diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpnca" "b/data_all_eng_slimpj/shuffled/split2/finalzzpnca" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpnca" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe Standard Model (SM) of particle physics has been greatly successful in forecasting a wide range of phenomena. However, with the ultimate discovery of the Higgs boson with approximately 125 GeV mass by CMS and ATLAS Collaborations at the LHC, the SM has obtained a significant achievement \\cite{1,2}. On the other hand, this model leaves some questions unanswered such as neutrino oscillations, the strong CP problem and matter-antimatter asymmetry, etc.\nThus, it is thought to be embedded in a more fundamental theory where its effects can be observed at higher energy scales.\n\n\nAmong the all observed elementary particles of the SM, the largest mass particle is the top quark with a mass of $173.0\\pm 0.4$ GeV \\cite{3}. Investigation of the interactions of the top quark is important not only for the dynamics of electroweak symmetry breaking but also for testing of SM and new physics beyond SM. Up to now, this heavy quark produced by the various processes at the Tevatron and LHC was examined in detail. In this case, in addition to detecting the top quark, it has been a tremendous motivation to examine the characteristics and potential of the top quark in both decay and production. The complicated experimental results of the LHC are accomplished by precise theoretical predictions within the framework of the SM and beyond the SM. Many of its properties are still poorly constrained such as the electric and magnetic dipole moments and the chromoelectric and chromomagnetic dipole moments. For this reason, important new insights on the properties of the top quark will be one of the tasks of the LHC. Especially, the anomalous $t\\bar{t}\\gamma$ couplings that can define the electromagnetic dipole moments of the top quark, which is the subject of this study, have been investigated extensively at lepton-lepton, hadron-hadron colliders and lepton-hadron colliders.\n\nOne of the significant events in the field of fundamental interactions currently defined by the SM is the violation of CP symmetry. CP violation in the SM is identified with a complex phase in the CKM matrix. However, this information from the CKM matrix for CP violation cannot define the matter-antimatter asymmetry in the universe. This asymmetry is one of the principal questions in the SM. Therefore, the measurement of large amounts of CP violation in\nthe top quark events in the examined processes can be a proof of new physics beyond the SM. Investigation of new physics beyond the SM, some of\nthe intrinsic properties of the top quark are examined in the context of its dipole moments such as the magnetic dipole moment arising from one-loop level and the corresponding electric dipole moment that is defined as a source of CP violation coming from the three-loop level in the SM \\cite{4,5}.\n\nThe value for the magnetic dipole moment of the top quark predicted by the SM is $0.02$. This value can be tested in the current and the upcoming experiments. In addition, the electric dipole moment of the top quark in the SM is suppressed with a value of less than $10^{-30} ({\\rm e \\hspace{0.8mm} cm})$. Besides,\nit is highly attractive for the investigation of new physics beyond the SM. If there is a sign of new physics beyond the SM in the examined processes at the LHC, then the top quark may have an the electric dipole moment higher than the SM value.\n\nIn the literature, there have been different proposals to observe the electric and magnetic dipole moments of the top quark. Studies at the Tevatron and the LHC were recommended to obtain the electromagnetic dipole moments of the top quark in measurements of the processes $p \\bar{p}\\rightarrow t\\bar{t}\\gamma$ \\cite{6}, $p p\\rightarrow t j\\gamma$ \\cite{7,8} and $p p\\rightarrow p \\gamma^{*} \\gamma^{*} p\\rightarrow p t \\bar{t} p$ \\cite{9}. The reactions $e^{-}e^{+}\\rightarrow t \\bar{t}$ \\cite{10}, $\\gamma e\\rightarrow \\bar{t} b \\nu_{e}$ \\cite{11}, $e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} e^{+} \\rightarrow \\bar{t} b \\nu_{e} e^{+}$ \\cite{11}, $\\gamma \\gamma \\rightarrow t \\bar{t}$ \\cite{12} and $e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} \\gamma^{*} e^{+} \\rightarrow e^{-}t \\bar{t}e^{+}$ \\cite{12} at the future $e^{-}e^{+}$ linear colliders and their operating modes of $e \\gamma$, $e \\gamma^{*}$, $\\gamma \\gamma$ and $\\gamma^{*} \\gamma^{*}$ were investigated to set the limits on the electric and magnetic dipole moments of the top quark. However, the reactions $ep\\rightarrow \\bar{t} \\nu_{e} \\gamma$ \\cite{13}, $ep\\rightarrow e \\gamma^{*} p\\rightarrow e t \\bar{t} X$ \\cite{14}, $ep\\rightarrow e \\gamma^{*} p\\rightarrow e t W X$ \\cite{14} and $ep\\rightarrow e \\gamma^{*} \\gamma^{*} p\\rightarrow e t \\bar{t} p$ \\cite{15} in phenomenological investigations on the future $ep$ colliders are considered. Finally, Ref. \\cite{16} studied the limits on the electromagnetic dipole moments of the top quark that are calculated from measurements of the semi-inclusive decays $b \\rightarrow s\\gamma$, and of $t \\bar{t} \\gamma$ production at the Tevatron and the LHC. Also, a complementary way to access the electric dipole moment of the top is through their indirect effects, such as the resulting, radiatively-induced the electric dipole moment of the electron. In summary, all of the current limits on the electric and magnetic dipole moments of the top quark are represented in Table I.\n\n\\begin{table}[h]\n\\caption{Sensitivity limits on the magnetic and electric dipole moments of top quark through different processes at $pp$, $e^{-} e^{+}$ and $ep$ colliders}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\\hline\n{\\bf Processes} & {\\bf $a_{V}$} & {\\bf $a_{A}$} \\\\\n\\hline\n$p p\\rightarrow t\\bar{t}\\gamma$ \\cite{6} & $ (-0.200, 0.200)$ & $(-0.100, 0.100) $ \\\\\n\\hline\n$p p\\rightarrow t j\\gamma$ \\cite{8} & $ (-0.220, 0.210)$ & $ (-0.200, 0.200) $ \\\\\n\\hline\n$pp \\to p\\gamma^*\\gamma^*p\\to pt\\bar t p $ \\cite{9} & $ (-0.4588, 0.0168)$ & $ (-0.0815, 0.0815) $ \\\\\n\\hline\n$e^+e^- \\to t\\bar t$ \\cite{10} & $ (-0.002, 0.002)$ & $ (-0.001, 0.001) $ \\\\\n\\hline\n$\\gamma e\\rightarrow \\bar{t} b \\nu_{e}$ \\cite{11} & $ (-0.027, 0.036)$ & $ (-0.031, 0.031) $ \\\\\n\\hline\n$e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} e^{+} \\rightarrow \\bar{t} b \\nu_{e} e^{+}$ \\cite{11} & $ (-0.054, 0.092)$ & $ (-0.071, 0.071) $ \\\\\n\\hline\n$\\gamma \\gamma \\rightarrow t \\bar{t}$ \\cite{12} & $ (-0.220, 0.002)$ & $ (-0.020, 0.020) $ \\\\\n\\hline\n$e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} \\gamma^{*} e^{+} \\rightarrow e^{-}t \\bar{t}e^{+}$ \\cite{12} & $ (-0.601, 0.015)$ & $ (-0.089, 0.089) $ \\\\\n\\hline\n$ep\\rightarrow \\bar{t} \\nu_{e} \\gamma$ \\cite{13} & $ (-0.204, 0.185)$ & $ (-0.193, 0.193) $ \\\\\n\\hline\n$ep\\rightarrow e \\gamma^{*} p\\rightarrow e t W X$ \\cite{14} & $ (-0.204, 0.185)$ & $ (-0.193, 0.193) $ \\\\\n\\hline\n$ep\\rightarrow e \\gamma^{*} p\\rightarrow \\bar{t} \\nu_{e} b p$ \\cite{14} & $ (-0.089, 0.085)$ & $ (-0.087, 0.087) $ \\\\\n\\hline\n$ep\\rightarrow e \\gamma^{*} \\gamma^{*} p\\rightarrow e t \\bar{t} p$ \\cite{15} & $ ( -0.468, 0.0177)$ & $ (-0.088, 0.088) $ \\\\\n\\hline\nRadiative $b\\to s\\gamma$ transitions \\cite{16} & $ (-2, 0.3)$ & $ (-0.5, 1.5) $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFor the present, the LHC has finalized its phase 2 and has closed for an upgrade between 2019 with 2020 years. In later times, it is going to operate at a center-of-mass energy of 14 TeV during the period 2021-2023 and is going to collect almost 300 fb$^{-1}$ of additional data for each detector. However, there will be a major upgrade of the LHC to High-Luminosity LHC (HL-LHC) between 2023 with 2026. Therefore, HL-LHC is anticipated to operate for ten years until 2036. At the end of this duration, it is estimated that each detector will collect approximately 3000 fb$^{-1}$ data. Other colliders other than HL-LHC are also discussed. Also, the High-Energy LHC (HE-LHC) with a center-of-mass energy of 27 TEV at CERN is designed. It will collect a dataset corresponding to an integrated luminosity of 10-15 ab$^{-1}$.\n\nFor the new physics research beyond the SM at LHC, $pp$ deep inelastic scattering processes that involve subprocesses of gluon-gluon, quark-quark and quark-gluon collisions are generally investigated in detail. However, due to proton remnants, these processes have not provided very clean environment. Pollution in this environment can occur certain uncertainties and make it tough to observe the signs which may arise from the new physics. Nevertheless, in the literature, exclusive and semi-elastic processes are much less examined. Both of the incoming protons in an exclusive process remains intact and do not dissociate into partons. In addition to this, only one of the incoming protons in a semi-elastic process dissociates into partons but the other proton remains intact. The exclusive and semi-elastic processes are $\\gamma^{*} \\gamma^{*}$ and $\\gamma^{*} p$, respectively. Among these processes, the cleanest channel is $\\gamma^{*} \\gamma^{*}$. The exclusive and semi-elastic have simpler final states with respect to $pp$ processes. Therefore, these processes compensate for the advantages of $pp$ processes such as having high center-of-mass energy and high luminosity.\n\nIn $\\gamma^{*} p$ processes, since one from the incoming protons decomposes into partons they contain more background than $\\gamma^{*} \\gamma^{*}$ processes.\nBesides, $\\gamma^{*} p$ processes have effective luminosity and much higher energy compared to $\\gamma^{*} \\gamma^{*}$ process. This may be significant because of the high energy dependencies of the cross sections containing the new physics parameters. For this reason, $\\gamma^{*} p$ processes are anticipated to have a high sensitivity to the anomalous couplings. Photons emitted from one of the proton beams in $\\gamma^{*} p$ collision at the LHC can be defined in the framework of the Equivalent Photon Approximation (EPA) \\cite{epa,epa1,epa2}. These photons in the EPA have low virtuality. Since protons emit quasi-real photons, they do not decompose into partons. The EPA has many advantages. It aids to obtain crude numerical predictions via easy formulas. In addition to this, the EPA can mainly simplify the experimental analysis because it provides an occasion one to directly get a rough cross-section for $\\gamma^{*} \\gamma^{*}\\rightarrow X$ subprocess via the investigation of the process $pp \\rightarrow p X p$. Here, $X$ denotes objects produced in the final state. In the literature, there are a lot of phenomenological studies which are based on the photon-induced processes at the LHC aimed at research for new physics beyond the SM \\cite{s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12,s13,s14,s15,s16,s17,s18,s19,s20,s21,s22,sa1,sa2,sa3,sa4,sa5,sa6,sa7,sa8,sa9,sa10,sa11}.\n\n\n\\section{Top quark pair production in $\\gamma^{*} p$ collisions}\n\n\n\\subsection{The anomalous $t\\bar t \\gamma$ couplings}\n\nA method for defining possible new physics beyond the SM in a model-independent way is effective Lagrangian approach. This approach is described by high-dimensional operators which cause the anomalous $t\\bar{t}\\gamma$ coupling. These operators can be defined below \\cite{Kamenik,Baur,Aguilar,Aguilar1}\n\n\n\\begin{equation}\n{\\cal L}_{t\\bar t\\gamma}=-e Q_t\\bar t \\Gamma^\\mu_{ t\\bar t \\gamma} t A_\\mu.\n\\end{equation}\n\n\nEq. (1) contains the SM coupling and contributions arising from dimension-six effective operators. Also, $e$ is the proton charge, $Q_t$ shows the top quark electric charge, $A_\\mu$ represents the photon gauge field. $\\Gamma^\\mu_{t\\bar t \\gamma}$ has the following form\n\n\\begin{equation}\n\\Gamma^\\mu_{t\\bar t\\gamma}= \\gamma^\\mu + \\frac{i}{2m_t}(a_V + i a_A\\gamma_5)q_\\nu \\sigma^{\\mu\\nu}\n\\end{equation}\n\n\\noindent where $m_t$ is the top quark mass, $q_\\nu$ describes the photon four-momentum, $\\gamma_5 q_\\nu$ term with $\\sigma^{\\mu\\nu}$ breaks the CP symmetry. Thus, $a_A$ parameter describes the strength of a possible CP violation process, which may be caused by new physics beyond the SM. Real $a_V$ and $a_A$ parameters are non-SM couplings and interested in the anomalous magnetic moment and the electric dipole moment of the top quark, respectively. The relations between these parameters and the electromagnetic dipole moments are described as follows\n\n\n\\begin{eqnarray}\na_V&=&Q_t a_t, \\\\\na_A&=&\\frac{2m_t}{e}d_t.\n\\end{eqnarray}\n\n\n\\subsection{The cross section of the process $pp\\rightarrow p \\gamma p \\rightarrow p t \\bar{t} X$}\n\nA quasi-real photon emitted from one of the two proton beams interacts with the incoming other proton beam, and $\\gamma^{*} p$ collisions occur. Symbolic diagram of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ is displayed in Fig. 1.\n\n\\begin{figure} [h]\n\\begin{center}\n\\includegraphics [width=8cm,height=6cm]{fig1}\n\\caption{Schematic diagram for the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ at the LHC.\n\\label{fig1}}\n\\end{center}\n\\end{figure}\n\n\nThe electromagnetic field of the colliding hadrons (protons or heavy ions) at the LHC can be seen as an incoming photon flux, distributed with some density\n$dN(\\frac{E_{\\gamma}}{E}, Q^{2})$. The EPA factorises the dependence on photon virtuality $Q^{2}$ from the cross-section of the photon-induced process to the equivalent photon flux $dN$. If the photon flux originates in a nucleon which is not considered as pointlike, the electric and magnetic form factors should be taken into account. These factors are defined via the matrix element of the electromagnetic current \\cite{gutt}\n\n\\begin{eqnarray}\n\\langle N(P')|j_{\\mu}^{em}|N(P)\\rangle=\\bar{N}(P')[\\gamma_{\\mu} F_{1}(Q^{2})-i\\frac{\\sigma_{\\mu\\nu}q^{\\nu}}{2m_{N}}F_{2}(Q^{2})]N(P)\n\\end{eqnarray}\nwhere $P$ and $P'$ are the 4 - momentum of the nucleon of mass $m_{N}$ before and after photon emission. $F_{1}$ and $F_{2}$ are the Dirac and Pauli form factors, respectively. $\\sigma_{\\mu\\nu}=\\gamma_{\\mu}\\gamma_{\\nu}-\\gamma_{\\nu}\\gamma_{\\mu}$, $q=P-P'$.\n\nThe Sachs form factors are expressed in terms of $F_{1}$ and $F_{2}$ electromagnetic functions\n\n\\begin{eqnarray}\nG_{E}(Q^{2})=F_{1}(Q^{2})-\\frac{Q^{2}}{4m_{N}}F_{2}(Q^{2}),\n\\end{eqnarray}\n\n\\begin{eqnarray}\nG_{M}(Q^{2})=F_{1}(Q^{2})+F_{2}(Q^{2})\n\\end{eqnarray}\n\nAt $Q^{2}=0$, these functions correspond to the total charge and to the magnetic momentum $\\mu_{p}$ of the proton, respectively;\n\n\\begin{eqnarray}\nG_{E}(0)=1,\n\\end{eqnarray}\n\n\\begin{eqnarray}\nG_{M}(0)=\\mu_{p}=2.79.\n\\end{eqnarray}\n\n\nIn the usual dipole approximation, the dependence on $Q^{2}$ of the form factors is explicit\n\n\n\\begin{eqnarray}\nG_{E}(Q^{2})=\\frac{G_{M}(Q^{2})}{\\mu_{p}}=(1+\\frac{Q^{2}}{Q_{0}^{2}})^{-4},\n\\end{eqnarray}\nwhere $Q_{0}^{2}=0.71 GeV^{2}$\n\nIn the EPA, the photon flux from a proton can then be written in terms of the form factors \\cite{bud}:\n\n\\begin{eqnarray}\n\\frac{d N_{\\gamma}}{d E_{\\gamma} d Q^{2}}=\\frac{\\alpha}{\\pi}\\frac{1}{E_{\\gamma}Q^{2}}[(1-\\frac{E_{\\gamma}}{E})(1-\\frac{Q_{min}^{2}}{Q^{2}})F_{E}+\\frac{E^{2}_{\\gamma}}{2E^{2}}F_{M}]\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nQ_{min}^{2}=\\frac{m_{p}^{2}E^{2}_{\\gamma}}{E(E-E_{\\gamma})}.\n\\end{eqnarray}\n\n$F_{E}$ and $F_{M}$ are functions of the electric and magnetic form factors. These are given below\n\n\\begin{eqnarray}\nF_{E}=\\frac{4 m_{p}^{2}G^{2}_{E}+Q^{2}G_{M}^{2}}{4 m_{p}^{2}+Q^{2}},\n\\end{eqnarray}\n\n\\begin{eqnarray}\nF_{M}=G_{M}^{2}.\n\\end{eqnarray}\n\nHere, the mass of the proton is $m_{p}=0.938$ GeV, $E$ represents the energy of the incoming proton beam.\n\nAfter integration over $Q^{2}$, equivalent photon spectrum can be given by\n\n\\begin{eqnarray}\n\\frac{d N_{\\gamma}}{d E_{\\gamma}}=\\frac{\\alpha}{\\pi E_{\\gamma}}\\{[1-\\frac{E_{\\gamma}}{E}][\\varphi(\\frac{Q_{max}^{2}}{Q_{0}^{2}})-\\varphi(\\frac{Q_{min}^{2}}{Q_{0}^{2}})]\n\\end{eqnarray}\n\nwhere the function $\\varphi$ is described as follows\n\n\\begin{eqnarray}\n\\varphi(\\theta)=&&(1+ay)\\left[-\\textit{In}(1+\\frac{1}{\\theta})+\\sum_{k=1}^{3}\\frac{1}{k(1+\\theta)^{k}}\\right]+\\frac{y(1-b)}{4\\theta(1+\\theta)^{3}} \\nonumber \\\\\n&& +c(1+\\frac{y}{4})\\left[\\textit{In}\\left(\\frac{1-b+\\theta}{1+\\theta}\\right)+\\sum_{k=1}^{3}\\frac{b^{k}}{k(1+\\theta)^{k}}\\right]. \\nonumber \\\\\n\\end{eqnarray}\nHere,\n\n\\begin{eqnarray}\ny=\\frac{E_{\\gamma}^{2}}{E(E-E_{\\gamma})},\n\\end{eqnarray}\n\\begin{eqnarray}\na=\\frac{1+\\mu_{p}^{2}}{4}+\\frac{4m_{p}^{2}}{Q_{0}^{2}}\\approx 7.16,\n\\end{eqnarray}\n\\begin{eqnarray}\nb=1-\\frac{4m_{p}^{2}}{Q_{0}^{2}}\\approx -3.96,\n\\end{eqnarray}\n\\begin{eqnarray}\nc=\\frac{\\mu_{p}^{2}-1}{b^{4}}\\approx 0.028.\n\\end{eqnarray}\n\nThe cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ can be calculated by integrating the cross section for the subprocess $\\gamma^{*} g \\rightarrow t \\bar{t}$ over the photon and quark spectra:\n\n\n\\begin{eqnarray}\n\\sigma(pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X)=\\int \\int (\\frac{d N_{\\gamma}}{d x_{1}}) (\\frac{d N_{g}}{d x_{2}}) d x_{1} d x_{2} \\hat{\\sigma}_{\\gamma^{*} g \\rightarrow t \\bar{t}}\n\\end{eqnarray}\nwhere $x_{1}=\\frac{E_{\\gamma}}{E}$, $x_{2}$ is the momentum fraction of the proton's momentum carried by the gluon. $\\frac{d N_{g}}{d x_{2}}$ is the parton distribution function of the gluon.\n\nAs seen in Fig. 2, the reaction $\\gamma^{*} g \\rightarrow t \\bar{t}$ has two Feynman diagrams.\n\n\n\\begin{figure} [h]\n\\centering\n\\includegraphics[width=8cm,height=5cm]{fig2}\n\\caption{Feynman diagrams for the subprocess $\\gamma^{*} g \\rightarrow t \\bar{t}$\n\\label{fig2}}\n\\end{figure}\n\nThe CalcHEP computer package was used to calculate the cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ including the anomalous $t\\bar{t}\\gamma$ vertex given in Eq. (2). Thus, we obtain numerically the cross sections as a function of the center-of-mass energies and effective couplings:\n\n\n- Total cross sections including an anomalous parameter at $\\sqrt{s}=14$ TeV:\n\n\\begin{eqnarray}\n\\sigma(a_V)&=&\\Bigl[(0.606)a^2_V +(0.658)a_V +0.481 \\Bigr] (pb), \\\\\n\\sigma(a_A)&=&\\Bigl[(0.606)a^2_A + 0.481 \\Bigr] (pb).\n\\end{eqnarray}\n\n- Total cross sections including an anomalous parameter at $\\sqrt{s}=27$ TeV:\n\n\\begin{eqnarray}\n\\sigma(a_V)&=&\\Bigl[(2.091)a^2_V +(2.082) a_V +1.537 \\Bigr] (pb), \\\\\n\\sigma(a_A)&=&\\Bigl[(2.091) a^2_A + 1.537 \\Bigr] (pb).\n\\end{eqnarray}\n\nTherefore,\n\n- Total cross section including two anomalous parameters at $\\sqrt{s}=14$ TeV:\n\n\\begin{eqnarray}\n\\sigma(a_V, a_A)&=&\\Bigl[(0.606)a^2_V +(0.606)a^2_A +(0.658)a_V +0.481 \\Bigr] (pb),\n\\end{eqnarray}\n\n- Total cross section including two anomalous parameters at $\\sqrt{s}=27$ TeV:\n\n\\begin{eqnarray}\n\\sigma(a_V, a_A)&=&\\Bigl[(2.091)a^2_V +(2.091)a^2_A +(2.082)a_V +1.537 \\Bigr] (pb).\n\\end{eqnarray}\n\n\n\nIn these equations, the independent terms from $a_V$ and $a_A$ parameters indicate the cross section of the SM. In addition, as can be understood from these equations, the linear terms of the anomalous couplings arise from the interference between the anomalous and the SM contribution, whereas the quadratic terms give purely anomalous contribution. Therefore, the total cross sections of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ with respect to the anomalous $a_V$ and $a_A$ couplings are represented in Figs. 3-6.\n\n\n\n\\begin{figure} [h]\n\\includegraphics{fig3}\n\\caption{The total cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ as a function of the anomalous $a_{A}$ coupling for center-of-mass energies of $\\sqrt{s}=14, 27$ TeV.\n\\label{fig3}}\n\\end{figure}\n\n\\begin{figure} [h]\n\\includegraphics{fig4}\n\\caption{The total cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ as a function of the anomalous $a_{V}$ coupling for center-of-mass energies of $\\sqrt{s}=14, 27$ TeV.\n\\label{fig4}}\n\\end{figure}\n\n\\begin{figure} [h]\n\\includegraphics{fig5}\n\\caption{The total cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ as a function of the anomalous $a_{A}$ and $a_{V}$ coupling for center-of-mass energy of $\\sqrt{s}=14$ TeV.\n\\label{fig5}}\n\\end{figure}\n\n\n\\begin{figure} [h]\n\\includegraphics{fig6}\n\\caption{The total cross section of the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ as a function of the anomalous $a_{A}$ and $a_{V}$ coupling for center-of-mass energy of $\\sqrt{s}=27$ TeV.\n\\label{fig6}}\n\\end{figure}\n\n\nIn Figs. 3-4, the total cross sections are calculated with considering that only one of the anomalous $a_V$ and $a_A$ couplings have changed while the other coupling is taken into account as zero. As seen from Figs. 3-6, the cross sections of the examined process show a clear dependence on the anomalous $a_V$ and $a_A$ couplings. From the above equations we understand that the anomalous parameters have different CP properties. The cross sections have even powers of the anomalous $a_A$ coupling and a nonzero value of $a_A$ coupling permits a constructive effect on the total cross section. On the other hand, the cross sections contain only odd powers of $a_V$ coupling. In Fig. 4, there are small intervals around $a_V$ in which the cross section that includes new physics beyond the SM is smaller than the SM cross section. Thus, $a_V$ coupling has a partially destructive effect on the total cross section. Fig. 4 represents that the deviation from the SM of the positive part of $a_V$ coupling is greater than the deviation of the negative part. So we expect the sensitivity of the positive part of $a_V$ coupling to be higher than the negative part.\n\n\n\\section{LIMITS ON THE TOP QUARK'S ELECTRIC AND MAGNETIC DIPOLE MOMENTS AT THE LHC, HL-LHC AND HE-LHC}\n\n\nTo obtain the sensitivity on the anomalous couplings, we consider $\\chi^{2}$ analysis with a systematic error\n\n\\begin{eqnarray}\n\\chi^{2}=\\left(\\frac{\\sigma_{SM}-\\sigma_{NP}(a_{A},a_{V})}{\\sigma_{SM}\\delta}\\right)^{2},\n\\end{eqnarray}\nwhere $\\sigma_{SM}$ is the SM cross section, $\\sigma_{NP}(a_{A},a_{V})$ is the total cross section containing contributions from the SM and new\nphysics, $\\delta=\\frac{1}{\\sqrt{\\delta_{stat}^{2}+\\delta_{sys}^{2}}}, \\delta_{stat}=\\frac{1}{\\sqrt{N_{SM}}}$ is the statistical error, $N_{SM}=L_{int}\\times BR \\times \\sigma_{SM}\\times b_{tag}\\times b_{tag}$; $L_{int}$ is the integrated luminosity and $b_{tag}$ tagging efficiency is 0.8. The top quark decays nearly $100\\%$ to $b$ quark and $W$ boson. For top quark pair production, we can categorize decay products according to the decomposition of $W$ boson. In our calculations, we consider pure leptonic and semileptonic decays of $W$ bosons in the final state. Thus, while branching ratios for pure leptonic decays of $W$ bosons are BR = 0.123, for semileptonic decays are BR = 0.228.\n\nThe inclusive $t\\bar{t}$ production cross section using 3.2 fb$^{-1}$ of $\\sqrt{s}=$ 13 TeV $pp$ collisions by the ATLAS detector at the LHC is measured \\cite{top}. The four uncertainties giving a total relative uncertainty of $4.4\\%$ have calculated in the process of determining the cross section of top pair production. These are experimental and theoretical systematic effects, the integrated luminosity and the LHC beam energy. In order to examine the limits on the electromagnetic dipole moments of the top quark, there are also theoretical studies that take into account systematic uncertainties. The processes $\\gamma \\gamma \\rightarrow t \\bar{t}$ and $e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} \\gamma^{*} e^{+} \\rightarrow e^{-}t \\bar{t}e^{+}$ with systematic uncertainties of $0, 5, 10\\%$ are discussed in Ref. \\cite{12}. In Ref. \\cite{14}, the processes $\\gamma e\\rightarrow \\bar{t} b \\nu_{e}$, $e^{-}e^{+}\\rightarrow e^{-}\\gamma^{*} e^{+} \\rightarrow \\bar{t} b \\nu_{e} e^{+}$, $ep\\rightarrow e \\gamma^{*} p\\rightarrow \\bar{t} \\nu_{e} b p$ are studied from $0\\%$ to $5\\%$ with systematic uncertainties. In Ref. \\cite{16}, a $10\\%$ total uncertainty for measurements of the process $\\gamma e \\rightarrow t \\bar{t}$ is considered. In the light of these discussions, systematic error values of $0, 3, 5\\%$ are assumed during statistical analysis.\n\nFigs. 7-8 indicate limit values obtained the anomalous $a_{A}$ and $a_{V}$ at $95\\%$ C.L. through the process $pp\\rightarrow p \\gamma p \\rightarrow p t \\bar{t} X$ at the LHC, the HL-LHC and the HE-LHC. We can easily compare the limits obtained from the LHC, the HL-LHC and the HE-LHC for various integrated luminosities.\n\n\\begin{figure} [!h]\n\\includegraphics{fig7}\n\\caption{$95\\%$ C.L. sensitivity limits of the $a_{A}$ coupling for various values of integrated luminosities through the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ at the LHC, the HL-LHC and the HE-LHC.\n\\label{fig7}}\n\\end{figure}\n\n\\begin{figure} [!h]\n\\includegraphics{fig8}\n\\caption{Same as in Fig. 7, but for the anomalous $a_{V}$ coupling.\n\\label{fig8}}\n\\end{figure}\n\n\nSimilarly, in Figs. 9-11, we present $95\\%$ C.L. contours for $a_{A}$ and $a_{V}$ couplings couplings for the process $pp\\rightarrow p \\gamma p \\rightarrow p t \\bar{t} X$ at the LHC, the HL-LHC and the HE-LHC for different integrated luminosities. We observe from these figures that the strongest constraint on the anomalous couplings comes from the HE-LHC with 15 ab$^{-1}$.\n\n\\begin{figure} [!h]\n\\includegraphics{fig9}\n\\caption{For semileptonic channel, contours at $95\\%$ C. L. for the anomalous $a_{A}$ and $a_{V}$ couplings for the process $pp\\rightarrow p \\gamma^{*} p \\rightarrow p t \\bar{t} X$ at the LHC.\n\\label{fig9}}\n\\end{figure}\n\n\\begin{figure} [!h]\n\\includegraphics{fig10}\n\\caption{Same as in Fig. 9, but for the HL-LHC.\n\\label{fig10}}\n\\end{figure}\n\n\\begin{figure} [!h]\n\\includegraphics{fig11}\n\\caption{Same as in Fig. 9, but for the HE-LHC.\n\\label{fig11}}\n\\end{figure}\n\nFor pure and semileptonic decay channels, the limits obtained at $68, 90, 95\\%$ C.L. on the electromagnetic dipole moments of the top quark via the process $pp\\rightarrow p \\gamma p \\rightarrow p t \\bar{t} X$ at the LHC-14 TeV with 300 fb$^{-1}$, the HL-LHC-14 TeV with 3000 fb$^{-1}$ and the HE-LHC-27 TeV with 15000 fb$^{-1}$ are presented in Tables II-XIX.\n\n\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the LHC via $t \\bar{t}$ production pure leptonic decay channel with integrated luminosities of $10, 30, 50, 100, 200$ and $300$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.2022$ &$ (-1.1235;0.0364)$ \\\\\n$10$&$3\\%$&$0.2176$ &$ (-1.1290;0.0419)$ \\\\\n$10$&$5\\%$&$0.2388$ &$ (-1.1372;0.0501)$ \\\\\n\\hline\n$30$&$0\\%$&$0.1536$ &$ (-1.1084;0.0213)$ \\\\\n$30$&$3\\%$&$0.1832$ &$ (-1.1171;0.0300)$ \\\\\n$30$&$5\\%$&$0.2150$ &$ (-1.1281;0.0409)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1352$ &$ (-1.1036; 0.0165)$ \\\\\n$50$&$3\\%$&$0.1733$ &$ (-1.1141;0.0269)$ \\\\\n$50$&$5\\%$&$0.2091$ &$ (-1.1259;0.0388)$ \\\\\n\\hline\n$100$&$0\\%$&$0.1137$ &$ (-1.0988;0.0117)$ \\\\\n$100$&$3\\%$&$0.1647$ &$ (-1.1115;0.0244)$ \\\\\n$100$&$5\\%$&$0.2044$ &$ (-1.1243;0.0371)$ \\\\\n\\hline\n$200$&$0\\%$&$0.0956$ &$ (-1.0954;0.0083)$ \\\\\n$200$&$3\\%$&$0.1598$ &$ (-1.1101;0.0230)$ \\\\\n$200$&$5\\%$&$0.1998$ &$ (-1.1234;0.0363)$ \\\\\n\\hline\n$300$&$0\\%$&$0.0864$ &$ (-1.0939;0.0068)$ \\\\\n$300$&$3\\%$&$0.1580$ &$ (-1.1096;0.0225)$ \\\\\n$300$&$5\\%$&$0.2011$ &$ (-1.1231;0.0360)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table II, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.2291$ &$ (-1.1334;0.0463)$ \\\\\n$10$&$3\\%$&$0.2464$ &$ (-1.1404;0.0532)$ \\\\\n$10$&$5\\%$&$0.2705$ &$ (-1.1507;0.0636)$ \\\\\n\\hline\n$30$&$0\\%$&$0.1740$ &$ (-1.1143;0.0271)$ \\\\\n$30$&$3\\%$&$0.2075$ &$ (-1.1253;0.0382)$ \\\\\n$30$&$5\\%$&$0.2435$ &$ (-1.1392;0.0520)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1532$ &$ (-1.1083;0.0211)$ \\\\\n$50$&$3\\%$&$0.1963$ &$ (-1.1215;0.0343)$ \\\\\n$50$&$5\\%$&$0.2369$ &$ (-1.1365;0.0494)$ \\\\\n\\hline\n$100$&$0\\%$&$0.1288$ &$ (-1.1021;0.0150)$ \\\\\n$100$&$3\\%$&$0.1865$ &$ (-1.1182;0.0311)$ \\\\\n$100$&$5\\%$&$0.2316$ &$ (-1.1344;0.0472)$ \\\\\n\\hline\n$200$&$0\\%$&$0.1083$ &$ (-1.0978;0.0106)$ \\\\\n$200$&$3\\%$&$0.1810$ &$ (-1.1164;0.0293)$ \\\\\n$200$&$5\\%$&$0.2263$ &$ (-1.1333;0.0461)$ \\\\\n\\hline\n$300$&$0\\%$&$0.0978$ &$ (-1.0958;0.0087)$ \\\\\n$300$&$3\\%$&$0.1790$ &$ (-1.1158;0.0287)$ \\\\\n$300$&$5\\%$&$0.2278$ &$ (-1.1329;0.0458)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table II, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.2831$ &$ (-1.1564;0.0693)$ \\\\\n$10$&$3\\%$&$0.3046$ &$ (-1.1666;0.0795)$ \\\\\n$10$&$5\\%$&$0.3343$ &$ (-1.1817;0.0945)$ \\\\\n\\hline\n$30$&$0\\%$&$0.2151$ &$ (-1.1281;0.0410)$ \\\\\n$30$&$3\\%$&$0.2564$ &$ (-1.1445;0.0574)$ \\\\\n$30$&$5\\%$&$0.3010$ &$ (-1.16491;0.0777)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1893$ &$ (-1.1191;0.0320)$ \\\\\n$50$&$3\\%$&$0.2427$ &$ (-1.1388;0.0517)$ \\\\\n$50$&$5\\%$&$0.2928$ &$ (-1.1609;0.0738)$ \\\\\n\\hline\n$100$&$0\\%$&$0.1592$ &$ (-1.1099;0.0228)$ \\\\\n$100$&$3\\%$&$0.2305$ &$ (-1.1340;0.0468)$ \\\\\n$100$&$5\\%$&$0.2862$ &$ (-1.1578;0.0707)$ \\\\\n\\hline\n$200$&$0\\%$&$0.1338$ &$ (-1.1033;0.0162)$ \\\\\n$200$&$3\\%$&$0.2237$ &$ (-1.1313;0.0442)$ \\\\\n$200$&$5\\%$&$0.2797$ &$ (-1.156;0.0691)$ \\\\\n\\hline\n$300$&$0\\%$&$0.1209$ &$ (-1.1004;0.0133)$ \\\\\n$300$&$3\\%$&$0.2213$ &$ (-1.1304;0.0433)$ \\\\\n$300$&$5\\%$&$0.2815$ &$ (-1.1557;0.0685)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the LHC via $t \\bar{t}$ production semileptonic decay channel with integrated luminosities of $10, 30, 50, 100, 200$ and $300$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.1732$ &$ (-1.11406;0.02694)$ \\\\\n$10$&$3\\%$&$0.1958$ &$ (-1.1214;0.0341)$ \\\\\n$10$&$5\\%$&$0.2231$ &$ (-1.1387;0.0440)$ \\\\\n\\hline\n$30$&$0\\%$&$0.1316$ &$ (-1.1028;0.0157)$ \\\\\n$30$&$3\\%$&$0.1717$ &$ (-1.1072;0.0264)$ \\\\\n$30$&$5\\%$&$0.2082$ &$ (-1.1175;0.0385)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1158$ &$ (-1.0993;0.0122)$ \\\\\n$50$&$3\\%$&$0.1654$ &$ (-1.1027;0.0246)$ \\\\\n$50$&$5\\%$&$0.2048$ &$ (-1.1108;0.0373)$ \\\\\n\\hline\n$100$&$0\\%$&$0.0974$ &$ (-1.0957;0.0086)$ \\\\\n$100$&$3\\%$&$0.1602$ &$ (-1.0982;0.0231)$ \\\\\n$100$&$5\\%$&$0.2021$ &$ (-1.1039;0.0363)$ \\\\\n\\hline\n$200$&$0\\%$&$0.0819$ &$ (-1.0932;0.0061)$ \\\\\n$200$&$3\\%$&$0.1574$ &$ (-1.0949;0.0223)$ \\\\\n$200$&$5\\%$&$0.2007$ &$ (-1.0990;0.0359)$ \\\\\n\\hline\n$300$&$0\\%$&$0.0740$ &$ (-1.0921;0.0050)$ \\\\\n$300$&$3\\%$&$0.1564$ &$ (-1.0935;0.0220)$ \\\\\n$300$&$5\\%$&$0.2003$ &$ (-1.0969;0.0382)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table VI, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.1962$ &$ (-1.1214;0.0343)$ \\\\\n$10$&$3\\%$&$0.2218$ &$ (-1.1306;0.0435)$ \\\\\n$10$&$5\\%$&$0.2528$ &$ (-1.1430;0.0559)$ \\\\\n\\hline\n$30$&$0\\%$&$0.1491$ &$ (-1.1072;0.0200)$ \\\\\n$30$&$3\\%$&$0.1945$ &$ (-1.1208;0.0337)$ \\\\\n$30$&$5\\%$&$0.2358$ &$ (-1.1361;0.0489)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1312$ &$ (-1.1027;0.0156)$ \\\\\n$50$&$3\\%$&$0.1874$ &$ (-1.1185;0.0313)$ \\\\\n$50$&$5\\%$&$0.232$ &$ (-1.1345;0.0474)$ \\\\\n\\hline\n$100$&$0\\%$&$0.1103$ &$ (-1.0982;0.0110)$ \\\\\n$100$&$3\\%$&$0.1814$ &$ (-1.1166;0.0295)$ \\\\\n$100$&$5\\%$&$0.2290$ &$ (-1.1333;0.0462)$ \\\\\n\\hline\n$200$&$0\\%$&$0.0928$ &$ (-1.0949;0.0078)$ \\\\\n$200$&$3\\%$&$0.1783$ &$ (-1.1156;0.0285)$ \\\\\n$200$&$5\\%$&$0.2274$ &$ (-1.1327;0.0456)$ \\\\\n\\hline\n$300$&$0\\%$&$0.0838$ &$ (-1.0935;0.0064)$ \\\\\n$300$&$3\\%$&$0.1772$ &$ (-1.1152;0.0281)$ \\\\\n$300$&$5\\%$&$0.2269$ &$ (-1.1325;0.0454)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table VI, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$10$&$0\\%$&$0.2425$ &$ (-1.1387;0.0516)$ \\\\\n$10$&$3\\%$&$0.2741$ &$ (-1.1523;0.0652)$ \\\\\n$10$&$5\\%$&$0.3124$ &$ (-1.1705;0.0834)$ \\\\\n\\hline\n$30$&$0\\%$&$0.1842$ &$ (-1.1175;0.0303)$ \\\\\n$30$&$3\\%$&$0.2403$ &$ (-1.1378;0.0507)$ \\\\\n$30$&$5\\%$&$0.2915$ &$ (-1.1603;0.0732)$ \\\\\n\\hline\n$50$&$0\\%$&$0.1621$ &$ (-1.1108;0.0236)$ \\\\\n$50$&$3\\%$&$0.2315$ &$ (-1.1344;0.0472)$ \\\\\n$50$&$5\\%$&$0.2867$ &$ (-1.1581;0.0710)$ \\\\\n\\hline\n$100$&$0\\%$&$0.1363$ &$ (-1.1039;0.0168)$ \\\\\n$100$&$3\\%$&$0.2242$ &$ (-1.1315;0.0444)$ \\\\\n$100$&$5\\%$&$0.2830$ &$ (-1.1563;0.0692)$ \\\\\n\\hline\n$200$&$0\\%$&$0.1146$ &$ (-1.0990;0.0119)$ \\\\\n$200$&$3\\%$&$0.2203$ &$ (-1.1300;0.0429)$ \\\\\n$200$&$5\\%$&$0.2810$ &$ (-1.1554;0.0683)$ \\\\\n\\hline\n$300$&$0\\%$&$0.1036$ &$ (-1.0969;0.0098)$ \\\\\n$300$&$3\\%$&$0.2189$ &$ (-1.1295;0.0424)$ \\\\\n$300$&$5\\%$&$0.2804$ &$ (-1.1552;0.068)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the HL-LHC via $t \\bar{t}$ production pure leptonic decay channel with integrated luminosities of $500, 1000, 1500, 2000, 2500$ and $3000$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.0760$ &$ (-1.0924;0.0053)$ \\\\\n$500$&$3\\%$&$0.1566$ &$ (-1.1092;0.0221)$ \\\\\n$500$&$5\\%$&$0.2004$ &$ (-1.1227;0.0357)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0639$ &$ (-1.0908;0.0037)$ \\\\\n$1000$&$3\\%$&$0.1555$ &$ (-1.1089;0.0218)$ \\\\\n$1000$&$5\\%$&$0.1999$ &$ (-1.1227;0.0355)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0577$ &$ (-1.0901;0.0030)$ \\\\\n$1500$&$3\\%$&$0.1551$ &$ (-1.1088;0.0217)$ \\\\\n$1500$&$5\\%$&$0.1997$ &$ (-1.1226;0.0355)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0537$ &$ (-1.0897;0.0026)$ \\\\\n$2000$&$3\\%$&$0.1550$ &$ (-1.1087;0.0216)$ \\\\\n$2000$&$5\\%$&$0.1995$ &$ (-1.1225;0.0355)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0508$ &$ (-1.0894;0.0023)$ \\\\\n$2500$&$3\\%$&$0.1548$ &$ (-1.1087;0.0216)$ \\\\\n$2500$&$5\\%$&$0.1995$ &$ (-1.1225;0.0354)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0485$ &$ (-1.0892;0.0021)$ \\\\\n$3000$&$3\\%$&$0.1548$ &$ (-1.1087;0.0216)$ \\\\\n$3000$&$5\\%$&$0.1995$ &$ (-1.1225;0.0354)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table VIII, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.0861$ &$ (-1.0939;0.0067)$ \\\\\n$500$&$3\\%$&$0.1774$ &$ (-1.1153;0.0287)$ \\\\\n$500$&$5\\%$&$0.2270$ &$ (-1.1324;0.0455)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0724$ &$ (-1.0919;0.0048)$ \\\\\n$1000$&$3\\%$&$0.1762$ &$ (-1.1149;0.0278)$ \\\\\n$1000$&$5\\%$&$0.2264$ &$ (-1.1324;0.0452)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0654$ &$ (-1.0910;0.0039)$ \\\\\n$1500$&$3\\%$&$0.1757$ &$ (-1.1148;0.0277)$ \\\\\n$1500$&$5\\%$&$0.2262$ &$ (-1.1323;0.0452)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0609$ &$ (-1.0905;0.0034)$ \\\\\n$2000$&$3\\%$&$0.1755$ &$ (-1.1147;0.0276)$ \\\\\n$2000$&$5\\%$&$0.2260$ &$ (-1.1323;0.0451)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0576$ &$ (-1.0901;0.0030)$ \\\\\n$2500$&$3\\%$&$0.1754$ &$ (-1.1147;0.0276)$ \\\\\n$2500$&$5\\%$&$0.2260$ &$ (-1.1323;0.0451)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0550$ &$ (-1.0899;0.0027)$ \\\\\n$3000$&$3\\%$&$0.1753$ &$ (-1.1147;0.0275)$ \\\\\n$3000$&$5\\%$&$0.2260$ &$ (-1.1323;0.0451)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table VIII, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.1064$ &$ (-1.0974;0.0103)$ \\\\\n$500$&$3\\%$&$0.2193$ &$ (-1.1296;0.0425)$ \\\\\n$500$&$5\\%$&$0.2805$ &$ (-1.1549;0.0681)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0895$ &$ (-1.0944;0.0073)$ \\\\\n$1000$&$3\\%$&$0.2177$ &$ (-1.1291;0.0420)$ \\\\\n$1000$&$5\\%$&$0.2798$ &$ (-1.1549;0.0678)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0809$ &$ (-1.0931;0.0059)$ \\\\\n$1500$&$3\\%$&$0.2172$ &$ (-1.1289;0.0418)$ \\\\\n$1500$&$5\\%$&$0.2796$ &$ (-1.1548;0.0677)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0752$ &$ (-1.0923;0.0051)$ \\\\\n$2000$&$3\\%$&$0.2169$ &$ (-1.1288;0.0417)$ \\\\\n$2000$&$5\\%$&$0.2794$ &$ (-1.1548;0.0676)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0712$ &$ (-1.0917;0.0046)$ \\\\\n$2500$&$3\\%$&$0.2168$ &$ (-1.1287;0.0416)$ \\\\\n$2500$&$5\\%$&$0.2794$ &$ (-1.1548;0.0676)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0680$ &$ (-1.0913;0.0042)$ \\\\\n$3000$&$3\\%$&$0.2167$ &$ (-1.1287;0.0416)$ \\\\\n$3000$&$5\\%$&$0.2794$ &$ (-1.15482;0.0675)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the HL-LHC via $t \\bar{t}$ production semileptonic decay channel with integrated luminosities of $500, 1000, 1500, 2000, 2500$ and $3000$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.0651$ &$ (-1.0910;0.0038)$ \\\\\n$500$&$3\\%$&$ 0.1556$ &$ (-1.1089;0.0220)$ \\\\\n$500$&$5\\%$&$0.1999$ &$ (-1.1227;0.0357)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0547$ &$ (-1.0898;0.0027)$ \\\\\n$1000$&$3\\%$&$0.1550$ &$ (-1.1088;0.0218)$ \\\\\n$1000$&$5\\%$&$0.1996$ &$ (-1.1226;0.0356)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0495$ &$ (-1.0893;0.0022)$ \\\\\n$1500$&$3\\%$&$0.1548$ &$ (-1.1087;0.0216)$ \\\\\n$1500$&$5\\%$&$0.1996$ &$ (-1.1226;0.0355)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0460$ &$ (-1.0890;0.0019)$ \\\\\n$2000$&$3\\%$&$0.1547$ &$ (-1.1086;0.0216)$ \\\\\n$2000$&$5\\%$&$0.1995$ &$ (-1.1225;0.0354)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0435$ &$ (-1.0888;0.0017)$ \\\\\n$2500$&$3\\%$&$0.1546$ &$ (-1.1086;0.0215)$ \\\\\n$2500$&$5\\%$&$0.1995$ &$ (-1.1225;0.0354)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0416$ &$ (-1.0887;0.0015)$ \\\\\n$3000$&$3\\%$&$0.1546$ &$ (-1.1086;0.0215)$ \\\\\n$3000$&$5\\%$&$0.1994$ &$ (-1.1225;0.0354)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table XI, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.0738$ &$ (-1.0921;0.0050)$ \\\\\n$500$&$3\\%$&$0.1763$ &$ (-1.1150;0.0280)$ \\\\\n$500$&$5\\%$&$0.2264$ &$ (-1.1324;0.0454)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0620$ &$ (-1.0906;0.0035)$ \\\\\n$1000$&$3\\%$&$0.1756$ &$ (-1.1147;0.0278)$ \\\\\n$1000$&$5\\%$&$0.2261$ &$ (-1.1322;0.0453)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0560$ &$ (-1.0900;0.0029)$ \\\\\n$1500$&$3\\%$&$0.1754$ &$ (-1.1147;0.0276)$ \\\\\n$1500$&$5\\%$&$0.2260$ &$ (-1.1322;0.0452)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0521$ &$ (-1.0896;0.0025)$ \\\\\n$2000$&$3\\%$&$0.1752$ &$ (-1.1146;0.0275)$ \\\\\n$2000$&$5\\%$&$0.2260$ &$ (-1.1322;0.0451)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0493$ &$ (-1.0893;0.0022)$ \\\\\n$2500$&$3\\%$&$0.1752$ &$ (-1.1146;0.0275)$ \\\\\n$2500$&$5\\%$&$0.2259$ &$ (-1.1322;0.0450)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0471$ &$ (-1.0891;0.0020)$ \\\\\n$3000$&$3\\%$&$0.1751$ &$ (-1.1146;0.0275)$ \\\\\n$3000$&$5\\%$&$0.2259$ &$ (-1.1322;0.0450)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Same as in Table XI, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$500$&$0\\%$&$0.0912$ &$ (-1.0947;0.0076)$ \\\\\n$500$&$3\\%$&$0.2178$ &$ (-1.1291;0.0424)$ \\\\\n$500$&$5\\%$&$0.2799$ &$ (-1.1549;0.0679)$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0767$ &$ (-1.0925;0.0054)$ \\\\\n$1000$&$3\\%$&$0.2170$ &$ (-1.1288;0.0420)$ \\\\\n$1000$&$5\\%$&$0.2795$ &$ (-1.1547;0.0678)$ \\\\\n\\hline\n$1500$&$0\\%$&$0.0693$ &$ (-1.0915;0.0044)$ \\\\\n$1500$&$3\\%$&$0.2167$ &$ (-1.1287;0.0417)$ \\\\\n$1500$&$5\\%$&$0.2793$ &$ (-1.1547;0.0676)$ \\\\\n\\hline\n$2000$&$0\\%$&$0.0644$ &$ (-1.0909;0.0038)$ \\\\\n$2000$&$3\\%$&$0.2166$ &$ (-1.1286;0.0415)$ \\\\\n$2000$&$5\\%$&$0.2793$ &$ (-1.1546;0.0675)$ \\\\\n\\hline\n$2500$&$0\\%$&$0.0609$ &$ (-1.0905;0.0034)$ \\\\\n$2500$&$3\\%$&$0.2165$ &$ (-1.1286;0.0415)$ \\\\\n$2500$&$5\\%$&$0.2792$ &$ (-1.1546;0.0675)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0582$ &$ (-1.0902;0.0031)$ \\\\\n$3000$&$3\\%$&$0.2164$ &$ (-1.1286;0.0415)$ \\\\\n$3000$&$5\\%$&$0.2792$ &$ (-1.1546;0.0675)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the HE-LHC via $t \\bar{t}$ production pure leptonic decay channel with integrated luminosities of $1000, 3000, 5000, 10000$ and $15000$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0460$ &$ (-1.1546;0.0021)$ \\\\\n$1000$&$3\\%$&$0.1488$ &$ (-1.0173;0.0217)$ \\\\\n$1000$&$5\\%$&$0.1918$ &$ (-1.0312;0.0356)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0349$ &$ (-0.9967;0.0012)$ \\\\\n$3000$&$3\\%$&$0.1487$ &$ (-1.0172;0.0216)$ \\\\\n$3000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0307$ &$ (-0.9964;0.0009)$ \\\\\n$5000$&$3\\%$&$0.1486$ &$ (-1.0172;0.0216)$ \\\\\n$5000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0258$ &$ (-0.9962;0.0006)$ \\\\\n$10000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$10000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0233$ &$ (-0.9960;0.0005)$ \\\\\n$15000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$15000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table XIV, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0521$ &$ (-0.9982;0.0027)$ \\\\\n$1000$&$3\\%$&$0.1685$ &$ (-1.0233;0.0277)$ \\\\\n$1000$&$5\\%$&$0.2173$ &$ (-1.0409;0.0453)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0396$ &$ (-0.9971;0.0015)$ \\\\\n$3000$&$3\\%$&$0.1684$ &$ (-1.0232;0.0273)$ \\\\\n$3000$&$5\\%$&$0.2172$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0348$ &$ (-0.9967;0.0012)$ \\\\\n$5000$&$3\\%$&$0.1683$ &$ (-1.0232;0.0276)$ \\\\\n$5000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0293$ &$ (-0.9964;0.0008)$ \\\\\n$10000$&$3\\%$&$0.1682$ &$ (-1.0232;0.0276)$ \\\\\n$10000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0264$ &$ (-0.9962;0.0007)$ \\\\\n$15000$&$3\\%$&$0.1682$ &$ (-1.0231;0.0276)$ \\\\\n$15000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table XIV, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0644$ &$ (-0.9996;0.0041)$ \\\\\n$1000$&$3\\%$&$0.2083$ &$ (-1.0373;0.0418)$ \\\\\n$1000$&$5\\%$&$0.2685$ &$ (-1.0633;0.0678)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0489$ &$ (-0.9979;0.0024)$ \\\\\n$3000$&$3\\%$&$0.2081$ &$ (-1.0372;0.0418)$ \\\\\n$3000$&$5\\%$&$0.2684$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0430$ &$ (-0.9973;0.0018)$ \\\\\n$5000$&$3\\%$&$0.2080$ &$ (-1.0372;0.0416)$ \\\\\n$5000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0362$ &$ (-0.9968;0.0013)$ \\\\\n$10000$&$3\\%$&$0.2079$ &$ (-1.0372;0.0416)$ \\\\\n$10000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0327$ &$ (-0.9966;0.0010)$ \\\\\n$15000$&$3\\%$&$0.2079$ &$ (-1.0372;0.0416)$ \\\\\n$15000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table} [!h]\n\\caption{Limits at $68\\%$ Confidence Level on the anomalous $a_{A}$ and $a_{V}$ couplings at the HE-LHC via $t \\bar{t}$ production semileptonic decay channel with integrated luminosities of $1000, 3000, 5000, 10000$ and $15000$ fb$^{-1}$ for systematic errors of $0,3\\%$ and $5\\%$.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0394$ &$ (-0.9970;0.0015)$ \\\\\n$1000$&$3\\%$&$0.1486$ &$ (-1.0172;0.0217)$ \\\\\n$1000$&$5\\%$&$0.1917$ &$ (-1.0312;0.0356)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0299$ &$ (-0.9964;0.0009)$ \\\\\n$3000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$3000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0263$ &$ (-0.9962;0.0007)$ \\\\\n$5000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$5000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0221$ &$ (-0.9960;0.0004)$ \\\\\n$10000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$10000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0200$ &$ (-0.9959;0.0003)$ \\\\\n$15000$&$3\\%$&$0.1485$ &$ (-1.0172;0.0216)$ \\\\\n$15000$&$5\\%$&$0.1917$ &$ (-1.0311;0.0356)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table XVII, but for $90\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0446$ &$ (-0.9975;0.0020)$ \\\\\n$1000$&$3\\%$&$0.1684$ &$ (-1.0232;0.0277)$ \\\\\n$1000$&$5\\%$&$0.2172$ &$ (-1.0408;0.0453)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0339$ &$ (-0.9966;0.0011)$ \\\\\n$3000$&$3\\%$&$0.1684$ &$ (-1.0232;0.0276)$ \\\\\n$3000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0298$ &$ (-0.9964;0.0008)$ \\\\\n$5000$&$3\\%$&$0.1682$ &$ (-1.0232;0.0276)$ \\\\\n$5000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0251$ &$ (-0.9961;0.0006)$ \\\\\n$10000$&$3\\%$&$0.1682$ &$ (-1.0232;0.0276)$ \\\\\n$10000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0226$ &$ (-0.9960;0.0005)$ \\\\\n$15000$&$3\\%$&$0.1682$ &$ (-1.0232;0.0276)$ \\\\\n$15000$&$5\\%$&$0.2171$ &$ (-1.0408;0.0452)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table} [!h]\n\\caption{Same as in Table XVII, but for $95\\%$ C.L.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nLuminosity($fb^{-1}$)&$\\delta_{sys}$&$|a_{A}|$ & $a_{V}$ \\\\\n\\hline\n$1000$&$0\\%$&$0.0551$ &$ (-0.9985;0.0030)$ \\\\\n$1000$&$3\\%$&$0.2081$ &$ (-1.0372;0.0418)$ \\\\\n$1000$&$5\\%$&$0.2684$ &$ (-1.0632;0.0678)$ \\\\\n\\hline\n$3000$&$0\\%$&$0.0419$ &$ (-0.9973;0.0017)$ \\\\\n$3000$&$3\\%$&$0.2079$ &$ (-1.0372;0.0418)$ \\\\\n$3000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$5000$&$0\\%$&$0.0368$ &$ (-0.9969;0.0013)$ \\\\\n$5000$&$3\\%$&$0.2079$ &$ (-1.0372;0.0416)$ \\\\\n$5000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$10000$&$0\\%$&$0.0310$ &$ (-0.9965;0.0009)$ \\\\\n$10000$&$3\\%$&$0.2078$ &$ (-1.0372;0.0416)$ \\\\\n$10000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n$15000$&$0\\%$&$0.0280$ &$ (-0.9963;0.0007)$ \\\\\n$15000$&$3\\%$&$0.2078$ &$ (-1.0372;0.0416)$ \\\\\n$15000$&$5\\%$&$0.2683$ &$ (-1.0632;0.0677)$ \\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\nIn Table II, the best limits obtained on the anomalous $a_{A}$ and $a_{V}$ couplings are $|a_{A}|=0.0864$ and $-1.09390$ is the regularization parameter, and $\\lambda$ is the tuning parameter.\n\nIt is noted that applying the MCP may lead to results not respecting the ``main effects, interactions\" hierarchy, which has been stressed in some recent studies \\cite{bien2013lasso}. The hierarchy postulates that the main effects corresponding to the identified interactions should be automatically identified. This can be achieved by replacing the MCP with for example sparse group penalties. However, we note that the computational cost of such penalties can be much higher. In addition, some published studies have demonstrated pure interactions without the presence of main effects \\cite{caspi2006gene,zimmermann2011interaction}. In data analysis, when it is necessary to reinforce the hierarchy, we can refit and add back the main effects corresponding to the identified interactions (if these main effects are not identified in the first place).\n\n\n\\subsection{Computation}\n\nFor optimizing the penalized objective function, we propose combining the majorize-minimization (MM) algorithm \\cite{hunter2005variable} with the coordinate descent (CD) algorithm \\cite{wu2008coordinate}. The MM is used to approximate the objective function using its quadratic majorizer, while the CD is used for iteratively updating the estimate.\n\nSpecifically, when $g(a,b)=ab$, it is easy to compute the gradient and hessian matrix for $Q_n(\\theta)$, and so approximation may not be needed. However when $g(a,b)=a+b$, computing the hessian matrix becomes difficult. With the estimate $\\theta^{(s)}$ at the beginning of the $s+1$th iteration, we approximate $Q_n(\\theta)$ by\n\\begin{eqnarray*}\nQ_n(\\theta; \\theta^{(s)}) &=& \\frac{1}{2}\\sum_{i=1}^n w_i\\left\\{\\frac{(1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta))^2}{|1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta^{(s)})|}+|1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta^{(s)})|\\right.\\\\\n&& \\left.+ \\frac{({1-y_i\\exp(-\\textbf{u}_i^\\top \\theta)})^2}{|{1-y_i\\exp(-\\textbf{u}_i^\\top \\theta^{(s)})}|} + |{1-y_i\\exp(-\\textbf{u}_i^\\top \\theta^{(s)})}| \\right\\}.\n\\end{eqnarray*}\nIt can be shown that $Q_n(\\theta; \\theta^{(s)})\\geq Q_n(\\theta)$, and the equality holds if and only if $\\theta^{(s)}= \\theta$. For the MCP, we use a quadratic approximation\n\\[\n\\varphi_{\\lambda}(\\theta; \\theta^{(s)}) = \\varphi_{\\lambda}(\\theta^{(s)})+ \\frac{1}{2 |\\theta^{(s)}|}\\varphi'_{\\lambda}(\\theta^{(s)})(\\theta^2-\\theta^{(s)2}).\n\\]\nBy ignoring terms not related to $\\theta$ in $Q_n(\\theta; \\theta^{(s)})+ \\varphi_{\\lambda}(\\theta; \\theta^{(s)})$, we have a smooth loss function $L_{n, \\lambda}(\\theta; \\theta^{(s)})$, which is\n\\begin{equation}\\label{app}\n\\sum_{i=1}^n w_i\\left\\{\\frac{(1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta))^2}{|1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta^{(s)})|}+ \\frac{({1-y_i\\exp(-\\textbf{u}_i^\\top \\theta)})^2}{|{1-y_i\\exp(-\\textbf{u}_i^\\top \\theta^{(s)})}|} \\right\\} + \\frac{1}{ |\\theta^{(s)}|}\\varphi'_{\\lambda}(\\theta^{(s)})\\theta^2.\n\\end{equation}\nTo solve the minimization problem $\\theta^{(s+1)}= \\arg \\min_{\\theta}L_{n, \\lambda}(\\theta; \\theta^{(s)})$, we employ the coordinate descent algorithm. In summary, the algorithm proceeds as follows:\n\n\\medskip\\noindent\nStep 1. Initialize $s=0$. Compute $\\theta^{(0)}$ as the Lasso estimate (which can be viewed as an extreme case of the MCP estimate).\n\n\\noindent\nStep 2. Apply the CD algorithm to minimize the loss function $L_{n, \\lambda}(\\theta; \\theta^{(s)})$ in (\\ref{app}). Denote the estimate as $\\theta^{(s+1)}$. Specially, the CD algorithm updates one coordinate at a time and treats the other coordinates as fixed. Define $u_{ij}$ as the $j$th component of $\\textbf{u}_i$. For $j \\in \\{1, \\cdots, p+q+pq\\}$, defined $\\vartheta_{i,-j}=\\sum_{tj}u_{it} \\theta_{t}^{(s)}$, then\n\\begin{eqnarray*}\n \\theta_j^{(s+1)} &=& \\arg\\min_{\\theta_j} \\left\\{\\sum_{i=1}^n w_i \\left[\\frac{(1-y_i^{-1}\\exp(\\vartheta_{i,-j} +u_{ij}\\theta_j ))^2}{|1-y_i^{-1}\\exp(\\textbf{u}_i^\\top \\theta^{(s)})|} \\right.\\right.\\\\\n && \\left.\\left.~~~~~~+ \\frac{({1-y_i\\exp(-\\vartheta_{i,-j}-u_{ij}\\theta_j )})^2}{|{1-y_i\\exp(-\\textbf{u}_i^\\top \\theta^{(s)})}|} \\right] + \\frac{1}{ |\\theta_j^{(s)}|}\\varphi'_{\\lambda}(\\theta_j^{(s)})\\theta_j^2\\right\\} ~.\n\\end{eqnarray*}\n\n\n\\noindent\nStep 3. Repeat Step 2 until convergence. We use the $L_2$-norm of the difference between two consecutive estimates less than $10^{-6}$ as the convergence criterion.\n\nThe proposed method involves tunings. For $\\gamma$, published studies \\cite{zhang2010nearly} suggest selecting from a small number of values or fixing it. In our simulation, we find that the estimation results are not sensitive to the value of $\\gamma$. We follow published studies and set $\\gamma=6$. The selection of $\\lambda$ will be described in the following sections.\n\n\n\n\\section{Simulation}\nBeyond evaluating performance of the proposed approach, we also use simulation to compare with the penalized weighted least squares (simply denoted as LS) and penalized weighted least absolute deviation (denoted as LAD) methods, which respectively have objective functions\n\\[\\sum_{i=1}^n w_i(\\log(y_i)-\\textbf{u}_i^\\top\\theta)^2+\\varphi_\\lambda(\\theta)\n~~\\mbox{and}\n~~\n\\sum_{i=1}^n w_i|\\log(y_i)-\\textbf{u}_i^\\top\\theta|+\\varphi_\\lambda(\\theta),\\]\nwhere $\\{w_i\\}_{i=1}^n$ and $\\varphi_\\lambda(\\theta)$ are the same as defined before. \\\\\n\n\\noindent{\\bf Simulation I.} In model\n$t_i=\\exp(\\textbf{x}_i^\\top\\alpha+\\textbf{z}_i^\\top\\beta+(\\textbf{x}_i\\otimes \\textbf{z}_i)^\\top\\xi)\\varepsilon_i,~ i=1,\\cdots,n,$ $\\textbf{z}_i$'s have a multivariate normal distribution with marginal means 0 and marginal variances 1. Denote the correlation coefficient between genes $j$ and $k$ as $\\rho_{jk}$. Consider the following correlation structures: (i) independent, where $\\rho_{jk}=0$ if $j\\ne k$, (ii) AR(0.2), where $\\rho_{jk}=0.2^{|j-k|}$; (iii) AR(0.8), where $\\rho_{jk}=0.8^{|j-k|}$; (iv) Band1, where $\\rho_{jk}=0.3$ if $|j-k|=1$ and $\\rho_{jk}=0$ otherwise; and (v) Band2, where $\\rho_{jk}=0.6$ if $|j-k|=1$, $\\rho_{jk}=0.3$ if $|j-k|=2$, and $\\rho_{jk}=0$ otherwise. We generate $\\textbf{x}_i$'s from the standard multivariate normal distribution. We set $n=200$, $q=5$, and $p=500$. The dimension of genetic effects and interactions is much larger than the sample size. There are a total of 35 nonzero effects: 5 main effects of the E factors, 10 main effects of the G factors, and 20 interactions. The nonzero coefficients are randomly generated from $Uniform(0.4,1.2)$. We consider two error distributions: (i) $\\log(\\varepsilon)$ follows $N(0,1)$, and (ii) $\\log(\\varepsilon)$ follows $Unif(-2,2)$. The event times are computed from the AFT model. The censoring times are generated from a uniform distribution, with a censoring rate about 20\\%.\n\n\n\\medskip\\noindent{\\bf Simulation II.} Data are first generated in the same manner as under Simulation I. To mimic discrete genetic data (for example SNPs), we dichotomize the simulated genetic data at -1 and 0.5 to create three levels.\n\n\n\\begin{table}[h]\n\\caption{Summary of Simulation II. In each cell, mean (sd) based on 200 replicates}\n\\vskip .1in\n\\label{table3}\n\\centering\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{c c c c c c}\\hline\n &&AUC&SE&TPR&FPR\\\\ \\hline\n\\multicolumn{6}{c}{$\\log(\\varepsilon)\\sim N(0,1)$} \\\\\n\\hline\nindependent&LARE&0.846(0.031)&19.53(3.321)&0.601(0.063)&0.098(0.013)\\\\\n&LPRE&0.837(0.032)&19.47(3.130)&0.572(0.171)&0.095(0.134)\\\\\n&LAD&0.833(0.029)&20.26(3.118)&0.564(0.117)&0.084(0.026)\\\\\n&LS&0.854(0.020)&20.78(2.641)&0.562(0.109)&0.076(0.013)\\\\\\hline\nAR(0.2)&LARE&0.868(0.034)&17.55(3.252)&0.739(0.082)&0.103(0.018)\\\\\n&LPRE&0.863(0.024)&16.68(3.671)&0.649(0.153)&0.062(0.027)\\\\\n&LAD&0.847(0.027)&19.57(2.947)&0.564(0.100)&0.078(0.026)\\\\\n&LS&0.860(0.024)&18.66(2.583)&0.628(0.086)&0.071(0.011)\\\\\\hline\nAR(0.8)&LARE&0.928(0.029)&7.655(2.611)&0.891(0.053)&0.062(0.027)\\\\\n&LPRE&0.898(0.032)&7.755(2.990)&0.871(0.076)&0.066(0.021)\\\\\n&LAD&0.911(0.022)&13.68(2.973)&0.758(0.098)&0.069(0.023)\\\\\n&LS&0.901(0.026)&12.74(2.417)&0.779(0.104)&0.063(0.019)\\\\\\hline\nBand1&LARE&0.868(0.033)&18.51(3.316)&0.673(0.080)&0.078(0.022)\\\\\n&LPRE&0.859(0.026)&17.78(3.560)&0.641(0.143)&0.059(0.023)\\\\\n&LAD&0.850(0.031)&19.27(3.676)&0.629(0.119)&0.085(0.025)\\\\\n&LS&0.864(0.022)&18.92(2.853)&0.616(0.074)&0.078(0.012)\\\\\\hline\nBand2&LARE&0.904(0.028)&10.82(2.571)&0.828(0.158)&0.060(0.017)\\\\\n&LPRE&0.875(0.031)&11.39(2.922)&0.787(0.102)&0.055(0.021)\\\\\n&LAD&0.872(0.033)&17.68(3.673)&0.685(0.108)&0.075(0.027)\\\\\n&LS&0.880(0.025)&16.92(3.114)&0.725(0.081)&0.075(0.014)\\\\\n\\hline\n\\multicolumn{6}{c}{$\\log(\\varepsilon)\\sim Unif(-2,2)$} \\\\\n\\hline\nindependent&LARE&0.840(0.032)&19.38(3.024)&0.634(0.073)&0.111(0.024)\\\\\n&LPRE&0.845(0.022)&20.46(2.898)&0.582(0.169)&0.094(0.035)\\\\\n&LAD&0.831(0.033)&21.29(3.453)&0.569(0.123)&0.081(0.027)\\\\\n&LS&0.847(0.021)&21.03(3.258)&0.557(0.087)&0.080(0.018)\\\\\\hline\nAR(0.2)&LARE&0.832(0.029)&18.63(3.286)&0.696(0.076)&0.093(0.019)\\\\\n&LPRE&0.850(0.022)&18.15(4.075)&0.616(0.082)&0.083(0.012)\\\\\n&LAD&0.835(0.028)&19.49(2.958)&0.583(0.127)&0.082(0.028)\\\\\n&LS&0.858(0.021)&20.52(3.063)&0.587(0.111)&0.076(0.018)\\\\\\hline\nAR(0.8)&LARE&0.913(0.031)&9.610(2.219)&0.833(0.128)&0.068(0.023)\\\\\n&LPRE&0.889(0.025)&8.732(2.770)&0.857(0.105)&0.052(0.016)\\\\\n&LAD&0.900(0.030)&15.85(2.980)&0.736(0.124)&0.072(0.026)\\\\\n&LS&0.895(0.026)&14.60(2.970)&0.732(0.108)&0.007(0.029)\\\\\\hline\nBand1&LARE&0.850(0.028)&14.23(3.010)&0.714(0.082)&0.097(0.020)\\\\\n&LPRE&0.856(0.023)&15.64(3.274)&0.624(0.120)&0.083(0.016)\\\\\n&LAD&0.844(0.030)&20.94(3.371)&0.543(0.114)&0.077(0.024)\\\\\n&LS&0.856(0.023)&19.70(2.899)&0.626(0.090)&0.076(0.011)\\\\\\hline\nBand2&LARE&0.868(0.032)&13.06(3.513)&0.782(0.163)&0.098(0.025)\\\\\n&LPRE&0.864(0.030)&12.23(3.713)&0.763(0.148)&0.057(0.024)\\\\\n&LAD&0.870(0.029)&17.23(3.555)&0.680(0.128)&0.073(0.027)\\\\\n&LS&0.869(0.033)&16.46(2.470)&0.704(0.093)&0.071(0.010)\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe evaluate the simulation results in two ways. First, we consider a sequence of $\\lambda$ values, evaluate identification performance at each value, and then compute the overall AUC (area under the ROC -- receiver operating characteristic -- curve). In addition, we also select the optimal $\\lambda$ using a cross validation approach and then compute the estimation squared error (SE), true positive rate (TPR), and false positive rate (TPR) at the optimal tuning. The summary based on 200 replicates is provided in Table 1 and 3 (Appendix), respectively. Simulation suggests that, when evaluated using AUC, the four methods have similar performance. Under Simulation I, the performance is also similar in terms of SE, TPR, and FPR. However, under Simulation II, the proposed LARE and LPRE can have better performance. In addition, it is also observed that LARE may outperform LPRE, at the cost of slightly higher computer time. Overall simulation suggests that the proposed approach, especially LARE, performs comparable to or better than the alternatives. Thus it provides a ``safe\" choice for practical data analysis.\n\n\n\\section{Analysis of lung cancer prognosis data}\n\nLung cancer is the leading cause of cancer death worldwide. Genetic profiling studies have been extensively conducting, searching for genetic risk factors associated with lung cancer prognosis. Here we analyze the TCGA (The Cancer Genome Atlas) data on the prognosis of lung adenocarcinoma. The TCGA data were recently collected and published by NCI and have a high quality. The prognosis outcome of interest is overall survival. The dataset contains records on 468 patients, among whom 117 died during follow-up. The median follow-up time is 8 months.\n\n\\begin{table}[h]\n\\caption{Analysis of lung cancer data with LARE: main genetic effects and G$\\times$E interactions. For the interactions, values in ``()'' are the stability results.}\n\\vskip .1in\n\\label{table5}\n\\centering\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{c c c c c c }\\hline\n\n &&\\multicolumn{4}{c}{Interactions}\\\\\n \\cline{3-6}\nGene &Main effects&Age&Gender&Smoking pack year&Smoking history\\\\\\hline\nADORA2B&\t-0.231\t&&&&\t-0.260(0.76)\\\\\nAKIRIN2&\t-0.281\t\\\\\nASB12&\t-0.241\t\\\\\nC5ORF45&\t-0.042\t\\\\\nC14ORF93&\t-0.472\\\\\nC16ORF93&\t-0.160&\t-0.293(0.91)\t\\\\\nCAND1&\t0.309\t&-2.181(0.95)\t\\\\\nCBWD2&\t0.234\t\\\\\nCDR2&\t0.210\t\\\\\nCIAPIN1&\t0.187&&\t&\t\t-0.179(0.85)\t\\\\\nDCP1B&\t0.448\t\\\\\nDYRK2&\t-1.41&\t0.758(0.66)\t\\\\\nEIF4EBP1&\t0.081&\t-0.001(0.81)\\\\\nEMB&\t0.224\t\\\\\nFDXR&\t0.293\t&&&\t-0.477(0.99)\t\\\\\nGALK2&\t-0.158\t&&&\t-0.240(0.75)\t\\\\\nGOLGA7&\t-0.146\t&-0.096(0.45)\t\\\\\nHERPUD2&\t0.121\t\\\\\nHOXC13&\t-0.248&\t-0.145(0.98)\t\\\\\nING1&\t-2.117\t&&&&\t\t2.154(0.97)\\\\\nINO80B&\t-0.164\t&&&\t-1.607(0.95)\t\\\\\nKIF21B&\t-0.391\t&-0.446(0.99)\t\\\\\nKLHDC1&\t-0.011\t&&&\t0.382(0.98)\\\\\nLIG4&\t-0.584\t&&\t0.299(0.80)\t\\\\\nLINC00471&\t0.236\t&&&&\t0.114(0.94)\\\\\nLINC00476&\t0.258\t&&&&\t0.056(0.55)\\\\\nLRRC45&\t-0.136\t&-0.083(0.93)\t\\\\\nMCAT&\t0.103\t&&&\t0.180(0.96)\t\\\\\nMVD&\t-0.348\t\\\\\nNCALD&\t0.376\t&&\t-0.605(0.70)\t\\\\\nOTUD1&\t0.189\t&&&\t0.038(0.34)\t\\\\\nPEX19&\t-0.444\t&&&&\t0.045(0.55)\\\\\nPHLPP1&\t-0.439\t\\\\\nPNPLA2&\t-0.193\t&0.014(0.55)\t\\\\\nPPM1A&\t-0.124\t&&&\t0.166(0.89)\t\\\\\nPPP2R2D&\t0.157&\t-0.234(0.67)\t\\\\\nRBM11&\t0.032\t&&\t-0.291(0.71)\t\\\\\nRNF6&\t-0.215\t&0.199(0.90)\t\\\\\nRNF126P1&\t0.225\t\\\\\nRPS27&\t0.134\t&&&&\t-0.155(0.22)\\\\\nSCAND2P&\t-0.002\t&&&\t0.329(0.35)\t\\\\\nSERTAD4&\t-0.356\t&&\t0.350(0.91)\t\\\\\nSGSM3&\t0.285\t&&&\t-0.039(0.46)\\\\\nSH3RF1&\t-0.096\t\\\\\nSLC25A2&\t-0.009\t&&\t-0.335(0.94)\t\\\\\nSPCS3&\t-0.310\t&&&&\t0.340(0.66)\\\\\nSPRED2&\t-0.260\t\\\\\nSRRM3&\t-0.317\t&&&&\t-0.244(0.70)\\\\\nTXN2&\t-0.339\t&&&\t0.012(0.46)\t\\\\\nUBE4B&\t0.418\t&-0.497(0.53)\t\\\\\nVPS13B&\t0.065\t&&&\t-0.108(0.99)\t\\\\\nZNF727&\t0.401&\t-0.254(0.78)\t\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nFour E factors are included in analysis: age, gender, smoking pack years, and smoking history. All four have been suggested as associated with lung cancer prognosis in the literature. Among them, age and smoking pack years are continuous and normalized prior to analysis. Gender and smoking history are binary. A total of 436 subjects have complete E measurements. Among them, 110 died during follow-up, and the median follow-up time is 23 months. For the 326 censored subjects, the median follow-up time is 6 months.\n\nMeasurements on 18,897 gene expressions are available. To improve stability and reduce computational cost, we conduct marginal prescreening based on genes' univariate regression significance (p-value less than or equal to 0.1) and interquartile range (above the median of all interquartile ranges). Similar procedures have been adopted in the literature. A total of 819 gene expressions are included in downstream analysis. For each gene expression, we normalize to have mean 0 and variance 1.\n\n\nWe apply the proposed approach and select the optimal $\\lambda$ using five-fold cross validation. The detailed identification and estimation results are presented in Tables 2 (LARE) and 5 (LPRE, Appendix). As previously described, it is possible that the main effects corresponding to the identified interactions are not identified. To respect the ``main effects, interactions\" hierarchy, we add back such main effects and re-fit. Beyond the proposed, we also apply the LS and LAD methods. The summary of applying different methods is provided in Table \\ref{table9} (Appendix). Detailed estimation and identification results using the alternatives are presented in Tables \\ref{table7} and \\ref{table8} (Appendix). Different methods identify different sets of main effects and interactions. It is interesting that all of the main effects and interactions identified by LPRE are identified by LARE. They may represent more reliable findings. The LAD method identifies much fewer effects.\n\nTo complement the identification and estimation analysis, we evaluate stability. Specifically, we randomly remove ten subjects and then analyze data. This procedure is repeated 200 times. We then compute the probability that an interaction term is identified. Such an evaluation has been conducted in the literature. The stability results are provided in Tables 2 and 5-7(Appendix). We can see that most of the identified interactions are relatively stable, with many having probabilities of being identified close to one.\n\n\n\\section{Discussion}\n\nThe identification of important G$\\times$E interactions remains a challenging problem. In this article, we have introduced using the relative error criteria as loss functions. A penalized approach has been adopted for estimation and selection. Simulation shows that the proposed approach has performance comparable to or better than the alternatives. Thus it may be provide a useful alternative for data analysis. A limitation of this study is that the asymptotic properties have not been established. In the analysis of a lung cancer dataset, the LARE and LPRE results are relatively consistent but different from the alternatives. The identified interactions are reasonably stable. More examination of the findings is needed in the future.\n\n\n\n\\begin{acknowledgement}\nWe thank the participants of Joint 24th ICSA Applied Statistics Symposium and 13th Graybill Conference in Colorado and organizers of this proceedings. This study was supported in part by the National Science Foundation of China (Grant No. 11401561), National Social Science Foundation of China (13CTJ001, 13\\&ZD148), NIH (CA016359, CA191383), and the U.S. VA Cooperative Studies Program of the Department of Veterans Affairs, Office of Research and Development.\n\\end{acknowledgement}\n\n\n\\bibliographystyle{spmpsci}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations} \n\\begin{Large}\n{Juhani Koivisto} \\footnote{Supported by V\\\"ais\\\"al\\\"a Foundation and the Academy of Finland, project 252293.}\n\\end{Large} \\\\ \\emph{Department of Mathematics and Statistics, University of Helsinki, Finland, email: juhani.koivisto@helsinki.fi}\n\n\n\\paragraph{Abstract.}\nWe consider $L^p$-cohomology of reflexive Banach spaces and give a spectral condition implying the vanishing of 1-cohomology with coefficients in uniformly bounded representations on a Hilbert space. \n\\paragraph*{Mathematics Subject Classification (2000):} 20F65\n\\paragraph*{Key words} Fixed point property, cohomology, Banach space, uniformly bounded representation, spectral criterion.\n\n\\section{Introduction}\nSince its introduction by David Kazhdan in \\cite{Kazhdan}, property $(T)$ and its generalizations as cohomological vanishing has become a fundamental concept in mathematics \\cite{BHV}. The aim of this paper is to extend the framework of W. Ballman and J. \\'Swi\\c atkowski \\cite{BS} to reflexive Banach spaces and as an application, to give a spectral condition implying vanishing of cohomology for uniformly bounded representations on a Hilbert space.\nAlong with W. Ballman and J. \\'Swi\\c atkowski, A. \\.{Z}uk \\cite{Zuk96} was among the first to also consider such criteria for unitary representations, both following fundamental work by H. Garland [G]. Since then, extending the spectral method beyond Hilbert spaces has been considered in \\cite{Monod, Cha, Dymara, Ers, Fish, L} and by Piotr W. Nowak \\cite{Nowak} extending the spectral method in \\cite{Zuk03} to reflexive Banach spaces. Appropriately extending the scheme in \\cite{Nowak} we similarly extend the spectral condition of \\cite{BS} to uniformly bounded representations on a Hilbert space. Motivation for such generalizations arises, among others, from Shalom's conjecture \\cite{OWR} stating that any hyperbolic group $\\Gamma$ admits a uniformly bounded representation $\\pi$ with ${H^1}(\\Gamma, \\pi) \\neq 0$ together with a proper cocycle in $Z^1(\\Gamma, \\pi)$. \\\\\nFor a finite graph $K$ with vertices $\\mathcal{V}_K$, consider the graph Laplacian $\\triangle_+$ on the space of real valued functions on $\\mathcal{V}_K$ defined by $$\\triangle_+f(v) = f(v)-Mf(v),$$ where $Mf(v)$ is the mean value of $f$ on the vertices adjacent to $v$. Denote by $\\lambda_1(K)$ the spectral gap of $\\triangle_+$ and its associated Poincar\\'e constant by $\\kappa_2(K, \\mathbb{R})=\\lambda_1(K)^{-1\/2}$. More generally $\\kappa_2(K, \\mathcal{H})= \\lambda_1(K)^{-1\/2}$ for any separable infinite-dimensional Hilbert space $\\mathcal{H}$, \\cite{Nowak}. \n\n\n\n\\begin{main}\nLet $X$ be a locally finite $2$-dimensional simplicial complex, $\\Gamma$ a discrete properly discontinuous group of automorphisms of $X$ and $\\pi : \\Gamma \\rightarrow \\mathrm{B}(\\mathcal{H})$ a uniformly bounded representation of $\\Gamma$ on a separable infinite-dimensional Hilbert space $\\mathcal{H}$. If for any vertex $\\tau$ of $X$ the link $X_\\tau$ is connected and $$\\sup_{g \\in \\Gamma} \\Vert \\pi_g\\Vert < \\dfrac{\\sqrt{2}}{ \\kappa_2(X_\\tau, \\mathcal{H})},$$ then $L^2H^1(X,\\pi) = 0$.\n\\end{main}\n\n\\subparagraph*{Structure of the paper} In Sections \\ref{SU} to \\ref{PI} the framework of \\cite{BS} for unitary representations on Hilbert spaces is extended to reflexive Bananch spaces and isometric representations: Section \\ref{SU} introduces the generalized set up; Section \\ref{projsec} and \\ref{L} deal with the dual of the twisted cochains; Section \\ref{DC} introduces differentials and codifferentials; Section \\ref{secloc} discusses localization of the problem and Section \\ref{PI} introduces the spectral set up in terms of Poincar\\'e inequalities and constants on the links. Section \\ref{LH} introduces $L^p$-cohomology as a natural extension to $L^2$-cohomology, which is then applied to uniformly bounded representations using the fact that they correspond to isometric representations on some reflexive Banach space.\n\n\\subparagraph*{Acknowledgements} I would like to thank Piotr W. Nowak for suggesting this topic, invaluable advice, and devotion without which this project would not have been possible. I would also like to thank V\\\"ais\\\"al\\\"a foundation, my advisor Ilkka Holopainen and the ''Analysis, metric geometry and differential and metirc topology'' project for financial support, Pekka Pankka and Izhar Oppenheim for helpful discussions and correspondence, and Antti Per\\\"al\\\"a for many enjoyable conversations on related topics. \n\n\\section{Set up} \\label{SU}\nIn this chapter notation is fixed. We recall the notation and some basic facts used by \\cite{BS} for weighted simplicial complexes and extend the notion of square integrable cochains to reflexive Banach spaces and $p > 1$. \n\\subsection{Weighted complexes}\nThroughout, let $X$ denote an $n$-dimensional locally finite simplicial complex. Following \\cite{BS} we use the following notation: $X(k)$ is the set of (unordered) $k$-simplexes of $X$; $\\Sigma(k)$ is the set of ordered $k$-simplexes of $X$. As usual we write $\\sigma = \\lbrace v_0, \\dots, v_k \\rbrace$ for a $k$-simplex and $\\sigma =(v_0, \\dots, v_k)$ for an ordered $k$-simplex. If the vertices of $\\tau \\in \\Sigma(l)$ are vertices of $\\sigma \\in \\Sigma(k)$, we say that $\\tau \\subset \\sigma$, and for $\\tau = (v_0, \\dots, \\hat{v}_i, \\dots, v_k)$, i.e. $v_i \\notin \\tau$, we denote by $[\\sigma : \\tau] = (-1)^i$ the sign of $\\tau$ in $\\sigma = (v_0, \\dots, v_k)$. As customary, we write $\\sigma_i$ for $(v_0, \\dots, \\hat{v}_i, \\dots, v_k)$. In addition to orientation we consider $X$ to be equipped with a weight $\\omega$, by which we mean a map from the oriented simplexes of $X$ to the integers such that for $\\sigma = (v_0, \\dots, v_k) \\in \\Sigma(k)$, $$\\omega(\\sigma) = \\omega(\\lbrace v_0, \\dots, v_k \\rbrace),$$ where $\\omega(\\lbrace v_0, \\dots, v_k \\rbrace)$ denotes the number of $n$-simplexes containing $\\lbrace v_0, \\dots, v_k\\rbrace$. In addition, we assume that $\\omega(\\sigma) \\geq 1$ for every simplex of $X$. Beginning from Section \\ref{secloc} and onwards, we consider $X$ locally through its links, where, by the link of $\\tau = (v_0, \\dots, v_l) \\in \\Sigma(l)$ denoted by $X_\\tau$, we mean the $(n-l-1)$-dimensional subcomplex consisting of all simplexes $\\lbrace w_0, \\dots, w_j\\rbrace$ disjoint from $\\tau$ such that $\\lbrace v_0, \\dots, v_l\\rbrace \\cup \\lbrace w_0, \\dots, w_j\\rbrace$ is a simplex of $X$. Since $X$ is locally finite, $X_\\tau$ is finite. Here as previously, $X_\\tau(j)$ denotes the $j$-simplexes of $X_\\tau$, $\\Sigma_\\tau(j)$ its oriented $j$-simplexes and so on. In particular, for $\\sigma \\in \\Sigma_\\tau(j)$ and $\\tau \\in \\Sigma(l)$ we denote by $\\sigma * \\tau \\in \\Sigma(j+l+1)$ the join of $\\sigma$ and $\\tau$ obtained by juxtaposing the two in that order. \\\\\nIn addition to the above, we assume throughout that $X$ is a $\\Gamma$-space where $\\Gamma$ is a discrete topological group acting properly and discontinuously by simplicial automorphisms on $X$. In other words, $\\Gamma$ permutes the simplexes of $X$ preserving their order and weights: that is for $\\sigma = (v_0, \\dots, v_k) \\in \\Sigma(k)$, $g \\cdot \\sigma = (g(v_0), \\dots, g(v_k)) \\in \\Sigma(k)$ and $\\omega (\\sigma) = \\omega(g \\cdot \\sigma)$. As usual, we denote by $\\Gamma \\sigma$ and $\\Gamma_\\sigma$ the $\\Gamma$-orbit and stabilizer of $\\sigma \\in \\Sigma(k)$, respectively, by $\\Sigma(k, \\Gamma) \\subset \\Sigma(k)$ some chosen set of representatives of $\\Gamma$-orbits in $\\Sigma(k)$, and by $\\vert \\cdot \\vert$ the counting measure on $\\Gamma$. In particular since $\\Gamma$ is discrete, stabilizers are finite and the Haar measure on $\\Gamma$ is $\\vert \\cdot \\vert$. Although the discreteness assumption can be avoided, it will be used when constructing projections in Section \\ref{projsec}. For the following frequently used facts we refer to \\cite{BS}:\n\n\n\n\n\n\\begin{proposition} \\label{combinatorial} \\cite{BS}\nLet $n$ be the dimension of $X$. Then, for $\\tau \\in \\Sigma(k)$ $$\\sum_{\\substack{\\sigma \\in \\Sigma(k+1) \\\\ \\tau \\subset \\sigma}}\\omega(\\sigma) = (n-k)(k+2)!\\omega(\\tau).$$ \n\\end{proposition} \\qed\n\n\\begin{proposition} \\label{switchingsums} \\cite{BS}\nFor $0 \\leq l < k \\leq n$, let $f=f(\\tau, \\sigma)$ be a $\\Gamma$-invariant function on the set of pairs $(\\tau, \\sigma)$, $\\tau \\in \\Sigma(l)$, $\\sigma \\in \\Sigma(k)$, such that $\\tau \\subset \\sigma$. Then $$\\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{\\substack{\\tau \\in \\Sigma(l) \\\\ \\tau \\subset \\sigma}} \\dfrac{f(\\tau, \\sigma)}{\\vert \\Gamma_\\sigma \\vert} = \\sum_{\\tau \\in \\Sigma(l, \\Gamma)}\\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}}\\dfrac{f(\\tau, \\sigma)}{\\vert \\Gamma_\\tau \\vert},$$ whenever either side is absolutely convergent. \\qed\n\\end{proposition} \nMore generally, Proposition \\ref{switchingsums} holds for locally compact unimodular groups \\cite{DJ00} replacing the counting measure with the Haar measure.\n\n\n\n\n\n\\subsection{Banach space setting, isometric representations and $p$-integrable cochains}\n\nThroughout, let $(E, \\Vert \\cdot \\Vert_E)$ denote a reflexive Banach space, $\\langle \\cdot, \\cdot \\rangle_E$ the natural pairing between $E$ and its continuous dual $E^*$, $\\simeq$ isomorphism, $\\cong$ isometric isomorphism and $p^*$ the adjoint index of $p$ such that $1\/p+1\/p^* = 1$. \nMoreover, let $\\pi: \\Gamma \\rightarrow \\mathrm{Iso}(E)$ denote an isometric representation of $\\Gamma$ on $E$ where $\\mathrm{Iso}(E)$ denotes the group of isometric linear automorphisms on $E$ and by $\\bar{\\pi}: \\Gamma \\rightarrow \\mathrm{Iso}(E^*)$ its corresponding contragradient representation given by $\\bar{\\pi}_g = \\pi^*_{g^{-1}}$ where $\\pi^*$ is the transpose of $\\pi$. For combinatorial purposes we also introduce antisymmetrization:\n\\begin{definition}\nFor $n \\geq 1$ we denote by $S_n$ the symmetric group of $n$ elements and by $\\mathrm{sign} \\colon S_n \\rightarrow \\lbrace -1, 1 \\rbrace$ the signature of the permutation: $1$ if $\\alpha \\in S_n$ is an even permutation of the $n$ elements and otherwise $-1$. For $f: \\Sigma(k) \\rightarrow E$ define its alternation point-wise as the linear idempotent map $$ \\mathrm{Alt} f(\\sigma) = \\dfrac{1}{(k+1)!} \\sum_{\\alpha \\in S_{k+1}} \\mathrm{sign}(\\alpha) \\alpha^*f(\\sigma),$$ where $\\alpha^*f(\\sigma) = f(v_{\\alpha(0)}, \\dots, v_{\\alpha(k)})$ for $\\sigma = (v_0, \\dots, v_k) \\in \\Sigma(k).$ As usual, we say that $f$ is alternating if $\\mathrm{Alt} f = f$, and symmetric if $\\mathrm{Alt} f = 0$.\n\\end{definition}\n\nReplacing inner product with dual pairing and unitary representations by isometric representations, we next introduce twising and cochains as in \\cite{BS}.\n\n\n\n\n\\begin{definition}\nLet $ \\mathcal{E}^{(k,p)}(X,E)$ denote the semi-normed vector space of $k$-cochains $f \\colon \\Sigma(k) \\rightarrow E $ for which the semi norm given by $$\\Vert f \\Vert_{(k,p)} = \\left( \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert f(\\sigma) \\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma\\vert}\\right)^{1\/p},$$ is finite.\n\\end{definition}\n\n\n\n\\begin{definition}\nFor $f \\in {\\mathcal{E}^{(k,p)}}(X,E)^*$, we denote by $$\\langle \\phi, f \\rangle_k = \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\langle \\phi(\\sigma), f(\\sigma) \\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma \\vert}$$ the dual pairing between $\\mathcal{E}^{(k,p)}(X,E)$ and ${\\mathcal{E}^{(k,p)}}(X,E)^*$.\n\\end{definition}\n\n\\begin{proposition} \\label{cochaindual}\n${\\mathcal{E}^{(k,p)}}(X,E)^*\\cong \\mathcal{E}^{(k,p^*)}(X,E^*)$. \\qed \n\\end{proposition} \n\n\\begin{definition}\nLet $f: \\Sigma(k) \\rightarrow E$. If for every $g \\in \\Gamma$ and every $\\sigma \\in \\Sigma(k)$ $$f(g \\cdot \\sigma) = \\pi_g \\cdot f(\\sigma),$$ then we say that $f$ is twisted by $\\pi$, or for short just twisted. \n\\end{definition}\n\n\\begin{definition}\nLet $C^{(k,p)}(X,E)$ denote the vector space of all alternating maps $f \\colon \\Sigma(k) \\rightarrow E$ twisted by $\\pi$.\n\\end{definition}\nThose alternating maps twisted by $\\pi$ whose $\\Vert \\cdot \\Vert_{(k,p)}$ norm is finite are called $p$-integrable mod $\\Gamma$ and we use the following notation:\n\\begin{definition}\nLet $L^{(k,p)}(X, E) = \\lbrace f \\in C^{(k,p)}(X,E) \\colon \\Vert f\\Vert_{(k,p)} < \\infty \\rbrace$ denote the vector subspace of all alternating $k$-cochains of $X$ twisted by $\\pi$. \n\\end{definition}\n\nIn particular, if $\\Gamma$ acts cocompactly on $X$, then $L^{(k,p)}(X,E) = C^{(k,p)}(X,E)$ since then $X\/\\Gamma$ is compact, the set of representatives $\\Sigma(k, \\Gamma)$ is finite, and $\\Vert f \\Vert_{(k,p)} < \\infty$ for all $f \\in C^{(k,p)}(X,E)$. \\\\ We end this section by proving that $L^{(k,p)}(X, E)$ is a normed space with respect to $\\Vert \\cdot \\Vert_{(k,p)}$. Towards this end we first show that $\\Vert \\cdot \\Vert_{(k,p)}$ is independent of the set of representatives when $f \\in L^{(k,p)}(X,E)$.\n\n\\begin{lemma} \\label{independentofrepresentatives}\nIf $f \\in L^{(k,p)}(X, E)$, then $\\Vert f \\Vert_{(k,p)}$ is independent of the choice of $\\Sigma(k, \\Gamma)$.\n\\end{lemma}\n\\begin{proof} Let $\\Sigma'(k, \\Gamma)$ be another set of representatives. Then,\n\\begin{align} \\Vert f \\Vert_{(k,p)}^p &= \\sum_{\\sigma' \\in \\Sigma'(k, \\Gamma)} \\Vert f(\\sigma') \\Vert^p_E \\dfrac{\\omega (\\sigma')}{(k+1)!\\vert \\Gamma_{\\sigma'} \\vert} = \\sum_{\\sigma' \\in \\Sigma'(k, \\Gamma)} \\Vert f(g' \\cdot \\sigma') \\Vert^p_E \\dfrac{\\omega (g' \\cdot \\sigma')}{(k+1)!\\vert \\Gamma_{g' \\cdot \\sigma'} \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert f(\\sigma) \\Vert^p_E \\dfrac{\\omega (\\sigma)}{(k+1)!\\vert \\Gamma_{\\sigma} \\vert}, \\nonumber \\end{align}\nchoosing for each $\\sigma' \\in \\Sigma'(k, \\Gamma)$ a $g' \\in \\Gamma$ such that $g' \\cdot \\sigma' = \\sigma \\in \\Sigma(k, \\Gamma)$ and observing that $f$ is twisted by $\\pi$ and both the norm and $\\omega$ are $\\Gamma$-invariant.\n\\end{proof}\n\n\\begin{proposition}\n$L^{(k,p)}(X,E) \\subseteq \\mathcal{E}^{(k,p)}(X,E)$ is a normed vector space.\n\\end{proposition} \n\\begin{proof} It suffices to show that the seminorm $\\Vert \\cdot \\Vert_{(k,p)}$ on $\\mathcal{E}^{(k,p)}(X,E)$ restricted to $L^{(k,p)}(X,E)$ is a norm. To this end, suppose $\\Vert f \\Vert_{(k,p)} = 0$ for $f \\in L^{(k,p)}(X,E)$. By Lemma \\ref{independentofrepresentatives} we may assume $f(\\sigma) = 0$ for all $\\sigma \\in \\Sigma(k, \\Gamma)$. Since $f(g \\cdot \\sigma) = \\pi_g f(\\sigma)$ and the action of $\\Gamma$ is transitive on the orbits it follows that $f(\\sigma)=0$ for all $\\sigma \\in \\Sigma(k)$. \\end{proof}\n\n\n\\section{Projecting $k$-cochains onto $L^{(k,p)}(X,E)$} \\label{projsec}\nIn order to extend the framework of \\cite{BS}, the dual space of the alternating and twisted cochains has at first to be identified up to isometric isomorphism. Following the scheme presented in \\cite{Nowak}, we begin by stepwise constructing a continuous projection $P_L$ from $\\mathcal{E}^{(k,p)}(X,E)$ onto $L^{(k,p)}(X,E)$\n\n\n\n\n\n\n\n\\begin{definition} \\label{projection1}\nDefine $\\widetilde{P}: \\mathcal{E}^{(k,p)}(X,E) \\rightarrow \\mathcal{E}^{(k,p)}(X,E)$ by \\begin{displaymath}\n\\widetilde{P}f (\\sigma) = \\left\\{ \\begin{array}{ll}\n\\displaystyle \\sum_{s \\in \\Gamma_\\sigma} \\pi_s f'(\\sigma) & \\textrm{if $\\sigma \\in \\Sigma(k, \\Gamma)$}\\\\\n\\displaystyle \\sum_{\\substack{h \\in \\Gamma \\\\ h \\cdot \\tau = \\sigma}} \\pi_{h} f'(\\tau) & \\textrm{if $\\sigma \\notin \\Sigma(k, \\Gamma)$ for $\\tau \\in \\Sigma(k, \\Gamma)$},\\\\\n\\end{array} \\right.\n\\end{displaymath}\nwhere $f': \\Sigma(k, \\Gamma) \\rightarrow E$ is the restriction of $f: \\Sigma(k) \\rightarrow E$ to $\\Sigma(k, \\Gamma)$.\n\\end{definition}\nThis map is well defined, in particular we note that $\\lbrace h \\in \\Gamma \\colon h \\cdot \\tau = \\sigma \\rbrace = h\\Gamma_\\tau$. As the following proposition shows, $\\widetilde{P}$ maps $k$-cochains to $k$-cochains twisted by $\\pi$.\n\n\\begin{proposition}\nFor $f \\in \\mathcal{E}^{(k,p)}(X,E)$, the $k$-cochain $\\widetilde{P}f: \\Sigma(k) \\rightarrow E$ is twisted by $\\pi$.\n\\end{proposition}\n\\begin{proof} Let $\\sigma \\in \\Sigma(k)$. Then either $\\sigma \\in \\Sigma(k, \\Gamma)$ or $\\sigma \\notin \\Sigma(k, \\Gamma)$. Suppose at first $\\sigma \\in \\Sigma(k, \\Gamma)$. If $g \\in \\Gamma_\\sigma$, then clearly $\\pi_g \\widetilde{P}f(\\sigma) = \\widetilde{P}f(g \\cdot \\sigma)$. On the other hand, if $g \\notin \\Gamma_\\sigma$ we get $$\\widetilde{P}f(g \\cdot \\sigma) = \\sum_{\\substack{h \\in \\Gamma \\\\ h \\cdot \\sigma = g \\cdot \\sigma}} \\pi_hf'(\\sigma),$$\nand $$ \\pi_g\\widetilde{P}f(\\sigma) = \\sum_{h \\in g \\Gamma_\\sigma} \\pi_h f'(\\sigma).$$\nBut $\\lbrace h \\in \\Gamma \\colon h \\cdot \\sigma = g \\cdot \\sigma \\rbrace = \\lbrace h \\in \\Gamma \\colon h \\in g \\Gamma_\\sigma \\rbrace$, so the claim holds for $\\sigma \\in \\Sigma(k, \\Gamma)$. \nSuppose $\\sigma \\notin \\Sigma(k, \\Gamma)$. If $g \\cdot \\sigma \\in \\Sigma(k, \\Gamma)$, then\n\\begin{align}\n\\widetilde{P}f(g \\cdot \\sigma) &= \\sum_{s \\in \\Gamma_{g \\cdot \\sigma}} \\pi_s f'(g \\cdot \\sigma) = \\sum_{s \\in g\\Gamma_{\\sigma}g^{-1}} \\pi_s f'(g \\cdot \\sigma) = \\sum_{h \\in \\Gamma_\\sigma} \\pi_{ghg^{-1}}f'(g \\cdot \\sigma) \\nonumber, \n\\end{align} as $\\Gamma_{g \\cdot \\sigma} = g \\Gamma_\\sigma g^{-1}$, and so\n\\begin{align}\n\\pi_g\\widetilde{P}f(\\sigma) &= \\sum_{\\substack{h \\in \\Gamma \\\\ hg \\cdot \\sigma = \\sigma}} \\pi_{gh}f'(g \\cdot \\sigma) = \\sum_{\\substack{h \\in \\Gamma \\\\ hg \\in \\Gamma_\\sigma}} \\pi_{gh}f'(g \\cdot \\sigma) = \\sum_{h \\in \\Gamma_\\sigma g^{-1}} \\pi_{gh}f'(g \\cdot \\sigma) \\nonumber \\\\ &= \\sum_{h \\in \\Gamma_\\sigma} \\pi_{ghg^{-1}}f'(g \\cdot \\sigma) = \\widetilde{P}f(g \\cdot \\sigma). \\nonumber \n\\end{align}\nOn the other hand, if $g \\cdot \\sigma \\notin \\Sigma(k, \\Gamma)$, write $\\widetilde{P}f(g \\cdot \\sigma) = \\sum_{h \\in A} \\pi_h f'(\\tau)$ where $A = \\lbrace h \\in \\Gamma \\colon h \\cdot \\tau = g \\cdot \\sigma \\rbrace = gB$ for $B = \\lbrace s \\in \\Gamma \\colon s \\cdot \\tau = \\sigma \\rbrace$ and $\\tau \\in \\Sigma(k, \\Gamma)$. Hence,\n\\begin{align}\n\\pi_g \\widetilde{P}f(\\sigma) &= \\sum_{\\substack{s \\in \\Gamma \\\\ s \\cdot \\tau = \\sigma}} \\pi_{gs}f'(\\tau) = \\sum_{h \\in g \\lbrace s \\in \\Gamma \\colon s \\cdot \\tau = \\sigma \\rbrace} \\pi_h f'(\\tau)= \\sum_{h \\in gB} \\pi_h f'(\\tau) \\nonumber \\\\ &= \\sum_{h \\in A} \\pi_h f'(\\tau) = \\widetilde{P}f(g \\cdot \\sigma), \\nonumber\n\\end{align} so the claim holds for $\\sigma \\notin \\Sigma(k, \\Gamma)$ as well. \\end{proof}\nRecalling that $\\Gamma$ is discrete, normalizing $\\widetilde{P}$ as below gives a projection onto the twisted cochains.\n\\begin{definition} \\label{projection2}\nDefine ${P}: \\mathcal{E}^{(k,p)}(X,E) \\rightarrow \\mathcal{E}^{(k,p)}(X,E)$ by $$Pf(\\sigma) = \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\widetilde{P}f(\\sigma).$$ \n\\end{definition}\n\n\\begin{proposition} \\label{projection11}\n${P}$ is a projection onto the twisted cochains.\n\\end{proposition}\n\\begin{proof} Clearly ${P}^2 = {P}$ and onto. Now, suppose $f$ is twisted and $\\sigma \\in \\Sigma(k)$. If $\\sigma \\in \\Sigma(k, \\Gamma)$, then, recalling the discreteness assumption \\begin{align}{P}f(\\sigma) &= \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\pi_sf'(\\sigma)= \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} f'(s \\cdot \\sigma) \\nonumber = \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} f'(\\sigma) = f'(\\sigma) =f(\\sigma). \\end{align} Similarly, for $\\sigma \\notin \\Sigma(k, \\Gamma)$\n\\begin{align}\n{P}f(\\sigma) &= \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{\\substack{h \\in \\Gamma \\\\ h \\cdot \\tau = \\sigma}} \\pi_{h} f'(\\tau) = \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{\\substack{h \\in \\Gamma \\\\ h \\cdot \\tau = \\sigma}} f(h \\cdot \\tau) = \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{\\substack{h \\in \\Gamma \\\\ h \\cdot \\tau = \\sigma}} f(\\sigma) = f(\\sigma), \\nonumber\n\\end{align} as $\\vert \\lbrace h \\in \\Gamma \\colon h \\cdot \\tau = \\sigma \\rbrace \\vert = \\vert h \\Gamma_\\tau \\vert = \\vert \\Gamma_\\tau \\vert$ and $\\vert \\Gamma_\\sigma \\vert = \\vert h \\Gamma_\\tau h^{-1}\\vert = \\vert \\Gamma_\\tau \\vert$.\\end{proof}\n\n\\begin{corollary}\n$P$ is continuous with $\\Vert {P}f \\Vert_{(k,p)} \\leq \\Vert f \\Vert_{(k,p)}$ for $f \\in \\mathcal{E}^{(k,p)}(X,E)$ with equality for $f \\in L^{(k,p)}(X,E)$. \n\\end{corollary}\n\\begin{proof} A straightforward consequence of Definition \\ref{projection2}, and the observation that ${P}f=f$ for $f \\in L^{(k,p)}(X,E)$.\\end{proof}\nThus, we have constructed a projection $P$ onto the cochains twisted by $\\pi$. However, cochains in the image are not necessarily alternating and hence not necessarily in $L^{(k,p)}(X,E)$. Antisymmetrizing $P$ fixes this. We begin with the following observation:\n\\begin{corollary} \\label{remainstwisted} \\label{propAlt}\nIf $f$ is twisted, then $\\mathrm{Alt} f$ is twisted.\n\\end{corollary}\n\\begin{proof} Suppose $f: \\Sigma(k) \\rightarrow E$ is twisted. Then, $\\mathrm{Alt}f(g \\cdot \\sigma) = \\mathrm{Alt}(\\pi_g f(\\sigma)) = \\pi_g (\\mathrm{Alt}f(\\sigma))$ for all $g \\in \\Gamma$ and $\\sigma \\in \\Sigma(k)$, where we used the fact that $f$ is twisted in the first equality and linearity of $\\pi_g$ in the last equality. Hence, $\\mathrm{Alt}f$ is twisted as well. \\end{proof}\n\\begin{corollary} \\label{althelp}\nSuppose $f \\in \\mathcal{E}^{(k,p)}(X,E)$, then $$\\Vert \\mathrm{Alt}f \\Vert_{(k,p)}^p \\leq (k+1)! \\Vert f \\Vert_{(k,p)}^p$$\n\\end{corollary}\n\\begin{proof} Since\n\\begin{align} \\Vert \\mathrm{Alt}f(\\sigma) \\Vert^p_E &= \\dfrac{1}{(k+1)!^p} \\Vert \\sum_{\\alpha \\in S_{k+1}} \\mathrm{sign}(\\alpha)\\alpha^*f(\\sigma) \\Vert^p_E \\leq \\dfrac{(k+1)!^p}{(k+1)!^p} \\sum_{\\alpha \\in S_{k+1}} \\Vert \\alpha^*f(\\sigma) \\Vert^p_E \\nonumber \\\\ &= \\sum_{\\alpha \\in S_{k+1}}\\Vert \\alpha^*f(\\sigma) \\Vert^p_E, \\nonumber \\end{align} it follows that \\begin{align}\\Vert \\mathrm{Alt}f \\Vert^p_{(k,p)} &= \\sum_{\\sigma \\in \\Sigma(k,\\Gamma)} \\Vert \\mathrm{Alt} f(\\sigma) \\Vert^p \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\leq \\sum_{\\alpha \\in S_{k+1}} \\sum_{\\sigma \\in \\Sigma(k,\\Gamma)} \\Vert \\alpha^* f(\\sigma) \\Vert^p \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= (k+1)! \\sum_{\\sigma \\in \\Sigma(k,\\Gamma)} \\Vert f(\\sigma) \\Vert^p \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\end{align}\nsince we sum over all representatives in the last equality, and for $\\sigma = (v_0, \\dots, v_k)$, $\\omega(v_{\\alpha(0)}, \\dots, v_{\\alpha(k)}) = \\omega(\\sigma)$ and $\\Gamma_{(v_{\\alpha(0)}, \\dots v_{\\alpha(k)})} = \\Gamma_\\sigma$ for all $\\alpha \\in S_{k+1}$. \\end{proof} \n\n\n\\begin{proposition}\nThe map $P_L : \\mathcal{E}^{(k,p)}(X,E) \\rightarrow \\mathcal{E}^{(k,p)}(X,E)$, given by $$P_L= \\mathrm{Alt} \\circ {P}$$ defines a projection onto $L^{(k,p)}(X,E)$. In other words, the diagram $$\n\\xymatrix{\n{\\mathcal{E}^{(k,p)}(X,E)} \\ar@{->>}[r]^P \\ar@{->>}[dr]_{{P_L}} & \n{\\lbrace \\mathrm{k-cochain}_{\\pi}\\rbrace} \\ar@{->>}[d]^{{\\mathrm{Alt}}} \\\\ {} & {{L^{(k,p)}(X,E)}}\n} $$ commutes. \n\\end{proposition}\n\\begin{proof} Clearly $P_L^2 = P_L$. By Proposition \\ref{projection11} $P$ is a projection onto the twisted cochains, and since $\\mathrm{Alt}$ preserves twisting by Corollary \\ref{remainstwisted}, $P_L$ is a projection onto $L^{(k,p)}(X,E)$.\\end{proof}\n\n\n\n\n\\begin{proposition} \\label{P_Lcontinuous}\n$P_L$ is continuous with $\\Vert P_Lf \\Vert_{(k,p)}^p \\leq (k+1)! \\Vert f \\Vert_{(k,p)}^p$. \n\\end{proposition}\n\\begin{proof} \\begin{align}\n\\Vert P_Lf \\Vert_{(k,p)}^p &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert P_Lf(\\sigma) \\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ \n&= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\left\\Vert \\mathrm{Alt}\\left( \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\pi_s f'(\\sigma) \\right)\\right\\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\left\\Vert \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\pi_s \\mathrm{Alt} f'(\\sigma) \\right\\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ & \\leq \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert^p} \\vert \\Gamma_\\sigma \\vert^p \\max_{s \\in \\Gamma_s} \\left\\lbrace \\Vert \\pi_s \\mathrm{Alt}f'(\\sigma)\\Vert^p_E \\right\\rbrace \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert \\mathrm{Alt}f'(\\sigma)\\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ \\leq} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} (k+1)! \\Vert f'(\\sigma)\\Vert^p_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= (k+1)! \\Vert f \\Vert_{(k,p)}^p, \\nonumber\n\\end{align}\nwhere we have used Corollary \\ref{althelp} in the last inequality.\n\n\\end{proof}\n\n\\begin{corollary} \\label{Lclosed}\n$\\displaystyle L^{(k,p)}(X,E) \\subseteq \\mathcal{E}^{(k,p)}(X,E)$ is closed. \\qed\n\\end{corollary} \n\n\\section{$L^{(k,p)}(X,E)^* \\cong L^{(k,p^*)}(X,E^*)$} \\label{L}\nHaving constructed a continuous projection from $\\mathcal{E}^{(k,p)}(X,E)$ onto $L^{(k,p)}(X,E)$ we show that the dual of $L^{(k,p)}(X,E)$ can be identified up to isometric isomorphism with $\\mathcal{E}^{(k,p^*)}(X,E^*) \/ \\mathrm{Ann}(L^{(k,p)}(X,E))$, cf. Corollary \\ref{fin2}, and finally that the latter is isometrically isomorphic to $L^{(k,p^*)}(X,E^*)$, cf. Proposition \\ref{isometric1} and \\ref{fin4} below. Towards this end, recall that by the annihilator of a subspace $M \\subseteq E$ we mean the vector space $\\mathrm{Ann}(M) = \\lbrace x \\in E^* \\colon \\langle y,x \\rangle_E = 0 \\,\\, \\forall y \\in M \\rbrace$ of all bounded linear functionals on $E$ that vanish on $M$. The following fact contains the idea of the proof:\n\n\\begin{proposition} \\label{dualcomposition1} \\label{firstdual}\n\\cite{Douglas} Suppose $E$ is a Banach space such that $E= M \\oplus N$ and denote by $P$ the corresponding projection onto $M$. Then, \n\\begin{enumerate}\n\\item $\\ker P^* = \\mathrm{Ann}(M)$ and $\\mathop{\\mathrm{im}}P^* = \\mathrm{Ann}(N)$;\n\\item $E^* \\simeq \\mathrm{Ann}(N) \\oplus \\mathrm{Ann}(M)$;\n\\item if $M$ is closed $M^* \\cong E^* \/ \\mathrm{Ann}(M).$ \\qed\n\\end{enumerate} \n\\end{proposition} \nLet $L^{(k,p)}_-(X,E)$ denote the closed complement of $L^{(k,p)}(X,E)$ in $\\mathcal{E}^{(k,p)}(X,E)$. That is $ L^{(k,p)}_-(X,E) = \\ker \\, {P}_{L}$, or in other words:\n\n\n\\begin{corollary} \\label{symmetry} \\label{fin1}\n$L_-^{(k,p)}(X,E) = \\lbrace f \\in \\mathcal{E}^{(k,p)}(X,E) \\colon \\mathrm{Alt}f(\\sigma) = 0 \\,\\, \\forall \\sigma \\in \\Sigma(k, \\Gamma) \\rbrace$ is a closed subspace of $\\mathcal{E}^{(k,p)}(X,E).$\n\\end{corollary} \\begin{proof} Given $f \\in L^{(k,p)}_-(X,E)$, $(I-P_L)f(\\sigma) = f(\\sigma)$ for all $\\sigma \\in \\Sigma(k)$, and hence for all $\\sigma \\in \\Sigma(k, \\Gamma)$\n\\begin{align}\n(I-P_L)f'(\\sigma) &= f'(\\sigma) - P_Lf'(\\sigma) = f'(\\sigma) - \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\pi_s \\mathrm{Alt}f'(\\sigma) = f'(\\sigma), \\nonumber \n\\end{align}\nimplying by linearity that $\\mathrm{Alt}f'(\\sigma) = 0$. Hence, $f$ is symmetric on representatives.\n\\end{proof}\n\n\n\n\n\\begin{corollary} \\label{fin2}\n$\\displaystyle L^{(k,p)}(X,E)^* \\cong \\mathcal{E}^{(k,p^*)}(X,E^*) \/ \\mathrm{Ann}(L^{(k,p)}(X,E)).$\n\\end{corollary}\n\\begin{proof} Since $L^{(k,p)}(X,E)$ is a closed subspace of $\\mathcal{E}^{(k,p)}(X,E)$ by Corollary \\ref{Lclosed}, the claim now follows from Proposition \\ref{firstdual}(3) and the fact that ${\\mathcal{E}^{(k,p)}(X,E)}^* \\cong \\mathcal{E}^{(k,p^*)}(X,E^*)$. \\end{proof}\nIt now remains to identify the annihilators, cf. Proposition \\ref{isometric1}, to prove isomorphism and finally isometry. As indicated by Proposition \\ref{dualcomposition1} this requires knowing $P_L^*$.\n\\begin{proposition} \\label{P_Ladjoint}\nLet $\\overline{P}_L : \\mathcal{E}^{(k,p^*)}(X,E^*) \\rightarrow \\mathcal{E}^{(k,p^*)}(X,E^*)$ be a projection as above. Then $\\overline{P}_L = P_L^*$.\n\\end{proposition}\n\\begin{proof} Assume first $k=1$, let $f \\in \\mathcal{E}^{(1,p)}(X,E)$ and $\\phi \\in \\mathcal{E}^{(1,p^*)}(X,E^*)$. For $\\sigma = (v_0, v_1) \\in \\Sigma(1, \\Gamma)$ we denote by $-\\sigma$ the simplex $(v_1,v_0)$. Now, \n\\begin{align}\n&\\langle P_Lf, \\phi \\rangle_1 = \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\langle P_Lf(\\sigma), \\phi(\\sigma) \\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{(*) \\\\ =} \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\left\\langle \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_s} \\pi_s \\left( \\dfrac{1}{2} f'(\\sigma) - \\dfrac{1}{2}f'(-\\sigma)\\right), \\phi'(\\sigma)\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\left\\langle \\pi_s \\left(f'(\\sigma) - f'(-\\sigma)\\right), \\phi'(\\sigma)\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\left\\langle \\pi_s f'(\\sigma), \\phi'(\\sigma) \\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\phantom{=} -\\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\left\\langle \\pi_s f'(-\\sigma), \\phi'(\\sigma)\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{(**) \\\\ =} \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\left\\langle \\pi_s f'(\\sigma), \\phi'(\\sigma) \\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\phantom{=} -\\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert}\\sum_{s \\in \\Gamma_\\sigma} \\left\\langle \\pi_s f'(\\sigma), \\phi'(-\\sigma)\\right\\rangle_E \n\\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\end{align} \\begin{align} &= \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\left\\langle f'(\\sigma), \\overline{\\pi}_s \\phi'(\\sigma) \\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\phantom{=} -\\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\left\\langle f'(\\sigma), \\overline{\\pi}_s\\phi'(-\\sigma)\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(1, \\Gamma)} \\dfrac{1}{2 \\vert \\Gamma_\\sigma \\vert} \\sum_{s \\in \\Gamma_\\sigma} \\left\\langle f'(\\sigma), \\overline{\\pi}_s \\left(\\phi'(\\sigma)-\\phi'(-\\sigma) \\right) \\right\\rangle_E \\dfrac{\\omega(\\sigma)}{2! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\langle f, \\overline{P}_L \\phi \\rangle_1,\\nonumber \n\\end{align} where $(*)$ and the last equality follow from the definition of $P_L$ and $\\overline{P}_L$, respectively when $k=1$. $(**)$ follows as we sum over all $\\sigma \\in \\Sigma(1, \\Gamma)$, so the sums where we switch the summation variable $\\sigma$ with $-\\sigma$ agree as $\\omega(\\sigma) = \\omega(-\\sigma)$. For $k>1$ the calculation goes similarly, denoting $\\sigma = (v_0, \\dots, v_k) \\in \\Sigma(k, \\Gamma)$ and arguing similarly,\n\\begin{align}\n&\\langle P_Lf, \\phi \\rangle_k = \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\langle P_Lf(\\sigma), \\phi(\\sigma) \\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =}\\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{s \\in \\Gamma_s} \\sum_{\\alpha \\in S_{k+1}} \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert} \\dfrac{1}{(k+1)!} (-1)^{\\mathrm{sgn}(\\alpha)} \\left\\langle \\pi_s f'((v_{\\alpha(0)}, \\dots, v_{\\alpha(k)})), \\phi'(\\sigma)\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =}\\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{s \\in \\Gamma_s} \\sum_{\\alpha \\in S_{k+1}} \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert} \\dfrac{1}{(k+1)!} (-1)^{\\mathrm{sgn}(\\alpha)} \\left\\langle \\pi_s f'(\\sigma), \\phi'((v_{\\alpha(0)}, \\dots, v_{\\alpha(k)}))\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &=\\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{s \\in \\Gamma_s} \\sum_{\\alpha \\in S_{k+1}} \\dfrac{1}{\\vert \\Gamma_\\sigma \\vert} \\dfrac{1}{(k+1)!} (-1)^{\\mathrm{sgn}(\\alpha)} \\left\\langle f'(\\sigma), \\overline{\\pi}_s \\phi'((v_{\\alpha(0)}, \\dots, v_{\\alpha(k)}))\\right\\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\langle f, \\overline{P}_L \\phi \\rangle_k, \\nonumber\n\\end{align} \nwhere the first and last equality follows by the definition of $P_L$ and linearity of the dual pairing, and the third similarly as in the case $k=1$ above.\\end{proof}\n\n\\begin{proposition} \\label{isometric1}\nThe following are equal as sets:\n\\begin{enumerate}\n\\item $\\mathrm{Ann}(L_-^{(k,p)}(X,E)) = L^{(k,p^*)}(X,E^*);$ \n\\item $\\mathrm{Ann}(L^{(k,p)}(X,E)) = L_-^{(k,p^*)}(X,E^*).$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof} Suppose $f \\in L_-^{(k,p)}(X,E)$ and $\\phi \\in \\mathcal{E}^{(k,p^*)}(X,E^*)$. Then,\n\\begin{align}\n\\langle f, \\phi \\rangle_k &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\langle f(\\sigma), \\phi(\\sigma) \\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\langle (I-P_L)f(\\sigma), \\phi(\\sigma) \\rangle_E \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &= \\langle f,\\phi \\rangle_k - \\langle P_L f,\\phi \\rangle_k = \\langle f,\\phi \\rangle_k - \\langle f,\\overline{P}_L \\phi \\rangle_k \\nonumber.\n\\end{align} Hence, $\\langle f, \\overline{P}_L \\phi \\rangle_E = 0$ for all $f \\in L_-^{(k,p)}(X,E)$ where $\\overline{P}_L \\phi \\in L^{(k,p^*)}(X,E^*)$. Thus, $L^{(k,p^*)}(X,E^*) \\subseteq \\mathrm{Ann}(L_-^{(k,p)}(X,E))$. On the other hand, suppose $\\phi \\in \\mathrm{Ann}(L_-^{(k,p)}(X,E))$, then $\\langle f, \\phi \\rangle_k = 0$ for all $f \\in L_-^{(k,p)}(X,E)$. Hence, for all $f \\in \\mathcal{E}^{(k,p^*)}(X,E^*)$, so $0= \\langle (I-P_L)f, \\phi \\rangle_k$ if and only if $\\langle f, \\phi \\rangle_k = \\langle P_Lf, \\phi \\rangle_k = \\langle f, \\overline{P}_L \\phi \\rangle_k$ for all $f \\in \\mathcal{E}^{(k,p^*)}(X,E^*)$. Thus, $\\phi = \\overline{P}_L \\phi$ implies that $\\phi \\in L^{(k,p^*)}(X,E^*)$, so $\\mathrm{Ann}(L_-^{(k,p)}(X,E)) \\subseteq L^{(k,p^*)}(X,E^*)$ proving the first claim. The proof of the second claim goes similarly. \\end{proof}\n\n\\begin{corollary} \\label{isometry1}\nThe following are isomorphic:\n\\begin{enumerate}\n\\item $ L^{(k,p)}(X,E)^* \\cong \\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}_-(X,E^*) \\simeq L^{(k,p^*)}(X,E^*);$\n\\item $ L^{(k,p)}_-(X,E)^* \\cong \\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}(X,E^*) \\simeq L^{(k,p^*)}_-(X,E^*).$\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof} The first isomorphic isomorphisms follow immediately combining Propositions \\ref{firstdual} and \\ref{isometric1}, and the latter isomorphisms by Proposition \\ref{dualcomposition1}.\\end{proof}\n\n\\begin{proposition} \\label{fin4}\nThe following are isometrically isomorphic:\n\\begin{enumerate}\n\\item $L^{(k,p)}(X,E)^* \\cong L^{(k,p^*)}(X,E^*);$\n\\item $L^{(k,p)}_-(X,E)^* \\cong L^{(k,p^*)}_-(X,E^*).$\n\\end{enumerate}\n\\end{proposition}\nProof. Consider the second claim. Consider $\\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}(X,E^*)$ consisting of the cosets $[\\phi] = \\phi + L^{(k,p^*)}(X,E^*)$ for $\\phi \\in \\mathcal{E}^{(k,p^*)}(X,E^*)$. We claim that if $\\phi \\in L^{(k,p^*)}_-(X,E^*)$, then $\\Vert [\\phi]\\Vert = \\Vert \\phi \\Vert_{(k,p^*)}$ where $\\Vert \\cdot \\Vert = \\inf \\lbrace \\Vert \\xi \\Vert_ {(k,p^*)} \\colon \\xi \\in [\\phi] \\rbrace$ is the quotient norm. On the other hand, $\\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}(X,E^*) \\simeq L^{(k,p^*)}_-(X,E^*)$ by Corollary \\ref{isometry1}, so $\\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}(X,E^*) \\cong L^{(k,p^*)}_-(X,E^*)$ and consequently $L^{(k,p)}_-(X,E)^* \\cong L^{(k,p^*)}_-(X,E^*)$. Towards this end, fix $\\phi \\in L^{(k,p^*)}_-(X,E^*)$. Thus, for $\\psi \\in L^{(k,p^*)}(X,E^*)$ we have\n\\begin{align}\n\\Vert \\phi + \\psi \\Vert_{(k,p^*)}^{p^*} &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert \\phi(\\sigma) + \\psi(\\sigma) \\Vert^{p^*}_{E^*} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert \\phi(\\sigma) - \\psi(-\\sigma) \\Vert^{p^*}_{E^*} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert \\phi(-\\sigma) - \\psi(-\\sigma) \\Vert^{p^*}_{E^*} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\Vert \\phi(-\\sigma) - \\psi(-\\sigma) \\Vert^{p^*}_{E^*} \\dfrac{\\omega(-\\sigma)}{(k+1)!\\vert \\Gamma_{-\\sigma} \\vert} \\nonumber \\\\ &\\substack{{} \\\\ =} \\Vert \\phi - \\psi \\Vert^{p^*}_{(k,p^*)}, \\nonumber\n\\end{align} \nwhere $-\\sigma = (v_1,v_0, v_2, \\dots, v_k)$ for $\\sigma = (v_0, \\dots, v_k)$. The second equality follows since $\\psi$ is alternating, the third equality since $\\phi$ is symmetric on representatives, the fourth since $\\omega$ is symmetric and $\\Gamma_\\sigma = \\Gamma_{- \\sigma}$, and the last equality holds since we sum over all $\\sigma \\in \\Sigma(k, \\Gamma)$, which contains all the oriented simplexes with the vertices of $\\sigma$. Thus, $\\Vert \\phi + \\psi \\Vert_{(k,p^*)}^{p^*} = \\Vert \\phi - \\psi \\Vert_{(k,p^*)}^{p^*}$ and consequently by the triangle inequality $$2\\Vert \\phi \\Vert_{(k,p^*)} = \\Vert 2 \\phi + \\psi - \\psi \\Vert_{(k,p^*)} \\leq \\Vert \\phi + \\psi \\Vert_{(k,p^*)} + \\Vert \\phi - \\psi \\Vert_{(k,p^*)},$$ and so $$\\Vert \\phi \\Vert_{(k,p^*)} \\leq \\dfrac{1}{2}(\\Vert \\phi + \\psi \\Vert_{(k,p^*)} + \\Vert \\phi - \\psi \\Vert_{(k,p^*)}) = \\Vert \\phi + \\psi \\Vert_{(k,p)}.$$ Now, taking the infimum over $\\psi \\in L^{(k,p^*)}(X,E^*)$ thus shows that the quotient norm of $[\\phi]$ is $\\Vert \\phi\\Vert_{(k,p^*)}$, proving the second claim. The first claim is proven similarly by considering the cosets in $\\mathcal{E}^{(k,p^*)}(X,E^*) \\, \/ L^{(k,p^*)}_-(X,E^*)$. \\begin{flushright}$\\Box$ \\end{flushright} \n\n\n\n\\section{Differentials and codifferentials} \\label{DC}\nHaving identified the dual of $L^{(k,p)}(X,E)$ up to isometric isomorphism we extend the notion of differentials and codifferentials as presented in \\cite{BS} to reflexive Banach spaces. \n\\begin{definition}\nCodifferentials and differentials. Define the codifferential $$d_k: \\mathcal{E}^{(k,p)}(X,E) \\rightarrow \\mathcal{E}^{(k+1,p)}(X,E)$$ point-wise by $$d\\phi(\\sigma) = \\sum_{i=0}^{k+1}(-1)^i\\phi(\\sigma_i),$$ \nand the differential $$\\delta_{k+1}: L^{(k+1,p^*)}(X,E^*) \\rightarrow L^{(k,p^*)}(X,E^*),$$ as the adjoint of $d$ given by $\\langle \\phi , d \\psi \\rangle_{k+1} = \\langle \\delta \\phi, \\psi \\rangle_k$ for all $\\psi \\in L^{(k,p)}(X,E)$ and $\\phi \\in L^{(k+1,p^*)}(X,E^*)$.\n\\end{definition}\nSimilarly, we denote by $\\bar{d}_k: \\mathcal{E}^{(k,p^*)}(X,E^*) \\rightarrow \\mathcal{E}^{(k+1,p^*)}(X,E^*)$ the map given by $\\bar{d}\\psi(\\sigma) = \\sum_{i=0}^{k}(-1)^i\\psi(\\sigma_i)$ for $\\psi \\in \\mathcal{E}^{(k,p^*)}(X,E^*)$, and likewise for the differential.\n\n\\begin{corollary}\n$$\\xymatrix@1{{\\cdots} \\ar[r]^-{\\delta_{k+2}} & L^{(k+1,p^*)}(X,E^*) \\ar[r]^-{\\delta_{k+1}} & L^{(k,p^*)}(X,E^*) \\ar[r]^-{\\delta_k} & {\\cdots} \n}$$\nis a chain complex over $\\mathbb{R}$ dual to the cochain complex \n$$\\xymatrix@1{{\\cdots\\,} & L^{(k+1,p)}(X,E) \\ar[l]_-{d_{k+1}} & L^{(k,p)}(X,E) \\ar[l]_-{d_k} & {\\cdots} \\ar[l]_-{d_{k-1}} \n}$$\n\\qed \\end{corollary} \nAs the following shows, both $d$ and $\\delta$ are bounded operators.\n\\begin{proposition}\nLet $\\phi \\in L^{(k,p)}(X,E)$. Then $d: L^{(k,p)}(X,E) \\rightarrow L^{(k+1,p)}(X,E)$ is a bounded operator and $$\\Vert d\\phi \\Vert^p_{(k+1,p)} \\leq (n-k)(k+2)^p\\Vert \\phi \\Vert^p_{(k,p)}.$$ \n\\end{proposition}\n\\begin{proof} Using the estimate \\begin{align} \\left\\Vert \\sum_{i=0}^{k+1} (-1)^i \\phi(s_i) \\right\\Vert_E^p &\\leq \\left( \\Vert \\phi(s_0) \\Vert_E + \\left\\Vert \\sum_{i=1}^{k+1} (-1)^i \\phi(s_i) \\right\\Vert_E \\right)^p \\leq \\left( \\sum_{i=0}^{k+1} \\Vert \\phi(s_i)\\Vert_E \\right)^p \\nonumber \\\\ &\\leq ((k+2) \\max \\lbrace \\Vert \\phi(s_0) \\Vert, \\dots, \\Vert \\phi(s_{k+1})\\Vert \\rbrace )^p \\nonumber \\\\ &\\leq (k+2)^p \\sum_{i=0}^{k+1} \\Vert \\phi(s_i) \\Vert_E^p , \\nonumber \\end{align} it follows that \\begin{align}\n\\Vert d \\phi \\Vert^p_{(k+1,p)} &= \\sum_{s \\in \\Sigma(k+1, \\Gamma)} \\Vert d \\phi (s) \\Vert_E^p \\dfrac{\\omega(s)}{(k+2)! \\vert \\Gamma_s \\vert} \\nonumber \\\\ &= \\sum_{s \\in \\Sigma(k+1, \\Gamma)} \\left\\Vert \\sum_{i=0}^{k+1}(-1)^i \\phi(s_i) \\right\\Vert_E^p \\dfrac{\\omega(s)}{(k+2)! \\vert \\Gamma_s \\vert} \\nonumber \\\\ &\\substack{{} \\\\ \\leq} \\sum_{s \\in \\Sigma(k+1, \\Gamma)} \\left( \\dfrac{(k+2)^p\\omega(s)}{(k+2)! \\vert \\Gamma_s \\vert} \\sum_{i=0}^{k+1}\\Vert \\phi(s_i)\\Vert_E^p \\right) \\nonumber \\end{align} \\begin{align} &\\substack{{} \\\\ =} \\sum_{s \\in \\Sigma(k+1, \\Gamma)} \\left( \\dfrac{(k+2)^p\\omega(s)}{(k+2)!(k+1)! \\vert \\Gamma_s \\vert} \\sum_{\\substack{t \\in \\Sigma(k) \\\\ t \\subset s}} \\Vert [s:t] \\phi(t)\\Vert_E^p \\right) \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{s \\in \\Sigma(k+1, \\Gamma)} \\left( \\dfrac{(k+2)^p\\omega(s)}{(k+2)!(k+1)! \\vert \\Gamma_s \\vert} \\sum_{\\substack{t \\in \\Sigma(k) \\\\ t \\subset s}} \\Vert \\phi(t)\\Vert_E^p \\right) \\nonumber \\\\\n&\\substack{(*) \\\\ =}\\sum_{t \\in \\Sigma(k, \\Gamma)} \\left( \\dfrac{(k+2)^p \\Vert \\phi(t)\\Vert^p_E}{(k+2)!(k+1)!\\vert \\Gamma_t \\vert} \\sum_{\\substack{s \\in \\Sigma(k+1) \\\\ t \\subset s}} \\omega(s) \\right) \\nonumber \\\\ \n&\\substack{(**) \\\\ =} \\sum_{t \\in \\Sigma(k, \\Gamma)} \\dfrac{(n-k)(k+2)!(k+2)^p\\omega(t)}{(k+2)!(k+1)! \\vert \\Gamma_t \\vert} \\Vert \\phi(t)\\Vert_E^p \\nonumber \\\\ &= (n-k)(k+2)^p\\Vert \\phi \\Vert^p_{(k,p)}, \\nonumber\n\\end{align}\nwhere we used Proposition \\ref{switchingsums} in $(*)$ followed by Proposition \\ref{combinatorial} in $(**)$.\n\\end{proof}\n\n\nSimilarly to \\cite{BS}, we also have the following useful point-wise expression for the differential.\n\\begin{proposition} \\label{codifferential}\nFor $\\phi \\in L^{(k,p^*)}(X, E^*)$ and $\\tau \\in \\Sigma(k-1)$ $$\\delta \\phi(\\tau) = \\sum_{\\substack{v \\in \\Sigma(0) \\\\ v * \\tau \\in \\Sigma(k)}} \\dfrac{\\omega(v * \\tau)}{\\omega(\\tau)} \\phi(v * \\tau).$$ \n\\end{proposition}\n\\begin{proof} Let $\\psi \\in L^{(k-1,p)}(X,E)$. The claim follows by a straightforward computation,\n\\begin{align}\n\\langle \\phi, d \\psi\\rangle_{k} &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma\\vert}\\langle \\phi(\\sigma), d \\psi(\\sigma)\\rangle_E \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma\\vert} \\langle \\phi(\\sigma), \\sum_{i=0}^{k-1}(-1)^i \\psi(\\sigma_i)\\rangle_E \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{\\omega(\\sigma)}{(k+1)! \\vert \\Gamma_\\sigma\\vert} \\langle \\phi(\\sigma), \\dfrac{1}{(k-1+1)!} \\sum_{\\substack{\\tau \\in \\Sigma(k-1) \\\\ \\tau \\subset \\sigma}} [\\sigma: \\tau] \\psi(\\tau)\\rangle_E \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{\\substack{\\tau \\in \\Sigma(k-1) \\\\ \\tau \\subset \\sigma}} [\\sigma :\\tau]\\dfrac{\\omega(\\sigma)\\langle \\phi(\\sigma), \\psi(\\tau)\\rangle_E}{(k+1)! \\vert \\Gamma_\\sigma\\vert k!} \\nonumber \n\\nonumber \n\\end{align}\n\\begin{align}\n\\phantom{\\langle \\phi, d \\psi\\rangle_{k} = \\,} &\\substack{(*) \\\\ =} \\sum_{\\tau \\in \\Sigma(k-1, \\Gamma)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}} [\\sigma :\\tau] \\dfrac{\\omega(\\sigma)\\langle \\phi(\\sigma), \\psi(\\tau)\\rangle_E}{(k+1)! \\vert \\Gamma_\\tau\\vert k!} \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(k-1, \\Gamma)} \\dfrac{\\omega(\\tau)}{\\omega(\\tau)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}} [\\sigma :\\tau] \\dfrac{\\omega(\\sigma)\\langle \\phi(\\sigma), \\psi(\\tau)\\rangle_E}{(k+1)! \\vert \\Gamma_\\tau\\vert k!} \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(k-1, \\Gamma)} \\dfrac{\\omega(\\tau)}{k! \\vert \\Gamma_\\tau \\vert} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}} [\\sigma :\\tau] \\dfrac{\\omega(\\sigma)\\langle \\phi(\\sigma), \\psi(\\tau)\\rangle_E}{(k+1)! \\omega(\\tau)} \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(k-1, \\Gamma)} \\dfrac{\\omega(\\tau)}{k! \\vert \\Gamma_\\tau \\vert} \\sum_{\\substack{v \\in \\Sigma(0) \\\\ v * \\tau \\in \\Sigma(k)}} [v\\tau : \\tau] \\dfrac{\\omega(v * \\tau)}{\\omega(\\tau)}\\langle \\phi(v * \\tau), \\psi(\\tau)\\rangle_E \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(k-1, \\Gamma)} \\dfrac{\\omega(\\tau)}{k! \\vert \\Gamma_\\tau \\vert} \\left\\langle \\sum_{\\substack{v \\in \\Sigma(0) \\\\ v * \\tau \\in \\Sigma(k)}} \\dfrac{\\omega(v * \\tau)}{\\omega(\\tau)} \\phi(v * \\tau), \\psi(\\tau) \\right\\rangle_E = \\langle \\delta \\phi, \\psi\\rangle_{k-1}, \\nonumber\n\\end{align} where we used Proposition \\ref{switchingsums} in $(*)$ above. \\end{proof}\n\n\\section{Localization and restriction} \\label{secloc}\nIn this section we recall the concept of localization following \\cite{BS} and develop the notion in the setting of reflexive Banach spaces. We also consider the concept of restriction, recently considered by I. Oppenheim in the context of $L^2$-cohomology \\cite{IO}. Proposition \\ref{averagenorms} and \\ref{propQ} are the key results. The former relates the norm of the average to the norm of the differential, whereas the latter gives a global vanishing condition in the kernel of the full codifferential. \n\\begin{definition}\nFor a weight $\\omega$, define the localized weight as $$\\omega_\\tau(\\sigma) = \\omega(\\tau * \\sigma)$$ for $\\sigma \\in \\Sigma_\\tau(j)$ and $\\tau \\in \\Sigma(l)$ such that $\\tau * \\sigma \\in \\Sigma(j+l+1)$. \n\\end{definition}\n\nIn other words, for $\\tau \\in \\Sigma(l)$, $\\omega_\\tau(\\sigma)$ is the number of $(n-l-1)$-simplexes in $X_\\tau$ containing $\\sigma \\in \\Sigma_\\tau(j)$. \n\n\n\\begin{lemma}\n$\\Gamma_\\tau$ acts by simplicial automorphisms on $X_\\tau$. \n\\end{lemma}\n\\begin{proof} Let $\\sigma \\in X_\\tau$ and suppose $g \\in \\Gamma_\\tau$. Since, $\\sigma \\subset \\tau * \\sigma$, the join of $\\sigma$ and $\\tau$, it follows that $g \\cdot \\sigma \\subset g \\cdot \\tau * g \\cdot \\sigma = \\tau * g \\cdot \\sigma$ as $\\Gamma$ acts by simplicial automorphisms on $X$ and $g \\in \\Gamma_\\tau$. Thus, $g \\cdot \\sigma$ is a simplex in $\\tau * g \\cdot \\sigma$, and so $g \\cdot \\sigma \\in X_\\tau$ since it is disjoint from $\\tau$. Hence, $\\Gamma_\\tau$ act by simplicial automorphisms on $X_\\tau$. \\end{proof}\n\n\\begin{lemma}\nFor $\\eta \\in X_\\tau$, $\\Gamma_{\\tau \\eta} = \\Gamma_\\tau \\cap \\Gamma_\\eta$.\n\\end{lemma}\n\\begin{proof} $\\Gamma_{\\tau \\eta} = \\lbrace g \\in \\Gamma_\\tau \\colon g \\cdot \\eta = \\eta \\rbrace = \\Gamma_\\tau \\cap \\Gamma_\\eta.$ \\end{proof}\n\n\\begin{definition}\nWe denote by \n\\begin{enumerate}\n\\item[i.] $\\pi_\\tau$ the restriction of $\\pi$ to $\\Gamma_\\tau$, that is $\\pi_\\tau = \\pi \\vert_{\\Gamma_\\tau}$;\n\\item[ii.] $d_\\tau$ the restriction of $d$ to $\\mathcal{E}^{(k,p)}(X_\\tau, E)$, that is $d_\\tau = d \\vert_{\\mathcal{E}^{(k,p)}(X_\\tau, E)}$;\n\\item[iii.] $\\delta_\\tau$ the restriction of $\\delta$ to $L^{(k+1,p^*)}(X_\\tau, E^*)$, that is $\\delta_\\tau = \\delta \\vert_{L^{(k+1,p^*)}(X_\\tau, E^*)}$. \n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\nLet $$ \\mathcal{E}^{(k,p)}(X_\\tau,E) = \\left\\lbrace f \\colon \\Sigma_\\tau(k) \\rightarrow E \\colon \\Vert f \\Vert^p_{(k,p)} < \\infty \\right\\rbrace$$ for $X_\\tau \\subset X$ denote the vector space of $p$-summable functions with semi-norm $$\\Vert f \\Vert_{(k,p)} = \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k, \\Gamma_\\tau)} \\Vert f(\\sigma) \\Vert^p_E \\dfrac{\\omega_\\tau(\\sigma)}{(k+1)! \\vert \\Gamma_{\\tau \\sigma}\\vert}\\right)^{\\dfrac{1}{p}}.$$\n\\end{definition}\n\n\n\\begin{definition}\nLet $L^{(k,p)}(X_\\tau, E)$ denote the subspace $$\\left\\lbrace f \\in \\mathcal{E}^{(k,p)}(X_\\tau,E)\\, \\colon \\, f \\, \\mathrm{alternating} \\, \\mathrm{and} \\, \\forall g \\in \\Gamma_\\tau, \\forall \\sigma \\in \\Sigma_\\tau(k), f(g \\cdot \\sigma) = \\pi_{\\tau g} \\cdot f(\\sigma) \\right\\rbrace$$ of simplicial $k$-cochains of $X_\\tau$ twisted by $\\pi_\\tau$. \n\\end{definition}\n\n\\begin{definition}\nFor $f \\in \\mathcal{E}^{(k,p)}(X,E)$ and $\\tau \\in \\Sigma(j)$ such that $k-j-1 \\geq 0$, the localization of $f$ to $X_\\tau$ is the function $f_\\tau \\in \\mathcal{E}^{(k,p)}(X_\\tau,E) \\in \\mathcal{E}^{(k-j-1, p)}(X_\\tau, E)$ defined by the localization map $${}_\\tau \\colon \\mathcal{E}^{(k,p)}(X,E) \\longrightarrow \\mathcal{E}^{(k-j-1,p)}(X_\\tau,E)$$ where for all $\\sigma \\in \\Sigma_\\tau(k-j-1)$, $f_\\tau(\\sigma) = f (\\tau * \\sigma)$. Similarly we define its dual $\\mathcal{E}^{(k,p^*)}(X,E^*) \\longrightarrow \\mathcal{E}^{(k,p^*)}(X_\\tau,E^*)$, also denoted by ${}_\\tau$.\n\\end{definition}\n\n\\begin{definition}\nFor $f \\in \\mathcal{E}^{(k,p)}(X,E)$ and $\\tau \\in \\Sigma(j)$ such that $k+j+1 \\leq n$, the restriction of $f$ to $X_\\tau$ is the function $f^\\tau \\in \\mathcal{E}^{(k,p)}(X_\\tau,E)$ defined by the restriction map $${}^\\tau \\colon \\mathcal{E}^{(k,p)}(X,E) \\rightarrow \\mathcal{E}^{(k,p)}(X_\\tau,E)$$ where for all $\\sigma \\in \\Sigma_\\tau(k)$, $f^\\tau (\\sigma) = f(\\sigma)$. Similarly we define its dual $\\mathcal{E}^{(k,p^*)}(X,E^*) \\longrightarrow \\mathcal{E}^{(k-j-1,p^*)}(X_\\tau,E^*)$, also denoted by ${}^\\tau$.\n\\end{definition}\n\nNext, we consider a number of local to global equalities that will be of use. We begin by the following useful local relation: \n\\begin{proposition} \\label{BS1.10}\nFor $f_\\tau \\in L^{(k,p)}(X_\\tau, E)$ $$\\Vert f_\\tau \\Vert^p_{(k,p)} = \\dfrac{1}{(k+1)!\\vert \\Gamma_\\tau \\vert} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\Vert f_\\tau(\\sigma) \\Vert^p_E \\omega(\\tau * \\sigma).$$\n\\end{proposition}\nProof. \\begin{align}\n\\Vert f_\\tau \\Vert^p_{(k,p)} &= \\sum_{\\sigma \\in \\Sigma_{\\tau}(k, \\Gamma_\\tau)} \\Vert f_\\tau(\\sigma) \\Vert^p_E \\dfrac{\\omega_\\tau(\\sigma)}{\\vert \\Gamma_{\\tau \\sigma}\\vert(k+1)!} \\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma_{\\tau}(k)} \\dfrac{\\Vert f_\\tau(\\sigma) \\Vert^p_E}{\\vert \\Gamma_\\tau \\sigma \\vert} \\dfrac{\\omega_\\tau(\\sigma)}{\\vert \\Gamma_{\\tau \\sigma}\\vert(k+1)!} \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma_{\\tau}(k)} \\Vert f_\\tau(\\sigma) \\Vert^p_E \\dfrac{\\vert \\Gamma_{\\tau\\sigma}\\vert}{\\vert\\Gamma_\\tau\\vert} \\dfrac{\\omega_\\tau(\\sigma)}{\\vert \\Gamma_{\\tau \\sigma}\\vert(k+1)!} = \\sum_{\\sigma \\in \\Sigma_{\\tau}(k)} \\Vert f_\\tau(\\sigma) \\Vert^p_E \\dfrac{\\omega_\\tau(\\sigma)}{\\vert \\Gamma_{\\tau}\\vert(k+1)!}, \\nonumber \\\\ &= \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert}\\sum_{\\sigma \\in \\Sigma_{\\tau}(k)} \\Vert f_\\tau(\\sigma) \\Vert^p_E {\\omega}(\\tau *\\sigma) \\nonumber\n\\end{align} where the second equality in terms of the $k$-simplexes of $X_\\tau$ follows from the $\\Gamma_\\tau$ invariance of the norm, and the third by the fact that $\\vert \\Gamma_\\tau \\sigma \\vert$, the size of the $\\Gamma_\\tau$ orbit of $\\sigma$, is $\\vert \\Gamma_\\tau \\vert \/ \\vert \\Gamma_{\\tau \\sigma} \\vert$. \\begin{flushright}$\\Box$ \\end{flushright}\n\n\n\\begin{proposition} \\label{oppenheim}\nLet $f \\in L^{(k,p)}(X,E)$. If $k + 1 \\leq n$, then \n$$(n-k) \\Vert f \\Vert_{(k,p)}^p = \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f^\\tau \\Vert_{(k,p)}^p.$$\n\\end{proposition}\n\\begin{proof} For $\\xi \\in \\Sigma(k+1)$ such that $\\tau \\subset \\xi$, denote by $\\xi-\\tau$ the $k$-simplex in $X_\\tau$ obtained by removing the vertex $\\tau$ from $\\xi$. Now,\n\\begin{align}\n\\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f^\\tau \\Vert_{(k,p)}^p &\\substack{{} \\\\ =} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\Vert f (\\sigma) \\Vert_E^p \\omega (\\tau * \\sigma) \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\sum_{\\substack{\\xi \\in \\Sigma(k+1) \\\\ \\tau \\subset \\xi}} \\dfrac{1}{(k+2)} \\Vert f (\\xi - \\tau) \\Vert_E^p \\omega (\\xi) \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\sum_{\\substack{\\xi \\in \\Sigma(k+1) \\\\ \\tau \\subset \\xi}} \\dfrac{1}{(k+2)! \\vert \\Gamma_\\tau \\vert} \\Vert f (\\xi - \\tau) \\Vert_E^p \\omega (\\xi) \\nonumber \\\\ &\\substack{(*) \\\\ =} \\sum_{\\xi \\in \\Sigma(k+1, \\Gamma)} \\sum_{\\substack{\\tau \\in \\Sigma(0) \\\\ \\tau \\subset \\xi}} \\dfrac{1}{(k+2)! \\vert \\Gamma_\\xi \\vert} \\Vert f (\\xi - \\tau) \\Vert_E^p \\omega (\\xi) \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\xi \\in \\Sigma(k+1, \\Gamma)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\sigma \\subset \\xi}} \\dfrac{1}{(k+2)! \\vert \\Gamma_\\xi \\vert} \\dfrac{1}{(k+1)!} \\Vert f (\\sigma) \\Vert_E^p \\omega (\\xi) \\nonumber \\\\ &\\substack{(**) \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{\\substack{\\xi \\in \\Sigma(k+1) \\\\ \\sigma \\subset \\xi}} \\dfrac{1}{(k+2)! \\vert \\Gamma_\\sigma \\vert} \\dfrac{1}{(k+1)!} \\Vert f (\\sigma) \\Vert_E^p \\omega (\\xi) \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{1}{(k+2)! \\vert \\Gamma_\\sigma \\vert} \\dfrac{1}{(k+1)!} \\Vert f (\\sigma) \\Vert_E^p (n-k)(k+2)!\\omega (\\sigma) \\nonumber \\\\ &= (n-k) \\Vert f \\Vert_{(k,p)}^p, \\nonumber\n\\end{align}\nwhere the first and second equality follow by Proposition \\ref{BS1.10}. noting that $f^\\tau (\\sigma) = f(\\sigma)$, writing $\\xi - \\tau$ as $\\sigma$ and accounting for ordering. $(*)$ and $(**)$ follows by switching sums by Proposition \\ref{switchingsums} and the second last equality follows by Proposition \\ref{combinatorial}. \\end{proof} \n\n\n\\begin{proposition} \\label{summing}\nLet $f \\in L^{(k,p)}(X, E)$ and $0 \\leq j < k$. Then, $$(k+1)! \\Vert f \\Vert^p_{(k,p)} = (k-j)! \\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\Vert f_\\tau \\Vert^p_{(k-j-1,p)}.$$\n\\end{proposition}\n\\begin{proof} \n\\begin{align}\n\\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\Vert f_\\tau \\Vert^p_{(k-j-1,p)} &\\substack{{} \\\\ =} \\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\sum_{\\eta \\in \\Sigma_\\tau(k-j-1)} \\dfrac{\\omega(\\tau *\\eta)}{(k-j)! \\vert \\Gamma_\\tau\\vert} \\Vert f_\\tau(\\eta)\\Vert^p_E \\nonumber \\\\ &\\substack{{} \\\\ =} \\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\sigma = \\tau * \\eta}} \\dfrac{\\omega(\\sigma)}{(k-j)!\\vert \\Gamma_\\tau \\vert} \\Vert f(\\sigma) \\Vert^p_E \\nonumber \\\\ &\\substack{(*) \\\\ =} \\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}} \\dfrac{(k-j)!}{(k+1)!} \\dfrac{\\omega(\\sigma)}{(k-j)!\\vert \\Gamma_\\tau \\vert} \\Vert f(\\sigma) \\Vert^p_E \\nonumber \\\\ &= \\sum_{\\tau \\in \\Sigma(j, \\Gamma)} \\sum_{\\substack{\\sigma \\in \\Sigma(k) \\\\ \\tau \\subset \\sigma}} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\tau \\vert} \\Vert f(\\sigma) \\Vert^p_E \\nonumber \\\\ &\\substack{(**) \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\sum_{\\substack{\\tau \\in \\Sigma(j) \\\\ \\tau \\subset \\sigma}} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\Vert f(\\sigma) \\Vert^p_E \\nonumber \\\\ &\\substack{(***) \\\\ =} \\sum_{\\sigma \\in \\Sigma(k, \\Gamma)} \\dfrac{(k+1)!}{(k-j)!} \\dfrac{\\omega(\\sigma)}{(k+1)!\\vert \\Gamma_\\sigma \\vert} \\Vert f(\\sigma) \\Vert^p_E \\nonumber \\\\ &= \\dfrac{(k+1)!}{(k-j)!} \\Vert f \\Vert_{(k,p)}^p, \\nonumber\n\\end{align}\nwhere the first and second equality follow by Proposition \\ref{BS1.10} and writing $\\tau * \\eta$ as $\\sigma \\in \\Sigma(k)$, respectively. On the other hand, $(*)$ follows since summing over all $\\sigma \\in \\Sigma(k)$ such that $\\tau \\subset \\sigma$ amounts to summing over each term in the previous sum $(k+1)! \/ ((k+1)-(j+1))! = (k+1)!\/(k-j)!$ times recalling that $\\omega$ is symmetric and $f$ alternating. $(**)$ follows by Proposition \\ref{switchingsums}, and finally $(***)$ follows since there are $(k+1)!\/(k-j)!$ terms independent of $\\tau$ in the sum over all $\\tau \\in \\Sigma(j)$ with vertices in $\\sigma$.\n\\end{proof}\n\n\\begin{corollary} \\label{combinedsumming}\nSuppose $f \\in L^{(k,p)}(X,E)$. If $1 < k + 1 \\leq n$, then $$ \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f^\\tau \\Vert_{(k,p)}^p = \\dfrac{n-k}{k+1} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f_\\tau \\Vert_{(k-1,p)}^p.$$\n\\end{corollary}\n\\begin{proof} Follows immediately by Proposition \\ref{oppenheim} and Proposition \\ref{summing} above in the case $j=0$, \n\\begin{align}\n\\dfrac{k!}{(k+1)!} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f_\\tau \\Vert_{(k-1,p)}^p = \\Vert f \\Vert^p_{(k,p)} = \\dfrac{1}{(n-k)} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert f^\\tau \\Vert^p_{(k,p)}. \\nonumber\n\\end{align} \\end{proof}\n\n\\begin{definition}\nDefine the localized average over a cochain $\\phi$ by the map $$M : L^{(k,p)}(X_\\tau,E) \\rightarrow L^{(k,p)}(X_\\tau,E)$$ $\\phi_\\tau \\mapsto M \\phi_\\tau = \\phi_\\tau^0$ where $$\\phi_\\tau^0 (\\sigma) = \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{-1} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\phi_\\tau(\\sigma).$$ Similarly, we define its dual as $\\overline{M} : L^{(k,p^*)}(X_\\tau,E^*) \\rightarrow L^{(k,p^*)}(X_\\tau,E^*)$. \n\\end{definition}\n\n\\begin{corollary} \\label{Mbound}\nThe map $M : L^{(k,p)}(X_\\tau,E) \\rightarrow L^{(k,p)}(X_\\tau,E)$ and its dual $\\overline{M}$ are bounded projections onto the space of constant maps. \n\\end{corollary}\n\\begin{proof} $M$ is well-defined. Towards this end, let $\\phi_\\tau \\in L^{(k,p)}(X_\\tau,E)$. Since $\\omega_\\tau$ is symmetric and $\\Gamma_\\tau$-invariant, and $\\phi_\\tau$ is alternating and twisted by $\\pi_\\tau$, $M \\phi_\\tau$ is alternating and twisted by $\\pi_\\tau$ as a finite weighted sum of such functions. Moreover,\n\\begin{align}\n\\Vert M \\phi_\\tau \\Vert^p_{(k,p)} &= \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\Vert \\phi^0_\\tau (\\eta) \\Vert^p_E \\nonumber \\\\ &= \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{-p} \\left\\Vert \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\nonumber \\\\ &= \\dfrac{1}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{1-p} \\left\\Vert \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\nonumber \\\\ &\\leq \\dfrac{C^p}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{1-p} \\left\\Vert \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\nonumber \\\\ &\\leq \\dfrac{C^p \\vert \\Sigma_\\tau(k) \\vert^p}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{1-p} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\left\\Vert \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\nonumber \\\\ &= \\dfrac{C^p \\vert \\Sigma_\\tau(k) \\vert^p}{(k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{1-p} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\dfrac{\\omega_\\tau(\\sigma)}{\\omega_\\tau(\\sigma)}\\left\\Vert \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\nonumber \\\\ &\\leq \\dfrac{C^p \\vert \\Sigma_\\tau(k) \\vert^p}{D (k+1)! \\vert \\Gamma_\\tau \\vert} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{1-p} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma)\\left\\Vert \\phi_\\tau(\\sigma) \\right\\Vert_E^p \\nonumber \\\\ &\\leq \\dfrac{C^p \\vert \\Sigma_\\tau(k) \\vert^p}{D} \\vert \\Sigma_\\tau(k) \\vert^{1-p} D^{1-p} \\Vert \\phi_\\tau \\Vert^p_{(k,p)} \\nonumber \\\\ &= \\dfrac{(C\/D)^p}{\\vert \\Sigma_\\tau(k) \\vert} \\Vert \\phi_\\tau \\Vert^p_{(k,p)} \\leq (C\/D)^p \\Vert \\phi_\\tau \\Vert^p_{(k,p)} \\nonumber\n\\end{align}\nwhere\n\\begin{enumerate}\n\\item[] $C = \\max \\lbrace \\omega_\\tau(\\sigma) \\colon \\sigma \\in \\Sigma_\\tau(k) \\rbrace$ which exists as $\\Sigma_\\tau (k)$ contains only finitely many $k$-simplexes;\n\\item[] $D = \\min \\lbrace \\omega_\\tau(\\sigma) \\colon \\sigma \\in \\Sigma_\\tau(k) \\rbrace$.\n\\end{enumerate}\nHence, $M \\phi_\\tau \\in L^{(k,p)}(X_\\tau,E)$ and $M$ is well-defined and bounded. Clearly $M$ is linear and \n\\begin{align} M^2 \\phi_\\tau &= M \\phi_\\tau^0 \\nonumber \\\\ &= \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{-1} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\phi_\\tau^0(\\sigma) \\nonumber \\\\ &= \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\right)^{-1} \\sum_{\\sigma \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\sigma) \\left( \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\right)^{-1} \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\phi_\\tau(\\eta) \\nonumber \\\\ &= \\left( \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\right)^{-1} \\sum_{\\eta \\in \\Sigma_\\tau(k)} \\omega_\\tau(\\eta) \\phi_\\tau(\\eta) \\nonumber \\\\ &= M \\phi_\\tau, \\nonumber\n\\end{align} so $M$ is a continuous projection onto $ \\lbrace f : \\Sigma_\\tau(k) \\rightarrow E \\colon f = \\, \\mathrm{constant} \\rbrace \\subseteq L^{(k,p)}(X_\\tau,E)$. Similarly for $\\overline{M}$ \\end{proof}\n\n\n\\begin{proposition} \\label{averagenorms}\nLet $0 \\leq j < k \\leq n$, $\\tau \\in \\Sigma(j)$ and $\\phi \\in L^{(k,p^*)}(X,E^*)$. Then,\n\\begin{enumerate}\n\\item if $j < k-1$, then $\\delta_\\tau \\phi_\\tau = (-1)^{j+1}(\\delta \\phi)_\\tau$;\n\\item if $j= k-1$, then $(-1)^k(n-k+1) \\phi^0_\\tau = \\delta \\phi(\\tau)$ and\n$$ \\Vert \\phi^0_\\tau \\Vert^{p^*}_{(0,p^*)} = \\dfrac{\\omega(\\tau)}{(n-k+1)^{p^*-1}\\vert \\Gamma_\\tau \\vert} \\Vert \\delta \\phi(\\tau) \\Vert^{p^*}_{E^*}.$$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof} $(1)$ As $\\phi \\in L^{(k,p^*)}(X,E^*)$, it follows that $\\phi_\\tau \\in L^{(k-j-1,p^*)}(X_\\tau,E^*)$ where $k-j-1 > 0$ so $\\delta_\\tau \\phi_\\tau \\in L^{(k-j-2,p^*)}(X_\\tau, E^*)$ and by Proposition \\ref{codifferential}, \n\\begin{align}\n\\delta_\\tau \\phi_\\tau (\\sigma) &\\substack{(*) \\\\ =} \\sum_{\\substack{v \\in \\Sigma_\\tau(0) \\\\ v * \\sigma \\in \\Sigma_\\tau(k-j-1)}} \\dfrac{\\omega_\\tau(v * \\sigma)}{\\omega_\\tau(\\sigma)} \\phi_\\tau(v * \\sigma) \\nonumber \\\\ &= \\sum_{\\substack{v \\in \\Sigma_\\tau(0) \\\\ v * \\sigma \\in \\Sigma_\\tau(k-j-1)}} \\dfrac{{\\omega}(\\tau * v * \\sigma)}{{\\omega}(\\tau * \\sigma)} \\phi(\\tau * v * \\sigma) \\nonumber \\\\ &\\substack{(**) \\\\ =} \\sum_{\\substack{v \\in \\Sigma_\\tau(0) \\\\ v * \\sigma \\in \\Sigma_\\tau(k-j-1)}} (-1)^{j+1} \\dfrac{{\\omega}(v * \\tau * \\sigma)}{{\\omega}(\\tau * \\sigma)} \\phi(v * \\tau * \\sigma) \\nonumber \\\\ &= \\sum_{\\substack{v \\in \\Sigma(0) \\\\ v * \\tau * \\sigma \\in \\Sigma(k)}} (-1)^{j+1} \\dfrac{{\\omega}(v * \\tau * \\sigma)}{{\\omega}(\\tau * \\sigma)} \\phi(v * \\tau * \\sigma) \\nonumber \\\\ &= (-1)^{j+1} \\delta \\phi(\\tau * \\sigma) = (-1)^{j+1} \\left( \\delta \\phi \\right)_\\tau (\\sigma) \\nonumber\n\\end{align}\nwhere $(*)$ follows by Proposition \\ref{codifferential} and $(**)$ holds since $\\omega$ is symmetric, $\\phi$ alternating and $\\tau \\in \\Sigma(j)$.\nAs for $(2)$, by Proposition \\ref{codifferential} together with the fact that $\\omega$ is symmetric and $\\phi$ antisymmetric it follows for the codifferential that\n\\begin{align}\n\\delta\\phi(\\tau) &= \\dfrac{1}{{\\omega}(\\tau)} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\omega(\\sigma * \\tau) \\phi (\\sigma * \\tau) = \\dfrac{1}{{\\omega}(\\tau)} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} (-1)^{j+1}\\omega(\\tau * \\sigma) \\phi (\\tau * \\sigma) \\nonumber \\\\ &= \\dfrac{1}{{\\omega}(\\tau)} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} (-1)^{k}\\omega(\\tau * \\sigma) \\phi (\\tau * \\sigma), \\nonumber\n\\end{align} \nsince $j=k-1$. Therefore, in terms of $\\phi_\\tau^0 \\in L^{(0,p^*)}(X_\\tau,E^*)$\n\\begin{align}\n\\delta \\phi(\\tau) = \\dfrac{(-1)^k}{{\\omega}(\\tau)} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\omega_\\tau(\\sigma) \\phi_\\tau (\\sigma) = \\dfrac{(-1)^k}{{\\omega}(\\tau)} \\left( \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\omega_\\tau(\\sigma)\\right) \\phi_\\tau^0. \\nonumber\n\\end{align}\nHowever, \n\\begin{align}\n\\dfrac{\\displaystyle \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\omega_\\tau(\\sigma)}{\\omega(\\tau)} = \\dfrac{\\displaystyle \\sum_{\\tau * \\sigma \\in \\Sigma(j+1)} {\\omega}(\\tau * \\sigma)} {\\omega(\\tau)} = \\dfrac{(n-j)(j+1)!}{(j+1)!} = (n-j) = (n-k+1) \\nonumber\n\\end{align}\nby Proposition \\ref{combinatorial}. The factor $(j+1)!$ in the denominator corresponds to the fact that we sum over one ordering as $\\tau$ is fixed. Therefore, $$\\delta \\phi(\\tau) = (-1)^k(n-k+1)\\phi_\\tau^0,$$ and once again by Proposition \\ref{BS1.10} this gives\n\\begin{align}\n\\Vert \\phi^0_\\tau \\Vert^{p^*}_{(0,p^*)} &= \\dfrac{1}{\\vert \\Gamma_\\tau \\vert} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\Vert \\phi^0_\\tau(\\sigma)\\Vert^{p^*}_{E^*} \\omega_\\tau(\\sigma) = \\dfrac{1}{\\vert \\Gamma_\\tau \\vert} \\sum_{\\sigma \\in \\Sigma_\\tau(0)} \\dfrac{\\Vert \\delta \\phi(\\tau) \\Vert^{p^*}_{E^*}}{(n-k+1)^{p^*}} \\omega_\\tau(\\sigma) \\nonumber \\\\ &= \\dfrac{1}{\\vert \\Gamma_\\tau \\vert} \\dfrac{\\Vert \\delta \\phi(\\tau) \\Vert^{p^*}_{E^*}}{(n-k+1)^{p^*}} (n-k+1) \\omega(\\tau) = \\dfrac{\\omega(\\tau) \\Vert \\delta \\phi(\\tau) \\Vert^{p^*}_{E^*}}{(n-k+1)^{p^*-1}\\vert \\Gamma_\\tau \\vert}. \\nonumber\n\\end{align} \\end{proof}\n\n\\begin{proposition} \\label{try1}\nLet $\\phi \\in L^{(k,p)}(X,E)$ and $\\tau \\in \\Sigma(k-1)$. Then,\n\\begin{enumerate}\n\\item if $k=1$, $$ d_\\tau \\phi_\\tau (\\sigma) = -(d \\phi)_\\tau (\\sigma) + \\phi (\\sigma).$$\n\\item if $k > 1$, $$ d_\\tau \\phi_\\tau (\\sigma) = (d \\phi)_\\tau (\\sigma) + \\sum_{i=0}^{k-1}(-1)^i\\phi (\\tau_i * \\sigma).$$\n\\end{enumerate} \n\\end{proposition}\n\\begin{proof} Suppose $k=1$. Then, for $\\sigma = x * y \\in \\Sigma_\\tau(1)$ \n\\begin{align}\nd_\\tau \\phi_\\tau (\\sigma) &= \\phi_\\tau (y) -\\phi_\\tau (x) = \\phi(\\tau * y) - \\phi(\\tau * x). \\nonumber\n\\end{align} \nOn the other hand, as $[\\tau * x * y : x * y] =1$\n\\begin{align}\nd \\phi (\\tau * x * y) &= [\\tau * x * y : x * y] \\phi(x*y) + [\\tau * x * y: \\tau * y] \\phi (\\tau * y) + [\\tau * x * y: \\tau * x] \\phi(\\tau * x) \\nonumber \\\\ &= \\phi(x*y) - \\phi (\\tau * y) +\\phi(\\tau * x), \\nonumber\n\\end{align}\ngives together with the expression for $d_\\tau \\phi_\\tau$ \n$$ d_\\tau \\phi_\\tau (\\sigma) = -d \\phi (\\tau * x * y) + \\phi (x * y) = -(d \\phi)_\\tau (\\sigma) + \\phi (\\sigma).$$\nSuppose $k>1$ and $\\tau \\in \\Sigma(k-1)$. Then, as previously\n\\begin{align}\nd_\\tau \\phi_\\tau (\\sigma) = \\phi_\\tau (y) - \\phi_\\tau (x), \\nonumber\n\\end{align}\nand the two rightmost terms are as previously the last two terms in $d \\phi (\\tau * \\sigma)$. \n\\end{proof}\n\n\\begin{proposition} \\label{diffrelnew}\nLet $\\phi \\in L^{(k,p)}(X,E)$ and $\\tau \\in \\Sigma(0)$, then $$(d \\phi)_\\tau (\\sigma) = - d_\\tau \\phi_\\tau (\\sigma) + \\phi(\\sigma).$$\n\\end{proposition}\n\\begin{proof} Let $\\sigma \\in \\Sigma(k)$. Then, similarly as in Proposition \\ref{try1}\n$$d_\\tau \\phi_\\tau (\\sigma) = \\sum_{i=0}^{k}(-1)^i \\phi_\\tau (\\sigma_i) = \\sum_{i=0}^{k}(-1)^i \\phi(\\tau * \\sigma_i) = - (d \\phi)_\\tau(\\sigma) + \\phi(\\sigma).$$ \\end{proof}\n\n\n\n\n\n\n\n\n\\begin{corollary} \\label{help1}\nSuppose $\\phi \\in L^{(k,p)}(X,E)$, $\\tau \\in \\Sigma(0)$ and $k+1 \\leq n$. Then, if $\\phi \\in \\ker d$, $$ \\Vert d_\\tau \\phi_\\tau \\Vert_{(k,p)} = \\Vert \\phi^\\tau \\Vert_{(k,p)}.$$\n\\end{corollary}\n\\begin{proof} By Proposition \\ref{diffrelnew} it follows since $\\phi \\in \\ker d$ that\n\\begin{align}\n\\Vert d_\\tau \\phi_\\tau \\Vert_{(k,p)}^p &= \\sum_{\\sigma \\in \\Sigma_\\tau (k, \\Gamma_\\tau)} \\Vert d_\\tau \\phi_\\tau (\\sigma)\\Vert_E^p \\dfrac{\\omega_\\tau(\\sigma)}{(k+1)! \\vert \\Gamma_{\\tau \\sigma}\\vert} \\nonumber \\\\ &= \\sum_{\\sigma \\in \\Sigma_\\tau (k, \\Gamma_\\tau)} \\Vert \\phi (\\sigma)\\Vert_E^p \\dfrac{\\omega_\\tau(\\sigma)}{(k+1)! \\vert \\Gamma_{\\tau \\sigma}\\vert} \\nonumber \\\\ &= \\Vert \\phi^\\tau \\Vert_{(k,p)}^p. \\nonumber\n\\end{align} \\end{proof}\n\n\\begin{corollary} \\label{try2}\nSuppose $\\phi \\in L^{(k,p)}(X,E)$ and $1 < k+1 \\leq n$. If $\\phi \\in \\ker d$, then $$ \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert d_\\tau \\phi_\\tau \\Vert_{(k,p)}^p = \\dfrac{n-k}{k+1} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert \\phi_\\tau \\Vert_{(k-1,p)}^p.$$\n\\end{corollary}\n\\begin{proof} Follows by Corollary \\ref{help1} and Proposition \\ref{combinedsumming}. \\end{proof}\n\n\\begin{corollary} \\label{BSspecial}\nSuppose $\\phi \\in L^{(1,p)}(X,E)$. If $\\phi \\in \\ker d$, then\n$$- (n-1) \\Vert \\phi \\Vert_{(1,p)}^p = \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\left( \\Vert d_\\tau \\phi_\\tau \\Vert_{(1,p)}^p - (n-1) \\Vert \\phi_\\tau \\Vert^p_{(0,p)} \\right).$$\n\\end{corollary}\n\\begin{proof} By a direct computation,\n\\begin{align}\n&\\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert d_\\tau \\phi_\\tau \\Vert_{(1,p)}^p - (n-1) \\Vert \\phi_\\tau \\Vert_{(0,p)}^p \\substack{(*) \\\\ =} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\dfrac{(n-1)}{2} \\Vert \\phi_\\tau\\Vert_{(0,p)}^p - (n-1) \\Vert \\phi_\\tau\\Vert_{(0,p)}^p \\nonumber \\\\ &= -\\dfrac{(n-1)}{2} \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert \\phi_\\tau \\Vert_{(0,p)}^p \\substack{(**) \\\\ =} - \\dfrac{(n-1)}{2} \\dfrac{2!}{(1-0)!} \\Vert \\phi \\Vert_{(1,p)}^p = -(n-1)\\Vert \\phi \\Vert_{(1,p)}^p, \\nonumber\n\\end{align}\nwhere in $(*)$ we used Corollary \\ref{try2} and in $(**)$ Proposition \\ref{summing}.\n\\end{proof}\n\n\\begin{corollary} \\label{try3}\nSuppose $\\phi \\in L^{(1,p)}(X,E)$. If $\\phi \\in \\ker d$, then\n$$\\sum_{\\tau \\in \\Sigma(0, \\Gamma)} \\Vert d_\\tau \\phi_\\tau \\Vert_{(1,p)}^p = (n-1) \\Vert \\phi \\Vert_{(1,p)}^p.$$\n\\end{corollary}\n\\begin{proof} Follows directly from Corollary \\ref{BSspecial} using Proposition \\ref{summing} once more. \\end{proof}\n\n\\begin{definition} \\label{Q}\nLet $\\phi \\in L^{(1,p)}(X,E)$. For $\\tau \\in \\Sigma(0)$ define a $p$-form on $L^{(0,p)}(X_\\tau, E)$ by\n$$Q_\\tau (\\phi_\\tau) = \\Vert d_\\tau \\phi_\\tau \\Vert^p_{(1,p)} - \\dfrac{(n-1)}{2} \\Vert \\phi_\\tau \\Vert_{(0,p)}^p.$$ Similarly, we define a $p^*$-form on $L^{(0,p^*)}(X_\\tau, E^*)$.\n\\end{definition}\n\n\\begin{corollary} \\label{propQ}\nSuppose $\\phi \\in L^{(1,p)}(X,E)$. If $\\phi \\in \\ker d$ then\n$$ \\sum_{\\tau \\in \\Sigma(0, \\Gamma)} Q_\\tau (\\phi_\\tau) = 0.$$\n\\end{corollary}\n\\begin{proof} Follows immediately by Corollary \\ref{try3} and Proposition \\ref{summing}. \\end{proof}\n\n\n\n\\section{Poincar\\'e inequalities on finite weighted graphs} \\label{PI}\nIn this section we recall some basic facts concerning Poincar\\'e inequalities on finite weighted graphs necessary for the spectral method. For details we refer to \\cite{gn, Nowak, ny}.\n\\begin{proposition} \\cite{Nowak} \\label{poink}\nSuppose $\\dim X = 2$. Then the link $X_\\tau$ of every vertex of $X$ is a finite graph. Hence, for any $p \\geq 1$ the $p$-Poincar\\'e inequality \\begin{align} \\sum_{\\sigma \\in \\Sigma(0, \\Gamma_\\tau)} \\Vert f_\\tau(\\sigma) - Mf_\\tau(\\sigma)\\Vert_E^p &\\dfrac{{\\omega_\\tau}(\\sigma)}{\\vert \\Gamma_{\\tau \\sigma} \\vert} \\nonumber \\\\ &\\leq \\kappa_p^p \\sum_{\\eta \\in \\Sigma_\\tau(1, \\Gamma_\\tau)} \\dfrac{1}{2} \\Vert f_\\tau(\\eta_0) - f_\\tau(\\eta_1) \\Vert_E^p \\dfrac{\\omega_\\tau(\\eta)}{\\vert \\Gamma_{\\tau \\eta} \\vert} \\nonumber \\end{align} holds for some $\\kappa_p>0$ and all $f \\colon \\Sigma_\\tau(0) \\rightarrow E$. Similarly for $f \\colon \\Sigma_\\tau(0) \\rightarrow E^*$. \n\\end{proposition} \\qed\n\nThe infimum of the above constants $\\kappa_p$ is known as the Poincar\\'e constant of the link $X_\\tau$, and denoted by $\\kappa_p(X_\\tau, E)$. In terms of the norms introduced previously:\n\n\\begin{corollary}\nLet $X$ be two dimensional. Then, for all $f \\in L^{(1,p)}(X,E)$ it holds that $$\\Vert f_\\tau - Mf_\\tau \\Vert_{(0,p)} \\leq \\kappa_{p}(X_\\tau,E) \\Vert {d}_\\tau f_\\tau \\Vert_{(1,p)},$$ for some $\\kappa_{p}(X_\\tau,E)$. Similarly for $f \\in L^{(1,p^*)}(X,E^*)$, $$\\Vert f_\\tau - \\overline{M}f_\\tau \\Vert_{(0,p^*)} \\leq \\kappa_{p^*}(X_\\tau,E^*) \\Vert \\overline{d}_\\tau f_\\tau \\Vert_{(1,p^*)},$$ for some $\\kappa_{p^*}(X_\\tau,E^*)$.\n\\end{corollary} \\qed\n\n\n\n\n\nSometimes it is useful to know how Poincar\\'e constants change under isomorphisms. The following is immediate:\n\\begin{proposition} \\label{BI}\nLet $T: E \\rightarrow F$ be a Banach space isomorphism. If $$ \\Vert x \\Vert_E \\leq \\Vert T(x) \\Vert_F \\leq C \\Vert x \\Vert_E,$$ then $\\kappa_p(X_\\tau, E) \\leq C \\kappa_p(X_\\tau, F)$.\n\\end{proposition} \\qed\n\nFor $1 < p< \\infty$ we denote by ${L}^p$ the Banach space ${L}^p(\\mu)$ of $p$-integrable functions on a standard Borel space $(Y, \\mathcal{B})$ with $\\sigma$-finite measure $\\mu$. As such, any separable infinite-dimensional Hilbert space $\\mathcal{H}$ is isometrically isomorphic to ${L}^2$. In particular, we have the following relation between the Poincar\\'e constant and spectral gap:\n\n\\begin{proposition} \\label{hilb1}\n\\cite{Nowak} Let $\\lambda_1$ be the smallest positive eigenvalue of the graph Laplacian $\\triangle_+ = (\\delta d)_\\tau$, defined by $$\\triangle_+f(v) = f(v) - \\dfrac{1}{\\omega_\\tau(v)} \\sum_{u \\in L_v}f(u),$$ where $L_v$ denotes the link of $v$ in $X_\\tau$, over the space $C^{(0,p)}(X_\\tau, \\mathbb{R})$ of real-valued functions on the vertices. For $L^2$ when $X$ is $2$-dimensional, $\\kappa_p(X_\\tau, L^2) = \\lambda_1^{-1\/2}$ and more generally $\\kappa_p(X_\\tau, L^p) = \\kappa_p(X_\\tau, \\mathbb{R})$.\n\\qed \\end{proposition} \n\n\n\n\n\n\\section{$L^p\\,$-cohomology and vanishing for uniformly bounded representations} \\label{LH}\nFollowing \\cite{BS} we introduce $L^p$-cohomology of $X$ with coefficients in $\\pi$ as a natural extension of the $L^2$-cohomology for unitary representations. In particular, if $\\pi$ is a unitary representation $L^2 H^k(X, \\pi)$, as described below, is the cohomology of the complex of $\\textrm{mod}$ $\\Gamma$ square integrable cochains of $X$ twisted by $\\pi$. The connection to property $(T)$ is as follows: if $X$ is a two dimensional contractible simplicial complex and $\\Gamma$ acts properly discontinuously and cocompactly by automorphisms on it, then $\\Gamma$ has property $(T)$ if and only if $L^2 H^1(X, \\pi) = 0$ for any unitary representation \\cite{hv}. As an application we derive a spectral condition for cohomological vanishing for square integrable cochains on a two dimensional simplicial complex twisted by a uniformly bounded representation. \n\\begin{definition}\nLet $$L^p H^k(X,\\pi) = \\ker \\left( d \\vert_{L^{(k,p)}(X,E)} \\right) \/ \\mathrm{im} \\left( d \\vert_{L^{(k-1,p)}(X,E)} \\right)$$ denote the $L^p$-cohomology groups of $X$ with coefficients twisted by $\\pi$. \n\\end{definition} \nAs the following shows, cohomological vanishing takes place when $\\delta$ is bounded from below:\n\\begin{proposition} \\label{VCH1}\nThe map $$d_{k-1} \\vert_{L^{(k-1,p)}(X,E)} : L^{(k-1,p)}(X,E) \\longrightarrow \\ker \\, d_k \\vert_{L^{(k,p)}(X,E)}$$ is onto if its adjoint $$\\delta_{k} : (\\ker \\, d_{k} \\vert_{L^{(k,p)}(X,E)})^* \\rightarrow L^{(k-1,p^*)}(X,E^*),$$ is bounded from below, that is $\\exists \\, K > 0$ such that for all $f \\in \\left( \\ker {d_k} \\vert_{L^{(k,p)}(X,E)} \\right)^*$\n$$\\Vert \\delta f \\Vert_{(k-1,p^*)} \\ge K \\Vert f \\Vert_{(k,p^*)}.$$ If in addition $d_{k-1}$ is injective, $d_{k-1}$ is onto if and only if $\\delta_k$ is bounded from below.\n\\end{proposition}\n\\begin{proof} Since $d_{k} \\circ d_{k-1} \\vert_{L^{(k-1,p)}(X,E)} = 0$, $\\mathrm{im} \\, d_{k-1} \\subseteq \\ker d_k \\vert_{L^{(k,p)}(X,E)}$ without further assumptions. Now, assume $\\delta_k$ is bounded from below. Then $\\delta_k$ is injective; towards a contradiction, suppose $f,g \\in (\\ker \\, d_{k} \\vert_{L^{(k,p)}(X,E)})^*$ such that $f \\neq g$ and $\\delta_k \\, f = \\delta_k \\, g$. Recalling that $\\Vert \\cdot \\Vert_{(k,p^*)}$ is a norm restricted to $L^{(k,p^*)}(X,E)$ leads to a contradiction\n\\begin{align}\n0 = \\Vert \\delta_k f - \\delta_k g \\Vert_{(k-1,p^*)} = \\Vert \\delta_k (f-g)\\Vert_{(k-1,p^*)} \\geq K \\Vert f - g\\Vert_{(k,p^*)} > 0. \\nonumber\n\\end{align}\nThus, $\\delta$ is injective. In particular, $\\ker \\, \\delta_k = \\lbrace 0 \\rbrace$ and since $L^{(k,p)}(X,E)$ is reflexive and $\\ker \\, d_k\\vert_{L^{(k,p)}(X,E)}$ is closed, the latter is also reflexive and \\begin{align} \\mathrm{im} \\, d_{k-1} &= \\mathrm{Ann}(\\ker \\, \\delta_k) \\nonumber \\\\ &= \\lbrace f \\in {(\\ker \\, d_{k} \\vert_{L^{(k,p)}(X,E)})^*}^* \\colon \\langle g,f \\rangle_{k} = 0, \\, \\forall g \\in \\ker \\, \\delta_k \\rbrace \\nonumber \\\\ &\\cong \\lbrace f \\in \\ker \\, d_{k} \\vert_{L^{(k,p)}(X,E)} \\colon \\langle g,f \\rangle_{k} = 0, \\, \\forall g \\in \\lbrace 0 \\rbrace \\rbrace \\nonumber \\\\ &= \\ker \\, d_{k} \\vert_{L^{(k,p)}(X,E)}, \\nonumber \\end{align} so $d_{k-1}$ is onto $\\ker d_k \\vert_{L^{(k,p)}(X,E)}$. Next, suppose $d_{k-1}$ is onto. Since $d_{k-1}$ is bounded, it is bounded from below by the open mapping theorem if $d_{k-1}$ is injective. \\end{proof}\nThis criteria is in fact related to the Poincar\\'e constants of the links as Proposition \\ref{bedlewoinequalityNOT} shows. This allows us to formulate a spectral condition for cohomological vanishing.\n\\begin{proposition} \\label{FIX1}\nSuppose $X$ is a $2$-dimensional locally finite simplicial complex such that for any vertex $\\tau$ of $X$ the link $X_\\tau$ is connected and $\\mathcal{H}$ a separable infinite-dimensional Hilbert space. Suppose there exists a constant $C$ such that \n\\begin{enumerate}\n\\item[i.] the map $C \\mathcal{I}: E \\rightarrow \\mathcal{H}$ where $\\mathcal{I}: E \\rightarrow \\mathcal{H}$ is the identity map, is a Banach space isomorphism with the property $\\Vert x \\Vert_E \\leq \\Vert C\\mathcal{I}(x) \\Vert_\\mathcal{H} \\leq C \\Vert x \\Vert_E$ for all $x \\in E$. Then, for $f \\in L^{(1,2)}(X,E) \\cap \\ker {d}$ $$\\kappa_2(X_\\tau, \\mathcal{H})^{-2} \\Vert Mf_\\tau\\Vert^2_{(0,2)} + Q_\\tau(f_\\tau) \\geq \\dfrac{1}{C^2}\\left( \\kappa_2(X_\\tau, \\mathcal{H})^{-2} - \\dfrac{C^2}{2}\\right) \\Vert f_\\tau \\Vert^2_{(0,2)};$$ \n\\item[ii.] the identity map $\\overline{\\mathcal{I}}: E^* \\rightarrow \\mathcal{H}$ is a Banach space isomorphism with the property $\\Vert x \\Vert_{E^*} \\leq \\Vert \\overline{\\mathcal{I}}(x) \\Vert_{\\mathcal{H}} \\leq C \\Vert x \\Vert_{E^*}$ for all $x \\in {E^*}$. Then, for $f \\in L^{(1,2)}(X,E^*) \\cap \\ker \\overline{d}$ $$\\kappa_2(X_\\tau, \\mathcal{H})^{-2} \\Vert \\overline{M}f_\\tau\\Vert^2_{(0,2)} + Q_\\tau(f_\\tau) \\geq \\dfrac{1}{C^2}\\left( \\kappa_2(X_\\tau, \\mathcal{H})^{-2} - \\dfrac{C^2}{2}\\right) \\Vert f_\\tau \\Vert^2_{(0,2)}.$$\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n$(i)$. As usual, write $\\Vert \\cdot \\Vert_{(k,2)}$ for the seminorm on $\\mathcal{E}^{(k,2)}(X,E)$ and write $\\Vert \\cdot \\Vert_{(k,2), \\mathcal{H}}$ for the seminorm on $\\mathcal{E}^{(k,2)}(X,\\mathcal{H})$. For $f \\in L^{(1,2)}(X,E) \\cap \\ker {d}$\n\\begin{align}\nQ_\\tau(f_\\tau) = \\Vert d_\\tau f_\\tau \\Vert_{(1,2)}^2 - \\dfrac{1}{2} \\Vert f_\\tau \\Vert^2_{(0,2)} \\geq \\Vert d_\\tau f_\\tau \\Vert_{(1,2), \\mathcal{H}}^2 - \\dfrac{C^2}{2} \\Vert f_\\tau \\Vert^2_{(0,2), \\mathcal{H}} \\nonumber\n\\end{align} \nOn the other hand, by the Poincar\\'e inequality and the Pythagorean identity $$\\Vert d_\\tau f_\\tau \\Vert_{(1,2), \\mathcal{H}}^2 \\geq \\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert f_\\tau\\Vert^2_{(0,2), \\mathcal{H}} - \\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert Mf_\\tau\\Vert^2_{(0,2), \\mathcal{H}}$$ Hence,\n\\begin{align}\nQ_\\tau(f_\\tau) \\geq \\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert f_\\tau\\Vert^2_{(0,2), \\mathcal{H}} - \\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert Mf_\\tau\\Vert^2_{(0,2), \\mathcal{H}} - \\dfrac{C^2}{2} \\Vert f_\\tau \\Vert^2_{(0,2), \\mathcal{H}}, \\nonumber \n\\end{align} and in terms of the $\\Vert \\cdot \\Vert_{(k,p)}$ norm \n\\begin{align}\n\\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert Mf_\\tau\\Vert^2_{(0,2)} + Q_\\tau(f_\\tau) \\geq \\dfrac{1}{C^2}\\left( \\kappa_2(X_\\tau, \\mathcal{H})^{-2} - \\dfrac{C^2}{2} \\right) \\Vert f_\\tau \\Vert^2_{(0,2)}. \\nonumber\n\\end{align} Similarly for $(ii)$.\n\n\\end{proof}\n\n\\begin{corollary} \\label{bedlewoinequalityNOT}\nAssuming Proposition \\ref{FIX1} holds such that $\\kappa_2(X_\\tau,\\mathcal{H}) \\leq \\kappa_2(X,\\mathcal{H})$ for every link $X_\\tau$ of $X$. Then, for $$\\kappa_2(X,\\mathcal{H}) < \\sqrt{2}C^{-1},$$ $\\delta$ and $\\overline{\\delta}$ are bounded from below.\n\\end{corollary}\n\\begin{proof}\nBy Proposition \\ref{FIX1}(i)\n\\begin{align}\n\\kappa_2(X_\\tau, \\mathcal{H})^{-2}\\Vert Mf_\\tau\\Vert^2_{(0,2)} + Q_\\tau(f_\\tau) \\geq \\dfrac{1}{C^2}\\left( \\kappa_2(X_\\tau, \\mathcal{H})^{-2} - \\dfrac{C^2}{2} \\right) \\Vert f_\\tau \\Vert^2_{(0,2)}. \\nonumber\n\\end{align} Thus, summing over the representatives $\\tau \\in \\Sigma(0, \\Gamma)$ gives, applying Propositions \\ref{summing}, \\ref{averagenorms}, and \\ref{propQ} to the three terms respectively, that\n\\begin{align}\n\\Vert \\delta f\\Vert^2_{(0,2)} \\geq \\left( \\dfrac{2 \\kappa_2(X,\\mathcal{H})}{C}\\right)^2 \\left( \\kappa_2(X_\\tau, \\mathcal{H})^{-2} - \\dfrac{C^2}{2} \\right) \\Vert f \\Vert_{(1,2)}^2. \\nonumber\n\\end{align} So, $\\delta$ is bounded from below for $\\kappa_2(X,\\mathcal{H}) < {\\sqrt{2}}{C^{-1}}$. Similarly for $\\overline{\\delta}$.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{theorem} \\label{theorem}\nLet $X$ be a locally finite $2$-dimensional simplicial complex, $\\Gamma$ a discrete properly discontinuous group of automorphisms of $X$ and $\\pi : \\Gamma \\rightarrow \\mathrm{B}(\\mathcal{H})$ a uniformly bounded representation of $\\Gamma$ on a separable infinite-dimensional Hilbert space $\\mathcal{H}$. Suppose the link $X_\\tau$ of every vertex $\\tau$ of $X$ is connected and the associated Poincar\\'e constants satisfy $$C < \\dfrac{\\sqrt{2}}{\\kappa_2(X_\\tau, \\mathcal{H})}$$\nfor $C = \\sup_{g \\in \\Gamma} \\Vert \\pi_g \\Vert$. Then, $L^2H^1(X,\\pi) = 0$.\n\\end{theorem}\n\n\\begin{proof}\nLet $E$ be the Banach space $(\\mathcal{H}, \\Vert \\cdot \\Vert_E)$ where $\\Vert \\cdot\\Vert_E = \\sup_{g \\in \\Gamma}\\Vert \\pi_g(\\cdot)\\Vert_\\mathcal{H}$. Now, $\\pi$ is an isometric representation on $E$ and have the dual diagrams: \\\\\n\\xymatrix{ \n{\\ker \\, \\overline{d}\\,} \\ar@{^{(}->}[r]^{\\overline{i}} \\ar@\/^3pc\/[d]^{\\delta \\circ i^* \\circ \\overline{i}} & {L^{(1,2)}(X,E^*)} \\ar@{->>}[d]^{i^*} \\\\\n{L^{(0,2)}(X,E^*)} \\ar[u]^{\\overline{d}} & {\\left( \\ker \\, d \\right)^*} \\ar[l]^{\\delta} \\\\ {L^{(0,2)}(X,E)} \\ar[r]^{d} & {\\ker d} \\ar@{^{(}->}[d]^{i} \\ar@\/^2pc\/[l]^{\\overline{d}^* \\circ \\overline{i}^* \\circ i} \\\\ {\\left( \\ker \\, \\overline{d} \\right)^*} \\ar[u]^{\\overline{d}^*} & {L^{(1,2)}(X,E)} \\ar@{->>}[l]^{\\overline{i}^*} \n}\n\\\\ We claim that $L^2H^1(X,E) = 0$, that is $d_0$ is onto $\\ker d_1$. By Proposition \\ref{VCH1} it is enough to prove that $d^*_0 = \\delta_1$ is bounded from below on $(\\ker {d}_1)^*$. Since Proposition \\ref{FIX1} holds for $C = \\sup_{g \\in \\Gamma} \\Vert \\pi_g \\Vert$, and $ \\kappa_2(X_\\tau, \\mathcal{H}) < \\sqrt{2} C^{-1}$, it follows by Corollary \\ref{bedlewoinequalityNOT} that $\\delta_1$ is bounded from below when restricted to $\\ker \\overline{d}$. Hence, \n$\\delta_1$ is bounded from below on the image of $i^* \\circ \\overline{i}$. Thus, if $i^* \\circ \\overline{i}$ is onto $(\\ker d_1)^*$, then $\\delta_1$ is bounded from below on $(\\ker d_1)^*$ and ${\\delta_1 \\,}^* = d_0$ is onto, by which the claim follows. By a similar argument ${\\overline{d^*_0}}$ restricted to $\\ker d_1$ is bounded from below, and thus $${\\overline{d}_0\\,}^* \\circ {\\overline{i}\\,}^* \\circ i \\colon \\ker d_1 \\rightarrow L^{(0,p)}(X,E)$$ is bounded from below. In particular, ${\\overline{i}\\,}^* \\circ i$ is bounded from below and hence $({\\overline{i}\\,}^* \\circ i)^* = i^* \\circ \\overline{i}$ is onto $(\\ker d_1)^*$. \\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Algorithm}\n\\label{sec:algo}\n\n\\begin{table}[t]\n\t\\tabcaption{Result of running example query shown in Fig.~\\ref{fig:query}}\n\t\\label{tab:result}\n\t\\resizebox{\\columnwidth}{!}{\n\t\n\t\t\\begin{tabular}{llllll}\n\t\t\t\\toprule\n\t\t\t\\bf Result & \\bf ProvPoly & \\bf PosBool & \\bf dervE & \\bf symE & \\bf Probability \\\\\n\t\t\t\\midrule\n\t\t\t(DEL,BAR) & $e_{3}e_{4} $ & $e_{3} \\land e_{4} $ & $e_{3} \\otimes e_{4} $ & $e_{3}e_{4} $& $0.480$ \\\\\n\t\t\t(DEL,JKF) & $e_{3}e_{5} $ & $e_{3} \\land e_{5} $ & $e_{3} \\otimes e_{5} $ & $e_{3}e_{5} $& $0.360$ \\\\ \n\t\t\t\\multirow{2}{*}{(SIN,MUN)} & $e_{1}e_{3} +$ & $(e_{1} \\land e_{3}) \\lor$ & $(e_{1} \\otimes e_{3}) \\oplus$ & $e_{1}e_{3} + e_{2}e_{3}$ & \\multirow{2}{*}{$0.564$} \\\\\n\t\t\t& $e_{2}e_{3}$ & $(e_{2}\\land e_{3})$ & $(e_{2}\\otimes e_{3})$ & $- e_{1}e_{2}e_{3}$ & \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\\end{table}\n\nWe extend our existing framework HUKA\\xspace~ \\footnote{\\url{https:\/\/github.com\/gaurgarima\/HUKA}} \\cite{huka} that maintains query\n\tresults along with their provenance for deterministic KG to probabilistic\n\tknowledge graphs. HUKA\\xspace supports provenance-aware query computation and\n\tresult maintenance over deterministic dynamic KGs. It captures the\n\t\\emph{how} provenance of query answers using the provenance semiring model.\n\tOur framework, \\textsc{HaPPI}\\xspace, on the other hand, employs the novel semiring\n\tintroduced in Sec.~\\ref{sec:framework} for symbolically computing the\n\tanswer probabilities. We next discuss how to construct these symbolic\n\tprobability expressions and, further, when a KG edge changes, how to\n\tmaintain the expressions.\n\n\n\\subsection{Construction of Probability Expressions}\n\\label{subsec:symConstruction}\n\nBy the virtue of the semiring framework \\cite{provenancesemirings}, the symbolic expression construction is piggybacked up the answer computation. Here, we discuss the construction of these expressions conceptually. \n\nSuppose we are given a graph query $Q$ of size $s$, i.e., there are $s$ triple patterns\nin the query. We want to compute the query over a probabilistic KG $G$. The\nresult set $R$ of query $Q$ is given as,\n\\begin{align*}\n\t%\n\tR = \\{\\langle a_{1}, \\mathcal{D}_{a_{1}}\\rangle, \\dots, \\langle a_{j},\n\t\\mathcal{D}_{a_{j}}\\rangle \\}\n\t%\n\\end{align*}\nwhere, $\\mathcal{D}_{a_{i}} = \\{D_1 \\dots D_l\\}$ is the collection of all\nderivations of an answer $a_{i}$. Each derivation $D_{i}: \\langle\ne_{p},\\dots,e_{q}\\rangle$ encodes a subgraph, involving edges $e_{p}, \\dots,\ne_{q}$ that result in the corresponding answer. We will use\n${e_{\\bullet}}$'s both for the edges and the corresponding indeterminates in\nour polynomial rings.\n\nSimilar to constructing a Boolean formula using the PosBool semiring, we first\nconstruct \\emph{derivation} expressions using our flat polynomial semiring.\nFrom the mapping established in Sec.~\\ref{sec:framework}, the conjuncts of a\nBoolean formula corresponds to the $\\otimes$ terms. Analogous to the\ndisjunction-of-conjunction form of Boolean formula, the derivation expression\nis addition ($\\oplus$) of multiplicative ($\\otimes$) terms, i.e., $\\oplus_{D\n\\in \\mathcal{D}_{a_{i}}} ( \\otimes_{e_{j} \\in D} e_{j} )$ corresponds to\n$\\vee_{D \\in \\mathcal{D}_{a_{i}}} ( \\wedge_{e_{j} \\in D} e_{j} )$. For each\nderivation $D_{k}$ of answer $a_{i}$, we multiply ($\\otimes$) edges involved in\nit, and iteratively add ($\\oplus$) expressions of all the derivations of that\nanswer.\n\nFor instance, the answer (SIN,MUN), in Table~\\ref{tab:result}, has two\nderivations; thus, it has $2$ conjuncts and $2$ multiplicative terms in Boolean\nformula (\\textsf{PosBool}) and derivation expression (\\textsf{dervE})\nrespectively. \nThe PosBool formula and\nderivation expression of all answers of Fig.~\\ref{fig:query} is given in\nTable~\\ref{tab:result}. Later, to compute the exact probability, existing\nsystems pass on the Boolean formula to compilation and counting tools, whereas\n\\textsc{HaPPI}\\xspace unfolds the derivation expression to get the equivalent probability\nsymbolic expression and evaluate it.\n\nWe handle one derivation at a time and incrementally construct the symbolic\nexpression. We iterate over all the derivations of an answer. At iteration\n$i$, we flatten out the polynomial constructed by multiplying edges involved in\nderivation $D_{i}$. Then we incrementally add ($\\oplus$) this resultant flat\npolynomial of $D_{i}$ to the symbolic expression computed at iteration $(i-1)$\nto get the new updated symbolic expression. After exhausting all the\nderivations, we get the final symbolic expression of the answer. To get the\nconcrete probability, we evaluate this expression by assigning\n$Pr(e_{\\bullet})$ to each variable $e_{\\bullet}$. For our example, using the\nedge probabilities shown in Fig.~\\ref{fig:kg}, the symbolic expressions\n\\textsf{symE} and the resulting probabilities are shown in\nFig.~\\ref{tab:result}. Here, only the probability of pair (SIN,MUN) is above\nthe query threshold $\\left(\\geq 0.5\\right)$.\n\nThe translation from \\textsf{dervE} to\n\\textsf{symE} expressions involves a complete simplification of the $\\otimes$\nand $\\oplus$ operators using their definitions. This is done \\emph{before}\nsubstituting concrete values in them. To see why this matters, consider the\nderivation expression $(e_{1} \\otimes e_{2}) \\oplus (e_{1} \\otimes e_{3})$ and\nlet each edge have probability $0.5$. If we \\emph{first} assign values\n(probabilities) to the indeterminates and then simplify the expression we get\n$0.25 + 0.25 - 0.0625 = 0.4375$ as the concrete probability. However, the\ncorrect value is $0.375$.\n\nOur framework can also handle self-joins. In case of graph queries, the\nsituation arises when an edge satisfies more than one triple pattern of a\nquery. The multiplication operator ($\\otimes$) ensures that the probability of\nan edge is considered only once irrespective of the number of triple patterns\nit satisfies in a derivation. \n\n\\subsection{Maintainability of \\textsc{HaPPI}\\xspace}\n\\label{subsec:maintain}\n\nBy virtue of \\emph{incremental} symbolic expression construction procedure,\n\\textsc{HaPPI}\\xspace is capable of maintaining query answers under edge update operations by using\nthe following \\emph{inverted} indexes:\n\n\\begin{itemize}\n\n\t\\item An inverted index, \\textsf{edgeToSymE}, that maps an edge to the\n\t\tcollection of \\emph{symbolic} expressions in which it participates.\n\t\n\t\n\n\t\\item An inverse map, \\textsf{symEval}, that associates a \\emph{symbolic}\n\t\texpression to its current evaluation. This map essentially holds the\n\t\tcurrent probability of all the answers.\n\n\\end{itemize}\n\nWe consider the following update operations: addition of an edge, deletion of\nan edge, change of probability value of an edge.\n\n\\subsubsection{Addition of an Edge}\n\\label{subsubsec:edgeAddition}\n\n\\begin{table}[t]\n\\tabcaption{Updated query result after insertion of edge $e_{6}$.}\n\\label{tab:updatedResult}\n\\centering\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{lll}\n\t\\toprule\n\t\\bf Result & $\\mathbf{symE}$ & \\bf Probability \\\\\n\t\\midrule\n\t(DEL,BAR) & $e_{3}e_{4} + e_{6}e_{4} - e_{3}e_{4}e_{6}$ & $0.5440$ \\\\ \n\t(DEL,JFK) & $e_{3}e_{5} + e_{6}e_{5} - e_{3}e_{5}e_{6}$ & $0.4080$ \\\\\n\t(SIN,MUN) & $(e_{1} e_{3} + e_{2}e_{3} - e_{1}e_{2}e_{3}) \\oplus (e_{1} \\otimes e_{6}) \\oplus (e_{2} \\otimes e_{6}) $ & $0.6395$ \\\\\n\t\\bottomrule\n\t\\end{tabular}\n\t}\n\t\\vspace{-6mm}\n\\end{table}\n\nA newly added edge may result in generation of either new answers or more\nderivations of an existing answer. A new \\emph{answer} generation is a simple\ncase of symbolic expression construction mentioned in\nSec.~\\ref{subsec:symConstruction}. However, the accommodation of new\n\\emph{derivations} involves updating symbolic expressions and, thus, the\nresulting probabilities of \\emph{affected} answers. We use the existing system\nHUKA\\xspace \\cite{huka} to efficiently identify and supply new derivations to our\n\\textsc{HaPPI}\\xspace framework. Here, we focus on updating the affected symbolic expressions.\n\n Suppose, we add an edge $e_{6}(DEL,MUN,A_{1},0.2)$ to\nthe example KG shown in Fig.~\\ref{fig:kg}. It would result in $2$ new\nderivations, $D_{1}: \\langle e_{1}, e_{6} \\rangle$ and $D_{2}: \\langle e_{2},\ne_{6} \\rangle$ of answer \\textit{(SIN,MUN)}. \\textsc{HaPPI}\\xspace \\emph{incrementally} updates the existing symbolic \nexpression of an \\emph{affected} answer by absorbing one\nderivation at a time. With existing expression $e_{1}e_{3} + e_{2}e_{3}-e_{1}e_{2}e_{3}$, the updated expression would be $(e_{1}e_{3} + e_{2}e_{3}-e_{1}e_{2}e_{3}) \\oplus e_{1}e_{6} \\oplus e_{2}e_{6}$. The updated expressions and the probabilities of all the answers is reported in\nTable~\\ref{tab:updatedResult}.\n\n\nOften queries specify a probability threshold to find answers with\nprobabilities above a certain threshold. In our example query, suppose the user\nwants flight routes with overall probability greater than $0.5$. Thus, instead\nof computing the exact probability after each edge insertion, we require a\nmechanism to quickly filter out answers that cannot pass the threshold, after a\nnew derivation. To this end, we devise a simple method based on the following observation. \n\n\\begin{observation}\n\t%\n\t\\label{obs:optimize}\n\t%\n\tAn \\emph{upper bound} on the probability computed by adding two symbolic\n\texpression $symE_{i}$ and $symE_{j}$ is\n\t%\n\t\n\t\\begin{align*}\n\t\tPr &\\left(symE_{i} \\oplus symE_{j} \\right) \\\\\n\t\t&\\leq Pr(symE_{i}) + Pr(symE_{j}) - Pr(symE_{i}) \\times Pr(symE_{j})\n\t\\end{align*}\n\t}\n\t%\n\\end{observation}\n\n\nThis observation gives an upper bound on the new probability value of an\nanswer. This upper bound that entirely ignores the dependence among the\ndifferent derivations of an answer, referred to as propagation score~\\cite{dissociation}, is often used for approximating the exact probability of answers. The\nupper bound computation is quite efficient as it involves straightforward\narithmetic computations.\n\nThus, for each \\emph{affected} answer, \\textsc{HaPPI}\\xspace follows a filter-and-refine\nmechanism: it first computes the upper bound, and then decides if the updated\nsymbolic expression computation is needed at that point in time. For instance,\nafter the insertion of edge $e_{6}$ (probability $0.2$), the upper bound of\nanswer (DEL,BAR) and (DEL,JFK) are $0.563$ and $0.437$ respectively. Thus, we\ncompute the exact probability of \\emph{only} (DEL,BAR) as it passes the query\nthreshold $0.5$.\n\nTo ensure correct operation of \\textsc{HaPPI}\\xspace to handle future edge update requests, we\ncannot altogether avoid the updated symbolic expression computation. We\ndefer it to improve the response time, i.e., the time taken to report the\nupdated answers of \\emph{affected} queries. \n\n\\subsubsection{Edge Probability Update}\n\\label{subsubsec:edgeProbUpdate}\n\nAn edge probability update operation affects \\emph{only} the answer\nprobabilities and not their symbolic expressions (assuming the new probability\nto be non-zero). Thus, instead of re-evaluating the symbolic expression of an\naffected answer from scratch, \\textsc{HaPPI}\\xspace computes only the offsets corresponding to\nthe new probability.\nSuppose, the probability of edge $e_{i}$ is updated from $Pr(e_{i})$ to\n$Pr(e_{i})'$. First, \\textsc{HaPPI}\\xspace fetches entry of $e_{i}$ in \\textsf{edgeToSymE} to\nget all the symbolic expressions in which $e_{i}$ participated. Then, for each\nsuch symbolic expression $symE_{j}$, it computes the offset. The \\emph{offset}\nis calculated by re-evaluating the monomials of the symbolic expression in\nwhich $e_{i}$ appeared. For instance, if the probability of $e_{2}$ is updated\nto $0.6$, then symbolic expression of (SIN,MUN) would be fetched from\n\\textsf{edgeToSymE}. The symbolic expression has $2$ relevant monomials,\n$e_{2}e_{3}$ and $e_{1}e_{2}e_{3}$, as shown in Table~\\ref{tab:result}. The\n\\emph{offset} is calculated as sum of valuations of all the relevant monomials.\nThese monomials are evaluated by assigning corresponding edge probabilities to\nall involved edges except $e_{i}$. The updated edge $e_{i}$ variable is\nsubstituted by value $Pr(e_{i})-Pr(e_{i})'$. Here, the relevant monomials\n$e_{2}e_{3}$ and $e_{1}e_{2}e_{3}$ are evaluated to $0.06$ and $0.048$\nrespectively, with $e_{1}=0.8$, $e_{2} = 0.1$ ($Pr(e_{2})-Pr(e_{2})' = 0.1$)\nand $e_{3}=0.6$. Thus, the offset is $0.108$ ($= 0.06+0.048$). Finally, the\nupdated probability value the answer is given as \\textit{newP} = \\textit{oldP}\n-- \\textit{offset}, where \\textit{oldP} is the probability of the answer before\nthe update.\n\n\n\\subsubsection{Deletion of an Edge}\n\\label{subsubsec:edgeDeletion}\n\nWhen an edge gets deleted, the derivations in which it participated becomes\n\\emph{invalid} as they cannot generate the corresponding answer anymore. There\nare two ways of computing the updated symbolic expression of $a_{i}$. Suppose\n$k$ out of $d$ derivations of an answer $a_{i}$ become \\emph{invalid}. The\nfirst one involves iterating over the monomials of the current symbolic\nexpression and dropping off the monomials containing the deleted edge. The\nother method involves removing the $k$ invalid derivations from the current\nderivation list and recomputing the symbolic expression with this updated list\nof derivations. As we will see in Sec.~\\ref{subsec:analysis}, the number of\nmonomials in a symbolic expression constructed from $d$ derivations can be\nexponential in $d$. When that bound is attained, the time involved in\nmanipulating an expression is as much, or more, than that for recomputing it\nfrom the scratch, i.e., $O(2^{d-k})$. Therefore, we adopt the latter strategy.\n\n\\subsection{Analysis}\n\\label{subsec:analysis}\n\n\\begin{table}[t]\n\t\\tabcaption{Parameters of a Query}\n\\centering\n\t\\resizebox{\\columnwidth}{!}{%\n\t\\begin{tabular}{ll}\n\t\t\\toprule\n\t\t\\bf Parameter & \\bf Description \\\\ \n\t\t\\midrule\n\t\t$s$ & Size of query, i.e., number of triple patterns \\\\\n\t\t$d$\t& Number of derivations of an answer \\\\\n\t\t$n$ & Total number of edges across all $d$ derivations of an answer \\\\\n\t\t$m$ & Number of monomials in a symbolic expression (flat polynomial) \\\\ \\bottomrule\n\t\\end{tabular}\n\t}\n\\label{tab:symbols}\n\\end{table}\n\nWe start by introducing the different parameters that characterize the queries and\ntheir answers. We refer to the number of triple patterns in a query as\n\\emph{query size}, $s$.\n\\comment{Our running query is of size $s=2$.}\nAn answer\n$a_{k}$ of an query can have $d$ derivation terms, each denoted as $D_{i}$, $1 \\leq\ni \\leq d$.\nFurther, let the number of edges involved across all $d$ derivations be\n$n$. \\comment{Here, in our example, answer \\textit{(SIN,MUN)} has $2$ derivations which\ninvolves a total of $3$ edges, i.e., $d=2$, $n=3$.} On simplifying the\nderivation expression $dervE_{a_{k}}$ of $a_{k}$, we get the corresponding\nsymbolic expression $symE_{a_{k}}$. The number of monomials in the symbolic\nexpression (\\emph{flat} polynomial) is denoted as $m$. \\comment{Effectively, a symbolic\nexpression is a flat polynomial over $n$ variables containing $m$ monomials.\n\\comment{For instance, the symbolic expression $e_{1}e_{3} + e_{2}e_{3} -\ne_{1}e_{2}e_{3}$ of answer \\textit{(SIN,MUN)} is a flat polynomial over\n$\\{e_{1},e_{2},e_{3}\\}$ and has $3$ monomials.}} \nTable~\\ref{tab:symbols} summarizes the parameters.\n\nThe cost of construction of symbolic probability expressions involves\nsimplifying a derivation expression to a polynomial in $Z[e_1,\\ldots,e_n]$ by\nusing the definitions of semiring operations $\\otimes$ and $\\oplus$. \nLet $f$ and $g$ be two flat polynomials and $|\\bullet|$ be the number\nof distinct monomials in the simplified form of polynomial\n$\\bullet$. Then, the bound on the number of distinct monomials is\n\n\\begin{align}\n |f\\otimes g| &\\le |f|\\times|g| \\\\\n |f \\oplus g| &\\le |f| + |g| + (|f|\\times|g|) \\label{eq:plussing}\n\\end{align}\n}\nThe time taken to compute $f \\times g$ is $O(|f| |g| (\\log|f|+\\log|g|))$, and\nthat for $f + g$ is $O(\\min\\{ |g| \\log|f|, |f| \\log|g| \\})$. The logarithmic\nterms correspond to searching for new entrants among existing monomials.\n\nWe compute the symbolic probability expressions incrementally by adding\nterms corresponding to one derivation at a time. In other words, at\nstep $i$, derivation $D_{i}$ is added ($\\oplus$) with the resultant\nsymbolic expression of step $i-1$. If $m_{i}$ is the size (number of\nmonomials) of the symbolic expression obtained after the $i^\\text{th}$\nstep, we have the recursive bound\n$m_{i+1} \\le 2 \\cdot m_i + 1$\nwith $m_1 = 1$. Therefore,\n$m_d \\le 2^d$.\n\nNotice that each update step can be analyzed as a\nspecial case of Eq.~\\eqref{eq:plussing} with $|g| = 1$. Given the bound\non $m_d$, the update step ($(d+1)^\\text{st}$ step) takes $O(d \\cdot 2^d)$\ntime. Thus, the total computation time is also $O(d \\cdot 2^d)$ in the\nworst case.\n\nInterestingly, we can get other bounds on $m_d$ as well. The symbolic\ncomputation is a flat polynomial on $n$ variables. Therefore, it can have at\nmost $2^n$ monomials. The update step takes $O(n \\cdot 2^n)$ time and the total\ncomputation time is $O(d \\cdot n \\cdot 2^n)$ in the worst case.\n\nTowards another bound, let us view monomials as sets of\nvariables. After either semiring operation, $\\otimes$ or $\\oplus$,\nevery monomial in the resultant polynomial is a superset of a monomial\nfrom the polynomials operated upon. Therefore, every monomial in our\nsymbolic probability expression is a superset of a monomial\ncorresponding to a single derivation. In cases where the smallest\nnumber of variables in a derivation (say, $t$) is large compared to\nthe total number of variables, the bound on the total number of\nmonomials, $\\Sigma_{i=t}^n{{n}\\choose{i}}$, is much better than\n$2^n$. In case where $n-t$ is very small, this bound degenerates to\n$O(n^{n - t})$ and the update step time is $O(n^{n - t + 1})$. To\nlocate answers where this bound takes effect, we found that using $s$\nas a proxy for $t$ works well in practice.\n\nIn the case where these $n$-dependent bounds are smaller than the\n$d$-dependent bounds, the update time is markedly lesser compared to\nthe total compute time. For example, if $n-t$ is very small and $n$\ndoes not change with incoming derivations, the time taken for each\nsubsequent update step is $O(n^{n - t + 1})$. This becomes a\nsuccessively smaller fraction of the total computation time as more\nderivations come in. This is in contrast to exponential time update\nsteps, where the time taken for each subsequent update step is the\nsame fraction of the total compute time till that point. These are the\nanswers where we expect the most advantage from our maintenance\nalgorithm.\n\n\n\\section{Proof of Theorem 1}\\label{appendix1}\nThe following facts about the function $flat$ are easy to prove and\nwill be used later.\n\n\\begin{lemma} \\label{lemma:flatfacts}\n For integer polynomials $f$ and $g$,\n \\begin{align*}\n \\overline{\\overline{f}} &= \\overline{f} \\\\\n \\overline{f + g} &= \\overline{f} + \\overline{g} \\\\\n \\overline{\\overline{f}g} &= \\overline{fg}\n \\end{align*}\n\\end{lemma}\n\nWe now prove our main theorem\n\\begin{theorem}\\label{thm:semiring}\n $(Z_F[p_1, p_2, \\ldots ,p_n], \\oplus, \\otimes, 0,1)$ is a commutative semiring.\n\\end{theorem}\n\n\\begin{proof}\n That $(Z_F[p_1, p_2, \\ldots ,p_n], \\otimes)$ is a commutative monoid\n with identity $1$ follows directly from the definitions.\n\n The operator $\\oplus$ is also clearly commutative and $0$ is an\n identity for it. To show that $\\oplus$ is associative, consider $f,\n g, h \\in Z_F[p_1, p_2, \\ldots ,p_n]$\n \\begin{align*}\n (f \\oplus g) \\oplus h &= (f + g - \\overline{fg}) \\oplus h\\\\\n &= f + g - \\overline{fg} + h \\\\\n &\\phantomrel{=}- \\overline{fh + gh - h\\times flat(fg}) \\\\\n &= f + g + h - \\overline{fg + fh + gh} \\\\\n &\\phantomrel{=} + \\overline{h\\times \\overline{fg}} \\\\\n &= f + g + h - \\overline{fg + fh + gh} + \\overline{fgh} \\\\\n &= (g \\oplus h) \\oplus f \\tag{by symmetry} \\\\\n &= f \\oplus (g \\oplus h)\n \\end{align*}\n This equality chain is just a repeated application of Lemma\n \\ref{lemma:flatfacts}.\n\nSince $0\\otimes f = f \\otimes 0 = 0$ follows directly by definition,\nwe are only left to show distributivity of $\\otimes$ over $\\oplus$. We\nprove a quick lemma before we prove distributivity.\n\n\\begin{lemma} For all $f \\in Z_F[p_1, p_2, \\ldots ,p_n]$,\n $$f\\otimes f = f$$\n\\end{lemma}\n\\begin{proof}\n Given the definition of $Z_F[p_1, p_2, \\ldots ,p_n]$ we prove this\n by structural induction.\n\n By definition of the $\\otimes$ operator, $1\\otimes 1 = 1$ and $p_i\n \\otimes p_i = p_i$ for all $0 \\le i \\le n$\n\n It is enough to show that if $g, h \\in Z_F[p_1, p_2, \\ldots ,p_n]$\n are such that $g \\otimes g = g$ and $h \\otimes h = h$ then\n $$(g \\otimes h) \\otimes (g \\otimes h) = g \\otimes h$$\n and\n $$(g \\oplus h) \\otimes (g \\oplus h) = g \\oplus h$$\n\n Liberally rewriting using Lemma \\ref{lemma:flatfacts} and using $g\n \\otimes g = \\overline{g^2} = g$ and $h \\otimes h = \\overline{h^2} =\n h$ from our hypothesis, we get\n \n \\begin{align*}\n (g \\otimes h) \\otimes (g \\otimes h) &= \\overline{\\overline{gh}\\times \\overline{gh}} \\\\\n &= \\overline{g^2 h^2} \\\\\n &= \\overline{\\overline{g^2} \\times \\overline{h^2}} \\\\\n &= \\overline{gh} = g \\otimes h\n \\end{align*}\n and\n \\begin{align*}\n (g \\oplus h) \\otimes (g \\oplus h) &= (g + h - \\overline{gh})\\otimes (g + h - \\overline{gh}) \\\\\n &= \\overline{g^2 + h^2 + 2gh - 2g\\overline{gh} - 2h\\overline{gh} + (\\overline{gh})^2} \\\\\n &= \\overline{g^2} + \\overline{h^2} + 2\\overline{gh} - 2\\overline{g\\overline{gh}} - 2\\overline{h\\overline{gh}} + \\overline{\\overline{gh}^2} \\\\\n &= g + h + 2\\overline{gh} - 2\\overline{g^2h} - 2\\overline{gh^2} + \\overline{\\overline{g^2h^2}} \\\\\n &= g + h + 2\\overline{gh} - 2\\overline{\\overline{g^2}h} - 2\\overline{g\\overline{h^2}} + \\overline{\\overline{g^2} \\times \\overline{h^2}} \\\\\n &= g + h + 2\\overline{gh} - 2\\overline{gh} - 2\\overline{gh} + \\overline{gh} \\\\\n &= g + h - \\overline{gh} \\\\\n &= g \\oplus h\n \\end{align*}\n\\end{proof}\n\nTo wrap up our proof of theorem~\\ref{thm:semiring}, we now prove left\ndistributivity of $\\otimes$ over $\\oplus$. For $f, g, h \\in Z_F[p_1,\n p_2, \\ldots ,p_n]$\n\\begin{align*}\n f \\otimes (g \\oplus h) &= \\overline{f(g + h - \\overline{gh})} \\\\\n &= \\overline{fg} + \\overline{fh} - \\overline{f\\overline{gh}} \\\\\n &= \\overline{fg} + \\overline{fh} - \\overline{\\overline{f^2} \\times \\overline{gh}} \\\\\n &= \\overline{fg} + \\overline{fh} - \\overline{fg \\times fh} \\\\\n &= \\overline{fg} + \\overline{fh} - \\overline{\\overline{fg} \\times \\overline{fh}} \\\\\n &= \\overline{fg} \\oplus \\overline{fh} = (f \\otimes g) \\oplus (f \\otimes h) \n\\end{align*} \\qed\n \n\\end{proof}\n\n\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Probabilistic Knowledge Graph}\n\nA probabilistic knowledge graph is a graph $G \\left(V, E, L, Id, Pr\\right)$ with vertex-set $V$ representing the entities, labeled edge-set $E$ with each edge $e$ represented as $\\langle u,l,v \\rangle$ with $l \\in L$ encoding the relation between two vertices $u$ and $v$. \nIt is also common to refer to an edge $e = \\langle u, l, v \\rangle$ in the knowledge graph as a \\emph{fact} where $u$, $l$, $v$ are subject, predicate and object of the fact respectively.\nEach edge is assigned a unique id, ${Id}: E \\to \\mathcal{N}$. Further, we associate with each edge a value between $0$ and $1$, $Pr: E \\to [0,1]$, representing the probability of the corresponding fact. We make the standard \\emph{edge independence assumption} where the existence of an edge is independent of the other edges in the KG.\nUnlike a deterministic KG, the presence of each fact in \nthe KG is a probabilistic event, the probability of which is referred to as the \\emph{existential probability} of the fact. \n\n\n\\subsubsection*{Possible World Semantics}\n\nEquivalently, we can interpret the probabilistic knowledge graph as a \ncollection of edge-induced subgraphs, called \\emph{possible worlds},\n$\\mathcal{G_{PW}} = \\{G_{1}, \\dots, G_{n}\\}$ of the knowledge graph $G$.\nThere is a probability distribution, $\\mathcal{P}$, defined over all possible worlds, such that $\\Sigma_{G_i \\in G_{PW}} \\mathcal{P}(G_i) = 1.$ With edge-independence assumption as before, $P(G_i)$ in terms of probability of its edges is\n\\begin{align}\n\\mathcal{P}(G_{i}) = \\prod_{e_{i}\\in E_{i}} Pr(e_{i}) \\times \\prod_{e_{i}\\in E\\setminus E_{i}} ( 1 - Pr(e_{i}))\n\\end{align}\nwhere $E_{i}$ is the set of edges present in $G_{i}$.\n\n\n\\subsection{Graph Query}\nA graph query is formulated as a graph pattern that a user intends to find in\nthe knowledge graph $G$. Similar to the triple representation of KG, a graph\nquery expresses the query graph pattern as a collection of triples. Each edge\nof the graph pattern corresponds to a triple pattern in the query. Similar to\ngraph triples, a \\emph{triple pattern} consists of subject, predicate, and\nobject. Both subject and object can be variables or be bound to one of the\nvertices of the KG. A predicate could be a variable or one of the labels of the\nedges of the KG. A graph query pattern can be realized as a query using the SPARQL query language.\n\n\nA graph query can be interpreted as a \\emph{conjunction} of triple patterns and\nthe aim is to find all possible bindings to the variables of the triple\npatterns as a whole. These conjunctive graph queries are popularly known as\n\\emph{Basic Graph Pattern} (BGP) queries~\\cite{sparqlnew}. Such SPARQL queries\ncan be expressed as relational SPJ (Select-Project-Join) queries\n\\cite{cyganiak2005}. \nIn this work we are handling a subset of SPARQL queries which does not include more sophisticated operators such as Union, Optional, etc. We see inclusion of these SPARQL queries as future work. \n\n\\subsection{Running Example} Throughout the paper we use the following running example. Consider an air-travel agency that provides a flight search engine. The search engine uses the knowledge graph shown in Fig.~\\ref{fig:kg} as its knowledge base. The nodes in the graph denote the airport codes of different cities it operates in: Singapore (SIN), New Delhi (DEL), Munich (MUN), Barcelona (BAR), and New York (JFK). An edge between two cities represents a direct flight with edge label representing the airline operating the flight.\n The operation of each flight is dependent on different factors, like environmental, financial, political, etc. Thus, the existence of the edges in the graph is a probabilistic event. Each edge of the KG, shown in Fig.~\\ref{fig:kg}, is annotated with its existential probability.\nSuppose a user wants to list down all the pairs of cities that have \\emph{one-stop} connecting flights between them with a joint probability greater than $0.5$. The corresponding query pattern, shown in Fig.~\\ref{fig:query}, and the equivalent SPARQL query, \n\n\\texttt{\n\t\\begingroup\n\t\t\\fontsize{8pt}{12pt}\n\t\t\\textbf{Select} \\textit{?city1} \\textit{?city2} \\textbf{Where} \\{ \\\\\n\t\t\\null \\hspace{0.8cm}\\textit{?city1} ?x1 \\textit{?city2}. \\\\\n\t\t\\null \\hspace{0.8cm}\\textit{?city2} ?x2 \\textit{?city3}. \\}\\\\\n\t\\endgroup\n\t}\n\nThis query is a collection of $2$ triple patterns, $\\langle ?city1, ?x1,\n?city2 \\rangle$ and $\\langle ?city2, ?x2, ?city3 \\rangle$. The variables in a\ntriple pattern has prefix, question-mark $?$. \n\nAll the answers to the query (along with their derivation polynomials) are listed in Table~\\ref{tab:result}. The answers will be further filtered out to report only matches with probability $>0.5$. \n\n\n\\subsection{Query Evaluation on Probabilistic Dataset}\n\nOn a deterministic graph, a graph query result is a collection of projected\nnodes of subgraphs that matches the query pattern. However, on a probabilistic\nKG, the query engine has to perform additional task of computing probability of\neach result item. This task is often referred to as \\emph{probabilistic\ninference}~\\cite{dalvidb}. Technically, \\emph{probability inference} involves matching the\nquery pattern over all \\emph{possible worlds} and computing the marginal \nprobability of matches. The result of a query $Q$ over probabilistic KG is\ngiven as,\n\\begin{align}\n\t%\n\tA(Q) = \\{ \\langle m_{i}, Pr(m_{i}) \\rangle, \\dots, \\langle m_{k}, Pr(m_{k}) \\rangle \\}\n\t%\n\\end{align}\nwhere each answer $m_{i}$ has a probability $Pr(m_{i})$ associated with it \nrepresenting the overall probability of it being part of the answer set.\n\n\nThe number of possible world of a KG will be exponential in number\nof edges, specifically $2^{|E|}$. Thus, the approach of enumerating all the\nworlds and evaluating the query on each of them is impractical. Instead, each\nanswer is associated with a Boolean formula and the probability of this Boolean\nformula to be \\emph{true} over all assignments gives the probability of the\ncorresponding result item. This Boolean formula is referred to as \\emph{lineage}.\nWe discuss how to compute lineage for query answers in Sec.~\\ref{subsec:prov}.\nFor now, we continue with discussing the techniques to compute probability of\nlineage.\n\nA na\\\"ive technique to compute probability is to enumerate all\nassignments of the Boolean formula and then count the satisfying\nones. This method of computing the probability is called\n\\emph{possible world computation}. Clearly, the possible worlds\ncomputation scales exponentially with the number of Boolean variables\nin a formula. The probability computation of a Boolean formula is\nequivalent to the weighted model counting problem. Thus, probabilistic\ninference is \\emph{\\#P-hard}.\n\nVarious heuristics are used to tackle this problem. These approaches\nfall under the category of \\emph{intensional query evaluation}. One\nof the most popular techniques used in intensional query evaluation is\nbased on converting a given Boolean formula into d-NNF forms, which\nare known to be more tractable. This methodology is known as\n\\emph{compilation} \\cite{compilation}. More details about different\nkinds of compilation techniques are given in a popular survey by\nBroeck and Suciu \\cite{survey}. Tools such as \\textsf{PS-KC}\\xspace\n\\cite{c2d}, D4 \\cite{d4} are based on this compilation strategy.\n\n\\subsection{Lineage Computation}\n\\label{subsec:prov}\n\nThe provenance semiring model piggybacks the lineage computation on to the\nquery processing. Thus, the lineage is computed using the semiring $\\left(PosBool(X_{\\bullet}), \\land,\n\\lor, 0, 1 \\right)$ over positive Boolean expressions.\nTuples are annotated with independent Boolean variables $X_{1} ,\\dots, X_{i}$.\nFor a query answer $m$, the lineage is a DNF (\\emph{disjunctive normal form})\nformula where each conjunct represents a possible derivation of $m$. A\nconjunct is constructed by applying the AND operator ($\\land$) on the Boolean\nvariables of the edges involved in that particular derivation.\nFor instance, in Table~\\ref{tab:result}, the lineage of answer of the query,\nshown in Fig.~\\ref{fig:kg}, is given as $PosBool$. The Boolean formula\ncorresponding to the answer (SIN,MUN) has $2$ conjuncts $\\left(e_{1} \\land\ne_{3}\\right)$ and $\\left(e_{2} \\land e_{3}\\right)$, each representing a\npossible derivation of the answer. \n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn this paper we have proposed a novel commutative semiring which\nenables us to symbolically compute probability of query answers over\nprobabilistic knowledge graphs. Further, we present a framework \\textsc{HaPPI}\\xspace\nthat uses the proposed semiring to support query processing and answer\nprobability maintenance over probabilistic KG. We have compared the\nefficiency of our proposed probability computation technique against\ntwo standard approaches used for probabilistic inference.\n\\textsc{HaPPI}\\xspace outperforms current\nsystems in a range of queries answer parameters containing almost\n$70\\%$ of the query answers. We have also shown that an\n\\emph{adaptive} approach that uses \\textsc{HaPPI}\\xspace in conjunction with a\nknowledge compilation based technique for large query answers is a\nsignificantly faster alternative for probabilistic inference.\n\n\\comment{\n\nWe have proposed a novel approach for probabilistic inference on\nKnowledge Graphs. This is backed by a robust theoretical\nfoundation. Our framework comprises of maintainable symbolic\ncomputations of probabilities of query answers in the semiring of\n\\emph{flat} integer polynomials. This computation method far\noutperforms current systems in a range of queries answer parameters\ncontaining almost $70\\%$ of the query answers we processed. We have\nmade a case for the use of this method in conjunction with a\ncompilation based technique for large query answers as the optimal way\nfor probabilistic inference.\n}\n\n\n\n\\section{Experiments}\n\\label{sec:expts}\n\n\nQuery processing over probabilistic KG involves two tasks: finding the answers\n\tof a posed query, and computing the probability of each answer of the query\n\talong with its provenance. Many systems such as HUKA\\xspace \\cite{huka},\n\tTripleProv\\xspace \\cite{tripleProv}, ProvSQL\\xspace \\cite{provsql} compute the graph\n\tquery along with provenance polynomials over deterministic data. Our \\textsc{HaPPI}\\xspace\n\tframework can be plugged into any of them. We have used HUKA\\xspace as the base\n\tsystem. In this section, therefore, we focus on the performance of the\n\tprobability computation task starting from the derivation lists of the\n\tquery answers. There is another important dimension of \\textsc{HaPPI}\\xspace, that of\n\tmaintainability since it is quite common for KGs to undergo changes.\n\tTherefore, we also test our system on maintenance time under these\n\toperations against the complete re-computation cost of the symbolic\n\texpressions.\n\n\n\\subsection{Setup}\n\n\\subsubsection*{Datasets}\n\nWe consider two widely used benchmark datasets in our experimental evaluations:\n\t\t(a)~\\textbf{YAGO2} \\cite{yago2,hoffart2011yago2}, an automatically\n\t\tbuilt ontology gathering facts from different sources like\n\t\tWikipedia,\n\t\tGeoNames,\n\t\tetc. It has\n\t\t$\\sim 23$\\,M facts over $\\sim 5.8$\\,M real-world entities.\n\t\n\t\t(b)~\\textbf{gMark} \\cite{gmark}, a synthetic dataset generated by\n\t\ta schema-driven data and workload generator, gMark\\xspace. We used the schema\n\t\tof LDBC SNB~\\cite{ldbc}\n\t\n\t\tto generate a graph with $0.9$\\,M nodes and $2.2$\\,M edges.\n\t\n\n\n\\subsubsection*{Query Collection}\n\nFor the YAGO2\\xspace dataset, we used a set of queries on which the RDF-3X was\noriginally validated~\\cite{neumann2010rdf}. We chose $3$ out of $6$ benchmark\nqueries since the other queries have answers with only a single derivation. The\nprobability of answers with a single derivation can be computed simply by\nmultiplying the probabilities of edges involved. This is a corner case that\ndoes not serve to make any comparison. The $3$ chosen queries are fairly large\nand complex, and have $7.25$ triple constraints on an average.\nFor the gMark\\xspace dataset, we generated queries of size between $3$\nand $7$. We generated $100$ queries out of which $11$ have answers with\nmultiple derivations. The average query size of these $11$ queries is\n$4.24$ triples.\n\nThe statistics of the datasets are reported in Table~\\ref{tab:yagoQ} and\nTable~\\ref{tab:gmarkQ}. Interestingly, given the size of gMark\\xspace KG, queries of\neven size $7$ have quite large ($\\approx 6000$) answer sets. This is due to the\nfact that none of the generated queries have bound variables, i.e., for all the\ntriple patterns, both subject and object are variables. Since each answer is\ndealt with independently, the variation across this large number of answers\nhelps us to evaluate the methods thoroughly.\n\n\\subsubsection*{Implementation}\n\nWe conducted all our experiments on a 32-core 2.1GHz CPU,\n512GB RAM machine with 1TB hard drive. Our implementation is single-threaded in Java. Our codebase is publicly available \\footnote{\\url{https:\/\/github.com\/gaurgarima\/HaPPI}}. The KGs YAGO2\\xspace and gMark\\xspace are realized as Neo4j property graphs of size $11$\\,GB and $211$\\,MB respectively. The inverted indexes\n\\textsf{edgeToSymE} and \\textsf{symEval} are of size $2.4$\\,MB and $2.3$\\,MB respectively for YAGO2\\xspace, and $805$\\,MB and $804$\\,MB\nrespectively for gMark\\xspace. \n\n\\subsubsection*{Choice of Baseline Systems}\n\nFor baseline system selection, we focused on only \\emph{intensional} technique\nbased systems since \\emph{extensional} techniques cannot handle\n\\emph{self-joins} (Sec.~\\ref{subsec:probWork}). We chose a recently proposed\nsystem ProvSQL\\xspace~\\cite{provsql} that implements \\textsf{PosBool} semiring based\nprobabilistic database on top of PostgreSQL.\nIt relies on $3$ standard ways to compute the probability: (a)~possible world\ncomputation, (b)~knowledge compilation, and (c)~Monte Carlo technique. We have\nnot considered the Monte Carlo approach as, unlike the other two approaches, it\ncomputes approximate probabilities. \nWe adopted the implementation of possible world (\\textsf{PS-PW}\\xspace) and knowledge compilation\n(\\textsf{PS-KC}\\xspace) used in publicly available ProvSQL\\xspace as our baselines.\n\n\\begin{table}[t]\n\t\\centering\n\t\\tabcaption{Details of answer set of YAGO2\\xspace queries.}\n\t\\footnotesize{\n\t\t\\begin{tabular}{ccccccc} \n\t\t\t\\toprule\n\t\t\t\\bf Query & \\bf Query & \\bf Size of & \\multicolumn{2}{c}{\\bf \\#Derivations, $d$} & \\multicolumn{2}{c}{\\bf \\#Distinct edges, $n$} \\\\ \n\t\t\n\t\t\t\\bf Id & \\bf Size, $s$ & \\bf Answer Set & min & max & min & max \\\\\n\t\t\t\\midrule\n\t\t\t$Q_{1}$ & 6 & 2 & 2 & 2 & 9 & 9 \\\\\n\t\t\t$Q_{3}$ & 4 & 728 & 2 & 14 & 6 & 30 \\\\\n\t\t\t$Q_{6}$ & 6 & 544 & 2 & 5 & 8 & 14 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{tab:yagoQ}\n\\end{table}\n\n\\begin{table}[t]\n\t\\tabcaption{Details of answer set of gMark\\xspace queries.}\n\t\\footnotesize{\n\t\t\\begin{tabular}{ccccccc} \n\t\t\t\\toprule\n\t\t\t\\bf Query & \\bf Query & \\bf Size of & \\multicolumn{2}{c}{\\bf \\#Derivations, $d$} & \\multicolumn{2}{c}{\\bf \\#Distinct edges, $n$} \\\\ \n\t\t\n\t\t\t\\bf Id & \\bf Size, $s$ & \\bf Answer Set & min & max & min & max \\\\\n\t\t\t\\midrule\n\t\t\t$Q_{4}$ & 7 & 6048 & 5 & 15 & 8 & 9 \\\\\n\t\t\t$Q_{6}$ & 4 & 1536 & 2 & 87 & 5 & 69 \\\\\n\t\t\t$Q_{9}$ & 5 & 88 & 2 & 25 & 7 & 52 \\\\\n\t\t\t$Q_{21}$ & 5 & 411 & 6 & 126 & 6 & 9 \\\\\n\t\t\t$Q_{23}$ & 3 & 2325 & 2 & 122 & 4 & 124 \\\\\n\t\t\t$Q_{32}$ & 3 & 32 & 2 & 2 & 6 & 6 \\\\\n\t\t\t$Q_{35}$ & 3 & 10 & 2 & 2 & 4 & 5 \\\\\n\t\t\t$Q_{38}$ & 3 & 4929 & 2 & 257 & 4 & 620 \\\\\n\t\t\t$Q_{46}$ & 7 & 6156 & 2 & 24580 & 8 & 8681 \\\\\n\t\t\t$Q_{54}$ & 6 & 400 & 2 & 6 & 7 & 16 \\\\\n\t\t\t$Q_{90}$ & 6 & 62 & 4 & 35 & 7 & 10 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{tab:gmarkQ}\n\n\\end{table}\n\n\\subsection{Probability Computation Time}\n\\label{sec:computation}\n\nThe \\textit{probability computation time} is the time taken to compute the\nprobability of query answers given a list of their derivation(s). Variance in\nprobability computation time across query answers happens due to the following\ncharacteristics: (i) $d$, the number of derivations, (ii) $n$, the total number\nof edges involved, and (iii) $s$, the size of query. We call this triple $(d,\nn, s)$ the \\emph{answer signature}.\n\n\n\n\\subsubsection*{Performance across Answer Signatures}\n\nSince answer signatures across a query show wide variations,\nwe try to understand the trends by grouping the answer signatures\ninto buckets. We expect, from\nSec.~\\ref{subsec:analysis}, that \\textsc{HaPPI}\\xspace scales exponentially\nwith $d$. Similarly, \\textsf{PS-PW}\\xspace scales exponentially with $n$. Hence,\nquery answers are first grouped on the basis of\n$d$, and then grouped further on the basis of $n$.\nThe bucket boundaries are chosen such\nthat the variation within a bucket is not very high.\nTable~\\ref{tab:baselineyago} and Table~\\ref{tab:baselinegmark} show\nthe detailed results across these buckets. \nThe \\emph{count} column shows the number of query answers each\nsignature bucket has.\nWhile the\nperformance of \\textsc{HaPPI}\\xspace can also depend on query size $s$, this\ndependence shows up for a very small set of answers (when $s$ is\nalmost as large as $n$). Hence, we do not show $s$ in the tables.\n\n\nFor YAGO2\\xspace queries, it can be seen that \\textsc{HaPPI}\\xspace massively outperforms both the\nsystems. The largest absolute time for any derivation is only $130\\mu$s for\n\\textsc{HaPPI}\\xspace. This is because $d$ and $n$ are not very large. As expected, \\textsf{PS-PW}\\xspace shows\nan exponential scaling with $n$. \\textsf{PS-KC}\\xspace shows a flat trend across buckets but a\nhigh variation within them. Knowledge compilation based techniques are\nsensitive to the precise Boolean formula whose probability is being computed\nand not just to its size parameters. \\comment{While there should still be a\ngeneral trend of computation time increase with size of answer parameters, this\nasymptotic trend does not show up within the range of our data.}\nThe trend across gMark\\xspace queries is more interesting. Up to $d = 12$ and $n <\n18$, \\textsc{HaPPI}\\xspace performs very well. When $n \\geq 18$ (for $d \\in [9,12]$), the time\nfor \\textsc{HaPPI}\\xspace shoots up to more than $4$ms. While \\textsf{PS-PW}\\xspace could not finish even after\n$30$s, \\textsf{PS-KC}\\xspace is faster than \\textsc{HaPPI}\\xspace in this range. When $d$ is even larger ($>\n13$), the time for \\textsc{HaPPI}\\xspace jumps to $41$ms. \\textsf{PS-KC}\\xspace remains more or less constant\nin the range of $4$ms.\n\n\n\n\\begin{table}[t]\n\t\\tabcaption{Probability computation times ($\\mu$seconds): YAGO2\\xspace.}\n\t\\resizebox{\\columnwidth}{!}{\n\t\t\\begin{tabular}{cc|c|c|c|c}\n\t\t\t\\toprule\n\t\t\t$\\mathbf{d}$ & $\\mathbf{n}$ & \\bf Count &\t\\bf \\textsc{HaPPI}\\xspace\t&\t\\bf \\textsf{PS-PW}\\xspace\t&\t\\bf \\textsf{PS-KC}\\xspace \t\\\\\n\t\t\t\\midrule\n\t\t\t$[2,5)$ & $-$ &\t1,225 &\t19.22 $\\pm$\t11.07 &\t444.72 $\\pm$\t518.29 &\t2,720.25 $\\pm$\t1,723.80 \\\\\n\t\t\t\\cmidrule(lr){1-6}\t\t\t\n\t\t\t\\multirow{2}{*}{$[5,9)$} & $< 13$ &\t7 &\t84.61 $\\pm$\t35.62 &\t2,624.87 $\\pm$\t359.38 &\t1,709.67 $\\pm$\t530.58 \\\\\n\t\t\t& $\\geq 13$ &\t39 &\t130.59 $\\pm$\t87.01 &\t19,265.49 $\\pm$\t19,102.84 &\t2,233.88 $\\pm$\t1,636.08 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\label{tab:baselineyago}\n\\end{table}\n\n\\begin{table}[t]\n\t\\centering\n\t\\tabcaption{Probability computation times ($\\mu$seconds): gMark\\xspace.}\n\t\\resizebox{\\columnwidth}{!} {\n\t\t\\begin{tabular}{cc|c|c|c|c}\n\t\t\t\\toprule\n\t\t\t$\\mathbf{d}$ & $\\mathbf{n}$ & \\bf Count &\t\\bf \\textsc{HaPPI}\\xspace\t&\t\\bf \\textsf{PS-PW}\\xspace\t&\t\\bf \\textsf{PS-KC}\\xspace \t\\\\\n\t\t\t\\midrule\n\t\t\t\\multirow{2}{*}{$[2,5)$} & $< 6$ & 4,558 & 5.69 $\\pm$ 3.05 & 10.88 $\\pm$ 5.74 & 4,811.58 $\\pm$ 3,200.32 \\\\\n\t\t\t&$\\geq 6$& 4,321 & 16.95 $\\pm$ 10.44 & 1,564.73 $\\pm$ 3,843.17 & 3,827.67 $\\pm$ 3,101.51 \\\\ \n\t\t\t\\cmidrule(lr){1-6}\t\t\t\n\t\t\t\\multirow{2}{*}{$[5,9)$} & $< 13$ & 5,523 & 87.67 $\\pm$ 64.81 & 900.61 $\\pm$ 918.17 & 4,202.18 $\\pm$ 2,784.27 \\\\ \t\t\t\t\t\t\t\n\t\t\t& $\\geq 13$ & 673 & 249.31 $\\pm$ 171.56 & 335,246.76 $\\pm$ 738,398.70 & 3,367.52 $\\pm$ 2,834.69 \\\\\n\t\t\t\\cmidrule(lr){1-6}\t\t\t\n\t\t\t\\multirow{2}{*}{$[9,13)$} & $< 18$ & 360 & 265.38 $\\pm$ 183.52 & 353,521.69 $\\pm$ 761,014.83 & 3,418.17 $\\pm$ 2,890.34 \\\\\n\t\t\t& $\\geq 18$ & 439 & 4,684.11 $\\pm$ 3,300.08 & time-out & 3,758.99 $\\pm$ 2,937.08 \\\\ \n\t\t\t\\cmidrule(lr){1-6}\t\t\t\n\t\t\t\\multirow{1}{*}{$[13,16)$} & $-$ & 2,415 & 41,697.01 $\\pm$ 32,091.19 & time-out & 3,924.66 $\\pm$ 3,176.96 \\\\\n\t\t\t\\bottomrule\t\t\t\t\t\t\n\t\t\\end{tabular}\n\t}\n\t\\label{tab:baselinegmark}\n\\end{table}\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.20]{plot\/yagoPW.pdf}\n\t\\includegraphics[scale=0.20]{plot\/gmarkPW.pdf}\n\\figcaption{Percentile gains of \\textsc{HaPPI}\\xspace against \\textsf{PS-PW}\\xspace.}\n\\label{fig:percentilePW}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale=0.20]{plot\/yagoC2d.pdf}\n\t\\includegraphics[scale=0.20]{plot\/gmarkC2d.pdf}\n\t\\figcaption{Percentile gains of \\textsc{HaPPI}\\xspace against \\textsf{PS-KC}\\xspace.}\n\t\\label{fig:percentileC2D}\n\\end{figure}\n\n\n\\subsubsection*{Percentile Gains}\n\nWe dig deeper to analyze the gains of \\textsc{HaPPI}\\xspace over the other two systems in more\ndetail. Timing averages of exponential systems tend to be dominated by corner\ncases. To guard against that, and to get a better insight, we employ a new\nmetric for comparison based on \\emph{percentile gains}. By reporting a\n\\emph{$p^\\text{th}$ percentile gain} of $x\\%$, we mean that for $p\\%$ of the\nanswers, \\textsc{HaPPI}\\xspace took at least $x\\%$ lesser time than the method compared\nagainst. The gain is computed using the following ratio:\n$$\\text{gain}_\\text{method} =\n(\\text{time}_\\text{method}-\\text{time}_\\textsc{HaPPI}\\xspace)\/\\text{time}_\\text{method}$$\nFor instance, Fig.~\\ref{fig:percentilePW} reports a $50^\\text{th}$ percentile\ngain of $82\\%$ for \\textsc{HaPPI}\\xspace over \\textsf{PS-PW}\\xspace for the bucket $d \\in [2,5)$, $n \\geq 1$.\nThis means that for $50\\%$ of the answers in this bucket, the probability\ncomputation time of \\textsc{HaPPI}\\xspace is at most $100 - 82 = 18\\%$ of the corresponding\ntime of \\textsf{PS-PW}\\xspace. A negative gain indicates that \\textsc{HaPPI}\\xspace is slower. \\comment{ For\nexample, if for a bucket having $100$ answers, the $75^\\text{th}$ percentile\ngain of \\textsc{HaPPI}\\xspace is $80\\%$, this means that for $75$ answers, the probability\ncomputation of \\textsc{HaPPI}\\xspace is at most $100 - 80 = 20\\%$ of the corresponding time of\nthe compared method. Note that if \\textsc{HaPPI}\\xspace is slower, gain is negative. }\n\\textsc{HaPPI}\\xspace shows superior performance across all signature buckets. \\comment\n{\\ab{The gains displayed are lesser than those the averages suggest. This is to\nbe expected, because timing averages of exponential systems tend to be\ndominated by the corner cases. However, the sizable gains even for the $90$th\npercentile showcase consistent and marginal better performance for individual\nanswers.} \\ab{Not clear -- can we cut?}}\nFig.~\\ref{fig:percentileC2D} shows that the better performance of \\textsc{HaPPI}\\xspace over\n\\textsf{PS-KC}\\xspace for small answer signatures is uniform inside the buckets with\nlittle deviation even for corner cases. The asymptotic deterioration of\n\\textsc{HaPPI}\\xspace is also visible. There is an exception to this trend. We discuss\nthe unexpected gains in the bucket $d\\in[13,16)$, $n \\geq 1$ in\nSec.~\\ref{sec:adaptive}. \n\n\n\\subsection{Adaptive Framework}\n\\label{sec:adaptive}\n\n\nWhile \\textsc{HaPPI}\\xspace outperforms \\textsf{PS-KC}\\xspace and \\textsf{PS-PW}\\xspace significantly for lower ranges of $d$ and\n$n$, it is quite slow for larger values.\nSince \\textsf{PS-KC}\\xspace takes roughly the same amount of time to compute probabilities for\nthe very smallest of query answers to the largest, an \\emph{adaptive} strategy\ninvolving both \\textsc{HaPPI}\\xspace and \\textsf{PS-KC}\\xspace seems to be the best. The \\emph{adaptive}\nstrategy utilizes the best of both the worlds: it employs \\textsc{HaPPI}\\xspace for lower\nranges of $d$ and $n$, and switches to \\textsf{PS-KC}\\xspace when these parameters become large.\nWe report $3$ bounds, $O(2^{d})$, $O(d \\cdot 2^{n})$ and\n$O(n^{s-n})$, on the probability computation of \\textsc{HaPPI}\\xspace\n(Sec.~\\ref{subsec:analysis}). We, thus, expect to use \\textsc{HaPPI}\\xspace for small values of\n$d$, $n$ or $s-n$.\n\nTo notice the $s-n$ dependence, we need to look at higher values of $d$ and\n$n$, since \\textsc{HaPPI}\\xspace would outperform \\textsf{PS-KC}\\xspace anyway for smaller values. For the\nrange $13 \\le d < 16$, this is precisely what happens. Closer inspection\nshowed that $2089$ out of $2415$ answers in this range had signatures $(5, 21,\n7)$ or $(7, 15, 9)$ and \\textsc{HaPPI}\\xspace outperformed \\textsf{PS-KC}\\xspace for all these answers. This\nexplains the aberrant percentile gain of \\textsc{HaPPI}\\xspace. It also empirically validates\nour analysis on $s-n$ values. \n\nWe now elaborate on our \\emph{adaptive} strategy. For each answer, we\nuse \\textsc{HaPPI}\\xspace if one of these $3$\nconditions is satisfied: (a)~$d < 10$, (b)~$n < 8$, or (c)~$n - s < 3$.\n\\comment{\n\\gaur{change the enumeration to inline}\n\n\\begin{enumerate}[label=(\\alph*)]\n\t%\n\t\\item $d < 10$, or\n\t\\item $n < 8$, or\n\t\\item $n-s < 3$.\n\t%\n\\end{enumerate}\n}\nWe refer to this signature range as the \\textsc{HaPPI}\\xspace domain of answers.\nWe employ \\textsf{PS-KC}\\xspace outside this domain. \n\nThe overall probability computation time of \\emph{adaptive} (\\textsc{HaPPI}\\xspace\/\\textsf{PS-KC}\\xspace) and\npure \\textsf{PS-KC}\\xspace techniques for YAGO2\\xspace and gMark\\xspace queries are reported under column\n`Computation Time' in Table~\\ref{tab:yagooverall} and\nTable~\\ref{tab:gmarkoverall} respectively. We have also reported the\npercentage of query answers for which \\textsc{HaPPI}\\xspace was used in the adaptive system.\nFor YAGO2\\xspace queries, our adaptive approach is on an average $303$ times faster\nthan pure \\textsf{PS-KC}\\xspace. For gMark\\xspace queries, we record an average speed-up of $317$\ntimes. The speed-up is, as expected, very high for queries $Q_{32}, Q_{35},\nQ_{54}$, where \\textsc{HaPPI}\\xspace is used for all the query answers. Importantly, the\nadaptive system gives a $21\\%$ speed-up for even $Q_{46}$ where \\textsc{HaPPI}\\xspace is used\nfor only about $37\\%$ of the answers and the answer signatures go up to\n$d=24580$ and $n=8681$.\n\n\n\n\n\\begin{table}[t]\n\\tabcaption{Performance of adaptive system (in $\\mu$seconds) for query computation and maintenance for YAGO2\\xspace query set.}\n\t\\centering\n\t\\resizebox{\\columnwidth}{!}{%\n\t\\begin{tabular}{c|r||ll|l||ll|l}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{\\bf QId} & \\multicolumn{1}{|c||}{\\bf \\textsc{HaPPI}\\xspace}\t\t\n\t\t& \\multicolumn{2}{|c|}{\\bf Computation Time} & \\multirow{2}{*}{\\bf Gain} & \\multicolumn{2}{|c|}{\\bf Maintenance Time} & \\multirow{2}{*}{\\bf Gain} \\\\ \t\t\n\t\n\t\t& \\multicolumn{1}{|c||}{\\bf answer \\%} & \\bf Adaptive & \\bf \\textsf{PS-KC}\\xspace & & \\bf Recompute & \\bf Incremental & \\\\\n\t\t\\midrule\n\t$Q_{1}$\t& $100.00$ & $2.7\\times 10^2$ & $6.9\\times 10^4$ & $0.99$ & $6.99\\times 10^2$ & $6.29\\times 10^2$ & $0.100$\\\\\n\t\n\t\t$Q_{3}$ & $99.86$ & $2.1\\times 10^4$ & $1.4\\times 10^6$ & $0.98$ & $3.26\\times 10^4$ & $2.43 \\times 10^4$ & $0.256$ \\\\\n\t\t\n\t\t$Q_{6}$\t& $100.00$ & $3.9\\times 10^3$ & $2.3\\times10^6$ & $0.99$ & $1.44\\times 10^4$ & $1.11 \\times 10^4$ & $0.226$\\\\ \t\t\n\t\t\\bottomrule\n\t\\end{tabular}\n\t}\n\t\\label{tab:yagooverall}\n\n\\end{table}\n\n\n\n\n\\begin{table}[t]\n\t\\tabcaption{Performance of adaptive system (in $\\mu$seconds) for query computation and maintenance for gMark\\xspace query set.}\n\n\t\\centering\n\t\\resizebox{\\columnwidth}{!}{\n\t\\begin{tabular}{c|r||ll|l||ll|l}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{\\bf QId} & \\multicolumn{1}{|c||}{\\bf \\textsc{HaPPI}\\xspace}\t\t\n\t\t& \\multicolumn{2}{|c|}{\\bf Computation Time} & \\multirow{2}{*}{\\bf Gain} & \\multicolumn{2}{|c|}{\\bf Maintenance Time} & \\multirow{2}{*}{\\bf Gain} \\\\ \t\t\n\t\n\t\t& \\multicolumn{1}{|c||}{\\bf answer \\%} & \\bf Adaptive & \\bf \\textsf{PS-KC}\\xspace & & \\bf Recompute & \\bf Incremental & \\\\\n\t\t\\midrule\n\t\t$Q_{4}$ & $100.00$ & $5.5\\times 10^5$ & $1.0\\times 10^7$ & $0.94$ & $6.2\\times 10^5$ & $5.6\\times 10^5$ & $0.097$\\\\\n\t\t\n\t\t$Q_{6}$ & $91.47$ & $4.1\\times 10^5$ & $3.9\\times 10^6$ & $0.89$ & $4.7\\times 10^5$ & $4.4\\times 10^5$ & $0.060$\\\\\n\t\t\n\t\t$Q_{9}$ & $94.32$ & $1.5\\times 10^4$ & $2.6\\times 10^5$ & $0.94$ & $2.0\\times 10^4$ & $1.8\\times 10^4$ & $0.095$\\\\\n\t\t\n\t\t$Q_{21}$ & $98.78$ & $6.8 \\times 10^4$ & $3.5\\times 10^5$ & $0.81$ & $1.4 \\times 10^5$ & $1.3\\times 10^5$ & $0.091$\\\\\n\t\t\n\t\t$Q_{23}$ & $98.88$ & $1.1\\times 10^5$ & $8.8\\times 10^6$ & $0.98$ & $9.1\\times 10^4$ & $7.8\\times 10^4$ & $0.14$ \\\\\n\t\t\n\t\t$Q_{32}$ & $100.00$ & $1.5\\times 10^2$ & $2.4\\times 10^5$ & $0.99$& $1.5\\times 10^3$ & $1.3\\times 10^3$ & $0.12$ \\\\\n\t\t\n\t\t$Q_{35}$ & $100.00$ & $5.1\\times 10^1$ & $8.0\\times 10^4$ & $0.99$ & $7.1\\times 10^2$ & $6.0\\times 10^2$ & $0.16$\\\\\n\t\t\n\t\t$Q_{38}$ & $92.45$ & $2.1\\times 10^6$ & $2.1\\times 10^7$ & $0.90$ & $2.3\\times 10^6$ & $2.2\\times 10^6$ & $0.05$ \\\\\n\t\t\n\t\t$Q_{46}$ & $36.92$ &$3.6\\times 10^7$ & $4.6\\times 10^7$ & $0.21$ & $3.7\\times 10^7$ & $3.6\\times 10^7$ & $0.005$\\\\\n\t\t\n\t\t$Q_{54}$ & $100.00$ &$4.7\\times 10^3$ & $8.5\\times 10^5$ & $0.99$ & $1.7\\times 10^4$ & $1.3\\times 10^4$ & $0.24$ \\\\\n\t\t\n\t\t$Q_{90}$ & $52.27$ & $1.2\\times 10^5$ & $1.5\\times 10^5$ & $0.20$ & $1.2\\times 10^5$ & $1.2\\times 10^5$ & $0.008$\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t}\n\t\\label{tab:gmarkoverall}\n\\end{table}\n\n\n\\subsection{Maintenance Time}\n\\label{sec:maintenance}\n\nNext we evaluate the maintainability of \\textsc{HaPPI}\\xspace. Notice that in our maintenance\nalgorithms (Sec.~\\ref{subsec:maintain}) we make use of the incremental build-up\nof the symbolic probability expressions in our semiring only for the case of\nedge addition. When an edge is deleted we simply recompute the probability\nexpression from scratch. Similarly, updating the probability value for an edge\ndoes not affect the already computed symbolic expression for the probability of\nany answer. Thus, we focus on handling edge insertions in the KG.\n\n\\subsubsection*{Addition of an Edge}\n\nWhen quoting maintenance times for our adaptive system we use \\textsc{HaPPI}\\xspace\nincrementally on the answers that lie in its domain and do a full\nre-computation with \\textsf{PS-KC}\\xspace on the rest. This is compared against\nre-computation with the adaptive system (\\textsc{HaPPI}\\xspace and \\textsf{PS-KC}\\xspace in their\nrespective domains) for all answers. We investigated the effect of\nthe addition of an edge on each individual answer. For every answer,\nwe randomly selected an edge that affects the answer. The sum total of\nthe times taken to incrementally update individual answers\n(Incremental) for each query is reported in\nTable~\\ref{tab:yagooverall} and Table~\\ref{tab:gmarkoverall}. This is\ncontrasted against the sum total of the re-computation time\n(Recompute) for each individual answer under the specific edge\naddition. \\comment{We also report the average number of derivations\n added to an answer for every query. These numbers are reported for\n the \\textsc{HaPPI}\\xspace domain of answers in the \\emph{adaptive} strategy outlined\n in Sec.~\\ref{sec:adaptive}.}\n\nWe also report, in the same tables, the percentage of answers for the\nentire query that are computed using \\textsc{HaPPI}\\xspace. Since \\textsf{PS-KC}\\xspace is not\nmaintainable, the necessitated re-computation on the answers for which\nthe adaptive system uses \\textsf{PS-KC}\\xspace markedly pulls down the overall gain for\nqueries with large answers. While we got $10$-$25\\%$ gains for queries\nthat use \\textsc{HaPPI}\\xspace for all their answers, we get more than $5\\%$ gains for\nqueries where at least $90\\%$ of the answers are computed by\n\\textsc{HaPPI}\\xspace. Thus, the proposed adaptive system inherits the maintainability\nof \\textsc{HaPPI}\\xspace.\n\nWe also investigated the maintainability of \\textsc{HaPPI}\\xspace. We compared, on the\n\\textsc{HaPPI}\\xspace domain of answers, incremental time (IncrTime) versus\nre-computation (RecompTime) with \\textsc{HaPPI}\\xspace in Table~\\ref{tab:insertyago}\nand Table~\\ref{tab:insertgmark}. Notice that addition of an edge can\nresult in a variable number of derivations being added to an\nanswer. Since the incremental algorithm adds one derivation at a time\nto our symbolic probability expression, its gains over re-computation\ngo down as the average number of added derivations per answer goes\nup. We report a gain of at least $20\\%$ for all queries where average\nnumber of derivations added is less than $4$ but greater than\n$2$. Note that for answers with just $2$ derivations, re-computation is\nalmost equivalent to the iterative step. Thus the gains ($10$-$35\\%$) are\nmuted for queries with smaller averages ($< 2$). Queries with\nhigher average number of derivations have more modest gains\n($8$-$11\\%$).\n\n\\begin{table}[t]\n\t\\tabcaption{Comparison of incremental update time and re-computation time (in $\\mu$seconds) for YAGO2\\xspace query answers in \\textsc{HaPPI}\\xspace domain.}\n\t\t\\centering\n\n\n\t{\\small\n\t\t\\begin{tabular}{l|l|ll|l}\n\t\t\t\\toprule\n\t\t\t\\bf QId & \\multicolumn{1}{|p{2.5cm}|}{\\bf Average number of derivations affected} & \\bf RecompTime & \\bf IncrTime & \\bf Gain \\\\ \n\t\t\t\\midrule\n\t\t$Q_{1}$ & $1.33$\t & $6.99 \\times 10^2$ & $6.29 \\times 10^2$ &\t$0.101$\t\\\\\n\t\t\t\n\t\t$Q_{3}$ & $1.37$\t & $2.95 \\times 10^4$ & $2.11 \\times 10^4$ &\t$0.284$\t\\\\ \n\t\t\t\n\t\t$Q_{6}$ & $1.55$\t & $1.44 \\times 10^4$ & $1.12 \\times 10^4$ &\t$0.226$\t\\\\\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\n\t}\n\t\\label{tab:insertyago}\n\\end{table}\n\n\\begin{table}[t]\n\t\\tabcaption{Comparison of incremental update time and re-computation time (in $\\mu$seconds) for gMark\\xspace query answers in \\textsc{HaPPI}\\xspace domain.}\n\t\\centering\n\n\n\t{\\small\n\t\t\\begin{tabular}{l|l|ll|l}\n\t\t\t\\toprule\n\t\t\t\\bf QId & \\multicolumn{1}{|p{2.5cm}|}{\\bf Average number of derivations affected} & \\bf RecompTime & \\bf IncrTime & \\bf Gain \\\\ \t\n\t\t\t\t\t\\midrule\n\t\t\t\n\t$Q_{4}$ & $6.815$ & $6.23\\times 10^5$ & $5.62\\times 10^5$ & $0.097$ \\\\\n\t\t\n\t$Q_{6}$ & $2.015$\t&$9.79\\times 10^4$ & $7.00\\times 10^4$ & $0.285$ \\\\\n\t\t\n\t$Q_{9}$ & $1.530$ & $6.98\\times 10^3$ & $5.46\\times 10^3$ & $0.217$ \\\\\n\t\t\n\t$Q_{21}$ & $6.848$ & $1.10 \\times 10^5$ & $9.70\\times 10^4$ & $0.118$ \\\\\n\t\t\n\t$Q_{23}$ & $1.572$ & $3.80\\times 10^4$ & $2.50\\times 10^4$ & $0.342$ \\\\\n\t\t\n\t$Q_{32}$ & $1.000$ &$1.48\\times 10^3$ & $1.30\\times 10^3$ & $0.122$ \\\\\n\t\t\n\t$Q_{35}$ & $1.433$ & $7.12\\times 10^2$ & $5.95\\times 10^2$ & $0.164$ \\\\\n\t\t\n\t$Q_{38}$ & $1.320$ & $3.29\\times 10^5$ & $2.27\\times 10^5$ & $0.310$ \\\\\n\t\t\n\t$Q_{46}$ & $2.633$ &$5.28\\times 10^5$ & $3.28\\times 10^5$ & $0.379$ \\\\\n\t\t\n\t$Q_{54}$ & $2.048$ &$1.67\\times 10^4$ & $1.27\\times 10^4$ & $0.239$ \\\\\n\t\t\n\t$Q_{90}$ & $4.267$ & $1.01\\times 10^4$ & $9.31\\times 10^3$ & $0.078$ \\\\\n\n\t\t\t\\bottomrule\t\t\t\t\t\t\n\t\t\\end{tabular}\n\n\t}\n\t\\label{tab:insertgmark}\n\n\\end{table}\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nKnowledge graphs (KGs) are central to many real-life systems such as search\nengines, social networks, medical assistants, question answering, etc. Some of\nthese KGs are automatically built by employing a suite of knowledge extractors\nand integrators. Differences in various extraction approaches, inherent\nambiguities of the extraction process itself, and variations in the credibility\nof data sources make the automatically extracted facts uncertain. The\nuncertainty is encoded by assigning a \\emph{confidence score} to each fact,\nleading to \\emph{probabilistic knowledge graphs} such as YAGO2~\\cite{yago2},\nNELL~\\cite{NELL-aaai15}, ReVerb~\\cite{reverb}, Probase~\\cite{probase}, etc.\n\nIn addition, despite the phenomenal progress in information extraction\ntechniques, erroneous facts invariably creep into KGs. This, in turn, results\nin query answers being erroneous. For instance, the query for ``a list of\nAfrican comedians'' over the NELL KG includes ``Jimmy Fallon'' and ``Ellen\nDeGeneres'' (as per NELL extraction round \\#1082). Although both are comedians,\nneither of them are from Africa. To determine the source of the error, or to\ndebug the KG, it is necessary to compute the \\emph{provenance} of each result.\n\n\\begin{figure}[t]\n\\centering\n \\centering\n \\includegraphics[scale=0.18]{plot\/yago_binning.pdf}\n\t\\hfill{}\n \\includegraphics[scale=0.18]{plot\/gmark_binning.pdf}\n\t\\figcaption{Distribution of YAGO2\\xspace and gMark\\xspace query answers according to their total derivations and involved edges}\n\\label{fig:heat}\n\\end{figure}\n\nProvenance represents the derivation process of an answer. For probabilistic\ndata, typically, it is modeled as a Boolean formula in disjunctive normal form\n(DNF), where each conjunct encodes a derivation. This Boolean formula,\nreferred to as \\emph{lineage}, is also used for probability computation of the\nanswer by counting all satisfying assignments -- equivalent to the model\ncounting problem which is known to be \\#P-hard ~\\cite{roth96}. The brute-force\nway of counting satisfying assignments, known as possible world computation, is\nto iterate over all possible assignments taking time exponential in the number\nof the Boolean variables involved. For provenance in probabilistic graphs, a\nBoolean variable is associated with each edge.\nA more nuanced approach to probability computation is the \\emph{knowledge\ncompilation} technique that translates the lineage formula to a more tractable\nBoolean circuit using SAT solvers. Although it cannot guarantee scalability,\nfor large answers the use of compilation tools (e.g., C2D~\\cite{c2d},\nD4~\\cite{d4}, dSharp~\\cite{dsharp}) is known to be practical.\n \nHowever, for answers with a small number of Boolean variables in their\nderivation formula, the overhead of a SAT solver invocation results in a\nconsiderably poor performance. Na\\\"ively using knowledge compilation tools may,\nthus, fail to take advantage of small size of the computation problem. We\ninvestigated the results of query workloads from gMark\\xspace~\\cite{gmark} as well as\nYAGO2\\xspace~\\cite{yago2} to understand the extent to which this behavior can affect\nthe overall performance. Fig.~\\ref{fig:heat} summarizes the distribution of\nanswers with different number of derivations (along x-axis) and edge counts\n(along y-axis), suitably bucketed for readability. We use color to indicate the\nabsolute count of answers in each region (darker the color, more the count),\nand also print the raw count. It can be seen that most graph query answers are\nconcentrated in the area where the derivation and edge counts are\nlow---precisely where knowledge compilation is not the best method. In fact, in\nthis region, they are out-performed by even a possible-world computation that\nemploys brute-force evaluation.\n\n\nIn this paper, we specifically target this region where a large number of\nanswers are found, and present an algorithm for probability computations that\nsignificantly speeds up the performance there. We implement this in a system\nnamed \\textbf{\\textsc{HaPPI}\\xspace} (\\underline{H}ow \\underline{P}rovenance of\n\\underline{P}robabilistic \\underline{I}nference). \nThematically to the provenance semiring model~\\cite{provenancesemirings}, we\nalso introduce a novel semiring which enables \\textsc{HaPPI}\\xspace to symbolically compute the\nanswer probability. The proposed semiring facilitates computing \\emph{how}\nprovenance of not just the query answer but also of its probability\ncomputation. Unlike a Boolean formula that simply represents how edges interact\nto generate an answer, we additionally capture the arithmetic involved to\ncompute the exact probability. This fine-grained provenance information allows\nfor efficient maintenance in \\textsc{HaPPI}\\xspace.\n\n\\textsc{HaPPI}\\xspace outperforms knowledge compilation tools as well as possible world\ncomputation for answers of the kind found in the highly populated bottom left\nregion of Fig.~\\ref{fig:heat}. Since knowledge compilation techniques work\nbest for answers with high derivation and edge counts, we also propose an\n\\emph{adaptive system} that uses \\textsc{HaPPI}\\xspace for small answers and compilation\ntechniques for larger answers. We show that this adaptive system produces\nsizeable gains in performance over either system used in isolation. Further,\nsince the adaptive system uses \\textsc{HaPPI}\\xspace for a large number of answers, it inherits\nthe maintainability of \\textsc{HaPPI}\\xspace.\n\nThe key \\emph{contributions} of this paper are four-fold: \n\n\\begin{enumerate}\n \n \\item A novel theoretical model (Sec.~\\ref{sec:framework}) based on a semiring to support efficient probability computation over probabilistic KGs.\n\n\t\\item A practical implementation (Sec.~\\ref{sec:algo}),\n\t\t\\textsc{HaPPI}\\xspace\\footnote{\\url{https:\/\/github.com\/gaurgarima\/HaPPI}}, extends a provenance-aware property graph system,\nHUKA\\xspace\\footnote{\\url{https:\/\/github.com\/gaurgarima\/HUKA}}. Our algorithm can be also used in conjunction with other works, like HUKA\\xspace \\cite{huka,gaur2017} and ProvSQL\\xspace \\cite{provsql},\nbased on the same underlying system to expand their support for probabilistic data.\n\n\t\t\\item A theoretical analysis as well as an empirical evaluation\n\t\t\thighlighting the easy maintainability of \\textsc{HaPPI}\\xspace under insertion of\n\t\t\tedges (i.e., new facts) to the KG.\n\t\t\t\n\t\t\\item Finally, a proposal for an \\emph{adaptive} framework that\n\t\t\tleverages the superior performance of \\textsc{HaPPI}\\xspace and knowledge\n\t\t\tcompilation tools at different regions of answer sizes. Our\n\t\t\textensive empirical evaluation, using queries over gMark\\xspace,\na synthetic KG, and, YAGO2\\xspace, a real-world KG, shows that this adaptive system\n\t\t\toutperforms any existing method used in isolation. \n\n\\end{enumerate}\n\n\\section{Related Work}\n\\label{sec:relwork}\n\n\\subsubsection*{Query Processing on Probabilistic Database}\n\\label{subsec:probWork}\n\nThe problem of query processing on probabilistic databases is quite\nwell-studied \\cite{Cavallo, dichotomy, probabilisticGM, greenModel}. A\nfundamental result, aka the \\emph{dichotomy} theorem, stating that either a\nquery can be computed in polynomial time or it is provably \\#P-hard, was stated\nin \\cite{dichotomy}. \n\nSolutions addressing probabilistic inference can be categorized into two\nclasses: (a)~\\emph{extensional}: those that compile the inference over\nprobabilistic data into a query plan, and (b)~\\emph{intensional}: those that\ndirectly manipulate the probability derivation expression of each tuple in the\noutput making use of its provenance.\nExtensional techniques \\cite{dalvidb, top-k, Dey, deshpande} are known to be more\nefficient than intensional evaluation for the class of queries they can handle.\nExtensional solutions cannot process arbitrary conjunctive queries. They cannot\nhandle \\emph{self-joins} as the safe plan construction is based on the\nassumption that the relations participating in the plan are independent. A self\njoin query violates this assumption. Few practical systems such as\nMystiQ~\\cite{mystiq} and Orion~\\cite{orion} have adopted extensional techniques\nfor simple queries. For hard queries, they rely on approaches based on\napproximation algorithms~\\cite{simulation,mcdb}. \n\nEvent tables, which eventually become a standard way of data modeling of\nintensional techniques, were introduced in~\\cite{fuhr, esteban}. Green et\nal.~\\cite{provenancesemirings} proposed a generalized semiring model to\nannotate probabilistic tuples with Boolean variables.\nTrio~\\cite{trio}, one of the early systems using intensional approach, relied\non data lineage to compute the probabilities of simple queries~\\cite{trio_prob}\nand Monte-Carlo simulation~\\cite{simulation} for complex ones. Other notable\nsystems equipped to process queries over probabilistic database are\nMayBMS~\\cite{maybms} and SPROUT~\\cite{sprout}. Recently, Senellart et al.\npresented ProvSQL\\xspace~\\cite{provsql}, a framework strictly adhering to the\nsemiring model~\\cite{provenancesemirings, m-semirings} to represent\nprobabilistic data. For probabilistic inference, it supports three standard\ntechniques: possible world computation, knowledge\ncompilation~\\cite{compilation} and Monte-Carlo simulation~\\cite{simulation}.\n\nProvSQL\\xspace can be considered as closest to our\nsystem in terms of data modeling as we also use the semiring based framework\nfor data modeling. Unlike our focus on knowledge graphs, ProvSQL\\xspace addresses the problem\nfor relational databases. \nFurther, we equipped our solution with an efficient symbolic probability computation technique using a novel semiring. Note that our proposal can be easily incorporated into the ProvSQL\\xspace framework. \n\n\\subsubsection*{Query Provenance}\n\nThe capability of provenance to provide better insight into the query result\nhas resulted into various provenance aspects. These include \\emph{why}\nprovenance~\\cite{whyandwhere} explaining why an answer is part of a query\nresult, \\emph{how} provenance~\\cite{provenancesemirings} providing information\nabout the derivation process of an answer, and \\emph{when}\nprovenance~\\cite{whenprovenance} for tracking temporal data. In this work, our\nintent is to track the provenance (derivation process) of a query pattern match as well as\nthat of probability computation. Complying with the provenance semiring model \\cite{provenancesemirings}, we propose a novel semiring which, unlike the $PosBool$\nsemiring, captures the \\emph{how} provenance of the probability\n\\emph{computation} as well.\n\n\n\\section{Theoretical Framework}\n\\label{sec:framework}\n\nWe propose a framework to produce a symbolic expression corresponding to each\nquery answer. The evaluation of this expression with the edge probabilities\nsubstituted will be the probability of the answer. This is in contrast to\ncomputation via PosBool \\cite{provenancesemirings}. There is a layer of\nprobability computation starting from the PosBool expression~\\cite{trio_prob,\nfuhr} which is not a mere substitution of values.\nThe major attraction of semiring frameworks is that the construction of their\nanswer representations can be piggybacked onto query evaluation. Importantly,\nour symbolic probability expressions lie in a semiring with a homomorphic\ncorrespondence with PosBool.\n\nWe adopt the \\emph{provenance semiring} model\n\\cite{provenancesemirings} to generate these expressions. Similar to\nthe $PosBool$ semiring, we annotate each edge $e_{i}$ with a random\nvariable $X_{i}$, indicative of the presence $(X_{i}=1)$ with\nprobability $p_i$ or absence $(X_{i}=0)$ with probability $(1 - p_i)$\nof the edge. In our framework, probabilities of events in $PosBool$\nare computed and stored as polynomials over indeterminates $\\{p_i\\}$.\n\nIn the $PosBool$ framework each derivation is associated\nwith the \\emph{conjunct} of Boolean variables associated to the\ninvolved edges. Here, the presence of each derivation $d_{i}$ of an\nanswer is interpreted as an event $D_{i}$ which, in turn, is defined by\nthe presence of edges involved in the derivation. The presence of an\nanswer will rely on the presence of at least one of these\nderivations. Thus, the probability of the answer is computed as\n$Pr(\\bigvee_{i} D_{i})$. For instance, in Table~\\ref{tab:result},\nanswer (SIN,MUN) has two derivations: $d_{1}= \\{e_{1},e_{3}\\}$ and\n$d_{2}= \\{e_{2},e_{3}\\}$. The probability of event $D_{1}$ is\nthe product of the probabilities of edges $e_{1}$ and $e_{2}$ and that\nof $D_{2}$ of $e_{2}$ and $e_{3}$. The probability of (SIN,MUN) being\na part of the result is computed as $Pr(D_{1} \\vee D_{2})$.\n\nAn incremental step in computing the probability of a PosBool event can be\ncomputing for a disjunct $Pr\\left(E_1 \\vee E_2\\right) = Pr\\left(E_1\\right) +\nPr\\left(E_2\\right) - Pr\\left(E_1 \\wedge E_2\\right)$ or for a conjunct\n$Pr\\left(E_1 \\wedge E_2\\right)$.\n\nNotice that, in either case, computing $Pr\\left(E_1 \\wedge E_2\\right)$\nseems to necessitate keeping track of the exact events $E_1$ and\n$E_2$. We show, however, that this is not necessary. Just the symbolic\nprobability expressions of the event $E_1$ and $E_2$ are enough for us\nto compute probabilities incrementally. The standard edge independence\nmodel is crucial for the correctness of this claim.\n\nWe start formally defining the semiring and presenting a semiring\nhomomorphism from $PosBool$ to our semiring.\nOur domain of interest is the polynomial ring with integer\ncoefficients $Z[p_1, \\ldots, p_n]$. We are interested in a subset of\nthis domain which we define as follows:\n\n\\begin{definition}[Flat Monomial; Flat Polynomial]\n\t%\n\tWe call a monomial \\emph{flat} if it is a product of distinct variables. A\n\t\\emph{flat} integer polynomial is defined to be a sum of flat monomials.\n\t%\n\\end{definition}\n\nThe function $flat$ flattens out a polynomial by reducing every\nexponent greater than 1 to 1. Formally, for $k \\in \\mathbb{Z}$\n \\begin{align}\n flat(k) &= k\\\\\n flat(f + g) &= flat(f) + flat(g) \\\\\n flat\\left(c \\cdot \\prod_{i} p_{i}^{k_i}\\right) &= c \\cdot \\prod_{i} p_{i} \n\\end{align}\n where $k_i > 0$ for all $i, 0 < i \\le n$ and $c \\in \\mathbb{Z}$\n\nWe use the notation $\\overline{f}$ for $flat(f)$ here on. \n\nWe propose two non-standard operators $\\oplus$\nand $\\otimes$ on the set of flat integer polynomials. For flat\npolynomials $f$ and $g$, the two operators are defined as follows:\n\n\\begin{itemize}\n\t\\item $f \\otimes g = \\overline{f \\times g}$\n\t\\item $f \\oplus g = f + g - (f \\otimes g)$\n\\end{itemize}\nwhere $+$, $-$ and $\\times$ are the standard operations on\npolynomials. Notice that the resultants of these operations are also\nflat polynomials.\n\nLet $Z_F[p_1, p_2, \\ldots ,p_n]$ denote the set of flat integer\npolynomials generated by $\\{1,p_1,p_2,\\ldots,p_n\\}$ and the operators\n$\\otimes$ and $\\oplus$. This is our domain of interest.\n\n\\begin{theorem}\\label{thm:semiring}\n $(Z_F[p_1, p_2, \\ldots ,p_n], \\oplus, \\otimes, 0,1)$ is a commutative semiring.\n\\end{theorem}\n\nWe defer the proof to Appendix \\ref{appendix1}. \n\nNext, we define a map $H$ from $(PosBool(X_1, X_2, \\ldots, X_n),$\n$\\lor, \\land, 0, 1)$ to $(Z_F[p_1, p_2, \\ldots, p_n], \\oplus, \\otimes,\n0,1)$. This is done inductively on the structure of $PosBool$.\n\nFor $\\theta_i, \\theta_j \\in PosBool(X_1,X_2,\\ldots,X_n)$,\n\\begin{align}\n\tH(X_{i}) &= p_{i} \\\\\n\tH(\\theta_{i} \\land \\theta_{j}) &= H(\\theta_{i}) \\otimes\n H(\\theta_{j}) \\\\\n\tH(\\theta_{i} \\lor \\theta_{j}) &= H(\\theta_{i}) \\oplus\n H(\\theta_{j}) \n\\end{align}\n\nAs defined, the above map is not yet guaranteed to be\nwell-defined. Elements of $PosBool$ obey relations beyond the usual\nsemiring axioms. For instance, consider that for arbitrary formulae\n$E, F$ in $PosBool$, $E \\lor (E \\land F) = E$ (absorption), or that\n$\\lor$ also distributes over $\\land$. These relations are precisely\nwhy, for example, there is no canonical map from $PosBool$ to the\n\\emph{WHY}-provenance semiring~\\cite{whyandwhere}. That $H$ is\nwell-defined would imply that these relations also hold in our\nsemiring $Z_F$. This is not obvious and turns out to be non-trivial to\nprove directly. \\comment {While we omit this proof and the proof of\n Theorem~\\ref{thm:symprob} for lack of space,} The well-definedness\nof $H$ and Theorem ~\\ref{thm:symprob} are best proved in tandem.\n\n\n\\begin{theorem} \\label{thm:symprob}\n For all formulae $E$ in $PosBool$\n $$ Pr(E) = H(E)\\left(Pr(X_1), \\ldots, Pr(X_n)\\right) $$\n where the RHS denotes the flat polynomial $H(E)$ evaluated at $p_i = Pr(X_i)$ for $1 \\le i \\le n$.\n \n\\end{theorem}\n\nThis asserts that the probability of a formula $E$ in $PosBool$ is,\nsymbolically, the flat polynomial associated to it by $H$.\n\n\n\\subsection{Proof Sketch of Theorem~\\ref{thm:symprob}}\n\\label{subsec:correctness}\n\nFor positive Boolean expressions $E_1$ and $E_2$ let $E_1\n\\overset{s.r}{\\leadsto} E_2$ mean that $E_1$ can be reduced to $E_2$\nusing only semiring axioms. Given that $H$ is defined structurally\nover $(\\wedge, \\vee)$ and that $Z_F$ is a semiring itself, it is clear\nthat\n\\begin{align}\\label{srreduce}\n E_1 \\overset{s.r}{\\leadsto} E_2 \\implies H(E_1) = H(E_2)\n\\end{align}\n\nWe reduce both the proof of well-definedness of $H$ and of\nTheorem \\ref{thm:symprob} to the special case of Theorem~\\ref{thm:symprob} for formulae in DNF form. Notice that any\nBoolean expression can be reduced to DNF form using only semiring\naxioms (a simple inductive argument shows we only need left and right\ndistributivity of $\\wedge$ over $\\vee$).\n\nFor formulae $E_1$ and $E_2$\nthat are equivalent in $PosBool$, let $F_1$ and $F_2$ be formulae in\nDNF form such that $E_1 \\overset{s.r}{\\leadsto} F_1$ and $E_2\n\\overset{s.r}{\\leadsto} F_2$. Note that $E_1$, $E_2$, $F_1$ and $F_2$ are all\nequivalent in $PosBool$. Consider the chain of equalities:\n\\begin{align}\n H(E_1) = H(F_1) \\overset{?}{=} Pr(F_1) = Pr(F_2) \\overset{?}{=}\n H(F_2) = H(E_2) \\label{eqn:chain}\n\\end{align}\nThe middle equality follows from the equivalence of $F_1$ and $F_2$.\nThe equalities marked $\\overset{?}{=}$ are the only ones that we have not\nshown yet. These are invocations of Theorem~\\ref{thm:symprob}\nfor formulae in DNF form. If these are established, we would have proved that\n$H$ is well-defined ($H(E_1) = H(E_2)$) and also Theorem~\\ref{thm:symprob} ($Pr(E_1) = Pr(F_1) = H(E_1)$). \n\nWe, thus, proceed to prove Theorem~\\ref{thm:symprob} for formulae in DNF\nform (without assuming well-definedness of $H$).\n\n\\begin{lemma}\n\t\\label{lemma:conjunct}\n For Boolean variables $\\theta_i$ in $PosBool$ \n $$ Pr(\\bigwedge \\limits_{i} \\theta_{i}) = H(\\bigwedge \\limits_{i}\n \\theta_{i}) $$\n\\end{lemma}\n\nThis is Theorem \\ref{thm:symprob} for pure conjuncts.\n\n\\begin{proof}\n Let $I = \\{\\,i \\mid X_i = \\theta_j \\text{ for some } j\\,\\}$. Recall\n that in a KG existence of different edges are independent\n events. Since this is what we are modeling all along, our random\n variables $X_i$ are also independent. Thus,\n $$Pr(\\bigwedge \\limits_{i} \\theta_{i}) = \\prod_{j \\in I} p_j $$\n Also,\n $$H(\\bigwedge \\limits_{i} \\theta_{i}) = \\underset{i}{\\otimes}\\,\n H(\\theta_i) = \\underset{i}{\\otimes}\\, p_i= \\prod_{j \\in I} p_i$$ The\n last equality is a consequence of how flattening of polynomials works.\n\t\\hfill{}\n\\end{proof}\n\n\\begin{lemma}\n For conjuncts of Boolean variables $C_i$\n \n$$Pr(\\bigvee \\limits_{i} C_{i}) = H(\\bigvee \\limits_{i}\n C_{i})$$\n\n This is Theorem \\ref{thm:symprob} for formulae in DNF form.\n\\end{lemma}\n\n\\begin{proof}\n We prove this by induction on the size of the disjunction. The base\n case is simply Lemma \\ref{lemma:conjunct}. Consider,\n {\\small\n \\begin{align*}\n Pr\\left(\\bigvee \\limits_{i = 1}^{n} C_i\\right) &= Pr \\left(\\left(\\bigvee \\limits_{i = 1}^{n-1} C_i\\right) \\vee C_n\\right) \\\\\n &= Pr \\left(\\bigvee \\limits_{i = 1}^{n-1} C_i\\right) + Pr(C_n) - Pr \\left(\\bigvee \\limits_{i = 1}^{n-1} {\\left(C_i\\wedge C_n\\right)}\\right)\n \\end{align*}\n }%\n Applying induction hypothesis\n{\\scriptsize \\begin{align*} \nPr\\left(\\bigvee \\limits_{i = 1}^{n} C_i\\right) &= H\\left(\\bigvee \\limits_{i = 1}^{n-1} C_i\\right) + H(C_n) - H \\left(\\bigvee \\limits_{i = 1}^{n-1} {\\left(C_i\\wedge C_n\\right)}\\right) \\\\\n\n\t &= H\\left(\\bigvee \\limits_{i = 1}^{n-1} C_i\\right) + H(C_n) - \\left( H \\left(\\bigvee \\limits_{i = 1}^{n-1} C_i \\right) \\otimes H\\left(C_n\\right) \\right) \\\\\n &= H\\left(\\bigvee \\limits_{i = 1}^{n-1} C_i\\right) \\oplus H(C_n)\n= H\\left(\\bigvee \\limits_{i = 1}^{n} C_i\\right)\n \\end{align*}\n}%\n\\end{proof}%\nHaving proved Theorem~\\ref{thm:symprob} for formulae in DNF form, we have the full equality chain in Eq.~\\eqref{eqn:chain}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{Introduction} \\label{sec:intro}\nA classical nova is a transient event involving an accreting white dwarf in a binary star system (e.g., \\citealt{2008clno.book.....B}). Once the pressure and temperature at the base of the accreted envelope reach a critical level, a thermonuclear runaway is triggered on the surface of the white dwarf, leading to the ejection of at least part of the envelope. \nTypically, $10^{-7}-10^{-4}$ M$_{\\odot}$ of material is ejected at velocities ranging between 500 and 5,000 km\\,s$^{-1}$ \\citep[e.g.,][]{Payne-Gaposchkin_1957,Gallaher_etal_1978, Yaron_etal_2005}. Remnants of the accreted envelope remain on the white dwarf's surface and continue nuclear burning for weeks to years after the thermonuclear runaway ends, bathing the ejecta with luminous ionizing radiation from within ($\\sim 10^{38}$ erg s$^{-1}$; \n\\citealt{Wolf_etal_2013}). Early in the nova's evolution, the ejecta are optically thick, and as the thermal emission from the white dwarf diffuses through the ejecta, the nova's spectral energy distribution peaks in the optical band. As the ejecta expand, their density drops, they become more optically-thin, and the peak of the nova's spectral energy distribution moves blueward \\citep{Gallagher_Code_1974}. \nWhen the white dwarf is finally revealed, the nova is considered a supersoft X-ray source (emitting at photon energies $<0.5$\\,keV), and it will remain in this state for days--years, until the residual fuel is all burnt \\citep{Krautter_2008,2011ApJS..197...31S,Page_Osborne_2014,Osborne_2015}. After a period of time, accretion will resume and the process restarts. All novae are theorized to recur, but some novae have been observed to erupt more than once during our observational records; these are known as recurrent novae.\n\n\\par\nThe discovery of GeV $\\gamma$-rays from nova V407~Cyg with the Large Area Telescope (LAT) on the \\textit{Fermi Gamma-Ray Space Telescope} (henceforth \\emph{Fermi}) has opened the door for a whole new realm of nova research \\citep{2010Sci...329..817A}. At first, the $\\gamma$-rays were thought to be the result of the ejecta interacting with the dense wind of V407~Cyg's Mira giant companion, and not a feature of typical nova systems \\citep[e.g.,][]{2011MNRAS.410L..52M, Nelson_etal_2012}. However, the discovery of $\\gamma$-rays from V959~Mon, V1324~Sco, and V339~Del with \\textit{Fermi}-LAT in the following years revealed that V407~Cyg was not a singular case \\citep{Ackermann+14}. Unlike V407~Cyg, these systems contain main-sequence companions, so the $\\gamma$-rays could not be coming from the ejecta interacting with a dense circumbinary medium. \n\n\\par\nSince 2013, even more novae with main sequence companions have been detected in the GeV $\\gamma$-ray band. These observations reveal that shocks are common in nova eruptions and that they are energetically important \\citep{Li_etal_2017,Aydi2020}. As the majority of \\textit{Fermi}-detected novae have dwarf (rather than giant) companions and low-density circumbinary material, the $\\gamma$-ray emitting shocks must be internal to the nova ejecta. From high-resolution radio imaging of the nova V959~Mon, it was found that these shocks may occur at the interface between a slow, dense, equatorial torus and a fast biconical wind \\citep{Chomiuk_etal_2014}. The shocks produced at these interfaces accelerate particles to relativistic speeds via the diffusive shock mechanism and lead to GeV $\\gamma$-ray emission \\citep{2015MNRAS.450.2739M}. These internal shocks have velocities $\\sim$1000 km s$^{-1}$, and consequently heat the post-shock gas to temperatures of $\\sim 10^7$ K, which emits relatively hard ($\\gtrsim$1\\,keV; compared to the supersoft component) X-rays. Even before $\\gamma$-rays were detected in novae, hard X-ray emission was observed and interpreted as an indication of shock interaction (e.g., \\citealt{2001ApJ...551.1024M,2008ApJ...677.1248M}).\n\nIn the last two decades, the X-ray Telescope (XRT; \\citealt{2005SSRv..120..165B}) on the \\emph{Neil Gehrels Swift Observatory} (hereafter \\textit{Swift}; \\citealt{2004ApJ...611.1005G}) has been instrumental in providing observations for novae in the 0.3\\,--\\,10\\,keV band at relatively high cadence (e.g., \\citealt{2019arXiv190802004P}).\nFor example, V407 Cyg showed hard X-ray emission during its first months of evolution concurrent with the $\\gamma$-ray producing phase; as with the $\\gamma$-rays, this is likely a result of the nova ejecta interacting with the secondary's wind \\citep{2010Sci...329..817A, Orlando_Drake12, 2012ApJ...748...43N}. \n\nThe hard X-ray behavior of classical novae with main sequence companions is less clear, especially while $\\gamma$-rays are being detected. \n\\textit{Swift} observed V1324~Sco while GeV $\\gamma$-rays were detected, but failed to detect any X-rays \\citep{Finzell_etal_2018}. \nBased on this non-detection, \\citet{2014MNRAS.442..713M} theorized that the X-ray emission from classical novae during the $\\gamma$-ray period would be absorbed by the initially dense ejecta.\nThese absorbed X-rays are then reprocessed and re-emitted as UV and optical photons, contributing to the luminosity in those bands (supporting this hypothesis, correlated $\\gamma$-ray and optical light curves have been observed in two novae to date; \\citealt{Li_etal_2017,Aydi2020}). Once the ejecta expand enough and the optical depth decreases, X-rays are allowed through.\n\n\nSimilar to V1324~Sco, there are hints from other novae that 1--10\\,keV X-rays were not detectable by \\emph{Swift} until $\\gtrsim$1 month after eruption (e.g., \\citealt{2016A&A...590A.123S}, \\citealt{2018ApJ...853...27M}), but the X-ray light curves were not explicitly discussed in the context of $\\gamma$-rays and shocks. \nInterestingly, harder X-rays ($>$10\\,keV) from novae have begun to be detected with \\emph{NuSTAR} concurrent with $\\gamma$-rays\n\\citep{Nelson_etal19, 2020NatAs...4..776A, 2020MNRAS.497.2569S}, but at surprisingly low fluxes (the implications of these observations will be discussed in Section \\ref{sec:disc_conc}). Despite the rapid response and agility of \\textit{Swift} that make it ideal for observations during the early weeks of nova eruptions (when novae are bright in GeV $\\gamma$-rays), no systematic study has been carried out of \\emph{Swift}-XRT observations of $\\gamma$-ray detected novae.\nIt is the goal of this paper to test if all classical novae are faint in the 1--10 keV X-ray band during $\\gamma$-ray detection.\n\n\n\n\n\n\n\\begin{deluxetable*}{lccccc}[t]\n\\tabletypesize{\\small}\n\\tablewidth{0 pt}\n\\tablecaption{Characteristics of $\\gamma$-ray detections of novae (2010--2018). \n\\label{table:gamma}}\n\\tablehead{Nova & Time$_{\\rm \\gamma-ray\\ start}$ & Time$_{\\rm \\gamma-ray\\ end}$ & $\\gamma$-ray Flux & Photon index & Reference\\\\\n & (MJD) & (MJD) & (10$^{-7}$ photon s$^{-1}$ cm$^{-2}$) & }\n\\startdata\nV392 Per & 58238\\tablenotemark{b} & 58246 & $2.2\\pm0.4$ & $2.0\\pm0.1$ & 10,11\\\\\nV906 Car & 58216\\tablenotemark{a} & 58239--58250\\tablenotemark{a} & $12.2\\pm0.4$ & $2.04\\pm0.02$ & 9 \\\\\nV357 Mus & 58129 & 58156 & $1.3\\pm0.2$ & $2.2\\pm0.1$ & 8, This work\\\\\nV549 Vel & 58037 & 58070 & $0.4\\pm0.2$ & $1.8\\pm0.2$ & 6,7 \\\\\nV5856 Sgr & 57700 & 57715 & $4.6\\pm0.5$ & $2.11\\pm0.05$ & 5 \\\\\nV5855 Sgr & 57686 & 57712 & $3.0\\pm0.8$ & $2.26\\pm0.12$ & 4 \\\\\nV407 Lup & 57657 & 57660 & $1.6\\pm0.7$ & $2.2\\pm0.3$ & 3, This work \\\\\nV5668 Sgr & 57105 & 57158 & $1.1\\pm0.2$ & $2.42\\pm0.13$ & 2 \\\\ \nV1369 Cen & 56634 & 56672 & $2.5\\pm0.4$ & $2.37\\pm0.09$ & 2 \\\\\nV339 Del & 56520 & 56547 & $2.3\\pm0.3$ & $2.26\\pm0.08$ & 1 \\\\\nV959 Mon & 56097 & 56119 & $4.8\\pm0.6$ & $2.34\\pm0.09$ & 1 \\\\\nV1324 Sco & 56093 & 56110 & $5.9\\pm0.9$ & $2.16\\pm0.09$ & 1 \\\\\nV407 Cyg & 55265 & 55287 & $5.8\\pm0.6$ & $2.11\\pm0.06$ & 1 \\\\\n\\enddata\n\\tablenotetext{}{References: 1= \\cite{Ackermann+14}; 2= \\cite{2016ApJ...826..142C}; 3= \\cite{2016ATel.9594....1C}; 4= \\cite{Nelson_etal19}; 5= \\cite{Li_etal_2017}; 6= \\cite{2017ATel10977....1L}; 7= \\cite{2020arXiv201010753L}; 8= \\cite{2018ATel11201....1L}; 9= \\cite{Aydi2020}; 10= \\cite{2018ATel11590....1L}; 11= Blochwitz, Linnemann et al.\\ 2020, in prep.}\n\\tablenotetext{a}{Due to \\emph{Fermi}-LAT downtime, the start time of $\\gamma$-ray detection for V906~Car was not captured, and the end time is only constrained to be within a date range. The $\\gamma$-ray flux is calculated over MJD 58216--58239.}\n\\tablenotetext{b}{Due to \\emph{Fermi}-LAT downtime, data are not available for MJD 58224-58238. When observations resumed on MJD 58238, V392 Per was immediately detected. The $\\gamma$-ray flux is calculated over MJD 58238--58246.}\n\\end{deluxetable*}\n\n\\par Previous studies have been carried out on large collections of novae in the supersoft X-ray phase using observations from \\emph{Swift}-XRT,\nincluding \\citet{2007ApJ...663..505N,2011ApJS..197...31S}, and \\citet{2019arXiv190802004P}. However, systematic studies of the harder X-ray component, or the X-ray behavior of $\\gamma$-ray detected novae, are lacking. \nIn this paper, we present a systematic study of 13 Galactic novae which have been detected by \\textit{Fermi}-LAT between 2010 and 2018 and have been observed by \\textit{Swift}-XRT. In Section~\\ref{sec:style} we discuss the sample selection and the multi-wavelength properties of the novae in our sample. \\textit{Fermi}-LAT data for most novae in our sample have already been published, but we present the first $\\gamma$-ray analysis of novae V407~Lup and V357~Mus.\nIn Section~\\ref{subsec:x-ray light ces} we present the \\textit{Swift}-XRT observations, emphasizing the hard X-ray emission during the $\\gamma$-ray detection phase.\nIn Section~\\ref{sec:disc_conc} we discuss what can be learned about nova shocks from observations concurrent with $\\gamma$-rays, and in Section~\\ref{sec:conc} we conclude.\n\n\n\\section{Our Sample of $\\gamma$-ray Detected Novae} \\label{sec:style}\n\n\\subsection{Sample Selection}\nIn this paper, we analyze all Galactic novae observed by \\emph{Fermi}-LAT between 2010 and 2018 that have a time-integrated detection $\\geq3 \\sigma$ significance over the period of $\\gamma$-ray emission. Details of the sample are listed in Table \\ref{table:gamma}. Despite hints that they produced $\\gamma$-ray emission, we do not include novae V745~Sco, V697~Car, or V1535~Sco in our sample because their \\emph{Fermi}-LAT detections were $<$3$\\sigma$ significance \\citep{Franckowiak_etal_2018}.\n\n\\subsection{$\\gamma$-ray Properties}\\label{sec:gamma}\nParameterizations of the $\\gamma$-ray light curves for our 13 novae are provided in Table \\ref{table:gamma}, taken from references listed therein. Time$_{\\rm \\gamma-ray\\ start}$ and time$_{\\rm \\gamma-ray\\ end}$ denote the time range during which $\\gamma$-rays are detected at $>$2$\\sigma$ significance when binning \\emph{Fermi}-LAT light curves with 1-day cadence. The $\\gamma$-ray flux column lists the average flux over this time period, fitting a single power law to the data over the energy range $>$100 MeV. Table \\ref{table:gamma} also lists the photon index for a single power law fit to the \\emph{Fermi}-LAT data with energy $>$100 MeV: \n\\begin{equation}\n \\dfrac{dN}{dE} \\propto E^{-\\Gamma},\n\\end{equation}\nwhere $N$ is the number of photons, $E$ is the photon energy, and $\\Gamma$ is the photon index. Although a single power law may not be the most physically motivated model, it is the simplest (most justified in cases of low S\/N), and most widely quoted in studies of the various novae. It is sufficient for estimating $\\gamma$-ray luminosities to the precision required for this study; modelling with a more complex exponentially-cutoff power law spectrum yields fluxes 75--85\\% that of a simple power law \\citep{Ackermann+14}.\n\nThe $\\gamma$-ray detections of novae V407~Lup and V357~Mus have been announced in \\cite{2016ATel.9594....1C} and \\cite{2018ATel11201....1L} respectively, but a full analysis of their light curves has not yet been published. We therefore provide this analysis here, in the following sub-sections\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{V407_Lup_optical_gamma_plot.pdf}\n \\caption{Top panel: the $\\gamma$-ray light curve ($>$100 MeV) of V407 Lup. Bottom panel: the $V$-band optical light curve of V407 Lup. $t_0$ is taken to be the time of discovery, 2016 Sep 24.}\n \\label{fig:V407 Lup gamma light curves}\n\\end{figure*}\n\n\\subsubsection{\\textit{Fermi}-LAT data reduction}\n\nWe downloaded the LAT data (Pass 8, Release 3, Version 2 with the instrument response functions of \\texttt{P8R3\\_SOURCE\\_V2}) from the data server at the \\textit{Fermi Science Support Center} (FSSC). \nFor data reduction and analysis, we used \\texttt{fermitools} (version 1.0.5) with \\texttt{fermitools-data} (version 0.17)\\footnote{\\url{https:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/software\/}}. For data selection, we used a region of interest $14^\\circ$ on each side, centered on the nova.\nEvents with the class \\texttt{evclass=128} (i.e., SOURCE class) and the type \\texttt{evtype=3} (i.e., reconstructed tracks FRONT and BACK) were selected. We excluded events with zenith angles larger than $90^\\circ$ to avoid contamination from the Earth's limb. The selected events also had to be taken during good time intervals, which fulfils the \\texttt{gtmktime} filter \\texttt{(DATA\\_QUAL$>$0)\\&\\&(LAT\\_CONFIG==1)}. \n\nNext, we performed binned likelihood analysis on the selected LAT data. For each nova, a $\\gamma$-ray emission model for the whole region of interest was built using all of the 4FGL cataloged sources located within $20^\\circ$ of the optical position \\citep{2019arXiv190210045T}. \nAs the two novae were the brightest $\\gamma$-ray sources in the fields (within at least 5 degrees according to the preliminary results), we only freed the normalization parameters for those cataloged sources located less than 1 degree from the targets.\nIn addition, the Galactic diffuse emission and the extragalactic isotropic diffuse emission were included by using the Pass 8 background models \\texttt{gll\\_iem\\_v07.fits} and \\texttt{iso\\_P8R3\\_SOURCE\\_V2\\_v1.txt}, respectively, which were allowed to vary during the fitting process. \n\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{V357_Mus_optical_gamma_plot.pdf}\n \\caption{Top panel: the $\\gamma$-ray light curve ($>$100 MeV) of V357 Mus. Bottom panel: the $V$-band optical light curve of V357 Mus. $t_0$ is taken to be the time of first observation in outburst, 2018 Jan 3.}\n \\label{fig:V357 Mus gamma light curve}\n\\end{figure*}\n\n\\subsubsection{$\\gamma$-rays from V407~Lup} \\label{sec:lup}\n\nNova V407 Lup (ASASSN-16kt) was discovered by the All-Sky Automated Survey for Supernovae (ASAS-SN) on 2016 September 24.0 UT at $V$ = 9.1 (\\citealt{2016ATel.9538....1S, 2018MNRAS.480..572A}). The nova was first detected in $\\gamma$-rays on the same day as its optical discovery but did not reach a significance of $\\geq$3$\\sigma$ until the next day, when the detection significance reached $4\\sigma$ (with a Test Statistic $TS = 16.6$). \nThe average flux over the detection duration was $(1.6 \\pm 0.7)\\,\\times\\,10^{-7}$ photon s$^{-1}$\\,cm$^{2}$. A single power-law fit gives a photon index of $2.2\\pm0.3$. \n\nThe $\\gamma$-ray light curve, shown in the top panel of Figure~\\ref{fig:V407 Lup gamma light curves}, shows a decrease in flux, completely fading below the LAT detection limit by 5 days after discovery. This makes V407~Lup's $\\gamma$-ray duration the shortest known to date for a nova (Table \\ref{table:gamma}). The bottom panel of Figure \\ref{fig:V407 Lup gamma light curves} shows the optical behavior of V407 Lup in the V-band during and shortly after the $\\gamma$-ray period.\nThis light curve is constructed from publicly available photometry from the American Association of Variable Star Observers (AAVSO; \\citealt{Kafka20}) and the Stony Brook\/SMARTS Atlas (SMARTS photometry can be found at Ref.\\footnote{\\url{http:\/\/www.astro.sunysb.edu\/fwalter\/SMARTS\/NovaAtlas\/}}; \\citealt{Walter_etal_2012}), along with \\citet{2016ATel.9550....1C} and \\citet{2016ATel.9564....1P}.\nThe optical light curve rapidly declines alongside the $\\gamma$-rays, exhibiting the shortest $t_2$ value of any nova in our sample (Table \\ref{table:characteristics}).\nThe light curve shows what could possibly be a lag in the $\\gamma$-ray emission compared to the optical. However, the date of the optical peak as estimated by \\citet{Aydi_etal_2018} is MJD 58656.4, implying that the first 3$\\sigma$ $\\gamma$-ray detection lags the optical peak by only 0.6 days. Based on the cadence of the optical and $\\gamma$-ray observations, which is around 0.5 days, this delay may be insignificant. A delay between the optical and $\\gamma$-ray emission, if it exists, would have significant implications on our understanding of shock formation and $\\gamma$-ray emission in novae (see e.g., \\citealt{2015MNRAS.450.2739M} and \\citealt{Aydi2020}).\n\n\n\\subsubsection{$\\gamma$-rays from V357~Mus} \\label{sec:mus}\n\nNova V357 Mus was discovered in the optical on 2018 Jan 14.5 UT at $\\sim$7 mag \\citep{cbet4473}. It was first detected in $\\gamma$-rays eight days later \\citep{2018ATel11201....1L}. The average $>$100 MeV flux over the detection period was $(1.3\\pm0.2) \\times 10^{-7}$ photon s$^{-1}$ cm$^{2}$, and the photon index from fitting a single power law was $\\Gamma = 2.2\\pm0.1$. The detection significance was $10\\sigma$ over this period (with a Test Statistic $TS = 98.6$).\n\nThe $\\gamma$-ray light curve is shown in the top panel of Figure~\\ref{fig:V357 Mus gamma light curve}. There may be variability of a factor of $\\sim$2 in the light curve, but the low S\/N makes it challenging to confidently measure this variability. The corresponding optical light curve is shown in the bottom panel of Figure \\ref{fig:V357 Mus gamma light curve}, with data from ASAS-SN \\citep{Shappee_etal_2014}, the Stony Brook\/SMARTS Atlas, and AAVSO. The nova was detected by ASAS-SN on the rise to optical maximum but quickly became so bright that it saturated the detectors. Observations resumed around day 10 when amateur observers found the nova and began taking data \\citep{cbet4473}. The nova likely reached a magnitude brighter than 6\\,mag at optical maximum (which was sometime between 0--10 days after the ASAS-SN pre-maximum detection). While the cadence and S\/N of the $\\gamma$-ray and optical light curves are not high enough to confirm, this nova may show evidence of correlated variation between the optical and $\\gamma$-ray light curves, similar to the two brightest $\\gamma$-ray novae V906 Car \\citep{Aydi2020} and V5856~Sgr \\citep{Li_etal_2017}. \n\n\n\n\\begin{longrotatetable}\n\\begin{deluxetable*}{lcccccccccc}\n\\tabletypesize{\\small}\n\\tablewidth{0 pt}\n\\tablecaption{\\label{table:characteristics} Nova Properties}\n\\tablehead{Name & $t_0$\\tablenotemark{a} & $t_0$\\tablenotemark{a} & Discovery Mag\\tablenotemark{b} & $V_{max}$ & Dust? & $t_2$ & Spec.\\ Class & FWHM & Distance & N(H) \\\\\n& (MJD) & (Date, UT) & (mag) & (mag) & (Y\/N) & (days) & & (km s$^{-1}$) & (kpc) & ($10^{21}$ cm$^{-2}$)}\n\\startdata\nV392~Per & 58237.47 (1) & 2018-04-29.47 (1) & $\\sim$6.2 (1) & 5.6 (2) & - & 3 (29) & Fe II (2) & 4700$\\pm200$ (2) & 4.1$^{+2.3}_{-0.4}$ (25) &3.4$\\pm$0.4 (2)\\\\\nV906~Car & 58193.03 (28) & 2018-03-16.03 (28) & $<$10\\tablenotemark{c} (3) & $\\sim$5.9 (28) & Y (28) & 44$\\pm2$ (28) & Fe II (28) & 1500$\\pm100$ (2) & 4.0$\\pm1.5$ (28) & 3.1$\\pm$0.4 (2) \\\\\nV357~Mus & 58121.24 (4) & 2018-01-3.24 (4) & 7.0 (5) & 7.0 (5) & - & 40$\\pm5$ (2) & Fe II (5) & 1200$\\pm100$ (2) & 3.2$\\pm0.5$ (2) & 4.2$\\pm$0.8 (2)\\\\\nV549~Vel & 58020.39 (6) & 2017-09-24.39 (6) & $\\sim$11.3 (6) & 9.1 (2) & - & 90 (2) & Fe II (7) & 2300$\\pm$200 (2) & $>$4 (2) & 9.0$\\pm$1.0 (2)\\\\\nV5856~Sgr & 57686.02 (8) & 2016-10-25.02 (8) & $\\sim$13.7 (9) & 5.4 (8) & - & 10 (8) & Fe II (8) & 1600$\\pm100$ (2) & 2.5$\\pm0.5$ (2) & 3.1$\\pm$0.4 (2)\\\\\nV5855~Sgr & 57681.38 (11) & 2016-10-20.84 (11) & 10.7\\tablenotemark{d} (11) & 7.5 (11) & - & 17$\\pm2$ (2) & Fe II (12) & 200$\\pm$200 (2) & 4.5 (11) & ---\\\\\nV407~Lup & 57655.00 (13) & 2016-09-24.00 (13) & $\\sim$9.1 (13) & $<$5.6 (14) & N (2) & 3$\\pm$1 (5) & He\/N (14) & 2900$\\pm$100 (2) & 4.2$\\pm$0.5 (2) & 9.0$\\pm$1.2 (2)\\\\\nV5668~Sgr & 57096.63 (15) & 2015-03-15.63 (15) & 6.0\\tablenotemark{d} (15) & 4.4 (16) & Y (17) & 75$\\pm$2 (2) & Fe II (2) & 1300$\\pm$100 (2) & 2.8$\\pm$0.5 (2) & 5.9$\\pm$0.8 (2)\\\\\nV1369~Cen & 56628.69 (18) & 2013-12-2.69 (18) & 5.5\\tablenotemark{d} (18) & $\\sim$3.3 (2) & Y (2) & 40$\\pm$2 (2) & Fe II (2) & 1200$\\pm$100 (2) & 1.0$\\pm$0.4 (2) & 0.6$\\pm$0.1 (2)\\\\\nV339~Del & 56518.58 (19) & 2013-08-14.58 (19) & 6.8\\tablenotemark{d} (19) & $\\sim$4.3 (2) & N (2) & 11$\\pm$1 (2) & Fe II (2) & 1700$\\pm$100 (2) & 4.9$\\pm$1 (2) & 1.7$\\pm$0.4 (2)\\\\\nV959~Mon & 56097.00 (20) & 2012-06-19.00 (20) & $\\sim$9.9 (2) & N\/A\\tablenotemark{f} (27) & N (2) & 10 (2) & He\/N (21) & 2000$\\pm$200 (2) & 1.4$\\pm$0.4 (26) & 3.4$\\pm$0.4 (2)\\\\\nV1324~Sco & 56069.80 (22) & 2012-05-22.80 (22) & 18.5\\tablenotemark{e} (23) & 9.8 (2) & Y (22) & 24 (29) & Fe II (22) & 1900$\\pm$200 (2) & $>$6.5 (29) & 10.1$\\pm$0.7 (2)\\\\\nV407~Cyg & 55265.81 (24) & 2010-03-10.81 (24) & 6.8\\tablenotemark{d} (24) & 7.1 (24) & - & 5.9 (24) & He\/N (24) & 1400$\\pm$100 (2) & 3.4$\\pm$0.5 (2) & 5.6$\\pm$0.8 (2)\\\\\n\\enddata\n\\tablenotetext{a}{Date of first observation in eruption.}\n\\tablenotetext{b}{$V$ band, unless otherwise noted.}\n\\tablenotetext{c}{Image was saturated.}\n\\tablenotetext{d}{Image was obtained in an unfiltered optical band.}\n\\tablenotetext{e}{Image was obtained in the $I$ band.}\n\\tablenotetext{f}{Optical maximum was during solar conjunction, so was missed}\n\\tablenotetext{}{References (1) \\citet{Munari_etal_2018}; (2) This work; (3) \\citet{Stanek_etal_2018}; (4) ASAS-SN data\n\\citet{Walter_etal_2018}; (5) \\citet{Aydi_etal_2018}; (6) \\citet{Stanek_etal_2017}; (7) \\citet{Luckas_etal_2017}; (8) \\citet{Li_etal_2017}; (9) AAVSO Alert 561; (10) \\citet{Munari_etal_2017}; (11) \\citet{Nelson_etal19}; (12) \\citet{Luckas_etal_2016}; (13) \\citet{2016ATel.9538....1S}; (14) \\citet{2018MNRAS.480..572A}; (15) \\citet{2016ApJ...826..142C}; (16) \\citet{2018ApJ...858...78G}; (17) \\citet{2015ATel.7748....1B}; (18) \\citet{2013CBET.3732....3W}; (19) \\citet{2013AAN...489....1W}; (20) \\citet{Ackermann+14}; (21) \\citet{2013ATel.4709....1M}; (22) \\citet{Finzell_etal_2018}; (23) \\citet{2012ATel.4157....1W}; (24) \\citet{2011MNRAS.410L..52M}; (25)\n\\citet{2018MNRAS.481.3033S}; (26)\n\\citet{2015ApJ...805..136L}; (27) \\citet{2014ASPC..490..217S}; (28) \\citet{Aydi2020}; (29) \\citet{Chochol_etal_2020}; (30) \\citet{Finzell_etal_2015}}\n\\end{deluxetable*}\n\\end{longrotatetable}\n\n\n\n\\subsection{Optical Properties}\\label{sec:optical}\nTable~\\ref{table:characteristics} presents the main characteristics of the novae in our sample,\nsome of which are compiled from the literature (with references given in parentheses following each table entry) and others estimated for the first time here. \nIt includes date of first detection in eruption ($t_0$) in MJD and UT, optical magnitude at $t_0$, peak magnitude in the $V$-band ($V_{max}$), and the time for the optical light curve to decline by two magnitudes from maximum ($t_2$). The peak magnitude and $t_2$ are determined from reports in the literature or derived in this work using publicly available photometry from the AAVSO, ASAS-SN, and the Stony Brook\/SMARTS Atlas. $t_2$ is measured as the duration between the first peak and the last time the nova reaches two magnitudes fainter than the peak.\n\nWe also list whether or not the nova formed dust based on reports in the literature or examining publicly available optical and near-infrared (NIR) photometry, particularly from SMARTS and AAVSO, to search for dust dips in the optical light and\/or IR excess. For some novae, we cannot tell if the nova has formed dust or not due to lack of multi-band photometric follow-up. \n\nWe give the spectroscopic class (\\eal{Fe}{II} or He\/N; \\citealt{Williams92}) and the Full Width at Half Maximum (FWHM) of Balmer emission lines after optical peak. The spectroscopic classes are based on previous reports in the literature or determined based on spectra obtained around optical peak ($\\lesssim t_2$). These spectra are either publicly available spectra from the Astronomical Ring for Access to Spectroscopy (ARAS\\footnote{\\url{http:\/\/www.astrosurf.com\/aras\/Aras_DataBase\/Novae.htm}}; \\citealt{Teyssier_2019}) or from our private database. The FWHM are measured from the same spectra by fitting a single Gaussian profile to the Balmer emission lines. Nova V959 Mon is an exception since this nova was discovered in optical 56 days after its $\\gamma$-ray detection by \\textit{Fermi}-LAT due to solar conjunction. The optical spectrum we use to determine the FWHM has been obtained 3 days after its optical discovery (around 60 days after optical peak, given that for most novae the $\\gamma$-ray detection occurs near optical peak). \n\nWe also use high-resolution optical spectroscopy to estimate the Galactic column density towards each nova. Again, these spectra are either from ARAS or from our private database, and are obtained near the light curve peak. We measure the equivalent widths of some diffuse interstellar bands (DIBs) and use the empirical relations of \\citet{Friedman_etal_2011} to derive an estimate of $E(B-V)$. $A_V$ is then derived assuming an extinction law of $R_V$ = 3.1. This $A_V$ is converted into an absorbing column density, $N(H)$, using the relation from \\citet{Bahramian_etal15}: $N(H)$ = ($2.81\\,\\pm\\,0.13$) $\\times 10^{21}\\, A_V$. \n\nWe also list distance estimates to the novae in our sample. For novae without an accurate distance estimate in the literature, we estimate the distance using our derived extinction values, along with the 3D Galactic reddening maps of \\citet{Chen_etal_2019}. \n\nIn our nova sample, one system is known to have a Mira giant secondary, namely V407~Cyg. The other 12 novae in the sample are likely systems with dwarf secondaries and will be designated as ``classical novae'' in the rest of the paper. However, it should be noted that V392~Per was recently found to have a mildly-evolved secondary star, with a binary orbital period of 3.4 days \\citep{Munari+20}, implying that V392~Per may be a ``bridge'' object between embedded novae with dense circumstellar material and classical novae with low-density surroundings. \n\n\n\\section{X-Ray Light Curves} \\label{subsec:x-ray light ces}\n\n\\subsection{\\emph{Swift}-XRT observations}\nThe \\textit{Swift}-XRT data products generator \\citep{2007A&A...469..379E,2009MNRAS.397.1177E} was used to produce X-ray (0.3--10\\,keV) light curves for all the novae in our sample. The same tool was also used to divide the XRT flux into soft (0.3--1.0\\,keV) and hard (1--10\\,keV) X-ray bands. \nOnce the data products were generated, the X-ray count rates were filtered to separate the significant detections from the upper limits. Observations with less than 3$\\sigma$ confidence on their count rates were considered upper limits. We quote 3$\\sigma$ upper limits throughout this paper, calculated using the uncertainty on the count rate.\n\nIn Figures~\\ref{fig:V392_Per}--\\ref{fig:V407_Cyg}, we present the \\textit{Swift} X-ray (0.3--10\\,keV) light curves of all the novae in our sample. In each figure's top panel, we plot the total XRT count rate, while the bottom panel distinguishes the light curves in the soft and hard bands. The time range of the \\emph{Fermi}-LAT $\\gamma$-ray detection is marked as a yellow bar, and the light curves focus on the first year following nova discovery.\n\nThere are \\emph{Swift} observations concurrent with \\emph{Fermi}-LAT $\\gamma$-ray detections for 9 of the 13 novae in our sample. Unfortunately, for some of the \\emph{Fermi}-detected novae, \\emph{Swift} observations were not obtained until long after discovery, and were therefore only detected after the end of the \\emph{Fermi} detection. Early observations were limited by solar conjunction for V392~Per, V959~Mon, and V549~Vel.\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth, height=0.41\\textheight]{Gamma_Novae_Hardness_Ratio_Evolution_SR_Obs.pdf}\n \\caption{The evolution of the X-ray emission as a function of time since discovery for our sample of \\textit{Fermi}-LAT-detected novae. Vertical tick marks represent times of \\emph{Swift}-XRT observations, and are color-coded according to the type of X-ray emission that was detected. Blue ticks denote hard X-ray emission, magenta ticks are soft X-ray emission, and black ticks represent X-ray non-detections. The durations of $\\gamma$-ray detections with \\emph{Fermi}-LAT are denoted with yellow rectangles. V407 Cyg is marked with an asterisk to note that this system has a red giant secondary, unlike the other novae in our sample.}\n \\label{fig:gamma schwarz}\n\\end{figure*}\n\nMany $\\gamma$-ray detected novae are very optically bright, and lead to optical loading of the XRT if observed in photon counting mode\\footnote{\\href{https:\/\/www.swift.ac.uk\/analysis\/xrt\/optical_loading.php}{https:\/\/www.swift.ac.uk\/analysis\/xrt\/optical\\_loading.php}}. Therefore, some \\emph{Swift}\/XRT observations early in our targets' eruptions were obtained in the less sensitive windowed timing mode. This affects observations of V906~Car, V357~Mus, V1369~Cen, V5668 Sgr, and V5856~Sgr. The supplementary online tables list information on each observation used, including the corresponding observation mode.\n\n\n\\subsection{Hardness ratio evolution} \\label{subsec:hardness ratio evolution}\nWe derive the hardness ratio ($HR$) for each \\textit{Swift}-XRT detection using the definition from \\citet{2011ApJS..197...31S}:\n\\begin{equation}\\label{eq:hr}\nHR = (H-S)\/(H+S) \n\\end{equation} \nwhere $S$ is the count rate in the 0.3\\,--\\,1.0\\,keV range and $H$ is the count rate in the 1\\,--\\,10\\,keV range. We also use similar criteria as \\citet{2011ApJS..197...31S} to classify the X-ray emission: we consider the X-ray emission ``hard\" if HR $> -0.3$, and ``soft\" if HR $< -0.3$. \n\nIn Figure~\\ref{fig:gamma schwarz} we present the evolution of the hardness ratio as a function of time since discovery for all the novae in our sample. The plot also shows the duration of the \\emph{Fermi}-LAT $\\gamma$-ray detection, represented as a yellow box, to compare with the \\emph{Swift} X-ray observations. Non-detections, denoted as black tick marks in Figure~\\ref{fig:gamma schwarz}, are defined as times when both the hard and soft bands were upper limits. If only one of the bands was detected, this epoch is counted as a detection, and the tick's color corresponds to the detected band.\n\nThe hardness ratio evolution of the novae is quite varied, but the main commonality is the lack of significant X-ray detection during the $\\gamma$-ray emission period (with the exception of V407 Cyg).\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:disc_conc}\n\n\\subsection{The drivers of hard X-rays in novae}\n\\label{subsec:hard_lum}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Luminosity_Plot_Hard_Xray_SR_Obs.pdf}\n \\caption{The luminosity in the hard X-ray band for all novae since the time the system was observed to be in eruption. When deriving these luminosities, we do not account for intrinsic absorption (e.g., absorption from the nova ejecta). Circles represent $\\geq$3$\\sigma$ hard X-ray detections and triangles represent upper limits. Filled-in symbols denote concurrent hard X-ray and $\\gamma$-ray emission, while open symbols were observed while $\\gamma$-rays are not detected. Note that the light curves have been edited to better highlight trends.}\n \\label{fig:hard_luminosity}\n\\end{figure*}\n\n\\par Hard X-ray emission from optically-thin plasma with a temperature of several keV has long been observed in novae\nand is commonly attributed to shocks \\citep[e.g.,][]{1994MNRAS.271..155O}. Figure \\ref{fig:hard_luminosity} compares the luminosity light curves in the 1--10\\,keV range for our nova sample. They have been smoothed to highlight bulk features; see Figures~\\ref{fig:V392_Per}--\\ref{fig:V407_Cyg} for full-cadence light curves. Distances are assumed as listed in Table~\\ref{table:characteristics}. Count rates are corrected for foreground absorbing columns consistent with the intervening interstellar medium, using the $N(H)$ values quoted in Table~\\ref{table:characteristics}. We do not account for intrinsic absorption (e.g., absorption from the nova ejecta). To make a rough conversion of the \\emph{Swift}-XRT count rate to X-ray flux, we assume a 5 keV thermal bremsstrahlung model (yielding the scale in units of erg s$^{-1}$ on the right y axis). The unabsorbed fluxes were then corrected for absorption using WebPIMMs as described above and then converted to luminosities scaled at a distance of 1 kpc.\n\n\\par This figure updates a similar plot from \\citet{2008ApJ...677.1248M}, with the goal of exploring the luminosities and durations of the hard X-ray emission from shocks. Figure 1 in \\citet{2008ApJ...677.1248M} shows hard X-ray luminosity as a function of time for 16 novae, but most had very limited time coverage so the duration of hard X-rays was unclear. In our sample, we see that typically the hard X-rays become detectable 1--2 months after the start of eruption, and last several months to $\\sim$a year. The notable exception is V407 Cyg, whose hard X-rays evolve much faster, starting shortly after day 10 (concurrent with $\\gamma$-ray detection).\nThis rapid evolution may be attributable to interaction with circumbinary material around the secondary (\\S \\ref{sec:v407}); it is notable that other novae with giant companions, RS~Oph \\citep{Sokoloski+06, Bode+06, 2008ApJ...677.1248M}, V745~Sco \\citep{2015MNRAS.454.3108P}, and V3890~Sgr \\citet{2020arXiv201001001P} were all detected in hard X-rays from the first pointed observations. However, RS Oph's last eruption occurred before the launch of Fermi, so we do not have information on its $\\gamma$-ray evolution. \n\n\\par The 1--10 keV X-ray luminosities of novae in Figure \\ref{fig:hard_luminosity} peak at $10^{33}-10^{34}$ erg s$^{-1}$. While we expect the bulk of the 1--10 keV luminosity to originate from shocked optically thin gas, in some cases it may be dominated by the hard tail of the supersoft component. For example, for moderate absorbing columns $N(H) \\lesssim 10^{22}$ cm$^{-2}$, as expected for the Galactic foreground (Table \\ref{table:characteristics}), a $10^{37.5}$ erg s$^{-1}$ blackbody of temperature $T_{BB} =$ 90 eV produces $\\sim$30 times as many counts in the 1--10 keV band\\footnote{\\url{https:\/\/heasarc.gsfc.nasa.gov\/cgi-bin\/Tools\/w3pimms\/w3pimms.pl}} as a $10^{34}$ erg s$^{-1}$ bremsstrahlung component of temperature 5 keV. Such a hot supersoft component is only expected for a near-Chandrasekhar mass white dwarf (e.g., \\citealt{Osborne_etal_2011}), and contamination of the 1--10 keV band depends sensitively on the temperature of the supersoft source. A more typical white dwarf ($T_{BB} \\approx$ 60 eV; \\citealt{Wolf_etal_2013}) of a similar luminosity contaminates the 1--10 keV band orders of magnitude less severely, contributing $\\lesssim$30\\% of the 1--10 keV flux.\n\n\\par A detailed analysis of when the supersoft source contributes significantly to the 1--10 keV band would require spectral fitting of the \\emph{Swift}-XRT data and is outside the scope of this paper. However, in the case of a moderate absorbing column, the hardness ratio is a powerful discriminant. For $N(H) = 5 \\times 10^{21}$ cm$^{-2}$, the count rate from a supersoft source should be $\\gtrsim 10 \\times$ higher in the 0.3--1 keV band compared to the 1--10 keV band. For example, the 1--10 keV X-rays observed from V407~Lup starting around day 150 are likely the hard tail of the supersoft source because the concurrent 0.3--1 keV X-rays are so much brighter (Figure \\ref{fig:V407_Lup}). At higher absorbing columns, the ratio of soft-to-hard X-rays from a supersoft source will be lower. In the case of V339 Del and V1369 Cen, some of the early 1--10 keV X-rays (day $\\sim$50 and day $\\sim$80, respectively) could be attributed to the supersoft source beginning to emerge from the absorbing nova ejecta, as the 0.3--1 keV flux is increasing during this time (Figures \\ref{fig:V1369_Cen} and \\ref{fig:V339_Del}). However, in many of the novae studied here, the 1--10 keV flux is significantly brighter than the 0.3--1 keV flux and relatively stable in time (e.g., Figures \\ref{fig:V906_Car}, \\ref{fig:V357_Mus}, \\ref{fig:V959_Mon}), implying that the early hard X-rays really are emitted from hot shocked gas.\n\n\\par It is also possible that accretion could be a source of hard X-rays, particularly at late times. This is mainly true for systems with highly magnetized white dwarfs ($B>10^6$\\,G), such as intermediate polars. In such systems, accretion is channeled by the strong magnetic field lines into an accretion column which then slams onto the white dwarf surface at high speeds, increasing the surface temperature and leading to hard X-ray emission (see~\\citealt{Warner_1995} for a review). Per example, the hard X-ray emission in Nova V407~Lup around 350 days after eruption is probably due to accretion resuming on the surface of the white dwarf (Figure \\ref{fig:V407_Lup}). This nova occurred in an intermediate-polar system where the white dwarf is highly magnetized (see \\citealt{2018MNRAS.480..572A} for more details). The luminosity of hard X-ray emission in intermediate polars is usually $\\sim$ 10$^{31}$\\,--\\,10$^{34}$\\,erg\\,s$^{-1}$ \\citep{Patterson_etal_1994,Pretorius_Mukai_2014}, which is consistent with the X-ray luminosity of V407~Lup around 2 years after eruption. \n\nNova V392~Per also shows hard X-ray emission, which is peculiarly constant over a period of more than 250 days (see also Murphy-Glaysher et al. in prep for a more detailed examination of the \\textit{Swift} data). As previously mentioned, this nova was recently found to have a mildly-evolved secondary star \\citep{Munari+20}. However, the origin of this constant and extended hard X-ray emission is not clear. While it could be accretion related, it is less likely to be due to shock interaction within the ejecta at this late stage. After the 1998 eruption of nova \nV2487~Oph, which is characterized by a $\\sim$ 1 day orbital period \\citep{Anupama_2013}, \\citet{Hernanz_Sala_2002} found hard X-ray emission more than two years after the eruption with comparable luminosity to that of nova V392~Per ($\\sim10^{33}$\\,erg\\,s$^{-1}$). \\citet{Hernanz_Sala_2002} attributed this \nlate X-ray emission to accretion resuming on the white dwarf. In addition, \\citet{Orio_etal_2001}'s study of \\textit{ROSAT} observations of a large number of novae identified late X-ray emission from several novae during quiescence, which they attributed to accretion. \n\n\n\\subsection{Novae with dwarf companions are not detected in 1--10\\,keV X-rays concurrent with $\\gamma$-rays}\n\\label{disc_1}\n\n\n\\par The internal shocks responsible for accelerating particles to relativistic speeds and producing $\\gamma$-ray emission have velocities of $\\sim$few thousand km\\,s$^{-1}$ and are expected to heat the post-shock gas to X-ray temperatures ($\\sim 10^7$\\,K; \\citealt{2015MNRAS.450.2739M}). Therefore, it is surprising that we do not detect \\emph{Swift} X-ray emission concurrent with GeV $\\gamma$-rays among the classical novae in our sample. Nine of the novae presented here have \\textit{Swift}-XRT observations during their \\emph{Fermi}-LAT detections, and all except V407~Cyg show no X-ray emission during this period. The other four novae did not have \\textit{Swift} observations concurrent with \\emph{Fermi}-LAT detections. In all cases for the classical novae, the first X-ray detection only occurs after the $\\gamma$-ray emission falls below the sensitivity limit of \\textit{Fermi}-LAT (Figure~\\ref{fig:gamma schwarz}).\n\n\\par The simplest explanation for the \\emph{Swift}-XRT non-detections during $\\gamma$-rays is that the shocks are deeply embedded within the nova ejecta due to their high density (among the highest for astrophysical events: $\\sim10^{10}$\\,cm$^{-3}$; see figure 1 in \\citealt{Metzger_etal_2016}). Such high densities imply substantial absorbing columns ahead of the shocks, which can absorb photons with energies $\\lesssim$10\\,keV. \nThe other explanation for the X-ray non-detections is that the thermal energy of the shocked material is sapped by cold regions around the shocks before it can be radiated, implying a suppression of the shock's temperature, i.e., the shocks do not reach X-ray energies \\citep{Steinberg_Metzger_2018}. This would lead to a suppression of the X-ray emission that can be detected by \\textit{Swift}. \n\n\\par To constrain the conditions in nova shocks, we compare the \\emph{Swift}-XRT upper limits on the 1--10 keV X-ray luminosity with concurrent GeV $\\gamma$-ray luminosities from \\emph{Fermi}-LAT (Figure \\ref{fig:gamma_xray_flux}). We convert the time-averaged \\emph{Fermi}-LAT $\\gamma$-ray count rates listed in Table \\ref{table:gamma} to $\\gamma$-ray fluxes assuming a single power-law spectrum and photon indices also listed in Table \\ref{table:gamma}. The fluxes are then converted to luminosites over the energy range 100 MeV--300 GeV, assuming the distances in Table \\ref{table:characteristics}. The resulting $\\gamma$-ray luminosities span a few $\\times 10^{34}$ to a few $\\times 10^{36}$ erg s$^{-1}$. The X-ray luminosities are as estimated for Figure \\ref{fig:hard_luminosity} (\\S \\ref{subsec:hard_lum}).\nV392 Per, V549 Vel, V407 Lup, and V959 Mon are not plotted in Figure \\ref{fig:gamma_xray_flux}, as there were no X-ray observations during their $\\gamma$-ray emitting periods. The $\\gamma$-ray luminosities are factors at least $10^{2}$ to $10^{4}$ times more luminous than the X-ray upper limits, with most of the novae clustered around $L_{\\gamma}\/L_X \\approx 10^{3}$. The ratios show a remarkable correlation, but no strong conclusions should be made as the plot is comparing a single data point in the X-rays per each nova to an averaged $\\gamma$-ray luminosity over the detection period. The \\emph{Swift} upper limit depth is heavily dependent on exposure time and the background count rate.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.08\\columnwidth]{Gamma_Xray_Flux2.pdf}\n \\caption{Comparison of hard X-ray and $\\gamma$-ray luminosities, for novae with concurrent X-ray and $\\gamma$-ray data. X-ray luminosities are in the 1--10 keV band and corrected for Galactic absorption (as given in Table \\ref{table:characteristics}) and represent the faintest (and most constraining) X-ray points during the $\\gamma$-ray detection period. Novae represented as black triangles denote that they are non-detections in the hard X-ray band, and 3$\\sigma$ upper limits are plotted. V407 Cyg is represented as an orange circle, as the only nova that had a $\\geq$3$\\sigma$ \\textit{Swift} detection during the $\\gamma$-ray detection period. $\\gamma$-ray luminosities are calculated in the 100 MeV--300 GeV band using parameters listed in Table \\ref{table:gamma}. Dashed lines guide the eye for estimating $L_{\\gamma}\/L_X$.}\n \\label{fig:gamma_xray_flux}\n\\end{figure}\n\n\\par Motivated by \\textit{Swift} non-detections, \nresearchers have begun searching for even harder X-rays during the $\\gamma$-ray bright phase using the \\textit{NuSTAR} satellite \\citep{2013ApJ...770..103H}, with instruments on board sensitive to photons with energies up to 79 keV. While softer X-rays are absorbed, harder X-rays $>$10 keV are expected to escape the dense ejecta, even in the early days of the eruption, due to the decreasing bound-free cross-section at high photon energies \\citep{2015MNRAS.450.2739M}.\nHarder X-rays have now been detected with \\emph{NuSTAR} from three classical novae concurrently with $\\gamma$-rays, namely V5855 Sgr, V906 Car, and YZ Ret (YZ Ret is not included in our sample as it erupted in 2020; \\citealt{Nelson_etal19, 2020MNRAS.497.2569S, Sokolovsky+20_ret20}).\nSpectral analysis of these \\emph{NuSTAR} data show low-luminosity hard X-ray emission ($\\sim 10^{33}-10^{34}$ erg~s$^{-1}$) originating from hot plasma ($kT \\approx 5-10$ keV) and absorbed by large column densities (N(H) $\\approx 10^{23}-10^{24}$ cm$^{-2}$).\n\n\\par Even with these \\emph{NuSTAR} detections corrected for internal absorption, the $L_{\\gamma}\/L_X$ ratio is still $\\gtrsim 10-10^{2}$. The high $L_{\\gamma}\/L_X$ observed with both \\textit{Swift} and \\textit{NuSTAR} is surprising because only a fraction of the shock's power should be going into producing $\\gamma$-rays given the predicted efficiency for particle acceleration \\citep{2015MNRAS.450.2739M}. Meanwhile, the high post-shock densities imply that the shocks should be radiative, and so the majority of the shock luminosity should be promptly transferred to radiative luminosity, which is naively expected to emerge in the X-ray band \\citep{2015MNRAS.450.2739M, Li_etal_2017, 2020NatAs...4..776A}. \n\\citet{Nelson_etal19} and \\citet{2020MNRAS.497.2569S} propose several scenarios that could yield a much higher $\\gamma$-ray luminosity compared to X-rays, including separate shocks producing the X-rays and $\\gamma$-rays, suppression of the X-rays by corrugated shock fronts \\citep{Steinberg_Metzger_2018}, remarkably efficient particle acceleration, or that modeling the shocks as radiative is an improper assumption.\n\n\\par Of the 12 classical novae investigated here, 10 eventually show 1--10 keV hard X-ray emission detectable by \\emph{Swift}-XRT.\nThis late emergence of the hard X-ray emission can be partially explained by a drop in the density of the ejecta as they expand---leading to a decrease in the absorbing column ahead of the shocks. But the faint \\emph{NuSTAR} detections imply that it is not only large absorbing columns that are leading to \\emph{Swift} non-detections; the X-ray luminosity is also intrinsically low.\n\n\n\n\n \n\n\n\n\\subsection{The exception: novae with giant companions are detected in X-rays concurrent with $\\gamma$-rays}\\label{sec:v407}\n\n\n\\par Although evolved giant companions are relatively rare in nova-hosting binaries, the first-ever nova detected by \\emph{Fermi}-LAT, V407~Cyg, was accompanied by a Mira giant donor \\citep{2010Sci...329..817A}. Previous to the nova eruption in 2010, V407~Cyg was well known as a D-type symbiotic star \\citep[e.g.,][]{Munari_etal90, Kolotilov_etal98, Kolotilov_etal03}. The giant donor's wind was dense, with a mass-loss rate of $\\sim 10^{-6}$ M$_{\\odot}$ yr$^{-1}$, resulting in a rich circumbinary medium \\citep{Chomiuk_etal_2012}.\n\n\\par During its 2010 nova eruption (discovered on 2010 March 10), V407~Cyg displayed faint but detectable X-rays in the first \\emph{Swift}-XRT observations of the nova eruption (four days after nova discovery; Figure~\\ref{fig:V407_Cyg}, \\citealt{Shore_etal11, Nelson_etal_2012}). Over the next $\\sim$20 days following the nova discovery, the X-ray flux rapidly brightened by a factor of $\\sim$10. During this same time period, V407~Cyg was detected as a GeV $\\gamma$-ray source by \\emph{Fermi}-LAT \\citep{2010Sci...329..817A}. V407~Cyg is the only nova in our sample with concurrent \\emph{Swift}-XRT and \\emph{Fermi}-LAT detections.\n\n\\par Both the X-rays and the $\\gamma$-rays in V407~Cyg are attributed to the interaction of the nova ejecta with the circumbinary medium \\citep{Orlando_Drake12, Martin_Dubus13}. The X-ray flux rises in the first three weeks because the absorbing column might have dropped, while the X-ray emission measure grows. The absorbing column, even at early times, is never much higher than $N(H) \\approx 10^{23}$ cm$^{-2}$ \\citep{Nelson_etal_2012}. This can be contrasted with the absorbing columns of $\\gtrsim {\\rm few} \\times 10^{23}$ cm$^{-2}$ for the internal shocks observed in classical novae with dwarf companions (e.g., \\citealt{Nelson_etal19, 2020MNRAS.497.2569S}). Therefore, V407~Cyg hints that X-rays can be detected concurrently with $\\gamma$-rays if the nova drives {\\it external} shocks (i.e., interaction with pre-existing circumbinary material), as opposed to more deeply-absorbed shocks internal to the nova ejecta.\n\n\\par This hypothesis is supported by two additional novae with giant companions that were marginally detected by \\emph{Fermi}-LAT between 2010 and 2018: V745~Sco and V1535~Sco \\citep{Franckowiak_etal_2018}. Hints of $\\gamma$-ray emission from V745~Sco were obtained at 2--3$\\sigma$ significance in the first two days of its 2014 nova eruption \\citep{2014ATel.5879....1C}. Bright hard X-ray emission was also observed during this time, with $N(H) = {\\rm few} \\times 10^{22}$ cm$^{-2}$ (\\citealt{Delgado_Hernanz19}; again, substantially lower than the absorbing columns observed for shocks in classical novae). Similarly, V1535~Sco was marginally detected in $\\gamma$-rays during the first seven days of its 2015 eruption \\citep{Franckowiak_etal_2018}, and hard X-rays were concurrently detected by \\emph{Swift}-XRT (on day 4; \\citealt{Linford_etal17}). Although these $\\gamma$-ray detections are marginal, they support a scenario where nova shocks with external circumbinary material (as occur in binaries with giant companions) are characterized by lower density, less embedded environments, in comparison with shocks that occur internal to nova ejecta in binaries with dwarf companions.\n\nIt is worth noting that the high $L_{\\gamma}\/L_X$ observed in V407~Cyg (see Figure~\\ref{fig:gamma_xray_flux}) could not be explained by high absorption or X-ray suppression in this case, given the less embedded environments of the shocks. However, a detailed analysis of the shocks in novae with evolved secondaries is outside the scope of this paper and will be the topic of future projects. \n\n\n\\subsection{Why are some $\\gamma$-ray detected novae never detected in X-rays?} \\label{no_detection_novae}\n\n\\par Out of the 13 novae in our sample, only two were never detected as X-ray sources with \\textit{Swift}, namely V1324~Sco (Figure~\\ref{fig:V1324_Sco}; \\citealt{Finzell_etal_2018}) and V5856~Sgr\\footnote{After further analysis of the WT data of V5856~Sgr, there is a possible X-ray detection on day 149, but the online generator did not find any detection. This is mainly affected by the estimate of background contribution for faint objects observed in WT mode.} (Figure~\\ref{fig:V5856_Sgr}; \\citealt{Li_etal_2017}). V5856~Sgr had only two \\textit{Swift} observations (15 and 149 days after discovery) which makes it difficult to draw conclusions about this nova as its X-ray emission could have been missed (as we might have missed the X-ray emission from e.g., V357~Mus if observations of it had been similarly sparse). However, V1324~Sco was followed with \\textit{Swift} between days 30 and $\\sim$ 500 after eruption and was still never detected. \n\n\\par There are a few reasons that might explain why V1324~Sco was not detected: lack of correlation between $\\gamma$-ray luminosity and X-ray luminosity, distance, and\/or absorption. While V1324 Sco was not detected in X-rays, it is among the brightest novae detected in $\\gamma$-rays. If X-ray luminosity does not scale with $\\gamma$-ray luminosity, this could explain the difference between the two.\n\n\n \n\n\\par Distance, however, appears to be an important factor to the detection of X-rays. V1324 Sco is the farthest nova of our sample ($\\gtrsim$6.5 kpc), and Figure \\ref{fig:hard_luminosity} shows that this translates to less sensitive upper limits on the hard X-ray luminosity.\nWe compared the flux of each nova's first 1--10 keV detection to what it would be at V1324 Sco's distance (also correcting for the additional interstellar absorption). \nThis analysis revealed that five novae would have been non-detections at the distance and $N(H)$ of V1324~Sco: V906 Car, V357 Mus, V5856 Sgr, V5855 Sgr, and V5668 Sgr. We therefore conclude that distance is probably the reason why V1324 Sco was not detected by \\emph{Swift}-XRT.\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{Hard_Xray_Multiplot.pdf}\n \\caption{The date of the first hard X-ray detection since $t_0$ for each nova plotted against $t_2$ (panel a), FWHM (measured from the Balmer lines a few days after optical maximum; panel b), duration of detectable $\\gamma$-rays (panel c), and apparent magnitude at optical peak ($V_{\\mathrm{max}}$; panel d) for the novae in our sample for which a measurement is available. The error in the first hard X-ray detection represents a lower limit on this quantity, extending to the date of the last non-detection; if the arrows extend to $t=t_0$, the first \\textit{Swift}-XRT observation was a 3$\\sigma$ detection. V407 Cyg is represented in dark gray in each panel as it has a giant companion (\\S \\ref{sec:v407}). The $r$ value denoted in the upper right corner of each panel is the Pearson correlation coefficient derived by using the grey and orange circles in each panel and excluding any weight for the error bars. V392 Per, V407 Lup, and V5855~Sgr are plotted in purple as the actual hard X-ray start for these novae is uncertain (see \\S\\ref{subsec:trends} for more details), therefore we exclude them from the correlation fitting. V1324~Sco and V5856~Sgr were never detected in X-rays by \\textit{Swift} (see \\S\\ref{no_detection_novae}) so they do not appear in the plots; V959 Mon does not appear in panel d as a maximum V-band magnitude could not be determined.}\n \\label{fig:trends}\n\\end{figure*}\n\n\\subsection{What determines when the hard X-rays appear?}\n\\label{subsec:trends}\n\n\\par Part of the intention of this project was to study a sample of $\\gamma$-ray detected novae in order to analyze possible trends in the data. \nIn Figure~\\ref{fig:trends}, we plot the date of the first hard X-ray detection against other nova properties described in \\S \\ref{sec:gamma} and \\S\\ref{sec:optical}: $t_2$, FWHM of Balmer emission lines (after optical peak), apparent magnitude at optical peak ($V_{\\mathrm{max}}$), and the duration of the $\\gamma$-ray detection to check for any correlations between these parameters. The Pearson correlation coefficient is shown in the top right corner of each panel. \n\n\\par Since the timing of \\emph{Swift} observations are different for each nova, it is challenging to draw conclusions about correlations between these parameters. For novae with extremely bright supersoft emission, it is possible that the harder shock component is contaminated by the supersoft component (see \\S\\ref{subsec:hard_lum} for more discussion). The cadence of novae V407~Lup, V392~Per, and V5855~Sgr was interrupted by solar conjunction and observation schedules. In addition, V407~Lup and V5855~Sgr were first detected during a bright supersoft phase which caused large uncertainties on the first hard X-ray start date for these novae as plotted in Figure~\\ref{fig:trends}. Because of these complications, we exclude these novae from the fitting done to derive the correlation coefficients.\n\n\\par Based on panel (a) in Figure~\\ref{fig:trends}, visual inspection indicates earlier hard X-ray emission for faster novae (characterized by smaller $t_2$)---particularly for novae with extensive \\emph{Swift} follow-up (with short error bars in Figure~\\ref{fig:trends}). The Pearson coefficient factor of $r = 0.39$ derived for novae with higher-quality data also implies that there may be a weak correlation. Interestingly, we find a weak anti-correlation between the time of first hard X-ray detection and FWHM ($r= -0.50$ shown in panel (b).\nA nova characterized by a faster optical light curve (short $t_2$) should typically have higher ejecta expansion velocities (large FWHM; \\citealt{Shafter_etal_2011}). In this case, the ejecta are expected to expand, drop in density, and become optically thin to the X-ray emitting shocked regions more rapidly than slower novae. There are hints that we may be observing these trends in Figure \\ref{fig:trends}, but a larger sample of novae will need to be observed in the future in order to confirm these hints.\n\n\\par In panel (c), a Pearson correlation coefficient of 0.81 implies a likely correlation between the duration of the $\\gamma$-ray detection and the first hard X-ray detection. To first order, this is expected given that none of the novae in our sample recorded \\emph{Swift} X-ray detections concurrent with the $\\gamma$-ray emission and were only detected after this period ended. But this correlation may hold important clues as to the drivers of shocks in novae, as the only other quantity that has been observed to potentially correlate with $\\gamma$-ray duration is $\\gamma$-ray fluence \\citep{2016ApJ...826..142C, Franckowiak_etal_2018}.\n\nPanel (d) shows some indication of a correlation between the peak brightness of the nova and the time of first hard X-ray detection (note that an anti-correlation here is a correlation with brightness, due to the ``flipped\" magnitude scale). However, again this correlation is weak and requires a larger sample or higher cadence data to test. \n\n\\par In summary, although there are intriguing hints at correlations, it is challenging to draw conclusions from the current sample---the number of novae detected in $\\gamma$-rays with dedicated multi-wavelength follow up is still small. Additional novae with high-cadence \\textit{Swift}-XRT and optical follow-up added to the current sample will allow us to draw better conclusions in the future. \n\n\n\\section{Summary and conclusions}\n\\label{sec:conc}\n\n\\par We have investigated the hard (1--10 keV) X-ray emission of 13 $\\gamma$-ray emitting novae using \\textit{Swift}-XRT. Novae have long been observed to emit X-rays from hot ($kT \\approx$ 1--10 keV) optically-thin plasma, presumably from shocked gas \\citep{1994MNRAS.271..155O, 2008ApJ...677.1248M}. The \\textit{Swift}-XRT light curves show evidence of hard X-ray emission from shocks in at least 7 out of the 13 novae studied, typically peaking several months after the start of eruption with luminosities $\\sim 10^{33}-10^{34}$ erg~s$^{-1}$.\n\nHowever, of the 9 novae with \\textit{Swift}-XRT observations during the $\\gamma$-ray detection phase (typically a few weeks around optical maximum), eight yielded X-ray non-detections during these early times.\nThe only nova showing X-ray emission concurrently with a \\textit{Fermi} $\\gamma$-ray detection is V407~Cyg, which has a giant secondary. We suggest that the non-detection of early X-ray emission from the other eight novae (all with dwarf companions) is due to a combination of large column densities ahead of the shocks absorbing the X-rays, and X-ray suppression by corrugated shock fronts \\citep[e.g.,][]{2015MNRAS.450.2739M,Steinberg_Metzger_2018}. The early X-ray detection of V407~Cyg (and possibly other novae with evolved companions) confirms that the shocks in symbiotic systems are external (between the nova ejecta and circumbinary material), rather than internal to the nova ejecta as claimed for novae with dwarf companions.\n\n\n\n\n\\par As more $\\gamma$-ray emitting novae are discovered and followed up at other wavelengths, we will be able to better constrain the physical parameters of the shocks and further investigate the conditions of their surrounding media. \n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to Tommy Nelson and Brian Metzger for conversations that inspired this work.\nACG, EA, LC, KVS, and JS are grateful for the support of NASA \\emph{Fermi} grant 80NSSC18K1746, \\emph{NuSTAR} grant 80NSSC19K0522, NSF award AST-1751874, and a Cottrell Scholarship of the Research Corporation. KLP acknowledges funding from the UK Space Agency.\nKLL is supported by the Ministry of Science and Technology of the Republic of China (Taiwan) through grants 108-2112-M-007-025-MY3 and 109-2636-M-006-017, and he is a Yushan (Young) Scholar of the Ministry of Education of the Republic of China (Taiwan).\n\nThis work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. We also acknowledge with thanks the Astronomical Ring for Access to Spectroscopy ARAS observers for their optical spectroscopic observations\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}