses difficult for or inaccessible to\nthe SQLS due to small separation, high flux ratio or a combination of\nthe two. We have used data from the UKIDSS 4th data release in this work.\nIn an earlier paper, we reported the discovery of the first\nlens found in this way (ULAS~J234311.9$-$005034, Jackson, Ofek \\& Oguri 2008).\nHere we describe a second detection of a lens system, of relatively\nlarge separation but with a relatively faint secondary. In section 2 we\ndescribe the survey selection and observations. In section 3 we discuss\nthe results, including the three objects rejected as lenses and the\nevidence that ULAS~J082016.1+081216 is a lens system. Finally, in\nsection 4 we revisit the survey selection in the light of the two lenses\ndiscovered by the MUSCLES programme, to assess its potential to discover\nnew lenses which are of smaller separation and\/or higher flux ratio.\n\n\\section{Sample selection and observations}\n\nObjects were selected from the fourth Data Release (DR4) of UKIDSS, and\ncompared against the SDSS quasar catalogue (SDSS DR5, Schneider et al.\n2007). Of the 77429 SDSS quasars, 6708 objects were identified, due mainly to the\nlimited area coverage of current UKIDSS. These were then inspected by\neye for extensions, although we are currently developing algorithms for\nsupplementing with objective selection from parameters fitted to the\nUKIDSS images. We identified 150 good candidates, of which 14 had\nalready been ruled out by other observations (mainly SQLS), and seven\n(not including ULAS~J234311.9$-$005034, Jackson 2008) \nwere known lenses. The survey\nrediscovered all known lenses in the current UKIDSS \nfootprint\\footnote{SDSS J080623.7+200632 (Inada et al. 2006), \nSDSS J083217.0+040405 (Oguri et al. 2008), \nSDSS J091127.6+055054 = RXJ0911+0551 (Bade et al. 1997), \nSDSS J092455.8+021925 (Inada et al. 2003), \nSDSS J122608.0$-$000602 (Inada et al. 2009, in prep), \nSDSS J132236.4+105239 (Oguri et al. 2008a), SDSS J135306.2+113805 (Inada et\nal. 2006).}. Of the 129 remaining objects, one, ULAS~J234311.9$-$005034, \nwas observed previously by us and found to be a lens (Jackson et al.\n2008). In this work we describe observations of four further objects\nfrom the candidate list.\n\nThese four objects were observed using the Keck-I telescope on Mauna Kea\non the night of 2009 February 17, using the LRIS-ADC double-beam imaging\nspectrograph (Oke et al. 1995). They were selected as the most\nconvenient objects for observation at the available time, which appeared\non subjective examination to be the most likely lenses, and which had\nestimated sizes which could be resolved by the seeing of the observations,\nroughly 1$^{\\prime\\prime}$.  The blue \narm of the spectrograph was used with a central wavelength of 430~nm, \nand the red arm with a central wavelength of 760~nm. A dichroic cutting \nbetween 560 and 570~nm was used to split the light between the two arms. \nA long slit of width 0\\farcs7 was used, with\na position angle chosen so as to cover the extended structure seen in\nthe UKIDSS images. A list of objects observed together\nwith integration times is given in Table 1, and UKIDSS images of the\nobserved objects are presented in Fig. 1.\n\n\\begin{table*}\n\\begin{tabular}{cccccc} \\hline\nObject & $z_{SDSS}$ & $r_{SDSS}$ & Exp. (blue)\/s & Exp. (red)\/s & Separation\/$^{\\prime\\prime}$\\\\ \\hline\nJ033248.5$-$002155 & 1.713 & 18.36 & 1800 & 1650 & 1.1 \\\\\nJ034025.5$-$000820 & 0.619 & 20.13 & 1600 & 1560 & 1.4 \\\\\nJ082016.1+081216 & 2.024 & 18.97 & 1450 & 1400 & 1.9 \\\\\nJ091750.5+290137 & 1.816 & 18.07 & 1540& 1400 & 1.0 \\\\ \\hline\n\\end{tabular}\n\\caption{Details of the Keck-I observations, showing the objects (with\nnames representing J2000 coordinates), the SDSS redshift and $r$ magnitude,\nimage separations (measured from the UKIDSS images) and the exposure times\nin the blue and red arms. All observations were\ncarried out on the night of 2009 February 17 using the LRIS\nspectrograph.}\n\\end{table*}\n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\psfig{figure=033248.5-002155.ps,width=4cm,angle=-90}&\n\\psfig{figure=034025.5-000820.ps,width=4cm,angle=-90}\\\\\n\\psfig{figure=082016.1+081216.ps,width=4cm,angle=-90}&\n\\psfig{figure=091750.5+290137.ps,width=4cm,angle=-90}\\\\\n\\end{tabular}\n\\caption{UKIDSS images of the objects observed. Images are in the $H$-band\nexcept for J091750.5+290137, which is in the $J$-band. All images have\nNorth at the top and East on the left, and each image is 12\\farcs8 on a side.}\n\\end{figure}\n\nData were reduced by bias removal, using the overscan strip at the edge\nof each chip, followed by extraction and flux calibration using\nstandard {\\sc iraf} software, distributed by the US National Optical\nAstronomy Observatory (NOAO). Flux calibration was performed using a\nspectrum of the standard star Hz2, obtained on a different night but\nusing the same instrumental setup. Wavelength calibration was done using\nspectra from Hg and Cd arc lamps, and the residuals indicate that this\nshould be accurate to a few tenths of a nanometre except at the edges of\nthe blue frames. \n\n\\section{Results}\n\nFlux-calibrated spectra for all four candidates (A and B images in each\ncase) are given in Fig. 2. In each case, we identify two objects along\nthe slit in each spectrum, and can clearly distinguish the two spectra.\nIn all four systems, we identify the primary (A) object as a quasar,\nwith a redshift that agrees with the SDSS redshift. In two cases \n(J033248.5$-$002155 and J091750.5+290137), we clearly identify the\nsecondary as an M dwarf, most likely with a spectral type around \ntype M5 (e.g. Bochanski et al. 2006). In the case of J034025.5$-$000820,\nthe identification of the object is less clear; it is hardly visible in\nthe blue, but the spectrum rises steeply to the red. There is a possible\nidentification of a break in the spectrum at around 640~nm, which if\nidentified with a galactic 400-nm break feature would imply that it is a\ngalaxy at roughly the same redshift as the quasar. In any case there is\nno sign of any emission lines which might lead us to conclude that we\nare dealing with a gravitational lens system.\n\n\\begin{figure}\n\\psfig{figure=allspec.ps,width=8.5cm}\n\\caption{Spectra of the four observed objects. Each panel shows the\nprimary object (a quasar in each case) together with the secondary. The\nsecondary is an M dwarf for two objects (J033248.5$-$002155 and\nJ091750.5+290137) and a quasar in one case (J082016.1+081216). The SDSS\nredshifts are given in parentheses. Cosmic rays have been interactively\nremoved from the spectra, and the area affected by the dichroic cut has\nbeen blanked. Atmospheric telluric absorption features are visible in\nthe spectra at 760 and 690 nm.}\n\\end{figure}\n\nIn the case of J082016.1+081216 (Fig. 3), we clearly see two objects with\nemission lines; Ly$\\alpha$, C{\\sc iv} and Mg{\\sc ii} are identifiable in\neach spectrum, and C{\\sc iii]} is hidden by the dichroic cut. Moreover, \nif we subtract a scaled version of the primary component, divided by a \nfactor 6, from the secondary component, we obtain a residual which is \nredder than either spectrum individually. This is what would be \nexpected from a two-image gravitational lens system, as the lensing galaxy\n(G) would be expected to lie very close to the fainter image (B) of the \nlens system, with the brighter (A) image some distance away. \nThe identical spectra, together with the identification of a\ngalactic residual in the fainter component, is convincing evidence that\nthis is a lens system and not, for example, a binary quasar. Unlike in\nthe case of ULAS~J234311.9$-$005034 (Jackson et al. 2008), there is no\nevidence of any differences in the spectra which might suggest\ndifferential reddening of the images within the lensing galaxy.\nLike ULAS~J234311.9$-$005034, ULAS~J082016.1+081216 is a radio-quiet\nquasar, having no radio identification at the level of 1~mJy\nin the FIRST 20-cm radio survey (Becker, White \\& Helfand 1995).\n\n\\begin{figure}\n\\psfig{figure=0820fig.ps,width=8.5cm,angle=-90}\n\\caption{Spectra of the J082016.1+081216 system. The figure shows the\nprimary component (interpreted as the brighter ``A'' image of the lens\nsystem) and the secondary component (consisting of the ``B'' image and\nthe lensing galaxy G) together with the residual (G) from the subtraction\nof one-sixth of the primary component from the spectrum of the secondary.\nThe residual is redder than either image. It contains a\npossible set of absorption lines at about 710~nm (inset) which can be\nidentified with Ca H and K at a wavelength of 393.3, 396.7~nm in the\nrest frame.\nAtmospheric telluric absorption features are visible in the spectra at\n760 and 690 nm.}\n\\end{figure}\n\n\nA final indication of lensing (Fig.~4) can be derived from fitting two images to the\nSDSS and UKIDSS data for J082016.1+081216. A clear trend for reduced\nseparation is seen between the optical and near infrared; this is\nexactly as would be expected if a relatively red lensing galaxy is lying\nbetween two blue quasar images, and close to the fainter quasar image.\nThe implication of Fig.~4 is that the separation of the two quasar\nimages is approximately 2\\farcs3, and that the lensing galaxy, which\nis likely to dominate the flux in the near-infrared, lies approximately\n1\\farcs8 from the brighter component. However, it cannot be detected\ndirectly from the UKIDSS images alone. We can test this by fitting two\nPSFs to the J-band UKIDSS image (which has the smallest pixel scale, \n0\\farcs2) separated by a fixed 2\\farcs27 separation implied by the blue\noptical images, and allowing a third Sersic component to be located in\nbetween them. A good fit is obtained using the {\\sc galfit} software\n(Peng et al. 2002), but is statistically\nindistinguishable from the 2-component fit, and the residuals for the\ntwo fits look very similar and noise-like.\n\n\\begin{figure}\n\\psfig{figure=sep_0820.ps,width=8cm,angle=-90}\n\\caption{Separation of the primary (A) and secondary (B+G) components in the \nfilters $ugrizJHK$ from SDSS and UKIDSS, against wavelength} \n\\end{figure}\n\nA redshift for the galaxy can be derived if we identify the\nabsorption lines seen in the difference spectrum \naround 710~nm with the Ca H and K doublet at 393.3 and\n396.7~nm. Fitting to these lines yields a galaxy redshift of 0.803$\\pm$0.001\nfor each line, which, together with an Einstein radius of 1\\farcs15 and\nan assumption of an isothermal model, predicts a galaxy of velocity\ndispersion $\\sigma\\simeq290$~km$\\,$s$^{-1}$. \nFrom the Faber-Jackson relation (Faber \\& Jackson 1976) as calibrated by\nRusin et al. (2003) and using the image separation together with\n$z_l=0.803$, we obtain an expected magnitude of $R\\simeq21.4$ for\na typical lensing galaxy. The magnitude of the galaxy implied by Fig. 3\nis about 0.07 times the total magnitude of the object, or $r\\simeq21.9$,\nwhich corresponds approximately to $R=21.6$. The good agreement with the\nobserved $R$ is further, though circumstantial, evidence for this object\nbeing a lens system.\n\nIf we assume an isothermal model for the galaxy, together with the\nobserved image flux ratio and separation, we obtain a likely time delay\nof approximately 350 days, assuming $H_0=70$kms$^{-1}$Mpc$^{-1}$, \nbetween variations of the A and B images. The\nrelatively long delay results from a combination of a high flux ratio\nand large separation.\n\n\\section{Discussion and conclusions}\n\nWe show that the use of the image quality together with the\ndepth of UKIDSS is likely to lead to discovery of lenses in a wider\nregion of parameter space than lenses selected using SDSS alone. \nThis is because the better image quality of UKIDSS should allow the \ndiscovery of both smaller-separation lenses and lenses of higher flux\nratio. To illustrate this, Fig. 5 shows the image separations and flux \nratios of lenses from the SQLS sample. For four-image lenses, the \nbrightness is dominated by an almost-unresolved pair of merging images, \nwith a third fainter image and a fourth, typically much fainter image. \nIn this case we take the flux ratio as the brightness of the third image\ndivided by that of the merging pair. Fig. 5 also shows the image separation\nand flux ratio distribution of lenses from the CLASS survey (Myers et al.\n2003, Browne et al. 2003), which has a resolution limit of 0\\farcs3 and\na flux ratio limit of 10:1, and of the two MUSCLES lenses found so far.\nThe lens presented here, ULAS~J082016.1+081216, has a flux ratio of 6,\nhigher than the limit of the SQLS main survey. In fact, of the SQLS \noptical lenses with separation $\\theta<4^{\\prime\\prime}$, this lens \nhas the highest flux ratio. Its nearest rival was found by a special \nimaging programme based on SDSS, rather than SDSS directly (Morgan et \nal. 2003).\n\nWe can extrapolate from the existing SQLS and CLASS surveys to attempt\nto estimate the lens yield of MUSCLES after all followup has been done. \nOnly eight of the 22 CLASS lenses lie in the part of the\nseparation\/flux-ratio diagram accessible to the main SQLS survey.\nAssuming that MUSCLES can detect lenses of up to 10:1 flux ratio, and\nwith separations $>0\\farcs6$ (cf. the SQLS survey limit of\n1$^{\\prime\\prime}$ for average seeing of 1\\farcs4 in SDSS), this implies\na potential yield of over 50 new lenses compared to the 30 in SQLS. The\nactual number may be somewhat less than this, as lenses with high flux\nratios {\\em and} lower separation will be harder to detect. There will\nalso be a reduction because the currently planned footprint of UKIDSS \nis around 4000 square degrees, compared to around 9000 degrees in the \nSDSS spectroscopic area. It is to be hoped that extensions to UKIDSS in\nthe future may remedy this, however. Moreover, many of the UKIDSS \ndetections of the SDSS quasars are at a level where high flux-ratio \nsecondaries may be harder to find. Nevertheless, a well-selected \nlens sample approximately 2 times greater than the existing SQLS\nsample has implications for studies of galaxy evolution.  For\nexample, the limits of Matsumoto \\& Futamase (2008), based on the SQLS\nsample alone together with the lens-redshift test, do not currently allow us\nto rule out the hypothesis of no evolution in lens galaxy number density\nor velocity dispersion. We expect that an increase \nin the statistical lens sample should allow this to be done.\n\n\\begin{figure}\n\\psfig{figure=sqls_lenses.ps,width=8.5cm,angle=-90}\n\\caption{Image separations and flux ratios from the SQLS lens sample\n(Inada et al. 2003a,b, 2005, 2006a,b, 2007, 2009, \nOguri et al. 2004, 2005, 2008a,b, Johnston et\nal. 2003, Pindor et al. 2004, 2006, \nMorokuma et al. 2007, Kayo et al. 2007, Ofek et al.\n2007, Morgan et al. 2003). The two MUSCLES lenses (Jackson et al. 2008\nand this work) are indicated as open circles. The UKIDSS median image\nquality (dot-dashed line) and SDSS (dashed line) are indicated, together\nwith the dynamic range and lens separation \nlimit of the SDSS statistical sample (dotted line). The\nprimary contribution of this survey is likely to be lenses at higher\nflux ratio and smaller separation. CLASS survey lenses, with a\nseparation limit of 0\\farcs3 and flux ratio limit of about 10 (2.5\nmagnitudes) are indicated by stars. One CLASS lens is just outside the\nplot, with a separation of 4\\farcs6 and flux ratio 0.86 magnitudes.}\n\\end{figure}\n\n\n\n\\normalsize\n\n\\section*{Acknowledgements}\n\nWe would like to thank the Kavli Institute for Theoretical Physics\nand the organizers of the KITP workshop ``Applications of Gravitational\nLensing'' for hospitality. This work began at this KITP\nworkshop. We thank an anonymous referee for useful comments. \nThe research was supported in part by the European\nCommunity's Sixth framework Marie Curie Research Training\nNetwork Programme, contract no. MRTN-CT-2004-505183, by the\nNational Science Foundation under grant no. PHY05-51164 and\nby the Department of Energy contract DE-AC02-76SF00515. This\nwork is based on data obtained as part of the UKIRT Infrared Deep\nSky Survey, UKIDSS (www.ukidss.org). Some of the data presented\nherein were obtained at the W.M. Keck Observatory, which is operated\nas a scientific partnership among the California Institute of\nTechnology, the University of California and the National Aeronautics\nand Space Administration. The Observatory was made possible\nby the generous financial support of the W.M. Keck Foundation.\nThe authors wish to recognize and acknowledge the very significant\ncultural role and reverence that the summit of Mauna Kea has always had\nwithin the indigenous Hawaiian community. We are most fortunate to have\nthe opportunity to conduct observations from this mountain. Funding for\nthe SDSS and SDSS-II has been provided by the Alfred P. Sloan\nFoundation, the Participating Institutions, the National Science\nFoundation, the US Department of Energy, the National Aeronautics and\nSpace Administration, the Japanese Monbukagakusho and the Max-Planck\nSociety, and the Higher Education Funding Council for England. The SDSS\nweb site is http:\/\/www.sdss.org\/. The SDSS is managed by the\nAstrophysical Research Consortium (ARC) for the Participating\nInstitutions. The Participating Institutions are the American Museum of\nNatural History, Astrophysical Institute Potsdam, University of Basel,\nUniversity of Cambridge, Case Western Reserve University, The University\nof Chicago, Drexel University, Fermilab, the Institute for Advanced\nStudy, the Japan Participation Group, The Johns Hopkins University, the\nJoint Institute for Nuclear Astrophysics, the Kavli Institute for\nParticle Astrophysics and Cosmology, the Korean Scientist Group, the\nChinese Academy of Sciences (LAMOST), Los Alamos National Laboratory,\nthe Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute\nfor Astrophysics (MPA), New Mexico State University, Ohio State\nUniversity, University of Pittsburgh, University of Portsmouth,\nPrinceton University, the United States Naval Observatory and the\nUniversity of Washington.\n\n\\section*{References}\n\n\\noindent Bade N., Siebert J., Lopez S., Voges W., Reimers D. 1997,  A\\&A, 317, L13. \n\n\\noindent Becker R.H., White 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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nSimplicial complexes are widely used in combinatorial and\ncomputational topology, and have found many applications in topological data\nanalysis and geometric inference. A variety of simplicial\ncomplexes have been defined, for example the \\v{C}ech complex, the Rips\ncomplex and the witness complex~\\cite{Alexa_topologicalestimation,DBLP:books\/daglib\/0025666}.\nHowever, the size of these structures grows\nvery rapidly with the dimension of the data set, and\ntheir use in real applications  has been quite limited so far.\n\nWe are aware of only a few works on the design of data structures for\ngeneral simplicial complexes. Brisson~\\cite{DBLP:conf\/compgeom\/Brisson89} and Lienhardt~\\cite{DBLP:journals\/ijcga\/Lienhardt94}\nhave introduced data structures to represent $d$-dimensional cell\ncomplexes, most notably subdivided manifolds. While those data\nstructures have nice algebraic properties, they are very redundant and\ndo not scale to large data sets or high dimensions.\nZomorodian~\\cite{DBLP:conf\/compgeom\/Zomorodian10} has\nproposed the tidy set, a compact data structure to simplify a\nsimplicial complex and compute its homology. Since the construction of\nthe tidy set requires to compute the maximal faces of the simplicial\ncomplex, the method is especially designed \nfor flag complexes. Flag complexes are a special type of simplicial\ncomplexes (to be defined later) whose combinatorial structure can be\ndeduced from its graph. In particular, maximal faces of a flag complex\ncan be computed without\nconstructing explicitly the whole complex.\nIn the same spirit, Attali et al.~\\cite{DBLP:conf\/compgeom\/AttaliLS11a}\nhave proposed the skeleton-blockers data structure. Again, the\nrepresentation is general but it requires to compute blockers, the\nsimplices which are not contained in the simplicial complex but whose\nproper subfaces are. Computing the blockers is difficult in general and\ndetails on the construction are given only for flag complexes, for\nwhich blockers can be easily obtained.\nAs of now, there is no data structure for general\nsimplicial complexes that scales to dimension and size. The best\nimplementations have been restricted to flag complexes.\n\nOur approach aims at combining both generality and scalability. We\npropose a tree representation for simplicial complexes. The nodes of\nthe tree are in bijection with the simplices (of all dimensions) of\nthe simplicial complex. In this way, our data structure, called a\n\\emph{simplex tree}, explicitly stores all the\nsimplices of the complex but does not represent explicitly all the\nadjacency relations between the simplices, two simplices being\nadjacent if they share a common subface. Storing all the simplices\nprovides generality, and the tree structure of our representation\nenables us to implement basic operations on simplicial complexes\nefficiently, in particular to retrieve incidence relations, \\emph{ie}\nto retrieve the faces that contain a given simplex or are contained\nin a given simplex.\n\nThe paper is organized as follows. In section~\\ref{subsec:DS}, we\ndescribe the simplex tree and, in section~\\ref{subsec:op_on_st}, we\ndetail the elementary operations on the simplex tree such as adjacency\nretrieval and maintainance of the data structure upon elementary\nmodifications of the complex. In section~\\ref{sec:examples}, we\ndescribe and analyze the construction of flag complexes, witness\ncomplexes and relaxed witness complexes. An algorithm for inserting\nnew vertices in the witness complex is also described. Finally,\nsection~\\ref{sec:experiments} presents a thorough experimental\nanalysis of the construction algorithms and compares our\nimplementation with the softwares {\\sc JPlex} and {\\sc\nDionysus}. Additional experiments are provided in\nappendix~\\ref{sec:additional_expe}. \n\n\n\\subsection{Background}\n\n\\label{subsec:background}\n\\paragraph{Simplicial complexes.}\nA {\\em simplicial complex} is a pair $\\mathcal{K}=(V,S)$ where $V$ is\na finite set whose elements are called the {\\em vertices} of $\\mathcal{K}$ and\n$S$ is a set of non-empty subsets of $V$ that is\nrequired to satisfy the following two conditions~:\n\\begin{enumerate}\n\\item $p\\in V \\Rightarrow \\{  p\\} \\in S$\n\\item $\\sigma\\in S, \\tau\\subseteq \\sigma \\Rightarrow\n\\tau\\in S$\n\\end{enumerate}\nEach element $\\sigma\\in S$ is called a {\\em simplex} or a \\emph{face} of $\\mathcal{K}$\nand, if $\\sigma\\in S$ has precisely $s+1$ elements ($s\\geq -1$),\n$\\sigma$ is called an $s$-simplex and the dimension of $\\sigma$ is\n$s$. The dimension of the simplicial complex $\\mathcal{K}$ is\nthe largest $k$ such that $S$ contains a $k$-simplex.\n\nWe define the \\emph{$j$-skeleton}, $j\\geq 0$, of a simplicial complex $\\mathcal{K}$ to be\nthe simplicial complex made of the faces of $\\mathcal{K}$ of dimension\nat most $j$. In particular, the $1$-skeleton of $\\mathcal{K}$ contains\nthe vertices and the edges of $\\mathcal{K}$. The $1$-skeleton has the\nstructure of a graph, and we will equivalently talk about the graph of\nthe simplicial complex.\n\nA \\emph{subcomplex} $\\mathcal{K}' = (V',S')$ of the simplicial complex $\\mathcal{K} =\n(V,S)$ is a simplicial complex satisfying $V' \\subseteq V$ and $S'\n\\subseteq S$. In particular, the $j$-skeleton of a simplicial complex is\na subcomplex.\n\n\\paragraph{Faces and cofaces.} \nA {\\em face} of a simplex $\\sigma=\\{p_0 ,\\cdots ,p_s\\}$ is a simplex whose vertices\nform a subset of $\\{p_0 ,\\cdots ,p_s\\}$. A {\\em proper} face\nis a face different from $\\sigma$ and the {\\em facets} of $\\sigma$ are\nits proper faces of maximal dimension.\nA simplex $\\tau\\in \\mathcal{K}$ admitting $\\sigma$ as a face is called a {\\em\ncoface} of $\\sigma$. The subset of simplices consisting of all the cofaces of a\nsimplex $\\sigma \\in \\mathcal{K}$ is called the {\\em star} of $\\sigma$.\n\nThe {\\em link}  of a simplex $\\sigma$ in a simplicial complex $\\mathcal{K} = (V,S)$ is\ndefined as the set of faces: \n$${\\rm Lk} (\\sigma) = \\{ \\tau\\in S | \\sigma\\cup \\tau \\in S, \\sigma \\cap \\tau = \\emptyset \\}$$ \n\n\\paragraph{Filtration.}\nA {\\em filtration} over a simplicial complex $\\mathcal{K}$ is an ordering of the\nsimplices of $\\mathcal{K}$ such that all prefixes in the ordering are subcomplexes\nof $\\mathcal{K}$. In particular, for two simplices $\\tau$ and $\\sigma$ in the simplicial\ncomplex such that $\\tau \\subsetneq \\sigma$, $\\tau$ appears before\n$\\sigma$ in the ordering. Such an ordering may be given by a real number associated to\nthe simplices of $\\mathcal{K}$. The order of the simplices is simply the order\nof the real numbers.\n\n\\section{Simplex Tree}\n\\label{sec:simplex-tree}\n\nIn this section, we introduce a new data structure which can represent\nany simplicial complex. This data structure is a\ntrie~\\cite{DBLP:conf\/soda\/BentleyS97} which explicitly represents all the simplices and allows\nefficient implementation of basic operations on simplicial complexes.\n\n\\subsection{Simplicial Complex and Trie}\n\\label{subsec:DS}\n\nLet  $\\mathcal{K} = (V,S)$ be a simplicial complex of dimension $k$.\nThe vertices are labeled from $1$ to $|V|$ and ordered accordingly.\n\nWe can thus\nassociate to each simplex of $\\mathcal{K}$ a word on the alphabet\n$1 \\cdots |V|$. Specifically, a $j$-simplex of $\\mathcal{K}$ is uniquely\nrepresented as the word of length $j+1$ consisting of the ordered set of the labels of\nits $j+1$ vertices.  Formally, let simplex $\\sigma = \\{v_{\\ell_0},\n\\cdots , v_{\\ell_j}\\} \\in S$, where $v_{\\ell_i}\\in V$, $\\ell_i\\in \\{1, \\cdots ,|V|\\}$ and\n$\\ell_0<\\cdots < \\ell_j$. $\\sigma$ is then represented by the word\n$[\\sigma] = [\\ell_0, \\cdots , \\ell_j]$. The last label of the word representation of a simplex $\\sigma$ will\nbe called the last label of $\\sigma$ and denoted by\n$last (\\sigma)$.\n\nThe simplicial complex $\\mathcal{K}$ can be defined as a collection\nof words on\nan alphabet of size $|V|$. To compactly represent the set of simplices\nof $\\mathcal{K}$, we store the corresponding words in a tree satisfying\nthe following properties:\n\n\\begin{enumerate}\n\\item The nodes of the simplex tree are in bijection with the\n  simplices (of all dimensions) of the complex. The\n  root is associated to the empty face.\n\\item Each node of the tree, except the root, stores the label of a\n  vertex. Specifically, a node associated to a simplex $\\sigma \\neq\n  \\emptyset$ stores the label $last (\\sigma)$. \n\\item The vertices whose labels are encountered along a path from the\n  root to a node associated to a simplex $\\sigma$, are the vertices of\n  $\\sigma$. \n Along such a path, the labels are\n  sorted by\n  increasing order and each label appears no more than once.\n\n\\end{enumerate}\n\nWe call this data structure the \\emph{Simplex Tree} of $\\mathcal{K}$. It may be seen as a\ntrie~\\cite{DBLP:conf\/soda\/BentleyS97} on the words\nrepresenting  the simplices of the complex (Figure~\\ref{fig:preft}). The depth of the root is $0$ and the depth of a node\nis equal to the dimension of the simplex it represents plus one.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=11cm]{simplextree_cof_horizontal.pdf}\n\\end{center}\n\\caption{A simplicial complex on $10$ vertices and its simplex\n  tree. The deepest node represents the tetrahedron of the complex. All\n  the positions of a given label at a given depth are linked in a\n  list, as illustrated in the case of label $5$.}\n\\label{fig:preft}\n\\end{figure}\n\nIn addition, we augment the data\nstructure so as to quickly locate all the instances of a given label\nin the tree. Specifically,\nall the nodes at a same depth $j$ which contain a same label $\\ell$ are\nlinked in a circular list $L_j(\\ell)$, as illustrated in Figure~\\ref{fig:preft}\nfor label $\\ell = 5$.\n\nThe children of the root of the simplex tree are called the\n{\\em top nodes}. The top nodes are in bijection with the elements of\n$V$, the vertices of $\\mathcal{K}$. Nodes which share the same\nparent (e.g. the top nodes) will be called  \\emph{sibling\n  nodes}.\n\nWe also attach to each set of sibling nodes a pointer to their parent\nso that we can access a parent in constant time.\n\nWe give a constructive definition of the simplex tree. Starting from\nan empty tree, we insert the words\nrepresenting the simplices of the complex in the following\nmanner. When inserting the word $[\\sigma]= [\\ell_0,\n\\cdots , \\ell_j]$ we start from the root, and follow the path containing\nsuccessively all labels $\\ell_0, \\cdots, \\ell_i$, where $[\\ell_0, \\cdots , \\ell_i]$\ndenotes the longest prefix of $[\\sigma]$  already stored in the simplex tree. We then append\nto the node representing $[\\ell_0, \\cdots , \\ell_i]$ a path consisting of\nthe nodes storing\nlabels $\\ell_{i+1}, \\cdots , \\ell_j$.\n\nIt is easy to see that the three properties above are\nsatisfied. Hence, if  $\\mathcal{K}$ consists of $|\\mathcal{K}|$\nsimplices\n (including the empty face), the associated\nsimplex tree contains exactly $|\\mathcal{K}|$ nodes.\n\n\nWe use dictionaries with size linear in the\nnumber of elements they store (like a red-black tree or a hash table)\nfor searching, inserting and removing elements\namong a set of sibling nodes. Consequently these additional structures do not\nchange the asymptotic memory complexity of the simplex tree. For the\ntop nodes, we simply use an array since the set of vertices $V$ is\nknown and fixed. Let $\\text{deg}(\\mathcal{T})$ denote the maximal outdegree of a node,\nin the simplex tree $\\mathcal{T}$, distinct from the root. Remark that $\\text{deg}(\\mathcal{T})$ is at most the maximal\ndegree of a vertex in the graph of the simplicial complex. \nIn the following, we will denote by $D_{\\text{m}}$ the\nmaximal number of operations needed to perform a search, an insertion\nor a removal in a dictionary of maximal size\n$\\text{deg}(\\mathcal{T})$ (for example, with red-black trees $D_{\\text{m}} =\nO(\\log(\\text{deg}(\\mathcal{T})))$ worst-case, with hash-tables $D_{\\text{m}} =\nO(1)$ amortized). Some algorithms, that we describe\nlater, require to intersect and to merge sets of\nsibling nodes. In order to compute fast set operations, we will prefer\ndictionaries which allow to traverse their elements in sorted order (e.g., red-black\ntrees). \nWe discuss the value of $D_{\\text{m}}$ at the end of this section in\nthe case where the points have a geometric structure.\n\nWe introduce two new notations for the analysis of the complexity of\nthe algorithms. Given a simplex $\\sigma \\in \\mathcal{K}$, we define $C_\\sigma$ to be\nthe number of cofaces of $\\sigma$. Note that $C_\\sigma$ only depends on the\ncombinatorial structure of the simplicial complex $\\mathcal{K}$. Let\n$\\mathcal{T}$ be the simplex tree associated to $\\mathcal{K}$. Given a label\n$\\ell$ and an index $j$, we define $\\mathcal{T}_\\ell^{> j}$ to be the number\nof nodes of $\\mathcal{T}$ at depth strictly greater than $j$ that store label\n$\\ell$. These nodes represent the simplices of dimension at least $j$ that\nadmit $\\ell$ as their last label. $\\mathcal{T}_\\ell^{> j}$ depends on the labelling of the\nvertices and is bounded by $C_{\\{v_\\ell\\}}$, the number of cofaces of\nthe vertex with label $\\ell$. For example, if $\\ell$ is the greatest\nlabel, we have $\\mathcal{T}_{\\ell}^{> 0} = C_{\\{v_\\ell\\}}$, and if\n$\\ell$ is the smallest label we have $\\mathcal{T}_{\\ell}^{> 0} = 1$ independently from the number of cofaces of $\\{v_\\ell\\}$.\n\n\\subsection{Operations on a Simplex Tree}\n\\label{subsec:op_on_st}\n\nWe provide algorithms for:\n\n\\begin{itemize}\n\\item {\\sc Search\/Insert\/Remove-simplex} to search, insert or remove a\n  single simplex, and {\\sc Insert\/Remove-full-simplex} to insert a\n  simplex and its subfaces or remove a simplex and its cofaces\n\\item {\\sc Locate-cofaces} to locate the cofaces of a simplex\n\\item {\\sc Locate-facets} to locate the facets of a simplex\n\\item {\\sc Elementary-collapse} to proceed to an elementary collapse\n\\item {\\sc Edge-contraction} to proceed to contract an edge\n\\end{itemize}\n\n\\subsubsection{Insertions and Adjacency Retrieval}\n\n\\paragraph{Insertions and Removals}\nUsing the previous top-down traversal, we can \\emph{search} and\n\\emph{insert} a word of length $j$ in $O(jD_{\\text{m}})$ operations. \n\nWe can extend this algorithm so as to insert a simplex and all its\nsubfaces in the simplex tree. Let $\\sigma$ be a simplex we want to\ninsert with all its subfaces. Let $[\\ell_{0}, \\cdots ,\\ell_j]$ be its\nword representation. For $i$ from $0$ to $j$ we insert, if not already\npresent, a node $N_{\\ell_i}$, storing label $\\ell_i$, as a child of the\nroot. We recursively call the algorithm on the subtree rooted at $N_{\\ell_i}$\nfor the insertion of the suffix $[\\ell_{i+1}, \\cdots ,\\ell_j]$.\nSince the number of subfaces of a simplex of dimension $j$ is\n$\\sum_{i=0\\cdots j+1} \\binom{j+1}{i} = 2^{j+1}$, this algorithm takes time\n$O(2^jD_{\\text{m}})$.\n\nWe can also remove a simplex from the simplex tree. Note that to\nkeep the property of being a simplicial complex, we need to remove all\nits cofaces as well. We locate them thanks to the algorithm described below.\n\n\\paragraph{Locate cofaces.}\nComputing the cofaces of a face is required to retrieve adjacency\nrelations between faces. In particular, it is useful when\ntraversing the complex or when removing a face. \nWe also need to compute the cofaces of a face when contracting an edge (described later) or during the construction of\nthe witness complex, described later in section~\\ref{subsec:witcpx}.\n\nIf $\\tau$ is represented by the word $[\\ell_0 \\cdots \\ell_j]$, the cofaces of $\\tau$ are\nthe simplices of $\\mathcal{K}$ which are represented by words of the form $[\\star \\ell_0\n\\star \\ell_1 \\star \\cdots \\star \\ell_j \\star]$, where $\\star$\nrepresents an arbitrary word on the alphabet, possibly empty.\n\nTo locate all the words of the form $[\\star \\ell_0 \\star\n\\ell_1 \\star \\cdots \\star \\ell_j \\star ]$ in the simplex tree, we first find all the words of the form $[\\star \\ell_0 \\star \\ell_1\n\\star \\cdots \\star \\ell_j ]$. Using the lists $L_i(\\ell_j)$ ($i >\nj$), we find\nall the nodes at depth at least $j+1$ which contain label $\\ell_j$. For\neach such node $N_{\\ell_j}$, we traverse the tree upwards from $N_{\\ell_j}$, looking\nfor a word of the form $[\\star \\ell_0 \\star \\ell_1 \\star \\cdots \\star \\ell_j\n]$. If the search succeeds, the simplex represented by $N_{\\ell_j}$ in the\nsimplex tree is a coface of $\\tau$, as well as all the simplices\nrepresented by the nodes in the subtree rooted at $N_{\\ell_j}$, which\nhave word representation of the form $[\\star \\ell_0 \\star \\ell_1 \\star \\cdots \\star \\ell_j\n\\star]$. Remark that the cofaces of a simplex are represented by\na set of subtrees in the simplex tree. The procedure searches only\nthe roots of these subtrees.\n\nThe complexity for searching the cofaces of a simplex $\\sigma$ of\ndimension $j$ depends on the\nnumber $\\mathcal{T}_{last(\\sigma)}^{>j}$ of nodes with label $last{ (\\sigma) }$ and depth at\nleast $j+1$. If $k$ is the dimension of the simplicial complex,\ntraversing the tree upwards takes $O(k)$ time.\nThe complexity of this procedure is thus $O(k\\mathcal{T}_{last(\\sigma)}^{>j})$.\n\n\\paragraph{Locate Facets.}\n\nLocating the facets of a simplex efficiently is the key point of\nthe incremental algorithm we use to construct witness complexes in\nsection~\\ref{subsec:witcpx}. \n\nGiven a simplex $\\sigma$, we want to access the nodes of the simplex\ntree representing the facets of $\\sigma$.  If the word\nrepresentation of $\\sigma$ is $[\\ell_0,\\cdots,\\ell_{j}]$, the word\nrepresentations of the facets of $\\sigma$ are \nthe words $[\\ell_0, \\cdots , \\widehat{\\ell}_i, \\cdots , \\ell_j]$, $0 \\leq i\n\\leq j$, where $\\widehat{\\ell}_i$ indicates that $\\ell_i$ is omitted. \nIf we denote, as before, $N_{\\ell_i}, i=0,\\cdots ,j$ the nodes representing the words $[\\ell_0, \\cdots ,\n\\ell_{i}],i=0,\\cdots ,j$ respectively, a traversal from the node representing $\\sigma$ up to the\nroot will exactly pass through the nodes $N_{\\ell_i}$, $i = j, \\cdots ,0$. When\nreaching the node $N_{\\ell_{i-1}}$, a search from $N_{\\ell_{i-1}}$ downwards for the word $[\\ell_{i+1}, \\cdots ,\n\\ell_{j}]$ locates (or proves the absence of) the facet $[\\ell_0, \\cdots\n, \\widehat{\\ell}_i, \\cdots , \\ell_j]$. See Figure~\\ref{fig:algo_facets} for a running example.\n\nThis procedure locates all the facets of the $j$-simplex $\\sigma$ in $O(j^2\nD_{\\text{m}})$ operations. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{algo_facets.pdf}\n\\end{center}\n\\caption{Facets location of the simplex $\\sigma = \\{2,3,4,5\\}$,\n  starting from the position of $\\sigma$ in the simplex tree. The\n  nodes representing the facets are colored in grey.}\n\\label{fig:algo_facets}\n\\end{figure}\n\n\\paragraph{Experiments.}\n\nWe report on the experimental performance of the facets and cofaces\nlocation algorithms. Figure~\\ref{fig:time_fac_cof}\nrepresents the average time for these operations on a simplex, as a\nfunction of the dimension of the simplex. We use the dataset {\\bf\n  Bro}, consisting of points in $\\mathbb{R}^{25}$, on top of which we\nbuild a relaxed witness complex with $300$ landmarks and $15,000$\nwitnesses, and relaxation parameter $\\rho = 0.15$. See\nsection~\\ref{sec:experiments} for a detailed description of the\nexperimental setup. We obtain a $13$-dimensional\nsimplicial complex with $140,000$ faces in less than $3$ seconds. \n\n\\setlength{\\tabcolsep}{2.5pt}\n\n\\begin{figure}\n\\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|r|r|r|r|r|}\n \\hline\n Dim.Face&0&1&2&3&4&5&6&7&8&9&10&11&12&13\\\\ \n \\hline\n$\\#$ Faces&300&2700&8057&15906&25271&30180&26568&17618&8900&3445&1015&217&30&2\\\\\n \\hline\n\\end{tabular}\n\n\n\\setlength{\\tabcolsep}{9pt}\n \\begin{tabular}{c c}\n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure0.pdf}\n    \\end{minipage}\n    &  \n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure1.pdf}\n    \\end{minipage}\\\\\n  \\end{tabular}\n\n\n  \n\n\\caption{Repartition of the number of faces per dimension (top) and average time to compute the facets (left) and the\n  cofaces (right) of a simplex of a given dimension.} \n\\label{fig:time_fac_cof}\n\\end{figure}\n\nThe theoretical\ncomplexity for computing the facets of a $j$-simplex $\\sigma$ is\n$O(j^2 D_{\\text{m}})$. As reported in Figure~\\ref{fig:time_fac_cof}, the\naverage time to search all facets of a $j$-simplex is well\napproximated by a quadratic function of the dimension $j$ (the standard\nerror in the approximation is $2.0\\%$).\n\nA bound on the complexity of computing the cofaces of a $j$-simplex\n$\\sigma$ is $O(k\\mathcal{T}_{last(\\sigma)}^{>j})$, where\n$\\mathcal{T}_{last(\\sigma)}^{>j}$ stands for the number of nodes in\nthe simplex tree that store the label $last{(\\sigma)}$ and have depth\nlarger than $j+1$. Figure~\\ref{fig:time_fac_cof} provides experimental results\nfor a random labelling of the vertices.  As can be seen, the time for\ncomputing the cofaces of a simplex $\\sigma$ is low, on average, when\nthe dimension of $\\sigma$ is either small ($0$ to $2$) or big ($6$ to\n$13$), and  higher for  intermediate dimensions ($3$ to $5$). The\nvalue $\\mathcal{T}_{last(\\sigma)}^{>j}$ in the complexity analysis\ndepends on both the labelling of the vertices and the number of cofaces of the\nvertex $v_{last{(\\sigma)}}$: these dependencies make the analysis of\nthe algorithm quite difficult, and we let as an open problem to fully understand\nthe experimental behavior of the algorithm as observed in\nFigure~\\ref{fig:time_fac_cof}\n(right).\n\n\n\\subsubsection{Topology preserving operations}\n\nWe show how to implement two topology preserving operations on a\nsimplicial complex represented as a simplex tree. Such simplifications\nare, in particular, important in topological data analysis.\n\n\\paragraph{Elementary collapse.} \n\nWe say that a simplex $\\sigma$ is collapsible through one of\nits faces $\\tau$ if $\\sigma$ is the only coface of $\\tau$, which can\nbe checked by computing the cofaces of $\\tau$. Such a pair $(\\tau,\\sigma)$\nis called a \\emph{free pair}. Removing both faces of a free pair is an\nelementary collapse.\n\nSince $\\tau$ has no coface other than $\\sigma$, either the node representing\n$\\tau$ in the simplex tree is a leaf (and so is the node\nrepresenting $\\sigma$), or it has the node\nrepresenting $\\sigma$ as its unique child. An elementary collapse of\nthe free pair $(\\tau,\\sigma)$ consists either in the removal of the\ntwo leaves representing $\\tau$ and $\\sigma$, or the removal of the\nsubtree containing exactly two nodes: the node representing $\\tau$ and\nthe node representing $\\sigma$.\n\n\\paragraph{Edge contraction.}\nEdge contractions are used in~\\cite{DBLP:conf\/compgeom\/AttaliLS11a} as\na tool for homotopy preserving simplification and\nin~\\cite{DBLP:journals\/corr\/abs-1208-5018} for computing the\npersistent topology of data points. \nLet $\\mathcal{K}$ be a simplicial complex and let $\\{v_{\\ell_a},v_{\\ell_b}\\}$ be an edge\nof $\\mathcal{K}$ we want to contract. We say that we contract $v_{\\ell_b}$ to $v_{\\ell_a}$ meaning that $v_{\\ell_b}$ is\nremoved from the complex and the link of $v_{\\ell_a}$ is augmented with\nthe link of $v_{\\ell_b}$. Formally, we define the map $f$ on the set of vertices $V$ which\nmaps $v_{\\ell_b}$ to $v_{\\ell_a}$\nand acts as the identity function for all other inputs:\n\n\\[\n  f(u) = \\left\\{\n          \\begin{array}{ll}\n            v_{\\ell_a} & \\qquad \\mathrm{if}\\quad u = v_{\\ell_b} \\\\\n            u & \\qquad \\mathrm{otherwise} \\\\\n          \\end{array}\n        \\right.\n\\]\n\nWe then extend $f$ to all simplices $\\sigma = \\{v_{\\ell_0}, \\cdots , v_{\\ell_j}\\}$\nof $\\mathcal{K}$ with $f(\\sigma) = \\{f(v_{\\ell_0}), \\cdots , f(v_{\\ell_j})\\}$. The\ncontraction of $v_{\\ell_b}$ to $v_{\\ell_a}$ is defined as the operation which replaces\n$\\mathcal{K} = (V,S)$ by $\\mathcal{K}' = (V \\setminus \\{v_{\\ell_b}\\} , \\{f(\\sigma) | \\sigma \\in S)\n\\}$. $\\mathcal{K}'$ is a simplicial complex. \n\nIt has been proved in~\\cite{DBLP:conf\/compgeom\/AttaliLS11a} that\ncontracting an edge $\\{v_{\\ell_a},v_{\\ell_b}\\}$\npreserves the homotopy type of a simplicial complex whenever the\n\\emph{link condition} is satisfied:\n\\[\n{\\rm Lk} (\\{v_{\\ell_a},v_{\\ell_b}\\}) = {\\rm Lk}( \\{v_{\\ell_a}\\}) \\cap\n{\\rm Lk}(\\{v_{\\ell_b}\\})\n\\]\nThis link condition can be checked using the {\\sc Locate-cofaces}\nalgorithm described above.\n\nLet $\\sigma$ be a\nsimplex of $\\mathcal{K}$. We\ndistinguish three cases~: 1. $\\sigma$ does not\ncontain $v_{\\ell_b}$ and remains unchanged; 2. $\\sigma$ contains both $v_{\\ell_a}$ and\n$v_{\\ell_b}$, and $f(\\sigma) = \\sigma \\setminus \\{v_{\\ell_b}\\}$; $|f(\\sigma)| = |\\sigma|\n-1$ and  $f(\\sigma)  $ is a strict subface of\n$\\sigma$; 3. $\\sigma$ contains $v_{\\ell_b}$ but not $v_{\\ell_a}$ and $f(\\sigma) = \\left(\\sigma\n\\setminus \\{v_{\\ell_b}\\}\\right) \\cup \\{v_{\\ell_a}\\}$, ($|f(\\sigma)| = |\\sigma|$).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=11cm]{merge_horizontal-h.pdf}\n\\end{center}\n\\caption{Contraction of vertex $3$ to vertex $1$ and the associated modifications of\n  the simplicial complex and of the simplex tree. The nodes which are\n  removed are marked with a red cross, the subtrees which are moved\n  are colored in blue.}\n\\label{fig:merge}\n\\end{figure}\n\nWe describe now how \nto compute the contraction of $v_{\\ell_b}$ to $v_{\\ell_a}$ when $\\mathcal{K}$ is represented as\na simplex tree. We suppose that the edge $\\{v_{\\ell_a},v_{\\ell_b}\\}$ is in the complex\nand, without loss of generality, $\\ell_a < \\ell_b$. All the simplices which do not contain\n$v_{\\ell_b}$ remain unchanged and we do not consider them. If a simplex $\\sigma$ contains\nboth $v_{\\ell_a}$ and $v_{\\ell_b}$, it will become $\\sigma \\setminus\n\\{v_{\\ell_b}\\}$, after edge contraction, which\nis a simplex already in $\\mathcal{K}$. \nWe simply remove $\\sigma$ from the simplex\ntree. Finally, if $\\sigma$ contains $v_{\\ell_b}$ but not $v_{\\ell_a}$, we need to remove $\\sigma$\nfrom the simplex tree and add the new simplex $\\left(\\sigma\n\\setminus \\{v_{\\ell_b}\\}\\right) \\cup \\{v_{\\ell_a}\\}$.\n\nWe consider each node $N_{\\ell_b}$ with label $\\ell_b$ in turn. To do\nso, we use the lists $L_j(\\ell)$ which link all nodes\ncointaining the label $\\ell$ at depth $j$. Let\n$\\sigma$ be the simplex represented by $N_{\\ell_b}$.\nThe algorithm traverses the tree upwards from $N_{\\ell_b}$ and collects the\nvertices of $\\sigma$.  Let $T_{N_{\\ell_b}}$\nbe the subtree rooted at $N_{\\ell_b}$. As $\\ell_a<\\ell_b$, if $\\sigma$\ncontains both $v_{\\ell_a}$ and $v_{\\ell_b}$, this will be true for all the simplices\nwhose representative nodes are in $T_{N_{\\ell_b}}$, and, if $\\sigma$\ncontains only $v_{\\ell_b}$, the same will be true for all the simplices\nwhose representative nodes are in $T_{N_{\\ell_b}}$. Consequently, if\n$\\sigma$ contains both $v_{\\ell_a}$ and $v_{\\ell_b}$, we remove the whole subtree $T_{N_{\\ell_b}}$\nfrom the simplex tree. Otherwise, $\\sigma$ contains only $v_{\\ell_b}$, all words\nrepresented in $T_{N_{\\ell_b}}$ are of the form $[\\sigma '] \\centerdot [\\sigma''] \\centerdot [\\ell_b]\n\\centerdot [\\sigma''']$ and will be turned into words $[\\sigma'] \\centerdot [\\ell_a] \\centerdot\n[\\sigma''] \\centerdot [\\sigma''']$ after edge contraction. We then have to move the subtree $T_{N_{\\ell_b}}$\n(except its root) from position $[\\sigma '] \\centerdot [\\sigma'']$ to position\n$[\\sigma ']\\centerdot [\\ell_a] \\centerdot [\\sigma'']$ in the simplex tree. If a subtree is\nalready rooted at this position, we have to merge $T_{N_{\\ell_b}}$ with this subtree as illustrated in\nFigure~\\ref{fig:merge}. In order to merge the subtree $T_{N_{\\ell_b}}$\nwith the subtree rooted at the node representing the word $[\\sigma\n']\\centerdot [\\ell_a] \\centerdot [\\sigma'']$, we can successively\ninsert every node of $T_{N_{\\ell_b}}$ in the corresponding set of\nsibling nodes, stored in a dictionary. See Figure~\\ref{fig:merge}.\n\nWe analyze the complexity of contracting an edge\n$\\{v_{\\ell_a},v_{\\ell_b}\\}$. For each node storing the label $\\ell_b$, we\ntraverse the tree upwards. This takes $O(k)$ time if the\nsimplicial complex has dimension $k$. As there are\n$\\mathcal{T}_{\\ell_b}^{>0}$ such nodes, the total cost is\n$O(k \\mathcal{T}_{\\ell_b}^{>0})$.\nWe also manipulate the subtrees rooted at the nodes storing label\n$\\ell_b$. Specifically,  either we remove such a subtree or we move a\nsubtree by changing its parent node.  In the latter case, we have\nto merge two subtrees. This is the more costly operation which takes,\nin the worst case, $O(D_{\\text{m}})$ operations per node in the\nsubtrees to be merged. As any node in such a subtree represents a coface of\nvertex $v_{\\ell_b}$, the total number of nodes in all the subtrees we\nhave to manipulate is at most $C_{\\{v_{\\ell_b}\\}}$, and  the\nmanipulation of the subtrees takes $O(C_{\\{v_{\\ell_b}\\}}\nD_{\\text{m}})$ time.\nConsequently, the time needed to contract the edge\n$\\{v_{\\ell_a},v_{\\ell_b}\\}$ is $O(k \\mathcal{T}_{\\ell_b}^{>0} + C_{\\{v_{\\ell_b}\\}}\nD_{\\text{m}})$.\n\n\\paragraph{Remark on the value of $D_{\\text{m}}$}: $D_{\\text{m}}$\nappears as a key value in the complexity analysis of the algorithms. Recall that $D_{\\text{m}}$ is\nthe\nmaximal number of operations needed to perform a search, an insertion\nor a removal in a dictionary of maximal size\n$\\text{deg}(\\mathcal{T})$ in the simplex\ntree. We suppose in the following that the dictionaries used are\nred-black trees, in which case $D_{\\text{m}} = O(\\log (\\text{deg}(\\mathcal{T})))$. As mentioned earlier, $\\text{deg}(\\mathcal{T})$ is bounded by the maximal\ndegree of a vertex in the graph of the simplicial complex. In the worst-case,\nif $n$ denotes the number of vertices of the simplicial complex, \nwe have\n$\\text{deg}(\\mathcal{T}) = O(n)$, and $D_{\\text{m}} = O(\\log(n))$.\nHowever, this bound can be improved in the case of simplicial\ncomplexes constructed on sparse data points\nsampled from a low dimensional manifold, an important case in\npractical applications.  Let $\\mathbb{M}$ be a $d$-manifold with bounded\ncurvature, embedded in $\\mathbb{R}^D$ and assume that the length of the longest (resp., shortest) edge of the\nsimplicial complex has length at most $r$ (resp., at least\n$\\epsilon$).\nThen, a volume argument shows that the maximal degree of a vertex in the simplicial complex\nis $\\Theta((r\/\\epsilon)^d)$. \nHence, when $r=O(\\epsilon)$, which is a typical situation when $S$ is an $\\epsilon$-net\nof $\\mathbb{M}$, the\nvalue of $D_{\\text{m}}$ is $O(d)$ with a constant depending only on\nlocal geometric quantities.\n\n\\section{Construction of Simplicial Complexes}\n\\label{sec:examples}\n\nIn this section, we detail how to construct two important types of\nsimplicial complexes, the flag and the witness complexes, using simplex trees.\n\n\\subsection{Flag complexes}\n \nA flag complex is a simplicial complex whose combinatorial structure is entirely\ndetermined by its $1$-skeleton. Specifically, a simplex is in the flag\ncomplex if and only if its\nvertices form a clique in the graph of the simplicial complex, or,\nin other terms, if and only if its\nvertices are pairwise linked by an edge.\n\n\\paragraph{Expansion.} \n\nGiven the $1$-skeleton of a flag complex, we call \\emph{expansion of\n  order $k$} the operation which reconstructs the $k$-skeleton of the flag complex. If the\n$1$-skeleton is stored in a simplex tree, the expansion of order $k$\nconsists in successively inserting all the\nsimplices of the $k$-skeleton into the simplex tree.\n\nLet $G = (V,E)$ be the graph of the simplicial complex, where\n$V$ is the set of vertices and $E \\subseteq V\\times V$ is the set of\nedges. For a vertex $v_{\\ell} \\in V$, we denote by\n\\[\n\\mathcal{N}^+(v_{\\ell}) = \\{\\ell' \\in \\{1, \\cdots ,|V|\\} \\mbox{ } | \\mbox{ }\n (v_{\\ell},v_{\\ell'}) \\in E \\wedge \\ell' > \\ell\\}\n\\]\nthe set of labels of the neighbors of\n$v_{\\ell}$ in $G$ that are bigger than $\\ell$. \nLet $N_{\\ell_j}$ be the node in the tree  that stores the label $\\ell_j$ and\nrepresents the word $[\\ell_0, \\cdots , \\ell_j]$.  The\n children of $N_{\\ell_j}$ store the labels in $\\mathcal{N}^+(v_{\\ell_0}) \\cap \\cdots \\cap\n\\mathcal{N}^+(v_{\\ell_j})$. Indeed, the children\nof $N_{\\ell_j}$ are  neighbors in $G$ of the vertices $v_{\\ell_i}$, $0 \\leq\ni \\leq j$, (by definition of a clique) and must have a bigger label\nthan $\\ell_0, \\cdots , \\ell_j$ (by construction of the simplex tree).\n\nConsequently, the sibling nodes of\n$N_{\\ell_j}$ are exactly the nodes that store the labels  in\n$A=\\mathcal{N}^+(v_{\\ell_0})~\\cap~\\cdots~\\cap~\\mathcal{N}^+(v_{\\ell_{j-1}})$,\nand the children of $N_{\\ell_j}$ are exactly the nodes that store the labels in $A\\cap\n\\mathcal{N}^+(v_{\\ell_j})$. See Figure~\\ref{fig:inter}.\n\nFor every vertex $v_{\\ell}$, we have an easy access to\n$\\mathcal{N}^+(v_{\\ell})$ since $\\mathcal{N}^+(v_{\\ell})$ is exactly\nthe set of labels stored in the children of the top node storing label\n$\\ell$. We easily deduce an in-depth expansion algorithm.\n\nThe time complexity for the expansion algorithm depends on our \nability to fastly compute intersections of the type $A \\cap\n\\mathcal{N}^+(v_{\\ell_j})$. In all of our experiments on the Rips\ncomplex (defined below) we have observed that the time taken by the expansion algorithm\ndepends linearly on the size of the output simplicial complex, for a\nfixed dimension. More details can be found in\nsection~\\ref{sec:experiments} and appendix~\\ref{sec:additional_expe}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{in-depth-inter-small.pdf}\n\\end{center}\n\\caption{Representation of a set of sibling nodes as intersection of neighborhoods.}\n\\label{fig:inter}\n\\end{figure}\n\n\\paragraph{Rips Complex.} Rips complexes are geometric flag complexes\nwhich are popular in computational topology due to their simple\nconstruction and their good approximation\nproperties~\\cite{DBLP:conf\/compgeom\/AttaliLS11,DBLP:conf\/compgeom\/ChazalO08}. Given a set of vertices $V$ in a\nmetric space and a parameter $r>0$, the Rips graph is defined as the\ngraph whose set of vertices is $V$ and two vertices are joined by an\nedge if their distance is at most $r$. The Rips complex\nis the flag complex defined on top of this graph. We will use this\ncomplex for our experiments on the construction of flag complexes.\n\n\\subsection{Witness complexes}\n\\label{subsec:witcpx}\n\n\\paragraph{The Witness Complex.}\nhas been first introduced in~\\cite{Alexa_topologicalestimation}.\nIts definition involves two given sets of points in a metric space, the set of landmarks\n$L$ and the set of witnesses $W$.\n\n\\begin{definition}\nA witness $w \\in W$ \\emph{witnesses} a simplex $\\sigma \\subseteq L$ iff:\n$$\\forall x \\in \\sigma \\mbox{ and } \\forall y \\in L\\setminus \\sigma\n\\mbox{ we have } \\text{d}(w,x) \\leq \\text{d}(w,y)$$\n\\end{definition}\n\nFor simplicity of exposition, we will suppose that no\nlandmarks are at the exact same distance to a witness. In this case, a\nwitness $w \\in W$ \\emph{witnesses} a simplex $\\sigma \\subseteq L$ iff\nthe vertices of $\\sigma$ are the $|\\sigma|$ nearest neighbors of $w$ in $L$.\nWe study later\nthe construction of the \\emph{relaxed witness complex}, which is a\ngeneralization of the witness complex which includes the case where\npoints are not in general position.\n\nThe \\emph{witness complex} ${\\rm Wit}(W,L)$ is the maximal\nsimplicial complex, with vertices in $L$, whose faces admit a witness\nin $W$. Equivalently, a simplex belongs to the witness complex if and\nonly if it\nis witnessed and  all its facets belong to  the witness\ncomplex. A simplex satisfying this property will be called \\emph{fully witnessed}. \n\n\\paragraph{Construction Algorithm.}\nWe suppose the sets $L$ and $W$ to be finite and give them labels $\\{1,\n\\cdots , |L|\\}$ and $\\{1, \\cdots , |W|\\}$ respectively. We describe\nhow to construct\nthe $k$-skeleton of the witness complex,  where $k$ may be any integer\nin $\\{1,\\cdots , |L|-1\\}$.\n\nOur construction algorithm is incremental, from lower to higher\ndimensions. At step $j$ we insert in the simplex tree the\n$j$-dimensional fully witnessed simplices.\n\nDuring the construction of the $k$-skeleton of the witness complex, we\nneed to access the nearest neighbors of the witnesses, in $L$. To do so, we compute the $k+1$ nearest neighbors\nof all the witnesses in a preprocessing phase, and store them in a\n$|W|\\times (k+1)$ matrix. Given an index $j \\in \\{0, \\cdots ,k\\}$ and a witness\n$w \\in W$, we\ncan then access in constant time the $(j+1)^{\\text{th}}$ nearest neighbor of $w$. \nWe denote this landmark by $s_j^w$. We maintain a list of \\emph{active witnesses}, initialized with\n$W$. \nWe insert the vertices of ${\\rm Wit}(W,L)$ in the simplex tree. For each\nwitness $w\\in W$ we insert a top node storing the label of the nearest\nneighbor of $w$ in $L$, if no such node already exists. $w$ is initially an\nactive witness and we make it point to the node mentionned above, representing the\n$0$-dimensional simplex $w$ witnesses.\n\nWe maintain the following loop invariants: \n\\begin{enumerate}\n\\item at the beginning of iteration $j$, the simplex tree\ncontains the $(j-1)$-skeleton of the witness complex\n${\\rm Wit}(W,L)$\n\\item the active witnesses are the elements of $W$ that\nwitness a\n$(j-1)$-simplex of the complex; each active witness $w$ points to the node\nrepresenting the $(j-1)$-simplex in the tree it witnesses.\n\\end{enumerate}\n\nAt iteration $j\\geq 1$, we traverse the list of active witnesses. Let\n$w$ be an active witness. We first retrieve the $(j+1)^{\\text{th}}$ nearest neighbor\n$s_j^w$ of $w$ from the nearest neighbors matrix (Step 1).  Let $\\sigma_j$ be the $j$-simplex \nwitnessed by $w$ and let us decompose the word representing $\\sigma_j$ into $[\\sigma_j] = [\\sigma']\\centerdot\n[s_j^w]\\centerdot[\\sigma'']$ (``$\\centerdot$'' denotes the concatenation of words).\nWe then look for the location in\nthe tree where $\\sigma_j$ might be inserted (Step 2).\nTo do so, we start at the node $N_w$ which represents the\n$(j-1)$-simplex  witnessed by $w$. Observe\nthat the word associated to the path from the root to $N_w$ is exactly $[\\sigma']\\centerdot\n[\\sigma'']$. We walk $|[\\sigma'']|$ steps up \nfrom $N_w$, reach the node representing $[\\sigma']$ and then search\ndownwards for the word\n$[s_w^j]\\centerdot[\\sigma'']$ (see Figure~\\ref{fig:algos_wc}, left). The cost of this operation is\n$O(jD_{\\text{m}})$.\n\nIf the node representing $\\sigma_j$ exists, $\\sigma_j$ has already been inserted; we\nupdate the pointer of $w$ and return. If the simplex tree contains neither this node\nnor its father, $\\sigma_j$ is not fully witnessed because the facet\nrepresented by its longest prefix is missing. We consequently remove $w$\nfrom the set of active witnesses.\nLastly, if the node is not in the tree but its father is, we check whether\n$\\sigma_j$ is fully witnessed.  To do so, we\nsearch for the $j+1$ facets of $\\sigma_j$ in the simplex tree (Step\n3). The cost of this \noperation is $O(j^2D_{\\text{m}})$ using the {\\sc Locate-facets}\nalgorithm described in section~\\ref{subsec:op_on_st}. If $\\sigma_j$ is fully witnessed, we\ninsert $\\sigma_j$  in the simplex tree and update the pointer of the active\nwitness $w$. Else, we remove $w$ from the list of active witnesses (see\nFigure~\\ref{fig:algos_wc}, right).\n\nIt is easily seen that the loop invariants are satisfied at the end of\niteration~$j$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=11cm]{algos_wc.pdf}\n\\end{center}\n\\caption{Third iteration of the witness complex construction. The\n  active witness $w$ witnesses the tetrahedron $\\{2,3,4,5\\}$ and points\nto the triangle $\\{2,4,5\\}$. (Left) Search for the potential position\nof the simplex $\\{2,3,4,5\\}$ in the simplex tree. (Right) Facets\nlocation for simplex $\\{2,3,4,5\\}$, and update of the pointer of the\nactive witness $w$.}\n\\label{fig:algos_wc}\n\\end{figure}\n\n\\paragraph{Complexity.}\nThe cost of accessing a neighbor of a witness using the nearest\nneighbors matrix is $O(1)$.\nWe access a neighbor (Step 1) and locate a node in the simplex tree\n(Step 2) at most $k|W|$ times. In total, the cost of Steps 1 and 2\ntogether is\n$O(|W|k^2 D_{\\text{m}})$. In Step 3, either we insert\na new node in the simplex tree, which happens exactly $|\\mathcal{K}|$ times\n(the number of faces in the complex), or we remove an active\nwitness, which happens at most $|W|$ times. The total cost of Step\n3 is thus $O( (|\\mathcal{K}| + |W|)k^2D_{\\text{m}})$. In conclusion, \nconstructing the $k$-skeleton of the witness complex takes time\n$$O((|\\mathcal{K}| + |W|)k^2D_{\\text{m}} + k|W|) = O((|\\mathcal{K}| + |W|)k^2D_{\\text{m}}).$$\n\n\\paragraph{Landmark Insertion.}\n\nWe present an algorithm to update the simplex tree under landmark\ninsertions. Adding new vertices in witness complexes is used\nin~\\cite{DBLP:journals\/dcg\/BoissonnatGO09} for manifold reconstruction.\nGiven the set of landmarks $L$, the set of witnesses $W$ and the $k$-skeleton of the\nwitness complex ${\\rm Wit}(W,L)$ represented as a simplex tree, we take a new landmark point\n$x$ and we update the simplex tree so as to construct the simplex tree\nassociated to ${\\rm Wit}(W,L\\cup \\{x\\})$. We assign to $x$ the biggest label $|L|+1$. We suppose to\nhave at our disposal an oracle that can compute the subset $W^x\n\\subseteq W$ of the witnesses that\nadmit $x$ as one of their $k+1$ nearest neighbors. \nComputing $W^x$ is known as the \\emph{reverse nearest neighbor} search\nproblem, which\nhas been intensively studied in the past few years~\\cite{DBLP:conf\/sigmod\/AchtertBKKPR06}.\nLet $w$ be a witness in $W^x$ and suppose $x$ is its $(i+1)^{\\text{th}}$\nnearest neighbor in $L\\cup \\{x\\}$, with $0\\leq i \\leq k$. Let\n$\\sigma_j \\subseteq L$\nbe the $j$-dimensional simplex witnessed by $w$ in $L$ and let\n$\\widetilde{\\sigma_j} \\subseteq L\\cup \\{x\\}$ be the $j$-dimensional simplex\nwitnessed by $w$ in $L\\cup \\{x\\}$. Consequently, $\\sigma_j = \\widetilde{\\sigma_j}$ for $j < i$\nand $\\sigma_j \\neq \\widetilde{\\sigma_j}$ for $j \\geq i$.\nWe equip each node $N$ of the simplex tree with a \\emph{counter of witnesses} which maintains\nthe number of witnesses that witness the simplex represented by $N$. As\nfor the witness complex construction, we consider all\nnodes representing simplices witnessed by elements of $W^x$,\nproceeding by increasing dimensions. \nFor a witness $w\\in W^x$ and a dimension  $j \\geq\ni$, we decrement the witness counter of $\\sigma_j$ and insert\n$\\widetilde{\\sigma_j}$ if and only if  its facets are in the simplex tree. We\nremark that $[\\widetilde{\\sigma_j}] = [\\sigma_{j-1}]\\centerdot [x]$\nbecause $x$ has the biggest label of all landmarks. We can thus access in time\n$O(D_{\\text{m}})$ the position of the word  $[\\widetilde{\\sigma_j}]$ since we\nhave accessed the node representing $[\\sigma_{j-1}]$ in the previous iteration of\nthe algorithm.\n\nIf the witness counter of a node is turned down to $0$, the simplex\n$\\sigma$ it represents is not witnessed anymore, and is consequently\nnot part of ${\\rm Wit}(W,L\\cup \\{x\\})$. We remove the nodes representing\n$\\sigma$ and its cofaces from the simplex tree, using {\\sc Locate-cofaces}.\n\n\\paragraph{Complexity.}\n\nThe update procedure is  a ``local''  variant of the witness complex construction,\nwhere, by ``local'',  we mean that we reconstruct only the star of vertex\n$x$. Let $C_x$ denote  the number of cofaces of $x$ in ${\\rm Wit}(W,L\\cup\n\\{x\\})$ (or equivalently the size of its star).  The same analysis as above shows that\nupdating the simplicial complex takes time $O( (|W^x| +\nC_x)k^2D_{\\text{m}})$, plus one call to the oracle to compute $W^x$.\n\n\n\\paragraph{Relaxed Witness Complex.}\nGiven a relaxation parameter $\\rho\\geq 0$ we define the \\emph{relaxed\n  witness complex}~\\cite{Alexa_topologicalestimation}:\n\n\\begin{definition}\nA witness $w\\in W$ \\emph{$\\rho$-witnesses} a simplex $\\sigma \\subseteq L$ iff:\n$$\\forall x \\in \\sigma \\mbox{ and } \\forall y \\in L\\setminus \\sigma\n\\mbox{ we have } d(w,x) \\leq d(w,y)+\\rho$$\n\\end{definition}\n\nThe \\emph{relaxed witness complex} $\\mbox{{\\rm Wit}}^{\\rho}$$(W,L)$ with parameter $\\rho$ is the maximal simplicial complex,\n  with vertices in $L$, whose faces admit a $\\rho$-witness in $W$. For\n  $\\rho=0$, the relaxed witness complex is the standard witness\n  complex. The parameter $\\rho$ defines a filtration on the witness\n  complex, which has been used in topological data analysis.\n\nWe resort to the same incremental algorithm as above. At each step\n$j$, we insert,\nfor each witness $w$, the $j$-dimensional\nsimplices which are $\\rho$-witnessed by $w$. \nDifferently from the standard witness complex, there may be more than\none $j$-simplex that is witnessed by a\ngiven witness $w\\in W$. Consequently, we do not\nmaintain a pointer from each active witness to the last inserted simplex\nit witnesses. We \nuse simple top-down insertions from the root of the simplex tree.\n\nGiven a witness $w$ and a dimension $j$, we generate all the $j$-dimensional simplices which are\n$\\rho$-witnessed by $w$. For the ease of exposition, we\nsuppose we are given the sorted list of\nnearest neighbors of $w$ in $L$, noted $\\{z_0 \\cdots z_{|L|-1}\\}$, and \ntheir distance to $w$,\n noted $m_i = \\text{d}(w,z_i)$, with\n$m_0\\leq \\cdots \\leq m_{|L|-1}$, breaking ties arbitrarily. Note that if one\nwants to construct only the $k$-skeleton of the complex, it is\nsufficient to know the list of\nneighbors of $w$ that are at distance at most $m_k +\\rho$\nfrom $w$. We preprocess this list of neighbors for all witnesses. \nFor $i \\in \\{0, \\cdots, |L|-1\\}$, we define the set $A_i$ of\nlandmarks $z$ such that $ m_i \\leq d(w,z) \\leq m_i+\\rho$. For $i \\leq\nj+1$, $w$ $\\rho$-witnesses all the $j$-simplices that contain\n$\\{z_0, \\cdots , z_{i-1}\\}$ and a $(j+1-i)$-subset of $A_i$, provided\n$|A_i|\\geq j+1-i$. We see that all $j$-simplices that are\n$\\rho$-witnessed by $w$ are obtained this way, and exactly once, when\n$i$ ranges from $0$ to $j+1$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{relax-algo.pdf}\n\\end{center}\n\\caption{Computation of the $\\rho$-witnessed simplices $\\sigma$ of\n  dimension $5$. If $z_3$ is the first neighbor of $w$ not in\n  $\\sigma$, then $\\sigma$ contains $\\{z_0,z_1,z_2\\}$ and any $3$-uplet\nof $A_3 = \\{z_4,\\cdots,z_8\\}$.}\n\\label{fig:relax-algo}\n\\end{figure}\n\nFor all $i \\in \\{0, \\cdots , j+1\\}$, we compute $A_i$ and generate\nall the simplices which contain $\\{z_0, \\cdots , z_{i-1}\\}$ and a \nsubset of $A_i$ of size $(j+1-i)$. In order to easily update $A_i$ when $i$ is\nincremented, we maintain two pointers to the list of neighbors, one\nto $z_i$  and the other to the end of $A_i$. We check in constant time if\n$A_i$ contains more than $j+1-i$ vertices, and compute all the\nsubsets of $A_i$ of cardinality $j+1-i$ accordingly. See Figure~\\ref{fig:relax-algo}.\n\n\\paragraph{Complexity.} \nLet $R_j$ be the number of $j$-simplices $\\rho$-witnessed by $w$.\nGenerating all those simplices takes $O(j + R_j)$ time. Indeed, for\nall $i$ from $0$ to $j+1$, we construct $A_i$ and check whether $A_i$\ncontains more than $j+1-i$ elements. This is done by a simple\ntraversal of  the list of neighbors of $w$, which takes $O(j)$ time. Then, when\n$A_i$ contains more than $j+1-i$ elements, we generate all subsets of\n$A_i$ of size $j+1-i$ in time $O( \\binom{|A_i|}{j+1-i} )$. As each\nsuch subset leads to a $\\rho$-witnessed simplex, the total cost\nfor generating all those simplices is $O(R_j)$.\n\nWe can deduce the complexity of the construction of the\nrelaxed witness complex. Let $\\mathcal{R} = \\displaystyle \\sum_{w\\in\n  W} \\sum_{j=0\\cdots k} R_j$ be the number of $\\rho$-witnessed\nsimplices we try to insert.  The construction of the relaxed witness complex takes\n$O(\\mathcal{R} k^2D_{\\text{m}})$ operations. This bound is quite pessimistic\nand, in practice, we observed that the construction time\nis sensitive to the size of the output complex.\nObserve that the quantity analogous to $\\mathcal{R}$ in the case of\nthe standard witness complex was $k|W|$ and that the complexity was better\ndue to our use of the notion of active witnesses.\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\begin{figure}\n\\begin{center}\n\\setlength{\\tabcolsep}{3.72pt}\n\\begin{tabular}{|l | r  r  r  r  r  r  r  r  r  r|}\n\\hline\nData &  $|\\mathcal{P}|$& $D$& $d$& $r$& $T_{\\mbox{g}}$& $|E|$& $T_{\\mbox{Rips}}$& $|\\mathcal{K}|$& $T_{\\mbox{tot}}$& $T_{\\mbox{tot}}\/|\\mathcal{K}|$\\\\\n\\hline\n{\\bf Bud} & 49,990& 3& 2& 0.11& 1.5& 1,275,930& 104.5& 354,695,000& 104.6& $3.0 \\cdot 10^{-7}$\\\\\n{\\bf Bro} & 15,000& 25& ?& 0.019& 0.6& 3083& 36.5& 116,743,000& 37.1& $3.2 \\cdot 10^{-7}$\\\\\n{\\bf Cy8}& 6,040& 24& 2& 0.4& 0.11& 76,657& 4.5& 13,379,500& 4.61& $3.4 \\cdot 10^{-7}$\\\\\n{\\bf Kl}   & 90,000& 5& 2& 0.075& 0.46& 1,120,000 & 68.1&  233,557,000& 68.5& $2.9 \\cdot 10^{-7}$\\\\\n{\\bf S4} & 50,000& 5& 4& 0.28& 2.2&\n1,422,490& 95.1& 275,126,000&  97.3 & $3.6 \\cdot 10^{-7}$\\\\\n\\hline\n\\end{tabular}\n\\setlength{\\tabcolsep}{3.6pt}\n\\begin{tabular}{|l|  r  r  r  r  r c r  r  r  r  r|}\n\\hline\nData &$|L|$ & $|W|$ & $D$ & $d$ & $\\rho$ && $T_{\\mbox{nn}}$ & $T_{\\mbox{$\\mbox{{\\rm Wit}}^{\\rho}$}}$ & $|\\mathcal{K}|$ & $T_{\\mbox{tot}}$ & $T_{\\mbox{tot}}\/|\\mathcal{K}|$\\\\\n\\hline\n{\\bf Bud}\n&10,000 & 49,990 & 3 & 2 & 0.12 && 1. &  729.6 & 125,669,000 & 730.6 & $0.58\n\\cdot 10^{-5}$\\\\\n{\\bf Bro}\n&3,000 & 15,000 & 25 & ? & 0.01 && 9.9 & 107.6 & 2,589,860 & 117.5 & $4.5 \\cdot 10^{-5}$\\\\\n{\\bf Cy8}&\n800 & 6,040  &  24  & 2  & 0.23 &&  0.38\n& 161 &  997,344  &  161.2  &  $16 \\cdot 10^{-5}$ \\\\\n{\\bf Kl}\n&10,000 & 90,000 & 5 & 2 & 0.11 && 2.2 & 572 & 109,094,000 & 574.2 & $0.53\\cdot 10^{-5}$\\\\\n{\\bf S4} &  50,000 & 200,000    &  5  & 4   &  0.06  && 25.1   & 296.7  &  163,455,000  &  321.8   & $0.20 \\cdot 10^{-5}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Data, timings (in s.) and statistics for the construction of\n Rips complexes (TOP)  and relaxed witness complexes  (BOTTOM). All\n complexes are constructed up to embedding dimension.}\n\\label{fig:table_rips_wc}\n\\end{figure}\nIn this section, we report on the performance of our algorithms on both\nreal and synthetic data, and compare them to existing software. More\nspecifically, we benchmark the construction of Rips complexes, witness complexes and relaxed\nwitness complexes. Our implementations are in {\\tt C++}. We use the {\\sc ANN}\nlibrary~\\cite{ann_mount} to compute the $1$-skeleton graph of the Rips\ncomplex, and to compute the lists of nearest neighbors of the\nwitnesses for the witness complexes. All\ntimings are measured on a Linux machine with $3.00$ GHz processor and $32$\nGB RAM. For its efficiency and flexibility, we use the {\\tt map} container\nof the {\\tt Standard Template Library}~\\cite{SGIguide} for storing\nsets of sibling nodes, except for the top nodes which are stored in an\narray. \n\nWe use a variety of both real and synthetic datasets. {\\bf Bud} is a\nset of points sampled from the surface of the {\\it Stanford Buddha}~\\cite{buddha_stanford_scan} in\n$\\mathbb{R}^3$. {\\bf Bro} is a set of $5\\times 5$ {\\it high-contrast patches}\nderived from natural images, interpreted as vectors in $\\mathbb{R}^{25}$, from the Brown database (with parameter $k=300$ and cut\n$30\\%$)~\\cite{DBLP:journals\/ijcv\/CarlssonISZ08,DBLP:journals\/ijcv\/LeePM03}. {\\bf Cy8} is a set of\npoints in $\\mathbb{R}^{24}$, sampled from the space of conformations of the\ncyclo-octane molecule~\\cite{martin2010top}, which is the union of two\nintersecting surfaces. {\\bf Kl} is a set of points sampled from the\nsurface of the figure eight Klein Bottle embedded in $\\mathbb{R}^5$. Finally\n{\\bf S4} is a set of points uniformly distributed on the unit $4$-sphere in\n$\\mathbb{R}^5$. Datasets are listed in Figure~\\ref{fig:table_rips_wc} with\ndetails on the sets of points $\\mathcal{P}$ or landmarks $L$ and\nwitnesses $W$,\ntheir size $|\\mathcal{P}|$, $|L|$ and $|W|$, the ambient dimension $D$, the intrinsic\ndimension $d$ of the object the sample points belong to (if known),\nthe parameter $r$ or $\\rho$, the dimension $k$ up to which we construct\nthe complexes, the time $T_{\\mbox{g}}$  to construct the Rips graph or\nthe time $T_{\\mbox{nn}}$ to compute the lists of nearest neighbors of\nthe witnesses, the number of edges $|E|$, the time for the\nconstruction of \nthe Rips\ncomplex $T_{\\mbox{Rips}}$ or for the construction of the witness complex $T_{\\mbox{$\\mbox{{\\rm Wit}}^{\\rho}$}}$, the size of the complex\n$|\\mathcal{K}|$, and the total construction time $T_{\\mbox{tot}}$ and\naverage construction time per face $T_{\\mbox{tot}} \/ |\\mathcal{K}|$.\n\nWe test our algorithms on these datasets, and\ncompare their performance with two existing softwares that are state-of-the-art. \nWe compare our implementation to the\n{\\sc JPlex}~\\cite{jplex_cite} library and the {\\sc Dionysus}~\\cite{dionysus_morozov}\nlibrary. The first is a Java package which can be used with\n{\\tt Matlab} and provides an implementation of the construction of\nRips complexes and witness complexes. The\nsecond is implemented in {\\tt C++} and provides an implementation of\nthe construction of Rips complexes. Both libraries are\nwidely used to construct simplicial complexes and to compute their\npersistent homology.\nWe also provide an experimental analysis of the memory\nperformance of our data structure compared to other\nrepresentations. Unless mentioned otherwise, all simplicial complexes\nare computed up to the embedding dimension.\n\nAll timings are\naveraged over $10$ independent runs. Timings are provided by the {\\tt clock} function from the {\\tt\n  Standard C Library}, and zero means that the measured time is below\nthe resolution of the {\\tt clock} function. Experiments are stopped\nafter one hour of computation, and data missing on plots means that\nthe computation ran above this time limit.\n\n\nFor readability, we do not report on\nthe performance of each algorithm on each dataset in this section, but the \nresults presented are a faithful sample of what we have\nobserved on other datasets. A complete set of experiments is reported\nin appendix~\\ref{sec:additional_expe}.\n\n\nAs illustrated in Figure~\\ref{fig:table_rips_wc}, we are able to\nconstruct and represent both Rips and relaxed witness complexes of up to several hundred million\nfaces in high dimensions, on all datasets.\n\n\n\\paragraph{Data structure in {\\sc JPlex} and {\\sc Dionysus}:} \nBoth {\\sc JPlex} and {\\sc Dionysus} represent the combinatorial\nstructure of a simplicial complex by its \\emph{Hasse diagram}.\nThe \\emph{Hasse diagram} of a\nsimplicial complex $\\mathcal{K}$ is the graph whose nodes are in\nbijection with the \nsimplices (of all dimensions) of the simplicial complex and where an edge\nlinks two nodes representing two simplices $\\tau$ and $\\sigma$ iff \n$\\tau \\subseteq \\sigma$ and $\\Dim{\\sigma} = \\Dim{\\tau}+1$.\n\n {\\sc JPlex} and {\\sc Dionysus} are\n libraries dedicated to topological data analysis, where only the\n construction of simplicial complexes and the computation of the\n facets of a simplex are necessary.\n\nFor a simplicial complex $\\mathcal{K}$ of dimension $k$ and a simplex $\\sigma\n\\in \\mathcal{K}$\nof dimension $j$, the Hasse diagram has\nsize $\\Theta(k|\\mathcal{K}|)$ and allows to compute {\\sc Locate-facets}$(\\sigma)$\nin time $O(j)$, whereas the simplex tree has size $\\Theta(|\\mathcal{K}|)$ and\nallows to compute {\\sc Locate-facets}$(\\sigma)$ in\ntime $O(j^2D_{\\text{m}})$.\n\n\n\\subsection{Memory Performance of the Simplex Tree}\n\n \\begin{figure}\n\n\n\\setlength{\\tabcolsep}{13pt}\n \\hspace{-0.5cm}\\begin{tabular}{c c}\n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure2.pdf}\n    \\end{minipage}\n    &  \n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure3.pdf}\n    \\end{minipage}\\\\\n  \\end{tabular}\n\n\\caption{Statistics and timings for the Rips complex (Left)\n  and the relaxed witness complex (Right) on {\\bf S4}.} \n\\label{fig:size_time}\n\\end{figure}\n\nIn order to represent the combinatorial structure of an arbitrary\nsimplicial complex, one needs to mark all maximal faces. Indeed,\nexcept in some special cases (like in flag complexes where all faces\nare determined by the $1$-skeleton of the complex), one cannot infer\nthe existence of a simplex in a simplicial complex $\\mathcal{K}$ from the\nexistence of its faces in $\\mathcal{K}$.\nMoreover, the number of\nmaximal simplices of a $k$-dimensional simplicial complex is at least\n$|V| \/(k+1)$. In the case, considered in this paper, where\nthe vertices are identified by their labels, a  \nminimal representation of the maximal simplices would then require at\nleast $\\Omega (\\log |V|)$ bits per\nmaximal face, for fixed $k$.\nThe simplex tree \nuses $O(\\log |V|)$ memory bits per face {\\em of any dimension}. The following\nexperiment compares the memory performance of the simplex tree with\nthe minimal representation described above, and with the\nrepresentation of the $1$-skeleton.\n\nFigure~\\ref{fig:size_time} shows results for both Rips \nand relaxed witness complexes associated to $10,000$ points from\n{\\bf S4}\nand various values of, respectively, the distance threshold $r$ and the\nrelaxation parameter $\\rho$. \nThe figure plots the total number of faces $|\\mathcal{K}|$, the number of\nmaximal faces $|\\mbox{m}\\mathcal{F}|$,\nthe size of the $1$-skeleton $|\\mathcal{G}|$ and the construction times\n$T_{\\mbox{Rips}}$ and $T_{\\mbox{$\\mbox{{\\rm Wit}}^{\\rho}$}}$.\n\nAs expected, the $1$-skeleton is significantly smaller than the two\nother representations. However, as explained earlier, a representation\nof the graph of the simplicial complex is only well suited for\nflag complexes.\n\nAs shown on the figure, the total number of faces and the number of maximal faces remain\nclose along the experiment. Interestingly, we catch the topology of {\\bf S4} when\n $r\\approx 0.4$ for the Rips complex and $\\rho \\approx 0.08$ for the relaxed witness\n  complex. For these ``good'' values of the parameters, the total\n  number of faces is not much bigger than the number of maximal\nfaces. Specifically, the total number of faces of the Rips complex is less than $2.3$\ntimes bigger than the number of maximal faces, and the ratio is less than $2$\nfor the relaxed witness complex.\n\n\n\\subsection{Construction of Rips Complexes}\n\\begin{figure}\n  \n\n\\setlength{\\tabcolsep}{13pt}\n \\hspace{-0.5cm}\\begin{tabular}{c c}\n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure4.pdf}\n    \\end{minipage}\n    &  \n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure5.pdf}\n    \\end{minipage}\\\\\n  \\end{tabular}\n\n\\caption{Statistics and timings for the construction of the Rips\n    complex on (Left) {\\bf Bud} and (Right) {\\bf Cy8}.} \n\\label{fig:plots_time_rips}\n\\end{figure}\n\n\nWe test our algorithm for\nthe construction of Rips complexes.\nIn Figure~\\ref{fig:plots_time_rips} we compare the performance of our\nalgorithm with {\\sc JPlex} and with {\\sc Dionysus} along two directions.\n\nIn the first experiment, we build the Rips complex on $49,000$ points\nfrom the dataset {\\bf Bud}. Our construction is\nat least $36$ times faster than {\\sc JPlex} along the experiment, and\nseveral hundred times faster for small values of the parameter $r$. Moreover, {\\sc\n  JPlex} is not able to handle the full dataset {\\bf Bud} nor big simplicial complexes due to memory\nallocation issues, whereas our method has no such problems. In our experiments, {\\sc\n  JPlex} is not able to compute complexes of more than $23$ million faces ($r=0.07$)\nwhile the simplex tree construction runs successfully until\n$r=0.11$, resulting in a complex of $237$ million faces.\nOur construction is at least $7$ times faster than {\\sc Dionysus}\nalong the experiment, and several hundred times faster for small\nvalues of the parameter $r$.\n\nIn the second experiment, we construct the Rips complex on the $6040$\npoints from {\\bf Cy8},\nwith threshold $r=0.4$, for different\ndimensions $k$. Again, our method outperforms {\\sc JPlex}, by a factor\n$11$ to $14$. {\\sc JPlex} cannot compute complexes of dimension higher than\n$7$ because it is limited by design to simplicial complexes of dimension smaller than $7$.\nOur construction is $4$ to $12$ times faster than {\\sc Dionysus}.\n\nThe simplex tree and the expansion algorithm we have described are\noutput sensitive. \nAs shown by our experiments, the construction time using a simplex tree depends linearly on the size of the output complex. Indeed, when the Rips graphs are dense enough\nso that the time for the expansion dominates the full\nconstruction, we observe that the average construction time per face is\nconstant and equal to\n$3.7\\times 10^{-7}$ seconds for the first experiment, and $4.1\\times\n10^{-7}$ seconds for the second experiment (with standard errors\n$0.20\\%$ and $0.14\\%$ respectively).\n\n\\subsection{Construction of Witness Complexes}\n\\begin{figure}\n\n\\setlength{\\tabcolsep}{13pt}\n \\hspace{-0.5cm}\\begin{tabular}{c c}\n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure6.pdf}\n    \\end{minipage}\n    &  \n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure7.pdf}\n    \\end{minipage}\\\\\n \n\\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure8.pdf}\n    \\end{minipage}\n    &  \n    \\begin{minipage}[b]{0.45\\linewidth}\n      \\centering\n      \\includegraphics{Algorithmica_ST-figure9.pdf}\n    \\end{minipage}\\\\\n\n  \\end{tabular}\n\n\n\n\n\n\n\n\n\n\\caption{Statistics and timings for the construction of: (TOP) the\n  witness complex and (BOTTOM) the relaxed witness complex, on\n  datasets (Left) {\\bf Bro} and (Right) {\\bf Kl}.}\n\\label{fig:non-relaxed-wc--new}\n\\end{figure}\n\nWe test our algorithms for the construction of witness complexes and\nrelaxed witness complexes. \n\nFigure~\\ref{fig:non-relaxed-wc--new} (top) shows the results of two\nexperiments for the full construction of witness complexes. The first\none compares the performance of the simplex tree algorithm and\nof {\\sc JPlex} on the dataset {\\bf Bro} consisting of $15,000$\npoints in dimension $\\mathbb{R}^{25}$. Subsets of different size of landmarks are selected at random\namong the sample points. \nOur algorithm is from several hundred to several thousand times faster\nthan {\\sc JPlex} (from small to big subsets of landmarks). Moreover, the simplex\ntree algorithm for the construction of the witness complex represent\nless than $1\\%$ of the total time spent, when more than $99\\%$ of the\ntotal time is spent computing the\nnearest neighbors of the witnesses.\n\nIn the second experiment, we construct the witness complex on $2,500$\nlandmarks from {\\bf Kl}, and sets of witnesses of different size.\nThe simplex tree algorithm outperforms {\\sc JPlex}, being tens of\nthousands times faster. {\\tt JPlex} runs above the one hour time limit\nwhen the simplex tree algorithm stays under\n$0.1$ second all along the experiment. Moreover, the simplex\ntree algorithm spends only about $10\\%$ of the time constructing the\nwitness complex, and $90\\%$ computing the\nnearest neighbors of the witnesses.\n\nFinally we test the full construction of the relaxed witness complex.\n{\\sc JPlex} does not provide an implementation of the relaxed\nwitness complex as defined in this paper; consequently, we were not\nable to compare the algorithms on the construction of the relaxed witness complex.\nWe test our algorithms along two directions, as illustrated in Figure~\\ref{fig:non-relaxed-wc--new} (bottom).\nIn the first experiment, we compute the $5$-skeleton of the relaxed\nwitness complex on {\\bf Bro}, with $15,000$ witnesses and $1,000$\nlandmarks selected randomly, for different values of the parameter\n$\\rho$. \nIn the second experiment, we construct the $k$-skeleton of the relaxed witness complex on\n{\\bf Kl} with $10,000$ landmarks, $100,000$ witnesses and fixed\nparameter $\\rho=0.07$, for various\n$k$. We are able to construct and store complexes of up to $260$ million faces.\nIn both cases the construction time is linear in the size of the\noutput complex, with a contruction time per face equal to $4.9 \\times\n10^{-6}$ seconds in the first experiment, and $4.0 \\times 10^{-6}$\nseconds in the second experiment (with standard errors $1.6\\%$ and\n$6.3\\%$ respectively).\n\n\\section*{Conclusion}\n\\label{sec:conclusion}\n\n We believe that the simplex tree is the first scalable and truly practical data structure\n to represent general simplicial complexes. The simplex tree is very\n flexible, can represent any kind of simplicial complexes and allow\n efficient implementations of all basic operations on simplicial\n complexes. \nFuthermore, since the simplex tree stores all simplices of the\nsimplicial complex, it has been successfully applied to represent\nfiltrations and to compute persistent homology~\\cite{boissonnat:hal-00761468}.\nWe plan to make our code publicly available and to use it for practical\n applications in data analysis and manifold learning.\nFurther developments also include more compact storage\nusing succinct representations of\ntrees~\\cite{DBLP:conf\/focs\/Jacobson89}.\n\n\n\n\\begin{acknowledgements}\n The authors thanks A.Ghosh, S. Hornus, D. Morozov and P.  Skraba for\n discussions that led to the idea of representing simplicial complexes\n by tries. They especially thank S. Hornus for sharing his notes with\n u\n. They also thank S. Martin and\n V. Coutsias for providing the cyclo-octane data set. \nThis research has been partially supported by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL Computational Geometry Learning).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}