diff --git a/data_all_eng_slimpj/shuffled/split2/finalzwja b/data_all_eng_slimpj/shuffled/split2/finalzwja new file mode 100644 index 0000000000000000000000000000000000000000..5fcf64983c257733b432677520bf38d9daf87762 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzwja @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{}\nEuropium compounds are known to show a variety of magnetic and thermal\nproperties due to existence of different valence \nstates~\\cite{novik82,gupta87,kitagawa02,tsutsui09,takikawa10}. In the\ndivalent compounds, Eu ions have 4$f^{7}$ configuration (Eu${}^{2+}$)\nwith atomic spin $S=7\/2$, angular momentum $L=0$, and total angular\nmomentum $J=7\/2$ showing a large magnetic moment, while\nin the trivalent compounds, Eu ions have 4$f^{6}$ configuration \n(Eu${}^{3+}$) with \n$S=L=3$, and $J=0$, thus no magnetic moment. Some of these compounds\nsuch as EuPd${}_{2}$Si${}_{2}$~\\cite{sampat81}, \nEuNi${}_{2}$(Si${}_{0.18}$Ge${}_{0.82}$)${}_{2}$~\\cite{matsuda09}, and \nEuRh${}_{2}$Si${}_{2}$~\\cite{mitsuda12} show the\nvalence instability with increasing temperature, pressure, and magnetic\nfield. In particular, EuNi${}_{2}$P${}_{2}$ has recently received much\nattention because it shows both heavy-fermion and mixed-valence\nbehaviors. In fact, M\\\"{o}ssbauer isomer-shift experiment reports a mixed\nvalence state even at zero temperature~\\cite{nagara85} and \nthe electronic specific heat coefficient $\\gamma$ shows a large value \nof 100 mJ\/(K${}^{2} \\cdot$mol)~\\cite{fisher95}. \n\nQuite recently, Hiranaka {\\it et. al.}~\\cite{hiranaka13} grew \nthe single crystalline\nEuNi${}_{2}$P${}_{2}$ and carried out systematic measurements of\nresistivity, specific heat, susceptibility, and thermal expansion. The\nlow-temperature data of resistivity and specific heat show the\nheavy-fermion behavior with a large electronic specific heat coefficient \n$\\gamma=93$ mJ\/(K${}^{2} \\cdot$ mol). \nThe susceptibility data show the Curie-Weiss\nbehavior with the effective Bohr magneton number $p_{\\rm eff}=7.4$\n$\\mu_{\\rm B}\/$Eu and the Weiss constant $\\Theta=-120$ K in the high\ntemperature regime, and becomes \nalmost constant ($\\approx 0.04$ emu\/mol) below 50 K. Anomalous 4$f$ \nelectronic contribution to the volume expansion $(\\Delta V\/V)_{4f}$\nis found to increase rapidly up to 100 K and tends to saturate with \nincreasing temperature. The new aspect which they found is that the \ntemperature variation of\n$(\\Delta V\/V)_{4f}$ scales well to that of the average Eu valence.\nBecause the present system shows both the heavy-fermion and mixed-valence\nbehaviors and the anomalous volume expansion continues up to 200 K, \nwhich is far above the experimentally suggested Kondo temperature \n$T_{\\rm K}$ ($\\sim 80$ K), the scaling relation may be controlled by \nan extra-parameter other than $T_{\\rm K}$, {\\it i.e.}, a valence\nfluctuation temperature $T_{v}$. \n\nIn this paper, we point out that the temperature dependence of volume\nthermal expansion and its scaling relation to the average 4$f$ electron\nnumber found in experiment are basically explained by a simple and \nphenomenological 2$\\gamma$-state model. \nThe model was first proposed by Weiss~\\cite{weiss63} to explain\nthe anomalous thermal expansion of $\\gamma$Fe and the Fe-Ni Invar\nalloys~\\cite{wasser90}. \nIt assumes the existence of two magnetic states, a low-spin\nsmall-volume state and a high-spin large-volume state. The former is\nassumed to be the ground state in $\\gamma$Fe, while the latter is\nassumed to be stabilized at the ground state in the Invar \nalloys~\\cite{wasser90,schilf99,lawson06}. One can describe the\nexcitations from the ground state to the independent excitations on\nsites with use of the Weiss model even in the \nitinerant electron system. \n\nWe remark that apart from the anomalous volume expansion, the Weiss\nmodel is somewhat similar to the interconfiguration fluctuation (ICF)\nmodel for some rare-earth\ncompounds~\\cite{nagara85,sales75,franz80,wada96,paramanik13}. The\npresent model however differes from the ICF model in the following\npoints.\n(1) The Weiss model assumes the nondegenerate metallic ground state\n({\\it i.e.}, a heavy-fermion state in the present case), while the ICF\nmodel assume the 4f atomic state with an integral 4f electron number as\nthe ground state. (2) The Weiss model does not make use of any other\nassumptions, while the ICF model introduces phenomenologically a\nfluctuation temperature ($T_{sf}$) corresponding to the width of the 4f\nlevel state. \n\nIn the following section, we apply the concept of the \n2$\\gamma$-state model in order to\nexplain a strong temperature dependence which is seen\nin the experimental data of \nEuNi${}_{2}$P${}_{2}$. \nWith use of the 2-state Weiss model, we will demonstrate that the model\nexplains an overall temperature dependence of thermal expansion, its\nscaling relation to the 4$f$ electron number, the specific heat, and the\nsusceptibility. We summalize the conclusion in the last section 3.\n\n\n\n\\section{Two-state Weiss model for EuNi${}_{2}$P${}_{2}$}\n\n\\subsection{Thermal expansion and scaling relation}\n\nExperimentally, a heavy-fermion state is realized in\nEuNi${}_{2}$P${}_{2}$ at low temperatures.\nWe therefore assume that the\nground state is a nonmagnetic heavy-fermion state with a small volume \nper atom $V_{\\rm L}$ as found experimentally, and call this state `the\nlow spin state' hereafter. We neglect the low energy excitations\nassociated with the heavy-fermions which yields the $T$-linear \nspecific heat. Instead, we take into account independent excitations to\nthe atomic Eu${}^{2+}$ state with \n$J=J_{\\rm H} \\, (=7\/2)$ and a large volume $V_{\\rm H}$.\nWe call the single-site excited state on each site \n`the high spin state'.\nThe volume per Eu atom is then characterized by $J$ as $V(J)$, and \nthe thermal average $V$ is given by \n\\begin{eqnarray}\nV = \\frac{\\displaystyle \\sum_{J}\\sum_{M=-J}^{J} V(J) \\, e^{-\\beta E_{J}}}\n{\\displaystyle \\sum_{J}\\sum_{M=-J}^{J} e^{-\\beta E_{J}}} \\, .\n\\label{vol0}\n\\end{eqnarray}\nHere $\\beta$ is the inverse temperature and $E_{J}$ denotes the energy\nof the system with the state $J$. \nNote that we allocated `$J=0$' to `the low-spin state' for\nconvenience. It does not means that the ground state is \nEu${}^{3+} (J=0)$. Instead, $E_{J=0}$ is defined by\nthe ground state energy per Eu atom $E_{\\rm L}$ and \n$V(J=0) \\equiv V_{\\rm L}$. Similarly, $E_{J_{\\rm H}} = E_{\\rm H}$\ndenotes the excitation energy per atom in the high-spin state \nand $V(J=J_{\\rm H}) \\equiv V_{\\rm H}$.\n\\begin{figure}\n\\includegraphics[scale=1.0]{vol.eps}\n\\caption{\nTemperature dependence of 4$f$ electron contribution to the thermal\n expansion ($\\Delta V\/V$). Closed circles: Experimental\n results~\\cite{hiranaka13}, \nsolid line: result of 2-state Weiss model for \n$\\Delta E\/k_{\\rm B}=150$ K.\n}\n\\label{figvol}\n\\end{figure}\n\nAfter simple calculations of the\nr.h.s. of Eq. (\\ref{vol0}), we obtain the following expression for \nvolume. \n\\begin{eqnarray}\nV = V_{\\rm L} + v_{0}\\frac{\\displaystyle w \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{vol1}\n\\end{eqnarray}\nHere $v_{0}=V_{\\rm H}-V_{\\rm L}$, $w=2J_{\\rm H}+1 \\, (= 8)$, and\n$\\Delta E = E_{\\rm H} - E_{\\rm L}$ is the excitation energy from \nthe low-spin state (L) to the high-spin state (H).\n$T_{v} \\equiv \\Delta E\/k_{\\rm B}$ is interpreted as a valence\nfluctuation temperature in the present system. \nThe thermal expansion\n$\\Delta V\/V_{\\rm L} = (V-V_{L})\/V_{L}$ is therefore given by\n\\begin{eqnarray}\n\\frac{\\Delta V}{V_{\\rm L}} = \\frac{\\Delta V(\\infty)}{V_{\\rm L}}\n\\frac{\\displaystyle (1+w) \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{vol2}\n\\end{eqnarray}\nHere $\\Delta V(\\infty)\/V_{\\rm L}$ denotes the volume change in the\nhigh-temperature limit. \n\nFigure 1 shows a numerical result of $\\Delta V\/V_{\\rm L}$ compared\nwith the experimental data. We adopted the experimental value\n$\\Delta V(\\infty)\/V_{\\rm L} = 7.3 \\times 10^{-3}$ at 300\nK~\\cite{hiranaka13} in the calculations. \nWith use of a characteristic temperature \n$T_{v} =\\Delta E\/k_{\\rm B}=150$ K, \nwe find that the formula (\\ref{vol2}) explains the overall feature of \nthe experimental thermal expansion. The deviation from the data at \nlow temperatures is attributed to the fact that the model does not \ntake into account the low-energy excitations associated with the \nheavy-fermion states.\n\\begin{figure}\n\\includegraphics[scale=1.0]{cv.eps}\n\\caption{\nSpecific heat in the present model (solid curve) and experimental data\n (closed circles)~\\cite{hiranaka13}.\nThin solid curve shows the electronic contribution $C^{({\\rm e})}_{v}$,\n which is calculated from Eq. (\\ref{cev}) with use of the same parameter\n $\\Delta E\/k_{\\rm B}=150$ K as in Fig. 1. Dashed curve is the lattice \ncontribution calculated by the Debye model. The Dulong-Petit parameter \n$A$ and the Debye temperature parameter $T_{\\rm D}$ are chosen to be \n$A=15.9$ in unit of the gas constant and $T_{\\rm D}=350$ K so that the \nexperimental data around 50 K and 200 K are reproduced. \n}\n\\label{figcv}\n\\end{figure}\n\nWe obtain in the same way the average 4$f$ electron number as\n\\begin{eqnarray}\nn_{f} = n_{f{\\rm L}} + n_{0}\\frac{\\displaystyle w \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{n4f}\n\\end{eqnarray}\nHere $n_{0}=n_{f{\\rm H}}-n_{f{\\rm L}}$, $n_{f{\\rm L}}$ ($n_{f{\\rm H}}$)\nbeing the electron number in the low-spin (high-spin) state. \nNote that $n_{f{\\rm L}}$ is the average $f$ electron number per Eu \natom in the heavy-fermion ground state ({\\it i.e.}, \n$n_{f{\\rm L}} \\neq 6$).\nEquation (\\ref{n4f}) implies that there is a scaling relation \nbetween the volume\nchange $\\Delta V\/V_{\\rm L}$ and that of 4$f$ electron \nnumber $\\Delta n_{f} = n_{f}-n_{f{\\rm L}}$:\n\\begin{eqnarray}\n\\frac{\\Delta V}{V_{\\rm L}} = \\frac{v_{0}}{n_{0}V_{\\rm L}} \\, \\Delta n_{f} .\n\\label{scalevn}\n\\end{eqnarray}\nThe relation is in agreement with the experimental \nfact~\\cite{hiranaka13}. According to\nthe M\\\"{o}ssbauer experiment, $n_{f}(T=0) \\sim 6.5$ and \n$n_{f}(T=\\infty) \\sim 6.75$~\\cite{nagara85}. \nUsing Eq. (\\ref{scalevn}) and these\nvalues, we obtain $n_{f{\\rm H}} \\sim 6.8$, which is rather close to the\natomic value 7.0.\n\n\\subsection{Specific heat}\n\nSimilar scaling relation is found between the electronic contribution \nto the volume thermal expansion coefficient \n$\\alpha^{({\\rm e})}_{v}= V_{\\rm L}^{-1} \\partial V\/\\partial T$ and \nthat to the heat capacity $C^{({\\rm e})}_{v}$. In fact, the\ninternal energy $E$ in the Weiss model is given by \n\\begin{eqnarray}\nE = E_{\\rm L} + \\Delta E \\, \\frac{\\displaystyle w \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{ener}\n\\end{eqnarray}\nThus we have a relation, \n\\begin{eqnarray}\nE - E_{\\rm L} = \\Delta E \\frac{V_{\\rm L}}{v_{0}} \n\\frac{\\Delta V}{V_{\\rm L}} \\, ,\n\\label{enern}\n\\end{eqnarray}\nand find a scaling relation,\n\\begin{eqnarray}\n\\alpha^{({\\rm e})}_{v} = \n\\frac{1}{\\Delta E} \\frac{v_{0}}{V_{\\rm L}} C^{({\\rm e})}_{v} \\, .\n\\label{alphav}\n\\end{eqnarray}\nHere $C^{({\\rm e})}_{v}$ is the electronic contribution in the present\nmodel, being given by \n\\begin{eqnarray}\nC^{({\\rm e})}_{v} = \\frac{\\displaystyle w \\, (\\beta \\Delta E)^{2} e^{-\\beta \\Delta E}}\n{\\displaystyle (1 + w \\, e^{-\\beta \\Delta E})^{2}} \\, .\n\\label{cev}\n\\end{eqnarray}\n\nIt should be noted however \nthat the relation (\\ref{alphav}) does not necessarily\nmean that the Schottky-like anomaly is visible in the experimental data\nof specific heat $C_{v}$, though the data of the volume expansion\ncoefficient show a clear anomaly around 40 K~\\cite{hiranaka13}.\nIn order to see this point, we consider here a simple Debye model as a\nlattice contribution $C^{({\\rm l})}_{v}$ to the specific heat:\n$C^{({\\rm l})}_{v} = A D(T_{\\rm D}\/T)$. Here $A (\\sim 15)$ is the\nDulong-Petit constant, $T_{\\rm D}$ is the Debye temperature and $D(x)$\nis the Debye function. The total specific heat is given by \n$C_{v} = C^{({\\rm e})}_{v} + C^{({\\rm l})}_{v}$.\nFigure 2 shows the calculated specific heat vs experimental data.\nThe present theory is consistent with the experimental data of the\nspecific heat, and we find that the anomalous contribution cannot be \nseen in the total $C_{v}$ because of a large lattice contribution of \n$C^{({\\rm l})}_{v}$ and its steep slope in the temperature region \nbetween 40 K and 80 K~\\cite{kake13}. \n\n\n\n\\subsection{Susceptibility}\n\nThe susceptibility in the 2-state Weiss model is obtained by adding\nthe Zeeman term to the Hamiltonian. The magnetization \n$\\langle M_{z} \\rangle$ per Eu atom under the magnetic field $H$ is \ncalculated from the following expression.\n\\begin{eqnarray}\n\\langle M_{z} \\rangle = \n\\frac{\\displaystyle \\sum_{JM} (M_{z})_{JMH} e^{-\\beta E_{JMH}}}\n{\\displaystyle \\sum_{JM} e^{-\\beta E_{JMH}}} .\n\\label{magz}\n\\end{eqnarray}\nHere the `$J=0$' state is the heavy-fermion ground state under the\nmagnetic field $H$. With use of the energy $E_{\\rm L}$ and the\nheavy-fermion ground-state susceptibility $\\chi_{\\rm L}$ which might \nbe inversely proportional to the Kondo temperature, \nthe energy under the magnetic field should be given by\n\\begin{eqnarray}\nE_{JMH} = E_{\\rm L} - \\frac{1}{2} \\chi_{\\rm L} H^{2} .\n\\label{enerj0h}\n\\end{eqnarray}\nThe magnetization per atom in the low spin state is therefore given by \n$(M_{z})_{JMH} = - \\partial E_{JMH}\/ \\partial H = \\chi_{\\rm L} H$.\nOn the other hand, the energy in the high spin state ($J=J_{\\rm H}$) \nis given by \n\\begin{eqnarray}\nE_{JMH} = E_{J} - g_{J}MH - \\frac{1}{2}\\alpha_{JM} H^{2} .\n\\label{enerjmh}\n\\end{eqnarray}\nHere $g_{J}$ is Land\\'{e}'s $g$ factor, $\\alpha_{JM}$ is a\nphenomenological Van Vleck constant~\\cite{takikawa10,nolting09}.\nThe magnetization is given by \n$(M_{z})_{JMH} = - \\partial E_{JMH} \/ \\partial H = \ng_{J}M + \\alpha_{JM} H$.\n\nFrom Eqs. (\\ref{magz}), (\\ref{enerj0h}), and (\\ref{enerjmh}), \nwe obtain the susceptibility.\n\\begin{eqnarray}\n\\chi = \\chi_{\\rm L} + \\Big(\\Delta \\chi + \\frac{p_{\\rm H}^{2}}\n{3k_{\\rm B}T} \\Big) \\frac{\\displaystyle w \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{chi0}\n\\end{eqnarray}\nHere $\\Delta \\chi = \\chi_{\\rm H} - \\chi_{\\rm L}$, and \n$\\chi_{\\rm H}$ is the susceptibility at $T=0$ in the high-spin \nstate defined by $\\sum_{M} \\alpha_{J_{\\rm H}M}\/(2J_{\\rm H}+1)$. \n$p_{\\rm H}$ is the effective Bohr\nmagneton number in the high-spin state, {\\it i.e.}, \n$p_{\\rm H}^{2}=g_{J}^{2}J_{\\rm H}(J_{\\rm H}+1)$.\nTaking the high-temperature limit, we obtain the effective \nBohr magneton number $p_{\\rm eff}$ as\n\\begin{eqnarray}\np_{\\rm eff} = \\sqrt{\\frac{w}{1+w}} \\, p_{\\rm H} \\, .\n\\label{peff}\n\\end{eqnarray}\nUsing the values $w=8$ and $p_{\\rm H}=\\sqrt{63}$, we obtain \n$p_{\\rm eff} = 7.48$ $\\mu_{\\rm B}$, which is in good agreement with the\nexperimental value $7.4$ $\\mu_{\\rm B}$~\\cite{hiranaka13}. \n\\begin{figure}\n\\includegraphics[scale=1.0]{chi.eps}\n\\caption{\nTemperature dependence of susceptibilities ($\\chi$).\nOpen circles: experimental data~\\cite{hiranaka13} for $H \/\/ [110]$,\nclosed circles: experimental data~\\cite{hiranaka13} for $H \/\/ [001]$,\nsolid line: present result based on the 2-state Weiss model \nfor $\\chi_{\\rm L} \\approx 0.037$ emu\/mol, $\\Delta \\chi =-0.044$ emu\/mol,\n $\\Theta=-120$ K, and $\\Delta E\/k_{\\rm B}=150$ K \n(see Eq. (\\ref{chimod})). \n}\n\\label{figchi}\n\\end{figure}\n\nThe susceptibility (\\ref{chi0}) however does not explain an overall\ntemperature variation of the\nexperimental data. We have to take into account the effect of \npolarization due to the RKKY-like magnetic interaction\nvia conduction electrons. This produces an effective magnetic field\naccording to the mean-field picture and should produce the Weiss constant\n$\\Theta$ in the susceptibility when $J=J_{\\rm H}$. \nWe reach then the following susceptibility.\n\\begin{eqnarray}\n\\chi = \\chi_{\\rm L} + \\Big(\\Delta \\chi + \\frac{C_{\\rm H}}\n{T - \\Theta} \\Big) \\frac{\\displaystyle w \\, e^{-\\beta \\Delta E}}\n{\\displaystyle 1 + w \\, e^{-\\beta \\Delta E}} \\, .\n\\label{chimod}\n\\end{eqnarray}\nHere $C_{\\rm H}=p_{\\rm H}^{2}\/3k_{\\rm B}$ is the Curie constant \nin the high-spin state. \nIt should be noted that the polarization effects\ndue to the magnetic field are negligible for the other\nquantities because they are of order of $H^{2}$. \n\nUsing the experimental values $\\Theta=-120$ K and \n$\\chi_{\\rm L} \\approx 0.037$ emu\/mol~\\cite{hiranaka13} and choosing\n$\\Delta \\chi$ as \n$\\Delta \\chi =-0.044$ emu\/mol, we can explain the global feature of\ntemperature dependence of EuNi${}_{2}$P${}_{2}$ as shown in Fig. 3. \nNote that we did not take into account the anisotropy due to \ntetragonal structure in the present analysis for simplicity. \n\n\n\n\\section{Summary}\n\nWe have shown in this paper that the phenomenological 2-state Weiss\nmodel with an emphasis of local excitations explains an overall feature\nof the volume expansion due to 4$f$ electrons as well as the scaling\nrelation between the 4$f$ electron contribution to the volume expansion \nand the average 4$f$ electron number. \nThese results indicate that the strong temperature dependence of thermal\nexpansion found in this system is mainly determined by local excitations\nassociated with valence fluctuations.\nWe also predicted that there is another scaling\nrelation between the thermal expansion coefficient due to the 4$f$\nelectrons and the associated specific heat. We have also shown that \ncalculated effective Bohr\nmagneton number in the high-temperature limit is in good agreement with\nthe experimental value 7.4 $\\mu_{\\rm B}\/$Eu. For explanation of\ntemperature dependence of susceptibility, we need inter-site magnetic\ninteractions via conduction electrons. \nModified susceptibility explains the temperature dependence of the \nexperimental data, and indicates that there are three temperature\nscales, the characteristic temperature \n$T_{v} = \\Delta E\/k_{\\rm B} \\ (\\sim 150 {\\rm K})$ \nfor on-site\nvalence fluctuations, the Weiss constant $\\Theta \\ (\\sim -120 {\\rm K})$ \nassociated with the\nRKKY type interactions via conduction electrons, and the Kondo\ntemperature $T_{\\rm K}$ \nfor the formation of heavy-fermions. \nThe results presented in this paper indicate that the heavy-fermion\nground state is collapsed by valence fluctuations which are \ncharacterized by\n$T_{v}$. In this sense, the characteristic temperature $T_{v}$ plays \nan essential role in the physics of EuNi${}_{2}$P${}_{2}$. \nIt has not yet been clarified theoretically whether or not the Kondo\ntemperature is well-defined in this system.\n\nAlthough the present model is consistent with the heavy-fermion ground\nstate as well as the effective Bohr magneton number in the high\ntemperature limit which is directly related to the local excitations\nof $4f{}^{7}$, it does not take into account the low energy excitations.\nThe temperature dependence associated with\nthe heavy-fermions is not described here. \nThe deviation of $\\Delta V\/V$ from the\nexperimental data below 40 K in Fig. 1 should be attributed to this fact.\nToo small a specific heat below 30 K found in Fig. 2 is due to the\nneglect of the low energy excitations. \nMicroscopic derivation of the phenomenological model and the inclusion\nof the low-energy excitations are left for \nfuture investigations.\n\nThe present work is supported by a Grant-in-Aid for\nScientific Research (25400404).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Methods}\n\t\n\\subsection*{Ethics Statement.}\n\nThe data collection process has been carried out using the Facebook\nGraph API \\cite{fb_graph_api}, which is publicly available. For the\nanalysis (according to the specification settings of the API) we only\nused publicly available data (thus users with privacy restrictions are\nnot included in the dataset). The pages from which we download data are\npublic Facebook entities and can be accessed by anyone. User content\ncontributing to these pages is also public unless the user's privacy\nsettings specify otherwise, and in that case it is not available to us.\n\t\n\\subsection*{Data collection.}\n\t\nDebate about social issues continues to expand across the Web, and\nunprecedented social phenomena such as the massive recruitment of people\naround common interests, ideas, and political visions are\nemerging. Using the approach described in Ref.~\\cite{bessi2014science},\nwe define the space of our investigation with the support of diverse\nFacebook groups that are active in the debunking of conspiracy theories.\n\nThe resulting dataset is composed of 67 public pages divided between\nconspiracy and science news. A second set, composed of two\ntroll pages, is used as a benchmark to fit our data-driven model. The\nfirst category (conspiracy theories) includes the pages that disseminate\nalternative, controversial information, often lacking supporting\nevidence and frequently advancing conspiracy theories. The second\ncategory (science news) includes the pages that disseminate scientific\ninformation. The third category (trolls) includes those pages that\nintentionally disseminate sarcastic false information on the Web.\n\t\nFor the three sets of pages we download all the posts (and their\nrespective user interactions) across a five-year timespan (2010 to\n2014). We perform the data collection process by using the Facebook\nGraph API \\cite{fb_graph_api}, which is publicly available and\naccessible through any personal Facebook user account. The exact\nbreakdown of the data is presented in the Supporting \nInformation (SI) Section~1.\n\n\\subsection*{Preliminaries and Definitions.}\n\nA tree is an undirected simple graph that is connected and has no simple\ncycles. An oriented tree is a directed acyclic\ngraph whose underlying undirected graph is a tree. A sharing tree in the\ncontext of our research is\nan oriented tree made up of the successive sharing of a news item through the\nFacebook system. The root of the sharing tree is the node that performs\nthe first temporal share. We define the size of the sharing tree as the\nnumber of nodes (and hence the number of news sharers) in the\ntree and the height of the sharing tree as the maximum path length\ndistant\n from the root. \n \nWe define the user polarization $\\sigma= 2\\varrho -1$, where $0\\leq \\varrho \\leq 1$ is the fraction of ``Likes'' a user executes on\nconspiracy related content, and hence $-1\\leq\n\\sigma\\leq 1$. From user polarization, we define the\nedge homogeneity, for any edge $e_{ij}$ between\nnodes $i$ and $j$, as\n\t\t\t$$\n\t\t\t\\sigma_{ij} = \\sigma_i\\sigma_j, \n\t\t\t$$ \nwith $-1 \\leq \\sigma_{ij} \\leq 1$. \nEdge homogeneity reflects the similarity level between the polarization of the two sharing nodes. A link in the sharing tree is homogeneous if its edge homogeneity is positive, otherwise it is non homogeneous.\nWe then define a sharing path to be any path from the root to one of the leaves of the sharing tree. A homogeneous path is a sharing path for which the edge homogeneity of each edge is positive, i.e., a sharing path whose edges are all homogeneous links.\n\n\\paragraph{Wald Test.} We use the Wald test to compare\nthe scaling parameters of two power law distributions. We define it as\n\\begin{eqnarray*}\n\t\tH_0 : \\hat{\\alpha_1} = \\hat{\\alpha_2}\\\\ H_1 :\n \\hat{\\alpha_1} \\neq \\hat{\\alpha_2}\n\\end{eqnarray*}\nwhere $\\hat{\\alpha_1}$ and $\\hat{\\alpha_2}$ are the estimated scaling\nparameters. The Wald statistic:\n$$\n\t\tW = \\frac{(\\hat{\\alpha_1} -\\hat{\\alpha_2})^2}{Var(\\hat{\\alpha_1})},\n$$\nfollows a $\\chi^2$ distribution with one degree of freedom. We reject\nthe null hypothesis $H_0$ and conclude that there is a significant\ndifference between the two scaling parameters if the $p$-value of $W$ is\nbelow a given significance level $\\alpha$.\n\n\\paragraph{Kolmogorov-Smirnov Test.} We use the Kolmogorov-Smirnov test \nto compare the empirical distribution functions of two samples. The\nKolmogorov-Smirnov statistic for two given cumulative distribution\nfunctions $F_1(x)$ and $F_2(x)$ is \n\t\t$$\n\t D = \\sup_x{|F_1(x) -F_2(x)|},\n\t\t$$\n\nwhich measures the maximum punctual distance between the two sample\ndistributions. If $D$ is bigger than a given critical value\n$D_{\\alpha}$\\footnote{The critical value $D_{\\alpha}$ depends on the\n sample sizes and on the considered significance level $\\alpha$, it can\n be computed as\n\t\t\t$$\n\t\t\tD_{\\alpha} = c(\\alpha)\\sqrt{\\frac{n_1 + n_2}{n_1n_2}},\n\t\t\t$$\nwhere $n_1$ and $n_2$ are the respective sample sizes and $c(\\alpha)$ is\na fixed value associated with the significance level $\\alpha$.} we\nreject the null hypothesis $H_0 : F_1(x) = F_2(x)$ and conclude that\nthere is a significant difference between the two sample distributions.\n\n\\section*{Results and discussion}\n\n\n\\subsection*{Anatomy of Cascades.}\n\nWe begin our analysis by characterizing the statistical signature of\ncascades as they relate to information type. We analyze the three\ntypes---science news, conspiracy rumors, and trolling---and find that\nsize and maximum degree are power-law distributed for all three. The\nmaximum cascade size values are 952 for science news, 2422 for\nconspiracy news, and 3945 for trolling, and the estimated exponents\nfor the power law distributions are 2.21 for science news, 2.47 for\nconspiracy theories, and 2.44 for trolling. Tree height values range\nfrom 1 to 5, with a maximum height of 5 for science news and conspiracy\ntheories and a maximum height of 4 for trolling. For further information\nsee SI Section~2.1.\n\nFigure~\\ref{fig:lf_tot} shows the probability density function (PDF) of\nthe cascade lifetime (using hours as time units) for science and\nconspiracy. We compute the lifetime as the length of time\nbetween the first user and the last user sharing a post. In both\ncategories we find a first peak at approximately 1--2 hours and a second\nat approximately 20 hours, indicating that the temporal sharing patterns\nare similar irrespective of the difference in topic. We also find that a\nsignificant percentage of the information diffuses rapidly (24.42\\% of\nthe science news and 20.76\\% of the conspiracy rumors diffuse in less\nthan two hours, and 39.45\\% of science news and 40.78\\% of conspiracy\ntheories in less than five hours). Only 26.82\\% of the diffusion of\nscience news and 17.79\\% of conspiracy lasts more than one day.\nKolmogorov-Smirnov test made us reject the hypothesis $H_0$\nthat the two distributions are equal.\n\nFigure~\\ref{fig:lf_size} shows lifetime as a function of cascade size.\nFor science news we have a peak in the lifetime corresponding to a\ncascade size value of $\\approx 200$, and higher cascade size values\ncorrespond to high lifetime variability. \nFor conspiracy related content the lifetime increases with cascade size.\n\nThese results suggest that news assimilation differs according to\ncategory. Science news is usually assimilated, i.e., it reaches a higher\nlevel of diffusion, quickly, and a longer lifetime does not correspond\nto a higher level of interest. \nConversely, conspiracy rumors are assimilated more\nslowly and show a positive relation between lifetime and size. \nFor both science and conspiracy news, we compute size as a function\nof lifetime and confirm that differentiation in the sharing patterns is\ncontent-driven, and that for conspiracy there is a positive\nrelation between size and lifetime. For a more detailed explanation, see\nSI Section~2.1.\n\n\\subsection*{Homogeneous Clusters.}\n\nWe next examine the social determinants that drive sharing patterns and\nwe focus on the role of homogeneity in friendship networks.\n\nFigure~\\ref{fig:polarization} shows the PDF of the mean edge homogeneity, computed for all cascades of science news and conspiracy theories. It shows that there are homogeneous links\nbetween consecutively sharing users. In particular, the average edge homogeneity\nvalue of the entire sharing cascade is always greater or equal to zero, indicating that either the information transmission occurs inside homogeneous clusters in which all links are homogeneous or it occurs inside mixed neighborhoods in which the balance between homogeneous and non homogeneous links is favorable towards the former ones. However, the probability of close to zero mean edge homogeneity is really small. \n\nTo further characterize the role of homogeneity in shaping sharing\ncascades, we compute cascade size as a function of mean edge homogeneity for both\nscience and conspiracy news, see Figure~\\ref{fig:size_polarization}. In science news, higher levels of mean edge homogeneity in the\ninterval (0.5, 0.8) correspond to larger cascades, but in conspiracy\ntheories lower levels of mean edge homogeneity ($\\sim 0.25$) correspond to larger\ncascades.\nNotice that, although viral patterns related to distinct contents differ, homogeneity is clearly the driver of information diffusion. In other words, different contents generate different echo chambers, characterized by the high level of homogeneity inside them.\n\nThe probability density function (PDF) of the edge homogeneity, computed for science and conspiracy news as well as the two taken together---both in the unconditional case and in the conditional case (in the event that the user that made the first share in the couple has a positive or negative polarization)---confirms the roughly null probability of a negative edge homogeneity (see SI Section~2.1).\n\n\n\n\n \n\nWe record the CCDF of the number of all sharing paths\\footnote{Recall that a sharing path is here defined as any path from the root to one of the leaves of the sharing tree. A homogeneous path is a sharing path for which the edge homogeneity of each edge is positive} on each tree \ncompared with the CCDF of the number of homogeneous paths for science and conspiracy news, and the two together. A Kolmogorov-Smirnov test and Q-Q plots confirm that for all three pairs of distributions considered there is no significant statistical\ndifference (see SI Section~2.2 for a more detailed analysis).\nIn SI Section~2.2 we report also the frequency of maximum length for all sharing paths and homogeneous paths, for both categories of content.\n \nWe confirm the pervasiveness of homogeneous paths, but we also find homogeneous paths in which there is a shift of $-1$ in the path length (with respect to the total path length $k$). \nNotice that the first publisher of a news is generally a page, hence the $(k-1)$-homogeneous paths are due to a discordant sharing in\nthe first step (i.e., when the product of the first sharer's user polarization\nand the sharer page category is negative). \n\nCascade lifetimes of science and conspiracy\nnews exhibit a probability peak in the first two hours, and that in\nthe following hours they rapidly decrease. Despite the similar\nconsumption patterns, cascade lifetime expressed as a function of\ncascade size differs greatly for the different content sets. \nThe PDF of the mean edge homogeneity indicates that there is homogeneity in the linking step\nof sharing cascades. \nThe distribution of the number of total and homogeneous sharing paths are very similar for both content categories.\n\nViral patterns related to contents belonging to different narratives differ, but homogeneity is\nclearly the driver of content diffusion.\n\n\\subsection*{The Model.}\n\nWe now introduce a percolation model of rumor spreading to account for\nhomogeneity and polarization. We consider $n$ users connected by a\nsmall-world network \\cite{watts1998collective}. \nThe model parameter space varies on a rewiring\nprobability $r$, mimicking the network density, and a news set of size $m$.\n\nEvery node has an opinion $\\omega_i$, $i\\in[1,n]$ uniformly distributed\nin $[0,1]$. Every news item has a fitness (degree of interest)\n$\\vartheta_j,\\,j\\in[1,m]$ uniformly distributed in $[0,1]$. \nAt each step the news items are diffused and initially shared by a\ngroup of first sharers. After the first step, the news recursively\npasses to the neighborhoods of previous step sharers, e.g., those of the\nfirst sharers during the second step. \nIf a friend of the previous step sharers has an opinion close to the fitness of the news, then she shares the news again. \n\nIn particular, when\n$$\n{|\\omega_i -\\vartheta_j|\\leq\\delta},\n$$\nuser $i$ shares news $j$; $\\delta$ is the sharing threshold.\n\nBecause $\\delta$ by itself cannot capture the homogeneous clusters\nobserved in the data, we model the connectivity pattern as a signed network \\cite{Quattrociocchi2014,leskovec2010signed} considering different fractions of homogeneous\nlinks and hence restricting diffusion of news only to homogeneous\nlinks. \nWe define $\\phi_{HL}$ as the fraction of homogeneous links in the network, $M$ as the number of total links, and $n_h$ as the number of homogeneous links, thus we have:\n$$\n\\phi_{HL} = \\frac{n_h}{M},\\,0\\leq n_h\\leq M.\n$$\nNotice that $0\\leq \\phi_{HL}\\leq 1$ and that $1 - \\phi_{HL}$, the fraction of non homogeneous links, is complementary to $\\phi_{HL}$. In particular, we can reduce the parameters space to $\\phi_{HL}\\in[0.5,1]$ as we would restrict our attention to either one of the two complementary clusters.\n\nThe model can be seen as a branching process where the sharing\nthreshold $\\delta$ and neighborhood dimension $z$ are the key parameters.\nMore formally, let the fitness $\\theta_{j}$ of the $j^{th}$ news and the\nopinion $\\omega_{i}$ of a the $i^{th}$ user be uniformly i.i.d. between\n$[0,1]$. Then the probability $p$ that a user $i$ shares a post $j$ is\ndefined by a probability $p=\\min(1,\\theta + \\delta) -\n\\max(0,\\theta-\\delta)\\approx2\\delta$, since $\\theta$ and $\\omega$ are\nuniformly i.i.d. In general, if $\\omega$ and $\\theta$ have\ndistributions $f(\\omega)$ and $f(\\theta)$, then $p$ will depend on\n$\\theta$,\n\\[\np_{\\theta}=f\\left(\\theta\\right)\\int_{\\max\\left(0,\\theta-\n\t\\delta\\right)}^{\\min\\left(1,\\theta+\n\t\\delta\\right)}f\\left(\\omega\\right)d\\omega. \n\\]\nIf we are on a tree of degree $z$ (or on a sparse lattice of degree\n$z+1$), the average number of sharers (the branching ratio) is defined\nby\n\\[\n\\mu=zp\\approx2\\delta\\, z\n\\]\nwith a critical cascade size $S=\\left(1-\\mu\\right)^{-1}$. If we assume\nthat the distribution of the number $m$ of the first sharers is\n$f\\left(m\\right)$, then the average cascade size is\n\\[\nS=\\sum_{m}f\\left(m\\right)m\\left(1-\\mu\\right)^{-1}=\n\\frac{\\left\\langle m\\right\\rangle _{f}}{1-\\mu}\n\\approx\\frac{\\left\\langle m\\right\\rangle _{f}}{1-2\\delta z}\n\\]\nwhere $\\left\\langle \\ldots\\right\\rangle _{f}=\\sum_{m}\\ldots f\\left(m\\right)$\nis the average with respect to $f$.\nIn the simulations we fixed neighborhood dimension $z = 8$ since the branching ratio $\\mu$ depends upon the product of $z$ and $\\delta$ and, without loss of generality, we can consider the variation of just one of them.\n\nIf we allow a probability $q$ that a neighbor of a user has a\ndifferent polarization, then the branching ratio becomes\n$\\mu=z\\left(1-q\\right)p$. If a lattice has a degree distribution\n$d\\left(k\\right)$ ($k=z+1$), we can then assume a usual percolation\nprocess that provides a critical branching ratio and that is linear in\n$\\left\\langle k^{2}\\right\\rangle _{d}\/\\left\\langle k\\right\\rangle _{d}$\n($\\mu\\approx\\left(1-q\\right)p\\left\\langle z^{2}\\right\\rangle\n\/\\left\\langle z\\right\\rangle $).\n\n\\subsection*{Simulation Results.}\n\nWe explore the model parameters space using $n = 5,000$ nodes and $m =\n1,000$ news items with the number of first sharers distributed as an (i)\ninverse Gaussian, (ii) log normal, (iii) Poisson, (iv) uniform\ndistribution, and as the real data distribution (from the science and\nconspiracy news sample). Parameters are chosen to fit the real data\ndistribution (for details see SI Section~3.1,~3.2). In Table~\\ref{tab1} we show a summary of relevant statistics (min value, first quantile, median, mean, third quantile, and max value) to compare the real data first sharers distribution with the fitted distributions\\footnote{For details on the parameters of the fitted distributions used see SI Section~3.2.}. The inverse Gaussian ($IG$), \nshows the best fit for the distribution of first sharers with respect to all the considered statistics. \n\nAlong with the first sharers distribution, we vary the sharing threshold $\\delta$ in the interval $[0.01, 0.05]$ and the fraction of homogeneous links $\\phi_{HL}$ in the interval $[0.5, 1]$. To avoid biases induced by statistical\nfluctuations in the stochastic process, each point of the parameter\nspace is averaged over 100 iterations. $\\phi_{HL}\\sim 0.5$ provides\na good estimate of real data values. In particular, consistently with the division of in two echo chambers (science and conspiracy), \nthe network is divided into two clusters in which news items remain\ninside and are transmitted solely within each community's echo chamber\n(see SI Section~3.2 for the details of the simulation\nresults).\n\nIn addition to the science and conspiracy content sharing trees, we\ndownloaded a set of 1,072 sharing trees of intentionally false\ninformation from troll pages. Frequently troll information, e.g.,\nparodies of conspiracy theories such as chem-trails containing the\nactive principle of Viagra, is picked up by habitual conspiracy theory\nconsumers. In SI Section~3.2 we report the same information as Table~\\ref{tab1} for trolling category. Also in this case we notice that the best fit is obtained by the inverse Gaussian distribution.\n\nWe computed the mean and standard deviation of \nsize and height of all trolling sharing trees, and reproduced the data \nusing our model\\footnote{Note that the real data values for the mean (and standard deviation) of size and height on the troll posts are respectively: $23.54\\, (122.32)$ and $1.78\\,(0.73)$.}. We used fixed \nparameters from trolling messages sample (the number of nodes in the system and the number of news items) and varied the fraction of homogeneous links $\\phi_{HL}$, the rewiring probability $r$, and sharing threshold $\\delta$. See SI Section~3.2 for the distribution of first sharers used and for additional simulation results of the fit on trolling messages.\n\n\nWe simulated the model dynamics with the best combination of parameters obtained from the simulations and the number first sharers distributed as an inverse Gaussian, figure~\\ref{fig5} shows the CCDF of size and the CDF of height. A summary of relevant statistics (min value, first quantile, median, mean, third quantile, and max value) to compare the real data size and height distributions with the fitted ones is reported in SI Section~3.2. We notice that the fit is good for all the statistics, with the exception of min and max value of size. For the min value, the presence of a zero is due to the fact that the inverse Gaussian is a real valued distribution function and in the simulations we considered the integer part of the number of first sharers, thus producing a number of never shared pieces of information. On the other hand, the high difference in the max value is probably due to the long tail of the data size distribution. \n\nWe find that the inverse Gaussian is the distribution that best fits the\ndata results both for science and conspiracy news, and for troll messages. For this reason, we performed one more simulation using the inverse Gaussian as distribution of the number of first sharers, 1,072 news items, 16,889 users, and the best parameters combination obtained in the simulations\n\\footnote{The best parameters combinations is $\t\\phi_{HL} = 0.56, \\, r = 0.01, \\, \\delta = 0.015$, in this case we have a mean size equal to $23.42\\,(33.43)$ and a mean height $1.28\\,(0.88)$, and it is indeed a good approximation, see Section~3.2.}.\nThe CCDF of size and the CDF of height for the above parameters\ncombination, as well as basic statistics considered, fit the real data ones from the trolling category.\n\n\\section*{Conclusions}\n\nDigital misinformation has become so pervasive in online social media\nthat it has been listed by the World Economic Forum (WEF) as one of the\nmain threats to human society. Whether a news item, either\nsubstantiated or not, is accepted as true by a user may be strongly\naffected by social norms or by how much it coheres with the user's\nsystem of beliefs \\cite{Zhu2010,Loftus2011}. Despite enthusiastic claims\nthat social media is generating a vast ``collective intelligence''\navailable to all \\cite{surowiecki2005wisdom}, many mechanisms cause\nfalse information to gain acceptance, which in turn generate false\nbeliefs that, once adopted by an individual, are highly resistant to\ncorrection \\cite{Garrett2013,Meade2002,koriat2000,Ayers98}. Using extensive\nquantitative analysis we show that social homogeneity is the primary\ndriver of content diffusion, and one frequent result is the formation of\nhomogeneous, polarized clusters (often called ``echo chambers''). We\nalso find that although consumers of science news and conspiracy\ntheories show similar consumption patterns with respect to content,\ntheir cascades differ. Social homogeneity appears to be the primary\ndriver of content diffusion, and each echo chamber has its own cascade\ndynamics. To mimic these dynamics, we introduce a data-driven\npercolation model of signed networks, i.e., networks composed of signed\nedges. Our analysis shows that for science and conspiracy news\na cascade's lifetime has a probability peak in the first two hours\nfollowed by a rapid decrease. Although the consumption patterns are\nsimilar, cascade lifetime as a function of the size differs greatly.\nThe PDF of the mean edge homogeneity indicates that\nhomogeneity is present in the linking step of sharing cascades. The\ndistribution of the number of total sharing paths and homogeneous\nsharing paths are similar in both content categories. Viral patterns\nrelated to distinct contents are different but homogeneity drives\ncontent diffusion. We simulate our data-driven percolation model by\nfixing the number of users and news items downloaded from troll pages\nand varying the other parameters. \nWe compare the simulated results with the data and find a high level of similarity.\n\n\n\n\\begin{acknowledgments}\nFunding for this work was provided by the EU FET project MULTIPLEX, no. 317532, SIMPOL, no. 610704, the FET project DOLFINS 640772 (H2020), SoBigData 654024 (H2020), and CoeGSS 676547 (H2020).\nThe funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Special thanks go to Delia\nMocanu,\"Protesi di Protesi di Complotto\", \"Che vuol dire reale\", \"La\nmenzogna diventa verita e passa alla storia\", \"Simply Humans\",\n\"Semplicemente me\", Salvatore Previti, Brain Keegan, Dino Ballerini,\nElio Gabalo and \"The rooster on the trash\" for their precious\nsuggestions and discussions.\n\n\\end{acknowledgments}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\\noindent Integral field units (IFUs) based on image-slicers are commonly used in astrophysical night-time slit-spectroscopy. They were initially proposed by \\citet{Bowen38} and have been developed since the work of \\citet{Weitzel96} and \\citet{Content97}. The working principle of image slicers is to slice a region of the focal-plane image into a series of small rectangles and rearrange them in the form of a long slit. This slit is composed of a given number of mini-slits (equal to the number of slicer mirrors). The width of the individual slicer mirrors determines the slit width that feeds the spectrograph, into which the IFU is integrated. \n\n In solar physics, IFUs based on this technology can be used to record fast-evolving solar features at high spatial and spectral resolution, even at near-infrared (NIR) wavelengths. \n As described in a review of instrumentation for solar spectropolarimetry by \\citet{Iglesias19}, there are few instruments currently in operation \\citep[see][]{SPINOR,FIRS,NIRIS,Collados12,jaeggli22} that can perform solar spectropolarimetry in the NIR ({\\itshape i.e.}, up to 1.7 $\\mu$m). \n \n In the case of GRIS \\citep[GREGOR Infrared Spectrograph, see][]{Collados12}, adding an IFU to the instrument makes possible to obtain spectra at all the points in a 2D field of view (FOV) and get the full Stokes profiles of\n the spectral lines in a given spectral interval. This makes possible to observe, at high spatial and spectral resolutions, the polarized spectrum of fast-evolving solar features at a cadence of a few seconds. \n The IFU added to GRIS was designed at the Instituto de Astrof\\'{i}sica de Canarias (IAC) as a \n first approach to\n the conceptual instrument called ``MUSICA'' \\cite{Calcines13b}, for EST instrumentation \\cite{Calcines13}.\n The IFU was built in collaboration with Winlight Systems (formerly Winlight Optics) within the framework of the SOLARNET and Getting Ready for EST (GREST) European projects. The IFU was taken to the GREGOR solar telescope \\cite{gregor12} and integrated into GRIS in 2017. From the first tests, it was clear that some work still needed to be done, and a second commissioning campaign was carried out in 2018 \\cite{cdt18}. During the scientific validation' procedure, a number of spectropolarimetric measurements at the solar disk center were recorded. \n After these campaigns, the IFU in GRIS was made available to the entire GREGOR observing community. \n\nBetween 2018 and 2019, more than 15 scientific teams observed with the IFU. It was in use for more than 70\\% of the telescope's observing time. More than 100 days of scientific data have been archived (i.e., selecting those obtained in good observing conditions). In 2020 and the first semester of 2021, the IFU was not offered because of the pandemic and changes in the telescope optics. Four scientific articles with results based on IFU observations have already been published \\cite{Anjali20,campbell21,tetsu21,nelson21}. \n\nThe instrument and the IFU are described in Section~\\ref{cdt:instdes}, including an analysis of performance. Some first light science observations and other examples are presented in Section~\\ref{cdt:sciobsv}. Our conclusions are given in Section~\\ref{cdt:concl}.\n\n\n\n\n\\section{Instrument description}\n\\label{cdt:instdes}\n\n\\subsection{GRIS}\n\nGRIS \\cite{Collados12} is a spectropolarimeter consisting of a 6~m focal-length Czerny--Turner spectrograph and a polarimeter built from the heritage of TIP-II \\cite{Collados07}.\n GRIS works in the wavelength range 1.0--2.3~$\\mu$m. Its 1k $\\times$ 1k NIR detector can be read at up to 36~fps. The polarimeter has two sets of Ferroelectric Liquid Crystals, one optimized to work at 1.0--1.3 $\\mu$m and the other at 1.5--1.8~$\\mu$m. GRIS has two observing modes, spectroscopy and spectropolarimetry, both with a long slit (0.26$^{\\prime\\prime}$~width~$\\times$~60$^{\\prime\\prime}$ length). It has a Slit Scan Unit (SSU) to do scans perpendicular to the slit. The SSU allows the scanning of a FOV up to 60$^{\\prime\\prime}$~$\\times$~64$^{\\prime\\prime}$ to be covered. The polarimeter is placed immediately after the SSU. As described in \\citet{Collados12}, the slit mask in the SSU is inclined $15^{\\circ}$ with respect to the entrance beam in order to reflect the light that arrives out of the slit towards \n a slit-jaw imaging system.\n This inclination does not alter the operation of the spectrograph and allows a field context image to be generated. \n Table~\\ref{cdt:spectable} shows the specifications of the slit mode in the first column. The IFU mode is discussed in the next section.\n \n\n\n\n\\begin{wstable}[ht]\n \\caption{GRIS slit and IFU specifications}\n \\label{cdt:spectable} \n\t \\begin{tabular}{l l p{0.5\\linewidth}} \\toprule\n Slit mode & IFU mode \\\\ \\colrule\n Number of slits: 1 & Number of slices: 8 \\\\\n\t Slit length: $<$ 60$^{\\prime\\prime}$ & Slice length: 6$^{\\prime\\prime}$ \\\\\n\t Slit width: 0.26$^{\\prime\\prime}$ & Slice width: 0.375$^{\\prime\\prime}$ (100 $\\mu$m) \\\\ \n Slit Scan (SSU) & 2D Scan (FOVSS) \\\\\n\t Moves in 1 direction & Moves in 2 directions \\\\\n\t Max. FOV: $60^{\\prime\\prime} \\times 64^{\\prime\\prime}$ & Max. FOV: $60^{\\prime\\prime}\\times 60^{\\prime\\prime}$ \\\\ \n\t Double sampling mode:& \\\\\n\t half slit width (see Section~\\ref{cdt:sciobsv}) & same \\\\ \n\t Spectral wavelength ranges: & \\\\\n\t 18 \\AA $\\:$ at 10830 \\AA \t & same\\\\\n\t 40 \\AA $\\:$ at 15650 \\AA \t & same\\\\\t \n\t Pixel scale: 0.135$^{\\prime\\prime}$ & same \\\\\n\t Simultaneous slit-jaw images acquisition \t & same \\\\\n\t Detector: 1k $\\times$ 1k\t & same \\\\\n\t Well depth: 16384 ADUs (14 bits)\\\\\n\t Gain: 19 e-\/ADU & same\\\\ \\botrule\n\n\t \\end{tabular}\n\\end{wstable}\n\n\n\n\n\n\\subsection{IFU}\n\\label{cdt:ifu}\n\nThe IFU has been coupled to GRIS to add an additional observing mode. Currently, it is possible to exchange the slit and IFU modes in a single day \\cite{Vega16}. In the slit mode, the slit is placed at the focal plane of the telescope and is part of the SSU. The light path exits the SSU and goes through the polarimeter to the rest of the spectrograph. In IFU mode, the arrangement is different because of the available space and the size of the optical components of the polarimeter. Figure~\\ref{cdt:path} shows the layout of the IFU mode. An entrance mask is placed at the telescope's focal plane to limit the FOV to the size of the image slicer. The mask is the first element of the Field of View Scan System (FOVSS). This new scanning system is described in Section~\\ref{cdt:scans}. A reimaging system is incorporated in order to place the IFU in a new focal plane. The reimaging system is based on a classic collimator-camera design and includes two mirrors (RS1 and RS2). The image slicer of the IFU (shown at the top of Figure~\\ref{cdt:path}) is at the focal plane generated by the RS2 mirror. \n\n\n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=15cm]{cdt_fig1.png}\n \\caption{Light path in the IFU mode. The telescope focus is at the bottom of the figure (Mask). The light goes through the FOVSS and the reimaging system up to the IFU and then continues towards the polarimeter. The light path distances ($a, b, c, d$) in the FOVSS are shown next to its components, the entrance mask and the three SMs. The components for the reimaging system are the two RS mirrors. Those for the IFU are the image slicer and the arrays of collimator and camera mirrors. The position of the output mini-slits is also shown. }\n \\label{cdt:path}\n\t \\end{center}\n \\end{figure}\n\n\n\n\n\n The IFU covers a FOV of $6^{\\prime\\prime}\\times 3^{\\prime\\prime}$ in a single exposure. \nIt is based on image-slicer technology with eight slicer mirrors of size 1.8~mm~$\\times$~0.1~mm, each. These mirrors cut a rectangular region of the image at the focal plane and reorganize it into a long slit formed by eight mini-slits. The whole IFU body is fabricated in Zerodur to reduce thermal sensitivity because the instrument works at room temperature. The reorganization of the image from the image slicer towards the output slit is accomplished by a reimaging system, which has a classic collimator-camera design. There is a pair of collimator and camera mirrors for each slice. The optical design evolved from the U-path described in \\citet{Calcines14} to a Z-path, using spherical mirrors for the collimator and camera. The system is designed to be telecentric and includes a mask at the pupil plane to reduce straylight. The eight beams have the pupil in the same position so that one pupil mask works for all of them. The output slit is arranged in two rows of mini-slits in order to minimize geometrical distortion and optical aberrations. The optical design is shown in Figure~\\ref{cdt:figzem}. The insert shows the output mini-slits illuminated with sunlight. There is red light at the mini-slits because the IFU is fed by wavelengths over 650~nm. \n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=13cm]{cdt_fig2.png}\n \\caption{IFU optical layout. The light paths generated by every slice of the image are represented in different colors. The insert shows a real image of the output mini-slits. The layout orientation is left-right reversed, compared to that in Figure~\\ref{cdt:path}, in order to show the mini-slits projected towards the reader. }\n \\label{cdt:figzem}\n\t \\end{center}\n \\end{figure}\n\n\nThe collimator and camera mirrors are grouped into arrays and their numbering is shown in Figure~\\ref{cdt:numslice}. It can be seen that the central mirrors in the image slicer correspond to the edges of the collimator and mini-slit arrays. After the mini-slits, the light path continues to the polarimeter and the rest of the spectrograph. Two flat mirrors are inserted in the path after the polarimeter in order to take the beam back to the spectrograph optical axis and to\n compensate for the difference in the path length with respect to the slit mode.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[height=5cm]{cdt_fig3.png}\n\\end{center}\n\\caption {Slicer, collimator and mini-slit numbering for identification purposes.}\n\\label{cdt:numslice} \n\\end{figure} \n\n\n\n \nFrom an operational point of view, observations in IFU mode are conducted in a similar way to those in slit mode, with the exception that the IFU provides the opportunity to observe a 2D region in one shot.\nIn addition, it has scanning capabilities with the FOVSS to record a larger solar area. \nBoth the SSU and the FOVSS can cover roughly the same area, although the IFU is most useful for fast small-area scans. The second column of Table~\\ref{cdt:spectable} shows the specifications of the IFU mode.\n\n\n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=8cm]{cdt_fig4a.png}\n \\includegraphics[width=8cm]{cdt_fig4b.png}\n \\caption{Pictures of the entrance mask in the IFU mode (left) and the slit mode (right). The IFU mode is showed with sunlight over the mask. Here the FOVSS, the IFU (Zerodur block) and the polarimeter (metallic cylinder) are partially visible. In the slit-mode picture the SSU and the polarimeter are clearly visible.}\n \\label{cdt:pcicssu}\n\t \\end{center} \n \\end{figure}\n\n\nAs described before, the entrance mask defines the FOV with precision in order to illuminate only the useful part of the image slicer. \nThe mask is placed at the entrance of the FOVSS (see Figure~\\ref{cdt:pcicssu}) and has the same inclination as the slit mask ($15^{\\circ}$). During the first campaigns it was a rectangle cut on the aluminum coating over an SiO$_{2}$ window. However the small particles of dust present over the inclined glass contaminated the spectra. Regular cleaning of the mask was required. It was replaced in 2019 by a true-hole mask fabricated in a Si wafer coated with protected aluminum. The wafers are as good as any optical mirror since they have an intrinsic flatness better than 2~$\\mu$m over 150~mm and the rectangle, machined using etching, has errors of the order of a few microns. This new mask improved the quality of the spectra. \n \n\n\\subsection{FOVSS}\n\\label{cdt:scans}\nThe FOVSS was designed to do 2D scans covering the total FOV up to $60^{\\prime\\prime}\\times 60^{\\prime\\prime}$ \\cite{Esteves18}. The scan size is configurable by the user. The FOVSS keeps the optical path length constant, regardless of the scanning position over the FOV. This condition is satisfied by means of three Scanning Mirrors (SM), which change their positions using three motors. Figure~\\ref{cdt:path} includes the layout of the FOVSS configuration and its components. \nThe entire assembly, with the exception of RS1, is mounted on a common translation stage (TS) which moves the system along the X-axis. The mask, plus the SM1 and SM2 set are mechanically connected to a second TS, which moves this set along the Y-axis. Finally, the SM2 and SM3 set is mechanically connected to a third TS which allows movement along the X-axis independently of the first TS. TS1 and TS2 allow the movement of the mask along the X and Y axes, respectively, to enable 2D scanning. TS3 uses the SM2 and SM3 set to compensate for the optical path length as a function of the system\nmovement while scanning. The total optical path length, determined by the sum of segments $b + c + d$ (shown in Figure~\\ref{cdt:path}), is always constant.\n\n\nFigure~\\ref{cdt:figifu} shows a representation of the image slicer and some possible scan positions over the FOV, which is illustrated by a real-scale image from the Sun (SDO\/HMI continuum). The image slicer is represented by a red rectangle, including the eight slicer mirrors. The left frame represents a single exposure over an area of 6$^{\\prime\\prime}$~$\\times$~3$^{\\prime\\prime}$. The black rectangles in the right frame represent an example of a 3~$\\times$~7 scan. The available scan movements are represented by the arrows.\n \n\n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=13cm]{cdt_fig5.png}\n \\caption{Example of the IFU mode in a single position (left frame). The image slicer is represented by a red rectangle (FOV of 6$^{\\prime\\prime}$~$\\times$~3$^{\\prime\\prime}$). The right frame is an example of a 3~$\\times$~7 scan (18$^{\\prime\\prime}$~$\\times$~21$^{\\prime\\prime}$). The available scan movements are represented by the arrows, and a FOV up to $60^{\\prime\\prime}$~$\\times$~$60^{\\prime\\prime}$ can be covered. The background image is from SDO\/HMI continuum.}\n \\label{cdt:figifu}\n\t \\end{center} \n \\end{figure}\n\n\n\n\n\n\nThe FOVSS allows scans to be made following different patterns, such as those shown in Figure~\\ref{cdt:raster}. The patterns called ``-vertical'' are also available horizontally, to give transposed scanning patterns. \n The initial scan position can be chosen between the geometrical center or one corner of the full scan pattern. Observers can choose the pattern that better fits the solar-feature dynamics they are studying.\n \n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=13cm]{cdt_fig6.png}\n \\caption{ Examples of the FOVSS scan patterns: (a) Raster-vertical, (b) Snake-vertical and (c) Spiral-horizontal. The arrows represent the movements that define the scan positions.}\n \\label{cdt:raster}\n\t \\end{center} \n \\end{figure}\n\n\n\n\n\n\\subsection{Performance of the IFU}\n\n\nAs described in Section~\\ref{cdt:ifu}, the IFU has a set of collimator and camera mirrors to arrange the FOV into a slit. The mirror arrangement and numbering (illustrated in Figure~\\ref{cdt:numslice}) show that the central mirrors in the image slicer correspond to the edges of the collimator and mini-slit arrays. This layout produces some light contamination on the adjacent mirrors. Figure~\\ref{cdt:pciifu} shows the IFU in operation and gives an idea of the small dimensions of its components.\n The size of each collimator and camera mirror is 4.2~mm~$\\times$~4.5~mm.\n One possible source of that contamination has been evaluated by studying the diffraction effects seen on the collimator mirrors due to the thin dimensions of the slicer mirror (100~$\\mu$m).\n The tests were done at a wavelength of 15650~\\AA, by illuminating only one slicer mirror at a time and measuring the light from all the output mini-slits. Table~\\ref{cdt:tablextalk} shows the result of the tests. The measurements are represented as percentages, normalized to the self-illumination of each slice (diagonal values). The background level at the detector is subtracted. The maximum value of the slice-to-slice contamination is 6.3\\% for the neighbors of slice number 7, and the median value is 3.5\\%. The pupil mask reduces this value significantly. The same tests done without the mask give a median value of 4.8\\%. Another effect, specific to the geometry of the mini-slits and collimator arrays (seen in Figure~\\ref{cdt:figzem}), is ``intra-row'' light contamination, resulting from the layout of the collimator array in two rows. The values that are affected for each row are underlined (see Table~\\ref{cdt:tablextalk}). The maximum value of the intra-row contamination is 1.5\\% from slice number 4 to number 8 and the median value is 1.2\\%. These measured values are compared with the modeled behavior (Regalado {\\itshape et al.}, in prep.), which estimates that at least 80\\% of the energy (from the first diffraction ring) is contained within a single slicer mirror. This value is lower than that obtained in the study by \\citet{Calcines14}, because that work was done at a wavelength of 10000~\\AA, which is a case with less diffraction. \n An additional mask at the mini-slit plane could help to reduce some straylight in the direction perpendicular to the slits. The inclusion of this mask has been considered; the manufacturing and fixing, however, are difficult owing to the discontinued form of mini-slits and the limited availability of space. It is planned for the future prototypes.\n\n\n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=8cm]{cdt_fig7a.png}\n \\includegraphics[width=8cm]{cdt_fig7b.png}\n \\caption{Pictures of the IFU in operation. The IFU is the glass block (Zerodur). The visible elements in the left picture are, from left to right: the RS2 mirror, the array of collimator mirrors, the M1 folding mirror and the array of camera mirrors. The right picture shows the mini-slits projected onto a piece of paper. The polarimeter has been moved out of the light path to take the pictures. }\n \\label{cdt:pciifu}\n\t \\end{center} \n \\end{figure}\n\n\n\n\n\n\n\\begin{wstable}[ht]\n \\caption{Light contamination measurements represented as percentages. Rows (I) denote the illuminated slicer mirror number and columns (M) the measured mirror number. The values that are affected for each row are underlined. The numbering is shown in Figure~\\ref{cdt:numslice}. }\n \\label{cdt:tablextalk}\n\t \\begin{tabular}{c r r r r r r r r p{0.5\\linewidth}} \\toprule\n {\\bf Mirrors} &\t {\\bf 4} & {\\bf 3} & {\\bf 2} & {\\bf 1} & {\\bf 5} & {\\bf 6} & {\\bf 7} & {\\bf 8} \\\\ \n {\\bf (I $\\setminus$ M)} &\t & & & & & & & \\\\ \\colrule\n\t{\\bf 4} & 100 & 3.0 & 0.8 & 0.3 & 0.1 & 0.1 & 0.4 & \\underline{1.5} \\\\\n\t{\\bf 3} & 3.5 & 100 & 3.3 & 1.0 & 0.5 & 0.5 & \\underline{1.1} & \\underline{1.2} \\\\\n\t{\\bf 2} & 0.5 & 3.4 & 100 & 3.3 & 0.9 & \\underline{1.5} & \\underline{1.2} & 0.3 \\\\\n\t{\\bf 1} & 0.1 & 0.8 & 3.4 & 100 & \\underline{4.2} & \\underline{1.6} & 0.5 & 0.3 \\\\\n\t{\\bf 5} & 0.0 & 0.2 & 0.8 & \\underline{3.5} & 100 & 4.7 & 0.8 & 0.5 \\\\\n\t{\\bf 6} & 0.1 & 0.4 & \\underline{1.6} & \\underline{1.5} & 3.3 & 100 & 5.2 & 0.8 \\\\\n\t{\\bf 7} & 0.2 & \\underline{1.2} & \\underline{1.2} & 0.5 & 0.6 & 3.7 & 100 & 6.3 \\\\\n\t{\\bf 8} & \\underline{1.1} & \\underline{1.1} & 0.4 & 0.1 & 0.5 & 0.6 & 2.8 & 100 \\\\ \\botrule\n\t \\end{tabular} \t \n \\end{wstable}\n\n\n\n\n\n\n\n\\section{Science observations}\n\\label{cdt:sciobsv}\n\n\n\nTypical observations in IFU mode are similar to those carried out in slit mode: the observer has to choose the exposure time and the number of accumulations. The scan option provides the possibility of covering a bigger FOV by controlling the number of scan steps. The only difference is that in IFU mode there are two dimensions for the movements, as explained previously.\n The scan can be automatically repeated in time to obtain temporal series.\n In order to calibrate the observations, flat-field images have to be taken\n every\n 1.5~h (maximum) and a series of polarimetric calibration images has to be recorded\n at least once per day.\n The latter is performed using the Polarimetric Calibration Unit \\cite{Hofmann12}. \n\n\nThe spatial sampling can be done in single- or double-mode. Single sampling is equivalent to taking a single exposure at every position in the FOV. In double sampling mode, two exposures are taken at every position, one displaced half-slice-width with respect to the other. With it, the spatial resolution due to the sampling is doubled. The two spectral images are adequately interlaced by the reduction pipeline.\n Figure~\\ref{cdt:doublesamp} illustrates how the double-sampling step size is 50~$\\mu$m for the slice width of 100~$\\mu$m ($0.375^{\\prime\\prime}$).\\\\\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=14cm]{cdt_fig8.png}\n \\caption {Representation of single sampling (left) and double sampling (right). The rectangles represent the slice mirrors and their apparent position with respect to the movements. Only two slice mirrors have been drawn for simplicity.}\n \\label{cdt:doublesamp} \n \\end{center} \n\\end{figure} \n\n\n\n\n\n\\begin{equation}\nt_{\\rm scan} \\simeq [4 * nac *(t_{\\rm int}+ t_{\\rm readout})+ t_{\\rm mov}]* sampling * nv * nh\n\\label{cdt:eqtscan}\n\\end{equation}\n\n\n\n\nThe scan time can be estimated using Eq.~\\ref{cdt:eqtscan}, where: \\\\\n\n\n\\noindent $nac$ = number of accumulations\\\\\n$t_{\\rm int}$ = integration time\\\\\n$t_{\\rm readout}$ = 30 ms\\\\\n$t_{\\rm mov}$ = 1.3 s approx.\\\\\n$sampling$ = 1 for single, 2 for double sampling (see the text)\\\\\n$nv$ = number of vertical steps (running perpendicular to the slices)\\\\\n$nh$ = number of horizontal steps (running parallel to the slices)\\\\\n\n\n\n\n\n \\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=8cm]{cdt_fig9.png}\n \\caption{Example of the raw image of IFU spectra in spectropolarimetric mode centered at 15650~\\AA. The wavelength range is 40~\\AA $\\:$ and Fe~{\\sc i} absorption lines are visible. The most intense one is Fe~{\\sc i}~15662.0~\\AA. Two series of eight spectra are shown for two orthogonal states of polarization, one at the top of the raw image and the other at the bottom. The apparent shift in wavelength between two consecutive spectra is an optical shift due to the arrangement of the mini-slits in two rows. }\n \\label{cdt:figdet}\n\t \\end{center}\n \\end{figure}\n\n\n\n\n\n\n\\subsection{Reduction pipeline}\n\n\n The reduction pipeline \\cite{Collados03} was updated to recognize the IFU mode and to calibrate the data in any observing mode. It is based on the procedures described in \\citet{Collados99}. \nThe current version, GRIS\\_V8 is used for the data reduction process with both slit and IFU modes.\n The pipeline does the dark current and flat corrections, as well as the wavelength and polarimetric calibrations. The pipeline also reorganizes the image of the spectra, from what is seen in the detector (see Figure~\\ref{cdt:figdet}) into a 3D data cube containing two spatial directions plus the spectral direction. The resulting file contains two additional dimensions, one for the Stokes parameters ($I$, $Q$, $U$, and $V$) and one for time, in the case of temporal series. The data has a spatial sampling of $0.135^{\\prime\\prime}\\times~0.1875^{\\prime\\prime}$, where the first number is the pixel scale and the second is, in double sampling mode, the slice width divided by 2. In single sampling mode it is $0.135^{\\prime\\prime}\\times~0.375^{\\prime\\prime}$. \n\n\n\n Figure~\\ref{cdt:figdet} shows an example of the raw image of IFU spectra seen at the detector. It was taken in spectropolarimetric mode, so there are two groups of eight spectra, corresponding to two orthogonal states of polarization. \n The central wavelength is 15650~\\AA $\\:$ and the range is 40~\\AA. There is an apparent shift in wavelength between two consecutive spectra. This effect is actually an optical shift due to the arrangement of the mini-slits in two rows (as seen in the insert of Figure~\\ref{cdt:figzem}). Some bad pixels and the readout structure (vertical lines) of the detector are also present in this raw image. The pipeline removes all these artifacts and corrects the optical shift. Five spectral absorption lines are visible, Fe~{\\sc i}~15662.0~\\AA $\\:$ being the most intense one. The wavelengths of the other Fe~{\\sc i} absorption lines are: 15645.0~\\AA, 15648.5~\\AA, 15652.9~\\AA $\\:$ and 15665.2~\\AA. \n\n\n\n\n\n\\subsection{Examples of observations}\n\nSome examples of the data obtained with the IFU are described below in order to show the instrument capabilities, versatility and performance.\n\n\n\n\n\\subsubsection{1~$\\times$~2 scan of solar granulation} \n\nA small region at the center of the solar disk was observed after the commissioning on September 2018.\nFigure~\\ref{cdt:figeg} shows an example of the time-series observations with a 1~$\\times$~2 scan ($6^{\\prime\\prime}\\times~6^{\\prime\\prime}$). This series shows the reconstructed images of the continuum intensity near 15650~\\AA , observed with a cadence of 18~s. The total duration of the series is about 90 minutes. The evolution of the granulation is clearly seen over the selected time range for the figure. \n The number at the top of each image is the elapsed time relative to the start of this temporal series. The formation of one exploding granule is distinguishable at the lower-left corner of the image at 273~s, and its evolution can be followed over the subsequent images. The analysis of the evolving magnetic fields is in progress (Dominguez-Tagle {\\itshape et al.}, in prep.). \n\n\n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=17.9cm]{cdt_fig10.png}\n \\caption{Example of a series of continuum intensity images near 15650~\\AA $\\:$ of the solar granulation, observed with a 1~$\\times$~2 scan ($6^{\\prime\\prime}\\times~6^{\\prime\\prime}$). The cadence is 18~s and the elapsed time relative to the start of this temporal series is printed at the top of each image. }\n \\label{cdt:figeg}\n \\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=15cm]{cdt_fig11.png}\n \\caption {Monochromatic images of parts of active region 12665 observed with a 3~$\\times$~3 scan ($18^{\\prime\\prime}\\times~9^{\\prime\\prime}$). The images are, from left to right and top to bottom, the continuum intensity and the Q, U, V monochromatic images at a wavelength located 0.5~\\AA $\\:$ to the blue from the center of the Fe~{\\sc i}~15648.52~\\AA $\\:$ spectral line. }\n \\label{cdt:monoimg}\n \\end{center}\n\\end{figure}\n\n\n\\subsubsection{3 $\\times$ 3 scan of solar active region} \n\n\nActive region 12665 (located at solar coordinates S06E24) was observed on July 2017, with a \n3~$\\times$~3 scan ($18^{\\prime\\prime}\\times~9^{\\prime\\prime}$). It was observed over 24 minutes in a temporal series with a cadence of 40~s. This FOV covers a region with a small umbra with a spiral-shaped penumbra.\nFigure~\\ref{cdt:monoimg} shows the continuum intensity map and the Q, U, V monochromatic images extracted from the polarized spectrum at 0.5~\\AA $\\:$ to the blue from the center of the Fe~{\\sc i}~15648.52~\\AA $\\:$ spectral line. This spectral line is specially suited for magnetic studies, because of its large magnetic sensitivity (Land\\'e factor $g_{\\rm eff}=2$). The data sets have been obtained with three modulation cycles. In each of these, a set of four images is recorded, from which the four Stokes parameters I, Q, U, and V can be retrieved. All images corresponding to the same modulation step are summed up in real time. \nThe signal to noise ratio achieved a value of 700 in the Q, U, and V continua ($I_c\/\\sigma_{Q,U,V}$, where $I_c$ \nis the continuum intensity and $\\sigma_{Q,U,V}$ is the Q, U, V continuum noise level -one standard deviation-).\n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=15cm]{cdt_fig12.png}\n \\caption {Example of polarized spectral profiles in a small wavelength range around the center of the Fe~{\\sc i}~15648.52~\\AA $\\:$ spectral line ($\\lambda_0$). The images correspond to the points indicated with a blue cross in Fig.~\\ref{cdt:monoimg}. The Zeeman splitting of the intensity profiles is apparent, as well as the amplitude and separation of the Zeeman components in the polarized profiles.}\n \\label{cdt:specprof}\n \\end{center}\n \\end{figure}\n\n\nFigure~\\ref{cdt:specprof} shows the I, Q, U, V profiles at the point indicated by the blue cross in Figure~\\ref{cdt:monoimg}. Only a spectral range of 4~\\AA $\\:$ around the Fe~{\\sc i}~15648.52~\\AA $\\:$ spectral line is shown, whereas the recorded spectral range is 40~\\AA. The Zeeman splitting in these selected points corresponds to about 2300~G. The amplitudes of the linearly polarized profiles indicate a non-negligible inclination of the magnetic field.\nThe red dashed lines in Figure~\\ref{cdt:monoimg} are used to display equivalent horizontal long-slit I, Q, U and V spectral images (see Figure \\ref{cdt:specimg}). The spatial variations of the field strength, polarity and orientation are clearly detected, as indicated by the separation and sign of the Zeeman components in the Q, U and V images.\n\n\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=15cm]{cdt_fig13.png}\n \\caption {Equivalent horizontal long-slit spectral images, corresponding to the red dashed lines displayed\n in Fig.~\\ref{cdt:monoimg}. The four images displays, from left to right and top to bottom, \n the I, Q, U, and V profiles, normalized to the continuum intensity of the quiet Sun at the\n solar disk center. The X-axis is centered on the wavelength of the Fe~{\\sc i}~15648.52~\\AA $\\:$ spectral line ($\\lambda_0$), and the Y-axis represents the horizontal length of each map.}\n \\label{cdt:specimg}\n \\end{center} \n\\end{figure}\n\n\n\n\\subsubsection{Published results based on IFU observations}\n\n\\begin{itemlist}\n\\item Quiet-Sun magnetic flux cancellations were studied by \\citet{Anjali20}, from observations of quiet-Sun granulation performed on November 2018. The 40-minute time-series observations have a cadence of 26.4~s, for a 1~$\\times$~2 scan ($6^{\\prime\\prime}\\times~6^{\\prime\\prime}$). The spectral range was centered on the Si~{\\sc i}~10827.108~\\AA $\\:$ spectral line, which is magnetically sensitive with a Land\\'e factor $g_{\\rm eff}=1.5$. \\\\\n\n\n\\item A plage region in active region 12723 was observed on October 2018, and the analysis of the magnetic field structures, published by \\citet{tetsu21} is partially based on observations with GRIS\/IFU. Three temporal series of 1~$\\times$~2 scan ($6^{\\prime\\prime}\\times~6^{\\prime\\prime}$) with a cadence of 26~s were performed by the IFU. The observations were centered on a spectral region containing the He~{\\sc i} triplet at 10830~\\AA $\\:$ and Si~{\\sc i}~10827.108~\\AA $\\:$ spectral line.\\\\\n\n\n\\item Magnetic fields in photospheric small-scale network regions were studied by \\citet{campbell21}, from observations performed on May 2019. A 3~$\\times$~3 scan ($18^{\\prime\\prime}\\times~9^{\\prime\\prime}$) of quiet Sun inter-network regions, very close to the disk center, was used for the observations. Their analysis is based on five spectral absorption lines, including Fe~{\\sc i}~15648.52~\\AA $\\:$ and Fe~{\\sc i}~15652.87~\\AA .\\\\\n\n\n\\item On September 2019, a 3~$\\times$~3 scan ($18^{\\prime\\prime}\\times~9^{\\prime\\prime}$) over the lead pore of active region 12748, was conducted by \\citet{nelson21} to study the line-of-sight magnetic field strength in pores. Their research is mostly based on a GRIS\/IFU time-series observations of Fe~{\\sc i}~15648.52~\\AA $\\:$ and Fe~{\\sc i}~15652.87~\\AA $\\:$ spectral lines, with a cadence of 67~s.\n\n\\end{itemlist}\n\n\\section{Conclusions}\n\\label{cdt:concl}\n\nAn IFU was designed and successfully commissioned on \n the GRIS spectrograph installed at the GREGOR solar telescope of the Observatorio del Teide (Tenerife).\n The IFU, based on image slicers, opens up new possibilities for NIR observations of fast-evolving features in the Sun. It has very good optical quality with low straylight, and gives unique performance, allowing faster small-area scans than a traditional slit spectropolarimeter.\n\n \n Based on the success of the first campaigns, the IFU has been offered to all observers who can be granted telescope observing time.\n So far, more than two years of operations have been completed and more than 15 teams of scientists have observed with this instrument. The response of observers has been very positive in terms of both feedback comments and the high demand for the instrument. The first scientific results of observations with the IFU are beginning to be published.\n\n\n\n\n\n\\subsection* {Acknowledgments}\nThis work was carried out with the funding of the Projects SOLARNET (FP7; funded by the European Commission's 7th Framework Program under grant agreement no. 312495), GREST (funded by the European Commission's H2020 Program under grant agreement no. 653982) and SOLARNET (H2020; funded by the European Commission's H2020 Program under grant agreement no. 824135). SDO data are provided by the Joint Science Operations Center-Science Data Processing. \n\nThe 1.5 m GREGOR solar telescope was built by a German consortium under the leadership of the Leibniz-Institut f\\\"{u}r Sonnenphysik in Freiburg with the Leibniz-Institut f\\\"{u}r Astrophysik Potsdam, the Institut f\\\"{u}r Astrophysik G\\\"{o}ttingen, and the Max-Planck-Institut f\\\"{u}r Sonnensystemforschung in G\\\"{o}ttingen as partners, and with contributions by the Instituto de Astrof\\'{i}sica de Canarias and the Astronomical Institute of the Academy of Sciences of the Czech Republic. \n\n\\vspace{0.5cm}\n\nPreprint of an article submitted for consideration in Journal of Astronomical Instrumentation \u00a9 2022 [copyright World Scientific Publishing Company] https:\/\/doi.org\/10.1142\/S2251171722500143\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\indent It is known that a connection $\\nabla$ on a Riemannian manifold $M$ is called a metric connection if there is a Riemannian metric $g$ on $M$ such that $\\nabla g=0$, otherwise it is non-metric. In 1924, Friedman and Schouten \\cite{FRSC} introduced the notion of semi-symmetric linear connection on a differentiable manifold. In 1932, Hayden \\cite{HAYDEN} introduced the idea of metric connection with torsion on a Riemannian manifold. In 1970, Yano \\cite{YANO} studied some curvature tensors and conditions for semi-symmetric connections in Riemannian manifolds. In 1975, Golab \\cite{GOLAB} defined and studied quarter symmetric linear connection on a differentiable manifold. A linear connection $\\overline{\\nabla}$ in an n-dimensional Riemannian manifold is said to be a quarter symmetric connection \\cite{GOLAB} if torsion tensor $T$ is of the form\n\\begin{equation}\\label{1.1}\nT(X,Y)= \\overline{\\nabla}_XY - \\overline{\\nabla}_YX - [X,Y] = A(Y) K(X)-A(X)K(Y)\n\\end{equation}\nwhere $A$ is a 1-form and $K$ is a tensor of type (1,1). If a quarter symmetric linear connection $\\overline{\\nabla}$ satisfies the condition\n\\begin{equation*}\n(\\overline{\\nabla}_Xg)(Y,Z)=0\n\\end{equation*}\nfor all $X$, $Y$, $Z\\in \\chi(M)$, where $\\chi(M)$ is a Lie algebra of vector fields on the manifold $M$, then $\\overline{\\nabla}$ is said to be a quarter symmetric metric connection. For a contact metric manifold admitting quarter symmetric connection, we can take $A=\\eta$ and $K=\\phi$ and hence (\\ref{1.1}) takes in the form\n\\begin{equation}\\label{1.2}\nT(X,Y)=\\eta(Y)\\phi X-\\eta(X)\\phi Y.\n\\end{equation}\nThe relation between Levi-Civita connection $\\nabla$ and quarter symmetric metric connection $\\overline{\\nabla}$\nof a contact metric manifold is given by\n\\begin{equation}\n\\label{1.3}\n\\overline{\\nabla}_XY=\\nabla_XY-\\eta(X)\\phi Y.\n\\end{equation}\n\\indent Quarter symmetric connection also studied by Ali and Nivas \\cite{ALNI}, Anitha and Bagewadi (\\cite{ANIBA} - \\cite{ANIBA2}), De and Uddin \\cite{DEUD}, Hui \\cite{HUI2}, Mishra and Pandey \\cite{MISHRA}, Mukhopadhya et al. \\cite{MRB}, Prakasha \\cite{PRAKAS}, Rastogi \\cite{RASTOGI}, Siddesha and Bagewadi \\cite{SIBA}, Yano and Imai \\cite{YANOI} and many others.\\\\\n\\indent In particular if $\\phi X=X$ and $\\phi Y=Y$, then quarter symmetric reduces to a semi-symmetric connection \\cite{FRSC}.\nThe semi-symmetric connection is the generalized case of quarter symmetric metric connection\nand it is important in the geometry of Riemannian manifolds.\\\\\n\\indent In 2003, Shaikh \\cite{SHAIKH2} introduced the notion of Lorentzian\nconcircular structure manifolds (briefly, $(LCS)_n$-manifolds),\nwith an example, which generalizes the notion of LP-Sasakian\nmanifolds introduced by Matsumoto \\cite{8} and also by Mihai and\nRosca \\cite{9}. Then Shaikh and Baishya (\\cite{SHAIKH3}, \\cite{SHAIKH4})\ninvestigated the applications of $(LCS)_n$-manifolds to the general\ntheory of relativity and cosmology. The $(LCS)_n$-manifolds is also\nstudied by Hui \\cite{SKH}, Hui and Atceken \\cite{HUI1}, Prakasha \\cite{PRAKAS2}, Shaikh and his\nco-authors (\\cite{SHAIKH1} - \\cite{SHAIKH9}) and many others.\\\\\n\\indent The present paper deals with the study of invariant submanifolds of $(LCS)_{n}$-manifolds with respect\nto quarter symmetric metric connection.\nSection 2 is concerned with some preliminaries which will be used in the sequel. $(LCS)_{n}$-manifolds with\nrespect to quarter symmetric metric connection\nis studied in section 3. In section 4, we study invariant submanifolds of $(LCS)_n$-manifolds with respect to\nquarter symmetric metric connection. It is proved that the mean curvature of an invariant submanifold\nwith respect to quarter symmetric metric connection and Levi-Civita connections are equal. In this section,\nwe construct an example of such notion to illustrate the result. Section 5 consists with the study of\nrecurrent invariant submanifolds of $(LCS)_n$-manifold with respect to quarter symmetric\nmetric connection. We obtain a necessary and sufficient condition of the second fundamental form of an\ninvariant submanifold of a $(LCS)_n$-manifold with respect\nto quarter symmetric metric connection to be recurrent, $2$-recurrent and generalized $2$-recurrent.\nSome equivalent conditions of invariant submanifold of\n$(LCS)_n$-manifolds with respect to quarter symmetric metric connection are obtained in this section.\n\\section{Preliminaries}\nThe covariant differential of the $p^{th}$ order, $p\\geq 1$, of a $(0,k)$-tensor field $T$, $k\\geq 1$,\ndefined on a Riemannian manifold $(\\widetilde{ M},g)$ with the Levi-Civita connection $\\nabla$ is denoted\nby $\\nabla^pT$. According to \\cite{ROTER} the tensor $T$ is said to be recurrent, respectively 2-recurrent, if\nthe following condition holds on $M$\n\\begin{eqnarray}\n\\label{2.1}(\\nabla T)(X_1,X_2,\\cdots,X_k ; X)T(Y_1,Y_2,\\cdots,Y_k) &=&\\\\\n\\nonumber (\\nabla T)(Y_1,Y_2,\\cdots,Y_k;X)T(X_1,X_2,\\cdots,X_k),\n\\end{eqnarray}\nrespectively\n\\begin{eqnarray}\n\\label{2.2}(\\nabla^2 T)(X_1,X_2,\\cdots,X_k ; X,Y)T(Y_1,Y_2,\\cdots,Y_k) &=&\\\\\n\\nonumber (\\nabla^2 T)(Y_1,Y_2,\\cdots,Y_k;X,Y)T(X_1,X_2,\\cdots,X_k),\n\\end{eqnarray}\nwhere $ X,Y,X_1,Y_1,\\cdots,X_k, Y_k \\in T \\widetilde{M}$. From (\\ref{2.1}) it follows that at a point\n$x$ $\\in \\widetilde{M}$ if the tensor $T$ is non-zero then there exists a unique 1-form $\\pi$\nrespectively, a (0,2) tensor $\\psi$ , defined on a neighbourhood $U$ of $x$, such that\n\\begin{equation}\\label{2.3}\n\\nabla T= T\\otimes \\pi, \\ \\ \\pi = d(\\log \\|T\\|)\n\\end{equation}\nrespectively,\n\\begin{equation}\\label{2.4}\n\\nabla^2T=T\\otimes\\psi,\n\\end{equation}\nholds on $U$, where $\\|T\\|$ denotes the norm of $T$ and $\\|T\\|^2=g(T,T)$.\\\\\nThe tensor is said to be generalized 2-recurrent if\n\\begin{eqnarray*}\n(\\nabla^2T)(X_1,X_2,\\cdots,X_k;X,Y)-(\\nabla T\\otimes\\pi)(X_1, X_2,\\cdots,X_k;X,Y)T(Y_1, Y_2,\\cdots,Y_k) =\\\\\n(\\nabla^2T)(Y_1, Y_2,\\cdots,Y_k;X,Y)-(\\nabla T\\otimes\\pi)(Y_1, Y_2,\\cdots,Y_k;X,Y)T(X_1,X_2,.....,X_k)\n\\end{eqnarray*}\nholds on $\\widetilde{M}$, where $\\phi $ is a 1-form on $\\widetilde{M}$. From this it follows that\nat a point $x\\in \\widetilde{M}$ if the tensor $T$ is non-zero, then there exists a unique $(0,2)$-tensor\n$\\psi$, defined on a neighbourhood $U$ of $x$, such that\n\\begin{equation}\\label{2.5}\n\\nabla^2T=\\nabla T\\otimes\\pi+T\\otimes\\psi\n\\end{equation}\nholds on $U$.\n\\par An $n$-dimensional Lorentzian manifold $\\widetilde{M}$ is a smooth connected\nparacompact Hausdorff manifold with a Lorentzian metric $g$, that\nis, $\\widetilde{M}$ admits a smooth symmetric tensor field $g$ of type (0,2)\nsuch that for each point $p\\in \\widetilde{M}$, the tensor $g_{p}:T_{p}\\widetilde{M}\\times\nT_{p}\\widetilde{M}$ $\\rightarrow\\mathbb{R}$ is a non-degenerate inner product of\nsignature $(-,+,\\cdots,+)$, where $T_{p}\\widetilde{M}$ denotes the tangent\nvector space of $\\widetilde{M}$ at $p$ and $\\mathbb{R}$ is the real number\nspace. A non-zero vector $v$ $\\in T_{p}\\widetilde{M}$ is said to be timelike\n(resp., non-spacelike, null, spacelike) if it satisfies $g_{p}(v,v)\n< 0$ (resp, $\\leq $ 0, = 0, $> 0$) \\cite{NIL}.\n\\begin{definition}\nIn a Lorentzian manifold $(\\widetilde{M},g)$ a vector field $P$ defined by\n\\begin{equation*}\ng(X,P)=A(X)\n\\end{equation*}\nfor any $X\\in\\Gamma(T\\widetilde{M})$, is said to be a concircular vector field \\cite{15} if\n\\begin{equation*}\n(\\widetilde{\\nabla}_{X}A)(Y)=\\alpha \\{g(X,Y)+\\omega(X)A(Y)\\},\n\\end{equation*}\nwhere $\\alpha$ is a non-zero scalar and $\\omega$ is a closed 1-form\nand $\\widetilde{\\nabla}$ denotes the operator of covariant\ndifferentiation with respect to the Lorentzian metric $g$.\n\\end{definition}\nLet $\\widetilde{M}$ be an $n$-dimensional Lorentzian manifold admitting a unit\ntimelike concircular vector field $\\xi$, called the characteristic\nvector field of the manifold. Then we have\n\\begin{equation}\n\\label{3.1}\ng(\\xi, \\xi)=-1.\n\\end{equation}\nSince $\\xi$ is a unit concircular vector field, it follows that\nthere exists a non-zero 1-form $\\eta$ such that for\n\\begin{equation}\n\\label{3.2}\ng(X,\\xi)=\\eta(X),\n\\end{equation}\nthe equation of the following form holds\n\\begin{equation}\n\\label{3.3}\n(\\widetilde\\nabla _{X}\\eta)(Y)=\\alpha \\{g(X,Y)+\\eta(X)\\eta(Y)\\},\n\\ \\ \\ (\\alpha\\neq 0)\n\\end{equation}\n\\begin{equation}\n\\label{3.4}\n\\widetilde\\nabla _{X}\\xi = \\alpha \\{X +\\eta(X)\\xi\\}, \\ \\ \\ \\alpha\\neq 0,\n\\end{equation}\nfor all vector fields $X$, $Y$, where $\\widetilde{\\nabla}$ denotes the\noperator of covariant differentiation with respect to the Lorentzian\nmetric $g$ and $\\alpha$ is a non-zero scalar function satisfies\n\\begin{equation}\n\\label{3.5}\n{\\widetilde\\nabla}_{X}\\alpha = (X\\alpha) = d\\alpha(X) = \\rho\\eta(X),\n\\end{equation}\n$\\rho$ being a certain scalar function given by $\\rho=-(\\xi\\alpha)$.\nLet us take\n\\begin{equation}\n\\label{3.6}\n\\phi X=\\frac{1}{\\alpha}\\widetilde\\nabla_{X}\\xi,\n\\end{equation}\nthen from (\\ref{3.4}) and (\\ref{3.6}) we have\n\\begin{equation}\n\\label{3.7} \\phi X = X+\\eta(X)\\xi,\n\\end{equation}\n\\begin{equation}\n\\label{3.8}\ng(\\phi X,Y) = g(X,\\phi Y),\n\\end{equation}\nfrom which it follows that $\\phi$ is a symmetric (1,1) tensor and\ncalled the structure tensor of the manifold. Thus the Lorentzian\nmanifold $\\widetilde{M}$ together with the unit timelike concircular vector\nfield $\\xi$, its associated 1-form $\\eta$ and an (1,1) tensor field\n$\\phi$ is said to be a Lorentzian concircular structure manifold\n(briefly, $(LCS)_{n}$-manifold), \\cite{SHAIKH2}. Especially, if we take\n$\\alpha=1$, then we can obtain the LP-Sasakian structure of\nMatsumoto \\cite{8}. In a $(LCS)_{n}$-manifold $(n>2)$, the following\nrelations hold \\cite{SHAIKH2}:\n\\begin{equation}\n\\label{3.9}\n\\eta(\\xi)=-1,\\ \\ \\phi \\xi=0,\\ \\ \\ \\eta(\\phi X)=0,\\ \\ \\\ng(\\phi X, \\phi Y)= g(X,Y)+\\eta(X)\\eta(Y),\n\\end{equation}\n\\begin{equation}\n\\label{3.10}\n\\phi^2 X= X+\\eta(X)\\xi,\n\\end{equation}\n\\begin{equation}\n\\label{3.11}\n\\widetilde{S}(X,\\xi)=(n-1)(\\alpha^{2}-\\rho)\\eta(X),\n\\end{equation}\n\\begin{equation}\n\\label{3.12}\n\\widetilde{R}(X,Y)\\xi=(\\alpha^{2}-\\rho)[\\eta(Y)X-\\eta(X)Y],\n\\end{equation}\n\\begin{equation}\n\\label{3.13}\n\\widetilde{R}(\\xi,Y)Z=(\\alpha^{2}-\\rho)[g(Y,Z)\\xi-\\eta(Z)Y],\n\\end{equation}\n\\begin{equation}\n\\label{3.14}\n(\\widetilde{\\nabla}_{X}\\phi)Y=\\alpha\\{g(X,Y)\\xi+2\\eta(X)\\eta(Y)\\xi+\\eta(Y)X\\},\n\\end{equation}\n\\begin{equation}\n\\label{3.15}\n(X\\rho)=d\\rho(X)=\\beta\\eta(X),\n\\end{equation}\n\\begin{equation}\n\\label{3.16}\n\\widetilde{R}(X,Y)Z =\\phi \\widetilde{R}(X,Y)Z +(\\alpha^{2}-\\rho)\\{g(Y,Z)\\eta(X)-g(X,Z)\\eta(Y)\\}\\xi\n\\end{equation}\nfor all $X,\\ Y,\\ Z\\in\\Gamma(T\\widetilde{M})$ and $\\beta = -(\\xi\\rho)$ is a scalar function,\nwhere $\\widetilde{R}$ is the curvature tensor and $\\widetilde{S}$ is the Ricci tensor of the manifold.\\\\\n\\indent Let $M$ be a submanifold of dimension $m$ of a $(LCS)_n$-manifold $\\widetilde{M}$ $(m0$ such that\n$$|f(y,\\xi)-f(y,\\xi')| \\leq L |\\xi-\\xi'|\\, \\quad \\text{ for a.e. }y \\in {\\mathbb{R}}^N \\text{ and all }\n\\xi,\\, \\xi' \\in {\\mathbb{R}}^{d \\times N}\\,.$$\n\\end{itemize}\nFor $\\e>0$, we define the functionals ${\\mathcal{F}}_\\e:L^1(\\O;{\\mathbb{R}}^d) \\to\n[0,+\\infty]$ by\n$${\\mathcal{F}}_\\e(u):=\\begin{cases} \\displaystyle \\int_\\O f\\left(\\frac{x}{\\e},\\nabla\nu\\right) dx & \\text{if }u \\in\nW^{1,1}(\\O;\\mathcal{M})\\,,\\\\[8pt]\n+\\infty & \\text{otherwise}\\,.\n\\end{cases}$$\n\\vskip5pt\n\nWe have proved in \\cite{BM} the following representation result on\n$W^{1,1}(\\O;{\\mathcal{M}})$.\n\n\\begin{theorem}[\\cite{BM}]\\label{babmilp=1}\nLet ${\\mathcal{M}}$ be a compact and connected smooth submanifold of ${\\mathbb{R}}^d$\nwithout boundary, and $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$~be a Carath\\'eodory function satisfying $(H_1)$ to\n$(H_3)$. Then the family $\\{{\\mathcal{F}}_\\e\\}_{\\e>0}$ $\\G$-converges for the\nstrong $L^1$-topology at every $u \\in W^{1,1}(\\O;{\\mathcal{M}})$ to ${\\mathcal{F}}_{\\rm\nhom} : W^{1,1}(\\O;{\\mathcal{M}}) \\to [0,+\\infty)$, where\n$${\\mathcal{F}}_{\\rm hom}(u):= \\int_\\O Tf_{\\rm hom}(u,\\nabla u)\\, dx\\,,$$\nand $Tf_{\\rm hom}$ is the tangentially homogenized energy density\ndefined for every $s\\in {\\mathcal{M}}$ and $\\xi\\in [T_s({\\mathcal{M}})]^N$ by\n\\begin{equation}\\label{Tfhom}\nTf_{\\rm hom}(s,\\xi)=\\lim_{t\\to+\\infty}\\inf_{\\varphi} \\bigg\\{\n- \\hskip -1em \\int_{(0,t)^N} f(y,\\xi+ \\nabla \\varphi(y))\\, dy : \\varphi \\in\nW^{1,\\infty}_0((0,t)^N;T_s(\\mathcal{M})) \\bigg\\}.\n\\end{equation}\n\\end{theorem}\n\\vskip5pt\n\nNote that the previous theorem is not really satisfactory since the domain of the $\\G$-limit is obviously larger than the Sobolev\nspace $W^{1,1}(\\O;{\\mathcal{M}})$. In view of the studies performed in\n\\cite{GM,Mucci}, the domain is exactly given by $BV(\\O;{\\mathcal{M}})$. \nUnder the additional (standard)\nassumption,\n\\begin{itemize}\n\\item[$(H_4)$]there exist $C>0$ and $00\\}\n\\text{ and }\\varphi=b \\text{ on }\\partial (tQ_\\nu)\\cap\\{x\\cdot \\nu \\leq 0\\}\\bigg\\}\\,,\n\\end{multline}\n$Q_\\nu$ being any open unit cube in ${\\mathbb{R}}^N$ centered at the\norigin with two of its faces orthogonal to $\\nu$.\n\\end{theorem}\n\n\nThe paper is organized as follows. We first review in Section 2\nstandard facts about of manifold valued Sobolev mappings and\nfunctions of bounded variation that will be used all the way\nthrough. The main properties of the energy densities $Tf_{\\rm hom}$ and\n$\\vartheta_{\\rm hom}$ are the object of\nSection 3. A locality property of the $\\Gamma$-limit is established in Section 4. The upper bound inequality\nin Theorem \\ref{babmil2} is the object of Section~5. The lower bound is obtained\n in Section 6 where the proof of the theorem is completed.\nFinally we state in the Appendix a relaxation\nresult for general manifolds and integrands which extends\n\\cite{AEL} and \\cite{Mucci}.\n\n\n\n\\section{Preliminaries}\n\nLet $\\O$ be a generic\nbounded open subset of ${\\mathbb{R}}^N$. We write ${\\mathcal{A}}(\\O)$ for the family of\nall open subsets of $\\O$, and $\\mathcal B(\\O)$ for the\n$\\sigma$-algebra of all Borel subsets of $\\O$. We also consider a\ncountable subfamily ${\\mathcal{R}}(\\O)$ of ${\\mathcal{A}}(\\O)$ made of all finite unions\nof cubes with rational\nedge length centered at rational points of ${\\mathbb{R}}^N$.\nGiven $\\nu \\in {\\mathbb{S}^{N-1}}$, $Q_\\nu$ stands for an open unit cube in ${\\mathbb{R}}^N$\ncentered at the origin with two of its faces orthogonal to $\\nu$ and\n$Q_\\nu(x_0,\\rho):= x_0 + \\rho \\,Q_\\nu$. Similarly $Q:=(-1\/2,1\/2)^N$\nis the unit cube in ${\\mathbb{R}}^N$ and $Q(x_0,\\rho):= x_0 + \\rho \\,Q$. We\ndenote by $h^\\infty$ the recession function of a generic scalar\nfunction $h$, {\\it i.e.},\n$$h^\\infty(\\xi):=\\limsup_{t\\to+\\infty}\\,\\frac{h(t\\xi)}{t}\\,.$$\n\n\nThe space of vector valued Radon measures in $\\O$ with finite total\nvariation is denoted by ${\\mathcal{M}}(\\O;{\\mathbb{R}}^m)$. We shall follow \\cite{AFP}\nfor the standard notation on functions of bounded variation. We only\nrecall Alberti Rank One Theorem which states that for $|D^c u|$-a.e.\n$x \\in \\O$, $$A(x):=\\frac{dD^cu}{d|D^cu|}(x)$$ is a rank one matrix.\n\n\n\\bigskip\n\nIn this paper, we are interested in Sobolev and $BV$ maps taking\ntheir values into a given manifold. We consider a connected smooth\nsubmanifold ${\\mathcal{M}}$ of ${\\mathbb{R}}^d$ without boundary. The tangent space of\n${\\mathcal{M}}$ at $s \\in {\\mathcal{M}}$ is denoted by $T_s({\\mathcal{M}})$, ${\\rm co}({\\mathcal{M}})$ stands\nfor the convex hull of ${\\mathcal{M}}$, and $\\pi_1({\\mathcal{M}})$ is the\nfundamental group of ${\\mathcal{M}}$. \n\nIt is well known that if $u \\in W^{1,1}(\\O;{\\mathcal{M}})$, then $\\nabla u(x)\n\\in [T_{u(x)}({\\mathcal{M}})]^N$ for ${\\mathcal{L}}^N$-a.e. $x \\in \\O$. The analogue\nstatement for $BV$-maps is given in Lemma \\ref{manifold} below.\n \n \n\\begin{lemma}\\label{manifold}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$,\n\\begin{align}\n\\label{aplimM}&\\tilde u(x) \\in {\\mathcal{M}}\\text{ for every } x \\in \\O\\setminus S_u\\,;\\\\[0.2cm]\n\\label{jumpM}&u^\\pm(x) \\in {\\mathcal{M}}\\text{ for every }x \\in J_u\\,;\\\\[0.2cm]\n\\label{gradM}&\\nabla u(x) \\in [T_{u(x)}({\\mathcal{M}})]^N \\text{ for } {\\mathcal{L}}^N\\text{-a.e. }x \\in \\O\\,;\\\\[0.2cm]\n\\label{cantM}&\\displaystyle A(x):=\\frac{dD^c u}{d|D^c u|}(x) \\in [T_{\\tilde u(x)}({\\mathcal{M}})]^N \\text{ for }|D^c u|\\text{-a.e. }x \\in \\O\\,.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nWe first show (\\ref{aplimM}). By definition of the space\n$BV(\\O;{\\mathcal{M}})$, $u(y)\\in{\\mathcal{M}}$ for a.e. $y\\in \\O$. Therefore for any\n$x\\in\\O\\setminus S_u$, we have $|u(y)-\\tilde u(x)|\\geq\n\\text{dist}(\\tilde u(x),{\\mathcal{M}})$ for a.e. $y\\in\\O$. By definition of\n$S_u$, this yields $\\text{dist}(\\tilde u(x),{\\mathcal{M}})=0$, {\\it i.e.},\n$\\tilde u(x)\\in {\\mathcal{M}}$. Arguing as for the approximate limit points,\none obtains (\\ref{jumpM}).\n\nNow it remains to prove \\eqref{gradM} and \\eqref{cantM}. We introduce the function $\\Phi:{\\mathbb{R}}^d\\to{\\mathbb{R}}$ defined by\n$$\\Phi(s)=\\chi(\\delta^{-1}\\text{dist}(s,{\\mathcal{M}})^2)\\, \\text{dist}(s,{\\mathcal{M}})^2\\,,$$\nwhere $\\chi\\in {\\mathcal{C}}_c^\\infty({\\mathbb{R}};[0,1])$ with $\\chi(t)=1$ for $|t|\\leq\n1$, $\\chi(t)=0$ for $|t|\\geq2$, and $\\delta>0$ is small enough so\nthat $\\Phi \\in {\\mathcal{C}}^1({\\mathbb{R}}^d)$. Note that for every $s \\in {\\mathcal{M}}$,\n$\\Phi(s)=0$ and\n\\begin{equation}\\label{ker}\n\\text{Ker}\\,\\nabla\\Phi(s)=T_s({\\mathcal{M}})\\,.\n\\end{equation}\nBy the Chain Rule formula in $BV$ (see, {\\it e.g., }\n\\cite[Theorem~3.96]{AFP}), $\\Phi\\circ u\\in BV(\\O)$ and\n\\begin{align*}\nD(\\Phi\\circ u)=&\\,\\nabla\\Phi(u)\\nabla u \\,\\mathcal{L}^N\\res \\, \\O+\\nabla \\Phi(\\tilde u) D^cu\n+\\big(\\Phi(u^+)-\\Phi(u^-)\\big)\\otimes\\nu_u\\,\\mathcal{H}^{N-1}\\res \\, J_u\\\\\n=&\\,\\nabla\\Phi(u)\\nabla u \\,\\mathcal{L}^N\\res \\, \\O+\\nabla \\Phi(\\tilde u)A|D^cu|\\,,\n\\end{align*}\nthanks to \\eqref{jumpM}. On the other hand, $\\Phi\\circ u=0$ a.e. in\n$\\O$ since $u(x)\\in{\\mathcal{M}}$ for a.e. $x\\in\\O$. Therefore we have that\n$D(\\Phi\\circ u)\\equiv 0$. Since $\\mathcal{L}^N\\res \\, \\O$ and $|D^c\nu|$ are mutually singular measures, we infer that\n$\\nabla\\Phi(u(x))\\nabla u(x)=0$ for $\\mathcal{L}^N$-a.e. $x\\in\\O$\nand $\\nabla \\Phi(\\tilde u(x))A(x)=0$ for $|D^cu|$-a.e. $x\\in\\O$.\nHence \\eqref{gradM} and \\eqref{cantM} follow from \\eqref{ker}\ntogether with \\eqref{aplimM}.\n\\end{proof}\n\n\nIn \\cite{B,BZ}, density results of smooth functions between\nmanifolds into Sobolev spaces have been established. In the\nfollowing theorem, we summarize these results only in $W^{1,1}$. Let\n$\\mathcal S$ be the family of all finite unions of subsets contained\nin a $(N-2)$-dimensional submanifold of ${\\mathbb{R}}^N$.\n\n\\begin{theorem}\\label{density}Let ${\\mathcal{D}}(\\O;{\\mathcal{M}}) \\subset\nW^{1,1}(\\O;{\\mathcal{M}})$ be defined by\n$${\\mathcal{D}}(\\O;{\\mathcal{M}}):=\\begin{cases}\nW^{1,1}(\\O;{\\mathcal{M}})\\cap{\\mathcal{C}}^\\infty(\\O;{\\mathcal{M}}) & \\text{if $\\,\\pi_1({\\mathcal{M}})=0$}\\,,\\\\[10pt]\n\\big\\{ u \\in W^{1,1}(\\O;{\\mathcal{M}})\\cap {\\mathcal{C}}^\\infty(\\O \\setminus \\Sigma;{\\mathcal{M}})\n\\text{ for some } \\Sigma \\in \\mathcal S \\big\\}\n & \\text{otherwise}\\,.\n\\end{cases}$$\nThen ${\\mathcal{D}}(\\O;{\\mathcal{M}})$ is dense in $W^{1,1}(\\O;{\\mathcal{M}})$ for the strong\n$W^{1,1}(\\O;{\\mathbb{R}}^d)$-topology.\n\\end{theorem}\n\nWe now present a useful projection technique (taken from \\cite{Dem}\nfor ${\\mathcal{M}}={\\mathbb{S}^{d-1}}$). It was first introduced in \\cite{HKL,HL}, and makes\nuse of an averaging device going back to \\cite{FF}. We sketch the\nproof for the convenience of the reader.\n\n\\begin{proposition}\\label{proj}\nLet ${\\mathcal{M}}$ be a compact connected $m$-dimensional smooth submanifold\nof ${\\mathbb{R}}^d$ without boundary, and let $v \\in W^{1,1}(\\O;{\\mathbb{R}}^d) \\cap\n{\\mathcal{C}}^\\infty(\\O\\setminus \\Sigma;{\\mathbb{R}}^d)$ for some $\\Sigma\\in\\mathcal{S}$\nsuch that $v(x) \\in {\\rm co}({\\mathcal{M}})$ for a.e. $x \\in \\O$. Then there\nexists $w \\in W^{1,1}(\\O;{\\mathcal{M}})$ satisfying $w=v$ a.e. in $\\big\\{x \\in\n\\O\\setminus\\Sigma: v(x) \\in {\\mathcal{M}} \\big\\}$ and\n\\begin{equation}\\label{1127}\n\\int_\\O |\\nabla w|\\, dx \\leq C_\\star \\int_\\O |\\nabla\nv|\\,dx\\,,\\end{equation} for some constant $C_\\star >0$ which only\ndepends on $d$ and ${\\mathcal{M}}$.\n\\end{proposition}\n\n\\begin{proof} According to \\cite[Lemma 6.1]{HL} (which holds for\n$p=1$), there exist a compact Lipschitz polyhedral set $X\\subset{\\mathbb{R}}^d$\nof codimension greater or equal to $2$, and a locally Lipschitz map $\\pi:{\\mathbb{R}}^d \\setminus X \\to\n{\\mathcal{M}}$ such that\n\\begin{equation}\\label{gradproj}\n\\int_{B^d(0,R)} |\\nabla \\pi(s)|\\, ds <+\\infty \\quad \\text{ for every\n}R<+\\infty\\,.\n\\end{equation}\nMoreover, in a neighborhood of ${\\mathcal{M}}$ the mapping $\\pi$ is smooth of\nconstant rank equal to $m$.\n\nWe argue as in the proof of \\cite[Theorem 6.2]{HL}. Let $B$ be an\nopen ball in ${\\mathbb{R}}^d$ containing ${\\mathcal{M}} \\cup X$, and let $\\d>0$ small\nenough so that the nearest point projection on ${\\mathcal{M}}$ is a well\ndefined smooth mapping in the $\\d$-neighborhood of ${\\mathcal{M}}$. Fix $\\sigma <\n\\inf\\{\\d,\\text{dist}({\\rm co}({\\mathcal{M}}),\\partial B)\\}$ small enough, and for $a\n\\in B^d(0,\\sigma)$ we define the translates\n$B_a:=a+B$ and $X_a:=a+X$,\nand the projection $\\pi_a:B_a \\setminus X_a \\to {\\mathcal{M}}$ by\n$\\pi_a(s):=\\pi(s-a)$. Since $\\pi$ has full rank and is smooth in a neighborhood of ${\\mathcal{M}}$,\nby the Inverse Function Theorem the number\n\\begin{equation}\\label{gradlambda}\n\\Lambda:=\\sup_{a \\in B^d(0,\\sigma)} {\\rm\nLip}\\big({\\pi_a}_{|{\\mathcal{M}}}\\big)^{-1}\n\\end{equation}\nis finite and only depends on ${\\mathcal{M}}$. Using Sard's lemma, one can show\nthat $\\pi_a \\circ v \\in W^{1,1}(\\O;{\\mathcal{M}})$ for\n${\\mathcal{L}}^d$-a.e. $a \\in B^d(0,\\sigma)$. Then Fubini's theorem together\nwith the Chain Rule formula yields\n\\begin{multline*}\n\\int_{B^d(0,\\sigma)}\n\\int_{\\O} |\\nabla (\\pi_a \\circ v)(x)|\\, d{\\mathcal{L}}^N(x) \\, d{\\mathcal{L}}^d(a)\n\\leq\\,\\\\\n\\leq \\int_\\O |\\nabla v(x)|\n\\left(\\int_{B^d(0,\\sigma)}|\\nabla \\pi(v(x)-a)| \\, d{\\mathcal{L}}^d(a) \\right)\\, d{\\mathcal{L}}^N(x)\n \\leq \\,\\\\\n \\leq \\left(\\int_{B} |\\nabla \\pi(s)|\\, d{\\mathcal{L}}^d(s)\\right)\n\\left(\\int_\\O |\\nabla v(x)|\\,d{\\mathcal{L}}^N(x)\\right)\\,.\n\\end{multline*}\nTherefore we can find $a \\in B^d(0,\\sigma)$ such that\n\\begin{equation}\\label{gradpia}\\int_\\O |\\nabla (\\pi_a \\circ v)|\\, dx\n\\leq C{\\mathcal{L}}^d\\left(B^d(0,\\sigma)\\right)^{-1} \\int_\\O |\\nabla\nv|\\,dx\\,,\\end{equation} where we used (\\ref{gradproj}). To conclude,\nit suffices to set $w:=\\big({\\pi_a}_{|{\\mathcal{M}}}\\big)^{-1} \\circ \\pi_a \\circ v$,\nand (\\ref{1127}) arises as a consequence of (\\ref{gradlambda}) and\n(\\ref{gradpia}).\n\\end{proof}\n\n\n\n\n\\section{Properties of homogenized energy densities}\n\nIn this section we present the main properties of the energy\ndensities $Tf_\\text{hom}$ and $\\vartheta_\\text{hom}$ defined in (\\ref{Tfhom})\nand (\\ref{thetahom}). In particular we will prove that\n$\\vartheta_\\text{hom}$ is well defined in the sense that the limit in\n(\\ref{thetahom}) exists.\n\n\\subsection{The tangentially homogenized bulk energy}\\label{thbe}\n\nWe start by considering the bulk energy density $Tf_{\\rm hom}$ defined in \\eqref{Tfhom}. \nAs in \\cite{BM} we first construct a new energy density\n$g:{\\mathbb{R}}^N\\times{\\mathbb{R}}^d\\times{\\mathbb{R}}^{d\\times N}\\to[0,+\\infty)$ satisfying\n$$g(\\cdot,s,\\xi)=f(\\cdot,\\xi)\\quad \\text{and}\\quad g_{\\rm hom}(s,\\xi)=Tf_{\\rm\nhom}(s,\\xi)\\quad \\text{for $s\\in{\\mathcal{M}}$ and $\\xi\\in[T_s({\\mathcal{M}})]^N$}\\,.$$\nHence upon extending $Tf_\\text{hom}$ by $g_\\text{hom}$ outside the set\n$\\big\\{(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}:\\; s \\in {\\mathcal{M}},\\, \\xi\n\\in [T_s({\\mathcal{M}})]^N\\big\\}$, we will tacitly assume $Tf_\\text{hom}$ to be\ndefined over the whole ${\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}$. We proceed\nas follow. \\vskip5pt\n\nFor $s \\in {\\mathcal{M}}$ we denote by $P_s:{\\mathbb{R}}^d \\to T_s({\\mathcal{M}})$ the orthogonal\nprojection from ${\\mathbb{R}}^d$ into $T_s({\\mathcal{M}})$, and we set\n$$\\mathbf{P}_s(\\xi):=(P_s(\\xi_1),\\ldots,P_s(\\xi_N)) \\quad \\text{for\n$\\xi=(\\xi_1,\\ldots,\\xi_N)\\in{\\mathbb{R}}^{d\\times N}\\,$.}$$ For $\\delta_0>0$\nfixed, let $\\mathcal U:=\\big\\{s\n\\in{\\mathbb{R}}^d\\,:\\,\\text{dist}(s,{\\mathcal{M}})<\\d_0\\big\\}$ be the $\\d_0$-neighborhood of\n${\\mathcal{M}}$. Choosing $\\delta_0>0$ small enough, we may assume that the\nnearest point projection $\\Pi: \\mathcal U \\to {\\mathcal{M}}$ is a well defined\nLipschitz mapping. Then the map $s \\in \\mathcal U \\mapsto\nP_{\\Pi(s)}$ is Lipschitz. Now we introduce a cut-off function $\\chi\n\\in {\\mathcal{C}}^\\infty_c({\\mathbb{R}}^d;[0,1])$ such that $\\chi(t)=1$ if $\\text{dist}(s,{\\mathcal{M}})\n\\leq \\delta_0\/2$, and $\\chi(s)=0$ if $\\text{dist}(s,{\\mathcal{M}}) \\geq 3\\delta_0\/4$,\nand we define\n$$\\mathbb{P}_s(\\xi):=\\chi(s) \\mathbf{P}_{\\Pi(s)}(\\xi)\\quad \\text{for $(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,$.}$$\nGiven the Carath\\'eodory integrand $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d\\times N}\n\\to [0,+\\infty)$ satisfying assumptions $(H_1)$ to $(H_3)$, we\nconstruct the new integrand $g: {\\mathbb{R}}^N\\times{\\mathbb{R}}^d\\times{\\mathbb{R}}^{d\\times\nN}\\to[0,+\\infty)$ as\n\\begin{equation}\\label{defig}\ng(y,s,\\xi):=f(y,\\mathbb{P}_s(\\xi))+|\\xi-\\mathbb{P}_s(\\xi)|\\,.\n\\end{equation}\n\n\nWe summarize in the following lemma the main properties of $g$.\n\n\\begin{lemma}\\label{defg}\nThe integrand $g$ as defined in (\\ref{defig}) is a Carath\\'eodory function satisfying\n\\begin{equation}\\label{idfg}\ng(y,s,\\xi)=f(y,\\xi)\\quad\\text{and}\\quad g^\\infty(y,s,\\xi)=f^\\infty(y,\\xi)\\quad \\text{for $s\n\\in {\\mathcal{M}}$ and $\\xi \\in [T_s({\\mathcal{M}})]^N\\,$,}\n\\end{equation}\nand\n\\begin{itemize}\n\\item[(i)] $g$ is $1$-periodic in the first variable;\n\\item[(ii)] there exist $0<\\alpha'\\leq \\b'$ such that\n\\begin{equation}\\label{pgrowth}\n\\alpha'|\\xi|\\leq g(y,s,\\xi)\\leq\n\\b'(1+|\\xi|)\\quad\\text{for every $(s,\\xi)\\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d\n\\times N}$ and a.e. $y \\in {\\mathbb{R}}^N$}\\,;\\end{equation}\n\\item[(iii)] there exist $C>0$ and $C'>0$ such that\n\\begin{equation}\\label{moduluscont}\n|g(y,s,\\xi)-g(y,s',\\xi)| \\leq C|s-s'|\\; |\\xi|\\,,\\end{equation}\n\\begin{equation}\\label{lipg}\n|g(y,s,\\xi) - g(y,s,\\xi')| \\leq C'|\\xi-\\xi'|\\end{equation} for every\n$s$, $s' \\in {\\mathbb{R}}^d$, every $\\xi \\in {\\mathbb{R}}^{d \\times N}$ and a.e. $y\n\\in {\\mathbb{R}}^N$;\n\\item[(iv)] if in addition $(H_4)$ holds, there exists $00$ such that\n\\begin{equation}\\label{grec}\n|g(y,s,\\xi) - g^\\infty(y,s,\\xi)| \\leq C''(1+|\\xi|^{1-q}) \\end{equation}\nfor every $(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}$ and a.e.\n$y \\in {\\mathbb{R}}^N\\,$.\n\\end{itemize}\n\\end{lemma}\n\\vskip5pt\n\nWe can now state the properties of $Tf_\\text{hom}$ and the relation\nbetween $Tf_\\text{hom}$ and $g_\\text{hom}$ through the homogenization procedure.\n\n\n\\begin{proposition}\\label{properties1}\nLet $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d\\times N} \\to [0,+\\infty)$ be a\nCarath\\'eodory integrand satisfying $(H_1)$ to $(H_3)$.\nThen the following properties hold:\n\\begin{itemize}\n\\item[(i)] for every $s \\in {\\mathcal{M}}$ and $\\xi \\in [T_s({\\mathcal{M}})]^N$,\n\\begin{equation}\\label{identhomform}\nTf_{\\rm hom}(s,\\xi)=g_{\\rm hom}(s,\\xi)\\,,\n\\end{equation}\nwhere\n$$g_{\\rm hom}(s,\\xi):=\\lim_{t\\to+\\infty}\\inf_{\\varphi}\\bigg\\{\n- \\hskip -1em \\int_{(0,t)^N} g(y,s,\\xi+\\nabla \\varphi(y))\\, dy: \\varphi\n\\in W^{1,\\infty}_0((0,t)^N;{\\mathbb{R}}^d) \\bigg\\}$$ is the usual homogenized\nenergy density of $g$ (see, e.g.,\n\\cite[Chapter~14]{BD});\\\\\n\\item[(ii)] the function $Tf_{\\rm hom}$ is tangentially\nquasiconvex, {\\it i.e.}, for all $s \\in {\\mathcal{M}}$ and all $\\xi \\in\n[T_s({\\mathcal{M}})]^N$,\n$$Tf_{\\rm hom}(s,\\xi) \\leq \\int_Q Tf_{\\rm hom}(s,\\xi + \\nabla \\varphi(y))\\, dy$$\nfor every $\\varphi \\in W^{1,\\infty}_0(Q;T_s({\\mathcal{M}}))$. In particular\n$Tf_{\\rm hom}(s,\\cdot)$ is rank one convex;\\\\\n\\item[(iii)] there exists\n$C>0$ such that\n\\begin{equation}\\label{pgT}\n\\a|\\xi|\\leq Tf_{\\rm hom}(s,\\xi) \\leq \\b(1+|\\xi|)\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{plipT}\n|Tf_{\\rm hom}(s,\\xi)-Tf_{\\rm hom}(s,\\xi')| \\leq\nC|\\xi-\\xi'|\n\\end{equation}\nfor every $s \\in {\\mathcal{M}}$ and $\\xi$, $\\xi' \\in [T_s({\\mathcal{M}})]^N$;\n\\item[(iv)] there exists $C_1>0$ such that\n\\begin{equation}\\label{hyp4}\n|Tf_{\\rm hom}(s,\\xi)-Tf_{\\rm hom}(s',\\xi)| \\leq\nC_1|s-s'|(1+|\\xi|)\\,,\n\\end{equation}\nfor every $s$, $s' \\in {\\mathbb{R}}^d$ and $\\xi \\in {\\mathbb{R}}^{d \\times N}$. In\nparticular $Tf_{\\rm hom}$ is continuous;\\\\\n\\item[(v)] if in addition $(H_4)$ holds, there exist $C_2>0$ and $00$,\nwe may find $k \\in {\\mathbb{N}}$ and $\\varphi \\in\nW_0^{1,\\infty}((0,k)^N;{\\mathbb{R}}^d)$ such that\n$$- \\hskip -1em \\int_{(0,k)^N} g(y,s,\\xi+\\nabla \\varphi)\\, dy \\leq g_\\text{hom}(s,\\xi)\n+ \\eta\\,.$$ We infer from \\eqref{pgrowth}\nthat $\\alpha'|\\xi|\\leq g_{\\rm hom}(s,\\xi)\\leq \\beta'(1+|\\xi|)$ and consequently\n\\begin{equation}\\label{varphi}\n- \\hskip -1em \\int_{(0,k)^N}|\\nabla \\varphi|\\, dy \\leq C(1+|\\xi|)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on $\\a'$ and $\\b'$.\nThen from (\\ref{identhomform}) and \\eqref{moduluscont} it follows\nthat\n\\begin{multline*}\nTf_\\text{hom}(s',\\xi)-Tf_\\text{hom}(s,\\xi)= g_\\text{hom}(s',\\xi)-g_\\text{hom}(s,\\xi)\\leq\\\\\n\\leq - \\hskip -1em \\int_{(0,k)^N} \\big(g(y,s',\\xi+\\nabla \\varphi)-g(y,s,\\xi+\\nabla\n\\varphi)\\big)\\, dy+ \\eta \\leq\\\\ \\leq C |s-s'|- \\hskip -1em \\int_{(0,k)^N}|\\xi\n+\\nabla \\varphi|\\, dy+\\eta\\leq C|s-s'|(1+|\\xi|)+\\eta\\,.\n\\end{multline*}\nWe deduce relation (\\ref{hyp4}) inverting the roles of\n$s$ and $s'$, and sending $\\eta$ to zero. In particular, we obtain\nthat $Tf_\\text{hom}$ is continuous as a consequence of (\\ref{hyp4}) and\n(\\ref{plipT}).\n\nTo show (\\ref{hyp5}), let us consider sequences $t_n \\nearrow +\\infty$, $k_n\n\\in {\\mathbb{N}}$ and $\\varphi_n\\in W^{1,\\infty}_0((0,k_n)^N;T_s({\\mathcal{M}}))$ such\nthat\n\\begin{equation}\\label{comptfhoninf}\nTf_\\text{hom}^{\\, \\infty}(s,\\xi)=\\lim_{n \\to\n+\\infty}\\frac{Tf_\\text{hom}(s,t_n\\xi)}{t_n}\\,,\n\\end{equation}\nand\n$$- \\hskip -1em \\int_{(0,k_n)^N}f(y,t_n \\xi + t_n \\nabla \\varphi_n)\\, dy \\leq\nTf_\\text{hom}(s,t_n\\xi)+\\frac{1}{n}\\,.$$ \nThen $(H_2)$ and\n(\\ref{pgT}) yield\n\\begin{equation}\\label{varphi_n}- \\hskip -1em \\int_{(0,k_n)^N}|\\nabla \\varphi_n|\\, dy \\leq\nC(1+|\\xi|)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on\n$\\a$ and $\\b$. Using $(H_4)$ and \\eqref{comptfhoninf}, we derive that\n\\begin{align*}\n Tf_\\text{hom}(s,\\xi) - Tf_\\text{hom}^{\\,\\infty}(s,\\xi)\n\\leq & \\liminf_{n \\to +\\infty} \\Bigg\\{- \\hskip -1em \\int_{(0,k_n)^N}\\bigg|\nf(y,\\xi+\\nabla\\varphi_n)- f^{\\,\\infty}(y,\\xi +\\nabla \\varphi_n)\\bigg|\\, dy\\, +\\\\\n&\\,+ - \\hskip -1em \\int_{(0,k_n)^N}\\bigg| f^{\\,\\infty}(y,\\xi+\\nabla\n\\varphi_n)-\\frac{ f(y,t_n \\xi + t_n \\nabla\n\\varphi_n)}{t_n}\\bigg|\\, dy\\Bigg\\}\\\\\n\\leq & \\liminf_{n \\to +\\infty} \\Bigg\\{C\n- \\hskip -1em \\int_{(0,k_n)^N}\\!(1+|\\xi+\\nabla \\varphi_n|^{1-q})\\, dy\\\\\n&+ \\frac{C}{t_n}- \\hskip -1em \\int_{(0,k_n)^N}\\!(1+t_n^{1-q}|\\xi+\\nabla\n\\varphi_n|^{1-q})\\, dy\\Bigg\\}\\,,\n\\end{align*}\nwhere we have also used the fact that $f^{\\,\\infty}(y,\\cdot)$ is\npositively homogeneous of degree one in the last inequality. Then\n(\\ref{varphi_n}) and H\\\"older's inequality lead to\n\\begin{equation}\\label{firstineq}\nTf_\\text{hom}(s,\\xi) - Tf_\\text{hom}^{\\,\\infty}(s,\\xi) \\leq C(1+|\\xi|^{1-q})\\,.\n\\end{equation}\nConversely, given $k\\in {\\mathbb{N}}$ and $\\varphi \\in\nW_0^{1,\\infty}((0,k)^N;T_s({\\mathcal{M}}))$, we deduce from $(H_2)$\nthat\n$$\\frac{f(\\cdot,t(\\xi+\\nabla\\varphi(\\cdot)))}{t} \\leq \\b(1+|\\xi+\\nabla\n\\varphi|) \\in L^1((0,k)^N)$$ whenever $t > 1$. Then Fatou's lemma\nimplies\n\\begin{equation*}\nTf_\\text{hom}^\\infty(s,\\xi)\n\\leq \\limsup_{t \\to +\\infty} - \\hskip -1em \\int_{(0,k)^N}\\frac{f(y,t\\xi+t\n\\nabla \\varphi)}{t}\\, dy \\leq - \\hskip -1em \\int_{(0,k)^N}f^\\infty(y,\\xi+ \\nabla \\varphi)\\, dy\\,.\n\\end{equation*}\nTaking the infimum over all admissible $\\varphi$'s and letting $k\\to+\\infty$, we\ninfer\n\\begin{equation}\\label{remdensinf}\nTf_\\text{hom}^\\infty(s,\\xi) \\leq T(f^\\infty)_\\text{hom}(s,\\xi)\\,.\n\\end{equation}\nFor $\\eta>0$ arbitrary small, consider $k \\in {\\mathbb{N}}$ and $\\varphi \\in\nW^{1,\\infty}_0((0,k)^N;T_s({\\mathcal{M}}))$ such that\n$$- \\hskip -1em \\int_{(0,k)^N}f(y,\\xi + \\nabla \\varphi)\\, dy \\leq\nTf_\\text{hom}(s,\\xi)+\\eta\\,.$$\nIn view of $(H_2)$ and (\\ref{pgT}), it turns\nout that \\eqref{varphi} holds \nwith constant $C>0$ only depending on $\\a$ and $\\b$. Then it\nfollows from \\eqref{remdensinf} that\n\\begin{multline*}\nTf_\\text{hom}^\\infty(s,\\xi) - Tf_\\text{hom}(s,\\xi) \\leq\nT(f^\\infty)_\\text{hom}(s,\\xi) - Tf_\\text{hom}(s,\\xi)\\leq\\\\\n\\leq - \\hskip -1em \\int_{(0,k)^N}|f^\\infty(y,\\xi + \\nabla \\varphi)- f(y,\\xi\n+ \\nabla \\varphi)|\\, dy+\\eta \\leq C- \\hskip -1em \\int_{(0,k)^N}(1+|\\xi + \\nabla \\varphi|^{1-q})\\, dy+\\eta\\,,\n\\end{multline*}\nwhere we have used $(H_4)$ in the last inequality. Using H\\\"older's\ninequality, relation (\\ref{varphi}) together with the arbitrariness of\n$\\eta$ yields\n\\begin{equation}\\label{secineq}Tf_\\text{hom}^\\infty(s,\\xi) -\nTf_\\text{hom}(s,\\xi) \\leq C(1+|\\xi|^{1-q})\\,.\\end{equation} Gathering\n(\\ref{firstineq}) and (\\ref{secineq}) we conclude the proof of\n(\\ref{hyp5}). \\prbox\n\n\\subsection{The homogenized surface energy}\\label{sectsurf}\n\n\\noindent We now present the homogenized surface energy density\n$\\vartheta_\\text{hom}$. We start by introducing some useful notations.\n\nGiven $\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$ and\n$(a,b) \\in {\\mathcal{M}} \\times {\\mathcal{M}} $, we denote by\n$$Q_\\nu:=\\Big\\{\\alpha_1\\nu_1+\\ldots+\\alpha_N\\nu_N\\;:\\;\\alpha_1,\\ldots,\\alpha_N \\in(-1\/2,1\/2)\\Big\\}\\,,$$\nand for $x \\in {\\mathbb{R}}^N$, we set $\\|x\\|_{\\nu,\\infty}:=\\sup_{i \\in\n\\{1,\\ldots,N\\}}|x\\cdot\\nu_i|$, $x_\\nu:=x\\cdot \\nu_1$ and\n$x':=(x\\cdot\\nu_2)\\nu_2 +\\ldots+(x\\cdot \\nu_N)\\nu_N$ so that $x$ can\nbe identified to the pair $(x',x_\\nu)$. Let $u_{a,b,\\nu}:Q_\\nu \\to\n{\\mathcal{M}}$ be the function defined by\n$$u_{a,b,\\nu}(x):=\\begin{cases}\na & \\text{if } x_\\nu>0\\,,\\\\[5pt]\nb & \\text{if } x_\\nu \\leq 0\\,.\n\\end{cases}$$\nWe introduce the class of functions\n$${\\mathcal{A}}_t(a,b,\\nu) :=\\Big\\{\\varphi \\in W^{1,1}(t Q_\\nu;{\\mathcal{M}}):\n\\varphi=u_{a,b,\\nu} \\text{ on }\\partial(tQ_\\nu)\\Big\\}\\,.$$\nWe have the following result.\n\n\\begin{proposition}\\label{limitsurfenerg}\nFor every $(a,b,\\nu_1)\\in{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}$, there exists\n\\begin{eqnarray*}\n\\vartheta_{\\rm hom}(a,b,\\nu_1) &: = & \\lim_{t\\to+\\infty}\\,\\inf_\\varphi\n\\left\\{\\frac{1}{t^{N-1}} \\int_{t Q_\\nu} f^\\infty(y,\\nabla\n\\varphi(y))\\, dy : \\varphi \\in {\\mathcal{A}}_t(a,b,\\nu) \\right\\}\\,,\n\\end{eqnarray*}\nwhere $\\nu=(\\nu_1,\\ldots,\\nu_N)$ is any orthonormal basis of ${\\mathbb{R}}^N$ with first element equal to $\\nu_1$ (the limit being independent of such a\nchoice).\n\\end{proposition}\n\nThe proof of Proposition \\ref{limitsurfenerg} is quite indirect and\nis based on an analogous result for a similar surface energy density\n$\\tilde \\vartheta_\\text{hom}$ (see \\eqref{surfen2} below). We will prove in Proposition \\ref{limsurf2}\nthat the two densities coincide.\n\\vskip5pt\n\n\nGiven $a$ and $b\\in {\\mathcal{M}}$, we introduce the family of geodesic curves\n between $a$ and $b$ by\n$$\\mathcal{G}(a,b):=\\bigg\\{\\g\\in {\\mathcal{C}}^{\\infty}({\\mathbb{R}};{\\mathcal{M}}):\\;\\g(t)=a\\text{ if } t\\geq 1\/2,\\, \\g(t)=b\\text{ if }\nt\\leq -1\/2\\,,\\;\\int_{\\mathbb{R}}|\\dot \\g|\\, dt=\\mathbf\nd_{\\mathcal{M}}(a,b)\\bigg\\}\\,,$$ where $\\mathbf d_{\\mathcal{M}}$ denotes the geodesic\ndistance on ${\\mathcal{M}}$. We define for $\\e>0$ and\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$,\n$$\\mathcal{B}_\\e(a,b,\\nu):=\\Big\\{u \\in W^{1,1}(Q_\\nu;{\\mathcal{M}})\\;:\\;u(x)=\\g(x_\\nu\/\\e)\n\\text{ on }\\partial Q_\\nu\\text{ for some }\n\\g\\in\\mathcal{G}(a,b)\\Big\\}\\,.$$\n\n\\begin{proposition}\\label{limsurf2}\nFor every $(a,b)\\in{\\mathcal{M}}\\times{\\mathcal{M}}$ and every orthonormal basis\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, there exists the limit\n\\begin{equation}\\label{surfen2}\n\\tilde\\vartheta_{\\rm hom}(a,b,\\nu) := \\lim_{\\e\\to 0}\\,\\inf_u \\left\\{\n\\int_{Q_\\nu} f^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right) dx : u \\in\n\\mathcal{B}_\\e(a,b,\\nu) \\right\\}\\,.\n\\end{equation}\nMoreover $\\tilde\\vartheta_{\\rm hom}(a,b,\\nu)$ only depends on $a$, $b$ and $\\nu_1$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows the scheme of the one in\n\\cite[Proposition~2.2]{BDV}. We fix $a$ and $b\\in {\\mathcal{M}}$. For every\n$\\e>0$ and every orthonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of\n${\\mathbb{R}}^N$, we set\n$$ I_\\e(\\nu)= I_\\e(a,b,\\nu):= \\inf \\left\\{ \\int_{Q_\\nu}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right) dx : u \\in\n\\mathcal{B}_\\e(a,b,\\nu) \\right\\}\\,.\n$$\nWe divide the proof into several steps.\\vskip5pt\n\n{\\bf Step 1.} Let $\\nu$ and $\\nu'$ be two orthonormal bases of\n${\\mathbb{R}}^N$ with equal first vector, {\\it i.e.}, $\\nu_1=\\nu'_1$. Suppose\nthat $\\nu$ is a rational basis, {\\it i.e.}, for all\n$i\\in\\{1,\\ldots,N\\}$ there exists $\\gamma_i\\in{\\mathbb{R}}\\setminus\\{0\\}$\nsuch that $v_i:=\\gamma_i\\nu_i\\in{\\mathbb{Z}}^N$. Similarly to Step 1 of the\nproof of \\cite[Proposition 2.2]{BDV}, we readily obtain that\n\\begin{equation}\\label{complimsupinf}\n\\limsup_{\\e \\to0}\\, I_\\e (\\nu')\\leq \\liminf_{\\e \\to0}\\, I_\\e(\\nu)\\,.\n\\end{equation}\n\n{\\bf Step 2.} Let $\\nu$ and $\\nu'$ be two orthonormal rational bases\nof ${\\mathbb{R}}^N$ with equal first vector. By Step~1 we immediately obtain\nthat the limits $\\displaystyle \\lim_{\\e\\to0}I_\\e(\\nu)$ and\n$\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu')$ exist and are equal.\n\n\\vskip5pt\n\n{\\bf Step 3.} We claim that for every $\\sigma>0$ there exists\n$\\delta>0$ (independent of $a$ and $b$) such that if $\\nu$ and\n$\\nu'$ are two orthonormal bases of ${\\mathbb{R}}^N$ with\n$|\\nu_i-\\nu'_i|<\\delta$ for every $i=1,\\ldots,N$, then\n$$\\liminf_{\\e\\to0} I_\\e(\\nu)-K\\sigma\\leq \\liminf_{\\e\\to0}I_\\e(\\nu')\\leq\\limsup_{\\e\\to0}I_\\e(\\nu')\\leq\n\\limsup_{\\e\\to0} I_\\e(\\nu)+K\\sigma$$ where $K$ is a positive\nconstant which only depends on ${\\mathcal{M}}$, $\\beta$ and $N$.\n\nWe use the notation $Q_{\\nu,\\eta}:=(1-\\eta)Q_\\nu$ where $0<\\eta<1$.\nLet $\\sigma>0$ be fixed and let $0<\\eta<1$ be such that\n\\begin{equation}\\label{condeta}\n\\eta<\\frac{1}{34}\\quad\\text{and}\\quad\\max\\bigg\\{1-(1-\\eta)^{N-1}\\,,\\;\n\\frac{(1-\\eta)^{N-1}(1-2\\eta)^{N-1}}{(1-3\\eta)^{N-1}}-(1-2\\eta)^{N-1}\\bigg\\}<\\sigma.\n\\end{equation}\nConsider $\\delta_0>0$ (that may be chosen so that $\\d_0 \\leq\n\\eta\/(2\\sqrt{N})$) such that for every $0<\\delta\\leq \\delta_0$ and\nevery pair $\\nu$ and $\\nu'$ of orthonormal basis of ${\\mathbb{R}}^N$\nsatisfying $|\\nu_i-\\nu'_i|\\leq \\delta$ for $i=1,\\ldots,N$, one has\n\\begin{equation}\\label{approxnu}\nQ_{\\nu,3\\eta}\\subset Q_{\\nu',2\\eta}\\subset Q_{\\nu,\\eta}\\,,\n\\end{equation}\nand $\\{x\\cdot\\nu_1'=0\\}\\cap \\partial Q_{\\nu,\\eta} \\subset \\{|x\\cdot\\nu_1|\\leq 1\/8\\}$.\n\nGiven $\\e>0$ small, we consider $u_\\e\\in\\mathcal{B}_{\\e}(a,b,\\nu')$\nsuch that\n$$\\int_{Q_{\\nu'}}f^\\infty\\left(\\frac{x}{\\e},\\nabla u_\\e\\right)dx\\leq I_{\\e}(\\nu')\n+\\sigma\\,,$$\n where $u_\\e(x)=\\g_\\e(x_{\\nu'}\/\\e)$ for $x\\in\\partial\nQ_{\\nu'}$. Now we construct\n$v_\\e\\in\\mathcal{B}_{(1-2\\eta)\\e}(a,b,\\nu)$ satisfying the boundary condition\n$v_\\e(x)=\\g_\\e\\big(x_\\nu\/(1-2\\eta)\\e\\big)$ for $x\\in\\partial\nQ_{\\nu}$. Consider $F_\\eta:{\\mathbb{R}}^N \\to {\\mathbb{R}}$,\n$$F_\\eta(x):=\\bigg(\\frac{1- 2\\|x'\\|_{\\nu,\\infty}}{\\eta}\\bigg)\\frac{x_{\\nu'}}{1-2\\eta}+\n\\bigg(\\frac{\\eta-1+\n2\\|x'\\|_{\\nu,\\infty}}{\\eta}\\bigg)\\frac{x_\\nu}{1-2\\eta}\\,,$$\nand define\n$$v_\\e(x):=\\begin{cases}\n\\displaystyle u_\\e\\bigg(\\frac{x}{1-2\\eta}\\bigg) & \\text{if }x\\in\nQ_{\\nu',2\\eta},\\\\[10pt]\n\\displaystyle \\g_\\e\\bigg(\\frac{x_{\\nu'}}{(1-2\\eta)\\e}\\bigg) &\\text{if }\nx\\in Q_{\\nu,\\eta}\\setminus Q_{\\nu',2\\eta}\\,,\\\\[8pt]\n\\displaystyle a & \\displaystyle\\text{if } x \\in Q_\\nu \\setminus Q_{\\nu,\\eta} \\text{ and }x_\\nu \\geq \\frac{1}{4}\\,,\\\\[8pt]\n\\displaystyle \\gamma_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg) & \\text{if }x \\in\nA_\\eta :=\\big\\{x: |x_\\nu|\\leq 1\/4\\big\\}\\cap(Q_\\nu\\setminus\nQ_{\\nu,\\eta})\\,,\\\\[8pt]\n\\displaystyle b & \\displaystyle\\text{if } x \\in Q_\\nu \\setminus Q_{\\nu,\\eta} \\text{ and\n}x_\\nu \\leq -\\frac{1}{4}\\,.\n\\end{cases}$$\nWe can check that $v_\\e$ is well defined for $\\e$ small enough\nand that $v_\\e\\in\\mathcal{B}_{(1-2\\eta)\\e}(a,b,\\nu)$. Therefore\n\\begin{align}\\label{Ebis}\nI_{(1-2\\eta)\\e}(\\nu)\\leq &\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla v_\\e\\bigg)\\,\ndx\\nonumber\\\\\n=&\\int_{Q_{\\nu',2\\eta}} f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\,dx+\\int_{Q_{\\nu,\\eta}\\setminus\nQ_{\\nu',2\\eta}}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\, dx\\nonumber\\\\\n&+\\int_{A_{\\eta}}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\, dx=:\\,I_1+I_2+I_3\\,.\n\\end{align}\nWe now estimate these three integrals. First, we easily get that\n\\begin{equation}\\label{E1bis}I_1= (1-2\\eta)^{N-1}\\int_{Q_{\\nu'}}f^\\infty\\left(\\frac{y}{\\e},\\nabla u_\\e\\right)\ndy\\leq I_\\e(\\nu')+\\sigma\\,. \\end{equation} In view of\n\\eqref{approxnu} we have $Q_{\\nu,\\eta}\\subset\n(1-\\eta)(1-2\\eta)(1-3\\eta)^{-1}Q_{\\nu'}=:D_{\\eta}$. Then we infer\nfrom the growth condition (\\ref{finfty1gc}) together with Fubini's\ntheorem that\n\\begin{eqnarray}\\label{E2bis}\nI_2 & \\leq & \\beta\\int_{D_\\eta\\setminus Q_{\\nu',2\\eta}}|\\nabla\nv_\\e|\\, dx = \\frac{\\beta}{(1-2\\eta)\\e} \\int_{(D_\\eta\\setminus\nQ_{\\nu',2\\eta})\\cap\\{|x_{\\nu'}|\\leq (1-2\\eta)\\e\/2\\}}\n\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{x_{\\nu'}}{(1-2\\eta)\\e}\\bigg)\\bigg|\\, dx\\nonumber\\\\\n&= & \\beta\\mathcal{H}^{N-1}\\big((D_\\eta\\setminus Q_{\\nu',2\\eta})\\cap\n\\{x_{\\nu'}=0\\}\\big)\\frac{1}{(1-2\\eta)\\e}\\int_{-(1-2\\eta)\\e\/2}^{(1-2\\eta)\\e\/2}\n\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{t}{(1-2\\eta)\\e}\\bigg)\\bigg|\\, dt\\nonumber\\\\\n&=& \\beta \\mathbf\nd_{\\mathcal{M}}(a,b)\\bigg(\\frac{(1-\\eta)^{N-1}(1-2\\eta)^{N-1}}{(1-3\\eta)^{N-1}}-(1-2\\eta)^{N-1}\\bigg)\\,.\n\\end{eqnarray}\nNow it remains to estimate $I_3$. To this purpose we first observe\nthat \\eqref{approxnu} yields \\begin{equation}\\label{Feta1} \\|\\nabla\nF_\\eta\\|_{L^\\infty(A_\\eta;{\\mathbb{R}}^N)}\\leq C\\,,\n\\end{equation}\nfor some absolute\nconstant $C>0$, and\n\\begin{equation}\\label{Feta2}|\\nabla\nF_\\eta(x)\\cdot \\nu_1|\\geq 1\\quad\\text{for a.e. $x\\in\nA_\\eta\\,$.}\n\\end{equation}\nHence, thanks the growth condition\n(\\ref{finfty1gc}), (\\ref{Feta1}) and (\\ref{Feta2}), we get that\n\\begin{multline*}\nI_3 \\leq \\beta\\int_{A_\\eta}|\\nabla v_\\e|\\, dx \\leq\n\\frac{C\\beta}{\\e} \\int_{A_\\eta}\\bigg|\\, \\dot\n\\g_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg)\\bigg |\\, dx \\leq \\frac{C \\beta}{\\e} \\int_{A_\\eta}\\bigg|\\, \\dot\n\\g_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg)\\bigg |\\, |\\nabla\nF_\\eta(x)\\cdot\\nu_1| \\,dx=\\\\\n=C\\beta\\int_{A'_\\eta}\\bigg(\\frac{1}{\\e}\\int_{-1\/4}^{1\/4}\\bigg|\\,\n\\dot \\g_\\e\\bigg(\\frac{F_\\eta(t\\nu_1+x')}{\\e}\\bigg)\\bigg |\\, |\\nabla\nF_\\eta(t\\nu_1+x')\\cdot\\nu_1|\\, dt\\bigg)\\, d\\mathcal{H}^{N-1}(x')\\,,\n\\end{multline*}\nwhere we have set $A'_\\eta:=A_\\eta\\cap\\{x_\\nu=0\\}$, and used Fubini's\ntheorem in the last equality. Changing variables\n$s=(1\/\\e)F_{\\eta}(t\\nu_1+x')$, we obtain that for\n$\\mathcal{H}^{N-1}$-a.e. $x' \\in A'_\\eta$,\n$$\\frac{1}{\\e}\\int_{-1\/4}^{1\/4}\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{F_\\eta(t\\nu_1+x')}{\\e}\\bigg)\\bigg|\\,\n|\\nabla F_\\eta(t\\nu_1+x')\\cdot\\nu_1|\\, dt\\leq \\int_{\\mathbb{R}} |\\dot\n\\g_\\e(s)|\\,ds=\\mathbf d_{\\mathcal{M}}(a,b)\\,.$$ Consequently,\n\\begin{equation}\\label{E3bis}I_3\\leq C\\beta\\, \\mathcal{H}^{N-1}(A'_\\eta)\\, \\mathbf\nd_{\\mathcal{M}}(a,b)=C\\beta \\big(1-(1-\\eta)^{N-1})\\, \\mathbf\nd_{\\mathcal{M}}(a,b)\\,.\\end{equation} In view of (\\ref{Ebis}), (\\ref{condeta})\nand estimates (\\ref{E1bis}), (\\ref{E2bis}) and (\\ref{E3bis}), we\nconclude that\n$$I_{(1-2\\eta)\\e}(\\nu)\\leq I_\\e(\\nu')+K\\sigma\\,,$$\nwhere $K=1+\\beta \\Delta(1+C)$, $\\Delta$ is the diameter of ${\\mathcal{M}}$ and\n$C$ is the constant given by (\\ref{Feta1}). Finally, letting\n$\\e\\to0$ we derive\n$$\\liminf_{\\e\\to 0}I_\\e(\\nu)\\leq \\liminf_{\\e\\to0}I_\\e(\\nu')+K\\sigma\\,, \\text{ and }\n \\limsup_{\\e\\to 0}I_\\e(\\nu)\\leq \\limsup_{\\e\\to0}I_\\e(\\nu')+K\\sigma\\,. $$\nThe symmetry of the roles of $\\nu$ and $\\nu'$ allows us to invert\nthem, thus concluding the proof of Step~3.\\vskip5pt\n\n{\\bf Step 4.} Let $\\nu$ and $\\nu'$ be two orthonormal bases of\n${\\mathbb{R}}^N$ with equal first vector. Similarly to Step~4 of the proof of\n\\cite[Proposition 2.2]{BDV}, by Steps 2 and 3 we readily obtain that\nthe limits $\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu)$ and\n$\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu')$ exist and are equal.\n\\end{proof}\n\n\\noindent{\\bf Proof of Proposition \\ref{limitsurfenerg}.}\nWe use the notation of the previous proof. Given $\\e>0$ and an\northonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, we set\n\\begin{align*}\nJ_\\e(\\nu)=J_\\e(a,b,\\nu) : = & \\inf \\left\\{ \\int_{Q_\\nu}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right)\\, dx : u \\in\n\\mathcal{A}_1(a,b,\\nu) \\right\\} \\\\\n=&\\inf\\bigg\\{\\e^{N-1}\\int_{\\frac{1}{\\e}Q_\\nu}f^\\infty(y,\\nabla\n\\varphi)\\,dy\\;:\\;\\varphi\\in \\mathcal{A}_{1\/\\e}(a,b,\\nu)\n\\bigg\\}\\,.\\end{align*}\nWe claim that\n\\begin{equation}\\label{idsurfen}\n\\lim_{\\e\\to0} J_{\\e}(\\nu)=\\lim_{\\e\\to 0} I_\\e(\\nu)\\,.\n\\end{equation}\nFor $0<\\e<1$ we set $\\tilde \\e=\\e\/(1-\\e)$, and we consider $u_{\\tilde\n\\e}\\in\\mathcal{B}_{\\tilde \\e}(a,b,\\nu)$ satisfying\n$$\\int_{Q_\\nu}f^{\\infty}\\left(\\frac{x}{\\tilde\\e},\\nabla u_{\\tilde\\e}\\right)dx\\leq I_{\\tilde \\e}(\\nu)+ \\e\\,, $$\nwhere $u_{\\tilde \\e}(x)=\\g_{\\tilde \\e}(x_\\nu\/\\tilde \\e)$ if\n$x\\in\\partial Q_\\nu$, for some $\\g_{\\tilde \\e}\\in\\mathcal{G}(a,b)$.\nWe define for every $x\\in Q_\\nu$,\n$$v_\\e(x):=\\begin{cases}\n\\displaystyle u_{\\tilde \\e}\\left(\\frac{x}{1-\\e}\\right) & \\text{if $x\\in Q_{\\nu,\\e}\\,$,}\\\\[10pt]\n\\displaystyle \\g_{\\tilde\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\n&\\text{otherwise}\\,.\n\\end{cases}$$\nOne may check that $v_\\e\\in\\mathcal{A}_1(a,b,\\nu)$, and hence\n$$J_\\e(\\nu)\\leq \\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right)dx=\\int_{Q_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right)dx\n+\\int_{Q_\\nu\\setminus Q_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{\\e},\\nabla\nv_\\e\\right)dx=I_1+I_2\\,.$$\nWe now estimate these two integrals.\nFirst, we have\n\\begin{equation}\\label{2016}I_1=(1-\\e)^{N-1}\\int_{Q_\\nu}f^\\infty\\left(\\frac{y}{\\tilde\\e},\\nabla\nu_{\\tilde\\e}\\right)dy \\leq (1-\\e)^{N-1}\\big(I_{\\tilde\\e}(\\nu)+\\e\\big)\\,.\\end{equation}\nIn view of the growth condition (\\ref{finfty1gc}),\n\\begin{align*}\nI_2&\\leq \\beta\\int_{Q_\\nu\\setminus Q_{\\nu,\\e}}\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}+\\frac{|x_\\nu||\\nabla(\\|x'\\|_{\\nu,\\infty})|}{(1-2\\|x'\\|_{\\nu,\\infty})^2}\\bigg)\\,dx\\\\\n&\\leq 2\\beta\\int_{(Q_\\nu\\setminus Q_{\\nu,\\e})\\cap\\{|x_\\nu|\\leq\n(1-2\\|x'\\|_{\\nu,\\infty})\/2\\}}\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\,dx\\,,\n\\end{align*}\nwhere we have used the facts that $\\dot \\g_{\\tilde\n\\e}(x_\\nu\/(1-2\\|x'\\|_{\\nu,\\infty}))=0$ in the set $\\{|x_\\nu|>\n(1-2\\|x'\\|_\\infty)\/2\\}$ and\n$\\|\\nabla(\\|x'\\|_{\\nu,\\infty})\\|_{L^\\infty(Q_\\nu;{\\mathbb{R}}^N)}\\leq 1$.\nSetting $Q'_\\nu=Q_\\nu\\cap\\{x_\\nu=0\\}$ and\n$Q'_{\\nu,\\e}=Q_{\\nu,\\e}\\cap\\{x_\\nu=0\\}$, we infer from Fubini's\ntheorem that\n\\begin{multline}\\label{2017}\nI_2\\leq 2\\beta\\int_{Q'_\\nu\\setminus\nQ'_{\\nu,\\e}}\\bigg(\\int_{-(1-2\\|x'\\|_{\\nu,\\infty})\/2}^{(1-2\\|x'\\|_{\\nu,\\infty})\/2}\n\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{t}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\, dt\\bigg)\\, d\\mathcal{H}^{N-1}(x')\\leq\\\\\n\\leq 2\\beta\\, \\mathcal{H}^{N-1}(Q'_\\nu\\setminus Q'_{\\nu,\\e})\\,\n\\mathbf d_{\\mathcal{M}}(a,b)\\leq 2\\beta \\mathbf d_{\\mathcal{M}}(a,b)\n\\big(1-(1-\\e)^{N-1}\\big)\\,.\n\\end{multline}\nIn view of the estimates (\\ref{2016}) and (\\ref{2017}) obtained for\n$I_1$ and $I_2$, we derive that\n\\begin{equation}\\label{2019}\n\\limsup_{\\e\\to0}J_\\e(\\nu)\\leq \\lim_{\\e\\to0}\nI_\\e(\\nu)\\,.\\end{equation}\n\nConversely, given $0<\\e<1$, we consider $\\tilde\nu_\\e\\in\\mathcal{A}_1(a,b,\\nu)$ such that\n$$\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla\\tilde u_\\e\\right)dx\\leq J_\\e(\\nu)+\\e\\,,$$\nand $\\g\\in\\mathcal{G}(a,b)$ fixed. We define for $x\\in Q_\\nu$,\n$$w_\\e(x):=\\begin{cases}\n\\displaystyle \\tilde u_\\e\\left(\\frac{x}{1-\\e}\\right) &\\text{if $x\\in Q_{\\nu,\\e}$,}\\\\[10pt]\n\\displaystyle\n\\g\\bigg(\\frac{x_\\nu}{(1-\\e)(2\\|x'\\|_{\\nu,\\infty}-1+\\e)}\\bigg)&\\text{otherwise.}\n\\end{cases}$$\nWe can check that $w_\\e\\in\\mathcal{B}_{(1-\\e)\\e}(a,b,\\nu)$, and arguing as previously we infer that\n\\begin{eqnarray*}I_{(1-\\e)\\e}(\\nu)\n& \\leq & \\int_{Q_{\\nu,\\e}}\\!f^\\infty\\left(\\frac{x}{(1-\\e)\\e}\\,,\\nabla\nw_\\e\\right)\\, dx+ \\int_{Q_\\nu\\setminus\nQ_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{(1-\\e)\\e}\\,,\\nabla\nw_\\e\\right)dx\\\\\n&\\leq & (1-\\e)^{N-1}\\big(J_\\e(\\nu)+\\e\\big) +2\\beta \\mathbf d_{\\mathcal{M}}(a,b)\\big(1-(1-\\e)^{N-1}\\big)\n\\,.\n\\end{eqnarray*}\nConsequently, $\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu)\\leq \\liminf_{\\e\\to0}J_\\e(\\nu)$, which, together with\n(\\ref{2019}), completes the proof of Proposition\n\\ref{limitsurfenerg}.\n\\prbox\n\\vskip5pt\n\nWe now state the\nfollowing properties of the surface energy density.\n\n\\begin{proposition}\\label{contsurfenerg}\nThe function $\\vartheta_\\text{hom}$ is continuous on ${\\mathcal{M}} \\times {\\mathcal{M}} \\times\n{\\mathbb{S}^{N-1}}$ and there exist constants $C_1>0$ and $C_2>0$ such that\n\\begin{equation}\\label{propsurf}\n|\\vartheta_{\\rm hom}(a_1,b_1,\\nu_1) - \\vartheta_{\\rm hom}(a_2,b_2,\\nu_1)| \\leq C_1\n(|a_1-a_2|+ |b_1-b_2|)\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{propsurf2}\n\\vartheta_{\\rm hom}(a_1,b_1,\\nu_1)\\leq C_2|a_1-b_1|\n\\end{equation}\nfor every $a_1,b_1,a_2,b_2\\in {\\mathcal{M}}$ and $\\nu_1 \\in {\\mathbb{S}^{N-1}}$.\n\\end{proposition}\n\n\\begin{proof}\nWe use the notation of the previous proof. By Proposition\n\\ref{limitsurfenerg} together with steps 3 and 4 of the proof of\nProposition \\ref{limsurf2}, we get that $\\vartheta_\\text{hom}(a,b,\\cdot)$\nis continuous on ${\\mathbb{S}^{N-1}}$ uniformly with respect to $a$ and $b$. Hence\nit is enough to show that (\\ref{propsurf}) holds to get the\ncontinuity of $\\vartheta_\\text{hom}$.\n\\vskip5pt\n\n{\\bf Step 1.} We start with the proof of \\eqref{propsurf}. Fix\n$\\nu_1 \\in {\\mathbb{S}^{N-1}}$ and let $\\nu=(\\nu_1,\\nu_2,\\ldots,\\nu_N)$ be any\northonormal basis of ${\\mathbb{R}}^N$.\nFor every $\\e>0$, let $\\tilde\n\\e:= \\e\/(1-\\e)$ and consider $\\g_{\\tilde \\e} \\in \\mathcal\nG(a_1,b_1)$ and $u_{\\tilde \\e} \\in \\mathcal B_{\\tilde\n\\e}(a_1,b_1,\\nu)$ such that $u_{\\tilde \\e}(x)=\\g_{\\tilde \\e}(x_\\nu\/\\tilde \\e)$ for\n$x\\in\\partial Q_\\nu$ and\n$$\\int_{Q_\\nu} f^\\infty\\left(\\frac{x}{\\tilde \\e},\\nabla u_{\\tilde\n\\e}\\right) dx \\leq I_{\\tilde \\e}(a_1,b_1,\\nu) + \\e\\,.$$\nWe shall now\ncarefully modify $u_{\\tilde \\e}$ in order to\nget another function $v_\\e \\in \\mathcal A_1(a_2,b_2,\\nu)$. We will\nproceed as in the proofs of Propositions \\ref{limitsurfenerg} and\n\\ref{limsurf2}. Let $\\g_a \\in \\mathcal G(a_2,a_1)$ and $\\g_b \\in\n\\mathcal G(b_2,b_1)$, and define\n$$v_\\e(x):=\\left\\{\n\\begin{array}{lll}\n\\displaystyle u_{\\tilde\\e}\\left(\\frac{x}{1-\\e}\\right) & \\text{ if } & x \\in\nQ_{\\nu,\\e}\\,,\\\\[0.3cm]\n\\displaystyle \\g_{\\tilde\\e}\\left(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}} \\right)\n& \\text{ if } & \\displaystyle x \\in A_1\\,,\\\\[0.3cm]\n\\displaystyle \\g_a\\left(\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2} \\right) &\n\\text{ if } & \\displaystyle x \\in A_2:=(Q_\\nu\\setminus\nQ_{\\nu,\\e})\\cap\\{x_\\nu\\geq \\e\/2\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_b\\left(\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2} \\right) &\n\\text{ if } & \\displaystyle x \\in A_3:=(Q_\\nu\\setminus Q_{\\nu,\\e})\\cap\\{x_\\nu\\leq -\\e\/2\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_a\\left(\\frac{2\\|x'\\|_{\\nu,\\infty}-1}{2x_\\nu}+\\frac{1}{2}\n\\right) & \\text{ if } & \\displaystyle x \\in A_4:=\\left\\{ 0 < x_\\nu \\leq\n\\frac{\\e}{2}\\,,\\;\n\\frac{1}{2}-x_\\nu \\leq \\|x'\\|_{\\nu,\\infty} < \\frac{1}{2}\\right\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_b\\left(\\frac{1-2\\|x'\\|_{\\nu,\\infty}}{2x_\\nu}+\\frac{1}{2}\n\\right) & \\text{ if } & \\displaystyle x \\in A_5:=\\left\\{ -\\frac{\\e}{2} < x_\\nu\n\\leq 0\\,,\\; \\frac{1}{2}+x_\\nu \\leq \\|x'\\|_{\\nu,\\infty} <\n\\frac{1}{2}\\right\\}\\,,\n\\end{array}\\right.$$\nwith\n$$ A_1:=\\left\\{ \\frac{1-\\e}{2} \\leq\n\\|x'\\|_{\\nu,\\infty} < \\frac{1}{2}\n\\text{ and } |x_\\nu|\\leq -\\|x'\\|_{\\nu,\\infty}+\\frac{1}{2}\\right\\}\\,.$$\nOne may check that the function $v_\\e$ has been constructed in such\na way that $v_\\e \\in {\\mathcal{A}}_1(a_2,b_2,\\nu)$, and thus\n\\begin{equation}\\label{Je}J_\\e(a_2,b_2,\\nu) \\leq\n\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx\\,.\n\\end{equation}\nArguing exactly as in the proof of Proposition \\ref{limsurf2}, one\ncan show that\n\\begin{equation}\\label{cunu}\\int_{Q_{\\nu,\\e}}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq I_{\\tilde\n\\e}(a_1,b_1,\\nu) + \\e\\,,\\end{equation}\nand\n\\begin{eqnarray}\\label{A1}\\int_{A_1}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq C\\mathbf\nd_{\\mathcal{M}}(a_1,b_1)(1-(1-\\e)^{N-1}))\\,.\n\\end{eqnarray}\nNow we only estimate the integrals over $A_2$ and $A_4$, the ones\nover $A_3$ and $A_5$ being very similar. Define the Lipschitz\nfunction $F_\\e:{\\mathbb{R}}^{N} \\to {\\mathbb{R}}$ by\n$$F_\\e(x):=\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2}\\,.$$\nUsing the growth condition (\\ref{finfty1gc}) together with Fubini's\ntheorem, and the fact that $A_2 \\subset\nF_\\e^{-1}\\big([-1\/2,1\/2)\\big)$, we derive\n\\begin{multline*}\n\\int_{A_2} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq\n\\b \\int_{A_2} |\\dot\\g_a (F_\\e(x))|\\, |\\nabla F_\\e(x)|\\, dx\n \\leq\\\\\n \\leq \\b \\int_{F_\\e^{-1}([-1\/2,1\/2))} |\\dot\\g_a\n(F_\\e(x))|\\, |\\nabla F_\\e(x)|\\, dx\n \\leq \\b \\int_{-1\/2}^{1\/2}|\\,\\dot \\g_a(t)| \\,\n{\\mathcal{H}}^{N-1}(F_\\e^{-1}\\{t\\})\\, dt\n\\,,\n\\end{multline*}\nwhere we used the Coarea\nformula in the last inequality.\nWe observe that for every $t\\in(-1\/2,1\/2)$, $F^{-1}_\\e\\{t\\}=\\partial Q_{\\nu,\\frac{\\e(1-2t)}{2}}$ so that\n${\\mathcal{H}}^{N-1}(F_\\e^{-1}\\{t\\})\\leq {\\mathcal{H}}^{N-1}(\\partial Q)$. Therefore\n\\begin{eqnarray}\\label{A2}\\int_{A_2}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq \\b\n{\\mathcal{H}}^{N-1}(\\partial Q)\\mathbf d_{\\mathcal{M}}(a_1,a_2)\\,.\\end{eqnarray} Define\nnow $G : {\\mathbb{R}}^N\\setminus\\{x_\\nu=0\\} \\to {\\mathbb{R}}$ by\n$$G(x):=\\frac{2\\|x'\\|_{\\nu,\\infty}-1}{2x_\\nu}+\\frac{1}{2}\\,.$$\nThe growth condition (\\ref{finfty1gc}) and Fubini's theorem yield\n\\begin{multline*}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\,\\leq \\\\\n\\leq \\b\n\\int_{0}^{\\e\/2} \\left(\\int_{G(\\cdot,x_\\nu)^{-1}([-1\/2,1\/2))}\n|\\dot\\g_a (G(x',x_\\nu))|\\, |\\nabla G(x',x_\\nu)|\\, d{\\mathcal{H}}^{N-1}(x')\n\\right)dx_\\nu\\,.\n\\end{multline*}\nAs $|\\nabla_{x'} G(x)| = 1\/x_\\nu$ and\n$|\\nabla_{x_\\nu} G(x)| \\leq 1\/x_\\nu$ for a.e. $x \\in A_4$, it\nfollows that $|\\nabla G(x)| \\leq 2 |\\nabla_{x'} G(x)|$ for a.e. $x\n\\in A_4$. Hence\n\\begin{multline*}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\,\\leq \\\\\n\\leq 2\\b\n\\int_{0}^{\\e\/2} \\left(\\int_{G(\\cdot,x_\\nu)^{-1}([-1\/2,1\/2))}\n|\\dot\\g_a (G(x',x_\\nu))|\\, |\\nabla_{x'} G(x',x_\\nu)|\\,\nd{\\mathcal{H}}^{N-1}(x') \\right)dx_\\nu\\,.\n\\end{multline*}\nFor every $x_\\nu \\in (0,\\e\/2)$ the\nfunction $G(\\cdot,x_\\nu):{\\mathbb{R}}^{N-1} \\to {\\mathbb{R}}$ is Lipschitz, and thus the\nCoarea formula implies\n\\begin{eqnarray}\\label{A4}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx & \\leq &\n2\\b \\int_0^{\\e\/2} \\left( \\int_{-1\/2}^{1\/2}|\\, \\dot\\g_a(t)|\\,\n{\\mathcal{H}}^{N-2}(\\{x': G(x',x_\\nu)=t\\})\\, dt \\right)dx_\\nu\\nonumber\\\\\n&\\leq & C\\e\\, \\mathbf d_{\\mathcal{M}}(a_1,a_2)\\,,\n\\end{eqnarray}\nwhere we used as previously the estimate ${\\mathcal{H}}^{N-2}(\\{x': G(x',x_\\nu)=t\\})\\leq {\\mathcal{H}}^{N-2}\\big(\\partial (\\frac{-1}{2},\\frac{1}{2})^{N-1}\\big)$.\nGathering (\\ref{Je}) to (\\ref{A4}) and considering the analogous estimates\nfor the integrals over $A_3$ and $A_5$ (with $b_1$ and $b_2$ instead\nof $a_1$ and $a_2$), we infer that\n$$J_\\e(a_2,b_2,\\nu)\n\\leq \\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx\n\\leq I_{\\tilde\\e}(a_1,b_1,\\nu) + C\\big(\\e + \\mathbf d_{\\mathcal{M}}(a_1,a_2) +\n\\mathbf d_{\\mathcal{M}}(b_1,b_2)\\big)\\,.$$ Taking the limit as $\\e \\to 0$, we\nget in light of Propositions \\ref{limitsurfenerg} and \\ref{limsurf2}\nthat\n$$\\vartheta_\\text{hom}(a_2,b_2,\\nu) \\leq \\vartheta_\\text{hom}(a_1,b_1,\\nu) + C \\big( \\mathbf\nd_{\\mathcal{M}}(b_1,b_2) + \\mathbf d_{\\mathcal{M}}(a_1,a_2) \\big)\\,.$$ Since the geodesic\ndistance on ${\\mathcal{M}}$ is equivalent to the Euclidian distance, we\nconclude, possibly exchanging the roles of $(a_1,b_1)$ and\n$(a_2,b_2)$, that (\\ref{propsurf}) holds. \\vskip5pt\n\n{\\bf Step 2.} We now prove \\eqref{propsurf2}. Given an arbitrary orthonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, let $\\g \\in\n\\mathcal G(a_1,b_1)$ and define $u_\\e(x):=\\g(x_\\nu\/\\e)$. Obviously\n$u_\\e \\in \\mathcal B_\\e(a_1,b_1,\\nu)$. Using (\\ref{idsurfen})\ntogether with the growth condition (\\ref{finfty1gc}) satisfied by\n$f^\\infty$, we derive that\n$$\\vartheta_\\text{hom}(a_1,b_1,\\nu_1) \\leq \\liminf_{\\e \\to\n0}\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla u_\\e\\right)dx \\leq\n\\liminf_{\\e \\to 0} \\frac{\\b}{\\e}\\int_{Q_\\nu}\n\\left|\\dot\\g\\left(\\frac{x\\cdot\\nu_1}{\\e}\\right)\\right|\\,\ndx=\\b\\mathbf d_{{\\mathcal{M}}}(a_1,b_1)\\,.$$ Then (\\ref{propsurf2}) follows\nfrom the equivalence between $\\mathbf d_{{\\mathcal{M}}}$ and the Euclidian\ndistance.\n\\end{proof}\n\n\n\\section{Localization and integral repersentation on partitions}\\label{BV}\n\nIn this section we first show that the $\\Gamma$-limit defines a measure. Then we prove an abstract representation on\npartitions in sets of finite perimeter. This two facts will allow us to obtain the upper bound on the $\\Gamma$-limit in the next section.\n\n\\subsection{Localization}\n\nWe consider an arbitrary given sequence\n$\\{\\e_n\\} \\searrow 0^+$ and we localize the functionals\n$\\{{\\mathcal{F}}_{\\e_n}\\}_{n\\in{\\mathbb{N}}}$ on the family ${\\mathcal{A}}(\\O)$, {\\it i.e.}, for\nevery $u \\in L^1(\\O;{\\mathbb{R}}^d)$ and every $A \\in {\\mathcal{A}}(\\O)$, we set\n$${\\mathcal{F}}_{\\e_n}(u,A):= \\begin{cases}\n\\displaystyle \\int_A\nf\\left(\\frac{x}{\\e_n},\\nabla u\\right) dx & \\text{if }u \\in\nW^{1,1}(A;{\\mathcal{M}})\\,,\\\\[8pt]\n+\\infty & \\text{otherwise}\\,.\n\\end{cases}$$\nNext we define for $u\\in L^1(\\O;{\\mathbb{R}}^d)$ and $A \\in {\\mathcal{A}}(\\O)$,\n\\begin{equation*}\n{\\mathcal{F}}(u,A):= \\inf_{\\{u_n\\}} \\bigg\\{ \\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}_{\\e_n}\n(u_n,A)\\, :\\, u_n \\to u \\text{ in }L^1(A;{\\mathbb{R}}^d) \\bigg\\}\\,.\n\\end{equation*}\nNote that ${\\mathcal{F}}(u,\\cdot)$ is an increasing set function for every\n$u\\in L^1(\\O;{\\mathbb{R}}^d)$ and that ${\\mathcal{F}}(\\cdot,A)$ is lower semicontinuous\nwith respect to the strong $L^1(A;{\\mathbb{R}}^d)$-convergence for every\n$A\\in {\\mathcal{A}}(\\O)$.\n\nSince $L^1(A;{\\mathbb{R}}^d)$ is separable, \\cite[Theorem~8.5]{DM} and a\ndiagonalization argument bring the existence of a subsequence (still\ndenoted $\\{\\e_n\\}$) such that ${\\mathcal{F}}(\\cdot,A)$ is the $\\G$-limit of\n${\\mathcal{F}}_{\\e_n}(\\cdot,A)$ for the strong $L^1(A;{\\mathbb{R}}^d)$-topology for\nevery $A \\in {\\mathcal{R}}(\\O)$ (or $A=\\O$). \\vskip5pt\n\nWe have the following locality property of the\n$\\G$-limit which, in the $BV$ setting, parallels \\cite[Lemma 3.1]{BM}.\n\n\\begin{lemma}\\label{measbis}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, the set function ${\\mathcal{F}}(u,\\cdot)$ is the\nrestriction to ${\\mathcal{A}}(\\O)$ of a Radon measure absolutely continuous\nwith respect to ${\\mathcal{L}}^N+|Du|$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u \\in BV(\\O;{\\mathcal{M}})$ and $A \\in {\\mathcal{A}}(\\O)$. By Theorem 3.9 in\n\\cite{AFP}, there exists a sequence $\\{u_n\\} \\subset\nW^{1,1}(A;{\\mathbb{R}}^d) \\cap {\\mathcal{C}}^\\infty(A;{\\mathbb{R}}^d)$ such that $u_n \\to u$ in\n$L^1(A;{\\mathbb{R}}^d)$ and $\\int_A |\\nabla u_n|\\, dx \\to |Du|(A)$. Moreover,\n$u_n(x)\\in {\\rm co}({\\mathcal{M}})$ for a.e. $x \\in A$ and every $n \\in {\\mathbb{N}}$.\nApplying Proposition \\ref{proj} to $u_n$, we obtain a new sequence\n$\\{w_n\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ satisfying\n$$\\int_A |\\nabla w_n|\\, dx \\leq C_\\star \\int_A |\\nabla u_n|\\, dx,$$ for\nsome constant $C_\\star>0$ depending only on ${\\mathcal{M}}$ and $d$. From\nconstruction of $w_n$, we have that $w_n \\to u$ in $L^1(A;{\\mathbb{R}}^d)$.\nTaking $\\{w_n\\}$ as admissible sequence, we deduce in light of the\ngrowth condition $(H_2)$ that\n\\begin{equation*}\n{\\mathcal{F}}(u,A) \\leq \\beta\\big({\\mathcal{L}}^N(A)+C_\\star|Du|(A)\\big)\\,.\n\\end{equation*}\n\nWe now prove that\n\\begin{equation*}\n{\\mathcal{F}}(u,A) \\leq {\\mathcal{F}}(u,B) + {\\mathcal{F}}(u,A \\setminus \\overline C)\n\\end{equation*}\nfor every $A$, $B$ and $C \\in {\\mathcal{A}}(\\O)$ satisfying $\\overline C\n\\subset B \\subset A$. Then the measure property of\n${\\mathcal{F}}(u,\\cdot)$ can be obtained as in the proof of \\cite[Lemma 3.1]{BM} with\nminor modifications. For this reason, we shall omit it.\n\nLet $R \\in {\\mathcal{R}}(\\O)$ such that $C \\subset\\subset R \\subset\\subset B$\nand consider $\\{u_n\\} \\subset W^{1,1}(R;{\\mathcal{M}})$ satisfying $u_n \\to u$\nin $L^1(R;{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{u_nbis}\n\\lim_{n \\to +\\infty} {\\mathcal{F}}_{\\e_n}(u_n,R) = {\\mathcal{F}}(u,R)\\,.\n\\end{equation}\nGiven $\\eta>0$ arbitrary, there exists a sequence $\\{v_n\\} \\subset\nW^{1,1}(A \\setminus \\overline C;{\\mathcal{M}})$ such that $v_n \\to u$ in\n$L^1(A\\setminus \\overline C;{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{v_nbis}\n\\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}_{\\e_n}(v_n,A \\setminus \\overline C) \\leq\n{\\mathcal{F}}(u,A \\setminus \\overline C) + \\eta\\,.\n\\end{equation}\nBy Theorem \\ref{density}, we can assume without loss of generality\nthat $u_n \\in {\\mathcal{D}}(R;{\\mathcal{M}})$ and $v_n \\in {\\mathcal{D}}(A \\setminus \\overline\nC;{\\mathcal{M}})$. Let $L:=\\text{dist}(C,\\partial R)$ and define for every $i \\in\n\\{0,\\ldots,n\\}$,\n$$R_i:=\\bigg\\{x \\in R:\\, \\text{dist}(x,\\partial R)\n>\\frac{iL}{n}\\bigg\\}\\,.$$\nGiven $i \\in \\{0,\\ldots,n-1\\}$, let $S_i:=R_i \\setminus\n\\overline{R_{i+1}}$ and consider a cut-off function $\\zeta_i \\in\n{\\mathcal{C}}^\\infty_c(\\O;[0,1])$ satisfying $\\zeta_i(x) = 1$ for $x\\in\nR_{i+1}$, $\\zeta_i(x) = 0$ for $x\\in \\O\\setminus R_{i}$ and $|\\nabla\n\\zeta_i|\\leq 2n\/L$. Define\n$$z_{n,i}:=\\zeta_i u_n + (1-\\zeta_i)v_n \\in W^{1,1}(A;{\\mathbb{R}}^d)\\,.$$\nIf $\\pi_1({\\mathcal{M}}) \\neq 0$, $z_{n,i}$ is smooth in $A\\setminus\n\\Sigma_{n,i}$ with $\\Sigma_{n,i} \\in \\mathcal S$, while $z_{n,i}$ is\nsmooth in $A$ if $\\pi_1({\\mathcal{M}}) = 0$. Observe that $z_{n,i}(x) \\in {\\rm\nco}({\\mathcal{M}})$ for a.e. $x \\in A$ and actually, $z_{n,i}$ fails to be\n${\\mathcal{M}}$-valued exactly in the set $S_i$. To get an admissible sequence,\nwe project $z_{n,i}$ on ${\\mathcal{M}}$ using Proposition \\ref{proj}. It yields\na sequence $\\{w_{n,i}\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ satisfying $w_{n,i}\n=z_{n,i}$ a.e. in $A \\setminus S_i$,\n\\begin{equation}\\label{L1}\n\\int_A|w_{n,i} -u |\\, dx \\leq \\int_A |z_{n,i} - u|\\, dx+C\n{\\mathcal{L}}^N(S_i)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on the diameter of ${\\rm\nco}({\\mathcal{M}})$, and\n$$\\int_{S_i} |\\nabla w_{n,i}|\\, dx \\leq C_\\star \\int_{S_i} |\\nabla z_{n,i}|\\, dx\n \\leq C_\\star \\int_{S_i}\\left(|\\nabla u_n| + |\\nabla v_n| +\n\\frac{n}{2L} |u_n - v_n|\\right)\\, dx\\,.$$ Arguing exactly as in the\nproof of \\cite[Lemma 3.1]{BM}, we now find an index $i_n \\in\n\\{0,\\ldots,n-1\\}$ such that\n\\begin{multline}\\label{nin}\n{\\mathcal{F}}_{\\e_n}(w_{n,i_n},A) \\leq\n{\\mathcal{F}}_{\\e_n}(u_n,R)+{\\mathcal{F}}_{\\e_n}(v_n,A \\setminus \\overline C)\\,+\\\\\n+ C_0 \\int_{R \\setminus \\overline C}|u_n - v_n|\\, dx +\n\\frac{C_0}{n}\\sup_{k \\in {\\mathbb{N}}}\\int_{R \\setminus \\overline C}(1+|\\nabla\nu_k|+ |\\nabla v_k|)\\, dx\\,,\n\\end{multline}\nfor some constant $C_0$ independent of $n$.\n\nA well\nknown consequence of the Coarea formula yields (see, {\\it e.g.},\n\\cite[Lemma 3.2.34]{Federer}),\n\\begin{equation}\\label{vanslice}\n{\\mathcal{L}}^N(S_{i_n}) = \\int_{i_n\nL\/n}^{(i_n+1)L\/n} {\\mathcal{H}}^{N-1}(\\{x \\in R: \\text{dist}(x,\\partial R)=t\\})\\, dt \\to 0 \\quad\\text{as $n \\to\n+\\infty$\\,.}\n\\end{equation}\nAs a consequence of (\\ref{L1}) and \\eqref{vanslice}, $w_{n,i_n}\n\\to u$ in $L^1(A;{\\mathbb{R}}^d)$. Taking the $\\liminf$ in\n(\\ref{nin}) and using (\\ref{u_nbis}) together with (\\ref{v_nbis}), we derive\n$${\\mathcal{F}}(u,A) \\leq {\\mathcal{F}}(u,R) + {\\mathcal{F}}(u,A \\setminus \\overline C)+\\eta \\leq {\\mathcal{F}}(u,B) + {\\mathcal{F}}(u,A \\setminus \\overline C)+\\eta\\,.$$\nThe conclusion follows from the arbitrariness of $\\eta$.\n\\end{proof}\n\n\\begin{remark}\\label{measconstr}\nIn view of Lemma \\ref{measbis}, for every $u\\in BV(\\O;{\\mathcal{M}})$, the set\nfunction ${\\mathcal{F}}(u,\\cdot)$ can be uniquely extended to a Radon measure\non $\\O$. Such a measure is given by\n$${\\mathcal{F}}(u,B):= \\inf \\big\\{{\\mathcal{F}}(u,A)\\,:\\,A\\in{\\mathcal{A}}(\\O),\\,B\\subset A\\big\\}\\,, $$\nfor every $B\\in\\mathcal{B}(\\O)$ (see, \\emph{e.g.}, \\cite[Theorem\n1.53]{AFP}).\n\\end{remark}\n\n\n\n\\subsection{Integral representation on partitions}\n\n\n\nBesides the locality of ${\\mathcal{F}}(u,\\cdot)$, another key point of the\nanalysis is to prove an abstract integral representation on\npartitions. Similarly to {\\it e.g.} \\cite[Lemma 3.7]{BDV}, using\n$(H_1)$ we easily obtain the translation invariance property of the\n$\\G$-limit, the proof of which is omitted.\n\n\\begin{lemma}\\label{invtrans}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, every $A \\in {\\mathcal{A}}(\\O)$ and every\n$y\\in{\\mathbb{R}}^N$ such that $y+A \\subset \\O$, we have\n$${\\mathcal{F}}(\\tau_yu,y+A)={\\mathcal{F}}(u,A) \\,,$$\nwhere $(\\tau_yu)(x):=u(x-y)$.\n\\end{lemma}\n\nWe are now in position to prove the integral representation of\nthe $\\G$-limit on partitions. \n\n\\begin{proposition}\\label{reppart}\nThere exists a unique function $K:{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}\\to[0,+\\infty)$\ncontinuous in the last variable and such that\n\\newline\n\\emph{(i)} $K(a,b,\\nu)=K(b,a,-\\nu)$ for every\n$(a,b,\\nu)\\in{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}$,\n\\newline\n\\emph{(ii)} for every finite subset $T$ of ${\\mathcal{M}}$,\n\\begin{equation}\\label{intsurfbor}\n{\\mathcal{F}}(u,S)=\\int_{S} K(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\nfor every $u\\in BV(\\O;T)$ and every Borel subset $S$ of $\\O\\cap S_u\\,$.\n\\end{proposition}\n\n\\begin{proof}\nIt follows the argument of \\cite[Proposition 4.2]{BDV} that is based\non the general result \\cite[Theorem 3.1]{AB}, on account to Lemmas\n\\ref{measbis}, \\ref{invtrans} and Remark \\ref{measconstr}. We omit\nany further details.\n\\end{proof}\n\n\n\\section{The upper bound}\n\n\n\n\\noindent We now adress the $\\G$-$\\limsup$ inequality. The upper\nbound on the diffuse part will be obtained using an extension of the\nrelaxation result of \\cite{AEL} (see Theorem \\ref{relax} in the\nAppendix) together with the partial representation of the $\\G$-limit\nalready established in $W^{1,1}$ (see Theorem~\\ref{babmilp=1}). The\nestimate of the jump part relies on the integral representation on\npartitions in sets of finite perimeter stated in Proposition\n\\ref{reppart}. \\vskip5pt\n\nIn view of the measure property of the $\\G$-limit, we may write for\nevery $u\\in BV(\\O;{\\mathcal{M}})$,\n\\begin{equation}\\label{decompupbd}\n{\\mathcal{F}}(u,\\O)={\\mathcal{F}}(u,\\O\\setminus S_u)+{\\mathcal{F}}(u,\\O\\cap S_u)\\,.\n\\end{equation}\nHence the desired upper bound ${\\mathcal{F}}(u,\\O)\\leq {\\mathcal{F}}_{\\rm hom}(u)$ will follow estimating separately the two terms in the right handside of \\eqref{decompupbd}.\n\n\\begin{lemma}\\label{upperboundBV}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, we have\n$${\\mathcal{F}}(u,\\O\\setminus S_u) \\leq \\int_\\O Tf_{\\rm hom}(u,\\nabla u)\\,\ndx+ \\int_\\O Tf^\\infty_{\\rm hom}\\left(\\tilde\nu,\\frac{dD^cu}{d|D^cu|}\\right)\\, d|D^cu|\\,.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $A \\in {\\mathcal{A}}(\\O)$ and $\\{u_n\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ be such that\n$u_n \\to u$ in $L^1(A;{\\mathbb{R}}^d)$. Since ${\\mathcal{F}}(\\cdot,A)$ is sequentially\nlower semicontinuous for the strong $L^1(A;{\\mathbb{R}}^d)$ convergence, it\nfollows from Theorem~\\ref{babmilp=1} that\n$${\\mathcal{F}}(u,A) \\leq \\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}(u_n,A)=\\liminf_{n \\to +\\infty}\\, \\int_A Tf_\\text{hom}(u_n,\\nabla u_n)\\, dx\\,.$$\nSince the sequence $\\{u_n\\}$ is arbitrary, we deduce\n$${\\mathcal{F}}(u,A) \\leq \\inf\\left\\{\\liminf_{n \\to +\\infty} \\int_A Tf_\\text{hom}(u_n,\\nabla u_n)\\, dx : \\{u_n\\} \\subset W^{1,1}(A;{\\mathcal{M}}),\n\\, u_n \\to u\\text{ in }L^1(A;{\\mathbb{R}}^d)\\right\\}.$$ According to\nProposition \\ref{properties1}, the energy density $Tf_\\text{hom}$ is a\ncontinuous and tangentially quasiconvex function which fulfills the\nassumptions of Theorem \\ref{relax}. Hence\n\\begin{equation}\\label{FUA}\n{\\mathcal{F}}(u,A) \\leq \\int_A Tf_\\text{hom}(u,\\nabla u)\\,\ndx+ \\int_A Tf^\\infty_\\text{hom}\\left(\\tilde u,\\frac{dD^cu}{d|D^cu|}\\right)\nd|D^cu| + \\int_{S_u \\cap A} H(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\n\\end{equation}\nfor some function $H:{\\mathcal{M}} \\times {\\mathcal{M}} \\times {\\mathbb{S}^{N-1}} \\to [0,+\\infty)$. By\nouter regularity, (\\ref{FUA}) holds for every $A \\in \\mathcal\nB(\\O)$. Taking $A=\\O \\setminus S_u$, we obtain\n$${\\mathcal{F}}(u,\\O \\setminus S_u) \\leq \\int_\\O Tf_\\text{hom}(u,\\nabla u)\\,\ndx+ \\int_\\O Tf^\\infty_\\text{hom}\\left(\\tilde\nu,\\frac{dD^cu}{d|D^cu|}\\right)\\, d|D^cu|\\,,\n$$ and the proof is complete.\n\\end{proof}\n\nTo prove the upper bound of the jump part, we first need to compare the energy density $K$ obtained in Proposition \\ref{reppart}\nwith the expected density $\\vartheta_\\text{hom}$.\n\n\\begin{lemma}\\label{upbdsurf}\nWe have $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)$ for every\n$(a,b,\\nu_1)\\in {\\mathcal{M}}\\times{\\mathcal{M}}\\times\\mathbb{S}^{N-1}$.\n\\end{lemma}\n\n\\begin{proof}\nWe will partially proceed as in the proof of Proposition\n\\ref{limsurf2} and we refer to it for the notation. Consider\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$. We shall\nprove that $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)$. Since\n$K$ and $\\vartheta_{\\rm hom}$ are continuous in the last variable,\nwe may assume that $\\nu$ is a rational basis, {\\it i.e.}, for all $i\n\\in \\{1,\\ldots,N\\}$, there exists $\\g_i \\in {\\mathbb{R}}\\setminus \\{0\\}$ such\nthat $v_i:= \\g_i \\nu_i \\in {\\mathbb{Z}}^N$, and the general case follows by\ndensity. \\vskip5pt\n\nGiven $0<\\eta<1$ arbitrary, by Proposition \\ref{limitsurfenerg} and\n\\eqref{idsurfen} we can find $\\e_0>0$,\n$u_0\\in\\mathcal{B}_{\\e_0}(a,b,\\nu)$ and\n$\\gamma_{\\e_0}\\in\\mathcal{G}(a,b)$ such that\n$u_0(x)=\\gamma_{\\e_0}(x\\cdot\\nu_1\/\\e_0)$ and\n$$\\int_{Q_{\\nu}} f^{\\infty}\\bigg(\\frac{x}{\\e_0},\\nabla u_0\\bigg)\\, dx\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)+\\eta\\,.$$\nFor every $\\lambda=(\\lambda_2,\\ldots,\\lambda_N)\\in{\\mathbb{Z}}^{N-1}$, we set\n$x_n^{(\\lambda)}:=\\e_n\\sum_{i=2}^N\\lambda_iv_i$ and\n$Q_{\\nu,n}^{(\\lambda)}:=x^{(\\lambda)}_n+(\\e_n\/\\e_0)Q_\\nu$. We define\nthe set $\\Lambda_n$ by\n\\begin{multline*}\n\\Lambda_n\n:=\\Bigg\\{\\lambda\\in{\\mathbb{Z}}^{N-1}\\;:\\;Q_{\\nu,n}^{(\\lambda)}\\subset\nQ_{\\nu} \\text{ and } x_n^{(\\lambda)}\\in \\sum_{i=2}^N\nl_i\\left(\\frac{\\e_n}{\\e_0}+\\e_n \\gamma_i\\right)\\nu_i+\\e_n P\\\\\n\\text{ for some }(l_2,\\ldots,l_N)\\in{\\mathbb{Z}}^{N-1}\\Bigg\\}\\,,\n\\end{multline*}\nwhere $$P:=\\Big\\{\\a_2v_2 + \\ldots + \\a_N v_N : \\, \\a_2,\\ldots,\\a_N\n\\in [-1\/2,1\/2)\\Big\\}.$$ Next consider\n$$u_n(x)=\\begin{cases}\n\\displaystyle u_0\\bigg(\\frac{\\e_0(x-x_n^{(\\lambda)})\\cdot\\nu_1}{\\e_n}\\bigg) & \\text{if $x\\in Q_{\\nu,n}^{(\\lambda)}$ for some $\\lambda\\in\\Lambda_n$}\\,,\\\\[10pt]\n\\displaystyle \\gamma_{\\e_0}\\bigg(\\frac{x\\cdot\\nu_1}{\\e_n}\\bigg) & \\text{otherwise}\\,.\n\\end{cases}$$\nNote that $u_n\\in W^{1,1}(Q_\\nu;{\\mathcal{M}})$, $\\{\\nabla u_n\\}$ is bounded in\n$L^1(Q_\\nu;{\\mathbb{R}}^{d \\times N})$, and $u_n\\to u^{a,b}_{\\nu_1}$ in\n$L^1(Q_\\nu;{\\mathbb{R}}^d)$ as $n\\to+\\infty$ with $u^{a,b}_{\\nu_1}$ given\nby \\begin{equation*}\nu_{\\nu_1}^{a,b}(x):=\\begin{cases}\na & \\text{if } x\\cdot\\nu_1 \\geq 0\\,, \\\\\nb & \\text{if } x\\cdot\\nu_1 <0\\,,\n\\end{cases}\n\\quad \\Pi_{\\nu_1}:=\\big\\{x\\in{\\mathbb{R}}^N : x\\cdot\\nu_1=0\\big\\}\\,.\n\\end{equation*}\nArguing as in Step 1 of the proof of \\cite[Proposition 2.2]{BDV}, we\nobtain that \n\\begin{equation}\\label{pasidee1}\n\\limsup_{n\\to+\\infty}\\,\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\leq\n\\int_{Q_\\nu} f^\\infty\\bigg(\\frac{x}{\\e_0},\\nabla u_0\\bigg)\\, dx \\leq\n\\vartheta_{\\rm hom}(a,b,\\nu_1)+\\eta\\,.\n\\end{equation}\nFor $\\rho>0$ define $A_\\rho:=Q_\\nu\\cap\\{|x\\cdot\\nu_1|<\\rho\\}$. By\nconstruction the sequence $\\{u_n\\}$ is admissible for\n${\\mathcal{F}}(u^{a,b}_{\\nu_1},A_\\eta)$ so that\n\\begin{multline}\\label{pasidee1b}\n{\\mathcal{F}}\\big(u^{a,b}_{\\nu_1},A_\\eta\\cap\\Pi_{\\nu_1}\\big)\\leq{\\mathcal{F}}(u^{a,b}_{\\nu_1},A_\\eta)\\leq\n\\liminf_{n\\to+\\infty} \\,\\int_{A_\\eta}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\leq \\\\\n\\leq\\beta\n\\mathcal{L}^N(A_\\eta)+\\liminf_{n\\to+\\infty}\\,\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla\nu_n\\bigg)\\, dx\\leq\n\\liminf_{n\\to+\\infty}\\,\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla\nu_n\\bigg)\\, dx +\\beta\\eta\\,,\n\\end{multline}\nwhere we have used $(H_2)$ and the fact that $\\nabla u_n=0$ outside\n$A_{\\e_n}$. On the other hand, Proposition~\\ref{reppart} yields\n\\begin{equation}\\label{pasidee2}\n{\\mathcal{F}}\\big(u^{a,b}_{\\nu_1},A_\\eta\\cap\\Pi_{\\nu_1}\\big)=\\int_{A_\\eta\\cap\\Pi_{\\nu_1}}K(a,b,\\nu_1)\\,\nd{\\mathcal{H}}^{N-1}= K(a,b,\\nu_1)\\,.\n\\end{equation}\nUsing $(H_4)$, the boundedness of $\\{\\nabla u_n\\}$ in\n$L^1(Q_\\nu;{\\mathbb{R}}^{d \\times N})$, the fact that\n$f^\\infty(\\cdot,0)\\equiv 0$, and H\\\"older's inequality, we derive\n\\begin{eqnarray}\\label{pasidee3}\n\\bigg|\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg) \\,dx-\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\,dx\\bigg|& \\leq & C\\int_{A_{\\e_n}}(1+|\\nabla u_n|^{1-q})\\,dx\\nonumber\\\\\n& \\leq & C\\big(\\e_n+\\e_n^q\\|\\nabla u_n\\|^{1-q}_{L^1(Q_\\nu;{\\mathbb{R}}^{d\n\\times N})}\\big)\\to 0\\,\n\\end{eqnarray}\nas $n \\to \\infty$. Gathering \\eqref{pasidee1}, \\eqref{pasidee1b}, \\eqref{pasidee2} and\n\\eqref{pasidee3}, we obtain $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm\nhom}(a,b,\\nu_1)+(\\beta+1)\\eta$ and the conclusion follows from the\narbitrariness of $\\eta$.\n\\end{proof}\n\nWe are now in position to prove the upper bound on the jump part of the energy. The argument is based on\nLemma \\ref{upbdsurf} together with an approximation procedure of \\cite{AMT}. In view of Lemma \\ref{upperboundBV}\nand \\eqref{decompupbd}, this will complete the proof of the upper bound ${\\mathcal{F}}(u,\\O)\\leq {\\mathcal{F}}_{\\rm hom}(u)$.\n\n\\begin{corollary}\\label{upbdjp}\nFor every $u\\in BV(\\O;{\\mathcal{M}})$, we have\n$${\\mathcal{F}}(u,\\O\\cap S_u)\\leq \\int_{\\O\\cap S_u} \\vartheta_{\\rm hom}(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,.$$\n\\end{corollary}\n\n\\begin{proof} First assume that $u$ takes a finite number of values, {\\it i.e.}, $u\\in BV(\\O;T)$ for some finite subset $T\\subset {\\mathcal{M}}$.\nThen the conclusion directly follows from Proposition \\ref{reppart} together with Lemma~\\ref{upbdsurf}.\n\nFix an arbitrary function $u\\in BV(\\O;{\\mathcal{M}})$ and an open set\n$A\\in{\\mathcal{A}}(\\O)$. For $\\delta_0>0$ small enough, let $\\mathcal\nU:=\\big\\{s \\in{\\mathbb{R}}^d\\,:\\,\\text{dist}(s,{\\mathcal{M}})<\\d_0\\big\\}$ be the\n$\\d_0$-neighborhood of ${\\mathcal{M}}$ on which the nearest point projection\n$\\Pi: \\mathcal U \\to {\\mathcal{M}}$ is a well defined Lipschitz mapping. We\nextend $\\vartheta_{\\rm hom}$ to a function $\\hat \\vartheta_{\\rm\nhom}$ defined in ${\\mathbb{R}}^d \\times {\\mathbb{R}}^d\n\\times\\mathbb{S}^{N-1}$ by setting\n$$\\hat \\vartheta_{\\rm hom}(a,b,\\nu):=\\chi(a)\\chi(b)\\vartheta_{\\rm hom}\\bigg(\\Pi(a),\\Pi(b), \\nu\\bigg)\\,, $$\nfor a cut-off function $\\chi\n\\in {\\mathcal{C}}^\\infty_c({\\mathbb{R}}^d;[0,1])$ satisfying $\\chi(t)=1$ if $\\text{dist}(s,{\\mathcal{M}})\n\\leq \\delta_0\/2$, and $\\chi(s)=0$ if $\\text{dist}(s,{\\mathcal{M}}) \\geq 3\\delta_0\/4$.\nIn view of Proposition \\ref{contsurfenerg}, we infer that\n$\\hat\\vartheta_{\\rm hom}$ is continuous and satisfies\n$$|\\hat\\vartheta_{\\rm hom}(a_1,b_1,\\nu)-\\hat\\vartheta_{\\rm hom}(a_2,b_2,\\nu)|\\leq C\\big(|a_1-a_2|+|b_1-b_2|\\big)\\,,$$\nand\n$$\\hat\\vartheta_{\\rm hom}(a_1,b_1,\\nu)\\leq C|a_1-b_1|\\,,$$\nfor every $a_1$, $b_1$, $a_2$, $b_2\\in{\\mathbb{R}}^d$, $\\nu\\in{\\mathbb{S}^{N-1}}$, and\nsome constant $C>0$. Therefore we can apply Step~2 in the proof of\n\\cite[Proposition 4.8]{AMT} to obtain a sequence $\\{v_n\\}\\subset\nBV(\\O;{\\mathbb{R}}^d)$ such that, for every $n\\in {\\mathbb{N}}$, $v_n\\in BV(\\O;T_n)$\nfor some finite set $T_n\\subset {\\mathbb{R}}^d$, $v_n\\to u$ in\n$L^\\infty(\\O;{\\mathbb{R}}^d)$ and\n\\begin{eqnarray*}\n\\limsup_{n\\to+\\infty}\\,\\int_{A\\cap S_{v_n}}\\hat\\vartheta_{\\rm\nhom}(v_n^+,v_n^-,\\nu_{v_n})\\, d{\\mathcal{H}}^{N-1} & \\leq & C|Du|(A\\setminus\nS_u)+\n\\int_{A\\cap S_{u}}\\hat\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\\\\n& = & C|Du|(A\\setminus S_u)+ \\int_{A\\cap S_{u}}\\vartheta_{\\rm\nhom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\,.\n\\end{eqnarray*}\nHence we may assume without loss of generality\nthat for each $n\\in {\\mathbb{N}}$, $\\|v_n - u\\|_{L^\\infty(\\O;{\\mathbb{R}}^d)} < \\d_0\/2$,\nand thus $\\text{dist}(v_n^\\pm(x),{\\mathcal{M}}) \\leq |v_n^\\pm(x)-u^\\pm(x)|< \\d_0\/2$ for\n${\\mathcal{H}}^{N-1}$-a.e. $x \\in S_{v_n}$. In particular, we can define\n$$u_n:=\\Pi(v_n)\\,, $$\nand then $u_n\\in BV(\\O;{\\mathcal{M}})$, $u_n\\to u$ in $L^1(\\O;{\\mathbb{R}}^d)$.\nMoreover, one may check that for each $n\\in\\mathbb{N}$,\n$S_{u_n}\\subset S_{v_n}$ so that\n${\\mathcal{H}}^{N-1}\\big(S_{u_n}\\setminus(J_{u_n}\\cap J_{v_n})\\big)\\leq\n{\\mathcal{H}}^{N-1}(S_{u_n}\\setminus J_{u_n})+ {\\mathcal{H}}^{N-1}(S_{v_n}\\setminus\nJ_{v_n})=0$, and\n$$u_n^\\pm(x)=\\Pi(v_n^\\pm(x)) \\quad \\text{ and } \\quad \\nu_{u_n}(x)=\\nu_{v_n}(x)\n\\quad\\text{ for every $x\\in J_{u_n}\\cap J_{v_n}$}\\,.$$\nConsequently,\n\\begin{multline}\\label{estilimsurf}\n\\limsup_{n\\to+\\infty}\\, \\int_{A\\cap S_{u_n}}\\vartheta_{\\rm\nhom}(u_n^+,u_n^-,\\nu_{u_n})\\, d{\\mathcal{H}}^{N-1} \\,\\leq \\\\\n\\leq \\limsup_{n\\to+\\infty}\\int_{A\\cap S_{v_n}}\n\\hat\\vartheta_{\\rm hom}(v_n^+,v_n^-,\\nu_{v_n})\\, d{\\mathcal{H}}^{N-1}\\,\\leq\\\\\n\\leq C|Du|(A\\setminus S_u)+ \\int_{A\\cap S_{u}}\\vartheta_{\\rm\nhom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\,.\n\\end{multline}\nSince $u_n$ takes a finite number of values, Proposition\n\\ref{reppart} and Lemma \\ref{upbdsurf} yield \n\\begin{equation}\\label{estisurfn}\n{\\mathcal{F}}(u_n,A\\cap S_{u_n})\\leq \\int_{A\\cap S_{u_n}}\\vartheta_{\\rm\nhom}(u_n^+,u_n^-,\\nu_{u_n})\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\nand, in view of Lemma \\ref{measbis},\n\\begin{equation}\\label{estidiffn}\n{\\mathcal{F}}(u_n,A\\setminus S_{u_n}) \\leq C{\\mathcal{L}}^N(A)\\,.\n\\end{equation}\nCombining \\eqref{estilimsurf}, \\eqref{estisurfn} and \\eqref{estidiffn}, we deduce\n\\begin{eqnarray*}\n\\limsup_{n\\to+\\infty} \\,{\\mathcal{F}}(u_n,A) & = & \\limsup_{n\\to+\\infty}\\big({\\mathcal{F}}(u_n,A\\setminus S_{u_n})+{\\mathcal{F}}(u_n,A\\cap S_{u_n})\\big)\\\\\n& \\leq & \\int_{A\\cap S_{u}}\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\,\nd{\\mathcal{H}}^{N-1}+ C\\big({\\mathcal{L}}^N(A)+|Du|(A\\setminus S_u)\\big)\\,.\n\\end{eqnarray*}\nOn the other hand, ${\\mathcal{F}}(\\cdot,A)$ is lower semicontinuous with\nrespect to the strong $L^1(A;{\\mathbb{R}}^d)$-convergence, and thus\n$\\displaystyle{\\mathcal{F}}(u,A)\\leq \\liminf_{n\\to+\\infty}\\,{\\mathcal{F}}(u_n,A)$ which leads to\n$${\\mathcal{F}}(u,A)\\leq \\int_{A\\cap S_{u}}\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}+\nC\\big({\\mathcal{L}}^N(A)+|Du|(A\\setminus S_u)\\big)\\,.$$ Since $A$ is\narbitrary, the above inequality holds for any open set $A\\in{\\mathcal{A}}(\\O)$\nand, by Remark \\ref{measconstr}, it also holds if $A$ is any Borel\nsubset of $\\O$. Then taking $A=\\O\\cap S_u$ yields the desired\ninequality.\n\\end{proof}\n\n\n\n\n\\section{The lower bound}\n\n\n\n\n\\noindent We adress in this section with the $\\G$-$\\liminf$\ninequality. Using the blow-up method, we follow the approach of\n\\cite{FM2}, estimating separately the Cantor part and the jump part,\nwhile the bulk part is obtained exactly as in the $W^{1,1}$ analysis, see\n\\cite[Lemma 5.2]{BM}.\n\n\\begin{lemma}\\label{lowerboundBV}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, we have ${\\mathcal{F}}(u,\\O)\\geq {\\mathcal{F}}_{\\rm hom}\n(u)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u \\in BV(\\O;{\\mathcal{M}})$ and $\\{u_n\\} \\subset W^{1,1}(\\O;{\\mathcal{M}})$ be such\nthat $${\\mathcal{F}}(u,\\O)= \\lim_{n \\to +\\infty}\\int_\\O f\n\\left(\\frac{x}{\\e_n},\\nabla u_n \\right)dx\\,.$$ Define the sequence\nof nonnegative Radon measures\n$$\\mu_n:=f \\left(\\frac{\\cdot}{\\e_n},\\nabla u_n \\right){\\mathcal{L}}^N\n\\res\\, \\O\\,.$$\nUp to the extraction of a subsequence, we can assume that\nthere exists a nonnegative Radon measure $\\mu \\in {\\mathcal{M}}(\\O)$\nsuch that $\\mu_n \\xrightharpoonup[]{*} \\mu$ in ${\\mathcal{M}}(\\O)$. By the\nBesicovitch Differentiation Theorem,\nwe\ncan split $\\mu$ into the sum of four mutually singular nonnegative measures\n$\\mu=\\mu^a + \\mu^j+\\mu^c+\\mu^s$ where $\\mu^a \\ll \\mathcal L^N$,\n$\\mu^j \\ll {\\mathcal{H}}^{N-1}\\res\\, S_u$ and $\\mu^c \\ll |D^c u|$. Since we\nhave $\\mu(\\O) \\leq {\\mathcal{F}}(u,\\O)$, it is enough to check that\n\\begin{equation}\\label{lambda^a}\n\\frac{d\\mu}{d{\\mathcal{L}}^N}(x_0) \\geq Tf_\\text{hom}(u(x_0),\\nabla u(x_0))\\quad\n\\text{ for }{\\mathcal{L}}^N\\text{-a.e. }x_0 \\in \\O\\,,\n\\end{equation}\n\\begin{equation}\\label{lambda^c}\n\\frac{d\\mu}{d|D^c u|}(x_0) \\geq Tf_\\text{hom}^\\infty\\left(\\tilde\nu(x_0),\\frac{dD^c u}{d|D^c u|}(x_0)\\right)\\quad \\text{ for }|D^c\nu|\\text{-a.e. }x_0 \\in \\O\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{lambda^j}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res\\, S_u}(x_0) \\geq\n\\vartheta_\\text{hom}(u^+(x_0),u^-(x_0),\\nu_u(x_0))\\quad \\text{ for\n}{\\mathcal{H}}^{N-1}\\text{-a.e. }x_0 \\in S_u\\,.\n\\end{equation}\nThe proof of (\\ref{lambda^a}) follows the one in \\cite[Lemma\n5.2]{BM} and we shall omit it. The proofs of (\\ref{lambda^c}) and\n(\\ref{lambda^j}) are postponed to the remaining of this subsection.\n\\end{proof}\n\n\n\\noindent {\\bf Proof of \\eqref{lambda^c}.} The lower bound on the\ndensity of the Cantor part will be achieved in three steps. We shall\nuse the blow-up method to reduce the study to constant limits, and\nthen a truncation argument as in the proof of \\cite[Lemma 5.2]{BM},\nto replace the starting sequence by a uniformly converging one.\n\\vskip5pt\n\n{\\bf Step 1.} Choose a point $x_0 \\in \\O$ such that\n\\begin{equation}\\label{cantor2}\n\\lim_{\\rho \\to 0^+}- \\hskip -1em \\int_{Q(x_0,\\rho)}|u(x)-\\tilde u(x_0)|\\, dx=0\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor3}\nA(x_0):=\\lim_{\\rho \\to\n0^+}\\frac{Du(Q(x_0,\\rho))}{|Du|(Q(x_0,\\rho))}\\in [T_{\\tilde\nu(x_0)}({\\mathcal{M}})]^N\\;\\text{ is a rank one matrix with }\\;|A(x_0)|=1\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor1}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\quad \\text{exists and is finite and}\\quad\n\\frac{d|Du|}{d|D^cu|}(x_0)=1\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor4}\n\\lim_{\\rho \\to 0^+}\\frac{|Du|(Q(x_0,\\rho))}{\\rho^{N-1}}=0 \\quad\\text{and}\\quad\n\\lim_{\\rho \\to 0^+}\\frac{|Du|(Q(x_0,\\rho))}{\\rho^N}=+\\infty\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor5}\n\\liminf_{\\rho \\to 0^+}\\,\\frac{|Du|(Q(x_0,\\rho) \\setminus\nQ(x_0,\\tau\\rho))}{|Du|(Q(x_0,\\rho))}\\leq 1-\\tau^N\\quad\\text{for every $0<\\tau<1$}\\,.\n\\end{equation}\nIt turns out that $|D^c u|$-a.e. $x_0\\in\\O$ satisfy these\nproperties. Indeed (\\ref{cantor1}) is immediate while\n(\\ref{cantor2}) is a consequence of the fact that $S_u$ is $|D^c\nu|$-negligible. Property (\\ref{cantor3}) comes from Alberti Rank One\nTheorem together with Lemma \\ref{manifold}, (\\ref{cantor4}) from\n\\cite[Proposition~3.92~(a),~(c)]{AFP} and (\\ref{cantor5}) from\n\\cite[Lemma~2.13]{FM2}. Write $A(x_0)=a \\otimes \\nu$ for some $a \\in\n{\\mathcal{M}}$ and $\\nu \\in {\\mathbb{S}^{N-1}}$. Upon rotating the coordinate axis, one may\nassume without loss of generality that $\\nu=e_N$. To simplify the\nnotations, we set $s_0:=\\tilde u(x_0)$ and $A_0:=A(x_0)$. \\vskip5pt\n\nFix $t\\in(0,1)$ arbitrarily close to $1$, and in view of\n\\eqref{cantor5}, find a sequence $\\rho_k \\searrow 0^+$ such\nthat\n\\begin{equation}\\label{cantor5bis}\n\\limsup_{k\\to+\\infty}\\,\\frac{|Du|(Q(x_0,\\rho_k) \\setminus\nQ(x_0,t\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\leq 1-t^N\\,.\n\\end{equation}\nNow fix $t<\\gamma<1$ and set $\\gamma':=(1+\\gamma)\/2$. Using (\\ref{cantor1}), we derive\n\\begin{multline}\\label{infmu}\n\\frac{d\\mu}{d|D^cu|}(x_0) =\\lim_{k \\to +\\infty}\\frac{\\mu(Q(x_0,\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\geq\n\\limsup_{k \\to +\\infty}\\,\\frac{\\mu(\\overline{Q(x_0,\\gamma'\\rho_k)})}{|Du|(Q(x_0,\\rho_k))}\\geq\\\\\n\\geq \\limsup_{k\\to+\\infty}\\,\\limsup_{n\\to+\\infty}\\,\n\\frac{1}{|Du|(Q(x_0,\\rho_k))}\\int_{Q(x_0,\\gamma'\n\\rho_k)}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)dx\\,.\n\\end{multline}\nArguing as in the proof of \\cite[Lemma 5.2]{BM} with minor\nmodifications, we construct a sequence $\\{\\bar v_{n}\\} \\subset\nW^{1,\\infty}(Q(0,\\rho_k);{\\mathbb{R}}^d)$ satisfying $\\bar v_{n} \\to\nu(x_0+\\cdot)$ in $L^1(Q(0,\\rho_k);{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{passuv}\n\\limsup_{n\\to+\\infty}\\, \\int_{Q(x_0,\\gamma'\n\\rho_k)}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\geq\n\\limsup_{n\\to+\\infty}\\, \\int_{Q(0,\\gamma\n\\rho_k)}g\\bigg(\\frac{x}{\\e_n}, \\bar v_n, \\nabla \\bar v_n\\bigg)\\,\ndx\\,,\n\\end{equation}\nwhere $g$ is given by \\eqref{defg}. Setting $w_{n,k}(x):=\\bar\nv_{n}(\\rho_k\\, x)$, a change of variable together with \\eqref{infmu}\nand \\eqref{passuv} yields\n\\begin{equation}\\label{c1}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty} \\limsup_{n\n\\to +\\infty}\\, \\frac{\\rho_k^N}{|Du|(Q(x_0,\\rho_k))} \\int_{\\gamma Q}\ng\\left(\\frac{\\rho_k\\, x}{\\e_n},w_{n,k}, \\frac{1}{\\rho_k} \\nabla\nw_{n,k}\\right)dx\\,.\n\\end{equation}\nThen we infer from (\\ref{cantor2}) that\n\\begin{equation}\\label{c2}\n\\lim_{k \\to +\\infty} \\lim_{n \\to +\\infty}\\int_Q|w_{n,k}-s_0|\\, dx=0\\,,\n\\end{equation}\nand\n\\begin{multline}\\label{c3}\n\\lim_{k \\to +\\infty} \\lim_{n \\to\n+\\infty}\\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))}\\int_Q\n\\bigg|w_{n,k}(x)-u(x_0+\\rho_k x)\\\\\n- \\int_Q \\big(w_{n,k}(y)-u(x_0+\\rho_k y)\\big)\\, dy\\bigg|\\, dx =0\\,.\n\\end{multline}\nBy \\eqref{c1}, \\eqref{c2} and (\\ref{c3}), we can extract a diagonal sequence $n_k\n\\to+\\infty$ such that $\\d_k:=\\e_{n_k}\/\\rho_k\\to 0$,\n$w_k:=w_{n_k,k}\\to s_0$ in\n$L^1(Q;{\\mathbb{R}}^d)$,\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\frac{\\rho_k^N}{|Du|(Q(x_0,\\rho_k))} \\int_{\\gamma Q}\ng\\left(\\frac{x}{\\d_k},w_k, \\frac{1}{\\rho_k} \\nabla w_k\\right)dx\\,,$$\nand\n\\begin{equation}\\label{c5}\n\\lim_{k \\to +\\infty}\\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))}\\int_Q\n\\bigg|w_k(x)-u(x_0+\\rho_k\\, x) - \\int_Q \\big(w_k(y)-u(x_0+\\rho_k\\, y)\\big)\\,\ndy\\bigg|\\,dx =0\\,.\n\\end{equation}\n\\vskip5pt\n\n{\\bf Step 2.} Now we reproduce the truncation argument used in Step\n2 of the proof of \\cite[Lemma~5.2]{BM} with minor modifications\n(make use of (\\ref{cantor4}) and \\cite[Lemma~2.12]{FM2} instead of\n\\cite[Lemma~2.6]{FM}, see \\cite{FM2} for details). Setting\n$a_k:=\\int_Q w_k(y)\\, dy$, it yields a sequence of cut-off functions\n$\\{\\zeta_k\\} \\subset {\\mathcal{C}}^\\infty_c({\\mathbb{R}};[0,1])$ such that\n$\\zeta_k(\\tau)=1$ if $|\\tau| \\leq s_k$, $\\zeta_k(\\tau)=0$ is\n$|\\tau|\\geq t_k$ for some\n$$\\|w_k-a_k\\|^{1\/2}_{L^1(Q;{\\mathbb{R}}^d)} s_k\\}}|w_k(x)-\\overline w_k(x)|\\, dx\\leq \\\\\n \\leq \\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))} \\int_{\\{|w_k-a_k|> s_k\\}}|w_k(x)-a_k|\\, dx= \\int_{\\{|w_k-a_k|> s_k\\}} |z_k(x)|\\, dx\\,.\n\\end{multline}\nBy Chebyshev inequality, we have\n\\begin{equation}\\label{ln}{\\mathcal{L}}^N(\\{|w_k-a_k|> s_k\\}) \\leq \\frac{1}{s_k}\\int_Q|w_k(x)-a_k|\\, dx\n\\leq \\|w_k-a_k\\|^{1\/2}_{L^1(Q;{\\mathbb{R}}^d)}\\to 0\\,,\n\\end{equation}\nand thus (\\ref{ei}), (\\ref{ln}) and the equi-integrability of $\\{z_k\\}$ imply $\\|\\overline z_k -\nz_k\\|_{L^1(Q;{\\mathbb{R}}^d)} \\to 0$. Therefore $\\overline z_k\n\\to v$ in $L^1(Q;{\\mathbb{R}}^d)$, and setting\n$\\a_k:=|Du|(Q(x_0,\\rho_k))\/\\rho_k^N \\to +\\infty$,\n\\begin{equation}\\label{tilde}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\frac{1}{\\a_k} \\int_{\\gamma Q} g\\left(\\frac{x}{\\d_k},s_0,\\a_k \\nabla\n\\overline z_k\\right)dx\\,.\n\\end{equation}\nUsing (\\ref{grec}) and the positive $1$-homogeneity of the recession function\n$g^\\infty(y,s,\\cdot)$, we infer that\n\\begin{align*}\n \\int_{\\gamma Q} \\left| \\frac{1}{\\a_k}\\, g\\left(\\frac{x}{\\d_k},s_0,\\a_k\n\\nabla \\overline z_k\\right) -\ng^\\infty\\left(\\frac{x}{\\d_k},s_0, \\nabla \\overline z_k\\right)\\right| dx &\\leq \\frac{C}{\\a_k} \\int_{\\gamma Q} (1+ \\a_k^{1-q} |\\nabla \\overline z_k|^{1-q})\\, dx\\\\\n&\\leq C\\big(\\a_k^{-1} + \\a_k^{-q} \\|\\nabla \\overline\nz_k\\|^{1-q}_{L^1(\\gamma Q;{\\mathbb{R}}^{d \\times N})}\\big) \\to 0\\,,\n\\end{align*}\nwhere we have used H\\\"older's inequality and the boundedness of\n$\\{\\nabla \\overline z_k\\}$ in $L^1(\\gamma Q;{\\mathbb{R}}^{d\n\\times N})$ (which follows from (\\ref{pgrowth}) and \\eqref{tilde}).\nConsequently,\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\int_{\\gamma Q} g^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\nz_k\\right)dx\\,.$$\n\\vskip5pt\n\n{\\bf Step 3.} Extend $\\theta$ continuously to ${\\mathbb{R}}$ by the values of\nits traces at $\\pm 1\/2$. Define $v_k(x)=v_k(x_N):=a \\theta * \\varrho_k (x_N)$ where\n$\\varrho_k$ is a sequence of (one dimensional) mollifiers. Then $v_k\n\\to v$ in $L^1(Q;{\\mathbb{R}}^d)$ and thus, since $\\overline u_k - v_k \\to\n0$ in $L^1(Q;{\\mathbb{R}}^d)$, it follows that (up to a subsequence)\n\\begin{equation}\\label{D}\nD\\overline u_k(\\tau Q) - Dv_k(\\tau Q) \\to 0\n\\end{equation}\nfor ${\\mathcal{L}}^1$-a.e. $\\tau \\in (0,1)$. Fix $\\tau\\in (t,\\gamma)$\nfor which \\eqref{D} holds. Since $\\|\\bar z_k -v_k\\|_{L^1(Q;{\\mathbb{R}}^d)}\\to 0$, one can use a standard cut-off\nfunction argument (see \\cite[p. 29--30]{FM2}) to modify the sequence $\\{\\overline z_k\\}$\nand produce a new sequence\n$\\{\\overline\\varphi_k\\} \\subset W^{1,\\infty}(\\tau Q;{\\mathbb{R}}^d)$ satisfying\n$\\overline \\varphi_k \\to v$ in $L^1(\\tau Q;{\\mathbb{R}}^d)$, $\\overline\n\\varphi_k=v_k$ on a neighborhood of $\\partial (\\tau Q)$ and\n\\begin{equation}\\label{may}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty} \\int_{\\tau Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\n\\varphi_k\\right)dx\\,.\n\\end{equation}\nA simple computation shows that\n\\begin{equation}\\label{D2}\nD\\overline u_k (\\tau Q)=\\frac{Du(Q(x_0,\\tau\n\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\quad \\text{ and }\\quad Dv_k(\\tau\nQ)=\\tau^N \\,A_k\\,,\n\\end{equation}\nwhere $A_k \\in {\\mathbb{R}}^{d \\times N}$ is the matrix given by\n$$A_k:= a \\otimes e_N \\frac{\\theta *\\varrho_k (\\tau\/2)- \\theta *\\varrho_k (-\\tau\/2)}{\\tau}\\,.$$\nWe observe that $A_k$ is bounded in $k$ since $\\theta$ has bounded\nvariation.\n\nLet $m_k:=[\\tau\/\\delta_k]+1\\in {\\mathbb{N}}$, and define for $x=(x',x_N)\\in \\delta_km_k Q$,\n$$\\varphi_k(x):=\\begin{cases}\n\\overline \\varphi_k(x)-A_kx & \\text{if $x\\in \\tau Q$}\\,,\\\\\nv_k(x_N)-A_k\\, x & \\text{if $|x_N|\\leq \\tau\/2$ and $|x'|\\geq \\tau\/2$}\\, ,\\\\\nv_k(\\tau\/2)-A_k(x',\\tau\/2) & \\text{if $x_N\\geq \\tau\/2$}\\,,\\\\\nv_k(-\\tau\/2)-A_k(x',-\\tau\/2) & \\text{if $x_N\\leq -\\tau\/2$}\\,.\n\\end{cases}$$\nOne may check that $\\varphi_k\\in W^{1,\\infty}(\\delta_km_kQ;{\\mathbb{R}}^d)$, $\\varphi_k$ is $\\delta_km_k$-periodic, and that\n\\begin{equation}\\label{periodization}\n \\limsup_{k \\to +\\infty} \\int_{\\tau Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\n\\varphi_k\\right)dx= \\limsup_{k \\to +\\infty} \\int_{\\delta_km_k Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,A_k+\\nabla\n\\varphi_k\\right)dx\\,.\n\\end{equation}\nSetting $\\phi_k(y):=\\tau^{N}\\delta_k^{-1}\\varphi_k(\\delta_k y)$ for $y\\in m_k Q$, we have $\\phi_k\\in W^{1,\\infty}_{\\#}(m_kQ;{\\mathbb{R}}^d)$, and a change of variables yields\n\\begin{align}\n\\nonumber\\int_{\\delta_km_k Q} g^\\infty\n\\left(\\frac{x}{\\d_k},s_0,A_k+\\nabla varphi_k\\right)dx&=\n\\tau^{-N}\\delta_k^Nm_k^N- \\hskip -1em \\int_{m_k Q} g^\\infty\n\\left(y,s_0,\\tau^{N}A_k+\\nabla\n\\phi_k\\right)dy \\\\\n\\label{compper} &\\geq \\tau^{-N}\\delta_k^Nm_k^N (g^\\infty)_{\\rm hom}(s_0,\\tau^N A_k)\\,,\n\\end{align}\nsince $(g^\\infty)_\\text{hom}$ can be computed as follows\n(see Remark \\ref{reminfty} and {\\it e.g., } \\cite[Remark~14.6]{BD}),\n\\begin{eqnarray*}\n(g^\\infty)_\\text{hom}(s,\\xi)\n= \\inf \\left\\{- \\hskip -1em \\int_{(0,m)^N} g^\\infty(y,s,\\xi +\\nabla \\phi(y))\\, dy : m\\in {\\mathbb{N}},\\,\n\\phi \\in W^{1,\\infty}_\\#((0,m)^N;{\\mathbb{R}}^d) \\right\\}\\,.\n\\end{eqnarray*}\nGathering \\eqref{may}, \\eqref{periodization} and \\eqref{compper}, we derive\n\\begin{equation*}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq\\limsup_{k \\to +\\infty}\n\\,(g^\\infty)_\\text{hom}(s_0,\\tau^N A_k)\\,.\n\\end{equation*}\nIn view (\\ref{D}), (\\ref{D2}), (\\ref{cantor5bis}) and (\\ref{cantor3}), we have\n\\begin{multline*}\n\\limsup_{k \\to +\\infty}|\\tau^N A_k -A_0|=\\limsup_{k \\to +\\infty}|Dv_k(\\tau Q) - A_0|=\\limsup_{k \\to +\\infty}|D\\overline u_k(\\tau Q)-A_0|=\\\\\n=\\limsup_{k \\to +\\infty}\\left| \\frac{Du(Q(x_0,\\tau \\rho_k))}{|Du|(Q(x_0,\\rho_k))}-A_0\\right|\n= \\limsup_{k \\to +\\infty}\n\\frac{|Du|(Q(x_0,\\rho_k)\\setminus Q(x_0,\\tau\\rho_k))}\n{|Du|(Q(x_0,\\rho_k))}\\leq 1- t^{N}\\,.\n\\end{multline*}\nBy Remark \\ref{reminfty}, $(g^\\infty)_\\text{hom}(s_0,\\cdot)$ is Lipschitz continuous, and consequently\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq (g^\\infty)_\\text{hom}(s_0,A_0)-C(1-t^{N})\\,.$$\nFrom the arbitrariness of $t$, we finally infer that\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq (g^\\infty)_\\text{hom}(s_0,A_0)\\,. $$\nSince $s_0 \\in {\\mathcal{M}}$ and $A_0 \\in [T_{s_0}({\\mathcal{M}})]^N$, Remark\n\\ref{reminfty} and \\eqref{remdensinf} yield\n$(g^\\infty)_\\text{hom}(s_0,A_0)= T(f^\\infty)_\\text{hom}(s_0,A_0)\\geq\nTf_\\text{hom}^\\infty(s_0,A_0)$, and the proof is complete. \\prbox\n\\vskip10pt\n\n\\noindent{\\bf Proof of \\eqref{lambda^j}.} The strategy used in that\npart follows the one already used for the bulk and Cantor parts. It\nstill rests on the blow up method together with the projection\nargument in Proposition~\\ref{proj}. \\vskip5pt\n\n{\\bf Step 1.} Let $x_0 \\in S_u$ be such that\n\\begin{equation}\\label{jump1}\n\\lim_{\\rho \\to 0^+}- \\hskip -1em \\int_{Q_{\\nu_u(x_0)}^\\pm(x_0,\\rho)} |u(x)-u^\\pm(x_0)|\\, dx=0\\,,\n\\end{equation}\nwhere $u^\\pm(x_0) \\in {\\mathcal{M}}$,\n\\begin{equation}\\label{jump2}\n\\lim_{\\rho \\to 0^+}\\frac{{\\mathcal{H}}^{N-1}(S_u \\cap Q_{\\nu_u(x_0)}(x_0,\\rho))}{\\rho^{N-1}}=1\\,,\n\\end{equation}\nand such that the Radon-Nikod\\'ym derivative of $\\mu$ with respect\nto ${\\mathcal{H}}^{N-1}\\res \\, S_u$ exists and is finite. By\nLemma~\\ref{manifold}, Theorem 3.78 and Theorem 2.83 (i) in\n\\cite{AFP} (with cubes instead of balls), it turns out that\n${\\mathcal{H}}^{N-1}$-a.e. $x_0\\in S_u$ satisfy these properties. Set\n$s_0^\\pm:=u^\\pm(x_0)$, $\\nu_0:=\\nu_u(x_0)$.\n\nUp to a further subsequence, we may assume that $(1+|\\nabla u_n|) {\\mathcal{L}}^N\n\\res\\, \\O \\xrightharpoonup[]{*} \\lambda$ in ${\\mathcal{M}}(\\O)$ for some nonnegative\nRadon measure $\\lambda \\in {\\mathcal{M}}(\\O)$. Consider a sequence $\\rho_k \\searrow 0^+$ satisfying $\\mu(\\partial\nQ_{\\nu_0}(x_0,\\rho_k))=\\lambda(\\partial\nQ_{\\nu_0}(x_0,\\rho_k))=0$ for each $k \\in {\\mathbb{N}}$. Using\n(\\ref{jump2}) we derive\n\\begin{multline*}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0)\\lim_{k \\to +\\infty}\\frac{\\mu(Q_{\\nu_0}(x_0,\\rho_k))}{{\\mathcal{H}}^{N-1}(S_u \\cap Q_{\\nu_0}(x_0,\\rho_k))}\n=\\lim_{k \\to +\\infty}\\frac{\\mu(Q_{\\nu_0}(x_0,\\rho_k))}{\\rho_k^{N-1}}=\\\\\n= \\lim_{k \\to +\\infty}\\lim_{n \\to\n+\\infty}\\frac{1}{\\rho_k^{N-1}}\\int_{Q_{\\nu_0}(x_0,\\rho_k)}\nf\\left(\\frac{x}{\\e_n},\\nabla u_n \\right)dx\\,.\n\\end{multline*}\nThanks to Theorem \\ref{density}, one can assume without loss of\ngenerality that $u_n \\in \\mathcal D(\\O;{\\mathcal{M}})$ for each $n \\in {\\mathbb{N}}$.\nArguing exactly as in Step 1 of the proof of \\cite[Lemma 5.2]{BM}\n(with $Q_{\\nu_0}(x_0,\\rho_k)$ instead of $Q(x_0,\\rho_k)$) we obtain\na sequence $\\{v_n\\} \\subset \\mathcal D(Q_{\\nu_0}(0,\\rho_k);{\\mathcal{M}})$ such\nthat $v_n \\to u(x_0+\\cdot)$ in $L^1(Q_{\\nu_0}(0,\\rho_k);{\\mathbb{R}}^d)$ as\n$n \\to +\\infty$, and\n$$\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\\limsup_{n \\to +\\infty}\\,\n\\frac{1}{\\rho_k^{N-1}}\\int_{Q_{\\nu_0}(0,\\rho_k)}\nf\\left(\\frac{x}{\\e_n},\\nabla v_n \\right)dx$$\n(note that the\nconstruction process to obtain $v_n$ from $u_n$ does not affect the\nmanifold constraint). Changing variables and setting\n$w_{n,k}(x)=v_n(\\rho_k\\, x)$ lead to\n$$\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to\n+\\infty}\\,\\limsup_{n \\to +\\infty} \\rho_k\\int_{Q_{\\nu_0}}\nf\\left(\\frac{\\rho_k \\, x}{\\e_n},\\frac{1}{\\rho_k}\\nabla w_{n,k}\n\\right)dx\\,.$$\nDefining\n$$u_0(x):=\\begin{cases}\ns_0^+ & \\text{ if } x\\cdot \\nu_0 > 0\\,,\\\\\ns_0^- & \\text{ if } x\\cdot \\nu_0 \\leq 0\\,,\n\\end{cases}$$\nwe infer from (\\ref{jump1}) that\n$$\\lim_{k \\to +\\infty}\\lim_{n \\to +\\infty}\n\\int_{Q_{\\nu_0}}|w_{n,k}-u_0|\\, dx=0\\,.$$ By a standard diagonal\nargument, we find a sequence $n_k \\nearrow +\\infty$ such that\n $\\d_k:=\\e_{n_k}\/\\rho_k\\to 0$, $w_k:=w_{n_k,k} \\in \\mathcal\nD(Q_{\\nu_0};{\\mathcal{M}})$ converges to $u_0$ in $L^1(Q_{\\nu_0};{\\mathbb{R}}^d)$, and\n\\begin{equation}\\label{dhnwj}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\rho_k\\int_{Q_{\\nu_0}}f\\left(\\frac{x}{\\d_k},\\frac{1}{\\rho_k}\\nabla\nw_k \\right)dx\\,.\n\\end{equation}\nAccording to $(H_4)$ and the positive $1$-homogeneity of $f^\\infty(y,\\cdot)$, we have\n\\begin{align}\\label{dhnwj2}\n\\nonumber\\int_{Q_{\\nu_0}}\\left|\\rho_k\\,\nf\\left(\\frac{x}{\\d_k},\\frac{1}{\\rho_k}\\nabla w_k \\right) -\nf^\\infty\\left(\\frac{x}{\\d_k},\\nabla w_k \\right)\\right| dx & \\leq C\\rho_k\\int_{Q_{\\nu_0}} (1 + \\rho_k^{q-1}\n|\\nabla w_k|^{1-q})\\, dx\\\\\n&\\leq C\\left(\\rho_k +\\rho_k^q \\|\\nabla\nw_k\\|^{1-q}_{L^1(Q_{\\nu_0};{\\mathbb{R}}^{d \\times N})}\\right)\\,,\n\\end{align}\nwhere we have used H\\\"older's inequality and $00$ and $\\b>0$ such that\n$$\\a |\\xi| \\leq f(x,s,\\xi) \\leq \\b(1+|\\xi|) \\quad \\text{ for every\n}(x,s,\\xi) \\in \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,;$$\n\n\\item[$(H_3')$] for every compact set $K \\subset \\O$, there exists a\ncontinuous function $\\omega : [0,+\\infty) \\to [0,+\\infty)$\nsatisfying $\\omega(0)=0$ and\n$$|f(x,s,\\xi) - f(x',s',\\xi)| \\leq \\omega(|x-x'| + |s-s'|)\n(1+|\\xi|)$$\nfor every $x$, $x' \\in \\O$, $s$, $s' \\in {\\mathbb{R}}^d$ and\n$\\xi \\in {\\mathbb{R}}^{d \\times N}$;\n\n\\item[$(H_4')$] there exist $C>0$ and $q \\in (0,1)$ such that\n$$|f(x,s,\\xi) - f^\\infty(x,s,\\xi)| \\leq C(1+|\\xi|^{1-q}), \\quad \\text{ for every\n}(x,s,\\xi) \\in \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,,$$ where\n$f^\\infty : \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N} \\to [0,+\\infty)$\nis the recession function of $f$ defined by\n$$f^\\infty(x,s,\\xi):=\\limsup_{t \\to +\\infty} \\frac{f(x,s,t\\xi)}{t}\\,\n.$$\n\\end{itemize}\n\nConsider the functional $F:L^1(\\O;{\\mathbb{R}}^d) \\to [0,+\\infty]$ given by\n$$F(u):=\\left\\{\n\\begin{array}{ll}\n\\displaystyle \\int_\\O f(x,u,\\nabla u)\\, dx & \\text{ if }u \\in\nW^{1,1}(\\O;{\\mathcal{M}}),\\\\[0.4cm]\n+\\infty & \\text{ otherwise}, \\end{array}\\right.$$ and its relaxation\nfor the strong $L^1(\\O;{\\mathbb{R}}^d)$-topology $\\overline F:L^1(\\O;{\\mathbb{R}}^d)\n\\to [0,+\\infty]$ defined by\n$$\\overline F(u):=\\inf_{\\{u_n\\}} \\left\\{ \\liminf_{n \\to +\\infty}\nF(u_n) : u_n \\to u \\text{ in }L^1(\\O;{\\mathbb{R}}^d)\\right\\}\\,.$$ Then the\nfollowing integral representation result holds:\n\n\\begin{theorem} \\label{relax}\nLet ${\\mathcal{M}}$ be a smooth compact and connected submanifold of ${\\mathbb{R}}^d$\nwithout boundary, and let $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$ be a continuous function satisfying $(H_1')$ to\n$(H_4')$. Then for every $u \\in L^1(\\O;{\\mathbb{R}}^d)$,\n\\begin{equation}\\label{reprel}\n\\overline F(u)= \\begin{cases} \\displaystyle\n\\begin{multlined}[8.5cm]\n\\,\\int_\\O f(x,u,\\nabla u)dx +\n\\int_{\\O\\cap S_u}K(x,u^+,u^-,\\nu_u)d{\\mathcal{H}}^{N-1}\\,+ \\\\[-15pt]\n+ \\int_\\O f^\\infty\\bigg(x,\\tilde u,\\frac{dD^cu}{d|D^cu|}\\bigg)\\,\nd|D^cu|\n\\end{multlined}\n& \\text{\\it if }\\,u \\in BV(\\O;{\\mathcal{M}})\\,,\\\\\n& \\\\\n\\,+\\infty & \\text{\\it otherwise}\\,,\n\\end{cases}\n\\end{equation}\nwhere for every $(x,a,b,\\nu) \\in \\O \\times {\\mathcal{M}} \\times {\\mathcal{M}} \\times {\\mathbb{S}^{N-1}}$,\n\\begin{multline*}\nK(x,a,b,\\nu) := \\inf_\\varphi \\bigg\\{\\int_{Q_\\nu}\nf^\\infty(x,\\varphi(y),\\nabla \\varphi(y))\\, dy : \\varphi \\in\nW^{1,1}(Q_\\nu;{\\mathcal{M}}),\\; \\varphi=a \\text{ on } \\{x\\cdot \\nu=1\/2\\},\\\\\n\\varphi=b \\text{ on }\\{x\\cdot \\nu=-1\/2\\} \\text{ {\\it and} } \\varphi \\text{\n\\it is $1$-periodic in the }\\nu_2,\\ldots,\\nu_{N} \\text{ directions}\n\\bigg\\}\\,,\n\\end{multline*}\n$\\{\\nu,\\nu_2,\\ldots,\\nu_N\\}$ forms any orthonormal basis of ${\\mathbb{R}}^N$,\nand $Q_\\nu$ stands for the open unit cube in ${\\mathbb{R}}^N$ centered at the\norigin associated to this basis.\n\\end{theorem}\n\n\n\\noindent{\\bf Sketch of the Proof.} The proof of the lower bound\n``$\\geq$\" in \\eqref{reprel} can be obtained as in \\cite[Lemma~5.2]{BM} and Lemma \\ref{lowerboundBV} using standard techniques to\nhandle with the dependence on the space variable. The lower bounds\nfor the bulk and Cantor parts rely on the construction of a suitable\nfunction $\\tilde f: \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$ replacing $f$ as we already pursued in Section\n\\ref{thbe}. On the other hand, the jump part rests on the projection\non ${\\mathcal{M}}$ of \\cite{HL} as in Proposition \\ref{proj} instead of the\nstandard projection on the sphere used in\n\\cite[Proposition~5.2]{AEL}. \\vskip5pt\n\n\\noindent\nTo obtain the upper bound, we localize as usual the functionals setting for every $u \\in\nL^1(\\O;{\\mathbb{R}}^d)$ and $A \\in {\\mathcal{A}}(\\O)$,\n$$F(u,A):=\\begin{cases}\n\\displaystyle \\int_A f(x,u,\\nabla u)\\, dx & \\text{ if }u \\in\nW^{1,1}(A;{\\mathcal{M}})\\,,\\\\\n+\\infty & \\text{ otherwise}\\,,\n\\end{cases}$$\n$$\\overline F(u,A):=\\inf_{\\{u_n\\}} \\left\\{ \\liminf_{n \\to +\\infty}\nF(u_n,A) : u_n \\to u \\text{ in }L^1(A;{\\mathbb{R}}^d)\\right\\}\\,.$$\nArguing as in the proof of Lemma \\ref{measbis}, we obtain that for every $u \\in BV(\\O;{\\mathcal{M}})$, the set function\n$\\overline F(u,\\cdot)$ is the restriction to ${\\mathcal{A}}(\\O)$ of a Radon\nmeasure absolutely continuous with respect to ${\\mathcal{L}}^N+|Du|$. Hence it uniquely extends into a Radon measure on $\\Omega$ (see Remark \\ref{measconstr}),\nand it suffices to prove that for any $u\\in BV(\\O;{\\mathcal{M}})$,\n\\begin{equation}\\label{jppartrel}\n\\overline F(u,\\Omega\\cap S_u)\\leq \\int_{\\Omega \\cap S_u}K(x,u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\n\\begin{equation}\\label{contpartrel}\n\\frac{d\\overline F(u,\\cdot)}{d {\\mathcal{L}}^N}(x_0)\\leq f(x_0,u(x_0),\\nabla u(x_0))\\quad \\text{for ${\\mathcal{L}}^N$-a.e. $x_0\\in \\Omega$}\\,,\n\\end{equation}\n\\begin{equation}\\label{cantpartrel}\n\\frac{d\\overline F(u,\\cdot)}{d |D^cu|}(x_0)\\leq f^\\infty\\bigg(x_0,\\tilde u(x_0),\\frac{dD^c u}{d|D^cu|}(x_0)\\bigg)\\quad \\text{for $|D^cu|$-a.e. $x_0\\in \\Omega$}\\,,\n\\end{equation}\n\\vskip5pt\n\n\\noindent{\\it Proof of \\eqref{jppartrel}.} Concerning the jump part, one can proceed as in \\cite[Lemma~6.5]{AEL}.\nA slight difference lies in the third step of its proof where one needs to approximate in energy\nan arbitrary $u\\in BV(\\O;{\\mathcal{M}})$ by a sequence $\\{u_n\\}\\subset BV(\\O;{\\mathcal{M}})$ such that for each $n$, $u_n$ assumes a finite number of values.\nThis can be performed as in the proof of Corollary \\ref{upbdjp} using the regularity properties of $K$ stated in \\cite[Lemma~4.1]{AEL} for ${\\mathcal{M}}=\\mathbb{S}^{d-1}$.\n\\vskip5pt\n\n\n\\noindent{\\it Proof of \\eqref{contpartrel}.} Let $x_0 \\in \\O$ be a Lebesgue\npoint for $u$ and $\\nabla u$ such that $u(x_0) \\in {\\mathcal{M}}$, $\\nabla u(x_0)\n\\in [T_{u(x_0)}({\\mathcal{M}})]^N$,\n$$\\lim_{\\rho \\to 0^+} - \\hskip -1em \\int_{Q(x_0,\\rho)} |u(x) - u(x_0)|(1+|\\nabla\nu(x)|)\\, dx=0\\,,\\quad \\lim_{\\rho \\to 0^+}\\frac{|D^s\nu|(Q(x_0,\\rho))}{\\rho^N}=0\\,,$$ and\n$$\\frac{d |Du|}{d{\\mathcal{L}}^N}(x_0) \\quad \\text{ and }\\quad \\frac{d\\overline\nF(u,\\cdot)}{d{\\mathcal{L}}^N}(x_0)$$ exist and are finite. Note that\n${\\mathcal{L}}^N$-a.e. $x_0 \\in \\O$ satisfy these properties. We select a sequence $\\rho_k\n\\searrow 0^+$ such that $Q(x_0,2\\rho_k)\\subset \\O$ and $|Du|(\\partial Q(x_0,\\rho_k)) =0$ for\neach $k \\in {\\mathbb{N}}$. Next consider a sequence of standard mollifiers\n$\\{\\varrho_n\\}$, and define $u_n :=\\varrho_n * u \\in\nW^{1,1}(Q(x_0,\\rho_k);{\\mathbb{R}}^d) \\cap {\\mathcal{C}}^\\infty(Q(x_0,\\rho_k);{\\mathbb{R}}^d)$. In the sequel, we shall argue\nas in the proof of Proposition \\ref{proj} and we refer to it for the notation. Fix $\\d>0$ small\nenough such that $\\pi:{\\mathbb{R}}^d\\setminus X\\to {\\mathcal{M}}$ is smooth\nin the $\\d$-neighborhood of ${\\mathcal{M}}$. \nSince $u_n$ takes its values in ${\\rm co}({\\mathcal{M}})$, we can reproduce the proof of Proposition \\ref{proj} to find $a_n^k \\in{\\mathbb{R}}^d$ with $|a_n^k|<\\delta\/4$\nsuch that setting $p_n^k:=(\\pi_{a_n^k}|_{{\\mathcal{M}}})^{-1}\n\\circ \\pi_{a_n^k}$, $w_n^k:=p_n^k\\circ u_n \\in W^{1,1}(Q(x_0,\\rho_k);{\\mathcal{M}})$ and\n\\begin{equation}\\label{Ank}\n\\int_{A_n^k} |\\nabla w_n^k|\\, dx \\leq C_* \\int_{A_n^k}|\\nabla u_n|\\,\ndx\\,,\n\\end{equation}\nwhere $A_n^k$ denotes the open set $A_n^k:=\\big\\{x \\in Q(x_0,\\rho_k) : {\\rm dist}(u_n(x),{\\mathcal{M}}) >\\d \/2\\big\\}$. \nFurthermore, since $ \\pi$ is smooth in the $\\d$-neighborhood\nof ${\\mathcal{M}}$ and $|a_n^k|<\\d\/4$,\nthere exists a constant $C_\\d>0$ independent\nof $n$ and $k$ such that\n\\begin{equation}\\label{dn}\n|\\nabla^2 p_n^k(s)|+|\\nabla p_n^k(s)| \\leq C_\\d \\text{ for every $s \\in {\\mathbb{R}}^d$ satisfying\n$\\text{dist}(s,{\\mathcal{M}})\\leq \\d\/2$}\\,,\n\\end{equation}\nand consequently,\n\\begin{equation}\\label{cAnk}\n|\\nabla w_n^k| \\leq C_\\d |\\nabla u_n| \\quad \\text{${\\mathcal{L}}^N$-a.e. in\n}Q(x_0,\\rho_k) \\setminus A_n^k\\,.\n\\end{equation}\nSince $u(x) \\in {\\mathcal{M}}$ for ${\\mathcal{L}}^N$-a.e. $x \\in \\O$, it follows that\n$${\\mathcal{L}}^N(A_n^k) \\leq\n\\frac{2}{\\d} \\int_{Q(x_0,\\rho_k)} \\text{dist}(u_n,{\\mathcal{M}})\\, dx \\leq\n\\frac{2}{\\d} \\int_{Q(x_0,\\rho_k)} |u_n-u|\\, dx \\xrightarrow[n \\to\n+\\infty]{} 0\\,,$$\nand then (\\ref{dn}) yields\n\\begin{multline*}\n\\int_{Q(x_0,\\rho_k)}|w_n^k - u|\\, dx = \\int_{\nA_n^k}|w_n^k - u|\\, dx + \\int_{Q(x_0,\\rho_k)\n\\setminus A_n^k}|p_n^k(u_n) - p_n^k(u)|\\, dx\\leq\\\\\n \\leq {\\rm diam}({\\mathcal{M}}) {\\mathcal{L}}^N(A_n^k) + C_\\d\n\\int_{Q(x_0,\\rho_k)}|u_n-u|\\, dx \\xrightarrow[n \\to +\\infty]{} 0\\,.\n\\end{multline*}\nHence $w_n^k \\to u$ in $L^1(Q(x_0,\\rho_k);{\\mathbb{R}}^d)$ as $n \\to +\\infty$\nso that we are allowed to take $w_n^k$ as competitor, {\\it i.e.},\n$$\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to +\\infty}\n\\int_{Q(x_0,\\rho_k)}f(x,w_n^k,\\nabla w_n^k)\\, dx\\,.$$\nAt this stage we can argue exactly\nas in \\cite[Lemma~6.4]{AEL} to prove that for any $\\eta>0$\nthere exists $\\lambda=\\lambda(\\eta)>0$ such that\n\\begin{multline}\\label{1numb}\n\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to\n+\\infty}\\bigg\\{\\int_{Q(x_0,\\rho_k)}f(x_0,u(x_0),\\nabla u_n)\\,\ndx+C\\int_{Q(x_0,\\rho_k)}|\\nabla u_n - \\nabla w_n^k|\\, dx\\, +\\\\\n+ C(\\eta +\\lambda \\rho_k) \\int_{Q(x_0,\\rho_k)}(1+|\\nabla u_n|)\\, dx\n+C\\lambda\\int_{Q(x_0,\\rho_k)}|w_n^k-u(x_0)|(1+|\\nabla\nw_n^k|)\\, dx \\bigg\\}\\,.\n\\end{multline}\nThe first and third term in the right handside of \\eqref{1numb} can be treated as in the proof of \\cite[Theorem~2.16]{FM2}. \nConcerning the remaining terms, we proceed as follows. \nUsing\n(\\ref{Ank}), (\\ref{dn}) and (\\ref{cAnk}), we get that\n\\begin{multline}\\label{2numb}\n\\int_{Q(x_0,\\rho_k)}|w_n^k-u(x_0)||\\nabla w_n^k|\\, dx\n\\leq{\\rm diam}({\\mathcal{M}}) \\int_{A_n^k}|\\nabla w_n^k|\\, dx\\,+\\\\\n+ \\int_{Q(x_0,\\rho_k)\n\\setminus A_n^k}|p_n^k(u_n) - p_n^k(u(x_0))| |\\nabla w_n^k| \\, dx\n \\leq C\\int_{A_n^k}|\\nabla u_n|\\, dx\\,+\\\\\n+ C_\\d \\int_{Q(x_0,\\rho_k) \\setminus A_n^k}|u_n - u(x_0)| |\\nabla\nu_n|\n\\,dx \n\\leq C_\\d \\int_{Q(x_0,\\rho_k)}|u_n - u(x_0)| |\\nabla u_n|\\, dx\\,,\n\\end{multline}\nwhere $C_\\d>0$ still denotes some constant depending on $\\d$ but independent of $k$ and $n$. Arguing\nin a similar way, we also derive\n\\begin{equation}\\label{3}\n\\int_{Q(x_0,\\rho_k)}|\\nabla u_n - \\nabla w_n^k|\\, dx \\leq C_\\delta\n\\int_{Q(x_0,\\rho_k)} |u_n -u(x_0)| |\\nabla u_n|\\, dx+\n\\int_{Q(x_0,\\rho_k) \\setminus A_n^k} |L_n^k \\nabla u_n|\\, dx\\,,\n\\end{equation}\nwhere $L_n^k:={\\rm Id} - \\nabla p_n^k(u(x_0)) \\in {\\rm\nLin}({\\mathbb{R}}^{d\\times d},{\\mathbb{R}}^{d \\times d})$. \nGathering \\eqref{1numb}, \\eqref{2numb} and (\\ref{3}) we finally\nobtain that\n\\begin{multline}\\label{4}\n\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to\n+\\infty}\\bigg\\{\\int_{Q(x_0,\\rho_k)}f(x_0,u(x_0),\\nabla u_n)\\,\ndx+C\\int_{Q(x_0,\\rho_k) \\setminus A_n^k} |L_n^k \\nabla u_n|\\, dx\\,+\\\\\n+C(\\eta +\\lambda \\rho_k) \\int_{Q(x_0,\\rho_k)}(1+|\\nabla u_n|)\\, dx\n+C_\\d\\lambda\\int_{Q(x_0,\\rho_k)}|u_n-u(x_0)|(1+|\\nabla u_n|)\\,\ndx \\bigg\\}\\,.\n\\end{multline}\nNow we can follow the argument in \\cite[Lemma~6.4]{AEL} to conclude that\n$$\\frac{d\\overline F(u,\\cdot)}{d{\\mathcal{L}}^N}(x_0) \\leq f(x_0,u(x_0),\\nabla\nu(x_0))\\,,$$\nwhich completes the proof of \\eqref{contpartrel}.\n\\vskip5pt\n\n\\noindent {\\it Proof of \\eqref{cantpartrel}.} Once again the proof\nparallels the one in \\cite[Lemma~6.4]{AEL}. We first proceed as in\nthe previous reasoning leading to \\eqref{4}. Then we can exactly\nfollow the argument of \\cite[Lemma~6.4]{AEL} to obtain\n\\eqref{cantpartrel}. \\prbox\n\n\n\\vskip15pt\n\n\\noindent{\\bf Acknowledgement. }The authors wish to thank Roberto\nAlicandro, Pierre Bousquet, Giovanni Leoni and Domenico Mucci for\nseveral interesting discussions on the subject. This work was initiated while\nV. Millot was visiting the department of {\\it Functional Analysis and Applications} at S.I.S.S.A.,\nhe thanks G. Dal Maso and the whole department for\nthe warm hospitality. The research of\nJ.-F. Babadjian was partially supported by the Marie Curie Research\nTraining Network MRTN-CT-2004-505226 ``Multi-scale modelling and\ncharacterisation for phase transformations in advanced materials''\n(MULTIMAT). V. Millot was partially supported by the Center for\nNonlinear Analysis (CNA) under the National Science Fundation Grant\nNo. 0405343.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}}\n\\else\n\\section{Introduction}\n\\label{sec:introduction}\n\\fi\n\n\\IEEEPARstart{O}{ne-class} novelty detection refers to the problem of determining if a test data sample is normal (known class) or anomalous (novel class). In real-world applications, novel data is difficult to collect since they are often rare or unsafe. Hence, one-class novelty detection considers training data from only a single known class. Most recent advances in one-class novelty detection are based on the deep Auto-Encoder (AE) style architectures, such as Denoising Auto-Encoder (DAE) \\cite{salehi2020arae,vincent2008extracting}, Variational Auto-Encoder (VAE) \\cite{kingma2013auto}, Adversarial Auto-Encoder (AAE) \\cite{makhzani2015adversarial, pidhorskyi2018generative}, Generative Adversarial Network (GAN) \\cite{goodfellow2014generative,perera2019ocgan,sabokrou2018adversarially,zhangp}, etc. Given an AE that learns the distribution of the known class, normal data are expected to be reconstructed accurately, while anomalous data are not. The reconstruction error of the AE is then used as a score for a test example to perform novelty detection. Although deep novelty detection methods achieve impressive performance, their robustness against adversarial attacks \\cite{goodfellow2015explaining,Szegedy2014Intriguing} lacks exploration.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{cover_pic.pdf}\n\t\\caption{Overview of the proposed adversarially robust one-class novelty detection idea (PLS). The vanilla Auto-Encoder (AE) and AE+PLS are trained with the known class defined as digit 8. AE+PLS reconstructs every adversarial data into the known class (digit 8) and thus produces preferred reconstruction errors for novelty detection, even under attacks.}\n\t\\label{fig:cover_pic}\n\\end{figure}\n\nAdversarial examples pose serious security threats to deep networks as they can fool them with carefully crafted perturbations. Over the past few years, many adversarial attack and defense approaches have been proposed for tasks such as image classification \\cite{guo2017countering,raff2019barrage,Xie_2019_CVPR,xu2017feature}, video recognition \\cite{lo2020defending,wei2019sparse}, optical flow estimation \\cite{ranjan2019attacking} and open-set recognition \\cite{shao2020open}. However, adversarial attacks or defenses have not been thoroughly investigated in the context of one-class novelty detection. We first show that present novelty detectors are vulnerable to adversarial attacks. Subsequently, we demonstrate that many state-of-the-art defenses \\cite{hendrycks2019selfsupervised,shi2021online,xie2020smooth,Xie_2019_CVPR} prove to be sub-optimal to properly defend novelty detectors against adversarial examples. This motivates us to design an effective defense strategy specifically for one-class novelty detection.\n\nTo this end, we propose to leverage task-specific knowledge to protect novelty detectors. These novelty detectors are only required to retain information about normal data, thereby resulting in poor reconstructions for anomalous data. This is favorable to the novelty detection problem. This can be achieved by constraining the latent space to make the features closer to a prior distribution \\cite{perera2019ocgan,park2020learning}. Also, it has been shown that adversarial perturbations can be removed in the feature space \\cite{Xie_2019_CVPR}. Therefore, one can largely manipulate the latent space of novelty detectors to devoid them of feature corruption introduced by adversaries, while maintaining the performance on clean input data. This property is unique to the novelty detection task, as most deep learning applications (e.g., image classification) require a model containing sophisticated semantic information, and a large manipulation on the latent space may limit the model capability, resulting in performance degradation.\n\nIn this paper, we propose a defense strategy, referred to as Principal Latent Space (PLS), to defend novelty detectors against adversarial examples. Specifically, PLS learns the incrementally-trained \\cite{ross2008incremental} cascade principal components in the latent space. This contains a cascade principal component analysis (PCA), which consists of a PCA operating on the vector dimension (i.e., channel) of a latent space \\cite{van2017neural} and the other PCA operating on the spatial dimension. We name these two PCAs as \\textit{Vector-PCA} and \\textit{Spatial-PCA}, respectively. First, Vector-PCA uses a learned \\textit{principal latent vector} to represent a latent space as the Vector-PCA space of a single-channel map. Since the principal latent vector is a pre-trained component that would not be affected by adversarial perturbations, most adversaries are removed at this step, and the remaining adversaries are enclosed within the small \\textit{Vector-PCA space}. Subsequently, Spatial-PCA uses learned \\textit{principal Vector-PCA maps} to represent the Vector-PCA space as the \\textit{Spatial-PCA space} and expel the remaining adversaries. Finally, the corresponding cascade inverse PCA transforms the Spatial-PCA space back to the original dimensionality, resulting in the \\textit{principal latent space}.\n\nWith PLS, the decoder could compute preferred reconstruction errors as novelty scores, even under adversarial attacks (see Fig.~\\ref{fig:cover_pic}). Additionally, we incorporate adversarial training (AT) \\cite{madry2018towards} with PLS to further exert PLS's ability. In contrast to typical defenses which often sacrifice their performance on clean data \\cite{tsipras2018robustness,xie2020adversarial}, the proposed defense strategy does not hurt the performance but rather improves it. The PLS module can be attached to any AE-style architectures (VAE, GAN, etc), so it is applicable to a wide variety of the existing novelty detection approaches, such as \\cite{kingma2013auto,makhzani2015adversarial,sabokrou2018adversarially,pidhorskyi2018generative,salehi2020arae} etc. We extensively evaluate PLS on eight adversarial attacks, three datasets and six different novelty detectors. We further compare PLS with commonly-used defense methods and show that it consistently enhances the adversarial robustness of novelty detectors by significant margins. To the best of our knowledge, this is one of the first adversarially robust novelty detection methods.\n\n\n\\section{Related work} \\label{sec2}\n\\noindent \\textbf{One-class novelty detection.}\nOne-class novelty detection is of great interest to the computer vision community. Earlier algorithms mainly rely on Support Vector Machines (SVM) formulation \\cite{scholkopf1999support,tax2004support}. With the advent of deep learning, AE-based approaches are dominating this area and achieve state-of-the-art performance \\cite{gong2019memorizing,park2020learning,perera2019ocgan,pidhorskyi2018generative,sabokrou2018adversarially,sakurada2014anomaly,salehi2020arae,xia2015learning,zhou2017anomaly}. ALOCC \\cite{sabokrou2018adversarially} considers a DAE \\cite{vincent2008extracting} as a generator and appends a discriminator to train the entire network by the generative adversarial framework \\cite{goodfellow2014generative}. GPND \\cite{pidhorskyi2018generative} is based on AAE \\cite{makhzani2015adversarial}, and it applies a discriminator to the latent space and the other discriminator to the output. OCGAN \\cite{perera2019ocgan} includes two discriminators and a classifier to train a DAE by the generative adversarial framework. ARAE \\cite{salehi2020arae} crafts adversarial examples from the latent space to adversarially train a DAE. Different from our work, ARAE's adversarial examples aim to pursue performance, and its adversarial robustness is not thoroughly evaluated (see Supplementary).\n\n\\noindent \\textbf{Adversarial attacks.}\nSzegedy et al. \\cite{Szegedy2014Intriguing} showed that carefully crafted perturbations can fool deep networks. Goodfellow et al. \\cite{goodfellow2015explaining} introduced the Fast Gradient Sign Method (FGSM), which leverages the sign of gradients to produce adversarial examples. Projected Gradient Descent (PGD) \\cite{madry2018towards} extends FGSM from single iteration gradient descent to an iterative version. MI-FGSM \\cite{dong2018boosting} generates more transferable adversarial attacks by a momentum mechanism. MultAdv \\cite{lo2020multav} produces adversarial examples via the multiplicative operation instead of the additive operation. Physically realizable attacks, which can be implemented in the physical scenarios, is also developed \\cite{Sharif16AdvML,zajac2019adversarial}. For example, Adversarial Framing (AF) \\cite{zajac2019adversarial} adds perturbations on the border of an image, while the remaining pixels are unchanged.\n\n\\noindent \\textbf{Adversarial defenses.}\nAt earlier time, a few studies aim to detect adversarial examples \\cite{hendrycks2016early,jere2020principal,li2017adversarial}. However, it is well-known that detection is inherently weaker than defense in terms of resisting adversarial attacks. Although several defense approaches based on image transformation are proposed afterward \\cite{guo2017countering,xu2017feature,bhagoji2017dimensionality}, they fail to defend against white-box attacks \\cite{carlini2017adversarial,obfuscated}. Recently, Adversarial Training (AT) has been considered one of the most effective defenses, especially in the white-box setting. Madry et al. \\cite{madry2018towards} formulated AT in a min-max optimization framework (PGD-AT), and this has been widely used as a benchmark. Xie et al. \\cite{Xie_2019_CVPR} includes the feature denoising block (FD) in networks to remove adversarial perturbations in the feature domain. SAT \\cite{xie2020smooth} uses smooth approximations of ReLU activation to enhance PGD-AT. Hendrycks et al. \\cite{hendrycks2019selfsupervised} added an auxiliary rotation prediction task \\cite{gidaris2018unsupervised} to improve PGD-AT (RotNet-AT). SOAP \\cite{shi2021online} takes self-supervised signals to purify adversarial examples during inference.\n\nTo the best of our knowledge, APAE \\cite{goodge2020robustness} might be the only present defense designed for anomaly detection. It uses approximate projection and feature weighting to reduce adversarial effects. However, its robustness is not fully tested and only anomalous data are perturbed in its evaluation (see Supplementary). Instead, we provide a generic framework for evaluating the adversarial robustness of novelty detectors and our proposed defense method.\n\n\n\\section{Attacking novelty detection models} \\label{sec3}\n\nWe consider several popular adversarial attacks \\cite{dong2018boosting,goodfellow2015explaining,lo2020multav,madry2018towards,papernot2017practical,zajac2019adversarial} and modify their loss objectives to suit the novelty detection problem setup. Here, we take PGD \\cite{madry2018towards} as an example to illustrate our attack formulation. The other gradient-based attacks can be extended by a similar formulation (see Supplementary).\n\nConsider an AE-based target model with an encoder $Enc$ and a decoder $Dec$, and an input image $\\mathbf{X}$ with the ground-truth label $y \\in \\{-1, 1\\}$, where ``$1$\" denotes the known class and ``$-1$\" denotes the novel classes. We generate the adversarial example $\\mathbf{X}_{adv}$ as follows:\n\\begin{equation}\n\\label{pgd_attack}\n\\mathbf{X}^{t+1} = Proj^{L_\\infty}_{\\mathbf{X}, \\ \\epsilon} \\big\\{ \\mathbf{X}^{t} + \\alpha \\cdot sign(\\bigtriangledown_{\\mathbf{X}^t} \\mathcal{L}(\\hat{\\mathbf{X}}^t, \\mathbf{X}^t, y)) \\big\\},\n\\end{equation}\nwhere, $\\hat{\\mathbf{X}}^t = Dec(Enc(\\mathbf{X}^t))$, $\\alpha>0$ denotes a step size, and $t \\in [0,t_{max}-1]$ is the number of attacking iterations, $\\mathbf{X} = \\mathbf{X}^0$ and $\\mathbf{X}_{adv} = \\mathbf{X}^{t_{max}}$. $Proj^{L_\\infty}_{\\mathbf{X},\\epsilon}\\{\\cdot\\}$ projects its element into an $L_\\infty$-norm bound with perturbation size $\\epsilon$ such that $\\parallel \\mathbf{X}^{t+1} - \\mathbf{X} \\parallel_{\\infty} \\leq \\epsilon$. $\\mathcal{L}$ corresponds to the mean square error (MSE) loss defined as follows:\n\\begin{equation}\n\\label{mse_loss}\n\\mathcal{L}(\\hat{\\mathbf{X}}^t, \\mathbf{X}^t, y) = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2.\n\\end{equation}\nGiven a test example, if it belongs to the known class, we maximize its reconstruction error (i.e., novelty score) by gradient ascent; while if it belongs to novel classes, we minimize its reconstruction error by gradient descent.\n\nPresent novelty detection methods are vulnerable to this attack (see Sec.~\\ref{robustness}); that is, normal data would be misclassified into novel classes, and anomalous data would be misclassified into the known class. Moreover, this attacking strategy is much stronger than the attacks introduced by \\cite{salehi2020arae}, which perturbs only normal data, and by \\cite{goodge2020robustness}, which perturbs only anomalous data (see Supplementary).\n\n\n\\section{Adversarially robust novelty detection} \\label{method}\nThe proposed defense strategy exploits the task-specific knowledge of one-class novelty detection. Specifically, we leverage the fact that a novelty detector's latent space can be manipulated to a larger extent as long as it retains the known class information. This property is especially useful to remove more adversarial perturbations in the latent space. Therefore, we propose to train a novelty detector by manipulating its latent space such that it can improve adversarial robustness while maintaining the performance on clean data. Note that these characteristics are specific to the novelty detection problem. The majority of visual recognition problems, such as image classification, require a model retaining multiple category information. Hence, a large manipulation on the latent space may hinder the model capability and thus degrade the performance.\n\nIn the following subsections, we first briefly review PCA to define the notations used in this paper, then discuss the proposed PLS in detail.\n\n\\subsection{Preliminary}\nPCA computes the principal components of a collection of data and uses them to conduct a change of basis on the data through a linear transformation. Consider a data matrix $\\mathbf{X} \\in \\mathbb{R}^{n \\times d}$, its mean $\\bm{\\mu} \\in \\mathbb{R}^{1 \\times d}$ and its covariance $\\mathbf{C} = (\\mathbf{X}-\\bm{\\mu})^\\top (\\mathbf{X}-\\bm{\\mu})$. $\\mathbf{C}$ can be written as $\\mathbf{C} = \\mathbf{U} \\bm{\\Lambda} \\mathbf{V}^\\top$ via Singular Vector Decomposition (SVD), where $\\mathbf{U} \\in \\mathbb{R}^{d \\times d}$ is an orthogonal matrix containing the principal components of $\\mathbf{X}$. Here, we define a mapping $h$ which computes the mean vector and the first $k$ principal components of the given $\\mathbf{X}$:\n\\begin{equation}\n\\label{hhh}\nh(\\mathbf{X}, k): \\mathbf{X} \\to \\{\\bm{\\mu}, \\tilde{\\mathbf{U}}\\},\n\\end{equation}\nwhere $\\tilde{\\mathbf{U}} \\in \\mathbb{R}^{d \\times k}$ keeps only the first $k$ columns of $\\mathbf{U}$. Now we define the forward and the inverse PCA transformation as a pair of mapping $(f: \\mathbb{R}^{n \\times d} \\to \\mathbb{R}^{n \\times k}$, $g: \\mathbb{R}^{n \\times k} \\to \\mathbb{R}^{n \\times d})$; $f$ performs the forward PCA: \n\\begin{equation}\n\\label{fff}\nf(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}}) = (\\mathbf{X}-\\bm{\\mu}) \\tilde{\\mathbf{U}},\n\\end{equation}\nand $g$ performs the inverse PCA:\n\\begin{equation}\n\\label{ggg}\ng(\\mathbf{X}_{PCA}; \\bm{\\mu}, \\tilde{\\mathbf{U}}) = \\mathbf{X}_{PCA} \\tilde{\\mathbf{U}}^\\top + \\bm{\\mu},\n\\end{equation}\nwhere $\\mathbf{X}_{PCA} = f(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}})$. Finally, we can write the PCA reconstruction of $\\mathbf{X}$ as $\\hat{\\mathbf{X}} = g(f(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}}); \\bm{\\mu}, \\tilde{\\mathbf{U}})$.\n\n\\subsection{Principal Latent Space (PLS)}\nThe proposed PLS contains two major components: (1) Vector-PCA and (2) Spatial-PCA. In Vector-PCA, we perform $(h, f, g)$ on the vector dimension as $(h_V, f_V, g_V)$, and in Spatial-PCA, we perform $(h, f, g)$ on the spatial dimension as $(h_S, f_S, g_S)$. Let $Enc$ be the encoder and $Dec$ be the decoder of a novelty detection model. Let us denote an adversarial image as $\\mathbf{X}_{adv}$, we have its latent space $\\mathbf{Z}_{adv} = Enc(\\mathbf{X}_{adv}) \\in \\mathbb{R}^{s \\times v}$, where $s = h \\times w$ is the spatial dimensionality obtained by the product of height and width, and $v$ is the vector dimensionality (i.e., the number of channels). Under adversarial attacks, $\\mathbf{Z}_{adv}$ would be corrupted by adversarial perturbations such that the decoder cannot compute reconstruction errors favorable to novelty detection. We define the proposed PLS as a transformation $PLS: \\mathbf{Z}_{adv} \\to \\mathbf{Z}_{PLS}$, which removes adversaries from $\\mathbf{Z}_{adv}$, where $\\mathbf{Z}_{PLS}$ is referred to as principal latent space. $PLS$ is implemented by our incrementally-trained cascade PCA. In the beginning, a sigmoid function replaces the encoder's last activation function to bound $\\mathbf{Z}_{adv}$ values between 0 and 1. The following procedures are described below.\n\nFirst, Vector-PCA computes the mean latent vector and the principal latent vector of $\\mathbf{Z}_{adv}$:\n\\begin{equation}\n\\label{hv}\n\\{\\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V\\} = h_V(\\mathbf{Z}_{adv}, k_V=1),\n\\end{equation}\nwhere, we always set $k_V$ to 1, so $\\tilde{\\mathbf{U}}_V$ is the first principal latent vector of $\\mathbf{Z}_{adv}$. Second, Vector-PCA transforms $\\mathbf{Z}_{adv}$ to its Vector-PCA space $\\mathbf{Z}_V \\in \\mathbb{R}^{s \\times 1}$:\n\\begin{equation}\n\\label{fv}\n\\mathbf{Z}_V = f_V(\\mathbf{Z}_{adv}; \\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V).\n\\end{equation}\nNext, Spatial-PCA computes the mean Vector-PCA map\\footnote{We use the word ``map\" to indicate they are on the spatial dimension.} and the principal Vector-PCA maps of $\\mathbf{Z}_V$:\n\\begin{equation}\n\\label{hs}\n\\{\\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S\\} = h_S(\\mathbf{Z}_V^\\top, k_S),\n\\end{equation}\nwhere, $k_S$ is a hyperparameter. Then, Spatial-PCA transforms $\\mathbf{Z}_V$ to its Spatial-PCA space $\\mathbf{Z}_S \\in \\mathbb{R}^{k_S \\times 1}$:\n\\begin{equation}\n\\label{fs}\n\\mathbf{Z}_S^\\top = f_S(\\mathbf{Z}_V^\\top; \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S).\n\\end{equation}\nFinally, the inverse Spatial-PCA and the inverse Vector-PCA transform $\\mathbf{Z}_S$ back to its original dimensionality:\n\\begin{equation}\n\\label{gs}\n\\hat{\\mathbf{Z}}_V^\\top = g_S(\\mathbf{Z}_S^\\top; \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S),\n\\end{equation}\n\\begin{equation}\n\\label{gv}\n\\mathbf{Z}_{PLS} = g_V(\\hat{\\mathbf{Z}}_V; \\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V),\n\\end{equation}\nwhere, $\\hat{\\mathbf{Z}}_V$ is the Spatial-PCA reconstruction of $\\mathbf{Z}_V$, and $\\mathbf{Z}_{PLS}$ is the resulting principal latent space. Fig.~\\ref{fig:big_pic} gives an overview of this procedure. The decoder then uses $\\mathbf{Z}_{PLS}$ to reconstruct the input adversarial example as $\\hat{\\mathbf{X}}_{adv} = Dec(\\mathbf{Z}_{PLS})$ for computing the novelty score.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{big_pic.pdf}\n\t\\caption{Overview of the proposed PLS. $f_V$: forward Vector-PCA, $f_S$: forward Spatial-PCA, $g_S$: inverse Spatial-PCA, $g_V$: inverse Vector-PCA, $h_V$ and $h_S$ are the mappings for computing principal components.}\n\t\\label{fig:big_pic}\n\\end{figure}\n\n\\subsection{Incremental training}\nThe \\textit{principal latent components} $\\{\\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V, \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S\\}$ are incrementally-trained along with the network weights by exponential moving average (EMA) during training, so we call this process incrementally-trained cascade PCA. Specifically, at training iteration $t$, these components are updated with following equations:\n\\begin{equation}\n\\label{train_V}\n\\{\\bm{\\mu}_V^t, \\tilde{\\mathbf{U}}_V^t\\} = (1-\\eta_V) \\{\\bm{\\mu}_V^{t-1}, \\tilde{\\mathbf{U}}_V^{t-1}\\} + \\eta_V \\cdot h_V(\\mathbf{Z}_{adv}^t),\n\\end{equation}\n\\begin{equation}\n\\label{train_S}\n\\{\\bm{\\mu}_S^t, \\tilde{\\mathbf{U}}_S^t\\} = (1-\\eta_S) \\{\\bm{\\mu}_S^{t-1}, \\tilde{\\mathbf{U}}_S^{t-1}\\} + \\eta_S \\cdot h_S(\\mathbf{Z}_V^{t \\top}),\n\\end{equation}\nwhere $\\eta_V$ and $\\eta_S$ are the EMA learning rates.\n\nConsider the model weights are trained by the mini-batch gradient descent with a batch size $b$, the latent dimensionality is shaped to $\\mathbf{Z}_{adv} \\in \\mathbb{R}^{bs \\times v}$, the resulting $\\mathbf{Z}_V \\in \\mathbb{R}^{bs \\times 1}$ is reshaped to $\\mathbf{Z}_V \\in \\mathbb{R}^{s \\times b}$ after the Vector-PCA $f_V$, and $\\hat{\\mathbf{Z}}_V \\in \\mathbb{R}^{s \\times b}$ is reshaped back to $\\hat{\\mathbf{Z}}_V \\in \\mathbb{R}^{bs \\times 1}$ after the inverse Spatial-PCA $g_S$. Hence, in a mini-batch, both $h_V$ and $h_S$ have $b$ times more data points to acquire better principal latent components at each training iteration. At iteration $t$, $(f_V, g_V)$ performs with the components $\\{\\bm{\\mu}_V^t, \\tilde{\\mathbf{U}}_V^t\\}$, and $(f_S, g_S)$ performs with the components $\\{\\bm{\\mu}_S^t, \\tilde{\\mathbf{U}}_S^t\\}$. When the training process ends, the well-trained components are denoted as $\\{\\bm{\\mu}_V^*, \\tilde{\\mathbf{U}}_V^*, \\bm{\\mu}_S^*, \\tilde{\\mathbf{U}}_S^*\\}$. During infernce, $(f_V, g_V)$ performs with $\\{\\bm{\\mu}_V^*, \\tilde{\\mathbf{U}}_V^*\\}$, and $(f_S, g_S)$ performs with $\\{\\bm{\\mu}_S^*, \\tilde{\\mathbf{U}}_S^*\\}$, while $h_V$ and $h_S$ do not operate (see Fig.~\\ref{fig:big_pic}). The entire process is differentiable during inference and thus does not cause obfuscated gradients \\cite{obfuscated}. This incremental training helps make sure the cascade PCA is aware of the network weight updates at each training step, encouraging mutual learning between the network weights and the principal latent components. The entire model and thus can be trained end-to-end.\n\n\\subsection{Defense mechanism}\nWe further elaborate on how the proposed PLS defends against adversarial attacks. Given an adversarial example $\\mathbf{X}_{adv}$, its latent space $\\mathbf{Z}_{adv}$ is adversarially perturbed. After Vector-PCA, each latent vector of $\\mathbf{Z}_{adv}$ is represented by a scaling factor of the learned principal latent vector $\\tilde{\\mathbf{U}}_V^*$ (with a bias term $\\bm{\\mu}_V^*$). The Vector-PCA space $\\mathbf{Z}_V$ stores these scaling factors on a single-channel map (i.e., on the spatial domain only). Since all the principal latent components are pre-trained parameters, they would not be affected by adversarial perturbations. Replacing the perturbed latent vectors by $\\tilde{\\mathbf{U}}_V^*$ removes the majority of the adversaries. The only place where the remaining adversaries can appear is the scaling factors of $\\tilde{\\mathbf{U}}_V^*$ on the single-channel map. In other words, these adversaries are enclosed within a small subspace, making them easier to expel.\n\nSubsequently, Spatial-PCA reconstructs this small subspace by a set of principal Vector-PCA maps $\\tilde{\\mathbf{U}}_S^*$ (with a bias term $\\bm{\\mu}_S^*$). Since $\\tilde{\\mathbf{U}}_S^*$ and $\\bm{\\mu}_S^*$ are adversary-free, the remaining adversaries are further removed. From another perspective, this step can be viewed as PCA-based denoising performing in the spatial domain of features. With the robust principal latent space $\\mathbf{Z}_{PLS}$, the decoder can obtain a preferred reconstruction error for novelty detection, even in the presence of an adversarial example. Additionally, we perform AT \\cite{madry2018towards} to train the model, further improving the robustness.\n\n\n\n\\section{Experiments}\n\nWe evaluate PLS on eight adversarial attacks, three datasets and six existing novelty detection methods. We further compare PLS with state-of-the-art defense approaches. An extensive ablation study is also presented. \n\n\\renewcommand{\\arraystretch}{0.6}\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{table*}[htp!]\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under various adversarial attacks.}\n\t\t\\label{table:main_results_1}\n\t\t\\begin{tabular}{r | r | c | ccccc | c}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDataset & Defense & Clean & FGSM \\cite{goodfellow2015explaining} & PGD \\cite{madry2018towards} & MI-FGSM \\cite{dong2018boosting} & MultAdv \\cite{lo2020multav} & AF \\cite{zajac2019adversarial} & Black-box \\cite{papernot2017practical} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.964 & 0.350 & 0.051 & 0.022 & 0.170 & 0.014 & 0.790 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.961 & 0.604 & 0.357 & 0.369 & 0.444 & 0.155 & 0.691 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.963 & 0.612 & 0.366 & 0.379 & 0.453 & 0.142 & 0.700 \\\\\n\t\t\tMNIST & SAT \\cite{xie2020smooth} & 0.947 & 0.527 & 0.295 & 0.306 & 0.370 & 0.142 & 0.652 \\\\\n\t\t\t\\cite{lecun2010mnist} & RotNet-AT \\cite{hendrycks2019selfsupervised} & \\textbf{0.967} & 0.598 & 0.333 & 0.333 & 0.424 & 0.101 & 0.695 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.940 & 0.686 & 0.504 & 0.506 & 0.433 & 0.088 & \\textbf{0.863} \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.925 & 0.428 & 0.104 & 0.105 & 0.251 & 0.022 & 0.730 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.967} & \\textbf{0.786} & \\textbf{0.678} & \\textbf{0.679} & \\textbf{0.701} & \\textbf{0.599} & 0.840 \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.892 & 0.469 & 0.088 & 0.047 & 0.148 & 0.112 & 0.562 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.890 & 0.518 & 0.368 & 0.348 & 0.327 & 0.253 & 0.540 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.886 & 0.524 & 0.379 & 0.359 & 0.335 & 0.252 & 0.535 \\\\\n\t\t\tF-MNIST & SAT \\cite{xie2020smooth} & 0.878 & 0.444 & 0.306 & 0.285 & 0.273 & 0.231 & 0.492 \\\\\n\t\t\t\\cite{xiao2017fashion} & RotNet-AT \\cite{hendrycks2019selfsupervised} & 0.891 & 0.527 & 0.375 & 0.351 & 0.312 & 0.240 & 0.541 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.876 & 0.639 & 0.475 & 0.475 & 0.327 & 0.274 & 0.611 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.861 & 0.510 & 0.174 & 0.174 & 0.220 & 0.135 & 0.513 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.909} & \\textbf{0.677} & \\textbf{0.600} & \\textbf{0.585} & \\textbf{0.573} & \\textbf{0.591} & \\textbf{0.696} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.550 & 0.186 & 0.034 & 0.018 & 0.025 & 0.035 & 0.227 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.546 & 0.236 & 0.145 & 0.139 & 0.107 & 0.096 & 0.223 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.546 & 0.237 & 0.147 & 0.141 & 0.109 & 0.103 & 0.222 \\\\\n\t\t\tCIFAR-10 & SAT \\cite{xie2020smooth} & 0.537 & 0.223 & 0.141 & 0.135 & 0.101 & 0.079 & 0.219 \\\\\n\t\t\t\\cite{krizhevsky2009learning} & RotNet-AT \\cite{hendrycks2019selfsupervised} & 0.547 & 0.236 & 0.139 & 0.107 & 0.075 & 0.092 & 0.224 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.546 & 0.270 & 0.131 & 0.141 & 0.096 & 0.070 & 0.231 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.552 & 0.259 & 0.097 & 0.097 & 0.077 & 0.112 & 0.255 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.578} & \\textbf{0.320} & \\textbf{0.245} & \\textbf{0.242} & \\textbf{0.201} & \\textbf{0.243} & \\textbf{0.331} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\n\\subsection{Experimental setup} \\label{sec51}\n\n\\noindent \\textbf{Datasets.}\nWe use three datasets for evaluation: MNIST \\cite{lecun2010mnist}, Fashion-MNIST (F-MNIST) \\cite{xiao2017fashion} and CIFAR-10 \\cite{krizhevsky2009learning}. MNIST consists of 28 $\\times$ 28 grayscale handwritten digits from 0 to 9. It contains 60,000 training data and 10,000 test data. F-MNIST is composed of 28 $\\times$ 28 grayscale images from 10 fashion product categories. It comprises of 60,000 training data and 10,000 test data. CIFAR-10 consists of 32 $\\times$ 32 color images from 10 different classes. There are 50,000 training and 10,000 test images in this dataset.\n\n\\noindent \\textbf{Evaluation protocol.}\nWe simulate a one-class novelty detection scenario by the following protocol. Given a dataset, each class is defined as the known class at a time, and a model is trained with the training data of this known class. During inference, the test data of the known class are considered normal, and the test data of the other classes (i.e., novel classes) are considered anomalous. We select the anomalous data from each novel class equally to constitute half of the test set, where the anomalous data within a novel class are selected randomly. Hence, our test set contains 50\\% anomalous data, where each novel class accounts for the same proportion. The area under the Receiver Operating Characteristic curve (AUROC) value is used as the evaluation metric, where the ROC curve is obtained by varying the threshold of the novelty score. For each dataset, we report the mean AUROC (mAUROC) across its 10 classes.\n\n\\noindent \\textbf{Attack setting.}\nWe test adversarial robustness against five white-box attacks, inclduing FGSM \\cite{goodfellow2015explaining}, PGD \\cite{madry2018towards}, MI-FGSM \\cite{dong2018boosting}, MultAdv \\cite{lo2020multav} and AF \\cite{zajac2019adversarial}, where PGD is the default attack if not otherwise specified. A black-box attack and two adaptive attacks \\cite{papernot2017practical,tramer2020adaptive} are also considered. All the attacks are implemented based on the formulation in Sec.~\\ref{sec3}.\n\nFor FGSM, PGD and MI-FGSM, we set $\\epsilon$ to $25\/255$ for MNIST, $16\/255$ for F-MNIST, and $8\/255$ for CIFAR-10. For MultAdv, we set $\\epsilon_m$ to $1.25$ for MNIST, $1.16$ for F-MNIST, and $1.08$ for CIFAR-10. For AF, we set $\\epsilon$ to $160\/255$, $120\/255$ and $80\/255$ for MNIST, F-MNIST and CIFAR-10, respectively. The framing width $w_{AF}$ is set to $1$. The number of attack iterations $t_{max}$ is set to $1$ for FGSM and $5$ for the other attacks.\n\n\\noindent \\textbf{Baseline defenses.}\nTo the best of our knowledge, APAE \\cite{goodge2020robustness} might be the only present defense designed for anomaly detection. In addition to APAE, we implement five commonly-used defenses, which are originally designed for classification tasks, in the context of novelty detection. They are PGD-AT \\cite{madry2018towards}, FD \\cite{Xie_2019_CVPR}, SAT \\cite{xie2020smooth}, RotNet-AT \\cite{hendrycks2019selfsupervised} and SOAP \\cite{shi2021online}, where FD, SAT and RotNet-AT incorporate PGD-AT. We use Gaussian non-local means \\cite{buades2005non} for FD, Swish \\cite{hendrycks2016gaussian} for SAT, and RotNet \\cite{gidaris2018unsupervised} for SOAP. These are their well-performing versions.\n\n\\noindent \\textbf{Benchmark novelty detectors.}\nWe apply PLS to six novelty detection methods, including a vanilla AE, VAE \\cite{kingma2013auto}, AAE \\cite{makhzani2015adversarial}, ALOCC \\cite{sabokrou2018adversarially}, GPND \\cite{pidhorskyi2018generative} and ARAE \\cite{salehi2020arae}, where the vanilla AE is the default novelty detector if not otherwise specified. PLS is added after the last layer of the novelty detection models' encoder.\n\nIn order to evenly evaluate the adversarial robustness of these approaches, we unify their AE backbones into the following archirecture. The encoder consists of four 3 $\\times$ 3 convolutional layers, where each of the first three layers are followed by a 2 $\\times$ 2 max-pooling with stride 2. We use a base channel size of 64, and increase the number of channels by a factor of 2. The decoder mirrors the encoder but replaces every max-pooling by a bilinear interpolation with a factor of 2. All the convolutional layers are followed by a batch normalization layer \\cite{ioffe2015batch} and ReLU. \n\n\n\\noindent \\textbf{Implementation details.}\nAll the models are trained by Adam optimizer \\cite{kingma2014adam} with initial learning rate $5e^{-5}$ and weight decay $1e^{-4}$, where the learning rate is decreased by a factor of 10 at the 20th and 40th epochs. The batch size is 128. For PLS, we set $k_V$ to $1$, $k_S$ to $8$, initial $\\eta_V$ to $0.1$ and initial $\\eta_S$ to $0.001$, where $\\eta_V$ and $\\eta_S$ are also decreased by a factor of 10 at the 20th and 40th epochs.\n\n\\setlength{\\tabcolsep}{5.5pt}\n\\begin{table*}[htp!]\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under PGD attack. Various novelty detectors are used.}\n\t\t\\label{table:main_results_2}\n\t\t\\begin{tabular}{r | r | c | cccccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDataset & Defense & Test type & AE & VAE \\cite{kingma2013auto} & AAE \\cite{makhzani2015adversarial} & ALOCC \\cite{sabokrou2018adversarially} & GPND \\cite{pidhorskyi2018generative} & ARAE \\cite{salehi2020arae} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.964 & 0.979 & 0.973 & 0.961 & 0.946 & 0.965 \\\\\n\t\t\t& No Defense & PGD & 0.051 & 0.087 & 0.056 & 0.141 & 0.128 & 0.133 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.357 & 0.521 & 0.427 & 0.312 & 0.582 & 0.341 \\\\\n\t\t\tMNIST & FD \\cite{Xie_2019_CVPR} & & 0.366 & 0.525 & 0.419 & 0.319 & 0.551 & 0.350 \\\\\n\t\t\t\\cite{lecun2010mnist} & SAT \\cite{xie2020smooth} & & 0.295 & 0.485 & 0.470 & 0.330 & 0.527 & 0.254 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.333 & 0.501 & 0.507 & 0.361 & 0.551 & 0.314 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.504 & 0.608 & 0.398 & 0.606 & 0.425 & 0.522 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.104 & 0.155 & 0.240 & 0.202 & 0.229 & 0.191 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.678} & \\textbf{0.739} & \\textbf{0.608} & \\textbf{0.693} & \\textbf{0.741} & \\textbf{0.695} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.892 & 0.914 & 0.912 & 0.901 & 0.915 & 0.901 \\\\\n\t\t\t& No Defense & PGD & 0.088 & 0.223 & 0.152 & 0.177 & 0.177 & 0.262 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.368 & 0.538 & 0.512 & 0.367 & 0.539 & 0.420 \\\\\n\t\t\tF-MNIST & FD \\cite{Xie_2019_CVPR} & & 0.379 & 0.533 & 0.513 & 0.370 & 0.542 & 0.428 \\\\\n\t\t\t\\cite{xiao2017fashion} & SAT \\cite{xie2020smooth} & & 0.306 & 0.504 & 0.499 & 0.332 & 0.530 & 0.351 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.375 & 0.542 & 0.509 & 0.365 & 0.524 & 0.396 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.475 & 0.509 & 0.313 & 0.477 & 0.386 & 0.548 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.174 & 0.366 & 0.300 & 0.246 & 0.398 & 0.310 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.600} & \\textbf{0.604} & \\textbf{0.599} & \\textbf{0.612} & \\textbf{0.626} & \\textbf{0.599} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.550 & 0.552 & 0.555 & 0.551 & 0.559 & 0.578 \\\\\n\t\t\t& No Defense & PGD & 0.034 & 0.073 & 0.051 & 0.037 & 0.027 & 0.087 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.145 & 0.177 & 0.195 & 0.146 & 0.182 & 0.157 \\\\\n\t\t\tCIFAR-10 & FD \\cite{Xie_2019_CVPR} & & 0.147 & 0.180 & 0.206 & 0.150 & 0.187 & 0.152 \\\\\n\t\t\t\\cite{krizhevsky2009learning} & SAT \\cite{xie2020smooth} & & 0.141 & 0.170 & 0.186 & 0.141 & 0.181 & 0.107 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.139 & 0.163 & 0.161 & 0.105 & 0.147 & 0.101 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.131 & 0.094 & 0.043 & 0.172 & 0.075 & 0.117 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.097 & 0.179 & 0.171 & 0.095 & 0.062 & 0.154 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.245} & \\textbf{0.247} & \\textbf{0.252} & \\textbf{0.244} & \\textbf{0.242} & \\textbf{0.245} \\\\\t\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\n\\subsection{Robustness} \\label{robustness}\n\n\\subsubsection{White-box attacks}\nThe robustness of one-class novelty detection against various white-box attacks is reported in Table~\\ref{table:main_results_1}, where the vanilla AE is used. Without a defense, mAUROC scores drop significantly under all the white-box attacks, which shows the vulnerability of novelty detectors to the adversarial examples. PGD-AT improves adversarial robustness to a great extent. FD makes a slight improvement upon PGD-AT in most cases. SAT and Rot-AT seem not effective upon PGD-AT in the context of novelty detection. SOAP performs well in some cases but not uniformly. Compared to other methods, APAE generally shows less robustness. The proposed method, PLS, significantly increases mAUROC with PGD-AT, leading the other defenses by a decent margin. Moreover, PLS is consistently better across all the five white-box attacks on three datasets.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{adaptive.pdf}\n\t\\caption{The mAUROC of PLS under PLS-knowledgeable attacks with varied trade-off parameters. (a) Knowledgeable A. (b) Knowledgeable B.}\n\t\\label{fig:adaptive}\n\\end{figure}\n\n\\noindent \\textbf{PLS-knowledgeable attacks.}\nAs discussed above, in a white-box attack, attackers are aware of the presence of the defense mechanism, i.e., PLS (it is differentiable at inference time, see Sec.~\\ref{method}). However, they count on only the novelty detection objective (i.e., MSE loss, see Eq.~\\eqref{mse_loss}) to generate adversarial examples. In this subsection, we follow the practice of the most recent adversarial defense studies such as \\cite{shi2021online}, to thoroughly evaluate the proposed defense mechanism. More precisely, we try to find an adaptive attack \\cite{papernot2017practical,tramer2020adaptive} by giving the full knowledge of the PLS defense mechanism to the attacker. We refer to this type of attack as \\textit{PLS-knowledgeable attack} in the paper.\n\nWe construct two PLS-knowledgeable attacks, Knowledgeable A and Knowledgeable B. They jointly optimize Eq.~\\eqref{mse_loss} and an auxiliary loss developed with the knowledge of PLS. Knowledgeable A attempts to minimize the $L_2$-norm between the latent space before and after the PLS transformation. The intuition is to void PLS such that the input and the output latent space of PLS become closer. In other words, Knowledgeable A replaces Eq.~\\eqref{mse_loss} with the following equation:\n\\begin{equation}\n\\label{adaptive_1}\n\\mathcal{L} = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2 - \\lambda_A \\parallel \\mathbf{Z}^t_{PLS} - \\mathbf{Z}_{adv}^t \\parallel_2, \n\\end{equation}\nwhere, $\\lambda_A$ is a trade-off parameter. Knowledgeable B attempts to maximize the $L_2$-norm between the latent space of the current adversarial example $\\mathbf{X}^t$ and its clean counterpart $\\mathbf{X}^0$ after the PLS transformation. The intuition is to keep the adversarial latent space away from the clean one. In other words, Knowledgeable B replaces Eq.~\\eqref{mse_loss} with the following equation:\n\\begin{equation}\n\\label{adaptive_2}\n\\mathcal{L} = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2 + \\lambda_B \\parallel \\mathbf{Z}^t_{PLS} - \\mathbf{Z}^0_{PLS} \\parallel_2, \n\\end{equation}\nwhere, $\\lambda_B$ is a trade-off parameter. When $\\lambda_A=0$ or $\\lambda_B=0$, the PLS-knowledgeable attacks reduce to the conventional white-box attacks.\n\nIn Fig.~\\ref{fig:adaptive}, we can observe that mAUROC monotonously increases as $|\\lambda_A|$ or $|\\lambda_B|$ increases. That is, these PLS-knowledgeable attacks cannot further reduce PLS's mAUROC, and the additional auxiliary loss terms would attenuate the MSE loss gradients. This indicates that attackers cannot straightforwardly benefit from the knowledge of PLS. Hence, the conventional white-box attack still has the greatest attacking strength. This result shows that it is not easy to find a stronger attack to break PLS, even with the full knowledge of the PLS mechanism.\n\n\\subsubsection{Black-box attacks}\nThe robustness against black-box attacks \\cite{papernot2017practical} is shown in the last column of Table~\\ref{table:main_results_1}. Here, we consider a naturally trained (i.e., train with only clean data) GPND as a substitute model and apply MI-FGSM, which has better transferability, to generate black-box adversarial examples for target models. As we can see, the defenses with PGD-AT degrade black-box robustness, which is identical to the observation in classification tasks \\cite{tramer2018ensemble}. SOAP, which is without using AT, shows better black-box robustness. PLS greatly improves the black-box robustness even with PGD-AT, and it is consistently better across all datasets. Naturally trained PLS achieves 0.907, 0.742 and 0.332 mAUROC on MNIST, F-MNIST and CIFAR-10, respectively, under the black-box attack.\n\n\\subsubsection{Generalizability}\nTable~\\ref{table:main_results_2} shows the adversarial robustness of various state-of-the-art novelty detection models. All of them are susceptible to adversarial attacks. We attach the PLS module to these models to protect them. We can see that PLS uniformly robustifies all of these novelty detectors and significantly outperforms the other defense approaches. This confirms that PLS is applicable to a wide variety of the present novelty detection methods, demonstrating its excellent generalizability.\n \n\n\\setlength{\\tabcolsep}{7pt}\n\\begin{table}\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under clean data.}\n\t\t\\label{table:clean_performance}\n\t\t\\begin{tabular}{r | ccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDefense & MNIST & F-MNIST & CIFAR-10 \\\\\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tNo Defense & 0.964 & 0.892 & 0.550 \\\\\n\t\t\tFD \\cite{Xie_2019_CVPR} & 0.965 & 0.892 & 0.551 \\\\\n\t\t\tSAT \\cite{xie2020smooth} & 0.949 & 0.883 & 0.543 \\\\\n\t\t\tRotNet-AT \\cite{hendrycks2019selfsupervised} & 0.963 & 0.897 & 0.554 \\\\\n\t\t\tSOAP \\cite{shi2021online} & 0.940 & 0.876 & 0.546 \\\\\n\t\t\tAPAE \\cite{goodge2020robustness} & 0.925 & 0.861 & 0.552 \\\\\n\t\t\tPLS (ours) & \\textbf{0.973} & \\textbf{0.922} & \\textbf{0.578} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\subsection{Performance on clean data}\nWe also evaluate the performance of PLS on clean data. In this experiment, all the models are naturally trained. As shown in Table~\\ref{table:clean_performance}, PLS improves the performance upon the original network architecture (No Defense), while, the other defenses do not make obvious improvements. This shows that PLS generalizes better for both clean data and adversarial examples. PLS enjoys this benefit because the principal latent components are learned from only the latent space of the known class. Due to this, when transforming the latent space of any novel class image, PLS projects it into the known class space defined by the principal latent component. This brings the transformed latent space closer to the latent space of the known class, resulting in the decoder trying to reconstruct it into a known class image. Subsequently, this produces high reconstruction error for the novel class images while barely affecting the reconstruction of the known class images.\n\n\n\n\\subsection{Ablation study} \\label{sec_ablation}\n\n\\noindent \\textbf{PLS components.}\nTable~\\ref{table:ablation} reports the results of different PLS variants. First, Vector-PCA alone significantly improves the robustness upon PGD-AT. This shows that the mechanism of replacing perturbed latent vectors by the incrementally-trained principal latent vector is effective. As discussed earlier, in PLS the adversaries can stay only on the scaling factors of the principal latent vector. Next, we further remove the adversaries with the help of denoising operation on the spatial dimension. We try to deploy a feature denoising block \\cite{Xie_2019_CVPR} after the forward Vector-PCA. This baseline is denoted as Vector-PCA+FD. This makes a slight improvement over Vector-PCA baseline. Finally, the complete PLS uses Spatial-PCA for this purpose instead, achieveing great mAUROC increase. This shows Spatial-PCA's advantage over FD in our case.\n\n\\setlength{\\tabcolsep}{7pt}\n\\begin{table}\n\t\\begin{center}\n\t\t\\caption{The mAUROC of PLS variants under PGD attack.}\n\t\t\\label{table:ablation}\n\t\t\\begin{tabular}{r | ccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDefense & MNIST & F-MNIST & CIFAR-10 \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\tPGD-AT \\cite{madry2018towards} & 0.357 & 0.368 & 0.145 \\\\\n\t\t\tVector-PCA & 0.566 & 0.499 & 0.215 \\\\\n\t\t\tVector-PCA+FD & 0.582 & 0.505 & 0.215 \\\\\n\t\t\tPLS (ours) & \\textbf{0.678} & \\textbf{0.600} & \\textbf{0.245} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{feature_diff.pdf}\n\t\\caption{Mean $L_2$-norm between the latent space of PGD adversarial examples and that of their clean counterpart on different defenses. The values are the mean over an entire dataset.}\n\t\\label{fig:feature_diff}\n\\end{figure}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{histogram.pdf}\n\t\\caption{Histograms of reconstruction errors. (a) No Defense under clean data. (b) No Defense under PGD attack. (c) PGD-AT under PGD attack. (d) PLS under PGD attack. Digit 0 of MNIST is set to normal data, and the other digits are anomalous.}\n\t\\label{fig:histogram}\n\\end{figure}\n\n\\noindent \\textbf{Stability of latent space.}\nWe compute the mean $L_2$-norm between the latent space of adversarial examples and that of their clean counterpart: $\\parallel \\mathbf{Z}_{adv} - \\mathbf{Z} \\parallel_2$. As can be seen in Fig.~\\ref{fig:feature_diff}, PLS's mean $L_2$-norm is three orders of magnitude smaller than the other defenses. This indicates that PLS's latent space are barely affected by adversaries, showing PLS's effectiveness in adversary removal.\n\n\\noindent \\textbf{Reconstruction errors.} For an AE-style novelty detection model, normal data and anomalous data are expected to get low and high reconstruction errors, respectively. The model follows this behavior given clean data, as shown in Fig.~\\ref{fig:histogram}(a). When an attacker attempts to maximize the reconstruction errors of normal data and minimize that of anomalous data, the model would make wrong predictions, shown in Fig.~\\ref{fig:histogram}(b). Fig.~\\ref{fig:histogram}(c) shows that PGD-AT pulls back the enlarged reconstruction errors of normal data, but they still overlap for the anomalous data. In Fig.~\\ref{fig:histogram}, it can be observed that PLS pushes the reconstruction errors of anomalous data with better margin. Although the reconstruction errors of normal data also increases, the gap between normal and anomalous data is retained resulting in PLS performing better under attacks.\n\n\\noindent \\textbf{Reconstructed images.} Fig.~\\ref{fig:visual} compares the reconstructed images of No Defense model and PLS under PGD attack. Digit 2 of MNIST is used as the known class. We can see that No Defense model captures the shape of the adversarial anomalous data and thus produces fair reconstructions. In other words, the reconstruction error gap between normal data and anomalous data is insufficiently large. Such observation is consistent with the quantitative results that it is not adversarially robust. In contrast, PLS reconstructs every data into the known class of digit 2. Hence, even under attacks, PLS can obtain very high reconstruction errors from anomalous data and low errors from normal data.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{visual.pdf}\n\t\\caption{Reconstructions under PGD attack with $\\epsilon = 0.3$. Digit 2 is set to normal data, and the other digits are anomalous.}\n\t\\label{fig:visual}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn this paper, we study the adversarial robustness in the context of one-class novelty detection problem. We show that existing novelty detection models are vulnerable to adversarial perturbations and then propose a defense method referred to as Principal Latent Space (PLS). Specifically, PLS purifies the latent space by the incrementally-trained cascade PCA process. Moreover, we construct a generic evaluation framework to fully test the effectiveness of the proposed PLS. We perform extensive experiments on multiple datasets with multiple existing novelty detection models and consider various attacks to show that PLS improves the robustness consistently across different attacks and datasets.\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\nThis work was supported by the DARPA GARD Program HR001119S0026-GARD-FP-052.\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNetwork slicing is an important technique in 5G to enable flexibility and customization. Based on software defined networks and network function virtualization techniques, network slicing can define various virtual network slices over a single physical network infrastructure \\cite{b1}. To realize the network slicing, resource allocation is an important part to guarantee service level agreements of slices. The network operator needs to allocate limited resources between slices. For example, enhanced Mobile Broad Band (eMBB) slice usually requires a high throughput, while Ultra Reliable Low Latency Communications (URLLC) slice needs a low latency and a high reliability. Compared with core network slicing, the RAN slicing is more complicated due to limited resources and dynamic channel states \\cite{b2}. In \\cite{b3}, a risk sensitive model for the resource allocation of URLLC slice is presented. A two-layer method is introduced in \\cite{b4} to realize an efficient and low complexity RAN slicing. \\cite{b5} shows a RAN slicing scheme by considering both rate and latency demands of various traffic types, and the scheme is tested in a industry 4.0 case. \n\nThe aforementioned works mainly focus on the radio resource allocation. However, the evolving network architecture requires a joint resource allocation scheme, such as radio and computation resources. Indeed, to support the emerging computation intensive applications, deploying mobile edge computing (MEC) servers becomes an ideal solution\\cite{b5-1}. The computation tasks can be processed in the MEC server instead of the central cloud, thus a lower delay is achieved. The work of \\cite{b6} shows that joint resources allocation slicing has a better performance than slicing one single resource, and \\cite{b7} introduced a mathematical model to jointly slice mobile network and edge computation resource. \n\nAlthough incorporating computation capability into RAN will bring significant benefits, it also leads to a higher network management complexity, especially when multiple slices are involved. To this end, machine learning methods provide a good opportunity for network management \\cite{b8}. For example, in reinforcement learning, the agent interacts with environment to maximize the long term reward based on Markov decision process (MDP), and the complexity of defining a dedicated optimization model is avoided. However, the reinforcement learning algorithms that are applied in most existing works, such as Q-learning and deep Q-learning, generally require a huge number of samples to train the algorithm, which consequently lead to a long convergence time. In addition, the low training efficiency will unavoidably affect the system performance, especially for tasks with tight delay budgets. \nTo this end, we propose a knowledge transfer based resource allocation (KTRA) method in this paper. Different with existing algorithms such as reinforcement learning or deep reinforcement learning, the proposed KTRA method has a knowledge transfer capability, and the agent can leverage the knowledge of other expert agents to improve its own performance on the target task \\cite{b9}. With the prior knowledge of experts, it requires less samples when exploring the target task, which means a higher exploration efficiency. The proposed KTRA is compared with Q-learning based resource allocation (QLRA), and the simulation shows that KTRA achieves a 18.4\\% lower delay for URLLC slice and a 30.1\\% higher throughput for eMBB slice as well as a faster convergence.\n\nThe rest of this work is organized as follows. Section \\ref{s2} presents the related work. Section \\ref{s3} introduces the system model and problem formulation, and Section \\ref{s4} defines the KTRA scheme and baseline algorithm. We show simulation results in Section \\ref{s5}, and conclude this work in Section \\ref{s6}. \n\n\n\\section{Related Work}\n\\label{s2}\nRecently machine learning techniques have been extensively studied for wireless network applications, and various techniques are proposed for resource allocation of 5G networks. \\cite{b10} proposed a reinforcement learning based method for joint power and radio resource allocation for URLLC and eMBB users. A Q-learning based solution is presented in \\cite{b11} to maximize the network utility by satisfying network slicing requests under network resources constraints. \\cite{b12} introduced a RAN slicing method to dynamically allocate radio and computation resources, and a constrained learning scheme is defined. A decentralized deep reinforcement learning method is presented in \\cite{b13} for network slicing, which ensures service level agreements under networking and computation resources constraints. The joint RAN slicing and computation offloading problem is investigated in \\cite{b14}, and multi-agent deep Q-learning is applied to maximize the communication and computation resources utilization. Furthermore, \\cite{b14-1} proposed a actor-critic network based deep reinforcement learning method for joint radio and computation resource allocation of virtualized RAN. \n\nAlthough various machine learning methods have been proposed for resource allocation of 5G, including reinforcement learning\\cite{b10,b11,b12}, deep reinforcement learning\\cite{b13,b14-1}, and multi-agent deep reinforcement learning\\cite{b14}, these methods usually require a huge amount of samples for training, which means a long exploration phase. Deep reinforcement learning is considered as a breakthrough, but the time-consuming network training is a well known issue \\cite{b15}. We propose a correlated Q-learning based method for radio resource allocation of network slicing in \\cite{b16}, but the knowledge transfer capability is still not considered. To this end, we propose a KTRA scheme for the joint radio and computation resources allocation of 5G networks in this work. Based on the knowledge transfer strategy, our proposed method can utilize the prior knowledge of expert agent to achieve a faster convergence speed or a higher average reward. \n\n\n\\section{System Model and Problem Formulation}\n\\label{s3}\n\\subsection{Network Architecture}\n\nThe proposed system model is shown as Fig.\\ref{fig1}. We assume the base station (BS) is equipped with a MEC server to process computation tasks of eMBB and URLLC slices. We apply a two-step resource allocation scheme. In the inter-slice phase, the BS intelligently allocates radio and computation resources between different slices. Then these resources are utilized within each slice in the intra-slice phase. For example, the URLLC slice operator will distribute radio resource between attached UEs, and use available MEC server capacity to process the computation tasks of its UEs. In this work, we mainly focus on the inter-slice resource allocation. Given this architecture, the delay experienced by a computation task is:\n\\begin{equation} \\label{eq1}\nd=d^{tx}+d^{rtx}+d^{que}+\\alpha d^{edge}+(1-\\alpha)d^{cloud},\n\\end{equation}\nwhere $d^{tx}$ and $d^{rtx}$ are the task transmission and retransmission delays, respectively. $d^{que}$ is the queuing delay in BS, which is considered as the scheduling delay. $d^{edge}$ is the processing delay in MEC server, and $d^{cloud}$ is the processing delay in central cloud. $\\alpha$ is a binary variable. $\\alpha=1$ if the task is computed in the BS, and $\\alpha=0$ means the task is forwarded to the central cloud. We presume the intelligent BS will decide whether to process the task in MEC server or offload it to the cloud. Eq. (\\ref{eq1}) shows that the delay is affected by both radio and computation resources. The radio resource allocation will affect the transmission delay $d^{tx}$, and computation resource allocation can change the processing delay $d^{edge}$. Meanwhile, the scheduling efficiency will affect the queuing delay $d^{que}$. Following we will explain the communication and computation model. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=9cm,height=7cm]{f1.jpg}\n\\caption{Proposed system architecture.}\n\\label{fig1}\n\\vspace{-10pt}\n\\end{figure}\n\n\\subsection{Communication Model}\nWe consider resource blocks (RBs) as the smallest time-frequency resource that is distributed to users. The transmission delay $d^{tx}$ between BS and user equipment (UE) is:\n\\begin{equation} \\label{eq2}\nd^{tx}=\\frac{L_{u}}{E_{j,u}},\n\\end{equation}\nwhere $L_{u}$ is the transmitted packet size of UE $u$, $E_{j,u}$ is the link capacity between the BS $j$ and the UE $u$. The link capacity depends on the number of RBs that are allocated to this transmission:\n\\begin{equation}\n\\resizebox{0.9\\hsize}{!}{$\\begin{split}\n \\label{eq3}\nE_{j,u}=&\\sum _{r\\in{\\mathcal{N}_{u}}}b_{RB}log(1+\\\\ &\\frac{p_{j,r}x_{j,u,r}g_{j,u,r}}{b^{RB}N_{0}+\\sum\\limits_{j'\\in \\mathcal{J}_{-j}}\\sum\\limits_{u'\\in \\mathcal{U}_{j'}}\\sum\\limits_{r'\\in \\mathcal{N}_{j'}}{p_{j',r'}x_{j',u',r'}g_{j',u',r'}}}),\n\\end{split}$}\n\\end{equation}\nwhere $\\mathcal{N}_{u}$ denotes the set of RBs that is allocated to UE $u$, $b_{RB}$ denotes the bandwidth of one RB, $N_{0}$ denotes the noise power density, $p_{j,r}$ denotes the transmission power of RB $r$ in BS $j$. $x_{j,u,r}$ is a binary variable. $x_{j,u,r}=1$ means the RB $r$ is allocated to UE $u$; otherwise $x_{j,u,r}=0$. $g_{j,u,r}$ denotes the channel gain between BS $j$ and UE $u$. $\\mathcal{J}_{-j}$ denotes the set of BSs except $j^{th}$ BS, $\\mathcal{U}_{j'}$ denotes the set of UEs in BS $j'$, and $\\mathcal{N}_{j'}$ denotes the set of total RBs in BS $j'$. \n\n\\subsection{Computation Model}\nFor a computation task from UEs, we assume it requires certain computation resources to complete the task (denoted by number of CPU cycles). Then the processing delay in MEC server $d^{edge}$ is:\n\\begin{equation} \\label{eq4}\nd^{edge}=\\frac{c_{u,q}}{\\beta C_{j}},\n\\end{equation}\nwhere $c_{u,q}$ denotes required computation resources of task $q$ from UE $u$, $\\beta$ denotes the proportion of computation resources allocated to this task ($0\\leq \\beta \\leq 1$), and $C_{j}$ denotes the total computation capacity of MEC server in BS $j$. \n\nOn the other hand, BS may decide to offload the task to central cloud computation servers. The $d^{cloud}$ in Eq. (\\ref{eq1}) is described as:\n\\begin{equation} \\label{eq4-1}\nd^{cloud}=d^{up}+d^{down}+d^{c,que}+d^{c,computation}, \n\\end{equation}\nwhere $d^{up}$ and $d^{down}$ are upload and download transmission delay of the computation task, respectively. $d^{c,que}$ is the cloud queuing delay, and $d^{c,computation}$ is cloud computation delay. The $d^{down}$ and $d^{c,computation}$ can be omitted because: i) the downloaded packet size after computation is usually much smaller than input packet; ii) the central cloud usually has a very high computation capacity \\cite{b16-2}. Then \nEq. (\\ref{eq4-1}) can be rewritten as:\n\\begin{equation} \\label{eq5}\nd^{cloud}=\\frac{1}{\\frac{B}{L_{s}}-\\lambda}+d^{c, que},\n\\end{equation}\nwhere $B$ is the backhaul capacity, $B\/L_{u}$ denote the service rate, and $\\lambda$ is the packet arrival rate. We apply the M\/M\/1 queue model to describe the upload delay $d^{up}=\\frac{1}{\\frac{B}{L_{s}}-\\lambda}$. Meanwhile, this work mainly focuses the RAN resource allocation, and it is reasonable to assume a fixed cloud queuing delay $d^{c, que}$. Finally, we assume: i) the task is preferred to be processed in the MEC server of the BS due to the potential benefit of MEC; ii) BS will offloaded the task to central cloud if the queuing time expires the preset target delay \\cite{b5-1}. \n\n\\subsection{Problem Formulation}\n\nHere we consider two typical slices: eMBB and URLLC slices. The eMBB slice intends to maximize the throughput, while the URLLC slice requires a lower latency. The intelligent BS needs to balance the requirements of both slices, then we define the problem formulation as following: \n\\begin{subequations}\\label{e2:main}\n\\begin{align}\n\\text{max} \\qquad & w^{embb}b^{embb,avg}_{j}+w^{urllc}(d^{tar}-d^{urllc,avg}_{j})& \\tag{\\ref{e2:main}} \\\\\n \\text{s.t.} \\qquad & b^{embb,avg}_{j}=\\frac{\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}b^{embb}_{j,u}}{|\\mathcal{M}^{embb}_{j}|} & \\label{e2:a} \\\\\n& b^{urllc,avg}_{j}=\\frac{\\sum\\limits_{v\\in\\mathcal{M}^{urllc}_{j}}d^{urllc}_{j,v}}{|\\mathcal{M}^{urllc}_{j}|} & \\label{e2:b} \\\\\n & (\\ref{eq1})\\, (\\ref{eq2})\\, (\\ref{eq3}) \\, (\\ref{eq4})\\, (\\ref{eq5}) & \\label{e2:c} \\\\\n&\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}{x_{j,u,r'}}+\\sum\\limits_{v\\in \\mathcal{M}^{urllc}_{j}}{x_{j,v,r'}}=1 & \\label{e2:d}\\\\\n \\sum\\limits_{r'\\in \\mathcal{N}_{j'}} & (\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}{x_{j,u,r'}}+\\sum\\limits_{v\\in \\mathcal{M}^{urllc}_{j}}{x_{j,v,r'}})\\leq |\\mathcal{N}_{j}|& \\label{e2:e}\\\\\n& C^{embb}_{j}+C^{urllc}_{j}\\leq C_{j} & \\label{e2:f}\n\\end{align}\n\\end{subequations}\nwhere $b^{embb,avg}_{j}$ and $d^{urllc,avg}_{j}$ denote the average throughput and latency of eMBB and URLLC slices, respectively, which are calculated by eq. (\\ref{e2:a}) and (\\ref{e2:b}). $w^{embb}$ and $w^{urllc}$ are weighting factors to balance two metrics and form a overall objective. $d^{tar}$ is the preset target delay of URLLC slice, which will bring a positive reward if $d^{tar}>d^{urllc,avg}_{j}$.\n$b^{embb}_{j,u}$ is the throughput of UE $u$ in the eMBB slice of BS $j$, and $d^{urllc}_{j,v}$ is the latency of UE $v$ in the URLLC slice. $\\mathcal{M}^{embb}_{j}$ and $\\mathcal{M}^{urllc}_{j}$ denote the UE set of eMBB and URLLC slices in BS $j$, respectively. In eq. (\\ref{e2:main}), a higher eMBB throughput and a lower URLLC latency are expected to maximize the objective. The eq. (\\ref{e2:d}) guarantees one RB can only be allocated to one UE. Eq. (\\ref{e2:e}) and (\\ref{e2:f}) mean that total allocated RBs and computation capacity should not exceed the available resources in the BS $j$. \n\n\\section{Knowledge Transfer based Radio and Computation Resource Allocation }\n\\label{s4}\nIn this section, we first introduce the proposed KTRA method, including the knowledge transfer strategy, MDP definition, and the knowledge transfer reinforcement learning. Then we introduce the Q-learning based resource allocation scheme as a baseline algorithm. \n\n\\subsection{Knowledge Transfer Strategy and MDP definition.}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm,height=5cm]{f2-1.jpg}\n\\caption{Learning strategy comparison .}\n\\label{fig2}\n\\vspace{-10pt}\n\\end{figure}\n\nFirst, we compare the knowledge transfer reinforcement learning with reinforcement learning to better explain the knowledge transfer strategy. The interaction between agent and task can be described by MDP $$, where $S$, $A$, $T$, $R$ are the set of states, set of actions, transition probability, and reward function, respectively. As shown in Fig.\\ref{fig2}, in reinforcement learning, the agent selects an action, receives reward and arrives new state. The agent starts from scratch to explore the task, since no prior knowledge is available in the learning phase.\n\nBy contrast, in knowledge transfer reinforcement learning, two agents are involved, namely learner and expert agents. Compared with single learning phase in RL, knowledge transfer reinforcement learning has two phases: knowledge transfer phase and learning phase. In knowledge transfer phase, considering different MDP definitions of two agents, a map function is needed to transform the expert agent's knowledge to the leaner agent. In the learning phase, learner agent utilizes the knowledge of expert agent to improve its own performance on target task. Learner agent is expected to achieve a higher exploration efficiency, since it already has some prior knowledge at the beginning of exploration. \n\nIn this work, we define a learner agent for joint radio and computation resources allocation, and an expert agent for radio resource allocation. The expert agent has no knowledge for computation resource allocation, but it's knowledge of radio resource allocation can be used by learner agent. Following we will define the state, action and reward of two agents. \n\n\\begin{itemize}\n \\item \\textbf{State}: The states of both expert and learner agents are $(q^{embb},q^{urllc})$. $q^{embb}$ denotes the number of tasks in the queue of eMBB slice, and $q^{urllc}$ is defined similarly. $(q^{embb},q^{urllc})$ represents the demands of two slices, and agent can select actions accordingly. \n \\item \\textbf{Action}: For the expert agent, it only implements the radio resource allocation, and then the action is defined as $(r^{embb},r^{urllc})$, which represents the number of radio resources allocated to eMBB and URLLC slices. For the learner agent, it considers both radio and computation resource allocation, and then the action is defined as $(r^{embb},r^{urllc},c^{embb},c^{urllc})$, where $c^{embb}$ and $c^{urllc}$ denote the computation capacity allocated to eMBB and URLLC slices, respectively. \n \\item \\textbf{Reward}: The reward is defined by the objective of slices, which is denoted by (\\ref{e2:main}). We also apply a penalty to guarantee the packet drop rate. \n\\end{itemize}\n\n\\subsection{Knowledge Transfer Reinforcement Learning}\nHere we will introduce the knowledge transfer reinforcement learning and define the map function for knowledge transfer. In reinforcement learning, to maximize the expected reward, the agent needs to improve its policy $\\pi(s_{t})$ to select the best action at state $s_{t}$, and arrives a new state $s_{t+1}$. Then the state value is defined to represent the potential reward of arriving a new state:\n\\begin{equation} \\label{eu_eqn}\nV_{\\pi}(s_{t+1}) =\\mathbb{E}_{\\pi}(\\sum_{n=0}^{\\infty}\\gamma^{n} r_{t+1}|s=s_{t+1}),\n\\end{equation}\nwhere $V_{\\pi}(s_{t+1})$ is the state value to describe the expected reward if the agent arrives $s_{t+1}$, $r_{t+1}$ is the reward at time $t+1$, and $\\gamma$ is the discount factor $(0<\\gamma<1)$. Furthermore, we need to define the state-action value to describe the expected reward of taking action $a_{t}$ under state $s_{t}$. In Q-learning, the Q-values are updated by:\n\\begin{equation} \\label{eq7}\n\\begin{aligned}\nQ^{new}(s_{t},a_{t}) &= Q^{old}(s_{t},a_{t})+\\\\\n&\\alpha(r+\\gamma \\max\\limits_{a} Q(s_{t+1},a)-Q^{old}(s_{t},a_{t})),\n\\end{aligned}\n\\end{equation}\nwhere $Q^{old}$ and $Q^{old}$ denote old and new Q-values, respectively, $a_{t}$ is the action at time $t$, $r$ is the reward, and $\\alpha$ is the learning rate ($0< \\alpha <1$). \n\n\\begin{algorithm}[!t]\n\t\\caption{KTRA algorithm}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Initialize:} Wireless network parameters and Q-table of the expert.\n\t\t\\FOR{$TTI=1$ to $T$}\n\t\t\\FOR{Every BS}\n\t\t\\STATE With probability $\\epsilon$ choose actions randomly, otherwise select $a_{l,t}$ by greedy policy.\n\t\t\\STATE BS allocates radio and computation resource between slices. Slices process computation tasks by allocated MEC server capacity, and allocate RBs by proportional fairness algorithm. \n\t\t\\STATE Calculating reward based on received metrics. \n\t\t\\STATE Updating state $s_{l,t}$, find $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$.\n\t\t\\STATE Updating Q-values by eq.(\\ref{eq8}).\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[!t]\n\t\\caption{QLRA algorithm}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Initialize:} Wireless network parameters. \n\t\t\\FOR{$TTI=1$ to $T$}\n\t\t\\FOR{Every BS}\n\t\t\\STATE With probability $\\epsilon$ choose actions randomly, otherwise select $a^{l,t}$ by greedy policy.\n\t \\STATE BS allocates radio and computation resource between slices. Slices process computation tasks by allocated MEC server capacity, and allocate RBs by proportional fairness algorithm.\n\t\t\\STATE Calculating reward based on received metrics. \n\t\t\\STATE Updating state $s_{t}$.\n\t\t\\STATE Updating Q-values by eq.(\\ref{eq7}).\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn Q-learning, the agent needs to explore the state-action space to find the optimal action sequence. This exploration usually takes a large number of iterations, because the agent needs to try huge amounts of action combinations without any prior knowledge. On the contrary, in knowledge transfer reinforcement learning, we use the prior knowledge of experts to improve the exploration efficiency\\cite{b17}. The Q-values are updated by:\n\\begin{equation} \\label{eq8}\n\\resizebox{0.89\\hsize}{!}{$\\begin{aligned}\nQ^{new}(s_{l,t},a_{l,t})= &Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))+Q^{old}(s_{l,t},a_{l,t})+\\\\\n&\\alpha(r+\\gamma \\max\\limits_{a} Q(s_{l,t+1},a)-Q^{old}(s_{l,t},a_{l,t})),\n\\end{aligned}$}\n\\end{equation}\nwhere $s_{l,t}$ and $a_{l,t}$ are learner's state and action at time $t$, respectively. $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$ is the mapped Q-values as an extra reward of selecting $a_{l,t}$ under state $s_{l,t}$, which aims to guide the exploration of learner agent. $\\mathcal{F}$ and $\\mathcal{F'}$ are state and action map functions, respectively. $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$ can be generated by:\n\\begin{equation} \\label{eq9}\n\\begin{aligned}\nQ^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))= Q^{e}(s_{e},a_{e}),\n\\end{aligned}\n\\end{equation}\nwhere $Q^{e}$ is the Q-values of expert agent, $s_{e}$ and $a_{e}$ are state and action of expert agent. The goal of eq. (\\ref{eq9}) is to find a specific Q-value of expert agent to represent the potential reward of taking $a_{l,t}$ under $s_{l,t}$ in learner. Thus we need to find similar states and actions in expert agent's Q-table. Note that the expert and learner agents have the same state definition, then we can always find a $s_{e}$ that satisfy $s_{l,t}=s_{e}$, then $\\mathcal{F}$ can be easily defined. Meanwhile, based on the definition of actions, for any $a_{l}=(r^{eM},r^{UR},c^{eM},c^{UR})$, we consider $a_{e}=(r^{eM},r^{UR})$ as a similar action for mapping, and $\\mathcal{F'}$ can be defined accordingly.\n\nThe KTRA algorithm is summarized as Algorithm 1. Noting that we apply a classic proportional fairness algorithm for intra-slice radio resource allocation, because this work mainly focus on inter-slice level scheduling \\cite{b16}, and there is no intra-slice computation resource allocation. \n\n\n\\subsection{Baseline Algorithms: QLRA}\nWe apply QLRA as a baseline algorithm. Q-learning is the most generally applied reinforcement learning algorithm. The MDP definition of QLRA is the same with KTRA, but there is no prior knowledge. QLRA is given in Algorithm 2. \n\n\\section{Performance Evaluation}\n\\label{s5}\n\\subsection{Parameter Settings}\nWe consider three different cases, including:\n\n\\begin{itemize}\n \\item \\textbf{Case I}: Q-learning based radio resource allocation. It works as an expert for the Case II. \n \\item \\textbf{Case II}: KTRA based joint radio and computation resource allocation. As a learner, BSs in Case II can utilize the Q-tables of expert agent as prior knowledge. \n \\item \\textbf{Case III}: QLRA based joint radio and computation resource allocation. It is considered as a baseline algorithm without any prior knowledge for the task. \n\\end{itemize}\n\nEach case contains 5 BSs with 500 m inter-site distance, and each BS is considered as an independent agent to implement the proposed strategy. For example, all BSs in Case II will implement the KTRA independently to achieve its own goal. We assume each BS has one eMBB slice with 5 UEs and one URLLC slice with 10UEs. The avaliable bandwidth of one BS is 20 MHz, which contains 100 RBs. We assume there are 13 resource block groups (RBGs) to reduce the allocation complexity. The first 12 RBGs contains 8 RBs each, while the last RBG has 4 RBs. 200 CPU cycles are required to process 1 bit data \\cite{b12}. Other parameters are shown as Table \\ref{tab2}.\n\n\\begin{table}[!t]\n\\vspace{-5pt}\n\\caption{Parameters Settings}\n\\centering\n\\renewcommand\\arraystretch{1.4}\n\\begin{tabular}{|p{4cm}<{\\centering}||p{3.8cm}<{\\centering}|}\n\\hline\n \\textbf{5G Networking} & \\textbf{Computation Settings}\\\\\n\\hline\n3GPP Urban Macro network & Computation capacity: 3 GHz\\\\ \n 2 OFDM symbols for each TTI & CPU cycles required per bit: 200\\\\\n\\cline{2-2}\n Tx\/Rx antenna gain: 15 dB. & \\textbf{Traffic Model} \\\\\n\\cline{2-2}\n \\quad \\, Number of subcarriers \\quad \\quad \\, in each RB: 12 & URLLC$\\backslash$eMBB traffic: Poisson distribution\\\\\n Subcarrier bandwidth: 15kHz & URLLC packet size: 50 Bytes \\\\ \n Transmission power: 40 dBm & eMBB packet size: 100 Bytes \\\\ \n \\cline{2-2}\n Backhaul capacity: 10 Mpbs& \\textbf{Problem Formulation}\\\\\n\\cline{1-2}\n \\textbf{Propagation Model} & eMBB$\\backslash$URLLC weight factor: 1 \\\\\n \\cline{1-1}\n 128.1+37.6log(distance(km)) & URLLC target delay: 2 ms \\\\\nLog-Normal shadowing: 8 dB.& Fixed cloud queuing delay: 1 ms \\\\\n \\cline{1-2}\n \\textbf{Retransmission Settings} & \\textbf{Learning Settings}\\\\\n\\cline{1-2}\nMax number of retransmissions: 1 & Learning rate: 0.9 \\\\\nRound trip delay: 4 TTIs & Discount factor: 0.5 \\\\\nProtocol: asynchronous HARQ. & Epsilon value: 0.05 \\\\\n\\hline\n\\end{tabular}\n\\label{tab2}\n\\vspace{-10pt}\n\\end{table}\n\n\\begin{figure*}[t!]\n\\centering\n\\label{f3}\n\\vspace{0pt}\n\\subfigure[ URLLC latency distribution \\text{[ms]} under 2 Mbps eMBB and URLLC traffic, and 3 GHz MEC server per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig4.jpg}\n}\n\\quad\n\\subfigure[ Average URLLC latency \\text{[ms]} under various URLLC traffic, 2 Mbps eMBB traffic and 3 GHz MEC server per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig5.jpg}\n}\n\\quad\n\\subfigure[eMBB throughput \\text{[Mbps]} per cell under various URLLC traffic, 2 Mbps eMBB traffic and 3 GHz MEC server.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig6.jpg}\n}\n\\quad\n\\subfigure[Average URLLC latency \\text{[ms]} under various MEC server capacities, 2 Mbps eMBB and URLLC traffic per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig7.jpg}\n}\n\\quad\n\\subfigure[eMBB throughput \\text{[Mbps]} per cell under various MEC server capacities, 2 Mbps eMBB and URLLC traffic.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig8.jpg}\n}\n\\quad\n\\subfigure[Convergence performance of KTRA and QLRA under 2 Mbps eMBB and URLLC traffic, and 2 GHz computation capacity per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig9.jpg}\n}\n\\caption{Simulation results comparison.}\n\\label{f3}\n\\end{figure*}\n\n\n\\subsection{Simulation Results}\n\nFirstly, Fig.\\ref{f3} (a) shows the empirical cumulative distribution function (ECDF) of URLLC latency of three cases under 2 Mbps eMBB and URLLC traffic, and 3 GHz MEC server per cell. The result shows that expert case has the highest delay distribution, and the reason is that we assume MEC servers are not deployed in expert. Thus all computation tasks need to be processed in the central cloud, which leads to a higher delay. Meanwhile, the proposed KTRA method outperforms QLRA by a better delay distribution, as indicated by a lower ECDF curve in Fig.\\ref{f3} (a), and it can be explained by the knowledge transfer capability of KTRA. \n\nFig.\\ref{f3} (b) presents the average delay experienced by URLLC UEs, and Fig.\\ref{f3} (c) shows the average eMBB throughput per cell. Here the eMBB traffic load is fixed to 2 Mbps per cell, and URLLC traffic varies from 1 Mbps to 4 Mbps. Without MEC servers, expert case still has the highest delay and the lowest throughput. Meanwhile, KTRA method achieves a lower URLLC delay and higher eMBB throughput than QLRA. KTRA has a 18.4\\% lower URLLC delay and 30.1\\% higher eMBB throughput under 2 Mbps URLLC traffic. \n\nShown by Fig.\\ref{f3} (d) and (e), we investigate the network performance under various MEC server capacities, which is indicated by CPU cycles per second. Considering expert case has no MEC capability, we focus on the performance of KTRA and QLRA to further present the advantage of proposed method. As expected, both algorithms have a lower URLLC delay and a higher eMBB throughput by increasing the MEC server capacity, because higher computation capacity means lower task processing delay. KTRA still outperforms QLRA by a 15.1\\% lower URLLC delay and a 33.8\\% higher eMBB throughput under 3 GHz MEC server capacity. \n\nFurthermore, we compare the convergence performance in Fig.\\ref{f3} (f). Based on prior knowledge of expert, KTRA has a significantly higher exploration efficiency, which is indicated by a shorter exploration period and a higher average reward. On the contrary, QLRA suffers a longer exploration phase and a lower average reward, because it needs to explore the task from scratch. To summarize, KTRA achieves a better performance in both network metrics (higher URLLC delay and lower eMBB throughput) and machine learning metrics (better convergence speed and higher average reward). \n\nFinally, the packet drop rate of KTRA and QLRA are 0.042\\% and 0.048\\%, respectively, under 2 Mpbs eMBB traffic and 4 Mbps URLLC traffic. The satisfying packet drop rate is because we apply a penalty in both algorithms to prevent dropping packet. \n\n\\section{Conclusion}\n\\label{s6}\nThe evolving network architecture requires more efficient solutions for network resource allocation. In this work, we propose a KTRA method for joint radio and computation resource allocation. Compared with existing works, the main difference is that the proposed method has a knowledge transfer capability. The proposed KTRA method is compared with Q-learning based resource allocation, and KTRA presents a 18.4\\% lower URLLC delay and 30.1\\% higher eMBB throughput as well as a faster convergence. In the future, we will consider the knowledge transfer of tasks with different state definition.\n\n\\section*{Acknowledgment}\nThis work is supported by Natural Sciences and Engineering Research Council of Canada (NSERC), Collaborative Research and Training Experience Program (CREATE) under Grant 497981 and Canada Research Chairs Program.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe ongoing technological progress in the fabrication and\ncontrol of nanoscale electronic circuits, such as quantum dots,\nhas stimulated detailed studies of various quantum-impurity\nmodels, where a few local degrees of freedom are coupled to a\ncontinuum. Of particular interest are models with\nexperimentally verifiable universal properties. One of the best\nstudied examples is the Anderson single impurity\nmodel,~\\cite{Anderson61} which describes successfully\nelectronic correlations in small quantum\ndots~\\cite{NgLee88,GlazmanRaikh88}. The experimental control of\nmost of the parameters of this model, e.g., the impurity energy\nlevel position or the level broadening due to hybridization\nwith the continuum, allows for detailed\ninvestigations~\\cite{DGG98,vanderWiel00} of the universal\nlow-temperature behavior of the Anderson model.\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{fig1.eps}\n\\caption{A schematic representation of the double-dot system,\n along with its reduction in the local-moment regime\n to an effective Kondo model with a tilted magnetic\n field.\n (a) The model system: two localized levels coupled\n by tunnelling matrix elements to one another and\n to two separate leads. A constant magnetic flux\n induces phase factors on those elements. Spinless\n electrons residing on the two levels experience a\n repulsive interaction.\n (b) The mapping onto a spinful generalized Anderson\n model, with a tilted magnetic field and different\n tunnelling elements for spin-up and spin-down\n electrons.\n (c) The low-energy behavior of the generalized\n Anderson model is mapped onto an anisotropic Kondo\n model with a tilted magnetic field,\n $\\vec{h}_{\\text{tot}}$.\n} \\label{fig:models}\n\\end{figure*}\n\nIn this paper we study the low-energy behavior of a generic\nmodel, depicted in Fig.~\\ref{fig:models}a, which pertains\neither to a single two-level quantum dot or to a double quantum\ndot where each dot harbors only a single level. The spin\ndegeneracy of the electrons is assumed to be lifted by an\nexternal magnetic field. Several variants of this model have\nbeen studied intensely in recent years, in conjunction with a\nplethora of phenomena, such as many-body resonances in the\nspectral density,~\\cite{Boese01} phase lapses in the\ntransmission phase,~\\cite{Silva02,Golosov06} charge\noscillations,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances in the conductance~\\cite{Meden06PRL,Karrasch06}.\nAlbeit being described by the same model, no clear linkage has\nbeen established between these seemingly different effects. The\nreason is in part due to the large number of model parameters\ninvolved, which so far obscured a clear physical picture. While\nsome exact statements can be made, these are restricted to\ncertain solvable limits,~\\cite{Boese01} and are apparently\nnongeneric~\\cite{Meden06PRL}. Here we construct a framework\nwhich encompasses all parameter regimes of the model, and\nenables a unified description of the various phenomena alluded\nto above, exposing their common physical origin. For the most\ninteresting regime of strong fluctuations between the two\nlevels, we are able to give: (i) explicit analytical\nconditions for the\n occurrence of transmission phase lapses;\n(ii) an explanation of the population inversion and the\n charge oscillations~\\cite{Gefen04,Sindel05,Silvestrov00}\n (including a Kondo enhancement of the latter);\n(iii) a complete account of the correlation-induced\n resonances~\\cite{Meden06PRL} as a disguised Kondo phenomenon.\n\nAfter introducing the details of the double-dot Hamiltonian in\nSec.~\\ref{sec:Model}, we begin our analysis by constructing a\nlinear transformation of the dot operators, \\emph{and} a\nsimultaneous (generally different) linear transformation of the\nlead operators, such that the 2$\\times$2 tunnelling matrix\nbetween the two levels on the dot and the leads becomes\ndiagonal (with generally different eigenvalues). As a result,\nthe electrons acquire a pseudo-spin degree of freedom which is\nconserved upon tunnelling between the dot and the continuum, as\nshown schematically in Fig.~\\ref{fig:models}b. Concomitantly,\nthe transformation generates a local Zeeman magnetic field. In\nthis way the original double-dot model system is transformed\ninto a generalized Anderson impurity model in the presence of a\n(generally tilted) external magnetic field. This first stage is\ndetailed in Sec.~\\ref{sec:ModelAnderson} and\nAppendix~\\ref{App:SVDdetails}.\n\nWe next analyze in Sec.~\\ref{sec:LocalMoment} the low-energy\nproperties of our generalized Anderson model. We confine\nourselves to the local moment regime, in which there is a\nsingle electron on the impurity. The fluctuations of the\npseudo-spin degree of freedom (which translate into charge\nfluctuations between the two localized levels in the original\nmodel) are determined by two competing effects: the polarizing\neffect of the local magnetic field, and the Kondo screening by\nthe itinerant electrons. In order to quantitatively analyze\nthis competition, we derive an effective low-energy Kondo\nHamiltonian, using Haldane's scaling\nprocedure,~\\cite{HaldanePRL78} together with the\nSchrieffer-Wolff~\\cite{Wolff66} transformation and Anderson's\npoor man's scaling~\\cite{Anderson70}. This portion of the\nderivation resembles recent studies of the Kondo effect in the\npresence of ferromagnetic leads,~\\cite{Martinek03PRL} although\nthe physical context and implications are quite different.\n\nAs is mentioned above, the tunnelling between the impurity and\nthe continuum in the generalized Anderson model is (pseudo)\nspin dependent. This asymmetry results in two important\neffects:\n(a) different renormalizations of the two local levels,\n which in turn generates an additional local magnetic\n field~\\cite{Martinek03PRL}. This field is not necessarily\n aligned with the original Zeeman field that is present\n in the generalized Anderson model.\n(b) An anisotropy of the exchange coupling between the\n conduction electrons and the local moment in the Kondo\n Hamiltonian.\nHowever, since the scaling equations for the anisotropic Kondo\nmodel~\\cite{Anderson70,AndersonYuvalHamann70} imply a flow\ntowards the \\emph{isotropic} strong coupling fixed point, the\nlow-energy behavior of the generalized Anderson model can be\nstill described in terms of two competing energy scales, the\nKondo temperature, $T^{}_{K}$, and the renormalized magnetic\nfield, $h_{\\text{tot}}$. Our two-stage mapping, double-dot\n$\\Rightarrow$ generalized Anderson model $\\Rightarrow$\nanisotropic Kondo model (see Fig.~\\ref{fig:models}), allows us\nto obtain analytic expressions for the original model\nproperties in terms of those of the Kondo model. We derive in\nSec.~\\ref{sec:observables} the occupation numbers on the two\nlocalized levels by employing the Bethe \\emph{ansatz} solution\nof the magnetization of a Kondo spin in a finite magnetic\nfield~\\cite{AndreiRMP83,WiegmannA83}. This solution also\nresults in a highly accurate expression for the conductance\nbased upon the Friedel-Langreth sum rule~\\cite{Langreth66}.\nPerhaps most importantly, it provides a single coherent picture\nfor the host of phenomena to which our model has been applied.\n\nExamples of explicit results stemming from our general analysis\nare presented in Sec.~\\ref{sec:results}. First, we consider the\ncase in which the tunnelling is isotropic, being the same for\nspin-up and spin-down electrons. Then the model is exactly\nsolvable by direct application of the Bethe \\emph{ansatz} to\nthe Anderson Hamiltonian~\\cite{WiegmannC83,WiegmannA83}. We\nsolve the resulting equations~\\cite{Okiji82,WiegmannC83}\nnumerically and obtain the occupation numbers for arbitrary\nparameter values of the model, and in particular, for arbitrary\nvalues of the local Zeeman field. By comparing with the\noccupation numbers obtained in Sec.~\\ref{sec:observables} from\nthe Kondo version of the model, we are able to test the\naccuracy of the Schrieffer-Wolff mapping onto the Kondo\nHamiltonian. We find that this mapping yields extremely precise\nresults over the entire local-moment regime. This exactly\nsolvable example has another virtue. It clearly demonstrates\nthe competition between the Kondo screening of the local spin,\nwhich is governed by $T_K$, and the polarizing effect of the\nlocal field $h_{\\text{tot}}$. This competition is reflected\nin the charging process of the quantum dot described by the\noriginal Hamiltonian. We next proceed to apply our general\nmethod to the features for which the anisotropy in the\ntunnelling is relevant, notably the transmission phase lapses\nand the correlation-induced resonances~\\cite{Meden06PRL}. In\nparticular, we derive analytical expressions for the occupation\nnumbers and the conductance employing the mapping onto the\nKondo Hamiltonian. These analytical expressions give results\nwhich are in a very good agreement with the data presented by\nMeden and Marquardt,~\\cite{Meden06PRL} which was obtained by\nthe functional and numerical renormalization-group methods\napplied to the original model.\n\nAs our treatment makes extensive usage of the exact Bethe\n\\emph{ansatz} solutions for the impurity magnetization in the\nisotropic Kondo and Anderson models with a finite magnetic\nfield, all relevant details of the solutions are concisely\ngathered for convenience in Appendix~\\ref{app:Bethe}.\n\n\\section{The double-dot system as a generalized Anderson model}\n\\label{sec:secII}\n\n\\subsection{The model\\label{sec:Model}}\n\nWe consider spinless electrons in a system of two distinct\nenergy levels (a `quantum dot'), labelled $i = 1, 2$, which are\nconnected by tunnelling to two leads, labelled $\\alpha = L, R$.\nThis quantum dot is penetrated by a (constant) magnetic flux.\nThe total Hamiltonian of the system reads\n\\begin{eqnarray}\n\\mathcal{H} = \\mathcal{H}_l + \\mathcal{H}_d + \\mathcal{H}_{ld} \\, ,\n\\label{IHAM}\n\\end{eqnarray}\nin which $\\mathcal{H}_{l}$ is the Hamiltonian of the leads,\n$\\mathcal{H}_{d}$ is the Hamiltonian of the isolated dot, and\n$\\mathcal{H}_{ld}$ describes the coupling between the dot and\nthe leads. The system is portrayed schematically in\nFig.~\\ref{fig:models}a.\n\nEach of the leads is modelled by a continuum of noninteracting\nenergy levels lying within a band of width $2D$, with a\nconstant density of states $\\rho$~\\cite{Comm-on-equal-rho}. The\ncorresponding Hamiltonian is given by\n\\begin{eqnarray}\n\\mathcal{H}_l = \\sum_{ k\\alpha} \\varepsilon^{}_{k}\n c_{k\\alpha}^{\\dagger} c^{}_{k\\alpha} \\, ,\n\\end{eqnarray}\nwhere $c^{\\dagger}_{ k\\alpha}$ ($c^{}_{ k\\alpha}$) creates\n(annihilates) an electron of wave vector $k$ on lead $\\alpha$.\nThe two leads are connected to two external reservoirs, held at\nthe same temperature $T$ and having different chemical\npotentials, $\\mu_L$ and $\\mu_R$, respectively. We take the\nlimit $\\mu_L \\!\\! \\to \\!\\! \\mu_R = 0$ in considering\nequilibrium properties and the linear conductance.\n\n\nThe isolated dot is described by the Hamiltonian\n\\begin{equation}\n\\mathcal{H}_d = \\left [\n \\begin{array}{cc}\n d^{\\dagger}_{1} & d^{\\dagger}_{2}\n \\end{array}\n \\right ] \\cdot \\hat{\\mathcal{E}}_d \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ] + U \\, n_{1} n_{2} \\, ,\n\\label{HDOT}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\hat{\\mathcal{E}}_{d} = \\frac{1}{2}\n \\left [\n \\begin{array}{cc}\n 2 \\, \\epsilon_0 + \\Delta &\n b \\, e^{i(\\varphi_{L}-\\varphi_{R})\/2} \\\\\n b \\, e^{-i(\\varphi_{L}-\\varphi_{R})\/2} &\n 2 \\, \\epsilon_0 -\\Delta\n \\end{array}\n \\right ] \\, .\n\\label{HPS}\n\\end{eqnarray}\nHere, $d^{\\dagger}_{i}$ ($d^{}_{i}$) creates (annihilates) an\nelectron on the $i$th level, $n_i \\equiv d^{\\dagger}_i d^{}_i$\nare the occupation-number operators (representing the local\ncharge), $U>0$ denotes the Coulomb repulsion between electrons\nthat occupy the two levels, $\\epsilon_{0} \\pm \\Delta \/2$ are\nthe (single-particle) energies on the levels, and $b\/2$ is the\namplitude for tunnelling between them. The phases $\\varphi_{L}$\nand $\\varphi_{R}$, respectively, represent the Aharonov-Bohm\nfluxes (measured in units of the flux quantum $2 \\pi \\hbar c\/e$)\nin the left and in the right hopping loops, such that the total\nflux in the two loops is $\\varphi \\equiv \\varphi_{L} +\n\\varphi_{R}$ [see Fig.~\\ref{fig:models}a].\n\nGauge invariance grants us the freedom to distribute the\nAharonov-Bohm phases among the inter-dot coupling $b$ and the\ncouplings between the dot levels and the leads. With the\nconvention of Eq.~(\\ref{HPS}), the coupling between the quantum\ndot and the leads is described by the Hamiltonian\n\\begin{eqnarray}\n\\mathcal{H}_{ld} = \\sum_{k}\n \\left [\n \\begin{array}{cc}\n c^{\\dagger}_{kL} & c^{\\dagger}_{kR}\n \\end{array}\n \\right ] \\cdot \\hat{A} \\cdot\n \\left [\n \\begin{array}{c}\n d_1 \\\\ d_2\n \\end{array}\n \\right ] + \\text{H.c.} \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\hat{A} =\n \\left [\n \\begin{array}{cc}\n a_{L1}e^{i\\varphi \/2} & a_{L2}\\\\\n a_{R1} & a_{R2}e^{i\\varphi \/2}\n \\end{array}\n \\right ]\\, , \\quad\n \\varphi = \\varphi_{L} + \\varphi_{R} \\, .\n\\label{AA}\n\\end{eqnarray}\nHere the real (possibly negative) coefficients $a_{\\alpha i}$\nare the tunnelling amplitudes for transferring an electron from\nthe level $i$ to lead $\\alpha$. Note that the Hamiltonian\ndepends solely on the total Aharonov-Bohm flux $\\varphi$ when\nthe interdot coupling $b$ vanishes. Also, the tunnelling matrix\n$\\hat{A}$ is assumed to be independent of the wave vector $k$.\nThis assumption considerably simplifies the analysis while\nkeeping the main physical picture intact.\n\n\n\\subsection{Mapping onto a generalized Anderson model}\n\\label{sec:ModelAnderson}\n\nThe analysis of the model defined in Sec.~\\ref{sec:Model}\nemploys an {\\it exact} mapping of the Hamiltonian of\nEq.~(\\ref{IHAM}) onto a generalized Anderson Hamiltonian, which\npertains to a single-level quantum dot, coupled to a\nspin-degenerate band of conduction electrons. We show in\nAppendix~\\ref{App:SVDdetails} that the model depicted in\nFig.~\\ref{fig:models}a is fully described by the Hamiltonian\n\\begin{widetext}\n\\begin{equation}\n\\mathcal{H} = \\sum_{k, \\sigma}\n \\varepsilon_k \\, c_{k\\sigma}^{\\dagger} c^{}_{k\\sigma}\n + \\sum_{\\sigma}\n \\Bigl (\n \\epsilon_0 - \\sigma \\frac{h}{2} \\cos \\theta\n \\Bigr )\n n_{\\sigma}\n - \\bigl ( d_{\\uparrow}^{\\dagger} d^{}_{\\downarrow} +\n d_{\\downarrow}^{\\dagger} d^{}_{\\uparrow} \\bigr ) \\,\n \\frac{h}{2} \\, \\sin \\theta\n + U n_{\\uparrow} n_{\\downarrow}\n + \\sum_{k, \\sigma} V^{}_{\\sigma}\n \\Bigl (\n c_{k\\sigma}^{\\dagger} d^{}_{\\sigma} + \\text{H.c.}\n \\Bigr ) \\, ,\n\\label{eq:Hand}\n\\end{equation}\n\\end{widetext}\nschematically sketched Fig.~\\ref{fig:models}b, which\ngeneralizes the original Anderson model~\\cite{Anderson61} in\ntwo aspects. Firstly, it allows for spin-dependent coupling\nbetween the dot and the conduction band. A similar variant of\nthe Anderson model has recently attracted much theoretical and\nexperimental attention in connection with the Kondo effect for\nferromagnetic\nleads~\\cite{Martinek03PRL,MartinekNRGferro,Pasupathy04,MartinekPRB05,\nComment-on-FM}. Secondly, it allows for a Zeeman field whose\ndirection is inclined with respect to the ``anisotropy'' axis\n$z$. For spin-independent tunnelling, one can easily realign\nthe field along the $z$ axis by a simple rotation of the\ndifferent operators about the $y$ axis. This is no longer the\ncase once $V_{\\uparrow} \\neq V_{\\downarrow}$, which precludes the\nuse of some of the exact results available for the Anderson\nmodel. As we show below, the main effect of spin-dependent\ntunnelling is to modify the effective field seen by electrons\non the dot, by renormalizing its $z$-component.\n\nThe derivation of Eq.~(\\ref{eq:Hand}) is accomplished by a\ntransformation known as the singular-value\ndecomposition,~\\cite{Golub96} which allows one to express the\ntunnelling matrix $\\hat{A}$ in the form\n\\begin{equation}\n\\hat{A} = R_l^{\\dagger} \\cdot\n \\left [\n \\begin{array}{cc}\n V_{\\uparrow} & 0 \\\\\n 0 & V_{\\downarrow}\n \\end{array}\n \\right ] \\cdot R^{}_d \\, .\n\\end{equation}\nHere $R_{l}$ and $R_{d}$ are unitary 2$\\times$2 matrices, which\nare used to independently rotate the lead and the dot operators\naccording to\n\\begin{eqnarray}\n \\left [\n \\begin{array}{c}\n d_{\\uparrow} \\\\ d_{\\downarrow}\n \\end{array}\n \\right ] \\equiv R_{d} \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ]\\, , \\quad\n \\left [\n \\begin{array}{c}\n c_{k\\uparrow} \\\\ c_{k\\downarrow}\n \\end{array}\n \\right ] \\equiv R_{l} \\cdot\n \\left [\n \\begin{array}{c}\n c_{kL} \\\\ c_{kR}\n \\end{array}\n \\right ]\\, .\n\\label{eq:SVDdef}\n\\end{eqnarray}\nTo make contact with the conventional Anderson impurity model,\nwe have labelled the linear combinations of the original\noperators [defined through Eqs.~(\\ref{eq:SVDdef})] by the\n``spin'' index $\\sigma = \\uparrow$ ($+1$) and $\\sigma =\n\\downarrow$ ($-1$).\n\nThe transformation (\\ref{eq:SVDdef}) generalizes the one in\nwhich the \\emph{same} rotation $R$ is applied to both the dot\nand the lead operators. It is needed in the present, more\ngeneral, case since the matrix $\\hat{A}$ generically lacks an\northogonal basis of eigenvectors. The matrices $R_d$ and $R_l$\ncan always be chosen uniquely (up to a common overall phase)\nsuch that~\\cite{Comment-on-uniqueness} (a) the tunnelling\nbetween the dot and the continuum is\n diagonal in the spin basis (so that the tunnelling\n conserves the spin);\n(b) the amplitudes $V_{\\uparrow} \\ge V_{\\downarrow} \\ge 0$\n are real; and\n(c) the part of the Hamiltonian of Eq.~(\\ref{eq:Hand})\n pertaining to the dot has only real matrix elements\n with $h \\sin \\theta \\geq 0$.\nThe explicit expressions for the rotation matrices $R_d$\nand $R_l$ as well as for the model parameters appearing\nin Eq.~(\\ref{eq:Hand}) in terms of those of the original\nHamiltonian are given in Appendix~\\ref{App:SVDdetails}.\n\nIt should be emphasized that partial transformations involving\nonly one rotation matrix, either $R_d$ or $R_l$, have\npreviously been applied in this context (see, e.g.,\nRefs.~~\\onlinecite{Boese01} and~~\\onlinecite{Glazman01}).\nHowever, excluding special limits, both $R_d$ and $R_l$ are\nrequired to expose the formal connection to the Anderson model.\nA first step in this direction was recently taken by Golosov\nand Gefen~\\cite{Golosov06}, yet only on a restricted\nmanifold for the tunnelling amplitudes $a_{\\alpha i}$. In the\nfollowing section we discuss in detail the low-energy physics\nof the Hamiltonian of Eq.~(\\ref{eq:Hand}), focusing on the\nlocal-moment regime. Explicit results for the conductance and\nthe occupations of the levels are then presented in\nSecs.~\\ref{sec:observables} and \\ref{sec:results}.\n\n\n\\section{The local-moment regime\\label{sec:LocalMoment}}\n\nThere are two limits where the model of Eq.~(\\ref{IHAM}) has an\nexact solution:~\\cite{Boese01} (i) when the spin-down state is\ndecoupled in Eq.~(\\ref{eq:Hand}), i.e., when $V_{\\downarrow} =\nh\\sin \\theta = 0$; (ii) when the coupling is isotropic, i.e.,\n$V_{\\uparrow} = V_{\\downarrow}$. In the former case,\n$n_{\\downarrow}$ is conserved. The Hilbert space separates then\ninto two disconnected sectors with $n_{\\downarrow} = 0$ and\n$n_{\\downarrow} = 1$. Within each sector, the Hamiltonian can be\ndiagonalized independently as a single-particle problem. In the\nlatter case, one can always align the magnetic field $h$ along\nthe $z$ axis by a simple rotation of the different operators\nabout the $y$ axis. The model of Eq.~(\\ref{eq:Hand}) reduces\nthen to a conventional Anderson model in a magnetic field, for\nwhich an exact Bethe {\\em ansatz} solution is\navailable~\\cite{WiegmannA83}. (This special case will be\nanalyzed in great detail in Sec.~\\ref{sec:ResultsIsotropic}.)\n\n\nIn terms of the model parameters appearing in the original\nHamiltonian, the condition $V_{\\downarrow} = 0$ corresponds to\n\\begin{equation}\n |a_{L1} a_{R2}| = |a_{R1} a_{L2}|, \\ \\ \\text{and}\n \\ \\ \\varphi = \\beta \\!\\!\\!\\! \\mod 2\\pi ,\n\\label{exact-a}\n\\end{equation}\nwhereas $V_{\\uparrow} = V_{\\downarrow} = V$ corresponds to\n\\begin{equation}\n |a_{L1}| = |a_{R2}| , \\\n |a_{L2}| = |a_{R1}| , \\\n \\text{and} \\\n \\varphi = (\\pi + \\beta) \\!\\!\\!\\! \\mod 2\\pi \\, .\n\\label{exact-b}\n\\end{equation}\nHere\n\\begin{equation}\n\\beta = \\left \\{\n \\begin{array}{cc}\n 0 & \\text{if} \\ \\ a_{L1} a_{L2} a_{R1} a_{R2} > 0 \\\\ \\\\\n \\pi & {\\rm if}\\ \\ a_{L1} a_{L2} a_{R1} a_{R2} < 0\n \\end{array}\n \\right .\n\\end{equation}\nrecords the combined signs of the four coefficients\n$a_{\\alpha i}$~\\cite{Comment-on-phi}.\n\nExcluding the two cases mentioned above, no exact solutions to\nthe Hamiltonian of Eq.~(\\ref{IHAM}) are known. Nevertheless,\nwe shall argue below that the model displays generic low-energy\nphysics in the ``local-moment'' regime, corresponding to the\nKondo effect in a finite magnetic field. To this end we focus\nhereafter on $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll\n-\\epsilon_0, U + \\epsilon_0$, and derive an effective\nlow-energy Hamiltonian for general couplings. Here\n$\\Gamma_{\\sigma} = \\pi \\rho V_{\\sigma}^2$ is half the\ntunnelling rate between the spin state $\\sigma$ and the leads.\n\n\n\\subsection{Effective low-energy Hamiltonian}\n\nAs is mentioned above, when $V_{\\uparrow} = V_{\\downarrow}$ one\nis left with a conventional Kondo effect in the presence of a\nfinite magnetic field. Asymmetry in the couplings,\n$V_{\\uparrow} \\neq V_{\\downarrow}$, changes this situation in\nthree aspects. Firstly, the effective magnetic field seen by\nelectrons on the dot is modified, acquiring a renormalized\n$z$-component. Secondly, the elimination of the charge\nfluctuations by means of a Schrieffer-Wolff\ntransformation,~\\cite{Wolff66} results in an anisotropic\nspin-exchange interaction. Thirdly, a new interaction term is\nproduced, coupling the spin and the charge. Similar aspects\nhave been previously discussed in the context of the Kondo\neffect in the presence of ferromagnetic\nleads,~\\cite{Martinek03PRL} where the source of the asymmetry\nis the inequivalent density of states for conduction electrons\nwith opposite spin~\\cite{Comment-on-FM}. Below we elaborate on\nthe emergence of these features in the present case.\n\nBefore turning to a detailed derivation of the effective\nlow-energy Hamiltonian, we briefly comment on the physical\norigin of the modified magnetic field. As is well known, the\ncoupling to the continuum renormalizes the bare energy levels\nof the dot. For $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll\n-\\epsilon_0, U + \\epsilon_0$, these renormalizations can be\naccurately estimated using second-order perturbation theory in\n$V_{\\sigma}$. For $V_{\\uparrow} \\neq V_{\\downarrow}$, each of\nthe bare levels $\\epsilon_{\\sigma} = \\epsilon_0 - \\frac{1}{2}\n\\sigma h \\cos \\theta$ is shifted by a different amount, which\nacts in effect as an excess magnetic field. Explicitly, for $T\n= 0$ and $D \\gg |\\epsilon_0|, U$ one\nobtains~\\cite{Martinek03PRL,Silvestrov00}\n\\begin{equation}\n\\Delta h_z =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{\\epsilon_0 + U}{|\\epsilon_0|} \\, .\n\\label{Delta-h_z}\n\\end{equation}\nAs $\\epsilon_0$ is swept across $-U\/2$, $\\Delta h_z \\propto\n\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}$ changes sign. Had\n$|\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}|$ exceeded $h$ this\nwould have dictated a sign-reversal of the $z$-component of the\ncombined field as $\\epsilon_0$ is tuned across the\nCoulomb-blockade valley. As originally noted by Silvestrov and\nImry,~\\cite{Silvestrov00} this simple but insightful\nobservation underlies the population inversion discussed in\nRefs.~\\onlinecite{Gefen04,Sindel05}\nand~\\onlinecite{Silvestrov00} for a singly occupied dot. We\nshall return to this important point in greater detail later\non.\n\n\nA systematic derivation of the effective low-energy Hamiltonian\nfor $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll -\\epsilon_0, U\n+ \\epsilon_0$ involves the combination of Anderson's poor-man's\nscaling~\\cite{Anderson70} and the Schrieffer-Wolff\ntransformation~\\cite{Wolff66}. For $|\\epsilon_0| \\sim U +\n\\epsilon_0$, the elimination of high-energy excitations\nproceeds in three steps. First Haldane's perturbative scaling\napproach~\\cite{HaldanePRL78} is applied to progressively reduce\nthe bandwidth from its bare value $D$ down to $D_{\\rm SW} \\sim\n|\\epsilon_0| \\sim U + \\epsilon_0$. Next a Schrieffer-Wolff\ntransformation is carried out to eliminate charge fluctuations\non the dot. At the conclusion of this second step one is left\nwith a generalized Kondo Hamiltonian [Eq.~(\\ref{H-Kondo})\nbelow], featuring an anisotropic spin-exchange interaction and\nan additional interaction term that couples spin and charge.\nThe Kondo Hamiltonian also includes a finite magnetic field\nwhose direction is inclined with respect to the anisotropy axis\n$z$. In the third and final stage, the Kondo Hamiltonian is\ntreated using Anderson's poor-man's scaling~\\cite{Anderson70}\nto expose its low-energy physics.\n\nThe above procedure is further complicated in the case where\n$|\\epsilon_0|$ and $U + \\epsilon_0$ are well separated in\nenergy. This situation requires two distinct Schrieffer-Wolff\ntransformations: one at $D_{\\rm SW}^{\\rm up} \\sim \\max \\{\n|\\epsilon_0|, U + \\epsilon_0\\}$ and the other at $D_{\\rm\nSW}^{\\rm down} \\sim \\min \\{ |\\epsilon_0|, U + \\epsilon_0\\}$.\nReduction of the bandwidth from $D_{\\rm SW}^{\\rm up}$ to\n$D_{\\rm SW}^{\\rm down}$ is accomplished using yet another\n(third) segment of the perturbative scaling. It turns out that\nall possible orderings of $|\\epsilon_0|$ and $U + \\epsilon_0$\nproduce the same Kondo Hamiltonian, provided that\n$\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$ and $h$ are\nsufficiently small. To keep the discussion as concise as\npossible, we therefore restrict the presentation to the case\n$|\\epsilon_0| \\sim U + \\epsilon_0$.\n\nConsider first the energy window between $D$ and $D_{\\rm SW}$,\nwhich is treated using Haldane's perturbative\nscaling~\\cite{HaldanePRL78}. Suppose that the bandwidth has\nalready been lowered from its initial value $D$ to some value\n$D' = D e^{-l}$ with $0 < l < \\ln (D\/D_{SW})$. Further reducing\nthe bandwidth to $D'(1 - \\delta l)$ produces a renormalization\nof each of the energies $\\epsilon_{\\uparrow}$,\n$\\epsilon_{\\downarrow}$, and $U$. Specifically, the\n$z$-component of the magnetic field, $h_z \\equiv\n\\epsilon_{\\downarrow} - \\epsilon_{\\uparrow}$, is found to obey\nthe scaling equation\n\\begin{equation}\n\\frac{d h_z}{d l} =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\left [\n \\frac{1}{1 - e^{l} \\epsilon_0\/D} -\n \\frac{1}{1 + e^{l} (U + \\epsilon_0)\/D}\n \\right ] .\n\\label{dh_z-dl}\n\\end{equation}\nHere we have retained $\\epsilon_0$ and $U + \\epsilon_0$ in the\ndenominators, omitting corrections which are higher-order in\n$\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$, and $h$ (these\ninclude also the small renormalizations of $\\epsilon_{\\sigma}$\nand $U$ that are accumulated in the course of the scaling). The\n$x$-component of the field, $h_x = h \\sin \\theta$, remains\nunchanged throughout the procedure. Upon reaching $D' = D_{\\rm\nSW}$, the renormalized field $h_z$ becomes\n\\begin{equation}\nh_z^{\\ast} = h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{D_{\\rm SW} + U + \\epsilon_0}\n {D_{\\rm SW} - \\epsilon_0} \\, ,\n\\end{equation}\nwhere we have assumed $D \\gg |\\epsilon_0|, U$.\n\nOnce the scale $D_{\\rm SW}$ is reached, charge fluctuations on\nthe dot are eliminated via a Schrieffer-Wolff\ntransformation,~\\cite{Wolff66} which generates among other\nterms also further renormalizations of $\\epsilon_{\\sigma}$.\nNeglecting $h$ in the course of the transformation, one arrives\nat the following Kondo-type Hamiltonian,\n\\begin{eqnarray}\n\\mathcal{H}_K &=& \\sum_{k, \\sigma}\n \\varepsilon^{}_k c_{k\\sigma}^{\\dagger} c^{}_{k\\sigma}\n + J_{\\perp} \\left ( S_x s_x + S_y s_y \\right )\n + J_z S_z s_z\n\\nonumber\\\\\n &+& v_{\\rm sc} S^z \\sum_{k, k', \\sigma}\\!\\!\n :\\! c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!:\n + \\sum_{k, k', \\sigma}\\! (v_+ + \\sigma v_-)\\!\n :\\! c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!:\n\\nonumber\\\\\n &-& \\tilde{h}_z S_z - \\tilde{h}_x S_x .\n\\label{H-Kondo}\n\\end{eqnarray}\nHere we have represented the local moment on the dot by\nthe spin-$\\frac{1}{2}$ operator\n\\begin{equation}\n\\vec{S} = \\frac{1}{2} \\sum_{\\sigma, \\sigma'}\n \\vec{\\tau}^{}_{\\sigma \\sigma'}\n d^{\\dagger}_{\\sigma} d^{}_{\\sigma'}\n\\end{equation}\n($\\vec{\\tau}$ being the Pauli matrices), while\n\\begin{equation}\n\\vec{s} = \\frac{1}{2} \\sum_{k, k'} \\sum_{\\sigma, \\sigma'}\n \\vec{\\tau}^{}_{\\sigma \\sigma'}\n c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma'}\n\\end{equation}\nare the local conduction-electron spin densities. The symbol\n$:\\!c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!\\!:\\ = c^{\\dagger}_{k\n\\sigma} c^{}_{k' \\sigma} - \\delta_{k, k'} \\theta(-\\epsilon_k)$\nstands for normal ordering with respect to the filled Fermi sea.\nThe various couplings that appear in Eq.~(\\ref{H-Kondo}) are given\nby the explicit expressions\n\\begin{equation}\n\\rho J_{\\perp} =\n \\frac{2 \\sqrt{\\Gamma_{\\uparrow} \\Gamma_{\\downarrow}}}\n {\\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{J-perp}\n\\end{equation}\n\\begin{equation}\n\\rho J_{z} =\n \\frac{\\Gamma_{\\uparrow} + \\Gamma_{\\downarrow}}\n {\\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{J-z}\n\\end{equation}\n\\begin{equation}\n\\rho v_{\\rm sc} =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {4 \\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\end{equation}\n\\begin{equation}\n\\rho v_{\\pm} =\n \\frac{\\Gamma_{\\uparrow} \\pm \\Gamma_{\\downarrow}}\n {4 \\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n - \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{v-pm}\n\\end{equation}\n\\begin{equation}\n\\tilde{h}_z = h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|} \\, ,\n\\label{h_z-tilde}\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{h}_x = h \\sin \\theta .\n\\label{h_x-tilde}\n\\end{equation}\n\nEquations~(\\ref{J-perp})--(\\ref{h_x-tilde}) are correct to\nleading order in $\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$,\nand $h$, in accordance with the inequality $\\Gamma_{\\uparrow},\n\\Gamma_{\\downarrow}, h \\ll |\\epsilon_0|, U + \\epsilon_0$. In\nfact, additional terms are generated in Eq.~(\\ref{H-Kondo})\nwhen $h$ is kept in the course of the Schrieffer-Wolff\ntransformation. However, the neglected terms are smaller than\nthe ones retained by a factor of $h\/\\min \\{|\\epsilon_0|, U +\n\\epsilon_0 \\} \\ll 1$, and are not expected to alter the\nlow-energy physics in any significant way. We also note that\n$\\tilde{h}_z$ accurately reproduces the second-order correction\nto $h_z$ detailed in Eq.~(\\ref{Delta-h_z}). As emphasized\nabove, the same effective Hamiltonian is obtained when\n$|\\epsilon_0|$ and $U + \\epsilon_0$ are well separated in\nenergy, although the derivation is notably more cumbersome. In\nunifying the different possible orderings of $|\\epsilon_0|$ and\n$U + \\epsilon_0$, the effective bandwidth in\nEq.~(\\ref{H-Kondo}) must be taken to be $D_0 \\sim \\min\n\\{|\\epsilon_0|, U + \\epsilon_0\\}$.\n\n\n\\subsection{Reduction to the Kondo effect in a finite\n magnetic field}\n\nIn addition to spin-exchange anisotropy and a tilted magnetic\nfield, the Hamiltonian of Eq.~(\\ref{H-Kondo}) contains a new\ninteraction term, $v_{\\rm sc}$, which couples spin and charge.\nIt also includes spin-dependent potential scattering,\nrepresented by the term $v_{-}$ above. As is well known,\nspin-exchange anisotropy is irrelevant for the conventional\nspin-$\\frac{1}{2}$ single-channel Kondo problem. As long as one\nlies within the confines of the antiferromagnetic domain, the\nsystem flows to the same strong-coupling fixed point no matter\nhow large the exchange anisotropy is. SU(2) spin symmetry is\nthus restored at low energies. A finite magnetic field $h$ cuts\noff the flow to isotropic couplings, as does the temperature\n$T$. However, the residual anisotropy is negligibly small if\n$h$, $T$ and the bare couplings are small. That is,\nlow-temperature thermodynamic and dynamic quantities follow a\nsingle generic dependence on $T\/T_K$ and $h\/T_k$, where $T_K$\nis the Kondo temperature. All relevant information on the bare\nspin-exchange anisotropy is contained for weak couplings in the\nmicroscopic form of $T_K$.\n\nThe above picture is insensitive to the presence of weak\npotential scattering, which only slightly modifies the\nconduction-electron phase shift at the Fermi energy. As we show\nbelow, neither is it sensitive to the presence of the weak\ncouplings $v_{\\rm sc}$ and $v_{-}$ in Eq.~(\\ref{H-Kondo}). This\nobservation is central to our discussion, as it enables a very\naccurate and complete description of the low-energy physics of\n$\\mathcal{H}_K$ in terms of the conventional Kondo model in a\nfinite magnetic field. Given the Kondo temperature $T_K$ and\nthe direction and magnitude of the renormalized field\npertaining to Eq.~(\\ref{H-Kondo}), physical observables can be\nextracted from the exact Bethe {\\em ansatz} solution of the\nconventional Kondo model. In this manner, one can accurately\ncompute the conductance and the occupation of the levels, as\ndemonstrated in Secs.~\\ref{sec:observables} and\n\\ref{sec:results}.\n\nTo establish this important point, we apply poor-man's\nscaling~\\cite{Anderson70} to the Hamiltonian of\nEq.~(\\ref{H-Kondo}). Of the different couplings that appear in\n$\\mathcal{H}_K$, only $J_z$, $J_{\\perp}$, and $\\tilde{h}_z$ are\nrenormalized at second order. Converting to the dimensionless\nexchange couplings $\\tilde{J}_z = \\rho J_z$ and\n$\\tilde{J}_{\\perp} = \\rho J_{\\perp}$, these are found to obey\nthe standard scaling\nequations~\\cite{AndersonYuvalHamann70,Anderson70}\n\\begin{eqnarray}\n\\frac{ d\\tilde{J}_z }{dl} &=& \\tilde{J}_{\\perp}^2 \\, ,\n\\label{scaling-J_z} \\\\\n\\frac{ d\\tilde{J}_{\\perp} }{dl} &=&\n \\tilde{J}_z \\tilde{J}_{\\perp} \\, ,\n\\label{scaling-J_perp}\n\\end{eqnarray}\nindependent of $v_{\\rm sc}$ and $v_{\\pm}$. Indeed, the\ncouplings $v_{\\rm sc}$ and $v_{\\pm}$ do not affect the scaling\ntrajectories in any way, other than through a small\nrenormalization to $\\tilde{h}_z$:\n\\begin{equation}\n\\frac{d \\tilde{h}_z}{d l} =\n D_0 \\, e^{-l}\n \\left (\n \\tilde{J}_z \\tilde{v}_{-} +\n 2 \\tilde{v}_{\\rm sc} \\tilde{v}_{+}\n \\right ) 8 \\ln 2 .\n\\label{scaling-h_z}\n\\end{equation}\nHere $\\tilde{v}_{\\mu}$ are the dimensionless couplings $\\rho\nv_{\\mu}$ ($\\mu = {\\rm sc}, \\pm$), and $l$ equals $\\ln (D_0\n\/D')$ with $D'$ the running bandwidth.\n\nAs stated above, the scaling equations\n(\\ref{scaling-J_z})--(\\ref{scaling-J_perp}) are identical to\nthose obtained for the conventional anisotropic Kondo model.\nHence, the Kondo couplings flow toward strong coupling along\nthe same scaling trajectories and with the same Kondo\ntemperature as in the absence of $v_{\\rm sc}$ and $v_{\\pm}$.\nStraightforward integration of\nEqs.~(\\ref{scaling-J_z})--(\\ref{scaling-J_perp}) yields\n\\begin{equation}\nT_K = D_0 \\exp\n \\left (\n -\\frac{1}{\\rho \\, \\xi} \\tanh^{-1}\\!\n \\frac{\\xi}{J_{z}}\n \\right )\n\\label{scaling-T_K-1}\n\\end{equation}\nwith $\\xi = \\sqrt{J_z^2 - J_{\\perp}^2}$. Here we have exploited\nthe hierarchy $J_z \\geq J_{\\perp} > 0$ in deriving\nEq.~(\\ref{scaling-T_K-1}). In terms of the original model\nparameters appearing in Eq.~(\\ref{eq:Hand}),\nEq.~(\\ref{scaling-T_K-1}) takes the form\n\\begin{equation}\nT_K = D_0 \\exp\n \\left [\n \\frac{\\pi \\epsilon_0 (U + \\epsilon_0)}\n {2U(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\ln\\!\n \\frac{\\Gamma_{\\uparrow}}{\\Gamma_{\\downarrow}}\n \\right ] \\, .\n\\label{scaling-T_K-2}\n\\end{equation}\nEquation~(\\ref{scaling-T_K-2}) was obtained within second-order\nscaling, which is known to overestimate the pre-exponential\nfactor that enters $T_K$. We shall not seek an improved\nexpression for $T_K$ encompassing all parameter regimes of\nEq.~(\\ref{eq:Hand}). More accurate expressions will be given\nfor the particular cases of interest, see\nSec.~\\ref{sec:results} below. Much of our discussion will not\ndepend, though, on the precise form of $T_K$. We shall only\nassume it to be sufficiently small such that the renormalized\nexchange couplings can be regarded isotropic starting at\nenergies well above $T_K$.\n\nThe other competing scale which enters the low-energy physics\nis the fully renormalized magnetic field: $\\vec{h}_{\\text{tot}}\n= h^x_{\\text{tot}}\\, \\hat{x} + h^z_{\\text{tot}}\\, \\hat{z}$.\nWhile the transverse field $h^x_{\\text{tot}}$ remains given by\n$h \\sin \\theta$, the longitudinal field $h^z_{\\text{tot}}$ is\nobtained by integration of Eq.~(\\ref{scaling-h_z}), subject to\nthe initial condition of Eq.~(\\ref{h_z-tilde}). Since the\nrunning coupling $\\tilde{J}_z$ is a slowly varying function of\n$l$ in the range where\nEqs.~(\\ref{scaling-J_z})--(\\ref{scaling-h_z}) apply, it can be\nreplaced for all practical purposes by its bare value in\nEq.~(\\ref{scaling-h_z}). Straightforward integration of\nEq.~(\\ref{scaling-h_z}) then yields\n\\begin{eqnarray}\nh_{\\text{tot}}^{z} &=&\n h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|}\n\\nonumber\\\\\n &+& 3 \\ln(2) \\, D_0\n \\frac{\\Gamma^2_{\\uparrow}-\\Gamma^2_{\\downarrow}}\n {\\pi^2} \\times\n \\frac{U (U + 2 \\epsilon_0)}\n {(U + \\epsilon_0)^2 \\epsilon_0^2} \\, ,\n\\label{h-total}\n\\end{eqnarray}\nwhere we have used Eqs.~(\\ref{J-z})--(\\ref{v-pm}) for $J_z$,\n$v_{\\rm sc}$, and $v_{\\pm}$. Note that the third term on the\nright-hand side of Eq.~(\\ref{h-total}) is generally much\nsmaller than the first two terms, and can typically be\nneglected.\n\nTo conclude this section, we have shown that the Hamiltonian of\nEq.~(\\ref{eq:Hand}), and thus that of Eq.~(\\ref{IHAM}), is\nequivalent at sufficiently low temperature and fields to the\nordinary \\emph{isotropic} Kondo model with a tilted magnetic\nfield, provided that $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}\n\\ll |\\epsilon_0|, U + \\epsilon_0$. The relevant Kondo\ntemperature is approximately given by\nEq.~(\\ref{scaling-T_K-2}), while the components of\n$\\vec{h}_{\\text{tot}} = h^x_{\\text{tot}}\\, \\hat{x} +\nh^z_{\\text{tot}}\\, \\hat{z}$ are given by $h^x_{\\text{tot}} = h\n\\sin \\theta$ and Eq.~(\\ref{h-total}).\n\n\n\\section{Physical observables\n\\label{sec:observables}}\n\nHaving established the intimate connection between the\ngeneralized Anderson Hamiltonian, Eq.~(\\ref{eq:Hand}), and the\nstandard Kondo model with a tilted magnetic field, we now\nemploy well-known results of the latter model in order to\nobtain a unified picture for the conductance and the occupation\nof the levels of our original model, Eq.~(\\ref{IHAM}). The\nanalysis extends over a rather broad range of parameters. For\nexample, when $U + 2\\epsilon_0 = 0$, then the sole requirement\nfor the applicability of our results is for $\\sqrt{\\Delta^2 +\nb^2}$ to be small. The tunnelling matrix $\\hat{A}$ can be\npractically arbitrary as long as the system lies deep in the\nlocal-moment regime. The further one departs from the middle of\nthe Coulomb-blockade valley the more restrictive the condition\non $\\hat{A}$ becomes in order for $\\vec{h}_{\\rm tot}$ to stay\nsmall. Still, our approach is applicable over a surprisingly\nbroad range of parameters, as demonstrated below. Unless stated\notherwise, our discussion is restricted to zero temperature.\n\n\n\\subsection{Conductance}\n\nAt zero temperature, a local Fermi liquid is formed in the\nKondo model. Only elastic scattering takes place at the Fermi\nenergy, characterized by the scattering phase shifts for the\ntwo appropriate conduction-electron modes. For a finite\nmagnetic field $h$ in the $z$-direction, single-particle\nscattering is diagonal in the spin index. The corresponding\nphase shifts, $\\delta_{\\uparrow}(h)$ and\n$\\delta_{\\downarrow}(h)$, are given by the Friedel-Langreth\nsum rule,~\\cite{Langreth66,Comment-on-Langreth}\n$\\delta_{\\sigma}(h) = \\pi \\qav{n_{\\sigma}}$, which when applied\nto the local-moment regime takes the form\n\\begin{equation}\n\\delta_{\\sigma}(h) = \\frac{\\pi}{2} + \\sigma \\pi M(h) \\, .\n\\label{phase-shift-1}\n\\end{equation}\nHere $M(h)$ is the spin magnetization, which\nreduces~\\cite{Comment-on-M_K} in the scaling\nregime to a universal function of $h\/T_K$,\n\\begin{equation}\n\\label{eq:MhUniversal}\nM(h) = M_K(h\/T_K) \\, .\n\\end{equation}\nThus, Eq.~(\\ref{phase-shift-1}) becomes\n$\\delta_{\\sigma}(h) = \\pi\/2 + \\sigma \\pi M_K(h\/T_K)$,\nwhere $M_K(h\/T_K)$ is given by Eq.~(\\ref{eq:MKfullWiegmann})\n\nTo apply these results to the problem at hand, one first needs\nto realign the tilted field along the $z$ axis. This is\nachieved by a simple rotation of the different operators about\nthe $y$ axis. Writing the field $\\vec{h}_{\\text{tot}}$ in the\npolar form\n\\begin{align}\n\\label{eq:htotExplicit}\n\\vec{h}_{\\text{tot}} & \\equiv h_{\\text{tot}}\n \\left (\n \\sin \\theta_h \\hat{x} +\n \\cos \\theta_h \\hat{z}\n \\right ) \\nonumber\\\\\n & \\approx h \\sin \\theta \\, \\hat{x} +\n \\left ( h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|}\n \\right ) \\hat{z} \\, ,\n\\end{align}\nthe lead and the dot operators are rotated according to\n\\begin{equation}\n \\left [\n \\begin{array}{c}\n \\tilde{c}_{k\\uparrow} \\\\\n \\tilde{c}_{k\\downarrow}\n \\end{array}\n \\right ] = R_{h} \\cdot\n \\left [\n \\begin{array}{c}\n c_{k\\uparrow} \\\\ c_{k\\downarrow}\n \\end{array}\n \\right ] = R_{h} R_{l} \\cdot\n \\left [\n \\begin{array}{c}\n c_{kL} \\\\ c_{kR}\n \\end{array}\n \\right ]\n\\end{equation}\nand\n\\begin{equation}\n \\left [\n \\begin{array}{c}\n \\tilde{d}_{\\uparrow} \\\\\n \\tilde{d}_{\\downarrow}\n \\end{array}\n \\right ] = R_{h} \\cdot\n \\left [\n \\begin{array}{c}\n d_{\\uparrow} \\\\ d_{\\downarrow}\n \\end{array}\n \\right ] = R_h R_{d} \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ]\\ , \\label{eq:RhRddef}\n\\end{equation}\nwith\n\\begin{equation}\nR_{h} = e^{i (\\theta_h\/2) \\tau_y} =\n \\left [\n \\begin{array}{cc}\n \\ \\ \\cos (\\theta_h\/2) &\\\n \\sin (\\theta_h\/2) \\\\\n - \\sin (\\theta_h\/2) &\\\n \\cos (\\theta_h\/2)\n \\end{array}\n \\right ] \\, . \\label{eq:Rhdef}\n\\end{equation}\nHere $R_{l}$ and $R_{d}$ are the unitary matrices used in\nEq.~(\\ref{eq:SVDdef}) to independently rotate the lead and the\ndot operators. Note that since $\\sin \\theta \\ge 0 $, the range\nof $\\theta_h$ is $\\theta_h \\in [0; \\pi]$.\n\nThe new dot and lead degrees of freedom have their spins\naligned either parallel ($\\tilde{d}_{\\uparrow}$ and\n$\\tilde{c}_{k \\uparrow}$) or antiparallel\n($\\tilde{d}_{\\downarrow}$ and $\\tilde{c}_{k \\downarrow}$) to\nthe field $\\vec{h}_{\\text{tot}}$. In this basis the\nsingle-particle scattering matrix is diagonal,\n\\begin{equation}\n\\tilde{S} =\n - \\left [\n \\begin{array}{cc}\n e^{i 2 \\pi M_K(h_{\\text{tot}}\/T_K)} &\n 0 \\\\\n 0 &\n e^{-i 2 \\pi M_K(h_{\\text{tot}}\/T_K)}\n \\end{array}\n \\right ] \\, .\n\\label{Scatt-mat-hat}\n\\end{equation}\nThe conversion back to the original basis set of left- and\nright-lead electrons is straightforward,\n\\begin{equation}\nS = R^{\\dagger}_{l} R^{\\dagger}_{h} \\tilde{S}\n R_{h}^{} R_{l}^{} \\equiv\n \\left [\n \\begin{array}{cc}\n r & t' \\\\\n t & r'\n \\end{array}\n \\right ] \\, ,\n\\label{Scatt-mat}\n\\end{equation}\nproviding us with the zero-temperature conductance\n$G = (e^2\/2 \\pi \\hbar) |t|^2$.\n\nEquations~(\\ref{Scatt-mat-hat}) and (\\ref{Scatt-mat})\nwere derived employing the mapping of Eq.~(\\ref{IHAM})\nonto an effective isotropic Kondo model with a tilted\nmagnetic field, in the $v_{\\rm sc}, v_{\\pm} \\to 0$\nlimit. Within this framework, Eqs.~(\\ref{Scatt-mat-hat})\nand (\\ref{Scatt-mat}) are exact in the scaling regime,\n$T_K\/D_0 \\ll 1$. The extent to which these equations\nare indeed valid can be appreciated by considering the special case\n$h \\sin \\theta = 0$, for which there exists an exact (and\nindependent) solution for the scattering matrix $S$ in terms of the\ndot ``magnetization'' $M = \\langle n_{\\uparrow} - n_{\\downarrow}\n\\rangle\/2$ [see Eq.~(\\ref{Scatt-mat-Langreth}) below]. That\nsolution, which is based on the Friedel-Langreth sum\nrule~\\cite{Langreth66} applied directly to a spin-conserving\nAnderson model, reproduces Eqs.~(\\ref{Scatt-mat-hat}) and\n(\\ref{Scatt-mat}) in the Kondo regime.\n\n\n\\subsubsection{Zero Aharonov-Bohm fluxes}\n\nOf particular interest is the case where no Aharonov-Bohm\nfluxes are present, where further analytic progress can\nbe made. For $\\varphi_L = \\varphi_R = 0$, the parameters\nthat appear in the Hamiltonian of Eq.~(\\ref{IHAM}) are\nall real. Consequently, the rotation matrices $R_{d}$\nand $R_{l}$ acquire the simplified forms given by\nEqs.~\\eqref{R_d-no-AB} and \\eqref{R_l-no-AB}\n(see Appendix~\\ref{App:SVDdetails} for details). Under\nthese circumstances, the matrix product $R_{h} R_{l}$\nbecomes $\\pm e^{i \\tau_y (\\theta_h + s_R \\, \\theta_l)\/2}\ne^{i \\pi \\tau_z (1 - s_R)\/4}$, and the elements of the\nscattering matrix [see Eq.~(\\ref{Scatt-mat})] are\n\\begin{align}\nt = t' =& - i \\sin[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\sin (\\theta_l + s_R \\theta_h) \\ ,\n\\nonumber\\\\\nr = (r')^{\\ast} =&\n - \\cos[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n\\nonumber\\\\\n &- i \\sin[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\cos (\\theta_l + s_R \\theta_h) \\ .\n\\end{align}\nHence, the conductance is\n\\begin{align}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\sin^2(\\theta_l + s_R \\theta_h) \\, ,\n\\label{G-no-flux}\n\\end{align}\nwhere the sign $s_R$ and angle $\\theta_l$ are given by\nEqs.~\\eqref{eq:sR} and \\eqref{eq:theta-no-AB}, respectively.\nAll dependencies of the conductance on the original model\nparameters that enter Eq.~(\\ref{IHAM}) are combined in\nEq.~(\\ref{G-no-flux}) into two variables alone, $\\theta_l + s_R\n\\theta_h$ and the reduced field $h_{\\text{tot}}\/T_K$. In\nparticular, $\\theta_l$ is determined exclusively by the\ntunnelling matrix $\\hat{A}$, while $s_R$ depends additionally\non the two dot parameters $\\Delta$ and $b$.\n\nThe conditions for a phase lapse to occur are particularly\ntransparent from Eq.~(\\ref{G-no-flux}). These lapses correspond\nto zeroes of $t$, and, in turn, of the conductance. There are\ntwo possibilities for $G$ to vanish: either $h_{\\text{tot}}$ is\nzero, or $\\theta_l + s_R \\theta_h$ equals an integer multiple\nof $\\pi$. For example, when the Hamiltonian of\nEq.~(\\ref{eq:Hand}) is invariant under the particle-hole\ntransformation $d_{\\sigma} \\to d_{\\sigma}^{\\dagger}$ and $c_{k\n\\sigma} \\to -c_{k \\sigma}^{\\dagger}$ (which happens to be the\ncase whenever $\\sqrt{ \\Delta^2 + b^2} = 0$ and $U + 2\\epsilon_0\n= 0$), then $h_{\\text{tot}}$ vanishes, and consequently the\nconductance vanishes as well. A detailed discussion of the\nramifications of Eq.~(\\ref{G-no-flux}) is held in\nSec.~\\ref{sec:ResultsAnisotropic} below.\n\n\n\\subsubsection{Parallel-field configuration}\n\nFor $h \\sin \\theta = 0$, spin is conserved by the\nHamiltonian of Eq.~(\\ref{eq:Hand}). We refer to this\ncase as the ``parallel-field'' configuration, since the\nmagnetic field is aligned with the anisotropy axis $z$.\nFor a parallel field, one can easily generalize the\nFriedel-Langreth sum rule~\\cite{Langreth66} to the\nHamiltonian of Eq.~(\\ref{eq:Hand}).~\\cite{MartinekNRGferro}\nApart from the need to consider\neach spin orientation separately, details of the\nderivation are identical to those for the ordinary\nAnderson model,~\\cite{Langreth66} and so is the\nformal result for the $T = 0$ scattering phase\nshift: $\\delta_{\\sigma} = \\pi \\Delta N_{\\sigma}$,\nwhere $\\Delta N_{\\sigma}$ is the number of\ndisplaced electrons in the spin channel $\\sigma$.\nIn the wide-band limit, adopted throughout our\ndiscussion, $\\Delta N_{\\sigma}$ reduces to the\noccupancy of the corresponding dot level,\n$\\langle n_{\\sigma} \\rangle$. The exact\nsingle-particle scattering matrix then becomes\n\\begin{equation}\nS = e^{i\\pi \\langle n_{\\uparrow} + n_{\\downarrow} \\rangle}\n R^{\\dagger}_{l} \\cdot\n \\left [\n \\begin{array}{cc}\n e^{i 2 \\pi M} & 0 \\\\\n 0 & e^{-i 2 \\pi M}\n \\end{array}\n \\right ] \\cdot R_{l} \\, ,\n\\label{Scatt-mat-Langreth}\n\\end{equation}\nwhere $M = \\langle n_{\\uparrow}-n_{\\downarrow} \\rangle\/2$\nis the dot ``magnetization.''\n\nEquation~(\\ref{Scatt-mat-Langreth}) is quite general. It covers\nall physical regimes of the dot, whether empty, singly occupied\nor doubly occupied, and extends to arbitrary fluxes $\\varphi_L$\nand $\\varphi_R$. Although formally exact, it does not specify\nhow the dot ``magnetization'' $M$ and the total dot occupancy\n$\\langle n_{\\uparrow} + n_{\\downarrow} \\rangle$ relate to the\nmicroscopic model parameters that appear in\nEq.~(\\ref{eq:Hand}). Such information requires an explicit\nsolution for these quantities. In the Kondo regime considered\nabove, $\\langle n_{\\uparrow} + n_{\\downarrow} \\rangle$ is\nreduced to one and $M$ is replaced by $\\pm\nM_K(h_{\\text{tot}}\/T_K)$. Here the sign depends on whether the\nfield $\\vec{h}_{\\text{tot}}$ is parallel or antiparallel to the\n$z$ axis (recall that $h_{\\text{tot}} \\ge 0$ by definition). As\na result, Eq.~(\\ref{Scatt-mat-Langreth}) reproduces\nEqs.~(\\ref{Scatt-mat-hat})--(\\ref{Scatt-mat}).\n\nTo carry out the rotation in Eq.~(\\ref{Scatt-mat-Langreth}), we\nrewrite it in the form\n\\begin{equation}\nS = e^{i\\pi \\langle n_{\\uparrow} + n_{\\downarrow} \\rangle}\n R^{\\dagger}_{l}\n \\left [\n \\cos (2 \\pi M) + i\\sin (2 \\pi M) \\tau_z\n \\right ]\n R_{l} \\, .\n\\end{equation}\nUsing the general form of Eq.~(\\ref{eq:phasel}) for the\nrotation matrix $R_{l}$, the single-particle scattering matrix\nis written as $S = e^{i\\pi \\langle n_{\\uparrow} +\nn_{\\downarrow} \\rangle} \\bar{S}$, where\n\\begin{eqnarray}\n\\bar{S} &=& \\cos (2 \\pi M)\n + i \\sin (2 \\pi M) \\cos \\theta_l\\, \\tau_z\n\\nonumber \\\\\n &+&\n i \\sin (2 \\pi M) \\sin \\theta_l\n \\left [\n \\cos \\phi_l\\, \\tau_x + \\sin \\phi_l\\, \\tau_y\n \\right ] \\, .\n\\end{eqnarray}\nThe zero-temperature conductance,\n$G = (e^2\/2 \\pi \\hbar)|t|^2$, takes then the exact form\n\\begin{equation}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2(2\\pi M) \\sin^2 \\theta_l \\, .\n\\label{G-parallel-field}\n\\end{equation}\n\nTwo distinct properties of the conductance are apparent form\nEq.~(\\ref{G-parallel-field}). Firstly, $G$ is bounded by\n$\\sin^2 \\theta_l $ times the conductance quantum unit $e^2\/2\n\\pi \\hbar$. Unless $\\theta_{l}$ happens to equal $\\pm \\pi\/2$,\nthe maximal conductance is smaller than $e^2\/2 \\pi \\hbar$.\nSecondly, $G$ vanishes for $M = 0$ and is maximal for $M = \\pm\n1\/4$. Consequently, when $M$ is tuned from $M \\approx -1\/2$ to\n$M \\approx 1\/2$ by varying an appropriate control parameter\n(for example, $\\epsilon_0$ when $\\Gamma_{\\uparrow} \\gg\n\\Gamma_{\\downarrow}$), then $G$ is peaked at the points where\n$M = \\pm 1\/4$. In the Kondo regime, when $M \\to \\pm\nM_K(h_{\\text{tot}}\/T_K)$, this condition is satisfied for\n$h_{\\text{tot}} \\approx 2.4 T_K$. As we show in\nSec.~\\ref{sec:ResultsAnisotropic}, this is the physical origin\nof the correlation-induced peaks reported by Meden and\nMarquardt.~\\cite{Meden06PRL} Note that for a given fixed\ntunnelling matrix $\\hat{A}$ in the parallel-field\nconfiguration, the condition for a phase lapse to occur is\nsimply for $M$ to vanish.\n\n\n\\subsection{Occupation of the dot levels}\n\nSimilar to the zero-temperature conductance, one can exploit\nexact results of the standard Kondo model to obtain the\noccupation of the levels at low temperatures and fields.\nDefining the two reduced density matrices\n\\begin{equation}\nO_d =\n \\begin{bmatrix}\n \\langle d^{\\dagger}_1 d^{}_1 \\rangle &\n \\langle d^{\\dagger}_2 d^{}_1 \\rangle \\\\\n \\langle d^{\\dagger}_1 d^{}_2 \\rangle &\n \\langle d^{\\dagger}_2 d^{}_2 \\rangle\n \\end{bmatrix}\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{O}_d =\n \\begin{bmatrix}\n \\langle\n \\tilde{d}^{\\dagger}_{\\uparrow}\n \\tilde{d}^{}_{\\uparrow}\n \\rangle &\n \\langle\n \\tilde{d}^{\\dagger}_{\\downarrow}\n \\tilde{d}^{}_{\\uparrow}\n \\rangle \\\\\n \\langle\n \\tilde{d}^{\\dagger}_{\\uparrow}\n \\tilde{d}^{}_{\\downarrow}\n \\rangle &\n \\langle\n \\tilde{d}^{\\dagger}_{\\downarrow}\n \\tilde{d}^{}_{\\downarrow}\n \\rangle\n \\end{bmatrix}\n\\, ,\n\\end{equation}\nthese are related through\n\\begin{equation}\nO_d^{} = R^{\\dagger}_{d} R^{\\dagger}_{h} \\tilde{O}_d^{}\n R^{}_{h} R^{}_{d} \\, .\n\\label{O_d-via-t-O_d}\n\\end{equation}\nHere $R_{h} R_{d}$ is the overall rotation matrix pertaining to\nthe dot degrees of freedom, see Eq.~\\eqref{eq:RhRddef}.\n\nAt low temperatures, the mapping onto an isotropic Kondo model\nimplies\n\\begin{equation}\n\\tilde{O}_d = \\begin{bmatrix}\n \\qav{\\tilde{n}_{\\uparrow}} & 0 \\\\\n 0 & \\qav{\\tilde{n}_{\\downarrow}} \\\\\n \\end{bmatrix} \\, ,\n\\label{t-O_d}\n\\end{equation}\nwhere\n\\begin{equation}\n\\qav{\\tilde{n}_{\\sigma}} = n_{\\text{tot}}\/2 +\n \\sigma \\tilde{M} \\, .\n\\label{eq:ntildeseparated}\n\\end{equation}\nHere we have formally separated the occupancies\n$\\qav{\\tilde{n}_{\\sigma}}$ into the sum of a spin component and\na charge component. The spin component involves the\nmagnetization $\\tilde{M}$ along the direction of the total\neffective field $\\vec{h}_{\\text{tot}}$. The latter is well\ndescribed by the universal magnetization curve\n$M_K(h_{\\text{tot}}\/T_K)$ of the Kondo model [see\nEq.~\\eqref{eq:MKfullWiegmann}]. As for the total dot occupancy\n$n_{\\text{tot}}$, deep in the local-moment regime charge\nfluctuations are mostly quenched at low temperatures, resulting\nin the near integer valance $n_{\\text{tot}} \\approx 1$. One can\nslightly improve on this estimate of $n_{\\text{tot}}$ by\nresorting to first-order perturbation theory in\n$\\Gamma_{\\sigma}$ (and zeroth order in $h$):\n\\begin{align}\nn_{\\text{tot}} & \\approx\n 1 + \\frac{\\Gamma_{\\uparrow} +\\Gamma_{\\downarrow}}{2 \\pi}\n \\left (\n \\frac{1}{\\epsilon_0}+\\frac{1}{U+\\epsilon_0}\n \\right )\n = 1 -2 \\rho v_{+} \\, .\n\\label{eq:n0PT}\n\\end{align}\nThis low-order process does not enter the Kondo\neffect, and is not contained in\n$M_K(h_{\\text{tot}}\/T_K)$.~\\cite{Comment-on-charge-fluc}\nWith the above approximations, the combination of\nEqs.~(\\ref{O_d-via-t-O_d}) and (\\ref{t-O_d}) yields\na general formula for the reduced density matrix\n\\begin{equation}\nO_d = n_{\\text{tot}}\/2 + M_K(h_{\\rm tot}\/T_K)\n R^{\\dagger}_{d} R^{\\dagger}_{h}\n \\tau_z R^{}_{h} R^{}_{d} \\, .\n\\label{O_d-general}\n\\end{equation}\n\n\\subsubsection{Zero Aharonov-Bohm fluxes}\n\nAs in the case of the conductance, Eq.~(\\ref{O_d-general})\nconsiderably simplifies in the absence of Aharonov-Bohm\nfluxes, when the combined rotation $R_{h} R_{d}$ equals\n$(s_R s_{\\theta})^{1\/2} e^{i \\tau_y (\\theta_h +\ns_{\\theta} \\theta_d)\/2} e^{i \\pi \\tau_z (1 - s_{\\theta})\/4}$\n[see Eqs.~\\eqref{eq:Rhdef} and \\eqref{R_d-no-AB}].\nExplicitly, Eq.~(\\ref{O_d-general}) becomes\n\\begin{eqnarray}\nO_d = n_{\\text{tot}}\/2 &+&\n M_K(h_{\\rm tot}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\tau_z\n\\nonumber \\\\\n &+& M_K(h_{\\rm tot}\/T_K)\n \\sin (\\theta_d + s_{\\theta} \\theta_h) \\tau_x \\, ,\n\\label{O_d-no-flux}\n\\end{eqnarray}\nwhere the sign $s_{\\theta}$ and angle $\\theta_d$ are given\nby Eqs.~\\eqref{eq:stheta} and \\eqref{eq:theta-no-AB},\nrespectively.\n\nSeveral observations are apparent from Eq.~(\\ref{O_d-no-flux}).\nFirstly, when written in the original ``spin'' basis\n$d^{\\dagger}_1$ and $d^{\\dagger}_2$, the reduced density matrix\n$O_d$ contains the off-diagonal matrix element $M_K(h_{\\rm\ntot}\/T_K) \\sin (\\theta_d + s_{\\theta} \\theta_h)$. The latter\nreflects the fact that the original ``spin'' states are\ninclined with respect to the anisotropy axis dynamically\nselected by the system. Secondly, similar to the conductance of\nEq.~(\\ref{G-no-flux}), $O_{d}$ depends on two variables alone:\n$\\theta_d + s_{\\theta} \\theta_h$ and the reduced field\n$h_{\\text{tot}}\/T_K$. Here, again, the angle $\\theta_d$ depends\nsolely on the tunnelling matrix $\\hat{A}$, while the sign\n$s_{\\theta}$ depends additionally on $\\Delta$ and $b$. Thirdly,\nthe original levels $d^{\\dagger}_1$ and $d^{\\dagger}_2$ have\nthe occupation numbers\n\\begin{subequations}\n\\label{eq:Actualn1n2}\n\\begin{align}\n\\qav{n_1} &=\n n_{\\text{tot}}\/2 + M_K(h_{\\text{tot}}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\, , \\\\\n\\qav{n_2} &=\n n_{\\text{tot}}\/2 - M_K(h_{\\text{tot}}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\, .\n\\end{align}\n\\end{subequations}\nIn particular, equal populations $\\langle n_1 \\rangle = \\langle\nn_2 \\rangle$ are found if either $h_{\\text{tot}}$ is zero or if\n$\\theta_d +s_{\\theta} \\theta_d$ equals $\\pi\/2$ up to an integer\nmultiple of $\\pi$. This provides one with a clear criterion for\nthe occurrence of population\ninversion,~\\cite{Gefen04,Sindel05,Silvestrov00} i.e., the\ncrossover from $\\qav{n_1} > \\qav{ n_2}$ to\n$\\qav{n_2} > \\qav{n_1}$ or vice versa.\n\n\\subsubsection{Parallel-field configuration}\n\\label{sec:occupany-PF}\n\nIn the parallel-field configuration, the angle $\\theta_h$ is\neither zero or $\\pi$, depending on whether the magnetic field\n$\\vec{h}_{\\text{tot}}$ is parallel or antiparallel to the $z$\naxis (recall that $h \\sin \\theta = h_{\\text{tot}} \\sin\n\\theta_h=0$ in this case). The occupancies $\\langle n_1\n\\rangle$ and $\\langle n_2 \\rangle$ acquire the exact\nrepresentation\n\\begin{subequations}\n\\label{eq:Actualn1n2-PF}\n\\begin{align}\n\\langle n_1 \\rangle &=\n n_{\\text{tot}}\/2 + M \\cos \\theta_d \\, , \\\\\n\\langle n_2 \\rangle &=\n n_{\\text{tot}}\/2 - M \\cos \\theta_d \\, ,\n\\end{align}\n\\end{subequations}\nwhere $n_{\\text{tot}}$ is the exact total occupancy of the dot\nand $M = \\langle n_{\\uparrow} - n_{\\downarrow} \\rangle\/2$ is\nthe dot ``magnetization,'' defined and used previously (not to\nbe confused with $\\tilde{M} = \\pm M$). As with the conductance,\nEqs.~(\\ref{eq:Actualn1n2-PF}) encompass all regimes of the dot,\nand extend to arbitrary Aharonov-Bohm fluxes. They properly\nreduce to Eqs.~(\\ref{eq:Actualn1n2}) in the Kondo regime, when\n$n_{\\text{tot}} \\approx 1$ [see Eq.~\\eqref{eq:n0PT}] and $M \\to\n\\pm M_K(h_{\\text{tot}}\/T_K)$. [Note that Eqs.~(\\ref{eq:Actualn1n2})\nhave been derived for zero Aharonov-Bohm fluxes.]\n\nOne particularly revealing observation that follows from\nEqs.~(\\ref{eq:Actualn1n2-PF}) concerns the connection between\nthe phenomena of population inversion and phase lapses in the\nparallel-field configuration. For a given fixed tunnelling\nmatrix $\\hat{A}$ in the parallel-field configuration, the\ncondition for a population inversion to occur is identical to\nthe condition for a phase lapse to occur. Both require that $M\n= 0$. Thus, these seemingly unrelated phenomena are synonymous\nin the parallel-field configuration. This is not generically\nthe case when $h_{\\text{tot}}^x\\neq 0$, as can be seen, for\nexample, from Eqs.~(\\ref{G-no-flux}) and (\\ref{eq:Actualn1n2}).\nIn the absence of Aharonov-Bohm fluxes, the conductance is\nproportional to $\\sin^2(\\theta_l + s_R \\theta_h)$. It therefore\nvanishes for $h_{\\text{tot}}^x\\neq 0$ only if $\\theta_l + s_R\n\\theta_h = 0\\!\\!\\mod\\!\\pi$. By contrast, the difference in\npopulations $\\langle n_1 - n_2 \\rangle$ involves the unrelated\nfactor $\\cos (\\theta_d + s_{\\theta}\\theta_h)$, which generally\ndoes not vanish together with $\\sin(\\theta_l + s_R \\theta_h)$.\n\nAnother useful result which applies to the parallel-field\nconfiguration is an exact expression for the $T = 0$\nconductance in terms of the population difference $\\langle n_1\n- n_2 \\rangle$. It follows from Eqs.~(\\ref{eq:Actualn1n2-PF})\nthat $M = \\qav{n_1 - n_2}\/( 2 \\cos \\theta_d)$. Inserting this\nrelation into Eq.~(\\ref{G-parallel-field}) yields\n\\begin{align}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2 \\left (\n \\frac{\\pi \\langle n_1-n_2 \\rangle}\n {\\cos \\theta_d}\n \\right )\n \\sin^2 \\theta_l \\, .\n\\label{G-parallel}\n\\end{align}\nThis expression will be used in Sec.~\\ref{sec:results} for\nanalyzing the conductance in the presence of isotropic\ncouplings, and for the cases considered by Meden and\nMarquardt~\\cite{Meden06PRL}.\n\n\n\\section{Results}\n\\label{sec:results}\n\nUp until this point we have developed a general framework for\ndescribing the local-moment regime in terms of two competing\nenergy scales, the Kondo temperature $T_K$ and the renormalized\nmagnetic field $h_{\\text{tot}}$. We now turn to explicit\ncalculations that exemplify these ideas. To this end, we begin\nin Sec.~\\ref{sec:ResultsIsotropic} with the exactly solvable\ncase $V_{\\uparrow} = V_{\\downarrow}$, which corresponds to the\nconventional Anderson model in a finite magnetic\nfield~\\cite{Boese01}. Using the exact Bethe \\emph{ansatz}\nsolution of the Anderson model,~\\cite{WiegmannA83} we present a\ndetailed analysis of this special case with three objectives in\nmind: (i) to benchmark our general treatment against\n rigorous results;\n(ii) to follow in great detail the delicate interplay\n between the two competing energy scales that\n govern the low-energy physics;\n(iii) to set the stage for the complete explanation of the\n charge oscillations~\\cite{Gefen04,Sindel05,Silvestrov00}\n and the correlation-induced resonances in the\n conductance of this device~\\cite{Meden06PRL,Karrasch06}.\n\nWe then proceed in Sec.~\\ref{sec:ResultsAnisotropic} to the\ngeneric anisotropic case $V_{\\uparrow} \\neq V_{\\downarrow}$. Here\na coherent explanation is provided for the ubiquitous phase\nlapses,~\\cite{Golosov06} population\ninversion,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances~\\cite{Meden06PRL,Karrasch06} that were reported\nrecently in various studies of two-level quantum dots. In\nparticular, we expose the latter resonances as a disguised\nKondo phenomenon. The general formulae of\nSec.~\\ref{sec:observables} are quantitatively compared to the\nnumerical results of Ref.~\\onlinecite{Meden06PRL}. The detailed\nagreement that is obtained nicely illustrates the power of the\nanalytical approach put forward in this paper.\n\n\n\\subsection{Exact treatment of $V_{\\uparrow}=V_{\\downarrow}$}\n\\label{sec:ResultsIsotropic}\n\nAs emphasized in Sec.~\\ref{sec:LocalMoment}, all tunnelling\nmatrices $\\hat{A}$ which satisfy Eq.~\\eqref{exact-b} give rise\nto equal amplitudes $V_{\\uparrow} = V_{\\downarrow} = V$ within\nthe Anderson Hamiltonian description of Eq.~\\eqref{eq:Hand}.\nGiven this extra symmetry, one can always choose the unitary\nmatrices $R_{l}$ and $R_{d}$ in such a way that the magnetic\nfield $h$ points along the $z$ direction [namely, $\\cos \\theta\n= 1$ in Eq.~\\eqref{eq:Hand}]. Perhaps the simplest member in\nthis class of tunnelling matrices is the case where $a_{L1} =\n-a_{L2} = a_{R1} = a_{R2} = V\/\\sqrt{2}$, $\\varphi_L = \\varphi_R\n= 0$ and $b=0$. One can simply convert the conduction-electron\noperators to even and odd combinations of the two leads,\ncorresponding to choosing $\\theta_l = \\pi\/2 + \\theta_d$.\nDepending on the sign of $\\Delta$, the angle $\\theta_d$ is\neither zero (for $\\Delta < 0$) or $\\pi$ (for $\\Delta > 0$),\nwhich leaves us with a conventional Anderson impurity in the\npresence of the magnetic field $\\vec{h} = |\\Delta| \\, \\hat{z}$.\nAll other rotation angle that appear in Eqs.~\\eqref{eq:phased}\nand \\eqref{eq:phased} (i.e., $\\chi$'s and $\\phi$'s) are equal\nto zero. For concreteness we shall focus hereafter on this\nparticular case, which represents, up to a simple rotation of\nthe $d^{\\dagger}_{\\sigma}$ and $c^{\\dagger}_{k \\sigma}$\noperators, all tunnelling matrices $\\hat{A}$ in this category\nof interest. Our discussion is restricted to zero temperature.\n\n\\subsubsection{Impurity magnetization}\n\\label{sec:ResTests}\n\nWe have solved the exact Bethe \\emph{anstaz} equations\nnumerically using the procedure outlined in\nAppendix~\\ref{app:Bethe}. Our results for the occupation\nnumbers $\\qav{n_{\\sigma}}$ and the magnetization $M =\n\\qav{n_{\\uparrow} - n_{\\downarrow}}\/2$ are summarized in\nFigs.~\\ref{fig:MethodCompare} and \\ref{fig:LargeFeature}.\nFigure~\\ref{fig:MethodCompare} shows the magnetization of the\nAnderson impurity as a function of the (average) level position\n$\\epsilon_0$ in a constant magnetic field, $h = \\Delta =\n10^{-3} U$. The complementary regime $\\epsilon_0 < -U\/2$ is\nobtained by a simple reflection about $\\epsilon_0 = -U\/2$, as\nfollows from the particle-hole transformation $d_{\\sigma} \\to\nd_{-\\sigma}^{\\dagger}$ and $c_{k \\sigma} \\to -c_{k\n-\\sigma}^{\\dagger}$. The Bethe \\emph{ansatz} curve accurately\ncrosses over from the perturbative domain at large $\\epsilon_0\n\\gg \\Gamma$ (when the dot is almost empty) to the local-moment\nregime with a fully pronounced Kondo effect (when the dot is\nsingly occupied). In the latter regime, we find excellent\nagreement with the analytical magnetization curve of the Kondo\nmodel, Eq.~\\eqref{eq:MKfullWiegmann}, both as a function of\n$\\epsilon_0$ and as a function of the magnetic field $\\Delta$\n(lower left inset to Fig.~\\ref{fig:MethodCompare}). The\nagreement with the universal Kondo curve is in fact quite\nsurprising in that it extends nearly into the mixed-valent\nregime. As a function of field, the Kondo curve of\nEq.~(\\ref{eq:MKfullWiegmann}) applies up to fields of the order\nof $h \\sim \\sqrt{\\Gamma U} \\gg T_K$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{fig2.eps}\n\\caption{(Color online) Magnetization of the isotropic\n case as a function of $\\epsilon_0$: exact Bethe\n \\emph{ansatz} curve and comparison with different\n approximation schemes.\n Black symbols show the magnetization $M$\n derived from the exact Bethe \\emph{ansatz}\n equations; the dashed (red) line marks the result\n of first-order perturbation theory in $\\Gamma$\n (Ref.~\\onlinecite{Gefen04}, divergent at\n $\\epsilon_0 = 0$); the thick (blue) line is the\n analytical formula for the magnetization in the\n Kondo limit, Eq.~\\eqref{eq:MKfullWiegmann}, with\n $T_K$ given by Eq.~\\eqref{eq:TKAndersonAccurate}.\n The model parameters are $\\Gamma\/U = 0.05$,\n $\\Delta\/U = 10^{-3}$ and $T = 0$.\n The upper right inset shows the same data but\n on a linear scale.\n The lower left inset shows the magnetization $M$\n as a function of the magnetic field $h = \\Delta$\n at fixed $\\epsilon_0\/U = -0.2$. The universal\n magnetization curve of the Kondo model well\n describes the exact magnetization up to\n $M \\approx 0.42$ (lower fields not shown),\n while first-order perturbation theory in\n $\\Gamma$ fails from $M \\approx 0.46$ downwards.}\n\\label{fig:MethodCompare}\n\\end{figure}\n\n\n\\subsubsection{Occupation numbers and charge oscillations}\n\\label{sec:ResCharging}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.5cm]{fig3.eps}\n\\caption{(Color online) The occupation numbers $\\qav{n_1}$\n [solid (blue) lines] and $\\qav{n_2}$ [dotted (red)\n lines] versus $\\epsilon_0$, as obtained from\n the solution of the exact Bethe \\emph{ansatz}\n equations. In going from the inner-most to the\n outer-most pairs of curves, the magnetic field\n $h = \\Delta$ increases by a factor of $10$\n between each successive pair of curves, with\n the inner-most (outer-most) curves corresponding\n to $\\Delta\/U = 10^{-5}$ ($\\Delta\/U = 0.1$).\n The remaining model parameters are\n $\\Gamma\/U = 0.05$ and $T=0$.\n Nonmonotonicities are seen in the process\n of charging. These are most pronounced for\n intermediate values of the field. The evolution\n of the nonmonotonicities with increasing field\n is tracked by arrows. The dashed black lines\n show the approximate values calculated from\n Eqs.~\\eqref{eq:Actualn1n2} and (\\ref{eq:n0PT})\n based on the mapping onto the Kondo Hamiltonian\n (here $\\theta_h = 0$ and $\\theta_d = \\pi$).}\n\\label{fig:LargeFeature}\n\\end{figure}\n\nFigure~\\ref{fig:LargeFeature} displays the individual\noccupation numbers $\\qav{n_1}$ and $\\qav{n_2}$ as a function of\n$\\epsilon_0$, for a series of constant fields $h = \\Delta$. In\ngoing from large $\\epsilon_0 \\gg \\Gamma$ to large $-(\\epsilon_0\n+ U) \\gg \\Gamma$, the total charge of the quantum dot increases\nmonotonically from nearly zero to nearly two. However, the\npartial occupancies $\\qav{n_1}$ and $\\qav{n_2}$ display\nnonmonotonicities, which have drawn considerable theoretical\nattention lately~\\cite{Silvestrov00,Gefen04,Sindel05}. As seen\nin Fig.~\\ref{fig:LargeFeature}, the nonmonotonicities can be\nquite large, although no population inversion occurs for\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$.\n\nOur general discussion in Sec.~\\ref{sec:LocalMoment} makes it\nis easy to interpret these features of the partial occupancies\n$\\qav{n_{i}}$. Indeed, as illustrated in\nFig.~\\ref{fig:LargeFeature}, there is excellent agreement in\nthe local-moment regime between the exact Bethe \\emph{ansatz}\nresults and the curves obtained from Eqs.~\\eqref{eq:Actualn1n2}\nand (\\ref{eq:n0PT}) based on the mapping onto the Kondo\nHamiltonian. We therefore utilize Eqs.~\\eqref{eq:Actualn1n2}\nfor analyzing the data. To begin with we note that, for\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$, there is no\nrenormalization of the effective magnetic field. The latter\nremains constant and equal to $h = \\Delta$ independent of\n$\\epsilon_0$. Combined with the fact that $\\cos(\\theta_d +\ns_\\theta \\theta_h) \\equiv -1$ in Eqs.~\\eqref{eq:Actualn1n2},\nthe magnetization $M = \\qav{n_{\\uparrow} - n_{\\downarrow}}\/2 =\n\\qav{n_{2} - n_{1}}\/2$ depends exclusively on the ratio\n$\\Delta\/T_K$. The sole dependence on $\\epsilon_0$ enters\nthrough $T_K$, which varies according to\nEq.~(\\ref{eq:TKAndersonAccurate}). Thus, $M$ is positive for\nall gate voltages $\\epsilon_0$, excluding the possibility of a\npopulation inversion.\n\nThe nonmonotonicities in the individual occupancies stem from\nthe explicit dependence of $T_K$ on the gate voltage\n$\\epsilon_0$. According to Eq.~(\\ref{eq:TKAndersonAccurate}),\n$T_K$ is minimal in the middle of the Coulomb-blockade valley,\nincreasing monotonically as a function of $|\\epsilon_0 + U\/2|$.\nThus, $\\Delta\/T_K$, and consequently $M$, is maximal for\n$\\epsilon_0 = -U\/2$, decreasing monotonically the farther\n$\\epsilon_0$ departs from $-U\/2$. Since $n_{\\text{tot}} \\approx\n1$ is nearly a constant in the local-moment regime, this\nimplies the following evolution of the partial occupancies:\n$\\qav{n_1}$ decreases ($\\qav{n_2}$ increases) as $\\epsilon_0$\nis lowered from roughly zero to $-U\/2$. It then increases\n(decreases) as $\\epsilon_0$ is further lowered toward $-U$.\nCombined with the crossovers to the empty-impurity and doubly\noccupied regimes, this generates a local maximum (minimum) in\n$\\qav{n_1}$ ($\\qav{n_2}$) near $\\epsilon_0 \\sim 0$ ($\\epsilon_0\n\\sim -U$).\n\nNote that the local extremum in $\\qav{n_i}$ is most pronounced\nfor intermediate values of the field $\\Delta$. This can be\nunderstood by examining the two most relevant energy scales in\nthe problem, namely, the minimal Kondo temperature\n$T_{K}^{\\text{min}} = T_{K}^{}|_{\\epsilon_0=-U\/2}$ and the\nhybridization width $\\Gamma$. These two energies govern the\nspin susceptibility of the impurity in the middle of the\nCoulomb-blockade valley (when $\\epsilon_0 = -U\/2$) and in the\nmixed-valent regime (when either $\\epsilon_0 \\approx 0$ or\n$\\epsilon \\approx -U$), respectively. The charging curves of\nFig.~\\ref{fig:LargeFeature} stem from an interplay of the three\nenergy scales $\\Delta$, $T_K^{\\text{min}}$ and $\\Gamma$ as\ndescribed below.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{fig4.eps}\n\\caption{(Color online) The exact conductance $G$\n [in units of $e^2\/(2\\pi\\hbar)$] versus\n $\\epsilon_0$, as obtained from the Bethe\n \\emph{ansatz} magnetization $M$ and\n Eq.~(\\ref{G-parallel}) with\n $\\theta_l = 3\\pi\/2$ and $\\theta_d = \\pi$.\n Here $\\Delta\/U$ equals $10^{-5}$ [full\n (black) line], $10^{-4}$ [dotted (red)\n line], $10^{-3}$ [dashed (green) line]\n and $0.1$ [dot-dashed (blue) line]. The\n remaining model parameters are\n $\\Gamma\/U = 0.05$ and $T = 0$. Once $\\Delta$\n exceeds the critical field $h_c^{} \\approx\n 2.4 T_K^{\\text{min}}$, the single peak at\n $\\epsilon_0 = -U\/2$ is split into two\n correlation-induced peaks, which cross\n over to Coulomb-blockade peaks at large\n $\\Delta$.}\n\\label{fig:isoCIR-1}\n\\end{figure}\n\nWhen $\\Delta \\ll T_K^{\\text{min}}$, exemplified by\nthe pair of curves corresponding to the smallest field\n$\\Delta = 10^{-5} U \\approx 0.24 T^{\\text{min}}_K$\nin Fig.~\\ref{fig:LargeFeature}, the magnetic field\nremains small throughout the Coulomb-blockade valley\nand no significant magnetization develops. The two\nlevels are roughly equally populated, showing a\nplateaux at $\\qav{n_{1}} \\approx \\qav{n_{2}}\n\\approx 1\/2$ in the regime where the dot is singly\noccupied. As $\\Delta$ grows and approaches\n$T_K^{\\text{min}}$, the field becomes sufficiently strong\nto significantly polarize the impurity in the vicinity\nof $\\epsilon_0 = -U\/2$. A gap then rapidly develops\nbetween $\\qav{n_{1}}$ and $\\qav{n_{2}}$ near\n$\\epsilon_0 = -U\/2$ as $\\Delta$ is increased. Once\n$\\Delta$ reaches the regime $T_K^{\\text{min}} \\ll\n\\Delta \\ll \\Gamma$, a crossover from $h \\gg T_K$\n(fully polarized impurity) to $h \\ll T_K$ (unpolarized\nimpurity) occurs as $\\epsilon_0$ is tuned away\nfrom the middle of the Coulomb-blockade valley. This\nleads to the development of a pronounced maximum\n(minimum) in $\\qav{n_1}$ ($\\qav{n_2}$), as marked by\nthe arrows in Fig.~\\ref{fig:LargeFeature}. Finally, when\n$h \\gtrsim \\Gamma$, the field is sufficiently large\nto keep the dot polarized throughout the local-moment\nregime. The extremum in $\\qav{n_i}$ degenerates into\na small bump in the vicinity of either $\\epsilon_0\n\\approx 0$ or $\\epsilon_0 \\approx -U$, which is\nthe nonmonotonic feature first discussed in\nRef.~\\onlinecite{Gefen04}. This regime is exemplified\nby the pair of curves corresponding to the largest\nfield $\\Delta = 0.1 U = 2\\Gamma$ in\nFig.~\\ref{fig:LargeFeature}, whose parameters match\nthose used in Fig.~2 of Ref.~\\onlinecite{Gefen04}.\nNote, however, that the perturbative calculations of\nRef.~\\onlinecite{Gefen04} will inevitably miss the\nregime $T_K^{\\text{min}} \\ll \\Delta \\ll \\Gamma$ where\nthis feature is large~\\cite{comm-Sindel-feature}.\n\n\n\\subsubsection{Conductance}\n\\label{Sec:isoCond}\n\n\\begin{figure}\n\\includegraphics[width=7cm]{fig5.eps}\n\\caption{(Color online) The exact occupation numbers\n $\\qav{n_i}$ and conductance $G$ [in units\n of $e^2\/(2 \\pi \\hbar)$] as a function of\n $\\epsilon_2$, for $T = 0$, $\\Gamma\/U = 0.2$\n and fixed $\\epsilon_1\/U = -1\/2$. The\n population inversion at $\\epsilon_2 =\n \\epsilon_1$ leads to a sharp transmission\n zero (phase lapse). Note the general\n resemblance between the functional dependence of\n $G$ on $\\epsilon_2$ and the correlation-induced\n resonances reported by Meden and\n Marquardt~\\cite{Meden06PRL} for\n $\\Gamma_{\\uparrow} \\neq \\Gamma_{\\downarrow}$\n (see Fig.~\\ref{fig:CIR}).}\n\\label{fig:isoCIR-2}\n\\end{figure}\n\nThe data of Fig.~\\ref{fig:LargeFeature} can easily be\nconverted to conductance curves by using the exact\nformula of Eq.~\\eqref{G-parallel} with $\\theta_l = 3\\pi\/2$\nand $\\theta_d = \\pi$.\nThe outcome is presented in Fig.~\\ref{fig:isoCIR-1}.\nThe evolution of $G(\\epsilon_0)$ with increasing\n$\\Delta$ is quite dramatic. When $\\Delta$ is small,\nthe conductance is likewise small with a shallow peak\nat $\\epsilon_0 = -U\/2$. This peak steadily grows with\nincreasing $\\Delta$ until reaching the unitary limit, at\nwhich point it is split in two. Upon further increasing\n$\\Delta$, the two split peaks gradually depart,\napproaching the peak positions $\\epsilon_0 \\approx 0$\nand $\\epsilon_0 \\approx -U$ for large $\\Delta$. The\nconductance at each of the two maxima remains pinned\nat all stages at the unitary limit.\n\nThese features of the conductance can be naturally\nunderstood based on Eqs.~\\eqref{G-parallel} and\n\\eqref{eq:Actualn1n2}. When $\\Delta \\ll T_K^{\\text{min}}$,\nthe magnetization $M \\approx \\Delta\/(2\\pi T_K)$ and the\nconductance $G \\approx (\\Delta\/T_K)^2 e^2\/(2\\pi \\hbar)$\nare uniformly small, with a peak at $\\epsilon_0 = -U\/2$\nwhere $T_K$ is the smallest. The conductance\nmonotonically grows with increasing $\\Delta$ until\nreaching the critical field $\\Delta = h_c^{} \\approx 2.4\nT_K^{\\text{min}}$, where $M|_{\\epsilon_0 = -U\/2} = 1\/4$\nand $G|_{\\epsilon_0 = -U\/2} = e^2\/(2 \\pi \\hbar)$. Upon\nfurther increasing $\\Delta$, the magnetization at\n$\\epsilon_0 = -U\/2$ exceeds $1\/4$, and the associated\nconductance decreases. The unitarity condition\n$M = 1\/4$ is satisfied at two gate voltages\n$\\epsilon^{\\pm}_{\\text{max}}$ symmetric about $-U\/2$,\ndefined by the relation $T_K \\approx \\Delta\/2.4$. From\nEq.~(\\ref{eq:TKAndersonAccurate}) one obtains\n\\begin{equation}\n\\epsilon_{\\pm}^{\\text{max}} = -\\frac{U}{2}\n \\pm \\sqrt{\n \\frac{U^2}{4} - \\Gamma^2 +\n \\frac{2\\Gamma U}{\\pi}\n \\ln \\left (\n \\frac{\\pi \\Delta}\n {2.4 \\sqrt{2 \\Gamma U}}\n \\right )\n } \\, .\n\\end{equation}\nThe width of the two conductance peaks,\n$\\Delta \\epsilon$, can be estimated for\n$T_K^{\\text{min}} \\ll \\Delta \\ll \\Gamma$ from the inverse\nof the derivative $d(\\Delta\/T_K)\/d\\epsilon_0$, evaluated\nat $\\epsilon_0^{} = \\epsilon_{\\text{max}}^{\\pm}$. It\nyields\n\\begin{equation}\n\\Delta \\epsilon \\sim \\frac{\\Gamma U}\n {\\pi | \\epsilon_{\\text{max}}^{\\pm} + U\/2 |} \\, .\n\\end{equation}\nFinally, when $\\Delta > \\Gamma$, the magnetization\nexceeds $1\/4$ throughout the local-moment regime. The\nresonance condition $M = 1\/4$ is met only as charge\nfluctuations become strong, namely, for either\n$\\epsilon_0 \\approx 0$ or $\\epsilon_0 \\approx -U$. The\nresonance width $\\Delta \\epsilon$ evolves continuously in\nthis limit to the standard result for the Coulomb-blockade\nresonances, $\\Delta \\epsilon \\sim \\Gamma$.\n\nUp until now the energy difference $\\Delta$ was kept\nconstant while tuning the average level position\n$\\epsilon_0$. This protocol, which precludes population\ninversion as a function of the control parameter, best\nsuits a single-dot realization of our model, where both\nlevels can be uniformly tuned using a single gate voltage.\nIn the alternative realization of two spatially separated\nquantum dots, each controlled by its own separate gate\nvoltage, one could fix the energy level\n$\\epsilon_1 = \\epsilon_0 + \\Delta\/2$ and sweep the other\nlevel, $\\epsilon_2 = \\epsilon_0 - \\Delta\/2$. This setup\namounts to changing the field $h$ externally, and is\nthus well suited for probing the magnetic response of\nour effective impurity.\n\nAn example for such a protocol is presented in\nFig.~\\ref{fig:isoCIR-2}, where $\\epsilon_1$ is held\nfixed at $\\epsilon_1 = -U\/2$. As $\\epsilon_2$ is\nswept through $\\epsilon_1$, a population inversion\ntakes place, leading to a narrow dip in the conductance.\nThe width of the conductance dip is exponentially\nsmall due to Kondo correlations. Indeed, one can\nestimate the dip width, $\\Delta \\epsilon_{\\text{dip}}$,\nfrom the condition $|\\epsilon_1 - \\epsilon_2| =\nT_K|_{\\epsilon_2 = \\epsilon_1}$, which yields\n\\begin{equation}\n\\Delta \\epsilon_{\\text{dip}} \\sim\n \\sqrt{U \\Gamma} \\exp\n \\left (\n -\\frac{\\pi U}{8 \\Gamma}\n \\right ) \\, .\n\\label{eq:CIRwidthIsotropic}\n\\end{equation}\n\n\\subsection{Anisotropic couplings, $\\Gamma_{\\uparrow}\n \\neq \\Gamma_{\\downarrow}$}\n\\label{sec:ResultsAnisotropic}\n\nAs demonstrated at length in Sec.~\\ref{sec:ResultsIsotropic},\nthe occurrence of population inversion and a transmission\nzero for $\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$\nrequires an external modulation of the effective\nmagnetic field. Any practical device will inevitably\ninvolve, though, some tunnelling anisotropy,\n$V_{\\uparrow} \\neq V_{\\downarrow}$. The latter provides\na different route for changing the effective magnetic\nfield, through the anisotropy-induced terms in\nEq.~\\eqref{h-total}. Implementing the same protocol\nas in Sec.~\\ref{sec:ResCharging} (that is, uniformly\nsweeping the average level position $\\epsilon_0$ while\nkeeping the difference $\\Delta$ constant) would now\ngenerically result both in population inversion and a\ntransmission zero due to the rapid change in direction\nof the total field $\\vec{h}_{\\text{tot}}$. As emphasized\nin Sec.~\\ref{sec:occupany-PF}, the two phenomena\nwill generally occur at different gate voltages\nwhen $V_{\\uparrow} \\neq V_{\\downarrow}$.\n\n\\subsubsection{Degenerate levels, $\\Delta = b = 0$}\n\nWe begin our discussion with the case where\n$\\Delta = b = 0$, which was extensively studied in\nRef.~\\onlinecite{Meden06PRL}. It corresponds to a\nparticular limit of the parallel-field configuration where\n$h = 0$. In the parallel-field configuration, the\nconductance $G$ and occupancies $\\qav{n_i}$ take the exact\nforms specified in Eqs.~(\\ref{G-parallel-field}) and\n(\\ref{eq:Actualn1n2-PF}), respectively. These expressions\nreduce in the Kondo regime to Eqs.~(\\ref{G-no-flux}) and\n(\\ref{eq:Actualn1n2}), with $\\theta_h$ either equal to\nzero or $\\pi$, depending on the sign of $h_{\\text{tot}}^z$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig6.eps}\n\\caption{The occupation numbers $\\qav{n_i}$ and conductance\n $G$ [in units of $e^2\/(2 \\pi \\hbar)$] as a\n function of $\\epsilon_0 + U\/2$ [in units of\n $\\Gamma_{\\text{tot}} = (\\Gamma_{\\uparrow} +\n \\Gamma_{\\downarrow})$],\n calculated from Eqs.~(\\ref{G-no-flux}) and\n (\\ref{eq:Actualn1n2}) based on the mapping\n onto the Kondo model. The model parameters\n are identical to those used in Fig.~2 of\n Ref.~\\onlinecite{Meden06PRL}, lower left panel:\n $h = \\varphi = 0$, $U\/\\Gamma_{\\text{tot}} = 6$,\n $\\Gamma_{\\uparrow}\/\\Gamma_{\\text{tot}} = 0.62415$\n and $T = 0$. The\n explicit tunnelling matrix elements are detailed\n in Eq.~(\\ref{eq:AMM}), corresponding to the\n rotation angles $\\theta_l = 2.1698$ and\n $\\theta_d = -0.63434$ (measured in radians).\n The angle $\\theta_h$ equals zero. The\n inset shows functional renormalization-group (fRG)\n data as defined in Ref.~\\onlinecite{Meden06PRL},\n corrected for the renormalization of the\n two-particle vertex~\\cite{Karrasch06,MedenThanks}.\n The small symbols in the inset\n show the conductance as calculated\n from the fRG occupation numbers using our\n Eq.~\\eqref{G-parallel}. The horizontal dotted\n lines in each plot mark the maximal conductance\n predicted by Eq.~\\eqref{G-parallel},\n $(e^2\/2 \\pi \\hbar) \\sin^2 \\theta_l$.}\n\\label{fig:CIR}\n\\end{figure}\n\nFigure~\\ref{fig:CIR} shows the occupation numbers and\nthe conductance obtained from Eqs.~(\\ref{G-no-flux})\nand (\\ref{eq:Actualn1n2}), for $\\Delta = b = 0$ and\nthe particular tunnelling matrix used in Fig.~2 of\nRef.~\\onlinecite{Meden06PRL}:\n\\begin{align}\n\\hat{A} = A_0\n \\begin{bmatrix}\n \\sqrt{0.27} & \\sqrt{0.16} \\\\\n \\sqrt{0.33} & -\\sqrt{0.24}\\\\\n\\end{bmatrix} \\, .\n\\label{eq:AMM}\n\\end{align}\nHere $A_0$ equals $\\sqrt{\\Gamma_{\\text{tot}}\/ (\\pi \\rho)}$,\nwith $\\Gamma_{\\text{tot}} =\n\\Gamma_{\\uparrow}+\\Gamma_{\\downarrow}$ being the combined\nhybridization width. The Coulomb repulsion $U$ is set equal to\n$6 \\Gamma_{\\text{tot}}$, matching the value used in the lower\nleft panel of Fig.~2 in Ref.~\\onlinecite{Meden06PRL}. For\ncomparison, the corresponding functional renormalization-group\n(fRG) data of Ref.~\\onlinecite{Meden06PRL} is shown in the\ninset, after correcting for the renormalization of the\ntwo-particle vertex~\\cite{Karrasch06,MedenThanks}. The accuracy\nof the fRG has been established~\\cite{Meden06PRL,Karrasch06} up\nto moderate values of $U\/\\Gamma_{\\text{tot}} \\sim 10$ through\na comparison with Wilson's numerical renormalization-group\nmethod~\\cite{NrgMethods}. Including the renormalization of\nthe two-particle vertex further improves the fRG data as\ncompared to that of Ref.~\\onlinecite{Meden06PRL}, as reflected,\ne.g., in the improved position of the outer pair of conductance\nresonances.\n\nThe agreement between our analytical approach and the fRG is\nevidently very good in the local-moment regime, despite the\nrather moderate value of $U\/\\Gamma_{\\text{tot}}$ used.\nNoticeable deviations develop in $\\qav{n_i}$ only as the\nmixed-valent regime is approached (for $\\epsilon_0 \\agt\n-\\Gamma_{\\text{tot}}$ or $\\epsilon + U \\alt\n\\Gamma_{\\text{tot}}$), where our approximations naturally break\ndown. In particular, our approach accurately describes the\nphase lapse at $\\epsilon_0 = -U\/2$, the inversion of population\nat the same gate voltage, the location and height of the\ncorrelation-induced resonances, and even the location and\nheight of the outer pair of conductance resonances. Most\nimportantly, our approach provides a coherent analytical\npicture for the physics underlying these various features, as\nelaborated below.\n\nBefore proceeding to elucidate the underlying physics, we\nbriefly quote the relevant parameters that appear in the\nconversion to the generalized Anderson model of\nEq.~(\\ref{eq:Hand}). Using the prescriptions detailed in\nAppendix~\\ref{App:SVDdetails}, the hybridization widths\n$\\Gamma^{}_{\\sigma} = \\pi \\rho V_{\\sigma}^2$ come out to be\n\\begin{equation}\n\\Gamma_{\\uparrow}\/\\Gamma_{\\text{tot}} = 0.62415 \\; ,\n\\;\\;\\;\\;\n\\Gamma_{\\downarrow}\/\\Gamma_{\\text{tot}} = 0.36585 \\, ,\n\\end{equation}\nwhile the angles of rotation equal\n\\begin{equation}\n\\theta_l = 2.1698 \\; , \\;\\;\\;\\;\n\\theta_d = -0.63434 \\, .\n\\end{equation}\nHere $\\theta_l$ and $\\theta_d$ are quoted in radians. Using the\nexact conductance formula of Eq.~(\\ref{G-parallel-field}), $G$\nis predicted to be bounded by the maximal conductance\n\\begin{equation}\nG_{\\text{max}} =\n \\frac{e^2}{2 \\pi \\hbar} \\sin^2 \\theta_l\n = 0.68210 \\frac{e^2}{2 \\pi \\hbar} \\, ,\n\\label{G-max}\n\\end{equation}\nobtained whenever the magnetization $M = \\qav{n_{\\uparrow} -\nn_{\\downarrow}}\/2$ is equal to $\\pm 1\/4$. The heights of the\nfRG resonances are in excellent agreement with\nEq.~(\\ref{G-max}). Indeed, as demonstrated in the inset to\nFig.~\\ref{fig:CIR}, the fRG occupancies and conductance comply\nto within extreme precision with the exact relation of\nEq.~(\\ref{G-parallel}). As for the functional form of the Kondo\ntemperature $T_K$, its exponential dependence on $\\epsilon_0$\nis very accurately described by Eq.~(\\ref{scaling-T_K-2}). In\nthe absence of a precise expression for the pre-exponential\nfactor when $\\Gamma_{\\uparrow} \\neq \\Gamma_{\\downarrow}$, we\nemploy the expression\n\\begin{equation}\nT_K = (\\sqrt{U \\Gamma_{\\text{tot}}}\/\\pi) \\exp\n \\left [\n \\frac{\\pi \\epsilon_0 (U + \\epsilon_0)}\n {2U(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\ln\\!\n \\frac{\\Gamma_{\\uparrow}}{\\Gamma_{\\downarrow}}\n \\right ] \\, ,\n\\label{T_K-anisotropic}\n\\end{equation}\nwhich properly reduces to Eq.~\\eqref{eq:TKAndersonAccurate} (up\nto the small $\\Gamma^2$ correction in the exponent) when\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow} = \\Gamma$.\n\nThe occupancies and conductance of Fig.~\\ref{fig:CIR} can be\nfully understood from our general discussion in\nSec.~\\ref{sec:LocalMoment}. Both quantities follow from the\nmagnetization $M$, which vanishes at $\\epsilon_0 = -U\/2$ due to\nparticle-hole symmetry. As a consequence, the two levels are\nequally populated at $\\epsilon_0 = -U\/2$ and the conductance\nvanishes [see Eqs.~(\\ref{G-parallel-field}) and\n(\\ref{eq:Actualn1n2-PF})]. Thus, there is a simultaneous phase\nlapse and an inversion of population at $\\epsilon_0 = -U\/2$,\nwhich is a feature generic to $\\Delta = b = 0$ and arbitrary\n$\\hat{A}$. As soon as the gate voltage is removed from $-U\/2$,\ni.e., $\\epsilon_0 = -U\/2 + \\delta \\epsilon$ with $\\delta\n\\epsilon \\neq 0$, a finite magnetization develops due to the\nappearance of a finite effective magnetic field\n$\\vec{h}_{\\text{tot}} = h^z_{\\text{tot}} \\hat{z}$ with\n\\begin{equation}\nh^{z}_{\\text{tot}} \\approx\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{1 + 2\\delta \\epsilon\/U}\n {1 - 2\\delta \\epsilon\/U}\n\\label{h-z-tot}\n\\end{equation}\n[see Eq.~(\\ref{eq:htotExplicit})]. Note that the sign of\n$h^{z}_{\\text{tot}}$ coincides with that of $\\delta \\epsilon$,\nhence $M$ is positive (negative) for $\\epsilon_0 > -U\/2$\n($\\epsilon_0 < -U\/2$). Since $\\cos \\theta_d > 0$ for the model\nparameters used in Fig.~\\ref{fig:CIR}, it follows from\nEq.~(\\ref{eq:Actualn1n2-PF}) that $\\qav{n_1} > \\qav{n_2}$\n($\\qav{n_1} < \\qav{n_2}$) for $\\epsilon_0 > -U\/2$ ($\\epsilon_0\n< -U\/2$), as is indeed found in Fig.~\\ref{fig:CIR}. Once again,\nthis result is generic to $\\Delta = b = 0$, except for the sign\nof $\\cos \\theta_d$ which depends on details of the tunnelling\nmatrix $\\hat{A}$.\n\nIn contrast with the individual occupancies, the conductance $G$\ndepends solely on the magnitude of $M$, and is therefore a\nsymmetric function of $\\delta \\epsilon$. Similar to the rich\nstructure found for $\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$\nand $\\Delta > 0$ in Fig.~\\ref{fig:isoCIR-1}, the intricate\nconductance curve in Fig.~\\ref{fig:CIR} is the result of the\ninterplay between $h_{\\text{tot}}^z$ and $T_K$, and the\nnonmonotonic dependence of $G$ on $|M|$. The basic physical\npicture is identical to that in Fig.~\\ref{fig:isoCIR-1}, except\nfor the fact that the effective magnetic field\n$h_{\\text{tot}}^z$ is now itself a function of the gate voltage\n$\\epsilon_0$.\n\n\nAs a rule, the magnetization $|M|$ first increases with\n$|\\delta \\epsilon|$ due to the rapid increase in\n$h_{\\text{tot}}^z$. It reaches its maximal value\n$M_{\\text{max}}$ at some intermediate $|\\delta \\epsilon|$\nbefore decreasing again as $|\\delta \\epsilon|$ is further\nincreased. Inevitably $|M|$ becomes small again once $|\\delta\n\\epsilon|$ exceeds $U\/2$. The shape of the associated\nconductance curve depends crucially on the magnitude of\n$M_{\\text{max}}$, which monotonically increases as a function\nof $U$. When $M_{\\text{max}} < 1\/4$, the conductance features\ntwo symmetric maxima, one on each side of the particle-hole\nsymmetric point. Each of these peaks is analogous to the one\nfound in Fig.~\\ref{fig:isoCIR-1} for $\\Delta < h_c$. Their\nheight steadily grows with increasing $U$ until the unitarity\ncondition $M_{\\text{max}} = 1\/4$ is met. This latter condition\ndefines the critical repulsion $U_c$ found in\nRef.~\\onlinecite{Meden06PRL}. For $U > U_c$, the maximal\nmagnetization $M_{\\text{max}}$ exceeds one quarter. Hence the\nunitarity condition $M = \\pm 1\/4$ is met at two pairs of gate\nvoltages, one pair of gate voltages on either side of the\nparticle-hole symmetric point $\\epsilon_0 = -U\/2$. Each of the\nsingle resonances for $U < U_c$ is therefore split in two, with\nthe inner pair of peaks evolving into the correlation-induced\nresonances of Ref.~\\onlinecite{Meden06PRL}. The point of\nmaximal magnetization now shows up as a local minimum of the\nconductance, similar to the point $\\epsilon_0 = -U\/2$ in\nFig.~\\ref{fig:isoCIR-1} when $\\Delta > h_c$.\n\nFor large $U \\gg \\Gamma_{\\text{tot}}$, the magnetization $|M|$\ngrows rapidly as one departs from $\\epsilon_0 = -U\/2$, due to\nthe exponential smallness of the Kondo temperature\n$T_K|_{\\epsilon_0 = -U\/2}$. The dot remains polarized\nthroughout the local-moment regime, loosing its polarization\nonly as charge fluctuations become strong. In this limit the\ninner pair of resonances lie exponentially close to $\\epsilon_0\n= -U\/2$ (see below), while the outer pair of resonances\napproach $|\\delta \\epsilon| \\approx U\/2$ (the regime of\nthe conventional Coulomb blockade).\n\nThe description of this regime can be made quantitative by\nestimating the position $\\pm \\delta \\epsilon_{\\text{CIR}}$ of\nthe correlation-induced resonances. Since $M \\to\nM_K(h^{z}_{\\text{tot}}\/T^{}_K)$ deep in the local-moment\nregime, and since $\\delta \\epsilon_{\\text{CIR}} \\ll\n\\Gamma_{\\text{tot}}$ for $\\Gamma_{\\text{tot}} \\ll U$, the\ncorrelation-induced resonances are peaked at the two gate\nvoltages where $h^{z}_{\\text{tot}} \\approx \\pm 2.4\nT^{}_K|_{\\epsilon_0 = -U\/2}$. Expanding Eq.~(\\ref{h-z-tot}) to\nlinear order in $\\delta \\epsilon_{\\text{CIR}}\/U \\ll 1$ and\nusing Eq.~(\\ref{T_K-anisotropic}) one finds\n\\begin{eqnarray}\n\\delta \\epsilon_{\\text{CIR}} &\\approx& 0.6\n \\frac{\\pi U}{\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow}}\n T_K|_{\\epsilon_0 = -U\/2}\n\\nonumber \\\\\n&=& 0.6 \\frac{U \\sqrt{U \\Gamma_{\\text{tot}}}}\n {\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow}}\n \\exp\\!\n \\left [\n \\frac{-\\pi U \\ln(\\Gamma_{\\uparrow}\/\n \\Gamma_{\\downarrow})}\n {8(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\right ] .\n\\label{eq:CIRwidth}\n\\end{eqnarray}\nHere the pre-exponential factor in the final expression for\n$\\delta \\epsilon_{\\text{CIR}}$ is of the same accuracy as that\nin Eq.~(\\ref{T_K-anisotropic}).\n\nWe note in passing that the shape of the correlation-induced\nresonances and the intervening dip can be conveniently\nparameterized in terms of the peak position $\\delta\n\\epsilon_{\\text{CIR}}$ and the peak conductance\n$G_{\\text{max}}$. Expanding Eq.~(\\ref{h-z-tot}) to linear order\nin $\\delta \\epsilon\/U \\ll 1$ and using\nEq.~(\\ref{G-parallel-field}) one obtains\n\\begin{equation}\nG(\\delta \\epsilon) = G_{\\text{max}} \\sin^2\\!\n \\left [\n 2 \\pi M_K\\!\n \\left (\n \\frac{2.4 \\delta \\epsilon}\n {\\delta \\epsilon_{\\text{CIR}}}\n \\right )\n \\right ] \\, ,\n\\end{equation}\nwhere $M_K(h\/T_K)$ is the universal magnetization curve of the\nKondo model [given explicitly by \\eqref{eq:MKfullWiegmann}].\nThis parameterization in terms of two easily extractable\nparameters may prove useful for analyzing future experiments.\n\nIt is instructive to compare Eq.~(\\ref{eq:CIRwidth}) for\n$\\delta \\epsilon_{\\text{CIR}}$ with the fRG results of\nRef.~\\onlinecite{Meden06PRL}, which tend to overestimate\n$\\delta \\epsilon_{\\text{CIR}}$. For the special case where\n$a_{L 1} = a_{R 1}$ and $a_{L 2} = -a_{R 2}$, an analytic\nexpression was derived for $\\delta \\epsilon_{\\text{CIR}}$ based\non the fRG~\\cite{Meden06PRL}. The resulting expression,\ndetailed in Eq.~(4) of Ref.~\\onlinecite{Meden06PRL}, shows an\nexponential dependence nearly identical to that of\nEq.~(\\ref{eq:CIRwidth}), but with an exponent that is smaller\nin magnitude by a factor of $\\pi^2\/8 \\approx\n1.23$~\\cite{Relating-the-Gamma's}. The same numerical factor\nappears to distinguish the fRG and the numerical\nrenormalization-group data depicted in Fig.~3 of\nRef.~\\onlinecite{Meden06PRL}, supporting the accuracy of our\nEq.~(\\ref{eq:CIRwidth}). It should be emphasized, however, that\nFig.~3 of Ref.~\\onlinecite{Meden06PRL} pertains to the\ntunnelling matrix of Eq.~(\\ref{eq:AMM}) rather than the special\ncase referred to above.\n\nWe conclude the discussion of the case where $\\Delta = b = 0$\nwith accurate results on the renormalized dot levels when the\ndot is tuned to the peaks of the correlation-induced\nresonances. The renormalized dot levels,\n$\\tilde{\\epsilon}_{\\uparrow}$ and\n$\\tilde{\\epsilon}_{\\downarrow}$, can be defined through the $T\n= 0$ retarded dot Green functions at the Fermi energy:\n\\begin{equation}\nG_{\\sigma}(\\epsilon = 0) =\n \\frac{1}{-\\tilde{\\epsilon}_{\\sigma}\n + i\\Gamma_{\\sigma}} \\, .\n\\label{renormalized-levels-def}\n\\end{equation}\nHere, in writing the Green functions of\nEq.~(\\ref{renormalized-levels-def}), we have made use of the\nfact that the imaginary parts of the retarded dot\nself-energies, $-\\Gamma_{\\sigma}$, are unaffected by the\nCoulomb repulsion $U$ at zero temperature at the Fermi energy.\nThe energies $\\tilde{\\epsilon}_{\\sigma}$ have the exact\nrepresentation~\\cite{Langreth66} $\\tilde{\\epsilon}_{\\sigma} =\n\\Gamma_{\\sigma} \\cot \\delta_{\\sigma}$ in terms of the\nassociated phase shifts $\\delta_{\\sigma} = \\pi \\qav{n_{\\sigma}}$.\nSince $M = \\pm 1\/4$ at the peaks of the correlation-induced\nresonances, this implies that $\\delta_{\\sigma} = \\pi\/2 \\pm\n\\sigma \\pi\/4$, where we have set $n_{\\text{tot}} =\n1$~\\cite{Comment-on-renormalized-levels}. Thus, the\nrenormalized dot levels take the form\n$\\tilde{\\epsilon}_{\\sigma} = \\mp\\sigma \\Gamma_{\\sigma}$,\nresulting in\n\\begin{equation}\n\\tilde{\\epsilon}_{\\uparrow} \\tilde{\\epsilon}_{\\downarrow}\n = -\\Gamma_{\\uparrow} \\Gamma_{\\downarrow} \\, .\n\\label{renormalized-levels}\n\\end{equation}\n\nThe relation specified in Eq.~(\\ref{renormalized-levels})\nwas found in Ref.~\\onlinecite{Meden06PRL},\nfor the special case where $a_{L 1} = a_{R 1}$ and\n$a_{L 2} = -a_{R 2}$~\\cite{Relating-the-Gamma's}.\nHere it is seen to be a generic feature of the\ncorrelation-induced resonances for $\\Delta = b = 0$\nand arbitrary $\\hat{A}$.\n\n\\subsubsection{Nondegenerate levels:\n arbitrary $\\Delta$ and $b$}\n\nOnce $\\sqrt{\\Delta^2 + b^2} \\neq 0$, the conductance and the\npartial occupancies can have a rather elaborate dependence on\nthe gate voltage $\\epsilon_0$. As implied by the general\ndiscussion in Sec.~\\ref{sec:LocalMoment}, the underlying\nphysics remains driven by the competing effects of the\npolarizing field $h_{\\text{tot}}$ and the Kondo temperature\n$T_K$. However, the detailed dependencies on $\\epsilon_0$ can be\nquite involving and not as revealing. For this reason we shall\nnot seek a complete characterization of the conductance $G$ and\nthe partial occupancies $\\qav{n_i}$ for arbitrary couplings.\nRather, we shall focus on the case where no Aharonov-Bohm\nfluxes are present and ask two basic questions: (i) under what\ncircumstances is the phenomenon of a phase lapse generic? (ii)\nunder what circumstances is a population inversion generic?\n\nWhen $\\varphi_L = \\varphi_R = 0$, the conductance and the\npartial occupancies are given by Eqs.~(\\ref{G-no-flux}) and\n(\\ref{eq:Actualn1n2}), respectively. Focusing on $G$ and on\n$\\qav{n_1 - n_2}$, these quantities share a common form, with\nfactorized contributions of the magnetization $M_K$ and the\nrotation angles. The factors containing\n$M_K(h_{\\text{tot}}\/T_K)$ never vanish when $h \\sin \\theta \\neq\n0$, since $h_{\\text{tot}}$ always remains positive. This\ndistinguishes the generic case from the parallel-field\nconfiguration considered above, where phase lapses and\npopulation inversions are synonymous with $M = 0$. Instead, the\nconditions for phase lapses and population inversions to occur\nbecome distinct once $h \\sin \\theta \\neq 0$, originating from\nthe independent factors where the rotation angles appear. For a\nphase lapse to develop, the combined angle $\\theta_l + s_R\n\\theta_h$ must equal an integer multiple of $\\pi$. By contrast,\nthe inversion of population requires that $\\theta_d +\ns_{\\theta}\\theta_h = \\pi\/2 \\!\\!\\mod\\!\\pi$. Here the dependence\non the gate voltage $\\epsilon_0$ enters solely through the\nangle $\\theta_h$, which specifies the orientation of the\neffective magnetic field $\\vec{h}_{\\text{tot}}$ [see\nEq.~(\\ref{eq:htotExplicit})]. Since the rotation angles\n$\\theta_l$ and $\\theta_d$ are generally unrelated, this implies\nthat the two phenomena will typically occur, if at all, at\ndifferent gate voltages.\n\nFor phase lapses and population inversions to be ubiquitous,\nthe angle $\\theta_h$ must change considerably as $\\epsilon_0$\nis swept across the Coulomb-blockade valley. In other words,\nthe effective magnetic field $\\vec{h}_{\\text{tot}}$ must nearly\nflip its orientation in going from $\\epsilon_0 \\approx 0$ to\n$\\epsilon_0 \\approx -U$. Since the $x$ component of the field\nis held fixed at $h_{\\text{tot}}^{x} = h \\sin \\theta > 0$, this\nmeans that its $z$ component must vary from $h_{\\text{tot}}^{z}\n\\gg h_{\\text{tot}}^{x}$ to $-h_{\\text{tot}}^{z} \\gg\nh_{\\text{tot}}^{x}$ as a function of $\\epsilon_0$. When this\nrequirement is met, then both a phase lapse and an inversion of\npopulation are essentially guaranteed to occur. Since\n$h_{\\text{tot}}^{z}$ crudely changes by\n\\begin{equation}\n\\Delta h_{\\text{tot}}^{z} \\sim\n \\frac{2}{\\pi} (\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow})\n \\ln ( U\/\\Gamma_{\\text{tot}} )\n\\end{equation}\nas $\\epsilon_0$ is swept across the Coulomb-blockade\nvalley, this leaves us with the criterion\n\\begin{equation}\n(\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow})\n \\ln ( U\/\\Gamma_{\\text{tot}} ) \\gg\n \\sqrt{\\Delta^2 + b^2} \\, .\n\\label{condition-for-PL}\n\\end{equation}\nConversely, if $\\sqrt{\\Delta^2 + b^2}\\gg (\\Gamma_{\\uparrow} -\n\\Gamma_{\\downarrow}) \\ln ( U\/\\Gamma_{\\text{tot}} )$, then\nneither a phase lapse nor an inversion of population will occur\nunless parameters are fine tuned. Thus, the larger $U$ is, the\nmore ubiquitous phase lapses\nbecome~\\cite{Golosov06,Meden06PRL}.\n\nAlthough the logarithm $\\ln(U\/\\Gamma_{\\text{tot}})$ can be made\nquite large, in reality we expect it to be a moderate factor of\norder one. Similarly, the difference in widths\n$\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}$ is generally expected\nto be of comparable magnitude to $\\Gamma_{\\uparrow}$. Under\nthese circumstances, the criterion specified in\nEq.~(\\ref{condition-for-PL}) reduces to $\\Gamma_{\\uparrow} \\gg\n\\sqrt{\\Delta^2 + b^2}$. Namely, phase lapses and population\ninversions are generic as long as the (maximal) tunnelling rate\nexceeds the level spacing. This conclusion is in line with that\nof a recent numerical study of multi-level quantum\ndots~\\cite{Karrasch06num}.\n\nFinally, we address the effect of nonzero $h = \\sqrt{\\Delta^2 +\nb^2}$ on the correlation-induced resonances. When $h \\gg\n\\Gamma_{\\uparrow} \\ln(U\/\\Gamma_{\\text{tot}})$, the effective\nmagnetic field $h_{\\text{tot}} \\approx h$ is large throughout\nthe local-moment regime, always exceeding\n$\\Gamma_{\\uparrow}$ and $\\Gamma_{\\downarrow}$. Consequently,\nthe dot is nearly fully polarized for all $-U < \\epsilon_0 <\n0$, and the correlation-induced resonances are washed out.\nAgain, for practical values of $U\/\\Gamma_{\\text{tot}}$ this\nregime can equally be characterized by $h \\gg\n\\Gamma_{\\uparrow}$~\\cite{Meden06PRL}.\n\n\n\nThe picture for $\\Gamma_{\\uparrow}\\ln(U\/\\Gamma_{\\text{tot}})\n\\gg h$ is far more elaborate. When $T_K|_{\\epsilon_0 = -U\/2}\n\\gg h$, the magnetic field is uniformly small, and no\nsignificant modifications show up as compared with the case\nwhere $h = 0$. This leaves us with the regime $T_K|_{\\epsilon_0\n= -U\/2} \\ll h \\ll \\Gamma_{\\uparrow}$, where various behaviors\ncan occur. Rather than presenting an exhaustive discussion of\nthis limit, we settle with identifying certain generic features\nthat apply when both components $|h \\cos \\theta|$ and $h\n\\sin\\theta$ exceed $T_K|_{\\epsilon_0 = -U\/2}$. To begin with,\nwhatever remnants of the correlation-induced resonances that\nare left, these are shifted away from the middle of the\nCoulomb-blockade valley in the direction where\n$|h_{\\text{tot}}^z|$ acquires its minimal value. Consequently,\n$h_{\\text{tot}}$ and $T_K$ no longer obtain their minimal\nvalues at the same gate voltage $\\epsilon_0$. This has the\neffect of generating highly asymmetric structures in place of\nthe two symmetric resonances that are found for $h = 0$. The\nheights of these features are governed by the ``geometric''\nfactors $\\sin^2 (\\theta_l + s_R \\theta_h)$ at the corresponding\ngate voltages. Their widths are controlled by the underlying\nKondo temperatures, which can differ substantially in\nmagnitude. Since the entire structure is shifted away from the\nmiddle of the Coulomb-blockade valley where $T_K$ is minimal,\nall features are substantially broadened as compared with the\ncorrelation-induced resonances for $h = 0$. Indeed, similar\ntendencies are seen in Fig.~5 of Ref.~\\onlinecite{Meden06PRL},\neven though the model parameters used in this figure lie on the\nborderline between the mixed-valent and the local-moment\nregimes.\n\n\n\\section{Concluding remarks}\n\\label{sec:conclusions}\n\nWe have presented a comprehensive investigation of the general\ntwo-level model for quantum-dot devices. A proper choice of the\nquantum-mechanical representation of the dot and the lead\ndegrees of freedom reveals an exact mapping onto a generalized\nAnderson model. In the local-moment regime, the latter\nHamiltonian is reduced to an anisotropic Kondo model with a\ntilted effective magnetic field. As the anisotropic Kondo model\nflows to the isotropic strong-coupling fixed point, this\nenables a unified description of all coupling regimes of the\noriginal model in terms of the universal magnetization curve of\nthe conventional isotropic Kondo model, for which exact results\nare available. Various phenomena, such as phase lapses in the\ntransmission phase,~\\cite{Silva02,Golosov06} charge\noscillations,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances~\\cite{Meden06PRL,Karrasch06} in the conductance, can\nthus be accurately and coherently described within a single\nphysical framework.\n\nThe enormous reduction in the number of parameters in\nthe system was made possible by the key observation that\na general, possibly non-Hermitian tunnelling matrix\n$\\hat{A}$ can always be diagonalized with the help of\ntwo simultaneous unitary transformations, one pertaining\nthe dot degrees of freedom, and the other applied to\nthe lead electrons. This transformation, known as the\nsingular-value decomposition, should have\napplications in other physical problems involving\ntunnelling or transfer matrices without any special\nunderlying symmetries.\n\nAs the two-level model for transport is quite general, it can\npotentially be realized in many different ways. As already\nnoted in the main text, the model can be used to describe\neither a single two-level quantum dot or a double quantum dot\nwhere each dot harbors only a single level. Such realizations\nrequire that the spin degeneracy of the electrons will be\nlifted by an external magnetic field. Alternative realizations\nmay directly involve the electron spin. For example, consider a\nsingle spinful level coupled to two ferromagnetic leads with\n\\emph{non-collinear} magnetizations. Written in a basis with a\nparticular \\emph{ad hoc} local spin quantization axis, the\nHamiltonian of such a system takes the general form of\nEq.~\\eqref{IHAM}, after properly combining the electronic\ndegrees of freedom in both leads. As is evident from our\ndiscussion, the local spin will therefore experience an\neffective magnetic field that is not aligned with either of the\ntwo magnetizations of the leads. This should be contrasted with\nthe simpler configurations of parallel and antiparallel\nmagnetizations, as considered, e.g., in\nRefs.~\\onlinecite{Martinek03PRL,MartinekNRGferro}\nand~~\\onlinecite{MartinekPRB05}.\n\nAnother appealing system for the experimental observation of the\nsubtle correlation effects discussed in the present paper is a\ncarbon nanotube-based quantum dot. In such a device both charging\nenergy and single-particle level spacing can be\nsufficiently large~\\cite{Buitelaar02} to provide a set of\nwell-separated discrete electron states. Applying external\nmagnetic field either perpendicular~\\cite{Nygard00} or\/and\nparallel~\\cite{HerreroSU4} to the nanotube gives great\nflexibility in tuning the energy level structure, and thus\nturns the system into a valuable testground for probing\nthe Kondo physics addressed in this study.\n\n\nThroughout this paper we confined ourselves to spinless\nelectrons, assuming that spin degeneracy has been lifted by an\nexternal magnetic field. Our mapping can equally be applied to\nspinful electrons by implementing an identical singular-value\ndecomposition to each of the two spin orientations separately\n(assuming the tunnelling term is diagonal in and independent of\nthe spin orientation). Indeed, there has been considerable\ninterest lately in spinful variants of the Hamiltonian of\nEq.~(\\ref{IHAM}), whether in connection with lateral quantum\ndots,~\\cite{Kondo2stage,HofstetterZarand04} capacitively\ncoupled quantum dots,~\\cite{KondoSU4,LeHuretal05,Galpinetal06}\nor carbon nanotube devices~\\cite{AguadoPRL05}. Among the\nvarious phenomena that have been discussed in these contexts,\nlet us mention SU(4) variants of the Kondo\neffect~\\cite{KondoSU4,LeHuretal05,AguadoPRL05}, and\nsinglet-triplet transitions with two-stage screening on the\ntriplet side~\\cite{Kondo2stage,HofstetterZarand04}.\n\nSome of the effects that have been predicted for the\nspinful case were indeed observed in lateral semiconductor\nquantum dots~\\cite{vanderWiel02,Granger05} and in carbon nanotube\nquantum dots~\\cite{HerreroSU4}. Still, there remains a\ndistinct gap between the idealized models that have been\nemployed, in which simplified symmetries are often imposed\non the tunnelling term, and the actual experimental systems\nthat obviously lack these symmetries. Our mapping should\nprovide a much needed bridge between the idealized models\nand the actual experimental systems. Similar to the present\nstudy, one may expect a single unified description\nencompassing all coupling regimes in terms of just\na few basic low-energy scales. This may provide valuable\nguidance for analyzing future experiments on such\ndevices.\n\n\\begin{acknowledgments}\nThe authors are thankful to V. Meden for kindly providing the\nnumerical data for the inset in Fig.~\\ref{fig:CIR}. VK is\ngrateful to Z.~A.~N\\'{e}meth for stimulating discussions of\nperturbative calculations. This research was supported by a\nCenter of Excellence of the Israel Science Foundation, and by a\ngrant from the German Federal Ministry of Education and\nResearch (BMBF) within the framework of the German-Israeli\nProject Cooperation (DIP). We have recently become aware of a\nrelated study by Silvestrov and Imry,~\\cite{SI-06} which\nindependently develops some of the ideas presented in this\nwork.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcqvz b/data_all_eng_slimpj/shuffled/split2/finalzzcqvz new file mode 100644 index 0000000000000000000000000000000000000000..a9cc139220139842e5ef06a6821aa6cb12409501 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcqvz @@ -0,0 +1,5 @@ +{"text":"\\section{}\n\n\n\\par\n\\medskip\nGraphene, a monolayer of graphite with carbons arranged in a two-dimensional honeycomb lattice has attracted great interest since its pioneering fabrication\\cite{r1,r2,r3,r4}. Its high mobility, current-carrying capacity and thermal conduction makes it a good candidate to replace silicon in future nanoelectronics\\cite{r5}. Graphene is a zero gap semiconductor, its Fermi surface consists of two non-equivalent Dirac points with linear energy dispersion in the Dirac region. The sublattice (chiral) symmetry of graphene guarantees the presence of the Dirac region where electrons behave as massless Dirac fermions, limits backscattering, introduces Andreev reflection and Klein tunnelling, etc\\cite{r6}. In order to control its gapless nature graphene was folded with cyclic boundary conditions in the short axis to form long cylinders (nanotubes\\cite{r7}), or long narrow strips with open perpendicular edges (nanoribbons\\cite{r8,r9}). In nanoribbons if the lattice geometry contains zigzag (zz) edges\\cite{r8,r9,r10,r11} the sublattice symmetry, which is responsible for the Dirac equation, guarantees the presence of $E=0$ edge states tied to the boundary\\cite{r8,r9}. We have studied Anderson localization\\cite{r12} in finite disordered graphene samples of various shapes known as graphene flakes\\cite{r13}, that is in confined lattices fabricated from graphene. \n\n\\par\n\\medskip\nIn two-dimensional(2D) disordered systems, with time-reversal and spin-rotation, all states are localized even for very small disorder, while in 3D localization occurs only above the Anderson metal-insulator transition\\cite{r14}. In 2D, however, the states have localization lengths which often exceed the system size. In other words, for weak disorder the states behave as diffusive, they extend from one end of the sample to the other, and the energy levels give chaotic (Wigner) level-spacing distribution\\cite{r15}. The observed level-repulsion and spectral rigidity are the main characteristics of quantum chaos, the Poisson exponential law which denotes localization requires unrealistically large system sizes to occur. In weakly disordered graphene flakes with zz edges for open boundary conditions (bc) the energy levels for a wide disorder region ($WW_{c}$ is localized (Poisson). The levels were unfolded by removing variations of the density of states $\\rho(E)$ averaged over $5000$ realizations of disorder. In this case $W_{c}\\simeq 2.3$.}\n\\end{figure}\n\n\\par\n\\medskip\nOur calculations are done for the nearest neighbour tight-binding Hamiltonian\n\\begin{equation}\n H=\\sum_{i}\\varepsilon_{i}c_{i}^{\\dag}c_{i} -\n \\sum_{}\\gamma _{i,j}(c_{i}^{\\dag}c_{j}+c_{j}^{\\dag}c_{i}),\n\\end{equation}\n$c_{i} (c_{i}^{\\dag})$ annihilates(creates) an electron at A or B\nsite $i$ of the honeycomb lattice, the diagonal disorder $\\varepsilon_{i}$ is constant in the range $[-W\/2,+W\/2]$, $W$ is its strength and ${}$ denotes nearest neighbours with hopping matrix elements $\\gamma_{i,j}=1$. For $W=0$ the band structure displays two non-equivalent valleys (act as pseudospin states) related by time-reversal\\cite{r5,r6} and in the Dirac region (at long length scales) Eq.(1) can be replaced by the continuous Dirac equation\\cite{r6}. The short-range diagonal disorder via $\\epsilon_{i}$ causes intervalley scattering and mixes the two-valleys which leads to localization. The decoupling of valleys by breaking time-reversal symmetry occurs for a smooth long-range disordered potential, this involves scattering within a single valley and gives weak antilocalization familiar from spin-orbit coupling\\cite{r18}. The disorder $\\varepsilon_{i}$ mixes valleys and destroys A,B sublattice symmetry. \n\n\\par\n\\medskip\nWe have cut the honeycomb lattice in various shapes, circular, stadium, square, etc. The shape of the flake turns out to be irrelevant for our study but the perimeter, as we shall see, it is not. In our computations the brick-wall lattice is mostly used, however we find no significant differences from the honeycomb lattice itself. The eigenvalues of Eq.(1) in the Dirac region are obtained via Lanczos numerical diagonalization, building a statistical ensemble of random $H$ matrices for every $W$. For $W=0$ quantum chaos for stadium flakes and integrability for circular flakes is expected, while if the lattice geometry contains zz edges the sublattice symmetry guarantees the presence of edge states. The edge states due to the symmetry of the bulk belong to one type (A or B) sublattice and they are protected against disorder by the lattice topology which is reflected in the Hilbert space structure. In the honeycomb lattice the edge states arise from quantum interference of an incoming wave from one bond which splits into two. In the presence of weak disorder the edge states spread almost uniformly in the Dirac region\\cite{r10}. For zero disorder from the tight-binding equations of the bulk\\cite{r8,r9} their amplitude $\\Psi (m)$ is nonzero only for $m=0$ and they are localized at the boundary having a fractal dimension of $1$. For nonzero $W$ they penetrate into the bulk and their fractal dimension varies. These massless degrees of freedom at the edges for finite $W$ are remnants of the sublattice symmetry which is broken by the diagonal disorder of Eq.(1). \n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.5]{rho-n-e-stadium-8940.eps}\n\\caption{The averaged density of states $\\rho(E)$ for circular graphene flakes with disorder $W$ obtained from more than one million eigenvalues of Eq.(1). For $W=1,2$ the edge states lie close to $E=0$ where $\\rho(E)$ is nonzero and almost constant. For strong disorder ($W=5$) their proportion becomes vanishingly small while in the opposite limit of pure ($W=0$) graphene: $\\rho(E)=\\frac{1}{\\pi \\sqrt{3}} |E|$ (continuous line) is linear. Inset: the averaged integrated density of states for weaker disorder $W=0.01, 0.1$ the $N(E)=\\int_{0^{+}}^E\\!\\rho(E')\\,\\,\\mathrm{d}E'$ (the $E=0$ states are not included) shows minigaps due to low $\\rho(E)$ values, while for higher $W$ the structure disappears.}\n\\end{figure}\n\n\\par\n\\medskip\nIn Fig.2 the averaged density of states $\\rho(E)$ for various $W$ is shown. In the absence of disorder ($W=0$) the zz edges contribute to zero energy, if they disappear the edge states also disappear. Their total number remains fixed for nonzero disorder\\cite{r10}, e.g. for $W=1,2$ the majority is in the almost constant $\\rho(E)$ region of Fig.2. Moreover, for weaker disorder $W=0.01, 0.1$ minibands and minigaps develop in the spectrum around the discrete $W=0$ states, this is due (inset of Fig.2) to the small\\ $\\rho(E)$ and the finite lattice. We have also computed $N(E)$ which counts the number of states from $E=0^{+}$ to $E$. In a log-log plot we find constant $N(E)\/E$ vs $E$ which implies constant $\\rho(E)$ for very low $E$, for slightly higher $E$ the $\\rho(E)$ decreases for $WW_{c}$ (no minigaps). For strong disorder ($W=5$) states fill up the Dirac region and make $\\rho(E)$ constant. \n\n\\par\n\\medskip\nWe have examined in detail the perimeter of the considered flakes, identified each type of lattice edge and its contribution to $\\rho(E)$. We find armchair and zz edges (the dangling bonds are rare), the most frequent edges are the zz\\cite{r10,r11}. The ratio of zz to armchair edges for circular flakes is $\\sim 3.8$ and it varies linearly with the averaged flake radius $R$ (the linear curve showed more oscillations for the zz edges). The ratio of zz edges over the total number of sites tends to zero inversely proportional to $R$, since their number is $\\sim R$ and the total number of sites is proportional to the area of the flake $\\sim R^{2}$, the density of states $\\rho(0)$ reaches a maximum before it vanishes as $1\/R$ for large $R$. The disorder shifts the edge states away from zero energy without changing their total number\\cite{r10}.\n\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.4]{ps-circle-e2-e1.eps}\n\\caption{The calculated spacing distribution $P(S)$ of the first two positive energies ($S=E_{2}-E_{1}$) for $WW_{c}$, $W=1, 5$ are obtained from stadium quarter flakes of sizes $N=924$, $2967$, and $8940$, for $50000$ realisations of disorder and energies in the window $[0,0.12]$ of the Dirac region. The two arrows indicate the flow towards the localized (Poisson) limit as the system size increases, the approach although slow is much faster than that for the square lattice. The Poisson limit verifies that in infinite graphene all states are localized.}\n\\end{figure}\n\n\\par\n\\medskip\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.4]{wave1-quartercircle-626.eps}\n\\caption{A critical edge state at nonzero $E=0.14$ for a quarter circular flake with disorder $W=2$ ($WW_{c}$ the approach to localization is faster than in other 2D (Fig.4). For weak disorder $WW_{c}$) states. This is shown in the Dirac region where the density of states is low and minibands appear with edge states. Our results are a signature that weakly disordered graphene is topological insulator-like, its conduction depends on the topology of the perimeter.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe formation of molecular clouds is one of the key steps for the star\nformation process. A large number of studies investigate the\ninternal dynamics of molecular clouds (e.g. see the review by MacLow\n\\& Klessen 2004), but only a few investigations address the problem of\ntheir formation itself (e.g. see the review by\nHennebelle, Mac Low \\& V\\'azquez-Semadeni 2008). This is partly due to the difficulty in\ntreating the large range of spatial scales relevant in this problem\nand partly due to uncertainties on the mechanisms at the origin of\ntheir formation. During the last decade, the idea has emerged that\nthe molecular clouds may be formed at the onset of a large scale\nconverging flow of atomic gas (e.g. Ballesteros-Paredes et al. 1999)\nwith the active role of thermal instability (Hennebelle \\& P\\'erault\n1999, Koyama \\& Inutsuka 2000, 2002, Audit \\& Hennebelle 2005, Heitsch\net al. 2005, V\\'azquez-Semadeni et al. 2006). Indeed, since the\ndensity of molecular clouds is much larger than the mean ISM density,\n large scale flows are a viable explanation to excite density\nenhancements.\nThe origin \nof these flows is however unclear and may be not unique.\n Most likely they arise from turbulent fluctuations \nor gravitational instability occuring at large scales. \n\nRecently large multi-dimensional non-magnetic\n numerical simulations have been\nperformed (Hennebelle \\& Audit 2007, V\\'azquez-Semadeni et al. 2007, \nHeitsch et al. 2008, the last two including self-gravity)\n with the aim of studying in detail the formation of \ndense gas from a large flow of warm neutral medium,\nresolving down to the star forming scales. \n\n\nIn this letter, we present \nthe first results of large adaptive mesh refinement (AMR), MHD simulations\nperformed with the codes RAMSES (Teyssier 2002, Fromang et al. 2006) \nand FLASH (Fryxell et al. 2000). \nThese are the first simulations which, starting from the WNM, \ninclude magnetic\nfield, cooling, self-gravity, and thanks to the AMR scheme, have \nsufficient spatial resolution to resolve individual, high\n density, clouds. \nThe molecular clouds we observe in these simulations are\n self-consistently generated by thermal instabilities, turbulence \nand gravitational contraction. \nWe stress that \nthe internal structure and the\n turbulent properties of these molecular clouds \nare not the result of an ad hoc assumption on the external\n turbulent driving. By performing these \nnumerical simulations, we expect to tackle unsolved and outstanding \nquestions such as what drives turbulence in molecular\nclouds; what is the gas density and temperature distribution, \nand what is the structure of the magnetic field in these objects{\\bf ?}\nIn this letter we report on the most important global\nproperties of the clouds. In subsequent articles (Banerjee et al. 2008), \nwe will give a more\ndetailed analysis of the cloud formation, structure and evolution,\nand on the efficiency of star formation in the simulations.\n\n \nIn section 2, we describe the numerical set up and the initial conditions\nwhereas in section 3 we present our results and preliminary \ncomparisons with observations. Section 4 concludes this paper.\n\n\n\\begin{figure}\n\\includegraphics[width=6.5cm]{hist_tot.ps}\n\\caption{Top and second panels: Column density and density PDF \nin the simulation. \nThird, fourth and fifth panels: Temperature, \nmagnetic intensity and magnetic intensity variance as a function of \ngas density. The {\\it solid, dotted, dashed, and\ndash-dotted} lines show the distributions at times $t=3.7$, 7.7,\n12.0 and 13.8 Myr, respectively. The straight line in\nthe fourth panel \nshows a distribution proportional to $n^{1\/2}$. }\n\\label{histo}\n\\end{figure}\n\n\n\n\\section{Numerical setup and initial conditions}\nThe numerical simulation presented in this letter has\nbeen performed with \n the AMR code RAMSES using the HLL solver. Ramses is a second order\nGodunov scheme and uses the constraint transport method to ensure\n ${\\rm div} B =0$ (see Fromang et al. 2006).\nStarting with an initial resolution of 256$^3$ cells, 2 levels of \nrefinement are allowed during the calculation leading to an effective \n1024$^3$ numerical simulation. The criterion used to refine the grid is a \nsimple density threshold of 50 cm$^{-3}$ for the first level and 200 cm$^{-3}$\nfor the second one. This ensures that the dense gas is uniformly \nresolved. With the box size being about 50 pc, this leads to \n a spatial resolution of about 0.05 pc. The total number of cells\nin the simulation is about $\\simeq 4 \\times 10^7$. About 25,000\ntimesteps have been performed for a total of 30,000 cpu hours. \n\n\nTo mimic a large scale converging \nflow (e.g. Audit \\& Hennebelle 2005),\na converging velocity field is imposed at the left and right faces \nof the simulation box, on top of which velocity modulations have been \nsuperimposed. The velocity of each incoming flow is twice\nthe sound speed of the WNM, leading to a total velocity difference of\nabout 40 km s$^{-1}$ within the box.\nThe amplitude of the modulation is about a factor of two, and it is\n periodic with a spatial frequency of 10 pc.\nThe boundary conditions are periodic for the 4 remaining faces.\nInitially, the density and temperature are respectively \n1 cm$^{-3}$ and about \n8000 K, which are also the values imposed at the left and right faces.\nThe velocity is initially equal to zero through the box.\nThe magnetic field is uniform initially and parallel to the x-axis, therefore\naligned with the incoming velocity field and has an intensity of about\n 5 $\\mu$G, corresponding to equipartition between magnetic and thermal pressure\ninitially.\nThe cooling is due to atomic species as described in Audit \\& Hennebelle \n(2005). Molecular cooling and H$_2$ formation are not treated at this stage.\nAs we will see, this nevertheless leads to reasonable temperature and \ndensity distributions. \n\n\\setlength{\\unitlength}{1cm}\n\\begin{figure*}\n\\begin{picture}(0,13.5)\n\\put(0,6.5){\\includegraphics[width=7cm]{ray_xy_00020_1.ps}}\n\\put(0,0){\\includegraphics[width=7cm]{dens_vit_xy_00020_1.ps}}\n\\put(9,6.5){\\includegraphics[width=7cm]{mag_xy_00020_1.ps}}\n\\put(9,0){\\includegraphics[width=7cm]{Temp_xy_00020_1.ps}}\n\\end{picture}\n\\caption{Top left panel: column density. Top right panel: Magnetic intensity \nand its xy-components (indicated as arrows) in the $z=0$ plane. Bottom left panel: density\n and velocity fields in the $z=0$ plane. Bottom right panel: temperature\n in the $z=0$ plane.}\n\\label{field}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Results}\nFigure~\\ref{histo} shows the column density and density pdf as \nwell as temperature, magnetic intensity and its variance as a function \nof density at time 3.71, 7.7, 12 and 13.79 Myr. Because of the mass \ninjection, the total mass increases continuously within the simulation box \nfrom about 3000 $M_s$ initially to roughly 10 times this value at \ntime 13.79 Myr. At time 3.71 and 7.7 Myr, the largest density reached in \nthe simulation is between a few times 10$^3$ cm$^{-3}$\n and $10^4 {\\rm ~cm}^{-3}$, whereas at time \n13.79 Myr gravity has taken over and triggers gravitational \ncollapse producing much higher densities. This indicates that the\ncloud should start\nforming stars roughly 12 Myr after the collision of the converging flow \nhas occurred ($t\\simeq$ 1 Myr in the simulation).\nAt later times, the cloud keeps forming stars while the total mass continues\nto increase. This is qualitatively in good agreement with the observations\nreported by Blitz et al. (2007) for the LMC, that the\nmasses of GMCs with little star formation activity are smaller than\nthose of GMCs with strong activity.\nWe note that the\nduration of 7 Myr, they infer for the first massive starless phase, is similar to the \ntimescale of our simulation roughly estimated between time 7.7 Myr and 13.79\nMyr. We stress, however, that precisely defining the\n``birth'' time of the molecular cloud in our simulation is an elusive\ntask since the mass is an increasing function of time. \nIn future papers we will address\nthis point from an observational perspective. \n\nThe mass weighted PDF of the column density distribution peaks at about $2 \\times 10^{21}$ cm$^{-2}$\nand drops rapidly for higher values. It is interesting to note that\nthis is similar to what has recently been inferred by Goldsmith et al. \n(2008) for the Taurus molecular cloud (see their figure 8).\n\nThe temperature drops rapidly for densities between \n3 and 30 cm$^{-3}$ where it reaches a value of about 50 K. It then slowly \ndecreases down to about 10 K for densities higher than 10$^4$ cm$^{-3}$. \nNote that since UV shielding and molecular cooling are not considered here, \n the temperature in the dense gas is\nprobably overestimated. This would imply \nthat the average density of the cold clumps should probably be a little\nhigher. Explicitly treating\nH$_2$ formation would have the same effect (Glover \\& MacLow 2007).\n\nFor densities smaller than $\\simeq 1000$ cm$^{-3}$, the magnetic \nintensity increases very smoothly with density whereas \nfor density larger than $\\simeq 1000$ cm$^{-3}$, it is roughly \nproportional to $\\sqrt{\\rho}$. Indeed, the lower density gas is\n magnetically sub-critical and mainly flows along the magnetic field\n lines without compressing magnetic flux. On the other hand, the \n high density gas is supercritical and the magnetic flux is compressed along\nwith the dense gas under the influence of the gravitational force.\nFuture studies will have to investigate whether ambipolar diffusion modifies \nthis behaviour significantly.\n The variance of the magnetic intensity increases \nsmoothly with density and ranges from about \none third to half the mean magnetic intensity. We note that this \nmagnetic intensity distribution appears to be very similar to what \nhas been inferred from observations (Troland \\& Heiles 1986, Crutcher 1999)\n\n\nFigure~\\ref{field} shows \nthe column density along the z-axis, the density and velocity field at \n$z=0$, and the temperature \nfield in the same plane at time $t=7.7$ Myr after the beginning of \nthe simulation. The column density reveals that \nthe cloud has a complex internal structure made of filaments and dense\nclumps of density between 100-1000 cm$^{-3}$,\n embedded in a more diffuse phase. This is even more clearly \napparent in the density and temperature cuts which show \nthat the clumps are relatively isolated and embedded into the warm diffuse\nphase.\nThis suggests that the molecular clouds are not\nhomogeneous isothermal media,\nbut rather, consist of a complex array of\nfilaments and clumps, with denser gas being colder, and therefore\nwith little or no excess of thermal pressure over the surrounding gas. \n\nInterestingly, the density of the \nwarm gas embedded in the molecular cloud\n is higher than in the outer medium ($n \\simeq 1 $ cm$^{-3}$)\n and can be as large as 3-4 cm$^{-3}$. \nIndeed, this gas has been previously shocked \n and it is in the process of cooling as it moves towards\nthe dense cold regions ({V\\'azquez-Semadeni}\\ et al. 2006; Hennebelle \\& Audit 2007).\nFrom a comparison between the column density and temperature distribution \nin the $z=0$ plane, \nwe note that the warm gas is deeply embedded in the molecular cloud. \nNote that in this work, the UV field is assumed to be constant.\n Although this is obviously not a good assumption for the \ndense gas, we see that since the filling factor of the cloud appears to be \nsmall, this is certainly a fair assumption for the WNM even when it is \ndeeply embedded inside the cloud. Moreover, \n the higher temperatures are sometimes found at the edge of the clumps \nat the onset of the accretion shocks which occurs when the WNM flow \nencounters a dense clump. This clearly indicates that \nthe dissipation of mechanical energy plays an active role in the heating \nof the warm phase. Altogether, this is in good agreement with the \npicture proposed by Hennebelle \\& Inutsuka (2006) except that the \nmechanical energy which heats the warm phase is the kinetic energy of\nthe shocks rather than the energy of the MHD waves (possibly underestimated \nin this work since the ion-neutral drift is not treated). \nNote that the finding of interclump medium being low density atomic hydrogen\n($n<4-10$ cm$^{-3}$) is consistent with the estimate of Williams et al. (1995) \n for the Rosette molecular cloud.\n\nFigure~\\ref{field} also shows the magnetic intensity in the $z=0$ plane\n as well \nas its xy-components. In the external medium the magnetic field remains\nmuch more uniform than in the dense regions, where its direction \nfluctuates significantly. \nSince the field was initially\nuniform, this implies that the turbulent\nmotions are able to significantly distort the field. \n\n\n\\section{Conclusion}\n\nWe have presented the results of AMR MHD simulations aiming to\ndescribe self-consistently the formation of a molecular cloud from a\nconverging flow of warm diffuse atomic hydrogen. Our simulations\nsuggest a complex multi-phase structure of the molecular clouds which\nconsists of cold and dense clumps embedded into a warm atomic phase.\nThe simulations reproduce reasonably well the observed variations of magnetic\n intensity with density and column density distribution.\nFinally, we suggest that star\nformation may start in the cloud while it is still accreting\nmaterial. This would imply that the mass of GMCs vary with\ntime, in\ngood agreement with recent observations (Blitz et al. 2007), and with the\nresults of previous non-magnetic simulations of the phenomenon.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThis expository paper aims to be a gentle introduction to the topology of configuration\nspaces, or equivalently spaces of little disks. The pantheon of topological spaces\nwhich beginning graduate students see is limited -- spheres, projective spaces,\nproducts of such, perhaps some spaces such as Lie groups, \nGrassmannians or knot complements. We would like\nfor Euclidean configuration spaces to be added to this list.\nWe aim for this article \nto be appropriate for someone who knows only basic homology and cohomology theory. The\none exception to this rule will be the light use of a spectral sequence argument,\nfor an upper bound. \n\nAny space from the pantheon has rich associated combinatorial and algebraic structure.\nFor example, the relationship between cohomology of\nprojective spaces and Grassmannians is encoded by the structure of symmetric polynomials.\nIn the case of configuration spaces, we are led to study graphs, trees, Jacobi and Arnold\nidentities, and ultimately the Poisson operad. One goal of this paper\nis to explain the topology which leads to the configuration pairing between\ngraphs and trees, developed \npurely combinatorially in \\cite{Sinh06.2}. That this pairing arises as that between canonical\nspanning sets for homology and cohomology\nof configuration spaces is a new result. Another goal is to prepare a reader\nfor further study of the theory of operads by giving a thorough understanding of the\ndisks operad from topology, the Poisson operad from algebra, and the fact that they are related\nthrough homology. We also bring in recently developed ingredients such as canonical \ncompactifications of configuration spaces and submanifolds defined by \ncollinearities. These new results and points of view,\nand our elementary development, differentiate this paper from expositions \nsuch as \\cite{CohF95, CLM76, FaHu01, CohF73, Arno69}.\n\nThe plan of the paper is as follows. First in \\refS{homology} \nwe associate a class in the homology of configuration spaces \nto any forest, as the fundamental class of a submanifold homeomorphic to a torus,\nand then develop relations between such classes. Then in \\refS{cohomology} we associate a cohomology class\n which is pulled back from a map to a torus to any graph. \\refS{pairing} gives the main new results, identifying the\nevaluation of our graphical cohomology classes on the forest homology classes with a combinatorially defined\npairing between graphs and trees. This pairing is useful in a number of contexts, for example in simultaneously \nunderstanding free Lie algebras and coalgebras \\cite{SiWa06}, but is not widely known. In \\refS{operads} we \ngive an informal ``examples-first'' development of operads, complementary to others in this volume, in \norder to be self-contained. In \\refS{diskshomology}, in particular \\refT{main2}, \nwe prove the well-known result that the homology of the little\ndisks operad is the graded Poisson operad. Instead of the usual practice of\nwaiving our hands at the operad structure maps, \nwe are able to provide a complete argument by arguing on cohomology instead.\nAt the end of most sections we give some (incomplete) historical notes.\n\n\n\nThe author would like to thank Ben Walter\nfor useful discussions, thank his students for feedback, and thank the organizers\nof both the Banff graduate workshop on homotopy theory in 2005 and\nthe Luminy graduate workshop on operads in 2009 for their\nencouragement in making this material accessible.\n\n\\tableofcontents\n\n\\section{Homology generators of configuration spaces}\\label{S:homology}\n\n\nIn this section we construct homology classes for configuration\nspaces, all represented by submanifolds homeomorphic\nto tori. \nThe term ``configuration space'' is used in different ways by different subfields.\nWe use the term as is standard in algebraic topology, as the space of distinct\nlabeled points in some ambient space. \n\n\\begin{definition}\nThe configuration space of $n$ distinct points in a space $X$, denoted ${\\rm Conf}_n(X)$,\nis the subspace of the product $X^{\\times n}$ defined as follows \n$$\\{(x_1, \\ldots, x_n) \\in X^{\\times n} | \\, x_i \\neq x_j \\, {\\rm if} \\, i \\neq j \\}.$$\n\\end{definition}\n\nWe will sometimes abbreviate $(x_1, \\cdots, x_n)$ as ${\\bf x}$. \nWe focus on the case in which $X$ is a Euclidean space $\\mathbb{R}^d$. This\nconfiguration space models all possible simultaneous positions of $n$ \ndistinct planets or particles. \nThis space as a whole may be visualized through linear algebra, starting\nwith the ambient Euclidean space $\\mathbb{R}^{nd}$ and removing the hyperplanes\nwhere some $x_i = x_j$. Indeed, the Euclidean configuration spaces are \nspecial important cases of complements of hyperplane arrangements.\n\nOur strategy to compute the homology and cohomology of these spaces\nis to ``just get our hands on things.'' When $n=2$\nwe have the following.\n\n\\begin{proposition}\\label{P:n=2}\nThe configuration space ${\\rm Conf}_2(\\mathbb{R}^d)$ is homotopy equivalent to $S^{d-1}$.\nThus the homology of ${\\rm Conf}_2(\\mathbb{R}^d)$ is free, rank one in dimensions $0$ and $d-1$,\nand zero otherwise.\n\\end{proposition}\n\n\\begin{proof}\nWe include $S^{d-1}$ into ${\\rm Conf}_2(\\mathbb{R}^d)$ as a deformation retract. With an\neye towards generalization, define the subspace $P_{\\tree{1}{2}}$ of ${\\rm Conf}_2(\\mathbb{R}^d)$\nas $\\{(x_1, x_2) \\, | \\, x_1 = -x_2 \\, {\\rm and} \\, |x_i| = 1\\}.$ The deformation\nretract onto this subspace sends $(x_1, x_2)$ to \n$\\left(\\frac{x_1 - m}{|x_1 - m|}, \\frac{x_2 - m}{|x_2 - m|}\\right)$, where $m = \\frac{x_1 + x_2}{2}$. \nThe homotopy between this retraction and the identity map\nis given by a straight-line homotopy. \n\\end{proof}\n\nWe also deduce that the generating cycle in $H_{d-1}\\left({\\rm Conf}_2(\\mathbb{R}^d)\\right)$ is the image of\nthe fundamental class of the sphere by the map which sends $v \\in S^{d-1}$\nto $(v, -v)$, parameterizing $P_{\\tree{1}{2}}$. \nDually, $H^{d-1}\\left({\\rm Conf}_2(\\mathbb{R}^d)\\right)$ is pulled back from the sphere\nby the given retraction. More geometrically, a generator of cohomology\nis Lefshetz dual to the submanifold $(x_1, x_2)$ such that $\\frac{x_1 - x_{2}}{|x_1 - x_{2}|}$\nis say the north pole\n$S^{d-1}$. This cohomology thus evaluates on some $d-1$ dimensional cycle by\ncounting with signs the number of configurations parameterized by that cycle\nfor which ``$x_1$ lies over $x_2$.'' \n\n\n \n \\medskip\n\n\nFor the general case the language of solar systems is suggestive, as Fred Cohen likes to point out.\nIn the $n=2$ case, the fundamental cycle had the ``planets'' $x_1$\nand $x_2$ ``orbiting'' their center of mass. For $n>2$ we can build further\ncycles by having that ``system'' orbit the common center of mass\nwith some other planet or system of planets. Each time we build a system, \nit is possible (if there are more planets around) to put\nthat in an orbit with another system to create a more complicated one. \nSuch systems, which are more difficult to formalize\nthan to visualize (see Figure~1), are naturally\nindexed by trees. \n\n\\begin{definition}\\label{D:trees}\n\\begin{enumerate}\n\\item An $S$-tree is an isotopy class of acyclic graph whose vertices\nare either trivalent or univalent, with a \ndistinguished univalent vertex called the root, embedded in the \nupper half-plane with the root at the origin -- for example $T =\n\\begin{xy}\n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-}; \n (7.5,4.5)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (3,6); (6,3)**\\dir{-},\n (6,6); (4.5,4.5)**\\dir{-},\n (0,4.2)*{\\scriptstyle 2},\n (3,4.2)*{\\scriptstyle 6},\n (3,7.2)*{\\scriptstyle 1},\n (6,7.2)*{\\scriptstyle 7},\n (7.5,5.7)*{\\scriptstyle 3},\n\\end{xy}$. \nUnivalent vertices other than the root are called leaves, and they\nare labeled by a subset $S$ of some set $\\mathbf{n} = \\{ 1, \\ldots, n \\}$.\nTrivalent vertices are also called internal vertices.\n\\item The height of a vertex in an $S$-tree, denoted $h(v)$, is the number of edges between \n$v$ and the root. Edges which connect a vertex to higher vertices are called outgoing.\n\\item To define a subtree of $T$, take some vertex $v$ and all of the \nvertices and edges above it. Restrict the ambient embedding in the upper half plane,\nand add a root edge from $v$ to the origin, to obtain a tree we call $T_v$.\nMoreover, let $T_v^L$ be the subtree associated to the left vertex over $v$,\nand similarly $T_v^R$ be the right subtree over $v$.\n\\item We say that $v$ is above or over $w$ if $w$ lies in the shortest path from\n$v$ to the root. Define a total order on the vertices of $T$ so that $v < w$ if $v$ \nlies over the left outgoing edge of $w$ and $v > w$ if it lies over the right \noutgoing edge of $w$. This total ordering can be realized as a left\nto right ordering of an appropriate planar embedding.\n\n\\end{enumerate}\n\\end{definition}\n\nWe now define the ``centers of mass'' for our systems and sub-systems.\n\n\\begin{definition}\nThe center $c({\\bf x}, T)$ of a configuration ${\\bf x}$ with respect to a tree\n$T$ is defined inductively by $c({\\bf x}, T_{v}) = \\frac{1}{2}\\left( c({\\bf x}, {T_v^L}) +\nc({\\bf x}, {T_v^R}) \\right)$, if $T$ has at least one internal vertex. If $T$ consists\nof only a leaf labeled by $i$, then $c({\\bf x}, T) = x_i$.\n\\end{definition}\n\nFinally, we can define the systems as ones where planets in a (sub)system\nare of a prescribed distance from the center of mass. Fix (for the moment)\nan $\\varepsilon < \\frac{1}{3}$.\n\n\\begin{definition}\\label{D:PT}\nGiven an $S$-tree $T$, the (planetary system) $P_T$ is the submanifold\nof all ${\\bf x} = (x_1, \\cdots, x_n)$ such that:\n\\begin{enumerate}\n\\item $c({\\bf x}, T) = 0$. \\label{p1}\n\\item For any vertex $v$ of $T$, \n$ d\\left(c({\\bf x}, {T_v^L}), c({\\bf x}, {T_v})\\right) = \\varepsilon^{h(v)} = \n d\\left(c({\\bf x}, {T_v}), c({\\bf x}, {T_v^R})\\right),$ where $d$ is the standard\n Euclidean distance function. \\label{p2}\n\\item If $i \\notin S$, $x_i$ is fixed as some point ``at infinity.'' \\label{p3}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{figure}[ht]\\label{F:1}\n\\psfrag{J}{$x_2$}\n\\psfrag{K}{$x_6$}\n\\psfrag{H}{$x_3$}\n\\psfrag{E}{$x_1$}\n\\psfrag{F}{$x_7$}\n\\psfrag{R}{$x_4$}\n\\psfrag{U}{$x_5$}\n$$\\includegraphics[width=12cm]{Figures\/PT.eps}$$\n\\caption{An illustration of $P_T$} \\end{figure}\n \n\nWe picture these submanifolds as in Figure~1, which illustrates the case of \n$T =\n\\begin{xy}\n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-}; \n (7.5,4.5)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (3,6); (6,3)**\\dir{-},\n (6,6); (4.5,4.5)**\\dir{-},\n (0,4.2)*{\\scriptstyle 2},\n (3,4.2)*{\\scriptstyle 6},\n (3,7.2)*{\\scriptstyle 1},\n (6,7.2)*{\\scriptstyle 7},\n (7.5,5.7)*{\\scriptstyle 3},\n\\end{xy}$\n\nOne configuration in this submanifold is illustrated by the $\\bullet$, which are labeled by\n $x_i$. The rest of the family is indicated by drawing some of \n the circular orbits where\n points in these configurations occur. The centers of this configuration,\nnamely the points $c({\\bf x}, {T_v})$, are indicated by $\\circ$.\n In any configuration in $P_T$, the points $x_4$ and $x_5$ occur \n where they are indicated.\n\nWe will use $P_{T}$ to define a homology class but to do so integrally, rather\nthan only with $\\mathbb{Z}\/2$ coefficients, it must be oriented. We orient $P_{T}$\nby parametrizing it through a map from a torus.\n\n\\begin{definition}\\label{D:param}\nBy abuse of notation, let $P_{T} : (S^{d-1})^{\\times|T|} \\to {\\rm Conf}_{n}(\\mathbb{R}^{d})$,\nwhere $|T|$ is the number of internal vertices of $T$, send\n$(u_{v_{1}}, \\ldots, u_{v_{|T|}})$ to $(x_{1}, \\ldots, x_{n})$, where\n$$x_{i} = \\sum_{v_{j} {\\; \\rm below \\; leaf} \\; i} \\pm \\varepsilon^{h_{j}} u_{v_{j}}.$$\nHere the sum is taken over all vertices $v_{j}$ which lie on the path\nfrom the leaf labeled by $i$ to the root vertex, and $h_{j}$ is the\nheight of $v_{j}$. The sign $\\pm$ is $+1$ if the path from \nthe leaf $i$ to the root goes through the left edge of $v_{j}$ and\n$-1$ if that path goes through the right edge of $v_{j}$.\n\\end{definition}\n\nWe may now orient $P_{T}$ by fixing an orientation of the sphere\nand using the product orientation for $ (S^{d-1})^{\\times|T|}$.\nWe will call the resulting homology class simply \n$T \\in H_{|T|(d-1)}({\\rm Conf}_{n}\\left(\\mathbb{R}^{d})\\right)$. Note that by its definition, $T$ is \nin the image of the map from the oriented bordism of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$.\nIn fact, because spheres are stably framed it is in the \nimage of the map from framed bordism, or equivalently stable homotopy.\n\nThe relations\nbetween these homology classes represent a fundamental blending of geometry and algebra.\n\n\n\\begin{proposition}\\label{P:relations}\nThe classes in $H_{*}({\\rm Conf}(\\mathbb{R}^{d}))$ given by trees satisfy the following relations:\n\\begin{align*}\n\\text{(anti-symmetry)} \\qquad \n & \\qquad\n\\begin{xy}\n (0,1.5); (1.5,0)**\\dir{-}; \n (3,1.5)**\\dir{-}, \n (1.5,-1.5); (1.5,0)**\\dir{-}, \n (-.4,2.7)*{\\scriptstyle T_1}, \n (3.8,2.7)*{\\scriptstyle T_2}, \n (1.5,-2.7)*{\\scriptstyle R}\n\\end{xy}\\ -\n (-1)^{d + |T_{1}||T_{2}|(d-1)} \\begin{xy}\n (0,1.5); (1.5,0)**\\dir{-}; \n (3,1.5)**\\dir{-}, \n (1.5,-1.5); (1.5,0)**\\dir{-}, \n (-.4,2.7)*{\\scriptstyle T_2}, \n (3.8,2.7)*{\\scriptstyle T_1}, \n (1.5,-2.7)*{\\scriptstyle R}\n\\end{xy} =\\ 0\\\\\n \\text{(Jacobi)} \\qquad \n & \\qquad \n \\begin{xy} \n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-};\n (6,3)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (-.4,4.2)*{\\scriptstyle T_1}, \n (3.2,4.2)*{\\scriptstyle T_2},\n (6.8,4.2)*{\\scriptstyle T_3}, \n (3,-2.7)*{\\scriptstyle R}\n \\end{xy} \\ + \\ \\begin{xy} \n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-};\n (6,3)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (-.4,4.2)*{\\scriptstyle T_2}, \n (3.2,4.2)*{\\scriptstyle T_3},\n (6.8,4.2)*{\\scriptstyle T_1}, \n (3,-2.7)*{\\scriptstyle R}\n \\end{xy} \\ + \\ \\begin{xy} \n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-};\n (6,3)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (-.4,4.2)*{\\scriptstyle T_3}, \n (3.2,4.2)*{\\scriptstyle T_1},\n (6.8,4.2)*{\\scriptstyle T_2}, \n (3,-2.7)*{\\scriptstyle R}\n \\end{xy} \n \\ =\\ 0\n\\end{align*}\nwhere $R$, $T_1$, $T_2$, and $T_3$ stand for arbitrary (possibly trivial) \nsubtrees which are not modified in these operations, and $|T_{i}|$ denotes\nthe number of internal vertices of $T_{i}$.\n\\end{proposition}\n\nWe call this relation the Jacobi identity because of the standard translation\nbetween $S$-trees and bracket expressions, under which this becomes\n$[[T_{1}, T_{2}], T_{3}] + [[T_{2}, T_{3}], T_{1}] + [[T_{3}, T_{1}], T_{2}] = 0$.\n\n\\begin{proof}\nThe anti-symmetry relation follows because the submanifolds defined\nby these two trees are the same, and their parametrizations differ only by\nthe antipodal map on one factor of $S^{d-1}$ from \\refD{param} and\nreordering by moving the factors labeled by vertices\nof $T_{1}$ after those of $T_{2}$\n\nThe Jacobi identity follows from the existence of Jacobi manifolds who\nbound the submanifolds in that relation. Letting $T$ be the first tree pictured in the Jacobi\nidentity above, consider the submanifold of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$\ndefined by conditions (\\ref{p1}) and (\\ref{p3}) from \\refD{PT}, as well as condition~(\\ref{p2}) \nfor vertices internal to $T_{1}$, $T_{2}$, $T_{3}$ or $R$. For the remaining\ntwo vertices we replace condition~(\\ref{p2}) by\n\\begin{enumerate}\n \\item[(2.1)] $\\sum_{i, j \\in \\{1, 2, 3\\}} \n d\\left(c({\\bf x}, {T_{i}}), c({\\bf x}, {T_{j}}) \\right) = 4\\varepsilon^{h} + 2\\varepsilon^{h+1}$,\n where $h$ is the height of the internal vertex immediately above the\n subtree $R$. \\label{j3}\n \\item[(2.2)] $d\\left(c({\\bf x}, {T_{i}}), c({\\bf x}, {T_{j}}) \\right) \\geq 2 \\varepsilon^{h+1}$, \n where $i,j \\in \\{1, 2, 3\\}$. \\label{j4}\n\\end{enumerate}\nCondition~(2.1) fixes the perimeter of the triangle with vertices at the centers\nof the sub-configurations associated to $T_{1}$, $T_{2}$, and $T_{3}$, and condition~(2.5)\nsays that triangle must have a minimum side length of at least $2\\varepsilon^{h+1}$.\n\nThese conditions determine a submanifold $J$\nwith boundary, whose boundary is where one of the three distances\n$d\\left(c({\\bf x}, {T_{i}}), c({\\bf x}, {T_{j}}) \\right)$ is equal to $2 \\varepsilon^{h+1}$. There\nare thus three boundary components.\nBy condition~(2.1), when some\n$d\\left(c({\\bf x}, {T_{i}}), c({\\bf x}, {T_{j}}) \\right) = 2 \\varepsilon^{h+1}$\nthe remaining center must be \ndistance roughly $2 \\varepsilon^{h}$ from the other two. So these \ncomponents of $\\partial J$ are close to being the \nsubmanifolds $P_{T}$ for the $T$ which occur in the\nJacobi identity -- see Figure~\\ref{F3}.\nWe get exactly those manifolds by replacing condition~(2.1) by\n\\begin{enumerate}\n\\item[(2.1')] $d\\left(c({\\bf x}, {T_{i}}), c({\\bf x}, {T_{j}}) \\right) = f({\\bf x})$.\n\\end{enumerate}\nHere $f({\\bf x})$ is an interpolation function. Its value is \n$4\\varepsilon^{h} + 2\\varepsilon^{h+1}$ when the $c{{\\bf x}, T_{i}}$ form a nearly equilateral\ntriangle. When the configuration in question is in $P_{T}$, its value is \nthe total length of the triangle with vertices\nat $c({\\bf x},{T_{i}})$, namely\n$$ \\sqrt{4 \\varepsilon^{2h} + \\varepsilon^{4h} - 4 \\varepsilon^{3h} \\cos(\\theta)} + \n\\sqrt{4 \\varepsilon^{2h} + \\varepsilon^{4h} + 4 \\varepsilon^{3h} \\cos(\\theta)} +\n2 \\varepsilon^{h+1},$$\nwhere $\\theta$ is the angle pictured in Figure~\\ref{F3}.\n\n\\begin{figure}[ht]\\label{F3}\n\\psfrag{A}{$c({\\bf x}, {T_{k}})$}\n\\psfrag{J}{$\\varepsilon^{h}$}\n\\psfrag{I}{$\\theta$}\n\\psfrag{H}{$c({\\bf x}, {T_{i}})$}\n\\psfrag{E}{ }\n\\psfrag{F}{$\\varepsilon^{h+1}$}\n\\psfrag{R}{$c({\\bf x}, {T_{j}})$}\n\\psfrag{U}{ }\n$$\\includegraphics[width=10cm]{Figures\/angle.eps}$$\n\\caption{ The geometry of $P_{T}$ for $T$ as in the Jacobi identity.}\n\\end{figure}\n\nWe argue by symmetry that the orientations of the\\ three \ncomponents of $\\partial J$ gives rise to the Jacobi identity exactly.\nAgain referring to Figure~2, for each boundary component we can define\nan inward normal vector through having $c({\\bf x}, {T_{i}})$ and $c({\\bf x}, {T_{j}})$ move\nradially outward, away from their center, and thus needing $c({\\bf x}, {T_{k}})$ move\nradially inward so that condition~(2.2) is satisfied. This normal vector\nis invariant under cyclic permutation of $T_{1}$, $T_{2}$ and $T_{3}$, as is\nthe definition of orientation for the $P_{T}$ for $T$ which appear in the\nJacobi identity. Thus, these orientations will either all agree or all\ndisagree with a chosen\norientation of $J$, meaning in either case that the Jacobi identity holds.\n\\end{proof}\n\nFinally, we allow for multiple planetary systems, \nfreeing the points which do not move in the definition\nof $P_{T}$, for example the points $x_{4}$ and $x_{5}$ in Figure~1.\n\n\\begin{definition}\n\\begin{itemize}\n\\item An $n$-forest is a collection of (by abuse) $S$-trees, with root vertices\nat the points $(0, 0)$, $(1, 0)$, \\ldots in the upper half-plane, where each\ninteger from $1$ to $n$ labels exactly one leaf.\n\\item If $F = \\bigcup T_{i}$ is a forest, let $P_{F}$ be the submanifold defined \nby conditions~(\\ref{p2}) and (\\ref{p3}) of \\refD{PT}, replacing condition~(\\ref{p1}) with \n$c({\\bf x}, {T_{i}}) = (i, 0, \\ldots, 0)$.\n\\item Parameterize $P_{F}$ by a map of the same name from a product\nover vertices of $F$ (ordered from left to right by the half-planar embedding\nof $F$) of spheres, which when restricted to the coordinates labeled by $T_{i}$ is\na translation of $P_{T_{i}}$, namely $P_{T_{i}} + (i, 0, \\ldots, 0)$.\n\\item By abuse let $F$ denote the homology class represented by $P_{F}$,\nagain using our fixed orientation of a sphere to orient the torus and thus its image\nunder $P_{F}$.\n\\end{itemize}\n\\end{definition}\n\nWe recover $P_{T}$ by letting $F$ be a forest which consists of $T$\nand a collection of one-leaf trees.\nWe can summarize our results so far as follows.\n\n\\begin{definition}\nLet ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ denote the quotient of the free module spanned by $n$-forests\nby anti-symmetry and Jacobi identities\nas in \\refP{relations} along with the following:\n\\begin{enumerate}\n\\item[] {\\it{(commutativity) \\, If $F_{1}$ and $F_{2}$ consist\nof the same trees, then $F_{1} = \\sigma^{(d-1)} F_{2}$, where $\\sigma$ is the\nsign of the permutation which relates the ordering of the internal vertices of\nthe trees in $F_{1}$ with those of $F_{2}$.}}\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\nSending a forest $F$ to the image of the fundamental class of $(S^{d-1})^{\\times |F|}$\nunder $P_{F}$ gives a well-defined homomorphism from ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ to \n$H_{*}\\left({\\rm Conf}_{n}(\\mathbb{R}^{d})\\right)$.\n\\end{theorem}\n\nOur main theorem will be that this map is an isomorphism (of {\\em operads}).\n\n\n\\subsection{Canonical realization after compactification}\n\nThere are a number of choices we have made in our definition of $P_{T}$,\nin particular the scale $\\varepsilon$. It is tempting to let $\\varepsilon$\ngo to zero, which is indeed possible with some recent ``compactification technology.''\nThere is a canonical completion of these configuration spaces due to \nFulton-MacPherson \\cite{FuMa94} and Axelrod-Singer \\cite{AxSi94},\nwhich we denote ${\\rm Conf}_{n}[\\mathbb{R}^{k}]$ in \\cite{Sinh04}. There we give the following\nelementary definition. \n\n\n\\begin{definition}\\label{D:compact}\n\\begin{itemize}\n\\item Define $\\alpha_{ij} \\colon {\\rm Conf}_n(\\mathbb{R}^{d}) \\to S^{d-1}$ as sending \n$(x_1, \\cdots, x_n)$ to $\\frac{x_j - x_i}{|x_j - x_i|}$. \n\\item Let $I = [0,\\infty]$, the one-point compactification of the nonnegative \nreals, and for $i,j,k$ distinct numbers between $1$ and $n$ \nlet $s_{ijk} \\colon {\\rm Conf}_n(\\mathbb{R}^{d}) \\to I = [0, \\infty]$ \nbe the map which sends $(x_1, \\cdots, x_n)$ to $\\left(|x_i - x_j|\/|x_i - x_k|\\right)$. \n\\item Let ${\\rm Conf}_{n}[\\mathbb{R}^{d}]$ be the closer of the image of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ \nin $$(\\mathbb{R}^{d})^{\\times n} \\times \\left(S^{d-1}\\right)^{\\times\\binom{n}{2}} \\times I^{\\times \\binom{n}{3}},$$\nunder the map which is the canonical inclusion on the first factor, the product over all $a_{ij}$ on the\nsecond factor, and the product over all $s_{ijk}$ on the third factor.\n\\end{itemize}\n\\end{definition}\n\nWith such an elementary definition, the hard work is to establish basic properties.\nThis completion is functorial for embeddings and yields a manifold with corners with\n${\\rm Conf}_{n}(\\mathbb{R}^{k})$ as its interior, and its strata are naturally indexed by trees (which are\nnot necessarily trivalent). The points added by the boundary may be viewed as \n``degenerate configurations'' in which some number of points now coincide in the\nlarge scale but have data which resolves them ``infinitesimally.'' See \\cite{Sinh04} for a\nmore thorough treatment.\n\n\nWhen the submanifold $P_{T}$ is included\nin ${\\rm Conf}_{n}[\\mathbb{R}^{k}]$, we may send $\\varepsilon$ in its definition to zero. We leave this\nas an exercise for the moment, since composing the map $P_{T}$ with the projections\n$\\alpha_{ij}$ will be a key calculation in \\refT{agree}.\nAfter this canonical homotopy of $P_{T}$ sending $\\varepsilon$ to zero, the\nresulting submanifold ends up being the stratum labeled by $T$ in the stratification\nof ${\\rm Conf}_{n}[\\mathbb{R}^{k}]$ as a manifold with corners. \nThus these strata represent the homology classes we have\nbeen constructing. The Jacobi manifolds are also homotopic to strata \nwhich labeled by trees with one four-valent vertex, and their boundaries as strata\ngive rise to the Jacobi identity. Indeed,\nthe manifold with boundary ${\\rm Conf}_{3}[\\mathbb{R}^{k}]$ is diffeomorphic to \nthe simplest Jacobi manifold.\nThus the compactification ${\\rm Conf}_{n}[\\mathbb{R}^{k}]$ ``wears its homology \non its strata.'' \n\n\\subsection{Historical notes}\n\nTo our knowledge, the approach to homology through ``orbital systems''\nfirst appeared implicitly in Fred Cohen's thesis work (\\cite{CohF73}, in particular Section~12 of part III). \nThis approach coincides with the ``twisted products''\nconstructions from Fadell and Husseini's book (\\cite{FaHu01}, Chapter VI), but in their approach trees and forests \ndo not explicitly appear. The role of trees and forests, and the explicit connection with the theory of operads,\nhas been in the air since the ``renaissance'' of that theory, in particular \\cite{GeJo94} which also emphasizes\nthe role of compactifications. One of our aims is to\nmake this basic theory which is well-known to experts as accessible as possible.\n\n\n\n\\section{The cohomology ring}\\label{S:cohomology}\n\nIn the previous section we constructed homology classes for the space\nof Euclidean configurations\nby mapping in fundamental classes of tori. \nIn this section we pull back cohomology from tori.\nIf homology classes in configuration spaces may be viewed as planetary\nsystems, cohomology classes may be view as recording planetary alignments.\n\n\\begin{definition}\\label{D:aij}\nRecall $\\alpha_{ij} \\colon {\\rm Conf}_n(\\mathbb{R}^{d}) \\to S^{d-1}$ as sending \n$(x_1, \\cdots, x_n)$ to $\\frac{x_j - x_i}{|x_j - x_i|}$. \nLet $\\iota \\in H^{d-1}(S^{d-1})$ denote the dual to the fundamental class, using\nour fixed orientation. \nLet $a_{ij}$ denote $\\alpha_{ij}^*(\\iota)$. \n\\end{definition}\n\nThe ring generated by these $a_{ij}$ can be represented graphically.\n\n\\begin{definition}\nConsider graphs with vertices labeled $1,\\ldots,n$, \nwith edges which are oriented and ordered. Let $\\Gamma(n)$\ndenote the free module generated by such graphs, which is a ring by taking\nthe union of edges of two graphs in order to multiply them (using\nthe order of multiplication to define the ordering on the union of edges). Map ${\\Gamma}(n)$\nto $H^*\\left({\\rm Conf}_n(\\mathbb{R}^{d})\\right)$ by sending a generator $\\linep{i}{j}$ to $a_{ij}$.\n\\end{definition}\n\nSo for example the graph $ \\linep{4}{2} \\linep{1}{3}$, \nwith $\\linep{4}{2}$ first in the ordering of edges, is mapped to the product $a_{42} a_{13}$.\nWe will see that the map from $\\Gamma(n)$ to $H^*\\left({\\rm Conf}_n(\\mathbb{R}^{d})\\right)$ is surjective.\nAs a base case, we show that after quotienting by the the relation $\\linep{i}{j} =\n(-1)^{d} \\linep{j}{i}$, this map is an isomorphism in degree $d-1$.\n\n\\begin{definition}\nLet $p_{i} : {\\rm Conf}_{n}(\\mathbb{R}^{d}) \\to {\\rm Conf}_{n-1}(\\mathbb{R}^{d})$ be the projection map which sends\n$(x_{1}, \\ldots, x_{n})$ to $(x_{1}, \\ldots, \\hat{x_{i}}, \\ldots, x_{n})$.\n\\end{definition}\n\n\\begin{lemma}\nThe projection map $p_{i}$ gives ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ the structure of a fiber bundle over\n${\\rm Conf}_{n-1}(\\mathbb{R}^{d})$, with fiber given by $\\mathbb{R}^{d}$ with $(n-1)$ points removed.\n\\end{lemma}\n\n\\begin{proof}\nFor simplicity let $i=n$. Consider a neighborhood $U_{{\\bf x}}$\nof ${\\bf x} = (x_{1}, \\ldots, x_{n-1})$\nof points $(y_{1}, \\ldots, y_{n})$ where $d(y_{j}, x_{j}) < \\epsilon$ for some fixed $\\epsilon$\nless than the minimum of the $\\frac{1}{5}d(x_{j}, x_{k})$. Construct a continuous family\nof homeomorphsims $h_{{\\bf y}}$ over $y \\in U_{{\\bf x}}$\nbetween $\\mathbb{R}^{d} - \\{y_{1}, \\ldots, y_{n-1}\\}$ and $\\mathbb{R}^{d} - \\{B(x_{1}, \\epsilon), \\ldots,\nB(x_{n-1}, \\epsilon)\\}$, which for good measure is the identity on \n$\\mathbb{R}^{d} - \\{B(x_{1}, 2\\epsilon), \\ldots,B(x_{n-1}, 2\\epsilon)\\}$. This may be done, for\nexample, by ``straight-line retractions.'' \nA trivialization of this fiber bundle, in other words a homeomorphism between \n$p_{n}^{-1}(U_{{\\bf x}})$ and $U_{{\\bf x}} \\times \\mathbb{R}^{d} - \\{x_{1}, \\ldots, x_{n-1}\\}$ \nrespecting $p_{n}$, is given by \n$$(y_{1}, \\cdots, y_{n}) \\mapsto (y_{1}, \\cdots, y_{n-1}) \\times h_{{\\bf x}}^{-1} \\circ h_{{\\bf y}}(y_{n}).$$\n\\end{proof}\n\nThe space $\\mathbb{R}^{d}$ with $(n-1)$ points removed retracts onto $\\bigvee_{n-1}S^{d-1}$.\nWe assemble these projection maps into a tower of fibrations first studied by \nFadell and Neuwirth \\cite{FaNe62}, which is central\nin the study of the topology of configuration spaces.\n\\begin{equation}\\label{tower}\n\\xymatrix@=18pt{ \n \\bigvee_{n-1}S^{d-1} \\ar@<.5ex>[r] & {\\rm Conf}_{n}(\\mathbb{R}^{d}) \\ar[d]^{p_{n}}\\\\\n \\bigvee_{n-2}S^{d-1} \\ar[r] & {\\rm Conf}_{n-1}(\\mathbb{R}^{d}) \\ar[d]^{p_{n-1}}\\\\\n & \\vdots \\ar[d]\\\\\n \\bigvee_{2}S^{d-1} \\ar[r] & {\\rm Conf}_{3}(\\mathbb{R}^{d}) \\ar[d]^{p_{n}}\\\\\n & {\\rm Conf}_{2}(\\mathbb{R}^{d}) \\simeq S^{d-1}\n }\n\\end{equation}\nThese fibrations split. Choice of sections of $p_{i}$ include adding a new $i$th\npoint ``at infinity'' or somehow ``doubling'' the $i$th point.\n\n\\begin{proposition}\nThe first non-trivial homology group $H_{d-1}({\\rm Conf}_{n}))$ is free of rank $\\binom{n}{2}$.\n\\end{proposition}\n\n\\begin{proof}\nWe give a proof only for $n>2$. We use the long exact sequences in homotopy of the fibrations \nconstituting the tower of Equation~(\\ref{tower}), which splits into short exact sequences because the\nmaps $p_{i}$ admit splittings. We base all of our configuration spaces\nat the configuration with $x_{i} = (i, 0, \\ldots, 0)$.\n\nFrom these short exact sequences, we deduce inductively \nthat ${\\rm Conf}_{i}(\\mathbb{R}^{d})$ is $(d-2)$-connected,\nand that $\\pi_{d-1}\\left({\\rm Conf}_{i}(\\mathbb{R}^{d})\\right)$ is free. Moreover, the rank of \n$\\pi_{d-1}\\left({\\rm Conf}_{i}(\\mathbb{R}^{d})\\right)$\nis that of $\\pi_{d-1}\\left({\\rm Conf}_{i-1}(\\mathbb{R}^{d})\\right)$ plus $i-1$, or ultimately $1 + 2 + \\cdots + (i-1)$,\nwhich is $\\binom{i}{2}$. Finally, the Hurewicz theorem applies to give that the homology\nis isomorphic to homotopy in this dimension.\n\\end{proof}\n\nWe now show that the homology and cohomology classes that we have constructed\nso far generate this first non-trivial group.\nLet $\\langle \\, , \\, \\rangle$ denote the standard pairing of cohomology and homology.\n\n\\begin{lemma}\nLet $i*\\dir{>}, \n \"b\";\"c\"**\\dir{-}?>*\\dir{>}, \n (3,-5),{\\ar@{. }@(l,l)(3,6)},\n ?!{\"a\";\"a\"+\/va(210)\/}=\"a1\",\n ?!{\"a\";\"a\"+\/va(240)\/}=\"a2\",\n ?!{\"a\";\"a\"+\/va(270)\/}=\"a3\",\n ?!{\"b\";\"b\"+\/va(120)\/}=\"b1\",\n \"a\";\"a1\"**\\dir{-}, \"a\";\"a2\"**\\dir{-}, \"a\";\"a3\"**\\dir{-},\n \"b\";\"b1\"**\\dir{-}, \"b\";(3,6)**\\dir{-},\n (3,-5),{\\ar@{. }@(r,r)(3,6)},\n ?!{\"c\";\"c\"+\/va(-90)\/}=\"c1\",\n ?!{\"c\";\"c\"+\/va(-60)\/}=\"c2\",\n ?!{\"c\";\"c\"+\/va(-30)\/}=\"c3\",\n ?!{\"b\";\"b\"+\/va(60)\/}=\"b3\",\n \"c\";\"c1\"**\\dir{-}, \"c\";\"c2\"**\\dir{-}, \"c\";\"c3\"**\\dir{-},\n \"b\";\"b3\"**\\dir{-}, \n\\end{xy}\\ + \\ \n\\begin{xy} \n (0,-2)*+UR{\\scriptstyle j}=\"a\", \n (3,3)*+UR{\\scriptstyle k}=\"b\", \n (6,-2)*+UR{\\scriptstyle \\ell}=\"c\", \n \"b\";\"c\"**\\dir{-}?>*\\dir{>}, \n \"c\";\"a\"**\\dir{-}?>*\\dir{>}, \n (3,-5),{\\ar@{. }@(l,l)(3,6)},\n ?!{\"a\";\"a\"+\/va(210)\/}=\"a1\",\n ?!{\"a\";\"a\"+\/va(240)\/}=\"a2\",\n ?!{\"a\";\"a\"+\/va(270)\/}=\"a3\",\n ?!{\"b\";\"b\"+\/va(120)\/}=\"b1\",\n \"a\";\"a1\"**\\dir{-}, \"a\";\"a2\"**\\dir{-}, \"a\";\"a3\"**\\dir{-},\n \"b\";\"b1\"**\\dir{-}, \"b\";(3,6)**\\dir{-},\n (3,-5),{\\ar@{. }@(r,r)(3,6)},\n ?!{\"c\";\"c\"+\/va(-90)\/}=\"c1\",\n ?!{\"c\";\"c\"+\/va(-60)\/}=\"c2\",\n ?!{\"c\";\"c\"+\/va(-30)\/}=\"c3\",\n ?!{\"b\";\"b\"+\/va(60)\/}=\"b3\",\n \"c\";\"c1\"**\\dir{-}, \"c\";\"c2\"**\\dir{-}, \"c\";\"c3\"**\\dir{-},\n \"b\";\"b3\"**\\dir{-}, \n\\end{xy}\\ + \\ \n\\begin{xy} \n (0,-2)*+UR{\\scriptstyle j}=\"a\", \n (3,3)*+UR{\\scriptstyle k}=\"b\", \n (6,-2)*+UR{\\scriptstyle \\ell}=\"c\", \n \"a\";\"b\"**\\dir{-}?>*\\dir{>}, \n \"c\";\"a\"**\\dir{-}?>*\\dir{>}, \n (3,-5),{\\ar@{. }@(l,l)(3,6)},\n ?!{\"a\";\"a\"+\/va(210)\/}=\"a1\",\n ?!{\"a\";\"a\"+\/va(240)\/}=\"a2\",\n ?!{\"a\";\"a\"+\/va(270)\/}=\"a3\",\n ?!{\"b\";\"b\"+\/va(120)\/}=\"b1\",\n \"a\";\"a1\"**\\dir{-}, \"a\";\"a2\"**\\dir{-}, \"a\";\"a3\"**\\dir{-},\n \"b\";\"b1\"**\\dir{-}, \"b\";(3,6)**\\dir{-},\n (3,-5),{\\ar@{. }@(r,r)(3,6)},\n ?!{\"c\";\"c\"+\/va(-90)\/}=\"c1\",\n ?!{\"c\";\"c\"+\/va(-60)\/}=\"c2\",\n ?!{\"c\";\"c\"+\/va(-30)\/}=\"c3\",\n ?!{\"b\";\"b\"+\/va(60)\/}=\"b3\",\n \"c\";\"c1\"**\\dir{-}, \"c\";\"c2\"**\\dir{-}, \"c\";\"c3\"**\\dir{-},\n \"b\";\"b3\"**\\dir{-}, \n\\end{xy}\\ =\\ 0,\n$$\nwhere ${j}$, ${k}$, and ${\\ell}$ stand for \nvertices in the graph which could possibly have other connections to\nother parts of the graph (indicated by the ends of edges abutting \n$j$, $k$, and $\\ell$)\nwhich are not modified in these operations. \n\\end{theorem}\n\n\\begin{proof}\nUsing the ring structure, it suffices to consider when \nthere there are no edges incident on $j$, $k$\nand $\\ell$, other than the two edges involved in the identity. \nIn this case the cohomology classes are all pulled \nback from $H^{2(d-1)}\\left({\\rm Conf}_{3}(\\mathbb{R}^{d})\\right)$ via a map \nwhich forgets all $x_{i}$ except $x_{j}$, $x_{k}$ and $x_{\\ell}$.\nSo we may assume that $n=3$ and $\\{j, k, \\ell\\} = \\{1, 2, 3\\}$.\n\nOur proof uses elementary intersection theory to compute some cup products.\nSince ${\\rm Conf}_{3}(\\mathbb{R}^{k})$ is a manifold, its cohomology is Lefshetz dual to \nits locally finite homology. \nConsider the submanifold of \n$(x_{1}, x_{2}, x_{3}) \\in {\\rm Conf}_{3}(\\mathbb{R}^{k})$ such that\n$x_{1}$, $x_{2}$, and $x_{3}$ are collinear.\nThis submanifold has three components. Let ${\\rm col}_{i}$ denote the component in \nwhich $x_{i}$ is in the middle. \nSince ${\\rm col}_{i}$ is a properly embedded submanifold of codimension $d-1$, \nonce oriented it represents a locally finite homology class, which through Lefshetz duality \ngives rise to a class in $H^{d-1}({\\rm Conf}_{3}(\\mathbb{R}^{d}))$. By \\refC{evald-1}, \nthis class is determined by its value on the classes $\\tree{j}{k}$, which in the context\nof Lefshetz duality means intersecting ${\\rm col}_{i}$ with \nvarious $P_T$.\n\nThese intersections can be understood directly. The submanifold \n${\\rm col}_{i}$ can only intersect $P_{\\tree{i}{j}}$ or $P_{\\tree{i}{k}}$, since otherwise $x_{i}$\nwould be ``at infinity'', and thus\ncould not be in the middle of a collinearity. Moreover, $P_{\\tree{i}{j}}$ and $P_{\\tree{i}{k}}$ each\nintersect ${\\rm col}_{i}$ exactly once, namely when $x_{i}$ is a negative multiple of\n$x_{j}$ (respectively, $x_{k}$). For purposes of our computation we only need\nthat these intersections differ in sign by $-1$, coming from orientation reversing\nof the line on which the three points lie. We deduce that the\ncohomology class represented by ${\\rm col}_{i}$ is $\\pm(a_{ij} - a_{ik})$.\n\nUnder Lefshetz duality, cup products are computed by\n(transversal) intersections. Since ${\\rm col}_{1}$ and ${\\rm col}_{2}$ are disjoint, the\ncohomology classes which they represent cup to zero. We have\n\\begin{align*}\n0 &= \\pm (a_{12} - a_{13})(a_{23} - a_{21}) \\\\\n&= a_{12}a_{23} - a_{12}a_{21} - a_{13}a_{23} + a_{13}a_{21} \\\\\n&= a_{12}a_{23} + 0 - (-1)^{d + (d+1)}a_{23}a_{31} +(-1)^{2d} a_{31}a_{12} \\\\\n&= a_{12}a_{23} + a_{23}a_{31} + a_{31}a_{12}.\n\\end{align*}\nWhen we translate back to the graphical language, this is exactly the Arnold\nidentity.\n\\end{proof}\n\nThis new proof through the disjointness of collinearity submanifolds is using\na fundamental geometric observation as the basis for a cohomology ring computation,\nakin to seeing the cohomology ring of projective spaces through the \nintersections of linear subspaces. \nUsing cochains defined through collinearities works better for\nthis calculation than our\noriginal cochains representing the classes $a_{ij}$, for which we have that\nthe quadratic polynomial in the Arnold identity is exact but not identically\nzero (unless $d=2$, where Kontsevich observed the vanishing using the\ndifferential forms ${\\rm dlog}\\;(x_{i} - x_{j})$.) The collinearity cochains are also invaraint\nunder the action of the full group of affine transformations in $\\mathbb{R}^{d}$.\n\nWe will see that there are no further relations among these graph classes\nin $H^{*}\\left({\\rm Conf}_{n}(\\mathbb{R}^{d})\\right)$, so that the image of $\\Gamma(n)$ \nwill be precisely the following module.\n\n\\begin{definition}\nLet ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ denote the quotient of $\\Gamma(n)$ by the\narrow-reversing relation and the Arnold identity.\n\\end{definition}\n\nInstead of starting with $\\Gamma(n)$, we can restrict to acyclic graphs.\n\n\\begin{proposition}\nAny element of ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ represented by a graph which has a cycle is zero.\n\\end{proposition}\n\n\\begin{proof}\nWe may use the Arnold identity inductively to reduce to graphs with shorter cycles.\nBut graphs with cycles of length two, that is which have more than one edge between\ntwo vertices, are zero.\n\\end{proof}\n\n\\subsection{Historical notes}\nAnalyzing projection maps and in particular assembling them into a tower has been a central\ntool in this area since its introduction by Fadell and Neuwirth \\cite{FaNe62}.\nThe calculation of cohomology of configurations in the plane is, famously, due to Arnold \\cite{Arno69}.\nIt was generalized in higher dimensions by Cohen \\cite{CLM76}, to the complements of other collections\nof subspaces defined by linear equations by Orlik and Solomon \\cite{OrSo80}, and to a myriad of\nother contexts. Note that what we call the Arnold identity, along with the fact that generators square to zero, \nare called the cohomological Yang-Baxter relations in Chapter V of \\cite{FaHu01}, which gives a complete\naccount of the spectral sequence approach to calculation. The graphical notation for this cohomology has\nbeen useful for a wide range of recent work, for example \\cite{KoSo00}.\n\n\n\\section{The homology-cohomology pairing}\\label{S:pairing}\n\nBuilding on the combinatorics which arose in the last two sections, we develop \na pairing between graphs and trees which coincides with the\nevaluation of cohomology on homology of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$.\n\n\n\\begin{definition}\\label{D:pairing}\nGiven an $n$-graph $G$ and an $n$-tree $T$, define the map \n$$\\beta_{G,T}:\\bigl\\{\\text{edges of } G\\bigr\\} \\longrightarrow \n\\bigl\\{\\text{internal vertices of } T\\bigr\\}$$ \nby sending an edge $\\linep{i}{j}$ in $G$ to the vertex at the nadir of \nthe shortest path in $T$ between the leafs with labels $i$ and $j$. \nDefine the mod-$2$ configuration pairing of $n$-graphs \n$G$ and $n$-trees $T$ by \n$$\\bigl\\langle G,\\, T\\bigr\\rangle = \n \\begin{cases} 1 & \\text{if $\\beta$ is a bijection} \\\\\n 0 & \\text{otherwise}.\n\\end{cases}$$\nDefine the dimension-$d$ configuration pairing by, in the first case above,\nintroducing the sign of the permutation relating the orderings of the edges\nof $G$ and internal vertices of $T$ given in \\refD{trees}\nwhen $d$ is even, or $(-1)^{k}$ where $k$\nis the number of edges $\\linep{i}{j}$ of $G$ for which leaf $i$\nis to the right of leaf $j$ under the planar embedding of $T$ when $d$ is odd.\n\n This definition extends to give a pairing\n between (possibly disconnected) $n$-graphs $G$ and $n$-forests $F$,\nwhich is zero if an edge of $G$ has endpoints which label leaves in two different \ncomponents of $F$ (so that $\\beta$ is not defined).\n\\end{definition}\n\n\n\\begin{figure}[ht]\n$$\\begin{aligned}\\begin{xy} \n (0,-2.5)*+UR{\\scriptstyle 1}=\"a\", \n (3.75,3.75)*+UR{\\scriptstyle 2}=\"b\", \n (7.5,-2.5)*+UR{\\scriptstyle 3}=\"c\", \n \"a\";\"b\"**\\dir{-}?>*\\dir{>} ?(.4)*!RD{\\scriptstyle e_1}, \n \"b\";\"c\"**\\dir{-}?>*\\dir{>} ?(.5)*!LD{\\scriptstyle e_2} \n \\end{xy}\\end{aligned}\\ \\longmapsto \\ \n \\begin{xy}\n (2,1.5); (4,3.5)**\\dir{-}, \n (0,3.5); (4,-.5)**\\dir{-} ?(.5)*!RU{\\scriptstyle \\beta(e_1)}; \n (4,-.5); (8,3.5)**\\dir{-} ?(.2)*!LU{\\scriptstyle \\beta(e_2)}, \n (4,-.5); (4,-2.5)**\\dir{-}, \n (0 ,4.7)*{\\scriptstyle 2}, \n (4,4.7)*{\\scriptstyle 1}, \n (8,4.7)*{\\scriptstyle 3}, \n (2,1.5)*{\\scriptstyle \\bullet},\n (4,-.5)*{\\scriptstyle \\bullet},\n \\end{xy} \\qquad \\qquad \\qquad \n \\begin{aligned}\\begin{xy} \n (0,-2.5)*+UR{\\scriptstyle 1}=\"a\", \n (3.75,3.75)*+UR{\\scriptstyle 2}=\"b\", \n (7.5,-2.5)*+UR{\\scriptstyle 3}=\"c\", \n \"a\";\"b\"**\\dir{-}?>*\\dir{>} ?(.4)*!RD{\\scriptstyle e_1}, \n \"b\";\"c\"**\\dir{-}?>*\\dir{>} ?(.5)*!LD{\\scriptstyle e_2} \n \\end{xy}\\end{aligned}\\ \\longmapsto \\ \n \\begin{xy}\n (2,1.5); (4,3.5)**\\dir{-}, \n (0,3.5); (4,-.5)**\\dir{-} ?(.8)*!RU{\\scriptstyle \\beta(e_1)}; \n (4,-.5); (8,3.5)**\\dir{-} ?(.2)*!LU{\\scriptstyle \\beta(e_2)}, \n (4,-.5); (4,-2.5)**\\dir{-}, \n (0 ,4.7)*{\\scriptstyle 1}, \n (4,4.7)*{\\scriptstyle 3}, \n (8,4.7)*{\\scriptstyle 2}, \n (4,-.5)*{\\scriptstyle \\bullet}, \n \\end{xy} $$\n \\caption{The map $\\beta_{G,T}$ for two different trees $T$. In the first\n case the configuration \n pairing is $-1$ if $d$ is odd or $1$ if $d$ is even, and in the second case it is zero.}\n\\end{figure}\n\n\n\\begin{theorem}\\label{T:agree}\nThe homology-cohomology pairing for ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ agrees with the configuration pairing.\nThat is, if we let $\\langle -, - \\rangle_{c}$ denote the combinatorially-defined configuration pairing,\nand let $\\langle -, - \\rangle_{H}$ denote the homology-cohomology pairing for\n${\\rm Conf}_{n}(\\mathbb{R}^{d})$, then $\\langle G, F \\rangle_{c} = \\langle G, F \\rangle_{H}$. \n\\end{theorem}\n\nHere we have continued the abuse of letting $G$ and $F$ denote both\ngraphs and trees and their images in cohomology and homology of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$.\n\n\\begin{proof}\nFor the homology pairing, we must evaluate a product of \nclasses $a_{ij}$ on a submanifold $P_{F}$,\nwhich by naturality of cap products is equal to computing the degree of the composite \n$$ \\pi_{G} \\circ P_{F} : \\prod_{v \\in F} S^{d-1} \\overset{P_{F}}{\\longrightarrow}\n {\\rm Conf}_{n}(\\mathbb{R}^{d}) \\overset{\\pi_{G}}{\\longrightarrow} \\prod_{e \\in G}S^{d-1}.$$\nHere $\\pi_{G}$ is the product over edges of $G$ which associates to\n$e = \\linep{i}{j}$ a factor of $\\pi_{ij}$.\nBy Definitions~\\ref{D:param} and \\ref{D:aij}, \nthis composite sends \n$(u_{v_{1}}, \\ldots, u_{v_{|F|}})$ to $(\\theta_{e_{1}}, \\ldots, \\theta_{e_{|G|}})$,\nwhere for $e = \\linep{i}{j}$, $\\theta_{e}$ is the unit vector in the direction of\n$$\\left( (n,0,\\cdots,0) + \\sum_{v_{k}<{\\rm leaf} \\; i} \\pm \\varepsilon^{h_{k}} u_{v_{k}} \\right) -\n\\left( (m, 0, \\dots, 0) + \\sum_{w_{\\ell}<{\\rm leaf} \\; j} \\pm \\varepsilon^{h_{\\ell}} u_{w_{\\ell}} \\right).$$\nHere $v_{k}$ is the vertex of height $k$ under leaf $i$, which is in the $n$th component\nof the forest $F$, and similarly $w_{\\ell}$\nis the vertex of height $\\ell$ under leaf $j$, which is in the $m$th component of $F$.\nIf leaves $i$ and $j$ are in the same component, common terms associated to \nvertices under both leaf $i$\nand leaf $j$ cancel, leaving $\\theta_{e}$ as the unit vector in the direction of\n$$\n\\varepsilon^{h_{n}} \\left(\\pm 2 u_{v_{n}} \\pm \\varepsilon (u_{v_{n+1}} - u_{w_{n+1}}) +\n \\varepsilon^{2} \\cdots \\right),$$\nwhere $v_{n}$ is the highest vertex under both leaf $i$ and $j$ (if it exists), which\nis also the nadir of the path between $i$ and $j$. \n\nConsider the homotopy of $P_{F}$, and thus this composite, in which \n$\\varepsilon$ approaches zero. Through this homotopy, \n$\\theta_{e}$ approaches either $(\\pm 1,0, \\cdots, 0)$ if leaves \n$i$ and $j$ are in different components, or otherwise $\\sigma_{e} u_{v_{n}}$.\nFrom \\refD{param} we see\nthat $\\sigma_{e}$ is $1$ if leaf $i$ is to the left of $j$ or $-1$ if it is to the right.\nTherefore, if there is some edge $\\linep{i}{j}$ in $G$ with leaves $i$\nand $j$ in different components of $F$, then $P_{F}$ is homotopic to the map between\ntori with at least one factor the constant map of $(\\pm 1, \\cdots, 0)$, and thus it is of degree zero.\nOtherwise, $P_{F}$ is homotopic to the map which sends\n$(u_{v_{1}}, \\ldots, u_{v_{|F|}}) \\in \\prod_{|F|} S^{d-1}$ to \n$$\\left(\\sigma_{e_{1}}u_{\\beta_{G,F}(e_{1})}, \\ldots, \n \\sigma_{e_{|G|}}u_{\\beta_{G,F}(e_{|G|})} \\right),$$\nwhose degree agrees with the definition of the configuration pairing through \n$\\beta_{G,F}$.\n\\end{proof}\n\nBecause the homology classes of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ represented by forests satisfy the \nanti-symmetry and Jacobi identities of \\refP{relations}, and the cohomology \nclasses represented by graphs \nsatisfy arrow-reversing and the Arnold identities, we have geometrically \nestablished the following fact, which is established combinatorially in \\cite{Sinh06.2}.\n\n\\begin{corollary}\\label{L:1}\nThe dimension-$d$ configuration pairing passes to a well-defined pairing\nbetween ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ and ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$.\n\\end{corollary}\n\nWe now outline the purely algebraic argument, given in \\cite{Sinh06.2}, that\nthis pairing between ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ and ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ is perfect.\nWe only give hints, leaving some fun for the reader.\n\n\\begin{lemma}\\label{L:3}\nThe module ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ is spanned by $n$-forests in which all trees are {\\em tall}\n(that is, the distance between the leaf with the minimal label and the root is maximal,\nand that leaf is leftmost in the planar ordering).\nThe module ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ is spanned by $n$-graphs whose components\nare {\\em long} (that is, each component is a linear graph, with one endpoint labeled by\nthe minimal label; edges are ordered consecutively and \noriented away from the minimal label).\n\\end{lemma}\n\n\\begin{figure}[ht]\\label{tallnlong}\n$$\\begin{aligned}\\begin{xy}\n (2,1.5); (4,3)**\\dir{-},\t\n (4,0); (8,3)**\\dir{-};\n (0,3); (6,-1.5)**\\dir{-};\n (12,3)**\\dir{-},\n (6,-1.5);(8,-3)**\\dir{.};\n (10,-4.5)**\\dir{-};\n (20,3)**\\dir{-},\n (10,-4.5);(10,-6)**\\dir{-},\n (0,4.2)*{\\scriptstyle{i_{1}}},\n (4,4.2)*{\\scriptstyle{i_2}},\n (8,4.2)*{\\scriptstyle{i_3}},\n (12,4.2)*{\\scriptstyle{i_4}},\n (20,4.2)*{\\scriptstyle{i_n}}\n \\end{xy}\\end{aligned} \\qquad \\qquad\n \\begin{aligned}\\begin{xy}\n (0,-3)*+UR{\\scriptstyle j_{1}}=\"1\",\n (4,4.5)*+UR{\\scriptstyle j_2}=\"2\",\n (8,-3)*+UR{\\scriptstyle j_3}=\"3\",\n (12,4.5)*+UR{\\scriptstyle j_4}=\"4\",\n (16,-3)*+UR{\\scriptstyle j_{n-1}}=\"5\",\n (20,4.5)*+UR{\\scriptstyle j_n}=\"6\",\n \"1\";\"2\"**\\dir{-}?>*\\dir{>},\n \"2\";\"3\"**\\dir{-}?>*\\dir{>},\n \"3\";\"4\"**\\dir{-}?>*\\dir{>},\n \"4\";\"5\"**\\dir{.},\n \"5\";\"6\"**\\dir{-}?>*\\dir{>},\n \\end{xy}\\end{aligned}$$\n \\caption{A tall tree and a long graph. Here $i_{1}$ and $j_{1}$ are minimal among\n indices in the tree and graph respectively.}\n\\end{figure}\n\n\\begin{proof}[Sketch of proof] \nFor the forests, use the Jacobi identity inductively to increase the distance from the minimally\nlabeled leaf to the root. For the graphs, use the Arnold identity to reduce the number\nof edges incident on a given vertex.\n\\end{proof}\n\nThe sets of tall forests and long graphs are in one-to-one correspondence with \npartitions of $\\mathbf{n}$, where\nboth the subsets and the constituent elements of each subset are ordered.\n\n\\begin{lemma}\\label{L:4}\nThe degree-$d$ configuration pairing of a tall forest and a long graph \nis equal to one if their associated ordered\npartitions agree, and is zero otherwise.\n\\end{lemma}\n\n\\begin{proof}[Sketch of proof] \nBy definition, the underlying unordered partitions must agree in order for the pairing\nto be non-zero. When looking at the configuration pairing between a single tall tree $T$\nand long graph $G$ which\nshare labels, look at the first place where their orderings differ to see how\n$\\beta_{G,T}$ fails to be a bijection.\n\n \n\n \n\n\n\n\n\\end{proof}\n\nThus, on tall forests and long graphs, the configuration pairing is essentially a Kronecker\npairing, showing that these spanning sets are linearly independent.\n\n\\begin{corollary}\\label{C:5}\nTall forests form a basis of ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$. Long graphs form a basis of ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$.\nBoth ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ and ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ are torsion-free.\n\\end{corollary}\n\nBecause tall forests and long graphs form bases, and the dimension-$d$\nconfiguration pairing is a Kronecker pairing on them, we deduce the main algebraic\nresult.\n\n\\begin{theorem}\\label{T:perfect}\nThe dimension-$d$ configuration pairing between ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ and ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ \nis perfect.\n\\end{theorem}\n\nAnd because the configuration pairing agrees with the homology pairing for \n${\\rm Conf}_{n}(\\mathbb{R}^{d})$ on the classes constructed by graphs and forests, we have the following.\n\n\\begin{corollary}\\label{C:inj}\nThe homomorphisms from ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ to $H_{*}\\left({\\rm Conf}_{n}(\\mathbb{R}^{d})\\right)$ and \n${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ to $H_{*}\\left({\\rm Conf}_{n}(\\mathbb{R}^{d})\\right)$ are injective.\n\\end{corollary}\n\nWe can now establish the first part of the main result of this paper.\n\n\\begin{theorem}\\label{T:main1}\nThe maps from ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ to $H_{*}({\\rm Conf}_{n}(\\mathbb{R}^{d}))$ and \n${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ to $H^{}{*}({\\rm Conf}_{n}(\\mathbb{R}^{d}))$ are isomorphisms.\n\\end{theorem}\n\n\\begin{proof}\nWe make light\nuse of the Leray-Serre spectral sequence. If in a fibration \n$F \\to E \\to B$, we have that $B$ is simply connected and either $F$ or $B$ has torsion-free\nhomology, then this spectral sequence says that $H^{*}(F) \\otimes H^{*}(B)$\nserves as an ``upper bound'' for $H^{*}(E)$. That is,\nif we let $P(X)(t) = \\sum \\; ({\\rm rank} \\; H_{i}(X)) t^{i}$, \nthe Poincar\\'e polynomial of $X$, \nthen $P(E) \\leq P(F) \\cdot P(B)$, where by $\\leq$ we mean that this inequality\nholds for all coefficients \nof $t^{i}$. Moreover, if the homology of $F$ and $B$ are free, equality \nis achieved only when that of $E$ is free.\n\nRecall the Fadell-Neuwirth tower of fibrations from Equation~(\\ref{tower}). If $d>2$ then the long exact sequence\nfor homotopy groups for the fibration $\\bigvee_{i-1} S^{d-1} \\to {\\rm Conf}_{i}(\\mathbb{R}^{d}) \\to {\\rm Conf}_{i-1}(\\mathbb{R}^{d})$ can be used to inductively establish that these configuration spaces are simply connected.\nThe Leray-Serre spectral sequence upper bound then yields the\ninequality $P( {\\rm Conf}_{i}(\\mathbb{R}^{d})) \\leq P( {\\rm Conf}_{i-1}(\\mathbb{R}^{d}) \\cdot (1 + (i-1) t^{d-1})$.\nFor $d=2$ these spaces are not simply connected - in fact their fundamental groups are pure \nbraid groups almost by definition, and in fact they are classifying spaces for pure braid groups \\cite{FoNe62} -\nbut Fred Cohen does the extra work at the beginning of Part III of \\cite{CLM76} necessary to show that this\nupper bound still holds.\n\n\nInductively we have \n$$P\\left( {\\rm Conf}_{n}(\\mathbb{R}^{d})\\right) \\leq \\prod_{i = 1}^{n-1} (1 + i t^{d-1}).$$ \nWe claim that this upper bound is sharp. Let $Q_{n}$ be polynomial\ndefined as $Q_{n}(t) = q_{i} t^{i(d-1)}$, where $q_{i}$ is the rank of\nthe submodule of ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ with $i$ edges. \nBy \\refC{inj}, $Q_{n}$ is a lower bound for \n$P( {\\rm Conf}_{n}(\\mathbb{R}^{d}))$, which we compute inductively. \nThe set of long graphs in ${{\\mathcal S}{\\it iop\\\/}^{d}}(i)$\nmaps to those of ${{\\mathcal S}{\\it iop\\\/}^{d}}(i-1)$ by taking a long graph, removing the\nvertex labeled $n$ and any edges connected to it, and then reconnecting the two\nadjacent vertices with a new edge if necessary. Given a long graph $G$ in \n${{\\mathcal S}{\\it iop\\\/}^{d}}(i-1)$ there is exactly one long graph in ${{\\mathcal S}{\\it iop\\\/}^{d}}(i)$\nwith the same number of edges which maps to it, namely the one in \nwhich vertex $n$ is added but not connected to an edge. Moreover,\nthere are $i-1$ long graphs in ${{\\mathcal S}{\\it iop\\\/}^{d}}(i)$ with one more edge\nwhich map to it, since one can choose which of the $i-1$ vertices\nin $G$ would have an edge connect to (as opposed to from) the $i$th vertex.\nWe deduce that $Q_{i} = Q_{i-1} \\cdot (1 + (i-1) t^{d-1})$.\n\nThus, the lower bound for $H^{*}({\\rm Conf}_{n}(\\mathbb{R}^{d}))$ given by the\nsubmodule ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$ matches the upper bound given by\nthe Leray-Serre spectral sequences for the tower of fibrations of \nEquation~(\\ref{tower}). We may inductively deduce\nthat $H^{*}({\\rm Conf}_{n}(\\mathbb{R}^{d}))$ is free, isomorphic to ${{\\mathcal S}{\\it iop\\\/}^{d}}(n)$.\nBy the Universal Coefficient Theorem and Theorems~\\ref{T:agree} and \n\\ref{T:perfect}, we have that $H_{*}({\\rm Conf}_{n}(\\mathbb{R}^{d}))$ is isomorphic\nto ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$.\n\\end{proof}\n\nOne can also obtain upper bounds by calculations in the cellular homology of the\none-point compactifications of these spaces, which is then dual to their chohomology,\nusing a cell structure first developed for $d=2$ by Fox and Neuwirth \\cite{FoNe62}.\n\nUsing the Leray-Serre spectral sequence in full force can lead to some\nof these results more quickly. For formal reasons, the spectral sequence\nfor each Fadell-Neuwirth fibration collapses, showing\nimmediately that the upper bound on cohomology groups \ngiven by each spectral sequence is sharp. One\ncan also use a symmetry argument to deduce the Arnold identity and thus\ndetermine the cohomology ring structure. Indeed, these fiber sequences are\nnice first examples to work with, since even though \nthe group structure mimics that of a trivial\n(product) fiber sequence, the cohomology ring of ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ differs\ngreatly from that of $\\prod_{i=1}^{n-1} \\left( \\bigvee_{i} S^{d-1} \\right)$.\n\nIn our approach, we not only have an understanding of the homology groups \nof ${\\rm Conf}_{n}(\\mathbb{R}^{d})$ and the cohomology ring up to isomorphism, but \nwe also have canonical spanning sets and an explicit understanding\nof the pairing between them, which enables hands-on calculations.\n\n\\subsection{Historical notes}\nThe pairing between graphs and trees we develop (for $d$ odd) was first noticed by Melan{\\c{c}}on\nand Reutenauer in a combinatorial study of free Lie superalgebras \\cite{MeRe96}. It was\nindependently identified as the pairing between canonical spanning sets for homology and cohomology\nof configuration spaces by Paolo Salvatore, Victor Tourtchine \\cite{Tour04}, and the author, all within the context of\nstudying spaces of knots. The present paper gives the first full account of this connection, to our knowledge. \nThe pairing is useful in related areas of algebra and topology, such as the study of Hopf invariants\n\\cite{SiWa08}. \n\n\n\\section{Operads}\\label{S:operads}\n\nAn operad encodes multiplication. Roughly speaking, an operad contains\ninformation needed to multiply in an algebra over that operad.\nFor example, in multiplying matrices one must supply an ordering of the \nmatrices to be multiplied, while in multiplying real numbers no such \nordering is needed. To Lie multiply (that is, take commutators of)\nsome matrices, one must not only order but parenthesize them.\n\nThe many definitions of an abstract operad \nare necessarily complicated. Even the elegant ``an operad is a monoid in the category\nof symmetric sequences,'' requires knowing what a symmetric sequence is and\nthen doing some work to relate that definition to standard examples. \nThorough introductions\nto the theory of operads are given elsewhere in this volume. \nWe prefer to be self-contained and to work with operads through trees, so we give\nour own development here. \nWe start with examples before giving the definition. For now\nwe work with the intuitive definition that an operad $\\Op$\nin a symmetric monoidal category $\\mathcal{C}$ \nis a sequence of objects indexed by natural\nnumbers so that the $n$th object $\\Op(n)$ \nparameterizes ways in which $n$ elements\nof some kind of algebra (that is, an algebra over that operad) can be multiplied.\n\n\n\\begin{examples}\\label{Ex:operads}\n\\begin{enumerate}\n\\item In any unital symmetric monoidal category ${\\mathcal{C}}$, the commutative operad \n${{\\mathcal C}{\\it om\\\/}}$ has ${{\\mathcal C}{\\it om\\\/}}(n) = {\\mathbbm{1}}_{{\\mathcal{C}}}$, since there is only one way to multiply\n$n$ things commutatively. (For vector spaces, ${\\mathbbm{1}}_{{\\mathcal{C}}}$ is the ground\nfield $\\mathbbm{k}$; for spaces, ${\\mathbbm{1}}_{{\\mathcal{C}}}$ is a point.) \n\\item In spaces, the associative operad ${{\\mathcal A}{\\it ss\\\/}}$ has ${{\\mathcal A}{\\it ss\\\/}}(n) = \\Sigma_{n}$,\nthe finite set of orderings of $n$ points, since the product of $n$ things is\ndetermined by their order if multiplication is associative. In vector spaces,\n${{\\mathcal A}{\\it ss\\\/}}(n) = \\mathbbm{k}[\\Sigma_{n}]$.\n\\item In vector spaces, the Lie operad ${{\\mathcal L}{\\it ie\\\/}}$ has ${{\\mathcal L}{\\it ie\\\/}}(n)$ spanned by $n$-trees\nmodulo the anti-symmetry and Jacobi identities of \\refP{relations} (with $d$ even).\nIn a Lie algebra one must parenthesize elements to multiply them, and \nour $n$-trees encode parenthesizations. The anti-symmetry\nand Jacobi identities are always respected in a Lie algebra, so they appear in the \ndefinition of this operad.\n\\item In spaces, the little $d$-disks operad has ${{\\mathcal D}{\\it isks\\\/}}^{d}(n)$ the space of $n$ \nlittle disks in $D^d$. Explicitly, ${{\\mathcal D}{\\it isks\\\/}}^{d}(n)$ is the subspace of ${\\rm Conf}_n({\\rm Int}D^d) \\times (0,1]^n$ of \n$(x_i) \\times (r_i)$ such that the balls $B(x_i, r_i)$ are contained in the interior of \n$D^d$ and have disjoint interiors. This space parameterizes some ways in which \nmaps from $S^{d}$, or rather $D^{d}$ with its boundary going to the basepoint,\ncan be multiplied. Given $n$ maps $f_{i}: (D^{d}, \\partial) \\to (X, *)$ \nand an element of this space we may multiply the $f_{i}$ by applying them on the\ncorresponding little disk, sending points outside any little disk to the basepoint of $X$.\nSee Figure~7.\n\\item We've already seen the degree-$d$\nPoisson operad in vector spaces, which has $n$th\nentry ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$. In a Poisson algebra one can both Lie multiply, represented\nby trees, and then multiply those results together, represented by placing those\ntrees together in a forest. One can of course also multiply first and then Lie multiply.\nBy the Leibniz rule, the bracket is a derivation with \nrespect to the multiplication, so we may always reduce to expressions which\ncorrespond to forests. For example (using bracket notation rather than forests and\nassuming $d$ is even),\n\\begin{align*}\n[x_{1}, [x_{2}, x_{3} \\cdot x_{4}]] &= [x_{1}, [x_{2}, x_{3}] \\cdot x_{4}]] + \n[x_{1}, x_{3} \\cdot [x_{2}, x_{4}]] \\\\\n&= [x_{1}, [x_{2}, x_{3}]] \\cdot x_{4} \n+ [x_{1}, x_{4}] \\cdot [x_{2}, x_{3}] + [x_{1}, x_{3}] \\cdot [x_{2}, x_{4}] + [x_{1}, [x_{2}, x_{4}]] \\cdot x_{3}.\n\\end{align*}\n\\item For any $X$ in ${\\mathcal{C}}$, the endomorphism operad of $X$ has ${{\\mathcal E}{\\it\\\/nd\\\/}}_{X}(n) =\n{\\rm Hom}_{{\\mathcal{C}}}(X^{\\odot n}, X)$. The endomorphism operad is often too large to understand\nexplicitly, much as groups of homeomorphisms are. But finding an interesting sub-operad of the\nendomorphism operad will endow $X$ with a multiplication, much as finding a sub-group\nof the self-equivalences of $X$ gives a group action.\n\\end{enumerate}\n\\end{examples}\n\n\nAs McClure and Smith\npoint out in \\cite{McSm04.2}, the axiomatic definitions of operads follow nicely\nfrom reflecting on what they do, just as the axioms of a group all follow\nfrom the notion that a group encodes symmetries through a group action. \nFor example, if one has a rule for multiplying four inputs and two inputs, then\none can make a rule for multiplying five inputs: first multiply two of them,\nthen take that result as an input with the remaining three original inputs in order\nto apply the rule for multiplying four inputs. Thus, for any operad there\nare maps $\\Op(4) \\odot \\Op(2) \\to \\Op(5)$. \n \n\nWe prefer to state\nthe definition using trees, and, because operads require maps which\ncommute, the language of categories and functors. In this language,\nwhat we just said about combining two- and four-input multiplications to \nmake a five-input multiplication is encoded in the first part of Figure~\\ref{F:basicmor}.\n\n\n\\begin{definition}\n\\begin{itemize}\n\\item An o-tree is a finite connected acyclic graph with a \ndistinguished vertex called the\nroot. Univalent vertices of an o-tree (not counting the root, if it is \nunivalent) are called leaves. \n\n\\item At each vertex, the edge which is closer to the root\nis called the output edge. The edges which are further from\nthe root are called input edges and are labeled from\n$1$ to $n$.\n\n\\item At each edge, the vertex of\nan edge which is further from the root is called its input vertex, and\nthe vertex closer to the root is called its output vertex. We say\nthat one vertex or edge lies over another if the latter is in the\npath to the root of the former. A non-root edge is called redundant if its \ninitial vertex is bivalent.\n\n\\item Given an o-tree $\\tau$ and an edge $e$, the\ncontraction of $\\tau$ by $e$ is the tree $\\tau'$ obtained by\nidentifying the input vertex of $e$ with its\noutput vertex, and removing $e$ from the set of\nedges. If the label of $e$ was $i$, the labels of the \n$k$ edges which were immediately over $e$ will be increased\nby $i-1$, and the labels of the edges which shared the output\nvertex of $e$ with labels greater than $i$ will be increased by $k-1$.\n\n\\item Let $\\Upsilon$ denote the category whose objects are o-trees, \nand whose morphisms are generated by contractions of edges\n(that is, there is a morphism from $\\tau$ to $\\tau'$ if $\\tau'$ is the contraction\nof $\\tau$ by $e$) and relabelings which are isomorphisms\n(that is, there is a morphism from $\\tau$\nto $\\tau'$, which could be $\\tau$ itself, if $\\tau'$ is obtained from $\\tau$ by\nrelabeling of its edges).\n\\end{itemize}\n\\end{definition}\n\n\nSee \\refF{basicmor} for some examples of objects and morphisms in $\\Upsilon$.\nLet $\\Upsilon_n$ denote the full subcategory\nof trees with $n$ leaves, which has a terminal object, namely the unique tree with one vertex\ncalled the $n$th corolla $\\gamma_n$. We allow for the tree\n$\\gamma_0$ which has no leaves, only a root vertex, and is\nthe only element of $\\Upsilon_0$.\nFor a vertex $v$ let $|v|$ denote the number of edges for which\n$v$ is terminal, usually called the arity of $v$.\n\n\\begin{definition}\\label{D:op}\nAn operad is a functor $\\Op$ from $\\Upsilon$ to a symmetric\nmonoidal category $({\\mathcal{C}}, \\odot)$ which satisfies the following axioms.\n\n\\begin{enumerate}\n\\item $\\Op(\\tau) \\cong \\odot_{v \\in \\tau} \\Op(\\gamma_{|v|})$. \\label{1}\n\\item If $e$ is a redundant edge and $v$ is its terminal vertex\nthen $\\Op(c_{\\{e\\}})$ is the identity map on\n$\\odot_{v' \\neq v} F(\\gamma_{v'})$ tensored with the isomorphism\n$({\\mathbbm{1}}_{{\\mathcal{C}}} \\odot -)$ under the decomposition of axiom (\\ref{1}). \\label{3}\n\\item If $S$ is a subtree of $\\tau$ and if $f_{S, S'}$ and $f_{\\tau, \\tau'}$\ncontract the same set of edges, then under the\ndecomposition of (\\ref{1}), $F(f_{\\tau,\\tau'}) = F(f_{S, S'}) \\odot id$. \\label{4}\n\\end{enumerate}\n\\end{definition}\n\nBy axiom~(\\ref{1}), the values of $\\Op$ are determined by \nits values on the corollas $\\Op(\\gamma_n)$, which corresponds\nto $\\Op(n)$ in the usual terminology. Because of the relabeling morphisms,\n$\\Op(n)$ has an action by the $n$th symmetric group. By axiom~(\\ref{4}),\nthe values of $\\Op$ on morphisms may be computed by\ncomposing morphisms on sub-trees, so we may identify some \nsubset of basic morphisms through which all morphisms\nfactor. In \\refF{basicmor} we illustrate some basic morphisms in $\\Upsilon$.\nThe first corresponds to what are known as $\\circ_i$ operations.\nThe second corresponds May's\noperad structure maps from Definition~1.1 of \\cite{May72}. \nThat $\\Op$ is a functor implies \nthe commutativity of diagrams involving these basic morphisms.\n\n\\begin{center}\n\\begin{figure}[ht]\\label{F:basicmor}\n\\begin{center}\n\\psfrag{C1}{}\n\\psfrag{C2}{}\n$$\\includegraphics[width=11cm]{Figures\/morphisms.eps}$$\n\\caption{Two morphisms in $\\Upsilon$ which give rise to standard operad structure maps.\nThe first corresponds to a $\\circ_i$ operation, the second to one of May's structure maps.}\n\\end{center}\n\\end{figure}\n\\end{center}\n\nFilling in what the operad structure maps are for our\nExamples from ~\\ref{Ex:operads} is a pleasant exercise, which we leave\nin part to the reader. \n\\begin{examples}\n\\begin{enumerate}\n\\item For ${{\\mathcal C}{\\it om\\\/}}$, all structure maps are the identity.\n\\item For ${{\\mathcal A}{\\it ss\\\/}}$, they are ``insertion and relabeling.''\n\\item For ${{\\mathcal L}{\\it ie\\\/}}$, the structure maps are defined\nby grafting trees, that is identifying the root of one with the leaf\nof another. These are well-defined because the Jacobi and anti-symmetry\nidentities are defined locally. \n\\item For ${{\\mathcal D}{\\it isks\\\/}}^{d}$ we give a full account.\nLet $T$ be a tree whose vertices consist of the root\nvertex $v_0$ and a terminal vertex $v_e$ for each root edge $e$. \nThus, $T \\to \\gamma_n$, where $n$ is the number of leaves of $T$,\ngives rise to one of May's structure maps as in \\refF{basicmor}. \nGiven a label $i \\in \\mathbf{n}$ let $v(i)$ be the initial vertex for the $i$th\nleaf, let $o(i)$ be the label of leaf $i$ within the ordering on edges\nof $v(i)$ and let $e(i)$ be the label of the root edge for\nwhich $v(i)$ is terminal.\nDefine ${{\\mathcal D}{\\it isks\\\/}}^{d}(T \\to \\gamma_n)$ as follows \n$$(x^v_i, r^v_i)^{v \\in V(T)}_{1 \\leq i \\leq \\#v} \\mapsto (y_j, \\rho_j)_{j \\in \\mathbf{n}} \\;\\;\\; {\\rm where}\n\\;\\;\\; y_j = x^{v_0}_{e(j)} + r^{v_0}_{e(j)} x^{v(j)}_{o(j)} \\;\\;\\; \n{\\rm and} \\;\\;\\; \\rho_j = r^{v_0}_{e(j)} r^{v(j)}_{o(j)}.$$\nSee Figure~6 for the standard picture.\n\\begin{figure}[ht]\\label{F:disks}\n$$\\includegraphics[width=12cm]{Figures\/disks1.eps}$$ \n\\caption{A structure map for the $2$-disks operad.} \n \\end{figure}\n\\item In the case of ${{\\mathcal P}{\\it ois\\\/}^{d}}$, the structure maps are essentially grafting as for \n${{\\mathcal L}{\\it ie\\\/}}$, but with an important additional wrinkle given by the Leibniz rule.\nIn order to be precise without unnecessary complication, it helps to switch\nfrom forests to more algebraic notation. We may associate to an $n$-forest \nan expression in variables $x_{1}, \\cdots, x_{n}$ with two binary products,\ndenoted $\\cdot$ and $[\\;,\\;]$. For example to the forest $F =\n\\begin{xy}\n (1.5,1.5); (3,3)**\\dir{-}, \n (0,3); (3,0)**\\dir{-}; \n (7.5,4.5)**\\dir{-}, \n (3,0); (3,-1.5)**\\dir{-}, \n (3,6); (6,3)**\\dir{-},\n (6,6); (4.5,4.5)**\\dir{-},\n (0,4.2)*{\\scriptstyle 2},\n (3,4.2)*{\\scriptstyle 6},\n (3,7.2)*{\\scriptstyle 1},\n (6,7.2)*{\\scriptstyle 7},\n (7.5,5.7)*{\\scriptstyle 3},\n\\end{xy}\n \\begin{xy} \n (0,1.5); (1.5,0)**\\dir{-}; \n (3,1.5)**\\dir{-}, \n (1.5,-1.5); (1.5,0)**\\dir{-}, \n (0,3.3)*{\\scriptstyle 4}, %\n (3,3.3)*{\\scriptstyle 5}, %\n \\end{xy}$ we associate the bracket expression $\\left[[x_{2}, x_{6}], [[x_{1}, x_{7}], x_{3}]\\right] \\cdot \n [x_{4}, x_{5}]$. More generally, bracket expressions may include multiplications\n by $\\cdot$ within brackets, but these may be reduced to expressions associated to forests\n after the Leibniz rule, $[X, Y\\cdot Z] = (-1)^{|X||Y|} Y \\cdot [X, Z] + [X, Y] \\cdot Z$, is imposed.\n\n\nLet $f: \\tau \\to \\gamma_{n}$ be a morphism in $\\Upsilon$ in which $\\tau$\nis a tree with one internal vertex over the $i$th root edge. \nDefine an operad structure on ${{\\mathcal P}{\\it ois\\\/}^{d}} = \\oplus {{\\mathcal P}{\\it ois\\\/}^{d}}(n)$ by sending $f$\nto the map $${{\\mathcal P}{\\it ois\\\/}^{d}}(n) \\otimes {{\\mathcal P}{\\it ois\\\/}^{d}}(m) \\to {{\\mathcal P}{\\it ois\\\/}^{d}}(n+m -1)$$ where $B_{1} \\otimes B_{2}$\nis sent to the bracket expression defined as follows. The variables in $B_{2}$ are re-labeled\nfrom $x_{i}$ to $x_{m}$. The variables $x_{j}$ in $B_{1}$ with $j>i$ are re-labeled\nby $x_{j+m-1}$. Finally $B_{2}$ is substituted for $x_{i}$\nin $B_{1}$. Note that in order to express this in terms of the $n$-forest basis, \nthe Leibniz rule would then need to be applied repeatedly.\n\n\\item For ${{\\mathcal E}{\\it\\\/nd\\\/}}_{X}$, structure maps are defined by composition.\n\\end{enumerate}\n \\end{examples}\n\nFinally, we give an anti-climactic definition of an algebra over an\noperad.\n\n\\begin{definition}\nAn algebra structure for $X$ over an operad $\\Op$ is a natural transformation\nof operads $\\Op \\to {{\\mathcal E}{\\it\\\/nd\\\/}}_{X}$. \n\\end{definition}\n\nBy adjointness, the maps $\\Op(n) \\to\n{\\rm Hom}_{{\\mathcal{C}}}(X^{\\odot n}, X)$ give rise to multiplication maps\n$\\Op(n) \\odot X^{\\odot n} \\to X$. Because of the relabeling \nmorphisms in $\\Upsilon$, these maps are equivariant with respect to the diagonal\nsymmetric group action on $\\Op(n) \\odot X^{\\odot n}$.\n\nAs for examples, algebras over ${{\\mathcal C}{\\it om\\\/}}$ are commutative algebras, and similar eponymous results\nhold for ${{\\mathcal A}{\\it ss\\\/}}$, ${{\\mathcal L}{\\it ie\\\/}}$, and ${{\\mathcal P}{\\it ois\\\/}^{d}}$. We will discuss the little disks operad \nin the next section. As for ${{\\mathcal E}{\\it\\\/nd\\\/}}_{X}$, the only general statement is that $X$ is an algebra \nover it.\n\n\\medskip\n\nWe leave historical remarks about the theory of operads for other papers in this volume.\n\n\\section{The homology of the little disks operad}\\label{S:diskshomology}\n\nThe little disks operad\nand its action on iterated loop spaces can trace their lineage to the proof of \none of the first theorems\nin homotopy theory, namely that $\\pi_{2}(X)$ is an abelian group. Stated in our\nlanguage, that proof gives a path of possible multiplications of $f$ and $g$, two\nelements of the second loop space $\\Omega^{2}(X)$, starting at $f \\cdot g$ and\nending at $g \\cdot f$. That path lies within the ``little rectangles'' sub-operad of the space\nof all multiplications. If we start with an arbitrary number of maps in any dimension, we are led to \nthe little disks action on a $d$-fold loop space.\n\n\\begin{definition}\nThe action of ${{\\mathcal D}{\\it isks\\\/}}^{d}$ on $\\Omega^{d}(X) = {\\rm Map}\\left((D^{d}, S^{d-1}),(X, *) \\right) $ \nis defined through maps\n$${{\\mathcal D}{\\it isks\\\/}}^{d}(n) \\times \\left(\\Omega^{d}(X)^{\\times n}\\right) \\to \\Omega^{d}X,$$\nwhich send $\\{ B_{i} \\} \\times \\{ f_{i} \\}$ to the map whose restriction to $B_{i}$ is\n$f_{i}$ composed with the canonical linear homeomorphism of $B_{i}$ with $D^{d}$.\nAt points in $D^{d}$ outside of any $B_{i}$, the resulting map is constant at the basepoint.\n\\end{definition} \n\nThus a $d$-fold loop space is an algebra over the little disks operad. Boardman-Vogt \\cite{BoVo73}\nand May \\cite{May72} showed that the converse is essentially true. That is if $X$\nhas an action of the little $d$-disks and $\\pi_{0}(X)$, which is necessarily then a monoid,\nis in fact a group then $X$ is homotopy equivalent to a $d$-fold loop space. \nThis result is known as a ``recognition principle,'' since it gives a criterion for\nrecognizing iterated loop spaces.\n\n\\begin{figure}[ht]\\label{F:disksaction}\n$$\\includegraphics[width=10cm]{Figures\/disks2.eps}$$ \n\\caption{Little $2$-disks acting on two maps.} \n \\end{figure}\n\nIn general, operad actions on spaces are reflected in their homology.\n\n\\begin{proposition}\nThe homology of any operad $\\Op$ of spaces will\nbe an operad of modules. Moreover, the homology of an algebra over\n$\\Op$ will be an algebra over $H_{*}(\\Op)$.\n\\end{proposition}\n\n\\begin{proof}\nThat $H_{*}(\\Op)$ is an operad is immediate by composing the K\\\"unneth map\n$$H_{*}(\\Op(r)) \\otimes H_{*}(\\Op(n_{1})) \\otimes \\cdots \\otimes H_{*}(\\Op(n_{r}))\n\\to H_{*}\\left(\\Op(r) \\times \\Op(n_{1}) \\times \\cdots \\times \\Op(n_{r})\\right)$$\nwith the induced map in homology of an operad structure map to get a corresponding\noperad structure map in homology. Moreover, a map $\\Op \\to {{\\mathcal E}{\\it\\\/nd\\\/}}_{X}$\ninduces a map $H_{*}(\\Op) \\to H_{*}({{\\mathcal E}{\\it\\\/nd\\\/}}_{X})$ which in turn maps to \n${{\\mathcal E}{\\it\\\/nd\\\/}}_{H_{*}(X)}$, again using the K\\\"unneth map.\n\\end{proof}\n\nWe can now state the main result of this paper in full.\n\n\\begin{theorem}\\label{T:main2}\nThe homology of the little $d$-disks operad ${{\\mathcal D}{\\it isks\\\/}}^{d}$ is \nthe degree $d$ Poisson operad ${{\\mathcal P}{\\it ois\\\/}^{d}}$. Thus the homology of $\\Omega^{d}(X)$\nis an algebra over ${{\\mathcal P}{\\it ois\\\/}^{d}}$.\n\\end{theorem}\n\nBefore establishing this theorem, we reflect on its significance, which is to \nendow the homology of iterated loop spaces a rich additional structure.\nThis homology has intrinsic interest, but may also be used to study homotopy groups.\nThe standard Hurewicz map $\\pi_{n}(X) \\to H_{n}(X)$ is often zero (for example,\nin all but one degree when $X$ is a sphere). But if we use \nadjointess to identify $\\pi_{n}(X)$ with $\\pi_{n-k}(\\Omega^{k} X)$ then \nthe $k$-looped Hurewicz map to $H_{n-k}(\\Omega^{k} X)$ can yield\nadditional information. For example, a theorem of Milnor and Moore \n(whose proof at the end of \\cite{MiMo65} is left as a nice exercise)\nstates that for rational homotopy and homology, the $1$-looped Hurewicz map is\ninjective for simply connected spaces.\n\nWe now bring in what we know about configuration spaces\nthrough the following.\n\n\\begin{lemma}\\label{L:equiv}\nThe space ${{\\mathcal D}{\\it isks\\\/}}^{d}(n)$ is homotopy equivalent to ${\\rm Conf}_{n}(\\mathbb{R}^{d})$.\n\\end{lemma}\n\n\\begin{proof}\nBecause ${\\rm Int}D^{d}$ is homeomorphic to $\\mathbb{R}^{d}$, their associated\nconfiguration spaces ${\\rm Conf}_{n}({\\rm Int}D^{d})$ and ${\\rm Conf}_{n}(\\mathbb{R}^{d})$\nare homeomorphic. The space of little disks ${{\\mathcal D}{\\it isks\\\/}}^{d}(n)$ projects onto\n${\\rm Conf}_{n}({\\rm Int}D^{d})$ by definition (mapping to the configuration defined\nby the centers of the disks). This projection defines a fiber bundle whose fibers,\ngiven by the set of possible radii, are convex spaces and thus contractible.\n\\end{proof}\n\n\\refT{main1} now implies \\refT{main2} at the level of underlying vector spaces,\nso we may focus on operad structure.\nTo establish the compatibility between the geometric insertion operad structure of the little\ndisks and the algebraic insertion operad structure of the Poisson operad is \neasier when considering homology classes\nof ${{\\mathcal D}{\\it isks\\\/}}^{d}$ represented by trees rather than forests. The key is to choose\nappropriately consistent lifts of the submanifolds $P_{F}$ to the spaces ${{\\mathcal D}{\\it isks\\\/}}^{d}(n)$\n(now the planets really look like planets, being represented by disks rather\nthan points). If for example $F$ is a forest and $T$ is a forest with one component,\na $\\circ_{i}$ map would send\n$\\widetilde{P}_{F} \\times \\widetilde{P}_{T}$ precisely to $\\widetilde{P}_{F'}$, where $F'$ is\nthe grafting of $T$ onto $F$. So in homology $F \\circ_{i} T$ is the grafting\nof $T$ onto the $i$th leaf of $F$ accordingly. \n\nTo manage the general case, we focus on cohomology instead of homology.\nIt is geometrically easier to establish the linear\ndual of \\refT{main2}, identifying the cohomology\nof ${{\\mathcal D}{\\it isks\\\/}}^{d}$ with the cooperad structure on ${{\\mathcal S}{\\it iop\\\/}^{d}}$.\nTranslating to homology is then a matter of pure algebra and combinatorics,\nwhich is carried out in \\cite{Sinh06.2}. A cooperad is a \nfunctor from $\\Upsilon^{\\text{op}}$ to a symmetric monoidal\ncategory which satisfies axioms dual to those of \\refD{op}. Note that \nin associating a dual cooperad to an operad, we are \nnot changing the symmetric monoidal product.\nFor example, a standard cooperad in the category of vector spaces \nis defined using the tensor product rather than the direct sum.\n\n\\begin{definition} \\label{D:mB}\nTo an o-tree $\\tau$ with $n$ leaves and\ntwo distinct integers $j,k \\in \\mathbf{n}$ let $v$ be the nadir\nof the shortest path between leaves labelled $i$ and $j$ and define\n$J_v(j), J_v(k)$ to be the labels of the branches of $v$ over\nwhich leaves $j$ and $k$ lie.\n\nThe module ${{\\mathcal S}{\\it iop\\\/}^{d}}$, forms a cooperad \nwhich associates to the morphism $\\tau \\to \\gamma_n$ \nthe homomorphism $g_\\tau$ sending $G \\in \\Gamma$ to \n$( {\\rm{sign}}\\ \\pi)^{d} \\bigotimes_{v_i} G_{v_i}$.\nHere $v_i$ ranges over internal\nvertices in $\\tau$ and $G_{v_i} \\in \\Gamma^{|v_i|}$, \nis defined by having for each edge in $G$, say from $j$ to $k$,\nan edge from $J_v(j)$ to $J_v(k)$ in $G_v$.\nThe edges of $G_{v_{i}}$ are ordered in accordance with that \nof the edges in $G$ which give rise to them, and $\\pi$ is the\npermutation relating this order on all of the edges in \n $\\bigotimes_{v_i} G_{v_i}$ to the ordering within $G$.\n\\end{definition}\n\nConsider for example when $\\tau$ is the first tree from Figure~5, with leaf\nlabeling given by the planar embedding.\nThe corresponding cooperad structure map would send a single\ngraph $G$ on five vertices to the tensor product of two graphs, $G_{r}$ with four\nvertices and $G_{v}$ with two vertices. The graph $G_{v}$ would have an\nedge between its two vertices if and only if $G$ had $\\linep{3}{4}$ as an edge.\nAny edge of $G$ of the form $\\linep{1}{3}$ or $\\linep{1}{4}$ would give\nrise to an edge $\\linep{1}{3}$ in $G_{1}$. The edge $\\linep{5}{4}$ in $G$\nwould give rise to $\\linep{4}{3}$ in $G_{1}$.\n\n\\begin{theorem}[Thm. 6.8 of \\cite{Sinh06.2}]\\label{T:coop}\nThe cooperad structure on ${{\\mathcal S}{\\it iop\\\/}^{d}}$ is linearly dual to that of ${{\\mathcal P}{\\it ois\\\/}^{d}}$\nthrough the configuration pairing.\n\\end{theorem}\n\nThe key to proving this theorem is that the configuration pairing can be\ndefined directly on bracket expressions. Looking at \\refD{pairing}, we use innermost pairs\nof brackets instead of nadirs of paths \nto define the analogue of the map $\\beta_{G,T}$. Remarkably,\nthe Leibniz rule is respected by this extended definition of this pairing.\nIndeed, the configuration pairing can be viewed as the central algebraic object in this area,\nand the anti-symmetry, Jacobi, Leibniz, and Arnold identities arise \nnaturally in describing its kernel.\n\nWhile the operad structure of ${{\\mathcal P}{\\it ois\\\/}^{d}}$ is more familiar, the cooperad structure\non ${{\\mathcal S}{\\it iop\\\/}^{d}}$ is simpler. The operad maps on ${{\\mathcal P}{\\it ois\\\/}^{d}(n)}$ require the Leibniz rule\nto be applied recursively to reduce to any standard basis, \nwhile the cooperad maps on ${{\\mathcal S}{\\it iop\\\/}^{d}}$ require no such reduction for many standard\nbases. While here we use this cooperad as a useful way to prove a theorem\nabout the corresponding operad, in work on Koszul duality cooperads play an equal role.\n\n\\begin{proof}[Proof of \\refT{main2}]\nBy \\refL{equiv} and \\refT{main1}, $H_{*}({{\\mathcal D}{\\it isks\\\/}}^d)$ and ${{\\mathcal P}{\\it ois\\\/}^{d}}$ are isomorphic as vector spaces,\nso it suffices to consider their operad structure. By \\refT{perfect} and \\refT{coop}, we may instead\nestablish that the cooperad structures on ${{\\mathcal S}{\\it iop\\\/}^{d}}$ and the cohomology\nof ${{\\mathcal D}{\\it isks\\\/}}^d$ agree.\n\nLet $f: \\tau \\to \\gamma_{n}$ be a morphism in $\\Upsilon$, where $\\gamma_{n}$ is a corolla,\nand let $\\prod_{w \\in \\tau} {{\\mathcal D}{\\it isks\\\/}}^{d}(|w|) \\to {{\\mathcal D}{\\it isks\\\/}}^{d}(n)$ be the corresponding operad\nstructure map. Using the ring structure on cohomology, it suffices to understand the pullback\nof a generator $a_{ij}$. By definition, we consider the composite \n $$\\pi_{ij} \\circ {{\\mathcal D}{\\it isks\\\/}}^{d}(f) : \\prod_{w \\in \\tau} {{\\mathcal D}{\\it isks\\\/}}^{d}(|w|) \\to {{\\mathcal D}{\\it isks\\\/}}^{d}(n) {\\rightarrow} S^{d-1}.$$ \n We apply a homotopy in which at time $t$ \n the disks in ${{\\mathcal D}{\\it isks\\\/}}^{d}(|w|)$ are all scaled by \nby $t^{h}$ where $h$ is the height of the vertex $w$.\nAs $t$ approaches zero, $\\pi_{ij} \\circ {{\\mathcal D}{\\it isks\\\/}}^{d}(f)$ approaches projection onto \nthe factor of ${{\\mathcal D}{\\it isks\\\/}}^{d}(|v|)$ where $v$ is the nadir\nof the shortest path between leaves labelled $i$ and $j$, followed by \n$\\pi_{J_{v}(i) J_{v}(j)}$, as in \\refD{mB}. Thus $a_{ij}$ pulls back to $a_{J_{v}(i) J_{v}(j)}$\nin the $v$th factor of ${{\\mathcal D}{\\it isks\\\/}}^{d}$, in agreement with the cooperad structure on ${{\\mathcal S}{\\it iop\\\/}^{d}}$.\n\\end{proof}\n\nTo recap, we have now shown that the homology of a $d$-fold loop space is a Poisson\nalgebra. The multiplication is the standard one given by loop-sum. The bracket, known\nas the Browder bracket, reflects\nsome ``higher commutativity'' of the loop-sum. If two homology\nclasses are represented by (pseudo-)manifolds $M, N \\to \\Omega^{d} X$, then their\nbracket will be represented by the map $S^{d-1} \\times M \\times N \\to \\Omega^{d} X$\nwhich when restricted to $v \\times M \\times N$ ``multiplies $M$ and $N$ in the direction\nof $v$.'' \n \n \\subsection{Historical notes}\nThe operad structure on spaces of little disks was determined by Cohen in his thesis \n\\cite{CLM76}. There both the non-equivariant and the much more delicate equivariant\nhomology of these spaces are determined, though the language of operads is not employed. \nEquivariant homology classes yield operations in the homology of iterated loop spaces \\cite{DyLa62},\nwhich are algebras over this operad, which is a main focus of \\cite{CLM76} and \\cite{May71}. \nThe simplest example\nis with coefficients modulo two, where for any homology class $x$, we have $[x, x] = 0$. But\nthis class can be ``divided by two, using only a hemisphere's worth of \nBrowder multiplication.'' The \nresult as an operation which ``sends $x$ to $\\frac{[x,x]}{2}$.'' \n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA fundamental evolutionary parameter of a galaxy is its size and the size distribution of galaxy populations can reveal important clues as to their assembly histories and underlying dark matter halos \\citep{Mo1998, Wechsler2018}. The comparison of galaxy structural properties such as sizes, stellar masses and luminosities further reveals tight scaling relations which likely dictate complex and diverse evolutionary pathways and afford the means to test the standard paradigm of galaxy formation and evolution. As an example, the size evolution of different galaxy types (i.e., early- and late-type galaxies) is found to be remarkably different, suggesting clearly distinct modes of stellar growth and dark matter halo assembly.\n\nThe empirical size-mass relationship, defined as $R\\textsubscript{eff} \\propto M^{\\alpha}$, and its evolution with redshift, $R\\textsubscript{eff} \\propto (1+z)^{\\beta}$, \nhave been investigated by several previous studies for both early- and late-type galaxies, sometimes with conflicting end results that can aid to test the basic theory of galaxy formation \\citep{Mo1998}. For instance, setting the benchmark at ``zero redshift'' using a complete sample of $\\sim$140,000 local galaxies (both early-type and late-type) from the Sloan Digital Sky Survey, \\cite{Shen2003} found a power-law slope of $\\alpha\\sim$0.4 for late-type galaxies more massive than M$_{*}>10^{10.6}M_{\\sun}$), while below this characteristic stellar mass the slope flattens down further to $\\alpha\\sim$0.15, implying a less rapid size evolution with stellar mass. On average, they find more massive galaxies tend to be characterised by larger radii than their less massive counterparts, implying a degree of ``inside out'' galaxy growth. Comparing this to their samples of early-type galaxies, the authors found that the relation for early-type galaxies displays a significantly steeper slope at fixed stellar mass, with $\\alpha\\sim$0.55, indicating a potentially separate and much faster evolutionary pathway. Despite the narrow redshift range of their sample (0.05$$-15) of the $z>6$ galaxy luminosity functions (LFs) allow one to determine the prevalence of the most abundant star-forming sources that are likely to be responsible for driving the reionization process \\citep{Atek2015, Kawamata2015, Livermore2017}. While extremely small sizes have been found for the faintest such galaxies (e.g., $\\sim$200 pc; \\citealt{Bouwens2017}), the associated uncertainty of these lensed measurements are driven by the assumed lens model and serve as the main source of uncertainty in the determination of the LF faint-end slope\n \\citep{Grazian2011, Oesch2015, Alavi2016, Bouwens2017, Atek2018}.\nIn addition to the detection of especially faint sources, gravitationally lensing also provides the chance to observe galaxies with high angular resolution.\nPioneering works e.g., \\cite{Marshall2007} presented study of super-resolving galaxies at $z\\sim$0.5.\n\\cite{Newton2011} explored 46 strongly lensed galaxies from the Sloan Lens ACS Survey (SLACS) at lower redshift 0.4$10^{9}M_{\\sun}$ at 1$10^{9} M_{\\sun}$,\n\\item has coverage by each of the central lens models derived by the five teams described above,\n\\item does not reside close to a particularly bright object or at the edge of the instrument detector,\n\\end{itemize}\n\nApplying the above set of criteria, we obtain a sample of 258 lensed galaxies at the aforementioned redshift, of which several objects have multiply-lensed images. \nIn the later case, we count such galaxies only once, providing a final sample of 255 sources. \nWe limit ourselves at this redshift range and leave the more complicated issue correlated with completeness at higher redshift to our future work.\nFollowing the criteria presented by \\cite{Williams2009}, we utilize rest-frame UVJ colors to distinguish between early-type (passive) and late-type (star-forming) galaxies, which we separate and present as a function of several redshift bins in Figure~\\ref{img:uvj}. \nThis color space allows for a separation of the two galaxy types, although we note beyond a redshift of 1.5 there are only a handful of early-type galaxies. In Figure~\\ref{img:uvmass}, we present the same samples as a function of rest-frame U-V color and magnification-corrected stellar mass, where it becomes clear that passive galaxies tend to be redder at all redshift bins compared to their star-forming counterparts.\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=2\\columnwidth]{UVJ.pdf}\n\\caption{The rest-frame UVJ color diagram for four redshift bins (each $\\Delta z$=0.5 wide). \nGalaxies are classified into two types, passive (early-) or star-forming (late-type). \nThe solid black lines in each panel indicates the selection criteria from \\citep{Williams2009}, which is used in this work.}\n\\label{img:uvj}\n\\end{figure*} \n\n\\begin{figure*}\n\\includegraphics[width=2\\columnwidth]{massuv.pdf}\n\\caption{The rest-frame U-V color as function of stellar mass for the same four redshift bins shown in Figure \\ref{img:uvj}. Both figures adopt the same color scheme.}\n\\label{img:uvmass}\n\\end{figure*} \n\n\n\n\\begin{table*}\n\\centering\n\\caption{Catalog of lensed galaxies at redshift 1$ 2\\times10^{10}$, so that we avoid the potentially flatter part of the size-mass distribution at lower stellar mass.\nFor the late-type galaxies, where our sample size is larger, we fit galaxies with intrinsic stellar mass $M_{*} > 3\\times10^{9}$ at $z<2.5$, and $M_{*} > 5\\times10^{9}$ at $2.50.3$ dex for the majority of the lens models at 2.5$3\\times10^{9}M_{\\sun}$ at 1$5\\times10^{9}M_{\\sun}$ at 2.5$7$, and its implication for the faint-end slope of the galaxy luminosity function.\n\n\n \n\\section*{Acknowledgements}\nThis work utilizes gravitational lensing models produced by PIs Brada\\v{c}, Natarajan \\& Kneib (CATS), Merten \\& Zitrin, Sharon, Williams, Keeton, Bernstein and Diego, and the GLAFIC group. This lens modeling was partially funded by the HST Frontier Fields program conducted by STScI. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. The lens models were obtained from the Mikulski Archive for Space Telescopes (MAST). \nLY is supported from the China Scholarship Council. LY and TT acknowledge support by NASA through grant JWST-ERS-1324.\nThe authors thank Marco Castellano, Adriano Fontana, Karl Glazebrook, Danilo Marchesini for several discussions that helped shaped the manuscript.\n\n\n\\section*{Data Availability} \n\nThe data underlying this article are available in the article itself and in its online supplementary material.\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe Internet of Things (IoT) is a heterogeneous domain, which makes it difficult to establish interoperability between all connected devices. One of the challenges is that IoT devices communicate via a variety of different connectivity protocols and transmit data in different formats \\cite{rahman2020comprehensive}. \n\nWith the Web of Things (WoT) architecture \\cite{wotArchitektur}, the World Wide Web Consortium (W3C) has presented a potential strategy for achieving semantic interoperability in the IoT. The distinctive aspect of the WoT architecture is that no new standard is created that IoT devices have to implement, but rather transmitted data and interaction affordances of already existing devices are described semantically in a uniform way that can be read by humans and machines. The WoT ecosystem also specifies a software stack, the so-called WoT Scripting API \\cite{ScriptingAPI}, which makes it simpler for programmers to create applications involving various IoT devices. The reference open source implementation of the WoT Scripting API is the JavaScript library node-wot\\footnote{\\url{https:\/\/github.com\/eclipse\/thingweb.node-wot}}.\n\n\nThe WoT focuses especially on the protocols located in the application layer of the internet protocol suite\\footnote{\\url{https:\/\/datatracker.ietf.org\/doc\/html\/rfc1122}} such as HTTP, MQTT and Websockets. The high-level communication standards of the application layer use transport layer protocols like the connection-oriented TCP or the connectionless UDP to transfer data between a destination and a source. TCP and UDP require the internet layer protocol, or network layer protocol in the OSI model, IP to transfer packages to other networks. The IP protocol interacts via the link layer with the MAC-based connectivity protocols WiFi and Ethernet.\n\n\nHowever, the IoT sector does not only consist of devices that implement the internet protocol suite and rely on WiFi or Ethernet connectivity in order to connect directly to the web but also of a large number of devices based on other connectivity standards, such as Bluetooth Low Energy (LE), ZigBee or Z-Wave \\cite{7460395}. Bluetooth LE, for example, is according to the 2020 Eclipse Developer Survey\\footnote{\\url{https:\/\/iot.eclipse.org\/community\/resources\/iot-surveys\/}} the third most relevant connectivity protocol in the IoT area only surpassed by WiFi and Ethernet.\n\nSince node-wot is focused on protocols that are based on IP, it is currently only possible to use the WoT abstraction and the associated advantages in a subset of all available IoT devices.\n\nWe aim to show that the communication patterns presented by the WoT architecture, i.e., the semantic description of devices and the communication model based on interaction affordances, are also suitable for non-IP-based connectivity protocols, which we will demonstrate using Bluetooth LE and the GATT protocol as an example. \n\nThe contributions presented in this paper are\n\\begin{itemize}\n \\item the design of ontologies to describe Bluetooth Low Energy devices and transmitted binary data,\n \\item the application of the WoT patterns to Bluetooth Low Energy and GATT,\n \\item the implementation and evaluation of Bluetooth Low Energy bindings and Thing Descriptions.\n\\end{itemize}\n\nWe will first give an example of the advantages that the application of WoT patterns on Bluetooth LE provides for programmers (section \\ref{section:Example}). After that we will discuss related work (section \\ref{section:RelatedWork}) and the background of Bluetooth, with focus on Bluetooth Low Energy (section \\ref{section:BackgroundOfBLE}). We define two vocabularies, one describing Bluetooth LE devices and the other describing the transmitted binary data, and map the abstract WoT operations to the concrete GATT methods (section \\ref{section:BLEforWoT}). We then explore the different ways of interacting with the Bluetooth protocol stack in Linux operating systems (section \\ref{section:BLEonLinux}) and implement and evaluate protocol bindings with three example Thing Descriptions for node-wot (section \\ref{section:Implementation} and \\ref{section:Evaluation}). Finally, we conclude our work and give an outlook on the future (section \\ref{section:conclusion}).\n\n\n\n\n\n\n\\section{Bluetooth LE using WoT}\n\\label{section:Example}\nWe want to demonstrate the advantages of the WoT abstraction with an example. We use a Bluetooth LE controlled lamp, that can be switched on or off and the current power status can be read. In the following, we consider the steps a programmer has to perform to read the power status, once by direct interaction with the lamp and once by using the WoT abstraction. \n\nThe first approach, without WoT abstraction, is shown in listing \\ref{ohne}. To read the power status, a programmer must first start the scan process and search for the IoT device using the MAC address (line 3). Once the device is found, the programmer can connect, access the GATT server, and select the appropriate service and characteristic (lines 4 to 8). The programmer is at this step able to read the bytes of the power characteristic (line 9). But the read bytes need to be decoded to get the actual value (line 10). Without the WoT abstraction, a programmer must know the MAC address, the IDs of the service and the characteristic, as well as the format of the transmitted binary data to access the power status.\n\n\\begin{lstlisting}[float, language=js, firstnumber=1, caption=Example pseudocode program reading the power status of the Bluetooth LE lamp without WoT abstraction., captionpos=b, label=ohne]\nadapter = defaultAdapter();\nadapter.startDiscovery();\ndevice = adapter.getDevice('BE:58:30:00:CC:11');\ndevice.connect();\nadapter.stopDiscovery();\ngattSrv = device.gatt();\nservice = gattSrv.getService('0000fff0-0000-1000-8000-00805f9b34fb');\nchar = service.getCharacteristic('0000fff3-0000-1000-8000-00805f9b34fb');\nbuffer = char.readValue();\nstatus = buffer.readIntLE(offset=0, bytelength=1);\ndevice.disconnect();\n\\end{lstlisting}\n\nThe WoT abstraction, on the other hand, uses a so-called Thing Description (TD) \\cite{TD}. The TD is a JSON-LD document that semantically describes the metadata and affordances of a device (a WoT Thing or Thing). The affordances of a Thing are formally grouped into three categories. Writeable and readable attributes are assigned to \\textit{properties}, executable and long-running processes to \\textit{actions}, and asynchronous notifications to \\textit{events}. Each affordance has a corresponding \\textit{forms} field in which protocol-specific information such as permitted operations, method names, and a Uniform Resource Identifier\\footnote{\\url{https:\/\/www.rfc-editor.org\/rfc\/rfc3986.html}} (URI) are defined.\n\nTo read the power status of the Bluetooth LE lamp using the WoT abstraction (listing \\ref{lst:mit}), a programmer simply needs to parse the TD (line 1), connect to the device and execute the abstract WoT method \\texttt{readproperty} with parameter 'power' (lines 2 and 3).\nThe interaction information, like MAC address, service ID, characteristic ID, and metadata of transmitted bytes are described in the TD. The programmer only needs to know the name of the property, which is in this case 'power'. The mapping of the abstract WoT operation (e.g. \\texttt{readproperty}) to corresponding protocol methods (e.g. \\texttt{read}) is performed by so-called WoT protocol bindings \\cite{BINDINGTemplate}.\n\nThe interaction with the BLE lamp using the WoT abstraction is much easier to realize for a programmer since a large part of the required information is already described in the TD.\n\n\\begin{lstlisting}[float, language=js, firstnumber=1, caption=Example pseudocode program reading the power status of the Bluetooth LE lamp with WoT abstraction., captionpos=b, label=lst:mit]\nthing = consume(TD);\nconnect(thing);\nstatus = thing.readProperty('power');\ndisconnect(thing);\n\\end{lstlisting}\n\n\\section{Related Work}\n\\label{section:RelatedWork}\nThe communication pattern of the Web of Things and the classification of device capabilities into properties, actions, and events have so far only been used for communication based on the network layer protocol IP. A number of different WoT protocol bindings for IP-based protocols have already been presented in the literature. The general method for developing protocol bindings was described by Mangas and Alonso \\cite{GARCIAMANGAS2019235}. The introduced approach was applied to HTTP, Websockets, MQTT, and CoAP. However, protocol bindings have also been developed for more specialized protocols. For example, Sciullo et al. \\cite{sciullo2020bringing} developed protocol bindings for the OPC UA and NETCONF protocols used in time-sensitive networks and presented additional vocabulary for semantic annotation within a Thing Description. There is also a W3C Editor's Draft for the Modbus \\cite{Modbusbindings} protocol, which can be used to manage hardware in industrial settings. \n\nAll these protocol bindings have in common that they are based on UDP or TCP and use the IP protocol. In contrast, we want to apply the WoT pattern to non-IP-based Bluetooth Low Energy communication. We focus especially on the application layer protocol GATT, which uses ATT in the transport layer and L2CAP in the network layer. \n\nnode-wot implements already a codec which deserializes and serializes binary data\\footnote{\\url{https:\/\/github.com\/eclipse\/thingweb.node-wot\/blob\/master\/packages\/core\/src\/codecs\/octetstream-codec.ts}} called \\texttt{octet-stream}. The \\texttt{octet-stream} codec takes the number of bytes (\\texttt{length}), the sign (\\texttt{signed}), the order of the bytes (\\texttt{byteorder}) and the character set (\\texttt{charset}) into account when operating on bytes. All parameters are passed as a comma-separated string. However, this information is not sufficient to adequately describe binary data transmitted by sensors. We therefore create a new codec as an extension to the existing one and introduce more relevant keywords (section \\ref{abschnittCodec} and table \\ref{tab:bdoOntolgy}). In addition, we do not want to pass the encoding or decoding information as a single string, but rather define each parameter using a separate RDF property in the Thing Description, which improves machine readability. \n\n\n\n\n\n\n\\section{Bluetooth Background}\n\\label{section:BackgroundOfBLE}\nFrom the beginning, the primary objective of Bluetooth has been to exchange data wirelessly between devices. For this purpose, the Bluetooth Special Interest Group (SIG) created a standardized specification \\cite{BluetoothLESpec}. Today there are two variants of the technology, Bluetooth Classic, also known as Bluetooth Basic Rate\/Enhanced Data Rate (BR\/EDR), and Bluetooth Low Energy (LE). Bluetooth Classic is used for direct data exchange between two devices, e.g. for audio streaming. Bluetooth LE was introduced with version 4.0 of the specification and was developed especially for small battery-operated devices.\nSince most IoT devices are small, standalone devices, we assume that the relevance of Bluetooth LE in the IoT sector is far greater than that of Bluetooth Classic. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=6cm]{BLEStack.png}\n \\caption{Each Bluetooth Low Energy device must implement the depicted protocol stack}\n \\label{fig:BLEStack}\n \\Description{The Bluetooth Low Energy protocol stack}\n\\end{figure}\n\nThe Bluetooth LE (BLE) protocol stack consists of two components, the host and the controller, which are connected via the host-controller interface (HCI), as shown in Figure \\ref{fig:BLEStack}. The host is typically part of the operating system and the controller is embedded logic on the Bluetooth controller hardware. The highest communication level in BLE, i.e. the application layer, is formed by the so-called profiles, such as GAP and GATT. Profiles are based on the transport layer and use protocols such as ATT. The transport layer protocols in turn utilize L2CAP, a network layer protocol. The functionalities of the different profiles and protocols are presented in more detail below \\cite{BluetoothLEPre}.\n\n\nThe \\textbf{Generic Access Profile (GAP)} deals with high-level device discovery and connection establishment. For this purpose, the device roles Broadcaster and Observer, as well as Peripheral (i.e. server) and Central (i.e. client) are defined. The combination of Broadcaster and Observer is a connectionless communication type, whereas communication between Peripheral and Central is connection-oriented and thus forms a point-to-point connection. However, a Peripheral supports only a single connection at a time. A Central, on the other hand, is able to initiate and maintain multiple connections.\n\nThe \\textbf{Attribute Protocol (ATT)} is used to read and write small data values (maximum 512 octets), but the protocol requires a connection for data exchange. Therefore, in the ATT environment, there are only the roles of a client and a server. The server organizes the data into attributes that are identified using so-called universally unique identifiers (UUIDs). The server can send responses to requests, receive data, and push asynchronous notifications to an ATT client. \n\nThe \\textbf{Generic Attribute Profile (GATT)} forms a superset of ATT, therefore a valid GATT client or server is also a valid ATT client or server. GATT specifies the structure of attributes contained on the server. GATT attributes are formatted in services, which in turn consist of characteristics. Characteristics hold a single data value and an indication of which GATT methods (see table \\ref{tab:mappingWoT2BLE}) are allowed. The data value of a characteristic can optionally be described in more detail by descriptors to indicate the meaning of the value to a client. \n\nThe \\textbf{Logical Link Control \\& Adaptation Protocol (L2CAP)} has essentially three tasks, the protocol multiplexing where packets are forwarded to the different higher layers, the control of the data flow between various layers, and segmentation as well as reassembly of packets that exceed the Maximum Transmission Unit (MTU). \n\n\n\n \n \n \n \n\n\n\n\n\n\\section{Bluetooth LE for the Web of Things}\n\\label{section:BLEforWoT}\nIn the IoT area, devices must have low energy consumption and provide data in a structured way. This can be achieved with Bluetooth LE in combination with GATT. For this reason, we focus on the description of the GATT profile with the technologies and communication patterns introduced by the Web of Things.\n\n\n\\subsection{Design}\n\\label{chapter:design}\nThe W3C has defined requirements for creating new protocol bindings \\cite{wotBinding}. The key considerations are the mapping of WoT operations to protocol methods, the definition of a URI scheme, and the specification of an appropriate media type.\nThese three points are covered in more detail below concerning our Bluetooth LE bindings.\n\n\\textbf{Default Mappings} In order to align and eventually integrate a new protocol into the WoT context, the required abstract WoT operations must first be mapped to the concrete operations of the new protocol. There are three categories of WoT operations. The first category is reading resources (e.g. \\texttt{readproperty}), the second is writing resources (e.g. \\texttt{writeproperty} or \\texttt{invokeaction}), and the third is receiving notifications asynchronously (e.g. \\texttt{subscribe}). The BLE GATT protocol has, among others, the methods \\texttt{read}, \\texttt{write} or \\texttt{write-without-response}, and \\texttt{notify}, which can be intuitively mapped to the three categories of WoT operations. However, a distinction must be made between the two GATT methods \\texttt{write} and \\texttt{write-without-response}. Both are capable of writing WoT resources, but the two methods differ in that \\texttt{write} expects a confirmation message from the server after a write operation, while \\texttt{write-without-response} requires no such confirmation. \nThus the GATT method chosen depends on the implementation of the attribute in the GATT server. Therefore, in Table \\ref{tab:mappingWoT2BLE} \\texttt{write} and \\texttt{write-without-response} are used interchangeably.\n\n\\begin{table}\n \\caption{Mapping of well-known abstract WoT operations to concrete BLE GATT methods}\n \\label{tab:mappingWoT2BLE}\n \\begin{tabular}{cc}\n \\toprule\n WoT Operation & BLE GATT Method\\\\\n \\midrule\n \\texttt{readproperty} & read \\\\\n \\texttt{writeproperty}& write | write-w\/o-response \\\\\n \\texttt{invokeaction}& write | write-w\/o-response\\\\\n \\texttt{readallproperties} & read \\\\\n \\texttt{writeallproperties}& write | write-w\/o-response\\\\\n \\texttt{readmultipleproperties}& read\\\\\n \\texttt{writemultipleproperties} & write | write-w\/o-response\\\\\n \\texttt{subscribeevent}& notify\\\\\n \\texttt{unsubscribeevent}& notify\\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\textbf{URI Scheme}\nThere is an expired Internet draft from the Internet Engineering Task Force (IETF) that addresses the issue of BLE URI schemes\\footnote{\\url{https:\/\/datatracker.ietf.org\/doc\/html\/draft-bormann-t2trg-ble-uri-00}}. It introduces a URI scheme that describes both GAP operations such as scan and connect and GATT operations such as read and write characteristics. However, the focus is on gateway interaction rather than direct communication, and the approach has not been followed up. \nThe draft proposes to get service links with a \\texttt{GET} request on \\textit{node\/services} and with another \\texttt{GET} request on \\textit{service\/characteristics} information about the characteristic can be obtained. \nBased on the considerations from the Internet Draft and the structuring of GATT attributes, we derived a custom URI scheme for GATT. Since we don't need information about services per se, but it is still advantageous for the implementation to know which characteristic is assigned to which service, we have defined the URI scheme in the form:\n\\begin{equation*}\n \\textrm{\\texttt{gatt:\/\/\/\/}}\n\\end{equation*}\nwith the following meaning:\n\\begin{description}\n \\item[\\texttt{gatt}] Identification of the transfer protocol\n \\item[\\texttt{}] MAC address of the Bluetooth device\n \\item[\\texttt{}] GATT service containing the characteristic\n \\item[\\texttt{}] GATT characteristic to interact with\n\\end{description}\nThe introduced URI scheme is suitable for uniquely identifying resources on GATT servers, and allows users to interact with the desired GATT characteristic.\n\n\\textbf{Media Type}\n\\label{abschnittCodec}\nIn the \\textit{forms} section of a Thing Description, a Media Type (also MIME Type or Content Type) can be specified that determines how sent data is encoded and received data is decoded. The encoding\/decoding is done using a suitable codec. Currently, there is no fitting combination of content type and codec that meets all requirements for the binary data transmitted by Bluetooth LE. The closest is \\textit{application\/octet-stream}, but unfortunately, this codec only fulfills parts of our requirements and can not encode or decode all binary data transmitted via Bluetooth LE. Therefore, we have extended \\textit{application\/octet-stream} and its codec to meet our needs. This can be done by adding a new user-defined subtype for the 'application' type since all unrecognized subtypes are interpreted as \\textit{application\/octet-stream} and its codec by definition\\footnote{\\url{https:\/\/www.rfc-editor.org\/rfc\/rfc1521}}. \nFor the Bluetooth LE bindings, we chose the new, non-standard subtype \\textit{x.binary-data-stream} and defined an associated codec that interprets the binary data using a newly created vocabulary. \n\n\\subsection{Vocabulary \/ Ontology}\n\\label{chapter:ontology}\nThe Thing Description is used to describe metadata and the interfaces of an existing Thing. However, to describe the possible interactions and transmitted binary data of a Bluetooth device, additional vocabulary is necessary. Therefore, we have developed two ontologies. One describes binary data in general, the other describes Bluetooth LE communication.\n\n\\textbf{Binary Data Ontology} The key vocabulary terms of the Binary Data Ontology\\footnote{\\url{https:\/\/freumi.inrupt.net\/BinaryDataOntology.ttl}} with the preferred prefix \\texttt{bdo} are shown in Table \\ref{tab:bdoOntolgy}. The bdo ontology is intended to provide maximum flexibility and describe all kinds of binary data. We want to use the bdo ontology to describe the data transmitted by Bluetooth LE devices, even those that do not comply with the Bluetooth standard (i.e. values of GATT characteristics should be in little-endian format as defined in Vol 3, Part G of \\cite{BluetoothLESpec}). The terms of the vocabulary are in our use case only useful within the properties, actions, or events parts of a Thing Description because they contain information about the data that is needed in the codec associated with \\textit{application\/x.binary-data-stream}.\nHowever, some Bluetooth devices expect not only the pure values but that a certain format is followed, for example, a certain start pattern followed by the actual value. For this case, the vocabulary provides the terms \\texttt{bdo:pattern} and \\texttt{bdo:variable}. With \\texttt{bdo:pattern} a byte pattern encoded as a hex string with variables can be specified. The data type and further meta-information about the variables are then defined with \\texttt{bdo:variable}. The user only has to provide the values of the variables, which are then inserted into the pattern before sending. The whole process is similar to URI variables. \nThe terms \\texttt{bdo:pattern} and \\texttt{bdo:variable} can also be used for received data, in which case only the bytes at the specified positions are decoded by the codec.\n\n\\begin{table*}\n \\caption{The key terms of the Binary Data Ontology}\n \\label{tab:bdoOntolgy}\n \\begin{tabular}{ccccc}\n \\toprule\n Vocabulary term & Description & Assignment & Type & Default Value\\\\\n \\midrule\n \\texttt{bdo:bytelength} & Number of octets in the data & required & integer & None\\\\\n \\texttt{bdo:signed} & Indicates if the data is signed & optional & boolean & false \\\\\n \\texttt{bdo:endianess} & Byte order of the binary data & optional & string & bdo:littleEndian\\\\\n \\texttt{bdo:offset} & Offset in number of octets & optional & integer & 0\\\\\n \\texttt{bdo:scale} & Scale of received integer value & optional & float & 1.0\\\\\n \\texttt{bdo:pattern} & Byte pattern of the binary data & optional & string & None\\\\\n \\texttt{bdo:variable} & Description of the variables in \\texttt{bdo:pattern} & required, if \\texttt{pattern} is used & --- & None \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{SBO3.png}\n \\caption{Classes and properties of the Simple Bluetooth Ontology}\n \\label{fig:SBO}\n \\Description{The Bluetooth Low Energy protocol stack}\n\\end{figure}\n\n\\textbf{Simple Bluetooth Ontology}\nThe communication and metadata of a Bluetooth Low Energy device is described using the Simple Bluetooth Ontology\\footnote{\\url{https:\/\/freumi.inrupt.net\/SimpleBluetoothOntology.ttl}} with preferred prefix \\texttt{sbo}. The sbo ontology provides classes and properties to describe the GAP role of a device, the structure of the data, the possible connection options, and low-level parameters of the Bluetooth LE link layer such as the length of the \\texttt{scanWindow}, the \\texttt{scanInterval} or the \\texttt{advertisingInterval}. GATT Characteristics additionally have a property called \\texttt{sbo:methodName} to specify allowed methods. In a Thing Description the \\texttt{sbo:methodName} property is used in the \\textit{forms} filed to indicate the required GATT method. Fig \\ref{fig:SBO} shows an overview of the most relevant classes and properties of the ontology.\n\n\\subsection{Bluetooth LE Thing Description Example}\nWith the design decisions and the ontologies presented in section \\ref{chapter:design} and section \\ref{chapter:ontology}, it is possible to appropriately describe a Bluetooth LE device and its transmitted data. An example Thing Description of the Bluetooth LE lamp introduced in section \\ref{section:Example} with a write property named \\texttt{power} that can switch the light off or on is shown in listing \\ref{lst:td}. \n\nLines 9 to 14 contain metadata. From the metadata, we can tell that the Bluetooth LE lamp is a peripheral, allows connections, and broadcasts an advertisement every 50 milliseconds.\n\nThe 'power' characteristic of the Bluetooth LE lamp requires a start and end pattern in addition to the actual value to switch the light on or off. This is represented in the Thing Description of the \\texttt{power} property via the term \\texttt{bdo:pattern} (line 20). The pattern is provided in the form of a hex string and contains a variable named \\textit{on} in curly brackets. Additional information about \\textit{on} is provided by \\texttt{bdo:variable} (line 21). From the annotation, it can be seen that \\textit{on} needs to be an integer with allowed values of 0 or 1 encoded in a single byte. For the not explicitly defined terms of the bdo ontology, the default values are used. This means that the value of \\textit{on} is converted to bytes in unsigned, little endian format without offset. \n\nIn the \\textit{forms} field, the method is specified as \\texttt{sbo:write} (line 31), which corresponds to a normal write operation that expects a confirmation response. Also, \\textit{application\/x.binary-data-stream} is selected as the codec for the operation (line 32). \n\nTo switch the Thing off or on, only a \\texttt{writeproperty} operation to the \\texttt{power} property containing the desired value of the variable \\textit{on} must be executed. The actual code to switch the lamp on would look like this: $$\\texttt{thing.writeproperty('power', \\{on: 1\\})}$$ The codec takes over the transformation of the integer value of the variable \\textit{on} into bytes and inserts the byte value into the pattern before the data is sent to the Thing.\n\n\\begin{lstlisting}[float, language=json,firstnumber=1, caption=Example TD of a Bluetooth RGB controller using bdo:pattern and bdo:variable to switch on and off, captionpos=b, label=lst:td]\n\"@context\": [\n 'https:\/\/www.w3.org\/2022\/wot\/td\/v1',\n {\n 'sbo': 'https:\/\/freumi.inrupt.net\/SimpleBluetoothOntology.ttl#',\n 'bdo': 'https:\/\/freumi.inrupt.net\/BinaryDataOntology.ttl#',\n 'rdf': 'http:\/\/www.w3.org\/1999\/02\/22-rdf-syntax-ns#',\n 'qudt': 'http:\/\/qudt.org\/schema\/qudt\/' },],\n ...\n'sbo:hasGAPRole': 'sbo:peripheral',\n'sbo:isConnectable': true,\n'sbo:hasGATTLayer': true,\n'sbo:hasAdvertisingInterval': {\n 'rdf:value': 50,\n 'qudt:unit': 'qudt:MilliSEC'},\n \n'properties': {\n 'power': {\n 'type': 'string',\n 'format': 'hex',\n 'bdo:pattern': '7e0004{on}00000000ef',\n 'bdo:variable': {\n 'on': {\n 'type': 'integer',\n 'minimum': 0,\n 'maximum': 1,\n 'bdo:bytelength': 1,\n } },\n 'forms': [{\n 'href': 'gatt:\/\/BE-58-30-00-CC-11\/0000fff0-0000-1000-8000-00805f9b\/0000fff3-0000-1000-8000-00805f9b34fb',\n 'op': 'writeproperty',\n 'sbo:methodName': 'sbo:write',\n 'contentType': 'application\/x.binary-data-stream',\n }], } },\n\n\\end{lstlisting}\n\n\\section{Using Bluetooth LE on Linux OS}\n\\label{section:BLEonLinux}\nA large part of the high-end and low-end IoT operating systems is Linux-based \\cite{bansal2020iot}. Therefore, we also want to focus on Linux. The official implementation of the Bluetooth protocol stack on Linux operating systems is BlueZ\\footnote{\\url{http:\/\/www.bluez.org\/}}. The BlueZ stack is divided into two parts, one part is integrated into the official Linux kernel since version 2.4.6, and the other part is included in the user space and is available as the BlueZ package\\footnote{\\url{https:\/\/packages.debian.org\/bullseye\/bluez}}. \nThe kernel handles low-level communication and security, the user space contains the central Bluetooth daemon \\texttt{bluetoothd} and other tools provided by BlueZ like \\texttt{btmgmt}\\footnote{\\url{https:\/\/manpages.debian.org\/testing\/bluez\/btmgmt.1.en.html}}. The communication between the user level and the kernel is realized by sockets \\cite{janc2016bluetooth}. \n\nThere are three possible methods for how a user space application can interact and manage the Bluetooth controller. The first variant is the direct communication with the raw HCI interface, the second is the use of the Bluetooth management interface (mgmt API) and the third is the use of the BlueZ D-Bus interface. \n\nCommands and events sent and received directly through the HCI interface are the lowest level of communication with a Bluetooth adapter. The HCI is in the true sense a hardware abstraction and not well suited for creating high-level applications, because even for simple Bluetooth operations a large amount of code has to be written \\cite{holtmann2006playing}. A detailed overview of the available HCI commands, functionalities, and events can be found in the Bluetooth specification in Vol. 4, Part E \\cite{BluetoothLESpec}. Due to the difficulty of this API, we do not consider it any further.\n\nThe Bluetooth management interface was introduced in kernel 3.4 to address problems caused by the old direct communication method with HCI sockets, such as the execution of blocking operations or synchronization between the kernel and user space when sending commands \\cite{mgmtAPI}. \nAvailable API calls to the Bluetooth management interface are, for example, the \\texttt{Set Local Name Command} to change the local name of a controller, or the \\texttt{Start Discovery Command} which starts the scan process and issues events when a device is found. These commands correspond internally to several HCI commands and are documented in the API specification\\footnote{\\url{https:\/\/git.kernel.org\/pub\/scm\/bluetooth\/bluez.git\/tree\/doc\/mgmt-api.txt}}. The aforementioned command line tool \\texttt{btmgmt} is based on the Bluetooth management interface.\n\nIn Linux systems, the so-called D-Bus\\footnote{\\url{https:\/\/www.freedesktop.org\/wiki\/Software\/dbus\/}} is used for interprocess communication. The Bluetooth daemon \\texttt{bluetoothd} provides a BlueZ API for communication using the D-Bus system. The communication approach using the bus offers the advantage that almost all common programming languages can interact with the D-Bus and thus also with the Bluetooth stack. In addition, the interface is well documented\\footnote{\\url{https:\/\/git.kernel.org\/pub\/scm\/bluetooth\/bluez.git\/tree\/doc}} and is mentioned in the \\textit{Bluetooth Technology for Linux Developers}\\footnote{\\url{https:\/\/www.bluetooth.com\/bluetooth-resources\/bluetooth-for-linux\/}} course as the preferred variant. The command line tool \\texttt{bluetoothctl}\\footnote{\\url{https:\/\/manpages.debian.org\/stretch\/bluez\/bluetoothctl.1.en.html}} is based on the D-Bus API.\n\nSince the D-Bus API has large documentation, is recommended by Bluetooth itself, and can be easily accessed in JavaScript via the D-Bus, we decided to use the D-Bus API for the implementation instead of the management API.\n\n\\begin{table*}[h]\n \\caption{Time data of bluetoothctl (N=25)}\n \\label{tab:bluetoothctlTime}\n \\begin{tabular}{cccc}\n \\toprule\n Device & Connect \/ ms & Disconnect \/ ms & read \/ ms \\\\\n \\midrule\n BLE RGB Controller & $837.76 \\pm 23.14$ & $2379.96 \\pm 48.05$& $100.91 \\pm 4.29$ \\\\\n Arduino GATT Server & $1084.15\\pm 37.21$ & $2326.93 \\pm 58.77$ & $84.13 \\pm 2.94$ \\\\\n Xiaomi Flower Care & $2764.45 \\pm 155.55$ & $2147.78 \\pm 46.13$ & $81.28 \\pm 2.79$ \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[h]\n \\caption{Time data of the BLE bindings (N=25)}\n \\label{tab:bindingsTime}\n \\begin{tabular}{cccc}\n \\toprule\n Device & Connect \/ ms & Disconnect \/ ms & read \/ ms \\\\\n \\midrule\n BLE RGB Controller & $918.22 \\pm 36.12$ & $2486.11 \\pm 76.33$& $103.91 \\pm 3.92$ \\\\\n Arduino GATT Server & $1326.47 \\pm 57.34$ & $2485.77 \\pm 67.42$ & $86.89 \\pm 2.48$ \\\\\n Xiaomi Flower Care & $3206.37 \\pm 255.54$ & $2263.17 \\pm 10.23$ & $84.51 \\pm 3.52$ \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\\section{Proof of Concept}\n\\label{section:Implementation}\n\\subsection{Implementation}\nTo demonstrate the practicality of the theoretical approaches presented in the previous sections, we implemented Bluetooth Low Energy protocol bindings\\footnote{\\url{https:\/\/github.com\/wintechis\/Bluetooth-Bindings}}. We decided to use JavaScript as the programming language for the implementation of the Bluetooth LE bindings. The use of JavaScript ensures compatibility with node-wot, the reference implementation of the WoT Scripting API. \nCurrently, only the client side is provided in the Bluetooth LE bindings, which allows communication with Bluetooth-enabled IoT devices of all kinds, but does not support creating custom GATT servers or exposing Bluetooth LE Things. \n\nWe do not implement a new Bluetooth LE library from scratch to interact with the D-Bus API, but instead, use the node-ble\\footnote{\\url{https:\/\/github.com\/chrvadala\/node-ble}} package as the basis for Bluetooth LE communication in our bindings. node-ble is built on BlueZ and already uses the BlueZ D-Bus API to control connection establishment, termination, and all other necessary interaction options (\\texttt{read}, \\texttt{write}, \\texttt{write-without-response}, and \\texttt{notify}). In addition to the low-level communication methods of node-ble, we have implemented other higher-level operations for general connection management. The provided operations can be used to start and stop the scanning process or to select which devices should stay connected, which should be reconnected each time, and which should be disconnected. \n\n\n\n\\subsection{Limitations}\nThe implementation of the Bluetooth LE bindings also has limitations. Because the Bluetooth LE bindings are based on an already existing Bluetooth LE library for JavaScript called node-ble. Since this library requires BlueZ and the associated D-BUS API, the protocol bindings are only usable on Linux-based operating systems and require a small setup for communication with the DBus daemon. The close relationship between BlueZ and node-ble also means that the Bluetooth LE bindings have all the advantages and disadvantages of the BlueZ implementation.\n\nBesides the requirement that BlueZ must be available, the Bluetooth LE bindings can only interact with GATT-based Bluetooth systems. Devices that broadcast their data via GAP advertisements cannot be used with the current implementation of the Bluetooth LE protocol bindings. \n\n\n\\section{Evaluation of the BLE bindings}\n\\label{section:Evaluation}\nIn this section, we evaluate the performance of the presented Bluetooth Low Energy bindings. To quantify the performance we first determined the latencies for different GATT operations using the WoT abstraction layer with our bindings and then performed the same GATT operations again with the well-established D-Bus-based tool \\texttt{bluetoothctl} and compared the determined timing results with each other. \n\nAll timing measurements were performed on an Intel NUC\\footnote{\\url{https:\/\/ark.intel.com\/content\/www\/de\/de\/ark\/products\/126147\/intel-nuc-kit-nuc8i5bek.html}} model NUC8i5BEK with Ubuntu 22.04 LTS (5.15.0-47) and BlueZ 5.65. Timings of the Bluetooth LE bindings were determined in Node.js with the command \\texttt{performance.now()}\\footnote{\\url{https:\/\/nodejs.org\/api\/perf_hooks.html\\#performancenow}}. \nWe modified the source code of \\texttt{bluetoothctl} in relevant places (\\textit{\/client\/main.c} and \\textit{src\/shared\/shell.c}) using the Linux system call \\texttt{gettimeofday} to determine the execution time of various GATT operations. The modified version of the \\texttt{bluetoothctl} software is available on GitHub\\footnote{\\url{https:\/\/github.com\/FreuMi\/bluez}}. Since both approaches are input and output-oriented, we measured elapsed real time rather than process time. \n\nThe latency measurements were performed with three different IoT devices at a constant distance of one meter from the Central in an environment without any other Bluetooth LE devices. The three test devices are an RGB Light Controller\\footnote{\\url{https:\/\/github.com\/arduino12\/ble_rgb_led_strip_controller}} with an advertising interval of 50ms, an Arduino\\footnote{\\url{https:\/\/create.arduino.cc\/projecthub\/monica\/getting-started-with-bluetooth-low-energy-ble-ab4c94}} with a custom GATT server and an advertising interval of 200 ms, and a Xiaomi Flower Care\\footnote{\\url{https:\/\/github.com\/vrachieru\/xiaomi-flower-care-api}} Sensor with an advertising interval of 2000 ms.\n\nThe timings were recorded for three operations, namely \\texttt{connect}, \\texttt{disconnect}, and \\texttt{read}.\nThe time for \\texttt{connect} covers the period between issuing the \\textit{connect} command until it is theoretically possible to interact with characteristics. This includes finding the device, establishing a connection, and exploring the available services and characteristics.\nThe time interval for the \\texttt{disconnect} operation starts after issuing the \\textit{disconnect} command and ends as soon as the session has been successfully disconnected. \nThe \\texttt{read} operation starts again with the \\textit{read} command and ends once the read value is available to the user.\n\nA total of 25 measurements were performed for each operation and the arithmetic means and associated standard errors of the means were calculated (Table \\ref{tab:bluetoothctlTime} and Table \\ref{tab:bindingsTime}).\n\nThe data shows that the introduced Bluetooth LE bindings are on average about 16 percent slower than bluetoothctl when establishing a connection and about 6 percent slower when disconnecting. The higher latency is partly due to the programming language, bluetoothctl is written in C, and partly due to the fact that the Bluetooth LE bindings have the additional abstraction layer of the Web of Things.\nThe read operations, on the other hand, are almost identical and only differ by about 3 percent.\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n \n\n \n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\n\\label{section:conclusion}\nThe Web of Things architecture's descriptive approach to create semantic interoperability between different incompatible networked devices is very promising. \n\nWe have shown in this work that it is possible to apply the property-, action-, and event-based communication model of the Web of Things to non-IP-based protocols using Bluetooth Low Energy interactions, in particular those involving servers whose attributes are formatted according to the GATT specification, as an example. We have designed ontologies that provide suitable vocabularies to describe metadata of Bluetooth Low Energy devices, various GATT methods, and the transmitted binary data, compared interaction possibilities with the Linux Bluetooth stack BlueZ, implemented protocol bindings with a matching codec for node-wot as a proof of concept, and evaluated the timing performance of the bindings. \n\nAfter laying the foundations for Bluetooth LE communication in the WoT context with this work, we want to extend the protocol bindings and ontologies in the next step. We plan to include not only GATT operations in our bindings but also other Bluetooth LE communication concepts and methods, such as listening to data that is sent via GAP advertisements and offering different Bluetooth LE security and encryption formats, which were out of scope in the Bluetooth LE bindings presented here. In addition, we want to use the technological basis and develop a WoT Servient-based gateway that can automatically consume Bluetooth LE devices described with a TD and expose them using an IP-based protocol.\n\n\\begin{acks}\nThis work was funded by the Bayerisches Verbundforschungsprogramm (BayVFP)\ndes Freistaates Bayern through the KIWI project (grant no. DIK0318\/03)\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdjnj b/data_all_eng_slimpj/shuffled/split2/finalzzdjnj new file mode 100644 index 0000000000000000000000000000000000000000..63c0ef0b9d92e9010b04d41a629a91b2716692ac --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdjnj @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\\title{Structure theorem for mod $p^m$ singular Siegel modular forms}\n\\author{Siegfried B\\\"ocherer and Toshiyuki Kikuta}\n\\maketitle\n\n\\noindent\n{\\bf 2020 Mathematics subject classification}: Primary 11F33 $\\cdot$ Secondary 11F46\\\\\n\\noindent\n{\\bf Key words}: Siegel modular forms, Congruences for modular forms, Fourier coefficients, mod $p^m$ singular. \n\n\\begin{abstract}\nWe prove that all mod $p^m$ singular forms of level $N$,\ndegree $n+r$, and $p$-rank $r$ with $n\\ge r$ \nare congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. \nMoreover, we prove that,\nthe levels of\ntheta series are of the form ``$p\\mbox{-power}\\times N$''. \nAdditionally, in some cases of mod $p$ singular forms with smallest possible weight,\nwe prove that the levels of theta series should be $p$.\n\\end{abstract}\n\n\\section{Introduction}\nFreitag \\cite{Frei} showed the following properties concerning singular (Siegel modular) forms. \n\\begin{itemize} \\setlength{\\itemsep}{-5pt}\n\\item A Siegel modular form of weight $k$ and degree $n$ is singular if and only if $k0\\}, \\]\nwhere $J_n:=\\smat{0_n}{1_n}{-1_n}{0_n}$. \nWe define the action of ${\\rm GSp}^+_n(\\R)$ on $\\hh_{n}$ by\n$gZ = (AZ + B)(CZ + D)^{-1}$ for $Z \\in \\hh_{n}$, $g \\in {\\rm GSp}^+_n(\\R)$.\nFor a holomorphic function $F:\\mathbb{H}_n\\longrightarrow \\mathbb{C}$ and a matrix $g=\\left( \\begin{smallmatrix} A & B \\\\ C & D \\end{smallmatrix}\\right)\\in {\\rm GSp}^+_n(\\R)$, \nwe define the slash operator in the usual way;\n\\[(F|_k\\; g)(Z):=n(g)^{\\frac{nk}{2}}\\det (CZ+D)^{-k}F(gZ).\\]\n\nLet $N$ be a natural number and $\\Gamma _n:={\\rm Sp}_n(\\mathbb{Z})$ the Siegel modular group (symplectic group with components in $\\Z$). \nIn this paper, we deal mainly with the congruence subgroup $\\Gamma _0^{(n)}(N)$ with level $N$ of $\\Gamma _n$ defined as \n\\begin{align*}\n&\\Gamma _0^{(n)}(N):=\\left\\{ \\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix}\\in \\Gamma _n \\: \\Big| \\: C\\equiv 0_n \\bmod{N} \\right\\}.\n\\end{align*}\nWe will also use the groups\n\\begin{align*}\n&\\Gamma _1^{(n)}(N):=\\left\\{ \\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix}\n\\in \\Gamma _n \\: \\Big| \\: C\\equiv 0_n \\bmod{N},\\\n\\det A \\equiv \\det D \\equiv 1 \\bmod{N} \\right\\},\\\\\n&\\Gamma ^{(n)}(N):=\\left\\{ \\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix}\\in \\Gamma _n \\: \\Big{|} \\: B\\equiv C \\equiv 0_n \\bmod{N},\\ A\\equiv D\\equiv 1_n \\bmod{N} \\right\\},\n\\end{align*}\nwhere $\\Gamma^{(n)}(N)$ is the so-called principal congruence subgroup of level $N$. We remark that \n\\[\\Gamma^{(n)}(N)\\subset \\Gamma _1^{(n)}(N)\\subset \\Gamma _0^{(n)}(N)\\subset \\Gamma _n.\\] \n\nFor a natural number $k$ and a Dirichlet character\n$\\chi : (\\mathbb{Z}\/N\\mathbb{Z})^\\times \\rightarrow \\mathbb{C}^\\times $, the space\n$M_k(\\Gamma _0^{(n)}(N),\\chi )$\nof Siegel modular forms of weight $k$ with\ncharacter $\\chi$ consists of all of holomorphic\nfunctions $F:\\mathbb{H}_n\\rightarrow \\mathbb{C}$ satisfying\n\\begin{equation*}\n(F|_{k}\\: g)(Z)=\\chi (\\det D)F(Z)\\quad \\text{for}\\quad g=\\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix}\\in \\Gamma_0^{(n)}(N).\n\\end{equation*}\nIf $n=1$, the usual condition in the cusps should be added.\nWhen $\\chi $ is a trivial character, we write simply $M_k(\\Gamma _0^{(n)}(N))$ for $M_k(\\Gamma_0^{(n)}(N) ,\\chi )$.\n\nLet $\\Gamma \\supset \\Gamma^{(n)}(N)$. \nSimilarly as above, we denote by $M_k(\\Gamma )$ the space consists of \nall of holomorphic functions $F:\\mathbb{H}_n\\rightarrow \\mathbb{C}$ satisfying\n\\begin{align*}\n(F|_{k}\\: g)(Z)=F(Z)\\quad \\text{for}\\quad g=\\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix}\\in \\Gamma.\n\\end{align*}\nIn this case also, we have to add the usual condition in the cusps when $n=1$.\n\nFor a prime $p$, let $\\Gamma $ be $\\Gamma _0^{(n)}(p^m)\\cap \\Gamma ^{(n)}(N) $ or $\\Gamma _0^{(n)}(p^m)\\cap \\Gamma _1^{(n)}(N)$, and $\\chi $ a Dirichlet character mod $p$.\nIn this paper, symbols $M_k(\\Gamma ,\\chi)$ and $M_k(\\Gamma )$ also for such $\\Gamma$ are sometimes used, but these are defined in the same way as above.\n\n\nNote that, for any Dirichlet character $\\chi $ mod $N$, we have \n\\[M_k(\\Gamma ^{(n)}(N))\\supset M_k(\\Gamma _1^{(n)}(N))\\supset M_k(\\Gamma _0^{(n)}(N),\\chi )\n. \\]\nWhen $F\\in M_k(\\Gamma)$ with $\\Gamma \\supset \\Gamma ^{(n)}(N)$, $N$ is called the ``level'' of $F$. Sometimes $\\Gamma $ itself is also called the level of $F$.\n\nAny $F \\in M_k(\\Gamma ^{(n)}(N))$ has a Fourier expansion of the form\n\\[\nF(Z)=\\sum_{0\\leq T\\in \\frac{1}{N}\\Lambda_n}a_F(T){\\boldsymbol e}({\\rm tr}(TZ)),\n\\quad Z\\in\\mathbb{H}_n,\n\\]\nwhere ${\\boldsymbol e}(x):=e^{2\\pi i x}$, \n\\[\n\\Lambda_n\n:=\\{ T=(t_{ij})\\in {\\rm Sym}_n(\\mathbb{Q})\\;|\\; t_{ii},\\;2t_{ij}\\in\\mathbb{Z}\\; \\},\n\\]\nand ${\\rm Sym}_n(R)$ is the set of symmetric matrices of size $n$ with components in $R$. \nIn particular, if $F \\in M_k(\\Gamma _1^{(n)}(N))$, the Fourier expansion of $F$ is given in the form \n\\[F(Z)=\\sum_{0\\leq T\\in \\Lambda_n}a_F(T){\\boldsymbol e}({\\rm tr}(TZ)). \\]\n\nIt is known that \n\\[M_k(\\Gamma _1^{(n)}(N))=\\bigoplus _{\\chi :(\\Z\/N\\Z)^\\times \\to \\C^\\times }M_k(\\Gamma_0^{(n)}(N) ,\\chi ),\\]\nwhere $\\chi $ runs over all the Dirichlet characters mod $N$. \nWe remark that, if $F\\in M_k(\\Gamma _0^{(n)}(N),\\chi )$, then we have\n\\[a_F(T[U])=(\\det U)^k\\chi (\\det U)a_F(T)\\]\nfor each $T\\in \\Lambda _n$ and $U\\in {\\rm GL}_n(\\Z)$. Here we write as $T[U]:={}^t U T U$. \nIn particular, if $\\chi (-1)=(-1)^k$, we have $a_F(T[U])=a_F(T)$ for \neach $T\\in \\Lambda _n$ and $U\\in {\\rm GL}_n(\\Z)$. \n\nLet $\\Phi$ be the Siegel $\\Phi$-operator defined by \n\\[\\Phi (F)(Z'):=\\lim _{t\\to \\infty }F\\mat{Z'}{0}{0}{it}, \\]\nwhere $F\\in M_k(\\Gamma_0^{(n)}(N) ,\\chi )$, $Z'\\in \\hh _{n-1}$, and $t\\in \\R$. \nAs is well-known, we have $\\Phi (F)\\in M_k(\\Gamma_0^{(n-1)}(N) ,\\chi )$ and the Fourier expansion of $\\Phi (F)$ is described as\n\\[\\Phi(F)(Z')=\\sum _{0\\le T\\in \\Lambda _{n-1}}a_F\\mat{T}{0}{0}{0}{\\boldsymbol e}({\\rm tr}(TZ')). \\]\nSuppose that $F\\in M_k(\\Gamma _0^{(n)}(N),\\chi )$ satisfies $\\Phi (F)\\neq 0$ (and then $F\\neq 0$). \nTaking $g\\in \\Gamma _0^{(n)}(N)$ as $g=\\smat{-1_n}{0_n}{0_n}{-1_n}$, \nwe have \n$F|_k g=(-1)^{nk}F=\\chi (-1)^{n}F$\nbecause of the transformation law of $F$. \nTherefore we have $\\chi (-1)^n=(-1)^{nk}$. \nOn the other hand, by the same property of $\\Phi (F)\\neq 0$, \nwe have $\\chi (-1)^{n-1}=(-1)^{(n-1)k}$. \nThese imply that $\\chi (-1)=(-1)^k$. \nIn this case (of $\\Phi(F)\\neq 0$), we have automatically $a_F(T[U])=a_F(T)$ for \n each $T\\in \\Lambda _n$ and $U\\in {\\rm GL}_n(\\Z)$. \n\nFor a subring $R$ of $\\mathbb{C}$, let $M_{k}(\\Gamma,\\chi )_{R}$ (resp. $M_{k}(\\Gamma )_{R}$)\ndenote the $R$-module of all modular forms in $M_{k}(\\Gamma, \\chi )$ (resp. $M_{k}(\\Gamma )$)\n whose Fourier coefficients are in $R$.\n\n\n\\subsection{Congruences for modular forms}\nLet $p$ be a prime and $\\Z_{(p)}$ the set of $p$-integral rational numbers. \nLet $F_i$ ($i=1$, $2$) be two formal power series of the form\n\\[F_i=\\sum _{T\\in \\frac{1}{N}\\Lambda _{n}}a_{F_i}(T){\\boldsymbol e}({\\rm tr}(TZ))\\]\nwith $a_{F_i}(T)\\in \\Z_{(p)}$ for all $T\\in \\frac{1}{N}\\Lambda _n$. \nWe write $F_1 \\equiv F_2$ mod $p^m$ if $a_{F_1}(T)\\equiv a_{F_2}(T)$ mod $p^m$ for all $T \\in \\frac{1}{N}\\Lambda _n$. \n\nLet $\\widetilde{M}_{k}(\\Gamma _0^{(n)}(N),\\chi )_{p^m}$ be the set of \n$\\widetilde{F}=\\sum _{T}\\widetilde{a_F(T)}{\\boldsymbol e}({\\rm tr}(TZ))$ with \n$F\\in M_{k}(\\Gamma _0^{(n)}(N),\\chi )_{\\Z_{(p)}}$, where $\\widetilde{a_F(T)}:=a_F(T)$ mod $p^m$. \nIf $m=1$, we write simply $\\widetilde{M}_{k}(\\Gamma _0^{(n)}(N),\\chi )$ for $\\widetilde{M}_{k}(\\Gamma _0^{(n)}(N),\\chi )_{p^m}$. \nNote that, $\\widetilde{M}_{k}(\\Gamma _0^{(n)}(N),\\chi )$ is a vector space over $\\F_p$.\n\nWe define the filtration weight as \n\\begin{align*}\n&\\omega^{n}_{N,\\chi,p^m} (F):=\\min \\{k \\;|\\; \\widetilde{F} \\in \\widetilde{M}_k(\\Gamma _0^{(n)}(N),\\chi )_{p^m} \\}. \n\\end{align*}\nWe write also $\\omega^{n}_{N,\\chi} (F):=\\omega^{n}_{N,\\chi,p} (F)$.\n\n\n\\begin{Def}\nLet $F\\in M_k(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$. \nWe say that $F$ is ``mod $p^m$ singular'' if \n\\begin{itemize} \\setlength{\\itemsep}{-5pt}\n\\item we have $a_F(T)\\equiv 0$ mod $p^m$ for any $T\\in \\Lambda _{n+r}$ with ${\\rm rank}(T)>r$, \n\\item there exists $T\\in \\Lambda _{n+r}$ with ${\\rm rank}(T)=r$ satisfying $a_F(T)\\not \\equiv 0$ mod $p$.\n\\end{itemize} \nWe call such $r$ ``$p$-rank'' of $F$. \nAdditionally if $n\\ge r$, we say that such $F$ is ``strongly mod $p^m$ singular''.\n\\end{Def}\n\\begin{Rem} \n\\begin{enumerate} \\setlength{\\itemsep}{-5pt}\n\\item\n Such a condition ``strongly\n singular\" also plays a crucial role in Freitag's book \\cite{Frei}\nfor the theory over ${\\mathbb C}$ for arbitrary congruence subgroups.\n\\item\nIf $F$ is nontrivial mod $p^m$ singular, then $F$ satisfies\n$\\Phi (F)\\neq 0$. \nThis implies that $\\chi (-1)=(-1)^k$ and $a_F(T[U])=a_F(T)$ \nfor each $T\\in \\Lambda _n$ and $U\\in {\\rm GL}_n(\\Z)$ (see Page 4). \nIn other words, when dealing with nontrivial mod $p^m$ singular modular forms, \nwe may assume that $\\chi (-1)=(-1)^k$.\n\\end{enumerate}\n\\end{Rem}\n\\begin{Thm}[B\\\"ocherer-Kikuta \\cite{Bo-Ki}]\n\\label{Thm:Bo-Ki}\nLet $n$, $r$, $k$, $N$ be positive integers and $p$ a prime with $p\\ge 5$. \nLet $\\chi $ be a quadratic Dirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_k(\\Gamma ^{(n+r)}_0(N),\\chi )_{\\Z_{(p)}}$ is mod $p^m$ singular of $p$-rank $r$. \nThen we have $2k-r\\equiv 0$ mod $(p-1)p^{m-1}$. \nIn particular, $r$ should be even. \n\\end{Thm}\n\nIt is a classical result by Serre \\cite{Se} that a congruence mod $p$ for two\nelliptic modular forms $f$ and $g$ for level one implies a congruence of their\nweights mod $p-1$. \nWe need a version for degree $n$ including levels and quadratic nebentypus:\n\\begin{Prop}\n\\label{weightcongruence}\nLet $p$ be a prime with $p\\ge 5$ and $N$ a positive integer with $p\\nmid N$. \nLet $\\chi$ and $\\chi'$ be two quadratic Dirichlet characters mod $p$ and\n$F\\in M_{k}(\\Gamma_0^{(n)}(p^m)\\cap \\Gamma_1^{(n)}(N),\\chi)$, $F'\\in M_{k'}(\n\\Gamma_0^{(n)}(p^m) \\cap \\Gamma_1^{(n)}(N),\\chi')$ be two modular forms satisfying $F\\equiv F'\\bmod p$.\nThen $k-k'=t\\cdot \\frac{p-1}{2}$ holds for some $t\\in {\\mathbb Z}$ and we have\n\\[\\chi=\\chi'\\iff t \\quad \\mbox{even}. \\]\n\\end{Prop} \n\\begin{proof} \nWe want to apply the results from B\\\"ocherer-Nagaoka \\cite{Bo-Na:3} to get the desired congruences for the weights.\nNote that in \\cite{Bo-Na:3} only the case of level $\\Gamma_1^{(n)}(N)$ with $N$ coprime to $p$ is covered.\n We may apply level change to $F$ and $F'$ to arrive at $G$ and $G'$ of\n level $\\Gamma_1^{(n)}(N)$ with $F\\equiv G$ mod $p$ and $G'\\equiv F'$ mod $p$ with weights $l$ and $l'$.\n\n If $\\chi=\\chi'$, we have $k\\equiv l$ mod $p-1$ and $k'\\equiv l'$ mod $p-1$\n and then, by \\cite{Bo-Na:3} $k\\equiv k'$ mod $p-1$, i.e. $t$ is even.\n \n If $\\chi\\not=\\chi'$, let us assume that $\\chi$ is nontrivial.\n Then $l\\equiv k+\\frac{p-1}{2}$ mod $p-1$ and the congruence\n $l\\equiv l'$ mod $p-1$ implies that $t$ has to be odd. This completes the proof. \n\\end{proof}\n\nTo formulate our results efficiently, we introduce the following\n(somewhat nonstandard). \n\\begin{Def}\n\\label{Eq_Char} \nSuppose that $k-k'=t\\cdot \\frac{p-1}{2}$ holds for some $t\\in {\\mathbb Z}$. \nFor a prime $p$ and a natural number $N$ coprime to $p$ let $\\chi$ and $\\chi'$ be two quadratic Dirichlet characters mod\n$pN$. \nWe write\n$$\\chi='\\chi'$$\nif $\\chi_N =\\chi'_N$, and $\\chi_p$ and $\\chi'_p$ are related as in\nProposition \\ref{weightcongruence}; i.e. \n\\[\\chi_p=\\chi'_p\\iff t \\quad \\mbox{even}. \\]\nHere $\\chi_N$ and $\\chi_p$ are the $N$-component and\n$p$-component of $\\chi$ (and the same for $\\chi'$).\nIn other words, we have\n\\[\\chi =' \\chi'\\iff \\chi =\\chi' \\left(\\frac{*}{p}\\right)^t, \\]\nwhere $(\\frac{*}{p})$ is the unique nontrivial quadratic character mod $p$. \n\nNote that this notation depends on $k$, $k'$ but it will always be clear form the \ncontext, which weights are involved.\n\\end{Def}\n\n\\subsection{Theta series for quadratic forms}\nLet $m$ be a positive integer. \nFor $S$, $T\\in \\Lambda _m$, we write $S\\sim T$ mod ${\\rm GL}_m(\\Z)$ if \nthere exists $U\\in {\\rm GL}_m(\\Z)$ such that $S[U]=T$.\nHere we put $S[U]:={}^tUSU$. \nWe say that $S$ and $T$ are ``${\\rm GL}_m(\\Z)$-equivalent'' if $S\\sim T$ mod ${\\rm GL}_m(\\Z)$. \nWe denote by $\\Lambda _m^+$ the set of all positive definite elements of $\\Lambda _m$. \nWe put $L:=\\Lambda _m$ or $\\Lambda ^+_m$. \nWe write $L\/{\\rm GL}_m(\\Z)$ for $L\/\\sim $ \nthe set of representatives of ${\\rm GL}_m(\\Z)$-inequivalence classes in $L$. \n\nLet $m$ be even. \nFor $S\\in \\Lambda _m^+$, we define the theta series of degree $n$ in the usual way:\n\\[\\theta _S^{(n)}(Z):=\\sum _{X\\in \\Z^{m,n}}{\\boldsymbol e}({\\rm tr}(S[X]Z))\\quad (Z\\in \\hh_{n}), \\] \nwhere $\\Z^{m,n}$ is the set of $m\\times n$ matrices with integral components and $S[X]:={}^tXSX$ and we write \n${\\boldsymbol e}(x):=e^{2\\pi i x}$. \nWe define the level of $S$ as \n\\[{\\rm level}(S):=\\min\\{N\\in \\Z_{\\ge 1} \\;|\\; N(2S)^{-1}\\in 2\\Lambda _m\\}. \\]\nThen $\\theta _S^{(n)}$ defines an element of $M_{\\frac{m}{2}}(\\Gamma _0^{(n)}(N),\\chi _S)$, \nwhere $N={\\rm level}(S)$, $\\chi _S$ is a Dirichlet character mod $N$ defined by \n\\[\\chi _S(d)={\\rm sign} (d)^\\frac{m}{2} \\left( \\frac{(-1)^\\frac{m}{2}\\det 2S}{|d|} \\right).\\] \nWe denote by ${\\rm cont}(S)$ the content of $S$ defined as\n\\[{\\rm cont}(S):=\\max\\{C\\in \\Z_{\\ge 1}\\;|\\; C^{-1}S\\in \\Lambda _n \\}. \\] \n\nFor fixed $S\\in \\Lambda^+ _{m}$ and $T\\in \\Lambda _n$, we put \n\\[A(S,T):=\\sharp\\{X \\in \\Z^{m,n} \\;|\\; S[X]=T \\}.\\]\nUsing this notation, we can write the Fourier expansion of the theta series in the\nform \n\\[\\theta _S^{(n)}(Z)=\\sum _{T\\in \\Lambda _n}A(S,T){\\boldsymbol e}({\\rm tr}(TZ)). \\]\n\n\n\nIn the above definitions, \n$S$ and $T$ are restricted to symmetric half integral matrices, \nbut the same symbols (such as $\\theta _S^{(n)}(Z)$ and $A(S,T)$) are used for symmetric matrices with rational components.\nThen $A(S,S)$ is the order of automorphism group of $S$;\n\\[A(S,S)=\\sharp \\{U\\in {\\rm GL}_m(\\Z)\\;|\\; S[U]=S\\}. \\]\nBy looking at minimal polynomials, one sees that ${\\rm GL}_m({\\mathbb Z})$ cannot\ncontain elements of order $p$ if $p>m+1$. \nFrom this we obtain the very useful statement that\n\\begin{align}\n\\label{Mink}\nA(S,S)\\not \\equiv 0 \\bmod{p} \\quad \\text{if}\\quad p> m+1\n\\end{align}\nfor any rational positive definite symmetric matrix $S$ of size $m$.\n\n\nFor later use, we introduce some results on the representation\nnumbers for binary quadratic forms: \n\\begin{Thm}[Dirichlet, Weber (see Kani \\cite{Kani}, Lemma 8, page 4)]\n\\label{Thm:Diri}\nLet $T$, $T_i\\in \\Lambda _2$ ($i=1,2$) be primitive forms, i.e. ${\\rm cont}(T)={\\rm cont}(T_i)=1$. \nAssume that $T_1$ and $T_2$ have a same discriminant $D<0$. Then we have the following statements. \n\\begin{enumerate} \\setlength{\\itemsep}{-5pt}\n\\item\nThere are infinitely many primes $l$ such that $A(T,l)>0$.\n\\item\nIf there exists a prime $l$ with $l\\nmid D$ such that $A(T_1,l)>0$ and $A(T_2,l)>0$, then $T_1\\sim T_2$ mod ${\\rm GL}_2(\\Z)$.\n\\end{enumerate}\n\\end{Thm}\n\n\n\\begin{Thm}[B\\\"ocherer-Nagaoka \\cite{Bo-Na:2}]\n\\label{Thm:Bo-Na}\nLet $S\\in \\Lambda _{2}^+$ be of level $p$. Let $n$ be a positive integer. \nThen there exists $F\\in M_{\\frac{p+1}{2}}(\\Gamma _n)_{\\Z_{(p)}}$ such that $F\\equiv \\theta ^{(n)}_S$ mod $p$. \n\\end{Thm}\nFor a quadratic form $S\\in \\Lambda _m^+$, we put $\\omega ^{n}_{N,\\chi ,p^m}(S):=\\omega ^{n}_{N,\\chi,p^m}(\\theta ^{(n)}_S)$,\n$\\omega ^{n}_{N,\\chi }(S):=\\omega ^{n}_{N,\\chi,p}(\\theta ^{(n)}_S)$.\n\n\n\\section{Conjectures and Results}\n\\label{Sec:3}\nWe begin by stating our conjecture in the most general situation which does not specify the weight and level. \n\\begin{Conj}\n\\label{Conj0}\nAny mod $p^m$ singular form is congruent mod $p^m$ to some true singular form. \n\\end{Conj}\nOur first result states that this conjecture is true in the strongly mod $p^m$ singular case (if $p$ is not small).\n\\begin{Thm}\n\\label{Thm:general}\nLet $n$, $k$, $N$ be positive integers, $r$ an even integer with $n\\ge r$. \nLet $p$ be a prime with $p>r+1$ and $\\chi $ a quadratic\nDirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_{k}(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$ is\nmod $p^m$ singular of $p$-rank $r$. \nThen we have the following statements. \n\\begin{enumerate} \\setlength{\\itemsep}{-5pt}\n\\item\nThere are finitely many $S\\in \\Lambda _r^+$ such that \n\\[F\\equiv \\sum_S c_S\\theta_S^{(n+r)} \\bmod{p^m} \\quad (c_S\\in \\Z_{(p)}) \\]\nand $\\widetilde{\\theta _{S}^{(n)}}\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^{m-\\nu}}$ (and hence $\\omega ^n_{N,\\chi ,p^{m-\\nu }}(S)\\le k$). \nHere $\\nu :=\\nu_p(c_S)$ and $\\nu _p$ is the additive valuation on $\\Q$ normalized so that $\\nu_p(p)=1$. \nMoreover, all $S$ involved satisfy $\\chi='\\chi_S$. \n\\item\nFor a suitable $e\\in {\\mathbb N}$, all of $S\\in \\Lambda _r^+$ appearing in (1) satisfy that ${\\rm level}(S)\\mid p^eN$. \n\\end{enumerate}\n\\end{Thm}\n\\begin{Rem}\n\\begin{enumerate} \\setlength{\\itemsep}{-5pt}\n\\item\nActually, each $c_S$ is described in terms of the primitive Fourier coefficient for $S$ of $F$. \nFor details, see the proof in Section \\ref{Sec:7}. \n\\item\nThe statement on $\\widetilde{\\theta _{S}^{(n)}}\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^{m-\\nu}}$ can be rephrased by $c_S\\theta_S^{(n)}\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^{m}}$.\nNote also that $\\nu =0$ if $m=1$. \n\\item\nWe emphasize that we do not know anything about $e$.\n\\end{enumerate}\n\\end{Rem}\nWe expect that the theorem above holds in the most general case:\n\\begin{Conj} \nTheorem \\ref{Thm:general} should hold for any prime (not only for $p>r+1$)\nand without the assumption $n\\geq r$. \n\\end{Conj}\n\n\nThe smallest weights (see Theorem \\ref{Thm:Bo-Ki}) where we can expect mod $p$ singular forms,\nwhich are not true singular forms, is of special interest. \nIn these cases, we expect also the power $e$ of $p$ in the levels of theta series to be the smallest possible: \n\\begin{Conj}\n\\label{Conj4}\nLet $n$ be a positive integer, $r$ an even integer, and $p$ a prime with $p\\ge n+r+3$. \nWe put \n\\[k=k(p,r):=\n\\begin{cases} \nr\/2+(p-1)\/2\\quad &\\text{if}\\quad r\\equiv 2 \\bmod{4},\\ p\\equiv -1 \\bmod{4} \\\\\nr\/2+p-1\\quad &\\text{if}\\quad r\\equiv 0 \\bmod{4}. \n\\end{cases}\\]\nLet $N$ be a positive integer and $\\chi $ a quadratic Dirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_{k}(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$ is mod $p$ singular of $p$-rank $r$. \nThen we have \n\\begin{align*}\nF\\equiv \\sum _{\\substack{S \\in \\Lambda _{r} \/ {\\rm GL}_r(\\Z) \\\\ {\\rm level}(S)\\mid pN}} c_S\\theta ^{(n+r)}_{S} \\bmod{p}\\quad (c_S\\in \\mathbb{Z}_{(p)}). \n\\end{align*} \n\\end{Conj}\nIn the special case of $N=1$, $r=2$, we can prove this conjecture: \n\\begin{Thm}\n\\label{Thm:r=2} \nLet $n$ be a positive integer and $p$ a prime with $p\\ge n+5$. \nSuppose that $F\\in M_{\\frac{p+1}{2}}(\\Gamma _{n+2})_{\\Z_{(p)}}$ is mod $p$ singular of $p$-rank $2$. \nThen we have \n\\begin{align*}\nF\\equiv \\sum _{\\substack{S \\in \\Lambda _{2} \/ {\\rm GL}_2(\\Z) \\\\ {\\rm level}(S)=p}} c_S\\theta ^{(n+2)}_{S} \\bmod{p}\\quad (c_S\\in \\mathbb{Z}_{(p)}). \n\\end{align*}\nMoreover for any such $F$ the degree one form $f:=\\Phi^{n+1}(F)$\nis nonzero mod $p$.\n\\end{Thm}\n\n\\begin{Rem}\nWe remark that Theorem \\ref{Thm:r=2}\nhas some potential for showing the nonvanishing of\nFourier coefficients mod $p$ in some interesting cases, e.g. \na degree $3$ modular form satisfying $\\Phi^2(F)\\equiv 0$ mod $p$\ncannot be mod $p$ singular. \nThis implies the existence of some nonvanishing\nrank 3-Fourier coefficients $a_F(T)\\bmod p$ of Klingen-Eisenstein series\n$F:=E^{3,2}_k$ attached to a cusp form $h$ of degree 2, provided that its\nFourier coefficients are in ${\\mathbb Z}_{(p)}$.\nNote that such Fourier coefficients are quite delicate,\ngiven by some critical values of\n$L$-series attached to $h$ and to $T\\in \\Lambda^+_3$, if $h$ is a Hecke eigenform\n(see \\cite{Bo,Kli} for more details on Klingen-Eisenstein series).\nTo cover more general\ncases, versions of this for congruences modulo prime ideals\nin the field generated by the Hecke eigenvalues of $h$ \nwill be necessary.\n\\end{Rem}\n\n\n\\section{Refinement of Freitag's expansion}\n\\label{Sec:4}\nIn this section, inspired by Freitag \\cite{FreiA,Frei}, we give a\nformal expansion of the ``singular part'' of the Fourier expansion of \nany modular form and apply it to mod $p^m$ singular forms.\n\n\nWe fix some notation. \nLet $M_n(R)$ be the set of all $n\\times n$ matrices whose components are in $R$. \nWe put $M^{(r)}_n(\\Z):=\\{M\\in M_n(\\Z)\\;|\\; {\\rm rank}(M)=r\\}$ and $M^{*}_n(\\Z):=M^{(n)}_n(\\Z)$. Similarly we write \n$\\Lambda _n^{(r)}:=\\{T\\in \\Lambda _n\\;|\\; {\\rm rank}(T)=r\\}$ ($\\Lambda _n^+=\\Lambda _n^{(n)}$).\nLet $F$ be a modular form of degree $n+r$ with Fourier expansion \n\\[F(Z)=\\sum _{T\\in \\Lambda _{n+r}}a(T){\\boldsymbol e}({\\rm tr}(TZ)).\\]\nWe write $a(S):=a\\smat{0}{0}{0}{S}$ when $S\\in \\Lambda _r$.\nWe define a subseries $F_{[r]}$ of $F$ as\n\\[F_{[r]}(Z):=\\sum _{T\\in \\Lambda _{n+r}^{(r)}} a(T){\\boldsymbol e}({\\rm tr}(TZ)). \\]\n\n\nIn B\\\"ocherer-Raghavan \\cite{Bo-Ra} (see page 82 and 83), the notion of ``primitive Fourier coefficient'' was introduced; \nwe denote it by $a^*(S)$ for $S$ positive definite. Namely, \n$a^*(S)$ is defined by the formula\n\\[a(S)=\\sum _{\\substack{G\\in {\\rm GL}_r(\\Z)\\backslash M^*_r(\\Z)\\\\S[G^{-1}]\\in \\Lambda _r}}a^*(S[G^{-1}]). \\] \nWe recall that by this formula, we can define a new ${\\rm GL}_r(\\Z)$-invariant function $a^*(S)$\nstarting from the ${\\rm GL}_r(\\Z)$-invariant function\n$T\\longmapsto a(T)$ on $\\Lambda_r$.\n\nAs explained in \\cite{Bo-Ra}, this can also be written as\n\\begin{align}\n\\label{Eq:Pri}\na(T)=\\sum _{S\\in \\Lambda _r^+ \/ {\\rm GL}_r(\\Z)} \\frac{1}{\\epsilon (S)}\\sum _{\\substack{ W\\in M_r^{*}(\\Z) \\\\ S[W]=T}} a^*(S), \n\\end{align}\nwhere $\\epsilon (S):=A(S,S)$. We use this version. \n\n \nUsing the Fourier coefficients $a(S)$ with $S\\in \\Lambda _r^+$ and their modification, \na slight refinement of Freitag's argument\ngives by formal rearrangement of the Fourier expansion our\ncrucial identity. Note that the statements are only claiming the equality\nof the Fourier coefficients on both sides (ignoring questions concerning\nconvergence!).\n\n\\begin{Lem}\n\\label{Lem:Refine}\nLet $\\chi $ be a Dirichlet character mod $N$ such that $\\chi (-1)=(-1)^k$. \nLet $F\\in M_k(\\Gamma ^{(n+r)}_0(N),\\chi )$. \nThen we have\n\\begin{align}\n\\label{Eq:0.1}\nF_{[r]}(Z)&=\\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\n\\frac{a^*(S)}{\\epsilon(S)}\\sum _{\\substack{(X_1,X_2)\\in \\Z^{r,r}\\times \\Z^{r,n}\\\\{\\rm rank}(X_1,X_2)=r}}{\\boldsymbol e}({\\rm tr}(S[(X_1,X_2)]Z)). \n\\end{align}\n\\end{Lem}\nNote that\n\\begin{align*}\n\\sum _{\\substack{(X_1,X_2)\\in \\Z^{r,r}\\times \\Z^{r,n}}}{\\boldsymbol e}({\\rm tr}(S[(X_1,X_2)]Z))&=\\sum _{\\substack{X\\in \\Z^{r,n+r}}}{\\boldsymbol e}({\\rm tr}(S[X]Z))=\\theta _S^{(n+r)}(Z).\n\\end{align*}\nHence if we can prove this lemma, then $F_{[r]}$ can be expressed by a infinite linear combination of (subseries of) theta series;\n\\begin{align}\n\\label{Eq:0.15}\nF_{[r]}=\\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z)}\\frac{a^*(S)}{\\epsilon(S)}(\\theta ^{(n+r)}_S)_{[r]}.\n\\end{align}\n\n\\begin{proof}[Proof of Lemma \\ref{Lem:Refine}]\nFor $X_1\\in \\Z^{r,r}$, $ X_2\\in \\Z^{r,n}$, we can take $W\\in M^*_r(\\Z)$ such that $(X_1,X_2)=W(G_1,G_2)$ and $\\smat{*}{*}{G_1}{G_2}\\in {\\rm GL}_{n+r}(\\Z)$.\nThis $W$ can be regarded as ``gcd'' of $X_1$ and $X_2$. \nWe observe that such $W$ is unique up to a factor in ${\\rm GL}_r(\\Z)$ from the right. \nWe switch this action of ${\\rm GL}_r(\\Z)$ to $(G_1,G_2)$. \nThen we can write the right hand side of (\\ref{Eq:0.1}) as\n\\begin{align}\n\\label{Eq:0.2}\n\\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z)}\\frac{a^*(S)}{\\epsilon(S)}\\sum _{W\\in M^*_r(\\Z)} \\sum _{\\substack{(G_1,G_2)\\in \n{\\rm GL}_r(\\Z)\\backslash \\Z^{r,r}\\times \\Z^{r,n}\\\\ \\smat{*}{*}{G_1}{G_2}\\in {\\rm GL}_{n+r}(\\Z)\n}}\n{\\boldsymbol e}({\\rm tr}(S[W][(G_1,G_2)]Z))\n\\end{align}\nwhere ${\\rm GL}_r(\\Z)\\backslash \\Z^{r,r}\\times \\Z^{r,n}=\\sim \\backslash \\Z^{r,r}\\times \\Z^{r,n}$ and $(X_1, X_2)\\sim (Y_1,Y_2)$ means that $(X_1, X_2)=G(Y_1,Y_2)$ for some $G\\in {\\rm GL}_r(\\Z)$.\n\n\n\nWe put $S[W]=T$ and we rewrite the summation over $S$ as over $T$.\nThen (\\ref{Eq:0.2}) becomes \n\\begin{align}\n\\label{Eq:0.3}\n\\sum _{T\\in \\Lambda _{r}^{+}}a(T) \\sum _{\n\\substack{(G_1,G_2)\\in \n{\\rm GL}_r(\\Z)\\backslash \\Z^{r,r}\\times \\Z^{r,n}\\\\ \\smat{*}{*}{G_1}{G_2}\\in {\\rm GL}_{n+r}(\\Z)\n}}\n{\\boldsymbol e}({\\rm tr}(T[(G_1,G_2)]Z))\n\\end{align} \nbecause of (\\ref{Eq:Pri}). \n\n\n\n\nIf we put $U:=\\smat{*}{*}{G_1}{G_2}$, then we have $T[(G_1,G_2)]=\\smat{0}{0}{0}T[U]$. \nTherefore we have $a(T)=a\\smat{0}{0}{0}{T}=a(T[(G_1,G_2)])$. \nThen (\\ref{Eq:0.3}) can be written as \n\\begin{align*}\n\\sum _{T\\in \\Lambda _{r}^{+}}&\\sum _{\n\\substack{(G_1,G_2)\\in \n{\\rm GL}_r(\\Z)\\backslash \\Z^{r,r}\\times \\Z^{r,n}\\\\ \\smat{*}{*}{G_1}{G_2}\\in {\\rm GL}_{n+r}(\\Z)\n}}\na(T[(G_1,G_2)]) {\\boldsymbol e}({\\rm tr}(T[(G_1,G_2)]Z))\\\\\n&= \\sum _{T\\in \\Lambda _{n+r}^{(r)}}a(T){\\boldsymbol e}({\\rm tr}(TZ))=F_{[r]}(Z). \n\\end{align*}\n\nHere the first equality in this formula follows from the fact that, if $T\\in \\Lambda _r^+$ and $(G_1,G_2)$ run as in the subscript, then $T[(G_1,G_2)]$ runs over all elements of $\\Lambda _{n+r}^{(r)}$.\n\\end{proof}\nNote that $F_{[r]}$ is not a modular form because some part of the Fourier expansion is missing. \n\nLet $Z_1\\in \\hh_r$, $Z_2\\in \\hh_n$. \nConsider the restriction of $F$ to $Z=\\smat{Z_1}{0}{0}{Z_2}$; \n\\[F\\mat{Z_1}{0}{0}{Z_2}=\\sum _{\\substack{\\smat{T}{*}{*}{*}\\in \\Lambda _{n+r} \\\\ T \\in \\Lambda _{r}}} \\phi _T (Z_2)\\e ({\\rm tr}(TZ_1)). \\]\nThen we have $\\phi _T(Z_2)\\in M_k(\\Gamma ^{(n)}_0(N),\\chi )$ for any $T\\in \\Lambda _{r}$. For the proof of this fact, we refer to Andrianov \\cite{And} (page 83 and 84). \n\nOn the other hand, we denote by $F^{\\sharp}$ the subseries of $F=\\sum _{\\mathfrak{T}\\in \\Lambda _{n+r}}a_F(\\mathfrak{T})\\e ({\\rm tr}(\\mathfrak{T}Z))$, characterized by \n\\[\\mathfrak{T}=\\mat{T}{*}{*}{*}\\quad \\text{with} \\quad T \\in \\Lambda _r^+. \\] \nNamely we put \n\\[F^{\\sharp}(Z):=\\sum _{\\substack{\\mathfrak{T}=\\smat{T}{*}{*}{*}\\in \\Lambda _{n+r} \\\\ T \\in \\Lambda _r^+}}a_F(\\mathfrak{T})\\e ({\\rm tr}(\\mathfrak{T}Z)). \\]\nThen we have\n\\[F^{\\sharp}\\mat{Z_1}{0}{0}{Z_2}=\\sum _{\\substack{\n\\smat{T}{*}{*}{*}\\in \\Lambda _{n+r} \\\\ T \\in \\Lambda _r^+}} \\phi _T (Z_2)\\e ({\\rm tr}(TZ_1)).\\]\nNote that still we have $\\phi _T (Z_2)\\in M_k(\\Gamma ^{(n)}_0(N),\\chi )$ for any $T\\in \\Lambda _r^+$. \n\nNow assume that $F$ is mod $p^m$ singular of $p$-rank $r$. \nThen from $F$ being mod $p^m$ singular we obtain \n$F^{\\sharp} \\smat{Z_1}{0}{0}{Z_2}\\equiv (F_{[r]})^{\\sharp}\\smat{Z_1}{0}{0}{Z_2}$ mod $p^m$. \nBy Lemma \\ref{Lem:Refine}, we have \n\\begin{align*}\n (F_{[r]})^{\\sharp}\\mat{Z_1}{0}{0}{Z_2}&\\equiv\n \\left( \\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\\frac{a^*(S)}{\\epsilon(S)}\n \\sum _{\\substack{(X_1,X_2)\\in \\Z^{r,r}\\times \\Z^{r,n}\\\\{\\rm rank}(X_1,X_2)=r}}\n {\\boldsymbol e}({\\rm tr}(S[X_1]Z_1)\n {\\boldsymbol e}({\\rm tr}(S[X_2]Z_2))\\right)^{\\sharp}\\\\\n &\\equiv\\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\\frac{a^*(S)}\n {\\epsilon(S)}\\sum _{\\substack{(X_1,X_2)\\in\n \\Z^{r,r}\\times \\Z^{r,n}\\\\{\\rm rank}(X_1)=r}}\n {\\boldsymbol e}({\\rm tr}(S[X_1]Z_1)\n {\\boldsymbol e}({\\rm tr}(S[X_2]Z_2))\\\\\n &\\equiv \\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\\frac{a^*(S)}\n {\\epsilon(S)}\\sum _{\\substack{X_1\\in \\Z^{r,r}\\\\{\\rm rank}(X_1)=r}}\n {\\boldsymbol e}({\\rm tr}(S[X_1]Z_1))\\sum _{\\substack{X_2\\in \\Z^{r,n}}}\n {\\boldsymbol e}({\\rm tr}(S[X_2]Z_2))\\\\\n &\\equiv\n \\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\n \\frac{a^*(S)}{\\epsilon(S)}(\\theta _S^{(r)}(Z_1))_{[r]}\n \\theta _S^{(n)}(Z_2)\\quad \\bmod p^m. \n\\end{align*} \nThis implies that\n\\[\\phi _T(Z_2)\\equiv \\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }\nA(S,T)\\frac{a^*(S)}{\\epsilon(S)}\\theta _S^{(n)}(Z_2) \\bmod{p^m} \\]\nfor any $T\\in \\Lambda _r^+$. \nHence we obtain that \n\\[\\sum _{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) }A(S,T)\\frac{a^*(S)}{\\epsilon(S)}\\theta _S^{(n)}(Z_2) \\bmod{p^m} \\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}. \\]\n\nIn the ordinary case (over $\\C$), the key in Freitag's setting would be\nthat $a(S)$ can be different from zero only if ${\\rm level}(S)\\mid N$. \nIn our mod $p^m$ setting, we get a condition on the filtration\nof $\\theta _S^{(n)}$ for $S\\in \\Lambda _r^+$ with $a^*(S)\\not \\equiv 0$ mod $p^m$. More precisely, we get the following property. \n\\begin{Prop}\n\\label{Prop:filt}\nLet $n$, $k$, $N$ be positive integers and $r$ an even integer. \nLet $p$ be a prime with $p> r+1$ and $\\chi $ a quadratic Dirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_k(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$ is mod $p^m$ singular of $p$-rank $r$. \nThen we have $a^*(S)\\theta _S^{(n)}$ mod $p^m$ $\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}$ for any $S\\in \\Lambda _r^+$. \nIn particular, if $S\\in \\Lambda _{r}^+$ satisfies $a^*(S)\\not \\equiv 0$ mod $p^m$, \nthen we have $\\theta _S^{(n)}$ mod $p^m$ $\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^{m-\\nu}}$ (and hence \n$\\omega _{N,\\chi,p^{m-\\nu }}^n(S)\\le k$) with $\\nu :=\\nu _p(a^*(S))$. \nMoreover $\\chi ='\\chi _S$ holds. \n\\end{Prop}\n\n\\begin{proof}\nSeeking a contradiction, we suppose that there exists $S$ such that the claim is not true. \nLet $S_0$ be one of $S$ such that $\\det S_0$ is minimal among such $S$. \nThen we consider \n\\begin{align*}\n\\phi _{S_0}-&\\sum _{\\substack{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) \\\\ \\det S<\\det S_0}}A(S,S_0)\\frac{a^*(S)}{\\epsilon(S)}\\theta _S^{(n)}\\\\\n&\\equiv\\sum _{\\substack{S\\in \\Lambda_{r}^{+}\/{\\rm GL_r}(\\Z) \\\\ \\det S\\ge \\det S_0}}A(S,S_0)\\frac{a^*(S)}{\\epsilon(S)}\\theta _S^{(n)}\\\\\n&\\equiv A(S_0,S_0)\\frac{a^*(S_0)}{\\epsilon(S_0)}\\theta _{S_0}^{(n)}\\bmod p^m. \n\\end{align*}\nHere the last congruence follows from the facts that\n$\\det S>\\det S_0$ implies $A(S,S_0)=0$ and $\\det S=\\det S_0$,\n$S\\not \\sim S_0$ mod ${\\rm GL}_r(\\Z)$ implies $A(S,S_0)=0$.\nBy the assumption, we have\n\\[\\phi _{S_0}-\\sum _{\\substack{S\\in {\\rm GL}_r(\\Z)\\backslash \\Lambda _r ^+ \\\\ \\det S<\\det S_0}}A(S,S_0)\\frac{a^*(S)}{\\epsilon(S)}\\theta _S^{(n)}\\bmod{p^m} \\in\n\\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}. \\] \nThis implies $A(S_0,S_0)\\frac{a^*(S_0)}{\\epsilon(S_0)}\n\\theta _{S_0}^{(n)}\\bmod{p^m} \\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}$. \nBy the fact (\\ref{Mink}), we have $p\\nmid A(S_0,S_0)$.\nThis shows $a^*(S_0)\\theta _{S_0}^{(n)}\\bmod p^m\\in\n \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}$. \nThis is a contradiction. \nHence we have $a^*(S)\\theta _S^{(n)} \\bmod p^m\\in\n \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^m}$ for any\n $S\\in \\Lambda _{r}^+$. \n In particular if $a^*(S)\\not \\equiv 0$ mod $p^m$, then we have $\\theta _S^{(n)}$ mod $p^m$ $\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )_{p^{m-\\nu}}$ with $\\nu :=\\nu _p(a^*(S))$, and therefore \n$\\omega _{N,\\chi,p^{m-\\nu }}^n(S)\\le k$. \nThe statement $\\chi ='\\chi _S$ follows from Proposition \\ref{weightcongruence}. \n\\end{proof}\nFor later use, we mention a simple consequence of Proposition \\ref{Prop:filt} for the mod $p$ case. \n\\begin{Cor}\n\\label{Cor:filt}\nLet $n$, $k$, $N$ be positive integers, $r$ an even integer with $n\\ge r$. \nLet $p$ be a prime with $p> r+1$ and\n$\\chi $ a quadratic Dirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_k(\\Gamma ^{(n+r)}_0(N),\\chi )_{\\Z_{(p)}}$ is mod $p$ singular of $p$-rank $r$. \n\\begin{enumerate} \\setlength{\\itemsep}{-5pt}\n\\item\nFor any $S\\in \\Lambda _{r}^+$ with $a^*(S)\\not \\equiv 0$ mod $p$, we have $\\widetilde{\\theta _S^{(r)}} \\in \\widetilde{M}_k(\\Gamma ^{(r)}_0(N),\\chi )$. \n\\item\nFor $S\\in \\Lambda _{r}^+$ with $a(S)\\not \\equiv 0$ mod $p$ such that $\\det S$ is minimal in such $S$, \nwe have $\\widetilde{\\theta _S^{(n)}} \\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )$ and therefore $\\omega _{N,\\chi }^n(S)\\le k$.\n\\end{enumerate}\n\\end{Cor}\n\\begin{Rem}\n The statement (2) can be proved also by the mod $p$ version of\n Freitag's original arguments in \\cite{Frei}. \nOur strategy can be viewed as a refinement of his method.\n\\end{Rem}\n\\begin{proof}[Proof of Corollary \\ref{Cor:filt}]\n(1) By Proposition \\ref{Prop:filt}, we have $\\widetilde{\\theta _S^{(n)}} \\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )$. \nHence the claim follows from $\\Phi ^{n-r}(\\theta _S^{(n)})=\\theta _S^{(r)}$ immediately. \\\\\n(2) If $S\\in \\Lambda _{r}^+$ satisfies such the minimality condition, we have $a^*(S)=a(S)$ by the definition of $a^*(S)$. \nThe claim follows from this fact. \n\\end{proof}\n\n\n\\section{Description by theta series for mod $p$ case}\n\\label{Sec:5}\nIn this section, we prove Theorem \\ref{Thm:general} (1) for $m=1$. \nTherefore, we treat with the case of mod $p$ in this section. \nThe goal is to show that all strongly mod $p$ singular modular forms\nare represented by a linear combination of finitely many theta series.\nIn our method, two new tools are important:\nthe existence of an ``abstract Sturm bound\"\nfor detecting mod $p$ singular forms\nand the refinement of Freitag's expansion as exposed in the previous section.\nThe nice thing is that we do not need to consider the\nexact level of the theta series. \n\n\n\\subsection{Abstract Sturm bounds}\nIn general, ``Sturm bounds\" give an\nexplicit finite set of $T\\in \\Lambda_n$ such that a modular \nform of degree $n$ with Fourier coefficients in ${\\mathbb Z}_{(p)}$ must be congruent\n mod $p$ to zero if \n the Fourier coefficients for all $T$ in that finite set are divisible by $p$,\nsee e.g. \\cite{RR, Stu}. \nSuch an explicit finite set would usually contain quadratic forms of all ranks.\nWe need a version, which involves only quadratic forms of\nrank $>r$ (finitely many) to detect mod $p$ singular forms.\nWe only need the existence of\nsuch a set, but we do not discuss explicit bounds for it.\n \n Let $\\chi $ be a quadratic Dirichlet character mod $N$. \nLet $M_{k,r}^{p\\text{-sing}}(\\Gamma _0^{(n+r)}(N),\\chi)$ be the submodule of\n$M_k(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$\nconsisting of all mod $p$ singular modular forms with $p$-rank $\\le r$. \nWe denote by $\\widetilde{M}_{k,r}^{p\\text{-sing}}(\\Gamma _0^{(n+r)}(N),\\chi)$ the set of reduction mod $p$ of elements of $M_{k,r}^{p\\text{-sing}}(\\Gamma _0^{(n+r)}(N),\\chi)$. \nThis is a subspace over $\\F_p$ of the vector space $\\widetilde{M}_k(\\Gamma _0^{(n+r)}(N),\\chi )$. \nWe consider the quotient space \n\\[V=V_{k,r}:=\\widetilde{M}_k(\\Gamma _0^{(n+r)}(N),\\chi )\/\\widetilde{M}_{k,r}^{p\\text{-sing}}(\\Gamma _0^{(n+r)}(N),\\chi ).\\] \n\nThen we have $\\dim V<\\infty$.\nFor fixed $T\\in \\Lambda_{n+r}$ with $\\mbox{rank}(T)>r$ we define a linear map\n$\\ell_T:V\\longrightarrow {\\mathbb F}_p$ by\n$$\\ell_T(\\widetilde{F}+ \\widetilde{M}_{k,r}^{p\\text{-sing}}(\\Gamma _0^{(n+r)}(N),\\chi )):=\\widetilde{a_F(T)}.$$\nClearly, the set $L$ of all such $\\ell_T$ is ``total'' for $V$, i.e.\nthe intersection of the kernels of all\n$\\ell_T$ is trivial.\nBy linear algebra ($V$ is of finite dimension!)\nwe can choose a finite subset\n${\\mathcal T}_{n+r,r}\n=\\{\\ell_{T_1},\\dots ,\n\\ell_{T_d}\\}$ of $L$\nwhich is still total for $V$.\n\\\\[0.3cm]\n{\\bf Conclusion:} {\\it In the situation above, there exist finitely many\n$T_1,\\dots ,T_d$ with all $T_j$ of rank larger than r such that\n for all $F\\in M_k(\\Gamma^{(n)}_0(N),\\chi )_{{\\mathbb Z}_{(p)}}$ the vanishing of\n $\\widetilde{a_F(T_j)}$\nfor all $T_j$ implies that $F$ is mod $p$ singular of $p$-rank $\\leq r$. }\n\\\\[0.3cm]\nWe call such $\\mathcal T_{n+r,r}$ a ``Sturm set'' for mod $p$ singular forms.\n\n\n\n\\subsection{Proof of Theorem \\ref{Thm:general} (1) for $m=1$}\n\\label{Subsec:4.3}\nFor the Sturm set ${\\mathcal T}_{r,r-1}$ $(\\subset \\Lambda_r^+)$ corresponding to $M_{k,r-1}^{p\\text{-sing}}(\\Gamma _0^{(r)}(N))$,\nwe take a natural number $M$ such that \n\\[M>\\max\\{\\det T\\;|\\;T\\in {\\mathcal T}_{r,r-1}\\}.\\] \nWe put \\[G:=F-\\sum_{\\substack{S\\in \\Lambda_r^+\/{\\rm GL}_r({\\mathbb Z})\\\\ \\det S1$. \nWe apply Theorem \\ref{Thm:Kita} of Kitaoka's original result for $n=1$ to $\\theta _S^{(1)}$ with $M=\\smat{*}{*}{p^j}{N'}$. \nThen we have \n\\[\\theta_S^{(1)}|M = \\kappa\\cdot \\theta _{S'}^{(1)}, \\]\nwhere $S'$ is the same as in the proof of Proposition \\ref{PropA}.\nThen $N'S'$ is half integral, $(N', {\\rm cont}(N'S'))=1$, and ${\\rm cont}(N'S')=p^\\alpha $ when $p^\\alpha \\;|| \\; {\\rm cont}(S)$. \nThen $\\frac{N'}{p^\\alpha }S'\\in \\Lambda _2^+$ is primitive.\nTherefore we can take a prime $l$ with $l\\nmid N'$ such that \n\\[A\\left(S',\\frac{l\\cdot p^\\alpha }{N'}\\right)=A\\left(\\frac{N'S'}{p^\\alpha },l\\right)\\not \\equiv 0 \\bmod{p}. \\] \nHowever, since $\\phi $ is of level $1$, we have also\n\\[A\\left(S',\\frac{l\\cdot p^\\alpha }{N'}\\right)\\equiv a_\\phi \\left(\\frac{l\\cdot p^\\alpha }{N'}\\right)=0 \\bmod{p}. \\] \nThis contradicts and we get $N'=1$. \n\\end{proof}\nHence ${\\rm level}(S)$ is a power of $p$ in the situation of Proposition \\ref{PropA-2}. \nThen $\\det (2S)$ also should be a power of $p$. \nBy the elementary divisor theorem, we can find $U$, $V\\in {\\rm GL}_2(\\Z)$ such that \n\\[U(2S)V=\\mat{p^s}{0}{0}{p^{s+t}}. \\] \nThen we have $U(2S)=\\smat{p^s}{0}{0}{p^{s+t}}V^{-1}$ and hence \n\\[U(2S){}^tU=p^s\\mat{1}{0}{0}{p^t}V^{-1}{}^tU. \\]\nIf we put $W:=V^{-1}{}^tU=\\smat{w_1}{w_2}{w_3}{w_4}$, then we can write \n\\[U(2S){}^tU=p^s\\mat{w_1}{p^tw_3}{w_2}{p^tw_4}. \\]\nSince $U(2S){}^tU$ is symmetric, we have $w_2=p^tw_3$ and \n\\[U(2S){}^tU=p^s\\mat{w_1}{p^tw_3}{p^tw_3}{p^tw_4}. \\]\nFrom these argument, we may assume that $S$ is of this form.\nIn particular, if $\\det (2S)$ is an odd power of $p$, by putting $b:=p^jw_3$, we may assume that $S$ is of the form\n\\[S=S(i,j)=p^i \\begin{pmatrix}a & bp^{j+1} \\\\ bp^{j+1} & dp^{2j+1}\\end{pmatrix}\\]\nwith $adp-b^2p^2=p$. \n\nNow we try to show that high powers of $p$ in the level of $S$ imply high filtration weight of $\\theta ^{(1)}_S$:\n\\begin{Prop}\n\\label{PropA-3}\nLet $p$ be an odd prime and $S\\in \\Lambda _2^+$. \nSuppose that $\\det(2S)$ is an odd power of $p$. \nThen we have\n\\[\\omega (\\theta ^{(1)}_S)\\ge p^{i+j} \\cdot \\frac{p+1}{2}, \\]\nwhere $i$, $j$ are determined by \n\\[S=S(i,j)=p^i \\begin{pmatrix}a & bp^{j+1} \\\\ bp^{j+1} & dp^{2j+1}\\end{pmatrix}\\]\nwith $adp-b^2p^2=p$. \n\\end{Prop}\nBefore proving this proposition, we confirm some notation and facts on filtration of modular forms mod $p$ for degree $1$ given by Serre \\cite{Se} and Swwinerton-Dyer \\cite{Sw}. \nLet $f=\\sum _{n=0}^{\\infty }a_f(n){\\boldsymbol e}(nz)$ be a formal power series with $a_f(n)\\in \\Z_{(p)}$. \nWe write $\\omega (f)$ for $\\omega _{1,1}^1(f)$. Let $V(p)$, $U(p)$ be the operators defined by\n\\begin{align*}\n&f|V(p)=\\sum _{n=0}^\\infty a_f(n){\\boldsymbol e}(pnz),\\\\\n&f|U(p)=\\sum _{n=0}^\\infty a_f(pn){\\boldsymbol e}(nz). \n\\end{align*}\nThen we have $\\omega (f|V(p))=p\\omega (f)$, $\\omega (f|U(p))\\le \\omega (f)$. \n\nWe put $\\omega (S(i,j)):=\\omega (\\theta _{S(i,j)}^{(1)})$. \n\n\\begin{proof}[Proof of Proposition \\ref{PropA-3}]\nUsing the facts that $\\theta _{S(i,j)}^{(1)}=\\theta _{S(i-1,j)}^{(1)}|V(p)$ and $\\omega (f|V(p))=p\\omega (f)$, \nwe have \n\\[\\omega (S(i,j))=p^i\\omega (S(0,j)). \\]\nFurthermore, by comparing the representation numbers of $A(S(0,j),pn)$ and \\\\\n$A(S(1,j-1),n)$, \nwe can easily confirm that \n\\[\\theta ^{(1)}_{S(0,j)}| U(p)=\\theta ^{(1)}_{S(1,j-1)}. \\]\nSince $\\omega (f)\\ge \\omega (f|U(p))$ in general, we have \n\\[\\omega (S(0,j))\\ge \\omega (S(1,j-1))=p\\omega (S(0,j-1)). \\]\nThis implies \n\\[\\omega (S(i,j))\\ge p^{i+j}\\omega (S(0,0))=p^{i+j}\\omega (S(0,0))=p^{i+j}\\cdot \\frac{p+1}{2}.\\]\nHere the last equality follows from $\\omega (S(0,0))=\\frac{p+1}{2}$ and this is due to Theorem \\ref{Thm:Bo-Na} (or Serre's result in \\cite{Se}). \nThis completes the proof. \n\\end{proof}\n\\begin{Cor}\n\\label{Cor1}\nLet $p$ be a prime and $S\\in \\Lambda _2^+$. \nSuppose that $\\theta _S^{(1)}\\equiv \\phi $ mod $p$ for some $\\phi \\in M_\\frac{p+1}{2}(\\Gamma _1)_{\\Z_{(p)}}$. \nThen we have ${\\rm level}(S)=p$. \n\\end{Cor}\n\\begin{proof}\nThe assumption $\\theta _S^{(1)}\\equiv \\phi $ mod $p$ implies \n\\begin{align}\n\\label{eq:filt_S}\n\\omega (S)=\\omega (S(i,j))\\le \\frac{p+1}{2}.\n\\end{align}\n\nOn the other hand, the level of $S$ is a power of $p$ because of Proposition \\ref{PropA-2}. \nThen $\\det (2S)$ is a power of $p$. \nNote here that, actually in this case, the power of $p$ is odd. \nBecause, the quadratic character $\\chi _S$ should have nontrivial $p$-component, otherwise $\\frac{p+1}{2}\\equiv 1$ mod $p-1$ should hold and this is impossible. \nHence we can assume that $S=S(i,j)$ as in Corollary \\ref{PropA-3}. \nThen we have \n\\[\\omega (S)=\\omega (S(i,j))\\ge p^{i+j}\\cdot \\frac{p+1}{2}. \\] \nCombing this with (\\ref{eq:filt_S}), we have $i=j=0$. Namely, we obtain ${\\rm level}(S)=p$. \n\\end{proof}\n\nWe can now prove our result for the case of $p$-rank $2$. \n\\begin{proof}[Proof of Theorem \\ref{Thm:r=2}]\nLet $\\{T_1,\\cdots,T_{h_p}\\}\\subset \\Lambda _{2}\/{\\rm GL}_2(\\Z)$ be a set of the representatives of ${\\rm GL}_{2}(\\Z)$-inequivalence classes \nof binary quadratic forms with level $p$. \nNote that $\\theta ^{(n)}_{T_i}\\in M_{1}(\\Gamma ^{(n)}_0(p),(\\frac{*}{p}))$, and there exists $G_i\\in M_{\\frac{p+1}{2}}(\\Gamma _n)$ such that $G_i \\equiv \\theta ^{(n)}_{T_i}$ mod $p$ by Theorem \\ref{Thm:Bo-Na}. \nThen $G_i$ is mod $p$ singular, because $\\theta ^{(n)}_{T_i}$ is true singular. \nThen we consider \n\\begin{align*}\nH:=F-\\sum _{i=1}^{h_p} \\frac{1}{2}a_F\\begin{pmatrix}0 & 0 \\\\ 0 & T_i\\end{pmatrix}G_i\\in M_{\\frac{p+1}{2}}(\\Gamma _{n+2})\n\\end{align*}\nThis $H$ is a mod $p$ singular of some $p$-rank $r'\\le 2$. \n\nNow we suppose that still $r'=2$. \nThen we can take $S\\in \\Lambda _2^+$ with $a_H\\smat{0}{0}{0}{S}\\not \\equiv 0$ mod $p$ such that $\\det S$ is minimal. \nBy Corollary \\ref{Cor:filt}, we have $\\widetilde{\\theta_S^{(1)}} \\in \\widetilde{M}_{\\frac{p+1}{2}}(\\Gamma _1)$. \nBy Corollary \\ref{Cor1}, we have ${\\rm level}(S)=p$ and hence $S\\sim T_j$ mod ${\\rm GL}_2(\\Z)$ for some $j$. \nHowever, from $a_H\\smat{0}{0}{0}{T_i}\\equiv 0$ mod $p$ for any $i$, we have $a_H\\smat{0}{0}{0}{S}\\equiv 0$ mod $p$. \nThis is a contradiction. \n\nTherefore we have $r'\\le 1$ or $H\\equiv 0$ mod $p$. \nIf $H\\not \\equiv 0$ mod $p$, then we have $p+1-r'\\equiv 0$ mod $p-1$ because of Theorem \\ref{Thm:Bo-Ki}. \nThis is impossible. This implies $H\\equiv 0$ mod $p$. \nThis shows the main statement of Theorem \\ref{Thm:r=2}. \n\n\nTo prove the final statement concerning $\\Phi^{n+1}(F)\\not \\equiv 0$ mod $p$ it is sufficient to\nassure the mod $p$ linear independence of the binary theta series in question: \nLet $\\{T_1,\\cdots,T_{h_p}\\}$ be as above. \nAs stated in Kani \\cite{Kani}, $\\theta _{T_j}^{(1)}$ ($j=1, \\cdots, h_p$) are linearly independent over $\\C$. \nNow we can prove that $\\theta _{T_j}^{(1)}$ ($j=1, \\cdots, h_p$) are linearly independent over $\\F_p$: \nLet $\\sum _{i=1}^{h_p} c_i \\theta _{T_j}^{(1)}\\equiv 0 \\bmod{p}$. \nThen, for any $n\\ge 0$, we have \n\\[\\sum _{i=1}^{h_p} c_i A(T_i,n)\\equiv 0 \\bmod{p}.\\] \nFor each $T_i$, we can find infinitely many primes $l$ such that $A(T_i,l)>0$. \nThen we have $A(T_j,l)=0$ for any $j\\neq i$. Since $A(T_i,l)=2,4,6\\not \\equiv 0$ mod $p$, we have $c_j \\equiv 0$ mod $p$. \nHence we have $c_i\\equiv 0$ mod $p$. This shows $c_i\\equiv 0$ mod $p$ for any $i$ with $1\\le i\\le h_p$. \nTherefore $\\theta _{T_j}^{(1)}$ ($j=1$, $\\cdots$, $h_p$) are linearly independent over $\\F_p$. \nThis completes the proof of Theorem \\ref{Thm:r=2}. \n\\end{proof}\n\n\\section{From mod $p$ to mod $p^m$} \n\\label{Sec:7}\nIn this section, using induction, we extend Theorem \\ref{Thm:general} (proved for $m=1$ in the previous sections) to the case of general $m$.\nThe main reason why we preferred to give a mod $p$ version first and then\nextend to arbitrary powers is the technical difficulty which one encounters,\nwhen one tries to give a mod $p^m$ analogue of our abstract Sturm bound.\nWe avoid such a problem in this way. \n\nWe start with proving the following weaker version of Theorem \\ref{Thm:general}.\n\\begin{Thm}[Weaker version of Theorem \\ref{Thm:general}]\n\\label{Thm:weak}\nLet $n$, $k$, $N$ be positive integers, $r$ an even integer with $n\\ge r$. \nLet $p$ be a prime with $p>r+1$ and $\\chi $ a quadratic\nDirichlet character mod $N$ with $\\chi (-1)=(-1)^k$. \nSuppose that $F\\in M_{k}(\\Gamma _0^{(n+r)}(N),\\chi )_{\\Z_{(p)}}$ is\nmod $p^m$ singular of $p$-rank $r$. \nThen there are finitely many $S\\in \\Lambda _r^+$ of level dividing $p^eN$ with\n$\\widetilde{\\theta _{S}^{(n)}}\\in \\widetilde{M}_k(\\Gamma ^{(n)}_0(N),\\chi )$ \n(only mod $p$) such that \n\\begin{align} \n\\label{xy} \nF\\equiv \\sum_S c_S\\theta_S^{(n+r)} \\bmod{p^m}\n \\quad (c_S\\in \\Z_{(p)}).\n\\end{align}\nMoreover, all $S$ involved satisfy $\\chi='\\chi_S$. \n\\end{Thm}\n\\begin{proof}[Proof of Theorem \\ref{Thm:weak}]\nWe prove the statement by induction on $m$.\nWe have already proved the statement for $m=1$. \nSuppose that the statements (1), (2) in Theorem \\ref{Thm:general} are true for any $m$ with $mr$, we have $a_H(T)\\equiv 0$ mod $p$ for any $T$ with ${\\rm rank}(T)>r$. \nThis means that $H$ is mod $p$ singular of some $p$-rank $r'\\le r$. \nBy $2k-r'\\equiv 2k-r \\equiv 0$ mod $p-1$ and $0\\le r'\\le r \\le p-1$, we have $r'=r$. \nTherefore $a_H(T)\\not \\equiv 0$ mod $p$ for some $T$ with ${\\rm rank}(T)=r$. \nAgain by our theorem for $m=1$, we have \n\\[H\\equiv \\sum _{\\substack{R\\in \\Lambda _r^+\\\\ {\\text{level}(R)}\\mid p^{e_2}N}}d_R\\theta _R^{(n+r)} \\bmod{p}\\quad \\text{with}\\quad d_R\\in \\Z_{(p)}, \\] \nand all $R$ involved satisfy $\\chi =' \\chi_R$. \nNoting $G:=\\sum _{S}c_S\\theta _S^{(n+r)}\\cdot {\\mathcal E}^{tp^{m_0-1}}\\equiv \\sum _{S}c_S\\theta _S^{(n+r)}$ mod $p^{m_0}$, we have\n\\begin{align*}\nF&\\equiv G+p^{m_0-1}H\\\\\n&\\equiv \\sum _{\\substack{S\\in \\Lambda _r^+\\\\ {\\text{level}(S)}\\mid p^{e_1}N}}\nc_S\\theta _S^{(n+r)}+p^{m_0-1}\\sum _{\\substack{R\\in \\Lambda _r^+\\\\ {\\text{level}(R)}\\mid p^{e_2}N}}d_R\\theta _R^{(n+r)} \\bmod{p^{m_0}}. \n\\end{align*} \nThis completes the proof of Theorem \\ref{Thm:weak}. \n\\end{proof}\n\n\\begin{proof}[Completion of the proof of Theorem \\ref{Thm:general}]\n We first claim that the coefficients $c_S$ in (\\ref{xy})\n coincide mod $p^m$ with $\\frac{a^*(S)}{\\epsilon(S)}$ provided that the\n summation in (\\ref{xy}) is restricted to pairwise ${\\rm GL}_r({\\mathbb Z})$-inequivalent elements of $\\Lambda^+_r$.\n To see this, we take an arbitrary $S_0\\in \\Lambda_r^+$.\n Then (using freely the notaion from Section \\ref{Sec:4})\n $$a(S_0)\\equiv \\sum_S c_S A(S,S_0)\\bmod p^m$$\n holds and passing to primitive Fourier coefficients gives\n $$a^*(S_0)\\equiv \\sum_S c_S A^*(S,S_0)\\bmod p^m.$$\n Now $A^*(S,S_0)=\\epsilon(S_0)$ if $S$ and $S_0$ are ${\\rm GL}_r(\\Z)$-equivalent and zero otherwise; this proves the claim from above.\n\n \n To complete the proof of Theorem \\ref{Thm:general},\n we have to prove\n $\\widetilde{\\theta _{S}^{(n)}}\\in \\widetilde{M}_k\n (\\Gamma ^{(n)}_0(N),\\chi )_{p^{m-\\nu}}$; this follows now directly\n from Proposition \\ref{Prop:filt}.\nThis completes the proof of Theorem \\ref{Thm:general}.\n\\end{proof}\n\n\\section{Appendix A: Modified $q$-expansion principle}\n\\label{Sec:8}\nTo specify the level of theta series in our theorems, we need some control over the behaviour of\ncongruences of modular forms, when we switch to other cusps.\nThe ``$q$-expansion principles\" available in the literature do not exactly provide the information\nnecessary for our purpose.\n\n\nWe fix a prime $p$ and a natural number $N$ coprime to $p$.\nFor $m\\geq 0$ let\n$R_m$ be a subring of ${\\mathbb C}$ containing \n${\\mathbb Z}[\ne^{\\frac{2\\pi i}{N\\cdot p^m}}, \ne^{\\frac{2\\pi i}{3}},\n\\frac{1}{6},\\frac{1}{N}]\n$. \n\n\\begin{Thm}\n\\label{Thm:q-exp}\nLet $p$ be a prime with $p\\ge n+3$ and $N$ a positive integer with $p\\nmid N$. \nLet $f\\in M_k(\\Gamma _0^{(n)}(p^m)\\cap \\Gamma^{(n)}(N))$. \nThen for all $\\gamma \\in \\Gamma _0^{(n)}(p^m)$ we have\n$$f\\in M_k(\\Gamma _0^{(n)}(p^m)\\cap \\Gamma^{(n)}(N))_{R_m} \\iff\nf|_k \\gamma\\in M_k(\\Gamma _0^{(n)}(p^m)\\cap \\Gamma^{(n)}(N))_{R_m}.$$\n\\end{Thm}\nThe case $m=0$ is the statement of the usual $q$-expansion principle in the formulation of\nPitale et al. \\cite{PiScSa} proved by Katz \\cite{Kat} for $n=1$ and Ichikawa \\cite{Ich} for $n>1$. \nThe main point in our version is that $f$ and $f| \\gamma$ share the same $p$-integrality property.\nNote that one only needs to prove one direction of the theorem.\nThe aim of Appendix A is to prove this property.\n\\subsection{Proof for $m=1$}\nWe will do induction on $m$.\nWe need the existence of a modular form ${\\mathcal E}\\in M_{l}(\\Gamma _0^{(n)}(p))_{\\Z_{(p)}}$ such that \n\\begin{align*}\n&{\\mathcal E}\\equiv 1 \\bmod{p}\\\\\n&{\\mathcal E}| \\omega _j \\equiv 0 \\bmod{p} \\quad (1\\le j \\le n ). \n\\end{align*}\nHere $\\omega _j$ can be any element $\\smat{*}{*}{C}{D}\\in \\Gamma _n$ with $C$ being of rank $i$ in $M_{n}(\\F_p)$.\nThe existence of such ${\\mathcal E}$ was explained in \\cite{Bo2017}, under the condition $p\\ge n+3$. \nWe mention that the congruence condition of ${\\mathcal E}$ can be rephrased as saying that for any $\\gamma \\in \\Gamma _n$, we have \n\\[{\\mathcal E}| \\gamma \\equiv 1 \\bmod{p} \\quad \\iff \\quad \\gamma \\in \\Gamma _0^{(n)}(p)\\]\nand ${\\mathcal E}| \\gamma \\equiv 0$ mod $p$ for $\\gamma \\not \\in \\Gamma _0^{(n)}(p)$. \n\nWe prove the statement for $m=1$. \nWe chose an appropriate power $\\alpha $ of ${\\mathcal E}$ and consider a trace from level $\\Gamma _0^{(n)}(p)\\cap \\Gamma ^{(n)}(N)$ to level $\\Gamma ^{(n)}(N)$:\n\\[F=f{\\mathcal E}^\\alpha +\\sum _{j\\bmod{p}}(f{\\mathcal E}^{\\alpha })|\\delta _j\\]\nwhere $\\delta _j$ are representatives of the left cosets not in $\\Gamma _0^{(n)}(p)\\cap \\Gamma ^{(n)}(N)$,\nin particular, ${\\mathcal E}|\\delta _j\\equiv 0$ mod $p$. \nApplying $\\gamma $, we see that $\\delta _j \\cdot \\gamma \\not \\in \\Gamma _0^{(n)}(p)$ and hence $(f {\\mathcal E}^\\alpha )|\\delta _j \\cdot \\gamma \\equiv 0$ mod $p$. \nWe observe that $F$ is of level $\\Gamma ^{(n)}(N)$ and hence we may apply the ordinary $q$-expansion principle: \nThen we have \n\\[F|\\gamma \\equiv (f|\\gamma ) {\\mathcal E}^\\alpha \\equiv f|\\gamma \\bmod{p}. \\]\nIn particular $F|\\gamma $ is $p$-integral and hence $f|\\gamma $ is also $p$-integral. \nThis completes the proof for $m=1$. \n\\qed\n\n\\subsection{Proof for $m\\ge 2$}\nWe prove it for $m\\ge 2$ by induction on the power $m$.\nWe suppose that the statement is true for the level $\\Gamma _0^{(n)}(p^{m-1})$. \nWe put \n\\[{\\mathcal F}:={\\mathcal E}(p^{m-1}z)=p^{-(m-1)\\frac{nl}{2}}{\\mathcal E}| \\begin{pmatrix}p^{m-1} \\cdot 1_n & 0_n \\\\ 0_n & 1_n\\end{pmatrix} \\in M_{l}(\\Gamma _0^{(n)}(p^{m}))_{\\Z_{(p)}}. \\]\nHere $l$ is the weight of ${\\mathcal E}$. \nTaking some positive integer $\\alpha $, we consider $f{\\mathcal F}^\\alpha $ and the trace of this from $\\Gamma _0^{(n)}(p^m)\\cap \\Gamma ^{(n)}(N)$ to $\\Gamma _0^{(n)}(p^{m-1})\\cap \\Gamma ^{(n)}(N)$; \n\\begin{align}\n\\label{Eq:tr}\nF={\\rm tr}(f{\\mathcal F}^\\alpha )&=f{\\mathcal F}^\\alpha +\\sum _{w}^{p-1}(f{\\mathcal F}^\\alpha )|\\gamma _w \\\\ \\nonumber\n&=f{\\mathcal F}^\\alpha +\\sum _{j=1}^{p-1}(f|\\gamma _w)({\\mathcal F}|\\gamma _w)^\\alpha , \n\\end{align}\nwhere $\\gamma _w=\\smat{1_n}{0_n}{p^{m-1}N_w}{1_n}$ and $w$ runs over a complete set of representatives of \nintegral symmetric matrices of size $n$ mod $p$ except the trivial coset $p\\cdot {\\rm Sym}_n(\\Z)$. \nNow we prove that $(f|\\gamma _w )({\\mathcal F}|\\gamma _w)^\\alpha \\equiv 0$ mod $p$. \nBy a direct calculation, we have \n\\begin{align}\n\\label{Eq:mt}\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}\n\\begin{pmatrix}\n1_n & 0_n \\\\ p^{m-1}Nw & 1_n\n\\end{pmatrix}\n&=\\begin{pmatrix}\n1_n & 0_n \\\\ Nw & 1_n\n\\end{pmatrix}\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}.\n\\end{align} \nThis implies \n\\begin{align*}\n{\\mathcal F}| \n\\begin{pmatrix}\n1_n & 0_n \\\\ p^{m-1}Nw & 1_n\n\\end{pmatrix}\n&=p^{-(m-1)\\frac{nl}{2}}\n{\\mathcal E}|\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}\n\\begin{pmatrix}\n1_n & 0_n \\\\ p^{m-1}Nw & 1_n\n\\end{pmatrix}\\\\\n&=p^{-(m-1)\\frac{nl}{2}}{\\mathcal E}|\\begin{pmatrix}\n1_n & 0_n \\\\ Nw & 1_n\n\\end{pmatrix}\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}.\n\\end{align*}\nSince ${\\mathcal E}|\\smat{1_n}{0_n}{Nw}{1_n}\\equiv 0$ mod $p$ and we can put ${\\mathcal E}|\\smat{1_n}{0_n}{Nw}{1_n}=pX$ for some $p$-integral modular form $X=X_w$. \nThen we have \n\\begin{align*}\n{\\mathcal F}| \n\\begin{pmatrix}\n1_n & 0_n \\\\ p^{m-1}Nw & 1_n\n\\end{pmatrix}&=p^{-(m-1)\\frac{nl}{2}}{\\mathcal E}\n|\\begin{pmatrix}\n1_n & 0_n \\\\ Nw & 1_n\n\\end{pmatrix}\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}\\\\\n&=p^{-(m-1)\\frac{nl}{2}} (p X) |\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n\n\\end{pmatrix}\\\\\n&=pX(p^{m-1}z)\\equiv 0 \\bmod{p}.\n\\end{align*}\nTherefore $(f|\\gamma _w )({\\mathcal F}|\\gamma _w)^\\alpha \\equiv 0$ mod $p$ follows if $\\alpha $ is large. \nBy the assumption that $f$ is $p$-integral, $F$ is also $p$-integral. \n\nNow let $\\delta \\in \\Gamma _0^{(n)}(p^m)$, we may apply induction because $F$ is of level $\\Gamma _0^{(n)}(p^{m-1}N)$. \nWe start form (\\ref{Eq:tr}) for $F$ and apply $\\delta $ to it:\nIf $f$ is $p$-integral, then $F$ is also $p$-integral, and then $F|\\delta $ is also $p$-integral. \nBy (\\ref{Eq:tr}), we have \n\\begin{align*}\nF|\\delta =(f|\\delta ){\\mathcal F}^\\alpha +\\sum _{j=1}^{p-1}(f{\\mathcal F}^\\alpha )|\\gamma _w \\delta \n\\end{align*} \nThe first term on the right hand side of (\\ref{Eq:tr}) is congruent mod $p$ to $f|\\delta $. \nWe prove $(f{\\mathcal F}^{\\alpha })|\\gamma _w\\delta \\equiv 0$ mod $p$. \nTo prove this, we must multiply (\\ref{Eq:mt}) from the right by $\\delta =\\smat{a}{b}{p^mc}{d}$. \nWe get \n\\[\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n \n\\end{pmatrix}\n\\gamma _w \\delta \n=\\begin{pmatrix}\na & p^{m-1}b \\\\ wNa+pc & wNp^{m-1}b+d\n\\end{pmatrix}\n\\begin{pmatrix}\np^{m-1}\\cdot 1_n & 0_n \\\\ 0_n & 1_n \n\\end{pmatrix}. \n\\] \nSince $p\\nmid wNa$ and by a similar argument as above, we obtain ${\\mathcal F}|\\gamma _w \\delta \\equiv 0$ mod $p$. \nThis shows $(f{\\mathcal F}^{\\alpha })|\\gamma _w\\delta \\equiv 0$ mod $p$ for large $\\alpha $.\nThis implies that $F|\\delta \\equiv f|\\delta $ mod $p$ and hence $f|\\delta $ is $p$-integral. \n\\qed\n\n\\begin{Rem} \nA minor variant of the proof above applies if $f$ is of ``quadratic nebentypus mod $p$\",\ni.e. $f$ satisfies (under the same conditions as in the theorem above)\n$f|_k \\gamma=(\\frac{\\det(d)}{p}) \\cdot f$ for \nall $\\gamma\\in \\Gamma_0^{(n)}(p^m)\\cap\\Gamma^{(n)}(N)$\nwith the character $(\\frac{*}{p})$.\nThe proof above needs to be modified only for the step ``$m=1$\", where one has to\nuse a function ${\\mathcal E}$ with nebentypus $(\\frac{*}{p})$.\n\\end{Rem}\n\\section{Appendix B: Kitaoka's formula}\n\\label{Sec:9} \nIn this Appendix, we generalize Kitaoka's theorem to higher degree to specify the level of theta series in our theorem. \nWe then apply this in practice and discuss levels.\n\\subsection{Generalization of Kitaoka's formula}\nWe freely switch here between the language of (positive integral) quadratic forms $S$ (or integral symmetric matrices) and the language of lattices $L$ in an euclidean space $({\\mathbb R}^m,\\left<\\ ,\\ \\right>)$. \nThe aim is to get a degree $n$ version of a result of Kitaoka; \nwe state it here only in the version sufficient for our application and allowing a rather short proof. \nA more sophisticated version generalizing the computation of Kitaoka would lead to a more general statement. \n\\begin{Thm}\n\\label{Thm:Kita}\nLet $L$ be an even integral positive definite lattice of rank $m=2k$ and level $N$. \nLet $d$ be a positive divisor of $N$ with $(d,\\frac{N}{d})=1$, and $c:=\\frac{N}{d}$. \nChoose integers $a$, $b$ such that $M=\\smat{a}{b}{c}{d}\\in {\\rm SL}_2(\\Z)$ and denote by $\\frak{M}$ the $n$-fold \ndiagonal embedding of $M$ into ${\\rm Sp}_n(\\Z)$, namely, \n$\\frak{M}:=\\smat{a\\cdot 1_n}{b\\cdot 1_n}{c\\cdot 1_n}{d\\cdot 1_n}$.\n\nThen, with a suitable constant $\\kappa $ \n\\begin{align}\n\\label{Eq:Kita}\n\\theta _L^{(n)}| _k\\frak{M}=\\kappa \\cdot \\theta _{L'}^{(n)}, \n\\end{align}\nwhere $L'$ is a lattice characterized by \n\\[L'\\otimes \\Z_p =\n\\begin{cases}\nL\\otimes \\Z_p\\ &\\text{if}\\ p\\nmid d, \\\\\n(L\\otimes \\Z_p)^*\\ &\\text{if}\\ p\\mid d, \n\\end{cases}\\]\nand $*$ denotes the dual. \n\\end{Thm}\nBefore we begin the proof, we need to prepare some more notation and recall the facts: \nKitaoka \\cite{Kita} proved the degree one version of this statement for a more general type of theta series\n$$\\theta^{(1)}_L(q)(z)=\\sum_{{\\mathfrak x}\\in L} q({\\mathfrak x}) e^{2\\pi i \\left<{\\mathfrak x},{\\mathfrak x}\\right>\\cdot z}$$\n allowing homogeneous harmonic polynomials $q$ on ${\\mathbb C}^m$ as coefficients in the theta series. \nAn inspection of his proof shows that the constants of proportionality do NOT depend on those harmonic polynomials. \n\nIbukiyama \\cite{Ibu} investigated holomorphic differential operators ${\\mathcal D}$ on $\\hh_n$, which are polynomials in the partial holomorphic derivatives, \nevaluated in ${\\rm diag}(z_1,\\cdots , z_n)$; Such a ${\\mathcal D}$ maps holomorhic functions $F$ on ${\\mathbb H}_n$ to holomorphic\nfunctions ${\\mathcal D}(F)$ defined on ${\\mathbb H}_1^n$\nsuch that, with the notation above \n\\[{\\mathcal D}(F|_k {\\frak M})={\\mathcal D}(F)|^{(z_1)} _{k_1}M\\cdots |^{(z_n)} _{k_n}M \\] \nholds for arbitrary $M\\in {\\rm SL}_2({\\mathbb R})$ with certain weights $k_i\\ge k$. \nThe upper index $z_i$ at the slash-operator indicates that $|_{k_i}M$ should be applied w.r.t. the variable $z_i$.\n\nHe also showed that from all the ${\\mathcal D}(F)$ one can recover the Taylor expansion of $F$ with respect to the non-diagonal coefficients of $Z=(z_{ij})\\in \\hh_n$. \nTo show the theorem, it is then enough, to show the equality of both sides after applying all such differential operators. \n\n To do this we need some more notation: \n For a fixed differential operator ${\\mathcal D}$ of Ibukiyama type and $w_1, \\cdots, w_n\\in \\C ^m$ changing the automorphy \n factor from $\\det ^k$ to $k_1,\\cdots,k_n$ in the \n variables $z_1,\\cdots ,z_n$ we define\n \\[{\\mathcal D}e^{2\\pi i \\sum \\leftz_{ij}}=P(w_1,\\cdots ,w_n )e^{2\\pi i \\sum \\leftz_{i}}\\]\n with $z_i:=z_{ii}$. \nThen $P$ is a harmonic polynomial in each of the variables $w_i$ and we may write it as a finite sum of pure tensors:\n\\[P(w_1, \\cdots ,w_n)=\\sum _{t}q_{1,t}(w_1)\\cdots q_{n,t}(w_n),\\]\nwhere the $q_{i,t}$ are harmonic polynomials in the variables $w_i$. \n\n\\begin{proof}[Proof of Theorem \\ref{Thm:Kita}]\nApplying ${\\mathcal D}$ to the right hand side of (\\ref{Eq:Kita}) gives \n\\[\\kappa \\cdot \\sum _t \\theta _{L'}^{(1)}(q_{1,t})(z_1)\\cdots \\theta _{L'}^{(1)}(q_{n,t})(z_n).\\]\nApplying ${\\mathcal D}$ to the left hand side gives \n\\[\\sum _t \\theta _{L}^{(1)}(q_{1,t})(z_1)\\cdots \\theta _{L}^{(1)}(q_{n,t})(z_n)|_{k_1}M\\cdots |_{k_n}M.\\]\nNow one has to use Kitaoka's result in degree $1$ to get our assertion. \n\\end{proof}\n\n\\subsection{Application}\nKitaoka's formula allows us to exclude certain congruences between modular forms and theta series of higher level (as far as the prime to $p$ components of the levels are concerned): \n\\begin{Prop}\n\\label{PropA}\nLet $n$ be an even positive integer and $p$ a prime with $p\\ge n+1$. \nLet $N$, $N'$ be coprime to $p$ and $S\\in \\Lambda_n^+$ with level $p^aN$.\nAssume that\n$$p^bN'\\mid p^aN\\quad\\mbox{and}\\quad \\phi \\equiv \\theta^{(n)}_S\\bmod p$$\nfor some $\\phi\\in M_k(\\Gamma_0^{(n)}(p^bN'),\\chi )_{{\\mathbb Z}_{(p)}}$ with $\\chi ^2=1$. \nThen $N=N'$ and $\\chi _N=(\\chi_S)_N$.\n\n\\end{Prop}\n\\begin{proof}\nWe want to apply the modified $q$-expansion principle. \nTo do so, we must modify $\\phi$ and $\\theta^{(n)}_S$ to arrive at the same weights and same (quadratic) $p$-component of nebentypus: \nWe recall from B\\\"ocherer-Nagaoka \\cite{Bo-Na:2007} that \nthere exist degree $n$ modular forms $\\mathcal E$\n for $\\Gamma_0^{(n)}(p)$ of weight $\\frac{p-1}{2}$ of nontrivial quadratic nebentypus.\nThen Proposition \\ref{weightcongruence} allows us to choose $t\\in {\\mathbb N}$ such that $\\theta^{(n)}_S\\cdot {\\mathcal E}^t$ and $\\phi$ have the same weight\n and the same $p$-component of nebentypus (this holds if $k\\geq \\frac{n}{2}$;\n the modification for the case of the opposite inequality is obvious).\n Then we can indeed apply Theorem \\ref{Thm:q-exp} to \n $\\frac{1}{p}\\left( \\phi-\\theta^{(n)}_S\\cdot {\\mathcal E}^t\\right)$. \n \n\n We may enlarge $b$ (if necessary) to get $a=b$.\n Assume that there is a prime $q$ different from $p$ with $q^r||N$ and $q^s || N'$ with $sn+1 $, we see that $A(S',S')\\not \\equiv 0$ mod $p$. \nOn the other hand, $\\phi| {\\mathfrak M}$ does not have a nonzero \nFourier coefficient at \n$S'$, because the $q$-part of the width of the cusp\n${\\mathfrak M}$ for $\\Gamma_0^{(n)}(p^aN')$ is $q^s$ with $sn+3$. Let $S\\in \\Lambda^+_n$.\nAssume that $\\theta _S^{(n)}\\equiv \\phi $ mod $p$ for some $\\phi \\in M_k(\\Gamma_0^{(n)}(N),\\chi )$ with $\\chi ^2=1$. \nThen ${\\rm level}(S)$ is of the form\n``$p$-power $\\times N'$'' with some $N'\\mid N$\nand $\\chi =' \\chi_S $.\n\\end{Cor}\n\n\n \n\n\n\\section*{Acknowledgment}\nThe authors would like to thank Professors G. Nebe and R. Schulze-Pillot for helpful advices, \nespecially on the order of automorphism groups for quadratic forms and on the generalization of Kitaoka's transformation formula.\nThey would also like to thank Professor T. Yamauchi for giving them information on $q$-expansion principles. \nThis work was supported by JSPS KAKENHI Grant Number 22K03259.\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Variational principle}\n\nA crucial\nrequirement for any theoretical model of coronal structures\nis to give account of the stability and evolution of far--from--equilibrium states which are responsible of the characteristic rich topology and dynamics of the solar corona.\nThis implies to \nconsider the coupling of thermal and mechanical equations.\nDifferent stability analysis of\nsolar structures can be found in the literature, generally\nrestricted to special types of perturbations and specific\nequilibrium models. These includes, models that consider adiabatic\nconfiguration such as the ones analyzed via the classical\ncriterion of \\citet{ber} or those that\npresuppose static equilibrium and analyze thermal stability. In\nthe application of Bernstein's\n criterion, the adiabatic assumption implies that the energy balance equation is not required\n and thus dissipation is\nimpossible. Also the assumption of static models is a strong, and\noften unjustified, restriction for open systems.\n\nIn this paper we apply an energy\nprinciple to analyze the stability of solar coronal loops when helical modes are present.\nThe principle was obtained in previous papers (Paper I: \\citep{cos1};\n\\citep{cos2}; see also \\citep{us0}) using a general\nprocedure of irreversible thermodynamics -based on firmly\nestablished thermodynamic laws- that can be understood as an\nextension of Bernstein's MHD principle to situations far from\nthermodynamic equilibrium. \n\nIn Paper I and in \\citep{cos2} we showed how to obtain the variational principle for solar\ncoronal structures\nfrom\nthe equations that describe the dynamics of the system. The method consists of obtaining a\nLyapunov function,\nalso known as generalized potential, that represents the\nmathematical expression of the stability conditions. The principle is subject to physically reasonable\nrequirements of hermiticity and antihermiticity over the matrices.\nFor a more detailed presentation see Paper I and the references therein.\n\n\n\n\n\n\\subsection{Solar coronal loops}\n\n\n\nMHD loop oscillations in the corona are known to be\nstrongly damped, mostly having decaying times of few periods\n$N_{p}\\approx 2 - 7 \\ periods.$ While thermal conduction, with the\ncontribution of\nradiative cooling mechanisms, could be the main\ncause of the damping of pure\nMHD slow magnetoacoustic mode oscillations they are unimportant for the MHD fast\nmodes.\nResonant absorption and phase mixing seem more promising\n in giving account of the rapid decay (\\citep{gh94}; hereafter HG,\n \\citep{go02}) of the ideal fast\n oscillations of these strongly inhomogeneous and structured plasma systems.\n Inhomogeneous\n equilibrium distributions of plasma density\nand temperature varying continuously across the magnetic field led to plasma waves with\ncontinuous intervals of eigenfrequencies. The\noccurrence of the Alfv\\'en ideal MHD continuum in a thin edge layer\nis derived from the highly\nanisotropic character of the fast magnetoacoustic waves\ngiving rise to a peak of the amplitudes where the perturbation\ndevelops large gradients and the absorption has maxims. However, there is another type of continuum commonly known as slow magnetosonic continuum associated to the inhomogeneity of the equilibrium parameters along the axis of the loop (see Paper I). This inhomogeneities are associated, for example, to changes in the\ndensity concentration at the loop basis. If the magnetic field is twisted the inhomogeneities led to the coupling of Alfv\\'en and slow magnetosonic continuum modes (\\citep{bel}). \n\n\nThe resonant absorption mechanism of\nwave heating consists on the non--dissipative transference of\nwave energy from the collective line-tied wave with fast discrete eigenvalues (kinetic energy of the fast\nradial component) to a local resonant mode in the Alfv\\'en continuum, (kinetic energy of the\nazimuthal component), which is then dissipated in an enhanced\nmanner. Then, the continuum oscillations are converted\ninto heat by dissipative processes; as the medium has large\ngradients in the Alfv\\'en speed, the oscillations of neighboring\nfield lines become out of phase and shear Alfv\\'en waves lead to\nenhanced viscous and ohmic dissipation (see \\citep{pr83} for the\nlinear regime and \\citep{nak97} for the nonlinear one). The mode\nconversion from the collective to the local mode occurs in a time\nthat is non--dissipative and generally much shorter than the\nsecond time scale which is related to the dissipative damping of the\nsmall--scale perturbations of the local mode in the resonance\nlayer (\\citep{rob00}; \\citep{van04}). \n\n\nThe whole temporal pattern description of\nmodes that exhibit a combination of global (discrete line--tied\nfast eigenmode) and localize (Alfv\\'en continuum mode) behaviour\nis known as quasi--mode. Moreover, the mixed nature of the modes\nis not only due to the temporal behaviour but also to the\nboundary value problem giving rise to a spatial behaviour which is\nalso of a mixed nature, i.e. coronal loops with line--tying\nconstraints cannot support pure waves: Alfv\\'en, slow or fast\nmagnetoacoustic modes. \nHG\nstudied the mixed spectral description of coronal loops (i.e. the\nresulting superposition of basic waves which adjust the line--tied\ncondition) without assuming a straight magnetic field and forcing\nthe loop to follow the photospheric velocity perturbations. They\nfound that pure Alfv\\'en and pure slow modes are obtained as\nsingular limiting cases of cluster spectra of Alfv\\'en--fast or\nslow--fast modes, where the fast components are localized in a\nphotospheric boundary associated to the line--tied condition: \nthe coronal part of the\nloop acting as a resonant cavity of large Alfv\\'en\ncomponents and fast components, with a small but\nrapidly varying amplitude, located in the photospheric boundary layer.\nThey found that heating of coronal loops by resonant absorption is\ndue to the line--tied Alfv\\'en continuum which no longer depends\non the poloidal magnetic field and that the corresponding\neigenmodes have a global ballooning feature which is characterized\nby an accumulation point given by the Alfv\\'en frequency. In \\citep{gh93} (hereafter\nGH), a variational principle, based in Bernstein's\nprinciple, was obtained to derive the Alfv\\'en and slow continuum\nfrequencies in a line--tied inhomogeneous cylinder. Stability\nconsiderations led them to conclude the global stability of\ncoronal loops.\n\nIn this paper, following results of Paper I we apply our energy principle to consider the stability and mode structure of loop inhomogeneous coronal models with non--vanishing helicity. Our principle has the advantages that it does not require a WKB approximation and that, as was mentioned, it allows the consideration of the coupling of the thermal and mechanical equations that are necessary to analyze far from equilibrium states.\n\n\\section{The MHD stability criterion for coronal structures}\n\n\n\nSolar coronal conditions with large Reynolds numbers are well fitted by\nideal MHD plasma models (i.e. infinite electrical\nconductivity $\\sigma \\gg 1 $ leading to vanishing viscosity and\nohmic dissipation). Thus, the fundamental equations considered are the\nmass conservation equation,\nthe perfect gas law or state equation for a fully\nionized $H$ plasma and the induction equation, with vanishing magnetic diffusivity due to the conductivity properties. The energy balance equation takes the form:\n\\begin{equation}\n\\frac{\\rho^{\\gamma}}{(\\gamma-1)}\\frac{D}{D\nt}(\\frac{p}{\\rho^{\\gamma}})=-\\nabla\\cdot\\vec{F_{c}}-L_{r}+H\\label{1}\n\\end{equation}\n$\\vec{F_{c}}$ is the heat flux due to particle conduction along\nthe loop, $L_{r}$ is the net radiation flux and $H$ the heating function\nwhich was chosen as in Paper I: $H = h \\rho + H_{0}$.\n Eq.~\\ref{1} expresses the fact that the gain in particle\nenergy (internal plus kinetic) is due to the\nexternal heating sources represented by the heating function, heat\nflow and radiation losses;\nall other heating sources were considered as vanishing terms\nimplying that the optically thin assumption holds. Note that the non--ideal contribution in the energy equation ($L$) is associated to the open character of the loop system.\n\nOnce the linearization around a nonlinear equilibrium or stationary state is performed, and after a straightforward manipulation procedure where the hermiticity requirements are fulfilled the generalized energy principle and the respective frequencies are obtained (Paper I and\n\\citep{cos2}) as:\n\\begin{equation}\n \\delta^{2} W_{p} =\\frac{1}{ 2}\\int ( \\vec{\\xi}^{*} \\beta F \\vec{\\xi}+T_{1}^{*} AT_{1}\n +T_{1}^{*}B\\vec{\\xi} -\\vec{\\xi}^{*}BT_{1})d^{3}x\\geq 0.\n \\label{2}\n\\end{equation}\n\\begin{equation}\n\\omega^{2} =- \\frac{\\int ( \\vec{\\xi}^{*} \\beta F\\vec{\\xi} +\nT_{1}^{*}AT_{1}+T_{1}^{*}B\\vec{\\xi}-\\vec{\\xi}^{*}BT_{1}^{*} )\nd^{3}x}{\\int (\\vec{\\xi^{*}} \\beta \\rho_{0}\\vec{\\xi} )d^{3}x}\n\\label{3}\n\\end{equation}\nwith the same normalization condition as in Paper I. \n$ F$ is the known Bernstein operator for the system, $\\xi$ and $T_{1}$ are the motion and temperature perturbations and operators $A$ and $B$ are as in Paper I.\nFor the non-dissipative cases ($L=0$ or equivalently $T_{1}=0$), last expressions (discarding the presence of factor $\\beta$ which appears in the equations to fit the Hermitian and anti--Hermitian conditions) are reduced to the\nwell--known Bernstein MHD energy principle and its respective frequencies. \n\n\\section{Application to an inhomogeneous loop model with non--vanishing helicity}\n\nOn one hand, the azimuthal component of the loop perturbation is\nbelieved to be one of the principle responsible of resonant\nabsorption and damping of ideal oscillations; on\nthe other, this component is associated to the storage of\nmagnetic energy in systems with non--vanishing helicity which\neventually is released by instabilities. Thus, we are interested in\nanalyzing the changes produced in the stability of\nnon--homogeneous loops subject to helical perturbations. This is, loops\nwith inhomogeneous distributions of plasma density and\ntemperatures subject to body modes and with non--vanishing helicity. In this case, the\nAlfv\\'en, slow and fast magnetoacoustic cylinder\nmodes cannot longer be associated to the azimuthal, longitudinal\nand radial components respectively. The\nobservational importance of helical modes cannot be neglected\nand it is poorly known how helicity affects important physical\nfeatures of mode oscillations (e.g., damping mechanisms,\nstability and periods). However, a mode classification can be accomplished\nvia the analysis of the mode variations, described in an orthogonal basis, while helicity is varied. The basis is formed by the orthogonal displacements:\nparallel and perpendicular to the magnetic field and the radial\n(and perpendicular to the surface of the tube) one, of\nobservational interest.\n\nThe fundamental modes are generally observationally and\nenergetically more important than their harmonics. For these\nglobal modes the inhomogeneous nature of the medium cannot be\nignored and it determines the structure of the disturbance which\ncannot be taken as sinusoidal, making the traditional normal mode\nanalysis useless for this treatment (sinusoidal dependence with constant coefficients), i.e. at least a WKB approximation, of weakly varying parameters compared to a typical wavelength, is required. Moreover, the occurrence of\neither an infinitely degenerate eigenvalue or an accumulation\npoint giving rise to a continuous spectrum are associated to inhomogeneities. \nWe consider two types of inhomogeneities: the inhomogeneity of the equilibrium parameters along the loop axis, and the inhomogeneity across the loop axis when the radius is varied. As a first order approximation we neglect the effect\nof gravitational stratification and thus confine the analysis to\ncharacteristic spatial scales lower than the pressure scale height\nin the solar corona. In order to analyze the stability and to\nobtain the frequencies and modes the physical quantities in\neq.~\\ref{2} and eq.~\\ref{3} must be calculated along the loop\nstructure.\n\n\n\n\n\\subsection{Mechanical equilibrium}\n\n\nTo determine an equilibrium configuration we assume\nforce--free equations. This assumption is justified for coronal conditions due to the fact that in plasmas with low\n$\\mathcal{\\beta}$ (gas pressure over the magnetic pressure) the\npressure gradient can be neglected in comparison to the Lorentz\nforce. For the chromosphere and the photosphere the force--free approximation may not be a good one. However, it is a widespread supposition \\citep{rud}:\n perturbed systems are believed to relax to new force-free, minimum energy states and chromospheric conditions seem to be well fitted to force--free models from $4. \\ 10^{5} \\ m$ \\citep{asc4} (Chapter 5).\n\n\n\nCoronal loops are generally modeled as thin cylindrical\nfluxtubes where the curvature and related forces can be neglected so\nthe cylindrical geometry can be applied. The fluxtube is assumed\nas line--tied to the photospheric plasma through its footpoints\nwhich are forced to follow the photospheric velocity\nperturbations. The random velocity field creates vorticity\ngenerally twisting the coronal fluxtubes. Thus, a relation between\nthe helical twist and the force--free parameter can be derived as\nfollows (e.g. \\citep{stu94}). The coronal loop model is obtained\nfrom the equations\n\n\n\\begin{equation}\n\\nabla \\times \\mathbf{B_{0}}=\\alpha(r) \\mathbf{B_{0}} \\ \\ \\ \\mathbf{j} \\times \\mathbf{B_{0}}=0. \\label{3}\n\\end{equation}\nAlso, since $\\mathbf{B_{0}}$ is force--free, $\\nabla p=0$ everywhere and thus has\na constant value along the loop. We consider a straight cylinder with a nonuniform distribution of density and temperature and a resulting uniform\ntwist over an initially non--rotated field $ \\textbf{B}=(0,0,B_{z})$ yielding\n the unperturbed\nmagnetic field \n\n$$\\mathbf{{B_{0}}}= (B_{r},B_{\\phi},B_{z})=B_{0}(0, \\frac{br}{\\Delta}\\frac{1}{\\Delta})$$\n\n\\noindent with $\\Delta=1+b^{2}r^{2}$ and $ b=2 \\Pi N_{t}\/L$ ($\nN_{t}$ number of turns over the cylinder length $ L$). Then,\n\\begin{equation}\n\\frac{B_{\\phi}}{B_{z}}=\\frac{r \\partial \\phi}{\\partial z}=\\frac{r 2 \\pi N_{t}}{L}=br \\\\\n\\alpha(r) =\\frac{2b}{\\Delta}\\label{4}\n\\end{equation}\nWe assume a given value of\nthe cylinder radius $r=R$, thus the line element results a function of\nthe coordinate\n$z$: $s=s(z)$. The dependence with the radial component will be taken into account by considering different values of the radius $R$.\n\n\\begin{equation}\nds^{2}=R^{2}d \\phi^{2}+dz^{2}=\\left( 1+R^{2} b^{2}\\right) dz^{2} =\\Delta dz^{2}\\label{5}\n\\end{equation}\n\n\n\n\n\\subsection{Thermal equilibrium}\n\nThe thermal equilibrium is obtained, as in Paper I, assuming $L=0$ in the balance energy equation \n(eq.~\\ref{1}) . The procedure\ndeveloped consists in obtaining the function of the temperature along the\narc element $s$ by integrating eq.~\\ref{1} with the constraint $L=0$ and\n replacing border conditions: the temperature at the bottom $T_{b}=10^{4}K$ and the temperature at the top $T_{t}=10^{6}K$. The known expression (see chapter 6 of\n\\citet{pri}) is obtained\n\\begin{equation}\n\\left[\\frac{dT}{ds}\\right]^{2}=\\frac{p^{2}\\chi}{2k_{B}^{2}k_{0}(\\alpha\n+\\frac{3}{2})}T^{\\alpha-\\frac{7}{2}} \\left[1 - (\n\\frac{T}{T_{t}})^{2-\\alpha}\\right]\\label{6}\n\\end{equation}\nwhich has to be inverted to obtain\n$T=f^{-1}(s)$ \\citep{ar95} as\n\\begin{equation}\n\\frac{dT}{ds}=\\mathcal{A}\\left[\\frac{d\\mathbb{B}_{v}}{dv}\\frac{dv}{dT}\\right]^{-1} \\ \\ \\ where \\ \\ \\ \\mathbb{B}_{v}(\\frac{1}{2},q)=\\int_{0}^{v}t^{p-1}(1-t)^{q-1}dt \\label{7}\n\\end{equation}\n\n\\noindent \n with \n\n\n$$p= \\frac{1}{2}; v=1-(\\frac{T}{T_{t}})^{2-\\alpha}; q=(\\frac{\\alpha}{2}+\\frac{3}{4}) (2-\\alpha)+1 $$\n\n\n$$ \\mathcal{A}=(2-\\alpha)T_{t}^{\\frac{\\alpha}{2}-\\frac{11}{4}}((p^{2}\\chi)\/(2k_{0}\n(\\alpha+\\frac{3}{2})k_{B}^{2}))^{\\frac{1}{2}}.$$\nWe use $\\alpha=-\\frac{1}{2}$ so $q=\\frac{6}{5}$\nto numerically calculate \n the modes, \\\\\n$ s=\\frac{1}{\\mathcal{A}}\\mathbb{B}_{v}(\\frac{1}{2}, \\frac{6} {5})\\rightarrow\\mathcal{A}=\\frac{5}{2}T_{t}^{3}(\\frac{p^{2}\\chi} {2k_{0}k_{B}^{2}})^{1\/2}.$\\\\\nAlso, from boundary conditions $\\upsilon =0$, thus the constant value of the heating function results \n$H_{0}= 7\np^{2} \\chi T^{\\alpha -2}_{t}\/\\left(8 k_{B}^{2}(\\alpha\n+\\frac{3}{2})\\right).$ \n\n\\subsection{The perturbation}\n\nTo calculate the stability and the structure of the modes the\ngeneral perturbation along the equilibrium magnetic field is\nwritten\n\\begin{equation}\n\\vec{\\xi}=[\\zeta_{r}(r,z)\n\\mathbf{e}_{t}+i \\zeta_{\\phi}(r,z) \\mathbf{e}_{\\phi}+\\zeta_{z}(r,z) \\mathbf{e}_{z}]e^{im\\phi} \\ \\ T_{1}=T_{1}(r,z)e^{im\\phi}\\label{9}\n\\end{equation}\nwith $r=R$.\nThe $\\phi$ dependence only appears in the exponents that multiply the perturbation; the integration with respect to this coordinate is straightforward.\nThen, representing the equilibrium functions of the different\nquantities with a 0 sub-index, defining \n\n\\noindent\n$$ \\mathbf{e}_{t}=(Rb \\mathbf{e}_{\\phi}+ \\mathbf{e}_{z} )\/ \\sqrt{\\Delta} \\ \\ \\ \\nabla_{\\parallel}=\\mathbf{e}_{t}(\\mathbf{e}_{t}\\cdot\\nabla)\\nonumber \\ \\ \\ \\rho_{t}=\\frac{m_{p}}{k_{B}T_{t}}$$\n\n\\noindent\n with $\\mathbf{e}_{\\phi}, \\mathbf{e}_{z}$ the cylindrical versors and $\\mathbf{e}_{t}$ the tangential versor, we obtain a non--dimensional expression for the energy principle\n of eq.~\\ref{2}:\n\\begin{equation}\n\\delta^{2} W_{p}= \\delta^{2} W_{c}+\\delta^{2} W_{m}+\\delta^{2} W_{hc}+\\delta^{2} W_{r} \n\\label{10}\n\\end{equation}\nwhere $\\delta^{2} W_{c}$ is the generalized potential energy associated to compressional terms, $\\delta^{2} W_{m}$ corresponds to the magnetic contributions, $\\delta^{2} W_{hc}$ corresponds to the heat conduction terms and $\\delta^{2} W_{r}$ to the radiative contributions. The explicit form of these functions are given in the Appendix. The Bernstein's generalized potential energy corresponds to the magnetic contribution and part of the compressional one. In the generalized version of the energy principle additional terms appear in the $\\delta^{2} W_{c}$ term and also $\\delta^{2} W_{hc}$ and $\\delta^{2} W_{r}$ are entirely new terms. \n\n\n\\section{Results and discussion}\n\nConvective motion of the photosphere is believed to provide the\nenergy that is storage in twisted magnetic coronal fields allowing\nthe presence of long--lived coronal structures until it is\nreleased by instabilities (\\citep{raa}; \\citep{vrs}). On the\nother hand, continuous spectra are generally associated to stability. \nAn accepted conjecture establishes that unstable\nmodes have a discrete spectrum (see\n \\citep{fre} or \\citep{pri}). There are two types of possible continuous spectra in this problem. The inhomogeneous character of the equilibrium parameters along the loop axis can lead to a continuum that couples to the Alfv\\'en continuum\n\\citep{bel}; e.g, when the disturbances considered are comparable to the inhomogeneous characteristic wavelength stable eigenvalues can give rise to a continuous\nspectrum ($L\/2$, the equilibrium structure in the $z$ component). This is the case studied in Paper I. On the other hand, GH\nestablished, for non vanishing helicity systems, that \nthere is a continuous spectrum associated with the line-tied Alfv\\'en\nresonance leading to the damping and heating by the resonant absorption mechanism and thus, directly relate to\nthe stability of loops. They also pointed out how to obtain the\nresonant singular limit $\\omega$, from the class of physically\npermissible solutions,\n\\begin{equation}\n\\omega(r)=\\frac{nB_{z}(r)}{\\int_{-L}^{L}\\sqrt{\\rho(z)}dz}.\n\\label{11}\n\\end{equation}\n This resonance results because of the absence\n of an explicit dependence on the azimuthal magnetic field component ($B_{\\varphi}$).\n\nThus, in order to understand in which conditions which mechanism can dominate\nand give account of the different \n scenarios i.e., the driving of the\n instability or the damping of mode oscillations, it is critical to gain\n knowledge about the dynamics and energetic contribution of twisted structures.\nYet, the implications of the twisting in theoretical and\nobservational descriptions are poorly known; e.g., there is no\nclarity about the modification of the dispersion relation and observational data\nare indirectly inferred.\n\n\nIn this paper we focused our attention to describe the changes in\nperiods, stability and mode structure of coronal loops when the\nhelicity, the magnetic field intensity and the radius are varied. For loops\nwith vanishing helicity it is well established that the Alfv\\'en\nline--tied resonance continuum is responsible of the damping of\nkink ($m=1$) quasi--modes via the transfer of energy from the\nradial component into the azimuthal one, i.e., from discrete\nglobal modes into the local continuum modes where phase mixing\ncan take place. Still, the twisting of the magnetic field leads to\nthe coupling of MHD cylindrical modes making difficult to provide\na classification in terms of the behavior of pure--like modes.\n\n\nIn order to calculate modes and frequencies we followed the\nschematic procedure described in Paper I and in\n\\citep{gal1}. We used a symbolic manipulation program to integrate\nthe equations. $\\delta^{2}W_{p}$ and the perturbations were\nexpanded in a six dimensional--Fourier basis on the independent\n coordinate $z$ that adjusts to\nborder conditions, i.e., the four perturbated components (eq.~\\ref{9})\nwere expanded in a six mode basis to\n obtain $24$ eigenvalues and eigenvectors for each of the helicity\n and the magnetic field values. \nOnly the first\n eighteen eigenvalues were considered (the others are more than two order of magnitude smaller\n and accumulate at zero; the eigenvectors are also\n vanishingly small). \nThus, a quadratic form for $\\delta^{2}W_{p}$\nwas obtained and was minimized with the Ritz variational\nprocedure. A matrix discrete eigenvalue problem subject to a\nnormalization constraint was obtained. From the resulting modes and the\ngeneralized potential energy (eq.~\\ref{2}): $\\delta^{2}W_{p}\\geq0$ the stability of each mode was determined.\n\n\nThe coronal loop parameters used were: $L=10^{10}cm$ (or $L=100Mm$),\n$T_{b}=10^{4}K$ $T_{t}=10^{6}K$ $n_{e}=10^{8}cm^{-3}$ electron\nnumber density $p_{t} =2k_{B}T_{t} \\;$;\n$\\rho_{t}=m_{p}p_{t}\/k_{B}T_{t}$. \nFrequencies and modes were calculated for\ntwo different values of the magnetic field: $B_{0}=10G$ and\n$B_{0}=100G$, and for three different values of the helicity $b=(3.1 \\ 10^{-8} ; \\ 3.1\n \\ 10^{-7}; \\ 1.9 \\ 10^{-6})$ which correspond to the adimentional values:\n $b_{a}=(2.8; \\ 28; \\ 170)$ with $N_{t}=(0.45; \\ 4.3; \\ 13.7)$, $N_{t}:$ the\n number of turns\nover the cylinder length. These helicity values defined as weak,\nmoderated and strong helicity respectively correspond to the\nclassification given in \\citep{asc4} (Chapter 5).\n The adimentional radius\nwas initially chosen as $R=0.01$.\nIn what\nfollows we summarize the conclusions obtained from the data analysis which are displayed in three\ntables.\n\n\n \n\n\n\n\nTable 1 shows the periods (in minutes) for weak, moderate and strong helicity\n for two values of the magnetic field intensity ($B_{0}=10G$ and $B_{0}=100G$\n(left and right panel respectively). S and U\nletters indicate the stable--unstable character of the modes.\nFrom the table we see that:\n\n\\noindent\nI) Weak helicity modes are stable. This is in accordance with the analytic results by \\citet{rud} who studied nonaxisymmetric oscillations of a thin twisted magnetic tube with fixed ends in a zero-beta plasma.\n\n\\noindent\nII) Higher modes have an accumulation point at zero, indicating the presence of\n a continuum spectra of stable modes (as in Paper I). Note that, calculus performed\n via discrete basis, as in our case, give spectra that are necessarily discrete.\n Thus, an\naccumulation of discrete eigenvalues \nsuggests a stable continuum spectrum. \n\n\n\n\\noindent\nIII) The $B_{0}=10G$ case has larger periods than the $B_{0}=100G$ one.\n For moderate and strong helicity the eigenvalues of the first panel follow a scaling\n law with that of the second one i.e., they scale with the magnetic field intensity exactly as the Alfv\\'en speed does $P_{10G}\\simeq 10 P_{100G}$. \n\n\\noindent\nIV) As HG and GH, we note a clustering of the spectra associated to the\n change from real to imaginary eigenvalues (and viceversa). There is a pronounced change (in the spacing of the periods or\/and in the stability) from the sixth mode to the seventh mode. This is noted by a double line in Table 1 and related to the importance of the parallel component with respect to the perpendicular component (see Table 2). \nUp to period number ten real -- imaginary eigenvalues of the first panel\n ($B_{0}=10G$) correspond to real--imaginary ones of the second panel ($B_{0}=100G$). Also,\nexcepting large order periods $n>10$, when\n the helicity is increased from weak to moderate the imaginary stable\n eigenvalues turn to imaginary unstable ones. For $B_{0}=10G$ and weak\n and moderate helicity cases there are five different groups of periods\n ($P_{1}-P_{6};P_{7}-P_{10};P_{11}-P_{12};P_{13}-P_{14};P_{15}-\nP_{18}$) (see also Table 2). The clustering is more difficult to establish i.e., the\ndifferences are less pronounced, with increasing magnetic field\nintensity and larger order periods.\n\n\\noindent\n In order to compare our results with those given by these authors\n we calculated the expression eq.~\\ref{11} for our modes. We found that, all periods excepting $P_{1}-P_{6}$ weak helicity modes satisfy \n the relation and thus, they belong to the Alfv\\'en continuum spectrum\njustifying the scaling law described in III.\n As HG, we conclude that the change in the real--complex character of the\n $P_{6}-P_{7}$ eigenvalues is associated to the existence of an accumulation\n point of the resonant Alfv\\'en continuum, however we find that this change is not necessarily related to a change \n in the stability as they claimed. Note that all modes with weak\n helicity are stable (even the imaginary ones); in all the other cases the imaginary character of the eigenvalues is associated to instability. \nYet, the continuum stable eigenvalue conjecture\n is here still valid \\citep{fre}, \\citep{pri}; it applies to a spectrum with an accumulation point in zero; we found stable modes for all the helicity values and for the two magnetic field values with $P_{n>14}$. Note that the analysis of stable modes is still of interest because depending on the relative characteristic times of stable and unstable modes the stable ones could be active \n and accessible to observations.\n\n\\noindent\nThe presence of at least one unstable mode means that the equilibrium\n state is unstable. Thus, taking into account the whole range of stable modes, we confirm previous results\nleading to conclude that field configurations with some\ndegree of twisting give a stabilizing effect allowing the storage\nof magnetic energy \\citep{raa2}, i.e., when the helicity is augmented the stable weak case turns to an unstable one suggesting a critical value. \n\n\\bigskip\n\nIn Paper I, we obtained only one unstable mode classified as slow\nmagnetoacoustic mode due to the almost longitudinal character (parallel to the\n magnetic field) of the wavevector perturbation\n and to the fact that the period did not changed with the\n intensity of the magnetic field, resembling acoustic waves with sound speeds,\n $v_{s}$, independent of the magnetic field. The characteristic\n unstable time obtained in Paper I was $\\tau_{u}=36 \\ min$, corresponding\n to a typical slow magnetoacoustic fundamental period with a characteristic\n wavelength of the order of the loop length $L\/2$. Also,\n we obtained a continuous set of stable modes classified as\n fast magnetoacoustic modes due to their large value component orthogonal\n to the magnetic field and to the fact that the eigenvalues scale with\n the intensity of\n the magnetic field as $P_{11G}\\simeq 10 P_{100G}$; thus resembling\n the dependence of the\n Alfv\\'en waves $v_{A} \\sim B_{0}$.\n\n\nTable 2 (First Panel) displays the resulting features associated to the\nrelative intensity of the parallel and perpendicular to the field components\n($(\\xi_{\\parallel},\\xi_{\\perp},\\xi_{r}) $ is an orthogonal basis)\nand their classification as slow--like (S) or fast--like (F). The relative\nphase between the components is also indicated in the table\nby P (in phase) and IP (inverted phase). Table 2 (Second Panel) also shows the\nintensity relationship between the cylindrical components. \nIn order to classify the modes and to compare \n with the slow and fast\nmagnetoacoustic modes obtained in Paper I, we calculated the cylindrical\n mode components and also the tangential and normal to the field\n components\n ($\\xi_{\\parallel}=\n(Rb\\xi_{\\phi}+\\xi_{z})\/\\Delta; \\xi_{\\perp}=\n(\\xi_{\\phi}-Rb\\xi_{z})\/\\Delta$).\n Our interest in the $\\xi_{\\parallel}$,\n $\\xi_{z}$, $\\xi_{r}$, $\\xi_{\\perp}$\n and $\\xi_{\\phi}$ components resides in that: First, when the helicity is weak, the $\\xi_{\\parallel}$\n component is expected to play the slow-mode role of $\\xi_{z}$\n in Paper I.\n Second, the $\\xi_{r}$ component is related to the\n fast modes and determines\n the resonant absorption mechanism when uniform cylindrical\n flux tubes are considered by\n the transferring of energy \n to the $\\xi_{\\phi}$ component. When helicity and inhomogeneous distribution of equilibrium parameters are present it is worth investigating the transferring\n of energy from the $\\xi_{r}$ component to the others. In this\n case the resonant damping of global oscillations\n will occur by conversion of kinetic energy of the\n radial component into kinetic energy of the\n $\\xi_{\\parallel}$ and $\\xi_{\\perp}$ components;\n both components forming the plane orthogonal to\n $\\xi_{r}$, and equal to the plane formed by $\\xi_{\\phi}$ and $\\xi_{z}$.\n\n\nFrom the analysis of the amplitude of the components of the $P_{1}-P_{6}$ modes with respect to the $P_{7}-P_{18}$ ones in the weak helicity case i.e., real and imaginary eigenvector respectively, we could classify the first ones as slow-like modes because: I) their tangential components $\\xi_{\\parallel}$\nare at least an order of magnitude larger\n than the normal ones $\\xi_{\\perp}$; II) as the helicity is\n weak $\\xi_{\\parallel}\\approx \\xi_{z}$ and \n $\\xi_{r}\\rightarrow 0; \\xi_{\\phi}\\rightarrow 0$, the\n wavevector is almost tangential to the magnetic field; III) they have \na larger characteristic time and a shorter characteristic speed than the imaginary eigenvectors. On the contrary, imaginary eigenvalues are associated to large values of\nthe $\\xi_{r}$ component and $\\xi_{\\perp}$ component (due to large values of $\\xi_{\\phi}$ (see Table 2 Second Panel)) , and small values of\nthe $\\xi_{\\parallel}$ and $\\xi_{z}$ components.\n As in Paper I, when the eigenvalues change form real to imaginary\n the period strongly diminishes and a change in\n the type of mode from the slow to fast magnetoacoustic type occurs. In opposition to Paper I where the\nacoustic mode has the same eigenvalue for both magnetic field\nintensities, here the modes are affected by the strengthening of\nthe magnetic field leading to an-order-of-magnitude shorter period than in the non--helicity case. \nThe $\\xi_{\\parallel}$ and $\\xi_{\\perp}$ components are in an inverted phase for real eigenvector modes and in phase for imaginary eigenvector modes.\n\n\n\n\nFor moderate helicity the overall description is similar but all the cases having non vanishing\n $\\xi_{\\phi}$ component and all the periods in the resonant\n line-tied continuum. As was mentioned, real--imaginary eigenvalues\n correspond to stable--unstable behavior.\n\n\n\n\nIn the strong helicity case, as the weak and moderate ones, we\nnote for $P_{1}-P_{6}$ larger, but comparable, values of the\n$\\xi_{\\parallel}$ component with respect to the $\\xi_{\\perp}$\ncomponent. In this case the two components of the mode are in phase.\nThis relationship between the $\\xi_{\\parallel}$ and $\\xi_{\\perp}$\ncomponents of Table 2 (FP), and their associated phases is found again in the\n modes with $P_{15} - P_{18}$. In spite that these features are associated to the slow\nmagnetoacoustic characterization, Table 2 (SP) shows that as $\\xi_{z}$ is\nvanishingly small, the strong helicity case cannot be classify as a\nslow mode. \n\n\\bigskip\n\nWhen helicity is present the mixed character of the modes manifests itself making difficult to identify the components that are involved in the damping mechanism. However, taking into account the resonant frequency of eq.~\\ref{11}, we noted that (HG) all the modes, except those with $P_{1}-P_{6}$\n periods of the weak helicity case, have resonant frequencies suggesting\n that resonant\n absorption in helical modes is associated to modes with significant values of\n $\\xi_{\\perp}$ component. If this argument is correct we can affirm\n that the damping mechanism of body helical modes\n is associated to the transfer of kinetic energy of the radial component into\n kinetic\n energy of the $\\xi_{\\perp}$ component which is not only related to the $\\xi_{\\phi}$\n cylindrical contribution but also to the $\\xi_{z}$ one\n by the expression\n $ \\xi_{\\perp}=(\\xi_{\\phi}-Rb\\xi_{z}) \\Delta$.\n\n\\bigskip\n\nWe also analyzed the change of the period as a function of the radius for different values of the helicity. \n We found that, for weak helicity, the increasing of the radius leads to a decrease of the periods. This is in accordance with observations, e.g., observed sausage modes are associated with thicker and denser loop structures and lower periods; while in other case (unstable cases) the increasing of the radius leads to an increase of the period. \n\n\\bigskip\n\nTable 3 -First and Second Panel- shows the variation of the radius $R$ with the twist\n$bR$ for weak and moderate helicity respectively. \n \\citep{rud} has conjectured that the line-tying condition at the tube ends should stabilize the tube and has suggested a critical value ($\\sim L b0$ to\n$\\delta^{2} W_{p}<0$. Figure~\\ref{fig:tres}a and Figure~\\ref{fig:tres}c\ndisplay the total energy composed by the compressional,\nradiative, thermal and magnetic energy contributions of $P_{6}$ mode in the weak and moderate case respectively. \nThe same features but for the $P_{7}$ mode are shown in Figure~\\ref{fig:tres}e and Figure~\\ref{fig:tres}g. Figure~\\ref{fig:tres}b and Figure~\\ref{fig:tres}d show the\nmagnetic energy content alone for $P_{6}$ mode in the weak and moderate case respectively. Figure~\\ref{fig:tres}f and Figure~\\ref{fig:tres}h show the magnetic energy content for $P_{7}$ mode and for the weak and moderate case respectively. It can be seen, in this and in all\nthe other cases, that the magnetic energy content has a determinant role on \n the stability--instability of the system, i.e., the stability changes when the magnetic generalized potential energy changes sign. Thus, a result of\nthis analysis is that the stability of twisted coronal loops is\nfundamentally determined by the storing of magnetic energy, being the\nother contributions less significant. Meanwhile, when the helicity is weak or vanishingly small and the magnetic contribution has a stabilizing effect the other non-dominant contributions, as the non-adiabatic ones, can play an important role. This makes possible, for example, the damping of fast excitations due to resonant absorption. Yet, even when one of these contributions is unstable, stable modes could be active for a while if their characteristic periods are shorter than the characteristic time of the instability. This is the case of Paper I, where we obtained a slow mode with an unstable characteristic times of $\\tau\\sim 36 \\ min$ coexisting with stable fast modes with periods about $P\\sim 1 \\ min$; moreover, we showed that the instability can be nonlinearly saturated giving rise to a limit-cycle solutions, i.e., an oscillation between\n parallel plasma kinetic energy and plasma internal energy where\n the magnetic energy plays no relevant role. \nThus, the\ncontribution to the stable--unstable character of the modes is\nmostly due to the magnetic energy content and not to other\nenergetic contributions. \nNote that as the balance energy equation takes into account non--adiabatic contributions, i.e., radiation, heat flow and heat function (with $L=0$ at the equilibrium), the resulting perturbations are not constrained to the force--free condition. So, one result of the analysis is that the pertubation energy contribution is mainly due to magnetic forces. \nThus, for these type of twisted magnetic field\nmodels, non--adiabatic perturbations (e.g. thermal perturbations) and resonant absorption seem unimportant to guarantee stability; a loop system with weak \n storage of magnetic energy (low values of the\nhelicity) could be released if the helicity is suddenly\nincreased, e.g., by footpoint motions. Meanwhile all the \"zoo\" of the coronal seismology can be active and accessible to observations.\n\n \n\n\\begin{figure*}\n\n \\includegraphics[width=4.3cm]{P6ewp.eps}\n \\includegraphics[width=4.3cm]{P6emwp.eps}\n \\includegraphics[width=4.3cm]{P6emp.eps}\n \\includegraphics[width=4.3cm]{P6emmp.eps}\n \\includegraphics[width=4.3cm]{P7ewp.eps}\n \\includegraphics[width=4.3cm]{P7emwp.eps}\n \\includegraphics[width=4.3cm]{P7emp.eps}\n \\includegraphics[width=4.3cm]{P7emmp.eps}\n \\caption{Energy content of the sixth and seventh mode for $B_{0}=10G$. a) Total potential energy and b) magnetic potential energy respectively for the sixth mode $P_{6}=1.23 \\ min$ and for weak helicity. c) Total potential energy and d) magnetic potential energy respectively for the sixth mode $P_{6}=1.23 \\ min$ and for moderate helicity. \n e) Total potential energy and f) magnetic potential energy respectively for the seventh mode $P_{7}=0.07 \\ min$ and for weak helicity. g) Total potential energy and h) magnetic potential energy respectively for the sixth mode $P_{7}=0.07 \\ min$ and for moderate helicity.}\n \\label{fig:tres}\n \\end{figure*}\n\n\n\\begin{table*}\n\\begin{tabular}{cccccccc}\n\\hline\n$P_{i}$&$weak $&$moderate$&$strong$&$ \\ \\ \\ $& $weak$&$moderate$&$strong$ \\\\ \\hline\n$P_{1}$&$1.921 \\ S $&$0.209 \\ S$&$0.525 \\ i \\ U$&$ \\ \\ \\ $&$0.159 \\ S$&$0.021 \\ S$&$0.052 \\ i \\ U$\\\\ \\hline\n$P_{2}$&$1.869 \\ S $&$0.204 \\ S$&$0.450 \\ S$&$ \\ \\ \\ $& $0.158 \\ S$&$0.020 \\ S$&$0.044 \\ S$\\\\ \\hline\n$P_{3}$&$1.535 \\ S $&$0.169 \\ S$&$0.430 \\ S$&$ \\ \\ \\ $& $0.154 \\ S$&$0.017 \\ S$& $0.042 \\ S$ \\\\ \\hline\n$P_{4}$&$1.533 \\ S $&$0.168 \\ S$&$0.424 \\ i \\ U$&$ \\ \\ \\ $&$0.153 \\ S$&$0.0167 \\ S$&$0.042 \\ i \\ U$\\\\ \\hline\n$P_{5}$&$1.306 \\ S $&$0.143 \\ S$&$0.206 \\ i \\ U$& $ \\ \\ \\ $&$0.151 \\ S$&$0.014 \\ S$&$0.020 \\ i \\ U$\\\\ \\hline\n$P_{6}$&$1.228 \\ S $&$0.135 \\ S$&$0.177 \\ i \\ U$&$ \\ \\ \\ $& $0.15 \\ S $&$0.013 \\ S$&$0.017 \\ i \\ U$\\\\ \\hline \\hline\n$P_{7}$&$0.068 \\ i \\ S$&$0.070 \\ i \\ U$&$0.125 \\ S$&$ \\ \\ \\ $& $0.0047 \\ i \\ S$&$0.007 \\ i \\ U$&$0.0125 \\ S$\\\\ \\hline\n$P_{8}$&$0.064 \\ i \\ S$&$0.066 \\ i \\ U$&$0.122 \\ S$&$ \\ \\ \\ $& $0.0046 \\ i \\ S$&$0.006 \\ i \\ U$&$0.012 \\ S$\\\\ \\hline\n$P_{9}$&$0.042 \\ i \\ S $&$0.044 \\ i \\ U$&$0.101 \\ S$&$ \\ \\ \\ $& $0.0044 \\ i \\ S$&$0.0043 \\ i \\ U$&$0.0101 \\ S$\\\\ \\hline\n$P_{10}$&$0.041 \\ i \\ S$&$0.043 \\ i \\ U $&$0.100 \\ S$&$ \\ \\ \\ $& $0.0043 \\ i \\ S$&$0.0042 \\ i \\ U$&$0.01 \\ S$\\\\ \\hline\n$P_{11}$&$0.033 \\ S $&$0.036 \\ S$&$0.989 \\ S$&$ \\ \\ \\ $& $0.0042 \\ i \\ S$& $0.0036 \\ S$&$0.099 \\ S$\\\\ \\hline\n$P_{12}$&$0.032 \\ S $&$0.035 \\ S$ &$0.096 \\ S$&$ \\ \\ \\ $& $0.0041 \\ i \\ S$&$0.0035 \\ S$ &$0.0096 \\ S$\\\\ \\hline\n $P_{13}$&$0.030 \\ i \\ S $&$0.031 \\ i \\ U$ &$0.085 \\ S$&$ \\ \\ \\ $&$0.003 \\ S$&$0.0031 \\ i \\ U$ &$0.0086 \\ S $\\\\ \\hline\n$P_{14}$ & $0.027 \\ i \\ S $&$0.029 \\ i \\ U$&$0.081 \\ S $&$ \\ \\ \\ $&$0.0026 \\ S$&$0.003 \\ i \\ U$&$0.0081 \\ S $\\\\ \\hline\n$P_{15}$&$0.025 \\ S $&$0.027 \\ S $&$0.077 \\ S $&$ \\ \\ \\ $&$0.0025 \\ S $&$0.0027 \\ S $&$0.0077 \\ S $\\\\ \\hline\n$P_{16}$&$0.024 \\ S $&$0.026 \\ S $&$0.076 \\ S $&$ \\ \\ \\ $&$0.002 \\ S $&$0.003 \\ S $&$0.0076 \\ S $\\\\ \\hline\n$P_{17}$&$0.02 \\ S $&$0.02 \\ S $&$0.063 \\ S $&$ \\ \\ \\ $&$0.0024 \\ S $&$0.0021 \\ S $&$0.0063 \\ S $\\\\ \\hline\n$P_{18}$&$0.018 \\ S $&$0.02 \\ S $&$0.059 \\ S $&$ \\ \\ \\ $&$0.0024 \\ S $&$0.0025 \\ S $&$0.006 \\ S $\\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:table1} Eighteen first periods associated to stable (S) and unstable (U) eigenvalues (minutes) for A) Left panel: $B_{0}=10G$ with A1) left column: weak helicity, A2) middle column: moderate helicity, A3) right column: strong helicity and B) Right panel: $B_{0}=100G$ with B1, B2, B3 the same as in A. Larger order modes were discarded.}\n\\end{table*}\n\n\n\\begin{table*}\n\\begin{tabular}{ccccccc}\n\\hline\n$P_{i}$&$weak $&$moderate$&$strong \\ \\ \\ \\ $&$weak $&$moderate$&$strong$\\\\ \\hline\n$P_{1}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp}\n \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 $\\\\ \\hline\n$P_{2}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 $\\\\ \\hline\n$P_{3}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 \\ $\\\\ \\hline\n$P_{4}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 $\\\\ \\hline\n$P_{5}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 $\\\\ \\hline\n$P_{6}$&$\\xi_{\\parallel} \\gg \\xi_{\\perp} \\mapsto 0 \\ S; \\ IP $&$\\xi_{\\parallel} > \\xi_{\\perp} \\ S; \\ IP $&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r}\\mapsto 0 $&$ \\xi_{z}\\gg \\xi_{\\phi} \\sim \\xi_{r} $&$ \\xi_{r} \\leq \\xi_{\\phi}; \\xi_{z} \\mapsto 0 $\\\\ \\hline \\hline\n$P_{7}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$ &$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0$&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z} \\mapsto 0 $&$\\xi_{z} > \\xi_{r} > \\xi_{\\phi} $\\\\ \\hline\n$P_{8}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z} > \\xi_{r}> \\xi_{\\phi} $ \\\\ \\hline\n$P_{9}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z} > \\xi_{r} > \\xi_{\\phi} $\\\\ \\hline\n$P_{10}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $& $\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z}> \\xi_{r} > \\xi_{\\phi} $\\\\ \\hline\n$P_{11}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi} \\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z} > \\xi_{r} > \\xi_{\\phi} $ \\\\ \\hline\n$P_{12}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ IP \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $ &$ \\xi_{r}> \\xi_{\\phi} > \\xi_{z} $ \\\\ \\hline\n $P_{13}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $&$\\xi_{r}\\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z} > \\xi_{r} > \\xi_{\\phi} $\\\\ \\hline\n$P_{14}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel}\\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$ &$\\xi_{\\perp} \\geq \\xi_{\\parallel} \\ IP \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{z} > \\xi_{r} > \\xi_{\\phi} $\\\\ \\hline\n$P_{15}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0$&$ \\xi_{r}> \\xi_{\\phi} > \\xi_{z} $ \\\\ \\hline\n$P_{16}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$ \\xi_{r}> \\xi_{\\phi} > \\xi_{z} $ \\\\ \\hline\n$P_{17}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$&$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0$&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$ \\xi_{r}> \\xi_{\\phi} > \\xi_{z} $ \\\\ \\hline\n$P_{18}$&$\\xi_{\\perp} \\gg \\xi_{\\parallel} \\mapsto 0 \\ F; \\ P $&$\\xi_{\\perp} > \\xi_{\\parallel} \\ F; \\ P$ &$\\xi_{\\parallel} \\geq \\xi_{\\perp} \\ P \\ \\ \\ \\ $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$\\xi_{r} \\sim \\xi_{\\phi}\\gg \\xi_{z}\\mapsto 0 $&$ \\xi_{r}> \\xi_{\\phi} > \\xi_{z} $\\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:table2} First Panel: Intensity relationship between the tangential and normal to the field components of the eighteen first periods for $B_{0}=10G$ and for weak (first column), moderate (second column) and strong helicity (third column) cases. The (P) indicates in phase and (IP) indicates inverted phase. Second Panel: Intensity relationship between the cylindrical components of the eighteen first periods for $B_{0}=10G$ and for weak (first column), moderate (second column) and strong helicity (third column) cases.}\n\\end{table*}\n\n\n\n\n\\begin{table*}\n\\begin{tabular}{cccccccc}\n\\hline\n$R$&$L $&$R\/2L$&$Twist=bR \\ \\ \\ \\ \\ \\ \\ \\ \\ $&$R$&$L $&$R\/2L$&$Twist=bR$ \\\\ \\hline\n$0.01$& $ 9.05 \\ 10^{7} $&$0.005 $&$ 0.028\\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.01$& $ 8.07 \\ 10^{7} $&$0.005 $&$ 0.28$\\\\ \\hline\n$0.02$&$ 9.04 \\ 10^{7} $&$ 0.01 $&$ 0.057 \\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.015$&$ 8.32 \\ 10^{7} $&$ 0.008 $&$ 0.43 $\\\\ \\hline\n$0.03$&$ 9.02 \\ 10^{7} $&$ 0.015 $&$ 0.085 \\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.02$&$ 7.86 \\ 10^{7} $&$ 0.011 $&$ 0.57$\\\\ \\hline\n$0.04 $&$ 8.99 \\ 10^{7} $&$ 0.02 $&$ 0.11\\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.025 $&$ 7.38 \\ 10^{7} $&$ 0.015 $&$ 0.71$\\\\ \\hline\n$0.05 $&$ 8.96 \\ 10^{7} $&$ 0.025 $&$ 0.14\\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.03 $&$ 6.88 \\ 10^{7} $&$ 0.02 $&$ 0.85$\\\\ \\hline\n$0.06 $&$ 8.9 \\ 10^{7} $&$ 0.03 $&$ 0.17\\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.04 $&$ 5.97 \\ 10^{7} $&$ 0.03 $&$ 1.13$\\\\ \\hline\n$0.1 $&$ 8.7 \\ 10^{7} $&$ 0.05 $&$ 0.28\\ \\ \\ \\ \\ \\ \\ \\ \\ $&$0.05 $&$ 5.2 \\ 10^{7} $&$ 0.04 $&$ 1.42$\\\\ \\hline\n \\hline\n\\end{tabular}\n\\caption{\\label{tab:table5} First Panel - Stable case: Variation of the Radius with the Twist for weak helicity $b=0.05$ and $B_{0}=10G$. Second Panel - Unstable case: Variation of the Radius with the Twist for moderate helicity $b=0.5$ and $B_{0}=10G$.}\n\\end{table*}\n\n\\section{Appendix: Generalized potential energy terms}\n\nFrom the procedure described above and extensively exemplified in Paper I we \ncan obtain -laboriously but in a straightforward way- the explicit terms for the energy principle given in eq.~\\ref{10}:\n$$\\delta^{2} W_{p}= \\delta^{2} W_{c}+\\delta^{2} W_{m}+\\delta^{2} W_{hc}+\\delta^{2} W_{r} $$\nwhere the right side of the equation corresponds to the compressional, magnetic, heat conduction and radiative contributions respectively.\nThe compressional term $ \\delta^{2} W_{c}=\\delta^{2} W_{c1}+\\delta^{2} W_{c2}$ has an additional contribution ($\\delta^{2} W_{c2}$) with respect to Bernsteins principle:\n$$ \\delta^{2} W_{B}= \\delta^{2} W_{c1}+ \\delta^{2} W_{m}$$\n\n $$\n \\delta^{2} W_{c1}=\\frac{1}{2}\\int_{-1}^{1}dz \\beta\\left\\{ T_{0} \\rho_{0} (1-m) \\frac{\\xi_{r}^{2}}{R^{2}}-\\frac{m}{R}T_{0}\\left(\\Delta\\frac{d\\rho_{0}}{ds}\\xi_{z}\\xi_{\\phi}+\\rho_{0}\\left(\\frac{\\xi_{r}\\xi_{\\phi}}{R} -\\right.\\right.\\right.$$\n\n$$ \\left. \\left. -\\frac{m}{R}\\xi_{\\phi}^{2}+\\frac{d\\xi_{\\phi}}{dz}\\right)\\right)+\\Delta \\frac{dT_{0}}{ds}\\left(\\Delta \\frac{d\\rho_{0}}{ds}\\xi_{z}^{2}+\\rho_{0}(\\frac{\\xi_{r}\\xi_{\\phi}}{R} -\\frac{m}{R}\\xi_{\\phi}\\xi_{z}+\\xi_{z}\\frac{d\\xi_{z}}{dz}) \\right) +\n$$\n$$\n+T_{0}\n\\left(\\Delta^{2}\\frac{d^{2}\\rho_{0}}{ds^{2}}\\xi_{z}^{2}+\\Delta\\frac{d\\rho_{0}}{ds}\n\\xi_{z}\\frac{d\\xi_{z}}{dz}+\\rho_{0}(\\frac{\\xi_{z}}{R}\\frac{d\\xi_{r}}{dz}-\n\\frac{m}{R}\\xi_{z}\\frac{d\\xi_{\\phi}}{dz} \n+\\xi_{z}\\frac{d^{2}\\xi_{z}}{dz^{2}})+\\right.\n$$\n$$\\left. \\Delta\\frac{d\\rho_{0}}{ds}(\\frac{\\xi_{r}\\xi_{\\phi}}{R} -\\frac{m}{R}\n\\xi_{\\phi}\\xi_{z}+\\xi_{z}\\frac{d\\xi_{z}}{dz}) \\right) \n$$\nThe magnetic contribution is:\n$$ \\delta^{2} W_{m}=C_{1}\\left\\{ \\beta \\Delta \\left (\\frac{m}{R}B_{\\phi}B_{z}\\xi_{r}\n\\xi_{\\phi}-B_{\\phi}B_{z}\\frac{d\\xi_{r}}{dz}\\xi_{z}\\right. \\right. $$\n$$\n-\\left. (\\frac{B_{\\phi}B_{z}}{R}+B_{\\phi}\\frac{dB_{z}}{dr})\n \\xi_{r}^{2}+(\\frac{m}{R}B_{\\phi}B_{z}\\xi_{r}\\xi_{\\phi}+B_{\\phi}\n \\frac{dB_{\\phi}}{dr} \\xi_{r}^{2})\\right) -$$$$\n-\\beta \\left( (B_{z}^{2}\\frac{d\\xi_{r}^{2}}{dz}+\\frac{m^{2}}{R^{2}}\nB_{\\phi}^{2}\\xi_{r}^{2})+(B_{\\phi}^{2}\\frac{d\\xi_{z}^{2}}{dz}+2B_{\\phi}+\n\\frac{dB_{\\phi}}{dr}\\xi_{r}\\frac{d^{2}\\xi_{z}}{dz}+\n \\frac{dB_{\\phi}^{2}}{dr}\\xi_{r}^{2}+B_{z}^{2}\\frac{d^{2}\n\\xi_{\\phi}}{dz})+\\right. \n$$$$\n\\left. \\left. \\left ( (B_{z}+R\\frac{dB_{z}}{dr})^{2}\\frac{\\xi_{r}^{2}}{R^{2}}-\n 2\\frac{m}{R^{2}}B_{z}(B_{z}+R\\frac{dB_{z}}{dr})\n\\xi_{r}\\xi_{\\phi}+(\\frac{m}{R}B_{z})^{2}\\xi_{\\phi}^{2}+\n(\\frac{m}{R}B_{\\phi})^{2}\n\\xi_{z}^{2}\\right) \\right) \\right\\} \n$$\nThe heat conduction term results:\n$$\\delta^{2} W_{hc}=-C_{2}\\left\\{5\\frac{T_{0}^{3\/2}}{\\Delta}\n\\frac{dT_{0}}{ds}T_{1}\\frac{dT_{1}}{dz}+T_{1}^{2}\n\\left( -T_{0}^{5\/2}(\\frac{mb}{\\Delta})^{2}\\frac{15}{4}T_{0}^{1\/2}\n\\frac{dT_{0}^{2}}{ds}+\\right.\\right. \n$$\n$$\n\\left. \\left. +\\frac{5}{2}T_{0}^{3\/2}\\frac{d^{2}T_{0}}{ds^{2}}\\right)\n+\\frac{1}{\\Delta^{2}}T_{0}^{5\/2}T_{1}\\frac{d^{2}T_{1}}{dz^{2}}\\right\\}\n$$\nThe new compressional contribution is expressed as:\n$$\\delta^{2} W_{c2}=-\\beta \\left(\\frac{m}{R} \\rho_{0}\\xi_{\\phi}T_{1}+\\Delta\\frac{d \\rho_{0}}{ds}\n\\xi_{z}T_{1}+\\rho_{0}\\xi_{z}\\frac{dT_{1}}{dz}\\right)\n$$\nand the term associated to radiation results:\n$$\n\\delta^{2} W_{r}=-\\alpha T_{1}^{2}\\rho_{0}^{2}T_{0}^{\\alpha-1}-\\beta \\left(\\frac{m}{R} \\rho_{0}\\xi_{\\phi}T_{1}+\\Delta\\frac{d \\rho_{0}}{ds}\n\\xi_{z}T_{1}+\\rho_{0}T_{1}\\frac{d\\xi_{z}}{dz}+\n\\frac{\\rho_{0}}{R} \\xi_{r}T_{1}\\right)$$\n\n\n\\noindent\nwhere the following changes were made:\n$$\\rho\\rightarrow\\frac{\\rho}{\\rho_{t}}; \\ \\\nT\\rightarrow\\frac{T}{T_{t}}; \\ \\\nB_{\\phi,z}\\rightarrow\\frac{B_{\\phi,z}}{B_{0}}; \\ \\ b\\rightarrow b S$$\n$$ r,z\\rightarrow\\frac{r,z}{S}; \\ \\ \\delta^{2}\nW_{p}\\rightarrow\\ \\delta^{2} W_{p}\/\\left(\\chi\nT_{t}^{\\alpha+1}\\rho_{t}^{2}L\/m_{p}^{2}\\right),$$\n$S=\\Delta L$ and the non--dimensional constants: $$C_{1}=\n\\rho_{t}^{2}T_{t}^{\\alpha+1}B_{0}^{2}\/(\\mu_{0}k_{B} T_{t}n_{e}); \\ \\\nC_{2}=c T_{t}^{\\frac{7}{2}-\\alpha}\/(S^{2} n_{e}^{2}).$$\nwere used. All the quantities were defined as in Paper I.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{MEAN FIELD THEORY AND MODEL POTENTIAL}\nIn their recent work on the band structure of Bose-Einstein condensates in a Kronig-Penney potential~\\cite{rf:KP}, Seaman {\\it et al}. have reported that when a swallow-tail energy loop is present in the band structure, the effective mass, which is known to be closely related to the dynamical instability of condensates in periodic potentials~\\cite{rf:effmass}, is always positive and the condensate is dynamically stable in the lower potion of the swallow-tail.\nIn this comment, however, we recalculate the band structure by solving the time-independent Gross-Pitaevskii equation and find a region where the effective mass is negative in the lower portion of the swallow-tail.\n\nWe start from the time-independent Gross-Pitaevskii equation,\n \\begin{eqnarray}\n \\left[-\\frac{1}{2m}\\frac{d^2}{d x^2}-\\mu+V(x)\n +g|\\Psi_0(x)|^2\\right]\\Psi_0(x) =0,\\label{eq:sGPE}\n \\end{eqnarray}\nwith a Kronig-Penney potential,\n \\begin{eqnarray}\n V(x)=V_0\\sum_{j}\\delta(x-ja),\\label{eq:delta}\n \\end{eqnarray}\nwhere $a$ is the lattice constant, $V_0$ is the potential strength, $\\mu$ is the chemical potential, $\\Psi_0(x)$ is the condensate wave function, and $g$ is the coupling constant of the interatomic interaction.\nNote that we set $\\hbar=1$.\nThe normalization condition is given by\n \\begin{eqnarray}\n \\int_{ja}^{(j+1)a}\\left|\\Psi_0(x)\\right|^2dx=N_{\\rm C},\\label{eq:normal}\n \\end{eqnarray}\nwhere $N_{\\rm C}$ is the number of condensate atoms per site.\nThe energy of the condensate per site is given by\n \\begin{eqnarray}\n E = \\int_{ja}^{(j+1)a}dx\\Psi_0^{\\ast}\n \\left[-\\frac{1}{2m}\\frac{d^2}{dx^2}+V(x)+\\frac{g}{2}|\\Psi_0|^2\n \\right]\\Psi_0.\\label{eq:mean_ener}\n \\end{eqnarray}\n\nThe condensate wave function can be written as $\\Psi_0(x)=\\sqrt{n_0}A(x){\\rm e}^{{\\rm i}S(x)}$, where $A(x)$ and $S(x)$ mean the amplitude and phase of the condensate. $A(x)$ is normalized by the density at the center of each site $n_0\\equiv |\\Psi_0\\left(\\left(j+1\/2\\right)a\\right)|^2$.\nThus, Eq. \\!(\\ref{eq:sGPE}) is reduced to\n \\begin{eqnarray}\n -\\frac{1}{2m}\\frac{d^2A}{dx^2}+\\frac{Q^2}{2m}A^{-3}\n +V(x)A-\\mu A+gn_0 A^3\\!=\\!0, \\label{eq:ampl}\n \\end{eqnarray}\n \\begin{eqnarray}\n A^2\\frac{dS}{dx}=Q.\\label{eq:conti}\n \\end{eqnarray}\nEquation \\!(\\ref{eq:conti}) is the equation of continuity and $Q$ describes the superfluid momentum.\n\n\\begin{figure}[b]\n \\includegraphics[scale=0.48]{firstband}\n\\caption{\\label{fig:1stband}\nFirst Bloch band of the energy of the condensate $E(K)$ for $V_0=1 gn_{\\rm av}\\xi_{\\rm av}$ and $a=5 \\xi_{\\rm av}$, where $n_{\\rm av}=N_{\\rm C}\/a$ and $\\xi_{\\rm av}=(mgn_{\\rm av})^{-1\/2}$.}\n \\end{figure}\n\\begin{figure}[thb]\n \\includegraphics[scale=0.47]{groupvelocity}\n\\caption{\\label{fig:group}\nGroup velocity $v_{\\rm g}(K)$ for $V_0=1 gn_{\\rm av}\\xi_{\\rm av}$ and $a=5 \\xi_{\\rm av}$, where $c_{\\rm s}=(gn_{\\rm av}\/m)^{1\/2}$.\n}\n \\end{figure}\n\\begin{figure}[t]\n \\includegraphics[scale=0.48]{effectivemass}\n\\caption{\\label{fig:mass}\nEffective mass $m^{\\ast}(K)$ for $V_0=1 gn_{\\rm av}\\xi_{\\rm av}$ and $a=5 \\xi_{\\rm av}$.}\n \\end{figure}\nAssuming that the condensate sits in the first Bloch band, one obtains the solution of Eq. \\!(\\ref{eq:ampl}) in the region $(j-1\/2)a70\\%$) of main sequence objects in the Galaxy and in the\nvicinity of the Sun \\citep{Reid2002,Covey2008,Bochanski2010}. \n\nHowever, even nearby M dwarfs are generaly faint at the visible\nwavelengths where most planet searches are conducted, and most\nexoplanet detection techniques \\--- with the notable exception of\nmicro-lensing \\citep{Dong_etal.2009} \\--- are currently restricted\nto relatively bright stars. This significantly limits the number of M\ndwarfs that can be targeted in exoplanet surveys. Doppler searches in\nparticular are usually restricted to stars with visual magnitudes\n$V<12$, and less than $\\sim$10\\% of late-K and early-M stars within\n30~pc are currently being monitored by the large-scale Doppler surveys\n\\citep{Butler2006,Mayor2009}. However, new surveys are pushing this\nlimit to fainter magnitudes \\citep{Apps2009}, and high-resolution\nspectrographs suitable for Doppler observations at near-infrared\nwavelengths, where M dwarfs are relatively brighter, are being\ndeveloped \\citep{Terada2008,Bean2010,Quirrenbach2010,Wang2010}. In any\ncase, only a fraction of all catalogued, nearby M dwarfs are bright\nenough to be included in radial-velocity monitoring programs. \n\nTransit surveys, on the other hand, can include much fainter\nstars \\citep{Irwin2009}. However, because they have a much lower\ndetection efficiency due to orbital inclination constraints, they\nrequire extensive lists (thousands) of targets in order to detect any\nsignificant number of transit events. For transit surveys, the Solar\nNeighborhood census and its estimated $\\approx$5,000 M dwarfs is\ntherefore too small, and transit programs would greatly benefit from\nextending their target lists to much larger distance limits.\n\nA fundamental obstacle to progress has been the lack of a large,\ncomplete, and uniform catalog of bright M dwarfs suitable as targets\nfor exoplanet programs. In particular, most catalogs and surveys of M\ndwarfs have focused on identifying the nearest objects, which are not\nnecessarily the brightest. Whereas the Hipparcos catalog\n\\citep{vanLeeuwen2007} provides a near-complete census of solar-mass\nstars within 100 parsecs of the Sun, the bright magnitude limit of the\ncatalog excludes all but the very nearest M dwarfs \\--- although it\nlists stars as faint as $V=12-13$, the Hipparcos catalog is complete\nonly to $V<8$.\n\nThe widely utilized {\\it Third Catalog of Nearby Stars}, or CNS3\n\\citep{gj1991}, which lists $\\approx$3,800 stars, though predating the\nHipparcos survey, has historically provided a more complete list of\nM dwarf candidates in the Solar Neighborhood. Many of the fainter\nstars in the CNS3 have (ground-based) parallax measurements from a\nvariety of sources \\citep{VLH95}. However, the CNS3 was largely\ncompiled based on a photometric analysis the high proper motion stars\ncatalogued by \\citet{lhs,nltt}, and in large part using photometric\ndata collected by \\citet{Gliese_1982} and\n\\citet{Weis1984,Weis1986,Weis1987}. The CNS3 has been largely used in\nrecent years to select M dwarf targets for exoplanet surveys\n\\citep{Marcy2001,Naef2003,Butler2004,Rivera2005,Endl2008,Bailey2009,Anglada2012a,Anglada2012b}. Unfortunately,\nthe catalog suffers from various sources of incompleteness. These\nmainly consist of: (1) limited availability of quality data for stars\nin the Luyten catalogs, at the time the CNS3 was compiled, (2)\nkinematic bias in the Luyten catalog due a relatively high\n($\\mu>0.18\\arcsec$ yr$^{-1}$) proper motion limit at the low end, and\n(3) incompleteness of the Luyten catalogs even for stars with proper\nmotions above the fiducial limit.\n\nMotivated mainly by the need to complete the census of the Solar\nNeighborhood, several surveys have since been conducted to identify\nthe low-mass stars (mostly M dwarfs) suspected to be missing from the\nCNS3. These have included a re-analysis of the proper motion catalogs\nof \\citet{nltt,lhs} in light of high quality photometric data\nprovided by the 2MASS survey \\citep{Cutri.etal.2003}. This has led to\nthe identification of hundreds of additional nearby star candidates\nthat had previously been overlooked\n\\citep{Reid_Cruz.2002,Reidetal.2004}. In addition, cross-matching\nagainst 2MASS and examination of Digitized Sky Survey images has\nuncovered significant ($>1\\arcmin$) errors in many of the coordinates\nquoted in the Luyten catalogs, which was hitherto preventing efficient\nfollow-up studies \\citet{Bakos.2002,SalimGould.2003}. \n\nParallelling these efforts, new proper motion surveys have been\nconducted, mainly to find the high proper motion stars missing from\nthe Luyten catalogs with a focus on completing the stellar census of\nthe Solar Neighborhood\n\\citep{Lepine2002,Lepine2003b,Deacon2005,Levine2005,Lepine2005,Subasavage2005a,Subasavage2005b,Lepine2008}. In\naddition, some surveys have also been reaching to lower proper motion\nlimits \\citep{Lepine2005,Reid2007,Boyd2011}, potentially extending the\ncensus of M dwarfs to larger distances. Recently, we have analyzed\ndata from theSUPERBLINK proper motion survey, which has a proper\nmotion limit $\\mu>0.04\\arcsec$ yr$^{-1}$, with an emphasis on the\nidentification of {\\em bright} M dwarfs, rather than just {\\em nearby}\nones; our search has turned up 8,889 candidate M dwarfs with infrared\nmagnitude $J<10$ \\citep{LepineGaidos.2011}. Of these, we found that\nonly 982 were previously listed in the Hipparcos catalog, and another\n898 in the CNS3. Most of the other 7009 stars were not commonly known\nobjects, and were identified as probable nearby M dwarfs for the first\ntime. With its high estimated completeness, especially in the northern\nsky, the \\citet{LepineGaidos.2011} census provides a solid basis for\nassembling an extensive and highly complete catalog of bright M\ndwarfs, suitable for exoplanet search programs.\n\n\nNot all M dwarfs, however, are equally suitable targets for planet\nsearches. Some M dwarfs have significant photometric variability\n(flares, spots) which are affecting transit searches\n\\citep{Hartman2011}; some display chromospheric emission affecting\nDoppler searches \\citep{Isaacson2010}. Because M dwarfs are relatively\nfaint stars, they often require considerable investement of observing\ntime on large telescopes to achieve exoplanet detection, and there is\nvalue in identifying subsets of M dwarfs that are intrinsically more\nlikely to host detectable planets \\citet{Herrero2011}. In\nparticular, one might be interested in selecting stars of higher\nmetallicity which may harbor more massive planets\n\\citep{Sousa_etal.2010}, or young stars with relatively luminous\nmassive planets which would be easier to detect through direct imaging\n\\citep{Mugrauer_etal.2010}. In addition, one would like to avoid\npossible contaminants (e.g. background giants) or problematic systems\n(e.g. very active stars) in order to optimize exoplanet survey\nefficiencies. \n\nDetermining physical properties of the M dwarfs is also important in\norder to better characterize the local populations of low-mass\nstars. This is especially true since proximity makes them brighter and\nthus more efficient targets for follow-up observations and detailed\nstudy. Some of the bright M dwarfs may be close enough\n(d$\\lesssim$20pc) to warrant inclusion in the parallax programs\ndevoted to completing the census of low-mass stars in the Solar\nNeighborhood \\citep{Henry_etal.2006}, in which case it is also\nimportant that the candidates first be vetted through spectral\ntyping. \n\n\\begin{deluxetable*}{llrrrrrrrrrcrrr}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{15} \n\\tablewidth{0pt} \n\\tablecaption{Survey stars: positions and photometry. \\label{table_photo}}\n\\tablehead{\n\\colhead{Star name} & \n\\colhead{CNS3\\tablenotemark{a}} & \n\\colhead{R.A.(ICRS)} &\n\\colhead{Decl.(ICRS)} &\n\\colhead{$\\mu_{R.A.}$} &\n\\colhead{$\\mu_{Decl.}$} &\n\\colhead{Xray\\tablenotemark{b}} &\n\\colhead{hr1\\tablenotemark{b}} &\n\\colhead{FUV\\tablenotemark{c}} &\n\\colhead{NUV\\tablenotemark{c}} &\n\\colhead{V} &\n\\colhead{V} &\n\\colhead{J\\tablenotemark{d}} &\n\\colhead{H\\tablenotemark{d}} &\n\\colhead{K$_s$\\tablenotemark{d}} \\\\\n\\colhead{} &\n\\colhead{} &\n\\colhead{(ICRS)} &\n\\colhead{(ICRS)} &\n\\colhead{$\\arcsec$ yr$^{-1}$} &\n\\colhead{$\\arcsec$ yr$^{-1}$} &\n\\colhead{cnts\/s} &\n\\colhead{} &\n\\colhead{mag} &\n\\colhead{mag} &\n\\colhead{mag} &\n\\colhead{flag} &\n\\colhead{mag} &\n\\colhead{mag} &\n\\colhead{mag} \n}\n\\startdata \nPM I00006+1829 & & 0.163528 & 18.488850 & 0.335 & 0.195& & & & 20.04& 11.28&T& 8.44& 7.79& 7.64\\\\\nPM I00012+1358S & & 0.303578 & 13.972055 & 0.025 & 0.144& & & & 19.85& 11.12&T& 8.36& 7.71& 7.53\\\\\nPM I00033+0441 & & 0.829182 & 4.686940 &-0.024 &-0.085& & & & 21.18& 12.04&T& 8.83& 8.18& 7.98\\\\\nPM I00051+4547 & Gl 2 & 1.295195 & 45.786587 & 0.870 &-0.151& & & & & 9.95&T& 6.70& 6.10& 5.85\\\\\nPM I00051+7406 & & 1.275512 & 74.105217 & 0.035 &-0.023& & & & & 10.63&T& 7.75& 7.15& 6.97\\\\\nPM I00077+6022 & & 1.927582 & 60.381760 & 0.340 &-0.027& 0.1700& -0.41& & & 14.26&P& 8.91& 8.33& 8.05\\\\\nPM I00078+6736 & & 1.961682 & 67.607124 &-0.045 &-0.091& & & & & 12.18&P& 8.35& 7.72& 7.51\\\\\nPM I00081+4757 & & 2.026727 & 47.950695 &-0.119 & 0.003& 0.2190& -0.27& 19.68& 18.91& 12.70&P& 8.52& 8.00& 7.68\\\\\nPM I00084+1725 & GJ 3008 & 2.113679 & 17.424309 &-0.093 &-0.064& & & & 19.24& 10.73&T& 7.81& 7.16& 6.98\\\\\nPM I00088+2050 & GJ 3010 & 2.224675 & 20.840403 &-0.065 &-0.247& 0.0899& -0.28& 21.07& 16.71& 13.90&P& 8.87& 8.26& 8.01\\\\\nPM I00110+0512 & & 2.769255 & 5.208822 & 0.241 & 0.061& & & 22.85& 20.58& 11.55&T& 8.53& 7.88& 7.69\\\\\nPM I00113+5837 & & 2.841032 & 58.617561 & 0.056 & 0.029& & & & & 11.21&T& 8.02& 7.31& 7.13\\\\\nPM I00118+2259 & & 2.970996 & 22.984573 & 0.142 &-0.221& 0.4110& 0.28& & 22.37& 13.09&P& 8.86& 8.31& 7.99\\\\\nPM I00125+2142En& & 3.139604 & 21.713478 & 0.189 &-0.290& & & & & 11.67&T& 8.84& 8.28& 8.04\\\\\nPM I00131+7023 & & 3.298130 & 70.398003 & 0.045 & 0.139& & & & 19.93& 11.37&T& 8.26& 7.59& 7.39\n\\enddata \n\\tablenotetext{a}{Designation in the Third Catalog of Nearby Stars \\citep{gj1991}}\n\\tablenotetext{b}{X-ray flux and hardness ratio from the ROSAT all-sky\n points source catalog \\citep{Voges.etal.1999,Voges.etal.2000}.}\n\\tablenotetext{c}{Far-UV and near-UV magnitudes in the GALEX fifth data release.}\n\\tablenotetext{d}{Infrared magnitudes from the Two Micron All-Sky\n Survey \\citep{Cutri.etal.2003}.}\n\\end{deluxetable*} \n\n\nSpectral classification and analysis for a significant fraction of the\nlow-mass stars in the CNS3 was performed as part of the Palomar-MSU\nspectroscopic survey \\citep{Reid1995,Hawley1996}, hereafter PMSU. The\nsurvey has notably provided formal spectral classification for 1,971\nof the fainter CNS3 stars, confirming 1,648 of them to be nearby M\ndwarfs. More recent spectroscopic follow-up surveys have mainly\nfocused on candidate nearby stars missing from the CNS3. Very high\nproper motion stars from the Luyten catalogs\n\\citep{GizisReid.1997,ReidGizis.2005}, or stars discovered in the more\nrecent proper motion surveys\n\\citet{Scholz2002,Lepine2003,Scholz2005,Reyle2006} have thus been\ntargeted. Most notably, the ``Meet the Cool Neighbours'' program\n(hereafter MCN) has determined spectral subtypes for several hundred M\ndwarfs identified from the Luyten catalogs but not listed in the CNS3\n\\citep{CruzReid.2002,Cruzetal.2003,Reidetal.2003,Reidetal.2004,Cruzetal.2007,Reid2007}. As\nwith the other more recent surveys, the MCN program placed an emphasis\non the identification and classification of nearby, very-cool M\ndwarfs, most of which are however relatively faint and unsuitable for\nexoplanet surveys. It should be noted that while large numbers of M\ndwarfs have also been identified and classified as part of the\nspectroscopic follow-up program of the Sloan Digital Sky Survey\n\\citep{Bochanski2005,Bochanski2010,West2011}, most of them are\nrelatively distant sources, and generally too faint for exoplanet\nsurveys.\n\nThis is why a significant fraction of the bright M dwarf candidates\npublished in \\citet{LepineGaidos.2011} had no available spectroscopic\ndata at the time of release. In order to assemble a comprehensive\ndatabase of M dwarfs targets suitable for exoplanet survey programs,\nwe are now conducting a spectroscopic follow-up survey of the\nbrightest M dwarf candidates from \\citep{LepineGaidos.2011}. Our goal\nis to provide a uniform catalog of spectroscopic measurements to\nconfirm the M dwarf classification, and initiate detailed studies of\ntheir physical properties, as well as the tayloring of exoplanet\nsearches. In this paper, we present the first results of our survey,\nwhich provides data for the 1,564 brightest M dwarf candidates north\nof the celestial equator, with apparent near-infrared magnitudes\nJ$<$9. Observations are described in \\S2. Our spectral classification\ntechniques are described in \\S3, and our model fitting and effective\ntemperature determinations are given in \\S4. Metallicity measurements\nare presented in \\S5. Activity diagnostics are presented and analyzed\nin \\S6. A kinematic study informed by our metallicity and activity\nmeasurements is presented in \\S7, followed by discussion and\nconclusions in \\S8.\n\n\\section{Spectroscopic observations}\n\n\\subsection{Target selection}\n\nTargets for the follow-up spectroscopic program were selected from the\ncatalog of 8,889 bright M dwarfs of \\citet{LepineGaidos.2011}. All\nstars are selected from the SUPERBLINK catalog of stars with proper\nmotions $\\mu>40$ mas yr$^{-1}$. Stars are identified as probable M\ndwarfs based on various color and reduced proper motions cuts; all\nselected candidates have, e.g., optical-to-infrared colors\n$V-J>2.7$. The low proper motion limit of the SUPERBLINK catalog\nexcludes nearly all background red giants. The low proper motion\nlimit however also results in a kinematic bias, whose effects are\ndiscussed in \\S2.3 below.\n\nWhile some astrometric and photometric data have already been compiled\nfor all the stars, most lack formal spectral classification. Spectral\nsubtypes have been estimated in \\citet{LepineGaidos.2011} only based on a\ncalibrated relationship between M subtype and $V-J$ color. However,\nthe $V$ magnitudes of many SUPERBLINK catalog stars are based on\nphotographic measurements (from POSS-II plates); the resulting $V-J$\ncolors have relatively low accuracy and are sometimes\nunreliable. Besides from affecting spectral type estimates, unreliable\ncolors can cause contamination of our sample of M dwarf candidates by\nbluer G and K dwarfs, which would have otherwise failed the color\ncut. These are strong arguments for performing systematic spectroscopic\nfollow-up observations, to provide reliable spectral typing and filter\nout G\/K dwarfs (or any remaining M giant contaminants).\n\nA subsample of the brightest of the M dwarfs candidates, with apparent\ninfrared magnitude J$<9$ was assembled for the first phase of this\nsurvey. We also restricted the sample to stars north of the celestial\nequator. This initial list contains a total of 1,564 candidates. All\nstars were indiscriminately targeted for follow-up observations,\nwhether or not they already had well-documented spectra. This would\nensure completeness and uniformity, and allows comparison of our\nsample with previous surveys. In particular, our target list\nincludes M dwarf classification standards from \\citet{KHM91} which\nprovide a solid reference for our spectral classification. The list\nincludes 557 stars that were previously observed in the PMSU\nspectroscopic survey, and 161 that were obseved and classified as part\nof the MCN spectroscopic program (including 82 stars observed in both\nthe PMSU and MCN).\n\nOf the 1,564 M dwarf candidates, we found that 286 had been observed\nat the MDM observatory by one of us (SL) prior to November 2008, as\npart of a separate spectroscopic follow-up survey of very nearby\n(d$<$20pc) stars \\citep{Alpert2011}. The remaining targets were\ndistributed between our observing teams at the MDM Observatory \n(hereafter MDM) and University of Hawaii 2.2-meter Telescope\n(hereafter UH), with the MDM team in charge of higher declination\ntargets ($\\delta>30$) and the UH team in charge of the lower\ndeclination range ($0<\\delta<30$). In the end we obtained spectra for\nall 1,564 stars from the initial target list.\n\nTo check for any possible systematic differences arising from using\ndifferent telescopes and instruments, we observed 146 stars at both\nMDM and UH. We call this subset the ``inter-observatory\nsubset''. Observations were obtained at different times at the two\nobservatories. Data were processed in the same manner as the rest of\nthe sample. \n\n\\begin{figure}\n\\vspace{-0.4cm}\n\\hspace{-0.8cm}\n\\includegraphics[scale=0.47]{f1.eps}\n\\caption{Examples of spectra collected at the MDM Obsevatory with\n either the MkIII or CCDS low-resolution spectrographs, and using the\n McGraw-Hill 1.3-meter or Hiltner 2.4-meter telescope. All spectra \n covered the 6,200\\AA-8,700\\AA\\ wavelength range with a resolution of\n 1.8\\AA-2.4\\AA\\ per pixel, and a signal-to-noise ratio\n 20$<$S\/N$<$30.\\label{spec_mdm}}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{-0.4cm}\n\\hspace{-0.8cm}\n\\includegraphics[scale=0.47]{f2.eps}\n\\caption{Examples of spectra collected at the University of Hawaii\n 2.2-meter telescope with the SNIFs spectrograph. Observations\n covered the 5,200\\AA-9,800\\AA\\ wavelength regime with a spectral\n resolution R$\\simeq$2000, and a signal-to-noise ratio\n 40$<$S\/N$<$50.\\label{spec_uh}}\n\\end{figure}\n\n\nThe full list of observed stars is presented in\nTable~\\ref{table_photo}. We used the standard SUPERBLINK catalog name\nas primary designation, however we also include the more widely used\ndesignations (GJ, Gl, and Wo numbers) for the 682 stars listed in the\nCNS3 \\citep{gj1991}. The CNS3 stars are often well-studied objects,\nwith abundant data from the literature; a majority of them (580) were\nclassified as part of the PSMU survey or MCN spectroscopic\nprogram. Another 56 stars on our table which are not in the CNS3 were\nhowever classified as part of the PMSU survey or the MCN program. The\nremaining 821 stars are not in the CNS3, and were also not classified\nas part of the PMSU survey or MCN progra; these are new\nidentifications for the most part, and little data existed about them\nuntil now.\n\nTable~\\ref{table_photo} lists coordinates and proper motion vectors\nfor all the stars, along with astrometric parallaxes whenever\navailable from the literature. The table also lists X-ray source\ncounts from the ROSAT all-sky point source catalogs\n\\citep{Voges.etal.1999,Voges.etal.2000}, far-UV ($FUV$) and near-UV\n($NUV$) magnitudes from the GALEX fifth data release, optical $V$\nmagnitude from the SUPERBLINK catalog \\citep{LepineShara.2005}, and\ninfrared $J$, $H$, and $K_s$ magnitudes from 2MASS\n\\citep{Cutri.etal.2003}. More details on how those data were compiled\ncan be found in \\citet{LepineGaidos.2011}. The optical ($V$ band)\nmagnitudes listed in Table~\\ref{table_photo} come from two sources\nwith different levels of accuracy and reliability. For 919 stars in\nTable~\\ref{table_photo}, generally the brightest ones, the $V$\nmagnitudes come from the Hipparcos and Tycho-2 catalogs. These are\ngenerally more reliable with typical errors smaller than $\\pm$0.1\nmagnitude; those stars are flagged ``T'' in\nTable~\\ref{table_photo}. The other 645 objects have their $V$\nmagnitudes estimated from POSS-I and\/or POSS-II photographic\nmagnitudes as prescribed in \\citet{Lepine2005}. Photographic\nmagnitudes of relatively bright stars often suffer from large errors\nat the $\\sim$0.5 magnitude level or more, in part due to photographic\nsaturation; those stars are labeled ``P'' in Table~\\ref{table_photo}.\n \n\\begin{deluxetable}{ccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{6} \n\\tablewidth{0pt} \n\\tablecaption{Definition of Spectral Indices.\\label{tab:si}}\n\\tablehead{\n\\colhead{Index} & \n\\colhead{Numerator} &\n\\colhead{Denominator} &\n\\colhead{Reference}\n}\n\\startdata \nCaH2 & 6814-6846 & 7042-7046 & \\citet{Reid1995} \\\\\nCaH3 & 6960-6990 & 7042-7046 & \\citet{Reid1995} \\\\\nTiO5 & 7126-7135 & 7042-7046 & \\citet{Reid1995} \\\\\nVO1 & 7430-7470 & 7550-7570 & \\citet{Hawley2002} \\\\\nTiO6 & 7550-7570 & 7745-7765 & \\citep{Lepine2003} \\\\\nVO2 & 7920-7960 & 8130-8150 & \\citep{Lepine2003} \n\\enddata \n\\end{deluxetable} \n\n\\subsection{Observations}\n\nSpectra were collected at the MDM observatory in a series of 22\nobserving runs scheduled between June, 2002 and April, 2012. Most of\nthe spectra were collected at the McGraw-Hill 1.3-meter telescope, but\na number were obtained at the neighboring Hiltner 2.4-meter\ntelescope. Two different spectrographs were used: the MkIII\nspectrograph, and the CCDS spectrograph. Both are facility\ninstruments which provide low- to medium-resolution spectroscopy in\nthe optical regime. Their operation at either 1.3-meter or 2.4-meter\ntelescopes is identical. Data were collected in slit spectroscopy\nmode, with an effective slit width of 1.0\\arcsec to 1.5\\arcsec. The\nMkIII spectrograph was used with two different gratings: the 300 l\/mm\ngrating blazed at 8000\\AA, providing a spectral resolution\nR$\\simeq$2000, and the 600 l\/mm grating blazed at 5800\\AA, which\nprovides R$\\simeq$4000. The two gratings were used with either one of\ntwo thick-chip CCD cameras ({\\it Wilbur} and {\\it Nellie}) both having\nnegligible fringing in the red. Internal flats were used to calibrate\nthe CCD response. Arc lamp spectra of Ne, Ar, and Xe provided\nwavelength calibration, and were obtained for every pointing of the\ntelescope to account for flexure in the system. The spectrophotometric\nstandard stars Feige 110, Feige 66, and Feige 34 \\citep{Oke1991} were\nobserved on a regular basis to provide spectrophotometric\ncalibration. Integration times varied depending on seeing, telescope\naperture, and target brightess, but were typically in the 30 seconds\nto 300 seconds range. Between 25 and 55 stars were observed on a\ntypical night. Spectra for a total of 901 bright M dwarf targets were\ncollected at MDM.\n\nAdditional spectra were obtained with the SuperNova Integral Field\nSpectrograph \\citep[SNIFS; ][]{Lantz2004} on the University of Hawaii\n2.2~m telescope on Mauna Kea between February 2009 and November 2012.\nSNIFS has separate but overlapping blue (3200-5600\\AA) and red\n(5200-10000\\AA) spectrograph channels, along with an imaging channel,\nmounted behind a common shutter. The spectral resolution is\n$\\sim$1000 in the blue channel, and $\\sim$1300 in the red channel; the\nspatial resolution of the 225-lenslet array is 0.4\\arcsec. SNIFS\noperates in a semi-automated mode, acquiring acquisition images to\ncenter the target on the lenslet array, and bias images and\ncalibration lamp spectra before and after each target spectrum. Both\ntwilight and dome flats were also obtained every night. Integration\ntimes depended on $J$ magnitude but were 54~s for the faintest ($J=9$)\ntargets. Up to 75 target spectra were obtained in one night. Spectra\nfor 655 bright M dwarf targets were collected at UH with SNIFS.\n\n\\begin{deluxetable*}{lcrrrrrrrrcrrr}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{14} \n\\tablewidth{0pt} \n\\tablecaption{Survey stars: spectroscopic data \\label{table_spectro}}\n\\tablehead{\n\\colhead{Star name} & \n\\colhead{Observatory} &\n\\colhead{Julian Date} &\n\\colhead{CaH2$_{c}$\\tablenotemark{a}} &\n\\colhead{CaH3$_{c}$} &\n\\colhead{TiO5$_{c}$} &\n\\colhead{VO1$_{c}$} &\n\\colhead{TiO6$_{c}$} &\n\\colhead{VO2$_{c}$} &\n\\colhead{Sp.Ty.\\tablenotemark{b}} &\n\\colhead{Sp.Ty.} &\n\\colhead{$\\zeta$} &\n\\colhead{log g\\tablenotemark{d}} &\n\\colhead{$T_{eff}$\\tablenotemark{d}} \\\\\n\\colhead{} &\n\\colhead{} &\n\\colhead{2,450,000+} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{index} &\n\\colhead{adopted} &\n\\colhead{} &\n\\colhead{} &\n\\colhead{K} \n}\n\\startdata \nPM I00006+1829 & MDM & 4791.75 &\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata& G\/K&\\nodata&\\nodata&\\nodata\\\\\nPM I00012+1358S & UH22 & 5791.05 & 0.706 & 0.864 & 0.788 & 0.967 & 0.944 & 0.970 & 0.14 & M0.0& 1.08 & 5.0 & 3790 \\\\\nPM I00033+0441 & UH22 & 5791.05 & 0.580 & 0.797 & 0.679 & 0.959 & 0.888 & 0.939 & 1.38 & M1.5& 0.93 & 4.5 & 3510 \\\\\nPM I00051+4547 & MDM & 5095.80 & 0.603 & 0.824 & 0.664 & 0.956 & 0.911 & 1.014 & 1.10 & M1.0& 1.10 & 4.5 & 3560 \\\\\nPM I00051+7406 & MDM & 5812.87 &\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata& G\/K&\\nodata&\\nodata&\\nodata\\\\\nPM I00077+6022 & MDM & 5838.82 & 0.364 & 0.620 & 0.392 & 0.905 & 0.630 & 0.798 & 4.60 & M4.5& 0.90 & 5.0 & 3140 \\\\\nPM I00078+6736 & MDM & 5099.83 & 0.534 & 0.789 & 0.623 & 0.905 & 0.806 & 0.929 & 2.03 & M2.0& 0.96 & 5.0 & 3500 \\\\\nPM I00081+4757 & MDM & 5098.84 & 0.410 & 0.700 & 0.420 & 0.871 & 0.684 & 0.814 & 3.80 & M4.0& 1.01 & 5.0 & 3280 \\\\\nPM I00084+1725 & UH22 & 5791.06 & 0.671 & 0.842 & 0.785 & 0.970 & 0.947 & 0.970 & 0.34 & M0.5& 0.91 & 4.5 & 3600 \\\\\nPM I00088+2050 & MDM & 4413.72 & 0.372 & 0.646 & 0.356 & 0.893 & 0.602 & 0.793 & 4.58 & M4.5& 0.99 & 5.0 & 3130 \\\\\nPM I00110+0512 & UH22 & 5050.03 & 0.653 & 0.839 & 0.706 & 0.962 & 0.893 & 0.925 & 0.86 & M1.0& 1.16 & 4.5 & 3660 \\\\\nPM I00113+5837 & MDM & 5811.90 &\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata&\\nodata& G\/K&\\nodata&\\nodata&\\nodata\\\\\nPM I00118+2259 & UH22 & 5050.03 & 0.439 & 0.729 & 0.424 & 0.923 & 0.694 & 0.798 & 3.50 & M3.5& 1.10 & 4.5 & 3260 \\\\\nPM I00125+2142En& UH22 & 5791.06 & 0.721 & 0.865 & 0.851 & 0.973 & 0.967 & 0.988 & -0.18 & M0.0& 0.80 & 4.5 & 3690 \\\\\nPM I00131+7023 & MDM & 5812.89 & 0.643 & 0.816 & 0.754 & 0.973 & 0.883 & 1.038 & 0.93 & M1.0& 0.88 & 4.5 & 3570 \n\\enddata\n\\tablenotetext{a}{All spectral indices are corrected for instrumental effects, see Section 3.2.} \n\\tablenotetext{b}{Numerical spectral subtype M evaluated from the corrected spectral band indices (not-rounded).}\n\\tablenotetext{c}{H$\\alpha$ spectral index, comparable to equivalent width.}\n\\tablenotetext{d}{Gravity and effective temperature from PHOENIX model fits.}\n\\end{deluxetable*} \n\nSpectroscopic data and results are summarized in\nTable~\\ref{table_spectro}. SUPERBLINK names are repeated in the first\ncolumn and provide a means to match with the entries in\nTable~\\ref{table_photo}. The second and third columns list the\nobservatory and Julian date of the observations. The various\nspectroscopic measurements whose values are listed in the remaining\ncolumns are described in detail in the sections below.\n\n\\subsection{Reduction}\n\n\\begin{figure}\n\\epsscale{1.15}\n\\plotone{f3.eps}\n\\caption{Histogram of the $V-J$ color distribution of survey stars\n with spectral morphologies inconsistent with red dwarf of subtype K5\n and later (in green). The distribution of $V-J$ colors from the full\n sample is shown in red. Stars with spectra inconsistent with red\n dwarfs are close to the blue edge of the survey, which indicates\n that they are most probably contaminants of the color-magnitude\n selection, and not the result of mis-acquisition at the\n telescope. \\label{contaminant}}\n\\end{figure}\n\n\\subsubsection{MDM data}\n\nSpectral reduction of the MDM spectra was performed using the CCDPROC\nand DOSLIT packages in IRAF. Reduction included bias and flatfield\ncorrection, removal of the sky background, aperture extraction, and\nwavelength calibration. The spectra were also extinction-corrected and\nflux-calibrated based on the measurements obtained from the\nspectrophotometric standards. We did not attempt to remove telluric\nabsorption lines from the spectra. Many spectra were\ncollected on humid nights or with light cirrus cover, which resulted\nin variable telluric features. However, telluric features generally do\nnot affect standard spectral classification or the measurement of\nspectral band indices, since all the spectral indices and primary\nclassification features avoid regions with telluric absorption. \n\nA more common problem at the MDM observatory was slit loss\nfrom atmospheric differential refraction. Although this problem could\nhave been avoided by the use of a wider slit, the concomitant loss of\nspectral resolution was deemed more detrimental to our science\ngoals. Instead, stars were observed as close to the meridian as\nobservational constraints allowed. In some cases, however, stars were\nobserved up to $\\pm$2 hours from the meridian, resulting in noticeable\nslit losses. Fluctuations in the seeing, which often exceeded the slit\nwidth, played a role as well. As a result, the spectrophotometric\ncalibration was not always reliable, since the standards were only\nobserved once per night to maximize survey efficiency. Flux\nrecalibration was\ntherefore performed using the following procedure. Spectral indices\nwere measured and the spectra were classified using the classification\nmethod outlined below (\\S 3.1). The spectra were then compared to\nclassification templates assembled from M dwarf spectra from the Sloan\nDigital Sky Survey (SDSS) spectroscopic database. Each spectrum\nwas divided by the classification template of the same alleged\nspectral subtype. In many cases, the quotients yielded a flat\nfunction, indicating that the spectrophotometric calibration was\nacceptable. Other quotients yielded residuals consistent with first or\nsecond order polynomials spanning the entire wavelength range, and\nindicating problems in the spectrophotometric calibration. A\nsecond-order polynomial was fit through the quotient spectra, and the\noriginal spectrum was normalized by this fit, correcting for\ncalibration errors. Spectral indices were then measured again, and the\nspectra reclassified; this yielded changes by 0.5 to 1.0 subtypes for\n$\\approx20\\%$ of the stars (no changes for the rest). The\nre-normalization was then performed again using the revised spectral\nsubtypes, and the procedure repeated until convergence for all stars.\n\nFinally, all the spectra were wavelength-shifted to the rest frames\nof their emitting stars. This was done by cross-correlating each\nspectrum with the SDSS template of the corresponding spectral\nsubtype. Spectral indices were again re-measured, and the stars\nre-classified. Any change in the spectral subtype prompted a repeat of\nthe flux-recalibration procedure outlined above, and the\ncross-correlation procedure was repeated using the revised spectral\ntemplate until converegence.\n\nA sequence of MDM spectra is displayed in Figure~\\ref{spec_mdm},\nwhich illustrates the wavelength regime and typical quality. Note the\ntelluric absorption features near 6,850\\AA, 7,600\\AA, and\n8,200\\AA. Signal-to-noise ratio is generally in the 30$<$S\/N$<$50\nrange near 7500\\AA.\n\n\\begin{figure*}\n\\vspace{-6.5cm}\n\\hspace{0.3cm}\n\\includegraphics[scale=0.87]{f4.eps}\n\\caption{Comparison between our spectral band index measurements for a\n subset of 484 stars observed at MDM and UH, and the band index\n measurements for the same stars as reported in the Palomar-MSU\n spectroscopic survey of \\citet{Reid1995}. Small but systematic offsets\n are observed, which are explained by differences in spectral\n resolution and spectrophotometric calibration between the different\n observatories. These offsets demonstrate that the spectral indices\n are instrument-dependent, but that the measurements can be corrected\n by observing large subsets of stars at the different\n obervatories. The red segments show the fits to the offsets, which\n are used to calibrate corrections for each observatory, here using\n the Palomar-MSU measurements as the standard.\\label{pmscomp}}\n\\end{figure*}\n\n\\subsubsection{SNIFS data}\n\nSNIFS data processing was performed with the SNIFS data reduction\npipeline, which is described in detail in \\citet{Bacon:2001} and\n\\citet{Aldering:2006}. After standard CCD preprocessing (dark, bias,\nand flat-field corrections), data were assembled into red and blue 3D\ndata cubes. The data cubes were cleaned for cosmic rays and bad\npixels, wavelength-calibrated using arc lamp exposures acquired\nimmediately after the science exposures, and spectrospatially\nflat-fielded, using continuum lamp exposures obtained during the same\nnight. The data cubes were then sky-subtracted, and the 1D spectra were\nextracted using a semi-analytic PSF model. We applied corrections to\nthe 1D spectra for instrument response, airmass, and telluric lines\nbased on observations of the Feige 66, Feige 110, BD+284211, or\nBD+174708 standard stars \\citep{Oke1991}. Because the SNIFS spectra\nare from an integral field spectrograph, operating without a slit,\ntheir spectrophotometry is significantly more reliable than the\nslit-spectra obtained at MDM. In fact, it is possible to perform\nsynthetic photometry by convolving with the proper filter response.\n\nAs with the MDM data, SNIFS spectra were shifted to the rest frames of\ntheir emitting stars by cross-correlation to SDSS templates\n\\citep{Bochanski2007} of the corresponding spectral subtype. Spectral\nindices were re-measured and the stars re-classified. This process was\nrepeated if there was a change in the spectral subtype determination.\n\nA sequence of UH SNIFS spectra are displayed in Figure~\\ref{spec_uh},\nwhich show the wavelength range and typical data\nquality. Signal-to-noise ratio is generally S\/N$\\approx$100 near\n7500\\AA.\n\n\\section{Spectral classification}\n\n\\subsection{Visual identification of contaminants}\n\nWe first examined all the spectra by eye to confirm the presence of\nmorphological features consistent with red dwarfs of spectral subtype\nK5 and later. The main diagnostic was the detection of broad CaH and\nTiO moleculars band near 7000\\AA. Of the 1564 stars observed, 1408 were\nfound to have clear evidence of CaH and TiO. The remaining 156 stars\ndid not show those molecular features clearly, within the noise\nlimit, and were therefore identified as probable contaminants in the\ntarget selection algorithm.\n\nMost of these contaminants appeared to be early to mid-type K dwarfs,\nwith a few looking like G dwarfs affected by interstellar reddening. A\nnumber of stars also displayed carbon features consistent with low gravity\nobjects, most probably K giants. We suspect that many of the G and\nK dwarfs have inaccurate V-band magnitudes, making them appear redder\nthan they really are, which would explain their inclusion in our\ncolor-selected sample. Interstellar reddening would also explain the\ninclusion of more distant G dwarfs in our sample, due to their redder\ncolors. An alternate explanation, however, might be that the targets\nwere mis-acquired in the course of the survey, and that the spectra\nrepresent random field stars. Indeed the very large proper motion of\nthe sources sometimes makes them difficult to identify at the\ntelescope, as they often have moved significantly from their positions\non finder charts. Stars in crowded field are particularly susceptible\nto this effect. To verify this hypothesis, we compared the $V-J$ color\ndistribution of the contaminants to the distribution of the full\nsurvey sample (Figure~\\ref{contaminant}, top panel); the fraction of\ncontaminant stars in each color bin is also shown (bottom panel). The\ntwo distributions are significantly different, with the contaminants\nbeing dominated by relatively blue stars, and their fraction quickly\ndrops as $V-J$ increases. We can only conclude that the contaminants\nare not mis-acquired stars, otherwise one would expect the two\ndistributions to be statistically equivalent. Rather, the majority of\nthe contaminants must have been properly acquired and are simply\nmoderately red FGK stars that slipped into the sample in the\nphotometric\/proper motion selection, as suggested first.\n\n\\begin{figure*}\n\\vspace{-6.5cm}\n\\hspace{0.3cm}\n\\includegraphics[scale=0.87]{f5.eps}\n\\caption{Comparison between the {\\em corrected} spectral band index\n measurements from MDM and the band index measurements of the same\n stars observed from UH. Observatory-specific corrections to the\n CaH2, CaH3, and TiO5 indices (see Figure~\\ref{pmscomp}) bring the\n MDM and UH values in close agreement. Small offsets are however\n observed for the TiO6, VO1, and VO2 indices, which we could not\n calibrate against Palomar-MSU survey stars. The horizontal red lines\n show the adopted offsets which are used to correct the MDM values to\n bring them in line with the UH ones. Offsets are again believed to\n be due to inter-observatory differences in the spectral resolution\n and spectrophotometric calibration.\\label{idx_comp}}\n\\end{figure*}\n\nWe find an overall contamination rate of $\\simeq$10\\% in our survey,\nalthough most of the contamination occurs among stars with relatively\nblue colors. The contamination rate is $\\simeq$26\\% for red dwarf\ncandidates in the $2.73.3$$ candidates. The 156 stars\nidentified as contaminants are included in Table~\\ref{table_photo} and\nTable~\\ref{table_spectro} for completeness and future\nverification. Spectroscopic measurements for these stars, such as band\nindices, subtypes, and effective temperatures, are however left blank.\n\n\\subsection{Classification by spectral band indices}\n\n\\subsubsection{M dwarf classification from molecular bandstrengths}\n\nThe spectra of M dwarfs are dominated by molecular bands from metal\noxides (mainly TiO, VO), metal hydrides (CaH, CrH, FeH), and metal\nhydroxides (CaOH). The most prominent of these in the optical-red\nwavelength range (5000\\AA-9,000\\AA) are the bands from titanium oxide\n(TiO) and calcium hydride (CaH). The resulting opacities from those\nbroad moleciular bands significantly affect the broadband colors and\nspectral energy distribution of M dwarfs\n\\citep{Jones1968,Allard2000,Kraw2000}. Early atmospheric models of M\ndwarfs showed that the strength of the TiO and CaH bands depends on\neffective temperature, but also on surface gravity and metal\nabundances \\citep{Mould.1976}. \n\nMolecular bands have historically been the defining diagnostic and\nclassification features of M dwarfs. For stars that have settled on\nthe main sequence, one can assume that the surface gravity is entirely\nconstrained by the mass and chemical composition. Leaving only the\neffective temperature and chemical abundances as general parameters in\nthe classification and\/or spectroscopic modeling. For local disk stars\nof solar metallicity, a classification system representing an\neffective temperature sequence can thus be established based on\nmolecular bandstrentghs.\n\nThe detection and measurement of TiO and CaH molecular bands thus\nforms the basis for the M dwarf classification system\n\\citep{JoyAbt1974}. Molecular bands become detectable starting at\nspectral subtype K5 and K7, the latest subtypes for K dwarfs (there\nare no K6, K8, or K9 subtypes). The increasing strength of the\nmolecular bands then defines a sequence running from M0 to M9. The\nstrength of both TiO and CaH molecular bands reach a maximum around\n$T_{eff}\\simeq$2700K. There is a turnaround in the correlation below\nthis point, and molecular bands become progressively weaker at lower\ntemperatures until they vanish \\citep{CruzReid.2002}. The reversal and\nweakening is thought to be due to the condensation of molecules into\ndust, and their settling below the photosphere \\citep{JonesTsuji1997}.\n\n\\begin{deluxetable*}{lcccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{7} \n\\tablewidth{0pt} \n\\tablecaption{Coefficients of the spectral index corrections, by\n observatory.\\label{tab:sic}}\n\\tablehead{\n\\colhead{{\\it IDX}} & \n\\colhead{$A_{IDX:{\\rm PMSU}}$} &\n\\colhead{$B_{IDX:{\\rm PMSU}}$} &\n\\colhead{$A_{IDX:{\\rm MDM}}$} &\n\\colhead{$B_{IDX:{\\rm MDM}}$} &\n\\colhead{$A_{IDX:{\\rm UH}}$} &\n\\colhead{$B_{IDX:{\\rm UH}}$}\n}\n\\startdata \nCaH2 & 1.00 & 0.00 & 0.95 & 0.011 & 0.92 & 0.004 \\\\ \nCaH3 & 1.00 & 0.00 & 0.90 & 0.070 & 1.00 & -0.028 \\\\\nTiO5 & 1.00 & 0.00 & 1.00 & 0.000 & 1.06 & -0.063 \\\\\nVO1 & \\nodata& \\nodata& 1.00 & 0.040 & 1.00 & 0.000 \\\\\nTiO6 & \\nodata& \\nodata& 1.00 & -0.021 & 1.00 & 0.000 \\\\\nVO2 & \\nodata& \\nodata& 1.00 & 0.005 & 1.00 & 0.000\n\\enddata \n\\end{deluxetable*} \n\nSequences of classification standards were compiled in \\citep{KHM91},\nwhich identified the main molecular bands in the yellow-red spectral\nregime, where TiO and CaH bandheads are most prominent. To better\nquantify the classification system, a number of ``band indices'' were\ndefined by \\citet{Reid1995}, which measure the ratio between on-band and\noff-band flux, for various molecular bandheads. Calibration of these\nband indices against classification standards provide a means to\nobjectively assign subtypes based on spectroscopic\nmeasurements. Originally, these band indices measured the strengths of\nvarious features near $7,000\\AA$, where the most prominent CaH and TiO\nfeatures are found in early-type M dwarfs. These bands, however,\nbecome saturated in late-type M dwarfs, which makes their use\nproblematic in later dwarfs. There is a VO band near $7,000$\\AA,\nlocated just between the main CaH and TiO features, which becomes\nprominent only at later subtypes; and band index measuring this\nfeature was introduced as a primary diagnosis for the so-called\n``ultra-cool'' M dwarfs, and provided a classification scale for\nsubtypes M7-M9 \\citep{KHS95}. Additional band indices associated with\nTiO and VO bands in the 7500\\AA-9000\\AA\\ range, where molecular\nabsorption develops at later subtypes, were introduced as secondary\nclassification features \\citep{Lepine2003}. These form the basis of\ncurrent classification methods based on optical-red spectroscopy.\n\nNearby low-mass stars associated with the local halo population have\nlong been known to show peculiar banstrength ratios, and in particular\nto have weak TiO compared to CaH\n\\citep{Mould.1976,Mould_McElroy.1978}. A system to identify and\nclassify the metal-poor M subdwarfs based on the strength and ratio of\nCaH and TiO was introduced by \\citet{Gizis1997}, and expanded by\n\\citet{Lepine2003b} and \\citet{Lepine2007}, as the spectroscopic\ncensus of M subdwarfs grew larger. In this system, the CaH\nbandstrengths are used as a proxy of $T_eff$ and determine spectral\nsubtypes, while the TiO\/CaH band ratio is used to evaluate\nmetallicity. For that purpose, the $\\zeta_{TiO\/CaH}$ parameter, which\nis a function of the TiO\/CaH band ratio, was introduced by\n\\citet{Lepine2007} as a possible proxy for metallicity, and a\ntentative calibration with [Fe\/H] was presented by \\citet{Woolf2009}.\nOne of the main issues in the current M dwarf classification scheme,\nis that both TiO and CaH bandstrengths are used to determine the\nspectral subtype, whereas TiO is now believed to be quite sensitive to\nmetallicity. This means, e.g., that moderately metal-rich M dwarfs may\nbe assigned later subtypes than Solar-metallicity ones. In addition,\nthe classification of young field M dwarfs may be affected by their\nlower surface gravities, which also tend to increase TiO\nbandstrengths and would thus yield to marginally later subtype\nassignments compared with older stars of the same $T_{eff}$. These\ncaveats must be considered when one uses M dwarf spectral subtypes as\na proxy for surface temperature.\n\n\\begin{figure*}\n\\vspace{-6.5cm}\n\\hspace{0.3cm}\n\\includegraphics[scale=0.87]{f6.eps}\n\\caption{Variation in the corrected spectral indices as a function \n of adopted spectral subtypes. The thin black lines show the\n adopted, revised calibrations for spectral\n classification. Blue circles represent the subset of classification\n standards from \\citet{KHM91} which we observed; the X-axis values of\n those data points are the formal spectral subtypes adopted in\n \\citet{KHM91}, while the Y-axis values are the spectral index\n measurements from our survey. For our own spectral typing scheme, we\n calculate the average of the subtype values for the ${\\rm\n CaH2}_{c}$, ${\\rm CaH3}_{c}$, ${\\rm TiO5}_{c}$, and ${\\rm\n TiO6}_{c}$ indices given by the adopted relationships. The two VO\n indices have relatively weak leverage on early type stars due to the\n shallow slope in the relationships, and are not used for assigning\n spectral subtypes of our (mostly early-type) survey\n stars.\\label{idx_spty}}\n\\end{figure*}\n\n\\subsubsection{Definition and measurement of band indices}\n\nThe strength of the TiO, CaH, and VO molecular bands are measured\nusing spectral band indices. These spectral indices measure the ratio\nbetween the flux in a section of the spectrum affected by molecular\nopacity to the flux in a neighboring section of the spectrum minimally\naffected by molecular opacity. The latter section defines a\npseudo-continuum of sorts, although M dwarf spectra do not have a\ncontinuum in the classical sense, because their spectral energy\ndistribution strongly deviates from that of a blackbody, and is\nessentially shaped by atomic and molecular line opacities.\n\nWe settle on a set of six spectral band indices: CaH2, CaH3, TiO5,\nTiO6, TiO7, VO1, and VO2. These band indixes, which we previously used\nin \\citet{Lepine2003} to classify spectra collected at MDM, measure\nthe strength of the most prominent bands of CaH, TiO, and VO in the\n$6000{\\rm \\AA}<\\lambda<8500{\\rm \\AA}$ regime. The spectral indices\nand are listed in Table~\\ref{tab:si} along with their definition. The\nCaH2, CaH3, and TiO5 indices are the same as those used in the\nPalomar-MSU survey, and were first defined in \\citep{Reid1995}. The TiO6,\nVO1, and VO2 indices were introduced by \\citet{Lepine2003} to better\nclassify late-type M dwarfs, whose CaH2 and TiO5 indices become\nsaturated at cooler tempratuers and are not as effective for accurate\nspectral classification of late-type M dwarfs. Each spectral band\nindex is calculated as the ratio of the flux in the spectral region of\ninterest (numerator) to the flux in the reference region\n(denominator), i.e.: \n\\begin{equation} IDX = \\frac{\\int_{num} I(\\lambda) d\\lambda}{\\int_{denom}\n I(\\lambda) d\\lambda} \n\\end{equation}\nBecause the wavelength range for some indices is relatively narrow\n(especially the denominator for CaH2, CaH3, and TiO5) it is important\nthat the spectra in which they are measured have their wavelengths\ncalibrated in the rest frame of the star, which is why special care\nwas made to correct all spectra for any significant redshift\/blueshift\n(see above). \n\nBecause the measured molecular bandheads are relatively sharp, and\nbecause the spectral indices measuring them are defined over\nrelatively narrow spectral ranges, the index values are potentially\ndependent on the spectroscopic resolution, and may thus depend\non the specific instrumental setup used for the observations. In\naddition, the index values may be affected by systematic errors in the\nspectrophotometric flux calibration, which can also be dependent on\nthe instrument and\/or observatory where the measurements were\nmade. One way to verify these effects is to compare spectral index\nmeasurements of the same stars obtained at different\nobservatories. Because of the significant overlap with the Palomar-MSU\nsurvey, we can use those stars as reference sample, and recalibrate\nthe spectral indices so that they are consistent to those reported in\n\\citet{Reid1995}. \n\nOur census have 557 stars in common with the PMSU spectroscopic\nsurvey; these stars are all identified with a flag (``P'') in the last\ncolumn of Table~\\ref{table_spectro}. We identify 206 stars from the\nlist observed at MDM and 281 stars from the list observed at UH which\nhave spectral index measurements\nreported in the PMSU survey. The differences between our measured\nCaH2, CaH3, and TiO5 and those reported in the PSMU catalog are\nplotted in Figure~\\ref{pmscomp}. Trends and offsets confirm the\nexistence of systematic errors, possibly due to differences in\nresolution and flux calibration. To verify the spectroscopic\nresolution hypothesis, we convolved the MDM spectra with a box kernel\n5-pixel wide; we found that indeed the MDM indices for the smoothed\nspectra had their offsets reduced by 0.01-0.02 units, bringing them\nmore in line with the PMSU indices. We also observe that the MDM\nmeasurements tend to have a larger scatter than the UH ones; we\nsuggest that this may be due to spectrophometric calibration issues\nwith some of the MDM spectra, as discussed in \\S2.3.1.\n\nTo achieve consistency in the measurements obtained at different\nobservatories, we adopt the values from the PMSU survey as a standard\nof reference, and calculate corrections to the measurements from MDM and\nUH by fitting linear relationships to the residuals. A corrections to a\nspectral band index is thus applied following the general function:\n\\begin{equation}\nIDX_{c} = A_{IDX:OBS}\\ IDX_{OBS} + B_{IDX:OBS} \\\\ \n\\end{equation}\nwhere $IDX_{OBS}$ represents the measured value of an index at the\nobservatory $OBS$, and ($A_{IDX:OBS}$,$B_{IDX:OBS}$) are the\ncoefficients of the transformation from the observed value to the\ncorrected one ($IDX_{c}$). Hence the corrected values of the indices\nCaH2, CaH3 and TiO5 for measurements done at MDM are defined as:\n\\begin{eqnarray}\n{\\rm CaH2_{c} = A_{\\rm CaH2:MDM} CaH2_{\\rm MDM} + B_{\\rm CaH2:MDM} } \\\\ \n{\\rm CaH3_{c} = A_{\\rm CaH3:MDM} CaH3_{\\rm MDM} + B_{\\rm CaH3:MDM} } \\\\ \n{\\rm TiO5_{c} = A_{\\rm TiO5:MDM} TiO5_{\\rm MDM} + B_{\\rm TiO5:MDM} } \n\\end{eqnarray}\nThe measurements from the PSMU survey are used as standards for these\nthree indices, and we thus have by definition: $A_{\\rm\n CaH2:PMSU}=A_{\\rm CaH3:PMSU}=A_{\\rm TiO5:PMSU}=1.0$, $B_{\\rm\n CaH2:PMSU}=B_{CaH3:PMSU}=B_{\\rm TiO5:PMSU}=0.0$.\nFor OBS=MDM and OBS=UH, the adopted correction coefficients are listed\nin Table~\\ref{tab:sic}. The corresponding linear relationships are\nshown as red segments in Figure~\\ref{pmscomp}. \n\nTo verify the consistency of the corrected spectral band index values,\nwe compare the corrected values for the stars observed at both MDM and\nUH (the inter-observatory subset). The differences are shown in\nFigure~\\ref{idx_comp}. We find the corrected values CaH2$_{c}$,\nCaH3$_{c}$, and TiO5$_{c}$ to be in good agreement, with no\nsignificant offsets beyond what is expected from measurement\nerrors. The corrected values of all three spectral indices are listed\nin Table~\\ref{table_spectro}.\n\nThe TiO6, VO1, and VO2 spectral index mesurements from MDM and UH are\nalso compared in Figure~\\ref{idx_comp}. Those were not measured in the\nPMSU survey, since the indices were introduced later\n\\citep{Lepine2003}, and thus are displayed here without any\ncorrection. Small but significant offsets between the MDM and UH\nvalues again suggest systematic errors due to differences in spectral\nresolution and flux calibration. This time we adopt the UH\nmeasurements as fiducials, and determine corrections to be applied to\nthe MDM data. The corrections are listed in Table~\\ref{tab:sic} and\nthe corresponding linear relationships are displayed in\nFigure~\\ref{idx_comp} as red segments. The corrected values of the\nthree spectral indices are also listed in Table~\\ref{table_spectro}.\n\nThe scatter between the MDM and UH values, after correction, as well\nas the scatter between the UH\/MDM and PMSU values, provide an\nestimate of the measurment accuracy for these spectral\nindices. Excluding a few outliers, the mean scatter is $\\approx$0.02\nunits (1$\\sigma$) for the CaH2, CaH3, TiO5, TiO6, and VO1 indices, and\n$\\approx$0.04 units for VO2. This assumes that M stars do not show any\nsignificant changes in their spectral morphology over time, and that\nthe spectral indices should thus not be variable.\n\n\\subsubsection{Spectral subtype assignments for K\/M dwarfs}\n\nBecause of the correlation between spectral subtype and the depth of\nthe molecular bands, it is possible to use the values of the spectral\nband indices to estimate spectral subtypes. This only requires a\ncalibration of the relationship between spectral index values and the\nspectral subtypes, in a set of stars which were classified by other\nmeans, e.g., classification standards. The system adopted in this\npaper uses the spectral indices listed in Table~\\ref{tab:si}, and\nfollows the methodology outlined in \\citep{Gizis1997} and\n\\citep{Lepine2003}. Relationships are calibrated for each spectral\nindex, and spectral subtypes are calculated from the mean values\nobtained from all relevant\/available spectral indices. The mean values\nare then be rounded to the nearest half integer, to provide formal\nsubtyping with half-integer resolution. The system is extended to\nlate-K dwarfs as well: an ``M subtype'' with a value $<0.0$ signifies\nthat star is a late-K dwarf: the star is classified as K7 for an index\nvalue $\\approx-1.0$ and as K5 for an index value $\\approx-2.0$ (note:\nthere is no K6 subtype for dwarf stars, and K7 is the subtype\nimmediately preceding M0).\n\nThe original spectral-index classification method for M\ndwarfs\/subdwarfs is based on a relationship between subtype and with\nthe CaH2 index, which measures one of the most prominent band at all\nspectral subtypes, and notably displays the deepest bandhead in\nmetal-poor M subdwarfs \\citet{Gizis1997,Lepine2003}. The original\nrelationship is: $\\left[ SpTy \\right]_{CaH2} = 10.71 - 20.63\n\\ {\\rm CaH2} + 7.91 \\ \\left( {\\rm CaH2} \\right)^2$. To verify this\nrelationship, we estimated spectral subtypes from our corrected\nindices CaH2$_{c}$ for 16 spectroscopic calibration standards from\n\\citet{KHM91}, which were observed as part of our survey, and span a\nrange of spectral subtypes from K7.0 to M6.0. We found small but\nsignificant differences in our estimated spectral subtypes and the\nvalues formally assigned by \\citet{KHM91}; subtypes estimated from\nthe \\citet{Gizis1997} relationship tend to systematically\nunderestimate the standard subtypes by $\\approx0.5$ units for stars\nlater than M3. To improve on the index classification method, we\nperformed a $\\chi^2$ polynomial fit to recalibrate the relationship,\nobtaining:\n\\begin{equation}\n\\left[ \\rm SpTy \\right]_{\\rm CaH2} = 11.50 - 21.71 \\ {\\rm CaH2}_{c} + 7.99\n \\ \\left( {\\rm CaH2} \\right)^2\n\\end{equation}\nwhich does correct for the observed offsets at later types. Using this\nthis relatioship as a starting point, and guided by the formal\nspectral subtype from the classification standards, we performed\nadditional $\\chi^2$ polynomial fits to calibrate an index-subtype\nrelationships for CaH3:\n\\begin{equation}\n\\left[ \\rm SpTy \\right]_{\\rm CaH3} = 18.80 - 21.68 \\ {\\rm CaH3}_{c}\n\\end{equation}\nWhere the corrected values of the spectral bands indices (see Eqs.2-5)\nare used. The relationships are slightly different from those quoted in\n\\citet{Gizis1997} and \\citet{Lepine2003} but are internally\nconsistent to each other, whereas an application of the older\nrelationships to our corrected band index measurements would yield\ninternal inconsistencies, with subtype difference up to 1 spectral\nsubtype between the relationships.\n\nThe ratio of oxides (TiO, VO) to hydrides (CaH,CrH,FeH) in M dwarfs is\nknown to vary significantly with metallicity\n\\citep{Gizis1997,Lepine2007}. In the metal-poor M subdwarfs, it is the\noxides bands that appear to be weaker, while hydride bands remain\nrelatively strong (in the most metal-poor ultrasubdwarfs, or usdM,\nthe TiO bands are almost undetectable). Therefore it makes sense to\nrely more on the CaH band as the primary subtype\/temperature\ncalibrator. The same $[\\rm SpTy]_{\\rm CaH2}$ and $[\\rm SpTy]_{\\rm\n CaH3}$ relationships should be used to determine spectral subtypes\nat all metallicity classes (i.e. in M subdwarfs as well as in M\ndwarfs).\n\nBecause the TiO and VO bands are also strong in the metal-rich M\ndwarfs, it is still useful to include these bands as secondary\nindicators, to refine the spectral classification. In the late-type M\ndwarfs, in fact, the CaH bandheads are saturating, and one has to rely\non the TiO and VO bands. In fact, the VO bands were originally used to\ndiagnose and calibrate ultracool M dwarfs of subtypes M7-M9 \\citep{KHS95}.\nThe main caveat in using the oxide bands for spectral classification\nis that this can potentially introduce a metallicity dependence on the\nestimated spectral subtype, with more metal-rich stars being assigned\nlater subtypes than what they would have based on the strength of\ntheir CaH bands alone. In any case, because our sample appears to be\ndominated by near-solar metallicity stars, we calibrate additional\nrelationships between subtype and the TiO5 and TiO6 bands indices. We\nfirst recalculate the subtypes by averaging the values of $\\left[ SpTy\n \\right]_{CaH2}$ and $\\left[ SpTy \\right]_{CaH2}$, and perform a\n$\\chi^2$ fit of the TiO5 and TiO6 indices to the mean subtypews\ncalculated from CaH2 and CaH3, finding:\n\\begin{equation}\n\\left[ \\rm SpTy \\right]_{\\rm TiO5} = 7.83 - 9.55 \\ {\\rm TiO5}_{c}\n\\end{equation}\n\\begin{displaymath}\n\\left[ \\rm SpTy \\right]_{\\rm TiO6} = 9.92 - 15.68 \\ {\\rm TiO6}_{c}\n+21.23 \\ \\left( {\\rm TiO6}_{c} \\right)^2 \n\\end{displaymath}\n\\begin{equation}\n- 16.65 \\ \\left( {\\rm TiO6}_{c} \\right)^3\n\\end{equation}\nwhere again the corrected band indices are used. The relatively sharp\nnon-linear deviation in the TiO6 distribution around M3 forces the use\nof a third order polynomial in the fit. \n\nWe also determine the relationships for the VO1 and VO2\nband indices. This this after recalculating the subtypes from the\naverage of $\\left[ \\rm SpTy \\right]_{\\rm CaH2}$ and $\\left[ \\rm SpTy\n \\right]_{\\rm CaH2}$, $\\left[ \\rm SpTy \\right]_{\\rm TiO5}$, and\n$\\left[ \\rm SpTy \\right]_{\\rm TiO6}$, a $\\chi^2$ fit again yields: \n\\begin{equation}\n \\left[ \\rm SpTy \\right]_{\\rm VO1} = 69.8 - 71.4 \\ {\\rm\n \\left[\\rm VO1\\right]_{c}} \n\\end{equation}\n\\begin{displaymath}\n\\left[ \\rm SpTy \\right]_{\\rm VO2} = 9.56 - 12.47 \\ {\\rm\n \\left[\\rm VO2\\right]_{c}} \n + 22.33 \\ \\left( {\\rm \\left[\\rm VO2\\right]_{c}} \\right)^2\n\\end{displaymath}\n\\begin{equation}\n - 19.59 \\ \\left( {\\rm \\left[\\rm VO2\\right]_{c}} \\right)^3.\n\\end{equation}\nThe VO indices however make relatively poor estimators of spectral\nsubtypes for our sample, mainly because the shallow slope at earlier\nsubtypes provides little leverage. The VO2 index also shows\nunexpectedly large scatter in the MDM spectra, including in the\nclassification standard stars, which we suspect is due the fact that\nthe index is defined very close to the red edge of the MDM spectral\nrange and is thus more subject to statistical noise and flux\ncalibration errors. We therefore do not include $\\left[ SpTy\n \\right]_{VO12}$ and $\\left[ SpTy \\right]_{VO2}$ in the final\ndetermination of the spectral subtypes.\n\n\\begin{deluxetable}{cc}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{2} \n\\tablewidth{160pt} \n\\tablecaption{Distribution by spectral subtype of the 1564 survey\n stars\\label{table_st}}\n\\tablehead{\n\\colhead{Spectral subtype} & \n\\colhead{N} \n}\n\\startdata \nG\/K\\tablenotemark{a} & 160 \\\\\nK7.0 & 27 \\\\\nK7.5 & 101 \\\\\nM0.0 & 177 \\\\\nM0.5 & 160 \\\\\nM1.0 & 152 \\\\\nM1.5 & 147 \\\\\nM2.0 & 141 \\\\\nM2.5 & 119 \\\\\nM3.0 & 125 \\\\\nM3.5 & 125 \\\\\nM4.0 & 72 \\\\\nM4.5 & 36 \\\\\nM5.0 & 13 \\\\\nM5.5 & 4 \\\\\nM6.0 & 2 \\\\\nM6.5 & 2 \\\\\nM7.0 & 1\n\\enddata \n\\tablenotetext{a}{Stars identified as earlier than K7.0 and\/or with no\n detected molecular bands.}\n\\end{deluxetable} \n\nFig.~\\ref{idx_spty} plots all the corrected spectral band indices as a\nfunction of the adopted spectral subtype. The relatively\nsmall scatter ($\\approx0.02$) in the distribution of $\\left[\\rm\n CaH2\\right]_{c}$, $\\left[\\rm CaH3\\right]_{c}$, $\\left[\\rm\n TiO5\\right]_{c}$ and $\\left[\\rm TiO6\\right]_{c}$ demonstrate\nthe internal consistency of the spectral type calibration for the four\nindices. All four relationships have an average slope $\\approx10$,\nwhich means that since those indices have an estimated measurement\naccuracy of $\\approx0.02$ units, the spectral subtypes calculated by\ncombining the four indices should be accurate to about $\\pm0.10$\nsubtype assuming that the measurement errors in the four indices are\nuncorrelated. While this would make it possible to classify the stars\nto within a tenth of a subtype, we prefer to follow the general\nconvention and assign spectral subtypes to the nearest half\ninteger. \n\nTo verify the consistency of the spectral classification, we\ncompare the spectral types evaluated independently for the list of 141\nstars observed at both MDM and UH. We find that 82\\% of the stars end\nup with the same spectral type assigments from both observatories,\ni.e. they have spectra assigned to the same half-subtype. All the\nother stars have classifications within 0.5 subtypes. This is\nstatistically consistent with a $1\\sigma$ error of $\\pm$0.18 on the\nspectral type determination, slightly larger than the assumed $0.1$\nsubtype precision estimated above. This suggests a $3\\sigma$ error of\nabout $\\pm$0.5, which justifies the more conservative use of\nhalf-subtypes as the smallest unit for our classification.\n\nThe resulting classifications based on the CaH and TiO band index\nmeasurements are listed in Table~\\ref{table_spectro}. The numerical\nspectral subtype measured from the average of the band indices is\nlisted to 2 decimal figures. These values are rounded to the nearest\nhalf integers to provide our more formal spectral classifications to a\nhalf-subtype precision. The non-rounded values are however useful for\ncomparison with other physical parameters as they provide a continuous\nrange of fractional values; these fractional subtype values are used\nin the analysis throughout the paper\n\nA histogram of the distribution of spectral subtypes is shown in\nFigure~\\ref{sptype_hist}, with the final tally compiled in\nTable~\\ref{table_st}. Most of the stars in our survey have\nsubtypes in the M0.0-M3.0 range. The sharp drop for stars of subtypes\nK7.5 and K7.0 is explained by the color selection used in the\n\\citet{LepineGaidos.2011} catalog ($V-J>2.7$) which was originally\nintended to select only M dwarfs; note that stars with subtypes\nearlier than K7.0 are also excluded from the graph, and are probably\ncontaminants of the color selection in any case. Our deficit of K7.0\nand K7.5 stars however spectra demonstrates that the adopted selection\ncriterion is efficient in excluding K dwarfs from the catalog. The\ndistribution of spectral subtypes also shows a marked drop in numbers\nfor subtypes M4 and later. This is a consequence of the relatively\nbright magnitude limit ($J<9$) of our subsample, combined with the low\nabsolute magnitudes of late-type M dwarfs, which excludes most\nlate-type stars from our survey, since these tend to be fainter than\nour magnitude limit, even relatively nearby ones. \n\n\\begin{figure}\n\\vspace{-2cm}\n\\hspace{-0.1cm}\n\\includegraphics[scale=0.42]{f7.eps}\n\\caption{Distribution of spectral subtype M, for the stars in our\n survey, with K5=-2 and K7=-1. Most stars are found to be early-M\n objects. The sharp drop at subtypes K7.5 and earlier is a\n consequence of our initial $V-J>2.7$ color cut, which was meant to\n select only stars cool enough and red enough to be M dwarf. The\n sharp drop for subtypes M4 and later is a consequence of the high\n magnitude limit ($J<9$) of our survey, which restricts the distance\n range over which late-M stars are selected. The magnitude limit also\n explains the slow drop in numbers from subtypes M0 to M3, whereas\n one would normally expect the lower-mass M3 stars to be more common\n than earlier-type objects in a volume limited\n sample.\\label{sptype_hist}}\n\\end{figure}\n\n\\subsection{Semi-automated classification using THE HAMMER}\n\n\\begin{figure}\n\\plotone{f8.eps}\n\\caption{Comparison of spectral types assigned by the spectral index\n method and those assigned by the Hammer code, which is used for\n stellar classification in the Sloan Digital Sky Survey. Subtypes\n generally agree to within the advertized precision of the Hammer ($\\pm$1\n subtype, illustrated by the slanted lines), although the Hammer\n subtypes are marginally later than the spectral index subtypes, by\n 0.27 subtype on average.\\label{spec_comp}}\n\\end{figure}\n\nTo verify the accuracy and consistency of spectral typing based the\nspectral-index method described above, we performed independent spectral\nclassification using the Hammer code \\citep{Covey2007}. The Hammer was\ndesigned to classify stars in the Sloan Digital Sky Survey\nSpectroscopic database, including M dwarfs \\citep{West2011}. The code\nworks by calculating a variety of spectral-type sensitive band\nindices, and uses a best-fit algorithm to identify the spectral\nsubtype providing the best match to those band indices. For late-K and\nM dwarfs, spectral subtypes are determined to within integer value\n(K5, K7, M0, M1, ..., M9).\n\nHowever, to ensure that the automatically determined spectral types were\naccurate we used the manual ``eye check'' mode of the Hammer (version\n1.2.5). This mode is typically used to verify that there are no\nincorrectly typed interlopers. The Hammer allows the user to compare\nspectra to a suite of template spectra to determine the best\nmatch. \\citet{West2011} have found that for late-type M dwarfs, the\nautomatic classifications were systematically one subtype earlier than\nthose determined visually. Our analysis confirms this offset, and we\ntherefore disregarded the automatically determined Hammer values to\nadopt the visually determined subtypes. \n\nThe resulting subtypes are listed in Table~\\ref{table_spectro}. \nSome 170 stars were not found to be good fits to any of the K5, K7, or\nM type templates, and thus identified as early-K or G dwarfs. This\nsubset includes all of the 156 stars that were visually identified as\nnon-M dwarfs on first inspection (see \\S3.1). The remaining 14 stars\nwere initially found to be consistent with late-K stars, and\nclassified as K7.0 and K7.5 objects using the spectral index method\ndescribed in \\S3.2 above; we investigated further to determine why the\nstars were classified as early K using the Hammer. On closer\ninspection, we found that 3 of the stars are indeed more consistent\nwith mid-K dwarfs that K7.0 or K7.5, and we thus overran the spectral\nindex classification and reclassified them as ''G\/K'' in\nTable~\\ref{table_spectro}. For the other 11 stars flagged as mid-K\ntype with the Hammer, we determined that the stars do show significant\nevidence for TiO absorption, which warrants that the stars retain\ntheir spectral-index classifiction of K7.0\/K7.5. \n\nIn the end Table~\\ref{table_spectro} lists 159 stars from our initial\nsample that are identified and early-type G and K dwarf contaminants,\nand most likely made our target list due to inaccurate or unreliable\n$V-J$ colors. The remaining 1405 stars are formally classified as\nlate-K and M dwarfs.\n\nA comparison of spectral subtypes determined from the spectral-index\nand Hammer methods is shown in Figure~\\ref{spec_comp}. Because the\nHammer yields only integer subtypes, we have added random values in\nthe $-0.4,0.4$ range to facilitate the comparion. Slanted lines in\nFigure~\\ref{spec_comp} show the range expected if the two\nclassification methods (spectral index, Hammer) agree to within 1.0\nsubtypes. There is however a mean offset of 0.26 subtypes between the\nspectral index and Hammer classifications, with the Hammer subtypes\nbeing on average slighlty later.\n\nFigure~\\ref{spec_comp} also reveals a number of outliers with large\ndifferences in spectral subtypes between the two methods. We found 48\nstars with differences in spectral subtyping larger than $\\pm$1.5. The\nspectra from these stars were examined by eye: except for one star, we\nfound the band-index classifications to agree much better with the\nobserved spectra than the Hammer-determined subtypes. The one\nexception is the star PM I11055+4331 (Gl 412B) which the band index\nmeasurements classify as M6.5; in this case the Hammer determined\nsubtype of M 5.0 appears to be more accurate. This exception likely\noccurs because of the saturation of the CaH2 and CaH3 subtypes in the\nlate-type star, which make the band-index classification less\nreliable.\n\n\\subsection{Comparison with the ``Meet the Cool Neighbors'' survey}\n\n\\begin{figure}\n\\plotone{f9.eps}\n\\caption{Comparison of spectral types assigned by our spectral index\n method and those assigned in the \"Meet the Cool Neighbors\" (MCN)\n spectroscopic follow-up program. This shows all 161 stars in common\n between our survey and the MCN program. Spectral index subtypes are\n show before rounding up to the nearest half-integer; the MCN\n subtypes have random values of $\\pm$0.2 to help in the\n comparison. The classifications generally agree to within $\\pm$0.5\n subtypes (slanted lines). Our subtypes are however marginally later\n (by 0.28 subtypes on average) than the MCN\n subtypes.\\label{mcn_comp}}\n\\end{figure}\n\nAfter the PMSU survey, the largest spectroscopic survey of M dwarfs in\nthe northern sky was the one presented in the ``Meet the Cool\nNeighbors'' (MCN) paper series\n\\citep{CruzReid.2002,Cruzetal.2003,Reidetal.2003,Reidetal.2004,Cruzetal.2007,Reid2007}. In\nthis section we compared the spectral clssification from the MCN\nsurvey with our own spectral type assignments. \n\nTo identify the stars in common between the two surveys, we first\nperformed a cross-correlation of the celestial coordinates of the\nstars listed in the MCN tables, to the coordinates listed in the\nSUPERBLINK catalog. This was performed for the 1077 stars in MCN which\nare north of the celestial equator. We found counterparts in the\nSUPERBLINK catalog to within 1\\arcsec for 860 of the MCN stars. Of the\n217 MCN stars with no obvious SUPERBLINK counterparts, 148 are\nclassified as ultracool M dwarfs (M7-M9) or L dwarfs (L0-L7.5) which\nmeans they are very likely missing from the SUPERBLINK catalog because\nthey are fainter than the V=19 completeness limit of the catalog. Of\nthe 71 remaining stars, close examination of Digitized Sky Survey\nscans failed to identify the stars at the locations quoted in MCN. A\ncloser examination of the fields around those stars identified 49\ncases where a high proper motion stars could be found within 3\\arcmin\nof the quoted MCN positions. These nearby high proper motion stars are\nall listed in SUPERBLINK, and have colors consistent with M dwarfs; we\ntherefore assumed that the quoted MCN positions are in error, and\nmatched those 49 MCN entries with the close by high proper motions\nstars from SUPERBLINK. Of the remaining 22 stars we found 6 that have\nproper motions below the SUPERBLINK limit of $\\mu>40$ mas yr$^{-1}$\nand another 5 stars with proper motions within the\nSUPERBLINK limit but that appear to have been missed by the SUPERBLINK\nsurvey. Finally, there were 11 MCN stars that we could not identify at\nall on the Digitized Sky Survey images, and we can only assume that\nthe positions quoted in MCN are too large for proper identification,\nand that the stars should be considered ``lost''.\n\nOf the 909 stars in the MCN program with SUPERBLINK counterparts, we\nfound only 219 which satisfy the magnitude limit ($J<9$) of our\npresent sample of very bright M dwarfs. Of those, 52 stars have colors\nbluer ($V-J<2.7$) than our sample limit; 48 of them are classified\nas F and G stars in MCN, consistent with their bluer colors. The other\n4 stars are classified as M dwarfs, although they have $V-J<2.7$\naccording to \\citet{LepineGaidos.2011}. We infer that our $V-J$ colors\nfor those stars are probably underestimated, which suggests that our\ncolor selection may be overlooking a small fraction of nearby M\ndwarfs. In addition, we found another 6 stars which have $V$\nmagnitudes and $V-J$ colors within our survey range, but were rejected\nby the additional infrared ($J-H$,$H-K$) color-color cuts used in\n\\citet{LepineGaidos.2011} to filter out red giants. All 6 stars are\nvery bright in the infrared, and it appears that at least one of the\n$H$ or $K$ magnitudes listed in the 2MASS catalog may be in error,\nmaking the stars appear to have $J-H$ and\/or $H-K$ colors more\nconsistent with giants. Four of the stars are classified as M dwarfs\nin MCN, the other two are late-K dwarfs. Overall, this makes a total\nof 8 M dwarfs from the MCN census that were overlooked in our\nselection out of the MCN subset of $\\approx$150 nearby M dwarfs. This\nsuggests that our color cuts, combined with magnitude measurement\nerrors, might be missing $\\sim5\\%$ of the very bright, nearby M dwarfs.\n\nIn the end, this leaves only 161 stars in common between the MCN\nprogram and our own spectroscopic survey. The stars are all classified\nas late-K and M dwarfs by MCN, with subtypes ranging from K5 to\nM5.5. All 159 stars are identified with a flag (''M'') in the last\ncolumn of Table~\\ref{table_spectro}; we note that 82 of these stars\nwere also observed as part of the PMSU survey. We compare the spectral\ntype assignments from both surveys in Figure~\\ref{mcn_comp}, where the\nM dwarf subtypes from MCN are plotted against the (non-rounded)\nsubtypes calculated from the spectral-band indices. To ease the\ncomparison, random values of $\\pm$0.2 are added\nto the MCN subtypes. Overall, our classifications agree to within\n$\\pm$0.5 subtypes with the MCN values. The MCN subtypes, however, tend\nto be marginally earlier on average, by 0.28 subtypes; this is in\ncontrast with the Hammer classifications (see above) which tend to be\nslightly later than our own. For the MCN subtypes, the effect is more\npronounced for the earlier M dwarfs ($<$M2.5), where the mean offset\nis 0.43 subtypes, whereas the mean offset is only 0.09 for the later\nstars. \n\nTo investigate the difference in spectral subtype assignments, we\ncompare the recorded values of the CaH2, CaH3, and TiO5 indices\nbetween the MCN program and our own survey. After a search of the\nvarious tables published in the MCN series of papers, we identified 54\nstars in common between the two programs, and for which values of\nCaH2, CaH3, and TiO5 were also recorded in both. The differences\nbetween the spectral index values are shown in\nFigure~\\ref{mcn_idx}. For our own survey, the corrected values of\nthese indices are used, i.e. CaH2$_c$, CaH3$_c$, and TiO5$_c$ as\ndefined in \\S3.2.2. We find that the CaH2 and TiO5 are estimated\nmarginally higher in the MCN program than they are in our survey, and\nthis very likely explains the difference in spectral typing: the \nhigher index values yield margnially earlier spectral subtypes. This\nagain emphasizes the variation in the spectral index measurements due\nto spectral resolution and other instrumental setups, and the need to\napply systematic corrections between observatories to obtain a uniform\nclassification system.\n\n\\begin{figure}\n\\hspace{1.0cm}\n\\includegraphics[scale=0.9]{f10.eps}\n\\caption{Differences between the spectral index values recorded in the\n ``Meet the Cool Neighbors'' (MCN) program and those measured in the\n present survey. The values are compared for a subset of 55 stars in\n common between the two surveys and for which values of CaH2, CaH3,\n and TiO5 were recorded. For the present survey, we use the corrected\n values (CaH2$_c$, CaH3$_c$, and TiO$_c$) as defined in \\S3.2.2. The\n MCN values tend to be marginally higher on average, especially for\n stars of earlier spectral subtypes. This explains the marginally\n earlier spectral subtype assigments in MCN compared with the\n ones presented in this paper (see\n Figure~\\ref{mcn_comp}).\\label{mcn_idx}}\n\\end{figure}\n\nWe note that smaller subsets of stars in our census may also have\nspectroscopic data published in the literature, from various other\nsources. This is especially the case for the 102 stars from the CNS3\nand stars with very large proper motions $mu>0.2\\arcsec$ yr$^{-1}$\nwhich have been more routinely targeted for follow-up spectroscopic\nobservations. Additional pectroscopic surveys of selected bright M\ndwarfs include, \\citet{Scholz2002}, \\citet{Scholz2005},\n\\citet{Reyle2006}, and \\citep{Riaz2006}, which all have a few stars in\ncommon with our catalog. Other surveys of nearby M dwarfs have mainly\nbeen targeting fainter stars\n\\citep{Bochanski2005,Bochanski2010,West2011}, and do not overlap with\nour present census.\n\n\\subsection{Color\/spectral-type relationships}\n\n\\begin{figure}\n\\vspace{-0.2cm}\n\\hspace{0.1cm}\n\\includegraphics[scale=0.8]{f11.eps}\n\\caption{Variation of the UV-to-optical $NUV-V$ color index and the\n optical-to-IR $V-J$ color index as a function of spectral subtype\n M. Stars with more reliable V magnitudes from the Tycho-2 catalog\n are shown in blue, stars with V magnitudes derived from less\n reliable photometric measurements are shown in red. The distribution\n of ultra-violet to optical colors ($NUV-V$) with spectral subtype\n shows two populations, one with a tight correlation, consistent with\n blackbody distribution and labeled ``quiescent'', and a scattered\n population of stars with clear $UV$ excess, labeled ``active''. The\n distribution of $V-J$ with subtype closely follows the relationship\n used by \\citet{LepineGaidos.2011} to predict subtypes from $V-J$\n colors (dashed line) except for stars of later subtypes which have\n redder colors than predicted. A revised relationship (full line) is\n fitted to the data. Outliers point to stars with bad $V$ magnitude\n measuremens.\\label{color_spty}}\n\\end{figure}\n\n\\begin{deluxetable*}{cccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{8} \n\\tablewidth{0pt} \n\\tablecaption{Colors and $T_{eff}$ for red dwarfs in our survey as a\n function of spectral subtype.\\label{color_subtype}}\n\\tablehead{\n\\colhead{subtype} & \n\\colhead{n$_{NUV-V}$\\tablenotemark{1}} & \n\\colhead{$\\overline{NUV-V}$\\tablenotemark{1}} &\n\\colhead{$\\sigma_{NUV-V}$\\tablenotemark{1}} &\n\\colhead{n$_{V-J}$} & \n\\colhead{$\\overline{V-J}$} &\n\\colhead{$\\sigma_{V-J}$} &\n\\colhead{$\\overline{T_{eff}}$} &\n\\colhead{$\\sigma_{T_{eff}}$}\n}\n\\startdata \nK 7.0 & 11 & 8.17& 0.21& 25 & 2.90 & 0.31 & 4073 & 98\\\\\nK 7.5 & 47 & 8.60& 0.31& 97 & 2.89 & 0.16 & 3883 & 82\\\\\nM 0.0 & 87 & 8.66& 0.36& 175 & 2.94 & 0.21 & 3762 & 71\\\\\nM 0.5 & 79 & 8.74& 0.30& 159 & 3.11 & 0.34 & 3646 & 48\\\\\nM 1.0 & 76 & 8.89& 0.39& 151 & 3.19 & 0.18 & 3565 & 44\\\\\nM 1.5 & 57 & 9.07& 0.38& 147 & 3.36 & 0.23 & 3564 & 39\\\\\nM 2.0 & 57 & 9.25& 0.47& 139 & 3.52 & 0.34 & 3518 & 57\\\\\nM 2.5 & 48 & 9.45& 0.50& 118 & 3.69 & 0.28 & 3500 & 61\\\\\nM 3.0 & 34 & 9.61& 0.38& 125 & 3.91 & 0.28 & 3423 & 62\\\\\nM 3.5 & 28 & 9.69& 0.33& 124 & 4.17 & 0.33 & 3320 & 66\\\\\nM 4.0 & 5 & 9.72& 0.35& 71 & 4.45 & 0.41 & 3204 & 76\\\\\nM 4.5 & 1 &\\nodata&\\nodata& 36 & 4.81 & 0.46 & 3119 & 43\\\\\nM 5.0 & 0 &\\nodata&\\nodata& 12 & 5.23 & 0.50 & 3014 & 61\n\\enddata\n\\tablenotetext{1}{Non-active (``quiescent'') red dwarfs only.}\n\\end{deluxetable*}\n\nSpectral subtypes were inititially estimated in\n\\citet{LepineGaidos.2011} based on $V-J$ colors alone. Here we verify\nthis assumption and re-evaluate the color-magnitude relationship for\nbright M dwarfs. The $V-J$ color index combines estimated optical ($V$)\nmagnitudes from the SUPERBLINK catalog to the infrared $J$\nmagnitudes of their 2MASS counterparts. The SUPEBLINK $V$ magnitudes\nare estimated either from the Tycho-2 catalog $V_T$ magnitudes, or\nfrom a combination of the Palomar photographic $B_J$ (IIIaJ), $R_F$\n(IIIaF), and $I_N$ (IVn) magnitudes, as described in\n\\citet{LepineShara.2005}. Values of $V$ are more accurate for the\nformer ($\\approx$0.1mag) than for the latter ($\\gtrsim$0.5mag);\nTable~\\ref{table_photo} indicates the source of the V magnitude.\n\nMean values and dispersion about the mean of the $V-J$ colors are\nlisted in Table~\\ref{color_subtype}, for each bin of half-integer\nsubtype; the table also lists how many stars of each type are in each\nbin. The $V-J$ colors of our stars are also plotted as a function of\nspectral subtype in Figure~\\ref{color_spty} (top panel). The adopted\ncolor-subtype relationship from \\citet{LepineGaidos.2011}\nis shown as a thick dashed line in Figure~\\ref{color_spty}. Stars with\nthe presumably more reliable Tycho-2 magnitudes are shown in blue,\nwhile stars with photographic $V$ magnitudes are shown in red. Stars\nwith Tycho-2 $V$ magnitudes appear to have marginally bluer colors\nat a given subtype; this however is an effect of the visual\nmagnitude limit of the Tycho-2 catalog, which includes only the\nbrightest stars in the $V$ band and is thus more likely to list bluer\nobjects. The \\citet{LepineGaidos.2011} relationship generally follows\nthe distribution at all subtypes, but with mean offsets up to\n$\\pm$0.4mag in $V-J$, especially at earlier and later subtypes. We\nperform a $\\chi^2$ fit to determine the following, improved\nrelationship:\n\\begin{displaymath}\n\\left[ \\rm SpTy \\right]_{V-J} = {\\rm -32.79 + 20.75 ( V - J ) - 4.04 ( V - J )^2}\n\\end{displaymath}\n\\begin{equation}\n{\\rm + 0.275 ( V - J )^3}\n\\end{equation}\nafter exclusion of 3-$\\sigma$ outliers. The relationship is shown in\nFigure~\\ref{color_spty} (solid line). There is a scatter of 0.7\nsubtype between $\\left[ SpTy \\right]_{V-J}$ and the subtype determined\nfrom spectral band indices. While the spectroscopic classification is\nmore accurate and reliable, photometrically determined spectral\nsubtypes using the equation above should still be accurate to $\\pm$0.5\nsubtype about 80\\% of the time, and to $\\pm$1.0 subtype 95\\% of the\ntime, which may be useful for a quick assessment of subtype when\nspectroscopic data is unavailable.\n\nWe also compare the near-UV to optical magnitude color $NUV-V$ for the\n714 stars in our sample which have counterparts in GALEX; the\ndistribution is shown in Figure~\\ref{color_spty} (bottom panel).\nWe find that stars become progressively redder as spectral subtype\nincreases, from $NUV-V=8$ at M0 to $NUV-V=10$ at M4. There is however\na significant fraction of M dwarfs which display much bluer $NUV-V$\ncolors at any given subtype. The excess in $NUV$ flux is strongly\nsuggestive of chromospheric activity (see \\S6 below for a more\ndetailed analysis). We separate the active stars from the more\nquiescent objects with the following condition:\n\\begin{equation}\n\\left[ \\rm NUV - V \\right] > 7.7 + 0.35 \\left[\\rm Spty\\right] \\\n\\end{equation}\nwhere $\\left[\\rm Spty \\right]$ is the mean spectral subtype calculated from\nequations 8-11. After excluding active stars, we calculate the mean\nvalues and scatter about the mean of $NUV-V$ for each half-integer\nspectral subtype. Again those are listed in Table~\\ref{color_subtype}\nfor reference; the table also lists the number of non-active stars\nused to calculate the mean. There is not a sufficient number of stars\nto calculate mean values and scatter at M4.5 (1 star) and M5.0 (0\nstar).\n\n\\section{Survey completeness}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{f12.eps}\n\\caption{Number of spectroscopically confirmed M dwarfs as a function\n of $sin(b)$, where $b$ is the Galactic latitude. Numbers from our\n census are shown as filled circles, with errorbars showing the\n Poisson noise. Distributions predicted from models with either\n uniform space density, or with decreases perpendicular to the\n Galactic plane following scale-heights of 200pc and 400pc, are shown\n for comparison. The observed distribution is largerly\n consistent with a uniform density in the local $d<70$pc\n volume, with no evidence for incompleteness at low Galactic\n latitude.\\label{compl}}\n\\end{figure}\n\nOur 1,564 spectroscopically confirmed M dwarfs are drawn from a\ncatalog with proper motion limit $\\mu<40$ mas yr$^{-1}$. The low\nproper motion limit of the SUPERBLINK catalog catches most of the\nnearby stars, but potentially overlooks nearby M dwarfs with small\ncomponents of motion in the plane of the sky \\--- either due to low\nspace motion relative to the Sun or to projection effects. The catalog\nmay also be affacted by other sources of incompleteness (e.g. missed\ndetection, faulty magnitude estimate) which means that at least some\nvery bright, nearby M dwarfs must be missing from our survey due to\nkinematics bias and other effects.\n\nTo evaluate the completeness of our census, we first consider the\nprimary source of incompleteness: the kinematics bias of the proper\nmotion catalog. As discussed in \\citet{LepineGaidos.2011}, the\ncompleteness depends on the local distribution of stellar motions, and\nincreases with distance from the Sun. To estimate the kinematics bias\nin our sample, we built a model reproducing the local distribution and\nkinematics of nearby M dwarfs. We first assumed the stars to have a\nuniform spatial distribution in the solar vicinity, and generated a\nrandom distribution of $10^{5}$ objects within a sphere of radius\n$d=70$pc centered on the Sun. We assigned transverse motions to all the\nstars, assuming a velocity-space distribution similar to that of the\nnearby (d$<$100pc) G dwarfs in the Hipparcos catalog\n\\citep{vanLeeuwen2007}. Because the distribution of stellar velocities\nis not isotropic, we assigned transverse motions for each simulated star\nbased on the statistical distribution of transverse motions for\nHipparcos stars with sky coordinates within $30^{deg}$ of the\nsimulated object. We also used the simplifying assumption that the\nlocal M dwarf population has a uniform distribution of absolute\nmagnitudes over the range $540$ mas yr$^{-1}$. We found that 93\\% of\nnearby M dwarfs with $J<9$, on average, have proper motions above the\nSUPERBLINK limit. M dwarfs with $J<9$ extend to a maximum distance of\n63 parsecs; most of the stars which fail the proper motion cut are in\nthe higher distance range (d$<$50pc), and are stars near the bright\nend of the luminosity distribution $M_J\\approx5-6$. For this reason,\nif we assume a luminosity function which increases at fainter absolute\nmagnitudes, the fraction of $J<9$ stars which fall within the proper\nmotion cut is increased, because more of the $J<9$ stars in the local\npopulation are now M dwarfs of lower luminosities and closer\ndistances, which are more likely to have high proper motions. The\nobserved field M dwarf luminosity function does indeed increase for\nearly-type dwarfs, to reach a peak at $M_J\\simeq8.0$\n\\citep{Reid2002,Bochanski2010}, which means that the 93\\% completeness\nestimated above must be a lower limit. To verify this, we tested a\nluminosity function where the number of stars increases linearly with\nabsolute magnitude and doubles from $M_J=5$ to $M_J=8$; with this\nmodel, our simulations showed that 96\\% of all $J<9$ M dwarfs, would\nhave $\\mu>40$ mas yr$^{-1}$, and thus be within the detection limit of\nSUPERBLINK. Overall, this suggests that only about $5\\%$ of all $J<9$\nM dwarfs on the sky will be overlooked in our census because of the\nproper motion bias.\n\nThe SUPERBLINK surveys does however suffer from various other sources\nof incompleteness, such as the inability to detect moving stars in\nsaturated regions of photographic plates (i.e. in the immediate\nvicinity of very bright stars), or a difficulty in detecting stars in\nvery crowded field. Also, the SUPERBLINK code has trouble detecting\nthe motions of relatively bright $V<12$ stars because of the saturated\ncores of their point spread functions. In practice, this is mitigated\nby incorporating data from the Tycho-2 catalog, which provides very\naccurate proper motion measurements for bright stars. However the\nTycho-2 catalog itself has some level of incompleteness in the\n$840$ mas yr$^{-1}$. The mean distributions from the\n400pc and 200pc scale-height models are shown in Figure~\\ref{compl},\nand do indeed predict a slight overdensity of objects at low Galactic\nlatitudes. Assuming that all the stars at high Galactic latitude are\ndetected in all the models, we find a $\\sim$5\\% excess of stars in the\n400pc scale-height model over the observed number, and a $\\sim$10\\%\nexcess of stars in the 200pc scale-height model. One might then argue\nthat the SUPEBLINK census potentially has an additional 5\\%-10\\%\nincompleteness level, due to incompleteness in the plane of the Milky\nway. This is most probably an overestimate, however, because there is\nno evidence that the local stellar distribution shows any significant\ndecrease with $Z$. Overall, the SUPERBLINK census appears to be\nessentially complete at low Galactic latitudes.\n\nProper-motion selection may introduce an additional bias against\nmetal-rich and\/or young stars, which tend to have lower components of\nmotion in the vicinity of the Sun. However this effect is expected to\nbe small, e.g. a few percent against [Fe\/H] = 0 relative to [Fe\/H] =\n-0.5 at 45~pc \\citep{Gaidos2012}. Nearly all likely members of the\nHyades \\citep{Perryman1998}, Ursa Majoris \\citep{King2003}, and TW\nHydrae \\citep{Reid2003} young nearby moving groups have proper motions\nthat exceed 40~mas yr$^{-1}$ and thus would not be selected against using\nthe current selection method.\n\nFinally, some bright M dwarfs may be overlooked because of the imposed\ncolor cuts, due to magnitude errors and uncertainties. As described in\n\\S3.4 above, a handful of previously known M dwarfs from the MCN\ncensus were overlooked in our target selection for precisely those\nreasons. As suggested in \\S3.4, it is possible that we may be\noverlooking 5\\% of bright M dwarfs because of magnitude errors. This,\ncombined with the estimated 96\\% completeness in the proper motion\nselection estimated above, suggests that our list of 1405 M dwarfs\nlikely includes $\\approx91\\%$ of all existing M dwarfs with infrared\nmagnitude $J<9$ as seen from Earth. There may still be $\\approx$140\nbright M dwarfs to be identified, although the precise number can only\nbe determined after these ``missing'' stars are found. \n\n\\section{Phoenix Model fits and $T_{eff}$ estimates}\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{f13.eps}\n\\caption{Effective temperatures for the bright M dwarfs in our survey\n determined by fits to the PHOENIX models. The gray points represent\n individual objects, while the large black points are the median\n values of the objects within each half subtype bin. The error bars\n are the interquartile ranges. Blue diamonds and red triangles\n show the T$_{eff}$ estimates for subsets of stars in our survey\n whose tempreatures were estimated from photometry\n \\citet{Casagrande2008} and model fits to infrared spectra\n \\citep{Rojas-Ayala2012} respectively. While both spectroscopic\n estimates show evidence of a plateau at M2-M3, the photometric\n estimates do not concur. \\label{spt_color}}\n\\end{figure}\n\nWe compared our spectra to a grid of 298 models of K- and M-dwarf\nspectra generated by the BT-SETTL version of PHOENIX \\citep{Allard2010}.\nBT-SETTL includes updated opacities (i.e. of H$_2$O), revised solar\nabundances \\citep{Asplund2009}, a refractory cloud model, and\nrotational hydrodynamic mixing. The models include effective\ntemperatures $T_{eff}$ of 3000-5000~K in steps of 100~K, $\\log g$ values\nof 4, 4.5, and 5, and metallicities of [M\/H] = -1.5, -1, -0.5, 0,\n+0.3, and +0.5. For each temperature, $\\log g$ and metallicity value,\nwe selected the model with $\\alpha$\/Fe that was closest to solar. \n\n\\begin{figure*}\n\\plotone{f14.eps}\n\\caption{Distribution of corrected CaH2+CaH3 vs TiO5 band index\n values for the stars in our survey (grey dots - with brightness\n levels correlating with spectral subtype). There is a tight correlation\n between the two indices, which are both also correlated\n with spectral subtypes, with earlier stars on the upper right of the\n diagram as shown. The distribution is used as a guide to calibrate\n the value of $\\zeta$, with $\\zeta=1$ assumed to trace the\n CaH2+CaH3\/TiO5 relationship for stars with average Galactic disk\n abundances (near-solar). The iso-$\\zeta$ contours from the earlier\n calibrations of \\cite{Lepine2007} and \\cite{Dhital2012} are shown as\n dotted and dashed lines, respectively. When applied to our corrected\n spectral index values, they both diverge from the observed\n distribution at earlier subtypes, with the \\cite{Lepine2007}\n overestimating the $\\zeta$ of late-K and early-M dwarfs, while the\n \\cite{Dhital2012} calibration yield underestimates. This emphasizes\n again the need to use properly recalibrated and corrected spectral\n index values (see Figures~\\ref{pmscomp}-\\ref{idx_comp}). Our\n revised, dataset-specific calibration is shown with the continuous\n lines. Known metal-rich amd metal-poor stars are denoted in red and\n blue, respectively.\\label{cah_tio}}\n\\end{figure*}\n\nThe spectral density of model calculations varies with wavelength but\nis everywhere vastly greater than the resolution of our spectra.\nModel spectra were thus convolved with a gaussian with FHWM of the same\nresolution as the spectra, and a corrective shift (typically less than\na resolution element) was found by cross-correlating the observed and\nmodel spectra. Normalized spectra were ratioed and $\\chi^2$\ncalculated using the variance spectrum of the observations. We\nrestricted the spectral range over which $\\chi^2$ is calculated to\n5600-9000\\AA\\ and excluded the problematic region 6400-6600\\AA\\ which\ncontains poorly-modeled TiO absorption \\citep{Reyle2011}. We also\nexcluded regions where the telluric correction is rapidly changing with\nwavelength, i.e. the slope, smoothed over 4 resolution elements or\n11.7\\AA, exceeds 1.37 \\AA$^{-1}$. The model with the smallest $\\chi^2$\nvalue was identified. For a more refined estimate of effective temperature,\nwe selected the 7 best-fit models and constructed 10,000 linear\ncombinations of them; the ``effective temperature'' of each is the\nweighted sum of the temperatures of the components. We again found\nthe model with the minimum $\\chi^2$. We calculated the standard\ndeviation of $T_{eff}$ among the combination models as a function of the\nmaximum allowed $\\chi^2$. We reported the maximum standard deviation\nas a conservative estimate of uncertainty. We also calculated formal\n95\\% confidence intervals for $T_{eff}$ based solely on the expected\ndistribution of $\\chi^2$ for $N-3$ degrees of freedom, where $N \\sim\n1100$ is the number of resolution elements used in the fit. The\nparameters of the best-fit model, and the refined $T_{eff}$, standard\ndeviation, and confidence intervals are reported in\nTable~\\ref{table_spectro}.\n \nValues of $T_{eff}$ calculated for individual stars are plotted in\nFigure~\\ref{spt_color} as a function of their spectral subtype (grey\ndots). Median values for stars within each half-subtype bin are\nplotted in black, with error bars showing the interquartile\nranges. Our model-fit algorithm prefers values that match the $T_{eff}$\nmodel grid, which have a 100K grid step (i.e., 3500K is preferred over\n3510K).\n\nOur results suggest the existence of a $T_{eff}$ plateau spanning\nM1-M3. To investigate this further, we compare our values to the\neffective temperatures reported in \\citet{Casagrande2008} for 18 of\nthe stars in our sample, and in \\citep{Rojas-Ayala2012} for another 49\nstars; our own spectral type determinations are combined to the\nT$_{eff}$ measured by the other authors. The values are compared in\nFigure~\\ref{spt_color}. We find that our mid-type plateau is\ncorroborated with the \\citep{Rojas-Ayala2012} values, but not with\nthose from \\citet{Casagrande2008}, whose values decrease more linearly\nwith spectral subtype. The effective temperatures in\n\\citet{Casagrande2008} are based on photometric measurements while\nthe \\citep{Rojas-Ayala2012} values are estimated by PHOENIX model fits\nto infrared spectra. It is interesting that the fits to the optical\nand infrared spectra yield T$_{eff}$ values which are in\nagreement. The disagreement with the photometric determinations\nhowever suggest that atmospheric models for M dwarfs are still not\nwell understood, in particular in the M1-M4 spectral regimes.\n\n\\section{The $\\zeta$-Parameter and Metallicity Estimates}\n\n\\begin{figure}\n\\epsscale{2.2}\n\\plotone{f15.eps}\n\\caption{The $\\zeta$ parameter as a function of spectral subtype. Top:\n difference in $\\zeta$ for the same stars measured at the two\n observatories (MDM and UH). The scatter provides an estimate of the\n measurement error on $\\zeta$, which is significantly larger at\n earlier subtypes. Bottom: adopted values of $\\zeta$ for all the\n stars in the survey. The larger scatter at subtypes M2 and earlier\n can be fully accounted by the measurement errors. The scatter at\n subtypes M3 and later is larger than the measurement error, and is\n thus probably intrinsic and is evidence for intrinsic metallicity\n scatter in the solar neighborhood.\\label{zeta_comp}}\n\\end{figure}\n\n\\subsection{Recalibration of the $\\zeta$ parameter}\n\nThe $\\zeta_{TiO\/CaH}$ parameter (denoted $\\zeta$ for short) is a\ncombination of the TiO5, CaH2, and CaH3 spectral indices which was\nshown to be correlated with metallicity in metal-poor M subdwarfs\n\\citep{Woolf2009}. The index was first described in\n\\citet{Lepine2007}, and a revised calibration has recently been\nproposed by \\citet{Dhital2012}. The index measures the relative\nstrength of the TiO molecular band around 7,000\\AA\\ with respect to the\nnearby CaH molecular band. In cool stars, in fact, the ratio between\nTiO and CaH is a function of both gravity and metallicity. The CaH\nband is noticeably stronger in giants \\citep{Mann2012}, and this\neffect can be used as affective means to separate out M giants from M\ndwarfs using optical spectroscopy. In the higher gravity M\ndwarfs\/subwarfs however, the TiO to CaH ratio is however believed to\nbe mostly affected by metallicity, although young stars may show\ngravity effects as well. \n\nThe $\\zeta$ parameter was originally introduced to rank metal-poor,\nmain-sequence M stars into three metallicity classes\n\\citep{Lepine2007}; stars with $0.5<\\zeta<0.825$ are formally\nclassified as subdwarfs (sdM), $0.2<\\zeta<0.5$ defines extreme\nsubdwarfs (esdM), while a $\\zeta<0.2$ identifies the star as an\nultrasubdwarf (usdM). However, it is conjectured that $\\zeta$ could\nbe used to measure metallicity differences in disk M dwarfs, i.e. at\nthe metal-rich end. Disk M dwarfs are generally found to have\n$0.9<\\zeta<1.1$, though it is unclear if variations in $\\zeta$\ncorrelate with metallicity for values within that range. Measurement\nof Fe lines in a subset of M dwarfs and subdwarfs does confirm that\nthe $\\zeta$ parameter is correlated with metallicity\n\\citep{Woolf2009}, with $\\zeta\\simeq1.05$ presumably corresponding to\nsolar abundances. However there is a significant scatter in the\nrelationship which raises doubts about the accuracy of $\\zeta$ as a\nmetallicity diagnostic tool. \n\nA important caveat is that the TiO\/CaH ratio is not sensitive to the\nclassical iron-to-hydrogen ratio Fe\/H, but rather depends on the\nrelative abundance of $\\alpha$-elements to hydrogen ($\\alpha$\/H)\nbecause O, Ca, and Ti are all $\\alpha$-elements. Variations in\n$\\alpha$\/Fe would thus weaken the correlation between $\\zeta$ and\nFe\/H. The $\\alpha$\/Fe abundance ratio is however relatively small in\nmetal-rich stars of the thin disk ($\\pm$0.05dex) disk stars, and are\nfound to be significant ($\\pm$0.2dex) mostly in more metal-poor stars\nassociated with the thick disk and halo \\citep{Navarro_etal.2011}. It\nis thus unclear whether typical $\\alpha$\/Fe variations would affect\nthe $\\zeta$ parameter significantly in our subset, which is dominated\nby relatively metal-rich stars.\n\nOn the other hand, it is clear that the index has significantly more\nleverage at later subtypes. This is because the strengths of both the\nTiO and CaH bands are generally greater, and their ratio can thus be\nmeasured with higher accuracy. The index is much leass reliable at\nearlier M subtypes however, and is notably inefficient for late-K\nstars.\n\nA more important issue is the specific calibration adopted for the\n$\\zeta$ parameter, which is a complicated function of the TiO5, CaH2,\nand CaH3 indices. The $\\zeta$ parameter itself is defined as:\n\\begin{equation}\n\\zeta = \\frac{1-{\\rm TiO5}}{1-[{\\rm TiO5}]_{Z_{\\odot}}},\n\\end{equation}\nwhich in turns depend on $[{\\rm TiO5}]_{Z_{\\odot}}$, itself a function\nof CaH2+CaH3. The function $[{\\rm TiO5}]_{Z_{\\odot}}$ represents the\nexpected value of the TiO5 index in stars of solar metallicity, for a\ngiven value of CaH2+CaH3. In \\citep{Lepine2007}, ${\\rm\n TiO5}]_{Z_{\\odot}}$ was defined as:\n\\begin{displaymath}\n[{\\rm TiO5}]_{Z_{\\odot}} = -0.050 - 0.118 \\ {\\rm CaH} + 0.670 \\ {\\rm CaH}^2 \n\\end{displaymath}\n\\begin{equation}\n- 0.164 \\ {\\rm CaH}^3\n\\end{equation}\nwhere ${\\rm CaH}={\\rm CaH2}+{\\rm CaH3}$. The more recent calibration\nof \\citet{Dhital2012}, on the other hand, uses:\n\\begin{displaymath}\n[{\\rm TiO5}]_{Z_{\\odot}} = -0.047 - 0.127 \\ {\\rm CaH} + 0.694 \\ {\\rm CaH}^2 \n\\end{displaymath}\n\\begin{equation}\n- 0.183 \\ {\\rm CaH}^3 - 0.005 \\ {\\rm CaH}^4 .\n\\end{equation}\nThe difference between the two calibrations is mainly in the treatment\nof late-K and early-type M dwarfs, as illustrated in\nFigure~\\ref{cah_tio}. When overlaid on the distribution of CaH2+CaH3\nand TiO5 values from our current survey, however, the two calibrations\nfail to properly fit the distribution of data points at the earliest\nsubtypes (high values of CaH2+CaH3 and TiO5). This results in the\n\\citet{Lepine2007} overestimating the metallicity at earlier subtypes,\nwhile the \\citet{Dhital2012} calibration tends to underestimate\nmetallicity.\n\nIn any case, the evidence presented in \\S3.3 and \\S3.4 which shows\nthat differences in spectral resolution and flux calibration can yield\ndifferences in the TiO5, CaH2, and CaH3 spectral indices of the same\nstars, also suggests that a calibration of the $\\zeta$ parameter may\nonly be valid for data from a particular observatory\/instrument. A\ngeneral calibration of $\\zeta$ may only be adopted after corrections\nhave been applied as described in Section 3.2. Because we do not have\nany star in common with the \\citet{Dhital2012} subsample, we cannot\nverify the consistency of their $\\zeta$ calibration to our data at\nthis time. In addition, because we have now applied a correction to\nour MDM spectral index measurements, the \\citet{Lepine2007}\ncalibration of $\\zeta$ may now be off, and should not be used for our\nsample.\n\nInstead, we recalibrate the $\\zeta$ parameter again, using our\ncorrected spectral index values. Our fit of $\\left[\\rm\nTiO5\\right]_{c}$ as a function of $\\left[{\\rm\n CaH}\\right]_{c}=\\left[{\\rm CaH2}\\right]_{c}+\\left[{\\rm\n CaH3}\\right]_{c}$ yields:\n\\begin{displaymath}\n[{\\rm TiO5}]_{Z_{\\odot}} = 0.622 - 1.906 \\ (\\left[{\\rm CaH}\\right]_{c}) -\n2.211 \\ (\\left[{\\rm CaH}\\right]_{c})^2 \n\\end{displaymath}\n\\begin{equation}\n- 0.588 \\ (\\left[{\\rm CaH}\\right]_{c})^3.\n\\end{equation}\nWe calculate the new $\\zeta$ values using the corrected values of the\nTiO5 index, i.e.:\n\\begin{equation}\n\\zeta = \\frac{1-{\\rm \\left[TiO5\\right]_{c}}}{1-[{\\rm TiO5}]_{Z_{\\odot}}}.\n\\end{equation}\nAll our values of $\\zeta$ are listed in Table~\\ref{table_spectro}.\n\nIn order to evaluate the accuracy of the $\\zeta$ measurements, we\ncompared the values of $\\zeta$ independently measured at both MDM and\nUH for the 146 stars in our inter-observatory subset. Values are\ncompared in Figure \\ref{zeta_comp} (top panel) which shows $\\Delta\n\\zeta=\\zeta_{MDM}-\\zeta_{UH}$ as a function of spectral subtype. We\nfind a mean offset $\\bar{\\Delta\\zeta}=-0.01$ and a\ndispersion $\\sigma_{\\Delta\\zeta}=0.10$. The small offset indicates\nthat the $\\zeta$ measurements are generally consistent between the two\nobservatories. The dispersion $\\sigma_{\\Delta\\zeta}$, on the other\nhand, provides an estimate of the measurement accuracy. Splitting the\nstars in three groups, we find the mean offsets and dispersions\n$(\\bar{\\Delta\\zeta},\\sigma_{\\Delta\\zeta})$ to be (-0.086,0.244) for\nsubtypes K7.0-M0.5, (-0.011,0.103) for subtypes M1.0-M2.5, and\n(0.008,0.036) for subtypes M3.0-M5.5. Assuming that stars do not show\nsignificant variability in those bands, we adopt the dispersions as\nestimates of the measurement errors on $\\zeta$ for that particular\nsubtype range. It is clear from Figure~\\ref{cah_tio} that early type\nstars should have larger uncertainties in $\\zeta$ because of the\nconvergence of the iso-$\\zeta$ lines. The best leverage for estimating\nmetallicities from the TiO and CaH bandheads is at later types when\nthe molecular bands and well developed.\n\nThe overall distribution of $\\zeta$ values as a function of spectral\nsubtype also shows a decrease in the dispersion as a function of\nspectral type (Figure~\\ref{zeta_comp}, bottom panel). In early-type\ndwarfs (K7.0-M0.5), the scatter in the $\\zeta$ values is relatively\nlarge, with $\\sigma_{\\zeta}\\simeq0.174$. It then drops to\n$\\sigma_{\\zeta}\\simeq0.100$ for subtypes M1.0-M2.5, and to\n$\\sigma_{\\zeta}\\simeq0.059$ for subtypes M3.0-M5.5. Note that the\nscatter in the M3.0-M5.5 bin is a factor 2 larger than the estimated\naccuracy of the $\\zeta$ for that range, as estimated above. We\nsuggests this to be evidence of an intrinsic scatter in the $\\zeta$\nvalues for the stars in our sample, which we allege to be the\nsignature of a metallicity scatter. If we subtract in quadrature the\n$0.035$ measurement error on $\\zeta$, we estimate the instrinsic\nscatter to be $\\approx$0.05 units in $\\zeta$. This intrinsic scatter,\nwhich presumably affects all subtypes equally, is unfortunately\ndrowned in the measurement error at earlier subtypes ($<$M3).\n\n\n\\subsection{Comparison with other metallicity diagnostics}\n\n\\begin{figure}\n\\vspace{-0.6cm}\n\\hspace{-0.5cm}\n\\includegraphics[scale=0.9]{f16.eps}\n\\caption{Comparison between the $\\zeta$ parameter values and\n independent metallicity measurements for subsets of M dwarfs in our\n survey. Top panel compares our $\\zeta$ to the Fe\/H estimated from\n the (V-K,M$_K$) calibration of \\citet{Neves2012} for the same\n stars. Bottom panel compares our $\\zeta$ values to the Fe\/H\n estimated from the infrared K-band index by\n \\citet{Rojas-Ayala2012}. Both distribution show weak positive\n correlations.\\label{zeta_feh}}\n\\end{figure}\n\n\\begin{figure*}\n\\plotone{f17.eps}\n\\caption{SDSS photometry of M dwarf stars synthesized from SNIFS\n spectra and transmission functions convolved with unit airmass.\n Upper left: r-Ks (where Ks is from the 2MASS point source catalog)\n against M spectral subtype (where K7=-1 and K5=-2). Points are\n colored by the $\\zeta$ parameter, which measures TiO\/CaH ratio and\n is a metallicity diagnostic in the optical. The $\\zeta$ values are\n undefined for late K stars, which are plotted as black points. The\n black curve is a running median (N=81). Upper right: difference of\n r-Ks with respect to the running median vs. spectral type, showing\n no obvious correlation with zeta. Lower left: g-r vs. r-z, showing\n an apparent correlation between these colors and $\\zeta$. The\n contours are the empirical function for $\\zeta$ derived by\n \\citet{West2011}. Lower right: SDSS g-r vs. r-z colors generated by\n the PHOENIX\/BT-SETTL model \\citep{Allard} for log g=5,\n Teff=3500K-4200K, and five different values of the metallicity as\n noted in the legend. The model predicts the more metal-rich M dwarfs\n to have {\\em bluer} g-r colors, while being redder in r-z. The color\n dependence on metallicity in most pronounced in late-type stars, and\n nearly vanishes at K7\/M0.\\label{model}}\n\\end{figure*}\n\nTo test our $\\zeta$ as a tracer of metallicity for dM stars from M3 to\nM6, we compared the values to two recent [Fe\/H] calibration techniques\nfor M dwarfs with solar metallicities. First, we used the photometric\ncalibration of \\citet{Neves2012}, which is based on\noptical-to-infrared V-K color and absolute magnitude M$_{K}$. The\nmethod is sensitive to small variations in V-K\/M$_{K}$ and thus\nrequires an accurate, geometric parallax. A total of 143 stars in our sample have\nparallaxes, and thus can have their metallicities estimated with the\nmethod. Figure~\\ref{zeta_feh} (top panel) plots the estimated $\\rm\n[Fe\/H]$ as a function of $\\zeta$ for those 143 stars. The distribution\nshows significant scatter, but we find a weak correlation of $\\rm\n[Fe\/H]$ with $\\zeta$, which we fit with the relationship:\n\\begin{equation}\n[Fe\/H]_{\\rm N12} = 0.750 \\zeta - 0.743.\n\\end{equation}\nStars are scattered about this relationship with a 1-$\\sigma$\ndispersion of 0.383 dex. One drawback of the photometric metallicity\ndetermination is that it assumes the star to be single. Unresolved\ndouble stars appear overluminous at a given color, and will thus be\ndetermined to be metal-rich. Also, young and active stars often appear\noverluminous in the color-magnitude diagram \\citep{Hawley2002}, and\ntheir metalicities based on V-K\/M$_{K}$ would also be\noverestimated. Multiplicity and activity could therefore contribute in\nthe observed scatter. Stars with $[Fe\/H]_{\\rm N12}>0.4$, in\nparticular, could be overluminous in the V-K\/M$_{K}$ diagram, as their\n$\\zeta$ does not suggest them to be metal-rich.\n\nNext, we retrieve metallicity measurements from\n\\citet{Rojas-Ayala2012}, who estimated $[Fe\/H]$ based on the\nspectroscopic calibration from infrared K-band atomic features. Their\nlist has 37 stars in common with our survey. The $[Fe\/H]$ values are\nplotted as a function of our $\\zeta$ values in the bottom panel of\nFigure~\\ref{zeta_feh}. Again there is significant scatter, but we also\nfind a weak correlation which we fit with the relationship:\n\\begin{equation}\n[Fe\/H]_{\\rm RA12} = 1.071 \\zeta - 1.096,\n\\end{equation}\nabout which there is a dispersion of 0.654 dex. The statistics are\nrelativey poor at this time, and more metallicity measurements in\nthe infrared bands would be useful.\n\n\nThe weak correlation found in both distribution is interesting in\nitself. Using a sample of stars spanning a wide range of metallicities\nand $\\zeta$ values, including metal-poor M subdwarfs, extreme\nsubdwarfs (esdM), and extremely metal-poor ultrasubdwarfs\n(usdM), \\citet{Woolf2009} determined a relationship of the form\n$[Fe\/H] = -1.685 + 1.632 \\zeta$, over the range $0.05<\\zeta<1.10$. All\nthe stars in the two distributions from the present survey have\n$\\zeta$ values between $\\sim$0.9 and $\\sim$1.2, and thus represent the\nmetal-rich end of the distribution. The weaker slopes we find in our\ncorrelations (0.75 and 1.07) may indicate that the relationship levels\noff at high metallicity end, which would make $\\zeta$ much less useful\nas a metallicity diagnostic tool in Solar-metallicity and metal-rich M\ndwarfs. The correlations are however weak, and more accurate\nmeasurement of $\\zeta$ and $Fe\/H$ would be needed to verify this\nconjecture.\n\n\\subsection{The one M subdwarf: PM I20050+5426 (V1513 Cyg)}\n\nThe primary purpose of the $\\zeta$ parameter is the identification of\nmetal-poor M subdwarfs, for which it has already proven effective. By\ndefinition, M subdwarfs are stars with $\\zeta<$0.82\n\\citep{Lepine2007}. Though we have a few stars with values of $\\zeta$\njust marginally under 0.82, only one star clearly stands out as a\ndefinite M subdwarf: the star PM I20050+5426 (= Gl 781) which boasts a\n$\\zeta=0.58$ well within the M subdwarf regime. The star also clearly\nstands out in Figure~\\ref{cah_tio} where it lies noticeably below the\nmain locus at TiO5$_{c}\\simeq$0.75.\n\nPM I20050+5426 is also known as V1513 Cyg, a star previously\nidentified as an M subdwarf by \\citet{Gizis1997}, and one clearly\nassociated with the Galactic halo \\citep{Fuchs1998}. The star is also\nnotorious for being a flare star, with chromospheric\nactivity due not to young age but to the presence of a low-mass\ncompanion on a close orbit \\citep{Gizis1998}. Our own spectrum indeed\nshows a relatively strong line of H$\\alpha$ in emission, which is\nextremely unusual for an M subdwarf. It is an interesting coincidence\nthat the brightest M subdwarf in the northern sky should turn out to\nbe a peculiar object.\n\nIn any case, because the TiO molecular bands are weaker in M subdwarfs\nthan they are in M dwarfs, the use of TiO spectral indices for\nspectral classification leads to underestimates of their spectral\nsubtype. The convention for M subdwarfs is rather to base the\nclassification on the strengths of the CaH bandheads\n\\citep{Gizis1997,Lepine2003,Lepine2007}. We adopt the same convention\nhere, and recalculate the subtype from the mean of Equations 6 and 7\nonly (CaH2 and CaH3 indices). We thus classify PM I20050+5426 as an\nsdM2.0, which is one half-subtype later than the sdM1.5 classification\nsuggested by \\citep{Gizis1997}.\n\n\\subsection{Photometric dependence on metallicity}\n\nA prediction of current atmospheric models is that metallicity\nvariations in M dwarfs yield significant variations in optical\nbroadband colors \\citep{Allard2000}. The metal-poor M subdwarfs have\nin fact long been known to have bluer V-I colors than the more\nmetal-rich field M dwarfs of the same luminosity\n\\citep{Monet_etal.1992,Lepine2003}. The bluer colors are due to\nreduced TiO opacities in the optical, which make the spectral energy\ndistribution of M subdwarfs closer to that of a blackbody, while it\nmakes the metal-rich M dwarfs display extreme red colors.\n\nInterestingly, the SDSS $g-r$ color index shows the opposite trend,\nand is bluer in the more metal-poor stars. This is because the TiO\nbands very strongly depress the flux in the 6000\\AA-7000\\AA\\ (r-band)\nrange, an effect which in fact makes the metal-rich M dwarfs\ndegenerate in $g-r$, as the increased TiO opacities in cooler stars\nbalance out the reduced flux in $g$ from lower T$_{eff}$. This effect\nis much weaker in metal-poor stars due to the reduced TiO opacity,\nwhich makes metal-poor stars go redder as they are cooler, as one\nwould normally expect. This has been observed in late-type M\nsubdwarfs, which have significantly redder color that field M dwarfs\n\\citep{LepineScholz2008}. The color dependence of M dwarfs\/subdwarfs\non metallicity is also predicted by atmospheric\nmodels. Figure~\\ref{model} (bottom-right panel) shows the predicted\n$g-r$ and $r-z$ colors from the PHOENIX\/BT-SETTL model of\n\\citet{Allard}. The models corroborate observations and predict redder\n$g-r$ colors in metal-poor stars.\n\nAlthough $ugriz$ photometry is not available for our stars (all of them\nare too bright and saturated in the Sloan Digital Sky Survey), it is\npossible to use the well-calibrated SNIFS spectrophotometry to\ncalculate synthetic broadband $riz$ magnitudes for the subset of stars\nobserved at UH. We first examine any possible correlation between the\noptical to infrared $r-K_S$ color (taken as a proxy for V-I) and the\n$\\zeta$ index. Figure~\\ref{model} plots $r-K_s$ as a function of\nspectral subtype, with the dots color-coded for the $\\zeta$ values of\ntheir associated M dwarf (top-left panel). We find a tight\nrelationship between $r-K_s$ and spectral subtype, which we fit using\na running median. The residuals are plotted in the top-right panel,\nand show no evidence of a correlation with $\\zeta$. There are a\nsignificant number of outliers with redder $r-K_s$ colors than the\nbulk of the M dwarfs: these likely indicate systematic errors in\nestimating the synthetic $r$ band magnitudes. The absence of any clear\ncorrelation suggests that an optical-to-infrared color such as $r-K_S$\nis not sensitive enough to detect small metallicity variations, at\nleast at the metal-rich end.\n\nThe synthetic $g-r$ and $r-z$ colors are plotted in Figure~\\ref{model}\n(lower-left panel). The redder stars ($r-z$>1.2) show a wide scatter\nin $g-r$, on the order of what is predicted for stars with a range of\nmetallicities $-0.5<[Fe\/H]<0.5$. Though we do not find a clear trend\nbetween the synthetic $g-r$ colors and the $\\zeta$ values measured in\nthe same stars, the high-$\\zeta$ stars (red and orange dots on the\nplot) do seem to have lower values of $g-r$ on average than the\nlow-$\\zeta$ ones (green dots). The trend is suggestive of a\nmetallicity link to both the $g-r$ colors and the $\\zeta$ values, and\nshould be investigated further with data of higher precision.\n\n\n\\section{Chromospheric activity}\n\n\\begin{deluxetable*}{lrrrrrrcccc}\n\\tabletypesize{\\scriptsize}\n\\tablecolumns{11} \n\\tablewidth{0pt} \n\\tablecaption{Survey stars: distances, kinematics, and activity.\\label{table_distance}}\n\\tablehead{\n\\colhead{Star name} & \n\\colhead{$\\pi_{trig}$} &\n\\colhead{$\\pi_{phot}$} &\n\\colhead{$\\pi_{spec}$} &\n\\colhead{U} &\n\\colhead{V} &\n\\colhead{W} &\n\\colhead{EWHA} &\n\\colhead{H$\\alpha$} &\n\\colhead{Xray} &\n\\colhead{UV} \\\\\n\\colhead{} &\n\\colhead{$\\arcsec$} &\n\\colhead{$\\arcsec$} &\n\\colhead{$\\arcsec$} &\n\\colhead{km s$^{-1}$} &\n\\colhead{km s$^{-1}$} &\n\\colhead{km s$^{-1}$} &\n\\colhead{\\AA} &\n\\colhead{active} &\n\\colhead{active} &\n\\colhead{active}\n}\n\\startdata \nPM I00006+1829 & \\nodata & \\nodata & \\nodata &\\nodata&\\nodata&\\nodata&\\nodata& -& -& -\\\\\nPM I00012+1358S & \\nodata & 0.030$\\pm$ 0.008& 0.028$\\pm$ 0.008& -13.9& 12.1&\\nodata& 0.41& -& -& -\\\\\nPM I00033+0441 & 0.0342$\\pm$ 0.0032& 0.031$\\pm$ 0.008& 0.030$\\pm$ 0.009& 8.5& -6.8&\\nodata& 0.39& -& -& -\\\\\nPM I00051+4547 & 0.0889$\\pm$ 0.0014& 0.078$\\pm$ 0.021& 0.083$\\pm$ 0.024& -38.2&\\nodata& -15.9& 0.47& -& -& -\\\\\nPM I00051+7406 & \\nodata & \\nodata & \\nodata &\\nodata&\\nodata&\\nodata&\\nodata& -& -& -\\\\\nPM I00077+6022 & \\nodata & 0.078$\\pm$ 0.031& 0.091$\\pm$ 0.027& -15.1&\\nodata& -4.4& -3.11& Y& -& -\\\\\nPM I00078+6736 & \\nodata & 0.046$\\pm$ 0.012& 0.055$\\pm$ 0.016& 5.1&\\nodata& -8.5& 0.39& -& -& -\\\\\nPM I00081+4757 & \\nodata & 0.071$\\pm$ 0.028& 0.061$\\pm$ 0.018& 7.9&\\nodata& 1.8& -2.93& Y& -& Y\\\\\nPM I00084+1725 & 0.0460$\\pm$ 0.0019& 0.040$\\pm$ 0.011& 0.040$\\pm$ 0.012& 11.2& 0.7&\\nodata& 0.41& -& -& -\\\\\nPM I00088+2050 & \\nodata & 0.078$\\pm$ 0.031& 0.080$\\pm$ 0.024& 9.5&\\nodata& -10.1& -4.64& Y& -& Y\\\\\nPM I00110+0512 & 0.0233$\\pm$ 0.0038& 0.032$\\pm$ 0.009& 0.031$\\pm$ 0.009& -48.5& -14.3&\\nodata& 0.35& -& -& -\\\\\nPM I00113+5837 & \\nodata & \\nodata & \\nodata &\\nodata&\\nodata&\\nodata&\\nodata& -& -& -\\\\\nPM I00118+2259 & \\nodata & 0.055$\\pm$ 0.022& 0.054$\\pm$ 0.016& -3.0&\\nodata& -16.7& 0.30& -& Y& -\\\\\nPM I00125+2142En& 0.0358$\\pm$ 0.0028& 0.023$\\pm$ 0.006& 0.024$\\pm$ 0.007& -5.9&\\nodata& -32.5& 0.44& -& -& -\\\\\nPM I00131+7023 & \\nodata & 0.037$\\pm$ 0.010& 0.037$\\pm$ 0.011& -6.2&\\nodata& 16.8& 0.37& -& -& -\n\\enddata\n\\end{deluxetable*} \n\n\\begin{figure}\n\\vspace{-0.3cm}\n\\hspace{-0.6cm}\n\\includegraphics[scale=0.60]{f18.eps}\n\\caption{Fraction of active stars as a function of the spectral\n subtype M. The rise at later subtypes is consistent with earlier\n studies of field M dwarf, which shows increased activity levels in\n mid-type M dwarfs. The fraction level at later subtype is however\n higher than that measured in the SDSS spectroscopic\n catalog.\\label{frac_active}}\n\\end{figure}\n\n\nTo evaluate the presence of H$\\alpha$ in emission in our M dwarfs, we\nused the H$\\alpha$ equivalent width index EWHA defined as:\n\\begin{equation}\nEWHA = 100{\\rm \\AA} \\left[ 1 - \\frac{ 14{\\rm \\AA}\n \\int_{6557.61}^{6571.61} S(\\lambda)\n d\\lambda}{ 100{\\rm \\AA} \\left( \\int_{6500}^{6550} S(\\lambda) d\\lambda +\n \\int_{6575}^{6625} S(\\lambda) d\\lambda \\right)} \\right],\n\\end{equation}\nwhere $S(\\lambda)$ is the observed spectrum. The EWHA index measures\nthe flux in a region (6557.61\\AA-6571.61\\AA), which includes the\n$H\\alpha$ line, in relation to a pseudo-continuum region spanning\n6500\\AA-6550\\AA\\ and 6575\\AA-6625\\AA; the calculation provides a value\nin units of wavelength (\\AA) like the traditional equivalent\nwidth. Note that for an $H\\alpha$ line in emission, values of the EWHA\nindex are negative, following convention. Assuming the W1-W2 region to\nmeasure the true spectral continuum, then the EWHA index would measure\nthe true equivalent width of $H\\alpha$. As it turns out, the $W1-W2$\nregion often has a higher mean flux than the $W3-W4$ region without\nthe $H\\alpha$ emission component, which means that the EWHA index\nsystematically underestimate the equivalent width of the $H\\alpha$\nline. The index is however reproducible and more convenient for\nautomated measurement than, e.g. manual evaluation of the equivalent\nusing interactive software such as IRAF.\n\nThe EWHA index was measured for all spectra in our sample, and used to\nflag active stars. Following \\citet{West2011}, we defined a\nstar to be chromospherically ``active'' if EWHA $<-0.75\\AA$, which\nusually corresponds to a clearly detectable $H\\alpha$ line in\nemission. Values of the EWHA index are listed in\nTable~\\ref{table_distance}. Under the above criterion, 171 M dwarfs in\nour survey are considered active.\n\n\\citet{Hawley1996} found that active stars (by their criterion,\nEW(H$\\alpha$) $> 1 \\AA$) have slightly redder $V-K$ colors for the\nsame value of TiO5 index, an effective temperature proxy. We\ncalculated a median ($N=30$) locus of $V-K$ vs. TiO5 index for our\nentire sample and found that 13 active stars are bluer than this\nlocus, while 45 are redder, seemingly confirming their result. The\nlarge scatter in $V-K$ colors however prevents us from quantifying\nthis offset more precisely.\n\nThe fraction of stars that are active at each spectral subtype is shown\nin Figure~\\ref{frac_active}, with error bars computed from the binomial\ndistribution. The increase in the active fraction with spectral type\nis consistent with previous studies\n\\citep{JoyAbt1974,Hawley1996,West2004,West2008,Kruse_etal.2010,West2011}. \nOur active fractions are higher at subtype M4-M6 than the\n\\citet{Hawley1996} results, even when using their criterion to define active\nstars (see above). This may be a result of a slightly different\ndefinition for EW, or a result of the relatively small number of\nobjects. Our active fractions are also higher than the \\citet{West2011}\nresults at each subtype. This discrepancy is likely caused by the\nmagnitude limit imposed in our survey: our objects are all nearby, and\nrelatively close to the Galactic plane (see Section 8), which makes\nthem statistically younger, as also suggested \\citet{West2011}. The\nactive fractions for our stars are closer to the active fractions for\nthe \\citet{West2011} stars in bins of stars closest to the Galactic\nplane.\n\nAnother chromospheric activity diagnostic in M dwarfs is the detection\nof X-rays. M dwarfs that are X-ray bright are often young, and this has\nbeen used to identify members of nearby young moving groups\n\\citep[e.g.,][]{Gaidos1998, Zuckerman2001, Torres2006}, and other\nyoung stars in the Solar Neighborhood \\citep{Riaz2006}. Most recently,\n\\citet{Shkolnik2009,Shkolnik2012} and \\citet{Schlieder2012} have used\nthe ratio of {\\it ROSAT} X-ray flux to 2MASS {\\it J} or {\\it K}-band\nflux to identify candidate members of young moving groups. This\ntechnique is particularly effective for objects $\\lesssim70$pc away,\nwhich includes all the stars in our sample. The\n\\citet{LepineGaidos.2011} catalog, from which our targets are drawn,\nwas already cross-matched to the {\\it ROSAT} All-Sky Bright Source\nCatalog \\citep{Voges.etal.1999} and the {\\it ROSAT} All-Sky Survey\nFaint Source Catalog \\citep{Voges.etal.2000}.\n\nWe have computed the X-ray flux for our survey stars from the measured\ncount rate and hardness ratio (HR1) using the prescription in\n\\citet{Schmitt1995}. Figure~\\ref{act_xray} shows the distribution of\nX-ray flux as a function of $V-K$ color, for the 290 M dwarfs with\n{\\it ROSAT} detections. Dots are color-coded according to the strength\nof the H$\\alpha$ emission, as measured by the EWHA index. As\nexpected, M dwarfs with strong H$\\alpha$ emission also tend to be more\nX-ray bright. Objects with $\\log F_X \\ F_K > -2.6$ (above the dashed\nline in Figure~\\ref{act_xray}) are considered bright enough in X-ray\nto qualify as chromospherically active, following the definition of\n\\citet{Schlieder2012}. Some 154 of the X-ray sources are active\nbased on their EWHA values, and all of them also qualify as active\nstars based on their X-ray fluxes. On the other hand, 53 M dwarfs\nidentified as active based on X-ray flux do not display significant\nH$\\alpha$ emission in our spectra; most of them tend to be earlier M\ndwarfs, in which H$\\alpha$ emission is not as easily detected as in\nlater type objects because of their higher continuum flux near\n$\\lambda6563\\AA$. There are also 22 stars in our survey which are active\nbased on $H\\alpha$ but are not detected by ROSAT. This suggests that\nonly two thirds of the ``active'' stars will be diagnosed as such from\nboth X-ray and H$\\alpha$ emission, while the other third will show\nonly either. This could be due to source confusion in the ROSAT X-ray\nsurvey, variability in either X-ray or H$\\alpha$ emission, or, in the\ncase of the X-ray flux, non-uniform sky coverage by ROSAT. We\ncalculated the luminosity ratio index $L_X\/L_{H\\alpha}$ of active\nstars, as defined by the criterion EW(H$\\alpha$) $>1 \\AA$ following\nthe procedure of \\citet{Hawley1996}, and adopting the relation $V-R\n\\approx 0.7 + 0.06 {\\rm SpTy}$ to estimate an $R$ magnitude and the\ncontinuum flux at the H$\\alpha$ line. We find that the ratio is\ninsensitive to bolometric magnitude and spectral type, and has a\nmedian value of 0.85. This is higher than the \\citet{Hawley1996}\naverage of $\\sim 0.5$, and may in part be due to Malmquist bias in the\nflux-limited ROSAT survey favoring the inclusion of the most X-ray\nluminous stars, as well as greater variation in the ratio because of\nthe elapsed time (two decades) between the ROSAT survey and\nour observations.\n\n\\begin{figure}\n\\vspace{-0.3cm}\n\\hspace{-0.6cm}\n\\includegraphics[scale=0.53]{f19.eps}\n\\caption{X-ray luminosity normalized by the flux in the infrared K$_s$\nband, plotted as a function of the optical-to-infrared color $V-K$,\nfor stars in our sample which have counterparts in the ROSAT all-sky\npoints source catalog. The color scheme shows the H$\\alpha$ equivalent\nwidth; active stars are found to have large X-ray flux, as expected\nfrom chromospheric activity.\\label{act_xray}}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{-0.3cm}\n\\hspace{-0.6cm}\n\\includegraphics[scale=0.53]{f20.eps}\n\\caption{Normalized near-UV flux as a function of the\n optical-to-infrared $V-K_s$ color. The color scheme shows the\n strength of the H$\\alpha$ equivalent width. Closed circles show\n stars identified as active based on X-ray emission, closed circles\n show stars with low or no detection in ROSAT. \\label{act_uv}}\n\\end{figure}\n\nActive stars can also be identified from ultra-violet\nexcess, as suggested in Section 3.4. \\citet{Shkolnik2011,Shkolnik2012}\nshowed that {\\it GALEX} UV fluxes can identify young M dwarfs in\nnearby moving groups, and can identify active stars to larger\ndistances. Figure~\\ref{act_uv} shows the {\\it GALEX} NUV to 2MASS\n{\\it J} flux ratio, for the objects with UV detections. The dashed\nline shows the selection criteria of \\citet{Shkolnik2011}. Dot colors\nrepresent the EWHA index values for the stars, while filled circles\nindicate objects with $\\log F_X \/ F_K > -2.6$, i.e. stars whose X-ray\nflux does not identify them as being active. Overall, there is a good\ncorrepondence between the different activity diagnostics. However,\nthere are some stars identified as active based on UV flux that are\nnot identified as such from their X-ray and\/or $H\\alpha$\nemission. Again this suggests that a complete identification of active\nM dwarfs in the solar vicinity may require a combination of diagnostic\nfeatures.\n\nIn any case our survey, which combines X-ray, UV, and $H\\alpha$\ndiagnostics, provides a valuable subset for identifying low-mass young\nstars in the Solar Neighborhood, and may potentially yield new members\nof young moving groups, or even the identification of new moving\ngroups. The last three columns in Table~\\ref{table_distance} display\nflags for stars found to be active from either H$\\alpha$, X-ray, or UV\nflux. The flag indicates activity by a ``Y''. Absence of a flag does not\nnecessarily indicate absence of activity: the GALEX survey does not\ncover the entire sky, and the ROSAT X-ray survey is not uniform in\nsensitivity, so a non-detection in either does not necessarily\nindicate quiescence. Activity diagnostics could also be\ntime-variable. H$\\alpha$ equivalent width is particular are know to be\nvariable on various timescales \\citep{Bell2012}. In any case, there is\na good correlation between the different diagnostics. We flag 175\nstars as active based on H$\\alpha$, 42 based on X-ray emission, and\n172 based on UV excess. Overall, 252 stars are assigned one or more\nactivity flags: 19 stars have all three flags on, 99 stars get two\nflags, and 137 get only one.\n\n\\section{Distances and kinematics}\n\n\\subsection{Spectroscopic distances}\n\\label{sec:distances}\n\n\\begin{figure}\n\\vspace{0.0cm}\n\\hspace{-0.2cm}\n\\includegraphics[scale=0.85]{f21.eps}\n\\caption{Absolute visual ($M_V$) and infrared ($M_J$) magnitudes for\n the 631 stars in our survey for which geometric parallax\n measurements exist in the literature. Top panels: absolute\n magnitudes against $V-J$ color, which follow the color-magnitude\n relationship used in \\citet{LepineGaidos.2011} to estimate\n photometric distances. Bottom panel: absolute magnitudes as a\n function of spectral subtype, based on the spectral-index\n classification described in this paper. Active stars are plotted\n in green, and are found to be overluminous at a given $V-J$ color\n and given spectral subtype, compared with non-active stars. The\n offset in notable for stars of earlier M subtypes (or bluer $V-J$\n colors).\\label{dist_cal}}\n\\end{figure}\n\nAstrometric parallaxes are available for 631 of the stars in our\nsample, spanning the full range of colors and spectral subtypes. We\ncombine these data with our spectroscopic measurements to re-evaluate\nphotometric and spectroscopic distances calibrations for M dwarfs in\nour census. Absolute visual magnitudes $M_V$ are calculated and are\nplotted against both $V-J$ color and spectral subtype in\nFigure~\\ref{dist_cal}. M dwarfs with signs of activity (H$\\alpha$, UV,\nX-ray) are plotted in green, other stars are plotted in\nblack. The solid red lines are the best-fit second order polynomials,\nwhen both active and inactive stars are used, and after elimination of\n3$\\sigma$ outliers. The equations for the fits, where\n$spT$ is the spectral type (K7 is -1 and M0=0, etc) are: \n\\begin{eqnarray}\nM_J = 1.194 + 1.823 (V-J) - 0.079 (V-J)^2\\\\\nM_J = 5.680 + 0.393 (SpT) + 0.040 (SpT)^2\n\\label{eqn:mj_vs_spt}\n\\end{eqnarray}\nwhere SpT are the spectral types, and with the least-squares fit\nperformed after exclusion of 3-sigma outliers. The 1$\\sigma$\ndispersion about these relationships are $\\pm0.61$ mag for\n(M$_J$,V-J), and $\\pm0.52$ mag for (M$_J$,SpT). The smaller scatter in\nthe spectroscopic relationship suggests that spectroscopic distances\nmay be marginally more accurate than the photometric ones. We suspect\nthat the larger uncertainty on the photographic $V$ magnitudes may be\nthe cause.\n\nThe most notable feature in the diagrams is that active stars appear\nto be systematically more luminous than non-active stars. This\ncorroborates the observation made previously by \\citet{Hawley2002}, as\npart of the PMSU survey. \\citet{Hawley2002} found that active stars\nwere more luminous by 0.48mag in a diagram of $M_K$ against TiO5\nindex, used as proxy for spectral subtype. For stars in our census, we\nfind that among stars of spectral subtype M2 and earlier, active stars\nare on average 0.46mag more luminous at a given subtype than\nnon-active stars; in the color-magnitude diagram, bluer stars of\ncolors $V-J<4.0$ which are active are on average 0.47mag more luminous\nthan non-active stars of the same color. Both values agree well with\nthe values quoted by \\citet{Hawley2002}. The systematic overluminosity\nof active stars is also responsible for some of the scatter in the\ncolor-magnitude and spectral-type magnitude relationships. If we\nexclude active stars, the scatter about the color-magnitude\nrelationship fals marginally to $\\pm0.58$, and the scatter in the\nspectral-type magnitude relationship falls to $\\pm0.49$. \n\nThe offset in absolute magnitude between active and non-active stars\nsuggests that spectroscopic and photometric distances would be more\naccurate for active stars in our census if their estimated absolute\nmagnitudes were made 0.46mag brighter that suggested by\nEquation~\\ref{eqn:mj_vs_spt}. We therefore adopt the following\nrelationships to be applied only on active stars of subtype M2.5 and\nearlier:\n\\begin{eqnarray}\n{[M_J]}_{early-active} = 0.734 + 1.823 (V-J) - 0.079 (V-J)^2\\\\\n{[M_J]}_{early-active} = 5.220 + 0.393 (SpT) + 0.040 (SpT)^2 \n\\label{eqn:mj_vs_spt2}\n\\end{eqnarray}\nAgain we define as ``active'' any star which qualifies as such base on\nany one of our criteria (H$\\alpha$, UV, X-ray). There are several\nreasons that would explain why active, early-type stars are more\nluminous at a given subtype. Activity in an early-type M dwarf could\nmean that the star is younger \\citep{Delfosse_1998}; early-type M\ndwarfs with ages $<100$Myr are known to be overluminous at a given\ncolor, due to lower surface gravity \\citep{Shkolnik2012}. Older stars\ncould remain active due to interaction with a close companion\n\\citep{Morgan_2012}, in which case the active stars would also appear\noverluminous due to this unresolved companion. Late-type M dwarfs,\nhowever, can remain active for long periods of time, and would thus\nnot require the star to be young or have a close companion.\n\nSplitting the active and inactive stars results in lowering the\nscatter of non-active stars in the subtype-magnitude\nrelationship (to $\\pm$0.5mag). Overall, our spectroscopic distances\nfor non-active M dwarfs provide a $1\\sigma$ uncertainty of $\\pm26\\%$\non the distance. For active stars, we find a scatter of $\\pm$0.6mag,\nwhich suggests distance uncertainties of $\\pm32\\%$. Our photometric\ndistances estimated from (V,V-J) have similar though perhaps slightly\nlarger uncertainties. Photometric and spectroscopic parallaxes,\nestimated from the above relationships for active and non-active\nstars, are listed in Table~\\ref{table_distance}.\n\nWe estimated the effect of two relevant sampling biases on the\ncalibration between $M_J$ and $V-J$ color. In Eddington bias,\nphotometric errors scatter more numerous, bluer, and intrinsically\nbrighter stars to redder apparent colors than redder stars are\nscattered to bluer apparent colors \\citep{Eddington1913}. The net\neffect is to make stars at a given apparent color appear more luminous\nthan they are. In Lutz-Kelker (LK) bias, a form of Malmquist bias,\nerrors in trigonometric prallax will scatter more numerous and more\ndistant stars with lower parallax to higher apparent parallax values,\nmaking them appear less luminous than they are \\citep{Lutz1973}. By\ntaking the derivative of $M_J$ with respect $V-J$ color, multiplying\nby the derivative of the number of stars in our $J$-magnitude-limited\ncatalog with respect to $M_J$, assuming that the errors in $V-J$ are\ngaussian-distributed with standard deviation $\\sigma_{V-J}$, and\nintegrating over the distribution, we find the Eddington bias in $M_J$\nto be;\n\\begin{equation}\n\\Delta_E = -\\ln 10 \\left[1.918 = 0.178\\left(V-J\\right)\\right]^2 \\left(0.6 -\n0.4\\gamma\\right) \\sigma_{V-J}^2,\n\\end{equation}\nwhere $\\gamma$ is the power-law index of a luminosity function for M\nstars which we take to be 0.325. Performing a similar derivation for\nthe effect of L-K bias on $M_J$, we find:\n\\begin{equation}\n\\Delta_{LK} = \\frac{15}{\\ln 10} \\sigma_{\\pi}^2,\n\\end{equation}\nwhere $\\sigma_{\\pi}$ is the fractional error in parallax. Using\npublished parallax errors for our {\\it Hipparcos} stars and adopting a\nconservative $\\sigma_{V-J} = 0.05$, we find that L-K bias usually\ndominates over Eddington bias and that 74\\% of our stars have a total\nbias of less than +0.2 magnitudes. A running median ($N = 100$)\nvs. $V-J$ color is highest ($\\sim 0.15$) for the bluest ($V-J =\n2.7$) stars and falling to less than +0.05 magnitudes for $V-J > 3.3$.\nAn analogous analysis can be performed for the bias in $M_J$\nvs. spectral type, with a similar result. To debias values of $M_J$,\nthese values should be {\\it subtracted} from our calibration but we do\nnot perform that operation here because of the small magnitude of the\neffect, which would overestimate distances by about 2\\% on\naverage. The correction would also seem negligible compared with the\nintrinsic scatter in our color-magnitude and subtype-magnitude\nrelationships are of order $\\pm0.5-0.6$, much larger than the L-K\ncorrection. \n\n\\begin{figure}\n\\vspace{-0.3cm}\n\\hspace{-0.2cm}\n\\includegraphics[scale=0.44]{f22.eps}\n\\caption{Top: distribution of spectroscopic distances for the stars in\n our survey, shown for three ranges of spectral subtypes. Early-type\n stars are clearly sampled over a larger volume, which explains why\n they dominate our survey. Bottom: distribution of Galactic scales\n heights for the same stars, assuming that the Sun is hovering 15~pc\n sbove the Galactic midplane. As expected from our magnitude-limited\n sample, stars of later spectral subtypes (and lower luminosity) are\n found at shorter distances. Our survey samples a region well within\n the Galactic thin disk.\\label{disthist}}\n\\end{figure}\n\nFigure \\ref{disthist} shows the distribution of photometric distances\nfor our complete sample using Equation \\ref{eqn:mj_vs_spt} and the\nM$_J$=M$_J$(V-J) color-magnitude relationship. The spectral subtypes are\nplotted in separate colors and demonstrates that the earlier-type\nstars, which are intrinsically brighter, are sampled to significantly\nlarger distances compared with the later-type stars. In the 20pc\nvolume, the M3-M4 stars still appear to dominate. We also plot the\nGalactic height of the stars in our sample, adopting a Galactic height\nof 15 pc for the Sun \\citep{Cohen1995,Ng1997,Binney1997}. It is clear\nthat our survey is largely contained within the Galactic thin disk,\nand barely extends south of the midplane. This is consistent with the\nrelative absence of metal-poor stars associated with the thick disk\nand halo.\n\n\\subsection{Kinematic analysis}\n\n\\begin{figure*}\n\\vspace{-0.3cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.82]{f23.eps}\n\\caption{Velocity-space projections for the M dwarfs in our\n survey. Velocies are calculated based on photometric distances and\n proper motions alone (no radial velocities used). Each star in our\n census is thus displayed in only one panel, which corresponds to the\n projection in which the radial velocity of the star has the smallest\n contribution. Stars with significant levels of H$\\alpha$ emission,\n i.e. chromospherically active M dwarfs, are plotted in\n red. \\label{uvw}}\n\\end{figure*}\n\nAccurate radial velocities are not available for most of the stars in\nour sample, which prevents us from calculating the full (U,V,W)\ncomponents of motion for each individual star. However, it is possible\nto use the distance measurements (for stars with parallaxes) or\nestimates (for stars with no parallax), and combine them to the proper\nmotions to evaluate with some accuracy at least two of these\ncomponents for each star. More specifically, we calculate the (U,V,W)\ncomponents by assuming that the radial velocities $R_V=0$. We then\nconsider the (X,Y,Z) positions of the stars in the Galactic reference\nframe, from the distances and sky coordinates. For stars with the\nlargest component of position in +X or -X, the radial velocity mostly\ncontibute to U, and has minimal influence on the values of V and\nW. Likewise stars with the largest component of position in +Y or -Y\n(+Z or -Z) make good tracers of the velocity distribution in U and W\n(U and V). We use this to assign any one of (U,V) or (U,W) or (V,W)\nvelocity component doublet to every M dwarf in our catalog.\nEstimated values of the components of velocity are listed in\nTable~\\ref{table_distance}. For each star, one of the components is\nmissing, which is the component that would most depend on the radial\nvelocity component based on the coordinates of the star. Again, the\nother two components are estimated only from proper motion and\ndistance. For the distance, we use the trigonometric parallaxes\nwhenever available; otherwise the spectroscopic distances as are used,\nbased on the raltionships described in the previous section.\n\nThe resulting velocity distributions are displayed in\nFigure~\\ref{uvw}. We measure mean values of the velocity components\nusing all allowable values and find:\n\\begin{displaymath}\n = -8.1~{\\rm km s^{-1}}, \\sigma_U = 32.8~{\\rm km s^{-1}},\n\\end{displaymath}\n\\begin{displaymath}\n = -17.0~{\\rm km s^{-1}}, \\sigma_V = 22.8~{\\rm km s^{-1}},\n\\end{displaymath}\n\\begin{displaymath}\n = -6.9~{\\rm km s^{-1}}, \\sigma_W = 19.3~{\\rm km s^{-1}}.\n\\end{displaymath}\nThe values are also largely consistent with those found from the PMSU\nsurvey and desribed in \\citep{Hawley1996}. They are also remarkably\nsimilar to the moments of the velocity components calculated by\n\\citet{Fuchs2009} for SDSS stars of the Galactic thin disk, and which\nare $ = -8.6, \\sigma_U = 32.4$, $ = -20.0, \\sigma_V = 23.0$, and\n$ = -7.1, \\sigma_W = 18.1$. The agreement suggests that our\ndistance estimates are reasonably accurate, and it corroborates\nearlier results about the kinematics of the local M dwarf population\nwhich indicate a larger scatter of velocities in $U$. The mean values\nof $$ and $$ are consistent with the offsets from the local\nstandard of rest as described in \\citet{DehnenBinney_1998}\n\nActive stars are found to have significantly smaller dispersions in\nvelocity space. All 252 stars with at least one activity flags\n(i.e. stars found to be active either from H$\\alpha$, X-ray flux, or\nUV excess) are plotted in red in Figure~\\ref{uvw}. Those active stars\nhave first and second moments:\n\\begin{displaymath}\n = -9.3~{\\rm km s^{-1}}, \\sigma_U = 25.2~{\\rm km s^{-1}},\n\\end{displaymath}\n\\begin{displaymath}\n = -13.4~{\\rm km s^{-1}}, \\sigma_V = 16.8~{\\rm km s^{-1}},\n\\end{displaymath}\n\\begin{displaymath}\n = -6.6~{\\rm km s^{-1}}, \\sigma_W = 15.3~{\\rm km s^{-1}}.\n\\end{displaymath}\nThe values are consistent with \\citep{Hawley1996}, who also reported\nthat active M dwarfs tend to have a smaller scatter compared with\ninactive M dwarfs. The smaller dispersion values suggest that these\nactive stars may be significantly younger than the average star in the\nSolar Neighborhood.\n\nIn any case, we also note that the velocioty space distribution is\nnon-uniform and shows evidence for substructure. Our M dwarf data\nshows velocity-space substructure as that observed and described in\n\\citet{Nordstrom2004,Holberg2008} for solar-type stars in the vicinity\nof the Sun. This substructure is sometimes referred to as ``streams''\nor ``moving groups'', although an analysis by \\citet{Bovy_Hog2010}\nshows that these groups do not represent coeval populations arising\nfrom star-formation episodes. The velocity-space substructure is more\nlikely transient and associated with gravitational perturbations which\nare the signature of the Galactic spiral arms\n\\citet{QuillenMinchev2005} and the Galactic bar \\citet{Minchev2012}. A\nsimple description of the velocity-space distribution in terms of\nmeans values and dispersions, or as velocity ellipsoid, is therefore only a\ncrude approximation of a more complex and structured distribution.\n\nFinally, we note that the stars of our catalog that were previously\npart of the CNS3 and stars with measured trigonometric parallaxes\n(e.g. from the Hipparcos catalog) tend to have larger velocity\ndispersions, with $(\\sigma_U,\\sigma_V,\\sigma_W)$=(38.4,25.9,23.1),\nwhile the newer stars have\n$(\\sigma_U,\\sigma_V,\\sigma_W)$=(26.2,19.4,15.3). The difference could\nbe due to systematic underestimation of the photometric\/spectroscopic\ndistances, but a more likely explanation is that the CNS3 and parallax\nsubsample suffers from proper motion selection. This is because most\nof the CNS3 stars and M dwarfs monitored with Hipparcos were selected\nfrom historic catalogs of high proper motion stars, which have a\nhigher limit than the SUPERBLINK proper motion catalog used in the\nLG2011 selection. This kinematic bias means that the current subset of\nM dwarfs monitored for exoplanet programs suffers from the same\nkinematic bias, which could possibly introduce age and metallicity\nselection effects.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nWe have now compiled spectroscopic data for a nearly complete list of\nM dwarfs in the northern sky with apparent magnitudes $J<9$. Our\nsurvey identifies a total of 1,403 very bright M dwarfs. Our new\ncatalog provides spectral subtypes and activity measurements\n($H\\alpha$ emission) for all stars, as well as a rough indicator of\nmetallicity in the guise of the $\\zeta$ parameter, which measures the\nratio of TiO to CaH bandstrengths. Only one of the stars in the survey\nis unambiguously identified as a metal-poor M subdwarf (PM I20050+5426\n= V1513 Cyg).\n\nOur target stars were identified from the all-sky catalog of bright M\ndwarfs presented in \\citet{LepineGaidos.2011}. As such, our\nspectroscopic survey suffers from the same selection effects and\ncompleteness issues. The completeness and bias of the SUPERBLINK\nproper motion survey, from which these stars were selected is\ndiscussed a length in \\citet{LepineGaidos.2011}. In the northern\nhemisphere, the SUPERBLINK catalog is complete for proper motions $\\mu\n>40$~mas$^{-1}$. We show in \\S4 that there is a kinematic bias in the\ncatalog which excludes stars with very low transverse motions (in the\nplane of the sky), but the low proper motion limit means that less\nthan 5\\% of stars within 65 parsecs of the Sun are in fact excluded in\nthe selection. In addition, we estimate that $\\approx$5\\% of the nearby,\nbright M dwarfs may have escaped our target selection scheme due to\nfaulty magnitudes. Therefore, we estimate that our census most likely\ninclude $>90\\%$ of all existing M dwarfs in the northern sky with\n$J<9$. Early-type K7-M1 dwarfs have absolute magnitudes\n$M_J\\approx5.5$, and our $J<9$ sample thus identifies them well to a\ndistance of about 50pc, as confirmed in Figure~\\ref{disthist}. On the\nother hand, later type M3-M4 dwarfs have $M_J\\approx8$ and thus only\nthose at very close distance range ($<$15~pc) will be included in the\ncatalog. Their completeness will however be very high because the\nproper motion bias excludes less than $1\\%$ of the stars within that\ndistance range. In any case, the different survey volumes for\nearly-type and late-type stars means that our survey favors the former\nover the latter by a factor of about 35 to 1. It is thus no surprise\nthat our spectroscopic catalog is dominated by early-type M dwarfs.\n\nAn important result of our spectroscopic analysis is the\nidentification of systematic errors in the spectral indices, which\nmeasure the strenghts of the CaH, TiO, and VO molecular\nbands. Systematic offsets between data obtained at MDM Observatory and\nat the University of Hawaii 2.2-meter telescopes, as well as offsets\nbetween these and the values measured for the sames stars in the\nPalomar-MSU survey of \\citet{Reid1995}, indicate that these spectral\nindices are susceptible to spectral resolution and spectrophotometric\ncalibration, such that using the raw measurements may result in\nsystematic errors in evaluating spectral subtypes and the metallicity\n$\\zeta$ parameter. In Section 3.2 we outline a procedure for\ncalculating corrected indices, based on a calibration of systematic\noffsets between two observatories. A proper calibration requires that\nlarge numbers of stars be re-observed every time a new observatory\nand\/or instrumental setup is used, in order to calibrate the offsets\nand correct the spectral indices. Only the corrected spectral indices\ncan be used reliably in the spectral subtype and $\\zeta$\nrelationships, which are calibrated with respect to the corrected\nvalues. We adopt the Palomar-MSU measurement as our standard of\nreference for the spectral indices, and correct our MDM and UH\nvalues accordingly.\n\nIn the end, this catalog provides a useful list of targets for\nexoplanet searches, especially those based on the radial velocity\nvariation method. Current methods and instruments require relatively bright\nstars to be efficient, and the stars presented in our spectroscopic\ncatalog all constitute targets of choice, having been vetted for\nbackground source contamination. Our accurate spectral types will be\nuseful to guide radial velocity surveys and selct stars of\ncomparatively lower masses.\n\nWe also provide diagnostics for chromospheric activity from H$\\alpha$\nemission, X-ray flux excess, and UV excess. Besides being useful to\nidentify more challenging sources for radial velocity surveys, they\nalso isolate the younger stars in the census. Follow-up radial\nvelocity observations could tie some of the stars to nearby moving\ngroups, and these objects would be prime targets for exoplanet\nsearches with direct imaging methods.\n\n\n\\acknowledgments\n\n{\\bf Acknowledgments}\n\nThis material is based upon work supported by the National Science\nFoundation under Grants No. AST 06-07757, AST 09-08419, and AST\n09-08406. We thank Greg Aldering for countless instances of\nassistance with SNIFS, the telescope operators of the UH 2.2m\ntelescope, and Justin Troyer for observing assistance. We thank Bob\nBarr and the staff at the MDM observatory for their always helpful\ntechnical assistance.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nBimetric theories are an intensively studied class of massive gravity theories considered as an alternative to general relativity (GR). On one hand they predict new phenomena, such as the graviton oscillation \\cite{DeFelice:2013nba, Narikawa:2014fua}. On the other hand, bimetric theories contain both a massless and a massive spin-2 field. It has been nontrivial to construct a consistent theory of massive gravity. The first bimetric model free of the so-called Boulware-Deser ghost was proposed by Hassan and Rosen \\cite{Hassan:2011zd}, based on the de Rham--Gabadadze--Tolley (dRGT) ghost-free massive gravity model \\cite{deRham:2010kj}. \n\nThe bigravity \\cite{Hassan:2011zd}, although allowing for a stable cosmological evolution, still requires an important fine-tuning of its parameters in order to be consistent. On one hand, it has been shown that to accommodate a stable evolution, the mass parameter $m$ (controlling the graviton potential terms) needs to be generically much larger than today's Hubble parameter, i.e.\\ $m \\gg H_0$ \\cite{Comelli:2012db, DeFelice:2014nja}. This condition forbids the graviton mass to account for the accelerated expansion of the Universe today. On the other hand, one needs another fine-tuning for (i) the Vainshtein mechanism \\cite{vmeca} to effectively screen extra forces on Solar System scales, for (ii) letting the theory be differentiable from GR by leaving nontrivial phenomenology, while (iii) satisfying the Higuchi bound $m_T > \\mathcal{O}(1) H_0$ \\cite{higuchi}, where $m_T$ is the mass of the tensor modes (proportional but not equal to $m$). Finally, the strong coupling is encountered at a rather low scale $\\Lambda_3=(M_{\\rm Pl}m^2)^{1\/3}$ easily by going early enough in the history of the Universe, which makes the need for a (partial) UV completion all the more important. \n\nIn response to these practical issues, it has been recently proposed to add a new chameleonlike degree of freedom to the theory \\cite{DeFelice:2017oym}. In this model, the constant coefficients appearing in the graviton potential are promoted to be general functions of the new scalar field $\\phi$, and matter is coupled to gravity through a $\\phi$-dependent effective metric. In this way, the effective graviton mass $m_T$ becomes environment dependent, so that $m_T^2$ scales as the local energy density of matter $\\rho$. This mechanism allows us to evade the need for the Vainshtein mechanism to screen the extra gravitational forces on Solar System scales, and lets the theory be viable against strong coupling, Higuchi bound, and instabilities up to the very early Universe. The scalar field also has a high enough mass to be possibly not detectable by fifth-force experiments \\cite{DeFelice:2017oym}. A possible cosmological application of the chameleonic extension of bigravity theory has been studied in \\cite{Aoki:2017ffl}.\n\nIn this work we study further the model presented in Ref.\\ \\cite{DeFelice:2017oym}. Indeed, notwithstanding the arguments in favor of stability and wider applicability that were given, it is important to study the compatibility of the theory versus the observed cosmic evolution. First, we present a detailed study of the scaling solutions, including the conditions for stability under homogeneous perturbations. Second, we present the stability conditions derived from studying the action that is quadratic in inhomogeneous linear perturbations around a flat Friedmann-Lema{\\^i}tre-Robertson-Walker (FLRW) spacetime. Finally, we present a viable set of parameters and initial conditions that upon numerical integration leads to a stable cosmological evolution, including radiation-dominated, matter-dominated, and de Sitter phases.\n\nThe text is organized as follows. In Sec.~\\ref{sec:review} we review the chameleon bigravity model presented in Ref.\\ \\cite{DeFelice:2017oym}, defining the action and the background equations obtained from its variation. In Sec.~\\ref{sec:hompert} we present the scaling solutions of the model, and their respective stability under homogeneous perturbations. In Sec.~\\ref{sec:noinstability} we discuss inhomogeneous linear perturbations of the model, and the derivation of the stability conditions in a general flat FLRW universe. In Sec.~\\ref{sec:numerics} we present the numerical integration, as well as the chosen parameters and initial conditions. Finally, we conclude in Sec.~\\ref{sec:conclusion} and briefly present future extensions of this work. \n\n\\section{Review of the model}\n\\label{sec:review}\n\n\\subsection{Action}\n\nThe chameleon bigravity model is defined by the total action $S_\\textrm{tot} = S_\\textrm{EH} + S_m + S_\\phi + S_\\textrm{mat}$ \\cite{DeFelice:2017oym}. In this model, the usual ghost-free bimetric theory \nis supplemented by a scalar field $\\phi$, coupled to both metrics via the promotion of the coefficients found in the graviton potential into the functions $\\beta_i(\\phi)$. The gravitational part of the action is given explicitly by\n\\begin{align}\nS_\\textrm{EH} &= \\frac{M_g^2}{2}\\int R[g]\\sqrt{-g} d^4x + \\frac{M_f^2}{2}\\int R[f]\\sqrt{-f} d^4x\\,,\\\\\nS_m &= M_g^2 m^2\\int \\sum_{i=0}^4 \\beta_i(\\phi)U_i[s]\\sqrt{-g} d^4x\\,, \\\\\nS_\\phi &= - \\frac{1}{2} \\int g^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi\\sqrt{-g} d^4x\\,,\n\\end{align}\nwhere $M_g$ and $M_f$ stand for the respective bare Planck masses of the gravitational $g$ and $f$ sectors. We also define $\\kappa\\equiv M^2_f\/M^2_g$ for later convenience. Just as in the usual bigravity case the construction of the potentials $U_i$ relies on powers of the metric square root $s^\\alpha_\\beta \\equiv (\\sqrt{g^{-1}f})^\\alpha_\\beta$ such that $s^\\alpha_\\gamma s^\\gamma_\\beta= g^{\\alpha\\delta}f_{\\delta\\beta}$. By defining $T_n \\equiv \\textrm{Tr}[s^n]$, we have\n\\begin{align}\nU_0 &= 1\\,,\\quad U_1 = T_1\\,,\\quad U_2 = \\frac{1}{2}[T_1^2 - T_2]\\,,\\nonumber\\\\\nU_3 &= \\frac{1}{6}[T_1^3-3T_2T_1 + 2T_3]\\,, \\nonumber\\\\\nU_4 &= \\frac{1}{24}[T_1^4 - 6T_1^2T_2 + 3T_2^2 + 8T_1T_3 - 6T_4]\\,.\n\\end{align}\nThe potentials $U_0$ and $U_4$ constitute the two cosmological constants of the metric sectors $g$ and $f$, respectively. The terms $\\beta_i(\\phi)U_i$ also play the role of potentials for the field $\\phi$. Finally, to implement the chameleon mechanism, the matter sector is coupled nonminimally to the metric $g_{\\mu\\nu}$, i.e.\\\n\\begin{equation}\nS_\\textrm{mat} = \\int \\mathcal{L}_\\textrm{mat}(\\psi,\\tilde{g}_{\\mu\\nu}) d^4x\\,,\n\\end{equation}\nwhere $\\psi$ stands for the different matter fields, $\\tilde{g}_{\\mu\\nu} = A^2(\\phi)g_{\\mu\\nu}$, and $A(\\phi)$ is a universal coupling function. In order to simplify the treatment, we adopt the choice of general functions $A(\\phi)$ and $\\beta_i(\\phi)$, following Ref.\\ \\cite{DeFelice:2017oym}. We thus set\n\\begin{align}\nA(\\phi) &= e^{\\beta\\phi\/M_g}\\nonumber\\,,\\\\\n\\beta_i(\\phi) &= -c_i e^{-\\lambda\\phi\/M_g}\\label{eq:toymodel}\\,,\n\\end{align}\nwith $i\\in\\{0, \\cdots, 4\\}$. These choices are sufficient to obtain a scaling solution described in Sec.~\\ref{sec:hompert}. We will use these specific functions for our numerical work. \n\n\\subsection{Background equations}\n\nIn order to study cosmological backgrounds, we choose a flat FLRW ansatz for both metrics, i.e.\n\\begin{align}\nds^2_g = - dt^2 + a^2(t)\\delta_{ij}dx^idx^j\\,,\\quad ds^2_f = \\xi^2(t)\\left[-c^2(t)dt^2 + a^2(t)\\delta_{ij}dx^idx^j\\right]\\,.\n\\end{align} \nUnder these assumptions, the computation of the metric square root $s^\\mu_\\nu$ becomes much simpler. We further define the Hubble parameters associated with each gravitational sector, $H\\equiv \\dot{a}\/a$ and $H_f\\equiv (a\\xi)\\dot{}\/(ac\\xi^2)$, where the dot stands for a derivative with respect to the cosmic time $t$. On such a FLRW background, the equations of motion become the two Friedmann equations\n\\begin{align}\n3H^2 &= \\frac{1}{M_g^2}\\left[\\rho A^4 + \\frac{1}{2}\\dot{\\phi}^2\\right] + m^2 R(\\xi,\\phi)\\,,\\label{eq:friedmannphysical}\\\\\n3 H_f^2 &= \\frac{m^2}{4\\kappa\\xi^3}U_{,\\xi}(\\xi,\\phi)\\,,\\label{eq:friedmannfiducial}\n\\end{align}\n(with $R$ and $U$ defined below) as well as the two dynamical equations\n\\begin{align}\n2\\dot{H} &= -\\frac{1}{M_g^2}\\left[\\left(\\rho+P\\right) A^4 + \\dot{\\phi}^2\\right] + m^2 \\xi (c-1) J(\\xi,\\phi)\\,,\\label{eq:backgroundeinsteinphysical}\\\\\n2 \\dot{H}_f &= m^2\\frac{1-c}{\\kappa\\xi^2} J(\\xi,\\phi) \\label{eq:backgroundeinsteinfiducial} \\,,\n\\end{align}\n(with $J$ defined below) and the equation of motion for the chameleon scalar field\n\\begin{equation}\n\\ddot{\\phi} + 3H\\dot{\\phi} = - \\alpha A^4\\left(\\rho - 3P\\right) + M^2_gm^2Q_{,\\phi}(\\xi,\\phi)\\,,\\label{eq:backgroundscalar}\n\\end{equation}\n(with $Q$ defined below). In these equations we have used \n\\begin{equation}\nR \\equiv U - \\xi U_{,\\xi}\/4\\,,\\quad J \\equiv R_{,\\xi}\/3\\,,\\quad Q \\equiv (c-1)R - cU\\,,\\quad U \\equiv - \\left(\\beta_4\\xi^4 + 4\\beta_3 \\xi^3 + 6 \\beta_2\\xi^2 + 4 \\beta_1\\xi + \\beta_0\\right)\\,,\n\\end{equation}\nand $\\rho$ and $P$ are, respectively, the total energy density and pressure of the matter fields. By combining the Friedmann (\\ref{eq:friedmannphysical}) and second Einstein (\\ref{eq:backgroundeinsteinphysical}) equations, one obtains an algebraic equation for $c$ in terms of other variables, \n\\begin{equation}\nc = \\frac{12 J \\left(H\\xi + \\dot{\\xi}\\right) }{\\xi \\left(12 H J + \\dot{\\phi} U_{,\\xi\\phi}\\right)}\\,.\\label{eq:fiduciallapse}\n\\end{equation}\n\nIn order to represent perfect fluids in the latter analysis, one can choose, for instance, to use $k$-essence scalar fields,\n\\begin{equation}\nS_{\\textrm{mat},\\alpha} = \\int P_\\alpha(X_\\alpha) \\sqrt{-\\tilde g}\\,d^4x\\,,\n\\end{equation}\nwhere $X_\\alpha \\equiv -\\frac{1}{2}{\\tilde g}^{\\mu\\nu}\\partial_\\mu\\psi_\\alpha\\partial_\\nu\\psi_\\alpha$ is the canonical kinetic term for a scalar field $\\psi_\\alpha$. One can then identify pressure $P_\\alpha$, energy density $\\rho_\\alpha$, and the sound speed squared $c_{s,\\alpha}^2$ in the Jordan frame as\n\\begin{equation}\n P_\\alpha \\equiv P_a(X_a)\\,,\\quad \\rho_a \\equiv 2 P_{a,X_a} X_a - P_a(X_a)\\,.\n \\quad c_{s,\\alpha}^2\\equiv \\frac{P_{\\alpha,X_\\alpha}}{2P_{\\alpha,X_\\alpha X_\\alpha}X_\\alpha + P_{\\alpha,X_\\alpha}}.\n \\label{eq:pressuredensitysoundspeed}\n\\end{equation}\n\n\\section{Stability condition of each era under homogeneous perturbations}\n\\label{sec:hompert}\n\n\\subsection{Scaling solutions}\n\\label{sec:scalingsol}\nIt is possible to find exact and approximate scaling solutions to Eqs.\\ (\\ref{eq:friedmannphysical})--(\\ref{eq:backgroundscalar}). We find that in radiation- and cosmological-constant-dominated eras there exist exact scaling solutions. In the matter-dominated era one can find an exact scaling solution only for $\\beta = 0$. When $0 < \\beta \\ll 1$ this turns into an approximate scaling solution. For a radiation-dominated or de Sitter Universe, on the other hand, the exact scaling solutions persist for any value of $\\beta$. \n\nFrom the Friedmann equation (\\ref{eq:friedmannfiducial}) for $f_{\\mu\\nu}$, we can show that both $\\xi =$ constant and $c=$ constant in any scaling solution. Assuming a power law behavior of the scale factor, all terms in the Friedmann equations (\\ref{eq:friedmannphysical}) and (\\ref{eq:friedmannfiducial}) should scale as $t^{-2}$. Then one can immediately see from the graviton potential terms that if $\\xi$ is constant, then \n\\begin{equation}\n\\frac{\\phi}{M_g} = \\frac{2}{\\lambda}\\ln{\\frac{t}{t_i}} = \\frac{n}{\\lambda}N_e\\,,\n\\end{equation}\nwhere we have used the standard scaling of the scale factor $a(t) \\sim t^{2\/n}$ (with $n = 4$ for radiation domination and $n = 3$ for matter domination, here with $\\beta = 0$) and introduced the $e$-folding number $N_e=\\ln{(a(t)\/a_i)}$. Here, $t_i$ ($>0$) is the initial time and $a_i=a(t=t_i)$. \nDenoting a derivative with respect to the $e$-folding time by a prime, one obtains\n\\begin{equation}\n\\frac{\\phi'}{M_g} = \\frac{\\dot{\\phi}}{M_g H} = \\frac{n}{\\lambda}\\,.\\label{eq:phidotscaling}\n\\end{equation}\nIn the case of an exponential increase of the scale factor, i.e.\\ in a purely de Sitter or $\\Lambda$-dominated universe, this last equation (\\ref{eq:phidotscaling}) can be extended with the value $n = 0$, since all background quantities (excepting the scale factor) can be taken as constant. Finally, we also have\n\\begin{equation}\n\\frac{H'}{H}=-\\frac{n}{2}\\,.\n\\end{equation}\nIn a radiation-dominated universe (and in de Sitter) the scaling expressions presented above can be shown to satisfy all background equations trivially.\n\nOn the other hand, in a matter-dominated universe, once we adopt the choices in Eq.~(\\ref{eq:toymodel}), we combine background equations to find the following condition including $\\beta$:\n\\begin{equation}\n\\beta\\left(\\lambda^2-\\frac{3c}{c+\\kappa\\xi^2}\\right)=0. \\label{eqn:beta=0}\n\\end{equation}\nAs $c$ and $\\kappa$ are positive, this condition with $\\beta\\ne 0$ can be satisfied only if $\\lambda \\leq \\sqrt{3}$. Since we are interested in the regime $\\lambda\\gg\\beta$ to have $m_T^2\\propto\\rho$ \\cite{DeFelice:2017oym}, the condition (\\ref{eqn:beta=0}) implies that there is no exact scaling solution in a matter-dominated era unless $\\beta=0$. However, if $\\beta$ is not zero but small enough then the system with $\\lambda\\gg\\beta$ exhibits an approximate scaling behavior. Therefore, we impose that $\\beta \\approx 0$ to allow for an approximate scaling solution. \n\n\n\\subsection{Stability under homogeneous perturbation of the scaling solutions}\n\nFor practicality, the chameleon scalar field and the Hubble expansion rate are rendered dimensionless using mass parameters of the theory, i.e.,\n\\begin{eqnarray}\n\\varphi \\equiv \\phi\/M_{g}\\,,\\quad h \\equiv \\frac{H}{m}\\,.\\label{eq:dimensionlessphihubble}\n\\end{eqnarray}\nThe equations are then written in terms of $\\ln{h}$, $\\varphi$, $\\xi$, and $c$. Homogeneous perturbations of the fields are defined as \n\\begin{equation}\n\\begin{cases}\n\\ln{h}=\\ln{h_0}-\\frac n2 N_e+\\epsilon h^{(1)}\\,, \\\\\n\\varphi=\\frac{nN_e}{\\lambda}(1+\\epsilon\\varphi^{(1)})\\,, \\\\\n\\xi={\\bar \\xi}+\\epsilon \\xi^{(1)}\\,, \\\\\nc=c^{(0)}+\\epsilon c^{(1)}\\,,\n\\end{cases}\n\\end{equation}\nwhere $\\epsilon$ is a small expansion parameter, $h_0$ is the initial background value of $h$, and $\\bar{\\xi}$ and $c^{(0)}$ are the constant values of $\\xi$ and $c$, respectively, for the scaling solutions. The background equations are then expanded to first order in $\\epsilon$. After using the zeroth order equations of motion to set, for instance, $c_0,\\kappa,c_4$, and the initial amount of matter (either radiation or dust) in terms of $c^{(0)}$ and the other background variables, one can solve the linearized equations for the variables $h^{(1)}$, $\\xi^{(1)}$, and $c^{(1)}$ in terms of $\\varphi^{(1)}$ and its derivatives. \n\nUpon making these replacements, one finds the dynamics is uniquely determined by a second-order equation for $\\varphi^{(1)}$. This can be written as\n\\begin{equation}\n\\varphi^{(1)}{}'' + \\left(1+\\frac{2}{N_e}\\right)\\varphi^{(1)}{}' + \\mathcal{A}_r\\varphi^{(1)}=0\\,,\n\\end{equation}\nduring radiation domination (with general $\\beta$), and\n\\begin{equation}\n\\varphi^{(1)}{}'' + \\left(\\frac{3}{2}+\\frac{2}{N_e}\\right)\\varphi^{(1)}{}' + \\mathcal{A}_m\\varphi^{(1)}=0\\,,\n\\end{equation}\nduring matter domination (with $\\beta=0$), where\n\\begin{align}\n\\mathcal{A}_r&=\\frac{1}{N_e}+\\frac{\\left[\\bar{c} d_{r1} \\lambda ^2+4 h_0^2 \\left(\\lambda ^2-4\\right)\\right] \\left[-6 \\bar{c}^3 d_{r1} d_{r2} \\lambda ^2-3 (\\bar{c}+4)\n\t\\bar{c}^2 d_{r1}^2 \\lambda ^2+32 \\left(\\bar{c}^2+5 \\bar{c}+2\\right) d_{r1} h_0^2+64 \\bar{c}^2 d_{r2} h_0^2\\right]}{2 h_0^2\n\t\\lambda ^2 \\left[\\bar{c}^3 d_{r1}^2 \\left(8-3 \\lambda ^2\\right)+16 \\bar{c}^2 \\left(d_{r1}^2+d_{r1} h_0^2+2 d_{r2} h_0^2\\right)+8 \\bar{c}\n\td_{r1} \\left(d_{r1}+10 h_0^2\\right)+32 d_{r1} h_0^2\\right]}\\,,\\label{eqn:mathcalAr}\\\\\n\\mathcal{A}_m&=\\frac{3}{2 N_e}+\\frac{\\left[\\bar{c} d_{m1} \\lambda ^2+3 h_0^2 \\left(\\lambda ^2-3\\right)\\right] \\left[-4 \\bar{c}^3 d_{m1} d_{m2} \\lambda ^2-4 \\bar{c}^2 d_{m1}^2\n\t\\lambda ^2+36 \\bar{c}^2 d_{m2} h_0^2+9 (7 \\bar{c}+3) d_{m1} h_0^2\\right]}{2 h_0^2 \\lambda ^2 \\left[\\bar{c}^3 d_{m1}^2 \\left(3-2\n\t\\lambda ^2\\right)+6 \\bar{c}^2 \\left(d_{m1}^2+2 d_{m2} h_0^2\\right)+3 \\bar{c} d_{m1} \\left(d_{m1}+7 h_0^2\\right)+9 d_{m1}\n\th_0^2\\right]}\\,,\\label{eqn:mathcalAm}\n\\end{align}\nwith $\\bar{c}=c^{(0)}-1$, and \n\\begin{align}\nd_{i1}&=c_1\\bar{\\xi}_i+2c_2\\bar{\\xi}_i^2+c_3\\bar{\\xi}_i^3\\,, \\label{eqn:def-di1} \\\\\nd_{i2}&=c_2\\bar{\\xi}_i^2+c_3\\bar{\\xi}_i^3\\,, \\label{eqn:def-di2}\n\\end{align}\nwith $i=r,m$. \nTo guarantee the stability during radiation and matter dominations, respectively, it is necessary and sufficient that \n\\begin{equation}\n\\mathcal{A}_r>0\\qquad{\\rm and}\\qquad\\mathcal{A}_m>0\\,.\\label{eq:hompertcondition}\n\\end{equation}\nHere, it is understood that $\\bar{\\xi}_r$ and $\\bar{\\xi}_m$ in (\\ref{eqn:def-di1}) and (\\ref{eqn:def-di2}) are the constant values of $\\xi$ in radiation- and matter-dominated epochs, respectively.\n\n\n\\section{Stability conditions of perturbations}\n\\label{sec:noinstability}\n\nOne can define the perturbations of the fields with respect to the spatially flat FLRW background as follows. In the Arnowitt-Deser-Misner (ADM) decomposition, the (perturbed) metrics are written as\n\\begin{equation}\nds_{g}^2 = - \\mathcal{N}^2 dt^2 + \\gamma_{ij}(\\mathcal{N}^i dt + dx^i)(\\mathcal{N}^j dt + dx^j)\\,,\\quad ds_{f}^2 = - \\tilde{\\mathcal{N}}^2 dt^2 + \\tilde{\\gamma}_{ij}(\\tilde{\\mathcal{N}}^i dt + dx^i)(\\tilde{\\mathcal{N}}^j dt + dx^j)\\\\\n\\end{equation}\nOne can then decompose the lapses, shifts, and 3D metrics separately as\n\\begin{eqnarray}\n&\\mathcal{N} = N(1 + \\Phi) \\,,\\quad \\mathcal{N}_{i} = N_{i} + {\\delta N}_i\\,,\\quad \\gamma_{ij} = a^2 \\delta_{ij} + {\\delta \\gamma}_{ij}\\,,\\nonumber\\\\\n&\\tilde{\\mathcal{N}} = \\tilde{N}(1+ \\tilde{\\Phi})\\,,\\quad \\tilde{\\mathcal{N}}_{i} = \\tilde{N}_i + {\\delta \\tilde{N}}_{i}\\,,\\quad\\tilde{\\gamma}_{ij} = \\tilde{a}^2 \\delta_{ij} + {\\delta\\tilde{\\gamma}}_{ij}\\,,\n\\end{eqnarray}\nwhere $\\Phi$, $\\tilde{\\Phi}$, ${\\delta N}_i$, ${\\delta \\tilde{N}}_i$, ${\\delta\\gamma}_{ij}$, and ${\\delta\\tilde{\\gamma}}_{ij}$ are the perturbations. In particular, we are free to choose $N = 1$ by the time reparametrization invariance, and we also have that $N_i = \\tilde{N}_i = 0$ in our particular background. One may use other equivalent definitions of the perturbations; for instance, as long as the background equations of motion are taken into account, any definitions that differ only at second order will be equivalent as far as the quadratic action is concerned. Finally, as perturbations are studied only linearly and on a spatially homogeneous and isotropic background, one can decompose the perturbations of the shifts and 3D metrics into $SO(3)$ scalar, vector, and tensor representations, i.e.\n\\begin{eqnarray}\n&{\\delta N}_i = N a (\\partial_i B + B_i)\\,,\\quad{\\delta\\gamma}_{ij} = a^2\\left[2\\delta_{ij}\\Psi + \\left(\\partial_i\\partial_j - \\frac{\\delta_{ij}}{3}\n\\Delta\n\\right)E + \\partial_{(i}E_{j)}+ h_{ij}\\right]\\,,\\nonumber\\\\\n&{\\delta \\tilde{N}}_i = \\tilde{N} \\tilde{a} (\\partial_i \\tilde{B} + \\tilde{B}_i)\\,,\\quad {\\delta\\tilde{\\gamma}}_{ij} = \\tilde{a}^2\\left[2\\delta_{ij}\\tilde{\\Psi} + \\left(\\partial_i\\partial_j - \\frac{\\delta_{ij}}{3}\n\\Delta\n\\right)\\tilde{E} + \\partial_{(i}\\tilde{E}_{j)}+ \\tilde{h}_{ij}\\right]\\,,\n\\end{eqnarray}\nwhere $h_{ij}$, $\\tilde{h}_{ij}$, $E_i$, $\\tilde{E}_i$, $B_i$, $\\tilde{B}_i$ obey tracelessness and transversality, i.e.\\ $\\delta^{ij}h_{ij} = \\partial^{i}h_{ij} = \\partial^{i}E_i = \\partial^{i}B_i = 0$ and $\\delta^{ij}\\tilde{h}_{ij} = \\partial^{i}\\tilde{h}_{ij} = \\partial^{i}\\tilde{E}_i = \\partial^{i}\\tilde{B}_i = 0$. The Laplacian is defined as $\\Delta \\equiv \\delta^{kl}\\partial_k\\partial_l$, and we use the notation $\\mathcal{O}_{(ij)} \\equiv \\frac{1}{2}(\\mathcal{O}_{ij} + \\mathcal{O}_{ji})$ to denote symmetrization of the indices. The latin indices of partial derivatives and perturbations can be raised and lowered with $\\delta^{ij}$ and $\\delta_{ij}$. The perturbations of the chameleon scalar field and matter fields are\n\\begin{equation}\n\\phi = \\bar{\\phi}+\\delta\\phi\\,,\\quad\\psi_\\alpha = \\bar{\\psi}_\\alpha+\\delta\\psi_\\alpha\\,.\n\\end{equation}\n\nThe full action is then expanded to second order in the linear perturbations just defined. In particular the perturbations to the metric square root can be computed along the lines of \\cite{Gumrukcuoglu:2011zh}. The treatment is separated into tensor, vector, and scalar sectors. For later use, we choose to represent the matter content of the Universe by two perfect fluids, thus labeled by $\\psi_\\alpha$, with $\\alpha \\in \\{1,2\\}$. \n\n\\subsection{Tensor perturbations}\n\nThe quadratic action for tensor perturbations (written in Fourier space) reduces to\n\\begin{equation}\n\\mathcal{L}^{(2)}_{\\textrm{T}} = \\frac{M_g^2 N a^3}{8 \n}\\delta^{ik}\\delta^{jl}\\left\\{\\frac{\\dot{h}_{ij}\\dot{h}_{kl}}{N^2} - \\frac{k^2\n}{a^2}h_{ij}h_{kl} + \\frac{\\kappa \\xi^2}{c}\\left[\\frac{\\dot{\\tilde{h}}_{ij}\\dot{\\tilde{h}}_{kl}}{N^2} - c^2\\frac{k^2\n}{a^2}\\tilde{h}_{ij}\\tilde{h}_{kl}\\right] - \\frac{\\kappa \\xi^2}{c + \\kappa \\xi^2} m_\\textrm{T}^2 h^{-}_{ij}h^{-}_{kl}\\right\\}\\,,\n\\end{equation}\nwhere $h^{-}_{ij} = h_{ij}- \\tilde{h}_{ij}$, $k^2=\\delta^{ij}k_ik_j$, $k_i$ is the comoving momentum of a perturbation mode, and \n\\begin{equation}\nm_\\textrm{T}^2 = \\frac{c + \\kappa \\xi^2}{\\kappa \\xi^2}m^2\\Gamma,\\quad \\Gamma=\\xi J+\\frac{c-1}{2}\\xi^2J_{,\\xi}.\n\\end{equation}\nIn obtaining this form, we have used both Friedmann equations. One obtains a simple no-ghost condition from the tensor sector, i.e.\n\\begin{equation}\nc\\geq 0\\,.\n\\end{equation}\nThe squared sound speeds of the tensor modes are $c_{T,1}^2 = 1$ and $c_{T,2}^2 = c^2$ for $h_{ij}$ and $\\tilde{h}_{ij}$, respectively. \n\nDue to the time dependence of the background geometry, the graviton mass cannot be defined without ambiguities of order $\\mathcal{O}(H)$ in general. On the other hand, in de Sitter spacetime with $\\xi =$ constant and $c = 1$, it is the combinations $h^{-}_{ij}$ and $h^{+}_{ij} = h_{ij}+ \\kappa\\xi^2\\tilde{h}_{ij}$ that are the two eigenmodes of the mass matrix. In such a case, one can simply rewrite the Fourier space action in the form\n\\begin{equation}\n\\mathcal{L}^{(2)}_{\\textrm{T,dS}} = \\frac{N a^3 M_g^2}{8(1+\\kappa\\xi^2)\n}\\delta^{ik}\\delta^{jl}\\left\\{\\frac{\\dot{h}^{+}_{ij}\\dot{h}^{+}_{kl}}{N^2} - \\frac{\nk^2\n}{a^2}h^{+}_{ij}h^{+}_{kl} + \\kappa\\xi^2\\left[\\frac{\\dot{h}^{-}_{ij}\\dot{h}^{-}_{kl}}{N^2} - \\frac{\nk^2\n}{a^2}h^{-}_{ij}h^{-}_{kl} - m_\\textrm{T}^2 h^{-}_{ij}h^{-}_{kl}\\right] \\right\\}\\,.\n\\end{equation}\nIn this case $m_\\textrm{T}$ is the mass of the massive mode, and\t both graviton sound speeds are equal to unity. \n\n\\subsection{Vector perturbations}\n\nAfter integrating out two nondynamical vectorial degrees of freedom (e.g.\\ $B_i$ and $\\tilde{B}_i$), the quadratic action for vector perturbations reduces to (in Fourier space)\n\\begin{equation}\n\\mathcal{L}^{(2)}_{\\textrm{V}} = \\frac{M_g^2Na^3}{8\n}\\frac{m^2\\kappa\\xi^2Jk^2\\delta^{ij}}{(c+1)\\kappa\\xi k^2\/a^2 + 2m^2(c+\\kappa\\xi^2)J}\n\\left[\\frac{\\dot{E}^{-}_i\\dot{E}^{-}_j}{N^2} - c^2_{\\textrm{V}}\\frac{\nk^2\n}{a^2}E^{-}_i E^{-}_j - m_\\textrm{V}^2 E^{-}_iE^{-}_j\\right]\\,,\n\\end{equation}\nwhere $E^{-}_i = E_i- \\tilde{E}_i$ is the only propagating (massive) vector mode, and\n\\begin{eqnarray}\nc^2_{\\textrm{V}} =\\frac{(c+1)\\Gamma}{2\\xi J}\\,,\\quad m^2_\\textrm{V} = m^2_\\textrm{T}\\,.\n\\end{eqnarray}\nThe associated no-ghost condition in the UV regime is, using $c>0$ and $\\xi>0$,\n\\begin{equation}\nJ\\geq 0\\,.\\label{eq:noghost_vector}\n\\end{equation}\nThe no-gradient-instability condition, $c^2_{V}\\geq 0$, implies\n\\begin{equation}\n\\Gamma\\geq 0\\,.\n\\end{equation}\n\n\\subsection{Scalar perturbations}\n\\label{sec:NGanddispersion}\n\nThe study of the quadratic action for the scalar perturbations requires more work than the vector and tensor sectors. Because of the size of the expressions, we do not give here the full Lagrangian. Instead, we give here the no-ghost conditions, which must be satisfied at all times during the numerical integration, and the squared sound speeds of the scalar sector, which must be positive at all times. \n\nWe start by integrating out four nondynamical degrees of freedom that enforce the Hamiltonian and (longitudinal part of) the momentum constraints (i.e., $\\Phi$, $\\tilde{\\Phi}$, $B$, $\\tilde{B}$). One can integrate out as well the would-be Boulware-Deser (BD) ghost, which is rendered nondynamical by the particular structure of the graviton potential term. One can further use the remaining gauge freedom to set, for instance, the spatially flat gauge, $\\Psi = E = 0$. Eventually, one finds that in addition to the two matter perturbation modes, one has two scalar degrees of freedom, one from the chameleon scalar and the other from the massive graviton. \n\nIn order to find both no-ghost conditions and dispersion relations, we take the subhorizon limit $k \\gg aH$. Indeed, we are solely interested in checking the presence or absence of instabilities in the UV, any IR instability being less problematic~\\cite{Gumrukcuoglu:2016jbh}. \n\n\\subsubsection{No-ghost conditions}\n\nIn the subhorizon limit, the action can be written schematically as\n\\begin{equation}\n\\mathcal{L}^{(2)}_{S,\\mathrm{s.h.}} = \\frac{N a^3}{2}\\left[\\frac{\\dot{\\mathcal{Y}}^{\\top}}{N} \\mathcal{K}\\frac{\\dot{\\mathcal{Y}}}{N} + \\frac{\\dot{\\mathcal{Y}}^\\top}{N}\\mathcal{F}\\mathcal{Y} - \n\\mathcal{Y}^\\top\\mathcal{F}\\frac{\\dot{\\mathcal{Y}}}{N} - \\mathcal{Y}^\\top\\mathcal{M}\\mathcal{Y}\\right],\n\\end{equation}\nwhere $\\mathcal{K}^\\top=\\mathcal{K}$, $\\mathcal{F}^\\top=-\\mathcal{F}$, $\\mathcal{M}^\\top=\\mathcal{M}$ are $4\\times4$ real matrices, and $\\mathcal{Y}$ is a vector containing the four remaining dynamical scalar perturbations, each of which may or may not be rescaled by a positive constant coefficient. The kinetic matrix $\\mathcal{K}$ can then be diagonalized, yielding the eigenvalues\n\\begin{align}\n\\kappa_1 &= \\frac{a^4 m^2 M^2_g}{8 H \\kappa} \\left\\{ 3m^2\\left( H - H \\kappa \\xi^2 + 2 H_f \\kappa \\xi^3 \\right) J^2 + 2\\kappa \\xi^2 J \\left[3 H_f H \\left(2H_f\\xi - 3 H\\right) - \\frac{1}{4}m^2 \\dot{\\phi}U_{,\\xi\\phi}\n\\right] \\right. \\nonumber \\\\\n& \\left. + 2 H \\kappa \\xi^2 \\left[3 H_f \\xi \\left(H - H_f\\xi\\right) J_{,\\xi} - 3H_f \\dot{\\phi} J_{,\\phi} - \\frac{1}{16}m^2M^2_g \nU_{,\\xi\\phi}^2\\right] \\right\\}\\,,\\\\\n\\kappa_2 &= 1\\,,\\\\\n\\kappa_3 &= \\frac{N^2\\left(\\rho_1 + P_1\\right)}{c^2_{s,1}\\dot{\\phi}^2_1}\\,,\\\\\n\\kappa_4 &= \\frac{N^2\\left(\\rho_2 + P_2\\right)}{c^2_{s,2}\\dot{\\phi}^2_2}\\,,\n\\end{align}\nup to overall positive constant coefficients. Because of some field redefinition used to diagonalize the kinetic matrix, the indices in $\\kappa_i$ are arbitrary, but roughly correspond to, respectively, the scalar graviton, the chameleon field, and both matter perturbations. While $\\kappa_2\\geq 0$ is trivial and $\\kappa_3,\\kappa_4\\geq 0$ translate into the null-energy conditions on matter fields, i.e., $\\rho_\\alpha + P_\\alpha \\geq 0$ (where $\\alpha$ is an index designing a specific matter field), $\\kappa_1\\geq 0$ yields a nontrivial no-ghost condition which will be checked at all times during the numerical integration. We also want to monitor the scalar sound speeds squared, which are read off from the dispersion relations in the subhorizon limit. \n\n\\subsubsection{Scalar sound speeds}\n\nThe scalar sound speed for high frequency modes can be found by studying the dispersion relation in the subhorizon limit. Two modes propagate with the usual squared sound speeds $c^2_{s,\\alpha}$ of perfect fluids and can thus be identified with the matter modes. The product and the sum of the two remaining scalar sound speeds squared, $c^2_{s,1}$, $c^2_{s,2}$, are given by\n\\begin{equation}\nc^2_{s,1}c^2_{s,2}=\\frac{\\Sigma_1}{\\Sigma}\\,,\n\\end{equation}\nand\n\\begin{equation}\nc^2_{s,1}+c^2_{s,2}=\\frac{\\Sigma_1 + \\Sigma_2}{\\Sigma} + 1\\,,\n\\end{equation}\nwhere\n\\begin{align}\n\\Sigma_1 = &\\kappa \\xi H J \\left[-16 M_g^2 \\dot{\\phi} \\left(J_{,\\phi} \\left\\{(6 c+2) H-(5 c+2) \\xi H_f\\right\\}-\\xi J_{,\\xi\\phi} \\left\\{(c+1) \\xi H_f-2 H\\right\\}\\right)\\right.\\nonumber\\\\\n&\\left.+8 \\dot{\\phi} {}^2 \\left(2 M_g^2 J_{,\\phi\\phi}+\\xi J_{,\\xi}\\right)+16 A(\\phi )^3 M_g^2 A'(\\phi ) J_{,\\phi} (3 (P+\\rho)-4 \\rho)+8 \\xi A(\\phi )^4 (P + \\rho) J_{,\\xi}\\right.\\nonumber\\\\\n&\\left.+M_g^2 \\left(\\xi \\left\\{c^2 m^2 M_g^2 U_{,\\xi\\phi}^2+16 \\xi J_{,\\xi\\xi} \\left(H-\\xi H_f\\right) \\left(H-c \\xi H_f\\right)\\right.\\right.\\right.\\\\\n&\\left.\\left.\\left.+16 J_{,\\xi} \\left[c \\left(-9 \\xi H_f H+5 \\xi ^2 H_f^2+6 H^2\\right)+2 \\xi H_f \\left(2 \\xi H_f-3 H\\right)\\right]\\right\\}\\right.\\right.\\nonumber\\\\\n&\\left.\\left.+16 (c-1)^2 m^2 \\xi M_g^2 J_{,\\phi}^2+8 m^2 M_g^2 J_{,\\phi} \\left\\{2Q_{,\\phi} - (c-1)c\\xi U_{,\\xi\\phi}\\right\\}\\right)\\right]\\nonumber\\\\\n&+4 \\xi J^2 \\left(-c \\kappa m^2 \\xi M_g^2 U_{,\\xi\\phi} \\dot{\\phi} +6 \\kappa H \\dot{\\phi} {}^2 +6 \\kappa A(\\phi )^4 H (P + \\rho)\\right. \\nonumber\\\\\n&\\left. +2 H M_g^2 \\left\\{2 \\kappa \\left[-3 (5 c+2) \\xi H_f H+4 (2 c+1) \\xi ^2 H_f^2+(9 c-3) H^2\\right]+3 (c-1) m^2 \\left(\\kappa \\xi ^2+1\\right) J_{,\\xi}\\right\\}\\right)\\nonumber\\\\\n&+16 (c+1) \\kappa \\xi H M_g^2 \\left(J_{,\\phi} \\dot{\\phi} +\\xi J_{,\\xi} \\left(\\xi H_f-H\\right)\\right){}^2 - 24 m^2 M_g^2 J^3 \\left(H \\left(3 c \\kappa \\xi ^2+c-2 \\kappa \\xi ^2-2\\right)-2 c \\kappa \\xi ^3 H_f\\right)\\,,\\\\\n\\Sigma_2 = & H m^2 M_g^4 \\kappa \\xi^2 (c-1)^2 J \\left(U_{,\\xi\\phi}-4J_{,\\phi}\\right)^2\\,,\\\\\n\\Sigma = & -M_g^2 J \\left\\{\\kappa \\xi ^2 H \\left[48 H_f J_{,\\phi} \\dot{\\phi} +48 \\xi H_f J_{,\\xi} \\left(\\xi H_f-H\\right)+m^2 M_g^2 U_{,\\xi\\phi}^2\\right]\\right. \\nonumber\\\\\n& \\left. +4 \\kappa \\xi ^2 J \\left[m^2 U_{,\\xi\\phi} \\dot{\\phi} -12 H_f H \\left(2 \\xi H_f-3 H\\right)\\right]-24 m^2 J^2 \\left(2 \\kappa \\xi ^3 H_f-\\kappa \\xi ^2 H+H\\right)\\right\\}\\,,\n\\end{align}\n$\\rho = \\rho_1 + \\rho_2$ and $\\rho = P_1 + P_2$. If one considers the vector sector no-ghost condition, $J>0$, then $\\Sigma_2 < 0$. The scalar sound speeds squared provide new stability conditions, as these need to be real and positive. We thus require that \n\\begin{equation}\n\\frac{\\Sigma_1}{\\Sigma}>0\\,,\\quad \\frac{\\Sigma_1+\\Sigma_2}{\\Sigma}+1>0\\,,\\quad \\left(\\frac{\\Sigma_1+\\Sigma_2}{\\Sigma}+1\\right)^2\\! -4\\frac{\\Sigma_1}{\\Sigma}>0\\,.\n\\end{equation}\n\nAlthough we do not give here the analytical expressions for the single squared sound speeds, which would be too large to write, we obtain their numerical value in the next section as part of our numerical example cosmology (see Fig.~\\ref{fig:conditions}). The reader may find a discussion on the respective contributions of the chameleon and the scalar graviton to the scalar squared sound speeds in Appendix~\\ref{sec:scalarsoundspeedcontribution}. \n\n\\section{Initial conditions and numerical results}\n\n\\label{sec:numerics}\n\n\\subsection{Set of equations}\n\nAlthough in principle one can obtain several background equations \\textemdash e.g.\\ both Friedmann equations, both second Einstein equations, the scalar equation of motion, or the combination Eq.\\ (\\ref{eq:fiduciallapse}),\\textemdash not all the equations will be directly integrated. For instance, this last equation can be used to fix the fiducial function $c$. Similarly, both Friedmann equations can be used to set two parameters or integration constants, as will be shown below. Of the equations cited above, only three will remain to be integrated: both second Einstein equations and the scalar equation of motion. In addition to finding the right set of equations, the choice of adequate initial conditions (ICs) is also essential. In what follows, a subscript $i$ stands for the quantity evaluated at initial time. \n\nAlthough in the previous section we were able to derive the results while keeping the functions $\\beta_j(\\phi)$, $j\\in \\{0,\\cdots,4\\}$, and $A(\\phi)$ completely general, these need to be specified for the sake of numerical integration. We will thus from now on use the example model defined in (\\ref{eq:toymodel}).\n\nSeveral definitions help render the equations more practical for the purpose of numerical integration. First of all we consider the equations of motion in $e$-fold time with its initial value being $N_{e,i}=0$. We then define dimensionless variables. We start by using the dimensionless chameleon scalar field, $\\varphi$, and Hubble parameter, $h$, as defined in Eq.\\ (\\ref{eq:dimensionlessphihubble}).\nFor the matter energy densities, we split the energy density of the matter fields (in the Jordan frame, for which $a_{{\\rm JF}}=A\\,a$) as\n\\begin{equation}\n\\rho_{{\\rm JF}}^{{\\rm TOT}} \\equiv \\frac{R_{ri}}{A^{4}a^{4}}+\\frac{R_{di}}{A^{3}a^{3}}+R_{\\Lambda i}\\,,\n\\end{equation}\nwhere the subscripts $r$, $d$, and $\\Lambda$, indicate the radiation, dust, and cosmological constant, respectively. We then define\n\\begin{equation}\nR_{ri} = r_{r}\\,a_{i}^{4}\\,M_{g}^{2}m^{2}\\,,\\quad R_{di} = r_{d}\\,a_{i}^{3}\\,M_{g}^{2}m^{2}\\,,\\quad R_{\\Lambda i} = r_{\\Lambda}\\,M_{g}^{2}m^{2}\\,,\n\\end{equation}\nwhere $r_r$, $r_m$, and $r_\\Lambda$ are dimensionless and constant throughout the evolution. Using these definitions, the Friedmann equation for the physical metric becomes\n\\begin{equation}\n3h^{2}=\\frac{1}{2}h^{2}{\\varphi'}^{2}+e^{-\\lambda\\varphi}\\left(c_{{0}}+3c_{{1}}\\xi+3c_{{2}}\\xi^{2}+c_{{3}}\\xi^{3}\\right)+e^{\\beta\\,\\varphi}r_{{d}}e^{-3N_e}+r_{{r}}e^{-4N_e}+e^{4\\beta\\,\\varphi}r_{{l}}\\,,\n\\end{equation}\nwhile the Friedmann equation for the fiducial metric can be written\n\\begin{equation}\n0 = 1 - e^{-\\lambda\\varphi}\\frac{\\bar{V}(\\xi)}{3h^2\\kappa\\xi} - \\frac{2\\lambda\\varphi'}{3}\\frac{\\bar{V}(\\xi)}{\\bar{J}(\\xi)} + \\frac{\\lambda^2(\\varphi')^2}{9}\\frac{\\bar{V}(\\xi)^2}{\\bar{J}(\\xi)^2} \\label{eq:fiducialfriedmann_numerical}\n\\end{equation}\nwhere as noted previously a prime denotes differentiation with respect to $N$, and we have defined\n\\begin{equation}\n\\bar{J}(\\xi) = c_1 + 2c_2\\xi + c_3\\xi^2\\,,\\quad\\bar{V}(\\xi) = c_1 + 3c_2\\xi + 3c_3\\xi^2 + c_4\\xi^3\\,.\n\\end{equation}\n\nIt is instructive to rewrite the physical Friedmann equation as\n\\begin{equation}\n1 = \\Omega^\\mathrm{EF}_\\Lambda + \\Omega^\\mathrm{EF}_d + \\Omega^\\mathrm{EF}_r + \\Omega^\\mathrm{EF}_k + \\Omega^\\mathrm{EF}_V\\,.\\label{eq:friedmann_densityparameters} \n\\end{equation}\nFor this, we have defined the Einstein frame density parameters\n\\begin{equation}\n\\Omega^\\mathrm{EF}_\\Lambda = \\frac{e^{4\\beta\\,\\varphi}r_{{l}}}{3h^2}\\,,\\quad\\Omega^\\mathrm{EF}_d = \\frac{e^{\\beta\\,\\varphi}r_{{d}}e^{-3N_e}}{3h^2}\\,,\\quad\\Omega^\\mathrm{EF}_r = \\frac{r_{{r}}e^{-4N_e}}{3h^2}\\,,\\quad \\Omega^\\mathrm{EF}_k = \\frac{(\\phi')^2}{6}\\,,\\quad \\Omega^\\mathrm{EF}_V = \\frac{e^{-\\lambda\\varphi}(c_{{0}}+3c_{{1}}\\xi+3c_{{2}}\\xi^{2}+c_{{3}}\\xi^{3})}{3h^2}\\,.\n\\end{equation}\nThe new subscripts $k$ and $V$ indicate contributions from the chameleon kinetic energy and from the graviton potential term, respectively. We can also define the Jordan frame density parameters, using the fact that \n\\begin{eqnarray}\nH_{{\\rm JF}} & \\equiv & \\frac{1}{a_{{\\rm JF}}^{2}}\\,\\frac{da_{{\\rm JF}}}{d\\eta}=\n\\frac{m\\,h}{A}\\left(\\beta\\varphi'+1\\right)\\,,\n\\end{eqnarray}\nwhere $\\eta$ is the conformal time defined by $\\eta \\equiv \\int_0^t \\frac{dt'}{a(t')}$. This allows us to write\n\\begin{equation}\n\\Omega_{r}^{{\\rm JF}}=\\frac{R_{ri}}{A^{2}a^{4}}\\,\\frac{1}{3M_{g}^{2}H_{{\\rm JF}}^{2}}=\\frac{r_{r}e^{-4N_e}}{3h^{2}(1+\\beta\\varphi')^{2}}\\,.\n\\end{equation}\nIn a similar way, \n\\begin{equation}\n\\Omega_{d}^{{\\rm JF}} = \\frac{r_{d}\\,e^{\\beta\\varphi-3N_e}}{3h^{2}(1+\\beta\\varphi')^{2}}\\,,\\quad \\Omega_{\\Lambda}^{{\\rm JF}} = \\frac{r_{\\Lambda}\\,e^{4\\beta\\varphi}}{3h^{2}(1+\\beta\\varphi')^{2}}\\,.\n\\end{equation}\nTherefore we can replace $r_{r},r_{d},r_{\\Lambda}$ with either Jordan frame or Einstein frame density parameters, evaluated at initial time, i.e.,\n\\begin{eqnarray}\nr_{r} & = & 3\\Omega_{r,i}^{{\\rm JF}}h_{i}^{2}(1+\\beta\\varphi_{i}')^{2} = 3\\Omega_{r,i}^{{\\rm EF}}h_{i}^{2}\\,,\\\\\nr_{d} & = & 3\\Omega_{d,i}^{{\\rm JF}}h_{i}^{2}(1+\\beta\\varphi_{i}')^{2} = 3\\Omega_{d,i}^{{\\rm EF}}h_{i}^{2}\\,,\\\\\nr_{\\Lambda} & = & 3\\Omega_{\\Lambda,i}^{{\\rm JF}}h_{i}^{2}(1+\\beta\\varphi_{i}')^{2}= 3\\Omega_{\\Lambda,i}^{{\\rm EF}}h_{i}^{2}\\,.\n\\end{eqnarray}\n\nIn terms of the new variables we have that Eq.\\ (\\ref{eq:fiduciallapse}) can be rewritten as\n\\begin{equation}\nc\\,\\xi=\\frac{3\\left(\\xi+\\xi'\\right)\\left(\\xi^{2}c_{{3}}+2\\,\\xi\\,c_{{2}}+c_{{1}}\\right)}{3(c_{{3}}\\xi^{2}+2c_{2}\\xi+c_{{1}})-(c_{{4}}\\xi^{3}+3c_{{3}}\\xi^{2}+3c_{{2}}\\xi+c_{{1}})\\lambda\\varphi'}\\,,\n\\end{equation}\nwhich defines $c$ in terms of the other dynamical variables. When using this definition in the fiducial second Einstein equation, this reduces the degree of the equation to 1, with respect to the variable of interest $\\xi$.\n\nThe set of dynamical equations to be integrated, the two second Einstein equations and the chameleon field equation, can be written as\n\\begin{equation}\n\\begin{cases}\nh'= h'(h,\\xi,\\varphi,\\varphi')\\,,\\\\\n\\varphi'' = \\varphi''(h,\\xi,\\varphi,\\varphi')\\,,\\\\\n\\xi' = \\xi'(h,\\xi,\\varphi,\\varphi')\\,,\n\\end{cases}\n\\end{equation}\nand, because of the choice of the variables\/parameters, they do not\nexplicitly depend on any scale, e.g., $M_{g}$ or $m$. \n\n\\subsection{Requirements on initial data}\n\nUsing a rescaling of the constants one can, without loss of generality, set the values of the ICs $\\varphi_{i}$, $\\xi_i$, and $h_i$. In detail, this can be done, for example, by (i) redefining $m^2$ to set $\\varphi_i = 0$, (ii) redefining the $c_j$ and $M_f$ to set $\\xi_i = 1$, and (iii) an additional overall rescaling of the constants $c_j$, which we use to set $h_i = 1$. Once this is done, we only need to give one supplementary IC, i.e.\\ $\\varphi'_i$. Then the total set of yet needed ICs and parameters is\n\\begin{gather*}\nc_{0},c_{1},c_{2},c_{3},c_{4},\\\\\nr_{r},r_{d},r_{\\Lambda},\\\\\n\\lambda,\\kappa,\\beta,\\varphi'_{i}.\n\\end{gather*}\nWe can use the two Friedmann equations to set two of the parameters (or ICs, in principle). Without loss of generality, we solve them in terms of $c_{0}$ and $\\kappa$ (by linear equations). \n\n\nThe initial conditions for the integration are set in a radiation-domination epoch, with the Universe obeying a scaling solution. These initial conditions allow us to recover a cosmology accommodating our Universe. In order to start with a radiation-domination phase, one simply needs to set $0<1-\\Omega_{r,i}^{{\\rm JF}}\\ll 1$. Since we also want to start from a scaling behavior during radiation domination, the remaining ICs and parameters are imposed so that the dynamics of the scale factor and the scalar field satisfy the scaling solution values found in Sec. \\ref{sec:scalingsol}, i.e.,\n\\begin{equation}\nh'_{i} \\approx -2h_{i}\\,,\\quad\n\\varphi'_{i} \\approx \\varphi'_{{\\rm sc}}=\\frac{4}{\\lambda}\\,,\\quad\n\\varphi''_{i} \\approx 0\\,,\\quad \\xi'_i\\approx 0\\,.\n\\end{equation}\n\nWe choose to replace the parameters $c_1$, $c_2$, $c_3$, and $c_4$ with new, more practical and transparent parameters. First, two of the constants can be chosen so that the condition (\\ref{eq:noghost_vector}) is always satisfied. This can be done, for example, by letting \n\\begin{equation}\nc_3 c_1 - c_2^2 = \\mathfrak{A}\\,,\\quad c_1 + 2c_2+ c_3 = \\mathfrak{B}\\,,\n\\end{equation}\nwhere both $\\mathfrak{A}$ and $\\mathfrak{B}$ are positive constants (and new parameters that replace two among $c_1$, $c_2$, and $c_3$), which is sufficient to guarantee that $J > 0$ for any $\\xi$. Second, one may use Eq.\\ (\\ref{eq:fiduciallapse}), while approximating $\\xi_i'\\approx 0$, to set $c$ at the initial time to a specific value instead of one of the $c_i$'s. Finally, we use the expression of the vector squared sound speed to replace the last parameter.\n\n\\subsection{Results}\n\nBased on the previous section, we describe here a set of parameters which allows for an evolution similar to the usual $\\Lambda$ cold dark matter models. The values used in our example are\n\\begin{gather}\nc_\\mathrm{in} = \\frac{101}{100}\\,,\\quad c_\\mathrm{V,in}^2 = 1\\,,\\quad \\mathfrak{A} = 1\\,,\\quad \\mathfrak{B} = 1\\,,\\nonumber\\\\\n\\Omega^\\mathrm{EF}_{\\Lambda i} = 1\\times 10^{-30}\\,,\\quad \\Omega^\\mathrm{EF}_{di} = 1\\times 10^{-5}\\,,\\quad\\Omega^\\mathrm{EF}_{ki} = \\frac{3}{200}\\,,\\quad \\Omega^\\mathrm{EF}_{Vi} = \\frac{1}{200}\\,,\\nonumber\\\\\n\\beta = 1\\times 10^{-2}\\,,\\quad\\lambda = \\frac{40}{3}\\,,\\label{eq:numerical_parameters}\n\\end{gather}\nwhere the subscripts ``in\" or ``i\" mean the respective initial value. The initial density parameter for radiation, $\\Omega^\\mathrm{EF}_{ri}$, is directly determined by the Friedmann equation (\\ref{eq:friedmann_densityparameters}) at initial time, and all other parameters are fully determined by this set of choices. The only fine-tuned value is $\\Omega_{\\Lambda i}^\\mathrm{EF}$, which we have chosen in order to have $\\Omega_{\\Lambda}$ of order unity today. In practice, this is the same as the cosmological constant problem today. \n\nThe simple choice of parameters in Eq.\\ (\\ref{eq:numerical_parameters}) is meant to show that it is possible to obtain a realistic cosmological evolution. It does not recover exactly today's observed values. However, it is possible, by an appropriate choice of constants \\textemdash and without fine-tuning anything other than the cosmological constant \\textemdash to obtain an evolution fitting more closely to data; e.g.\\ one can reproduce today's abundances and other data. This, along with the constraints on the model from today's observational data, will be studied further in a future work.\n\nFor the sake of exposition, we present the evolution\\footnote{The number of steps and $e$-fold time range chosen for the integration are \ninitial time = 0, final time = 25, and number of steps = 3199.} of the density parameters in Fig.\\ \\ref{speciesratio}, while the evolution of other relevant variables is presented in Fig.\\ \\ref{fig:varevolution}. The evolution, starting from a radiation-dominated era, moves on to a matter-dominated era, finally attaining a final de Sitter phase. Given our set of initial density parameters, the system stays $\\mathcal{O}(10)$ $e$-folds in each era before settling to a de Sitter epoch (roughly from $0$ to $12$ $e$-folds for radiation domination, from $12$ to $19$ $e$-folds for matter domination, from $19$ to the end for the de Sitter era). However, by arranging these density parameters, one can achieve very different numbers of $e$-folds spent in each era. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=12cm]{speciesratiosplit.pdf\n\t\\caption{Evolution of the density parameters of the different species versus number of $e$-folds of evolution. The thick solid line stands for radiation, the thick dashed line stands for dust, the dashed-dotted line stands for the cosmological constant, the thin dotted line stands for the scalar field kinetic energy, and the thin solid line stands for the contribution from the graviton potential term.}\\label{speciesratio}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=12cm]{varevolution.pdf\n\t\\caption{Evolution of the time derivative of the Hubble parameter, the ratio of fiducial and physical scale factors $\\xi$, the chameleon field $\\varphi$, and its second time derivative. The system starts from a radiation-dominated era (from $N_e = 0$ to roughly $N_e= 12$), then goes through a matter-domination phase (roughly from $N_e = 12$ to roughly $N_e= 19$), and finishes in a de Sitter era.}\\label{fig:varevolution}\n\\end{figure}\n\nIn order to have a handle on the precision of the numerical integration, we check all along the evolution to which extent the Friedmann equations are satisfied. For this purpose, one may define, for instance,\n\\begin{equation}\n\\mathcal{C}_1 = \\frac{1 - \\sum_\\alpha\\Omega^\\mathrm{EF}_\\alpha}{1 + \\sum_\\alpha \\left|\\Omega^\\mathrm{EF}_\\alpha\\right|}\\,,\\quad \\mathcal{C}_2 = \\frac{ 1 - e^{-\\lambda\\varphi}\\frac{\\bar{V}(\\xi)}{3h^2\\kappa\\xi} - \\frac{2\\lambda\\varphi'}{3}\\frac{\\bar{V}(\\xi)}{\\bar{J}(\\xi)} + \\frac{\\lambda^2(\\varphi')^2}{9}\\frac{\\bar{V}(\\xi)^2}{\\bar{J}(\\xi)^2}}{1 + \\left|e^{-\\lambda\\varphi}\\frac{\\bar{V}(\\xi)}{3h^2\\kappa\\xi}\\right| + \\left|\\frac{2\\lambda\\varphi'}{3}\\frac{\\bar{V}(\\xi)}{\\bar{J}(\\xi)}\\right| + \\left|\\frac{\\lambda^2(\\varphi')^2}{9}\\frac{\\bar{V}(\\xi)^2}{\\bar{J}(\\xi)^2}\\right|}\\,,\n\\end{equation}\ninspired by both Friedmann equations (\\ref{eq:friedmann_densityparameters}) and (\\ref{eq:fiducialfriedmann_numerical}), and where $\\alpha$ stands for any of the species, i.e., indices $\\{r,d,\\Lambda,k,V\\}$. The evolution of these two constraints is presented in Fig.\\ \\ref{fig:constraintevolution}. Both constraints are seen to be satisfied up to order $\\mathcal{O}(10^{-5})$. In our implementation, this has been achieved by using constraint damping, i.e., adding the constraint equations into the dynamical equations of motion (after normalizing the constraints by an appropriate factor), in order to damp any unwanted deviation.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=12cm]{constraints_new.pdf\n\t\\caption{Evolution of the first and second constraints. Constraint damping is efficient during most of the integration time.}\\label{fig:constraintevolution}\n\\end{figure}\n\nIn addition to the Friedmann equations, we also present in Fig.\\ \\ref{fig:conditions} the evolution of the sound speeds and the fiducial lapse $c$. Together with the no-ghost conditions, which are found to be satisfied all along the evolution, the positivity of these shows that the background is stable under cosmological perturbations.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=12cm]{conditions_new.pdf\n\t\\caption{Evolution of some consistency conditions. Here are presented the evolution of the two scalar squared sound speeds ($c^2_{s,i}$ with $i\\in\\{1,2\\}$) and of the fiducial lapse $c$. Both the sound speeds and the fiducial lapse tend rapidly to $1$ in a $\\Lambda$-dominated universe. Respective contributions from the scalar graviton and the chameleon scalar field to the scalar sound speeds are discussed in the Appendix.}\\label{fig:conditions}\n\\end{figure}\n\nFinally, in order to demonstrate the purpose of the new scaling brought by the scalar field dependence in the graviton potential, we plot the Higuchi condition along the evolution in Fig.\\ \\ref{fig:higuchi}. The generalized Higuchi bound $\\frac{m^2_\\mathrm{T}}{H^2} > \\mathcal{O}(1)$ is seen to be well satisfied during the three eras, and in particular both at late and early times. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=12cm]{higuchiv0.pdf\n\t\\caption{Evolution of the Higuchi condition. The ratio of the tensor mass to the Hubble expansion rate has to be $> \\mathcal{O}(1)$ in order for the model to be stable. The condition is thus satisfied all along the evolution.}\\label{fig:higuchi}\n\\end{figure}\n\n\\section{Discussion and conclusions}\n\\label{sec:conclusion}\n\nFollowing the recent proposal \\cite{DeFelice:2017oym} of an extended massive bigravity theory supplemented by a chameleon scalar field as a means to cure or evade the fine-tunings of the original theory and improve its applicability, we have found it important to study further its validity and implications. For this reason, in this work we have explored the stability conditions of the model and confirmed its intended behavior by integrating numerically the equations of motion. In particular, we have numerically confirmed that at all times, the Higuchi ghost is never present: indeed, the presence of this ghost represented one of the most serious problems for a viable phenomenology of the original bigravity theory. In our model though, we have here shown that if no Higuchi ghost is present at one scale then the same ghost will not appear during the whole evolution of the Universe including the early epoch. This set of such allowed initial conditions is not of zero measure in general, so that we do not need to fine-tune the parameters of the theory.\n\nThe study of the action quadratic in perturbations with respect to a general flat FLRW background leads, in the UV, to no-ghost conditions for the tensor, vector, and scalar sectors. In addition to this, we have found the explicit action for the tensor and vector linear perturbations and for the scalar linear perturbations in the UV. From these the propagation speeds at short scales are easily extracted, thus leading to additional no-instability conditions. It is found as expected that the theory propagates four tensor, two vector, and two scalar degrees of freedom (not including matter degrees of freedom), thus corresponding to the expected massive spin-2, massless spin-2, and chameleon scalar of the theory. \n\n\nIn order to show the typical background time evolution, we have numerically integrated the background equations by using a choice of initial parameters consistent with an initial radiation-dominated era of the Universe. As supplementary input for the initial conditions, we have required that the stability conditions be satisfied and that the parameters of the theory are in the regime of interest for the expected scaling behaviors. The evolution displays an initial radiation-dominated era, followed by matter domination and a de Sitter era. The no-instability conditions are satisfied all along the evolution, and, in our implementation, the constraint equations show a numerical error of order $\\mathcal{O}(10^{-5})$ at most. This stable evolution comforts us into arguing that it may be possible to find a region of the parameter space allowing a close match with our cosmological observations.\n\nThe recent binary neutron star merger observation, the first gravitational and electromagnetic wave multimessenger detection \\cite{GBM:2017lvd}, has allowed us to set stringent bounds on the speed difference between gravitational and electromagnetic waves (see, e.g., \\cite{Creminelli:2017sry,Ezquiaga:2017ekz,Baker:2017hug,Sakstein:2017xjx}). Although in our model one of the gravitons propagates with a slightly modified sound speed $c$ (see the lower panel of Fig.~\\ref{fig:conditions}), the physical metric remains unaffected and the interactions between the two metrics are suppressed by the smallness of $m^2\\beta_i$, $i\\in \\{1,2,3\\}$. This implies that the propagation of gravitational waves in our model is essentially the same as that of photons as far as $m^2\\beta_i$ are small enough compared to the typical (squared) energy scales of the gravitational waves produced astrophysically. As a result, the constraint on our model from GW170817 is essentially the same as those from the previous GW observations (e.g., \\cite{bib:LIGO})~\\cite{Baker:2017hug}. Concretely, the constraint is of the form of an upper bound on the mass of the graviton (which was not improved by GW170817) of $m_T < 1.2 \\times 10^{-22}$~eV. While this bound has to and can be satisfied today, the scalar field dependence of the graviton mass in our model allows without problem for a larger mass at early times, rendering the cosmological evolution stable all the time. Therefore, our model can be considered as a unique testing ground of gravitational wave phenomenologies in bimetric theories of gravity. For example, it is intriguing to investigate the possible modification of the waveform of the gravitational wave signal due to the influence of the massive graviton. \n\nAs a clear avenue for future extension, the evolution of cosmological perturbations and an improved understanding of the viable parameter space will be considered in a future work. Furthermore, it may be interesting to study the detailed working of the screening mechanism for the chameleon scalar field and scalar graviton modes.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdwua b/data_all_eng_slimpj/shuffled/split2/finalzzdwua new file mode 100644 index 0000000000000000000000000000000000000000..1a345905bb82738520e2cf3f529d72b4c0a50fb2 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdwua @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe experiments CDF and D\\O, taking data at the Tevatron\nproton-antiproton collider located at the Fermi National Accelerator\nLaboratory, have made several direct experimental measurements of the\ntop-quark pole mass, \\ensuremath{M_{\\mathrm{t}}}. The pioneering measurements were based on about\n$100~\\ensuremath{\\mathrm{pb}^{-1}}$ of \\hbox{Run~I}\\ (1992-1996) data~\\cite{Mtop1-CDF-di-l-PRLa, \n Mtop1-CDF-di-l-PRLb,\n Mtop1-CDF-di-l-PRLb-E, Mtop1-D0-di-l-PRL, Mtop1-D0-di-l-PRD,\n Mtop1-CDF-l+j-PRL, Mtop1-CDF-l+j-PRD, Mtop1-D0-l+j-old-PRL,\n Mtop1-D0-l+j-old-PRD, Mtop1-D0-l+j-new1, Mtop1-CDF-all-j-PRL,\n Mtop1-D0-all-j-PRL} \nand include results from the \\ensuremath{\\ttbar\\rightarrow\\had}\\ (all-j), the \\ensuremath{\\ttbar\\rightarrow\\ljt}\\ (l+j), and the \n\\ensuremath{\\ttbar\\rightarrow\\dil}\\ (di-l) decay channels\\footnote{Here $\\ell=e$ or $\\mu$. Decay \nchannels with explicit tau lepton identification are presently under \nstudy and are not yet used for measurements of the top-quark mass.}. \nThe \\hbox{Run~II}\\ measurements summarized here are the most recent results in the \nl+j, di-l, and all-j channels using $1.9-2.8~\\ensuremath{\\mathrm{fb}^{-1}}$ of data and improved \nanalysis techniques~\\cite{\nMtop2-CDF-di-l-new,\nMtop2-CDF-l+j-new,\nMtop2-CDF-all-j-new, \nMtop2-CDF-trk-new,\nMtop2-D0-l+ja-final,\nMtop2-D0-l+j-new,\nMtop2-D0-di-l-jul08-1,\nMtop2-D0-di-l-jul08-2}. \n\\vspace*{0.10in}\n\nThis note reports the world average top-quark mass obtained by\ncombining five published\n\\hbox{Run~I}\\ measurements~\\cite{Mtop1-CDF-di-l-PRLb, Mtop1-CDF-di-l-PRLb-E,\n Mtop1-D0-di-l-PRD, Mtop1-CDF-l+j-PRD, Mtop1-D0-l+j-new1,\n Mtop1-CDF-all-j-PRL} with four preliminary \\hbox{Run~II}\\ CDF\nresults~\\cite{Mtop2-CDF-di-l-new, Mtop2-CDF-l+j-new, Mtop2-CDF-all-j-new,\nMtop2-CDF-trk-new} and three preliminary\n\\hbox{Run~II}\\ D\\O\\ results~\\cite{Mtop2-D0-l+ja-final, Mtop2-D0-l+j-new,\nMtop2-D0-di-l-jul08-1, Mtop2-D0-di-l-jul08-2}.\nThe combination takes into account the\nstatistical and systematic uncertainties and their correlations using\nthe method of references~\\cite{Lyons:1988, Valassi:2003} and\nsupersedes previous\ncombinations~\\cite{Mtop1-tevewwg04,Mtop-tevewwgSum05,\n Mtop-tevewwgWin06,Mtop-tevewwgSum06, Mtop-tevewwgWin07, Mtop-tevewwgWin08}.\n\\vspace*{0.10in}\n\nThe input measurements and error categories used in the combination are \ndetailed in Sections~\\ref{sec:inputs} and~\\ref{sec:errors}, respectively. \nThe correlations used in the combination are discussed in \nSection~\\ref{sec:corltns} and the resulting world average top-quark mass \nis given in Section~\\ref{sec:results}. A summary and outlook are presented\nin Section~\\ref{sec:summary}.\n \n\\section{Input Measurements}\n\\label{sec:inputs}\n\nFor this combination twelve measurements of \\ensuremath{M_{\\mathrm{t}}}\\ are used: five\npublished \\hbox{Run~I}\\ results, and seven preliminary\n\\hbox{Run~II}\\ results, all reported in Table~\\ref{tab:inputs}. In general,\nthe \\hbox{Run~I}\\ measurements all have relatively large statistical\nuncertainties and their systematic uncertainty is dominated by the\ntotal jet energy scale (JES) uncertainty. In \\hbox{Run~II}\\ both CDF and\nD\\O\\ take advantage of the larger \\ensuremath{t\\overline{t}}\\ samples available and employ\nnew analysis techniques to reduce both these uncertainties. In\nparticular the JES is constrained using an in-situ calibration based\non the invariant mass of $W\\rightarrow qq^{\\prime}$ decays in the l+j and\nall-j channels. The \\hbox{Run~II}\\ D\\O\\ analysis in the l+j channel\nconstrains the response of light-quark jets using the in-situ $W\\rightarrow\nqq^{\\prime}$ decays. Residual JES uncertainties associated with\n$\\eta$ and $p_{T}$ dependencies as well as uncertainties specific to\nthe response of $b$-jets are treated separately. Similarly, the\n\\hbox{Run~II}\\ CDF analyses in the l+j and all-j channels also constrain the\nJES using the in-situ $W\\rightarrow qq^{\\prime}$ decays. Small residual JES\nuncertainties arising from $\\eta$ and $p_{T}$ dependencies and the\nmodeling of $b$-jets are included in separate error categories. The\n\\hbox{Run~II}\\ CDF and D\\O\\ di-l measurements use a JES determined from external\ncalibration samples. Some parts of the associated uncertainty are\ncorrelated with the \\hbox{Run~I}\\ JES uncertainty as noted below.\n\\vspace*{0.10in}\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\small\n\\begin{tabular}{|l||rrr|rr||rrrr|rrr|}\n\\hline \n & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{2-13}\n & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & all-j & l+j & di-l & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l \\\\\n\\hline\n$\\int \\mathcal{L}\\;dt$ & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 2.7 & 1.9 & 2.1 & 1.9 & 1.0 & 1.2 & 2.8 \\\\\n\\hline\n\\hline \nResult & 186.0 & 176.1 & 167.4 & 180.1 & 168.4 & 172.2 & 171.2 & 176.9 & 175.3 & 171.5 & 173.0 & 174.4 \\\\\n\\hline \n\\hline \niJES & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.9 & 0.0 & 1.9 & 0.0 & 0.7 & 1.4 & 0.0 \\\\\naJES & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.8 & 0.8 & 1.2 \\\\\nbJES & 0.6 & 0.6 & 0.8 & 0.7 & 0.7 & 0.4 & 0.4 & 0.6 & 0.0 & 0.0 & 0.1 & 0.3 \\\\\ncJES & 3.0 & 2.7 & 2.6 & 2.0 & 2.0 & 0.3 & 1.7 & 0.6 & 0.6 & 0.0 & 0.0 & 0.0 \\\\\ndJES & 0.3 & 0.7 & 0.6 & 0.0 & 0.0 & 0.0 & 0.1 & 0.1 & 0.0 & 0.8 & 0.0 & 1.6 \\\\\nrJES & 4.0 & 3.4 & 2.7 & 2.5 & 1.1 & 0.4 & 1.9 & 0.6 & 0.1 & 0.0 & 0.0 & 0.0 \\\\\nlepPt & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.2 & 0.1 & 0.0 & 1.1 & 0.0 & 0.0 & 0.0 \\\\\nSignal & 1.8 & 2.6 & 2.8 & 1.1 & 1.8 & 0.3 & 0.8 & 0.5 & 1.6 & 0.5 & 0.5 & 0.5 \\\\\nBG & 1.7 & 1.3 & 0.3 & 1.0 & 1.1 & 0.4 & 0.4 & 1.0 & 1.6 & 0.4 & 0.4 & 0.6 \\\\\nFit & 0.6 & 0.0 & 0.7 & 0.6 & 1.1 & 0.2 & 0.6 & 0.5 & 1.4 & 0.3 & 0.2 & 0.3 \\\\\nMC & 0.8 & 0.1 & 0.6 & 0.0 & 0.0 & 0.5 & 0.9 & 0.5 & 0.6 & 0.0 & 0.0 & 0.0 \\\\\nUN\/MI & 0.0 & 0.0 & 0.0 & 1.3 & 1.3 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n\\hline \nSyst. & 5.7 & 5.3 & 4.9 & 3.9 & 3.6 & 1.3 & 2.9 & 2.6 & 3.0 & 1.5 & 1.7 & 2.1 \\\\\nStat. & 10.0 & 5.1 & 10.3 & 3.6 & 12.3 & 1.0 & 2.7 & 3.3 & 6.2 & 1.5 & 1.3 & 3.2 \\\\\n\\hline \n\\hline \nTotal & 11.5 & 7.3 & 11.4 & 5.3 & 12.8 & 1.7 & 4.0 & 4.2 & 6.9 & 2.1 & 2.2 & 3.9 \\\\ \n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption[Input measurements]{Summary of the measurements used to determine the\n world average $\\ensuremath{M_{\\mathrm{t}}}$. Integrated luminosity ($\\int \\mathcal{L}\\;dt$) is in\n \\ensuremath{\\mathrm{fb}^{-1}}, and all other numbers are in $\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$. The error categories and \n their correlations are described in the text. The total systematic uncertainty \n and the total uncertainty are obtained by adding the relevant contributions \n in quadrature.}\n\\label{tab:inputs}\n\\end{table}\n\n\nThe D\\O\\ Run~IIa l+j analysis is using the JES determined from the\nexternal calibration derived using $\\gamma$+jets events as an\nadditional Gaussian constraint to the in-situ calibration. Therefore\nthe total resulting JES uncertainty has been split into the part\ncoming solely from the in-situ calibration and the part coming from\nthe external calibration. To do that, the measurement without external\nJES constraint has been combined iteratively with a pseudo-measurement\nusing the BLUE method that would use only the external calibration so\nthat the combination gives the actual total JES uncertainty. The\nsplitting obtained in this way is used to assess the iJES and dJES\nuncertainties~\\cite{Mtop2-D0-comb}.\n\nA new analysis technique from CDF is included (trk).\nThis measurement uses both the mean\ndecay-length from B-tagged jets and the mean lepton transverse momentum\nto determine the top-quark mass in l+j candidate events.\nWhile the statistical sensitivity is not as good as the more\ntraditional methods, this technique has the advantage that since it\nuses primarily tracking information, it is almost entirely independent of\nJES uncertainties. As the statitistics of this sample continue to\ngrow, this method could offer a nice cross-check of the top-quark mass\nthat's largely independent of the dominant JES systematic uncertainty\nwhich plagues the other measurements. The statistical correlation\nbetween an earlier version of the trk analysis and a\ntraditional \\hbox{Run~II}\\ CDF l+j measurement was\nstudied using Monte Carlo signal-plus-background pseudo-experiments\nwhich correctly account for the sample overlap and was found to be\nconsistent with zero (to within $<1\\%$) independent of the assumed\ntop-quark mass.\n\\vspace*{0.10in}\n\nThe two D\\O\\ \\hbox{Run~II}\\ lepton+jets results~\\cite{Mtop2-D0-l+ja-final,\nMtop2-D0-l+j-new} are derived from Run~IIa and Run~IIb datasets,\nrespectively, and are labelled as such. The D\\O\\ \\hbox{Run~II}\\ dilepton\nresult is itself a combination of two results\nusing different techniques but partially overlapping dilepton data\nsets~\\cite{Mtop2-D0-di-l-jul08-1,Mtop2-D0-di-l-jul08-2}.\n\\vspace*{0.10in}\n\nTable~\\ref{tab:inputs} also lists the uncertainties of the results,\nsub-divided into the categories described in the next Section. The\ncorrelations between the inputs are described in\nSection~\\ref{sec:corltns}.\n\n\n\n\\section{Error Categories}\n\\label{sec:errors}\n\nWe employ the same error categories as used for the previous world\naverage~\\cite{Mtop-tevewwgWin08}, plus one new category (lepPt). They\ninclude a detailed breakdown of the various sources of uncertainty and\naim to lump together sources of systematic uncertainty that share the\nsame or similar origin. For example, the ``Signal'' category\ndiscussed below includes the uncertainties from ISR, FSR, and\nPDF---all of which affect the modeling of the \\ensuremath{t\\overline{t}}\\ signal. Some\nsystematic uncertainties have been broken down into multiple\ncategories in order to accommodate specific types of correlations.\nFor example, the jet energy scale (JES) uncertainty is sub-divided\ninto several components in order to more accurately accommodate our\nbest estimate of the relevant correlations. Each error category is\ndiscussed below.\n\\vspace*{0.10in}\n\n\\begin{description}\n \\item[Statistical:] The statistical uncertainty associated with the\n \\ensuremath{M_{\\mathrm{t}}}\\ determination.\n \\item[iJES:] That part of the JES uncertainty which originates from\n in-situ calibration procedures and is uncorrelated among the\n measurements. In the combination reported here it corresponds to\n the statistical uncertainty associated with the JES determination\n using the $W\\rightarrow qq^{\\prime}$ invariant mass in the CDF \\hbox{Run~II}\\\n l+j and all-h measurements and D\\O\\ Run~IIa and Run~IIb l+j\n measurements. Residual JES uncertainties, which arise\n from effects\n not considered in the in-situ calibration, are included in other\n categories.\n \\item[aJES:] That part of the JES uncertainty which originates from\n differences in detector $e\/h$ response between $b$-jets and light-quark\n jets. It is specific to the D\\O\\ \\hbox{Run~II}\\ measurements and is\n taken to be uncorrelated with the D\\O\\ \\hbox{Run~I}\\ and CDF measurements.\n \\item[bJES:] That part of the JES uncertainty which originates from\n uncertainties specific to the modeling of $b$-jets and which is correlated\n across all measurements. For both CDF and D\\O\\ this includes uncertainties \n arising from \n variations in the semi-leptonic branching fraction, $b$-fragmentation \n modeling, and differences in the color flow between $b$-jets and light-quark\n jets. These were determined from \\hbox{Run~II}\\ studies but back-propagated\n to the \\hbox{Run~I}\\ measurements, whose rJES uncertainties (see below) were \n then corrected in order to keep the total JES uncertainty constant.\n \\item[cJES:] That part of the JES uncertainty which originates from\n modeling uncertainties correlated across all measurements. Specifically\n it includes the modeling uncertainties associated with light-quark \n fragmentation and out-of-cone corrections.\n \\item[dJES:] That part of the JES uncertainty which originates from\n limitations in the calibration data samples used and which is\n correlated between measurements within the same data-taking\n period, such as \\hbox{Run~I}\\ or \\hbox{Run~II}, but not between\n experiments. For CDF this corresponds to uncertainties associated\n with the $\\eta$-dependent JES corrections which are estimated\n using di-jet data events. For D\\O\\ this includes uncertainties in the\n calorimeter response to light-quark jets, and $\\eta$- and\n $p_{T}$-dependent uncertainties\n constrained using \\hbox{Run~II}\\ $\\gamma+$jet data samples.\n \\item[rJES:] The remaining part of the JES uncertainty which is \n correlated between all measurements of the same experiment \n independent of data-taking period, but is uncorrelated between\n experiments. For CDF, this is dominated by uncertainties in the\n calorimeter response to light-quark jets, and also includes small \n uncertainties associated with the multiple interaction and underlying \n event corrections.\n \\item[lepPt:] The systematic uncertainty arising from uncertainties\n in the scale of lepton transverse momentum measurements. This is an\n important uncertainty for CDF's track-based measurement. It was not\n considered as a source of systematic uncertainty in the \\hbox{Run~I}\\\n measurements or in measurements at D\\O.\n \\item[Signal:] The systematic uncertainty arising from uncertainties\n in the modeling of the \\ensuremath{t\\overline{t}}\\ signal which is correlated across all\n measurements. This includes uncertainties from variations in the ISR,\n FSR, and PDF descriptions used to generate the \\ensuremath{t\\overline{t}}\\ Monte Carlo samples\n that calibrate each method. It also includes small uncertainties \n associated with biases associated with the identification of $b$-jets.\n \\item[Background:] The systematic uncertainty arising from uncertainties\n in modeling the dominant background sources and correlated across\n all measurements in the same channel. These\n include uncertainties on the background composition and shape. In\n particular uncertainties associated with the modeling of the QCD\n multi-jet background (all-j and l+j), uncertainties associated with the\n modeling of the Drell-Yan background (di-l), and uncertainties associated \n with variations of the fragmentation scale used to model W+jets \n background (all channels) are included.\n \\item[Fit:] The systematic uncertainty arising from any source specific\n to a particular fit method, including the finite Monte Carlo statistics \n available to calibrate each method.\n \\item[Monte Carlo:] The systematic uncertainty associated with variations\n of the physics model used to calibrate the fit methods and correlated\n across all measurements. For CDF it includes variations observed when \n substituting PYTHIA~\\cite{PYTHIA4,PYTHIA5,PYTHIA6} (\\hbox{Run~I}\\ and \\hbox{Run~II}) \n or ISAJET~\\cite{ISAJET} (\\hbox{Run~I}) for HERWIG~\\cite{HERWIG5,HERWIG6} when \n modeling the \\ensuremath{t\\overline{t}}\\ signal. Similar\n variations are included for the D\\O\\ \\hbox{Run~I}\\ measurements. The D\\O\\ \n \\hbox{Run~II}\\ measurements use ALPGEN~\\cite{ALPGEN} to model the \\ensuremath{t\\overline{t}}\\ signal and the\n variations considered are included in the Signal category above.\n \\item[UN\/MI:] This is specific to D\\O\\ and includes the uncertainty\n arising from uranium noise in the D\\O\\ calorimeter and from the\n multiple interaction corrections to the JES. For D\\O\\ \\hbox{Run~I}\\ these\n uncertainties were sizable, while for \\hbox{Run~II}, owing to the shorter\n integration time and in-situ JES determination, these uncertainties\n are negligible.\n\\end{description}\nThese categories represent the current preliminary understanding of the\nvarious sources of uncertainty and their correlations. We expect these to \nevolve as we continue to probe each method's sensitivity to the various \nsystematic sources with ever improving precision. Variations in the assignment\nof uncertainties to the error categories, in the back-propagation of the bJES\nuncertainties to \\hbox{Run~I}\\ measurements, in the approximations made to\nsymmetrize the uncertainties used in the combination, and in the assumed \nmagnitude of the correlations all negligibly effect ($\\ll 0.1\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$) the \ncombined \\ensuremath{M_{\\mathrm{t}}}\\ and total uncertainty.\n\n\\section{Correlations}\n\\label{sec:corltns}\n\nThe following correlations are used when making the combination:\n\\begin{itemize}\n \\item The uncertainties in the Statistical, Fit, and iJES\n categories are taken to be uncorrelated among the measurements.\n \\item The uncertainties in the aJES and dJES categories are taken\n to be 100\\% correlated among all \\hbox{Run~I}\\ and all \\hbox{Run~II}\\ measurements \n on the same experiment, but uncorrelated between \\hbox{Run~I}\\ and \\hbox{Run~II}\\\n and uncorrelated between the experiments.\n \\item The uncertainties in the rJES and UN\/MI categories are taken\n to be 100\\% correlated among all measurements on the same experiment.\n \\item The uncertainties in the Background category are taken to be\n 100\\% correlated among all measurements in the same channel.\n \\item The uncertainties in the bJES, cJES, Signal, and Generator\n categories are taken to be 100\\% correlated among all measurements.\n\\end{itemize}\nUsing the inputs from Table~\\ref{tab:inputs} and the correlations specified\nhere, the resulting matrix of total correlation co-efficients is given in\nTable~\\ref{tab:coeff}.\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{|ll||rrr|rr||rrrr|rrr|}\n\\hline \n & & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{3-14}\n & & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & & l+j & di-l & all-j & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l \\\\\n\\hline\n\\hline\nCDF-I & l+j & 1.00& & & & & & & & & & & \\\\\nCDF-I & di-l & 0.29& 1.00& & & & & & & & & & \\\\\nCDF-I & all-j & 0.32& 0.19& 1.00& & & & & & & & & \\\\\n\\hline\nD\\O-I & l+j & 0.26& 0.15& 0.14& 1.00& & & & & & & & \\\\\nD\\O-I & di-l & 0.11& 0.08& 0.07& 0.16& 1.00& & & & & & & \\\\\n\\hline\n\\hline\nCDF-II & l+j & 0.32& 0.18& 0.20& 0.19& 0.07& 1.00& & & & & & \\\\\nCDF-II & di-l & 0.45& 0.28& 0.33& 0.22& 0.11& 0.34& 1.00& & & & & \\\\\nCDF-II & all-j & 0.17& 0.11& 0.16& 0.10& 0.05& 0.15& 0.19& 1.00& & & & \\\\\nCDF-II & trk & 0.16& 0.08& 0.07& 0.13& 0.05& 0.17& 0.12& 0.05& 1.00& & & \\\\\n\\hline\nD\\O-II & l+j\/a & 0.11& 0.05& 0.03& 0.08& 0.03& 0.09& 0.04& 0.03& 0.09& 1.00& & \\\\\nD\\O-II & l+j\/b & 0.12& 0.06& 0.04& 0.09& 0.03& 0.10& 0.05& 0.03& 0.09& 0.24& 1.00 & \\\\\nD\\O-II & di-l & 0.05& 0.04& 0.02& 0.04& 0.04& 0.04& 0.05& 0.03& 0.03& 0.31& 0.16 & 1.00\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption[Global correlations between input measurements]{The resulting\n matrix of total correlation coefficients used to determined the\n world average top quark mass.}\n\\label{tab:coeff}\n\\end{table}\n\nThe measurements are combined using a program implementing a numerical\n$\\chi^2$ minimization as well as the analytic BLUE\nmethod~\\cite{Lyons:1988, Valassi:2003}. The two methods used are\nmathematically equivalent, and are also equivalent to the method used\nin an older combination~\\cite{TM-2084}, and give identical results for\nthe combination. In addition, the BLUE method yields the decomposition\nof the error on the average in terms of the error categories specified\nfor the input measurements~\\cite{Valassi:2003}.\n\n\\section{Results}\n\\label{sec:results}\n\nThe combined value for the top-quark mass is:\n\\begin{eqnarray}\n \\ensuremath{M_{\\mathrm{t}}} & = & 172.4 \\pm 1.2~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }\\,,\n\\end{eqnarray}\nwith a $\\chi^2$ of 6.9 for 11 degrees of freedom, which corresponds to\na probability of 81\\%, indicating good agreement among all the input\nmeasurements. The total uncertainty can be sub-divided into the \ncontributions from the various error categories as: Statistical ($\\pm0.7$),\ntotal JES ($\\pm0.8$), Lepton scale ($\\pm0.1$), Signal ($\\pm0.3$), Background ($\\pm0.3$), Fit\n($\\pm0.1$), Monte Carlo ($\\pm0.3$), and UN\/MI ($\\pm0.02$), for a total\nSystematic ($\\pm1.0$), where all numbers are in units of \\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }.\nThe pull and weight for each of the inputs are listed in Table~\\ref{tab:stat}.\nThe input measurements and the resulting world average mass of the top \nquark are summarized in Figure~\\ref{fig:summary}.\n\\vspace*{0.10in}\n\n\nThe weights of many of the \\hbox{Run~I}\\ measurements are negative. \nIn general, this situation can occur if the correlation between two measurements\nis larger than the ratio of their total uncertainties. This is indeed the case\nhere. In these instances the less precise measurement \nwill usually acquire a negative weight. While a weight of zero means that a\nparticular input is effectively ignored in the combination, a negative weight \nmeans that it affects the resulting central value and helps reduce the total\nuncertainty. See reference~\\cite{Lyons:1988} for further discussion of \nnegative weights.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{topmass_tev0708.eps}\n\\end{center}\n\\caption[Summary plot for the world average top-quark mass]\n {A summary of the input measurements and resulting world average\n mass of the top quark.}\n\\label{fig:summary} \n\\end{figure}\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\small\n\\begin{tabular}{|l||rrr|rr||rrrr|rrr|}\n\\hline \n & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{2-13}\n & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & l+j & di-l & all-j & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l\\\\\n\\hline\n\\hline\nPull & $+0.5$ & $-0.4$ & $+1.2$ & $+1.5$ & $-0.3$ & $-0.1$ & $-0.3$ & $+1.1$ & $+0.4$ \n & $-0.5$ & $+0.3$ & $+0.5$ \\\\\nWeight [\\%]\n & $- 3.4$ & $- 0.6$ & $- 0.6$ & $+ 1.2$ & $+ 0.2$ & $+46.1$ & $+ 3.7$ & $+ 5.2$ & $+ 0.02$ \n & $+23.7$ & $+21.5$ & $+ 3.0$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption[Pull and weight of each measurement]{The pull and weight for each of the\n inputs used to determine the world average mass of the top quark. See \n Reference~\\cite{Lyons:1988} for a discussion of negative weights.}\n\\label{tab:stat} \n\\end{table} \n\nAlthough the $\\chi^2$ from the combination of all measurements indicates\nthat there is good agreement among them, and no input has an anomalously\nlarge pull, it is still interesting to also fit for the top-quark mass\nin the all-j, l+j, and di-l channels separately. We use the same methodology,\ninputs, error categories, and correlations as described above, but fit for\nthe three physical observables, \\ensuremath{\\MT^{\\mathrm{all-j}}}, \\ensuremath{\\MT^{\\mathrm{l+j}}}, and \\ensuremath{\\MT^{\\mathrm{di-l}}}.\nThe results of this combination are shown in Table~\\ref{tab:three_observables}\nand have $\\chi^2$ of 4.9 for 9 degrees of freedom, which corresponds to a\nprobability of 84\\%.\nThese results differ from a naive combination, where\nonly the measurements in a given channel contribute to the \\ensuremath{M_{\\mathrm{t}}}\\ \ndetermination in that channel, since the combination here fully accounts\nfor all correlations, including those which cross-correlate the different\nchannels. Using the results of \nTable~\\ref{tab:three_observables} we calculate the chi-squared consistency\nbetween any two channels, including all correlations, as \n$\\chi^{2}(dil-lj)=0.08$, $\\chi^{2}(lj-allj)=1.7$, and \n$\\chi^{2}(allj-dil)=1.9$. These correspond to \nchi-squared probabilities of 78\\%, 19\\%, and 17\\%, respectively, and indicate \nthat the determinations of \\ensuremath{M_{\\mathrm{t}}}\\ from the three channels are consistent with \none another.\n\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{|l||c|rrr|}\n\\hline\nParameter & Value (\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }) & \\multicolumn{3}{|c|}{Correlations} \\\\\n\\hline\n\\hline\n$\\ensuremath{\\MT^{\\mathrm{all-j}}}$ & $177.5\\pm 4.0$ & 1.00 & & \\\\\n$\\ensuremath{\\MT^{\\mathrm{l+j}}}$ & $172.2\\pm 1.2$ & 0.14 & 1.00 & \\\\\n$\\ensuremath{\\MT^{\\mathrm{di-l}}}$ & $171.5\\pm 2.6$ & 0.17 & 0.32 & 1.00 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption[Mtop in each channel]{Summary of the combination of the 12\nmeasurements by CDF and D\\O\\ in terms of three physical quantities,\nthe mass of the top quark in the all-jets, lepton+jets, and di-lepton channels. }\n\\label{tab:three_observables}\n\\end{table}\n\n\\section{Summary}\n\\label{sec:summary}\n\nA preliminary combination of measurements of the mass of the top quark\nfrom the Tevatron experiments CDF and D\\O\\ is presented. The\ncombination includes five published \\hbox{Run~I}\\ measurements and \nseven preliminary \\hbox{Run~II}\\ measurements. Taking into\naccount the statistical and systematic uncertainties and their\ncorrelations, the preliminary world-average result is: $\\ensuremath{M_{\\mathrm{t}}}= 172.4 \\pm\n1.2~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$, where the total uncertainty is obtained assuming Gaussian\nsystematic uncertainties and adding them plus the statistical\nuncertainty in quadrature. While the central value is somewhat higher\nthan our 2007 average, the averages are compatible as appreciably more\nluminosity and refined analysis techniques are now used.\n\\vspace*{0.10in}\n\nThe mass of the top quark is now known with a relative precision of\n0.7\\%, limited by the systematic uncertainties, which are dominated by\nthe jet energy scale uncertainty. This systematic is expected to\nimprove as larger data sets are collected since new analysis\ntechniques constrain the jet energy scale using in-situ $W\\rightarrow\nqq^{\\prime}$ decays. It can be reasonably expected that with the full\n\\hbox{Run~II}\\ data set the top-quark mass will be known to better than\n0.7\\%. To reach this level of precision further work is required to\ndetermine more accurately the various correlations present, and to\nunderstand more precisely the $b$-jet modeling, Signal, and Background\nuncertainties which may limit the sensitivity at larger data sets.\nLimitations of the Monte Carlo generators used to calibrate each fit\nmethod also become more important as the precision reaches the\n$\\sim1~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$ level; these warrant further study in the near future.\n\n\\clearpage\n\n\\bibliographystyle{tevewwg}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nQuantum field theory required the finest element of mathematical physics for its development, pushing at time the development of new aspects of mathematics. \nThe analyticity requirements have been instrumental in the theory of the holomorphic functions of many variables and their analyticity domains, \nwith such high points as the edge of the wedge theorem of Bogolubov or the Bros--Iagolnitzer transform used for the definition of the analytic \nwave front. Type III factors are everywhere and the instrument of their classification stems from the KMS condition, introduced for the study of \nfinite temperature theory, which was found to coincide with the Tomonoga condition. Perturbative expansions and their renormalization require \nsubtle combinatorics and the introduction of Hopf algebras allows one to clarify and make more explicit the classical proofs of renormalizability. \nUsing the formalism of groupoids may be useful to reduce the burden of controlling the effect of the symmetry factors. Evaluating Feynman \nintegrals requires numbers which can be periods, with the action of a motivic Galois group and links with many conjectures in algebraic geometry.\nConstructive theory has been able to show that some of these theories can be given a precise mathematical sense, but has failed to address \nthe most relevant ones for our understanding of the physical world, the four dimensional gauge theories. \n\nIn fact, due to the specificity of renormalization, the perturbative series is the richest source of information on quantum field theory.\nOther approaches deal necessarily with a regularized version of the theory which lacks many of the structural properties of the final theory. A precise \ndefinition will therefore seek to define the relevant Green functions from their perturbative series, but the procedure cannot be straightforward, \nsince it has been long recognized that quantum field theories (QFT for short) cannot depend holomorphically on the coupling parameter in a full \nneighborhood of the origin and therefore the perturbative series is at most asymptotic. A number of works have tended to show that the growth of \nthe terms of the formal perturbative series is slow enough to allow the definition of a Borel transform around the origin. This is however \nonly the first step in the definition of a sum for the perturbative series. One must also be able to analytically continue this Borel \ntransform up to infinity and furthermore verify that near infinity, this analytically continued Borel transform does not grow \nfaster than any exponential function.\n\nDefining an analytic \ncontinuation seems a formidable task, but in the cases where the functions obey equations, analog equations for their Borel transform can be \ndeduced and, with the help of alien calculus, used to constrain the possible singularities of the Borel transform. Singularities may appear on \nthe positive real axis, preventing a straightforward application of the Laplace transform. One could resort to lateral summation, using a \nshifted integration axis, but this means that the reality properties of the solution are lost. This can be an advantage in some situations, \nwhere the imaginary part of a perturbed energy signals the possible decay of a metastable state to the continuum, but unitarity could be at risk. However a suitable average of the analytic continuations circumventing the singularities on the real axis can be used to define a sum which both is real and respects the equations.\n\nIn this work, we will therefore argue that quantum field theory cannot dispense with the whole body of work on summation methods which has been \nthoroughly expanded by Jean Ecalle~\\cite{Ecalle81,Ecalle81b,Ecalle81c}. As in previous works~\\cite{BeCl14}, we will base our considerations on computations in a specific model, \nbut we think that they reveal phenomena at work in most exactly renormalizable field theories.\n\nAlthough the insight about these summation methods comes from the study of Borel transforms which form an algebra under convolutive products, \nmuch can be done while remaining in the domain of more or less formal series. In particular, formal transseries solutions allow to \nrecognize the possible forms of alien derivatives. It is then possible to express the solution in terms of transmonomials, special functions \nwith simple alien derivatives, so that we never have to explicitly refer to the Borel transforms in computations. The Borel-transformed functions are however \nwhat justify the different computations. \n\nThis article is divided as follows: first we give a short introduction to some concepts of resurgence theory that are of importance for our work. \nThen some previous results of \\cite{BeSc08,BeCl13,BeCl14} that are to be developed further are recalled. We then compute the leading terms \nfor every exponentially small terms in the anomalous dimension in the subsequent section. They are shown to sum up to a known analytic function of the \nnonperturbatively small quantity \\(e^{-r}\\). Alien calculus is then used to deduce from the computed terms the singularities of the Borel transform, while \nthe first singularity of the Borel transform is used to constrain a free parameter in the previous paragraphs.\nFinally, we apply the same process to the two-point function of the theory and see that the nonperturbative terms can get multiplied by powers of the momentum. The resulting function has a singularity for a timelike momentum, which fixes a nonperturbative mass scale, while the euclidean side is completely tame. The way such a result could appear in the process of Borel summation gives further indications on the analyticity domain of the theory.\n\n\n\\section{Elements of resurgence theory}\n\n\\subsection{Borel transform and Borel resummation}\n\nA very nice introduction to the subject of Borel transform and resurgence theory is \\cite{Sa14}. Here we will follow the presentation of \\cite{Bo11}, \nwith some additional material needed for our subject.\n\nA simple definition of the Borel transform is to say that it is a morphism between two rings of formal series, defined as being linear and its value on monomials:\n\\begin{eqnarray}\n \\mathcal{B}: a\\mathbb{C}[[a]] & \\longrightarrow & \\mathbb{C}[[\\xi]] \\\\\n\\tilde{f}(a) = a\\sum_{n=0}^{+\\infty}c_na^n & \\longrightarrow & \\hat{f}(\\xi) = \\sum_{n=0}^{+\\infty}\\frac{c_n}{n!}\\xi^n. \\nonumber\n\\end{eqnarray}\nThis definition is extended to noninteger powers by\n\\begin{equation*}\n \\mathcal{B}\\left(a^{\\alpha+1}\\right)(\\xi) := \\frac{\\xi^{\\alpha}}{\\Gamma(\\alpha+1)}.\n\\end{equation*}\nIn the case where $\\alpha$ is an integer, we find back the above definition.\n\nThen the point-like product of formal series becomes the convolution product of formal series, or of functions when the Borel transform is the \ngerm of an analytic function. It is therefore consistent to define the Borel transform of a constant as the formal unity of the convolution product (i.e.,\nthe Dirac ``function''):\n\\begin{equation}\n \\mathcal{B}(1) := \\delta.\n\\end{equation}\n\nIn the following, we will restrict ourselves to the cases where $\\hat f$ is the germ of an analytic function at the origin, endlessly continuable on \n$\\mathbb{C}$, i.e., that on any line $L$ of $\\mathbb{C}$ a representative of the germ $\\hat f$ has a countable number of singularities and is continuable along \nany path obtained by following $L$ and avoiding the singularities by going over or below them. Moreover, we will assume that these singularities are \nalgebraic, that is to say that they are not more singular than a pole. More precisely, if $\\xi_0$ is a singularity of $\\hat\\phi$, we shall have\n\\begin{equation}\n \\exists\\alpha\\in\\mathbb{R}\\mid |(\\xi-\\xi_0)^{-\\alpha}\\hat\\phi_r(\\xi)|\\underset{\\xi\\rightarrow\\xi_0}{\\longrightarrow}0.\n\\end{equation}\nThe supremum of the $\\alpha$ for which the above holds will be called the order of $\\hat\\phi$ at $\\xi_0$. Notice that if $\\alpha$ is positive, \nit is important that the condition is not directly on \\(\\hat\\phi\\) but on \\(\\hat\\phi_r\\), obtained from \\(\\hat\\phi\\) by subtracting a suitable polynomial in~\\(\\xi\\). {A typical example for \\(\\alpha\\) a positive integer is \\(\\hat\\phi_r(\\xi) \\simeq (\\xi-\\xi_0)^\\alpha \\ln(\\xi-\\xi_0) \\), but there are no reason for \\(\\hat\\phi\\) itself to have a zero at \\(\\xi_0\\). Alternatively, the singularity can be characterized by the behavior of the difference of the function and its analytic continuation after looping around \\(\\xi_0\\), but this does not work for poles.}\nIt is even possible to have singularities with an infinite order, but in our applications, the order will always be a finite rational number.\n\nLet us call $\\widehat{RES}$ the space of germs of analytic functions at the origin endlessly continuable on $\\mathbb{C}$ and $\\widetilde{RES}\\subset a\\mathbb{C}[[a]]$ the set of formal series whose image under the Borel transform is in\n$\\widehat{RES}$. When working with elements of $\\widehat{RES}$ we will say that we are in the convolutive model, while $\\widetilde{RES}$ will be called the \nformal model. We will also say that we are in the Borel plane and the physical plane, respectively.\n\nThere exists an inverse to the Borel transform: the Laplace transform. {For $\\phi\\in\\widehat{RES}$ we write $\\hat\\phi$ for the analytic continuation of the Borel transform of $\\phi$. The definition of the Laplace transform \non $\\hat\\phi$} involves a certain direction $\\theta$ in the complex plane:\n\\begin{equation}\n \\mathcal{L}_{\\theta}[\\hat\\phi] := \\int_0^{e^{i\\theta}\\infty}\\hat\\phi(\\zeta)e^{-\\zeta\/a}\\d\\zeta.\n\\end{equation}\nIt is well defined for at least some values of \\(a\\) if \\(\\hat\\phi\\) is smaller than some exponential in the direction \\(\\theta\\). The resummation operator in the direction $\\theta$ is the composition of the Borel transform and the Laplace transform in the direction \\(\\theta\\):\n\\begin{equation}\n S_{\\theta} := \\mathcal{L}_{\\theta}\\circ\\mathcal{B}.\n\\end{equation}\nIf $\\hat\\phi$ does not have any singularities in the directions \\(\\theta\\) between $\\theta'$ and $\\theta''$ included and satisfies suitable exponential bounds at infinity in this sector, different $\\mathcal{L}^{\\theta}[\\hat\\phi]$ coincide wherever they are both defined through Cauchy's theorem, so that they define a single analytic function on the sector delimited \nby $\\theta' -\\pi\/2$ and $\\theta''+ \\pi\/2$, which is a possible resummation of the formal series \\(\\tilde \\phi\\). We see that the different resummations are defined in sectors whose limits depend on the singularities of the Borel transform, but their definition domains have nontrivial intersections. \n\nIn the following, we will need more general objects than the elements of \\(\\widehat{RES}\\) and the corresponding $\\widetilde{RES}$. \nWe define simple resurgent symbols with an additional variable \\(\\omega \\in \\mathbb{C}\\) with the meaning that the corresponding object in the formal or geometric models get multiplied by \\(e^{-\\omega\/a}\\), so that: \n\\begin{equation*}\n \\dot\\phi^{\\omega} := e^{-\\omega\/a}\\phi\\in\\dot{\\widetilde{RES}} \\supset \\widetilde{RES}\n\\end{equation*}\nIn the formal model, linear combinations of simple resurgent symbols are simple examples of transseries. One can define more general transseries, but this would lead us away from our topic. {A very pleasant introduction to transseries, very accessible to physicists is} \\cite{Ed09}.\n\n\nFinally, the operator $S_{\\theta}$ extends to \\(\\dot{\\widehat{RES}}\\) through the formula\n\\begin{equation}\\label{SthetaDot}\n S_{\\theta}[\\hat\\phi^{\\omega}](a) := e^{-\\omega\/a}\\mathcal{L}_{\\theta}[\\hat\\phi](a)\n\\end{equation}\nand by linear extension.\n\n\\subsection{Stokes automorphism and Alien derivative}\n\nNow let $\\phi\\in\\widetilde{RES}$ be such that $\\hat\\phi\\in\\widehat{RES}$ has singularities in the direction $\\theta$. Then we define the lateral \nresummations $S_{\\theta\\pm}$ as the usual one but with the Laplace transform \\(\\mathcal{L}_{\\theta\\pm}\\) involving integrals avoiding the singularities by going above (for \n$\\mathcal{L}_{\\theta+}$) or below (for $\\mathcal{L}_{\\theta-}$) all of them. They correspond to the limit of \\(S_{\\theta'}\\) when \\(\\theta'\\) tends to \\(\\theta\\) either from above or from below.\\footnote{{\nIn most applications, the possible arguments of the positions of the singularities form a discrete set, so that the different \\(S_{\\theta'}\\) define the same analytic function for an open set of values of \\(\\theta'\\) and we do not really need to take the limit in the definition of \\(S_{\\theta\\pm}\\). However, it is possible to have singularities for example at the positions of all Gaussian integers \\(\\mathbb{Z} + i \\mathbb{Z}\\), in which case the limiting procedure is unavoidable.}}\n\\begin{equation}\n S_{\\theta\\pm}[\\phi](a) := \\int_0^{e^{i(\\theta\\pm\\varepsilon)}\\infty}\\hat\\phi(\\zeta)e^{-\\zeta\/a}\\d\\zeta.\n\\end{equation}\nThe extension of the lateral resummations to the elements of $\\dot{\\widetilde{RES}}$ is similar to the extension of the regular ones given by equation~(\\ref{SthetaDot}).\n\nNow, the key point is that the lateral resummation are linked by the so-called Stokes automorphism in the direction $\\theta$, written \n$\\mathfrak{G}_{\\theta}$.\n\\begin{equation} \\label{def_Stokes}\n \\mathcal{L}_{\\theta+} \\circ \\mathfrak{G}_{\\theta} = \\mathcal{L}_{\\theta-}.\n\\end{equation}\nWe clearly have $\\mathfrak{G}_{\\theta}:\\dot{\\widehat{RES}}\\longrightarrow\\dot{\\widehat{RES}}$. Since both \\(\\mathcal{L}_{\\theta+}\\) and \\(\\mathcal{L}_{\\theta-}\\) are algebra morphisms from the convolutive algebra in the Borel plane to the algebra of functions, \\(\\mathfrak{G}_{\\theta}\\) is an automorphism of the convolution algebra \\(\\dot{\\widehat{RES}}\\). This automorphism encodes how the \nfunction ``jumps'' when the direction of integration crosses a line of singularities (called a Stokes line), already in the extended convolutive model. It can be decomposed in homogeneous components that shift the exponents of the extended models by the complex numbers \\(\\omega\\) which belong to the direction \\(\\theta\\): they are called the lateral alien operators \\(\\Delta^+_\\omega\\). The action of the alien operator \\(\\Delta_\\omega^+\\) on \\({\\hat\\phi}^\\sigma\\) is linked to the singularity of \\({\\hat\\phi}^\\sigma\\) in \\(\\omega\\) and carries the index \\(\\omega+\\sigma\\). The fact that \\(\\mathfrak{G}_{\\theta}\\) is an automorphism translates in the following relations for its components:\n\\begin{equation} \\label{rel_lateral}\n\t\\Delta^+_\\omega (\\hat f \\star \\hat g) = \\sum_{\\omega' + \\omega'' = \\omega} \\Delta^+_{\\omega'} f \\star \\Delta^+_{\\omega''} g,\n\\end{equation}\nwhere the sum includes the cases where \\(\\omega'\\) or \\(\\omega''\\) is 0, and \\(\\Delta^+_0\\) is defined to be the identity.\n\nSince the relation \\eqref{rel_lateral} is not very simple, we use the logarithm of the Stokes automorphism, which is a derivation. The homogeneous components of this logarithm are therefore also derivations, which are called alien derivatives. More precisely\n\\begin{equation} \\label{def_alien}\n \\mathfrak{G}_{\\theta} = \\exp\\left(\\sum_{\\omega\\in\\Gamma_{\\theta}}\\Delta_{\\omega}\\right),\n\\end{equation}\nwith $\\Gamma_{\\theta}$ the singular locus of the function of interest in the direction $\\theta$. The alien derivatives and lateral alien \nderivatives are linked by the equivalent relations\n\\begin{align*}\n & \\Delta_{\\omega_n}^+ = \\sum_{p=1}^n\\sum_{\\omega_1+\\cdots+\\omega_p=\\omega_n}\\frac{1}{p!}\\Delta_{\\omega_1}\\cdots\\Delta_{\\omega_p} \\\\\n & \\Delta_{\\omega_n} = \\sum_{p=1}^n\\sum_{\\omega_1+\\cdots+\\omega_p=\\omega_n}\\frac{(-1)^{p-1}}{p}\\Delta_{\\omega_1}\\cdots\\Delta_{\\omega_p}\n\\end{align*}\nwith all the $\\omega_i$s on the same half-line from the origin to infinity.\n\nAs their names suggest, the alien derivatives are indeed derivations for the convolution product. We shall not give a proof here (we refer the \nreader to \\cite{Sa14} for such a proof), but this fact shall not come as a surprise. Indeed, it is well-known that the operator $\\mathcal{A}$ \nacting on smooth function as a translation\n\\begin{equation*}\n \\mathcal{A}[f](x) = f(x+1)\n\\end{equation*}\ninduces an automorphism on the space of functions. And we can write\n\\begin{equation*}\n \\mathcal{A} = \\exp\\left(\\frac{\\d}{\\d x}\\right)\n\\end{equation*}\nthanks to the Taylor expansion for analytic functions, so that it appears as the exponential of a derivation. Since the Stokes automorphism can be viewed as a translation across a Stokes line, it is understandable that its \nlogarithm is a derivative.\n\nThe alien operators are a priori defined in the convolutive model, but it is convenient to extend them to $\\dot{\\widetilde{RES}}$ by\n\\begin{equation}\n \\mathcal{B}[\\Delta_{\\omega}\\phi] := \\Delta_{\\omega}\\hat\\phi,\n\\end{equation}\nand similarly for $\\Delta_{\\omega}^+$. The alien derivatives, so extended to formal series, become derivatives for the point-like product, since the point-like product \nof functions becomes the convolution product in the Borel plane. \n\nThe alien derivatives have other useful properties. The most important one is \n\\begin{equation} \\label{commutation_alien_usuelle}\n \\Delta_{\\omega}\\partial_z = \\partial_z\\Delta_{\\omega}\n\\end{equation}\nin the formal model, with $z=1\/a$. A proof of this result\ncan be found in \\cite{Sa14}, and another in the general case (which is of interest for us) is in \\cite{Sa06}. The commutation with the ordinary derivative is not so simple if we consider the alien derivative to act in \\(\\widehat{RES}\\) and not in the dotted model. In fact, in many texts, what we denote simply as \\(\\Delta_\\omega\\) is denoted \\(\\dot\\Delta_\\omega\\).\n\n\\subsection{Real resummations}\n\nStokes operators can be a part of the study of the monodromy around singular points of a differential equation, but it may happen that the Borel transform $\\hat\\phi$ has singularities in the direction $\\theta$ in which we are interested to perform a \nresummation, typically the direction $\\theta=0$ for a series with only real coefficients. In this case, the lateral resummations get imaginary parts. This can be a nice feature when the imaginary part of \nthe energy corresponds to the decay probability through tunneling of a state, but generally, we would like to obtain a real solution for a physical \nproblem. The issue is that the simple mean of the two different lateral summations is real, but it is nevertheless not satisfying: we would like this \nreal resummation to satisfy the same equations as the formal solution and this can only be ensured if the convolution product of the means is the mean of the convolution products. \nThe only way to ensure this is to take a suitable combination of the analytic continuations of the function \nalong all the paths that can be taken, going above or below each of the singularities.\n\nIt has been known for a long time that such a real solution is given by the so-called median resummation, a fact that have been shown explicitly in \n \\cite{AnSc13}. A possible expression for this median resummation is\n\\begin{equation}\n S_{\\text{med}} := S_{\\theta-}\\circ\\mathfrak{G}_{\\theta}^{1\/2} = S_{\\theta+}\\circ\\mathfrak{G}_{\\theta}^{-1\/2}\n\\end{equation}\nwhere the power of the Stokes automorphism is defined from a natural extension of the definition \\eqref{def_alien}:\n\\begin{equation}\n \\mathfrak{G}_{\\theta}^{\\nu} := \\exp\\left(\\nu\\sum_{\\omega\\in\\Gamma_{\\theta}}\\Delta_{\\omega}\\right).\n\\end{equation}\nSince the Stokes operator \\(\\mathfrak{G}_{\\theta}\\) is an automorphism, its powers are also automorphisms and the median resummation respects products as a composition of operations preserving products. \n\nReturning to the definition of the Stokes automorphism, it can be seen that \\(\\Delta^+_\\omega\\) corresponds to taking the difference between two possible analytic continuation of the Borel transform beyond the point \\(\\omega\\). The different alien derivatives can also be computed as combinations of different analytic continuations of the Borel transform, going above or below the different singularities (but without ever going backwards). \nThe median summation likewise is a suitable average of the different possible analytic continuations of the Borel transform. \nWhen we go beyond a singularity \\(\\omega\\), we must take a different combination of analytic continuations of the Borel transform: the function we will integrate in the Laplace transform has therefore singularities at the points~\\(\\omega\\) that cannot be avoided and result in nonperturbative contributions to the resummed function.\n\nThe square root of the Stokes operator is simple, but since it gives quite an important weight to paths which cross the real axis a large number of times, the obtained average may grow faster than the lateral values. \nIn~\\cite{Ec92}, Ecalle shows how one could circumvent this problem through accelerations, which allow to reduce the ambiguity between lateral summations from \\(1{\/}\\!\\exp(z)\\) to \\(1{\/}\\!\\exp(\\exp(\\dots(z)\\dots)) \\), with theoretically any finite composition of exponentials, through the control of `emanated' resurgence.\nHowever, other averages are possible, the organic averages, still compatible with the convolution product, which are essentially no larger than the lateral determinations and therefore allow us to avoid this whole procedure. \nIn any cases, these averages still define from the Borel transform a function which is real on the real axis and has singularities on the real axis so that the sum is only defined in the positive half-plane, since it is impossible to relate this integral to others on different integration axis.\n\nAt the approximation level we will reach in the present work, such subtleties will not have a clear effect. However, they can become important if we are to improve on our treatment of these nonperturbative contributions.\n\n\\section{Rehearsal}\nWe are still working with the model used in our previous investigations~\\cite{BeSc08,BeCl13,BeCl14}, the massless supersymmetric Wess--Zumino \nmodel. Even if it is far from a realistic particle physics model, the fact that we only deal with two-point functions and their simple dependence on a unique kinetic \ninvariant gives a more tractable situation than more realistic theories. Nevertheless, the presence of singularities on all integer \npoints for the Borel transform of the renormalization group \\(\\beta\\)-function is probably the generic case in massless exactly \nrenormalizable QFTs, the kind we would like to better understand for their relevance in the description of our universe.\n\nOur former studies are all based on the same simple Schwinger--Dyson equation, solved through the combination of the extraction of the \nanomalous dimension of the field from the Schwinger--Dyson equation and the use of the renormalization group equation to obtain the full \npropagator from this anomalous dimension. We will limit ourselves to the simplest one of the Schwinger--Dyson equations, since it allows us to \nretain a degree of explicitness in the apparent explosion of different series appearing in the object named the {\\em display} by Jean Ecalle, \nan object which collects all information on the alien derivatives of a function. A proper extension of the arguments put forward \nin~\\cite{BeSc12} should prove that any higher order correction to this Schwinger--Dyson equation will only change higher order terms in the \nindividual components of this display, letting its main characteristic unchanged. The factorial growth of the number of high order terms \nbeyond the large \\(N\\), planar limit would present a further challenge, but we will see that there are plenty of questions to solve before.\n\nThe fundamental insight in~\\cite{BeCl14} is that it is in the Borel plane, where the alien derivatives \nhave a clear meaning as singular parts of a function at a given point, that general properties are easier to prove. \nHowever, most computations are easier to carry on in the form of \ntransseries, where the computations look like mechanical operations on formal objects. However, one important component in the \ncomputation scheme of~\\cite{BeCl14} was the contour integral representation of the propagator, with its characteristic property that the possible contours \nchange when considering different points in the Borel plane. It would be interesting to \ngive an interpretation of these contour integrals in the formal scheme and recover how computations can be done using the \nexpansion of the Mellin transform, with subtracted pole parts, at integer points. However, we will see that the simple approximations used for this \nwork do not need such a development.\n\nWe start with the renormalization group equation (RGE) for the two-point function\n\\begin{equation} \\label{renorm_G}\n \\partial_L G(a,L) = \\gamma(1 + 3a\\partial_a )G(a,L),\n\\end{equation}\nwhere we have used $\\beta=3\\gamma$, which can be proved by superspace \\cite{Piguet} or Hopf \\cite{CoKr00} techniques. {A derivation of this equation for the solution of a Schwinger--Dyson equation is detailed in~\\cite{BeSc08}, see also~\\cite{Cl15}.}\nHere $L=\\ln (p^2\/\\mu^2)$ is the kinematic parameter. Expanding $G$ in this parameter \\(L\\),\n\\begin{equation} \\label{rep_G_serie}\n G(a,L) = \\sum_{k=0}^{+\\infty}\\frac{\\gamma_k(a)}{k!}L^k\n\\end{equation}\n(with $\\gamma_1:=\\gamma$)\\footnote{{Be aware that other authors~\\cite{BrKr99,KrYe2006} use different conventions but the same $\\gamma_k$ notations.}} gives a simple recursion on the $\\gamma_k$s\n\\begin{equation} \\label{renorm_gamma_old}\n \\gamma_{k+1} = \\gamma(1+ 3a\\partial_a)\\gamma_k.\n\\end{equation}\nTherefore, at least in principle, it is enough to know $\\gamma$ to rebuild the two-point function.\n\nOn the other hand, we also have the (truncated) Schwinger--Dyson equation, graphically depicted as\n\\begin{equation}\\label{SDnlin}\n\\left(\n\\tikz \\node[prop]{} child[grow=east] child[grow=west];\n\\right)^{-1} = 1 - a \\;\\;\n\\begin{tikzpicture}[level distance = 5mm, node distance= 10mm,baseline=(x.base)]\n \\node (upnode) [style=prop]{};\n \\node (downnode) [below of=upnode,style=prop]{}; \n \\draw (upnode) to[out=180,in=180] \n \tnode[name=x,coordinate,midway] {} (downnode);\n\\draw\t(x)\tchild[grow=west] ;\n\\draw (upnode) to[out=0,in=0] \n \tnode[name=y,coordinate,midway] {} (downnode) ;\n\\draw\t(y) child[grow=east] ;\n\\end{tikzpicture}.\n\\end{equation}\n{The L.H.S. is the two-point function while the R.H.S. contains two dressed propagators, which are equal to the free propagator multiplied by the two-point function.} Computing the loop integral allows to write this equation as\n\\begin{equation} \\label{SDE_old}\n \\gamma(a) = a\\left.\\left(1+\\sum_{n=1}^{+\\infty}\\frac{\\gamma_n}{n!}\\frac{\\text{d}^n}{\\text{dx}^n}\\right)\\left(1+\\sum_{m=1}^{+\\infty}\\frac{\\gamma_m}{m!}\\frac{\\text{d}^m}{\\text{dy}^m}\\right)H(x,y)\\right|_{x=y=0} =: a\\mathcal{I}(H(x,y)).\n\\end{equation}\nwith $H$ known as the Mellin transform of the one-loop integral:\n\\begin{equation} \\label{def_H}\n H(x,y) := \\frac{\\Gamma(1-x-y)\\Gamma(1+x)\\Gamma(1+y)}{\\Gamma(2+x+y)\\Gamma(1-x)\\Gamma(1-y)}.\n\\end{equation}\nThe idea of \\cite{Be10a}, which was fully exploited in \\cite{BeCl13} is to replace the one-loop Mellin transform by a truncation \ncontaining its singularities. Let us define \n\\begin{equation} \\label{form_F}\n F_k := \\mathcal{I}\\left(\\frac1{k+x}\\right)=\\frac{1}{k}\\biggl(1+\\sum_{n=1}^{+\\infty}\\left(-\\frac{1}{k}\\right)^n\\gamma_n\\biggr).\n\\end{equation}\n(which gives the contributions of the poles $1\/(k+x)$ or $1\/(k+y)$ of $H$) and \n\\begin{equation} \\label{def_Lk}\n L_k:=\\mathcal{I}\\left(\\frac{Q_k(x,y)}{k-x-y}\\right)=\\sum_{n,m=0}^{+\\infty}\\frac{\\gamma_n\\gamma_m}{n!m!}\\left.\\frac{\\text{d}^n}{\\text{d}x^n}\\frac{\\text{d}^m}{\\text{d}y^m}\\frac{Q_k(x,y)}{k-x-y}\\right|_{x=0,y=0}\n\\end{equation}\nwhich contains the contributions from the {part of \\(H\\) singular on the line \\(k-x-y=0\\)}. Here $Q_k$ is a suitable expansion of \nthe residue of $H$ at this singularity, a polynomial in the product \\(xy\\).\n\nIt was shown in \\cite{Be10a} that these $F_k$ and $L_k$ obey renormalization group derived equations. Here we are only interested \nin the equations of $L_k$ which are\n\\begin{equation} \\label{equa_L}\n (k-2\\gamma - 3\\gamma a\\partial_a)L_k = Q_k(\\partial_{L_1}\\partial_{L_2})G(a,L_1)G(a,L_2)\\Bigr|_{L_1=L_2=0} = \\sum_{i=1}^kq_{k,i}\\gamma_i^2.\n\\end{equation}\nThe Schwinger--Dyson equations can also be written in terms of these functions. Since we are looking for a resummation of the two-point\nfunction along the positive real axis, and since it was shown in \\cite{BeCl14} that the $L_k$ are responsible for the singularities \nof the Borel transform of $\\gamma$ (and therefore of the two-point function) on the real axis, we will only take care of the terms \nof the Schwinger--Dyson equation involving $L_k$. They have the very simple form\n\\begin{equation} \\label{SDE_phys_plan}\n \\gamma(a) = a\\sum_{k=1}^{+\\infty}L_k(a) + (\\text{contributions from }F_k){=a+O(a^2)}.\n\\end{equation}\nIn order to simplify the results of \\cite{BeCl14}, we consider $\\gamma$ and all the other quantities as formal series in $r:=1\/(3a)$ rather than \nin $a$. We then perform a Borel transform in $r$, according to the conventions most used in the mathematical literature. The perturbative domain is then the one\nfor large values of \\(r\\) and typical nonperturbative contributions will be of the form \\(e^{-nr}\\) for some integer \\(n\\). The advantage being that the singularities of the Borel transform $\\hat\\gamma$ are now located in $\\mathbb{Z}^*$ rather than \n$\\mathbb{Z}^*\/3$. Let us notice that we now have $\\hat\\gamma(0)=1\/3$, but what is most relevant is that \\(\\hat\\beta(0) = 3 \\hat\\gamma(0) = 1\\).\n\nAs explained in \\cite{BeCl14}, a perturbative analysis (i.e., for $\\xi$ small) of the Borel-transformed renormalization group equation suggests to\nwrite the Borel transform $\\hat G:=\\mathcal{B}(G-1)$ as a loop integral\n\\begin{equation} \\label{param_G}\n \\hat{G}(\\xi,L) = \\oint_{\\mathcal{C}_{\\xi}}\\frac{f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta\n\\end{equation}\nwhere $\\mathcal{C}_{\\xi}$ is any contour enclosing $\\xi$ and the origin. Writing the \nrenormalisation group equation \\eqref{renorm_G} in the Borel plane and in term of the $f(\\xi,\\zeta)$ function we get\n\\begin{equation} \\label{renorm_f}\n (\\zeta-\\xi)f(\\xi,\\zeta) = \\hat{\\gamma}(\\xi) + \\int_0^{\\xi}\\hat{\\gamma}(\\xi-\\eta)f(\\eta,\\zeta)\\d\\eta + \\int_0^{\\xi}\\hat{\\beta}'(\\xi-\\eta)\\eta f(\\eta,\\zeta)\\d\\eta.\n\\end{equation}\n\n\\section{Resummations of the anomalous dimension}\n\nWe purposefully put a plural in the ``resummations'' of the title of this section to emphasize that two distinct resummations will be performed here.\nFirst the median resummation and its transseries analysis deliver exponentially small terms, then we sum the dominant terms of the obtained transseries.\n\n\\subsection{Transseries solution}\n\nWe want to compute the leading coefficient of \\(e^{-nr}\\) in the transseries expansion of $\\beta$. Let us start by writing \nthe Schwinger--Dyson equation \\eqref{SDE_phys_plan} and the renormalization group-like equation \\eqref{equa_L} with the variable \\(r\\).\nWe obtain, {while singling out the lowest order term coming from \\(F_1\\),} \n\\begin{equation} \\label{SDE_utile}\n \\beta =\\frac 1 r + \\frac 1 r \\sum_{k=1}^{+\\infty}L_k + (\\text{contributions from }F_k)\n\\end{equation}\nand \n\\begin{equation} \\label{eqLk}\nk {L}_k = \\frac23 {\\beta} {L}_k - r \\beta \\partial_r {L}_k\n \t+ \\sum_{i=1}^k q_{k,i}{\\gamma}_i^2.\n\\end{equation}\nWe are interested in the freedom in the solutions of this system of equations: the perturbative solution is uniquely defined, but since it is a system of differential equations, it must have a space of solutions. Using the fact that the two dominant terms of \\(\\beta\\) are \\(r^{-1} - 2\/3r^{-2}\\), the dominant orders of the linearized equation for \\(L_k\\) are:\n\\begin{equation}\n k L_k = - \\partial_r L_k + \\frac 2 3 r^{-1}(L_k + \\partial_r L_k) \n\\end{equation}\nThe dominant order of the solution is:\n\\begin{equation}\\label{dominant}\n \tL_k = m_k r^{\\frac 2 3 (1-k) } e^{-k r }\n\\end{equation}\nOne can check that the additional terms coming from substituting this value of \\(L_k\\) in the system of equations are smaller by at least \\(r^{-2}\\), so that they cannot change the exponent \\(\\frac23(1-k)\\) of this solution, but only multiply this solution by a power series in \\(r^{-1}\\).\nSince a possible deformation of the solution proportional to \\(e^{-kr}\\) signals the possibility of a nonzero \\(\\Delta_k\\), we recover the results of~\\cite{BeCl14} on the possible forms of the alien derivations of \\(\\beta\\), now written in the formal model instead of the convolutive one.\n\nThe computation of the \\(r^{-1}\\) corrections was carried out in~\\cite{BeCl13} in the case of \\(L_1\\) and involves summations over the effect of all the other \\(L_k\\) as well as over the \\(F_k\\). The language was different, but the resulting computations are totally equivalent to what would be the computation of the terms proportional to \\(m_1\\) in a full solution. \nSuch a computation nevertheless involves summations over \\(k\\) which give highly nontrivial combinations of multizeta values, some of which cancel and the others can be expressed as product of zeta values. In our following work~\\cite{BeCl14}, the introduction of the contour integral representation of the propagator of equation~(\\ref{param_G}) gave a simple interpretation of these results and a prospective way of carrying the computations up to larger orders.\nWe will not consider these corrections here, since the simple study of the dominant terms gives already quite interesting results.\n\nThe nonlinear nature of the system of equations~(\\ref{SDE_utile},\\ref{eqLk}) means that the general solution will have terms with any product of the \\(m_k\\) as coefficient. The behavior of exponentials under differentiation and multiplication ensures that such a term will also have a factor \\(e^{-nr}\\) with \\(n\\) the sum of the indices of the \\(m_k\\) in the coefficient.\nWe therefore have that \\(e^{-r}\\) only appears with the coefficient \\(m_1\\), but \\(e^{-2r}\\) can have the coefficients \\(m_2\\) or \\(m_1^2\\), \\(e^{-3r}\\), the coefficients \\(m_3\\), \\(m_2 m_1\\) or \\(m_1^3\\),\\dots\\ \nThe question then is to know which is the larger possible term for a given coefficient \\(\\prod m_k\\) in the evaluation of \\(\\beta\\). It turns out that the larger possible terms come from the product \\(r \\beta \\partial_r L_k\\) if \\(L_k\\) was the source of the largest term in \\(\\beta\\) with the same coefficient. \nThe end result is that the largest power of \\(r\\) coming in \\(L_k\\) for some product of \\(m_j\\) including \\(m_k\\) is \\(\\frac2 3 \\sum (1-j)\\), giving a term with an exponent less for \\(\\beta\\). The nice point is that the dominant term among the ones with the factor \\(e^{-nr}\\) is the one proportional to \\(m_1^n\\), which has no additional powers of \\(r\\) for \\(L_1\\) and just the factor \\(r^{-1}\\) for \\(\\beta\\).\n\nWe therefore can parameterize the sum of the dominant terms in \\(L_1\\) with\n\\begin{equation}\\label{l1trans}\n\tL_1 = \\sum_{n=1}^\\infty c_n m_1^n e^{-nr}\n\\end{equation}\nUsing that at this order, \\(\\beta\\) is \\(r^{-1} + r^{-1} L_1\\), the equation~(\\ref{eqLk}) for \\(k=1\\) gives the following recurrence relation for the \\(c_n\\):\n\\begin{equation}\\label{relation_c}\n\t(1-n) c_n = \\sum_{p=1}^{n-1} c_{n-p}\\, p\\, c_p\n\\end{equation}\nIf we define a formal series \\(S\\) by\n\\begin{equation} \\label{series_S}\n S(x) := \\sum_{n\\geq1}c_nx^n.\n\\end{equation}\nwe obtain that a first transseries solution for \\(L_1\\) is given by\n\\begin{equation}\n\tL_1 = S(m_1 e^{-r})\n\\end{equation}\n\n\n\\subsection{Summation of the transseries}\n\nThe previous subsection introduced the formal series \\(S\\), and we need to know its properties, in particular its radius of convergence.\nThe inductive formula \\eqref{relation_c} for the $c_n$'s implies a differential equation for $S(x)$ (seen as a formal series):\n\\begin{equation*}\n\t\\frac {S(x)} {x} - S'(x) = S(x)S'(x).\n\\end{equation*}\nDividing by \\(S(x)\\) and regrouping terms depending on \\(S\\), one obtains:\n\\begin{equation*}\n\t\\frac{S'(x)}{S(x)} + S'(x) = \\frac{1}{x}.\n\\end{equation*}\nThe left hand side is the logarithmic derivative of the function $F(x):=S(x)e^{S(x)}$ so that we have\n\\begin{equation*}\n\tS(x) e^{S(x)} = k x,\n\\end{equation*}\nfor some $k\\in\\mathbb{R}$. From the definition of \\(m_1\\) in~\\eqref{dominant} and the comparison with the formula~\\eqref{l1trans}, we see that \\(c_1=1\\), which also fixes \\(k=1\\). The presence of a minimum of the function \\(u \\rightarrow u e^u\\) for \\(u=-1\\) with the value \\(-1\/e\\) gives rise to \na singularity of \\(S(x)\\) for \\(x = -\\frac 1 {e}\\) of the square root type. Since it is the singularity nearest to the origin, it implies that the convergence radius of the series is \\(\\frac1 {e}\\).\n\nIn fact, the above function inversion problem has been studied and the solution of the case \\(k=1\\) is known as (the principal branch of) Lambert's \\(W\\)-function.\nUsing the initial condition $S'(0)=1$ and the fact that $W'(0)=1$ we find that the series $S(x)$ of \\eqref{series_S} is actually\n\\begin{equation}\n S(x)=W(x).\n\\end{equation}\nAn explicit series representation of the Lambert \\(W\\)-function is known :\n\\begin{equation*}\n W(x) = \\sum_{n\\geq1}\\frac{(-n)^{n-1}}{n!}x^n.\n\\end{equation*}\nThis formula can be deduced from the Lagrange inversion formula. The convergence radius $1\/e$ is then a simple consequence of the Stirling formula for the factorial.\n\nAll in all we have shown that the {sum of the lowest order terms in all nonperturbative sectors of the anomalous dimension} is\n\\begin{equation} \\label{result_anormal_dimension}\n \\gamma^\\mathrm{res}(r) = r^{-1} W( m_1 e^{-r}) + \\mathcal{O}(r^{-2})\n\\end{equation}\nand is defined in the region $|m_1 e^{-r}|<1\/e$ of the complex plane. \n\n{The construction presented here is a particular example of ``transasymptotic analysis'', which suggests that similar formulae exist at any order in $e^{-r}$. See for example \\cite{Costin2001} and references therein.} {An interesting aspect of the analysis in~\\cite{Costin2001} is that they show that the singularity of this lowest order resummed solution signals a singularity of the full solution in its vicinity.} \n\n\n\\subsection{Links with the alien calculus}\n\nIn the preceding sections, we studied a possible transseries deformation of the perturbative solution for the \\(\\beta\\)-function, but to what use can it be put for the evaluation of the function? In particular, could it be possible to have a determination of \\(m_1\\)? \nPart of the response comes from the idea of the bridge equations. Since the Stokes automorphism and its powers respect products and commutes with the derivation with respect to \\(r\\), the functions remain solution of the equations when transformed by these automorphisms. Since in the formal model, the alien derivation \\(\\Delta_n\\) gives rise to a factor \\(e^{-nr}\\), the solutions after the action of a Stokes automorphism will be in the form of a transseries.\n\nSince the most general transseries solution is a function of the parameters \\(m_k\\) appearing in the linear deformations of the solution~\\eqref{dominant}, \nalien derivatives can be expressed through derivations acting on these parameters, giving bridges between alien calculus and ordinary calculus. The alien derivation \\(\\Delta_n\\) is, a priori, any combination of operations which lower the weights by \\(n\\), so that for example, \\(\\Delta_1\\) has not only a term proportional to \\(\\partial\/\\partial m_1\\), but also \\(m_1 \\partial\/\\partial m_2\\) and many others. Nevertheless, the same reasons which made the terms proportional to \\(m_1^n\\) dominate imply that the coefficient \\(f\\) of \\(\\partial\/\\partial m_1\\) is the most important part of \\(\\Delta_1\\). A value for this coefficient \\(f\\) could be extracted in~\\cite{BeCl13} from the comparison of the asymptotic behavior of the perturbative series for \\(\\gamma\\) deduced from the singularities of the Borel transform and the known coefficients of this series. \n\nThe dominant term in the singularity at the point~\\(n\\) necessarily comes from the coefficient of \\(m_1^n\\) and can only be extracted by the \\(f^n (\\partial\/\\partial m_1)^n\\) term in \\(\\Delta_1^n\\). The \\(n!\\) coming from the iterated differentiations is compensated by the \\(1\/n!\\) factor in front of \\(\\Delta_1^n\\) in the definition of \\(\\Delta^+_n\\), so that we obtain, from the relation between alien derivatives and singularities of the Borel transform, that \\(\\hat L_1\\) has a pole in \\(n\\) with residue \\(c_n f^n\\), while the only divergent part for \\(\\hat \\beta\\) is proportional to \\(- c_n f^n \\log(|\\xi - n|) \\).\n\nThese singularities of the Borel transform are transmitted to the median average. In turn, these singularities of the integrand produce nonperturbative contributions to the result of resummation, so that the factors like \\(f\\), determined through alien calculus, can be used to fix the unknown coefficients in the transseries expansion. What is important is that the consistency of all the steps of the resummation procedure with the products and derivation ensures that the result respects the original equations and must therefore be of a form compatible with the transseries solution. The influence of the singularity at 1 will therefore be sufficient to obtain the dominant part for the singularities for all \\(n\\). \n \nAlthough we have been able to compute nonperturbative terms, with coefficients which could be computed from the perturbative expansion of the anomalous dimension, the situation seems quite complicated. \nIndeed the resurgent analysis seems to make the situation go from bad to worst: instead of a unique formal series, we end up with formal series multiplying \\(e^{-rn}\\) for each \\(n\\), and with furthermore coefficients which are polynomials in \\(r^{-2\/3}\\) and \\(\\log r\\) of degrees growing with \\(n\\), with many undetermined coefficients. \nMoreover, each of these series are actually divergent and need some form of resummation.\nHowever, we may remember that in many cases, divergent series are not so bad news, and as Poincar\\'e has put it, they are ``convergent in the sense of astronomers'': the first few terms give a fairly accurate approximation of the final result, as is the case for example in quantum electrodynamics.\nOur position will therefore be to use the information we have and forget for the time being about all the unknown quantities. We would of course prefer to have arguments proving that indeed what we neglect is negligible, but it is the best we can do at the moment.\n \n{We reshuffle the transseries and write them as series in $r$ whose terms are series in $e^{-r}$. This operation is inspired by a remark of Stingl \\cite{St02}, page 70 about\nphysical} considerations on what the ``true'' observables {are,} were put forward to provide a justification to this manipulation.\nThe take home message could be that it is important to keep all the terms of a convergent series but series with 0 convergence radius could be truncated without remorse. A physicist way of dealing with such a situation would be to look at how the results change when we add terms from the formal series, but we would need at least one more term.\n\nA more mathematical view could come from transmonomials~\\cite{Sa07}, which are special functions with simple properties under the action of alien derivatives.\nThis could lead to an expansion of the function where transmonomials get multiplied by ``alien constants'', functions on which all alien derivatives give zero and therefore easily computable from their power series expansion. The simplest transmonomial \\(\\mathcal U^1\\), with the only non zero alien derivative \\(\\Delta_1 {\\mathcal U}^1 =1 \\) would replace \\(e^{-r}\\) in all our transseries expressions and take care of the dominant terms at large orders neglected in the naive approach.\n\nTo conclude this short exposition of the idea of resumming the transseries, let us emphasize that in other contexts, methods using grouping of terms of nonsummable families have been put to good use to produce convergent expressions. \nArbitrary groupings can produce arbitrary results, but some well defined procedures have been shown to reproduce the results of a Borel summation.\nA prime example is the arborification procedure presented in Ecalle's work on mould calculus~\\cite{Ecalle1992}, which separates terms in smaller parts to be regrouped in other objects. {Our procedure might be seen as an \narborification where only ladder trees give nonzero contributions. The reader can be referred to \\cite{FaMe12}, Section 6} for a clear introduction of the arborification--coarborification \nin the context of linearization problems. Other cases appear in the study of Dulac's problem~\\cite{Ec92}. The main advantage of such procedures is that they allow practical computation. It is thus possible that the somewhat ad hoc computation presented above can also be justified.\n\n\\section{Properties of the Green function}\n\n\\subsection{Nonperturbative mass scale generation}\n\nThe mass of a particle is given by the position of a pole of the two-point function as a function of the invariant \\(p^2\\) of the momentum. In our case, there is always a pole for \\(p^2=0\\), reflecting the fact that we started from a massless theory, since the function \\(G\\) we are studying is a multiplication factor for the free propagator, the same for all states of the supermultiplet. We define a mass scale as the value of the external momentum $p$ for which the Green function has a singularity. The expansion of \\(G\\) in powers of the logarithm \\(L= \\log(p^2\/\\mu^2)\\) is ill suited for such an analysis. We expect the sign of \\(p^2\\) to fundamentally change the situation, the pole for a physical particle being for a timelike \\(p\\), while \\(L\\) only change by \\(i\\pi\\) when going from timelike to spacelike momenta.\n\nThis is why we will rather use the integral representation \\eqref{param_G}:\n\\begin{equation*}\n \\hat{G}(\\xi,L)=\\oint_{\\mathcal{C}_{\\xi}}\\frac{f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta.\n\\end{equation*}\nIt was shown in \\cite{BeCl14} that the function $\\zeta\\mapsto f(\\xi,\\zeta)$ has singularities at $\\zeta=\\xi$ and at \\(\\zeta=0\\). Therefore we can expand the contour \n$\\mathcal{C}_{\\xi}$ to infinity without changing the value of the integral. This being done, we can make the lateral alien derivative go \nthrough the integral and obtain:\n\\begin{equation*}\n \\Delta_n^+\\hat{G}(\\xi,L)=\\oint_{\\mathcal{C}}\\frac{\\Delta_n^+f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta,\n\\end{equation*}\nwhere the lateral alien derivative acts on the $\\xi$ variable. Taking the lateral alien derivative of the renormalisation group equation \n\\eqref{renorm_f} we get\n\\begin{equation} \\label{renorm_f_lat_alien_der}\n (\\zeta-(\\xi+n))\\Delta_n^+f(\\xi,\\zeta) = \\frac 1 3 \\Delta_n^+\\hat\\beta(\\xi) + \\frac 1 3 \\sum_{i=0}^n \\bigl(\\Delta_{n-i}^+\\hat{\\beta}\\star\\Delta_i^+ f \\bigr)(\\xi,\\zeta) + \\sum_{i=0}^n \\bigl(\\Delta_{n-i}^+\\hat{\\beta}'\\star \\text{Id}. \\Delta_i^+f\\bigr)(\\xi,\\zeta).\n\\end{equation}\nWe are looking for the dominant term in $\\Delta_n^+f(\\xi,\\zeta)$ seen as a function of $\\xi$. It will come from the dominant term in \\(\\Delta_n^+ \\hat\\beta\\), which is constant in \\(\\xi\\), while all other terms give higher powers of \\(\\xi\\). It will therefore be given by a term \\(f_{n,0}(\\zeta)\\), which satisfies\n\\begin{equation*}\n (\\zeta-n)f_{n,0}(\\zeta) = c_n f^n.\n\\end{equation*}\nPlugging this into the integral representation of $\\Delta_n^+\\hat{G}(\\xi,L)$ we get\n\\begin{equation*}\n \\Delta_n^+\\hat{G}(\\xi,L) = \\frac{c_n f^n}{n}\\left(e^{nL}-1\\right)+\\mathcal{O}(\\xi^{2\/3}).\n\\end{equation*}\nThe transseries expansion of \\(G\\) deduced from these singularities of \\(\\hat G\\) is then \n\\begin{equation*} \n G^\\mathrm{res}(r,L)= 1+\\frac 1 r \\sum_{n=1}^{+\\infty}\\frac {c_n} {n} (f e^{-r})^n \\Bigl(e^{nL} -1 \\Bigr) + \\text{higher orders}\n\\end{equation*}\nNow, as we have done \nfor the anomalous dimension in the previous section we can simply sum the above series, without worrying on the other terms. \nIf we neglect 1 with respect to \\(e^{nL} = (p^2\/\\mu^2)^n\\), we obtain a function of \\(f e^{L-r}\\) with Taylor coefficients \\(c_n\/n\\).\nSince the \\(c_n\\) are the Taylor coefficients of Lambert's \\(W\\)-function, these are the coefficients of the primitive of the \\(W\\)-function divided by \\(x\\).\nThis is a function which grows only logarithmically for positive arguments, but goes to 0 has a three half power of the variable at \\(-1\/e\\).\n$f$ was numerically computed in \\cite{BeCl13} to be $0.208143(4)$. This shows that for an euclidean momentum where \\(e^L\\) is positive, we are in a situation where the function grows really slowly and the large terms of the series, proportional to \\( (p^2)^n \\), combine to a simple logarithmic correction to the propagator. \nSince propagators can be Wick rotated to the euclidean domain in loop computations, this means that nothing prevents us, at this approximation level, from defining consistently the renormalized theory. \nOn the other hand, we have for a finite value of \\(p^2\\) in the timelike domain a singularity of the propagator which defines a mass scale\n\\begin{equation}\n M_{NP}(r)^2 = \\frac{\\mu^2}{f}e^{r-1}.\n\\end{equation}\nLet us notice that we find that the nonperturbative mass goes to infinity as $r$ goes to infinity, which corresponds to $a$ going to $0^+$. \nThis was to be expected. However this mass scale is not renormalization group invariant, since a renormalization group invariant mass scale should involve a factor \\(r^{2\/3}\\). We hope that a more careful analysis can give back this factor so that we obtain a fully consistent analysis of this nonperturbative mass scale.\n\nLet us finally remark that the detour by the contour integral representation of \\(G\\), which is very valuable if we wanted to consider all the corrections proportional to powers of \\(L\\) in the expansion of \\(G\\), is not really necessary at this level of approximation. We could have simply deduced that a term proportional to \\((p^2)^n\\) appears when considering a \\(e^{-nr}\\) term in the \\(\\beta\\)-function. \n\n\\subsection{Analyticity domain: a necessary acceleration?}\n\nWe want to see how the resummed two-points function $G^{\\text{res}}$ could be obtained through Laplace transforms, to better understand its analyticity domain. \nWe make the bold approximation that the size of the Borel transform at a point can be approximated by the contribution of the nearest singularity. We have seen that the singularity in the point \\(n\\) is dominated by the contribution \\(1\/n! \\Delta_1^n \\hat G\\) in the lateral derivative, which has the factor \\(c_n (p^2)^n\\). \nThe coefficients \\(c_n\\) have also a power like behavior so that the domain in \\(r\\) where the Laplace transform of \\(\\hat G\\) is well defined shrinks when \\(p^2\\) grows.\nSince we would like that our theory defines the two-point function for any values of the momentum, we cannot define it by a simple Laplace transform. {Indeed, even if the reformulation of the Schwinger--Dyson equation we use does not make it explicit, the proper definition of the two-point function is necessary to compute the loop integral appearing in the definition of the \\(\\beta\\) function. It is therefore important to have at all stages computation which are uniform in \\(p\\).}\n\nThe fact that the dominant terms in the transseries representation sum up to an analytic function of \\(p^2\\), with a well behaved extension to any positive values of \\(p^2\\) is of no relevance here: the Laplace transform is well defined only if \\(p^2\\) is small enough that we are in the convergence domain of the sum of the dominant terms. \nWe need\n \\begin{equation*}\n |e^{-r}|\\leq e^{\\kappa}=|c|^{-1} e^{-L+1},\n\\end{equation*}\nin other terms that the real part of \\(r\\) should be larger than \\(-\\kappa\\), which grows like \\(L\\).\n\nIn order to be able to do the Laplace integral, we could think of doing a Borel transform with respect to \\(p^2\\), but we cannot see how it could be possible to use the Borel-transformed two-point function as an ingredient of the Schwinger--Dyson equations. A possible way out is rather through acceleration. This formally corresponds to making a Laplace transform followed by the Borel transform of the function expressed in terms of a new variable, which is a growing function of the old one. If the Laplace transform was well defined, one sees that there exists a kernel \\(K(\\xi_1,\\xi)\\) such that the new Borel transform \\(\\hat f_1\\) of a function \\(f\\) is given by:\n\\begin{equation}\n\t\\hat f_1(\\xi_1) = \\int_0^\\infty K(\\xi_1,\\xi) \\hat f(\\xi) d\\xi.\n\\end{equation}\nFor fixed \\(\\xi_1\\), the kernel \\(K(\\xi_1,\\xi)\\) vanishes faster than any exponential when \\(\\xi\\) goes to infinity, allowing this acceleration transform to remain defined in cases where the Laplace transform in the original variable was not possible.\nIn fact, since the Borel transform remains of exponential growth, but only with a coefficient which can be arbitrarily large, the acceleration transform can be defined whenever its kernel has a slightly faster decay than the exponential. This is the case already for the kernel associated with the change of variable \\(r \\to r_1 = e^r\\), for which the kernel behaves like \\(\\exp(-\\xi \\log\\xi)\\) for large \\(\\xi\\) and fixed \\(\\xi_1\\).\n\nThe terms we kept of the transseries expression of the two-point function translate in the following value for its accelerated Borel transform in terms of the Borel transform \\(\\hat W\\) of the Lambert function:\n\\begin{equation}\n\t \\hat G_1(\\xi_1, p^2) \\simeq \\hat W ( \\xi_1 p^2 )\n\\end{equation}\nSince the Lambert function is holomorphic in a neighborhood of the origin, its Borel transform is an entire function, so that \\(\\hat G_1\\) is well defined {at this approximation level}. However, if the final Laplace transform is to be valid for any value of \\(p^2\\), it must be done in directions such that \\(\\hat W\\) is smaller than any exponential, and this in turn restrict the angular width of the domain in \\(r_1=e^r\\) where the field theory can be defined by resummation.\nIn turn, this implies that the imaginary part of \\(r\\) is bounded: the limits of the analyticity domain are lines of fixed imaginary part, which correspond to circle arcs tangent to the real axis in the original coupling \\(a\\).\nThe ensuing analyticity domain is quite similar to the one proposed by 't Hooft~\\cite{Ho79} for any sensible quantum field theory.\n\nIt remains to know whether for some functional of the two-point function singularities of the Borel transform in the variable \\(\\xi_1\\) could appear, ensuring that there are non trivial alien derivatives in this second Borel plane {and thus proving the unavoidable character of acceleration. We must remember that the approximation of the system of equations obeyed by the renormalization group function used in this work is a rather crude one. Conceptually, we do not have a clearly defined differential system, but a functional equation involving the two-point function with its dependence on the momentum. For the perturbative solution, we could transform it to an infinite system of differential equations by considering the function to be given by its Taylor series at the origin in the variable \\(L\\), but this approximation is not suitable for the computation of the properties of the analytic continuation of the Borel transform. We had to introduce terms associated with the poles of the Mellin transform to obtain deformation parameters naturally associated with each of the terms \\(e^{nr}\\), with \\(n\\) any nonzero integer. But our system of differential equations has other components, since we also need the coefficients \\(\\gamma_k\\). This suggests that the terms we discussed in~\\cite{BeCl14} do not exhaust the possible transseries deformations of the solution, but the change in the representation of the two-point function which would allow to pinpoint these additional possible terms goes far beyond the ambition of this work.}\n\n{In fact, this acceleration procedure should not be viewed with too much fear. It can be seen as a tool to transform the difficult problem of bounding the analytic continuation of the Borel transform into the much simpler problem of finding some kind of formal solutions which allow to characterize the possible alien derivatives in a second Borel plane. In any case, what happens in the different Borel planes are somehow unrelated, so that the process of analytic continuation, analysis of the singularities and eventually averaging of the Borel transform is totally independent of the fact that it will be followed by a Laplace transform or an acceleration transform.}\n\n\\section*{Conclusion}\n\nUsing to a bigger extent the power of alien calculus and transseries expansions we have been able to go much further than in our previous work~\\cite{BeCl14}. The dominant terms of the singularities of the Borel-transformed anomalous dimension of the theory has been computed.\nThis computation was carried out by using transseries expansions and considering the effect \nof lateral alien derivatives on the Schwinger--Dyson and the renormalization group equation of the theory. Then, using a suitable form \nof the median resummation, this gave the first order in every `instantonic' sectors of the anomalous dimension of our theory.\n\nFollowing a procedure suggested by Stingl, we have kept in this transseries solution the terms of a formal series in \n$e^{-r}$, forgetting the terms in (negative) powers of \\(r\\). This series of the dominant terms turned out to be convergent and its sum evaluated.\n\nThe same analysis could also be transferred to the two-point function of the theory. Indeed, the two-point function is \nessentially determined by the anomalous dimension through the renormalization group. \nThe important point is that the singularity of the anomalous dimension in the point \\(n\\) of the Borel plane, or a term \\(e^{-nr}\\) in the transseries expansion, gives rise to a term proportional to \\((p^2)^n\\) for the two-point function. What looked like negligible contributions to the anomalous dimension becomes dominant in the two-point function for large \\(p^2\\). \nWhile an explicit value of the factor multiplying \\(p^2\\) could not be \nobtained, the functional dependence can be obtained. In the euclidean domain, which corresponds to \\(p^2\\) positive with our conventions, the asymptotic behavior of the Lambert function means that {this series of powers of \n\\(p^2\\) has a finite radius of convergence.} These powers of \\(p^2\\) sum up to something of merely logarithmic growth. \nThis final asymptotic behavior may however appear only for so large values of the momentum that it would be totally invisible in usual nonperturbative studies, where the ratio between the largest and the smallest scales that can be studied is rather limited.\nNevertheless the two-point function has, at this approximation level, a quite regular behavior at the smallest scales, in contradistinction to the divergence for some finite scale obtained with finite order approximations of the \\(\\beta\\)-function: many general arguments for the triviality of such a quantum field theory\nwith a positive \\(\\beta\\)-function break down. We must however remain cautious, since we are just scratching the surface of the kind of beyond perturbative theory analysis resummation theories allow, and many new phenomena may be revealed by a more careful study.\n\nThe simple analytic dependences on \\(p^2\\) of all the terms in the expansion of the two-point function make it easy to study its analytic continuation to the timelike domain, with negative \\(p^2\\). In this case, a singularity appears which is of square root type. This singularity defines a nonperturbative mass scale for the theory, but our computation is not fully satisfying since this mass scale is not fully renormalization group invariant. \n\nThe fact that even our fairly simple computation revealed a nonperturbative mass scale for the theory is quite remarkable. \nAlso we did not have to choose the functional dependence of the two-point function, but it was provided by our computation. In our case, singularities of the Borel transform were associated with ultraviolet divergent contribution to the two-point function: \nin an asymptotically free theory like QCD, we would obtain infrared divergent contributions, but likewise it could be possible to sum this contributions to obtain well behaved propagators up to the lowest scales. Since we do not decide a priori their functional form, we may have signs of confinement in the form of states having singularities different than poles in the timelike domain. \n\nConsiderations about the way that the sum of the terms of the transseries could come from the Laplace integral give indications on the growth \nof the Borel transform, which is harder to check: changing the \nvariable on which the final Laplace transform is done corresponds to an integral transform (dubbed acceleration by Ecalle) with a kernel \nthat have faster than exponential decay at infinity. A suitable change of variables therefore allows to define a new germ of analytic function \nnear the origin, the singularities of which can be studied by a new set of alien derivations. If all such possible accelerations do not present \nsingularities, the first Borel transform should be suitable to directly define the sum, giving an indirect check on the growth of the Borel transform. \n\nLimits on the analyticity domain for complex values of the \ncoupling constant devised by 't Hooft~\\cite{Ho79} and Stingl~\\cite{St02} suggests, contrarily to what Stingl said in some papers, that at least \none such acceleration should be needed in the case of nonabelian gauge theories. \nThis question is linked to the shape of the analyticity domain in the coupling constant of the theory, that we did not study. However, in the case of the two-point function, if \\(r\\) is given an imaginary part of \\(i\\pi\\), \\(e^{-r}\\) changes sign and the singularity which was for timelike momenta enters the euclidean domain. \nThis should pose serious problems to the continuation of the theory to such values of the coupling, so that \\(r\\) should be limited to a band of finite extension in the imaginary direction, which converts to the horn-shaped domains proposed by\n't Hooft when converting back to the coupling proportional to \\(r^{-1}\\). \n\nIt should certainly be interesting to compute the higher order corrections to the transseries solutions of the system of equations studied here and try to deduce their full system of alien derivations. \nAn effective computation however appears to be quite a formidable task, but in what are certainly simpler cases, mould calculus has been shown to provide for quite explicit results, expressing results in terms of resurgent monomials~\\cite{Sa07}.\nWith this strategy, one could avoid as much as possible to work explicitly in the Borel plane, even if the analytic continuation of functions in the Borel planes (plural if acceleration is needed) is the ultimate justification of the computations one may attempt.\n\nMoreover, our study could also be carried out when including additional terms involving higher-loops primitively divergent diagrams \nin the Schwinger--Dyson equation. This should allow to expand the results of \\cite{BeSc12}, which only considered the asymptotic behavior of the perturbative series, that is, the singularities closest to the origin of the Borel transform.\n\nIn this work, we limited ourselves to the two-point functions, which have a simple dependence on a unique Lorentz invariant: a general study of quantum field theory would certainly benefit from a careful investigation of the analytic properties of the Borel-transformed Green functions in all their variables.\n\nFinally, let us notice that the probable usefulness of mould calculus in the realm of quantum field theory expands the \nlist of elements of Ecalle's theory of resurgence which should be used in physics: we have used alien calculus, median resummation \nand it seems very likely that acceleration will be needed in the next steps of our program. Bridge equations are nowadays a common tool of some \nphysicists (see e.g., \\cite{AnScVo12}) and we argued that we might also use mould calculus while resurgent monomials will come into the game. Proper use of these tools could well provide solutions to old questions in quantum field theory.\n\\bibliographystyle{unsrturl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCompressive Sensing (CS) is a method in signal processing which aims to reconstruct signals from a relatively small number of measurements. \nIt has been shown that sparse signals can be reconstructed with a sampling rate far less than the Nyquist rate by exploiting the sparsity \\cite{Donoho06}.\n\nIn this paper, we focus on Binary Compressive Sensing (BCS) which restricts the signals of interest to binary $\\{ 0, 1 \\}$-valued signals, which are widely used in engineering applications, such as fault detection \\cite{Bickson11}, single-pixel image reconstruction~\\cite{Duarte08}, and digital communications~\\cite{Wu11}.\nRelated works are as follows: Nakarmi and Rahnavard \\cite{Nakarmi12} designed a sensing matrix tailored for binary signal reconstruction. Wang et al.~\\cite{Wang13} combined $\\ell_1$ norm with $\\ell_\\infty$ norm to reconstruct sparse binary signals. Nagahara \\cite{Nagahara15} exploited the sum of weighted $\\ell_1$ norms to effectively reconstruct signals whose entries are integer-valued and, in particular, binary signals and bitonal images. Keiper et al.~\\cite{Keiper17} analyzed the phase transition of binary Basis Pursuit.\n\nWe note that most of the previous work on BCS are based on convex optimization. Indeed, convex optimization based algorithms allow performance guarantee via rich mathematical tools. However, they are found to be notoriously slow in large-scale applications compared to greedy methods such as the Orthogonal Matching Pursuit (OMP) \\cite{Donoho08}. On the other hand, greedy methods like OMP are fast but often have a worse recovery rate than convex optimization methods. In this work, we propose a fast BCS algorithm with a high recovery rate. Taking the binariness of signals into account, our algorithm is a gradient descent method based on the smoothed $\\ell_0$ norm \\cite{Mohimani09}. Through numerical experiments, we show that the proposed algorithm compares favorably against previously proposed CS and BCS algorithms in terms of recovery rate and speed.\n\nThe rest of the paper is organized as follows. We give a short review on CS\/BCS algorithms in Section \\ref{sec:BCS} and present our algorithm in Section \\ref{sec:BSSL0}. In Section \\ref{sec:Experiments}, we present experimental results which compare the performance of the proposed algorithm with other algorithms. We conclude this paper with some remarks in Section \\ref{sec:conclusion}.\n\n\n\\subsection*{Notations:}\n\nFor a vector $\\mathbf{v} = (v_1, \\cdots, v_N)^\\top$ and $1 \\leq p \\leq \\infty$, the $\\ell_p$ norm of $\\mathbf{v}$ is denoted by $\\| \\mathbf{v} \\|_p$. The number of non-zero entries in $\\mathbf{v}$ is denoted by $\\|\\mathbf{v}\\|_0$. The probability of an event $E$ is denoted by $\\mathbb{P}(E)$. Let $[N]=\\{1,\\cdots,N\\}$ for $N \\in \\mathbb{N}$. We denote by $\\textbf{1}_N$ the $N$-dimensional vector with all entries equal to $1$.\n\n\n\\section{Binary Compressive Sensing (BCS)}\n\\label{sec:BCS}\n\nIn the standard CS scheme, one aims to recover a sparse signal from its linear measurements. The constraints posed by the measurements can be formulated as\n\\begin{equation}\n\\Phi \\, \\mathbf{z} = \\mathbf{y},\n\\quad \\mathbf{z} \\in \\mathbb{R}^N,\n \\label{eq:measurement}\n\\end{equation}\nwhere $\\Phi \\in \\mathbb{R}^{m \\times N}$, $m \\ll N$, is the measurement matrix \nand $\\mathbf{y} = \\Phi \\mathbf{x} \\in \\mathbb{R}^m$ is the measurement of a \\emph{sparse} signal $\\mathbf{x} \\in \\mathbb{R}^N$. CS algorithms exploit the fact that $\\mathbf{x}$ is sparse and seek a sparse solution $\\mathbf{z}$ satisfying (\\ref{eq:measurement}).\n\n\nThe BCS scheme considers binary signals for $\\mathbf{x}$. Note that a binary signal $\\mathbf{x}$ is sparse if and only if its complementary binary signal $\\widetilde{\\mathbf{x}} := \\textbf{1}_N - \\mathbf{x}$ is dense, i.e., is almost fully supported. As the measurement matrix $\\Phi$ is known, the equation (\\ref{eq:measurement}) converts equivalently to\n\\begin{equation}\n\\Phi \\widetilde{\\mathbf{z}} = \\widetilde{\\mathbf{y}},\n \\label{eq:measurement_dense}\n\\end{equation}\nwhere $\\widetilde{\\mathbf{z}} := \\textbf{1}_N - \\mathbf{z}$ and $\\widetilde{\\mathbf{y}} := \\Phi \\widetilde{\\mathbf{x}} = \\Phi \\textbf{1}_N - \\mathbf{y}$. This shows that reconstructing a sparse signal $\\mathbf{z}$ under the constraint (\\ref{eq:measurement}) is equivalent to reconstructing a dense signal $\\widetilde{\\mathbf{z}}$ under the constraint (\\ref{eq:measurement_dense}). For this reason, in contrast to the case of generic signals, binary signals that are dense can be recovered as well as those that are sparse.\n\n\nTwo types of models for binary signals have been considered in the literature (e.g., \\cite{Donoho10,Wang13,Nagahara15}):\n(i) $\\mathbf{x}$ is a deterministic vector which is binary and sparse, i.e., most of its entries are $0$ and only few are $1$; (ii) $\\mathbf{x}$ is a random vector whose entries are independent and identically distributed (i.i.d.) with probability distribution $\\mathbb{P}(x_j = 1) = p$ for some fixed $0 \\leq p \\leq 1$. If $p$ is small, a realization of $\\mathbf{x}$ is likely a sparse binary signal.\n\nIn this work, we shall consider the second model which can accommodate dense binary signals as well as sparse binary signals.\n\nBelow we give a short review of CS\/BCS methods that are related to our work.\n\n\\subsection{$\\ell_0$ minimization (L0)}\n\nA naive approach to finding sparse solutions is the $\\ell_0$ minimization,\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_0 & & \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\tag{$P_0$}\n\\label{eq:P_0}\n\\end{equation}\nThis method works generally for continuous-valued signals that are sparse, i.e., signals whose entries are mostly zero. However, solving the $\\ell_0$ minimization requires a combinatorial search and is therefore NP-hard \\cite{Natarajan95}.\n\n\n\\subsection{Smoothed $\\ell_0$ minimization (SL0)}\n\nSmoothed $\\ell_0$ minimization (SL0) \\cite{Mohimani09} replaces the $\\ell_0$ norm in (\\ref{eq:P_0}) with a non-convex relaxation:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & \\sum_{i = 1}^N \\left ( 1 - \\exp \\left (\\frac{- z_i^2}{2 \\sigma^2} \\right ) \\right ) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\label{eq:SL0}\n\\end{equation*}\nThis is motivated by the observation \n\\[ \\lim_{\\sigma \\to 0} \\exp \\left( \\frac{-t^2}{2 \\sigma^2} \\right) = \\begin{cases}\n 1 & \\text{if~} t = 0 \\\\\n 0 & \\text{if~} t \\neq 0,\n \\end{cases}\n\\]\nwhich implies that for any $\\mathbf{z} = (z_1, \\dots, z_N )^\\top \\in \\mathbb{R}^N$,\n\\begin{equation}\n\\lim_{\\sigma \\to 0} \\sum_{i = 1}^N \\left( 1 - \\exp \\left( \\frac{-z_i^2}{2 \\sigma^2} \\right) \\right) = \\| \\mathbf{z} \\|_0 .\n\\label{eq:SL0convergence}\n\\end{equation}\nNoticing that $\\mathbf{z} \\mapsto \\sum_{i = 1}^N \\big( 1 - \\exp \\big( \\frac{-z_i^2}{2 \\sigma^2} \\big) \\big)$ is a smooth function for any fixed $\\sigma > 0$, Mohimani et al.~\\cite{Mohimani09} proposed an algorithm based on the gradient descent method. The algorithm iteratively obtains an approximate solution by decreasing $\\sigma$.\n\nMohammadi et al.~\\cite{Mohammadi14} adapted the SL0 algorithm particularly to non-negative signals. Their algorithm, called the Constrained Smoothed $\\ell_0$ method (CSL0), incorporates the non-negativity constraints by introducing some weight functions into the cost function. Empirically, CSL0 shows better performance than SL0 in the reconstruction of non-negative signals.\n\n\n\\subsection{Basis Pursuit (BP)}\n\nA well-known and by now standard relaxation of (\\ref{eq:P_0}) is the $\\ell_1$-minimization, also known as the \\emph{Basis Pursuit} (BP) \\cite{Chen01}:\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\tag{$P_1$}\n\\label{eq:P_q}\n\\end{equation}\nSimilar to (\\ref{eq:P_0}), this method works generally for continuous-valued signals $\\mathbf{x} \\in \\mathbb{R}^N$ that are sparse.\n\n\n\\subsection{Boxed Basis Pursuit (Boxed BP)}\n\nDonoho et al.~\\cite{Donoho10} proposed the Boxed Basis Pursuit (Boxed BP) for the reconstruction of \\emph{k-simple bounded signals}:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in [0,1]^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\end{equation*}\nThe intuition behind Boxed BP is straightforward: the $\\ell_1$ norm minimization promotes sparsity of the solution while the restriction $\\mathbf{z} \\in [0,1]^N$ reduces the set of feasible solutions. Recently, Keiper et al.~\\cite{Keiper17} analyzed the performance of Boxed BP for reconstructing binary signals.\n\n\n\n\\subsection{Sum of Norms (SN)}\nWang et al. \\cite{Wang13} introduced the following optimization problem which combines the $\\ell_1$ and $\\ell_{\\infty}$ norms:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 + \\lambda \\, \\| \\mathbf{z} - \\tfrac{1}{2} \\, \\textbf{1}_N \\|_{\\infty} && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\end{equation*}\nMinimizing $\\|\\mathbf{z}\\|_1$ promotes sparsity of $\\mathbf{z}$ while minimizing $\\| \\mathbf{z} - \\frac{1}{2} \\, \\textbf{1}_N \\|_{\\infty}$ forces the entries $|z_i - \\frac{1}{2}|$ to be small and of equal magnitude (see Fig.~\\ref{fig:ell1-ellinfty}).\nThe two terms are balanced by a tuning parameter $\\lambda >0$.\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\centering\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\def8{8}\n\\filldraw [fill=blue!20, draw=blue!20] (4,0) -- (0,4) -- (-4,0) -- (0,-4) -- (4,0);\n\\draw[<->] (-9,0) -- (9,0) node[below] {$z_1$};\n\\draw[<->] (0,-9) -- (0,9) node[above] {$z_2$};\n\\draw[-, thick] (-0.5,-9) -- (8.5,9);\n\\draw (4,0) circle (2mm) [fill=black];\n\\node[below] at (5,-1) {\\scriptsize $(1,0)$};\n\\node[left] at (7,6) {\\scriptsize $\\Phi \\mathbf{z} = \\mathbf{y}$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\def8{8}\n\\filldraw [fill=blue!20, draw=blue!20] (4,0) -- (4,4) -- (0,4) -- (0,0) -- (4,0);\n\\draw[<->] (-9,0) -- (9,0) node[below] {$z_1$};\n\\draw[<->] (0,-9) -- (0,9) node[above] {$z_2$};\n\\draw[-, thick] (-0.5,-9) -- (8.5,9);\n\\draw (4,0) circle (2mm) [fill=black];\n\\node[below] at (5,-1) {\\scriptsize $(1,0)$};\n\\draw (2,2) circle (2mm) [fill=black];\n\\node[left] at (2.2,2) {\\scriptsize $(\\frac{1}{2},\\frac{1}{2})$};\n\\node[left] at (7,6) {\\scriptsize $\\Phi \\mathbf{z} = \\mathbf{y}$};\n\\end{tikzpicture}\n \\caption{Left: the minimization of $\\|\\mathbf{z}\\|_1$ finds sparse solutions. Right: the minimization of $\\| \\mathbf{z} - \\frac{1}{2} \\cdot \\textbf{1}_N \\|_{\\infty}$ forces the entries $|z_i - \\frac{1}{2}|$ to be small and of equal magnitude.}\n \\label{fig:ell1-ellinfty}\n\\end{figure}\n\n\n\\begin{figure}[t]\n \\centering\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\draw[<->] (-5,0) -- (9,0) node[below] {$t$};\n\\draw[<->] (0,-1) -- (0,9);\n\\draw[-, thick] (-5,6.5) -- (0,1.5);\n\\draw[-, thick] (0,1.5) -- (6,4.5);\n\\draw[-, thick] (6,4.5) -- (9,7.5);\n\\draw (6,4.5) circle (2mm) [fill=black];\n\\draw (0,1.5) circle (2mm) [fill=black];\n\\node[left] at (0,1.5) {\\scriptsize $(0,p)$};\n\\node[right] at (6,4.5) {\\scriptsize $(1,1-p)$};\n\\end{tikzpicture}\n\\caption{The function $f$ given in (\\ref{eq:ftn_f}).}\n \\label{fig:SAV}\n\\end{figure}\n\n\n\\subsection{Sum of Absolute Values (SAV)}\n\nNagahara \\cite{Nagahara15} proposed the following method for reconstruction of discrete signals whose entries are chosen independently from a set of finite alphabets $\\alpha = \\{\\alpha_1, \\alpha_2, \\dots, \\alpha_L\\}$ with a priori known probability distribution. In the special case $\\alpha = \\{0, 1\\}$ of binary signals, SAV is formulated as,\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & (1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y},\n\\end{aligned}\n\\end{equation*}\nwhere $p = \\mathbb{P}(x_j = 1)$, $j \\in [N]$, is the probability distribution of the entries of $\\mathbf{x}$.\nIf $p \\approx 0$, i.e., if $\\mathbf{x}$ is sparse, then $(1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1 \\approx \\| \\mathbf{z} \\|_1$ so that SAV performs similar to BP. We note that\n\\begin{align*}\n(1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1\n= \\sum_{i=1}^N f(z_i),\n\\end{align*}\nwhere\n\\begin{align}\n\\label{eq:ftn_f}\nf(t) :=\n\\begin{cases}\n - t + p & \\text{if~} t < 0, \\\\\n (1-2p) \\, t + p & \\text{if~} 0 \\leq t < 1, \\\\\n t - p & \\text{if~} t \\geq 1 .\n \\end{cases}\n\\end{align}\n\n\n\\section{Box-Constrained Sum of Smoothed $\\ell_0$}\n\\label{sec:BSSL0}\nL0 and SL0 utilize the $\\ell_0$ norm and its smoothed version respectively, however, they do not take into account that $\\mathbf{x}$ is binary.\nOn the other hand, Boxed BP, SN, and SAV utilize the $\\ell_1$ norm in one way or another and are specifically adjusted to the binary setting.\nA natural question arises: Can we achieve a better recovery rate for binary signals by adjusting L0 and SL0 to the binary setting?\n\nWe note that Boxed BP takes into account the binariness of $\\mathbf{x}$ by imposing the restriction $\\mathbf{x} \\in [0,1]^N$. It is straightforward to apply this trick to L0 and SL0, and we will call the resulting algorithms \\emph{Boxed L0} and \\emph{Boxed SL0} respectively.\nBoxed L0 is still NP-hard like L0, but Boxed SL0 shows a clear improvement over SL0 while requiring a similar amount of run time (Fig.~\\ref{fig:Lt}). However, the recovery rate of Boxed SL0 is significantly worse than Boxed BP or SN.\n\nIn this paper, we aim to adapt the SAV method and the restriction $\\mathbf{x} \\in [0,1]^N$ to SL0, in order to achieve a better performance.\nA straightforward adaptation leads to the following formulation. For $\\sigma > 0$ small,\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in [0,1]^N}{\\text{min}}\n & & F_{\\sigma}(\\mathbf{z}) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}, \\label{eq:straightforward_adapt}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n& F_{\\sigma}(\\mathbf{z}) \\triangleq (1-p) \\sum_{i = 1}^N \\left (1 - e^{-z_i^2\/(2 \\sigma^2)} \\right ) \\\\\n& \\quad \\quad \\quad + p \\sum_{i = 1}^N \\left ( 1 - e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) \\\\\n& = \\sum_{i = 1}^N \\left ( 1 - (1-p) \\, e^{-z_i^2\/(2 \\sigma^2)} - p \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) \\label{eq:SWl0}\n\\end{split}\n\\end{equation}\nand $p = \\mathbb{P}(x_j = 1),~\\forall j \\in [N]$.\nNote that by (\\ref{eq:SL0convergence}), we have\n\\[ \\lim_{\\sigma \\to 0} F_{\\sigma}(\\mathbf{z}) =(1 - p) \\, \\| \\mathbf{z} \\|_0 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_0 \\] so that $F_0 (\\mathbf{z})$ can be approximated by $F_{\\sigma}(\\mathbf{z})$ with small $\\sigma > 0$.\n\n\nNext, we will use a weight function to incorporate the restriction $\\mathbf{z} \\in [0,1]^N$ into the function $F_{\\sigma}(\\mathbf{z})$. For integers $k \\geq 1$, let\n\\begin{align*}\n&\\begin{split}\nw_{k}(t) & \\triangleq \\begin{cases}\n 1 & \\text{if~} 0 \\leq t \\leq 1 \\\\\n k & \\text{otherwise}.\n\t\\end{cases}\n \t\\end{split} \\\\\n \\end{align*}\n\n \n \nFor $\\sigma > 0$ and integers $k \\geq 1$, we define\n\\begin{align*}\n& F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) \\\\\n&\\triangleq \\sum_{i = 1}^N w_{k}(z_i) \\left ( 1 - (1-p) \\, e^{-z_i^2\/(2 \\sigma^2)} - p \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) . \\nonumber\n\\end{align*}\nNote that since $1 - (1-p) \\, e^{-t^2\/(2 \\sigma^2)} - p \\, e^{-(t-1)^2\/(2 \\sigma^2)} > 0$ for all $t \\in \\mathbb{R}$, minimizing $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ forces $w_k(z_i)$ to be small so that all $z_i$'s lie within $[0,1]$.\nIn this way, the restriction $\\mathbf{z} \\in [0,1]^N$ is incorporated into the cost function.\nOur optimization problem now reads as follows:\nFor $\\sigma > 0$ small and $k \\in \\mathbb{N}$ large,\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y} .\n\\end{aligned}\n\\end{equation*}\nTo solve this problem, we propose an algorithm which is based on the gradient descent method and is implemented similarly as algorithms in \\cite{Mohimani09,Mohammadi14}.\nA major difference in our algorithm is that the cost function $\\sum_{i = 1}^N \\big( 1 - \\exp \\big(\\frac{- z_i^2}{2 \\sigma^2} \\big) \\big)$ of SL0 is replaced with $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ which is designed specifically for binary signals by adapting the formulation of SAV \\cite{Nagahara15}.\n\n\n\n\t\\begin{algorithm}\n\t\t\\caption{Box-Constrained Sum of Smoothed $\\ell_0$ (BSSL0)}\n\t\t\t\t\\begin{algorithmic}[1]\n\t\t\t\\State \\textbf{Data:} Measurement matrix $\\Phi \\in \\mathbb{R}^{m \\times N}$, observation $\\mathbf{y} \\in \\mathbb{R}^m$, probability distribution prior $p = \\mathbb{P}(x_j = 1)$.\n\t\t\t\\State \\textbf{Parameters:} \n \n Iters and $L$ are the number of iterations in the outer and inner loops respectively, $\\mu$ is a step-size parameter for gradient descent, and $d$ is a decreasing factor for $\\sigma$.\n \n\t\t\t\\State \\textbf{Initialization:} $\\hat{\\mathbf{x}}=\\Phi^\\top(\\Phi \\Phi^\\top)^{-1} \\mathbf{y}$, $\\sigma = 2 \\max |\\hat{\\mathbf{x}}|$, \\\\ \\quad \n \n \n $k = 1 + N p\/\\text{Iters}$;\n\t\t\t\\For{$1 : \\text{Iters}$}\n\t\t\t\\For{$1 : L$}\n\t\t\t\\State $\\hat{\\mathbf{x}} \\leftarrow \\hat{\\mathbf{x}} - \\sigma^2 \\mu \\nabla F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$;\n\\quad \\emph{\\% gradient descent}\n\t\t\t\\State $\\hat{\\mathbf{x}} \\leftarrow \\hat{\\mathbf{x}} - \\Phi^\\top(\\Phi \\Phi^\\top)^{-1}(\\Phi \\hat{\\mathbf{x}} - \\mathbf{y})$; \\quad \\emph{\\% projection}\n\t\t\t\\EndFor\n\t\t\t\\State $\\sigma = \\sigma \\times d$;\n\t\t\n \\State$k = k + N p\/\\text{Iters}$;\n\t\t\t\\EndFor\n\t\t\t \\State $\\hat{\\mathbf{x}} \\leftarrow \\textbf{round}(\\hat{\\mathbf{x}})$; \\quad \\emph{\\% round to a binary vector}\n\t\t\\end{algorithmic}\n\t\t\\label{algorighm:BSSL0}\n\t\\end{algorithm}\t\t\n\nThe proposed algorithm is comprised of two nested loops. In the outer loop, we slowly decrease $\\sigma$ and iteratively search for an optimal solution from a coarse to a fine scale by decreasing $\\sigma$ by a factor of $0 < d < 1$. As $\\sigma$ decreases, we also gradually increase $k$ so that a larger penalty is put on solutions that have entries outside the range $[0,1]$.\nThe inner loop performs a gradient descent of $L$ iterations for the function $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$, where $\\sigma$ and $k$ are given from the outer loop. In each iteration of the gradient descent, the solution is projected into the set of feasible solutions $\\{ \\mathbf{z} : \\Phi \\mathbf{z} = \\mathbf{y} \\}$.\n\t\n\nNumerical experiments in Section \\ref{sec:Experiments} show that for binary signals the proposed algorithm outperforms all other algorithms (BP, Boxed BP, SN, SAV, SL0, and Boxed SL0).\n\n\nAs already mentioned, our algorithm is implemented similarly as SL0 \\cite{Mohimani09,Mohammadi14}. The parameters used in our algorithm are exactly the same as in \\cite{Mohimani09} except $k$ and $p$. As justified in \\cite[Section IV-B]{Mohimani09}, we set the initial estimate of $\\mathbf{x}$ as the minimum $\\ell_2$ norm solution of $\\Phi \\mathbf{z} = \\mathbf{y}$, i.e., $\\hat{\\mathbf{x}}=\\Phi^\\top(\\Phi \\Phi^\\top)^{-1} \\mathbf{y}$.\nThe initialization value for $\\sigma$ is discussed in \\cite[Remark 5 in Section III]{Mohimani09}. Also, the choice of the step-size $\\sigma^2 \\mu$ for gradient descent is justified in \\cite[Remark 2 in Section III]{Mohimani09} and the choice of $k$ in \\cite[Lemma 1]{Mohammadi14}.\n\t\n\nThe gradient of $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ used in Algorithm \\ref{algorighm:BSSL0} is given by\n\\begin{align*}\n\\begin{split}\n \\nabla F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) &= \\left( \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_1}, \\dots, \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_N} \\right)^\\top ,\n \\end{split}\n\\end{align*}\nwhere\n\\begin{align*}\n \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_i} &= \\frac{w_{k}(z_i)}{\\sigma^2} \\left ((1-p) \\, z_i \\, e^{-z_i^2\/(2 \\sigma^2) } \\right. \\\\\n & \\qquad \\left. + \\; p \\, (z_i -1) \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right )\n\\quad a.e.\n\\end{align*}\nThis is derived using the fact that $w_{k}'(t) = 0$ for all $t$ except $t = 0,1$; we have set $w_{k}'(0) = w_{k}'(1) = 0$ in the implementation.\nLet us point out that the discontinuity of $w_{k}(t)$ at $t = 0,1$ does not deteriorate the performance of gradient descent.\nOne can replace the function $w_{k}(t)$ with a smooth function, however, at the cost of increased run time.\n\n\n\\section{Numerical Experiments} \\label{sec:Experiments}\n\nIn this section, we compare the performance of our algorithm BSSL0 with other CS\/BCS algorithms described in Section \\ref{sec:BCS}.\nThe MATLAB codes for the experiments are available in \\cite{myCode}.\n\n\n\n\\subsection{Experiment 1: Binary Sparse Signal Reconstruction}\n\nIn this experiment, we tested BSSL0 with randomly generated binary signals and compared it with other CS\/BCS algorithms. \nRandom Gaussian matrices are considered for the measurement matrix $\\Phi \\in \\mathbb{R}^{40 \\times 100}$, that is, all entries of $\\Phi$ are drawn independently from the standard normal distribution.\nThe parameter $p$ is varied from $0$ to $1$ by step-size $0.05$, and a binary signal $\\mathbf{x} \\in \\{ 0 , 1 \\}^{100}$ is generated by drawing its entries independently with $\\mathbb{P}(x_i = 1) = p$ and $\\mathbb{P}(x_i = 0) = 1 -p$. For $\\Phi$ and $\\mathbf{x}$, we compute the measurement vector $\\mathbf{y} = \\Phi \\mathbf{x}$ and run the respective algorithms introduced in section II (BP, Boxed BP, SN, SAV, SL0, Boxed SL0, and BSSL0) to obtain a solution vector $\\mathbf{z}$ as a approximated reconstruction of $\\mathbf{x}$. Additionally, we consider the Orthogonal Matching Pursuit (OMP) \\cite{Tropp2007} which is a fast greedy algorithm for sparse signal reconstruction.\nThe following are considered for the performance evaluation: (i) \\textbf{Failure of Perfect Reconstruction (FPR)}: $0$ if $\\mathbf{z} = \\mathbf{x}$ (successfully recovered the signal perfectly) and $1$ if $\\mathbf{z} \\neq \\mathbf{x}$ (failed to recover perfectly); (ii) \\textbf{Noise Signal Ratio (NSR)}: NSR = $\\frac{\\| \\mathbf{x} - \\mathbf{z} \\|_2}{\\|\\mathbf{x}\\|_2}$; (iii) \\textbf{Run time}.\nFor each $p$, experiments are repeated $10,000$ times and the results are averaged. For SN, we set the parameter $\\lambda$ to be $100$ as fine-tuned in \\cite{Wang13}. For BSSL0, we set $\\sigma_{\\text{min}} = 0.1$, $d = 0.5$, $\\mu = 2$, and $L = 1000$.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=1\\linewidth]{result.eps}\n\\caption{Results for Experiment 1.}\n\\label{fig:Lt}\n\\end{figure}\nIn Fig.~\\ref{fig:Lt}, BSSL0 shows a better recovery rate than other CS\/BCS algorithms and also shows a run time comparable to SL0.\n\n\n\\subsection{Experiment 2: Bitonal Image Reconstruction}\n\nAs in \\cite{Nagahara15}, we considered reconstruction of the $37\\times 37$-pixel bitonal image given in Fig.~\\ref{fig:img_orig} (left).\nFollowing the same setup in \\cite{Nagahara15}, we added to each pixel a random Gaussian noise with mean-zero and standard deviation of $0.1$, as shown in Fig.~\\ref{fig:img_orig} (right).\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{img_orig.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_noise.eps}\n\\caption{Original image (left) and the image corrupted by Gaussian noise (right).}\n\\label{fig:img_orig}\n\\end{figure}\n\n\\noindent\nThe noisy image is represented by a real-valued $37 \\times 37$ matrix $X$\nand we apply the discrete Fourier transform (DFT) to obtain\n\\[\n\\hat{X} = WXW \\;\\; \\in \\mathbb{C}^{37 \\times 37} ,\n\\]\nequivalently,\n\\[\n\\mathrm{vec}(\\hat{X}) = (W\\otimes W) \\, \\mathrm{vec}(X) \\;\\; \\in \\mathbb{C}^{1369} ,\n\\]\nwhere $W = [ \\omega^{k, \\ell} ]_{k,\\ell =0}^{K-1}$ with $K=37$ and $\\omega = e^{-2 \\pi i \/ K}$ is the $K$-point DFT matrix.\nAs in \\cite{Nagahara15}, we randomly subsampled $\\mathrm{vec}(\\hat{X}) \\in \\mathbb{C}^{1369}$ to obtain a half-sized vector $\\mathbf{y} \\in \\mathbb{C}^{685}$ and set the measurement matrix $\\Phi$ as the corresponding $685 \\times 1369$ submatrix of $W\\otimes W$.\nFig.~\\ref{fig:reconstruction} shows the reconstructed images by BP, SN, SAV, and BSSL0, all with entrywise rounding off to $\\{ 0 , 1 \\}$.\nFor SN, an optimal tuning parameter $\\lambda$ was searched from $50$ to $1000$ by stepsize $50$ and the value $\\lambda = 800$ was chosen. For SAV and BSSL0, as in \\cite{Nagahara15}, we chose the parameter $p = \\mathbb{P}(x_j = 0) = 0.5$ as a rough estimate for the sparsity of the bitonal image (see \\cite{Nagahara15}).\nWe set $\\sigma_\\text{min} = 0.01$, $d = 0.9$, $\\mu = 2$, and $L = 3$ for the parameters of BSSL0.\nThe respective run time for BP, SN, SAV, and BSSL0 are also given in Tab.~\\ref{tab:timesonsumption}.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{img_BP.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_SN.eps}~ \\\\\n\\includegraphics[width=0.48\\linewidth]{img_SAV.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_BSSL0.eps}\n\\caption{Reconstructed images by BP (upper left), SN (upper right), SAV (lower left), and the proposed method BSSL0 (lower right).}\n\\label{fig:reconstruction}\n\\end{figure}\n\\begin{table}[H]\n\\caption{The Run Time Comparison}\n\\begin{center}\n\\label{tab:timesonsumption}\n \\begin{tabular}{ | l | l | l | l | }\n \\hline\n \\text{Algorithm} & Run Time\\\\ \\hline\n Basis Pursuit & 185.2044 seconds \\\\ \\hline\n SN & 406.1007 seconds \\\\ \\hline\n SAV & 191.5366 seconds \\\\ \\hline\n \\textbf{BSSL0} (proposed) & \\textbf{0.92577 seconds} \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this work, we proposed a fast algorithm (BSSL0) for reconstruction of binary signals which is based on the gradient descent method and smooth relaxation techniques. We showed that for binary signals our algorithm outperforms other CS\/BCS methods in terms of the recovery rate and speed. Future work includes a detailed analysis of BSSL0 in stability\/robustness and extensions to ternary and finite alphabet signals.\n\n\n\\section*{Acknowledgment}\nT.~Liu and D.~G.~Lee acknowledge the support of the DFG Grant PF 450\/6-1. The authors are grateful to Robert Fischer and G\\\"otz E.~Pfander for their helpful suggestions. The authors thank anonymous reviewers for their comments.\n\n\n\n\n\n\n\n\\IEEEpeerreviewmaketitle\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\subsection{Different methods to determine the density}\nThe density sets a crucial scale for our problem. Its precise determination is mandatory for quantitative precision. We will discuss two different methods for its determination and show that the results agree within our precision. For $T=0$, we also find agreement with the Ward identity $n=\\rho_0$.\n\nThe first method is to derive flow equations for the density. This has the advantage that the occupation numbers for a given momentum $\\vec{p}$ are mainly sensitive to running couplings with $k^2=\\vec{p}^2$. In the grand canonical formalism, the density is defined by\n\\begin{equation}\nn=-\\frac{\\partial}{\\partial \\mu}\\frac{1}{\\Omega}\\Gamma[\\varphi]{\\Big |}_{\\varphi=\\varphi_0,\\mu=\\mu_0}\n\\end{equation}\nWe can formally define a $k$-dependent density $n_k$ by\n\\begin{equation}\nn_k=-\\frac{\\partial}{\\partial \\mu}\\frac{1}{\\Omega}\\Gamma_k[\\varphi]{\\Big |}_{\\varphi=\\varphi_0,\\mu=\\mu_0}=-(\\partial_\\mu U)(\\rho_0,\\mu_0).\n\\end{equation}\nThe flow equation for $n_k$ is given in Eq.\\ \\eqref{eqFlowprescriptionnk} and the physical density obtains for $k=0$. The term $\\partial_\\mu \\zeta \\big{|}_{\\rho_0,\\mu_0}$ that enters Eq.\\ \\eqref{eqFlowprescriptionnk} is the derivative of the flow equation \\eqref{eqFlowpotentialMatrix} for $U$ with respect to $\\mu$. To compute it, we need the $\\mu$-dependence of the propagator $G_k$ in the vicinity of $\\mu_0$. Within a systematic derivative expansion, we use the expansion of $U(\\rho,\\mu)$ and the kinetic coefficients $Z_1$ and $Z_2$ to linear order in $(\\mu-\\mu_0)$, as described in section \\ref{sec:Derivativeexpansionandwardidentities}. Here, $Z_1(\\rho,\\mu)$ and $Z_2(\\rho,\\mu)$ are the coefficient functions of the terms linear in the $\\tau$-derivative and linear in $\\Delta$, respectively. \nNo reasonable qualitative behavior is found, if the linear dependence of $Z_1$ and $Z_2$ on $(\\mu-\\mu_0)$ is neglected. Also, the scale dependence of $\\alpha$ and $\\beta$ are quite important. The flow equations for $\\alpha$ and $\\beta$ can be obtained directly by differentiating the flow equation of the effective potential with respect to $\\mu$ and $\\rho$, cf. Eq.\\ \\eqref{eqflowalphabeta}. The situation is more complicated for the kinetic coefficients $Z_1^{(\\mu)}=\\partial_\\mu Z_1(\\rho_0,\\mu_0)$ and $Z_2^{(\\mu)}=\\partial_\\mu Z_2(\\rho_0,\\mu_0)$. Their flow equations have to be determined by taking the $\\mu$-derivative of the flow equation for $Z_1(\\rho, \\mu)$ and $Z_2(\\rho,\\mu)$. As discussed in section \\ref{sec:Derivativeexpansionandwardidentities}, we use in this paper the approximation $Z_1^{(\\mu)}=Z_2^{(\\mu)}=2V=2V_1(\\rho_0,\\mu_0)$.\n\nAs a check of both our method and our numerics, we also use another way to determine the particle density. This second method is more robust with respect to shortcomings of the truncation, but less adequate for high precision calculations as needed e.g. to determine the condensate depletion. The second method determines the pressure $p=-U(\\rho_0,\\mu_0)$ as a function of the chemical potential $\\mu_0$. Here, the effective potential is normalized by $U(\\rho_0=0,\\mu_0)=0$ at $T=0$, $n=0$. The flow of the pressure can be read of directly from the flow equation of the effective potential and is independent of the couplings $\\alpha$ and $\\beta$. We calculate the pressure as a function of $\\mu$ and determine the density $n=\\frac{\\partial}{\\partial \\mu}p$ by taking the $\\mu$-derivative numerically. It turns out that $p$ is in very good approximation given by $p=c\\,\\mu^2$, where the constant $c$ can be determined from a numerical fit. The density is thus linear in $\\mu$. \n\nAt zero temperature and for $\\tilde{v}=0$, we can additionally use the Ward identities connected to Galilean symmetry, which yield $n=\\rho_0$. We compare our methods in figure \\ref{densitycompared} and find that they give numerically the same result. We stress again the importance of a reliable method to determine the density, since we often rescale variables by powers of the density to obtain dimensionless variables.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig5.eps}\n\\caption{Pressure and density as a function of the chemical potential at $T=0$. We use three different methods: $n=-\\partial_\\mu U_{\\text{min}}$ from the flow equation (triangles), $n=\\rho_0$ as implied by Galilean symmetry (stars) and $n=\\partial_\\mu p$, where the pressure $p=-U$ (boxes) was obtained from the flow equation and phenomenologically fitted by $p=56.5 \\mu^2$ (solid lines). Units are arbitrary and we use $a=3.4\\cdot 10^{-4}$, $\\Lambda=10^3$.}\n\\label{densitycompared}\n\\end{figure}\n\n\\subsection{Quantum depletion of condensate}\nWe want to split the density into a condensate part $n_c$ and a density for uncondensed particles or \"depletion density\" $n_d=n-n_c$. For our model the condensate density is given by the \"bare\" order parameter\n\\begin{equation}\nn_c=\\bar{\\rho}_0=\\bar{\\rho}_0(k=0).\n\\end{equation}\nIn order to show this, we introduce occupation numbers $n(\\vec{p})$ for the modes with momentum $\\vec{p}$ with normalization\n\\begin{equation}\n\\int_{\\vec{p}}n(\\vec{p})=n.\n\\end{equation}\nOne formally introduces a $\\vec{p}$ dependent chemical potential $\\mu(\\vec{p})$ in the grand canonical partition function\n\\begin{equation}\ne^{-\\Gamma_{\\text{min}}[\\mu]}=\\text{Tr}e^{-\\beta(H-\\Omega_3\\int_{\\vec{p}}\\mu(\\vec{p})n(\\vec{p}))},\n\\end{equation}\nwith three dimensional volume $\\Omega_3=\\int_{\\vec{x}}$. Then one can define the occupation numbers by\n\\begin{equation}\nn(\\vec{p})=-\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma[\\varphi,\\mu(\\vec{p})]{\\Big |}_{\\varphi=\\varphi_0,\\mu(\\vec{p})=\\mu_0}.\n\\end{equation}\nThis construction allows us to use $k$-dependent occupation numbers by the definition\n\\begin{equation}\nn_k(\\vec{p})=-\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma_k[\\varphi,\\mu]{\\Big |}_{\\varphi=\\varphi_0,\\mu(\\vec{p})=\\mu_0}.\n\\end{equation}\nOne can derive a flow equation for this occupation number $n_k(\\vec{p})$ \\cite{WetterichOccupationNumbers}:\n\\begin{eqnarray}\n\\nonumber\n\\partial_k n_k(\\vec{p}) &=& -\\frac{1}{2}\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\text{Tr}\\left\\{(\\Gamma^{(2)}+R_k)^{-1}\\partial_k R_k\\right\\}\\\\\n& & +\\frac{\\partial}{\\partial \\bar \\rho}\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma[\\varphi,\\mu](\\partial_k \\bar\\rho_0).\n\\label{flowofnp}\n\\end{eqnarray}\n\nWe split the density occupation number into a $\\delta$-distribution like part and a depletion part, which is regular in the limit $\\vec{p}\\rightarrow0$\n\\begin{equation}\nn_k(\\vec{p})=n_{c,k}\\,\\delta(\\vec{p})+n_{d,k}(\\vec{p}).\n\\end{equation}\nOne can see from the flow equation for $n_k(\\vec{p})$ that the only contribution to $\\partial_k n_{c,k}$ comes from the second term in equation \\eqref{flowofnp}. Within a more detailed analysis \\cite{WetterichOccupationNumbers} one finds\n\\begin{equation}\n\\partial_k n_{c,k}=\\partial_k\\bar{\\rho}_{0,k}.\n\\end{equation}\nWe therefore identify the condensate density with the bare order parameter\n\\begin{equation}\nn_c=\\bar{\\rho}_0=\\frac{\\rho_0}{\\bar{A}}=\\bar{\\varphi}_0^2.\n\\end{equation}\nCorrespondingly, we define the $k$-dependent quantities\n\\begin{equation}\nn_{c,k}=\\bar{\\rho}_{0,k},\\quad n_k=n_{c,k}+n_{d,k}\n\\end{equation}\nand compute $n_d=n_d(k=0)$ by a solution of its flow equation. \n\nEven at zero temperature, the repulsive interaction connected with a positive scattering length $a$ causes a portion of the particle density to be outside the condensate. From dimensional reasons, it is clear, that $n_d\/n=(n-n_c)\/n$ should be a function of $an^{1\/3}$. The prediction of Bogoliubov theory or, equivalently, mean field theory, is $n_d\/n=\\frac{8}{3\\sqrt{\\pi}}(an^{1\/3})^{3\/2}$. We may determine the condensate depletion from the solution to the flow equation for the particle density, $n=n_{k=0}$, and $n_c=\\bar{\\rho}_0=\\bar{\\rho}_0(k=0)$. \n\nFrom Galilean invariance for $T=0$ and $\\tilde{v}=0$, it follows that\n\\begin{equation}\n\\frac{n_d}{n}=\\frac{\\rho_0-\\bar{\\rho}_0}{\\rho_0}=1-\\frac{1}{\\bar{A}},\n\\end{equation}\nwith $\\bar{A}=\\bar{A}(k=0)$. This gives an independent determination of $n_c$. \nIn figure \\ref{figDepletiond3} we plot the depletion density obtained from the flow of $n$ and $\\bar{\\rho}_0$ over several orders of magnitude. Apart from some numerical fluctuations for small $an^{1\/3}$, we find that our result is in full agreement with the Bogoliubov prediction.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig6.eps}\n\\caption{Condensate depletion $(n-n_c)\/n$ as a function of the dimensionless scattering length $a n^{1\/3}$. For the solid curve, we vary $a$ with fixed $n=1$, for the dashed curve we vary the density at fixed $a=10^{-4}$. The dotted line is the Bogoliubov-Result $(n-n_c)\/n=\\frac{8}{3\\sqrt{\\pi}}(a n^{1\/3})^{3\/2}$ for reference. We find perfect agreement of the three determinations. The fluctuations in the solid curve for small $a n^{1\/3}$ are due to numerical uncertainties. Their size demonstrates our numerical precision.}\n\\label{figDepletiond3}\n\\end{figure}\n\n\\subsection{Quantum phase transition}\n\\label{ssec:Quantumphasdiagram}\nFor $T=0$ a quantum phase transition separates the phases with $\\rho_0=0$ and $\\rho_0>0$.\nIn this section, we investigate the phase diagram at zero temperature in the cube spanned by the dimensionless parameters $\\tilde{\\mu}=\\frac{\\mu}{\\Lambda^2}$, $\\tilde{a}=a\\Lambda$ and $\\tilde{v}=\\frac{V_\\Lambda}{S_\\Lambda^2}\\Lambda^2$. This goes beyond the usual phase transition for nonrelativistic bosons, since we also include a microscopic second $\\tau$-derivative $\\sim\\tilde{v}$, and therefore models with a generalized microscopic dispersion relation.\nFor non-vanishing $\\tilde{v}$ (i.e. for a nonzero initial value of $V_1$ with $V_2=V_3=0$ in section \\ref{sec:Derivativeexpansionandwardidentities}), the Galilean invariance at zero temperature is broken explicitly. For large $\\tilde{v}$, we expect a crossover to the \"relativistic\" $O(2)$ model. If we send the initial value of the coefficient of the linear $\\tau$-derivative $S_\\Lambda$ to zero, we obtain the limiting case $\\tilde{v}\\rightarrow\\infty$. The symmetries of the model are now the same as those of the relativistic O(2) model in four dimensions. The space-time-rotations or Lorentz symmetry replace Galilean symmetry. \n\nIt is interesting to study the crossover between the two cases. Since our cutoff explicitly breaks Lorentz symmetry, we investigate in this paper only the regime $\\tilde{v}\\lesssim1$. Detailed investigations of the flow equations for $\\tilde{v}\\rightarrow\\infty$ can be found in the literature \\cite{Papenbrock:1994kf, Berges2000ew, PhysRevA.60.1442, PhysRevB.68.064421, PhysRevD.67.065004, Bervillier2007}. The phase diagram in the $\\tilde{\\mu}-\\tilde{v}$ plane with $\\tilde{a}=1$ is shown in figure \\ref{figQPTsigmava1}. The critical chemical potential first increases linearly with $\\tilde{v}$ and then saturates to a constant. The slope in the linear regime as well as the saturation value depend linearly on $\\tilde{a}$ for $\\tilde{a}<1$. \n\nAt $T=0$, the critical exponents are everywhere the mean field ones ($\\eta=0$, $\\nu=1\/2$). This is expected: It is the case for $\\tilde{v}=0$ \\cite{Wetterich:2007ba, Uzunov1981, SachdevQPT}, and for $\\tilde{v}=\\infty$ the theory is equivalent to a relativistic $O(2)$ model in $d=3+1$ dimensions. This is just the upper critical dimension of the Wilson-Fisher fixed point \\cite{PhysRevLett.28.240}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig7.eps}\n\\caption{Quantum phase diagram in the $\\tilde{\\mu}$-$\\tilde{v}$ plane for $\\tilde{a}=1$.}\n\\label{figQPTsigmava1}\n\\end{figure}\n\nFrom section 3 we know that for $\\tilde{v}=0$ the parameter $\\tilde{a}$ is limited to $\\tilde{a}<\\frac{3\\pi}{4}\\approx 2.356$. For $\\tilde{v}=0$ and a small scattering length $a\\rightarrow0$, a second order quantum phase transition divides the phases without spontaneous symmetry breaking for $\\mu<0$ from the phase with a finite order parameter $\\rho_0>0$ for $\\mu>0$. It is an interesting question, whether this quantum phase transition at $\\mu=0, \\tilde{v}=0$ also occurs for larger scattering length $a$. We find in our truncation that this is indeed the case for a large range of $a$, but not for $\\tilde{a}>1.55$. Here, the critical chemical potential suddenly increases to large positive values as shown in Fig. \\ref{figQPTsigmaofa}. For $\\tilde{v}>0$ this increase happens even earlier. (For a truncation with $V_1\\equiv0$, the phase transition would always occur at $\\mu=0$.) We plot the $\\tilde{\\mu}-\\tilde{a}$ plane of the phase diagram for different values of $\\tilde{v}$ in figure \\ref{figQPTsigmaofa}. The form of the critical line can be understood by considering the limits $\\tilde{v}\\rightarrow0$ as well as $\\tilde{a}\\rightarrow0$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig8.eps}\n\\caption{Quantum phase diagram in the $\\tilde{\\mu}$-$\\tilde{a}$ plane for $\\tilde{v}=1$ (dotted), $\\tilde{v}=0.01$ (dashed) and $\\tilde{v}=0$ (solid).}\n\\label{figQPTsigmaofa}\n\\end{figure}\n\nFor a fixed chemical potential, the order parameter $\\rho_0$ as a function of $a$ goes to zero at a critical value $a_c$ as shown in Fig. \\ref{figQPTrhoofa}. This happens in a continuous way and the phase transition is therefore of second order. For $\\mu\\rightarrow0$, we find $a_c=1.55 \\Lambda^{-1}$. We emphasize, however, that $a_c$ is of the order of the microscopic distance $\\Lambda^{-1}$. Universality may not be realized for such values, and the true phase transition may depend on the microphysics. For example, beyond a critical value for the repulsive interaction, the system may form a solid. Ultracold atom gases correspond to metastable states which may lose their relevance for $a\\rightarrow\\Lambda^{-1}$. For $\\tilde{v}>0$ and $\\mu\\ll\\Lambda^2$ the phase transition occurs for $a_c \\Lambda \\ll 1$ such that universal behavior is expected.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig9.eps}\n\\caption{Quantum phase transition for fixed chemical potential $\\mu=1$, with $\\Lambda=10^3$. The density $\\rho_0=n$ as a function of the scattering length $a$ goes to zero at a critical $a_c \\Lambda=1.55$, indicating a second order quantum phase transition at that point.}\n\\label{figQPTrhoofa}\n\\end{figure}\n\n\n\t\t\t\n\\subsection{Thermal depletion of condensate}\n\\label{ssec:Phasediagramatnonzerotemperature}\nSo far, we have only discussed the vacuum and the dense system at zero temperature. A non vanishing temperature $T$ will introduce an additional scale in our problem. For small $T\\ll n^{2\/3}$ we expect only small corrections. However, as $T$ increases the superfluid order will be destroyed. Near the phase transition for $T \\approx T_c$ and for the disordered phase for $T>T_c$, the characteristic behavior of the boson gas will be very different from the $T\\rightarrow0$ limit.\n\nFor $T>0$ the particle density $n$ receives a contribution from a thermal gas of bosonic (quasi-) particles. It is no longer uniquely determined by the superfluid density $\\rho_0$. We may write \n\\begin{equation}\nn=\\rho_0+n_T\n\\label{eqDensityatT}\n\\end{equation}\nand observe, that $n_T=0$ is enforced by Galilean symmetry only for $T=0, V_\\Lambda=0$. The heat bath singles out a reference frame, such that for $T>0$ Galilean symmetry no longer holds. In our formalism, the thermal contribution $n_T$ appears due to modifications of the flow equations for $T\\neq0$. We start for high $k$ with the same initial values as for $T=0$. As long as $k\\gg \\pi T$ the flow equations receive only minor modifications. For $k \\approx \\pi T$ or smaller, however, the discreteness of the Matsubara sum has important effects. We plot in Fig. \\ref{fignoftemperature} the density as a function of $T$ for fixed $\\mu=1$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig10.eps}\n\\caption{Density $n\/{\\mu^{2\/3}}$ (solid) and order parameter $\\rho_0\/{\\mu^{2\/3}}$ (dashed) as a function of the temperature $T\/\\mu$. The units are arbitrary with $a=2\\cdot 10^{-4}$ and $\\Lambda=10^3$. The plot covers only the superfluid phase. For higher temperatures, the density is given by the thermal contribution $n=n_T$ only.}\n\\label{fignoftemperature}\n\\end{figure}\n\nIn Fig. \\ref{fignofsigma} we show $n(\\mu)$, similar to Fig. \\ref{densitycompared}, but now for different $a$ and $T$. For $T=0$ the scattering length sets the only scale besides $n$ and $\\mu$, such that by dimensional arguments $a^2\\mu=f(a^3 n)$. Bogoliubov theory predicts \n\\begin{equation}\nf(x)=8\\pi x(1+\\frac{32}{3\\sqrt{\\pi}}x^{1\/2}).\n\\end{equation}\nThe first term on the r.h.s. gives the contribution of the ground state, while the second term is added by fluctuation effects. For small scattering length $a$, the ground state contribution dominates. We have then $\\mu\\sim a$ for $n=1$ and $\\mu\/n$ can be treated as a small quantity. For $T\\neq 0$ and small $a$ one expects $\\mu=g(T\/n^{2\/3})an$. The curves in Fig. \\ref{fignofsigma} for $T=1$ show that the density, as a function of $\\mu$, is below the curve obtained at $T=0$. This is reasonable, since the statistical fluctuations now drive the order parameter $\\rho_0$ to zero. At very small $\\mu$, the flow enters the symmetric phase. The density is always positive, but for simplicity, we show the density as a function of $\\mu$ in figure \\ref{fignofsigma} only in those cases, where the flow remains in the phase with spontaneous $U(1)$ symmetry breaking. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig11.eps}\n\\caption{Density $n$ for different temperatures and scattering length. We plot $n(\\mu)$ in arbitrary units, with $\\Lambda=10^3$, and for a scattering length $a=2\\cdot10^{-4}$ (solid and dotted), $a=10^{-4}$ (dashed and dashed-dotted). The temperature is $T=0$ (solid and dashed) and $T=1$ (dotted and dashed-dotted).}\n\\label{fignofsigma}\n\\end{figure}\n\nFor temperatures above the critical temperature, the order parameter $\\rho_0$ vanishes at the macroscopic scale and so does the condensate density $n_c=\\bar{\\rho_0}=\\frac{1}{\\bar{A}}\\rho_0$. The density is now given by a thermal distribution of particles with nonzero momenta. \nUp to small corrections from the interaction $\\sim aT$, it is described by a free Bose gas, \n\\begin{equation}\nn= \\frac{T^{3\/2}}{(4\\pi)^{3\/2}}g_{3\/2}(e^{\\beta \\mu}),\n\\end{equation}\nwith the \"Bose function\"\n\\begin{equation}\ng_p(z)=\\frac{1}{\\Gamma(p)}\\int_0^\\infty dx\\,x^{p-1}\\frac{1}{z^{-1}e^x-1}.\n\\end{equation}\n\nIn figure \\ref{figrhooftemperature} we show the dimensionless order parameter $\\rho_0\/n$ as a function of the dimensionless temperature $T\/n^{2\/3}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig12.eps}\n\\caption{Order parameter $\\rho_0\/n$ as a function of the dimensionless temperature $T\/(n^{2\/3})$ for scattering length $a=10^{-4}$. Here, we varied $T$ keeping $\\mu$ fixed. Numerically, this is equivalent to varying $\\mu$ with fixed $T$.\\label{figrhooftemperature}}\n\\end{figure}\nThis plot shows the second order phase transition from the phase with spontaneous $U(1)$ symmetry breaking at small temperatures to the symmetric phase at higher temperatures. The critical temperature $T_c$ is determined as the temperature, where the order parameter just vanishes - it is investigated in the next section. Since we find $(\\bar{A}-1)\\ll1$, the condensate fraction $n_c\/n=\\bar{\\rho_0}\/n=\\rho_0\/(\\bar{A}n)$ as a function of $T\/n^{2\/3}$ resembles the order parameter $\\rho_0\/n$. We plot $\\bar{A}$ as a function of $T\/n^{2\/3}$ in Fig. \\ref{Abaroftemperature}. Except for a narrow region around $T_c$, the deviations from one remain indeed small. Near $T_c$ the gradient coefficient $\\bar{A}$ diverges according to the anomalous dimension, $\\bar{A}\\sim \\xi^\\eta$, with $\\eta$ the anomalous dimension. The correlation length $\\xi$ diverges with the critical exponent $\\nu$, $\\xi\\sim |T-T_c|^{-\\nu}$, such that\n\\begin{equation}\n\\bar{A}\\sim |T-T_c|^{-\\eta\\nu}.\n\\end{equation}\nHere, $\\eta$ and $\\nu$ are the critical exponents for the Wilson Fisher fixed point of the classical three-dimensional $O(2)$ model, $\\eta=0.0380(4)$, $\\nu=0.67155(27)$ \\cite{Pelissetto2002549, Berges2000ew, PhysRevA.60.1442, PhysRevB.68.064421, PhysRevD.67.065004, Bervillier2007}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig13.eps}\n\\caption{Order parameter divided by the condensate density $\\bar{A}=\\rho_0\/n_c$, as a function of the dimensionless temperature $T\/(n^{2\/3})$, and for scattering length $a=10^{-4}$. Here, we varied $T$ keeping $\\mu$ fixed. Numerically, this is equivalent to varying $\\mu$ with fixed $T$. The plot covers only the phase with spontaneous symmetry breaking. For higher temperatures, the symmetric phase has $\\rho_0=n_c=0$. The divergence of $\\bar{A}$ for $T\\rightarrow T_c$ reflects the anomalous dimension $\\eta$ of the Wilson-Fisher fixed point.}\n\\label{Abaroftemperature}\n\\end{figure}\n\nIn figure \\ref{figrhoofsigma} we plot $\\rho_0\/n$ as a function of the chemical potential $\\mu$ for different temperatures and scattering lengths. We find, that $\\rho_0\/n=1$ is indeed approached in the limit $T\\rightarrow0$, as required by Galilean invariance. All figures of this section are for $\\tilde{v}=0$. The modifications for $\\tilde{v}\\neq0$ are mainly quantitative, not qualitative. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig14.eps}\n\\caption{Order parameter divided by the density, $\\rho_0\/n$, as a function of the chemical potential. We use arbitrary units with $\\Lambda=10^3$. The curves are given for a scattering length $a=2\\cdot10^{-4}$ (solid and dotted), $a=10^{-4}$ (dashed and dashed-dotted) and temperature $T=0.1$ (solid and dashed) and $T=1$ (dotted and dashed-dotted). At zero temperature, Galilean invariance implies $\\rho_0=n$, which we find within our numerical resolution.}\n\\label{figrhoofsigma}\n\\end{figure}\n\n\\subsection{Critical temperature}\nThe critical temperature $T_c$ for the phase transition between the superfluid phase at low temperature and the disordered or symmetric phase at high temperature depends on the scattering length $a$. By dimensional reasoning, the temperature shift $\\Delta T_c=T_c(a)-T_c(a=0)$ obeys $\\Delta T_c\/T_c\\sim an^{1\/3}$. The proportionality coefficient cannot be computed in perturbation theory \\cite{Andersen:2003qj}. It depends on $\\tilde{v}$ and we concentrate here on $\\tilde{v}=0$. Monte-Carlo simulations in the high temperature limit, where only the lowest Matsubara frequency is included, yield $\\Delta T_c\/T_c=1.3 \\,a n^{1\/3}$ \\cite{PhysRevLett.87.120401, PhysRevLett.87.120402}. Within the same setting, renormalization group studies \\cite{PhysRevLett.83.1703, blaizot:051116, blaizot:051117, PhysRevA.69.061601, PhysRevA.70.063621} yield a similar result, for composite bosons see \\cite{Diehl:2007th}. Near $T_c$, the long wavelength modes with momenta $\\vec{p}^2\\ll(\\pi T)^2$ dominate the \"long distance quantities\". Then a description in terms of a classical three dimensional system becomes valid. This \"dimensional reduction\" is achieved by \"integrating out\" the nonzero Matsubara frequencies. \nHowever, both $\\Delta T_c\/T_c$ and $n$ are dominated by modes with momenta $\\vec{p}^2\\approx(\\pi T_c)^2$ such that corrections to the classical result may be expected.\n\nWe have computed $T_c$ numerically by monitoring the zero of $\\rho_0$, as shown in Fig. \\ref{figrhooftemperature}, $\\rho_0(T\\rightarrow T_c)\\rightarrow0$. Our result is plotted in Fig. \\ref{Tcofa}. In the limit $a\\rightarrow0$ we find for the dimensionless critical temperature $T_c\/(n^{2\/3})=6.6248$, which is in good agreement with the expected result for the free theory $T_c\/(n^{2\/3})=\\frac{4\\pi}{\\zeta(3\/2)^{2\/3}}=6.6250$. For the shift in $T_c$ due to the finite interaction strength, we obtain\n\\begin{equation}\n\\frac{\\Delta T_c}{T_c}=\\kappa \\,a n^{1\/3}, \\quad \\kappa=2.1.\n\\end{equation}\nWe expect that the result for $\\kappa$ depends on the truncation and may change somewhat if additional higher order couplings are included.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig15.eps}\n\\caption{Dimensionless critical temperature $T_c\/(n^{2\/3})$ as a function of the dimensionless scattering length $an^{1\/3}$ (points). We also plot the linear fit $\\Delta T_c\/T_c=2.1\\, a n^{1\/3}$ (solid line).}\n\\label{Tcofa}\n\\end{figure}\n\n\\subsection{Zero temperature sound velocity}\nThe macroscopic sound velocity $v_S$ is a crucial quantity for the hydrodynamics of the gas or liquid. It is accessible to experiment. As a thermodynamic observable, the adiabatic sound velocity is defined as\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}\\bigg{|}_s\n\\end{equation}\nwhere $M$ is the particle mass (in our units $1\/M=2$), $p$ is the pressure, $n$ is the particle density, and $s$ is the entropy per particle. It is related to the isothermal sound velocity $v_T$ by\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\left(\\frac{\\partial p}{\\partial n}\\bigg{|}_T+\\frac{\\partial p}{\\partial T}\\bigg{|}_n\\frac{\\partial T}{\\partial n}\\bigg{|}_s\\right)=v_T^2+2\\frac{\\partial p}{\\partial T}\\bigg{|}_n \\frac{\\partial T}{\\partial n}\\bigg{|}_s\n\\label{eqSoundAdiabaticIsothermal}\n\\end{equation}\nwhere we use our units $2M=1$. One needs the \"equation of state\" $p(T,n)$ and\n\\begin{equation}\ns(T,n)=\\frac{S}{N}=\\frac{1}{n}\\frac{\\partial p}{\\partial T}\\bigg{|}_\\mu.\n\\end{equation}\nBy dimensional analysis, one has\n\\begin{equation}\np=n^{5\/3}{\\cal F}(t,c), \\quad t=\\frac{T}{n^{2\/3}}, \\quad c=a n^{1\/3},\n\\end{equation}\nwith ${\\cal F}(0,c)=4\\pi c$ (in Bogoliubov theory), and ${\\cal F}(t,0)=\\frac{\\zeta(5\/2)}{(4\\pi)^{3\/2}}t^{5\/2}$ (free theory), such that for small $c$\n\\begin{equation}\n{\\cal F}=\\frac{\\zeta(5\/2)}{(4\\pi)^{3\/2}}t^{5\/2}+g(t)c.\n\\end{equation}\n\nAt zero temperature the second term in Eq.\\ \\eqref{eqSoundAdiabaticIsothermal} vanishes, such that $v_S=v_T$. For the isothermal sound velocity one has \n\\begin{equation}\nv_T^2=2\\frac{\\partial p}{\\partial n}\\bigg{|}_{T}=2 \\frac{\\partial p}{\\partial \\mu}\\bigg{|}_T\\left(\\frac{\\partial n}{\\partial \\mu}\\bigg{|}_T\\right)^{-1}.\n\\end{equation}\nWe can now use \n\\begin{equation}\n\\frac{\\partial p}{\\partial \\mu}\\big{|}_T=-\\frac{dU_{\\text{min}}}{d\\mu}=-\\partial_\\mu U(\\rho_0)=n\n\\end{equation}\nand infer\n\\begin{equation}\nv_T^2=2\\left(\\frac{\\partial\\, \\text{ln}\\,n}{\\partial \\mu}\\right)^{-1}.\n\\end{equation}\n\nOne may also define a microscopic sound velocity $c_S$, which characterizes the propagation of (quasi-) particles. At zero temperature, where we can perform the analytic continuation to real time, we can calculate the microscopic sound velocity from the dispersion relation $\\omega(p)$ (with $p=|\\vec{p}|$). In turn, the dispersion relation is obtained from the effective action by setting $\\text{det}(G^{-1})=0$, where $G^{-1}$ is the full inverse propagator. We perform the calculation explicitly at the end of section \\ref{sec:Derivativeexpansionandwardidentities} and find\n\\begin{equation}\nc_S^{-2}=\\frac{S^2}{2\\lambda\\rho_0}+V.\n\\end{equation}\n\nThe Bogoliubov result for the sound velocity is in our units\n\\begin{equation}\nc_S^2=2\\lambda\\rho_0=16\\pi an.\n\\end{equation}\nIn three dimensions, the decrease of $S$ is very slow and the coupling $V$ remains comparatively small even on macroscopic scales, cf. Fig. \\ref{figFlowKinetic}. We thus do not expect measurable deviations from the Bogoliubov result for the sound velocity at $T=0$. In figure \\ref{figSound}, we plot our result over several orders of magnitude of the dimensionless scattering length and, indeed, find no deviations from Bogoliubovs result.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig16.eps}\n\\caption{Dimensionless sound velocity $c_s\/(n^{1\/3})$ at zero temperature, as a function of the scattering length $an^{1\/3}$. Within the plot resolution the curves obtained by varying $a$ with fixed $n$, by varying $n$ with fixed $a$, and the Bogoliubov result, $c_s=\\sqrt{16\\pi}(an)^{1\/2}$, coincide.}\n\\label{figSound}\n\\end{figure}\n\nWe finally show that for $T=0$ the macroscopic and microscopic sound velocities are equal, $v_S=v_T=c_S$. For this purpose, we use \n\\begin{equation}\n\\frac{\\partial n}{\\partial \\mu}\\big{|}_T=-\\frac{d}{d\\mu}\\left(\\partial_\\mu U(\\rho_0)\\right)=-\\partial_\\mu^2U(\\rho_0)-\\partial_\\rho\\partial_\\mu U(\\rho_0)\\frac{d\\rho_0}{d\\mu}.\n\\end{equation}\nFrom the minimum condition $\\partial_\\rho U=0$, it follows\n\\begin{equation}\n\\frac{d\\rho_0}{d\\mu}=-\\frac{\\partial_\\rho\\partial_\\mu U}{\\partial_\\rho^2 U}=-\\frac{\\alpha}{\\lambda}.\n\\end{equation}\nCombining this with the Ward identities from section \\ref{sec:Derivativeexpansionandwardidentities}, namely $\\partial_\\mu^2 U=-2V\\rho_0$ and $\\alpha=\\partial_\\rho\\partial_\\mu U=-S$, valid at $T=0$, it follows that the macroscopic sound velocity equals the microscopic sound velocity\n\\begin{equation}\nv_S^2(T=0)=c_S^2.\n\\end{equation}\n\n\n\n\\chapter{Introduction}\n\\label{ch:Introduction}\n\\pagenumbering{arabic}\n\\input{Introduction}\n\n\n\\section{Flow equations to solve an integral}\n\\label{ch:Flowequationstosolveanintegral}\n\\input{Flowequationstosolveanintegral}\n\n\n\\section{Functional integral representation of quantum field theory}\n\\label{ch:Functionalintegralrepresentationofquantumfieldtheory}\n\\input{Functionalintegralrepresentationofquantumfieldtheory}\n\n\t\n\\chapter{The Wetterich equation}\n\\label{ch:TheWetterichequation}\n\\input{TheWetterichequation}\n\n\t\t\n\\chapter{Generalized flow equation}\n\\label{ch:Generalizedflowequation}\n\\input{Generalizedflowequation}\n\n\\chapter{Truncations}\n\\label{ch:Truncations}\n\\input{Truncations}\n\t\t\n\\chapter{Cutoff choices}\n\\label{ch:Cutoffchoices}\n\\input{Cutoffchoices}\n\n\\chapter{Investigated models}\n\\label{ch:Variousphysicalsystems}\n\\input{Variousphysicalsystems}\n\n\\chapter{Symmetries}\n\\label{ch:Symmetries}\n\\input{Symmetries}\n\n\\chapter{Truncated flow equations}\n\\label{ch:Truncationsandflowequations}\n\\input{Truncationsandflowequations}\n\n\n\\chapter{Few-body physics}\n\\label{ch:Few-bodyphysics}\n\\input{Fewbodyphysics}\n\n\t\\section{Repulsive interacting bosons}\n\t\\label{sec:Repulsiveinteractingbosons}\n\t\\input{Repulsiveinteractingbosons}\n\t\n\n\t\\section{Two fermion species: Dimer formation}\n\t\\label{sec:Twofermionspecies:Dimerformation}\n\t\\input{TwofermionspeciesDimerformation}\n\n\t\\section{Three fermion species: Efimov effect}\n\t\\label{sec:Threefermionspecies:ThomasandEfimoveffect}\n\t\\input{ThreefermionspeciesThomasandEfimoveffect}\n\n\n\\chapter{Many-body physics}\n\\label{ch:Many-bodyphysics}\n\n\t\\section{Bose-Einstein Condensation in three dimensions}\n\t\\label{sec:Bose-EinsteinCondensationinthreedimensions}\n\t\\input{BoseEinsteinCondensationinthreedimensions}\n\t\\input{Thermodyn}\n\t\t\n\t\\section{Superfluid Bose gas in two dimensions}\n\t\\label{sec:SuperfluidBosegasintwodimensions}\n\t\\input{SuperfluidBose2d}\n\n\t\t\t\n\t\\section{Particle-hole fluctuations and the BCS-BEC Crossover}\n\t\\label{sec:Particle-holefluctuationsandtheBCS-BECCrossover}\n\t\\input{ParticleHole}\n\t\t\t\n\t\\section{BCS-Trion-BEC Transition}\n\t\\label{sec:BCS-Trion-BECTransitionlong}\n\t\\input{BCS-Trion-BECTransitionlong}\n\t\n\\chapter{Conclusions}\n\\label{ch:Conclusions}\n\\input{Conclusions}\n\n\\begin{appendix}\n\\chapter{Some ideas on functional integration and probability}\n\\label{ch:Foundationsofquantumtheory:someideas}\n\\input{functionalprobabilities}\n\n\\chapter{Technical additions}\n\\label{ch:Technicaladditions}\n\n\\section{Flow of the effective potential for Bose gas}\n\\label{sec:FlowoftheeffectivepotentialforBosegas}\n\\input{FlowoftheeffectivepotentialforBosegas}\n\n\\section{Flow of the effective potential for BCS-BEC Crossover}\n\\label{sec:FlowoftheeffectivepotentialforBCSBECcrossover}\n\\input{FlowoftheeffectivepotentialforBCSBECcrossover}\n\n\\section{Hierarchy of flow equations in vacuum}\n\\label{sec:Hierarchyofflowequationsinvacuum}\n\\input{Hierarchyofflowequationsinvacuum}\n\n\\end{appendix}\n\n\n\\input{dissSFbib}\n\n\\chapter*{Danken...}\nm\\\"{o}chte ich ganz besonders Prof.\\ Dr.\\ Christof Wetterich f\\\"ur die hervorragende Betreuung und die ausgezeichnete Zusammenarbeit. Von den vielen Gespr\\\"achen und guten Diskussionen habe ich sehr profitiert.\\\\\n\nF\\\"ur tolle Zusammenarbeit und viele interessante Diskussionen danke ich auch ganz herzlich Michael Scherer, Richard Schmidt, Sergej Moroz, Dr.\\ Sebastian Diehl, Dr.\\ Hans-Christian Krahl, Dr.\\ Philipp Strack, Prof.\\ Dr.\\ Holger Gies, Prof.\\ Dr.\\ Jan Pawlowski, Prof.\\ Dr.\\ Markus Oberthaler, Prof.\\ Dr.\\ Selim Jochim, dessen Arbeitsgruppe sowie insbesondere auch allen Teilnehmern des Seminars ,,Kalter Quantenkaffee''. \\\\\n\nProf. Dr. Holger Gies danke ich auch f\\\"ur die bereitwillige \\\"{U}bernahme des Zweitgutachtens und der damit verbundenen M\\\"{u}hen.\\\\\n\nAnne Doster danke ich f\\\"ur die sehr sch\\\"{o}ne, gemeinsam verbrachte Zeit, willkommene und notwendige Ablenkung sowie viel Verst\\\"{a}ndniss und Unterst\\\"{u}tzung. Sehr dankbar f\\\"{u}r sehr Vieles bin ich auch meinen Eltern, Geschwistern und Freunden.\n\\end{document}\n\n\n\n\n\\subsubsection{From the lattice to field theory}\n\nOne way to approach the infinitely many degrees of freedom of a continuous field theory comes from a discrete lattice of space-time points. Consider a lattice of points\n\\begin{equation}\n\\vec x_{ijk} = a \\begin{pmatrix}i \\\\ j \\\\ k\\end{pmatrix}, \\quad i,j,k\\in \\mathbb{Z}\n\\end{equation}\nat times\n\\begin{equation}\nt_n = b\\, n, \\quad n\\in \\mathbb{Z}.\n\\end{equation}\nFor every set of indices $(ijk,n)$ the field $\\varphi(x_{ijk},t_n)$ has some value, e.~g. $\\varphi(x_{ijk},t_n)\\in \\mathbb{C}$ for a complex scalar. The partition sum, i.~e. the sum over all possible configurations weighted by the corresponding functional probability is given by\n\\begin{equation}\nZ = \\left( \\prod_{(ijk,n)\\in \\mathbb{Z}^4} \\int d \\varphi(\\vec x_{ijk},t_n)\\right)\\, e^{-S(\\varphi(\\vec x_{ijk},t_n))}.\n\\label{eq:PartitionfunctionSTlattice}\n\\end{equation}\nThe action $S$ depends on the values of $\\varphi(\\vec x_{ijk},t_n)$ for the different lattice points. Eq.\\ \\eqref{eq:PartitionfunctionSTlattice} describes a theory on a discrete space-time lattice. From the probabilities $e^{-S}$ we can calculate all sorts of expectation values, correlation functions and so on. \n\nOur theory becomes a continuum field theory in the limit where $a\\to 0$ and $b\\to 0$. The partition function reads then \n\\begin{equation}\nZ = \\lim_{a,b\\to0} \\left(\\prod_{(ijk,n)\\in \\mathbb{Z}^4}\\int d\\varphi(\\vec x_{ijk},t_n)\\right) e^{-S(\\varphi(\\vec x_{ijk},t_n))}.\n\\end{equation}\nThis can also be written as\n\\begin{equation}\nZ = \\int D\\varphi \\,e^{-S[\\varphi]}.\n\\end{equation}\nThe functional integral $\\int D\\varphi$ might be defined by the limiting procedure above. The microscopic action $S$ is now a functional of the field configuration $\\varphi(\\vec x, t)$, where space and time are now continuous, $(\\vec x, t)\\in \\mathbb{R}^4$. \n\n\n\\subsubsection{Expectation values, correlation functions}\n\nWith our formalism we aim for a statistical description of fields. Important concepts are expectation values of operators and correlation functions. For simplicity, we denote the field degrees of freedom by $\\tilde \\Phi_\\alpha$. The collective index $\\alpha$ labels both continuous degrees of freedom such as position or momentum and discrete variables such as spin, flavor or simply ``particle species''. The field $\\tilde \\Phi_\\alpha$ might consist of both bosonic and fermionic parts. The fermionic components are described by Grassmann numbers while the bosonic components correspond to ordinary ($\\mathbb{C}$) numbers. As an example we consider a theory with a complex scalar field $\\varphi$ and a fermionic complex two-component spinor $\\psi=(\\psi_1,\\psi_2)$. It is useful to decompose the complex scalar into real and imaginary parts\n\\begin{equation}\n\\varphi = \\frac{1}{\\sqrt{2}}(\\varphi_1+i \\varphi_2).\n\\end{equation}\nIn momentum space the Nambu spinor of fields reads\n\\begin{equation}\n\\Phi(q) = \\left( \\varphi_1(q), \\varphi_2(q), \\psi_1(q), \\psi_2(q), \\psi_1^*(-q), \\psi_2^*(-q)\\right)\n\\end{equation}\nThe index $\\alpha$ stands in this case for the momentum $q$ and the position in the Nambu-spinor, e.q. $\\Phi_\\alpha = \\psi_1(q)$ for $\\alpha=(q,3)$.\n\nThe field expectation value is given by\n\\begin{equation}\n\\Phi_\\alpha = \\langle \\tilde \\Phi_\\alpha\\rangle= \\frac{1}{Z}\\int D\\tilde \\Phi \\,\\tilde\\Phi_\\alpha \\, e^{-S[\\tilde\\Phi]},\n\\label{eq3:fieldexpectationvalue}\n\\end{equation}\nwith\n\\begin{equation}\nZ = \\int D\\tilde \\Phi\\, e^{-S[\\tilde \\Phi]}.\n\\end{equation}\nQuite similar one defines correlation functions as\n\\begin{equation}\nc_{\\alpha\\beta\\gamma\\dots} = \\langle\\tilde\\Phi_\\alpha\\tilde \\Phi_\\beta\\tilde\\Phi_\\gamma\\dots\\,\\rangle = \\frac{1}{Z}\\int D\\tilde\\Phi \\,\\tilde\\Phi_\\alpha \\tilde\\Phi_\\beta\\tilde\\Phi_\\gamma\\dots e^{-S[\\tilde\\Phi]}.\n\\end{equation}\nAs an example let us consider the two-point function. It is sensible to decompose it into a connected and a disconnected part like\n\\begin{equation}\n\\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle = \\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle_c + \\langle\\tilde\\Phi_\\alpha\\rangle \\langle\\tilde\\Phi_\\beta\\rangle.\n\\end{equation}\nThe connected part is the (full) propagator \n\\begin{equation}\nG_{\\alpha\\beta} = \\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle_c.\n\\end{equation}\nAlthough we discussed here the statistical formulation of the theory (``imaginary time'') the concepts of expectation values and correlation functions are also useful for the real-time formalism. Formally, the main difference is that the weighting factor $e^{-S[\\tilde\\Phi]}$ becomes complex after analytic continuation\n\\begin{equation}\ne^{-S[\\tilde\\Phi]}\\to e^{iS_t[\\tilde\\Phi]},\n\\end{equation} \nwhere $S_t[\\tilde\\Phi]$ is now the real-time action.\n\n\n\\subsubsection{Functional derivatives, generating functionals}\n\nTo calculate expectation values and correlation functions it is useful to work with sources, functional derivatives and generating functionals. We first explain what a functional derivative is. In some sense it is a natural generalization of the usual derivative to functionals, i.\\ e.\\ to objects that depend on an argument which is itself a function on some space. The basic axiom for the functional derivative is\n\\begin{equation}\n\\frac{\\delta}{\\delta f(x)} f(y) = \\delta(x-y) \\quad \\text{or} \\quad \\frac{\\delta}{\\delta f(x)}\\int_y f(y)g(y) = g(x).\n\\label{eq:axiomfunctionalderivative}\n\\end{equation} \nHere we use a notation where the precise meaning of $\\delta(x-y)$ and $\\int_x$ depends on the situation. For example when we consider a space with $3+1$ dimensions we have\n\\begin{equation}\n\\delta(x-y) = \\delta^{(4)}(x-y) = \\delta(x_0-y_0) \\delta^{(3)}(\\vec x-\\vec y)\n\\end{equation}\nand\n\\begin{equation}\n\\int_x = \\int dx_0 \\int d^3x.\n\\end{equation}\nIt should always be clear from the context what is meant. Eq.\\ \\eqref{eq:axiomfunctionalderivative} is the natural extension of the corresponding rule for vectors $x,y\\in \\mathbb{R}^n$\n\\begin{equation}\n\\frac{\\partial}{\\partial x_i} x_j = \\delta_{ij}\\quad \\text{or} \\quad \\frac{\\partial}{\\partial x_i}\\sum_j x_j y_j = y_i.\n\\end{equation}\nIn addition to Eq.\\ \\eqref{eq:axiomfunctionalderivative} the functional derivative should obey the usual derivative rules such as product rule, chain rule etc. Using the abstract index notation introduced before Eq.\\ \\eqref{eq3:fieldexpectationvalue} we write the axiom in Eq.\\ \\eqref{eq:axiomfunctionalderivative} as\n\\begin{equation}\n\\frac{\\delta}{\\delta f_\\alpha} f_\\beta = \\delta_{\\alpha\\beta} \\quad \\text{or}\\quad \\frac{\\delta}{\\delta f_\\alpha} \\sum_\\beta f_\\beta g_\\beta = g_\\alpha. \n\\end{equation}\n\nWith this formalism at hand we can now come back to our task of calculating expectation values and correlation functions. We introduce the source-dependent partition function by the definition\n\\begin{equation}\nZ[J]=\\int D\\tilde\\Phi \\, e^{-S[\\tilde \\Phi]+J_\\alpha \\tilde\\Phi_\\alpha}.\n\\label{eq3:sourcedeppartfunction}\n\\end{equation}\nExpectation values are obtained as functional derivatives\n\\begin{equation}\n\\Phi_\\alpha = \\langle\\tilde\\Phi_\\alpha\\rangle = \\frac{1}{Z} \\frac{\\delta}{\\delta J_\\alpha} Z[J],\n\\label{eq1:expectvalue}\n\\end{equation}\nand similarly correlation functions\n\\begin{equation}\nc_{\\alpha\\beta\\gamma\\dots} = \\frac{1}{Z} \\frac{\\delta}{\\delta J_\\alpha} \\frac{\\delta}{\\delta J_\\beta} \\frac{\\delta}{\\delta J_\\gamma}\\dots Z[J].\n\\end{equation}\nThe connected part of the correlation functions can be obtained more direct from the Schwinger functional\n\\begin{equation}\nW[J] = \\ln Z[J].\n\\label{eq3:Schwingerfunctaslog}\n\\end{equation}\nFor example the propagator $G$, the connected two-point function, is given by\n\\begin{equation}\nG_{\\alpha\\beta} = \\frac{\\delta}{\\delta J_\\alpha} \\frac{\\delta}{\\delta J_\\beta} W[J].\n\\end{equation}\nDue to these properties one calls $Z[J]$ ($W[J]$) the generating functional for the (connected) correlation functions. \n\n\\subsubsection{Microscopic actions in real time and analytic continuation}\n\nIn this subsection we discuss the relation between the real-time and the imaginary-time action as well as the analytic continuation in more detail. For concreteness we consider a nonrelativistic repulsive Bose gas in three-dimensional homogeneous space and in vacuum ($\\mu=0$). It is straightforward to transfer the discussion also to other cases. \n\nIn real time the microscopic action is given by\n\\begin{equation}\nS_t = -\\int dt \\int d^3 x \\left\\{\\varphi^*(-i\\partial_t-\\Delta-i\\epsilon)\\varphi+\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:microscopicactionrealtime}\n\\end{equation}\nThe overall minus sign is to match the standard convention. After Fourier transformation the term quadratic in $\\varphi$ that determines the propagator reads\n\\begin{equation}\nS_{t,2} = \\int \\frac{d \\omega}{2\\pi} \\int \\frac{d^3p}{(2\\pi)^3} \\, \\varphi^*\\left(\\omega-\\vec p^2+i\\epsilon\\right)\\varphi.\n\\label{eq3:quadraticpartrealtimeforier}\n\\end{equation}\nIn the basis with the complex fields $\\varphi$, $\\varphi^*$ the inverse microscopic propagator reads\n\\begin{equation}\nG(p)^{-1}\\,\\delta(p-p^\\prime) = \\frac{\\delta}{\\delta \\varphi^*(p)} \\frac{\\delta}{\\delta \\varphi(p^\\prime)} S_{t,2} = \\left(\\omega-\\vec p^2+i\\epsilon\\right)\\, \\delta (p-p^\\prime).\n\\label{eq3:propagatorrealtime}\n\\end{equation}\nFrom $\\det G^{-1}(p)=0$ we obtain for $\\epsilon\\to0$ the dispersion relation $\\omega=\\vec p^2$. \n\nFor the action in Eq.\\ \\eqref{eq3:microscopicactionrealtime} one can determine the field theoretic expectation values and correlation functions using the formalism described in the previous subsection with the complex weighting function\n\\begin{equation}\ne^{iS_t[\\varphi]}.\n\\label{eq3:weightingfactorrealtime}\n\\end{equation}\nIn Eqs. \\eqref{eq3:microscopicactionrealtime}, \\eqref{eq3:quadraticpartrealtimeforier} and \\eqref{eq3:propagatorrealtime} the small imaginary term $i\\epsilon$ is introduced to enforce the correct frequency integration contour (Feynman prescription). In Eq.\\ \\eqref{eq3:weightingfactorrealtime} it leads to a Gaussian suppression for large values of $\\varphi^*\\varphi$,\n\\begin{equation}\ne^{iS_t[\\varphi]} = e^{i \\text{Re} S_t[\\varphi]} e^{-\\epsilon \\int_x \\varphi^*\\varphi},\n\\end{equation}\nwhich makes the functional integral convergent. Let us now consider the analytic continuation to imaginary time\n\\begin{equation}\nt\\to e^{-i\\alpha}\\tau,\\quad 0\\leq \\alpha <\\pi\/2.\n\\end{equation}\nFor $\\alpha\\to \\pi\/2$ we have $t\\to - i \\tau$ and \n\\begin{equation}\n(-i\\frac{\\partial}{\\partial t}-i\\epsilon)\\to \\frac{\\partial}{\\partial \\tau}, \\quad \\int dt\\to -i\\int d\\tau.\n\\end{equation}\nThe weighting factor in Eq.\\ \\eqref{eq3:weightingfactorrealtime} becomes\n\\begin{equation}\ne^{-S[\\varphi]}\n\\end{equation}\nwith\n\\begin{equation}\nS[\\varphi] = \\int d\\tau \\int d^3 x \\left\\{\\varphi^*(\\partial_\\tau-\\Delta)\\varphi+\\frac{1}{2}\\lambda (\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:imaginarytimemicroscopicaction}\n\\end{equation}\n\n\n\\subsubsection{Matsubara formalism}\n\nIn statistical physics one is often interested in properties of the thermal (and chemical) equilibrium. For quantum field theories the thermal equilibrium is conveniently described using the Matsubara formalism. In this section we give a short account of the formalism and refer for a more detailed discussion to the literature \\cite{Mahan1981}. \n\nThe grand canonical partition function is defined as\n\\begin{equation}\nZ=\\text{Tr}\\, e^{-\\beta(H-\\mu N)}.\n\\label{eq3:grandcanparttrace}\n\\end{equation}\nHere we use $\\beta =\\frac{1}{T}$ and recall our units for temperature with $k_B=1$. The trace operation in Eq.\\ \\eqref{eq3:grandcanparttrace} sums over all possible states of the system, including varying particle number. The operator $H$ is the Hamiltonian and $N$ the particle number operator. The factor\n\\begin{equation}\ne^{-\\beta(H-\\mu N)}\n\\label{eq3:imaginarytimeevoloperator}\n\\end{equation}\nis quite similar to an unitary time evolution operator $e^{i\\Delta t H}$ evolving the system over some time interval $\\Delta t = t_2-t_1$. Indeed, we can define $\\tilde H = H-\\mu N$ and evolve the system from time $t_1=0$ to the imaginary time $t_2=-i\\beta$ with the operator in Eq.\\ \\eqref{eq3:imaginarytimeevoloperator}. If we take a (generalized) torus with circumference $\\beta$ in the imaginary time direction as our space-time manifold we can use the functional integral formulation of quantum field theory to write Eq.\\ \\eqref{eq3:grandcanparttrace} as\n\\begin{equation}\nZ = \\int D \\tilde\\varphi e^{-S[\\tilde\\varphi]}\n\\end{equation}\nwhere $S$ is an action with imaginary and periodic time. From the imaginary time action described in the last subsection it is obtained by replacing also the Hamiltonian $H$ by $H-\\mu N$. For our Bose gas example this results in\n\\begin{equation}\nS[\\varphi] = \\int_0^\\beta d\\tau \\int d^3 x \\left\\{\\varphi^*(\\partial_\\tau-\\Delta-\\mu)\\varphi+\\frac{1}{2}\\lambda (\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:matsubaraaction}\n\\end{equation}\nSince time is now periodic, the Fourier transform leads to discrete frequencies. The quadratic part of $S$ in Eq.\\ \\eqref{eq3:matsubaraaction} reads in momentum space\n\\begin{equation}\nS_2[\\varphi] = T\\sum_{n=-\\infty}^\\infty \\int \\frac{d^3 p}{(2\\pi)^3} \\varphi^*(i\\omega_n+\\vec p^2-\\mu)\\varphi\n\\label{eq3:matsubactionquadraticpart}\n\\end{equation}\nwith the Matsubara frequency $\\omega_n=2\\pi T n$. In the limit $T\\to 0$ the summation over Matsubara frequencies becomes again an integration\n\\begin{equation}\nT \\sum_n \\to \\int \\frac{d\\omega}{2\\pi}.\n\\end{equation}\nFor the Fourier decomposition in Eq.\\ \\eqref{eq3:matsubactionquadraticpart} we used the boundary condition\n\\begin{eqnarray}\n\\nonumber\n\\varphi(\\tau=\\beta, \\vec x) &=& \\varphi(\\tau=0,\\vec x),\\\\\n\\varphi^*(\\tau=\\beta, \\vec x) &=& \\varphi^*(\\tau=0,\\vec x),\n\\end{eqnarray}\nas appropriate for bosonic fields. For fermionic or Grassmann-valued fields $\\psi$ a careful analysis (see e.g. \\cite{WegnerGrassmannVariable}) leads to the boundary conditions\n\\begin{eqnarray}\n\\nonumber\n\\psi(\\tau=\\beta,\\vec x) &=& -\\psi (\\tau=0,x)\\\\\n\\psi^*(\\tau=\\beta,\\vec x) &=& -\\psi^*(\\tau=0,\\vec x).\n\\end{eqnarray}\nIn this case the Matsubara frequencies appearing in Eq.\\ \\eqref{eq3:matsubactionquadraticpart} are of the form\n\\begin{equation}\n\\omega_n = 2\\pi T \\left(n+\\frac{1}{2}\\right), \\quad n\\in \\mathbb{Z}.\n\\end{equation}\n\\subsubsection{Functional integral with probability measure}\nIn this section we reconsider the functional integral formulation of quantum field theory and formulate an representation with a (quasi-) probability distribution. Let us start with a simple Gaussian theory\n\\begin{equation}\nS= \\sum_{\\alpha,\\beta} \\varphi_\\alpha^* \\left(P_{\\alpha\\beta}+i \\epsilon \\delta_{\\alpha\\beta} \\right) \\varphi_\\beta.\n\\label{eq:Gauusianaction}\n\\end{equation}\nWe use here an abstract index notation where e.g. $\\alpha$ stands for both continuous variables such as position or momentum and internal degrees of freedom. In principle, the field $\\varphi$ may have both bosonic and fermionic components, the latter are Grassmann-valued. For simplicity we assume in the following that $\\varphi$ is a bosonic field. The formalism can be extended to cover also the case of fermions with minor modifications. \n\nWe included in Eq.\\ \\eqref{eq:Gauusianaction} a small imaginary part $\\sim i\\epsilon$ to make the functional integral convergent and to enforce the correct frequency integration contour (Feynman prescription). Although $\\epsilon$ is usually taken to be infinitesimal, we work with an arbitrary positive value here and take the limit $\\epsilon \\to 0_+$ only at a later point in our investigation. \nFor simplicity, we will often drop the abstract index and use a short notation with e.~g.\n\\begin{equation}\nS=\\varphi^* (P+ i \\epsilon ) \\varphi.\n\\end{equation}\nThe operator $P$ is the real part of the inverse microscopic propagator. As an example we consider a relativistic theory for a scalar field where $P$ reads in position space\n\\begin{equation}\nP(x,y)=\\delta^{(4)}(x-y)\\left(-g^{\\mu\\nu}\\frac{\\partial}{\\partial y^\\mu}\\frac{\\partial}{\\partial y^\\nu}-m^2\\right).\n\\end{equation}\nAnother example is the nonrelativistic case with \n\\begin{equation}\nP(x,y)=\\delta^{(4)}(x-y)\\left(i\\frac{\\partial}{\\partial y_0} + \\frac{1}{2M}\\vec \\nabla_y^2\\right).\n\\end{equation}\n\nFor a Gaussian theory the microscopic propagator coincides with the full propagator. The latter is obtained for general actions $S$ from\n\\begin{eqnarray}\n\\nonumber\ni G_{\\alpha\\beta} &=& \\langle \\varphi_\\alpha \\varphi^*_\\beta\\rangle_c\\\\\n&=& (-i)^3 \\frac{\\delta}{\\delta J^*_\\alpha}\\frac{\\delta}{\\delta J_\\beta} W[J]\n\\end{eqnarray}\nwith\n\\begin{equation}\ne^{-i W[J]} = Z[J] = \\int D \\varphi e^{i S[\\varphi]+i\\int \\{J^*\\varphi+\\varphi^* J\\}}.\n\\label{eq:Schwingerfunctional}\n\\end{equation}\nFor $\\alpha=(x_0,\\vec x)$ and $\\beta=(y_0,\\vec y)$ the object $G_{\\alpha\\beta}$ can be interpreted as the {\\itshape probability amplitude} for a particle to propagate from the point $\\vec y$ at time $y_0$ to the point $\\vec x$ at time $x_0$. More general, one might label by $\\alpha$ some single-particle state $|\\varphi_\\alpha\\rangle$ at time $t_\\alpha$ and with $\\beta$ the state $|\\varphi_\\beta\\rangle$ at time $t_\\beta$. The propagator $G_{\\alpha\\beta}$ describes then the probability amplitude for the transition between the two states.\nHowever, the description of an actual physical experiment involves the transition probability given by the modulus square of the probability amplitude (no summation over $\\alpha$ and $\\beta$)\n\\begin{equation}\nq_{\\alpha\\beta} = |G_{\\alpha\\beta}|^2= G^*_{\\alpha\\beta} G_{\\alpha\\beta}.\n\\end{equation}\nThis transition probability can also directly be obtained from functional derivatives of\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_1,J_2]= \\int D\\varphi_1 \\int D \\varphi_2 e^{i S[\\varphi_1]} e^{-i S^*[\\varphi_2]} \\\\\ne^{i \\int \\{J_1^* \\varphi_1+\\varphi_1^* J_1\\}} e^{-i \\int \\{J_2^* \\varphi_2+\\varphi_2^* J_2\\}}.\n\\label{eq:tildedpartfunction}\n\\end{eqnarray}\nWe note that Eq.\\ \\eqref{eq:tildedpartfunction} contains twice the functional integral over the field configuration $\\varphi$. One may also write this as a single functional integral over fields that depend in addition to the position variable $\\vec x$ on the contour time $t_c$ which is integrated along the Keldysh contour \\cite{Keldysh}. For our purpose it will be more convenient to work directly with the expression in Eq.\\ \\eqref{eq:tildedpartfunction}.\n\nFor $\\langle \\varphi \\rangle = \\langle \\varphi^* \\rangle = 0$ we can write\n\\begin{equation}\nq_{\\alpha\\beta} =\\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_1)_\\beta} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\frac{\\delta}{\\delta (J_2^*)_\\beta} \\tilde Z[J_1,J_2].\n\\end{equation}\nThis is immediately clear since $\\tilde Z[J_1,J_2] = Z[J_1] Z^*[J_2]$ and\n\\begin{eqnarray}\n\\nonumber\ni G_{\\alpha\\beta} &=& (-i)^2 \\frac{1}{Z[J_1]} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_1)_\\beta} Z[J_1]\\\\\n-i G^*_{\\alpha\\beta} &=& i^2 \\frac{1}{Z^*[J_2]} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\frac{\\delta}{\\delta (J_2^*)_\\beta} Z^*[J_2].\n\\end{eqnarray}\nfor\n\\begin{equation}\n\\langle \\varphi \\rangle = \\frac{-i}{Z} \\frac{\\delta}{\\delta J^*} Z[J] = \\langle \\varphi^* \\rangle = \\frac{-i}{Z} \\frac{\\delta}{\\delta J} Z[J]=0.\n\\end{equation}\nUsually one obtains $q_{\\alpha\\beta}$ by first calculating $G_{\\alpha\\beta}$ and then taking the modulus square thereof. The way we go here seems to be more complicated from a technical point of view, but has the advantage that it will allow for an intuitive physical interpretation. \n\nWe first concentrate on $\\tilde Z[J_1,J_2]$. This object plays a similar role as the partition function in statistical field theory. In some sense it is a sum over microscopic states weighted with some ``probability''. However, in contrast to statistical physics, the summation does not go over states of a system at some fixed time $t$ but over field configurations that depend both on the position variable $\\vec x$ and the time variable $t$. The summation seems to go over a even larger space since the functional integral appears twice\n\\begin{equation}\n\\int D\\varphi_1 \\int D \\varphi_2\n\\end{equation}\nso that the configuration space seems to be the tensor product of twice the space that contains the field configurations in space-time $\\varphi(\\vec x,t)$. In addition the ``probability weight''\n\\begin{equation}\ne^{i S[\\varphi_1]} e^{-i S^*[\\varphi_2]}\n\\end{equation}\nis not positive semi-definite and has even complex values in general. This last two features (``doubled'' configuration space and missing positivity) prevent us from interpreting quantum field theory in a similar way as statistical field theory.ting quantum field theory in a similar way as statistical field theory.\n\nAn idea to overcome these difficulties is to partially perform the functional integral in Eq.\\ \\eqref{eq:tildedpartfunction}. For this purpose we make a change of variables of the form\n\\begin{eqnarray}\n\\nonumber\n\\varphi_1 = \\frac{1}{\\sqrt{2}}\\phi + \\frac{1}{\\sqrt{2}}\\chi, &\\quad& J_1 = \\frac{1}{\\sqrt{2}}J_\\phi +\\frac{1}{\\sqrt{2}} J_\\chi, \\\\\n\\varphi_2 = \\frac{1}{\\sqrt{2}}\\phi - \\frac{1}{\\sqrt{2}}\\chi, &\\quad& J_2 = -\\frac{1}{\\sqrt{2}}J_\\phi + \\frac{1}{\\sqrt{2}} J_\\chi.\n\\label{eq:12phichitranslation}\n\\end{eqnarray}\nFor $\\tilde Z$ this gives then\n\\begin{equation}\n\\tilde Z = \\int D\\phi\\,\\, v[\\phi,J_\\chi]\\,\\,e^{i\\int \\{J_\\phi^* \\phi+\\phi^* J_\\phi\\}}\n\\label{eq:ZwithJ1J2}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nv[\\phi, J_\\chi] &=& \\int D \\chi\\, e^{iS[(\\phi+\\chi)\/\\sqrt{2}]}\\, e^{-i S^*[(\\phi-\\chi)\/\\sqrt{2}]}\\\\\n&& \\times\\, e^{i \\int \\{J_\\chi^* \\chi+\\chi^* J_\\chi\\}}.\n\\label{eq:vphiJchigeneralcase}\n\\end{eqnarray}\nWe note that $v[\\phi,J_\\chi]$ as a functional of $\\phi$ and $J_\\chi$ is real. This follows from comparison with the complex conjugate together with the change of variables $\\chi\\to-\\chi$. If it is also positive, we can interpret this object as a probability for the field configurations $\\phi(x)$. We call $v$ the {\\itshape functional probability} for the field configuration $\\phi$. \n\nBefore we discuss the general properties of $v[\\phi,J_\\chi]$ in more detail, we consider it explicitly for a Gaussian action $S[\\varphi]$ as in Eq.\\ \\eqref{eq:Gauusianaction}. In that case we can perform the functional integral\n\\begin{eqnarray}\n\\nonumber\nv[\\phi,J_\\chi] &=& \\int D \\chi e^{-\\epsilon \\{\\phi^*\\phi+\\chi^*\\chi\\}} e^{i\\{\\phi^* P \\chi +\\chi^* P \\phi\\}}\\\\\n\\nonumber\n&& \\times e^{i\\{J_\\chi^* \\chi + \\chi^* J_\\chi\\}}\\\\\n&=& e^{-\\epsilon \\phi^*\\phi}\\, e^{-\\frac{1}{\\epsilon}(J_\\chi^*+\\phi^* P)(J_\\chi+P\\phi)}.\n\\label{eq:vphijGaussian}\n\\end{eqnarray}\nThe last line holds up to a multiplicative factor that is irrelevant for us here. For $\\tilde Z$ we are left with\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_\\phi,J_\\chi] &=& \\int D\\phi \\,\\,v[\\phi, J_\\chi]\\, \\,e^{i\\int\\{J_\\phi^*\\phi+\\phi^*J_\\phi\\}}\\\\\n\\nonumber\n&=& \\int D \\phi \\, e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi} \\, e^{i\\{J_\\phi^* \\phi +\\phi^* J_\\phi\\}}\\\\\n&&\\times \\, e^{-\\frac{1}{\\epsilon}\\{J_\\chi^* P \\phi+\\phi^* P J_\\chi\\}}\\, e^{-\\frac{1}{\\epsilon}J_\\chi^*J_\\chi}.\n\\label{eq:partitionfunctionphiJphiJchi}\n\\end{eqnarray}\nFor $J_\\phi=J_\\chi=0$ the integrand in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is strictly positive. Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} can therefore be interpreted in a similar way as the partition function in statistical field theory. The probability measure is\n\\begin{equation}\nv=e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}.\n\\label{eq:probmeasuregaussian}\n\\end{equation}\n\nWe can distinguish three different classes of field configurations $\\phi$. In the simplest case the norm vanishes,\n\\begin{equation}\n|\\phi|^2=\\phi^*\\phi=\\sum_\\alpha \\phi^*_\\alpha\\phi_\\alpha\\to 0.\n\\end{equation} \nThe functional probability for this case is of order $1$. The second class contains field configurations where the norm is nonzero, $\\phi^*\\phi\\neq 0$, but where $\\phi$ satisfies the on-shell condition, i.~e. $\\phi^*P^2\\phi=0$. The functional probability for this case is of order $e^{-\\epsilon}$ (for $\\phi^*\\phi\\sim1$). Finally, in the third class the norm is nonzero and the field configuration is off-shell, i.~e.\n\\begin{equation}\n\\phi^*\\phi\\sim1,\\quad \\phi^*P^2\\phi \\sim 1.\n\\end{equation}\nThe functional probability in Eq.\\ \\eqref{eq:probmeasuregaussian} for this case is only of the order $e^{-1\/\\epsilon}$. This shows that off-shell configurations are strongly suppressed in the limit $\\epsilon\\to0$ compared to the trivial case $|\\phi|=0$ and the on-shell fields with\n\\begin{equation}\n\\phi^* P^2\\phi = 0.\n\\label{eq:onshellcond}\n\\end{equation}\nHowever, for $\\epsilon>0$ the probability for off-shell configurations is not strictly zero and they give also contributions to $\\tilde Z$. This is in contrast to classical statistics where only states that fulfill the equation of motion are included. (At nonzero temperature states with different energies are weighted according to a thermal distribution.) \n\nThere are more differences between the partition function in classical statistics and the quantum partition function in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi}. In classical statistics the averaging over some phase space is directly linked to time averaging by the ergodic hypothesis. Indeed, this hypothesis says that a mean value calculated by taking the average of some quantity over the accessible phase space is equal to the average of that quantity over a -- sufficiently long -- time interval. \nIn classical statistics time plays an outstanding role. The formalism breaks space-time symmetries such as Lorentz- or Galilean symmetry explicitly. For the case of quantum field theory this point is different. The theory in the vacuum (zero temperature and density, $T=n=0$) is symmetric under Lorentz symmetry (or Galilean symmetry in the nonrelativistic case). \n\nThe summation over possible field configurations in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is not related to time averaging. We directly interpret it in the following way. Every physical experiment or ``measurement-like situation'' corresponds to a different microscopic field configuration $\\phi$. This field configuration does not necessarily have to fulfill the on-shell condition Eq.\\ \\eqref{eq:onshellcond} but has a probability $v$ that strongly favors on-shell fields. There is, however, a subtlety in this interpretation. When the action $S$ is given as an integral over the $3+1$ dimensional spacetime $\\Omega$, then $v$ describes the probability for a field configuration $\\phi(x_0,\\vec x)$ on this manifold $\\Omega$. Since we experience only one universe with one configuration one might ask why we should take the sum over different configurations weighted with some probability. To answer that question it is important to realize that our information about the field configuration $\\phi(x_0,\\vec x)$ is limited. \n\nFirst we can investigate only limited regions in space-time (around our own ``world-line''). Regions that are too far away either in the spatial or the temporal sense are not accessible. However, in the framework of a local field theory, the experiments in some region of space-time depend on the other (not accessible) regions only via the boundary conditions. Second, and more important, we have only access to the field configuration in some ``momentum range''. No experiment has an arbitrary large resolution and can resolve infinitely small wavelength. Therefore the true microscopic field configuration is inevitably hidden from our observation. In a Gaussian or non-interacting theory this issue seems to be not so important since different momentum modes decouple from each other. In a theory with interactions this is different, however. Modes with different momenta $p^\\mu$ (and different values of $p^\\mu p_\\mu$) are coupled via the interaction. The microscopic regime does influence the macroscopic states. \n\nOur interpretation of Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is therefore that the functional integral sums over possible microscopic configurations $\\phi(x)$ with probability (up to a factor) given by\n\\begin{equation}\ne^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\n\\end{equation}\nfor the Gaussian theory considered above and more general (for $J_\\chi=0$) by\n\\begin{equation}\nv[\\phi]=v[\\phi,J_\\chi=0]=\\int D\\chi e^{iS[(\\phi+\\chi)\/\\sqrt{2}]}\\,e^{-i S^*[(\\phi-\\chi)\/\\sqrt{2}]}.\n\\end{equation}\nIn this general case, the functional probability for nonzero source terms is given by Eq.\\ \\eqref{eq:vphiJchigeneralcase}. As argued there, it is always real. This is made more explicit in the expression\n\\begin{eqnarray}\n\\nonumber\nv[\\phi,J_\\chi] &=& \\int D \\chi \\,\\text{cos}\\left(S_1[\\phi+\\chi]-S_1[\\phi-\\chi]\\right)\\\\\n&& \\times \\, \\text{exp}\\left(-S_2[\\phi+\\chi]-S_2[\\phi-\\chi]\\right)\n\\label{eq:vphiJchiexplrealform}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nS_1[\\varphi] &=& \\text{Re}\\, S[\\varphi\/\\sqrt{2}]+ \\frac{1}{\\sqrt{2}}\\left\\{J_\\chi^*\\varphi+\\varphi^*J_\\chi\\right\\}\n\\end{eqnarray}\nand\n\\begin{equation}\nS_2[\\varphi]=\\text{Im}\\, S[\\varphi\/\\sqrt{2}].\n\\end{equation}\nWe note that the functional integral in Eq.\\ \\eqref{eq:vphiJchiexplrealform} converges when $S_2[\\varphi]$ increases with $\\varphi^*\\varphi$ fast enough. For $S_2[\\varphi]\\sim \\varphi^*\\varphi$ as in our Gaussian example, the convergence is quite good. Although for arbitrary actions $S[\\varphi]$ the ``probability'' $v[\\phi,J_\\chi]$ does not have to be positive, this is expected to be the case for many choices of $S[\\varphi]$. \n\nWhen $v[\\phi,J_\\chi]$ is negative for some choices of $\\phi$, this indicates that different values for $\\phi$ do not directly correspond to independent physical configurations. One might come to positive definite probabilities when the space of possible fields is restricted to a physically subspace. However, $v[\\phi,J_\\chi]$ as defined above can in any case be seen an quasi-probability for $\\phi$. This is in some respect similar to Wigner's representation of density matrices \\cite{Wigner}. \n\nLet us make another comment on the case of non-Gaussian actions $S[\\varphi]$. When $S[\\varphi]$ contains terms of higher then quadratic order in the fields $\\varphi$ the form of the action is subject to renormalization group modifications. Usually the true microscopic action $S[\\varphi]$ is not known. Measurements have only access to the effective action $\\Gamma[\\varphi]$ which already includes the effect of quantum fluctuations. (Measurements at some momentum scale $k^2=|p_\\mu p^\\mu|$ might probe the average action or flowing action $\\Gamma_k[\\varphi]$ \\cite{Wetterich1993b}.) The microscopic action $S$ is connected to $\\Gamma$ by a renormalization group flow equation \\cite{Wetterich1993b}, however it is in most cases not possible to construct $S$ from the knowledge of $\\Gamma$. Typically many different microscopic actions $S$ lead to the same effective action $\\Gamma$. It may therefore often be possible that a microscopic action $S$ exists that is consistent with experiments and allows for a positive probability $v[\\phi,J_\\chi]$. \n\nFinally we comment on the general properties of $v[\\phi,J_\\chi]$. Since it is defined as a functional integral over a (local) complex action one expects that $v[\\phi,J_\\chi]$ is local to a similar degree as the the effective action $\\Gamma[\\varphi]$ or the Schwinger functional $W[J]$ defined in Eq.\\ \\eqref{eq:Schwingerfunctional}. For general non-Gaussian microscopic actions $S[\\varphi]$ the functional $v[\\phi,J_\\chi]$ may be quite complicated and not necessarily local in the sense that it can be written in the form\n\\begin{equation}\nv[\\phi,J_\\chi] = e^{-\\int_x{\\cal L}_v}\n\\end{equation}\nwhere ${\\cal L}_v$ is a local ``Lagrange density'' that depends only on $\\phi(x)$, $J_\\chi(x)$ and derivatives thereof at the space-time point $x$. \n\nSince $v[\\phi,J_\\chi]$ is similarly defined as the effective action $\\Gamma[\\varphi]$ or the Schwinger functional $W[J]$ we expect that it respects the same symmetries as the microscopic action $S[\\varphi]$ when no anomalies are present. For example, when $S[\\varphi]$ is invariant under some $U(1)$ symmetry transformation $\\varphi \\to e^{i\\alpha} \\varphi$, we expect that $v[\\phi,J_\\chi]$ has a corresponding symmetry under the transformation\n\\begin{equation}\n\\phi \\to e^{i\\alpha} \\phi, \\quad J_\\chi \\to e^{i\\alpha} J_\\chi.\n\\end{equation} \n\n\\subsubsection{Correlation functions from functional probabilities}\nIn this subsection we use the expression for $\\tilde Z$ in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} to derive functional integral representations of some correlation functions. In the following we denote by $\\langle \\cdot \\rangle$ the ``expectation value'' in the quantum field theoretic sense, e.~g. for an operator $A[\\varphi]$\n\\begin{equation}\n\\langle A[\\varphi] \\rangle = \\frac{1}{Z} \\int D \\varphi e^{i S[\\varphi]} \\,A[\\varphi].\n\\end{equation}\nIn contrast, we use $\\langle\\langle\\cdot\\rangle\\rangle$ to denote the expectation value with respect to the functional integral over $\\phi$, i.~e.\n\\begin{equation}\n\\langle\\langle A[\\phi]\\rangle\\rangle = \\frac{1}{\\tilde Z} \\int D \\phi \\,\\,e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi} \\,A[\\phi],\n\\label{eq:gaussiandistroperator}\n\\end{equation}\nor more general\n\\begin{equation}\n\\langle\\langle A[\\phi]\\rangle\\rangle = \\frac{1}{\\tilde Z} \\int D \\phi\\,\\, v[\\phi]\\, A[\\phi]. \n\\end{equation}\nFor the discussion of the correlation functions it is useful to express $\\tilde Z$ in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} again in terms of $J_1$ and $J_2$. Using Eq.\\ \\eqref{eq:12phichitranslation} we find\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_1,J_2] &=& \\int D\\phi \\,\\,e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&& \\times\\, e^{-\\frac{1}{\\sqrt{2}\\epsilon}\\left\\{J_1^*(P-i\\epsilon)\\phi+\\phi^*(P-i\\epsilon)J_1\\right\\}}\\\\\n\\nonumber\n&& \\times \\,e^{-\\frac{1}{\\sqrt{2}\\epsilon}\\left\\{J_2^*(P+i\\epsilon)\\phi+\\phi^*(P+i\\epsilon)J_2\\right\\}}\\\\\n&& \\times \\,e^{-\\frac{1}{2\\epsilon}\\left\\{J_1^*J_1+J_1^*J_2+J_2^*J_1+J_2^*J_2\\right\\}}.\n\\end{eqnarray}\nWe start with the modulus square of the quantum field theoretic one-point function (no summation convention used in the following)\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\tilde Z \\\\\n\\nonumber\n&=& \\frac{1}{\\tilde Z} \\int D\\phi \\,\\, e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&& \\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\beta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\beta}\\phi_\\beta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n\\nonumber\n&=& \\frac{1}{2\\epsilon} \\left[\\sum_{\\beta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\beta}\\langle\\langle\\phi_\\beta \\phi^*_\\gamma\\rangle\\rangle(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n&=& 0.\n\\label{eq:vanishingexpectationvalue}\n\\end{eqnarray}\nIn the last line of Eq.\\ \\eqref{eq:vanishingexpectationvalue} we used the standard property of the Gaussian distribution Eq.\\ \\eqref{eq:gaussiandistroperator}\n\\begin{equation}\n\\langle\\langle\\phi_\\beta\\phi^*_\\gamma\\rangle\\rangle=\\epsilon (P^2+\\epsilon^2)_{\\beta\\gamma}^{-1}.\n\\end{equation}\nNext we turn to the two-point function or ``transition probability''\n\\begin{eqnarray}\n\\nonumber\nq_{\\alpha\\beta} &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha}\\frac{\\delta}{\\delta (J_2)_\\alpha}\\frac{\\delta}{\\delta (J_1)_\\beta}\\frac{\\delta}{\\delta (J_2^*)_\\beta} \\tilde Z[J_1,J_2]\\\\\n\\nonumber\n&=& \\frac{1}{\\tilde Z}\\int D\\phi \\,\\, e^{\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&&\\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\eta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n\\nonumber\n&&\\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\kappa,\\lambda}\\frac{1}{\\epsilon}(P+i\\epsilon)_{\\beta\\kappa}\\phi_\\kappa\\phi^*_\\lambda(P-i\\epsilon)_{\\lambda\\beta}-\\delta_{\\beta\\beta}\\right]\\\\\n&=& \\left\\langle\\langle\\rho_\\alpha\\rho_\\beta\\right\\rangle\\rangle-\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle.\n\\label{eq:qxyconnexptvalue}\n\\end{eqnarray}\nThis expression is the (connected) two-point correlation function of the operator\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2\\epsilon^2}\\sum_{\\gamma,\\eta}(P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}\n\\end{equation}\nwith respect to averaging over the possible field configurations $\\phi(x)$. Note that $\\rho_\\alpha$ is real and positive for all field configurations $\\phi$. Indeed, we can write with $P^\\dagger =P$\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2\\epsilon^2} \\left|\\sum_\\eta (P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta\\right|^2\n\\end{equation}\nshowing this more explicit. The multiplicative normalization of $\\rho$ is somewhat arbitrary and could be changed by rescaling the fields according to $\\phi\\to\\phi^\\prime=c\\phi$. \nNote that for on-shell modes with $\\phi^* P^2\\phi=0$ the operator $\\rho$ reads\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2} \\phi^*_\\alpha\\phi_\\alpha.\n\\end{equation}\n\nAlthough the description of $q_{\\alpha\\beta}$ as a connected correlation function of the operators $\\rho_\\alpha$ and $\\rho_\\beta$ is appealing, its meaning as a transition probability is not yet completely clear. In a typical experiment one asks for the probability to find a particle both at the space-time point $y=(y_0,\\vec y)$ and at the space-time point $x=(x_0,\\vec x)$. We denote the probability for this by $p(x \\cap y)$. Quite generally, one would calculate this quantity as a sum over all field configurations $\\phi$ weighted by the product\n\\begin{equation}\np[\\phi]\\,p(x|\\phi] \\,p(y|\\phi].\n\\end{equation}\nHere $p(x|\\phi]$ gives the probability for the event ``particle measured at $x$'' under the condition that the field configuration $\\phi$ is realized. The expression $p[\\phi]$ is the probability for the field configuration $\\phi$. In combination, we find\n\\begin{eqnarray}\np(x \\cap y) &=& \\int D\\phi\\, p[\\phi]\\,p(x|\\phi] \\,p(y|\\phi].\n\\end{eqnarray}\nLet us now compare this to our expression for $q_{\\alpha\\beta}$ in Eq.\\ \\eqref{eq:qxyconnexptvalue}. If we identify $\\alpha=x=(x_0,\\vec x)$ and $\\beta=y=(y_0,\\vec y)$ and neglect for the moment the second term in the last line of Eq.\\ \\eqref{eq:qxyconnexptvalue}, we can write\n\\begin{equation}\nq(x,y)=\\int D\\phi\\, v[\\phi]\\, \\rho(x)\\, \\rho(y).\n\\end{equation}\nThe expressions for $p(x\\cap y)$ and $q(x,y)$ are proportional when the probability for the field configuration $\\phi$ is\n\\begin{equation}\np[\\phi] \\sim v[\\phi]\n\\end{equation}\nand the probability to find a particle at $x=(x_0,\\vec x)$ for the field configuration $\\phi$ is given by\n\\begin{equation}\np(x|\\phi] \\sim \\rho(x). \n\\end{equation}\nThe subtraction of the term $\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle$ in Eq.\\ \\eqref{eq:qxyconnexptvalue} provides for the two events ``particle measured at $y$'' and ``particle measured at $x$'' not to be in a coincidence. Instead, there has to be a ``causal connection'' between them. Only in that case would we speak of ``two measurements on the same particle''. Moreover, fluctuations at different space-time points that are uncorrelated would not show the characteristics of particles at all. Let us assume for definiteness that we use a cloud chamber as a particle detector. The vapor would only condense if neighboring points in space are stimulated during a small but nonzero period of time. Stimulations at random points in space-time would not lead to the detection of a particle. The disconnected part of the two point function $\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle$ should therefore be seen as part of the nontrivial vacuum structure in quantum field theory. \n\nTo end this subsection let us comment of the general, not necessary Gaussian case. We can obtain the quantum field theoretic one-point function from\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\tilde Z\\\\\n\\nonumber\n&=& \\frac{1}{2 \\tilde Z} \\left(\\frac{\\delta}{\\delta (J_\\phi^*)_\\alpha}+\\frac{\\delta}{\\delta (J_\\chi^*)_\\alpha}\\right)\\\\\n&&\\times \\left(-\\frac{\\delta}{\\delta (J_\\phi)_\\alpha}+\\frac{\\delta}{\\delta (J_\\chi)_\\alpha}\\right) \\tilde Z[J_\\phi,J_\\chi]. \n\\end{eqnarray}\nWith Eq.\\ \\eqref{eq:ZwithJ1J2} this gives\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{2\\tilde Z} \\int D\\phi\\\\\n&& \\times\\, \\left[\\phi^*_\\alpha\\phi_\\alpha+\\frac{\\delta}{\\delta (J_\\chi^*)_\\alpha}\\frac{\\delta}{\\delta (J_\\chi)_\\alpha}\\right] \\, v[\\phi,J_\\chi]. \n\\end{eqnarray}\nHere we used that $v[\\phi,J_\\chi]$ and $|\\langle \\phi_\\alpha\\rangle|^2$ have to be real. The general expression for the two-point function $q_{\\alpha\\beta}$ is somewhat more complicated, but straightforward to obtain in an analogous way as the calculations above. \n\n\\subsubsection{Conservation laws for on-shell excitations}\nAlthough particles are created and annihilated in quantum field theory, these processes are constraint by several conservation laws. For example, in quantum electrodynamics, the electric charge is a conserved quantum number. Electrons and positrons can only be created in pairs such that the total charge remains constant. In a formalism where particles are described as excitations of fields, one must show that these excitations fulfill the usual conservation constraints. \n\nIn quantum field theory, conserved quantities such as charge or also energy are associated to a continuous symmetry via Noether's theorem. However, only the combination of a symmetry together with some field equation leads to a conservation law. For example, for a field that satisfies the on-shell condition\n\\begin{equation}\nP\\phi = (-\\partial_\\mu\\partial^\\mu-m^2)\\phi=0\n\\end{equation}\none can easily show that the current\n\\begin{equation}\nj^\\mu=i(\\partial^\\mu\\phi^*)\\phi-i\\phi^*(\\partial^\\mu\\phi)\n\\label{eq:currentrelativistic}\n\\end{equation}\nis conserved, i.~e. $\\partial_\\mu j^\\mu=0$. This current is directly linked to the symmetry of the action\n\\begin{equation}\nS[\\varphi]=\\int_x \\varphi^*(-\\partial_\\mu\\partial^\\mu-m^2)\\varphi\n\\end{equation}\nunder global U(1) transformations $\\varphi\\to e^{i\\alpha}\\varphi$, $\\varphi^*\\to e^{-i\\alpha}\\varphi^*$. As discussed in the last section, the functional probability $v[\\phi]$ is invariant under the same symmetries as the microscopic action $S[\\varphi]$ if no anomalies are present. This implies that there should be conservation laws associated with these symmetries for on-shell excitations, that fulfill a field equation as Eq.\\ \\eqref{eq:onshellcond}. \nWe emphasize again that e.~g. the current in Eq.\\ \\eqref{eq:currentrelativistic} is not conserved for general field configurations with $P\\phi\\neq0$. However, if particles correspond to on-shell field excitations, the usual conservation laws are indeed fulfilled. \n\n\\subsubsection{Conclusions}\nIn this appendix we discussed a (quasi-) probability representation of quantum field theory based on the functional integral. We showed for a Gaussian theory of bosonic fields that the functional integral can be reordered such that an interpretation in terms of real and positive probabilities for field configurations (``functional probabilities'') is possible. Our formalism is also applicable to the more general case of non-Gaussian microscopic actions where it may be necessary to work also with negative (quasi-) probabilities. We believe that a description using only positive probabilities is possible in many cases. However, it is not excluded that for some physical theories negative probabilities are needed. This would be highly interesting and demonstrate -- once again -- the extraordinariness of quantum theory. In any case the (quasi-) probability representation developed here might be useful as a theoretical tool, for example in studies of non-equilibrium quantum field dynamics. The formalism can be extended with minor modifications to fermionic or Grassmann valued fields.\n\nThe concept of functional probabilities addresses both classical field configurations and particles. The former are described by a nonzero expectation value $\\langle\\langle\\phi\\rangle\\rangle$ while particles correspond to on-shell excitations, described by the connected two-point function $\\langle\\langle\\phi\\phi\\rangle\\rangle-\\langle\\langle\\phi\\rangle\\rangle\\langle\\langle\\phi\\rangle\\rangle$. For quadratic microscopic actions as in Eq.\\ \\eqref{eq:Gauusianaction} the functional probability is local (Eq.\\ \\eqref{eq:probmeasuregaussian}). This does no longer have to be the case once interactions are included. For example in a perturbation theory for weak interactions it should be possible to derive explicit expressions beyond the Gaussian case. Higher order correlation functions can then be studied which might shed more light on the question of locality. Interesting features of quantum mechanics as entanglement and the implications of Bells inequalities \\cite{Bell} can then be studied in this framework. \n\n\\section{Scale-dependent Bosonization}\n\nLet us consider a scale-dependent Schwinger functional for a theory formulated in terms of the field $\\tilde \\psi$\n\\begin{equation}\ne^{W_k[\\eta]}=\\int D\\tilde\\psi\\, e^{-S_\\psi[\\tilde\\psi]-\\frac{1}{2}\\tilde\\psi_\\alpha(R_k^\\psi)_{\\alpha\\beta}\\tilde\\psi_\\beta+\\eta_\\alpha\\tilde\\psi_\\alpha}.\n\\label{eq:scaledepSF}\n\\end{equation}\nAgain we use the abstract index notation where e.g. $\\alpha$ stands for both continuous variables such as position or momentum and internal degrees of freedom. We now multiply the right hand side of Eq.\\ \\eqref{eq:scaledepSF} by a term that is for $R_k^\\varphi=0$ only a field independent constant. It has the form of the functional integral over the field $\\tilde\\varphi$ with a Gaussian weighting factor\n\\begin{equation}\n\\int D \\tilde\\varphi \\, e^{-S_\\text{pb}-\\frac{1}{2}\\tilde \\varphi_\\epsilon(R_k^\\varphi)_{\\epsilon\\sigma}\\tilde\\varphi_\\sigma+j_\\epsilon\\tilde\\varphi_\\epsilon},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\nonumber\nS_\\text{pb} &=& \\frac{1}{2}\\left(\\tilde\\varphi_\\epsilon-\\chi_\\tau Q^{-1}_{\\tau\\epsilon}\\right)Q_{\\epsilon\\sigma}(\\tilde\\varphi_\\sigma-Q^{-1}_{\\sigma\\rho}\\chi_\\rho).\\\\\n&=& \\frac{1}{2}\\left(\\tilde\\varphi-\\chi Q^{-1}\\right)Q(\\tilde\\varphi-Q^{-1}\\chi),\n\\label{eq:Spb}\n\\end{eqnarray}\nand $\\chi$ depends on the ``fundamental field'' $\\tilde \\psi$. We will often suppress the abstract index as in the last line of Eq.\\ \\eqref{eq:Spb}. We assume that the field $\\tilde\\varphi$ and the operator $\\chi$ are bosonic. Without further loss of generality we can then also assume that $Q$ and $R_k^\\varphi$ are $k$-dependent symmetric matrices.\n\nAs an example, we consider an operator $\\chi$ which is quadratic in the original field $\\tilde\\psi$,\n\\begin{equation}\n\\chi_\\epsilon = H_{\\epsilon\\alpha\\beta}\\tilde\\psi_\\alpha\\tilde\\psi_\\beta.\n\\end{equation}\nThe Schwinger functional reads now\n\\begin{equation}\ne^{W_k[\\eta,j]} = \\int D\\tilde \\psi\\,D\\tilde\\varphi \\, e^{-S_k[\\tilde \\psi, \\tilde\\varphi]+\\eta\\tilde\\psi+j\\tilde\\varphi}\n\\label{eq:SFwithbosonfi}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nS_k[\\tilde\\psi,\\tilde\\varphi] &=& S_\\psi[\\tilde\\psi]+\\frac{1}{2}\\tilde\\psi R_k^\\psi\\tilde \\psi + \\frac{1}{2}\\tilde\\varphi (Q+R_k^\\varphi)\\tilde\\varphi\\\\\n&&+ \\frac{1}{2}\\chi Q^{-1}\\chi -\\tilde\\varphi \\chi. \n\\label{eq:actionferbos}\n\\end{eqnarray}\nIn the integration over $\\tilde\\varphi$, we can easily shift the variables to obtain\n\\begin{eqnarray}\n\\nonumber\ne^{W_k[\\eta,j]} &=& \\int D \\tilde\\psi \\, e^{-S_\\psi[\\tilde \\psi]-\\frac{1}{2}\\tilde\\psi R_k^\\psi\\tilde\\psi+\\eta \\tilde\\psi}\\\\\n\\nonumber\n&& \\times e^{\\frac{1}{2}(j+\\chi)(Q+R_k^\\varphi)^{-1}(j+\\chi)-\\frac{1}{2}\\chi Q^{-1}\\chi}\\\\\n&&\\times \\int D\\tilde\\varphi\\, e^{-\\frac{1}{2}\\tilde\\varphi(Q+R_k^\\varphi)\\tilde\\varphi}.\n\\label{eq:Schingerfas}\n\\end{eqnarray}\nThe remaining integral over $\\tilde\\varphi$ gives only a ($k$-dependent) constant. For $R_k^\\varphi=0$ and $j=0$ we note that $W_k[\\eta,j]$ coincides with $W_k[\\eta]$ in Eq.\\ \\eqref{eq:scaledepSF}.\n\nWe next derive identities for correlation functions of composite operators which follow from the equivalence of the equations\\eqref{eq:SFwithbosonfi} and \\eqref{eq:Schingerfas}. Taking the derivative with respect to $j$ we can calculate the expectation value for $\\tilde \\varphi$\n\\begin{eqnarray}\n\\nonumber\n\\varphi_\\epsilon &=& \\langle\\tilde\\varphi_\\epsilon\\rangle = \\frac{\\delta}{\\delta j_\\epsilon} W_k[\\eta,j]\\\\\n&=& (Q+R_k^\\varphi)^{-1}_{\\epsilon\\sigma}\\,\\left(j_\\sigma+H_{\\sigma\\alpha\\beta}\\langle\\tilde\\psi_\\alpha\\tilde\\psi_\\beta\\rangle\\right).\n\\label{eq:varphiintermsofpsi}\n\\end{eqnarray}\nThis can also be written as\n\\begin{equation}\n\\langle\\chi\\rangle = Q\\varphi - l\n\\label{eq:expvchi}\n\\end{equation}\nwith the modified source $l$\n\\begin{equation}\nl_\\epsilon = j_\\epsilon-(R_k^\\varphi)_{\\epsilon\\sigma}\\varphi_\\sigma.\n\\end{equation}\nFor the connected two-point function\n\\begin{equation}\n(\\delta_j\\delta_j W_k)_{\\epsilon\\sigma}= \\frac{\\delta^2}{\\delta j_\\epsilon \\delta j_\\sigma} W_k =\\langle\\tilde\\varphi_\\epsilon\\tilde\\varphi_\\sigma\\rangle_c\n\\end{equation}\nwe obtain from Eq.\\ \\eqref{eq:Schingerfas} \n\\begin{eqnarray}\n\\nonumber\n&&(Q+R_k)(\\delta_j\\delta_j W_k)(Q+R_k) \\\\\n\\nonumber\n&&= \\langle(j+\\chi)(j+\\chi)\\rangle -\\langle(j+\\chi)\\rangle\\langle(j+\\chi)\\rangle+(Q+R_k^\\varphi)\\\\\n&&= \\langle\\chi\\chi\\rangle-\\langle\\chi\\rangle\\langle\\chi\\rangle+(Q+R_k^\\varphi)\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n\\nonumber\n\\langle\\chi_\\epsilon\\chi_\\sigma\\rangle &=& \\left[(Q+R_k^\\varphi) (\\delta_j\\delta_j W_k)(Q+R_k^\\varphi)\\right]_{\\epsilon\\sigma} \\\\\n&&+ (Q\\varphi-l)_\\epsilon(Q\\varphi-l)_\\sigma -(Q+R_k^\\varphi)_{\\epsilon\\sigma}.\n\\label{eq:idk12}\n\\end{eqnarray}\nSimilarly, the derivative of Eq.\\ \\eqref{eq:expvchi} with respect to $j$ yields\n\\begin{eqnarray}\n\\langle\\tilde\\varphi_\\epsilon\\chi_\\sigma\\rangle = \\langle\\tilde\\varphi_\\epsilon\\tilde\\varphi_\\tau\\rangle (Q+R_k^\\varphi)_{\\tau\\sigma}-\\varphi_\\epsilon j_\\sigma -\\delta_{\\epsilon\\sigma}\\\\\n\\nonumber\n= \\varphi_\\epsilon (Q\\varphi)_\\sigma + \\left[(\\delta_j\\delta_j W_k)(Q+R_k^\\varphi)\\right]_{\\epsilon\\sigma}-\\varphi_\\epsilon l_\\sigma-\\delta_{\\epsilon\\sigma}.\n\\label{eq:idk13}\n\\end{eqnarray}\n\nWe now turn to the scale-dependence of $W_k[\\eta,j]$. In addition to $R_k^\\psi$ and $R_k^\\varphi$ also $Q$ and $H$ are $k$-dependent. For $H$ we assume\n\\begin{equation}\n\\partial_k H_{\\epsilon\\alpha\\beta} = (\\partial_k F_{\\epsilon\\rho}) H_{\\rho\\alpha\\beta}\n\\end{equation}\nwhere we take the dimensionless matrix $F$ to be symmetric for simplicity. For the operator $\\chi$ this gives\n\\begin{equation}\n\\partial_k \\chi_\\epsilon = \\partial_k H_{\\epsilon\\alpha\\beta}\\tilde\\psi_\\alpha\\tilde\\psi_\\beta = \\partial_k F_{\\epsilon\\rho} \\chi_\\rho.\n\\end{equation}\nFrom Eqs. \\eqref{eq:SFwithbosonfi} and \\eqref{eq:actionferbos} we can derive (for fixed $\\eta$, $j$)\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\langle\\tilde\\psi(\\partial_k R_k^\\psi)\\tilde\\psi\\rangle - \\frac{1}{2}\\langle\\tilde\\varphi(\\partial_k R_k^\\varphi+\\partial_k Q)\\tilde\\varphi\\rangle\\\\\n\\nonumber\n&&-\\frac{1}{2}\\langle\\chi\\left(\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_k F)Q^{-1}\\right)\\chi\\rangle\\\\\n&&+\\langle\\tilde\\varphi(\\partial_k F)\\chi\\rangle.\n\\end{eqnarray}\nNow we insert Eqs. \\eqref{eq:idk12} and \\eqref{eq:idk13}\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\psi (\\partial_k R_k^\\psi)\\psi - \\frac{1}{2}\\varphi (\\partial_k R_k^\\varphi) \\varphi\\\\\n\\nonumber\n&&-\\frac{1}{2} \\text{STr}\\,\\{ (\\delta_\\eta\\delta_\\eta W_k)(\\partial_k R_k^\\psi) \\}\\\\\n\\nonumber\n&&- \\frac{1}{2}\\text{Tr}{\\big \\{} (\\delta_j\\delta_j W_k)(\\partial_k R_k^\\varphi){\\big \\}}\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{Tr} {\\big \\{}{\\big [}Q(\\partial_k Q^{-1})R_k^\\varphi+R_k^\\varphi(\\partial_k Q^{-1})Q\\\\\n\\nonumber\n&&\\,\\,\\,+R_k^\\varphi(\\partial_k Q^{-1})R_k^\\varphi+ R_k^\\varphi Q^{-1}(\\partial_k F)(Q+R_k)\\\\\n\\nonumber\n&&\\,\\,\\,+(Q+R_k^\\varphi)(\\partial_kF)Q^{-1}R_k^\\varphi{\\big ]}(\\delta_j\\delta_j W_k){\\big \\}}\\\\\n\\nonumber\n&&+\\frac{1}{2}l\\left[(\\partial_k Q^{-1})Q+Q^{-1}(\\partial_k F)Q\\right]\\varphi\\\\\n\\nonumber\n&&+\\frac{1}{2}\\varphi \\left[Q(\\partial_k Q^{-1})+Q(\\partial_k F)Q^{-1}\\right]l\\\\\n\\nonumber\n&&-\\frac{1}{2}l \\left[\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_kF)Q^{-1}\\right]l\\\\\n\\nonumber\n&&+\\frac{1}{2} \\text{Tr} \\left\\{\\left[\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_k F)Q^{-1}\\right]R_k^\\varphi\\right\\}\\\\\n&&+\\frac{1}{2}\\text{Tr} \\left\\{Q\\partial_k Q^{-1}\\right\\}.\n\\label{eq:longflowW}\n\\end{eqnarray}\nThe supertrace $\\text{STr}$ contains the appropriate minus sign in the case that $\\psi_\\alpha$ are fermionic Grassmann variables.\n\nEquation \\eqref{eq:longflowW} can be simplified substantially when we restrict the $k$-dependence of $F$ and $Q$ such that\n\\begin{equation}\n\\partial_k F = -Q(\\partial_k Q^{-1}) = -(\\partial_k Q^{-1})Q.\n\\label{eq:restrSQ}\n\\end{equation}\nIn fact, one can show that the freedom to choose $F$ and $Q$ independent from each other that is lost by this restriction, is equivalent to the freedom to make a linear change in the source $j$, or at a later stage of the flow equation in the expectation value $\\varphi$. With the choice in Eq.\\ \\eqref{eq:restrSQ} we obtain\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\psi (\\partial_k R_k^\\psi)\\psi -\\frac{1}{2}\\varphi(\\partial_k R_k^\\varphi)\\varphi\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{STr}{\\big \\{}(\\partial_k R_k^\\psi)(\\delta_\\eta\\delta_\\eta W_k){\\big \\}}\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{Tr}{\\big \\{} \\left[\\partial_k R_k^\\varphi-R_k^\\varphi(\\partial_k Q^{-1})R_k^\\varphi\\right](\\delta_j\\delta_j W_k){\\big \\}}\\\\\n\\nonumber\n&&+\\frac{1}{2}l(\\partial_k Q^{-1})l+\\frac{1}{2}\\text{Tr}{\\{}\\partial_kQ^{-1}(Q-R_k^\\varphi){\\}}.\n\\label{eq:shortflowW}\n\\end{eqnarray}\nThe last term is independent of the sources $\\eta$ and $j$ and is therefore irrelevant for many purposes.\n\n\n\\section{Flowing action}\n\nThe average action or flowing action is defined by subtracting from the Legendre transform\n\\begin{equation}\n\\tilde\\Gamma_k[\\psi,\\varphi] = \\eta \\psi + j \\varphi - W_k[\\eta,j]\n\\end{equation}\nthe cutoff terms\n\\begin{equation}\n\\Gamma_k[\\psi,\\varphi]=\\tilde\\Gamma_k[\\psi,\\varphi]-\\frac{1}{2}\\psi R_k^\\psi\\psi -\\frac{1}{2}\\varphi R_k^\\varphi\\varphi.\n\\label{eq:defflowingaction}\n\\end{equation}\nAs usual, the arguments of the effective action are given by\n\\begin{equation}\n\\psi_\\alpha=\\frac{\\delta}{\\delta \\eta_\\alpha}W_k \\quad \\text{and} \\quad \\varphi_\\epsilon=\\frac{\\delta}{\\delta j_\\epsilon} W_k.\n\\end{equation}\nBy taking the derivative of Eq.\\ \\eqref{eq:defflowingaction} it follows\n\\begin{equation}\n\\frac{\\delta}{\\delta \\psi_\\alpha}\\Gamma_k = \\pm \\eta_\\alpha - (R_k^\\psi)_{\\alpha\\beta} \\psi_\\beta,\n\\end{equation}\nwhere the upper (lower) sign is for a bosonic (fermionic) field $\\psi$. Similarly,\n\\begin{equation}\n\\frac{\\delta}{\\delta \\varphi_\\epsilon}\\Gamma_k = j_\\epsilon - (R_k^\\varphi)_{\\epsilon\\sigma} \\varphi_\\sigma=l_\\epsilon.\n\\end{equation}\nIn the matrix notation\n\\begin{eqnarray}\n\\nonumber\nW_k^{(2)} &=& \\begin{pmatrix}\\delta_\\eta\\delta_\\eta W_k, && \\delta_\\eta\\delta_j W_k \\\\ \\delta_j\\delta_\\eta W_k, && \\delta_j\\delta_j W_k\\end{pmatrix},\\\\\n\\nonumber\n\\Gamma_k^{(2)} &=& \\begin{pmatrix}\\delta_\\psi\\delta_\\psi \\Gamma_k, && \\delta_\\psi\\delta_\\varphi \\Gamma_k \\\\\\delta_\\varphi\\delta_\\psi \\Gamma_k, && \\delta_\\varphi\\delta_\\varphi \\Gamma_k\\end{pmatrix},\\\\\nR_k&=&\\begin{pmatrix} R_k^\\psi, && 0 \\\\ 0,&& R_k^\\varphi \\end{pmatrix},\n\\end{eqnarray}\nit is straight forward to establish\n\\begin{equation}\nW_k^{(2)} \\,\\tilde\\Gamma_k^{(2)} =1,\\quad\\quad W_k^{(2)} = (\\Gamma_k^{(2)}+R_k)^{-1}.\n\\end{equation}\n\nIn order to derive the exact flow equation for the average action we use the identity\n\\begin{equation}\n\\partial_k \\tilde \\Gamma_k{\\big |}_{\\psi,\\varphi} = -\\partial_k W_k{\\big |}_{\\eta,j}.\n\\end{equation}\nThis yields the central result of this chapter\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\Gamma_k &=& \\frac{1}{2}\\text{STr}\\, \\left\\{(\\Gamma_k^{(2)}+R_k)^{-1}\\left(\\partial_k R_k-R_k(\\partial_k Q^{-1})R_k\\right)\\right\\}\\\\\n&&-\\frac{1}{2}\\Gamma_k^{(1)} \\left(\\partial_k Q^{-1}\\right)\\Gamma_k^{(1)}+\\gamma_k\n\\label{eq:flowequationGamma}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\gamma_k=-\\frac{1}{2} \\text{Tr}\\left\\{ (\\partial_k Q^{-1})(Q-R_k)\\right\\}.\n\\end{equation}\nAs it should be, this reduces to the standard flow equation for a framework with fixed partial bosonization in the limit $\\partial_k Q^{-1}=0$. The additional term is quadratic in the first derivative of $\\Gamma_k$ with respect to $\\varphi$ -- we recall that $\\partial_k Q^{-1}$ has non-zero entries only in the $\\varphi$-$\\varphi$ block. Furthermore there is a field independent term $\\gamma_k$ that can be neglected for many purposes. At this point a few remarks are in order.\n\n(i) For $k\\to 0$ the cutoffs $R_k^\\psi$, $R_k^\\varphi$ should vanish. This ensures that the correlation functions of the partially bosonized theory are simply related to the original correlation functions generated by $W_0[\\eta]$, Eq.\\ \\eqref{eq:scaledepSF}, namely\n\\begin{eqnarray}\n\\nonumber\nW_0[\\eta, j] &=& \\ln \\left(\\int D\\tilde \\psi \\, e^{-S_\\psi[\\tilde \\psi]+\\eta\\tilde\\psi+jQ^{-1}\\chi}\\right)+\\frac{1}{2}j\\, Q^{-1} \\,j+\\text{const.},\\\\\nW_0[\\eta,j=0] &=& W_0[\\eta] +\\text{const.}\n\\end{eqnarray}\nKnowledge of the dependence on $j$ permits the straightforward computation of correlation functions for composite operators $\\chi$.\n\\newline\n\n(ii) For solutions of the flow equation one needs a well known ``initial value'' which describes the microscopic physics. This can be achieved by letting the cutoffs $R_k^\\psi$, $R_k^\\varphi$ diverge for $k\\to\\Lambda$ (or $k\\to\\infty$). In this limit the functional integral in Eqs. \\eqref{eq:SFwithbosonfi}, \\eqref{eq:actionferbos} can be solved exactly and one finds\n\\begin{equation}\n\\Gamma_\\Lambda[\\psi,\\varphi] = S_\\psi[\\psi] +\\frac{1}{2}\\varphi Q_\\Lambda \\varphi +\\frac{1}{2}\\chi[\\psi]Q_\\Lambda^{-1} \\chi[\\psi]-\\varphi \\chi[\\psi].\n\\label{eq:averageactionmicrosc}\n\\end{equation}\nThis equals the ``classical action'' obtained from a Hubbard-Stratonovich transformation, with $\\chi$ expressed in terms of $\\psi$. \n\n(iii) In our derivation we did not use that $\\chi$ is quadratic in $\\psi$. We may therefore take for $\\chi$ an arbitrary bosonic functional of $\\psi$. It is straightforward to adapt our formalism such that also fermionic composite operators can be considered.\n\nThe flow equation \\eqref{eq:flowequationGamma} has a simple structure of a one loop expression with a cutoff insertion -- $\\text{STr}$ contains the appropriate integration over the loop momentum -- supplemented by a ``tree-contribution'' $\\sim \\left(\\Gamma_k^{(1)}\\right)^2$. Nevertheless, it is an exact equation, containing all orders of perturbation theory as well as non-perturbative effects. The simple form of the tree contributions will allow for easy implementations of a scale dependent partial bosonization. The details of this can be found in \\cite{FloerchingerWetterich2009exact}.\n\n\\section{General coordinate transformations}\n\nIt is sometimes useful to perform a change of coordinates in the space of fields during the renormalization flow. In this section we discuss the transformation behavior of the Wetterich equation \\eqref{eq4:Wettericheqn} under such a change of the basis for the fields. We follow here the calculation in \\cite{Wetterich1996, Gies:2001nw} For simplicity we restrict the discussion to bosonic fields. It is straightforward to transfer this to fermions as well \\cite{Wetterich1996, Gies:2001nw}. Similarly, one might also consider a general coordinate transformation for the generalized flow equation \\eqref{eq:flowequationGamma}. \n\nLet us consider a transformation of the form\n\\begin{equation}\n\\Phi \\to \\Psi[\\Phi].\n\\label{eq2:Fieldmap}\n\\end{equation}\nHere we denote by $\\Phi$ the original fields. The functional $\\Psi[\\Phi]$ is a $k$-dependent map of the old coordinates to the new ones. We assume that the map in Eq.\\ \\eqref{eq2:Fieldmap} is invertible and write the inverse\n\\begin{equation}\n\\Phi[\\Psi].\n\\end{equation}\nIn terms of the fields $\\Psi$ the definition of the flowing action reads\n\\begin{equation}\n\\Gamma_k[\\Psi] = J_\\alpha \\Phi_\\alpha [\\Psi]-W_k[J]-\\frac{1}{2} \\Phi[\\Psi] R_k \\Phi[\\Psi]\n\\end{equation}\nwhere $J$ is determined by\n\\begin{equation}\n\\Phi_\\alpha[\\Psi]=\\frac{\\delta W_k}{\\delta J_\\alpha}.\n\\end{equation}\nIn the limit $k\\to0$ the flowing action is a Legendre transform with respect to the old fields $\\Phi$ but not with respect to the new fields $\\Psi$. This implies for example that $\\Gamma[\\Psi]$ is not necessarily convex with respect to the fields $\\Psi$. In addition, only the fields $\\Phi$ are expectation values of fields as in Eq.\\ \\eqref{eq1:expectvalue}. The relation of the fields $\\Psi$ to the microscopic fields $\\tilde \\Phi$ is more complicated. The field equation reads in terms of the new fields\n\\begin{equation}\n\\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} = J_\\beta \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\alpha} - \\frac{\\delta}{\\delta \\Psi_\\alpha} \\Delta S_k.\n\\end{equation}\nNote that the cutoff term\n\\begin{equation}\n\\Delta S_k = \\frac{1}{2}\\Phi_\\alpha[\\Psi](R_k)_{\\alpha\\beta} \\Phi_\\beta[\\psi]\n\\end{equation}\nis not necessarily quadratic in the fields $\\Psi$. The matrix $R_k$ obtains from\n\\begin{eqnarray}\n\\nonumber\n(R_k)_{\\alpha\\beta} &=& \\frac{\\delta}{\\delta \\Phi_\\alpha}\\frac{\\delta }{\\delta \\Phi_\\beta} \\Delta S_k\\\\\n&=& \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\alpha} \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\mu}\\frac{\\delta }{\\delta \\Psi_\\nu} \\Delta S_k\\right) + \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu}\\Delta S_k\\right).\n\\label{eq3:defRk}\n\\end{eqnarray}\nSimilarly we obtain for the matrix $\\Gamma_k^{(2)}$\n\\begin{eqnarray}\n\\nonumber\n(\\Gamma_k^{(2)})_{\\alpha\\beta} &=& \\frac{\\delta}{\\delta \\Phi_\\alpha}\\frac{\\delta }{\\delta \\Phi_\\beta} \\Gamma_k\\\\\n&=& \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\alpha} \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\mu}\\frac{\\delta }{\\delta \\Psi_\\nu} \\Gamma_k\\right) + \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu}\\Gamma_k\\right).\n\\label{eq3:defGamma2}\n\\end{eqnarray}\nWe now come to the scale dependence of $\\Gamma_k$. It is given by\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi]=\\partial_k \\Gamma_k[\\Psi]{\\big |}_\\Phi - \\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\label{eq2:scaledepGammaphipsi}\n\\end{equation}\nFor the first term on the right hand side of Eq.\\ \\eqref{eq2:scaledepGammaphipsi} we can use the Wetterich equation\\eqref{eq4:Wettericheqn} and obtain\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi] = \\frac{1}{2} \\text{Tr} (\\Gamma_k^{(2)}+R_k)^{-1} \\partial_k R_k-\\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\label{eq3:Wettericheqwithcoordtransf}\n\\end{equation}\nWe emphasize that $\\Gamma_k^{(2)}$ and $R_k$ are now somewhat more complicated objects then usually. They are defined by Eqs. \\eqref{eq3:defRk} and \\eqref{eq3:defGamma2}. One might also define the transformed matrices\n\\begin{eqnarray}\n\\nonumber\n(\\widehat \\Gamma_k^{(2)})_{\\mu\\nu} &=& \\frac{\\delta}{\\delta \\Psi_\\mu} \\frac{\\delta}{\\delta \\Psi_\\nu} \\Gamma_k + \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu} \\Gamma_k\\right),\\\\\n(\\widehat R_k)_{\\mu\\nu} &=& \\frac{\\delta}{\\delta \\Psi_\\mu} \\frac{\\delta}{\\delta \\Psi_\\nu} \\Delta S_k + \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu} \\Delta S_k\\right),\n\\label{eq3:hatteddef}\n\\end{eqnarray}\nand similar\n\\begin{eqnarray}\n(\\widehat{\\partial_k R_k})_{\\mu\\nu} &=& \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} (\\partial_k R_k)_{\\alpha\\beta}.\n\\label{eq3:hatted2}\n\\end{eqnarray}\nIn Eq.\\ \\eqref{eq3:hatteddef} the second functional derivatives are supplemented by connection terms as appropriate for general (non-linear) coordinate systems. With Eqs.\\ \\eqref{eq3:hatteddef} \\eqref{eq3:hatted2} the flow equation for $\\Gamma_k$ reads\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi] = \\frac{1}{2} \\text{Tr} (\\widehat \\Gamma_k+\\widehat R_k)^{-1} \\widehat{\\partial_k R_k} - \\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\end{equation}\nUnfortunately this equation has lost its one-loop structure due to the connection terms in Eq.\\ \\eqref{eq3:hatteddef}. An important exception is a linear coordinate transformation\n\\begin{equation}\n\\Psi_\\alpha[\\Phi] = \\Xi_\\alpha + M_{\\alpha\\beta} \\Phi_\\beta.\n\\end{equation}\nIn that case the terms $\\frac{\\delta^2 \\Psi}{\\delta \\Phi\\delta \\Phi}$ vanish and the one-loop structure is preserved.\n\n\n\\subsection{Particle-hole fluctuations}\n\\label{sec:ParticleHole}\nThe BCS theory of superfluidity in a Fermi gas of atoms is valid for a small attractive interaction between the fermions \\cite{PhysRev.104.1189, Bardeen:1957kj, Bardeen:1957mv}. In a renormalization group setting, the features of BCS theory can be described in a purely fermionic language. The only scale dependent object is the fermion interaction vertex $\\lambda_\\psi$. The flow depends on the temperature and the chemical potential. \nFor positive chemical potential ($\\mu>0$) and small temperatures $T$, the appearance of pairing is indicated by the divergence of $\\lambda_\\psi$.\n\nIn general, the interaction vertex is momentum dependent and represented by a term\n\\begin{eqnarray}\n\\Gamma_{\\lambda_\\psi}&=&\\int_{p_1,p_2,p_1^\\prime,p_2^\\prime}\\lambda_{\\psi}(p_1^\\prime,p_1,p_2^\\prime,p_2)\\nonumber\\\\\n& &\\times\\psi_1^{\\ast}(p_1^\\prime)\\psi_1(p_1)\\psi_2^{\\ast}(p_2^\\prime)\\psi_2(p_2)\n\\label{eq:momentumdepvertex}\n\\end{eqnarray}\nin the effective action. In a homogeneous situation, momentum conservation restricts the expression in Eq.\\ \\eqref{eq:momentumdepvertex} to three independent momenta, $\\lambda_{\\psi}\\sim \\delta(p_1^\\prime+p_2^\\prime-p_1-p_2)$. The flow of $\\lambda_\\psi$ has two contributions which are depicted in Fig. \\ref{fig:lambdaflow}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{PHfigure1.eps}\n\\caption{Running of the momentum dependent vertex $\\lambda_{\\psi}$. Here $\\tilde{\\partial}_k$ indicates derivatives with respect to the cutoff terms in the propagators and does not act on the vertices in the depicted diagrams. We will refer to the first loop as the particle-particle loop (pp-loop) and to the second one as the particle-hole loop (ph-loop).}\n\\label{fig:lambdaflow}\n\\end{figure}\nThe first diagram describes particle-particle fluctuations. For $\\mu>0$ its effect increases as the temperature $T$ is lowered. For small temperatures $T\\leq T_{c,\\text{BCS}}$ the logarithmic divergence leads to the appearance of pairing, as $\\lambda_\\psi\\to \\infty$. \n\nIn the purely fermionic formulation the flow equation for $\\lambda_{\\psi}$ has the general form \\cite{Ellwanger1994137, Aoki2000, PTPS.160.58, PTP.105.1}\n\n\\begin{equation}\\label{eq:lambdapsi2}\n\\partial_k \\lambda_{\\psi}^{\\alpha}=A^{\\alpha}_{\\beta\\gamma}\\lambda_{\\psi}^{\\beta}\\lambda_{\\psi}^{\\gamma}\\,, \n\\end{equation}\nwith $\\alpha, \\beta, \\gamma$ denoting momentum as well as spin labels. A numerical solution of this equation is rather involved due to the rich momentum structure. The case of the attractive Hubbard model in two dimensions, which is close to our problem, has recently been discussed in \\cite{strack:014522}.\nThe BCS approach concentrates on the pointlike coupling, evaluated by setting all momenta to zero. For $k \\rightarrow 0,\\ \\mu_0 \\rightarrow 0,\\ T \\rightarrow 0$ and $n \\rightarrow 0$ this coupling is related the scattering length, $a= \\frac{1}{8\\pi} \\lambda_{\\psi}(p_i=0)$. In the BCS approximation only the first diagram in Fig. \\ref{fig:lambdaflow} is kept, and the momentum dependence of the couplings on the right-hand side of Eq.\\ \\eqref{eq:lambdapsi2} is neglected, by replacing $\\lambda_{\\psi}^{\\alpha}$ by the pointlike coupling evaluated at zero momentum. In terms of the scattering length $a$, Fermi momentum $k_F$ and Fermi temperature $T_F$, the critical temperature is found to be \n\\begin{equation}\n \\frac{T_c}{T_F}\\approx 0.61 e^{\\pi\/(2 a k_F)}\\,.\n\\end{equation}\nThis is the result of the original BCS theory. However, it is obtained by entirely neglecting the second loop in Fig. \\ref{fig:lambdaflow}, which describes particle-hole fluctuations. At zero temperature the expression for this second diagram vanishes if it is evaluated for vanishing external momenta. Indeed, the two poles of the frequency integration are always either in the upper or lower half of the complex plane and the contour of the frequency integration can be closed in the half plane without poles. \n\nThe dominant part of the scattering in a fermion gas occurs, however, for momenta on the Fermi surface rather than for zero momentum. For non-zero momenta of the \"external particles\" the second diagram in Fig. \\ref{fig:lambdaflow} - the particle-hole channel - makes an important contribution. \n\nSetting the external frequencies to zero, we find that the inverse propagators in the particle-hole loop are \n\\begin{equation}\\label{eq:loopmom1}\nP_\\psi(q)=i q_0 +(\\vec{q}-\\vec{p}_1)^2-\\mu\\,,\n\\end{equation}\nand \n\\begin{equation}\\label{eq:loopmom2}\nP_\\psi(q)=i q_0 +(\\vec{q}-\\vec{p}_2^{\\,\\prime})^2-\\mu.\n\\end{equation}\nDepending on the value of the momenta $\\vec{p}_1$ and $\\vec p_2^{\\,\\prime}$, there are now values of the loop momentum $\\vec q$ for which the poles of the frequency integration are in different half planes so that there is a nonzero contribution even for $T=0$.\n\nTo include the effect of particle-hole fluctuations one could try to take the full momentum dependence of the vertex $\\lambda_\\psi$ into account. However, this leads to complicated expressions which are hard to solve even numerically. \nOne therefore often restricts the flow to the running of a single coupling $\\lambda_\\psi$ by choosing an appropriate projection prescription to determine the flow equation. In the purely fermionic description with a single running coupling $\\lambda_\\psi$, this flow equation has a simple structure. The solution for $\\lambda_{\\psi}^{-1}$ can be written as a contribution from the particle-particle (first diagram in Fig. \\ref{fig:lambdaflow}, pp-loop) and the particle-hole (second diagram, ph-loop) channels \n\\begin{equation}\\label{PHComp}\n \\frac{1}{\\lambda_{\\psi}(k=0)}=\\frac{1}{\\lambda_{\\psi}(k=\\Lambda)} + \\mbox{pp-loop} + \\mbox{ph-loop}\\,.\n\\end{equation}\nSince the ph-loop depends only weakly on the temperature, one can evaluate it at $T=0$ and add it to the initial value $\\lambda_\\psi(k=\\Lambda)^{-1}$. Since $T_c$ depends exponentially on the \"effective microscopic coupling\"\n\\begin{equation}\n \\left(\\lambda_{\\psi,\\Lambda}^{\\text{eff}}\\right)^{-1}=\\lambda_{\\psi}(k=\\Lambda)^{-1} + \\text{ph-loop}\\,,\n\\end{equation}\nany shift in $\\left(\\lambda_{\\psi,\\Lambda}^{\\text{eff}}\\right)^{-1}$ results in a multiplicative factor for $T_c$. The numerical value of the ph-loop and therefore of the correction factor for $T_c\/T_F$ depends on the precise projection description.\n\nLet us now choose the appropriate momentum configuration. For the formation of Cooper pairs, the relevant momenta lie on the Fermi surface, \n\n\\begin{equation}\\label{absmom}\n\\vec{p}^2_1=\\vec{p}^2_2=\\vec{p}^{{\\,\\prime}2}_1=\\vec{p}^{{\\,\\prime}2}_2=\\mu\\,,\n\\end{equation}\nand point in opposite directions\n\n\\begin{equation}\\label{oppmom}\n \\vec{p}_1=-\\vec{p}_2,\\ \\vec{p}^{\\,\\prime}_1=-\\vec{p}^{\\,\\prime}_2\\,.\n\\end{equation}\nThis still leaves the angle between $\\vec{p}_1$ and $\\vec{p}^{\\,\\prime}_1$ unspecified. Gorkov's approximation uses Eqs. \\eqref{absmom} and \\eqref{oppmom} and projects on the $s$-wave by averaging over the angle between $\\vec{p}_1$ and $\\vec{p}^{\\,\\prime}_1$. One can shift the loop momentum such that the internal propagators depend on $\\vec{q}^2$ and $(\\vec{q}+\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2$. In terms of spherical coordinates the first propagator depends only on the magnitude of the loop momentum $q^2=\\vec{q}^2$, while the second depends additionally on the transfer momentum $\\tilde{p}^2=\\frac{1}{4}(\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2$ and the angle $\\alpha$ between $\\vec{q}$ and $(\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)$,\n\n\\begin{equation}\n (\\vec{q}+\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2=q^2+4\\tilde{p}^2+4\\,q\\, \\tilde{p}\\,\\text{cos}(\\alpha)\\,.\n\\end{equation}\nPerforming the loop integration involves the integration over $q^2$ and the angle $\\alpha$. The averaging over the angle between $\\vec{p}_1$ and $\\vec{p}_1^{\\,\\prime}$ translates to an averaging over $\\tilde{p}^2$. Both can be done analytically \\cite{Heiselberg} for the fermionic particle-hole diagram and the result gives the well-known Gorkov correction to BCS theory, resulting in\n\n\\begin{equation}\nT_c=\\frac{1}{(4e)^{1\/3}}T_{c,\\text{BCS}}\\approx \\frac{1}{2.2} T_{c,\\text{BCS}}\\,.\n\\end{equation}\n\nIn our treatment we will use a numerically simpler projection by choosing $\\vec{p}^{\\,\\prime}_1=\\vec{p}_1$, and $\\vec{p}_2=\\vec{p}^{\\,\\prime}_2$, without an averaging over the angle between $\\vec{p}^{\\,\\prime}_1$ and $\\vec{p}_1$. The size of $\\tilde p^2 = \\vec{p}^2_1$ is chosen such that the one-loop result reproduces exactly the result of the Gorkov correction, namely $\\tilde p = 0.7326 \\sqrt{\\mu}$. Choosing different values of $\\tilde p$ demonstrates the dependence of $T_c$ on the projection procedure and may be taken as an estimate for the error that arises from the limitation to one single coupling $\\lambda_{\\psi}$ instead of a momentum dependent function.\n\n\\subsection{Bosonization}\nIn Sec. \\ref{sec:BCS-BECCrossover} we describe an effective four-fermion interaction by the exchange of a boson. In this picture the phase transition to the superfluid phase is indicated by the vanishing of the bosonic ``mass term'' $m^2 = 0$. Negative $m^2$ leads to the spontaneous breaking of U(1)-symmetry, since the minimum of the effective potential occurs for a nonvanishing superfluid density $\\rho_0>0$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{PHfigure2.eps}\n\\caption{Flow of the boson propagator.}\n\\label{fig:bosonexchangeloop}\n\\end{figure}\nFor $m^2 \\geq 0$ we can solve the field equation for the boson $\\varphi$ as a functional of $\\psi$ and insert the solution into the effective action. This leads to an effective four-fermion vertex describing the scattering $\\psi_1(p_1)\\psi_2(p_2)\\to \\psi_1(p_1^{\\,\\prime})\\psi_2(p_2^{\\,\\prime})$\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=\\frac{-h^2}{i(p_1+p_2)_0+\\frac{1}{2}(\\vec p_1+\\vec p_2)^2+m^2}.\n\\label{eq:lambdapsieff}\n\\end{equation}\nTo investigate the breaking of U(1) symmetry and the onset of superfluidity, we first consider the flow of the bosonic propagator, which is mainly driven by the fermionic loop diagram. For the effective four-fermion interaction this accounts for the particle-particle loop (see r.h.s. of Fig. \\ref{fig:bosonexchangeloop}). In the BCS limit of a large microscopic $m_\\Lambda^2$ the running of $m^2$ for $k\\to0$ reproduces the BCS result \\cite{PhysRev.104.1189, Bardeen:1957kj, Bardeen:1957mv}.\n\nThe particle-hole fluctuations are not accounted for by the renormalization of the boson propagator. Indeed, we have neglected so far that a term\n\n\\begin{equation}\\label{eq:fourfermionvertex}\n \\int_{\\tau,\\vec{x}}\\lambda_{\\psi}\\psi_1^{\\ast}\\psi_1\\psi_2^{\\ast}\\psi_2\\,,\n\\end{equation}\nin the effective action is generated by the flow. This holds even if the microscopic pointlike interaction is absorbed by a Hubbard-Stratonovich transformation into an effective boson exchange such that $\\lambda_\\psi(\\Lambda)=0$. The strength of the total interaction between fermions\n\n\\begin{equation}\\label{eq:lambdapsieff2}\n\\lambda_{\\psi,\\text{eff}}=\\frac{-h^2}{i(p_1+p_2)_0+\\frac{1}{2}(\\vec p_1+\\vec p_2)^2+m^2} + \\lambda_{\\psi}\n\\end{equation}\nadds $\\lambda_\\psi$ to the piece generated by boson exchange. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{PHfigure3.eps}\n\\caption{Box diagram for the flow of the four-fermion interaction.}\n\\label{fig:boxes}\n\\end{figure}\nIn the partially bosonized formulation, the flow of $\\lambda_\\psi$ is generated by the box-diagrams depicted in Fig. \\ref{fig:boxes}. We may interpret these diagrams and establish a direct connection to the particle-hole diagrams depicted in Fig. \\ref{fig:lambdaflow} on the BCS side of the crossover and in the microscopic regime. There the boson gap $m^2$ is large. In this case, the effective fermion interaction in Eq.\\ \\eqref{eq:lambdapsieff2} becomes momentum independent. Diagrammatically, this is represented by contracting the bosonic propagator. One can see, that the box-diagram in Fig. \\ref{fig:boxes} is then equivalent to the particle-hole loop investigated in Sec. \\ref{sec:ParticleHole} with the pointlike approximation $\\lambda_{\\psi,\\text{eff}}\\to-\\frac{h^2}{m^2}$ for the fermion interaction vertex. As mentioned above, these contributions vanish for $T=0$, $\\mu<0$ for arbitrary $\\vec p_i$. Indeed, at zero temperature, the summation over the Matsubara frequencies becomes an integral. All the poles of this integration are in the upper half of the complex plane and the integration contour can be closed in the lower half plane. We will evaluate $\\partial_k \\lambda_\\psi$ for $\\vec p_1=\\vec p_1^{\\,\\prime}=-\\vec p_2=-\\vec p_2^{\\,\\prime}$, $|\\vec p_1|=\\tilde p = 0.7326 \\sqrt{\\mu}$, as discussed in the Sec. \\ref{sec:ParticleHole}. For $\\mu>0$ this yields a nonvanishing flow even for $T=0$.\n\nAnother simplification concerns the temperature dependence. While the contribution of particle-particle diagrams becomes very large for small temperatures, this is not the case for particle-hole diagrams. For nonvanishing density and small temperatures, the large effect of particle-particle fluctuations leads to the spontaneous breaking of the U(1) symmetry and the associated superfluidity. In contrast, the particle-hole fluctuations lead only to quantitative corrections and depend only weakly on temperature. This can be checked explicitly in the pointlike approximation, and holds not only in the BCS regime where $T\/\\mu \\ll 1$, but also for moderate $T\/\\mu$ as realized at the critical temperature in the unitary regime. We can therefore evaluate the box-diagrams in Fig. \\ref{fig:lambdaflow} for zero temperature. We note that an implicit temperature dependence, resulting from the couplings parameterizing the boson propagator, is taken into account.\n\nAfter these preliminaries, we can now incorporate the effect of particle-hole fluctuations in the renormalization group flow. A first idea might be to include the additional term \\eqref{eq:fourfermionvertex} in the truncation and to study the effects of $\\lambda_{\\psi}$ on the remaining flow equations. On the initial or microscopic scale one would have $\\lambda_{\\psi}=0$, but it would then be generated by the flow. This procedure, however, has several shortcomings. First, the appearance of a local condensate would now be indicated by the divergence of the effective four-fermion interaction\n\n\\begin{equation}\n \\lambda_{\\psi,\\text{eff}}=-\\frac{h^2}{m^2}+\\lambda_{\\psi}\\,.\n\\end{equation}\nThis might lead to numerical instabilities for large or diverging $\\lambda_{\\psi}$. The simple picture that the divergence of $\\lambda_{\\psi,\\text{eff}}$ is connected to the onset of a nonvanishing expectation value for the bosonic field $\\varphi_0$, at least on intermediate scales, would not hold anymore. Furthermore, the dependence of the box-diagrams on the center of mass momentum would be neglected completely by this procedure. Close to the resonance the momentum dependence of the effective four-fermion interaction in the bosonized language as in Eq.\\ (\\ref{eq:lambdapsieff2}) is crucial, and this might also be the case for the particle-hole contribution.\n\nAnother, much more elegant way to incorporate the effect of particle-hole fluctuations is provided by the method of bosonization \\cite{Gies:2001nw, Gies:2002kd, Pawlowski2007a}, see also chapter \\ref{ch:Generalizedflowequation}. For this purpose, we use scale dependent fields in the average action. The scale dependence of $\\Gamma_k[\\chi_k]$ is modified by a term reflecting the $k$-dependence of the argument $\\chi_k$ \\cite{Gies:2001nw, Gies:2002kd}\n\n\\begin{equation}\\label{eq:scalefield}\n \\partial_k \\Gamma_k[\\chi_k]=\\int\\frac{\\delta \\Gamma_k}{\\delta \\chi_k}\\partial_k\\chi_k+\\frac{1}{2}\\mathrm{STr}\\left[ \\left(\\Gamma_k^{(2)}+R_k \\right)^{-1} \\partial_k R_k\\right] \\,.\n\\end{equation}\n\nFor our purpose it is sufficient to work with scale dependent bosonic fields $\\bar\\varphi$ and keep the fermionic field $\\psi$ scale independent. In practice, we employ bosonic fields $\\bar\\varphi_k^*$, and $\\bar\\varphi_k$ with an explicit \nscale dependence which reads in momentum space\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\bar \\varphi_k(q) & = & (\\psi_1\\psi_2)(q) \\partial_k \\upsilon\\,,\\\\\n\\partial_k \\bar \\varphi_k^*(q) & = & (\\psi_2^\\dagger\\psi_1^\\dagger)(q) \\partial_k \\upsilon.\n\\label{eq:scaledependenceoffields}\n\\end{eqnarray}\nIn consequence, the flow equations in the symmetric regime get modified\n\n\\begin{eqnarray}\n \\partial_k \\bar{h} &=& \\partial_k \\bar{h}{\\big |}_{\\bar \\varphi_k}-\\bar{P}_{\\varphi}(q)\\partial_k \\upsilon\\,,\\\\\n \\partial_k \\lambda_{\\psi} &=& \\partial_k \\lambda_{\\psi}{\\big |}_{\\bar \\varphi_k}-2\\bar{h}\\partial_k \\upsilon.\n\\label{eq:modifiedflowequations}\n\\end{eqnarray}\nHere $q$ is the center of mass momentum of the scattering fermions. In the notation of Eq.\\ \\eqref{eq:lambdapsieff} we have $q=p_1+p_2$ and we will take $\\vec q=0$, and $q_0=0$. The first term on the right hand side in Eq.\\ \\eqref{eq:modifiedflowequations} gives the contribution of the flow equation which is valid for fixed field $\\bar \\varphi_k$. The second term comes from the explicit scale dependence of $\\bar \\varphi_k$. The inverse propagator of the complex boson field $\\bar{\\varphi}$ is denoted by $\\bar{P}_{\\varphi}(q)=\\bar A_\\varphi P_\\varphi(q)=\\bar A_\\varphi (m^2+i Z_\\varphi q_0+\\vec q^2\/2)$, cf. Eq.\\ \\eqref{eq:Bosonpropagator}. \n\nWe can choose $\\partial_k \\upsilon$ such that the flow of the coupling $\\lambda_{\\psi}$ vanishes, i.e. that we have $\\lambda_{\\psi}=0$ on all scales. This modifies the flow equation for the renormalized Yukawa coupling according to\n\n\\begin{equation}\n \\partial_k h = \\partial_k h{\\big |}_{\\bar \\varphi_k}-\\frac{m^2}{2h}\\partial_k \\lambda_{\\psi}{\\big |}_{\\bar \\varphi_k}\\,,\n \\label{eq:modfiedflowofh}\n\\end{equation}\nwith $\\partial_k h{\\big |}_{\\bar \\varphi_k}$ the contribution without bosonization and $\\partial_k \\lambda_\\psi{\\big |}_{\\bar \\varphi_k}$ given by the box diagram in Fig. \\ref{fig:boxes}. Since $\\lambda_\\psi$ remains zero during the flow, the effective four-fermion interaction $\\lambda_{\\psi,\\text{eff}}$ is now purely given by the boson exchange. However, the contribution of the particle-hole exchange diagrams is incorporated via the second term in Eq.\\ \\eqref{eq:modfiedflowofh}. \n\nIn the regime with spontaneously broken symmetry we use a real basis for the bosonic field\n\\begin{equation}\n \\bar{\\varphi}=\\bar{\\varphi}_0+\\frac{1}{\\sqrt{2}}(\\bar{\\varphi}_1+i\\bar{\\varphi}_2),\n\\end{equation}\nwhere the expectation value $\\bar{\\varphi}_0$ is chosen to be real without loss of generality. The real fields $\\bar \\varphi_1$ and $\\bar \\varphi_2$ then describe the radial and the Goldstone mode, respectively. To determine the flow equation of $\\bar{h}$, we use the projection description\n\n\\begin{equation}\\label{eq:projectiononh}\n \\partial_k \\bar{h}=i\\sqrt{2}\\Omega^{-1}\\frac{\\delta}{\\delta\\varphi_2(0)}\\frac{\\delta}{\\delta\\psi_1(0)}\\frac{\\delta}{\\delta\\psi_2(0)}\\partial_k \\Gamma_k\\,,\n\\end{equation}\nwith the four volume $\\Omega=\\frac{1}{T}\\int_{\\vec{x}}$. Since the Goldstone mode has vanishing ``mass'', the flow of the Yukawa coupling is not modified by the box diagram (Fig. \\ref{fig:boxes}) in the regime with spontaneous symmetry breaking.\n\nWe emphasize that the non-perturbative nature of the flow equations for the various couplings provides for a resummation similar to the one in Eq.\\ \\eqref{PHComp}, and thus goes beyond the treatment by Gorkov and Melik-Barkhudarov \\cite{Gorkov} which includes the particle-hole diagrams only in a perturbative way. Furthermore, the inner bosonic lines $h^2\/P_\\varphi(q)$ in the box-diagrams represent the center of mass momentum dependence of the four-fermion vertex. This center of mass momentum dependence is neglected in Gorkov's pointlike treatment, and thus represents a further improvement of the classic calculation. Actually, this momentum dependence becomes substantial -- and should not be neglected in a consistent treatment -- away from the BCS regime where the physics of the bosonic bound state sets in. Finally, we note that the truncation \\eqref{eq:truncation} supplemented with \\eqref{eq:fourfermionvertex} closes the truncation to fourth order in the fields except for a fermion-boson vertex $\\lambda_{\\psi\\varphi}\\psi^\\dagger\\psi\\varphi^*\\varphi$ which plays a role for the scattering physics deep in the BEC regime \\cite{DKS} but is not expected to have a very important impact on the critical temperature in the unitarity and BCS regimes.\n\n\n\\subsection{Critical temperature}\nTo obtain the flow equations for the running couplings of our truncation Eq.\\ \\eqref{eq:truncation} we use projection prescriptions similar to Eq.\\ \\eqref{eq:projectiononh}. The resulting system of ordinary coupled differential equations is then solved numerically for different chemical potentials $\\mu$ and temperatures $T$. For temperatures sufficiently small compared to the Fermi temperature $T_F=(3\\pi^2n)^{2\/3}$, $T\/T_F\\ll 1$ we find that the effective potential $U$ at the macroscopic scale $k=0$ develops a minimum at a nonzero field value $\\rho_0>0$, $\\partial_\\rho U(\\rho_0)=0$. The system is then in the superfluid phase. For larger temperatures we find that the minimum is at $\\rho_0=0$ and that the ``mass parameter'' $m^2$ is positive, $m^2=\\partial_\\rho U(0)>0$. The critical temperature $T_c$ of this phase transition between the superfluid and the normal phase is then defined as the temperature where one has\n\\begin{equation}\n\\rho_0=0,\\quad \\partial_\\rho U(0)=0\\quad \\text{at} \\quad k=0.\n\\end{equation}\nThroughout the whole crossover the transition $\\rho_0\\to0$ is continuous as a function of $T$ demonstrating that the phase transition is of second order.\n\nIn Fig. \\ref{fig:tcrit2} we plot our result obtained for the critical temperature $T_c$ and the Fermi temperature $T_F$ as a function of the chemical potential $\\mu$ at the unitarity point with $a^{-1}=0$. From dimensional analysis it is clear that both dependencies are linear, $T_c, T_F\\sim \\mu$, provided that non-universal effects involving the ultraviolet cutoff scale $\\Lambda$ can be neglected. That this is indeed found numerically can be seen as a nontrivial test of our approximation scheme and the numerical procedures as well as the universality of the system. Dividing the slope of both lines gives $T_c\/T_F=0.264$, a result that will be discussed in more detail below. \n\\begin{figure}\n \\centering\n\t\\includegraphics[width=0.45\\textwidth]{PHfigure4.eps}\n\t\\caption{Critical temperature $T_c$ (boxes) and Fermi temperature $T_F=(3\\pi^2 n)^{2\/3}$ (triangles) as a function of the chemical potential $\\mu$. For convenience the Fermi temperature is scaled by a factor 1\/5. We also plot the linear fits $T_c=0.39\\mu$ and $T_F=1.48\\mu$. The units are arbitrary and we use $\\Lambda=e^7$.}\n\t\\label{fig:tcrit2}\n\\end{figure}\nWe emphasize that part of the potential error in this estimates is due to uncertainties in the precise quantitative determination of the density or $T_F$. \n\n\n\\subsection{Phase diagram}\nThe effect of the particle-hole fluctuations shows most prominently in the result for the critical temperature. With our approach we can compute the critical temperature for the phase transition to superfluidity throughout the crossover. The results are shown in Fig. \\ref{fig:tcrit}. We plot the critical temperature in units of the Fermi temperature $T_c\/T_F$ as a function of the scattering length measured in units of the inverse Fermi momentum, i.~e. the concentration $c=a k_F$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{PHfigure5.eps}\n\\caption{Dimensionless critical temperature $T_c\/T_F$ as a function of the inverse concentration $c^{-1}=(a k_F)^{-1}$. The black solid line includes the effect of particle-hole fluctuations. We also show the result obtained when particle-hole fluctuations are neglected (dot-dashed line). For comparison, we plot the BCS result without (left dotted line) and with Gorkov's correction (left dashed). On the BEC side with $c^{-1}>1$ we show the critical temperature for a gas of free bosonic molecules (horizontal dashed line) and a fit to the shift in $T_c$ for interacting bosons, $\\Delta T_c\\sim c$ (dotted line on the right). The black solid dot gives the QMC results \\cite{bulgac:090404, bulgac:023625, burovski:160402}.}\n\\label{fig:tcrit}\n\\end{figure}\nWe can roughly distinguish three different regimes. On the left side, where $c^{-1}\\lesssim-1$, the interaction is weakly attractive. Mean field or BCS theory is qualitatively valid here. In Fig. \\ref{fig:tcrit} we denote the BCS result by the dotted line on the left ($c^{-1}<0$). However, the BCS approximation has to be corrected by the effect of particle-hole fluctuations, which lower the value for the critical temperature by a factor of $2.2$. This is the Gorkov correction (dashed line on the left side in Fig. \\ref{fig:tcrit}). The second regime is found on the far right side, where the interaction again is weak, but now we find a bound state of two atoms. In this regime the system exhibits Bose-Einstein condensation of molecules as the temperature is decreased. The dashed horizontal line on the right side shows the critical temperature of a free Bose-Einstein condensate of molecules. In-between there is the unitarity regime, where the two-atom scattering length diverges ($c^{-1} \\rightarrow 0$) and we deal with a system of strongly interacting fermions.\n\nOur result including the particle-hole fluctuations is given by the solid line. This may be compared with a functional renormalization flow investigation without including particle-hole fluctuations (dot-dashed line) \\cite{Diehl:2007th}. For $c\\to 0_-$ the solid line of our result matches the BCS theory including the correction by Gorkov and Melik-Barkhudarov \\cite{Gorkov},\n\\begin{equation}\n\\frac{T_c}{T_F}=\\frac{e^C}{\\pi}\\left(\\frac{2}{e}\\right)^{7\/3} e^{\\pi\/(2c)}\\approx 0.28 e^{\\pi\/(2c)}.\n\\end{equation}\nIn the regime $c^{-1}>-2$ we see that the non-perturbative result given by our RG analysis deviates from Gorkov's result, which is derived in a perturbative setting. \n\nOn the BEC-side for very large and positive $c^{-1}$ our result approaches the critical temperature of a free Bose gas where the bosons have twice the mass of the fermions $M_B=2M$. In our units the critical temperature is then\n\\begin{equation}\n\\frac{T_{c,\\text{BEC}}}{T_F}=\\frac{2\\pi}{\\left[6\\pi^2 \\zeta(3\/2)\\right]^{2\/3}}\\approx 0.218.\n\\end{equation} \nFor $c\\to 0_+$ this value is approached in the form\n\\begin{equation}\n\\frac{T_c-T_{c,\\text{BEC}}}{T_{c,\\text{BEC}}}=\\kappa a_M n_M^{1\/3}=\\kappa \\frac{a_M}{a}\\frac{c}{(6\\pi^2)^{1\/3}}.\n\\end{equation}\nHere, $n_M=n\/2$ is the density of molecules and $a_M$ is the scattering length between them. For the ratio $a_M\/a$ we use our result $a_M\/a=0.718$ obtained from solving the flow equations in vacuum, i.~e. at $T=n=0$, see section \\ref{DimerDimer}. For the coefficients determining the shift in $T_c$ compared to the free Bose gas we find $\\kappa=1.55$. \n\nFor $c^{-1}\\gtrsim0.5$ the effect of the particle-hole fluctuations vanishes. This is expected since the chemical potential is now negative $\\mu<0$ and there is no Fermi surface any more. Because of that there is no difference between the new curve with particle-hole fluctuations (solid in Fig. \\ref{fig:tcrit}) and the one obtained when particle-hole contributions are neglected (dot-dashed in Fig. \\ref{fig:tcrit}). Due to the use of an optimized cutoff scheme and a different computation of the density our results differ slightly from the ones obtained in \\cite{Diehl:2007th}.\n\nIn the unitary regime ($c^{-1}\\approx 0$) the particle-hole fluctuations still have a quantitative effect. We can give an improved estimate for the critical temperature at the resonance ($c^{-1}=0$) where we find $T_c\/T_F=0.264$. Results from quantum Monte Carlo simulations are $T_c\/T_F = 0.15$ \\cite{bulgac:090404, bulgac:023625, burovski:160402} and $T_c\/T_F = 0.245$ \\cite{akkineni:165116}. The measurement by Luo \\textit{et al.} \\cite{Luo2007} in an optical trap gives $T_c\/T_F = 0.29 (+0.03\/-0.02)$, which is a result based on the study of the specific heat of the system.\n\n\n\\subsection{Crossover to narrow resonances}\nSince we use a two channel model (Eq.\\ \\eqref{eqMicroscopicAction}) we can not only describe broad resonances with $h_\\Lambda^2\\to \\infty$ but also narrow ones with $h_\\Lambda^2\\to0$. This corresponds to a nontrivial limit of the theory which can be treated exactly \\cite{Diehl:2005an, Gurarie2007}. In the limit $h_\\Lambda\\to 0$ the microscopic action Eq.\\ \\eqref{eqMicroscopicAction} describes free fermions and bosons. The essential feature is, that they are in thermodynamic equilibrium so that they have equal chemical potential. (There is a factor 2 for the bosons since they consist of two fermions.) For vanishing Yukawa coupling $h_\\Lambda$ the theory is Gaussian and the macroscopic propagator equals the microscopic propagator. There is no normalization of the ``mass''-term $m^2$ so that the detuning parameter in Eq.\\ \\eqref{eqMicroscopicAction} is $\\nu=\\mu_M(B-B_0)$ and\n\\begin{equation}\nm^2=\\mu_M(B-B_0)-2\\mu.\n\\end{equation}\n\nTo determine the critical density for fixed temperature, we have to adjust the chemical potential $\\mu$ such that the bosons are just at the border to the superfluid phase. For free bosons this implies $m^2=0$ and thus\n\\begin{equation}\n\\mu=\\frac{1}{2}\\mu_M (B-B_0)=-\\frac{1}{16\\pi}h^2 a^{-1}.\n\\label{eq:ChemicalpotetialatTc}\n\\end{equation}\nIn the last equation we use the relation between the detuning and the scattering length\n\\begin{equation}\\label{eq:detuningandscatlenth}\na=-\\frac{h^2}{8\\pi \\mu_M(B-B_0)}.\n\\end{equation}\nThe critical temperature $T_c$ is now determined from the implicit equation\n\\begin{equation}\n\\int \\frac{d^3 p}{(2\\pi)^3}\\left\\{\\frac{2}{e^{\\frac{1}{T_c}(\\vec p^2-\\mu)}+1}+\\frac{2}{e^{\\frac{1}{2T_c}\\vec p^2}-1}\\right\\}=n.\n\\label{eq:TcNarrowresonanceimplicit}\n\\end{equation}\n\nWhile the BCS-BEC crossover can be studied as a function of $B-B_0$ or $\\mu$, Eq.\\ \\eqref{eq:detuningandscatlenth} implies that for $h_\\Lambda^2\\to 0$ a finite scattering length $a$ requires $B\\to B_0$. For all $c\\neq 0$ the narrow resonance limit implies for the phase transition $B=B_0$ and therefore $\\mu=0$. (A different concentration variable $c_\\text{med}$ was used in \\cite{Diehl:2005an, Diehl:2005ae}, such that the crossover could be studied as a function of $c_\\text{med}$ in the narrow resonance limit, see the discussion at the end of this section.) For $\\mu=0$ Eq.\\ \\eqref{eq:TcNarrowresonanceimplicit} can be solved analytically and gives\n\\begin{equation}\n\\frac{T_c}{T_F}=\\left(\\frac{4\\sqrt{2}}{3(3+\\sqrt{2})\\pi^{1\/2}\\zeta(3\/2)}\\right)^{2\/3}\\approx 0.204.\n\\end{equation} \nThis result is confirmed numerically by solving the flow equations for different microscopic Yukawa couplings $h_\\Lambda$ and taking the limit $h_\\Lambda\\to 0$. In Fig. \\ref{fig:narrowbroad}, we show the critical temperature $T_c\/T_F$ as a function of the dimensionless Yukawa coupling $h_\\Lambda\/\\sqrt{k_F}$ in the ``unitarity limit'' $c^{-1}=0$ (solid line). For small values of the Yukawa coupling, $h_\\Lambda\/\\sqrt{k_F} \\lesssim 2$ we enter the regime of the narrow resonance limit and the critical temperature is independent of the precise value of $h_\\Lambda$. The numerical value matches the analytical result $T_c\/T_F\\approx 0.204$ (dotted line in Fig. \\ref{fig:narrowbroad}). For large Yukawa couplings, $h_\\Lambda\/\\sqrt{k_F} \\gtrsim 40$, we recover the result of the broad resonance limit as expected. In between there is a smooth crossover of the critical temperature from narrow to broad resonances.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{PHfigure6.eps}\n\\caption{The critical temperature divided by the Fermi temperature $T_c\/T_F$ as a function of the dimensionless Yukawa coupling $h_\\Lambda\/\\sqrt{k_F}$ for $c^{-1}=0$ (solid line). One can clearly see the plateaus in the narrow resonance limit ($T_c\/T_F \\approx 0.204$, dotted line) and in the broad resonance limit ($T_c\/T_F \\approx 0.264$, dashed line).}\n\\label{fig:narrowbroad}\n\\end{figure}\n\nWe use here a definition of the concentration $c=a k_F$ in terms of the vacuum scattering length $a$. This has the advantage of a straightforward comparison with experiment since $a^{-1}$ is directly related to the detuning of the magnetic field $B-B_0$, and the ``unitarity limit'' $c^{-1}=0$ precisely corresponds to the peak of the resonance $B=B_0$. However, for a nonvanishing density other definitions of the concentration parameter are possible, since the effective fermion interaction $\\lambda_{\\psi,\\text{eff}}$ depends on the density. For example, one could define for $n\\neq 0$ a ``in medium scattering length'' $\\bar a=\\lambda_{\\psi,\\text{eff}}\/(8\\pi)$, with $\\lambda_{\\psi,\\text{eff}}=-h^2\/m^2$ evaluated for $T=0$ but $n\\neq 0$ \\cite{Diehl:2005an}. The corresponding ``in medium concentration'' $c_\\text{med}=\\bar a k_F$ would differ from our definition by a term involving the chemical potential, resulting in a shift of the location of the unitarity limit if the latter is defined as $c_\\text{med}^{-1}=0$. While for broad resonances both definitions effectively coincide, for narrow resonances a precise statement how the concentration is defined is mandatory when aiming for a precision comparison with experiment and numerical simulations for quantities as $T_c\/T_F$ at the unitarity limit. For example, defining the unitarity limit by $c_\\text{med}^{-1}=0$ would shift the critical temperature in the narrow resonance limit to $T_c\/T_F=0.185$ \\cite{Diehl:2005an}.\n\\subsubsection{Vacuum flow equations and their solution for $d=3$}\n\nThe vacuum is defined to have zero temperature $T=0$ and vanishing density $n=0$, which also implies $\\rho_0=0$. The interaction strength $\\lambda$ at the scale $k=0$ determines the four point vertex at zero momentum. It is directly related to the scattering length $a$ for the scattering of two particles in vacuum, which is experimentally observable. We therefore want to replace the microscopic coupling $\\lambda_\\Lambda$ by the renormalized coupling $a$. In our units ($2M=1$), one has the relation \n\\begin{equation}\na=\\frac{1}{8\\pi}\\lambda(k=0, T=0, n=0).\n\\end{equation}\nThe vacuum properties can be computed by taking for $T=0$ the limit $n\\rightarrow0$. We may also perform an equivalent and technically simple computation in the symmetric phase by choosing $m^2(k=\\Lambda)$ such, that $m^2(k\\rightarrow0)=0$. This guarantees that the boson field $\\varphi$ is a gap-less propagating degree of freedom. \n\nWe first investigate the model with a linear $\\tau$-derivative, $S_\\Lambda=1$, $V_\\Lambda=0$. Projecting the flow equation \\eqref{eq4:Wettericheqn} to the truncation in Eqs. \\eqref{eqSimpleTruncation}, \\eqref{eq10:truncationU}, we find the following equations:\n\\begin{eqnarray}\n\\nonumber \\partial_t m^2 & = & 0\\\\\n\\partial_t \\lambda & = & \\left(\\frac{\\lambda^2}{6}\\right)\\frac{{\\left( k^2 - m^2 \\right) }^{3\/2}}{k^2\\,\n {\\pi }^2\\,S}\\,\\Theta(k^2-m^2).\n\\label{eqflowvacuummlambda}\n\\end{eqnarray}\nThe propagator is not renormalized, $\\partial_t S=\\partial_tV=\\partial_t \\bar{A}=0$, $\\eta=0$, $\\partial_t\\alpha=0$, and one finds $\\partial_tn_k=0$. The coupling $\\beta$ is running according to\n\\begin{equation}\n\\partial_t \\beta = \\left(\\frac{1}{3}\\alpha\\lambda^2-\\frac{1}{3}k^2\\beta\\lambda\\right)\\frac{\\left(k^2 - m^2\\right)^{3\/2}}{k^4\\,\\pi^2 \\,S}\\Theta(k^2 - m^2).\n\\label{eqflowvacuumbeta}\n\\end{equation}\nSince $\\beta$ appears only in its own flow equation, it is of no further relevance in the vacuum. Also, no coupling $V$ is generated by the flow and we have therefore set $V=0$ on the r.h.s. of Eqs. \\eqref{eqflowvacuummlambda} and \\eqref{eqflowvacuumbeta}. \n\nInserting in Eq. \\eqref{eqflowvacuummlambda} the vacuum values $m^2=0$ and $S=1$, we find\n\\begin{equation}\n\\partial_t\\lambda=\\frac{k}{6\\pi^2}\\lambda^2.\n\\end{equation}\nThe solution \n\\begin{equation}\n\\lambda(k)=\\frac{1}{\\frac{1}{\\lambda_\\Lambda}+\\frac{1}{6\\pi^2}(\\Lambda-k)}\n\\end{equation}\ntends to a constant for $k\\rightarrow0$, $\\lambda_0=\\lambda(k=0)$. The dimensionless variable $\\tilde{\\lambda}=\\frac{\\lambda k}{S}$ goes to zero, when $k$ goes to zero. This shows the infrared freedom of the theory. For fixed ultraviolet cutoff, the scattering length\n\\begin{equation}\na=\\frac{\\lambda_0}{8\\pi}=\\frac{1}{\\frac{8\\pi}{\\lambda_\\Lambda}+\\frac{4}{3\\pi}\\Lambda},\n\\end{equation}\nas a function of the initial value $\\lambda_\\Lambda$, has an asymptotic maximum\n\\begin{equation}\na_{\\text{max}}=\\frac{3\\pi}{4\\Lambda}.\n\\label{eqscatteringbound}\n\\end{equation}\nThe relation between $a$ and $\\lambda_\\Lambda$ is shown in fig. \\ref{figscatteringbound}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig4.eps}\n\\caption{Scatt\\-er\\-ing length $a$ in dependence on the microscopic interaction strength $\\lambda_\\Lambda$ (solid). The asymptotic maximum $a_{\\text{max}}=\\frac{3 \\pi}{4\\Lambda}$ is also shown (dashed).}\n\\label{figscatteringbound}\n\\end{figure}\n\nAs a consequence of Eq.\\ \\eqref{eqscatteringbound}, the nonrelativistic bosons in $d=3$ are a \"trivial theory\" in the sense that the bosons become noninteracting in the limit $\\Lambda\\rightarrow\\infty$, where $a\\rightarrow0$. The upper bound \\eqref{eqscatteringbound} has important practical consequences. It tells us, that whenever the \"macrophysical length scales\" are substantially larger than the microscopic length $\\Lambda^{-1}$, we deal with a weakly interacting theory. As an example, consider a boson gas with a typical inter-particle distance substantially larger than $\\Lambda^{-1}$. (For atom gases $\\Lambda^{-1}$may be associated with the range of the Van der Waals force.) We may set the units in terms of the particle density $n$, $n=1$. In these units $\\Lambda$ is large, say $\\Lambda=10^3$. This implies a very weak interaction, $a\\lesssim 2.5\\cdot10^{-3}$. In other words, the scattering length cannot be much larger than the microscopic length $\\Lambda^{-1}$. For such systems, perturbation theory will be valid in many circumstances. We will find that the Bogoliubov theory indeed gives a reliable account of many properties. Even for an arbitrary large microphysical coupling $(\\lambda_\\Lambda\\rightarrow\\infty)$, the renormalized physical scattering length $a$ remains finite.\n\nLet us mention, however, that the weak interaction strength does not guarantee the validity of perturbation theory in all circumstances. For example, near the critical temperature of the phase transition between the superfluid and the normal state, the running of $\\lambda(k)$ will be different from the vacuum. As a consequence, the coupling will vanish proportional to the inverse correlation length $\\xi^{-1}$ as $T$ approaches $T_c$, $\\lambda \\sim T^{-2}\\xi^{-1}$. Indeed, the phase transition will be characterized by the non-perturbative critical exponents of the Wilson-Fisher fixed point. Also for lower dimensional systems, the upper bound \\eqref{eqscatteringbound} for $\\lambda_0$ is no longer valid - for example the running of $\\lambda$ is logarithmic for $d=2$. For our models with $V_\\Lambda\\neq0$, the upper bound becomes dependent on $V_\\Lambda$. It increases for $V_\\Lambda>0$. In the limit $S_\\Lambda\\rightarrow0$, it is replaced by the well known \"triviality bound\" of the four dimensional relativistic model, which depends only logarithmically on $\\Lambda$. Finally, for superfluid liquids, as $^4\\text{He}$, one has $n\\sim\\Lambda^3$, such that for $a\\sim \\Lambda^{-1}$ one finds a large concentration $c$.\n\nThe situation for dilute bosons seems to contrast with ultracold fermion gases in the unitary limit of a Feshbach resonance, where $a$ diverges. One may also think about a Feshbach resonance for bosonic atoms, where one would expect a large scattering length for a tuning close to resonance. In this case, however, the effective action does not remain local. It is best described by the exchange of molecules. The scale of nonlocality is then given by the gap for the molecules, $m_M$. Only for momenta $\\vec{q}^20$. At the microscopic scale $k=\\Lambda$ the minimum of the effective potential $U$ is then at $\\rho_{0,\\Lambda}=\\mu\/\\lambda>0$. \n\nThe superfluid density $\\rho_0$ is connected to a nonvanishing ``renormalized order parameter'' $\\varphi_0$, with $\\rho_0=\\varphi_0^*\\varphi_0$. It is responsible for an effective spontaneous breaking of the U(1)-symmetry. Indeed, the expectation value $\\varphi_0$ points out a direction in the complex plane so that the global U(1)-symmetry of phase rotations is broken by the ground state of the system. Goldstone's theorem implies the presence of a gapless Goldstone mode, and the associated linear dispersion relation $\\omega\\sim|\\vec{q}|$ accounts for superfluidity. The Goldstone physics is best described by using a real basis in field space by decomposing the complex field $\\varphi=\\varphi_0+\\frac{1}{\\sqrt{2}}(\\varphi_1+i\\varphi_2)$. Without loss of generality we can choose the expectation value $\\varphi_0$ to be real. The real fields $\\varphi_1$ and $\\varphi_2$ describe then the radial and Goldstone mode. respectively. For $\\mu=\\mu_0$ the inverse propagator reads in our truncation\n\\begin{equation}\nG^{-1}=\\bar{A}\\begin{pmatrix} \\vec{p}^2+V p_0^2+U^\\prime+2\\rho U^{\\prime\\prime}, & -S p_0 \\\\ S p_0, & \\vec{p}^2+V q_0^2+U^\\prime \\end{pmatrix}.\n\\label{eqprop}\n\\end{equation}\nHere $\\vec{p}$ is the momentum of the collective excitation, and for $T=0$ the frequency obeys $\\omega=-ip_0$.\nIn the regime with spontaneous symmetry breaking, $\\rho_0(k)\\neq 0$, the propagator for $\\rho=\\rho_0$ has $U^\\prime=0$, $2\\rho U^{\\prime\\prime}=2\\lambda \\rho_0\\neq 0$, giving rise to the linear dispersion relation characteristic for superfluidity. This strongly modifies the flow equations as compared to the vacuum flow equations once $k^2 \\ll 2\\lambda \\rho_0$. For $n\\neq0$ the flow is typically in the regime with $\\rho_0(k)\\neq0$. In practice, we have to adapt the initial value $\\rho_{0,\\Lambda}$ such that the flow ends at a given density $\\rho_0(k_\\text{ph})=n$. For $k_\\text{ph}\\ll n^{1\/2}$ one finds that $\\rho_0(k_\\text{ph})$ depends only very little on $k_\\text{ph}$. As mentioned above, we will often choose the density to be unity such that effectively all length scales are measured in units of the interparticle distance $n^{-1\/2}$. \n\nIn contrast to the vacuum with $T=\\rho_0=0$, the flow of the propagator is nontrivial in the phase with $\\rho_0>0$ and spontaneous U(1) symmetry breaking. In Fig. \\ref{figFlowKinetic} we show the flow of the kinetic coefficients $\\bar{A}$, $V$, $S$ for a renormalized or macroscopic interaction strength $\\lambda=1$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig3.eps}\n\\caption{Flow of the kinetic coefficients $\\bar{A}$ (solid), $S$ (dashed), and $V$ (dashed-dotted) at zero temperature $T=0$, density $n=1$, and vacuum interaction strength $\\lambda=1$.}\n\\label{figFlowKinetic}\n\\end{figure}\nThe wavefunction renormalization $\\bar{A}$ increases only a little at scales where $k\\approx n^{1\/2}$ and saturates then to a value $\\bar{A}>1$. As will be explained below, we can directly infer the condensate depletion from the value of $\\bar{A}$ at macroscopic scales. The coefficient of the linear $\\tau$-derivative $S$ goes to zero for $k\\rightarrow 0$. The frequency dependence is then governed by the quadratic $\\tau$-derivative with coefficient $V$, which is generated by the flow and saturates to a finite value for $k\\rightarrow 0$. \n\nThe flow of the interaction strength $\\lambda(k)$ for different values of $\\lambda=\\lambda(k_\\text{ph})$ is shown in Fig. \\ref{figFlowLambda}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig4.eps}\n\\caption{Flow of the interaction strength $\\lambda(k)$ at zero temperature $T=0$, density $n=1$, for different initial values $\\lambda_\\Lambda$. The dotted lines are the corresponding graphs in the vacuum $n=0$. The vertical line labels our choice of $k_\\text{ph}$. The lower plot shows $\\lambda(k)\/k$ for the same parameters, demonstrating that $\\lambda(k)\\sim k$ for small $k$.}\n\\label{figFlowLambda}\n\\end{figure}\nWhile the decrease with the scale $k$ is only logarithmic in vacuum, it becomes now linear $\\lambda(k)\\sim k$ for $k\\ll n^{1\/2}$. It is interesting that the ratio $\\lambda(k)\/k$ reaches larger values for smaller values of $\\lambda_\\Lambda$. \n\n\\subsection{Quantum depletion of condensate}\n\nAs $k$ is lowered from $\\Lambda$ to $k_\\text{ph}$, the renormalized order parameter or the superfluid density $\\rho_0$ increases first and then saturates to $\\rho_0=n=1$. This is expected since the superfluid density equals the total density at zero temperature. In contrast, the bare order parameter $\\bar{\\rho}_0=\\rho_0\/\\bar{A}$ flows to a smaller value $\\bar{\\rho}_0<\\rho_0$. As argued in section \\ref{sec:Bose-EinsteinCondensationinthreedimensions}, the bare order parameter is just the condensate density, such that\n\\begin{equation}\nn-n_C=\\rho_0-\\bar{\\rho}_0=\\rho_0(1-\\frac{1}{\\bar{A}})\n\\end{equation}\nis the condensate depletion. Its dependence on the interaction strength $\\lambda$ is shown in Fig. \\ref{figDepletion}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig5.eps}\n\\caption{Condensate depletion $(n-n_c)\/n$ as a function of the vacuum interaction strength $\\lambda$. The dashed line is the Bogoliubov result $(n-n_c)\/n=\\frac{\\lambda}{8\\pi}$ for reference.}\n\\label{figDepletion}\n\\end{figure}\nFor small interaction strength $\\lambda$ the condensate depletion follows roughly the Bogoliubov form\n\\begin{equation}\n\\frac{n-n_c}{n}=\\frac{\\lambda}{8\\pi}.\n\\end{equation}\nHowever, we find small deviations due to the running of $\\lambda$ which is absent in Bogoliubov theory. For large interaction strength $\\lambda\\approx 1$ the deviation from the Bogoliubov result is quite substantial, since the running of $\\lambda$ with the scale $k$ is more important. \n\n\\subsection{Dispersion relation and sound velocity}\n\nWe also investigate the dispersion relation at zero temperature. The dispersion relation $\\omega(p)$ follows from the condition\n\\begin{equation}\n\\text{det}\\, G^{-1}(\\omega(p),p)=0\n\\label{eqdispersionfromprop}\n\\end{equation}\nwhere $G^{-1}$ is the inverse propagator after analytic continuation to real time $p_0\\rightarrow i\\omega$. As was shown at the end of section\\ref{sec:Derivativeexpansionandwardidentities} the generation of the kinetic coefficient $V$ by the flow leads to the emergence of a second branch of solutions of Eq.\\ \\eqref{eqdispersionfromprop}. In our truncation the dispersion relation for the two branches $\\omega_+(\\vec{p})$ and $\\omega_-(\\vec{p})$ are\n\\begin{eqnarray}\n\\nonumber\n\\omega_\\pm(\\vec{p})&=&{\\Bigg (}\\frac{1}{V}(\\vec{p}^2+\\lambda \\rho_0)+\\frac{S^2}{2V^2}\\\\\n&&\\pm{\\Bigg (}\\left(\\frac{1}{V}(\\vec{p}^2+\\lambda\\rho_0)+\\frac{S^2}{2V^2}\\right)^2-\\frac{1}{V^2}\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0){\\Bigg )}^{1\/2}{\\Bigg )}^{1\/2}.\n\\label{eqdispersionrelation}\n\\end{eqnarray}\nIn the limit $V\\rightarrow 0$, $S\\rightarrow 1$ we find that the lower branch approaches the Bogoliubov result $\\omega_-\\rightarrow \\sqrt{\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0)}$ while the upper branch diverges $\\omega_+\\rightarrow \\infty$ and thus disappears from the spectrum. The lower branch is dominated by phase changes (Goldstone mode), while the upper branch reflects waves in the size of $\\rho_0$ (radial mode). \n\nIn principle, the coupling constants on the right-hand side of Eq.\\ \\eqref{eqdispersionrelation} also depend on the momentum $p=|\\vec{p}|$. Since an external momentum provides an infrared cutoff of order $k\\approx |\\vec{p}|$ we can approximate the $|\\vec{p}|$-dependence by using on the right-hand side of Eq.\\ \\eqref{eqdispersionrelation} the $k$-dependent couplings with the identification $k=|\\vec{p}|$. Our result for the lower branch of the dispersion relation is shown in Fig. \\ref{figDispersionlinear}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig6.eps}\n\\caption{Lower branch of the dispersion relation $\\omega_{-}(p)$ at temperature $T=0$ and for the vacuum interaction strength $\\lambda=1$ (solid curve), $\\lambda=0.5$ (upper dashed curve), and $\\lambda=0.1$ (lower dashed curve). The units are set by the density $n=1$. We also show the Bogoliubov result for $\\lambda=1$ (upper dotted curve) and $\\lambda=0.5$ (lower dotted curve). For $\\lambda=0.1$ the Bogoliubov result is identical to our result within the plot resolution.}\n\\label{figDispersionlinear}\n\\end{figure}\nWe also plot the Bogoliubov result $\\omega=\\sqrt{\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0)}$ for comparison. For small $\\lambda$ our result is in agreement with the Bogoliubov result, while we find substantial deviations for large $\\lambda$. Both branches $\\omega_+$ and $\\omega_-$ are shown in Fig. \\ref{figDispersion} on a logarithmic scale. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig7.eps}\n\\caption{Dispersion relation $\\omega_{-}(p)$, $\\omega_{+}(p)$ at temperature $T=0$ and for vacuum interaction strength $\\lambda=1$ (solid), $\\lambda=0.5$ (long dashed), and $\\lambda=0.1$ (short dashed). The units are set by the density $n=1$.}\n\\label{figDispersion}\n\\end{figure}\nSince we start with $V=0$ at the microscopic scale $\\Lambda$ we find $\\omega_+(\\vec{p})\\rightarrow\\infty$ for $|\\vec{p}|\\rightarrow \\Lambda$.\n\nThe sound velocity $c_S$ can be extracted from the dispersion relation. More precisely, we compute the microscopic sound velocity for the lower branch $\\omega_-(\\vec{p})$ as $c_S=\\frac{\\partial \\omega}{\\partial p}$ at $p=0$. In our truncation we find\n\\begin{equation}\nc_S^2=\\frac{2\\lambda\\rho_0}{S^2+2\\lambda\\rho_0 V}.\n\\end{equation}\nOur result for $c_S$ at $T=0$ is shown in Fig. \\ref{figsoundvelocity} as a function of the interaction strength $\\lambda$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig8.eps}\n\\caption{Dimensionless sound velocity $c_S\/n^{1\/2}$ as a function of the vacuum interaction strength (solid). We also show the Bogoliubov result $c_S=\\sqrt{2\\lambda \\rho_0}$ for reference (dashed).}\n\\label{figsoundvelocity}\n\\end{figure}\nFor a large range of small $\\lambda$ we find good agreement with the Bogoliubov result $c_S^2=2\\lambda \\rho_0$. However, for large $\\lambda$ or result for $c_S$ exceeds the Bogoliubov result by a factor up to 2. \n\n\n\\subsection{Kosterlitz-Thouless physics}\n\n\\subsubsection{Superfluidity and order parameter}\nAt nonzero temperature and for infinite volume, long range order is forbidden in two spatial dimensions by the Mermin-Wagner theorem. Because of that, no proper Bose-Einstein condensation is possible in a two-dimensional homogeneous Bose gas at nonvanishing temperature. However, even if the order parameter vanishes in the thermodynamic limit of infinite volume, one still finds a nonzero superfluid density for low enough temperature. The superfluid density can be considered as the square of a renormalized order parameter $\\rho_0=|\\varphi_0|^2$ and the particular features of the low-temperature phase can be well understood by the physics of the Goldstone boson for a phase with effective spontaneous symmetry breaking \\cite{Wetterich:1991be}. The renormalized order parameter $\\varphi_0$ is related to the expectation value of the bosonic field $\\bar{\\varphi}_0$ and therefore to the condensate density $\\bar{\\rho}_0=\\bar{\\varphi}_0^2$ by a wave function renormalization, defined by the behavior of the bare propagator $\\bar{G}$ at zero frequency for vanishing momentum\n\\begin{equation}\n\\varphi_0=\\bar{A}^{1\/2}\\bar{\\varphi}_0,\\quad \\rho_0=\\bar{A}\\bar{\\rho}_0,\\quad \\bar{G}^{-1}(\\vec{p}\\rightarrow 0)=\\bar{A}\\vec{p}^2.\n\\end{equation}\nWhile the renormalized order parameter $\\rho_0(k)$ remains nonzero for $k\\rightarrow 0$ if $T0$ for nonzero temperature $00$.\n\n\\subsubsection{Critical temperature}\nThe flow equations permit a straightforward computation of $\\rho_0(T)$ for arbitrary $T$, once the interaction strength of the system has been fixed at zero temperature and density. We have extracted the critical temperature as a function of $\\lambda=\\lambda(k_\\text{ph})$ for different values of $k_\\text{ph}$. The behavior for small $\\lambda$,\n\\begin{equation}\n\\frac{T_c}{n}=\\frac{4\\pi}{\\text{ln}(\\zeta\/\\lambda)}\n\\label{Tcperturbative}\n\\end{equation}\nis compatible with the free theory where $T_c$ vanishes for $k_\\text{ph}\\rightarrow0$ and with the perturbative analysis in Ref. \\cite{Popov1983, PhysRevB.37.4936, MarkusHolzmann01302007}. We find that the value of $\\zeta$ depends on the choice of $k_\\text{ph}$. For $k_\\text{ph}=10^{-2}$ we find $\\zeta=100$, while $k_\\text{ph}=10^{-4}$ corresponds to $\\zeta=225$ and $k_\\text{ph}=10^{-6}$ to $\\zeta=424$. In Fig. \\ref{figtcoflambda} we show or result for $T_c\/n$ as a function of $\\lambda$ for these choices. We also plot the curve in Eq.\\ \\eqref{Tcperturbative} with the Monte-Carlo result $\\zeta=380$ from Ref. \\cite{PhysRevLett.87.270402, PhysRevA.66.043608, PhysRevB.69.144504, PhysRevB.59.14054}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig9.eps}\n\\caption{Critical temperature $T_c\/n$ as a function of the interaction strength $\\lambda$. We choose here $k_\\text{ph}=10^{-2}$ (circles), $k_\\text{ph}=10^{-4}$ (boxes) and $k_\\text{ph}=10^{-6}$ (diamonds). For the last case the bound on the scattering length is $\\lambda<\\frac{4\\pi}{\\text{ln}(\\Lambda\/k_\\text{ph})}\\approx 0.78$. We also show the curve $\\frac{T_c}{n}=\\frac{4\\pi}{\\text{ln}(\\zeta\/\\lambda)}$ (dashed) with the Monte-Carlo result $\\zeta=380$ \\cite{PhysRevLett.87.270402, PhysRevA.66.043608, PhysRevB.69.144504, PhysRevB.59.14054} for reference.}\n\\label{figtcoflambda}\n\\end{figure}\nWe find that $T_c$ vanishes for $k_\\text{ph}\\to0$ in the interacting theory as well. This is due to the increase of $\\zeta$ and, for a fixed microscopical interaction, to the decrease of $\\lambda(k_\\text{ph})$. Since the vanishing of $T_c\/n$ is only logarithmic in $k_\\text{ph}$, a phase transition can be observed in practice. We find agreement with Monte-Carlo results \\cite{PhysRevLett.87.270402} for small $\\lambda$ if $k_\\text{ph}\/\\Lambda\\approx 10^{-7}$. The dependence of $T_c\/n$ on the size of the system $k_\\text{ph}^{-1}$ remains to be established for the Monte-Carlo computations.\n\nThe critical behavior of the system is governed by a Kosterlitz-Thouless phase transition. Usually this is described by considering the thermodynamics of vortices. In Refs. \\cite{Grater:1994qx, VonGersdorff:2000kp} it was shown that functional renormalization group can account for this ``nonperturbative'' physics without explicitly taking vortices into account. The correlation length in the low-temperature phase is infinite. In our picture, this arises due to the presence of a Goldstone mode if $\\rho_0>0$. The system is superfluid for $T0$) the transition is smoothened. In order to see the jump, as well as essential scaling for $T$ approaching $T_c$ from above, our truncation is insufficient. These features become visible only in extended truncations that we will briefly describe next. \n\nFor very small scales $\\frac{k^2}{T}\\ll 1$, the contribution of Matsubara modes with frequency $q_0=2\\pi T n$, $n\\neq 0$, is suppressed since nonzero Matsubara frequencies act as an infrared cutoff. In this limit a dimensionally reduced theory becomes valid. The long distance physics is dominated by classical two-dimensional statistics, and the time dimension parametrized by $\\tau$ no longer plays a role. \n\nThe flow equations simplify considerably if only the zero Matsubara frequency is included, and one can use more involved truncations. Such an improved truncation is indeed needed to account for the jump in the superfluid density. In Ref. \\cite{VonGersdorff:2000kp} the next to leading order in a systematic derivative expansion was investigated. It was found that for $k\\ll T$ the flow equation for $\\rho_0$ can be well approximated by\n\\begin{equation}\n\\partial_t \\rho_0=2.54\\,T^{-1\/2}(0.248\\, T-\\rho_0)^{3\/2}\\,\\theta(0.248 \\,T-\\rho_0).\n\\label{eq:improvedflowofrho}\n\\end{equation}\nWe switch from the flow equation in our more simple truncation to the improved flow equation \\eqref{eq:improvedflowofrho} for scales $k$ with $k^2\/T<10^{-3}$. We keep all other flow equations unchanged. A similar procedure was also used in Ref. \\cite{DrHCK}.\n\nIn Fig. \\ref{figFlowofnrho} we show the flow of the density $n$, the superfluid density $\\rho_0$ and the condensate density $\\bar{\\rho}_0$ for different temperatures.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig10.eps}\n\\caption{Flow of the density $n$ (solid), the superfluid density $\\rho_0$ (dashed), and the condensate density $\\bar{\\rho}_0$ (dotted) for chemical potential $\\mu=1$, vacuum interaction strength $\\lambda=0.5$ and temperatures $T=0$ (top), $T=2.4$ (middle) and $T=2.8$ (bottom). The vertical line marks our choice of $k_\\text{ph}$. We recall $n=\\rho_0$ for $T=0$ such that the upper dashed and solid lines coincide.}\n\\label{figFlowofnrho}\n\\end{figure}\nIn Fig. \\ref{figrhodnoftemperature} we plot our result for the superfluid fraction of the density as a function of the temperature for different scales $k_\\text{ph}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig11.eps}\n\\caption{Superfluid fraction of the density $\\rho_0\/n$ as a function of the dimensionless temperature $T\/n$ for interaction strength $\\lambda=0.5$ at different macroscopic scales $k_\\text{ph}=1$ (upper curve), $k_\\text{ph}=10^{-0.5}$, $k_\\text{ph}=10^{-1}$, $k_\\text{ph}=10^{-1.5}$, $k_\\text{ph}=10^{-2}$, $k_\\text{ph}=10^{-2.5}$ (bottom curve). We plot the result obtained with the improved truncation for small scales (solid) as well as the result obtained with our more simple truncation (dotted). (The truncations differ only for the three lowest lines.)}\n\\label{figrhodnoftemperature}\n\\end{figure}\nOne can see that with the improved truncation the jump in the superfluid density is indeed found in the limit $k_\\text{ph}\\rightarrow 0$. Fig. \\ref{figcondensatedensity} \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig12.eps}\n\\caption{Condensate fraction of the density $\\bar{\\rho}_0\/n$ as a function of the dimensionless temperature $T\/n$ for interaction strength $\\lambda=0.5$ at macroscopic scale $k_\\text{ph}=10^{-2}$ (solid curve) and $k_\\text{ph}=10^{-4}$ (dashed curve). For comparison, we also plot the superfluid density $\\rho_0\/n$ at $k_\\text{ph}=10^{-2}$ (dotted). These results are obtained with the improved truncation.}\n\\label{figcondensatedensity}\n\\end{figure}\nshows the condensate fraction $\\bar{\\rho}_0\/n$ and the superfluid density fraction $\\rho_0\/n$ as a function of $T\/n$. We observe the substantial $k_\\text{ph}$ dependence of the condensate fraction, as well as an effective jump at $T_c$ for small $k_\\text{ph}$. We recall that the infinite volume limit $k_\\text{ph}=0$ amounts to $\\bar{\\rho}_0=0$ for $T>0$. \n\nThe Kosterlitz-Thouless description is only valid if the zero Matsubara frequency mode ($n=0$) dominates. For a given nonzero $T$ this is always the case if the the characteristic length scale goes to infinity. In the infinite volume limit the characteristic length scale is given by the correlation length $\\xi$. The description in terms of a classical two dimensional system with U(1) symmetry is the key ingredient of the Kosterlitz-Thouless description and holds for $\\xi^2 T\\gg 1$. In the infinite volume limit this always holds for $Tk_\\text{ph}^2$. \n\nFor very small temperatures $Tk_\\text{ph}^2$ and $T\\Lambda_\\text{UV}^2$. \n\nFor bosons with a pointlike repulsive interaction we found in section \\ref{sec:Repulsiveinteractingbosons} that the scattering length is bounded by the ultraviolet scale $a<3\\pi\/(4\\Lambda)$. This is an effect due to quantum fluctuations similar to the ``triviality bound'' for the Higgs scalar in the standard model of elementary particle physics. For a given value of the dimensionless combination $an^{1\/3}$ we cannot choose $\\Lambda\/n^{1\/3}$ larger then $3\\pi\/(4 an^{1\/3})$. For our numerical calculations we use $\\Lambda\/n^{1\/3}\\approx 10$. Other momentum scales are set by the temperature and the chemical potential. The lowest nonzero Matsubara frequency gives the momentum scale $\\Lambda_T^2=2\\pi T$. For a Bose gas with $a=0$ one has $T_c\/n^{2\/3}\\approx 6.625$ such that $\\Lambda_{T_c}\/n^{1\/3}\\approx 6.45$. The momentum scale associated to the chemical potential is $\\Lambda_\\mu^2=\\mu$. For small temperatures and scattering length one finds $\\mu\\approx 8\\pi a n$ and thus $\\Lambda_\\mu\/n^{1\/3}\\approx \\sqrt{8\\pi a n^{1\/3}}$.\n\nWe finally note that the thermodynamic relations for intensive quantities can only involve dimensionless ratios. We may set the unit of momentum by $n^{1\/3}$. The thermodynamic variables are then $T\/n^{2\/3}$ and $\\mu\/n^{2\/3}$. The thermodynamic relations will depend on the strength of the repulsive interaction $\\lambda$ or the scattering length $a$, and therefore on a ``concentration'' type parameter $a n^{1\/3}$.\n\n\n\\subsubsection{Density, superfluid density, condensate and correlation length}\nLet us start our discussion with the density. In the grand canonical formalism it is obtained by taking the derivative of the thermodynamic potential with respect to $\\mu$\n\\begin{equation}\nn=-\\frac{1}{V}\\frac{\\partial}{\\partial \\mu}\\Omega_G = \\frac{\\partial p}{\\partial \\mu}{\\big |}_T.\n\\end{equation}\nWe could compute the $\\mu$-derivative of $p$ numerically by solving the flow equation for U with neighboring values of $\\mu$. As desribed above, we use also another method which employs a flow equation directly for $n$. Since we often express dimensionful quantities in units of the interparticle distance $n^{-1\/3}$, it is crucial to have an accurate value for the density $n$. Comparison of the numerical evaluation and the solution of a separate flow equation for $n$ shows higher precision for the latter method and we will therefore employ the flow equation. We plot in Fig. \\ref{fig:densityofmu} the density in units of the scattering length, $na^3$, as a function of the dimensionless combination $\\mu a^2$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig1.eps}\n\\caption{Density in units of the scattering length $n a^3$ as a function of the (rescaled) chemical potential $\\mu a^2$. We choose for the temperatures $T a^2=2\\cdot 10^{-4}$ (solid curve), $T a^2=4 \\cdot 10^{-4}$ (dashed-dotted curve) and $T a^2= 6 \\cdot 10^{-4}$ (dashed curve). For all three curves we use $a\\Lambda=0.1$.}\n\\label{fig:densityofmu}\n\\end{figure}\n\nFor a comparison with experimentally accessible quantities we have to replace the interaction parameter $\\lambda$ in the microscopic action \\eqref{microscopicaction} by a scattering length $a$ which is a macroscopic quantity. For this purpose we start the flow at the UV-scale $\\Lambda_\\text{UV}$ with a given $\\lambda$, and then compute the scattering length in vacuum ($T=n=0$) by following the flow to $k=0$, see section \\ref{sec:Repulsiveinteractingbosons}. This is a standard procedure in quantum field theory, where a ``bare coupling'' ($\\lambda$) is replaced by a renormalized coupling ($a$). For an investigation of the role of the strength of the interaction we may consider different values of the ``concentration'' $c=an^{1\/3}$ or of the product $\\mu a^2$. While the concentration is easier to access for observation, it is also numerically more demanding since for every value of the parameters one has to tune $\\mu$ in order to obtain the appropriate density. For this reason we rather present results for three values of $\\mu a^2$, i.~e. $\\mu a^2= 2.6\\times 10^{-5}$ (case I), $\\mu a^2=0.0040$ (case II) and $\\mu a^2=0.044$ (case III). The prize for the numerical simplicity is a week temperature dependence of the concentration $c=a n^{1\/3}$ for the three different cases, as shown in Fig. \\ref{fig:an13}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig2.eps}\n\\caption{Concentration $c=an^{1\/3}$ as a function of temperature $T\/(n^{2\/3})$ for the three cases investigated in this paper. Case I corresponds to $an^{1\/3}\\approx 0.01$ (crosses), case II to $an^{1\/3}\\approx 0.05$ (dots) and case III has $an^{1\/3}\\approx 0.01$ (stars).}\n\\label{fig:an13}\n\\end{figure}\nHere and in the following figures case I, which corresponds to $an^{1\/3}\\approx 0.01$, is represented by the little crosses, case II with $an^{1\/3}\\approx 0.05$ by the dots and case III with $an^{1\/3}\\approx 0.1$ by the stars. It is well known that the critical temperature depends on the concentration $c=an^{1\/3}$. From our calculation we find $T_c\/(n^{2\/3})=6.74$ with $c=0.0083$ at $T=T_c$ in case I, $T_c\/(n^{2\/3})=7.16$ with $c=0.044$ at $T=T_c$ in case II and finally $T_c\/(n^{2\/3})=7.75$ with $c=0.088$ at $T=T_c$ in case III.\n\nThis values can are obtained by following the superfluid fraction of the density $n_S\/n$, or equivalently the condensate part of the density $n_C\/n$ as a function of temperature. For small temperatures $T\\to0$ all of the density is superfluid, which is a consequence of Galilean symmetry. However, in contrast to the ideal gas, not all particles are in the condensate. For $T=0$ this condensate depletion is completely due to quantum fluctuations. With increasing temperature both the superfluid density and the condensate decrease and vanish eventually at the critical temperature $T=T_c$. That the melting of the condensate is continuous shows that the phase transition is of second order. We plot our results for the superfluid fraction in Fig. \\ref{fig:superfluidfraction} and for the condensate in Fig. \\ref{fig:condensatefraction}. For small temperatures, we also show the corresponding result obtained in the framework of Bogoliubov theory \\cite{Bogoliubov} (dashed lines). This approximation assumes a gas of non-interacting quasiparticles (phonons) with dispersion relation\n\\begin{equation}\n\\epsilon(p)=\\sqrt{2\\lambda n \\vec p^2+\\vec p^4}.\n\\end{equation} \nIt is is valid in the regime with small temperatures $T\\ll T_c$ and small interaction strength $an^{1\/3}\\ll 1$. For a detailed discussion of Bogoliubov theory and the calculation of thermodynamic observables in this framework we refer to ref. \\cite{PitaevsikiiStringari2003}. Our curves for the superfluid fraction match the Bogoliubov result for temperatures $T\/n^{2\/3}\\lesssim 1$ in all three cases I, II, and III. For larger temperatures there are deviations as expected. For the condensate density, there is already notable a deviation at small temperatures for case III with $an^{1\/3}\\approx 0.1$. This is also expected, since Bogoliubov theory gives only the first order contribution to the condensate depletion in a perturbative expansion for small $an^{1\/3}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig3.eps}\n\\caption{Superfluid fraction of the density $n_S\/n$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III. For small $T\/n^{2\/3}$ we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines).}\n\\label{fig:superfluidfraction}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig4.eps}\n\\caption{Condensate fraction of the density $n_C\/n$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III. For small $T\/n^{2\/3}$ we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines).}\n\\label{fig:condensatefraction}\n\\end{figure}\nFor temperatures slightly smaller than the critical temperature $T_c$ one expects that the condensate density behaves like \n\\begin{equation}\nn_c(T)=B^2 \\left(\\frac{T_c-T}{T_c}\\right)^{2\\beta}\n\\label{eq:scalingnc}\n\\end{equation}\nwith $\\beta=0.3485$ the critical exponent of the three-dimensional XY-universality class \\cite{Pelissetto2002549}. Indeed, the condensate density is given by $n_C=\\bar\\varphi_0^*\\bar\\varphi_0$\nwhere $\\bar \\varphi_0$ is the expectation value of the boson field which serves as an order parameter in close analogy to e.~g. the magnetization $\\vec M$ in a ferromagnet. Eq.\\ \\eqref{eq:scalingnc} is compatible with our findings, although our numerical resolution does not allow for a precise determination of the exponent $\\beta$. \n\nWith our method we can also calculate the correlation length $\\xi$. For temperatures $TT_c$ there is only one correlation length $\\xi^{-1}=m=\\frac{1}{\\bar A}\\frac{\\partial U}{\\partial \\bar \\rho}$, which also diverges for $T\\to T_c$. From the theory of critical phenomena one expects close to $T_c$ the behavior\n\\begin{eqnarray}\n\\nonumber\n\\xi_R &=& f_R^- \\left(\\frac{T_c-T}{T_c}\\right)^{-\\nu} \\quad \\text{for}\\quad TT_c$ as a function of the temperature $T\/n^{2\/3}$ in Fig. \\ref{fig:correlationlength}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig5.eps}\n\\caption{Correlation length $\\xi_R n^{1\/3}$ for $TT_c$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III.}\n\\label{fig:correlationlength}\n\\end{figure}\n\n\n\\subsubsection{Entropy density, energy density, and specific heat}\nThe next thermodynamic quantity we investigate is the entropy density $s$ and the entropy per particle $s\/n$. We can obtain the entropy as\n\\begin{equation}\ns=\\frac{\\partial p}{\\partial T}{\\big |}_\\mu.\n\\end{equation}\nWe compute the temperature derivative by numerical differentiation, using flows with neighboring values of $T$ and show the result in Fig. \\ref{fig:entropypp}. For small temperatures our result coincides with the entropy of free quasiparticles in the Bogoliubov approximation (dashed lines in Fig. \\ref{fig:entropypp}). \nAs it should be, the entropy per particle increases with the temperature. For small temperatures, the slope of this increase is smaller for larger concentration $c$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig6.eps}\n\\caption{Entropy per particle $s\/n$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I, II, and III. For $T\/n^{2\/3}<5$ we also plot the results obtained within the Bogoliubov approximation (dashed lines).}\n\\label{fig:entropypp}\n\\end{figure}\n\nFrom the entropy density $s$ we infer the specific heat per particle,\n\\begin{equation}\nc_v=\\frac{T}{n}\\frac{\\partial s}{\\partial T}{\\bigg |}_n,\n\\end{equation}\nas the temperature derivative of the entropy density at constant particle density.\nUsing the Jacobian, we can write\n\\begin{equation}\n\\frac{\\partial s}{\\partial T}{\\big |}_n = \\frac{\\partial(s,n)}{\\partial(T,n)}=\\frac{\\partial(s,n)}{\\partial(T,\\mu)}\\frac{\\partial(T,\\mu)}{\\partial(T,n)}.\n\\end{equation}\nFor the specific heat this gives\n\\begin{equation}\nc_v=\\frac{T}{n}\\left(\\frac{\\partial s}{\\partial T}{\\big |}_\\mu -\\frac{\\partial s}{\\partial \\mu}{\\big |}_T\\frac{\\partial n}{\\partial T}{\\big |}_\\mu \\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right)^{-1} \\right).\n\\end{equation}\nOur result for the specific heat per particle is shown for different scattering lengths in Fig. \\ref{fig:specificheatpp}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig7.eps}\n\\caption{Specific heat per particle $c_v$ as a function of the dimensionless temperature $T\/n^{2\/3}$. The dashed lines show the Bogoliubov result for $c_v$ which coincides with our findings for small temperature. However, the characteristic cusp behavior cannot be seen in a mean-field theory.}\n\\label{fig:specificheatpp}\n\\end{figure}\nWhile this quantity is positive in the whole range of investigated temperatures, it is interesting to observe the cusp at the critical temperature $T_c$ which is characteristic for a second order phase transition. This behavior cannot be seen in a mean-field approximation, where fluctuations are taken into account only to second order in the fields. Only for small temperatures, our curve is close to the Bogoliubov approximation, shown by the dashed lines in Fig. \\ref{fig:specificheatpp}. \n\nIn fact, close to $T_c$ the specific heat is expected to behave like\n\\begin{eqnarray}\n\\nonumber\nc_v &\\approx& b_1-b_2^- \\left(\\frac{T_c-T}{T_c}\\right)^{-\\alpha} \\quad \\text{for} \\quad TT_c,\n\\end{eqnarray}\nwith the universal critical exponent $\\alpha$ of the $3$-dimensional $XY$ universality class, $\\alpha=-0.0146(8)$ \\cite{Pelissetto2002549}. The critical region, where the law $c_v~\\sim |T-T_c|^{-\\alpha}$ holds, may be quite small.\nOur numerical differentiation procedure cannot resolve the details of the cusp. \n\nIn the grand canonical formalism, the energy density $\\epsilon$ is obtained as\n\\begin{equation}\n\\epsilon = -p +Ts+\\mu n.\n\\end{equation}\nWe plot $p\/(n^{5\/3})$ as a function of temperature in Fig. \\ref{fig:pressure} and the energy density $\\epsilon\/(n^{5\/3})$ is plotted in Fig. \\ref{fig:energy}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig8.eps}\n\\caption{Pressure in units of the density $p\/n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines).}\n\\label{fig:pressure}\n\\end{figure}\nWe have normalized the pressure such that it vanishes for $T=\\mu=0$. Technically we subtract from the flow equation of the pressure the corresponding expression in the limit $T=\\mu=0$. This procedure has to be handled with care and leads to an uncertainty in the offset of the pressure, i.~e. the part that is independent of $T\/n^{2\/3}$ and $\\mu\/n^{2\/3}$. \n\nFor zero temperature, the pressure is completely due to the repulsive interaction between the particles. For nonzero temperature, the pressure is increased by the thermal kinetic energy, of course.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig9.eps}\n\\caption{Energy per particle $\\epsilon\/n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines).}\n\\label{fig:energy}\n\\end{figure}\nFor the energy and the pressure we find some deviations from the Bogoliubov result already for small temperatures in cases II and III. These deviations may be partly due to the uncertainty in the normalization process described above. For weak interactions $an^{1\/3}=0.01$ as in case I, the Bogoliubov prediction coincides with our result. \n\n\n\\subsubsection{Compressibility}\nThe isothermal compressibility is defined as the relative volume change at fixed temperature $T$ and particle number $N$ when some pressure is applied\n\\begin{equation}\n\\kappa_T=-\\frac{1}{V}\\frac{\\partial V}{\\partial p}{\\big |}_{T,N}=\\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_T.\n\\label{eq:isothermalkomp}\n\\end{equation}\nVery similar, the adiabatic compressibility is\n\\begin{equation}\n\\kappa_S=-\\frac{1}{V}\\frac{\\partial V}{\\partial p}{\\big |}_{S,N}=\\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_{s\/n}\n\\end{equation}\nwhere now the entropy $S$ and the particle number $N$ are fixed. Let us first concentrate on the isothermal compressibility $\\kappa_T$. To evaluate it in the grand canonical formalism, we have to change variables to $T$ and $\\mu$. \nWith $\\partial p\/\\partial \\mu{\\big |}_{n,T}=n$ and $\\partial p\/\\partial n{\\big |}_T=n \\partial \\mu\/\\partial n{\\big |}_T$ one obtains\n\\begin{equation}\n\\kappa_T=\\frac{1}{n^2}\\frac{\\partial n}{\\partial \\mu}{\\big |}_{T}.\n\\end{equation}\nThis expression can be directly evaluated in our formalism by numerical differentiation with respect to $\\mu$. \n\nThe approach to the adiabatic compressibility is similar. Using again the Jacobian we have\n\\begin{eqnarray}\n\\nonumber\n\\kappa_S &=& \\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_{s\/n}=\\frac{1}{n}\\frac{\\partial(n,s\/n)}{\\partial(p,s\/n)}\\\\\n&=& \\frac{1}{n}\\frac{\\partial (n,s\/n)}{\\partial (\\mu,T)}\\frac{\\partial(\\mu,T)}{\\partial(p,s\/n)}.\n\\end{eqnarray}\nWe need therefore\n\\begin{equation}\n\\frac{\\partial(n,s\/n)}{\\partial(\\mu,T)}=\\frac{1}{n}\\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_{T}\\frac{\\partial s}{\\partial T}{\\big |}_{\\mu}-\\frac{\\partial n}{\\partial T}{\\big |}_{\\mu}\\frac{\\partial s}{\\partial \\mu}{\\big |}_{T}\\right)\n\\end{equation}\nand also\n\\begin{eqnarray}\n\\nonumber\n\\frac{\\partial(p,s\/n)}{\\partial(\\mu,T)} &=& \\frac{1}{n}{\\bigg (}\\frac{\\partial p}{\\partial \\mu}{\\big |}_{T}\\frac{\\partial s}{\\partial T}{\\big |}_\\mu - \\frac{\\partial p}{\\partial \\mu}{\\big |}_{T} \\frac{s}{n}\\frac{\\partial n}{\\partial T}{\\big |}_\\mu\\\\\n&& -\\frac{\\partial p}{\\partial T}{\\big |}_{\\mu} \\frac{\\partial s}{\\partial \\mu}{\\big |}_T+\\frac{\\partial p}{\\partial T}{\\big |}_\\mu \\frac{s}{n} \\frac{\\partial n}{\\partial \\mu}{\\big |}_T {\\bigg )}\\\\\n\\nonumber\n&=& \\left(\\frac{\\partial s}{\\partial T}{\\big |}_\\mu-2\\frac{s}{n}\\frac{\\partial n}{\\partial T}{\\big |}_\\mu+\\frac{s^2}{n^2}\\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right).\n\\end{eqnarray}\nIn the last equations we used the Maxwell identity $\\frac{\\partial n}{\\partial T}{\\big |}_\\mu=\\frac{\\partial s}{\\partial \\mu}{\\big |}_T$. Combining this we find\n\\begin{equation}\n\\kappa_S=\\frac{\\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_T \\frac{\\partial s}{\\partial T}{\\big |}_\\mu-\\left(\\frac{\\partial n}{\\partial T}{\\big |}_\\mu\\right)^2\\right)}{\\left(n^2 \\frac{\\partial s}{\\partial T}{\\big |}_\\mu-2 s n \\frac{\\partial n}{\\partial T}{\\big |}_\\mu+s^2 \\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right)}.\n\\end{equation}\nSince $\\partial s\/\\partial T{\\big |}_\\mu=(\\partial^2 p\/\\partial T^2){\\big |}_\\mu$ we need to evaluate a second derivative numerically.\nWe plot the isothermal and the adiabatic compressibility in Figs. \\ref{fig:Isothermalcompressibility} and \\ref{fig:Adiabaticcompressibility}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig10.eps}\n\\caption{Isothermal compressibility $\\kappa_T\\, n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:Isothermalcompressibility}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig11.eps}\n\\caption{Adiabatic compressibility $\\kappa_S\\, n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:Adiabaticcompressibility}\n\\end{figure}\n\nFor the isothermal compressibility the temperature dependence is qualitatively different than in Bogoliubov theory already for small temperatures, while there seem to be only quantitative differences for the adiabatic compressibility. The perturbative calculation of the compressibility is difficult since it is diverging in the non-interacting limit $an^{1\/3}\\to 0$. \n\n\n\\subsubsection{Isothermal and adiabatic sound velocity}\nThe sound velocity of a normal fluid under isothermal conditions, i.~e. for constant temperature $T$ is given by\n\\begin{equation}\nv_T^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}{\\big |}_T.\n\\label{eq:singlefluidisothermalsound}\n\\end{equation} \nWe can obtain this directly from the isothermal compressibility\n\\begin{equation}\nM v_T^2=(n \\kappa_T)^{-1}\n\\end{equation}\nas follows from Eq.\\ \\eqref{eq:isothermalkomp}. We plot our result for $v_T^2$ in Fig. \\ref{fig:SingleFluidisothermalsound}, recalling our units $2M=1$ such that $v_T^2$ stands for $2Mv_T^2$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig12.eps}\n\\caption{Isothermal velocity of sound as appropriate for single fluid $v_T^2\/n^{2\/3}=1\/(\\kappa_T\\, n^{5\/3})$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:SingleFluidisothermalsound}\n\\end{figure}\nThis plot also covers the superfluid phase where the physical meaning of $v_T^2$ is partly lost. This comes since the sound propagation there has to be described by more complicated two-fluid hydrodynamics. In addition to the normal gas there is now also a superfluid fraction allowing for an additional oscillation mode. We will describe the consequences of this in the next section.\n\nFor most applications the adiabatic sound velocity is more important then the isothermal sound velocity. Keeping the entropy per particle fixed, we obtain\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}{\\big |}_{s\/n}\n\\label{eq:singlefluidadiabaticsound}\n\\end{equation}\nand therefore\n\\begin{equation}\nM v_S^2 = (n\\kappa_S)^{-1}.\n\\end{equation}\nOur numerical result is plotted in Fig. \\ref{fig:SingleFluidadiabaticsound}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig13.eps}\n\\caption{Adiabatic velocity of sound as appropriate for single fluid $v_S^2\/n^{2\/3}=1\/(\\kappa_S \\,n^{5\/3})$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:SingleFluidadiabaticsound}\n\\end{figure}\nAgain the plot covers both the superfluid and the normal part, but only in the normal phase the object $v_S^2$ has its physical meaning as a sound velocity.\n\n\n\\subsubsection{First and second velocity of sound}\nFor temperatures $00$ (dashed). The inset is a magnification of the little box and shows the energy of the first excited Efimov state.}\n\\label{fig:Efimov}\n\\end{figure}\nFor small Yukawa couplings $\\bar h^2\/\\Lambda\\ll 1$, or narrow resonances we find that the range of scattering length where the trimer is the lowest excitation of the vacuum increases linear with $\\bar h^2$ \\cite{PhysRevLett.93.143201, gogolin:140404}. More explicit, we find $a^{-1}_{c1}=-0.0015\\,\\bar h^2$, $a^{-1}_{c2}=0.0079\\,\\bar h^2$. However, for very broad Feshbach resonances $\\bar h^2\/\\Lambda\\gg1$ the range depends on the ultraviolet scale $a^{-1}_{c1},a^{-1}_{c2}\\sim \\Lambda$. We show this behavior in Fig. \\ref{fig:ascaling}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TFfig3.eps}\n\\caption{Interval of scattering length $a^{-1}_{c1}0$, while the other half has formally $\\bar g^2(k)<0$. We may use the mapping discussed after Eq.\\ \\eqref{eq:subst} to obtain an equivalent picture with positive $\\bar g^2$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\linewidth]{TFfig4.eps}\n\\caption{Limit cycle in the renormalization group flow at the unitarity point $a^{-1}=0$, and for energy at the fermion threshold $\\tilde{\\mu}=\\mu+E=0$. We plot the rescaled gap parameter of the trimer $\\tilde m^2(t)$ (solid) and the rescaled Yukawa coupling $\\tilde g^2(t)$ (dashed). The dotted curves would be obtained from naive continuation of the flow after the point where $\\tilde g^2=0$.}\n\\label{fig:limitcycla}\n\\end{figure}\n\nWe may understand the repetition of states by the following qualitative picture. The coupling $\\tilde m^2$, that is proportional to the energy gap of the trimer, starts on the ultraviolet scale with some positive value. The precise initial value is not important. The Yukawa-type coupling $\\bar g$ vanishes initially so that the trion field $\\chi$ is simply an auxiliary field which decouples from the other fields and is not propagating. However, quantum fluctuations lead to the emergence of a scattering amplitude between the original fermions $\\psi$ and the bosons $\\varphi$. We describe this by the exchange of a composite fermion $\\chi$. \nThis leads to an increase of the coupling $\\tilde g^2$ and a decrease of the trion gap $\\tilde m^2$. At some scale $t_1=\\text{ln}(k_1\/\\Lambda)$ with $k_1^2\\approx -\\mu$ the coupling $\\tilde m^2$ crosses zero which indicates that a trion state $\\chi$ becomes the lowest energy excitation of the vacuum. Indeed, would we consider the flow without modifying the chemical potential, this would set an infrared cutoff that stops the flow at the scale $k_1\\approx\\sqrt{|\\mu|}$ and the trion $\\chi$ would be the gapless propagating particle while the original fermions $\\psi$ and the bosons $\\varphi$ are gapped since they have higher energy. \n\nFollowing the flow further to the infrared, we find that the Yukawa coupling $\\tilde g^2$ decreases again until it reaches the point $\\tilde g^2=0$ at $t=t_1^\\prime$ (see Fig. \\ref{fig:limitcycla}). Naive continuation of the flow below that scale would lead to $\\tilde g^2<0$ and therefore imaginary Yukawa coupling $\\tilde g$. However, since the trion field $\\chi$ decouples from the other fields for $\\tilde g=0$, we are not forced to use the same field $\\chi$ as before. We can simply use another auxiliary field $\\chi_2$ with very large gap $m^2_{\\chi_2}=\\bar m_{\\chi_2}^2\/\\bar A_{\\chi_2}$ to describe the scattering between fermions and bosons on scales $t2\\, l_\\textrm{vdW}$. Therefore the zero-range approximation might be questionable. Since the precise value of $h$ is not known, we use the dependence of our results on $h$ as an estimate of their uncertainty within our truncation \\eqref{eq:action}.\nThe initial values of the couplings $m_\\chi^2$ and $g_i$ are parameters in addition to the scattering lengths which have to be fixed from experimental observation. For equal interaction between atoms $\\psi$ and bosons $\\varphi$ in the UV, the parameter to be fixed is\n\\begin{equation}\n\\lambda^{(3)}=-\\frac{g^2(\\Lambda)}{m_\\chi^2(\\Lambda)}\n\\end{equation}\nwith $g=g_1=g_2=g_3$. Pointlike interactions at the microscopic scale may be realized by $m_\\chi^2(\\Lambda)\\to \\infty$. \n\n\nWe solve the flow equations \\eqref{eq:flowbosonprop}, \\eqref{eq:flowofh}, \\eqref{eq:flowofm} and \\eqref{eq:flowofg} numerically. We find $m_\\chi^2=0$ at $k=0$\nfor some range of $\\lambda^{(3)}$ and $\\mu\\leq 0$ for large enough values of the scattering lengths $a_{12}$, $a_{23}$ and $a_{31}$. This indicates the presence of a bound state of three atoms $\\chi\\widehat{=}\\psi_1\\psi_2\\psi_3$. The binding energy $E_T$ of this bound state is given by the chemical potential, $E_T=3|\\mu|$ with $\\mu$ fixed such that $m_\\chi^2=0$. To compare with the recently performed experimental investigations of $^6$Li \\cite{ottenstein:203202, Huckans2008}, we adapt the initial value $\\lambda^{(3)}$ such that the appearance of this bound state corresponds to a magnetic field $B=125\\, \\text{G}$, the point where strong three-body losses have been observed. Using the same initial value of $\\lambda^{(3)}$ also for other values of the magnetic field, all microscopic parameters are fixed. We can now proceed to the predictions of our model.\n\nFirst we find that the bound state of three atoms exists in the magnetic field region from $B=125\\, \\text{G}$ to $B=498\\, \\text{G}$. The binding energy $E_T$ is plotted as the solid line in the lower panel of Fig. \\ref{fig:EnergiesLi}. We choose here $h^2=100\\, a_0^{-1}$, as appropriate for $^6$Li in the (1,2)-channel close to the resonance, while the shaded region corresponds to $h^2\\in(20\\, a_0^{-1},300\\, a_0^{-1})$. If one includes the contribution to the flow of the atom-dimer interaction arising from box-diagrams by means of a refermionization procedure (see section \\ref{ssect:SU3symmetricmodel}), the flow equations for the Yukawa couplings $g_i$, as in Eq.\\ \\eqref{eq:flowofg}, receive an additionial contribution\n\\begin{eqnarray}\nm_\\chi^2\\left(-\\frac{\\partial_t \\lambda_{ij}^{(3)}}{2 g_j}-\\frac{\\partial_t \\lambda_{il}^{(3)}}{2 g_l}+\\frac{g_i\\partial_t \\lambda_{jl}^{(3)}}{2 g_j g_l}\\right).\n\\end{eqnarray}\nHere we define $(i,j,l)=(1,2,3)$ and permutations thereof and we find\n\\begin{eqnarray}\n\\partial_t\\lambda_{ij}^{(3)}=\\frac{k^5 h_1 h_2 h_3 h_l(9k^2-7\\mu+\\frac{4 m_{\\varphi l}^2}{A_{\\varphi l}})}{6\\pi^2 A_{\\varphi l}(k^2-\\mu)^3(3k^2-2\\mu+\\frac{2 m_{\\varphi l}^2}{A_{\\varphi l}})^2}.\n\\end{eqnarray}\nThis leads to a reduction of the trion binding energy $E_T$ and the result is shown as the dashed line in Fig. \\ref{fig:EnergiesLi}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Lifig3.eps}\n\\caption{{\\itshape Upper panel:} Scattering length $a_{12}$ (solid), $a_{23}$ (dashed) and $a_{31}$ (dotted) as a function of the magnetic field $B$ for $^6$Li. These curves were calculated by P.~S.~Julienne\nand taken from Ref.\n{\\itshape Lower panel:} Binding energy $E_T$ of the three-body bound state $\\chi\\widehat{=}\\psi_1\\psi_2\\psi_3$. The solid line shows the result without the inclusion of box diagrams contributing to the atom-boson interaction and it corresponds to the initial value $h^2=100\\, a_0^{-1}$, while the shaded region gives the result in the range $h^2=20\\, a_0^{-1}$ (upper border) to $h^2=300\\, a_0^{-1}$ (lower border). The dashed line corresponds to the calculated binding energy $E_T$ when the refermionization of the atom-boson interaction is taken into account.}\n\\label{fig:EnergiesLi}\n\\end{figure}\n\n\nAs a second prediction, we present an estimate of the three-body loss coefficient $K_3$ that has been measured in the experiments by Jochim {\\itshape et al.} \\cite{ottenstein:203202} and O'Hara {\\itshape et al.} \\cite{Huckans2008}. For this purpose it is important to note that the fermionic bound state particle $\\chi$ might decay into states with lower energies. These may be some deeply bound molecules not included in our calculation here. In order to include these decay processes we introduce a decay width $\\Gamma_\\chi$ of the trion. We first assume that such a loss process does not depend strongly on the magnetic field $B$ and therefore work with a constant decay width $\\Gamma_\\chi$. \nThe decay width $\\Gamma_\\chi$ appears as an imaginary part of the trion propagator when continued to real time\n\\begin{equation}\nG_\\chi^{-1}=\\omega-\\frac{\\vec p^2}{3}-m_\\chi^2+i\\frac{\\Gamma_\\chi}{2}.\n\\end{equation}\nWe now perform the calculation of the loss for the fermionic energy gap $\\mu=0$, which corresponds to the open channel energy level. In the region from $B=125\\, \\text{G}$ to $B=498\\, \\text{G}$ the energy gap of the trion is then negative $m_\\chi^2<0$.\nThe three-body loss coefficient $K_3$ for arbitrary $\\Gamma_\\chi$ is obtained as follows. The amplitude to form a trion out of three fermions with vanishing momentum and energy is given by $\\sum_{i=1}^3 h_i g_i\/m_{\\varphi i}^2$.\nThe amplitude for the transition from an initial state of three atoms to a final state of the trion decay products (cf. Fig \\ref{fig:treeLossProcess}) further involves the trion propagator that we evaluate in the limit of small momentum $\\vec p^2=(\\sum_i \\vec p_i)^2\\to0$, and small on-shell atom energies $\\omega_i=\\vec p_i^2$, $\\omega=\\sum_i \\omega_i\\to 0$. A thermal distribution of the initial momenta will induce some corrections. Finally, the loss coefficient involves the unknown vertices and phase space factors of the trion decay -- for this reason our computation contains an unknown multiplicative factor $c_K$. In terms of $p$ given by Eq.\\ \\eqref{eq:decayprob} we obtain the three-body loss coefficient\n\\begin{equation}\nK_3=c_K \\, p.\n\\end{equation}\n\nOur result as well as the experimental data points \\cite{ottenstein:203202} are shown in Fig. \\ref{fig:Losscoefficient}. The agreement between the form of the two curves is already quite remarkable.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Lifig4.eps}\n\\caption{Loss coefficient $K_3$ in dependence on the magnetic field $B$ as measured in \\cite{ottenstein:203202} (dots). The solid line is the fit of our model to the experimental curve. We use here a decay width $\\Gamma_\\chi$ that is independent of the magnetic field $B$.}\n\\label{fig:Losscoefficient}\n\\end{figure}\n\nWe have used three parameters, the location of the resonance at $B_0=125\\,\\text{G}$ which we translate into $\\lambda^{(3)}$, the overall amplitude $c_K$ and the decay width $\\Gamma_\\chi$. They are essentially fixed by the peak at $B_0=125\\,\\text{G}$. The extension of the loss rate away from the peak involves then no further parameter. Our estimated decay width $\\Gamma_\\chi$ corresponds to a short lifetime of the trion of the order of $10^{-8}\\,\\textrm{s}$.\n\nOur simple prediction involves a rather narrow second peak around $B_1\\approx500\\,\\text{G}$, where the trion energy becomes again degenerate with the open channel, cf. Fig. \\ref{fig:EnergiesLi}. The width of this peak is fixed so far by the assumption that the decay width $\\Gamma_\\chi$ is independent of the magnetic field. This may be questionable in view of the close-by Feshbach resonance and the fact that the trion may actually decay into the associated molecule-like bound states which have lower energy. We have tested several reasonable approximations, which indeed lead to a broadening or even disappearance of the second peak, without much effect on the intermediate range of fields $150\\,\\text{G}0, \\nonumber\\\\\n\\partial_k \\lambda &=& \\partial_kU''_k{\\big |}_{\\rho_0}. \n\\end{eqnarray}\nTaking a derivative of Eq.\\ \\eqref{2J} with respect to $\\rho$ one obtains for $\\tilde T=0$\n\\begin{eqnarray}\n\\nonumber\nk\\partial_k U_k^\\prime \\!&=&\\! \\eta_{A_\\varphi}(U_k^\\prime-\\rho U_k^{\\prime\\prime})+\\frac{\\sqrt{2}k}{3\\pi^2 Z_\\varphi}\\left(1-\\frac{2}{d+2}\\eta_{A_\\varphi}\\right)\\\\\n\\nonumber\n&&\\!\\!\\times\\! \\left[ 2\\rho (U_k^{\\prime\\prime})^2\\left(s_{\\text{B,Q}}^{(1,0)}\n +3 s_{\\text{B,Q}}^{(0,1)}\\right)+4\\rho^2 U_k^{\\prime\\prime} U_k^{(3)}s_{\\text{B,Q}}^{(0,1)}\\right]\\\\\n&&+\\frac{k}{3\\pi^2}h_\\varphi^2\\,l(\\tilde \\mu)\\, s_{\\text{F,Q}}^{(1)}.\n\\label{eq:Flowu1}\n\\end{eqnarray}\nThe threshold functions $s_{\\text{B,Q}}^{(0,1)}$, $s_{\\text{B,Q}}^{(1,0)}$, and\n$s_{\\text{F,Q}}^{(1)}$ are defined in App. \\ref{sec:FlowoftheeffectivepotentialforBCSBECcrossover} and\ndescribe again the decoupling of the heavy modes. They can be obtained\nfrom $\\rho$ derivatives of $s^{(0)}_{\\text{B}}$ and $s^{(0)}_{\\text{F}}$. Setting $\\rho=0$\nand $\\tilde T\\to0$, we can immediately infer from Eq.\\ \\eqref{eq:Flowu1} the\nrunning of $m^2$ in the symmetric regime.\n\\begin{equation}\nk\\partial_k m^2=k\\partial_k U_k^\\prime =\\eta_{A_\\varphi}m^2 \n+\\frac{k}{3\\pi^2} h^2\\,l(\\tilde \\mu) \\,s^{(1)}_{\\text{F,Q}}(w_3=0).\n\\label{eq:flowmphi}\n\\end{equation}\nOne can see from Eq.\\ \\eqref{eq:flowmphi} that fermionic fluctuations lead to a\nstrong renormalization of the bosonic ``mass term'' $m^2$. In the course\nof the renormalization group flow from large scale parameters $k$\n(ultraviolet) to small $k$ (infrared) the parameter $m^2$\ndecreases strongly. When it becomes zero at some scale $k>0$ the flow\nenters the regime where the minimum of the effective potential $U_k$ is at\nsome nonzero value $\\rho_0$. This is directly related to spontaneous\nbreaking of the $U(1)$ symmetry and to local order. If $\\rho_0\\neq0$ persists\nfor $k\\to0$ this indicates superfluidity.\n\nFor given $\\bar A_\\varphi,Z_\\varphi,h_\\varphi$, Eq.\\ \\eqref{2J} is a nonlinear\ndifferential equation for $U_k$, which depends on two variables $k$\nand $\\rho$. It has to be supplemented by flow equations for\n$\\bar A_\\varphi,Z_\\varphi,h$. The flow equations for the wave\nfunction renormalization $Z_\\varphi$ and the gradient coefficient\n$\\bar A_\\varphi$ cannot be extracted from the effective potential, but are\nobtained from the following projection prescriptions,\n\\begin{eqnarray}\n\\nonumber\n\\partial_t \\bar Z_\\varphi &=& -\\partial_t \n\\frac{\\partial}{\\partial q_0} (\\bar P_\\varphi)_{12}(q_0,0){\\big |}_{q_0=0} ,\\\\\n\\partial_t \\bar A_\\varphi &=& \\partial_t 2 \\frac{\\partial}{\n \\partial \\vec q\\,^2} (\\bar P_\\varphi)_{22}(0,\\vec q){\\big |}_{\\vec q=0},\n\\end{eqnarray} \nwhere the momentum dependent part of the propagator is defined by\n\\begin{equation}\n\\frac{\\delta^2 \\Gamma_k}{\\delta\\bar\\varphi_a(q)\n\\delta\\bar\\varphi_b(q^\\prime)}\\Big|_{\\varphi_1 =\n\\sqrt{2\\rho_0}, \\varphi_2=0} = (\\bar P_\\varphi)_{ab}(q)\\delta(q+q^\\prime). \n\\end{equation}\nThe computation of the flow of the gradient coefficient is rather involved,\nsince the loop depends on terms of different type, $\\sim (\\vec q \\cdot\\vec\np)^2,~ \\vec q\\,^2$, where $\\vec p$ is the loop momentum. An outline of the calculation and explicit expressions can be found in \\cite{Diehl:2008}.\n\n\n\\section{BCS-Trion-BEC Transition}\n\nNow we turn to the truncation used to investigate the model with three fermion species in Eq.\\ \\eqref{eq8:microscopicactiontrionmodel}. For this model the focus will be on the few-body problem where the approximation scheme can be simpler in some respects. We use the following truncation for the average action\n\\begin{eqnarray}\n\\nonumber\n\\Gamma_k&=&\\int_x {\\bigg \\{} \\psi^\\dagger\\left(\\partial_\\tau-\\Delta-\\mu\\right)\\psi+\\varphi^\\dagger\\left(\\partial_\\tau-\\Delta\/2+m_\\varphi^2\\right)\\varphi\\\\\n\\nonumber\n&&+h\\,\\epsilon_{ijk}\\,\\left(\\varphi_i^*\\psi_j\\psi_k-\\varphi_i\\psi_j^*\\psi_k^*\\right)\/2+\\lambda_\\varphi\\left(\\varphi^\\dagger \\varphi\\right)^2\/2\\\\\n\\nonumber\n&&+\\chi^*\\left(\\partial_\\tau-\\Delta\/3+m_\\chi^2\\right)\\chi+g\\left(\\varphi_i^*\\psi_i^*\\chi-\\varphi_i\\psi_i\\chi^*\\right)\\\\\n&&+\\lambda_{\\varphi\\psi}\\left(\\varphi_i^*\\psi_i^*\\varphi_j\\psi_j\\right){\\bigg \\}}.\n\\label{eq:triontruncation}\n\\end{eqnarray}\nHere we use as always natural nonrelativistic units with $\\hbar=k_B=2M=1$, where $M$ is the mass of the original fermions.\nThe integral in Eq.\\ \\eqref{eq:triontruncation} goes over homogeneous space and over imaginary time as appropriate for the Matsubara formalism $\\int_x=\\int d^3x \\int_0^{1\/T}d\\tau$. On the level of the three-body sector, the symmetry of the problem would allow also for a term $\\sim \\psi^\\dagger \\psi\\varphi^\\dagger \\varphi$ in Eq.\\ \\eqref{eq:triontruncation}. This term plays a similar role as for the case of two fermion species, where it was investigated in \\cite{DKS}. The qualitative features of the three-body scattering are dominated by the term $\\sim\\lambda_{\\varphi\\psi}$ in Eq.\\ \\eqref{eq:triontruncation}. The quantitative influence of a term $\\sim \\psi^\\dagger \\psi \\varphi^\\dagger \\varphi$ on the flow equations was also investigated in \\cite{Moroz2008}.\n\nAt the microscopic scale $k=\\Lambda$, we use the initial values of the couplings in Eq.\\ \\eqref{eq:triontruncation} $g=\\lambda_\\varphi=\\lambda_{\\varphi\\psi}=0$ and $m^2_\\chi\\to\\infty$. Then the fermionic field $\\chi$ decouples from the other fields and is only an auxiliary field which is not propagating. However, depending on the parameters of our model we will find that $\\chi$, which describes a composite bound state of three original fermions $\\chi=\\psi_1\\psi_2\\psi_3$, becomes a propagating degree of freedom in the infrared. The initial values of the boson energy gap $\\nu_\\varphi$ and the Yukawa coupling $h$ will determine the scattering length $a$ between fermions and the width of the resonance, see below. The pointlike limit (broad resonance) corresponds to $\\nu_\\varphi\\to\\infty$, $h^2\\to\\infty$ where the limits are taken such that the effective renormalized four fermion interaction remains fixed. In Eq.\\ \\eqref{eq:triontruncation} we use renormalized fields $\\varphi=\\bar A_\\varphi^{1\/2}(k)\\, \\bar{\\varphi}$, $\\psi=\\bar A_\\psi^{1\/2}(k)\\,\\bar \\psi$, $\\chi = \\bar A_\\chi^{1\/2}(k)\\,\\bar \\chi$, with $\\bar A_\\varphi(\\Lambda)=\\bar A_\\psi(\\Lambda)=\\bar A_\\chi(\\Lambda)=1$, and renormalized couplings $m_\\varphi^2=\\bar{m}_\\varphi^2\/\\bar{A}_\\varphi$, $h=\\bar{h}\/(\\bar{A}_\\varphi^{1\/2}\\bar A_\\psi)$, $\\lambda_\\varphi=\\bar \\lambda_\\varphi\/\\bar A_\\varphi^2$, $m_\\chi^2=\\bar m_\\chi^2\/\\bar A_\\chi$, $g=\\bar g \/(\\bar A_\\chi^{1\/2}\\bar A_\\varphi^{1\/2}\\bar A_\\chi^{1\/2})$, and $\\lambda_{\\varphi\\psi}=\\bar \\lambda_{\\varphi\\psi}\/(\\bar A_\\varphi \\bar A_\\psi)$.\n\nTo derive the flow equations for the couplings in Eq.\\ \\eqref{eq:triontruncation} we have to specify an infrared regulator function $R_k$. Here we use the particularly simple function\n\\begin{equation}\nR_{k}=r(k^2-\\vec p^2)\\theta(k^2-\\vec p^2),\n\\label{eq:cutofftrions}\n\\end{equation}\nwhere $r=1$ for the fermions $\\psi$, $r=1\/2$ for the bosons $\\varphi$, and $r=1\/3$ for the composite fermionic field $\\chi$. This choice has the advantage that we can derive analytic expressions for the flow equations and that it is optimized in the sense of \\cite{Litim2000b}.\n\\subsubsection{Symmetries as a guiding principles}\n\nHow should one choose a truncation? The choice of the appropriate ansatz for the flowing action is certainly one of the most important points for someone who wants to work with the flow equation method in praxis. Besides the necessary physical insight there is one major guiding principle: symmetries. As will be discussed in chapter \\ref{ch:Symmetries} the flowing action $\\Gamma_k$ respects the same symmetries as the microscopic action $S$ if no anomalies of the functional integral measure are present and if the cutoff term $\\Delta S_k$ is also invariant. In the notation of Eq.\\ \\eqref{eq6:truncationexpansion} this implies that the coefficient $g_i$ of an operator ${\\mathcal O}_i[\\Phi]$ that is not invariant under all symmetries will not be generated by the flow equation such that $g_i=0$ for all $k$. As an example we consider the microscopic action of a Bose gas\n\\begin{equation}\nS=\\int \\varphi^* (\\partial_\\tau-\\Delta-\\mu)\\varphi +\\frac{1}{2}\\lambda_\\varphi (\\varphi^*\\varphi)^2.\n\\label{eq6:actionBosegas}\n\\end{equation}\nIt is invariant under the global U(1) symmetry\n\\begin{eqnarray}\n\\nonumber\n\\varphi &\\to& e^{i\\alpha} \\varphi \\\\\n\\varphi^* &\\to& e^{-i\\alpha} \\varphi^*. \n\\end{eqnarray}\nThis implies that only operators that are invariant under this transformation may appear in the flowing action $\\Gamma_k[\\Phi]$. For example, the part that describes homogeneous fields, the effective potential is of the form\n\\begin{equation}\n\\Gamma_k = \\ldots + \\int_x U(\\varphi^*\\varphi)\n\\end{equation}\nwhere $U(\\rho)$ is a function of the U(1)-invariant combination $\\rho=\\varphi^*\\varphi$, only. The action in Eq.\\ \\eqref{eq6:actionBosegas} has more symmetries such as translation, rotation or, at zero temperature, Galilean invariance. \n\nA useful strategy to find a sensible truncation is to start from the microscopic action $S$ or an effective action $\\Gamma$ calculated in some (perturbative) approximation scheme such as for example mean-field theory. One now renders the appearing coefficients to become $k$-dependent ``running couplings'' and adds also additional terms after checking that they are allowed by the symmetries of the microscopic action. \n \n\\subsubsection{Separation of scales}\n\nSome symmetries are realized only in some range of the renormalization group flow. For example, Galilean symmetry is broken explicitly by the thermal heat bath for $T>0$. Nevertheless, for $k^2\\gg T$ the flowing action $\\Gamma_k$ (or its real-time version obtained from analytic continuation) will still be invariant under Galilean boost transformations. This comes since the scale parameter $k$ sets the infrared scale on the right hand side of the flow equation. As long as $k^2\/T$ is large, the flow equations are essentially the same as for $T\\to0$. In other words, the flow only ``feels'' the temperature once the scale $k$ is of the same order of magnitude $k^2\\approx T$. On the other side, for $k^2\\ll T$ the flow equations may simplify again. Now they are equivalent to those obtained in the large temperature limit $T\\to \\infty$. Different symmetries may apply to the action in this limit. This separation of scales is often very useful for practical purposes. In different regimes of the flow different terms are important, while others might be neglected. For example, the universal critical properties such as the critical exponents or amplitude ratios can be calculated in the framework of the classical theory, i.~e. in the large temperature limit $T\\to\\infty$. The flow equations in this limit are much simpler then the ones obtained for arbitrary temperature $T$. \n\nThe scale-separation is also useful for the fixing of the initial coupling constants at the initial scale $k=\\Lambda$. If this scale is much larger then the temperature $\\Lambda^2\\gg T$ and the relevant momentum scale for the density, the inverse interparticle distance $\\Lambda\\gg n^{1\/3}$, the initial flow is the same as in vacuum where $T=n=0$. One can then also use the same initial values for most of the couplings and only change the temperature and the chemical potential appropriately to describe points in the phase diagram that correspond to $T>0$ and $n>0$. \n\n\\subsubsection{Derivative expansion}\n\nA central part of a truncation is the form of the propagator. It follows from the second functional derivative of the flowing action. For the example of a Bose gas one has in the normal phase\n\\begin{equation}\nG_k^{-1}(p) \\,\\delta(p-p^\\prime) = \\frac{\\delta}{\\delta \\varphi^*(p)} \\frac{\\delta}{\\delta \\varphi(p^\\prime)} \\Gamma_k[\\Phi].\n\\label{eqn6:derexpansion}\n\\end{equation}\nThe inverse propagator $G_k^{-1}$ may be a quite complicated function of the momentum $p$ which consists of the spatial momentum and the (Matsubara-) frequency, $p=(p_0, \\vec p)$. From rotational invariance it follows that $G_k^{-1}$ depends on the spatial momentum only in the invariant combination $\\vec p^2$. At zero temperature it follows from Galilean invariance that $G_k^{-1}$ is a (analytic) function of the combination $ip_0+\\vec p^2$, provided that Galilean invariance is not broken by the cutoff. \n\nUsing a derivative expansion, one truncates the flowing action in the form\n\\begin{equation}\n\\Gamma_k = \\int_x \\varphi^* (Z\\partial_\\tau-A\\vec \\nabla^2 - V \\partial_\\tau^2+\\ldots)\\varphi + U(\\varphi^*\\varphi).\n\\end{equation}\nThe ``kinetic coefficients'' $Z$, $A$, $V$ etc.\\ depend on the scale parameter $k$ and for more advanced approximations also on the U(1) invariant combination $\\rho=\\varphi^* \\varphi$. One can improve the expansion in Eq.\\ \\eqref{eqn6:derexpansion} by promoting the coefficients $Z$, $A$, $V$ etc.\\ to functions of $p_0$ and $\\vec p^2$. \n\nIn praxis one usually neglects terms higher then quadratic in the momenta. Nevertheless, derivative expansion often leads to quite good results. The reason is the following. On the right hand side of the flow equation the cutoff insertion $R_k$ in the propagator $(\\Gamma^{(2)}+R_k)^{-1}$ suppresses the contribution of the modes with small momenta. On the other side, the cutoff derivative $\\partial_k R_k$ suppresses the contribution of very large momenta provided that $R_k(q)$ falls of sufficiently fast for large $q$. Effectively mainly modes with momenta of the order $k^2$ contribute. It would therefore be sensible to use on the right hand side of the flow equations the coefficients\n\\begin{equation}\nZ(p_0 = k^2,\\vec p^2=k^2), A(p_0=k^2,\\vec p^2=k^2), \\ldots.\n\\end{equation}\nOne main effect of the external frequencies and momenta in $Z(p_0,\\vec p^2)$ etc.\\ is to provide an infrared cutoff scale of order $\\text{Max}(p_0,\\vec p^2)$. Such an infrared cutoff scale is of course also provided by $R_k$ itself and one might therefore also work with the $k$-dependent couplings\n\\begin{equation}\nZ(p_0=0,\\vec p^2=0), A(p_0=0,\\vec p^2=0),\\ldots.\n\\end{equation}\nWe emphasize that it is important that the cutoff $R_k(q)$ falls off sufficiently fast for large $q$. If this is not the case, the derivative expansion might lead to erroneous results since the kinetic coefficients as appropriate for small momenta and frequencies are then also used for large momenta and frequencies. Only when the scale derivative $\\partial_k R_k$ provides for a sufficient ultraviolet cutoff does the derivative expansion work properly.\n\\subsubsection{Two-body problem}\n\nThe two-body problem is best solved in terms of the bare couplings. Their flow\nequations read\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\bar{m}_\\varphi^2 &=& \\frac{\\bar{h}_\\varphi^2}{6\\pi^2\n k^3}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\nonumber\n\\partial_k \\bar Z_\\varphi &=& -\\frac{\\bar{h}_\\varphi^2}{6\\pi^2\n k^5}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\nonumber\n\\partial_k \\bar A_\\varphi &=& -\\frac{\\bar{h}_\\varphi^2}{6\\pi^2 k^5}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\partial_k \\bar{h}_\\varphi &=&0.\n\\label{eq:vacuumflow}\n\\end{eqnarray}\nThe flow in the two-body sector is driven by fermionic diagrams only. There\nis no renormalization of the Yukawa coupling $\\bar h$. The equations \\eqref{eq:vacuumflow} are solved by direct\nintegration with the result\n\\begin{eqnarray}\n\\nonumber\n\\bar{m}_\\varphi^2(k) &=&\n\\bar{m}_\\varphi^2(\\Lambda)\\\\\n\\nonumber\n&&-\\theta(\\Lambda^2+\\mu)\\frac{\\bar{h}_\\varphi^2}{6\\pi^2}{\\bigg\n [}\\sqrt{\\Lambda^2+\\mu}\\,\\left(1-\\frac{\\mu}{2\\Lambda^2}\\right)\n-\\frac{3}{2}\\sqrt{-\\mu}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]} \\\\\n\\nonumber\n&&+\\theta(k^2+\\mu)\\frac{\\bar{h}_\\varphi^2}{6\\pi^2}{\\bigg\n [}\\sqrt{k^2+\\mu}\\,\\left(1-\\frac{\\mu}{2k^2}\\right) \n-\\frac{3}{2}\\sqrt{-\\mu}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\ \n\\nonumber\n\\bar Z_\\varphi(k) &=&\n\\bar Z_\\varphi(\\Lambda)\\\\\n\\nonumber\n&& -\\theta(\\Lambda^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{\\Lambda^2+\\mu}\n\\,\\frac{\\left(5\\Lambda^2+2\\mu\\right)}{\\Lambda^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber\n&& +\\theta(k^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{k^2+\\mu}\n\\,\\frac{\\left(5k^2+2\\mu\\right)}{k^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber \n\\bar A_\\varphi(k) &=& \\bar A_\\varphi(\\Lambda)\\\\\n\\nonumber\n&& -\\theta(\\Lambda^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{\\Lambda^2+\\mu}\\,\n\\frac{\\left(5\\Lambda^2+2\\mu\\right)}{\\Lambda^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber\n&& +\\theta(k^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{k^2+\\mu}\\,\n\\frac{\\left(5k^2+2\\mu\\right)}{k^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}.\\\\\n\\label{eq:vacuumsolutions}\n\\end{eqnarray}\nHere, $\\Lambda$ is the initial ultraviolet scale. Let us discuss the initial value for the boson mass. It is given by\n\\begin{equation}\n\\bar{m}_\\varphi^2(\\Lambda)=\\nu(B) -2\\mu+\\delta\n\\nu(\\Lambda). \\label{eq:new41} \n\\end{equation}\nThe detuning $\\nu(B)= \\mu_{\\text{M}} (B-B_0)$ describes the energy level of the microscopic state represented by the field $\\varphi$ with respect to the fermionic state $\\psi$. At a Feshbach resonance, this energy shift can be tuned by the magnetic field $B$, $\\mu_{\\text{M}}$ denotes the magnetic moment of the field $\\varphi$, and $B_0$ is the resonance position. Physical observables such as the scattering length and the binding energy are obtained from the effective action and are therefore related to the coupling constants at the infrared scale $k=0$. The quantity $\\delta\\nu(\\Lambda)$ denotes a renormalization counter term that has to be adjusted conveniently, see below.\n\n\n\\subsubsection{Renormalization}\n\nWe next show that close to a Feshbach resonance the microscopic parameters \n$\\bar m_{\\varphi,\\Lambda}\\equiv\\bar m_\\varphi^2(k=\\Lambda)$ and $\\bar\nh_{\\varphi,\\Lambda}^2\\equiv\\bar h_\\varphi^2(k=\\Lambda)$\nare related to $B-B_0$ and $a$ by two simple relations\n\\begin{equation}\n\\bar m_{\\varphi,\\Lambda}^2=\\mu_\\text{M}(B-B_0)-2\\mu+\\frac{\\bar h_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda\n\\label{eq:mvarphiLambda}\n\\end{equation}\nand \n\\begin{equation}\na=-\\frac{\\bar h_{\\varphi, \\Lambda}^2}{8\\pi \\mu_\\text{M}(B-B_0)}.\n\\label{eq:hvarphiLambda}\n\\end{equation}\nAway from the Feshbach resonance the Yukawa coupling may depend on $B$, $\\bar\nh_\\varphi^2(B)=\\bar h_\\varphi^2+c_1(B-B_0)+\\dots$. Also the microscopic\ndifference of energy levels between the open and closed channel may show\ncorrections to the linear $B$-dependence,\n$\\nu(B)=\\mu_\\text{M}(B-B_0)+c_2(B-B_0)^2+\\dots$ or $\\mu_\\text{M}\\to\n\\mu_\\text{M}+c_2(B-B_0)+\\dots$. Using $\\bar h_\\varphi^2(B)$ and\n$\\mu_\\text{M}(B)$ our formalism can easily be adapted to a more general\nexperimental situation away from the Feshbach resonance. The relations in\nEqs. \\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} hold for all\nchemical potentials $\\mu$ and temperatures $T$. For a different choice of the\ncutoff function the coefficient $\\delta\\nu(\\Lambda)$ being the term linear in $\\Lambda$ in Eq.\\ \\eqref{eq:mvarphiLambda} might be modified.\n\nWe want to connect the bare parameters $\\bar m_{\\varphi,\\Lambda}^2$ and $\\bar\nh_{\\varphi,\\Lambda}^2$ with the magnetic field $B$ and the scattering length $a$ for\nfermionic atoms as renormalized parameters. In our units, $a$ is related to\nthe effective interaction $\\lambda_{\\psi,\\text{eff}}$ by\n\\begin{equation}\na=\\frac{\\lambda_{\\psi,\\text{eff}}}{8\\pi}.\n\\end{equation}\nThe fermion interaction\n$\\lambda_{\\psi,\\text{eff}}$ is determined by the molecule exchange process in\nthe limit of vanishing spatial momentum\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{\\bar\n h_{\\varphi,\\Lambda}^2}{\\bar{P}_\\varphi(\\omega,\\vec{p}^2=0,\\mu)}. \n\\label{eqlambdaeffmoleculeexchange}\n\\end{equation}\nEven though \\eqref{eqlambdaeffmoleculeexchange} is a tree-level process, it is not an approximation, since $\\bar{P}_\\varphi\\equiv\\bar P_{\\varphi}|_{k\\to0}$ denotes the full bosonic propagator which includes all fluctuation effects. The frequency in Eq.\\ \\eqref{eqlambdaeffmoleculeexchange} is the sum of the frequency of the incoming fermions which in turn is determined from the on-shell condition\n\\begin{equation}\n\\omega=2\\omega_\\psi=-2\\mu.\n\\label{eqinitialvalueofmass}\n\\end{equation} \n\nOn the BCS side we have\n$\\mu=0$ and find with \n\\begin{equation}\n\\bar P_\\varphi(\\omega=0,\\vec q=0)=\\bar m_\\varphi^2(k=0)\\equiv \\bar m^2_{\\varphi,0}\n\\end{equation}\nthe relation\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{\\bar m_{\\varphi,0}^2},\n\\end{equation}\nwhere $\\bar m_{\\varphi,0}^2=\\bar m_\\varphi^2(k=0)$. For the bosonic mass terms at $\\mu=0$, we can read off from\nEqs. \\eqref{eq:vacuumsolutions} and \\eqref{eq:new41} that \n\\begin{equation}\n \\bar{m}_{\\varphi,0}^2=\\bar{m}_{\\varphi,\\Lambda}^2\n -\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2} \\Lambda \n = \\mu_{\\text{M}}(B-B_0)+\\delta \\nu(\\Lambda)\n -\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda.\n\\end{equation}\nTo fulfill the resonance condition $a\\to\\pm\\infty$ for $B=B_0$, $\\mu=0$,\nwe choose\n\\begin{equation}\n\\delta \\nu(\\Lambda)=\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda.\n\\end{equation}\nThe shift $\\delta\\nu (\\Lambda)$ provides for the additive UV renormalization\nof $\\bar{m}_\\varphi^2$ as a relevant coupling. It is exactly canceled by the\nfluctuation contributions to the flow of the mass. This yields the general\nrelation \\eqref{eq:mvarphiLambda} (valid for all $\\mu$) between the bare mass\nterm $\\bar m_{\\varphi,\\Lambda}^2$ and the magnetic field. On the BCS side we\nfind the simple vacuum relation\n\\begin{equation}\n\\bar m_{\\varphi,0}^2=\\mu_\\text{M}(B-B_0).\n\\end{equation}\nFurthermore, we obtain for the fermionic\nscattering length\n\\begin{equation}\na=-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{8\\pi \\mu_{\\text{M}}(B-B_0)}.\n\\label{eqscatterinlengthandmageticfield}\n\\end{equation}\nThis equation establishes Eq.\\ \\eqref{eq:hvarphiLambda} and shows that\n$\\bar{h}_{\\varphi,\\Lambda}^2$ determines the width of the resonance. We have thereby\nfixed all parameters of our model and can express $\\bar m_{\\varphi,\\Lambda}^2$\nand $\\bar h_{\\varphi,\\Lambda}^2$ by $B-B_0$ and $a$. The relations\n\\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} remain valid also at\nnonzero density and temperature. They fix the ``initial values'' of the flow\n($\\bar h_\\varphi^2\\to \\bar h_{\\varphi,\\Lambda}^2$) at the microscopic scale\n$\\Lambda$ in terms of experimentally accessible quantities, namely $B-B_0$ and\n$a$.\n\nOn the BEC side, we encounter $\\mu<0$ and thus $\\omega>0$. We therefore need\nthe bosonic propagator for $\\omega\\neq 0$. Even though we have computed\ndirectly only quantities related to $\\bar P_\\varphi$ at $\\omega=0$ and\nderivatives with respect to $\\omega$ ($Z_\\varphi$), we can obtain information\nabout the boson propagator for nonvanishing frequency by using the semilocal\n$U(1)$ invariance described in section \\ref{sec:Derivativeexpansionandwardidentities}. In momentum space,\nthis symmetry transformation results in a shift of energy levels\n\\begin{eqnarray}\n\\nonumber\n\\psi(\\omega, \\vec{p}) &\\to& \\psi(\\omega-\\delta,\\vec{p})\\\\\n\\nonumber\n\\varphi(\\omega,\\vec{p}) &\\to& \\varphi(\\omega-2\\delta,\\vec{p})\\\\\n\\mu &\\to& \\mu+\\delta.\n\\end{eqnarray}\nSince the effective action is invariant under this symmetry, it follows for\nthe bosonic propagator that\n\\begin{equation}\n\\bar{P}_\\varphi(\\omega,\\vec{p},\\mu)\n=\\bar{P}_\\varphi(\\omega-2\\delta,\\vec{p},\\mu+\\delta).\n\\end{equation}\nTo obtain the propagator needed in Eq.\\ \\eqref{eqlambdaeffmoleculeexchange}, we\ncan use $\\delta=-\\mu$ and find as in Eq.\\ \\eqref{eqscatterinlengthandmageticfield}\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}\n=-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{\\bar{P}_\\varphi(\\omega=0,\\vec{p}^2=0,\\mu=0)}\n=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{\\mu_\\text{M}(B-B_0)}.\n\\end{equation}\nThus the relations \\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} for\nthe initial values $\\bar m_{\\varphi,\\Lambda}$ and $\\bar h_{\\varphi,\\Lambda}^2$\nin terms of $B-B_0$ and $a$ hold for both the BEC and the BCS side of the\ncrossover.\n\n\n\\subsubsection{Binding energy}\n\nWe next establish the relation between the molecular binding energy\n$\\epsilon_\\text{M}$, the scattering length $a$, and the Yukawa coupling $\\bar\nh_{\\varphi,\\Lambda}^2$. From Eq.\\ \\eqref{eq:vacuumsolutions}, we obtain for\n$k=0$ and $\\mu\\leq 0$\n\\begin{eqnarray}\\label{mphiFinal}\n\\nonumber\n\\bar{m}_{\\varphi,0}^2 &=& \\mu_{\\text{M}} (B-B_0)-2\\mu\\\\\n\\nonumber\n&& +\\frac{\\bar h_{\\varphi,\\Lambda}^2}{6\\pi^2} {\\Bigg [}\\Lambda-\\sqrt{\\Lambda^2+\\mu}\n\\left(1-\\frac{\\mu}{2\\Lambda^2}\\right)\\\\\n&& +\\frac{3}{2}\\sqrt{-\\mu} \\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\Bigg ]}.\n\\end{eqnarray}\nIn the limit $\\Lambda\/\\sqrt{-\\mu}\\to\\infty$ this yields\n\\begin{equation}\n\\bar m_{\\varphi,0}^2 = \\mu_{\\text{M}} (B-B_0)-2\\mu+\\frac{\\bar{h}_{\\varphi,\\Lambda}^2\\sqrt{-\\mu}}{8\\pi}.\n\\end{equation}\nTogether with Eq.\\ \\eqref{eqscatterinlengthandmageticfield}, we can deduce\n\\begin{equation}\na=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{8\\pi \\left( \\bar m_{\\varphi,0}^2 +2\\mu \n-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2\\sqrt{-\\mu}}{8\\pi}\\right)},\n\\end{equation}\nwhich holds in the vacuum for all $\\mu$. On the BEC side where $\\bar\nm_{\\varphi,0}^2=0$ this yields\n\\begin{equation}\n a=\\frac{1}{\\sqrt{-\\mu}\\left(1+\\frac{16 \\pi}{\\bar h_{\\varphi,\\Lambda}^2}\\sqrt{-\\mu}\\right)}.\n\\label{ScattLength}\n\\end{equation}\nThe binding energy of the bosons is given by the difference between the\nenergy for a boson ${\\bar{m}_\\varphi^2}\/{\\bar{Z}_\\varphi}$ and the energy\nfor two fermions $-2\\mu$. On the BEC side, we can use $\\bar{m}_{\\varphi,0}^2=0$ and\nobtain\n\\begin{equation}\n\\epsilon_{\\text{M}}=\\frac{\\bar{m}_\\varphi^2}{\\bar{Z}_\\varphi}+2\\mu\\Big|_{k\\to0}=2\\mu.\n\\label{eq:bindingenergy}\n\\end{equation}\nFrom Eqs.\\ \\eqref{ScattLength} and \\eqref{eq:bindingenergy} we find a relation\nbetween the scattering length $a$ and the binding energy $\\epsilon_\\text{M}$\n\\begin{equation}\n\\frac{1}{a^2}=\\frac{-\\epsilon_\\text{M}}{2}\n+(-\\epsilon_\\text{M})^{3\/2} \\frac{4\\sqrt{2}\\pi}{\\bar h_{\\varphi,\\Lambda}^2}\n+(-\\epsilon_\\text{M})^2\\frac{(8\\pi)^2}{\\bar h_{\\varphi,\\Lambda}^4}.\n\\label{eq:scatteringlengthandbindingenergy}\n\\end{equation}\nIn the broad resonance limit $\\bar{h}_{\\varphi,\\Lambda}^2\\to\\infty$, this is\njust the well-known relation between the scattering length $a$ and the binding\nenergy $\\epsilon_{\\text{M}}$ of a dimer (see for example \\cite{BraatenHammer})\n\\begin{equation}\n\\epsilon_{\\text{M}}=-\\frac{2}{a^2}=-\\frac{1}{M a^2}.\n\\label{eq:scatteringlengthbroad}\n\\end{equation}\nThe last two terms in Eq.\\ \\eqref{eq:scatteringlengthandbindingenergy} give\ncorrections to Eq.\\ \\eqref{eq:scatteringlengthbroad} for more narrow\nresonances.\n\nThe solution of the two-body problem turns out to be exact as expected. In our\nformalism, this is reflected by the fact that the two-body sector decouples\nfrom the flow equations of the higher-order vertices: no higher-order\ncouplings such as $\\lambda_\\varphi$ enter the set of equations\n(\\ref{eq:vacuumflow}). Extending the truncation to even higher order vertices or\nby including a boson-fermion vertex $\\psi^\\dagger \\psi \\varphi^\\ast\\varphi$\ndoes not change the situation.\n\n\\subsubsection{Dimer-Dimer Scattering}\n\\label{DimerDimer}\n\nSo far we have considered the sector of the theory up to order\n$\\varphi^\\ast\\psi\\psi$, which is equivalent to the fermionic two-body problem\nwith pointlike interaction in the limit of broad resonances. Higher-order\ncouplings, in particular the four-boson coupling\n$\\lambda_\\varphi(\\varphi^\\ast\\varphi)^2$, do not couple to the two-body\nsector. Nevertheless, a four-boson coupling emerges dynamically from the\nrenormalization group flow. \nIn vacuum we have $\\rho_0=0$ and $\\lambda_\\varphi$ is defined as $\\lambda_\\varphi=U^{\\prime\\prime}_k(0)$. The flow equation for $\\lambda_\\varphi$ can be found by taking the $\\rho$-derivative of Eq.\\ \\eqref{eq:Flowu1}\n\\begin{eqnarray}\n\\nonumber\nk \\partial_k \\lambda_\\varphi &=& 2 \\eta_{A_\\varphi} U_k^{\\prime\\prime} - \\frac{\\sqrt{2} k^3}{3\\pi^2 S_\\varphi}\\left(1-\\frac{2}{d+2}\\eta_{A_\\varphi}\\right)\\\\\n\\nonumber\n&&\\times 2 (U_k^{\\prime\\prime})^2 \\left(s_{B,Q}^{(1,0)}+3 s_{B,Q}^{(0,1)}\\right)+\\frac{h_\\varphi^4}{3\\pi^2 k^3} s_{F,Q}^{(2)}\\\\\n\\nonumber\n&=& 2\\eta_{A_\\varphi} \\lambda_\\varphi + \\frac{\\sqrt{2} k^5 \\lambda_\\varphi^2}{3\\pi^2\\, S_\\varphi\\, (m_\\varphi^2+k^2)^2}(1-2\\eta_{A_\\varphi}\/5)\\\\\n&&-\\frac{h_\\varphi^4\\,\\theta(\\mu+k^2)\\,(\\mu+k^2)^{3\/2}}{4\\pi^2k^6}.\n\\end{eqnarray} \nThere are contributions from fermionic and bosonic vacuum fluctuations, but no contribution from higher $\\rho$ derivatives of $U$. The fermionic diagram generates a four-boson coupling even for zero initial value. This coupling then feeds back into the flow equation via the bosonic diagram.\n\nThe scattering lengths are related to the corresponding couplings by the relation (cf. \\cite{Diehl:2007th})\n\\begin{eqnarray}\n \\frac{a_{\\text{M}}}{a}=2\\,\n \\frac{\\lambda_\\varphi}{\\lambda_{\\psi,\\text{eff}}}, \n \\quad \\lambda_{\\psi,\\text{eff}}= 8\\pi a.\n\\end{eqnarray}\nOmitting the bosonic fluctuations, a direct integration yields the mean field result $a_{\\text{M}}\/a=2$. This value is lowered when the bosonic fluctuations are taken into account. With our truncation and choice of cutoff one finds $a_{\\text{M}}\/a =0.718$. The calculation can be improved by extending the truncation to\ninclude a boson-fermion vertex $\\lambda_{\\varphi\\psi}$ which describes the\nscattering of a dimer with a fermion \\cite{DKS}. Inspection of the diagrammatic structure\nshows that this vertex indeed couples into the flow equation for $\\lambda_\\varphi$.\n\nThe ratio $a_M\/a$ has been computed by other methods. Diagrammatic approaches give $a_{\\text{M}}\/a =0.75(4)$\n\\cite{PhysRevB.61.15370}, whereas the solution of the 4-body Schr\\\"{o}dinger\nequation yields $a_{\\text{M}}\/a =0.6$ \\cite{PhysRevLett.93.090404}, confirmed in QMC\nsimulations \\cite{PhysRevLett.93.200404} and with diagrammatic techniques \\cite{Brodsky2005}.\n\\section{Bose gas in three dimensions}\n\\label{sec:Bosegasinthreedimensions}\nA gas of non-relativistic bosons with a repulsive pointlike interaction is one of the simplest interacting statistical systems. Since the first experimental realization \\cite{Andersonetal1995, PhysRevLett.75.1687, PhysRevLett.75.3969} of Bose-Einstein condensation (BEC) \\cite{Einstein1924, Einstein1925, Bose1924} with ultracold gases of bosonic atoms, important experimental advances have been achieved, for reviews see \\cite{RevModPhys.71.463, RevModPhys.73.307, morsch:179, bloch:885, PethickSmith2002, PitaevsikiiStringari2003}. Thermodynamic observables like the specific heat \\cite{1367-2630-8-9-189} or properties of the phase transition like the critical exponent $\\nu$ \\cite{Donneretal2007} have been measured in harmonic traps. Still, the theoretical description of these apparently simple systems is far from being complete.\n\nFor ultracold dilute non-relativistic bosons in three dimensions, Bogoliubov theory gives a successful description of most quantities of interest \\cite{Bogoliubov}. This approximation breaks down, however, near the critical temperature for the phase transition, as well as for the low temperature phase in lower dimensional systems, due to the importance of fluctuations. One would therefore like to have a systematic extension beyond the Bogoliubov theory, which includes the fluctuation effects beyond the lowest order in a perturbative expansion in the scattering length. Such extensions have encountered obstacles in the form of infrared divergences in various expansions \\cite{Beliaev1958, Gavoret1964, Nepomnyashchii1975}. Only recently, a satisfactory framework has been found to cure these problems \\cite{PhysRevLett.78.1612, PhysRevB.69.024513, Wetterich:2007ba}.\nIn this thesis, we extend this formalism to a nonvanishing temperature. We present a quantitative rather accurate picture of Bose-Einstein condensation in three dimensions and find that the Bogoliubov approximation is indeed valid for many quantities. The same method is also applied for two spatial dimensions (see section \\ref{sec:Bosegasintwodimensions}) and can also be applied for one dimension.\n\nFor dilute non-relativistic bosons in three dimensions with repulsive interaction we find an upper bound on the scattering length $a$. This is similar to the \"triviality bound\" for the Higgs scalar in the standard model of elementary particle physics. As a consequence, the scattering length is at most of the order of the inverse effective ultraviolet cutoff $\\Lambda^{-1}$, which indicates the breakdown of the pointlike approximation for the interaction at short distances. Typically, $\\Lambda^{-1}$ is of the order of the range of the Van der Waals interaction. For dilute gases, where the interparticle distance $n^{-1\/3}$ is much larger than $\\Lambda^{-1}$, we therefore always find a small concentration $c=a n^{1\/3}$. This provides for a small dimensionless parameter, and perturbation theory in $c$ becomes rather accurate for most quantities. For typical experiments with ultracold bosonic alkali atoms one has $\\Lambda^{-1}\\approx 10^{-7} \\,\\text{cm}$, $n^{1\/3}\\approx 10^4 \\,\\text{cm}^{-1}$, such that $c\\lesssim 10^{-3}$ is really quite small.\n\nBosons with pointlike interactions can also be employed for an effective description of many quantum phase transitions at zero temperature, or phase transitions at low temperature $T$. In this case, they correspond to quasi-particles, and their dispersion relation may differ from the one of non-relativistic bosons, $\\omega=\\frac{\\vec{p}^2}{2M}$. We describe the quantum phase transitions for a general microscopic dispersion relation, where the inverse classical propagator in momentum and frequency space takes the form $G_0^{-1}=-S\\omega-V\\omega^2+\\vec{p}^2$ (in units where the particle mass $M$ is set to $1\/2$). We present the quantum phase diagram at $T=0$ in dependence on the scattering length $a$ and a dimensionless parameter $\\tilde{v}\\sim V\/S^2$, which measures the relative strength of the term quadratic in $\\omega$ in $G_0^{-1}$. In the limit $S\\rightarrow 0$ ($\\tilde{v}\\rightarrow\\infty$) our model describes relativistic bosons.\n\n\\subsubsection{Lagrangian}\n\nOur microscopic action describes nonrelativistic bosons, with an effective interaction between two particles given by a contact potential. It is assumed to be valid on length scales where the microscopic details of the interaction are irrelevant and the scattering length is sufficient to characterize the interaction. The microscopic action reads\n\\begin{equation}\nS[\\varphi]=\\int_x \\,{\\Big \\{}\\varphi^*\\,(S\\partial_\\tau-V\\partial_\\tau^2-\\Delta-\\mu)\\,\\varphi\\,+\\,\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2{\\Big \\}},\n\\label{microscopicaction}\n\\end{equation}\nwith \n\\begin{equation}\nx=(\\tau,\\vec{x}), \\,\\,\\int_x=\\int_0^{\\frac{1}{T}}d\\tau\\int d^3x.\n\\end{equation}\nThe integration goes over the whole space as well as over the imaginary time $\\tau$, which at finite temperature is integrated on a circle of circumference $\\beta=1\/T$ according to the Matsubara formalism. We use natural units $\\hbar=k_B=1$. We also scale time and energy units with appropriate powers of $2M$, with $M$ the particle mass. In other words, our time units are set such that effectively $2M=1$. In these units time has the dimension of length squared. For standard non-relativistic bosons one has $V=0$ and $S=1$, but we also consider quasiparticles with a more general dispersion relation described by nonzero $V$.\n\nAfter Fourier transformation, the kinetic term reads\n\\begin{equation}\n\\int_q \\varphi^*(q)(i S q_0+V q_0^2+\\vec{q}^2)\\varphi(q),\n\\label{eqMicroscopicFourier}\n\\end{equation}\nwith\n\\begin{eqnarray}\nq=(q_0,\\vec{q}),\\quad \\int_q=\\int_{q_0}\\int_{\\vec{q}},\\quad \\int_{\\vec{q}}=\\frac{1}{(2\\pi)^3}\\int d^3q.\n\\end{eqnarray}\nAt nonzero temperature, the frequency $q_0=2\\pi T n$ is discrete, with\n\\begin{equation}\n\\int_{q_0}=T \\sum_{n=-\\infty}^\\infty,\n\\end{equation}\nwhile at zero temperature this becomes\n\\begin{equation}\n\\int_{q_0}=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty dq_0.\n\\end{equation}\nThe dispersion relation encoded in Eq.\\ \\eqref{eqMicroscopicFourier} obtains by analytic continuation\n\\begin{equation}\nS\\omega+V\\omega^2=\\vec{q}^2\/2M.\n\\end{equation}\n\nIn this thesis, we consider homogeneous situations, i.e. an infinitely large volume without a trapping potential. Many of our results can be translated to the inhomogeneous case in the framework of the local density approximation. One assumes that the length scale relevant for the quantum and statistical fluctuations is much smaller than the characteristic length scale of the trap. In this case, our results can be transferred by taking the chemical potential position dependent in the form $\\mu\\left(\\vec{x})=2M(\\mu-V_t(\\vec{x}\\right))$, where $V_t(\\vec{x})$ is the trapping potential.\n\nThe microscopic action \\eqref{microscopicaction} is invariant under the global $U(1)$ symmetry which is associated to the conserved particle number,\n\\begin{equation}\n\\varphi\\rightarrow e^{i\\alpha}\\varphi.\n\\end{equation}\nOn the classical level, this symmetry is broken spontaneously when the chemical potential $\\mu$ is positive. In this case, the minimum of $-\\mu\\varphi^*\\varphi+\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2$ is situated at $\\varphi^*\\varphi=\\frac{\\mu}{\\lambda}$. The ground state of the system is then characterized by a macroscopic field $\\varphi_0$, with $\\varphi_0^*\\varphi_0=\\rho_0=\\frac{\\mu}{\\lambda}$. It singles out a direction in the complex plane and thus breaks the $U(1)$ symmetry. Nevertheless, the action itself and all modifications due to quantum and statistical fluctuations respect the symmetry. For $V=0$ and $S=1$, the situation is similar for Galilean invariance. At zero temperature, we can perform an analytic continuation to real time and the microscopic action \\eqref{microscopicaction} is then invariant under transformations that correspond to a change of the reference frame in the sense of a Galilean boost. It is easy to see that in the phase with spontaneous $U(1)$ symmetry breaking also the Galilean symmetry is broken spontaneously: A condensate wave function, that is homogeneous in space and time, would be represented in momentum space by\n\\begin{equation} \n\\varphi(\\omega,\\vec{p})=\\varphi_0 \\,(2\\pi)^4\\, \\delta^{(3)}(\\vec{p})\\delta(\\omega).\n\\end{equation}\nUnder a Galilean boost transformation with a boost velocity $2\\vec{q}$, this would transform according to\n\\begin{eqnarray}\n\\nonumber\n\\varphi(\\omega,\\vec{p})\\rightarrow&&\\varphi(\\omega-\\vec{q}^2,\\vec{p}-\\vec{q})\\\\\n&&=\\varphi_0\\,(2\\pi)^4\\,\\delta^{(3)}(\\vec{p}-\\vec{q})\\delta(\\omega-\\vec{q}^2).\n\\end{eqnarray} \nThis shows that the ground state is not invariant under such a change of reference frame. This situation is in contrast to the case of a relativistic Bose-Einstein condensate, like the Higgs boson field after electroweak symmetry breaking. A relativistic scalar transforms under Lorentz boost transformations according to\n\\begin{equation}\n\\varphi(p^\\mu)\\rightarrow\\varphi((\\Lambda^{-1})^\\mu_{\\,\\,\\nu}\\,p^\\nu),\n\\end{equation}\nsuch that a condensate wave function\n\\begin{eqnarray}\n\\nonumber\n\\varphi_0\\,(2\\pi)^4 \\,\\delta^{(4)}(p^\\mu)\\rightarrow&&\\varphi_0\\,(2\\pi)^4\\,\\delta^{(4)}((\\Lambda^{-1})^\\mu_{\\,\\,\\nu}\\,p^\\nu)\\\\\n&&=\\varphi_0\\,(2\\pi)^4\\, \\delta^{(4)}(p^\\mu)\n\\end{eqnarray}\ntransforms into itself. We will investigate the implications of Galilean symmetry for the form of the effective action in chapter \\ref{ch:Symmetries}. An analysis of general coordinate invariance in nonrelativistic field theory can be found in \\cite{Son2006}.\n\n\n\\section{Bose gas in two dimensions}\n\\label{sec:Bosegasintwodimensions}\n\nBose-Einstein condensation and superfluidity for cold nonrelativistic atoms can be experimentally investigated in systems of various dimensions \\cite{morsch:179, bloch:885, PethickSmith2002}. Two dimensional systems can be achieved by building asymmetric traps, resulting in different characteristic sizes for one ``transverse extension'' $l_T$ and two ``longitudinal extensions'' $l$ of the atom cloud \\cite{PhysRevLett.87.130402, PhysRevLett.92.173003, 0953-4075-38-3-007, 0295-5075-57-1-001, Koehl2005, C.Orzel03232001, spielman:080404, PhysRevLett.93.180403, Hazibabic2006}. For $l \\gg l_T$ the system behaves effectively two-dimensional for all modes with momenta $\\vec{q}^2\\lesssim l_T^{-2}$. From the two-dimensional point of view, $l_T$ sets the length scale for microphysics -- it may be as small as a characteristic molecular scale. On the other hand, the effective size of the probe $l$ sets the scale for macrophysics, in particular for the thermodynamic observables.\n\nTwo-dimensional superfluidity shows particular features. In the vacuum, the interaction strength $\\lambda$ is dimensionless such that the scale dependence of $\\lambda$ is logarithmic \\cite{lapidus:459}. The Bogoliubov theory with a fixed small $\\lambda$ predicts at zero temperature a divergence of the occupation numbers for small $q=|\\vec{q}|$, $n(\\vec{q})\\sim n_C\\, \\delta^{(2)}(\\vec{q})$ \\cite{Bogoliubov}. In the infinite volume limit, a nonvanishing condensate $n_c=\\bar{\\rho}_0$ is allowed only for $T=0$, while it must vanish for $T>0$ due to the Mermin-Wagner theorem \\cite{PhysRevLett.17.1133, PhysRev.158.383}. On the other hand, one expects a critical temperature $T_c$ where the superfluid density $\\rho_0$ jumps by a finite amount according to the behavior for a Kosterlitz-Thouless phase transition \\cite{Berezinskii1971, Berezinskii1972, 0022-3719-6-7-010, PhysRevLett.39.1201}. We will see that $T_c\/n$ (with $n$ the atom-density) vanishes in the infinite volume limit $l\\to \\infty$. Experimentally, however, a Bose-Einstein condensate can be observed for temperatures below a nonvanishing critical temperature $T_c$ -- at first sight in contradiction to the theoretical predictions for the infinite volume limit.\n\nA resolution of these puzzles is related to the simple observation that for all practical purposes the macroscopic size $l$ remains finite. Typically, there will be a dependence of the characteristic dimensionless quantities as $\\bar{\\rho}_0\/n$, $T_c\/n$ or $\\lambda$ on the scale $l$. This dependence is only logarithmic. While $\\lambda(n=T=0, l\\to \\infty)=0$, $(\\bar{\\rho}_0\/n)(T\\neq0, l\\to \\infty)=0$, $(T_c\/n)(l\\to0)=0$, in accordance with general theorems, even a large finite $l$ still leads to nonzero values of these quantities, as observed in experiment. \n\nThe description within a two-dimensional renormalization group context starts with a given microphysical or classical action at the ultraviolet momentum scale $\\Lambda_\\text{UV}\\sim l_T^{-1}$. When the scale parameter $k$ reaches the scale $k_\\text{ph}\\sim l^{-1}$, all fluctuations are included since no larger wavelength are present in a finite size system. The experimentally relevant quantities and the dependence on $l$ can be obtained from $\\Gamma_{k_\\text{ph}}$. For a system with finite size $l$ we are interested in $\\Gamma_{k_\\text{ph}}$, $k_\\text{ph}=l^{-1}$. If statistical quantities for finite size systems depend only weakly on $l$, they can be evaluated from $\\Gamma_{k_\\text{ph}}$ in the same way as their thermodynamic infinite volume limit follows from $\\Gamma$. Details of the geometry etc. essentially concern the appropriate factor between $k_\\text{ph}$ and $l^{-1}$. \n\nThe microscopic model we use for the two-dimensional Bose gas is basically the one for the three-dimensional case in Eq.\\ \\eqref{microscopicaction}. The difference is that now $\\vec x$ and the space-integral are two-dimensional\n\\begin{equation}\nx=(\\tau,\\vec{x}), \\,\\,\\int_x=\\int_0^{\\frac{1}{T}}d\\tau\\int d^2x,\n\\end{equation}\nand similarly in momentum space. The dimensionless interaction parameter $\\lambda$ in \\eqref{microscopicaction} describes now a reduced two-dimensional interaction strength and is directly related to the scattering length in units of the transverse extension $a\/l_T$. The few-body physics and the logarithmic scale-dependence of $\\lambda$ is discussed in section \\ref{sec:Repulsiveinteractingbosons}. \n\n\n\\section{BCS-BEC Crossover}\n\\label{sec:BCS-BECCrossover}\n\nBesides the bosons we also investigate systems with ultracold fermions. A qualitative new feature for fermions in comparison to bosons is the antisymmetry of the wavefunction and the tightly connected ``Pauli blocking''. Due to the antisymmetry of the wavefunction it is not possible to have two identical fermions in the same state. This feature has many interesting consequences. For example, a $s$-wave interaction between two identical fermions is not possible. This in turn implies that a gas of fermions in the same spin- (and hyperfine-) state has many properties of a free Fermi gas provided the $p$-wave and higher interactions are suppressed. The situation changes for a Fermi gas with two spin or hyperfine states. $S$-wave interactions and pairing are now possible. In the simplest case the densities of the two components are equal. Depending on the microscopic interaction the system has different properties. For a repulsive interaction one expects Landau Fermi liquid behavior (for not too small temperature) where many qualitative properties are as for the free Fermi gas \\cite{Landau1957}. For weak attractive interaction the theory of Baarden, Cooper and Schrieffer (BCS) \\cite{Bardeen:1957kj, Bardeen:1957mv} is valid. Cooper-pairs are expected to form at small temperatures and the system is then superfluid. On the other hand, for strong attractive interaction one expects the formation of bound states of two fermions. These bound states are then bosons and undergo Bose-Einstein condensation (BEC) at small temperatures. Again, the system shows superfluidity. As first pointed out by Eagles \\cite{PhysRev.186.456} and Leggett \\cite{Leggett1980} there is a smooth and continuous crossover (BCS-BEC crossover) between the two limits described above. \n\nExperimental realizations of this crossover can be realized using Feshbach resonances. The detailed mechanism how these resonances work can be found in the literature, e.~g. \\cite{PitaevsikiiStringari2003, PethickSmith2002}. It is important that the scattering length $a$ which serves as a measure for the $s$-wave interaction can be tuned to arbitrary values. As an example we consider the case of $^6$Li where the resonance was investigated in Refs. \\cite{PhysRevA.66.041401, PhysRevLett.94.103201} and is shown in Fig.\\ \\ref{fig:Feshbach}. For magnetic fields in the range around $B=1200\\, \\text{G}$ the scattering length is relatively small and negative. In this regime the many-body ground state is of the BCS-type. Fermions with different spin and with momenta on opposite points on the Fermi surface form pairs. These Cooper pairs are (hyperfine-) spin singlets and have small or vanishing momentum. They are condensed in a Bose-Einstein condensate (BEC). The system is superfluid and the U(1) symmetry connected with particle number conservation is spontaneously broken. The macroscopic wavefunction of the BEC can be seen as an order parameter which is quadratic in the fermion field $\\varphi_0\\sim \\langle\\psi_1\\psi_2\\rangle$. Increasing the temperature, the system will at some point undergo a second order phase transition to a normal state where the order parameter vanishes, $\\varphi_0=0$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{Feshbach.eps}\n\\caption{Scattering length $a$ in units of the Bohr radius $a_0$ as a function of the magnetic field $B$ for the lowest hyperfine states of $^6$Li \\cite{PhysRevA.66.041401, PhysRevLett.94.103201}.}\n\\label{fig:Feshbach}\n\\end{figure}\n\n\nIn the magnetic field range around $B=600 \\,\\text{G}$ in Fig. \\ref{fig:Feshbach} the scattering length $a$ is small and positive. There is now a bound state of two fermions in the spectrum and the ground state of the many-body system is BEC-like. Pairs of fermions with different spin constitute bound states (dimers) which are pairs in position space. The interaction between these dimers is repulsive and proportional to the scattering length between fermions. When this repulsive interaction is weak the dimers are completely condensed in a BEC at zero temperature (no quantum depletion of the condensate). Again the order parameter is the macroscopic wavefunction of this condensate which is quadratic in the fermion fields $\\varphi_0\\sim\\langle\\psi_1\\psi_2\\rangle$. The phase transition between the superfluid state at small temperatures and the normal state is of second order, again. \n\nNow we come to the magnetic field in the intermediate crossover regime $700\\, \\text{G} \\lesssim B \\lesssim 1000 \\,\\text{G}$. The scattering length is now large and positive or large and negative with a divergence at $B\\approx 834\\, \\text{G}$ \\cite{PhysRevLett.94.103201}. Since the two-body scattering properties are solely governed by the requirement of unitarity of the scattering matrix for $a\\to\\pm \\infty$, the point $B=834\\, \\text{G}$ is also called the ``unitarity point''. Due to the divergent scattering length one speaks of strongly interacting fermions. Perturbative methods for small coupling constants fail in the crossover regime. Non-perturbative methods show that the ground state is superfluid and governed by a order parameter $\\varphi_0\\sim \\langle \\psi_1\\psi_2\\rangle$ as before. \n\nThe crossover from the BCS- to the BEC-like ground state is conveniently parameterized by the inverse scattering length in units of the Fermi momentum $c^{-1}=(a k_F)^{-1}$ where the Fermi momentum is determined by the density $n=\\frac{1}{3\\pi^2}k_F^3$ (in units with $\\hbar=k_B=2M=1$). The dimensionless parameter $c^{-1}$ varies from large negative values on the BCS side to large positive values on the BEC side of the crossover. It crosses zero at the unitarity point. We will also use the Fermi energy which equals the Fermi temperature in our units $E_F=T_F=k_F^2$.\n\nThe quantitatively precise understanding of BCS-BEC crossover physics is a challenge for theory. Experimental breakthroughs as the realization of molecule condensates and the subsequent crossover to a BCS-like state of weakly attractively interacting fermions have been achieved \\cite{PhysRevLett.92.040403, PhysRevLett.92.120403, PhysRevLett.92.150402, PhysRevLett.93.050401, C.Chin08202004, PhysRevLett.95.020404}. Future experimental precision measurements could provide a testing ground for non-perturbative methods. An attempt in this direction are the recently published measurements of the critical temperature \\cite{Luo2007} and collective dynamics \\cite{altmeyer:040401, wright:150403}.\n\nA wide range of qualitative features of the BCS-BEC crossover is already well described by extended mean-field theories which account for the contribution of both fermionic and bosonic degrees of freedom \\cite{Nozieres1985, PhysRevLett.71.3202}. In the limit of narrow Feshbach resonances mean-field theory becomes exact \\cite{Diehl:2005an, Gurarie2007}. Around this limit perturbative methods for small Yukawa couplings \\cite{Diehl:2005an} can be applied. Using $\\epsilon$-expansion \\cite{nussinov:053622, nishida:050403, nishida:063617, nishida:063618, arnold:043605, chen:043620} or $1\/N$-expansion \\cite{Sachdev06} techniques one can go beyond the case of small Yukawa couplings.\n\nQuantitative understanding of the crossover at and near the resonance has been developed through numerical calculations using various quantum Monte-Carlo (QMC) methods \\cite{PhysRevLett.91.050401, PhysRevLett.93.200404, bulgac:090404, bulgac:023625, burovski:160402, akkineni:165116}. Computations of the complete phase diagram have been performed from functional field-theoretical techniques, in particular from $t$-matrix approaches \\cite{Haussmann1993, PhysRevLett.85.2801, PhysRevB.61.15370, PhysRevLett.92.220404, PhysRevB.70.094508}, Dyson-Schwinger equations \\cite{Diehl:2005an,Diehl:2005ae}, 2-partice irreducible (2-PI) methods \\cite{haussmann:023610}, and renormalization-group flow equations \\cite{Birse2005,Diehl:2007th,Diehl:2007ri,Gubbels:2008zz}. These unified pictures of the whole phase diagram \\cite{Sachdev06, Haussmann1993, PhysRevLett.85.2801, PhysRevB.61.15370, PhysRevLett.92.220404, PhysRevB.70.094508, Diehl:2005an, Diehl:2005ae, haussmann:023610, Diehl:2007th, Diehl:2007ri, Gubbels:2008zz}, however, do not yet reach a similar quantitative precision as the QMC calculations.\n\nIn this thesis we discuss mainly the limit of broad Fesh\\-bach resonances for which all thermodynamic quantities can be expressed in terms of two dimensionless parameters, namely the temperature in units of the Fermi temperature $T\/T_F$ and the concentration $c=ak_F$. In the broad resonance regime, macroscopic observables are to a large extent independent of the concrete microscopic physical realization, a property referred to as universality \\cite{Diehl:2005an, Sachdev06, Diehl:2007th}. This universality includes the unitarity regime where the scattering length diverges, $a^{-1}=0$ \\cite{PhysRevLett.92.090402}, however it is not restricted to that region. Macroscopic quantities are independent of the microscopic details and can be expressed in terms of only a few parameters. In our case this is the two-body scattering length $a$ or, at finite density, the concentration $c=ak_F$. At nonzero temperature, an additional parameter is given by $T\/T_F$. \n\nFor small and negative scattering length $c^{-1}<0, |c|\\ll 1$ (BCS side), the system can be treated with perturbative methods. However, there is a significant decrease in the critical temperature as compared to the original BCS result. This was first recognized by Gorkov and Melik-Barkhudarov \\cite{Gorkov}. The reason for this correction is a screening effect of particle-hole fluctuations in the medium \\cite{Heiselberg}. There has been no systematic analysis of this effect in approaches encompassing the full BCS-BEC crossover so far.\n\nIn section \\ref{sec:Particle-holefluctuationsandtheBCS-BECCrossover}, we present an approach using the flow equation described in chapter \\ref{ch:TheWetterichequation}. We include the effect of particle-hole fluctuations and recover the Gorkov correction on the BCS side. We calculate the critical temperature for the second-order phase transition between the normal and the superfluid phase throughout the whole crossover.\n\nWe also calculate the critical temperature at the point $a^{-1}=0$ for different resonance widths $\\Delta B$. As a function of the microscopic Yukawa coupling $h_\\Lambda$, we find a smooth crossover between the exact narrow resonance limit and the broad resonance result. The resonance width is connected to the Yukawa coupling via $\\Delta B=h_\\Lambda^2\/(8\\pi\\mu_M a_b)$ where $\\mu_M$ is the magnetic moment of the bosonic bound state and $a_b$ is the background scattering length.\n\n\n\\subsubsection{Lagrangian}\n\nWe start with a microscopic action including a two-component Grassmann field $\\psi=(\\psi_1,\\psi_2)$, describing fermions in two hyperfine states. Additionally, we introduce a complex scalar field $\\varphi$ as the bosonic degrees of freedom. In different regimes of the crossover, it can be seen as a field describing molecules, Cooper pairs or simply an auxiliary field. Using the resulting two-channel model we can describe both narrow and broad Feshbach resonances in a unified setting. Explicitly, the microscopic action at the ultraviolet scale $\\Lambda$ reads\n\n\\begin{eqnarray}\n\\nonumber\nS[\\psi, \\varphi] & = & \\int_0^{1\/T} d\\tau \\int d^3x{\\Big \\{}\\psi^\\dagger(\\partial_\\tau-\\Delta-\\mu)\\psi\\\\\n\\nonumber\n& & +\\varphi^*(\\partial_\\tau-\\frac{1}{2}\\Delta-2\\mu+ \\nu_\\Lambda)\\varphi\\\\\n& & - h_\\Lambda(\\varphi^*\\psi_1\\psi_2+h.c.){\\Big \\}}\\,,\n\\label{eqMicroscopicAction}\n\\end{eqnarray}\nwhere we choose nonrelativistic natural units with $\\hbar=k_B=2M=1$, with $M$ the mass of the atoms.\nThe system is assumed to be in thermal equilibrium, which we describe using the Matsubara formalism. In addition to the position variable $\\vec{x}$, the fields depend on the imaginary time variable $\\tau$ which parameterizes a torus with circumference $1\/T$. The variable $\\mu$ is the chemical potential. The Yukawa coupling $h$ couples the fermionic and bosonic fields. It is directly related to the width of the Feshbach resonance. The parameter $\\nu$ depends on the magnetic field and determines the detuning from the Feshbach resonance. Both $h$ and $\\nu$ get renormalized by fluctuations, and the microscopic values $h_\\Lambda$, and $\\nu_\\Lambda$ have to be determined by the properties of two body scattering in vacuum. For details, we refer to \\cite{Diehl:2007th} and section \\ref{sec:Twofermionspecies:Dimerformation}. \n\nMore formally, the bosonic field $\\varphi$ appears quadratically in the microscopic action in Eq.\\ \\eqref{eqMicroscopicAction}. The functional integral over $\\varphi$ can be carried out. This shows that our model is equivalent to a purely fermionic theory with an interaction term\n\\begin{eqnarray}\n\\nonumber\nS_{\\text{int}} & = & \\int_{p_1,p_2,p_1^\\prime,p_2^\\prime} \\left\\{-\\frac{h^2}{P_\\varphi(p_1+p_2)}\\right\\}\\psi_1^{\\ast}(p_1^\\prime){\\psi_1}(p_1)\\\\\n&& \\times \\psi_2^{\\ast}(p_2^\\prime){\\psi_2}(p_2)\\,\\delta(p_1+p_2-p_1^\\prime-p_2^\\prime),\n\\label{lambdapsieff}\n\\end{eqnarray}\nwhere $p = (p_0,\\vec p)$ and the classical inverse boson propagator is given by\n\\begin{equation}\n\tP_{\\varphi}(q)= i q_0 + \\frac{\\vec{q}^2}{2}+ \\nu_\\Lambda-2\\mu\\,.\n\\label{eq:Bosonpropagator}\n\\end{equation}\n\nOn the microscopic level the interaction between the fermions is described by the tree level expression\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{h^2}{-\\omega+\\frac{1}{2}\\vec q^2+\\nu_\\Lambda-2\\mu}.\n\\end{equation}\nHere, $\\omega$ is the real-time frequency of the exchanged boson $\\varphi$. It is connected to the Matsubara frequency $q_0$ via analytic continuation $\\omega=-iq_0$. Similarly, $\\vec q=\\vec p_1+\\vec p_2$ is the center of mass momentum of the scattering fermions $\\psi_1$ and $\\psi_2$ with momenta $\\vec p_1$ and $\\vec p_2$, respectively.\n\nThe limit of broad Feshbach resonances, which is realized in current experiments, e.g. with $\\mathrm{^6Li}$ and $\\mathrm{^{40}K}$ corresponds to the limit $h\\to\\infty$, for which the microscopic interaction becomes pointlike, with strength $-h^2\/\\nu_\\Lambda$. \n\n\n\\section{BCS-Trion-BEC Transition}\n\\label{sec:BCS-Trion-BECTransitionshort}\n\nIn the last section we discussed the interesting BCS-BEC crossover that is realized in a system consisting of two fermion species. We restricted ourselves to the case where the density for both components is equal. Interesting physics is also found if this constraint is released. The phase diagram of the imbalanced Fermi gas shows also first order phase transitions and phase separation, see \\cite{KetterleZwierlein2007, Chevy2007} and references therein. \n\nAnother interesting generalization is to take a third fermion species into account. A very rich phase diagram can be expected for the general case where the total density is arbitrarily distributed to the different components. Even the simpler case where the densities for all three components are equal is far less understood as the analogous two-component case. For simplicity we restrict much of the discussion to the case where all properties of the three components apart from the hyperfine-spin are the same. In particular, we assume that they have equal mass, chemical potential and scattering properties. We label the different hyperfine states by 1, 2 and 3. The $s$-wave scattering length $a_{12}$ for scattering between fermions of components 1 and 2 is the same as for scattering between fermions of species 2 and 3 or 3 and 1, $a_{12}=a_{23}=a_{31}=a$. \n\nClose to a common resonance where $a\\to\\pm\\infty$ one expects the three-body problem to be dominated by the Efimov effect \\cite{Efimov1970, Efimov1973}. This implies the formation of a three-body bound state (the ``trion''). Directly at the resonance an infinite tower of three-body bound states, the Efimov-trimers, exists. We refer to the Efimov trimer with the lowest lying energy as trion. The few-body physics is discussed in more detail in section \\ref{sec:Threefermionspecies:ThomasandEfimoveffect}. \n\nThe many-body phase diagram is far less understood. Not too close to the resonance one expects a superfluid ground state which is similar to the BCS ground state for $a<0$ or a BEC-like ground state for $a>0$. However, there are also some important differences. While in the two-component case the order parameter is a singlet under the corresponding SU(2) spin symmetry, the order parameter for the three component case with SU(3) spin symmetry is a (conjugate) triplet. In the superfluid phase the spin symmetry is therefore broken spontaneously. Due to some similarities with QCD this was called color superfluidity \\cite{PhysRevB.70.094521, paananen:053606, paananen:023622, cherng:130406, zhai:031603, Bedaque2007}.\n\nBetween the extended BCS and BEC phase one can expect the ground state to be dominated by trions. Since trions are SU(3) singlets, the spin symmetry is unbroken in this regime such that there have to be true quantum phase transitions at the border to the BCS and BEC regimes. Such a trion phase has first been proposed for fermions in an optical lattice by Rapp, Zarand, Honerkamp and Hofstetter \\cite{rapp:160405, rapp:144520}, see also \\cite{Wilczek2007}. We will further discuss the many-body physics in section \\ref{sec:BCS-Trion-BECTransitionlong}. To the knowledge of the author, there have been no experiments addressing the many-body issues so far. Only recently, experiments with $^6$Li probing the few-body physics found interesting phenomena \\cite{ottenstein:203202, Huckans2008}. For the case of $^6$Li the assumption of equal scattering properties for the three different species are not fulfilled. We will present a more general model where SU(3) symmetry is broken explicitly and where the parameters can be chosen to describe $^6$Li in section \\ref{sec:Threefermionspecies:ThomasandEfimoveffect}. We also discuss the experiments and show how their results can be explained in our framework. The remainder of this section is devoted to the discussion of the microscopic model in the SU(3) symmetric case. \n\n\\subsubsection{Lagrangian}\n\nAs our microscopic model we use an action similar to the one for the BCS-BEC crossover in Eq.\\ \\eqref{eqMicroscopicAction}\n\\begin{eqnarray}\n\\nonumber\nS&=&\\int_x {\\bigg \\{} \\psi^\\dagger\\partial_\\tau-\\Delta-\\mu)\\psi+\\varphi^\\dagger(\\partial_\\tau-\\frac{1}{2}\\Delta-2\\mu+\\nu_\\varphi)\\varphi\\\\\n\\nonumber\n&&+\\chi^*(\\partial_\\tau-\\frac{1}{3}\\Delta-3\\mu+\\nu_\\chi)\\chi\\\\\n\\nonumber\n&&+\\frac{1}{2} h\\,\\epsilon_{ijk}\\,\\left(\\varphi_i^*\\psi_j\\psi_k-\\varphi_i\\psi_j^*\\psi_k^*\\right)\\\\\n\\nonumber\n&&+g\\left(\\varphi_i^*\\psi_i^*\\chi-\\varphi_i\\psi_i\\chi^*\\right){\\bigg \\}}.\n\\label{eq8:microscopicactiontrionmodel}\n\\end{eqnarray}\nThe (Grassmann valued) fermion field has now three components $\\psi=(\\psi_1,\\psi_2,\\psi_3)$ and similar the boson field $\\varphi=(\\varphi_1,\\varphi_2,\\varphi_3)\\hat{=} (\\psi_1\\psi_2,\\psi_2\\psi_3,\\psi_3\\psi_1)$. In addition we also include a single component fermion field $\\chi$. This trion field represents the totally antisymmetric combination $\\psi_1\\psi_2\\psi_3$. One choose the parameters such that $g=0$ and $\\nu_\\chi\\to\\infty$ at the microscopic scale. The trion field $\\chi$ is then only an non-dynamical auxiliary field. \n\nWe assumed in Eq.\\ \\eqref{eq8:microscopicactiontrionmodel} that the fermions $\\psi_1$, $\\psi_2$, and $\\psi_3$ have equal mass $M$ and chemical potential $\\mu$. We also assume that the interactions are independent of the spin (or hyperspin) so that our microscopic model is invariant under a global SU(3) symmetry transforming the fermion species into each other. While the fermion field $\\psi=(\\psi_1,\\psi_2,\\psi_3)$ transforms as a triplet ${\\bf 3}$, the boson field $\\varphi=(\\varphi_1,\\varphi_2,\\varphi_3)$ transforms as a conjugate triplet $\\bar {\\bf 3}$. The trion field $\\chi$ is a singlet under SU(3). In concrete experiments, for example with $^6\\text{Li}$ \\cite{ottenstein:203202}, the SU(3) symmetry may be broken explicitly since the Feshbach resonances of the different channels occur for different magnetic field values and have different widths. In addition to the SU(3) spin symmetry our model is also invariant under a global U(1) symmetry $\\psi\\to e^{i\\alpha}\\psi$, $\\varphi\\to e^{2i\\alpha}\\varphi$, and $\\chi\\to e^{3i\\alpha}\\chi$. The conserved charge related to this symmetry is the total particle number. Since we do not expect any anomalies the quantum effective action $\\Gamma=\\Gamma_{k=0}$ will also be invariant under $\\text{SU(3)}\\times \\text{U(1)}$.\n\nApart from the terms quadratic in the fields that determine the propagators, Eq.\\ \\eqref{eq8:microscopicactiontrionmodel} contains the Yukawa-type interactions $\\sim h$ and $\\sim g$. The energy gap parameters $\\nu_\\varphi$ for the bosons and $\\nu_\\chi$ for the trions are sometimes written as $m_\\varphi^2=\\nu_\\varphi-2\\mu$, $m_\\chi^2=\\nu_\\chi-3\\mu$, absorbing an explicit dependence on the chemical potential $\\mu$.\n\nIn Eq.\\ \\eqref{eq8:microscopicactiontrionmodel}, the fermion field $\\chi$ can be ``integrated out'' by inserting the $(\\psi,\\varphi)$-dependent solution of its field equation into $\\Gamma_k$. For $m_\\chi^2\\rightarrow \\infty$ this results in a contribution to a local three-body interaction, $\\lambda_{\\varphi\\psi}=-g^2\/m_\\chi^2$. Furthermore one may integrate out the boson field $\\varphi$, such that (for large $m_\\varphi^2$) one replaces the parts containing $\\varphi$ and $\\chi$ in $\\Gamma_k$ by an effective pointlike fermionic interaction\n\\begin{equation}\n\\Gamma_{k,\\text{int}}=\\int_x\\frac{1}{2}\\lambda_\\psi(\\psi^\\dagger\\psi)^2+\\frac{1}{3!}\\lambda_3 \\left(\\psi^\\dagger \\psi\\right)^3,\n\\end{equation}\nwith\n\\begin{equation}\n\\lambda_\\psi= -\\frac{h^2}{m_\\varphi^2}, \\quad \\quad \\lambda_3=-\\frac{h^2 g^2}{m_\\varphi^4 m_\\chi^2}.\n\\label{eq:subst}\n\\end{equation}\nWe note that the contribution of trion exchange to $\\lambda_{\\varphi\\psi}$ or $\\lambda_3$ depends only on the combination $g^2\/m_\\chi^2$. The sign of $g$ can be changed by $\\chi\\rightarrow - \\chi$, and the sign of $g^2$ can be reversed by a sign flip of the term quadratic in $\\chi$. Keeping the possible reinterpretation by this mapping in mind, we will formally also admit negative $g^2$ (imaginary $g$).","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe are in the age of renaissance of hadron\nspectroscopy, initiated by the announcement of the pentaquark baryon\n\\cite{nakano}, which is followed by the discovery of \nmany other possible exotic hadrons with a mass larger than 2 GeV \ncontaining heavy quarks\\cite{Quigg:2005tv}.\nThese experimental developments prompted the intensive theoretical\nstudies of QCD dynamics with new as well as old ideas on the structure\nand dynamics of the exotic hadrons, such as\nchiral dynamics\\cite{Nowak:1992um},\nmulti-quark states with diquark correlations or\nmolecular states and hybrids\\cite{Quigg:2005tv}. \n\nSuch a controversy on the structure of hadrons is\nalso the case for the scalar mesons below 1 GeV:\nthe existence of the $I=0$ and $J^{PC}=0^{++}$ meson, i.e.,\nthe $\\sigma(400-600)$, has been reconfirmed \\cite{PDG,leutwyler} \nafter around twenty years not only in $\\pi$$\\pi\\ $ scattering \nbut also in various decay\nprocesses from heavy-quark systems, {\\it e.g.} , \nD $\\to \\pi \\pi \\pi$ and $\\Upsilon(3S) \\to \\Upsilon \\pi \\pi$ \n\\cite{KEK,E791decay,Ishida,Bugg}.\nMoreover, the resonance of a scalar meson with $I=1\/2$\nis also reported to exist in the K-$\\pi$ system \nwith a mass $m_{\\kappa}$ of about \n800 MeV \\cite{Bugg,E791,BES}.\nThis meson is called the $\\kappa$ meson and may constitute\nthe nonet scalar state together with the $\\sigma$ meson.\nSee Fig.\\ref{fig:nonet}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{figure1.eps}\n\\caption{\nScalar meson nonet. \nThe $\\sigma$ and $f_0(980)$ mesons may be ideal mixing\nstates of singlet state \n$\\frac{1}{\\sqrt{3}} ( u \\bar{u} + d \\bar{d} + s \\bar{s} )$\nand octet state \n$\\frac{1}{\\sqrt{6}} ( u \\bar{u} + d \\bar{d} - 2 s \\bar{s} )$. \nThere is , however, experimental evidence that the $\\sigma$ meson\nconsist of only $u \\bar{u}$\nand $d \\bar{d}$ components. Hence, we take the $\\sigma$ wave function given\nin the figure.\n}\n\\label{fig:nonet}\n\\end{center}\n\\end{figure}\n\nThe problem is the nature of these low-lying scalar mesons\n\\cite{close-tornqvist}:\nthey cannot be ordinary $q\\bar{q}$ mesons as described in the \nnon-relativistic constituent quark model\nsince in such a quark model, \nthe $J^{PC}$=$0^{++}$ meson is realized in the $^{3}P_{0}$ state, \nwhich implies that the mass of the $\\sigma$ meson must be as high as 1.2\n$\\sim$ 1.6 GeV.\nThus, the low-lying scalar mesons below 1 GeV \nhave been a source of various ideas of exotic structures, as mentioned above:\nthey may be four-quark states such as $qq\\bar{q}\\bar{q}$ \\cite{Alfold},\nor $\\pi\\pi$ or K$\\pi$ molecules\nas the recent high-lying exotic hadrons can be.\nThese mesons may be {\\em collective} $q\\bar{q}$ states described\nas a superposition of many {\\em atomic} $q\\bar{q}$ \nstates \\cite{NJL,HK85}.\nA mixing with glueball states is also possible\n\\cite{Lee,McNeile,Narison,giacosa}. \n\nIn the previous work\\cite{ScalarSIGMA1,ScalarSIGMA2,ScalarSIGMA3,ScalarSIGMA4},\nwe have presented a lattice calculation for the $\\sigma$ meson,\nby full lattice QCD simulation\non the $8^{3}\\times16$ lattice using the plaquette action and\nWilson fermions: We have shown that the disconnected diagram \nplays an essential role in order to make the $\\sigma$ meson mass \nlight. \nThe importance of the disconnected diagram suggests that the\nwave function of the $\\sigma$ meson may have a significant \nfour-quark, a collective $q$-$\\bar{q}$\nor an even glueball component, although the smallness of the \nlattice requires caution in giving a definite conclusion.\nIn contrast to the $\\sigma$ meson, the $\\kappa$ is a flavor non-singlet state\nwith which a glueball state cannot mix.\nIn previous reports \\cite{ScalarSIGMA4,ScalarKAPPA},\nwe reported also a preliminary analysis on the $\\kappa$ meson \nusing the dynamical fermion for \nthe $u(d)$ quark but using the valence approximation for the $s$ quark,\nwhich shows that \nthe $I=1\/2$ scalar meson has a mass as large as about 1.8 GeV and \ncannot be identified with the $\\kappa$ meson observed in experiments.\n\nThe lattice volume in the previous investigations was admittedly \ntoo small to yield a definite conclusion at all, and \n the lattice cutoff was not appropriately chosen to accommodate\nlarge masses : $m_{\\kappa}a>1$, where $a$ is the lattice spacing.\nHence, we present a simulation with weaker couplings\non a larger lattice than any other previous simulations\nalthough in the quenched level.\nWe perform quenched level simulations on the \n$\\kappa$ meson so as {\\it to clarify the structure of the mysterious \nscalar meson rather than to reproduce the experimental\nvalue of the mass}; a precise quenched-level simulation should\ngive a rather clear perspective on whether\nthe system can fit with the simple constituent-quark model\npicture or not. \n\n\\section{Simulation}\n\nWe perform a quenched QCD calculation using the Wilson fermions,\nwith the plaquette gauge action, on a relatively large lattice\n($20^3 \\times 24$). \n\nThe values of the hopping parameter for the $u\/d$ quark are \n$h_{u\/d} = 0.1589, 0.1583$ and 0.1574, while \n$h_s = 0.1566$ and 0.1557 for the $s$ quark.\nUsing these hopping parameters except for $h_s=0.1557$, \nCP-PACS collaboration performed a quenched QCD calculation of \nthe light meson spectrum \nwith a larger lattice ($32^3 \\times 56$) \\cite{CP-PACS}, which we refer to for comparison.\nThe gauge configurations are generated \nby the heat bath algorithm at $\\beta = 5.9$. \nAfter 20000 thermalization iterations, we start to calculate \nthe meson propagators. On every 2000 configurations,\n80 configurations are used for the ensemble average.\n\nWe emply the point-like source and sink for the $\\kappa^{+}$ meson\n\\begin{equation}\n\\hat{\\kappa}(x) \\equiv \\sum_{c=1}^3\\sum_{\\alpha=1}^4 \n{\\bar{s}_\\alpha^c(x) u_\\alpha^c(x)} \\ \\ ,\n\\label{eq:kappa_operator}\n\\end{equation}\nwhere $u(x)$ and $s(x)$ are the Dirac operators \nfor the $u\/d$ and $s$ quarks, and \nthe indices $c$ and $\\alpha$ \ndenote the color and Dirac-spinor indices, respectively.\nThe point source and sink in Eq.(\\ref{eq:kappa_operator}) lead a positive spectral\nfunction $\\rho(m^2)$ in the correlation function\n $ \\langle \\hat{\\kappa}(t) \\hat{\\kappa}(0) \\rangle =\n\\int dm \\rho(m^2){\\rm exp}(-mt) $.\nThe result obtained here is thus an upper bound of $\\kappa$ mass,\nbecause our result should include excited states.\n\n\n\nFirst, we check finite lattice volume effects by comparing\nour results for \nthe $\\pi$ and $\\rho$ masses as well as the mass ratio $m_{\\pi}\/m_{\\rho}$ \nwith those of the CP-PACS group. \nThe results are summarized in Table \\ref{table:pi_rho}. \nOur result for the $\\rho$ meson mass\nis only slightly ($<$ 5 $\\%$) larger than the CP-PACS's result.\nThe resulting larger value is reasonable because\nthe smaller lattice size gives rise to a mixture of higher mass states.\nWe rather emphasize that the deviation between our results and the\nlarger lattice result (CP-PACS) is so small in spite of the large difference \nin the lattice size. \n\\begin{center}\n\\begin{table*}[h]\n\\caption{Summary of results for $\\bar{q}q$ type mesons. }\n\\label{table:pi_rho}\n\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & 0.1589 & 0.1583 & 0.1574 & 0.1566 & 0.1557 \\\\ \\hline\n$m_{\\pi}$ & 0.2064(62) & 0.2691(36) & 0.3401(29) & 0.3935(28) & 0.4478(28) \\\\\n$m_{\\rho}$ & 0.442(13) & 0.461(06) & 0.496(05) & 0.527(04) & 0.563(03) \\\\\n$m_{\\pi}\/m_{\\rho}$ & 0.467(21) & 0.584(10) & 0.686(05) & 0.746(03) & 0.796(03) \\\\ \\hline\n$m_{\\sigma_v}$ & 1.12(74) & 0.84(23) & 0.886(98) & 0.857(52) & 0.897(35) \\\\\n\\hline\n\\multicolumn{6}{c}{CP-PACS}\\cite{CP-PACS} \\\\ \\hline \\hline\n$m_{\\pi}$ & 0.20827(33) & 0.26411(28) & 0.33114(26) & 0.38255(25) & $-$ \\\\\n$m_{\\rho}$ & 0.42391(132) & 0.44514(96) & 0.47862(71) & 0.50900(60) & $-$ \\\\\n$m_{\\pi}\/m_{\\rho}$ & 0.491(2) & 0.593(1) & 0.692(1) & 0.752(1) & $-$ \\\\ \\hline\n\\end{tabular}\n\\label{table:qqbar}\n\\end{table*}\n\\end{center}\nIn Fig.~\\ref{fig:extrapolation}, \nwe show $m_{\\pi}^2$, $m_{\\rho}$ and $m_{\\sigma_v}$ in the lattice unit\nas a function of the inverse hopping parameter $1\/h_{u\/d}$ for the $u\/d$ quark. \nThe chiral limit ($m_{\\pi}^2 = 0$) is obtained\nat $h_{u\/d}=0.1598(1)\\equiv h_{\\rm crit}$ ($1\/h_{\\rm crit}$=6.2581). \nWe find the lattice spacing $a$ = 0.1038(33) [fm] in the chiral limit\nfrom the value $m_{\\rho}a$ = 0.406(13) at this point \nwith the physical $\\rho$ meson mass being used for $m_{\\rho}$.\nNote that these values are consistent with\nthe CP-PACS's result, $h_{\\rm crit}$ = 0.1598315(68) and $a$ = 0.1020(8) [fm], \nwithin the error bars. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure2.eps}\n\\caption{$m_{\\pi}^2$, $m_{\\rho}$ and $m_{\\sigma_v}$ in the lattice\nunit as a function of the inverse $h_{u\/d}$. \nThe chiral limit is obtained at $h_{\\rm crit}$ = 0.1598(1).}\n\\label{fig:extrapolation}\n\\end{center}\n\\end{figure}\n\nIn Table \\ref{table:pi_rho}, the mass of the \nvalence $\\sigma$ for each\nhopping parameter is shown;\nthe valence $\\sigma$, which is denoted as $\\sigma_v$,\nis defined as the scalar \nmeson described solely with the connected propagator. \nThe mass ratio $m_{\\sigma_v}\/m_\\rho$ varies from 2.5 ($h_{u\/d}=0.1589$) \nto 1.6 ($h_{u\/d}=0.1557$), which is consistent with our previous \nresults \\cite{ScalarSIGMA4}. \nIn other words, without the disconnected part of \nthe propagator the ``$\\sigma$\" mass becomes heavy. \n\nThe propagators of the $K$, $K^*$ and $\\kappa$ mesons \nare calculated with the same configurations \nusing the $s$-quark hopping parameter, $h_s$ = 0.1566 and 0.1557.\nFor $h_s$ = 0.1557, the effective mass plots of the $K^*$ and $\\kappa$ \nmesons are shown in Figs.~\\ref{fig:K*} and \\ref{fig:kappa}. \nThe masses of the $K$, $K^{*}$ and $\\kappa$ mesons, which are \nextracted from the effective mass plots \\cite{DeGrand}\n, are \nsummarized in Tables \\ref{table:ratio1566} and \n\\ref{table:ratio1557}.\nErrors are estimated by jack-knife method.\nWe find that the effective masses of the $K$ and $K^*$ mesons have \nonly small errors and are taken to be reliable, \nwhile that of the $\\kappa$ meson suffers from large errors, especially at\nlarger time regions.\nTo avoid possible large errors coming from the data at large $t$, \nwe fit the effective mass of the $\\kappa$ meson\nonly in the time range $5 \\le t \\le 7, 8$ \nwhere the effective masses are almost constant with small errors.\nSince the effective mass of the $K^*$ meson is reliable,\nwe show the mass of the $\\kappa$ in terms of the ratio to $m_{K^*}$:\nTable \\ref{table:mass_ratio} gives the mass ratios $m_{K}\/m_{K^{*}}$\nand $m_{\\kappa}\/m_{K^{*}}$ at the chiral \nlimit together with $m_{\\phi}\/m_{K^*}$ for $h_s=0.1566$ and 0.1577. \nFor example, $m_\\kappa\/m_{K^*}=0.89(29)\/0.4649(69)=1.92(61)$ at \n$h_s=0.1566$ in Table \\ref{table:mass_ratio}. \nThese calculated mass ratios are shown in Fig.~\\ref{fig:mass_ratio}.\nAll the mass ratios are almost independent of $h_s$.\nAlthough the error bar for $m_\\kappa\/m_{K^*}$ is \nlarge, the behavior as a function of $h_s$ is reasonable. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure3.eps}\n\\caption{Effective mass plots \nof $K^*$ for $s$ quark hopping parameter $h_s$ = 0.1557.}\n\\label{fig:K*}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure4.eps}\n\\caption{Effective mass plots \nof the $\\kappa$ meson for the $s$ quark \nhopping parameter $h_s$ = 0.1557.}\n\\label{fig:kappa}\n\\end{center}\n\\end{figure}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the $K$, $K^{*}$ and $\\kappa$ mesons at\n$h_s$ = 0.1566.}\n\\label{table:ratio1566}\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & $h_{\\rm crit}^{1)}$ & 0.1589 & 0.1583 & 0.1574 \\\\ \\hline\n$m_{K}$ & 0.2829(23) & 0.3138(33) & 0.3368(30) & 0.3677(29) \\\\\n$m_{K^{*}}$ & 0.4649(69) & 0.4821(57) & 0.4941(49) & 0.5117(42) \\\\\n$m_{\\kappa}$ & 0.89(29) & 0.88(23) & 0.81(12) & 0.814(81) \\\\\n\\hline\n\\multicolumn{5}{c}{CP-PACS} \\cite{CP-PACS} \\\\ \\hline \\hline\n$m_{K}$ & $-$ & 0.30769(28) & 0.32833(26) & $-$ \\\\\n$m_{K^{*}}$ & $-$ & 0.46724(84) & 0.47749(74) & $-$ \\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ $h_{\\rm crit}$ = 0.1598(1).\n\\end{table}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the $K$, $K^{*}$ and $\\kappa$ mesons at\n$h_s$ = 0.1557.}\n\\label{table:ratio1557}\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & $h_{\\rm crit}^{1)}$ & 0.1589 & 0.1583 & 0.1574 \\\\ \\hline\n$m_{K}$ & 0.3188(25) & 0.3474(31) & 0.3684(29) & 0.3971(28) \\\\\n$m_{K^{*}}$ & 0.4835(61) & 0.5006(52) & 0.5126(44) & 0.5299(37) \\\\\n$m_{\\kappa}$ & 0.89(21) & 0.88(16) & 0.828(96) & 0.833(72) \\\\\n \\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ $h_{\\rm crit}$ = 0.1598(1). \\\\\n\\end{table}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the mass ratios\n$m_K\/m_{K^*}$ and $m_\\kappa\/m_{K^*}$\ntogether with $m_\\phi\/m_{K^*}$ at chiral limit for $u\/d$ quarks. \n}\n\\label{table:mass_ratio}\n\\begin{tabular}{c|c|c||c|c}\n\\hline\n\\hline\n$h_s$ & 0.1566 & 0.1557 & 0.1563(3) & 0.1576(2) \\\\\n$1\/h_s$ & 6.3857 & 6.4226 & 6.396(13) & 6.3452(80) \\\\ \\hline\n$m_{\\phi}\/m_{K^*}$ & 1.135(10) & 1.164(10) & 1.143$^{1)}$ & $-$ \\\\\n$m_K\/m_{K^{*}}$ & 0.6086(79) & 0.6593(63)& 0.623(11) & 0.5556$^{1)}$ \\\\\n$m_{\\kappa}\/m_{K^{*}}$ & 1.92(61) & 1.84(43) & 1.89(55) & 2.00(80) \\\\\n\\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ inputs for calculation of physical value of $h_s$. See the text. \n\\end{table}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure5.eps}\n\\caption{The ratios $m_K\/m_{K^*}$ and $m_{\\kappa}\/m_{K^*}$\nat chiral limit, and $m_{\\phi}\/m_{K^*}$ for $s$ quark hopping parameters \n$h_s$ = 0.1566 and 0.1557.}\n\\label{fig:mass_ratio}\n\\end{center}\n\\end{figure}\n\nWe have searched for the physical value of the $s$ quark hopping \nparameter $h_s$ in the following two ways, both of \nwhich are found to give similar results:\n1)~ By tracing a regression line \nfor $m_{\\phi}\/m_{K^*}$ (Fig.~\\ref{fig:mass_ratio}), we have $h_s$ = 0.1563(3)\n(or $1\/h_s$ = 6.396(13)) for $m_{\\phi}\/m_{K*}$=1019[MeV]\/892.0[MeV]=1.143 \n(input), taken from the PDG \\cite{PDG}.\nThis hopping parameter gives the mass ratio \n$m_{\\kappa}\/m_{K^*}$ = $1.89(55)$.\n2)~ We have also determined the hopping parameter\nso as to reproduce the mass ratio \n$m_K\/m_{K^*}$ = 495.6[MeV]\/892[MeV] = 0.5556, with\n$m_K= 495.6$ [MeV] being the average value of the Kaon masses given in the PDG \n\\cite{PDG}.\nThe resulting value is found to be $h_s$ = 0.1576(2) \n(or $1\/h_s$ = 6.3452(80)), which in turn gives the mass ratio \n$m_{\\kappa}\/m_{K^*}$ = 2.00(80).\nThe mass ratios obtained using methods 1) and 2) are \nalso presented in Table \\ref{table:mass_ratio}. \nBoth methods give almost identical results for the masses \nof the $\\kappa$, that are about twice that of the $K^*$.\n\n\\section{Concluding remarks}\n\nThe motivation of our lattice study is \nto reveal the nature of the scalar meson nonet,\nand the results should be important especially \nin clarifying how the $\\kappa$ meson with\na reported low mass $\\sim 800$ MeV obtained from experiments\ncan be compatible with the \nvalence or constituent quark model:\nthe $\\kappa$ is a\n$P$-wave $q\\bar{q}$ bound state in the non-relativistic\nquark model, and the $\\kappa$ meson constitutes\na nonet together with the $\\sigma$ meson and the $a_0$ mesons.\n\nThere have not been many lattice studies of $\\kappa$ meson.\nRecently, estimations of the $\\kappa$ meson have been reported\nby two groups.\nPrelovsek {\\it et al.} \\cite{Sasa2} \nhave presented a rough estimate of the mass of\nthe $\\kappa$ as $1.6$ GeV, which is obtained using the average quark mass\nof the $u$ and $s$ quarks from the dynamical simulations\nwith the degenerate $N_f=2$ quarks on a $16^3\\times 32$ lattice.\nMathur {\\it et al.} have studied $u\\bar{s}$ meson with the overlap fermion in the quenched approximation\n and obtained a mass of the $u\\bar{s}$ scalar meson to be 1.41 $\\pm$ 0.12 GeV \\cite{Mathur}.\nThe UKQCD Collaboration has\nstudied to some extent the $\\kappa$ meson using the dynamical\n$N_{f}$=2 sea quarks and a valence strange quark\non a $16^3\\times 32$ lattice \\cite{UKQCDkappa};\nthey estimated the $\\kappa $ mass as about 1.1 GeV,\nwhich is much smaller than those in \\cite{ScalarSIGMA4,ScalarKAPPA,Sasa2} \nbut still far from the experimental value $\\sim$800 MeV.\n\nIn this paper, we have presented the lattice simulation results\nin the quenched approximation\nfor the $\\kappa$ meson; the results on the $\\pi$, $\\rho$, $K$\nand $K^*$ mesons are also shown for comparison.\n\nWe have first checked that \nthe masses of the $\\pi$, $\\rho$, $K$ and $K^*$ mesons \nobtained in our simulation are in good agreement\nwith those on a larger lattice ($32^3\\times 56$) \\cite{CP-PACS}; \nour results are only within five percent larger than the latter.\nOur estimated value of the mass of the $\\kappa$ is $\\sim$ 1.7 GeV, which is \nlarger than twice the experimental mass $\\sim 800$ MeV. \nThis result was expected on the basis of our experience \nin calculating the $\\sigma$ meson.\nThe relatively heavy mass of the $\\kappa$ may\nbe at least partly attributed to the absence of the disconnected diagram in \nthe $\\kappa$ propagator; the $\\kappa$ propagator is composed of only \na connected diagram.\nWhile the disconnected diagram was\nessential for realizing the low-mass $\\sigma$ \\cite{ScalarSIGMA4}, \nit does not exist for the $\\kappa$; therefore, the mass of the $\\kappa$\nis not made lighter by the disconnected diagram.\nIndeed, the mass of the valence \n$\\sigma_v$ described solely with the connected propagator \nis far larger than the experimental value\n$\\sim 500$-$600$ MeV, as seen in Table \\ref{table:qqbar}. \n\nOur lattice study and the quark model analysis\\cite{QM85} suggest \nthat the simple two-body constituent-quark picture \nof the $\\kappa$ meson does not agree well with \nthe experimentally observed $\\kappa$.\nNote that the quench simulation is a clean theoretical experiment\nin which a virtual intermediate like $qq\\bar{q}\\bar{q}$ is\nhighly suppressed \\cite{Alfold}.\nTherefore, \nif its existence with the reported low mass is experimentally established, \nthe dynamical quarks may play an essential role for making\nthe $\\kappa$ mass so lighter\nor \nthe $\\kappa$ may contain\nan unconventional state such as\na $qq\\bar{q}\\bar{q}$\\cite{tetra} or $K\\pi$ molecular\nstate\\cite{torn-phys-rep}, which are missing in the \ncalculation here. \n\nIn order to establish this possible scenario, \nthe systematic errors should be much reduced \nin future simulations. \nOur statistics here is reasonably high \n(80 configurations separated by 2000 sweeps), \nand the standard meson masses have small error bars; see Fig.\\ref{fig:K*}. \nOn the contrary, as seen in Fig.\\ref{fig:kappa},\nthe effective mass of $\\kappa$ suffers from large errors \nfor large $t$,\nwhich may be due to a small overlap of the physical states.\nThis is not surprising because $\\kappa$ is a P-wave meson, and\nexpected to be extended.\nChoosing more adequate\nextrapolation operators and with much higher statistics,\nwe can study the dynamics of\nhadrons by comparing results in \nthe quenched lattice QCD,\nfull lattice QCD and various effective theories\/models that include \nthe constituent quark models with and without \nthe tetra-quark structure, chiral effective theories.\n\n{\\bf Acknowledgment}\nT.K. is supported by Grants-in-Aid for Scientific Research from \nthe Ministry of Education, Culture, Sports, Science and Technology\n(No. 17540250) and \nfor the 21st century COE ``Center for Diversity and Universality in\nPhysics'' program of Kyoto University.\nThe work is partially supported by \nGrants-in-Aid for Scientific Research from\nthe Ministry of Education, Culture, Sports, Science and Technology\nNos. 13135216 and 17340080.\nThe calculation was carried out on SX-5 at RCNP, Osaka University and\non SR-8000 at KEK.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfxht b/data_all_eng_slimpj/shuffled/split2/finalzzfxht new file mode 100644 index 0000000000000000000000000000000000000000..01b36de1018a60b9b87c7143cc864f4db18ffb74 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfxht @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nVarious approaches to quantum gravity, such as string theory and\nloop quantum gravity as well as black hole physics, predict a\nminimum measurable length of the order of the Planck length,\n$\\ell_{p}=\\sqrt{\\frac{G\\hbar}{c^{3}}}\\sim10^{-35}m$. In the presence\nof this minimal observable length, the standard Heisenberg\nUncertainty Principle attains an important modification leading to\nthe so-called Generalized Uncertainty Principle (GUP). As a result,\ncorresponding commutation relations between position and momenta are\ngeneralized too \\cite{1}. In recent years a lot of attention has\nbeen attracted to extend the fundamental problems of physics in this\nframework (see for instance\n\\cite{21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,p,PRD,PhysA,PLB}).\nSince in the GUP framework one cannot probe distances smaller than\nthe minimum measurable length at finite time, we expect it modifies\nthe Hamiltonian of systems too. Recently it has been shown that the\nGUP affects Lamb shift, Landau levels, reflection and transmission\ncoefficients of a potential step and potential barrier \\cite{9}. In\naddition, they speculated on the possibility of extracting\nmeasurable predictions of GUP in the future experiments. In this\nwork we will follow the procedure introduced in the Ref. \\cite{9},\nbut we are going to address the effect of GUP on the\nRamsauer-Townsend (RT) effect. The RT effect can be observed as long\nas the scattering does not become inelastic by excitation of the\nfirst excited state of the atom. This condition is best fulfilled by\nthe closed shell noble gas atoms. Physically, the RT effect may be\nthought of as a diffraction of the electron around the rare-gas\natom, in which the wave function inside the atom is distorted in\nsuch a way that it fits on smoothly to an undistorted wave function\noutside. The effect is analogous to the perfect transmission found\nat particular energies in one-dimensional scattering from a square\nwell. The one-dimensional treatment of scattering from a square well\nand also three-dimensional treatment using the partial waves\nanalysis can be found in \\cite{14}. We generalize the\none-dimensional treatment of the scattering from a square well to\nthe GUP framework. We also address the condition for interference in\nthe Fabry-Perot interferometer in the framework of GUP.\n\n\n\\section{A Generalized Uncertainty Principle}\nQuantum mechanics with modification of the usual canonical\ncommutation relations has been investigated intensively in the last\nfew years (see \\cite{PRD} and references therein). Such works which\nare motivated by several independent streamlines of investigations\nin string theory and quantum gravity, suggest the existence of a\nfinite lower bound to the possible resolution $\\Delta X$ of\nspacetime points. The following deformed commutation relation has\nattracted much attention in recent years \\cite{1}\n\\begin{equation}\n[X, P]=i\\hbar(1+\\beta P^2),\n\\label{eq1}\n\\end{equation}\nand it was shown that it implies the existence of a minimal\nresolution length $\\Delta X=\\sqrt{\\langle X^2 \\rangle -\\langle X\n\\rangle^2}\\ge\\hbar\\sqrt\\beta$. This means that there is no\npossibility to measure coordinate $X$ with accuracy smaller than\n$\\hbar\\sqrt\\beta$. Since in the context of the string theory the\nminimum observable distance is the string length, we conclude that\n$\\sqrt{\\beta}$ is proportional to this length. If we set $\\beta=0$,\nthe usual Heisenberg algebra is recovered. The use of the deformed\ncommutation relation (\\ref{eq1}) brings new difficulties in solving\nthe quantum problems. A part of difficulties is related to the break\ndown of the notion of locality and position space representation in\nthis framework \\cite{1}. The above commutation relation results in\nthe following uncertainty relation:\n\\begin{eqnarray}\n \\Delta X \\Delta P \\geq \\frac{\\hbar}{2}\n\\left( 1 +\\beta (\\Delta P)^2 +\\gamma \\right),\n\\label{eq2}\n\\end{eqnarray}\nwhere $\\beta$ is the GUP parameter and $\\gamma$ is a positive\nconstant that depends on the expectation value of the momentum\noperator. In fact, we have $\\beta=\\beta_0\/(M_{Pl} c)^2$ where\n$M_{Pl}$ is the Planck mass and $\\beta_0$ is of the order of unity.\nWe expect that these quantities are only relevant in the domain of\nthe Planck energy $M_{Pl} c^2\\sim 10^{19}$GeV. Therefore, in the low\nenergy regime, the parameters $\\beta$ and $\\gamma$ are irrelevant\nand one recovers the well-known Heisenberg uncertainty principle.\nThese parameters, in principle, can be obtained from the underlying\nquantum gravity theory such as string theory. Moreover, the\ncomparison between Eqs.~(\\ref{eq1}) and (\\ref{eq2}) shows that\n$\\gamma=\\beta\\langle P\\rangle^2$. Now, let us define \\cite{9}\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{ll}\nX = x,\\\\\\\\ P = p \\left( 1 + \\frac{1}{3}\\beta\\, p^2 \\right),\n\\end{array}\n\\right.\n\\label{eq4}\n\\end{eqnarray}\nwhere $x$ and $p$ obey the canonical commutation relations\n$[x,p]=i\\hbar$. One can check that using Eq.~(\\ref{eq4}),\nEq.~(\\ref{eq1}) is satisfied up to ${\\cal{O}}(\\beta)$. Also, from\nthe above equation we can interpret $p$ as the momentum operator at\nlow energies ($p=-i\\hbar \\partial\/\\partial{x}$) and $P$ as the\nmomentum operator at high energies. Now, consider the following form\nof the Hamiltonian:\n\\begin{eqnarray}\nH=\\frac{P^2}{2m} + V(x),\n\\label{eq5}\n\\end{eqnarray}\nwhich using Eq.~(\\ref{eq4}) can be written as\n\\begin{eqnarray}\nH=H_0+\\beta H_1+{\\cal{O}}(\\beta^2),\n\\label{eq6}\n\\end{eqnarray}\nwhere $H_0=\\frac{\\displaystyle p^2}{\\displaystyle2m} + V(x)$ and\n$H_1=\\frac{\\displaystyle p^4}{\\displaystyle3m}$.\n\nIn the quantum domain, this Hamiltonian results in the following\ngeneralized Schr\\\"odinger equation in the quasi-position\nrepresentation\n\\begin{eqnarray}\n-\\frac{\\hbar^2}{2m}\\frac{\\partial^2\\psi(x)}{\\partial\nx^2}+\\beta\\frac{\\hbar^{4}}{3m}\\frac{\\partial^{4}\\psi(x)}{\\partial\nx^{4}} +V(x)\\psi(x)=E\\psi(x),\n\\label{eq7}\n\\end{eqnarray}\nwhere the second term in the left side is due to the generalized\ncommutation relation (\\ref{eq1}). This equation is a fourth-order\ndifferential equation which in principle admits four independent\nsolutions. Therefore, solving this equation in $x$ space and\nseparating the physical solutions is not an easy task. With these\npreliminaries, in the next section we solve equation (\\ref{eq7}) for\na quantum well to address the RT effect and the Fabry-Perot\ninterferometer resonance condition in the presence of the minimal\nobservable length.\n\n\\section{The Ramsauer-Townsend effect with GUP}\nWe choose the following geometry of the quantum well (see\nFig.~\\ref{fig1})\n\\begin{equation}\nV(x)=\\left\\{\\begin{array}{ll} -V_{0}&\\quad \\quad0< x < a,\\\\ \\\\\n\\quad0&\\quad\\quad{\\rm elsewhere},\\end{array}\\right. \\label{eq8}\n\\end{equation}\nwhere $V_{0}$ is a positive constant and we assume $E>0$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{gup3.eps}\n\\caption{ The geometry of a quantum well.}\\label{fig1}\n\\end{figure}\n\nThe eigenfunctions of a particle in this potential well satisfy the\ngeneralized Schr\\\"{o}dinger equation (\\ref{eq7}). We need to find\nthe solutions in three different regions which are indicated in\nFig.~(\\ref{fig1}). To proceed further, we rewrite Eq.~(\\ref{eq7}) in\nthese regions separately as\n\\begin{equation}\nd^{2}\\psi(x)+q^{2}\\psi(x)-\\ell_{p}^{2}d^{4}\\psi(x)=0, \\label{eq12}\n\\end{equation}\nfor $0)^2 + v_\\theta^2 + v_\\phi^2\\Big) \\right>\/\\left<\\rho\\right>}\n\\label{eq:vaniso}\n\\end{equation}\nwhere $\\left<\\right>$ represents an angle average, $v_r$ is the radial velocity, $v_\\theta$ is the velocity component in the polar direction, $v_\\phi$ is the velocity component in the azimuthal direction, and $\\rho$ is the density. \n\nWith the introduction of rotation, a positive angular momentum gradient can be established, leading to inhibited convection, according to the Solberg-H{\\o}iland stability criterion. To quantify this criterion we calculate the condition at the equator for stability in the vertical direction, $R_{\\mathrm{SH}}$, consistent with \\citet{heger:2000}:\n\n\\begin{equation}\n R_{\\mathrm{SH}} := \\frac{g}{\\rho}\\Bigg[\\Bigg(\\frac{d\\rho}{dr}\\Bigg)_{\\mathrm{ad}}-\\frac{d\\rho}{dr}\\Bigg] + \\frac{1}{r^3}\\frac{d}{dr}(r^2\\omega)^2 \\geq 0\n\\label{eq:SHI}\n\\end{equation}\nwhere $g$ is the local gravitational acceleration, $\\rho$ is the density, $(d\\rho\/dr)_{\\mathrm{ad}}$ is the radial density gradient at constant entropy and composition, $r$ is the distance from the axis of rotation, and $\\omega$ is the rotational velocity. \n\nTo examine the shape of the shock front, $R_S(\\theta,\\phi)$, we represent it as a linear combination of spherical harmonics, $Y_l^m(\\theta,\\phi)$:\n\n\\begin{equation}\n R_S(\\theta,\\phi) = \\sum_{l=0}^\\infty \\sum_{m=-l}^{l} a_l^m Y_l^m(\\theta,\\phi)\n\\end{equation}\n\\begin{equation}\n Y_l^m = \\sqrt{\\frac{2l+1}{4\\pi}\\frac{(l-m)!}{(l+m)!}}P_l^m(\\cos(\\theta)) e^{im\\phi}\n\\end{equation}\nwhere $P_l^m$ are the associated Legendre polynomials \\citep{burrows:2012,takiwaki:2012}. However, because of the 2D nature of our simulations $\\phi = 0$ and all $m = 0$ as well; thus the coefficients $a_l^0$ are\n\n\\begin{equation}\n a_l^0 = \\int_0^\\pi d\\theta \\sin(\\theta)R_S(\\theta)Y_l^0(\\theta) .\n\\label{eq:sho_coeff}\n\\end{equation}\nIt follows that $a_0^0$ corresponds to the average shock radius.\n\\subsection{Rotation's Influence on Shock Front Evolution}\n\nWhile our focus in the present work is on the GW signals up to 300 ms postbounce, we briefly discuss the impact of rotation on the evolution of the shock front as it propagates outward. In certain cases, independent of the mechanism, the shock front may require over 300 ms to revive and complete a successful explosion. Because our simulations are only run until 300 ms postbounce, we refrain from asserting which progenitors successfully explode. Rather, we remark on how the average shock radii develop with time.\n\nOf our 15 simulations, only the nonrotating 20 \\ensuremath{\\mathrm{M}_\\odot}\\xspace and 60 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitors show substantial shock expansion. The effect rotation has on reviving the shock is not a simple one. \nIn one respect, one expects greater centrifugal support to lead to a larger shock front. However, there are two factors that inhibit the shock from propagating outward. The first is the inhibited convection due to the positive angular momentum gradient within the progenitor. Weaker convection results in less efficient neutrino heating \\citep{dolence:2013, murphy:2013} and less positive support from turbulence in the gain region \\citep{couch:2015a, mabanta:2018}. The second rotational element that inhibits explosions is the lack of neutrino production. Rotation centrifugally supports matter that is infalling during the initial collapse of a star. As such, the collapsing material does not settle as deeply into the gravitational potential of the stellar core, thereby releasing less gravitational binding energy. This process results in a lower neutrino luminosity and slower contraction of the PNS \\citep{summa:2018}. \nThese two dominant effects, weaker convection and reduced neutrino luminosity, can create an unfavorable scenario for a supernova explosion that is revived by neutrino heating.\n\nDespite rotation inhibiting certain aspects of a successful explosion, some of our rotating models ($\\Omega_0 = 3$ rad s$^{-1}$, 20 \\ensuremath{\\mathrm{M}_\\odot}\\xspace, and 60 \\ensuremath{\\mathrm{M}_\\odot}\\xspace) have advancing shock radii. With longer simulation times, these could lead to explosion. In these cases, it seems that rotation could be sufficiently rapid to overcome the deleterious effects on convection and reduced neutrino luminosity.\nSimilar nonmonotonic behavior is reported by \\citet{summa:2018} in their 2D simulations.\nHence, the introduction of rotation involves competing forces that can enhance or diminish the shock. Figure \\ref{fig:shock} shows the average shock radius evolution versus time (postbounce) over our entire parameter space.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width = 0.5\\textwidth]{M1_shock_mass_raster.pdf}\n \\caption{Shock radius evolution of the four progenitor models versus time (postbounce). As different progenitors evolve at different rates, they may not have enough time to revive their shock front within the 300 ms interim. As such, only the nonrotating 20 \\ensuremath{\\mathrm{M}_\\odot}\\xspace and 60 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitors show substantial shock expansion. }\n \\label{fig:shock}\n\\end{figure} \n\n\\subsection{Comparison with CFC GR}\n\n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{bounce_richers.pdf}\n \\caption{GW strain vs. time (postbounce) for a 12 \\(M_\\odot\\) progenitor \\citep{woosley:2007} with $\\Omega_0 = 3$ rad s$^{-1}$. Plotted in the dashed line is the GW strain from \\citet{richers:2017} using the CFC \\texttt{CoCoNuT} code, and the solid line is our result using the effective relativistic potential coupled with Newtonian dynamics. While the different grids and treatment of hydrodynamics lead to differences in the strain in the early postbounce phase, we qualitatively verify our gravitational treatment by obtaining a nearly exact bounce signal. }\n \\label{fig:bounce_cfc}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering \n \\includegraphics[width=0.49\\textwidth]{hd3_bounce_test.pdf}\n \\includegraphics[width=0.49\\textwidth]{hdj_bounce_final.pdf}\n \\caption{(Left) GW bounce signal from all 10 progenitor masses with $\\Omega_0 = 3 \\text{ rad s}^{-1}$. By applying Equation (\\ref{eq:omega}), we assign a radially dependent, angular velocity to our progenitors. Because the central density profiles of each progenitor are different---namely, a less compact $12\\,M_\\odot$ and more compact $40\\,M_\\odot$---the progenitor cores are endowed with different amounts of angular momenta. (Right) Modified bounce signals after adjusting rotation rates to yield similar angular momenta ($\\sim 2.4\\times10^{49} \\text{erg s}$) of the inner $1.75\\,M_\\odot$ of matter. As predicted by \\citet{dimm:2008} and \\citet{abdik:2010,abdik:2014}, the GW bounce signals depend on the inner core angular momentum at bounce, not the original ZAMS mass.}\n \\label{fig:bounce}\n\\end{figure*}\n\nIn multidimensional simulations of CCSNe, the treatment of gravity must offer a balance between numerical accuracy and computational cost. The CFC offers a nearly identical GW signal, compared with full GR, while reducing simulation time \\citep{ott:2007}. Figure \\ref{fig:bounce_cfc} offers a qualitative check of our effective GR potential compared with CFC \\citep{richers:2017}. We incorporate an identical deleptonization profile \\citep{lieb:2005} and SFHo EOS \\citep{steiner:2013} for a 12 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitor \\citep{woosley:2007}. Moreover, we match the differential rotation parameter and rotation profile by selecting an $A = 634$ km and $\\Omega_0 = 3$ rad s$^{-1}$. For this comparison, we match the neutrino physics of \\citet{richers:2017}'s simulation by using a ray-by-ray, three-species, neutrino leakage scheme \\citep{oconnor:2010,couch:2014}. We capture a nearly identical bounce signal and similar strain up to 5 ms postbounce. \\par\nHowever, after the initial bounce signal ring-down, it is clear that the different computational treatments of hydrodynamics and grid geometry result in differences in the GW strains. Although not exact, the efficiency of the effective GR potential offers a reasonable method to accurately model the GW signal from CCSNe to within 10\\% and allows for larger sweeps of parameter space \\citep{muller:2013}.\n\n\n\n\\subsection{ZAMS Influence on Gravitational Bounce Signal}\n\nWhile different progenitors $\\gtrsim$$8\\, M_\\odot$ will experience widely varied evolution, once their iron cores reach the effective Chandrasekhar mass \\citep{baron:1990} and collapse commences, the physics of the collapse becomes somewhat universal.\nIn particular, the mass of the homologously collapsing inner core is fixed more by microphysics than by the macrophysics of varied stellar evolution. \nThis nearly identical inner core mass across the ZAMS parameter space yields similar core angular momenta, for identical rotation rates. Hence, the core bounce signal is nearly indistinguishable between progenitor masses. For further verification of our gravitational treatment, we perform 12 additional simulations using neutrino leakage---from collapse---until 8 ms after core bounce, in order to replicate this bounce signal degeneracy, using the \\citet{Suk:2016} progenitors. Outlined by \\citet{ott:2012}, neutrino leakage has a small effect on the GW bounce and early postbounce signal. \nMoreover, our results are consistent with 3D, fully GR predictions given by \\citet {ott:2012} that similar core angular momenta yield similar GW bounce signals. \n\nFigure \\ref{fig:bounce} displays the bounce signals for all 10 progenitor masses, ranging from 12 \\ensuremath{\\mathrm{M}_\\odot}\\xspace to 120 \\ensuremath{\\mathrm{M}_\\odot}\\xspace. The left panel is for uniform rotational velocity prescriptions at $\\Omega_0 = 3\\text{ rad s}^{-1}$. As previously highlighted, the angular momentum of the inner core is the main contributor to the gravitational bounce signal. While many of the waveforms have similar amplitudes, there are two clear outliers: the $12\\,M_\\odot$ and $40\\,M_\\odot$ progenitors. The $12\\,M_\\odot$ and $40\\,M_\\odot$ progenitors, respectively, have lower and higher compactness values at collapse, by nearly a factor of 2. Because we endow each progenitor with angular velocity, and not specific angular momentum, the more compact $40\\,M_\\odot$ progenitor will receive more angular momentum, compared with the remaining progenitors, thereby affecting the resulting GW bounce signal. As outlined by \\citet{dimm:2008}, once a star is sufficiently rotating, the centrifugal support slows the bounce, diminishing the GW bounce amplitude and widening out the bounce peak of the waveform. \n\nThe inverse is true for the $12\\,M_\\odot$ case. Because it has a less compact inner core at collapse, using Equation (\\ref{eq:omega}) leads to less initial angular momentum, thereby producing a lower amplitude bounce signal. After modifying the initial rotation rates of both progenitors, to match the progenitor core angular momenta (right panel of Figure \\ref{fig:bounce}), the change produces nearly identical GW bounce signals. \n\nHence, our results from exploring the bounce signal over a wide range of progenitor masses support the results of previous studies of the angular momentum dependence of the GW signal \\citep{dimm:2008,abdik:2010,abdik:2014} but also serve as a cautionary note for future groups who perform rotating CCSN simulations with a wide variety of progenitor models. \nIt is worth noting that other rotational treatments exist beyond the simple angular velocity law, such as specifying a radial, specific angular momentum profile \\citep[eg.,][]{oconnor:2011} or using the rotational profile from the rotating stellar evolution models directly \\citep{summa:2018}. The profiles used by \\citet{oconnor:2011} lead to a roughly uniform rotation rate within a mass coordinate of 1 $M_\\odot$ and $\\Omega(r)$ decreasing with $r^{2}$ outside this mass coordinate. \\citet{summa:2018} utilize two different rotation schemes: one that matches the \\citet{heger:2005} models seen in Figure \\ref{fig:ovsr} and one that is solid body out to $\\sim 1500$ km and then falls as $r^{-3\/2}$.\n\n\\subsection{Rotational Influence on Accretion-phase GW Emission}\n\nOur results in the previous section support the efficacy of our effective GR potential for accurately modeling the GW signals from CCSNe. While the effective GR potential has been shown to overestimate peak frequency from GWs compared with GR, it produces similar GW amplitudes and accurately captures PNS compactness during the accretion-phase \\citep{muller:2013}. Thus we now turn to exploring the rotational effects on the GW signal during the accretion-phase, up to 300 ms after bounce.\n\nWhile the consistency of the inner core mass for a collapsing iron core creates a setting where envelope mass has little effect on the bounce signal, the postbounce dynamics of the explosion largely depend on the mass surrounding the PNS. For nonrotating CCSNe, the shock front propagates outward and loses energy due to dissociation of iron nuclei and neutrino cooling. In the case of rotation, the initial progenitor and resulting shock front become more oblate. Rotation can affect the GW emission in three respects: the postshock convection is damped, the SASI becomes restricted, and it slows the rate at which the PNS peak vibrational frequency increases.\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{vaniso_rad.pdf}\n \\caption{Spherically averaged anisotropic velocity of the postshock region for the 12 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitor. Brighter colors correspond to increased convection in the postshock region according to Equation (\\ref{eq:vaniso}). As rotational velocity increases, convective activity is inhibited. Traced in red is the radius of the PNS.}\n \\label{fig:vaniso}\n\\end{figure}\n\n\\begin{figure}[]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{SHI_panel_invert.pdf}\n \\caption{Slices along the equator of the 12 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitor at each rotational velocity. Colors correspond to the Solberg-H{\\o}iland stability criterion, $R_{\\mathrm{SH}}$, from Equation (\\ref{eq:SHI}). As rotational velocity increases, not only does the convectively stable band in the core grow (seen in blue), but the amount of convection within the postshock region (seen in red) decreases as well. The differences in shock radius evolution between Figure \\ref{fig:vaniso} and this figure arise because Figure \\ref{fig:vaniso} uses an angular average over the domain, whereas this figure uses equatorial slices.}\n \\label{fig:SHI}\n\\end{figure}\n\n \\begin{figure*}[htp]\n \\centering \n \\includegraphics[width=\\textwidth]{tdwf_region_20.pdf}\n \\caption{Time domain waveforms for the 20 $M_\\odot$ progenitor. Each panel corresponds to the region from which the GWs are emitted. The large contribution in the top panel indicates the main source of GWs during the accretion-phase is from the vibrating PNS. The lower panel displays the inhibited convective signal $\\sim 50$--$100 $ ms postbounce that is characteristic of this quiescent phase.}\n \\label{fig:region}\n\\end{figure*}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{sasi_axis_norm.pdf}\n \\caption{Coefficients from spherical harmonic decomposition of the shock front, outlined in Equation (\\ref{eq:sho_coeff}). The $a_1^0\/a_0^0$ and $a_2^0\/a_0^0$ terms describe the overall dipole and quadrupole nature of the shock front, respectively. As the SASI is one of the main contributors to the creation of asymmetries in the shock front, the lower $a$ values correspond to a less prolate shock, or one with diminished SASI.}\n \\label{fig:sasi}\n\\end{figure}\n\n As the $\\Omega_0$ value increases in our models, a positive angular momentum gradient is established within the postshock region, partially stabilizing it to convection via the Solberg-H{\\o}iland instability criterion \\citep{endal:1978,fryer:2000}. We quantify the reduced convection in Figure \\ref{fig:vaniso}. Brighter colors correspond to higher values of the anisotropic velocity as outlined in Equation (\\ref{eq:vaniso}). As expected, the convection in the gain region is reduced with increasing rotational velocity. To tie this inhibited convection to the Solberg-H{\\o}iland instability criterion, we follow the prescription of Section 2.3.2 of \\citet{heger:2000}. We quantify this instability criterion as outlined in Equation (\\ref{eq:SHI}) by taking slices along the equator and tracking its evolution. Figure \\ref{fig:SHI} displays the $R_{\\mathrm{SH}}$ value along the equator of the 12 \\ensuremath{\\mathrm{M}_\\odot}\\xspace progenitor for all four rotational velocities. As the $\\Omega_0$ increases, the propensity for convection (colored red) within the postshock region clearly decreases. This inhibited convection results in weakened turbulent mass motion within the gain region, thereby reducing the GW amplitude at later times.\n \nFurthermore, we recast our analysis by focusing on regions within the CCSN that emit GWs. The lower panel of Figure \\ref{fig:region} displays the inhibited convective signal with increasing rotation, as the GW signal in the gain region becomes increasingly muted. The typical convective signals in the early postbounce regime are then quickly washed out by the postbounce ring-down of the PNS, as rotation increases.\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{ccsn2D_M1_all.pdf}\n\\centering\n\\caption{Time domain waveforms over our entire parameter space. For all four progenitor masses, the rotational muting of the accretion-phase GW signal is clear. While there is some weak dependence in the character of the accretion-phase GW signals with progenitor ZAMS mass, the rotational muting occurs for all progenitors.}\n\\label{fig:ccsn_all}\n\\end{figure*}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[trim=80 0 0 0, scale=0.38]{gws_2x2_line_test_size.pdf}\n \\caption{Spectrograms for the $12 M_\\odot$ progenitor over all four rotational velocities. The key aspects revealed by the spectrogram are the rotational muting of GWs and the flattening of the signal from the surface g-mode of the PNS. This flattening is a product of the enlarged radius of the PNS due to centrifugal effects and can be characterized by the dynamical frequency ($f_{dyn} = \\sqrt{G \\overline{\\rho}}$), overlaid in gray.}\n \\label{fig:2x2}\n\\end{figure}\n\n\\begin{figure*}[t!]\n \\centering \n \\includegraphics[width=\\textwidth]{tbe6tbe300_combined_M1_long_referee.pdf}\n \\caption{ASD plot of all progenitors for all rotation rates from $t_{be}+6$ ms $\\rightarrow t_{be}+300$ ms, with an assumed distance of 10 kpc. The rotational muting of the fundamental PNS g-mode is displayed as the peak frequency ($\\sim 800$ Hz) becomes less prevalent, with increasing rotation rate. Likewise, the low-frequency signals ($\\sim40$ Hz) from the gain region become more audible, with increasing rotational velocity. The damping of the vibrational modes of the PNS allows the slower postshock convection to contribute more to the overall GW signal. Plotted in the black dashed line is the design sensitivity curve for aLIGO in the zero-detuning, high-sensitivity configuration \\citep{barsotti:2018}. The cyan dashed line is the predicted KAGRA detuned, sensitivity curve \\citep{komori:2017}. The purple dashed line is the design sensitivity curve for AdV \\citep{abbott:2018}.}\n \\label{fig:spetra_long}\n\\end{figure*}\n\nUnder nonrotating conditions, the shock can grow unstable due to nonradial deformations exciting a vortical-acoustic cycle that leads to the growth of large-scale shock asymmetries, that is, the SASI \\citep{blondin:2003, blondin:2006, scheck:2008,marek:2009a}.\nIn 2D simulations, the SASI excites large, oscillatory flows along both poles that drive changes in entropy capable of causing postshock convection. It is worth noting in 3D simulations that the SASI can excite `spiral' modes that correspond to nonzero $m$ values \\citep{blondin:2007,kuroda:2016}. The high degree of nonlinearity among the hydrodynamic flows, neutrino interactions, and gravitational effects can yield matter flow that is quadrupolar, thereby resulting in GW emission. However, when the shock becomes restricted in the polar direction, due to centrifugal effects, SASI development is inhibited. To quantify the role of SASI, we decompose the shock front into coefficients based on the spherical harmonics, $Y_l^m$, according to Equation (\\ref{eq:sho_coeff}). Figure \\ref{fig:sasi} illustrates the evolution of the $a_1^0$ and $a_2^0$ coefficients over time. Both coefficients quantify the deviation of the shock from spherical symmetry. Specifically, the $a_1^0$ term describes the overall dipole nature of the shock, and the $a_2^0$ term describes its quadrupole nature. Both coefficients are normalized by the mean shock radius, $a^0_0$. Clearly, both approach zero with increasing rotational velocity. Physically, this effect corresponds to a shock that is becoming less prolate. To further illustrate this transition, we direct the reader to Figures \\ref{fig:vaniso} and \\ref{fig:SHI}. Figure \\ref{fig:vaniso} takes an angular average to calculate $v_{\\mathrm{aniso}}$, whereas Figure \\ref{fig:SHI}, by contrast, uses equatorial slices to calculate the Solberg-H{\\o}iland stability criterion. The boundary between the white and colored region in both panels then acts as a proxy for average shock radius and equatorial shock radius, respectively. Thus, as rotational velocity increases, average shock radius decreases, while increasing the equatorial shock radius. Put more simply, the rotation in our 2D simulations acts to create less prolate shock fronts. Hence, because SASI plays a significant role in creating a shock that is extended along the axis of rotation, we conclude that the effect of SASI is reduced as rotational velocity increases in our 2D simulations.\nWhile we expect the SASI activity to contribute uniquely to the GW spectrum, depending on progenitor mass, the rotational muting of the GWs is universal across ZAMS mass parameter space, as illustrated in Figure \\ref{fig:ccsn_all}. \nBoth \\citet{burrows:2007} and \\citet{moro:2018} point out the partial suppression of SASI, but the former does not focus on the gravitational radiation emitted and the latter only examines a single, slow rotating, progenitor. Our work provides strong support for the rotational muting of accretion-phase GWs, over such a wide region of parameter space of 2D CCSN simulations. \n \nWith respect to PNSs, a variety of oscillatory modes exist that could be of interest to current and future GW astronomers: fundamental f-modes, pressure based p-modes, and gravity g-modes---due to chemical composition and temperature gradients \\citep{unno:1989}. The typical frequency of the PNS f-mode is around 1 kHz, and p-modes have frequencies greater than f-modes, which are of little use to GW astronomers, with the current detector capabilities \\citep{ho:2018}. \nThe frequencies of g-modes, however, are on the order of hundreds of hertz, falling squarely within the detectability range of current GW detectors \\citep{martynov:2016}. \nThe top panel of Figure \\ref{fig:region} displays the contribution of the vibrating PNS to the majority of the GW signal during the accretion-phase, with $h_+D$ normalized strain amplitudes around 50 cm. \nThese g-modes are thought to be excited by downflows from postshock convection or internal PNS convection \\citep{marek:2009b,murphy:2009,muller:2013}. \nFigure \\ref{fig:2x2} shows a spectrogram for the $12 \\, M_\\odot$ progenitor over all rotational speeds, where lighter colors represent greater strain amplitudes, $h_+$. The dominant yellow band that extends from 100 to 1000 Hz represents this contribution. \nOverlaid in gray is the dynamical frequency that is characterized by the average density of the PNS, $\\overline{\\rho}$, and gravitational constant, $G$, $f_\\mathrm{dyn} = \\sqrt{G \\overline{\\rho}}$, that evolves synchronously with the g-mode contribution. The synchronized evolution of $f_\\mathrm{dyn}$ and the frequency at which the PNS emits gravitational radiation are no coincidence. As both are fundamentally related to the mass and radius of the PNS, we expect that both are affected similarly when introducing rotation. The initial progenitor rotation will centrifugally support the PNS, thereby leaving it with a larger average radius. Similar to two tuning forks of different lengths, the PNS with a larger radius will emit at a lower frequency, compared with a smaller PNS. This ``flattening'' of the emitted frequency is displayed in Figure \\ref{fig:2x2}. Furthermore, Figure \\ref{fig:2x2} provides a different lens through which the rotational muting is displayed, via the progressively darker panels with increasing rotational velocity. We note that more robust peak GW frequency calculations exist \\citep[e.g.,][]{muller:2013,moro:2018}, but we find that the simple $f_\\mathrm{dyn}$ relation gives a good estimate of the PNS peak frequency.\n\nWe also Fourier transform the accretion-phase GW signal, as displayed in Figure \\ref{fig:spetra_long} and scale the magnitude of the Fourier coefficients by $\\sqrt{f}$ in order to produce ASD plots. These plots commonly display the sensitivity curves of current and next-generation GW detectors. We define $t_{\\mathrm{be}}$ similar to \\citet{richers:2017} as the third zero crossing of the gravitational strain. We focus on the signal later than $t_{\\mathrm{be}} + 6$ ms in order to remove the bounce signal and early postbounce oscillation contribution to the signal. \nThe dominant contributions are the prompt convection, SASI, and surface g-modes of the PNS---as displayed by a peak frequency ranging from 700 to 1000 Hz. Universally, the prevalence of the peak frequency decreases with increasing rotational velocity. It is worth noting this peak could shift to higher frequencies with longer simulation times.\n\nWhen incorporating magnetic fields into CCSN simulations, other instabilities may arise that can compromise stability in the postshock region and possibly affect the behavior of the PNS. The $\\alpha$--$\\Omega$ dynamo and MRI are two such mechanisms that can reexcite postshock convection; however, work from \\citet{bonanno:2005} suggests that the $\\alpha$--$\\Omega$ dynamo is unimportant on dynamical timescales. MRI has the potential to drive convection in the postshock region, yet as the strength and geometry of magnetic fields in 3D simulations are largely still unknown, we exclude them from our simulations \\citep{cerda-duran:2007}.\n\n\\subsection{Observability of the Accretion-phase Signal}\n\nOverlaid on our ASD plots is the expected sensitivity of future GW observatories. \nIn the black, cyan, and purple dashed lines we have plotted the sensitivity curves of design sensitivity for aLIGO in the zero-detuning, high-sensitivity configuration, the predicted KAGRA detuned sensitivity curve, and design sensitivity for AdV, respectively \\citep{komori:2017,abbott:2018,barsotti:2018}.\nThese curves represent the incoherent sum of the principal noise sources to the best understanding of the respective collaborations. While these curves do not guarantee the performance of the detectors, they act as good guides for their anticipated sensitivities nonetheless. \n\nBeyond the decreased prevalence of the peak frequency, an interesting trend emerges in Figure \\ref{fig:spetra_long} as rotation increases. \nWe separate the GW signals by region within the star. The top row of Figure \\ref{fig:spetra_long} corresponds to GWs originating from the inner 50 km of the supernova, and the GW signal in the bottom row originates from radial distances between 50 and 150 km from the supernova center. In the top row, we note the first peak of emission, around 80 Hz, is independent of rotation. We point to the bright, higher $v_\\mathrm{aniso}$ region in Figure \\ref{fig:vaniso} within the first 25 ms postbounce that is present for all rotational velocities.\nFocusing on the bottom row, we highlight a noticeable difference in the amplitude of the low-frequency contributions, particularly around 40 Hz. The nonrotating progenitors have undetectable low-frequency signals for all three detectors, whereas rotating progenitors create measurable signals at low frequencies. \nThis enhanced low-frequency signal may provide an observable feature that can help determine progenitor angular momentum information. \n\nThe amplitude of low-frequency GWs in the 50--150 km region of the supernova increases with rotational velocity, but this trend does not occur within the inner 50 km. As such, we restrict the low-frequency GW contribution to the gain region. We note the two main physical mechanisms in this region correspond to postshock convection and the SASI. While both mechanisms are reduced in strength due to rotational effects, they do not completely cease. This fact is displayed in Figure \\ref{fig:vaniso}, as the region between 50 and 150 km is nonzero. For the nonrotating case, the high convective velocities (bright yellow) create higher frequency GWs within the 100 km region of interest. As rotation velocity increases, convective velocities decrease enough to cease exciting the vibrational modes of the PNS. These slower convective flows thereby reduce the total amount of power produced by the GWs and push the peak GW frequency---from the gain region---to lower frequencies. Performing an order-of-magnitude estimate on the source of the low-frequency signal, from Figure \\ref{fig:vaniso}, we find $v_{\\mathrm{aniso}} \\sim 1 \\times 10^9$ cm s$^{-1}$ for $\\Omega_0 = 0$ rad s$^{-1}$ and $v_{\\mathrm{aniso}} \\sim 5 \\times 10^8$ cm s$^{-1}$ for $\\Omega_0 = 3$ rad s$^{-1}$. As the region of interest is $\\sim 10^7$ cm, we yield an estimated frequency of emission $f_{\\mathrm{low}}$ around $\\sim 100$ Hz and $\\sim 50$ Hz, respectively. These quantitative frequency estimates are reflected in the ASD as the contribution from peak frequency ($\\sim 100$ Hz) from the gain region decreases, while the contribution $\\sim 40$ Hz increases.\n\n\n\n\n\\section{Summary and Conclusion}\n\\label{sec:summary}\n\nThe strength of this project is its ability analyze GWs hundreds of milliseconds postbounce from multiple progenitors while accurately accounting for rotation and neutrinos. The wide breadth of parameter space we examine allows us to reveal certain rotational effects on the GW signal in the context of a controlled study. We have explored the influence of rotation on the GW emission from CCSNe for four different progenitors and four different core rotational speeds. \nWe point out that there exists a roughly linear relation between compactness, $\\xi$, and the differential rotation parameter, $A$, as defined in Equation (\\ref{eq:omega}). \nUsing this relation, we calculate appropriate $A$ values for each progenitor mass, based on their individual compactness parameters of the \\citet{Suk:2016} progenitors. Of our 15 simulations, only two nonrotating progenitors have average shock radii that show substantial shock expansion, while the remaining rotating progenitors do not because of rotationally inhibited convection in the gain region and less neutrino production. In agreement with other recent work \\citep[e.g.,][]{summa:2018}, we find a complex interplay between centrifugal support and neutrino heating as successful explosions do not display a monotonic relationship with rotation.\n\nWhile there are more accurate treatments of gravity, we utilize the effective GR potential in order to streamline calculations, granting us the ability to explore larger sections of parameter space. \nWe find that our results utilizing this approximation match very closely the CCSN bounce signal of CFC gravity with GR hydrodynamics \\citep{richers:2017}. \n\nThe main contributors to the GW signal (10--300 ms postbounce) are postbounce convection, the SASI, and the surface g-modes of the PNS \\citep{moro:2018}. By establishing a positive angular momentum gradient, the convection is suppressed according to the Solberg-H{\\o}iland stability criterion \\citep{endal:1978,fryer:2000}. The more oblate shock front inhibits the bipolar sloshing of the SASI. Since the SASI and convection are the principal drivers exciting the g-modes of the PNS, vibrational emission from the PNS is also inhibited by rotation. \nWe, therefore, find that rotation in 2D CCSN simulations results in the muting of GW emission.\nThis result is consistent across progenitors with different ZAMS masses. \n\nBefore the PNS g-mode signal is completely muted, as rotation gradually increases, this signal is pushed to lower peak frequencies and can be characterized by its dynamical frequency. This observation is no coincidence as both fundamentally depend on the radius and mass of the PNS. With more centrifugal support, the PNS has a larger radius. This larger radius causes the surface of the PNS to emit at lower frequencies, thereby producing a ``flatter,'' lower frequency signal.\n\nWe reveal a novel rotational effect on the GW signal during the accretion-phase. We notice that the nonrotating progenitors all produce low-frequency signals ($\\sim 40$ Hz) that are below the plausible detection threshold of the aLIGO and KAGRA detectors, whereas the progenitors with larger angular velocities produce measurable GW signals in this frequency range. We attribute this increase of low-frequency emission to the SASI and postshock convection. For nonrotating progenitors, the convective velocity within the postshock region is high, emitting GWs $\\sim 100$ Hz. As rotational velocity increases, the PNS GW contribution is reduced. Likewise, as the convection slows, the mass within the gain region emits at lower GW frequencies. The slower convective flows reduce the total amount of GW power and push the peak GW frequency from the gain region to lower values. Whereas previous rotating core-collapse GW studies have focused on the bounce signal as a means to determine rotational features, or have focused on late time signals without rotation, our study unifies both facets and opens the door to measuring GW signals beyond the bounce phase that encode progenitor, angular momentum information. \nWe postpone asserting quantitative relations between low-frequency emission and progenitor angular momentum until we incorporate more detailed microphysics.\n\n\nWhile our approximations have allowed us to make large sweeps of parameter space, they leave room for us to include more robust microphysics. In an ideal situation, we would compute 3D simulations, including full GR, magnetohydrodynamics, and GR Boltzmann neutrino transport that incorporates velocity dependence and inelastic scattering on electrons and nucleons. These additions would allow for more accurate gravitational waveforms and allow other phenomena to occur, for example the $m\\ne 0$ (spiral) modes of the SASI. \\citet{andresen:2019} recently highlighted the rotational effects on GWs in 3D. Inherent to its 3D nature, their study finds the strongest GW amplitudes at high rotation velocities due to these spiral modes. The 2D geometry of our study, however, allows us to observe the relative strength of the convective signal, without interference from $m\\ne 0$ modes, as we extend beyond the case of a single rotational velocity.\nWhile the physical origin of this muting that damps the convection and the SASI is not constrained only to 2D, in 3D, as \\citet{andresen:2019} point out, other nonaxisymmetric instabilities can contribute to significant GW emission at late times, negating this rotational muting effect.\nThus, once again, we are reminded of the key role of 3D simulations in the study of the CCSN mechanism.\n\n\\acknowledgements\n\nWe would like to thank Jess McIver for pointing us to the aLIGO and AdV sensitivity curves. M.A.P. was supported by a Michigan State University Distinguished Fellowship. \nS.M.C. is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics,\nunder award Nos. DE-SC0015904 and DE-SC0017955 and the \\textit{Chandra\nX-ray Observatory} under grant No. TM7-18005X.\nThis research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) that are responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering, and early testbed platforms, in support of the nation's exascale computing imperative.\nThe software used in this\nwork was in part developed by the DOE NNSA-ASC OASCR Flash Center at\nthe University of Chicago. \n\n \\software{FLASH (see footnote 7) \\citep{fryxell:2000,fryxell:2010}, Matplotlib\\footnote[8]{\\url{https:\/\/matplotlib.org\/}} \\citep{hunter:2007},\n NuLib\\footnote[9]{\\url{http:\/\/www.nulib.org}}\n \\citep{oconnor:2015},\n NumPy\\footnote[10]{\\url{http:\/\/www.numpy.org\/}} \\citep{vanderwalt:2011}, SciPy\\footnote[11]{\\url{https:\/\/www.scipy.org\/}} \\citep{jones:2001}}\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{{\\rmfamily \\scshape vitaLITy}}\n\\label{sec:system}\n\n\\begin{figure*}[!t]\n \\centering\n \\setlength{\\belowcaptionskip}{-10pt}\n \\includegraphics[width=\\linewidth]{figures\/vitality-architecture.pdf}\n \\caption{The {\\rmfamily \\scshape vitaLITy}{} architecture. (1) DBLP data is filtered by relevant venues. (2) Author and title metadata from DBLP is augmented with abstracts, keywords, and citations from custom scrapers. (3) Data is cleaned (e.g., to resolve duplicate keywords, etc). (4) GloVe and Specter document embeddings are created. (5) Data is exported to a variety of formats for subsequent open-source use. (6) The server exposes a RESTful API that can ultimately be called upon in rendering the interactive system.}\n \\label{fig:architecture}\n\\end{figure*}\n\nWe present {\\rmfamily \\scshape vitaLITy}{}, a system designed to complement existing tooling for conducting academic literature reviews by supporting serendipitous discovery of relevant literature. \n\n\\subsection{Data} \n\\label{sec:data}\nFigure~\\ref{fig:architecture} outlines the pipeline for curating the paper corpus.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/venues.pdf}\n \\caption{Results from a Twitter survey with \\texttt{24} users on venues where VIS researchers publish; (numbers in parentheses) are aggregated counts of the \\texttt{24} responses; \\st{struckthrough venues} were not in DBLP and hence currently not available in {\\rmfamily \\scshape vitaLITy}{}; * venues were added as also-relevant venues by the authors after the survey; ``Vis.'' is short for Visualization.}\n \\label{fig:venues}\n\\end{figure}\n\n\n\\noindent\\textbf{1. Filter:} We conducted an open-ended crowd-sourced survey on Twitter asking visualization researchers about venues (e.g., journals, conferences, workshops) where they publish. We received responses from \\texttt{24} users (current roles: 17 Ph.D. students, 4 Faculty, 2 Industry researchers, and 1 Postdoctoral scholar; self-reported visualization literacy out of 5: $\\mu$=4, $\\sigma$=1.142, median=4). We supplemented the list with \\texttt{six} additional venues based on our own knowledge that were not captured in the survey. Figure~\\ref{fig:venues} outlines the final list of \\texttt{38} venues within our corpus. Next, we downloaded the November 2020 release~\\footnote{\\url{https:\/\/dblp.org\/xml\/release\/dblp-2020-11-01.xml.gz}} of DBLP~\\cite{dblp} and filtered it for the aforementioned \\texttt{38} venues. From the resultant subset, we chose the \\attr{Title}, \\attr{Authors}, \\attr{Source} (venue), \\attr{Year} (published), and \\attr{URL} attributes and added a unique \\attr{ID} for tracking. Note that the {\\rmfamily \\scshape vitaLITy}{} dataset and hence the UI show more than 38 venues because DBLP (a) utilizes multiple descriptors to represent different tracks at the same venue (e.g., Eurographics (Area Papers), Eurographics (State of the Art Papers), Eurographics (Short Papers), etc.) and (b) splits some venues across different versions (e.g., Interact, Interact (1), Interact (2), etc.).\n\n\\noindent\\textbf{2. Scrape:} The DBLP dataset does not include abstracts, keywords, and number of citations for papers. Thus, we developed a \\emph{scraper} module that, given a list of publication URLs, scrapes the corresponding publisher's webpage (e.g., IEEE Xplore, ACM Digital Library) and extracts the \\attr{Abstract}, \\attr{Keywords}, and \\attr{CitationCounts} from it.\n\n\\noindent\\textbf{3. Clean:} We performed data cleaning and transformation operations. To aid search, we encoded all text attributes to ASCII and converted \\attr{Authors} and \\attr{Keywords} into a JSON array from a comma separated list. We de-duplicated \\attr{Keywords} by matching their lowercase forms; we combined similar keywords (e.g., HCI \\& Human-Computer Interaction, Visuali\\textbf{z}ation \\& Visuali\\textbf{s}ation) through manual inspection. We dropped \\texttt{1497} papers with \\emph{null} \\{\\attr{Title}, \\attr{Authors}, \\attr{Abstract}\\} values, and very short or very long \\attr{Title} ($<$5, $>$250 characters) and \\attr{Abstract} ($<$50, $>$2500 characters) to create effective word embeddings. We retained DBLP's strategy in disambiguating author names (e.g., \\emph{J. Thompson} and \\emph{J. Thompson 001}). At the end of this step, the dataset has \\texttt{8} attributes (columns) and \\texttt{59,232} papers (rows).\n\n\\noindent\\textbf{4. Embed:} We next curated a dataframe of \\attr{Title}, \\attr{Abstract}, \\attr{Authors}, \\attr{Source}, \\attr{Year}, and \\attr{Keywords} and created the GloVe \\cite{pennington2014glove} and Specter\\cite{cohan2020specter} document embeddings. To create the document embeddings for GloVe, we used TF-IDF weightings (instead of mean vectors) and SIF weightings that have been shown to remove noise through PCA reduction \\cite{arora2017simple}. We used the public API to create the Specter embeddings~\\cite{cohan2020specter}. With these document embeddings, we used UMAP to construct 2-D document representations used in the Visualization Canvas (see Figure \\ref{fig:umap}).\n\n\\noindent\\textbf{5. Export:} We export the consolidated dataset as JSON and a MongoDB dump for different open-source use.\n\n\\noindent\\textbf{6. Serve:} We also developed a server that exposes a RESTful API to (a) load the {\\rmfamily \\scshape vitaLITy}{} document corpus, (b) perform similarity search by a list of seed papers as input, (c) perform similarity search by a working title and abstract as input, and (d) download metadata of (saved) papers as a JSON array. The similarity search by seed papers (b) supports querying by 2-D UMAP as well as n-D document embeddings for both GloVe and Specter. We used MongoDB to maintain the 2-D indexes and Facebook Research's faiss library~\\cite{faiss} to maintain the n-D indexes. \\revised{For one seed paper as input, we utilize existing APIs to compute the Euclidean (2-D; MongoDB) and L2 (n-D; Faiss) distances between the input paper and other papers, compute their reciprocals, and normalize them between 0-1 for use as the similarity scores (1 = most similar). For more than one seed paper as input, we first compute the average vector from all input papers and then follow the same procedure as above to compute the similarity scores. \n} The {\\rmfamily \\scshape vitaLITy}{} UI interfaces with this server, described next.\n\n\\subsection{System Overview}\n\\label{sec:system_overview}\n\nThe system, shown in Figure~\\ref{fig:teaser}, is comprised of a \\emph{Paper Collection View} (A), \\emph{Similarity Search View} (B), \\emph{Visualization Canvas} (C), \\emph{Meta View} (D), and \\emph{Saved Papers View} (E), described in turn below. \n\n\n\\smallskip\n\\noindent\\textbf{Paper Collection View} shows the entire corpus of papers in an interactive tabular layout. (1) shows an overview (number of visible papers) and UI controls to perform a global search (\\faSearch), show hidden columns ([Column~\\faPlus]), add all papers to the input list of papers in the \\emph{Similarity Search} table ([\\faPlusCircle~All]), and save all papers to the ``cart'' in the \\emph{Saved Papers View} ([\\faSave~All]). (2) shows the attributes along with UI controls to filter (range sliders for Quantitative attributes, multiselect dropdowns for Nominal attributes), hide a column (\\faEyeSlash), and define a column on hover (\\faQuestionCircle). (3) shows an interactive table of all papers with options to see detail \n(\\faInfoCircle), \nlocate in the UMAP (\\faMapMarker), add to the input list of papers for similarity search (\\faPlusCircle), and save to the ``cart'' (\\faSave). Search and filter capabilities are designed to be an intuitive entry-point into the dataset of academic articles (\\textbf{DG 2}).\n\n\n\\smallskip\n\\noindent\\textbf{Similarity Search View} shows options to find papers similar to (a) one or more input papers (Figure~\\ref{fig:search-by-papers}, \\textbf{DG 1}) or (b) a work-in-progress title and abstract (Figure~\\ref{fig:search-by-abstract}, \\textbf{DG 3}). {\\rmfamily \\scshape vitaLITy}{} supports setting the dimensions (2-Dimensional, n-Dimensional), number of similar papers to return, and the word embedding approach (e.g., Specter) to compute similarity.\n\n\\smallskip\n\\noindent\\textbf{Visualization Canvas} shows a 2-D UMAP projection of the embedding space of the entire paper collection (Figure~\\ref{fig:umap}, \\textbf{DG 4}): hovering on a point highlights it, shows the corresponding title in a fixed tooltip below, and automatically scrolls the collection (table) to bring the corresponding paper (row) into the viewport; clicking on a point (de)selects it and shows it in the tooltip below with additional options to \\faInfoCircle, \\faPlusCircle, \\faSave; clicking on \\faTimes~deselects all selected points; pressing Shift enables lasso-mode to select multiple points using a free-form lasso operation; zooming and panning support helps navigate the UMAP to specific regions; clicking on \\faDotCircleO~re-centers and fits the UMAP in the viewport. By default, each point in the UMAP is colored based on the state of the corresponding paper (``Default''): Unfiltered (unfiltered and visible in the main paper collection table; dark-grey), Filtered (filtered out and not visible in the paper collection table; light-grey), Similarity Input (added to the \\emph{By Papers} section in the Similarity Search View; pink), Similarity Output (in the \\emph{Output Similar} table; orange), and Saved (added to the Saved Papers table; red). Other options to color include \\attr{Source}, \\attr{Year}, \\attr{CitationCounts}, and \\attr{Similarity Score}.\n\n\\smallskip\n\\noindent\\textbf{Meta View} shows aggregated summaries of \\attr{Keywords}, \\attr{Authors}, \\attr{Source}, \\attr{Year} with respect to the \\emph{Paper Collection View} (A). Figure~\\ref{fig:meta} shows how a filter in the main table (\\attr{Authors}=\\emph{John T. Stasko}) updates the Meta Views with the distribution of keywords (a) associated with their research, their co-authors (b), venues where they have published (c), and in which years (d).\n\n\\smallskip\n\\noindent\\textbf{Saved Papers View} shows a table with the papers added to the ``cart'' with an additional option to export them as a JSON (Figure~\\ref{fig:search-by-papers}d).\n\n\\begin{figure}[!t]\n \\centering\n \\setlength{\\belowcaptionskip}{-10pt}\n \\includegraphics[width=\\columnwidth]{figures\/umap.pdf}\n \\caption{Interactive 2-D scatterplot of the UMAP projection.}\n \\label{fig:umap}\n \\end{figure}\n\n\\begin{figure*}[!t]\n \\centering\n \\setlength{\\belowcaptionskip}{-10pt}\n \\includegraphics[width=\\linewidth]{figures\/meta.pdf}\n \\caption{The Meta View showing aggregated summaries of (a) \\attr{Keywords}, (b) \\attr{Co-authors}, (c) \\attr{Source}, and (d) \\attr{Year} associated with \\emph{John T. Stasko}.}\n \\label{fig:meta}\n \\end{figure*}\n\n\\begin{comment}\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/info.pdf}\n \\caption{Clicking on \\faInfoCircle~opens a modal that describes all attributes and also lets the user to select, save, and open the paper's Google Scholar listing.\\ali{this fig can be removed if we don't have enough space}}\n \\label{fig:info}\n \\end{figure}\n\\end{comment}\n\n\\subsection{Implementation}\nThe \\emph{filter}, \\emph{scrape}, \\emph{clean}, \\emph{embed}, \\emph{export}, and \\emph{serve} modules are all implemented in Python. The \\emph{UI} is developed in React and uses the regl-based WebGL library\\footnote{https:\/\/github.com\/flekschas\/regl-scatterplot} to render the UMAP. MongoDB provides the document corpus to the UI and maintains the 2-D indexes while faiss~\\cite{faiss} maintains the n-D indexes for efficient similarity search.\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\medskip\n\\noindent\\textbf{Quality of Search Results. }\nAcross our (relatively small) sample of participants, there was variability in terms of perceived relevance of Similarity Search results. \nSome participants felt that, like Google Scholar, relevant results were lost among a sea of irrelevant papers, while others felt that the results were highly relevant. \nIn general, participants perceived results from SPECTER embeddings to be more relevant than GloVe, suggesting that further exploration of alternative transformer-based approaches (e.g., BERT~\\cite{devlin2018bert}, or training a custom model on the target document corpus) could yield better search results. \nFurthermore, given the disparity in perceived quality and disparity in participants' perception of when this approach could be useful in their literature review process, future work could develop additional guidelines that assess the specific role of document retrieval based on semantic similarity.\n\n\n\n\n\\medskip\n\\noindent\\textbf{Relevance \\& Space. }\nPresuming {\\rmfamily \\scshape vitaLITy}{} is able to provide serendipitous discovery of relevant literature, the process doesn't abruptly come to a successful end. \nAuthors still need to manage goals in their writing that may be at odds with one another: i.e., the tradeoff of relevance or salience of related work and the commodity of space. \nFrom this perspective, {\\rmfamily \\scshape vitaLITy}{} is best viewed as a way to identify critical gaps or serve as kindling for a new literature review. \nIn its current form, {\\rmfamily \\scshape vitaLITy}{} shows (1) similarity score, and (2) citation counts as the primary cues of relevance or salience of a given paper.\nIt still requires substantial knowledge from the author to (1) read an abstract or paper and determine its actual relevance to a given topic, and (2) assess the credibility of the work, author(s), and venue. \nSubsequent versions of {\\rmfamily \\scshape vitaLITy}{} could focus on innovating solutions to support these and other parts of the literature review process.\n\n\n\n\n\\medskip\n\\noindent\\textbf{Future Work. } \nBased on our use of {\\rmfamily \\scshape vitaLITy}{} and participant feedback, we identify a number of potential future directions. \\revised{First, as mentioned in the Related Work, with citation and user activity data, {\\rmfamily \\scshape vitaLITy}{} could expand its functionality to citation or read\/view recommendation using SPECTER.}\n\\revised{Second}, current similarity scores in the projected 2-D space (UMAP) are based on the reciprocal of the distance measure and might yield different results compared to distances in the N-D embedding space.\nThese scores and their context may not be especially intuitive for users. \nHence, future work could refine the similarity score formulation and \/ or presentation in {\\rmfamily \\scshape vitaLITy}{} to provide users an accessible framework to interpret results.\n\\revised{Third}, the Saved Papers Cart currently exports a file in JSON format with the papers. \nAt least two improvements could be made within this view, including exporting files in .bibtex format for easy incorporation in \\LaTeX{} bibliographies.\nFurthermore, it could be useful to users to provide a meta analysis of the saved papers, e.g., via topic modeling. \nHow can these papers be summarized? \n\\revised{Fourth, while our research prototype of {\\rmfamily \\scshape vitaLITy}{} is intended to be complementary to existing search strategies, future work could expand {\\rmfamily \\scshape vitaLITy}{} to a more comprehensive search tool, incorporating the benefits of e.g., citation networks.}\n\\revised{Lastly, {\\rmfamily \\scshape vitaLITy}{} is modular, scalable, and extensible: it applies the virtual scrolling principle in the UI table views (preventing unnecessary rendering of objects not visible in the viewport), renders the UMAP using WebGL, and uses a library (faiss) that performs efficient similarity search of dense vectors with an option to leverage GPUs. The \\emph{scraper} module currently uses DBLP as the source of raw data but can be extended to support other digital libraries, e.g., JSTOR (https:\/\/www.jstor.org\/).\nHence, augmenting the system with additional venues (and allowing users to define which venues are relevant to load in their specific literature review) is a feasible next step to expand {\\rmfamily \\scshape vitaLITy}{} to other research domains.}\n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\nBased on the final form of {\\rmfamily \\scshape vitaLITy}{}, developed from formative feedback with visualization researchers, we next describe the summative evaluation of {\\rmfamily \\scshape vitaLITy}{} in a qualitative study.\nWe recruited 6 Computer Science PhD students (1 female, 5 male; avg. 3.3 yrs. into PhD program) whose primary research area is within the field of visualization.\nNone of the participants were involved in the formative study. \nSessions lasted approximately 45 minutes.\nParticipation was voluntary with no compensation.\n\n\\subsection{Task \\& Procedures}\n\\label{sec:task}\nAfter obtaining informed consent,\nparticipants were asked to reflect on a topic for which they had recently or were currently conducting a literature review. \n{\\rmfamily \\scshape vitaLITy}{} was loaded with all \\texttt{59,232} papers, described in Section~\\ref{sec:data}, running locally on the study investigator's machine. \nParticipants connected to the study virtually via Microsoft Teams. \nThey were asked to \\textbf{recreate or continue their literature review using {\\rmfamily \\scshape vitaLITy}}, which they interacted with by using Microsoft Teams's ``Request Control'' feature on the study investigator's machine.\nWe utilized a think aloud protocol to capture users' impressions and qualitative feedback on the system. \nSessions were screen-recorded for subsequent analysis.\n\nParticipants chose the following topics for their literature reviews: \\emph{multiple comparisons problem}, \\emph{interpretable machine learning}, \\emph{misinformation}, \\emph{network visualization}, \\emph{scrollytelling visualization}, and \\emph{transformation \/ similarity between two visualizations}.\n\n\n\\subsection{Findings}\n\\label{sec:findings}\nIn this section, we discuss qualitative findings from our evaluation of {\\rmfamily \\scshape vitaLITy}{} for each of its primary features (Figure~\\ref{fig:sus-scores-specific}), and lastly summarize participants' general impressions of the system (Figure~\\ref{fig:sus-scores-generic}).\n\n\n\n\\subsubsection{Paper Collection View}\nParticipants felt that the Paper Collection View was a good ``entry point'' into the paper corpus in {\\rmfamily \\scshape vitaLITy}, containing familiar data that users expected to see, e.g., authors, abstracts, etc (S02). \nSearching by keyword was familiar and produced expected outcomes.\nFor instance, S03 identified some papers previously read as well as an interesting new one, which led them to iterate on their search query to find other papers by the same author.\n\nHowever, some users expressed the desire for the keyword search features to support more robust or customizable queries. \nAs a case in point, S01 conducted a global search for multiple keyword variations ``uncertainty visualisations'' $\\rightarrow$ ``uncertainty visualizations'' $\\rightarrow$ ``uncertainty visualization'', which returned 0, 8, and 49 hits respectively.\nFuzzy string matching would be a useful feature to support in subsequent iterations of {\\rmfamily \\scshape vitaLITy}{} (S06).\nAnother small usability issue that arose was lack of feedback upon clearing filters. \nFor instance, some participants would backspace to delete text; however, the system would only remove filters by selecting the `x' icon next to the filter (S01, S05). \nFurthermore, S06 suggested it would be useful for {\\rmfamily \\scshape vitaLITy}{} to expand the searchable text beyond titles and abstracts: \\textit{``Google Scholar searches body text too.''}\n\n\n\n\\subsubsection{Similarity Search}\n\\noindent\\textbf{By Paper. } The ability to start with a seed paper(s) and identify other relevant literature was appreciated, with varying opinions about the quality and relevance of results.\nMany participants were able to identify interesting and relevant papers; e.g., S04 identified a relevant paper from two key authors that they were not aware had collaborated.\nS05 indicated a significant finding of a paper that \\textit{``did something similar to what [they] were considering doing.''}\nCompared to searching by keywords, S05 said \\textit{``the papers [they are] seeing now are a lot more relevant. Some of these papers [they have] been reviewing. Some of them are kind of new.''}\nS05 later acknowledged the utility of the similarity score: \\textit{``It seems reasonable\u2026 Those on top tended to be more relevant to what [they were] looking for.''}\nS06 commented that the similarity score was good feedback on the precision and quality of the search itself: \\textit{``some would return like 0.0001 and [they] could see that [their] search was wrong.''}\n\nNot all feedback about the similarity search was positive, however. \nS01 was uncertain about the quality of the results, stating they \\textit{``could find a few papers that came up that slipped [their] mind, but [they] didn't find any new papers that [they] hadn't already cited. [...] [they] have some confidence that it would work, but for this particular context, [they] did not find anything new.''}\nIn response to some search queries, participants expressed disappointment with the results. \nFor instance, using a single seed paper as input to similarity search, S02 indicated \\textit{``these do not seem to be good results. The 2-D search does not seem to be good with GloVe. The N-D results were much better.''}\nS02 then added additional papers as input to the similarity search and again noted \\textit{``some match, but some do not. [...] [they] could have expected better search results.''} \nS02 ultimately suggested to explore other transformer embeddings, e.g., BERT.\n\n\n\n\n\\smallskip\n\\noindent\\textbf{By Abstract. } While not all participants had an abstract prepared to utilize the Similarity Search by Abstract feature, they nonetheless saw value in it. \nS01, for instance, indicated that if they are \\textit{``starting a new project [...] [they] can write up some words in the form of an abstract to see if this has been done.''}\n\nS06 interestingly appropriated the abstract search in response to perceived shortcomings of traditional search features. \nFor instance, after searching by keyword, applying filters, and iteratively revising queries to try to capture multiple keywords, S06 felt dissatisfied with the limitations of searching by keyword in {\\rmfamily \\scshape vitaLITy}: \\textit{``Maybe [they] should use word embeddings because it might have more flexibility, and [they] can pass more information in [their] search.''}\nThey wrote a quick abstract paragraph during the study session and observed that the results showed \\textit{``a lot of foundational literature.''}\nThey iterated, adding additional details to the abstract and expressed \\textit{``Wow, this shows much better results now than the short abstract.''}\nBy the end of the study session, S06 identified several papers they had already cited as well as a few key new ones: \\textit{``For 15 minutes, [they] found two papers [they] might be interested in. It's a really useful process. Otherwise [they] might spend a lot of time scanning PDFs, which is not a very pleasant experience.''}\n\n\n\n\n\n\\subsubsection{Visualization Canvas}\nMany participants found the projection visualization of the embedded space to be a useful way to identify conceptually ``nearby'' relevant papers. \nS05 suggested the visualization \\textit{``provides a nice overview of the selected papers, and [they] could see to drill down into more details or look for clusters.''}\nS01 appreciated the ability to select nearby papers in the embedded space via lasso, indicating \\textit{``It's like a mystery. [They] feel like if [they] spent some time on this, [they] might stumble upon a paper that was relevant that was published in a different domain [...] It might be especially useful if [they] worked on a different topic that [they] had not worked on in the past.''}\nS04 echoed this sentiment and added that the feature to locate a given paper on the visualization was helpful for orienting.\n\nHowever, this impression was not universal. \nWhile S06 appreciated searching by abstract, they preferred to examine results in tabular format, because \\textit{``personally [they are] not super familiar with these visualizations, dimensionality reduction, so it's harder to interpret how to assess this information.''}\nS03 was skeptical about the accuracy of the projection, stating \\textit{``the algorithm might be bad, or the projection. It doesn't accurately depict similarity between papers.''}\n\nSome participants suggested variations, such as spacing out papers in the visualization and connecting them by edges where the weight reflects the similarity with other papers in the visualization (S04).\nS06 suggested for lasso selection, it would be useful to see \\textit{``factors that can cluster similar papers.''}\nFurthermore, S04 suggested additional interactivity to filter out papers on different ``layers'' in the visualization, e.g., those that are part of similarity search, saved papers, etc. \nS02 suggested a minor tweak: \\textit{``when [they] do this similarity search, it should automatically zoom to show the paper(s) that were the beginning search point and the papers that it found, rather than this zoomed out view where [they] have to look for the orange or red dots.''}\n\n\n\n\n\\subsubsection{Meta View}\nThe Meta View went relatively unused compared to other features of {\\rmfamily \\scshape vitaLITy}.\nHowever, some participants did express ideas to improve its utility. \nFor instance, S03 expressed that they would have preferred if the Meta View \\textit{``[did not display] keywords for the stuff above [Paper Collection View], but for what [they] have selected [Similarity Search input, Saved papers].''}\nS05 suggested that the Meta View could offer additional keyword \\emph{recommendations} based on semantically similar keywords, to help users identify other potential search terms. \nOthers indicated a desire for further integration of the Meta View such that selecting a keyword could highlight papers in the visualization (S04) or filter the Paper Collection View (S06). \n\n\n\n\n\\subsubsection{Saved Papers Cart}\nThe Saved Papers Cart was also not used as often as some of the other views. \nSome preferred their existing workflow of downloading PDFs directly (S06), while others appreciated the ``cart'' analogy and the accompanying mindfulness to \\textit{``fill the cart with relevant papers''} (S02) as an alternative to manually maintaining \\textit{``a word document to keep track of the titles''} (S03). \n\n\\begin{figure}[!t]\n \\centering\n \\setlength{\\abovecaptionskip}{4pt}\n \\includegraphics[width=\\columnwidth]{figures\/sus-scores-features.pdf}\n \\caption{Usability scores of {\\rmfamily \\scshape vitaLITy}{} features.}\n \\label{fig:sus-scores-specific}\n \\end{figure}\n \n \n\\begin{figure}[!t]\n \\centering\n \\setlength{\\abovecaptionskip}{4pt}\n \\setlength{\\belowcaptionskip}{-10pt}\n \\includegraphics[width=\\columnwidth]{figures\/sus-scores.pdf}\n \\caption{Overall SUS scores of {\\rmfamily \\scshape vitaLITy}{}.}\n \\label{fig:sus-scores-generic}\n \\end{figure}\n\n\\subsubsection{Summary \\& Workflow}\n\n\\noindent\\textbf{Overall Impressions. }\nUsers believed that {\\rmfamily \\scshape vitaLITy}{} would be useful in a variety of contexts.\nSeveral users believed {\\rmfamily \\scshape vitaLITy}{} would be helpful in identifying gaps in their literature review (S01, S04). \nFor instance, S04 indicated \\textit{``it's very helpful to actually find a set of papers that are semantically relevant to one paper. If [they] identify a paper that [they] missed in the lit review, [they] can find other papers similar to that one to be sure [they] don't miss anything else.''}\nParticipants felt that it could help avoid ``embarassment'' of reviewers pointing out missing related work (S01, S06). \n\n\\revised{The individual SUS scores per participant were S01=72.5, S02=77.5, S03=45, S04=72.5, S05=70, S06=92.5 for an overall average SUS score=72.5 (Figure~\\ref{fig:sus-scores-generic}).} While participants generally liked using {\\rmfamily \\scshape vitaLITy}{}, several expressed that, given the large number of features, customization of the screen real estate would have been beneficial (S02, S04, S06).\nFor instance, S02 indicated \\textit{``when [they] had already filtered by keywords, [they are] only focusing on this view [Visualization Canvas]. It's very small in the screen space. [They] want to hide the Meta View and maybe even the [Paper Collection View], so [they] can easily zoom and pan and lasso. The Similarity Search panel could also be bigger.''}\nOthers echoed formative feedback, wanting to see the citation network in {\\rmfamily \\scshape vitaLITy}, e.g., which papers cite others (S01, S06).\n\n\n\\smallskip\n\\noindent\\textbf{Workflow. }\nSome participants viewed {\\rmfamily \\scshape vitaLITy}{} as a complementary component to their existing literature review workflow. \nFor instance, S01 indicated they would \\textit{``interleave this with a Google Scholar search. If [they] found a few relevant papers, [they] would go to Google Scholar to see the references in that paper and who has cited that paper.''}\nS06 indicated preference to continue their existing approach of beginning a literature review with Google Scholar and use {\\rmfamily \\scshape vitaLITy}{} at a later stage of the research, e.g., \\textit{``when [they] want to do some sanity checks [...] [after they] have [their] abstract, papers [they] have already cited, and based on that [they] can do a more narrow search for papers [they] might be missing,''} while others preferred to use {\\rmfamily \\scshape vitaLITy}{} as early in the lit review process that you are able to \\textit{``structure the related work sections [...] and [identify] those 2-3 themes''} (S05).\n\nOthers felt that {\\rmfamily \\scshape vitaLITy}{} suffered from many of the same problems that existing tooling has. \nFor instance, S01 said \\textit{``[The] target is one unknown paper among hundreds. A lot of the papers [they] find because coauthors tell [them] about them.''}\nS03 indicated they would use the tool primarily in the same ways as Google Scholar, e.g., \\textit{``[they] would just search for keywords.''}\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion} \n\nWe introduced a visualization system, {\\rmfamily \\scshape vitaLITy}, designed to promote serendipitous discovery of relevant academic literature. \nDesigned and developed with formative input from data visualization researchers, {\\rmfamily \\scshape vitaLITy}{} allows users to search and explore academic literature using a document-level transformer-based approach to identify semantically similar literature.\nIn addition, we contributed a dataset about \\texttt{59,232} academic articles with metadata (titles, abstracts, authors, keywords, citation counts, etc.) across \\texttt{38} venues common in data visualization research, along with open-source scrapers to expand and customize the corpus of literature searchable in {\\rmfamily \\scshape vitaLITy}.\nWe demonstrated how {\\rmfamily \\scshape vitaLITy}{} can complement existing academic literature review practices through a series of usage scenarios and shared feedback from 6 data visualization researchers from a qualitative study. \nParticipants expressed excitement to incorporate {\\rmfamily \\scshape vitaLITy}{} in their workflow, to identify gaps in their academic literature searches or to kickstart the literature review of a new topic. \n\\revised{While our initial prototype and evaluation focused on the data visualization field, we have open-sourced our system and scraper framework to enable expansion of the {\\rmfamily \\scshape vitaLITy}{} approach to other venues and academic communities.\nWe invite those who are interested to augment the {\\rmfamily \\scshape vitaLITy}{} system and data for their academic interests.}\n\\section{Usage Scenarios}\n\\label{sec:case_study}\n\nA common thread among the authors' prior research deals with \\textbf{human bias in data visualization}, and in particular, the authors have focused on defining~\\cite{wall2018four}, detecting~\\cite{cho2017,WallBias,WallFormative,WallMarkov}, and mitigating~\\cite{WallDesign,Lumos,LRG} cognitive biases.\nThus, we find it fitting to demonstrate the usage of {\\rmfamily \\scshape vitaLITy}{} through a series of usage scenarios in the context of a literature review on bias in visualization.\n\n\n\\subsection{Usage Scenario 1: Identifying Missing Papers}\nMaya is a data visualization PhD student working on their dissertation on the topic of ``Mitigating Bias in Data Visualization.'' \nThey are wrapping up the related work and preparing to submit their thesis. \nBefore submitting, Maya wants to check for potential gaps in the literature review and ensure there is no critical missing work. \nMaya decides to use {\\rmfamily \\scshape vitaLITy}{} to explore the visualization literature.\n\nMaya wants to be systematic about their search. \nThey begin by taking some of the key papers related to bias in visualization, including the following~\\cite{dimara2017attraction,dimara2018task,dimara2019mitigating,WallBias,WallDesign,cho2017,wesslen2019investigating,gotz2016adaptive}. \nMaya has already examined the papers cited from these works and written about the relevant ones in their dissertation. They locate these key papers in {\\rmfamily \\scshape vitaLITy}{} and ``select'' them [add them as input to Similarity Search] (Figure~\\ref{fig:search-by-papers}a), then map them in the Visualization (Figure~\\ref{fig:search-by-papers}b).\n\nStarting with N-Dimensional Specter embedding, Maya searches for similar papers (Figure~\\ref{fig:search-by-papers}c). \nThe first result, ``A Formative Study of Interactive Bias Metrics in Visual Analytics Using Anchoring Bias''~\\cite{WallFormative} (similarity score \\texttt{0.4355}), is cited in one of the papers~\\cite{WallDesign} so Maya was already aware. \nScanning down the list, the fourth result is ``CogTool-Explorer: A Model of Goal-Directed User Exploration That Considers Information Layout''~\\cite{teo2012cogtool}, a paper Maya is not aware of. \nPublished at CHI in 2012, this paper describes a method for modeling and predicting user interactive behavior. \nIntrigued by the relevance of precursory work in HCI to predict interactive behavior~\\cite{teo2012cogtool} to work on modeling user bias~\\cite{WallBias}, Maya saves this paper to the ``cart''. \n\nContinuing to examine the list of output papers, the next result also proves relevant with a similarity score of \\texttt{0.2593}: a BELIV paper titled ``Just the Other Side of the Coin? From Error to Insight Analysis''~\\cite{smuc2016just} which models errors and insights in cognitive processing. \nSeveral others also catch Maya's eye relevant to the design of bias mitigation strategies, including research about introducing visualization ``difficulties'' in design to aid comprehension and recall~\\cite{hullman2011benefitting} and even use of so-called ``transparent deception'' in visualization if and when it is aligned with certain user goals~\\cite{ritchie2019lie}.\nMaya saves these papers and exports them for further review (Figure~\\ref{fig:search-by-papers}d). \n\nFurthermore, Maya notices a particularly relevant paper, ``Priming and Anchoring Effects in Visualization''~\\cite{valdez2018priming}, which they forgot about, so adds it to the input similarity search and re-computes the output. \nThey find ``Pushing the (Visual) Narrative: the Effects of Prior Knowledge Elicitation in Provocative Topics''~\\cite{heyer2020pushing}, discussing persuasive visualization designs, which again Maya finds relevant for designing bias mitigation interventions. \nMaya continues iterating on their exploration of the literature, augmenting their dissertation related work section and filling in gaps, especially from the CHI community. \n\n\n\n\n\\subsection{Usage Scenario 2: Analysis of Keyword Quality}\nKatherine is a visualization researcher who focuses on topics related to bias and decision making. \nShe has primarily relied on keyword searches supported by IEEE Xplore, ACM Digital Library, etc. for identifying relevant literature in the past. \nBeginning with a set of known papers about bias in visualization (i.e., the same set from the previous scenario~\\cite{gotz2016adaptive,WallBias,WallDesign,cho2017,wesslen2019investigating,dimara2017attraction,dimara2018task,dimara2019mitigating}), she identifies several relevant keywords, including\n\\emph{human biases}, \\emph{bias mitigation}, \\emph{bias mitigation strategies}, \\emph{bias alleviation}, \\emph{debiasing}, \\emph{cognition}, \\emph{cognitive bias}, \\emph{cognitive biases}, \\emph{cognitive heuristics}, \\emph{heuristics}, \\emph{decision making}, \\emph{decision-making}, \\emph{human decision-making}, \\emph{sensemaking capabilities}, \\emph{uncertainty}, \\emph{anchoring bias}, and \\emph{attraction effect}.\nShe disregards several others that she believes are too broad, e.g., \\emph{visualization}, \\emph{information visualization}, \\emph{data visualization}, \\emph{human-centered computing}, \\emph{visual analytics}, etc. \nShe notes the multiplicity of some keywords defined by authors. \n\nKatherine conducts a similarity search using {\\rmfamily \\scshape vitaLITy}{} (yielding the same output as the previous scenario for Maya's literature review). \nShe notes a number of papers that she would have been unable to identify given only these keyword searches. \nFor instance, ``Designing Information for Remediating Cognitive Biases in Decision-Making''~\\cite{zhang2015designing} contains keywords \\emph{Human Computer Interaction (hci)} and \\emph{Human-centered Computing} and would have been missed by targeted bias-related keywords and likely lost among a sea of other papers by searching for more generic HCI keywords.\nSimilarly, ``A Lie Reveals the Truth: Quasimodes for Task-Aligned Data Presentation''~\\cite{ritchie2019lie} contains very broad keywords, including \\emph{Visualization}, \\emph{Empirical Studies In Visualization}, and \\emph{Human-centered Computing}.\nOther papers not directly related to bias, but still relevant, are even less likely to contain keyword matches. \nFor instance, ``Observation-Level Interaction with Statistical Models for Visual Analytics''~\\cite{Endert2011} describes data- or ``observation''-level interactions users perform with data based on perceived relationships and interests in the data, a topic of precursory relevance to bias research in data visualization. \nHowever, it contains keywords with no overlap to the bias-related search terms: \\emph{Principal Component Analysis}, \\emph{Data Models}, \\emph{Data Visualization}, \\emph{Visual Analytics}, \\emph{Analytical Models}, and \\emph{Layout}.\n\n\nNotably, Katherine observes that some venues expose only index terms from e.g., IEEE or ACM, while others also expose author-defined keywords. \nThis provides different levels of granularity in the ability to search for literature by keyword. \nHence, Katherine finds that alternative approaches based on document-level embeddings can be a fruitful way to identify literature when keyword searches prove insufficient or inconsistent across venues.\n\n\n\\subsection{Usage Scenario 3: Beginning a New Project} \n\\begin{comment}\n\\emily{Remco approved our use of this abstract.\nAlso, need to consider how to frame this; seems odd to use fictional Ella when it's a real paper w\/ real author names... but would also be weird to use real author names since they didn't actually use {\\rmfamily \\scshape vitaLITy}; or maybe the disclaimer before the quote is enough?}\nElla is a big data researcher, new to the area of data visualization. \nShe wants to study how prominent cognitive biases are when people view progressive visualizations of big data.\nShe plans to conduct a series of experiments in which participants will complete tasks using progressive visualizations. \nShe writes out her idea in the form of the abstract below.\nNote: This abstract was used with permission from the authors of a recent 2021 TVCG paper titled ``Impact of Cognitive Biases on Progressive Visualization''~\\cite{procopio2021impact}, which was not yet in our document corpus.\nHence, it serves as a dramatized exemplar of using {\\rmfamily \\scshape vitaLITy}{} with a working abstract. \n\n\\begin{quotation}\n\\textit{Progressive visualization is fast becoming a technique in the visualization community to help users interact with large amounts of data. \nWith progressive visualization, users can examine intermediate results of complex or long running computations, without waiting for the computation to complete. \nWhile this has shown to be beneficial to users, recent research has identified potential risks. \nFor example, users may misjudge the uncertainty in the intermediate results and draw incorrect conclusions or see patterns that are not present in the final results. \nIn this paper, we conduct a comprehensive set of studies to quantify the advantages and limitations of progressive visualization. \nBased on a recent report by Micallef et al., we examine four types of cognitive biases that can occur with progressive visualization: uncertainty bias, illusion bias, control bias, and anchoring bias. \nThe results of the studies suggest a cautious but promising use of progressive visualization \u2013 while there can be significant savings in task completion time, accuracy can be negatively affected in certain conditions. \nThese findings confirm earlier reports of the benefits and drawbacks of progressive visualization and that continued research into mitigating the effects of cognitive biases is necessary.}\n\\end{quotation}\n\\arpit{Is this quotation required to be in-text? We can could come up with a set of figures encompassing either or both of the abtract search + umap stuff discussed in the scenario, e.g.Figure~\\ref{fig:search-by-abstract-remco}.}\n\nElla enters her working title and abstract in {\\rmfamily \\scshape vitaLITy}{} and examines the resulting literature. \nShe maps several of the output papers on the Visualization and realizes they are fairly close together in the Specter embedding space. \nShe selects them on the map and clicks \\faInfoCircle~ to examine them more closely. \nShe finds two recent papers by the author Emanuel Zgraggen: (1) a 2018 CHI paper titled ``Investigating the Effect of the Multiple Comparisons Problem in Visual Analysis''~\\cite{zgraggen2018investigating} that describes an experiment to quantify spurious findings users make from visualizations, and (2) a 2017 TVCG paper titled ``How Progressive Visualizations Affect Exploratory Analysis''~\\cite{zgraggen2016progressive}. \nShe finds these works to be relevant points of comparison and saves them for a closer reading. \n\nAfter reading the saved papers more carefully, Ella believes there is significant innovation in her project with respect to quantifying specific types of bias. \nShe begins writing about the papers she found to form a Related Work section of her paper and continues to refine her experimental design.\n\\emily{surprisingly there weren't that many results that were ultimately cited in Remco's paper. We could leave it at this \/ I can possibly mention a couple other papers; Or this could be a more comprehensive example, where there are a few papers identified from the abstract search, and then maybe move on to keyword \/ author search?}\n\n\\arpit{Yeah, Remco's abstract based search scenario ended abruptly at first read (I was expecting a lot more). Also we are not directly commenting that they did cite vitality recommended papers or that they missed a few relevant ones which is somehow making this exemplar incomplete and hence weakish. this section abruptly ended for me. Based on your points, I think mentioning a couple more and then moving on to keyword\/author search will be nice for the exemplar to stand-out.}\n\n\\arpit{OR, we could stop at what we have here and let the next paragraph take-up that role. I think our own abstract based search scenario should be there since we have control over the resultant citations and vitality is already doing a great job recommending nice relevant citations}\n\n\n\\emily{Yeah, I struggled with this one because there weren't that may {\\rmfamily \\scshape vitaLITy}{} recommended papers that were cited... lots of stuff wasn't cited, and it felt odd to point to a bunch of gaps in someone else's lit review, ya know? Leaning toward just keeping ours instead...}\n\n\\hrule\n\n\n\\emily{version 2: our abstract}\n\\end{comment}\n\n\nIn this scenario, we showcase how {\\rmfamily \\scshape vitaLITy}{} facilitated our own literature review for the present work. \nAfter using traditional approaches based on keyword searches or citations from known papers, we found {\\rmfamily \\scshape vitaLITy}{} helped us identify a plethora of additional literature we were previously unaware of.\nWe used the Similarity Search by Abstract feature of {\\rmfamily \\scshape vitaLITy}{} with our paper title and abstract (Figure~\\ref{fig:search-by-abstract}a). \n\nThe first returned result is a 2011 Computer Graphics Forum paper titled ``PaperVis: Literature Review Made Easy''~\\cite{chou2011papervis} that utilizes a node-link visualization approach to support literature review and creates a topic hierarchy based on semantically meaningful topics (Figure~\\ref{fig:search-by-abstract}b). \nThe next paper similarly focuses on creating iterative citation networks to facilitate creation and sharing of bibliographies~\\cite{dattolo2018visualbib}.\nIn general, after searching the output, a few themes emerge: (1) visualization systems that focus on citation networks (e.g.,~\\cite{wilkins2015evolutionworks,heimerl2015citerivers}), systems that focus on clustering or similarity (typically by matching keywords, e.g.,~\\cite{wang2019vispubcompas}, or using topic modeling, e.g.,~\\cite{alexander2014serendip}), and (3) systems that focus on both citation networks and similarity measures (e.g.,~\\cite{nakazawa2018analytics}).\nOther notable topics also surfaced, including a design space~\\cite{felix2017taking} and analyses of keywords utilized in the visualization community~\\cite{isenberg2016visualization}, a system for supporting dissemination of curated survey results~\\cite{beck2015visual}, analysis of the contextual \\emph{reasons} for citations~\\cite{yoon2020conference}, and an emergent design space for considering visualizations of literature collections~\\cite{hinrichs2015speculative}.\n\nReflecting on these findings, we believe traditional methods for searching literature left many gaps in our literature review for two primary reasons: (1) many of these works are distributed across several publication venues (e.g., IV, PacificVis, Interact, VAST, TVCG), and (2) many of these papers received relatively little traction since their original publication 5-10 years ago. \n\n\n\n\n\n\n\n\\subsection{Usage Scenario 4: Getting to Know VIS}\nRosa is a new PhD student joining a lab that conducts research in data visualization. \nTo become acquainted with the field, her advisor suggests that Rosa browse through some of the prominent literature in {\\rmfamily \\scshape vitaLITy}.\nUpon loading the system, Rosa observes that it contains \\texttt{59,232} papers in the Paper Collection View. \nInspecting the Meta View, she observes those papers are described by \\texttt{49,278} keywords, written by \\texttt{82,391} authors from \\texttt{55} different venues, across \\texttt{47} years.\nAmong the top keywords are \\emph{Human-centered Computing} and \\emph{Human Computer Interaction (hci)}, describing \\texttt{13,833} and \\texttt{8,365} papers respectively. \nThe lineage of data visualization becomes apparent to Rosa when she notices that the fifth most common keyword is \\emph{Computer Graphics}, followed by \\emph{Data Visualization}. \nOther common keywords that catch Rosa's eye describe topics such as \\emph{Machine Learning}, \\emph{Information Retrieval}, \\emph{Artificial Intelligence}, \\emph{Interaction Design}, and \\emph{Animation}, among others. \n\nRosa enters \\emph{Data Visualization} as a filter in the Keywords column of the Paper Collection View, then filters to show only papers in the past 10 years to focus on the \\texttt{2,032} most relevant recent works in the field. \nInterestingly, these papers appear in a fairly dense area near the center of the Visualization. \nIn the Meta View, she notes a few authors whose names she recognizes, including Kwan-liu Ma who authored \\texttt{58} of the papers with the keyword \\emph{Data Visualization} since 2010. \nShe also notices Daniel Keim, John T. Stasko, Niklas Elmqvist, and Hanspeter Pfister, among others. \nShe next filters the Paper Collection View to see only John T. Stasko's papers (\\texttt{80}) and removes the other filters (Figure~\\ref{fig:meta}a-d). \nThe Meta View reveals that his work is associated with the following keywords: \\emph{data visualization, visualization, human-center computing, visual analytics, human computer interaction (hci)} (a). Some of his common co-authors include Zhicheng Liu, Carsten Gorg, and Youn Ah Kang (b).\nHe publishes primarily at TVCG (\\texttt{21}) and VAST (\\texttt{15}) (c), with 2007 his most productive year (\\texttt{11} publications) followed by 2008 (\\texttt{10} publications) then 2011, 2012, and 2014 each with \\texttt{6} publications (d).\n\n\n\n\\begin{figure*}[!t]\n \\centering\n \\setlength{\\belowcaptionskip}{-10pt}\n \\includegraphics[width=\\linewidth]{figures\/search-by-papers.pdf}\n \\caption{\\textbf{Search by a list of seed papers: Scenario 1}. Based on a list of known relevant bias papers (a), Maya observes the clustering of similar papers in the Visualization (b). She examines the similar papers more closely to gauge their relevance (c) and exports relevant saved papers (d).}\n \\label{fig:search-by-papers}\n \\end{figure*}\n\n\\begin{figure}[!t]\n \n \\centering\n \\includegraphics[width=\\linewidth]{figures\/search-by-abstract.pdf}\n \\caption{\\textbf{Search by Abstract: {\\rmfamily \\scshape vitaLITy}{}'s own Literature Review.} The authors using {\\rmfamily \\scshape vitaLITy}{}'s working title and abstract (a) to find similar papers (b) to assist in its own literature review.}\n \\label{fig:search-by-abstract}\n \\end{figure}\n\n \n \n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\n- evaluate the system via a case study comparison of our technique to an existing survey paper, perhaps Dimara's interaction paper~\\cite{dimara2019interaction}\n\n\n\\subsection{Existing Survey Methodology}\n\n- in addition to systematic lit reviews conducted for papers, we also examined several survey papers (e.g., ~\\cite{})\n- these surveys describe their methodology for lit reivew, including collecting an initial set of seed papers, keyword search, inclusion criteria, venues considered, etc.\n\n\n\\subsection{Existing Survey Results}\n\n- total number of papers found, high level topics covered\n\n\n\\subsection{Our Survey Methodology}\n\n- need to give our system a catchy name to refer to it here ;) \n\n- describe our alternative approach, including both keyword and exemplar paper search\n\n\n\\subsection{Our Survey Results}\n\n- total number of papers found, clusters of topics that formed\n\n\n\n\\subsection{Comparing Survey Results}\n\n- using two alternative methodologies, we achieved (hopefully at least close to) comparable results using less manual effort\n\n- set comparison: if the Dimara survey is set A and our survey is set B, how many papers are in the intersection? how many in the set difference A-B and B-A? \n\\section{Usage Scenarios: Bias in Visualization}\n\\label{sec:case_study}\n\n\\subsection{Motivation}\nA common thread among the authors' prior research deals with \\textbf{human bias in data visualization}, and in particular, the authors have focused on defining~\\cite{wall2018four}, detecting~\\cite{cho2017,WallBias,WallFormative,karduni2018can}, and mitigating~\\cite{WallDesign} cognitive biases.\nThrough a number of prior discussions, we identified topics in visualization research that were relevant to our own work: e.g., Bayesian cognitive modeling~\\cite{kim2019bayesian}, guidance~\\cite{ceneda2017characterizing}, and mixed-initiative~\\cite{Horvitz1999} visual analytics (e.g.,~\\cite{BrownDisFunction2012,WallPodium,cook2015mixed}), to name a few.\nHowever, seldom do authors of these topics utilize the language of \\emph{bias}. \n\nKnowledge of these topics motivated the present work.\nHow could we identify these relevant topics in our literature reviews? \nA simple keyword search would not be fruitful due to a lack of common language. \nThus, we find it fitting to put {\\rmfamily \\scshape vitaLITy}{} to the test in the domain of \\textbf{bias in data visualization}.\nIn addition to identifying these known topics of interest, can our approach bring to light any additional topics of interest?\n\n\n\n\\subsection{Topics of Interest}\nPrior to beginning our literature review, we identified three topics of interest we hoped our literature review would uncover: (1) Bayesian cognitive modeling, (2) guidance, and (3) mixed-initiative visual analytics. \n\nBayesian cognitive modeling is often used in data visualization to compare how a user updates their beliefs in the presence of new and uncertain data to a normative benchmark~\\cite{kim2019bayesian,karduni2020bayesian}. \nWith respect to bias, this approach can be one promising direction toward characterization of bias (e.g., it can provide a normative framework in which to interpret observed user decisions).\nWhile this approach has been recently proposed~\\cite{wu2017towards}, to our knowledge it has not been used in empirical studies of bias in data visualization.\n\nThe topics of guidance and mixed-initiative visual analytics can similarly be relevant in the context of bias mitigation. \nThese topics motivated a recent design space of bias mitigation techniques in visualization~\\cite{WallDesign}.\nIn particular, techniques used to generally guide users through knowledge gaps in data analysis~\\cite{ceneda2017characterizing} can be applied to guiding users toward a less biased analysis process. \nIn doing so, systems are likely to employ mixed-initiative techniques~\\cite{Horvitz1999} wherein systems assume some responsibility and control from users. \n\n\n\\subsection{Strategy}\nWe began our search by ``blinding'' ourselves to the topics of interest. \nHence, our search strategies were initially restricted to include use of ``obvious'' keywords (e.g., bias, decision making) and known papers on bias in visualization (e.g., the authors' own papers and select others that utilize the language of bias~\\cite{dimara2017attraction,dimara2018task,dimara2019mitigating,gotz2016adaptive}). \n\n\n\\subsection{Findings}\n\n\\subsubsection{Keyword Search}\nglobal keyword search for ``bias'' returns 1639\/59413 hits\n\ngiven the scope of venues included, this revealed some relevant papers of interest (that likely would have been lost \/ difficult to filter for if searching via google scholar; less common venue, but still important for vis); \n- e.g., Two Implications and Dual-Process Theories of Reasoning. (venue: Diagrams)\n- The Nudge Deck: A Design Support Tool for Technology-Mediated Nudging.(venue: Conference on Designing Interactive Systems)\n\nSome of the usual suspects identified: \n- FairSight: Visual Analytics for Fairness in Decision Making.\n- Selection Bias Tracking and Detailed Subset Comparison for High-Dimensional Data. (follow up paper to Gotz adaptive contextualization paper)\n- Recent work on perceptual bias: Biased Average Position Estimates in Line and Bar Graphs: Underestimation, Overestimation, and Perceptual Pull.\n\n\n\\subsubsection{Similarity Search}\n\nbegin with my own paper~\\cite{WallBias} as a seed\nsome obvious results:\n- A Formative Study of Interactive Bias Metrics in Visual Analytics Using Anchoring Bias\n- lots of DECISIVe hits (Cognitive Biases in Visualization book chapters)\n- Adaptive Contextualization: Combating Bias During High-Dimensional Visualization and Data Selection.\n\nsome less obvious results: \n- Internalization, qualitative methods, and evaluation. (venue: BELIV)\n- An Analysis of Machine- and Human-Analytics in Classification. (interestingly describes machine v. human+machine)\n- Evaluating visualization using cognitive measures.\n- Designing Theory-Driven User-Centric Explainable AI.\n(can explanation techniques be used to mitigate bias in machine reliance?)\n- Taking Development Seriously: Modeling the Interactions in the Emergence of Different Word Learning Biases.\n- Bounded rationality leads to optimal decision-making and learning under uncertainty: Satisficing, prospect theory, and comparative valuation breaking the speed-accuracy tradeoff.\n- BiDots: Visual Exploration of Weighted Biclusters.\n- An Information-theoretic Framework for Visualization.\n\n\n\nthen adding my formative study paper and adaptive contextualization: \n- some weird ones come up: \n- A review of monocular visual odometry.\n- Wall cavitation caused by projectile impact.\n\n(by the end of the 25 output, we are getting similarity scores of 0)\n\nthen adding ~\\cite{cho2017}, we get \n- Measuring Cognitive Load using Eye Tracking Technology in Visual Computing\n\n\nnew query: add all of my own papers as input to similarity search\noutput: User Evaluations of Interactive Multimodal Data Presentation. (suggests vis + sonification is promising...; this was actually rated really similar for some reason, interesting)\n- Towards an instrument for measuring sensemaking and an assessment of its theoretical features. (sensemaking motivated a lot of my early work)\n- Casual Information Visualization: Depictions of Data in Everyday Life. (I have since gone into some recent work on causality...)\n\ntried similar query for each of us -- interestingly, similarity search based on all of our individual papers leads to some sensemaking papers for each of us (except Arpit, just based on the papers that are scraped of yours)\n\n\nif we use ALL of our papers as input (20 in total): \ntop 3 interestingly seem irrelevant: \n- Wall cavitation caused by projectile impact.\n- BiDots: Visual Exploration of Weighted Biclusters.\n- A review of monocular visual odometry.\n\nNext ones seem better: \n- The Impact of User Characteristics and Preferences on Performance with an Unfamiliar Voice User Interface.\n- Towards Deeper Understanding of User Experience with Ubiquitous Computing Systems: Systematic Literature Review and Design Framework.\n- User Evaluations of Interactive Multimodal Data Presentation.\n- Towards an instrument for measuring sensemaking and an assessment of its theoretical features.\n- A Computational Model of the Role of Attention in Subitizing and Enumeration.\n- Casual Information Visualization: Depictions of Data in Everyday Life.\n- Effects of Sensemaking Translucence on Distributed Collaborative Analysis\n- Trust in AutoML: exploring information needs for establishing trust in automated machine learning systems\n- Demonstrational Interaction for Data Visualization.\n\n\n\n\n\\emily{- Not a gold standard by any stretch, but would be interesting to compare to the references captured in my (Emily)'s dissertation; I tried to be reasonably complete. What did I miss?}\n\n\\begin{figure*}[!t]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/search-by-papers.pdf}\n \\caption{Search by Papers. \\arpit{To tweak to replicate the eventual bias case study.}}\n \\label{fig:search-by-papers}\n \\end{figure*}\n\n\\begin{figure*}[!t]\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/search-by-abstract.pdf}\n \\caption{Search by Abstract. \\arpit{For this one, it'll be cool to do a search with our own final abstract that results in some similar papers that we actually cite in our related work.}}\n \\label{fig:search-by-abstract}\n \\end{figure*}\n\\subsubsection{Positive Quotes}\n\n- searching nearby papers in the embedded space, after lassoing: S01 said \"This is interesting. It seems there are some papers here that might be useful.\"\n\"I feel like I would spend a lot of time on this. It's like this mystery, I feel like if I spent some time on this, I might stumble upon a paper that was relevant that was published in a different domain. And that's basically like half of the papers that I cite.\"\n\"It might be especially useful if I worked on a different topic that I hadn't worked on in the past.\"\n\n- Search by abstract: S01 said \"If I'm starting a new project and I don't know if this has been done, I can write up some words in the form of an abstract to see if this is done.\"\n\n- would you use this? S01 \"Yeah, I think it could be very useful if I have a gap in my lit search. There are some papers I should cite, and if i don't cite them, then reviewers will be like 'you should cite these papers', and this seems like it would yield such results. I have some confidence. This garden of forking paths paper I had not search for but it came up because of this similarity search. And if I were in a new domain where I didn't have a lot of background knowledge, this would be useful. I would interleave this with a Google Scholar search. If I found a few relevant papers, I would go to Google Scholar to see the references in that paper and who has cited that paper. And that's a workflow that I'm familiar with.\"\n\n- would you use this: S02 \"I can see myself using a tool like this, especially for someone like me who has been in the field for a bit and knows where to begin the search process. But I wish fine tuning search results were more intuitive. I could find some relevant papers, then use the JSON output to reduce the search space within another tool like Google Scholar.\"\n\n- specter embeddings: S02, after less successful 2-D search w\/ glove: \"I'm going to try 2-D search with Specter. This search seems to be a little bit better.\"\n\n- favorite features: S02 \"I think the table view at the top is a really good starting point. It works really well. It shows the authors, etc. that I expect to see. It's a good entry point for the user. The saved paper cart is also a neat addition. I will always be mindful of my goal to fill the cart with relevant papers.\"\n\n- in response to keyword search results, S03 \"I've read this one before [Verifi]. This one I don't care about, it's more machine learning. I'd like to find more social science.\" \n\"Social Media\u2026 sounds interesting, I want to save this paper to read it later.\"\n\"Nudging, this is interesting. I should take a look at the abstract.\"\n\"Now I just want to check out the other papers from this author\" [Zhicong Lu]\n\"I think I'll read some of these.\"\n\n- saved paper cart: S03 \"I liked the fact that I'm able to save some of the papers. Typically I wouldn't do it that way. When I search for papers that might be relevant, I just use a word document or something to keep track of the titles and maybe download the paper as well. It's nice to be able to save the papers.\"\n\n- after doing n-D similarity search, S04: \"Ohhh, interesting, I never knew that these two people had a collaboration.\" -- in response to interesting relevant network paper [Cody Dunne and Ben Shneiderman]\n\n- S04 \"I liked the visualization, especially the lasso select. It's really cool and really helpful.\" \n\"Also, locate is a really good feature, you can actually identify the papers in different colors.\"\n\n- would you use it? S04 \"I definitely see myself using this app. I really like the idea of this visualization, it's really cool, because it's very helpful to actually find a set of papers that are semantically relevant to one paper. Especially when I can randomly identify a paper that I missed in the lit review, and find other papers similar to that one to be sure you aren't missing anything.\"\n\n- relevance of similarity output: S05 \"The papers I'm seeing now are a lot more relevant, some of the papers I've been reviewing. Hmm, some of them are kind of new, but let's see if they're relevant to my topic.\"\n\"I'm trying to find if there are new papers worth exploring.\"\n\"This one seems interesting, so I would like to add this to my list and maybe save it as well.\"\n\n- found something really similar to what she was thinking of writing about: S05 \"I'm noticing some papers that I was considering to review to write my discussion section. I wanted to review other related domains and use this analytical framework to conduct research in that domain. One possibility was to use it for evaluations of different structures of storytelling visualizations. I'm seeing a paper that did something similar to what I was considering doing.\"\n\n- S05 \"I want to see 2-D similarity results now -- ohh, this is interesting: Design Patterns for Data Comics.\"\n\n- distinction between 2-D and n-D similarity: S05 \"I do see a difference, I think 2-D search may be giving me more relevant results.\" \n\"Might be better for more in-depth exploration. If there's a very specific topic you're looking for. For example, I was looking for analytical frameworks within storytelling. And it seemed like 2-D similarity search gave better results.\"\n\"n-D seemed to give better results that were exploratory. I found some interesting papers I would like to read more.\" \n\n- S05 \"This was actually useful, because there were some papers that caught my eye that I hadn't seen before.\"\n\n- similarity search S05 \"I really liked the similarity search. I liked that you showed the similarity score. It seems reasonable\u2026 Those on top tended to be more relevant to what I was looking for.\"\n\n- umap S05 \"I guess this [umap] provides a nice overview of the selected papers, and I could see to drill down into more details or look for clusters.\"\n\n- would you use it? S05 \"Yes. I would probably use it when I have some high-level ideas about how I would structure the related sections -- when I know what those 2-3 themes are. I would use this to find more relevant papers.\"\n\n- S06: Hid author column and resized column to see full titles\u2026 seemed very intuitive to him, he expected it would do this\n\n- Appropriation of abstract search: S06 after searching in abstracts, applying filters for relevant venues, etc. felt unsatisfied by results; so used search by abstract instead: \"Maybe I should use word embedding because it might have more flexibility and I can pass more information in my search.\" \n\"It shows a lot of foundational literature\u2026\"\n[after first iteration results were only ok, added more content to the abstract and was much more satisfied with results]\n\"Wow, this shows much better results now than the short abstract.\"\n\"Presentation-oriented visualization techniques -- hmm looks interesting.\"\n\"Oh this looks interesting, but I haven't looked at.\" [Comparing the Effectiveness of Visualizations of Different Data Distributions]\n\"I should take a note.\" [writes down paper info]\n\"This one, I think I cited this one.\"\n\"I'm also writing down this one. It's look at EEG, wow. It looks interesting because I cited a similar mathematical paper, they want to approximate the same thing.\" [Comparing Similarity Perception in Time Series Visualizations]\n\n- satisfaction w\/ search: S06 \"Yes, I was mostly looking for papers that I haven't looked at before. For 15 minutes, I found two papers I might be interested in. It's a really useful process. Otherwise I might spend a lot of time scanning PDFs, which is not a very pleasant experience.\"\n\n- lots of good features: S06 \"Search by abstract was super useful. Personally I'm not super familiar with these visualizations, dimensionality reduction, so it's harder to interpret how to assess this information.\"\n\"The similarity measures here [in similarity output] were really useful. But some would return like 0.0001 and I could see that my search was wrong.\"\n\"For lasso selection, I want to see factors that can cluster similar papers. I want to just see similarity output, not select everything.\" \n\"Changing columns was useful too, so I can get a better sense of the data. If the search gets improved then search by abstract will be much better. Google Scholar searches body text too. But for Google Scholar the similarity [or relevance?] of results is not so good.\"\n\n- when it's useful (at the end of lit review) S06: \"For searching papers, I would continue to use Google Scholar when I start the lit review. But I think vitaLITy could be super useful at the later stage of your research. You want to do some sanity checks on your literature review. At that point of time, I have my abstract, more details, papers I have already cited, and based on that, I can do a more narrow search for papers I might be missing. When you're finalizing your literature review. It can help people evaluate their process and understand how they are doing, how they can improve. Sometimes you get reviews that you didn't cite this or this or this and it's sort of embarrassing.\"\n\n\n\n\n\n\\subsubsection{Negative Quotes} \n- in reasoning about the quality of similarity output: \"I see why these papers are not giving the best results. It seems in the word embedding space, they are not close together.\" (S01)\n- did you find useful papers: S01 said \"I could find a few papers that came up that slipped my mind. But I didn't find any new papers that I hadn't already cited. I got some sense that the search was giving me useful results. I have some confidence that it would work, but for this particular context, I did not find anything new.\"\n\n- some resistance to doing things a different way: S01 said \"I'm having some of the same difficulties that I have when I do a Google Scholar search. I need to find one paper. My target is one unknown paper among hundreds. It's very rare that I find multiple useful papers when I try random keywords on Google Scholar. A lot of the papers I find because coauthors tell me about a relevant paper. A lot of the papers that we cite are not directly relevant, but it's adjacent. That kind of makes it complicated to search if I had already filtered on a particular keyword.\"\n- S03 \"I prefer Google Scholar. Would I use it? Maybe\u2026 I suspect I would use it just like I use Google Scholar. I wouldn't use similarity search. I would just search for keywords. I'm not sure how Google Scholar does it, but I like this global search to look in abstracts, titles, etc.\"\n\n- didn't use umap much: S01 \"This visualization [scatterplot] was interesting, but I spent very little time on this. I'm not sure why, but it almost seemed like I would need very specific input to find relevant results, so I ended up spending more time on filtering that I would want a similarity search on.\"\n\n- relevance of glove output: S02 \"The original paper I'm looking for Regroup, these do not seem to be good results. The 2-D search does not seem to be good with Glove. The n-D results were much better.\" [switched to specter], \"Maybe if I give a few other papers as examples, it may help.\" [adds 2 more papers]\n\"Some matches, but some do not.\"\n- S05 \"I'm reviewing the results, some seem relevant, some not so much.\"\n\n- quality of similarity search results: S02 \"When I look by author or keywords, I find relevant papers. But when I try to fine tune, it loses track. I could have expected better search results.\"\n\n- accuracy \/ utility of umap projection, S03 \"not sure these are so similar\" -- appear far apart (did n-D search rather than 2-D)\n \"I don't think I need to see all the papers at once. I mostly want to see all the papers I have selected and their geolocation and whether they're similar or not so similar. But the algorithm might be bad. Or the projection. It doesn't accurately depict similarity between papers.\"\n \n - too much packed into one tool: S04 \"The unique part of this interface is the visualization of the space of papers. I wish it could be bigger canvas space so I can select papers. The interface is a little small and too compact that makes me feel like too many things are changing in front of my face and I lost track.\"\n\"I feel like the similarity search and visualization can be a separate thing on its own. And the meta thing and table search can be on its own.\"\n\n- didn't use saved papers cart S06: \"I realized looking at the questions that I never used the saved papers cart. Even using Google Scholar I usually just prefer to save the PDF.\"\n\n\n\n\n\n\\subsubsection{Desired Features} \n\n- citation networks: S01 \"The other thing, in this case, I forgot to mention this, but a way I conduct literature reviews is to look at the cited papers of a relevant paper and then go in a search. That is missing. Google Scholar makes it easier, but it's not great.\"\n- S06 \"I think it's hard to implement, I've tried it before, but maybe it could show the citation map. Sometimes I find a really old paper that has some really core ideas and I want to see more recent papers that cite it.\"\n\n- more embedding options: S02 \n\"Other than Glove, you can consider to use transformer embeddings like BERT. It uses context vectors.\"\n\n- indicators about credibility of venue \/ author: S03 \"If I'm able to get a sense of how credible the source is, e.g., this particular conference. When it comes to some other domains or venues, we have a hard time getting an indicator of whether this is a good conference or not until we read the paper, or if we know the author is highly cited and respected.\"\n\n- alternate way to visualize umap: S04 \"Is there any possible way to visualize this in terms of networks? If you can separate out the nodes in space and add edge weights to reflect similarity. The biggest problem is overlapping nodes, so I don't have a full picture. Or at least put the red dots in the top layer. Now you have three layers: filtered, unfiltered, saved. Maybe I can use the legend as a filter to show only that layer.\"\n\n- metal panel integration: S04 \"I didn't feel like I used the meta panel a lot. Is it possible that I select a keyword that it could query all papers that have this keyword in the search \/ umap?\"\n\n- suggested keywords: S05 \"For keywords, if I could look for multiple keywords\u2026\"\n\"Can it suggest semantically similar keywords?\"\n\n\n\n\\subsubsection{Small Usability Issues}\n\n- fuzzy string search could be useful: S01 searched for \"uncertainty visualisations\" -> \"uncertainty visualizations\" -> \"uncertainty visualization\" \n- S01 also tried multiple keywords; may be beneficial to have more robust and customizable search mechanism\n- SOMEONE used the search by abstract as a workaround (interestingly)\n\n- \"Wall Cavitation\" paper seems to come up frequently (in our usage scenarios and in participant sessions) [I think just for Glove?]; something about abstract length (very short) makes it seem like a good match for lots of searches\n\n- feedback about queries: S01 \"I did find a lack of feedback on the queries.\"\nS05: \"Apart from being a little laggy, it was pretty straightforward. But I kept forgetting to clear out the search [clear filters].\"\n\n- customizable layout: S02 \"I found myself when I had already filtered by keywords, I am only focusing on this view [scatterplot], it's very small in the screen space. I want to hide the meta view and maybe even the main table view, so I can easily zoom and pan and lasso. The similarity search panel could also be bigger.\"\n- S04: \"I wish this interface [umap] could be a little bigger.\"\n- S06 \"I think the window [top table] is a little bit short. Can I expand this?\" [no, could not... although was able to very intuitively resize \/ customize column widths]\n\n- zoom level on umap: S02 \"When I do this similarity search, it should automatically zoom to show the paper(s) that were the beginning search point and the papers that it found, rather than this zoomed out view where I have to look for the orange or red dots.\"\n\n- button layout: S02 \"The glove \/ specter and n-D \/ 2-D buttons should be closer together so it's easier for me to realize where to make changes related to the search process.\"\n\n- keywords in meta view: S03 \"Keyword feature can be useful if it's not displaying keywords for the stuff above but for what I have selected.\"\n\n- S06 \"Most of the features were really good as-is.\" \n\"Maybe the interface might be able to support more natural interactions -- more connected between these panels. E.g., if I filter keyword here [meta view] it would filter here [main table].\"\n\"When I search the abstract, it might be useful to support non-exact match, because you usually don't expect an exact string match.\"\n\"Match the conferences were not matched -- like 20 Eurographics [variations].\"\n\"Some focus on sections. If I'm working on the table, this could be bigger, or let users change the proportion of the panels.\"\n\n\\section{Formative Study}\n\\label{sec:formative}\n\nWe conducted a formative study to better understand the needs of researchers as they perform literature reviews. \nParticipants were 4 Computer Science PhD students (3 female, 1 male; avg. 2.75 yrs. into PhD program) who had prior experience conducting literature reviews in the field of visualization.\nSessions lasted approximately 45 minutes. \nParticipation was voluntary with no compensation. \n\nWe presented the first two participants with an initial version of the literature review tool. \nAfter incorporating feedback in the next iteration of the system, we worked with the next two participants using the updated system.\nFinally, we incorporated feedback from all formative study participants in {\\rmfamily \\scshape vitaLITy}{}, presented in the next section.\n\n\n\\subsection{Current Workflow}\n\\label{sec:workflow}\n\nAfter obtaining informed consent, we asked participants to describe their typical workflow for conducting literature reviews via a semi-structured interview.\nParticipants expressed some haphazard nature to the beginning of their processes, e.g., \\textit{``someone tells [them] about a paper, and [they] look up the citations and branch out from there''} (P1) or \\textit{``use a starting point from an advisor''} (P2).\nFrom there, there are some commonalities in processes.\n\nParticipants all utilized keyword searches on Google Scholar (P1-4).\nAs a fairly comprehensive database, participants did not worry whether a venue or paper would be present, and they appreciated the ``cited by'' feature to identify more recent relevant papers.\nHowever, participants also expressed that keyword searches on Google Scholar result in many irrelevant papers that require a lot of manual filtering for relevance.\nFor instance, P4 viewed Google Scholar as a last resort, expressing they really only use it \\textit{``if [they] don't have a better starting point seed paper.''} \nEchoing some of the motivation for this work, P1 indicated, \\textit{``if a keyword is used differently in different fields, [they] have to read a lot of abstracts to determine whether it's relevant or not.''}\n\nWhile Google Scholar seems to be the default search tool, there are others that participants integrate in various parts of their workflow when conducting literature reviews. \nFor instance, P4 indicated regular use of bibliography management tools like Mendeley and Zotero.\nAmong our relatively small sample in this formative study, participants did not mention some other elements in their workflow that we anticipated, e.g., DBLP, manual scripting \/ web scraping, etc. \n\n\n\n\n\\subsection{Preliminary Feedback}\n\\label{sec:prelim_feedback}\n\nNext, participants used a preliminary version of our literature review tool. \nThe preliminary tool included \\texttt{17,926} papers from the following venues over the past \\texttt{39} years (1982-2020): \\{\\emph{CGA, CGF, EuroVis, Graphics Interface, Information Visualization, Interact, Journal of Visualization, PacificVis, SciVis, TVCG, VAST, VIS}\\} and supported two main mechanisms for searching the corpus: keyword search and similarity search (described in greater detail in the next section).\nAfter using the tool, we asked participants for additional feedback about the current implementation, possible improvements, and any new capabilities that they could envision to better support their literature review process. \n \nParticipants appreciated the ability to start their search with a seed paper or papers (P1 said they got \\textit{``pages and pages of results which is what [they] would get on Google Scholar, but these are actually more relevant''}). \nP2 searched based on the seminal paper on hypothetical outcome plots (HOPs)~\\cite{hullman2015hypothetical} and observed \\textit{``it pulled up lots of uncertainty vis papers, which were not in the title -- cool!''}\nbut expressed that there was still a lot of noise when searching by keywords.\n\n\nParticipants \\revised{suggested several} new features: being able to visualize connections between papers (e.g., by citations, co-authors, etc. - P1), adding critical information on citation count as a mechanism for determining importance of a paper (P1), making the overview interactive (with brushing and linking, summarizing dynamic regions, etc. - P2), and being able to type in a custom abstract or paper idea as the basis of the similarity search (e.g., to identify relevant literature for a paper idea that hasn't been fully fleshed out yet - P2). \nParticipants also steered away from one of the features in the tool: the word cloud. \nP4 indicated \\textit{``it wasn't clear how it was related to what [they] had selected.''}\n\nOverall, participants indicated that a tool like this in their workflow could supplement tools like Google Scholar for serendipitous exploration. \nP4 suggested it would be beneficial in the early ``discovery'' phases of literature review, with the caveat that the data on included venues needed to be sufficiently comprehensive. \nAs a result of this feedback, we updated the system to address these ideas, including scraping data from additional venues, adding citation counts, adding brushing and linking between views, and searching by a custom abstract. \nWe did not add features based on citation networks in our system\\revised{; instead, we focused on leveraging transformer models to serve as a complementary literature search technique to existing tools that address these needs}. \\removed{due to space and complexity constraints in the tool.}\n\n\n\\subsection{Design Goals}\n\\label{sec:design_goals}\n\nCollectively, these interviews led us to the following set of four design goals for our literature review system.\n\n\\smallskip\n\\noindent\\textbf{DG 1. Serendipity:} Enable serendipitous identification of semantically related articles that do not necessarily have shared keywords through visual exploration.\n\n\\smallskip\n\\noindent\\textbf{DG 2. Familiarity:} Facilitate a familiar search functionality to what users are currently accustomed to, such as keyword and author search.\n\n\\smallskip\n\\noindent\\textbf{DG 3. Novelty:} Afford users to find semantically related articles by searching based on the author's own ideas in the form of unpublished sentences \/ abstract.\n\n\\smallskip\n\\noindent\\textbf{DG 4. Overview:} Enable users to interact with a visual overview of a group of papers.\n\n\n\\section{Related Work} \n\\label{sec:related_work}\n\n\\subsection{Literature Review Methodologies}\nLiterature reviews and surveys are an essential part of scientific disciplines. \nThey are broadly defined as systematic ways of collecting and synthesizing research on a specific topic \\cite{baumeister1997writing,snyder2019literature}. \nThere are a variety of different guidelines and methodologies, such as systematic reviews \\cite{moher2009preferred}, narrative reviews \\cite{baumeister1997writing}, and integrative reviews \\cite{torraco2005writing}. \nThese guidelines and methods mostly vary in how they organize, synthesize, and analyze a set of selected articles through a combination of quantitative and qualitative methods \\cite{snyder2019literature}. \nThese methodologies often include multiple stages, the first of which is related to identifying a strategy for searching and selecting a set of related literature. \nFor example, Hannah Snyder states that \\textit{``a search strategy for identifying relevant literature must be developed. \nThis includes selecting search terms and appropriate databases and deciding on inclusion and exclusion criteria. \nHere, a number of important decisions must be made that are crucial and will eventually determine the quality and rigor of the review''}\\cite{snyder2019literature}. \n\nSimilarly, within the visualization community, defining search strategies and keywords are described as the primary step for conducting literature reviews \\cite{mcnabb2019write}.\nMany visualization survey papers include explicit excerpts about their selection criteria that describe keywords, databases, and the search process of each survey paper \\cite{tong2018storytelling,fuchs2016systematic,roberts2018visualising}. \nFor example, in their survey of glyph visualization techniques, Fusch et al. employ a ``snow ball'' sampling technique in which they start by searching the keyword ``glyph'' within various libraries, select all the findings, filter based on their exclusion criteria, and then look at the related work of the selected papers to find more papers \\cite{fuchs2016systematic}. \n\nAlthough keyword search is the most prevalent method for searching literature, it comes with some limitations:\n\\begin{itemize}[nosep]\n \\item Often it won't yield papers that do not include a specific keyword but might be very related to the topic at hand.\n \\item Within different communities, different keywords are used to represent a common concept.\n \n\\end{itemize}\n\nAs a result, selecting sufficiently broad yet relevant keywords can be a challenge.\n{\\rmfamily \\scshape vitaLITy}{} offers a visual system that complements traditional keyword search-based methods to enhance literature searches\n{\\rmfamily \\scshape vitaLITy}{} implements a state-of-the-art transformer-based document similarity search that can find semantically similar documents that may not always share the same set of keywords.\n\n\n\n\n\n\n\\subsection{Visualization of Academic Articles}\nVisual analytics research has been effective in incorporating many machine learning and natural language processing models (e.g., topic modeling or word embeddings) into vis systems for exploratory analysis of large corpora of text documents \\cite{Endert2012,liu2018bridging,dou2013hierarchicaltopics,el2017progressive}. A common task is identifying similar documents \\cite{endert2017state}. Early visualization papers on document similarities used representations of a corpus' similarity matrix through dot plots \\cite{church1993dotplot} or histograms \\cite{freire2008visualizing}. More recent vis systems have considered more author assigned keyword-based approaches like constructive text similarity \\cite{abdul2017constructive} and GlassViz \\cite{benito2020glassviz}.\nAlternative approaches have considered word embeddings including for iterative lexicon construction \\cite{park2017conceptvector} that provide related ability to query documents. \n\nOne key application area for incorporating visual techniques to help users find similar and relevant documents is in searches for academic articles. Several prominent article databases have implemented such systems to find relevant articles. Text Analyzer by JSTOR extracts the most important topics and keywords from entered papers and recommends other relevant documents to users (\\url{https:\/\/www.jstor.org\/analyze\/}). Pubmed uses a word-based technique to help users retrieve the most similar papers ( \\url{https:\/\/pubmed.ncbi.nlm.nih.gov\/help\/#pubmedhelp.Computation_of_Weighted_Relev}). Open Knowledge graph uses similarity scores provided by Pubmed and develops a circle packing visualization to help users understand groups of related research relevant to their search terms (\\url{https:\/\/openknowledgemaps.org\/}).\n\nWithin the visualization community, several works highlight the importance of understanding and visualizing academic literature. \nFelix et al., introduce a design space and highlight how different keyword summarization techniques might impact users' understanding of related literature \\cite{felix2017taking}. \nUsing the open source vis literature dataset (VisPubData), Isenberg et al. introduce KeyVis and analyze keywords utilized in the visualization community~\\cite{isenberg2016visualization,Isenberg:2017:VMC}. \nOthers introduce relevant systems for supporting dissemination of curated survey results~\\cite{beck2015visual}, \\revised{visualization of lead-lag analysis of text corpora~\\cite{liu2014exploring}}, analysis of the contextual \\emph{reasons} for citations~\\cite{yoon2020conference}, and an emergent design space for considering visualizations of literature collections~\\cite{hinrichs2015speculative}.\nIn general, within visualization systems on academic literature we can observe three themes: (1) visualization systems that focus on citation networks (e.g.,~\\cite{wilkins2015evolutionworks,heimerl2015citerivers,chou2011papervis,dattolo2018visualbib}), systems that focus on clustering or similarity (typically by matching keywords, e.g.,~\\cite{wang2019vispubcompas}, or using topic modeling, e.g.,~\\cite{alexander2014serendip,isenberg2016visualization}), and (3) systems that focus on both citation networks and similarity measures (e.g.,~\\cite{nakazawa2018analytics,chen2006citespace}).\n\\revised{In the latter category, CiteSpace II introduces a technique to computationally define \\emph{co-citation clusters}.}\n\nInspired by these works, our paper introduces (1) a more comprehensive public dataset of visualization literature, and (2) utilizes state-of-the-art document embedding techniques using transformers to enable serendipitous discovery of articles.\n\n\n\n\n\n\n\n\\subsection{Word Embeddings and Transformers}\nDocument similarity is a classic problem in natural language processing and information retrieval \\cite{Jurafsky2021}. \nWord embeddings provide an approach in which words (or documents) that have similar meanings have similar (vector) representations.\n\\revised{Recent advances in word embeddings have yielded significant improvements in standard similarity benchmarks like STS or SentEval \\cite{arora2017simple,reimers-gurevych-2019-sentence,cohan2020specter}.} \nBeginning with word2vec \\cite{mikolov2013efficient}, many extensions of learned dense representations of word vectors have followed including GloVe \\cite{pennington2014glove}, fasttext \\cite{bojanowski2017enriching}, skipthought \\cite{kiros2015skip}, ELMo \\cite{peters2018deep}, and BERT \\cite{devlin2018bert}. \\revised{More recently, specialized transformer models like SPECTER \\cite{cohan2020specter} have been developed to specialize in domains like academic literature. SPECTER combines self-supervised pre-training on transformer architectures (e.g., BERT-like) on academic abstracts and is ``citation-informed'' to enhance performance for tasks like academic literature recommendation and topic classification.}\n\n\\revised{SPECTER provides four advantages over past word embedding approaches for {\\rmfamily \\scshape vitaLITy}{}'s task. First, it incorporates contextual embeddings (via BERT\/transformer architecture) that enable different vector representations depending on the context (e.g., ``bias'' in different contexts). Second, the model was pre-trained on academic titles and abstracts (sciBERT \\cite{beltagy2019scibert}). This enables the model to have transfer learning gains from pre-training with a BERT-like \\cite{devlin2018bert} transformer architecture but with specialization for academic literature recommendation. Third, it incorporates a triplet-loss pre-training objective that enables it to use citations as an inter-document incidental supervision signal for fine-tuning. By incorporating both text pre-training with citation fine-tuning, the model achieved state-of-the-art performance for academic literature recommendations as well as six additional tasks like citation prediction, user activity (view or read), and topic classification. Tasks like citation prediction or user activity were out of scope of {\\rmfamily \\scshape vitaLITy}{}'s design due to data limitations, but future work could easily incorporate such tasks with additional citation or activity data. Fourth, the model is available out-of-the-box without fine-tuning as well as in model deployment through a publicly released API. This API enables fast and efficient real time scoring in {\\rmfamily \\scshape vitaLITy}{}.} \n\n\n\n\n\\section{Introduction}\n\nVisualization research is inherently interdisciplinary, borne out of fields such as Computer Graphics and Human-Computer Interaction, with heavy influence from fields outside of computing such as Perceptual Psychology and Cognitive Science. \nFurthermore, visualization is applied to explore data and support data-driven decision making problems in domains ranging from enterprise analytics to medicine. \nAs a result of the multi-faceted nature of the field, there may be parallel research efforts that can be difficult to become aware of, even with a comprehensive methodology for conducting literature reviews. \n\nOne challenge of interdisciplinary research is when different fields use similar terminology to study different problems. \nFor instance, \\emph{transformer} in electronics refers to a device that transfers energy between circuits~\\cite{kulkarni2017transformer}; while in computing, \\emph{transformer} refers to a type of neural network based on attention mechanisms, commonly applied to unstructured text data~\\cite{vaswani2017attention}.\nAs a result, keyword searches often yield irrelevant work. \nFurther, sifting through all hits from a keyword search may still miss critical work.\nFor instance, the recent wave of work on \\emph{bias} in visualization (e.g.,~\\cite{dimara2017attraction,dimara2019mitigating,WallBias,wall2018four,cho2017,valdez2018priming,LRG,Lumos}) seldom mentions \\emph{uncertainty} (e.g.,~\\cite{hullman2015hypothetical,hullman2019authors}).\nYet, as the seminal work on bias in Cognitive Science points out, bias emerges when people make decisions under uncertainty~\\cite{Tversky1974}; hence, there is a critical need to examine uncertainty literature that may fundamentally address similar problems using different terminology.\nAs a result, conducting a simple keyword search for ``bias'' (i.e., matching tokens in a paper title or abstract) \nto identify relevant work may neglect pockets of influential research.\nHowever, these challenges are not unique to data visualization research or even computing. \nThey extend to virtually all interdisciplinary research.\n\nCurrent prevalent practices for conducting literature reviews tend to utilize two common search strategies: (1) keyword search and (2) examination of back-references from a snowballing set of seed papers, usually through searching Google Scholar or DBLP. \nThese approaches can successfully identify a large number of relevant citations, but can suffer from at least two key limitations: thoroughness and efficiency. \nThat is, they may fail to unearth related papers that use different terminology, and they require significant manual effort to gauge relevancy of potentially thousands of hits. \nIn other words, a prominent challenge, then, in conducting literature reviews or surveys is to effectively identify research of significance to a given topic based on similarity of topics, irrespective of matching exact keywords.\n\nTo address these challenges, we introduce {\\rmfamily \\scshape vitaLITy}{}, an open-source visualization system designed to support a flexible exploration of research articles. \nInspired by work on \\emph{insight} in visualization (i.e., ``eureka'' or ``aha'' moments~\\cite{chang2009defining}), we similarly aim to support \\emph{serendipity} with {\\rmfamily \\scshape vitaLITy}, \\revised{operationally intended to describe the goal that users may ``stumble upon'' relevant literature, when other search approaches might otherwise fail}. \n\\revised{{\\rmfamily \\scshape vitaLITy}{} incorporates SPECTER \\cite{cohan2020specter}, a state-of-the-art document-level contextual embedding model for scientific document recommendation.\nUnlike many pre-trained language models that use a general corpus like Wikipedia or the Common Crawl \\cite{mikolov2013efficient,pennington2014glove,devlin2018bert}, SPECTER was pre-trained on academic literature (sciBERT \\cite{beltagy2019scibert}) and fine-tuned with citations which provides out-of-the-box state-of-the-art performance for academic literature recommendations and topic classification.\n\n\n\nIn summary, this work presents the following contributions: \n\\begin{enumerate}[nosep]\n \\item results of a formative interview study in which visualization researchers identified key challenges in current literature review practices (Section~\\ref{sec:formative}),\n \\item a dataset of scraped metadata from \\texttt{59,232} academic articles (\\textbf{\\url{https:\/\/figshare.com\/articles\/dataset\/VitaLITy_A_Dataset_of_Academic_Articles\/14329151}}~\\cite{Narechania2021}, CC0 License), including paper titles, keywords, and abstracts from \\texttt{38} popular venues for visualization research (Section~\\ref{sec:data}),\n \\item an open-source tool, {\\rmfamily \\scshape vitaLITy}{} (\\textbf{\\url{http:\/\/vitality-vis.github.io}}, MIT License), for supporting discovery of relevant articles while conducting literature reviews (Section~\\ref{sec:system_overview}),\n \\item usage scenarios describing potential workflows in which {\\rmfamily \\scshape vitaLITy}{} might be used in different ways to support serendipitous discovery of relevant academic literature\n \n \n \n \n \n (Section~\\ref{sec:case_study}), and \n \\item results of a summative evaluation of {\\rmfamily \\scshape vitaLITy}{} (Section~\\ref{sec:evaluation}).\n\\end{enumerate}\n\n\n\n\n\n\n\\begin{comment}\nThe remainder of this paper is organized as follows. \nIn Section~\\ref{sec:related_work}, we describe existing literature review methodologies and word embedding techniques. \nWe next describe a formative interview study that led to a set of design goals for the system in Sections~\\ref{sec:design_considerations}-\\ref{sec:system}.\nWe present results of a benchmarking evaluation in Section~\\ref{sec:evaluation} and a case study application of our system in the domain of bias in visualization in Section~\\ref{sec:case_study}.\nWe discuss challenges, limitations, and future work and conclude with final remarks in Sections~\\ref{sec:discussion}-\\ref{sec:conclusion}.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\\input{sections\/related_work}\n\\input{sections\/formative}\n\\input{sections\/system}\n\\input{sections\/bias}\n\\input{sections\/evaluation}\n\\input{sections\/discussion}\n\\input{sections\/conclusion}\n\n\n\n\\bibliographystyle{abbrv-doi}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Section_I_Introduction}\n\n\nRealizing the promises of quantum computing requires the ability to manipulate and measure complex states of quantum devices with high fidelity. An outstanding challenge towards reaching this goal is the realization of fast and high-fidelity entangling gates with large on-off ratio. An approach to turn on and off entangling interactions is to frequency tune pairs of qubits in and out of resonance. However, noise in the control parameter allowing to tune the qubit frequency introduces an additional source of dephasing, which can be mitigated by operating the qubits at sweet spots where they are first-order insensitive to this noise channel \\citep{vionManipulatingQuantumState2002}.\n\nIn superconducting qubits such as the transmon \\cite{kochChargeinsensitiveQubitDesign2007a}, tuning the qubit frequency is most commonly accomplished by threading a loop with magnetic flux. While for static dc flux bias there are a few sweet spots per single flux period $\\Phi_0$, it was recently shown that tailored ac modulation of the flux or direct voltage drive on the qubits extends these few static sweet spots to a larger class of dynamical sweet spots \\citep{huangEngineeringDynamicalSweet2020,didierFluxControlSuperconducting2019,guoDephasinginsensitiveQuantumInformation2018,didierACFluxSweet2019,valeryDynamicalSweetSpot2021}. This gives more flexibility in choosing the operating points to both maximize dephasing times and facilitate two-qubit gates. \n\nIn the presence of a continuous ac drive, the static computational basis of the qubits is replaced by a set of eigenstates of the periodic Floquet Hamiltonian, also known as Floquet states. Protocols for initialization, readout, single-qubit operations and entangling gates on these Floquet qubits have been theoretically proposed \\citep{huangEngineeringDynamicalSweet2020} and experimentally investigated \\citep{mundadaFloquetengineeredEnhancementCoherence2020} showing improvements in the dephasing time of the qubits. Floquet qubits can be frequency-tuned in large frequency range by changing the parameters of the drive. This tunability can be used, for example, to implement single-qubit phase gates, but also to bring together pairs of Floquet qubits to activate SWAP-type interactions \\citep{huangEngineeringDynamicalSweet2020}. On the other hand, for X-type single-qubit gates, or for two-qubit gates such as the cross-resonance \\citep{rigettiFullyMicroTunable2010}, a second drive is introduced to induce transitions between the Floquet-qubit states \\cite{huangEngineeringDynamicalSweet2020}. \nMoreover, as shown in Refs.~\\cite{huangEngineeringDynamicalSweet2020,mundadaFloquetengineeredEnhancementCoherence2020}, the readout of the Floquet qubit can be performed in a two-step process: the Floquet qubit is first mapped to the laboratory-frame qubit by adiabatically turning off the drive. At that point, a usual dispersive readout is performed by driving a cavity coupled to the qubit \\citep{Blais2021}.\n\nIn this work, we exploit the Many-Mode Floquet theory (MMFT) introduced in Refs.~\\citep{hoFloquetLiouvilleSupermatrixApproach1986, shirleySolutionSchrodingerEquation1965a, shillSEMICLASSICALMANYMODEFLOQUET1983} to provide an analytical description of the dynamics of the Floquet qubit during such gates, which involve more than one drive frequency. Generalized spectra describing the system during the operation of the gate can be used to optimize gate parameters, as well as to understand the dynamics of higher-energy states.\n\nUsing this approach, which allows for multiple simultaneous drives, we also show how it is possible to use a driven coupler to engineer a longitudinal interaction between the Floquet qubit and a readout cavity. Thanks to both the longitudinal nature of this interaction -- which is known to lead to fast qubit measurements \\citep{didierFastQuantumNondemolition2015} -- and to the fact that there is no need to map the Floquet qubit back to the undriven qubit states before the readout, we find from numerical simulations that this approach can lead to fast and high-fidelity Floquet qubit readout. A superconducting circuit design for this longitudinal Floquet readout is proposed.\n\n\nThe paper is structured as follows. In \\cref{Section_II_Floquet_Framework}, we review the Floquet and MMFT frameworks. In \\cref{Section_III_Single_qubit_operations} we show how X-type gates can be implemented on Floquet states by adding a second drive to the qubit and we apply MMFT to the driven Floquet qubits during the gate.\nWe then demonstrate the feasibility of dynamical longitudinal readout of Floquet states with an additional drive in \\cref{Section_IV_Readout_of_floquet_states}, and compare our analytical results to full numerics. Finally, in \\cref{Section_V_Initialization_of_arbitrary_states} we explore the timescales necessary for the initialization of a Floquet qubit.\n\n\n\\section{Floquet Framework} \n\\label{Section_II_Floquet_Framework}\n\n\n\\subsection{Floquet qubits}\n\\label{Section_II_Subsection_I_Floquet_Qubits}\n\nDriven quantum systems are part of a larger class of systems evolving under a time-periodic Hamiltonian with period $T=2\\pi\/\\omega_{d}$ and which are efficiently described by the Floquet formalism \\citep{grifoniDrivenQuantumTunneling1998b,chuFloquetTheoremGeneralized2004b}, where a system Hamiltonian $H_{\\text{s}}$ with Hilbert-space dimension $d$ is replaced with the time-dependent Floquet Hamiltonian (with $\\hbar = 1$): \n\\begin{equation}\n\\label{General_Floquet_Hamiltonian}\n H_F(t) = H_{\\text{s}}+V(t)-i\\frac{\\partial}{\\partial t}, \n\\end{equation}\nwith $V(t)=V(t+T)$ the periodic drive on the system. Based on the symmetry of the Hamiltonian $H_F$ under time translation $t\\rightarrow t+T$, the Floquet theorem states the existence of a full set of solutions to the time-dependent Schrodinger equation so that $\\forall t,~ H_F(t)\\ket{\\psi_n(t)} = 0, (n=1,2,...,d)$. To complete the analogy with the static Hamiltonian, these solutions are related to the eigenvalue problem for the Floquet Hamiltonian $\\forall t,~ H_F(t)\\ket{\\phi_n(t)} = \\epsilon_{n}\\ket{\\phi_n(t)}, (n=1,2,...,d)$ with: \n\\begin{equation}\n\\label{General_Floquet_Modes}\n \\forall t,\\forall n\\leq d, ~\\ket{\\psi_{n}(t)} = e^{-i\\epsilon_nt}\\ket{\\phi_{n}(t)},\n\\end{equation}\nwhere the \\textit{Floquet modes} $\\ket{\\phi_{n}(t)}$ are $T$-periodic in time, and the \\textit{quasienergies} $\\epsilon_{n}$ are real-valued coefficients which are invariant under translation by multiples $k$ of the drive frequency $\\omega_{d}$. The term \\textit{quasienergies} thus refers to representatives of equivalence classes, often chosen in the first Brillouin zone $[-\\omega_{d}\/2, \\omega_{d}\/2]$. With appropriately designed driving protocols, one can convert an undriven Hamiltonian $H_{s}$ into the dressed $H_F(t)$ and continuously map the energies to the quasienergies and the eigenstates to the corresponding Floquet states, hence the name \\textit{Floquet qubit} when the time-dependent dressed states are used to define the two-level system.\\\\\n\n\nIn this context, dynamical protection consists of operating the Floquet qubit at extrema of the quasienergy difference with respect to the drive parameters subject to noise \\citep{didierACFluxSweet2019, didierFluxControlSuperconducting2019, huangEngineeringDynamicalSweet2020}. As shown by \\textcite{huangEngineeringDynamicalSweet2020}, dynamical sweet spots represent manifolds in parameter space, in contrast with the few isolated static sweet spots that are found in the absence of a drive. This allows for an increased freedom in the parameter choice that can be used to operate the Floquet qubit while being protected from low-frequency noise, which translates to high coherence times. This property is compatible with single and two-qubit gate operations, which justifies the promising role Floquet qubits could play in quantum information processing.\\\\\n\n\n\\subsection{Many-Mode Floquet Theory}\n\\label{Section_II_Subsection_II_Many_Mode_Floquet_Theory}\n\nFloquet qubits describe a subgroup of driven systems evolving under the dynamics of a $T$-periodic Hamiltonian as introduced in \\cref{General_Floquet_Hamiltonian}. Here, the Hamiltonian often includes the effect of only one drive on a static system or else multiple drives at different harmonics of one characteristic frequency. In some cases such as certain gates on Floquet qubits, one can face Hamiltonians with two distinct time-dependent terms:\n\\begin{equation}\n\\label{General_Floquet_Hamiltonian_Two_Tones}\n H_F(t) = H_{\\text{s}}+V_1(t)+V_2(t)-i\\frac{\\partial}{\\partial t},\n\\end{equation}\nwith $V_1,V_2$ respectively $2\\pi\/\\omega_1$ and $2\\pi\/\\omega_2$ periodic in time. The Floquet qubit is generated by the first driving term and the system Hamiltonian, while the second term is typically only switched on during the gate without any \\emph{a priori} link between the frequencies $\\omega_1$ and $\\omega_2$. We will limit ourselves to two distinct frequencies and their harmonics, even if the scheme used in studies of Hamiltonians with multiple drives is more general \\citep{fainshteinNONLINEARSUSCEPTIBILITIESLIGHT1992, dorrMultiphotonProcessesIntense1991a, crowleyTopologicalClassificationQuasiperiodically2019, boyersExploring2DSynthetic2020, longNonadiabaticTopologicalEnergy2020}.\n\nIn the rotating-wave approximation (RWA), the two distinct frequencies typically result in a single effective frequency on the system at the difference $(\\omega_1 - \\omega_2)$, but Floquet analysis aims at describing the dynamics of driven systems without such simplification.\nWe first notice that the extension of Floquet theory to commensurate frequencies is straightforward using the greatest common divisor $\\omega_{\\mathrm{GCD}}=\\mathrm{GCD}(\\omega_1, \\omega_2)$ as the new frequency of a single-tone Floquet system for the duration of the gate \\citep{poertnerBichromaticDressingRydberg2020}. This result translates into the definition of Floquet states and quasienergies for \\cref{General_Floquet_Hamiltonian_Two_Tones} which create the continuous connection between Floquet states before and after the gate. These intermediary states can be numerically evaluated, but complexity arises as the new period of the system can be orders of magnitude greater than the distinct timescales $2\\pi\/\\omega_1$ and $2\\pi\/\\omega_2$. In practice, this leads to long simulation times.\n\nA generalization of these ideas for multiple incommensurate frequencies exists under the name of Many-Mode Floquet Theory\n\\citep{chuFloquetTheoremGeneralized2004b, hoSemiclassicalManymodeFloquet1985a,hoSemiclassicalManymodeFloquet1983}. The main idea here \nis to look for a generalization of the $N$ Floquet states and quasienergies by considering a Fourier basis with two dimensions rather than only one:\n\\begin{align}\n\\label{General_Fourier_Basis_Two_Tones}\n \\forall n\\leq N,~\\ket{\\phi_n(t)} &= \\sum_{k_1, k_2}e^{i(k_1\\omega_1+k_2\\omega_2)t}\\ket{\\phi_{n,k_1,k_2}}\\\\\n \\forall n\\leq N,~\\epsilon_{n, k_1, k_2} &= \\epsilon_{n, 0, 0} + k_1\\omega_1 + k_2\\omega_2.\n\\end{align}\nMMFT has been implemented into a numerical solver using truncated Fourier bases \\citep{poertnerValidityManymodeFloquet2020a}, which we do not make use of in this work. However, the generalization of quasienergies and Floquet Modes provides an analytical approach to understand the dynamics of driven systems with additional drives such as naturally occurs in Floquet qubits.\n\n\n\n\\section{Single-qubit operations} \n\\label{Section_III_Single_qubit_operations}\n\nApproaches to realize $X$, $\\sqrt{X}$ and single-qubit phase gates on Floquet qubits were proposed in Ref.~\\citep{huangEngineeringDynamicalSweet2020} with fidelities obtained from numerical simulations exceeding $99.99\\%$ and gate durations on the order of tens of nanoseconds. Our focus here is on the $X$ and $\\sqrt{X}$ gates based on adding a secondary drive to the Floquet qubit to induce Rabi oscillations between the Floquet states. \n\nWe extend the principle of the Floquet spectrum to driven Floquet qubits, widely used in static systems with a single drive frequency. This approach has recently been used by \\textcite{Petrescu2021} to extract gate parameters maximizing the gate rate while minimizing higher order $ZZ$-terms. The generalized Floquet spectrum is defined with respect to the second drive frequency acting on the periodic Floquet System, typically the drive inducing a X-Gate on a Floquet qubit. We characterize the generalized avoided crossings appearing at resonances in the spectrum of the Floquet qubit.\n\n\\subsection{X-Gate in the RWA}\n\\label{Section_III_Subsection_I_XGate_in_the_RWA}\n \nWe use as logical states the eigenstates of a two-level system (TLS) and, without loss of generality, we will set a transition frequency $\\omega_0\/2\\pi = 5.02~\\text{GHz}$ and a near-resonant Rabi drive with amplitude $\\varepsilon_{d1}\/2\\pi = 0.21$ GHz and with a detuning $\\Delta = \\omega_{0}-\\omega_{d1}$:\n\\begin{equation}\n\\label{TLS_Floquet_Hamiltonian}\n H(t) = \\frac{\\omega_0}{2}\\sigma_z + \\varepsilon_{d1}\\cos(\\omega_{d1}t)\\sigma_x.\n\\end{equation}\nGoing to a frame rotating at $\\omega_{d1}$ and applying the RWA by assuming $\\varepsilon_{d1}\\ll~\\omega_{d1}$, the Floquet states and quasi-energies of the Hamiltonian of \\cref{TLS_Floquet_Hamiltonian} take the form:\n\\begin{equation}\n\\begin{split}\n\\label{TLS_Floquet_Modes}\n \\epsilon_{0,1} &= \\pm\\sqrt{\\left(\\frac{\\Delta}{2}\\right)^2+\\varepsilon_{d1}^2}, \\\\\n \\ket{\\phi_{0,1}(t)} &= \\frac{e^{+i\\omega_{d1}t\/2}}{\\sqrt{\\varepsilon_{d1}^2+(\\epsilon_{0,1}-\\frac{\\Delta}{2})^2}}\\begin{pmatrix} |\\varepsilon_{d1}|e^{-i\\omega_{d1}t} \\\\ \\epsilon_{0,1}-\\frac{\\Delta}{2}\\end{pmatrix}.\n\\end{split}\n\\end{equation}\nFurther imposing $|\\Delta|\\ll\\varepsilon_{d1}$, the Floquet states $\\ket{\\phi_{0,1}}$ are located near the equatorial plane of the Bloch sphere. Following \\textcite{huangEngineeringDynamicalSweet2020}, the addition of a second drive along the $Z$-axis in the laboratory frame with a frequency $\\omega_{d2}$ chosen close to the quasienergy difference induces Rabi oscillations of the Floquet qubit. With this addition the Hamiltonian now reads:\n\\begin{equation}\n\\label{TLS_FLoquet_with_Drive}\n H(t)= \\frac{\\omega_0}{2}\\sigma_z+\\varepsilon_{d1}\\cos\\left(\\omega_{d1} t\\right)\\sigma_x+\\varepsilon_{d2}\\cos\\left(\\omega_{d2} t\\right)\\sigma_z,\\\\\n\\end{equation}\nwhere $\\varepsilon_{d2}$ is the amplitude of the second drive. As an example, for a transmon qubit, a drive along the $Z$ axis is realized by flux pumping the qubit's SQUID loop \\cite{kochChargeinsensitiveQubitDesign2007a}.\n\nIn \\cref{Figure_XGate_Floquet_Qubits}(a) we plot results from an integration of the Schr\\\"odinger equation showing a full population transfer between the states $\\ket{\\phi_0(t)}, \\ket{\\phi_1(t)}$ with fidelity 99.99\\% and ramp times of the order of $20~\\text{ns}$, corresponding to the pulses represented in \\cref{Figure_XGate_Floquet_Qubits}(b). There, the green curve corresponds to the amplitude $\\varepsilon_{d1}$ of the first drive and is used to establish the Floquet logical states. A second, flux, tone (blue curve) is switched on for the duration of the gate and its frequency $\\omega_{d2}$ is close to twice the amplitude $\\varepsilon_{d1}$ of the Floquet qubit drive.\n\nThe analysis in this section relies on the RWA and is therefore only valid in the limit $|\\Delta|\\ll\\varepsilon_{d1}$. An analogous application of the RWA would be further necessary when treating the second drive, for example in order to express the gate rate of the Floquet-qubit X-gate. Corrections beyond the RWA, applicable also in the more general case of off-resonant drives can be derived \\cite{mirrahimi2015dynamics}. Instead, in the following subsection we rely on the exact Floquet two-tone numerical method to obtain the gate rate.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/Final_Fig\/Fig1_quasiphase_0810.pdf}\n \\caption{a) Population of the Floquet modes $\\ket{\\phi_0(t)}$ and $\\ket{\\phi_1(t)}$ as a function of time under the Hamiltonian \\cref{TLS_FLoquet_with_Drive}. \n b) Typical drive amplitude as a function of time. The first drive $\\varepsilon_{d1}$ (green line) is used to generate the Floquet qubit states while the second drive $\\varepsilon_{d2}$ (blue line) drives Rabi oscillations between these levels.\n c) Four quasiphase spectra for different numerators $p=1, 2, 3, 4$ in the ratio $\\omega_{d2}\/\\omega_{d1}=p\/q$. The colored dots are obtained from diagonalization of the propagator associated with \\cref{TLS_FLoquet_with_Drive} after a common period $2\\pi\/\\omega_{\\mathrm{GCD}}$ and by sweeping the values of $q$. Because of the folded space, several crossings are observed. However, a unique anticrossing corresponding to the resonance of the second drive with the Floquet qubit is observed (vertical line).}\n \\label{Figure_XGate_Floquet_Qubits}\n\\end{figure}\n\n\\subsection{Two-tone Floquet analysis}\n\\label{Section_III_Subsection_II_Two_tone_floquet_analysis}\n\nHere, we propose an alternative approach to analyze a gate on the Floquet states based on a two-tone Floquet analysis. A Floquet spectrum is obtained from the Hamiltonian of \\cref{TLS_FLoquet_with_Drive} with respect to the second drive frequency $\\omega_{d2}$ without requiring any of the previous RWAs. To do so, we regroup the time-dependent terms into a single quasi-periodic drive $V(t)$.\nIf the frequencies $\\omega_{d1}$ and $\\omega_{d2}$ are commensurate, then the periodicity of $V(t)$ is given by the greatest common divisor $\\omega_{\\mathrm{GCD}} = \\mathrm{GCD}(\\omega_{d1}, \\omega_{d2})$. \n\nIn the presence of two tones, the quasienergy spectrum is probed by sweeping one of the drive frequencies.\nHowever, because the quasienergies are only defined modulo $\\omega_{\\mathrm{GCD}}$ and because this quantity will strongly depend on the chosen $\\omega_{d2}$, it is not possible to define a continuous quasienergy spectrum. \n\nAs explained in \\cref{Section_II_Floquet_Framework}, when the drive frequencies can be written as an irreducible fraction $\\omega_{d1}\/\\omega_{d2} = p\/q$, the frequency $\\omega_{\\mathrm{GCD}}$ can be expressed as $\\omega_{\\mathrm{GCD}} = \\omega_{d1}\/p = \\omega_{d2}\/q$. Here, $\\omega_{d1}$ is taken as a fixed parameter such that \neach numerator $p$ corresponds to a distinct first Brillouin zone. For each numerator $p$, we can introduce a discrete quasienergy spectrum satisfying $\\omega_{d2}=\\omega_{d1}\\times q\/p$ for $q\\in\\mathbb{N}$. To compare quasienergy spectra corresponding to different numerators $p$, we normalize the Floquet quasienergy spectrum $\\epsilon_{1\/2}(\\omega_{d2})$ defined over the range $[-\\omega_{\\mathrm{GCD}}\/2,\\omega_{\\mathrm{GCD}}\/2]$ to obtain the Floquet quasiphase spectrum defined as $\\phi^F_{1\/2}(\\omega_{d2}) = \\epsilon_{1\/2}(\\omega_{d2})\\times 2\\pi\/\\omega_{\\mathrm{GCD}}$ over $[-\\pi, \\pi]$.\\\\\n\n\nIn \\cref{Figure_XGate_Floquet_Qubits}(c), we plot the quasiphase spectra associated with the Hamiltonian \\cref{TLS_FLoquet_with_Drive} for commensurate ratios $\\omega_{d2}\/\\omega_{d1}$ with small numerators. The different subplots illustrate the discrete quasiphase spectra for different values of the numerator $q$. The difference between the two quasiphases $\\phi^F_{1}$ and $\\phi^F_{2}$ exhibits a local minimum over all the subplots around $\\omega_{d2}\/\\omega_{d1}\\approx 0.04$ which is linked to an avoided crossing characterizing the resonance of the second drive with the Floquet qubit and thus yields the gate rate.\nThe analytical approximation obtained in the previous subsection shows good agreement when the parameter choice satisfies the RWA conditions. A precise numerical estimate of the size of this anticrossing, valid without any RWA, is obtained by increasing the maximum allowed numerator at the cost of longer simulations. \nA detailed procedure for extracting the local minimum corresponding to the avoided crossing can be found in \\cref{Appendix_I_Numerical_approach}. \nNotably, we mitigate the need for intensive numerical simulations by taking advantage of Dysolve \\citep{shillitoFastDifferentiableSimulation2020a}, a recent semi-analytic solver capturing the effects of oscillatory terms in the system Hamiltonian and whose performances are discussed in \\cref{Appendix_I_Numerical_approach}.\n\nThe validity of the approach presented here goes beyond the two-level approximation used in this work and can easily be extented to, for example, the higher energy levels of Floquet qubits.\n\n\n\\section{Longitudinal Floquet qubit readout}\n\\label{Section_IV_Readout_of_floquet_states}\n\nWe now turn to the readout of the Floquet qubit. In Refs.~\\cite{huangEngineeringDynamicalSweet2020,mundadaFloquetengineeredEnhancementCoherence2020,Deng2015}, this is realized by adiabatically mapping the Floquet logical states back to the original undriven qubit states, followed by a usual dispersive qubit readout \\cite{Blais2021}. This two-step process, adiabatic mapping following by readout, leads to a longer measurement time than strictly necessary. Here, we introduce an approach to directly measure the Floquet qubit without the additional step of an adiabatic mapping. Moreover, we show how it is possible to engineer a longitudinal coupling between the Floquet qubit and a readout mode by using a modulated transversal coupling. Because of its longitudinal nature, this readout can reach a large signal-to-noise ratio (SNR) in a measurement time that is small compared to the usual dispersive readout of circuit QED \\cite{didierFastQuantumNondemolition2015}. This approach bears similarities with the stroboscopic measurements of Ref.~\\citep{eddinsStroboscopicQubitMeasurement2018} and the Kerr-cat qubit readout of Ref.~\\citep{grimmKerrCatQubitStabilization2020}. \n\n\n\\subsection{Engineered longitudinal coupling}\n\\label{Section_IV_Subsection_I_Derivation_of_the_readout}\n\nOur approach is based on the usual capacitive, or transversal, coupling between a laboratory-frame qubit and a readout cavity. In the laboratory frame, the Hamiltonian reads\n\\begin{equation}\n \\label{TLS_Readout_Theorical_Hamiltonian}\n H_\\mathrm{lab}(t) = \\frac{\\omega_0}{2}\\sigma_z+\\varepsilon_{d1}\\cos(\\omega_1t)\\sigma_x+\\omega_r\\hat{a}^\\dag\\hat{a}+g(t)(\\hat{a}+\\hat{a}^\\dag)\\sigma_x,\n\\end{equation}\nwhere we have added to the driven qubit (first two terms) a cavity of frequency $\\omega_r$ and annihilation operator $\\hat a$ coupled to the qubit with a strength $g(t)$ which we allow to be time-dependent. In the regime where the detuning between the drive and the qubit $\\Delta$ is small compared to the drive amplitude $\\varepsilon_{d1}$, the laboratory frame $\\sigma_x$ acts as $\\sigma_z$ on the Floquet qubit corresponding to a longitudinal coupling to the cavity mode.\n\nTo make this more apparent, we move to the interaction frame defined by the transformation\n\\begin{equation}\n\\label{TLS_Readout_Propagator}\n U(t;0) = e^{-i\\omega_r t \\hat{a}^\\dag\\hat{a} }\n \\sum_{j\\in\\{0,1\\}} \\ket{\\phi_j(t)}\\bra{\\phi_j(0)}e^{-i\\epsilon_j t},\n\\end{equation}\n\\noindent where, in the limit $\\Delta\/\\varepsilon_{d1} \\ll 1$, the interaction-picture Hamiltonian takes the form\n\\begin{equation}\n \\label{TLS_Readout_Interaction_Picture}\n H_\\mathrm{int}(t) \\approx g(t)\\cos(\\omega_{0}t)(\\hat{a}e^{i\\omega_rt}+\\hat{a}^\\dag e^{-i\\omega_rt})\\sigma_z^{F}(0).\n\\end{equation}\nHere, we have introduced the interaction-picture Pauli matrices $\\sigma_k^{F}(t)$ with $k=x,y,z$ acting on the basis of the Floquet modes $\\left\\{\\ket{\\phi_0(t)},\\ket{\\phi_1(t)}\\right\\}$. Choosing the time-dependent coupling to be of the form $g(t) = \\tilde g \\cos(\\omega_m t)$ with a modulation frequency $\\omega_m = \\omega_r-\\omega_0$ (and\/or $\\omega_r+\\omega_0$) yields the longitudinal coupling Hamiltonian \\cite{didierFastQuantumNondemolition2015}\n\\begin{equation}\n\\label{TLS_Readout_Interaction_Picture_Modulated}\n H_\\mathrm{int}(t) \\approx \\frac{\\tilde{g}}{2} (\\hat{a}+\\hat{a}^\\dag)\\sigma_z^{F}(0).\n\\end{equation}\nAs discussed in Ref.~\\cite{didierFastQuantumNondemolition2015}, evolution under this Hamiltonian leads to an optimal separation of the cavity pointer states where the initial cavity vacuum state is displaced 180 degrees out of phase depending on the state of the qubit.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/Final_Fig\/Fig2_0825.pdf}\n \\caption{a) Pointer-state separation $D(t)$ as a function of time for longitudinal Floquet readout (full blue line), dispersive readout without state mapping (full green line), dispersive readout with the necessary state mapping with a ramp time of $T_\\mathrm{map}=30~\\text{ns}$ (dashed green line). b) Pointer state separation $D(t)$ over the steady-state separation $D(\\infty) = \\tilde g \/\\kappa$ as obtained from numerical integration under \\cref{TLS_Readout_Theorical_Hamiltonian} as a function of time and for different tilt angles $\\Delta\/\\varepsilon_{d1}$. As expected, for small $\\Delta\/\\varepsilon_{d1}$ the pointer states follow the ideal longitudinal dynamics expected from \\cref{TLS_Readout_Interaction_Picture_Modulated}. c) Pointer state separation $D(t)$ over the steady-state separation $D(\\infty)$ as found from numerical simulation of the system dynamics under the Hamiltonian of \\cref{Eq:3KNO}. Here, $D(\\infty)$ is numerically evaluated at long times and at small $\\Delta\/\\varepsilon_{d1}$, corresponding to the average value of the bottom-right corner in panel (c). As in panel (b), the pointer state dynamics follow the expected behavior for small with $\\Delta\/\\varepsilon_{d1}$. The parameters used in panel (c) are $\\{ \\bm{\\omega}_a, \\bm{\\omega}_b, \\bm{\\omega}_c\\}\/2\\pi = \\{8.2, 5.2,7.78\\}~\\text{GHz}$, $\\{\\bm{\\alpha}_b\/2,\\bm{\\alpha}_c\/2\\}\/2\\pi =\\{-0.17,0.4\\}~\\text{GHz}$, $\\{g_a,g_b\\}\/2\\pi = \\{0.2,0.2\\}~\\text{GHz}$, $\\tilde{\\epsilon}_{d1}\/2\\pi = 0.7~\\text{GHz}$ and $\\kappa\/2\\pi = 0.05~\\text{GHz}$.\n }\n \\label{Figure_Pointer_state_separation}\n\\end{figure}\n\nTo compare this Floquet longitudinal readout to the approach based on an adiabatic map followed by a dispersive readout of Refs.~\\cite{huangEngineeringDynamicalSweet2020,mundadaFloquetengineeredEnhancementCoherence2020,Deng2015}, we show in \\cref{Figure_Pointer_state_separation}(a) the measurement pointer state separation \\cite{Blais2021} $D(t)=\\left|\\langle \\hat{a}\\rangle_0(t) -\\langle \\hat{a}\\rangle_1(t)\\right|$. This quantity is an helpful proxy for the signal-to-noise ratio (SNR) assuming a unit measurement-chain efficiency, as obtained from the expression \\citep{bultinkGeneralMethodExtracting2018}\n\\begin{equation}\n \\label{General_SNR_optimal}\n \\text{SNR}(T) = \\sqrt{2\\kappa\\int_0^TD(t)^2 dt }.\n\\end{equation}\nIn this expression, $T$ is the measurement time, $\\kappa$ the decay rate of the cavity. For longitudinal coupling, this separation takes the simple form \\cite{didierFastQuantumNondemolition2015}\n\\begin{equation}\n \\label{TLS_readout_Pointer_state_displacement}\n D(t) = \\frac{\\tilde{g}}{\\kappa} \\left( 1 - e^{-\\kappa t\/2}\\right).\n\\end{equation}\n\nThe full blue lines in \\cref{Figure_Pointer_state_separation} correspond to longitudinal readout while the green lines to dispersive readout for which an expression equivalent to \\cref{TLS_readout_Pointer_state_displacement} can be obtained \\cite{didierFastQuantumNondemolition2015}. Comparing the full blue and full green lines, we see that longitudinal readout leads to much faster separation of the pointer states than dispersive readout even when ignoring the adiabatic mapping stage. When taking into account the required mapping stage (dashed green line), the advantage of the longitudinal approach over dispersive becomes even clearer. Here, we have used a mapping time of $T_\\mathrm{map}=30~\\text{ns}$ as in Ref.~\\cite{huangEngineeringDynamicalSweet2020}. Finally, as a reference, the dashed blue line corresponds to a situation where the mapping stage is followed by a longitudinal readout. Although this would lead to a faster separation of the pointer state at short times as compared to the standard dispersive readout, we see that the main gain in the longitudinal Floquet readout introduced here comes from the fact that mapping to the laboratory frame qubit is no longer required.\n\nAs a further verification, \\cref{Figure_Pointer_state_separation}(b) shows the pointer state separation $D(t)$ as obtained from numerical integration of the system dynamics under the laboratory-frame Hamiltonian in \\cref{TLS_Readout_Theorical_Hamiltonian} as a function of time and for different ratios $\\Delta\/\\varepsilon_{d1}$. In the laboratory frame, we take the modulated coupling to be of the form $g(t) = \\tilde{g} [\\cos(\\omega_r t-\\omega_0 t)+\\cos(\\omega_r t + \\omega_0 t)]$. In each simulation, the initial state of the cavity is chosen to be vacuum and the Floquet qubit state $\\ket{\\phi_0(0)}$ or $\\ket{\\phi_1(0)}$. For ratios $\\Delta\/\\varepsilon_{d1}<0.01$ (horizontal green dashed line), we find the expected exponential increase up to the steady-states $D(\\infty)$ in agreement with the analytical result of \\cref{TLS_readout_Pointer_state_displacement} shown in panel (a). On the other hand and as expected from the discussion below \\cref{TLS_Readout_Theorical_Hamiltonian}, when the Floquet qubit is too far away from resonance $\\Delta\/\\varepsilon_{d1}>0.1$ (horizontal red dashed line), the separation between the pointer states does not follow the trajectory predicted by \\cref{TLS_readout_Pointer_state_displacement} and the readout is suboptimal.\n\n\n\\subsection{Superconducting circuit implementation}\n\\label{Section_IV_Subsection_III_Circuit_implementation}\n\nA possible realization of this longitudinal Floquet readout with superconducting quantum circuits is illustrated in \\cref{Figure_Appendix_B_CQED}. Here, a transmon qubit ($\\hat b$) is interaction with a readout cavity ($\\hat a$) via a flux-tunable coupler ($\\hat c$). This system can be modeled as a triplet of coupled Kerr oscillators \\cite{Petrescu2021}\n\\begin{align}\n H = H_a + H_b + H_c(t) + H_g + H_d(t), \\label{Eq:3KNO}\n\\end{align}\nwhere $H_a = \\bm{\\omega}_a \\hat{\\bm{a}}^\\dagger \\hat{\\bm{a}}$ corresponds to the linear readout resonator, and $H_b = \\bm{\\omega}_b \\hat{\\bm{b}}^\\dagger \\hat{\\bm{b}} + (\\bm{\\alpha}_b\/2) \\hat{\\bm{b}}^{\\dagger 2} \\hat{\\bm{b}}^2$ to the transmon-like qubit with negative anharmonicity $\\bm{\\alpha}_b$. The coupler Hamiltonian takes the same form $H_c = \\bm{\\omega}_c(t) \\hat{\\bm{c}}^\\dagger \\hat{\\bm{c}} + (\\bm{\\alpha}_c\/2) \\hat{\\bm{c}}^{\\dagger 2} \\hat{\\bm{c}}^2$, except that it is parametrically modulated with $\\bm{\\omega}_c(t)=\\bm{\\omega}_c + \\delta \\bm{\\omega}_c (t)$ using a time-dependent flux. The capacitive interactions are modeled by a linear off-diagonal Hamiltonian coupling the bare modes $H_g = \\sum_{\\alpha < \\beta} g_{\\alpha \\beta} \\hat{\\bm{\\alpha}}^\\dagger \\hat{\\bm{\\beta}} + \\text{H.c.}$, where the summation runs over the mode indices $a,b,c$. As shown in \\cref{Appendix_II_Coupler_mediated_interaction}, switching to a normal-mode representation, we can eliminate these bilinear terms to obtain the desired modulated coupling $g(t)$ of \\cref{TLS_Readout_Theorical_Hamiltonian} between the normal modes corresponding to the qubit and the readout resonator. This is achieved by modulating the coupler frequency at one or both of the sidebands $\\omega_a \\pm \\omega_b$. Finally, the drive on the qubit takes the usual form $H_d(t) = -i \\varepsilon_{d1}(t) (\\hat{\\bm{b}}- \\hat{\\bm{b}}^\\dagger)$.\n\nThe coupling strength $\\tilde{g}$ depends on the three capacitive couplings, on the placement of the coupler frequency and on the amplitude of the modulation. Here, we choose this frequency to satisfy the constraint $\\bm{\\omega}_a < \\bm{\\omega}_c < \\bm{\\omega}_b$ to avoid excessive asymmetry. \\Cref{Figure_Pointer_state_separation}(c) shows the pointer state separation under the evolution generated by \\cref{Eq:3KNO} and with similar conditions and parameters to those used in panel (b). At $\\Delta\/\\varepsilon_{d1}$ small, we verify that the cavity pointer state displacement induced in the cavity by the readout of the Floquet States in the simulation of the full system \\cref{Eq:3KNO} matches that of Hamiltonian \\cref{TLS_Readout_Theorical_Hamiltonian}. Importantly, the fast separation of the pointer states at short time is clearly observed.\n\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{Figures\/Final_Fig\/Fig3_cqed_0708.pdf}\n \\caption{Possible realization of the longitudinal Floquet readout. A driven transmon qubit (green) is coupled to a readout cavity (blue) via a flux modulated coupler (gray).}\n \\label{Figure_Appendix_B_CQED}\n\\end{figure}\n\n\\section{Initialization of arbitrary Floquet states}\n\\label{Section_V_Initialization_of_arbitrary_states}\n\nIn this section, we consider the timescale needed to initialize a Floquet qubit state with high fidelity. In particular, we point out that the adiabatic state transfer protocols of Refs.~\\citep{mundadaFloquetengineeredEnhancementCoherence2020, desbuquoisControllingFloquetState2017a} are not optimal in the regime advantageous for longitudinal Floquet readout, \\emph{i.e.},\\ small $\\Delta\/\\varepsilon_{d1}$. Instead, we introduce an instantaneous ramping protocol which leads to high preparation fidelity in that regime.\n\n\\subsection{Adiabatic Regime}\n\\label{Section_V_Subsection_I_Adiabatic_regime}\n\nWe first consider the timescale needed to initialize a Floquet qubit with a given fidelity in the adiabatic regime. More precisely, we consider an initial laboratory frame state $\\alpha\\ket{0} + \\beta\\ket{1}$ and evaluate the fidelity of the Floquet-state preparation protocol by projecting on the expected resulting Floquet state $\\ket{\\psi(T_\\mathrm{ramp})} = \\alpha\\ket{\\phi_0(t)} + \\beta\\ket{\\phi_1(t)}$ after some ramp-up time $T_\\mathrm{ramp}$ of the drive amplitude. Without loss of generality, we take $\\alpha=1$ and $\\beta=0$, and compute the preparation fidelity $\\mathcal{F} = |\\braket{\\psi(T_\\text{ramp})}{\\phi_0(T_\\text{ramp})}|^2$ as a function of the ramp time $T_\\text{ramp}$ and for various ratios $\\Delta\/\\varepsilon_{d1}$ for the drive profile illustrated in \\cref{Figure_Initialization_Floquet_Qubits}(a).\n\nUsing a binary search in the range $\\left[1~\\text{ns},3000~\\text{ns}\\right]$, we extract in \\cref{Figure_Initialization_Floquet_Qubits}(c) the minimal value for $T_\\text{ramp}$ corresponding to a fidelity $\\mathcal{F}$ larger than 99\\% (plain green), 99.9\\% (hatched green) and 99.99\\% (dotted green) for each ratio $\\Delta\/\\varepsilon_{d1}$. We characterize the boundary of these empirical regions (dashed lines in log-log scale) by fitting an empirical law\n\\begin{equation}\n\\label{Initialization_Empirical_Law}\nT_\\text{ramp} \\times \\left|\\frac{\\Delta}{\\varepsilon_{d}}\\right|\\geq C_1,\n\\end{equation}\nwhere we find $C_1=18.9~\\text{ns}$ for a 99\\% fidelity, $C_1=28.4~\\text{ns}$ for 99.9\\%, and $C_1=36.4~\\text{ns}$ for 99.99\\%, respectively. Extrapolating this proportionality relation closer to resonance $\\Delta=0$, we obtain the divergence of the adiabatic ramping time already observed in the context of driven two-body quantum systems~\\citep{desbuquoisControllingFloquetState2017a}. In particular, for the small $\\Delta\/\\varepsilon_{d1}$ used in the longitudinal readout of the previous section, we find that the initialization is limited by the adiabatic lower bound $T_\\text{ramp} = C_1\/0.01 = 1.9~\\text{ms}$ for a 99\\% fidelity and $2.8~\\text{ms}$ for a 99.9\\% fidelity.\n\n\\begin{figure}[b!]\n \\centering\n \\includegraphics[width=\\columnwidth]{Figures\/Final_Fig\/Fig4_initialization_0810.pdf}\n \\caption{\n Ramp profile for a) adiabatic and b) instantaneous preparation pulses, as well as illustrative paths on the Bloch sphere. The grey area can be used to compare the timescales involved in the two subplots. c) Initialization fidelity versus title angle $\\Delta\/\\varepsilon_{d1}$ and ramp time $T_\\mathrm{ramp}$. The different highlighted areas correspond to sectors where an initialization fidelity higher than 99\\% (plain), 99.9\\% (hatched) and 99.99\\% (dotted) can be obtained in the adiabatic limit (green) and the instantaneous (blue) regimes.\n }\n \\label{Figure_Initialization_Floquet_Qubits}\n\\end{figure}\n\n\\subsection{Instantaneous regime}\n\\label{Section_V_Subsection_II_Instantaneous_regime}\n\nBecause of the long preparation time required with small $\\Delta\/\\varepsilon_{d1}$ which is optimal for the longitudinal readout of \\cref{Section_IV_Readout_of_floquet_states}, we now consider an alternative in the form of an instantaneous ramping protocol. Here, the main idea consists in preparing an initial superposition of the laboratory states $\\alpha\\ket{0} + \\beta\\ket{1}$ that matches the instantaneous eigenstate $\\ket{\\phi_0(0)}$ of the desired time-dependent Hamiltonian. An abrupt increase of the drive amplitude $\\varepsilon_{d1}(t)$, as illustrated in \\cref{Figure_Initialization_Floquet_Qubits}(b), then connects the eigenstates $\\ket{\\phi_0(0)}$ of the instantaneous Hamiltonian $H(t_0)$ and the Floquet states $\\ket{\\phi_0(t)}$ of the time-dependent Hamiltonian $H(t)$. The same idea can, in principle, be also used for the reverse mapping.\n\nComputing again the fidelity of the protocol as a function of the ramp time and ratio $\\Delta\/\\varepsilon_{d1}$, we find in \\cref{Figure_Initialization_Floquet_Qubits}(c) that the high-fidelity region (plain blue) is now delimited in parameter space by an upper bound in log-log scale rather than by a lower bound as was the case for the adiabatic protocol:\n\\begin{equation}\n\\label{Initialization_Empirical_Law_Instantaneous}\nT_\\text{ramp}\\times \\frac{\\Delta}{\\varepsilon_{d1}}\\leq C_2,\n\\end{equation}\nwhere $C_2=0.18~\\text{ns}$ for a 99\\% fidelity, $C_2=0.06~\\text{ns}$ for 99.9\\%, and $C_2=0.03~\\text{ns}$ for 99.99\\%, respectively. For the desired ratio $\\Delta\/\\varepsilon_{d1}=0.01$ which led to a fast longitudinal readout, this corresponds to a ramp time of up to $18~\\text{ns}$ (resp.~$6~\\text{ns}$) to reach 99\\% (resp.~99.9\\%) fidelity. For larger ratios $\\Delta\/\\varepsilon_{d1}>0.3$, we were unable to reach convergence of the simulations, which indicates that the timescales involved are less than $1~\\text{ns}$.\n\n\n\\section{Summary}\n\nWith the objective of identifying optimal gate parameters for Floquet qubits, we have shown how to define the quasiphase spectra of a static system with two distinct drives and how to extract gate parameters from such spectra. To compensate for the computational cost of this approach, we use the semi-analytic Dysolve method for the integration of the unitary dynamics in our system \\cite{shillitoFastDifferentiableSimulation2020a}. In this way, we find a tenfold improvement in simulation time as compared to the QuTiP solver \\cite{johanssonQuTiPPythonFramework2013}, opening up a path toward precise quasiphase spectra of complex quantum systems with two drives and a larger Hilbert space. Additionally, we introduce longitudinal Floquet readout which, in contrast with previous methods, does not require mapping the Floquet qubit to the laboratory-frame qubit before the measurement. This approach for readout of Floquet qubits completes the existing procedures for initialization, single-qubit gates and two-qubit gates, showing that quantum information processing on Floquet qubits is possible without having to come back to the underlying static undriven system. These results open new possibilities to further optimize gates and operations on Floquet qubits using the analytical understanding of the extended Floquet theory when only a few uncorrelated driving frequencies are involved. In future work, we will apply this framework to two-Floquet qubit gates with the objective of identifying optimal gate parameters with an approach that is free of approximations.\n\n\\section*{Acknowledgments}\nWe thank Marie Lu, Jean-Loup Ville, and Joachim Cohen for a collaboration on a related topic and Ziwen Huang, Jens Koch for stimulating discussions. This work was undertaken thanks to funding from NSERC, the Canada First Research Excellence Fund and the U.S. Army Research Office Grant No.~W911NF-18-1-0411. This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The PKS 1830-211 gravitationally lensed quasar}\\label{sec:introduction}\nPKS 1830-211 is a high redshift (z=2.5 \\citep{1999ApJ...514L..57L}) Flat Spectrum Radio Quasar (FSRQ) which has been detected in all wavelengths from radio to high-energy gamma-rays. It is a known gravitationally lensed object with two compact images of the quasar nucleus visible in the radio \\citep{1991Natur.352..132J} and optical \\citep{2005A&A...438L..37M} passbands. The Einstein ring, well visible\nat radio frequencies, comes from the imaging of the quasar jet \\citep{1992ApJ...401..461K}. \nThe quasar source is lensed by a foreground galaxy at z=0.89 \\citep{1996Natur.379..139W}. The angular size of the \nEinstein ring and separation of compact images is roughly 1 arcsecond so that it cannot be resolved with high-energy instruments such as H.E.S.S. (50 GeV-50 TeV range) or {\\it Fermi-LAT} (100 MeV-100 GeV range). PKS 1830-211 is seen as a bright, high-energy source by the {\\em Fermi-LAT} instrument and had several flaring periods during the decade of {\\em Fermi-LAT} observations. \nPKS 1830-211 is listed in the 1FHL \\citep{2013ApJS..209...34A} and the 3FHL \\citep{2017ApJS..232...18A} catalogues with a \nphoton index above 10 GeV of $3.55 \\pm 0.34$ which corresponds to the average \"low-state\" spectrum. \nNo significant curvature in the spectrum was detected. Photons up to 35 GeV, potentially detectable by H.E.S.S. have been observed by {\\em Fermi-LAT} \\citep{2017ApJS..232...18A}. \nObservations of these very high energy photons and the measurement of the very high energy tail of the spectrum would give useful constraints on EBL at redshift z=2.5.\n\nSince the components of the lens cannot be resolved at high or very high energy, the evidence for lensing was searched indirectly on the observed light curve. Due to the different travel paths, the \nlight curves of the two compact components of the lens have a relative time delay, measured in the radio \\citep{1998ApJ...508L..51L} and microwave \\citep{2001ASPC..237..155W} pass-bands, of $26\\pm5$ days. \n\\cite{2011A&A...528L...3B} have studied the first three years of the {\\em Fermi-LAT} light curve with cepstral and autocorrelation methods. \nEvidence for a delay of $27.5\\pm1.3$ days was found with a 3 $\\sigma$ significance. The time delay between the compact images of PKS 1830-211 was also studied by the {\\em Fermi-LAT} collaboration \\citep{2015ApJ...799..143A}. They selected several flaring periods and calculated the autocorrelation function of the light curve. No significant peak was found. A possible peak of \n$\\sim 20$ days was found with a 1-day binning of the data, which could be attributed to the $\\sim 20$ days separation between two flaring events and perhaps to gravitational lensing. \n\\citet{2015ApJ...809..100B} have argued that the time delay measured by high-energy instruments could be very different than the value measured by radio telescopes. The delay measured by \n\\cite{1998ApJ...508L..51L} is obtained from the emission of the compact images. \nSince the jet of the PKS 1830-211 source is imaged close to the Einstein ring, the time difference\nbetween the intial burst and its lensed image can be much smaller if the source of high-energy emission is located inside the jet.\n\nPKS 1830-211 is monitored by {\\em Fermi-LAT} and its light curve is posted on the internet\n\\footnote{\\tiny https:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/access\/lat\/msl\\_lc\/source\/PKS\\_1830-211}\non a daily basis. \nH.E.S.S. observations of PKS 1830-211 were triggered by an alert posted by the {\\em Fermi-LAT} team on August 2, 2014 \\citep{2014ATel.6361....1K}. The flare seen by the {\\em Fermi-LAT} instrument started on July 27 and lasted $\\sim$4 days. The H.E.S.S. observations are described in Section \\ref{sec:observations} and data analysis in Section \\ref{sec:analyses}. The H.E.S.S. limits are compared to the \n{\\em Fermi-LAT} signal in Section \\ref{sec:limits} and discussed in Section \\ref{sec:conclusion}. \n\\begin{figure}\n\\centering\n\\includegraphics[height=6cm]{fig1-2.pdf}\n\\caption{$\\theta^2$ plot of PKS 1830-211 obtained with the {\\em Mono} reconstruction. The background, shown by crosses, is estimated with the ring background method.\n}\n\\label{fig:hess-obs1}\n\\end{figure} \n\n \n\n\\section{H.E.S.S. observations}\\label{sec:observations}\nThe very-high-energy (50 GeV-50 TeV range) gamma-ray observatory of the H.E.S.S. collaboration consists of\nfive Imaging Atmospheric Cherenkov Telescopes (IACTs) located in the Khomas Highland \nof Namibia ($23 ^{\\circ}$ 16' 18'' S, $16 ^{\\circ}$ 30' 1'' E), 1800 m above sea level. \nFrom January 2004 to October 2012, the array was a four-telescope instrument, with telescopes labeled CT1-4. \n Each of the telescopes, located at the corners of a square with a side length of 120 m has an effective mirror \nsurface area of 107~m$^{2}$, and is able to detect cosmic gamma-rays in the energy \nrange 0.1 -- 50 TeV .\nIn October 2012, a fifth telescope CT5, with an effective mirror surface area of 600~m$^{2}$ and an improved camera \\citep{2014NIMPA.761...46B} was installed at the center of the original square, giving access to energies below 100 GeV \\citep{2017A&A...600A..89H}. \n\nPKS 1830-211 was observed by the five telescopes of the H.E.S.S. IACT array between August 12 2014 and August 26 2014, to allow for the detection of delayed flares with time delays ranging from 20 to 27 days. \nThe observations were taken at an average zenith angle of 12 degrees. \n\n\\section{Data analyses}\\label{sec:analyses}\n This paper is based on a sample of 12.4 hours of high quality data. Data selection cuts have been described in\n \\citep{2017A&A...600A..89H}.\n Data were next analyzed with the Model analysis \\citep{2009APh....32..231D} and cross-checked with the ImPACT analysis \\citep{2014APh....56...26P}, the two methods giving compatible results. The two analyses use different calibration chains. \n With both reconstruction chains, data of CT5 were analyzed either alone ({\\em Mono} reconstruction) or combined with the CT1-4 data ({\\em Combined} reconstruction). The {\\em Mono} reconstruction has an energy threshold of 67 GeV. \nThe {\\em Combined} reconstruction has a higher threshold of 144 GeV,\n but a larger effective area. \n\nA point source is searched at the location of PKS 1830-211. \nFig. \\ref{fig:hess-obs1} shows the distribution of the squared angular distance $\\theta^2$ of candidate photons from the target position. This distribution, obtained in the {\\em Mono} analysis, is compared to \nthe background from hadrons mis-identified as photons. The background is calculated with the {\\em ring background} method \\citep{2007A&A...466.1219B}, other methods giving similar results. \n\nTable \\ref{tab:results} summarizes the number of candidate photons in the signal region, the expected background and the significance of the excess, calculated with Li and Ma formula 17 \\citep{1983ApJ...272..317L}. \n\\begin{table}\n\\caption{Analysis results of observations of PKS 1830-211 by H.E.S.S.}\n\\label{tab:results}\n\\centering\n\\begin{tabular}{c c c c}\n\\hline \\hline\nReconstruction& $N_\\mathrm{ON}$ & $N_\\mathrm{background}$ & significance ($\\sigma$) \\\\\n\\hline\n{\\em Mono} & 1641 & 1649.2 & -0.2 \\\\\n{\\em Combined} & 935 & 954.4 & -0.6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nNo significant excess of photons over background is seen by H.E.S.S. at the position of PKS 1830-211. A similar search using the {\\em Combined} analysis also gives a negative result.\n\nBecause of the very soft spectrum measured by {\\em Fermi LAT} in the low state, PKS~1830-211 has a chance of being detectable by H.E.S.S. only during flares. \nThe delayed flare lasts only less than about 4 days, however, due to the uncertainties on the date of the flare, it could have happened at any time \nbetween August 17 = MJD 56886 (time delay of 20 days) and August 24 = MJD 56893 (radio time delay of 27 days) as explained in Section \\ref{sec:introduction}. \nFig. \\ref{fig:hess-obs2b} shows the evolution over time of significance, binned by 28-minute runs.\nNo significant daily photon excess was detected during the H.E.S.S. observation period. \n\n\\begin{figure}\n\\centering\n\\includegraphics[height=4.5cm]{fig2-2.pdf}\n\\caption{\nSignificance of the H.E.S.S. signal versus date, obtained with the {\\em Mono} analysis. The red arrow shows the expected date of the delayed flare for a lensing time delay \nof 27 days. \n}\n\\label{fig:hess-obs2b}\n\\end{figure} \n\n\n\n\\section{Flux upper limits and comparison to the {\\em Fermi-LAT} spectra}\\label{sec:limits}\nThe non-detection by H.E.S.S. \ntranslates into 99\\% confidence level (C.L.) upper limits on the average very-high-energy flux of PKS 1830-211 during H.E.S.S. observations. These upper limits are shown in Fig. \\ref{fig:hess-ul}. Red (resp. blue) arrows \nshow the limits obtained from the {\\em Mono} (resp. {\\em Combined}) analysis and the corresponding solid lines show the effect of deabsorption using the Extragalactic Background Light (EBL) model of \\citet{2012MNRAS.422.3189G}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{fig3-1.pdf}\n\\caption{99\\% C.L. upper limits (arrows) on the PKS 1830-211 flux between 67 GeV and 1 TeV for the August 2014 H.E.S.S. observations. A constant photon index of -3 was assumed. The solid lines show the effect of EBL deabsorption, assuming the EBL model of \\citet{2012MNRAS.422.3189G}. \n}\n\\label{fig:hess-ul}\n\\end{figure} \n\nH.E.S.S. upper limits are compared to {\\em Fermi-LAT} GeV spectra on Fig \\ref{fig:hess-fermi-comp}. \nThe {\\em Fermi-LAT} observations have been analyzed with Fermi Science Tools v10-r0p5\nand Pass8 data, in the Enrico framework \\citep{2013arXiv1307.4534S}. The spectral data from 26-30 July 2014 (flare) are well \ndescribed by a power-law spectrum with an index of $n_{flare}= -2.36 \\pm 0.17$ for photon energies $> 1$ GeV. The relatively high low-energy cut was used to avoid contamination from the Galactic plane. \n The flare spectrum is much harder than the spectrum measured in the low state of PKS 1830-211, but $n_{flare}$ is compatible with the photon indices of previous flare spectra, as measured by {\\em Fermi-LAT} \\citep{2015ApJ...799..143A}. The \nspectrum of PKS 1830-211 obtained from the {\\em Fermi-LAT} observations within the H.E.S.S. observation window is well described by a power-law with an index of $n_{low}=-2.97 \\pm 0.44$ above 1 GeV. The value of $n_{low}$ is compatible with the value published in the 3FHL catalogue.\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{fig4-2-mod2.pdf}\n\\caption{Comparison of H.E.S.S. flux 99 \\%C.L. upper limits (red solid line: {\\em Mono} analysis, long dashed line: {\\em Combined} analysis) to the measured\nspectra in the GeV region obtained with the {\\em Fermi-LAT} data. The horizontal lines show the spectral resolution of the analyses. The lower blue butterfly is the GeV spectrum of PKS 1830-211 during H.E.S.S. observations. The upper blue butterfly is the \ncorresponding spectrum during the July 2014 flare. The absorption of the July flare spectrum by EBL is calculated with models by \\citet{2012MNRAS.422.3189G} (black dash-dotted line),\\citet{2010ApJ...712..238F} (dotted line) and \\citet{2008A&A...487..837F} (black solid line). \n}\n\\label{fig:hess-fermi-comp}\n\\end{figure} \n\n\nA proper comparison between H.E.S.S. upper limits and the Fermi signal has to take into account the effect of the absorption of the flux of PKS 1830-211 by the \n EBL and the difference between the flare duration and H.E.S.S. exposure. Since no significant curvature of the spectrum was measured bt the Fermi-LAT collaboration, the unabsorbed spectrum was modeled by a powerlaw.\n The effect of light absorption by EBL from PKS 1830-211 has been estimated with the models of \n \\citet{2012MNRAS.422.3189G} (black dash-dotted line), \\citet{2010ApJ...712..238F} (dotted line) and \\citet{2008A&A...487..837F} (black solid line). Fig. \\ref{fig:hess-fermi-comp} shows that there \n is a substantial difference between the predictions of these models for a source at redshift 2.5 such as PKS 1830-211. \n Note that EBL absorption could also be affected by the lens environment.\n Light from lensed AGN is expected to be more absorbed than average, due to the presence of galaxies along the line of sight. Indeed, absorption from the intervening galaxy has been detected by \\cite{2005A&A...438..121D} in the X-ray spectrum of PKS 1830-211. However \\cite{2014ApJ...790..147B} and \\cite{2016A&A...595A..14B} have argued that gravitational lensing \n could help gamma-rays from a distant source avoiding excess absorption. \n The {\\em Fermi-LAT} flare spectrum from Fig \\ref{fig:hess-fermi-comp} is a 4-night average while the H.E.S.S. exposure amounts to 10 nights of data taking. The steady source upper limits from Fig \\ref{fig:hess-ul} are thus a factor $\\sim\\sqrt{10\/4}$ too constraining, which is corrected for in Fig \\ref{fig:hess-fermi-comp}. \n \n \n\\section{Conclusion}\\label{sec:conclusion} \n \nNo significant delayed flare from PKS 1830-211 was detected by either H.E.S.S. or {\\em Fermi-LAT}. The flare did not repeat or was too faint to be detected.\nFig \\ref{fig:hess-fermi-comp} shows however that the detection of a strong flare would have been possible \nclose to the {\\em Mono} analysis energy threshold if the level of EBL absorption was at or below the absorption predicted\nby the model of \\citet{2008A&A...487..837F}.\nDue to its lensed nature, observation of flaring event of PKS 1830-211 in the TeV passband could be useful to constrain EBL models at redshift as large as 2.5. \nThe detection of the lensing time delay in future very high energy observations\n\n would help pinpoint the spatial origin of the high-energy emission \\citep{2015ApJ...809..100B}.\n It would also permit more exotic applications such as constraining photon mass \\citep{2017ApJ...850..102G} or testing Lorentz Invariance Violation \\citep{2009MNRAS.396..946B}.\n\n\n\n\\section*{Acknowledgments}\n{\\small\nThe support of the Namibian authorities and of the University of Namibia in facilitating the construction\n and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), \n the Max Planck Society, the German Research Foundation (DFG), the Helmholtz Association, the Alexander von Humboldt Foundation,\nthe French Ministry of Higher Education, Research and Innovation, the Centre National de la Recherche Scientifique (CNRS\/IN2P3 and CNRS\/INSU), \nthe Commissariat \\`a l' Energie Atomique et aux Energies Alternatives (CEA), \nthe U.K. Science and Technology Facilities Council (STFC), the Knut and Alice Wallenberg Foundation, the National Science Centre, \n Poland grant no. 2016\/22\/M\/ST9\/00382, the South African Department of Science and Technology and National \nResearch Foundation, the University of Namibia, the National Commission on Research, Science \\& Technology of Namibia (NCRST), the Austrian Federal Ministry of Education, Science and Research \nand the Austrian Science Fund (FWF), the Australian Research Council (ARC), the Japan Society for the Promotion of Science and by the University of Amsterdam. \nWe appreciate the excellent work of the technical\n support staff in Berlin, Zeuthen, Heidelberg, Palaiseau, Paris, Saclay, T\\\"uebingen and in Namibia in the construction and operation of the equipment. This work benefited from services provided by the\nH.E.S.S. Virtual Organisation, supported by the national resource providers of the EGI Federation\n}\n\n\n\\bibpunct{(}{)}{;}{a}{}{,}\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgdkg b/data_all_eng_slimpj/shuffled/split2/finalzzgdkg new file mode 100644 index 0000000000000000000000000000000000000000..2281ac98b2d011e142bdd6e08038110b279b16d4 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgdkg @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\abrev{Over the last two decades,\nmany new challenging problems in the field of computational partial differential equations (PDEs) have been motivated by\nthe rapidly developing area of uncertainty quantification.\nEfficient numerical solution of PDE problems with parametric or uncertain inputs\nis one of these challenges.\nSeveral numerical methods have been developed and analyzed in this context.\nIn particular, the stochastic Galerkin finite element method\n(SGFEM)~\\cite{ghanemspanos91, MR2084236, MR3202242} has emerged as\nan efficient and rapidly convergent alternative to traditional Monte Carlo sampling.\nHowever, the implementation of the SGFEM requires the solution of huge (although highly-structured) linear systems.\nFor realistic applications, such linear systems can only be solved using iterative methods\nequipped with effective, bespoke preconditioners.}\n\\abrev{To that end, a range of linear algebra techniques have been employed including the}\nmultigrid and multilevel methods~\\cite{ghanem03}, \\cite{elmanfurnival07}, \\cite{vandewalle07},\n\\cite{brezina14},~\\cite{lee16},~\\cite{pultarova17},~\\cite{elmansu18},\ndomain decomposition methods~\\cite{ghanem09}, \\cite{loisel14}, \\cite{waad14},\nhierarchical methods~\\cite{PellissettiG_00_ISS}, \\cite{ghanem14}, \\cite{ghanem14a},\nas well as Krylov methods~\\cite{ghanemkruger96}, \\cite{PellissettiG_00_ISS},\n\\cite{elmanpowell2009}, \\cite{jin09}, \\cite{vandewalle10}, \\cite{Ullmann10}.\n\nIn this work, we focus on Krylov methods; in particular, for\n\\abrev{a parametric elliptic PDE problem}\nwith solution approximated by the SGFEM, we\n\\abrev{employ} iterative methods of Krylov subspace type for which\nwe design and analyze a suitable class of preconditioners. \n\nAs a model problem we consider the parametric steady-state diffusion equation\n\\abrev{subject to homogeneous Dirichlet boundary conditions}: \n\\begin{equation}\n\\begin{aligned}-\\nabla\\cdot\\left(a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla u({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\right) & =f({\\bm{x}}}\\def \\y{{\\bm{y}}), & & {\\bm{x}}}\\def \\y{{\\bm{y}}\\in \\Omega,\\ \\y\\in\\bGamma,\\\\\nu({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) & =0, & & {\\bm{x}}}\\def \\y{{\\bm{y}}\\in\\partial \\Omega,\\ \\y\\in\\bGamma,\n\\end{aligned}\n\\label{eq:strong:form}\n\\end{equation}\nwhere\n$\\Omega\\subset\\mathbb{R}^{d}$ ($d=1,2,3$) is a bounded (spatial)\ndomain with Lipschitz polygonal boundary $\\partial\\Omega$, \n$f\\in H^{-1}(\\Omega)$, and\n$\\bGamma := \\prod_{m=1}^{\\abrev{\\infty}}\\Gamma_m$ is the parameter domain with bounded\nintervals $\\Gamma_m\\subset\\RR$, $m\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}$.\nWe also note that $\\nabla$ denotes the spatial gradient operator $\\nabla_{{\\bm{x}}}\\def \\y{{\\bm{y}}}$.\n\nThe SGFEM applied to problem~\\Refx{eq:strong:form} generates approximations in tensor product spaces $X \\otimes S$,\nwhere $X$ is a finite element space associated with the physical domain~$\\Omega$, and\n$S$ is a space of multivariate polynomials over a finite-dimensional manifold\n\\abrev{$\\bGamma_M \\subset \\bGamma$\n(here, $M \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}$ refers to the number active parameters in the SGFEM approximation)}.\nA typical SGFEM discretization of problem \\Refx{eq:strong:form}\nyields a structured linear system $A{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}=\\f$ \\abrev{with the coefficient matrix}\n\\[\n A=\n \\begin{bmatrix}\n A_{11} & A_{12} &\\cdots & A_{1\\Ny}\\\\\n A_{21} & A_{22} &\\cdots & A_{2\\Ny}\\\\\n \\vdots& \\vdots & \\ddots & \\vdots\\\\\n A_{\\Ny1}& A_{\\Ny2}& \\cdots & A_{\\Ny \\Ny}\n \\end{bmatrix},\n\\]\n\\abrev{where the}\nblocks $A_{ij}$ are $N_\\x}\\def\\Ny{N_\\y \\times N_\\x}\\def\\Ny{N_\\y$ matrices\n\\abrev{with $N_\\x}\\def\\Ny{N_\\y = \\dim(X)$ and $\\Ny = \\dim(S)$}.\n\\abrev{If the parametric coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ in~\\Refx{eq:strong:form} is represented via\na (truncated or infinite) series expansion that is \\emph{affine} in parameters, e.g.,\n\\[\n a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) = a_0({\\bm{x}}}\\def \\y{{\\bm{y}})+\\sum_{m=1}^{\\infty} a_m({\\bm{x}}}\\def \\y{{\\bm{y}}) y_m,\\quad\n {\\bm{x}}}\\def \\y{{\\bm{y}} \\in \\Omega,\\ \\y \\in \\bGamma,\n\\]\nthen it is well known (see, e.g., \\cite[Chapter~9]{lord14})\nthat the system matrix $A$ can be written as}\na sum of Kronecker products:\n\\begin{equation}\n A=G_0\\otimes K_0+\\sum_{m=1}^{\\abrev{M}} G_m\\otimes K_m.\n \\label{eq:kron:sum}\n\\end{equation}\n\\abrev{Here,\n$G_m \\in \\RR^{\\Ny \\times \\Ny}$ are the (parametric) matrices built from polynomial basis functions in $S$,}\nand $K_m \\in \\RR^{N_\\x}\\def\\Ny{N_\\y \\times N_\\x}\\def\\Ny{N_\\y}$ are the (spatial) stiffness matrices\n\\abrev{associated with coefficients $a_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ in the series expansion of~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$}.\n\n\\abrev{The numerical solution of stochastic Galerkin linear systems\npresents significant challenges.\nOn the one hand,}\nit is evident from the structure of $A$ indicated above that such matrices can reach\nhuge sizes very quickly.\n\\abrev{For example, if $S$ is the space of complete polynomials of degree $\\le k$ in $M$ parameters,\nthen $\\Ny={M+k \\choose k}$}\ngrows very fast with $M$ and $k$.\nOn the other hand, in the case of affine-parametric expansion of the coefficient~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ as given above,\nthe matrix $A$ is block-sparse due to the sparsity of matrices $G_m$ in~\\Refx{eq:kron:sum}.\nThis feature, however, is not guaranteed for other parametric representations of~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$\n(see~\\S\\ref{sec:numerics:non-affine} for one example of such a representation).\nThus, the availability of effective preconditioning techniques is of paramount importance, in order to\nenable the application of the SGFEM to a range of parametric PDE problems.\n\nIn an early effort to provide an efficient solver technique for stochastic Galerkin linear systems,\nthe \\emph{mean-based preconditioner} \\abrev{was proposed by Ghanem and Kruger in~\\cite{ghanemkruger96}\nand subsequently analyzed by Powell and Elman in~\\cite{elmanpowell2009}}.\nIn the notation employed above, this preconditioner is defined as\n\\begin{equation}\n \\label{eq:mb}\n P_0 := G_{0}\\otimes K_{0}.\n\\end{equation}\nIt has been shown that \nunder certain standard boundedness conditions on the diffusion coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$,\nthe performance of \\abrev{the conjugate gradient (CG) method} equipped\nwith the preconditioner $P_0$ is independent of the \\abrev{dimensions $N_\\x}\\def\\Ny{N_\\y$ and $\\Ny$ of the\nunderlying spatial and parametric approximation spaces}.\nThis is essentially due to the mean component $a_0({\\bm{x}}}\\def \\y{{\\bm{y}})$\nstrongly dominating other terms in the expansion of~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$.\nWhen this is not the case, the performance deteriorates and dependence on \\abrev{$\\Ny$} may arise\n\\abrev{(e.g., via dependence on the number $M$ of active parameters and\/or the polynomial degree $k$\nof parametric approximations)}.\n\nAn alternative \\abrev{approach} that takes into account contributions from\n\\abrev{all component matrices in~\\Refx{eq:kron:sum}}\nwas suggested by Ullmann in~\\cite{Ullmann10}.\nThis preconditioner, which we denote by $P_\\otimes$, is also defined as a Kronecker product:\n\\begin{equation}\n \\label{eq:kron}\n P_{\\otimes} := G\\otimes K_{0},\n\\end{equation}\nwhere the matrix \\abrev{$G \\in \\RR^{\\Ny \\times \\Ny}$} is constructed in order to minimize the Frobenius\nnorm of the difference between the system matrix and the preconditioner, i.e.,\n$$G:=\\arg\\min_{Q}\\norm{A - Q\\otimes K_{0}}{F}.$$\nWhile the eigenvalue bounds for the preconditioned system derived in~\\cite{Ullmann10}\nare not sharp\n\\abrev{and one cannot generally expect the iteration counts of the $P_{\\otimes}$-preconditioned CG\nto be independent of the dimension $\\Ny$ of the parametric approximation space~$S$, the\n\\emph{Kronecker product preconditioner}} $P_\\otimes$ outperforms the mean-based preconditioner $P_0$,\n\\abrev{particularly in the case of the approximation space $S$ comprising polynomials\nof large degree $k$.}\n\n\\rev{\nA preconditioning strategy that exploits the hierarchical structure of stochastic Galerkin matrices\nwas proposed by Soused{\\' i}k and Ghanem in~\\cite{ghanem14}.\nIn this strategy, the inverses of submatrices\nare \\emph{approximated} by inverses of their diagonal blocks\nin the action of a hierarchical symmetric block Gauss--Seidel preconditioner.\nThis preconditioner is further enhanced in~\\cite{ghanem14} by performing the matrix-vector multiplications in its action\nusing only a subset of component matrices in~\\Refx{eq:kron:sum} selected according to the size of the norm of stiffness matrices $K_m$.\nIn particular, a monotonic decay of the norms of $K_m$ effectively results in \\emph{truncating} the sum in~\\Refx{eq:kron:sum}.\nExtensive numerical experiments\nfor a model problem with truncated lognormal diffusion coefficient have demonstrated the effectiveness and competitiveness\nof this combined preconditioning strategy (called the \\emph{truncated hierarchical preconditioning})\nin terms of both convergence of iterations and computational cost.\nThe results of these experiments have also shown that\ntruncated versions of the non-hierarchical symmetric block Gauss--Seidel preconditioner and\nthe approximate hierarchical Gauss--Seidel preconditioner\nare largely comparable in terms of convergence of~iterations.}\n\nIn this paper, we \\rev{study} a preconditioning technique based on\ntruncating the \\abrev{sum of Kronecker products in~\\Refx{eq:kron:sum} as follows:}\n\\[\n P_r := G_0\\otimes K_0 + \\sum_{m=1}^r G_m\\otimes K_m.\n\\]\nWe will refer to this class of solvers as \\emph{truncation preconditioners.}\n\\abrev{While it includes the mean-based preconditioner as a special case,\nby capturing additional significant components of the stochastic Galerkin matrix $A$ \\rev{one aims}\nto improve the preconditioner's performance\nretaining its optimality with respect to the discretization parameters (i.e., $N_\\x}\\def\\Ny{N_\\y$, $M$ and $k$).}\n\\rev{\nTruncation preconditioners of this type were considered in~\\cite[Section~4.2]{KubinovaP_20_BPS}\nand analyzed therein for two extreme cases, namely $r = 0$ and~$r = M-1$.\nWhile the preconditioning matrix $P_r$ has a block-diagonal structure in the case of the \\emph{tensor-product} polynomial space~$S$\n(with appropriately ordered basis functions; see~\\cite{KubinovaP_20_BPS}),\nthis property, in general, does not hold for the matrix $P_r$ if $S$ is the space of \\emph{complete} polynomials.\nTherefore, in the latter case, considered in the present paper, a practical implementation of the truncation preconditioner $P_r$\nrequires an additional technique to ensure efficient application of the action of $P_r^{-1}$ on a given vector.\nTo this end, similarly to the strategy in~\\cite{ghanem14},\nwe propose to use a symmetric block Gauss--Seidel approximation $\\tilde P_r$ of the truncation preconditioner $P_r$.\n}\n\n\\rev{\nFocusing on the case of affine-parametric representation of~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$,\nour main goal in this paper is to perform spectral analysis of the preconditioned matrices and\nestablish optimality of the preconditioners $P_r$ and $\\tilde P_r$\nwith respect to all discretization parameters.\nBy doing this we fill a gap in the theoretical analysis of preconditioning techniques\nfor the numerical solution of stochastic Galerkin linear systems.\n}\n\nThe paper is structured as follows. In the next section we present a\ndetailed problem formulation, including\n\\abrev{specific assumptions on the parametric diffusion coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$},\nthe variational formulation of~\\Refx{eq:strong:form}, as well as\nthe definitions and properties of the matrices $A_{ij}$, $G_m$, $K_m$.\nIn section~\\ref{sec:preconditioners}, we introduce \\rev{and analyze} a class of\npreconditioners $P_r$ based on truncation of the series representation\nof the parametric diffusion coefficient~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$.\nA symmetric block Gauss--Seidel\napproximation to $P_r$ is introduced and analyzed in section~\\ref{sec:modified:precond}.\nSection~\\ref{sec:numerics} includes a range of numerical results, with\nconclusions and potential extensions summarized in section~\\ref{sec:conclusions}.\n\n\n\\section{Problem formulation} \\label{sec:problem}\n\nIn this section we outline some standard\nbackground results and assumptions and introduce the variational formulation\nrequired for the numerical solution of \\Refx{eq:strong:form} via the SGFEM.\n\n\n\n\\subsection{Functional analytic framework} \\label{sec:func:frame}\n\n\n\\abrev{Let $y_m\\in\\Gamma_m$ be} the images of independent bounded random variables with cumulative\ndensity function $\\pi_{m}(y_{m})$ and probability density function\n$p_{m}(y_{m})=\\dd\\pi_{m}(y_{m})\/\\dd y_{m}$. The\njoint cumulative density function and the joint probability density\nfunction of the associated multivariable random variable $\\y\\in\\bGamma$ are defined,\nrespectively, as\n\\[\\pi(\\bm{y}):=\\prod_{m=1}^{\\infty}\\pi_{m}(y_{m})\\quad \\text{and}\\quad\n p(\\bm{y}):=\\prod_{m=1}^{\\infty}p_{m}(y_{m}).\n\\]\nWithout loss of generality, we assume for all $m\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}$ that $\\Gamma_{m}:=[-1,1]$\\abrev{; additionally, we assume} that $p_{m}$ is even. Note that each $\\pi_{m}$ is a probability measure\non $(\\Gamma_{m},\\mathcal{B}(\\Gamma_{m}))$, where $\\mathcal{B}(\\Gamma_{m})$\nis the Borel $\\sigma$-algebra on $\\Gamma_{m}$. Accordingly, $\\pi$\nis a probability measure on $(\\bGamma,\\mathcal{B}(\\bGamma))$, where\n$\\mathcal{B}(\\bGamma)$ is the Borel $\\sigma$-algebra on $\\bGamma$.\nThen $L_{\\pi_{m}}^{\\abrev{2}}(\\Gamma_{m})$, $L_{\\pi}^{\\abrev{2}}(\\bGamma)$\nrepresent the weighted Lebesgue spaces with associated inner products\n\\begin{align*}\n \\langle f,g\\rangle_{\\pi_{m}} & := \\int_{\\Gamma_{m}}p_{m}(y_{m})f(y_{m})g(y_{m})\\dd y_{m}, \\qquad\n f,g\\in L_{\\pi_{m}}^{2}(\\Gamma_{m}),\n \\\\[3pt]\n \\langle f,g\\rangle_{\\pi} & := \\int_{\\bGamma}p(\\bm{y})f(\\bm{y})g(\\bm{y})\\dd\\bm{y}, \\qquad\n f,g\\in L_{\\pi}^{2}(\\bGamma).\n\\end{align*}\nFinally, a space relevant to the\nweak formulation of problem (\\ref{eq:strong:form}) is\n$L_{\\pi}^{2}(\\bGamma;H_{0}^{1}(\\Omega)),$\nwhich is the space of strongly measurable functions\n$v:\\Omega\\times\\bGamma\\to\\mathbb{R}$ such that \n\\[\n \\norm{v}{L_{\\pi}^{2}(\\bGamma;H^1_0(\\Omega))} :=\n \\norm{\\norm{v(\\cdot,\\bm{y})}{H^1_0(\\Omega)}}{L_{\\pi}^{2}(\\bGamma)} :=\n \\bigg[\n \\int_\\bGamma\n p(\\y) \\norm{v(\\cdot,\\bm{y})}{H^1_0(\\Omega)}^2\\dd\\y\n \\bigg]^{1\/2}\n < +\\infty,\n\\]\nwhere $H_{0}^{1}(\\Omega)$ is the usual Sobolev\nspace of functions in $H^{1}(\\Omega)$ that vanish at the boundary $\\partial\\Omega$\nin the sense of traces. It is known that $L_{\\pi}^{2}(\\bGamma;H_{0}^{1}(\\Omega))$\nis isometrically isomorphic to the following tensor product Hilbert space (see \\cite[Remark C.24]{schwabgittelson2011}):\n\\begin{equation}\nV:=L_{\\pi}^{2}(\\bGamma)\\otimes H_{0}^{1}(\\Omega).\\label{eq:tensorspace}\n\\end{equation}\nWe will use the space $V$ to \\abrev{derive} the variational formulation of problem~(\\ref{eq:strong:form})\nin~\\S\\ref{sec:weak:and:Galerkin} below.\nBefore doing this, let us make some specific assumptions on the parametric diffusion coefficient~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$.\n\n\n\\subsection{The parametric diffusion coefficient} \\label{sec:diff:coeff}\n\nWe will assume that the diffusion coefficient $a$ is \\emph{affine-parametric}, i.e.,\n\\begin{equation}\na({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)=a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})+\\sum_{m=1}^{\\infty}a_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_{m},\\quad{\\bm{x}}}\\def \\y{{\\bm{y}}\\in \\Omega,\\ \\y\\in\\bGamma,\\label{eq:affine}\n\\end{equation}\nwith $a_{m}\\in L^{\\infty}(\\Omega)$, $m \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$, and with the series\nconverging uniformly in $L^{\\infty}(\\Omega)$.\nThe representation~\\Refx{eq:affine} is motivated by the Karhunen--Lo{\\`e}ve\nexpansion of a second-order random field $a$ with given mean $\\mathbb{E}[a]$\nand covariance function $\\hbox{\\rm Cov}[a]({\\bm{x}}}\\def \\y{{\\bm{y}},{\\bm{x}}}\\def \\y{{\\bm{y}}')$.\nIn this case, $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ is represented as in~(\\ref{eq:affine})\nwith $a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})=\\mathbb{E}[a]$ and $a_{m}(\\bm{x})=\\sqrt{\\lambda_{m}}\\varphi_{m}(\\bm{x})$\n($m=1,2\\ldots$), where $\\{(\\lambda_{m},\\varphi_{m})\\}_{m=1}^{\\infty}$\nare the eigenpairs of the integral operator $\\int_{\\Omega}\\hbox{\\rm Cov}[a]({\\bm{x}}}\\def \\y{{\\bm{y}},{\\bm{x}}}\\def \\y{{\\bm{y}}')\\varphi({\\bm{x}}}\\def \\y{{\\bm{y}}')\\dd{\\bm{x}}}\\def \\y{{\\bm{y}}'$\nsuch that $\\lambda_{1}\\geq\\lambda_{2}\\geq\\ldots>0$, and $y_{m}$\n($m=1,2\\ldots$) are the images of pairwise uncorrelated random variables\nwith zero mean and unit variance (see, e.g.,~\\cite[\\S2.3]{ghanemspanos91}).\n\nAs is the case for deterministic diffusion problems, the standard\nconditions for well-posedness of the weak formulation of problem~\\Refx{eq:strong:form} are the positivity and boundedness of the diffusion\ncoefficient~$a$.\nIn order to ensure that the coefficient~$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ given by~(\\ref{eq:affine})\nsatisfies these conditions,\nwe assume that (cf.~\\cite[Proposition~2.22]{schwabgittelson2011})\n\\begin{equation}\na_{0}^{\\min}\\leq a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})\\leq a_{0}^{\\max}\\quad\\text{a.e. in }\\Omega\\label{eq:assumption:on:a0}\n\\end{equation}\nfor some constants $0 < a_0^{\\min} \\le a_0^{\\max} < \\infty$\nand that \n\\begin{equation}\n \\tau:=\\frac{1}{a_{0}^{\\min}} \\bigg\\| \\sum_{m=1}^{\\infty}|a_{m}| \\bigg\\|_{\\infty}<1,\n \\label{eq:assumption:on:ai-1}\n \\end{equation}\nwhere $\\|\\cdot\\|_{\\infty}$ denotes the norm in $L^{\\infty}(\\Omega)$.\nIn this case, an elementary calculation shows~that\n\\begin{equation}\n0M},\\quad\n k \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0,\\ \\ M \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I},\n\\]\nwhere $|{\\bm{\\alpha}}|=\\sum_{m\\in\\spp{\\bm{\\alpha}}}\\alpha_{m}$.\nWe denote the corresponding finite-dimensional subspaces of $L_{\\pi}^{2}(\\bGamma)$~as\n\\begin{equation}\n \\label{eq:skm}\n S_k^M:=\\spn\\seq{\\psi_{{\\bm{\\alpha}}} : {\\bm{\\alpha}}\\in\\I_{k}^{M}}.\n\\end{equation}\nThus, $S_k^M$ is the space of complete polynomials of degree $\\le\\,k$ in $M$ variables;\nits dimension is given by\n\\[\n \\Ny:=\\dim(S_k^M)={\\rm card\\,}{\\I_{k}^{M}}={M+k \\choose k}.\n\\]\nFurthermore, \nthere exists a bijection $\\bm{\\kappa}:\\{1,2,\\ldots,\\Ny\\}\\to\\I_{k}^{M}$, so that we can also\ndescribe $S_k^M$ via the span\n$S_k^M=\\spn\\seq{\\psi_{\\bm{\\kappa}(j)} :\\, 1\\leq j\\leq \\Ny}.$\n\nWe can now define the following finite-dimensional subspace of $V$:\n\\begin{equation}\n \\label{eq:VhkM}\n V_{hk}^M \\,{:=}\\, X_{h} {\\otimes} S_k^M \\,{=}\\,\n \\spn\\!\\big\\{\\varphi_{ij}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) \\,{:=}\\, \\phi_i({\\bm{x}}}\\def \\y{{\\bm{y}})\\psi_{\\bm{\\kappa}(j)}(\\y) :\\,\\!\n 1 \\,{\\leq}\\, i \\,{\\leq}\\, N_\\x}\\def\\Ny{N_\\y,\\ 1 \\,{\\leq}\\, j \\,{\\leq}\\, \\Ny\\big\\}.\n\\end{equation}\nThe resulting discrete formulation of \\Refx{eq:weak:form} reads: find $u_{hk}^M\\inV_{hk}^M$\nsuch that\n\\begin{equation}\n\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(u_{hk}^M,v)=\\F(v)\\quad \\forall\\, v \\in V_{hk}^M.\\label{eq:sGFEM}\n\\end{equation}\nUsing the definition of\n$V_{hk}^M$ in \\Refx{eq:VhkM} we write the Galerkin approximation $u_{hk}^M$~as\n\\begin{equation}\nu_{hk}^M({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)=\\sum_{i=1}^{N_\\x}\\def\\Ny{N_\\y}\\sum_{j=1}^{\\Ny}u_{ij}\\varphi_{ij}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y).\\label{eq:approx:sol}\n\\end{equation}\n The Galerkin projection onto the finite-dimensional space $V_{hk}^M$\n defined via the choice of finite index set $\\I_k^M$\n can be shown to correspond to a discrete weak formulation\n involving a truncation at $m=M$ of the parametric diffusion\n coefficient $a$ given in~\\Refx{eq:affine}. More precisely, if we let\n\\[\n a_M({\\bm{x}}}\\def \\y{{\\bm{y}},\\y):=a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})+\\sum_{m=1}^Ma_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_{m},\\quad{\\bm{x}}}\\def \\y{{\\bm{y}}\\in \\Omega,\\ \\y\\in\\bGamma,\n\\]\nthen the associated bilinear form\n\\[\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_M(v,w) :=\\int_{\\bGamma}p(\\y)\\int_{\\Omega}a_M({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla\n v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla w({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y\n\\]\nsatisfies (see, e.g.,~\\cite[p.~A349]{BespalovPS_14_ENA})\n\\begin{equation}\n\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_M(v,w)=\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,w)\\quad\\forall v,w\\in V_{hk}^M.\\label{eq:AM}\n\\end{equation}\n\\abrev{Using representation~(\\ref{eq:approx:sol}) of the Galerkin solution,\nand setting $v=\\varphi_{st}$ in~(\\ref{eq:sGFEM})\nfor $s=1,\\ldots,N_{{\\bm{x}}}\\def \\y{{\\bm{y}}}$ and $t=1,\\ldots,N_{\\y}$, we obtain the following\n linear system:}\n\\[\n \\sum_{i=1}^{N_\\x}\\def\\Ny{N_\\y}\\sum_{j=1}^{\\Ny} u_{ij}\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(\\varphi_{ij},\\varphi_{st})=\\F(\\varphi_{st}).\n\\]\n\\abrev{This system becomes, using \\Refx{eq:AM} and the separable form \\Refx{eq:VhkM} of\n$\\varphi_{ij}$, $\\varphi_{st}$ and of\neach term in the series expansion~(\\ref{eq:affine}) of the diffusion coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$,}\n\\[\n \\sum_{m=0}^M\\sum_{i=1}^{N_{{\\bm{x}}}\\def \\y{{\\bm{y}}}} \\sum_{j=1}^{N_{\\y}}\n u_{ij} \\int_{\\Omega} a_{m}\\nabla\\phi_{i}\\cdot\\nabla\\phi_{s} \\dd{\\bm{x}}}\\def \\y{{\\bm{y}}\n \\int_{\\bGamma} y_{m} \\psi_{\\bm{\\kappa}(j)} \\psi_{\\bm{\\kappa}(t)}p \\dd\\y =\n \\int_{\\Omega}f\\phi_{s}\\dd{\\bm{x}}}\\def \\y{{\\bm{y}}\\int_{\\bGamma}\\psi_{\\bm{\\kappa}(t)}p\\dd\\y,\n\\]\nwhere we set $y_0=1$.\nTherefore, the discrete formulation~(\\ref{eq:sGFEM}) yields a\nlinear system $A{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}=\\f$ with block structure. Specifically, the\ncoefficient matrix $A$, the solution vector ${\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}$, and the right-hand\nside vector $\\f$ are given by \n\\begin{equation}\nA=\\left[\\begin{array}{cccc}\nA_{11} & A_{12} & \\cdots & A_{1N_{\\y}}\\\\\nA_{21} & A_{22} & \\cdots & A_{2N_{\\y}}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\nA_{N_{\\y}1} & A_{N_{\\y}2} & \\cdots & A_{N_{\\y}N_{\\y}}\n\\end{array}\\right],\\quad {\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}=\\left[\\begin{array}{c}\n{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}_{1}\\\\\n{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}_{2}\\\\\n\\vdots\\\\\n{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}_{N_{\\y}}\n\\end{array}\\right], \\quad \\f=\\left[\\begin{array}{c}\n\\f_{1}\\\\\n\\f_{2}\\\\\n\\vdots\\\\\n\\f_{N_{\\y}}\n\\end{array}\\right],\n\\label{eq:A:u:f}\n\\end{equation}\nrespectively, where \n\\[\nA_{tj}=\\langle\\psi_{\\bm{\\kappa}(j)},\\psi_{\\bm{\\kappa}(t)}\\rangle_{\\pi}\\,K_{0}+\\sum_{m=1}^{M}\\langle y_{m}\\psi_{\\bm{\\kappa}(j)},\\psi_{\\bm{\\kappa}(t)}\\rangle_{\\pi}\\,K_{m},\\quad t,j=1,\\ldots,N_{\\y}\n\\]\nwith finite element (stiffness) matrices $K_{m}$, $m=0,1,\\ldots,M$,\ndefined by \n\\begin{align*}\n\\left[K_{m}\\right]_{si} & := \\int_{\\Omega}a_{m}\\nabla\\phi_{i}\\cdot\\nabla\\phi_{s}\\,\\dd{\\bm{x}}}\\def \\y{{\\bm{y}},\\quad s,i=1,\\ldots,N_{{\\bm{x}}}\\def \\y{{\\bm{y}}},\n\\\\[3pt]\n{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}_{j} & := \\left[u_{1j}\\;u_{2j}\\;\\ldots\\;u_{N_{{\\bm{x}}}\\def \\y{{\\bm{y}}}j}\\right]^{T},\\quad j=1,\\ldots,N_{\\y},\n\\end{align*}\nand \n\\[\n\\left[\\f_{t}\\right]_{s}:=\\langle1,\\psi_{\\bm{\\kappa}(t)}\\rangle_{\\pi}\\,\\int_{\\Omega}f\\phi_{s}\\,\\dd{\\bm{x}}}\\def \\y{{\\bm{y}},\\quad\n s=1,\\ldots,N_{{\\bm{x}}}\\def \\y{{\\bm{y}}},\\ t=1,\\ldots,N_{\\y}.\n\\]\nUsing Kronecker products for matrices, it is convenient to write the\ncoefficient matrix $A$ in the following \\abrev{compact} form \n\\begin{equation}\nA=G_{0}\\otimes K_{0}+\\sum_{m=1}^{M}G_{m}\\otimes K_{m},\\label{eq:decomp:A}\n\\end{equation}\nwhere \n\\begin{equation}\n\\left[G_{0}\\right]_{tj}:=\\langle\\psi_{\\bm{\\kappa}(j)},\\psi_{\\bm{\\kappa}(t)}\\rangle_{\\pi},\\quad\\left[G_{m}\\right]_{tj}:=\\langle y_{m}\\psi_{\\bm{\\kappa}(j)},\\psi_{\\bm{\\kappa}(t)}\\rangle_{\\pi}\\label{eq:G:matrices}\n\\end{equation}\nfor $m=1,\\ldots,M$ and $t,j=1,\\ldots,N_{\\y}$.\n\nThe stochastic Galerkin matrix $A$ is symmetric and positive definite.\nFurthermore, as it follows from the theorem below, $A$ is block sparse\nwith no more than $2M+1$ nonzero block matrices per row.\n\n\\begin{theorem} \\label{thm:G:matrices} {\\rm \\cite[Theorems 9.58, 9.59]{lord14}} \nConsider the matrices $G_{m}$ defined\n in~\\Refx{eq:G:matrices}\nfor $m=0,1,\\ldots,M$. The matrix $G_{0}$ is the $N_{\\y}\\times N_{\\y}$\nidentity matrix and each matrix $G_{m}$ for $m=1,2,\\ldots,M$ has\n\\rev{at most} two nonzero entries per row. More precisely, \n\\[\n\\left[G_{m}\\right]_{tj}\n\\begin{cases}\nc_{\\gamma_{m}+1}^{m}, & \\text{if \\ }\\gamma_{m}=\\beta_{m}-1\\text{ \\ and \\ }\\gamma_{\\ell}=\\beta_{\\ell}\\ \\ \\forall\\,\\ell\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}\\setminus\\{m\\},\\\\[2pt]\nc_{\\gamma_{m}}^{m}, & \\text{if \\ }\\gamma_{m}=\\beta_{m}+1\\text{ \\ and \\ }\\gamma_{\\ell}=\\beta_{\\ell}\\ \\ \\forall\\,\\ell\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}\\setminus\\{m\\},\\\\[2pt]\n0, & \\text{otherwise,}\n\\end{cases}\n\\]\nwhere $\\bm{\\gamma}=\\bm{\\kappa}(t)$, $\\bm{\\beta}=\\bm{\\kappa}(j)$ and\n$c_{\\gamma_{m}}^{m}$ \\abrev{are the coefficients arising in the three-term\n recurrence \\Refx{eq:3term} which defines\nthe orthonormal polynomials $P_j^m$.}\n\\end{theorem}\n\n\n\\section{Truncation preconditioners} \\label{sec:preconditioners}\n\n\\rev{In this section, we consider \na class of preconditioners that are induced by bilinear\nforms associated with truncations of the series representation of the parametric diffusion coefficient $a$\n(cf.~\\cite{ghanem14, KubinovaP_20_BPS}).}\nWe will show that these \\emph{truncated} bilinear forms are equivalent to the bilinear form\narising in the variational formulation of our PDE.\nThe immediate consequence of this fact is that \\rev{the resulting \\emph{truncation preconditioners}} will be\n\\abrev{optimal in some sense to be described below (see Definition~\\ref{def:optimal})}.\n\n\\subsection{Equivalent bilinear forms and preconditioning} \\label{sec:equiv:bilinear:forms}\n\nA generic approach to preconditioner design for\ndiscretizations of variational problems is based on\napproximating the bilinear forms arising in the formulation of\nthe problem. \\abrev{For symmetric and coercive problems, the\nwell-known concept\nof equivalence of bilinear forms translates into spectral equivalence\nbetween the coefficient matrix and the preconditioner induced by the\napproximating bilinear form; in turn, spectral equivalence enables both the design and\nanalysis of effective preconditioning techniques.}\nWe summarize this approach in Proposition~\\ref{prop:equiv} below,\nwhich requires the following two definitions.\n\n\\begin{definition}\n We say that \\abrev{positive definite} symmetric bilinear forms\n $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}, \\B :V\\times V\\rightarrow\\RR$ are equivalent if there exist positive constants\n $\\uptheta,\\Uptheta$ such that for all $v\\in V$ there holds\n $$\\uptheta\\B(v,v)\\leq \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v)\\leq \\Uptheta\\B(v,v).$$\n\\end{definition}\n\n\n\\begin{definition}\n We say that symmetric positive definite matrices $A,B\\in\\RR^{n\\times n}$ are spectrally equivalent\n if\n there exist positive constants $\\uptheta,\\Uptheta$ independent of\n $n$ such that for all $\\bv\\in\\RR^n$ there holds\n $$\\uptheta \\bv^TB\\bv\\leq \\bv^TA\\bv \\leq \\Uptheta\\bv^TB\\bv.$$\n \\abrev{In this case, we write $A\\sim B$}.\n\\end{definition}\n\n\\begin{remark\n The relation $\\sim$ is an equivalence relation. In particular,\n transitivity will be relevant in our subsequent discussion.\n\\end{remark}\n\n\\abrev{Bilinear form equivalence is connected to the well-known concepts of operator and\nspectral equivalence (see~\\cite{dyakonov66},~\\cite{fmp90}) as well as\nnorm-equivalent preconditioners (see~\\cite{loghinwathen04}). In this\ncontext, the following result is key to our subsequent analysis.}\n\n\\begin{proposition} \\label{prop:equiv}\n \\abrev{Let $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R},\\B$ denote positive definite symmetric bilinear forms on\n $V\\times V$ which are equivalent.}\n Let $V_n=\\spn\\seq{\\varphi_1,\\ldots,\\varphi_n}\\subset V$\n and let $A,B\\in\\RR^{n\\times n}$ be defined as follows\n $$A_{ij}=\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(\\varphi_j,\\varphi_i),\\qquad B_{ij}=\\B(\\varphi_j,\\varphi_i).$$\n Then $A\\sim B$ and the spectrum of\n $B^{-1}A$ satisfies\n \\begin{equation}\n \\Lambda(B^{-1}A)\\subset[\\uptheta,\\Uptheta],\\label{eq:eigbound}\n\\end{equation}\n where $\\uptheta,\\Uptheta$ are the constants of equivalence for\n $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R},\\B$. \n\\end{proposition}\n\nThe above result motivates the following definition.\n\n\\begin{definition}\\label{def:optimal}\nLet $A,B\\in\\RR^{n\\times n}$ satisfy \\Refx{eq:eigbound} with\nconstants $\\theta,\\Theta$ independent of $n$. Then $B$ is said to be\nan optimal preconditioner for $A$ with respect to the problem size $n$.\n\\end{definition}\n\nPreconditioner optimality translates into performance optimality. In\nparticular, it is well-known that the preconditioned Conjugate\nGradient algorithm applied to the linear system $A{\\bf u}}\\def \\f{{\\bf f}}\\def \\b{{\\bf b}}\\def \\bv{{\\bf v}}\\def \\w{{\\bf w}}\\def \\z{{\\bf z}}\\def \\d{{\\bf d}=\\f$ with optimal\npreconditioner $B$ converges in a number of steps independent of $n$.\nOur aim is to construct optimal preconditioners with respect to\n the problem size for the coefficient matrix in \\Refx{eq:A:u:f}. We do this by first adapting the result of Proposition \\ref{prop:equiv} to the parametric elliptic problem~\\Refx{eq:strong:form}.\nWe will need the following auxiliary result.\n\n\\begin{lemma}\\label{lem: equiv tool} Let\n $p:\\boldsymbol{\\Gamma}\\to\\mathbb{R}_{+}$ and\n assume that $b_i:\\Omega\\times\\bGamma \\,{\\to \\RR_{+}}$ $(i=1,2)$ satisfy\n $$0<\\beta_i^{\\min}\\leq b_i({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\leq \\beta_i^{\\max}\\quad\n {\\text{a.e. in $\\Omega\\times\\bGamma \\ni ({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$}}.$$\n Define the bilinear\n forms $\\B_i:V\\times V \\to \\RR$ via\n $$\\B_i(v,w)=\\int_{\\bGamma}p(\\y)\\int_{\\Omega}b_{i}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla\n v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla w({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y,\\quad {i = 1,2}.$$\n Then the bilinear forms $\\B_i$ are equivalent:\n $$\\uptheta\\B_2(v,v)\\leq \\B_1(v,v)\\leq\n \\Uptheta\\B_2(v,v)\\quad \\forall\\, v \\in V,$$\n where\n $$\\uptheta=\\frac{\\beta_1^{\\min}}{\\beta_2^{\\max}},\\qquad \\Uptheta=\\frac{\\beta_1^{\\max}}{\\beta_2^{\\min}}.$$\n\\end{lemma}\n\n\\begin{proof}\n For any $v\\in V$, we have \n \\begin{align*}\n \\B_1(v,v) \n & =\\int_{\\boldsymbol{\\Gamma}}p(\\y)\\int_{\\Omega}\n \\left(\\frac{b_{1}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)}{b_{2}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)}\\right)b_{2}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y\n \\\\[4pt]\n & \\leq\\left(\n {\\operatorname*{ess\\;sup}_{({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\in \\Omega\\times\\boldsymbol{\\Gamma}}}\\;\n \\frac{b_{1}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)}{b_{2}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)}\\right)\n \\int_{\\boldsymbol{\\Gamma}}p(\\y)\\int_{\\Omega}b_{2}({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y\n \\\\[4pt]\n & \\leq\\frac{\\beta_{1}^{\\max}}{\\beta_{2}^{\\min}}\\, \\B_2(v,v).\n \\end{align*}\n The lower bound follows analogously. \n\\end{proof}\n\nThe boundedness required in the above lemma for\n\\abrev{$b_i({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$} holds for the \\abrev{parametric diffusion coefficient}\n$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ (see~\\Refx{eq:assumption:on:a}).\n\\abrev{In the next \\rev{subsection} we show}\nthat \\abrev{assumptions~\\Refx{eq:assumption:on:a0},~\\Refx{eq:assumption:on:ai-1},\nwhich guarantee~\\Refx{eq:assumption:on:a}, also yield}\nboundedness of \\abrev{\\emph{truncated} expansions} of the \\abrev{coefficient} $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$.\n\\abrev{That result} will \\abrev{enable our main goal---\\rev{the analysis\nof truncation preconditioners} for stochastic Galerkin matrices}.\n\n\\subsection{Spectral analysis} \\label{sec:spectral:analysis}\n\nIn analogy with (\\ref{eq:assumption:on:ai-1}), we define\n\\begin{equation}\n \\label{eq:tauess}\n \\tau_0 := 0,\\quad\n \\tau_r:=\\frac{1}{a_{0}^{\\min}} \\bigg\\| \\sum_{m=1}^{r} \\abs{a_m} \\bigg\\|_{\\infty},\\ \\ r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}.\n\\end{equation}\nNote that $(\\tau_r)_{r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0}$ is a monotonic increasing\nsequence, which is bounded from above (cf. \\Refx{eq:assumption:on:ai-1}),~i.e.,\n\\[\n 0 \\le \\tau_r \\leq \\tau_{r+1} \\le \\tau<1,\\quad r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0.\n\\]\nFirst, we establish the boundedness of \\emph{truncated} expansions\nof the parametric diffusion coefficient~$a$.\n\n\\begin{lemma}\\label{lem:asbound}\n \\abrev{Assume that \\Refx{eq:assumption:on:a0} and \\Refx{eq:assumption:on:ai-1} hold.\n Let $r\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$ and define $a_r:\\Omega\\times\\bGamma\\, \\to \\RR$ to be the\n finite~sum\n \\begin{equation}\n a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\; \\abrev{:=}\\; a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})+\\sum_{m=1}^ra_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_m.\n \\label{eq:form of a (param)-1}\n \\end{equation}\n Then $a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ is positive and bounded almost everywhere\n in~$\\Omega\\times\\bGamma$.\n}\n\\end{lemma}\n\n\\begin{proof}\n The case $r=0$ follows from the boundedness and positivity\n assumptions~\\Refx{eq:assumption:on:a0} on\n $a_0({\\bm{x}}}\\def \\y{{\\bm{y}})$. Consider now $r\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}$. Since $\\abs{y_m}\\leq 1$, using the definition~\\Refx{eq:tauess} of\n $\\tau_r$ we obtain\n \\[\n \\left|a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)-a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})\\right|\n =\\bigg|\\sum_{m=1}^ra_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_m\\bigg|\\leq\\sum_{m=1}^r\\left|a_m({\\bm{x}}}\\def \\y{{\\bm{y}})\\right|\\leq a_0^{\\min}\\tau_r\n \\qquad\n \\text{a.e. in $\\Omega \\times \\bGamma$}.\n \\]\n Hence,\n\\begin{equation} \\label{eq:bound:ar:a0}\n a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})-a_0^{\\min}\\tau_r \\leq a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\leq a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})+a_0^{\\min}\\tau_r\n\\end{equation}\n and using the boundedness of $a_0({\\bm{x}}}\\def \\y{{\\bm{y}})$ (see~\\Refx{eq:assumption:on:a0}), we get\n \\begin{equation}\n \\eta_r^{\\min}:=a_{0}^{\\min}-a_0^{\\min}\\tau_r \\leq a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\leq a_{0}^{\\max}+a_0^{\\min}\\tau_r=:\\eta_r^{\\max}\n \\quad\n \\text{a.e. in $\\Omega \\times \\bGamma$}.\\label{eq:etadef}\n \\end{equation}\nThe proof concludes by noting that $\\eta_r^{\\min} > 0$ since $\\tau_r \\leq \\tau < 1$\n (cf. \\Refx{eq:tauess} and \\Refx{eq:assumption:on:ai-1}).\n\\end{proof}\n\nCombining Lemmas~\\ref{lem: equiv tool} and~\\ref{lem:asbound}, we obtain the following result.\n\n\\begin{theorem}\\label{thm: B equiv BN}\n Let $a:\\Omega\\times\\bGamma \\to \\RR_{+}$ be a parametric diffusion coefficient\n given by \\Refx{eq:affine} and let $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R} : V \\times V \\to \\RR$ be the associated bilinear form defined in~\\Refx{eq:A-form}.\n Assume~\\Refx{eq:assumption:on:a0} and~\\Refx{eq:assumption:on:ai-1} hold.\n Let $a_r$ be given by~\\Refx{eq:form of a (param)-1} and define the associated bilinear form as\n \\[\n \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,w) := \\int_{\\boldsymbol{\\Gamma}}p(\\y)\\int_{\\Omega}a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla w({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y.\n \\]\n Then $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}$ and $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$ are equivalent for any $r\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$.\n\\end{theorem}\n\n\\begin{proof}\n By \\Refx{eq:assumption:on:a}, the diffusion coefficients $a$ is bounded.\n Furthermore, by Lem\\-ma~\\ref{lem:asbound}, the coefficient $a_r$, $r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$, is bounded as well.\nConsequently, by Lemma \\ref{lem: equiv tool}, the bilinear forms are\n equivalent. In particular,\n $$\\uptheta_r\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v)\\leq \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v)\\leq\n \\Uptheta_r\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v),$$\n where\n \\begin{equation}\n \\uptheta_r := \\frac{a_{\\min}}{\\eta_r^{\\max}}=\\frac{(1-\\tau)a_0^{\\min}}{a_{0}^{\\max}+a_0^{\\min}\\tau_r},\\qquad\n \\Uptheta_r := \\frac{a_{\\max}}{\\eta_r^{\\min}}=\\frac{a_{0}^{\\max}+a_0^{\\min}\\tau}{(1-\\tau_r)a_0^{\\min}}\\label{eq:thetas}\n\\end{equation}\nwith $\\eta_r^{\\min}$ and $\\eta_r^{\\max}$ defined in~\\Refx{eq:etadef}.\n\\end{proof}\n\n\\begin{remark}\n The equivalence of the bilinear form $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_0$ associated with the pa\\-ra\\-me\\-ter-free term $a_0({\\bm{x}}}\\def \\y{{\\bm{y}})$ in~\\Refx{eq:affine}\n and the bilinear form $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}$ is well known (see, e.g., {\\rm \\cite[eq.~(2.5)]{BespalovPRR_19_CAS}}).\n Theorem~{\\rm \\ref{thm: B equiv BN}} extends this result to the case of arbitrary finite truncation\n of the affine-parametric coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$.\n\\end{remark}\n\n\\begin{remark}\n The constants $\\uptheta_r,\\, \\Uptheta_r$ in~\\Refx{eq:thetas}\n depend on \\abrev{$\\tau$},\n $a_0^{\\min}$, $a_0^{\\max}$ and indirectly on $r$, via $\\tau_r$.\n\\end{remark}\n\nTheorem~\\ref{thm: B equiv BN}, combined with Proposition\n\\ref{prop:equiv}, indicates that the bilinear form $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$ induces a family of\npreconditioners for the SGFEM matrix $A$ in~\\Refx{eq:A:u:f}.\nThis is made precise in the next theorem which is the main result of this~section.\n\n\\begin{theorem}\\label{thm:truncprec}\n Let $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R},\\, \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$ be defined as in Theorem {\\rm \\ref{thm: B equiv BN}} and\n assume \\Refx{eq:assumption:on:a0} and~\\Refx{eq:assumption:on:ai-1} hold.\n Let $\\{\\varphi_{ij}\\}$ be the tensor-product basis\n for the finite dimensional space $V_{hk}^M$ in~\\Refx{eq:VhkM}\n and $A = \\left[\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(\\varphi_{ij},\\varphi_{st})\\right]$ be the associated SGFEM matrix.\n \\abrev{For a fixed $r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$}\n define the preconditioner $P_r$ via\n $$P_r:=\\left[\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(\\varphi_{ij},\\varphi_{st})\\right].$$\n Then $P_r\\sim A$ and the spectrum of $P_r^{-1}A$ satisfies\n \\begin{equation}\n \\Lambda(P_r^{-1}A)\\subset[\\uptheta_r,\\Uptheta_r]\\label{eq:bounds}\n\\end{equation}\nwith $\\uptheta_r,\\, \\Uptheta_r$ defined in \\Refx{eq:thetas}.\n\\end{theorem}\n\n\\begin{remark}\nTo obtain alternative spectral bounds under the assumption in~\\Refx{eq:assumption:on:ai-1:KubPul},\nwe can define (cf.~\\Refx{eq:tauess})\n\\[\n \\tilde\\tau_0 := 0,\\quad\n \\tilde\\tau_r := \\bigg\\| a_{0}^{-1} \\sum_{m=1}^{r} \\abs{a_m} \\bigg\\|_{\\infty},\\ \\ r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}.\n\\]\nNote that $(\\tilde\\tau_r)_{r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0}$ is a monotonic increasing sequence bounded from above by~$\\tilde\\tau$.\nThen, instead of~\\Refx{eq:etadef} we obtain by using~\\Refx{eq:assumption:on:a:KubPul}\n\\[\n \\frac{1 - \\tilde\\tau_r}{1 + \\tilde\\tau}\\, a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) \\leq\n (1-\\tilde\\tau_r) a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\leq\n a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) \\leq\n (1+\\tilde\\tau_r) a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\leq\n \\frac{1 + \\tilde\\tau_r}{1 - \\tilde\\tau}\\, a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\ \\\n \\text{a.e. in }\\Omega\\times\\bGamma.\n\\]\nThis implies the equivalence of bilinear forms $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R},\\; \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_0,\\; \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$ as follows:\n\\[\n (1-\\tilde\\tau_r) \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_0(v,v) \\le \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v) \\le (1+\\tilde\\tau_r) \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_0(v,v)\\quad\n \\forall\\, v \\in V\n\\]\nand\n\\[\n \\frac{1-\\tilde\\tau}{1+\\tilde\\tau_r}\\, \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v) \\le \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v) \\le \\frac{1+\\tilde\\tau}{1-\\tilde\\tau_r}\\, \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v)\\quad\n \\forall\\, v \\in V.\n\\]\nTherefore, by Proposition~{\\rm \\ref{prop:equiv}}, the following spectral bounds hold:\n\\[\n \\Lambda(P_0^{-1}P_r) \\subset [ 1 - \\tilde\\tau_r, 1 + \\tilde\\tau_r ]\n \\text{ \\ and \\ }\n \\Lambda(P_r^{-1}A) \\subset \\bigg[ \\frac{1-\\tilde\\tau}{1+\\tilde\\tau_r}, \\frac{1+\\tilde\\tau}{1-\\tilde\\tau_r} \\bigg].\n\\]\n\\end{remark}\n\n\\abrev{The preconditioners $P_r$, that we will refer to as \\emph{truncation preconditioners},\nare induced by the bilinear form $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$, $r\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$.\nTherefore, by~\\Refx{eq:AM}, there holds $P_r=A$ for all $r\\geq M$. In practice, the values of\n$r$ are expected to be small in order to allow for sparse\napproximations of $A$ which can be efficiently implemented. Note also that $P_r$ can be written as a sum of Kronecker products, just\nas was the case for the SGFEM matrix $A$ (cf.~\\Refx{eq:decomp:A}):\n\\begin{equation}\n P_r=G_0\\otimes K_0+\\sum_{m=1}^rG_m\\otimes K_m.\\label{eq:Pr}\n\\end{equation}\n}\n\n\\abrev{The result of Theorem~\\ref{thm:truncprec}\nindicates that the performance of the preconditioned Conjugate\nGradient method will be independent of discretization parameters,\nbut may depend on the choice of truncation parameter $r$. Since\n$$\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v)=\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v)+\\R(v,v)$$\nwith\n$$\\R(v,v):=\\int_{\\boldsymbol{\\Gamma}}p(\\y)\\int_{\\Omega}(a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)-a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y))\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\cdot\\nabla v({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\dd{\\bm{x}}}\\def \\y{{\\bm{y}} \\dd\\y,$$\na smallness assumption of the form\n\\begin{equation}\n \\label{eq:resbound}\n \\abs{\\R(v,v)}\\leq \\varepsilon_r \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v),\\quad 0<\\varepsilon_r<1\n\\end{equation}\nwould allow for the following equivalence of $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}$ and $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$\n\\begin{equation}\n \\label{eq:equiv}\n (1-\\varepsilon_r)\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v)\\leq \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r(v,v)\\leq (1+\\varepsilon_r) \\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}(v,v).\n\\end{equation}\nSince\n$$a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)-a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)=\\sum_{m=r+1}^\\infty a_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_m,$$\nby the definition of $\\R,$ assumption \\Refx{eq:resbound} holds for\nsufficiently large $r$. As a result,~\\Refx{eq:equiv} implies the\n\\rev{following} eigenvalue bounds\n$$\\Lambda(P_r^{-1}A)\\subset\\left[\\frac{1}{1+\\varepsilon_r},\\frac{1}{1-\\varepsilon_r}\\right].$$\nThis suggests that the closer $a_r$ approximates $a$, the tighter the\npreconditioned spectrum will be clustered around unity. We will\ninvestigate this conclusion numerically in section~\\ref{sec:numerics}.\n}\n\n\n\\section{Modified truncation preconditioners} \\label{sec:modified:precond}\n\nAny practical implementation of a preconditioner requires an efficient\ntechnique for applying the action of its inverse on a given\nvector. Standard approaches include constructing sparse\nfactorizations, or employing multigrid or multilevel techniques;\ndomain decomposition methods represent yet another\napproach. The potential for parallelism could also be a deciding\nfactor in the choice of solution method.\n\nThe preconditioner $P_r$ \\rev{introduced} in the previous section is block-sparse,\nwith sparsity deteriorating with increasing $r$\n(cf. Theorem~\\ref{thm:G:matrices}). In the case when $r=1$, the\nstructure can be shown to be block-tridiagonal under a certain\npermutation---this is not an ideal situation, as it requires\nadditional techniques to ensure an efficient application of the\npreconditioner. For this reason, we replace $P_r$ with its\ncorresponding symmetric block Gauss--Seidel (SBGS) approximation:\n\\begin{equation}\n \\abrev{\\tilde{P}_r :=} \\left(G_0\\otimes K_0+\\sum_{m=1}^rL_m\\otimes\n K_m\\right)(G_0\\otimes K_0)^{-1}\\left(G_0\\otimes K_0+\\sum_{m=1}^rL_m^T\\otimes\n K_m\\right),\\label{eq:prt}\n\\end{equation}\nwhere $L_m+L_m^T=G_m$.\n\\abrev{The matrix $\\tilde{P}_r$ thus}\nrepresents a sparse approximation to $P_r$\ninvolving block-triangular and block-diagonal matrices. In the remainder of this section we prove that \\abrev{$P_r \\sim \\tilde{P}_r$}\n\\abrev{and} provide complexity considerations, including\na discussion of implementation.\n\n\n\\subsection{Analysis of SBGS approximation of $P_r$} \\label{sec:SBGS:analysis}\n\n\\rev{In this subsection we assume that the ordering of multi-indices in the index set $\\I_{k}^{M}$ is such that\nthe matrices $L_m$ in~\\Refx{eq:prt} have at most one nonzero entry per row and per column\n(this property holds, e.g., for lexicographic or anti-lexicographic ordering as well as for \nascending or descending ordering by the total degree of the associated complete polynomials in~$S_k^M$).\nLet us define}\n\\begin{equation}\n S_r \\;\\abrev{:=}\\; \\sum_{m=1}^rL_m\\otimes K_m,\\quad D_0 \\;\\abrev{:=}\\; G_0\\otimes K_0,\n \\label{eq:Sr:D0}\n\\end{equation}\nso that\n\\begin{equation}\n P_r \\;\\abrev{\\stackrel{\\Refx{eq:Pr}}{=}} D_0+S_r+S_r^T\n \\label{eq:Pr:mod}\n\\end{equation}\nand\n$$\\tilde{P}_r \\;\\abrev{\\stackrel{\\Refx{eq:prt}}{=}}\\; (D_0+S_r)D_0^{-1}(D_0+S_r^T)=P_r+S_r D_0^{-1}S_r^T.$$\nOur spectral analysis\nfocuses on deriving bounds for the generalized Rayleigh~quotient\n\\begin{equation}\n \\frac{\\bv^T\\tilde{P}_r\\bv}{\\bv^TP_r\\bv}=1+\\frac{\\bv^T S_r\n D_0^{-1}S_r^T\\bv}{\\bv^T(D_0+S_r+S_r^T)\\bv},\\qquad \\bv\\in\\RR^{N_\\x}\\def\\Ny{N_\\y\\Ny}\\setminus\\seq{\\bf 0}.\n \\label{eq:Rayleigh}\n\\end{equation}\nSince the lower bound is 1, we restrict our attention to deriving an\nupper bound for the second term on the right-hand side of~\\Refx{eq:Rayleigh}, which we write using the change\nof variable $\\w=D_0^{1\/2}\\bv$ as\n\\begin{equation*}\n \\rho(\\w)\\;\\abrev{:=}\\; \\frac{\\w^T\\tilde{S}_r\\tilde{S}_r^T\\w}{\\w^T(I+\\tilde{S}_r+\\tilde{S}_r^T)\\w}.\n\\end{equation*}\nHere,\n\\begin{equation}\n \\tilde{S}_r\\;\\abrev{:=}\\; D_0^{-1\/2}S_rD_0^{-1\/2}=\\sum_{m=1}^rL_m\\otimes\\tilde{K}_m\n \\label{eq:Srt}\n\\end{equation}\nwith $\\tilde{K}_m\\;\\abrev{:=}\\; K_0^{-1\/2}K_mK_0^{-1\/2}$, using the fact that $G_0=I_{\\Ny}$.\nHence,\n\\begin{equation}\n \\rho(\\w)\\leq\\max_{\\w\\neq{\\bf\n 0}}\\frac{\\w^T\\tilde{S}_r\\tilde{S}_r^T\\w}{\\w^T\\w}\\cdot\\max_{\\w\\neq{\\bf 0}}\\frac{\\w^T\\w}{\\w^T(I+\\tilde{S}_r+\\tilde{S}_r^T)\\w}=\n \\frac{\\sigma_{\\max}^2(\\tilde{S}_r)}{\\lambda_{\\min}(I+\\tilde{S}_r+\\tilde{S}_r^T)},\n \\label{eq:rho:bound}\n \\end{equation}\n where $\\sigma_{\\max}(\\cdot)$ and $\\lambda_{\\min}(\\cdot)$ denote, respectively,\nthe largest singular value and the smallest eigenvalue of a~matrix.\nIn the next lemma, we provide bounds for $\\sigma_{\\max}(\\tilde{S}_r)$ and $\\lambda_{\\min}(I+\\tilde{S}_r+\\tilde{S}_r^T)$ in order to\nconclude the derivation of the upper bound on~$\\rho$.\n\n\\begin{lemma} \\label{lm:aux:bounds}\n\\abrev{Suppose that \\Refx{eq:assumption:on:a0} and \\Refx{eq:assumption:on:ai-1} hold\n\\rev{and $\\tau_r$ is defined in~\\Refx{eq:tauess}}.\n}\nLet $\\tilde{S}_r$ be defined \\abrev{by~\\Refx{eq:Srt}}. Then\n\\begin{equation}\n \\lambda_{\\min}(I+\\tilde{S}_r+\\tilde{S}_r^T) \\geq \\rev{1 - \\tau_r}\n \\label{eq:bound:lambda}\n\\end{equation}\nand\n \\begin{equation}\n \\sigma_{\\max}(\\tilde{S}_r)\\leq\n \\abrev{\\frac{1}{a_0^{\\min}}\\sum_{m=1}^r \\norm{a_m}{\\infty}}.\n \\label{eq:bound:sigma}\n \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince\n\\[\n I+\\tilde{S}_r+\\tilde{S}_r^T =\n I+\\sum_{m=1}^rG_m\\otimes \\tilde{K}_m =\n D_0^{-1\/2} \\bigg( D_0+\\sum_{m=1}^rG_m\\otimes K_m \\bigg) D_0^{-1\/2}\n \\,{=}\\, D_0^{-1\/2} P_r D_0^{-1\/2},\n\\]\nthe eigenvalues of $I+\\tilde{S}_r+\\tilde{S}_r^T$ are the eigenvalues\nof $D_0^{-1}P_r =P_0^{-1}P_r$.\n\\rev{To find the bounds on the spectrum of $P_0^{-1}P_r$, recall the inequalities in~\\Refx{eq:bound:ar:a0}\nand the lower bound for $a_0({\\bm{x}}}\\def \\y{{\\bm{y}})$ in~\\Refx{eq:assumption:on:a0}, which together imply~that\n\\[\n (1-\\tau_r) a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\leq a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) \\leq (1 + \\tau_r) a_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})\\quad\\text{a.e. in }\\Omega\\times\\bGamma.\n\\]\nHence, the bilinear forms $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_0$ and $\\eus{A}}\\def \\B{\\eus{B}}\\def \\F{\\eus{F}}\\def \\R{\\eus{R}_r$ are equivalent and by Proposition~\\ref{prop:equiv} there~holds\n\\begin{equation} \\label{eq:spectrum:0:r}\n \\Lambda(P_0^{-1}P_r) \\subset \\left[ 1 - \\tau_r, 1 + \\tau_r \\right].\n\\end{equation}\n}\nThis proves~\\Refx{eq:bound:lambda}.\n\nOn the other hand, since $\\sigma_{\\max}$ \\abrev{defines} a norm, we use the triangle inequality to~estimate\n\\begin{equation}\n \\sigma_{\\max}(\\tilde{S}_r) \\leq\n \\sum_{m=1}^r\\sigma_{\\max}(L_m\\otimes \\tilde{K}_m) \\leq\n \\sum_{m=1}^r\\sigma_{\\max}(L_m)\\sigma_{\\max}(\\tilde{K}_m).\n \\label{eq:bound:aux1}\n\\end{equation}\nNow, $\\sigma^2_{\\max}(L_m)=\\lambda_{\\max}(L_mL^T_m)$; since $L_mL_m^T$\nis diagonal for \\abrev{every} $m$\n\\abrev{(due to $L_m$ having \\rev{at most one nonzero entry per row and per column})},\nit follows that for all $m$\n\\begin{equation}\n \\sigma_{\\max}(L_m)=\\max_{i,j} \\abrev{[G_m]}_{ij} \\le \\max_{k} c_k^m \\leq 1\n \\label{eq:bound:aux2}\n\\end{equation}\n\\abrev{with $c_k^m$\nbeing bounded by 1 (cf. Theorem~\\ref{thm:G:matrices} and inequalities~\\Refx{eq:cmj}).}\nFinally, since the eigenvalues of $\\tilde{K}_m$ are the eigenvalues of\n$K_0^{-1}K_m$, we find~\\abrev{that}\n\\[\n \\sigma_{\\max}(\\tilde{K}_m)=\\max_{i}\\abs{\\lambda_i(K_0^{-1}K_m)}\\leq\n \\frac{\\norm{a_m}{\\infty}}{a_0^{\\min}}\n\\]\n\\abrev{and then inequality~\\Refx{eq:bound:sigma} follows from~\\Refx{eq:bound:aux1} and~\\Refx{eq:bound:aux2}.\nThis finishes the~proof.}\n\\end{proof}\n\nWe summarize our discussion in the following result.\n\n\\begin{proposition} \\label{prop:tildebounds}\nSuppose that \\Refx{eq:assumption:on:a0} and \\Refx{eq:assumption:on:ai-1} hold\n\\rev{and $\\tau_r$ is defined in~\\Refx{eq:tauess}}.\n Let $P_r$ be defined in \\Refx{eq:Pr} and let $\\tilde{P}_r$ be its\n symmetric block Gauss--Seidel approximation~\\Refx{eq:prt}.\n Then $\\tilde{P}_r\\sim \\abrev{P_r}$ and the spectrum of \\abrev{$P_r^{-1} \\tilde{P}_r$} satisfies\n \\begin{equation}\n \\label{eq:tildebounds}\n \\abrev{\\Lambda(P_r^{-1}\\tilde{P}_r) \\subset [1,\\, 1+\\delta_r]},\n \\end{equation}\n where\n \\begin{equation}\n \\delta_r := \\rev{\\frac{1}{1-\\tau_r}}\n \\bigg( \\frac{1}{a_0^{\\min}} \\sum_{m=1}^r \\norm{a_m}{\\infty} \\bigg)^2.\n \\label{eq:deltar}\n\\end{equation}\n\\end{proposition}\n\n\\abrev{The proof of the upper bound in~\\Refx{eq:tildebounds}\nis completed by substituting the estimates~\\Refx{eq:bound:lambda}, \\Refx{eq:bound:sigma} into\n\\Refx{eq:rho:bound} and then using the resulting bound for $\\rho(\\w)$ in~\\Refx{eq:Rayleigh}.}\n\nThe following \\abrev{result} is a straightforward consequence of\n\\abrev{Theorem~\\ref{thm:truncprec}, Proposition~\\ref{prop:tildebounds} and the}\ntransitivity of spectral equivalence.\n\n\\begin{theorem}\n Let $\\tilde{P}_r$ be the symmetric block Gauss--Seidel approximation \\Refx{eq:prt}\n to $P_r$. Let\n $\\uptheta_r, \\Uptheta_r$ be defined in \\Refx{eq:thetas} and let\n $\\delta_r$ be defined in \\Refx{eq:deltar}. Then $\\tilde{P}_r\\sim A$ and the spectrum of\n $\\tilde{P}_r^{-1}A$ satisfies\n \\begin{equation}\n \\label{eq:tildebounds1}\n \\abrev{\\Lambda(\\tilde{P}_r^{-1}A) \\subset \\bigg[\\frac{\\uptheta_r}{1+\\delta_r},\\, \\Uptheta_r\\bigg]}.\n \\end{equation}\n\\end{theorem}\n\n\n\\subsection{Implementation. Complexity considerations} \\label{sec:implement}\n\nOur proposed solution method for solving the linear system\n\\Refx{eq:sGFEM} is\nthe preconditioned Conjugate Gradient (PCG) method, for which the main\ncomputational effort at each step comprises a matrix-vector product with the matrix\n$A$ and the solution of a linear system with the preconditioning\nmatrix.\nSince the main computational cost is associated with the\nlatter operation, we discuss this in detail.\nWe will denote by ${\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(operation)$ the complexity, i.e., number of flops required to perform \\abrev{an} $operation$.\nThe number of nonzeros of a matrix will be denoted by $\\nnz(\\cdot)$.\n\nAs indicated previously, the action of the inverse of $P_r$ needs to\nbe approximated, due to its sparse (but non-diagonal) block structure.\nWe achieve this by replacing $P_r$ with its SBGS approximation $\\tilde{P}_r$.\nLet us consider the implementation of the action of $\\tilde{P}_r^{-1}$ onto a given vector~$\\bv$;\nfor general $r \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$, this can be achieved as follows\n(see~\\Refx{eq:Sr:D0}--\\Refx{eq:Pr:mod} for the definitions of the\n respective matrices):\n\\begin{enumerate}\n\\item solve $(D_0+S_r)\\w=\\bv$;\n\\item solve $(D_0+S_r^T)\\z=D_0\\w$.\n\\end{enumerate}\nBoth of the above steps involve the solution of a block-triangular\nsystem, with the main computational cost arising from solving linear\nsystems with the diagonal blocks $K_0$. Specifically, since $P_r$ has\nat most $2r+1$ nonzero block matrices per row, we find \n$${\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(\\tilde{P}_r^{-1}\\bv)\\approx (2rN_\\y)\n\\nnz(K_0)+2N_\\y{\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b),$$\nfor some vector $\\b$ of size $N_{\\bm{x}}}\\def \\y{{\\bm{y}}$.\nBelow we consider two special cases.\n\n\n\\subsubsection{Special case: $r=0$}\nThe preconditioner $P_0$ is the mean-based preconditioner introduced in \\cite{ghanemkruger96}.\nThe complexity associated with the action of the inverse on a given vector is\n\\[\n {\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}({P}_0^{-1}\\bv)=N_\\y{\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b).\n\\]\n\n\\subsubsection{Special case: $r=1$}\nThe structure of $P_r$ simplifies\ngreatly when $r=1$.~In particular, $G_1$ has a block-diagonal\nstructure under a certain permutation~\\cite[\\S9.5]{lord14}:\n\\[\n G_1 = \\diag( T_{k+1},\\,T_k,\\ldots,T_k,\\ldots,T_1,\\ldots,T_1),\n\\]\nwhere $T_j\\in\\RR^{j\\times j}$ $(j=1,\\ldots,k+1)$ are tridiagonal, with zero main diagonal. Note in\nparticular that $T_1=0$. As a result, $P_1$ will have a block-diagonal\nstructure, where each diagonal block is a block-tridiagonal matrix of\nsize $jN_{\\bm{x}}}\\def \\y{{\\bm{y}}$. Specifically,\n$$P_1=\\bigoplus_{j=1}^{k+1}\\bigoplus_{i=1}^{n_j}\\left(I\\otimes\n K_0+T_j\\otimes K_1\\right),$$\nwhere, assuming $M>1$,\n$$n_j={k+M-j-1 \\choose M-2}.$$\nGiven this structure, the\ncomplexity for the implementation of the\naction of $\\tilde{P}_1^{-1}$ on a given vector $\\bv$ is\n$${\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(\\tilde{P}_1^{-1}\\bv)\\approx\n2\\left(\\sum_{j=2}^{k+1}jn_j\\right)\\left(\\nnz(K_0)+{\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b)\\right)+n_1 {\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b).$$\nUnder the assumption that ${\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b)$ dominates the computation, we deduce that\nthe implementation of $\\tilde{P}_1$ is at most twice as expensive as the\nimplementation of $P_0$.\n\n\n\\subsubsection{Kronecker preconditioner}\nWe end this section with a\ndiscussion of the complexity required for an implementation of the\nKronecker preconditioner \\cite{Ullmann10}. Since\n$$P_\\otimes=G\\otimes K_0=(G\\otimes I_{N_\\x}\\def\\Ny{N_\\y})(I_{\\Ny}\\otimes K_0)=(G\\otimes\nI_{N_\\x}\\def\\Ny{N_\\y})P_0,$$\nthe complexity is given by\n\\[\n {\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}({P}_\\otimes^{-1}\\bv) = N_\\y {\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(K_0^{-1}\\b) + N_{{\\bm{x}}}\\def \\y{{\\bm{y}}} {\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(G^{-1}\\d)\n\\]\nfor some vector $\\d$ of size $\\Ny$. Thus, the complexity exceeds that\nof $P_0$ by a computational cost dependent on the sparsity of $G$. In\nparticular, it was estimated in \\cite{Ullmann10} that this additional\ncost would amount to ${\\eus{F}\\!\\ell}}\\def\\flp{{\\eus{F}\\!\\ell p}}\\def \\nnz{\\textup{nnz}(G^{-1}\\d) \\sim O((2M+1)^2)$ operations,\nexcluding the cost of performing a Cholesky factorization of $G$.\n\n\n\n\\section{Numerical experiments} \\label{sec:numerics}\n\\abrev{\nIn this section, we investigate the effectiveness of\n\\rev{the preconditioning strategies considered in~\\S\\S\\ref{sec:preconditioners}--\\ref{sec:modified:precond}}.\nIn particular, we verify the theoretical optimality of truncation preconditioners $P_r$ and $\\tilde P_r$ ($r \\ge 1$)\nwith respect to discretization parameters\nand compare their performance with that of\nthe mean-based preconditioner $P_0$ and\nthe Kronecker product preconditioner $P_{\\otimes}$ defined in \\Refx{eq:kron}\n(see \\cite{Ullmann10} for details and analysis).\n}\n\nWe chose to use test problems\nsatisfying the descriptions and assumptions in this paper, as well as\nproblems outside the theoretical framework. Thus, we solved model\n problem~\\Refx{eq:strong:form} using the following choices of\n parametric diffusion coefficient:\n\\begin{itemize}\n \\item $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ has the affine representation\n \\Refx{eq:affine}, with the coefficients $a_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ satisfying\n \\Refx{eq:assumption:on:a0} and \\Refx{eq:assumption:on:ai-1};\n \\item $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ is a lognormal diffusion coefficient, i.e.,\n $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)=\\exp(b({\\bm{x}}}\\def \\y{{\\bm{y}},\\y))$, where $b({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ is assumed to have the affine representation\n \\Refx{eq:affine} but with unbounded parameters $y_m$\n \\rev{(we note that this choice of coefficient $a$ is not covered by our theoretical~analysis)}.\n\\end{itemize}\n\nIn all our tests, we chose $\\Omega=(0,1)^2$ and $f({\\bm{x}}}\\def \\y{{\\bm{y}})=1$. \nWe used the MATLAB toolbox\nS-IFISS~\\cite{SIFISS} to generate SGFEM discretizations of our model\nproblem for a range of discretization parameters. We used uniform\nsubdivisions of $\\Omega$ into square elements of edge length~$h$, with\n$h$ ranging between $2^{-3}$ to $2^{-7}$.\nThe discretization parameters $k,M$ had ranges $1\\leq k\\leq 6$, $1\\leq M\\leq 8$.\nWe solved the resulting linear systems using the preconditioned CG method with\ntolerance $tol = 10^{-6}$ and zero initial guess.\n\n\n\\subsection{Test problem 1: affine-parametric diffusion coefficient} \\label{sec:numerics:affine}\n\n\\abrev{For this test problem, the diffusion coefficient $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$ had\n the affine-parametric form~\\Refx{eq:affine}, with the coefficients\n$a_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ in the expansion chosen such that they exhibit either slow or fast decay.\nAs indicated at the end of section~\\ref{sec:preconditioners}, this is expected to affect the performance of the\ntruncation preconditioners $P_r$. In particular, our numerical\nexperiments will highlight the dependence on the truncation parameter $r$.\n}\n\nFollowing~\\cite[Section~11.1]{MR3154028}, we select the expansion coefficients $a_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ ($m \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0$) in~\\Refx{eq:affine}\nto represent planar Fourier modes of increasing total order,~i.e.,\n\\begin{equation}\n\ta_{0}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\,{=}\\, 1,\\ \n\ta_{m}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\,{=}\\, \\bar\\alpha m^{-\\tilde\\sigma} \\cos\\big(2\\pi\\beta_{1}(m)x_{1}\\big) \\cos\\big(2\\pi\\beta_{2}(m)x_{2}\\big),\n\t\\ {\\bm{x}}}\\def \\y{{\\bm{y}} \\,{=}\\, (x_{1},x_{2}) \\,{\\in}\\, \\Omega.\n\t\\label{eq:ex5}\n\\end{equation}\nHere, $\\tilde{\\sigma}>1$, $0<\\bar{\\alpha}<1\/\\zeta(\\tilde{\\sigma})$, where $\\zeta$ denotes the Riemann zeta function,\nand $\\beta_1,\\,\\beta_2$ are given by\n\\begin{align*}\n\t\\beta_{1}(m)=m-k(m)\\left(k(m)+1\\right)\/2,\\qquad \\beta_{2}(m)=k(m)-\\beta_{1}(m)\n\\end{align*}\nwith $k(m)=\\left\\lfloor -1\/2+\\sqrt{1\/4+2m}\\right\\rfloor$.\nFurthermore, we assume the parameters $y_m$ ($m \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}$) in~\\Refx{eq:affine} to be the images\n of independent uniformly distributed mean-zero random variables on $\\Gamma_m = [-1,1]$.\n In this case, $p_m(y_m) = 1\/2$, $y_m \\in \\Gamma_m$, and \n the orthonormal polynomial basis of $L^2_{\\pi_m}(\\Gamma_m)$ consists of scaled Legendre polynomials.\nNote that with these settings, conditions~\\Refx{eq:assumption:on:a0} and~\\Refx{eq:assumption:on:ai-1}\nare satisfied with constants $a_{0}^{\\min}=a_{0}^{\\max}=1$ and\n$\\tau \\le \\bar{\\alpha} \\zeta(\\tilde{\\sigma})$, respectively.\n\n\\abrev{The choice $\\tilde{\\sigma}=2$ in \\Refx{eq:ex5} yields coefficients\n$a_m$ with slow decay of the amplitudes $\\bar\\alpha m^{-\\tilde\\sigma}$, while the fast decay corresponds to the choice $\\tilde{\\sigma}=4$.\nIn each case, we select the factor $\\bar\\alpha$ such that $\\bar{\\alpha} \\zeta(\\tilde{\\sigma}) = 0.9999$,\nwhich gives $\\bar{\\alpha} \\approx 0.6079$ for $\\tilde{\\sigma}=2$ and\n$\\bar{\\alpha} \\approx 0.9239$ for $\\tilde{\\sigma}=4$.}\nThe magnitudes of the expansion coefficients $a_m$ for increasing $m$\nfor slow and fast decay are displayed in\nTable~\\ref{tab:magnitude:decay}.\n\n\\begin{table}[!b]\n\t\\centering{}%\n{\\small\n\t\\begin{tabular}{c|ccccccc}\n\t\t$m$ & 0 & 1 & 2 & 3 & 4 & 5 & 6\\\\\n \\hline \n {\\Large\\strut}\n slow decay ($\\tilde{\\sigma}=2)$ & {1.0000} & {0.6079} & {0.1520} & {0.0675} & {0.0380} & {0.0243} & {0.0169}\\\\\n \\hline \n {\\Large\\strut}\n fast decay ($\\tilde{\\sigma}=4$) & {1.0000} & {0.9239} & {0.0577} & {0.0114} & {0.0036} & {0.0015} & {0.0007}\\\\\n\t\\end{tabular}\n\t\\caption{Magnitudes $\\left\\Vert a_{m}\\right\\Vert _{\\infty}$ of expansion coefficients~\\Refx{eq:ex5} for test problem~1.\n\t\n\t\t}\n\t\\label{tab:magnitude:decay}\n}\n\\end{table}\n\nWhile the magnitudes of $a_m$ ($m = 1,2,\\ldots$)\n(and hence, the importance of the corresponding parameters $y_m$, $m=1,2,\\ldots$)\ndecay significantly faster for $\\tilde\\sigma = 4$\n(e.g., $\\|a_1\\|_{\\infty} \\approx 16 \\|a_2\\|_{\\infty}$ for $\\tilde\\sigma = 4$, whereas\n$\\|a_1\\|_{\\infty} \\approx 4 \\|a_2\\|_{\\infty}$ for $\\tilde\\sigma = 2$),\nwe observe that the magnitude of $a_1$ is much closer\nto the magnitude of the mean field $a_0$ in the case of fast decay\nthan in the case of slow decay.\nThis suggests that for fast decay there holds $a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\approx\na_1({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)$, which in turn implies that equivalence \\Refx{eq:equiv} may\nhold for $r=1$ and with a small $\\varepsilon_1$. We therefore expect the performance of $P_1$ to\nbe superior in the fast decay case. This is indeed confirmed by the\niteration counts in Table~\\ref{tab:ex5:ideal:precond}. The results\nalso confirm the optimal performance of $P_r$ with respect to $k$.\n\nA similar behavior can be observed also for the case where the\npreconditioners $P_r$ are replaced by their symmetric block Gauss--Seidel approximations\n$\\tilde{P}_r$. Table~\\ref{tab:ex5:modified:precond} diplays the\ncorresponding iteration counts for a range of $r$, as well as for the\nmean-based preconditioner $P_0$ and Kronecker preconditioner\n$P_{\\otimes}$, for both fast and slow decay cases.\n\n\\begin{table}[!t]\n\\begin{center}\n{\\small\n\t\\begin{tabular}{r| *7{c} | *7{c}}\n\t\\multirow{2}{*}{$k$}& \\multicolumn{7}{c|}{fast decay} & \\multicolumn{7}{c}{slow decay} \\\\\n\t\\cline{2-15}\n\t{\\large\\strut} & $P_{0}$ & $P_{1}$ & $P_{2}$ & $P_{3}$ & $P_{4}$ & $P_{5}$ & $P_{6}$ &\n\t $P_{0}$ & $P_{1}$ & $P_{2}$ & $P_{3}$ & $P_{4}$ & $P_{5}$ & $P_{6}$\\\\\n\t\\hline \n\t1 & 13 & 4 & 3 & 3 & 2 & 2 & 2 & 10 & 6 & 4 & 4 & 4 & 3 & 3\\\\\n\t2 & 16 & 5 & 4 & 3 & 3 & 2 & 2 & 12 & 7 & 5 & 5 & 4 & 4 & 3\\\\\n\t3 & 21 & 6 & 4 & 3 & 3 & 2 & 2 & 14 & 7 & 6 & 5 & 4 & 4 & 4\\\\\n\t4 & 24 & 6 & 4 & 3 & 3 & 3 & 2 & 15 & 8 & 6 & 5 & 4 & 4 & 4\\\\\n\t\\end{tabular}\n\t\\caption{PCG iterations counts for test problem~1;\n\t$h=2^{-4}, M=8$.}\n\t\\label{tab:ex5:ideal:precond}\n}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!b]\n\\setlength\\tabcolsep{5.7pt} \n\\begin{center}\n{\\small\n\t\\begin{tabular}{r| *8{c} | *8{c}}\n\t\\multirow{2}{*}{$k$} & \\multicolumn{8}{c|}{fast decay} & \\multicolumn{8}{c}{slow decay} \\\\\n\t\\cline{2-17}\n\t{\\Large\\strut}& $P_{\\otimes}$ & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$ & $\\tilde P_{3}$ & $\\tilde P_{4}$ & $\\tilde P_{5}$ & $\\tilde P_{6}$ &\n\t $P_{\\otimes}$ & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$ & $\\tilde P_{3}$ & $\\tilde P_{4}$ & $\\tilde P_{5}$ & $\\tilde P_{6}$\\\\\n\t\\hline\n\t1 & 12 & 13 & 7 & 6 & 6 & 6 & 6 & 6\t\t & 9 & 10 & 6 & 5 & 5 & 5 & 5 & 5\\\\\n\t\t2 & 16 & 16 & 8 & 7 & 7 & 7 & 7 & 7\t\t & 12 & 12 & 7 & 6 & 6 & 6 & 5 & 5\\\\\n\t\t3 & 20 & 21 & 9 & 9 & 8 & 8 & 8 & 8\t\t & 14 & 14 & 8 & 7 & 6 & 6 & 6 & 6\\\\\n\t\t4 & 24 & 24 & 10 & 9 & 9 & 9 & 9 & 9\t\t & 15 & 15 & 9 & 7 & 7 & 6 & 6 & 6\\\\\n\t\t5 & 26 & 27 & 11 & 10 & 10 & 10 & 10 & 10\t & 16 & 16 & 9 & 7 & 7 & 7 & 6 & 6\\\\\n\t\t6 & 29 & 29 & 12 & 11 & 11 & 11 & 11 & 11\t & 17 & 17 & 10 & 8 & 7 & 7 & 7 & 7\\\\\n\t\\end{tabular}\n \\caption{PCG iterations counts for test problem~1;\n $h=2^{-4}, M=8$.\n }\n\t\\label{tab:ex5:modified:precond}\n }\n\\end{center}\n\\end{table}\n\n\\abrev{The results in Tables~\\ref{tab:ex5:ideal:precond}\n and~\\ref{tab:ex5:modified:precond} indicate that\nthe iteration counts corresponding to the approximations $\\tilde{P}_r$\nof the preconditioners $P_r$ are higher. This is expected given the\ntheoretical deterioration of the spectral bounds in\n\\Refx{eq:tildebounds1} as compared to the bounds in \\Refx{eq:bounds}}.\nHowever, all truncation preconditioners require fewer iterations than their mean-based and Kronecker product counterparts.\nThis improvement in the iteration counts is more pronounced\nin the case of fast decay than in the case of slow decay, which is consistent, in particular,\nwith how the magnitudes of expansion coefficients $a_m$ change with $m$\n(see Table~\\ref{tab:magnitude:decay} and the associated discussion above).\nFor example, the numbers of iterations for the truncation preconditioner $P_1$ (resp., $\\tilde P_1$)\nare less than those for the mean-based preconditioner $P_0$ by factors of 3 to 4\n(resp., by factors of about 2 to 2.5) in the case of fast decay.\nThis is because in this case, the expansion coefficient $a_1$ has approximately the same magnitude as the mean field.\nIn the case of slow decay, however, both $P_1$ and $\\tilde P_1$ outperform $P_0$\nin terms of the number of iterations only by a factor between 1.5 and 2.\nIt is worth recalling here that the computational cost for the truncation preconditioner $\\tilde P_1$\nis about twice the cost of the mean-based preconditioner $P_0$ (see~\\S\\ref{sec:implement});\nthus, in terms of the overall computational complexity, $\\tilde P_1$ performs at least the same as $P_0$;\nin the fast decay case, the overall computational cost for modified truncation preconditioners\nis significantly lower than that for $P_0$.\n\nIf more expansion coefficients are retained in $P_r$ ($r \\ge 2$),\nthen the iteration counts naturally (and consistently) decrease;\nin particular, they decrease quicker in the case of fast decay of coefficient amplitudes\nthan in the case of slow decay of the amplitudes; see Table~\\ref{tab:ex5:ideal:precond}.\nThis is again in agreement with what one might expect and reflects different decay patterns\nof $\\|a_m\\|_\\infty$ in each of these cases, as shown in Table~\\ref{tab:magnitude:decay}.\nHowever, when applying the corresponding modified truncation preconditioners $\\tilde P_r$ ($r \\ge 2$)\nand increasing the number $r$ of retained expansion coefficients, the iteration counts decrease very slowly\n(in the case of fast decay they even stagnate for $r \\ge 2$ in most cases);\nsee Table~\\ref{tab:ex5:modified:precond}. \\abrev{This indicates that\n no significant improvement is obtained by including additional terms\n $a_m$ in the definition of $P_r$.}\n\nThe above set of experiments demonstrates that the modified truncation preconditioners $\\tilde P_r$\nprovide sufficiently accurate approximations of stochastic Galerkin matrices and thus\ncan be used as effective practical preconditioners for linear systems arising from SGFEM approximations\nof the model problem~\\Refx{eq:strong:form} with \\emph{affine-parametric} representation of the diffusion coefficient.\nDepending on the decay of magnitudes of the expansion coefficients,\none may choose larger values of $r$ to improve the efficiency of the\nsolver (e.g., in the case of slow decay). However, in most cases, we recommend to choose $r=1$ or $r=2$.\n\nWe end the discussion of our first test problem with a numerical\nconfirmation of optimality of the modified truncation preconditioners\nwith respect to discretization parameters $h$ and $M$.\nWe consider again the cases of fast ($\\tilde{\\sigma}=4$) and slow ($\\tilde{\\sigma}=2$)\ndecay of coefficient amplitudes $\\bar\\alpha m^{-\\tilde\\sigma}$ in~\\Refx{eq:ex5} and\nemploy three preconditioners:\nthe mean-based $P_0$ and the modified truncation preconditioners\n$\\tilde P_1$ and $\\tilde P_2$.\n\\abrev{Note that other modified\ntruncation preconditioners (for $r>2$) yield similar performance and\nthe corresponding results are not included here. We chose to work with two values of $M \\in\n\\{4,\\,8\\}$ and several uniform subdivisions into squares of side\nlengths ranging from $h=2^{-3}$ to $h=2^{-7}$, \nwhile keeping fixed the polynomial degree $k = 3$.}\n\\abrev{The results of these computations are presented in Table~\\ref{tab:ex5:hM-optimality}.\nThese results show that indeed,\nthe iteration counts do not change as $M$ increases from 4 to 8\n(for the same value of $h$) and\nas the spatial mesh gets sufficiently refined (for the same value of~$M$).\n}\n\n\\begin{table}[!t]\n\\begin{center}\n{\\small\n\t\\begin{tabular}{c | *3{c} | *3{c} || *3{c} | *3{c}}\n\t\t\t\t & \\multicolumn{6}{c||}{fast decay} & \\multicolumn{6}{c}{slow decay} \\\\\n\t\\cline{2-13}\n\t\\multirow{2}{*}{$h$} & \\multicolumn{3}{c|}{{\\large\\strut} $M=4$} & \\multicolumn{3}{c||}{$M=8$}\n\t & \\multicolumn{3}{c|}{$M=4$} & \\multicolumn{3}{c}{$M=8$} \\\\\n\t\\cline{2-13}\n\t{\\Large\\strut} & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$ & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$\n\t\t\t & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$ & $P_{0}$ & $\\tilde P_{1}$ & $\\tilde P_{2}$\\\\\n\t\\hline\n\t\t{\\large \\strut}\\!\\!\n\t\t$2^{-3}$ & 18 & 8 & 8\t\t & 18 & 8 & 8 \t& 13 & 7 & 6\t\t & 13 & 7 & 6 \\\\\n\t\t$2^{-4}$ & 21 & 9 & 9 \t & 21 & 9 & 9\t& 14 & 8 & 7 \t & 14 & 8 & 7\\\\\n\t\t$2^{-5}$ & 23 & 10 & 9 \t & 23 & 10 & 9\t& 14 & 8 & 7 \t & 15 & 8 & 7\\\\\n\t\t$2^{-6}$ & 24 & 10 & 10\t & 24 & 10 & 10\t& 15 & 8 & 7\t \t & 15 & 8 & 7\\\\\n\t\t$2^{-7}$ & 24 & 10 & 10\t & 24 & 10 & 10\t& 15 & 8 & 7\t \t & 15 & 8 & 7\\\\\n\t\\end{tabular}\n\t\\caption{PCG iterations counts for test problem~1\n\tfor various $h$ and $M$, for $k=3$.}\n\t\\label{tab:ex5:hM-optimality}\n}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Test problem 2: non-affine parametric diffusion coefficient} \\label{sec:numerics:non-affine}\n\nConsider again model problem \\Refx{eq:strong:form}, now\nwith the following truncated \\emph{lognormal} diffusion coefficient\n\\begin{equation}\n a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) := \\exp\\!\\big(b({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)\\big) = \\exp\\!\\bigg(b_0({\\bm{x}}}\\def \\y{{\\bm{y}}) + \\sum_{m=1}^N\n b_m({\\bm{x}}}\\def \\y{{\\bm{y}})y_m\\bigg),\\ \\ {\\bm{x}}}\\def \\y{{\\bm{y}} \\in \\Omega,\\ \\ \\y \\in \\bGamma := \\prod_{m=1}^N\\Gamma_m,\n \\label{eq:ex5log}\n\\end{equation}\nwhere $b_0,\\,b_m \\in L^{\\infty}(\\Omega)$ for all $m=1,\\ldots,N$ and \nthe parameters $y_m \\in \\Gamma_m := \\RR$\nare the images of independent normally distributed random variables with zero mean and unit variance.\nAccordingly, $p_m$ now denotes the standard Gaussian probability density function,\nand the joint probability density function is $p(\\y)=\\prod_{m=1}^N p_m(y_m)$.\nThe well-posedness of weak formulation~\\Refx{eq:weak:form} in this case has been studied in~\\cite{Charrier_12_SWE}.\n\nAs a polynomial basis of $L^2_\\pi(\\bGamma)$ we choose the set of scaled Hermite polynomials\n$\\{\\psi_{{\\bm{\\alpha}}}:\\bGamma\\rightarrow\\RR : {\\bm{\\alpha}}\\in\\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0^N\\}$\northonormal with respect to the inner product $\\langle\\cdot,\\cdot\\rangle_{\\pi}$.\nIn this basis, the diffusion coefficient~\\Refx{eq:ex5log} has the representation\n\\[\n a({\\bm{x}}}\\def \\y{{\\bm{y}},\\y):=\\sum_{{{\\bm{\\alpha}} \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0^N}} a_{{\\bm{\\alpha}}}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\psi_{{\\bm{\\alpha}}}(\\y)\n\\]\nwith (cf.~\\cite[p.~926]{Ullmann10})\n\\begin{equation}\n a_{\\bm{\\alpha}}({\\bm{x}}}\\def \\y{{\\bm{y}}) = \\langle a({\\bm{x}}}\\def \\y{{\\bm{y}},\\cdot), \\psi_{{\\bm{\\alpha}}}\\rangle_{\\pi} =\n \\mathbb{E}[a({\\bm{x}}}\\def \\y{{\\bm{y}},\\cdot)] \\prod_{m\\in\\spp{\\bm{\\alpha}}}\\frac{b_{m}^{\\alpha_{m}}(\\boldsymbol{x})}{\\sqrt{\\alpha_{m}}}\n \\quad\n \\forall\\,{\\bm{\\alpha}} \\in \\mathbb{N}}\\def\\RR{\\mathbb{R}}\\def\\PP{\\mathbb{P}}\\def\\I{\\mathbb{I}_0^N,\n \\label{eq:t-alpha}\n\\end{equation}\nwhere\n\\[\n \\mathbb{E}[a({\\bm{x}}}\\def \\y{{\\bm{y}},\\cdot)] =\n \\int_{\\bGamma} \\exp\\sep{b({\\bm{x}}}\\def \\y{{\\bm{y}},\\y)} p(\\y) \\dd\\y =\n \\exp\\bigg(b_{0}({\\bm{x}}}\\def \\y{{\\bm{y}})+\\frac{1}{2}\\sum_{m=1}^N b_{m}^{2}({\\bm{x}}}\\def \\y{{\\bm{y}})\\bigg) > 0\\ \\ \\text{a.e. in $\\Omega$}.\n\\]\n\nLet $M\\norm{a_{{\\bm{\\alpha}}_j}}{\\infty},\\quad i < j.\n\\]\nUnlike in the affine case, this ordering is not sufficient to ensure\npositivity of the truncated diffusion coefficient\n\\begin{equation}\n a_r({\\bm{x}}}\\def \\y{{\\bm{y}},\\y) :=\n \\sum_{\\ell=0}^r a_{{\\bm{\\alpha}}_\\ell}({\\bm{x}}}\\def \\y{{\\bm{y}})\\psi_{{\\bm{\\alpha}}_\\ell}(\\y),\\quad 0 \\le r \\le {\\rm card\\,}{\\I_{2k}^M} - 1.\n \\label{eq:lognormal:truncation}\n\\end{equation}\nConsequently, the resulting truncation preconditioner\n\\begin{equation}\n P_r :=\n \\sum_{\\ell=0}^{r}G_{{\\bm{\\alpha}}_\\ell}\\otimes K_{{\\bm{\\alpha}}_\\ell}\n \\label{eq:lognormal:Pr}\n\\end{equation}\nis not guaranteed to be positive definite.\nHowever, as in the affine case, we replace $P_{r}$\nby its symmetric block Gauss--Seidel approximation $\\tilde{P}_{r}$.\nAs demonstrated in Proposition~\\ref{prop:lognormal:SBGS} below, the modified truncation preconditioner $\\tilde{P}_{r}$ is positive definite,\nprovided that the mean field $a_{\\bf 0}({\\bm{x}}}\\def \\y{{\\bm{y}}) = \\mathbb{E}[a({\\bm{x}}}\\def \\y{{\\bm{y}},\\cdot)]$ is included in the\ntruncation~\\Refx{eq:lognormal:truncation}.\nIt is important to note that the complexity associated with the action of $\\tilde{P}_{r}^{-1}$\nremains unchanged from the affine case; cf.~\\S\\ref{sec:implement}.\n\n\\begin{proposition} \\label{prop:lognormal:SBGS}\nLet $a$ be the lognormal diffusion coefficient given by~\\Refx{eq:ex5log}.\nLet $0 \\le r \\le {\\rm card\\,}{\\I_{2k}^M} - 1$ and assume that the truncated diffusion coefficient $a_r$ in~\\Refx{eq:lognormal:truncation}\nsatisfies $\\mathbb{E}\\left[a_r\\right] = \\mathbb{E}\\left[a\\right] = a_{\\bf 0} > 0$ a.e. in $\\Omega$.\nThen the truncation preconditioner $P_r$ given by~\\Refx{eq:lognormal:Pr} can be represented as\n\\begin{equation}\n P_r = D + L + L^T,\n \\label{eq:lognormal:Pr:decomp}\n\\end{equation}\nwhere $D$ is a block-diagonal symmetric positive definite matrix and\n$L$ is a strictly lower block-triangular matrix.\nAs a consequence, the SBGS approximation of $P_r$ defined~by\n\\[\n \\tilde P_r := (D + L)\\, D^{-1} (D + L^T)\n\\]\nis a symmetric positive definite matrix.\n\\end{proposition}\n\n\\begin{proof}\nDenote by $\\I_{\\rm \\,even}$ the set of multi-indices in $\\I_{2k}^M$ with even entries, i.e.,\n\\[\n \\I_{\\rm \\,even} := \n \\left\\{\n {\\bm{\\alpha}} = (\\alpha_1,\\ldots,\\alpha_M) \\in \\I_{2k}^M;\\; \\alpha_m\\ \\text{is even for all}\\ m = 1,\\ldots,M\n \\right\\}.\n\\]\nLet ${\\bm{\\alpha}} \\in \\I_{\\rm \\,even}$.\nIt follows from~\\Refx{eq:t-alpha} that $a_{\\bm{\\alpha}}({\\bm{x}}}\\def \\y{{\\bm{y}}) \\ge 0$ a.e. in $\\Omega$.\nTherefore, the stiffness matrices $K_{\\bm{\\alpha}}$ for ${\\bm{\\alpha}} \\in \\I_{\\rm \\,even}$ are symmetric positive semi-definite;\nin particular, the matrix $K_{\\bf 0}$ is symmetric positive definite.\nSince all diagonal elements of the matrix $G_{\\bm{\\alpha}}$ are nonnegative, \neach nonzero diagonal block of $G_{\\bm{\\alpha}} \\otimes K_{\\bm{\\alpha}}$ is a symmetric positive semi-definite matrix.\nIn particular, the block-diagonal matrix $G_{\\bf 0} \\otimes K_{\\bf 0} = I \\otimes K_{\\bf 0}$ is symmetric positive definite.\n\nNow let ${\\bm{\\alpha}} \\in \\I_{2k}^M \\setminus \\I_{\\rm \\,even}$.\nIn this case, all diagonal elements of the matrix $G_{\\bm{\\alpha}}$ are zeros.\nIndeed, for any $j = 1,\\ldots,N_\\y$, one has\n\\[\n \\left[G_{{\\bm{\\alpha}}}\\right]_{jj} =\n \\big\\langle \\psi_{{\\bm{\\alpha}}} \\psi_{\\bm{\\kappa}(j)}, \\psi_{\\bm{\\kappa}(j)}\\big\\rangle _{\\pi} =\n \\big\\langle \\psi_{{\\bm{\\alpha}}}, \\psi^2_{\\bm{\\kappa}(j)}\\big\\rangle _{\\pi} = 0,\n\\]\nbecause there exists $m^* \\in \\{1,\\ldots,M\\}$ such that $\\alpha_{m^*}$ is odd and the associated univariate Hermite polynomial\nis an odd function.\nTherefore, in this case, all diagonal blocks of $G_{\\bm{\\alpha}} \\otimes K_{\\bm{\\alpha}}$ are zero matrices.\n\nOverall, by combining the above observations\nand using the assumption that $\\mathbb{E}\\left[a_r\\right] = \\mathbb{E}\\left[a\\right]$,\nwe conclude that the diagonal blocks of the truncation preconditioner $P_r$ in~\\Refx{eq:lognormal:Pr}\nare symmetric positive definite matrices. This proves~\\Refx{eq:lognormal:Pr:decomp}.\n\nIt is now easy to see that the SBGS approximation of $P_r$ is a positive definite matrix.\nIndeed, for any nonzero vector $\\bv$ there holds\n\\[\n \\bv^T \\tilde P_r \\bv = \\bv^T (D + L)\\, D^{-1} (D + L^T) \\bv = \\w^T D^{-1} \\w > 0\n\\]\nwith nonzero $\\w := (D + L^T) \\bv$.\n\\end{proof}\n\n\\begin{table}[!t]\n\t\\centering{}%\n\t\\begin{tabular}{c|c|c}\n\t\t$\\ell$ & ${\\bm{\\alpha}}_\\ell$ & $\\norm{a_{{\\bm{\\alpha}}_\\ell}}{\\infty}$\\tabularnewline\n\t\t\\hline \n\t\t0 & (0,0,0,0,0,0) & 3.20\\tabularnewline\n\t\t1 & (1,0,0,0,0,0) & 1.75\\tabularnewline\n\t\t2 & (2,0,0,0,0,0) & 0.68\\tabularnewline\n\t\t3 & (0,1,0,0,0,0) & 0.44\\tabularnewline\n\t\t4 & (1,1,0,0,0,0) & 0.24\\tabularnewline\n\t\t5 & (3,0,0,0,0,0) & 0.21\\tabularnewline\n\t\t6 & (0,0,1,0,0,0) & 0.19\\tabularnewline\n\t\t7 & (0,0,0,1,0,0) & 0.11\n\t\\end{tabular}\n\t\\caption{Multi-indices of first 8 largest magnitudes $\\norm{a_{{\\bm{\\alpha}}}}{\\infty}$ for test problem 2; $M = k = 6$.}\n\t\\label{tab:magnitude:non-affine}\n\\end{table}\n\nIn numerical experiments, we set $N = 20$ and chose $b_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ in~\\Refx{eq:ex5log}\nto be the coefficients $a_m({\\bm{x}}}\\def \\y{{\\bm{y}})$ in test problem~1 as defined in~\\Refx{eq:ex5}\nwith $\\tilde{\\sigma}=2$ and $\\bar{\\alpha}=0.547$.\nIn Table~\\ref{tab:magnitude:non-affine}, for $M = k = 6$, we show first eight multi-indices in the sequence\n$\\seq{{\\bm{\\alpha}}_\\ell}$\nand the corresponding coefficient magnitudes $\\norm{a_{{\\bm{\\alpha}}_\\ell}}{\\infty}$.\nWe see that in this example, the coefficient with the largest magnitude is the mean field, i.e., $a_{{\\bm{\\alpha}}_0} = a_{\\bf 0}$.\nWhile the distribution of indices inducing the ordering does not display any obvious pattern,\nwe note a fast decay with $\\ell$ in the magnitudes recorded, which is similar to the affine~case.\n\nTable~\\ref{tab:numeric:non-affine} displays the PCG iteration counts corresponding to solving linear systems arising from SGFEM\ndiscretizations of the described test problem.\nWe used the following discretization parameters: $h \\,{=}\\, 2^{-4}$, $M \\,{=}\\, 6$, and $k \\,{\\in}\\, \\{1,\\ldots,6\\}$.\nIn our experiments, we employed the modified truncation preconditioners $\\tilde P_r$ with $r \\in \\{1,\\ldots,6\\}$,\nalongside the mean-based ($P_0$) and Kronecker ($P_\\otimes$) preconditioners.\nThese experiments included cases where preconditioners $P_{r}$ defined by~\\Refx{eq:lognormal:Pr} were not positive definite\n(in Table~\\ref{tab:numeric:non-affine},\nthe iteration counts for such cases are shown in~boldface).\n\n\\begin{table}[!b]\n\t\\centering{}%\n\t\\begin{tabular}{r|cccccccc}\n\t\t $k$ & $P_{\\otimes}$ & $P_{0}$ & $\\tilde{P}_{1}$ & $\\tilde{P}_{2}$ & $\\tilde{P}_{3}$ & $\\tilde{P}_{4}$ & $\\tilde{P}_{5}$ & $\\tilde{P}_{6}$\\\\\n\t\t\\hline \n\t\t1 & 12 & 12 & 6 & 7 & 6 & 6 & 6 & 6\\\\[-1pt]\n\t\t2 & 18 & 19 & 8 & 10 & 9 & 9 & 8 & 8\\\\[-1pt]\n\t\t3 & 25 & 26 & \\bf{10} & 12 & 11 & 11 & 10 & 10\\\\[-1pt]\n\t\t4 & 32 & 34 & \\bf{13} & 15 & 13 & 13 & 12 & 11\\\\[-1pt]\n\t\t5 & 40 & 43 & \\bf{17} & 19 & 16 & 17 & \\bf{13} & \\bf{12}\\\\[-1pt]\n\t\t6 & 49 & 52 & \\bf{24} & 22 & 19 & 20 & \\bf{14} & \\bf{14}\n\t\\end{tabular}\n\t\\caption{PCG iterations counts\n\t for test problem 2; $h=2^{-4}$, $M=6$.}\n\t\\label{tab:numeric:non-affine}\n\\end{table}\n\nThe results in Table \\ref{tab:numeric:non-affine} indicate that the numbers of iterations by the modified truncation preconditioners\nare significantly lower than those corresponding to the mean-based and Kronecker preconditioners\n(it is worth noting here that while the computational cost for $P_0$ and $\\tilde P_r$ remains unchanged from the affine case,\nthe cost for $P_{\\otimes}$\nin this test problem will be significantly higher than in the affine case,\ndue to the density of the matrix $G$ in~\\Refx{eq:kron} for the lognormal diffusion coefficient).\nFor all preconditioners, the experiments show that the iteration counts grow with $k$,\nalthough this growth is much less pronounced for\ntruncation preconditioners.\nFurthermore, while we see only a negligible improvement with increasing $r$ for $k = 1,\\ldots,4$,\nthis becomes more pronounced for higher polynomial degrees ($k = 5,6$).\n\n\n\\section{Summary and future work} \\label{sec:conclusions}\n\nEfficient solution of large coupled linear systems is a key ingredient in successful implementation\nof the stochastic Galerkin finite element method.\n\\rev{Truncation preconditioners\nrepresent} a competitive alternative to\nexisting solvers relying on the mean-based and Kronecker preconditioners.\nOur theoretical analysis shows that for elliptic problems with \\emph{affine-parametric} coefficients,\ntruncation preconditioners are optimal with respect to discretization parameters.\nOur numerical experiments confirm this, while also demonstrating the improvement in the iteration count\nwhen compared with the mean-based and Kronecker preconditioners.\n\nOn a practical note, the superior efficiency of \\rev{considered} solvers requires,\ncrucially, suitable fast (possibly parallel) implementation of the corresponding symmetric block Gauss--Seidel approximations,\nwhich were also analyzed and shown to be optimal.\nFor simplicity, we considered a model diffusion problem, however, the analysis included in this work can be\nextended in a straightforward manner to the general case of elliptic PDE with parametric or uncertain inputs, under standard assumptions.\n\nWe have also\n\\rev{applied truncation preconditioners in}\nthe case of \\emph{non-affine} (specifically, lognormal) diffusion coefficient.\nThe numerical experiments suggest this is a promising approach.\nTheoretical analysis of truncation preconditioners for this class of parametric problems will be the focus of future research on the~topic.\n\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn algebraic geometry, we often encounter singularities\nwhich are quotients of other singularities by algebraic groups.\nOrbifold singularities are finite quotients of smooth points, \ntoric singularities are abelian quotients of smooth points~\\cite{CLS11}, and\nterminal $3$-fold singularities are finite quotients of \nhypersurface singularities~\\cite{Rei87}.\nFurthermore,\nmany interesting\nfactorial singularities are \n${\\rm SL}_n(\\mathbb{K})$-quotients of smooth points~\\cite{Bra21, Bra21b}.\nIn most cases, the respective group is reductive~\\cite{Nag61}. \nIndeed, the reductivity assumption is what \nensures that the quotient is of finite type. \nReductive quotients preserve normal singularities\nand rational singularities~\\cite{Bou78}.\nRecently, together with Greb and Langlois,\nthe authors proved that reductive quotients preserve the singularities\nof the minimal model program~\\cite{BGLM21}, the so-called klt type singularities~\\cite{Kol13}.\n\nIn this article, we study a central topic in algebraic geometry:\nhow to improve the singularities of an algebraic variety\nby taking appropriate covers. \nWe focus on the singularities of the minimal model program.\nTo tackle this question, we need to comprehend what type of covers will indeed \nimprove the singularities that we are studying.\nThis will lead us to the concepts of $G$-covers\nand $G$-quasi-torsors.\n\n\\subsection{G-covers of klt singularities}\nLet $G$ be an algebraic group.\nA $G$-cover of a singularity $(X;x)$ is an\nalgebraic singularity $(Y;y)$ endowed with a $G$-action fixing $y$ \nso that $X$ is isomorphic to the quotient \n$Y\/\\!\\!\/G$ and $x$ is the image of $y$ (see Definition~\\ref{def:G-cover-local}). \nIn this setting, we say that $(Y;y)$ is a {\\em $G$-cover} of\n$(X;x)$ and we say that $(X;x)$ is a {\\em $G$-quotient} of $(Y;y)$.\nOne can think about a $G$-cover of a singularity as a degenerate principal $G$-bundle\nover the singularity having maximal degeneration at the \ndistinguished singular point.\n$G$-covers often occur in singularity theory;\nwhen replacing a singularity with\nits universal cover~\\cite{Bra20, BFMS22, LLM19}, \n when taking the index one cover with respect to a $\\mathbb{Q}$-Cartier divisor~\\cite{KM98}, and \nwhen taking the Cox ring of the singularity~\\cite{Bra19, BM21}.\n$G$-covers are often useful to compute invariants of singularities~\\cite{Mor20d}.\nThus, it is natural to ask whether \na class of singularities is preserved by $G$-covers.\nOf course, the answer to this question depends on the choice of $G$.\nThe first question that we settle on in this article is\nwhether the class of klt type singularities\nis closed under reductive covers. \nOur first theorem is a negative answer to this question. \nWe show the existence of a $3$-dimensional toric singularity\nadmitting a $5$-dimensional $\\mathbb{P}{\\rm GL}_3(\\mathbb{K})$-cover which is not of\nklt type. \n\n\\begin{introthm}\\label{introthrm:gl2-cover}\nThere exists a $3$-fold toric singularity $(X;x)$\nthat admits a $\\mathbb{P}{\\rm GL}_3(\\mathbb{K})$-cover \n$(Y;y)\\rightarrow (X;x)$\nfrom a $5$-dimensional \nsingularity $(Y;y)$\nwhich is not of klt type.\n\\end{introthm}\n\nIn Proposition~\\ref{prop:gln-cover-free}, \nwe give further examples in this direction in which\nthe group $\\mathbb{P}{\\rm GL}_n(\\mathbb{K})$ acts\nfreely on an open subset $Y$. \nHowever, the singularities are of higher dimension in these cases.\n\nThe previous theorem is the local analog of the well-known fact that\nprojective bundles over Fano type varieties may not be of Fano type.\nIndeed, there exist projective bundles over Fano type surfaces\nthat are not Mori dream spaces~\\cite{GHPS12}. \nHowever, it is known that split projective bundles over Fano\ntype varieties are Fano type~\\cite{BM21}. \nFurthermore, finite covers of Fano type varieties \nare again Fano type varieties, under a restrictive hypothesis\nin case there is ramification in codimension one (see, e.g.,~\\cite[Lemma 3.18]{Mor20c}).\nThese two facts motivate the proof of the following theorem.\n\n\\begin{introthm}\\label{introthm:torus-finite-cover}\nLet $(X;x)$ be a klt type singularity. \nLet $G$ be a finite extension of a torus and \n$Y\\rightarrow X$ be a \n$G$-quasi-torsor.\nThen $(Y;y)$ is a klt type singularity.\n\\end{introthm}\n\nA {\\em $G$-quasi-torsor} is a special kind of $G$-cover which behaves\nlike a $G$-torsor outside codimension two subsets of $X$ \\emph{and} $Y$. \n$G$-quasi-torsors are also called {\\em almost principal fiber bundles} in the literature.\nHence, the class of klt type singularities is preserved\nunder reductive quotients \nand under $G$-quasi-torsors, whenever $G$ is a finite extension of a torus.\nWe emphasize that the condition \non the ramification is necessary: \neven finite covers of a smooth point\nwith codimension one ramification\nmay not be of klt type (see Example~\\ref{ex:cod-1-ram}).\nNote that $G$ being a finite extension of a torus \nis equivalent to asking that the derived subgroup of its connected component is trivial~\\cite{Hum75}.\nIt is an open problem to decide\nwhether the previous statement holds\nfor $G$ a reductive group (see Question~\\ref{quest:G-qt-klt-type}).\nWith the previous theorem,\nwe have found the right type of covers that can improve\nour klt type singularity: finite quasi-torsors and torus quasi-torsors. \nWhenever these are torsors, i.e., finite \\'etale covers and toric bundles, the class of singularities of our variety will not change.\nThus, we are mostly interested in the finite quasi-torsors and $\\mathbb{T}$-quasi-torsors that are not torsors.\nThese are the covers for which the \\'etale class of a singularity may change. \nMoreover, these are exactly the covers detected by the\nregional fundamental group of the singularity\nand by the local Cox ring of the singularity (see~\\cite{Bra20} and Definition~\\ref{def:local-Cox-ring}).\n\n\\subsection{Torus covers of klt varieties}\nAs mentioned above,\none way to improve the singularities\nof a variety is to produce\n$\\mathbb{T}$-quasi-torsors.\nFor instance, all toric varieties\nare quotients of a smooth affine variety\nby the action of an abelian linear algebraic group.\nIn a similar vein, the local Cox ring of a singularity often simplifies the singularity.\nA natural way to obtain a $\\mathbb{T}$-quasi-torsor of a \nvariety is to mimic the Cox ring construction.\nFor example, \nif we consider Weil divisors\n$W_1,\\dots,W_k$ in $X$\nspanning the subgroup $N$ of ${\\rm WDiv}(X)$, then we can define the sheaf\n\\[\n\\mathcal{R}(X)_N := \\bigoplus_{(m_1,\\dots,m_k)\\in \\mathbb{Z}^k}\n\\mathcal{O}_X(m_1W_1+\\dots+m_kW_k).\n\\] \nThen, the relative spectrum \n\\[\nY := {\\rm Spec}_X( \\mathcal{R}(X)_N) \\rightarrow X, \n\\]\nadmits a natural $\\mathbb{T}$-cover structure over $X$\nwhich is a $\\mathbb{T}$-quasi-torsor.\nHere, $\\mathbb{T}$ is a $k$-dimensional algebraic torus \nand the action of $\\mathbb{T}$ on $Y$ is induced by the $\\mathbb{Z}^k$-grading of the sheaf $\\mathcal{R}(X)_N$.\nNote that $Y\\rightarrow X$ is a $\\mathbb{T}$-torsor precisely at the points at which all the $W_i$'s are Cartier divisors.\nThe variety $Y$ will be called a {\\em relative Cox space} of $X$.\nIndeed, this relative version of the Cox space\nlocally behaves like the Cox space of the singularities of $X$.\n\nOur next theorem states that every\n$\\mathbb{T}$-quasi-torsor over a normal variety is equivariantly isomorphic to a relative Cox space.\n\n\\begin{introthm}\\label{introthm:cox-space-vs-g-covers}\nLet $X$ be a normal variety.\nLet $Y\\rightarrow X$ be a $\\mathbb{T}$-quasi-torsor.\nThen, we can find Weil divisors $W_1,\\dots,W_k$ on $X$\nfor which there\nis a $\\mathbb{T}$-equivariant isomorphism\n\\[\nY \\simeq {\\rm Spec}_X\\left( \\bigoplus_{(m_1,\\dots,m_k)\\in \\mathbb{Z}^k}\n\\mathcal{O}_X(m_1W_1+\\dots+m_kW_k)\n\\right).\n\\]\n\\end{introthm}\n\nTheorem~\\ref{introthm:torus-finite-cover} implies that a $\\mathbb{T}$-quasi-torsor over a klt type singularity is again of klt type. \nOn the other hand, Theorem~\\ref{introthm:cox-space-vs-g-covers}, implies that a relative Cox ring of a normal variety\nis equivariantly isomorphic to a $\\mathbb{T}$-quasi-torsor.\nCombining these two results, we obtain that a relative Cox ring of a klt type variety \nalso has klt type singularities.\n\n\\begin{introthm}\\label{introthm-finite-torus-cover-klt-var}\nLet $X$ be a variety with klt type singularities.\nLet $Y\\rightarrow X$ be a relative Cox ring.\nThen $Y$ has klt type singularities.\n\\end{introthm}\n\nIn summary, the class of $\\mathbb{T}$-quasi-torsors\nof klt type varieties agrees with the class of relative Cox spaces.\nFurthermore, the Cox spaces again have klt type singularities.\nIn Example~\\ref{ex:sing-improve}, we show that the singularities\ncan indeed improve by taking the relative Cox ring. \n\nIn~\\cite[Theorem 1.1]{GKP16}, the authors show that any sequence of finite quasi-torsors over a variety with klt type singularities\nis eventually a sequence of finite torsors.\nThis means that all but finitely many of the finite quasi-torsors are torsors, i.e., \nfinite Galois \\'etale covers.\nIt is natural to ask if a similar principle holds\nfor $\\mathbb{T}$-quasi-torsors.\nIn this direction, we prove a torus version\nof the theorem due to Greb, Kebekus, and Peternell.\n\n\\begin{introthm}\\label{introthm:iteration-torus-quasi-torsors}\nLet $X$ be a variety with klt type singularities.\nConsider a sequence of morphisms:\n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\] \nsuch that each $\\phi_i\\colon X_i\\rightarrow X_{i-1}$\nis a $\\mathbb{T}$-quasi-torsor.\nThen, there exists $j$ such that\nfor every $i\\geq j$\nthe morphism $\\phi_j$ is a \n$\\mathbb{T}$-torsor.\n\\end{introthm}\n\n\\subsection{Iteration of torus and finite covers} \nOur previous theorem states that \nany sequence of $\\mathbb{T}$-quasi-torsors \nover a variety with klt type singularities\nis eventually a sequence of $\\mathbb{T}$-torsors.\nIt is natural to investigate what happens for sequences of $\\mathbb{T}$-quasi-torsors\nand finite quasi-torsors, i.e.,\nto study mixed sequences of torus and finite covers.\nIn this direction, we will show that \nall but finitely many of the finite quasi-torsors are indeed torsors.\n\n\\begin{introthm}\n\\label{thm:iteration1}\nLet $X$ be a variety with klt type singularities.\nConsider a sequence of morphisms:\n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{i+2}&\n}\n\\]\nsuch that each\n$\\phi_i$ is either a finite quasi-torsor or a torus quasi-torsor.\nThen, all but finitely many of the finite quasi-torsors are torsors, i.e., finite \\'etale Galois morphisms.\n\\end{introthm}\n\nNote that the previous theorem gives a generalization of~\\cite[Theorem 1.1]{GKP16}.\nIndeed, if we require that each $\\phi_i$ is a finite quasi-torsor, then we recover this statement.\nIt is natural to wonder whether in the context of the previous theorem we can further obtain that all but finitely many of the quasi-torsors are torsors. \nFirst, notice that such a statement holds trivially for smooth varieties. \nIndeed, by the purity of the branch locus, every finite quasi-torsor over a smooth variety is a finite torsor.\nOn the other hand, by Theorem~\\ref{introthm:cox-space-vs-g-covers}, every $\\mathbb{T}$-quasi-torsor over a smooth variety is a $\\mathbb{T}$-torsor.\nHence, in order to produce interesting sequences of quasi-torsors we need to consider singular varieties.\nToric singularities are arguably the simplest kind of klt singularities because of their combinatorial nature.\nThe following theorem shows that even for varieties with toric singularities, we may produce infinite sequences of finite quasi-torsors and $\\mathbb{T}$-quasi-torsors\nso that infinitely many of the $\\mathbb{T}$-quasi-torsors are not torsors.\n\n\\begin{introthm}\n\\label{thm:iteration2}\nFor each $n\\geq 2$,\nthere exists an $n$-dimensional\nprojective variety $X^n$\nwith toric singularities\nand an infinite sequence of morphisms:\n\\[\n \\xymatrix@R=2em@C=2em{\nX^n=X^n_0 &\nX^n_1\\ar[l]_-{\\phi_1} &\nX^n_2\\ar[l]_-{\\phi_2} &\nX^n_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX^n_i\\ar[l]_-{\\phi_i} &\nX^n_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{i+2}&\n}\n\\]\nsuch that the following conditions hold:\n\\begin{enumerate}\n \\item each $\\phi_i$ is either a finite quasi-torsor or a $\\mathbb{T}$-quasi-torsor, \n \\item infinitely many of the $\\phi_i$'s are finite torsors, and \n \\item infinitely many of the $\\phi_i$'s are $\\mathbb{T}$-quasi-torsors that are not torsors.\n\\end{enumerate}\n\\end{introthm}\n\nNote that (2) in the previous theorem is implied by\nTheorem~\\ref{introthm:iteration-torus-quasi-torsors}.\nThus, the importance relies on (3).\nIt shows that a full generalization of Greb-Kebekus-Peternell to the case of $\\mathbb{T}$-quasi-torsors and finite quasi-torsors is not feasible.\nOur final statement in this subsection says that this failure can be fixed if we restrict ourselves to a special class of $\\mathbb{T}$-quasi-torsors.\nA $\\mathbb{T}$-quasi-torsor $Y\\rightarrow X$ is said to be {\\em factorial} if the variety $Y$ is factorial.\n\n\\begin{introthm}\n\\label{thm:iteration3}\nLet $X$ be a variety with klt type singularities.\nConsider a sequence of morphisms\n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nsuch that each $\\phi_i$ is either\na finite quasi-torsor\nor a factorial $\\mathbb{T}$-quasi-torsor.\nThen, all but finitely many of the $\\phi_i$ are torsors.\n\\end{introthm}\n\n\\subsection{Factorial models}\nIn this section, motivated by the previous statement, we study factorial covers of klt varieties. \nIn~\\cite[Theorem~1.5]{GKP16}, the authors prove that a variety $X$ with klt type singularities admits\na quasi-\\'etale finite Galois cover\n$Y\\rightarrow X$ for which\n$\\pi_1(Y^{\\rm reg})$ is isomorphic to $\\pi_1(Y)$.\nIn particular, every \\'etale cover of $Y^{\\rm reg}$ extends to an \\'etale cover\nof $Y$.\nOur next aim is to improve this result\nby considering both; finite quasi-torsors and torus quasi-torsors. \nBy doing so, we can also improve the local class groups of the variety $Y$ obtained by Greb, Kebekus, and Peternell.\nWe show that any variety with klt type singularities\nis a $G$-quotient of a variety with canonical factorial singularities\nfor which its \\'etale fundamental group\nagrees with the \\'etale fundamental group of its smooth locus.\n\n\\begin{introthm}\\label{introthm:factorial-cover}\nLet $X$ be a variety\nwith klt type singularities.\nThen, there exists a variety $Y$ \nsatisfying the following conditions:\n\\begin{enumerate}\n \\item the natural epimorphism\n $\\hat{\\pi}_1(Y^{\\rm reg})\\rightarrow \\hat{\\pi}_1(Y)$ of \\'etale fundamental groups\n is an isomorphism, \n \\item for every finite quasi-\\'etale morphism $Y'\\rightarrow Y$ the variety $Y'$ has canonical factorial singularities,\n \\item $Y$ admits the action of a reductive group $G$, \n \\item the group $G$ is the extension of an algebraic torus by a finite solvable group, and\n \\item the isomorphism $X\\simeq Y\/\\!\\!\/G$ holds.\n\\end{enumerate}\nIn particular, $Y$ itself has canonical factorial singularities.\n\\end{introthm}\n\nIn general, this factorial variety is highly non-unique.\nThe previous theorem can be regarded as a generalization of~\\cite[Theorem~1.5]{GKP16}. Theorem~\\ref{introthm:factorial-cover}.(1) follows from~\\cite[Theorem~1.1]{GKP16}.\nAn equivalent statement of the latter does not hold for combinations\nof finite and torus covers (c.f.~Theorem~\\ref{thm:iteration2}).\nHowever, the statement is still valid if we restrict ourselves to finite quasi-torsors \nand factorial torus quasi-torsors.\nThus, we may still apply Theorem~\\ref{thm:iteration3}.\nWe also observe that the singularities of the variety $Y$ \nproduced in the previous theorem can not be improved\nby taking finite quasi-\\'etale covers and relative Cox rings.\nIndeed, every finite quasi-torsor\nor torus quasi-torsor over $Y$ is a torsor.\n\nA classic topic in algebraic geometry \nis deciding when a variety which is locally a quotient\nis indeed globally a quotient.\nFulton asked whether varieties with\nfinite quotient singularities are finite quotients of smooth varieties.\nIn~\\cite{EHKV01}, the authors prove that a variety with \nfinite quotient singularities is the quotient of a smooth variety\nby a linear algebraic group.\nIn~\\cite{KV04}, it is proved that a variety with finite quotient singularities admits a finite flat surjection from a smooth variety.\nIn~\\cite[Theorem 1.2]{GS15}, the authors show that a variety with finite abelian quotient singularities that is globally the quotient of a smooth variety by a torus is globally the quotient of a smooth variety by a finite group.\nIn this last paper, the language of stacks and Cox rings is used.\nIn this direction, we prove the following positive result \nin the case of locally toric singularities.\n\n\\begin{introthm}\\label{introthm:toric-quot}\nLet $X$ be a variety with locally toric singularities. \nThen, $X$ admits a torus quasi-torsor\nwhich is a smooth variety.\nIn particular, $X$ is the quotient of a smooth variety by the action of a torus.\n\\end{introthm}\n\nIn the previous theorem, \na singularity $x\\in X$ is said to be \nlocally toric if \nthere exists a toric variety $T$ and a closed invariant point $t\\in T$ such that\n$X_x\\simeq T_t$.\nHere, $X_x$ (resp. $T_t$) is the spectrum of the local ring $\\mathcal{O}_{X,x}$ (resp. $\\mathcal{O}_{T,t}$).\nThe previous theorem can be regarded as a generalization of the fact that \na toric variety is the quotient \nof an open subset of $\\mathbb{A}^n$ by a torus action.\nA point $x\\in X$ is said to be {\\em formally toric} if\nthere exists a toric variety $T$\nand a closed invariant point $t\\in T$\nsuch that \n$\\hat{X}_x\\simeq \\hat{T}_t$.\nThe statement of the previous theorem does not hold if we replace the condition\non locally toric singularities\nwith formally toric singularities (see Example~\\ref{ex:thm7-analytically-local}).\n\n\\subsection{Normal singularities}\n\nThroughout the introduction, \nwe focused on varieties with klt type singularities.\nIn this last part, we discuss what class of singularities is the optimal class for which the previous theorems work.\nWe recall the following theorem due to Stibitz (see~\\cite[Theorem 1]{Sti17}).\n\n\\begin{introthm}\nLet $X$ be a normal variety. \nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item Every sequence of finite quasi-torsors \n \\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nis eventually a sequence of torsors.\n\\item For every point $x\\in X$ the image of the homomorphism\n$\\hat{\\pi}_1^{\\rm reg}(X;x)\\rightarrow \\hat{\\pi}_1^{\\rm reg}(X)$\nis finite.\n\\end{enumerate}\n\\end{introthm}\n\nThe group $\\hat{\\pi}_1^{\\rm reg}(X)$ is the \\'etale fundamental group of the smooth locus of $X$. \nOn the other hand, $\\hat{\\pi}_1^{\\rm reg}(X;x)$, called \\emph{\\'etale regional fundamental group}, is the profinite completion of the fundamental group of the smooth locus around the singularity (see, e.g.,~\\cite{Bra19}). The regional fundamental group\nof klt type singularities is finite, so the previous theorem recovers~\\cite[Theorem 1.1]{GKP16}.\nMotivated by the previous result, we prove the following theorem regarding $\\mathbb{T}$-quasi-torsors.\n\n\\begin{introthm}\\label{introthm:optimal-T}\nLet $X$ be a normal variety. \nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item Every sequence of $\\mathbb{T}$-quasi-torsors \n \\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nis eventually a sequence of torsors.\n\\item For every point $x\\in X$ the group\n${\\rm Cl}(X;x)$ is finitely generated.\n\\end{enumerate}\n\\end{introthm}\n\nDue to Theorem~\\ref{thm:iteration2}, we know that the similar statement for finite quasi-torsors and torus quasi-torsors fails, even for toric singularities. \n However, in view of Theorem~\\ref{thm:iteration3}, we can expect a similar statement to hold for finite quasi-torsors and factorial $\\mathbb{T}$-quasi-torsors. \nIn order to state the following theorem, we need to introduce the concept of {\\em partial quasi-\\'etale Henselizations}.\n\n\\begin{introdef}\n{\\em \nLet $X$ be an algebraic variety and $x\\in X$ be a point. The {\\em partial quasi-\\'etale Henselization} of $X$ at $x$, denoted by $X^{ph}_x$, is the spectrum of the colimit of all \nquasi-\\'etale covers $\\mathcal{O}_{X,x}\\rightarrow R$ that extend to quasi-\\'etale covers of $X$ itself.\n}\n\\end{introdef} \nWith the previous definition, we can \nstate the theorem that describes the optimal class of singularities\nfor which every sequence of\nfinite quasi-torsors and factorial $\\mathbb{T}$-quasi-torsors is eventually \\'etale.\n\n\\begin{introthm}\\label{introthm:optimal-mixed}\nLet $X$ be a normal variety. \nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item Every sequence of\nfinite quasi-torsors and \nfactorial $\\mathbb{T}$-quasi-torsors \n \\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nis eventually a sequence of torsors.\n\\item For every point $x\\in X$, the following two conditions are satisfied:\n\\begin{enumerate}\n \\item[(a)] The image $\\hat{\\pi}_1^{\\rm reg}(X;x)\\rightarrow \\hat{\\pi}_1^{\\rm reg}(X)$ is finite, and\n \\item[(b)] the Class group ${\\rm Cl}(X^{ph}_x)$ is finitely generated.\n\\end{enumerate}\n\\end{enumerate}\n\\end{introthm}\nThe proofs of Theorem~\\ref{introthm:optimal-T}\nand Theorem~\\ref{introthm:optimal-mixed} are quite similar to those of the statements for klt type singularities.\nWe will prove these statements in Subsection~\\ref{subsec:normal-singularities}.\n\n\\subsection*{Acknowledgements}\nThe authors would like to thank Olivier Benoist and Ofer Gabber for helpful discussions.\nThe authors would like to thank Burt Totaro for \nproviding Example~\\ref{ex:semisimple}.\n\n\\section{Preliminaries}\n\nIn this section, \nwe recall the definitions\nof the singularities\nof the minimal model program.\nWe also recall the definition\nof $G$-quotients\nand $G$-quasi-torsors,\nand prove some \npreliminary results.\nWe work over an algebraically closed field $\\mathbb{K}$ of characteristic zero.\nAll the considered varieties\nare normal\nunless stated otherwise.\nA {\\em reductive group} $G$ is a linear algebraic group $G$ for which\nthe unipotent radical is trivial.\n\n\\subsection{Singularities of the MMP}\nIn this subsection, we recall the definitions of the singularities of the MMP.\n\n\\begin{definition}\n{\\em \nA {\\em log pair} $(X,\\Delta)$ consists of the data of a quasi-projective variety $X$\nand an effective $\\mathbb{Q}$-divisor\n$\\Delta$ for which\n$K_X+\\Delta$ is $\\mathbb{Q}$-Cartier.\nLet $x\\in X$ be a closed point.\nWe write $(X,\\Delta;x)$ for the log pair $(X,\\Delta)$ around $x$. \nWhen we write statements about $(X,\\Delta;x)$, we mean that such statement holds for $(X,\\Delta)$\non a sufficiently small neighborhood of $x$.\n}\n\\end{definition}\n\n\\begin{definition}\n{\\em \nLet $(X,\\Delta)$ be a log pair.\nLet $\\pi\\colon Y\\rightarrow X$ be a projective birational morphism \nfrom a normal quasi-projective variety $Y$.\nLet $E\\subset Y$ be a prime divisor.\nWe let $\\Delta_Y$ be the strict transform of $\\Delta$ on $Y$.\nWe fix canonical divisors $K_Y$ on $Y$\nand $K_X$ on $X$ for which \n$\\pi_* K_Y=K_X$.\nThe {\\em log discrepancy} of\n$(X,\\Delta)$ at $E$,\ndenoted by $a_E(X,\\Delta)$, is the rational number:\n\\[\n1+{\\rm coeff}_E(K_Y-\\pi^*(K_X+\\Delta)).\n\\] \nHence, the following equality holds: \n\\[\n\\pi^*(K_X+\\Delta) = \nK_Y+\\Delta_Y + (1-a_E(X,\\Delta))E.\n\\] \nThe log discrepancy $a_E(X,\\Delta)$ only depends on $E$ and does not depend on $Y$.\nWe say that $(X,\\Delta)$ is a {\\em Kawamata log terminal pair} \n(or {\\em klt pair} for short) \nif the inequality\n\\[\na_E(X,\\Delta)>0\n\\] \nholds for every prime divisor $E$ over $X$.\nWe say that the pair $(X,\\Delta)$\nis {\\em log canonical} (or {\\em lc} for short) if the inequality \n\\[\na_E(X,\\Delta)\\geq 0\n\\]\nholds for every prime divisor $E$ over $X$.\n}\n\\end{definition}\n\n\\begin{definition}\n{\\em \nWe say that $(X,\\Delta_0)$ is of {\\em klt type} \nif there exists a boundary $\\Delta\\geq \\Delta_0$ on $X$ for which the pair $(X,\\Delta)$ is klt.\nWe say that $(X,\\Delta_0)$ is of {\\em lc type}\nif there exists a boundary $\\Delta\\geq \\Delta_0$ on $X$ for which the pair\n$(X,\\Delta)$ is lc.\n}\n\\end{definition}\n\nThe following proposition is proved in~\\cite[\\S 4]{BGLM21}.\n\n\\begin{proposition}\\label{prop:klt-etale}\nThe klt type condition is an \\'etale condition.\nMore precisely, let $X$ be an algebraic variety, if for every point $x\\in X$ we can find an \\'etale neighborhood $U_x\\rightarrow X$ \nand a boundary $\\Delta_x$ \nfor which $(U_x,\\Delta_x)$ is klt, then there exists a boundary $\\Delta$ on $X$ for which\n$(X,\\Delta)$ is klt.\n\\end{proposition}\n\n\\subsection{$G$-quotients and $G$-quasi-torsors}\nIn this section, we recall the \ndefinitions of \n$G$-quotients\nand $G$-quasi-torsors.\n\n\\begin{definition}\\label{def:G-cover-global}\n{\\em\nLet $(X,\\Delta)$ be a pair.\nLet $G$ be a reductive group acting on $(X,\\Delta)$. \nAssume that the quotient $Y:=X\/\\!\\!\/G$ exists. \nThen, we say that $Y$ is a {\\em $G$-quotient} of $X$.\nWe also say that $X$ is a {\\em $G$-cover} of the variety $Y$.\n}\n\\end{definition}\n\n\\begin{definition}\\label{def:G-cover-local}\n{\\em \nLet $(X,\\Delta;x)$ be a singularity of a pair.\nAssume that $X$ is an affine variety.\nLet $G$ be a reductive group acting on \n$(X,\\Delta)$ and fixing $x$, i.e., \nwe have that \n$g^*\\Delta =\\Delta$ and $g(x)=x$ for each $g\\in G$.\nLet $(X,\\Delta;x) \\rightarrow (Y;y)$ be the quotient morphism where $y$ is the image of $x$.\nWe say that $(X,\\Delta;x)\\rightarrow (Y;y)$\nis a {\\em $G$-quotient} around $x$. \nThe morphism $X\\rightarrow Y$ will be called a {\\em $G$-quotient}.\nWe say that $Y$ is the {\\em $G$-quotient} of $X$\nand that $X$ is the {\\em $G$-cover} of $Y$.\n}\n\\end{definition}\n\nNow, we turn to define better behaved quotients. \nWe introduce the concept\nof {\\em $G$-quasi-torsors}.\n\n\\begin{definition}\\label{def:quasi-torsor} \n{\\em \nLet $(Y,\\Delta_Y)$ be a log pair.\nLet $X$ be a variety with \nthe action of $G$ reductive\nfor which\n$Y\\simeq X\/\\!\\!\/G$.\nWe say that the quotient morphism\n$\\phi\\colon X\\rightarrow Y$ is a \n{\\em $G$-quasi-torsor} for $(Y,\\Delta_Y)$ if the following conditions are satisfied:\n\\begin{enumerate}\n \\item there are codimension two open subsets $U_Y \\subset Y$\n and $U_X =\\phi^{-1}(U_Y)\\subset X$ for which \n \\[\n \\phi|_{U_X} \\colon U_X\\rightarrow U_Y\n \\] \n is a $G$-torsor, and \n \\item the global invertible homogeneous functions on $X$ descend to $Y$ via the induced\n homomorphism \n $\\mathcal{O}(X)^G \\simeq \\mathcal{O}(Y) \\hookrightarrow \\mathcal{O}(X)$.\n\\end{enumerate}\n}\n\\end{definition}\n\nIn general, the $G$-quotient $Y$ does not come with a naturally defined boundary.\nHowever, in some cases, it is possible to introduce such boundary\nand compare the log discrepancies \non $X$ with those on $Y$.\nThe following lemma is well-known to the experts\n(see, e.g.,~\\cite[Proposition 2.11]{Mor21}).\n\n\\begin{lemma}\\label{lem:klt-type-finite-cover-quotient}\nLet $(X;x)$ be a klt type singularity.\nThen, the following statements hold:\n\\begin{enumerate} \n\\item \nLet $G$ be a finite group acting on $(X;x)$.\nThe $G$-quotient $(Y;y)$ is of klt type. \n\\item \nLet $G$ be a finite group\nand $(Y;y)\\rightarrow (X;x)$ be \na $G$-quasi-torsor.\nThen, $(Y;y)$ is of klt type.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{definition}\\label{def:ab-quasi-torsor}\n{\\em \nWe say that a $G$-quasi-torsor\nis an {\\em abelian quasi-torsor} if $G$ is an abelian group. \nWe say that a $G$-quasi-torsor\nis a {\\em torus quasi-torsor} if \n$G$ is a torus.\nIn this case, we also write $\\mathbb{T}$-quasi-torsor or\n$\\mathbb{T}$-torsor.\nA quasi-torsor $Y\\rightarrow X$ is said to be a {\\em factorial} quasi-torsor if $Y$ is factorial. \n}\n\\end{definition}\n\nThe following lemma follows from the definitions. \n\n\\begin{lemma}\\label{lem:quot-qetale-torus-cover}\nLet $Y\\rightarrow X$ be a \n$\\mathbb{T}$-quasi-torsor\nand $\\mathbb{T}_0 \\leqslant \\mathbb{T}$ be a sub-torus. \nLet $Y\\rightarrow Y'$ be the quotient of $Y$\nby $\\mathbb{T}_0$\nand $Y'\\rightarrow X$ be the induced morphism.\nThen, both $Y\\rightarrow Y'$\nand $Y'\\rightarrow X$ are \ntorus quasi-torsors.\n\\end{lemma}\n\n\\subsection{Cox rings} In this subsection, we recall some statements about Cox rings for singularities and pairs.\nFirst, we define the concept of affine local Cox rings.\n\n\\begin{definition}\\label{def:local-Cox-ring}\n{\\em \nLet $(X;x)$ be a singularity.\nAssume that ${\\rm Cl}(X;x)$ is finitely generated.\nLet $N\\leqslant {\\rm WDiv}(X)$ be a free finitely generated subgroup surjecting onto ${\\rm Cl}(X;x)$\nand $N^0$ be the kernel of the surjection\n$\\pi\\colon N\\rightarrow {\\rm Cl}(X;x)$.\nConsider a group homomorphism\n$\\chi\\colon N^0\\rightarrow \\mathbb{K}(X)^*$ for which\n\\[\n{\\rm div}(\\chi(E))=E\n\\] \nfor all $E\\in N^0$.\nWe call such $\\chi$ a {\\em character}.\nLet $\\mathcal{S}$ be the sheaf of divisorial algebras associated to $N$\nand $\\mathcal{I}$ be the ideal subsheaf generated by\nsections $1-\\chi(E)$ where $E\\in N^0$.\nThen, we define the {\\em affine local Cox ring} of $(X,\\Delta)$ at $x$ to be\n\\[\n{\\rm Cox}(X;x)^{\\rm aff}_{N,\\chi} := \\bigoplus_{[D]\\in {\\rm Cl}(X;x)} \n\\frac{\\bigoplus_{D'\\in \\pi^{-1}([D])} \\mathcal{S}_{D'}(X) }{\\mathcal{I}(X)}.\n\\] \n}\n\\end{definition}\n\nNow, we define the concept of relative Cox ring for a log pair.\n\n\\begin{definition}\\label{def:rel-Cox-ring}\n{\\em \nLet $(X,\\Delta)$ be a log pair.\nLet $W_1,\\dots,W_k$ be orbifold Weil divisors on $(X,\\Delta)$.\nLet $N$ be the subgroup of ${\\rm WDiv}(X)$ spanned by $W_1,\\dots, W_k$.\nWe define the sheaf \n\\[\n\\mathcal{R}(X)_N : =\n\\bigoplus_{D \\in N} \\mathcal{O}_X(D) \\simeq \n\\bigoplus_{(m_1,\\dots,m_k)\\in \\mathbb{Z}^k} \\mathcal{O}_X(m_1W_1+\\dots+m_kW_k).\n\\] \nThe ring $\\mathcal{R}(X)_N$ is called a \n{\\em relative Cox ring} of $X$.\nThe relative spectrum \n\\[\nY:= {\\rm Spec}_X(\\mathcal{R}(X)_N) \\rightarrow X,\n\\]\nis called a {\\em relative Cox space} of $X$.\nWe may also call $\\mathcal{R}(X)_N$ the relative Cox ring associated to $N$\nand $Y$ the {\\em relative Cox space}\nassociated to $N$.\n}\n\\end{definition}\n\nNote that in the definition of the local Cox ring, we \nquotient by a certain ideal $\\mathcal{I}(X)$\nwhich comes from a character $\\chi$.\nHowever, in our definition of the relative Cox ring we do not perform such a quotient.\nExample~\\ref{ex:torsion-quotient} shows some pathology that would happen otherwise. \nThe definition of the relative Cox ring \ndoes not depend on the choice of $W_i$ in its linear equivalence class.\n\n\\begin{lemma}\\label{lem:replacement-lin-equiv}\nLet $X$ be an algebraic variety.\nLet $W_1,\\dots,W_k$ be Weil divisors on $X$ spanning $N$ in ${\\rm WDiv}(X)$.\nFor each $i\\in \\{1,\\dots,k\\}$, let $W'_i\\sim W_i$.\nLet $N'$ be the subgroup \nin ${\\rm WDiv}(X)$ spanned by the Weil divisors $W'_i$.\nThen, we have a $\\mathbb{T}$-equivariant isomorphism\n\\[\n\\mathcal{R}(X)_N \\simeq \\mathcal{R}(X)_{N'}.\n\\]\n\\end{lemma}\n\nThe following is proved in~\\cite[Proposition 4.10]{BM21} for the case of klt type singularities.\nThe general case follows from the theory of polyhedral divisors~\\cite{AH06}.\nIt states that in the affine setting \na torus quasi-torsor\nis the same as a relative Cox space. \n\n\\begin{lemma}\\label{lem:G-cover-is-local-Cox}\nLet $X$ be a normal affine variety \nand $x\\in X$ a closed point.\nLet $Y\\rightarrow X$ be a \n$\\mathbb{T}$-quasi-torsor\nover $X$.\nThen, up to shrinking $X$ around $x$, we can find a finitely generated subgroup\n$N\\leqslant {\\rm WDiv}(X)$ for\nwhich the isomorphism\n\\[\nY \\simeq {\\rm Spec}\\left( \n\\bigoplus_{D\\in N}H^0(X,\\mathcal{O}_X(D))\n\\right) \n\\]\nholds.\n\\end{lemma}\n\nFurthermore, the Cox ring \nin the local setting has klt type singularities (see, e.g.,~\\cite[Theorem 3.23]{BM21}).\n\n\\begin{lemma}\\label{lem:klt-type-torus-cover}\nLet $X$ be an affine variety and $(X;x)$ be a klt type singularity.\nLet $N\\leqslant {\\rm WDiv}(X,\\Delta)$ be a free finitely generated subgroup\nand $N^0 := \\ker(N\\rightarrow {\\rm Cl}(X;x))$.\nLet $\\chi \\colon N^0\\rightarrow \\mathbb{K}(X)^*$ be a character.\nThen, the spectrum of the affine local Cox ring \n\\[\n{\\rm Cox}(X;x)^{\\rm aff}_{N,\\chi} \n\\] \nhas klt type singularities.\n\\end{lemma}\n\nThe following lemma will be used in the comparison of quasi-torsors and relative Cox rings.\n\n\\begin{lemma}\\label{lem:isom-implies-lin-equiv}\nLet $W$ and $W'$ be two Weil divisors on a normal variety $X$.\nAssume that there is a $\\mathbb{G}_m$-equivariant\nisomorphism\n\\[\n{\\rm Spec}_X\\left(\\bigoplus_{m\\in \\mathbb{Z}}\\mathcal{O}_X(mW)\\right) \\simeq \n{\\rm Spec}_X \\left(\\bigoplus_{m\\in \\mathbb{Z}}\\mathcal{O}_X(mW')\\right).\n\\] \nThen, we have that $W\\sim W'$ on $X$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows verbatim from the proof of~\\cite[Construction~1.4.1.1]{ADHL15}. Since the conditions there are different from ours (but lead to the same conclusion), we recall the argument. Let $W-W'={\\rm div}(f)$ and define a homomorphism\n$$\n\\eta \\colon \\langle W_\\mathbb{Z} \\rangle \\to \\mathbb{K}(X)^*\\qquad \\text{ and } \\qquad kW \\mapsto f^k.\n$$\nThen we obtain an equivariant isomorphism between the sheaves $\\bigoplus_{m\\in \\mathbb{Z}}\\mathcal{O}_X(mW)$ and $\\bigoplus_{m\\in \\mathbb{Z}}\\mathcal{O}_X(mW')$ by mapping $f \\in \\mathcal{O}_X(kW)$ to $\\eta(kW) \\cdot f \\in \\mathcal{O}_X(kW')$.\n\\end{proof}\n\n\\section{G-covers of klt type singularities}\n\nIn this section, we study $G$-covers of klt type singularities.\nAs seen in Example~\\ref{ex:cod-1-ram}, \nwe need to focus on those $G$-covers that\nare unramified over codimension one points.\nFirst, we will show that \nsemisimple covers of klt type singularities may not be of klt type.\nThe following is a generalization of Theorem~\\ref{introthrm:gl2-cover}\nto higher-dimensional toric singularities.\n\n\\begin{theorem}\\label{thm:gln-cover-general}\nFor any $n\\geq 2$, \nthere exists a $(n+1)$-dimensional\ntoric singularity $(X;x)$ \nthat admits a \n$\\mathbb{P}{\\rm GL}_{r}(\\mathbb{K})$-cover \n$Y\\rightarrow X$, satisfying the following conditions:\n\\begin{enumerate}\n \\item we have $r=3$ if $n\\in \\{2,3\\}$ and $r=n$ otherwise,\n \\item the singularity $(Y;y)$ has dimension $(n+r-1)$, and\n \\item the singularity $(Y;y)$ is not of klt type.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFirst, we choose an appropriate projective toric variety, depending on the number $n$.\nIf $n=2$, we choose a smooth projective toric surface $T$ of Picard rank $4$ and fix $r=3$.\nIf $n=3$, we choose a smooth projective toric threefold $T$ of Picard rank $3$\nand fix $r=3$.\nFor $n\\geq 4$, we choose a smooth projective toric $(n-1)$-fold $T$\nof Picard rank $n$ and we fix $r=n$.\nIn any of the previous cases,\nby~\\cite[Theorem 1.1]{GHPS12}, we can find a vector bundle $\\mathcal{E}$\nof rank $r$ over $T$\nsuch that the Cox ring of \n$\\mathbb{P}(\\mathcal{E})$ is not finitely generated.\nWe fix $G:=\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$ to be the projective linear group acting on $\\mathbb{P}(\\mathcal{E})$.\nNote that $\\pi\\colon \\mathbb{P}(\\mathcal{E})\\rightarrow T$ is a quotient for the $G$-action.\nLet $A_T$ be an ample toric divisor on $T$. \nLet $m$ be a positive integer, \n\\[ \n\\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(1)\\otimes \\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(\\pi^*A_T\/m) \n\\] \nis an ample $\\mathbb{Q}$-line bundle on\n$\\mathbb{P}(\\mathcal{E})$, for $m$ large enough, which is $G$-invariant.\nThus, the affine variety \n\\[\nY = {\\rm Spec}\n\\left( \n\\bigoplus_{n\\in \\mathbb{Z}} \nH^0\\left(\n\\mathbb{P}(\\mathcal{E}), \n\\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(n)\\otimes\n\\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(n\\pi^*A_T\/m)\n\\right) \n\\right) \n\\] \nadmits a $G$-action which fixes the vertex $y\\in Y$ of the $\\mathbb{G}_m$-action induced by the $\\mathbb{Z}$-grading.\nObserve that an element \n\\[\nf\\in H^0\\left(\\mathbb{P}(\\mathcal{E}), \n\\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(n)\n\\otimes\n\\mathcal{O}_{\\mathbb{P}(\\mathcal{E})}(n\\pi^*A_T\/m)\\right)\n\\] \nis preserved by the action of $G$\nif and only if it is constant along the fibers\nof $\\mathbb{P}(\\mathcal{E})\\rightarrow T$.\nIn other words, the $G$-invariant elements \nhave the form \n$f=\\pi^*g$ for some\n$g\\in H^0(T,\\mathcal{O}_T(nA_T\/m))$.\nWe conclude that there is an isomorphism \n\\[\nY\/\\!\\!\/G \\simeq \n{\\rm Spec}\\left( \n\\bigoplus_{n\\in \\mathbb{Z}}H^0(T,\\mathcal{O}_T(nA_T\/m))\n\\right)=X.\n\\] \nThus, the quotient $Y\/\\!\\!\/G$ is isomorphic to the cone over a $\\mathbb{Q}$-ample toric divisor on a\nsmooth projective toric variety. \nHence, $(X;x)$ is a toric singularity \nof dimension $n$.\nIt suffices to check\nthat $(Y;y)$ is not of klt type. \n\nWe proceed by contradiction.\nAssume that $(Y;y)$ is of klt type. \nLet $\\widetilde{Y}\\rightarrow Y$ be the blow-up of $Y$ at the maximal ideal of $y$. Then, the exceptional divisor $E$ of $\\phi\\colon \\widetilde{Y}\\rightarrow Y$ is isomorphic\nto $\\mathbb{P}(\\mathcal{E})$. \nSince $\\mathbb{P}(\\mathcal{E})$ is smooth, we conclude that $\\widetilde{Y}$ has $\\mathbb{Q}$-factorial singularities.\nLet $\\Delta_Y$ be the effective divisor\nthrough $y$ for which $(Y,\\Delta_Y;y)$ has klt singularities.\nLet $\\Delta_{\\widetilde{Y}}$ be the strict transform of $\\Delta_Y$ on $\\widetilde{Y}$.\nWe write\n\\[\n\\phi^*(K_Y+\\Delta_Y)=\nK_{\\widetilde{Y}}+\\Delta_{\\widetilde{Y}}+(1-a)E,\n\\]\nfor some positive number $a$.\nNote that $\\Delta_{\\widetilde{Y}}$ is ample over $Y$ as $\\rho(\\widetilde{Y}\/Y)=1$.\nWe conclude that $K_{\\widetilde{Y}}+(1-a)E$ is antiample over $Y$, \nso $K_Y+E$ is antiample over $Y$ as well.\nSince $E$ is smooth, we conclude that\nthe pair $(Y,E)$ is plt.\nThus, the pair $K_E+\\Delta_E=(K_Y+E)|_E$, obtained by performing adjunction to $E$, is log Fano.\nIn particular, the projective variety\n$\\mathbb{P}(\\mathcal{E})\\simeq E$ is of Fano type.\nThus, the Cox ring of $\\mathbb{P}(\\mathcal{E})$ is finitely generated by~\\cite[Corollary 1.9]{BCHM10}.\nThis leads to a contradiction.\nWe conclude that $(Y;y)$\nis not a klt type singularity.\n\\end{proof}\n\nIn the previous theorem, the action is not free outside the point $y\\in Y$.\nWe show that this can be improved in the following statement.\n\n\\begin{proposition}\\label{prop:gln-cover-free}\nFor any $n\\geq 2$, there exists a $(n+1)$-dimensional toric singularity \n$(X;x)$ that admits a \n$\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$-cover \n$Y\\rightarrow X$, satisfying the following conditions:\n\\begin{enumerate}\n \\item we have that $r=3$ if $n\\in \\{2,3\\}$ and $r=n$ otherwise, \n \\item the germ $(Y;y)$ has dimension $n+r^2$, \n \\item the action of\n $\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$ on $Y$\n is free on a dense open set, and\n \\item the singularity $(Y;y)$ is not of klt type.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nLet $T$ be a $n$-dimensional smooth projective toric variety.\nLet $\\mathbb{P}(\\mathcal{E})$ be the rank $r$ vector bundle over $T$\nconsidered in the proof of Theorem~\\ref{thm:gln-cover-general}.\nHence, the variety $\\mathbb{P}(\\mathcal{E})$ is not a Mori dream space.\nLet $Y_0\\rightarrow T$ be the associated principal\n$\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$-bundle.\nConsider a $\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$-equivariant projectivization\n$Y_0\\hookrightarrow \\bar{Y}$\nwith a relatively ample line bundle\n$\\mathcal{O}_{\\bar{Y}}(1)$ over $T$.\nObserve that the action of $\\mathbb{P}{\\rm GL}_r(\\mathbb{K})$ on \n$\\bar{Y}$ is free on the open subset $Y_0$.\nWe claim that $\\bar{Y}$ is not a Mori dream space. \nLet $\\rho\\colon \\mathbb{P}{\\rm GL}_r(\\mathbb{K})\\rightarrow {\\rm Aut}(\\mathbb{P}^r)$\nbe the standard representation. \nThen $\\bar{Y}\\times \\mathbb{P}^r$ admits the action of\n$G$ given by $g\\cdot (u,v)=(g^{-1}u,\\rho(g)v)$.\nThe quotient of $\\bar{Y}\\times \\mathbb{P}^r$ by $G$ is isomorphic $\\mathbb{P}(\\mathcal{E})$.\nIf $\\bar{Y}$ is a Mori dream space, then $\\mathbb{P}(\\mathcal{E})$ is also a Mori dream space by~\\cite[Theorem 1.1]{Oka16}.\nThis leads to a contradiction.\nWe conclude that $\\bar{Y}$ is not a Mori dream space.\nThe rest of the proof proceeds as in Theorem~\\ref{thm:gln-cover-general},\nby replacing \n$\\mathbb{P}(\\mathcal{E})$\nwith $\\bar{Y}$.\n\\end{proof}\n\nNow, we turn to prove that \n$G$-covers of klt type singularities\nare again of klt type,\nprovided that $G$ is a finite extension of a torus. \nThe following is a generalization of Theorem~\\ref{introthm:torus-finite-cover}\nwhich allows ramification over codimension one points. \n\n\\begin{theorem}\\label{thm:torus-finite-cover-klt-sing}\nLet $(X,\\Delta;x)$ be a klt type singularity.\nLet $G$ be a finite extension of a torus.\nLet $\\pi\\colon Y\\rightarrow X$ be a\n$G$-quasi-torsor.\nThen, the variety $Y$ is of klt type.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:klt-type-finite-cover-quotient}, we know that klt type singularities\nare preserved under finite covers\nand finite quotients.\nHence, we may assume that $G\\simeq \\mathbb{G}_m^k$ for some $k$.\nBy Lemma~\\ref{lem:G-cover-is-local-Cox}, we know that there exists a \nfinitely generated subgroup\n$N\\leqslant {\\rm WDiv}(X,\\Delta)$\nsuch that the isomorphism\n\\[\nY \\simeq {\\rm Spec}\\left( \\bigoplus_{D\\in N} H^0(X,\\mathcal{O}_X(D)) \\right)\n\\]\nholds.\nLet $\\pi\\colon N\\rightarrow {\\rm Cl}(X,\\Delta;x)$ be the induced homomorphism. \nLet $N_0$ be the kernel of $\\pi$.\nWe can choose a homomorphism\n$\\chi\\colon N_0\\rightarrow \\mathbb{K}(X)^*$\nso that\n${\\rm div}(\\chi(E))=E$ for every $E\\in N_0$.\nThen, we can define the local-affine Cox ring of $(X,\\Delta;x)$ associated to the data\n$N,\\chi$ as in Definition~\\ref{def:local-Cox-ring}.\nWe denote this ring by\n\\[\n{\\rm Cox}(X,\\Delta;x)^{\\rm aff}_{N,\\chi}\n\\] \nand \nwe denote by $Y_0$ its spectrum.\nBy Lemma~\\ref{lem:klt-type-torus-cover}, we know that $Y_0$ has klt type singularities.\nApplying Lemma~\\ref{lem:G-cover-is-local-Cox}\n to the torus cover \n$Y\\rightarrow Y_0$, \nwe can find a free finitely generated subgroup $N_1\\leqslant {\\rm CaDiv}(Y_0)$ \nfor which the isomorphism \n\\[\nY \\simeq {\\rm Spec}\\left( \n\\bigoplus_{D\\in N_1} H^0(Y_0,\\mathcal{O}_{Y_0}(D))\n\\right) \n\\]\nholds. \nObserve that we can choose the divisors of $N_1$ to be Cartier on $Y_0$.\nIndeed, these divisors correspond to the divisors of $N$, which become Cartier on $Y_0$.\nThus, we conclude that the \ntorus quotient \n$Y\\rightarrow Y_0$\nis a principal torus cover.\nHence, the variety $Y$ has klt type singularities, since the klt type property\nis locally \\'etale by Proposition~\\ref{prop:klt-etale}.\n\\end{proof}\n\n\\section{Weil divisors modulo Cartier divisors}\n\nIn this section, we study the group \nof Weil divisors modulo Cartier divisors, which is called the \\emph{local class group} in~\\cite{BdFFU15}.\nIn general, the group \n${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is not finitely generated.\nThis group is trivial if and only if $X$ is locally factorial.\nIn~\\cite{BGS11}, the authors prove that the \n$\\mathbb{Q}$-factorial and factorial locus of an algebraic variety\nare open.\nIn~\\cite[Section 14]{Kol20}, the author studies the non-$\\mathbb{Q}$-Cartier loci\nof Weil divisors.\nWe recall the following proposition due to Koll\\'ar (see~\\cite[Proposition 138]{Kol20}).\n\n\\begin{proposition}\\label{prop:kollar-Cartier}\nLet $X$ be a normal proper variety.\nLet $Z\\subset X$ be an irreducible variety.\nThere exists a dense open subset $Z^0\\subset Z$ such that the following holds.\nLet $D$ be a Weil divisor that is Cartier at the generic point $\\eta_Z$ of $Z$. \nThen, the divisor $D$ is Cartier at every closed point of $Z^0$.\n\\end{proposition}\n\nDue to the previous proposition, \nwe can prove the following theorem \nusing Noetherian induction.\n\n\\begin{theorem}\\label{thm:wdiv-cdiv}\nLet $X$ be a normal variety. \nThere are finitely many closed points\n$x_1,\\dots,x_r\\in X$ such that the homomorphism \n\\[\n{\\rm WDiv}(X)\/{\\rm CaDiv}(X) \\rightarrow \\bigoplus_{i=1}^r {\\rm Cl}(X_{x_i}),\n\\]\nis a monomorphism.\n\\end{theorem}\n\n\\begin{proof}\nLet $U_1,\\dots,U_s$ be an affine open cover of $X$.\nObserve that the homomorphism\n\\[\n{\\rm WDiv}(X)\/{\\rm CaDiv}(X)\\rightarrow \n\\bigoplus_{i=1}^s {\\rm WDiv}(U_i)\/{\\rm CaDiv}(U_i),\n\\]\ninduced by restricting\n$D\\mapsto (D|_{U_1},\\dots,D|_{U_s})$, \nis a monomorphism.\nHence, it suffices to prove the statement for an affine variety.\n\nWithout loss of generality, we may assume that $X$ is\naffine. Let $\\bar{X}$ be its closure in a projective space.\nBy Proposition~\\ref{prop:kollar-Cartier}, there exists an open set $\\bar{X}^0\\subset \\bar{X}$ \nso that every Weil divisor on $\\bar{X}$ is Cartier\nat every closed point of $\\bar{X}^0$.\nLet $Z_1,\\dots,Z_k$ be the irreducible components of $\\bar{X}\\setminus \\bar{X}^0$.\nFor each $i\\in \\{1,\\dots,k\\}$, we choose\n$Z_i^0$ as in the statement of Proposition~\\ref{prop:kollar-Cartier}.\nThen, we proceed inductively with the \nirreducible components\nof each $Z_i\\setminus Z_i^0$. \n\nWe obtain a finite set of irreducible subvarieties \n$Z_1,\\dots,Z_{r_0} \\subset \\bar{X}$ \nand dense open subsets \n$Z_i^0 \\subset Z_i$\nso that the following set-theoretic equality holds\n\\[\n\\bar{X} = \\bigcup_{i=1}^{r_0} Z_i^0.\n\\]\nWe may assume that there exists $r\\leq r_0$ \nfor which \n\\begin{equation}\\label{eq:covering-affine}\nX=\\bigcup_{i=1}^r Z_i^0\\cap X,\n\\end{equation}\nand each intersection $Z_i^0\\cap X$ is non-empty\nfor $i\\in \\{1,\\dots,r\\}$.\nFor each $i\\in \\{1,\\dots,r\\}$, we choose a closed point $x_i \\in Z_i^0 \\cap X$.\nThe homomorphism \n\\begin{equation}\\label{eq:hom-cl} \n{\\rm WDiv}(X)\/{\\rm CaDiv}(X) \\rightarrow \\bigoplus_{i=1}^r {\\rm Cl}(X_{x_i}),\n\\end{equation} \nis well-defined. Indeed, Cartier divisors are mapped to the zero element on the right-hand side.\n\nIt suffices to prove that~\\eqref{eq:hom-cl} is a monomorphism. \nLet $D$ be a Weil divisor on $X$.\nLet $\\bar{D}$ be the closure of $D$ on $\\bar{X}$.\nAssume that $[D_{x_i}]=0 \\in {\\rm Cl}(X_{x_i})$ for every $i\\in \\{1,\\dots,r\\}$.\nThen, $\\bar{D}$ is Cartier at the generic point \n$\\eta_{Z_i}$ of $Z_i$ for every $i\\in \\{1,\\dots,r\\}$. \nBy Proposition~\\ref{prop:kollar-Cartier}, we conclude that $\\bar{D}$ is Cartier at every closed point of \n$Z_i^0$ for every $i\\in \\{1,\\dots,r\\}$. \nIn particular, $D=\\bar{D}\\cap X$ is Cartier\nat every closed point of\n$Z_i^0\\cap X$. \nBy equality~\\eqref{eq:covering-affine}, we conclude that \n$D$ is Cartier at every closed point of $X$.\nThis means that \n$[D]=0\\in {\\rm WDiv}(X)\/{\\rm CaDiv}(X)$.\nThis finishes the proof of the theorem.\n\\end{proof}\n\nIf the variety has rational singularities,\nthen we conclude that the group\n${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is finitely generated. \nIn particular, we have the following statement.\n\n\\begin{theorem}\\label{thm:wdiv-cdiv-klt-type}\nLet $X$ be a variety with klt type singularities. \nThen, the group ${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is finitely generated.\n\\end{theorem}\n\n\\begin{proof}\nLet $X$ be a variety with klt type singularities. Let $x_1,\\dots,x_r\\in X$ be closed points. \nBy~\\cite[Theorem 3.27]{BM21}, we know that \n$\\bigoplus_{i=1}^r {\\rm Cl}(X_{x_i})$ is a finitely generated abelian group.\nBy Theorem~\\ref{thm:wdiv-cdiv}, we conclude that ${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is a finitely generated abelian group.\n\\end{proof}\n\nWe have the following corollary from the previous theorem.\n\n\\begin{corollary}\\label{cor:wdiv-cdiv-pairs}\nLet $(X,\\Delta)$ be a klt type pair.\nThen, the group\n${\\rm WDiv}(X,\\Delta)\/{\\rm CaDiv}(X)$ is finitely generated.\n\\end{corollary}\n\n\\section{Torus covers of klt type varieties}\n\nIn this section, we study torus covers.\nWe establish a characterization theorem\nfor torus quasi-torsors over varieties with klt type singularities. \nWe will start with the following lemma that will be used in this section.\n\n\\begin{lemma}\\label{lem:torus-inv-lin-equiv-div}\nLet $X$ be a variety that admits a $\\mathbb{T}$-action.\nLet $W$ be a Weil divisor on $X$.\nWe can find a $\\mathbb{T}$-invariant \nWeil divisor $W'$ on $X$ for which \n$W\\sim W'$.\n\\end{lemma}\n\n\\begin{proof}\nBy Sumihiro's equivariant completion~\\cite{Sum74}, we may assume that $X$ is an affine $\\mathbb{T}$-variety, where $\\mathbb{T}$ is an $n$-dimensional torus.\nBy~\\cite[Proposition 1.6]{AH06}, we can find $\\mathbb{T}$-invariant divisors \n$D_1,\\dots,D_k$ such that \n$X\\setminus \\bigcup_{i=1}^k D_i \\simeq \\mathbb{T}\\times U$\nfor some variety $U$.\nWe may assume that $U$ is smooth. Let $U\\hookrightarrow Y$ be a smooth\nprojectivization. By~\\cite[Exercise 12.6.(b)]{Har77}, we know that\n${\\rm Cl}(Y\\times \\mathbb{P}^n)\\simeq {\\rm Cl}(Y)\\times \\mathbb{Z}$.\nThe class group of $Y\\times \\mathbb{P}^n$ is generated by\n$Y\\times H$ and \ndivisors\nof the form $\\Gamma_1 \\times \\mathbb{P}^n,\\dots, \\Gamma_r\\times\\mathbb{P}^n$,\nwhere $\\Gamma_i \\subset Y$ are prime divisors\nand $H\\subset \\mathbb{P}^n$ is a hyperplane.\nHence, the Class group of $\\mathbb{T}\\times U$ is generated by torus invariant divisors. \nThis implies that the Class group of $X$ is generated by torus invariant divisors.\n\\end{proof}\n\nNow, we can prove that every\ntorus quasi-torsor is a relative Cox ring.\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:cox-space-vs-g-covers}]\nLet $X$ be a normal variety.\nLet $Y\\rightarrow X$ be a $\\mathbb{T}$-quasi-torsor. \nWe will prove the statement by induction on $k$ the dimension of $\\mathbb{T}$.\nFirst, we will show that the statement holds for $k=1$.\n\nLet $\\pi\\colon Y\\rightarrow X$ be a\n$\\mathbb{G}_m$-quasi-torsor. \nLet $U\\subset X$ be the largest open subset of $X$\nfor which there is a $\\mathbb{G}_m$-equivariant \nisomorphism \n\\begin{equation}\\label{eq:local-Cox-U}\n\\pi^{-1}(U) \\simeq {\\rm Spec}_U\\left(\n\\bigoplus_{m\\in \\mathbb{Z}} \\mathcal{O}_U(mW)\n\\right) \n\\end{equation} \nfor a certain Weil divisor $W$ on $U$.\nBy Lemma~\\ref{lem:G-cover-is-local-Cox}, \nwe know that $U$ is not empty. \nWe claim that $U=X$. \nBy contradiction, assume that \n$U\\subsetneq X$. \nLet $x\\in X$ be a closed point\ncontained in the complement of $U$. \nBy definition, we can find a Weil divisor $W$ on $X$\nfor which the isomorphism~\\eqref{eq:local-Cox-U} holds.\nBy Lemma~\\ref{lem:G-cover-is-local-Cox}, \nwe may find an affine neighborhood $V$ of $X$\nand a Weil divisor $W'$ on $V$ for which there\nis a $\\mathbb{G}_m$-equivariant isomorphism \n\\begin{equation}\\label{eq:isom-V}\n\\pi^{-1}(V) \\simeq {\\rm Spec}_V\\left( \n\\bigoplus_{m\\in \\mathbb{Z}} \\mathcal{O}_V(W')\n\\right).\n\\end{equation} \nIn particular, there is a $\\mathbb{G}_m$-equivariant\nisomorphism \n\\[ \n{\\rm Spec}_{U\\cap V}\\left( \n\\bigoplus_{m\\in \\mathbb{Z}} \\mathcal{O}_{U\\cap V}(mW|_{U\\cap V)}\n\\right) \n\\simeq \n{\\rm Spec}_{U\\cap V}\\left( \n\\bigoplus_{m\\in \\mathbb{Z}} \\mathcal{O}_{U\\cap V}(mW'|_{U\\cap V)}\n\\right) \n\\]\nover $U\\cap V$.\nBy Lemma~\\ref{lem:isom-implies-lin-equiv}, we conclude that\n$W|_{U\\cap V} \\sim W'|_{U\\cap V}$ holds in $U\\cap V$.\nWrite \n\\[\nW|_{U\\cap V}-W'_{U\\cap V} = {\\rm div}(f)|_{U\\cap V}\n\\]\nfor some $f\\in \\mathbb{K}(X)$.\nWe can replace $W'$ with $W'+{\\rm div}(f)$. \nBy Lemma~\\ref{lem:replacement-lin-equiv}, this replacement\npreserves the equivariant isomorphism~\\eqref{eq:isom-V}.\nThus, we may assume that \n$W|_{U\\cap V} = W'|_{U\\cap V}$.\nHence, we can find a Weil divisor \n$W''$ on $U\\cup V$ for which there\nis a $\\mathbb{G}_m$-equivariant isomorphism \n\\[\n\\pi^{-1}(U\\cup V) \\simeq {\\rm Spec}_X\\left( \n\\bigoplus_{m \\in \\mathbb{Z}} \\mathcal{O}_{U\\cup V}(W'')\n\\right).\n\\]\nThis contradicts the maximality of $U$.\nWe conclude that $U=X$.\nThus, the statement of the theorem holds for $k=1$. \n\nNow, let $Y\\rightarrow X$ be a \n$\\mathbb{T}$-quasi-torsor. \nLet $Y\\rightarrow Y_0$ be the quotient \nby a sub-torus $\\mathbb{T}_0\\leqslant \\mathbb{T}$ of dimension $k-1$.\nBy Lemma~\\ref{lem:quot-qetale-torus-cover}, \nwe conclude that\n$\\pi_1\\colon Y\\rightarrow Y_0$ is a \na $\\mathbb{T}_0$-quasi-torsor\nand $\\pi_0\\colon Y_0\\rightarrow X$\nis a $\\mathbb{G}_m$-quasi-torsor. \nBy induction on the dimension, we can find \nWeil divisors\n$W_1,\\dots, W_{k-1}$ on $Y_0$\nfor which\n\\[\nY \\simeq {\\rm Spec}_{Y_0}\\left( \n\\bigoplus_{(m_1,\\dots,m_{k-1})\\in \\mathbb{Z}^{k-1}}\n\\mathcal{O}_{Y_0}(m_1W_1 +\\dots+m_{k-1}W_{k-1})\n\\right) \n\\] \nand $W$ on $X$ for which \n\\[\nY_0 \\simeq {\\rm Spec}_{X}\\left( \n\\bigoplus_{m\\in \\mathbb{Z}}\\mathcal{O}_X(mW)\n\\right).\n\\] \nBoth isomorphisms are torus equivariant.\nBy Lemma~\\ref{lem:torus-inv-lin-equiv-div}\nand Lemma~\\ref{lem:replacement-lin-equiv},\nwe may assume that each \n$W_i$, with $i\\in\\{1,\\dots,k-1\\}$\nis torus invariant.\nSince $Y_0\\rightarrow X$ contains\nno horizontal $\\mathbb{G}_m$-invariant divisors, \nwe conclude that \n$W_i=\\pi_0^* W_{i,X}$\nfor some Weil divisors $W_{i,X}$.\nHere, the pull-back is defined by restricting to the smooth locus.\nWe set\n\\[\nY' := {\\rm Spec}_X\\left( \n\\bigoplus_{(m_1,\\dots,m_k)\\in\\mathbb{Z}^k}\n(m_1W_{X,1}+\\dots+m_{k-1}W_{X,k-1}+m_kW_k)\n\\right). \n\\]\nNote that $Y'$ has a $\\mathbb{T}_0$-quotient\n$Y'_0$ obtained by considering the graded subring\ngiven by $m_i=0$ for every $i\\in \\{1,\\dots,k-1\\}$.\nThis quotient is isomorphic to $Y_0$. \nHence, we have a commutative diagram \n\\[\n \\xymatrix@R=2em@C=2em{\nY\\ar[d]_-{\\pi_1} & Y'\\ar[d]^-{\\pi_1'} \\\\\nY_0\\ar[d]_-{\\pi_0}\\ar[r]^-{\\phi} & Y_0'\\ar[dl]^-{\\pi_0'} \\\\\nX.\n }\n\\] \nBy construction, we have that \n\\[\nW_i = \\pi_0^* W_{X,i} = \\phi^* {\\pi_0'}^* W_{X,i},\n\\]\nholds for every $i\\in \\{1,\\dots,k-1\\}$.\nWe conclude that $Y'$ is $\\mathbb{T}$-equivariantly isomorphic to $Y$.\nThis finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm-finite-torus-cover-klt-var}]\nThis statement is local.\nHence, it follows from\nTheorem~\\ref{introthm:torus-finite-cover}\nand \nTheorem~\\ref{introthm:cox-space-vs-g-covers}.\n\\end{proof}\n\nIn order to prove Theorem~\\ref{introthm:iteration-torus-quasi-torsors},\nwe will need the following lemma. \n\n\\begin{lemma}\\label{lem:two-qt}\nLet $\\phi \\colon Y \\rightarrow X$ be a\n$\\mathbb{T}_\\phi$-quasi-torsor \nand \n$\\psi\\colon Z\\rightarrow Y$ \nbe a $\\mathbb{T}_\\psi$-quasi-torsor.\nThe following statements hold:\n\\begin{enumerate}\n \\item the composition\n $\\phi\\circ \\psi\\colon Z\\rightarrow X$ is a torus quasi-torsor,\n \\item if $\\phi$ corresponds to the subgroup\n $N_Y\\leqslant {\\rm WDiv}(X)$ and $Z\\rightarrow X$ corresponds to the subgroup $N_Z\\leqslant {\\rm WDiv}(X)$, then $N_Y\\leqslant N_Z$, and \n \\item the torus quasi-torsor $\\psi$ is a torsor\n if and only if for every closed point $x\\in X$ the images \n $N_Y\\rightarrow {\\rm Cl}(X_x)$ and $N_Z\\rightarrow {\\rm Cl}(X_x)$ agree. \n In particular, if the images of $N_Y$ and $N_Z$ agree on\n \\[{\\rm WDiv}(X)\/{\\rm CaDiv}(X)\\] then $\\psi$ is a torsor.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe start by showing that if $\\phi \\colon Y \\to X$ and $\\psi \\colon Z \\to Y$ are two torus quasi-torsors with acting tori $\\mathbb{T}_\\phi$ and $\\mathbb{T}_\\psi$ respectively, \nthen $\\phi \\circ \\psi \\colon Z \\to X$ is a torus quasi-torsor as well. \nBy Theorem~\\ref{introthm:cox-space-vs-g-covers}, $\\psi$ corresponds to a relative Cox ring, \ni.e. to a sheaf of graded algebras as in~\\cite[Section~4.2.3]{ADHL15}. \nThus, we can lift the action of $\\mathbb{T}_\\phi$ on $Y$ to $Z$ by~\\cite[Proposition~4.2.3.6]{ADHL15}.\nThe quotient by the action of $\\mathbb{T}_\\phi \\times \\mathbb{T}_\\psi$ is $\\phi \\circ \\psi$. This is a quasi-torsor, since $\\phi$ and $\\psi$ are so. \nAgain by Theorem~\\ref{introthm:cox-space-vs-g-covers}, $\\phi \\circ \\psi$ corresponds to a relative Cox ring with respect to a subgroup $N \\leqslant \\operatorname{WDiv}(X)$. This shows $(1)$.\nThe previous construction also shows $(2)$.\n\nFor $(3)$, note that $\\psi$ is a torsor if and only if \nevery element of $\\phi^*N_Z$ is Cartier in $Y$.\nSince $N_Y\\leqslant N_Z$, this happens if and only if\nthe image of $N_Z$ equals the image of $N_Y$ in every local Class group of $X$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:iteration-torus-quasi-torsors}]\nLet $X$ be a variety with klt type singularities.\nConsider a sequence of morphisms:\n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\] \nsuch that each $\\phi_i\\colon X_i\\rightarrow X_{i-1}$\nis a $\\mathbb{T}$-quasi-torsor.\nWe write $\\psi_i=\\phi_i\\circ\\dots\\circ \\phi_1$.\nBy Lemma~\\ref{lem:two-qt}.(1), we know that each \n$\\psi_i$ is a $\\mathbb{T}$-quasi-torsor\ncorresponding to a subgroup $N_i\\leqslant {\\rm WDiv}(X)$.\nBy Lemma~\\ref{lem:two-qt}.(2), we know that there is a sequence of subgroups\n\\[\nN_1 \\leqslant N_2 \\leqslant \\dots \\leqslant N_i \\leqslant \\dots \n\\] \nBy Theorem~\\ref{thm:wdiv-cdiv-klt-type}, we know that ${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is a finitely generated abelian group. \nIn particular, for some $i_0$, we have that the image\nof every $N_i$, with $i\\geq i_0$, stabilizes in \n${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$. \nBy Lemma~\\ref{lem:two-qt}.(3), we conclude that each \n$\\phi_i$, with $i\\geq i_0$ is a torus torsor.\nThis finishes the proof of the theorem.\n\\end{proof} \n\n\\section{Iteration of torus and finite covers} \n\nIn this section, we study the iteration\nof quasi-\\'etale $G$-covers, where $G$ is either finite\nor a torus.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:iteration1}]\nWe consider a sequence\n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nas in the statement of Theorem~\\ref{thm:iteration1}. We claim that if $\\phi_i$ is a $\\mathbb{T}$-cover and $\\phi_{i+1}$ a finite $G_{i+1}$-cover (for any $i\\geq 1$), then we have a variety $X_{i}'$, a commutative diagram \n\\[\n \\xymatrix@R=2em@C=2em{\n X_{i+1} \\ar[r]^-{\\phi_{i+1}} \\ar[d]_-{\\phi_{i+1}'} & X_i \\ar[d]^-{\\phi_i} \\\\\n X_{i}' \\ar[r]^-{\\phi_{i}'} & X_{i-1} ,\n }\n\\]\nwhere $\\phi_{i}'$ (resp. $\\phi_{i+1}'$) is a $G_{i+1}$-cover (resp. $\\mathbb{T}$-cover), which is \\'etale if and only if $\\phi_{i+1}$ (resp. $\\phi_{i}$) is \\'etale. This claim holds since by~\\cite[Proof of Proposition 5.1]{BM21}, we have an exact sequence\n\\[\n\\xymatrix@C=15pt{\n\\mathbb{Z}^{\\dim(\\mathbb{T})} \\ar[r] & \\pi_1(X_i^{\\rm reg}) \\ar[r]^{\\varphi} & \\pi_1(X_{i-1}^{\\rm reg}) \\ar[r] &\n1,\n}\n\\]\nwhere $\\mathbb{T}$ is the general fiber of $\\phi_i$. Thus if the finite cover $\\phi_{i+1}$ corresponds to the normal subgroup $N \\leqslant \\pi_1(X_{i}^{\\mathrm{reg}})$, then we get $\\phi_{i}'$ as the finite cover of $X_{i-1}$ corresponding to the image of $N$ in $\\pi_1(X_i^{\\mathrm{reg}})$ under the above homomorphism $\\varphi$. By the fiber product $X_{i+1}=X_i \\times_{X_{i-1}} X_i'$ (which preserves \\'etaleness, finiteness and GIT-quotients), we get the commutative diagram. \n\nThus, if for all $j \\in \\mathbb{N}$, there exists a $k\\geq j$ such that $\\phi_k$ is a finite quasi-\\'etale but not \\'etale cover, by reordering of the $\\phi_i$ according to the claim just proven, we can construct an infinite sequence\n\\[\n \\xymatrix@R=2em@C=2em{\nX'=X_0 &\nX'_1\\ar[l]_-{\\phi'_1} &\nX'_2\\ar[l]_-{\\phi'_2} &\nX'_3\\ar[l]_-{\\phi'_3} & \n\\dots \\ar[l]_-{\\phi'_4} &\nX'_i\\ar[l]_-{\\phi'_i} &\nX'_{i+1}\\ar[l]_-{\\phi'_{i+1}}& \n\\dots \\ar[l]_-{\\phi'_{i+2}}&\n}\n\\]\nwhere the $\\phi'_i$ are finite Galois quasi-\\'etale but not \\'etale covers. But this is a contradiction to~\\cite[Theorem 1.1]{GKP16}. \nThus, there are only finitely many finite Galois quasi-\\'etale and not \\'etale covers in the sequence, i.e., there are only finitely many finite quasi-torsors in this sequence that are not finite torsors.\n\\end{proof}\n\nIn what follows, we turn to prove Theorem~\\ref{thm:iteration2}.\nTo do so, we will use the following lemmata.\n\n\\begin{lemma}\\label{lem:square-diagram}\nLet $\\phi\\colon Y\\rightarrow X$ be a $\\mathbb{T}$-quasi-torsor of a klt type variety corresponding to the subgroup\n$N$ of ${\\rm WDiv}(X)$.\nLet $\\pi\\colon X'\\rightarrow X$ be a finite torsor.\nLet $Y'=Y\\times_X X'$ and $Y'\\rightarrow X'$ be the associated $\\mathbb{T}$-quasi-torsors so that we have a commutative diagram:\n\\[\n \\xymatrix@R=2em@C=2em{\n Y \\ar[d]_-{\\phi} & Y' \\ar[l]\\ar[d]^-{\\phi'} \\\\\n X & X'\\ar[l]_-{\\pi}. \n }\n\\]\nThen, $\\phi'$ is the $\\mathbb{T}$-quasi-torsor associated to the subgroup $\\pi^*N$ of ${\\rm WDiv}(X')$.\n\\end{lemma} \n\n\\begin{proof}\nWe consider the dual diagram\n\\[\n \\xymatrix@R=2em@L=2em{\n \\bigoplus_{D \\in N} \\mathcal{O}_X(D) \\ar[r]& \\left(\\bigoplus_{D \\in N} \\mathcal{O}_X(D)\\right) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X'} \\\\\n \\mathcal{O}_X\\ar[r] \\ar[u] & \\mathcal{O}_{X'}. \\ar[u]\n }\n\\]\nThe top right entry equals $\\bigoplus_{D \\in N} \\left(\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X'}\\right)$. We have $\\mathcal{O}_{X'}(\\pi^*D)=\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X'}$ by definition of the pullback. So the claim follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:dominating-qt}\nLet $Y\\rightarrow X$ be a torus quasi-torsor corresponding to\nthe subgroup $N_Y$ of ${\\rm WDiv}(X)$.\nLet $Z\\rightarrow X$ be a torus quasi-torsor corresponding to the subgroup $N_Z$ of ${\\rm WDiv}(X)$.\nIf $N_Z\\geqslant N_Y$ and $N_Z\/N_Y$ is torsion free, then \nthere is an induced \ntorus quasi-torsor $Z\\rightarrow Y$.\n\\end{lemma}\n\n\\begin{proof}\nThe condition that $N_Z\/N_Y$ is torsion free means that we have a direct product representation $N_Z=N_Y \\oplus N'$ with a subgroup $N'$ of $N_Z$ isomorphic to $N_Z\/N_Y$. The downgrading of $\\mathcal{O}_Z$ from $N_Z$ to $N'$ gives an action of a subtorus $\\mathbb{T}_{N'} \\leqslant \\mathbb{T}_{N_Z}$ on $Z$. By construction, $Z\/\\!\\!\/\\mathbb{T}_{N'}=Y$. Now the statement follows from Lemma~\\ref{lem:quot-qetale-torus-cover}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:iteration2}]\nThe proof of the theorem will consist of three steps. We briefly explain the steps here. In the first step, we will produce a singular variety with a special toric divisor.\nIn the second step, we will produce an infinite sequence of finite torsors for such a singular variety. We show that the rank of the group of Weil divisors modulo Cartier divisors diverges in this sequence.\nFinally, we will use this divergence property to produce the infinite sequence of $\\mathbb{T}$-quasi-torsors that are not torsors.\\\\\n\n\\noindent\\textit{Step 1:} For each $n\\geq 2$, we construct a $n$-dimensional projective variety with a single isolated toric singularity and infinite \\'etale fundamental group.\\\\\n\nLet $Z^n$ be a smooth projective variety with infinite \\'etale fundamental group.\nLet $z\\in Z^n$ be a smooth point.\nIn local coordinates around $z\\in Z$,\nthe formal completion $\\hat{\\mathcal{O}}_{Z,z}$ corresponds to the standard fan \n$\\langle e_1,\\dots, e_n \\rangle \\subset \\mathbb{R}^n$.\nFor each $n\\geq 2$, we consider the blow-up given by the fan decomposition\n\\[\n\\Sigma_n:=\\{ \n\\langle \\bar{e_1}, e_2, e_3,\\dots, e_n, v\\rangle,\n\\langle e_1, \\bar{e_2},e_2,e_3, \\dots, v\\rangle, \\dots ,\\langle e_1,\\dots, e_{n-1},\\bar{e_n},v\\rangle \n\\},\n\\]\nwhere $v=2e_1+e_2+e_3+\\dots+e_n$.\nWe let $Y^n \\rightarrow Z^n$ to be the corresponding blow-up. \nObserve that $Y^n$ has a unique isolated toric singularity.\nWe let $y^n\\in Y^n$ be such isolated toric singularity.\nNote that the local Class group\nof $Y^n$ at $y^n$ is \n$\\mathbb{Z}_2$.\nFor $n=2$, this point is a rational double point.\nBy construction, there is a divisor $T^n\\subset Y^n$ which is a normal projective toric variety \nand $y^n$ is contained in $T^n$.\nIndeed, this toric variety corresponds to the primitive lattice generator $v\\in \\Sigma_n(1)$.\nSince $Y^n$ has klt singularities and $Z^n$ is smooth, we conclude that\n$\\pi_1(Y^n)\\simeq \\pi_1(Z^n)$.\nIn particular, the \\'etale fundamental group of $Y^n$ is infinite.\\\\\n\n\\noindent\\textit{Step 2:} We construct a sequence of finite \\'etale Galois covers of $Y^n$ \nand study their groups of Weil divisors modulo Cartier divisors.\\\\\n\nLet \n\\[\n \\xymatrix@R=2em@C=2em{\nY^n=Y^n_0 &\nY^n_1\\ar[l]_-{f_1} &\nY^n_2\\ar[l]_-{f_2} &\nY^n_3\\ar[l]_-{f_3} & \n\\dots \\ar[l]_-{f_4} &\nY^n_i\\ar[l]_-{f_i} &\nY^n_{i+1}\\ar[l]_-{f_{i+1}}& \n\\dots \\ar[l]_-{f_{i+2}}&\n}\n\\]\nbe an infinite sequence of finite \\'etale Galois covers. \nLet $k_i:={\\rm deg}(Y^n_{i+1}\\rightarrow Y^n_i)$\nbe the degree of the cover.\nThen, the variety $Y^n_i$ is $n$-dimensional and it has $k_0\\cdots k_{i-1}$ isolated singularities.\nWe denote these singularities as \n\\[\ny^n_{i,(m_0,\\dots,m_{i-1})} \\in Y^n_i,\n\\]\nwhere $1\\leq m_j \\leq k_j$ for each $j$.\nWe can order the singularities in such a way that\n\\[\nf_i^{-1}(y_{i-1,(m_0,\\dots,m_{i-1})})\n= \n\\bigcup_{m=1}^{k_i} y^n_{i,(m_0,\\dots,m_{i-1},m)}.\n\\]\nSince $T^n$ has trivial fundamental group, we conclude that \n$f_i^*\\dots f_1^*T^n$ is the disjoint union \nof $k_0\\cdots k_{i-1}$ toric varieties\nisomorphic to $T^n$.\nWe write\n$T^n_{i,(m_0,\\dots,m_{i-1})}$ \nwith $1\\leq m_j\\leq k_j$ for such toric divisors.\nBy construction,\nthe toric divisor\n$T^n_{i,(m_0,\\dots,m_{i-1})}$ contains the singular point \n$y^n_{i,(m_0,\\dots,m_{i-1})}$.\nWe claim that \n\\begin{equation}\\label{eq:isom-weil-mod-cart} \n{\\rm WDiv}(Y_i^n)\/{\\rm CaDiv}(Y_i^n) \\simeq \\bigoplus_{i=1}^{k_0\\cdots k_{i-1}} \\mathbb{Z}_2.\n\\end{equation} \nFirst, observe that $T^n_{i,(m_0,\\dots,m_{i-1})}$ is not a Cartier divisor.\nIndeed, if this was the case then\n$T^n_{i,(m_0,\\dots,m_{i-1})}$ would be analytically Cartier around\n$y^n_{i,(m_0,\\dots,m_{i-1})}$.\nThis implies that $T^n$ is analytically Cartier around $y^n$, leading to a contradiction.\nOn the other hand\n$2T^n_{i,(m_0,\\dots,m_{i-1})}$ is Cartier in $Y^n_i$ as\nit is the pull-back of $2T^n$ on a neighborhood of the only singular point that it contains.\nWe conclude that each \n$T^n_{i,(m_0,\\dots,m_{i-1})}$ is $2$-torsion in the abelian group\n${\\rm WDiv}(Y^n)\/{\\rm CaDiv}(Y^n)$.\nLet $J\\subset ([1,k_0] \\cap \\mathbb{Z}) \\times\\dots\\times ([1,k_{i-1}]\\cap \\mathbb{Z})$ be a subset. \nAssume that we have a relation of the form \n\\[\n\\sum_{j\\in J} T^n_{i,j} =0 \\in {\\rm WDiv(Y^n)}\/{\\rm CaDiv}(Y^n).\n\\]\nThis means that the divisor\n$\\sum_{j\\in J} T^n_{i,j}$ is Cartier in $Y^n_i$.\nLet $j_0\\in J$ be a fixed element.\nFor each $j_k \\neq j_0$ in $J$, we have that $T^n_{i,j_k}$ is Cartier at $y^n_{i,j_0}$. \nWe conclude that $T^n_{i,j_0}$ is Cartier at $y^n_{i,j_0}$.\nHence, it is a Cartier divisor. This leads to a contradiction. \nThen, the isomorphism~\\eqref{eq:isom-weil-mod-cart} holds.\\\\\n\n\\noindent\\textit{Step 3:} In this step, we construct an infinite sequence of finite torsors and\ntorus quasi-torsors of $Y^n$.\\\\\n\nFor each $i\\geq 1$, we denote by $N_i$\nthe group of ${\\rm WDiv}(Y^n_i)$ generated by\n\\[\n\\{ T^n_{i,(m_0,\\dots,m_{i-1})} \\mid \n1\\leq m_0\\leq k_0, \\dots, \n1\\leq m_{i-2}\\leq k_{i-2}, \n\\text{ and }\n1\\leq m_{i-1} \\leq k_{i-1}-1\n\\}.\n\\]\nFor each $i\\geq 0$, \nwe define $X_{2i}^n$ to be the relative Cox ring of $Y_i^n$ with respect to $N_i$.\nWe define $X_{2i+1}$\nto be $X_{2i}\\times_{Y_i^n} Y_{i+1}^n$.\nWe define $\\phi_{2i+1}\\colon X_{2i+1}^n \\rightarrow X_{2i}^n$ to be the induced morphism.\nThus, we have a commutative diagram as follows:\n\\[\n \\xymatrix@R=2em@C=2em{\n X_{2i}^n \\ar[d] & X_{2i+1}^n \\ar[l]_-{\\phi_{2i+1}} \\ar[d] \\\\\n Y^n_i & Y^n_{i+1}\\ar[l]_-{f_{i+1}} ,\n }\n\\]\nBy Lemma~\\ref{lem:square-diagram},\nthe torus quasi-torsor\n$X_{2i+1}^n \\rightarrow Y^n_{i+1}$ is induced by the subgroup\n$f_{i+1}^* N_i \\leqslant {\\rm WDiv}(Y^n_{i+1})$.\nBy construction, we have that\n$f_{i+1}^*N_i \\leqslant N_{i+1}$.\nBy Lemma~\\ref{lem:dominating-qt} there is a corresponding quasi-torsor $\\phi_{2(i+1)} \\colon X_{2(i+1)}\\rightarrow X_{2i+1}^n$.\n\nWe claim that $\\phi_{2(i+1)}$ is not a torus torsor.\nLet $C$ be the Class group\nof $Y_{i+1}$ at the point\n$y_{i+1,(k_0,\\dots,k_{i-1},1)}$.\nBy the isomorphism~\\eqref{eq:isom-weil-mod-cart},\nwe know that $C\\simeq \\mathbb{Z}_2$.\nNote that the image of $N_{i+1}$ in $C$ is isomorphic to $\\mathbb{Z}_2$.\nIndeed, the image of the divisor $T^n_{i+1,(k_0,\\dots,k_{i-1},1)}$ generates $C$.\nOn the other hand,\nthe image of $f_{i+1}^*N_i$ in $C$ is trivial\nsince no divisor among the generators of $N_i$ \npass through $y_{i,(k_0,\\dots,k_{i-1})}$.\nBy Lemma~\\ref{lem:two-qt}, we conclude that $\\phi_{2(i+1)}$ is a torus quasi-torsor which is not a torsor.\nWe deduce that there exists an infinite sequence \n\\[\n \\xymatrix@R=2em@C=2em{\nX^n=X^n_0 &\nX^n_1\\ar[l]_-{\\phi_1} &\nX^n_2\\ar[l]_-{\\phi_2} &\nX^n_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX^n_i\\ar[l]_-{\\phi_i} &\nX^n_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{i+2}&\n}\n\\]\nsatisfying the following conditions:\n\\begin{itemize}\n\\item $X^n$ is a $n$-dimensional projective variety with a single isolated toric singularity,\n\\item each $\\phi_i$, with $i$ odd, is a finite torsor, and\n\\item each $\\phi_i$, with $i\\geq 2$ even, is a $\\mathbb{T}$-quasi-torsor which is not a torsor.\n\\end{itemize}\nThis finishes the proof.\n\\end{proof}\n\nNow, we turn to prove Theorem~\\ref{thm:iteration3}.\nWe will need the following two lemmata.\n\n\\begin{lemma}\\label{lem:pullback-cl}\nLet $f\\colon X'\\rightarrow X$ be a finite $G$-torsor.\nLet $x\\in X$ and $x'\\in f^{-1}(x)$ be two closed points.\nThen, the induced homomorphism\n$f^*\\colon {\\rm Cl}(X;x)\\rightarrow {\\rm Cl}(X';{x'})$\nis a monomorphism.\n\\end{lemma}\n\n\\begin{proof}\nLet $W$ be a Weil divisor through $x \\in X$ such that $f^*W$ is principal near $x'$. We can even assume that there is a $G$-invariant open around $x'$ where $f^*W=\\mathrm{div}(h)$. By~\\cite[Thm.~II.3.1]{AGIV}, $h$ is a semiinvariant, i.e. $g^*h=\\chi(g) h$, where $\\chi(g) \\in \\mathbb{K}^*$, for every $g$ in $G$. But the induced action via $\\chi \\colon G \\to \\mathbb{K}^* \\subseteq \\mathbb{K}[X]^*$ on $X$ is ramified if it is nontrivial. This can not happen since $f$ is \\'etale. Therefore, $h$ is $G$-invariant and defines $W=\\mathrm{div}_{U}(h)$ on some $x \\in U \\subseteq X$. The claim follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:factorial-qt}\nLet $Y\\rightarrow X$ be the $\\mathbb{T}$-quasi-torsor\nassociated to the group $N\\leqslant {\\rm WDiv}(X)$.\nThe variety $Y$ is locally factorial if and only if\n$N\\rightarrow {\\rm Cl}(X;x)$ is surjective for every\nclosed point $x\\in X$.\nIn particular, if $N\\rightarrow {\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is surjective, then $Y\\rightarrow X$ is a factorial $\\mathbb{T}$-quasi-torsor. \n\\end{lemma}\n\n\\begin{proof}\nThe statement follows from~\\cite[Theorem~1.3.3.3]{ADHL15} applied to the spectrum $X_x$ of the local ring $\\mathcal{O}_{X,x}$, where we view $N$ as a subgroup of ${\\rm WDiv}(X_x)$ by restriction. Then the aforementioned theorem says that the stalk $\\mathcal{R}(X)_{N,x}$ is factorial if and only if $N\\rightarrow {\\rm Cl}(X;x)$ is surjective. In fact,~\\cite[Theorem~1.3.3.3]{ADHL15} only states one direction, but it directly follows from applying~\\cite[Theorem~1.3.3.1]{ADHL15} to the smooth locus, which gives an equivalence. It is clear that $Y$ is locally factorial if and only if all the stalks $\\mathcal{R}(X)_{N,x}$ are factorial.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:iteration3}]\nConsider a sequence \n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\]\nas in the statement of the theorem.\nThis means that every $\\phi_i$ is either a factorial\n$\\mathbb{T}$-quasi-torsor or a finite quasi-torsor.\nBy Theorem~\\ref{thm:iteration1}, we may assume, after possibly truncating our sequence, that every finite quasi-torsor in this sequence is a finite torsor, i.e., a finite Galois \\'etale cover.\nProceeding as in the proof of Theorem~\\ref{thm:iteration1}, we obtain a sequence of finite torsors:\n\\[\n \\xymatrix@R=2em@C=2em{\nX'=X'_0 &\nX'_1\\ar[l]_-{\\phi'_1} &\nX'_2\\ar[l]_-{\\phi'_2} &\nX'_3\\ar[l]_-{\\phi'_3} & \n\\dots \\ar[l]_-{\\phi'_4} &\nX'_i\\ar[l]_-{\\phi'_i} &\nX'_{i+1}\\ar[l]_-{\\phi'_{i+1}}& \n\\dots \\ar[l]_-{\\phi'_{i+2}}&\n}\n\\]\nso that each $X_i\\rightarrow X'_i$ is a relative Cox ring\nwith respect to the subgroup\n$N_i\\leqslant {\\rm WDiv}(X_i')$.\nBy Lemma~\\ref{lem:two-qt}, we know that \nevery $\\mathbb{T}$-quasi-torsor over a factorial variety is\nindeed a torsor.\nWe write $\\psi_i = \\phi_i \\circ \\dots \\circ \\phi_1$.\nBy~\\cite[Theorem 3.4.(1)]{bf84}, there exists a locally closed decomposition \n$X'=\\bigsqcup_{j\\in J} Y'_j$\nsuch that the class group\nof $X'_i$ at $x$ is independent of $x\\in \\psi_i^{-1}(Y'_j)$.\nApplying Lemma~\\ref{lem:pullback-cl} to closed points of $Y'_j$, we conclude that there exists $i_0\\in \\mathbb{Z}_{>0}$, \nsuch that \n\\begin{equation}\\label{eq:isoms-cl} \n{\\phi'_i}^*\\colon {\\rm Cl}({X'_{i}};y)\\rightarrow \n{\\rm Cl}({X'_{i+1}};x)\n\\end{equation} \nis an isomorphism for every\n$x\\in X'_i$,\nevery $y\\in {\\phi'_i}^{-1}(x)$, and $i\\geq i_0$.\nIt suffices to show that whenever $X_i$ is factorial\nand $X_{i+1}\\rightarrow X_i$ is a finite \\'etale Galois cover in our sequence, the variety $X_{i+1}$ is again factorial for every $i\\geq i_0$. \nIn this case, we have a commutative diagram \n\\[\n \\xymatrix@R=2em@C=2em{\n X_i \\ar[d]_-{\\pi_i} & X_{i+1}\\ar[d]^-{\\pi_{i+1}} \\ar[l]_-{\\phi_{i}} \\\\\n X_i' & X_{i+1}\\ar[l]_-{\\phi_i'}\n }\n\\]\nwhere $\\pi_i$ is the relative Cox ring over $X'_i$ with respect to the subgroup $N_i\\leqslant {\\rm WDiv}(X_i')$.\nBy Lemma~\\ref{lem:square-diagram}, the $\\mathbb{T}$-quasi-torsor\n$X_{i+1}\\rightarrow X_i$ is induced by\n${\\phi_i'}^*N_i \\leqslant {\\rm WDiv}(X_{i+1})$.\nBy Lemma~\\ref{lem:factorial-qt}, we know that for\nevery closed point $x\\in X_i'$, the induced homomorphism\n\\begin{equation}\\label{eq:surj-local-class}\nN_i \\rightarrow {\\rm Cl}({X_i'};x)\n\\end{equation} \nis surjective.\nBy isomorphism~\\eqref{eq:isoms-cl} and surjectivity~\\eqref{eq:surj-local-class}, we conclude that for every closed point $x\\in X'_{i+1}$ we have that \n${\\phi'_i}^*N_i\\rightarrow {\\rm Cl}(X'_{i+1};x)$ is surjective. By Lemma~\\ref{lem:factorial-qt}, we conclude that $X_{i+1}$ is a factorial variety. \nThis finishes the proof.\n\\end{proof} \n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:factorial-cover}]\nDue to Theorem~\\ref{thm:iteration3}, we may find a variety $Y$ \nsatisfying the following properties:\n\\begin{enumerate}\n \\item[(i)] every finite quasi-torsor over $Y$ is a finite torsor, \n \\item[(ii)] every factorial $\\mathbb{T}$-quasi-torsor over a finite torsor of $Y$ is a $\\mathbb{T}$-torsor, \n \\item[(iii)] $Y$ admits the action of a reductive group $G$,\n \\item[(iv)] the group $G$ is an extension of an algebraic torus by a finite solvable group, and \n \\item[(v)] the isomorphism $X\\simeq Y\/\\!\\!\/G$ holds.\n\\end{enumerate}\nNote that condition (i) implies that the natural epimorphism\n\\[\n\\hat{\\pi}_1(Y^{\\rm reg})\\rightarrow\n\\hat{\\pi}_1(Y)\n\\]\nis an isomorphism.\nOtherwise, we could find a finite Galois quasi-\\'etale cover of $Y$ that ramifies over the singular locus.\nThis shows that (1) in the statement of the theorem holds.\n\nLet $Y'\\rightarrow Y$ be a finite quasi-\\'etale morphism.\nBy condition (i) this morphism is indeed\na finite \\'etale morphism. \nAssume that $Y'$ is not factorial at the point $y'$.\nBy~\\cite[Theorem 3.7]{GKP16}\nthere exists a finite \\'etale Galois morphism\n$Y''\\rightarrow Y$ \nsuch that $Y''$ admits a finite \\'etale Galois morphism to $Y'$.\nBy Lemma~\\ref{lem:pullback-cl}, we conclude that $Y''$ is not factorial.\nThus, $Y''$ admits a factorial $\\mathbb{T}$-quasi-torsor that is not a $\\mathbb{T}$-torsor.\nIndeed, we can take the relative Cox ring of $Y''$ with respect to a subgroup $N$ of ${\\rm WDiv}(Y'')$\nthat surjects onto\n${\\rm WDiv}(Y'')\/{\\rm CaDiv}(Y'')$.\nThis contradicts condition (ii).\nWe conclude that (2) in the statement of the theorem holds. \nNote that (iii)-(v) are the same than\n(3)-(5) in the statement of the theorem.\nThis finishes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:toric-quot}]\nLocally toric singularities are klt type singularities.\nThen, we can apply Theorem~\\ref{thm:iteration1} to deduce that \n$X$ admits a torus quasi-torsor $Y$ that is factorial.\nBy Theorem~\\ref{introthm-finite-torus-cover-klt-var}, the variety $Y$ has klt type singularities, hence canonical factorial singularities.\nThe local Cox ring of a locally toric singularity is a locally toric singularity.\nHence, the variety $Y$ has factorial locally toric singularities.\nHowever, a factorial toric singularity is smooth.\nWe conclude that $Y$ is a smooth variety.\n\\end{proof}\n\n\\subsection{Normal singularities}\n\\label{subsec:normal-singularities}\nIn this subsection, we show that some of the proofs explained above naturally generalize to normal singularities with some minor considerations.\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:optimal-T}]\nIf each Class group ${\\rm Cl}(X;x)$ is finitely generated, then by Theorem~\\ref{thm:wdiv-cdiv}, we know that\n${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is finitely generated.\nThen, the proof is verbatim from the proof of Theorem~\\ref{introthm:iteration-torus-quasi-torsors}.\nThis shows that $(2)$ implies $(1)$.\n\nNow, we turn to prove that $(1)$ implies $(2)$.\nOn the other hand assume that some local Class group ${\\rm Cl}(X;x_0)$ is not finitely generated. We consider an infinite sequence of divisors\n$\\{W_i\\}_{i\\in \\mathbb{N}}$ in ${\\rm WDiv}(X)$\nsuch that the image of $N_k:=\\langle W_1,\\dots,W_k\\rangle$ in ${\\rm Cl}(X;x_0)$ strictly contains $N_{k-1}$.\nLet $X_i\\rightarrow X$ be the $\\mathbb{T}$-quasi-torsor associated to $N_i$. \nBy construction, the quotients \n$N_{i+1}\/N_i$ are free.\nThen, by Lemma~\\ref{lem:dominating-qt}, we have associated \ntorus quasi-torsors \n\\[\n \\xymatrix@R=2em@C=2em{\nX=X_0 &\nX_1\\ar[l]_-{\\phi_1} &\nX_2\\ar[l]_-{\\phi_2} &\nX_3\\ar[l]_-{\\phi_3} & \n\\dots \\ar[l]_-{\\phi_4} &\nX_i\\ar[l]_-{\\phi_i} &\nX_{i+1}\\ar[l]_-{\\phi_{i+1}}& \n\\dots \\ar[l]_-{\\phi_{i+2}}&\n}\n\\] \nBy Lemma~\\ref{lem:two-qt}.(3), no \n$\\phi_i$ is a $\\mathbb{T}$-torsor.\n\\end{proof}\n\nNow, we turn to give a proof of Theorem~\\ref{introthm:optimal-mixed} that discusses optimal normal singularities for which sequences of quasi-torsors are eventually torsors.\nWe will use the following lemma.\n\n\\begin{lemma}\\label{lem:rat1-sing-cover}\nLet $f\\colon X'\\rightarrow X$ be a quasi-\\'etale finite Galois morphism. Let $x\\in X$ and $x'\\in f^{-1}(x)$ be finite points.\nThen, if ${\\rm Cl}(X';x')$ is finitely generated, then \n${\\rm Cl}(X;x)$ is finitely generated.\n\\end{lemma}\n\n\\begin{proof}\nLet $W$ be a Weil divisor through $x\\in X$ such that $f^*W$ is Cartier.\nThen, passing to a $G$-invariant affine we may assume $f^*W={\\rm div}(h)$. The regular function $h^{|G|}$ is $G$-invariant. \nHence $mW\\sim 0$ around $x$.\nWe conclude that the kernel of\n$f^*\\colon {\\rm Cl}(X;x)\\rightarrow {\\rm Cl}(X';x')$ is torsion.\nSo, if ${\\rm Cl}(X;x)$ is not finitely generated, then \n$f^*{\\rm Cl}(X;x)$ is not finitely generated and the claim follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{introthm:optimal-mixed}]\nFirst, assume that conditions $(a)$ and $(b)$ are satisfied. \nIn the proof of Theorem~\\ref{thm:iteration3}, we used the argument of Theorem~\\ref{thm:iteration1}\nto deduce that the finite quasi-torsors are eventually torsors. \nThe same argument \ngoes through in the present case by \nreplacing~\\cite[Theorem 1.1]{GKP16}\nwith~\\cite[Theorem 1]{Sti17}.\n\nOn the other hand, in the proof of Theorem~\\ref{thm:iteration3}, we used\nthe constructibility of the functor ${\\rm Cl}$ in the \\'etale topology.\nWe argue that this still holds in the present setting.\nBy Lemma~\\ref{lem:rat1-sing-cover}, every local Class group ${\\rm Cl}(X;x)$ is finitely generated.\nLet $f\\colon X'\\rightarrow X$ be a resolution of singularities. \nIn this case we have that\n$R^1f_*(\\mathcal{O}_{X'})=0$ \nas the local Class groups are finitely generated.\nThen, we can apply~\\cite[Theorem 3.1.(4)]{GKP16} to conclude that the Class group functor is constructible in the \\'etale topology.\nNow, the proof is verbatim from the proof of Theorem~\\ref{thm:iteration3}.\n\nNow, assume that condition $(1)$ is not satisfied.\nBy~\\cite[Theorem 1]{Sti17}, there is an infinite sequence of finite quasi-torsors of $X$ that are not torsors.\nOn the other hand, assume that condition $(2)$ is not satisfied.\nThen, there exists a finite quasi-\\'etale cover $X'\\rightarrow X$ and a point $x'$ for which ${\\rm Cl}(X';x')$ is not finitely generated.\nHence, proceeding as in the proof of Theorem~\\ref{introthm:optimal-T}, we conclude that there exists an infinite sequence of $\\mathbb{T}$-quasi-torsors\nover $X'$ that are not torsors.\n\\end{proof} \n\n\\section{Examples and Questions}\n\nIn this section, we collect some examples related to the theorems of the article\nand some questions that lead to further research.\n\n\\begin{example}\\label{ex:cod-1-ram}\n{\\em \nIn this example, we show that \nfinite covers of klt singularities ramified\nover codimension one points may not be klt.\nLet $D=\\{(y,z)\\mid y^3+z^m=0 \\}\\subset \\mathbb{A}^2_{y,z}$.\nThen the singularity\n\\[\nX:=\\{ (x,y,z) | x^2+y^3+z^m=0\\} \n\\]\nis a double cover of $\\mathbb{A}^2_{y,z}$ ramified along $D$.\nThe singularity $(X;(0,0,0))$ is Du Val\nfor $m\\in \\{4,5\\}$.\nOtherwise, it is not a klt surface singularity.\n}\n\\end{example}\n\n\\begin{example}\\label{ex:torsion-quotient}\n{\\em \nIn the construction of the local Cox ring of a singularity $(X;x)$, \nwe choose a homomorphism\n$N\\rightarrow {\\rm Cl}(X;x)$ with kernel $N_0$ \nand a character $\\chi \\colon N^0\\rightarrow \\mathbb{K}(X)^*$ for which\n${\\rm div}(\\chi(E))=E$\nfor all $E\\in N^0$.\nIf $X$ is a projective variety for which ${\\rm WDiv}(X)\/{\\rm CaDiv}(X)$ is torsion, \nthen we can consider a surjective homomorphism\n$N\\rightarrow {\\rm WDiv}(X)\/{\\rm CaDiv}(X)$\nwith kernel $N^0$\nand a character as before $\\chi\\colon N^0\\rightarrow \\mathbb{K}(X)^*$.\nIf $\\mathcal{I}$ is the ideal sheaf \ngenerated by $1-\\chi(E)$ with $E\\in N^0$, \nthen the morphism\n\\[\nY:={\\rm Spec}_X(\\mathcal{R}(X)_N\/\\mathcal{I})\\rightarrow X\n\\]\nis finite and ramifies over codimension one points. A priori, it is not clear how to control the divisors over which the previous morphism ramifies.\nHence, it is not clear whether $Y$ has klt type singularities provided that $X$ has klt type singularities.\nThis means that the concept of relative Cox ring with quotients induced by characters is not well-behaved from the singularities perspective.\n}\n\\end{example}\n\n\\begin{example}\\label{ex:sing-improve}\n{\\em \nThe local Cox ring of a klt type singularity $(X;x)$ is non-trivial whenever\n${\\rm Cl}(X;x)$ is non-trivial.\nIf $(Y;y)\\rightarrow (X;x)$ is the spectrum of the local Cox ring of $(X;x)$,\nthen we expect that the equations defining $(Y;y)$ are somewhat simpler than the equations defining $(X;x)$.\nAlthough, whenever ${\\rm Cl}(X;x)$ has non-trivial free part, the dimension of $(Y;y)$ is larger than the dimension of $(X;x)$.\nFor instance, if $(X;x)$ is a toric singularity, then\n${\\rm Cox}(X;x)$ is a smooth point of dimension $\\dim X +\\rho(X_x)$.\n}\n\\end{example}\n\n\\begin{example}\\label{ex:thm7-analytically-local}\n{\\em \nConsider the affine variety\n\\[\nX=\\{(x,y,z,w)\\mid xy+zw+z^3+w^3=0\\}.\n\\]\nThe variety $X$ is canonical with isolated singularities.\nFurthermore, $X$ is factorial at $x:=(0,0,0,0)$.\nHowever, $X$ is not analytically factorial at $x$. \nLet $Y\\rightarrow X$ be a $\\mathbb{T}$-quasi-torsor.\nThen, $\\pi\\colon Y\\rightarrow X$ is a $\\mathbb{T}$-torsor on an affine neighborhood of $x\\in X$, by Lemma~\\ref{lem:two-qt}.\nHence, $Y$ is singular along $\\pi^{-1}(x)$.\nThis leads to a contradiction.\n}\n\\end{example}\n\n\\begin{example}\n\\label{ex:semisimple} \n{\\em \nOver a smooth point, every finite quasi-torsor is a torsor\nand every torus quasi-torsor is a torsor.\nIn this example, we show the existence of a ${\\rm SL}_n(\\mathbb{K})$-quasi-torsor over a smooth germ that is not a torsor.\nWe refer the reader to~\\cite{Pop92} for more examples in this direction.\n\nIn what follows, we let $n\\geq 2$.\nLet $W$ be the space of linear transformations from \n$\\mathbb{C}^{n+1}$ to $\\mathbb{C}^n$.\nNote that $W$ has dimension $n^2+n$.\nLet ${\\rm SL}_n(\\mathbb{C})$ act on $W$ by acting on the range of the linear function.\nThe action is free exactly at all the points corresponding to surjective linear transformations.\nThe closure of the orbit of an element contains $0\\in W$ if and only if the corresponding linear transformation does not have full rank.\nLet $U\\subset W$ be the open set consisting of surjective linear transformations.\nNote that the ${\\rm SL}_n(\\mathbb{C})$-action naturally extend to a ${\\rm Gl}_n(\\mathbb{C})$-action.\nThe space of surjective linear transformations, up to the ${\\rm GL}_n(\\mathbb{C})$-action, is parametrized by their kernels.\nHence, the quotient $U\/\\!\\!\/{\\rm GL}_n(\\mathbb{C})\\simeq \\mathbb{P}^{n}$.\nFurthermore, we have that \n$U\/\\!\\!\/{\\rm SL}_n(\\mathbb{C})\\simeq \\mathbb{A}^{n+1}-\\{0\\}$.\nIt follows that \n$W\/\\!\\!\/{\\rm SL}_n(\\mathbb{C})\\simeq \\mathbb{A}^n$.\nAs explained above, the action is free on $U$\nso $W\\rightarrow \\mathbb{A}^n$ is a ${\\rm SL}_n(\\mathbb{C})$-torsor over $\\mathbb{A}^n-\\{0\\}$.\nOn the other hand, the fiber over $\\{0\\}$ is given by the vanishing of at least $n$ minors, so its codimension in $W$ is at least $2$.\nThus, $W\\rightarrow \\mathbb{A}^n$ gives a ${\\rm SL}_n(\\mathbb{C})$-quasi-torsor which is not a torsor.\n}\n\\end{example}\n\n\\begin{remark}\n{\\em \nOne of the reasons that makes the understanding of finite quasi-torsors of a singularity $(X;x)$ easier than other kinds of quasi-torsors\nis the existence of an algebraic object that detects them.\nThe same holds for torus quasi-torsors.\n\nThe previous example might point in the direction that this is not true anymore for arbitrary reductive groups, which might render the study of their (quasi-)torsors a lot more complicated.\n}\n\\end{remark}\n\n\\begin{question}\n{\\em \nIn Theorem~\\ref{introthrm:gl2-cover}, we showed that there exists a $3$-fold toric singularity $(T;t)$ that admits a $\\mathbb{P}{\\rm GL}_3(\\mathbb{K})$-cover from a $5$-dimensional singularity which is not of klt type.\nThis cover is unramified over codimension points over $T$ so the pathology in Example~\\ref{ex:cod-1-ram} does not happen.\nThis naturally leads to the following question.\nIs there a $G$-cover over a surface klt singularity that is unramified over codimension one points and is not of klt type?\n}\n\\end{question}\n\n\\begin{question}\\label{quest:G-qt-klt-type}\n{\\em \nIn Theorem~\\ref{introthm:torus-finite-cover}, we showed that if\n$(X;x)$ is a klt type singularity\nand $G$ is a finite extension of a torus, then a $G$-quasi-torsor over $(X;x)$ is of klt type. \nDoes this statement still hold if we only assume that $G$ is a reductive group?\nWe expect that there are counter-examples for this statement if $G$ is a unipotent group.\n}\n\\end{question}\n\n\\bibliographystyle{habbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\n\nIn 1985, Falconer \\cite{Falconer85} (implicitly) conjectured that if $A\\subset\\mathbb{R}^d$, with $d\\ge 2$, is a Borel set of Hausdorff dimension at least $d\/2$, then the set of distances\n\\[\n\\mathrm{dist}(A,A) = \\{ |x-y|:x,y\\in A\\}\n\\]\nhas Hausdorff dimension $1$. He also showed that the value $d\/2$ would be sharp. The conjecture remains wide open in every dimension, but several deep advances have been obtained; we discuss in some detail what is known in the plane, and refer to \\cite{Erdogan05} for some of the known results in higher dimensions. Throughout the paper, $\\hdim$, $\\pdim$, $\\lbdim$, $\\ubdim$, and $\\mlbdim$ denote, respectively Hausdorff, packing, lower box-counting, upper box-counting, and modified lower box-counting dimensions. See \\S\\ref{subsec:dimension} for the definitions, and \\cite{Falconer14} for further background on fractal dimensions.\n\nRelying on earlier work of Mattila \\cite{Mattila87}, and using deep harmonic-analytic techniques, Wolff \\cite{Wolff99} showed that if $\\hdim(A)>4\/3$, then $\\mathrm{dist}(A,A)$ has positive Lebesgue measure. As remarked in \\cite{Wolff99}, $4\/3$ appears to be the limit of these methods. Nevertheless, Iosevich and Liu \\cite{IosevichLiu16} recently obtained an improvement for a large class of cartesian products.\n\nAssuming only that $\\hdim(A)\\ge 1$, Bourgain \\cite{Bourgain03} (relying on earlier work of Katz and Tao \\cite{KatzTao01}) used sophisticated additive-combinatorial arguments to prove that\n\\[\n\\hdim(\\mathrm{dist}(A,A)) > 1\/2+\\varepsilon,\n\\]\nfor some small absolute constant $\\varepsilon>0$.\n\nTo the best of our knowledge, the following stronger version of Falconer's conjecture might hold: if $\\hdim(A)\\ge 1$, then there exists $x\\in A$ such that the \\textbf{pinned distance set}\n\\[\n\\mathrm{dist}(x,A) = \\{ |x-y|:y\\in A\\}\n\\]\nhas Hausdorff dimension $1$. Although the method of Wolff does not appear to say anything about pinned distance sets, Peres and Schlag \\cite{PeresSchlag00} employed the transversality method to prove that, under the stronger assumption $\\hdim(A)>3\/2$, for all $x$ outside of a set of dimension at most $3-\\hdim(A)$, the pinned distance set $\\mathrm{dist}(x,A)$ has positive Lebesgue measure.\n\nVery recently, Orponen \\cite{Orponen17} approached the problem from a different angle. Recall that a set $A\\subset\\mathbb{R}^d$ is called \\textbf{$(s,C)$-Ahlfors regular}, or $s$-Ahlfors regular with constant $C$, if there exists a measure $\\mu$ supported on $A$, such that $C^{-1} r^s \\le \\mu(B(x,r)) \\le C r^s$ for all $x\\in \\textrm{Supp}(\\mu)$ and all $r\\in (0,1]$. Orponen showed that if $A\\subset\\mathbb{R}^2$ is $(s,C)$-Alhfors regular for some $s\\ge 1$ and any $C>1$, then the \\emph{packing} dimension of $\\mathrm{dist}(A,A)$ is $1$. In fact, a small modification of his method shows that also the lower box-counting of $\\mathrm{dist}(A,A)$ equals $1$.\n\nIn this article, we improve upon Orponen's result in several directions: we obtain results on the existence of many large \\emph{pinned} distance sets, we weaken slightly the hypothesis of Ahlfors-regularity, we show that the \\emph{modified} lower box-counting dimension of the distance set is 1, and we are able to consider the set of distances between two different sets.\n\nOur first main result is a discretized version for large subsets of (discrete) Ahlfors-regular sets: we say that a set $A\\subset \\mathbb{R}^d$ is \\textbf{discrete $(s,C)$-Ahlfors regular at scale $2^{-N}$} if\n\\[\nC^{-1} 2^{(N-k)s} \\le |B(x,2^{-k})\\cap A| \\le C 2^{(N-k)s}\\quad\\text{for all }x\\in A, k\\in [N],\n\\]\nwhere $[N]=\\{0,1,\\ldots,N-1\\}$. For a bounded set $F\\subset \\mathbb{R}^d$, we denote by $\\mathcal{N}(F,\\varepsilon)$ the number of $\\varepsilon$-grid cubes hit by $F$.\n\n\\begin{theorem} \\label{thm:many-large-pinned-dist-sets}\nGiven $s>1, C>1, t\\in (0,1)$, there exist $\\varepsilon=\\varepsilon(s,C,t)>0$ and $N_0=N_0(s,C,t)\\in\\mathbb{N}$ such that the following holds:\n\nIf $N\\ge N_0$, and $A\\subset [0,1]^2$ is a subset of a discrete $(s,C)$-Ahlfors regular set at scale $2^{-N}$, then\n\\[\n| x\\in A:\\mathcal{N}(\\mathrm{dist}(x,A),2^{-N}) < 2^{tN}| \\le 2^{(s-\\varepsilon)N}.\n\\]\n\\end{theorem}\nAn inspection of the proof shows that we can take $\\varepsilon=(1-t)\/C'$ for some effective $C'=C'(s,C)>0$. The value of $N_0$ does not appear to be effective from the current proof.\n\nTheorem \\ref{thm:many-large-pinned-dist-sets} fails rather dramatically for $s=1$, as witnessed by the example described in \\cite[Eq. (2) and Figure 1]{KatzTao01}. Namely, given $N\\gg 1$, let\n\\[\nA_N = \\left\\{ (x,y): x\\in \\{ i 2^{-N\/2}:0\\le i < 2^{N\/2}-1 \\}, y\\in \\{ j 2^{-N}: 0\\le j < 2^{N\/2} \\}\\right\\}.\n\\]\nThis is a discrete $1$-Ahlfors regular set at scale $2^{-N}$, yet one can check that $\\mathcal{N}(\\mathrm{dist}(x,A_N),2^{-N}) = O(2^{N\/2})$ for all $x\\in A_N$. In the proof of Theorem \\ref{thm:many-large-pinned-dist-sets}, the role of the assumption $s>1$ is to ensure that the set of directions determined by pairs of points in $A$ is dense ``with high multiplicity'', see \\S\\ref{subsec:conical-density} below. This obviously fails for each of the sets $A_N$ and, more generally, for many discrete $1$-Ahlfors regular sets. We thank an anonymous referee for pointing out this ``almost counter-example'' to Theorem \\ref{thm:many-large-pinned-dist-sets}.\n\n\nWe obtain several corollaries from Theorem \\ref{thm:many-large-pinned-dist-sets} . Firstly, for sets of full Hausdorff dimension inside an Ahlfors-regular set, nearly all pinned distance sets have full lower box-counting dimension:\n\\begin{corollary} \\label{cor:pinned-dist-set-large-dev}\nFor every $t\\in (0,1)$, $s>1$, $C>0$ there is $\\varepsilon=\\varepsilon(s,C,t)>0$ such that the following holds. Let $A$ be a bounded subset of a $(s,C)$-Ahlfors regular set in $\\mathbb{R}^2$. Then\n\\[\n\\hdim\\{x\\in A: \\lbdim(\\mathrm{dist}(x,A))0$, then\n\\[\n\\lbdim(\\mathrm{dist}(x,A))=1 \\quad\\text{for $\\mathcal{H}^s$-almost all $x\\in A$}.\n\\]\nIn particular, this holds if $A$ is itself $(s,C)$-Ahlfors regular.\n\\end{corollary}\nIn the above corollary, $\\mathcal{H}^s$ denotes $s$-dimensional Hausdorff measure. It is also easy to deduce a statement purely about box-counting dimensions:\n\\begin{corollary} \\label{cor:dist-set-box-dim}\nLet $A$ be a bounded subset of a $(s,C)$-Ahlfors regular set in $\\mathbb{R}^2$, with $\\lbdim(A)=s>1$ (resp. $\\ubdim(A)=s>1$). Then\n\\[\n\\lbdim(\\mathrm{dist}(A,A)) = 1 \\quad(\\text{resp. } \\ubdim(\\mathrm{dist}(A,A)=1)).\n\\]\n\\end{corollary}\nWe underline that the Hausdorff dimension of sets satisfying the hypothesis of the above corollary may be arbitrarily small, or even zero.\n\nOur second main result concerns the set of distances between two, possibly disjoint, sets $A, B\\subset\\mathbb{R}^2$. Although here we do not get a discretized result, we do get large \\emph{modified} lower box-counting dimension of the distance set (which we recall is smaller than both lower box dimension and packing dimension, and unlike the former is countably stable). Moreover, while for one of the sets we still need to assume Ahlfors-regularity, for the other we only require that the Hausdorff dimension strictly exceeds $1$.\n\\begin{theorem} \\label{thm:mlbdim-distance-sets-AR}\nLet $A,B\\subset \\mathbb{R}^2$ be two Borel sets such that $\\hdim(A)>1$ and $B$ is $(s,C)$-Ahlfors regular for some $s>1$. Then\n\\[\n\\mlbdim(\\mathrm{dist}(A,B)) = 1.\n\\]\nIn particular, if $A$ is $s$-Ahlfors regular with $s>1$, then its distance set has full modified lower box-counting dimension.\n\\end{theorem}\nIn fact, we are able to somewhat weaken the assumptions on $A$ and $B$, see Theorem \\ref{thm:mlbdim-distance-sets} below and the remark after the proof.\n\nThe proof of Theorem \\ref{thm:mlbdim-distance-sets-AR} also yields the following:\n\\begin{corollary} \\label{cor:pinned-dist-set-upper-box-dim}\nLet $A,B\\subset \\mathbb{R}^2$ be two Borel sets such that $\\hdim(A)>1$ and $B$ is $(s,C)$-Ahlfors regular for some $s>1$. Then\n\\[\n\\hdim(\\{ x\\in A: \\ubdim(\\mathrm{dist}(x,B))<1 \\})\\le 1.\n\\]\nIn particular, this applies to $A=B$.\n\\end{corollary}\nCompared with Corollary \\ref{cor:pinned-dist-set-large-dev}, we lower the size of the exceptional set (from zero measure to Hausdorff dimension $1$), at the price of dealing with upper box-counting dimension instead of lower box-counting dimension.\n\nFor the proofs, we follow some of the ideas of Orponen \\cite{Orponen17}, but there are substantial differences. A key step in his approach is a projection theorem for entropy in the Ahlfors regular case, see \\cite[Proposition 3.8]{Orponen17}, which is applied at all scales. It is unclear whether such a result continues to hold after removing even very small pieces of the initial regular set. Hence, in order to make the method robust under passing to large subsets (which is essential to the proof of Theorem \\ref{thm:many-large-pinned-dist-sets}), we needed a different device to handle the entropy of projections. This more flexible device is the theory of CP-processes and projections developed in \\cite{HochmanShmerkin12}, which we review in Section \\ref{sec:preliminaries}. Very roughly speaking, a CP-process is a measure-valued dynamical system which consists in zooming in dyadically towards a typical point of the measure. Thus, this paper is another example of an application of ergodic-theoretic ideas to problems that, a priori, have nothing to do with dynamics or ergodic theory.\n\nAs noted by Orponen already in \\cite{Orponen12}, in the study of distance sets the spherical projections $\\sigma_x(y)=(x-y)\/|x-y|$ play a key role (the reason is that they arise when linearizing the distance function). An important fact in Orponen's approach is that spherical projections of sets of dimension at least $1$ are dense. For the proof of Theorem \\ref{thm:many-large-pinned-dist-sets} we require a discrete quantitative version of this (established in \\S\\ref{subsec:conical-density}), while for Theorem \\ref{thm:mlbdim-distance-sets-AR} we rely instead on a recent result of Mattila and Orponen \\cite{MattilaOrponen15}, see also \\cite{Orponen16}.\n\nThe paper is organized as follows. In Section \\ref{sec:preliminaries} we set up notation, recall different notions of dimensions, and review the parts of the theory of CP-processes that we will require. In Section \\ref{sec:Ahlfors-regularity} we discuss a notion of regularity weaker than Ahlfors-regularity. Theorem \\ref{thm:many-large-pinned-dist-sets} and its corollaries are proved in Section \\ref{sec:pinned-dist-sets}, while Theorem \\ref{thm:mlbdim-distance-sets-AR} is proved in Section \\ref{sec:distances-between-sets}.\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\n\\subsection{Notation}\n\nWe use $O(\\cdot)$ notation: $A=O(B)$ means $0\\le A\\le C B$ for some constant $B$; if $C$ is allowed to depend on any parameters, this are denoted as subscripts; e.g. $A = O_d(B)$ means $0 \\le A \\le C(d) B$. Finally, $A=\\Omega(B)$ means $B=O(A)$, and likewise with subscripts.\n\nGiven a metric space $X$, we denote the family of all Borel probability measures on $X$ by $\\mathcal{P}(X)$, and the family of all Radon measures on $X$ by $\\mathcal{M}(X)$. When $X$ is compact, $\\mathcal{P}(X)$ is endowed with the weak topology, which is metrizable. If $f:X\\to Y$ and $\\mu\\in\\mathcal{M}(X)$, the \\emph{push-down measure} $f\\mu$ is defined as $f\\mu(A)=\\mu(f^{-1}A)$. We note this is sometimes denoted $f_\\#\\mu$.\n\n\nIf $\\mu\\in\\mathcal{M}(X)$ and $\\mu(A)>0$, then $\\mu|_A$ is the restriction of $\\mu$ to $A$ and, provided also $\\mu(A)<\\infty$, we denote by $\\mu_A$ the restriction normalized to be a probability measure, that is\n\\[\n\\mu_A(B) =\\frac{1}{\\mu(A)}\\mu(A\\cap B).\n\\]\n\nWe work in an ambient dimension $d$; this will always be $1$ or $2$ in this paper. We denote by $\\mathcal{D}_k^{(d)}$ the partition of $\\mathbb{R}^d$ into half-open dyadic cubes\n\\[\n\\left\\{ [j_1 2^{-k}, (j_1+1)2^{-k})\\times \\cdots\\times [j_d 2^{-k}, (j_d+1)2^{-k}): j_1,\\ldots,j_d\\in\\mathbb{Z} \\right\\}.\n\\]\nWhen $d$ is clear from context, we simply write $\\mathcal{D}_k$. If $x\\in \\mathbb{R}^d$, we denote the unique element of $\\mathcal{D}_k^{(d)}$ containing $x$ by $D_k(x)$. In addition to the Euclidean metric, on $\\mathbb{R}^d$ we consider the dyadic metric $\\rho$ defined as follows: $\\rho(x,y)=2^{-\\ell}$, where $\\ell=\\max\\{ k: D_k(x)=D_k(y)\\}$.\n\nLogarithms are always to base $2$. We denote Shannon entropy of the probability measure $\\mu$ with respect to the finite measurable partition $\\mathcal{F}$ by $H(\\mu,\\mathcal{F})$, and the conditional entropy with respect to the finite measurable partition $\\mathcal{G}$ by $H(\\mu,\\mathcal{F}|\\mathcal{G})$. That is,\n\\begin{align*}\nH(\\mu,\\mathcal{F}) &= \\sum_{F\\in\\mathcal{F}} -\\mu(F)\\log\\mu(F),\\\\\nH(\\mu,\\mathcal{F}|\\mathcal{G}) &= \\sum_{G\\in\\mathcal{G}:\\mu(G)>0} \\mu(G) H(\\mu_G,\\mathcal{F}).\n\\end{align*}\nHere and below we follow the usual convention $0\\cdot \\log(0)=0$. We denote by $H_k(\\mu)$ the normalized entropy $H(\\mu,\\mathcal{D}_k)\/k$, and note that if $\\mu\\in\\mathcal{P}([0,1)^d)$, then $0\\le H_k(\\mu)\\le 1$. The following are some standard properties of entropy that will get used in the sequel:\n\\begin{enumerate}\n \\item If $|\\mathcal{F}|\\le N$, then $H(\\mu,\\mathcal{F})\\le \\log N$.\n \\item If $\\mathcal{F},\\mathcal{G}$ are finite partitions such that each element of $\\mathcal{F}$ intersects at most $N$ elements of $\\mathcal{G}$ and vice versa, then\n\\[\n|H(\\mu,\\mathcal{F})-H(\\mu,\\mathcal{G})| \\le\\log N.\n\\]\n \\item (Concavity of entropy). If $\\mu,\\nu$ are probability measures, $t\\in [0,1]$ and $\\mathcal{F}$ is a finite measurable partition, then\n \\[\n H(t\\mu+(1-t)\\nu,\\mathcal{F}) \\ge t H(\\mu,\\mathcal{F})+(1-t) H(\\nu,\\mathcal{F}).\n \\]\n\n\\end{enumerate}\n\n\nGiven two integers $A< B$ we denote $[A,B]=\\{A,A+1,\\ldots, B-1\\}$. When $A=0$, we simply write $[B]=\\{ 0,1,\\ldots, B-1\\}$.\n\n\\subsection{Notions of dimension} \\label{subsec:dimension}\nIn this section we quickly review the notions of dimension of sets and measures we will require. For further background on dimensions of sets, see e.g. Falconer's textbook \\cite{Falconer14}, while for dimensions of measures and their relationships, we refer to \\cite{FLR02}.\n\nRecall that $\\mathcal{N}(F,\\varepsilon)$ is the number of $\\varepsilon$-grid cubes that intersect a bounded set $F\\subset\\mathbb{R}^d$. The \\textbf{upper and lower box-counting dimensions} of $F$ are defined as\n\\begin{align*}\n\\lbdim(F) &= \\liminf_{\\varepsilon\\downarrow 0} \\frac{\\log \\mathcal{N}(F,\\varepsilon)}{-\\log\\varepsilon},\\\\\n\\ubdim(F) &= \\limsup_{\\varepsilon\\downarrow 0} \\frac{\\log \\mathcal{N}(F,\\varepsilon)}{-\\log\\varepsilon}.\n\\end{align*}\nThese dimensions are not countably stable. After making them countably stable in the natural way, one gets \\textbf{modified lower box-counting dimension} $\\mlbdim$ and \\textbf{packing dimension}:\n\\begin{align*}\n\\mlbdim(F) &= \\inf\\{ \\sup_i \\lbdim(F_i): F\\subset\\cup_i F_i \\},\\\\\n\\pdim(F) &= \\inf\\{ \\sup_i \\ubdim(F_i): F\\subset\\cup_i F_i \\}.\n\\end{align*}\nThe inequalities $\\hdim(F)\\le \\mlbdim(F)\\le \\lbdim(F)\\le \\ubdim(F)$ and $\\mlbdim(F)\\le \\pdim(F)\\le \\ubdim(F)$ always hold, while $\\lbdim$ and $\\pdim$ are not comparable in general.\n\nWe move on to dimensions of measures. Let $\\mu\\in\\mathcal{P}(\\mathbb{R}^d)$. The \\textbf{lower and upper entropy dimensions} are defined as\n\\begin{align*}\n\\ledim(\\mu) &= \\liminf_{k\\to\\infty} H_k(\\mu),\\\\\n\\uedim(\\mu) &= \\limsup_{k\\to\\infty} H_k(\\mu).\n\\end{align*}\n\nThe \\textbf{Hausdorff dimension} of $\\mu\\in\\mathcal{M}(\\mathbb{R}^d)$ is\n\\[\n\\hdim(\\mu) = \\inf\\{\\hdim(A):\\mu(A)>0\\}.\n\\]\nWe note that this is sometimes called the \\emph{lower} Hausdorff dimension. Finally, we recall that $\\mu\\in\\mathcal{M}(\\mathbb{R}^d)$ is called \\textbf{exact dimensional} if there exists $s\\ge 0$ (the \\emph{exact dimension} of $\\mu$) such that\n\\[\n\\lim_{r\\downarrow 0} \\frac{\\log \\mu(B(x,r))}{\\log r} = s\\quad\\text{for $\\mu$-almost all }x.\n\\]\n\nFor any $\\mu\\in\\mathcal{P}(\\mathbb{R}^d)$ it holds that\n\\[\n\\hdim(\\mu) \\le \\ledim(\\mu) \\le \\uedim(\\mu),\n\\]\nwith strict inequalities possible, see \\cite[Theorem 1.3]{FLR02}. However, for measures of exact dimension $s$, there is an equality $\\hdim(\\mu)=\\uedim(\\mu)=s$.\n\n\\subsection{Global sceneries, entropy and projections}\nIn this section we recall some results from \\cite{Hochman14, Orponen17} (similar ideas go back to \\cite{HochmanShmerkin12}). We write $\\delta(\\omega)$ or $\\delta_\\omega$ to denote the point mass at $\\omega$ (often $\\omega$ will be a measure).\n\nWe denote the topological support of $\\mu\\in\\mathcal{P}([0,1)^d)$ in the $\\rho$-metric by $\\textrm{Supp}_\\rho(\\mu)$. Note that $x\\in\\textrm{Supp}_\\rho(\\mu)$ if and only if $\\mu(D_n(x))>0$ for all $n\\in\\mathbb{N}$. Given $Q\\in\\mathcal{D}_n^{(d)}$, let $T_Q$ be the homothety that maps $Q$ onto $[0,1)^d$, and define\n\\[\n\\mu^Q = T_Q\\left( \\frac{\\mu|_{Q}}{\\mu(Q)}\\right).\n\\]\nIf $x\\in\\textrm{Supp}_\\rho(\\mu)$, we also write $\\mu^{x,n}=\\mu^{D_n(x)}$ for short. That is, $\\mu^{x,n}$ is the normalized restriction of $\\mu$ to $D_n(x)$, renormalized back to the unit cube.\n\nGiven $\\mu\\in\\mathcal{P}([0,1)^d)$, $x\\in\\textrm{Supp}_\\rho(\\mu)$, and an integer interval $[A,B]$, we write\n\\begin{align*}\n\\langle \\mu,x\\rangle_{[A,B]} &= \\frac{1}{B-A} \\sum_{n=A}^{B-1} \\delta(\\mu^{x,n}),\\\\\n\\langle \\mu \\rangle_n &= \\int \\delta(\\mu^{x,n})\\,d\\mu(x) = \\sum_{Q\\in\\mathcal{D}_n} \\mu(Q) \\delta(\\mu^Q),\\\\\n\\langle \\mu \\rangle_{[A,B]} &= \\int \\langle \\mu,x\\rangle_{[A,B]} \\,d\\mu(x) = \\frac{1}{B-A} \\sum_{n=A}^{B-1} \\langle\\mu\\rangle_n.\n\\end{align*}\nThe second equality in the last line follows from interchanging the order of sum and integration; it will be convenient to alternatively use either definition of $\\langle \\mu \\rangle_{[A,B]}$.\n\nThe following simple but important fact is proved in \\cite[Lemma 3.4]{Hochman14}. It allows to recover the global entropy of a measure from local entropies.\n\\begin{lemma} \\label{lem:local-entropy-from-global-entropy}\nLet $\\mu\\in\\mathcal{P}([0,1)^d)$. Then\n\\[\n\\left|H_N(\\mu) - \\int H_q(\\eta)\\, d\\langle \\mu\\rangle_{[0,N]}(\\eta) \\right| = O_d(q\/N).\n\\]\n\\end{lemma}\nIn the above lemma, one should think that the value of $q$ is fixed, and $N$ tends to infinity (possibly along a subsequence).\n\n\nThe following is a variant of a result of Orponen \\cite{Orponen17}, which in turn adapts ideas of Hochman \\cite{Hochman14}.\n\\begin{prop} \\label{prop:projected-entropies-from-local-entropies}\nFix $2< q< N$. Let $\\mu\\in\\mathcal{P}([0,1)^2)$, and let $U$ be an open set containing $\\textrm{Supp}(\\mu)$. Suppose that $f: U\\to \\mathbb{R}$ is a $C^1$ map such that, for some fixed $v\\in S^1$,\n\\[\n\\|Df(x)-v\\| \\le 2^{-q} \\quad\\text{for all } x\\in\\textrm{Supp}(\\mu).\n\\]\nThen, if $\\Pi_v(x)= v \\cdot x $ denotes the orthogonal projection of $x$ onto a line in direction $v$,\n\\[\nH_N(f\\mu) \\ge \\int H_q(\\Pi_v\\eta) \\,d\\langle \\mu\\rangle_{[0,N]}(\\eta) - O(q\/N) - O(1\/q).\n\\]\nThe constants in the $O$ notation are absolute.\n\\end{prop}\n\\begin{proof}\nOrponen \\cite[Lemma 3.5]{Orponen17} showed that if $1\\le q0$.\n\nBy concavity of entropy, if $\\widetilde{D}\\in \\mathcal{D}_j^{(2)}$, then\n\\[\nH_N(f\\mu) \\ge \\mu(\\widetilde{D}) H_N(f\\mu_{\\widetilde{D}}).\n\\]\nApplying \\eqref{eq:projected-entropies-0} to each $\\nu=\\mu_{\\widetilde{D}}$ with $\\widetilde{D}\\in\\mathcal{D}_j^{(2)}$, $j\\in [q]$, and adding up, and then averaging over $j$, we get\n\\begin{equation} \\label{eq:projected-entropies-1}\nH_N(f\\mu) \\ge \\frac{1}{N} \\sum_{i=0}^{N-q} \\sum_{D\\in\\mathcal{D}_i^{(2)}} \\mu(D) \\,\\frac{1}{q} H(f\\mu_D,\\mathcal{D}_{i+q}|\\mathcal{D}_i).\n\\end{equation}\nOn the other hand, by \\cite[Lemma 3.12]{Orponen17}, the almost linearity hypothesis on $f$ ensures that\n\\begin{equation} \\label{eq:projected-entropies-2}\n|H(f\\mu_D,\\mathcal{D}_{i+q}|\\mathcal{D}_i) - H(\\Pi_v\\mu_D,\\mathcal{D}_{i+q}|\\mathcal{D}_i)| = O(1)\n\\end{equation}\nfor any $D\\in\\mathcal{D}_i^{(2)}$. (This was stated in \\cite{Orponen17} for $i$ a multiple of $q$ but the proof in general is identical.)\n\nFinally, as observed in \\cite[Remark 3.6]{Orponen17}, the linearity of $\\Pi_v$ implies that\n\\begin{equation} \\label{eq:projected-entropies-3}\n\\frac{1}{q} H(\\Pi_v\\mu_D,\\mathcal{D}_{i+q}|\\mathcal{D}_i) \\ge H_q(\\Pi_v\\mu^D) - O(1\/q).\n\\end{equation}\n\nPutting together \\eqref{eq:projected-entropies-1}, \\eqref{eq:projected-entropies-2} and \\eqref{eq:projected-entropies-3} yields the claim.\n\\end{proof}\n\nWe will apply the above proposition to functions $f$ of the form $\\phi_x(y)=\\tfrac12|x-y|$. Let $\\sigma(x,y)=(x-y)\/|x-y|\\in S^1\\subset \\mathbb{R}^2$ be the direction generated by $x\\neq y$, and note that $Df_x(y)=\\sigma(x,y)$. Hence, we have the following corollary of Proposition \\ref{prop:projected-entropies-from-local-entropies}.\n\n\\begin{corollary} \\label{cor:lower-bound-entropy-pinned-dist}\nFix $x\\in\\mathbb{R}^2$, $D\\in\\mathcal{D}_k^{(2)}$, $v\\in S^1$ and $q\\gg 1$ such that $|\\sigma(x,y)-v| \\le 2^{-q}$ for all $y\\in D$. Then for any $\\mu\\in\\mathcal{P}([0,1)^2)$ supported on $D$ and any $N\\ge q$,\n\\[\nH_N(\\phi_x\\mu) \\ge \\int H_q(\\Pi_v\\eta) \\,d\\langle \\mu\\rangle_{[0,N]}(\\eta) - O(q\/N) - O(1\/q).\n\\]\n\\end{corollary}\n\n\\subsection{CP processes}\n\nFollowing \\cite{Furstenberg08}, we consider CP processes on the tree $([0,1)^d,\\rho)$ rather than on Euclidean cubes; the dyadic metric helps avoid technicalities with functions that would not be continuous on Euclidean space (due to dyadic hyperplanes) but are on the tree, notably entropy. We will denote the induced weak topology on $\\mathcal{P}([0,1)^d)$ by $\\widetilde{\\rho}$, and the weak topology induced by this on $\\mathcal{P}(\\mathcal{P}([0,1)^d))$ by $\\widehat{\\rho}$. Slightly abusing notation, we will also denote by $\\widetilde{\\rho}$ the product topology $\\widetilde{\\rho}\\times \\rho$ on $\\mathcal{P}([0,1)^d)\\times [0,1)^d$, and by $\\widehat{\\rho}$ the corresponding weak topology on $\\mathcal{P}(\\mathcal{P}([0,1)^d)\\times [0,1)^d)$. We note that all these topological spaces are compact and metrizable. To avoid any ambiguity, we will occasionally denote the topology under consideration with a subscript.\n\nWe let $S:[0,1)^d\\to [0,1)^d$, $S(x)=2x\\bmod 1$ be the doubling map.\n\\begin{definition}[CP magnification operator]\nLet\n\\[\n\\Xi = \\left\\{ (\\mu,x)\\in \\mathcal{P}([0,1)^d)\\times [0,1)^d: x\\in\\textrm{Supp}_\\rho(\\mu) \\right\\}.\n\\]\nWe define the \\textit{CP magnification operator} $M$ on $\\Xi$ by\n\\[\nM(\\mu,x)= (\\mu^{x,1}, Sx).\n\\]\n\\end{definition}\nNote that $M^n(\\mu,x)=(\\mu^{x,n},S^n x)$.\n\n\nWe now define CP distributions (we refer to probability measures on ``large'' probability spaces such as $\\Xi$ as distributions). This definition goes back to \\cite{Furstenberg08}; see \\cite{HochmanShmerkin12} and \\cite{KSS15} for some variants and generalizations.\n\n\\begin{definition}[CP distributions]\nA distribution $Q$ on $\\Xi$ is \\emph{adapted}, if there is a disintegration\n\\begin{equation} \\label{eq:adapted}\n\\int f(\\nu,x) \\,\\mathrm{d}Q(\\nu,x) = \\iint f(\\nu,x)\\,\\textrm{d}\\nu(x) \\,\\textrm{d}\\overline{Q}(\\nu),\n\\end{equation}\nfor all $f\\in C_{\\widetilde{\\rho}}(\\mathcal{P}([0,1)^d)\\times [0,1)^d)$, where $\\overline{Q}$ is the projection of $Q$ onto the measure component.\n\nA distribution on $\\Xi$ is a \\emph{CP distribution} (CPD) if it is $M$-invariant (that is, $MQ=Q$) and adapted.\n\\end{definition}\n\nNote that adaptedness can be interpreted in the following way: in order to sample a pair $(\\mu,x)$ from the distribution $Q$, we have to first sample a measure $\\mu$ according to $\\overline{Q}$, and then sample a point $x$ using the chosen distribution $\\mu$. From now on we will denote by $Q$ both the CPD acting on $\\Xi$ and its measure component acting on $\\mathcal{P}([0,1)^d)$, since by adaptedness the latter determines the former.\n\n\nAn easy consequence of the Ergodic Theorem applied to CP distributions is that if $P$ is a CPD which is ergodic under the action of $M$, then $P$-a.e. $\\nu$ is exact dimensional, and has dimension\n\\[\n\\dim P = \\int H_q(\\eta)\\,dP(\\eta)\n\\]\nfor any $q\\in\\mathbb{N}$ (see e.g. \\cite[Equation (2.7)]{Furstenberg08}). Let $P = \\int P_\\mu \\, d P(\\mu)$ be the ergodic decomposition of $P$ (that is, each $P_\\mu$ is $M$-invariant and ergodic, and $\\mu\\mapsto P_\\mu$ is a Borel mapping). By general properties of Markov processes, $P_\\mu$ is again a CPD for $P$-almost all $\\mu$, see e.g. \\cite[Remark before Proposition 5.2]{Furstenberg08}. Hence, if $P$ is a (non-necessarily ergodic) CPD, then $P$-a.e. $\\nu$ is still exact-dimensional, but $\\dim\\nu$ needs no longer be $P$-a.e. constant.\n\n\\begin{definition}\nIf $P$ is a CP distribution, we define its \\textbf{lower dimension} $\\dim_*P$ as the $P$-essential infimum of $\\dim\\nu$.\n\\end{definition}\n\nWe turn to the behavior of entropy under projections. For this, we recall some results from \\cite{HochmanShmerkin12} on CP-processes and projections. Recall that $\\Pi_v(x)=\\langle v,x\\rangle$, $v\\in S^1$. Elementary properties of entropy imply that\n\\begin{equation} \\label{eq:continuity-of-projected-entropy}\n|H_q(\\Pi_v\\eta)-H_q(\\Pi_{v'}\\eta)| \\le O(1\/q) \\text{ if } |v-v'|\\le 2^{-q},\n\\end{equation}\nwith the $O(\\cdot)$ constant independent of $\\eta$. Indeed, $H_q(\\Pi_v\\eta)=\\frac1q H(\\eta, \\Pi_v^{-1}\\mathcal{D}_q)$ and likewise with $v'$. But if $|v-v'|\\le 2^{-q}$, then each element of $\\Pi_v^{-1}\\mathcal{D}_q$ hits $O(1)$ elements of $\\Pi_{v'}^{-1}\\mathcal{D}_q$ and vice-versa, so \\eqref{eq:continuity-of-projected-entropy} follows.\n\nThe following result is a consequence of \\cite[Theorem 8.2]{HochmanShmerkin12}. It will act as our projection theorem for entropy.\n\\begin{theorem} \\label{thm:projections-CPD}\nLet $P$ be a (not necessarily ergodic) CP-distribution. Write $\\mathcal{E}_q:S^1\\to [0,1]$, $v\\mapsto \\int \\min(H_q(\\Pi_v\\eta),1)\\,dP(\\eta)$. Then:\n\\begin{enumerate}\n\\item The function $\\mathcal{E}_q$ satisfies\n\\[\n|\\mathcal{E}_q(v)-\\mathcal{E}_q(v')| \\le O(1\/q) \\text{ if } |v-v'|\\le 2^{-q};\n\\]\n\\item The limit $\\mathcal{E}(v):=\\lim_{q\\to\\infty} \\mathcal{E}_q(v)$ exists for all $v$ and $\\mathcal{E}(v)$ is lower semicontinuous;\n\\item $\\mathcal{E}(v)\\ge \\min(\\dim_*P,1)$ for almost all $v$;\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nThe first claim is immediate from \\eqref{eq:continuity-of-projected-entropy}. Let\n\\[\n\\widetilde{\\mathcal{E}}_q(v)=\\int H_q(\\Pi_v\\eta)\\,dP(\\eta).\n\\]\nSince $\\Pi_v\\eta$ is supported on an interval of length $\\le\\sqrt{2}$, $|\\widetilde{\\mathcal{E}}_q(v)-\\mathcal{E}_q(v)|\\le 1\/q$ for all $v\\in S^1$. In the case $P$ is ergodic, the latter claims are a particular case of \\cite[Theorem 8.2]{HochmanShmerkin12}. More precisely, in \\cite{HochmanShmerkin12}, the stated convergence is $\\widetilde{\\mathcal{E}}_q(v)\\to \\mathcal{E}(v)$, but by our observation, this immediately yields $\\mathcal{E}_q(v)\\to \\mathcal{E}(v)$. The general case follows by considering the ergodic decomposition of $P$ (notice that an integral of lower semicontinuous functions is lower semicontinuous by Fatou's Lemma).\n\\end{proof}\n\n\\subsection{Global tangents}\n\nWe want to be able to estimate the entropy of projections of a given measure $\\mu\\in\\mathcal{P}([0,1)^2)$, but the tools we have at our disposal concern typical measures for a CP process. Following \\cite{Hochman13}, we handle this by passing to suitable tangent objects.\n\n\nGiven $\\mu\\in\\mathcal{P}([0,1)^d)$, the set of accumulation points of $\\langle\\mu\\rangle_{[0,N]}$ in the $\\widehat{\\rho}$ metric will be denoted $\\mathcal{T}(\\mu)$. Unlike in \\cite{Hochman13}, our tangent distributions are global, rather than local but, as the next lemma shows, they are still CP processes:\n\\begin{lemma} \\label{lem:limits-are-CPDs}\nLet $\\mu_n$ be a sequence in $\\mathcal{P}([0,1)^d)$. Suppose\n\\[\n\\langle \\mu_{N_j}\\rangle_{[0,N_j]} \\underset{\\widehat{\\rho}}{\\to} P,\n\\]\nfor some subsequence $(N_j)$. Then $P$ is a CPD (in the sense that the adapted distribution with measure marginal $P$ is a CPD).\n\nIn particular, if $\\mu\\in\\mathcal{P}([0,1)^d)$, then any element of $\\mathcal{T}(\\mu)$ is a CPD.\n\\end{lemma}\n\\begin{proof}\nBoth the claim and the proof are similar to those of \\cite[Propositions 5.2]{Furstenberg08}. For $\\nu\\in\\mathcal{P}([0,1)^d)$, write\n\\[\n\\langle \\nu \\rangle^*_{[A,B]} = \\frac{1}{B-A}\\sum_{n=A}^{B-1} \\int \\delta(M^n(\\nu,x)) d\\nu(x).\n\\]\nNote that the measure component of $\\langle \\nu \\rangle^*_{[A,B]}$ is $\\langle\\nu\\rangle_{[A,B]}$, and that $\\langle \\nu \\rangle^*_{[A,B]}$ is always adapted.\n\nNow suppose $\\langle \\mu_{N_j}\\rangle^*_{[0,N_j]}\\to P$ in the $\\widehat{\\rho}$ topology. Since adaptedness is a closed property (it is tested on equalities of continuous functions), $P$ is adapted.\n\nSince we are using the dyadic metric and $M$ is adapted, $M$ is well defined and continuous at $P$-a.e. $(\\mu,x)$ (notice that $x\\in\\textrm{Supp}_\\rho(\\mu)$ for $P$-a.e. $(\\mu,x)$ by adaptedness). Using standard properties of weak convergence (see e.g. \\cite[Theorem 2.7]{Billingsley99}) we conclude that\n\\begin{align*}\nMP &= M\\left(\\lim_{j\\to\\infty} \\langle \\mu_{N_j}\\rangle^*_{[0,N_j]}\\right) \\\\\n&= \\lim_{j\\to\\infty} M(\\langle \\mu_{N_j}\\rangle^*_{[0,N_j]})\\\\\n&= \\lim_{j\\to\\infty} \\langle \\mu_{N_j}\\rangle^*_{[1,N_j+1]}\\\\\n&= \\lim_{j\\to\\infty} \\langle \\mu_{N_j} \\rangle^*_{[0,N_j]} = P.\n\\end{align*}\n\\end{proof}\n\n\n\\section{Ahlfors regularity and weak regularity}\n\\label{sec:Ahlfors-regularity}\n\nThe following definition introduces a notion of regularity that, as we will see, extends the concept of Ahlfors-regularity in a suitable sense.\n\\begin{definition}\n\\begin{enumerate}\n\\item A measure $\\mu\\in\\mathcal{P}([0,1)^d)$ is said to be \\textbf{$s$-rich at resolution $(N,q,\\delta)$} if\n\\[\n\\langle \\mu \\rangle_{[0,N]}\\{\\eta: H_q(\\eta) < s-\\delta\\} < \\delta.\n\\]\n\\item\nA measure $\\mu\\in\\mathcal{P}([0,1)^d)$ is said to be \\textbf{weakly $s$-regular} if for every $\\delta>0$ there is $q\\in\\mathbb{N}$ such that $\\mu$ is $s$-rich at resolution $(N,q,\\delta)$ for all sufficiently large $N$ (depending on $q$ and $\\delta$).\n\\end{enumerate}\n\\end{definition}\nNote that if a measure is weakly $s$-regular then it is weakly $t$-regular for all $t0$, then $\\mu_A$ is weakly $s$-regular.\n\\end{lemma}\n\\begin{proof}\nThis is essentially a consequence of the Lebesgue density theorem (which for the dyadic metric is an immediate consequence of the convergence of conditional expectation given the dyadic filtration). Fix $\\delta>0$, and let $q$ be such that $\\mu$ is $s$-rich at resolution $(N,q,\\delta)$ for all sufficiently large $N$. Write $\\Omega_\\kappa=\\{ \\eta: H_q(\\eta)>s-\\kappa\\}$. Then we have\n\\begin{equation} \\label{eq:consequence-s-rich}\n\\int_B \\langle \\mu,x\\rangle_{[0,N]}(\\Omega_\\delta)\\,d\\mu(x) > \\mu(B)-\\delta\n\\end{equation}\nfor any Borel set $B$, provided $N$ is large enough depending on $\\delta$ and $q$ only. By the density point theorem, for $\\mu$ almost all $x\\in A$, the sequences $\\mu_A^{x,n}$ and $\\mu^{x,n}$ are $\\widetilde{\\rho}$-asymptotic (i.e. $\\widetilde{\\rho}(\\mu_A^{x,n},\\mu^{x,n})\\to 0$). In particular, if $N$ is large enough (depending on $\\delta$), we then have $\\mu_A(B)>1-\\delta$, where\n\\[\nB=\\left\\{ x: \\langle \\mu_A,x \\rangle_{[0,N]}(\\Omega_{2\\delta}) \\ge \\langle \\mu,x \\rangle_{[0,N]}(\\Omega_{\\delta})\\right\\}.\n\\]\nHere we used that $H_q$ is continuous on $(\\mathcal{P}([0,1)^d),\\widetilde{\\rho})$. Recalling \\eqref{eq:consequence-s-rich} we conclude that, always assuming $N$ is large enough,\n\\begin{align*}\n\\langle \\mu_A \\rangle_{[0,N]}(\\Omega_{2\\delta}) &\\ge \\frac{1}{\\mu(A)}\\int_B \\langle \\mu_A,x \\rangle_{[0,N]}(\\Omega_{2\\delta})d\\mu(x)\\\\\n&\\ge \\frac{1}{\\mu(A)}(\\mu(B)-\\delta) \\ge 1-(1+\\mu(A)^{-1})\\delta.\n\\end{align*}\nThis gives the claim.\n\\end{proof}\n\nRecall that $\\mu\\in\\mathcal{P}(\\mathbb{R}^d)$ is called $(s,C)$-Ahlfors regular if $C^{-1} r^s \\le \\mu(B(x,r)) \\le C r^s$ for all $x\\in \\textrm{Supp}(\\mu)$ and all $r\\in (0,1]$. If this holds only for $r\\in [2^{-N},1]$, we say that $\\mu$ is \\textbf{$(s,C)$-Ahlfors regular at scale $2^{-N}$}. We also say that a set $A$ is $(s,C)$-Ahlfors regular if the restriction $\\mathcal{H}^s|_A$ is a positive finite $(s,C)$-Ahlfors regular measure.\n\nGiven a discrete $(s,C)$-Ahlfors regular set at scale $2^{-N}$ contained in $[0,1]^d$, we can construct a measure $\\mu$ in the following manner:\n\\begin{equation} \\label{eq:AR-measure-from-AR-set}\n\\mu = \\mu^A = \\frac{1}{|A|}\\sum_{D\\in\\mathcal{D}_N} |A\\cap D| \\,\\mathcal{L}_D,\n\\end{equation}\nwhere $\\mathcal{L}$ denotes $d$-dimensional Lebesgue measure. Reciprocally, from a measure $\\mu$ supported on $[0,1]^d$ which is $(s,C)$-Ahlfors regular at scale $2^{-N}$, one can construct the set\n\\[\nA = A^\\mu = \\{ x_L(D) : D\\in\\mathcal{D}_N, \\mu(D)>0 \\},\n\\]\nwhere $x_L(D)$ is the left-endpoint of $D$. One then has the following easy lemma:\n\n\\begin{lemma} \\label{lem:Ahlfors-set-to-measure}\n\\begin{enumerate}\n\\item If $\\mu\\in\\mathcal{P}([0,1]^d)$ is $(s,C)$-Ahlfors regular at scale $2^{-N}$, then $A^\\mu$ is discrete $(s,O(C))$-Ahlfors regular at scale $2^{-N}$.\n\\item Conversely, if $A\\subset [0,1]^d$ is discrete $(s,C)$-Ahlfors regular at scale $2^{-N}$, then $\\mu^A$ is $(s,O(C))$-Ahlfors regular at scale $2^{-N}$.\n\\end{enumerate}\nThe implicit constants depend only on the ambient dimension $d$.\n\\end{lemma}\n\\begin{proof}\nSuppose $\\mu$ is $(s,C)$-Ahlfors regular at scale $2^{-N}$ and fix $k\\in [N]$. If $y\\in A^\\mu$, then $\\mu(B(y,3\\cdot 2^{-N}))\\in (\\Omega(C)2^{-sN},O(C)2^{-sN})$ and likewise with $k$ in place of $N$. If $B(x,2^{-k})\\cap A=\\{y_1,\\ldots,y_m\\}$, then $\\{ B(y_j,3\\cdot 2^{-N})\\}$ is a covering of $\\textrm{Supp}(\\mu)\\cap B(x,2^{-k})$ with bounded overlapping, so the first claim follows. The proof of the second claim is analogous, so is omitted.\n\\end{proof}\n\nWe will see that $s$-Ahlfors regular measures are weakly $s$-regular. The following quantitative version of this will be crucial later.\n\\begin{lemma} \\label{lem:Ahlfors-regular-is-rich}\nGiven $\\varepsilon, q, N, C$ such that $\\log C\/q<\\varepsilon$ and $q<\\varepsilon N$, the following holds.\n\nLet $\\nu$ be $(s,C)$-Ahlfors regular at scale $2^{-N}$. Then if $\\mu\\in\\mathcal{P}([0,1)^d)$ is supported on $\\textrm{Supp}(\\nu)$ and $H_N(\\mu)> s-\\varepsilon$, then $\\mu$ is $s$-rich at resolution $(N,q,\\sqrt{\\varepsilon}\/C')$, where $C'>0$ depends only on $d$.\n\\end{lemma}\n\\begin{proof}\nAny constants implicit in the $O$ notation are allowed to depend on $d$ only. Since $q<\\varepsilon N$ and\n\\begin{equation} \\label{eq:decomp-scales}\n\\langle \\mu\\rangle_{[0,N]} = \\frac{N-q}{N}\\langle \\mu\\rangle_{[0,N-q]} + \\frac{q}{N} \\langle\\mu\\rangle_{[N-q,N]},\n\\end{equation}\nit is enough to show that $\\mu$ is $s$-rich at resolution $(N-q,q,\\sqrt{\\varepsilon}\/C')$.\n\n\nWrite $A=\\textrm{Supp}(\\nu)$. We begin by noting that for any $D\\in\\mathcal{D}_n^{(d)}$ with $n\\in [N-q]$, the set $A$ meets at most $O(C) 2^{sq}$ cubes $D'\\subset D, D'\\in\\mathcal{D}_{n+q}$. Indeed, let $D_1,\\ldots, D_m$ be the sub-cubes of $D$ in $\\mathcal{D}_{n+q}$ that hit $A$, and pick $x_i\\in D_i\\cap A$. The family $B(x_i,2^{-(n+q)})$ has overlapping bounded by $O(1)$ and each member is contained in $D(2^{-n})$, the $(2^{-n})$-neighborhood of $D$. On the other hand, $D(2^{-n})\\subset B(x_1, (\\sqrt{d}+1) 2^{-n})$. Hence\n\\[\nO(C) 2^{-sn} \\ge \\nu(D(2^{-n})) \\ge \\sum_{i=1}^m \\nu(B(x_i,2^{-(n+q)})) \\ge (O(C))^{-1} 2^{-sn} 2^{-sq} m,\n\\]\ngiving the claim. In particular, we see that $H_q(\\mu^{x,n}) \\le s + O(\\log C\/q) \\le s+O(\\varepsilon)$ for any $x\\in \\textrm{Supp}(\\mu)$ and any $n\\in [N-q]$.\n\nWe know from Lemma \\ref{lem:local-entropy-from-global-entropy}, the assumption and \\eqref{eq:decomp-scales} that\n\\[\n\\int H_q(\\eta)\\, d\\langle \\mu\\rangle_{[0,N-q]}(\\eta) \\ge \\int H_q(\\eta)\\, d\\langle \\mu\\rangle_{[0,N]}(\\eta) - \\frac{q}{N-q} \\ge s - O(\\varepsilon)\n\\]\nwhich, since $H_q(\\eta) \\le s+O(\\varepsilon)$ for $\\langle \\mu\\rangle_{[0,N-q]}$ a.e. $\\eta$, we can rewrite as\n\\[\n\\int s-H_q(\\eta)+C'\\varepsilon\\, d\\langle \\mu\\rangle_{[0,N-q]}(\\eta) \\le O(\\varepsilon),\n\\]\nwhere the constant $C'$ was chosen so that the integrand is positive. The lemma now follows from Markov's inequality.\n\\end{proof}\n\nAs an immediate consequence, we deduce that a class of measures, including $s$-Ahlfors regular measures, are indeed weakly $s$-regular.\n\\begin{corollary}\nIf $\\mu$ is supported on an $s$-Ahlfors regular set and $\\ledim\\mu=s$, then $\\mu$ is weakly $s$-regular. In particular, this is the case for $\\nu_A$ when $\\nu$ is $s$-Ahlfors regular and $\\nu(A)>0$.\n\\end{corollary}\n\\begin{proof}\nFix $\\delta>0$ and take $q$ large enough that $\\log(C)\/q < \\delta^2$. Since $\\ledim\\mu=s$, we know that $H_N(\\mu)>s-\\delta^2$ for large enough $N$. If $N$ is also large enough that $N > \\delta^{-2} q$, then the previous lemma says that $\\mu$ is rich at resolution $(N,q,O(\\delta))$.\n\nFor the latter claim, note that $\\nu_A$ has exact dimension $s$ (as a consequence of the density point theorem), so that $\\ledim\\nu_A=s$.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm:many-large-pinned-dist-sets}, and consequences}\n\\label{sec:pinned-dist-sets}\n\n\\subsection{Discrete conical density lemmas}\n\\label{subsec:conical-density}\n\nIn the proof of Theorem \\ref{thm:many-large-pinned-dist-sets} we will require some discrete conical density results. These are similar to those in \\cite[Section 3]{SSS13}.\n\nWe say that a set $A\\subset [0,1]^2$ is \\textbf{$k$-discrete} if $|A\\cap D|\\le 1$ for all $D\\in\\mathcal{D}_k^{(2)}$. Also, let $X(a,\\beta,v)$ be the two-sided cone with center $a\\in\\mathbb{R}^2$, opening $\\beta\\in (0,\\pi\/2)$ and direction $v\\in S^1$. The following is a discrete analog of \\cite[Lemma 15.13]{Mattila95}.\n\\begin{lemma} \\label{lem:discrete-dense-radial-projection}\nGiven $\\beta>0$, there is a constant $C=C(\\beta)>0$ such that the following holds. If $A$ is $k$-discrete and for each $a\\in A$ there is a direction $v$ such that\n\\[\nX(a,\\beta,v) \\cap A\\setminus \\{a\\} = \\emptyset,\n\\]\nthen $|A|\\le C 2^k$.\n\\end{lemma}\n\\begin{proof}\nWe begin with a simplification. Choose a finite set $\\{ v_j\\}$ with $O_\\beta(1)$ elements such that for every $v\\in S^1$ there exists $v_j$ with\n\\[\nX(a,\\beta\/2,v_j)\\subset X(a,\\beta,v).\n\\]\nHence, if $A$ is as in the statement, for every $a\\in A$ we can pick $j(a)$ such that\n\\[\nX(a,\\beta\/2,v_{j(a)}) \\cap A\\setminus\\{a\\}= \\emptyset.\n\\]\nLet $A_j = \\{ a\\in A: j(a)=j\\}$. Some $A_j$ has $\\ge |A|\/O_\\beta(1)$-elements. Moreover, by passing to a further refinement with $|A|\/O_\\beta(1)$ elements, we can assume that the elements of $A_j$ are $(2^{-k})$-separated. This shows that it is enough to prove the following statement: if $v_0$ is a fixed direction, and $A\\subset [0,1]^2$ is a $(2^{-k})$-separated set such that\n\\[\nX(a,\\beta\/2,v_0) \\cap A\\setminus\\{a\\}= \\emptyset \\quad\\text{for all }a\\in A,\n\\]\nthen $|A|\\le O_\\beta(2^k)$.\n\nLet $\\Pi(x)=\\Pi_{v_0^\\perp}(x)= x \\cdot v_0^{\\perp}$, where $v_0^\\perp$ is a unit vector perpendicular to $v_0$. It follows from our assumptions on $A$ that $|\\Pi(a)-\\Pi(a')| \\sin(\\beta\/2)\\ge 2^{-k}$ for any distinct $a,a'\\in A$. In particular, $\\Pi|_A$ is injective and its range has $O_\\beta(2^k)$ elements so $|A| \\le O_\\beta(2^k)$, as claimed.\n\\end{proof}\n\nFor sets which are dense in a discrete $s$-Ahlfors regular set, we obtain the following consequence.\n\\begin{lemma} \\label{lem:dense-radial-proj-discr-AR}\nGiven $s\\in (1,2), C>1, \\kappa\\in (0,(s-1)\/(2s)), \\beta\\in (0,\\pi\/2)$, the following holds for all large enough $N$ (depending on $s,C,\\kappa,\\beta$ only):\n\nLet $A$ be a discrete $(s,C)$-Ahlfors regular set at scale $2^{-N}$, and suppose $B\\subset A$ satisfies $|B|> 2^{(1-\\kappa) s N}$. Then there exists a subset $E\\subset B$ with $|E|\\le 2^{(1-\\kappa)sN}$ such that for all $x\\in B\\setminus E$ and any $v\\in S^1$ there exists $y\\in B$ such that $y\\in X(x,\\beta,v)$ and $|x-y|\\ge 2^{-2s(s-1)^{-1}\\kappa N}$.\n\\end{lemma}\n\\begin{proof}\nWrite $\\kappa'=2s(s-1)^{-1}\\kappa$ and note that $\\kappa'\\in (0,1)$. We say that a point $x\\in B$ is \\emph{well surrounded} if for every $v\\in S^1$ there is $y\\in B$ such that $y\\in X(x,\\beta,v)$ and $|x-y|\\ge 2^{-\\kappa' N}$.\n\nLet $E\\subset B$ be the set of all points in $B$ which are \\emph{not} well surrounded, and suppose $|E|>2^{(1-\\kappa)sN}$. Let $E_1$ be a maximal $(2^{-\\kappa' N})$-separated subset of $E$. Since each ball of radius $2^{-\\kappa' N}$ contains $O(C) 2^{(1-\\kappa')sN}$ points of $A\\supset E$, it follows that $|E_1|> \\Omega(C) 2^{(\\kappa'-\\kappa)s N}$. Note that $(\\kappa'-\\kappa)s> \\kappa$, and let $C'=C'(\\beta)$ be the constant given by Lemma \\ref{lem:discrete-dense-radial-projection}. Provided $N$ is large enough that $\\Omega(C) 2^{(\\kappa'-\\kappa)s N}> C' 2^{\\kappa N}$, it follows from Lemma \\ref{lem:discrete-dense-radial-projection} and the definitions that $E_1$ contains a well surrounded point. This contradiction proves the lemma.\n\\end{proof}\n\n\\subsection{Pinned distance sets in discrete regular sets}\n\nThe core of the proof of Theorem \\ref{thm:many-large-pinned-dist-sets} consists in showing the existence of \\emph{one} large pinned distance set. We state and prove the corresponding statement separately:\n\\begin{prop} \\label{prop:discrete-Falconer}\nGiven $s>1, C>1, t\\in (0,1)$, there exist $\\varepsilon=\\varepsilon(s,C,t)>0$ and $N_0=N_0(s,C,t,\\varepsilon)\\in\\mathbb{N}$ such that the following holds: if $N\\ge N_0$, and $A\\subset[0,1]^2$ is a subset of a discrete $(s,C)$-Ahlfors regular set at scale $2^{-N}$, such that $|A| \\ge 2^{(s-\\varepsilon)N}$, then there exists $x\\in A$ such that\n\\[\n\\mathcal{N}(\\mathrm{dist}(x,A),2^{-N}) \\ge 2^{tN}.\n\\]\n\\end{prop}\nBefore embarking on the proof of this proposition, we show how to deduce Theorem \\ref{thm:many-large-pinned-dist-sets} from it.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:many-large-pinned-dist-sets} (assuming Proposition \\ref{prop:discrete-Falconer})]\nLet $\\varepsilon$ and $N_0$ be as given by Proposition \\ref{prop:discrete-Falconer}, and take $N\\ge N_0$. Let\n\\[\nB = \\{ x\\in A: \\mathcal{N}(\\mathrm{dist}(x,A),2^{-N})<2^{tN}\\}.\n\\]\nIn particular, if $x\\in B$, then $\\mathcal{N}(\\mathrm{dist}(x,B),2^{-N})<2^{tN}$. By Proposition \\ref{prop:discrete-Falconer} applied to $B$, $|B|\\le 2^{(s-\\varepsilon)N}$, as claimed.\n\\end{proof}\n\nThe rest of this section is devoted to the proof of Proposition \\ref{prop:discrete-Falconer}. Suppose the claim is false. Then we can find sequences $N_j\\to\\infty$, $\\varepsilon_j\\to 0$, $A_j, B_j\\subset [0,1]^2$ such that $B_j$ is discrete $(s,C)$-Ahlfors regular at scale $2^{-N_j}$, $A_j\\subset B_j$, $|A_j|\\ge 2^{(s-\\varepsilon_j)N_j}$, and $\\mathcal{N}(\\mathrm{dist}(x,A_j),2^{-N_j}) < 2^{tN_j}$ for all $x\\in A_j$.\nLet\n\\[\n\\mu_j = \\frac{1}{|A_j|} \\sum_{D\\in\\mathcal{D}_{N_j}} |A_j\\cap D|\\mathcal{L}_D.\n\\]\n\nBy passing to a subsequence if needed, we may assume that $\\langle\\mu_j\\rangle_{[0,N_j]}$ converges to some $P\\in\\mathcal{P}(\\mathcal{P}([0,1)^2))$ in the $\\widehat{\\rho}$ topology. By Lemma \\ref{lem:limits-are-CPDs}, $P$ is a CPD. We underline that $P$ needs not be ergodic under $M$; if it was, the next lemma would hold automatically. It is only in this lemma that the hypothesis of Ahlfors-regularity gets used.\n\n\\begin{lemma} \\label{lem:lower-dim-regular-CPD}\n$\\dim_*P\\ge s$\n\\end{lemma}\n\\begin{proof}\nSince $P$-a.e. measure is exact-dimensional, it is enough to show that $\\uedim\\nu\\ge s$ for $P$-a.e. $\\nu$. In turn, by Borel-Cantelli this will follow if we can show that for every $\\widetilde{\\delta}>0$, if $q$ is sufficiently large (depending on $\\widetilde{\\delta}$), then\n\\begin{equation} \\label{eq:supported-on-large-measures}\nP\\{\\eta: H_q(\\eta) \\ge s-\\widetilde{\\delta}\\} > 1-\\widetilde{\\delta}.\n\\end{equation}\n\nSince $\\varepsilon_j\\to 0$, we know that $|A_j| \\ge 2^{(s-\\delta\/2) N_j}$ for large enough $j$. Since $B_j$, and hence $A_j$, hits at most $C$ points in each dyadic square of side length $2^{-N_j}$, a calculation shows that, if $j$ is large enough, then\n\\[\nH_{N_j}(\\mu_j) \\ge s-\\delta.\n\\]\nLemma \\ref{lem:Ahlfors-regular-is-rich} (together with Lemma \\ref{lem:Ahlfors-set-to-measure}(2) applied to $B_j$) can then be invoked to conclude that, given $q$ is taken large enough in terms of $\\delta$, and then $j$ is taken large enough in terms of $q$, the measure $\\mu_j$ is $s$-rich at resolution $(N_j,q,\\sqrt{\\delta}\/C')$ for some universal $C'>0$. Since $H_q$ is continuous on $(\\mathcal{P}([0,1)^d),\\widetilde{\\rho})$, the set $\\{\\eta: H_q(\\eta) \\ge s-\\sqrt{\\delta}\/C'\\}$ is compact, so we can pass to the limit to obtain our claim \\eqref{eq:supported-on-large-measures}.\n\\end{proof}\n\n\nAt this point we fix a small $\\delta>0$. In the end, a contradiction will be obtained provided $\\delta$ was taken sufficiently small.\n\nLet $\\mathcal{E}_q, \\mathcal{E}$ be the functions given by Theorem \\ref{thm:projections-CPD}. We fix $v\\in S^1$ such that $\\mathcal{E}(v)= 1$ (this is possible because $\\dim_*P>1$). Pick $q$ large enough that $1\/q\\le \\delta$ and (recalling the definition of $\\mathcal{E}_q$)\n\\begin{equation} \\label{eq:expected-projected-entropy-large}\n\\int \\min(H_q(\\Pi_v\\eta),1)\\,dP(\\eta) > 1-\\delta.\n\\end{equation}\n\nWe take $j$ large enough that $|A_j| \\ge 2^{(1-\\delta\/2)s N_j}$. We know from Lemma \\ref{lem:dense-radial-proj-discr-AR} that, again assuming $j$ is large enough, there is a set $E_j$ with $|E_j| \\le 2^{(1-\\delta)s N_j}$ such that if $x\\in A_j\\setminus E_j$, then there is $y\\in A_j$ with $y\\in X(x,2^{-q-1},v)$ and $|x-y|\\ge 2^{-K\\delta N_j}$, with $K>0$ depending only on $s$.\n\nWrite $M_j = \\lfloor (K+1)\\delta N_j\\rfloor$ and note that if $y\\in X(x,2^{-q-1},v)$ and $|x-y|\\ge 2^{-K\\delta N_j}$, then $y\\in X(x',2^{-q},v)$ for all $x'\\in D_{M_j}(x)$, again provided $j$ is large enough (the point is that the diameter of $D_{M_j}(x)$ is very small compared to $|x-y|$). If $D_{j,1},\\ldots, D_{j,L_j}\\in \\mathcal{D}_{M_j}^{(2)}$ is an enumeration of the the squares containing some point of $A_j\\setminus E_j$, the previous observations show that, if $j$ is sufficiently large, then:\n\\begin{enumerate}\n\\item For each $k\\in\\{1,\\ldots, L_j\\}$, there is $y_{j,k}\\in A_j$ such that $y_{j,k}\\in X(x,2^{-q},v)$ for all $x\\in D_{j,k}$.\n\\item\n\\begin{equation} \\label{eq:good-projections-bad-set}\n\\mu_j\\left(\\bigcup_{k=1}^{L_j} D_{j,k}\\right) > 1 - O_C(2^{-\\delta N_j}) > 1-\\delta.\n\\end{equation}\n\\end{enumerate}\n\nSince $P$-a.e. measure is exact dimensional and has dimension $>1$, $P$-a.e. measure gives no mass to lines, hence the function $\\eta\\mapsto H_q(\\Pi_v\\eta)$ is continuous $P$-almost everywhere. Consequently, if $j$ is large enough we deduce from \\eqref{eq:expected-projected-entropy-large} that\n\\[\n\\int \\min(H_q(\\Pi_v\\eta),1) d\\langle \\mu_j \\rangle_{[0,N_j]}(\\eta) > 1-\\delta.\n\\]\nSince $M_j\/N_j\\le (K+1)\\delta$, it follows that\n\\begin{equation} \\label{eq:good-projections-global}\n\\int \\min(H_q(\\Pi_v\\eta),1) d\\langle \\mu_j \\rangle_{[M_j,N_j]}(\\eta) > 1-(K+2)\\delta.\n\\end{equation}\nOn the other hand, note that for any $\\eta\\in\\mathcal{P}([0,1)^2)$, and any $1\\le M\\le n$, there is a decomposition\n\\[\n\\langle\\eta\\rangle_n = \\sum_{D\\in\\mathcal{D}_M} \\eta(D) \\langle\\eta_D\\rangle_n.\n\\]\nHence, if we denote $\\nu_{j,k}=(\\mu_j)_{D_{j,k}}$, adding up over $n=M_j,M_{j+1},\\ldots, N_j$ yields\n\\begin{equation} \\label{eq:good-projections-split-into-squares}\n\\langle \\mu_j \\rangle_{[M_j,N_j]} = \\sum_{k=1}^{L_j} \\mu(D_{j,k}) \\langle \\nu_{j,k} \\rangle_{[M_j,N_j]} + Q,\n\\end{equation}\nwhere $Q$ has total mass at most $\\delta$ by \\eqref{eq:good-projections-bad-set}.\n\nIt follows from \\eqref{eq:good-projections-global} and \\eqref{eq:good-projections-split-into-squares} that\nfor large enough $j$ there exists a square $D_{j,k}$ with\n\\[\n\\int \\min(H_q(\\Pi_v\\eta),1) d\\langle \\nu_{j,k} \\rangle_{[M_j,N_j]}(\\eta) > 1-(K+3)\\delta.\n\\]\nFrom now on we fix such a good square $D_{j,k}$ for each $j$, denote it simply by $D_j$ and forget about the other squares. We also denote $\\nu_j=\\nu_{j,k}$ and $y_j=y_{j,k}$. Recall that this is the point in $A_j$, whose existence we established earlier, such that $y_j \\in X(x,2^{-q},v)$ for all $x\\in D_j$.\n\nUsing again that $M_j\/N_j\\le (K+1)\\delta$, we get\n\\[\n\\int H_q(\\Pi_v\\eta) d\\langle \\nu_{j} \\rangle_{[0,N_j]}(\\eta) > 1-(2K+4)\\delta.\n\\]\nWe have arranged things so that the hypotheses of Corollary \\ref{cor:lower-bound-entropy-pinned-dist} are met. Since $1\/q<\\delta$, we conclude that, provided $j$ is large enough that $q\/N_j<\\delta$,\n\\[\nH_{N_j}(\\phi_{y_j}\\nu_j) \\ge 1 -O_s(\\delta),\n\\]\nwhere $\\phi_y(x)=\\frac12|x-y|$. In particular, since $\\nu_j$ is supported on a $(2^{-N_j})$-neighborhood of $A_j$, this shows that\n\\[\n\\mathcal{N}(\\mathrm{dist}(y_j,A_j),2^{-N_j}) \\ge 2^{(1-O_s(\\delta))N_j}\n\\]\nprovided $j$ is large enough (depending on $\\delta$). This contradicts with\n\\[\n\\mathcal{N}(\\mathrm{dist}(y_j,A_j),2^{-N_j}) < 2^{t N_j} \\text{ for all $j$}\n\\]\nif $\\delta$ is small enough, yielding the result.\n\n\\subsection{Proof of Corollaries \\ref{cor:pinned-dist-set-large-dev} and \\ref{cor:dist-set-box-dim}}\n\nIt is now easy to deduce Corollaries \\ref{cor:pinned-dist-set-large-dev} and \\ref{cor:dist-set-box-dim}.\n\n\\begin{proof}[Proof of Corollary \\ref{cor:pinned-dist-set-large-dev}]\n\nLet $A$ be as in the statement. Write $A_N$ for the collection of centers of squares in $\\mathcal{D}_N$ hitting $A$, so that in particular $A$ is contained in the $2^{-N}$-neighborhood of $A_N$. By Lemma \\ref{lem:Ahlfors-set-to-measure}, the sets $A_N$ are contained in a $(s,C')$-discrete Ahlfors regular set at scale $2^{-N}$, for some $C'=O(C)$.\n\nLet $\\varepsilon=\\varepsilon(s,C',t)>0$ be the value given by Theorem \\ref{thm:many-large-pinned-dist-sets}. By the theorem, if $N$ is large enough, then there is a set $B_N\\subset A_N$ with $|B_N|< 2^{(s-\\varepsilon)N}$ such that if $x\\in A_N\\setminus B_N$, then\n\\[\n\\mathcal{N}(\\mathrm{dist}(x,A_N),2^{-N}) \\ge 2^{tN}.\n\\]\nLet\n\\[\nB=\\limsup_N B_N(2^{-N}) = \\bigcap_{N=1}^\\infty \\bigcup_{M=N}^\\infty B_M(2^{-M}).\n\\]\nFix $s'>s-\\varepsilon$. Since $|B_M|< 2^{(s-\\varepsilon)M}$, we see that for each $N$ the set $B$ can be covered by a sequence of balls containing $2^{(s-\\varepsilon)M}$ balls of radius $2^{-M}$ for each $M\\ge N$. It follows that $\\mathcal{H}^{s'}(B)<0$ so that, letting $s'\\downarrow s-\\varepsilon$, we get $\\hdim(B)\\le s-\\varepsilon$.\n\n\nOn the other hand, it follows from the previous observations that if $x\\in A\\setminus B$, then\n\\[\n\\mathcal{N}(\\mathrm{dist}(x,A),2^{-N}) \\ge 2^{tN} \\quad\\text{for large enough } N,\n\\]\nso $\\lbdim(\\mathrm{dist}(x,A))\\ge t$. This gives the first claim.\n\nNow suppose $\\mathcal{H}^s(A)>0$. It is enough to check that if $t\\in (0,1)$, then\n\\[\n\\lbdim(\\mathrm{dist}(x,A))\\ge t\n\\]\nfor $\\mathcal{H}^s|_A$-almost all $x$. Suppose otherwise. Then there is a set $B\\subset A$ such that $\\mathcal{H}^s(B)>0$ (in particular, $\\hdim(B)\\ge s$), and $\\lbdim(\\mathrm{dist}(x,B))0$ be the number given by Theorem \\ref{thm:many-large-pinned-dist-sets}. If $N$ is large enough, $|A_N|\\ge 2^{(s-\\varepsilon)N}$, so Theorem \\ref{thm:many-large-pinned-dist-sets} says that there is $x=x_N\\in A_N$ such that $\\mathcal{N}(\\mathrm{dist}(x,A_N),2^{-N}) \\ge 2^{tN}$. But\n\\[\n\\mathcal{N}(\\mathrm{dist}(A,A), 2^{-N}) \\ge \\frac13 \\mathcal{N}(\\mathrm{dist}(x,A_N),2^{-N})\n\\]\nby the triangle inequality, so the claim follows.\n\\end{proof}\n\n\\section{Distances between two sets}\n\\label{sec:distances-between-sets}\n\nNow we investigate the set of distances between two sets. The following result immediately implies Theorem \\ref{thm:mlbdim-distance-sets-AR}.\n\\begin{theorem} \\label{thm:mlbdim-distance-sets}\nLet $A,B\\subset \\mathbb{R}^d$ be two Borel sets such that $\\hdim(A)>1$ and $\\mu(B)>0$ for some weakly $1$-regular $\\mu$ which also satisfies $\\hdim(\\mu)>1$. Then\n\\[\n\\mlbdim(\\mathrm{dist}(A,B)) = 1.\n\\]\n\\end{theorem}\n\n\n\nWe begin the proof of Theorem \\ref{thm:mlbdim-distance-sets}, by showing that it is enough to prove the corresponding claim for lower box counting dimension.\n\n\\begin{lemma} \\label{lem:simplif-box-dim}\nSuppose that, under the assumptions of Theorem \\ref{thm:mlbdim-distance-sets},\n\\begin{equation} \\label{eq:lbd-distance-set}\n\\lbdim(\\mathrm{dist}(A,B)) = 1.\n\\end{equation}\nThen Theorem \\ref{thm:mlbdim-distance-sets} holds.\n\\end{lemma}\n\\begin{proof}\nLet $A,B$ be as in the statement of Theorem \\ref{thm:mlbdim-distance-sets}. Without loss of generality, $A$ and $B$ can be taken to be compact. Moreover, by Frostman's Lemma, we may further assume that $\\hdim(A\\cap B(x,r))>1$ for any open ball $B(x,r)$ for which $A\\cap B(x,r)\\neq \\emptyset$ (more precisely, let $\\nu$ be a measure supported on $A$ such that $\\nu(B(x,r)) \\le C\\, r^s$ for some $s>1$, and replace $A$ by the support of $\\mu$). Finally, we may assume that $\\textrm{Supp}(\\mu)=B$ simply by replacing $B$ by $\\textrm{Supp}(\\mu_B)$.\n\nAfter these reductions, suppose $\\mlbdim(\\mathrm{dist}(A,B))=t<1$, and partition $\\mathrm{dist}(A,B)$ into countably many Borel sets $D_j$, so that $\\lbdim(D_j)\\le t$ for all $j$. By Baire's Theorem (and since we are assuming that $A$ and $B$ are compact), $\\mathrm{dist}^{-1}(D_j)$ has nonempty interior in $A\\times B$ for some $j$. Hence $\\mathrm{dist}^{-1}(D_j)$ contains a set of the form $A_0\\times B_0$ where, by our assumptions, $\\hdim(A_0)>1$ and $\\mu(B_0)>0$. This contradicts \\eqref{eq:lbd-distance-set}.\n\\end{proof}\n\nRecall that the direction determined by two different vectors $x,y\\in\\mathbb{R}^2$ is denoted by $\\sigma(x,y)$. In the next Lemma we perform a further regularization of the set $B$; this step uses a recent result of Mattila and Orponen \\cite{MattilaOrponen15}.\n\\begin{lemma} \\label{lem:simplif-uniform-angle}\nIn order to prove Theorem \\ref{thm:mlbdim-distance-sets}, it is enough to prove the following.\n\nLet $A, B,\\mu$ be as in the statement of the theorem, and further assume that $A, B$ are compact and disjoint and that there exists a set $\\Theta\\subset S^1$ of positive length such that for each $v\\in \\Theta$,\n\\[\n\\mu_B\\{y: \\sigma(x,y)=v \\text{ for some }x\\in A\\} > 1-\\delta,\n\\]\nfor some $\\delta\\in (0,1)$. Then\n\\[\n\\lbdim(\\mathrm{dist}(A,B)) > 1-\\varepsilon(\\delta),\n\\]\nwhere $\\varepsilon(\\delta)\\downarrow 0$ as $\\delta\\downarrow 0$.\n\\end{lemma}\n\\begin{proof}\nSuppose there exist $A,B,\\mu$ as in Theorem \\ref{thm:mlbdim-distance-sets} with\n\\[\n\\lbdim(\\mathrm{dist}(A,B)) < 1.\n\\]\nIn light of Lemma \\ref{lem:simplif-box-dim}, to derive a contradiction it is enough to show that, given $\\delta>0$, we can find subsets $A_0, B_0$ of $A, B$ (depending on $\\delta$), so that the pair $(A_0,B_0)$ satisfies the assumptions in the present lemma.\n\nWe start by noticing that we can easily make $A, B$ disjoint by taking appropriate subsets so we assume that they are already disjoint as given. By \\cite[Corollary 1.5]{MattilaOrponen15}, for $\\mu$-almost every $y\\in B$, the set $\\Theta_y=\\{ \\sigma(x,y):x\\in A\\}$ has positive length. Notice that the set \\[\n\\Upsilon=\\{ (v,y):v\\in\\Theta_y\\}\n\\]\nis Borel (we leave the routine verification to the reader). Thus, by Fubini, $(\\gamma\\times \\mu)(\\Upsilon)>0$ (where $\\gamma$ is Lebesgue measure on $S^1$). Let $(v_0,y_0)$ be a $(\\gamma\\times \\mu)$-density point of $\\Upsilon$ (for its existence, see e.g. \\cite[Corollary 2.14]{Mattila95}). We can then find compact neighborhoods $\\Theta_0$ of $v_0$ and $B_0$ of $y_0$, such that\n\\[\n(\\gamma\\times \\mu)\\{ (v,y)\\in \\Theta_0\\times B_0\\cap \\Upsilon\\} \\ge (1-\\delta\/2)\\gamma(\\Theta_0)\\mu(B_0).\n\\]\nApplying Fubini once again, we conclude that for $v$ in a set $\\Theta$ of positive measure (contained in $\\Theta_0$),\n\\[\n\\mu\\{ y\\in B_0: v\\in \\Theta_y\\} > (1-\\delta)\\mu(B_0) .\n\\]\nReplacing $B$ and $\\mu$ by $B_0$ and $\\mu_{B_0}$ concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:mlbdim-distance-sets}]\nWe will prove the claim of Lemma \\ref{lem:simplif-uniform-angle} with $\\varepsilon(\\delta)=O(\\delta)$. Hence, let $A, B, \\mu, \\Theta$ and $\\delta$ be as in that lemma. We also assume, as we may, that $B\\subset [0,1)^2$.\n\nLet $t=\\lbdim(\\mathrm{dist}(A,B))$. Our goal is then to show that $t>1-O(\\delta)$. Recall that $\\mathcal{N}(X,2^{-N})$ stands for the number of cubes in $\\mathcal{D}_N$ hit by the set $X$. Let $N_j\\to\\infty$ be a sequence such that\n\\begin{equation} \\label{eq:convergence-of-box-counting}\n\\frac{\\log\\mathcal{N}(\\mathrm{dist}(A,B),2^{-N_j})}{N_j} \\to t.\n\\end{equation}\nBy passing to a subsequence if needed, we may assume that $\\langle \\mu_B \\rangle_{[0,N_j]}$ converges, in the $\\widehat{\\rho}$ topology, to a distribution $P$ which, as we have seen in Lemma \\ref{lem:limits-are-CPDs}, is a CPD. Moreover, using weak $1$-regularity of $\\mu$, the same argument from Lemma \\ref{lem:lower-dim-regular-CPD} shows that $\\dim_*P \\ge 1$.\n\nLet $\\mathcal{E}_q$ and $\\mathcal{E}$ be as in Theorem \\ref{thm:projections-CPD}. By the last part of that theorem, we know that $\\mathcal{E}(v)=1$ for almost all $v$. Thus, since $\\Theta$ has positive measure, we can fix $v$ such that $\\mathcal{E}(v)=1$ and $v\\in\\Theta$.\n\nFrom this point on, the proof is similar to that of Proposition \\ref{prop:discrete-Falconer} but simpler as we do not need quantitative estimates. Since $\\mathcal{E}_q\\to \\mathcal{E}$ pointwise, we can fix $q=q(P,\\delta)$ such that $\\mathcal{E}_q(v)>1-\\delta^2$ and $1\/q<\\delta$. Recalling the definition of $\\mathcal{E}_q$, we see from Markov's inequality that\n\\[\nP(\\{\\eta: H_q(\\Pi_v\\eta)>1-\\delta\\})>1-\\delta.\n\\]\n\nNow since $A$ and $B$ are compact and disjoint, there exists $k$ (depending on $A,B,q$) such that if $x\\in A, y\\in B$ and $\\sigma(x,y)=v$, then\n\\begin{equation} \\label{eq:direction-almost-constant-on-square}\n|\\sigma(x,y')-v| \\le 2^{-q} \\quad\\text{if } y'\\in D_k(y).\n\\end{equation}\nNext, let $B_0$ be the union of $D_k(y)$ over all $y$ such that $\\sigma(x,y)=v$ for some $x\\in A$. Note that $\\mu(B_0)>1-\\delta$ by hypothesis. Let $D_1,\\ldots, D_\\ell$ be the cubes in $\\mathcal{D}_k$ that make up $B_0$, and pick $y_i\\in D_i, x_i\\in A$ such that $\\sigma(x_i,y_i)=v$ (i.e. if there are many such pairs we select one; this can be done in a Borel manner although we do not require this). Arguing exactly as in the proof of Proposition \\ref{prop:discrete-Falconer}, for each sufficiently large $j$ we find a cube $D_{i}$ (with $i$ depending on $j$) such that\n\\begin{equation} \\label{eq:cube-with-good-projections}\n\\langle \\mu_{D_{i}} \\rangle_{[0,N_j]}(\\{\\eta: H_q(\\Pi_v\\eta)>1-2\\delta\\})\\ge 1-O(\\delta).\n\\end{equation}\nHence, there is a value of $i$ such that the above happens infinitely often. From now on we fix that value of $i$, and write $M_j\\to\\infty$ for the corresponding subsequence of $N_j$.\n\nWrite $\\phi_x(y)=\\frac12|x-y|$. It follows from \\eqref{eq:direction-almost-constant-on-square} and Corollary \\ref{cor:lower-bound-entropy-pinned-dist} that if $j$ is large enough, then\n\\[\nH_{M_j}(\\phi_{x_i}\\mu_{D_i}) \\ge 1-O(\\delta).\n\\]\n\nSince $\\phi_{x_i}\\mu_{D_i}$ is supported on $\\frac12\\mathrm{dist}(A,B)$ and $M_j$ is a subsequence of $N_j$, we conclude from \\eqref{eq:convergence-of-box-counting} that\n\\[\nt=\\lbdim(\\mathrm{dist}(A,B))> 1-O(\\delta),\n\\]\nwhich is what we wanted to show.\n\\end{proof}\n\n\\begin{remark}\nThe proof of \\cite[Corollary 1.5]{MattilaOrponen15} goes through under the assumption of positive $1$-capacity rather than Hausdorff dimension $>1$ (or finite $I_1$ energy for the corresponding statement for measures that occurs in the proof). Hence, the assumptions in Theorem \\ref{thm:mlbdim-distance-sets} can be weakened to positive $1$-capacity of $A$ and $I_1(\\mu)<+\\infty$ instead of $\\hdim\\mu>1$ (we still need to assume that $\\mu$ is weakly $1$-regular). This gives many examples of (pairs of) sets of dimension $1$ to which the results apply.\n\\end{remark}\n\n\\begin{proof}[Proof of Corollary \\ref{cor:pinned-dist-set-upper-box-dim}]\nLet $A_0= \\{ x\\in A:\\ubdim(\\mathrm{dist}(x,B))=1\\}$. The proof of Theorem \\ref{thm:mlbdim-distance-sets} shows that $A_0$ is nonempty (we begin with a sequence $N_j\\to\\infty$ such that $\\langle \\mu_B \\rangle_{[0,N_j]}$ converges; the rest of the proof is identical). This implies that $\\hdim(A\\setminus A_0)\\le 1$, for otherwise there would be $x\\in A\\setminus A_0$ such that $\\ubdim(\\mathrm{dist}(x,B))=1$.\n\\end{proof}\n\n\\section*{Acknowledgments}\n\n\nI thank Tuomas Orponen for useful discussions, and two anonymous referees for a careful reading and many helpful suggestions that have made this a better paper.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction \\label{intro}}\nStar-forming regions -- such as the Orion Nebula -- are home to\nvarious phenomena associated with the early stages of stellar evolution.\nSome of the more prominent features in the visible part of the spectrum \nare\nthe arcs associated with gas flows known as Herbig Haro (HH) flows\n\\citep{rei01}.\nMany of these flows have been identified in the Orion Nebula and have had\nboth their radial \\citep{doi04} and tangential \\citep{doi02} velocities\nmeasured. The origins of these flows have in a few cases been associated\nwith IR sources embedded within the Orion Molecular Cloud 1 South (OMC-1S)\n\\citep{doi02}.\nHowever,\nthere are many flows that have not been paired with any source (X-ray,\nradio, or near-IR) -- including HH~529 \\citep{ode03}.\nThis flow\ncontains at least three curved shocks which appear in [O~{\\sc iii}] WFPC2 \nimages\n\\citep{ode96} and extend approximately $36\\arcsec$\nfrom the centre of the inferred source of the optical outflow (OOS) at\n$\\alpha$, $\\delta$ (J2000) = $5^{\\rm h}35^{\\rm m}14\\fs56$,$-5^{\\rm\no}$23\\arcmin54\\arcsec \\citep{ode03,doi04}.\\footnote{\n\\footnotesize{\nIn addition to HH~529, many other HH flows (HH~269, HH~202\nand HH~203\/204) appear to originate in the OOS region -- supplying ample\nevidence for OOS housing HH flow driver(s).\n\\citet{smi04} have detected an infrared source (IR source 2 in their \nTable~2; $\\alpha$, $\\delta$\n= $5^{\\rm h}35^{\\rm m}14\\fs40$,$-5^{\\rm o}$23\\arcmin51\\farcs0) which lies\nwithin $3\\arcsec$ of the predicted location of the OOS. \\citet{zap04} \nhave also observed this\nsource at 1.3cm.\nWithin the OOS region,\n\\citet{bal00b} have identified another\nnear-IR source (`s' in their Fig.~20) which was concurrently labelled\nHC209 \\citep{hil00}: $\\alpha$, $\\delta$ = $5^{\\rm h}35^{\\rm\nm}14\\fs57$,$-5^{\\rm o}$23\\arcmin50\\farcs8.\nRecently, an X-ray source (F421,\n\\citet{fei02}) has been found to be coincident with this near IR source.\nHowever, there is still not definitive proof as to the particular\ndriving source as neither of these sources lies directly in line with the \nflow of HH~529.\n}\n}\nThis is 0.08~pc in the plane of sky given a distance to the nebula of 460~pc \n(\\citet{bal00b}, hereafter BOM).\n\nThe radial (line-of-sight) velocity is $-44$~km~s$^{-1}$\n\\citep{doi04}.\nThis radial velocity is quoted as ``systemic''\n-- relative to the [O~{\\sc iii}] nebular component, which itself has a\nheliocentric\nvelocity of $+18\\pm2$~km~s$^{-1}$ \\citep{doi04}. Coupling this with the\nheliocentric radial velocity of the PDR ($+28$~km~s$^{-1}$,\n\\citet{gou82}), we obtain\na radial velocity relative to the source embedded within OMC-1: $-54$~km~s$^{-1}$.\nThe \naverage proper motion velocity is $54$~km~s$^{-1}$ \\citep{doi04} which leads \nto a total velocity of $76$~km~s$^{-1}$ (with respect to OMC-1S) at an angle of\n$45^{\\rm o}$ out of the plane of the sky.\n\nUsing this geometry, a distance from the embedded source to the leading edge of \nthe\neastern-most shock can be calculated: 0.12~pc ($36\\arcsec\\times\\sqrt{2}$).\nAssuming that the source lies within OMC-1S and that $\\theta^1$~Ori~C\nis itself $\\sim0.25$~pc from the main ionization front \\citep{wen95,ode01b}, this \nwould place\nthe HH~529 system on\nthe far side (i.e., further from the observer)\nof $\\theta^1$~Ori~C.\nIt is remarkable that the flow\nhas emerged from the cloud into the ionized zone.\n \nThe dynamical age of the HH~529 system can be calculated from the average \nproper motion\n(25 mas yr$^{-1}$).\nAssuming that the proper motion has remained\nconstant over the $36\\arcsec$ from its point of origin, we\nfind the dynamical age of the eastern-most visible feature of HH~529 to be \nroughly 1500\nyears. All shock model timescales will need to be consistent with this dynamical age\nin order for the model to be valid (\\S~\\ref{anal.model}).\n\n\nAs was recognized by \\citet{ode97a}, the fact that this and other Orion \nnebula\nHH flows\nappear strongly in [O~{\\sc iii}] (atypical of most HH flows which show \nmuch\nlower ionization)\nsuggests that these shocks are photoionized. We examine the physical\nconditions of HH~529 by comparing our high-resolution echelle spectra with\n(matter-bounded) photoionization models of this feature.\nOther studies of non-photoionized HH flows show evidence for\na decrease in the amount of Fe depletion in some of the shocks, as \ndetermined\nfrom [Fe~{\\sc ii}] lines \\citep{boh01,bec96}.\nThis has been linked to grain\ndestruction as matter originating from the molecular cloud passes\nthrough the shocks.\nIn this paper, we assess the Fe depletion using\na set of [Fe~{\\sc iii}]\nlines in the eastern-most feature of HH~529.\n\n \n\n\n\n\n\n\n\n\n\n\\section{Observations \\label{observe}}\n\nSpectra were obtained using the echelle spectrograph on the 4m Blanco \ntelescope at CTIO (see \\citet{bal00} for details) covering the spectral \nrange from the near-UV ($3500$\\AA) to the near-IR ($7500$\\AA). Three \nsets \nof red \nand blue spectra were \nobtained on two dates in 1997 and 1998. One of the three slit positions \n(x2, see Fig.~\\ref{slitpos}) intentionally\noverlaps with the eastern-most visible feature of HH~529.\nWavelength and flux calibrations were performed as in \\citet{bal00}.\nWe also used archival flux-calibrated (\\citet{ode99} using \\citet{bal91}) HST WFPC2 \nassociations\n(F487N, F502N, F547N, F631N, F656N, F658N, F673N) and\nBally mosaics of these associations (less F487N).\nAs there were a series of discrepant exposure times in the image headers of the\nBally mosaics, the flux calibration had to be redone -- again using the ground-based\nspectroscopic results of \\citet{bal91} -- to determine the relevant exposure times.\nWith these exposure times in hand, all WFPC2 pixel brightnesses (from both Bally\nmosaics and archival WFPC2 associations) have been accurately converted to absolute\nfluxes\/surface brightnesses, matching the ground-based flux calibration of \\citet{bal91}.\n\n\nLooking at the spatially-resolved `x2' echelle spectra, we have noticed \ntwo distinguishing features associated with the shock feature: a \nwide ($5\\arcsec$) velocity-shifted \ncomponent and a narrow ($2\\farcs5$) velocity bridge seemingly connecting \nthe \nnebula and the shock.\nSuch a bridge\nfeature -- which can also be seen in Fig.~6 of BOM --\nappears only to be associated with \nthe leading optically visible shock.\nAs this feature is intrinsically narrow\n($1\\farcs5$ from WFPC2 images), we have \nbeen able to determine\nthe effective seeing\nfor the red and blue spectra\nby measuring the\nwidth (along the slit) of the He~{\\sc i}~5876 bridge feature -- \na line that is found in both the red and blue spectra\n(see Fig.~\\ref{5876bridge}). The seeing was slightly different\non each of the two observing nights: $2\\arcsec$ for the red observations and \n$2\\farcs5$ for the blue.\nWe can also see from this figure\nthat the blue and red slits are aligned along their\nlengths to an accuracy of $\\sim0\\farcs2$. However, there\nare small differences in the absolute observed flux, most\nprobably as a result of a position\ndifference in the transverse direction, along the shock feature.\nThese deviations will be \naddressed in \\S~\\ref{bluered}.\n\nA direct comparison of spatial variation in ground-based\nand space-based observations over the same wavelength range was made to confirm\nthat the flux calibration of the echelle spectra is robust and\nthat the slit alignment and orientation are correct.\nThe echelle spectra were extracted over the same wavelength bandpass as the F656N\nWFPC2 filter.\nWith knowledge of the approximate slit position (from a Polaroid of the slit\nagainst the nebular background), the F656N flux-calibrated image was used to re-create the\nexpected spatial variation along the slit.\nThis re-created profile was\nconvolved with an appropriate-width\n($2\\arcsec$) Gaussian to simulate the seeing of the ground-based observations\n(see Fig.~\\ref{wfpc2slit}).\nThis processing allows for direct comparison between ground- and space-based\nobservations.\n(Note that there has been no continuum (or line contamination)\nsubtraction from either the echelle spectrum or the WFPC2\nreproduction, so the surface brightnesses in Fig.~\\ref{wfpc2slit} are not those\nof H$\\alpha$.)\nThe slit's position on the F656N WFPC2 image was adjusted -- while maintaining\nthe slit orientation, $PA = 116^{\\rm o}$ -- so as to emulate\nmore accurately the ground-based echelle slit spatial variation.\nThis \nrequired only a slight adjustment ($<1\\arcsec$) of the slit from its original position on \nthe WFPC2 image.\nUsing the slit position determined from this analysis, we compared all\nWFPC2 filters with their respective portions of the ground-based spectra,\nresulting in accurate reproductions of both the spatial variation and\nabsolute flux.\n\nThe high-resolution echelle spectra allow us to analyze the spatial variation of the nebula\nand shock separately -- offering insight not possible from the WFPC2 photometry.\nFor example, the slit variation of the [O~{\\sc iii}]~5007 and\n[O~{\\sc ii}]~3726 shock fluxes are shown in Fig.~\\ref{oiii_slit}. \nDifferences in variation across the slit between these two ions may be \nindicative of a higher density at the eastern-most edge of the shock:\na higher density would lead to more recombinations and a slightly \nhigher ionization fraction for O$^{+}$.\n\nThese WFPC2 and shock component analyses suggest that\n10 pixels ($-0\\farcs5$ to $+4\\farcs0$)\nalong the slit should be extracted in order to\nobtain the best contrast between the background nebular component\nand the velocity-shifted shock component\n(referred to hereafter as simply the `nebular' and `shock' components).\nFollowing this extraction, and with the nebular\nline identifications from \\citet{bal00} as a guide\\footnote{\nAll ID wavelengths are from Atomic Line List v2.04 \n(http:\/\/www.pa.uky.edu\/\\~{}peter\/atomic\/, maintained by \nP.~A.~M.~van~Hoof), except [O~{\\sc ii}] \\citep{bla04}.\n}, the `x2' spectral features were fit \nwith two \nGaussian components representing the nebular and shock components,\nas was done by \\citet{doi04}. \nEight parameters were used in the fit: FWHM, peak\nwavelength and area for both components, and two parameters to fit the\ncontinuum baseline level and slope. The result of such a fit is shown in\nFig.~\\ref{doublegauss}.\n\nFor cases where the shock component had a low\nsignal-to-noise ($S\/N < 5.2$),\nthe lines were re-fit with a constrained double \nGaussian.\nThe strong nebular component of the constrained fit had no \nconstraints while the weaker shock\ncomponent's FWHM was fixed using the weighted average of the \nstronger lines' FWHM ($28.3$~km~s$^{-1}$). The constrained velocity of the shock\ncomponent was set using the weighted mean of the \nH~{\\sc i} shock components ($-42.1$~km~s$^{-1}$), and was maintained as a constant \nrelative to the H~{\\sc i} gas.\nBecause of the\nionization\/velocity structure along the line-of-sight \\citep{bal00},\nthe actual velocity \ndifferences between the weak shock component and the strong nebular\ncomponent depend on the ion.\nIf the $S\/N$ of the shock component\nimproved and remained above $2.6$, the constrained fit was \nused. Otherwise the double Gaussian fit was used for all lines with $S\/N_{shock}\n> 2.6$.\n\nIf the double and constrained Gaussian fits resulted in an undetectable \nshock component ($S\/N < 2.6$), a five-component (FWHM, peak \nwavelength, area, continuum baseline and slope) single Gaussian fit was \nused for the nebular line.\nThe results of the line-fitting models are shown in Table~\\ref{lines} with\nnebular (neb) and shock (sh) components included in separate consecutive rows \nfor each ID wavelength.\nColumn descriptions are included in the table\n\nThe shock component can be seen most prominently in the \nmedium-ionization forbidden lines (e.g., [O~{\\sc iii}]) as well as in the \nHe~{\\sc i} and H~{\\sc i} permitted \nlines. Although the shock component can also be seen in the \nlow-ionization lines ([O~{\\sc ii}], [N~{\\sc ii}]), its \nstrength relative to the nebular line is much weaker\n(see Column~(9) of Table~\\ref{lines}).\nOf lines normally\nassociated with the ionization front (IF) of photoionized gas, some [S~{\\sc ii}] can be seen \nvery weakly in the shock component,\nwhereas others ([N~{\\sc i}]) are too weak to be detected.\nAs will be discussed, the presence of [S~{\\sc ii}] \ndoes not imply an ionization front in the shock.\n\nUnfortunately, the [O~{\\sc i}] sky lines\\footnote{These lines are identified as such from \nsky spectra and other nebular spectra (at positions which did not\nhave a velocity-shifted feature) that were taken on the same evening.}\nlie close to the wavelength where the shock component would be.\nUsing a triple Gaussian fit for the nebula, shock and sky components, we\ncan determine if there is a detectable shock component for the [O~{\\sc i}]\n6300 line.\nThe sky line FWHM, wavelength and area constraints are set by the sky line in\nthe 1SW echelle spectrum taken on the same evening;\nthe shock is constrained as in the constrained\ndouble Gaussian case.\nFollowing the fit of the three components in [O~{\\sc i}] 6300, the shock component\nhas a null detection ($S\/N\\ll2.6$) lying well below our detection limit.\nNeither is there a detectable bridge component as seen with the other shock lines. It can\nbe safely said that [O~{\\sc i}] (as with [N~{\\sc i}]) line emission in the\nshock lies below the detection limit for these spectra ($i.e., S\/N < 2.6$).\n\nAt first sight this seemed at odds with the BOM HH~529 [O~{\\sc i}] observations\ndepicted in their Figure 6 (WFPC2 631N image and Keck HIRES spectrum).\nHowever, their detection of [O~{\\sc i}] with the 631N filter\nis not definitive due to contamination from\n the [S~{\\sc iii}] line ($\\lambda6312$) \\citep{ode99}.\nBOM's original HIRES spectrum shows a strong [S~{\\sc iii}] velocity-shifted feature\n($v\\sim-39$~km~s$^{-1}$) associated with the eastern-most shock of HH~529\n(O'Dell, private communication, 2005). We also detect this in our spectrum and have\ndetermined quantitatively that [S~{\\sc iii}] would explain the presence of the shock in the\nWFPC2 631N image.\nFurthermore, the [O~{\\sc i}] velocity contour plot displayed in BOM Fig.~6 is\nactually an inadvertent copy\nof the [O~{\\sc iii}] plot (O'Dell, private communication, 2005).\nThe correct [O~{\\sc i}]\ncontours are similar to the [S~{\\sc ii}] contours in the west but have no velocity-shifted\nfeature in the east.\n\n\n\n\n\n\n\n\n\n\\subsection{Blue\/red line strengths \\label{bluered}}\n\n\nSince the red and the blue spectra were taken on different nights, there\nis a slight pointing uncertainty (see Fig.~{\\ref{5876bridge}}) which makes \ncomparison between the\nred and blue spectra more difficult.\nTo study the uncertainties involved in inter-spectral comparison, we\nidentified lines that are found in both the red and blue spectra.\nSix such lines had both a nebular and a measurable\nshock component ([Fe~{\\sc iii}]~5270,\n[Cl~{\\sc iii}]~5518, \n[Cl~{\\sc iii}]~5538, Si~{\\sc iii}~5740, [N~{\\sc ii}]~5755, and\nHe~{\\sc i}~5876). Table~\\ref{redblue} summarizes the results from the\n(constrained) double Gaussian line-fitting for these seven lines\nprior to applying the reddening correction.\nThe blue\/red ratios for the nebular and\nshock components are each shown separately\nin Column~(8) of Table~\\ref{redblue}, in the same rows as the\nblue results.\n\nThe nebular lines measured from the blue spectrum\nare not any stronger than the red on average ($B\/R^{weighted}_{avg}\\sim1.02\\pm0.04$).\nHowever, the average blue\/red ratio ($B\/R^{weighted}_{avg}\\sim0.85\\pm0.04$)\nindicates otherwise for the shock.\nThis difference in blue\/red ratios is not unexpected, as there is no \nreason to expect a correlation\nbetween surface brightnesses in the nebula and shock.\nUsing these results, we make an\nacross-the-board\nadjustment to all the red shock lines such\nthat the shock line strengths match between the red\nand blue (0.85 adjustment) -- allowing for a complete (blue\/red)\nanalysis of the shock.\nNo such correction is made to the nebular feature, whose blue\/red ratio is \nconsistent with 1.0.\n\n\\subsection{Reddening}\nIt is expected that the reddening of both the nebular and\nshock components is the same, being dominated by foreground material.\nHowever, prior to making the correction discussed in \\S~\\ref{bluered}, the\nnebula and shock had drastically different H$\\alpha$\/H$\\beta$\nBalmer decrements: $4.99\\pm0.04$ and $6.72\\pm0.32$,\nrespectively. After adjusting the line strengths so there is congruity \nbetween\nthe red and blue\nlines (\\S~\\ref{bluered}) in the red and blue spectra and accounting for that \nuncertainty, these values\nbecome $5.1\\pm0.1$ and $5.7\\pm0.4$ for the \nnebula and shock, \nrespectively. \nThis \njustifies\nthe use of the blue\/red correction in \\S~\\ref{bluered} and the\nuse of the same reddening correction for both nebula and shock: \n$E_{B-V} = 0.3655$ \\citep{mar06}. The surface brightnesses are corrected \nfor reddening \nas in \\citet{mar06} and these dereddened values are included in Column~(7) of \nTable~\\ref{lines}.\n\n\n\n\n\n\n\n\\section{Analysis}\n\n\\subsection{Velocity \\label{analvel}}\nFigure~\\ref{shockvel} plots all the velocities determined from the shock\ncomponents of the Gaussian fits. They are quite consistent, as expected\nsince unlike the expanding nebular gas,\nthere should be no velocity gradient in the shocked gas.\nThe shocked H~{\\sc i} lines are shifted by $-42.1\\pm1.2$~km~s$^{-1}$ \nrelative to \nthe nebular H~{\\sc i} lines\n(see Table~\\ref{lines} and Fig.~\\ref{shockvel}), or $-54.1\\pm1.2$~km~s$^{-1}$ \nrelative to the PDR \nin the molecular cloud, and hence, relative to the OOS embedded within the cloud. \nThis agrees with the radial velocity measurements made \nby \\citet{doi04} for the roughly coincident position 167-359 HH~529:\n$-52$ to $-54$~km~s$^{-1}$ relative to the PDR\/OMC-1.\n\nThe [Fe~{\\sc iii}]~5270 shock component\n(with $S\/N\\sim10$)\nappears to\nbe discrepant in Fig.~\\ref{shockvel}, with\nvelocities of \n$-32.9\\pm1.1$ (blue) and $-31.9\\pm1.1$~km~s$^{-1}$ (red).\nThis anomaly has an impact on the apparent nebular\nvelocity gradient of [Fe~{\\sc iii}] lines\n(see Fig.~10 in \\citet{bal00}) and is taken up in Appendix~\\ref{velgrad}.\n\n\\subsection{Temperature and density \\label{tempdensity}}\nTemperatures (in K) and densities (in cm$^{-3}$)\nare calculated from emission line ratios using the NEBULAR routines\nincluded within the iraf STSDAS package. \nThese are summarized in Table~\\ref{temden_nebshock},\nwith \nthe respective\ntransition probabilities and collision strengths used in the calculations.\n\n\nThe $T_e$([O~{\\sc iii}]) and $T_e$([N~{\\sc ii}]) diagnostic \nlines can be seen in both the nebula and the shock, while the [O~{\\sc i}]\ntemperature diagnostic lines can only be seen in the nebula.\nThe nebula temperature from the blue\n[O~{\\sc iii}] lines is\n$T_e^{\\rm neb}\\mbox{([O~{\\sc iii}])}\\sim8536^{+35}_{-33}$,\nwhereas for the red [N~{\\sc ii}] lines, \nthe temperature is higher, $T_e^{\\rm neb}\\mbox{([N~{\\sc ii}])}\\sim10672^{+53}_{-52}$.\nAlthough these temperatures come from the blue and red spectra respectively\nand therefore represent two slightly different lines-of-sight, the temperature\nrise with depth in the nebula is what is generally seen for other lines-of-sight,\nand is largely the result of a hardening\nof the radiation field as photons close to the ionization\nlimit are attenuated preferentially.\nTo complete the nebular temperature analysis, we have found\n$T_e^{\\rm neb}\\mbox{([O~{\\sc i}])}\\sim8005^{+580}_{-408}$.\n\nIn the shock, the lines are weaker (in the case of [N~{\\sc ii}], much weaker)\nand therefore the calculated temperatures have much larger uncertainties.\nThe [O~{\\sc iii}] temperature is $8366^{+252}_{-214}$, and that found from \nthe [N~{\\sc ii}]\ntemperature diagnostic lines is consistent (within $1\\sigma$): $8784^{+1184}_{-729}$.\nSince the shock is matter-bounded\n(see \\S~\\ref{linestrength}), \nO$^{++}$ ([O~{\\sc iii}]) and N$^{+}$ ([N~{\\sc ii}]) are not\ndistinct zones and\nthe attenuation seen in the nebula is not possible.\n\nThe electron density can be calculated from the\ndiagnostic lines ([O~{\\sc ii}] 3726, 3729; [S~{\\sc ii}] 6716, 6731;\n[Cl~{\\sc iii}] 5517, 5537) which are seen in the nebula and weakly in the shock.\nIn the nebula, these three sets of density diagnostic lines cover slightly different \nionization zones along a particular line-of-sight, but in the shock -- because of the\nlack of distinct ionization zones -- the densities are expected to characterize the same\nzone.\nHowever, because of the disparity between red and blue slit positions, the calculated \ndensities are also being defined along slightly different lines-of-sight.\n\nFor the nebula, we get\n$N_e^{\\rm neb}\\mbox{[O~{\\sc ii}]}\\sim1939^{+50}_{-50}$ \n($N_e^{\\rm neb}\\mbox{[O~{\\sc ii}]}\\sim2164$ using entire slit) from the blue [O~{\\sc ii}] \nlines.\nThe red [S~{\\sc ii}] lines yield a much higher density,\n$N_e^{\\rm neb}\\mbox{([S~{\\sc ii}])}\\sim5896^{+404}_{-366}$\n($N_e^{\\rm neb}\\mbox{([S~{\\sc ii}])}\\sim5638$ using entire slit),\nand the [Cl~{\\sc iii}] lines yield an even higher density,\n$N_e^{\\rm neb}\\mbox{([Cl~{\\sc iii}])}\\sim12074^{+1300}_{-1118}$.\n\nIt has been noted in \\citet{est04} that the use of \\citet{zei82} \ntransition probabilities and \\citet{pra76} collisions strengths drastically \nincreases the calculated $N_e$([O~{\\sc ii}]). Upon further \ninvestigation, we find that a change in the transition \nprobabilities alone will bring about the same result. Using these older \natomic data, we \nalmost double the measured density: $N_e^{\\rm neb}\\mbox{([O~{\\sc ii}])}\\sim3811$,\nbringing it more in line with the densities as measured from \nother indicators.\nAnother reason for questioning the atomic data comes from the [O~{\\sc ii}] \ntemperature \n-- which we overestimate slightly due to the \nshocked component impinging on the nebular component in the line pairs at 7320 and \n7330. Using the density as \ncalculated from [O~{\\sc ii}] 3726\/3729 (2000 cm$^{-3}$),\n$T_e^{\\rm neb}\\mbox{([O~{\\sc ii}])}\\sim20000$~K.\nHowever, with the larger density (4000 cm$^{-3}$) and the old atomic \ndata, $T_e^{\\rm neb}\\mbox{([O~{\\sc ii}])}\\sim15000$~K. An even larger density is required to \nreduce the temperature to 10000~K. Note that these densities from [O~{\\sc ii}] and\n[S~{\\sc ii}] are probably larger than in the more relevant [O~{\\sc iii}] zone, because of a \nfalloff of density in the expanding gas.\nA similar result appears when we use older transition probability data for the \n$N_e$([Cl~{\\sc iii}]) calculation. The density is reduced to a more consistent value:\n$N_e^{\\rm neb}\\mbox{([Cl~{\\sc iii}])}\\sim7247^{+575}_{-519}$.\nTo round out our discussion of density, we have looked at the density dependence of \n[Fe~{\\sc iii}] (following \\citet{kee01}) and O~{\\sc ii} (following \\citet{pei05}) lines.\nThe results are consistent with the densities we see in the rest of the nebula:\n$N_e^{\\rm neb}\\mbox{([Fe~{\\sc iii}])}\\sim4700^{+800}_{-800}$ and \n$N_e^{\\rm neb}\\mbox{(O~{\\sc ii})}\\sim6700^{+100}_{-100}$.\n\n\nThe density of the shock is also calculated, but as the low-ionization\nlines are weak, this calculated density is very uncertain. The blue [O~{\\sc ii}]\nlines yield $N_e^{\\rm sh}\\mbox{([O~{\\sc ii}])}\\sim2898^{+8429}_{-1997}$,\nthe red [S~{\\sc ii}]\nlines yield a density\nnear the limits of this diagnostic ratio, \n$N_e^{\\rm sh}\\mbox{([S~{\\sc ii}])}\\sim13183^{+10000}_{-11183}$,\n[Cl~{\\sc iii}] lines yield\n$N_e^{\\rm sh}\\mbox{([Cl~{\\sc iii}])}\\sim21715^{+39170}_{-9641}$,\nand the [Fe~{\\sc iii}] lines yield\\footnote{\nThe lower limit is set using [Fe~{\\sc iii}]~4986 which is not observed in the shock.\nThis indicates that $I_{4986}$ is below the detection limit\n($I_{\\lambda}\/I_{6678}\\sim0.01$, or $I_{\\lambda}\/I_{4658}\\sim0.05$), resulting\nin a minimum density of 3200 \\citep{kee01}.\n}\n$N_e^{\\rm sh}\\mbox{([Fe~{\\sc iii}])}\\sim7300^{+8000}_{-4100}$.\n(The O~{\\sc ii} lines are too weak to yield a consistent estimate\nof the shock density.)\nUse of the older atomic data again\nresults in a higher [O~{\\sc ii}] density, $N_e^{\\rm sh}\\mbox{([O~{\\sc ii}])}\\sim7304$,\nand a lower [Cl~{\\sc iii}] density, $N_e^{\\rm sh}\\mbox{([Cl~{\\sc iii}])}\\sim10911$.\nThe shock density appears to be larger (by roughly a factor of two) than that of the nebula,\nbut given the large uncertainties, a density identical to that\nof the nebula is also allowed by the line ratios. Density will be revisited\nin a discussion of\nshock models in \\S~\\ref{anal.model}.\n\n\\subsection{Relative line strengths and ionization structure \\label{linestrength}}\n\nTo maximize the shock-to-nebula ratio, the echelle spectra were extracted over only \nhalf the slit.\nEven then, the echelle spectra\nmaintain a weaker shock component as compared to the nebular component (see Column~(9) of \nTable~\\ref{lines}),\nindicative of a lower\ndensity, or more probably, a shorter emitting column in the shock.\nSince the illumination of the shock is roughly the same as that of the nebula, if the\nshock were optically thick,\nthe shock-to-nebula\nratio would be close to one for all lines, barring minor changes due to differences in\ndensity (near the critical density) or changes due to abundance (see \\S~\\ref{cloudyx2_discussion}).\nHere, the shock-to-nebula ratio is clearly lower than one,\nand so the shock is matter-bounded.\n\nThe relative strength\nvaries from $0.2$ for the medium-ionization lines (e.g., [O~{\\sc iii}]) to \nless than $0.03$\nfor the low-ionization lines (e.g., [N~{\\sc ii}]) to below the detection\nlimit for the lines usually\nassociated with the ionization front (e.g., [N~{\\sc i}])\nand is plotted as a function of ionization potential \nin Fig.~\\ref{ratio}.\nIn the case of a shortened emitting column,\nthe ionization potential serves as an indicator of\nionization fraction \n(where higher ionization potential\nindicates higher ionization fraction)\nwhile the shock-to-nebula ratio is a measure of the\noptical thickness of the shock\nto the relevant ionizing radiation.\nH~{\\sc i} is presented as a standard for shock\/nebula ionization comparison\nas its\noriginating ion (H$^{+}$) has an ionization fraction of roughly one\nthroughout\nboth the shock and the nebula.\nThe ratios of the medium-ionization species\n([O~{\\sc iii}], [Ar~{\\sc iii}], [Ne~{\\sc iii}]) all lie\nabove H~{\\sc i} as they have a higher net ionization fraction in the shock than\nin the nebula column.\nHowever, none of these ratios is unity either.\nThus, for example,\nin the shock there is not a complete O$^{++}$ zone, preceding a distinct\nO$^{+}$ zone.\nThe ratios of the low-ionization species\n([O~{\\sc ii}], [N~{\\sc ii}], [S~{\\sc ii}]) lie below H~{\\sc i} as they have a\nlower ionization fraction in the shock than in the nebula.\nIn fact, they must arise from trace ionization stages in a more highly ionized zone (e.g.,\ntrace O$^{+}$ in the O$^{++}$ zone). This is in contrast\nto the nebular column in which \nlines arise from distinct ionization zones.\nThe lack of an ionization front tracer ([N~{\\sc i}]) in \nthe shock component provides further corroboration for a\nmatter-bounded shock. \n\nThe critical densities associated with the [O~{\\sc ii}], [S~{\\sc ii}] and \n[Cl~{\\sc iii}] line transitions need to be considered as these lie within \nthe expected density range of the shock and so collisional de-excitation\ncould contribute to the \nrelative weakness of the shock lines. However, the weak [N~{\\sc ii}] \nlines have critical densities of $7.8\\times10^4$ and \n$1.2\\times10^7$~cm$^{-3}$ which lie well above the model-predicted density \nas discussed in \\S~\\ref{anal.model}.\nThe predominant cause of weakness is the lack of parent ions in this highly-ionized \nmatter-bounded geometry.\n\n\n\n\n\n\n\\subsection{Temperature fluctuations \\label{profiles}}\n\n\n\nTemperature fluctuations ($t^2$),\nfirst defined\/introduced by\n\\citet{pei67},\nhave been popular in explaining the\ndifferences in abundances found from forbidden lines as compared to those\nfound from permitted lines.\nAlthough these fluctuations have been deduced to exist, \ntheir deduced size ($t^2\\sim0.02$) has not been explained.\n\\citet{fer01} has suggested a possible link with additional photoelectric\nheating from grains.\nOther suggestions -- large scale variations in $T_e$, or the\npresence of regions either shielded from direct illumination by $\\theta^1$ Ori C\nor heated by shocks (from SNe mainly) -- might explain temperature fluctuations in \nthe nebula, but not in a small-scale shock.\n\n\nO~{\\sc ii} permitted and [O~{\\sc iii}] forbidden lines can be used to \ninfer a value of $t^2$ as has been done by \\citet{est98} and \n\\citet{est04} for the nebula. We apply this to the shock too, adjusting our\npermitted line analysis to allow for deviations from LTE \\citep{pei05}.\nFirst, we must confirm that the nebular and shock O~{\\sc ii} permitted lines\nform following recombination \\citep{gra76}.\nThe shock-to-nebula ratios of the O~{\\sc ii} and\n[O~{\\sc iii}] lines are the same, and\nmuch larger than the shock-to-nebula ratios of the\n[O~{\\sc ii}] lines.\nAlso, note that the velocities of the O~{\\sc ii} lines are consistent with the velocities\nof [O~{\\sc iii}] in the nebula (Table~\\ref{lines}). \n These two observations both\nconfirm that the O~{\\sc ii} lines are actually a result of recombinations \nfrom O$^{++}$ and not a result of direct starlight excitation of O$^{+}$,\nvalidating the use of these lines in the determination of the \nO$^{++}$\/H$^+$ ratio. We have\nused O~{\\sc ii} recombination line multiplet~1 and [O~{\\sc iii}] collisionally-excited\nlines 4363, 4959 and 5007 with the NEBULAR\\footnote{\nThe collisionally-excited line\nresults were calculated using the three-zone model in IRAF.\nIn this case only the low- and medium-ionization zones (those of O$^{0}$\/O${+}$ and O$^{++}$) \nare of interest.\nThe adopted densities of the nebula and shock are $N_e = 6000$ and $10000$, respectively.\nThe temperatures are those determined\nfrom the [N~{\\sc ii}] and [O~{\\sc iii}] temperature diagnostic lines (refer to \n\\S~\\ref{tempdensity}) for the low- and medium-ionization zones, respectively.\n} routines in IRAF (as in \\citet{est98,est04})\nto determine $t^2$ for the nebula and the shock. Not all permitted lines of \nO~{\\sc ii} multiplet~1 are observed, so individual (or pairings of) recombination lines\nare used to predict the complete multiplet's relative surface brightness (see \nTable~\\ref{tempfluct}), following \\citet{pei05} (their equations 3 and 4).\nUsing case A and case B O~{\\sc ii} recombination coefficients from \n\\citet{sto94} and case B H~{\\sc i} recombination coefficients from \\citet{sto95},\nO$^{++}$\/H$^{+}$ is calculated (see Table~\\ref{tempfluct}).\n\nThe O$^{++}$\/H$^{+}$ abundances from recombination and \ncollisionally-excited lines and the inferred $t^2$ are summarized in \nTable~\\ref{RLCEL} for both the nebula and the shock (along with the O$^{+}$\/H$^{+}$, O$^{0}$\/H$^{+}$\nand total O\/H abundances).\nOur nebular $t^2$, $0.009\\pm0.004$, is much lower than what\nhas been deduced from another line-of-sight (for the same O$^{++}$ ion), \n$t^2\\sim0.020\\pm0.002$ \\citep{est04} -- which did not correct O~{\\sc ii} lines for deviations \nfrom LTE.\nDespite the presence of detectable O~{\\sc ii}\nlines in the shock, the uncertainties are large enough that there is only \na $1\\sigma$ ``detection'' of $t^2$ in the shock, $t^2 = 0.010\\pm0.010$.\nIf the grains are depleted in the shock, a detectable $t^2$\nsuggests that the grains may not be the main contributor to $t^2$. This will be \nfollowed up in \\S~\\ref{tempfluc2}.\n\n\n\n\n\\section{Models \\label{anal.model}}\n\nThe HH object has been shown to be photoionized, so we can model\nthe emission using the radiative-collisional equilibrium code, Cloudy. \nAs the [Fe~{\\sc iii}] lines figure prominently in our discussion, we have improved the description of the\nFe$^{++}$ atom in Cloudy from a two-level to a 14-level atom, using collision \nstrengths and transition probabilities from\n\\citet{zha96} and \\citet{qui96}\nrespectively. This allows all multiplet \nlines associated with $\\lambda4658$ and $\\lambda5270$\nto be included in the determination of Fe abundance.\nAlso, as the accuracy of the atomic data for O$^{+}$ has been questioned \n(\\S~\\ref{tempdensity}, \\citet{est04}) we have \nreplaced the up-to-date transition probabilities \\citep{wie96} with the \nolder ones \\citep{zei82}.\n\n\n\\citet{bal00} showed that the incident continuum radiation (from the ionizing star, $\\theta^1$~Ori~C) is \nbest represented by a Mihalas stellar atmosphere model. However, to test the robustness \nof our result, we also developed models using a Kurucz stellar atmosphere. Note that\nthe issues with the Kurucz atmosphere (primarily with its inability to accurately\npredict the high ionization line [Ne~{\\sc iii}]~3869)\nare not that relevant to our discussion of low- and medium-ionization species.\n\n\n\n\nSince the shock has\na \nsmall\ncovering factor compared to the nebula,\nspherical geometry is not assumed and\nan inner radius is not set.\nThe sound-crossing time \nfor the HH feature ($\\sim10^3$ years) is roughly the same order as the \ndynamical timescale of the flow ($1500$ years), so instead of assuming a \nconstant pressure\n(as would be the case in a nebular model),\nwe assume a constant density.\nAlso, as the flow has only been in existence for\n1500 years ($5\\times10^{10}$ s), it is important to check the \nvalidity of a photoionization equilibrium code. The\nlongest timescale from the Cloudy shock model comes from \nH-recombination: $2\\times10^8$s -- well within the limit of the flow's age.\nThe incident surface flux of ionizing photons, $\\phi(H)$ should be close to \nthe value derived\nfor nebular models (log$\\phi(H)\\sim13.0$, $e.g.$~\\citet{bal91})\nas the shock\nis roughly the same distance from $\\theta^1$~Ori~C as the nebula (see \\S~\\ref{intro}).\nHowever, the electron density \nis probably significantly higher in the shock than in the nebula as evident from the \nobserved $\\lambda6731\/\\lambda6716$ ratios.\nSince the shock has been shown to be matter-bounded and\nhomogeneous with respect to its ionization structure\n(\\S~\\ref{linestrength}), the shock model can be developed simply as a finite thickness\ntruncated nebula (i.e., with a pre-defined stopping thickness).\nThis thickness can be predicted\nfrom the length ($10\\arcsec$) and width ($2\\arcsec$) of the shock in\nthe plane of the sky (from [O~{\\sc iii}] WFPC2 image) and its assumed \ncylindrically-symmetric geometry. Adopting a distance to the nebula of\n460 pc (BOM), the predicted median depth ($3\\arcsec$)\ntranslates to a thickness of 0.007 pc ($2\\times10^{16}$ cm). \n\nThe parameters are varied from these initial values, using observed surface\nbrightness of He~{\\sc i}~6678, and line ratios indicating temperature, \ndensity and ionization (see Table~\\ref{constraint}) to determine the best-fit\nmodels.\nIn the case of an\noptically thin model, the surface brightness varies as $n_H^2 t$, where $n_H$ is \nthe hydrogen density and $t$ is the model thickness. Adjusting the model thickness\ndoes not result in (much of) a change to any of the other constraint ratios as the\nionization fractions of most species are constant through the entire model.\nTherefore, $t$ is not completely independent, leaving\n$T_{\\star}$,\n$\\phi(H)$ and $n_H$ as the three independent parameters.\n\n\n\nA series of models were developed, two of which are summarized in Table~\\ref{cloudy}:\none with a Mihalas stellar atmosphere and Cloudy Orion abundances \n(from \\citet{bal91,rub91,ost92});\nand one with a Kurucz stellar atmosphere and \\citet{est04} Orion abundances \n(see Table~\\ref{abund_table}).\nAfter determining the best-fit parameters for\nboth of these models, the Fe abundance was adjusted to fit\nthe series of [Fe~{\\sc iii}] lines using the Cloudy $optimize$ routine. Some \nimplications\nof the derived abundances will be discussed in \\S~\\ref{depletion}.\n\n\n\n\n\\section{Discussion \\label{cloudyx2_discussion}}\n\nThe echelle observations (from Table~\\ref{lines}) and\nthe model predictions\nare summarized in\nTable~\\ref{modeltab}\nas $I_{\\lambda}\/I_{6678}$. If there is no model prediction (i.e., the particulars of\nthe line formation are not included in the model) then the\nobservations are not included in the table.\n\nIt is informative to compare the model predictions with the echelle\nobservations for not only the constraint ratios, but\nall\nlines predicted by the model. This will further test the robustness of the model.\nSpecial note\nshould be taken of lines predicted to be seen in\nthe \nshock,\nbut not observed.\nOf such cases,\nmany of them appear around or below the detection limit\n($I_{\\lambda}\/I_{6678}\\sim0.01$).\nMany of those lines predicted to be above this limit \n(He~{\\sc i}~3705, [S~{\\sc iii}]~3722, H~{\\sc i}~3722, He~{\\sc i}~3889, He~{\\sc i}~4009,\n[S~{\\sc ii}]~4076, C~{\\sc ii}~4267, O~{\\sc ii}~4341,\n[O~{\\sc ii}]~7320, [O~{\\sc ii}]~7331) appear as blended line\nfeatures in the spectrum\nand therefore are not included in Table~\\ref{modeltab}.\nThere are another three undetected-but-predicted shock lines:\nO~{\\sc ii}~4093,\nO~{\\sc ii}~4111,\nO~{\\sc ii}~4277.\nEach of these is a complete multiplet prediction requiring a series of multiplet\ncorrection factors to predict the observed multiplet component lines.\nAfter applying these correction factors to the shock model lines, \ntheir predicted flux would lie below the observed detection limit.\nAs discussed in \\S~\\ref{observe}, the velocity-shifted [O~{\\sc i}] lines are\nsky lines and not associated with the shock, explaining the disagreement\nbetween observation and model at [O~{\\sc i}]~6300.\n\n\n\n\n\\subsection{Depletion \\label{depletion}}\n\nThe Orion nebula is\nthought to have a depleted gas-phase abundance of Fe of roughly a\nfactor of 10 (with respect to solar) due to the presence of grains.\nFrom a preliminary analysis, this does not appear to be the case for the \nshock.\nThe ionization fraction of Fe$^{++}$ remains roughly constant through the slab\n(Fe$^{++}\\sim0.2$, Figure~\\ref{ionizeiron}) with no well-defined\nFe$^{++}$ zone, and yet the \n[Fe~{\\sc iii}]\nlines appear quite strong relative to the nebula lines (see \nFigure~\\ref{ratio}).\nThis may indicate an ``undepletion'' of Fe (possibly up to the solar level).\n\nA series of [Fe~{\\sc iii}] lines ($\\lambda4658$, $\\lambda5270$, etc.)\\ is predicted\nusing the higher resolution Fe$^{++}$ ion (\\S~\\ref{anal.model})\nand numerous [Fe~{\\sc ii}] lines are predicted\nusing the 371-level Fe$^{+}$ ion. These [Fe~{\\sc ii}] lines have been shown to\nhave large contributions from continuum pumping \\citet{ver00}\nand therefore, cannot be used as indicators of Fe abundance, but the modelled [Fe~{\\sc iii}]\nlines scale linearly with the Fe abundance.\nThe iron abundances determined from matching the observed and modelled [Fe~{\\sc iii}] \nlines in both\nshock models appear to be roughly consistent with the nebular gas-phase Fe abundance\n(see Table~\\ref{depletion_table})\nindicating that\nthe seemingly high shock [Fe~{\\sc iii}] line strengths can mostly be\nexplained by differences in the\nmodels' parameters,\nnot needing to resort to an order of magnitude change in the abundance. However, \nif the nebular Fe\/H gas-phase abundance is as low as 6.23 \\citep{est04}, the extreme\nprediction of Model B would suggest a three-fold increase in Fe\/H gas-phase abundance indicating\na partial destruction of grains in the shock.\n\nAn analysis of the Fe abundance of Orion B stars \\citep{cun94} and a follow-up analysis\nof Orion F\nand G stars \\citep{cun98} imply that the total abundance of Fe is consistent from \nstar to star within the\nOrion association, but that there may be a slight total Fe depletion \nwith respect to solar (-0.16~dex, \\citet{cun98}).\nThe Fe depletions obtained from our shock analyses are greater, ranging\n from -0.8 to -1.0 dex with respect to solar \n-- on \nthe order of the depletions\nfound in the nebula \\citep{bal91, rub97, est98, est04}. Assuming that the\ntotal Orion Fe abundance is on the order of that found from the Orion association\nstars, \nthe majority of the iron in the shock, as in the nebula, must be locked up in grains.\n\nA number of Si lines are also seen in the shock. \nAlthough there is no Cloudy prediction for these Si lines, the \nobservations can still be analyzed using ionization models from Cloudy and \nline information from \\citet{gra76}.\nThe shock-to-nebula ratio is high ($\\sim0.15$) for Si~{\\sc \nii}~3856,\n5056,\n6347\n(and 6371), but these\nlines have been shown to form due to starlight \nexcitation \\citep{gra76} in the Si$^+$ gas.\nThe Si$^{+}$ ionization fraction predicted from the Cloudy models (0.03)\nis much less than that for Fe$^{++}$ (0.2), but the Si~{\\sc ii} lines\nare not linearly dependent on Si abundance so these lines alone can not\nbe used to determine Si abundance.\n\nSince $\\sim20\\%$ of O atoms are thought to be in dust grains \\citep{est04},\nthe gas-phase abundance of O can be analyzed to determine the extent of dust destruction.\nThe total O\/H in the nebula and in the shock is summarized in Table~\\ref{RLCEL}.\nNote that O\/H for the shock component ($8.73\\pm0.05$) is an upper limit and the actual value is most\nlikely closer to that of O$^{++}$\/H$^{+}$ ($8.69\\pm0.05$).\nThe shock [O~{\\sc ii}] and [O~{\\sc iii}] line profiles across the extracted part of the slit\npeak at different spatial positions (see Fig.~\\ref{oiii_slit}),\nindicating that these lines are tracing physically different\nlines of sight and that a simple addition of O$^{++}$ and O$^{+}$ may overestimate the O\/H abundance.\nOur observed nebula O\/H abundance ($8.48\\pm0.01$ or $8.52\\pm0.03$ using recombination lines)\ndeviates slightly from other Orion\nnebula observations, which\nfind O\/H$\\sim8.60-8.65$ \\citep{bal91,rub91,ost92,est04}.\nThe shock O\/H abundance should be compared to an average\/typical O\/H nebula abundance, as the \nshock originates in a different region of the nebula.\nFor our observations of the shock, the uncertainty in O\/H is large enough that no definitive\nstatement can be made with regards to dust destruction in the shock, except that there\nmay be a small ``undepletion'' of gas-phase O to parallel the ``undepletion'' of gas-phase Fe.\n\n\n\\citet{smi05} have imaged the bow shocks of HH~529 with T-ReCS at $11.7\\mu$, seeing what they\nrefer to as ``most likely thermal dust emission'' associated with the eastern-most shock.\nAlthough supporting the argument of \\citet{smi05},\nour evidence for the existence of grains in this one HH object is\nanomalous when compared with the 21 HH objects studied by\n\\citet{boh01}. For both their high-excitation\/fast-moving ($v~>~85$~km~s$^{-1}$)\nand low-excitation\/slow-moving ($v~\\leq~50$~km~s$^{-1}$)\nHH objects, the derived Fe depletion is never more than $-0.4$~dex\nsuggesting that the grains are most likely destroyed in the HH objects regardless\nof their velocity. It is of interest that for HH~529 -- measured to have a velocity \nof\n$76$~km~s$^{-1}$ relative to OMC-1 -- the depletion is on the order of that of\nthe nebula ($-1.0$~dex);\nthere is no evidence for the complete\ndestruction of grains in the eastern-most visible\nshock of HH~529. This is more along the lines\nof what one would expect: a slow-moving flow would not be expected\nto destroy grains, whereas a fast-moving flow would. \\citet{boh01} suggest\nthat the molecular cloud material currently associated with\ntheir slow-moving shock may have had its grains destroyed in \nan earlier pass through a faster-moving shock.\nFollowing this argument,\nthe material associated with HH~529 must\nnot have ever passed through a high-excitation\/fast-moving shock.\nThis is slightly inconsistent with the set of HH~529 velocities\nmeasured by \\citet{doi02,doi04}, many of which suggest\nthe material may have been travelling faster than 85~km~s$^{-1}$.\nA full Fe abundance analysis of all HH~529 shocks could offer further\ninsight into grain destruction in Herbig Haro objects.\n\n\\subsection{Temperature fluctuations \\label{tempfluc2}}\nThe $t^2$ deduced to exist in the nebula is $0.009\\pm0.004$\nand in the shock is $0.010\\pm0.010$\n(\\S~\\ref{profiles}) (which are both within $2\\sigma$ of zero).\nTwo suggested explanations for the existence of $t^2$ --\nlarge scale variations in $T_e$ or the presence of shielded or\nheated regions -- can not apply for the small column\ncovered by the shock.\nHowever, since the grains still appear to be present in the shock\na $t^2$ detection suggests a third explanation: that the grains may be the \nmain contributor to $t^2$.\nConversely, an ``effective'' $t^2$ may \nbe introduced if the\neffective recombination coefficients, collision strengths and\/or transition probabilities are\ninaccurate, or if there were\nsome other contributions to the line emission besides solely\nrecombination or collisional excitation.\n\n\\section{Conclusions \\label{conclusions}}\n\nHigh-resolution spectroscopy of the Orion nebula across the \nHerbig Haro object HH~529 has allowed for a comparison of that local part \nof the nebula with the velocity-shifted spectrum of the flow. The radial\nvelocity (as measured from the H~{\\sc i} emission lines), $-42.1\\pm1.2$~km~s$^{-1}$ \nis consistent with the $-40$ to $-42$~km~s$^{-1}$ range\nas measured by \\citet{doi04} for \na slightly different line-of-sight. In addition, there is ample evidence \nto suggest that this flow has been photoionized. Herbig Haro objects \nusually have a strong low-ionization line spectrum. In this case, the fact \nthat we see strong medium-ionization lines and much weaker low-ionization \nlines indicates that we have a photoionized shock, as first suggested \nby \\citet{ode97a}. The distinguishing shock-to-nebula ratios as a function \nof ionization fraction, or ionizing potential as in Figure~\\ref{ratio}, \nfurther support this hypothesis, leading us to model the shock as a \nmatter-bounded photoionization region.\n\nThe\nshock component was modelled using the \nphotoionization equilibrium code, Cloudy.\nBoth Mihalas and Kurucz stellar atmosphere models were investigated to ensure\nthe robustness of our conclusions.\nA series of ``best-fit'' models covering a range of stellar temperatures,\ndensities, and $\\phi(H)$ fluxes has allowed us to determine that the \ndepletion\nof Fe (relative to solar) in the nebula \nalso exists in the shock.\nThe higher density of the photoionized shock allows for the formation of\nrelatively strong [Fe~{\\sc iii}] lines without necessitating a reduction\nof the Fe depletion. The Fe depletion for the shock is roughly the same as for the \nOrion nebula, an order of magnitude relative to solar (-1.0 dex).\nThe total Fe abundance of the Orion association stars may be slightly depleted\n(-0.16 dex, \\citet{cun98}), but not to the extent of the gas-phase Fe in the \nnebula and shock.\nThis suggests that if the total Fe abundance in the nebula and shock is\nof the same order as that found from the Orion association stars, grains \nmust be present \nin the Herbig Haro flow to account for the depletion of gas-phase Fe.\n\\citet{boh01} suggests that grains are destroyed in many HH objects as the material\npasses through high-excitation\/fast-moving shocks. From our results,\nwe infer that the \neastern-most shock of\nHH~529 never reached the velocities necessary to destroy the majority of the\ngrains despite the presence of fast-moving shocks elsewhere in HH~529.\nThis supports the observations of 11.7$\\mu$ thermal dust emission in the eastern-most\nshock of HH~529 \\citep{smi05}.\nFurther information about grain destruction in HH~529\ncan be obtained from parallel Fe abundance analyses for the remainder\nof the HH~529 photoionized shocks.\n\n\n\n\nTemperature fluctuations in the Orion nebula have been used to explain discrepancies\nin abundances found from recombination lines\nversus abundances found from collisionally-excited lines.\nUsing solely lines originating from the O$^{++}$ gas, we derive $t^2$ for the \nnebula\n($t^2 = 0.009\\pm0.004$) and the shock ($t^2 = 0.010\\pm0.010$). \\citet{est04} have \npublished\na series of $t^2$ for a number of ions, including O$^{++}$ ($0.020\\pm0.002$), as \nwell as an average\nfrom their series of ions ($0.022\\pm0.002$). The interesting result is that the \nshock maintains\na $t^2$ similar to the nebula (albeit with a large uncertainty)\ndespite being much thinner.\nThese observations, if corroborated with higher $S\/N$ data, may draw\ninto question some of the theories that have been expounded surrounding an \nexplanation for these\ninferred $t^2$ fluctuations.\nGrains appear to be present in the shock,\nsuggesting that the grains may still somehow be contributing to $t^2$.\nThe measurement of a non-zero $t^2$ in a \nmatter-bounded shock would more likely\nsupport the argument for an ``effective'' $t^2$ resulting from uncertainties in \nthe atomic data and\/or \nmissing contributions to the line emission.\nHigher $S\/N$ O~{\\sc ii} spectra of the shock will reduce the \nuncertainty of the inferred\n$t^2$, and allow for more definitive conclusions to be made.\n\\acknowledgements\nThis work was supported by the Natural Sciences and Engineering\nResearch Council of Canada. Line wavelengths were obtained from the\n Atomic Line List\\footnote{Atomic Line List v2.04 is available at:\nhttp:\/\/www.pa.uky.edu\/\\~{}peter\/atomic\/.} maintained by P.~A.~M.~van~Hoof. \nCalculations were performed with version 05.07 of Cloudy, last described by \n\\citet{fer98}.\nThe authors wish to thank C.~R.~O'Dell for his clarification of a portion of\nthe BOM data, and referee M.~Peimbert for his detailed review of this paper.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\IEEEPARstart{O}{ver} a century ago, Charles Darwin alluded to an experimental paradigm that involved direct observation of the brain's physical mechanisms (nervous matter)~\\cite{darwin1871descent}, where such observations~\\cite{roysher} would serve as the physical basis for dichotomising species-specific behaviour.\\comments{according to the stimulus adminstered at time \\textit{t}. The main caveat with Darwin's approach is the assumption that the physical manifestations of mental activity, presumably reflected in the movements of the brain's nervous matter, can be measured.} In the early 90s, an indirect and non-invasive measurement of mental activity over uniformly-spaced time points became possible through functional Magnetic Resonance Imaging (fMRI), which allows paramagnetic deoxyhemoglobin to act as an endogenous Blood Oxygenation-Level Dependent (BOLD) contrast~\\cite{ogawa3}. This article pursues Darwin's proposed experimental paradigm by Locally Linear Embedding (LLE)~\\cite{lle1} the BOLD time-series to produce precise summaries of cerebral activity that may optimise the classification of different brain states, such as mental disorders.\n\\comments{measured BOLD response (over time) in a topology~\\cite{bourbaki} defined over Cartesian space with the Pythagorean distance metric, which is expected to produce precise measurements of mental activity that will assist in dichotomising healthy animals from those with mental disorders.}\n\n\n\n\\par Logothetis \\textit{et al} found that an increase in invasively-measured neural activity directly and monotonically reflects local BOLD signal increases and, for short stimulus presentations, there is a linear relationship between BOLD and neural responses~\\cite{nikos2}. This suggests the unobservable neural activity is spatially-localised in anatomical space. Locally Linear Embedding of fMRI data in space and time can, in principle~\\cite{archibald1914,compkern,vsepr} (see Appendix~\\ref{app:LLE}), summarise these \\textit{local} measurements of neuronal mass activity~\\cite{natmrirev}--via the notion of \\textit{analytic capacity}~\\cite{melnikov}--to disclose information about \\textit{global} (i.e., whole-brain) activity patterns\n\n\n\\section{Methods: Testing Locally Linear Embedding (LLE) with Cross Validation}\n\\par The discriminatory power of Locally Linear Embedding was compared to the original fMRI, both before and after applying Principal Component Analysis (PCA)~\\cite{pcabook} for artefact reduction~\\cite{nikos2}. This comparison was performed on eleven datasets containing cohorts with different mental disorders, using a combination of Leave One Out Cross Validation (LOOCV) on the training set, with greedy feature selection based on Fisher discriminability~\\cite{fisher1,fisher2}. The purpose of using LOOCV and feature selection is to find the time points that discriminate patients from controls on the respective dataset. The feature selection step initially starts with an empty candidate set of time points and proceeds to select the time point with the highest discriminatory power~\\cite{sfs}. Then, time points that improve discrimination in conjunction with those already in the candidate set are added to this set in incremental fashion; this process is terminated when there are no more time points that can be added to the candidate set to improve discriminability. Note that the selection of time points is based upon cross validation and does not induce any biased sampling.\n\n\\par To illustrate the patterns that best discriminate between groups, a paired two-sample t-test between the patient and control groups is performed to both threshold and identify the statistically-significant differences (\\textit{p} $<$ 0.05 uncorrected) in space (at the time points identified by the greedy feature selection). Per Mill's Methods of Induction (Method of difference), the functional differences (depicted by the statistically significantly-different regions) at the respective time point are therefore a necessary part of the cause of the phenomena that distinguish the subject groups~\\cite{sol1}, which in this case pertain to a neuropsychiatric disorder. Neuropsychiatric disorders are diagnosed using clinical assessments that include: evaluating the background demographics, collecting first and third party observations, and a structured psychiatric interview with the subject~\\cite{psychassess}. In detail:\n\n\\comments{removed by dr friston: This process reflects the intuition that differences in mental activity between patients and controls can occur, and therefore be identified, at any time point during the scan, where these time points can then be objectively thresholded to identify statistically significant physical differences. Per Mill's Methods of Induction (Method of difference), the functional differences (depicted by the statistically significantly-different regions) at the respective time point are therefore a necessary part of the cause of the phenomena that distinguish the subject groups~\\cite{sol1}, which in this case pertain to a neuropsychiatric disorder. Given the simplicity of the feature selection and that cohorts' resting-state fMRI are not synchronised -- i.e., cohorts are not administered identical stimuli at the same time during the scan-- the predictive power of the diagnostic volumes determined on resting-state data is anticipated to vary. Given that differences in mental activity can occur at any time, it follows that the number of diagnostic time points can vary between datasets with the same experimental design and subject groups, especially for resting-state data.}\n\n\n\nEvery subject's fMRI time-series is treated as a four-dimensional array $\\mathbf{X}\\in\\mathbb{R}^{ L \\times W \\times H \\times T}$ with $V=LWH$ $T$-dimensional voxel waveforms $\\mathbf{x}_i \\in \\mathbb{R}^T$ for $i=1,\\ldots,V$. Assume each subject scan $\\mathbf{X}_i$ is associated with a binary-valued class label $y_i$ representing the diagnosis and that, for any subject scan, every voxel waveform $\\mathbf{x} \\in \\mathbb{R}^T$ is generated by a vector $\\mathbf{z}\\in\\mathbb{R}^d$ corresponding to a point on the manifold. Our approach to fMRI-based diagnosis involves two stages:\n\n\\noindent \\textbf{fMRI reconstruction} takes the subject's fMRI as input and outputs a reconstructed fMRI that is more informative than the original. Formally, this reconstruction is a mapping $\\mathcal{L}: \\mathbb{R}^{L\\times W\\times H \\times T} \\to \\mathbb{R}^{L\\times W \\times H \\times d}.$ \n\nIn other words, Locally Linear Embedding reduces a time-series of length \\textit{T} to a smaller number of spatial modes of dimensionality \\textit{d}; these modes contain all the information used for the subsequent step.\\\\\n\\textbf{Classification} builds a classifier that takes the subject's reconstructed fMRI as input, and outputs a class label $y_i\\in\\{0,1\\}$. The classifier is therefore a mapping $\\mathcal{C}: \\mathbb{R}^{L\\times W\\times H \\times d} \\to \\{0,1\\}$.\n\nThe reconstructed, or reduced, fMRI data produced from step 1 is hereon referred to as $\\mathbf{Z}$. All reconstructions initially vectorise the fMRI data to produce a two-dimensional array $\\mathbf{X}\\in\\mathbb{R}^{V\\times T}$, and conclude by reshaping the resulting two-dimensional reconstruction $\\mathbf{Z}\\in\\mathbb{R}^{V \\times d}$ into a four-dimensional array $\\mathbf{Z}\\in\\mathbb{R}^{L\\times W \\times H \\times d}$\n\n\\paragraph{Principal Component Analysis (PCA)~\\cite{bishop,pcabook}} reconstructs $\\mathbf{X}$ by finding an orthogonal rotation that minimises the reconstruction cost\n\n\\begin{equation*}\\label{eqn:pca}\n\\min_{\\mathbf{B} =[\\mathbf{b}_1,\\ldots,\\mathbf{b}_T]} \\sum_{i=1}^V || \\mathbf{x}_{i} - \\mathbf{\\bar{x}} - \\mathbf{B}\\mathbf{z}_i||_2^2\n\\end{equation*}\n\n\\noindent where $\\bar{\\mathbf{x}} \\in \\mathbb{R}^T$ is the mean over all voxel waveforms, and $\\mathbf{z}_i = \\mathbf{B}^\\intercal(\\mathbf{x}_i-\\mathbf{\\bar{x}}) \\in \\mathbb{R}^T$. To find the optimal $\\mathbf{B}$, compute the right-hand matrix for the singular value decomposition (SVD) of $\\mathbf{X} = \\mathbf{UDB}^\\intercal$, which contains the $T$ right-singular vectors of $\\mathbf{X}$~\\cite{htf}. It follows that $\\mathbf{Z} = \\mathbf{XB} \\in \\mathbb{R}^{V \\times T}$ is the rotated matrix that minimises the reconstruction cost of the subject's fMRI, where every column of $\\mathbf{Z}$ is a \\textit{principal component}. It is assumed the first $d$ principal components (columns) of $\\mathbf{Z}$ capture the ``systematic structure\", where the confounding factors are relegated to the remaining \\textit{T} - \\textit{d} principal components, which produces the two-dimensional reconstructed fMRI $\\mathbf{Z}\\in\\mathbb{R}^{V\\times d}$.\n\n\\par Using PCA's ``systematic structure\" for distinguishing humans with different neurological disorders has been met with caution~\\cite{classcalhoun}, largely because PCA's application to fMRI has some subjective components~\\cite{pcafmri}. The main limitation behind the PCA reconstruction is that it assumes the lower-dimensional manifold is a linear subspace.\\comments{inability to measure the \\textit{local} spatiotemporal covariance between voxels, which which requires a graphical model to define the spatial neighbours of every voxel.} We demonstrate that introducing the Cauchy stress tensor~\\cite{cauchy} on the Cartesian space with the Pythagorean distance metric enables three-dimensional measurements over time, thereby revealing the local (group) action in the physical system.\n\n\n\\paragraph{Locally Linear Embedding (LLE)~\\cite{lle1,LLE2}} reconstructs $\\mathbf{X}$ by constructing the Cauchy stress tensor~\\cite{cauchy} at every voxel $i$ for $i=1,\\ldots,V$, which is achieved by minimising the reconstruction cost of its waveform $\\mathbf{x}_i$ in terms of its spatially-adjacent neighbours:\t\n\\begin{equation} \\label{eqn:objfunc}\n\\min || \\mathbf{x}_{i} - \\sum_{j \\in \\mathcal{N}(i)} w_{i,j}\\mathbf{x}_{j}||_2^2, \\;\\;\\;\\;\\; \\textrm{where} \\;\\; \\sum_{j \\in {\\cal N}(i)} w_{i,j} = 1\n\\end{equation}\n\n\\noindent where the neighbourhood set ${\\cal N}(i)$ for voxel $i$ is the complement of its $K$ spatial neighbours on the surface of the sphere with radius \\textit{r}, and $\\mathbf{w}_i = [w_{i,1},\\ldots,w_{i,|{\\cal N}(i)|}] \\in \\mathbb{R}^{|{\\cal N}(i)|}$ are the reconstruction weights containing the spatially-invariant geometric properties of the Cauchy stress tensor at voxel \\textit{i}.\n\nTo determine the \\comments{second-order invariant properties of the Cauchy stress tensor}\\textit{analytic capacity}~\\cite{melnikov} at every voxel location \\textit{i}, LLE first subtracts the \\textit{K} spatial patterns of the voxels on the boundary of the sphere centred around voxel \\textit{i} to determine the separation distance from the origin of the tensor at the respective voxel. Then, it computes the local (symmetric) spatiotemporal covariance matrix:\n\\begin{equation}\n\\begin{split} \n\\mathbf{G}_i &= \\mathbf{C}_i^\\intercal\\mathbf{C}_i\\\\ &= [(\\mathbf{x}_j-\\mathbf{x}_i),...,(\\mathbf{x}_{j+|\\mathcal{N}(i)|} - \\mathbf{x}_i)]^\\intercal [(\\mathbf{x}_j - \\mathbf{x}_i),..., (\\mathbf{x}_{j+|\\mathcal{N}(i)|} - \\mathbf{x}_i)] \\\\&+ \\xi \\mathbf{I}_{|{\\mathcal{N}}(i)|}\n\\end{split}\n\\end{equation}\nwhere\\comments{ that conditions the $\\mathbf{C}_i$ is a Hilbert space.} $\\xi\\mathbf{I}_{|{\\cal N}(i)|}$ is a non-negative regularisation term to enforce positive-definiteness (for this study, $\\xi=0$). LLE calculates the reconstruction weights by finding the unique minimum-norm solution~\\cite{pinv} to the constrained least-squares problem defined by:\n\\begin{align}\n\\mathbf{G}_i \\mathbf{w}_i = \\mathbf{1}_{|{\\cal N}(i)|} &\\iff \\mathbf{G}_i^+ \\mathbf{1}_{|{\\cal N}(i)|} = \\mathbf{w}_i\n\\end{align}\n\\noindent where $\\mathbf{1}_{|{\\cal N}(i)|} \\in \\mathbb{R}^{|{\\cal N}(i)|}$ is a vector of ones and the $j^{th}$ element of $\\mathbf{w}_{i,j}$ can be thought of as the average height of a curve representing the \\textit{mean transit time of the indicator}~\\cite{lasseningvar} of voxel $j$ from voxel $i$ over the duration of the scan. Since $\\mathbf{G}_i$ represents the squared distance of the surface forces from voxel $i$, the reconstruction weights $\\mathbf{w}_i$ are Lebesgue measures~\\cite{lebesgue} summarising the \\textit{analytic capacity}, or spatially-invariant geometry~\\cite{gregory}, of the space-filling curve~\\cite{hilbert}, where the constraint ensures that the areas between the imaginary surface (acting as the origin that divides the body) and curves (defined by the stress vectors) are 1 in each of the $|\\mathcal{N}(i)|$ directions. In practice the weights can be brittle~\\cite{mlle} due to any number of reasons. Modified Locally Linear Embedding (MLLE) therefore computes the $1 \\leq s_i \\leq \\textit{K}$ linearly-independent (orthogonal) vectors\\footnote{When using MLLE it is possible for $d > K$, thus the optimal number of weight vectors $s_i$ for each voxel $i$ is determined by setting $d=1$ so that \\textit{K - 1} $\\leq s_i \\leq K$--i.e., $s_i$ is set to span as large of a basis as possible. After this step the desired dimensionality \\textit{d} is then input to the eigensolver.} $\\mathbf{Q}_i \\in\\mathbb{R}^{K \\times K}$ of $\\mathbf{G}_i$ using the eigendecomposition $\\mathbf{G}_i = \\mathbf{Q}_i^\\intercal\\mathbf{A}\\mathbf{Q}_i$, thereby allowing the definition of multiple weight vectors for each voxel. Assuming the columns (eigenvectors) $[\\mathbf{q}_1,\\ldots,\\mathbf{q}_K] = \\mathbf{Q}_i$ are sorted in descending order of their respective eigenvalues $\\lambda_1^{(i)},\\ldots,\\lambda_K^{(i)}$, MLLE uses the first $s_i$ columns to compute multiple local weight vectors for a single voxel:\n\\begin{multline*}\n \\mathbf{w}_i^{(\\ell)} = (1-\\alpha_i)\\mathbf{w}_i + \\mathbf{Q}_i\\mathbf{H}_i^{(\\ell)}\\;\\;\\textrm{where:}\\;\\; \\alpha_i = \\frac{1}{\\sqrt{s_i}}||\\mathbf{Q}_i^\\intercal \\mathbf{1}_K ||_2^2\\\\\n \\mathbf{H}_i^{(\\ell)}\\in\\mathbb{R}^{s_i},\\;\\;\\mathbf{H}_i = \\mathbf{I} - 2\\mathbf{h}\\mathbf{h}^\\intercal\\in\\mathbb{R}^{s_i\\times s_i},\\;\\;\\mathbf{h} \\in \\mathbb{R}^{s_i},\\;\\textrm{and}\\\\ s_i = \\max_{\\ell}\\Bigg\\{\\ell \\leq K- d, \\frac{\\sum_{p=K-\\ell+1}^K\\lambda_p^{(i)}}{\\sum_{p=1}^{K-\\ell}\\lambda_p^{(i)}} < \\eta \\Bigg\\}\n\\end{multline*}\n\\noindent where $\\mathbf{h} = \\frac{\\mathbf{h}_0}{||\\mathbf{h}_0||}$ if $\\mathbf{h}_0 = \\alpha_i\\mathbf{1}_{s_i} - \\mathbf{Q}_i^\\intercal\\mathbf{1}_K \\neq \\mathbf{0}\\in\\mathbb{R}^{s_i}$ (else $\\mathbf{h} = \\mathbf{h}_0 = \\mathbf{0}\\in\\mathbb{R}^{s_i}$), $\\eta = \\boldsymbol\\rho_{\\lceil V\/2 \\rceil}$, $\\boldsymbol{\\hat\\rho}_i = \\frac{\\sum_{p=d+1}^K\\lambda_p^{(i)}}{\\sum_{p=1}^d \\lambda_p^{(i)}}$, $\\boldsymbol\\rho = \\mathrm{sort}(\\boldsymbol{\\hat\\rho},\\mathrm{ascending})$, and $\\boldsymbol{\\hat\\rho},\\boldsymbol\\rho \\in\\mathbb{R}^{V}$.\nSince every measure's invariant properties are determined in a square-integrable space~\\cite{schmidt}, LLE performs a global least-squares optimisation based on Gau{\\ss}' Principle of Least Constraint~\\cite{gauss} to calculate the vectors $\\mathbf{z}_1,\\ldots, \\mathbf{z}_V$ corresponding to points on the manifold:\n\\begin{multline}\\label{eqn:llered}\n\\mathbb{E}[\\mathbf{Z}] = \\min_{ \\mathbf{Z} = [\\mathbf{z}_1,\\ldots,\\mathbf{z}_V]} \\sum_{i=1}^V \\sum_{\\ell=1}^{s_i} || \\mathbf{z}_{i} - \\sum_{j \\in \\mathcal{N}(i)} w_{i,j}^{(\\ell)}\\mathbf{z}_{j}||_2^2, \\;\\;\\;\\\\ \\textrm{ such that} \\;\\;\\; \\mathbf{Z}\\mathbf{Z}^\\intercal = \\mathbf{I}\n\\end{multline}\n\\noindent where $d \\leq T$ is the dimensionality parameter selected by the user and $\\mathbf{Z}\\in\\mathbb{R}^{V \\times d}$ is the two-dimensional reconstructed fMRI. The global optimisation therefore calculates the points on the manifold~\\cite{sherr,seunglee} that act as the four-dimensional orthogonal basis that best retains the geometry of the stress vectors. These represent the second order invariant properties of each voxel's Cauchy stress tensor. A detailed explanation of this optimisation is provided below.\n\n\\par Define $\\mathbf{\\hat{W}}_i \\in \\mathbb{R}^{V \\times s_i}$ as the local sparse adjacency matrix, where:\n\\begin{multline*}\n\\mathbf{\\hat{W}}_i(\\mathcal{N}(i),:) = \\mathbf{w}_i,\\;\\; \\mathbf{\\hat{W}}_i(i,:) = -\\mathbf{1}_{s_i}^\\intercal,\\;\\;\\textrm{and} \\\\\\mathbf{\\hat{W}}_i(j,:) = 0,\\;\\; \\forall j \\not\\in \\{\\mathcal{N}(i)\\cup i \\}\n\\end{multline*}\nThe optimisation in Equation~\\ref{eqn:llered} can be written as a minimisation of the expected reconstruction cost, or error:\n\\begin{equation}\n\\mathbb{E}[\\mathbf{Z}] = \\sum_{i=1}^V ||\\mathbf{Z}\\hat{\\mathbf{W}}_i||_2^2 = \\mathrm{trace}(\\mathbf{Z}\\sum_{i=1}^V\\hat{\\mathbf{W}}_i\\hat{\\mathbf{W}}_i^\\intercal\\mathbf{Z}^\\intercal) = \\mathrm{trace}(\\mathbf{Z}\\boldsymbol\\Phi\\mathbf{Z}^\\intercal)\n\\end{equation}\n\\noindent where $\\mathbf{\\hat{W}}_i\\mathbf{\\hat{W}}_i^\\intercal$ is the orthogonal projection for voxel $i$, and $\\boldsymbol\\Phi =\\sum_{i=1}^V\\mathbf{\\hat{W}}_i\\mathbf{\\hat{W}}_i^\\intercal$ is the sparse, symmetric and positive-definite \\textit{alignment matrix}, and therefore admits the eigendecomposition~\\cite{drmackay}:\n\n\\begin{equation}\n\\mathbf{Z}\\boldsymbol\\Phi\\mathbf{Z}^\\intercal=\\mathbf{Z}\\boldsymbol{\\Lambda}\\mathbf{Z}^\\intercal\n\\end{equation}\nwhere $\\boldsymbol{\\Lambda} \\in \\mathbb{R}^{(d+1) \\times (d+1)}$ is the diagonal matrix containing the ${d+1}$ smallest eigenvalues of $\\mathbf{Z}\\boldsymbol\\Phi\\mathbf{Z}^\\intercal$, and $\\mathbf{Z} \\in \\mathbb{R}^{V \\times (d+1)}$ are the corresponding eigenvectors; Rayleigh's variational principle~\\cite{rayleigh,courant} enables calculation of these bottom $(d+1)$ eigenvectors. Each eigenvector represents a degree of freedom in space and time, where the $(d+1)$ eigenvector is the global unit vector that fills three-dimensional space. The global unit vector is discarded to enforce the constraint that the manifolds have mean zero.\n\n\\noindent \\textbf{Note:} To avoid degenerate solutions, LLE requires the manifolds to be centred around the origin in \\textit{both} space and time -- i.e., $\\sum_{i}^V \\mathbf{Z}_{i,:} = \\mathbf{0}\\in\\mathbb{R}^d$ and $\\sum_{i}^d \\mathbf{Z}_{:,i} = \\mathbf{0} \\in\\mathbb{R}^{V}$-- \\textit{and} also have outer products with unit covariance -- i.e., $\\mathbf{Z}\\mathbf{Z}^\\intercal = \\mathbf{I}$. Centring the manifolds about the origin ensures they are of the same scale, which is superficially similar to the common practice of signal, or count rate, normalisation~\\cite{neuronuclear}. The unit covariance constraint imposes the requirement that the reconstruction errors of the extracted manifolds are measured on the same scale. \n\n\n\n\n\n\n\n\n\n\\begin{table*}[t]\n\\tabcolsep 0.25pt\n {\\fontsize{5.5}{7}\\selectfont\n \\centering\n \\caption[Caption for LOF]{Results.}\\label{tbl:results} \n \\begin{tabular}{rrrrrrrrrrrrrrrrrr}\n \\toprule\n \\multicolumn{1}{c}{\\multirow{2}[-5]{*}{Dataset}} & \\multicolumn{1}{c}{\\multirow{2}[-5]{*}{Partition}} & \\multicolumn{4}{c}{Specificity} & \\multicolumn{4}{c}{Sensitivity} & \\multicolumn{4}{c}{Precision} & \\multicolumn{4}{c}{Accuracy} \\\\\n \\midrule\n \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{Chance} & \\multicolumn{1}{c}{Original} & \\multicolumn{1}{c}{LLE} & \\multicolumn{1}{c}{PCA} & \\multicolumn{1}{c}{Chance} & \\multicolumn{1}{c}{Original} & \\multicolumn{1}{c}{LLE} & \\multicolumn{1}{c}{PCA} & \\multicolumn{1}{c}{Chance} & \\multicolumn{1}{c}{Original} & \\multicolumn{1}{c}{LLE} & \\multicolumn{1}{c}{PCA} & \\multicolumn{1}{c}{Chance} & \\multicolumn{1}{c}{Original} & \\multicolumn{1}{c}{LLE} & \\multicolumn{1}{c}{PCA} \\\\\n{Beijing} & Training & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{73.3\\% $\\pm$ 15.8\\%} & \\multicolumn{1}{l}{80\\%\\;\\,\\, $\\pm$ 14.3\\%} & \\multicolumn{1}{l}{66.7\\% $\\pm$ 16.9\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{53.3\\% $\\pm$ 17.9\\%} & \\multicolumn{1}{l}{93.3\\% $\\pm$ 8.9\\%} & \\multicolumn{1}{l}{80\\% \\;\\;\\,$\\pm$ 14.3\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{66.7\\% $\\pm$ 16.9\\%} & \\multicolumn{1}{l}{82.4\\% $\\pm$ 13.6\\%} & \\multicolumn{1}{l}{70.6\\% $\\pm$ 16.3\\%} & \\multicolumn{1}{l}{50\\% \\;\\;\\,$\\pm$ 17.9\\%} & \\multicolumn{1}{l}{63.3\\% $\\pm$ 17.2\\%} & \\multicolumn{1}{l}{\\textbf{86.7\\% $\\pm$ 12.2\\%}} & \\multicolumn{1}{l}{73.3\\% $\\pm$ 15.8\\%} \\\\\n (Peking\\_3) & Holdout & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{75\\%} & \\multicolumn{1}{l}{87.5\\%} & \\multicolumn{1}{l}{25\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{50\\%} & \\multicolumn{1}{l}{25\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{66.7\\%} & \\multicolumn{1}{l}{66.7\\%} & \\multicolumn{1}{l}{14\\%} & \\multicolumn{1}{l}{67\\%} & \\multicolumn{1}{l}{83.3\\%} & \\multicolumn{1}{l}{\\textbf{75\\%}} & \\multicolumn{1}{l}{25\\%} \\\\\n \\multirow{2}[0]{*}{COBRE} & Training & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{57.7\\% $\\pm$ 13.6\\%} & \\multicolumn{1}{l}{92.3\\% $\\pm$ 7.3\\%} & \\multicolumn{1}{l}{96.2\\% $\\pm$ 5.3\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{60\\%\\;\\;\\, $\\pm$ 13.4\\%} & \\multicolumn{1}{l}{88\\% \\;\\;\\,$\\pm$ 8.9\\%} & \\multicolumn{1}{l}{52\\%\\;\\;\\, $\\pm$ 13.7\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{57.7\\% $\\pm$ 13.6\\%} & \\multicolumn{1}{l}{91.7\\% $\\pm$ 7.6\\%} & \\multicolumn{1}{l}{92.9\\% $\\pm$ 7.1\\%} & \\multicolumn{1}{l}{51\\%\\;\\;\\, $\\pm$ 13.7\\%} & \\multicolumn{1}{l}{54.9\\% $\\pm$ 13.7\\%} & \\multicolumn{1}{l}{\\boldmath{}\\textbf{90.2\\% $\\pm$ 8.2\\%}\\unboldmath{}} & \\multicolumn{1}{l}{74.5\\% $\\pm$ 6.1\\%} \\\\\n & Holdout & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{60\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{70\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{60\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{63.6\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{50\\%} & \\multicolumn{1}{l}{65\\%} & \\multicolumn{1}{l}{\\textbf{100\\%}} & \\multicolumn{1}{l}{80\\%} \\\\\n {Mind Research} & Training & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{88.2\\% $\\pm$ 11.5\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{84.6\\% $\\pm$ 12.9\\%} & \\multicolumn{1}{l}{23.1\\% $\\pm$ 15.1\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{84.6\\% $\\pm$ 12.9\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{56.7\\% $\\pm$ 17.7\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{\\boldmath{}\\textbf{86.7\\% $\\pm$ 12.2\\%}\\unboldmath{}} & \\multicolumn{1}{l}{66.7\\% $\\pm$ 16.9\\%} \\\\\n Network (MRN) & Holdout & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{67\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{71\\%} & \\multicolumn{1}{l}{14\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{63\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{56\\%} & \\multicolumn{1}{l}{N\/A} & \\multicolumn{1}{l}{\\textbf{68.8\\%}} & \\multicolumn{1}{l}{62.5\\%} \\\\\n \\multirow{2}[0]{*}{Stanford} & Training & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{73.3\\% $\\pm$ 15.8\\%} & \\multicolumn{1}{l}{80\\%\\;\\,\\, $\\pm$ 14.3\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{66.7\\% $\\pm$ 16.9\\%} & \\multicolumn{1}{l}{86.7\\% $\\pm$ 12.2\\%} & \\multicolumn{1}{l}{33.3\\% $\\pm$ 16.9\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{71.4\\% $\\pm$ 16.2\\%} & \\multicolumn{1}{l}{81.3\\% $\\pm$ 14\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{50\\%\\;\\;\\, $\\pm$ 17.9\\%} & \\multicolumn{1}{l}{70\\% \\;\\;\\,$\\pm$ 16.4\\%} & \\multicolumn{1}{l}{\\boldmath{}\\textbf{83.3\\% $\\pm$ 13.3\\%}\\unboldmath{}} & \\multicolumn{1}{l}{66.7\\% $\\pm$ 16.9\\%} \\\\\n & Holdout & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{80\\%} & \\multicolumn{1}{l}{80\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{40\\%} & \\multicolumn{1}{l}{60\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{67\\%} & \\multicolumn{1}{l}{75\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{50\\%} & \\multicolumn{1}{l}{60\\%} & \\multicolumn{1}{l}{\\textbf{70\\%}} & \\multicolumn{1}{l}{50\\%} \\\\\n University of & Training & \\multicolumn{1}{l}{100\\%} & 69.2\\% $\\pm$ 20.2\\% & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{42.9\\% $\\pm$ 21.7\\%} & \\multicolumn{1}{l}{71.4\\% $\\pm$ 19.8\\%} & \\multicolumn{1}{l}{14.3\\% $\\pm$ 15.3\\%} & \\multicolumn{1}{l}{0\\%} & 42.9\\% $\\pm$ 21.7\\% & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{100\\%} & 65\\% \\;\\;\\,$\\pm$ 20.9\\% & 60\\%\\;\\;\\, $\\pm$ 21.5\\% & \\boldmath{}\\textbf{90\\%\\;\\;\\, $\\pm$ 13.1\\%}\\unboldmath{} & 70\\%\\;\\;\\, $\\pm$ 20.1\\% \\\\\n Michigan (UM\\_2) & Holdout & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{66.6\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{83.3\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{25\\%} & \\multicolumn{1}{l}{50\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{20\\%} & \\multicolumn{1}{l}{100\\%} & \\multicolumn{1}{l}{0\\%} & \\multicolumn{1}{l}{60\\%} & \\multicolumn{1}{l}{30\\%} & \\multicolumn{1}{l}{\\textbf{80\\%}} & \\multicolumn{1}{l}{50\\%} \\\\\n \\bottomrule\n \\end{tabular\n }\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n\n\n\\end{table*}\n\n\\begin{table*}[h]\n \n \\centering\n \\caption{Dataset Summary} \\label{tbl:dsinfo}%\n \\footnotesize\n \\tabcolsep 1pt\n {\\fontsize{6.75}{7}\\selectfont\n \\begin{tabular}{rrrrrrrrrrrcrrr}\n \\toprule\n \\multirow{2}[0]{*}{Dataset} & \\multicolumn{1}{c}{Experimental} & \\multirow{2}[0]{*}{Pulse Sequence} & \\multicolumn{1}{c}{\\multirow{2}[0]{*}{Patient Disorder}} & \\multicolumn{1}{c}{\\multirow{2}[0]{*}{Source}} & \\multicolumn{3}{c}{\\multirow{2}[-5]{*}{Training Data}} & \\multicolumn{3}{c}{\\multirow{2}[-5]{*}{Holdout Data}} & \\multicolumn{1}{c}{\\multirow{2}[-5]{*}{fMRI Dimensionality}} & \\multicolumn{2}{c}{\\multirow{2}[-5]{*}{Parameters}} & \\multicolumn{1}{c}{\\multirow{2}[-5]{*}{Age Range}} \\\\\n &\\multicolumn{1}{c}{Design} & & & & total & patients & controls & total & patients & controls & ($L \\times W \\times H \\times T $) & LLE ($r$,$d$) & PCA ($d$) & [min,mean,max] \\\\\n \\toprule\n Beijing (Peking\\_3) & resting-state & EPI & ADHD & ADHD200 & 30 & 15 & 15 & 12 & 4 & 8 & $57\\times 68 \\times 42 \\times 236$ & (2,236) & 236 & [11, 13.24, 16] \\\\\n COBRE & resting-state & EPI & Schizophrenia & COBRE & 51 & 25 & 26 & 20 & 10 & 10 & $53 \\times 62 \\times 52 \\times 150$ & (2,150) & 150 & [18, 37.45, 65] \\\\\n Mind Research Network (MRN) & block-design & EPI & Schizophrenia & MCIC & 30 & 13 & 17 & 16 & 7 & 9 & $57 \\times 68 \\times 38 \\times 177$ & (2,177) & 177 & [18, 34.15, 60] \\\\\n Stanford & resting-state & Spiral-IN\/OUT EPI & Autism & ABIDE & 30 & 15 & 15 & 10 & 5 & 5 & $63 \\times 75 \\times 42 \\times 180$ & (2,60) & 60 & [7.52, 9.95, 12.93] \\\\\n University of Michigan (UM\\_2) & resting-state & Spiral-IN EPI & Autism & ABIDE & 20 & 7 & 13 & 10 & 4 & 6 & $57 \\times 68 \\times 63 \\times 300$ & (2,222) & 222 & [13.1, 16.26, 28.8] \\\\\n \\bottomrule\n \\end{tabular\n}\n\\end{table*\n\n\\subsection[Feature \\& Parameter Selection]{Feature \\& Parameter Selection}\\label{app:fs}\n\\textit{Hyper-parameter Selection} involves selecting the algorithm parameters that will be used to generate the reconstructed fMRI from LLE and PCA, respectively. Both PCA and LLE require one to specify the number of reconstruction dimensions $d$, where LLE's additional hyper-parameter, $K=(1+2r)^3-1$, selects the neighbouring voxels whose coordinates are on the boundary of a sphere with radius $r \\in\\mathbb{Z}_{+},r\\geq1$ that is centred around the respective voxel. For both PCA and LLE, the number of reconstruction dimensions is bounded by the number of time points. In the case of PCA, the optimal \\textit{d} is often chosen by the proportion of variance captured by the first \\textit{d} eigenvectors, expressed as $\\frac{\\sum_{i=1}^d\\lambda_i}{\\sum_{j=1}^T\\lambda_j}$. For LLE, however, there is no analogous interpretation because the \\textit{d} eigenvectors are uniformly spaced ``time points\" on the {\\it world line}~\\cite{mink}. Thus cross-validation procedures are implemented (see following subsection for details) on the training data to systematically select the best hyper-parameters, iterating over $d \\in\\{1,\\ldots,T\\}$, where $d$ is generated on a log-scale from 1 to $T$. \n\n\n \\paragraph[]{Sequential Forward Selection (SFS)\\footnote{Implemented using MATLAB's (R2013a) \\textit{sequentialfs} function}~\\cite{sfs}} is a nonparametric method for measurement (feature) selection that starts with an empty ``candidate\", or near-optimal, set of ``time points\". The method first finds the ``time point\", defined as the second moment~\\cite{bills}, with the highest classification accuracy and adds the corresponding image volume to the set. The method then finds an \\textit{additional} ``time point\" \\comments{\\deleted{(or volume)}} that \\textit{strictly} improves the classification accuracy in \\textit{conjunction} with the volume(s) whose ``time points\" are already in the ``candidate\" set, and terminates when no such volume can be found. Thus, SFS produces a near-optimal set of ``time points\" (volumes) that distinguish the subject groups with high classification accuracy, where this set is determined by adding ``time points\" to the near optimal set in a one-by-one fashion. It follows that the near-optimal set of ``time points\" represent points in time during the scan that enable discrimination of patients from controls with high classification accuracy. Note that SFS is only used on the training data (see Section~\\ref{sect:evalcrit}). \n\n\\subsection{Classifier}\nA slight abuse of notation is introduced by redefining variable $\\mathbf{z} \\in \\mathbb{R}^{Vc}$ as the one-dimensional representation of the \\textit{c} diagnostic volumes from the respective subject's reconstructed fMRI $\\mathbf{Z}$ produced in the previous step. Here, it is assumed there are \\textit{c} diagnostic time points and that $\\mathcal{Z}$ is a random variable containing the collection of cohorts' reconstructed fMRI.\n\n\\paragraph[]{Fisher's Linear Discriminant (LDA)\\footnote{Implemented using MATLAB's (R2013a) \\textit{classify} function with the `diaglinear' argument to estimate the positive diagonal covariance matrix}~\\cite{fisher1,fisher2}} is a linear classification rule~\\cite{htf} that assumes both the patient and control class densities (at each voxel location) can be represented as multivariate Gaussians in three-dimensional space, each with some intrinsic curvature~\\cite{gauss2,gauss3}. Each class density is expressed as\n\\begin{equation*}\n{f_k(\\mathbf{z}) = \\frac{1}{(2\\pi)^{\\frac{Vc}{2}}|\\boldsymbol\\Sigma_k|^{\\frac{1}{2}}}}{\\Large e^{-\\frac{1}{2}(\\mathbf{z}-\\boldsymbol{\\mu}_k)^\\intercal\\boldsymbol\\Sigma_k^{-1}(\\mathbf{z}-\\boldsymbol{\\mu}_k)}}\n\\end{equation*}\n\\noindent where it is assumed the $k$ classes have a common covariance matrix-- i.e., $\\boldsymbol\\Sigma_k = \\boldsymbol\\Sigma,\\,\\forall k$. This assumption allows the log ratio between the posterior distribution of each class to form a decision boundary that lies between patients (class 1) and controls (class 0), written as $P(Y=0|\\mathcal{Z}=\\mathbf{z}) = P(Y=1|\\mathcal{Z}=\\mathbf{z})$, which is linear in $\\mathbf{z}$. LDA therefore calculates the $(Vc)$-dimensional hyperplane that best discriminates, or separates, the diagnostic volumes that have been determined using SFS with LOOCV on the respective dataset, for each class' reconstructed fMRI. It follows these diagnostic volumes contain spatial locations with sufficiently different patient and control class densities, where statistically-significantly different regions possess the requisite margin between the class densities such that they are perceptible~\\cite{neuronuclear}.\n\n\n\\subsection{Evaluation Criteria}\\label{sect:evalcrit}\n\\par Each dataset was split such that the proportion of patients in the training partition was near-equal to the proportion of controls, and the holdout dataset contained at least 10 cohorts. In all but two cases, the aforementioned criteria could not be met because the datasets were unbalanced. For these situations, the training dataset was constructed such that it was a representative sample of the overall data, with the remaining cohorts being assigned to the holdout dataset. The holdout set therefore contained group proportions that could deviate from the training set. The cohorts used to define the training and holdout partition for each dataset are provided in Table 3 of the Supplementary Information\n\n\\par Performance was compared to the original fMRI, both before and after applying PCA for artefact reduction~\\cite{nikos2}, and chance, which is defined as the proportion of the majority class on the respective dataset. The performance of the reconstruction methods are evaluated using a combination of Leave-One-Out Cross Validation (LOOCV) and holdout data classification performance~\\cite{htf}. LOOCV is used on the training data to find both the best reconstruction parameter $d$, and the diagnostic volumes that produce the highest accuracy for this $d$. The holdout data classification accuracies use the parameters found from performing LOOCV on the training data. Note: the cohorts in the holdout set are \\textit{never} involved in determining the optimal hyper-parameter $d$, \\textit{or} the diagnostic volumes produced from this $d$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[htp!]\n\\centering\n\\begin{subfigure}[t]{7.25in}\n\\centering\n\\includegraphics{fig3a_cobremaps.pdf}\n\\subcaption{Center of Biomedical Research Excellence (COBRE). Cohorts were at rest for the duration of the scan.}\\label{subfig:cobrevol}\n\\end{subfigure}\\\\\n\\vfill \\begin{subfigure}[t]{7.25in}\n\\centering\n\\includegraphics{fig3b_unmmap.pdf}\n\\subcaption{Mind Research Network (MRN). Cohorts performed the Sternberg Item Recognition Paradigm (SIRP) task.}\\label{subfig:mrnvol}\n\\end{subfigure}\n\\caption{\\small Statistical maps illustrating the individual differences in mental activity (schizophrenic patients versus healthy controls) for the discriminative time points determined on the training partition. }\\label{fig:schiz}\n\\end{figure*} \n\\begin{figure*}[ht!]\n\n\\begin{subfigure}[t]{7.25in}\n\\centering\n\\includegraphics{fig4_pek3maps.pdf}\n\\subcaption*{Beijing (Peking\\_3) University. Cohorts were at rest for the duration of the scan.}\n\\end{subfigure}\n\\caption{\\small Statistical maps illustrating the individual differences in mental activity (ADHD patients versus healthy controls) for the discriminative time points determined on the training partition.}\\label{fig:adhdvol}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\centering\n\\begin{subfigure}[t]{7.25in}\n\\centering\n\\includegraphics{fig5a_stanmaps.pdf}\n\\subcaption{Stanford University. Cohorts were at rest for the duration of the scan.}\\label{subfig:stanvol}\n\\end{subfigure}\n\\begin{subfigure}[t]{7.25in}\n\\centering\n\\includegraphics{fig5b_um2maps.pdf}\n\\subcaption{University of Michigan (UM\\_2). Cohorts were at rest for the duration of the scan.}\\label{subfig:um2vol}\n\\end{subfigure}\n\\caption{\\small The statistical maps illustrating the individual differences in mental activity (ASD patients versus healthy controls) for the discriminative time points determined on the training partition.} \\label{fig:asdvol}\n\\end{figure*}\n\n\\section{Results}\n\\subsection*{Performance \\& Visualisation}\n\\par Tables~\\ref{tbl:results} and \\ref{tbl:dsinfo} demonstrate that Locally Linear Embedded fMRI can distinguish various mental disorders from healthy controls with high discriminatory power ($>$ 80\\%); results for six additional resting-state datasets are provided in Table 2 of the Supplementary Information. The datasets contained healthy controls and patients with either Schizophrenia, Attention-Deficit Hyperactivity Disorder (ADHD), or Autism Spectrum Disorder (ASD). Ten of these datasets contained cohorts in the resting-state, with the remaining containing schizophrenic patients and healthy controls performing the Sternberg Item Recognition Paradigm (SIRP) task~\\cite{sternberg}. For a description of the data sources and technical details, please consult the appendix.\n\n Given that performance on the training partition uses Leave One Out Cross Validation (LOOCV) to individually predict each subject's diagnosis, which is analogous to performing \\textit{n} Bernoulli trials~\\cite{bernoulli} (where \\textit{n} is the number of cohorts), the training data's performance metrics can be interpreted as the mean of successes over \\textit{n} binomially-distributed observations. To calculate the error of these estimates, we follow Laplace's approach of employing a normal distribution to estimate the error of binomially-distributed observations~\\cite{laplace}. Given that discrimination performance on the holdout partition is determined in a one-time fashion, variance estimates are not applicable. \n\n\\par The Harvard-Oxford Subcortical\/Cortical and Cerebellum atlases~\\cite{fsl} are used to identify the statistically-significant differences between patients and controls in the diagnostic volumes for the respective dataset, shown in Figures~\\ref{fig:schiz},~\\ref{fig:adhdvol}, and~\\ref{fig:asdvol} (Figures 1, 2, 3, 4, 5, and 6 for datasets in the Supplementary Information). Each dataset's figure contains six different views of the statistically-significant differences at the time reflected by the respective time point(s); the coloured voxels at these time points denote statistically significant (\\textit{p} < 0.05 uncorrected) physical differences between the patient and control groups, where these groups include cohorts from \\textit{both} the training and holdout partitions. The proportion of significantly different voxels in each region, calculated by dividing the number of significantly different voxels by the total number of voxels in the respective region defined by the atlas, are provided in Tables 4 and 5 of the Supplementary Information.\n\n\\subsection*{Neurobiological Interpretation}\n\n\\paragraph{Schizophrenia}\n\n\\par David Ingvar \\& G\\\"oran Franz\\`en found that healthy controls exhibit increased flows in the prefrontal regions and decreased flow in the post central regions, and schizophrenic patients exhibited the reversed pattern, with low flows prefrontally and high flows postcentrally~\\cite{lancetschiz}. Furthermore, they noticed that a lower flow in the premotor and frontal regions was associated with symptoms of indifference, activity and autism, and a higher postcentral flow over the temporo-occipito-parietal regions was associated with disturbed cognition~\\cite{lancetschiz}. Inspecting the statistical maps for the volumes in Figure~\\ref{fig:schiz}, the significantly different areas seem to further substantiate Ingvar \\& Franz\\`en's observations, as there are significant differences in the various temporo-occipital-parietal regions in all of the volumes. \n\n\n\n\n\n\\par The Sternberg Item Recognition Paradigm (SIRP) task evaluates cohorts' short-term, or working, memory. Each of the seven tasks during the scan involves the subject memorising a set of objects, followed by presentation of a new object whose membership in this set is identified by a `yes` or `no`. The tasks therefore evaluate the hypothesised information processing differences between schizophrenics and healthy controls~\\cite{classcalhoun}. It has been shown that the prefrontal and medial temporal regions are involved in encoding information, and it is believed the interactions between these regions are central to retrieval of stored information~\\cite{memoryreview}. Figure~\\ref{subfig:mrnvol}, especially volume 136, illustrates significant differences in the areas associated with the prefrontal and medial temporal regions; this is further supported by the fact that the accuracy on the holdout data rose from 68.8\\% to 75\\% when using only volume 136. Thus, it is possible that schizophrenia indeed affects the physical mechanisms associated with retrieving stored information, as these mechanisms are central to the SIRP task. The reconstruction method therefore successfully reveals physical differences associated with task performance between patients and controls, which are different from the resting-state differences for the same subject groups. \n\n\n\n\n\n\n\\paragraph{Attention-Deficit Hyperactivity Disorder (ADHD)}\n\t\n\t\n\n\n\n\t\n\\par Similar to the goals of Ingvar \\& Franz\\`en~\\cite{lancetschiz}, previous work used PET scans to compare the regional cerebral blood flow of children with Attention-Deficit Hyperactivity Disorder (ADHD) to healthy controls, where it was found that the disorder was associated with hypoperfusion in the striatal and posterior periventricular regions~\\cite{lancetadhd}; these results provide biological evidence that is consistent with the canonical model for ADHD as a fronto-striatal deficient disorder. Figure~\\ref{fig:adhdvol} shows significant differences in the various occipital, striatal, cerebellar and ventral regions of the brain. \n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[htp!]\n\\centering\n\\begin{subfigure}[b]{2in}\n\\centering\n\\includegraphics[scale=0.85]{fig1a_tensor.pdf}\n\\subcaption{\\small An illustration~\\cite{clrs} of the Cauchy stress tensor of a spherical body with radius $r=1$.}\n\\end{subfigure}\\quad\\quad\\quad\n\\begin{subfigure}[b]{4.5in}\n\\centering\n\\includegraphics[scale=0.85]{fig1b_spatialpatterns.pdf}\n\\subcaption{\\small The relative spatiotemporal patterns represent the surface forces relative to the \\textit{deflexion axis}~\\cite{dadmackay1,dadmackay2} at voxel $i$. The stress vectors (Lebesgue measures~\\cite{lebesgue}) are the minimum-norm solution to the unitarily-constrained least-squares problem $\\mathbf{G}_i\\mathbf{w}_i=\\mathbf{1}_{|\\mathcal{N}(i)|}$, where $\\mathbf{G}_i \\in\\mathbb{R}^{|\\mathcal{N}(i)|\\times|\\mathcal{N}(i)|}$ is the local spatiotemporal covariance matrix (Lebesgue, $\\ell^2$, or square-integrable~\\cite{schmidt} space).}\n\\end{subfigure} \n\\caption{The local geometry of the Cauchy stress tensor and its relative spatiotemporal patterns on topology~\\cite{bourbaki} $\\mathcal{X}$, which is defined on the Cartesian space with the Pythagorean distance metric.}\n\\label{fig:blackbody}\n\\end{figure*}\n\\paragraph{Autism Spectrum Disorder (ASD)}\n\tSimilar to Ingvar \\& Franz\\`en's observations that schizophrenic patients and healthy controls had normal hemisphere flows~\\cite{lancetschiz}, studies using PET to compare the regional cerebral blood flow of Autism Spectrum Disorder (ASD) patients to healthy controls observed normal metabolism and blood flow. Hypoperfusion in the temporal lobes, centred in the associative auditory and adjacent multimodal cortex~\\cite{autismpsych}, was observed in autistic children. Furthermore, this temporal hypoperfusion was individually identifiable in 75\\% of autistic children~\\cite{autismpsych}. Figure~\\ref{fig:asdvol} illustrates statistical differences in many areas of the temporal lobe. \n\n\n\t\n\n\n\t\n\n\nIn summary, Locally Linear Embedding appears to have conserved spatiotemporal patterns in resting-state fMRI (and task-related responses) that are consistent with literature on regionally-specific abnormalities of cerebral activity in the psychiatric conditions used to assess classification performance.\n\n\n\\section{Discussion}\n\n\\par The multidisciplinary nature of this work undoubtedly introduces difficulty when discussing its motivations, which is the deployment of this methodology in a clinical setting. Such a goal imposes some conditions. First, while the results support deployment, the methodology must be further evaluated by trained clinicians well-versed in the etymology of the disease under investigation. More importantly, however, experimental designs are compulsory when discovering biological markers for disease; that is, patients must be subject to the same stimulus presentation at the same time during the scan in order to homogenise comparisons. With respect to resting-state fMRI, it is felt that a consensus is required to glean neurobiological insight that can generalise, which makes further discussion more appropriate for future work.\n\n\\par This exposition focused on Locally Linear Embedding as a promising and effective form of dimensionality reduction as a pre-processing step for the analysis of fMRI time-series. The approaches and aims of this form of pre-processing share a close relationship with other approaches in imaging neuroscience. These approaches include Independent Component Analysis, the use of Support Vector Machines (and regression) to classification problems (and prediction), and algorithms based on adaptive smoothing. In future work, it will be interesting to explore the formal connections between other approaches and assess their relative sensitivity in the context of the classification problems considered above.\n\nOne hundred and fourty-eight years after Darwin ascertained that mental activity invokes physical mechanisms in the brain~\\cite{darwin1871descent}, the brain of man and ant alike are among the marvellous collection of atoms in the world.\n\n\n\n\n\n\n\n{\\setcounter{secnumdepth}{-1}\n\\section{Acknowledgements}\n\n\\par This work is dedicated to the late Sam Roweis (1972-2010), Donald MacCrimmon MacKay (1922-1987) and his son David John Cameron MacKay (1967-2016). Additionally, many thanks to colleagues Ruitong Huang \\& Dr. Csaba Szepesv\\`ari, Joshua T. Vogelstein, Dr. Vincent D. Calhoun, Dr. Neil Lawrence, Dr. Bert Vogelstein, Dr. Michael Milham, Dr. Karl J. Friston, Dr. Klaas Enno Stephan, Dr. Nikos K. Logothetis, Dr. Christof Koch, and Dr. Yifan Hu of AT\\&T Labs' Information Visualisation department for his assistance with GraphViz, which was used to depict the Cauchy Stress Tensor in three dimensions. \n\n\n\n\n\n\n\n}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgexs b/data_all_eng_slimpj/shuffled/split2/finalzzgexs new file mode 100644 index 0000000000000000000000000000000000000000..6fd1b138832ced89a47d39876533723619e0df63 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgexs @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\IEEEPARstart{S}{elf-localization} is an essential capability of autonomous mobile robots, and localization algorithms based on inexpensive commercial vision sensors have become useful and widespread.\nDespite this success, long-term \\emph{metric} localization, where the goal is to continuously estimate the \\mbox{6-dof} pose of the vehicle with respect to a visual map, remains challenging in the presence of appearance change caused by illumination variations over the course of a day, changes in weather conditions, or seasonal shifts.\nThis difficulty is largely due to simplifying assumptions such as brightness constancy and feature descriptor invariance that, when violated, cause visual localization systems to fail.\nIdeally, we would like our systems to function across this `appearance gap', immune to variations in environmental conditions.\n\nLong-term maps based on multiple visual `experiences' of an environment have proven to be effective tools for metric localization through daily and seasonal appearance change~\\cite{Churchill2013-ng,Linegar2015-xs,Paton2016-bz,Paton2018-qa}.\nIn~\\cite{Paton2016-bz}, consecutive visual experiences are recorded in a spatio-temporal pose graph, and localization against a privileged mapping experience proceeds by recalling a relevant experience and tracing through a chain of relative transformations in the graph.\nThis process is often aided by a prior on the vehicle's topological location in the graph, whether from dead reckoning, place recognition, or GNSS, which serves to limit the number of candidate vertices for metric localization.\nHowever, the number of intermediate `bridging' experiences required for reliable long-term localization can be very large, and methods for compressing experience graphs are necessary for this approach to scale.\n\nRecent work in~\\cite{Clement2018-vm,Porav2018-xb} has explored deep image-to-image translation~\\cite{Isola2017-uc,Zhu2017-qc} as a means of directly bridging the `appearance gap' and localizing with fewer experiences.\nHowever these methods rely at least in part on well-aligned training images, which are difficult to obtain at scale in the real world.\nMoreover, the losses used to train these models are not explicitly connected to a target localization pipeline, and provide few assurances that the learned image transformations will ultimately improve localization performance.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.97\\textwidth]{overview}\n \\caption{We learn an image transformation that improves visual feature matching performance over day-night cycles by maximizing the response of a differentiable proxy network trained to predict the number of inlier feature matches returned by a conventional non-differentiable feature detection\/matching algorithm. The learned transformation can then be used as a pre-processing step in a visual localization pipeline to improve its robustness to appearance change.}\n \\label{fig:pipeline}\n \\vspace{-12pt}\n\\end{figure*}\n\nWe address these limitations by learning an image transformation optimized for a given combination of localization pipeline, sensor, and operating environment.\nRather than translating between arbitrary appearance conditions, we learn to map images to a \\emph{maximally matchable} representation (i.e., one which maximizes the number of inlier feature matches) for a given feature detection\/matching algorithm.\nSpecifically, we learn a drop-in replacement for the standard RGB-to-grayscale colorspace mapping used to pre-process RGB images for use with conventional feature detection\/matching algorithms, which typically operate on single-channel images (\\Cref{fig:pipeline}).\nThis formulation builds upon prior work on color constancy theory~\\cite{Ratnasingam2010-cr}, does not require well-aligned images for training, and naturally admits a self-supervised training approach as training targets can be generated on the fly by the localization pipeline.\nOur main contributions are:\n\\begin{enumerate}\n \\item a technique for improving the robustness of a conventional visual localization pipeline to appearance change using a learned image pre-processing step;\n \\item a method for optimizing the performance of a non-differentiable localization pipeline by approximating the pipeline using a deep neural network;\n \\item experimental results on synthetic and real long-term vision datasets showing that our method enables continuous 6-dof metric visual localization across day-night cycles using a single mapping experience; and\n \\item an open-source implementation of our method using PyTorch~\\cite{paszke2017automatic}.\\footnote{\\url{github.com\/utiasSTARS\/matchable-image-transforms}}\n\\end{enumerate}\n\n\\section{Related Work}\nAppearance robustness in metric visual localization has previously been studied from the perspective of illumination invariance, with methods such as~\\cite{Clement2017-gx, Corke2013-hl, McManus2014-op, Paton2017-fi} making use of hand-engineered image transformations to improve feature matching over time for a given feature detector and descriptor.\nSimilarly, affine models and other simple analytical transformations have been used to improve the robustness of direct visual localization to illumination change~\\cite{Engel2015-il,Park2017-zx}.\nOther approaches such as~\\cite{McManus2015-vj,Linegar2016-cn,Krajnik2017-tz,Zhang2018-ib} have focused on learning feature descriptors that are robust to certain types of appearance change in autonomous route following applications.\nHowever, \\cite{McManus2015-vj,Linegar2016-cn} produce correspondences that are only weakly localized, and \\cite{Krajnik2017-tz,Zhang2018-ib} require sets of true and false point correspondences to train feature descriptors, which are challenging to obtain at scale over long periods.\n\nDeep image-to-image translation~\\cite{Isola2017-uc,Zhu2017-qc} has recently been applied to the problem of metric localization across appearance change.\nIn~\\cite{Gomez-Ojeda2018-ai} the authors train a convolutional encoder-decoder network to enhance the temporal consistency of image streams captured in environments with high dynamic range.\nHere the main source of appearance change is the camera itself as it automatically modulates its imaging parameters in response to the local brightness of a static environment.\nOther work has tackled the problem of localization across \\emph{environmental} appearance change, with \\cite{Clement2018-vm} learning a many-to-one mapping onto a privileged appearance condition and \\cite{Porav2018-xb} learning multiple pairwise mappings between appearance categories such as day and night.\nImage-to-image translation has also been applied to the related task of appearance-invariant place recognition~\\cite{Latif2018-ui,Anoosheh2019-fc}, which typically relies on patch matching or whole-image statistics to identify images corresponding to nearby physical locations rather than estimating the 6-dof pose of the vehicle.\nWhile \\cite{Porav2018-xb,Gomez-Ojeda2018-ai} include loss terms to maximize gradient information, these heuristics are not directly tied to the performance of the localization pipeline.\nMoreover, \\cite{Clement2018-vm,Porav2018-xb,Gomez-Ojeda2018-ai} require well-aligned training images exhibiting appearance variation, which are difficult to obtain at scale in the real world, and it is not clear how categorical appearance mappings such as \\cite{Porav2018-xb,Latif2018-ui,Anoosheh2019-fc} should be applied to continuous appearance change in long-term deployments.\n\nSurrogate-based methods for approximating computationally expensive or non-differentiable objective functions are common in the numerical optimization literature~\\cite{Koziel2011}.\nNeural network surrogates in particular have found applications in a variety of domains including computer graphics~\\cite{Grzeszczuk:1998:FNN:3009055.3009178} and computational oceanography~\\cite{VANDERMERWE2007462}, where high-fidelity physics simulations are available but expensive to compute.\nOur method of learning a differentiable loss function is similar in spirit to Generative Adversarial Networks (GANs)~\\cite{Goodfellow2014-df} in that a complex discriminator\/loss function is trained using a comparatively simple analytical loss function.\nIt also bears resemblances to perceptual losses~\\cite{Johnson2016-lu}, where the loss function is derived from the feature activations of a network trained on a proxy task such as image classification.\n\n\\section{Learning Matchable Colorspace Transformations}\nOur goal in this work is to learn a nonlinear transformation $f: \\Real^3 \\rightarrow \\Real $ mapping the RGB colorspace onto a grayscale colorspace that explicitly maximizes a chosen performance metric of a vision-based localization pipeline.\nWe investigate two approaches to formulating such a mapping: 1) a single function to be applied as a pre-processing step to all incoming images, similarly to \\cite{Clement2017-gx,McManus2014-op,Paton2017-fi}; and 2) a parametrized function tailored to the specific image pair to be used for localization, where the parameters of this function are derived from the images themselves.\nAdditionally, the functional form of either mapping may be specified analytically (e.g., from physics) or learned from data using a function approximator such as a neural network.\n\nIn order to find an optimal colorspace transformation for a given application, we require an appropriate objective function to optimize, which should ideally be tied to the performance of the target localization pipeline.\nAn intuitive choice of objective could be to directly minimize pose estimation error for the entire pipeline relative to ground truth if it is available.\nIn the absence of accurate ground truth data, we might instead choose to maximize the number or quality of feature matches in the front-end of a feature-based localization pipeline.\nWe adopt the latter approach in this work, since high-quality 6-dof ground truth is difficult to obtain over long time scales.\n\nAlthough it is straightforward to choose a target performance metric to optimize, the most commonly used localization front-ends in robotics rely on non-differentiable components such as stereo matching, nearest-neighbors search, and RANSAC~\\cite{Fischler1981-ue}, which are incompatible with the gradient-based optimization schemes commonly used in deep learning.\nIn this work we \\emph{learn} an objective function by training a deep convolutional neural network (CNN) to act as a \\emph{differentiable proxy} to the localization front-end.\nSpecifically, we train a siamese CNN to predict the number of inlier feature matches for a given image pair, where the training targets are generated using a conventional non-differentiable feature detector\/matcher algorithm based on \\texttt{libviso2} features~\\cite{Geiger2011-xe}.\nThis proxy network can then be used to define a fully differentiable objective function, allowing us to train a nonlinear colorspace mapping using gradient-based methods.\nFinally, the trained image transformation can be used as a pre-processing step in a conventional visual localization pipeline, enabling it to operate more reliably under appearance change.\n\\Cref{fig:pipeline} summarizes our full data pipeline pictorially.\n\n\\subsection{Differentiable Matcher Proxy} \n\\label{sec:matcher}\n\nWe consider the task of training a CNN $\\MatcherNet$, with parameters $\\MatcherNetParams$, to predict the number of inlier feature matches returned by a non-differentiable feature detector\/matcher $\\Matcher$ for a given image pair $ (\\Image_1, \\Image_2) $.\nThis training objective is a convenient choice for our intended application as it is closely tied to the ability of our visual localization pipeline to operate across appearance change, however it is not by any means the only choice.\nFor example, we could also train a CNN to predict a measure of localization accuracy such as geodesic distance from ground truth on the $\\LieGroupSE{3}$ manifold, similarly to the estimator correction framework proposed in~\\cite{Peretroukhin2018-qc}.\nImportantly, this formulation admits a self-supervised training approach as training targets can be generated automatically by $\\Matcher$.\n\n\\Cref{fig:pipeline} (right-hand side) summarizes the training setup for this task.\nA pair of single-channel images is fed into a conventional feature detection\/matching algorithm (e.g., SURF~\\cite{Bay2008-zi}, ORB~\\cite{Rublee2011-hd}, or \\texttt{libviso2}~\\cite{Geiger2011-xe}), and a summary statistic is computed such as the quantity of RANSAC-filtered inlier feature matches.\nThis summary statistic forms the training target for a CNN whose task is to predict the statistic for the same image pair.\nGiven enough training pairs, the network should learn a set of convolutional filters that correspond to the types of features and patterns that best predict the performance of $\\Matcher$ in a given environment.\nCritically, the proxy network is fully differentiable and can provide a gradient signal to train a nonlinear image transformation.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\columnwidth]{matcher_net}\n \\caption{Network architecture for $\\MatcherNet$. Each block is denoted by stride, kernel size, input channels, and output channels. The left and right branches share weights. The final output is produced by a fully-connected layer (implemented as a convolution operator), which projects the feature map to a scalar value.}\n \\label{fig:matcher_net}\n \\vspace{-12pt}\n\\end{figure}\n\nOur matcher proxy network (\\Cref{fig:matcher_net}) is a siamese network built from convolutional and residual~\\cite{He2016-zg} blocks using batch normalization~\\cite{Ioffe2015-bs} and PReLU non-linearities~\\cite{He2015-zg}.\nEach image in the input pair is processed by one of two feature detection branches, which share weights to ensure that both images are mapped onto a common feature space.\nThe outputs of the feature detection branches are concatenated along the channel dimension to be further processed by the remainder of the network.\nEach non-residual convolution block downsamples the feature map by a factor of two, allowing for salient features to be learned at multiple scales.\nThe final output is produced by a fully-connected layer, which projects the feature map to a scalar value.\nWe train $\\MatcherNet$ to fit $\\Matcher$ in a least-squares sense by minimizing the mean squared error of the predicted match counts for a minibatch of $N$ image pairs:\n\\begin{align} \\label{eq:matcher_loss}\n \\CNNLoss(\\MatcherNetParams) & = \\frac{1}{N} \\sum_{i=1}^N \\left( \\MatcherNet(\\Image_1^i, \\Image_2^i) - \\Matcher(\\Image_1^i, \\Image_2^i) \\right)^2.\n\\end{align}\n\n\\subsection{Physically Motivated Transformations}\n\\label{sec:logrgb}\nPrior work in \\cite{Ratnasingam2010-cr} has shown that under the assumptions of a single black-body illuminant and an infinitely narrow sensor response function, an appropriately weighted linear combination of the log-responses of a three-channel (e.g., RGB) camera represents a projection onto an invariant one-dimensional chromaticity space that is independent of both the intensity and color temperature of the illuminant, and depends only on the imaging sensor and the materials in the scene:\n\\begin{align} \\label{eq:color_constant}\n \\InvariantImage^i_j = \\log \\Image^i_j({\\lambda_2}) - \\alpha \\log\\Image^i_j({\\lambda_1}) - \\beta \\log\\Image^i_j({\\lambda_3}),\n\\end{align}\nwhere $ \\Image^i_j({\\lambda_k}) $ is the image of sensor responses at wavelength $\\lambda_k$, the weights $(\\alpha, \\beta)$ are subject to the constraints\n\\begin{align} \\label{eq:color_constant_constraints}\n \\frac{1}{\\lambda_2} & = \\frac{\\alpha}{\\lambda_1} + \\frac{\\beta}{\\lambda_3} & \\mathrm{and} & & \\beta & = (1-\\alpha),\n\\end{align}\nand the indices $k$ are chosen such that $\\lambda_1 < \\lambda_2 < \\lambda_3$ (i.e., red, green and blue channels, respectively).\n\nThe image formed from this pixel-wise linear combination of log-responses can then be rescaled to produce a valid grayscale image that can be further processed by a localization pipeline.\nGrayscale images generated using this procedure are somewhat resistant to variations in lighting and shadow, and have been shown to improve stereo localization quality in the presence of shadows and changing daytime lighting conditions \\cite{Clement2017-gx,McManus2014-op,Paton2017-fi}, but have not been successful in adapting to nighttime navigation with headlights.\n\nGiven the constraints defined by~\\Cref{eq:color_constant_constraints}, the weights $(\\alpha, \\beta)$ are completely specified as a function of the imaging sensor.\nHowever, in practice, these constraints are relaxed and the parameters $(\\alpha, \\beta)$ are tuned to a specific environment, sensor, and feature matcher, where the theoretical values do not perform optimally.\nIndeed, \\cite{Clement2017-gx,Paton2017-fi} used two sets of parameters tuned to maximize the stability of SURF features~\\cite{Bay2008-zi} in regions where grassy or sandy materials dominate.\n\nWe argue that environmental appearance is best thought of as continuous rather than categorical, and that a better approach to selecting the transformation parameters should take into account the content of the specific scene being imaged, rather than using the same parameters at every location within a large and potentially heterogeneous operating environment.\nAccordingly, we train a second encoder network $\\EncoderNet$, with parameters $\\EncoderNetParams$, to predict the optimal values of the transformation parameters (i.e., which yield the most inlier feature matches) for a given RGB image pair.\n\nFurthermore, we relax the constraints in~\\Cref{eq:color_constant_constraints} and generalize~\\Cref{eq:color_constant} to be of the form\n\\begin{align} \\label{eq:color_constant_generalized}\n \\InvariantImage^i_j & = \\alpha^i \\log \\Image^i_j({\\lambda_1}) + \\beta^i \\log\\Image^i_j({\\lambda_2}) + \\gamma^i \\log\\Image^i_j({\\lambda_3}),\n\\end{align}\nwhere the parameters are computed for the $i^\\mathrm{th}$ image pair as\n\\begin{align} \\label{eq:encoder_output}\n \\TransformParams^i & = \\bbm \\alpha^i & \\beta^i & \\gamma^i \\ebm^T = \\EncoderNet(\\Image^i_1, \\Image^i_2),\n\\end{align}\nand~\\Cref{eq:color_constant_generalized} is applied to both $\\Image^i_1$ and $\\Image^i_2$ using the same set of parameters.\nDue to the need to rescale $\\InvariantImage^i_j$ to form a valid single-channel image, a degree of freedom in $\\TransformParams^i$ is lost and $(\\alpha, \\beta, \\gamma)$ represent the relative mixing proportions of the three color channels.\nWe enforce $\\Norm{ \\TransformParams^i }_1 = 1$ using a normalization layer to ensure a consistent range of outputs.\n\nOur encoder network $\\EncoderNet$ follows a similar siamese architecture to $\\MatcherNet$, but takes pairs of 3-channel images as inputs and outputs a 3-dimensional vector.\nWe train $\\EncoderNet$ to maximize the mean number of inlier feature matches as predicted by $\\MatcherNet$, or equivalently, to minimize its negation:\n\\begin{align} \\label{eq:encoder_loss}\n \\CNNLoss(\\EncoderNetParams) & = -\\frac{1}{N} \\sum_{i=1}^N \\MatcherNet(\\InvariantImage_1^i, \\InvariantImage_2^i),\n\\end{align}\nwhere $\\InvariantImage_1^i, \\InvariantImage_2^i$ are computed from input RGB images $\\Image_1^i, \\Image_2^i$ using \\Cref{eq:color_constant_generalized,eq:encoder_output}.\n\nRather than rescaling using the minimum and maximum response of each output image, we rescale by the joint mean $ \\mu^i $ and standard deviation $ \\sigma^i $ of the output pair and apply a clamping operation to map the output onto the range $ [0, 1] $:\n\\begin{align} \\label{eq:rescale_instancenorm}\n \\InvariantImage^i_j & \\leftarrow \\frac{1}{2} \\left[ \\frac{\\InvariantImage^i_j - \\mu^i }{3 \\sigma^i} \\right]_{-1,1} + \\frac{1}{2},\n\\end{align}\nwhere we have used the notation $\\left[ \\cdot \\right]_{a,b} = \\min(\\max(\\cdot, a), b)$.\nThis rescaling scheme allows the model to saturate parts of the output images while still using the full range of valid pixel values.\nMoreover, it avoids introducing significant sparsity in the gradients through the $\\min(\\cdot)$ and $\\max(\\cdot)$ operators, which improves the flow of gradient information during training.\n\n\n\\subsection{Learned Nonlinear Transformations}\n\\label{sec:learned}\nWhile the assumption of a single black-body illuminant in~\\cite{Ratnasingam2010-cr} is reasonable for daytime navigation where the dominant light source is the sun, it does not hold in many common navigation scenarios such as nighttime driving with headlights.\nMoreover, the assumption of an infinitely narrow sensor response is unrealistic for real cameras.\nAs an alternative to the physically motivated colorspace transformation outlined in \\Cref{sec:logrgb}, we investigate the possibility of learning a bespoke nonlinear mapping that maximizes matchability for a particular combination of imaging sensor, estimator and environment.\nWe parametrize this mapping using a small neural network $\\TransformerNet$, with parameters $\\TransformerNetParams$, operating independently on each pixel of each input RGB image.\nWe structure $\\TransformerNet$ as a multilayer perceptron (MLP) implemented using $1 \\times 1$ convolutions and PReLU nonlinearities.\n\nWe consider two versions of this MLP-based transformation, both with and without incorporating an additional pairwise context feature obtained from encoder network $\\EncoderNet$ using \\Cref{eq:encoder_output}.\nIn the case where $\\EncoderNet$ is used, the input to $\\TransformerNet$ becomes the concatenation of the input RGB image and the parameters $\\TransformParams^i$ along the channel dimension, and the first convolutional layer of $\\TransformerNet$ is modified accordingly.\nWe train $\\TransformerNet$ and $\\EncoderNet$ (if used) jointly by minimizing a similar loss function to \\Cref{eq:encoder_loss}, where in place of \\Cref{eq:color_constant_generalized}, we have\n\\begin{align} \\label{eq:learned_invariant_transform}\n \\InvariantImage_j^i & = \\begin{cases} \\TransformerNet(\\Image_j^i, \\TransformParams^i) & \\text{if $\\EncoderNet$ is used,} \\\\ \\TransformerNet(\\Image_j^i) & \\text{if $\\EncoderNet$ is not used}. \\end{cases}\n\\end{align}\nSimilarly to the physically motivated transformations described in~\\Cref{sec:logrgb}, we rescale $\\InvariantImage_j^i$ to fill the range of valid grayscale values by applying~\\Cref{eq:rescale_instancenorm}.\n\n\\begin{figure*}\n \\centering\n \n \n \n \n \n \n \\begin{subfigure}{0.32\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/all_matcher_target_est}\n \\caption{\\texttt{VKITTI} (Morning vs. Sunset)}\n \\label{fig:matcher_target_est:vkitti_morning-sunset}\n \\end{subfigure}\n ~\n \\begin{subfigure}{0.31\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/all_matcher_target_est}\n \\caption{\\texttt{InTheDark}}\n \\label{fig:matcher_target_est:inthedark}\n \\end{subfigure}\n ~\n \\begin{subfigure}{0.31\\textwidth}\n \\includegraphics[width=\\textwidth]{oxford\/all_matcher_target_est}\n \\caption{\\texttt{RobotCar}}\n \\label{fig:matcher_target_est:oxford}\n \\end{subfigure}\n \\caption{Estimated vs. actual match counts for $\\MatcherNet$ after ten epochs of pre-training on each dataset, colour-coded by relative density. Match counts are aggregated over all test sequences and include self-matches. The dashed red line corresponds to perfect agreement of $\\MatcherNet$ with \\texttt{libviso2}. In each case, the match count predictions produced by $\\MatcherNet$ are very strongly correlated with the true match counts.}\n \\label{fig:matcher_target_est}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\begin{subfigure}{0.95\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/0020_grid_horiz_cropped}\n \\caption{\\texttt{VKITTI\/0020} (Sunset to Morning, cropped)}\n \\label{fig:vkitti_0020_grid}\n \n \\end{subfigure}\n ~\n \\begin{subfigure}{0.95\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/inthedark_000041_grid}\n \\caption{\\texttt{InTheDark\/0041} (Night to Day)}\n \\label{fig:inthedark_000041_grid}\n \\end{subfigure}\n \\caption{Sample input RGB pairs and corresponding outputs of each RGB-to-grayscale transformation.}\n \\label{fig:output_grid}\n \\vspace{-12pt}\n\\end{figure*}\n\n\\section{Experiments}\nWe conducted experiments on synthetic and real-world long-term vision datasets to validate and compare each approach.\nSpecifically, we evaluated the ability of our matcher proxy network to capture the performance of \\texttt{libviso2} feature matching across viewpoint and appearance changes, as well as the effect of each image transformation on feature matching and localization performance.\nWhen evaluating feature matching, we assumed that we had a prior on the vehicle's topological location in the map, such that we could reliably identify the nearest vertex in the pose graph.\nThis is typical for autonomous visual route-following systems such as~\\cite{Paton2018-qa}, where the topological prior is derived by dead reckoning from a previous successful localization, or using place recognition or GNSS in the event that the system becomes lost.\n\nWe refer to the generalized color-constancy model of~\\cite{Ratnasingam2010-cr} (\\Cref{sec:logrgb}) as ``SumLog'' and ``SumLog-E'' , where the latter uses \\Cref{eq:encoder_output} to derive the parameters $\\TransformParams^i$ per image pair, and the former uses a constant $\\TransformParams$ that maximizes inlier feature matches over the training set (similarly to~\\cite{Clement2017-gx,Paton2017-fi}).\nAnalogously, we refer to the learned multilayer perceptron models (\\Cref{sec:learned}) as ``MLP'' and ``MLP-E'', where the latter incorporates $\\EncoderNet$ and the former does not .\nWe refer to the standard RGB-to-grayscale transformation as ``Gray''.\\footnote{We refer specifically to the ITU-R 601-2 luma transform implemented by the \\texttt{Pillow} library: $L = 0.299 R + 0.587 G + 0.114 B$.}\n\nTraining proceeds in two stages.\nFirst, we pre-train $\\MatcherNet$ using standard grayscale images.\nTraining labels are generated using the open-source \\texttt{libviso2} library~\\cite{Geiger2011-xe} to detect and match features, and the eight-point RANSAC algorithm to reject outlier matches.\nSecond, we train $\\EncoderNet$ and\/or $\\TransformerNet$ using the matchability loss defined in \\Cref{eq:encoder_loss}.\nTo ensure that $\\MatcherNet$ accurately predicts feature match counts for the output images, which differ significantly from standard grayscale images, we continue to train $\\MatcherNet$ in an alternating fashion using the output images at each iteration.\nAll models are implemented in \\mbox{PyTorch}~\\cite{paszke2017automatic} and trained for 10 epochs with a batch size of 8, using the Adam optimizer~\\cite{Kingma2015-wl} with default parameters and a learning rate of $10^{-4}$. \nWe rescale all images to a height of 192 pixels for both training and testing.\n\n\\subsection{Datasets}\nWe evaluated our approach using both synthetic and real-world datasets exhibiting severe illumination change.\n\n\\paragraph{Virtual KITTI} \\label{sec:vkitti_dataset}\nThe Virtual KITTI (\\texttt{VKITTI}) dataset~\\cite{Gaidon2016-by} is a synthetic reconstruction of a portion of the KITTI vision benchmark~\\cite{Geiger2013-ky}, consisting of five sets of non-overlapping trajectories with \\mbox{RGB-D} imagery rendered under a variety of simulated illumination conditions.\nThis dataset is a convenient validation tool as it provides perfect data association and a range of daytime illumination conditions.\nFor each trajectory, we train models using image pairs from the others.\nSince each trajectory is non-overlapping, this spatial split allows us to evaluate how well our method generalizes to unseen environments.\nFurther, since \\texttt{VKITTI} provides corresponding images from identical viewpoints, we augmented the training data to ensure generalizability to viewpoint changes and dynamic objects by associating each training image in one condition with a window of nearby images in the other.\n\n\\paragraph{UTIAS In The Dark}\nThe UTIAS In The Dark (\\texttt{InTheDark}) dataset~\\cite{Paton2016-bz} provides stereo imagery of a 250 m outdoor loop traversed repeatedly over a 30-hour period using on-board headlights to illuminate the scene at night.\nWe use the multi-experience localization system in \\cite{Paton2016-bz} to obtain corresponding image pairs with overlapping fields of view but non-identical poses.\nWe train our models using left-camera images from 66 traversals and test on 7 held-out traversals spanning a full day-night cycle (listed in \\Cref{tab:match_stats}).\nThis temporal split allows us to evaluate how well our method generalizes to multiple unseen illumination conditions.\n\n\\paragraph{Oxford RobotCar}\nWe further evaluate our method using three sequences from the Oxford RobotCar (\\texttt{RobotCar}) dataset~\\cite{Maddern2016-ng}, captured along the same 10~km route in overcast, nighttime, and sunny conditions.\\footnote{``Overcast'', ``Night'', and ``Sunny'' refer to \\texttt{2014-12-09-13-21-02}, \\texttt{2014-12-10-18-10-50}, and \\texttt{2014-12-16-09-14-09}, respectively.}\nWe find corresponding images using the GNSS\/INS poses, and split the trajectory into two non-overlapping segments: the first 70\\% of the trajectory is used for training and the remaining 30\\% is used for testing.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/0020_matches}\n \\caption{\\texttt{VKITTI\/0020} (Sunset to Morning)}\n \\label{fig:vkitti_0020_matches}\n \n \\end{subfigure}\n ~\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/run_000041_matches}\n \\caption{\\texttt{InTheDark\/0041} (Night to Day)}\n \\label{fig:inthedark_000041_matches}\n \\end{subfigure}\n \\caption{Box-and-whiskers plots of inlier \\texttt{libviso2} feature matches for corresponding image pairs with each RGB-to-grayscale transformation applied. Orange lines indicate the median values.}\n \\label{fig:matches_box_whiskers}\n \\vspace{-12pt}\n\\end{figure}\n\n\\begin{table}[]\n \\setlength{\\tabcolsep}{2.7pt}\n \\centering\n \\caption{Actual inlier feature matches using \\texttt{libviso2} and each RGB-to-grayscale transformation. The highest mean number of matches for each sequence is highlighted in bold.}\n\n \\begin{tabular}{@{}llccccc@{}}\n \\toprule\n \\multicolumn{2}{l}{} & \\multicolumn{5}{c}{\\textbf{Inlier Feature Matches $\\mu (\\sigma)$}} \\\\ \\cmidrule{3-7}\n \\multicolumn{2}{l}{\\textbf{Test Sequence}} & Gray & SumLog & SumLog-E & MLP & MLP-E \\\\ \\midrule\n \\multicolumn{2}{l}{\\texttt{VKITTI\/0001}} & & & & & \\\\\n & Sunset-Morning & 262 (82) & \\textbf{726 (136)} & 689 (157) & 661 (108) & 623 (116) \\\\\n & Overcast-Clone & 444 (58) & \\textbf{790 (129)} & 767 (125) & 747 (106) & 770 (107) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0002}} & & & & & \\\\\n & Sunset-Morning & 240 (22) & \\textbf{812 (67)} & 803 (70) & 774 (65) & 702 (62) \\\\\n & Overcast-Clone & 290 (41) & 739 (67) & 757 (76) & 755 (65) & \\textbf{764 (84)} \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0006}} & & & & & \\\\\n & Sunset-Morning & 125 (33) & 669 (44) & \\textbf{735 (43)} & 711 (37) & 642 (41) \\\\\n & Overcast-Clone & 142 (33) & 647 (39) & \\textbf{666 (43)} & 566 (47) & 546 (38) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0018}} & & & & & \\\\\n & Sunset-Morning & 234 (53) & \\textbf{817 (36)} & 816 (37) & 745 (37) & 731 (40) \\\\\n & Overcast-Clone & 311 (46) & 548 (52) & \\textbf{555 (50)} & 450 (42) & 486 (39) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0020}} & & & & & \\\\\n & Sunset-Morning & 210 (39) & \\textbf{762 (69)} & 758 (71) & 756 (62) & 675 (69) \\\\\n & Overcast-Clone & 287 (78) & 716 (71) & 716 (72) & \\textbf{718 (62)} & 708 (64) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{InTheDark}} & & & & & \\\\\n & Map~~(08:54) & - & - & - & - & - \\\\\n & \\texttt{0006} (09:46) & \\textbf{178 (48)} & 125 (56) & 165 (51) & 138 (52) & 158 (53) \\\\\n & \\texttt{0027} (18:36) & 9 (9) & 52 (29) & \\textbf{57 (26)} & 44 (25) & 48 (28) \\\\\n & \\texttt{0041} (21:48) & 10 (16) & 33 (25) & \\textbf{39 (31)} & 33 (29) & 35 (29) \\\\\n & \\texttt{0058} (05:48) & 99 (34) & 95 (47) & \\textbf{114 (43)} & 97 (44) & 113 (40) \\\\\n & \\texttt{0071} (09:18) & \\textbf{181 (44)} & 126 (52) & 167 (47) & 141 (48) & 161 (48) \\\\\n & \\texttt{0083} (14:01) & 53 (21) & 76 (39) & \\textbf{83 (37)} & 82 (37) & \\textbf{83 (37)} \\\\\n & \\texttt{0089} (16:43) & 45 (22) & 67 (35) & \\textbf{70 (36)} & 63 (31) & 68 (33) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{RobotCar}} & & & & & \\\\ \n& Overcast-Night & 11(3) & \\textbf{12 (2)} & \\textbf{12 (3)} & 11 (2) & 11 (2) \\\\\n& Overcast-Sunny & 26 (12) & 25 (12) & 25 (12) & 24 (9) & \\textbf{71 (41)} \\\\ \\bottomrule\n \\end{tabular}\n \\label{tab:match_stats}\n \\vspace{-12pt}\n\\end{table}\n\n\\subsection{Feature Matcher Approximation}\nWe train $\\MatcherNet$ in a self-supervised manner to predict the number of inlier feature matches for overlapping image pairs captured under different illumination conditions from nearby but different poses.\nTraining labels are generated on the fly for each image pair using the open-source \\texttt{libviso2} library~\\cite{Geiger2011-xe} in monocular flow matching mode with default parameters, using the eight-point RANSAC algorithm to reject outlier matches.\nIn practice we train $\\MatcherNet$ to minimize \\Cref{eq:matcher_loss} over all combinations of input images.\n\n\\Cref{fig:matcher_target_est} plots actual and estimated match counts for each dataset after ten epochs of pre-training on Gray images, aggregated over all test sequences.\nThese include self-matches (same viewpoint, same appearance) and non-self-matches (different viewpoint, different appearance), which appear as clusters.\nIn each case the test-time match counts predicted by $\\MatcherNet$ are strongly correlated with the true performance of \\texttt{libviso2}.\nThis indicates that our approach generalizes well and that $\\MatcherNet$ is capturing salient properties of feature matching rather than memorizing training examples.\n\n\\begin{table*}[]\n \\centering\n \\caption{Maximum distances travelled on dead reckoning for each test sequence of the UTIAS In The Dark dataset, based on various thresholds of inlier feature matches against the ``Map'' sequence. The best results are highlighted in bold.}\n \\begin{threeparttable} \n \\begin{tabular}{@{}ll*{17}c@{}}\n \\toprule\n & & \\multicolumn{17}{c}{\\textbf{Maximum Distance on Dead Reckoning (m)}} \\\\ \\cmidrule{3-19}\n & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 10 Inliers}} & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 20 Inliers}} & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 30 Inliers}} \\\\ \n \\multicolumn{2}{l}{} & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} & & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} & & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} \\\\ \\cmidrule{3-7} \\cmidrule{9-13} \\cmidrule{15-19}\n \\multicolumn{2}{l}{\\texttt{InTheDark}} & & & & & & & & & & & & & & & & & \\\\\n & Map~~(08:54) & - & - & - & - & - & & - & - & - & - & - & & - & - & - & - & - \\\\\n & \\texttt{0006} (09:46) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & 0.1 & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} \\\\\n & \\texttt{0027} (18:36) & 7.5 & 0.7 & \\textbf{0.0} & 0.3 & 0.3 & & 25.9 & 3.6 & \\textbf{0.7} & 2.5 & 2.2 & & 101.1 & 10.5 & \\textbf{0.7} & 8.3 & 7.8 \\\\\n & \\texttt{0041} (21:48) & 14.9 & 0.5 & \\textbf{0.2} & 0.9 & 1.4 & & 46.3 & 8.4 & \\textbf{3.2} & 7.2 & 7.4 & & 104.5 & 15.1 & \\textbf{10.0} & 18.6 & 15.7 \\\\\n & \\texttt{0058} (05:48) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & 0.2 & \\textbf{0.0} & 0.2 & \\textbf{0.0} & & \\textbf{0.0} & 0.6 & 0.3 & 0.6 & \\textbf{0.0} \\\\\n & \\texttt{0071} (09:18) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} \\\\\n & \\texttt{0083} (14:01) & \\textbf{0.0} & 0.2 & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & 0.4 & 0.5 & 0.3 & 0.2 & \\textbf{0.0} & & 2.2 & 4.0 & \\textbf{0.3} & 0.4 & 0.5 \\\\\n & \\texttt{0089} (16:43) & 0.3 & 0.3 & 0.2 & \\textbf{0.0} & \\textbf{0.0} & & 4.8 & 1.0 & 3.7 & 1.2 & \\textbf{0.8} & & 6.1 & 6.1 & 4.7 & \\textbf{2.9} & \\textbf{2.9} \\\\ \\addlinespace\n \\multicolumn{2}{l}{\\texttt{RobotCar}} & & & & & & & & & & & & & & & & & \\\\ \n& Map (Overcast) & - & - & - & - & - & & - & - & - & - & - & & - & - & - & - & - \\\\\n& Night & 27.0 & 7.6 & \\textbf{3.7} & 27.3 & 27.3 & & 307.1 & 307.1 & \\textbf{245.9} & 400.8 & 398.9 & & 740.5 & \\textbf{541.5} & 740.5 & 697.3 & 740.5 \\\\\n& Sunny & 1.3 & 1.3 & 1.5 & 1.8 & \\textbf{0.9} & & 36.1 & 65.1 & 48.0 & 39.4 & \\textbf{6.3} & & 129.1 & 139.8 & 109.2 & 124.3 & \\textbf{16.5} \\\\ \\bottomrule\n \\end{tabular}\n \\begin{tablenotes}\n \\item[1] G: Gray\n \\item[2] S: SumLog \n \\item[3] M: MLP \n \\end{tablenotes}\n \\label{tab:inthedark:loc_succ}\n \\end{threeparttable}\n \\vspace{-12pt}\n\\end{table*}\n\n\\subsection{Feature Matching Across Appearance Change}\n\\Cref{fig:output_grid} shows the outputs of each image transformation for sample RGB image pairs in the \\texttt{VKITTI\/0020} Morning and Sunset sequences (\\Cref{fig:vkitti_0020_grid}) and the challenging sequence \\texttt{InTheDark\/0041} (\\Cref{fig:inthedark_000041_grid}).\nWe see that each model produced image pairs that are visually more consistent than standard Gray images, and that local illumination variations such as shadows, uneven lighting, and specular reflections were minimized by optimizing \\Cref{eq:encoder_loss}.\n\n\\Cref{fig:matches_box_whiskers} visually compares the distributions of actual \\texttt{libviso2} feature matches for each transformation, while \\Cref{tab:match_stats} summarizes the results numerically.\nEach model significantly increased the mean number of inlier matches across most test sequences, with the greatest improvements generally obtained from the SumLog and SumLog-E transformations.\nSequences \\texttt{InTheDark\/0006} and \\texttt{InTheDark\/0071} are exceptions in that the standard Gray transformation performed best.\nThese sequences were recorded under similar conditions to the ``Map'' sequence, so feature matching can be expected to perform optimally on Gray images.\nWe saw little improvement in match counts on the \\texttt{RobotCar\/}Overcast-Night experiment, which we attribute to motion blur in the nighttime images making feature matching exceptionally difficult.\nIn contrast, the {MLP-E} method more than doubled the mean number of feature matches in the Sunny experiment.\n\nWe note that the pairwise encoder did not confer any significant benefit on the \\texttt{VKITTI} sequences.\nThese results are consistent with \\cite{Clement2017-gx,Corke2013-hl,McManus2014-op,Paton2017-fi}, where one or two sets of parameter values were sufficient to achieve good performance across varying daytime conditions.\nIn contrast, the encoder network provided a noticeable performance boost on most \\texttt{InTheDark} and \\texttt{RobotCar} sequences.\nWe attribute this difference to more variation in illumination and terrain, as a single transformation is less likely to perform well under more varied conditions.\nWe also note that the MLP-E transformation frequently performed similarly to the SumLog-E transformation, suggesting that, in spite of key assumptions being broken, a linear combination of log-responses as proposed by~\\cite{Ratnasingam2010-cr} may in fact be an optimal solution for this problem, and that a careful choice of weights is the key to obtaining good cross-appearance feature matching over day-night cycles.\n\n\\subsection{Impact on Localization Performance}\nWe evaluate localization performance in an autonomous route-following context by examining the maximum distances in each sequence that would have been navigated using dead reckoning (e.g., visual odometry) as a result of failing to localize against the map.\nThese results are summarized in \\Cref{tab:inthedark:loc_succ} for thresholds of 10, 20, and 30 inlier feature matches against the ``Map'' sequence.\nA typical criterion for requiring manual intervention is dead reckoning in excess of 10 meters, depending on the accuracy of the underlying dead reckoning system.\nBased on a relatively conservative threshold of 20 inlier feature matches against the ``Map'' sequence, we see that \\texttt{InTheDark\/0027} (evening) and \\texttt{InTheDark\/0041} (night) presented significant difficulty for localization, which was substantially alleviated using any of the four image transformations.\nIn particular, the SumLog-E transformation yielded maximum dead reckoning distances below four meters across all illumination conditions.\nWe see similar improvements using the SumLog-E method with a threshold of 10 inliers on the \\texttt{RobotCar} dataset.\nTogether, these results imply that near-continuous 6-dof visual localization over a full day-night cycle is achievable using only a single mapping experience and a simple image pre-processing step, representing a dramatic reduction in data requirements to scale experience-based localization to long deployments.\n\nWith more conservative thresholds (e.g., 30 inliers), localization failures became more common, as expected.\nHowever, the proposed image transformations continued to provide substantially more robust localization performance compared to standard grayscale images.\nFor example, we achieved a maximum distance on dead reckoning of 10.0 m using the SumLog-E method on sequence \\texttt{InTheDark\/0041} with a threshold of 30 inliers.\nIn contrast, using the standard Gray method would have required the system to rely continuously on dead reckoning for approximately 40\\% of the route.\n\n\\section{Conclusions and Future Work}\nThis paper presented a method for learning pixel-wise RGB-to-grayscale colorspace mappings that explicitly maximize the number of inlier feature matches for a given input image pair, feature detector\/matcher and operating environment.\nOur key insight is that by training a deep neural network to predict the performance of a conventional non-differentiable feature detector\/matcher, we can define a fully differentiable loss function that can be used to learn image transformations optimized for localization performance.\nWe evaluated our approach using both physically motivated and data-driven transformations and demonstrated substantially improved feature matching and localization performance on synthetic and real long-term vision datasets exhibiting severe illumination change, allowing experience-based localization to scale to long deployments with dramatically fewer bridging experiences.\nWe consistently achieved the best performance using a physically motivated weighted sum of log-responses with weights derived from a pairwise context encoder network.\nIn future work we plan to explore alternative loss functions such as pose estimation error, the use of feature locations or photometric consistency as a more granular supervision signal, and the impact of context feature dimension on MLP-based transformations.\nWe also intend to investigate the impact of our method on feature matching across seasonal appearance change.\n\n\\bibliographystyle{ieeetr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nZero-range processes \\cite{cc1,cc2,cc3,cc4,cc5,cc6} have attracted attention of many researchers\nsince they provide an exactly solvable example of \nfar-from-equilibrium dynamics and of condensate formation.\nMany questions concerning the dynamics of the model\ncan be addressed and solved analytically. It is known that a \nzero-range process has a steady state and that this state \nis described by the partition function of the balls-in-boxes (B-in-B) \nmodel \\cite{bbj,bbj2}, also called the Backgammon model. \nThe B-in-B model has two phases: a fluid and a condensed \none, separated by a critical point at which the system \nundergoes a phase transition and the condensate is formed. \nUnlike the Bose-Einstein condensation,\nthe B-in-B condensation takes place in real space\nrather than in momentum space. \nTherefore it mimics such processes like mass transport \\cite{cc1}, \ncondensation of links in complex networks \\cite{cc2,bbw} or phase separation \\cite{ph1,ph2}.\n\nThe zero-range process (ZRP) \\cite{evans} discussed in this paper describes\na gas of identical, indistinguishable \nparticles hopping between the neighboring nodes of a network.\nThe state of such a system is characterized by the topology of \nthe network, which is fixed during the process, and\nby the particle distribution which is given by\nthe occupation numbers of particles $\\{m_i\\}$ on all nodes \n$i=1,\\ldots, N$ of the network.\nThe total number of particles $M=m_1+m_2+\\ldots + m_N$ \nis conserved during the process. \nThe zero-range dynamics is characterized by the outflow rates $u(m)$ of particles from network nodes,\nwhich depend only on the occupation number $m$ of the node from which the particle hops.\nWe shall assume that these ultra local hopping rates\nare identical for each node. The stream of particles \noutgoing from a node is equally distributed among all its\nneighbors. So if we denote by $k_i$ the number of neighbors of the node $i$, called also its degree, \nthen the hopping rate from the $i$-th node to each of its neighbors is equal to $\\frac{1}{k_i} u(m_i)$.\nThe outflow rate $u(m)$ is a semi-positive function which is \nequal to zero for $m=0$. For a given graph the function\n$u(m)$ entirely defines the dynamics of the system. \n\nWe consider here the ZRP on a network being a connected simple\ngraph. In this case, the ZRP has a unique steady state,\nin which the probability $P(m_1,\\dots,m_N)$ \nof finding the distribution of balls $\\{m_1,\\ldots,m_N\\}$ \nfactorizes into a product of some weight functions\n$p_i(m_i)$ for individual nodes, except that there is a global constraint reflecting the conservation of particles. \n\nOn a $k$-regular network, that is when all node degrees are equal to $k$, \nall weight functions $p_i(m)$ are identical, $p_i(m)\\equiv p(m)$, \nand thus the probability $P$ is invariant under permutations \nof the occupation numbers. When the density\n$\\rho=M\/N$ of balls per node exceeds a certain critical value $\\rho_c$ depending on the functional form of $p(m)$, \na single node attracts an extensive number of balls called the condensate. The relative occupation of that node does not disappear\nin the thermodynamic limit $N,M\\to\\infty$, with fixed $\\rho$. The larger the density $\\rho$, the larger is the\nnumber of balls in the condensate. In other words,\nthe system undergoes a phase transition at $\\rho=\\rho_c$ \nbetween the fluid (low density) and the condensed (high density)\nphase. The permutation symmetry of $\\{m_i\\}$ is respected in the\nfluid phase while it is broken in the condensed phase where one node\nbecomes evidently distinct from the $N\\!-\\!1$ remaining ones.\nThe symmetry of the partition function reduces to the subgroup of permutations\nof $N\\!-\\!1$ occupation numbers. This mechanism \nhas been extensively studied in the B-in-B model \\cite{bbj,bbj2,bbjw}.\nThe value of the critical\ndensity depends on the weight function\n$p(m)$ which can be translated to the asymptotic properties of $u(m)$ for $m\\to\\infty$.\nIf $u(m)$ tends to infinity\nthen also $\\rho_c=\\infty$ and the condensation does not occur regardless\nof the density of balls. The system is\nin the fluid phase for any finite density $\\rho$. Intuitively,\nthis means that there exists an effective repulsive force \npreventing a node from being occupied by many balls\nand they distribute uniformly on the whole graph.\nOn the contrary, if $u(m) \\to 0$, \nthe critical density is $\\rho_c=0$ and therefore the system is in the condensed phase\nfor any $\\rho>0$. The larger the number of particles on a node, the smaller\nare the chances for balls to escape from it since $u(m)$ becomes very small\nfor large $m$. This can be seen as the existence of an effective attraction between particles.\n\nThe most interesting case is when $u(m)$ goes to some positive constant $u_\\infty$ with $m\\to\\infty$. \nOne can show that the probability distribution\n$P(m_1,\\ldots,m_N)$ does not depend on $u_\\infty$ \\cite{bbjw}\nbut on how fast $u(m)$ approaches the constant value.\nTherefore without loss of generality we can choose $u_\\infty=1$ and concentrate on the asymptotic behavior\nhaving the form: $u(m)=1+b\/m$, with $b$ being some positive number.\nIf $0\\le b \\le 2$, then the critical density $\\rho_c$ is infinite.\nThe effective attraction between balls is too weak to form the condensate.\nHowever, if $b>2$, then $\\rho_c$ has a finite value. \nIn this case the attraction is strong enough to trigger \nthe condensation above the critical density $\\rho_c$. \n\nSo far we have discussed the criteria of the condensation on $k$-regular networks, where it arises\nas a result of the spontaneous symmetry breaking. All those facts are well known \\cite{evans,evans2}.\nOn the other hand, the permutation symmetry can \nbe explicitly broken if the weight functions $p_i(m)$ \nare not identical for all nodes. For instance, this happens when the network \non which the process takes place is inhomogeneous, that is when\nthe degrees $k_1,\\dots,k_N$ vary. A particularly important example are complex networks \\cite{cn},\nfor which the distribution of degrees has usually a long tail, and thus there are\nmany nodes with relatively high degree. They are, however, not easy for analytical studies although \nsome predictions are possible \\cite{jdn}.\nBelow we shall argue that to gain some insight into the static and dynamical properties\nof the ZRP on such networks it is sufficient to study some simplified models.\n\nIn the remaining part of the paper we \nconsider only the most favorable situation when\nthe weight of only one node is different from \nthe remaining ones. Such an inhomogeneity of the weights\ncan be introduced either by an inhomogeneity of the outflow\nrates $u_i(m)$ or by an inhomogeneity of the degree distribution.\nWe focus here on the latter situation, when a graph has one node of \ndegree $k_1$ which differs from all\nremaining degrees $k_2=\\ldots =k_N\\equiv k$. As we shall see,\nin this case the quantity $\\mu = \\log(k_1\/k)$ plays the role\nof an external field breaking the permutation symmetry.\n\nIn particular, we discuss the dynamics of the condensate on such inhomogeneous networks.\nThis is a relatively new topic and, in contrast to the stationary properties, less understood.\nAlthough the emergence of the condensate has been studied for homogeneous and inhomogeneous systems \\cite{evans} and numerically for scale-free networks in \\cite{jdn}, studies of the dynamics of an existing condensate\nare rare \\cite{god}.\nFor instance, one question which may be asked is what is the typical life time of the condensate,\nthat is how much time does it take to ``melt'' the condensate at one node and rebuild it at another node.\nTo provide an answer to this problem is the main goal of the present article.\n\nThe rest of the paper is organized as follows. In Sec.~II we \ndiscuss the static properties of the ZRP on inhomogeneous networks. \nWe consider some particular graph topologies: $k$-regular graphs,\nstar graphs and $k$-regular graphs with a single inhomogeneity\nintroduced by a vertex of degree $k_1>k$. Their advantage is that\nall calculations can be done exactly or at least with an excellent approximation.\nIn all cases we calculate the effective occupation number distribution\n$\\pi(m)$ and use it to derive information about the condensate\ndynamics. In Sec.~III we derive\nanalytic expressions for the life time of the condensate.\nWe concentrate on the role of\ninhomogeneity and typical scales at which it becomes relevant.\nAll analytical results are cross-checked by Monte Carlo simulations.\nThe last section is devoted to a summary of our results.\n\n\\section{Steady state -- statics}\nA zero-range process on a connected simple graph has a steady state\nwith the following partition function $Z(N,M,\\{k_i\\})$ \n\\cite{evans}:\n\\begin{equation}\nZ(N,M,\\{k_i\\}) = \n\\sum_{m_1=0}^M \\cdots \\sum_{m_N=0}^M \\delta_{ \\sum_{i=1}^N m_i, M}\n\\prod_{i=1}^N p(m_i) k_i^{m_i},\t\\label{part}\n\\end{equation}\nwhere $\\delta_{i,j}$ denotes the discrete delta function and the weight function $p(m)$ is related to the hop rate $u(m)$ through the formula:\n\\begin{equation}\np(m)= \\prod_{n=1}^m \\frac{1}{u(n)}, \\;\\;\\; p(0)=1.\t\n\\label{pbyu}\n\\end{equation}\nWe shall denote $Z(N,M,\\{k_i\\})$ in short by $Z(N,M)$, having in mind its dependence on the degrees. \nThe partition function (\\ref{part}) contains the whole \ninformation about the static properties of the system in the steady state.\nThe only trace of graph topology in the formula is through\nthe nodes degrees. The dynamics, however, depends also on other topological \ncharacteristics but they become important only in refined treatments.\nIn a sense, the degree sequence is the first-order approximation also for the dynamics. \n\nThe probability $P(m_1,\\dots,m_N)$ of a given configuration \n$\\{m_i\\}$ reads:\n\\begin{eqnarray}\nP(m_1,\\dots,m_N) &=& \\frac{1}{Z(N,M)} \n\\prod_{i=1}^N p(m_i) k_i^{m_i} \\nonumber \\\\\n&=& \\frac{1}{Z(N,M)} \\prod_{i=1}^N \\tilde{p}_i(m_i),\n\\end{eqnarray}\nwhere we have defined ``renormalized'' weights\n$\\tilde{p}_i(m_i) = p(m_i)k_i^{m_i}$, being now node-dependent.\nThe most important quantity characterizing the steady state is\nthe probability $\\pi_i(m)$ that the\n$i$-th node is occupied by $m$ particles:\n\\begin{eqnarray}\n\\pi_i(m_i) = \\sum_{m_1} \\cdots \\sum_{m_{i-1}} \n\\sum_{m_{i+1}} \\cdots \\sum_{m_N} P(m_1,\\dots,m_N) \\times \\nonumber \\\\\n\\times \\delta_{\\sum_{j=1}^N m_j, M} = \\frac{Z_i(N-1,M-m_i)}{Z(N,M)} \\tilde{p}_i(m_i),\n\\label{pigeneral}\n\\end{eqnarray}\nwhere $Z_i(N-1,M-m)$ denotes the partition function \nfor $M-m$ particles occupying a graph consisting \nof $N-1$ nodes with degrees $\\{k_1,\\dots,k_{i-1},k_{i+1},\\dots,k_N\\}$. \nWe shall call $\\tilde{p}_i(m)$ ``bare'' occupation probability\nwhile $\\pi_i(m)$ ``dressed'' or effective occupation probability\nof the node $i$. We also define the average occupation probability:\n\\begin{equation}\n\\pi (m) = (1\/N) \\sum_i \\pi_i(m).\n\\end{equation}\nFor a $k$-regular graph, that is for $k_i\\equiv k$,\nthe occupation probability $\\pi_i(m)=\\pi(m)$ is the same for every\nnode and all the formulas above reduce to those discussed \nin \\cite{bbj}. In general, the partition function can be calculated recursively:\n\\begin{eqnarray}\n& & Z(N,M,\\{k_1,\\dots,k_N\\}) = \\nonumber \\\\\n&=& \\sum_{m_N} \\tilde{p}_N(m_N) \\sum_{m_1,\\dots,m_{N-1}} \\delta_{\\sum_{i=1}^{N-1} m_i, M-m_N} \\prod_{i=1}^{N-1} \\tilde{p}_i(m_i) \\nonumber \\\\\n& =& \\sum_{m_N=0}^{M} \\tilde{p}_N(m_N) Z(N-1,M-m_N,\\{k_1,\\dots,k_{N-1}\\}). \\nonumber \\\\\n\\label{znmrec}\n\\end{eqnarray}\nFor $N=1$ the partition function simply reads \n$Z(1,M,k_1) = \\tilde{p}_1(M)$. The recursive use of the\nformula (\\ref{znmrec}) allows one to compute \nthe partition function within a given numerical accuracy.\nUsing this method we were able to push the computation as far as to\n$N$ of order $500$. For identical weights, one can find\na more efficient recursion relation by splitting the system into\ntwo having similar size, which allows to study much larger systems.\nThe computation of the partition function can be used together\nwith Eq.~(\\ref{pigeneral}) to determine \nnumerically the node occupation distribution $\\pi_i(m)$. \nThis gives an exact result with a given accuracy and is more efficient than the corresponding \nMonte Carlo simulations of the ZRP. The dynamics, however, is not accessible in this way.\n\nAs mentioned in the introduction we are going to examine the effect of\ntopological inhomogeneity on the properties of the ZRP.\nWe shall consider an almost $k$-regular graph \nwith one node, say number one, having a degree bigger than the rest of nodes:\n$k_1>k$ and $k_2=\\dots=k_N=k$. \nThe simplest realization of such a graph is a star or a wheel graph.\nIn general, such a graph can be constructed from any $k$-regular\ngraph by a local modification. We proceed as follows. First we\n derive exact formulas for the particle distribution\nin a steady state on a $k$-regular graph which shall later\nserve us as a reference point. Then we consider the particular\nexample of a star topology as the simplest example\nof a single defect and finally an arbitrary $k$-regular graph with a singular node $k_1>k$.\nThe system has now the following weights: \n$\\tilde{p}_1(m)= k_1^m p(m)$ for the singular and $\\tilde{p}_i(m)= k^m p(m)$ for the regular nodes. \nThey differ by an exponential factor\n$\\tilde{p}_1(m)\/\\tilde{p}_i(m)= (k_1\/k)^m = e^{\\mu m}$\nwhere $\\mu =\\log(k_1\/k)>0$, which clearly favors the situation\nin which the singular node has much more particles\nthan the regular ones. To make things as simple as possible, \nand to concentrate on the effect of inhomogeneity\nwe assume that the outflow rate $u(m)=1$ is constant\nand independent of $m$. In this case $p(m)=1$ is also constant,\nsimplifying calculations.\nAll other functions with the asymptotic behavior \n$u(m)\\rightarrow 1$ would lead basically to the same qualitative \nbehavior. This is because\nin this case $p(m)$ would have a power-law tail which is much\nless important for the large $m$-behavior than the exponential\nfactor $e^{\\mu m}$ introduced by the inhomogeneity.\nWe shall briefly comment on this towards the end of the paper.\n\n\\subsection{$k$-regular graph}\nWith the assumption $p(m)=1$,\nthe partition function $Z(N,M)$ from Eq.~(\\ref{part}) for the steady state of the ZRP on a $k$-regular graph reads:\n\\begin{eqnarray}\nZ_{\\rm reg}(N,M) &=& \\sum_{m_1=0}^M \\cdots \\sum_{m_N=0}^M \n\\delta_{\\sum_{i} m_i, M} \\;\nk^{\\sum_i m_i} \\nonumber \\\\\n&=& k^M \\frac{1}{2\\pi i} \\oint {\\rm d} z \\, \nz^{-M-1} \\left(\\sum_{m=0}^M z^m\\right)^N.\n\\label{cint}\n\\end{eqnarray}\nWe used an integral representation of the discrete delta function\nwhich allowed us to decouple the sums over $m_1,\\dots, m_N$ for\nthe price of having the integration over $z$.\nThe sum over $m$ can be done yielding $1\/(1-z)$. Using\nthe expansion:\n\\begin{equation}\n\\left(\\frac{1}{1-z}\\right)^N = \n\\sum_{m=0}^\\infty \\binom{-N}{m} (-z)^m \n= \\sum_{m=0}^\\infty \\binom{N+m-1}{m} z^m\n\\end{equation}\nand Cauchy's theorem we see that the contour\nintegration over $z$ selects only the term with $m=M$\nfrom the integrand in (\\ref{cint}), so we obtain:\n\\begin{equation}\nZ_{\\rm reg}(N,M) = k^M \\binom{N+M-1}{M}. \\label{zreg}\n\\end{equation}\nInserting this into Eq.~(\\ref{pigeneral})\nwe find the occupation number distribution\n\\begin{eqnarray}\n\\pi(m) &=& \\pi_i(m) = \\binom{M+N-m-2}{M-m} \/ \\binom{M+N-1}{M} \\nonumber \\\\\n&\\propto & \\frac{(M+N-m-2)!}{(M-m)!}. \\label{pik}\n\\end{eqnarray}\nThe distribution $\\pi(m)$ is identical for all nodes and independent on $k$. \nIt falls faster than exponentially for large $m$, therefore the condensate\nnever appears.\n\nIn particular one can apply these formulas to the complete \ngraph which is just a $k$-regular graph with $k=N-1$.\nIn Fig.~\\ref{f1} we see the comparison between the\ntheoretical expression (\\ref{pik}) and results of\nnumerical Monte Carlo simulations for a $4$-regular graph with $N=20$ nodes and \ntwo different numbers of balls, $M$. \nThe simulations of the\nZRP were organized in sweeps consisting on $N$ steps each. In a single\nstep a node was chosen at random and if it was non-empty\na particle was picked and moved to a neighboring node.\nFor each graph the process was initiated from a random\ndistribution of particles. After some thermalization,\n measurements of $\\pi(m)$\nwere done on $10^4$ configurations generated every sweep.\nThe results were averaged over these configurations and then\nover $5\\times 10^4$ independent graphs drawn at random\nfrom the ensemble of $k$-regular graphs. \n\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\includegraphics[width=8cm]{f1.eps}\n\\caption{The ``experimental'' distribution $\\pi(m)$ compared to \nthe theoretical prediction (\\ref{pik}) for regular graphs \nwith $k=4$ and $N=20$ and for $M=20$ (+) and $M=40$ (x) balls.}\n\\label{f1}\n\\end{figure}\n\n\\subsection{Star graph}\nWe consider first a special case of a single \ninhomogeneity graph, namely the star graph having $N-1$ nodes of \ndegree $k_2=\\ldots=k_N=1$ connected to the central node with $k_1=N-1$.\nThe partition function $Z_{\\rm star}(N,M)$ is\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) &=& \n\\sum_{m_1=0}^\\infty \\cdots \\sum_{m_N=0}^\\infty \\delta_{\\sum_{i=1}^N m_i , M}\n(N-1)^{m_1} \\hspace{5mm} \\nonumber \\\\\n&=& \\sum_{m=0}^M (N-1)^m \\binom{M+N-m-2}{M-m}, \\label{zstar}\n\\end{eqnarray}\nas follows from Eq.~(\\ref{zreg}). It is convenient to change\nthe summation index from $m$ to $j = M-m$ which can be interpreted as\na deficit of particles counted relatively to the full occupation:\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) = (N-1)^M \\sum_{j=0}^M (N-1)^{-j} \\binom{N+j-2}{j} . \\nonumber \\\\\n\\end{eqnarray}\nLet us assume that $N\\gg 1$. The summands in the last expression are strongly\nsuppressed when $j$ increases so the sum can \nbe approximated by changing the upper limit from $M$ to $\\infty$. We obtain\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) \n&\\cong & (N-1)^M \\sum_{j=0}^\\infty \\left(\\frac{-1}{1-N}\\right)^j \n\\binom{-(N-1)}{j} \\nonumber \\\\\n&=& (N-1)^M \\left(1-\\frac{1}{N-1}\\right)^{1-N} \\nonumber \\\\\n&=& (N-1)^M \\left( \\frac{N-1}{N-2} \\right)^{N-1}.\n\\end{eqnarray}\nUsing Eq.~(\\ref{pigeneral}) \nand the partition function $Z_{\\rm reg}$ \ncalculated in Eq. (\\ref{zreg}) of the previous subsection we can determine the distribution of particles\n$m$ at the central (singular) node,\n\\begin{eqnarray}\n&\\pi_1(m) &= \\frac{Z_{\\rm reg}(N-1,M-m)}{Z_{\\rm star}(N,M)} (N-1)^m \\nonumber \\\\\n&=& (N-1)^{m-M} \\binom{M+N-m-2}{M-m} \\left( \\frac{N-2}{N-1} \\right)^{N-1}. \\nonumber \\\\\n\\label{pi1star}\n\\end{eqnarray}\nSimilarly, we can determine the distribution of particles\non any external (regular) node $i$:\n\\begin{equation}\n\\pi_{i}(m) = \\frac{(N-2)(N-1)^{-m}}{N-1-(N-1)^{-M}} \\approx \\frac{N-2}{(N-1)^{m+1}}. \\label{pistext}\n\\end{equation}\nWe see that $\\pi_{i}(m)$ decays exponentially \nwith $m$ while $\\pi_1(m)$ grows exponentially for $m\\ll M$: \n\\begin{equation}\n\\pi_1(m) \\propto e^{ m\\left( \n-\\frac{1}{2M}+\\frac{1}{2(M+N-2)} +\\log\\frac{M(N-1)}{M+N-2} \\right)}.\n\\end{equation}\nThe growth slows down for $m$ approaching $M$. \nAt $m=M$, $\\pi_1(m)$ reaches its maximal value:\n\\begin{equation}\n\\pi_1(M) = \\left(\\frac{N-2}{N-1}\\right)^{N-1},\n\\end{equation}\nwhich tends to $e^{-1}$ when $N$ goes to infinity.\nIn Fig.~\\ref{f2} we compare the theoretical distributions \n(\\ref{pi1star}) and (\\ref{pistext}) with Monte Carlo \nsimulations of the ZRP for $N=20$ nodes and $M=20, 30, 40$. \nThe agreement is very good.\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\includegraphics[width=8cm]{f2.eps}\n\\caption{The ``experimental'' and the theoretical (solid lines) \nparticle number distributions for the star graph, \nfor $N=20$, and $M=20$ (crosses), $30$ (empty squares) and $40$ (filled squares). \nThe theoretical distributions were calculated according to the\nformula (\\ref{pi1star}) for the central node (rising curves) \nand according to Eq.~(\\ref{pistext}) for external nodes (falling curve, the Monte Carlo data (points) plotted for $N=20$ and $M=30$). \n}\n\\label{f2}\n\\end{figure}\n\nIt is instructive to calculate the mean number \nof particles at the central node. To simplify calculations\nwe make use of the distribution $\\widehat{\\pi}_1(j)\\equiv \\pi_1(M-j)$ of\nthe deficit of particles defined above:\n\\begin{equation}\n\\widehat{\\pi}_1(j) = \n(1-\\alpha)^{1-N} (-\\alpha)^j \\binom{-(N-1)}{j}.\n\\label{remain}\n\\end{equation}\nThe parameter $\\alpha=1\/(N-1)$ is just the ratio of any external node degree to the degree of the central node and measures\nthe level of inhomogeneity. \nThe overall prefactor $(1-\\alpha)^{1-N}$ is independent of $j$. It is just a normalization constant\nthat results from summing the $j$-dependent part of the expressions (\\ref{remain}),\n\\begin{equation}\nS(\\alpha) = \\sum_{j=0}^\\infty \n(-\\alpha)^j \\binom{-(N-1)}{j} = \n\\frac{1}{(1-\\alpha)^{N-1}} .\n\\label{auxs}\n\\end{equation}\nAs before we changed the upper limit from\n$M$ to infinity because $\\alpha \\ll 1$ for the star graph and hence\nthe summands are strongly suppressed for large $j$.\nThe average deficit at the central node is\n\\begin{equation}\n\\left = \n\\sum_{j=0}^M \\widehat{\\pi}_1(j) j = \n\\alpha \\frac{{\\rm d} \\log S(\\alpha)}{{\\rm d}\\alpha} = \\frac{N-1}{N-2}, \n\\label{m1star}\n\\end{equation}\nas follows from Eq.~(\\ref{auxs}). For large $N$ it tends to \none, so we have\n\\begin{equation}\n\\left = M - \\left \\cong M-1.\n\\end{equation}\nWe see that on average almost all balls are concentrated at the central\nnode and only one ball is in the rest of the system.\nWe can also determine the range of fluctuations around $\\left$\nby calculating the variance. Taking again advantage of the generating function (\\ref{auxs}) one finds\n\\begin{eqnarray}\n\\left<(m_1-\\left)^2\\right> = \\left<(j_1-\\left)^2\\right> \\nonumber \\\\\n= \\frac{{\\rm d}^2 \\log S(e^{-\\mu})}{{\\rm d}\\mu^2} = \\left(\\frac{N-1}{N-2}\\right)^2,\n\\end{eqnarray}\nwhere we used the parameter $\\mu=-\\log\\alpha=\\log (N-1)$.\nFor large $N$ the result tends to one, so\nwe can draw the following picture. \nFor any $N\\gg 1$ we observe a condensation of \nparticles at the central node regardless of their density $\\rho=M\/N$. \nThe critical density is equal to zero and the system is always in the condensed phase. \nThe condensate residing at the central node contains $M-1$ particles, with very small fluctuations,\nwhile the other nodes are almost empty. \n\n\\subsection{Single inhomogeneity}\nA very particular property of the star graph is that the\ninhomogeneity increases with its size.\nTherefore, it is interesting to consider the situation\nwhen the inhomogeneity $\\alpha = k\/k_1$ is arbitrary and independent of $N$. \nThe single inhomogeneity graph we consider here has one node of degree \n$k_1$ and $N-1$ nodes of degree $k$. Again, we assume that $k_1>k$.\nThe partition function (\\ref{part}) takes now the form\n\\begin{equation}\nZ_{\\rm inh}(N,M) = \n\\sum_{m_1=0}^M k_1^{m_1} \n\\sum_{m_2,\\dots,m_N=0}^M \\delta_{\\sum_i m_i,M} \n\\; k^{\\sum_{i=2}^N m_i}.\n\\label{zsh1}\n\\end{equation}\nThe sum over $m_2,\\dots,m_N$ is equal to the partition function \n$Z_{\\rm reg}(N-1,M-m_1)$ given by Eq.~(\\ref{zreg}).\nThe whole formula looks almost identical to that\nfor the star graph except that\nnow the degree $k_1$ does not need to be much greater than \n$k$ and therefore the substitution of $M$\nby $\\infty$ has to be done carefully in a manner\nincorporating finite-size corrections. As before, we first\nchange variables from $m_1$ to $j = M-m_1$. Using\nEq.~(\\ref{auxs}) we can cast the formula (\\ref{zsh1}) into the \nfollowing form:\n\\begin{eqnarray}\nZ_{\\rm inh}(N,M) &=& \nk_1^M \\sum_{j=0}^M \\alpha^j \n\\binom{N+j-2}{j} \\nonumber \\\\\n&=& k_1^M \\left[ (1-\\alpha)^{1-N} - c(M) \\right].\n\\label{zinh}\n\\end{eqnarray}\nThe correction $c(M)$ is equal to the sum over $j$ from $M+1$ to infinity.\nIt corresponds to the surplus which has to be\nsubtracted from the infinite sum represented by the first\nterm in square brackets. In the limit $M\\to\\infty$ it can be estimated\nas follows:\n\\begin{equation}\nc(M) = \\sum_{j=M+1}^\\infty \\alpha^j \\binom{N+j-2}{j} \n\\approx \\frac{1}{(N-2)!} \\int_M^\\infty {\\rm d} j \\, e^{F(j)},\n\\label{intc}\n\\end{equation}\nwhere \n\\begin{equation}\nF(j) = j \\log\\alpha + \\log\\left((N+j-2)!\\right) - \\log(j!).\n\\end{equation}\nUsing Stirling's formula we can calculate the \nintegral (\\ref{intc}) by the saddle-point method.\nTaking into account only leading terms we have\n\\begin{eqnarray}\n& &\\int_M^\\infty {\\rm d} j e^{F(j)} \\approx e^{F(j_*)} \\times \\nonumber \\\\\n& &\\times \\sqrt{\\frac{-\\pi}{2F''(j_*)}} \\mbox{erfc}\\left( (M-j_*)\n\\sqrt{-F''(j_*)} \\right),\n\\end{eqnarray}\nwhere erfc denotes the complementary error function,\n\\begin{equation}\n\\mbox{erfc}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty {\\rm d} y e^{-y^2} ,\n\\end{equation}\nand $j_*$ is determined from the saddle equation $F'(j_*)=0$:\n\\begin{equation}\nj_* \\approx \\frac{\\alpha (N-2)}{1-\\alpha}.\n\\end{equation}\nCollecting all terms we eventually find\n\\begin{eqnarray}\nc(M) &\\approx &\n\\frac{\\alpha^{\\frac{\\alpha(N-2)}{1-\\alpha}}}{1-\\alpha} \\frac{((N-2)\/(1-\\alpha))!}\n{(\\alpha(N-2)\/(1-\\alpha))!}\\sqrt{\\frac{\\pi \\alpha (N-2)}{2}} \\times \\nonumber \\\\\n&\\times & \\frac{1}{(N-2)!} \\mbox{erfc}\\left( \\frac{M(1-\\alpha)-\\alpha(N-2)}{\\sqrt{\\alpha(N-2)}}\\right)\t.\n\\label{cmfin}\n\\end{eqnarray}\nIn order to keep formulas shorter we used here the notation\n$x! \\equiv \\Gamma(1+x)$ also for non-integer arguments.\nThe complete partition function \n$Z_{\\rm inh}(N,M)$ is given by the right-hand side of Eq.~(\\ref{zinh}) \nwith $c(M)$ given by Eq.~(\\ref{cmfin}). \nWe can now calculate $\\pi_1(m)$, that is the distribution\nof balls at the singular node,\n\\begin{equation}\n\\pi_1(m) = \\frac{Z_{\\rm reg}(N-1,M-m)}{Z_{\\rm inh}(N,M)} k_1^m,\n\\end{equation}\nwhere $Z_{\\rm reg}$ is the partition function (\\ref{zreg}) for a \nregular graph with degree $k$. Using Eqs.~(\\ref{zreg}) and (\\ref{zinh})\nwe obtain\n\\begin{equation}\n\\pi_1(m) = \\binom{M+N-m-2}{M-m} \\frac{\\alpha^{M-m}}{(1-\\alpha)^{1-N} - c(M)}. \\label{pi1inh}\n\\end{equation}\nIn Fig.~\\ref{f3} we show the theoretical ball distributions for graphs \nwith $k=4$, $N=20$ and various $M$ for singular nodes with $k_1=8$ and $k_1=16$, respectively,\nand compare them with the corresponding results obtained\nby Monte Carlo simulations. The agreement, although very good for the presented plots, is the better, the smaller is the ratio $\\alpha=k\/k_1$.\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\psfrag{a}{$\\alpha=\\frac{1}{2}$}\n\\psfrag{a2}{$\\alpha=\\frac{1}{4}$}\n\\includegraphics[width=8cm]{f3a.eps}\n\\includegraphics[width=8cm]{f3b.eps}\n\\caption{The distribution of balls at the singular node for graphs \nwith $k=4$, $N=20$, $k_1=8$ (top) and $k_1=16$ (bottom). \nThe total number of balls is $M=20,40$ and $80$ from left to \nright curve, respectively. Points represent numerical data while solid \nlines show Eq.~(\\ref{pi1inh}).\n}\n\\label{f3}\n\\end{figure}\nNeglecting an inessential normalization, we see that\nEq.~(\\ref{pi1inh}) has the asymptotic behavior\n\\begin{equation}\n\\pi_1(m) \\propto \\left(\\frac{k_1}{k}\\right)^m \n\\binom{M+N-m-2}{M-m} \\sim \\exp(G(m)),\n\\end{equation}\nwhere \n\\begin{eqnarray}\nG(m) &=& \\left(M+N-m-\\frac{3}{2}\\right)\\log(M+N-m-2)- \\nonumber \\\\\n&-& m \\log\\alpha - \\left(M-m+\\frac{1}{2}\\right) \\log(M-m).\n\\label{gfunc}\n\\end{eqnarray}\nThe number of particles of the condensate can be \nestimated using the saddle point equation\n$G'(m_*)=0$ for $m_*>0$. Neglecting terms \nof order $1\/M^2$ we find\n\\begin{equation}\nm_* \\cong M - \\frac{\\alpha}{1-\\alpha} (N-2).\n\\end{equation}\nAlternatively one can calculate the number of particles\nof the condensate as the mean of the distribution $\\pi_1(m)$.\nAdapting the same trick as in the previous section,\n\\begin{eqnarray}\n\\left &=& M - \\left = \nM - \\alpha \\frac{{\\rm d} \\log S(\\alpha)}{{\\rm d}\\alpha} \\nonumber \\\\\n&=& M - \\frac{\\alpha}{1-\\alpha} (N-1) \\approx m_*,\n\\label{m1inh}\n\\end{eqnarray}\nas follows from Eqs.~(\\ref{auxs}) and (\\ref{m1star}). The criterion for\ncondensation is that the central node contains an extensive\nnumber of balls. In the limit $N,M\\to \\infty$ and fixed density\n$\\rho=M\/N$ it amounts to the condition $\\left>0$ \nleading to the critical density\n\\begin{equation}\n\\rho_c = \\frac{\\alpha}{1-\\alpha}.\n\\end{equation}\nThe condensation takes place when $\\rho>\\rho_c$ exactly like \nin the Single Defect Site model \\cite{evans}. \nThe critical density decreases with decreasing ratio $\\alpha=k\/k_1$ or, equivalently, \nwith increasing ``external field'' $\\mu=\\log(k_1\/k)$. The singular node attracts\n$N(\\rho-\\rho_c)+\\rho_c$ balls on average as \nfollows from Eq.~(\\ref{m1inh}).\nIt is also easy to find that the distribution of balls \n$\\pi_i(m)$ at any regular node falls exponentially,\n\\begin{equation}\n\\pi_i(m) \\sim \\left(\\frac{k}{k_1}\\right)^m = \\alpha^m = e^{-\\mu m},\n\\end{equation}\nthus the condensate never appears on it. \nA regular node contains on average $\\left = \\rho_c$ balls independently of the total density $\\rho$ of balls in the system\nas long as it exceeds $\\rho_c$.\n\n\\section{Dynamics of the condensate}\n\nLet us now turn to a discussion of the dynamics of the condensate.\nFrom the previous section we know that the condensate\nspends almost all time at the node with highest degree. However,\n occasionally it ``melts'' and disappears from the singular node for a short while.\nWe know that the probability of such an event is very small, so we expect \nthe life-time of the condensate to be very large. \nFollowing the ideas of \\cite{god}, let us imagine that we monitor only\nthe number of particles at the singular node, which fluctuates in time. \nThe temporal sequence of occupation numbers at this node\nperforms a sort of one-dimensional random walk \nand can be viewed as a Markov chain. Using a mean-field approximation\none can derive effective detailed balance equations for \nthe incoming and the outgoing flow of particles for this node.\nThe approximation is based on the assumption that the\nremaining part of the system is quickly thermalized, much faster\nthan the typical time scale of the melting process on the monitored node.\nThus the balance equations are written for the singular node and a single mean-field node having \nsome typical properties. \nFor this mean-field dynamics one can derive many quantities of interest.\nIn particular it is convenient to calculate the average \ntime $\\tau_{mn}$ it takes to decrease the occupation number of\nthe monitored node from $m$ to $n$,\nor more precisely, the average first passage time for the Markov process\ninitiated at $m$ to pass $n$. This quantity was first derived \nin \\cite{god} for the ZRP on a complete graph with outflow rates \n$u(m) = 1+b\/m$. The formula derived there,\n\\begin{equation}\n\\tau_{mn} = \\sum_{p=n+1}^{m} \\frac{1}{u(p)\\pi(p)} \\sum_{l=p}^{M} \\pi(l),\n\\label{Tmngen}\n\\end{equation}\ncan be easily adapted to the case discussed in our paper by \nsetting $u(p)=1$ and using the distribution $\\pi_1(p)$ of the singular node\nin place of $\\pi(p)$ in the original formula.\nEquipped with the formula for $\\tau_{mn}$ we are in principle able to calculate the \ntypical melting time $\\tau$. What is yet missing is the condition for $n$ at\nwhich the condensate can be considered as completely melted. \nWe shall choose the simplest possible criterion and define the\n``typical'' melting time $\\tau$ as $\\tau_{m0}$, that is the time needed to completely empty \nthe monitored node beginning from $m$ equal to the average occupation \nof the node in the steady state. \n\nIt was shown \\cite{god} that for the complete graph and $u(m) = 1+b\/m$, the melting time\nis approximately given by\n\\begin{equation}\n\\tau\\propto (\\rho-\\rho_c)^{b+1} M^b,\n\\end{equation}\nwhere $\\rho_c$ is the critical density above which the condensate\nis formed. The power-law increase with $M$ can be attributed\nto the power-law fall of $\\pi(m)\\sim m^{-b}$, characteristic for homogeneous systems\nwith $u(m)=1+b\/m$.\nThe key point of our paper is that for inhomogeneous networks the melting time does no longer \nfollow a power-law but instead increases {\\em exponentially} with $M$ due to the occurrence\nof the inhomogeneity which can be regarded as an external field $\\mu = \\log (k_1\/k)$, breaking the symmetry.\n\nBefore we do the calculations let\nus make a general remark about the dependence of $\\tau_{mn}$ on $m$ and $n$. A quick inspection of Eq.~(\\ref{Tmngen}) \ntells us that significant contribution to the sum over $p$ comes from terms \nfor which $\\pi(p)$, respectively $\\pi_1(p)$, is small. As we know from the previous section, in the condensed phase\n$\\pi_1(p)$ is many orders of magnitude greater for large $p$ than for\nsmall $p$. Therefore when $m$ is of order $M$, and $n$ is of order $1$, \nthe time $\\tau_{mn}$ varies very slowly with $m$ and, on the other hand, it is\nvery sensitive to $n$. We thus put $m=M$ for simplicity and concentrate on\n$\\tau_{Mn}$.\n\n\n\\subsection{Star graph}\nWe assume $u(p)=1$ as before. \nInserting the expression (\\ref{pi1star}) for the particle occupation distribution $\\pi_1(m)$ for the central node \nof the star into Eq.~(\\ref{Tmngen}) in place of $\\pi(m)$ we obtain\n\\begin{equation}\n\\tau_{mn} = \\sum_{p=n+1}^{m} \\sum_{l=p}^{M} (N-1)^{l-p} \n\\frac{(M+N-l-2)!(M-p)!}{(M+N-p-2)!(M-l)!} .\n\\label{tmnstar}\n\\end{equation}\nFrom Sec.~II~B we know that the condensate contains \n$m =\\left \\approx M$ balls in the steady state and that fluctuations \nare very small. This justifies the choice $m=M$ we made above.\nChanging the summation variables similarly as in the previous section\nwe find:\n\\begin{eqnarray}\n\\tau_{Mn} = (N-2)! \\sum_{r=0}^{M-n-1} \\frac{r!}{(N-2+r)!} (N-1)^r \\times \\nonumber \\\\\n\t\\times \\sum_{q=0}^r (N-1)^{-q} \\frac{(N-2+q)!}{q!(N-2)!}.\n\\end{eqnarray}\nIn the second sum we can move the upper limit to infinity using exactly\nthe same approximation as in Sec.~II~B:\n\\begin{equation}\n\\tau_{Mn} \\approx \\left(\\frac{N-1}{N-2}\\right)^{N-1}(N-2)! \n\\sum_{r=0}^{M-n-1} \\frac{r!(N-1)^r}{(N-2+r)!} .\n\\end{equation}\nAfter a variable change $r\\to M-n-1-r$, the remaining sum can be approximated as\n\\begin{eqnarray}\n\t\\sum_{r=0}^{M-n-1} \\frac{(M-n-1-r)!}{(M+N-n-3-r)!} (N-1)^{-r} \\approx \\nonumber \\\\\n\t\\approx \\frac{(M-n-1)!}{(M+N-n-3)!}\t \\sum_{r=0}^\\infty \\left(\\frac{M+N-n-3}{(M-n-1)(N-1)}\\right)^r, \\hspace{-5mm} \\nonumber \\\\\n\t\\label{eq:sumr0}\n\\end{eqnarray}\nsuch that we finally arrive at the formula:\n\\begin{eqnarray}\n\\tau_{Mn} \\approx \\left(\\frac{N-1}{N-2}\\right)^{N}(N-2)! (N-1)^{M-n-1} \\times \\nonumber \\\\\n\\times \\frac{M-n-1}{M-n-2} \\frac{(M-n-1)!}{(M+N-n-3)!}.\n\\label{TMnstar}\n\\end{eqnarray}\nWe see that the presence of $(N-1)^{-n}$ makes the time $\\tau_{Mn}$\nindeed very sensitive to $n$.\nIn Fig.~\\ref{f4} we see $\\tau_{M0}$ compared to computer simulations.\nThis complicated formula has a simple behavior in the limit of\nvery large systems and for $n=0$. In the limit of large $M$ and \nfor $N$ being fixed, the time $\\tau_{Mn}$ grows exponentially with $M$,\n\\begin{equation}\n\\tau_{M0} \\sim (N-1)^M = e^{\\mu M},\t\\label{tm0ap1}\n\\end{equation}\nwith $\\mu=\\log(k_1\/k)=\\log(N-1)$, while for fixed density $\\rho=M\/N$ \nand $N\\to\\infty$ it increases faster than exponentially,\n\\begin{equation}\n\\tau_{M0} \\sim e^{\\rho N \\log N}. \\label{tm0ap2}\n\\end{equation}\nThe approximate formulas (\\ref{tm0ap1}) and (\\ref{tm0ap2}) \ncan be alternatively obtained using a kind of Arrhenius law \\cite{arh,god},\nwhich states that the average life time is inversely proportional to the \nminimal value of the occupation number distribution:\n\\begin{equation}\n\\tau_{m0} \\sim 1\/\\pi_1({\\rm min}),\n\\end{equation}\nwhere one thinks about the condensate's melting as of tunneling \nthrough the potential barrier in a potential $V(m)=-\\log \\pi_1(m)$. In our case the potential $V(m)$ grows monotonically with $m$ going to zero, so the ball rather bounces from the wall at $m=0$ than tunnels through it, but the reasoning is the same.\nFrom Eq.~(\\ref{pi1star}) we have $\\pi_1({\\rm min})\\sim (N-1)^{-M}$ for \nfixed $N$ and large $M$ and we thus get again Eq.~(\\ref{tm0ap1}), while \nfor fixed density $\\pi_1({\\rm min})$ falls over-exponentially \nwhich results in Eq.~(\\ref{tm0ap2}).\n\\begin{figure}\n\\psfrag{x}{$M$}\n\\psfrag{y}{$\\tau_{M0}$}\n\\includegraphics[width=8cm]{f4.eps}\n\\caption{The average lifetime $\\tau_{M0}$ for the star graph with \n$N=10$ calculated from Eq.~(\\ref{TMnstar}) (solid line) and found in \ncomputer simulations (points).}\n\\label{f4}\n\\end{figure}\n\nSo far we have discussed the singular node. It is quite surprising that\nthe formula (\\ref{Tmngen}) works also well for regular nodes.\n If we blindly substitute $\\pi_1(k)$ by $\\pi_i(k)$ we get the following expression for $\\tau_{i,mn}$:\n\\begin{eqnarray}\n& &\\tau_{i,mn} = \\nonumber \\\\\n& &\\frac{(N-1)^n-(N-1)^m+(m-n)(N-2)(N-1)^M}{(N-2)^2(N-1)^{M-1}}. \\nonumber \\\\\n\\end{eqnarray}\nFor $m$ fixed, the transition time decreases \nalmost linearly with $n$. The typical occupation of the regular node\nis much smaller than $M$, so we concentrate on $m\\ll M,n=0$ and $N\\gg 1$.\nThe approximate formula reads:\n\\begin{equation}\n\\tau_{i,m0} \\approx m \\frac{N-1}{N-2}\t\\label{tm0ext},\n\\end{equation}\nand grows very slowly in comparison to the life time of the condensate \nat the singular node. This linear growth can be easily understood as \nthe minimal time needed for $m$ particles to hop out from a regular node.\nOne must remember that in practice we cannot observe transitions for large $m$, because the \nprobability of having such states is extremely small as it stems from \nEq.~(\\ref{pistext}) for $\\pi_i(m)$.\n\n\\subsection{Single inhomogeneity}\nNext we consider the single inhomogeneity graph from Sec.~II~C. \nIn the condensed phase the occupation $m$ fluctuates quickly \naround $\\left$ and even if it is smaller than $M$ we can\nassume that $\\tau_{mn}\\approx \\tau_{Mn}$ because the transition time $\\tau_{Mm}$\nfrom $M$ to $m$ balls is very small in comparison to $\\tau_{Mn}$ .\nTherefore we shall concentrate again on $\\tau_{Mn}$. \nFrom Eqs.~(\\ref{pi1inh}) and (\\ref{Tmngen}) we have\n\\begin{equation}\n\\tau_{Mn} = \\sum_{p=n+1}^M \\sum_{l=p}^M \\alpha^{p-l} \n\\frac{\\binom{M+N-l-2}{M-l}}{\\binom{M+N-p-2}{M-p}}.\n\\end{equation}\nChanging variables we get\n\\begin{eqnarray}\n\\tau_{Mn} = \\sum_{p=n+1}^M \\frac{(M-p)!}{(M+N-p-2)!} \\times \\nonumber \\\\\n\\times \\sum_{q=0}^{M-p}\\alpha^{-q} \\frac{(M+N-p-q-2)!}{(M-p-q)!}.\n\\end{eqnarray}\nThe sum over $q$ can be approximated by an\nintegral which can then be estimated by the saddle-point method.\nThe saddle point is $q_*=\\alpha (N-2)\/(1-\\alpha)$ as in \nSec.~II~C and therefore all calculations are almost identical. In \nthis way we obtain\n\\begin{equation}\n\\sum_q \\dots \\approx \\alpha^p \\times \\alpha^{\\alpha\\frac{N-2}{1-\\alpha}-M}\n\\frac{\\left(\\frac{N-2}{1-\\alpha}\\right)!}{\\left(\\frac{\\alpha(N-2)}{1-\\alpha}\\right)!}\n\\sqrt{\\frac{2\\pi \\alpha (N-2)}{(1-\\alpha)^2}},\n\\label{factor}\n\\end{equation}\nwhere we used again the notation \n$x!\\equiv \\Gamma(1+x)$.\nThe only dependence on $p$ in this expression is through the factor $\\alpha^p$.\nThus to calculate $\\tau_{Mn}$ it suffices to evaluate the sum\n\\begin{equation}\n \\sum_{p=n+1}^M \\alpha^p \\frac{(M-p)!}{(M+N-p-2)!}. \\label{eq:sumpn}\n\\end{equation}\nBecause every term in the sum is proportional to $1\/\\pi_1(p)$ from Eq.~(\\ref{pi1inh}), in the condensed phase the function under the sum has a minimum at the saddle point $p_*=m_*$. As this minimum is very deep, the effective contribution to the sum can be split into two terms: for small $p\\ll m_*$ and for $p\\gg m_*$. The ``small-$p$'' part can be evaluated like in Eq.~(\\ref{eq:sumr0}) by pushing the upper limit to infinity and approximating the ratio of factorials by some number to power $p$. To calculate the ``large-$p$'' part it is sufficient to take the last two terms in Eq.~(\\ref{eq:sumpn}), namely for $p=M$ and $p=M-1$, because they decrease quickly. The complete formula for $\\tau_{Mn}$ is finally given by\n\\begin{eqnarray}\n\t\\tau_{Mn} \\approx \\alpha^{\\alpha\\frac{N-2}{1-\\alpha}-M} \\frac{\\left(\\frac{N-2}{1-\\alpha}\\right)!}{\\left(\\frac{\\alpha(N-2)}{1-\\alpha}\\right)!}\n\t\\sqrt{\\frac{2\\pi \\alpha (N-2)}{(1-\\alpha)^2}} \\times \\nonumber \\\\ \n\t\\times \\left[ \\frac{M!}{(M+N-2)!} \\left(\\alpha\\frac{M+N-2}{M}\\right)^{n+1} \\times \\right. \\nonumber \\\\\n\t\\left. \\times \\left( 1-\\alpha\\frac{M+N-2}{M}\\right)^{-1} + \\frac{\\alpha^{M-1}(\\alpha(N-1)+1)}{(N-1)!} \\right]. \\nonumber \\\\\n\t\\label{compl}\n\\end{eqnarray}\nIn Fig.~\\ref{f5} we compare this theoretical formula with $\\tau_{M0}$ from numerical simulations. \nEquation (\\ref{compl})\nsimplifies in the limit of large systems. When \none allows $M\\to\\infty$ while keeping $N$ and $\\alpha$ fixed,\nthen the life time grows exponentially,\n\\begin{equation}\n\\tau_{M0} \\sim \\left(\\frac{1}{\\alpha}\\right)^M = e^{\\mu M}. \\label{tm0inh}\n\\end{equation}\nFor $\\mu=\\log(N-1)$, that is for a star graph, it reduces to the formula (\\ref{tm0ap1}).\nIn the limit of fixed density $\\rho=M\/N>\\rho_c$ \nand for $N,M\\to\\infty$:\n\\begin{equation}\n\\tau_{M0} \\sim e^{N\\left[ -\\log(1-\\alpha)+\n\\rho\\log(\\rho\/\\alpha)-(1+\\rho)\\log(1+\\rho)\\right]}. \\label{last}\n\\end{equation}\nWe see that the life time grows exponentially only if \\mbox{$k_1>k$}, that is for positive external field $\\mu=\\log(k_1\/k)$.\nAs before, we can explore the limit when the single inhomogeneity graph reduces to a star graph. \nInserting $\\mu=-\\log\\alpha=\\log(N-1)$ into Eq.~(\\ref{last}) we recover Eq.~(\\ref{tm0ap2}) as the leading term\nfor large $N$.\n\\begin{figure}\n\\psfrag{x}{$M$}\n\\psfrag{y}{$\\tau_{M0}$}\n\\psfrag{y2}{\\hspace{-5mm}$\\log \\tau_{M0}$}\n\n\\includegraphics[width=8cm]{f5.eps}\n\\caption{Comparison between the ``experimental'' results (points) and the\ntheoretical prediction (\\ref{compl}) (solid line) for $\\tau_{M0}$ of a $4$-regular graph\nwith single inhomogeneity $k_1=16$. The graph size is $N=20$. The inset compares $\\tau_{M0}$ calculated from Eq.~(\\ref{compl})\n(solid line) with that from Eq.~(\\ref{tm0inh}) (dashed line), valid in the large-$M$ limit. The prefactor\nin Eq.~(\\ref{tm0inh}) is chosen to match the $M\\to\\infty$ limit of the two formulas.}\n\\label{f5}\n\\end{figure}\n\n\\section{Conclusions}\nIn this paper we have studied static and dynamical properties of the condensation\nin zero-range processes on inhomogeneous networks. We have focused on the case where the network\nis almost a $k$-regular graph except that it has a single node of degree $k_1$\nlarger than $k$. This type of network could be a rude prototype \nof inhomogeneities encountered in scale-free networks having\na single hub with very high degree and many nodes of much smaller degrees.\nIndeed, from the point of view of the hub, \nthe remaining nodes look as if they formed an almost homogeneous system.\nWe have shown that the distribution of balls $\\pi_1(m)$ at the singular node has a maximum at $m\\approx N(\\rho-\\rho_c)$ where $\\rho_c$ is the critical density above which the condensate is formed. The average occupation of regular nodes is equal to $\\rho_c$ and the condensate never appears on them. However, the condensate is not a static phenomenon. It fluctuates and it sometimes\nmelts and disappears from the singular node. Then the surplus of balls distributes uniformly on all other nodes. After a while the condensate reappears and its typical life time $\\tau$ grows exponentially like $e^{\\mu M}$, where $M$ is the number of balls and $\\mu=\\log (k_1\/k)$ plays the role of an external field explicitly breaking the permutational symmetry of the system. This behavior is qualitatively distinct from that observed in homogeneous systems with a power-law distribution of balls, where $\\tau$ grows only like a power of $M$, and the symmetry is spontaneously broken. Thus the transition $\\mu=0\\to \\mu> 0$ changes dramatically all properties of the system.\n\nIn all above calculations we assumed for simplicity that the hop rate $u(m)=1$ and thus for the homogeneous system there would be no condensation. However, in the case $u(m)=1+b\/m$ which produces the power-law in $\\pi(m)$ for regular graphs, in an inhomogeneous system, apart from the condensation on the singular node, we would expect a second condensate on some regular node if the number of particles would be large enough to exceed the critical density for the homogeneous sub-system. Thus we could expect the presence of two critical densities $\\rho_{1c}$ and $\\rho_{2c}$ and two condensates having completely distinct properties. We leave this interesting question for future research.\n\n\\section{Acknowledgments}\nWe acknowledge support by the EC-RTN Network ``ENRAGE'' under \ngrant No. MRTN-CT-2004-005616, an Institute Partnership grant \nof the Alexander von Humboldt Foundation,\na Marie Curie Host Development Fellowship under \nGrant No. IHP-HPMD-CT-2001-00108 (L.B. and W.J.), \na Marie Curie Actions Transfer of Knowledge project ``COCOS'',\nGrant No. MTKD-CT-2004-517186 and a Polish Ministry of Science\nand Information Society Technologies Grant 1P03B-04029 (2005-2008)\n(Z.B.). B.W. thanks the German Academic Exchange Service (DAAD) for a fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nBy comparing the black hole physics with the thermodynamics \nand discovering of the black hole evaporation by Hawking, \nit was shown that the black hole entropy is proportional to the\nhorizon area\\cite{Bekenstein1,Hawking}.\n\\be\nS_{BH} = \\frac{A_H}{4}\n\\ee\nin unit $\\hbar = c = G = 1$.\nIn Euclidean path integration approach it was shown the tree level \ncontribution of the gravitation action gives the black hole \nentropy\\cite{Gibbons}.\nHowever the exact statistical origin of the Bekenstein-Hawking \nblack hole entropy is unclear.\n\nRecently many efforts have been concentrated to understand \nthe statistical origin of black hole thermodynamics, specially the \nblack hole entropy by various methods (for review \nsee \\cite{Bekenstein2}): \n't Hooft was calculated the entropy of a quantum field propagating\nthe outside of the black hole. After the regularization he obtained\n $S = 1\/4 A_H$ (the brick wall method)\n \\cite{tHooft,Pa1,uglum,barbon}.\nAnother approach is to identify the black hole entropy \nwith the entanglement entropy $S_{ent}$.\nEntanglement entropy arises from ignoring the degree of freedom\nof a proper region of space: $S = - Tr \\rho \\ln \\rho$.\nIt is found that the entropy is proportional to the area of the \nboundary\\cite{Ent}.\nIn fact the entanglement entropy and the brick wall method are \nequivalent.\nFrolov and Novikov argued that the black hole entropy can be \nobtained by identifying the dynamical degrees of freedom \nwith the states of all fields which are located inside the \nblack hole\\cite{Frolov}. \nThe leading term of the entropy obtained by those methods \nis proportional to the surface area of the horizon.\nHowever the proportional coefficient diverges as the cut off goes to \nzero.\nThe conical approach also gives similar result with \nothers\\cite{conical}. \nThe divergence is because of an \ninfinite number of states near the horizon,\nwhich can be explained by the equivalence principle \\cite{Barbon}.\nAn alternative approach by Frolov is to identify the black hole \nentropy with the thermodynamic one. In this approach the entropy\nis finite\\cite{Frolov2}.\nHowever they all treat the only spherical symmetrical black hole.\n\nIf the black hole has a rotation, what is changed ?\nIt is well known that in a rotating black hole spacetime a particle\nwith a zero angular momentum dropped from infinity is dragged \njust by the influence of gravity so that it acquires an angular \nvelocity in the same direction in which the black hole rotates. \nThe dragging becomes more and more extreme the nearer one approaches\nthe horizon of the black hole.\nThis effect is called the dragging of inertial frames\\cite{Misner}. \n\nThus the field at equilibrium with the rotating black hole\nmust also be rotating. The rotation is not rigid but\nlocally is different.\nSo the velocity of the radiation does not exceed the velocity of light. \nHowever we do not know how to treat the equilibrium state with \na locally different angular velocity. More precisely there \nis no global static coordinates. So we assume that \nthe radiation has a rigid rotation $\\Omega_0$ small than or equal to\nthe extremum value of the local rotation. In a rotating black hole\nthe extremum value of it is $\\Omega_H$, which is the \nangular velocity of the event horizon.\n\nRecently we considered the black hole entropy by the \nbrick wall method in the charged Kerr black hole in\\cite{minho} \nand showed the entropy is proportional to the event horizon \nin Hartle-Hawking states.\nIn this paper to more deep understand the black hole entropy we shall\ninvestigate the black hole entropy by the brick wall method\nin various stationary black holes: \nthe Kaluza-Klein black hole \\cite{kaluza} which is the solution of \nthe 4-dimensional effective theory reduced from the 5-dimensional \nKaluza-Klein theory, and the Sen black hole\\cite{sen} which \nis the solution of the Einstein-Maxwell-dilaton-antisymmetric tensor\ngauge field theory came from the heteroitic string theory, \nand the Kerr-Newman black hole\\cite{kerr} which is the solution of \nthe Einstein-Maxwell theory.\n\nIn order to understand the equilibrium state of the radiation \n(the field) in the rotating black hole spacetime in Sec.2 we will \nfirst consider the rotating heat bath in the flat spacetime.\nIn Sec.3 we will consider the radiation in equilibrium\nstate in Rindler spacetime with rotation, which is the most simple\nspacetime having the event horizon and a rotation. \nIn Sec.4 we will investigate the entropy of the quantum field\nin the stationary black hole background.\nWe find the condition to give the finite value to the free energy\nand the entropy. \nIn Sec.5 we calculate the entropy in Hartle-Hawking state for the \nrotating black holes.\nFinal section is devoted to the summary.\n\n\n\\section{A Rotating Heat Bath }\n\nLet us consider a massless scalar field with a constant angular \nvelocity $\\Omega_0$ about $z-$axis at thermal equilibrium \nwith a temperature $T = 1\/\\beta$ in Minkowski \nspacetime, of which line element in cylindrical coordinate is given by\n\\begin{equation}\nds^2 = -dt^2 + r^2 d \\phi^2 + dr^2 + dz^2. \\label{metric1}\n\\end{equation}\nIn this spacetime the positive frequency field mode can be written as\n $\\Phi_{q,m} (x) = f_{q m } (r,z) e^{- i \\omega t + i m \\phi } $,\n where $q$ denotes a quantum number and $m$ is the \n azimuthal quantum number.\n\nFor such a equilibrium ensemble of the states of the scalar \nfield the partition function is given by\n\\begin{equation}\nZ = \\sum_{n_q, m} e^{- n_q ( \\omega_q - m \\Omega_0 ) \\beta } \n\\end{equation}\nand the free energy is given by \n\\begin{equation}\n \\beta F = \\sum_m \\int_0^\\infty d \\omega g(\\omega, m) \n \\ln \\left( 1 - e^{- \\beta (\\omega - m \\Omega_0)} \\right),\n\\end{equation}\nwhere $g(\\omega ,m)$ is the density of state for \na fixed $\\omega$ and $m$.\n\nFollowing 't Hooft we assume that all possible modes of \na scalar field vanish\nat $r = r_1 $ ( $r_1$ is very small.)\nand at $r = L$. In the WKB approximation with \n$\\Phi = e^{i S(r) - i \\omega t + i m \\phi + i k z }$ the radial wave \nnumber $K( x, \\omega,m) = \\partial_r S $ is given by\n\\begin{equation}\nK^2 ( x, \\omega, m) = \\omega^2 - \\frac{m^2}{r^2} - k^2. \\label{con1}\n\\end{equation}\nThis expression denotes the ellipsoid in momentum phase space at a \nfixed frequency $\\omega$.\nThe total number of modes with energy less than $\\omega$ and a fixed $m$\nis obtained by integrating over the volume of phase space, which is \ndetermined by (\\ref{con1})\n\\begin{eqnarray}\n\\nonumber\n\\Gamma (\\omega,m) &=& \n \\sum_m \\int d \\phi dz \\int_{r_1}^L dr \\frac{1}{\\pi} \n \\int dk K(x,\\omega,m) \\\\\n &=& \\frac{1}{\\pi} \\sum_m \\int d \\phi dz \\int_{r_1}^L dr \n \\int dk \\left( \\omega^2 - \\frac{m^2}{r^2} - k^2 \\right)^{1\/2}.\n \\label{Pvol}\n\\end{eqnarray}\nThe integration over $k$ must be carried out over the phase space that\nsatisfies $ K^2 \\ge 0$. \n$\\Gamma ( \\omega, m)$ can be obtained by investigating\nthe shape of the expression (\\ref{con1}) in momentum phase space.\nThus the free energy, after the integration by parts, becomes\n\\begin{eqnarray}\n\\nonumber\n\\beta F &=& - \\beta \\sum_m \\int_0^\\infty d \\omega \\Gamma ( \\omega,m)\n \\frac{1}{e^{\\beta ( \\omega - m \\Omega_0 )} - 1 } \\\\\n &=& - \\frac{\\beta}{2} \\int_0^\\infty d \\omega \n \\int_{- r \\omega}^{r \\omega} d m\n ( \\omega^2 - \\frac{m^2}{r^2} ) \n \\frac{1}{e^{\\beta ( \\omega - m \\Omega_0 )} - 1 }, \n\\end{eqnarray}\nwhere we assume that the azimuthal quantum number $m$ is a continuous \nparameter. \nBy making the change of variable $ m = r \\omega u $ we obtain \nthe free energy \n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi dz \\int_{r_1}^L \n\\frac{r}{( 1 - v^2)^2} dr, \\label{free1}\n\\end{equation}\nwhere $N$ is a constant and $v = r \\Omega_0$.\nNote that as $L$ goes to $1\/\\Omega_0$ this partition function diverges\nas $\\gamma^4$, where $\\gamma = ( 1 - v^2 )^{-1\/2}$.\n\nFrom the expression (\\ref{free1}) it is easy to obtain expressions \nfor the energy $E$, angular momentum $J$, and entropy $S$ of radiation\n\\begin{eqnarray}\nJ & = & \\langle m \\rangle_{av} = \\frac{1}{\\beta} \n \\frac{ \\partial}{\\partial \\Omega_0} (\\beta F) \n = 4 N \\frac{1}{\\beta^4} \\Omega_0\n \\int r^2 \\gamma^6 r dr d \\phi dz, \\\\\nE & = & \\langle \\omega \\rangle_{av} = \\Omega_0 \\cdot J - \n \\frac{\\partial}{\\partial \\beta } ( \\beta F) = N \\frac{1}{\\beta^4} \n \\int ( 3 + v^2 ) \\gamma^4 r dr d\\phi dz, \\\\\nS & = & \\beta^2 \\frac{ \\partial}{\\partial \\beta } F = \n4 N \\frac{1}{\\beta^3} \\int \\gamma^4 r dr d \\phi dz. \\label{entropy1}\n\\end{eqnarray}\nThese coincides with those in ref.\\cite{zurek}.\nSimilarly to the free energy $F$ these expressions $J, E,$ and $S$ \n diverge as $ L \\rightarrow 1\/\\Omega_0$.\nThe divergence is related to the rigid rotation.\nIn rigid rotating system the velocity of the comoving observer \ngrows as one move from the origin to infinity.\nSo beyond some point the velocity exceeds the velocity of the light. \nThis is unphysical. Thus a rotating system cannot have the size \ngreater than $1\/\\Omega_0$.\nTherefore to obtain a finite value for $J, E,$ and $S$, we must take\n$L < 1\/\\Omega_0$. In such a finite system $\\omega > m \\Omega_0$.\n \n\n\nNow let us consider above problem in co-moving coordinate that are\nrotating with angular velocity $\\Omega_0$. The line element in comoving\nframe is given by\n\\begin{equation}\nds^2 = - (1 - \\Omega_0^2 r^2 )dt^2 + 2 \\Omega_0 r d \\phi' dt \n+ dr^2 + dz^2, \\label{metric2}\n\\end{equation}\nwhere we have used $\\phi' = \\phi - \\Omega_0 t$.\nIn this coordinate the positive frequency field mode is written as\n$\\Phi_{q m} (x) = {\\bar f}_{qm} (r,z) e^{ - i \\omega' t + i m \\phi' }$.\n\nBecause in comoving frame the field has no rotation the free energy is \ngiven by\n\\begin{equation}\n\\beta F = \\int_0^\\infty d \\omega' g' ( \\omega') \\ln \n \\left( 1 - e^{- \\beta \\omega'} \\right), \\label{free3}\n\\end{equation}\nwhere $g' (\\omega')$ is the density of state for a fixed $\\omega'$. \nIn WKB approximation the Klein-Gordon equation $\\Box \\Phi = 0$ yields\nthe constraint\\cite{Mann} \n\\begin{equation}\ng^{ab} k_a k_b = 0 \\label{con2}\n\\end{equation}\nor\n\\begin{equation}\n- ( \\omega' - \\Omega_0 m)^2 + ( \\frac{1}{r^2} m^2 + k^2 + p^2 ) = 0,\n\\label{con3}\n\\end{equation}\nwhere $p = \\frac{\\partial S}{\\partial r}$.\nIn region where $\\Omega_0 r < 1$, for a fixed $\\omega'$, this expression\nrepresents the ellipsoid in momentum space. Therefore the total number \nof modes with energy less than $\\omega'$ is given by\n\\begin{eqnarray}\n\\Gamma' ( \\omega' ) &=& \\frac{1}{\\pi} \\sum_m d \\phi dz \\int dr\n\\int dk \\left(\n( \\omega' - m \\Omega_0)^2 - \\frac{m^2}{r^2 } - k^2 \\right)^{1\/2} \n\\label{star} \\\\\n&=& \\frac{4}{3} \\int d \\phi dz \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2} ~\\omega^{'3}, \\label{pvol}\n\\end{eqnarray}\nwhich is the volume of the ellipsoid.\nThe expression (\\ref{star}) is just the same form as Eq.(\\ref{Pvol})\nwhen $\\omega \\rightarrow \\omega - m \\Omega_0$.\nThe phase volume (\\ref{pvol}) diverges as $L \\rightarrow 1\/\\Omega_0$.\nInserting the expression (\\ref{pvol}) into (\\ref{free3}) and integrating\nwe get\n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi dz \\int_{r_1}^L dr\n \\frac{r}{( 1- \\Omega_0^2 r^2 )^2}. \\label{free4}\n\\end{equation}\nThis expression is the same with Eq(\\ref{free1}).\nFrom this we get the energy $E'$ and the entropy $S$:\n\\begin{eqnarray}\nE' &=& \\langle \\omega' \\rangle_{av} = \n - \\frac{\\partial}{\\partial \\beta } (\\beta F ) = 3 \n \\frac{N}{\\beta^4} A \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2}, \\\\\nS &=& \\beta^2 \\frac{\\partial}{\\partial \\beta }( \\beta F) = \n 4 \\frac{N}{\\beta^3} A \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2},\n\\end{eqnarray}\nwhere $A = \\int d \\phi dz$.\nIt is noted that the entropy $S$ is the same with Eq.(\\ref{entropy1}) \nand the energy $E'$ is satisfied with $E' = E - \\Omega_0 J$.\nThis fact show that the coordinate transformation to comoving frame\nonly change the energy and not change the entropy in WKB approximation.\nThus in the case of calculating the entropy or the free energy \nit is convenient to choose the comoving frame.\nIt is noted that in co-moving frame the divergence is related to the \ntime component $g_{tt}$ of the metric (\\ref{metric2}).\n\n\n\\section{A Thermal Bath in Rindler Spacetime with a Rotation}\n\nIn this section we will consider the thermal equilibrium \nstate of the scalar field with the mass $\\mu$ and an uniform \nrotation about $z-$axis in the Rindler spacetime. \nThe line element of the Rindler spacetime in cylindrical coordinates \nis given by \n\\begin{equation}\nds^2 = - \\xi^2 d \\eta^2 + d \\xi^2 \n + r^2 d \\phi^2 + d r^2. \\label{metric3} \n\\end{equation}\nIn this spacetime the event horizon is at $\\xi = 0$, and \n$\\xi = constant$ represent the trajectory of \nthe uniform acceleration\\cite{birrell}.\nThe importance of the Rindler space-time is that in the large\nblack hole mass limit the metric of the black space-time reduces\nto that of the Rindler space-time\\cite{uglum}.\n\nAs in Sec.2, the WKB approximation with \n$\\Phi (x) = e^{- i \\omega t + i m \\phi + i S(\\xi, r)}$\nyields\n\\begin{equation}\nK^2 ( \\xi, r,\\omega, m) = \n\\frac{\\omega^2}{\\xi^2} - \\frac{1}{r^2} m^2 - p_r^2 - \\mu^2,\n \\label{con4}\n\\end{equation}\nwhere $ K = \\partial_\\xi S$ and $p_r = \\partial_r S$.\nIn this section we will calculate the free energy by \nusing the slightly different method with that in section 2.\n\nIt is important to note that in WKB approximation the density \nof state $g(\\omega,m)$ is determined by the constraint (\\ref{con4}),\nand that the free energy is singular at $ \\omega = m \\Omega_0$. \nIn particular if $\\omega - m \\Omega_0 < 0 $ \nthe free energy becomes an imaginary number.\nHowever in the WKB approximation we can easily see \n$\\bar{\\omega} = \\omega - m \\Omega_0 > 0$ in the region such that\n$ \\xi -\\Omega_0 r > 0$. But in the region such that $ \\xi - \\Omega_0\nr < 0$ it is possible that $\\omega - m \\Omega_0 < 0$.\n( More details are in Sec.4.)\nTherefore to obtain the finite value for the free energy \nwe must require the system to be in the region such \nthat $ \\xi - \\Omega_0 r >0$. Then the free energy is written as\n\\begin{eqnarray}\n\\nonumber\n\\beta F &= & \\sum_m \\int_{m \\Omega_0}^\\infty d \\omega g( \\omega, m) \n\t \\ln \\left( 1 - e^{- \\beta (\\omega - m \\Omega_0)} \\right) \\\\\n \\nonumber\n\t &=& \\int_0^\\infty d \\omega \\sum_m g(\\omega + m \\Omega_0,m) \\ln \n\t \\left( 1 - e^{- \\beta \\omega} \\right) \\\\\n\t &=& - \\beta \\int_0^\\infty d \\omega \\frac{1}{e^{\\beta \\omega } \n\t -1 }\n\t \\int d m \\Gamma ( \\omega + m \\Omega_0,m),\n\\end{eqnarray}\nwhere we have integrated by parts and assumed that the quantum number\n$m$ is a continuous variable. \nThe total number of modes with energy less than $\\omega$ \nis obtained by integrating over the volume of phase space\n\\begin{eqnarray} \n\\nonumber\n\\Gamma(\\bar{\\omega}) &=& \\int d m \\Gamma (\\omega + m \\Omega_0,m) \\\\\n\\nonumber\n &=& \\int dm\n \\int d \\phi dr \\int_{r_1}^L d \\xi \\frac{1}{\\pi}\n \\int d p_r K(\\xi, r, \\omega + m \\Omega_0 ,m) \\\\\n &=& \\frac{1}{\\pi} \\int dm \\int d \\phi dr \\int_{r_1}^L d \\xi\n \\int d p_r \\left(\n \\frac{\\omega^2}{\\xi^2} + \\frac{2}{ \\xi^2}\n m \\Omega_0 \\omega + \n \\frac{m^2 \\Omega_0^2}{\\xi^2}\n - \\frac{1}{r^2} m^2 - p_r^2 - \\mu^2\n \\right)^{1\/2}. \\label{pvol2}\n\\end{eqnarray}\nThe integrations over $m$ and $p_r$ must be carried out over the \nphase space that satisfies $ K^2 ( \\omega + m \\Omega_0,m) \\ge 0$.\nAfter the integration we obtain \nthe number of states with energy less than $\\omega$, \nwhich is given by \n\\begin{equation}\n\\Gamma (\\omega) = \\frac{4}{3} \\int d^3 x \n\t\\frac{\\xi r}{\\sqrt{( \\xi^2 - \\Omega_0^2 r^2 )}}\n\t\\left( \\frac{\\omega^2}{ \\xi^2 - \\Omega_0^2 r^2 } \n\t - \\mu^2 \\right)^{3\/2}.\n\\end{equation}\nThus the free energy becomes\n\\begin{equation}\n\\beta F = - \\frac{4}{3} \\beta \\int d^3 x \n\\int_{\\mu \\sqrt{ \\xi^2 - \\Omega_0^2 r^2 } }^\\infty \nd \\omega \\frac{1}{e^{\\beta \\omega } - 1 } \n\t\\frac{\\xi r}{\\sqrt{( \\xi^2 - \\Omega_0^2 r^2 )}}\n\t\\left( \\frac{\\omega^2 }{ \\xi^2 - \\Omega_0^2 r^2 } \n\t - \\mu^2 \\right)^{3\/2}.\n\\end{equation}\nFor a massless scalar field ( $\\mu =0$ ) the free energy becomes\n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi d r \\int_{\\xi_1}^L d \\xi\n\t\\frac{\\xi r}{( \\xi^2 - \\Omega^2 r^2 )^2}. \\label{Free}\n\\end{equation}\nFrom this we get the energy $E$, the angular momentum $J$, \nand the entropy $S$ of the field\n\\bea\nJ &=& \\bra m \\ket_{av} = 4 \\frac{N}{\\beta^4} \\Omega_0\n\\int \\frac{r^2}{( \\xi^2 - \\Omega_0^2 r^2 )^3} \\xi r d \\xi dr dz, \\\\\nE &=& \\bra E \\ket_{av} = \\frac{N}{\\beta^4} \n\\int \\frac{3 \\xi^2 + \\Omega_0^2 r^2 }{ ( \n \\xi^2 - \\Omega_0^2 r^2 )^3 } \\xi r d \\xi dr dz, \\\\\nS &=& 4 \\frac{N}{\\beta^3} \\int \\frac{1}{( \n \\xi^2 - \\Omega_0^2 r^2 )^2} \\xi r d \\xi dr dz. \\label{ent2}\n\\eea\nIt is noted that the thermodynamic quantities $F, E$, and $S$ \nare divergent\nas $\\xi \\rightarrow \\Omega_0 r$ rather than the event horizon.\nOnly in $\\Omega_0 = 0$ case the divergence occurs at the horizon\n$\\xi = 0$. Such a fact can be easily understand in the co-moving \nframe, of which line element is given by \n\\begin{eqnarray}\nds^2 &=& - \\xi^2 d \\eta^2 + r^2 ( d \\phi' + \\Omega_0 d \\eta )^2 + \n d \\xi^2 + d r^2 \\\\\n\\nonumber\n &=& - ( \\xi^2 - \\Omega_0^2 r^2 ) d \\eta^2 + 2 \\Omega_0 r^2 d \\eta \n d \\phi' + r^2 d {\\phi'}^2 + d \\xi^2 + d r^2, \n\\end{eqnarray}\nwhere we used $\\phi' = \\phi - \\Omega_0 \\eta $.\nIn this spacetime the event horizon is at $\\xi = 0$. In addition to the \nevent horizon there is a stationary limit surface \nat $\\xi = \\Omega_0 r$,\nwhere the Killing vector $\\partial_\\eta$ becomes null. That \nsurface is the elliptic hyper-surface\\cite{letaw}. In the \ninterval $ 0 < \\xi < \\Omega_0 r$, the Killing vector is spacelike.\nWe can also show that the entropy in the co-moving frame\nis the same form with Eq.(\\ref{ent2}).\n{\\it These facts imply that the divergence of the thermodynamic \nquantities is deeply related to the stationary\nlimit surface in the co-moving frame rather than the event horizon.}\n\n\n\n\\section{A Entropy of a Scalar Field in a Rotating Black Hole }\n\n\\subsection{General Formalism}\n\nLet us consider a scalar field with mass $\\mu$\nin thermal equilibrium at temperature $1\/\\beta$ in the \n rotating black hole background, \n of which line element is generally given by\n\\be\nds^2 = g_{tt}(r, \\theta) d t^2 \n + 2 g_{t \\phi}( r,\\theta) dt d \\phi \n + g_{\\phi \\phi}(r, \\theta) d\\phi^2 \n + g_{rr} (r,\\theta) d r^2 \n + g_{\\theta \\theta}(r, \\theta) d\\theta^2. \n \\label{Metric} \\\\\n\\ee \nThis metric has two Killing vector fields: the timelike \nKilling vector $\\xi^\\mu = (\\partial_t)^\\mu$ and the axial\nKilling vector $\\psi^\\mu =(\\partial_\\phi)^\\mu$.\nThe metrics, we concern, of the Kaluza-Klein, the Sen, and \nthe Kerr-Newman black holes are in the appendix.\nThe properties of those metrics are \n\\be\ng_{tt} g_{\\phi \\phi} - g^2_{t \\phi} = - \\Delta(r) \\sin^2 \\theta\n\\rightarrow 0 \n\\ee\nand\n\\be\n\\left( g_{tt} g_{\\phi \\phi} - g^2_{t \\phi}\\right) g_{rr} \n\\rightarrow finite\n\\ee\nas one approaches the horizon. \nAnother property is that\nthere are two important surfaces (the event horizon and the\nstationary limit surface), and the two surfaces does not coincide.\nOn the stationary limit surface the Killing vector $\\xi^\\mu$ vanishes,\nand the Killing vector $\\xi^\\mu + \\Omega_H \\psi^\\mu$ is null on\nthe horizon, where $\\Omega_H$ is the angular velocity of the\nhorizon.\n\nThe equation of motion of the field with mass $\\mu$ and \narbitrary coupled to the scalar curvature $R(x)$ is\n\\be\n\\left[ \\Del_\\mu \\Del^\\mu - \\xi R - \\mu^2 \\right] \n\\Psi = 0, \\label{equation}\n\\ee\nwhere $\\xi$ is an arbitrary constant.\n$\\xi = 1\/6$ and $\\mu =0$ case corresponds to the conformally\ncoupled one.\nWe assume that the scalar field is rotating with a constant \nazimuthal angular velocity $\\Omega_0$. \nThe associated conserved quantities are\nangular momentum $J$. \nThe free energy of the system is then given by \n\\be\n F = \\frac{1}{\\beta} \\sum_m \\int_0^\\infty d \\E g(\\E, m) \\ln \n \\left( 1 - e^{- \\beta( \\E - m \\Omega_0 )} \\right), \n\\ee\nwhere $g(\\E,m)$ is the density of state for a given $\\E$ and $m$. \n\n\nTo evaluate the free energy we will follow \nthe brick wall method of 't Hooft \\cite{tHooft}.\nFollowing the brick wall method we impose a small radial cut-off $h$\nsuch that \n\\begin{equation}\n\\Psi (x) = 0 ~~~~{\\rm for }~~~ r \\leq r_H + h,\n\\end{equation}\nwhere $r_H$ denotes the coordinate of the event horizon.\nTo remove the infra-red divergence we also introduce another \ncut-off $ L \\gg r_H$ such that \n\\be\n\\Psi (x) = 0~~~~ {\\rm for} ~~~r \\geq L.\n\\ee\nIt is noted that the brick wall is spherically symmetric.\nIn the WKB approximation with $\\Psi = \ne^{- i \\E t + i m \\phi + i S(r, \\theta)}$\nthe equation (\\ref{equation}) yields \nthe constraint \\cite{Mann}\n\\begin{equation}\n p_r^2 = \\frac{1}{g^{rr}} \\left[\n - g^{tt} \\E^2 + \n 2 g^{t \\phi} \\E m - g^{ \\phi \\phi } m^2 \n - g^{ \\theta \\theta } p_\\theta^2 \n - V(x) \\right], \\label{Con1}\n\\end{equation}\nwhere $ p_r = \\partial_r S$, $ p_\\theta = \\partial_\\theta S$, and\n$V(x) = \\xi R(x) + \\mu^2$.\nIn WKB approximation it is important to note that\nthe number of state for a given $\\E$ is determined by \n$p_\\theta, p_r$ and $m$.\nThe number of mode with energy less than $\\E$ and with a fixed\n$m$ is obtained by integrating over $p_\\theta$ in phase space. \n\\bea\n\\nonumber\n\\Gamma (\\E,m ) &=& \\frac{1}{\\pi} \\int d \\phi d \\theta \\int dr\n\\int d p_\\theta p_r ( \\E, m,x) \\\\\n&=& \\frac{1}{\\pi} \\int d \\phi d \\theta \\int dr\n\\int d p_\\theta \n \\left[ \\frac{1}{g^{rr}} \\left(\n - g^{tt} \\E^2 + \n 2 g^{t \\phi} \\E m - g^{ \\phi \\phi } m^2 \n - g^{ \\theta \\theta } p_\\theta^2 \n - V(x) \\right) \n \\right]^{\\frac{1}{2}}. \n\\eea\nThe integration over $p_\\theta$ must be carried over the phase space\nsuch that $p_r \\geq 0$.\n\nIn this point we need some remarks.\nIn a rotating system, in general, there is a superradiance effect,\nwhich occurs when $ 0 < \\E < m \\Omega_0$.\nFor this range of the frequency the free energy $F$ becomes a complex\nnumber. In case $\\E = m \\Omega_0$ the free energy is divergent.\nTherefore to obtain a real finite value for the free energy $F$,\nwe must require that $\\E > m \\Omega_0$. ( For $ 0 < \\E < m \\Omega_0$\nthe free energy diverges. See below.) This requirement say that\nwe must restrict the system to be in the region \nsuch that $g_{tt}^{'} \\equiv g_{tt} + 2 \\Omega_0 g_{t \\phi} \n+ \\Omega_0^2 g_{\\phi \\phi} < 0$. \nIn this region $ \\E - m \\Omega_0 >$,\nso the free energy is a finite real value.\nIt is easily showed as follows.\nLet us define $ E = \\E - m \\Omega_0$.\nThen it is written as \n\\bea\n\\nonumber\nE &=& \\left(\n \\frac{g^{t \\phi}}{ g^{tt} } - \\Omega_0 \n \\right) m\n + \\frac{1}{- g^{tt}} \n \\left[\n \\left( \n g^{t \\phi} m \n \\right)^2 \n + \\left( - g^{tt} \\right)\n \\left( V + g^{\\phi \\phi} m^2 + \n g^{rr} p_r^2 + g^{\\theta \\theta }p_\\theta^2 \n \\right)\n \\right]^{1\/2} \\\\\n &=& \\left( \n \\Omega - \\Omega_0 \n \\right) m + \\frac{ -\\D }{g_{\\phi \\phi}}\n \\left[\n \\frac{1}{-\\D} m^2 + \\frac{ g_{\\phi \\phi} }{- \\D }\n \\left( V + \\frac{p_r^2 }{g_{rr}}\n + \\frac{p_\\theta^2}{g_{\\theta \\theta} } \n \\right)\n \\right]^{1\/2}, \\label{Con2}\n\\eea\nwhere we used \n\\be\ng^{tt} = \\frac{g_{\\phi \\phi}}{ \\D},~~\ng^{t \\phi} = \\frac{ - g_{t \\phi}}{\\D},~~\ng^{\\phi \\phi} = \\frac{ g_{tt}}{\\D},\n\\ee\nand $ \\Omega = -\\frac{ g_{t \\phi}}{g_{\\phi \\phi}} $.\nHere $ - \\D = g_{t \\phi}^2 - g_{tt}g_{\\phi \\phi}$.\nFrom Eq.(\\ref{Con2}), for all $m, p_r$ and $p_\\theta$, \none can see the condition such that $ E >0$ is\n\\be\n \\frac{ \\sqrt{-\\D } }{ g_{\\phi \\phi}} \\pm \n\\left( \\Omega - \\Omega_0 \\right) > 0\n\\ee\nor\n\\be\ng_{tt}^{'} \\equiv g_{tt} + 2 \\Omega_0 g_{t \\phi} \n+ \\Omega_0^2 g_{\\phi \\phi} < 0.\n\\ee\n Therefore in the region such that $ - g_{tt}^{'} >0$ \n ( called region I) the free energy is a real, but\n in the region such that $- g_{tt}^{'} < 0$ (called region II)\n the free energy is complex.\nHowever in the region I the integration over the momentum \nphase space is convergent. \nBut in the region II the integration over the momentum \nphase is divergent.\n These facts become more apparent if we investigate the momentum \n phase space.\n In the region I \nthe possible points of $p_i$ satisfying $ \\E - \\Omega_0 p_\\phi = E$ \nfor a given $E$ are located on the following surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} +\n \\frac{- {g'}_{tt}}{- \\cal D} \\left(\n p_\\phi + \\frac{g_{t \\phi } \n + \\Omega_0 g_{\\phi \\phi}}{{g'}_{tt}} \n E \\right)^2 \n = \\left( \\frac{ E^2}{- {g'}_{tt} } \n - V \\right), \\label{ellipsoid}\n\\end{equation}\nwhich is the ellipsoid, {\\it a compact surface}. \nHere $p_\\phi = m$.\nSo the density of state $g(E)$ for a given $E$ is finite and \nthe integrations \nover $p_i$ give a finite value.\nBut in the region II \nthe possible points of $p_i$ are located on the following surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} -\n \\frac{ {g'}_{tt}}{- \\cal D} \\left( \n p_\\phi + \\frac{g_{t \\phi } + \n \\Omega_0 g_{\\phi \\phi}}{{g'}_{tt}} \n\t E \\right)^2 \n = - \\left( \n \\frac{ E^2 }{ {g'}_{tt}} \n + V \\right),\n\\end{equation}\nwhich is the hyperboloid, {\\it a non-compact surface}. So \n$g(E)$ diverges and the integration over $p_i$ diverges. \nIn case of $g^{'}_{tt} = 0$, the possible points are \ngiven by the surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} =\n\\frac{p_\\phi - \\left( \\frac{ g_{\\phi \\phi} E^2 }{ \\cal D } + V \n \\right)\/ \\left( \\frac{ 2 g_{t \\phi} }{ \\cal D } E \n\t \\right) \n }{ \\frac{- {\\cal D} }{ 2 g_{t \\phi} E } \n },\n\\end{equation}\nwhich is elliptic paraboloid and also $non-compact$. Therefore \nthe value of the $p_i$ integration are divergent.\nActually the surface such that ${g'}_{tt} = 0$ is the velocity of the\nlight surface (VLS). Beyond VLS (in region II) the co-moving \nobserver must move more rapidly than the\nvelocity of light. \nThus we will assume that the system is in the region I.\n( For the possible region I see Sec. 4.2.)\nFor example, in the case of $\\Omega_0 = 0$ the points \nsatisfying ${g'}_{tt} =0$ are on the stationary limit \nsurface.\nThe region of the outside (inside) of the stationary \nlimit surface corresponds to the region I (II).\nIn the rotating system in Sec. 2 the region I is $ r < 1\/\\Omega_0$\nand $ r > 1\/\\Omega_0$ corresponds to the region II.\nIn the Rindler spacetime with a rotation\n$ \\xi > \\Omega_0 r$ corresponds to the region I, and $\\xi < \\Omega_0 r$\nto the region II.\n\n\n\n\n\nWith the assumption that the system is in the region I \nwe can obtain the free energy as follows\n\\begin{eqnarray}\n\\nonumber\n \\beta F &=& \\sum_m \\int_{m \\Omega_0}^\\infty d \\E g(\\E, m) \\ln \n \\left( 1 - e^{- \\beta( \\E - m \\Omega_0 )} \\right) \\\\\n\\nonumber\n &=& \\int_0^\\infty d\\E \\sum_m g(\\E + m \\Omega_0 , m) \\ln \n \\left( 1 - e^{- \\beta \\E } \\right) \\\\\n &=& - \\beta \\int_0^\\infty d\\E \\frac{1}{e^{\\beta \\E} - 1}\n \\int d m \\Gamma (\\E + m \\Omega_0, m),\n\\end{eqnarray}\nwhere we have integrated by parts and we assume that the quantum \nnumber $m$ is a continuous variable. \nThe integrations over $m$ and $p_\\theta$ yield \n\\begin{equation}\nF = - \\frac{4 }{3} \n\\int d \\phi d \\theta \\int_{r_H + h}^L dr\n\\int_{V(x) \\sqrt{- {g'}_{tt}}}^\\infty d\\E \n\\frac{1}{e^{\\beta \\E} - 1 }\n\\frac{ \\sqrt{g_4}}{\\sqrt{ - {g'}_{tt} }} \\left( \n\\frac{\\E^2}{ - {g'}_{tt} } - V(x) \\right)^{3\/2 }. \n\\label{freeenergy}\n\\end{equation} \nIn particular when $\\Omega_0 = 0$ and non-rotating case \n$g_{t \\phi} = 0$, \nthe free energy (\\ref{freeenergy}) \ncoincides with the expression in ref.\\cite{tHooft,barbon} and \nit is proportional to the volume of the optical space in the limit\n$V(x) = 0$ \\cite{optical}.\nIt is easy to see that the integrand diverges as \n$ r_H + h $ or $L$ approach the surface such that $g_{tt}^{'} = 0$. \nIn that case the contribution of the $V(x)$ can be negligible. \n\nFor a massless and minimally coupled scalar field case\n($\\mu = \\xi = 0$) the free energy reduces to \n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3 } \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 } \n = - N \\int_0^\\beta d \\tau \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\sqrt{g_4} \n \\frac{1}{ \\beta_{local}^4}, \\label{freeenergy2}\n\\end{equation}\nwhere $\\beta_{local} = \\sqrt{ - g^{'}_{tt } } \\beta$ is \nthe reciprocal of the local Tolman temperature \\cite{Tolman}\nin the comoving frame.\nThis form is just the free energy of a gas of \nmassless particles at local\ntemperature $1\/\\beta_{local}$.\n\n\nFrom this expression (\\ref{freeenergy2}) \nit is easy to obtain expressions for \nthe total energy $U$, angular momentum $J$, and entropy $S$ of \na scalar field\n\\begin{eqnarray}\nJ &=& \\langle m \\rangle =\n - \\frac{1}{\\beta} \\frac{\\partial}{\\partial \\Omega_0}\n( \\beta F) = \\frac{4 N}{\\beta^4} \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 }\n\\frac{ g_{\\phi \\phi}}{( - {g'}_{tt})} \n\\left( \\Omega_0 - \\Omega \\right), \\\\\nU &=& \\langle \\E \\rangle = \\Omega_0 J + \\frac{\\partial}{\\partial \\beta} \n( \\beta F) = \\frac{N}{\\beta^4} \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 }\n\\left[ \n3 + 4 \\frac{ \\Omega_0 \\left( \\Omega_0 - \\Omega \\right)\n g_{\\phi \\phi } }{( - {g'}_{tt})} \\right], \\\\\nS &= & \\beta^2 \\frac{\\partial}{\\partial \\beta } F = \n \\beta ( U - F - \\Omega_0 J) = \n 4 \\frac{N}{\\beta^3 } \\int d \\theta d \\phi \n \\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 },\n\\end{eqnarray}\nwhich are also divergent as one approach the surface such \nthat $ g_{tt}^{'} =0$.\n\n\n\\subsection{The region such that $ - g_{tt}^{'} > 0$.}\n\nIn this section we study where is the possible region I\nfor three black hole, the Kaluza-Klein,\nand the Sen, the Kerr-Newman black holes,\nfor $ \\Omega_0 = \\Omega_H, \\Omega_0\n< \\Omega_H$ and the extreme case with $\\Omega_0 = \\Omega_H$. \n\n\\subsubsection{The Kaluza-Klein black hole }\nA) $\\Omega_0 =\\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case the position of the light \nof velocity surface is exactly found. \nIn such a case $g^{'}_{tt}$ can be written as\n\\bea\ng^{'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t\\phi} + \n\t \\Omega_H^2 g_{\\phi \\phi} \\\\\n\\nonumber\n&=& \\frac{\\mu^2}{B \\Sigma} ( x - \\bar{r}_H ) \n \\left\\{\n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 }\n ( 1 - v^2) x^3 + \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n \\left( 2 - \\bar{r}_- (1 - v^2) \\right) x^2 \n \\right. \\\\\n\\nonumber\n & &~~ +\n \\left[ -1 + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n\t \\left(\n 4 + y^2 ( 1- v^2) \\cos^2 \\theta - 2 \\bar{r}_- \n\t \\right) \n \\right] x \\\\\n\\nonumber\n& & ~~\\left.\n + \\left[\n \\bar{r}_- + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n\t \\left(\n -4 \\bar{r}_H - \\bar{r}_- y^2 ( 1- v^2) \\cos^2 \\theta \n \\right) \n \\right] \n \\right\\} \\\\\n &\\equiv& \n \\frac{\\mu^2}{B \\Sigma} ( x - \\bar{r}_H ) \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } ( 1- v^2)\n \\left( \n x^3 + a_1 x^2 + a_2 x + a_3 \n \\right)\n\\eea\nfor $\\theta \\neq 0$, where $x = \\frac{r}{\\mu },\ny = \\frac{a}{\\mu}, \\bar{r}_H = \\frac{r_H}{\\mu}$, and \n$ \\bar{r}_- = \\frac{ r_-}{\\mu}$.\nFrom this we can see that there are two VLS.\nOne is the horizon ($r = r_H$), and another \nlight of velocity surface (call outer VLS) is given by\\cite{Table}\n\\be\nr_{VLS} = 2 \\mu \\sqrt{-Q} \\cos \\left( \n \\frac{1}{3} \\Theta \\right)\n - \\frac{1}{3} a_1 \\mu,\n\\ee \nwhere\n\\be\n\\Theta = {\\rm arccos} \\left( \n\t \\frac{ P}{ \\sqrt{ - Q^2}} \\right)\n\\ee\nwith \n\\be\nQ = \\frac{ 3 a_2 - a_1^2}{9},~~~\nP = \\frac{9 a_1 a_2 - 27 a_3 - 2 a_1^3 }{54}.\n\\ee\nIn case of the slowly rotating black hole ($a$ is small)\nthe VLS is approximately given by\n\\be\nr_{VLS} \\sim 2 \\mu \\frac{r_H}{a\\sqrt{1 - v^2}\\sin\\theta} - \n\\frac{1}{3}\n\\left( \\frac{2 }{1 - v^2} - \\frac{r_-}{\\mu} \\right) \\mu,\n\\ee\nwhich is an open, roughly, cylindrical surface.\nAs $v \\rightarrow 1$ or $a \\rightarrow 0$ the VLS become more \ndistant, which came from the fact that as $v \\rightarrow 1$ or\n$ a \\rightarrow 0$ the coordinate angular velocity\n$\\frac{d \\phi}{d t} = - \\frac{g_{t \\phi}}{g_{\\phi \\phi}}$ becomes\nvanish.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig1.eps,height=8cm, angle=0}}\n\\vspace{0.1cm}\n{\\footnotesize Figure 1: The position of the outer velocity of \nlight surface for the Kaluza-Klein black hole. }\n\\end{figure}\nFor $\\theta = 0$ it is always that $g_{tt}^{'} <0$ for $r >r_H$. \nAs $ a \\rightarrow \\mu$ the outer VLS approaches horizon. See\nFig.1.\n\n\nB) $ \\Omega_0 < \\Omega_H $ {\\it case}:\nIn this case $g^{'}_{tt} = 0 $ is a fourth order polynomial equation \nin $r$ for a given $\\theta$.\nThe region I corresponds to $ r_{in} < r < r_{VLS}$. \nAt $\\theta = \\pi\/2$\n$r_{in}$ is between the stationary limit surface and \nthe event horizon, \nand at $\\theta = 0$ $r_{in}$ contacts with the event horizon.\n Actually the inner VLS $r_{in}$ places between the stationary\n limit surface and the event horizon for all $\\theta$.\nThe particular point is that as $ \\Omega_0 \\rightarrow \\Omega_H$, \n$r_{in}$ approaches the horizon.\nHowever it does attach the horizon only when $\\Omega_0 = \\Omega_H$.\nWhile, the outer velocity of light surface locates at the very far \ndistance from the horizon, and it is a roughly cylindrical \nsurface as in case $\\Omega_0 = \\Omega_H$. \nFor the position of the inner VLS see Fig.2.\n\n\nC){ {\\it the extreme black hole case with }} $\\Omega_0 = \\Omega_H $: \nThe extreme black hole for the Kaluza-Klein black hole occurs when \n$ \\mu^2 = a^2$. In this case the inner horizon and outer horizon are \nat the same place.\nAt $\\theta = 1\/2 \\pi$, $g_{tt}^{'}$ is written as\n\\be\ng^{'}_{tt} = \\frac{\\mu^2}{B \\Sigma} \n( x - \\bar{r}_H )^2 x \\left( \n x + \\frac{2}{1 - v^2} \\right)\n\\frac{ 1 - v^2 }{4},\n\\ee\nwhich shows that the possible region such that $g^{'}_{tt} < 0 $ \ndoes not exist at $\\theta = 1\/2 \\pi$.\nTherefore in the extreme black hole case it is impossible to consider\nthe brick wall model of 't Hooft.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig2.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 2: The position of $ r_{in}$ at $ \\theta = 0.5 \\pi$ for the \nKaluza-Klein black hole. $v = 0.5$.\n\\end{figure}\n\n\n\\subsubsection{The Sen black hole}\n\nA) $\\Omega_0 = \\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case \n$g^{'}_{tt}$ can be written as\n\\bea\ng^{'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t\\phi} + \n\t \\Omega_H^2 g_{\\phi \\phi} \\\\\n\\nonumber\n&=& \\frac{\\mu^2}{ \\Sigma} ( x - \\bar{r}_H ) \n \\left\\{\n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma}\n x^3 + \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n \\left( 2 \\cosh 2 \\gamma - \\bar{r}_- \\right) x^2 \n \\right. \\\\\n \\nonumber\n & &~~ +\n \\left[ -1 + \\frac{ y^2 \\sin^2 \\theta }{ \\bar{r}_H^2 } \n + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n\t \\left(\n y^2 \\cos^2 \\theta - 2 \\bar{r}_- \\cosh 2 \\gamma\n\t \\right) \n \\right] x \\\\\n\\nonumber\n& & ~~\\left.\n + \\left[\n \\bar{r}_- + \\frac{ y^2 \\sin^2 \\theta }{ \\bar{r}_H } \n - \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n\t \\left(\n \\bar{r}_- y^2 \\cos^2 \\theta \n \\right) \n \\right] \n \\right\\} \\\\\n &\\equiv& \n \\frac{\\mu^2}{ \\Sigma} ( x - \\bar{r}_H ) \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n \\left( \n x^3 + a_1 x^2 + a_2 x + a_3 \n \\right)\n\\eea\nfor $\\theta \\neq 0$, where $x = \\frac{r}{\\mu },\ny = \\frac{a}{\\mu}, \\bar{r}_H = \\frac{r_H}{\\mu}$, and \n$ \\bar{r}_- = \\frac{ r_-}{\\mu}$.\nThen the exact position of the inner VLS and outer VLS are\nare given by\n\\be\nr_{in} = r_H,~~~ \nr_{VLS} = 2 \\mu \\sqrt{-Q} \\cos \\left( \n \\frac{1}{3} \\Theta \\right)\n - \\frac{1}{3} a_1 \\mu.\n\\ee \nThe position of the outer VLS for small $a$ is approximately given by\n\\be\nr_{VLS} \\sim \\frac{2 \\mu r_H \\cosh^2 \\gamma}{a\n \\sin \\theta }\n - \\frac{ 1}{3} \\left( 2 \\cosh(2\\gamma) - \\frac{r_-}{\\mu} \\right) \\mu,\n \\ee\nwhich is an open, roughly, cylindrical surface.\nAs $ a \\rightarrow 0 $ the VLS goes to the infinity, and it \ndisappears when $a = 0$.\nAs $ \\gamma$ or $a $ is increasing the VLS approaches the horizon.\nAt $ \\theta = \\frac{1}{2} \\pi$, similarly to the \nKaluza-Klein black hole, $g^{'}_{tt} < 0$ for $ r > r_H$.\nSee Fig.3.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig3.eps , height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 3: The position of the outer velocity of light surface for the \nSen black hole. $\\gamma = 5.0$.\n\\end{figure}\n\n\nB) $ \\Omega_0 < \\Omega_H $ {\\it case}:\nIn this case $g^{'}_{tt} = 0 $ is also a fourth order equation \nin $r$ for a given $\\theta$.\nSimilarly to the Kaluza-Klein black hole the region I is \n$ r_{in} < r < r_{VLS}$. \nAt $\\theta = 0 $ the inner VLS $r_{in}$ is at the horizon,\nand at $\\theta = \\pi\/2$\n$r_{in}$ locates at the between the stationary limit surface and \nthe event horizon. See Figure 4. \nAs $ \\Omega_0 \\rightarrow \\Omega_H$, $r_{in} $ approaches \nto the horizon. Only when $\\Omega_0 = \\Omega_H$ it coincides with the\nevent horizon.\nThe outer velocity of light surface, in case of small $a$, locates\nat the very far distance from the horizon, \nand it is a roughly cylindrical \nsurface.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig4.eps, height= 8cm, angle=0}}\n\\vspace{0.1cm}\nFigure 4: The position of the inner velocity of light surface for \nthe Sen black hole. $\\gamma = 5.0, \\theta = 0.5 \\pi$. \n\\end{figure}\n\n\nC){ {\\it the extreme black hole case with }} $\\Omega_0 = \\Omega_H $: \nThe extreme black hole for the Kaluza-Klein black hole occurs when \n$ \\mu^2 = a^2$. \nIn this case the inner horizon \nand outer horizon are at the same place.\nAt $\\theta = \\frac{1}{2} \\pi $ $g_{tt}^{'}$ is written as\n\\be\ng^{'}_{tt} = \\frac{\\mu^2}{ \\Sigma} \n( x - \\bar{r}_H )^2 x \\left( \n x + 2 \\cosh 2 \\gamma \\right)\n\\ee\nwhich shows that the possible region such that $g^{'}_{tt} < 0 $ \ndoes not exist at $\\theta = 1\/2 \\pi$.\nTherefore in the extreme black hole case it is impossible to consider\nthe brick wall model of 't Hooft.\n\n \n\n\\subsubsection{The Kerr-Newman black hole}\n\n\nA) $\\Omega_0 = \\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case we can exactly find the \nposition of the light of velocity surface. \nIn such a case ${g'}_{tt}$ can be written as\n\\begin{eqnarray}\n{g'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t \\phi} + \\Omega_H^2 \n\t g_{\\phi \\phi} \\\\\n\\nonumber\n\t &=& \\frac{M^2}{\\Sigma} ( x - \\br_H) \\left\\{\n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta ~x^3 + \\bar{r}_H\n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta ~x^2 \\right. \\\\\n\\nonumber\n\t & & ~ + \\left[ -1 + \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left(\n\t y^2 + y^2 \\cos^2 \\theta + \\bar{r}_H^2 \\right) \n\t \\right] x \\\\\n &+ & \\left. \\left[ \n 2 \\left( 1 - \\bar{\\Omega}_H y \\sin^2 \\theta \\right)^2 - \\br_H + \n \\br_H \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left( \\br_H^2 + \n y^2 + y^2 \\cos^2 \\theta \\right)\n \\right] \\right\\} \\\\\n\t &\\equiv& \\frac{M^2}{\\Sigma} ( x - \\br_H ) \n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left(\n\t x^3 + a_1 x^2 + a_2 x + a_3 \\right)\n\\end{eqnarray}\nfor $\\theta \\neq 0$, where\n$x = r\/M, y = a\/M, z = e\/M, \\bar{\\Omega}_H = M \\Omega_H, \\br_H = r_H \/M $.\nThen the exact position of the outer light of velocity surface \nis given by \n\\be\n r_{VLS} = 2 M \\sqrt{\n - Q} \\cos \\left( \\frac{1}{3} \\Theta \\right) - \\frac{1}{3} a_1 M.\n \\label{sol}\n \\ee\nFor small $a$ Eq. (\\ref{sol}) is approximately given by \n \\be\n r_{VLS} \\sim \\frac{1}{\\Omega_H \\sin \\theta } - \\frac{r_H}{3 },\n \\ee\n which is an open, roughly, cylindrical surface.\n For $\\theta = 0$ it is always that ${g'}_{tt} < 0 $ for \n $r > r_H$. \n As $ a \\rightarrow 0$, $r_{VLS}$ goes to infinity, and\n as $ a \\rightarrow \\sqrt{M^2 + e^2 }$ it approaches \n the event horizon. See Fig.5.\n The inner VLS $r_{in}$ is the event horizon.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig5.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 5: The position of the outer light of velocity surface\nfor the Kerr-Newman black hole. $e = 0.0$. \n\\end{figure}\n\nB) $\\Omega_0 < \\Omega_H$ {\\it case}:\nIn this case, similarly to other black holes, \nthe inner VLS $r_{in}$ approaches to the horizon as \n$\\Omega_0 \\rightarrow \\Omega_H$. See Fig.6. \nThe inner VLS is a compact surface, which shrink to horizon as\n$\\Omega_0 \\rightarrow \\Omega_J$. See Fig.7.\nThe outer VLS is at far place, which disappears when $\\Omega_0 = 0$.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig6.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 6: The position of the inner light of surface for the \nKerr-Newman black hole. $\\theta = 0.5 \\pi$. \n\\end{figure}\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig7.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 6: The shape of the inner light of surface for the \nKerr-Newman black hole. $a = 0.8 M, e = 0 $. \n\\end{figure}\n\nC) {\\it the extreme black hole case with} $\\Omega_0 =\\Omega_H$: \n For extreme Kerr-Newman black hole case, which occurs when\n $M^2 = a^2 + e^2 $, \n $g_{tt}^{'}$ at $ \\theta = \\frac{1}{2} \\pi$ \n is written as \n \\be\ng_{tt}^{'}=\\frac{M^2}{\\Sigma} \\frac{y}{1 + y^2}\n( x - 1)^2 \n\\left(x + 1 - \\frac{1}{y} \\right)\n\\left(x + 1 + \\frac{1}{y} \\right)\n\\ee\nFrom this we obtain \n the position of VLS at $\\theta = \\frac{\\pi}{2} $ as\n\\bea\n r &=& M ~~~~~~{\\rm for ~} \\frac{1}{2} M~\\leq a \\leq M~ {\\rm and }~ a = 0, \\\\\n r &=& \\left( -1 + \\frac{M}{a} \\right) M ~~~~~{\\rm for ~} \n 0 < a < \\frac{1}{2} M. \n\\eea\nThe second case corresponds to the extreme black hole that\nis slowly rotating and has many charge. (In this case \n $ e > \\sqrt{3}\/2M \\approx 0.866 M $). \nIn particular in case of $e \\leq \\sqrt{3}\/2 M $ ( $a = M$ for $ e = 0$) \nthe horizon and the light of velocity surface \nare at the same position. \n Therefore in case of the extreme black hole with $a \\geq 1\/2 M $\n it is impossible to consider the brick wall model of 't Hooft. \n\n\n\n\n\\section{The Entropy in the Hartle-Hawking Vacuum}\n\nThe Hartle-Hawking vacuum state is one that\nthe angular velocity $\\Omega_0$ is equal to that of the \nevent horizon, and the temperature $\\beta$ is equal to the Hawking\ntemperature, where the Hawking temperature and the angular velocity of \nthe horizon are defined as\\cite{Wald}\n\\be\nT_H = \\frac{\\kappa}{2 \\pi},~~~ \\Omega_H = \\lim_{r \\rightarrow r_H}\n\\left( - \\frac{g_{t \\phi}}{g_{\\phi \\phi}} \\right).\n\\ee\nHere $\\kappa$ is the surface gravity of the horizon.\n\nFirst of all let us assume that $\\Omega_0 = \\Omega_H $.\nIn this case, as stated in Sec.4, \nthe possible region I is $r_H < r < L < r_{VLS}$.\nThe outer brick wall must locate inside the outer VLS.\nThis fact was already pointed out by Frolov and Thorne \\cite{Thorne} \nto remove the singular structure of the Hartle-Hawking vacuum and\nmodify it.\nNow recall that in general ${g'}_{tt}|_{r = r_H} =0$.\nThis came from that ${g'}_{tt}$ is the same form as \n$\\chi^\\mu \\chi_\\mu = (\\xi^\\mu + \\Omega_H \\psi^\\mu)(\n\\xi_\\mu + \\Omega_H \\psi_\\mu)$, and $\\chi^\\mu$ is null on the\nhorizon.\nSo it follows that ${g'}_{tt} = ( r - r_H) G(r, \\theta)$, where\n$G(r,\\theta)$ is a non-vanishing function at $r = r_H$ except \nthe extremal case.\n( We can not consider the extreme black hole case.) \n\nTherefore for the three black holes the leading behaviors of the free \nenergy $F$ for very small $h$ are then given by \n\\bea\n\\beta F &\\approx& - \\frac{N}{\\beta^3}\n\\int d \\theta d \\phi \\int_{r_H + h}^L dr \\frac{\n\\sqrt{g_4}}{ ( - g_{tt}^{'} )^2 } \\\\\n&=& - \\frac{N}{\\beta^3}\n\\int d \\theta d \\phi \\int_{r_H + h}^L dr \\frac{\nD(r)}{( r - r_H)^2 G^2(r, \\theta) },\n\\eea\nwhere $D(r,\\theta) = \\sqrt{g_4}$. Since $D(r,\\theta) $ and \n$G(r,\\theta)$ are non-vanishing functions at $r = r_H$ we can expand it\nabout $r= r_H$ as follows.\n\\bea\nD(r,\\theta) &=& D(r_H,\\theta) + D^{'} (r_H, \\theta) (r -r_H) +\nO((r- r_H )^2 ),\\\\\n\\frac{1}{ G^2(r, \\theta)} &=&\n\\frac{1}{ G^2(r_H, \\theta)} + \n\\left( \\frac{1}{G^2(r_H,\\theta)} \\right)^{'} \n+ O((r - r_H)^2 ),\n\\eea\nwhere $'$ denotes the partial derivative for $r$.\nSo the free energy is approximately given by\n\\bea\n\\nonumber\n\\beta F &\\approx & - \\frac{N}{\\beta^3}\n\\int d \\phi d \\theta \\int dr \n\\left\\{ \\frac{D(r_H, \\theta)}{G^2(r_H,\\theta)} \n\\frac{ 1}{(r - r_H)^2 } +\n\\left( \\frac{ D(r_H,\\theta) }{G^2(r_H,\\theta) } \\right)^{'} \n\\frac{1}{(r - r_H)}\n + O((r - r_H)^0) \n\\right\\} \\\\\n&=& \n- \\frac{2 \\pi N}{\\beta^3} \\left\\{\n\\frac{1}{h} \\int d \\theta \n \\frac{D(r_H, \\theta) }{G^2(r_H,\\theta) } \n- \\ln (h) \\int d \\theta \n\\left( \\frac{ D(r_H,\\theta) }{G^2(r_H,\\theta) } \\right)^{'} \n+ ... \\right\\},\n\\label{gen}\n\\eea\nwhich show that \ngenerally, in addition to the linear divergence term in $h$, \nthere is a logarithmic one in the case of rotating black hole.\n\nIf we written the free energy in terms of the proper distance \ncut-off $\\epsilon$, it become very simple form.\n \\bea\n \\nonumber\n\\beta F &\\approx& - \n\\frac{N}{ \\beta^3} \\int_{r = r_H} d \\phi d \\theta \n\\sqrt{g_{\\theta \\theta} g_{\\phi \\phi}} \n \\int_{r_H + h}^L dr \\sqrt{g_{rr}}\n\\left( \\frac{g_{\\phi \\phi}}{g^2_{t \\phi} -g_{tt}g_{\\phi \\phi}}\n\\right)^{3\/2} \\\\\n&\\approx& - \\frac{N}{ 2 ( \\kappa \\beta)^3 } \n \\frac{A_H}{\\epsilon^2},\n \\label{free2}\n\\eea\nwhere $A_H$ is the area of the event horizon,\n and $\\epsilon$ is the \nproper distance from the horizon to $r_H + h$.\n\\be\n\\epsilon = \\int_{r_H}^{r_H + h} dr \\sqrt{g_{rr}}.\n\\ee\nHowever the proper distance cut-off is dependent on the \ncoordinate $\\theta$, which is the general property of the \nrotating black hole.\n\nFrom the free energy $F$ we obtain the leading behaviors of \nthe entropy $S$ as\n\\bea\n\\nonumber\nS &=& \\beta^2 \\frac{\\partial}{\\partial \\beta} F \\\\\n &\\approx& \n \\frac{N}{\\beta^3} \\left( A ~ \\frac{1}{h} + B \\ln (h) + finite \n \\right),\n\\eea\nwhere $A$ and $B$ are in $c$-number in Eq.(\\ref{gen}),\n or\n\\be \nS \\approx \\frac{4 N}{ 2 ( \\kappa \\beta)^3 } \n\\frac{A_H}{\\epsilon^2}. \n\\label{Entropy} \n\\ee\nThe entropy $S$ is linearly and logarithmically divergent \nas $h \\rightarrow 0$. \nThe divergences arise because the density of state for a given $E$\n diverges as $h$ goes to zero.\n\nNow we take $T$ as the Hartle-Hawking temperature \n$T_H = \\frac{ \\kappa}{2 \\pi}$. \nThen the entropy becomes\n\\be\nS_H \\approx \\frac{N 8 \\pi^3 }{\\kappa^3} \\left(\nA~ \\frac{1}{h} + B \\ln (h) + finite \\right),\n\\ee\nor\n\\be\nS_H \\approx \\frac{N}{4 \\pi^3}\n\\frac{ A_H}{ \\epsilon^2}. \\label{result} \n\\ee\nThe entropy of a scalar field in Hartle-Hawking state \ndiverges quadratically \nin $\\epsilon^{-1}$ as the system approaches the horizon.\nOr it diverges in $h^{-1}$ and $ \\ln (h) $.\nIn case $ a= 0$ our result (\\ref{result}) agrees with the \nresult calculated by 't Hooft \\cite{tHooft} and \nwith one in ref.\\cite{ohta}.\nThese facts imply that the leading behaviors of entropy (\\ref{result})\nis general form.\n\n\n\\section{Summary and Conclusion}\n\nBy using the brick wall method we have calculated the entropies \nof the rotating systems with a rotation $\\Omega_0$ \nat thermal equilibrium with temperature $T$ in the \nrotating black holes. \nIn WKB approximation to get the real finite free energy and entropy \nthe system must be in the region I.\nAs the system approaches the VLS ( $r_{in}$ and $r_{VLS}$)\nthe thermodynamic quantities become divergent. From this fact \n{\\it we conclude that the divergence of the thermodynamic quantities \nincluding the entropy is related to the stationary limit\nsurface in the co-moving frame}. In spherical symmetric black hole\nthe stationary limit surface and the event horizon are coincide.\nOnly when $\\Omega_0 =\\Omega_H$ the system can be approach the horizon.\nThe entropy for this case is linearly and logarithmically divergent \nas the ultraviolet cut-off goes to zero.\nTo remove such a divergence, in addition to the renormalization of the \ngravitational constant, we need the renormalization of the \ncurvature square term\\cite{ohta}. But after the renormalization\nthe entropy does not proportional to the area of the event horizon.\nIf we use the proper distance cut-off \nthe entropy is proportional to the horizon area $A_H$. But \nthe cut-off depends on the coordinate $\\theta$.\n\nAnother particular point is that in the extremal black hole case\nwe can not consider the brick wall method of 't Hooft except for\nthe case $ 0 < a < 1\/2 M$ in Kerr-Newman black hole.\n\n\\section*{Appendix} \nFor the three rotating black holes the metrics, the surface \ngravities, , and the proper distances $\\epsilon$ \nare given as follows:\n\n1) the Kaluza-Klein black hole \\cite{Frolov} \n\\begin{eqnarray}\n\\nonumber\nds^2 &=& - \\frac{\\Delta - a^2 \\sin^2 \\theta }{B \\Sigma}dt^2\n -2 a \\sin^2 \\theta \\frac{1}{\\sqrt{1 - v^2 }} \n \\frac{Z}{B} dtd \\phi \\\\\n & & ~ + \\left[\n B \\left( r^2 + a^2 \\right) + a^2 \\sin^2 \\theta \n \\frac{Z}{B} \\right]\n \\sin^2 \\theta d \\phi^2 + \n \\frac{ B \\Sigma}{\\Delta} dr^2 + B \\Sigma d \\theta^2,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\Delta = r^2 - 2 \\mu r + a^2 ,~~~\n\\Sigma = r^2 + a^2 \\cos^2 \\theta,~~~\nZ = \\frac{2 \\mu r}{\\Sigma},~~~\nB = \\left( 1 + \\frac{v^2 Z}{1 - v^2 } \\right)^{\\frac{1}{2}}.\n\\end{equation}\nThe physical mass $M$, the charge $Q$, the angular momentum $J$,\nand the horizon\nare expressed by the parameters $v,\\mu,$ and $a$ as\n\\be\nM = \\mu \\left[\n 1 + \\frac{ v^2 }{2 ( 1 - v^2 )} \\right],~~~\n Q = \\frac{\\mu v}{ 1 - v^2 },~~~\n J = \\frac{\\mu a}{\\sqrt{1 -v^2}},\n r_H = \\mu + \\sqrt{ \\mu^2 - a^2}.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Kaluza-Klein} &=& \n \\frac{ \\sqrt{( 1 - v^2) ( \\mu^2 - a^2)}}{ r_H^2 + a^2},\\\\\n\\epsilon_{Kaluza-Klein} &=& \n 2 \\left( \\frac{B(r_H) \\Sigma (r_H) }{ 2 r_H - 2 \\mu} \\right)^{1\/2}\n \\sqrt{h}. \n\\end{eqnarray}\n\n\n2) the Sen black hole\\cite{sen}:\n\\bea\nds^2 &=& - \\frac{\\Delta - a^2 \\sin^2 \\theta}{\\Sigma}dt^2\n- \\frac{4 \\mu r a \\cosh^2 \\gamma \\sin^2 \\theta}{\\Sigma} dt d \\phi \\\\\n& &~+ \\frac{\\Sigma}{\\Delta} dr^2 + \\Sigma d\\theta^2\n+ \\frac{\\Lambda}{\\Sigma} \\sin^2 \\theta d \\phi^2,\n\\eea\nwhere\n\\bea\n\\Delta &=& r^2 - 2 \\mu r + a^2 ,~~~\n\\Sigma = r^2 + a^2 \\cos^2 \\theta + 2 \\mu r \\sinh^2 \\gamma, \\\\\n\\Lambda &= & \\left( r^2 + a^2 \\right) \\left(\n r^2 + a^2 \\cos^2 \\theta \\right) + 2 \\mu r a^2\n \\sin^2 \\theta \\\\\n & &~ + 4 \\mu r \\left( r^2 + a^2 \\right) \\sinh^2 \\gamma \n + 4 \\mu^2 r^2 \\sinh^4 \\gamma.\n\\eea\nThe mass $M$, the charge $Q$, the angular momentum $J$, and the\nhorizon are given \nby parameters $\\mu,\\beta$, and $a$ as\n\\be\nM = \\frac{\\mu}{2} \\left( 1 + \\cosh 2 \\gamma \\right),~~\nQ = \\frac{\\mu}{\\sqrt{2}} \\sinh 2 \\gamma,~~\nj = \\frac{ a \\mu }{2} \\left( 1 + \\cosh 2 \\gamma \\right),~~\nr_H = \\mu + \\sqrt{\\mu^2 - a^2}.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Sen} &=& \n \\frac{ \\sqrt{ ( 2 M^2 - e^2 )^2 - 4 J^2 }\n\t\t }{ 2 M \\left[\n\t 2 M^2 - e^2 + \\sqrt{ ( 2 M^2 - e^2 )^2 - 4 J^2 }\n\t\t\t \\right] }, \\\\\n\\epsilon_{Sen} &=& \n 2 \\left( \\frac{ r_H^2 + a^2 \\cos^2 \\theta + 2 \\mu r_H \\sinh^2 \n \\gamma }{ 2 r_H - 2 \\mu } \\right)^{1\/2} \\sqrt{h}.\n\\end{eqnarray}\n\n\n3) the charged Kerr black hole \\cite{kerr}\n\\begin{eqnarray}\n\\nonumber\nds^2 &= & - \\left( \n\t\t\\frac{ \\Delta - a^2 \\sin^2 \\theta }{\\Sigma}\n\t\t\\right) \n\t\tdt^2 - \\frac{2 a \\sin^2 \\theta ~( r^2 + a^2 - \\Delta)}{\n\t\t\\Sigma } dt d\\phi \\\\\n & &~ + \\left[ \\frac{(r^2 +a^2 )^2 - \\Delta a^2 \\sin^2 \\theta }{\n \\Sigma} \\right] \\sin^2 \\theta d \\phi^2 + \\frac{\\Sigma}{\\Delta} dr^2 +\n \\Sigma d \\theta^2, \n\\end{eqnarray} \nwhere \n\\begin{equation}\n\\Sigma = r^2 + a^2 \\cos^2 \\theta, ~~~~~ \n\\Delta = r^2 + a^2 + e^2 - 2 M r,\n\\end{equation}\nand $e,a,$ and $M$ are charge, angular momentum per unit mass, and\nmass of the spacetime respectively. \nThe event horizon is \n\\be\n r_H = M + \\sqrt{M^2 - a^2 - e^2 }.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Kerr}&=& \\frac{ \\sqrt{M^2 - a^2 -e^2}}{ 2 M \n\t\\left[ M + \\sqrt{M^2 - a^2 - e^2} \\right] - e^2}, \\\\\n\\epsilon_{Kerr} &= &\n 2 \\left( \\frac{ r_H^2 + a^2 \\cos^2 \\theta }{ 2 r_H - 2 M } \n\\right)^{1\/2} \\sqrt{h}. \n\\end{eqnarray}\n\n\n\\begin{flushleft}\n{\\bf Acknowledgment}\n\\end{flushleft}\n\nThis work is partially supported by Korea Science and Engineering \nFoundation.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\n\nAll graphs in this paper are finite and have neither loops nor parallel edges. We denote by $\\mathbb{N}$ the set of positive integers.\nA class $\\mathcal{C}$ of graphs is said to have the \\emph{Erd\\H{o}s-P\\'osa property} if there exists a function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$, called a \\emph{gap function}, such that\nfor every graph $G$ and a positive integer $k$, $G$ contains either\n\\begin{itemize}\n\\item $k+1$ pairwise vertex-disjoint subgraphs in $\\mathcal{C}$, or\n\\item a vertex set $T$ of $G$ such that $\\abs{T}\\le f(k)$ and $G- T$ has no subgraphs in $\\mathcal{C}$.\n\\end{itemize} \nErd\\H{o}s and P\\'osa~\\cite{ErdosP1965} showed that the class of all cycles has this property with a gap function $\\mathcal{O}(k\\log k)$.\nThis breakthrough result sparked an extensive research on finding min-max dualities of packing and covering for various graph families and combinatorial objects. \nErd\\H{o}s and P\\'osa also showed that the gap function cannot be improved to $o(k\\log k)$ using a probabilistic argument, and Simonovits~\\cite{Simonovits1967} provided a construction \nachieving the lower bound. \nThe result of Erd\\H{o}s and P\\'osa has been strengthened for cycles with additional constraints; for example, long cycles~\\cite{RobertsonS1986, BirmeleBR2007, FioriniH2014, MoussetNSW2016, BruhnJS2017}, directed cycles~\\cite{ReedRST1996, KakimuraK2012}, cycles with modularity constraints~\\cite{Thomassen1988, HuyneJW2017} or cycles intersecting a prescribed vertex set~\\cite{KakimuraKM2011, PontecorviW2012, BruhnJS2017, HuyneJW2017}. \nNot every variant of cycles has the Erd\\H{o}s-P\\'osa property; for example, Reed~\\cite{Reed1999} showed that the class of odd cycles does not satisfy the Erd\\H{o}s-P\\'osa property. \n\n\nWe generally say that a graph class $\\mathcal{C}$ has the \\emph{Erd\\H{o}s-P\\'osa property} under a graph containment relation $\\le_{\\star}$ \nif there exists a gap function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that\nfor every graph $G$ and a positive integer $k$, $G$ contains either\n\\begin{itemize}\n\\item $k+1$ pairwise vertex-disjoint subsets $Z_1,\\ldots , Z_k$ such that each subgraph of $G$ induced by $Z_i$ contains a member of $\\mathcal{C}$ under $\\le_{\\star}$, or\n\\item a vertex set $T$ of $G$ such that $\\abs{T}\\le f(k)$ and $G- T$ contains no member of $\\mathcal{C}$ under $\\le_{\\star}$.\n\\end{itemize} \nHere, $\\le_{\\star}$ can be a graph containment relation such as subgraph, induced subgraph, minor, topological minor, induced minor, or induced subdivision. \nAn edge version and directed version of the Erd\\H{o}s-P\\'osa property can be similarly defined. \nIn this setting, the Erd\\H{o}s-P\\'osa properties of diverse undirected and directed graph families have been studied for graph containment relations such as minors~\\cite{RobertsonS1986}, immersions~\\cite{Liu2015, GianKRT2016}, and (directed) butterfly minors~\\cite{AmiriKKW2016}. It is known that the edge-version of the Erd\\H{o}s-P\\'osa property also holds for cycles~\\cite{diestel2012}. Raymond and Thilikos~\\cite{RaymondT16} provides an up-to-date overview on the Erd\\H{o}s-P\\'osa properties for a range of graph families.\n\nIn this paper, we study the Erd\\H{o}s-P\\'osa property for cycles of length at least $4$ under the induced subgraph relation.\nAn induced cycle of length at least $4$ in a graph $G$ is called a \\emph{hole} or a \\emph{chordless cycle}.\nConsidering the extensive study on the topic, it is somewhat surprising that \nwhether the Erd\\H{o}s-P\\'osa property holds for cycles of length at least $4$ under the induced subgraph relation has been left open till now. \nThis question was explicitly asked by Jansen and Pilipczuk~\\cite{JansenP17} in their study of the \\textsc{Chordal Vertex Deletion} problem,\nand was also asked by Raymond and Thilikos~\\cite{RaymondT16} in their survey. \nWe answer this question positively.\n\n\n\\begin{THM}\\label{thm:main}\nThere exist a constant $c$ and a polynomial-time algorithm \nwhich, given a graph $G$ and a positive integer $k$, finds either $k+1$ vertex-disjoint holes or a vertex set of size at most $ck^2\\log k$ hitting every hole of $G$.\n\\end{THM}\n\n\n\nOne might ask whether Theorem~\\ref{thm:main} can be extended to the class of cycles of length at least $\\ell$ for fixed $\\ell\\ge 5$.\nWe present a complementary result that for every fixed $\\ell\\ge 5$, the class of cycles of length at least $\\ell$ does not satisfy the Erd\\H{o}s-P\\'osa property under the induced subgraph relation.\n\n\\begin{THM}\\label{thm:main2}\nLet $\\ell\\ge 5$ be an integer. \nThen the class of cycles of length at least $\\ell$ does not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation.\n\\end{THM}\n\nTheorem~\\ref{thm:main} is closely related to the \\textsc{Chordal Vertex Deletion} problem.\nThe \\textsc{Chordal Vertex Deletion} problem asks whether, for a given graph $G$ and a positive integer $k$, there exists a vertex set $S$ of size at most $k$ such that $G-S$ has no holes; \n\tin other words, $G-S$ is a chordal graph.\n\tIn parameterized complexity, whether or not \\textsc{Chordal Vertex Deletion} admits a polynomial kernelization was one of major open problems since it was first mentioned by Marx~\\cite{Marx10}.\n\t A \\emph{polynomial kernelization} of a parameterized problem is a polynomial-time algorithm \n\t that takes an instance $(x,k)$ and outputs an instance $(x', k')$ such that \n\t (1) $(x,k)$ is a \\textsc{Yes}-instance if and only if $(x', k')$ is a \\textsc{Yes}-instance, and\n\t (2) $k'\\le k$, and $\\abs{x'}\\le g(k)$ for some polynomial function $g$.\n\nJansen and Pilipczuk~\\cite{JansenP17}, and independently Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17}, presented polynomial kernelizations for the \\textsc{Chordal Vertex Deletion} problem. In both works, an approximation algorithm for the optimization version of this problem emerges as an important subroutine. Jansen and Pilipczuk~\\cite{JansenP17} obtained an approximation algorithm of factor $\\mathcal{O}({\\sf opt}^2 \\log {\\sf opt} \\log n)$ using iterative decomposition of the input graph and linear programming. Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17} obtained an algorithm of factor $\\mathcal{O}({\\sf opt}\\log^2 n)$ based on divide-and-conquer. \nAs one might expect, the factor of an approximation algorithm for the \\textsc{Chordal Vertex Deletion} is intrinsically linked to the quality of the polynomial kernels \nattained in \\cite{JansenP17} and~\\cite{AgrawalLMSZ17}.\nWe point out that the polynomial-time algorithm of Theorem of~\\ref{thm:main} can be easily converted into an approximation algorithm of factor $O({\\sf opt}\\log {\\sf opt})$. \n\n\n\n\n\\begin{THM}\\label{thm:main3}\nThere is an approximation algorithm of factor $O({\\sf opt}\\log {\\sf opt})$ for {\\sc Chordal Vertex Deletion}. \n\\end{THM}\n\nIt should be noted that an $\\mathcal{O}(\\log^2 n)$-factor approximation algorithm \nwas presented recently by Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17b}, which outperforms \nthe approximation algorithm of Theorem~\\ref{thm:main3}. \n\nOur result has another application on packing and covering for weighted cycles.\nFor a graph $G$ and a non-negative weight function $w:V(G)\\rightarrow \\mathbb{N}\\cup \\{0\\}$, \nlet $\\pack(G, w)$ be the maximum number of cycles (repetition is allowed) such that each vertex $v$ used in at most $w(v)$ times, and\nlet $\\cover(G, w)$ be the minimum value $\\sum_{v\\in X} w(v)$ where $X$ hits all cycles in $G$.\nDing and Zang~\\cite{DingZ2002} characterized \\emph{cycle Mengerian graphs} $G$, which satisfy the property that for all non-negative weight function $w$, $\\pack(G,w)=\\cover(G,w)$.\nUp to our best knowledge, it was not previously known whether $\\cover(G,w)$ can be bounded by a function of $\\pack(G,w)$.\n\nAs a corollary of Theorem~\\ref{thm:main}, we show the following.\n\n\\begin{COR}\nFor a graph $G$ and a non-negative weight function $w:V(G)\\rightarrow \\mathbb{N}\\cup \\{0\\}$, \n$\\cover(G,w)\\le \\mathcal{O}(k^2\\log k)$, where $k=\\pack(G,w)$.\n\\end{COR}\n\n\n\n\nThe paper is organized as follows. Section~\\ref{sec:prelim} provides basic notations and previous results that are relevant to our result. \nIn Section~\\ref{sec:overview}, we explain how to reduce the proof of Theorem~\\ref{thm:main} to a proof under a specific premise, in which \nwe are given a shortest hole $C$ of $G$ \nsuch that $C$ has length more than $c\\cdot k\\log k$ for some constant $c$ and $G-V(C)$ is chordal.\nIn this setting, we introduce further technical notations and terminology. An outline of our proof will be also given in this section.\nWe present some structural properties of a shortest hole $C$ and its neighborhood in Section~\\ref{sec:lemmas}.\nIn Sections~\\ref{sec:hittingsunflower} and \\ref{sec:tulip}, we prove the Erd\\H{o}s-P\\'osa property for different types of holes intersecting $C$ step by step, and we conclude Theorem~\\ref{thm:main} at the end of Section~\\ref{sec:tulip}.\nSection~\\ref{sec:lowerbound} demonstrates that the class of cycles of length at least $\\ell$, for every fixed $\\ell\\ge 5$, does not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation. \nSection~\\ref{sec:applications} illustrates the implications of Theorem~\\ref{thm:main} to weighted cycles and to the \\textsc{Chordal Vertex Deletion} problem.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\n\nFor a graph $G$, we denote by $V(G)$ and $E(G)$ the vertex set and the edge set of $G$, respectively.\nLet $G$ be a graph. \nFor a vertex set $S$ of $G$, let $G[S]$ denote the subgraph of $G$ induced by $S$, and \nlet $G-S$ denote the subgraph of $G$ obtained by removing all vertices in $S$.\nFor $v\\in V(G)$, we let $G-v:=G-\\{v\\}$.\nIf $uv\\in E(G)$, we say that $u$ is a \\emph{neighbor} of $v$. \nThe set of neighbors of a vertex $v$ is denoted by $N_G(v)$, and the \\emph{degree} of $v$ is defined as the size of $N_G(v)$.\nThe \\emph{open neighborhood} of a vertex set $A\\subseteq V(G)$ in $G$, denoted by $N_G(A)$, is the set of vertices in $V(G)\\setminus A$ having a neighbor in $A$. The set $N_G(A)\\cup A$ is called the \\emph{closed neighborhood} of $A$, and denoted by $N_G[A]$. \nFor convenience, we define these neighborhood operations for subgraphs as well; that is, for a subgraph $H$ of $G$, \nlet $N_G(H):=N_G(V(H))$ and $N_G[H]:=N_G[V(H)]$.\nWhen the underlying graph is clear from the context, we drop the subscript $G$. \nA vertex set $S$ of a graph is a \\emph{clique} if every pair of vertices in $S$ is adjacent, and \nit is an \\emph{independent set} if every pair of vertices in $S$ is non-adjacent.\nFor two subgraphs $H$ and $F$ of $G$, the \\emph{restriction} of $F$ on $H$ is defined as the graph $F[V(F)\\cap V(H)]$.\n\n\nA \\emph{walk} is a non-empty alternating sequence of vertices and edges of the form $(x_0,e_0,\\ldots ,e_{\\ell-1},x_{\\ell})$, beginning and ending with vertices, such that for every $0\\leq i\\leq \\ell-1$, $x_{i}$ and $x_{i+1}$ are endpoints of $e_i$. \n A \\emph{path} is a walk in which vertices are pairwise distinct. \nFor a path $P$ on vertices $x_0,\\ldots , x_{\\ell}$ with edges $x_ix_{i+1}$ for $i=0,1, \\ldots , \\ell-1$, we write $P=x_0x_1 \\cdots x_{\\ell}$.\nIt is also called an $(x_0, x_{\\ell})$-path.\nWe say $x_{i}$ is the $i$-th neighbor of $x_0$, and similarly, $x_{\\ell-i}$ is the $i$-th neighbor of $x_{\\ell}$ in $P$.\nA \\emph{cycle} is a walk $(x_0, e_0, \\ldots, e_{\\ell-1}, x_{\\ell})$ in which vertices are pairwise distinct except $x_0=x_{\\ell}$. \nFor a cycle $C$ on $x_0,x_1, \\ldots , x_{\\ell}$ with edges $x_ix_{i+1}$ for $i=0,1, \\ldots , \\ell-1$ and $x_{\\ell}x_0$, \nwe write $C=x_0x_1 \\cdots x_{\\ell}x_0$. If a cycle or a path $H$ is an induced subgraph of the given graph $G$,\nthen we say that $H$ is an induced cycle or an induced path in $G$, respectively. \n\nA subpath of a path $P$ starting at $x$ and ending at $y$ is denoted as $xPy$. \nIn the notation $xPy$, we may replace $x$ or $y$ with $\\mathring{x}$ or $\\mathring{y}$, to obtain a subpath starting from the neighbor of $x$ in $P$ closer to $y$ or ending at the neighbor of $y$ in $P$ closer to $x$, respectively.\nFor instance, $xP\\mathring{y}$ refers to the subpath of $P$ starting at $x$ and ending at the neighbor of $y$ in $P$ closer to $x$. \nGiven two walks $P=(v_0,e_0,\\ldots ,e_{p-1},v_{p})$ and $Q=(u_{0},f_{0},\\ldots ,f_{q-1},u_{q})$ such that $v_p=u_0$, the \\emph{concatenation} of $P$ and $Q$ is \ndefined as the walk $(v_0,e_0,\\ldots ,e_{p-1},v_{p}(=u_0),f_{0},\\ldots ,f_{q-1},u_{q})$, which we denote as $P\\odot Q$.\nNote that for two internally vertex-disjoint paths $P_1$ and $P_2$ from $v$ to $w$, \n$vP_1w\\odot wP_2v$ denotes the cycle passing through $P_1$ and $P_2$.\n\nGiven a graph $G$, the distance between two vertices $x$ and $y$ in $G$ is defined as the length of a shortest $(x,y)$-path and denoted as $\\dist_G(x,y)$. If $x=y$, then we define $\\dist_G(x,y)=0$, and $\\dist_G(x,y)=\\infty$ if there is no $(x,y)$-path in $G$. The distance between two vertex sets $X,Y\\subseteq V(G)$, written as $\\dist_G(X,Y)$, is the minimum $\\dist_G(x,y)$ over all $x\\in X$ and $y\\in Y$. If $X=\\{x\\}$, then we write $\\dist_G(X,Y)$ as $\\dist_G(x,Y)$. For a vertex subset $S$ of $G$, a vertex set $U$ is the \\emph{$r$-neighborhood} of $S$ in $G$ if it is the set of all vertices $w$ such that $\\dist_G(w, S)\\le r$. \nWe denote the $r$-neighborhood of $S$ in $G$ as $N^r_G[S]$. When the underlying graph $G$ is clear from the context, we omit the subscript $G$.\n\nGiven a cycle $C=x_0x_1 \\cdots x_{\\ell}x_0$, an edge $e$ of $G$ is a \\emph{chord} of $C$ if both endpoints of $e$ are contained in $V(C)$ \nbut $e$ is not an edge of $C$. A graph is \\emph{chordal} if it has no holes.\nA vertex set $T$ of a graph $G$ is called a \\emph{chordal deletion set} if $G-T$ is chordal.\n\nGiven a vertex set $S\\subseteq V(G)$, a path $P$ is called an \\emph{$S$-path} if $P$ connects two distinct vertices of $S$ and all internal vertices, possibly empty, are not in $S$. An $S$-path is called \\emph{proper} if it has at least one internal vertex. An $(A,B)$-path of a graph $G$ is a path $v_0v_1\\cdots v_{\\ell}$ such that $v_0\\in A$, $v_{\\ell}\\in B$ and all, possibly empty, internal vertices are in $V(G)\\setminus (A\\cup B)$. Observe that every path from $A$ to $B$ contains an $(A,B)$-path. If $A$ or $B$ is a singleton, then we omit the bracket from the set notation. A vertex set $S$ is an \\emph{$(A,B)$-separator} if $S$ disconnects all $(A,B)$-paths in $G$.\n\nWe recall the Menger's Theorem.\n\\begin{THM}[Menger's Theorem; See for instance \\cite{diestel2012}]\\label{thm:menger}\nLet $G$ be a graph and $A,B\\subseteq V(G)$. Then the size of a minimum $(A,B)$-separator in $G$ equals the maximum number of vertex-disjoint $(A,B)$-paths in $G$. \nFurthermore, one can output either one of them in polynomial time.\n\\end{THM}\nA bipartite graph is a graph $G$ with a vertex bipartition $(A,B)$ in which each of $G[A]$ and $G[B]$ is edgeless.\nA set $F$ of edges in a graph is a \\emph{matching} if no two edges in $F$ have a common endpoint.\nA vertex set $S$ of a graph $G$ is a \\emph{vertex cover} if $G-S$ has no edges.\nBy Theorem~\\ref{thm:menger}, \ngiven a bipartite graph with a bipartition $(A,B)$, \none can find a maximum matching or a minimum vertex cover in polynomial time.\n\n\n\nThe following result is useful to find many vertex-disjoint cycles in a graph of maximum degree $3$. We define $s_k$ for $k\\in \\mathbb{N}$ as\n\\begin{align*}\ns_k=\n\\begin{cases}\n4k(\\log k + \\log \\log k +4) \\quad &\\text{if } k\\geq 2\\\\\n2 &\\text{if } k=1.\n\\end{cases}\n\\end{align*}\n\\begin{THM}[Simonovitz~\\cite{Simonovits1967}]\\label{thm:simonovitz}\nLet $G$ be a graph all of whose vertices have degree $3$ and let $k$ be a positive integer. If $\\abs{V(G)}\\geq s_k$, then $G$ contains at least $k$ vertex-disjoint cycles. Furthermore, such $k$ cycles can be found in polynomial time.\n\\end{THM}\n\nLastly, we present lemmas which are useful for detecting a hole.\n\n\\begin{LEM}\\label{lem:twopaths}\nLet $H$ be a graph and $x,y\\in V(H)$ be two distinct vertices. Let $P$ and $Q$ be internally vertex-disjoint $(x,y)$-paths such that $Q$ contains an internal vertex $w$ having no neighbor in $V(P)\\setminus \\{x,y\\}$. If $Q$ is an induced path, then $H[V(P)\\cup V(Q)]$ has a hole containing $w$.\n\\end{LEM}\n\\begin{proof}\nLet $x_1$ and $x_2$ be the neighbors of $w$ in $Q$. As $xPy\\odot yQx$ is a cycle, $H[V(P)\\cup V(Q)]-w$ is connected.\nLet $R$ be a shortest $(x_1, x_2)$-path in $H[V(P)\\cup V(Q)]-w$. As the only neighbors of $w$ contained in $(V(P)\\cup V(Q)) \\setminus \\{w\\}$ are $x_1$ and $x_2$, \n$w$ has no neighbors in the internal vertices of $R$. Note that $R$ has length at least two since $x_1,x_2\\in V(Q)$ and $Q$ is an induced path.\nTherefore, $x_1Rx_2\\odot x_2wx_1$ is a hole containing $w$, as required.\n\\end{proof}\n\nA special case of Lemma~\\ref{lem:twopaths} \nis when there is a vertex $w$ in a cycle $C$ such that $w$ has no neighbors in the internal vertices of $C-w$ and the \nneighbors of $w$ on $C$ are non-adjacent. \nIn this case, $C$ has a hole containing $w$ by Lemma~\\ref{lem:twopaths}.\n\nOne can test in polynomial time whether a graph contains a hole or not.\n\\begin{LEM}\\label{lem:detectinghole}\nGiven a graph $G$, one can test in polynomial time whether it has a hole or not.\nFurthermore, one can find in polynomial time a shortest hole of $G$, if one exists.\n\\end{LEM}\n\\begin{proof}\nWe guess three vertices $v,w,z$ where $vw, wz\\in E(G)$ and $vz\\notin E(G)$, and \ntest whether there is a path from $v$ to $z$ in $G-(N_G[w]\\setminus \\{v,z\\})$.\nIf there is such a path, then we choose a shortest path $P$ from $v$ to $z$. As $w$ has no neighbors in the set of internal vertices of $P$, \n$V(P)\\cup \\{w\\}$ induces a hole. Clearly if $G$ has a hole, then we can find one by the above procedure.\n\nTo find a shortest one, for every such a guessed tuple $(v,w,z)$, we keep the length of the obtained hole.\nThen it is sufficient to output a hole with minimum length among all obtained holes.\n\\end{proof}\n\n\n\n\\section{Terminology and a proof overview.}\\label{sec:overview} \n\nThe proof of Theorem~\\ref{thm:main} begins by finding a sequence of shortest holes. Let $G$ be the input graph and let $G_1=G$. \nFor each $i=1,2,\\ldots$, we iteratively find a shortest hole $C_i$ in $G_i$ and set $G_{i+1}:=G_i - V(C_i)$. If the procedure fails to find a hole at $j$-th iteration, then $G_j$ is a chordal graph. \nThis iterative procedure leads us to the following theorem, which is the core component of our result.\n\nFor $k\\in \\mathbb{N}$, we define\n$\\mu_k=76s_{k+1}+3217k+1985$.\n\n\\begin{THM}\\label{thm:core}\nLet $G$ be a graph, $k$ be a positive integer and $C$ be a shortest hole of $G$\nsuch that $C$ has length strictly larger than $\\mu_k$ and $G-V(C)$ is chordal. Given such $G$, $k$, and $C$, one can find in polynomial time either $k+1$ vertex-disjoint holes or a vertex set $X\\subseteq V(G)$ of size at most $\\mu_{k}$ hitting every hole of $G$.\n\\end{THM}\n\n\nIt is easy to derive our main result from Theorem~\\ref{thm:core}.\n\n\\smallskip\n\n\\begin{proofof}{Theorem~\\ref{thm:main}}\nWe construct sequences $G_1,\\ldots, G_{\\ell+1}$ and $C_1,\\ldots , C_{\\ell}$ \nsuch that \n\\begin{itemize}\n\\item $G_1=G$, \n\\item for each $i\\in \\{1, 2, \\ldots, \\ell\\}$, $C_i$ is a shortest hole of $G_i$, and \n\\item for each $i\\in \\{1, 2, \\ldots, \\ell\\}$, $G_{i+1}=G_i-V(C_i)$.\n\\item $G_{\\ell+1}$ is chordal.\n\\end{itemize}\nSuch a sequence can be constructed in polynomial time repeatedly applying Lemma~\\ref{lem:detectinghole} to find a shortest hole.\nIf $\\ell \\geq k+1$, then we have found a packing of $k+1$ holes. \nHence, we assume that $\\ell \\leq k$. \n\nWe prove the following claim for $j=\\ell+1$ down to $j=1$. \n\\begin{quote}\nOne can find in polynomial time either $k+1$ vertex-disjoint holes, or \na chordal deletion set $T_{j}$ of $G_{j}$ of size at most $(\\ell+1-j)\\mu_k$. \n\\end{quote}\n\n\nThe claim trivially holds for $j=\\ell+1$ with $T_{\\ell+1}=\\emptyset$ because $G_{\\ell+1}$ is chordal.\nLet us assume that for some $j\\leq \\ell$, we obtained a chordal deletion set $T_{j+1}$ of $G_{j+1}$ of size at most $(\\ell-j)\\mu_k$.\nThen in $G_{j}-T_{j+1}$, $C_{j}$ is a shortest hole, and $\\left( G_{j}-T_{j+1} \\right)-V(C_{j})$ is chordal.\nIf $C_{j}$ has length at most $\\mu_k$, then we set $T_{j}:=T_{j+1}\\cup V(C_{j})$. Clearly, $\\abs{T_j}\\leq (\\ell-j+1)\\mu_k$.\nOtherwise, by applying Theorem~\\ref{thm:core} to $G_{j}-T_{j+1}$ and $C_{j}$, \none can find in polynomial time either $k+1$ vertex-disjoint holes or a chordal deletion set $X$ of size at most $\\mu_k$ of $G_{j}-T_{j+1}$. \nIn the former case, we output $k+1$ vertex-disjoint holes, and we are done. If we obtain a chordal deletion set $X$, then we set $T_{j}:=T_{j+1}\\cup X$. Observe that the set $T_{j}$ is a chordal deletion set of $G_{j}$ and $\\abs{T_{j}}\\le (\\ell-j+1)\\mu_k$ as claimed. \n\nFrom the claim with $j=1$, we conclude that \nin polynomial time, one can find either $k+1$ vertex-disjoint holes, \nor a chordal deletion set of $G_1=G$ of size at most $\\ell\\mu_k\\le k\\mu_k=\\mathcal{O}(k^2\\log k)$. \n\\end{proofof}\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \\foreach \\y in {4,5}{\n\t\\draw (4.5,2)--(a\\y);\n }\n \\draw(3,2)--(4.5,2);\n\n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (8,2) node [w] {};\n \\draw (4.5,2) node [w] {};\n\n\\draw[dashed, rounded corners] (4, 2.3)--(5, 2.3)--(4, -.5)--(2.5, 2.3)--(4, 2.3);\n\n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (4, 2.7) {$Z_v$};\n \\node at (4, -.8) {$v$};\n\n \\end{tikzpicture} \\caption{The set of vertices adjacent to all vertices of $C$ is denoted by $D$, and for each $v\\in V(C)$, $Z_v$ \n denotes the set $\\{v\\}\\cup (N(v)\\setminus V(C)\\setminus D)$. Using the fact that $C$ is chosen as a shortest hole and it is long, we will prove in Lemma~\\ref{lem:consecutive} that each vertex in $N(C)\\setminus D$ has at most $3$ neighbors on $C$ and they are consecutive in $C$. }\\label{fig:setting}\n\\end{figure}\n\n\\medskip\n\nThe rest of this section and Sections~\\ref{sec:lemmas}-\\ref{sec:tulip} are devoted to establish Theorem~\\ref{thm:core}. \nThroughout these sections, we shall consider the input tuple $(G,k, C)$ of Theorem~\\ref{thm:core} as fixed. \n\nLet us introduce the notations that are frequently used (see Figure~\\ref{fig:setting}).\nA vertex $v\\in N(C)$ is \\emph{$C$-dominating} if $v$ is adjacent to every vertex on $C$.\nWe reserve $D$ to denote the set of all $C$-dominating vertices.\nFor each vertex $v$ in $C$, we denote by $Z_v:=\\{v\\}\\cup (N(v)\\setminus V(C)\\setminus D )$, and \nfor a subset $S$ of $V(C)$, we denote by $Z_S:=\\bigcup_{v\\in S}Z_v$.\nWe also define\n\\begin{itemize}\n\\item $G_{deldom}:=G-D$ and $G_{nbd}:=G[N[C]\\setminus D]$. \n\\end{itemize}\nFor a subpath $Q$ of $C$, the subgraph of $G$ induced by $Z_{V(Q)}$ is called a \\emph{$Q$-tunnel}. \nBy definition of $Z_{V(Q)}$, $Q$-tunnel is an induced subgraph of $G_{nbd}$. \nWhen $q, q'$ are endpoints of $Q$, we say that $Z_{q}$ and $Z_{q'}$ are \\emph{entrances} of the $Q$-tunnel.\n\nWe distinguish between two types of holes, namely \\emph{sunflowers} and \\emph{tulips}. \nA hole $H$ is said to be a \\emph{sunflower} if $V(H)\\subseteq N[C]$, that is, its entire vertex set is placed within the closed neighborhood of $C$.\nA hole that is not a sunflower is called a \\emph{tulip}. Every tulip contains at least one vertex not contained in $N[C]$.\nAlso, we classify holes depending on whether one contains a $C$-dominating vertex or not.\nA hole is \\emph{$D$-traversing} it contains a $C$-dominating vertex (which is a vertex of $D$), and \\emph{$D$-avoiding} otherwise.\n\nIn the remainder of this section, we present a proof outline of Theorem~\\ref{thm:core}. \nHere are three basic observations, necessary to give the ideas of our proofs.\n\\begin{itemize}\n\\item (Lemma~\\ref{lem:consecutive})\nFor every vertex $v$ of $N(C)$, either it has at most $3$ consecutive neighbors in $C$ or it is $C$-dominating.\n\\item (Lemma~\\ref{lem:farnonadj})\nLet $x,y$ be two vertices in $C$ such that $\\dist_C(x,y)\\geq 4$. Then there is no edge between $Z_x$ and $Z_y$. \n\\item (Lemma~\\ref{lem:dominating})\n$D$ is a clique.\n\\end{itemize}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \n \n \\draw[dashed](8,2) [in=30,out=150] to (3,2);\n\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (8,2) node [w] {};\n \n \n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (2, -.8) {$x$};\n\t\\node at (9, -.8) {$y$};\n \\node at (2.6, 2.4) {$v$};\n \\node at (8.4, 2.4) {$w$};\n \n \\end{tikzpicture} \\caption{Illustration of Lemma~\\ref{lem:farnonadj}: if $\\dist_C(x,y)\\ge 4$, then there are no edges between $Z_x$ and $Z_y$.\n For instance, suppose $v$ and $w$ are adjacent. \n Note that $v$ and $w$ have at most $3$ consecutive neighbors in $C$.\n If the distance from $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$ is at least $1$, then we can find a hole shorter than $C$ using the shortest path from $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$.\n Otherwise, we have $\\dist_C(x,y)=4$ and $\\abs{N(v)\\cap V(C)}=\\abs{N(w)\\cap V(C)}=3$ and thus, the longer path between $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$ creates a hole shorter than $C$.}\\label{fig:distance}\n\\end{figure}\n\n\n\\subsection{$D$-avoiding sunflowers.} \n\nThe set of $D$-avoiding sunflowers is categorized into two subgroups, \\emph{petals} and \\emph{full sunflowers}. Roughly speaking, petals are \\emph{seen} by a small number of consecutive vertices on $C$, while a full one is \\emph{seen} by every vertex of $C$. For the precise definition, we introduce the notion of support. \n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\t \\draw(3,3) [in=120,out=-120] to (3,0);\n\t \\draw(3.4,3) [in=120,out=-120] to (3,0);\n\t \\draw(4.1,3) [in=120,out=-140] to (3,0);\n\t \\draw(4.5,3) [in=120,out=-140] to (3,0);\n\t \\draw(4.5,2) -- (3,0);\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \\foreach \\y in {4,5}{\n\t\\draw (4.5,2)--(a\\y);\n }\n \\draw (3,2)--(3,3);\n \\draw(4.5,3)--(4.5,2);\n\t\\draw(3,3)--(3.4,3);\\draw[dotted](3.4,3)--(4.1,3);\\draw(4.1,3)--(4.5,3);\n\t \n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (3,3) node [w] {};\n \\draw (8,2) node [w] {};\n \\draw (4.5,2) node [w] {};\n \\draw (4.5,3) node [w] {};\n\n\t \\draw (3.4,3) node [w] {};\n \\draw (4.1,3) node [w] {};\n \n\\draw[dashed, rounded corners] (4, 3.3)--(5,3.3)--(5, 2.3)--(4, -.5)--(2.5, 2.3)--(2.5,3.3)--(4, 3.3);\n\n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (4, 3.7) {$H$};\n\n \\node at (2, -.8) {$v_1$};\n \\node at (3, -.8) {$v_2$};\n \\node at (4, -.8) {$v_3$};\n \\node at (5, -.8) {$v_4$};\n\n \\end{tikzpicture} \\caption{The cycle $H$ is a petal having the support $\\{v_1, v_2, v_3, v_4\\}$. A petal can be arbitrarily long.}\\label{fig:petal}\n\\end{figure}\n\nFor a subgraph $H$ of $G$, the \\emph{support} of $H$, denoted by $\\operatorname{\\textsf{sp}} (H)$, is the set of all vertices $v\\in V(C)$ such that $(Z_v\\cup D)\\cap V(H)\\neq\\emptyset$. \nObserve that if $H$ contains a vertex of $D$, then trivially $\\operatorname{\\textsf{sp}}(H)=V(C)$.\nFor a $D$-avoiding sunflower $H$, we say that \n\\begin{itemize}\n\\item it is a \\emph{petal} if $\\abs{\\operatorname{\\textsf{sp}}(H)}\\leq 7$, and \n\\item it is \\emph{full} if $\\operatorname{\\textsf{sp}}(H)=V(C)$.\n\\end{itemize}\nSee Figure~\\ref{fig:petal} for an illustration of a petal.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:petal}.]} We first obtain a small hitting set of petals, unless $G$ has $k+1$ vertex-disjoint holes.\nFor this, we greedily pack petals and mark their supports on $C$.\nClearly, if there are $k+1$ petals whose supports are pairwise disjoint, then we can find $k+1$ vertex-disjoint holes. \nThus, we can assume that there are at most $k$ petals whose supports are pairwise disjoint. \nWe take the union of all those supports and call it $T_1$. By construction, for every petal $H$, $\\operatorname{\\textsf{sp}}(H)\\cap T_1\\neq\\emptyset$.\nThen we take the $6$-neighborhood of $T_1$ in $C$ and call it $T_{petal}$.\nIt turns out that \n\\begin{itemize}\n\\item[($\\ast$)] for every petal $H$, $\\operatorname{\\textsf{sp}} (H)$ is fully contained in $T_{petal}$, \n\\end{itemize} \nand in particular, $V(H)\\cap T_{petal}\\neq \\emptyset$. The size of $T_{petal}$ is at most $19k$.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:allisfull}.]} \nSomewhat surprisingly, we show that every $D$-avoiding sunflower that does not intersect $T_{petal}$ is a full sunflower. \nIt is possible that there is a sunflower with support of size at least $8$ and less than $\\abs{V(C)}$.\nWe argue that if such a sunflower $H$ exists, then there is a vertex $v\\in V(C)\\cap V(H)$ and a petal whose support contains $v$. \nBut the property $(\\ast)$ of $T_{petal}$ implies that $T_{petal}$ contains $v$, which implies that such a sunflower should be hit by $T_{petal}$.\nTherefore, it is sufficient to hit full sunflowers for hitting all remaining $D$-avoiding sunflowers.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:sunflowerwithout}.]} \nWe obtain a small hitting set of full sunflowers, when $G$ has no $k+1$ vertex-disjoint holes.\nFor this, we consider two vertex sets $Z_v$ and $Z_w$ for some $v$ and $w$ on $C$, \nand apply Menger's theorem for two directions, say `north' and `south' hemispheres around $C$, between $Z_v$ and $Z_w$ in the graph $G_{nbd}$.\nWe want to argue that if there are many paths in both directions, then we can find many vertex-disjoint holes. \nHowever, it is not clear how to link two families of paths.\n\nTo handle this issue, we find two families of paths whose supports cross on constant number of vertices. \nSince $C$ is much larger than the obtained hitting set $T_{petal}$ for petals, we can find $25$ consecutive vertices that contain no vertices in $T_{petal}$.\nLet $v_{-2}, v_{-1}, v_0, \\ldots, v_{22}$ be such a set of consecutive vertices.\nLet $\\mathcal{P}$ be the family of vertex-disjoint paths from $Z_{v_0}$ to $Z_{v_{20}}$ whose supports are contained in $\\{v_{-2}, v_{-1}, v_0, v_1, \\ldots, v_{20}, v_{21}, v_{22}\\}$, and let $\\mathcal{Q}$ be the family of vertex-disjoint paths from $Z_{v_5}$ to $Z_{v_{16}}$ whose supports do not contain $v_8$ and $v_{13}$. \nNon-existence of petals with support intersecting $\\{v_{-2}, v_{-1}, \\ldots, v_5\\}$ implies that \npaths in $\\mathcal{P}$ and $\\mathcal{Q}$ are `well-linked' at $Z_{v_0}$ except for few paths, and a symmetric argument holds at $Z_{v_{20}}$. \nThis allows us to link any pair of paths from $\\mathcal{P}$ and $\\mathcal{Q}$. \nIf one of $\\mathcal{P}$ and $\\mathcal{Q}$ is small, then we can output a hitting \nset of full sunflowers using Menger's theorem. The size of the obtained set $T_{full}$ will be at most $3k+14$.\n\n\\subsection{$D$-traversing sunflowers}\n[Subsection~\\ref{subsec:sunflowerwith}.] \nIt is easy to see that every $D$-traversing hole $H$ contains exactly one vertex of $D$ (since $D$ is a clique), and every vertex of $V(C)\\cap V(H)$ is a neighbor of the vertex in $D\\cap V(H)$. Let $v\\in V(C)\\cap V(H)$ and $d\\in D\\cap V(H)$ be such an adjacent pair.\nWe argue that $H$ contains a subpath $Q$ in $N[C]\\setminus D$ that starts from $v$ and is contained in $Z_{\\{v, v_2, v_3\\}}$ for some three consecutive vertices $v, v_2, v_3$ of $C$, such that\n\\begin{itemize}\n\\item $G[V(Q)\\cup \\{v, v_2, v_3, d\\}]$ contains a $D$-traversing sunflower containing $d$ and $v$.\n\\end{itemize}\n In other words, even if $H-d$ has large support, we can find another $D$-traversing sunflower $H'$ \n containing $d$ and $v$ where $H'-d$ has support on small number of vertices. \n The fact that $H$ and $H'$ share $v$ is important as we will take one of $d$ and $v$ as a hitting set for such $H'$, and this will hit $H$ as well.\n\nTo this end, we create an auxiliary bipartite graph in which one part is $D$ and the other part consists of sets of $3$ consecutive vertices $v_1, v_2, v_3$ of $C$, and we add an edge \nbetween $d\\in D$ and $\\{v_1, v_2, v_3\\}$ if $G[Z_{\\{v_1, v_2, v_3\\}}\\cup \\{d\\}]$ contains a $D$-traversing hole.\nWe argue that if this bipartite graph has a large matching, then we can find many vertex-disjoint holes, \nand otherwise, we have a small vertex cover. The union of all vertices involved in the vertex cover suffices \nto cover all $D$-traversing sunflowers. The hitting set $T_{trav:sunf}$ will have size at most $15k+9$.\n\n\n\\subsection{$D$-avoiding tulips.} \nWe follow the approach of constructing a subgraph of maximum degree $3$ used in proving the Erd\\H{o}s-P\\'osa property for various types of cycles: roughly speaking, if there is a cycle after removing the vertices of degree $3$ in the subgraph constructed so far, we augment the construction by adding some path or cycle.\nSimonovitz~\\cite{Simonovits1967} first proposed this idea and proved that if the number of degree $3$ vertices is $s_{k+1}$, then there are $k+1$ vertex-disjoint cycles. \nIf a maximal construction has less than $s_{k+1}$ vertices of degree 3, then we can hit all cycles of the input graph by taking all vertices of degree $3$ and a few more vertices.\n\\medskip \n\n\\noindent [Subsection~\\ref{subsec:tuliphive}.] The major obstacle for employing Simonovitz' approach is that for our purpose, we need to guarantee that every cycle of such a construction gives a hole.\nFor this reason, we will carefully add a path so that every cycle in a construction contains some hole.\nWe arrive at a notion of an $F$-extension, which is a path to be added iteratively with $C$ at the beginning.\nBy adding $F$-extensions recursively, we will construct a subgraph such that all vertices have degree $2$ or $3$ and it contains $C$.\nFor a subgraph $F$ of $G_{deldom}$ such that all vertices have degree $2$ or $3$ and it contains $C$< \nan $F$-extension is a proper $V(F)$-path $P$ in $G_{deldom}$, but has additional properties that \n\\begin{enumerate}[(i)]\n\\item both endpoints of $P$ are vertices of degree $2$ in $F$,\n\\item one of its endpoints should be in $C$, and \n\\item $P$ has at least one endpoint in $C$ whose second neighbor on $P$ has no neighbors in $F$, \nand the path obtained from $P$ by removing its endpoints is induced.\n\\end{enumerate}\nAn almost $F$-extension is a similar object, but its endpoints on $F$ is the same. Note that an almost $F$-extension is a cycle and is not an $F$-extension.\nWe depict an (almost) $F$-extension in Figure~\\ref{fig:wextension}. \nWhen we recursively choose an $F$-extension to add, we apply two priority rules:\n\\begin{itemize}\n\\item We choose a shortest $F$-extension among all possible $F$-extensions. \n\\item We choose an $F$-extension $Q$ with maximum $\\abs{V(Q)\\cap V(C)}$ among all shortest $F$-extensions. \n\\end{itemize}\nFollowing these rules, we recursively add $F$-extensions until there are no $F$-extensions. \n\nLet $W$ be the resulting graph.\nThe properties (ii), (iii) together with Lemma~\\ref{lem:twopaths} guarantee that the subgraph induced by the vertex set of every cycle of $W$ contains a hole. Therefore, in case when $W$ contains $s_{k+1}$ vertices of degree $3$, we can find $k+1$ vertex-disjoint holes by Theorem~\\ref{thm:simonovitz}. \nWe may assume that it contains less than $s_{k+1}$ vertices of degree $3$. \nLet $T_{branch}$ be the set of degree $3$ vertices in $W$.\nWe also separately argue that we can hit all of almost $W$-extensions by at most $5k+4$ vertices.\nLet $T_{almost}$ be the hitting set for almost $W$-extensions.\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n\n \\foreach \\y in {3.5,4.5}{\n\t\\draw[dashed] (3,2)--(\\y, 1.5);\n }\n \n \\draw (-.1,0.3)--(-.1,-0.3);\n \\draw (.1,0.3)--(.1,-0.3);\n \\draw (4-.1,0.3)--(4-.1,-0.3);\n \\draw (4.1,0.3)--(4.1,-0.3);\n \n \\draw (2,0)--(2,1)--(3,2)--(4,2.5);\n \\draw (4,2.5)--(6,2.5);\n \\draw (2,0) node [w] {};\n \\draw (2,1) node [w] {};\n \\draw (3,2) node [w] {};\n \\draw (4,2.5) node [w] {};\n \\draw (6,2.5) node [w] {};\n \n \\node at (7, 2) {$W$};\n \\node at (-2, 2) {$U$};\n\n \\node at (2, -.5) {$v$};\n \\node at (2.7, 2.3) {$w$};\n \n\t\\draw[rounded corners] (-1,0)--(-1.5, 1)--(-1.5, 4)--(-.5,4)--(-.5,1)--(-1,0);\n \\node at (3.2, 3) {$P$};\n\n\n \\node at (-2, -1) {$C$};\n\n \\end{tikzpicture} \\caption{A brief description of the construction $W$. \n Each extension contains at least one endpoint in $C$ whose second neighbor in the extension \n has no neighbor in $W$ hitherto constructed.\n For instance, $P$ is a $W$-extension, and $v$ is the vertex in $V(C)\\cap V(P)$, and its second neighbor $w$ in $P$ has no neighbors in $W$.\n The subgraph $U$ depicts an almost $W$-extension.}\\label{fig:wextension}\n\\end{figure}\n\n\n\nNow, let $T_{ext}$ be the union of \n\\[T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}\\] and \n\\[N^{20}_C[(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})\\cap V(C)].\\] \nNote that \n\\[ \\abs{T_{ext}}\\le 41(19k+(3k+14)+(15k+9)+(s_{k+1}-1)+(5k+4))\\le 41(s_{k+1}+42k+26). \\]\n\nFurthermore, $C-(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})$ contains at most $s_{k+1}+42k+26$ connected components, and thus $C-T_{ext}$ does as well.\n\\medskip\n\n\\noindent [Subsection~\\ref{subsec:cfragment}] We discuss the patterns of the remaining tulips in $G_{deldom}-T_{ext}$.\nSince we will add all components of $C-T_{ext}$ having at most $35$ vertices to the deletion set for $D$-avoiding tulips, \nwe focus on components of $C-T_{ext}$ with at least $36$ vertices.\nLet $H$ be a $D$-avoiding tulip in $G_{deldom}-T_{ext}$.\nLet $Q=q_1q_2 \\cdots q_m$ be a connected component of $C-T_{ext}$, and we consider the $Q$-tunnel $R$.\n\nWe argue that there is no edge $vw$ in $H$ such that \n$v$ is in the $Q$-tunnel, and $w$ is not in $N[C]$. \nSee Figure~\\ref{fig:qtunnel} for an illustration.\nSuppose there is such a pair, and let $q\\in V(Q)$ be a neighbor of $v$. We mainly prove that since $q$ is sufficiently far from degree $3$ vertices of $W$ in $C$, $w$ has no neighbors in $W$ \n(this is why we take the $20$-neighborhood of $V(C)\\cap (T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})$ in $C$).\nThis is formulated in Lemma~\\ref{lem:distance2}. \nNote that $qvw$ is a path where $q\\in V(C)$ and $w$ has no neighbors in $W$, and furthermore, $H$ contains a vertex in $V(C)\\setminus T_{ext}$ which is a vertex of degree $2$ in $W$.\nTherefore, if we traverse in $H$ following the direction from $v$ to $w$, we meet some vertex having a neighbor which is a vertex of degree $2$ in $W$. This gives a $W$-extension or an almost $W$-extension. But it is a contradiction as there is no $W$-extension, and $T_{ext}$ hits all of almost $W$-extensions.\nSo, there are no such edges $vw$.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n \\draw[rounded corners] (6,2.5)--(-1,2.5)--(-1,0);\n\n\t\\draw[fill=black] (-2,-.3)--(-2,.3)--(1,.3)--(1,-.3)--(-2,-.3);\n\t\\draw[fill=black] (4,-.3)--(4,.3)--(7,.3)--(7,-.3)--(4,-.3);\n\t\n\t\\draw[red, dotted, very thick, rounded corners] (1,0)--(1,1)--(4,1)--(4,0);\n\t\n\t\\draw[rounded corners] (0, .5)--(2,.5)--(2.3, -.2)--(2.6,.5)--(3.5,.5)--(3.5,1.5)--(4, 2);\n\t \\draw (3.45,.5) node [w] {};\n \\draw (3.5,1.5) node [w] {};\n \\node at (3.45, 0) {$v$};\n \\node at (3, 1.5) {$w$};\n\n \\node at (7, 2) {$W$};\n\n \\node at (-2, -1) {$C$};\n \\node at (2.5, -1) {$Q$};\n \\node at (5.5, -1) {$T_{ext}$};\n\n \\end{tikzpicture} \n \\caption{A description of the set $T_{ext}$ and a component $Q$ of $C-T_{ext}$. \n When there is an edge $vw$ where $v\\in Z_{V(Q)}\\setminus V(Q)$ and $w\\notin N[C]$, \n we will prove in Lemma~\\ref{lem:tunnellemma1} that $w$ has no neighbors in $W$.\n In particular, if there is a $D$-avoiding tulip containing such an edge, then we can find a $W$-extension or an almost $W$-extension starting with $qvw$ for some $q\\in V(Q)$.\n }\\label{fig:qtunnel}\n\\end{figure}\n\nThis argument leads to an observation that \nif $H$ contains some vertex $q_i$ with $6\\le i\\le m-5$, then the restriction of $H$ on $R$ should be a path from $Z_{q_1}$ to $Z_{q_m}$. Let $P$ be the restriction of $H$ on $R$. \nWe additionally remove $\\{q_j:1\\le j\\le 15, m-14\\le j\\le m\\}$ and assume $H$ is not removed.\nThen there are two ways that $P$ can be placed inside the $Q$-tunnel $R$: either\nthe endpoints of $P$ are in the same connected component of $R-(T_{ext}\\cup V(Q))$ or not.\nIn the former case, we could reroute this path so that this part does not contain a vertex of $Q$.\nSo, we could obtain a $D$-avoiding tulip containing less vertices of $C$.\nHowever, since $G-V(C)$ is chordal, \nthere should be some subpath $Q'$ of $C-T_{ext}$ such that \nthe restriction of $H$ on the $Q'$-tunnel is of the second type.\nWe will show that such a path can be hit by removing $5$ more vertices in $Q'$.\nThis will give a vertex set $T_{avoid:tulip}$ of size at most $35(s_{k+1}+42k+26)$ hitting all the remaining $D$-avoiding tulips.\n\n\\subsection{$D$-traversing tulips}\n[Subsection~\\ref{subsec:Dtulip}.]\nThis case can be handled similarly as the case of $D$-traversing sunflowers. \nIt turns out that $T_{ext}$ hits every $D$-traversing tulip that contains precisely two vertices of $C$. \nUsing a matching argument between $D$ and the set of three consecutive vertices of $C$, we show that an additional set $T_{trav:tulip}$ of size at most $25k+9$ \nplus $T_{avoid:tulip}$ \nhits all the remaining $D$-traversing tulips unless $G$ contains $k+1$ vertex-disjoint holes.\n\nIn total, we can output in polynomial time either $k+1$ vertex-disjoint holes in $G$, or a vertex set of size at most \n\\begin{align*}\n&\\abs{T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}} \\\\\n&\\le 41(s_{k+1}+42k+26)+ 35(s_{k+1}+42k+26)+25k+9 \\\\\n\t\t\t\t\t\t\t\t\t\t&\\le 76(s_{k+1}+42k+26)+25k+9= 76s_{k+1}+3217k+1985\n\t\t\t\t\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\t\t\t\t\t hitting all holes.\n\n\n\\section{Structural properties of $G$}\\label{sec:lemmas}\n\nIn this section, we present structural properties of a graph $G$ with a shortest hole $C$.\nIn Subsection~\\ref{subsec:distance}, we derive a relationship between the distance between $Z_v$ and $Z_w$ in $G_{deldom}$ for two vertices $v, w\\in V(C)$ and the distance between $v$ and $w$ in $C$.\nBriefly, we show that the distance between $Z_v$ and $Z_w$ in $G_{deldom}$ is at least some constant times the distance between $v$ and $w$ in $C$. \nWe also prove that every connected subgraph in $G_{nbd}$ has a connected support.\nIn Subsection~\\ref{subsec:dominating}, we obtain some basic properties of $C$-dominating vertices.\nRecall that we assume that the length of $C$ exceeds $\\mu_k$.\n\n\\subsection{Distance lemmas}\\label{subsec:distance}\n\nThe following lemma classifies vertices in $N(C)$ with respect to the number of neighbors in $C$.\n\n\\begin{LEM}\\label{lem:consecutive}\nFor every vertex $v$ of $N(C)$, either it has at most $3$ neighbors in $C$ that are consecutive in $C$ or it is $C$-dominating. \n\\end{LEM}\n\\begin{proof}\nLet us write $N_i:=\\{v\\in N(C):\\abs{N(v)\\cap V(C)}=i\\}$ for $i\\geq 1$.\nWe first show that $N(C)=N_1\\uplus N_2\\uplus N_3\\uplus D$.\nLet $v\\in N(C)\\setminus D$.\n\nWe claim that $v$ has no two neighbors $w_1$ and $w_2$ in $C$ \nsuch that \n\\begin{itemize}\n\\item[($\\ast$)] there is a $(w_1, w_2)$-subpath $Q$ of $C$ where $Q$ has length at least $2$ and at most $\\abs{V(C)}-3$ and $v$ has no neighbor in the internal vertices of $Q$.\n\\end{itemize}\nIf there is such a path $Q$, then $w_1vw_2\\odot w_2Qw_1$ is a hole of length at most $\\abs{V(C)}-1<\\abs{V(C)}$, which contradicts the assumption that $C$ is a shortest hole. So the claim holds.\n\nThis implies that $v$ has no neighbors $z_1$ and $z_2$ with $\\dist_C(z_1, z_2)\\ge 3$. Indeed, if such neighbors exist, then let $Q$ be a $(z_1,z_2)$-subpath of $C$ containing at least one internal vertex non-adjacent to $v$. Since $v\\notin D$, such $Q$ exists. Note that the length of $Q$ is at least 3 and at most $\\abs{V(C)}-3$. Then there exist two neighbors $w_1$ and $w_2$ of $v$ in $V(Q)$ satisfying $(\\ast)$, a contradiction. \nTherefore, the neighbors of $v$ in $C$ are contained in three consecutive vertices of $C$. \nThis implies that $v\\in N_1\\uplus N_2\\uplus N_3$.\n\nFurthermore, if $v\\in N(C)\\setminus D$ has exactly two neighbors with distance $2$ in $C$, then $G$ contains a hole of length $4$, a contradiction. \nTherefore, such a vertex has at most $3$ neighbors in $C$ that are consecutive in $C$, as required.\n\\end{proof}\n\nThe next lemma is illustrated in Figure~\\ref{fig:distance}.\n\\begin{LEM}\\label{lem:farnonadj}\nLet $x$ and $y$ be two vertices in $C$ such that $\\dist_C(x,y)\\geq 4$. Then there is no edge between $Z_x$ and $Z_y$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:consecutive}, the neighbors of any vertex in $N(C)\\setminus D$ lie within distance at most two, and thus $x$ has no neighbors in $Z_y$ and $y$ has no neighbors in $Z_x$. \nSuppose $v\\in Z_x\\setminus \\{x\\}$ and $w\\in Z_y\\setminus \\{y\\}$ are adjacent. \nLet $P$ and $Q$ be $(x,y)$-subpaths of $C$ such that $P$'s length is not greater than $Q$'s. \n\nWe may assume that $Q$ does not contain a common neighbor of $v$ and $w$. \nIf the length of $Q$ is at least five, then $N(v)\\cap V(Q)$ is included in $N_C^2(x)\\cap V(Q)$ and likewise we have \n$N(w)\\cap V(Q)\\subseteq N_C^2(y)\\cap V(Q)$. Thus, $Q$ does not contains a common neighbor of $v$ and $w$ by Lemma~\\ref{lem:consecutive}. Otherwise, both $P$ and $Q$ have length four and at least one of the two paths satisfy the assumption.\n\nSince the length of $Q$ is at most $\\abs{V(C)}-4$, a shortest $(v,w)$-path $Q'$ in $G[\\{v,w\\}\\cup V(Q)]-vw$ has length at most $\\abs{V(C)}-2$. \nMoreover, $Q'$ has length at least three due to the above assumption. \nIt remains to observe that the closed walk $vQ'w\\odot wv$ is a hole strictly shorter than $C$, a contradiction.\n\\end{proof}\n\nWe prove a generalization of Lemma~\\ref{lem:farnonadj}.\n\\begin{LEM}\\label{lem:generalfarnonadj}\nLet $m$ be a positive integer, and let $P$ be a $V(C)$-path in $G_{deldom}$ with endpoints $x$ and $y$.\nIf $P$ has length at most $m+2$, then $\\dist_C(x,y)\\leq 4m-1$.\n\\end{LEM}\n\\begin{proof}\nWe prove by induction on $m$. Lemma~\\ref{lem:farnonadj} settles the case when $m=1$.\nLet us assume $m\\ge 2$.\nLet $P=p_1p_2 \\cdots p_n$ be a $V(C)$-path of length at most $m+2$ from $p_1=x$ and $p_n=y$ such that \nall of $p_2, \\ldots, p_{n-1}$ are contained in $V(G_{deldom})\\setminus V(C)$, and suppose that $\\dist_C(x,y)\\geq 4m$.\nFor $\\dist_C(x,y)\\ge 4m\\ge 4$, Lemma~\\ref{lem:farnonadj} implies that $p_2$ is not adjacent to $p_{n-1}$.\nTherefore, $P$ contains at least $5$ vertices.\nWe distinguish cases depending on whether $\\{p_3, \\ldots, p_{n-2}\\}$ contains a vertex in $N(C)$ or not.\n\n\\medskip\n\\noindent {\\bf Case 1.} $\\{p_3, \\ldots, p_{n-2}\\}$ contains a vertex in $N(C)$. \\\\\nWe choose an integer $i\\in \\{3, \\ldots, n-2\\}$ such that $p_i\\in N(C)$, and\nchoose a neighbor $z$ of $p_i$ in $C$.\nSince there is a $V(C)$-path from $p_1$ to $z$ of length $i$, by induction hypothesis, $\\dist_C(p_1, z)< 4(i-2)$.\nBy the same reason, we have $\\dist_C(z, p_n)<4(n-i+1-2)=4(n-i-1)$.\nTherefore, we have \n\\[\\dist_C(p_1, p_n)\\le \\dist_C(p_1, z)+\\dist_C(z, p_n)<4(n-3)\\le 4m,\\]\na contradiction.\n\n\\medskip\n\\noindent {\\bf Case 2.} $\\{p_3, \\ldots, p_{n-2}\\}$ contains no vertices in $N(C)$.\n\\\\\nLet $Q$ be a shortest path from $N(p_2)\\cap V(C)$ to $N(p_{n-1})\\cap V(C)$ in $C$, and let $q, q'$ be its endpoints.\nObserve that $p_2Pp_{n-1}$ and $p_2q\\odot qQq' \\odot q'p_{n-1}$ are two paths from $p_2$ to $q'$ where there are no edges between their internal vertices.\nTherefore, $G[V(Q)\\cup (V(P)\\setminus \\{p_1, p_n\\})]$ is a hole.\n\nSince $\\dist_C(x,y)\\le \\frac{\\abs{V(C)}}{2}$, we have $m\\le \\frac{\\abs{V(C)}}{8}$. Therefore, the hole\n$G[V(Q)\\cup (V(P)\\setminus \\{p_1, p_n\\})]$ has length at most \n\\[ \\abs{V(Q)}+\\abs{V(P)}\\le \\frac{\\abs{V(C)}}{2}+ \\frac{\\abs{V(C)}}{8} +1 < \\abs{V(C)}.\\]\nThis contradicts the assumption that $C$ is a shortest hole of $G$.\n\n\\medskip\nThis concludes the proof.\n\\end{proof}\n\nNext, we show that every connected subgraph in $G_{nbd}$ has a connected support.\nThe following observation is useful.\n\n\\begin{LEM}\\label{lem:threeconsecutive}\nLet $a,b$ be vertices of $C$ with $\\dist_C(a,b)\\in \\{2,3\\}$ and \nlet $S$ be the set of internal vertices of the shortest $(a,b)$-path of $C$.\nThen there is no edge between $Z_a\\setminus Z_S$ and $Z_b\\setminus Z_S$.\n\\end{LEM}\n\\begin{proof}\nSuppose there is an edge between $x\\in Z_a\\setminus Z_S$ and $y\\in Z_b\\setminus Z_S$.\nIf $x$ is adjacent to $b$, then $x\\neq a$, and by Lemma~\\ref{lem:consecutive}, $x$ has a neighbor in $S$, contradicting the assumption that $x\\notin Z_S$.\nTherefore, $x$ is not adjacent to $b$.\nFor the same reason, $y$ is not adjacent to $a$.\nTherefore, the distance between $N(x)\\cap V(C)$ and $N(y)\\cap V(C)$ in $C$ is $2$ or $3$, \nand the vertex set of the shortest path from $N(x)\\cap V(C)$ to $N(y)\\cap V(C)$ in $C$ with $\\{x,y\\}$ induces a hole of length $5$ or $6$. \nThis contradicts with the assumption that $C$ is a shortest hole in $G$ and it has length greater than $6$.\n\\end{proof}\n\n\\begin{LEM}\\label{lem:connectedsupport}\nLet $H$ be a connected subgraph in $G_{nbd}$.\nThen $C[\\operatorname{\\textsf{sp}}(H)]$ is connected.\n\\end{LEM}\n\\begin{proof}\nSuppose $\\operatorname{\\textsf{sp}}(H)$ is not connected.\nThen $H$ contains an edge $xy$ such that $\\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])$ are contained in distinct components of $C[\\operatorname{\\textsf{sp}}(H)]$.\nNotice that $\\operatorname{\\textsf{sp}}(G[\\{x\\}])=N(x)\\cap V(C)$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])=N(y)\\cap V(C)$, and it follows that $x,y\\notin V(C)$ by Lemma~\\ref{lem:consecutive}. \nWe choose $a\\in \\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $b\\in \\operatorname{\\textsf{sp}}(G[\\{y\\}])$ with minimum $\\dist_C(a,b)$.\nBy Lemma~\\ref{lem:farnonadj}, we have $\\dist_C(a,b)\\le 3$, and since \n$\\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])$ are contained in distinct components of $C[\\operatorname{\\textsf{sp}}(H)]$, \nwe have $\\dist_C(a,b)\\ge 2$.\nLet $S$ be the set of internal vertices of the shortest $(a,b)$-path in $C$.\nBy the choice, $x\\in Z_a\\setminus Z_S$ and $y\\in Z_b\\setminus Z_S$.\nThen by Lemma~\\ref{lem:threeconsecutive}, \nthere is no edge between $Z_a\\setminus Z_S$ and $Z_b\\setminus Z_S$.\nThis contradicts the assumption that $x$ is adjacent to $y$.\n\\end{proof}\n\nThe following lemma provides a structure of a $(Z_x, Z_y)$-path in $G_{nbd}$ for two vertices $x,y\\in V(C)$.\n\n\\begin{LEM}\\label{lem:pathsupport}\nLet $x,y$ be two distinct vertices in $C$ and $P_1$, $P_2$ be two $(x,y)$-paths in $C$.\nLet $Q$ be a $(Z_x, Z_y)$-path in $G_{nbd}$ such that $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$.\nThen either $Q$ is contained in $Z_{V(P_1)}$ or $Z_{V(P_2)}$.\n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:connectedsupport}, \n$\\operatorname{\\textsf{sp}}(Q)$ is connected, and thus $\\operatorname{\\textsf{sp}}(Q)$ contains either $V(P_1)$ or $V(P_2)$.\nWithout loss of generality, we assume that $\\operatorname{\\textsf{sp}}(Q)$ contains $V(P_1)$.\nBy the definition of a $(Z_x, Z_y)$-path, $Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nLet $s$ and $t$ be the two endpoints of $Q$ contained in $Z_x$ and $Z_y$, respectively.\n\n\nWe claim that $Q$ contains no vertex of $V(G_{nbd})\\setminus Z_{V(P_1)}$, which immediately implies the statement.\nSuppose for contradiction that $Q$ contains a vertex $v\\in V(G_{nbd})\\setminus Z_{V(P_1)}$. Clearly, we have $v\\neq s$ and $v\\neq t$.\nLet $u$ be a vertex in $C$ such that $v\\in Z_u$. Observe that $u\\neq x$ and $u\\neq y$, as\n$Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nLet $Q_s$ and $Q_t$ be the $(s,v)$- and $(t,v)$-subpath of $Q$, respectively.\n\nBy Lemma~\\ref{lem:connectedsupport}, $\\operatorname{\\textsf{sp}}(Q_s)$ contains an $(x,u)$-subpath of $C$.\nAssume $\\operatorname{\\textsf{sp}}(Q_s)$ contains the $(x,u)$-subpath of $C$ containing $y$.\nThis means that $Q_s$ contains a vertex of $Z_y$, other than $t$, contradicting the fact that $Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nTherefore, \n$\\operatorname{\\textsf{sp}}(Q_s)$ contains the vertex set of the $(x,u)$-subpath of $C$ avoiding $y$. \nSimilarly, $\\operatorname{\\textsf{sp}}(Q_t)$ contains the vertex set of the $(y,u)$-subpath of $C$ avoiding $x$. \nNow, observe that $\\operatorname{\\textsf{sp}}(Q)=\\operatorname{\\textsf{sp}}(Q_s)\\cup \\operatorname{\\textsf{sp}}(Q_t) \\supseteq V(P_2)$ and also by assumption, we have $V(P_1)\\subseteq \\operatorname{\\textsf{sp}}(Q)$. \nConsequently, we have $\\operatorname{\\textsf{sp}}(Q)=V(C)$, a contradiction. This completes the proof.\n\\end{proof}\n\n\n\n\n\nThe following lemma is useful to find a hole with a small support.\n\\begin{LEM}\\label{lem:overlay}\nLet $P$ and $Q$ be two vertex-disjoint induced paths of $G_{nbd}$ such that\n\\begin{itemize}\n\\item there are no edges between $V(P)$ and $V(Q)$, and \n\\item $\\operatorname{\\textsf{sp}}(P)\\neq V(C)$ and $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$.\n\\end{itemize}\nIf $\\abs{\\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)}\\geq 3$ and $x,y,z\\in \\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)$ are three consecutive vertices on $C$, then $Z_{\\{x,y,z\\}}$ contains a hole. \n\\end{LEM}\n\\begin{proof}\nSince $x,y,z\\in \\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)$ and there is no edge between $V(P)$ and $V(Q)$, $P$ and $Q$ contains no vertex of $\\{x,y,z\\}$.\nLet $P'$ be a shortest $(Z_x,Z_z)$-subpath of $P$, and \nlet $Q'$ be a shortest $(Z_x,Z_z)$-subpath of $Q$.\nAs $\\operatorname{\\textsf{sp}}(P)\\neq V(C)$ and $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$, \n$V(P')$ and $V(Q')$ are contained in $Z_{\\{x,y,z\\}}$ by Lemma~\\ref{lem:pathsupport}.\nBy the preconditions, $P'$ and $Q'$ are vertex-disjoint and there are no edges between $P'$ and $Q'$. \nThus, by Lemma~\\ref{lem:twopaths}, $G[V(P')\\cup V(Q')\\cup \\{x,z\\}]$ contains a hole, which is in $Z_{\\{x,y,z\\}}$.\n\\end{proof}\n\n\\subsection{$C$-dominating vertices}\\label{subsec:dominating}\n\nWe recall that $D$ is the set of $C$-dominating vertices.\nWe observe that $D$ is a clique because $G$ does not contain a hole of length 4.\n\n\\begin{LEM}\\label{lem:dominating}\nThe set $D$ is a clique. Furthermore, every hole contains at most one vertex of $D$.\n\\end{LEM}\n\\begin{proof}\nNote that $G$ contains no hole of length $4$. This implies that any two vertices of $D$ are adjacent, which proves the first statement. To see the second statement, suppose that $H$ is a hole containing two distinct vertices $u,v$ of $D$ and let $x\\in V(H)\\cap V(C)$ (there are no holes in $G-V(C)$). Then $\\{x,u,v\\}$ forms a triangle, contradicting the assumption that $H$ is a hole. \n\\end{proof}\n\n\\begin{LEM}\\label{lem:neighborofdominating}\nIf $H$ is a $D$-traversing hole, then it contains at most two vertices of $C$. Furthermore, every vertex of $V(H)\\cap V(C)$ is adjacent to the unique $C$-dominating vertex on $H$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:dominating}, $H$ contains exactly one vertex of $D$, say $v$. \nIf there is a vertex $x\\in V(C)\\cap V(H)$, then $x$ is adjacent to $v$ as $v$ is $C$-dominating. Therefore, any vertex of $V(C)\\cap V(H)$ is adjacent to $v$ on $H$. Since $H$ is a cycle, $H$ contains at most two vertices of $C$.\n\\end{proof}\n\n\n\n\n\n\\section{Hitting all sunflowers}\\label{sec:hittingsunflower}\n\nIn this section, we obtain a hitting set for sunflowers, unless $G$ contains $k+1$ vertex-disjoint hols.\nLike in the previous section, we assume that $(G,k,C)$ is given as an input such that $C$ is a shortest hole of $G$ of length strictly greater than $\\mu_k$, \nand $G-V(C)$ is chordal.\n\n\n\\subsection{Hitting all petals.} \\label{subsec:petal}\n\n\\begin{LEM}\\label{lem:petalcover}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{petal}\\subseteq V(C)$ of at most $19k$ vertices such that \n\\begin{itemize}\n\\item for every petal $H$, we have $\\operatorname{\\textsf{sp}}(H)\\subseteq T_{petal}$.\n\\end{itemize} \n\\end{LEM}\n\\begin{proof}\nSet $X:=\\emptyset$, $\\mathcal{C}=\\emptyset$, and $counter:=0$ at the beginning. We recursively do the following until the counter reaches $k+1$. For every set of nine consecutive vertices $v_0, v_1, v_2, \\ldots, v_7, v_8$ of $C$ with $\\{v_1, v_2, \\ldots, v_7\\}\\cap X=\\emptyset$, we test if $G[Z_{\\{v_1, v_2, \\ldots, v_7\\}}\\setminus Z_{\\{v_0, v_8\\}}]$ contains a hole $H$, and if so, add vertices in $\\{v_1, v_2, \\ldots, v_7\\}$ to $X$, and add $H$ to $\\mathcal{C}$ and increase the counter by 1. If the counter reaches $k+1$, then we stop. \nIf the counter does not reach $k+1$, then\nwe have $\\abs{X}\\leq 7k$. In this case, we set $T_{petal}$ as the $6$-neighborhood of $X$ in $C$.\n\nBy construction, any hole $H \\in \\mathcal{C}$ has a support that is fully contained in the considered set $\\{v_1, v_2, \\ldots, v_7\\}$. \nObserve that we choose this set to be disjoint from $X$ constructed thus far. \nTherefore, holes in $\\mathcal{C}$ are pairwise vertex-disjoint; otherwise, their supports have a common vertex. \nThis implies that if the counter reaches $k+1$, then we can output $k+1$ vertex-disjoint holes.\n\nAssume the counter does not reach $k+1$. In this case, we claim that for every petal $H$, $\\operatorname{\\textsf{sp}}(H)\\subseteq T_{petal}$.\nLet $H$ be a petal. By the definition of a petal, there is a set of $7$ consecutive vertices $w_1, w_2, \\ldots, w_7$ in \n$C$ such that $\\operatorname{\\textsf{sp}}(H)\\subseteq \\{w_1, w_2, \\ldots, w_7\\}$. If the set $\\{w_1, w_2, \\ldots, w_7\\}$ is disjoint from $X$, then the above procedure must have considered this set and added it to $X$, a contradiction. \nTherefore, $\\{w_1, w_2, \\ldots, w_7\\}\\cap X\\neq \\emptyset$. Then during the step of adding 6-neighborhood of $X$ to $T_{petal}$, \n$\\{w_1, w_2, \\ldots, w_7\\}$ is added to $T_{petal}$, and thus we have $\\operatorname{\\textsf{sp}}(H)\\subseteq \\{w_1, w_2, \\ldots, w_7\\} \\subseteq T_{petal}$ as claimed.\n\\end{proof}\n\nIn what follows, we reserve $T_{petal}$ to denote a vertex subset of $V(C)$ that contains the support of every petal. \n\n\\subsection{Polarization of $D$-avoiding sunflowers.}\\label{subsec:allisfull}\n\n\nWe show that every $D$-avoiding sunflower in $G-T_{petal}$ is full. \nThis will imply that, in order to hit every $D$-avoiding sunflower it is sufficient to find a hitting set for full sunflowers.\n\n\n\n\\begin{LEM}\\label{lem:sppdichotomy}\nEvery $D$-avoiding sunflower $H$ in $G-T_{petal}$ is full, that is, $\\operatorname{\\textsf{sp}}(H)=V(C)$. \n\\end{LEM}\n\\begin{proof}\nSuppose $H$ is a $D$-avoiding sunflower in $G-T_{petal}$ such that $8\\le \\abs{\\operatorname{\\textsf{sp}}(H)}< \\abs{V(C)}$. \nBy Lemma~\\ref{lem:connectedsupport}, $\\operatorname{\\textsf{sp}}(H)$ is a subpath of $C$. \nLet $\\operatorname{\\textsf{sp}}(H)=v_1v_2 \\cdots v_{\\ell}$. Choose $x\\in V(H)\\cap Z_{v_1}$ and $y\\in V(H)\\cap Z_{v_{\\ell}}$, and let $P$ and $Q$ be the two $(x,y)$-paths on $H$. \nAs each of $C[\\operatorname{\\textsf{sp}}(P)]$ and $C[\\operatorname{\\textsf{sp}}(Q)]$ is connected by Lemma~\\ref{lem:connectedsupport}, \nwe have $\\operatorname{\\textsf{sp}}(P)=\\operatorname{\\textsf{sp}}(Q)= \\operatorname{\\textsf{sp}}(H)$.\n\n\n\nRecall that $V(H)\\cap V(C)\\neq \\emptyset$ and $V(H)\\cap V(C)$ must be contained in $\\operatorname{\\textsf{sp}}(H)$. We argue that any $v_i\\in \\operatorname{\\textsf{sp}}(H)$ with $i\\in \\{4, 5, \\ldots, \\ell-3\\}$ does not lie on $H$. Suppose $v_i\\in V(H)\\cap V(C)$ for some $4\\leq i\\leq \\ell-3$. Notice that both $x$ and $y$ are distinct from $v_i$. Therefore, $v_i$ belongs to exactly one of $P$ and $Q$. Without loss of generality, we assume $v_i\\in V(P)$. Since $v_i\\in \\operatorname{\\textsf{sp}}(Q)$, $Z_{v_i}\\cap V(Q)\\neq \\emptyset$ and thus we can choose a vertex $v'_i$ from the set $Z_{v_i}\\cap V(Q)$. Lemma~\\ref{lem:consecutive} and $v'_i\\notin D$ imply that $v'_i$ is not adjacent to $v_1$ or $v_{\\ell}$, and thus $v'_i\\notin Z_{v_1}$ and $v'_i\\notin Z_{v_{\\ell}}$. This means that $v'_i$ is distinct from $x$ and $y$, especially $v'_i$ is an internal vertex of $Q$. However, $v_iv'_i\\in E(G)$ is a chord of $H$, a contradiction. \n\nTherefore, $v_i \\notin V(H)$ for every $4\\leq i\\leq \\ell-3$. At least one of $\\{v_1,v_2,v_3\\}$ and $\\{v_{\\ell-2},v_{\\ell-1},v_{\\ell}\\}$ intersects with $V(H)$, and we assume that $\\{v_1,v_2,v_3\\}$ intersects with $V(H)$ without loss of generality (a symmetric argument works in the other case). From each of $P$ and $Q$, choose the first vertex (starting from $x$) that lies in $Z_{v_4}$ and call them $p\\in V(P)$ and $q\\in V(Q)$ respectively; the existence of such vertices follows from $v_4\\in \\operatorname{\\textsf{sp}}(P)=\\operatorname{\\textsf{sp}}(Q)$. Let $H'$ be the cycle $pPx\\odot xQq\\odot qv_4p$.\n\nSince $p$, $q$ are the first vertices contained in $Z_{v_4}$, \n$v_4$ has no neighbors in $(V(xPp)\\cup V(xQq))\\setminus \\{p, q\\}$, and thus $H'$ is a hole.\nBy Lemma~\\ref{lem:consecutive}, we have $\\operatorname{\\textsf{sp}}(H')\\subseteq \\{v_1, v_2, \\ldots, v_6\\}$, i.e. $H'$ is a petal. \nHowever, $\\{v_1,v_2,v_3\\}\\cap V(H)\\neq \\emptyset$ and $V(H)\\cap T_{petal}=\\emptyset$ implies $\\{v_1,v_2,v_3\\}\\setminus T_{petal}\\neq \\emptyset$. This contradicts \nthe assumption $T_{petal}$ contains the support of every petal. This completes the proof. \n\\end{proof}\n\n\n\\subsection{Hitting all $D$-avoiding sunflowers.}\\label{subsec:sunflowerwithout}\nIn this subsection we focus on full sunflowers.\n\n\\begin{PROP}\\label{prop:hitsunflower}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{full}\\subseteq V(G)\\setminus T_{petal}$ of at most $3k+14$ vertices such that $T_{petal}\\cup T_{full}$ hits all full sunflowers. \n\\end{PROP}\n\nOur strategy is to find two collections of many vertex-disjoint paths so that we can link the paths to obtain many vertex-disjoint holes.\nThe following lemma explains how to do this.\nNote that since $C$ has length greater than $\\mu_k$, $V(C)\\setminus T_{petal}$ contains $25$ vertices that are consecutive in $C$.\n\n\\begin{LEM}\\label{lem:connectingpaths}\nLet $v_{-2}v_1v_0v_1 \\cdots v_{20}v_{21}v_{22}$ be a subpath of $C$ that does not intersect $T_{petal}$, \nand $X$ be the $(v_0, v_{20})$-subpath of $C$ containing $v_1$ \nand $Y$ be the $(v_5, v_{15})$-subpath of $C$ containing $v_4$.\nLet $\\mathcal{P}$ be a collection of vertex-disjoint $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(X)}]$, and \nlet $\\mathcal{Q}$ be a collection of vertex-disjoint $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(Y)}]$. \nGiven such $\\mathcal{P}$ and $\\mathcal{Q}$, if $\\abs{\\mathcal{P}}\\ge k+13$ and $\\abs{\\mathcal{Q}}\\ge 3k+15$, then one can output $k+1$ vertex-disjoint holes in polynomial time.\n\\end{LEM}\n\\begin{proof}\nWe begin with the observation that there is no petal whose support contains a vertex of $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}$. \nThis is because $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}\\cap T_{petal}=\\emptyset$ by assumption and \n$T_{petal}$ contains the support of every petal of $G$.\nWe may assume that every path in $\\mathcal{P}$ is induced, \nand similarly every path in $\\mathcal{Q}$ is induced.\n\nWe take a subset $\\mathcal{P}_1$ of $\\mathcal{P}$ with $\\abs{\\mathcal{P}_1}=k+1$ that consists of paths containing no vertices of $\\{v_0, v_1, \\ldots, v_5\\}\\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. Such a collection $\\mathcal{P}_1$ exists because the paths of $\\mathcal{P}$ are vertex-disjoint and at most 12 of them \nintersect with $\\{v_0, v_1, \\ldots, v_5\\}\\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. \nSimilarly we take a subset $\\mathcal{Q}_1$ of $\\mathcal{Q}$ with $\\abs{\\mathcal{Q}_1}=3k+3$ \nthat consists of paths containing no vertices of $\\{v_0, v_1, \\ldots, v_5\\} \\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. \n\nFor each $P\\in \\mathcal{P}_1$, let $\\ell(P)$ and $r(P)$ be the endpoints of $P$ contained in $Z_{v_0}$ and $Z_{v_{20}}$, respectively.\nFor each $Q\\in \\mathcal{Q}_1$, let $a(Q)$ be the vertex of $Z_{v_0}\\cap V(Q)$ that is closest to $Z_{v_5}\\cap V(Q)$ in $Q$, and \nlet $b(Q)$ be the vertex in $Z_{v_{20}}$ that is closest to $Z_{v_{15}}\\cap V(Q)$ in $Q$.\nBy definition, no internal vertex of the subpath of $Q$ from $a(Q)$ to the vertex in $V(Q)\\cap Z_{v_5}$ contains a neighbor of $v_0$, and \nsimilarly no internal vertex of the subpath of $Q$ from $b(Q)$ to the vertex in $V(Q)\\cap Z_{v_{15}}$ contains a neighbor of $v_{20}$. See Figure~\\ref{fig:pathsystems} for an illustration.\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(5,0);\\draw[thick, dotted](5,0)--(6,0);\n \\draw (6,0)--(13,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(13, 0)--(14,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (14,-.5) node [w] {};\n \t\\draw[dashed](15, -1)--(14,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,13}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n\n \\foreach \\y in {-1, 4, 7, 12}{\n\\draw[dashed, rounded corners] (\\y, 3.3)--(\\y+.5, 3.3)--(\\y, -.5)--(\\y-.5, 3.3)--(\\y, 3.3);\n}\n\n \\node at (-1, 4) {$Z_{v_0}$};\n \\node at (4, 4) {$Z_{v_5}$};\n \\node at (7, 4) {$Z_{v_{15}}$};\n \\node at (12, 4) {$Z_{v_{20}}$};\n \\node at (-1, -.8) {$v_0$};\n \\node at (4, -.8) {$v_5$};\n \\node at (7, -.8) {$v_{15}$};\n \\node at (12, -.8) {$v_{20}$};\n\n \\node at (-2, -1) {$C$};\n\n \\node at (1.5, 2.8) {$P$};\n \\node at (1.5, 1.6) {$Q$};\n\n \\node at (-2, 2.4) {$\\ell(P)$};\n \\node at (13, 2.4) {$r(P)$};\n\n\t \\node at (-1.8, 0.8) {$a(Q)$};\n \\node at (12.8, 0.8) {$b(Q)$};\n\n\t \\draw (-1,2.4)--(12,2.4);\n \\draw[rounded corners] (4,1.2)--(-2,1.2)--(-3,0.7);\\draw[dashed] (-3, 0.7)--(-4,0.2);\n \\draw[rounded corners] (7,1.2)--(13,1.2)--(14,0.7);\\draw[dashed] (14, 0.7)--(15,0.2);\n \\draw (-1,2.4) node [w] () {};\n \\draw (4,1.2) node [w] () {};\n \\draw (7,1.2) node [w] () {};\n \\draw (12,2.4) node [w] () {};\n \\draw (-1,1.2) node [w] () {};\n \\draw (12,1.2) node [w] () {};\n\n \\end{tikzpicture} \\caption{Paths $P\\in \\mathcal{P}$ and $Q\\in \\mathcal{Q}$ in Lemma~\\ref{lem:connectingpaths}. The vertices $\\ell(P)$ and $a(Q)$ are completely adjacent, otherwise, we can find a hole $Z_{\\{v_0, v_1, v_2\\}}$, which is a petal. For the same reason, $r(P)$ is completely adjacent to $b(Q)$.}\\label{fig:pathsystems}\n\\end{figure}\n\n\n\n\n\\begin{CLAIM}\\label{claim:ef}\nLet $w\\in \\{\\ell(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{a(Q):Q\\in \\mathcal{Q}_1\\}$. If $w\\neq z$, then $wz\\in E(G)$.\nLet $w\\in \\{r(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{b(Q):Q\\in \\mathcal{Q}_1\\}$. If $w\\neq z$, then $wz\\in E(G)$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $w\\in \\{\\ell(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{a(Q):Q\\in \\mathcal{Q}_1\\}$ such that $w\\neq z$ and they are not adjacent. \nLet $P_w\\in \\mathcal{P}_1$ and $Q_z\\in \\mathcal{Q}_1$ such that \n$\\ell(P_w)=w$ and $a(Q_z)=z$. Let $Q_z'$ be the subpath of $Q_z$ from $z$ to the vertex in $Z_{v_5}$.\nNote that $v_0$ has no neighbors in $V(P_w)\\setminus \\{w\\}$ and $V(Q_z')\\setminus \\{z\\}$.\nIn case when $P_w$ and $Q_z'$ meet somewhere in $\\bigcup_{i\\in \\{1, 2\\}}Z_{v_i}$, \nwe obtain a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$ by Lemma~\\ref{lem:twopaths}.\nWhen $P_w$ and $Q_z'$ do not meet in $\\bigcup_{i\\in \\{1, 2\\}}Z_{v_i}$, \nthere is a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$ by Lemma~\\ref{lem:twopaths} since $v_2$ has a neighbor in both $P_w$ and $Q_z'$. \nIn both cases, there is a petal with support contained in $\\{v_{i}: -2\\le i\\le 4\\}$, a contradiction. \nWe conclude that $wz\\in E(G)$.\nThe proof of the latter statement is symmetric.\n\\end{proofofclaim}\n\nFor every $P\\in \\mathcal{P}_1$, $\\ell(P)$ is the unique vertex of $Z_{v_0}\\cap V(P)$. Therefore, for fixed $P\\in \\mathcal{P}_1$,\nthere is at most one path $Q\\in \\mathcal{Q}_1$ such that $V(Q)\\cap V(P)\\cap Z_{v_0}\\neq \\emptyset$. \nSimilarly, there is at most one path of $\\mathcal{Q}_1$ intersecting with $P$ at a vertex of $Z_{v_{20}}$. \nWe construct a new collection $\\mathcal{Q}_2$ so that\n\\begin{quote}\nfor every $Q\\in \\mathcal{Q}_1$, $\\mathcal{Q}_2$ contains the subpath $a(Q)Qb(Q)$ if and only if \n$Q$ does not intersect with any $P\\in \\mathcal{P}_1$ at a vertex of $Z_{v_0}\\cup Z_{v_{20}}$.\n\\end{quote}\nObserve that $\\mathcal{Q}_2$ contains at least $k+1$ paths because each path of $\\mathcal{P}_1$ can make \nat most two paths of $\\mathcal{Q}_1$ drop out. For our purpose, taking precisely $k+1$ paths is sufficient. \nLet $\\mathcal{P}_1=\\{P_1,\\ldots , P_{k+1}\\}$ and $\\mathcal{Q}_2=\\{Q_1, Q_2, \\ldots, Q_{k+1}\\}$.\nFor each $i\\in \\{1, 2, \\ldots, k+1\\}$, we create a cycle $C_i$ from the disjoint union of $P_i\\in \\mathcal{P}_1$ and $Q_i\\in \\mathcal{Q}_2$ \nby adding two edges $a(Q_i)\\ell(P_i)$ and $b(Q_i)r(P_i)$. \nSuch edges exist by Claim~\\ref{claim:ef}.\n\nWe observe that each $C_i$ contains a hole. To see this, take $v\\in Z_{v_{10}}\\cap V(P_i)$. \nAs $Q_i\\in \\mathcal{Q}_2$ is a path of $G[Z_{V(Y)}]$, Lemma~\\ref{lem:consecutive} implies that $v$ is not adjacent to any vertex of $Q_i$. \nNote that $v$ is an internal vertex of \nthe induced path $P_i$. Therefore, $G[V(C_i)]$ contains a hole by Lemma~\\ref{lem:twopaths}. \n\nLastly, we verify that two holes contained in distinct cycles of $\\{C_i:1\\le i\\le k+1\\}$ are vertex-disjoint. To prove this, it is sufficient to show that \nfor two integers $a,b\\in \\{1, 2, \\ldots, k+1\\}$, no internal vertex of $P_a\\in \\mathcal{P}_2$ is an internal vertex of $Q_b\\in \\mathcal{Q}_2$. \nSuppose the contrary, that is, $w$ is an internal vertex of $P_a\\in \\mathcal{P}_2$ and $Q_b\\in \\mathcal{Q}_2$ simultaneously for some $a,b$.\nSince $Q_b$ is a path of $G[Z_{Y}]$ and $w\\notin Z_{v_0}\\cup Z_{v_{20}}$ is an internal vertex of $P_a$, we have $w\\in Z_{\\{v_1, v_2, v_3, v_4,v_5\\}}\\cup Z_{\\{v_{15}, v_{16}, v_{17}, v_{18},v_{19}\\}}$. \nWithout loss of generality, we assume $w\\in Z_{\\{v_1, v_2, v_3, v_4,v_5\\}}$. \nIn fact, $w$ cannot be in $Z_{v_5}$ since otherwise, the path $Q'_b\\in \\mathcal{Q}_1$ having $Q_b$ as a proper subpath contains a vertex of $Z_{v_5}$ as an internal vertex; \nviolating the definition of $(Z_{v_5},Z_{v_{15}})$-path. \nNow observe that $Q_b$ contains, as a subpath, a $Z_{v_0}$-path $Q'$ having all internal vertices in $Z_{\\{v_1, v_2, v_3, v_4\\}}\\setminus Z_{\\{v_0, v_5\\}}$. \nLet $x,y$ be the endpoints of $Q'$. Since $v_0$ is adjacent to $x,y$ but is not adjacent to any internal vertices of $Q'$,\n$G[\\{v_0\\}\\cup V(Q')]$ contains a hole by Lemma~\\ref{lem:twopaths}, a contradiction. A symmetric argument holds for the case \n$w\\in Z_{\\{v_{15}, v_{16}, v_{17}, v_{18},v_{19}\\}}$. Therefore, $\\{C_i:1\\le i\\le k+1\\}$ \nis a vertex-disjoint holes of $G$, which completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:hitsunflower}]\nNote that $\\abs{T_{petal}}\\le 19k$ and $\\abs{V(C)}>\\mu_k> 25\\abs{T_{petal}}$ by assumption. Thus, there are $25$ consecutive vertices on $C$ having no vertices in $T_{petal}$.\nWe choose a subpath $v_{-2}v_{-1}v_0v_1 \\cdots v_{20}v_{21}v_{22}$ of $C$ that contains no vertices in $T_{petal}$.\n Let $P_1$ be the $(v_0, v_{20})$-subpath of $C$ containing $v_1$ \n and $P_2$ be the $(v_5, v_{15})$-subpath of $C$ containing $v_4$.\n\nWe apply Menger's Theorem for $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(P_1)}]$,\nand then for $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(P_2)}]$. \nWe have one of the following.\n\\begin{itemize}\n\\item The first application of Menger's Theorem outputs a vertex set $X$ with $\\abs{X}\\leq k+12$ hitting all $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(P_1)}]$.\n\\item The second application of Menger's Theorem outputs a vertex set $X$ with $\\abs{X}\\leq 3k+14$ hitting all $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(P_2)}]$.\n\\item The first algorithm outputs at least $k+13$ vertex-disjoint paths, and the second algorithm outputs at least $3k+15$ vertex-disjoint paths. \n\\end{itemize} \n\nIn the third case, by Lemma~\\ref{lem:connectingpaths}, we can construct $k+1$ vertex-disjoint holes in polynomial time.\n\nSuppose we obtained a vertex set $X$ in the first case. We claim that $T_{petal}\\cup X$ hits all full sunflowers. Suppose that there is a full sunflower $H$ avoiding every vertex of $T_{petal}\\cup X$. By definition, $\\operatorname{\\textsf{sp}} (H)=V(C)$. In particular, $H$ contains at least one vertex of $Z_{v_{10}}$, say $w$. \nLet $F$ be the connected component of the restriction of $H$ on $G[Z_{V(P_1)}]$ containing $w$. \nClearly $F$ is a path. \nWe argue that its endpoints are contained in $Z_{\\{v_0, v_{20}\\}}$ because of Lemma~\\ref{lem:connectedsupport}.\n\nSuppose that the endpoints of $F$ are contained in distinct sets of $Z_{v_0}$ and $Z_{v_{20}}$, respectively. \nLet $F'$ be a subpath of $F$ that is a $(Z_{v_0}, Z_{v_{20}})$-path. Note that $F'$ is a $(Z_{v_0}, Z_{v_{20}})$-path of $G[Z_{V(P_1)}]$ \nbecause $F$ is a path of $G[Z_{V(P_1)}]$.\nBut it contradicts with the fact that $X$ hits all such paths. \n\nSuppose that both endpoints of $F$ are contained in one of $Z_{v_0}$ or $Z_{v_{20}}$, say $Z_{v_0}$.\nLet $F_1$ and $F_2$ be the two subpaths of $F$ from $w$ to its endpoints.\nThen by Lemma~\\ref{lem:connectedsupport}, both $\\operatorname{\\textsf{sp}}(F_1)$ and $\\operatorname{\\textsf{sp}}(F_2)$ contain the $(v_0, v_{10})$-subpath of $P_1=v_0v_1 \\cdots v_{20}$. \nThis implies that $\\operatorname{\\textsf{sp}}(F_1-w)\\cap \\operatorname{\\textsf{sp}}(F_2-w)$ contains $\\{v_0, v_1, v_2\\}$.\nSince there are no edges between $F_1-w$ and $F_2-w$, Lemma~\\ref{lem:overlay} implies that \nthere is a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$.\nThis is a contradiction because we assumed $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}\\cap T_{petal}=\\emptyset$ while \n$T_{petal}$ contains the support of every petal of $G$.\nTherefore, $T_{petal}\\cup X$ hits every full sunflower. The case when both endpoints of $F$ are contained on $Z_{v_{20}}$ \nfollows from a symmetric argument.\n\nThe second case when we obtain the vertex set $X$ with $\\abs{X}\\leq 3k+14$ can be handled similarly. Hence, in the first or second case, we can output a required vertex set $T_{full}$ of size at most $3k+14$ hitting every full sunflower in polynomial time.\n\\end{proof}\n\n\n\\subsection{Hitting all $D$-traversing sunflowers.}\\label{subsec:sunflowerwith}\n\nOur proof builds on the observation that any $D$-traversing sunflower entails another $D$-traversing sunflower $H'$ where the support of the path $H'- D$ is `small'. Then we exploit the min-max duality of vertex cover and matching on bipartite graphs\nin order to cover such $D$-traversing sunflowers with small support.\n\nThe following lemma describes how to obtain such a sunflower $H'$. \nWe depict the setting of Lemma~\\ref{lem:smallsunflower} in Figure~\\ref{fig:traversingsunflower}.\n\n\n\\begin{LEM}\\label{lem:smallsunflower}\nLet $v_1v_2 \\cdots v_5$ be a subpath of $C$ such that $\\{v_1,\\ldots , v_5\\}\\cap T_{petal}=\\emptyset$ and let $P=p_1p_2 \\cdots p_m$ be a path in $G[D\\cup Z_{\\{v_1, v_2, \\ldots, v_5\\}}]$ such that\n\\begin{enumerate}[(i)]\n\\item $p_1$ is a $C$-dominating vertex and $p_2=v_3$, \n\\item $p_m\\in Z_{\\{v_1, v_5\\}}\\setminus \\{v_1, v_5\\}$, \n\\item all internal vertices of $P$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$, and \n\\item $E(G[V(P)])\\setminus E(P)\\subseteq \\{p_1p_m\\}$; that is, $G[V(P)]$ is either an induced path $p_1p_2 \\cdots p_m$ or an induced cycle $p_1p_2 \\cdots p_mp_1$, \n\\item if $G[V(P)]$ is an induced cycle, then $m\\ge 4$.\n\\end{enumerate}\nThen there exists a $D$-traversing sunflower $H$ containing $p_1$ and $p_2$ such that $V(H)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}} \\cap (V(P)\\cup \\{v_1\\})$\n or $ V(H)\\setminus \\{p_1\\} \\subseteq Z_{\\{v_3, v_4, v_5\\}}\\cap (V(P)\\cup \\{v_5\\})$. \n\\end{LEM}\n\\begin{proof}\nWe claim the following:\n\\begin{quote}\nIf $p_m\\in Z_{v_1}$, then $P-p_1$ is contained in $Z_{\\{v_1, v_2, v_3\\}}$. Likewise, \nif $p_m\\in Z_{v_5}$, then $P-p_1$ is contained in $Z_{\\{v_3, v_4, v_5\\}}$.\n\\end{quote} \n\nWe only prove the first statement; the proof of the second statement will be symmetric.\nLet us assume $p_m\\in Z_{v_1}$. We observe that $P$ contains no vertex in $Z_{v_5}$ because \nall internal vertices of $P$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$.\n\nWe first show that $P$ contains no vertex in $Z_{v_4}\\setminus Z_{v_3}$.\nSuppose the contrary and let $w\\in V(P)\\cap (Z_{v_4}\\setminus Z_{v_3})$. \nSince both $C[\\operatorname{\\textsf{sp}}(p_2Pw)]$ and $C[\\operatorname{\\textsf{sp}}(wPp_m)]$ are connected by Lemma~\\ref{lem:connectedsupport}, \n$P$ contains a $Z_{v_3}$-subpath $P'$ whose internal vertices are all contained in $Z_{v_4}\\setminus Z_{v_3}$.\nThen $v_3$ is not adjacent to any internal vertex of $P'$ by Lemma~\\ref{lem:twopaths}, and thus $G[V(P')\\cup \\{v_3\\}]$ contains a hole, which is a petal.\nThis contradicts the fact that $v_3\\notin T_{petal}$, because by the construction of $T_{petal}$ in Lemma~\\ref{lem:petalcover}, $T_{petal}$ fully contains the support of every petal.\nHence, $P$ contains no vertex of $Z_{v_4}\\setminus Z_{v_3}$ and we have $V(P)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$.\n\n\nWe claim that there is a $D$-traversing sunflower as claimed. \nIf $p_1p_m\\in E(G)$, then $G[V(P)]$ is a hole as claimed by (iv) and due to the previous claim.\nHence, we may assume $p_1$ is not adjacent to $p_m$.\nObserve that $v_1p_1p_2$ is an induced path.\nAlso, $G[(V(P)\\setminus \\{p_1\\})\\cup \\{v_1\\}]$ is a path from $v_1$ to $v_3$, and it does not contain $v_2$. Indeed, \nif the path $G[(V(P)\\setminus \\{p_1\\})\\cup \\{v_1\\}]$ contains $v_2$, then $\\{p_1,p_2=v_3,v_2\\}$ forms a triangle, a contradiction to (iv). \nNow, $p_1$ has no neighbors in $V(P)\\setminus \\{p_1, p_2\\}$ as we assumed that $p_1$ is not adjacent to $p_m$.\nTherefore, we can apply Lemma~\\ref{lem:twopaths} with the induced path $v_1p_1p_2$ with $p_1$ as an internal vertex and \nthe path $p_2Pp_m\\odot p_mv_1$. It follows that there is a hole in $G[V(P)\\cup \\{v_1\\}]$ containing $p_1$, $p_2$ \nsuch that $V(H)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$. The statement follows immediately.\n\\end{proof}\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n\t\n\t\n \n \\draw(5,3.5) [in=80,out=150] to (.5,0);\n \\draw(.5,0)--(.5,1.5);\n \\node at (0, 1.5) {$p_m$};\n \n \t\\draw (5, 3.5)--(4.5,0)--(4, 2)--(3,2.7)--(2, 2)--(1.5, 0.5)--(.5, 1.5);\n \\draw (4,2) node [w] {};\n \\draw (3,2.7) node [w] {};\n \\draw (2,2) node [w] {};\n \\draw (1.5,.5) node [w] {};\n \\draw (.5,1.5) node [w] {};\n\n \\node at (3, 1.5) {$P$};\n \n \\foreach \\y in {-1.5,0.5, ..., 11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\node at (0.5, -.8) {$v_1$};\n \\node at (2.5, -.8) {$v_2$};\n \\node at (4.5, -.8) {$v_3=p_2$};\n \\node at (6.5, -.8) {$v_4$};\n \\node at (8.5, -.8) {$v_5$};\n\n \\foreach \\y in {0.5, 8.5}{\n\\draw[dashed, rounded corners] (\\y, 3.3)--(\\y+.5, 3.3)--(\\y, -.5)--(\\y-.5, 3.3)--(\\y, 3.3);\n}\n\n \\node at (0.5, 4) {$Z_{v_1}$};\n \\node at (8.5, 4) {$Z_{v_5}$};\n\n \n\\draw[rounded corners] (-3-.5+7,4)--(-3-.5+7,3)--(0-.5+7,3)--(0-.5+7,5)--(-3-.5+7,5)--(-3-.5+7,4);\n \\draw (-2+7, 3.5) node [w] {};\n \\node at (5.5, 3.5) {$p_1$};\n\n\n \\node at (-1.5-.5+7, 4.5) {$D$};\n \n \\end{tikzpicture} \\caption{Obtaining another sunflower in Lemma~\\ref{lem:smallsunflower}.\n As $v_1p_1p_2$ is an induced path and $p_1$ has no neighbors in the set of internal vertices of $p_2Pp_m\\odot p_mv_1$, $G[V(P)\\cup \\{v_1\\}]$ contains a hole.}\\label{fig:traversingsunflower}\n\\end{figure}\n\n\\begin{LEM}\\label{lem:small}\nLet $H$ be a $D$-traversing sunflower in $G-T_{petal}$ containing a $C$-dominating vertex $d$. Then there exist three consecutive vertices $x,y,z$ on $C$ and a $D$-traversing sunflower $H'$ containing $d$ such that $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:neighborofdominating}, $H$ contains at most $2$ vertices of $C$, and every vertex in $V(H)\\cap V(C)$ is neighboring the \n(unique) vertex of $V(H)\\cap D$ on $H$.\nLet $P$ be the connected component of $H-(V(C)\\cup \\{d\\})$.\nNote that $P$ contains no vertices of $D$.\nLet $a\\in V(H)\\cap V(C)$.\nIf the support of $P$ is contained in $N_C[a]$, then we are done as $\\abs{N_C[a]}\\le 3$ and we can take $H'=H$ and $\\{x,y, z\\}=N_C[a]$.\nWe may assume that the support of $P$ contains a vertex in $C$ whose distance to $a$ in $C$ is $2$ by Lemma~\\ref{lem:connectedsupport}.\nLet $v_1, v_2, v_3, v_4, v_5$ be the consecutive vertices of $C$ where $a=v_3$.\nThis assumption implies that $P$ contains a vertex in either $Z_{v_1}$ or $Z_{v_5}$.\n\nLet $w$ be the vertex of $Z_{\\{v_1, v_5\\}}\\cap V(H)$ that is closest to $N_H(a)\\setminus \\{d\\}$.\nLet $Q$ be the $(d,w)$-subpath of $H$ containing $a$.\nWe verify the preconditions of Lemma~\\ref{lem:smallsunflower} with $(p_1, p_2, P)=(d, a, Q)$.\nThe first condition is clear.\nNote that $H$ contains neither $v_2$ nor $v_4$; otherwise, $dav_2$ or $dav_4$ is a triangle in $H$, contradicting the assumption that $H$ is a hole.\nThus $Q$ contains neither $v_2$ nor $v_4$.\nIf $w$ is $v_1$ or $v_5$, then the neighbor of $w$ in $Q$ is also in $Z_{\\{v_1, v_5\\}}$, contradicting the choice of $w$.\nThus, $w\\in Z_{\\{v_1, v_5\\}}\\setminus \\{v_1, v_5\\}$.\nClearly, all internal vertices of $Q$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$; otherwise by Lemma~\\ref{lem:connectedsupport}, \n$Q$ must contain an internal vertex from $Z_{\\{v_1, v_5\\}}$, contradicting the choice of $w$.\nThe last two conditions are satisfied because $H$ is a hole.\nThen $(d,a,Q)$ meets the preconditions of Lemma~\\ref{lem:smallsunflower}. \n\nTherefore, there exists a $D$-traversing sunflower $H'$ containing $d$ and $a$ such that $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$ or $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{v_3, v_4, v_5\\}}$.\nAs $a=v_3\\in V(H)$, we have $V(H)\\cap \\{v_1, v_2, v_3\\}\\neq\\emptyset$ or $V(H)\\cap \\{v_3, v_4, v_5\\}\\neq\\emptyset$, respectively.\n\\end{proof}\n\nBased on Lemma~\\ref{lem:small}, we prove the following.\n\n\n\n\n\\begin{PROP}\\label{prop:skew}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{trav:sunf}\\subseteq (D\\cup V(C))\\setminus T_{petal}$ of size at most $15k+9$ such that $T_{petal}\\cup T_{trav:sunf}$ hits all $D$-traversing sunflowers.\n\\end{PROP}\n\\begin{proof}\nLet $C=v_0v_1 \\cdots v_{m-1}v_0$. All additions are taken modulo $m$. \nWe create an auxiliary bipartite graph $\\mathcal{G}_i=(D\\uplus \\mathcal{A}_i, \\mathcal{E}_i)$ for each $0\\leq i \\leq 4$, such that\n\\begin{itemize}\n\\item $\\mathcal{A}_i=\\{\\{v_{5j+i},v_{5j+i+1},v_{5j+i+2}\\}: j=0, 1, \\ldots ,\\lfloor \\frac{m}{5} \\rfloor -1\\}$,\n\\item there is an edge between $d\\in D$ and $\\{x,y,z\\}\\in \\mathcal{A}_i$ if and only if \nthere is a hole $H$ containing $d$ such that $V(H)\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$ (thus, $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$).\n\\end{itemize}\nClearly, the auxiliary graph $\\mathcal{G}_i$ can be constructed in polynomial time using Lemma~\\ref{lem:detectinghole}. \n\nNow, we apply Theorem~\\ref{thm:menger} to each $\\mathcal{G}_i$ and outputs either a matching of size $k+1$ or a vertex cover of size at most $k$.\n\nSuppose that there exists $i\\in \\{0,1,\\ldots, 4\\}$ such that $\\mathcal{G}_i$ contains a matching $M$ of size at least $k+1$. \nWe argue that there are $k+1$ vertex-disjoint holes in this case. Let $e=(d,\\{x,y,z\\})$ and $e'=(d',\\{x',y',z'\\})$ be two distinct edges of $M$. \nBy construction, \nthere exist two holes $H$ and $H'$ such that \n\\begin{itemize}\n\\item $H$ contains $d$ and $V(H)\\setminus \\{d\\} \\subseteq Z_{\\{x,y,z\\}}$,\n\\item $H'$ contains $d'$ and $V(H')\\setminus \\{d'\\} \\subseteq Z_{\\{x',y',z'\\}}$.\n\\end{itemize}\nRecall that any vertex of $N(C)\\setminus D$ has at most three neighbors on $C$, which are consecutive by Lemma~\\ref{lem:consecutive}. On the other hand, the distance between $\\{x,y,z\\}$ and $\\{x',y',z'\\}$ on $C$ is at least three by the construction of the family $\\mathcal{A}_i$. Therefore the two sets $Z_{\\{x,y,z\\}}$ and $Z_{\\{x',y',z'\\}}$ are disjoint, \nwhich implies \n$H$ and $H'$ are vertex-disjoint. We conclude that one can output $k+1$ vertex-disjoint holes when there is a matching $M$ of size $k+1$ in one of $\\mathcal{G}_i$'s.\n\nConsider the case when for every $0\\le i\\leq 4$, $\\mathcal{G}_i$ admits a vertex cover $S_i$ of size at most $k$. For $S_i$, let $S^*_i$ be the vertex set \n\\[ (S_i\\cap D) \\cup \\bigcup_{\\{x,y,z\\}\\in S_i\\cap \\mathcal{A}_i} \\{x,y,z\\} \\]\nand let $T_{trav:sunf}:=\\left( \\bigcup_{i=0}^4 S^*_i \\right) \\cup \\{v_{5\\lfloor \\frac{m}{5} \\rfloor+i}:-2\\le i\\le 6\\}$. Notice that $\\abs{T_{trav:sunf}}\\leq 15k+9$. \n\n\\begin{CLAIM}\nThe vertex set $T_{petal}\\cup T_{trav:sunf}$ hits all $D$-traversing sunflowers. \n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $G-(T_{petal}\\cup T_{trav:sunf})$ contains a $D$-traversing sunflower $H$ having a vertex $d\\in D$. \nBy Lemma~\\ref{lem:small}, \nthere exist $x,y,z$ that are consecutive vertices on $C$ and a $D$-traversing sunflower $H'$ containing $d$ such that $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$. Clearly, we have either \n\\begin{itemize}\n\\item $d$ is adjacent to $\\{x,y,z\\}$ in one of the bipartite graphs $\\mathcal{G}_i$, or\n\\item $\\{x,y,z\\}\\subseteq \\{v_{5\\lfloor \\frac{m}{5} \\rfloor+i}:-2\\le i\\le 6\\}$.\n\\end{itemize}\nIn the first case, $S^*_i$ contains $\\{d\\}$ or $\\{x,y,z\\}$, as $S_i$ is a vertex cover of $\\mathcal{G}_i$. Since $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $H$ contains $d$, $S^*_i$ contains a vertex of $H$, which contradicts the assumption that $H$ is a $D$-traversing sunflower in $G-(T_{petal}\\cup T_{trav:sunf})$. In the second case, \n$T_{trav:sunf}$ contains $\\{x,y,z\\}$, which again contradicts that $H$ is a $D$-traversing sunflower in $G-(T_{petal}\\cup T_{trav:sunf})$.\n\\end{proofofclaim}\n\nThis completes the proof. \n\\end{proof}\n\n\n\n\\section{Hitting all tulips}\\label{sec:tulip}\n\n\n\nIn this section, \nwe show that one can find in polynomial time either $k+1$ vertex-disjoint holes or a vertex set hitting all tulips.\nAgain, we assume that $(G,k,C)$ is given as an input such that $C$ is a shortest hole of $G$ of length \nstrictly greater than $\\mu_k$ and $G-V(C)$ is chordal.\n\nThe first few sections will focus on $D$-avoiding tulips. In Subsection~\\ref{subsec:Dtulip}, we settle the case of $D$-traversing tulips.\nSubsection~\\ref{subsec:final} will establish the main theorem for holes in general, Theorem~\\ref{thm:core}.\nFor $D$-avoiding tulips, it is sufficient to consider the graph $G_{deldom}=G-D$. \n\n\\subsection{Constructing a nested structure of partial tulips}\\label{subsec:tuliphive}\n\nWe recursively construct a subgraph of $G_{deldom}$ in which all vertices have degree $2$ or $3$ and it contains $C$. \nA subgraph of $G$ is called a \\emph{$(2,3)$-subgraph} if its all vertices have degree $2$ or $3$.\nFor a $(2,3)$-subgraph $F$, a vertex $v$ of degree $3$ in $F$ is called a \\emph{branching point} in $F$,\n and other vertices are called \\emph{non-branching points}. \n\nGiven a $(2,3)$-subgraph $F$ of $G_{deldom}$ containing $C$, an $(x,y)$-path $P$ of $G_{deldom}$ is a \\emph{$F$-extension} if it satisfies the following.\n\\begin{enumerate}[(i)]\n\\item $x$ and $y$ are distinct non-branching points of $F$.\n\\item $\\{x,y\\}\\cap V(C)\\neq \\emptyset$.\n\\item $P$ is a proper $V(F)$-path and $P-V(F)$ is an induced path of $G_{deldom}$.\n\\item There exists a vertex $v\\in V(P)$ such that $\\dist_P(v,\\{x,y\\}\\cap V(C))=2$ and $v\\notin N[F]$. \n\\end{enumerate} \nNote that by condition (iv), the length of an $F$-extension is at least $4$.\n\nA cycle $H$ of $G_{deldom}$ is an \\emph{almost $F$-extension} \nif it satisfies the following. \n\\begin{enumerate}[(i)]\n\\item[(i')] $\\abs{V(H)\\cap V(C)}=1$ and the vertex in $V(H)\\cap V(C)$ is a non-branching point of $F$ in $C$. \n\\item[(ii')] $H-V(C)$ is an induced path of $G_{deldom}$ and contains no vertex of $F$.\n\\item[(iii')] There exists a vertex $v\\in V(H)$ such that $\\dist_H(v,V(H)\\cap V(C))=2$ and $v\\notin N[F]$. \n\\end{enumerate} \nWe call the vertex in $V(H)\\cap V(C)$ the \\emph{root} of the almost $F$-extension $H$. \n\nIt is not difficult to see that given a $(2,3)$-subgraph $F$ containing $C$, \nthere is a polynomial-time algorithm to find a shortest $F$-extension $P$ or correctly decides that there is no $F$-extension. \nFor this, we exhaustively guess five vertices $x,y,x',y',v$ such that \n\\begin{itemize}\n\\item $x$ and $y$ are non-branching points of $F$ such that $x\\in V(C)$,\n\\item $x'$ and $y'$ are neighbors of $x$ and $y$ in $V(G_{deldom})\\setminus V(F)$, respectively, \n\\item $v$ is a neighbor of $x'$ in $V(G)\\setminus N[F]$.\n\\end{itemize}\nSince we are looking for a $(x,y)$-path $P$ where $\\mathring{x}P\\mathring{y}$ is induced, \nwe check whether there is a path from $v$ to $y'$ in $G_{deldom}-((V(F)\\cup N[x'])\\setminus \\{v\\})$. \nIf there is such a path, then we find a shortest one $Q$. Then $xx'v\\odot vQy'\\odot y'y$ is an $F$-extension. \nAmong all possible choices of five vertices $x,y,x',y',v$, we find a shortest $F$-extension using these five vertices.\nClearly if there is an $F$-extension, then we can find a tuple of such five vertices that outputs a shortest $F$-extension in the above procedure.\n\nThroughout this section, we heavily rely on the structure of a maximal subgraph obtained by adding a sequence of $F$-extensions exhaustively. We additionally impose a tie breaking rule for the choice of $F$-extensions.\n\n\n\\begin{description}\n\\item [Initialize] $W_1=C$, $B_1=\\emptyset$, and $i=1$.\n\\item [At step $i$] We perform the following.\n\\begin{enumerate}\n\\item Find a shortest $W_i$-extension $P_{i}$ such that\n\\begin{itemize}\n\\item[] (\\textbf{Tie break}) $\\abs{V(P_i)\\cap V(C)}$ is maximum.\n\\end{itemize}\nIf no $W_i$-extension exists, then terminate. Let $x_i$, $y_i$ be the endpoints of $P_{i}$ otherwise. \n\\item Set $W_{i+1}:= (V(W_i)\\cup V(P_{i}), E(W_{i})\\cup E(P_{i}))$. \n\\item Set $B_{i+1}:=B_{i} \\cup \\{x_i,y_i\\}$ and increase $i$ by one.\n\\end{enumerate}\n\\end{description}\n\nNotice that every vertex of $W_i$ has degree 2 or 3. Let $W_1, W_2, \\ldots , W_{\\ell}$ be \nthe sequence of subgraphs constructed exhaustively until there is no $W_{\\ell}$-extension. Let $W=W_{\\ell}$ and $T_{branch}=B_{\\ell}$. \nThroughout this section, we fix those sequences $W_1, W_2, \\ldots , W_{\\ell}=W$ and $P_1, P_2, \\ldots, P_{\\ell-1}$, and $B_1, B_2, \\ldots, B_{\\ell}=T_{branch}$. \nClearly, the construction of $W$ requires at most $n$ iterations, and thus we can construct these sequences in polynomial time.\n\n\nThe first observation is that if $T_{branch}$ has size at least $s_{k+1}$, then $G[V(W)]$ contains $k+1$ vertex-disjoint holes. \nIn fact, the construction of $W$ is calibrated so that every cycle of $W$ contains a hole of $G$. \nFor this, the condition (iv) of $W$-extension is crucial. Due to the next lemma, we may assume that $\\abs{T_{branch}}< s_{k+1}$.\n\n\\begin{LEM}\\label{lem:manybranching}\nIf $W$ has at least $s_{k+1}$ branching points, then there are $k+1$ vertex-disjoint holes and they can be detected in polynomial time. \n\\end{LEM}\n\\begin{proof}\nBy Theorem~\\ref{thm:simonovitz}, $\\abs{T_{branch}}\\geq s_{k+1}$ implies that $W$ has at least $k+1$ cycles, and such a collection of cycles can be found in polynomial time.\n We shall prove that for each cycle $H$ of $W$, there is a hole in the subgraph of $G$ induced by $V(H)$. Clearly, this immediately establishes the statement.\n We fix a cycle $H$ of $W$. We may assume that $H\\neq C$.\n Recall that for each $i$, $P_i$ is a $W_i$-extension added to $W_i$.\n\nLet $i$ be the minimum integer such that $E(H)\\subseteq E(W_{i+1})$. We claim that $P_{i}$ is entirely contained in $H$ as a subgraph. \nNotice that for any $W_{j}$-extension $P_j$, every branching point $v\\in T_{branch}$ which is an internal vertex of $P_j$ has been added at iteration $j'>j$. \nTherefore, if $P_{i}$ is not entirely contained in $H$ as a subgraph, then for some $i'>i$, there exists a subpath of $P_{i'}$ \nsuch that $E(P_{i'})\\cap E(H)\\neq \\emptyset$, contradicting the choice of $i$. \n\nLet $x,y$ be the endpoint of $P_i$. \nLet $v$ be an internal vertex of $P_{i}$ that is not contained in $N[W_i]$.\nSuch a vertex exists by the condition (iv) of the definition of a $W$-extension. \nLet $Q:=H-v$. Since $\\mathring{x}P_i\\mathring{y}$ is induced, the neighbors of $v$ in $P_i$ are not adjacent, and $v$ has no neighbors in the set of internal vertices of $Q$. \nTherefore, by Lemma~\\ref{lem:twopaths}, $G[V(H)]$ contains a hole, as claimed.\n\\end{proof}\n\nIn the next step, we exhaustively find almost $W$-extensions and cover them if there are no $k+1$ vertex-disjoint holes. We show that if there are two almost $W$-extensions with roots $x_1$ and $x_2$ and $\\dist_C(x_1, x_2)\\ge 5$, then these two almost $W$-extensions do not intersect. This is because if they meet, then we can obtain a $W$-extension, contradicting the maximality of $W$. Using this, we can deduce that if there are $5k+5$ almost $W$-extensions with distinct roots, then there are $k+1$ vertex-disjoint holes. \n\n\n\n\n\\begin{PROP}\\label{prop:almostpacking}\nThere is a polynomial-time algorithm that\nfinds either $k+1$ vertex-disjoint holes or a vertex set $T_{almost}\\subseteq V(C)\\setminus T_{branch}$ of size at most $5k+4$ such that $T_{almost}\\cup T_{branch}$ hits all almost $W$-extensions. \n\\end{PROP}\n\\begin{proof}\nLet $C=v_0v_1 \\cdots v_{m-1}v_0$. All additions are taken modulo $m$. \nWe greedily construct a collection of almost $W$-extensions $\\mathcal{Y}=\\{Y_1, Y_2, \\ldots , Y_t\\}$ (not necessarily vertex-disjoint) with distinct roots $v_{a_1}, v_{a_2}, \\ldots, v_{a_t}\\in V(C)\\setminus T_{branch}$, and stop if $t$ reaches $5k+5$. \nTo construct such a collection, we do the following for each vertex $v\\in V(C)\\setminus T_{branch}$:\n\\begin{enumerate}\n\\item Choose three vertices $w_1, w_2, w_3$ such that $w_1, w_3\\in Z_v\\setminus \\{v\\}$, $w_2\\notin N[W]$, and $w_1$ is adjacent to $w_2$ but not adjacent to $w_3$.\n\\item Test whether there is a path from $w_2$ to $w_3$ in $G_{deldom}-((V(W)\\cup N[w_1])\\setminus \\{w_2\\})$.\nIf there is such a path $P$, then we add the cycle $H=w_1w_2\\odot w_2Pw_3\\odot w_3vw_1$ \nto $\\mathcal{Y}$.\n\\end{enumerate}\nIt is not difficult to verify that there is an almost $W$-extension with root $v$ if and only if \nthe algorithm outputs such a cycle $H$.\n\n\nWe claim that if $v_{a_p}$ and $v_{a_q}$ have distance at least $5$ in $C$, \nthen $Y_p$ and $Y_q$ do not meet.\n\n\\begin{CLAIM}\\label{claim:distancealmost}\nLet $p,q\\in \\{1, 2, \\ldots, t\\}$. \nIf $\\dist_C(v_{a_p}, v_{a_q})\\ge 5$, \nthen $V(Y_p)\\cap V(Y_q)=\\emptyset$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose for contradiction that $Y_p$ and $Y_q$ meet at a vertex $z$.\nLet $Y_p=p_1p_2 \\cdots p_rv_{a_p}p_1$ and $Y_q=q_1q_2 \\cdots q_sv_{a_q}q_1$.\nFor convenience let $p_0:=v_{a_p}$ and $q_0:=v_{a_q}$.\nBy the condition (iii') of an almost $W$-extension, \nwe may assume that $p_2, q_2\\notin N[W]$.\nLet $t_1$ be the minimum integer such that $p_{t_1}$ has a neighbor in $Y_q$.\nWe choose a neighbor $q_{t_2}$ of $p_{t_1}$ in $Y_q$ with minimum $t_2$.\nLet $R:=p_0Y_pp_{t_1}\\odot p_{t_1}q_{t_2}\\odot q_{t_2}Y_qq_0$.\nIt is not difficult to see that $R$ is an induced path.\n\nSince $\\dist_C(p_0, q_0)\\ge 5$, the length of $R$ is at least $4$ by Lemma~\\ref{lem:generalfarnonadj}. Therefore, \n$R$ contains either $p_2$ or $q_2$.\nIt implies that $R$ is a $W$-extension, contradicting the maximality of $W$.\n\\end{proofofclaim}\n\nSuppose $t\\geq 5k+5$. There exists $M=\\{b_1, b_2, \\ldots, b_{k+1}\\}\\subseteq \\{a_1, a_2, \\ldots, a_{t}\\}\\setminus \\{v_i: m-4 \\le i\\le m-1\\}$ \nsuch that for all $b_i, b_j\\in M$, $b_i\\equiv b_j\\pmod 5$. \nAs we exclude the vertices of $\\{v_i: m-4 \\le i\\le m-1\\}$, for every $b_i, b_j\\in M$, $\\dist_C(v_{b_i}, v_{b_j})\\ge 5$.\nBy Claim~\\ref{claim:distancealmost}, \nfor $i, j\\in \\{1, 2, \\ldots, k+1\\}$, corresponding almost $W$-extensions $Y_{b_{i}}$ and $Y_{b_{j}}$ are vertex-disjoint.\nThus, we can output $k+1$ vertex-disjoint holes in polynomial time. \nOtherwise, $\\abs{\\mathcal{Y}}\\le 5k+4$, and thus the set of all roots $\\{v_{a_1}, v_{a_2}, \\ldots, v_{a_t}\\} \\subseteq V(C)\\setminus T_{branch}$ \nof $\\mathcal{Y}$ \ncontain at most $5k+4$ vertices. Clearly, the set of all roots \nhits every element of $\\mathcal{Y}$, that is, every almost $W$-extension.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{$Q$-tunnels}\\label{subsec:cfragment}\n\nWe define \n\\begin{align*}\nT_{ext}:=&T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}\\cup\\\\\n &N_C^{20}[(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})\\cap V(C)].\n\\end{align*}\nNote that \n\\[ \\abs{T_{ext}}\\le 41(19k+(3k+14)+(15k+9)+(s_{k+1}-1)+(5k+4))\\le 41(s_{k+1}+42k+26). \\]\nSince $\\abs{T_{petal}\\cup T_{full}\\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost}}\\le s_{k+1}+42k+26$, $C-(T_{petal}\\cup T_{full}\\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost})$ contains at most $s_{k+1}+42k+26$ connected components, and so does $C-T_{ext}$.\nLet $\\mathcal{Q}$ be the set of connected components of $C-T_{ext}$ and we call each element of $\\mathcal{Q}$ a \\emph{$C$-fragment}.\n\n \n We want to show that for every $D$-avoiding tulip $H$ not hit by $T_{ext}$ and for every $Q\\in \\mathcal{Q}$, \n if $H$ contains a vertex of $Q$ far from the endpoints of $Q$, then $H$ must traverse the $Q$-tunnel from one entrance to the other entrance. \n To argue this, we show that $H$ contains no edge $vw$ where $v\\in Z_{V(Q)}$ and $w\\notin N[C]$.\n We first need to show that in such a case, we have $w\\notin N[W]$. \n The next lemma states a useful distance property of $W$.\n See Figure~\\ref{fig:distancelemma} for an illustration.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n \n \\draw (-.1,0.3)--(-.1,-0.3);\n \\draw (.1,0.3)--(.1,-0.3);\n \\draw (4-.1,0.3)--(4-.1,-0.3);\n \\draw (4.1,0.3)--(4.1,-0.3);\n \n \\draw (2,0)--(2,1)--(3,2)--(6,2.5);\n \\draw (2,0) node [w] {};\n \\draw (2,1) node [w] {};\n \\draw (3,2) node [w] {};\n \\draw (6,2.5) node [w] {};\n \n \\node at (7, 2) {$P_i$};\n\n \\node at (2, -.5) {$v_0$};\n \\node at (1.5, 1) {$v_1$};\n \\node at (2.7, 2.3) {$v_2$};\n \\node at (5.5, 2.8) {$v_3$};\n \n\n\n \\node at (-2, -1) {$C$};\n\n \\end{tikzpicture} \\caption{The setting in Lemma~\\ref{lem:distance2} where $v_0$ is a vertex of $C-T_{ext}$ and there is a path of length $3$ from $v_0$ to $V(W)\\setminus V(C)$ whose internal vertices are in $V(G_{deldom})\\setminus V(W)$. By Lemma~\\ref{lem:distance2}, $v_2$ should have a neighbor in $C$. }\\label{fig:distancelemma}\n\\end{figure}\n\n\n\\begin{LEM}\\label{lem:distance2}\nLet $Q\\in \\mathcal{Q}$ be a $C$-fragment, and \n$v\\in V(Q)$ and $u\\in V(W)$ with $v\\neq u$. \nThen every $V(W)$-path $R=v_0v_1 \\cdots v_{s}$ from $v_0=v$ to $v_s=u$ satisfies one of the following.\n\\begin{enumerate}[(1)]\n\\item $u$ is a vertex of $C$ such that $\\dist_C(u, V(Q))\\le 4$, \n\\item $R$ has length $3$ and $v_2$ is adjacent to a vertex of $C$, \n\\item $R$ has length at least 4.\n\\end{enumerate}\n\\end{LEM}\n\\begin{proof}\nRecall that sequences $W_1, W_2, \\ldots , W_{\\ell}=W$ and $P_1, P_2, \\ldots, P_{\\ell-1}$, and $B_1, B_2, \\ldots, B_{\\ell}=T_{branch}$ \nrespectively denote the sequences of subgraphs, $W_i$-extensions and branching points during the construction of $W$.\n\nSuppose $u\\in V(C)$. If a $V(W)$-path $R$ between $u$ and $v$ has length at most $3$, \nthen there is an edge between $Z_u$ and $Z_v$ or we have $Z_u\\cap Z_v\\neq \\emptyset$.\nThen Lemma~\\ref{lem:farnonadj} implies that $\\dist_C(u,v)\\le 3$, and $R$ satisfies (1). \nTherefore, we may assume $u\\notin V(C)$. In particular, the following claim for every $i\\in \\{1, \\ldots, \\ell-1\\}$ establishes the statement immediately. \nWe prove by induction on $i$: \n\\begin{itemize}\n\\item[$(\\ast)$] if $u$ is an internal vertex of $P_i$, then every $V(W)$-path $R$ between $v$ and $u$ satisfies (2) or (3).\n\\end{itemize}\n\n\nLet $P_{i}=u_0u_1 \\cdots u_p$. \nBy definition of a $W_{i}$-extension, we may assume $u_0\\in V(C)$ and $u_2\\notin N[W_{i}]$. \nSuppose there exists a $V(W)$-path from $v$ to an internal vertex of $P_i$ violating (3). \nSuch a path has length at most $3$.\nLet $s\\in \\{1,2,3\\}$ be the minimum integer such that \nthere is a $V(W)$-path of length $s$ between $v$ and an internal vertex of $P_i$. \nWe choose the minimum integer $j\\in \\{1, 2, \\ldots, p-1\\}$ such that \nthere is a $(v,u_j)$-path $R$ of length $s$.\nLet $R:=v_0v_1 \\cdots v_s$ with $v_0=v$ and $v_s=u_j$, and \n $R_1=u_0P_iu_j\\odot R$. \n\nWe verify that $R_1$ is a $W_i$-extension. \n\\begin{CLAIM}\\label{claim:r1}\n$R_1$ is a $W_i$-extension containing $u_2$. \n\\end{CLAIM}\n\\begin{proofofclaim}\nBy the choice of $s$ and $u_j$, every vertex in $\\mathring{v_0}R\\mathring{u_j}$ has no neighbors in $\\mathring{u_0}P_i\\mathring{u_j}$. Therefore, $\\mathring{u_0}R_1\\mathring{v_0}$ is an induced path. Also, $v_0$ is a non-branching point of $W_i$. \nHence, $R_1$ satisfies the conditions (i)-(iii) of $W_i$-extension. \nFor (iv), it is sufficient to show that $j\\ge 2$.\nSuppose $j=1$. \nThen $R_1$ has length at most $4$, and by Lemma~\\ref{lem:generalfarnonadj} with $m=2$, we have $\\dist_C(v, u_0)\\le 7$.\nSince $u_0\\in T_{branch}\\cap V(C)$, this contradicts the fact that $v\\in V(Q)$ and thus $\\dist_C(v, T_{branch}\\cap V(C))\\ge 20$. \nWe conclude that $j\\ge 2$ and thus $R_1$ contains $u_2$. Since $P_i$ meets (iv) as a $W_i$-extension, we have $u_2\\notin N[W_i]$.\nTherefore, $R_1$ satisfies all four conditions for being a $W_i$-extension.\n\\end{proofofclaim}\n\nNext, we show that $R$ has length exactly $3$. \nWhen $R$ has length $1$ or $2$, \nwe derive a contradiction from the fact that $P_i$ is taken as a $i$-th $W_i$-extension.\n\n\\begin{CLAIM}\\label{claim:length3}\n$s=3$; that is, $R$ has length $3$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nFirst assume that $R$ has length $1$.\nIf $j2$ such that $v_i$ has a neighbor that is a non-branching point of $W$. Clearly $2 i+1$ because we have $p_{i+1}\\notin N[C]$ due to the choice of $i$. \nObserve that $p_{\\ell}wp_i$ is an induced path with $w$ as an internal vertex and $w$ is not adjacent to any internal vertex \nof $p_iPp_{\\ell}$. Now Lemma~\\ref{lem:twopaths} applies, implying that $G[V(p_iPp_{\\ell})\\cup \\{w\\}]$ has a hole $H'$ containing $w$. \nBy Claim~\\ref{claim:nopoint}, $H'$ contains no point of $W$ other than $w$. \n\nObserve that $H'$ qualifies as an almost $W$-extension if $p_{\\ell}\\neq d$; especially we have $p_{i+1}\\notin N[W]$ by Claim~\\ref{claim:secondvertex}.\nTherefore $T_{almost}$ hits $H'$. On the other hand, $T_{almost}\\cap (V(H')\\setminus \\{w\\})\\subseteq T_{ext}\\cap V(H)=\\emptyset$, which \nimplies $w\\in T_{almost}$. Then by the construction of $T_{ext}$, we have $x\\in T_{ext}$, a contradiction. \nIf $p_{\\ell}=d$, then $H'-dw$ is a path certifying an edge in an auxiliary bipartite graph. Therefore either one of $\\{d,w\\}$ is contained in the vertex cover \nor $w=v_{5\\lfloor{\\frac{m}{5}}\\rfloor+a}$ with $0\\leq a \\leq 4$. In both cases, $x$ is included in $T_{trav:tulip}$, a contradiction.\nThis completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of our main result}\\label{subsec:final}\n\nWe prove Theorem~\\ref{thm:core}. \n\nWe apply Lemma~\\ref{lem:petalcover}, Proposition~\\ref{prop:hitsunflower}, and Proposition~\\ref{prop:skew}.\nOver all, we can in polynomial time either output $k+1$ vertex-disjoint holes or vertex sets $T_{petal}, T_{full}, T_{trav:sunf}$ hitting petals, full sunflowers, and $D$-avoiding sunflowers, respectively.\n\n\n\n\nWe construct $W$ with the set $T_{branch}$ of branching points as described in Subsection~\\ref{subsec:tuliphive}.\nBy Lemma~\\ref{lem:manybranching}, \nif $W$ has at least $s_{k+1}$ branching points, then there are $k+1$ vertex-disjoint holes and they can be detected in polynomial time. \nWe apply Proposition~\\ref{prop:almostpacking}.\nIf it outputs \n$k+1$ vertex-disjoint holes in $G$, then we are done.\nWe may assume it outputs\na vertex set $T_{almost}$ of at most $5k+4$ vertices where $T_{almost}$ hits all almost $W$-extensions.\n\nLet $T_{ext}$ be the union of $T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}$ and the $20$-neighborhood of $V(C)\\cap (T_{petal}\\cup T_{full} \\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost})$.\n\nBy Proposition~\\ref{prop:Davoid}, we can in polynomial time either find $k+1$ vertex-disjoint holes or find a set \n$T_{avoid:tulip}\\subseteq V(G)\\setminus T_{ext}$ of at most $35(s_{k+1}+42k+26)$ vertices such that $T_{ext}\\cup T_{avoid:tulip}$ hits all $D$-avoiding tulips.\nBy Lemma~\\ref{lem:dominatingtulip2}, we can either find $k+1$ holes \nor find a set $T_{trav:tulip}\\subseteq V(G)\\setminus (T_{ext}\\cup T_{avoid:tulip})$ of size $25k+9$ such that \n$T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}$ hits all $D$-traversing tulips. \nTherefore, we can either find $k+1$ vertex-disjoint holes, or output a vertex set with at most \n\\begin{align*}\n&\\abs{T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}} \\\\\n&\\le 41(s_{k+1}+42k+26)+ 35(s_{k+1}+42k+26)+25k+9 \\\\\n\t\t\t\t\t\t\t\t\t\t%\n\t\t\t\t\t\t\t\t\t\t&\\le 76s_{k+1}+3217k+1985\n\t\t\t\t\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\tvertices hitting all holes.\nThis completes the proof of Theorem~\\ref{thm:core}.\n\n\n\\section{Cycles of length at least $5$ do not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation}\\label{sec:lowerbound}\n\nIn this section, we show that the class of cycles of length at least $\\ell$ for every fixed $\\ell\\ge 5$ \nhas no Erd\\H{o}s-P\\'osa property under induced subgraph reltation.\n\nA \\emph{hypergraph} is a pair $(X,\\mathcal{E})$ such that $X$ is a set of elements and $\\mathcal{E}$ is a family of non-empty subsets of $X$, called \\emph{hyperedges}.\nA subset $Y$ of $X$ is called a \\emph{hitting set} if for every $F\\in \\mathcal{E}$, $Y\\cap F\\neq \\emptyset$.\nFor positive integers $a,b$ with $a\\geq b$, \nlet $(a,b)$-uniform hypergraph, denote it by $U_{a,b}$, be the hypergraph $(X, \\mathcal{E})$ such that \n$\\abs{X}=a$ and $\\mathcal{E}$ is the set of all subsets of $X$ of size $b$.\nIt is not hard to observe that in $U_{2k-1, k}$, every two hyperedges intersect and\nthe minimum size of a hitting set of $U_{2k-1, k}$ is precise $k$.\n\n\n\\begin{THMMAIN2}\nLet $\\ell\\ge 5$ be a positive integer. \nThen the class of cycles of length at least $\\ell$ has no Erd\\H{o}s-P\\'osa property under induced subgraph relation.\n\\end{THMMAIN2}\n\n\\begin{proof}\nSuppose for contradiction that there is a function $f_{\\ell}:\\mathbb{N}\\rightarrow \\mathbb{N}$ such that\nfor every graph $G$ and a positive integer $k$, either\n\\begin{itemize}\n\\item $G$ contains $k+1$ pairwise vertex-disjoint holes of length at least $\\ell$ or\n\\item there exists $T\\subseteq V(G)$ with $\\abs{T}\\le f_{\\ell}(k)$ such that $G- T$ contains no holes of length at least $\\ell$.\n\\end{itemize} \nLet $x=\\max \\{f_{\\ell}(1)+1, \\ell\\}$. From the hypergraph $U_{2x-1, x}=(X,\\mathcal{E})$, \nwe construct a graph $G$ on the vertex set $S\\uplus \\bigcup_{F\\in \\mathcal{E}}Y_F$, where\n\\begin{itemize}\n\\item $S=\\{s_v:v\\in X\\}$ is an independent set of size $\\abs{X}$,\n\\item $Y_F=\\{y_v:v\\in F\\}$ is an independent set of size $x$ for each $F\\in \\mathcal{E}$.\n\\end{itemize}\nThe edge set of $G$ is created as follows.\n\\begin{itemize}\n\\item For each hyperedge $F\\in \\mathcal{E}$ with $F=\\{v_i:1\\le i\\le x\\}$, we add the edge set\n\\[\\{y_{v_1}s_{v_1},s_{v_1}y_{v_2},\\ldots , y_{v_x}s_{v_x},s_{v_x}y_{v_1}\\}.\\]\n\\item For each pair of two distinct hyperedges $F_1, F_2\\in \\mathcal{E}$, we add all possible edges between $Y_{F_1}$ and $Y_{F_2}$.\n\\end{itemize}\nNote that for each $F\\in \\mathcal{E}$, $G[Y_F\\cup S]$ contains precisely one hole, which has length $2x(\\ge \\ell)$. We denote this hole as $C_F$.\nFigure~\\ref{fig:construction} depicts the construction.\n\n\nWe verify that every hole of length at least $\\ell$ is one of the holes in $\\{C_F:F\\in \\mathcal{E}\\}$.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\foreach \\y in {0, 2, 4, 6, 8}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n\n \\foreach \\y in {0, 2, 4}{\n \\draw (\\y-2,2.5) node [w] (b\\y) {};\n }\n\n \\foreach \\y in {4, 6, 8}{\n \\draw (\\y+2,2.5) node [w] (c\\y) {};\n }\n \n \\draw (b0)--(a0)--(b2)--(a2)--(b4)--(a4)--(b0);\n \\draw (c4)--(a4)--(c6)--(a6)--(c8)--(a8)--(c4);\n\n\t \\draw(b0) [in=150,out=30] to (c4); \n\t \\draw(b0) [in=150,out=30] to (c6); \n\t \\draw(b0) [in=150,out=30] to (c8); \n\t \\draw(b2) [in=150,out=30] to (c4); \n\t \\draw(b2) [in=150,out=30] to (c6); \n\t \\draw(b2) [in=150,out=30] to (c8); \n\t \\draw(b4) [in=150,out=30] to (c4); \n\t \\draw(b4) [in=150,out=30] to (c6); \n\t \\draw(b4) [in=150,out=30] to (c8); \n \n\\draw[rounded corners] (0,1)--(9.5,1)--(9.5,-1)--(-1.5,-1)--(-1.5,1)--(0,1);\n\n\\draw[rounded corners] (0,.5)--(4.5,.5)--(4.5,-.5)--(-.5,-.5)--(-.5,.5)--(0,.5);\n\\draw[rounded corners] (4,.7)--(8.5,.7)--(8.5,-.7)--(4-.5,-.7)--(4-.5,.7)--(4,.7);\n\n \\node at (-2, 0) {$S$};\n \n \\end{tikzpicture} \\caption{An illustration of two holes constructed from two hyperedges.}\\label{fig:construction}\n\\end{figure}\n\n\n\\begin{CLAIM}\\label{claim:chordlesscycle}\nEvery hole of length at least $\\ell$ is exactly one of the holes in $\\{C_F:F\\in \\mathcal{E}\\}$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $C$ is a hole of length at least $\\ell\\ge 5$.\nWe show that $V(C)\\subseteq V(C_F)$ for some $F\\in \\mathcal{E}$. Clearly, it implies the claim as each $C_F$ is a hole.\n\nSuppose for contradiction that $C$ is not contained in one of $\\{C_F:F\\in \\mathcal{E}\\}$. Then there are two distinct \nhyperedges $F, F'\\in \\mathcal{E}$ such that $V(C)\\cap Y_F\\neq \\emptyset$ and $V(C)\\cap Y_{F'}\\neq \\emptyset$. \nLet $v\\in V(C)\\cap Y_F$ and $v' \\in V(C)\\cap Y_{F'}$. Due to construction of $G$, we have $vv'\\in E(G)$. Furthermore, \nthis also implies that for every $F''\\in \\mathcal{E}\\setminus \\{F,F'\\}$, we have $V(C)\\cap Y_{F''}=\\emptyset$. \n\nSince $S$ is independent, among the vertices of $V(C)\\setminus \\{v,v'\\}$ there are at least $\\lfloor (\\abs{V(C)}-2)\/2 \\rfloor$ vertices \nof $Y_{F}\\cup Y_{F'}$. Suppose $V(C)\\setminus \\{v,v'\\} \\setminus S$ has two vertices $w$ and $w'$. \nIf both of $w$ and $w'$ are in $Y_F$, then $v'$ is adjacent to at least three vertices of $C$, a contradiction. \nTherefore, we may assume that $w\\in Y_F$ and $w'\\in Y_{F'}$. Then $G[\\{v,v',w,w'\\}]$ is a cycle of length four, \ncontradicting the assumption that $C$ is a hole of length at least $\\ell(\\ge 5)$. If $V(C)\\setminus \\{v,v'\\} \\setminus S$ \ncontains a unique vertex, say $w\\in Y_F$, observe that $\\abs{V(C)}=5$ and $wv'$ is a chord of $C$, a contradiction.\n\\end{proofofclaim}\n\nBy Claim~\\ref{claim:chordlesscycle}, $\\{C_F:F\\in \\mathcal{E}\\}$ is precisely the set of all holes of length at least $\\ell$ in $G$.\nOne can observe that two holes in $\\{C_F:F\\in \\mathcal{E}\\}$ intersect because $(X,\\mathcal{E})$ is the hypergraph $U_{2x-1, x}$, \nin which every two hyperedges intersect.\nTherefore, by the property of the function $f_{\\ell}$, \nthere exists a vertex subset $T\\subseteq V(G)$ with $\\abs{T}\\le f_{\\ell}(1)